Those who have had much occasion to use the. mathematical instruments constructed to facilitatenbsp;the arts of drawing, surveying, amp;c. have long complained that a treatise was wanting to explainnbsp;their use, describe their adjustments, and givenbsp;such an idea of their construction, as might enablenbsp;them to select those that arc best adapted to theirnbsp;respective purposes.

This complaint has been the more general, as there are few active stations in life whose professors are not often obliged to have recourse to mathematical instruments. To the civil, the military,nbsp;and the naval architect, their use must be familiar;nbsp;and they^ arc of equal, if not of more importancenbsp;to the engineer, and the surveyor; they are thenbsp;means by which the abstract pails of the mathematics are rendered useful in life, they connectnbsp;theory with practice, and reduce speculation tonbsp;use.

Monsieur Bions treatise on the constmetion of rnatl^matical instruments, which was translatednbsp;into English by Mr. Stone, and published in 17^3,nbsp;is the only regular treatise¹ we have upon thisnbsp;subject; the numerous, improvements that havenbsp;been made in instruments since that time, have

I do not speak of Mr. Robertson�s work, as it is confined wholly to the instruments contained in a case of drawing instnj-ments.

rendered this work but of little use. It has beea my endeavour by the following Essays to do awaynbsp;this complaint; and I have spared no pains tonbsp;render them intelligible, and make them useful.nbsp;Though the materials, of which they are composed, lie in common, yet it is presumed, thatnbsp;essential improvements will be found in almostnbsp;every part.

These Essays begin by defining the necessary terms, and stating a few of those first principlesnbsp;on which the whole of the work is founded: theynbsp;then proceed to describe the mathematical drawing instruments; among these, the reader will findnbsp;an account of an improved pair of triangular compasses, a small pair of beam compasses with a micrometer screw, four new parallel rules, and other articles not hitherto described: these are follow'ed bynbsp;a large collection of useful geometrical problems;nbsp;I flatter myself, that the practitioner will find manynbsp;that are new, and which are well adapted to lessennbsp;labour and promote accuracy. In describing thenbsp;manner of dividing large quadrants, I have firstnbsp;given the methods used by instrument makers,nbsp;previous to the publication of that of Mr. Bird,nbsp;subjoining his mode thereto, and endeavouring tonbsp;render it more plain to the artist by a different arrangement. This is succeeded by geometrical andnbsp;mechanical methods of describing circles of everynbsp;possible magnitude; for the greater part of whichnbsp;I am indebted to Joseph Priestley, Esq. of Bradford,nbsp;Yorkshire, whose merit has been already noticednbsp;by an abler pen than mine.¹ From this, I proceed to give a short view of elliptic and other compasses, and a description of Suardis geometric pen,nbsp;an instrument not known in this country, and

whose curious properties will exercise the ingenuity of mcchanies and mathematicians.

Trigonometry is the next subject; but as this work was not designed to teach the elements ofnbsp;this art, I have contented myself with stating thenbsp;general principles, and giving the ^canons for calculation, subjoining some useful and curious problems, which, though absolutely necessary in manynbsp;cases that occur in county and marine surveying,nbsp;have been neglected by every practical writer onnbsp;this subject, except Mr. Mackenzie ¹ and B.Domi.\nbsp;Some will also be found, that are even unnoticednbsp;by the above-mentioned authors.

Our next article treats of surveying, and it is presumed the reader will find it a complete, thoughnbsp;concise system thereof. The several instrumentsnbsp;now in use, and the methods of adjusting them,nbsp;are described in order; and I think it will appearnbsp;evident, from a view of those of the best constrvic-tion, that large estates maybe surveyed and plotted with greater accuracy than heretofore.

The great improvements that have been made within these few years in the art of dividing, havenbsp;rendered observers more accurate and more attentive to the necessary adjustments of their instruments, which are not now considered as perfect,nbsp;unless they are so constructed, that the person whonbsp;uses them can either correct or allow for the errorsnbsp;to which they are liable. Among the various improvements which the instruments of science havenbsp;received from Mr. liamsden, we are to reckonnbsp;those of the theodolite here described; the sur-veyorwill find also the description of a small quadrant that should be constantly used with the

chain, improvements in the circumferentor, plain-table, protractor, amp;:c. In treating of surveying, I thought to have met with no difficulty ; havingnbsp;had however no opportunity of practice myself,nbsp;I had recourse to books; a multiplicity have beennbsp;written upon this subject, but they are for thenbsp;most part imperfect, irregulai¹, and obscure. Inbsp;have endeavoured (with what success must be leftnbsp;to the reader�s judgment) to remove their obscurities, to rectify their errors, and supply their defi-ficiencies; but whatever opinion he may form ofnbsp;my endeavours, I can venture to say, he will benbsp;highly gratified with the valuable communicationsnbsp;of Mr. Gale,¹' and Mr. Milne, here inserted, andnbsp;which I think will contribute more to the improvement of the art of surveying, than any thingnbsp;it has received since its original invention.

The reader will, I hope, excuse me, if I stop a moment to give him some account of Mr. Gale's,nbsp;improvements; they consist, first, in a new method of plotting, which is performed by scales ofnbsp;equal parts, without a protractor, from the northings and southings, eastings and westings, takennbsp;out of the table which forms the appendix to thisnbsp;work;'!' this method is much more accurate thannbsp;that in common use, because any small inaccuracynbsp;that might happen in laying down one line is naturally corrected in the next; whereas, in thenbsp;common method of plotting by scale and protractor, any inaccuracy in a former line is naturallynbsp;cotntmmicated to all the succeeding lines. Thenbsp;next improvement consists in a new method of determining the area, with superior accuracy, from

-j- The table is printed separate, that it may be purchased, or not,' as the surveyor sees convenient.

the northings, southings, castings, and westings, without any regard to the plot or draught, by annbsp;easy computation.

As the measuring a strait line with exactness is one of the greatest difficulties in surveying, I wasnbsp;much surprised to find many land surveyors usingnbsp;only a chain; a mode in which errors are multipliednbsp;wnthoiit a possibility of their being discovered, ornbsp;corrected. I must not forget to mention here,nbsp;that I have inserted in this part Mr. Break's method of surveying and planning by the plain table,nbsp;the bearings being taken and protracted at thenbsp;same instant in the field upon one sheet of paper;nbsp;thus avoiding the trouble and inconvenience ofnbsp;shifting the paper: this is followed by a smallnbsp;sketch of maritime surveying; the use of the pan-tographer, or pantagraph; the art of levelling, andnbsp;a few astronomical problems, with the manner ofnbsp;using Hadley�s quadrant and sextant; even herenbsp;some suggestions will be found that arc new andnbsp;useful.

I have now to name another gentleman, wfeo has contributed to render this work more perfect thannbsp;it would otherwise have been, and it is with pleasure I return my best thanks to Mr. Landman',nbsp;Professor of fortification and artillery to the Royalnbsp;Academy at Woolwich, for his communications,nbsp;more particularly for the papers from which thenbsp;course of practical geometry on the ground wasnbsp;extracted. If the professors of useful sciencesnbsp;w'ouid thus liberally co-operatc for their advancement, the progress thereof would he rapid and extensive. This course wall be found useful not onlynbsp;to the military officer, but would make a usefulnbsp;and cutcrtaiuiiig part of every gentleman�s education. I found it necessary to abridge the papersnbsp;Mr. Landman lent me, and leave out the calcula-

tions, as the work had already swelled to a larger size than was originally intended, though printednbsp;on a page unusually full.

The work finishes with a small tract on perspective, and a description of two instruments desighed to promote and facilitate the practice of that useful art. It is hoped, that the publication of thesenbsp;will prevent the public from being imposed uponnbsp;by men, who, under the pretence of secresy, enhance the value of their contrivances. I knew annbsp;instance where 40l. was paid for an instrumentnbsp;inferior to the most ordinary of the kind that arenbsp;sold in the shops. Some pains have been taken,nbsp;and no small expense incurred, to offer somethingnbsp;to the public superioj- in construction, and easiernbsp;to use, than any instrument of the kind that hasnbsp;been hitherto exhibited.

I have been anxious and solicitous not to neglect any thing that might be useful to the practitioner, or acceptable to the intelligent. In a work which embraces so many subjects, notwithstand-' ing all the care that has been taken^ many defectsnbsp;may still remain; I shall therefore be obliged tonbsp;any one who will favour me with such hints ornbsp;observations, as may tend towards its improvement.

A list of the authors I have seen is subjoined to this Preface. I beg leave to return my thanks tonbsp;the following gentlemen for their hints and valuable communications, the Rev. Mr. Hawkins,nbsp;J. Priestley, Esq. Mr. Gale, yix. Milne, Dr. Rotherham, Mr. Heywood, Mr. Landman, and Mr, Beck,nbsp;a very ingenious artist.

THE EDITOR.

The fi rst edition of these Essays haloing, like the rest of the late ingenious Author s works, receivednbsp;much share of public approbatmi and encouragement-, and being myself a joint proprietor with mynbsp;brother, S. Jones, of the copyright of all his publications, 1 conceive, that I cannot employ the few leisure moments, after the business of the day, better,nbsp;than by revising, correcting, and improving thosenbsp;works that require reprinting. The present is the firstnbsp;of my editing: considerable errors in the former edition have been corrected-, more complete explanationsnbsp;of instruments given, arid many particulars of new andnbsp;useful articles, not noticed by the Author, with notes,nbsp;ksc. are inserted in their proper places. The additionsnbsp;and amendments are, upon the whole, such, as I presume, without any pretension to superior abilities onnbsp;my part, will again render the work deserving of thenbsp;notice of all students and practitioners in the differentnbsp;professional branches of practical geometry. Thenbsp;principal additions I have made are the following:

Description of a new pair of pocket Compasses, containing the ink and pencil points in its twonbsp;legs�Improved Perambulator�ay W iscr�Improved Surveying Cross�Improved Circumferentor�Complete portable Theodolite�Great Theodolite, by Ramsden�Pocket box Sextant�Artificial Horizon�Pocket Spirit Levels�A Pair ofnbsp;Perspective Compasses�Keith's im[)rovcd Parallelnbsp;Scale�New Method of Surveying and keeping anbsp;Pheld Book�Gunner�s Callipers�Gunner�s Qua-di-anf�Gunner's Level, amp;e.

iry;.

LIST

OF AUTHORS,

Dalrymple . . . Essay on. Nautical Surveying, .... London, Eckhardt .... Description d�un Graphometre,.. A la Haye,

llobertson . .. Treatise of Mathematical Instruments, ....................London,

WORKS

liy the late Avinoi!. of these Essays, ?iow published and sold If W. mid S. Jones, IMborn^ London.

I. AN ESSAY ON ELECTRICITY, explaining clearly and fully the Principles of that useful Science, describing the various Instruments that have been contrived either to illustrate thenbsp;Theory, or render the Practice of it entertaining. The difterentnbsp;Modes in which the Electrical Fluid may be applied to the human Frame for Medical Purposes, are distinctly and clearlynbsp;pointed out, and the necessary Apparatus explained. To whichnbsp;is added AN ESSAY ON MAGNETISM. Third Edition, 8vo.nbsp;Price 6s. illustrated with seven Plates.

�II. AN ESSAY ON VISION, briefly explaining the Fabric of the Eye, and the nature of vision; intended for the Service ofnbsp;those whose Eyes arc weak and impaired, enabling them to formnbsp;an accurate Idea of the State of their Sight, the Means of preserving it, together with proper Rnles for ascertaining whennbsp;Spectacles are necessary, and how to choose them without injuring the Sight.

III. nbsp;nbsp;nbsp;ASTRONOMICAL AND GEOGRAPHICAL ESSAYS, containing, 1. A full and comprehensive View, on anbsp;new Plan, of the general Principles of Astronomy, with a largenbsp;Account of the Discoveries of Mr. Herschel. 2. The Use of thenbsp;Celestial and Terrestrial Globes, exemplified in a greater Varietynbsp;of Problems than are to be found in any other Work; they arenbsp;arranged under distinct Heads, and interspersed with much curious but relative Information. 3. The Description and Lise ofnbsp;*mall Orreries or Planetaria, Src. 4. An Introduction to Practical Astronomy, by a Set of easy'and entertaining Problems.nbsp;Second Edition, 8vo. Price 10s. �d. in Boards, illustrated withnbsp;twenty-one Plates.

IV. nbsp;nbsp;nbsp;AN INTRODUCTION TO PRACTICAL ASTRONOMY, or the Use of the Quadrant and Equatorial, beingnbsp;extracted from the preceding Work. Sewed, with two Plates,nbsp;2s. 6d.

V. AN APPENDIX to the GEOMETRICAL AND GRAPHICAL ESSAYS, containing the following Table bynbsp;Mr- John Gale, viz-, a Table of the Northings, Southings, Eastings, and Westings, to every Degree and fifteenth Minute of thenbsp;Ouadrant, Radius from 1 to 100, with all the intermediatenbsp;Numbers, computed to three Places of Decimals-

In the Press,

MICROSCOPE,

PHILOSOPHY,

In Five Volumes, 8vo. The Second Edition, with many Plates, considerable Alterations and Improvements j containingnbsp;more complete Explanations of the Instruments, Machines, See.nbsp;and the Description of many others not inserted in the formernbsp;Edition, by W. Jones.

CONTENTS.

Psgf

A List of the Principal Instruments and their Prices^ as made and sold by \Y, and S. JoseSj Holborn, London .. 5iG

GEOMETRICAL

ESSAY

Cteometry originally signified, according to the etymology of the name, the art of measuringnbsp;the earth; but is now the science that treats of^nbsp;and considers the properties of magnitude in general�In other words, extension and figure arenbsp;the objects of geometry. It is a science, in whichnbsp;human reason has the most ample field, and cannbsp;go deeper, and with more certainty, than in anynbsp;other. It is divided into two parts, theoreticalnbsp;and practical.

Theoretical geometry considers and treats of first principles abstractedly. Practical geometrynbsp;applies these considerations to the purposes of life.nbsp;By practical geometry many operations are performed of the utmost importance to society andnbsp;the arts. � The effects thereof are extendednbsp;through the principal operations of human skill:nbsp;it conducts the soldier in the field, the seaman on

The invention of geometry has been, by all the most eminent writers on the science, attributednbsp;to the Egyptians; and, that to the frequent inundations of the river Nile upon the country, we owenbsp;the rise of this sublime branch of human knowledge; the land-marks and boundaries being innbsp;this way destroyed, the previous knowledge of thenbsp;figure and dimensions was the only method of ascertaining individual property again. But, surely,nbsp;it is not necessary to gratify learned curiositynbsp;by such accounts as these; for geometry is an artnbsp;that must have grown with man; it is, in a greatnbsp;measure, natural to the human mind; we were'nbsp;born spectators of the universe, which is the kingdom of geometry, and arc continually obliged tonbsp;judge of heights, measure distances, ascertain thenbsp;figure, and estimate the bulk of bodies.

' The first definition in geometry is a foint, which is considered by geometricians, as that which hasnbsp;no parts or magnitude.

A plane angle is an opening, or corner, made by' two strait lines meeting one another.

upon another CD, makes angles ABC, ABD, on each side equal to one another; each of thesenbsp;angles is called a right angle-, and the line A B isnbsp;said to he perpendicular to the line C D.

It is usual to express an angle by three letters, that placed at the angular point being ahvays irrnbsp;the middle; as B is the angle of A B C..

A line A Vgt;,fig. Q,, plate A, cutting another line CD in E, Avill make the opposite angles equalsnbsp;namely, the angle A E C equal to B E D, andnbsp;A E D equal to B E C.

A line A B, fig. 3, plate A, standing any-way upon another CD, makes two angles CBA, ABDjnbsp;which, taken together, are equal to two rightnbsp;angles.

An equilateral triangle is that which has three equal sides,nbsp;nbsp;nbsp;nbsp;^

In a right-angled triangle, the side opposite to the right angle is called the hypothenuse.

In the same triangle, opposite to the greater side is the greater angle; and opposite to thenbsp;greater angle is the greater side.

If any side of a plane triangle be produced, the outward angle will be equal to both the inwardnbsp;remote angles.

The three angles of any plane triangle taken together, are equal to two right angles.

Parallel lines are those which have no inclination towards each other, or which arc every-where equidistant.

All plane figures, bounded by four right lines, are called quadrangles, or quadrilaterals.

A square is a quadrangle, whose sides arc all equal, and its angles allnbsp;right angles.

A rhombus is a quadrangle, whose sides are all equal, but its angles notnbsp;right angles.

A rhomboid is a parallelogram, whose j 7 angles are not right angles.nbsp;nbsp;nbsp;nbsp;/_/

A right line joining any two opposite angles of a four-sided figure, is called the diagonal.

Polygons having five sides, are pentagons ^ those having six sides, hexagons-, with seven sides,nbsp;heptagons-, and so on.

The base of any figure is that side on which it is supposed to stand, and the altitude is thenbsp;perpendicular falling thereon from the oppositenbsp;angle.

If a triangle and parallelogram have equal bases and equal altitudes, the triangle is half the parallelogram.

A circle is a plane figure, bounded by a curve line called the circumference, every part whereof isnbsp;equally distant from a point within the same figure,nbsp;called the center.

Any right line drawn from the center to the circumference of a circle, is called a radius.

The circumference of every circle is supposed to be divided into 300 equal parts, called degrees;nbsp;each degree into 6o equal parts, called minutes,^ amp;c.

A quadrant of a circle will therefore contain 00 degrees, being a fomth part of 300.

Equal angles at the centers of all circles, will intercept equal numbers of degrees, minutes, amp;c.nbsp;in their circumferences.

The measure of every plane angle is an arch of a circle, wdiose center is the angular point, and isnbsp;said to be of so many degrees, minutes. See. as arcnbsp;contained in its measuring arch.

All right angles, therefore, are of go degrees, or contain QO degrees, because their measure is anbsp;quadrant.

The three angles of every plane triangle taken together, contain 180 degrees, being equal to twonbsp;right angles.

In a right-angled plane triangle, the sum of its two acute angles is go degrees.

The complement of an arch, or of an angle, is its difference from a quadrant c-r a right angle.

The supplement of an arch, or of an angle, is its difference from a semicircle, or two rightnbsp;angles.

The magnitudes of arches and angles are determined by certain strait lines, appertaining to a circle, called chords, sines, tangents, amp;c.

A sector is any part of a circle hounded by an arch, and two radii drawn to its extremities.

The sine of an arch is a line drawn from either end of it, perpendicular to a diameter meeting thenbsp;other end.

The versed sine of an arch is that part of the diameter intercepted between the sine and the end of the said arch.

The tangent of an arch is a line proceeding from either end, perpendicular to the radius joining it;nbsp;its length is limited by a line drawn from thenbsp;center, through the other end.

The secant of an arch is the line proceeding from the center, and limiting the tangent of thenbsp;same arch.

The cosine and co-tangmt, amp;c. of any arch is ^he sine and tangent, amp;c. of its complement.

Thus \VLfig. A, plate 4, FO is the chord of the arch F V O, and F R is the sine of the archesnbsp;T V, F A D; R V, R D arc the versed sines of thenbsp;arches FV, FAD.

The chord of 60�, the sine of go�, the versed sine of 90�, the tangent of 45, and the secant ofnbsp;0.0, are all equal to the radius.

It is obvious, that in making use of these lines, Ave must always use the same radius, otherwisenbsp;there would be no settled proportions betweennbsp;them.

Whosoever considers the whole extent and depth of geometry, will find that the main designnbsp;of all its speculations is mensuration. To this thenbsp;Elements of Euclid are almost entirely devoted,nbsp;and this has been the end of the most labourednbsp;geometrical disquisitions, of either the ancientsnbsp;or moderns.

Now the whole mensuration of figures may be reduced to the measure of triangles, ivhich arenbsp;always the half of a rectangle of the same basenbsp;and altitude, and, consequently, their area is obtained by taking the half of the product of thynbsp;base multiplied by the altitude.

By dividing a polygon into triangles, and taking the value of these, that of the polygon is obtained; by considering the circle as a polygon, Avith an infinite number of sides, we obtain thenbsp;nicasure thereof to a sufficient degree of accuracy.

The theory of triangles is, as it were, the hinge upon which all geometrical knowledge turns.

All triangles are more or less similar, according as their angles are nearer to, or more remote from,nbsp;equality.

The similitude is perfect, when all the angles of the one are equal respectively to those of thenbsp;other; the sides are then also proportional.

The angles and the sides determine both the relative and absolute size, not only of triangles,nbsp;but of all things.

Strictly speaking, angles only determine the relative size; equiangular triangles may be of verynbsp;unequal magnitudes, yet perfectly similar.

But, when they are also equilateral, the one having its sides equ d to the homologous sides ofnbsp;the other, they are not only similar and equian-gled, but are equal in every respect.

The angles, therefore, determine the relative species of the triangle; the sides, its absolute size,nbsp;and, consequently, that of every other figure, asnbsp;all are resolvible into triangles.

Yet the essence of a triangle seems to consist much more in the angles than the sides; for thenbsp;angles are the true, precise, and determined boundaries thereof; their equation is always fixed andnbsp;limited to two right angles.

The sides have no fixed equation, but may be extended from the infinitely little, to the infinitelynbsp;great, without the triangle changing its naturenbsp;and kind,

It is in the theory of isoperimetrical figures'¹ that we feel how efficacious angles are, and hownbsp;inefficacious lines, to determine not only the kind,nbsp;but the size of the triangles, and all kinds of figures.

For, the lines still subsisting the same, we see how a square decreases, in proportion as it isnbsp;changed into a more oblique rhomboid; and thusnbsp;acquires more acute angles. The same observation holds good in all kinds of figures, whethernbsp;plane or solid.

Of all isopcrimctrical figures, the plane triangle and solid triangle, or pyramid, are the least capacious; and, amongst these, those have the leastnbsp;capacity, whose angles are most acute.

But curved surtaces, and curved bodies, and, among curves, the'circle and sphere, are tliosenbsp;whose capacity are the largest, being formed, ifnbsp;we may so speak, of the most obtuse angles.

The theory of geometry may, therefore, be reduced to the doctrine of angles, for it treats oniyj' of the boundaries of things, and by angles the ultimate bounds of all things are formed. It is thenbsp;angles which give them their figure.

Angles are measured by the circle; to these we may add parallels, which, according to the signification of the term, are the source of all geometrical similitude and comparison.

The taking and measuring of angles is the chief operation in practical geometry, and of great usenbsp;and extent in surveying, navigation, geography,nbsp;astronomy, amp;c. and the instruments generallynbsp;used for this purpose, arc quadrants, sextants, theodolites, circumferentors, amp;c. as described in thenbsp;following pages. It is necessary for the learnernbsp;first to be acquainted with the names and usesnbsp;of the drawing instruments; which are as follow.,

Common Names of the principal Instruments, as represented in Plates 1, 2, and 3.

B, a pair of best drawing compasses; h, the plain point with a joint; c, the ink point; d, thenbsp;dotting point; e, the pencil or crayon point;nbsp;PQ, additional pieces fitting into the place ofnbsp;the moveable point h, and to which the other partsnbsp;are fitted.

F, a pair of bow compasses for ink; G, a ditto for a pencil; FI, a pair of ditto with a plain pointnbsp;for stepping minute divisions; h, a screw to onenbsp;of the legs thereof, which acts like the spring legnbsp;of the hair compasses.

I K, the drawing pen; I, the upper part; I, the protracting pin thereof; K, the lower, or pen part.

V, a pair of portable compasses which contabis the ink and pencil points within its two legs,

�3, the double barred ditto.

D, nbsp;nbsp;nbsp;the cross barj-cd parallel rule. Of thesenbsp;rules, that figured at C is the most perfect.

Fig. 7 and 8, two views of a pair of proportionable compasses, with an adjusting screw.

Fig. g, a pair of sectoral compasses. In this instrument are combined the sector, beam elliptical, and calliper compasses; fig. Q, a, the square fornbsp;ellipses; be, the points to work therein; de, thenbsp;calliper points.

The strictness of geometrical demonstration admits of no other instruments, than a rule and anbsp;pair of compasses. But, in proportion as thenbsp;practice of geometry was extended to the differentnbsp;arts, either connected with, or dependent upon it,nbsp;new instruments became necessary, some to answer peculiar purposes, some to facilitate operation, and others to promote accuracy.

It is the business of this work to describe these instruments, and explain their various uses. Innbsp;performing this task, a difficulty arose relative tonbsp;the arrangement of the subject, whether eachnbsp;instrument, with its application, should be dc-

scribed separately, or whether the description should be introduced under those problems, fornbsp;whose performance they were peculiarly designed.nbsp;After some consideration, I determined to adoptnbsp;neither rigidly, but to use either the one or thenbsp;other, as they appeared to answer best the purposes of science.

As almost every artist, whose operations are connected with mathematical designing, furnishesnbsp;himself with a case of drawing instrument s suitednbsp;to his peculiar purposes, the]^ are fitted up in various modes, some containing more, others, fewernbsp;instruments. The smallest collection put into anbsp;case, consists of a plane scale, a pair of compassesnbsp;with a moveable leg, and two spare points, whichnbsp;may be applied occasionally to the compasses; onenbsp;of these points is to hold ink; the other, a portenbsp;crayon, for holding a piece of black-lead pencil.

A pair of large compasses with a moveable point, an ink point, a pencil point, and one fornbsp;dotting; either of these points may be inserted innbsp;the compasses, instead of the moveable, leg.

The plain scale, the protractor, and parallel rule, are sometimes so constructed, as to form butnbsp;one instrument; but it is a construction not to bo

recommended, as it injures the plain scale, and lessens the accuracy of the protractor. In a casonbsp;with the best instruments, the protractor and plainnbsp;scale are always combined. The instruments innbsp;niost general use are those of six inches; instruments are seldom made longer, but often smaller.nbsp;Those of six inches are, however, to be preferred,nbsp;in general, before any other size; they will eftectnbsp;all that can be performed with the shorter ones,nbsp;while, at the same time, they are better adapted tonbsp;large work.

Large collections are called, magazine cases of instruments-, these generally contain

A pair of six inch compasses with a moveable leg, an ink point, a dotting point, the crayonnbsp;point, so contrived as to hold a whole pencil, twonbsp;additional pieces to lengthen occasionally one legnbsp;of the compasses, and thereby enable them tonbsp;measure greater extents, and describe circles of a.nbsp;larger radius.

A pair of small hair compasses, with a head similar to those of the bow compasses.

A pair of sectoral compasses, forming, at the same time, a pair of beam and calliper compasses.

Compasses are made either of silver or brass,-but with steel points. The joints should always be framed of different substances; thus, one sidcynbsp;or part, should be of silver or brass, and the othernbsp;of steel. The difference in the texture and poresnbsp;of the two metals causes the parts to adhere lessnbsp;together, diminishes the wear, and promotes uniformity in their motion. The truth of the wmrknbsp;is ascertained by the smoothness and equality of thenbsp;motion at the joint, for all shake and irregularity isnbsp;a certain sign of imperfection. The points shouldnbsp;be of steel, so tempered, as neither to be easilynbsp;bent or blunted; not too fine and tapering, and yetnbsp;meeting closely when the compasses are shut.

As an instrument of art, compasses are so w'ell known, that it would be superfluous to enumeratenbsp;their various uses; suffice it then to say, that theynbsp;are used to transfer small distances, measure givennbsp;spaces, and describe arches and circles.

If the arch or circle is to be described obscurely^ fhc steel points are best adapted to the purpose:nbsp;if it is to be in ink or black lead, either the di'aw-ing pen, or crayon points are to be used.

To use a fair of compasses. Place the thumb and middle finger of the right hand in the oppositenbsp;hollows in the shanks of the compasses, then pressnbsp;the compasses, and the legs w'ill open a little way;nbsp;this heilig done, push the innermost leg with thenbsp;third finger, elevating, at the same time, the furthermost, with the nail of the middle finger, tillnbsp;the compasses are sufficiently opened to receivenbsp;the middle and third finger; they may then be extended at pleasure, by pushing the furthermost legnbsp;outwards with the middle, or pressing it inwardsnbsp;�with the fore finger. In describing circles, ornbsp;arches, set one foot of the compasses on the center, and then roll the head of the compasses between the middle ami fore finger, the other pointnbsp;pressing at the same time upon the paper. Theynbsp;shoidd be held as upright as possible, and carenbsp;should be taken not to press forcibly upon them,nbsp;but rather to let them act by their own weight;nbsp;the legs should never be so far extended, as to formnbsp;an obtuse angle with the paper or plane, on whichnbsp;they are used.

The ink and crayon points have a joint just under that part which fits into the compasses, by this they may be always so placed, as to be set nearlynbsp;perpendicular to the paper; the end of the shanknbsp;of the best compasses is framed so as to form anbsp;strong spring, to bind firmly the movcnble points,nbsp;and prevent them from shaking. This is foundnbsp;to be a more ctrcctual method than that by a screw.

B, p/a/-e 1, represents a pair of the best compasses, wdth the plain point, c, the ink, f thenbsp;dotting, c, the crayon point.

In small cases, the crayon and ink points ar� joined by a middle piece, with a socket at eachnbsp;end to receive the points, which, by this meanSjnbsp;only occupy one place in the case.

Two additional pieces, fig. P3 Q, flate 1, are often applied to these compasses; these, by lengthening the leg enable them to strike larger circles, or measure greater extents, than they wouldnbsp;otherwise perform, and that without the inconveniences attending longer compasses. When compasses are furnished with this additional piece, thenbsp;moveable leg has a Jointj as at h, that it may benbsp;placed perpendicular to the paper;

Of the hair compasses, fig. L, plate 1. They are so named, on account of a contrivance in the shanknbsp;to set them with greater accuracy than can be effected by the motion of the joint alone. One ofnbsp;the steel points is fastened near the top of the compasses, and may be moved very gradually by turning the screw n, either backwards or forwards.

To use these compasses. 1. Place the leg, t� �which the screw is annexed, outermost; 2. Set thenbsp;fixed leg on that point, from whence the extent isnbsp;to be taken; 3. Open the compasses as nearly asnbsp;possible t� the required distance, and then makenbsp;the points accurately coincide therewith by turning the screw.

Of the bow compasses, fig. F, plate 1. These are a small pair, usually with a point for ink; they arenbsp;used to describe small ai'ches or circles, which theynbsp;do much more conveniently than large compasses,nbsp;not only on account of their size, but also from thenbsp;shape of the head, which rolls with great ease between the fingers. It is, for this I'cason, custo-ma^'y to put into magazine cases of instruments,nbsp;a small pair of hair compasses, fig. H, plats 1, with anbsp;head similar to the bows; these are principally

Rsedt for repeating divisions of a small but equal extent, a practice that has acquired the name ofnbsp;stepping.

Of the drawing pen and protracting pin, fig. I K, plate 1. The .pen part of this instrument is usednbsp;to draw strait lines; it consists of two blades wdthnbsp;steel points fixed to a handle, the blades are sonbsp;bent, that the ends of the steel points meet, and yetnbsp;leave a sufficient cavity for the ink; the blades maynbsp;be opened more or less by a screw, and, beingnbsp;properly set, will draw a line of any assigned thickness. One of the blades is framed with a joint,nbsp;that the points may be separated, and thus cleanednbsp;more conveniently; a small quantity only of inknbsp;should be put at one time into the drawing pen,nbsp;and this should be placed in the cavity, betweennbsp;the blades, by a common pen, or the feeder; thenbsp;drawing pen acts better, if the feeder, or pen, bynbsp;which the ink is Inserted, be made to pass throughnbsp;the blades. To use the drawing pen, first feed itnbsp;with ink, then regulate it to the thickness of thenbsp;required line by the screw. In drawing lines, incline the pen a small degree, taking care, however,nbsp;that the edges of both the blades touch the paper,nbsp;beeping the pen close to the rule and in the samenbsp;direction during the whole operation; the bladesnbsp;should always be wiped very clean, before the pennbsp;is put away.

These directions are equally applicable to the mk point of the compasses, only observing, thatnbsp;when an arch or circle is to be described, of morenbsp;than an inch radius, the point should be so bent,nbsp;that the blades of the pen may be nearly perpendicular to the paper, and both of them touch it atnbsp;the same time.

Tlhe protracting. pin k, is only a short piece rf steel wire, with a very fine point, fixed at one end

of the Upper part of the handle of the drawing pen. It is used to mark the intersection of lines,nbsp;or to set oft� divisions from the plotting scale, andnbsp;protractor.

The feeder, fig. O, plate 1, is a thin flat piece of metal; it sometimes forms one end of a eap to fitnbsp;on a pencil, or it is framed at the top of the tracing point, as in the figure. It serves to place thenbsp;ink between the blades of the drawing pens, or tonbsp;pass between them when the ink does not flownbsp;freely. The tracing point, or polntrel, is a pointednbsp;piece of steel fitted to a brass handle; it is usednbsp;to draw obscure lines, or to trace the out-lines ofnbsp;a drawing or print, when an exact copy is required,nbsp;an article that will be fully explained in the coursenbsp;of this work; it forms the bottom part of thenbsp;feeder O.

Of triangular compasses. A pair of these are represented ^X.fig. N, plate 1. They consist of a pair of compasses, to whose head a joint and socket isnbsp;fitted for the reception of a third leg, which maynbsp;be moved in almost every direction.

These compasses, though exceedingly useful, are but little known; they are very serviceable innbsp;copying all kinds of drawings, as from two fixednbsp;points they will always ascertain the exact positionnbsp;of a third point.

Fig. \1, plate 3, represents another kind, which has some advantages over the preceding. l. Thatnbsp;there are many situations so oblique, that the thirdnbsp;point cannot be ascertained by the former, thoughnbsp;it may by these. 2. It extends much further thannbsp;the other, in proportion to its size. 3. The pointsnbsp;are in all positions perpendicular to the paper.

Of wholes and halves, fig. R, plate 1. A name given to these compasses, because that whennbsp;the longer legs are opened to any given line, the

Fig. V, represents a new pair of very curious and portable compasses, which may be considerednbsp;as a case of instruments in itself. The ink andnbsp;pencil points slide into the legs by spring socketsnbsp;at a; the ink, or pencil point, is readily placed,nbsp;hy only sliding either out of the socket, reversingnbsp;it, and sliding in the plain point in its stead.

Proportional compasses. These compasses arc of two kinds, one plain, represented j^. A, 1;nbsp;the other with an adjusting screw, ot which therenbsp;are two views, one edgewaySj 8, plate 3, thenbsp;other in the front, Jig. 7, plate 3: the principlenbsp;On which they both act is exactly the same; thosenbsp;with an adjusting screw are more easily set to anynbsp;given division or line, and are also more firmlynbsp;fixed, when adjusted.

There is a groove in each shank of these compasses, and the center is moveable, being constructed to slide with regularity in these grooves, and when properly placed, is fixed by a nut andnbsp;screw; on one side of these grooves are placed twonbsp;scales, one for lines, the other for circles. By thenbsp;scale of lines, a right line may be divided into anynbsp;Dumber of equal parts expressed on the scale. Bynbsp;the scale for circles, a regular polygon may be inscribed in a circle, provided the sides do not exceed the numbers on the scale. Sometimes arenbsp;added a scale for superficies and a scale for solids.

To inscribe in a circle a regular polygon of Xl sides. 1*. Shut the compasses. 2. Unscrew thenbsp;milled nut, and set the division on the slider tonbsp;coincide with the l'2th division on the scale ofnbsp;circles. 3. Fasten the milled nut. 4. Open thenbsp;compasses, so that the longer legs may take the radius, and the distance between the shorter legsnbsp;will be the side of the required polygon.

To use the proportional compasses with an adjust^ ing screw. The application being exactly the samenbsp;as the simple one, we have nothing more to describe than the use and advantage of the adjustingnbsp;screw. 1. Shut the legs close, slacken the screwsnbsp;of the nuts g and �; move the nuts and slider k tonbsp;the division wanted, as near as can be readily donenbsp;by the hand, and screw fast the nut f: then, bynbsp;turning the adjuster h, the mark on the slider knbsp;may be brought exactly to the division: screw fastnbsp;the nut^. 2. Open the compasses; gently liftnbsp;the end e of the screw of the nut � out ot the holenbsp;in the bottom of the nut move the beam roundnbsp;its pillar a, and slip the point e into the hole innbsp;the pin n, which is fixed to the under leg; slackennbsp;the screw of the nut/; take the given line betweennbsp;the longer points of the compasses, and screw fastnbsp;the nut �; then may the shorter points of the compasses be used, without any danger of the legsnbsp;changing their position; this being one of the inconveniences that attended the proportional compasses, before this ingenious contrivance.

Fig. 10, plate 3, represents a pair of beam com-fasses-, they ai'e used for taking off and transferring divisions from a diagonal or nonius scale, describing large arches, and bisecting lines or arches. It is the instrument upon which Mr. Bi!,-d princi-

fjally depended, in dividing those instruments, 'vhose accuracy has so much contributed to thenbsp;progress of astronomy. These compasses consistnbsp;of a: long beam made of brass or wood, furnished'nbsp;With two brass boxes, the one fixed at the end, thenbsp;other sliding along the beam, to any part of whichnbsp;it may be firmly fixed by the screw P. An adjusting screw and micrometer are adapted to thenbsp;box A at the end of the beam; by these, the pointnbsp;connected therewith may be moved with extremenbsp;regularity and exactness, even less than the thousandth part of an inch.

passes, with a micrometer and adjusting screw, for accurately ascezlaining and laying down small distances.

Fig. 1 plate 3, represents a scale of equal parts, constructed by Mr. Sisson; that figured here contains two scales, one of three chains, the other ofnbsp;four chains in an inch, being those most frequentlynbsp;used; each of these is divided into 10 links, whichnbsp;are again subdivided by a nonius into single links;nbsp;the index carries the protracting pin for settingnbsp;off the lengths of the several station lines on thenbsp;plan. By means of an instrument of this kind,nbsp;the length of a station line may be laid down onnbsp;paper with as much exactness as it can be measured on land.

Parallel lines occur so continually in even- species,of mathematical drawing, that it is no wonder so many instruments have been contrived to delineate them with more expedition than could benbsp;effected by the general geometrical methods: of

the various contrivances for this purpose, the fol--lowing are those most approved.

'1. The common parallel rule, fig. A, plate 2. This consists of two strait rules, which are connectednbsp;together, and always maintained in a parallel position by the two equal and parallel bars, whichnbsp;move very freely on the center, or rivets, by whichnbsp;they are fastened to the strait rules.

2. nbsp;nbsp;nbsp;The double parallel rule, fig. B, plate 2. Thisnbsp;instrument is constructed exactly upon the samenbsp;principles as the foregoing, but with this advantage, that in using it, the moveable rule maynbsp;always be so placed, that its ends may be exactlynbsp;over, or even with, the ends of the fixed rule,nbsp;whereas, in the former kind, they are always shifting away from the ends of the fixed rule.

This instrument consists of two equal flat rules, and a middle piece; they are connected togethernbsp;by four brass bars, the ends of two bars are ri-vetted on the middle line of one of the strait rules;nbsp;the ends of the other two bars are rivetted on thenbsp;middle line of the other strait rule; the other endsnbsp;of the brass bars are taken two and two, and rivetted on the middle piece, as is evident from thenbsp;figure; it would be needless to observe, that thenbsp;brass bars move freely on their rivets, as so manynbsp;centers.

3. nbsp;nbsp;nbsp;Of the improved double parallel 7-ule, fig. C,nbsp;plate 2. The motions of this rule are more regularnbsp;than those of the preceding one, but with some-W'hat more friction; its construction is evidentnbsp;from the figure; it w'as contrived by the ingeniousnbsp;mechanic, Mr. Haywood.

4. nbsp;nbsp;nbsp;The cross barred parallel rule, fig. D, plate 2.nbsp;In this, two strait rules are joined by two brass bars,nbsp;which cross each other, and turn on their inter-

'Section as on a center; one end of eaeh bar moves on a eenter, the other slides in a groove, as thenbsp;rules reeede from each other.

As the four parallel rules above described, are all used in the same way, one problem will servenbsp;for them all; ex. gr. a right line being given tonbsp;draw a line parallel thereto by either of the foregoing instruments:

Set the edge of the uppermost rule to the given line; press the edge of the tower rule tight to thenbsp;paper with one hand, and, with the other, movenbsp;the upper rule, till its edge coincides with the given point; and a line drawn along thenbsp;through the point, is the line required.

5. Of the rolling parallel rule. This instrument was contrived by Mr. Eckhardt, and the simplicitynbsp;of the consti'uction does credit to the inventor; itnbsp;must, however, be owned, that it requires somenbsp;practice and attention to use it with success.

Fig. E. plate 2, represents this rule; it is a rectangular parallelogram of black ebony, with slips of ivory laid on the edges of the rule; and dividednbsp;into inches and tenths. The rule is supported bynbsp;two small wTeels, whieh are connected togethernbsp;by a long axis, the wheels being exactly of thenbsp;same size, and their rolling surfaces being parallelnbsp;to the axis; when they are rolled backwards ornbsp;forwards, the axis and rule will move in a directionnbsp;parallel to themselves. The wheels are somewhatnbsp;indented, to prevent their sliding on the paper;nbsp;small ivory cylinders are sometimes affixed to thenbsp;rollers, as in this figure; they are called rollingnbsp;scales. The circumferences of these are so adjusted,nbsp;that they indicate, with exactness, the parts of annbsp;inch moved through by the rule.

In rolling these rules, one hand only must be '^sed, and the fingers should be placed nearly in

the middle of the rule, that one end may not have a tendency to move faster than the other. The^nbsp;wheels only should touch the paper when the rulenbsp;is moving, and the surface of the paper should benbsp;smooth and flat.

In using the rule with the rolling scales, to draw a line parallel to a given line at any determined distance, adjust the edge of the rule to the givennbsp;line, and pressing the edge down, raise the wheels-a little from the paper, and you may turn the cylinders round, to bring the first division to the index ; then, if you move the rule towards you, looknbsp;at the ivory cylinder on the left hand, and thenbsp;numbers will shew in tenths of an inch, how muchnbsp;the rule moves. If you move the rifle from you,nbsp;then it will be shewn by the numbers on the rightnbsp;hand cylinder.

To raise a perpendicular from a given point on a given line. Adjust the edge of the rule to thenbsp;line^ placing any one of the divisions on the edgenbsp;of the rule to the given point; then roll the rule �nbsp;to any distance, and make a dot or point on thenbsp;papei', at the same division on the edge of the rulenbsp;through this point di-awthe pcrpendieular.

To let fall a perpendicular from any given point to a given line. Adjust the rule to the given line,nbsp;and roll it to the given point; then, observingnbsp;what division, or point, on the edge of the,rulenbsp;the given point comes to, roll the rule back againnbsp;to the given line, and the division, or point, on thenbsp;edge of the rule will shew the point on the givennbsp;line, to which the perpendicular is to be drawn.

By this method of drawing perpendiculars, squares and parallelograms may be easily drawn ofnbsp;any dimensions.

To divide any given line into a number of eq^ual parts. Draw a right line from either of the ex-

treme points of the given line, making any angle with it. By means of the rolling scales, dividenbsp;that line into as many inches, or parts of� an inch,nbsp;as will equal the number of parts into which thenbsp;given line is to be divided. Join the last point ofnbsp;division, and the extreme point of the given line:nbsp;to that line draw parallel lines through the othernbsp;points of division, and they will divide the givennbsp;line into equal parts.

6. Of the square parallel rule. The evident advantages of the T square and drawing board over every other kind of parallel rule, gave rise to a variety of contrivfinces to be used, when a drawingnbsp;board was not at hand, or could not, on accountnbsp;of the size of the paper, be conveniently used;nbsp;among these are, 1. The square parallel rule^nbsp;2. The parallel rule and protractor, both of whichnbsp;I contrived some years since, as substitutes to thenbsp;T square. The square parallel rule, besides itsnbsp;Use as a parallel rule, is peculiarly applicable tonbsp;the mode of plotting recommended by Mr. Gale.nbsp;Its use, as a rule for drawing parallel lines at givennbsp;distances from each other, for raising perpendiculars, forming squares, rectangles, amp;c. is evidentnbsp;from a view of the figure alone; so that what haS'nbsp;been already said of other rules, will be sufficientnbsp;to explain how this may be used. It is also evident, that it will plot with as much accuracy asnbsp;the beam, fig. 11,plated.

Tig. TGli, plate�1, represents this instrument; the two ends F G are lower than the rest of thenbsp;rule, that weights may be laid on them to steadynbsp;the rule, when both hands are wanted. The two-^les are fastened together by the brass ends, thenbsp;frame is made to slide regularly betwmen the twonbsp;rules, carrying at the same time, the rule H in a po-*dion at right angles to the edges of the rule E G.

There are slits in the frame a, b, with marks to coincide with the respective scales on the rules,nbsp;while the frame is moving up or down; c is anbsp;point for pricking off with certainty divisions fromnbsp;the scales. The limb, or rule H, is made to takenbsp;off, that other rules with different scales of equalnbsp;parts may be applied; when taken off, this instrument has this further advantage, that if the distances, to which the parallels are to be drawn,nbsp;exceed the limits of the rules, the square part,nbsp;when taken oft', may be used with any strait rule,nbsp;by applying the perpendicular part against it.

7. Fig. DLL,, plate 2, represents the protract^ hig parallel rule-, the uses of this in drawing parallel lines in different directions, are so evidentnbsp;from an inspection of the figure, as to render anbsp;particular description unnecessary. It answers allnbsp;the purposes of the T square and bevil, and is peculiarly useful to surveyors for plotting and protracting, which will be seen, when we come tonbsp;treat of those branches.

M N O is a parallel rule upon the same principle as the former: it was also contrived by Mr. Haywood. It is made either of wood edged withnbsp;ivory, or of brass, and any scales of equal partsnbsp;placed on it to the convenience of the purchaser.nbsp;Each of the rules M N turns upon a center; itsnbsp;use as a parallel rule is evident fiom the figure,nbsp;but it would require more pages than can be sparednbsp;to describe all the uses of which it is capable; itnbsp;forms the best kind of callipers, or guage; servesnbsp;for laying down divisions and angles with peculiarnbsp;accuracy; answers as a square, or bevil; indeed,nbsp;scarce any artist can use it, without reaping considerable advantage from it, and finding uses pe^nbsp;culiar to his own line of business.

�8. Of the German parallel rule, fig. 1, plate 3, This was, probably, one of the first instruments invented to facilitate the drawing of parallel lines.nbsp;It has, however, only been introduced within thesenbsp;few years among the English artists; and, as thisnbsp;introduction probably came from some Germannbsp;Work, it has thence acquired its name. It consists of a square and a strait rule, the edge of thenbsp;Square is moved in use by one hand against thenbsp;rule, w'hich is kept steady by the other, the edgenbsp;having been previously set to the given line; itsnbsp;Use and construction is obvious from fig. 1, plate 3.nbsp;Simple as it is in its principle, it has undergonenbsp;some variations, two of which I shall mention;nbsp;the one by Mr. Jackson, of Tottenham; the othernbsp;by Mr. Marquois, of London.

Mr. Jacksons, fig. �, plate 3, consists of tv^o equal triangular pieces of brass, ivory, or wood,nbsp;AB C, D E F, right angled at B and E; the edgesnbsp;AB and AC are divided into any convenient numr-ber of equal parts, the divisions in each equal;nbsp;B C and D E are divided into the same number ofnbsp;equal parts as AC, one side of DEE may be divided as a protractor.

To draw a line G LI, fig. V, plate 1, parallel to a ^iven line, through a point P, or at a given distance.

Place the edge DE upon the given line IK, and let the instrument form a rectangle, then slide thenbsp;dipper piece till it come to the given point or distance, keeping the other steady with the left hand,nbsp;Rnd draw the line G H. By moving the piecenbsp;equal distances by the scale on B C, any numbernbsp;of equidistant parallels may be obtained.

If the distance, fig. X, plate 1, between the given and required lines be considerable, place A B ^pon IK, and E F against A C, then slide thenbsp;pieces alternately till D E comes to the required

point; in this manner it is easy also to constrttct any square or rectangle. See.

From any given point or angle P, jig. S, plate to let fall or raise a perpendicular on a given lineinbsp;Place either of the edges AC upon GP, and slidenbsp;AB upon it, till it comes to the point P, andnbsp;draw PH.

divide a line into any proposed number of equal parts, fig. T, plate 2. Find the proposednbsp;number in the scale BC, and let it terminate at G,nbsp;then place the rules in a rectangular form, andnbsp;move the whole about the point G, till the sidenbsp;D E touches H; now move D one, two, or threenbsp;divisions, according to the number and size of thenbsp;required divisions, down B C, and make a dot at I,nbsp;where DE cuts the line for the first part; thennbsp;move one or more divisions as before, make a second dot, and thus proceed till the whole be completed.

Of Marquous parallel scales. These consist of a right angled triangle, whose hypothenusc,or longestnbsp;side, is three times the length of the shortest, andnbsp;two rectangular scales. It is from this relativenbsp;length of the hypothenuse that these scales derivenbsp;their peculiar advantages, and it is this alone thatnbsp;renders them different from the common German,nbsp;parallel rule; for this we are much indebted tonbsp;Mr. Marquois.

What has been already said of the German rule, applying equally to those of Mr. Marquois�s, I shallnbsp;proceed to explain their chief and peculiar excellence. On each edge of the rectangular rule arenbsp;placed two scales, one close to the edge, the othernbsp;within this. The outer scale, Mr. Marquois termsnbsp;the artificial scale, the inner one, the natural scale:nbsp;the divisions on the efuter are always three timesnbsp;longer than those bn the inner scale, as, to derive

fiRy advantage from this invention, they must always bear the same proportion to each other, that the shortest side of the right angled triangle doesnbsp;to the longest. The triangle has a line near thenbsp;niiddle of it, to serve as an index, or pointer; whennbsp;in use, this point should be placed so as to coincide with the O division of the scales; the numbers on the scales are reckoned both ways fromnbsp;this division; consequently, by confining the ride,nbsp;and sliding the triangle either way, parallel linesnbsp;may be drawn on either side of a given line, at anynbsp;distance pointed out by the index on the triangle.nbsp;The advantages of this contrivance are; 1. Thatnbsp;the sight is greatly assisted, as the divisions on thenbsp;outer scale are so much larger than those of thenbsp;inner one, and yet answer the same purpose, fornbsp;the edge of the right angled triangle only movesnbsp;through one third of the space passed over by thenbsp;index. 2. That it promotes accuracy, for all errornbsp;in the setting of the index, or triangle, is diminished one third.

Mr. Marquois recommends the young student to procure two rules of about two feet long, havingnbsp;One of the edges divided into inches and tenths,nbsp;find several triangles with their hypothenusc in different proportions to their respective perpendiculars. Thus, if you would make it answer for anbsp;Scale of twenty to an inch, the hypothenusc mustnbsp;be twice the length of the perpendicular; if a scalenbsp;of 30 be required, three times; of 40, four times;nbsp;of 50, five times, and so on. Tims also for intermediate proportions; if a scale of 25 is wanted,nbsp;the hypothenusc must be in the proportion of 25nbsp;to 2; if 35j of 7 to 42, amp;c. Or a triangle may benbsp;foi'ined, in which the hypothenusc may be so setnbsp;^ to bear any required proportion with the perpendicular.

This is an instrument used to protract, or lay down an angle containing any number of degrees,nbsp;or to find how many degrees are contained in anynbsp;given angle. There are two kinds put into casesnbsp;of mathematical drawing instruments; one in thenbsp;form of a semicircle, the other in the form of a parallelogram. The circle is undoubtedly the onlynbsp;natural measure of angles; when a strait line isnbsp;therefore used, the divisions thereon are derivednbsp;from a cirele, or its properties, and the strait linenbsp;is made use of for some relative eonvenience: it isnbsp;thus the parallelogram is often used as a protractor, instead of the semicircle, because it is in somenbsp;cases more convenient, and that other scales, amp;c.nbsp;may be placed upon it.

The semicircular protractor, fig. 2, plate 3, is divided into 180 equal parts or degrees, which are numbered at every tenth degree each way, for thenbsp;conveniency of reckoning either from the rightnbsp;towards the left, or from the left towards the right;nbsp;or the more easily to lay down an angle fromnbsp;either end of the line, beginning at each end udthnbsp;10, 20, amp;c. and proceeding to 180 degrees. Thenbsp;edge is the diameter of the semicircle, and thenbsp;mark in the middle points out the center. Fig. 3,nbsp;plate 3, is a protractor in the form of a parallelo-g?-arn: the divisions ure here, as in the semicircularnbsp;one, numbered both ways; the blank side represents the diameter of a circle. The side of thenbsp;protractor to be applied to the paper is made flat,nbsp;and that whereon the degrees are marked, is chamfered or sloped away to the edge, that an anglenbsp;may be more easily measured, and the divisions setnbsp;off with greater exactness.

Protractors of horn arc, from their transparency, very convenient in measuring angles, and raisingnbsp;perpendiculars. When they are out of use, theynbsp;sliould be kept in a book to prevent their warping.

Upon some protractors the numbers denoting the angle for regular polygons are laid down, tonbsp;avoid the trouble of a reference to a table, or thenbsp;operation of dividing; thus, the number 5, for anbsp;pentagon is set against 72�; the number 6, foranbsp;hexagon, against 00�; the number 7, for a heptagon, against 51^�; and so on.

Protractors for the purposes of surveying will he described in their proper place.

Application of the protractor to use. 1. A number of degrees being given, to protract, or lay dovjn annbsp;angle, whose measure shall be equal thereto.

Thus, to lay down an angle of 6o degrees from the point A of the line K^,fig. \ A, plate 3, applynbsp;the diameter of the protractor to the line AB, sonbsp;that the center thereof may coincide exactly wnthnbsp;the point A; then with a protracting pin makenbsp;a fine dot at C against 00 upon the limb of thenbsp;protractor; now remove the protractor, and drawnbsp;a line from A through the point C, and the anglenbsp;CAB contains the given number of degrees.

2. nbsp;nbsp;nbsp;To find the number of degrees contained in anbsp;given angle BAC.

Place the center of the protractor upon the an-gnlar point A, and the fiducial edge, or diameter, exactly upon the line AB; then the degree uponnbsp;the limb that is cut by the line C A will be thenbsp;measure of the given angle, which, in the presentnbsp;instance, is found to be 6o degrees.

3. nbsp;nbsp;nbsp;From a given point A, in the line AB, to erectnbsp;a perpendicular to that line.

Apply the protractor to the line AB, so that the center may coincide with tbe point A, and the di'

vision marked QO may be cut by the line A B, then SL line DA drawn against the diameter of the protractor will be perpendicular to A B.

Further uses of this instrument will occur in most parts of this work, particularly its use in constructing polygons, which will be found undernbsp;their respective heads. Indeed, the general usenbsp;being explained, the particular application must benbsp;left to the practitioner, or this work would be unnecessarily swelled by a tedious detail and continual repetitions.

The divisions laid down on the plain scale arc of two kinds, the one having more immediatenbsp;relation to the circle and its properties, the othernbsp;being merely concerned with dividing straitnbsp;lines.

It has been already observed, that though arches of a circle are the most natural measures of an angle, yet in many cases right lines were substituted,nbsp;as being more convenient; for the comparison ofnbsp;one right line with another, is more natural andnbsp;easy, than the comparison of a right line with anbsp;curve; hence it is usual to measure the quantitiesnbsp;of angles not by the arch itself, which is describednbsp;on the angular point, but by certain lines describednbsp;about that arch. See definitions.

The-lines laid down on the plain scales for the measuring of angles, or the protracting scales, are,nbsp;1. A line of chonJs marked cho. �2. A line of sinesnbsp;marked sin. of tangents marked tan. of semitan-tangents marked st. and of secants marked sec,nbsp;this last is often upon the same line as the sines,nbsp;because its gradations do not begin till the sinesnbsp;end.

There arc two other scales, namely, the rhimibs, marked eu. and long, marked lon. Scales ofnbsp;latitude and hours are sometimes put upon thenbsp;plain seale; but, as dialling is now but very littlenbsp;studied, they are only made to order.

The divisions used for measuring strait lines are called scales of equal parts, and are of variousnbsp;lengths for the convenience of delineating any figure of a large or smaller size, according to thenbsp;fancy or purposes of the draughts-man. They are,nbsp;indeed, nothing more than a measure in miniaturenbsp;for laying down upon paper, amp;c. any known measure, as chains, yards, feet, amp;c. each part on thenbsp;scale answering to one foot, one yard, amp;Ci and thenbsp;plan will be larger or smaller, as the scale containsnbsp;a smaller or a greater number of parts in an inch.nbsp;Hence a variety of scales is usefxil to lay down linesnbsp;cf any required length, and of a convenient proportion with respect to the size of the drawing.nbsp;If none of the scales happen to suit the purpose,nbsp;Recourse should be had to the line of lines on thenbsp;sector; for, by the different openings of that instrument, a line of any length may be divided intonbsp;many equal parts as any person chuses.

Scales of equal parts are divided into two kinds, the one simple, the other diagonally divided.

Six of the simply divided scales are generally placed one above another upon the same rule;nbsp;they are divided into as many equal parts as thenbsp;wngth of the rule will admit of; the numbersnbsp;placed on the right hand, shew how many parts innbsp;inch each scale is divided into. The uppernbsp;Scale is sometimes shortened for the sake of introducing another, called the line of chords.

The first of the larger, or primary divisions, on every scale is subdivided into 10 equal parts, whichnbsp;�niall parts are those which give a name to the

scale; thus It is called a scale of �20, when IQ of these divisions are equal to one inch. If, therefore, these lesser divisions be taken as units, andnbsp;each represents one league, one mile, one chain,nbsp;or one yard^ amp;c. then will the larger divisions benbsp;so many tens; but if the subdivisions are supposednbsp;to be tens, the larger divisions will be hundreds.

To illustrate this, suppose it were required to set off from either of the scales of equal parts fl,nbsp;30, or 300 parts, either miles or leagues. Set onenbsp;foot of your compasses on 3, among the larger oVnbsp;primary divisions^ and open the other point till itnbsp;falls on the 6th subdivision, reckoning backwardsnbsp;or towards the left hand. Then will this extentnbsp;represent fl, 30, or 300 miles or leagues, amp;e. andnbsp;bear the same proportion in the plan as the linenbsp;measured does to the thing represented.

To adapt these scales to feet and inches, the first primary division is often duodecimally dividednbsp;by an upper line; therefore, to lay down any number of feet and inches, as for instance, eight feetnbsp;eight inches, extend the compasses from eight ofnbsp;the larger to eight of the upper small ones, andnbsp;that distance laid down on the plan will represent'nbsp;eight feet eight inches.

Of the scale of equal parts diagonally divided. The use of tliis scalp is the same as those alreadynbsp;described. But by it a plane may be more accurately divided than by the former; for any one ofnbsp;the larger divisions may by this be subdivided into-100 equal [rails; and, therefore, if the scale contains 10 of the larger divisions, any number undernbsp;1000 may be laid down with accuracy.

The diagonal scale is seldom placed on the same' side of the rule with the other plotting scales.nbsp;The first division of the diagonal scale, if it be �

�^qual pails, and at the opposite end there is usu-fidy half an inch divided iato 100 equal parts. If the scale be six inches long, one end has com-tnonly half an inch, the other a quarter of an inchnbsp;subdivided into lOOequal parts.

The nature of this scale will be better understood by considering its construction. For this purpose1

First. Draw eleven parallel lines at equal distances; divide the upper of these lines into such a number of equal parts, as the scale to be expressednbsp;IS intended to contain; from each of these divisions draw perpendicular lines through the eleven,nbsp;parallels.

Secondly. Subdivide the first of these divisions into ten equal parts, both in the upper and lower lines.

Thirdly. Subdivide again each of these subdi-'^'sions, by drawing diagonal lines from the 10th below to the Qth above; from the 8th below tonbsp;the 7th above; and so on, till from the first belownbsp;to the 0 above; by these lines each of the smallnbsp;divisions is divided into ten parts, and, conse-'luently, the whole first space into 100 equal parts;nbsp;^or, as each of the subdivisions is one tenth part ofnbsp;the whole first space or division, so each parallelnbsp;above it is one tenth of such subdivision, and, consequently, one hundreth part of the whole firstnbsp;space; and if there be ten of the larger divisions,nbsp;^iie thousandth part of the whole space.

If, therefore, the larger divisions be accounted as units, the first subdivisions will be tenth partsnbsp;of an unit, and the second, marked by the diagonal upon the parallels, hundredth parts of thenbsp;nnjt. But, if we suppose the larger divisions tonbsp;oc tens, the first subdivisions will be units, and th^nbsp;aaconcl tenths. If tiie larger are hundreds, then

The numbers, therefore, 576, 57,6, 5,76, are all expressible by the same extent of the compasses-;nbsp;thus, setting one foot in the number five of thenbsp;larger divisions, extend the other along the sixthnbsp;jjarallel to the seventh diagonal. For, if the fivenbsp;larger divisions be taken for 500, seven of the firstnbsp;subdivisions will be 70, which upon the sixth parallel, taking in six of the second subdivisions fornbsp;units, makes the whole number 576. Or, if thenbsp;five larger divisions be taken for five tens, or 50,nbsp;seven of the first subdivisions will be seven units,nbsp;and the six second subdivisions upon the sixth parallel, will be six tenths of an unit. Lastly, if thenbsp;five larger divisions be only esteemed as five units,nbsp;then will the seven first subdivisions be sevennbsp;tenths, and the six second subdivisions be the sixnbsp;hundredth parts of an unit.

Of the use of the scales of equal parts. Though what I have already said on this head may benbsp;deemed sufficient, I shall not scruple to introducenbsp;a few more examples, in order to render the youngnbsp;practitioner more perfect in the management ofnbsp;an instrument, that will be continually occurringnbsp;to him in practical geometry. He will have already observed, that by scales of equal parts linesnbsp;may be laid down, or geometrical figures constructed, whose right-lined sides shall be in the-same proportion as any given numbers.

Example 1. To take off the nuniher A,7Q from a diagonal scale. Set one foot of the compassesnbsp;on the point where the fourth vertical line cutsnbsp;the seventh horizontal line, and extend the othernbsp;foot to the point where the ninth diagonal cuts thenbsp;seventh horizontal line.

E.xatnple 2. To take off the nuniher 76,4. Observe the points where the sixth horizontal cuts

t�ie seventh vertical and fourth diagonal line, the extent between these points will represent thenbsp;number 76,4.

In the first example each primary division is taken for one, in the second it is taken for ten.

Example 3. To lay down a line of 7,85 chains hy the diagonal scale. Set one point of yournbsp;compasses where the eighth parallel, counting upwards, cuts the seventh vertical line; and extendnbsp;the other point to the intersection of the samenbsp;eighth parallel with the fifth diagonal. Set oflFthenbsp;.extent 7,85 thus found on the line.

Example 4. To measure hy the diagonal scale 0 line that is already laid down. Take the extentnbsp;of the line in yfour compasses, place one foot onnbsp;the first vertical line that will bring the other footnbsp;among the diagonals; move both feet upwards tillnbsp;one of them fall into the point where the diagonalnbsp;from the nearest tenth cuts the same parallel as isnbsp;cut by the other on the vertical line; then onenbsp;foot shews the chain, and the other the hundredthnbsp;parts or odd links. Thus, if one foot is on thenbsp;eighth diagonal of the fourth parallel, while thenbsp;other is on the same parallel, but coincides withnbsp;the twelfth vertical, we have 12 chains, 48 links,nbsp;or 12,48 chains.

Example 5. Three adjacent parts of any right-lined triangle being given, to form the plan thereof. Thus, suppose the base of a triangle, fig. 15, plate 3,nbsp;^0 chains, the angle ABC equal 36 deg. and anglenbsp;^AC equal 41 deg.

Draw the line AB, and from any of the ^ales of equal parts take off 40, and set itnbsp;on the same line from A to B for the base ofnbsp;fhe triangle; at the points A, B, make the anglenbsp;�^BC equal to 30 degrees, and BAG to 41, andnbsp;foe triangle will be formed; then take in your

compasses the length of the side AC, and applj it to the same scale, and you will find its lengthnbsp;to be 24 chains; do the same by the side BC, andnbsp;you will find it measure 27 chains, and the protractor will shew that the angle ACB containsnbsp;103 degrees.

Example Q. Given the base AB,J?^. \ 6, plate of a triangle ^^21 yards, the angle CAB 44�, 30',nbsp;and the side AC 20S yards, to constitute the said trUnbsp;angle, and find the length of the other sides.

Draw the line AB at pleasure, then take 327 parts from the scale, and lay it from A to B; getnbsp;the center of the protractor to the point A, laynbsp;off 44� 30', and by that mark draw AC; thennbsp;take with the compasses from the same scalenbsp;208, and lay it from A to C, and join C B, andnbsp;the parts of the triangle in the plan will bear thenbsp;same proportion to each other as the real parts innbsp;the field do.

1. Of the line of chords. This line is used to set off an angle from a given point in any right line,nbsp;or to measure the quantity of an angle alreadynbsp;laid down.

Thus to draw a line AC, fig. \i, plates, that shall make with the line AB an angle, contain-,nbsp;ing a given number of degrees, suppose 40 degrees.

Open your compasses to the extent of 6o de- � grees upon the line of chords, (which is always

equal to the radius of the circle of projection,) nnd setting one foot in the angular point, withnbsp;quot;that extent describe an ai�ch; then taking the ex�-tent of 40 degrees from the said chord line, set itnbsp;off from the given line on the arch described; anbsp;right line drawn from the given point, throughnbsp;the point marked upon the arch, will form thenbsp;required angle.

The degrees contained in an angle already laid down, are found nearly in the same manner; fornbsp;instance, to measure the angle CAB. From thenbsp;center describe an arch with the chord of 6o de^nbsp;grees, and the distance CB, measured on thenbsp;chords, will give the number of degrees containednbsp;in the angle.

If the number of degi'ecs are more than 00, they must be measured upon the chords at twice-:nbsp;thus, if 120 degrees were to be practised, 6o may

taken from the chords, and those degrees be laid off twice upon the arch. Degrees taken fromnbsp;the chords are always to be counted from the be�-ginning of the scale.

Of the rhumb line. This is, in fact, a line of chords constructed to a quadrant divided intonbsp;^ight parts or points of the compass, in order tonbsp;facilitate the work of tlie navigator in laying dowiinbsp;� ship�s course.

Thus, supposing the ship�s course to be N N E, And it be required to lay down that angle. Drawnbsp;the line A B,/^. 14, fhte 3, to represent the me-^ifllan, and about the point A sweep an arch withnbsp;the chord of 6o degrees; then take the extent tonbsp;the second rhumb, from the line of rhumbs, andnbsp;^ct it off on the arch from B to e, and draw thenbsp;hne A e, and the angle BAe will represent thenbsp;chip�s course. The second rhumb was taken, be �nbsp;Gause NNE is the second point from the north.

Of the line of longitudes. The line of longitudes is a line divided into sixty unequal parts, and so applied to the line of chords, as to shewnbsp;by inspection, the number of equatorial milesnbsp;contained in a degree on any parallel of latitude.nbsp;The graduated line of chords is necessary, in order to shew the latitudes; the line of longitudenbsp;shews the quantity of a degree on each parallel innbsp;sixtieth parts of an equatorial degree, that is,nbsp;miles. The use of this line will be evident fromnbsp;the following example. A ship in latitude 44�nbsp;12'N. sails E. 79 miles, required her difference ofnbsp;longitude. Opposite to 44i, nearest equal to thenbsp;latitude on the line of chords, stands 43 on the linenbsp;of longitude, which is therefore the number of milesnbsp;in a degree of longitude in that latitude. Whencenbsp;as 43 : 60 : : 79 : 110 miles the difference of longitude.

'Vhe lines of tangents, semitangents, and secants serve to find the centers and poles of projectednbsp;circles in the sLereographical projection of thenbsp;sphere.

The line of sines is principally used for the orthographic projection of the sphere; but as the application of these lines is the same as that of similar lines on the sector, we shall refer the reader to thenbsp;explanation of those lines in our description ofnbsp;that instrument.

The lines of latitudes and hours are used conjointly, and serve very readily to mark the hour lines in the construction of dials; they are generally on the most complete sorts of scales and sectors; for the uses of which sec treatises on dialling,

Amidst the variety of mathematical instruments that have been contrived to facilitate the art of

drawing, there is none so extensive in its use, or of sueh general application as the sector. It is annbsp;Universal scale, uniting, as it were, angles and parallel lines, the rule and the compass, which arenbsp;the only means that geometry makes use of fornbsp;measuring, whether in speculation or practice.nbsp;The real inventoivof this valuable instrument isnbsp;Unknown; yet of so much merit has the inventionnbsp;appeared, that it was claimed by Galileo, and disputed by nations.

This instrument derives its name from the tenth definition of the third book of Euclid, where henbsp;defines the sector of a circle. It is formed of twonbsp;oqual rules, (Jig- 4 and 5, plate 3,) AB, DB,nbsp;called legs; these legs are moveable about thenbsp;center C of a joint d e f, and will, consequently,nbsp;by their different openings, represent every possible variety of plane angles. The distance of thenbsp;extremities of these rules are the subtenses ornbsp;chords, or the arches they describe.

Sectors are made of different sizes, but tbeir length is usually denominated from the length ofnbsp;the legs when the sector is shut. Thus a sectornbsp;cf six inches, when the legs are close together,nbsp;forms a rule of 12 inches when opened; and anbsp;boot sector is two feet long when opened to itsnbsp;greatest extent. In describing the lines usuallynbsp;placed on this instrument, I refer to those commonlynbsp;laid down on the best six-inch brass sectors. Butnbsp;as the principles are the same in all, and the differences little more than in the number of subdi-''isions, it is to be presumed that no difficulty willnbsp;occur in the application of what is here said tonbsp;Sectors of a larger radius.

. Of this instrument. Dr. Priestley thus speaks his Treatise on Perspective. � Besides thenbsp;^^^all sector in the common pocket cases of iastru-

ments, I. would advise a person vdio proposes to learn to draw, to get another of one foot radius.nbsp;Two sectors are in many cases exceedingly useful,nbsp;if not absolutely necessary; and I would not advise a person to be sparing of expense in procuring a very good instrument, the uses of which arenbsp;so various and important.�

The scales, or lines graduated upon the faces of the instrument, and which are to be used as sectoral lines, proceed from the center; and are,nbsp;1. Two scales of equal parts, one on each leg,nbsp;marked lin. or l. each of these scales, from thenbsp;great extensiveness of its use, is called the line ofnbsp;lines. �2. Two lines of chords, marked cho. or c,nbsp;3. Two lines of secants, marked sec. or s. A linqnbsp;of polygons, marked pol. Upon the other face,nbsp;the sectoral lines arc, 1. Twm lines of sines, markednbsp;sin. or s. 2. Two lines of tangents, marked tan,nbsp;3. Between the lines of tangents and sines, there isnbsp;another line of tangents to a lesser radius, to supply the defect of the former, and extending fromnbsp;45� to 75�.

Each pair of these lines (except the line of polygons) is so adjusted as to make equal angles at the center, and consequently at whatever distancenbsp;the sector be opened, the angles wall be alwaysnbsp;respectively equal. That is, the distance betweennbsp;10 and 10 on the line of lines, will be equal to 6onbsp;and 6o on the line of chords, QO and go on thenbsp;line of sines, and 45 and 45 on the line of tangents.

Besides the sectoral scales, there are others on each face, placed parallel to the outward edges,nbsp;and used as those of the common plain scale.nbsp;There are on the face, �^. 5, 1, A line of inches.nbsp;lt;2. A line of latitudes. 3. A line of hours. 4. Anbsp;line of inclination of meridians. 5. A line of

chords. On the face, fig. 4, three logarithmic scales, namely, one of numbers, one of sines, andnbsp;one of tangents; these arc used when the sector isnbsp;hilly opened, the legs forming one line; theirnbsp;Use will be explained when we treat of trigonometry.

read agt;ul estimate the divisions on the sectoral lines. The value of the divisions on most of thenbsp;lines are determined by the figures adjacent tonbsp;them; these proceed b}quot; tens, which constitute thenbsp;divisions of the first order, and are numbered accordingly; but the value of the divisions on thenbsp;line of lines, that are distinguished by figures, 15nbsp;entirely arbitrary, and may represent any value thatnbsp;is given to them; hence the figures 1, 3, 3,4, amp;c.nbsp;may denote either 10, 20, 30, 40j or 100, 200,nbsp;300, 400, and soon.

The line of lines is divided into fen equal part^ uunribered 1, 2, 3, to 10; these may be called di-t^isions of the first order; each of these are againnbsp;Subdivided into 10 other equal parts, wlfieh maynbsp;called divisions of the second order; and eachnbsp;uf these is divided into two equal parts, formingnbsp;divisions of the third order.

The divisions on all the scales are contained between four parallel lines; those of the firstnbsp;Order extend to the most distant; those of thenbsp;third, to the least; those of the second to the in-rnbsp;termediate parallel.

When the whole line of lines represents 100,. divisions of the first order, or those to whichnbsp;tfic figures are annexed, represent tens; those ofnbsp;the second order, units; those of the third order,nbsp;the halves of these units. If the whole line repre-�^cnt ten, then the divisions of the first order are;nbsp;Units; those of the second, tenths, and the thirds,

twentieths.

In the line of tangents, the divisions to tvhich the numbers are affixed, are the degrees expressed bynbsp;those numbers. Every fifth degree is denoted bynbsp;a line somewhat longer than the rest; betweennbsp;every number and each fifth degree, there are fournbsp;divisioirs, longer than the intermediate adjacentnbsp;ones, these are whole degrees; the shorter ones,nbsp;or those of the third order, are 30 minutes.

From the center, to 6o degrees, the line of sines is divided like the line of tangents; from 6o to 70,nbsp;it is divided only to every degree; from 7o to 80,nbsp;to every two degrees; from 80 to QO, the divisionnbsp;must be estimated by the eye.

The divisions on the line of chords are to be estimated in the same manner as the tangents.

The lesser Vine of tangents is graduated every two degrees from 45 to 50; but from 50 to 6o, tonbsp;every degree; from 6o to the end, to half degrees.

The line of secants from O to 10, is to be estimated by the eye; from 20 to 50 it is divided to every two degrees; from 50 to 6o, to every degree;nbsp;and from 6o to the end, to every half degree.

of sectoral lines, (ex. gr. those of the line of lines,) forming the angle AC B; divide each of these linesnbsp;into four equal parts, in the points H, D, F;nbsp;I, E, G; draw the lines H I, D E, F G, AB. Thennbsp;because C A, C B, arc equal, their sections are alsonbsp;equal, the triangles are equiangular, having a common angle at C, and equal angles at the base; andnbsp;therefore, the sides about the equal angles will bonbsp;proportional; for as CH to C A, so is H I to AB,nbsp;0nd, therefore, as CA to C H, so is A B to Hd, and.

consequently, as C H to H I, so is CA to AB; and thence if CH be one fourth of CA, H I will benbsp;one fourth of AB, and so of all other sections.

Hence, as in all operations on the sectoral lines, there are two triangles, both isosceles aud equi-angled; isosceles, because the pairs of sectoralnbsp;lines are equal by construction; equiangled, because of the common angle at the center; thenbsp;sides encompassing the equal angles are, therefore, proportional.

Hence also, if the lines CA, CB, represent the line of chords, sines, or tangents; that is, if CA,nbsp;AB be the radius, and the line CF the chord, sine,nbsp;or tangent of any proposed number of degrees,nbsp;then the line F G will be the chord, sine, or tangent, of the same number of degrees, to the radius AB.

It is necessary to explain, in this place, a few terms, either used by other writers in their description of the sector, or such as we may occasionally use ourselves.

The solution of questions on the sector is said to be simple, when the work is begun and ended onnbsp;the same line; compound, when the operation be-ghis on one line, and is finished on the other.

The operation varies also by the manner in '^hich the compasses are applied to the sector.nbsp;If a measure be taken on any of the sectoral lines,nbsp;beginning at the center, it is called a lateral distance. But if the measure be taken from any pointnbsp;in one line, to its corresponding point on the linenbsp;�f the same denomination, oti the other leg, it isnbsp;Sillied a transverse or parallel distance.

The divisions of each sectoral line are bounded by three parallel lines; the innermost of these isnbsp;that on which the points ot the compasses are tonbsp;be placed, because this alone is the line whichnbsp;goes to the center, and is alone, therefore, the sectoral line.

We shall now proceed to give a few general instances of the manner of operating with the sector, and then proceed to practical geometry, exemplifying its use further in the progress of the work,nbsp;as occasion offers.

Multiplication hy the line of lines. Make the lateral distance of one of the factors the parallelnbsp;distance of 10; then the parallel distance of thenbsp;other factor is the product.

Example. Multiply 5 by 6, extend the compasses from the center of the sector to 5 on the primary divisions, and open the sector till this distance become the parallel distance from lO to 10nbsp;on the same divisions; then the parallel distancenbsp;from 6 to 6, extended from the center of the sec-toi', shall reach to 3, which is now to be reckonednbsp;30. At the same opening of the sector, the parallel distance of 7 sh^h reach from the center tonbsp;35, that of 8 shall reach from the center to 40, amp;c.

Division hy the line of lines. Make the lateral distance of the dividend the parallel distance ofnbsp;the divisor, the parallel distance of 10 is the quotient. Thus, to divide 30 by 5, make the lateralnbsp;distance of 30, viz. 3 on the primaiy divisions, thenbsp;parallel distance of-5 of the same divisions; thennbsp;the parallel distance of lo, extended from thenbsp;center, shall reach to 6.

Proportion by the line of Ihies. INlake the lateral distance of the second tenn the parallel distancenbsp;of the first term; the parallel distance of the thirdnbsp;term is the fourth proportional. Example, To

find a fourth proportional to 8, 4, and 6, take the lateral distance of 4, and make it the parallel distance of 8; then the parallel distanee of �, extended from the center, shall reach to the fourthnbsp;proportional 3.

In the same manner a third proportional is found to two numbers. Thus, to find a thirdnbsp;proportional to 8 and 4, the sector remaining asnbsp;in the former example, the parallel distance of 4,nbsp;extended from the center, shall reach to the thirdnbsp;proportional 2. In all these cases, if the numbernbsp;to be made a parallel distance be too great for thenbsp;sector, some aliquot part of it is to be taken, andnbsp;the answer multiplied by the number by whiebnbsp;the first number was divided. Thus, if it werenbsp;required to find a fourth proportional to 4, 8,nbsp;and 6; because the lateral distance of the secondnbsp;term 8 cannot be made the parallel distance of thenbsp;first term 4, take the lateral distance of 4, viz. thenbsp;half of 8, and make it the parallel distance of thenbsp;first term 4; then the parallel distance of the thirdnbsp;terra 6, shall reach from the center to 6, viz. thenbsp;half of 13. Any other aliquot part of a numbernbsp;r^*ay be used in the'same way. In like manner, ifnbsp;the number proposed be too small to be made thenbsp;parallel distanee, it may be multiplied by somenbsp;Clumber, and the answer is to be divided by thenbsp;-�^ame number.

To protract angles by the line of chor-ds. Case 1. When the given degrees arc under 6o.nbsp;nbsp;nbsp;nbsp;1. With

^�y radius AB, yfg. 14, plate 3, on A as a center, describe the areh B G. 3. Make the same radiusnbsp;^ transverse distance between 6o and 6o on thenbsp;fine of chords. 3. Take out the transverse distance of the given degrees, and lay this on thenbsp;arch from B towards G, which v, ill mark out thenbsp;^regular distanee required.

Case 2. When the given degrees are more than 6o.nbsp;nbsp;nbsp;nbsp;1. Open the sector, and describe the arch as

before. 2. Take � or -j- of the given degrees, and take the transverse distance of this i or i, and laynbsp;it off from B towards G, twice, if the degrees werenbsp;halved, three times if the third was used as anbsp;transverse distance.

Case 3. When the required angle is less than 6 degrees; supposes. 1. Open the sector to thenbsp;given radius, and describe the arch as before.nbsp;2. Set off the radius from B to C. 3. Set off thenbsp;chord of 57 degrees backward from C to f, whichnbsp;will give the arc f b of three degrees.

Given the radius of a circle, (suppose equal to 'two inches f required the sine and tangent of 28� 30'nbsp;to that radius.

Solution. Open the sector so that the transverse distance of 90 and go on the sines, or of 45 and 45 on the tangents, may be equal to the givennbsp;radius, viz. two inches; then will the transversenbsp;distance of 28� 30', taken from the sines, be thenbsp;length of that sine to the given radius; or if takennbsp;from the tangents, will be the length of that tangent to the given radius.

Make the given radius, two inches, a transverse distance to O and o, at the beginning of the linenbsp;of secants; and then take the transverse distancenbsp;of the degrees wanted, viz. 28� 30'.

Make the given radius, suppose two inehes, a transverse distance to 45 and 45 at the beginningnbsp;of the scale of upper tangents; and then the required number 60� OO' may be taken from thisnbsp;scale.

The scales of upper tangents and secants do not run quite to 76 degrees; and as the tangent andnbsp;secant may be sometimes wanted to a greaternbsp;number of degrees than can be introduced on thenbsp;sector, they may be readily found by the help ofnbsp;the annexed table of the natural tangents and secants of the degrees above 7 5; the radius of thenbsp;Circle being unity.

Measure the radius of the circle used upon any scale of equal parts. Multiply the tabular number by the parts in the radius, and the productnbsp;'V1II give the length of the tangent or secantnbsp;bright, to be taken from the same scale of equal

parts.

Example. Required the length of the tangent and secant of 80 degrees to a circle, whose radius,nbsp;measured on a scale of 25 parts to an inch, is 475 ofnbsp;those parts?

So the length of the tangent on the twenty-fifth scale wiil be 269! nearly. And that of the secant about 27 3 5.

Or thus. The tangent of any number of degrees may be taken from the sector at once; if the radius of the circle can be made a transverse distance to the complement of those degrees on thenbsp;lower tangent.

Make two inches a transverse distance to 12 degrees on the lower tangents; then the transverse distance of 45 degrees will be the tangent of 78nbsp;degrees.

In like manner the secant of any number of degrees may be taken from the sines, if the radius of the circle can be made a transverse distance to thenbsp;co-sine of those degrees. Thus making twonbsp;inches a transverse distance to the sine of twelvenbsp;degrees; then the transverse distance of 90 and gonbsp;will be the secant of 78 degrees.

From hence it will be easy to find the degrees answering to a given line, expressing the length

a tangent or secant, which is too long to be �neasured on those scales, when the sector is set tonbsp;the given radius.

Thus, for a tangent, make the given line a transverse distance to 45 and 45 on the lower tan-,nbsp;gents; then take the given radius, and apply it tonbsp;the lower tangents; and the degrees where it becomes a transverse distance is the co-tangent ofnbsp;the degrees answering to the given- line.

And for a secant; make the given line a trans-'quot;crse distance to yo and go on the sines. Then the degrees answering to the given radius, applied

a transverse distance on the sines, will be the co-sine of the degrees answering to the given secant line.

Given the length of the sine, tangent, or secant, of degrees-, to fnd the length of the radius to thatnbsp;^^ne, tangent, or secant.

_ Make the given length a transverse distance to Its given degrees on its respective scale: then.

In the lower tangents. The transverse distance cf 45 and 45, near the end of the sector, will benbsp;radius sought.

In the upper tangents. The transverse distance Cl 45 and 45, taken towai'ds the center of thenbsp;sector on the line of upper tangents, will be thenbsp;Center sought.

In the secant. The transverse distance of 0 and 0, cr the beginning of the secants, near the center ofnbsp;be sector, will be the radius sought.

Given the radius and any line representing a sme, ^^ngent or secant-, to find the decrees correspondinz

cerned.

Take the given line between the eompasses; apply the two feet transversely to the seale concerned, and slide the feet along till they both restnbsp;on like divisions on both legs; then will those divisions shew the degrees and parts correspondingnbsp;to the given line.

To find the length of a versed sine to a given number of degrees, and a given radius.

Make the transverse distanee of go and gO on the sines, equal to the given radius.

If the given degrees are less than QO, the difference between the sine complement and the radius gives the versed sine.

If the given degrees are more than go, the sum of the sine complement and the radius gives thenbsp;versed sine.

To open the legs of the sector, so thett the corresponding double scales of lines, chords, sines, and tangents, may make each a right angle.

On the lines, make the lateral distance lo, a distance between eight on one leg, and six on thenbsp;other leg.

On the sines, make the lateral distance pO, a transverse distance from 45 to 45; or from 40 tonbsp;50; or from 30 to 60; or from the sine of any degrees to their complement.

Or on the sines, make the lateral distance of 45 a transverse distance between 30 and 30.

Science may suppose, and the mind conceive things as possible, and easy to be effected, innbsp;which art and practice often find insuperable dif-

Hculties. � Pure science has to do only with ideas; but in the application of its laws to the usenbsp;of life, we are constrained to submit to the imper fections of matter and the influence of accident.�nbsp;Thus practical geometry shews how to performnbsp;what theory supposes; in the theory, however, itnbsp;is sufficient to have only a right conception of thenbsp;objects on which the demonstrations are founded;nbsp;drawing or delineations being of no further usenbsp;than to assist the imagination, and lessen the exertions of the mind. But in practical geometry, wenbsp;not only consider the things as possible to be ef-fceted, but are to teach the ways, the instruments,nbsp;and the operations by which they may be actuallynbsp;performed. It is not sufficient here to shew�, thatnbsp;a right line may be drawn between two points, ornbsp;a circle described- about a center, and then demonstrate their properties; but we must actuallynbsp;delineate them, and exhibit the figures to thenbsp;Senses: and it will be found, that the drawing ofnbsp;a strait line, which occurs in all geometrical operations, and which in theory is conceived as easynbsp;to be effected, is in practice attended with considerable difficulties.

To draw a strait line between two points upon a plane, w�e lay a rule so that the strait edge thereofnbsp;^iiay Just pass by the two points; then movingnbsp;a fine pointed needle, or drawing pen, along thisnbsp;edge, we draw a line from one point to the other,nbsp;wnich for common purposes is sufficiently exact;nbsp;Dnt w�here great accuracy is required, it W�ill benbsp;found extremely difficult to lay the rule equally,nbsp;with respect to both the points, so as not to benbsp;nearer to one point than the other. It is difficultnbsp;also so to carry the needle or pen, that it shall neither incline more to one side than the other ot the

rule; anti thirdly? it is very difficult to find a rule that shall be perfectly strait. If the two points benbsp;very far distant, it is almost impossible to, drawnbsp;the line with accuracy and exactness; a circularnbsp;line may be described more easily, and more exactly, than a strait, or any other line, though evennbsp;then many difficulties occur, when the circle isnbsp;required to be of a large radius.

It is from a thorough consideration of these difficulties, that geometricians will not allow those lines to be geometrical, which in their descriptionnbsp;require the sliding of a point along the edge of anbsp;rule, as in the ellipse, and several other curvenbsp;lines, whose properties have been as fully investigated, and as clearly demonstrated, as those ofnbsp;the circle.

From hence also we may deduce some of those maxims which have been introduced into practicenbsp;by Bird and Smeaton, which will be seen in theirnbsp;proper place. And let no one consider these reflections as the effect of too scrupulous exactness,nbsp;or as an unnecessary aim at precision; for as thenbsp;foundation of all our knowledge in geography,nbsp;navigation and astronomy, is built on observation,nbsp;and all observations are made with instruments, itnbsp;follows, that the truth of the observations, and thenbsp;accuracy of the deductions therefrom, will principally depend on the exactness with which the instruments are made and divided; and that thesenbsp;sciences will advance in proportion as those arenbsp;less difficult in their use, and more perfect in thenbsp;performance of their respective operations.

There is scarce any thing which proves more clearly the distinction between mind and body,nbsp;and tire superiority of the one over the other, thannbsp;a reflection on the rigid exactness of speculative

geometry, and the inaccuraey of practice, that is not directed by theory on one hand, and its approximation to perfection on the other, whennbsp;guided by a just theory.

In theory, most figures may be measured to an almost infinite exactness, yet nothing can be morenbsp;inaccurate and gross, than the ordinary methods ofnbsp;mensuration; but an intelligent practice finds anbsp;medium, and corrects the imperfection of our mechanical organs, by the resources of the mind. Ifnbsp;we were more perfect, there would be less roomnbsp;for the exertions of our mind, and our knowledgenbsp;Would be less.

If it had been easy to measure all things with exactness, we should have been ignorant of manynbsp;curious properties in numbers, and been deprivednbsp;of the advantages we derive from logarithms, sines,nbsp;tangents, amp;c. If practice were perfect, it is doubtful whether we should have ever been in possessionnbsp;of theory.

We sometimes consider with a kind of envy, the Oiechanical perfection and exactness that is to benbsp;found in the works of some animals; but this perfection, which does honor to the Creator, doesnbsp;little to them; they are so perfect, only becausenbsp;they are beasts.

The imperfection of our organs is abundantly made up by the perfection of the mind, of whichnbsp;We are ourselves to be the artificers.

If any wish to see the difficulties of rendering practice as perfect as theory, and the wonderful resources of the mind, in order to attain this degreenbsp;cf perfection, let him consider the operations ofnbsp;General Roy, at Hounslow-hcath; operations thatnbsp;cannot be too much considered, nor too muchnbsp;praised by every practitioner in the art of geo-*^ctry. See Fhilosoph. Trans, vol. 80, et seq.

Problem 1. To erect a perpendicular at or near the end of a given right line, C ^,Jig. 5, plate 4.

Method 1. On C, with the radius C D, describe a faint arc ef on D; with the same radius, erossnbsp;ef at G, on G as a center; with the same radius, describe the arc D E F; set off the extent D G twice,nbsp;that is from Igt; to E, and from E to F. Join thenbsp;points D and F by a right line, and it will be thenbsp;perpendicular required.

Method 2, fig. 5. On any point G, with the radius D G, describe an arc FED; then a rulenbsp;laid on C and G, will eut this arc in F, a line join ing the points F and D will be the required perpendicular.

with any radius describe the are rnm, euttingthc line AC in r. 2. From the point r, with the samenbsp;radius, cross the arc in n, and from the point n,nbsp;cross it in m. 3. From the points n and m, withnbsp;the same, or any other radius, describe two arcsnbsp;cutting each other in S. 4. Through the pointsnbsp;S and C, draw the line S C, and it will be the perpendicular required.

Method 4. By the line of lines on the sector, fig. y, plate A. 1. Take the extent of the given line A C.nbsp;2. Open the sector, till this extent is a transversenbsp;distance between 8 and 8.nbsp;nbsp;nbsp;nbsp;3. Take out the trans

verse distance between 6 and 6, and from A with that extent sw'eep a faint are at B. 4, Take outnbsp;the distance between 10 and 10, and with it fromnbsp;C, cross the former arc at B. 5. A line drawn,nbsp;through A and B, will be the perpendicular required; the numbers 6, 8, 10, are used as multiples of 3, 4, 5.

By this method, a perpendicidar may he easily an'd accurately erected on the ground.

line, and A the given point. 1. At any point with the radius DA, dcsci�ibe the arc EAB,nbsp;2. With a rule on E and D, cross this arc at B.

Problem 2. To raise a perpendicular from the middle, or any other�^oint G, of a given line AB,nbsp;fiS- 85 plate 4.

1. On G, with any convenient distance within the limits of the line, mark or set off the points n and m.nbsp;2- Prom n and m with any radius greater than G A,nbsp;describe two arcs intersecting at C. 3. Join CG bynbsp;^ line, and it will be the perpendicular required.

Problem 3. From a given point C,fig. Q, plate 4, out of a given line AB, to let fall a perpendicidar.

When the point is nearly opposite to the middle, nf the line, this problem is the converse of the pre-^^dingone. Therefore, 1. From C, with any radius, describe the arc n m. 2. From n m, withnbsp;Ihe same, or any other radius, describe two arcs intersecting each other at S. 3. Through the pointsnbsp;9 S draw the line C S, which will be the requirednbsp;line.

When the point is nearly opposite to the end of feline, it is the converse of Method 5, Problem 1,nbsp;fiS- 7, plate 4.

. 1- Draw a faint line through B, and any convenient point E, of llx', line AC. 2. Bisect this line 3. From D with the radius DE describenbsp;nn arc cutting AC at A. 4. Through A and Bnbsp;oraw the line AB, and it will be the perpendicular

required.

�Another method. 1. From K,fig. Q, plate 4, or miy other point in AB, wdth the radius AC, dc-ii^eribe the arc CD. 2. from any other point n,nbsp;^dh the radius n C, describe another arc cutting

Problem 4. Through a given point C, to draw a line parallel to a given strait line AB, Jig. lo,nbsp;plate 4.

1. On any point D, (within the given line, or without it, and at a convenieat distance from C,)nbsp;describe an arc passing through C, and cutting thenbsp;given line in A. 2. With the same radius describenbsp;another arc cutting AB at B. 3. Make BE equalnbsp;to AC. 4. Drav/ a line CE through the pointnbsp;C and E, and it will be the required parallel.

This problem answers whether the required line is to be near to, or far from the given line;nbsp;or whether the point D be situated on AB, or anynbsp;where between it and the required line.

Problem 5. -At the given point D, to make an angle equal to a given an^e ABC,y?g. 11, plate A.

1. From B, with any radius, describe the arc nm, cutting the legs BA BC, in the pointsnbsp;n and m. 1. Draw the line D r, and from thenbsp;point D, with the same radius as before, describenbsp;the arc r s. 3. Take the distance m n, and applynbsp;it to the arc r s, from r to s. 4. Through thenbsp;points D and s draw the line D s, and the anglenbsp;rDs will be equal to the angle mBn, or ABC,nbsp;as required.

Problem 6, To extend with accuracy a short strait line to any assignable length-, or, through twonbsp;given points at a small distance from each other tonbsp;draw a strait line.

It frequently happens that a line as short as that between A and quot;amp;, jig. 11, plate A, is required to benbsp;extended to a considerable length, M'hich is scarcenbsp;attainable by the help of a rule alone; but may benbsp;performed by means of this problem, without error.nbsp;Let the given line be AB, or the two points A

and B; then from A as a center, describe an arch C B D; and from the point B, lay oft B C equalnbsp;to BD; and from C and D as centers, with anynbsp;radius, describe two arcs intersecting at E. Fromnbsp;the point A describe the arc PEG, making E Fnbsp;lt;iqual to E G; then from F and G as centers, describe two arcs intersecting at H, and so on; thennbsp;^ strait line from B drawm through E will pass innbsp;continuation through El, and in a similar mannernbsp;the line may be extended to any assignable length.

Problem 7. To bisect or divide a given strait iine AB into two equal parts. Jig. I?! opiate 4.

1. On A and B as centers, with any radius greater than half AB, describe arcs intersectingnbsp;each other at C and D. 1. Draw the line CD,nbsp;^Dd the point E', where it cuts A B, will be thenbsp;Diiddle of the line.

If the line to be bisected be near the extreme edge of any plane, describe two pair of arcs of different radii above the given line, as at C and Ejnbsp;then a line C produced, will bisect AB in F.

Idy the line of lines on the sector. 1. Take AB In the compasses. 2. Open the sector till this extentnbsp;a transverse distance between 10 and 10. 3. Thenbsp;extent from 5 to 5 on the same line, set olF fromnbsp;A or B, gives the half required r by this means anbsp;given line may be readily divided into 2, 4, 8, 16,nbsp;quot;^2, 64, 128, amp;c. equal parts.

. Problem 8. To divide a given strait line AB any number of equal parts, for instance, five.nbsp;Method \,fig. \A, plate A. 1. Through A, onenbsp;�5itreinity of the line AB, draw AC, making anynbsp;^�gie therewith. 2. Set off on this line from A tonbsp;�IS many equal parts of any length as AB is to

be divided into. 3. Join HB. 4. Parallel to U B draw lines through the .points D, E, F, G,nbsp;and these will divide the line A B into the partsnbsp;required.

draw L D, forming any angle with AB. 2. Take any point D either above or below AB, andnbsp;through D, draw D K parallel to AB. 3. On Dnbsp;set off five equal parts DF, FG, GH, HI, IK.nbsp;4, Through A and K draw AK, cutting BD innbsp;L. 5. Lines drawn through L, and the pointsnbsp;F, G, H, I, K, will divide the line AB into thenbsp;required number of parts.

ends of the line AB, draw two lines AC, BD, parallel to each other. 2. In each of these lines,nbsp;beginning at A and B, set off as many equal partsnbsp;less one, as AB is to be divided into, in the presentnbsp;instance four equal parts, AI, IK, KL, LM, onnbsp;AC; and four, BE, E F, FG, GH, on BD.nbsp;3. Draw lines from M to E, from L to F, K to G,nbsp;I to H, and AB will be divided into five equalnbsp;parts.

Fourth method, fig. l6, plate A. 1. Draw any two lines C E, D F parallel to each other. 2. Setnbsp;off on each of these lines, beginning at C and D,nbsp;any numbea of equal parts. 3. Join each point innbsp;C E with its opposite point in D F. � 4. Take thenbsp;extent of the given line in your compasses. 5. Setnbsp;one foot of the compasses opened to this extent innbsp;D, and move the other about till it crosses NG in 1.nbsp;6. Join DI, which being equal to AB, transfernbsp;the divisions of D I to AB, and it will be dividednbsp;as required. H M is a line of a different lengthnbsp;to be divided in the same number of pails.

The foregoing methods are introduced on account not only of their own peculiar advantages.

fcut because they also arc the foundation of several luechanical methods of division.

Problem 9., To cut ojj' from a given line AB odd part, as \d, \th, \th, rth, amp;c. of that line,nbsp;�fiS' 18, plate 4.

1. Draw through either end A, a line AC, forming any angle with AB. 2. Make AC equalnbsp;fo AB. 3. Through C and B di'aw the line CD.nbsp;Make BD equal to CB. 5. Bisect AC in a.nbsp;A rule on a and D will cut oft' a B equal id ofnbsp;AB.nbsp;nbsp;nbsp;nbsp;^

If it be required to divide AB into five equal parts, 1. Add unity to the given number, andnbsp;^alve it, 5 1=6, �=3.nbsp;nbsp;nbsp;nbsp;2. Divide AC into

three parts; or, as AB is equal to AC, setoff Ab ^qual Aa. 3. A rule on D, and b will cut offnbsp;�P rth part of AB. 4. Divide Ab into fournbsp;^^ual parts by two bisections, and AB will be divided into five equal parts.

To divide AB into seven equal parts, 7 1=8, 1. Now divide A C into four parts, or bisect a C in c, and c C will be the ith of AC. 2. Anbsp;*�1110 on c and D cuts off c B -fth of AB. 3. Bisectnbsp;Ac, and the extent c B will divide each half intonbsp;three equal parts, and consequently the whole linenbsp;mto seven equal parts.

^To divide AB into nine equal parts, g l=:io ^nbsp;nbsp;nbsp;nbsp;Here, 1. Make Ad equal to Ab, and d C

'vill be fth of A C. 2. A rule on D and d cuts dB i of AC. 3. Bisect Ad. 4. Halve eachnbsp;^^f these bisections, and Ad is divided into fournbsp;equal parts. 5. The extent d B will bisect eachnbsp;eit these, and thus divide AB into nine equalnbsp;parts.

If any odd number can be subdivided, as Q by 3, then first divide the given line into three parts,nbsp;^Hdtake the third as a new line, and find the third

Method 2. Let DB, fg. 19, plate A, be the given line. 1. Make two equilateral trianglesnbsp;ADB, CDB, one on each side of the line D B.nbsp;2. Bisect AB in G. 3. Draw GG, which willnbsp;cut off H B equal ^d of D B. 4. Draw D F, andnbsp;make GF equal to D G. 5. Draw HF whichnbsp;cuts oft' Bh equal i of AB or DB. 6. Ch cutsnbsp;D B in i one fifth part. 7- F i cuts AB in k equalnbsp;^th of DB. 8. Ck cuts Db in 1 equal j.th ofnbsp;DB. 9. FI cuts AB in m equal 4th of DB.nbsp;10. Cm cuts D B at n equal 4th part thereof.

Method 3. Let K^,fig. 12, plate 5, be the given line to be divided into its aliquot parts 4, i. 1. On AB erect the square ABCD. 2. Drawnbsp;the two diagonals AC, D B, which will cross eachnbsp;other at E. 3. Through E draw F E G parallelnbsp;to AD, cutting AB in G. 4. Join DG, and thenbsp;line will cut the diagonal AC at H. 5. Throughnbsp;H draw IHK parallel to AD. 6. Draw DKnbsp;crossing AC in L. 7- And through L drawnbsp;M L N parallel to AD, and so proceed as far asnbsp;necessary. AG is i, AK 4, AN 4 of AB.¹

Method A. Let AB, fig. 13, plate 5, be the given line to be subdivided. 1. Through A and B drawnbsp;CD, FE parallel to each other. 2. Make CA,nbsp;AD, FB, FE equal to each other. 3. Drawnbsp;C E, which shall divide AB into two equal pansnbsp;at G. 4. Draw AE, DB, intersecting each othernbsp;at H. 5. Draw C H intersecting AB at I, making AI 4d of AB. 6. Draw D F cutting AE innbsp;K. 7. Join CK, which will cut AB in L, making AL equal 4 of AB. 8. Then draw D g, cut

Corollary. Hence a given line may be accu-rately divided into any prime number whatsoever,

6y first cutting oti' the odd part, then dividing the I'cniainder by continual bisections.

Problem lO. An easy, simple, and very useful ^etjiog of laying down a scale for dividing lines intonbsp;number of equal farts, or for reducing flans tonbsp;any size less than the original.

If the scale is for dividing lines into two equal parts, constitute a triangle, so that the hypothesise may be twice the length of the perpentlicular.

it be three times for dividing them into three i^ual parts; four, for four parts, and so on; fig. 11,nbsp;^tate represents a set of triangles so consti-

To find the third of the line by this scale. Take any line in your compasses, and set oflFnbsp;^his extent from A towards i, on the line markednbsp;iie third; then close the compasses so as to strikenbsp;arc that shall touch the base AC, and thisnbsp;'listance will be the i of the given line. Similarnbsp;fo this is what is termed the angle of reduction,nbsp;i'quot; proportion, described by some foreign writers,nbsp;Rnd which wc shall introduce in its proper place.

Problem ll. To divide hy the sector a given strait line into any number of equal farts.

. Case 1. Where the given line is to be divided into a number of equal parts that may be ob-f^nned by a continual bisection.

In this case the operations are best performed by �^ontinual bisection; let it be required to dividenbsp;plate 5, into l6 equal parts. 1. Makenbsp;a transverse distance between 10 and 10 onnbsp;he line of lines. 2. Take out from thence thenbsp;istance between 5 and 5, and set it from A or Bnbsp;^ 8? and AB will be divided into two equal parts.nbsp;* Make A 8 a transverse distance between 10 and

10, and 4 the transverse distance between 5 and 5, will bisect 8 A, and 8 B at 4 and 12; and thus AB-is divided into four equal parts in the points 4, 8,nbsp;and 12.nbsp;nbsp;nbsp;nbsp;4. The extent A 4, put between 10 and

10, and then the distance between 5 and 5 applied from A to 2, from 4 to 6, from 8 to 10, and fromnbsp;12 to 14, will bisect each of those parts, and dividenbsp;the whole line into eight equal parts. 5. To bisect each of these, we might take the extent ofnbsp;A 2, and place it between 10 and 10 as before;nbsp;but as the spaces are too small for that purpose,nbsp;take three of them in the compasses, and open thenbsp;sector at 10 and 10, so as to accord with this measure. 6. Take out the transverse measure betweennbsp;5 and 5, and one foot of the compasses in A willnbsp;give the point 3, in 4 will fall on 7 and 1, on 8nbsp;will give 5 and 11, on 12 gives 9 and 15, and onnbsp;B will give 13. Thus we have, in a correct andnbsp;easy manner divided AB into 16 equal parts by anbsp;continual bisection.

If it were required to bisect each of t he foregoing divisions, it would be best to open the sector at 10 and 10, with the extent of live of the divisions already obtained; then take out the transverse distance between 5 and 5, and set it off fromnbsp;the other divisions, and they will thereby be bisected, and the line divided into 32 equal parts.

Let the given line be AB,plate A, to be divided into 14 equal parts, a number which is notnbsp;a multiple of 2.

1. Take the extent AB, and open the sector to it on the terms 10 and 10, and the transverse distance of 5 and 5, set from A or B to 7, will dividenbsp;AB into two equal parts, each of which are to benbsp;subdivided into 7, which maybe done by dividing

^7 into 6 and 1, or 4 and 3, which last is preferable to the first, as by it the operation may be Dnished with only two bisections.

Therefore open the sector in the terms 7 and 7} with the extent A 7 ; then take out the transverse distance between 4 and 4, this laid off fromnbsp;-^3 gives the point 4, from 7 gives 3 and 11, andnbsp;from B gives 10.

3. Make A4 a transverse distance at 10 and 19� then the transverse distance between 5 and 5nbsp;bisects A b, and loB in 2 and 12, and gives thenbsp;point 6 and 8; then one foot in 3 gives 1 and 5,nbsp;^od from 11,13 and Q; lastly, from 4 it gives 6,nbsp;and from 10, 8; and thus the line AB is dividednbsp;into 14 equal parts.

Problem 12. To make a scale of equal parts ^^ntaining any given number in an inch.

. Example. To construct a scale of feet and inches In such a manner, that 25 of the smallest parts shallnbsp;ho equal to one inch, and 12 of them representnbsp;i^ne foot.

S'ven numbers by 4, the products will be 100, and 48.nbsp;nbsp;nbsp;nbsp;2. Take one inch between your compasses,

^nd make it a transverse distance between lOO and ^tgt;0, and the distance between 48 and 48 willnbsp;^99�^ of these 25 parts in an inch; this extent setnbsp;from A to I, fig. 2, plate 5, from 1 to 2, amp;c. tonbsp;B divides AB into a scale of 12 feet^ 3. Setnbsp;W one of these parts from A to a, to be subdi-Vided into 12 parts to represent inches. 4. Tonbsp;his end divide this into three parts; thus take thenbsp;^xtent A 2 of two of these parts, and make it anbsp;�[ansverse distance between 9 and 9.nbsp;nbsp;nbsp;nbsp;5. Set the

istance between 6 and 6 from b to e, the same ^^fent from 1 gives g, and from e gives n, thusnbsp;ividing A a into three equal parts in the points n

and g. 6. By two bisections each of these may be subdivided into tour equal parts, and thus thenbsp;whole space into 1'2 equal parts.

When a small number of divisions are required^ as 1, 2f or 3, instead of taking the transverse dis'-tanee near the center of the sector, the divisionnbsp;will be more accurately performed by using thenbsp;following method.

Th us, if three parts are required from A, of which the whole line contains 00, make AD, Jig. 4,nbsp;plate 5, a transverse distance between yo and QO;nbsp;then take the distance between 87 and 87, whichnbsp;set off from D to E backwards, and AE will contain the three desired parts.

Example 1. Supposing a scale of six inches to contain 140 poles, to open the sector so that itnbsp;may ansAver for such a scale; divide 140 by 2,nbsp;which gives 70, the half of 6 equal to 3; becausenbsp;T40 was too large to be set off on the line of lines.nbsp;Make three inches a transverse distance betweennbsp;70 and 70, and your sector becomes the requirednbsp;scale.

Example 2gt;. To make a scale of seven inches that shall contain 180 fathom;nbsp;therefore make 34 a transverse between Q and 9,nbsp;and you have the required scale.

Problem 13. To cut a given line AD, j%-, 14, plate 5, into tzvo jinequal parts that shall have anynbsp;given proportmi, ex. gr. of C to D.

1. Draw AG forming any angle with AD� 2. From A on AG set otF AC equal to C, andnbsp;C E equal to D. 3. Draw E D, and parallel to itnbsp;C B, which will cut AD at B in the required proportion.

^ divide by the sector the line AB, JJg. 1, plate 5, in the proportion of 3 to 2. Now as 3 andnbsp;2 vvoLild fell near the center, multiply them by �1,nbsp;thereby forming 6 and 4, which use instead of 3nbsp;^/id 2. As the parts are to be as 6 to 4, the wholenbsp;line will be 10; therefore make AB a transversenbsp;distance between 10 and 10, and then the transverse distance between 6 and �, set off from B to

is Iths of AB; or the distance between 4 and 4 will give Ae l-ths of AB; therefore AB is cut innbsp;the proportion of 3 to 2.

Example 2. To cut AB,Q., plate 5, in the proportion of 4 to 5 ; here we may use the numbers themselves; therefore with A B open the sector at 9 and 9, the sum of the two numbers; thennbsp;the distance between 5 and 5 set off� from B to c.nbsp;Or between 4 and 4, set offquot; A to c, and it dividesnbsp;in the required proportion.

Eo/e. If the numbers be too small, use their rtjuimultiples; if too large, subdivide them.

Corollary, From this problem we obtain another mode of dividing a strait line into any number

ProbLkm 14. To estimate the proportion he~ ^^een two or more given lines, AB, C D, E F,nbsp;�l^S'9, plate 5.

Make AB a transverse distance between 10 and to, then take the extents severally of CD and EF,nbsp;^od carry them along the line of lines, till bothnbsp;points rest exactly upon the same number; in thenbsp;byst it will be found to be 85, in the second 67,nbsp;therefore AB is to CD as 100 to 85, to EF' asnbsp;too to 67, and of C D to E F as 85 to �7.

lines take DG equal to the first term A, and DC, DII, each equal to the second term B. 3. Joinnbsp;G H, and draw C F parallel thereto; then D Fnbsp;will be the third proportional required, that is,nbsp;D G ( A) to D C, (B,) so is D H (B) to D F.

By the sector. 1. Make AB, fig. 5, plate 5, a transverse distance between 100 and 100. 2. Findnbsp;the transverse distance of E F, which suppose 50,nbsp;.3. Make EF a transverse distance between 100 andnbsp;100.nbsp;nbsp;nbsp;nbsp;4, Take the extent between 50 and 50, and

Problem i6. To find a fourth propor-tional to three given right lines A, B, Q,fig. l6, plate 5:

1. From the point a draw two right lines, making any angle whatever. 2. In these lines make ab equal to the first term A, ac equal to the second B, and a d equal to the third C. 3. Join be,nbsp;and draw de parallel thereto, and ae will be thenbsp;fourth proportional required; that is, a b (A) is tonbsp;ac (B,) so is ad (C) to ae.

By the sector. 1. Make the line A a transverse distance between 100 and 100.nbsp;nbsp;nbsp;nbsp;2. Find the transverse measure of B, w'hich is 6o.nbsp;nbsp;nbsp;nbsp;3. Make c the

third line a transverse measure between 100 and 100.nbsp;nbsp;nbsp;nbsp;4. The measure between 6o and 60 will be

Problem 17. To find a mean proportional he-t-iveen two given strait lines A and B,fig. 17, plate 5.

1. Draw any right line, in which take C E equal to A, and EA equal to B. � 2. Bisect AC in B, andnbsp;with BA, or BC as a radius, describe the semicircle ADC. 3. From the point E draw EGnbsp;perpendicular to AC, and it will be the mean proportional required.

By the sector. Join the lines together, (suppose them 40 and 90) and get the sum of them, I3�1 then find the half of this sum 65,- and half

the dlfFerence 25. Open the line of lines, so that they may be at right angles to each other; thennbsp;take with the compasses the lateral distance 65,nbsp;^nd apply one foot to the half difference 25, andnbsp;the other foot will reach to �O, the mean proportional required; for 40 to 6o, so is 6o to QO.

Problem 18. To cut a given line AB into extreme and mean proportion, Jig. 18, plate 5.

1. Extend AB to C. 2. At A erect a perpendicular AD, and make it equal to AB. 3. Set the half of AD or AB from A to F. 4. With thenbsp;I'adius F D describe the arc D G, and AB will benbsp;divided into extreme and mean proportion. A Gnbsp;is the greater segment.

By the sector. Make AB a transverse distance between 6o and 6o of the line of chords. 2. Takenbsp;Out the transverse distance between the chord ofnbsp;36, which set from A to G, gives the greatestnbsp;segment.

Or make AB, a transverse distance between 54 and 54 of the line of sines, then is the distance between 30 and 30 the greater segment, and 18 andnbsp;18 the lesser segment.

Problem IQ. To divide a given strait line in the same proportion as another given strait line is di-quot;^ided, 'Jig. 10, plate 5.

Let AB, or CD be two given strait lines, the first divided into 100, the second into 6o equalnbsp;parts; it is required to divide EF into 100, andnbsp;OH into 6o equal parts.

Make EF a transverse distance in the terms 100 and 100, then the transverse measure between gonbsp;^ud go set from E to gO, and from F to 10; thenbsp;measure between 80 and 80 set from E to 80, andnbsp;from F to 20, and the measure between 70 and70nbsp;act from E gives 70, from F 30. The distancenbsp;between 6o and 6o, gives 6o and 40; andnbsp;lastly, the transverse measure between 50 and 50

bisects the given line in the point 50, and we shall, by five transverse extents, have divided the linenbsp;EF into 10 equal parts, each of which are to benbsp;subdivided into 10 smaller divisions by problemsnbsp;11 and 12.

To divide G H into 6o parts, as we have supposed CD to be divided, make GH a transverse distance in the terms of 6o, then work as hefore.

Problem 20. To find the angular point of tzvo fiven lines AK, cP, fig. 22, plate 5, which inclinenbsp;to each other without producing either of them.

1. Draw the parallel lines Av, Ge, K^. 2. Take any number of times the extents, AN, GO, K P,nbsp;and set them on their respective lines, as from N tonbsp;y and V, from o to |3 and i, from P to � and andnbsp;a line through e, u, and another through a, p, y,nbsp;will tend to the same point as the lines AB, c P.

�plate 5, draw any two parallel lines, as GH and F E. 2. Set off the extent B D twice, from B tonbsp;G, and D to H; and the extent AC twice, fromnbsp;A to F, and from C to E. 3. A line passingnbsp;through F and G will intersect another line passing through E and H in I, the angular point required.

The extent FA, G B, may be multiplied or di -vided, so as to suit peculiar circumstances.

Corollary. Hence, if any two lines he given that tend to the same angular point, a third, or morenbsp;lines may be drawn that shall tend to the samenbsp;point, and yet pass through a given point.

Solution by the sector. Case 1. When the proposed point e is between the two given lines VL and a \gt;,fig. 23, plate 5.

Through e draw a line av, cutting ah at a, and VL at V, then from any other point b, in ab, the

f-ailher the better, draw b x parallel to a V, and cutting V L in x, make a v a transverse distaneenbsp;between lOO and 100, on the line of lines; takenbsp;l^he extent e v, an-d find its transverse measure,nbsp;which suppose to be 60; now make xb a transversenbsp;distance between lOO and 100, and take out thenbsp;li�ansverse distance of the terms 6o, which set offnbsp;from X to f, then a line drawn through the pointsnbsp;Cand f, shall tend to the same inaccessible point q

Case %. When the proposed point e is without the given lines ab and VL, fig. '24.

Through e draw any line ev, cutting ab in a, and VL in v; and from any other point b in ab drawnbsp;xT parallel to ev, and cutting VL in x, makenbsp;We a transverse distanee in the terms of 100; findnbsp;the transverse measure of av, which suppose 72;nbsp;tfiakexb a transverse measure of 72, and take outnbsp;the distance between the terms of 100, which setnbsp;off from X to f, and a line ef drawn through e andnbsp;fj will tend to the same point with the line ab.

If the given or required lines fall so near each other, t hat neither of them can be measured on thenbsp;terms of 100, then use any other number, as 80,nbsp;7o, 6o, amp;c. as a transverse measure, and work withnbsp;that as you did with 100.

This problem is of considerable use in many geometrical operations, but particularly so in perspective; for \ve may consider VL as a vanishingnbsp;hne, and the other two lines as images on the picture; hence having any hnage given on the picturenbsp;that tends to an niaccessible vanishing point, asnbsp;luany more images of lines tending to the samenbsp;Pcint as may be required, are readily drawn. Thisnbsp;problem is more fully illustrated, and all the various eases investigated, in another part of thisnbsp;Work.

Problem 21. Upon a given right line AB, Jig. IQ, plate 5, to make an equilateral triangle.

1. From A and B, with a radius equal to AB, describe arcs cutting in C. 2. Draw AC andnbsp;BC, and the figure ACB is the triangle required.

Problem 22. To make a triangle, whose three sides shall he respectively equal to three given lines,

A, nbsp;nbsp;nbsp;B, C, Jig- 20, plate 5, provided any two of themnbsp;he greater than the third.

B, nbsp;nbsp;nbsp;with a radius equal to C, describe an arc at A.nbsp;3. On C, with a radius equal to A, describe another arc, cutting the former at A. 4. Draw thenbsp;lines AC and AB, and the figure AB C will be thenbsp;triangle required.

1. From the point B draw BD perpendicular, and equal to AB. 2. On A and D, with the radius AB, describe arcs cutting in C. 3. Drawnbsp;AC and CD, and tlie figure ABCD is the required square.

Problem 24. To describe a rectangle or parallelogram, whose length and breadth shall be equal to two given lines A and ^,fig. 2, plate 6.

1. Draw C D equal to A, and make D E perpendicular thereto, and equal to B. 2. On the points E and C, with the radii A and B, describenbsp;arcs cutting in F. 3. Join CFand EF; thennbsp;C D E F will be the rectangle required.

1. On B, with the radius AB, describe an arc at D. 2. On A, with the same radius, describe

an arc at C. 3. On C, but still with the same �'adius, make the intersection D. 4. Draw thenbsp;lines AC, DC, DB, and you have the requirednbsp;figure.

Make the angle A C D equal to the given angle, nnd set off CD equal to AB, and AC equal tonbsp;AD; then from A, with the distance AB, describenbsp;^n arc at B; intersect this arc with the extent AD,nbsp;set off from D; join AB, B D, and the, figure isnbsp;Completed.

Problem 26. Having the diagoiml AD, and four sides AB, B D, D C, AC, to construct a trape-^iuni, fig, 5, plate 6.

Draw an occult line AD, and make it equal to the given diagonal. Take AB in the compasses,nbsp;^nd from A strike an arc at B; intersect this arcnbsp;fi'om D with the extent DB, and draw AB, DB;nbsp;now with the other two lines AC, CD, and fromnbsp;A and D make an intersection at C; join DC, AC,nbsp;�nd the figure is completed.

Problem 27- Having the four sides and one an-to construct a trapezium, fig. 5, plate 6.

A make the angle CAB equal to the given an-and AC equal to the given side AC; then, ynh the extent B D from B, describe an arc at D,nbsp;intersect this from C with the extent C D; join thenbsp;Several lines, and the figure is obtained.

1. Draw any chord AB, and bisect it with the nhord CD. 2. Bisect CD by the chord EF,nbsp;nnd their intersection o will be the center of the

*quot;lrcle.

Problem 29. To describe the circumference of a circle through any three given points AB C,fg. 7,nbsp;plate 6, provided they are not in a strait line.

1. From the micldlc point B draw the chords BA and B C. 2. Bisect these chords with thenbsp;perpendicular lines n O, n O, 3. From the pointnbsp;of intersection O, and radius OA, OB, and OC,nbsp;you may describe the required circle ABC. Bynbsp;this problem a portion of the circumference ofnbsp;a circle may be finished, by assuming threenbsp;points.

Problem 30. To draw a tangent to a given circle, that shall pass through a given point A, fig. 8nbsp;and Q, plate 6.

Case 1. When the point A is in the circumference of the circle, j'z^. 8, plate 6.

1. From the center O, draw the radius OA. 2. Through the point A, draw C D perpendicularnbsp;to OA, and it will be the required tangent.

1. From the center O draw OA, and bisect it in n. 2. From the point n, with the radius n A,nbsp;or n O, describe the semicircle AD O, cutting thenbsp;given circle in D. 3. Through the points A andnbsp;D, draw AB, the tangent required.

Problem 31. To cut off from a circle, fig. 10, plate 6, a segment containing any proposed angle,nbsp;ex. gr. 120�.

Let F be the point from whence it is required to draw a chord which shall contain an angle of 120�.

2. nbsp;nbsp;nbsp;From F draw FA, making an angle of 60 degrees, with the tangent Fll, and FCA is thenbsp;segment required.

Pros LEM 32. On a given line AB to describe ibe segment of a circle capable of contahan? a (rivennbsp;^ngleji^,^{^ plate Q.

Draw AC and B C, making the angles ABC, � Ac, each equal to the given angle. Draw ADnbsp;PWpendicular to AC, and BD perpendicular tonbsp;^ C; -with the center D, and radius DA, or DB,nbsp;^escribe the segment AEB, and any angle made

this segment will be equal to the given angle. A more easy solution of this problem will be givennbsp;when we come to apply it to practice.

Problem 33. l/o describe an arc of a circle ibat shall contain any number of degrees, without coni-fasses, or without finding the center of the circle,nbsp;fig-Vl,plateQ.

Geometrically by finding points through which ^he arc is to pass, let AB be the given chord.

At any point F, in AF, make the angle E F G, ^htial to the given angle. 3. Through B draw BEnbsp;pRi'allel to F G, and the intersection gives thenbsp;point E, in the same manner as many points D,nbsp;A'? amp;c. may be found, as will be necessary to complete the arc.

�creaftcr.

Problem 34. To inscribe a circle in a given triangle^ A B C,nbsp;nbsp;nbsp;nbsp;13, plate �.

F Bisect the angles A and B with the lines AO and BO. 2. From the point of intersection O,nbsp;lot fidl iPp perpendicular O N, and it will be thenbsp;^3-fiius of the required circle.

E Draw the diameters AB and CE at right ^^igles to each other. 2. Bisect DB at G. 3. Onnbsp;with the radius GC, describe the are CF,

The two sides DC, F D of the triangle F D C, enable us to inscribe a hexagon or decagon in thenbsp;same circle; for DC is the side of the hexagon,nbsp;D F that of the decagon.

Problem 36. To inscribe a square or an octagon in a given circle, jig. l6, plate 6.

1. Draw the diameters AC, B D, at right anglesnbsp;to each other. 2. Draw the lines AD, BA, BC,nbsp;C D, and you obtain the required square.

Bisect the arc AB of the square in the point F, and the line AE being carried eight times round,nbsp;will form the octagon required.

Problem 37- bn a given circle to inscribe an equilateral triangle, an hexagon, or a dodecagon,nbsp;fig- plate Q.

1. From any point A as a center, with a distance equal to the radius AO, describe the arc FOB.

Carry the radius AO six times round the circumference, and you obtain a hexagon.

Bisect the arc A B of the hexagon in the point n, and the line An being carried twelve times round

Problem 38. Another method to inscribe a dodecagon in a circle, or to divide the circumference of ^ given circle into 12 equal parts, each of 30 degrees,nbsp;Jig. 13, plate Q.

1. Draw the two diameters 1, 7? 4, lO, perpendicular to each other. 2. With the radius of the circle and on the four extremities 1, 4, 7? lOj RSnbsp;^enters, desciibe arcs through the center of thenbsp;�Circle; these arcs will cut the circumference in thenbsp;points required, dividing it into 12 equal parts, atnbsp;Pie points mai'ked with the numbers.

Problem 3Q. To find the angles at the center, ^nd the sides of a regidar polygon.

Divide 360 by the number of sides in the proposed polygon, thus�Ogives 72, for the angle at the center of a pentagon. To find the anglenbsp;formed by the sides, subtract the angle at the center from 180, and the remainder is the angle re-^luired; thus 72� from 180�, gives 108� for the an gt;nbsp;Sie of a pentagon.

This table is constructed by dividing 300, tfi� degrees in a circumference, by the number of sidesnbsp;in each polygon, and the quotients are the anglesnbsp;at the centers; the angle at the center subtractednbsp;from 180 degrees, leaves the angle at the circumference.

Problem 40* In a given circle to inscribe any' regular polygon, jig. 14, plate 6.

1. At the center c make an angle equal to the center of the polygon, as contained in the preceding table, and join the angular points AB.nbsp;2. The distance AB will be one side of the polygon,nbsp;which being carried round the circumference, thenbsp;proper number of times will complete the figure.

1. Divide the diameter AB into as many equal parts as the figure has sides. 2. From the centernbsp;O raise the perpendicular Om. 3. Make mnnbsp;equal to three fourths of Om. 4. From n draw nC,nbsp;through the second division of the diameter. 3.nbsp;Join the points AC, and the line AC will be thenbsp;side of the required polygon, in this instance anbsp;pentagon.

Problem 41. About any given triangle AB C, to circumscribe a circle, jig. IX, plate 6.

1. Bisect any two sides AB, BC, by the perpendiculars mo, no. 2. From the point of intersection o, with the distance OA or OB, describe the required circle.

1. Inscribe a pentagon within the circle, 2. Through the middle of each side draw' the linesnbsp;O A, OB, O C, O D, and O E. 3. Through thenbsp;point n draw the tangent AB, meeting OA andnbsp;O B in A and B, 4. Through the points A and m,

di'aw the line AmC, meeting OC in C. like manner draw the lines C D, D E, E B, andnbsp;AB C D E will be the pentagon required.

Problem 43. About a given square to circum-~ sc7'ibe a circle., jig. 22, flute �.

1. Draw the two diagonals AD, B C, intersecting each other atO. 2. From P, with the distance OA or OB, describe the circle ABCD, whichnbsp;will circumscribe the square.

From the point O, with the distance OA or OB, describe the circle ABCDEF. 3. Carrynbsp;-^B six times round the circumference, and it willnbsp;^nn the required hexagon.

Problem 45. On a given line AB to form a regular polygon of any frofosed number of sides, fg. 14, flute 6.

1. Make the angles CAB, CBA each equal to Iialf the angles at the circumference; sec the preceding table. 2. From the point of intersection c,nbsp;''quot;ith the distance C A, describe a circle. 3. Applynbsp;the chord ABto the circumference the proposednbsp;dumber of times, and it will form the required polygon.

1- On the extremities of the given line AB direct tire indefinite perpendiculars AF and BE..

Produce AB both v/ays to s and w, and bisect the angle nAs and oBw by the lines AH, BC.nbsp;3- Make All and BC each equal to AB, and,nbsp;draw the line H C. 4. Make ov equal to on, andnbsp;through V draw GD parallel to H C. 5. Draw

HG and CD parallel to AF and BE, and make cE equal to cD. 6. Through E draw EF, parallel to AB, and join the points GF and DE^nbsp;and ABCDEFGH will be the octagon required.

Problem 47. On a ^'tvsn right line AB,_/?^. 26, plate 6, to describe a regidar pentagon.

2. nbsp;nbsp;nbsp;Bisect AB in n. 3. On n, with distance nm,nbsp;cross AB produced in O. 4. On A and B, withnbsp;radius AO, describe arcs intersecting at D. 5. Onnbsp;D, with radius AB, describe the arc E C, and on Anbsp;and B, with the same extent, intersect this arc atnbsp;E and C. 6. Join AE, ED, DC, CB, and younbsp;corhplete the figure.

Problem 48. Upon a given right line AB, fig. 27, plate 6, to describe a triangle similar to thenbsp;triangle C D E.

1. At the end A of the given line AB, make an angle FAB, ecjual to the angle E C D. 2. Atnbsp;B make the angle AB F equal to the angle C D E.

3. nbsp;nbsp;nbsp;Draw the two sides AF, B F, and ABF willnbsp;be the required triangle.

'Problem 49. To describe a polygon similar to a given polygon ABCDEF, one of its sides ab beingnbsp;given, fig. 28, plate 6.

1. Draw AC, AD, AF. 2. Set off ab on AB, from a to r. 3. Draw rg parallel to BE, meetingnbsp;AF in g. 4. Through the point g draw gh, parallel to F C, meeting AC in h. 5. Through thenbsp;point h draw the parallel hi. 6. Through i drawnbsp;i k parallel to E D, and the figure A r g h i k willnbsp;be similar to the figure ABCDEF.

]. Measure each side of the figure AB C D E with the scale GFI. 2. Make ab as many partsnbsp;of a smaller scale ICL, as AB was of the larger.

8. be as many of K L, as B C of G H, and ac of K L, AC, amp;c. by which means the figure will benbsp;reduced to a smaller one.

Pro BLEM 51. To change a triangle into another equal extent, hit differe7it height, fig. 1, 2, 3, 4,nbsp;tlate 7.

Let AB C be the given triangle, D a point at the given height.nbsp;nbsp;nbsp;nbsp;'

Case 1. Where the point T),fig. 1 and2, y'/^/e IS either in one of the sides, or in the prolongation of a side. 1. Draw a line from D to the opposite angle C. 2. Draw a line AE parallel theretonbsp;from A, the summit of the given triangle. 3. Joinnbsp;E, and B D E is the, required triangle.

^ Case 2. When the point D, fig. 3 and 4, plate 7, neither in one of the sides, nor in the prolongation thereof. 1. Draw an indefinite line BDa,nbsp;from B through the point D. 2. Draw from A,nbsp;the summit of the ^iven triangle, a line A a, parallel to the base BC, and cutting the line BD in a.nbsp;3. Join a C, and the triangle B a C is equal to thenbsp;triangle BAG; and the point D heing in the samenbsp;ine with B a. 4. By the preceding case, find anbsp;triangle from D, equal to BaC; i. e. join DC,nbsp;dr^v a E parallel thereto, then join D E, andnbsp;-o D E is the required triangle.

Corollary. If it be required to change the ti'i-BAC into an equal triangle, of which the yoight and angle B D E are given; 1. Di-aw thenbsp;oiclefinite line BDA, making the required anglenbsp;With B C. 2. Take on B D^a a point D at thenbsp;K�ven height; and, 3. Construct the triangle bynbsp;he toregoing rules. ,,

Pr OBLEM52. To make an isosceles triangle AE B, jig. 5, 'plate 7, equal to the scalene triangle

1. Bisect the base in D. 2. Erect the perpendicular DE. 3. Draw CE parallel to AB. 4. Draw AEj EB, and AEB is the required triangle.

Problem 53. To make an equilateral triangle equal to a given scalene triangle Kamp;Q.,Jig.T, plate 7.

1, On the base AB make an equilateral triangle ABD. 2. Prolong BD towards E. 3. Drawnbsp;CE parallel to AB. 4. Bisect DE at I, on D Inbsp;describe the semicircle D F E. 5. Draw B P, thenbsp;mean proportional between B E, B D. 6. ^dthnbsp;B F from B, describe the arc F G H; with thenbsp;same radius from G, intersect this arc at drawnbsp;B H, G H, and B G H is the triangle required.

Corollary. If you want an equilateral triangle equal to a rectangle, or to an isosceles triangle;nbsp;iind a scalene triangle respectively equal to each,nbsp;and then work by the foregoing problem.

Problem 54. To reduce a rectilinear figure ABCDE,//^'. 8 and g, plate J, to another equal tonbsp;it, hut with one side less.

1. Join the extremities E, C, of two sides D E, D C, of the same angle D. 2. From D draw anbsp;line D F parallel to E C. 3. Draw E F, and younbsp;obtain a new polygon ABFE, equal to ABCDE,nbsp;but with one side less.

Corollary. Hence every rectilinear figure may be reduced to a triangle, by reducing it successively to a figure with one side less, until it isnbsp;brought to one wifh only three sides.

For example; let it be required to reduce the polygon ABCDEF, fig. lo and ll, plate 7, intonbsp;a triangle I AH, with its summit at A, in the circumference of the polygon, and its base on thenbsp;base thereof prolonged.

� � Draw the diagonal D F. 1. Draw E G parallel to DF. 3. Draw F G, which gives us a new polygon, ABCGF, with one side less. 4. To reduce ABCGF, draw AG, and parallel theretonbsp;FH; then join AH, and you obtain a polygonnbsp;ABCH, equal to the preceding one ABCGF.nbsp;5. The polygon ABCH having a side AH, whichnbsp;iRay serve for a side of the triangle, you have onlynbsp;to reduce the part AB C, by drawing AC, andnbsp;parallel thereto BI; join A I, and you obtain thenbsp;required triangle lAH.

N. B. In figure 10 the point A is taken at one of the angular points of the given polygon; innbsp;figure 11 it is in one of the sides, in which casenbsp;there is one reduction more to be made, than whennbsp;is at the angular point.

Corollary. As a triangle may be changed into Another of any given height, and with the anglenbsp;the base equal to a given angle; if it be re-lt;^iuired to reduce a polygon to a triangle of a givennbsp;height, and the angle at the base also given, younbsp;^ust first reduce it into a triangle by this4)roblem,nbsp;�^nd then change that triangle into one, with thenbsp;data, as given by the problem 51.

Corollary. If the given figure is a parallelogram, 12, plate draw the diagonal E C, and DFnbsp;parallel thereto; join EF, and the triangle EBFnbsp;equal to the parallelogram EBCD.

1 � If the figures to be added are triangles ol the ''ame height as AMB, BNC, COD, DPE,nbsp;14, plate 7, make a line AE equal to the sumnbsp;their bases, and constitute a triangle A MEnbsp;^hereon, whose height is equal to the given height,nbsp;^nd AM E will be equal to the given triangles.

2. nbsp;nbsp;nbsp;If the given figures are triangles of differentnbsp;heights, or different polygons, they must first benbsp;reduced to triangles of the same height, and thennbsp;these may be added together.

3. nbsp;nbsp;nbsp;If the triangle, into which they are to benbsp;summed up, is to be of a given height, and with anbsp;given angle at the base, they must first be reducednbsp;into one triangle, and then that changed into another by the preceding rules.

4. nbsp;nbsp;nbsp;The triangle obtained may be changed intonbsp;a parallelogram by the last corollary.

r. To multiply AMB, 13, plate by a given number, for example, by 4; or more accurately, to find a triangle that shall be quadruple thenbsp;triangle AM B. Lengthen the base AE, so thatnbsp;it may be four times AB; join M E, and the triangle AM E will be quadruple the triangle AMB,nbsp;2. By reducing any figure to a triangle, we maynbsp;obtain a triangle which may be multiplied in the

1. If the two triangles BAG, dac, fg. 15, plate 7, are of the same height, take from the basenbsp;B C of the first a part D C, equal to the base d cnbsp;of the other, and join AD; then will the trianglenbsp;ABD be the difference between the two triangles.

If the two triangles be not of the same height, they must be reduced to it by the preceding rules,nbsp;and then the difference may be found as above;nbsp;or if a polygon is to be taken from another, and anbsp;triangle found equal to the remainder, it may benbsp;easily effected, by reducing them to triangles ofnbsp;the same height.

A triangle may be taken from a polygon by drawing a line within the polygon, from a givennbsp;point F on one of its sides. To effect this, let usnbsp;suppose the triangle, to be taken from the polygonnbsp;ABCDE, fig. l6, flate 7, has been changed intonbsp;a triangle MOP, fig. 26, whose height above itsnbsp;kase M P is equal to that of the given point F,nbsp;above AB offig. 16; this done, on AB (prolongednbsp;if necessary) lay off AG equal to O P, join F G,nbsp;and the triaingle AFG is equal to the trianglenbsp;Mop. There are, however, three eases in thenbsp;Solution of this problem, which we shall thereforenbsp;notice by themselves.

If the base MP does not exceed AB, 16, plate 7, the point G will fall thereon, and the pro^nbsp;klem will be solved.

But if the base M P exceeds the base AB, G Mil be found upon AB prolonged, 17 and 18,nbsp;plate 7 ; join F B, and draw G H parallel thereto;nbsp;from the situation of this point arise the other twonbsp;biases.

Case 1. When the point H, fig. 17, plate 7, is found on the side BC, contiguous to the side AB,nbsp;joiiiFH, and the quadrilateral figure FAB Hisnbsp;oqual to the triangle M O P.

Casel. When H, 18, plate 7, meets BC prolonged, from F draw F C and HI parallelnbsp;^hereto; then join FI, and the pentagon FABnbsp;^ I is equal to the triangle MOP.

. 1. To divide the triangle AM E,j%. 13, platej, ^^do four equal parts; divide the base into fournbsp;Mual parts by the points B, C, D; draw MD,nbsp;M C, MB, and the triangle fall be divided intonbsp;^Piir equal parts.

2. nbsp;nbsp;nbsp;If the triangle AME, fig. ig, flat e 7, is tonbsp;be divided into four equal parts from a point m,nbsp;in one of its sides, change it into another Ame,nbsp;with its summit at m, and then divide it into fournbsp;equal parts as before; if the lines of division arenbsp;contained in the triangle AME, the problem isnbsp;solved; but if some of the lines, as mD, terminatenbsp;without the triangle, join mE, and draw dD parallel thereto; join md, and the quadrilateralnbsp;m C E d is equal to CmD |.th of AM E, and thenbsp;triangle is divided into four equal parts.

3. nbsp;nbsp;nbsp;To divide the polygon ABCDEP, fig. 20,nbsp;plate 7, into a given number of equal parts, ex. gr.nbsp;four, from a point G, situated in the side A F;nbsp;1. Change the polygon into a triangle A G M,nbsp;whose summit is at G. 2. Divide this trianglenbsp;into as many equal triangles AGH, HGI, IGK,nbsp;K G M, as the polygon is required to be dividednbsp;into. 3. Subtract from the polygon a part equalnbsp;to the triangle AGH; then a part equal to thenbsp;triangle AG I, and afterwards a part equal to thenbsp;triangle AGK, and the lines GH, GR, GO,nbsp;drawn from the point G, to make these subtractions, will divide the polygon into four equalnbsp;parts, all which will be sufficiently evident fromnbsp;consulting the figure.

NO , �^ NO , .

4. nbsp;nbsp;nbsp;4

Observing, 1. That in the two figures �21 and ^2., A is placed between N and O; and that APnbsp;IS taken on AO, prolonged if necessary.

2. nbsp;nbsp;nbsp;In jig. 23, where the point O is situated between N and A, AP should be put on the sidenbsp;Opposite to AO.

3. nbsp;nbsp;nbsp;In 24, where the point N is situated between A and O, if AP be smaller than AN, itnbsp;must be taken on the side opposite to AN.

4. nbsp;nbsp;nbsp;In jig. 25, where the point N is also placednbsp;between A and O, if AP be greater than A N, itnbsp;must be placed on AN prolonged.

Now by problem 17 make MN in all the five figures a mean proportional between NO, NP,nbsp;^nd carry this line from L to B, and from L to b.nbsp;and NO will be : AB ;: AB : NBnbsp;NO :nbsp;nbsp;nbsp;nbsp;A b : : Ab : NB.

Problem 56. Two lines EF, GH, 27, 28, 29, plate 7, intersecting each other at A, beingnbsp;given, to draw jroni C a tliird line B D, which shallnbsp;form with the other two a triangle DAB equal to anbsp;given triangle X.

1. From C draw CN parallel to EF. 2. Change the triangle X into another C N O, whose summit

at the point C. 3. Find on GH a point B, so that N O : AB :: AB : N B, and from this pointnbsp;^3 draw the line C B, and DAB shall be the re-tiuired triangle.

Scholium. This problem may be used to cut off one rectilinear figure from another, by drawing anbsp;fine from a given point.

Thus, if from a point C, without or within the triangle E A Yi,jig. 30, a right line is required tonbsp;bo drawn, that shall cut off a part DAB, equal tonbsp;the triangle 'iL,fig. 30 ; it is evident this may benbsp;offected by the preceding problem.

If it be required to draw a line BD from a. point c, which shall cut off from the quadrilateralnbsp;figure E F G H� a portion DEGnbsp;nbsp;nbsp;nbsp;31, equal

to the triangle Z; if you are sure, that the ri^ht line BD will cut the two opposite sides EF, GH,nbsp;prolong EF, GH, till they meet; then form anbsp;triangle Z, equal to the two triangles Z and F AG;nbsp;and then take Z from AEH by a line B D fromnbsp;the point C, which is effected by the preceding-problem.

If it be required to take from a polygon Y, a part DFIHB equal to a triangle X; and thatnbsp;the line BD is to cut the two sides EF, GH;nbsp;prolong these sides till they meet in A; then makenbsp;a triangle Z, equal to the triangle X, and the figure AFIH; and then retrench from the trianglenbsp;E A G the triangle DAB equal to Z, by a linenbsp;B D, from a given point C.

As all rectilinear figures may be reduced to triangles, we may, by this problem, take one rectilinear figure from another by a strait line drawn from a given point.

Pros LEVI 57. To nuihe a triangle equal to any given quadrilateral figure KQCY), fig. �iZ, plate quot;J.

1. Draw the diagonal BD. 2. Draw CE parallel thereto, intei�secting AD produced in E. 3. Join AC, and ACE is the required triangle.

Probleim 58. To make a rectangle, or parallelogram equal to a given triangle ACE, fig. 33, plate 7.

1. Bisect the base AE in D. 2. Through C draw CB parallel to AD. 3. Draw CD, BA,nbsp;parallel to each other, and either perpendicular to*nbsp;A E, or making any angle with it. And the rectangle or parallelogram ABCD will be equal to th.e'nbsp;given triangle.

Draw the radius OB, and tangent AB pcrpen-lt;igt;cular thereto; make AB equal to three times the diameter of the circle, and f more; join AO,nbsp;Rnd the triangle AOB will be nearly equal thenbsp;given circle.

Problem 6o. To make a square equal to a given rectangle, A B C D, Jig. 3 5, plate 7 �

Produce one side AB, till B L be equal to the other BC. lt;1. Bisect AL in O. 3. With thenbsp;^listance AO, describe the semicircle LFA. 4.nbsp;Produce B C to F. 5. On BF make the squarenbsp;PFGH^ which is equal to the rectangle ABCD.

Problem 6i. To make a square equal to the-^um of any numher of squares taken together, ex. gr. rqual to three given squares, whose sides are equal tonbsp;^he lines AB C,fig. 36, plate.

1. Draw the indefinite lines ED, DF, at right angh's to each other. 2. Make D G equal to A,nbsp;^nd D H equal to B. 3. Join G and FI, and GHnbsp;be the side of a square, equal the two squaresnbsp;'' hose sides are A and B. 4. Make DL equal GH,nbsp;DK equal C, and join KL; then will KL be thenbsp;^ide of a square equal the three given squares. Or,nbsp;after the same manner may a square be constructednbsp;'^qnal to any number of given squares.

Problem62. To describe a figure equal to the su7n *f any gijnen 7iumber of similar figures, fig. 30, plate 7.

This problem is similar to the foregoing: 1. Form right angle. 2. Set off thereon two homologousnbsp;*gt;ides of the given figures, as from D to G, and from

Strictly, this can only be solved but by an approximation; the area, oi of the circle is yet a desiderata in mathematics. See lluttvfi'xnbsp;andnbsp;nbsp;nbsp;nbsp;JJiciiQnarf^ a Vols. 4to. 1796. EdU'.

D to H. 3. Draw GH, and thereon describe a figure similar to one of the given ones, and it willnbsp;be equal to their sum. In the same manner younbsp;may go on, adding a greater number of similarnbsp;figures together.

If the similar figures be circles, take the radii or diameters for the homologous lines.

Problem 63. To make a square equal to the difference of two given squares, whose sides, are AB,nbsp;CD, Jig.o7,plate7.

1. On one end B of the shortest line raise a perpendicular B F. *2. With the extent C D from A, cut B F in F, and B F will be the side of the required square.

Problem 64. To make a figure which shall he similar to, and contain a given figure, a certain numbernbsp;of times. Let MN he an homologous side of thenbsp;given figure, fig. plate 7 �

1. Draw the indefinite line BZ. 2. At any point D, raise DA perpendicular to BZ. 3. Makenbsp;B D equal to JVTN, and BC as many times a multiple of B D, as the required figure is to be of thenbsp;given one. 4. Bisect BC, and describe the semicircle BAG. 5. Draw AC, BC. 6. Make AEnbsp;equal MN. 7- Draw EF parallel to BC, andnbsp;E F will be the homologous side of the requirednbsp;figure.

Problem 65. To reduce a complex figure from one scale to another, mechanically, hy means of squares.nbsp;Fig. 3g, plate 7 �

1. Divide the given figure by cross bnes into as many squares as may be thought necessary. 2. Divide another paper into the same number ofnbsp;squares, either greater, equal, or less, as required.nbsp;3. Draw in every square what is contained in the

correspondent square of the given figure, and you will obtain a copy tolerably exact.

Problem 66. To enlarge a map or plan, attd '^lake it twice, three, four, or five, ^c. times largernbsp;than the original, fig. Xl, plate

1. Draw the indefinite line ab. 2. Raise a perpendicular at a. 3. Divide the original plan into Squares by the preceding problem. 4. Take thenbsp;Side of one of the squares, which set off from a to d,nbsp;^Rd on the perpendicular from a to e, finish thenbsp;�quare a e f d, which is equal to one of the squaresnbsp;�fthe proposed plan. 5. Take the diagonal de,nbsp;it off from a to g, and from a to 1; completenbsp;fbe square a 1 n g, and it will be double the squarenbsp;^ e fd. To find one three times greater, take d g,nbsp;Rnd with that extent form the square amoh, whichnbsp;^dl be the square required. With d h you maynbsp;^orm a square that will contain the given one aefdnbsp;JORr times. The line d 1 gives a square five timesnbsp;larger than the original square.

1. Divide the given plan into squares by problem 65. nbsp;nbsp;nbsp;2. Draw a line, on which set off from

to B one side of one of these squares. 3. Divide ^his line into two equal parts at F, and on F as anbsp;Center, with FA or FB, describe the semicircle

AHb.

4. To obtain I the given square, at F erect the perpendicular FH, and draw the right line AHgt;nbsp;'''hich will be the side of the required square.

For Id, divide AB into three parts; take one �f these parts, set it off from A to C, at C raise thenbsp;perpendicular C I, through I draw AI, and it willnbsp;q the side of the required square. 6. For ith, divide AB into four equal parts, set off one of thesenbsp;roin B to E, at E make E G perpendicular to AB,

Problem 68i To make a map or plan tn proportion to a given one, e. gr. as three to five, fig. 6, plate 8.

The original plan being divided into squares, 1. draw AM equal to the side of one of thesenbsp;squares. 2. Divide AM into five equal parts.nbsp;Q. At the third divison raise the perpendicular CD,nbsp;and draw AD, which will be the side of the required square.

Problem 6g. To reduce figures hy the angle of reduction. Let tih he the given side on �which it isnbsp;required to construct a figure similar to ACDEjjnbsp;fig. 1, �2,3, plate 8.

1. Form an angle LMN at pleasure, and set off the side AB from M to I. 2. From I, with ab,nbsp;cut ML in K. 3. Draw the line IK, and several lines parallel to, and on both sides of it.nbsp;The angle LM N is called the angle of proportionnbsp;or redviction. 4. Draw the diagonal lines B C,nbsp;AD, AE, B D. 5. Take the distance B C, andnbsp;set it off from M towards Lon ML. 6. Measurenbsp;its corresponding line KI. 7- From b describenbsp;the arc n o. 8. Now take AC, set it ofl�on M L,nbsp;and find its correspondent line fg. g. Fi'om a,nbsp;with the radius f g, cut the former arc no in c, anclnbsp;thus proceed till you have completed the figure.

Problem 70. To enlarge a figure hy the angle of reduction. Let abede he the given figure, andnbsp;AB the given side, fig. 3, 1, and A,plate 8.

1. Form, as in the preceding problem, the angle LMN, by setting off ab from M to H, and from H with AB, cutting M L in 1.nbsp;nbsp;nbsp;nbsp;2. Draw'

H I, and parallels to, and on both sides of it. 3. Take the diagonal be, set it off from M towardsnbsp;L, and take off it� corresponding line qr. 4. With

C[r as a radius on B describe the are mn. 5. Take the same correspondent line to ac, and on A cutnbsp;in c, and so on for the other sides.

Problem 71- To cut ojffrom any given arc of o circle a third, a fifth, a seventh, ^c. odd parts,nbsp;ernd thence to divide that arc into any number of equalnbsp;parts, jig. 7, plate 8.

Example 1. To divide the arc AK B into three equal parts, CA being the radius, and C the center of the arc.

Bisect AB in K, draw the two radii CK, C B, ^�Dd the chord AB; produce AB at pleasure, andnbsp;make BL equal AB; bisect AC at G; then anbsp;/ule on G and L will cut CB in E, and BE willnbsp;he dd, and C E vds of the radius C B; on CB withnbsp;C E describe the arc Eed; lastly, set off the extentnbsp;E e or D e from B to a, and from a to b, and thenbsp;^rc AK B will be divided into three equal parts.nbsp;Corollary. Hence having a sextant, quadrant,nbsp;accurately divided, h, the chord of any arc setnbsp;upon any other arc of | that radius will cut offnbsp;arc similar to the first, and containing the samenbsp;dumber of degrees.

Also td^ |th, fth, amp;c. of a larger chord will Constantly cut similar arcs on a circle whose radiusnbsp;id, Jth, fth, amp;c. of the radius of the first arc.nbsp;Example 2. Let it be required to divide the arcnbsp;akb into five equal parts, or to find the fthnbsp;part of the arc AB.

Having bisected the given arc AB in K, and drawn the three radii CA, C K, C B, and havingnbsp;found the fifth part of I B of the radius C B, withnbsp;*quot;^lt;11118 CI describe the arc I n M, which will be

bisected in n, by the line CK; then take the extent I n, or its equal M n^ and set it off twice from A to B; that is, iirst, from A to d, and from d to o,nbsp;and oB will be |th of the arc AB. Again, set offnbsp;the same extent from B to ni, and from m to c, andnbsp;the arc, AB will be accurately divided into fivenbsp;equal parts.

Example 3. To divide the given arc AB into seven equal parts. AB being bisected as before,nbsp;and the radii CA, CK, CB, drawn, find by problem 9 the. seventh part of P B of the radius C B,nbsp;and with the radius C P describe the arc P r N;nbsp;then set off the extent P r tw'ice from A to 3, andnbsp;from 3 to 6, and 6 B will be the seventh part ofnbsp;the given arc AB; the compasses being kept tonbsp;the same opening Pr, set it from B to 4, from 4nbsp;to 1; then the extent A1 will bisect 1, 3 into 2,nbsp;and 4,6 into 5: and thus divide the given arc intonbsp;seven equal parts.

In Jig, 8, let the arc B D be ith part, of the given circle, and AB the radius of the circle. Divide AB into eight equal parts, then on center A, with radius AC, describe the arc CE, bisect thenbsp;arc B D in a, and set off this arc Ba from C to b,nbsp;and from b to c; then through A and c draw Ace,nbsp;cutting BD in e, and Be will be Ith part of thenbsp;given circle.

Corollary 1. Hence we have a method of finding the seventh part of any given angle; for, if from the extremities of the given arc radii benbsp;di'awn to the center, and one of these be dividednbsp;into eight equal parts, and seven of these parts benbsp;taken, and another arc described therewith, thenbsp;greater arc will be to the lesser, as 8 to 7 j and sonbsp;of any other proportion.

Corollary 2. If an arc be described with the vadius, it will be equal to the |th part of a circlenbsp;whose radius is AB, and to the seventh part of anbsp;circle whose radius is AC, and to the sixth partnbsp;cf a circle whose radius is AL, amp;c.

Corollary 3. Hence also a pentagon may be derived from a hexagon, Jig. 9, plate 8. Let thenbsp;given circle beABCDEF, in which a pentagonnbsp;is to be inscribed; wdth the radius AC set off APnbsp;equal ith of the circle, divide AC into six equalnbsp;parts; then c G will be five of these parts; withnbsp;1�adius CG describe the arc GH, bisect AF in q,nbsp;and make G P and P H each equal to A q; thennbsp;through C and H draw th? semi-diameter C w,nbsp;Cutting the given circle in w, join Aw, and it willnbsp;one side of the required pentagon.

Corollary 4. Hence as radius divides a circle into six equal parts, each equal 60 degrees, twicenbsp;I�adius gives 120 degrees, or the third part of thenbsp;Circumference.

Onee radius gives 60 degrees, and that arc bisected gives 30 degrees, which, added to 60, di-''ides the circumfei'ence into four equal parts; whence we divide it into two, three, four, five, sixnbsp;^^lual parts; the preceding corollary divides intonbsp;five equal parts, the arc of a quadrant bisected divides it into eight equal parts. By problem 71 wenbsp;iifitain the seventh part of a circle, and by this me-thod divide it into any number of equal parts, evennbsp;^ prime number; for the odd unit may be cut offnbsp;the preceding problem, and the remaining partnbsp;�c subdivided by continual bisection, till anothernbsp;prune number arises to be cut off in the samenbsp;manner.

Problem 73. To divide a given right line, or arc of a circle, into any number of equal parts bynbsp;�quot;e help of a pair of beam, or other compasses, the

distance of �ivhose points shall not he nearer to each other than the given line, fig. i\, plate 8.

From this problem, pviblislicd by Clavius, the Jesuit, in l6l 1, in a treatise on the construetion ofnbsp;a dialling instrument, it is presumed that he was thenbsp;original inventor of that species of division, callednbsp;Nonius, and which by many modern mathematicians has been called the scale of Vernier.

Let A B be the given line, or circular arch, to be divided into a number of equal parts. Producenbsp;them at pleasure; then take the extent AB, andnbsp;set it off on the prolonged line, as many times asnbsp;the given line is to be divided into smaller parts,nbsp;BC, CD, DE, EF, FG. Then divide the wholenbsp;line AG into as many equal parts as are required innbsp;AB, asGH, HI, IK, KL, LA, each of whichnbsp;contains the given line, and one of those partsnbsp;into which the given line is to be divided. Fornbsp;AG is to AL as AF to AB; in other words, ALnbsp;is contained five times in AG, as AB in AF;nbsp;therefore, since AG contains AF, and ith of AF,nbsp;AL will contain AB, and fth of AB; thereforenbsp;BL is the fth of AB. Then as GH containsnbsp;AB plus, FH, which is fths of AB, El will benbsp;fths of AB, DK -Hh, C L |ths. Therefore, if wenbsp;set off the interval G H from F and H, we obtainnbsp;two pails at L and I, set offquot; from the points nearnbsp;E, gives three parts between D K, from tbe fournbsp;points at D and K, gives four parts at C L, andnbsp;the next setting olfquot; one more of these parts; sonbsp;that, lastly, the extent G H set off' from the pointsnbsp;between C and L, divides the given line as required.

To divide a given line AC, or arc of any circle, into any number of equal parts, suppose 30, fig. 11,nbsp;plate 8.

1. Divide it into any number of equal parts less than 30, yet so that they may be aliquot parts

of 30; as for example, AG is divided into six equal parts, AB^ BC, CD, DE, EF, EG, each of whichnbsp;are to be subdivided into fiv-e equal parts. 2. Di-'^alc the first part AB into five parts, by means ofnbsp;the interval AL or GH, as taught in this problem.

now, one foot of the compasses be put into the point A, (the extent AL remaining between themnbsp;'^Haltered) and then into the point next to A, andnbsp;On to the next succeeding point, the whole linenbsp;^G will be divided by the other foot of the compasses into 30 equal parts.

^ Or if the right lino, or arc, be first divided into hve equal parts, each of these must be subdividednbsp;into six parts, which may be effected by bisectingnbsp;^ach part, and then dividing the halves into threenbsp;parts.

Or it may be still better to bisect three of the orst five parts, and then to divide four of thesenbsp;^oto three, which being set off from every point,nbsp;complete the division required.

Corollary. It frequently happens that so many �J^all divisions arc required, that, notwithstandingnbsp;Peir limited number, they can be hardly takennbsp;. *^twcen the points of the compasses without en'or;nbsp;this case use the following method.

If the whole number of smaller parts can be Subdivided, take so many of the small parts in thenbsp;given line, as each is to be subdivided into, yet sonbsp;^at tUcy may together make up the whole of thenbsp;given line. For if the first of these parts be cutnbsp;^uto as many smaller parts as the proposed numbernbsp;^^quires every one of them to contain, and thenbsp;^^ttie is also done in the remaining parts, we shallnbsp;tain the given number of smaller parts.

. it 84 parts are to be taken in the proposed line, bisect it, and each half will contain 42; bisectnbsp;^se again, and you have four parts, each of which,nbsp;H

la to contain '21; and these, divided into three, give 12 parts, each of which is to contain sevennbsp;parts; subdivide these into seven each.

But if the proposed number of small parts cannot be thus subdivided, it will be necessary to take a number a little less or greater, that will be capable of subdivision; for if the superfluous parts arcnbsp;rejected, or those wanting, added, we shall obtain,nbsp;the proposed number of parts. Thus if 74 partsnbsp;arc to be cut from any given line of 80 parts;, 1.nbsp;Bisect the given line, and each ^ will contain 40.nbsp;�2. Bisect these again, and you have four parts tonbsp;contain 20 each. 3. Each of these bisected, younbsp;have eight parts to contain 10 each. 4. Bisect,nbsp;these, and you obtain l6 parts, each to be dividednbsp;into five parts. 6. Reject six of the parts, andnbsp;the remainder is the 74 parts proposed.

Or if 72 parts be proposed; divide the line into 24 equal parts, and each of these into three parts,nbsp;and you obtain 72; to which adding two, younbsp;will obtain the number of 74.

Problem 74. To cut off from the circumference of any given arc of a circle any number of degrees andnbsp;minutes, Jig. 3, plate 9.

1. Let it be proposed to cut off from any arc 57 degrees; with the radius of the given arc, or circle,nbsp;describe a separate arc as AB, and having set offnbsp;the radius from A to C, bisect AC in E, then AEnbsp;and EC will be each of them an arc of 30 degrees.nbsp;Make CB equal to AE, and AB will be a quadrant, or 90 degrees, and will also be divided intonbsp;three equal parts; next, divide each of these intonbsp;three by the preceding rules, and the quadrantnbsp;wall be divided into nine equal parts, each containing 10 degrees. Lastly, divide the first of thesenbsp;into 10 degrees, then set one foot of the compassesnbsp;into the seventh singk degree, and extend the

*ther to the 50tli, and tl^ie distance between ths points of the compasses will contain 57 degrees,nbsp;'vhich transfer to the given arc. Or at two'ope-rations; first, take 50 degrees, and then from thenbsp;first 10 take seven, and you have 57 degrees.

'2. Let it be required to cut off from any given Ere of a circle 45 degrees, 53 minutes, J/g. 3,nbsp;plate 9.

Divide the arc of 53 degrees of the quadrant AB, whose radius is AC (or rather its equal arcnbsp;iigt;to 60 equal parts, first into 5, and thennbsp;�oe of these into three; or first into three, andnbsp;*^ite of these into five. Again, one of these bisected, and this bisection again bisected, gives thanbsp;fioth part of an arc of 53 degrees.

_Por the 5th part of the arc F G is F H, containing 12 parts; its third part is FI, containing fiiur parts; the 5 of FI is N, which contains twonbsp;parts; and FN again bisected in K, leaves P^K thenbsp;60th part of the arc FG; consequently, FK comprehends 53 minutes; therefore, add the arc FKnbsp;to 45 degrees, and the arc AF will contain 45 degrees 53 minutes.

_ Corollary. If we describe a separate arc LM ^ith the radius AC, and set off thereon the extentnbsp;L M of 61 degrees of the given arc AB, and dividenbsp;D M into 60 equal parts; thus, first into two, thennbsp;both of these into three, and then the first of thesenbsp;three into lo by the former rules, which gives thenbsp;fioth part of the arc L M. And as one divisionnbsp;the arc L M contains by construction one degree of the quadrant AB, and one sixtieth part of anbsp;fiogree more, that is one minute, therefore two di-Visions of L M contain two degrees, and two mi-*tutes over; three divisions exceed three degrees bynbsp;three minutes, and,so of all the rest.

Whence if one division of L M be set off fronl any degree on AB, it will add one minute to thatnbsp;degree; two adds two minutes; three, three minutes, and so on.

When the division is so small that the compasses will hardly take it in without error, take two,nbsp;three, four, or more of the parts on L, set them offnbsp;from as many deg-rees back from the degree intended, and you will obtain the degree and minutenbsp;required.

Reduce the whole into 3d parts, which gives us 1040; find the greatest multiple of 3 less than 1040,nbsp;which may be bisected; this number will be foundnbsp;in a double geometrical progression, whose firstnbsp;term is 3, as in the margin; 768 the ninth number,nbsp;is the number sought, as in the margin. Subtractnbsp;768 from 1040, the remainder is �272, 1,hen 3nbsp;find how many degrees and minutes this 6nbsp;remainder contains by the rule of three. 12nbsp;As 1040 is to 360 degrees, so is 272 to 24nbsp;g4� 9' 23quot;. Now set off 94� 9' 23quot; upon 48nbsp;the circle to bo divided, and divide the 96nbsp;remaining part of that circle by continual 192nbsp;bisections, till you come to the number 3, 384nbsp;which will be one of the required divisions 7�8nbsp;of the 34� equal parts, by ivhich dividing the arcnbsp;of 04� 9' 23quot; you will have the whole circle divided into 346f equal parts; for there will be 250nbsp;divisions in the greatest arc, and gof in the other.

Example 2. Let it be required to divide a circle into 179 equal parts. Find the greatest numbernbsp;not exceeding 179? which may be continually bi-

sectecl to unity, which you will find to be 128. Subtract 128 from 170, the I'cmainder is 51; thennbsp;find what part of the circle this remainder willnbsp;Occupy by the following proportion, as 179 'S tonbsp;36o, so is 51 to 102� 34' llquot;; set oft'from thenbsp;circle an arc of 102� 34' llquot;, and divide the re-Diaining part of the circle by continual bisections.nbsp;Seven of which will he unity in this example; bynbsp;^diich means this part of the circle will be dividednbsp;^nto 128 equal parts, and the remaining 51 may benbsp;obtained by using as many of the former bisectionsnbsp;�IS the space will contain, so that the whole circumference will be divided into 179 equal parts.

Rxcnnple 3. Let it be required to divide a circle into 29I equal parts, to represent the days of thenbsp;moon�s age.

Ilcduce the given number into halves, which gives 59 parts; seek the greatest number, not exceeding 59, which may be continually bisected tonbsp;Unity, which you will find to be 32. Subtract thisnbsp;from 59, the remainder is 27; and find, as before,nbsp;the angle equal to the remainder by this proportion, as 59 is to 360, so is 27 to 164� 44' 14quot;; setnbsp;eff 164� 44' 44quot;, divide the remaining part of thenbsp;circle by continual bisections, which will dividenbsp;this portion into 32 parts, and from that, the restnbsp;mto 27I, making 29I parts, as required.

Example 4. Let it be required to divide a circle into 365� 5' 49quot; equal parts, the length of a tropi-cal year.

Reduce the whole into minutes, which will be �^'25949; then seek the greatest nmltiple of 1440,nbsp;the minutes in a solar day, that may be halved, andnbsp;IS at the same time less than 525040; this you willnbsp;find to be 368040, which subtracted from 525040,

that is to contain this number, use the following proportion; as 525049 is to 157300 multipliednbsp;by 360, so is 157309 to 107� 40' 27quot; 49'quot;,

Now set off an angle of 107� 40' 27quot; 49'quot; upon the cirele to be divided, and divide the remainingnbsp;part of that cirele by continual bisections, till younbsp;come to the number 1440, which in this case isnbsp;unity, or one natural solar day; by which, dividingnbsp;the arc of 107� 40' 27quot; 49'quot;, the whole circle willnbsp;be divided into 305� 5' 49quot;; for there will be 256nbsp;divisions, or days, in the greater arc, and 100� 5'nbsp;49quot; in the lesser arc.

Example 5. Let it be required to divide a.circle into 3051 equal parts; which is the quantity of anbsp;Julian year.

Reduce the whole into four parts, which gives us 1401; 1024 is the greatest multiple of 2, lessnbsp;than 1401; when subtracted from 1461, we havenbsp;for a remainder 437, � Then by the following proportion, as 1461 is to 437 X 360, so is 407 tonbsp;107� 40' 46quot; 49'quot;, the degrees to be occupied bynbsp;this remainder.

Set oft'an angle of 107� 40' 47''quot; upon the circle to be divided, and divide the remainder by continualnbsp;bisections, until you arrive at unity, by which dividing the arc, you will have the whole circlenbsp;divided into 305? parts.

Let AB be the quadrant, C the center thereof. With the radius AC describe the two arcs AD,nbsp;B E, and the quadrant will be divided into threenbsp;equal parts, each equal to 30 degrees; then dividenbsp;each of these into five parts, by the preceding rules,nbsp;and the quadrant is divided into 15 equal parts;nbsp;bisect these parts, and then subdivide as already

directed, and the quadrant will be divided into 90 degrees. Other methods will be sexm explained more at large.

Take the smaller with a pair of compasses, and 'vith this opening step the greater. With the rc-^Dainder, or surplus, step one of the former steps;nbsp;'vith the remainder ot this, step one the lastnbsp;�teps, setting down the number of steps each time,nbsp;�^bout five times will measure angles to five seconds.

Then to find the fraction, expressing what part' of the whole the smaller part is.

9. 7. 8. 2. 5. Then 5X2-1-1 = 11, and 11 X 8 -f- 5 = 93, and 93 X 7 11 = �62, and 662nbsp;^9 93=6031; so that -��h is the fraction re quired.

If we call the number of steps a b c d e begllifting at the last, the rule may run thus; multiply ^ by b, and add 1; multiply that sum by c, aminbsp;^dda; multiply this sum by d, and add the preceding sum; multiply this sum by e, and add thenbsp;preceding sum; then the two last sums are thenbsp;f^orins of the fraction.

Txample 1. Suppose the steps are 3.5. 1.9. then 9X1-1-1 = 10, 10 X 5 -b 9 = 59, and 59 Xnbsp;8 -)- 10 = 187; hence the terms are tVt. Nownbsp;^80'� X 59 = 10620, this divided by 187 givesnbsp;86 , with a remainder of J.48, 1-18 X 60 = 8880,nbsp;gives 47, amp;c. so that the measure requirednbsp;56� 47'29quot;.

Example 3. Take a semicircle three inches radius and let the angle be 2 in 1; then the steps will be4, 2,1,2, 3, 2. the answer 41� lO' 50quot; q'quot;.

The whole circle, continued and stepped with the same opening, gave 8. 1. 3. 2. 3, for the same,nbsp;yet the answers agreed to the tenth of a second.

By the same method any given line may be measured, and the proportion it bears to any othernbsp;strait line found. Or it will give the exact valuenbsp;of any strait line, ex. gr. the opening of a pair ofnbsp;compasses by stepping any known given line withnbsp;it, ami this much nearer than the eye can discern, hynbsp;comparing it with any other line, as a foot, a yard, ^c.

This method will be found more accurate than by scales, or even tables of sines, tangents, amp;c. because the measure of a chord cannot be so nicelynbsp;determined by the eye with extreme exactness.

There may be some apparent difficulty attending the rule when put in practice, it being impossiblenbsp;to assign any example which another person cannbsp;repeat with perfect accuracy, on account of thenbsp;inequality in the scales, by which the same steps,nbsp;or line, will be measured by different persons.nbsp;There will, therefore, be always some small variation in the answer; it is however, demonstrablynbsp;true, that the answer given by the problem is mostnbsp;accurately the measure of the given angle, althoughnbsp;you can never delineate another angle, or line, exactly equal to the given one, first measured bynbsp;way of example, and this arising from the inequality of our various scales, our inattention in mea-.-Euring, and the imperfection of our eyes. Hence,nbsp;though to all appearance two angles may appearnbsp;perfectly equal fo each other, this method will givenbsp;the true measure of each, and assign the minutestnbsp;difference between them.

Figure \, pleite^, will illustrate clearly this method; thus, to measure the angle AC B, take AB hetweeu your compasses, and step Ba, ah, be,nbsp;there will be c D over.

Take e D, and with it step Ae, ef, and you will have fB over; with this opening step Ag, gh,nbsp;and you will have, h e over, and so on.

We shall here give the principal methods used hy instrument-makers, before the publication ofnbsp;hdr. Bird'?, method by the Board of Longitude,nbsp;leaving it to artists to judge of their respectivenbsp;Merits, and to use them separately, or combinenbsp;them together, as occasion may require; avoidingnbsp;^ minute detail of particulars, as that ivill be foundnbsp;quot;'vhen we come to describe Mr. Bird's method.nbsp;It will be necessary, however, previously to mention a few circumstances, which, though in common use, had not been described until Mr. Bird'snbsp;and Mr. Ludlams comment thereon were pub-hslied.

In all mathematical instruments, divided by hand, and not by an engine, or pattern, the circles,

lines, which bound the divisions arc not those tvhich are actually divided by the compasses.�

. � A faint circle is drawn veiy near the bounding Circle; it is this that is originally divided. It hasnbsp;heen termed the primitive circle.quot;

� The divisions m:ide upon this circle are faint struck with the beam compasses; finenbsp;points, or conical holes, are made by the pricknbsp;punch, or pointing tool, at the points where thesenbsp;arcs cross the primitive circle; these are callednbsp;^^iginal points.quot;

� The visible divisums are transferred from the ^^'ginal points to the gpitec between the bounding

circles, and are cut by the beam compasses; they are therefore always arcs of a circle, though so-short, as not to be distinguished from strait lines.�

Method 1. The faint or primitive arc is first struck; the exact measure of the radius thereofnbsp;is then obtained upon a standard scale with a nonius division of 1000 parts of an inch, which ifnbsp;the radius exceed 10 inches, may be obtained tonbsp;five places of figures. This measure is tl^e chordnbsp;of 6o. The other chords necessary to be laid offnbsp;are computed by the subjoined proportion,¹ andnbsp;then taken off from the standard scale to be laidnbsp;down on the quadrant.

Set off the chord of 6o�, then add to it the chord of 30, and you obtain the goth degree.

Mr. Bird, to obtain 90�, bisects the chord of 60�, and then sets off the same chord from 30 tonbsp;90quot;, and not of 30 from 00� to 90�. Some of thenbsp;advantages that arise from this method are these;nbsp;for whether the chord of 30 be taken accuratelynbsp;Or not from the scale of equal parts, yet the arc ofnbsp;00 will be truly bisected, (see remarks on bisectionnbsp;hereafter) and if the radius unaltered be set offnbsp;from the point of bisection, it will give 90 true;nbsp;but if the chord 30, as taken from the scale, benbsp;laid off from 60 to 90, then an error in that chordnbsp;will make an equal error in the place of 90�.

Sixty degrees is divided into three parts by setting off the computed chord of 20 degrees, and the whole quadrant is divided to every 10 degrees, by setting off the same extent ffom thenbsp;other points.

Thirty degrees, bisected by the computed chord cf 15, gives 15�, which stepped from the points

As the radius is to the given angle, so is the measure of the radius to half the required chord.

^ The computed chord of 6� being laid off, divides 30 degrees into five parts; and set oft'from the other divisions, subdivides the quadrant intonbsp;Single degrees.

Thus with five extents of the beam compasses, ^Dd none of them less than six degrees, the quadrant is divided into go degrees.

Fifteen degrees bisected, gives 7^ 30', whieh set pft from the other divisions, divides the quadrantnbsp;half degrees.

The chord of 6� 40' divides 20� into three pans, �iid set off from the rest of the divisions, dividesnbsp;the whole instrument to every ten minutes.

Method 2. The chords are here supposed to be ^^iiiputed as before, and taken off from the noniusnbsp;scale.

1. nbsp;nbsp;nbsp;Radius bisected divides the quadrant intonbsp;three parts, each equal to 30 degrees.

2. nbsp;nbsp;nbsp;The chord of 10� gives nine parts, each equalnbsp;to 10 degrees.

3. nbsp;nbsp;nbsp;Thirty degrees bisected and set off, gives 18nbsp;P^rts, each equal to five degrees.

4. nbsp;nbsp;nbsp;Thirty degrees into five, by the chord of 6�;nbsp;then set oft' as before gives go parts, each equal tonbsp;1 degree.

any division whatsoever, and yet preserve a sufficient extent between the points of his compasses.

If the quadrant be divided as above, to every 15. degrees, and then the computed arc of 16 degreesnbsp;set off, this arc may be divided by continual bisection into single degrees. If from the arc of 45�,nbsp;2� 20' be taken, or l� lo' from 22� 30', we maynbsp;obtain every fifth minute by continual bisection.nbsp;If to the arc of 7� lO' be added the arc of 62 minutes, the arc of every single minute may be hadnbsp;by bisection.

In 1767 the Commissioners of Longitude proposed an handsome reward to Mr. Bird, on condition, among other things, that he should publish an account of his method of dividing astronomicalnbsp;instruments; which was accordingly done: and anbsp;tract, describing his method of dividing, was written by him, and published by order of the Commissioners of Longitude in the same year; somenbsp;defects in this publication were supplied by thenbsp;Rev. Mr. Ludlam, one of the gentlemen who attended Mr. Bird to be instructed by him in hisnbsp;method of dividing, in consequence of the Board�s

I shall use my endeavours to render this method elearer to the practitioner, by combining andnbsp;arranging the subject of both tracts.

1 � One of the first requisites is a scale of inches, *ach inch being subdivided into 10 equal parts.

a nonius, in which 10.1 inches is divided into ^00 equal parts, thus shewing the lOOOndth part

an inch. By the assistance of a magnifying glass of one inch focus, the SOOOndth part may be

3. nbsp;nbsp;nbsp;Six beam compasses are necessary, furnishednbsp;^ith magnifying glasses of not more than one inchnbsp;mens. The longest beam is to measure the radius

chord of 60; the second for the chord of42.40; l^be third for the chord of 30; the fourth for 10.20;nbsp;^be fifth fpr 4.40; the sixth for the chord of 15nbsp;^t'grees.

4. nbsp;nbsp;nbsp;Compute the chords by the rules given, andnbsp;l^ake their computed length from the scale in thenbsp;uiftcrent beam compasses.

5. nbsp;nbsp;nbsp;Let these operations be performed in thenbsp;evening, and let the scale and the different beamnbsp;^pnipasses be laid upon the instrument to be di-'aded, and remain there till the next morning.

6. nbsp;nbsp;nbsp;The next morning, before sun-rise, examinenbsp;^bc compasses by the scale, and rectify them, ifnbsp;they are either lengthened or shortened by anynbsp;change in the temperature of the air.

7 � The quadrant and scale being of the same cmperature, describe the faint arc /gt; d, or primi-We circle; then with the compasses tliat are set

to the radius and with a fine prick punch, make a point at a, which is to be the 0 point of the quadrant ; see fig. 3, plate g.

8. With the same beam compasses unaltered lay off from a to e the chord of 00�, making a finenbsp;point at e.

10. nbsp;nbsp;nbsp;Then from the point c, with the beam compasses containing 6o, mark the point r, which isnbsp;that of go degrees.

11. nbsp;nbsp;nbsp;Next, with the beam compasses containingnbsp;15�, bisect the arc er in n, which gives 7 5�.

12. nbsp;nbsp;nbsp;Lay off from n towards r the chord ofnbsp;10� 20', and from r towards n the chord of 4� 40';nbsp;these tAvo ought to meet exactly at the point g ofnbsp;85� 20'.

13. nbsp;nbsp;nbsp;Now as in large instruments each degree isnbsp;generally subdivided into 12 equal parts, of fivenbsp;minutes each, wc shall find that 85� 20' containsnbsp;10.24 such parts, because 20' equal 4 of thesenbsp;parts, and 85X12 makes 1020; noAV 1024 is anbsp;number divisible by continual bisection.

The last computed chord was 42� 40', with which ag was bisected in o, and ao, og, werenbsp;bisected by trials. Though Mr. Bird seems tonbsp;have used this method himself, still he thinks itnbsp;more adviseablc to take the computed chord ofnbsp;21� 20', and by it find the point g; then proceednbsp;by continual bisections till you have 1024 parts.nbsp;Thus the arc 85� 20', by ten bisections, will givenbsp;ii.s the arcs 42� 40', 21� 80', 10� 40', 5� 20', 2� 40',nbsp;l�2�', 40', 20', 10', 5'.

14. nbsp;nbsp;nbsp;To fill up the space between g and r, 85�nbsp;2o', and gO�, which is 4� 40', or 4 X 12 8 equal tonbsp;50 divisions; the chord of 64 divisions was laid offnbsp;from g towards d, and divided like the rest by con-tinuai bisections, as was also from a towards b.

It the work is ivell performed, you will again find points 30, 45, �0, 75, and QO, without anynbsp;�^nsible difference. It is evident that these arcs.nbsp;Well as those of 15�, are multiples of the arcnbsp;5'; for one degree contains 12 arcs of 5' each,nbsp;which 15� contains 180; the arc of 30� containsnbsp;^60; the arc of 60�, 720; that of 75�, QOO; and,nbsp;therefore, 90� contains 1080.

_ Mr. Graham, in 17.25 applied to the quadrant *^iyided into 90�, or rather into 1080 parts of fivenbsp;P^�Dutes each, another quadrant, which he dividednbsp;I'lto 96 equal parts, subdividing each of these intonbsp;equal parts, forming in all 1536. This arc is anbsp;severe check upon the divisions of the other; butnbsp;^ird says, that if his instructions be strictly fol-lowed, the coincidence between them will be surprising, and their difference from the truth exceed-^gly small.

The arc of 96� is to be divided first into three ^Ual parts, in the same manner as the arc of 90�;nbsp;^ych third contains 512 divisions, which number isnbsp;^�visible continually by 2, and gives 16 in eachnbsp;90th part of the whole.

'I'he next step is to cut the linear divisions front points obtained by the foregoing rules. Fornbsp;fhis purpose a pair of beam compasses is to benbsp;Dsed, both of whose points are conical and verynbsp;^harp. Draw a tangent to the arc b d, suppose atnbsp;It will intersect the arc x y in q, this will be thenbsp;distance between the points of the beam compassesnbsp;^ cut the divisions nearly at right angles to the arc.nbsp;-I he point of the beam compasses next the rightnbsp;liand is to be placed in the point r, the other pointnbsp;fall freely into the arc x y; then pressing gentlynbsp;Dpon the screw head which fastens the socket, cutnbsp;divisions with the point towards the right hand.

1. nbsp;nbsp;nbsp;Chuse any part of the arc where there is anbsp;coincidence of the QO and 96 arcs; for example,nbsp;at e, the point of t)0�. Draw the faint arc st andnbsp;i k, which may be continued to any length towardsnbsp;A; upon these the nonius divisions are to be di-,nbsp;vided in points. The original points for the nonius of the 90th arc are to be made upon the arcnbsp;st; the original points for the nonius of the 96thnbsp;arc are to be made upon the arc i k.

Because 90 is to 9�, as 15 to 16, there will be a coincidence at 15� and l6pts, 30� and 32pts, 45�nbsp;and 48pts; 6o� and 64pts, 90� and g�pts.

2. nbsp;nbsp;nbsp;Draw a tangent line to the primitive circle asnbsp;before, intersecting the arc d, which gives the distance of the points of the beam compasses, withnbsp;which the nonius of the 90th arc must be cut.

3. nbsp;nbsp;nbsp;Let us suppose then that the nonius is tonbsp;subdivide the divisions of the limb to half a minute,nbsp;which is effected by making 10 divisions of thenbsp;nonius equal to 11 divisions of the limb; measurenbsp;the radius of the arc, and compute the chord of 16,nbsp;or rather 32 of the nonius dffision, which may easily be obtained by the following proportion; ifnbsp;10 divisions of the nonius plate make 55 minutesnbsp;of a degree, what will 32 of those divisions make ?nbsp;the answer is 2� 56', the chord of which must benbsp;computed and taken from the scale of equal parts;nbsp;but as different subdivisions by the nonius may benbsp;required, let n be the number of nonius divisions,nbsp;m the number of minutes taken in by the nonius b,nbsp;l6, 32' or 64, and X the arc sought; then as n :

4. Lny ofFwith the beam compasses, having the length of the tangent QO between the points, thenbsp;poiiit q froin c, q being a point in the arc st, andnbsp;��^11 original point in the primitive circle, and thenbsp;ehord of 32 from q towards the left hand, (thenbsp;^hqrd of 32 being the chord which subtends 32 di-on the nonius plate, or the chord of 2� 50';nbsp;l^^liis chord to be computed from the radius withnbsp;^vhich the faint arc s t was struck, and taken offnbsp;|he scale of equal parts,) and divide by continualnbsp;quot;sections; ten of those divisions, counting fromnbsp;�1 lo the left, will be the required points.

The nonius belonging to the 96th arc is subject *^0 no difficulty, as the number should always benbsp;32, amp;c. that the extremes may be laid off fromnbsp;J^hti divisions of the limb without computation.nbsp;*0 be more particular, the length of the tangentnbsp;or radius, with which the divisions of thenbsp;quot;^^nius of the g6 arc are to be cut, must be foundnbsp;the way before directed for the nonius of thenbsp;arc; the ark i k standing instead of the arc stinbsp;'t^ving the tangental distance between the pointsnbsp;pf the compasses from one of the original pointsnbsp;the primitive 96, lay off a point on the faint arcnbsp;^ k. towards the left hand; count from that pointnbsp;the primitive 96 cirele 17 points to the leftnbsp;quot;p'ld, and lay off from thence another point on thenbsp;.quot;quot;U arc i k; the distance between those two pointsnbsp;the faint arc i k is to be subdivided by bisee-^'oiis into 16 parts, and those parts jiointed; from

Ludlam thinks, that instead of laving the l^^tcTit of the nonius single, it would be better tonbsp;�f It oft double or quadruple; thus, instead of 17nbsp;points to the left hand, count 34 or �S, and thatnbsp;oth Ways, to the right and to the left, and lay oft

a point from each extreme on the faint ate ik; subdivide tlic u hole between these extreme pointsnbsp;hv continual bisections, till you get l6 points toge-tlier on the left hand side of the middle, answeringnbsp;to the extent of the nonius. No more of the subdivisions arc to be completed than are necessary tonbsp;�)btain the middle portion of l6 points as before.

The nonius points obtained, next process is to transfer them on the nonius plate, which plate isnbsp;chamfered on both edges; on the inner edge isnbsp;the nonius for the QO arc, on the outer edge is thenbsp;nonius for theg6arc,iu the middle between tlic twonbsp;cliamfcrs is a flat part parallel to the under surfacenbsp;of the nonius plate; upon the flat part the faintnbsp;line of the next operation, is to be drawn.nbsp;7h Jind the place -where the nonius is to begin uponnbsp;the chamfered edge of the nonius plate, measure thenbsp;ilistance of the center of the quadrant from thenbsp;axis of the telescope; this distance from the axisnbsp;of the telescope at the eye end, will be the placenbsp;for the first division of the nonius; then draw anbsp;faint line from the center on the flat part of thenbsp;nonius plate.

Fasten the nonius plate to the are with two pair of hand vices; then with one point of the beamnbsp;eonipasscs in the center of the quadrant, and thenbsp;other at the middle of the nonius plate, draw anbsp;faint arc from end to end; where this arc cuts thenbsp;taint line before-mentioned, make a fine point;nbsp;from this point lay off on each side another point,nbsp;which may be at any distance in the arc, onlynbsp;care must be taken that they be equally distantnbsp;from the middle point; from the two last make anbsp;faint intersection as near as possible to cither ofnbsp;the chamfered edges of the nonius plate; throughnbsp;this intcfscction the first division of the noniusnbsp;must be ctttv

Let US suppose that we have go inehes to divide mto goo equal parts, take the third of this number,nbsp;f�' 300; now, the first power of 1 aliove this isnbsp;'��12; therefore, take Wths of an inch in one pairnbsp;of beam compasses, Vv in another, tc? in a third,nbsp;^iid T� in a fourth; then lay the scale fi-om whichnbsp;these measures were taken, the scale to be divided,nbsp;^�id the beam compasses near together, in a roomnbsp;hieing the north; let them lie there the wholenbsp;Oight; the next morning correct your compasses,nbsp;^iid lay off Vs three times; then with the compasses Ts% Vs, Ts, bisect these three spaces as expeditiously as possible; the space 64 is so smallnbsp;that there is no danger from any partial or unequalnbsp;expansion, therefore the remainder may be finishednbsp;hy continual bisections. The linear divisions arenbsp;to be cut from the points with the beam compasses

The nonius of this scale is Wths of an inch longj 'vhich is to be divided into 100 equal parts, asnbsp;too is to 101, so is 250 : 258,56 tenths of an inch,nbsp;the integer being rW. Suppose the scale to benbsp;�lumbered at every inch from left to right; thennbsp;make a fine point exactly against Vs, to the left of o,nbsp;from this lay off 258,56 to the right hand, whichnbsp;divide after the common method.

If you are not furnished with a scale long enough o lay off 258,50, then set ofi� VV, and add 8,56^nbsp;iGin a diagonal scale.

. The pointing tool consisted of a steel wire tV 'dch diameter, inserted into a brass wire i inch

diameter, the brass part 2i long, the steel part stood out f, whole length 3^ inches. The anglenbsp;of the conical point about 20 or 25 degrees, somewhat above a steel temper; the top of the brassnbsp;part was rounded off, to receive the pressure ofnbsp;the finger; the steel point should be first turned,nbsp;hardened, and tempered, and then whetted on thenbsp;oil-stone, by turning the pointril round, and atnbsp;the same time drawing it along the oil-stone, notnbsp;against, but from tiie point; this will make a sharpnbsp;point , and also a kind of very fine teeth along thenbsp;slant side of the cone, and give it the nature of anbsp;very fine countersink.

In striking the primitive circle by the beam compasses, the cutting point raises up the metal anbsp;little on each side the arc; the metal so thrownnbsp;up forms what is called the bur. When an arcnbsp;is struck across the primitive circle, this bur willnbsp;be in some measure thrown down; but if that circle be struck again ever so lightly, the bur will benbsp;raised up again; the arcs struck across the primitive circle have also their bur. Two such rasuresnbsp;or trenches across each other, will of course havenbsp;four salient, or prominent angles within; and asnbsp;the sides of the trenches slope, so do also the linesnbsp;which terminate the four solid angles. You maynbsp;therefore,/cr/ what you cannot see; when the conical points bear against all four solid angles, theynbsp;will gukle, and keep the point of the tool in thenbsp;center of decussation, while keeping the tool up-right, pressing it gently wdth one hand, and turning it round with the other, you make a conicalnbsp;hole, into which you can at anytime put the pointnbsp;of the beam compasses, and ffcl, as well as see,nbsp;when it is lodged there.

1. nbsp;nbsp;nbsp;The points of the beam compasses shouldnbsp;pever be brought nearer together than two or threenbsp;inches, except near the end of the line or arc to benbsp;divided; and there spring dividers with rouijclnbsp;iiioveable points had best be used.

2. nbsp;nbsp;nbsp;The prick-punch, used to mark the points,nbsp;should be very sharp and round, the conical pointnbsp;heilig formed to a very* acute angle; the point tonbsp;he made by it ought not to exceed the one thousandth of an inch. When lines, or divisions, arcnbsp;h) be traced from these points, a magnifying glass

i an inch focus must be used, which will render ^he impression or scratch made by the beam compasses sufficiently visible; and if the impression benbsp;*iot too faint, feeling will contribute, as well asnbsp;teeing, towards making the points properly.

3. nbsp;nbsp;nbsp;The method of iinding the principal poijiitsnbsp;hy computing the chords, is preferable to othernbsp;piethods; as by taking up much less time, therenbsp;is much less risk of any error from the expansionnbsp;�^fthe instrument, or beam compasses.

4. nbsp;nbsp;nbsp;To avoid all possible error from expansion,nbsp;^r. Bird never admitted more than one person,nbsp;pnd him only as an assistant; nor suffered any fire

Mr. Bin! guards, by this method, against any ^riequality that might possibly happen among thenbsp;original points, by first setting out a few capitalnbsp;points, distributed equally through the arc, leavingnbsp;the intervals to be filled up afterwards; he couldnbsp;dy this method check the distant divisions withnbsp;��ospect to each other, and shorten the time of thenbsp;5�lost essential operations,

5. Great care is to be observ'ed in pointing intersections, which is more difficult than in pointing from a single line, made by one point of the compasses. For in bisections, the place to benbsp;pointed is laid off from the right to the left, andnbsp;trom the left to the right. If any error arises fromnbsp;an alteration of the compasses, it will be shewnnbsp;double; even if the chord be taken a little toonbsp;long, or too short, it will not occasion any inequality, provided the point be made in the middle,nbsp;between the two short lines traced by the compasses.

Now, as Mr. Lndlam observes, if the bisecting chord be taken exactly, the two fore-mentionednbsp;faint arcs will intersect each other in the primitive circle, otherwise the intersection will fallnbsp;above or below it. In either case, the eye, assistednbsp;by a magnifier, can accurately distinguish on thenbsp;primitive circle the middle between these twonbsp;arcs, and a point may be made by the pointing tool.

In small portions of the primitive circles, the two faint arcs will intersect in so acute an angle,nbsp;that they will run into one another, and form asnbsp;it were a single line; yet even here, though thenbsp;bisecting chord be not exact, if the intersection benbsp;pointed as before, the point will fall in the middlenbsp;of the portion to be bisected.

If, in the course of bisecting, you meet with a hole already rnade with the pointril, the point ofnbsp;the compasses should fall exactly into that hole,nbsp;both from the right and left haqd, apd you maynbsp;readily feel what you cannot see, whether itnbsp;fit or no; if it fits, the point of the Compasses willnbsp;have a firm Rearing against the bottom of thenbsp;conical hole, and strike a solid blow against it; ifnbsp;it docs not exactly coincide with the center of thenbsp;hole, the slant part of the point will slide down the

Slant side of the hole, drawing, or pushing the otlier point of the compasses from its place.

Mr. Bird's method of transferring the diyLsions by the beam compasses from the original pointe, isnbsp;founded on this maxim, tluit a right line camiot henbsp;ra/ upon brass, so as aceuraiely to pass through .twonbsp;given points-, but that a circle may be describednbsp;from any center to pass with accuracy through anbsp;given point. It j.s exceeding diflicult, in the. tirstnbsp;place, to fix the rule accurately, and keep it firmlynbsp;to the two points; and, secondly, supposing itnbsp;could be held properly, yet, as the very point ofnbsp;the knife which enters the metal and ploughs itnbsp;out, cannot bear against the rule, but some othernbsp;part above that point will bear against it; it follows, that if the knife be held in a difierent situation to or from the rule, it will throw the cuttingnbsp;point out or in; besides, any hardness or inequality of the metal will turn the knife out of itsnbsp;course, for the rule does not oppose the knife innbsp;departing from it, and the force of the hand cannot hold it to it. For these reasons it is almostnbsp;impossible to draw a knife a second time againstnbsp;the rule, and cut within the same line as before.

On the other hand, an arc of a circle maj' al-�'vays be described by the beam compasses so as to pass through a given point, proviilcd both pointsnbsp;**f the compasses be conical. Let one point ofnbsp;the compasses be set in the given point or conicalnbsp;bole in the brass plain; make the other point,nbsp;'^'hatever be its distance, the central, or still point;nbsp;'''�ith the former point cut the arc, and it will benbsp;to pass through the given point in the brassnbsp;pl'iiu, and the operation may be repeated s.'itelv,nbsp;^�nd theetrokc be strengthened by degrees, as thenbsp;lUoving }X)int is not likely to be shilted out of itenbsp;direction, nor the cutting point to be broken.

The visible divisions on a large quadrant are ahva3�s the arcs of a circle, though so short as notnbsp;to be distinguished from strait lines; they shouldnbsp;be perpendicular to the arc that bounds them, andnbsp;therefore the still or eentral point of the beamnbsp;compasses must be somewhere in the tangent tonbsp;that arc; the bounding circle of the visible divisions, and the primitive circle should be very nearnbsp;each other, that the arc forming the visible divisions may be as to sense perpendicular to bothnbsp;circles, and each visible division shew the originalnbsp;point from which it was cut.

Another maxim of Mr. Bird's, attributed to Mr. Graham, That it is possible practically to bisectnbsp;tin arc, or right line, but not to trisect, quinquisect,nbsp;0�c. The advantages to be obtained by bisectionnbsp;have been already seen; we have now to shew thenbsp;objections against trisecting, amp;c.

1. nbsp;nbsp;nbsp;That as the points of trisection in the primitive circle must be made by pressing the point ofnbsp;the* beam compasses down into the metal, the leastnbsp;cxtubcrance, or hard particle, will cause a deviationnbsp;in the first impression of a taper point, and forcenbsp;the point of the compasses out of its place;nbsp;when a point is made by the pointing tool, thenbsp;tool is kept turning round while it is pressed down,nbsp;find therefore drills a conical hole.

2. nbsp;nbsp;nbsp;Alueh less force is necessary to make anbsp;.scratch or faint arc, than a hole by a pressurenbsp;downwards of the point of the compasses.

3. nbsp;nbsp;nbsp;So much time must be spent in trials, that anbsp;partial expansion would probably take place; and,nbsp;perhaps, many fidse marks, or holes made, whichnbsp;might occasion consiclerable error.

Another maxim of Mr. Bird's was this, that stepping was liable to great uncertainties, and notnbsp;to blt;? trusted; that is, if the chord of t�� was as^

sumed, and laid down five times in succession, by tnrning the compasses over upon the primitive cir-yet tlie arcs so marked would not, in his opi-?iion he equal.

It consists of a scale of inches, each divifled into tenths, and numbered at every inch from the leftnbsp;to the right, thus, O, 1, �1,3, amp;.c. in the order of thenbsp;Natural numbers. The nonius scale is below this,nbsp;^ut contiguous to it, so that one common line tcr-tninates the bottoms of the divisions on the scalenbsp;of inches, and the tops of the divisions on tirenbsp;Oonius; this nonius scale contains in length 101nbsp;tenths of an inch, this length is divided into KXlnbsp;e^ual parts, or visible divisions; the left hand di-yision of this scale is set otf trom a point t-e- of annbsp;hich to the left of O, on the scale of inches; therefore, the right hand end of the scale reaches to,nbsp;ynd coincides with the 10th inch on the scale ofnbsp;hiches. Every tenth division on this nonius scalenbsp;is figured from the right to the left, thus, 100. 00.nbsp;^0. 70. 60. 50. 40. 30. 20. 10. O. and thus O onnbsp;fhe nonius coincides with 10 on the inches; andnbsp;loo on the nonius falls against the first subdivisionnbsp;(oftenths) to the left hand ofo on the inches; andnbsp;these two, viz. the first and last, are the only twonbsp;strokes that do coincide in the two scales.

^ o take ofi' any given number of inches, deci-and millesimals of an inch; for example, 42,7(54j observe, that one point of the beam compass must stand in a (pointed) division on the no-*fius, and the other point of the compasses in anbsp;^Wilted division on the scale of inches.

The left hand jxiint ot the compasses must stand in that division on the nonius which expresses the number of millesimal parts; this, in ournbsp;example, is 64.

To find where the other point must stand in the scale of inches and tenths, add 10 to the givennbsp;number of inches and tenths (exclusive of the twonbsp;millesimal figures;) from this sum subtract thenbsp;two millesimal figures, considered ncuv as unitsnbsp;and tenths, and the remainder will shew� in whatnbsp;division, on the scale of inches, the other point ofnbsp;the compasses must stand; thus, in our example,nbsp;add 10 to 42,7, ^md the sum is 52,7; fi'om thisnbsp;subtract 6,4 and the remainder is 46,3. Set thennbsp;one point of the compasses in the 64th division onnbsp;the nonius, and the other point in 46,3 on the scalenbsp;of inches, and the two points will comprehend between them 42,764 inches.

This will be plain, if we consider that the junction of the two scales is at the 10th inch on the scale of inches; therefore, the compasses willnbsp;comprehend 36,3 inches on the scale of inches;nbsp;but it will likewise comprehend 64 divisions onnbsp;the nonius scale. Each of these divisions is onenbsp;tenth and one millesimal part of an inch; therefore, 64 divisions is 64 tenths, and 64 millesimalnbsp;parts, or 6,404 inches; to 6,404 inches taken onnbsp;the nonius, add 36,3 inches taken on the scale ofnbsp;inches, and the whole length is42,704 inches; andnbsp;thus the whole is taken from two scales, viz, inchesnbsp;mid the nonius; each subdivision in the former isnbsp;-ik of an inch, each subdivision in the latter is ik-l-TksTyth of an inch. By taking a proper number ofnbsp;pach sort of subdivisions (the lesser and the greater)nbsp;tlie length sought is obtained.

The business of taking a given length will be expedited, and carried on with fur less danger of

injuring the scale, if the proposed length be lirst of all taken, nearly, on the scale of inches onle,nbsp;guessing the inillesirnal parts; thus, in our case.nbsp;We ought to take off from the scale of inches 42,7,nbsp;und above half a tenth more; for then if one pointnbsp;set in the proper division on the nonius, thenbsp;other point will fall so near the proper division onnbsp;the scale of inches, as to point it out; and thenbsp;point of the compasses may be brought, by thenbsp;Regulating screw, to fall exactly into the true di-''^ision.

If, when the points of the compasses arc set, they Rio not comprehend an integral number of millc^-^unal parts, they will not precisely fall into anynbsp;I Wo divisions, but will either exceed, or fall short;nbsp;R'it the exact distance of the points of the compassesnbsp;he 42,7045; if, as before, the left hand point benbsp;*�Ot in the 64th division of the nonius, then thenbsp;f^ght hand point will exceed 46,3 among thenbsp;*Uehes; if the left hand point be carried one division more to the left, and stand in 65 of the nonius,nbsp;^hcn the right hand point will fall short of 40,2 innbsp;scale of inches; the excess in the former casenbsp;oeing ecjual to the defect in the latter. By ob-scrving whether the diflerence be equal, or asnbsp;great again in one ease as the other, we mav csti-|Wate to jcl part of a millesimal. Sec Mr. Bird'snbsp;-I I�Act, p. lt;2,.

It niay be asked, why should the nonius scale oonimcnce at the lOlh inch; why not at 0, and sonbsp;he nonius scale lay wholly on the left hand of thenbsp;Seale of inches ? and, in this ease, both scales mightnbsp;0 in one right line, and not one under the other;nbsp;Rit? in such a case, a less distance than 10 inchesnbsp;Riuuld not always be found upon the scale, as aprnbsp;pears from the rule before given, d'hc number 10nbsp;not, m this case, lie added ba the inches and

tenths, and then the subtraction before directed Avoiild not always be possible.

Yet, upon this principle, a scale in one continued line may be constructed for laying oft'inches, tenths, and hundredths of an inch, for any lengthnbsp;above one inch; at the head of the scale of inches,nbsp;to the left hand of 0, and in the same line, set offnbsp;eleven tenths of an inch (or the multiple,) whichnbsp;subdivide in Mr. Biras way, into ten equal parts.nbsp;Such a compound scale would be far more exactnbsp;than the common diagonal scale; for the divisionsnbsp;being pointed, you may feel far more nicely thannbsp;you can see, when the points of the beam compassesnbsp;are set to the exact distance. But to return tonbsp;Mr. Bird's. Tract.

The nature of Mr. Bird's scale being known, there will be no difficulty in understanding hisnbsp;directions how to divide it. A scale of this kindnbsp;is far preferable to any diagonal scale; not onlynbsp;on account of the extreme difficulty of drawing thenbsp;diagonals exactly, but also because there is nonbsp;check upon the errors in that scale; here the uniform manner in which the strokes of one scale separate from those of the other, is some evidence ofnbsp;the truth of both; but Mr. Bird�s method of assuming a much longer Ijne than what is absolutelynbsp;necessary for the scale, subdividing the whole bynbsp;a continual bisection, and pointing the divisions asnbsp;before explained, and guarding against partial expansions of the metal, is sure to render the divisions perfectly equal. The want of such a scalenbsp;of equal parts (owing, perhaps, to their ignorancenbsp;of constructing it) is one reason why Mr. Birdsnbsp;method of dividing is not in so great estimationnbsp;among mathematical instrument makers, as itnbsp;justly deserves.

As it Is my intention to collect in this place '�vhatever is valuable on this subject, I cannotnbsp;retrain from inserting the following remark ofnbsp;Air. Smeaton\, though it militates strongly againstnbsp;�One of Mr. Bird�s maxims. He advises us to compute from the measured radius the chord of l6 de-�'quot;008 only, and to take it from an excellent plainnbsp;�^'ule, and lay it off five times in succession fromnbsp;primary point of 0 given, this would give 80nbsp;'^^egrees; then to bisect each of these arcs, and tonbsp;% oft' one of them beyond the 80th, which wouldnbsp;.?'ve the 88th degree; then proceed by bisection,nbsp;you come to an arc of two degrees, which laidnbsp;from the 88th degree, will give the QO degrees;nbsp;^hen proceed again by bisection, till you have re-fUiced the degrees into quarters, or every fifteennbsp;Jffnutcs. Here Mr. Smeaton would stop, beingnbsp;Apprehensive that divisions, when over close, cannot be accurately obtained even by bisection.

If it were necessary to have subdivisions upon fhe limb equivalent to five minutes, he advises usnbsp;fo compute the chord of '21� 20' only, and to lay itnbsp;^ff four times from the primary point; the lastnbsp;'voiiltl give 85� 20', and then to supply the re-mainder from the bisected divisions as they rise,nbsp;gt;iof from other computed chords.

Air. Bird asserts, that after he had proceeded by Ihc bisections from the arc of 85� 20', the severalnbsp;points of 30. 6o. 75. Qo. fell in without sensible inequality^ and so indeed they might, though theynbsp;'''ere not equally true in their places; for whatevernbsp;error was in them would be communicated to allnbsp;Connected with, or taking their departure from

JL is not the same thing whether you twice take. a measure as nearly as you can, and lay it offnbsp;!5eparately, or lay off two openings of the compasses in succession unaltered-, for though thenbsp;game opening, carefully taken off trom the samenbsp;scale a second time, will doubtless fall into thenbsp;holes made by the first, without sensible error;nbsp;yet, as the sloping sides of the conical cavitiesnbsp;made by the first points, will conduct the pointsnbsp;themselves to the center, there may be an error,nbsp;w hich, though insensible to the sight, w'ould havenbsp;been avoided by the more simple process of layingnbsp;off the opening twice, without altering the compasses.

As the whole of the QO arc may now be divided by bisection, it is equally unexceptionable tvithnbsp;thei)6 arc; and, consequently, if another arc ofnbsp;00, upon a different radius, was laid down, theynbsp;would be real checks upon each other.

Mr. Ramsden, in laying down the oi-iginal divisions on his dividing engine, divided his circle first into five parts, and each of these into three;nbsp;these parts were then bisected four times; butnbsp;being apprehensive some error might arise fromnbsp;qumquisection, and trisection, in order to examinenbsp;the accuracy of the divisions, he described anothernbsp;circle tV inch within the former, by continual bisections, but found no sensible difference betw'cennbsp;the two sets of divisions. It appears also, thatnbsp;Mr. Bird, notwithstanding all his objections to,nbsp;and declamations against the practice of stepping,nbsp;so'inciuncs used it himself.

It will be necessary to give the young practitioner some account of the nature and use of that ^dininible contrivance commonly called a nonius,nbsp;Igt;y which the divisions on the limbs of instrumentsnbsp;^I�C subdivided.

The nonius depends on this simple circumstance, that if any line be divided into equal parts, the length of each part will be greater, the fewernbsp;divisions there are in the original; on the contrary,nbsp;the length of each division will be less in proportion, as the divisions arc more numerous.

Thiys, let us suppose the limb of Hadley�s quadrant divided to every 20 minutes, which are the *^iiiallest divisions on the quadrant; the two extreme strokes on the nonius contain seven degrees,nbsp;'gt;r �21 of the afore-mentioned small divisions, butnbsp;that it is divided only into 20 parts ; each of thesenbsp;parts will be longer than those on the arc, in thenbsp;proportion of 21 to 20; that is to say, they will benbsp;�aiie-twentieth part, or one minute longer than thenbsp;divisions on the arc; consequently, if the first, ornbsp;Jndex division of the nonius, be set precisely opposite to any degree, the relative position of the no-ifius and the arc must be altered one minute be-ii^rc the next division on the nonius will coincidenbsp;With the next division on the arc, the second division will require a change of two minutes; the-tliird, of three minutes, and so on, till the 20thnbsp;stroke on the nonius arrive at the next 20 minutes^nbsp;^11 the arc; the index division will then have movednbsp;exactly 20 minutes from the division whence it se^nbsp;^iit, and the intermediate divisions of each minut�nbsp;have been regularly pointed out by the divisions ofnbsp;nonius.

To render this still plainer, we must observe that the index, or counting division of the nonius^nbsp;is distinguished by the iiiarb 0, which is placed oiinbsp;the extreme right hand division; the mmibersnbsp;running regularly on thus, 20, 15, 10, 5, O.

The index division points out the entire degrees and odd 20 minutes, subtended by the objects observed; but the intermediate divisions arc shewnnbsp;bv the other strokes of the nonius; thus, looknbsp;among the strokes of the nonius for one that standsnbsp;directly opposite to, or perfectly coincident withnbsp;some one division on the limb; this division reckoned on the nonius, shews the number of minutesnbsp;to be added to what is jgt;ointcd out by the indexnbsp;division.

To illustrate this subject, let us suppose two cases. The first, when the index division perfectlynbsp;coincides with a division on the limb of the ejuad-runt; here there is no difficulty, for at whatsoevernbsp;di\'ision it is, that division indicates the requirednbsp;angle. If the index divisions stand at 40 degrees,nbsp;40 degrees is the measure of the required angle.nbsp;If it coincide with the next division beyond 40 onnbsp;the right hand, 40 degrees 20 minutes is the angle.nbsp;If w ith the second division beyond 40, then 40 degrees 40 minutes is the angle, and so in every othernbsp;instance.

The second case is, when the index line docs not coincide with any division on the limb. Wenbsp;are, in this instance, to look for a division on thenbsp;nonius that shall stand directly opposite to one onnbsp;the limb, and that division gives us the, odd mi-lUites, to be added to those pointed out by thenbsp;index division: thus, suppose the index divisionnbsp;does not coincide with 40 degrees, but that thenbsp;next division to it is the first coincident division,nbsp;then is the required angle 40 degrees 1 minute.

If it had been the second division, the angle would tiave been 40 degrees 2 minutes, and so on to 20nbsp;luinutes, when the index division coincides withnbsp;first 20 minutes from 40 degrees. Again, letnbsp;suppose the index division to stand betweennbsp;30 degrees, and 30 degrees 20 minutes, and thatnbsp;l6th division on the nonius coineides exactlynbsp;quot;^ith a division on the limb, then the angle isnbsp;30 degrees l6 minutes. Further, let the indexnbsp;^hvision stand between 35 degrees 20 minutes,nbsp;35 degrees 40 minutes, and at the same timenbsp;12th division on the nonius stand directly op-Positeto a division on the arc, then the angle will.nbsp;^*^35 degrees 32 minutes.

1. Find the value of each of the divisions, or ^Gbdivisions, of the limb to which the nonius isnbsp;Applied. 2. Divide the quantity of minutes ornbsp;^oconds thus found, by the number of divisionsnbsp;the nonius, and the quotient will give the va-^0 of the nonius division.

Thus, suppose each subdivision of the limb be 0 minutes, and that the nonius has 15 divisions,nbsp;len 4| giveg two minutes for the value of thenbsp;^pnuis. If the nonius has 10 divisions, it wouldnbsp;three minutes; if the limb be divided tonbsp;oveiy 12 minutes, and the nonius to 24 parts,nbsp;?on 12 minutes, or 720 seconds divided by 24,nbsp;feives 30 seconds for the required value.

As there are many cases where ares are required to be drawn of a radius too large for any ordinarynbsp;compasses, Mr. Hejzvood and myself contrivednbsp;several instruments for this purpose; the mostnbsp;perfect of these is delineated at fg. 5^ plate \ l.nbsp;It is an instrument that must give great satisfaction to every one who uses it, as it is so extensivenbsp;in its nature, being capable of describing arcs fromnbsp;an infinite radius, or a strait line, to those of twonbsp;or three inches diameter. When it was firstnbsp;contrived, both Mr. Heyvuood and myself werenbsp;ignorant of what had been done by that ever tonbsp;be celebrated mechanician. Dr. Hooke.

Since the invention tliereof, I have received some very valuable communications from differentnbsp;gentlemen, who saw and admired the simplicitynbsp;of its construction; among others, from Mr. Ni~nbsp;chohoii, author of several very valuable works;nbsp;Dr. Rotherham^ Earl Stanhope, and J. Priestley,nbsp;Esq. of Bradford, Yorkshire; the last gentlemannbsp;has favoured me with so complete an investigationnbsp;of the subject, and a description of so many admirable contrivances to answer the purpose of thenbsp;artist, that any thing I could say would be altogether superfluous; I shall, therefore, be very briefnbsp;in my description of the instrument, representednbsp;Jig. 5, plate 11, that I may not keep the readernbsp;from Mr. Priestley s valuable essay, subjoiningnbsp;Dr. I-looke\ account of his own contrivance to thatnbsp;of ours. Much is always to be gained from annbsp;attention to this great man; and I am sure mynbsp;reader will think his time well employed in perusing the short extract 1 shall here insert.

The branches A and B, Jig. b, plate 11, carry two independent equal wheels C, D. The pencil,nbsp;or point E, is in a line drawn between the centernbsp;of the axis of the branches, and equidistant fromnbsp;oach; a weight is to be placed over the pencilnbsp;'''hen in use. When all the wheels have theirnbsp;^xes in one line, and the instrument is moved ianbsp;rotation, it will describe an infinitely small circle;nbsp;'n this case the instrument will overset. Whennbsp;the two w'hcels C, D, have their horizontal axesnbsp;parallel to each other, a right line, or infinitelynbsp;J^rge circle will be described; when these axes arenbsp;*oclined to each other, a circle of finite magnitude will be described.

The distance between one axis and the center, (or pencil,) being taken as unity, or the commonnbsp;��adius, the numbers 1, 2, 3, 4, amp;c. being soughtnbsp;^or in the natural tangents, will give arcs of indignation for setting the nonii, and at which circles of the radii of the said numbers, multipliednbsp;nnto the common radius, will be described.

Extracts from Dr. Hooke, on the Difficulty, amp;c. of Dr a-dving Ares of Great Circles. � This thing,nbsp;says he, is so difficult, that it is almost impossible,nbsp;especially where exactness is required, as I wasnbsp;sufficiently satisfied by the difficulties that occurred in striking a part of the arc of a circle ofnbsp;6o feet for the radius, for the gage of a tool fornbsp;grinding telescope glasses of that length; wherebynbsp;it was found, that the beam compasses made withnbsp;all care and circumspection imaginable, and usednbsp;^vith as great care, would not perform the operation; nor by the way, an angular compass, such asnbsp;described by Guido Ubaldus, by Cla-vius, and bynbsp;Blagrave, See.

� The Royal Society met; I discoursed of my instrument to draw a great circle, and producednbsp;an instrument I had pi-ovidcd for that purpose;nbsp;and therewith, by the direction of a wire aboutnbsp;100 feet long, I shewed how to draw a circle otnbsp;that radius, which gave great satisfaction. Sec-Again, at the last meeting I endeavoured to explainnbsp;the difficulties there are in iriaking considerablenbsp;discoveries either in nature or art; and yet, whennbsp;they are discovered, they often seem so obvious andnbsp;plain, that it seems more difficult to give a reasonnbsp;why they were not sooner discovered, than hoWnbsp;they came to be detected now: how easy it was,nbsp;we nowi think, to find out a method of printing

letters, and yet, except what may have happened in China, there is no specimen or history of anynbsp;thing of that kind done in this part of the world.nbsp;How obvious was the vibration of pendulous bodies? and yet, we do not find that it was madenbsp;Use of to divide the spaces of time, till Galileo discovered its isochronous motion, and thought ofnbsp;that proper motion for it, amp;c. And though itnbsp;uiay be difficult enough to find a way before it benbsp;shewn, every one will be ready enough to saynbsp;'vhen done, that it is easy to do, and was obviousnbsp;to be thought of and invented.�

To illustrate this, the Doctor produced an instrument somewhat similar to that described, 5, plate 11, as appears from the journal of the Royalnbsp;Society, where it is said, that Dr. Hooke producednbsp;instrument capable of describing very large circles, by the help of two rolling circles, or trucklesnbsp;Ui the two ends of a rule, made so as to be turnednbsp;their sockets to any assigned angle. In anothernbsp;place he had extended his views relative to this instrument, that he had contrived it to draw the arcnbsp;uf a circle to a center at a considerable distance,nbsp;^here the center cannot be approached, as fromnbsp;Ihe top of a pole set up in the midst of a wood, ornbsp;^i�om the spindle of a vane at the top of a tower, ornbsp;^��oin a point on the other side of a river ; in allnbsp;'^�hich cases the center cannot be conveniently ap-Pi�oached, otherwise than by the sight. This henbsp;performed by two telescopes, so placed at thenbsp;truckles, as thereby to see through both of themnbsp;the given center, and by thus directing them tonbsp;the center, to set the truckles to their true inclination, so as to describe by their motion, any partnbsp;nf such a circle as shall be desired.

In the projection of the sphere, perspective and architecture, as well as in many other branches ofnbsp;practical mathematics, it is often required to drawnbsp;arcs of circles, whose radii are too great to admitnbsp;the use of common, or even beam compasses; andnbsp;to draw lines tending to a given point, whosenbsp;situation is too distant to be brought upon the plan.nbsp;The following essay is intended to furnish somenbsp;methods, and describe a few instruments that maynbsp;assist the artist in the performance of both thesenbsp;problems.

The methods and instruments I shall propose for this purpose, will chiefly depend on the followingnbsp;propositions, which I shall premise as principles.

Principle 1. The angles in the same segment of a circle, are equal one to another.

Let AC DB,ji?^-. \,plate 10, be the segment of a circle; the angles formed by lines drawn fromnbsp;the extremities A and B, of the base of the segment,nbsp;to any points C and D in its arc, as the anglesnbsp;ACB, ADB, are equal.

This is the 31 st proposition of Euclid'% third book of the Elements of Geometry.

Principle 2. If upon the ends AB, Jig-1, plate 10, of a right line AB as an axis, two circles or rollersnbsp;CD and EF be firmly fixed, so that the saidnbsp;line shall pass through the centers, and at rightnbsp;angles to the plains of the circles; and the wholenbsp;be suftcred to roll upon a plain without sliding ;

1- If the rollers CD and EF be equal in clia-rneter, the lines, described upon the plain by their circumferences, will be parallel right lines; andnbsp;quot;the axis AB, and every line DF, drawn betweennbsp;contemporary points of contact of the rollers andnbsp;plain, will be parallel among themselves.

2. If the rollers CD and EF be unequal, then lines formed by their circumferences upon thenbsp;plain will be concentric circles; and the axis AB,nbsp;^nd also the lines D F, will, in every situation,nbsp;I^nd to the common center of those circles.

Prmciple 3. If there be two equal circles or rol-A and B,Jig. Z,plate 10, each separately fixed In its own axis, moveable on pivots; and thesenbsp;nxes placed in a proper frame, so as to be in thenbsp;�nme plain, and to maintain the situation givennbsp;Ihem with respect to each other; and if the apparatus be rolled upon a plain without sliding:

1. nbsp;nbsp;nbsp;If the axes C D and E F, be placed in a parallel situation, the circumferences of the rollersnbsp;A and B will trace upon the plain strait lines;nbsp;''vhich will be at right angles to the axes C D

and EF.

2. nbsp;nbsp;nbsp;If the axes CD and EF, continuing as before in the same plain, be inclined to each other,

as if produced to meet in some point G, the rollers A and B will describe in their motions uponnbsp;1^0 plain arcs of the same, or of concentric cir-cles, whose center is a point H, in that plain perpendicularly under the point of intersection G ofnbsp;Ibe two axes.

I shall not stop to demonstrate the truth of the last principles, it will easily appear on soeingnbsp;Ihc operations performed.

In the performance of some of the following problems, an instrument not unlike jig. 4, flatsnbsp;10, will be found useful. It consists of two rulers,nbsp;moveable on a common center, like a carpenter�snbsp;rule, with a contrivance to keep them fixed at. anynbsp;required angle. The center C must move on anbsp;very fine axis, so as to lie in a line with thenbsp;fiducial edges CB, CD of the rulers, and projectnbsp;as little as possible before them. The fiducialnbsp;edges of the legs represent the sides of any givennbsp;angle, and their intersection or center C its angular point.

A more complete instrument of this kind, adapted to various uses, will be described hereafter.

N. B. A pin fixed in the lower rule, passes through a semicircular groove in the upper, andnbsp;has a nut A which screws upon it, in order to fixnbsp;the rulers or legs, when placed at the desirednbsp;angle.

Problem 1. Given the threefoints A, B and C, supposed to he in the circumference of a circle toonbsp;large to he described by a fair of coinpasses �, to findnbsp;any number of other foints in that circumference.

1. Join AC, which bisect with the line FM G at right angles; from B, draw BD parallel tonbsp;AC, cutting FG in E; and making E D=E Bnbsp;D will be a point in the same circumference, innbsp;^vhich arc A, B and C.

By joining AB, and bisecting it at idght angles with IK; and from C drawing Ca parallel to AB,nbsp;cutting IK in L, and making L a=LCj a willnbsp;be another of the required points,

Continuing to draw from the point, last found, lines alternately parallel to AC and AB; thosenbsp;lines will be cut at right angles by F G and I Ivnbsp;�^speetivcly; and by making tlie parts equal onnbsp;^'�'ich side of F (t and I K, they become chords ofnbsp;the circle, in which arc the original points A, B,nbsp;C, and, of consccpience, determine a seriesnbsp;of points on each side of the circumference.

It is plain from the construction, (which is too evident to require a formal demonstration,) thatnbsp;the arcs AD, A a, C c, amp;c. intercepted betweennbsp;the points A and D, A and a, C and c, amp;e. aa'enbsp;oqual to the arc B C, and to one another.

In like manner, joining B C, and bisecting it 3t right angles with P Q; drawing A c' parallel tonbsp;^ C, and making R c'=AR, (c') is another of thenbsp;*�equired points; and, from (o') the point lastnbsp;found, drawing c' a' parallel to C A, and makingnbsp;^ N=N c', (a') is another point in the same cii--oumfcrencc; and the arcs comprehended betweennbsp;Cc' and A a' are equal to that between AB.nbsp;^ence, by means of the perpendiculars P Q andnbsp;h G, any number of points in the cireumfci-encenbsp;^f the circle, passing through the given ones. A,nbsp;and C may be found, whose distance is equal tonbsp;in the same manner, as points at the distancenbsp;of Be were found by the help of the perpendi-culars I K and F G.

-^gain, if A, C and (e) or A, C and (c) be taken the three given points, multiples of the arcnbsp;^C naay be found in the same manner as those ofnbsp;khe arc AB were found as above described.

. 2. Another method of performing this problem, as follows, Jig. 6, plate 10. Produce CB andnbsp;A; and with a convenient radius on C, describenbsp;arc D E; on which set off the parts F G, G E,nbsp;^0. each equal to D F; draw C G, C E, amp;c. con-

tinned out beyond G and E if necessary; take the distance AB, and with one ,foot of the compassesnbsp;in A, strike an arc to cut C G produced in H;nbsp;and El is a point in the circumference of the circlenbsp;that passes through the given points ABC; withnbsp;the same opening AB, and center H, strike annbsp;arc to cut C E produced in I, which will be another of the required points, and the process maynbsp;be continued as far as is necessary.

The reason of this construction is obvious ; for since the angles, BCA, ACH, HCI, amp;c. arenbsp;equal, they must intercept equal arcs BA, AH, HInbsp;of the circumference.

If it were required to find a number of points K, L, amp;c. on the other side, �whose distances werenbsp;equal to B C, lay down a number of angles CAK,nbsp;KAL, amp;c. each equal to BAG, and make the distances CK, KL, amp;c. each equal to B C.

This problem is much easier solved by the help of the bevel above described, as follows. See jig. 5.

Bring the center of the bevel to the middle B, of the three given points A, B and C, and holdingnbsp;it there, open or shut the instrument till thenbsp;fiducial edges of the legs lie upon the other twonbsp;points, and fix them there, by means of the screwnbsp;Aj (jig- 4^; this is called setting the bevel tq thenbsp;given points. Then removing the center of thenbsp;bevel, to any part between B and A or C, the legsnbsp;of it being at the same time kept upon A and C,nbsp;that center will describe (or be alw'ays found in)nbsp;the arc which passes through the given points, andnbsp;will, by that means, ascertain as many others asnbsp;may be required within the limits of A and C.

In order to find points without tliose limits, proceed thus: the bevel being set above described,nbsp;bring the center to C, and mark the distance CBnbsp;^'pon the amp;�/ leg; remove the center to B, andnbsp;mark the distance BA on the same leg; then placing the center on A, bring the right leg upon B,nbsp;^md the first mark will fall upon (a) a point in thenbsp;circumference of the circle, passing through A, Bnbsp;^nd C, whose distance from A is equal to the distance B C. Removing the center of the bevel tonbsp;the point (a) last found, and bringing the right legnbsp;to A, the second mark will find another point (aquot;)nbsp;io the same circumference, whose distance a a' isnbsp;equal AB. Proceeding in this manner, any number of points may be found, whose distances onnbsp;the circumference are alternately B C and BA.

In the same manner, making similar marks on thenbsp;nbsp;nbsp;nbsp;leg, points on the other side, as at (c')

and (cquot;) are found, whose distances C c, c' cquot;, are equal to BA, B C respectively.

It is almost unnecessary to add, that intermediate points between any of the above are given by the bevel, in the same manner as between the original points.

Problem 1. Fig. 7, ^Jate 10. Three 'points, k, P and C, being given, as in the last problem, to findnbsp;�� fourth point D, situated in the circumference of thenbsp;circle passing through A, B and C, and at a giverinbsp;number of degrees distant from any of these points ;nbsp;A for instance.

Make the angles AB D, and AC D, each equal to one half of the angle, which contains the givennbsp;j^nnnber of degrees, and the intersection of thenbsp;tines B D, C D gives the point D required.

Por, an angle at the circumference being equal In half that at the center, the arc AD will con^

'tain t.\vice tbc mimber of degrees contained by either of the angles ABD or A CD.

Problem 3. Fig. 8, plate 10. Given three points, as in the former problems, to draw a line fromnbsp;any of them, tending to the center of the circle, whichnbsp;passes through them all.

Let A, B and C be the given points, and let it be rcquii'cd to draw AD, so as, if continued, itnbsp;would pass through the center of the circle containing A, B and C.

Make the angle BAD equal to the complement of the angle B CA, and AD is the line required.

For, supposing AE a tangent to the point A, then is EAD a right angle, and EAB=B CA;nbsp;whence, BAD^right angle, less the ^BCA,nbsp;or the complement of B C A.

Corollary 1. AD being drawn, lines from B and C, or any other points in the same cirele, arc'nbsp;'easily found; thus, make ABG=BAD, whichnbsp;gives BG; then make BCF=CBG, whichnbsp;gives C F; or C F may be had without the intervention of B G, by making AC F=C AD.

Corollary 2. A tangent to the circle, at any of the points (A for instance), is thus found.

By the level. Set the bevel to the three given points A, B and C, (fg. 8,) lay the center on A,nbsp;and the right leg to the point C; and the pthcrnbsp;leg will give the tangent AG'. Draw AD perpendicular to AGquot; for the line required.

For BAE being=B CA, the Z. EA C is the supplement to Z. A B C, or that to which thenbsp;bevel is set; hence, when one leg is applied tonbsp;C, and the center brought to A, the direction ofnbsp;the other leg must be in that of the tangent G E.

^ Problem 4. Fig. Q, plate 10. Three points given, as in the former problems, to draw fromnbsp;a given fourth point a line tending to the center of anbsp;^�^ecle passing through the three first points.

Lo4 the three points, through which the circle supposed to pass, be A, B and C, and the givennbsp;lourth point D; it is required to draw tlirough Dnbsp;R line Dd tending to the center of the said circle.nbsp;Prom A and B, the two points nearest D, drawnbsp;the last problem, the lines A a, Bb, tending tonbsp;Pie said center; join AB, and from any point E,nbsp;taking in B b, (the farther from B the better) drawnbsp;PP parallel to Aa, catting AB in F; join ADnbsp;und BD, and draw FG parallel to AD, cuttingnbsp;in G; join GE, and through D parallelnbsp;thereto, draw Dd for the line required.

For, (continuing Dd and Bb till they meet in since A a and Bb, if prodvxced, would meet innbsp;the center, and F E is parallel to Aa, we havenbsp;B P : BA ;; BE ; radius; also, since AD and FGnbsp;ni'e parallel, B F : BA ;; B G ; B D; thereforenbsp;B G ; B D ;; BE: radius; but from the parallelnbsp;hnes Dd and GE, we have BG : BD :: BE : BO;nbsp;hence BO is the radius of the circle passingnbsp;through A, B and C.

Fy the bevel. On D with radius DA describe arc AK; set the bevel to the three given pointsnbsp;F, B and C, and bring its center (always keepingnbsp;the legs on A and C) to fall on the arc AK, as at H;

A and H severally, with any convenient radius, strike two arcs crossing each other at I; and thenbsp;required line D d will pass through the pointsnbsp;t and D.

For a line drawn from A to II will be a common chord to the circles AHK and ABC; and thenbsp;jne ID bisecting it at right angles, must passnbsp;through both their centers.

Problem 5. Fig. g, plate 10. Three points being given, as before, together with a fourth point,nbsp;to find two other points, such, that a circle passingnbsp;through them and the fourth point, shall he concentricnbsp;to that passing through three given points.

Let A, B and C be the three given points, and D the fourth point; it is required to find twonbsp;other points, as N and P, such, that a circle passing through N, D and P shall have the Same center with that passing through A, B and C.

The geometrical construction being performed as directed by the last problem, continue E G tonbsp;L, making E L=E B; and through B and L drawnbsp;BLM, cutting Dd produced in M; make ANnbsp;and B P severally equal to M D, and N and P arenbsp;the points required.

For, since LE is parallel to MO, we have BE : LE :: BO : MO; but BE=:LE by the con^nbsp;struction; therefore, M O = B O = radius of thenbsp;circle passing through A, B and C, and M is in thenbsp;circumference of that circle. Also, N, D and Pnbsp;being points of the radii, equally distant fromnbsp;A, M and B respectively, they will be in the circumference of a circle concentric to that passingnbsp;through A, M and B, or A, B and C,

By the bevel. Draw Aa and C c tending to the center, by problem 3; set the bevel to the threenbsp;given points A, B and C; bring the center of thenbsp;bevel to D, and move it upon that point till its legsnbsp;cut olF equal parts AN, CQ of the lines Aa andnbsp;C c; and N and Q will be the points reqt^red.

For, supposing lines drawn from A to �C, and from N to Q, the segments ABC and N D Q willnbsp;be similar ones; and consequently, the anglesnbsp;contained in them w'ill be equal.

Problem 6. Fig. lO, plate lo. Three points, A; B and C, lying in the circumference of a circle,

ieitig given as before: and a fourth �point D, U find another point F, suchy that a circle passing throughnbsp;and D shall touch the other passing through A, Bnbsp;and C, at any of these pomts-, as for instance, B.

Draw BE a tangent to the arc ABC, by problem 3, corollary 2; and join BD; draw B F, making the angle D B F=E B D, with the distance BD; on D strike an arc to cut BF in F;nbsp;and F is the point sought.

Since DF=DB, the DFB = DBF; but Egt;BFz=DB E by construction; therefore, D F Bnbsp;^D B E, and E B is a tangent to the arc B D Fnbsp;^t B; but EB is also a tangent to the arc ABCnbsp;(by construction) at the same point; hence, thenbsp;B D F touches AB C as required.

Problem 7. Fig. \3, plate \0. Two lines tend-to a distant point being given, and also a point in *gt;ne of them; to find two other points, (one of zvhichnbsp;^niist he in the other given line,) such, that a circlenbsp;passing throigh those three points, may have its centernbsp;at the point of intersection of the given lines.

Let the given lines be AB and C D, and E the given point in one of them; it is required to findnbsp;two other points, as I and FI, one of which (I)nbsp;shall be in the other line, such, that a circle HIEnbsp;passing through the three points, shall have itsnbsp;Pienter at O, where the given lines, if produced,nbsp;'vould meet.

Prom E, the given point, draw E H, crossing ^B at right angles in F; make FH = FE, and FInbsp;onfe of the required points. From any point Dnbsp;m C D, the farther from E the better, draw G Dnbsp;parallel-to AB, and make the angle HE I equalnbsp;to half the angle GDE; and El will cut AB in I,nbsp;the other required point.

Por, since EH crosses A B at right angles, and PI P is equal to F E, IH will be equal to IE, and

the H E r = Z. E H r; also, since G D is parallel to AB, the GDE = Z FOE = double Z IH E = double Z H E T; but 11 E I = haltnbsp;ZGDE by couslructiou; hence, the points Egt;nbsp;1 and 11 arc in the circle whose center is O.

By the bevel. Draw EH at right angles to AB, and make FH = FE as before; set the bevel tonbsp;the angle G D O, and keeping its legs on thenbsp;points H and E, bring its center to the line AB,nbsp;which will give the point I.

Problem 8. Fig. 13, plate TO. Tzvo lines tending to a distant point being given, to find the distance of that point.

Let AB and CD be the Evo 2:ivcn lines, tend-ing to a distant point O; and let it be required to lind the distance of that point, from any pointnbsp;(E for instance) in cither of the given lines.

From E draw EF perpendicular to AB; and from D (a point taken any where in C D, the farther from E the better) draw D G parallel to AB.nbsp;On a scale of inches and parts measure the lengthsnbsp;of GE, ED and EF separately; then say, as thenbsp;length of GE is to ED, so is EP' to EO, thenbsp;distance sought.

P'or the triangles E G D and E F O are similar, and from thence the rule is manifest.

I shall now proceed to give some idea of a few instruments for these purposes, whose rationale de-.nbsp;pends on the principles laid down in the beginningnbsp;of this cssav.

11, plate 10, is a sketch of an instrument grounded upon principle l,p. 134, by which thenbsp;�ros of circles of any radius, without the limits at-^nbsp;finable by a common pair of compasses, may benbsp;described.

_ It consists of a ruler AB, composed of two pieces ^ivetted together near C, the center, or axis,nbsp;I�-nd of a triangular part CFED. The axis is anbsp;hollow socket, fixed to the triangular part, aboutnbsp;''^hich another socket, fixed to the arm C B of thenbsp;^uler AB, turns. These sockets are open in thenbsp;Itont, for part of their length upwards, as repre-^c^nted in the section at I, in order that the pointnbsp;a tracer or pen, fitted to slide in the socket,nbsp;hiay be more easily seen.

The triangular part is furnished with a graduated arc D E, by which, and the vernier at B, the angle D C B may be determined to a minute. Anbsp;groove is made in this arc, by which, and bynbsp;nut and screw at B, or some similar contri-''auce, the ruler AB may be fixed in any requirednbsp;Position.

A scale of radii is put on the arm C B, by which roe instrument may be set to describe arcs of givennbsp;'^'roles, not less than 20 inches in diameter. Innbsp;order to set the instrument to any given radius,nbsp;roe number expressing it in inches on CB isnbsp;nrought to cut a fine line drawn on C D, parallel,nbsp;^od near to the fiducial edge of it, and the armsnbsp;^stened in that position by the screw at B,

. I Wo heavy pieces of lead or brass, G, G, made jo form of the sector of a circle, the angular partsnbsp;_ cing of steel and wrought to a true upright, edge,nbsp;^ shewn at H, are used with this instrument,nbsp;oose arms arc made to bear against those edges

when the arcs arc drawn. The under sides of these sectors are lurnished with fine short points,nbsp;to prevent them from sliding.

The fiducial edges of the arms CA and C D are each divided from the center C into �200 equalnbsp;parts.

The instrument might be furnished with small Castors, like the pentagraph; but little buttonsnbsp;fixed on its underside, near A, E and D, will enable it to slide with sufficient ease.

Place the sectors G, G, with their angular edges over the two extreme points; apply the arms of thenbsp;bevel to them, and bring at the same time its center C (that is, the point of the tracer, or pen, putnbsp;into the socket) to the third point, and there fix thenbsp;arm C B; then, bringing the tracer to the left handnbsp;sector, slide the bevel, keeping the arms constantlynbsp;bearing against the two sectors, till it comes to thenbsp;right hand sector, by which the required arc willnbsp;be described by the motion of its center C.

If the arc be wanted in some part of the drawing �without the given points, find, by problem 1, p.nbsp;136, other points in those parts where the arc isnbsp;required. By this means a given arc may benbsp;lengthened as far as is requisite.

Fix the arm C B so that the part of its edge, corresponding to the given radius, always reckoned in inches, may lie over the fine line drawn on C Pnbsp;for that purpose: bring the center to the pointnbsp;through which the arc is required to pass, and

pose the bevel in the direction it is intended to be drawn; place the sectors G, G, exactly to the divisions 100 on each arm, and strike the arc asnbsp;iibovc described.

3. nbsp;nbsp;nbsp;The hevel being set to strike arcs of a givennbsp;^quot;�idius, as directed in. the last paragraph, to dra%vnbsp;other arcs whose radii shall have a given proportionnbsp;^0 that of the first arc.

Suppose the bevel to be set for describing arcs 50 inches radius, and it be required to drawnbsp;^rcs of 6o inches radius, with the bevel so set.

. Say, as 50, the radius to whieh the bevel is set, to 6o, the radius of the arcs required; so is thenbsp;constant number 100 to 120, the number on thenbsp;^rrns C A and CD, to which the sectors must benbsp;Pla^ced, in order to describe arcs of 6o inchesnbsp;^udius.

B. When it is said that the bevel is set to uraw arcs of a particular radius, it is always understood that the sectors G, G, are to be placed atnbsp;yo. 100 on CA and CD, when those arcs arenbsp;drawn.

4. nbsp;nbsp;nbsp;..^dn arc AC B (fig. 11, plate lOj being given,nbsp;draw other arcs concentric thereto, which shall

Through the extremities A and B of the given arc dj�aw lines A P, BP tending to its center, by pro-3, p. 140. Take the nearest distance of thenbsp;S^ven point P from the arc, and set it from A to P,nbsp;from B to P. Hold the center of tlie level onnbsp;^ gt; (any point near the middle of the given are) andnbsp;ring its arms to pass through A and B at tlie sarnenbsp;and there fix them. Plaee the sectors to thenbsp;Ppmts P and P, and with the bevel, set as beforenbsp;draw an arc, which will pass through P',nbsp;^ given point, and be concentric to the given

L A,

5. nbsp;nbsp;nbsp;Through a point A, (fig. 14, plate lOj in the �nbsp;fti-veti line AB, to strike an arc of a given radiusynbsp;and ivhose center shall lie in that line, produced ifnbsp;necessary.

Through A, at right angles to AB, draw CD; lay the center of the bevel, set as above, on A, andnbsp;the arm CA, on the line AC, and draw a line AEnbsp;along the edge C D of the other arm. Divide thenbsp;angle DAE into two equal parts by the line AF^nbsp;place the bevel so, that its center being at A, thenbsp;arm C D shall lie on AF; while in this situation,nbsp;place the sectors at No. 100 on each arm, and thennbsp;strike the arc.

Place the center of the bevel on the middle of the arc, and open or shut the arms, till No. lOOnbsp;on C A and C D fall upon the arc on each sidenbsp;the center; the radius will be found on C B (innbsp;inches) at that point of it, where it is cut by thenbsp;line drawn on CD.

If the extent of the arc be not equal to that between the two Numb. 100, make use of the Numb. 50, in which case the radius found on C B will benbsp;double of that sought; or the arc may be lengthened, by problem 1, till it be of an extent sufficient to admit the two Numbers 100.

Many more instances of the use of this instrument might be given; but from what has been already done, and an attentive perusal of the foregoing problems, the principle of them may benbsp;easily conceived.

An instrument for drawing lines that arc para!-is called a parallel ruler; one for drawing ^ines tending to a point, as such lines arc oblicpienbsp;to each other, may, by analogy, be called an o/'-ruler.

Fig. IT,plate 10, represents a simple contrivance ^orthis purpose; it consists of a cylindrical or pris-^Tiatical tube AB, to one end of which is fixed thenbsp;^�oller A; into this tube there slides another C Bnbsp;'�f six or eight flat sides. The tubes slide stiffly,nbsp;as to remain in the position in which they arcnbsp;placed. Upon the end C, screw different rollers,nbsp;^11 of them something smaller than A.

In order to dcscrili arcs, a drawing pen E, and ^ tracer may be put on the pin D, arid are retainednbsp;there by a screw G; the pen is furnished with anbsp;loveable arm E F, having a small ball of brass P'nbsp;the end, whose use is to cause the pen to pressnbsp;^''hh tlue force upon the paper, the degree ofnbsp;'yhioh can be regulated by placing the arm in dif-I^i�ent positions.

The ruler AB being set to any given line, by *'olling it along other lines may be drawn, all ofnbsp;'fhich will tend to some one point in the givennbsp;or a continuation of it, whose distance willnbsp;greater, as the distance between the rollers Anbsp;and C is increased; and as the iliameter of C ap-pi'oachcs that of A; all which is evident fromnbsp;P'^inciple 2, page 134..

. It also appears from the said principle, that du-the motion of the ruler, any point in its axis will accurately describe the arc of a circle,nbsp;^ aving the said point of intersection for its center;

The rollers, as C, which screw upon the end of the inner tube are numbered, 1, 2, 3, amp;c. and asnbsp;many scales are drawn on that tube as there arenbsp;rollers, one belonging to each, and numbered accordingly. These, scales shew the distance in inches, of the center or point of intersection, reckoned from the middle of the pin D, (agreeing to thenbsp;point of the pen or tracer;) thus.

No. 1, will describe circles, or serve for drawing lines tending to a point, whose radius or distancenbsp;from D, is from 1200 inches to 6oo inches, according as the tube is drawn out.

If it should be required to extend the radius or distance farther than 1200 inches, by using another ruler, it might be carried to 2400 inches; butnbsp;lines in any common sized drawing, which tendnbsp;to a point above 100 feet distance, may be esteemed as parallel.

Fig. \B, plate 10. This is nothing more than the last instrument applied to a flat ruler, in thenbsp;manner the rolling parallel rulers arc made.

C D is an hexagonal axis, moveable on pivots in the heads A and F fixed upon a flat ruler; onnbsp;this axis the smaller roller B, is made to slidenbsp;through one half of its length; the larger roller A,nbsp;screwed on the other end of the axis, and can

diameters. Scales adapted to each of the rollers A, are either put on the flat sides of the axisnbsp;from C to E, or drawn on the corresponding partnbsp;of the flat ruler; and the scales and rulers distinguished by the same number-, at F is a screw tonbsp;raise or lower the end C of the axis, till the rulernbsp;goes parallel to the paper on which the drawingnbsp;made; and at G there is a socket, to which anbsp;drawing pen and tracer is adapted for describingnbsp;urcs.

in using these instruments, the fingers should placed about the middle part between the rol-^rs; and the ruler drawn, or pushed at right angles to its length. The tube AB, 14, and one,nbsp;Or both of the edges of the flat ruler, 18, arenbsp;divided into inches and tenths.

This instrument is constructed upon the ^l^iird principle mentioned in page 135, of thisnbsp;�frssay.

plate 10, is composed of five rulers; *our of them DE, DF, GE and GF, formingnbsp;^ frupezium, are moveable on the joints D, E, Fnbsp;G; the fifth ruler DI, passes under the jointnbsp;, � and through a socket carrying the oppositenbsp;Juint G. The distances from the center of thenbsp;joint D, to that of the joints E and F, are exactlynbsp;as arc the distances from G to the samenbsp;Joints. The miers DE and DF pass beyondnbsp;UG joints E and F, where a roller is fixed to each;nbsp;ue rollers arc fixed upon their axes, which movenbsp;^oely, but steadily on pivots, so as to admit of nonbsp;lake by which the inclination of the axes can benbsp;^ried. The ruler I D passing beyond the jointnbsp;5 carries a third roller A, like the others, whose

axis lies precisely in the direction of that ruler; the axes of B and C extend to K and L.

A scale is put on the ruler D I, from H to G, shewing, by the position of the socket G thereon,nbsp;the length of the radius of the arc in inches, thatnbsp;would be described by the end I, in that positionnbsp;of the trapezium. When the socket G is broughtnbsp;to the end of the scale near I, the axes of the twonbsp;rollers B and C, the ruler D I, and the axis of thenbsp;roller A, are precisely parallel; and in this position, the end I, or any other point in D I, will describe strait lines at right angles to DI; but onnbsp;sliding the socket G towards H, an inclination isnbsp;given to the axes of B and C, so as to tend tonbsp;some point in the line I D, continued beyond D,nbsp;whose distance from I is shewn by the scale.

A proper socket, for holding a pen or tracer, is made to put on the end I, for the purpose of describing arcs; and another is made for fixing on any part of the ruler D I, for the morenbsp;convenient description of concentric arcs, where anbsp;number are wanted.

It is plain from this description, that the middle ruler D I in this instrument, is a true oblique ruler,nbsp;by which lines may be drawn tending to a point,nbsp;whose distance from I is shewn by the position ofnbsp;the socket G on the scale; and the instrument isnbsp;made sufficiently large, so as to ansAver this purpose as w'cll as the other.

^nFig. l6, plate lo, the part, intended to be used in drawing lines, lies within the trapezium,nbsp;which is made large on that account; but this isnbsp;not necessary; and �^. 15, plate lo, will give an

idea of a like instrument, wlicre the trapezium �lay be made mueh smaller, and consequentlynbsp;less cumbersome.

B E C represents such a trapezium, rollers, socket, and scale as aliove described, but muchnbsp;I�liialler. Elere the ruler E D is eontiuuc'd a sufficient length beyond D, asto A, where the thirdnbsp;roller is fixed; a pen or tracer may be titted tonbsp;^fie end E, or made to slide between D and A, fornbsp;purpose of drawing arcs.

�f'o describe an eUlj^se, the transverse and conjugate lt;^xes being given.nbsp;nbsp;nbsp;nbsp;:

Method 1. By the line of sines on the sector, ^Pcn the sector with the extent AG of the semi-^�insverse axis in the terms of go and gO; takenbsp;�iit the transverse distance of 70 and 70, (io andnbsp;and SO for every tenth sine, and set them otFnbsp;lom G to A, and from G to B; then draw linesnbsp;hrough these points perpendicular to AB. Makenbsp;C a transverse distance between go and go, andnbsp;�et off each tenth sine from G towards C, andnbsp;fom G towards D, and through these points drawnbsp;'*^cs parallel to AB, which will intersect the perpendiculars to AB in the points A, a, b, c, d, c, f,nbsp;figt; b k, 1, m, n, o, p, q, B, for half the ellipse,nbsp;irough which points and the intersections of thenbsp;other half, a curve being drawn with a steadynbsp;^�d, will complete the ellipse.

Method ^.^\\\\ the elliptical compasses, y?g. 3, 1apply the transverse axis of the ellipticalnbsp;ooiDpasscs to the line AB, and discharge thenbsp;^^ews of both the sliders; set the Beam over the

transverse axis AB, and slide it backwards and forwards until the pencil or ink point coincidenbsp;with the point A, and tighten the screw of that slidernbsp;which moves on the conjugate axis; now turn thenbsp;beam, so as to lay over the conjugate axis C D,nbsp;and make the pencil or ink point coincide withnbsp;the point C, and then fix the screw, whichnbsp;is over the slider of the transverse axis of the compasses; the compasses being thus adjusted, movenbsp;the ink point gently from A, through C to B, andnbsp;it will describe the semi-ellipse ACB; reverse thenbsp;elliptical compasses, and describe the other semiellipse B DA, These compasses were contrivednbsp;by my Father in 1748; they are superior to thenbsp;trammel which describes the whole ellipse, asnbsp;these will describe an ellipse of any excentricity,nbsp;which the others will not.

Through any given point F to describe an ellipse, the transverse axis AB being given.

Apply the transverse axis of the elliptical compasses to the given line AB, and adjust it to the point A; fix the conjugate screw, and turn thenbsp;beam to F, sliding it till it coincide therewith, andnbsp;proceed as in the preceding problem.

Fig. 'I,plate 11, represents another kind of elliptical apparatus, acting upon the principle of the oval lathes; the paper is fixed upon the boardnbsp;AB, the pencil C is set to the transverse diameternbsp;by sliding it on the bar D E, and is adjusted to thenbsp;conjugate diarheter by the screw G; by turningnbsp;the board AB, an ellipse will be described by thenbsp;pencil. Fig. 2, A, plate 11, is the trammel, innbsp;which the pins on the under side of the board AB,nbsp;move for the description of the ellipse.

Ellipses are described in a very pleasing manner by thenbsp;nbsp;nbsp;nbsp;pen, Jig. \, plate 11; this part of

Fo clescrihe a parabola, ivliose parameter shall he ^'}�lt;-al to agiveuUne. Pig. \ J, plate \3.

I^raw a line to represent the axis, in which make equal to kalf the given parameter. Open thenbsp;Rector, so that AB may be the transverse distancenbsp;between QO and go on the line of sines, and setnbsp;every tenth sine from A towards B; andnbsp;through the points thus found, draw lines at rightnbsp;^'''gles to the axis AB. Make the lines A a, 10 b,nbsp;c, 30 d, 40 e, amp;c. respectively equal to thenbsp;^hords of go�, 80�, 70�, 6o , 50�, amp;c. to the radiusnbsp;and the points abode, amp;c. will be in thenbsp;i^rabolic curve: for greater exactness, interme-diate points may be obtained from the intermediate degrees; and a curve drawn through tliesenbsp;points and the vertex B, will be the parabola re-ijnircd: if the whole curve be wanted, the samenbsp;operation must be performed on the other side ofnbsp;the axis.

As the chords on the sector run no further than 00�, those of 70, 80 and go, may be found bynbsp;t^^ing the transverse distance of the sines of 35 ,nbsp;�^9 j 45�, to the radius AB, and applying thosenbsp;distances twice along the lines, '20 c, 10 b, amp;e.

^Ig. A, plate 11, is an instrument for describing ^.parabola; the figure wall render its use sufficiently evident to every geometrician. ABCDnbsp;a Wooden frame, whose sides AC, BD are pa-I'allcl to each other; E F G H is a square frame ofnbsp;^ass or wood, sliding against the sides AC, BDnbsp;^1 the exterior frame; H a socket sliding on thenbsp;ar E P of the interior frame, and carrying thenbsp;Pencil I; K a fixed point in the board, (the situa-.'dll of which may be varied occasionally); E a Knbsp;^Bircad equal in length to E F, one end thereofnbsp;' ixcd at E, the other to the piece K, going overnbsp;d pencil at a. Bring the frame, so that the pen-

cil may be in a line with the point K; then slide it in the exterior frame, and the pencil will describenbsp;one part ot a parabola. If the frame E F G H benbsp;turned about, so that E F may be on the othernbsp;side of the point K, the remaining part of the pa-�nbsp;rabola may be completed.

To describe an hyperbola, the vertex A, and asymptotes BH, BI being given. Fig. \8,pl. 13.

Draw AI, AC, parallel to the asymptotes. Make AC a transverse distance to 45, and 45, onnbsp;the upper tangents of the sector, and apply from Bnbsp;as many of these tangents taken transversely asnbsp;may be thought convenient; as B D 50�, D E 55�,nbsp;and so on; and through these points draw D d,nbsp;E c, amp;c. parallel to AC.

Make AC a transverse distance between 45 and 45 of the lower tangents, and take the transversenbsp;rlistance of the cotangents before used, and laynbsp;them on those parallel lines; thus making Ddnbsp;equal 40�, E C to 35�, E F to 80�, amp;c. and thesenbsp;points will be in the hyperbolic curve, and a linenbsp;drawn through them will be the hyperbola required.

To assist the hand in drawing curves through a number of points, artists make use of what is termednbsp;the bow, consisting of a spring of hard wood, ornbsp;steel, so adapted to a firm strait rule, that it maynbsp;be bent more or less by three screws passingnbsp;through the strait rule.

A set of spirals cut out in brass, are extremely convenient for the same purpose; for there are fewnbsp;curve lines of a short extent, to which some partnbsp;of these wall not apply.

Fig. 6, plate 11, represents an instrument for drawing spirals; A the foot by which it is affixednbsp;to the paper, B the pencil, a, b, c, d a running linenbsp;going over the cone G and cylinder H, the ends

Ki N O, the thread carries the pencil progressively from the cone to the cylinder, and thus describesnbsp;^ spiral. The size of the spiral may be varied, bynbsp;placing the thread in different grooves, by puttingnbsp;fr on the furtbcrniv:.3t cone, or by putting on anbsp;larger cone.

The geometric pen is an instrument in which, a circular motion, a right line, a circle, an ellipse, and a great variety of geometrical figures,nbsp;^lay be described.

This curious instrument was invented and de-frribed by John Baftist Suardi, in a work entitled ^uovo Istromenti per la Descriz^one di diversenbsp;^urve Antichi e Moderne, amp;c.

Though several writers have taken notice of the curves arising from the compound motion of twonbsp;circles, one moving round the other, yet no onenbsp;?ceras to have realized the principle, and reducednbsp;to practice, before J. B. Suardi. It has latelynbsp;happily introduced into the steam engine bynbsp;Messieurs JVatt and Bolton-, a proof, among manynbsp;ethers, not only of the use of these speculations,nbsp;ent of the advantages to be derived from the highernbsp;parts of the mathematics, in the hands of an inge-mous mechanic. There never was, perhaps, anynbsp;instrument which delineates so many curves as thenbsp;geometric'pen; the author enumerates 1273, asnbsp;possible to be described by it in the simple form,nbsp;and with the few wheels appropriated to it for thenbsp;present work.

Pig 1, plate 11, represents the geometric pen-, \ B, C, the stand by which it is supported; the

legs A, B, C, arc contrived to fold one Avitlnn the other, for the eonvenience of packing.

A strong axis D is fitted to the top of the frame; to the lower part of this axis any of the wheelsnbsp;(as i) maybe adapted; when screwed to it theynbsp;arc immoveable.

E G is an arm contrived to turn round upon tbe main axis D; two sliding boxes arc fitted to thisnbsp;arm; to these boxes any of the wheels belonging tonbsp;the geometric pen may be fixed, and then movednbsp;so that the w heels may take into each other, andnbsp;the immoveable wheel i; it is evident, that bynbsp;making the arm E G revolve round the axis D,nbsp;these wheels will be made to revolve also, and thatnbsp;the number of their revolutions will depend on thenbsp;proportion between the teeth.

fg is an ann carrying the pencil; this arm slides backwards and forwards in the box c d, innbsp;order that the distance of the pencil from the center of the wheel h may be easily varied; the boxnbsp;c d is fitted to the axis of the wheel h, and turnsnbsp;round -with it, carrving the arm f g along with it;nbsp;it is evident, therefore, that the revolutions will benbsp;fewer or greater, in proportion to the difFereneenbsp;between the numbers of the teeth in the wheels hnbsp;and i; this bar and socket are easily removed fornbsp;changing the wheels.

When two wheels only are used, the bar f g moves in the same direction with the bar EG;nbsp;but it another wheel is introduced between them,nbsp;they move in contrary directions.

The number of'teeth in the wheels, and conse-cpiently, the relative velocity of the epicycle, or arm fg, may be varied in infinitum.

The construction and application of this instru-fncnt is so evident from the figure, that nothing more need be pointed out than the combinations

To render the description as concise as possible, I shall in future describe the arm E G by the letter A, and f g by the letter B.

To describe fig. 1, plate 12. The radius of A must be to that of B, as lO to 5 nearly, their velo-^�Ities, or the numbers of teeth in the wheels, to benbsp;^yoal, the motion to be in the same direction.

If the length of B be varied, the looped figure, 'Iclineated at12, will be produced.

A circle may be described by equal wheels, and ^ny radius, but the bars, must move in contrarynbsp;directions.

To describe the two level figures, ecc yfjquot;. 11, plate 12. Let the radius of A to B be as 10 to 3�,nbsp;the velocities as 1 to 2, the motion in the samenbsp;direction.

To describe hy this circular motion, a strait AND AN ELLIPSE. For a Strait line, equalnbsp;mdii, the velocity as 1 to 2, the motion in a con-trary direction; the same data will give a varietynbsp;cgt;f ellipses, only the radii must be unequal; the cl-^�pses may be described in any direction; seenbsp;10, plate 13.

Pig. 13, plate 12, with seven leaves, is to be mrmed when the radii are as 7 to 2, velocity asnbsp;^ to 3 motion in contrary directions.

The six triangular figures, seen at fig. 2,4, 6, 8, 03 10, are all produced by the same wheels, bynbsp;unly Varying the length of the arm B, the velocitynbsp;should be as 1 to 3, the arms are to niove in con-tfary directions.

F/O-, 3, plale 12, with eight Icai c.s, is tormccl by equal radii, velocities as 5 to 8, A and B to movenbsp;the same way; if an intermediate wheel is added,nbsp;and thus a motion produced in a contraiy direction, the pencil will delineate Jig. l6, plate 12.

The ten-leaved figure, fig. 15, plate 12, is produced by equal radii, velocity as 3 to 10, directions of the motions contrary to each other.

Hitherto the velocity of the epicycle has been the greatest; in the three following figures thenbsp;curves are produced when the velocity of the epicycle is less than that of the primum mobile.

For//y. 7, the radius of A to B to be as 2 to 1, the velocity as 3 to 2; to be moved the same ivay,

Yovjig. 14, the radius of A, somewhat less than the diameter given to B, the velocity as 3 to 1; tonbsp;be moved in a contrary direction.

For Jig. 5, equal radii, velocity as 3 to 1; moved the same way. These instances are sufficient tonbsp;shew how much may be performed by this instrument; with a few additional pieces, it may benbsp;made to describe a cycloid, with a circular base,nbsp;spirals, and particularly the spiral of Archimedes, he.

To know how to divide land into any number of equal, or unequal parts, according to any assigned proportion, and to make proper allowancesnbsp;for the different qualities of the land to be divided,nbsp;form a material and useful branch of surveying.

In dividing of land, numerous cases arise; in some it is to be divided by lines parallel to eachnbsp;other, and to a given fence, or road; sometimes,nbsp;they are to intersect a given line; the division is

often to be made according to the particular directions of the parties concerned. In a subject which has been treated on so often, novelty is,nbsp;perhaps, not to be desired, and scarcely expected.

considerable improvement has been made in this branch of surveying since the time of Spe'ulelLnbsp;^r. Talbot, whom we shall chiefly follow, hasnbsp;arranged the subject better than those who preceded him, and added thereto two or three problems ; his work is well worth the surveyor�s peru-Some problems also in the foregoing part ofnbsp;this work should be considered in this place.

Problem 1. To divide a triangle in a given-^atio^ by right lines drawn from any angle to the op-posite side thereof.

t. Divide the opposite side in the proposed ratio. 2-. Draw lines from the several points of divisionnbsp;the given angle, and then divide the triangle as

Thus, to divide the triangle ABC, 13^ plate 8, containing 26i acres, into three parts, innbsp;proportion to the numbers 40, 20, 10, the lines ofnbsp;wision to proceed from the angle C to AB, whosenbsp;.CRgth is 28 chains; now, as the ratio of 40, 20, 10,nbsp;the same as 4, 2, 1, whose sum is 7, divide ABnbsp;'bto seven equal parts; draw C a at four of thesenbsp;parts, C b at six of them, and the triangle is di-'oed as required.

-Arithmetically. As 7, the sum of the ratios, is ?nbsp;nbsp;nbsp;nbsp;28 chains; so is 4, 2, to l6, 8, and 4

.1 know how many acres in each part, sayg as la ^ fratios is to the whole quantity ofnbsp;bj so is each ratio to the quantity of acres;

1. Divide the triangle into the given proportion, from the angle opposite the given point, 2. Reduce this triangle by problem 51, so as to passnbsp;through the given point.

Thus, to divide the field ABC, \A, plate 8, of sev^en acres, into two parts, in the proportion ofnbsp;2 to 5, for two different tenants, from a pond b,nbsp;in B C, but so that both may have the benefit ofnbsp;the pond.

1. nbsp;nbsp;nbsp;Divide B C into seven equal parts, make B anbsp;= 5, then C a=2, draw A a, and the field is divided in the given ratio. 2. To reduce this to thenbsp;point b, draw Ab and a c parallel thereto, join c b,nbsp;and it will be the required dividing line.

Operation in the field. Divide B C in the ratio required, and set up a mark at the point a, andnbsp;also at the pond b-, at A, with the theodolite, ornbsp;other instrument, measure the angle bAa; at clnbsp;lay off the same angle A a c, which will give thenbsp;point c in the side A c, from whence the fencenbsp;must go to the pond.

2. nbsp;nbsp;nbsp;To-divide ABC,15, plate 8, into threenbsp;equal parts from the pond c. 1. Divide AB intonbsp;three equal parts, A a, a b, b B, and C a C b willnbsp;divide it, as required, from the angle A; reducenbsp;these as above directed to cd and ce, and theynbsp;will be the true dividing lines.

Problem 3. To divide a triangular field in any required ratio, hy lines drawn parallel to one side.,nbsp;^nd cutting the others.

Let ABC, fig. 16, plate 8, be the given triangle to be divided into three equal parts, by lines parallel to AB, and cutting AC, BC.

Jiule 1. Divide one of the sides that is to be cut the parallel lines, into the given ratio. 2. Findnbsp;iDean proportional between this side, and the firstnbsp;division next the parallel side. 3. Draw a linenbsp;parallel to the given side through the mean proportional. 4. Proceed in the same manner withnbsp;^he remaining triangle.

Example 1. Divide B C into three equal parts �o 1^5 D P, P C. 2. Find a mean proportional between B C and DC. 3. Make C G equal to thisnbsp;^ean proportional, and draw GH parallel to AB.nbsp;t roeeed in the same manner with the remainingnbsp;triangle C FI G, dividing G C into two equal partsnbsp;I, finding a mean proportional between C Gnbsp;^Rd CI; and then making C L equal to this meannbsp;proportional, and drawing LM parallel to AB,nbsp;tRc triangle will be divided as required.

^ square, or rectangle, a rhombus, or rhomboides, be divided into any given ratio, by lines cuttingnbsp;opposite parallel sides, by dividing the sides intonbsp;^proposed ratio, and joining the points of division.nbsp;Problem 4. To divide a right-lined figure intonbsp;proposed ratio, by lines proceeding from one angle.nbsp;Problem 5. To divide a right-lined figure intonbsp;proposed ratio, by lines proceeding from a givennbsp;in one of the sides.

�problem 6. To divide a right-lined figure into ^^y proposed ratio, by right lines proceeding from anbsp;S^�^enpo'inf ivithin the said figure or field.

, tt Would be needless to enter into a detail of R mode of performing the three foregoing pro-

blems, as the subject has been already sufficiently treated of in pages 84, 85, amp;c.

Problem 7-¹ � li is required to divide any given quantity of ground into any given number of farts,nbsp;and in frofortmi as any given numbers.quot;

Rule. � Divide the given piece after the rule of �'ellowship, by dividing the whole content by thenbsp;sum of the numbers expressing the proportions ofnbsp;the several shares, and multiplying the quotientnbsp;severally by the said proportional numbers for thenbsp;respective shares required.�

Example. It is required to divide 300 acres of land among A, B, C and D, whose claims uponnbsp;it are respectively in proportion as the numbers 1,nbsp;3,6, 10, or whose estates may be supposed lOOLnbsp;300l. 600l. and lOOOl. per annum. .

The sum of these proportional numbers is 20, by which dividing 300 acres, the quotient is 15nbsp;acres, which being multiplied by each of the numbers 1, 3, 6, 10, wc obtain for the several sharesnbsp;as follows:

Now let us suppose the land, to be laid off for each person�s share, is of the following different

Problem 7, an erroneons rule given by a late writer, introduced here to prevent the practitioner being led into error.--From Talbot�s Complete Art of Land Measuring, Problem page 205, and Appendix, page 410,

�Values per acre, viz. A�s � 5s. B�s=8s. C�s = 12s. D�s = 15s. an acre; whose sum, 40s. divided by 4,nbsp;their number, quotes 10s. for the mean value pernbsp;^cre. And, according to a late author, we mustnbsp;augment or diminish each share as follows :

�^vliich is less than 300, by more than 38 acres, and shews the rule to be absolutely false; for whennbsp;each person�s share is laid out as above, there regain 38 acres unapplied; I suppose for the use ofnbsp;the surveyor.

But suppose we change the value of each person�s land, and call A�s 15s. B�s r2s.- C�s 8s. and �h's 5s. then we shall have

{15 nbsp;nbsp;nbsp;:nbsp;nbsp;nbsp;nbsp;10nbsp;nbsp;nbsp;nbsp;:;nbsp;nbsp;nbsp;nbsp;15nbsp;nbsp;nbsp;nbsp;:nbsp;nbsp;nbsp;nbsp;10nbsp;nbsp;nbsp;nbsp;A�s share

12:10:: nbsp;nbsp;nbsp;45nbsp;nbsp;nbsp;nbsp;:nbsp;nbsp;nbsp;nbsp;37,5nbsp;nbsp;nbsp;nbsp;B�s share

8 nbsp;nbsp;nbsp;*.nbsp;nbsp;nbsp;nbsp;10nbsp;nbsp;nbsp;nbsp;;:nbsp;nbsp;nbsp;nbsp;gt;00nbsp;nbsp;nbsp;nbsp;;nbsp;nbsp;nbsp;nbsp;112,5nbsp;nbsp;nbsp;nbsp;C�s share

5 nbsp;nbsp;nbsp;:nbsp;nbsp;nbsp;nbsp;10nbsp;nbsp;nbsp;nbsp;150nbsp;nbsp;nbsp;nbsp;;nbsp;nbsp;nbsp;nbsp;300nbsp;nbsp;nbsp;nbsp;D�sshare

how must the surveyor manage here, as he must ttialce 400 acres of 300; for here D�s share onlynbsp;akes the w'hole 300, where is he to find the 100nbsp;^orcs for the other three shares? But enough ofnbsp;fos; see it truly and methodically performed innbsp;next problem.

Problem 8. It is required to divide any given quantity of land among any given number of persons,nbsp;in proportion to their several estates, and the value ofnbsp;the land that falls to each persons share.

Rule. Divide the yearly value of each person�s estate by the value per acre of the land that is allotted for his share, and take the sum of the quotients, by which divide the whole given quantitynbsp;of land, and this quotient will be a common multiplier, by which multiply each particular quotient,nbsp;and the product will be each particular share of thenbsp;land.

Or say, as the sum of all the quotients is to the whole quantity of land, so is each particularnbsp;quotient to its proportional share of the land.

Example. Let 300 acres of land be divided among A, B, C, D, whose estates are tool. 300l.nbsp;600l. and lOOOl. respectively per annum; and thenbsp;value of the land allotted to each is 5, 8, 12, andnbsp;15 shillings an acre, as in the example to the lastnbsp;problem.

T-i, 100 nbsp;nbsp;nbsp;�nbsp;nbsp;nbsp;nbsp;300nbsp;nbsp;nbsp;nbsp;�� ^ �00 rr, o A

sum of the shares = 299,999 acres, 300 very nearly, and is a proof of the whole,

Let us now change the values of land as in the last problem, and see what each will have for hisnbsp;share. Suppose A�s 15, B�s 12, C�s 8, and D�s 5

{0,97826 nbsp;nbsp;nbsp;Xnbsp;nbsp;nbsp;nbsp;6,666 =nbsp;nbsp;nbsp;nbsp;6,5217nbsp;nbsp;nbsp;nbsp;A�s share

0,97826 nbsp;nbsp;nbsp;Xnbsp;nbsp;nbsp;nbsp;25,nbsp;nbsp;nbsp;nbsp;=nbsp;nbsp;nbsp;nbsp;24,4565nbsp;nbsp;nbsp;nbsp;B�s share

0,97826 nbsp;nbsp;nbsp;Xnbsp;nbsp;nbsp;nbsp;75,nbsp;nbsp;nbsp;nbsp;=nbsp;nbsp;nbsp;nbsp;73,3695nbsp;nbsp;nbsp;nbsp;C�s share

0,97826 nbsp;nbsp;nbsp;Xnbsp;nbsp;nbsp;nbsp;200,nbsp;nbsp;nbsp;nbsp;=nbsp;nbsp;nbsp;nbsp;195,6520nbsp;nbsp;nbsp;nbsp;D�s share

Example 2. Let 500 acres be divided among persons, whose estates areas follow; viz. A�snbsp;'iol- B�s 20I. C�s lOl. D�s lOOl. E�s 400I. and F�snbsp;lOOOl. per annum, and the value of the land mostnbsp;^nvenient for each is A�s, B�s and C�s, each 7s.

lo�s. E�s 15s. and F�s 12 s. an acre; now each estate divided by the value of his share of the land,nbsp;quot;�^11 stand thus ;

by which, multiplying each quotient, we shall have for each share as follows, viz.

I B = nbsp;nbsp;nbsp;10,9918

N. B. If any single share should contain land of several different values, then use the means tonbsp;divide his estate by.

Also if there be different quantities as well as values, find what each quantity is worth at its value, and add their sums together; then say, as thenbsp;sum of the quantities is to this sum; so is one acrenbsp;to its mean value to he made use of.

Now having found each person�s share, they may be laid out in any form required, by the directions and problems given in the next section.nbsp;But when several shares contain land of the samenbsp;value, it is best to lay out their sum in the mostnbsp;convenient form, and then subdivide it.

For dividing of commons, amp;c. Surveyors generally measure the land of different value in separate parcels, and find the separate value thereof, whichnbsp;added in different sums, gives the whole content,nbsp;and whole value.

By problem 1, they find each man�s propor-tipnal share of the whole value, and then lay out

for each person, a quantity of land equal in value to his share; this they cftect by first laying out anbsp;yiuantity by guess, and then casting it u}) and finding its value; and if such value be equal to hisnbsp;^hare of the whole value, the dividing line is right;nbsp;if otherwise, they shift the dividing line a little,nbsp;till by trial they find a quantity just equal in valuenbsp;to the value of the required share.

If any single share contains land of several different values, each is measured separately, and their several values found; and if the sum of themnbsp;I\eequal to the value sought, the division is right;

As the quantity of land is generally given in ^cres, roods, and perches, it is necessary, first, tonbsp;deduce them to square links, which may be jicr-formed by the following rule.

Jinks.

Rule 1. To the acres annex five cyphers on the *�*ght hand, and the whole will be links. 2. Placenbsp;cyphers to the right of the roods, and flividcnbsp;this by 4, the quotient will be links. 3. Placenbsp;four cyphers on the right hand of the perches, di-this by l6, the quotient will be links. 4.nbsp;fhese sums added together, give the sum ofnbsp;�hnare links in the given quantity.

This is no other than to determine the side of a Square that shall contain any desired number ofnbsp;^ores; reduce, therefore, the given number of

acres to square links, and the square root thereof will be the side of the square required.

Problem 1. To lay out any desired quantity of land in form, of a parallelogram, having either itsnbsp;base or altitude giveyi.

Divide the content, or area, by the given base, and the quotient is the altitude; if divided by thenbsp;altitude, the quotient is the base; by the samenbsp;rule a rectangle may be laid out.

Problem 3. To lay out any desired quantity of land inform of a parallelogram, vohose base shall benbsp;2, 3, 4, times greater than its altitude.

Divide the j^area by the number of times the base is to be greater than the altitude, and extractnbsp;the square I'oot of the quotient; this square rootnbsp;will be the requii'ed altitude, which being multiplied by the number of times that the base is to benbsp;greater than the altitude, will give the length ofnbsp;the required base.

Problem 4. To lay out a given quantity of land in form of a triangle, having either the base or thenbsp;perpendicular given.

Divide the area by half the given base, if the base be given; or by half the given perpendicular,nbsp;if the perpendicular be given; and the quotientnbsp;will be the perpendicular, or base required.

1. Find in the following table, the area of a polygon of the sam.e name with that required, thenbsp;side of which is l. 2. Divide the proposed areanbsp;by that found in the table. 3. Extract the squarenbsp;root of the quotient, and the root is the side of thenbsp;polygon required.

Plain trigonometry is the art of measuring atid. Computing the sides of plain triangles, or of suchnbsp;'''hose sides are right lines.

As this work is not intended to teach the ele-^^nts of the mathematics, it w'ill be sufficient for just to point out a few of the principles, and.nbsp;S�ve the rules of plain trigonometry, for thosenbsp;lt;^ascs that occur in surveying. In most of thosenbsp;it is required to find lines or angles, whosenbsp;^ctual admeasurement is difficult or impracticable;nbsp;are discovered by the relation they bear tonbsp;. er given lines or angles, a calculation being in-^ffiuted for that purpose; and as the comparisonnbsp;One right line with another right line, is morenbsp;l^ouvenient and easy, than the comparison of a rightnbsp;to a curve; it has been found advantageous to

measure the quantities of angles, not by the arc itself, which is described on the angular point, but by certain lines described about that arc.

If any three parts¹ of a plain triangle be given, any required part may be found both by construction and calculation.

If two angles of a plain triangle are known in degrees, minutes, amp;c. the third angle is found bynbsp;subtracting their sum from 180 degrees.

In a right-angled plain triangle, if either acute angle (in degrees) be taken from go degrees, thenbsp;remainder will express the other acute angle.

When the sine of an obtuse angle is required, subtract such obtuse angle from 180 degrees, andnbsp;take the sine of the remainder, or supplement.

If two sides of a triangle are equal, a line bisecting the contained angle, will be perpendicular to the remaining side, and divide it equally.

Before the required side of a triangle can be found by calculation, its opposite angle must firstnbsp;be given, or found.

The required part of a triangle must be the last term of four proportionals, w'rittcn in order undernbsp;one another, whereof the three first are given ornbsp;known.

In four proportional quantities, either of them may be made the last term; thus, let A, B, C, D,nbsp;be proportional quantities.

As first to second, so is third to fourth, i\.;B; ;C:D. As second to first, so is fourth to third, B: A;: D: C.nbsp;As third to fourth, so is first to second,C;D:: A:B.nbsp;As fourth to third, so is second to first,D: C:: B; A.

Against the three first terms of every proportion, or stating, must be written their respective values taken from the proper tables.

This is imperfectly stated by several writers. One of the given parts must be a side. A triangle consists of six parts, vi¹'nbsp;three sides and three angles. Edit,

If the value of the first term be taken from the sum of the second and third, the remainder willnbsp;hu the value of the fourth term or thing required ;nbsp;because the addition and subtraction of logarithms,nbsp;corresponds with the multiplication and divisionnbsp;of natural numbers.

If to the complement of the first value, be added the second and third values, the sum, rejecting thenbsp;borrowed index, will be the tabular number expressing the thing required; this method is gene-rally used when radius is not one of the proportionals.

The complement of any logarithm, sine, or tan-?ient, in the common tables, is its diftercnce from the radius 10.000.000 , or its double 20.000.000.

1. nbsp;nbsp;nbsp;The following proportion is to be used wdicnnbsp;two angles of a triangle, and a side opposite to onenbsp;uf them, is given to find the other side.

to the sine of the angle opposite the required Side; so is the given side to the required side.

2. nbsp;nbsp;nbsp;When two sides and an angle opposite tonbsp;one of them is given, to find another angle; usenbsp;the following rule :

. As the side opposite the given angle, is to the opposite the required angle; so is the sine ofnbsp;the given angle, to the sine of the required angle.

The memory will be assisted in the foregoing oases, by observing that when a side is w�anted, thenbsp;Pi'oportion must begin with an angle-, and w'hen annbsp;is wanted, it must begin with a side,

3. nbsp;nbsp;nbsp;When two sides of a triangle and the inclu-ocl angle are given, to find the ether angles apd

As the sum of tlie two given sides is to their differcuce; so is the tangent of half the sum ofnbsp;the two unknown angles, to the tangent of halfnbsp;their difference.

Half the difference thus found, added to half their sum, gives the greater of the two angles,nbsp;which is the angle opposite the greatest side. Ifnbsp;the third side is wanted, it may be found by solution 1.

4. The following steps and proportions are to be used when the three sides of a triangle are given,nbsp;and the angles required.

Let ABC, plate Q, Jig- 30, be the triangle; make the longest side AB the base; from C thenbsp;angle opposite to the base, let fall the perpendicular C D on AB, this will divide the base intonbsp;two segments AD, B D.

As the base AB, or sum of the two segments, is to the sum of the other sides (AC-fB C); so isnbsp;the difference of the sides (AC�B C), to the difference of the segments of the base (AD�D B).

Half the difference of the segments thus found, added to the half of AB, gives the greater segment AD, or subtracted, leaves the less DB.

In the triangle AD C, find the angle AC D by solution 2; for the two sides AD and AC arcnbsp;known, and the right angle at D is opposite tonbsp;one of them.

Then in the triangle ABC, you have two sides AB, BC, and the angle at A, opposite to one ofnbsp;them, to find the angles C and B.

r 175 ]

There are three of these lines usually put on the Sector, they arc often termed the Gunter�s lines,nbsp;arc made use of for readily working proportions ; when used, the sector is to be quite openednbsp;dkc a strait rule.

If the 1 at the beginning of the scale, or at the left hand of the first interval, be taken for unity,nbsp;then the 1 in the middle, or that which is at thenbsp;^�d of the first interval and beginning of the second will express the number 10; and the ten atnbsp;the end of the right-hand of the second interval ornbsp;*^nd of the scale, will represent the number lOO.nbsp;^fthe first is 10, the middle is 100, and the lastnbsp;1000; the primary and intermediate divisions ianbsp;^uch interval, are to be estimated according tonbsp;the value set on their extremities.

In working proportions with these lines, attention must be paid to the terms, whether arithmetical or trigonometrical, that the first and third term may be of the same name, and the secondnbsp;^nd fourth of the same name. To work a proportion, take the extent on its proper line, fromnbsp;the first term to the third in your compasses, andnbsp;applying one point of the compasses to the se-^'ond, the other applied to the right or left, according as the fourth term is to be more or lessnbsp;than the second, will reach to the fourth.

Example 1. If 4 yards of cloth cost 18 shillings, tvhat will 32 yards cost ? This is solved by the line

numbers; take in your compasses the distance between 4 and 32, then apply one foot thereof onnbsp;^he same line at 18, and the other w'ill reach 144*.

Eximij^�e 2. As radius to the h_\ pothcnuse t2(,t; so is the sinc of the angle opposite the base 30''nbsp;17'* to the base. In this example, radius, or thenbsp;sinc of go, and the sine of 30� 17^ taken from thenbsp;line of sines, and one foot being then applied tonbsp;120 on the line of numbers^ and the other foot onnbsp;the left ill reach to 6o| the length of the requirednbsp;base. The foot was applied to the left, because thenbsp;legs of a right-angled triangle arc less than thenbsp;bypothenuse.

Example^. As the cosine of the latitude 31^ 30', (equal the sine of 38� 30') is to radius, so isnbsp;the sine of the sun�s declination 20� 14', to thenbsp;sinc of the sun�s amplitude. Take the distancenbsp;between the sines of 38� 30' and 20� 14' in yournbsp;compasses; set one foot on the radius, or sine ofnbsp;g0�, and the other will rc4vch to 3301�, the sun�snbsp;amplitude required.

The following problems, though of the greatest use, and sometimes of absolute necessity to thenbsp;surveyor, are not to be found in any of the common treatises on surveying. The maritime surveyor can scarce proceed without the knowledgenbsp;of them; nor can a kingdom, province, or countynbsp;be accurately surveyed, unless the surveyor is wellnbsp;acquainted with the use and application of them.nbsp;Indeed, no man should attempt to survey a county,nbsp;or a sea coast, wdio is not master of these problems.nbsp;The second problem, which is peculiarly usefulnbsp;for determining the exact situation of sands, ornbsp;rocks, within sight of three places upon land,nbsp;whose distances are W'cll know n, was first proposed

Mr. To^vonly, and solved by Mr. Collins, Philosophical Transactions, No. 69. There is � no problem more useful in surveying, than that bynbsp;Miich we find a station, by observed angles ofnbsp;three or more objects, whose reciprocal distancesnbsp;known; 'but distance, and bearing from thenbsp;place of observation arc unknown.

� Previous to the resolution of these problems, Another problem for the easy finding the segmentnbsp;a circle, capable of containing a given angle,nbsp;is necessary, as will be clear from the followingnbsp;observation.

� Two objects can only be seen under the same from some part of a circle passing throughnbsp;hose objects, and the place of observation.

, � If the angle under which those objects appear, 0 less than 90�, the place of observation will benbsp;Somewhere in the greater segment, and those objects will be seen under the same angle from everynbsp;of the segment.

� If the angle, under which those objects are be more than 90�, the place of observationnbsp;be somewhere in the lesser segment, and thosenbsp;Objects will be seen under the same angle fromnbsp;part of that segment*.� Hence, from thenbsp;�huation of three known objects, we are able tonbsp;^termine the station point Avith accuracy.

� 9� plate 9, a segment of a circle, capable of contain-�S a grjgi-i angle.

Method 1. Bisect B C in A. 2. Through the of bisection, draw the indefinite right line

^ perpendicular to B C. 3. Upon B C, at the P^^nt C, constitute the angles DCB, FCB, G C Bj

H C B, respectively equal to the diftcrence of the anqlcs of the intended segments and QO degrees:nbsp;the angle to be formed on the same side with thenbsp;segment, if the angle be less than gO; but on thenbsp;opposite, if the angle is to be greater than go degrees. 4. The points D, F, G, H, where the angular lines CD, CF, C G, Cbl, amp;c. intersectnbsp;the line DE, will be the centers of the intendednbsp;segments.

Thus, if the intended segment is to contain an angle of 120�, constitute on B C, at C, (on the opposite side to which you intend the segment to benbsp;described,) the angle D C B equal to 30�, the difference between go� and 120�: then on center D,nbsp;and i-adius DC, describe the segment C, 120, B,nbsp;in every part of which, the two points C and Bnbsp;will subtend an angle of 120 degrees.

If I want the segment to contain 80 degrees at O, on B C make an angle B C G, equal 10 degrees,nbsp;and on the same side of B C as the intended segment; then on G, with radius GC, describe segment C 80 B, in every part of which C and B willnbsp;subtend an angle of 80 degrees.

Method. 2. Bv the sector. Bisect B C as before, and draw the indefinite line D E, make AC radius,nbsp;and with that extent open the sector at 45 on thenbsp;line of tangents, and set off on the line D E, thenbsp;tangent of the difference between the observednbsp;angle and go degrees, on the same side as the intended segment, if the observed angle is less thannbsp;gO; on the contrary side if more than gO degrees.^

If the. angle is of go degrees, A is the center of the circle, and AC the radius.

The intersection of the line DE with any circle, is the center of a segment, corresponding half the angle of the sco-rnent in the lirst circle-Ilcace, if the difterenec between the observed

and 90�, be more than the scale of tangents contains, fmd-the center to double the angle ob-'\^rved, and the point where the circle cuts the iu-Gefinite line, wdll he the required center.

Problem 1. To determine the position of a pointy f'oin ^whence three points or a triangle can be dis-eovered, whose distances are kno%vn.

, I'he point is either without, or within the given triangle, or in the direction of two points of thenbsp;triangle.

Case l. When the three given objects form a tri-and the point or station whose position is re-i^eired^ h' without the triangle.

Txample. Suppose I want to determine the position of a rocW),fg. 17, plate g, from thenbsp;'^hore; the distances of the three points A, C, B,nbsp;rather the three sides AC, C B, AB, of the tri-^quot;gle A, B, C, being given.

. In the first place, the angles AD C, C D B, must ^ 'tieasured by an Hadley�s sextant or theodolite;nbsp;hen the situation of the point D may he readilynbsp;*^Ond, either by calculation or construction.

construction. Method 1. On AC, fg. 18, 9, describe by the preceding problem, a cir-^ capable of containing an angle equal to thenbsp;ADC; on C B, a segment containing annbsp;?^g�e equal to the angle C D B; and the point ofnbsp;^�^fersection D is the place required.

^�^d the intersection E, describe a circle AE B D ; ^trough E, C, draw' EC, and produce it to inter-Ihe circle at D; join AD, B D, and the distan-^^^1^3 CD, B D, will be the required distances.

o-lculation. In the triangle AB C, arc given ^ three sides, to find the angle BAC. In the

Iriiingle AE B, are given the angle BAE, the angles ABE, A EB, and the side AB, to findnbsp;AE and B E.

In the triangle AE D, wc have the side AE, and the angles AE D, AD E, and consequentlynbsp;D FA, to find the sides AD.

The angle ADE, added to the angle AEG, and then taken from 180�, gives the angle DAE.nbsp;The angle CAE, taken from the angle DAE,nbsp;gives the angle CAD, and hence D C. Lastly,nbsp;the angle AEC, taken from AEB, gives DEB,nbsp;and consequcntlv, in the triangle DEB, we havenbsp;E B, the angle D E B, and the angle- E D B, tonbsp;find B D.

In this method, when the angle B D C is less than that of BAG, the point C will be above thenbsp;])oint E; but the calculation is so similar to thenbsp;foregoing, as to require no particular explanation-

When the points E and C fall too nearly toge-thcr, to produce E C towards D with certainty, irhe first method of construction is the most ac-euratc.

Case 2. When the given place or station D, fig-38, plate 9, is voithout the trmrgle made hy the three given objects AB C, but in a line with one of the sidesnbsp;produced.

Measure the angle AD B, then the problem may be easily resolved, cither by construction ovnbsp;calculation.

By constrnctio7i. Subtract the measured angle ADB from the angle CAB, and you obtain thenbsp;angle AB D; then at B, on the side BA, draw thenbsp;angle AB D, and it will meet the produced sidenbsp;CA at D; and DA, DC, DB, will be the required distances.

By calculatio7i. In the triangle AB D, the angl�quot; D is obtained by observation,, the ^ BAD is ths

AB D, three angles and one side given to find the *sngth of the other two sides, which arc rcadilvnbsp;obtained by the preceding canons.

Case 3. When the sialion �point is in one of the ^ides of the given triangle, fig. 1, plate 13.

-By construction. 1. Measure the angle B D Q. Make the angle BAE equal to tlie observednbsp;3. Draw C D parallel to E A, and D is thenbsp;station point required.

ddy calculation. Find the angle B in the triangle AB C, then the angles B and B D C being known,nbsp;obtain D C B; and, consequeatly, as sin. anglenbsp;^ t) C to B C, so is sin. angle D C B to B D.

Example. Being at sea, near a strait shore, I observed three objects. A, B, C, which were truly ^aid down on my chart; I wished to lay down thenbsp;place of a sunken rock D; for this purpose thenbsp;ingles AD B, BDC, were observed with Fladley�snbsp;Quadrant.

ddy construction. Method 1. On AB, fig. 3, plate 13^ describe the segment of a circle, capable

containing the observed angle ADB. On BC describe the segment of a circle, capable ot con-fainiiig the angle BDC; the point D will be atnbsp;^lic intersection of the arcs, and by joining DA,nbsp;�*^2, DC, you obtain the required distances.

Method 2. Make the angle KQ^,fig. 4, plate ^3, equal to ADB, and the angle EAC equal tonbsp;�ql^C; and from the point of intersection E,nbsp;through B, draiv a line to E D, to intersect thenbsp;gcA�C; join A, D; and D, C, and DA,

By calculation. 1. In the triangle CAE, 75:]^. 4, plate 13, we have all the angles, and the side AC,nbsp;to find AE. 2. In the triangle ABE, AB, AE,nbsp;and the included angle arc given, to lind the angles AE B, AB E. 3. In the triangle B D C, the

Cases. JVlien the station falls u'ithin the triangle, formed by the three given objects, fig. 1(), plate 13.

Let ABC represent three towers, whose distance from each other is known; to find the distance from the tower D, measure the angles AD C, BDC, ADB.

Construction. On two of the given sides AC, AB, fiig. �10 plate 13, describe segments of circles capable of containing the given angles, and the point Dnbsp;of their intersection will be the required place.

jhiother method. Or, we may proceed as in some of the foregoing cases; making the angle ABE,nbsp;fig. 21, plate 13, equal to the angle ADE, andnbsp;B A E equal B D E; and describe a circle throughnbsp;the three points A, B, E, and join E, C, by the linenbsp;EC; and the point D, where BC intersects thenbsp;circle E, A, D, B, E, will be the required station.

Case 6. IVhsn the statmi �point Yy,jig. 7, plate 1.3, falls zviihmi the triangle ABC, but the point Cnbsp;falls iozvards D.

Thus, let AB C,fig. f plate 13, represent three towers, whose resjjcctive distances from each othernbsp;are known; required their distance from the pointnbsp;D. Measure tlie angles ADC, BDC, and to jirovenbsp;the truth of the observations, measure also A D B.

plate 13, describe a circle capable of containing the angle BDC, and on AB, one capable of con-

tauling tlie angle AD B, and the point of intersection will be the place required.

Oi�, it may be constructed by method 2, case 1, which gives ns the pointnbsp;nbsp;nbsp;nbsp;plate 13, eom-

P^ired with fg. 39, plate 9. The calculation is '^poii principles so exactly like those in method 2,nbsp;t-'ase 1, that a further detail would be superfluous.

The point of station D found Instrument ally. The point D may be readily laid dow n on a draught,nbsp;% drawing on a loose transparent paper indefinitenbsp;^%ht lines DA, DB, DC, at angles equal tonbsp;'t'hosc observed; which being placed on the draughtnbsp;�o as each line may pass over, or coincide wnth, itsnbsp;^'cspcctivc object, the angular point D will thennbsp;Coincide wnth the place of observation. Or,nbsp;Provide a graduated semicircle of brass, aboutnbsp;inches in diameter, having three radii withnbsp;chamfered edges, each about 20 inches long, (ornbsp;as long as it may be judged the distance of thenbsp;�fations of the three given objects may require)nbsp;of wdiich radii to be a continuation of the dia-�^leter that passes through the beginning ot the decrees of the semicircle, but immovcably fixed to it,nbsp;the other two moveable round the center, so as tonbsp;he set and screwx'd fast to the semicircle at anynbsp;�'^gle. In the center let there be a small socket,nbsp;hole, to admit a pin for marking the central pointnbsp;the draught. When the sloped edges of thenbsp;two moveable radii are set and screwed fast to thenbsp;seinicireJe^ at the respective degrees and minutesnbsp;^hthe two observed angles, and the whole instrument moved on the draught until the edges of thenbsp;three radii are made to lie along the three stasime-trie points, each touching its respective point, thenbsp;f-ciitcr of the semicircle will then be in tlie pointnbsp;^t'ltion D; w'hich may be marked on the draught,nbsp;through the socket, wdth a pin. Sych an instru-

mcnt as this may be called a station-pointer �, and would prove convenient for finding the point ofnbsp;station readily and accurately, except when thenbsp;given objects were near; when the breadth of thenbsp;arc, and of the radii, and of the brass about thenbsp;center of the semicircle, might hinder the pointsnbsp;from being seen, or the radii so placed as to comprehend a very small angle between them.

'Fhe three succeeding problems may occur at sea, in finding the distances and position of the rocks, sands,nbsp;(ffc. from the shore, in many cases of maritime sur-�veying\ they are also very serviceable in making anbsp;map of a country, from a series of triangles derivednbsp;from one or more measured bases.

Problem 1. Given the distance of tivo objects K^, fg. 5, plate Q, and the angles ADB, BDC,nbsp;B CA, to fnd the distance of the two stations D, C,nbsp;from the objects A, B.

By construction. Assume d c any number at pleasure, and make the angles b d c, adc, amp;c. respectively equal to the angles BDC, ADC, amp;c. and join a b; it is plain that this figure must be similarnbsp;to that required; therefore draw AB,_^^. 4, platenbsp;g, equal the given distance, and make AB C equalnbsp;to a b c, BAG to bac, and so on respectively;nbsp;join the points, and you have the distances required.

By calculation, in the triangle adc, we have d e, ad c, and a c d, to find a d, a e; in b c d, wcnbsp;have in like manner the three angles, and d c, tonbsp;find d b, d c.

In the triangle a d b, we have a d, b d, and the angle a d b, to find a b. Hence by the nature ofnbsp;similar figures, as ab to AB dc : DC :: adnbsp;� AD ;; bd : BD be ; BC.

Problem 2. The distances of three objects A, B, C, from each other, and the angles ADC, CDE,

Assume any line d c, at pleasure, make the angle ^ d e equal the angle C D E, and angle c c d equalnbsp;to the angle C E D; also the angle c d a, equal tanbsp;the angle CDA, and the angle ceb equal to thenbsp;^ogle CEB; produce ad, be to intersect eachnbsp;other at f, and join c f.

_ It is evident that the figures c d f e, C D F E, are similar; therefore, on AC, fg. 7, piate g, describenbsp;a segment of a circle, containing an angle AFCnbsp;�^fiGal to a f c; and on C B a segment capable ofnbsp;Containing an angle C F B, equal the angle c f b;nbsp;the point of intersection F draw F A, F B,nbsp;C; make the angle F C D equal the angle f c d,nbsp;and F C E equal the angle fc c, which completesnbsp;^he construction; then by assuming d e equal tonbsp;^ay number, the rest may be found as before.

This method fails when AD is jiarallcl to BE, 8, plate g; therefore, having described the segments ADC, B C, draw C F, to cut off a segmentnbsp;equal to the angle CDF, and the right line CG,nbsp;cut off a segment equal to the angle CEG;nbsp;will be in the right line D E; therefore, joinnbsp;^P, and produce the line eaeh way, till it intersects the segments, and the points D, E, will benbsp;ac stations required.

Problem 3. Four points B, C, D, F, fig. g, t�eite g, or the four sides of a quadrilateral figure^nbsp;its angles being given, and the angles BA C,nbsp;AE, AE D, D E F, known by observation, to findnbsp;f station point AE, and, consequently, the length ofnbsp;AC,ED,EF.

^y eonstruction. 1. On B C describe the seg^ mcnt of a circle, to contain an angle equal tonbsp;.jAC- �2. From C draw the chord CM, so thatnbsp;m angle B C M may be equal to the supplement

of the angle BAE. 3. On DP� describe the segment of a, circle capable of containing an angle equal to D E P�; join MN, cutting the two circlesnbsp;at A. and E, the required points.

By calculation. In the triangle B C M, the angle B C M, (the supplement of BAE,) and the anglenbsp;B M C, (equal BAG,) and the side B C are given,nbsp;whence it is easy to find A1C. In the same manner, DN in the triangle DNF maybe found;nbsp;but the angle AI C D (equal angle BCD, less angle BCM) is known with the legs M C, CD;nbsp;consequently, AI D, and the angle M D C, will benbsp;readily fouqd.

The angle MDN (equal angle CDF, less C D M, less F D N,) and M D, D N, are known;nbsp;whence we find M M, and the angles D M N,nbsp;DNM.

The angle CMx\, (equal DMC added to D M N,) the angle AIAC, (equal MAB added tonbsp;BAC,) and the side M C are given; therefore, bynbsp;calculation, MA, and AC will also be known.

In the triangle E D N, the side D N, and the angles E and N are given; whence we find E N,nbsp;E D, and, consequently, A E equal M N, lessnbsp;M A, less E N.

And in the triangle ABC, the angle A, with its sides BC, AC, arc known; hence AB, andnbsp;angle. B CA, are found.

In the triangle EFD, the angle E, with the sides E D, D F, being known, E F and the anglenbsp;E D P' will be found.

Lastly, in the triangle ACD, the angle A CD, (equal BCD, less BCA,) and AC, CD, arcnbsp;given; hence AD is found, as in the same mannernbsp;E C in the triangle E C D.

Note. That in this problem, and in problems I and 2, if the two stations fail in a right line with

either of the given objects, the problem is indeterminate. As to the other cases of this problem, they fall in with what has already been said.

The solution of this problem is general, and �iay be used for the two preceding ones: for suppose C D the same point in the last figure, it givesnbsp;the solution of problem 2; but if B, C, be supposed the same points D F, you obtain the solutionnbsp;of problem 1.

Problem 4. Having the distance and magnetic ^^arings of two points A and ^,Jig. 10, plate Q, pro-^^��acled at any station S, 7iot very ohiique to AB, tonbsp;Jind its distance froyn these points ly the needle.

At S, with a good magnetic needle, take the, hearings of A and B in degrees, and parts of a degree; then from these points, draw out their rcs-peetive bearings in the opposite direction towardsnbsp;S; that is, if A bears exactly north, draw a linenbsp;h'om the point A exactly south; if it bears east,nbsp;lO or 20 degrees southward, draw the line west,nbsp;lO or 20 degrees northwaixl; and so for any othernbsp;bearing, draw the opposite bearing of B in thenbsp;^ame manner, and S, the intersection of these twonbsp;points, will be the point of station, and SA, SB,nbsp;the distances required.

This is an easy method of finding the distance of any station from two places, whose distancesnbsp;have been accurately determined before, and willnbsp;be found very convenient in the course of a sur-and on many occasions sufficiently exact,nbsp;provided the places are not too remote from thenbsp;station, nor the intersection of the bearings toonbsp;oblique. If the needle be good, a distance of 20nbsp;hfiles is not too far, when the angle subtended bynbsp;^be two places, is not less than 50 degrees, ornbsp;than 140.

In surveys of kingdoms, provinces, counties, amp;c. where signals, churches, amp;c. at a distance arenbsp;used for points of observation, it very often happens that the instruinent cannot be placed exactlynbsp;at the center of the signal or mark ot observation;nbsp;consequently, the angle observed will be eithernbsp;greater, less, or equal to that which woxdd havenbsp;have been found at the center. This problemnbsp;shews how to reduce them to the center; the correction seldom amounts to more than a few seconds, and is, therefore, seldom considered, unlessnbsp;where great accuracy is required.

The observer may be considered in three different positions with respect to the center, and the objects; for he is either in a line with the center,nbsp;and one of these objects, or in an intermediate one,nbsp;that is, a line from this center to the observer produced, would pass between the objects; or he is,nbsp;lastly, in an oblique direction, so that a line fromnbsp;the center to him would pass without the objects.

In the first position, jig. \ \, plate g, where the observer is at 0, between the center and one of thenbsp;objects, the exterior angle m o n is greater than thenbsp;angle men, at the center, by the angle emo;nbsp;therefore, taking emo from the observed angle,nbsp;you have that at the center.

If the observer is at a,y?f. 11, plate Q, the exterior angle m a n is greater than that of m e n at the center, by the value of n; therefore, taking thisnbsp;from the observed angle, you obtain the angle atnbsp;the center. But if the observer is further from thenbsp;objects than the center, as at i, the observed anglenbsp;jn i n is less than that at the center men, by thenbsp;angle m; tbercforc, by adding ra to the observed

{ingle, you obtain the angle m e n at the center. In the same manner, if the observer is at u, wenbsp;should add the angle n to the observed m u n, innbsp;order to have the angle m e n at the center.

C�j^2. When the observer is at o,Jig. 11, plate 05 draw ao, and the exterior angle d exceeds thenbsp;^ngle u at the center by the angle m, and the exterior angle c exceeds the angle at the center a,nbsp;Vthe angle n; therefore, m o n exceeds the anglenbsp;the center, by the value of the two angles mnbsp;nnd II; these, therefore, must be subtracted fromnbsp;to obtain the central angle. On the contrary,nbsp;the observer is at a, the two angles m and nnbsp;^�iust be adtled to the observed angle.

Case 3. Fig. 13, plate Q. When the observer is o, having measured the angles m o n, m o c, thenbsp;angle i is exterior to the two triangles m o i, n c i;nbsp;therefore, to render m e n, equal to m i n, wenbsp;^tlust add the angle n; and to render the exteriornbsp;^Ugle min, equal to the observed angle m o n,nbsp;must take away the angle m; therefore, add-m to the observed angle, and subtracting nnbsp;h�om the total, we obtain the central angle men.nbsp;Prom wdiat has been said, it is clear, that in thenbsp;case, you are to add or subtract from the observed angles, that of the angles m or n, whichnbsp;not in the direction of the observer.

In the third position, you add to the observed that of the two m or n, which is of thenbsp;^urnc side with the observer, and subtract the

To know the position of the observer, care must ^ taken to measure the distance of the instru-� rument from the center, and tlie angle this cea-'�t makes with the objects.

An inspection of the figures is suffieient to shew how the value of the angles m n may be obtained.nbsp;Thus, in the triangle m o e, we have the angle atnbsp;o, the distance o e, and the distances e m, o m,nbsp;j(which are considered as equal,) given.

After the reduction of the observed angles to the center of each respective station, it is generally necessary to reduce the parts of one, or ofnbsp;several triangles to the same level.

Case 1. Let us suppose the three points A, P, E, fig. 15, plate Q, to be equally distant from the center of the earth, and that the point R is highernbsp;thafi these points by the distance or quantity RE;nbsp;now it is required to reduce the triangle AP R tonbsp;that APE.

By the following rule, you may reduce the angles RAP, RPA, which have their summitsnbsp;in the plain of reduction, to the angle EPA.nbsp;EAP.

Rule The cosine of the reduced angle is equal to the cosine of the observed angle, divided bynbsp;the cosine of the angle of elevation.

These two angles being known, the third E is consequently known; we shall, howev^cr, give anbsp;rule for finding AEP, independent of the othernbsp;two.

Rule. The cosine of the reduced angle is equal to the cosine of the observed angle, lessened bynbsp;the rectangle of the sines of the angles of elevation, divided by the rectangle of the cosine of thenbsp;Same angles.

The reduction of the sides can be no difficulty. Case 2, Hg. 17, plale 9. Let AR r be the tri-to be reduced to the plain AE e, the pointsnbsp;E, e, of the vertical lines R E, r e, being supposednbsp;Equally distant from the center of the earth.

Prolong the plain AE c to P, that is, till it meets the line Rr produced to P; and the value ofnbsp;PAe wall be found by the following formuhe.

^iid half difference of PAR and PAr, wc obtain value of each of the angles; the value ofnbsp;PAE and PAe, may be then obtained by the firstnbsp;the two preceding rules, and the diftcrcnce be-Eveen them is the angle sought.

Ect C, Jig. 16, plate g, be the center of the '^arth, let AB be the side of a triangle reduced tonbsp;common horizon by the preceding methods ; ifnbsp;E be required to reduce this to the plain D E, asnbsp;these planes are parallel, the angles will remain thenbsp;Same; therefore, the sides only are to be reduced,nbsp;\hc mode of performing which is evident from thenbsp;hgure.

, Method of referring a series of triangles to a mc-^Mian line, and ano ther line perpendicular to it.

This method will be found somewhat similar to Used by Mr. Gale, and described at length innbsp;artiele of surveying; it is a mode that shouldnbsp;t�c adopted wherever extreme accuracy is required, for whatever care is taken to protract anbsp;Series of triangles, the protractor, the points of thenbsp;^mnpasses, the thickness of the line, the inequalitynbsp;ut the paper, amp;c. will produce in the fixing of thenbsp;^uits of a triangle an error, whieh, though smallnbsp;first, will have its influence on those that sue-

ceecl, and become very sensible, in proportion aa tbc number of triangles is augmentctl. TJiis multiplication of errors is avoided by the followingnbsp;problem.

Let AB, �^�. lA, plate be the meridian, CD the perpendicular, and the triangles o a d, d a e,nbsp;deg, egi, gil, those that have been observed;nbsp;from the point o, (which is always supposed to benbsp;on a meridian, or whose relation to a meridian isnbsp;known) observe the angle Boa, to know hownbsp;much the point a declines from the meridian.

In the right-angled triangle oBa, we have the angle Boa, and the right angle, and consequently, the angle o a B, together with the side o a,� tonbsp;find O B, and B a.

For the point d, add the angle B o a to the observed angle a o d, for the /_doh, or its equal o d m, and the complement is the angle m o d,nbsp;whence as before, to find o m and m d.

For the point G, add the angles m d o, o d a, ad c, and edg, which subtract from 3�O, to obtain the angle g d r, of the right-angled trianglenbsp;g d r; hence we also readily, as in the precedingnbsp;triangles, obtain rg=mt, which added to mo,nbsp;gives to the distance from the meridian. Thennbsp;we obtain r d, from which taking d in, you obtainnbsp;rm, equal gt, the distance from the perpendicular.

For the point e, take the right angle rdf, from the two angles rdg, gd c, and the remainder isnbsp;the angle f d c of the right-angled triangle d f c;nbsp;hence we obtain fed and d f, which added to d b,nbsp;gives b f, equal to x e, the distance from the meridian ; from the same right-angled triangle wenbsp;obtain f e, which added to f n, equal to d m, givesnbsp;e n, the distance from the perpendicular.

GEOMETRICAL PROBLEMS.

Por the point i, add together the angles rgd, egh and from the sum subtract the rightnbsp;O-ngle r g h, and you obtain the angle g h i of thenbsp;i-ight -angled triangle h g i, and consequently thenbsp;^i^gle i; hence also we get h i, equal t whichnbsp;added to t o, gives o p distance from the meridianjnbsp;and g h, from which subtracting g t, we obtain t h,nbsp;equal p i distance from the perpendicular.

Por the point 1, the angle g h i, added to the angle 1 g i, gives the angle 1 g k of the right-anglednbsp;triangle g k 1, and of course the angle g 1 k, whencenbsp;obtain k 1 or t y, which added to t o, gives o ynbsp;distance from the meridian; hence we also obtainnbsp;� k, which taken from g t, gives k t, equal 1 y distance from the perpendicular.

. If, before the operation, no fixed meridian was ^iven, one may be assumed as near as possiblenbsp;troin the point o; for the errqr in its position w'illnbsp;dot at all influence the respective position of thenbsp;triangles.

mathematical student, who may have a desire to proceed on a com-i-e- ^t)urse of the mathematical sciences, the foliowina eminent authors are

hy Simpson, 8vo. I755; Maclaurin, 8vo* 1771; Bonnycastle�s.. ^?duction, ramo. 1796.

^ticlid�s Elements, by Dr. Simson, 8vo. 1791; Elements, by 1768, and Emerson, 8vo. 1765.nbsp;by Simpson, 8vo. 176^; �merson, 8vo. 17885 Traite, bynbsp;4�^o* Paris, 1786; and a Treatise now in the press, 8vo. by T. Keith,,nbsp;by Hamilton, 4to. 1758; Hutton, 8vo, 17875 Elements, bynbsp;Kewton, 8vo. 1794-

Simpson, 8vo. 1776, and Maclaurin, 4to. 2, vols, 17425 Intro-� P-owe, 8vo, 1767, and Emerson, SvO, 1768. andnbsp;nbsp;nbsp;nbsp;Tahlesj by Taylor, to every second of the quadrant, 4to, 1792�

�Hid a nbsp;nbsp;nbsp;be added, Emerson�s Mathematical IVorks, in lO vols. 8vo*

SURVEYING.

lines. 2. Finding the position of strait lines with respect to each other. 3. Laying down, or planning upon paper these positions and measures.nbsp;4. Obtaining the superficial measure of the landnbsp;to be surveyed.

We may, therefore, define /am/ surveying to be the art which teaches us to find how many timesnbsp;any customary measure is contained in a givennbsp;piece of ground, and to exhibit the true boundaries thereof in a plan or map.

A station Vine is a strait line, whose length is accurately ascertained by a chain, and the bearing determined by some graduated instrument.

An offset is the distance of any angular point in the boundary from the station line, measured bynbsp;a line perpendicular thereto.

The curvature of the earth within the limits of an ordinary survey, is so inconsiderable, that itsnbsp;surface may be safely considered by the land surveyor as a plain. In a large extent, as a province,nbsp;or a kingdom, the curvature of the earth�s surfacenbsp;becomes very considerable, and due allowancenbsp;must be made for it.

All plains, how many sides soever they consist of, may be reduced into triangles, and may therefore be considered as composed thereof; and, consequently, what is required in surveying, arc suchnbsp;instruments as will measure the length of a side,nbsp;and the quantity of an angle of a triangle.

C 19s 1

A few general observations, or hints, can only expected in this place; for, after all that we cannbsp;the surveyor nmst depend on his own judgment for contriving his work, and his own skill innbsp;jliscriminating, among various methods, that whichnbsp;is best.

The first business of the surveyor is to take ^iich a general view of the ground to be surveyed,nbsp;will fix a map thereof in his mind, and thencenbsp;j^terrnine the situations for his station lines, andnbsp;places where his instruments may be used tonbsp;greatest advantage.

. Having settled the plan of operations, his next ^siness is to examine his instruments, and seenbsp;mm they arc all in proper order, and accuratelynbsp;^^Usted. He should measure carefully his chain,nbsp;if there be any errors therein, correct them;nbsp;P^'eparc staves, marks, amp;c. for distinguishing thenbsp;*^mval stations.

, -The fewer stations that can be made use of, the ss will be the labour of the survey; it will also benbsp;accurate, less liable to mistakes while in the-or errors when plotting the work at home,nbsp;he station lines should always be as long asnbsp;possible, where it can be done without renderingnbsp;offsets too large; where grejit accuracy is required, these lines should be repeatedly measured,nbsp;firgj- point being the careful mensurationnbsp;� Po^ station lines; the second, to determine thenbsp;Uation of places adjoining to them. For everynbsp;lon line is the basis of the succeeding opera-os, and fixes the situation of the diflerent parts,nbsp;av � 1 surveyor should so contrive his plan, as tonbsp;the multiplication of small errors, and par-

ticularly those that by communication will extend themselves through the whole operation. If thenbsp;estate be large, or if it be subdivided into a greatnbsp;number of fields, it would be improper to surveynbsp;the tields singly, and then put them together, nornbsp;could a survey be accui'ately made by taking allnbsp;the angles and boundaries that inclose it. If possible, dx upon, for station points, two or morenbsp;eminent situations in the estate, from whence thenbsp;principal parts may be seen; let thefc be as farnbsp;distant from each other as possible; the work willnbsp;be more accurate when but few of these stationsnbsp;are made use of, and the station lines will be morenbsp;convenient, if they are situated near the boundaries of the estate.

Marks should be erected at the intersection of all hedges with the station line, in order to kno%vnbsp;where to measure from, when the fields arc surveyed. All necessary angles between the mainnbsp;stations should be taken, carefully measuring in anbsp;right line the distance from each station, notingnbsp;down, while measuring, those distances where thenbsp;lines meet a hedge, a ditch, amp;c If any remarkable object be situated near the station line, itsnbsp;perpendicular distance therefrom should be ascertained; in the. same manner, all offsets from thenbsp;ends of hedges, ponds, houses, amp;c. from the mainnbsp;station line should be obtained, care being takennbsp;that all observations from the station line, as thenbsp;measure of angles, amp;e. be always made fromnbsp;points in the station line.

When the main stations, and every thing adjoining to them, have been found, then the estate may be subdivided into two or three parts by ncAVnbsp;station lines, fixing the stations where the bestnbsp;views can be olrtained; these station lines must benbsp;accurately measured, and the places where they

intersect hedges be exactly ascertained, and all the ^^^cessary offsets determined.

this effected, proceed to survey the adjoining ^^Ids, by observing the angles that the sides makenbsp;'^�hh the station lines at their intersections thcre-''^'th; the distances of each corner of the fieldnbsp;ffom these intersections, and that of all necessary

Every thing that could be determined from these stations being found, assume more internalnbsp;^tjitions, and thus continue to divide and sab-r^t'ide, till at last you obtain single fields, repeat-J.'ig the same operations, as well for the inner asnbsp;the exterior work, till all be finished.

Every operation performed, and every observation made, is to be carefully noted down, as the hata for fixing the situations upon the plan. Thenbsp;should be closed as often as convenient, andnbsp;as few lines as possible; what is performed innbsp;clay should be carefully laid down everynbsp;^ight, in order not only to discover the regularnbsp;P^'ocess of the work, but to find whether any cir-^tirnstance has been neglected, or any error com-ypitted, noticing in the field-book, how one fieldnbsp;'es by another, that they may not be displaced innbsp;draft.

^^an estate be so situated, that the whole can-be surveyed together, because one part caught be seen from the other, divide it into three or pcirts, and survey them separately, as if theynbsp;. lands belonging to different persons, and atnbsp;join them together.

it is thus necessary to lay down the work as proceed, it will be proper to find the mostnbsp;^nvenient scale for this purpose; to obtain this,nbsp;the whole length of the estate in chains,nbsp;consider how many inches long your plan

is to be, and from these conditions, you will ascertain how many chains you have in an inch, and th cnee choose your scale.

In order that the surveyor may prove his w'ork, and see daily that it goes on right, let him choosenbsp;some conspicuous object that may be seen fromnbsp;all, or greater part of the estate he is surveying,nbsp;and then measure with accuracy the angle thisnbsp;object makes, with two of the most convenientnbsp;stations in the first round, entering them in thenbsp;field-book or sketch where they were taken; whennbsp;jmu plot your first round, you will find the truenbsp;situation of this object by the intersection of thenbsp;angles. Measure the angle this object makes,nbsp;with one of your station lines in the second, third,nbsp;Etc. rounds: these angles, when plotted one day after another, will intersect each other in the place ofnbsp;the object, if the work be right; otherwise somenbsp;mistake has been committed, which must be corrected before the work is carried any further.

Fields plotted from measured lines only, are always plotted nearest the truth, when those lines form at their meeting, angles nearly approachingnbsp;to right angles.

The three angles of every triangle sliould al-'�ways, if possible, be measured.

As it is impossible to avoid some degree of erroiquot; in taking of angles, we should be careful so to order our operations, that this error may have thenbsp;least possible influence on the sides, the exactnbsp;measure of which is the end of the operations.

Now, in a right-lined triangle, it is necessary to have at least one side measured mediately or im*'

oiediately; the choice of the base is therefore the fundamental operation; the determinations willnbsp;he most accurate to find one side, when the basenbsp;equal to the side required; to find two sides, annbsp;�equilateral triangle is most advantageous.

In general, when the base cannot be equal to the side or sides sought, it should be as long asnbsp;possible, and the angles at the base should benbsp;equal.

In any particular case, where only two angles ef a triangle can be actually observed, they shouldnbsp;he each of them as near as possible to 45�; at anynbsp;�quot;iitc their sum should not differ much from 00�, fornbsp;the less the computed angle differs from 00�; thenbsp;less chance there will be of any considerable errornbsp;the intersection.

USED IN surveying, AND THE METHOD OP APPLY'ING THEM TO PRACTICE, AND EXAMINING THEIR ADJUSTMENTS.

The variety of instruments that arc now made Dse of in surveying is so great, and the improve-Dients that have been made within these few yearsnbsp;so numerous, that a particular description ofnbsp;�^ach is become necessary, that by seeing their respective merits or defects, the purchaser may benbsp;enabled to avail himself of the one, and avoid thenbsp;ether, and be also enabled to select those that arcnbsp;eest adapted to his purposes.

The accuracy of geometrical and trigonometri-cal mensuration, depends in a great degree on the exactness and perfection of the instruments madenbsp;Use of; if these are defective in construction, ornbsp;difficult in use, the surveyor will either be subjectnbsp;e error, or embarrassed with continual obstacles,nbsp;Ihe adjustments, by which they are to be ren-

dered fit for observation, be troublesome and inconvenient, they will be taken upon trust, and the instrument will be used without examination, andnbsp;thus subject the surveyor to errors, that he cannbsp;neither account for, nor correct.

In the present state of science, it may be laid down as a maxim, that every instrument shouldnbsp;be so contrived, that the observer may easily examine and rectify the principal parts; for howevernbsp;careful the instrument-maker may be, howevernbsp;perfect the execution thereof, it is not possiblenbsp;that any instrument should long remain accurately fixed in the position in which it came out of thenbsp;maker�s hands, and therefore the principal partsnbsp;should be moveable, to be rectified occasionallynbsp;by the observer.

Fewer or more of which will be wanted, according to the extent of his work, and the accuracy required.

_ Besides these, a number of measuring rods, iron P'ns, or arrows, See. will be found very convenient,nbsp;^iid two or three offset staves, which are straitnbsp;pieces of wood, six feet seven inches long, andnbsp;^bout an inch and a quarter square; they shouldnbsp;^e accurately divided into ten equal parts, each ofnbsp;'''hich will be equal to one link. These are used

Five or six staves of about five feet in length, ^*^d one inch and an half in diameter, the uppernbsp;P'lrt painted white, the lower end shod with iron,nbsp;to he struck into the g-routid as marks.

J-wenty or more iron arrows, ten of which arc ^hvays wanted to use with the chain, to count thenbsp;^^imber of links, and preserve the direction of thenbsp;phain, so that the distance measured may be really

The pocket measuring tapes, in leather boxes, S'6 often very convenient and useful. They are

made to the different lengths of one, two, three, four poles, or sixty-six feet and 100 feet; divided, on one side into feet and inches, and on thenbsp;other into links of the chain. Instead of the latter, are sometimes placed the centesimals of anbsp;yard, or three feet into 100 equal parts.

The length of a strait line must be found mechanically by the chain, previous to ascertaining any ilistance by trigonometry: on the exactnessnbsp;of this mensuration the truth of the operationsnbsp;will depend. The surveyor, therefore, cannot benbsp;too careful in guarding against, rectifying, ornbsp;making allowances for every possible error; andnbsp;the chain should be examined previous and subsequent to every operation.

For the chain, however useful and necessary, is not infallible, it is liable to many errors. 1. Innbsp;itself. 2. In the method of using it. 3. In thenbsp;uncertainty of pitching the arrows; so that thenbsp;surveyor, who wishes to obtain an accurate survey,nbsp;will depend as little as possible upon it, using itnbsp;only where absolutely necessary as a basis, andnbsp;then with every possible precaution.

If the chain be stretched too tight, the ring� will give, the arrow incline, and the measured basenbsp;will appear shorter than it really is; on the othei'nbsp;band, if it be not drawn sufficiently tight, thenbsp;measure obtained will be too Ions:. I have beennbsp;informed by an accurate and very intelligent surveyor, that when the chain has been much usedj

�ic has generally found it necessary to shorten it every second or third day. Chains made ofnbsp;strong wire are preferred.

Gutiters chain is the measure universally adopted in tins kingdom for the purpose of land surveying,nbsp;being exceedingly well adapted for the mensuration of land, and aft'ording very expeditious methods of casting up what is measured. It is sixty-t^ix feet, or four poles in length, and is divided intonbsp;loo links, each link with the rings between themnbsp;is 7.92 inches long, every tenth link is pointed outnbsp;by pieces of brass of different shapes, for the morenbsp;readily counting of the odd links.

The English acre is 4840 square yards, and Gunter�s chain is 22 yards in length, and dividednbsp;iuto 100 links; and the square chain, or 22 multiplied by 22, gives 484, exactly the tenth part of

acre; and ten chains squared are equal to one ^ere; consequently, as the chain is divided intonbsp;loo links, every superficial chain contains 100nbsp;lUultiplicd by 100, that is 10.000 square links;nbsp;^nd 10 superficial chains, or one acre, containsnbsp;100.000 square links.

If, therefore, the content of a field, cast up in Square links, be divided by 100.000, or (which isnbsp;tbc same thing) if from the content we cut off thenbsp;bve last figures, the remaining figure towards thenbsp;�uft hand gives the content in acres, and consequently the number of acres at first sight; thenbsp;J'emaining decimal fraction, multiplied by 4, givesnbsp;*bc roods, and the decimal part of this last product multiplied by 40, gives the poles or perches.

Thus, if a field contains 16,54321 square links, i^e see immediately that it contains 16 acres,nbsp;^4321 multiplied by 4, gives 2,17284, or 2 roodsnbsp;und 17284 parts; these, rmdtiplicd by 40, producenbsp;b;9l36o, or 6 poles, 91300 parts.

Directions for using the chain. Marks are first to be set up at the places whose distances are to benbsp;obtained; the place where you begin may be callednbsp;your first station; and the station to which younbsp;measure, the second station. Two persons are tonbsp;hold the chain, one at each end; the foremost, ornbsp;chain leader, must be provided with nine arrows,nbsp;one of which is to be put down perpendicularly atnbsp;the end of the chain when stretched out, and tonbsp;be afterwards taken up by the follower, by way ofnbsp;keeping an account of the number of chains.nbsp;When the arrows have been all put down, thenbsp;leader must wait till the follower brings him thenbsp;arrows, then proceeding onwards as before, butnbsp;without leaving an arrow at the tenth extention ofnbsp;the chain. In order to keep an account of thenbsp;number of times which the arrows are thus exchanged, they should each tie a knot on a string,nbsp;carried for that purpose, and which may be fastened to the button, or button-hole of the coat;nbsp;they should also call out the number of those exchanges, that the surveyor may have a check onnbsp;them.

It is very necessary that the chain bearers should proceed in a strait line; to this end, the second,nbsp;and all the succeeding arrows, should always benbsp;so placed, that the next foregoing one may be innbsp;a line with it, the place measured from, and thatnbsp;to which you are advancing; it is a very goodnbsp;method to set up a stall� at every ten chains, as wellnbsp;for the purpose of a guide to preserve the rectilinear direction, as to prevent mistakes.

All distances of offsets from the chain line to any boundary which are less than a chain, arenbsp;most conveniently measured by the offset staff;nbsp;the measure must always be obtained in a direction perpendicular to the chain.

The several problems that may be solved by the chain alone, will be found in that part of thenbsp;�'vork, which treats of practical geometry on the

ground.¹

aUADRANT, FOR ADJUSTING AND REGULATING THE MEASURES OBTAINED BY THE CHAIN WHEN USED ON HILLY GROUND, INVENTED BY R. King, surveyor.

There are two circumstances to be considered in the measuring of lines in an inclined situation:nbsp;the first regards the plotting, or laying down thenbsp;Pleasures on paper; the second, the area, or superficial content of the land. With respect to thenbsp;first, it is evident that the oblique lines will benbsp;longer than the horizontal ones, or base; if, thcrc-tore, the plan be laid down according to such.nbsp;Pleasures, all the otlier parts thereof would benbsp;^hereby pushed out of their true situations; hencenbsp;It becomes necessary to reduce the hypothenusal

The best method of surveying by the chain, and now generally used by the more skilful surveyors, I judge, a sleetch of here quot;�''ll be acceptable to many readers. It consists of forming thenbsp;^states into triangles, and applying lines within them parallelnbsp;contiguous to ever)quot; fence and line to be laid down, withnbsp;Onsets from these lines when necessary. The peculiar advan-j S� of this method is, that, after three lines are measured andnbsp;^'d down, every other line proves Itselfnbsp;nbsp;nbsp;nbsp;a

'^Pon application. Tims, if the trian-S 0 a � c be laid down, and the points and c given in the sides, when thenbsp;de has been measured for the pur-pose.of taking a fence contiguous to it,

in^ extremities being given. This method cannot be used Woods, where the principal lines could not be observed, or innbsp;roads or very detached parts of estates; in such casesnbsp;oourse must be had to the theodolite, or other angular instru-Edit.

With respect to the area, there is a differcnco among surveyors; some contending that it shouldnbsp;be made according to the hypothcnusal; others,nbsp;according to the horizontal lines: but, as idl havenbsp;agreed to the necessity of the reduction for the firstnbsp;purpose, we need not enter minutely into theirnbsp;reasons here; for, even if we admit that in somenbsp;cases more may be grown on the hypothcnusalnbsp;plain than the horizontal, even then the areanbsp;should be given according to both suppositions, asnbsp;the hilly and uneven ground requires more labournbsp;in the working.

wooden square, which slides upon an offset staff, and may be fixed at any height by means of thenbsp;screw C, which di'aws in the diagonal of the staff,nbsp;thus embracing the four sides, and keeping thenbsp;limb of the square perpendicular to the staff; thenbsp;staff should be pointed with ii'on to prevent wear;nbsp;when the staff is fixed in the ground on the station,nbsp;line, the square answers the purpose of a crossnbsp;staff, and may, if desired, have sights fitted to it.

The quadrant is three inches radius, of brass, ia furnished with a spirit level, and is fastened to thenbsp;limb D E of the square, by the screw G.

When the several lines on the limb of the quadrant have their first division coincident with their respective index divisions, the axis of the level isnbsp;parallel to the staff.

The first line next the edge of the quadrant is numbered from right to left, and is divided intonbsp;100 parts, which shews the number of links in thenbsp;horizontal line, which are completed in lOO linksnbsp;on the hypothcnusal line, and in proportion fornbsp;any lesser number.

The second, or middlemost line, shews the number of links the chain is to be drawn forward,nbsp;to render the hypothcnusal measure the same asnbsp;the horizontal.

. The third, or uppermost line, gives the perpen-^^icular height, when the horizontal line is equal to ICX).

To use the quadrant. Lay the staff along the chain line on the ground, so that the plain of thenbsp;M^adrant may be upright, then move the quad-*�ant till the bubble stands in the middle, and on,nbsp;the several lines you will have, 1. The horizontalnbsp;length gone forward in that chain. 2. The linksnbsp;to be drawn forward to complete the horizontalnbsp;chain. 3, The perpendicular height or descentnbsp;triade in going forward one horizontal chain.

surveying land, which cannot possibly be plan-with any degree of accuracy without having the horizontal line, and this is not to be obtainednbsp;oy any instrument in use, without much loss ofnbsp;tune to the surveyor. For with this, he has onlynbsp;to lay his staff on the ground, and set the quadrantnbsp;the bubble is in the middle of the space, whichnbsp;is very soon performed, and he saves by it morenbsp;tunc in plotting his survey, than he can lose in thenbsp;bold; for as he completes the horizontal chain asnbsp;be goes forwards, the offsets are always in their rightnbsp;Tffes, and the field-book being kept by horizon-�tt measure, his lines are always sure to close.

If the superficial content by the hypothcnusal jbeasurc be required for an,y particular purpose, henbsp;that likewise by entering in the margin of hisnbsp;- cid-book the links drawn forward in each chain,nbsp;thus the hypothcnusal and horizontal

The third line, wliieh is the pcrpctidiculai' height, may be used with success in finding thenbsp;height of timber; thus measure with a tape ofnbsp;100 feet, the surface of the ground from the rootnbsp;of the tree, and find, by the second line, how muchnbsp;the tape is to be drawn forward to complete thenbsp;distance of 100 horizontal feet; and the line ofnbsp;perpendiculars shews how many feet the foot ofnbsp;the tree is above or below the place where the 100nbsp;feet distance is completed.

Then inverting the quadrant by means of sights fixed on the staff, place the staff in such a position, as to point to that part of the tree whosenbsp;height you want; and slide the quadrant till thenbsp;bubble stands level, you will have on the line ofnbsp;perpendiculars on the quadrant, the height of thatnbsp;part of tlic tree above the level of the place wherenbsp;you are; to whieh add or subtract the perpendicular height of the place from the foot of thenbsp;tree, and y ou obtain the height require;^

Fig. Q, plate 17. represents the perambulator, which consists of a wheel of wood A, shod or linednbsp;with iron to prevent the wear; a short axis is fixednbsp;to this wheel, whicli communicates, by a long pinion rod in one of the sides of the carriage B,nbsp;motion to the wheel-work C, included in the boxnbsp;part of the instrument.

In this instrument, the circumference of the wheel A, Is eight feet three inches, or half a pole;nbsp;one revolution of this wheel turns a single-threaded WQrin once round; the worm takes into

^ wheel of 80 teeth, and turns it once round in 80 revolutions; on the socket of this wheel is fixednbsp;index, which makes one revolution in 40 poles,nbsp;one furlong; on the axis of this worm is fixednbsp;Another worm with a single thread, that takes intonbsp;^ wheel of 40 teeth; on the axis of this wheel isnbsp;Another worm with a single thread, turning aboutnbsp;^ wheel of l6o teeth, whose socket carries an in-that makes one revolution in 80 furlongs, ornbsp;^0 miles. On the dial plate, seej^^. 7, there arenbsp;three graduated circles, the outermost is dividednbsp;into 220 parts, or the yards in a furlong; the nextnbsp;into 40 parts, the number of poles in a furlong;nbsp;the third into 80 parts, the number of furlongs innbsp;to miles, every mile being distinguished by itsnbsp;proper Roman figure.

Idiis wheel is much superior to those hitherto ttinde, 1.Because the worms and wheels act with-tilit shake, and, as they have only very light indicesnbsp;to carry, move with little or no friction, and are,nbsp;therefore, not liable to wear or be soon out of or -^iir; which is not the case with the general num-j i^r of those that are made, in which there i^ anbsp;^iig train of wheels and pinions, and consequentlynbsp;iiiuch shake and friction. 2. The divisions on thenbsp;graduated circles are at a much greater distance,nbsp;may therefore be subdivided into feet, if re-qmred. 3. The measure shewn by the indices isnbsp;jin� more accurate, as there is no shake nor anynbsp;OSS of time in the action of one part or another,nbsp;oe instrument is sometimes made with a doublenbsp;beel for steadiness when using, and also with anbsp;connected to the wheel-work, to strike thenbsp;^\?^or of miles gone over.

^ -^his instrument is very useful for measuring . ^os, commons, and every thing where expedi-is required; one objection is however made

to it, namely, tliat it gives a measure somewhat too long by entering into hollows, and going overnbsp;vSmall hills. This is certainly the case; the measuring wheel is not an infallible mode of ascertaining the horizontal distance between any twonbsp;places; but then it ma}'' with propriety be askednbsp;whether any other method is less fallible? whether,nbsp;upon the whole, and in the circumstances to whichnbsp;the measuring wheel is usually appropriated, thenbsp;chain is not equally uncertain, and the measurenbsp;ol)taincd from it as liable to error, as that from thenbsp;wheel.

The. waj-wiser is a similar kind of instrument, but generally applied to carriages for measuringnbsp;the roads or distance travelled. The best methodnbsp;for constructing such a one is represented in fg. 8,nbsp;plate 17- A piece of plate iron A is screwed tonbsp;the inside nave of the wheel; this being of a cur-vilineal shape, in every revolution of the coach-wheel B it pushes against the sliding bar C, which,nbsp;at the other end, withinside of the brass box oinbsp;\^'lieel-work D, is cut with teeth, and therebynbsp;communicates motion to the w'hecl-w'ork in thenbsp;box. d'he bar is rc-acted upon by a sjwingin thenbsp;box, so as to driv'c it out again for the fresh irn- ,nbsp;])ulsc from the iron piece on the nave, at everynbsp;revolution. As the wheels of carriages differ innbsp;size, the wheel-work is calculated to register thenbsp;number of revolutions, and shew by three indicesnbsp;on the dial plate to the amount of �20,000. Innbsp;any distane.e, or journey performed, the length ofnbsp;feet and inches in the circumference of the wdicelnbsp;jnust be first accurately measured, and that multiplied by the number shewn on the dial of thenbsp;way-wiser gives the distance run.

By means of rods, universal joint, amp;c. it is often made to act within the carriage, so that the person

luay at any moment, without the trouble of getting �ut, see the number of the revolutions of thenbsp;'vheel. If the instrument is to be always appliednbsp;to one wheel, a table may easily be constructed tonbsp;shew the distance in miles and its parts by inspection only.

The pedometer is exactly the same kind of in-�'trument as the way-wiser. The box containing the wheels is made of the size of a watch case, andnbsp;goes into the fob, or breeches pocket; and, bynbsp;J^neans of a string and hook fastened to the waistband or at the knee, the number of steps a mannbsp;takes in his regular paces are registered, from thenbsp;Action of the string upon the internal wheel-work,nbsp;^t every step, to the amount of 30,000. It is necessary to ascertain the distance walked, that thenbsp;average length of one pace be previously known,nbsp;and that multiplied by the number of steps registered on the dial plate.

The cross consists of two pair of sights, placed at right angles to each other: these sights arenbsp;Sometimes pierced out in the circumference of anbsp;^ick tube of brass about 2� inches diameter, seenbsp;3, plate 14. Sometimes it consists of fournbsp;a*gbts strongly fixed upon a brass cross; this is,nbsp;quot;'hen in use, screwed on a staff having a sharpnbsp;point at the bottom to stick in the ground; one ofnbsp;his kind is represented at Jig. 2, plate 14. Thenbsp;�nr sights screw off to make the instrument con-

^ Or five feet in length (for both the crosses) un-'^wsinto three parts to go into a portmantcau,amp;c. -the surveying cross is a very useful instrumentnbsp;Cf placing of Offsets, or even for measuring small

pieces of ground; its accuracy depends on ,the sights being exactly at right angles to each other.nbsp;It maybe proved by looking at one object throughnbsp;two of the sights, and observing at the same time,nbsp;without moving the instrument, another objeetnbsp;through the other two sights; then turning thenbsp;cross upon the staff, look, at the same objectsnbsp;through the opposite sights; if they are accuratelynbsp;in the direction of the sights, the instrument isnbsp;correct.¹

It is usual, in order to ascertain a crooked line by off sets, first to measure a base or station line innbsp;the longest direction of the piece of ground, andnbsp;while measuring, to find by the cross the placesnbsp;where perpendiculars would fall from the severalnbsp;corners and bends of the boundary; this is donenbsp;by trials, fixing the instrument so, that by one pairnbsp;of sights both ends of the line may be seen; andnbsp;by the other pair, the corresponding bend or corner; then measuring the length of the said perpendicular. To be more particular, let A, h, i, k, 1, in,nbsp;jig. 35, fldte 0, be a crooked hedge or river; measure a strait line, as A B, along the side of thenbsp;foregoing line, and while measuring, observe whennbsp;you are opposite to any bend or corner of thenbsp;hedge, as at c, d, e; from thence measure the per-pendieular offsets, as at c h, d i, amp;c. with- thenbsp;offset staff, if they are not too long; if so, with thenbsp;chain. The situation of the offsets are readilynbsp;found, as above directed, by the cross, or King^

I 'have made some additions to the box cross staff, which have been found useful and convenient for the pocket, wherenbsp;great accuracy is not required. See fig. (i. A compass and needle at the top A to give the bearings, aud a moveable graduatetfnbsp;base at B, by rack-work and pinion C, to give an angle to 5' ofnbsp;a degree by the nonius divided on the box above. Thus the surveyor may have a small theodolite, circumferentor, and cros�nbsp;staff all in one instrument. Edit.

Of surveying with the chain and cross. What has been denominated by many writers, surveyingnbsp;hy the chain only, is in fact surveying by the crossnbsp;^nd chain; for it is necessary to use the cross, ornbsp;Optical square, for determining their perpendicular lines, so that all that has been said, even bynbsp;these men, in favour of the chain alone, is foundednbsp;I'l fallacy. To survey the triangular field ABC,nbsp;22, plate Q, by the chain and cross: 1. Set upnbsp;��'aarks at the corners of the field. 2. Beginning,nbsp;Suppose at A, measure on in a right line till younbsp;ure arrived near the point D, where a perpendicular will fall from the angle, let the chain lie in thenbsp;direction or line AB. 3. Fix the cross over AB,nbsp;so as to see through one pair of sights the mark atnbsp;A or B, and through the other, the mark at C; ifnbsp;it does not coincide at C with the mark, the crossnbsp;Uiust be moved backw'ards or forwards, till bynbsp;trials one pair of tlie sights exactly coincide withnbsp;the mark at C, and the other with A or B. 4. Observe how many chains and links the point D isnbsp;A, suppose 3.0. 3. which must be entered innbsp;the field-book. 5. Measure the perpendicular D C,nbsp;^43. 7. finish the measure of the base line, and thenbsp;'^'ork is done. This mode is used at present by manynbsp;^U7-veyors, probably because there Is no .check wherebynbsp;discover their errors, which must be very great, ifnbsp;survey is of any extent.

. To plot this, make AB equal 11.41. AD equal 3.0. 3. on the point D erect the perpendicularnbsp;l^C, and make it equal 6.43. then draw AC,nbsp;quot; C, and the triangle is formed.

This instrument has the two principal glasses of Hadley ?, quadrant, was contrived by iny father; itnbsp;is in most, if not in all respects, superior to thenbsp;common surveying cross, because it requires nonbsp;staff', may be used in the hand, and is of course ofnbsp;great use to a military officer. It consists of twonbsp;plain mirrors, so disposed, that an object seen bynbsp;reflexion from both, will appear to coincide withnbsp;another object seen by direct �vision whenever thenbsp;two objects subtend a ri^ht angle from the centernbsp;of the instrument, and serves therefore to raise ornbsp;let fall perpendiculars on the ground, as a squarenbsp;does on paper, of which we shall give some examples. Its application to the purposes of surveyingnbsp;will be evident from these, and what has been already said concerning the cross.

Fig. 4, plate 14, is a representation of the Instrument without its cover, in order to render the construction more evident. There is a cover with a slit or sight for viewing the objects; the objectnbsp;seen directly, always coincides with the object seennbsp;by reflexion, when they are at right angles to eachnbsp;other¹.

From a given point in a given line, to raise a perpendicular. 1. The observer is to stand withnbsp;this instrument over, the given point, causing anbsp;person to stand with a mark, or fixing one at somenbsp;convenient distance on the given line, 2. An assistant must be placed at a convenient distance,nbsp;with a mark somewhere near the line in which itnbsp;is supposed the perpendicular will fall; then if onnbsp;looking at one of the objects, the other be seen in

See the description of a considerable improvement upon it, after the description of the Hadley�s sextant,nbsp;nbsp;nbsp;nbsp;19. eiut.

o line with it, the place where the markof the assistant is fixed is the required point.

Prom a given point over a given line to let fall ^ perpendicular. Every strait line is limited andnbsp;determined by two points, through which it is supposed to pass; so in a field the line is detenninednbsp;% two fixed objects, as steeples, trees, marksnbsp;A'reeted for the purpose, amp;c. For the present ope-�'ation, the two objects that determine must be onnbsp;'^ne side the point where the perpendicular falls;

in other words, the observer must not be between the objects, he must place himself over the dne, in which he will always be when the two objects coincide; he must move himself backwardsnbsp;On this line, till the mark, from whence the perpendicular is to be let fall, seen by direct vision,nbsp;coincides with one of the objects which determinenbsp;the given line seen by reflexion, and the instrument will be over the required point.

Required the distance from the steeple A, fig. p'O, plate 9, to B. Let the observer stand with hisnbsp;instrument at B, and direct an assistant to movenbsp;^bout C with a staff as a mark, until he secs itnbsp;coincide by direct vision with the object at A ; letnbsp;^dn fix the staff there; then let the observer walknbsp;nlongthe line until A and B coincide in the instru-*^^ont, and BD will be perpendicular to AC;nbsp;measure the three lines B C, D C, B D, and thennbsp;following proportion will give the required dis-hmee, for as D C is to D B, so is B C to AB.

Second method. I. Make ^0,, fig. 21, plate p. Perpendicular to BA. 2. Divide BC into fournbsp;�^final parts. 3. Make C D perpendicular to CB.nbsp;4. BringFEA into one line, and the distancenbsp;�orn C to F will be j of the distance from B to A.

This instrument consists of a compass box, a magnetic needle, and two plain sights, perpendicular to the meridian line in the box, by whichnbsp;the bearings of objects are taken from one stationnbsp;to another. It is not much used in England wherenbsp;land is valuable; but in America where land is notnbsp;so dear, and where it is necessary to survey largenbsp;tracts of ground, overstocked with wood, in anbsp;little time, and where the surveyor must take anbsp;multitude of angles, in which the sight of the twonbsp;lines forming the angle may be hindered by underwood, the circumferentor is chiefly used.

The circumferentor, see Jig. 1, plate 15, consists of a brass arm, about 14 or 15 inches long, with sights at each end, and in the middle thereofnbsp;a circular box, with a glass cover, of about 5lnbsp;inches diameter; within the box is a brass graduated circle, the upper surface divided into 300nbsp;degrees, and numbered 10. 20. 30. to 300; everynbsp;tenth degree is cut down on the inner edge ofnbsp;the circle. The bottom of the box is divided into four parts or quadrants, each of which is subdivided into 00 degrees numbered from the meridian, or north and south points, each way to thenbsp;east and west points; in the middle of the box isnbsp;placed a steel pin finely pointed, called the centernbsp;pin, on which is placed a magnetic needle, thenbsp;quality of which is such, that, allowing for thenbsp;difference between the astronomic and magneticnbsp;meridians, however the instrument may be movednbsp;about, the bearing or angle, which any line makesnbsp;with the magnetic meridian, is at once shewn bynbsp;the needle.

At each end of the brass rule, and perpendicular thereto, sights are fixed; in each sight there is

a large and siiiall aperture, or slit, one over the other, these are alternate; that is, if the aperturenbsp;uppermost in one sight, it will be lowest innbsp;hhe other, and so of the small ones; a fine piecenbsp;of sewing silk, or a horse hair runs through thenbsp;^iiiddle of the large slit. Under the compass boxnbsp;a socket to fit on the pin of the staff; the instrument ma}' be turned round on this pin, ornbsp;^*xcd in any situation by the milled screw; it maynbsp;�ilso be readily fixed in an horizontal direction bynbsp;ball and socket of the staff, moving for thisnbsp;P^rrpose the box, till the ends of the needle are.nbsp;^qviidistant from the bottom, and traverse or playnbsp;'^hth freedom.

. Occasional %'ariatiotis in the construction of this ^'istrument, are, 1. In the sights, which are some-^^nies made to turn down upon an hinge, innbsp;�rder to lessen the bulk of the instrument, andnbsp;* Crider it more convenient for carriage; sometimesnbsp;^l^ey arc made to slide on and off with a dovetail;nbsp;Sometimes to fit on with a screw and two steadynbsp;pins. lt;2. In the box, which in some instrumentsnbsp;a brass cover, and very often a spring is placednbsp;''quot;ifhin the box to throw the needle off the cen-pin, and press the cap close against the glass,nbsp;preserve the point of the center pin from beingnbsp;�iunted by the continual friction of the cap of thenbsp;needle. 3. In the needle itself, which is made ofnbsp;different forms. 4. A further variation, and fornbsp;^�nc best, will be noticed under the account of thenbsp;'unproved circumferentor.

The surveying compass represented at fg. 3, I�r15, is a species of circumferentor, which hasnbsp;. itherto been only applied to military purposes;

eonyig^g of a square box, within which there is ^^nrass circle divided into 300 degrees; in tlienbsp;^ntcr of the box ii a pin to carry a. magnetic

needle; a telescope is fixed to one side of tbs box, in such a manner as to be parallel to thenbsp;north and south line; the telescope has a verticalnbsp;motion for viewing objects in an inclined plain ;nbsp;at the bottom of the box is a socket to receive anbsp;stick or staft for supporting the instrument.

To tise the circumferentor. Let ABC, Ip, plate Q, be the angle to be measured. 1. The instrument being fixed on the staff, place its centernbsp;over the point B. �1. Set it horizontal, by movingnbsp;the bail in its socket till the needle is parallel tonbsp;the bottom of the compass box. 3. Turn that endnbsp;of the compass box, on which the N. or fleur denbsp;lis is engraved, next the eye. 4. Look along BA,nbsp;and observe at what degree the needle stands, suppose 30.nbsp;nbsp;nbsp;nbsp;5. Turn the instrument round upon the

pin of the ball and socket, till you can see the object C, and suppose the needle now to stand atnbsp;125.nbsp;nbsp;nbsp;nbsp;6. Take the former number of obseiwed de

If, in two observations to find the measure of an angle, the needle points in the first on one sidenbsp;360�, and in the second on the other, add whatnbsp;one wants of 360quot;, to what the other is past 360�,nbsp;and the sum is the required angle.

This general idea of the use of the circumferentor, it is presumed, will be sufficient for the present; it will be more partieularly treated of hereafter.

When in the use of the circumferentor, yon look through the upper sights, from the ending ofnbsp;the station to the beginning, it is called a backnbsp;sight-, but when you look through the lower slitnbsp;from the beginning of the station towards the end,nbsp;it is termed the fore sight. A theodolite, or an/nbsp;instrument which is not set by the needle, mnst

fixed in its place, by taking back and fore sights every station, for it is by the foregoing stationnbsp;that it is set parallel; but as the needle pi'cservesnbsp;^ts parallelism throughout the whole suiwey, whosoever works by the circumferentor, need take nonbsp;^ore than one sight at every station.

There IS, indeed, a difference betvvccn the mag� and astronomic or true meridian, Avhich isnbsp;'�Slied the �variation of the 7ieedle. This variation isnbsp;different at ditl'ercnt places, and is also different atnbsp;^merent times; this difference in the variation isnbsp;trailed the variation of the variation; 1gt;ut the in-'-^ease and decrease thereof, botli with respect tonbsp;^^me and place, proceeds by such very small increments or decrements, as to be altogether insig-^rificant and insensible, within the small limits ofnbsp;^*1 ordinary survey, and the short time requirednbsp;the performance thereof.*

The excellency and defects of the preceding in-^^ruiiient both originate in the needle; from the l^cgular direction thereof, arise all its advantages;

Unsteadiness of the needle, the difficulty of ascertaining with exactness the point at which it set-. cs, are some of its principal defects. In this ^^proved construction these are obviated, as will

t}jg present variation, or, more properly, the decimation of illnbsp;nbsp;nbsp;nbsp;is near 22� W. of the north at London; or two jioints

rnay be allowed on an instrument to the E. to fix the mcluiatian, or dip, was about �J'tP of thenbsp;ag W below the horizon in the yearnbsp;nbsp;nbsp;nbsp;The inclination,

fepjju� to develope any law by which its jiosition can be de-�^d for any future time. Edit.

be evident from the following description. One o( these instruments is represented at Jig.2, plate 15.

A pin of about three quarters of an inch diame-ter goes through the middle of the box, and forms as it were a vertical axis, on which the instrumentnbsp;may be turned round horizontally; on this axisnbsp;an index AB is fastened, moving in the inside otnbsp;the box, having a nonius on the outer end to cut,nbsp;Rnd subdivide the degrees on the graduated circle.nbsp;By the help of this' index, angles may be takennbsp;with much greater accuracy than by the needlenbsp;alone; and, as an angle may be ascertained by thenbsp;index with or without the needle, it of course removes the difficulties, which would otherwise arise,nbsp;if the needle should at any time happen to be actednbsp;upon, or drawn out of its ordinary position by extraneous matter; there is a pin beneath, wherebynbsp;the index may be fastened temporarily to the bottom of the box, and a screw, as usual, to fix thenbsp;whole occasionally to the pin of the ball and socket,nbsp;so that the body of the instrument, and the index,nbsp;may be either turned round together, or the onenbsp;turned round, and the other remain fixed, as occasion shall require. A further improvement isnbsp;that of preventing all horizontal motion of the hall igt;tnbsp;the socket-, the ball has a motion in the socket everynbsp;possible way, and every one of these possible motions is necessary, except the horizontal one, whichnbsp;is here totally destroyed, and every other possibl^'nbsp;motion left perfectly free.*

* The instrument is made to turn into a vertical position, and by the addition of a spirit level to take altitudes and depressions-The index AB has been found to interfere too much with th�nbsp;free play of the needle. In the year 1701 I contrived an externanbsp;nonius piece a. Jig. 6, to move against and round the graduatenbsp;circle b, either with or without rack-work or pinion. The c fnbsp;cle and compass plate are fixed, and the nonius piece and outsio�nbsp;rim and sight carried round together when in use. This h^snbsp;he�n generally approved of. Edit.

this purpose, let1, jgt;lau 18, represent field to be surve^�cd. 1. Set up the circuinfcren-^Gr at any corner, as at A, and therewith take thenbsp;bourse or bearing, or the angle that such a linenbsp;^Gakes with the magnetic meridian shewn by thenbsp;^'eeclle, of the side AB, and measure the lengthnbsp;thereof with the chain.

If the circumferentor be a common one, having Go index in the box, the course or bearing is takennbsp;A� simply turning the sights in a direct line fromnbsp;^ to B, and when the needle settles, it will pointnbsp;Gwt on the graduated limb the course or numbernbsp;of degrees which the line bears from the magneticnbsp;Gteridian.

But If the circumferentor has an index in the Gox, it is thus used. 1. Bring the index to the northnbsp;point on the graduated limb, ^nd fix it there, bynbsp;fiistening the body of the instrument and the undernbsp;P^tt together by the pin for that purpose, andnbsp;film the instrument about so that the needle shallnbsp;^Gttle at the same point; then fasten the undernbsp;pint of the instrument to the ball and socket, andnbsp;biking the pin out, turn the sights in a direct linenbsp;loin A towards B, so will the coarse and bearingnbsp;G pointec], out on the graduated circle, both by thenbsp;Gcedlc and by the index. I�liis done, fasten thenbsp;Gdy of the instrument to the under part aa;ain,nbsp;. iG having set the instrument up at B, turn thenbsp;�'.?fits in a direct line back from B to A, and therenbsp;�isten the under part of the instrument to the ballnbsp;ggu Socket; then take out the pin which fastens thenbsp;,1 Gy of the instrument to the under part, and turnnbsp;10 Sights ill a direct line towards C, and proceed

in the same manner all round the survey; so will the courses or bearings of the several lines benbsp;pointed out both by the needle and the index, unless the needle should happen to be drawn out ofnbsp;itseourse by extraneous matter; but, in this case,nbsp;the index will not only shew the course or bearing,nbsp;but will likewise shew how much the needle is sonbsp;drawn aside. After this long digression to explainnbsp;more minutely the use of the instruments, we maynbsp;proceed. 2. Set the circumferentor up at B, takenbsp;the course and bearing of B C, and measure thenbsp;length thereof, and so proceed with the sides C D,nbsp;D E, E F, �' G, GA, all the way round to thenbsp;place of beginning, noting the several courses ornbsp;bearings, and the lengths of the several sides in anbsp;field-book, which let us suppose to be as the following :

dV. B. By north 7� west, is meant seven degi'ccs to the wcstwxird, or left hand, of the nortli, aSnbsp;shewn by the needle; by north 5 5� 15' east, nfty-five degrees fifteen minutes to the eastward, or rightnbsp;hand of the north, as shewn by the needle.

In like manner by south 02� 30' east, is meant gixty-two degrees and thirty minutes to the eastward, or left hand of the south; and by southnbsp;40� west, forty degrees to the westward, or rightnbsp;hand of the south.

The 2] chains, 18 chains 20 links, amp;c. arc the lengths or distances of the respective sides, as niea-snred by the chain.

Fig. A, plate 15, represents a small circumferentor, or theodolite; it is a kind that was much used lgt;y General Roy, for delineating the smaller parts ofnbsp;'I survey. The diameter is 4 inches. It is betternbsp;to have the sight pieces double, as shewn in Jig. 2.

The error to which an instrument is liable, quot;�here the whole dependance is placed on the nee-'lle, soon rendered some other invention necessarynbsp;to measure angles with accuracy; among these, thenbsp;oommon theodolite, with four plain sights, took thenbsp;lead, being simple in construction, and easy in use.

The common theodolite is represented Jig. 3, plate 14; it consists of a brass graduated circle, anbsp;loveable index AB; on the top of the index is anbsp;pompass with a magnetic needle, the compass box

Covered with a glass, two sights, C, D, are fixed ^o the index, one at each end, perpendicular to thenbsp;- plain of the instrument. There arc two more sights

P, which are fitted to the graduated circle at the Points of and 180�; they all take on and offnbsp;�Or the convcniency of packing. In each sightnbsp;^here is, as in the circumferentor, a large and anbsp;^Hiall aperture placed alternately, the large aper-^Ore in one sight being always opposed to the nar-^Ow aperture in the other; underneath the brassnbsp;'^'*'ole, and in the center thereof, is a sprang to fitnbsp;the pin of the ball and socket, which fixes on anbsp;staff.

A he circle is divided into degrees, which are 1 Numbered one way to 300�, usually from thenbsp;0 t to the right, supposing yourself at the center

of the instrument; on the end of the index is A nonius division, by which the degrees on the liinbnbsp;arc subdivided to five minutes; the divisions oHnbsp;the ring of the compass box are numbered innbsp;contrary direction to those of the limb.

As much of geometrical mensuration depends on the accuracy of the instrument, it behoves everynbsp;surveyor to examine them carefully; different methods will be pointed out in this work, accordingnbsp;to the nature of the respective instruments. In thatnbsp;under consideration, the index should move regularly when in use; the theodolite should always benbsp;placed truly horizontal, otherwise the angles measured by it will not be true; of this position younbsp;may judge with sufficient accuracy by the needle,nbsp;for if this be originally well balanced, it '^�ill not benbsp;parallel to the compass plate, unless the instrumentnbsp;be horizontal; two bubbles, or spirit levels, arenbsp;sometimes placed in a compass box at right anglesnbsp;to each other, in order to level the instrument, butnbsp;it appears to me much better to depend on thenbsp;needle; 1. Because the bubbles, from their size,nbsp;arc seldom accurate. 2. Because the operatoiquot;nbsp;cannot readily adjust them, or ascertain when theynbsp;indicate a true Idvel.

To examine the instrument; on an extensive plain set three marks to form a triangle; with yournbsp;iJicodolite take the three angles of this triangle,nbsp;and if these, when added together, make 180�, younbsp;may be certain of the justness of your instrument.

To examin^.thc needle; observe accurately where the ncpdle settles, and then remove it froiunbsp;that situation, by placing a piece of steel or magnet near it; if it afterwards settles at the same-point, it is so far right, and you may judge it to benbsp;perfectly so, if it settles properly in all situations ofnbsp;the box. If in any situation of the box a deviation

objects to w aceuracyr is rc(}uired, the angles should always benbsp;ak.en twice over, oftner where great accuracy isnbsp;jOaterial, and the mean of the observation mustnbsp;^0 taken for the true angle.

. ^ 0 measure an angle with the theodolite. Let j. B C, Jig. ip, plate Q, represent two stationnbsp;place the theodolite over the angular point,nbsp;�Od direct the fixed sights along one of the lines,nbsp;sp )'ou see through the sights the mark A; at thisnbsp;i�e\v the instrument fast; then turn the moveablenbsp;till through its sights you see the othernbsp;arlv C; then the degrees cut by the index uponnbsp;, graduated limb, or i-ing of the instrument,nbsp;the quantity of the angle.

1 �^ be fixed sights are always to be directed to the j^'^^i^ntion, and those on the index to the next,nbsp;su � beginning of the degrees are towards thenbsp;^^.'�eyor, when the fixed sights are directed to annbsp;and the figured or N. point towards him innbsp;Acting the index, then that end of the index to-

Marcis the survT�vor will point, out the angle, and llic south end of the needle the bearing; the ap-plieation of the instrument to various cases thatnbsp;may occur in surveying, will be evident from whatnbsp;we shall say on that subject in the course of thi;^nbsp;work.

The tabular part of this instrument is usually made of two well-scaisoncd boards, forming a pa-rallelogram of about J 5 inches long, and 12 inchesnbsp;broad; the size is occasionally varied to suit thenbsp;intentions of the operator.

The aforesaid parallelogram is framed with a ledge on each side to supj)ort a box frame, ivhichnbsp;frame coniines the ])apc,r on the table, and keepsnbsp;it close thereto; the frame is therefore so contrived,nbsp;that it may be taken off and put on at pleasure,nbsp;either side upwai'ds. Each side of the frame isnbsp;graduated; one side is usually divided into scalesnbsp;ot' ecjual parts, for drawing lines parallel or per-pendieidar to the edges of the table, and also fotnbsp;more convenient]v sliifting the paper; the othetnbsp;t�aee, or side of the frame, is divided into 3()0�, fromnbsp;a brass center in the middle of the table, in ordernbsp;that angles may lie measured asivith a theodolite;nbsp;on the same face of the frame, and on two ot thenbsp;edges, are graduated 180�; the center of thesenbsp;degrees is exactly in the middle between the tv�Onbsp;emls, and about dth part of the breadth from onenbsp;of the sides.

A magnetic needle and compass box, covered with a glass and spring ring, slides in a dovetailnbsp;the under side of the table, and is fixed there by �nbsp;fmger screw, it serves to point out the dircctitgt;i�gt;-rmdhe a check upon the sights.

There is tIso a brass index somewhat larger than the diagonal of the table, at each end of which anbsp;sight is fixed; the vertical hair, and the middle ofnbsp;the edge of the index, are in the same plain; thisnbsp;edge is chamfered, and is usually called the fiducialnbsp;edge of the index. Scales of different parts in annbsp;inch arc usually laid down on one side of the index.

Under the tabic is a sprang to fit on the pin of the ball and socket, by which it is placed upon anbsp;three-legged staff.

To place the paper on the table. Take a sheet of paper that will cover it, and wet it to make it expand, then spread it flat upon the table, pressingnbsp;down the frame upon the edges to stretch it, andnbsp;heep it in a fixed situation; when the paper is drynbsp;It will by contracting become smooth and flat.

To shift the paper on the pgt;lam table. When the paper on the table is full, and there is occasion fornbsp;Hiore, draw a line in any manner through the farthest point of the last station line, to which thenbsp;'Vork can be conveniently laid down; then takanbsp;''ffthe sheet of paper, and fix another on the table;nbsp;draw a line upon it in a part most convenient fornbsp;fhe rest of the work; then fold, or cut the oldnbsp;^hect of paper by the line drawn on it, apply thenbsp;pdge to the line on the new sheet, and, as they lienbsp;that position, continue the last station line uponnbsp;new paper, placing upon it the rest of thenbsp;Measures, beginning where the old sheet left off,nbsp;'^nd so on from sheet to sheet.

To fasten all the sheets of paper together, and fhus form one rough plan, join the aforesaid linesnbsp;^vcaratelyr together, in the same manner as whennbsp;fhe lines were transferred from the old sheets tonbsp;new ouc. But if the joining lines upon thenbsp;^|d and new sheets have not the same inclination tonbsp;side of the table, the needle will not point tonbsp;a 2

the original degree when the table is I'ectificd. It the needle therefore should respect the same degree of the compass the easiest way of drawingnbsp;the line in the same position is to draw them bothnbsp;parallel to the same sides of the table, by means ofnbsp;the scales of equal parts on the two sides.

7o use the plain table. Fix it at a convenient part of tlie ground, and make a point on the papernbsp;to rejjrcsent that part of the ground.

Run a fine steel pin or needle through this ]X)int into the table, against which you must apply the liducial edge of the index, moving it roundnbsp;till you perceive some remarkable object, or marknbsp;set up for that purpose. Then draAV a line fromnbsp;the station point, along the fiducial edge of thenbsp;index.

Now set the sights to another mark, or object, and draw that station line, and so proceed till younbsp;have obtained as many angular lines as are ncccs-sarv from this station.

The next requisite, is the measure or distance from the station to as many objects as may be necessary by tlie chain, taking at the same time thenbsp;offsets to the required corners or crooked parts ofnbsp;the hedges; setting off all the measures upon theii'nbsp;rcsjicetive lines up'on the table.

Now remove tlic table to some other station, whose distance from the foregoing was previouslynbsp;measured; then lay down the objects which appear from tlience, and continue these operationsnbsp;till your work is finished, measuring such lines asnbsp;arc necessary, and determining as many as younbsp;can by intersecting lines of direction, drawn froiUnbsp;different stations.

It seems to be the universal opinion of the best surveyors, that the plain table is not an instrumentnbsp;to be trusted to iu large surveys, or on hilly situa-

s-nd labour, but the acknowledgement of an error; which they are not sure they can amend.

In uneven ground, where the table cannot in all �fations be set horizontal, or uniformly in anyonenbsp;place, it is impossible the work should be true innbsp;all parts.

Tlie contraction and expansion of the paper according to the state of moisture in the air, is a Source of many errors in plotting; for betweennbsp;^ dewy morning and the heat of the sun at noon,nbsp;there is a great difference, which may in some dc-gree be allowed for in small work, but cannot benbsp;remedied in surveys of considerable, extent.

To remedy some of the inconveniences, and correct some of the errors to which the commonnbsp;plain table is liable, that which we are now goingnbsp;f-o describe has been constructed. It is usuallynbsp;Called Beightons plain table, though differing innbsp;Wiany respects from that described by him in thenbsp;^^Inlosophical Transactions.

It is a plain board, if) inches square, with a frame of box or brass round the edge, for the purpose of being graduated. On the sides, AB, C D,nbsp;^re two grooves and holdfasts for confining firmly,

easily removing the paper; they are disengaged ~y turning the screws under the table from thenbsp;^'ght towards the left, and drawn down and made

to press on the paper by turning the screws the contrary way. When the holdfasts are screwednbsp;down, their surface is even with that of the table.nbsp;There are two pincers under the table, to holdnbsp;that part of the paper, which in some cases liesnbsp;beyond the table, and prevent its flapping aboutnbsp;with the wind.

The compass box is made to fit either side of the table, and is fixed by two screw's, and doesnbsp;not, when fixed, project above one inch and annbsp;half from the side of the table.

There is an index wdth a semicircle, and telescopic sight, EFG; it is sometimes so constructed, as to answ'er the purpose of a parallel rule. The figure renders the whole so evident,nbsp;that a greater detail w'ould be superfluous.

The papers, or charts for this table, are to bs either of fine thin pasteboard, fine paper pasted onnbsp;cartridge paper, or two papers pasted together,nbsp;cut as square as possible, and of such a length thatnbsp;they may slide in easily, betw'een the uprightnbsp;parts, and under the flat part of the holders.

Any one of these charts may be put into the table at any of the four sides, be fixed, taken out, andnbsp;changed at pleasure; any two of them maybe joinednbsp;together on the table, by making each of themnbsp;meet exactly at the middle, whilst near one half otnbsp;each will hang over the sides of the table; or, bynbsp;doubling them both ways through the middle, fournbsp;of them may be put on at one time meeting in thenbsp;center of the table. For this purpose, each chartnbsp;is always to be crossed quite through the middle;nbsp;by these means the great trouble and inaccuracynbsp;in shifting the papers is removed.

The charts thus used, are readily laid together by corresponding numbers on their edges, andnbsp;thus make up the whole map in one view'; and,

�ieing in squares, are portable, easily copied, cn~ ^�'irged, or contracted.

The line of sight in viewing objects may, if that method be preferred, be always over the center ofnbsp;^lie table, and the station lines drawn parallel tonbsp;those measured on the land. Underneath the ta~nbsp;^le is a sprang to tit on the socket of a staff, withnbsp;parallel plates and adjusting screws.

Mr. Searle contrived a plain tabic, wdiose size (which renders it convenient, w'hile it multipliesnbsp;^very error) is only five inches square, and consists of two parts, the table and the frame; thenbsp;^mtiie, as usual, to tighten the paper observednbsp;^ipon. In the center of the table is a serew, onnbsp;which the index sight turns; this screw is tightenednbsp;^fter taking an observation.

In proportion as science advances, we find our-''^clvcs standing upon higher ground, and arc enabled to see further, and distinguish objects better lhan those that went before us; thus the greatnbsp;advances in dividing of instruments have renderednbsp;observers more accurate, and more attentive to thenbsp;^�ecessary adjustments of their instruments. Instruments are not now considered as perfect, unlessnbsp;they arc so constructed, that the, person who usesnbsp;them may either correct, or allow for the errors tonbsp;which they are liable.

Theodolites wdth telescopic sights arc, without �^ioubt, the most accurate, commodious, and uni-�'^t^rsal instruments for the purposes of surveying,nbsp;^i^d have been recommended as such by the mostnbsp;'^-''Pert practitioners and best writers on the sub-Gardiner, [lammond., Cniiii, Stone, WxJd,nbsp;�r^,u,�g,on, amp;c.

The leading requisites in a good theodolite are, 1. That the parts be firmly connected, so thatnbsp;they may always preserve the same figure. 2. Thenbsp;circles must be truly centered and accurately graduated. 3. The extremity of the line of sightnbsp;should describe a true circle.

Fig 1, plate l6, represents a theodolite of the second best kind; the principal parts are, 1. A telescope and its level, C, C, D. 2. The verticalnbsp;arc, BB. 3. The horizontal limb and compass,nbsp;AA. 4. The staff with its parallel plates, E.nbsp;The limb A A is generally made about sevennbsp;inches in diameter.

An attentive view of the instrument, or drawing, compared with what has been said before, will shew that its perfect adjustment consists innbsp;the following particulars.

1. The horizontal circle A A must be truly level. 2. The plain of the vertical circle B B must be trulynbsp;perpendicular to the horizon. 3. The line ofnbsp;sight, or line of collimation,¹ must be exactly innbsp;the center of the circles on which the telescopenbsp;turns. 4. The level must be parallel to the line ,nbsp;of collimation.

Of the telescope CC. Telescopic sights not only enable the operator to distinguish objects better,nbsp;but direct the sight with much greater accuracynbsp;than is attainable with plain sights; hence also wenbsp;can make use of much finer subdivisions. Thenbsp;telescope, generally applied to the best instruments, is of the achromatic kind, in order to obtain a larger field, and greater degree of magnifying power. In the focus of the eye-glass are two

The line of collimation is the line of vision, cut by the intersecting point of the cross hairs in the telescope, answering to the visual line, by which we directly point at objects withnbsp;plain sights.

fine hairs, or wires, at right angles to each other, whose intorscctiou is in tlic plain of the ver-tiea] arc. The object glass may be moved tonbsp;diftcrent distances from the eye glasses, by turn-the milled nut a, and may by this means benbsp;�'^ccoinmodatcd to the eye of the observer, and thenbsp;^fistance of the object. The screws for movingnbsp;^'^d adjusting the cross hairs are sunk a littlenbsp;''�dhin the eye tube, and at about one inch fromnbsp;the eye end: there are four of these screws, twonbsp;which are exactly opposite to each other, andnbsp;right angles to the other two. By casing onenbsp;the screws, and tightening the opposite one,nbsp;the wire connected with it may be moved in opposite directions. On the outside of the telescopenbsp;''^re two metal rings, which are, ground perfectlynbsp;these rings ai-e to lay on the supporters c, c,nbsp;^^lled Y�s, which arc fixed to the vertical arc.

Of the vertical arc B B. This arc is firmly fixed ^0 a long axis which is at right angles to the plainnbsp;of the arc. This axis is sustained by, and movea-�^0 on the two supporters, which are fixed firmlynbsp;file horizontal plate: on the upper part of thenbsp;Cortical arc arc the two Y�s for holding the teles-^'ope; the inner sides of these Y�s arc so framed,nbsp;to be tangents to the cylindric rings of the tc-wscope, and therefore bear only on one part.

lelcscopc is confined to the Y�s by two loops, 'fiiieh turn on a joint, and ma}' therefore be rea-_ fiy fjpened and turned back, when the two pinsnbsp;taken out.

One side of the vertical arc is graduated to ^very half degree, which are subdiviclcd to everynbsp;Minute of a degree bv the nonius. It is numberednbsp;*^^^h Way from O to 90�, towards tlic eye end, fornbsp;fugles of altitude; from O to b(f, towards the ob-

side of the vertical arc are two ranges of division?, the lowermost for taking the upright height otnbsp;timber in 100th parts of the distance the instrument is placed at from tlie tree at the time of observation. The uppermost cirele is for reducingnbsp;hypothenusal lines to horizontal, or to shew thenbsp;difference between the hypotheuase and base ot 3nbsp;right-angled triangle, always supposing the hypo-tbenuse to consist of 100 equal parts; consequently, it gives by inspection the number of link?nbsp;to be deducted from each chain�s length, in measuring up or down any ascent or descent, in ordernbsp;to reduce it to a true horizontal distance, similarnbsp;to those on King?, quadrant, p. 205.

The vertical arc is cut with teeth, or a rack, and may be moved regularly and with ease, bynbsp;turning the milled nut b; there is sometimesnbsp;placed about the nonius a steady pin, by which itnbsp;may be fixed when at the o, or zero point of thenbsp;divisions.

Of the compass. The compass is fixed to the upper horizontal plate; the ring of the compass i?nbsp;divided into 300�, which are numbered in a direction contraiy to those on the horizontal bmb-The bottom of the box is divided into four parts,nbsp;or quadrants, each of which is subdivided to everynbsp;10 degrees, numbered from the meridian, or northnbsp;and south points each way to the east and westnbsp;points. In the middle of the box is a steelnbsp;finely pointed, on which is placed the magnetic�'nbsp;needle; there is a wire trigger for throwing thenbsp;needle off its point when not in use.

Of the horizontal limb AA. This limb consists of two plates, one moveable on the other; thenbsp;outside edge of the upper plate is chamfered, tonbsp;serve as an index to the degrees on the lower.nbsp;The upper plate, together with the compass, vet'

*-ical arc, and telescope, arc easily turned round by ^ pinion fixed to the screw c; d is a nut for fixingnbsp;index to any part of the limb, and therebynbsp;flaking it so secure, that there is no danger of itsnbsp;keing moved out of its place, while the instrumentnbsp;removed from one station to another. The ho-|''zontal limb is divided to half degrees, and num-kered from the right hand towards the left, ] 0,nbsp;^*^3 30, amp;c. to 300; the divisions are subdivided bynbsp;kke nonius scale to every minute of a degree.

On the upper plate, opposite to the nonius, are few divisions similar to those on the vertical arc,nbsp;Saving the 100th parts for measuring the diameternbsp;�f trees, buildings, amp;c.

The whole instrument fits on the conical ferril ^f a strong brass headed stalF, with three substan-k^^l wooden legs; the top, or head of the staffsnbsp;Consists of two brass plates E, parallel to eachnbsp;^ther; four screws pass through the upper plate,nbsp;Hiid rest on the lower plate; by the action of thesenbsp;��^rews the situation of the plate may be varied, sonbsp;to set the horizontal limb truly level, or in anbsp;Pparallel to the horizon; for this purpose, anbsp;pin is fixed to the underside of the plate,nbsp;pin is connected with a ball that fits into anbsp;Rochet in the lower plate; the axis of the pin andnbsp;^'1 are so framed, as to be always perpendicular tonbsp;plate, and, consequently, to the horizontal

amp; depends on the accuracy of the instruments, y-bsolutely necessary that the surveyor shouldnbsp;- Very expert in their adjustments, without whichnbsp;, ^ cannot expect the instruments will properlynbsp;the purposes they were designed for, ornbsp;hiy surveys will have the requisite exactness.

The necessary adjustments to the theodolite� which we have just described, are, 1. That the linenbsp;of sight, or collimation, be exactly in the centernbsp;of the cylindric rings round the telescope, antinbsp;which lie in the Y�s. 2. That the level be paral'nbsp;lel to this line, or the axis of the above-mentionednbsp;rings. 3. The horizontal limb must be so set�nbsp;that when the vertical arc is at zero, and the np'nbsp;per part moved round, the bubble of the level ^vntnbsp;remain in the middle of the open space.

Previous to the adjustments, place the instrn' ment upon the staff, and set the legs thereof firmlynbsp;upon the ground, and at about three feet Iron*nbsp;each other, so that the telescope may be at a jumper height for the eye, and that two of the screwsnbsp;on the staff that are opposite to each other maynbsp;nearly in the direction of some conspicuous and

To adjust the line of coUhnatlon. Having set np the theodolite agreeable to the foregoing direction�nbsp;direct the telescope to some distant object, pl^'nbsp;cing it so that the horizontal hair, or wire, inn)'nbsp;exactly coincide with some well defined part of th^nbsp;object; turn the telescope, that is, so that thnnbsp;tube of the spirit level D may be uppermost, andnbsp;observe whether the horizontal hair still coincide�nbsp;with the object; if it does, the hair is in its righ^nbsp;position; if not, correct half the difference hfnbsp;moving the hair, or wire, which motion is effectednbsp;by easing one of the screws in the eye tube, aiWnbsp;tightening the other; then turn the telcseop�nbsp;round to its former position, with the tube of jh**nbsp;spirit level lowermost, and make the hair colnciii*^nbsp;with the object, by moving the vertical arc;nbsp;verse the telescope again, and if the hair does ndtnbsp;coincide with the same part of the object, you must

t'ppeat the foregoing operat ion, till in both posi-boris it perfectly coincides with the same part of object.

he precise situation oflhc horizontal hair being hiis ascertained, adjust the vertical hair in thenbsp;^�jnie manner, lading it for tins purpose in an ho-^lEontal position; the spirit tube will, during thenbsp;'quot;�hjustment of the vertical liair, be at right anglesnbsp;its former position. When the two wires arenbsp;^hus adjusted, their intersection will coincide cx-'�lt;itlynbsp;nbsp;nbsp;nbsp;same point of tlic object, while the

^^iescopc is turned quite round; and the hairs are '^ot properly adjusted, till this is elicctcd.

^djmhnenl of the level. To render the level faraliel to tlie line of eollimation, place the verti-arc over one pair of the staff screws, then raisenbsp;j'Ue. of ;he screws, and depress the other, till thenbsp;^'ibble of the level is stationary in the middle ofnbsp;he glass; now take the telescope out of the Y�s,nbsp;j�^^d turn it end for end, that is, let the eye endnbsp;.''�y ivhere the object end was placed; and if, whennbsp;this situation, the bubble remains in the middlenbsp;before, the levul is well adjusted; if it^does not,nbsp;hstciulto which the bubble runs is too high; thenbsp;position thereof must be corrected by turningnbsp;quot;'hh a screw-driver one or both of the screws whichnbsp;through the end of the tube, till the bubblenbsp;moved half the distance

grce, in order more convcnicntlv to make the bu^-* blc continue in its place when the vertical arcnbsp;at o, and the horizontal limb turn round.

quot;^ro adjust the level of the horizontal limh. Plac� the level so that it may be in a line with twonbsp;the staff screws, then adjust it, or cause the bubble to become stationary in the middle of thenbsp;open space by' means of these screws. Turn thenbsp;horizontal limb half round, and if the bubblenbsp;mains in the middle as before, the level is well ah'nbsp;jiiste^l; if not, correct half the error by the screW'Snbsp;at the end of the level, and the other half by thenbsp;staff screws. Now return the horizontal limb tonbsp;its former position, and if it remains in the middle?nbsp;the errors are corrected; if not, the process ofnbsp;tering must be pursued till the error is annihilated-Sec this adjustment in the description of Ramsdeti Snbsp;theodolite.

When the bubble is adjusted, the horizontal limb may always be levelled by means of the stai^nbsp;screws.

Among the improvements the instruments of science have received from Mr. Ramsden, and thenbsp;perfection wdth which he, has constructed them,nbsp;arc to rank those of the theodolite; in the presentnbsp;instance, he has happily combined elegance a^dnbsp;neatness ot form, with accuryicy of construction;nbsp;and the surveyor will contemplate with plcasutonbsp;this instrument, and the various methods by whie*�nbsp;the parts concur to give the most accurate result-The principal parts of this instrument arc hoquot;'-ever so similar to the foregoing, that a descriptio�^,nbsp;thereof must, in some degree, be a repetition o-*

F P represents the horizontal limb, of six, seven, or eight inches in diameter, but gene-rally of seven inches, so called, because when iunbsp;^^se, it ought always to be placed parallel to thenbsp;horizon. It consists, like the former, of two plates,nbsp;the edges of these two are chamfered, so that thenbsp;divisions and the nonius are in the same plain,nbsp;^vhich is oblique to the plain of the instrument.nbsp;Pile limb is divided into half degrees, and subdi-''idedbythe nonius to every minute; it is nuin-hered to 300� from the north towards the cast; besides these, the tangents to 100 of the radius arcnbsp;I'lid down thereon.

The upper plate is moved by turning the pinion G: on this plate are placed, at right anglesnbsp;to each other, two spirit levels for adjusting morenbsp;Accurately the horizontal limb.

NOP' is a solid piece fitted on the upper hori-^'Ontal plate, by means of three capstan headed Screws, passing through three similar screws. Bynbsp;the action of these, the vertical arc may be set perpendicular to the horizontal limb, or be made tonbsp;ttiove in a vertical plain. On this solid piece, arenbsp;fixed two stout supports, to carry the axis of thenbsp;t'crfical arc, which arc is moveable by the pinionnbsp;On the upper part of the vertical arc, arc thenbsp;and loops to support and confine the telescope; the Y�s arc tangents to the cylindric rings

the telescope, which rings are turned, and then ground as true as possible, and are prevented fromnbsp;inoving backwards or forwards, by means of twonbsp;shoulders. The telescope is achromatic, andnbsp;^�fiout twelve inches in length, and may be adjusted to the eye of the observer, or the distancenbsp;^fithe object, by turning the milled nut B. The

hairs are adjusted by tire screws in the eye tub� at A. Under tlie telescope is fixed a spirit levelnbsp;the distance of whose ends from the telescope maynbsp;be regulated by the screws c, c.

Bczicath tlie horizontal limb thei'e is a second or auxiliarv telescope, which has both an horizontal and vertical motion; it is moved horizontallynbsp;bv the milled screw H, and when directed to anynbsp;object, is fixed in its situation by another millednbsp;sci'cw; it moves vci'tically on the axis; there is annbsp;adjustment to this axis, to make the line of colli-mation move in a vertical plain. By the horizontal motion, this telescope is easily set to what Bnbsp;called the backset stations; the under telescopenbsp;keeping in view the back object, while the uppetnbsp;one is directed to the fore object. Underneathnbsp;the lower telescope is a clip to fasten occasionallynbsp;the main axis; this clip is tightened bv the fingci'nbsp;screw L, aiid when tightened, a small motion ofnbsp;the adjusting screw K will move the telescope anbsp;few degrees, in oi�der to set it witli great accuracy.nbsp;Beneath these is the staff, the nature of whichnbsp;will be sufilcicntly evident from what was saidnbsp;thereon in the description of the last theodolite,nbsp;or by inspection of the figure.

To adjust jhe levels of the horizontal plate. 1* Place the instrument on its staff, with the leg�nbsp;thereof at such a distance from each other, as willnbsp;give the instrument a firm footing oji the ground.nbsp;2. Set the nonius to 3�0, and move the instrument round, till one of the levels is cither innbsp;right line with two of the screws of the parallelnbsp;plates, or else parallel to such a line. 3. By meansnbsp;of the two last mentioned screws, cause the bubble in the level to become stationary in the middlenbsp;of the glass. 4. Turn the horizontal limb by thenbsp;milled nut half round, or till the nonius is at 1^0,

find if the bubble remains in the middle as before, the level is' adjusted; if it does not, correct thenbsp;position of the level, by turning one or both thenbsp;screws which pass through its ends, till the bubblenbsp;has moved half the distance it ought to come, tonbsp;j'cach the middle, and cause it to move the othernbsp;half by turning the screws of the parallel plates.nbsp;Return the horizontal limb to its former posi-and if the adjustments have been well made,nbsp;bubble will remain in the middle; if otherwise,nbsp;process of altering must be repeated till it bearsnbsp;this proof of accuracy. �. Now regulate thenbsp;screws of the staff head, so that the bubble remainnbsp;the middle while the limb is turned quitenbsp;^ound. 7. Adjust the other level by its own proper screws, to agree with that already adjusted.

To adjust the level under the telescope. 1. The horizontal plate being levelled, set the index ofnbsp;^hc nonius of the vertical arc to o, pull out thenbsp;pins, and open the loops which confine thenbsp;Rlescope. 1. Adjust the bubble by its own screws.nbsp;Reverse the level, so that its right hand endnbsp;now be placed to the left; if the bubble con-jnues to occupy the middle of the glass it is in itsnbsp;^jSht position; if not, correct one half of the cr-by the capstan screws under the plate, andnbsp;other half by the screws under the level.nbsp;Reverse the level, and correct, if there is anynbsp;^hcasion, continuing the operation till the errornbsp;^anishes, and the bubble stands in the middle innbsp;positions.

. adjust the Tine of collmation. 1. Direct the jescope, so that the horizontal wire may coincidenbsp;�oine well defined part of a remote object,nbsp;�t Urn the telescope so that the bubble may benbsp;Ppermost; if the wire does not coincide with thenbsp;pai�t of the object as before, correct half the

difference by moving the vertical circle, and the other half by moving the wire, which is eftcctednbsp;by the screws in the eye tube of the telescope;nbsp;and so on repeatedly, till the difference whollynbsp;dl-^appears. Lastly, adjust the vertical wire in thenbsp;same manner; when the two wires are properlynbsp;adjusted, their intersection will coincide exactlynbsp;with the same point of an object, while the tcleS'nbsp;cope is turned quite round.

To accommodate those who may not wish to go to the price �f the foregoing instruments, othersnbsp;have been made with less work, in order to be afforded at a lower price; one of these is representednbsp;�-tfg. 5, plate 15. It is clear from the figure, thatnbsp;the difference consists principally in the soliditynbsp;of the parts, and in there being no rack-work tonbsp;give motion to the vertical arc and horizontal limb-The mode of using it is the same with the other,nbsp;and the adjustment for the line of collimation, andnbsp;the -level under the telescope, is perfectly similarnbsp;to the same adjustments in the instruments alreadynbsp;described, a further description is therefore unnecessary.

A small one, about four inches in diameter, was invented by the late Mr. Benj. Martin,nbsp;which the telescope, about six inches in length,nbsp;with a level, has no vertical motion, but the horizontal motion is given by a pinion; and it maynbsp;be turned into a vertical position to take ang^�'�nbsp;of altitudes or depressions. The divisions by th�nbsp;nonius arc to five minutes.

It generally happens, that the observer has occasion to take the vertical and horizontal angles at Ihc same time, by portable as well as by larger theo-'^lolites; the following is,therefore,recommended asnbsp;themost complete and portable instrument hithertonbsp;i^ade, and is in truth almost the best theodolite innbsp;^niniaturc. Its construction renders it somewhatnbsp;P^ore expensive than those before described. Itnbsp;the one that the late author alluded to in a note,nbsp;P^gc 315 of the former edition of this book, butnbsp;not time to describe it.

-F/^.7, �gt;/.14,is a representation of theinstrument. '^quot;he graduated limb and index plate A, A, arenbsp;^bout four inches in diameter, and move by racknbsp;^'id pillion B; it reads off by mc.ans of the noniusnbsp;three minutes of a degree; If the observernbsp;'should not object to very fine divisions, it may benbsp;J-c two minutes of a degree. The achromatic te-^^scope C is about six inches in length, and con-^^'ns a small spirit bubble at C, partly sunk intonbsp;tube; it turns upon a long axis, and isnbsp;�^Oved very accurately by rack and pinion on thenbsp;at D. This arc is necessarily of a short length,nbsp;admits about 30 degrees motion on each sidenbsp;Oj for altitudes or depressions. The staffjnbsp;^hich from one piece opens into a tripod, is aboutnbsp;yp feet in length, and has the parallel plates ofnbsp;^^hustment at the top. A small screw from thesenbsp;�crews into a socket under the limb A, and by annbsp;^^tcrnal rim of metal, the horizontal motion onlynbsp;the theodolite is produced, when the plates arenbsp;P^Cperly set by the screws. The telescope restsnbsp;P*^!! a cradle, and by opening the two semicircles:

(1, (1-, it inav be reversed, in order to adjust the spirit level, or prove its truth to the axis ofnbsp;the telescope. I'hc adjustments of this littlenbsp;instrnment, being in all respects similar to,nbsp;and made, as in those just described, they will benbsp;evident to the reader, and quite unnecessary tonbsp;repeat here.

The instrument, exclusive of its staff, packs into a jiocket mahogany ease, of 6 inches innbsp;length, inches in breadth, and 3? inches innbsp;depth.

In surveys of very great extent and importance, or in great trigonometrical operations, a largernbsp;instrument is required, in order that the subdivisions may be greater in number, or the angle takennbsp;more accurately, to five, two, or even one secondnbsp;of a degree. Several plans have been suggested,nbsp;but I do not at present see any better principle tonbsp;adopt than that of the great one by Mr. Ramsdeiinbsp;hereafter to be described. A proportionate reduction of its size, as wmll as simplifying its machinery and movements, necessary only for thenbsp;grand purpose that it was applied to, w�ould accommodate the practitioner with as complete annbsp;instrument as he could desire. The diameter ofnbsp;the horizontal circle I w�ould recommend to benbsp;from about 15 to 20 inches, and the other partsnbsp;in proportion. The price, according to the workmanship, would be from about 00 to 120 guineas,nbsp;stand, eases, amp;c. included.

In the preceding impression of this wmrk, the ingenious author, now deceased, made the fi�on-tis})icce plate a representation of a new� theodolitenbsp;of his own contrivance, the adjustments of whichnbsp;he thought to be more perfect than those of anynbsp;other, and annexed the description and mode ofnbsp;its adj ustments to his preface. Future trials, how-

on his own part, as well as by the hands of others, gave me reason to conclude that it wasnbsp;ii�t answerable to the intended improvements;nbsp;^^though more complicated and costly than thatnbsp;Represented on Jig. 2, plate l6, yet it was less sus-peptible of accuracy, and not so simple and easynbsp;the adjustments; I have, therefore, thoughtnbsp;proper to dispense with it here, and substitute anbsp;^hort description of the largest, most aceurate andnbsp;elegant theodolite ever made.

It is hardly necessary to acquaint the intelligent Reader, that the theodolite is a kind of generalnbsp;Angular instrument, not useful merely to ascertainnbsp;ingles for the surveyor, but also for many purposes in practical astronomy, and other sciencesnbsp;^hat have trigonometry as their fundamental basis.nbsp;Soine years ago it was found necessary to in-�dtute a course of trigonometrical operations innbsp;^his country and in P'rance, in order to determinenbsp;'^'ith precision the distance between the. Royalnbsp;Observatories of Greenwich and Paris. The latenbsp;O-eneral Roy was deputed as the chief manager innbsp;^bis country. A very accurate theodolite to takenbsp;^'igles, and other instruments, were essentially nc-R^^ssary; and the General was fortunate enough tonbsp;obtain the best articles and assistance that wasnbsp;afforded in any mathematical undertakingnbsp;^Rbatever.

. The frontispiece to this book gives a general �'quot;lew of the theodolite, the reader must not cx-P^ct from this a complete representation of all thenbsp;^Jtinuter parts. In the General�s account in thenbsp;quot;ilosophical Transactions, containing ti� quartonbsp;and six large plates, replete with explanatorynbsp;rSercs, he confined himself only to the dcscrib-of the principal paits; and the limits of tins

work will only admit of a summary description to convey some idea of its plan, and to shew the ingenious operator the superior utility and accuracynbsp;of the instrument. If a further and more particular knowledge be desired, it will be best obtainednbsp;by a reference to page 135, vol. 80 of the Transactions before cited. A A. a brass circle three feetnbsp;diameter; B the principal, or transit achromaticnbsp;telescope of 36 inches focal length, and 2^ inchesnbsp;aperture, admitting its adjustment by inversionnbsp;on its supports, as performed by the transits innbsp;tixed observatories; C a small Ian thorn fixed tonbsp;an horizontal bar for giving light to the axis ofnbsp;the telescope upon an illuminator that reflectsnbsp;light on the wire in nocturnal observations; D anbsp;semicircle of six inches radius, attached to the axisnbsp;of the transit. Each degree being divided intonbsp;two parts, or 3C/, and one revolution of the micrometer head moving the wire in the field of thenbsp;microscope at ^^three^minutes; therefore lOrevolu-tions produce 3(/, which are shewn by a scale ofnbsp;10 notches in the upper part of the field of thenbsp;microscope, each notch corresponding to threenbsp;minutes, or 180 seconds, and the head being divided into three minutes, and each minute intonbsp;12 parts; therefore 12 jiarts is equal to fivenbsp;seconds. When the angles of altitude and depression to be determined, arc very small, they arcnbsp;measured by the motion of an horizontal wire innbsp;the focus of the eye glass of the telescope atr/. Twonbsp;.spirit levels are used to this telescope; one to levelnbsp;the axis, making the long conical axis of the instrument truly vertical, not shewn in the plate?nbsp;and the other level E is suspended on a rod attached to the telescope, and serves to make it horizontal when vertical angles are to be taken.

1 he vertical bar F cxtericling across the top of the is supported by two braces G G tliat come fromnbsp;coiie, id above the plain of the instrument.nbsp;The great divided circle is attached by 10 brassnbsp;conical tubes, or radii, to a large vertical conicalnbsp;of 24 inches in height, called the exteriornbsp;�ixis. Within the base of this hollow axis, a castnbsp;�teel collar is strongly driven; and on iis top isnbsp;inserted a thick bell-metal plate, with slopingnbsp;cheeks, which by means of five screws can benbsp;^^ised or depressed a little.

The instrument rests on three feet, one of ^''hich is shewn at I, united at the center by anbsp;strong round plate of bell-metal, upon which risesnbsp;�quot;Another vertical hollow cone, going into thenbsp;^ther, H, and is called the interior axis; a castnbsp;�^cel pivot in its top, with sloping cheeks, passesnbsp;through the bell-metal plate at the top of the cx-tci'ior axis, being ground to fit one another. Thenbsp;^'^h-nuetal base of this interior axis is also groundnbsp;fit the steel collar of that without it. Whennbsp;P^it together, the circle is to be lifted up by layingnbsp;field of its radii, and the exterior placed upon thenbsp;interior axis, the cheeks at top, adjusted to theirnbsp;Proper bearings, will then turn round smoothlynbsp;steadily, and free from any central shake; thenbsp;Sreat circle, exterior axis, and upper telescope,nbsp;ficrefore, arc moveable, independent ot the lowernbsp;parts.nbsp;nbsp;nbsp;nbsp;,

The feet of the mahogany stand K. form a stpiare about three feet four inches at bottom, and bynbsp;lie separation of the legs, make an octagon at thenbsp;or the first plain; in the center of which isnbsp;Opening, nine inches in diameter. On thenbsp;^^P of this lies another mahogany ot;tagoual plain,nbsp;I 1�athcr greater dimensions than the lonncr, v. ith

a circular curb about I an inch within the plain of its sides. This hath in its center an open conicalnbsp;brass socket, three inches in diameter; and onnbsp;four of its opposite sides there are fixed fournbsp;screws, acting against pieces of brass on the topnbsp;of the stand. The plain, with every thing upon it,nbsp;may, therefore, be moved in four opposite directions, until the plummet L, is brought to coincidenbsp;with the station points underneath, in order to_nbsp;level the stand. The third or uppermost plain oinbsp;mahogany is part of the instiTimcnt, being connected hy screws, and carrying the handlesnbsp;whereby it is lifted up for use. In the middle ofnbsp;this bottom to the instrument there is another conical brass socket, i inches in diameter, that turnsnbsp;easily on that in the center of the octagon underneath. In the cover of this socket is an holenbsp;concentric tp the instrument, to admit the threadnbsp;or wire to pass, which suspends the plummet at L.nbsp;There is a small box with a winch handle at M?nbsp;that serves occasionally to raise or lower the pluna-met. To the three feet there are screws, such aSnbsp;at N, for levelling the instrument; and also threenbsp;blocks of box wood, and thi'ce brass conical I'ol'nbsp;lers under the feet screws, fixed to the lower suf'nbsp;face of the mahogany, to give the whole a pet'nbsp;fectly easy motion; O, O, O, are three of the fournbsp;screws attached to the octagonal plain, for accU'nbsp;ratcly centering the instrument by the pluiun^Rfnbsp;P and Q represent two positions of screw�nbsp;give a circular motion to the entire machine, butnbsp;these having been found to act by jerks from thenbsp;great weight, another apparatus or clamp, see �/�nbsp;2, was adjusted, attached to the curbs, consistingnbsp;of a brass cock fixed to, and projecting from thenbsp;curb of the instrument; the cock being acted

^ipon by two screws working in opposite directions, and which are clamped to the curb of the Octagon.

The curb upon which the feet of the instrument rest, carries the mahogany balustrade R R, fitted to receive a mahogany cover, that guardsnbsp;the whole instrument. In this cover arc fournbsp;Small openings, one for cjich of the vertical microscopes S, S, one for the clamp of the circle,nbsp;^rid one for the socket of the Hook's joint, l^hisnbsp;Cover secures the circles and its cones from dirt,nbsp;^nd serves conveniently for laying any thing upon,nbsp;tfiat may be wanted near at hand; and particu-fin'ly lanthorns used at night, for reading off thenbsp;fiivisions on the limb of the instrument.

There is a lower telescope T, lying exactly un-ficr the center of the instrument, and directed through one of the openings on the balustrade,nbsp;^md used only for terrestrial objects, requires butnbsp;^ Small elevation, and has an axis of 17 inches innbsp;length, supported by the braces attached to thenbsp;^^yt. Jt is moved by rack-work, by turning thenbsp;kmion at V. There is a small horizontal motionnbsp;that can be given to the right hand of the axis ofnbsp;the end of this telescope. The whole instrumentnbsp;ceing nicely levelled, the upper telescope at zero,nbsp;likewise on its object, the lower telescopenbsp;help of this adjustment is brought accuratelynbsp;t^ the same object, from the point of commence-mcnts from which the angles are to be measured.

There are three flat arms, one of which is represented at U, fixed by screws to the edge of the cll-p^etal plate. These arms are also braced tonbsp;feet of the instrument, rising as they projectnbsp;^ritwards towards the circumference of the circle,nbsp;Smng beyond it about ll: inches. One arm, lyingnbsp;leetly over one of the feet, is that to wTich is at-mmd wheels and screws moved by Hook's joint,

not seen in the figure, and also a clamp to tlie circle. It is this that produces the aecurate motionnbsp;of the circle. The other two arms, one of whichnbsp;also lies over a foot, and the other directly opposite to it, become the diameter of the circle, having their extremities terminated on a kind ofnbsp;blunted triangular figure, forming the bases ofnbsp;pedestals, whereon stand the vertical microscopesnbsp;S, S. The arms, braces, base, amp;c. are every wherenbsp;pierced, in order to lessen weight without diminishing strength.

The angles are not read off in this instrument by a nonius as commor/ to others, but with microscopes, and which form the most essential part ofnbsp;the instrument. But a short account of them cannbsp;be given here, an adequate idea can only be obtained by a reference to the Philosophical Transactions, page 145 and 149. The horizontal microscope for the vertical angles has been alreadynbsp;mentioned. The two vertical ones S, S, are usednbsp;for reading off the divisions on the opposite sidesnbsp;of the circle immediately under them. Each microscope contains two slides, one over the other,nbsp;their contiguous surfaces in the foci of the eyeglasses. The upper one is a very thin brass plate,nbsp;at its lower surface is attached a fixed wire, havingnbsp;no other motion than what is necessary for adjustment, by the left hand screw to its proper dot,nbsp;hereafter to be explained. The other slide is ofnbsp;steel of one entire piece, directly under the former, of sufficient thickness to permit a micrometer screw of about 72 threads in an inch tonbsp;be formed of it. To its upper surface is fixed thenbsp;immoveable wire, which changes its place by thenbsp;motion of the micrometer head. This head is divided i nto 60 equal parts, each of which representsnbsp;one second or angular motion of the telescope.nbsp;This steel slide is attached by a chain to the

spr'mg of a watcli coiled up witbin a small baiTcl 'Adjacent to it in tbe frame. By this no time wbat-pvci- is lost, tbe smallest motion of tbe bead beingnbsp;��istantly sbevvn by a proportionable motion ofnbsp;the wire in tbe field of tbe microscope.

Each microscope is supported between its pil-and can be a little raised or depressed in respect to tbe plain of tbe circle by two levers. By ^his motion distinctness is obtained of tbe wires,nbsp;?'^d by tbe motion of tbe proper screw of tbe abject lense, wbicb follows that given to tbe wholenbsp;iTiicroscope, the scale is so adjusted, that 13 revo~nbsp;�^tions of the head shall move the wire over 15',nbsp;one grand division of the limb, equal to QOOquot;,nbsp;degree on the circle being only divided intonbsp;^hur parts. To effect this, at the same time thenbsp;jjxed wire must bisect the dot on a gold tongue,nbsp;de moveable wire must also bisect the dot at 180�nbsp;the limb, as well as a first notch in the magni-scale at the bottom of the plate. In this ad-Eistineat there is another circumstance to be at-tcned to, viz. that 6o on the micrometer headnbsp;^^^ould stand nearly vertical, so as to be conveni-^ntlynbsp;nbsp;nbsp;nbsp;^ seconds of inclination are of no

^'^risequencc, because tbe dart, or index, being bought to that position, whatever it may be, must

^'de with their respective dots, and the first notch �^11 the micrometer head should happen to benbsp;^�merneath, or so far from the vertex side as to benbsp;^en with difficulty, then the gold tongue is to benbsp;a little by capstan head screws, which actnbsp;each other on the opposite extremities ofnbsp;�W�llnbsp;nbsp;nbsp;nbsp;Thus, by repeated trials the purpose

to ti effected, viz. the 6o, to which the dart is ^ set, will stand in a place easily seen. It isnbsp;to be expected that each microscope will givvi

just goo seconds for the run of 15 minutes; without loss of time this cannot be done; besides, two observers will adjust the microscopes diflFerently-After several trials of the runs in measuring 15 ifli-nutes in the different parts of the limb, one mi'nbsp;croscope gave 896quot;, while the other gave at a medium 901''; in a year afterwards, the former gavenbsp;900quot;, while the latter gave 894quot;. These diffenbsp;renees were allowed for in the estimation of anglesnbsp;for computation.

The gold tongue mentioned is extremely thin,-and goes close to the surface of the circle. This, contrivance of a tongue with a dot was to guardnbsp;against any error from any accidental motion givennbsp;to the instrument between the observations, andnbsp;if any, it immediately detected them. This was alsonbsp;a severe cheek upon the divisions of the instrument. General Roy observes, that it rarely hap'nbsp;pens that two observers, reading off with the opposite microscopes, differ more than half a secondnbsp;from each other at the first reading; and judges,nbsp;that in favourableweather for repeating the observation with the telescope, a wonderful degree of accuracy in the measure of the angles may be obtained.*

For the auxiliary apparatus, such as the lOOfeet steel chain, portahle scaffold, tripod ladder, commonnbsp;flag-staff, tripod for white lights, portable crane,nbsp;the reader will see the account of in the Transactions before cited. The horizontal angles takennbsp;by the instrument as regulated by the General,nbsp;since deceased, are to the tenth of a second.

^ The weight of the whole instrument was about 200 pounds, and the price, as I base been informed, about 350 guineas. By the completion of th*^nbsp;measurements and the necessary calculations, the difference of the two mcii-dians made 9' as before fixed by Dr. Maskehne.

A second instrument has since been made, and is now using by Col-Cap. Mudge^ and Mr. Dcdby, from whose skill and ingenuity it is expecte ^ very accurate survey of this country will be made. In this instrument, tnbsp;great circle is divided to 10 minutes, improvements made -in the microscopc^inbsp;amp;c. by Mr. Ratnsden. See Fkiloiophicai TranMctiom for 1795-

DESCRIPTION, USE, and method of adjustingnbsp;HADLEY�S QUADRANT.^

�� At the appointed time, when it pleased the Supreme Dispenser of every good gift to restorenbsp;'ght to a bewildered world, and more particularlynbsp;manifest his wisdom in the simplicity, as wellnbsp;in the grandeur of his works, he opened thenbsp;^^orious scene with the revival of sound astro-^lomy.� This observation of an excellent philo-'^^pher and physician-f- is verified in every instancenbsp;the progress of seience; in bach of which wenbsp;^^y trace some of the steps of that vast plan ofnbsp;divine Providence to which all things are con-'^Crging, namely, the bringing all his creatures tonbsp;^ state of truth, goodness, and consequent happi-I�sss; an end worthy of the best and wisest ofnbsp;�eings, and whieh we may perceive to be gradu-effecting, by the advancement of knowledge,nbsp;diffusion of liberty, and the removal of error,nbsp;Ihat truth and virtue may at last shine forth in allnbsp;beauty of their native colours.

. It is thus that the discovery of the compass gave to the present art of navigation; and whennbsp;art grew of moz'e importance to mankind,nbsp;divine Providence blessed them with the inventionnbsp;Hadley s quadrajit, and in our own day and ournbsp;time has further improved both it and the ait

^ * This accoimt of Hadley's quadrant, amp;c. is extracted fVom ^rnall tract I published thereon sometime since, 8vo.

of navisrat ion, by the present method of finding the longitude, which enables the mariner to ascertain with certainty his situation on the unvarie^lnbsp;face of the ocean,

Hadley�s quadrant or sextant is the only knoW�i instrument, on which the mariner can depend fofnbsp;determining with accuracy and precision his lati'nbsp;tude and longitude. It is to the use of this instrH'nbsp;ment that navigation is indebted for the very greatnbsp;and rapid advances it has made within these fequot;^nbsp;years. It is easy to manage, and of extensive uscgt;nbsp;rc(|uiring no peculiar steadiness of hand, nor anynbsp;such fixed basis as is necessary to other astronO'nbsp;inical instruments. It is not the science of navi'nbsp;gation only Avhich is indebted to this instrumenbnbsp;but its uses arc so extensive in astronomy, that dnbsp;may, accompanied with an artificial horizon, withnbsp;propriety be called a portable observatory ,an�. in thi^nbsp;work we shall exemplify its application to surveying'

Mankind arc ever desirous of knowing to w'hom they are indebted for any peculiar or useful discovery; it is the tribute of gratitude, and a rewardnbsp;to merit. In the present instance there is no dil'nbsp;ficulty in giving the information; the respectivenbsp;claims of the inventors arc easily decided. Thenbsp;first thought originated with the celebratednbsp;Hooke, it w'as completed by Sir Isaac Newton, andnbsp;published by Mr. Hadley.

Notwithstanding, however, the manifest snpf' riority of this instrument over those that were mnbsp;use at the time of its publicationj it was many'nbsp;years before the sailors could be persuaded tenbsp;adopt it, and lay aside their imperfect and inaccurate instruments: so great is the difficulty to remove prejudice, and emancipate the mind fromnbsp;the slavery of opinion.

No instrument has undergone, since the origi-�ial invention, more ehanges than the quadrant of Jiadley, of the various alterations, many have hadnbsp;better foundation than the coneeit and caprieenbsp;the makers, who by these attempts have oftennbsp;*'endered the instrument more complicated in con-'^truction and more difficult in use, than it was innbsp;original state.

It is not my intention under this head to enu-^ticrate all the advTintaa-cs of this instrument; but uarely to point out one or two of those essentialnbsp;properties which distinguish it from every othernbsp;'ostrument of the kind, and rank it among one ofnbsp;the greatest improvements in the practice of navi-S^^tion.

It is an essential property of this instrument, derived from the laws of reflection, that half decrees on the arc answer to whole ones in the an-Sles measured: hence an octant, or the eighthnbsp;P^rt of a circle, or 45 degrees on the arc, serves tonbsp;^tieasure go degrees; and sextants will measure annbsp;^^gular distance of 120 degrees, though the arc ofnbsp;instrument is no more than 6o degrees.* Itnbsp;from this property that foreigners term that in-^truijjent an octant, which we usually call a qua-and which in effect it is. This propertynbsp;^duces indeed considerably the bulk of the in-^'uinent; but at the same time it calls for thenbsp;*tiost accuracy in the divisions, as every errornbsp;the arc is doubled in the observation.

plj] , a concise explanation of the theory, amp;c, gee my panama this instrument, 8vo. Edit.

Another essential, and indeed an invaluable pfO' perty of this instrument, whereby it is renderednbsp;peculiarly advantageous in marine observations, iS;nbsp;that it does not require any peculiar steadiness oinbsp;the hand, nor is liable to be disturbed by thenbsp;ship�s motion; for, provided the mariner can seenbsp;distinctly the two objects in the field of his ip'nbsp;strument, no motion nor vacillation of the shipnbsp;will Inncl^r his observation.

Thirdly, the errors to which it is liable are easily discovered, and readily rectified, while the apph'nbsp;cation and use of it is facile and plain.

The principal requisites in a good sextant ot quadrant, arc, 1. Tkat it be strong, and so constructed as not to bend across the plain. 2. Thatnbsp;it be accurately divided. 3. That the surfaces ofnbsp;the glasses be perfectly plain and parallel to eachnbsp;other. 4. That the index turn upon a long axis-5. That the motion be free and easy in every part,nbsp;and yet without the least shake or jerk.

Fig. 1, plate ig, represents a quadrant, or octant, of the common construction. The following parts are those which require the particular attcU'nbsp;tion of the observer.

The quadrant consists of an arc B C, firmly attached to two radii, or bars, AB, AC, which arc strengthened and bound together by the twonbsp;firaces LM.

Of the index. The index D is a flat bar of ^'�ass, that turns on the center of the octant; atnbsp;the lower end of the index there is an oblongnbsp;Opening, to one side of this opening a nonius sealcnbsp;fixed to subdivide the divisions of the are; atnbsp;tfie bottom or end of the index there is a piece ofnbsp;�^rass, which bends under the arc, carrying anbsp;Spring to make the nonius scale lie close to thenbsp;divisions; it is also furnished with a serew to fixnbsp;^fie index in any desired position.

The best instruments have an adjusting screw htted to the index, that it may be moved morenbsp;slowly, and with greater regularity and accuracynbsp;than by the hand. It is proper however to ob-�^fve, that the index must be previously fixednbsp;^'�ear its right position by the above-mentionednbsp;^'-'rew, before the adjusting screw is put in motion,nbsp;'=�^lt;5BC,/^r.4.

The circular arcs on the arc of the quadrant drawn from the center on which the indexnbsp;Urns: the smallest exccntricity in the axis of thenbsp;^'ulex would be productive of considerable errors.

-*� he position of the index on the arc, after an ob^ ^�''rvation, points out the number of degrees andnbsp;^�^inutcs contained in the observed angle.

. Of tJig Index-glass E. Upon the indexes and near ^ axis, is fixed a plain speculum, or mirror ofnbsp;V quicksilvered. It is set in a brass frame, andnbsp;.^^P^^ced so that the fiice of it is perpendicular tonbsp;� plain of the instrument; this mirror being

tixccl lo tlie iiulcx, moves along with it, and haS its direction changed by the motion thereof.

This glass is designed to reeeive the image ot the sun, or any other objeet, and reflect it uponnbsp;either of the two horizon glasses F and G, according to the nature of the observation.

The brass frame with the glass is fixed to the index by the screw c; the other screw serves tonbsp;rc])lace it in a perpendicular position, if by anynbsp;accident it has been deranged, as will be seennbsp;hereafter.

Tire index glass is often divided into two parts, the one silvered, the other black, with-a smallnbsp;screen in front. A single black surtace has indeednbsp;some advantages; but if the glasses be well selected, there is little danger to be apprehended otnbsp;error, from a want of parallelism; more is to benbsp;feared from the surfaces not being flat.

Of the horizon glasses F, G. On the radius AB of the octant, arc two small spcculums. Thenbsp;surface of the upper one is parallel to the indelt;nbsp;glass, wdien the counting division of the index is atnbsp;o on the arc; but the surface of the low'er one i�nbsp;perpendicular to the index glass, when the index i*'�nbsp;at o degrees on the arc: these mirrors receive thenbsp;reflected rays from the object, and transmit thciunbsp;to the observer.

I'he horizon glasses are not entirely quicksih vered; the upper one F, is only silvered onnbsp;lower part, or that half next the quadrant, the_nbsp;other halt being transparent, and the back part oinbsp;the frame is cut away, that nothing may impedenbsp;the sight through the unsilvered part of thenbsp;The edge of the foil of this glass is nearly panifle*nbsp;to the plane of the instrument, and ought to henbsp;very sharp, and without a flaw.

The other horizon glass G is silveredatboth ends; the middle there is a transparent slit, throughnbsp;�which the horizon, or other object, Inay be seen.

Each of these glasses is set in a brass frame, to which there is an axis; this axis passes through thenbsp;Wood-work, and is fitted to a lever on the undernbsp;Side of the quadrant; by this lever the glass maynbsp;turned a few degrees on its axis, in order to setnbsp;parallel to the index glass. The lever has anbsp;poiitrivance to turn it slowly, and a button to fixnbsp;To set the glasses perpendicular to the planenbsp;the quadrant, there are two sunk screws, onenbsp;�^^forc and one behind each glass; these screwsnbsp;pass through the plate, on which the frame is fixed,nbsp;another plate, so that by loosening one, andnbsp;^'^htening the other of these screws, the directionnbsp;the frame, with its mirror, may be altered, andnbsp;thus be set perpendicular to the plane of the in-striiincnt. For the lever, amp;c. sec^V. 13.

the shades, or dark glasses, K. There are two fed Of dark glasses, and one green one; they arcnbsp;'^�^ed to prevent the bright rays of the sun, or thenbsp;Share of the moon, from hurting the eye at thenbsp;.line of observation. They are each of them setnbsp;a brass frame, which turns on a center, so thatnbsp;may be used separately, or together, as thenbsp;I'ghtness of the sun may require. The greennbsp;Siass niay be used also alone, if the sun benbsp;faint; it is also used for taking the altitudenbsp;f-he moon, and in ascertaining her distancenbsp;a fixed star.

yV hen these glasses are used for the fore obser-they are fixed as at K 'va fig. \; when used the back observation, they arc removed to N.nbsp;isnbsp;nbsp;nbsp;nbsp;vanes H, 1. Each of these vanes

the P.^^�f'^fated piece of brass, designed to direct �'ght parallel to the plane of the quadrant,nbsp;s g

That which is fixed at I is used for the fore, the other for the back observation.

The vane I has two holes, one exactly at the height of the quicksilvered edge of the horizonnbsp;glass, the other somewhat higher, to direct thenbsp;sight to the middle of the transparent part of thenbsp;mirror, for those objects which arc bright enoughnbsp;to be rctlccted from the unsilvered part of thenbsp;mirror.'

Of the divisions on the lirnh of the quadrant, and of the nordus on the index. For a description of thesenbsp;divisions, see page 127-

Directions to hold the instrument. It is rccoiU' mended to support the weight of the instrumentnbsp;by the right hand, and reserve the left to goveriinbsp;the index. Place the thumb of the right handnbsp;against the edge of the quadrant, under the swcl-ling jxirt on which the fore sight I stands, extend'nbsp;ing the fingers across the back of the quadrant,nbsp;so as to lay hold on the opposite edge, placing thenbsp;fore finger above, and the other fingers below thenbsp;swelling part, or near the fore horizon glass; thusnbsp;you ma)� support the instrument conveniently, innbsp;a vertical position, by the right hand only; bynbsp;resting the thumb of the left hand against the side,nbsp;or the fingers against the middle bar, you maynbsp;move the index gradually either way.

In the back observation, the instrument should be su])portcd by the left hand, and the index benbsp;governed by the right.

Of the axis of vision, or Vine of sight. Of the two objects which are made to coincide by this instm-ment, the one is seen directly by a ray passingnbsp;through, the other, by a ray reflected from,nbsp;same point of the horizon glass to the eye. Thisnbsp;ray is ealled the visual ray; but when it is considered merely as a line drawn from the middle oi

The axis of a tube, or telescope, used to direct the sight, is also called the axis of vision.

The quadrant, if it be held as before directed, ^ay be easily turned round between the fingersnbsp;^nd thumb, and thus nearly on a line parallel tonbsp;the axis of vision; thus the plane of the quadrantnbsp;t�ll pass through the two objects when an obsei*-''ation is made, a circumstance absolutely necessary, and Avhich is more readily cftectcd when thenbsp;instrument is furnished with a telescope; withinnbsp;the telescope are two parallel wires, which bynbsp;turning the eye glass tube may be brought parallel to the plane of the quadrant, so that by bringing the object to the middle between them, younbsp;nre certain of having the axis of vision parallel tonbsp;the plane of the quadrant.

It is a pcculiai* excellence of Hadley�s quadrant, that the errors to which it is liable are easily detected, and soon rectified; the observer may,nbsp;therefore, if he will be attentive, always put hisnbsp;instrument in a state fit Tor accurate observation.nbsp;?-he importance of this instrument to navigationnbsp;is Self-evident; yet much of this importance depends on the accuracy with which it is made, andnbsp;the necessary attention of the observer; and onenbsp;'''Quid hardly think it possible that any observernbsp;'quot;'ould, to save the trifling sum of one or two gui-iinas, prefer an imperfect instrument to one thatnbsp;rightly constructed and accurately made; ornbsp;jhat any consideration should induce him to neg-the adjustments of an instrument, on whose

truth he is so highly interested. But such is the nature of man! he is too apt to be lavish on baubles, and penurious in matters of consequence;nbsp;active about trifles, indolent where his welfare andnbsp;happiness are concerned. The adjustments fornbsp;the fore observation are, l.To set the fore horizonnbsp;glass parallel to the index glass; this adjustmentnbsp;is of the utmost importance, and should always benbsp;made previous to actual observation. The seuondnbsp;is, to see that the plane of this glass is perpendicular to the plane of the quadrant. 3. To seenbsp;that the index glass is perpendicular to the planenbsp;of the instrument.

To adjust the fore horizon glass. This rectifiea-tion is deemed of such importance, that it is usual to speak of it as if it included all the rest, and to

to plaee the horizon glass, that the index may shew upon the arc the true angle between the objeets:nbsp;for this purpose, set the index line of the noniusnbsp;exactly against o on the limb, and fix it there bynbsp;the screw at the under side. Now look throughnbsp;the sight I at the edge of the sea, or some verynbsp;distant well-defined small object. The edge ofnbsp;the sea will be seen directly through the unsilverednbsp;part of the glass, but by reflection in the silverednbsp;part. If the horizon in the silvered part exactlynbsp;meets, and forms one continued line with that seennbsp;through the unsilvered part, then is the instrumentnbsp;said to be adjusted, and the horizon glass to benbsp;parallel to the index glass. But if the horizonsnbsp;do not coincide, then loosen the milled nut onnbsp;the under side of the quadrant, and turn the horizon glass on its axis, by means of the adjustingnbsp;lever, till you have made them perfectly coincide;nbsp;then fix the lever firmly in the situation thus

obtained, by tightening the milled nut. This *idjustmcnt ought to be repeated before and afternbsp;every astronomical observation.

So important is this rectification, that experienced observers, and those who are aiesirous of¹ being very accurate, will not be content with thenbsp;preceding mode of adjustment, but adopt anothernbsp;Hiethod, which is usually called finding the indexnbsp;error; a method preferable to the foregoing botlinbsp;for ease and accuracy.

7b find the index error. Instead of fixing the index at o, and moving the horizon glass, till thenbsp;image of a distant object coincides with the samenbsp;object seen directly ; let the horizon glass remainnbsp;fixed, and move the index till the image and objectnbsp;coincide; then observe whether the index divisionnbsp;On the nonius agrees with the o line on the arc;nbsp;if it docs not, the number by which they differ is anbsp;quantity to be added to the observation, if thenbsp;index line is beyond the o on the limb; but if thenbsp;index line of the nonius stands between o and QOnbsp;degrees, then this error is to be subtracted fromnbsp;the observation.¹

We have already observed, that the part of the lt;irc beyond o, towards the right hand, is callednbsp;the arc of excess; and that the nonius, when atnbsp;that part, must be read the contrary way, or, whichnbsp;i^ the same thing, you may read them off in thenbsp;visual way, and then their complement to 20 min.nbsp;tvill be the real number of degrees and minutes tonbsp;be added to the observation.

This adjustmeBt may be made more accurately, or the ^rror better found, by using the sun instead of the horizon; butnbsp;^bis method requires another set of dark glasses, to darken thenbsp;direct rays of -the sun; such a set is applied to the best instru-^'ents, and this method of adjustment is explained in the foUow-description of the sextant.

7� maJie the index glass and fore horizon ghss perpendicular to the plane of the instrument. Thoughnbsp;these adjustments are neither so necessary nornbsp;important as the preceding one; yet, as after beingnbsp;once performed, they do not require to be repeatednbsp;for a considerable time, and as they add to thenbsp;accuracy of observation, they ought not to benbsp;neglected; and further, a knowledge of them enables the mariner to examine and form a propernbsp;judgment of his instrument.

To adjust the index glass. This adjustment consists in setting the plane of the index glass perpendicular to that of the instrument.

Method 1. By means of the two adjusting tools, represented at fig. 2 and 3, which are twonbsp;wooden frames, with two lines on each, exactlynbsp;at the same distance from the bottom.

Place the quadrant in an horizontal position on a table, put the index about the middle of the arc,nbsp;turn back the dark glasses, place one of the above-mentioned tools near one end of the arc, the othernbsp;at the opposite end, the side with the lines towardsnbsp;the index glass; then look down the index glass,nbsp;directing the sight parallel to the plane of the in- quot;nbsp;strument, you will see one of the tools by directnbsp;vision, the other by reflection in the mirror; bynbsp;moving the index a little, they may be broughtnbsp;exactly together. If the lines coincide, the mirrornbsp;is rightly flxed; if not, it must be restored to itsnbsp;proper situation by loosening the screw c, andnbsp;tightening the screw d; or, vice versa, by tightening the screw c, and releasing the screw d.

Method 2. Hold the quadrant in an horizontal position, w'ith the index glass close to the eye;nbsp;look nearly in a right line down the glass, and innbsp;such a manner; that you may see the arc of thenbsp;quadrant by direct view, and by reflection at the

�^ame lime. If they join in one direct line, and the arc seen by reflection forms an exact planenbsp;with the arc seen by direct view, the glass is per-pwidicular to the plane of the cpiadrant; if not,nbsp;the error must be rectified by altering the position of the screws behind the frame, as directednbsp;^bovc.

7o ascertain v.'hether the fore horizon glass, be t^rperuiicular to the plane of the instrument, diavingnbsp;adjusted the index and horizon glasses agreeablenbsp;to the foregoing directions, set the index divisionnbsp;of the nonius exactly against o on the limb; holdnbsp;the plane of the quadrant parallel to the horizon,nbsp;and observe the image of any distant object atnbsp;laud, or at sea the horizon itself; if the image ofnbsp;the horizon at the edge of the silvered part coincide with five object seen directly, the glass isnbsp;perpendicular to the plane of the instrument. Ifnbsp;d fall above or below, it must be adjusted. If thenbsp;Lnage seen by reflection be higher than the objectnbsp;dself seen directly, release the fore screw andnbsp;fightcn the back screw; and, znce versa, if thenbsp;onage seen by reflection be lower, release the backnbsp;^crew and screw up the foi'C one; and thus proceed till both arc of an equal height, and that bynbsp;Lioving the index you can make the image andnbsp;the object appear as one.

Or, adjust the fore horizon glass as directed in P^^gc 2,62; then incline the quadrant on one sidenbsp;much as possible, provdded the horizon continues to be seen in both parts of the glass. If,nbsp;d�hen the instrument is thus inclined, the edge ofnbsp;tile sea continues to form one unbroken line, thenbsp;hcadrant is perfectly adjusted; but if the reflectednbsp;fcrizon be; separated from that seen by directnbsp;t'l^ion, the speculum is not perpendicular to tlienbsp;plane of the quadrant. And if the oUserver is m-

dined to the right, with the face of the quadrad upwards, and tiie reflected sea appears higher thannbsp;the real sea, you must slacken the screw bctd'*^nbsp;the horizon glass, and tighten that which is hU'nbsp;hind it; but if the reflected sea appears lower, thenbsp;contrary must be performed.

Care must be always taken in these adjustment* to loosen one screw before the other is screwed upgt;nbsp;and to leave the adjusting screws tight, or so as tonbsp;draw wdth a moderate force against each other.

This adjustment jiiay be also made by the sun� moon, or star; in this case the quadrant may henbsp;held in a vertical position; if the image seen bynbsp;reflection appears to the right or left of the objectnbsp;seen directly, then the glass must be adjusted *1*nbsp;before by the two screws.

The back observation is so called, because the back is turned upon one of the two objects who?enbsp;angular distance is to be measured.

The adjustment consists in making the reflected image of the object behind the observer coincidlt;^nbsp;wdth another seen directly before him, at the sain�nbsp;time that the index division of the nonius is di'nbsp;rectly against the o divison of the arc.

The method, therefore, of adjusting it consist� in measuring the distance of two objects nearlynbsp;180 degrees apart from each other; the arc passingnbsp;through each object must be measured in both it�nbsp;parts, and if the sura of the parts be 36o degree��nbsp;the speculums are adjusted; but, if not, the axi*nbsp;of the horizon glass must be moved till this sninnbsp;is obtained.

Set the index as far behind o as twice tl:ie dip* of gt;tlie horizon amounts to; then look at the horizonnbsp;through the slit jiear G, and at the same time thenbsp;opposite edge of the sea will appear by reflectionnbsp;inverted, or upside down. By moving the levernbsp;�f the axis, if necessaiy, the edges may be madenbsp;quot;^0 coincide, and the quadrant is adjusted.

There is but one position in which the quadrant *^'in be held with the limb downwards, withoutnbsp;JJausingthc refleeted horizon to cross the part seennbsp;direct vision.

If, on trial, this position be found to lac that in quot;'��^�hich the plane of the quadrant is perpendicularnbsp;I� the horizon, no fartlicr adjustment is neccssarynbsp;the fore-mentioned one; but if the horizonsnbsp;^5*^gt;ss each other when the quadrant is held up-T'Rlit, observe which part of the reflected horizonnbsp;*s lowest.

It the right-hand part be lowest, the sunk ^pi�ew which is before the horizon glass must benbsp;^^ghtened after slackening that which is behind thenbsp;but if the right hand is highest, the con-trary must be performed: this adjustment is,nbsp;flowever, of much less iirqjortance than the pre-'^efling as it does not so much aflect the anglenbsp;��ieaaured.

Th c occasions on which the back observation is be used arc, when the altitude of the sun or anbsp;^b'lr is to be taken, and the fore horizon is brokennbsp;l*)� adjacent shores; or when the angular distancenbsp;etween tlie moon and sun, or a star, exceeds 90

This is according- to the height of the observer�s eye above ^nbsp;nbsp;nbsp;nbsp;lloi�orl.wii's Njvigiitioii.

degrees, and is required to be measured for obtaining the longitude at sea: but there are objection^ to its use in both cases; for if a known land lie anbsp;few miles to the north or southward of a ship, th�nbsp;latitude may be known from its bearing and di�'nbsp;tance, without having recourse to observation�nbsp;and again, if the distance of the land in miles exceed the number of minutes in the dip, as is almostnbsp;always the case in coasting along an open shorC)nbsp;the horizon will not be broken, and the fore observation may be used; and lastly, if the land be toOnbsp;near to use the fore observation, its extreme point�nbsp;will in general be so far asunder, as to prevent thenbsp;adjustment, by taking away the back horizon,nbsp;the case of measuring the angular distances of thenbsp;heavenly bodies, the very great accuracy require^!nbsp;in these observations, makes it a matter of impof'nbsp;tance that the adjustments should be well made�nbsp;and frequently examined into. But the quantity,nbsp;of the dip is varied by the pitching and rolling ^nbsp;the ship; and this variation, which is perceptiblenbsp;in the measuring altitude by the fore observation�nbsp;is doubled in the adjustment for the back observa'nbsp;tion, and amounts to several minutes. It is like'nbsp;wise exceeding difficult, in a ship under way, tonbsp;hold the quadrant for any length of time, so thatnbsp;the two horizons do not cross each other, andnbsp;the night the edge of the sea cannot be accuratelynbsp;distinguished. All these circumstances concur tonbsp;render the adjustment uncertain; the fore observation is subject to npne of these inconveni'�nbsp;encies.*

t269]

DESCRIPTION, USE, and method of adjustingnbsp;HADLEY�S SEXTANT.

As the taking the angular distances of the ^^oon ajjj the sini, ora star, is one of the best andnbsp;accurate methods of discovering the longi-it was necessary to enlarge the arc of thenbsp;*^ctant to the sixth part of a circle; but as the ob-^^rvations for determining the longitude must benbsp;!^ade with the utmost accuracy, the framing of thenbsp;^'Htriinicnt was also altered, that it might benbsp;1�^ndered more adequate to the solution of thisnbsp;^'^Portant problem. Hence arose the presentnbsp;^\!i^*ti�uction of the sextant, in the description ofnbsp;�gt;ch, it is presumed that the foregoing pages havenbsp;read, as otherwise Ave should be obliged tonbsp;^*^Peat the same observations.

Sextants are mostly executed (some trifling ^^lations excepted) on two plans; in the one, allnbsp;ij^^l ^adjustments arc left to the observer: he has itnbsp;bis power to examine and rectify every part ofnbsp;instrument. This mode is founded on thisnbsp;� beral principle, that the parts of no instrumentnbsp;be so fixed as to remain accurately in the samenbsp;j^�sition they had when first put out of the maker�snbsp;^bds; and that therefore the principal parts shouldnbsp;ev ^bade moveable, that their positions may benbsp;mined and rectified from time to time by thenbsp;Observer.

b the second construction, the principal ad-ind ^bTit, or that of the horizon glass for the is rejected; and this rejection isnbsp;buded upon two reasons:

1. That ffom tlic nature of the acljuhtnicnt it frequently happens that a sextant will alter evennbsp;during the time of an observation, without an/nbsp;apparent cause whatever, or without being sensi'nbsp;blc at what period of the observation such alteration took place, and consequently the observernbsp;unable to allow for the error of adjustment.

'2. That this adjustment is not in itself suffi' cicntly exact, it being impossible to adjust a sextant with the same accuracy by the coincidence olnbsp;two images of an object, as by the contact of th^^nbsp;limbs thereof; and hence experienced and accurate observers have always directed the index crroi'nbsp;to be found and allowed for, which renders thenbsp;adjustment of the horizon glass in this directionnbsp;useless; for it is easy to place it nearly parallel tonbsp;the index glass, when the instrument is made, andnbsp;then to fix it firmly in that position by screws.nbsp;The utility of this method is contirmed by experience: ma,ny sextants, whose indices had beennbsp;tletermincd previous to their being carried out tonbsp;India, have been found to remain the same at thcitnbsp;return.

Notwithstanding the probable certainty of th^ hprizon glass remaining permanently in its situation, the observer ought from time to timenbsp;examine the index error of his instrument, to seenbsp;if it remains the same; or make the proper allowances for it, if any idteration should have takennbsp;place.

One material point in the formation of a sextant is so to construct it, that it may support its ownnbsp;weight, and not be liable to bend by any incline'nbsp;tion of the plane of the instrument, as every flexurenbsp;would alter the relative position of the mirrors, onnbsp;which the determination of an angle depends.

i^esides the errors now mentioned, to which the �cxt�nit or quadrant, amp;c. is liable, there is anothernbsp;which seems inseparable from the constructionnbsp;and materials of the instrument, and have beennbsp;Noticed by several skilful observers. It arisesnbsp;h�om the bending and elasticity of the index, andnbsp;resistance it meets with in turning round itsnbsp;^^�ntcr.

-l o obviate this error, let the observer be care-always to finish his observations, by movdng the ^�^flex in the same direction which was used innbsp;Getting it to o for adjusting, or iii finding the indexnbsp;ror.

The direction of the motion is indeed indifFe-but as the common practice in observing is finish the observations by a motion of the indexnbsp;that direction which increases the angle, thatnbsp;in the fore observation from o towards pt), itnbsp;^ �^uld be well if the observer would adopt it as anbsp;l?^Ueral rule, to finish the motion of the index, bynbsp;pushing it from him, or turning the screw in thatnbsp;*. *'ection which carries it farther from him. Bynbsp;�Uishing the motion of the index, we mean thatnbsp;fip last motion of the index should be for somenbsp;[!^uiutes of a degree at least in the required dircc-fion.

.��anicd, as not to be liable to bend. The arc AA, divided into 120�, each degree is divided intonbsp;parts, of course equal to 20 minutes, whichnbsp;again subdivided by the nonius into every halfnbsp;''Uute, or 30 seconds: see the nature of thenbsp;and the general rule for estimating thenbsp;^ thereof, in the preceding part of these Essays,nbsp;second division or minute on the nonius, is

cut longer than the intermediate ones. Th^ nonius is numbered at every fifth of these long'll'nbsp;divisions, from the right towards the left, withnbsp;5, 10, 15, and 20, the first division towards th^nbsp;right hand being to be considered as the indexnbsp;division.

In order to observe with accuracy, and mak�? the images come precisely in contact, an adjustingnbsp;screw B is added to the index, which may be movednbsp;with greater accuracy than it can by hand; butnbsp;this screw docs not act until the index is fixed bynbsp;the finger screw C. Care should be taken not tonbsp;force the adjusting screw when it arrives at eithci*nbsp;extremity of its adjustment. When the index isnbsp;to be moved any considerable quantity, the screWnbsp;C at the back of the sextant must be loosened;nbsp;but u'hcn the index is brought nearly to the di-visionrcquired,this back scrcwshouldbc tightened,nbsp;and then the index may be moved gradually bynbsp;tlic adjusting screw. A small shade is sometimesnbsp;fixed to that part of the index where the nonius isnbsp;divided, this being covered with white paper, reflects the light strongly upon the divisions.

There are four tinged glasses at D, each �of whidt Ls set in a separate frame turning on a center: they'nbsp;arc used to screen and save the eye from the-brightness of the solar image, and the glare of tb*^nbsp;moon, and may be used separately, or together^nbsp;as occasion recpxires.nbsp;nbsp;nbsp;nbsp;_ ,

There are three more such glasses placed bchii^d the horizon glass at E, to weaken tlic rays of fbenbsp;sun or moon, when they are viewed directlynbsp;through the horizon glass. The paler glassnbsp;sometimes used in observing altitudes at sea, ^nbsp;take off the strong glare of the horizon.nbsp;nbsp;nbsp;nbsp;,

The frame of the index glass I, is firmly by a strong cock to the center plate of the inJ^^'

quot;^hc horizon glass F, is fixed in a frame that turns ^'1 the axes or pivots, which move in an exteriornbsp;frame: the holes in which the pivots move may benbsp;frghtened by four screws in the exterior frame : Gnbsp;a screw by which the horizon glass may be setnbsp;PPi�pcndieular to the plane of the instrument;nbsp;should this screw become loose, or move too easy,nbsp;F may be easily tightened by turning the capstannbsp;headed screw H, Avhich is on one sicle the socket,nbsp;through which the stem of the finger screw.

The sextant is furnished with a plain tube, fg-7, quot;without any glasses; and to render the objects stillnbsp;Hiore distinct, it has also two achromatic telescopes.nbsp;Jig, shewing the objects erect, or in theirnbsp;Natural position ; the longer one. Jig. Ӓ, shewsnbsp;them inverted. It has a large field of view, andnbsp;'^thcr advantages; and a little use will soonnbsp;accustom the observer to the inverted jiosition,nbsp;a'Hl the instrument will be as readily managed bynbsp;as by tbe plain tube only. By a telescope,, thcnbsp;^ontact of the images is more perfectly clistin-Sifrshed; and by the place of. the images in thenbsp;^cld of the telescope, it is easy to perceive whethernbsp;he sextant is held in the proper plane for observa-By sliding the tube that contains the eyenbsp;glasses in the inside of the other tube, the imagenbsp;the object-is suited to different eyes, and madenbsp;� appear perfectly distinct and well defined.

. The telescopes arc to be screwed into a circular _ 'g atK; this ring rests on twxj points againstnbsp;exterior ring, and is held thereto by twm screw's;nbsp;, turning one, and tightening the other, thenbsp;axis of tire telescope may be set parallel to thenbsp;P ane of the sextant. The exterior ring is fixed

on a triangular brass stem tliat slides in a socket, and by means of a screw at the back of the seX'nbsp;tant, may be raised or lowered so as to move thenbsp;center of the telescope to point to that part of thenbsp;horizon glass which shall be judged the most ftnbsp;1�or observation. Fig. 8, is a eireular head, withnbsp;tinged glasses to screw on the eye end of either otnbsp;the telescopes, or the plain tube. The glasses atenbsp;contained in a circular plate, which has four holes;nbsp;three of these are fitted with tinged glasses, thenbsp;fourth is open. By pressing the finger against thenbsp;projecting edge of this circular plate, and turningnbsp;it round, the open hole, or any of the tingednbsp;glasses, may be brought between the eye glassnbsp;the telescope and the eye.

Fig. g, a small screw driver. Fig. 10, a mag' nifying glass, to read oft' the divisions by,

To^Jind the index error of the sextant. To fir'd the index error, is, in other words, to shew whatnbsp;number of degrees and minutes is indicated b)'nbsp;the nonius, when the direct and reflected imaglt;-''^nbsp;of an object coincide with each other.nbsp;nbsp;nbsp;nbsp;.

The most general and most certain method rh ascertaining this error, is to measure the diametrquot;quot;nbsp;of the sun, by bringing the limb of its image tj*nbsp;coincide with the limb of the sun itself seen di'nbsp;rcctly, ho/h on tlie quadrantal arc, and on the arenbsp;of excess.

If the diameter taken b}quot; moving the index fiquot;quot; ward on the (piadrantal arc be greater than th�''nbsp;taken on the arc of excess, then half the diftcreneenbsp;is to be snblracted-, but if the diameter takennbsp;the arc of excess be greater than that by thenbsp;drantal arc, half the difference is to be added,nbsp;the numbers shewn on the arc be the same in bonbsp;cases, the glasses arc truly parallel, and there is

index error; but if the numbers be different, then hidf the diftercnec is the index eiTor.*

. It is however to be observed, that when thcindex on the arc of excess, or to the right of o, thenbsp;complement of the numbers shewn on the noniusnbsp;lo 20 ought to be set down.

Several observations of the sun�s diameter should I�o made, and a mean taken as the result, whichnbsp;'^ill give the index error to very great exactness.

Example. Let the numbers of minutes shewn cy the index to the right of zero, when the limbsnbsp;clquot; the two images are in contact, be 20 minutes,nbsp;the odd number shewn by the nonius be 5,nbsp;^^0 complement of this to 20 is 15, which, addednbsp;lo 20, gives 35 minutes; and, secondly, that thenbsp;^limber shewn by the index, when on the left ofnbsp;^cro, and the opposite limbs are in contact, be 20nbsp;^*iinutes, and by the nonius p' 30quot;, which makesnbsp;^Cgethcr 2p' 30quot;; this subtracted from 35' givesnbsp;? 3oquot;, which divided by 2, gives 2' 45quot; for thenbsp;^^dex error; and because the greatest of the twonbsp;^Umbers thus found, was, when the index was tonbsp;die right of the o, this index error must be addednbsp;P the number of degrees shewn on the arc at thenbsp;of an observation.

To set the horizon glass perpendicular to the plane '?/' the sextant. Direct the telescope to the sun, anbsp;Or any other well-defined object, and bringnbsp;�le direct object and reflected image to coincidenbsp;pearly ^vith each other, by moving the index;nbsp;set the two images parallel to the plane of the

f, * other words, the difference between the degree and mi-the^ �^own by the index: first, when the lower reflected limb of is exactly in contact with the upper limb of the sun;nbsp;waj*^^�odly, when the upper edge of the image is in contactnbsp;*�^0 lower edge of the object, divided by 2, Will be the index

sextant, by turning the screw, and the images wiH ])ass exactly over each other, and the mirror wiUnbsp;then be adjusted in this direction.

To set the axis of the telescope parullel to the plan^ of the sextant. We have alreatly observed, that innbsp;measuring angular distances, the line of sight, ornbsp;plane of observxition, should be parallel to the planenbsp;of the instrument, as a deviation in that respectnbsp;will occasion great errors in the observation, andnbsp;this is most sensible in large angles: to avoidnbsp;these, a telescope is made use of, in whose fieldnbsp;there arc placed two wii-es parallel to each other?nbsp;and ccjuidistant from the center. These wiresnbsp;may be placed parallel to the plane of the sextant,nbsp;by turning the eye glass tube, and, consecpicntly,nbsp;by bringing the object to the middle betweennbsp;them, the observer may be certain of having thenbsp;axis of vision parallel to the plane of the quadrant.

To adjust the telescope. Screw the telescope in its place, and turn the eye tube round, that thenbsp;tvircs in the focus of the eye glass may be paradednbsp;to the planeof the instrument; then seek two oh- .nbsp;jeets, as the sun and moon, or the moon and anbsp;.�tar, whose distance should, for this jntrposc,nbsp;cced 90 degrees; the distance of the sun andnbsp;moon is to be taken great, because the same deviation of the axis will cause a greater error, andnbsp;will consequently be more easily discovered-Move the index, so as to bring the limbs of th^nbsp;sun and moon, if they are made use of, exactlynbsp;contact with that wire which is nearest to the phi��^nbsp;of the sextant; fix the index there; then, by�d-tering a little the position of your instruinenbnbsp;make the images appear on the wire furthest frojdnbsp;the sextant. Jf the nearest limbs be now preciselynbsp;in contact, as they were before, then the axis 0

i'liHbs of the two objects appear to separate at the wire that is furthest from the plane of the instru-jiient, it shews that the object end of tlic tclcseopenbsp;inclines towards the plane of the instrument, andnbsp;inust be rectified by tightening the screw nearestnbsp;ifie sextant, having previously unturned the screwnbsp;inithest from it. If the images overlap eachnbsp;other at the wire furthest from the sextant, tlicnbsp;object end of the telescope is inclined from thenbsp;plane of the sextant, and the highest screw mustnbsp;l^c turned towards the right, and the lowest screwnbsp;lowards the left; by repeating this operation anbsp;few times, the contact will be precisely the samenbsp;both wires, and consequently the axis of thenbsp;telescope will be parallel to the plane of the instrument.

'l^o examine the glasses of a sextant, or quadrant. t � To find whether the two surfaces of auy one ofnbsp;the reflecting glasses be parallel, apply your eyenbsp;�d one end of it, and observe the image of somenbsp;object reflected very obliquely from it; if thatnbsp;itnage appears singly, and well defined about thenbsp;�^^Iges, it is a proof that the surfaces arc parallel;

the contrary, if the edge of the reflected image �Tpcars misted, as if it threw a shadow from it,nbsp;separated like two edges, it is a proof that the,nbsp;surfaces of the glass are inclined to eachnbsp;pther: if the image in the speculum, particularlynbsp;^ffiiat image be the sun, be viewed through a smallnbsp;telescope, the examination will be more perfect.

2. To find whether the surface of a reflecting glass be plane. Chusetwo distant objects, nearlynbsp;pn a level with each other; hold the instrumentnbsp;^0 an horizontal position, view the left hand objectnbsp;^'rectly through the transparent ]3art of the hori-fon glass, ancl move the index till the reflectednbsp;ynage of the other is seen below it in the silvered

part; make the two images unite just at the line of separation, then turn the instrument roundnbsp;slowly on its own plane, so as to make the unitednbsp;images move along the line of separation of thenbsp;horizon glass. If the images continue unitednbsp;without reeeding from eaeh other, or varying theirnbsp;respeetive position, therefleeting surfaee is a goodnbsp;plane. The observer must be careful that henbsp;does not give the instrument a motion about thenbsp;axis of vision, as that will cause a separation if thenbsp;planes be perfect.

3. To find if the two surfaces of a red or darkening glass are parallel and perfectly plane. It is difficult, nay, almost impossible, to procure thenbsp;shades perfectly parallel and good; they willnbsp;therefore, according to their different combinations, give diff erent altitudes or measures of thenbsp;sun and moon.

The best way to discover the error of the shades, is to take the sun�s diameter with a piece ofnbsp;smoaked glass before your telescope, all the vanesnbsp;being removed; then take away the smoakednbsp;glass, and view the sun through each shade andnbsp;the several combinations thereof. If the twonbsp;images still remain in contact, the glasses are good;nbsp;but if they separate, the error is to be attributed tonbsp;the dark glasses, which must either be changed,nbsp;or the error found in each combination must benbsp;allowed for in the observations. If you use thenbsp;same dark glasses in tlie observation as in the adjustment, there will be no error in the observednbsp;angle,¹

The darlc glasses are generally left to turn in their cells; so that, after one observation has been made, turning either of thetonbsp;Ijalf round, after another observation, half the difference thusnbsp;given is the error. Edit.

[ 279 ]

No instrument can be so conveniently used for taking angles in maritime surveying as Hadley'snbsp;^extant. It is used with equal facility at the mastnbsp;^^eac], as upon deck, by which its sphere of obser-yation is much extended; for, supposing manynbsp;islands to be visible from the mast head, and onlynbsp;iiitc from deck, no useful observation can be madenbsp;any other instrument. But by this, angles maynbsp;taken at the mast head from the one visiblenbsp;object with great exactness; and further, takingnbsp;�iigles from heights, as hills, or a ship�s mastnbsp;^^^ad, is almost the only way of exactly describingnbsp;the figure and extent of the shoals.

It has been objected to the use of Hadley�s sextant for surveying, that it does not measure the horizontal angles, by which alone a plan can benbsp;haid down. This observation, however true innbsp;theory, may be obviated in practice by a littlenbsp;i^aution.

If an angle be measured between an object on elevation, and another near to it in a hollowqnbsp;the difference between the base, which is the horizontal angle, and the hypothenuse, which is thenbsp;^^gle observed, may be very great; but if thesenbsp;objects are measured, not from each other, butnbsp;from some very distant object, the difference between the angles of each from the very distantnbsp;object, will be very near the. same as the liorizon-tiil angle. This may be still further corrected, bynbsp;^Oeasuring the angle not bctw�ccu an object on anbsp;phinc and an object on an elevation, but bctw'ccnnbsp;the object on a plane and some object in the samenbsp;. Wcetion as the elevated object, of which the eyenbsp;Sufficiently able to judge.

TIoiv to observe the hori%o7ital angle, or atiguliit distance, betiveen two objects. First adjust thenbsp;sextant, and if' the objects are not small, fix onnbsp;a sharp top, or corner, or some small distinct partnbsp;in each to observe; then, having set the index tonbsp;o deg. hold the sextant horizontally, as abovenbsp;directed, and as nearly in a plane passing throughnbsp;the two objects as you can; direct the sightnbsp;tKrough the tube to the left hand object, till it isnbsp;seen directly through the transparent part of thenbsp;horizon glass; keeping that object still in sight,nbsp;then move the index till the other object is seennbsp;by reflection in the silvered part of the horizonnbsp;glass; then bring both objects together by thenbsp;index, and by the inclination of the plane of thenbsp;sextant when necessary, till they unite as one, ornbsp;appear to join in one vertical line in the middle ofnbsp;the line which divides the transparent and reflect'nbsp;ing parts of the horizon glass; the two objectsnbsp;thus coinciding, or one appearing directly belownbsp;the other, the index then shews on the limb thenbsp;angle whicli the two objects subtend at the nakednbsp;eye. This angle is always double the inclinationnbsp;of the planes of the twm reflecting glasses to onenbsp;another; and, therefore, every degree and minutenbsp;the index is actually moved from o, to bring thenbsp;two objects together, the angle subtended bynbsp;them at the eye will be twice that number of degrees and minutes, and is accordingly numberednbsp;so on the arc of the sextant; which is really annbsp;are of 6o degrees only, but graduated into 120nbsp;degrees, as before observed.

The angle found in this manner between two objects that arc near the observer, is not precise;nbsp;and maybe reckoned exact only when the objectsnbsp;are above half a mile ofi'. For, to get the anglonbsp;truly exact, the objects should be viewed from tho

center of the index glass, and not where the sight vane is placed; therefore, except the ohjcctsaresonbsp;demote, that the distance between the index glassnbsp;sight vane vanishes, or is as nothing comparednbsp;it, the angle will not be quite exact. This inaccuracy in the angle between near objects is callednbsp;parallax of the instrument, and is the anglenbsp;which the distance between the index glass andnbsp;^ight vane subtends at any near object. It is sonbsp;��nall, that a surveyor will seldom have occasionnbsp;1-0 regard it; but if it shall happen that great accuracy is required, let him choose a distant objectnbsp;Exactly in a line with each of the near ones, andnbsp;^^he the angles between them, and that will benbsp;the true angle between the near objects. Or, observe the angle between near objects, when thenbsp;^extant has been first properly adjusted by a distant object; then adjust it by the left hand object,nbsp;^vhich will bring the index on tlie arc of excessnbsp;heyond o degrees; add that excess to the anglenbsp;found between the objects, and the sum will benbsp;the true angle between them. If one of the objects is near, and the other distant, and no rc-^lotc object to be found in a line with the nearnbsp;^'ic, adjust the sextant to the near object, and thennbsp;Jake the angle between them, and the parallaxnbsp;'''111 be found.

1 � Set up such marks at A and C, as may he �cen when yon are standing at B. 21 Set o onnbsp;Jhe index to coincide with o on the qnadrantalnbsp;3. Hold the sextant in an horizontal ])osj-J^u, look through the sight and horizon glass atnbsp;'C mark A, and observe whether the image is di-under the object; if not, move the indexnbsp;Bd they coincide. 4. if the index divdsion of the

nonius is on the arc of excess, the indicated quantity is to be added to the observed angle; but it it be on the quadrantal arc, the quantity indicated 1�nbsp;to be subti�actcd: let us suppose five to be added.

5. nbsp;nbsp;nbsp;Now direct the sight through the transparentnbsp;part of the horizon glass to A, keep that ohjcct i�nbsp;view, move the index till the object C is seen bynbsp;reflection in the silvered part of the horizon glass.

6. nbsp;nbsp;nbsp;The objects being now both in view, move thenbsp;index till they unite as one, or appear in one vertical line, and the index will shew the angle subtended at the eye by the two objects: supposenbsp;75.20, to which add 5 for the index error, andnbsp;you obtain 75.25, the angle required; if the anglenbsp;be greater than 120�, which seldom happens iirnbsp;practice, it may be subdivided by marks, and thennbsp;measured.

No instrument can be more convenient or expeditious than the sextant, for setting of offsets. Adjust the instrument, and set the index to QO degrees; walk along the station line with the octantnbsp;in your hand, always directing the sight to thenbsp;farther station staff; let the assistant walk alongnbsp;the boundary line; then, if you wish to makenbsp;offset from a given point in the station line, stop atnbsp;that place, and wait till you see your assistant bynbsp;reflection, he is then at the point in the boundarynbsp;through whieh that offset passes; on the othernbsp;hand, if you wish an offset from a given point ornbsp;bend in the boundary, let the assistant stop at thatnbsp;place, and do you walk on in the station line tinnbsp;you see the assi.stant by reflection in the octanflnbsp;and that will be the point where an offset from thonbsp;proposed point or bend will fall.

The manner of using this instrument for too solution of those astronomical problems that arenbsp;necessary in surveying, will be shewn in its propernbsp;place.

11, plate 19, is a representation of a very con-'^enient pocket sextant, and contains a material im-Pi'ovement on the relieeting cross staff before described, nbsp;nbsp;nbsp;plate 14. In military operations.

Well as ti'igonometrical ones, it has been found c*f very essential service. AB a round brass boxnbsp;^hree inches in diameter, and one inch deep. ACnbsp;is the index turning an index glass within the box.nbsp;^5 a, are the two outside ends of the screws thatnbsp;i^ciifine an horizon glass also \yithin the box.

angle is observed by the sight being directed t^^rough an hole in the side of the box about D,nbsp;upon and through the horizon glass and the se-I'Ond opening at E, and the angle is read off tonbsp;minute by the divided arc and nonius F, G, H.nbsp;�oy sliding a pin projecting on the side of the box,nbsp;� dark glass is brought before the sight hole, notnbsp;shewn in the figure; by pushing the pin at b, anbsp;. ^J�k screen for the sun is interposed between thenbsp;I�^dcx and horizon glasses. I is an endless screw,nbsp;�onictimcs applied to give a very accurate motion,nbsp;'he the tangent screw to the index of a sextant,nbsp;a racked arc and pinion may be applied at

Make a mark upon the object, if acccssibkh equal to the heiglit of your eye from the ground.nbsp;Set the index to any of the angles from this table�nbsp;and walk from the object, till the top is broughtnbsp;by the glasses to coincide with the mark; then, itnbsp;the angle be greater than 45�, multiply the dis'nbsp;tance by the corresponding figure to the angl�dnbsp;in the table; if it be less, divdde, and the product�nbsp;or quotient, will be the height of the object abovenbsp;the mark. If the object be inaccessible, set thenbsp;index to the greatest angle in the table that thenbsp;least distance from the object will admit of, whe'hnbsp;by moving backwards and forwards, till the top ojnbsp;the object is brought to a level with the eye, and^nbsp;at this place set up a mark equal to the height ofnbsp;the eye. Then set the index to any of the lessernbsp;angles, and go backwards in a line from the object, till the top is made to appear on the levefnbsp;with the eye, or mark before set; set here anotheinbsp;mark, measure the distance between the two luarks�nbsp;and this divided by the diftercncc of the figuresnbsp;in the last colump, against the angle made usCnbsp;of, the quotient will give the height of the objectnbsp;above the height of the eye, or mark. Fornbsp;distance, multiply the height of the object by thenbsp;numbers against either of the angles made use o�

ilild the product'will be the (fistance of the object from the place where such angle was used.

If the index is set at 45� the distance is the freight of the object, and vice versa. The indexnbsp;to 00� becomes a reflecting cross staif, and isnbsp;^^sed according to the directions in page 282.

The sextants, as before described by the author, �'f the best kind, arc made of brass, or other me-fr^h The radii now most approved of arc fromnbsp;�''ix to ten inches, their arcs accurately divided bynbsp;an engine, and the nonii shewing the angles tonbsp;'frh 15, or even 10 seconds; but the flue divisionsnbsp;tlie latter are liable to be obliterated by thenbsp;fr'oqucnt cleaning of the instrument.

In many cases it happens that altitudes are to be laken on land by the sextant; which, for want ofnbsp;a natural horizon, can only be obtained by an ar-frficial one. There have been a variety of thesenbsp;^ort of instruments made, but the kind now to benbsp;frescribed is allowed to be the only one that cannbsp;fre depended upon. Fig. 12, IQ, representnbsp;Ifre horizon fixed up for use. A is a wood ornbsp;^etal framed roof containing two true parallelnbsp;�jasses of about 5 by 3| inches, fixed not toonbsp;frght in the frames of the roof. This servesnbsp;shelter from the air a wooden trough fillednbsp;j'^ith quicksilver. In making an observation bynbsp;fr with the sextant, the reflected image of thenbsp;moon, or other object, is brought to coin-*^Jde with the same object reflected from thenbsp;amp;^asses of the sextant; half the angle shewn uponnbsp;limt, is the altitude above the horizon ornbsp;�^'el required. It is necessary in a set of obser-^Pfrons that the roof be always placed the same

way. When done with, the roof folds up flat' ways, and, with the quicksilver in a bottle,nbsp;is packed into a portable flat case.

The difficulties that occuf in measuring with accuracy a strait line, render this method of sut'nbsp;veying altogether insufficient for measuring anbsp;piece of ground of any extent; it would be notnbsp;only extremely tedious, but liable to many errorsnbsp;that could not be detected; indeed there are verynbsp;few situations where it could be used withoutnbsp;King s surveying quadrant, or some substitute fotnbsp;it. The method is indeed in itself so essentiallynbsp;defective, that those who have praised it most,nbsp;have been forced to call in some instrument, asnbsp;the surveying cross and optical square, to thcifnbsp;aid. Little more need be said, as it is evident, aSnbsp;well from the nature of the subject, as from thonbsp;practice of the most eminent surveyors, that thenbsp;measuring of fields by the chain can only be proper for level ground and small inclosures; and thatnbsp;even then, it is better to go round the field andnbsp;measure the angles thereof, taking offsets fromnbsp;the station lines to the fences. That this work ,nbsp;may not be deemed imperfect, we shall introducenbsp;an example or two selected from some of the bestnbsp;writers on the subject; observing, however,nbsp;felds that are plotted from measured lines, are ah^'^d^nbsp;f hi ted nearest to the truth, xvhen those lines for fnbsp;their function angles that approach nearly to a rig^dnbsp;angle.

Example 1. To survey the triangular nctn AB C,fg. 22, plate g, by the chain and cross. Setnbsp;up marks at the corners, then begin at one of them,nbsp;and measure from A to B, till you imagine tha

}Ygt;u arc near the point D, where a perpendicular ^yould fall from the angle C, then letting the chainnbsp;lie in the line AB, fix the cross at D, so as to seenbsp;Birough one pair of the sights the marks at A andnbsp;then look through the other pair towards C,nbsp;if you see the marks there, the cross is at itsnbsp;��ight place; if not, you must move it backwardsnbsp;��nd forwards on the line AB, till you see the marksnbsp;C, and thus find the point D; place a mark atnbsp;set down in your field book the distance AD,nbsp;''nd complete the measure of AB, by measuringnbsp;ii'om D to B, 11.41. Set down this measure, thennbsp;Return to D, and measure the perpendicular D C,nbsp;li-43. Having obtained the base and perpendi-^pdar, the area is readily found: it is on this prin-^hgt;le that irregular fields may be surveyed by thenbsp;^dain and cross; the iheodoVite, or Hadley s sextant,nbsp;play even here be used to advantage for ascertain-��ig perpendicular lines. Some authors have givennbsp;^he method of raising perpendiculars by the chainnbsp;^nly; the principle is good, but the practice is toonbsp;PPerosc, tedious, and cycn inaccurate to be usednbsp;surveying; for the method, see Geometry on thenbsp;Ground.

Measure either of the diagonals, as AC, and the f\Vo perpendiculars DE, BF, as in the last pro-which gives you the above data for completing the figure.

�^3 plate 13. Having set up marks or station ' wherever it may be necessary, walk ovfr

the ground, and consider how it can lie most com� vcniently divided into triangles and trapcziun)Sgt;nbsp;and then measure them by the last two problems.

It is best to subdivide the field into as few separate triangles as possible, but rather into trape ziums, by drawing diagonals from corner to corner,nbsp;so that the jjcrpendicular may fall within the fi'nbsp;gure; thus the figure is divided into two trapeziums AB C G, G D E F, and the triangle GCD*nbsp;Measure the diagonal AC, and the two perpendiculars GM, BN, then the base GC, and thenbsp;perpendicular Dq; lastly, the diagonal Ob', andnbsp;the two pcrj)endiculars, p E, O G, and you havenbsp;obtained sulReicnt for your purpose.

We have already given our oi)inion of this instrument, and shewn how far only it can be depended upon where accuracy is required; that there are many eases wdiere it may be used to advantage, there is no doubt; that it is an expeditious mode of surveying, is allow'ed by all. I shall�nbsp;therefore, here lay down the general modes of surveying w'ith it, leaving it to the' practitioaer tonbsp;select those best adapted to his peculiar circumstances, recommending him to use the modes laidnbsp;dow n in example 3, in'preference to others, wherenbsp;they may be readily applied. He wdll also be nnbsp;better judge than I can be, of the advantages ofnbsp;Mr. Break'?, method of using the plain table.

Example 1. To take by the plain table the plot of a piece of land AB C D E, fig. 30, plate p, �dnbsp;one station near the middle, from wFence all thenbsp;corners niay be seen.

plan of the field, Jig. 36, is to be drawn; go round the field and set up objects at all the cornersnbsp;thereof, then put up and level your plain table,nbsp;turning it about till the south point of the needlenbsp;points to the N. point, or 300� in the compassnbsp;hox; screw the table fast in that position, and thennbsp;^raw a line P p parallel to one of the sides for anbsp;Meridian line. Now choose some point on thenbsp;paper for your station line, and make there a finenbsp;hole with a small circle of black lead round it;nbsp;Ihis is to represent the station point on the land,nbsp;^hd to this the edge of the index is to be appliednbsp;''^hen directed to an object.

Thus, apply the edge of the index to the point ^3 and direct the sight to the object at A', whennbsp;^uis is cut by the hair, draw a blank line along thenbsp;chamfered edge of the index from � towards A,nbsp;'^^ter this move the index round the point 0 as anbsp;^^nter, till you have successively observed throughnbsp;^he sights, the several marks at A, B, C, D, E;nbsp;^ud when these marks coincide with the sights,nbsp;draw blank or obscure lines by the edge of thenbsp;^udex to 0. Now measure the distance from thenbsp;�^^tion point on the ground to each of the objects,nbsp;^quot;^d set off by your scale, which should be as largenbsp;your paper will admit of, these measures onnbsp;heir respective lines: join the points AB, B C,nbsp;V O, D E, EA, by lines for the boundaries of thenbsp;which, if the work be properly executed,nbsp;''dl be truly represented on the paper.

N. B. It is necessary, before the lines are mea-^d, to find by a plumb-line the place on the ground under the mark 0 on the paper, and tonbsp;P an arrow at that point.

not so near the middle as in the foregoing instance* go round the field, and set up your objects at allnbsp;the corners, then plant the table where they maynbsp;all be conveniently seen; and if in any place a nearnbsp;object and one more remote are in the same line?nbsp;that situation is to be preferred. Thus, in thenbsp;present case, as at �, ^ coincides with h, and rnbsp;with Z'; the table is planted thereon, making thenbsp;lengthway of the table correspond to that of thenbsp;field. Make your point-hole and circle to reprC'nbsp;.sent the place of the table on the land, and appl)nbsp;the edge of the index thereto, so as to see throughnbsp;the slit the mark at a cut by the hair; then withnbsp;your pointrel draw a blank line from � towardsnbsp;do the same by viewing through the sights the several marks c, d, a,/, g, keeping the edge of thenbsp;index always close to �, and drawing blank line�nbsp;from � towards each of these marks.

Find by a plumb-line the place on the ground under � on the paper, and from this point men'nbsp;sure the distances first to g, and proceed on in thenbsp;same line to h, writing down their lengths as ynj*nbsp;come to each; then go to a, and measure from dnbsp;to �, then set off from your scale the respectivenbsp;distance of each on its proper line; after this*nbsp;measure to c, and continue on the line to h,nbsp;set off their distances; then measure from � to d*nbsp;and from c to � ; and, lastly from � to f, argt;nbsp;sett off their distances. Then, draw lines innbsp;from each point thus found to the next for bouii'nbsp;daries, and a line to cross the whole for a mcridia''nbsp;line.nbsp;nbsp;nbsp;nbsp;.

F'lg. 33, maybe supposed to be two fields, the table to be planted in the N. E. angle of th^nbsp;lower field, where the other angles of both ficl =nbsp;may be observed and measured to.nbsp;nbsp;nbsp;nbsp;.

'Within or without, taking at the same time offsets to the boundaries; suppose inside.

Set up marks at a, b, c, d, at a small distance rom the hedges, but at those places which younbsp;rotend to make your station points.

Then, beginning at O, plant your instrument and havdng adjusted it, make a fine pointnbsp;� 1 on that part of the paper, where it will benbsp;j^ost probable to get the whole plan, if not toonbsp;in one sheet; place the index to � I, andnbsp;ifect the sight to the mark at 0 6, draw a blanknbsp;�Oefrom 0 1 to 0 6, then direct the index to thenbsp;near the middle of the field, and afterwards tonbsp;naark at 0 2, then dig a hole in the groundnbsp;'^�tder 0 1 in the plan, and taking up the table.nbsp;Up an object in it exactly upright^ and measurenbsp;*^�rn it towards � 2, and find that perpendicularnbsp;^Sainst 218, the offset to the angle at the boun-is 157 links, which set off in the plan; thennbsp;jj'easuring on at 375 the oflset is but 63 and con-juues the same to 698, at both which set off 6 innbsp;plan; then measure on to 0 2, and find thenbsp;''^nole 1041 links, which set off in the blank linenbsp;ra\vn for it, and mark it 0 2; then taking outnbsp;, ^ object, plant the table to have � 2 over thenbsp;. ^ 0, when placed parallel to what it was on � 1;

the edge of the index-ruler touching both ^fions, the hair must cut the object at � 1, andnbsp;^ screw it fast.

^ -Now setting up objects in the by-angles a and h, turn the index to view that at a, and draw anbsp;^^nk line from � 2 towards it; then do the samenbsp;Ij^^^^rds h, Q 3 and the tree, which last crossingnbsp;d�.? '^�'avvn towards it from � 1, the intersectionnbsp;^^tJiines the place of the tree, which being re-as seen from all the stations, mark it innbsp;^ plan; then measure to a 412 links, and from b

353, and set them ofF in the blank lines drawn towards them; then set off the distance to thenbsp;boundaries in the two station lines produced, vh-151 in the next produced backwards, and 15 i�^nbsp;the first produced forwards; after this, draw thenbsp;boundaries from the an�:le where the first offsetnbsp;was made to the next, and so on round by a, Anbsp;through the 151 to the next angle; then, takinSInbsp;up the table, fix again the object as before, an^lnbsp;measure on the � 3, which set oft' 504 links, andnbsp;the offset to the angle in the boundary 27, andnbsp;then di�aw the boundary from it through the l5 tnnbsp;the angle at meeting that last drawn.

Now, taking out the object at � 3, plant thn table so as to have � 3 over the hole, whe�*nbsp;placed parallel to what it was at the former sta'nbsp;ti�ns, and screwed fast; then turn the index tf�nbsp;make the edge touch the place of the tree and 0nbsp;3 in the plan, and finding the hair cuts the tree�nbsp;turn the index to view � 4, and draw a blank h^^nbsp;towards it; then taking up the tabic, fix the obje^.nbsp;as before, and measure on to � 4, which set onnbsp;471 links, and the offset 23, and draw the bouii'nbsp;dary from the last angle through it to the next?nbsp;tiicn measure on in the station line produced t^nbsp;the next boundary 207 links, and the distance enbsp;� 4, from the nearest place in the same boundarynbsp;173, both which set off and draw the bounda^'ynbsp;from this last through the 207 to the angle.

Now taking out the mark, plant the table fo liave � 4 over the hole, when screwed fast in th*^nbsp;same parallelism as at the other stations; then�nbsp;after viewing again the tree, turn the index tnnbsp;view � 5, and draw a blank line towards it; th*^^nbsp;taking up the table, fix the object as before,nbsp;measuring on towards � 5, at 225 the neare ^nbsp;])Iace of the boundary is distant 121, which

off bearing forwards, as the figure shews; at 388 the perpendicular offset is Q, and at 712 it is 12,nbsp;both which set off in your plan; then measure onnbsp;O 5, and s�t it off at 912 links.

Take out the mark, plant the table to have O 5 Over the hole, when screwed fast in the same pa-^^Helism as before; then set up objects in thenbsp;hy-angles c and d, and after viewing the tree,nbsp;^^rn the index to view the objects at c, and � �,nbsp;^nd draw a blank line towards each; then measurenbsp;c 159, and from d 245, both which set off innbsp;your plan, and also the distance to the boundarynbsp;the next station line produced backward 95;nbsp;^bd now make up the boundary round by the se-t^eral offsets to the angles c and d\ then takingnbsp;your table, fix the object as before, and measuring towards 0 d, find at 102 the offset is 32,nbsp;'''hich set off; measure on to 0 6, and set it offnbsp;708, and the offset from it to the boundary isnbsp;^6 links.

Finding the blank line drawn from � 1 to in-^^rsect the point-hole here made for 0 6, do not P^^nt the table at � 6, but begin measuring fromnbsp;towards � 1, and finding at right angles to thenbsp;0 �, the offset to the angle is 42, set thatnbsp;in your plan; then measuring to � 1, 582,nbsp;'^nich, measuring the same by the scale in thenbsp;proves the truth of the work; the offset isnbsp;also 42, which set off, and draw the boun-ary from d, round by the several offsets, throughnbsp;.�ISlast to the angle; then measure on in the sta-line produced to the next boundary 88 links,nbsp;^ud set that off also, and draw the boundary fromnbsp;angle at the first offset, taken through it atnbsp;^ angle at meeting the last boundary; and thennbsp;9- meridian line be drawm, as in the former, thenbsp;plan is completed.

But if O 6 had not met in the intersection, of its distance from O 1 been too much, or too little?nbsp;you would very likely have all your work, exceptnbsp;the offsets, to measure and plot over again.

� The plain table surveyors, says Mr. Gardinefi when they find their work not to close right, donbsp;often close it wrong, not only to save time andnbsp;labour, but tbe acknowledging an error to thenquot;nbsp;assistants, which they are not sure they can amend?nbsp;because in many cases it is not in their power?nbsp;and may be more often the fault of the instrumentnbsp;than the surveyor-, for in uneven land, where thenbsp;table cannot at all stations be set horizontal, or ii^nbsp;any other one plane, it is impossible the wof^^nbsp;should be true in all parts; but to prevent greatnbsp;errors, at every � after the second, view wherevernbsp;it is possible, the object at some former �, beside*nbsp;that which the table was last planted at; becausenbsp;� if the edge of the index ruler do* not quite touch,nbsp;or but very little covers that 0 in the plan, whilstnbsp;it touches the 0 you are at, the error may henbsp;amended before it is more increased, and if itnbsp;ries much, it may be examined by planting aga�*^nbsp;the table at the former station, or stations.

If a field is so hilly, that you cannot, withord increasing the number of stations, see more thaijnbsp;one objeet backward, and another forward, an�nbsp;there is nothing fit within the field, as the sup'nbsp;posed tree in fig. 3�2, then set up an object canbsp;purpose to be viewed from all the stations, if P^�'nbsp;sible, for such a rectifier.

The lengthening and shortening of the papet? as the weather is moister or drier, often cause*nbsp;small error in plotting on the plain table; for bC'nbsp;tween a dewy morning, and the sun shining hnnbsp;at noon day, there is great difference, and earnnbsp;should be taken to allow for it; but that canno

te done in large surveys, and so ought not to be expected; indeed, those working by the degrees,nbsp;'vithout having their plan on it, are not liable tonbsp;this error, though they arc to the former; butnbsp;both ways are liable to another error, which is,nbsp;that the station lines drawn, or the degrees taken,nbsp;not in the line between the objects, nor pa-^�illel thereto; neither will this error be small innbsp;�hort distances, and may be great, if each 0 onnbsp;the plan, or the center used with the degrees, isnbsp;^ot exactly over the station holes; but to be mostnbsp;^Xact, it is the line of their sights that should benbsp;^hrectly over the hole.

Mr. Beighton made such improvements to his plain tables, by a conical ferril fixed on the samenbsp;staves as his theodolite, that the above errors, except that of the paper, are thereby remedied; fornbsp;l^he line of the sights, in viewing, is always overnbsp;the center of the table, which is as readily setnbsp;perpendicular over the hole, as the center of thenbsp;theodolite, and the station lines drawn parallel tonbsp;those measured on the land; and the tabic is setnbsp;horizontal with a spirit level by the same fournbsp;Screws that adjust the theodolite; therefore somenbsp;choose to have both instruments, that they maynbsp;either, as they shall think most convenient.

Let fig. 32 now be a wood, to be measured and plotted on the ouisiJe ; if on coming round to thenbsp;hfst o, the lines meet as they ought, the plan willnbsp;ho as truly made, as if done on the inside; butnbsp;here having no rectifier of the work as you go on,nbsp;you must trust to the closing of the last measurednbsp;hoe; and if that does not truly close with the first,nbsp;must go over the work again; and, without anbsp;better instrument than the common plain table,

you cannot be sure of not making an error in this case. .nbsp;nbsp;nbsp;nbsp;�

Suppose the table planted at � 1 on the outside, with paper fixed on it, and objects set up all the other stations on the outside, and dry blank-lines drawn from 0 1 on the paper towards 0 �nbsp;and 0 2; these done, take up the table, and setnbsp;up an object at 0 1; then, measuring from it towards 0 2, you find at 20 the offset to the firstnbsp;angle is 38, then at 280 the offset to the next angle is 26 links, both of which set off in the plan;nbsp;then at 394 the perpendicular offset to the nextnbsp;angle is 2o6; then at 698 the distance of the samenbsp;angle is 366 bearing backward, as may be seennbsp;in the figure, that by the intersection of thesenbsp;two offset lines the angle may be more trulynbsp;plotted; then measure the distance from this angle to the next 323, and from that to 698 place innbsp;the station line 280, which is the perpendiculatnbsp;offset; then by the intersection of these, that angle will be well plotted; then at 776 the offsetnbsp;to angle a is 48, and at 1012 the offset to h is 22,nbsp;both which set off in the plan, and at 1306 yonnbsp;make � 2; now draw the boundaries from thonbsp;first offset to the next, to the angle b. Asnbsp;there is no difficulty in taking the offsets fromnbsp;the other station lines, we shall not proceed fat'nbsp;ther in plotting it on the outside; for a sight ofnbsp;the figure is sufficient.

Some surveyors would plant their table at n place between 394 and 698 in the first stationnbsp;line, and take the two angles, which are herenbsp;plotted by the intersection of lines, as the by'quot;nbsp;angles a and h were taken at � 2 within thj?nbsp;field; but if the boundary should not be a strmfnbsp;line from one angle to the other, then their distance should be measured, and offsets takennbsp;the several bends in it.

You plot an inaccessible distance in the same manner as the tree in jig. 32; for if you couldnbsp;*^ome no nearer to it than the station line, yet younbsp;^^^ght with a scale measure its distance from Onbsp;Or 0 2, or any part of the line between themnbsp;m the plan, the same as if you measured it withnbsp;chain on the land; observing to make thenbsp;stations at such a distance from one another, thatnbsp;the lines drawn towards the tree may intersectnbsp;^s^ch other as near as possible to right angles,nbsp;drawing a line from each to Q 1;* writing downnbsp;^he degree the north end of the needle points to,nbsp;it should point to the same degree at each sta-tion; remove the table, and set up a mark at Q 1.

The imperfections in all the common methods �^fusing the plain table, are so various, so tedious,nbsp;^'^d liable to such inaccuracies, that this instru-so much esteemed at one time, is now dis-^^garded by all those who aim at correctness innbsp;their work. Mr. Break has endeavoured to remedy the evils to which this instrument is liable,nbsp;�y adopting another method of using it; a methodnbsp;quot;'hich I think does him considerable honour, andnbsp;'yhich I shall therefore extract from his completenbsp;of Land Surveying j for the informationnbsp;t the practitioner.

E.xample 1. To take the plot of afield ABCDEF, ^S- \ \, plate \3,from one station therein.

t-hoose a station from whence you can see every ^^fner of the field, and place a mark at each,nbsp;f'.^'^bering these with the figures 1, 2, 3, 4, amp;c.nbsp;�'t this station erect your plain table covered with

several the mark � always denotes a station or place instrument is planted. ITie dotted lines leading fromnbsp;�la-tipn to another, are the station lines; the black lines, thenbsp;� nitlaries; the dotted lines from the boundart^ to the station

paper, and bring the south point of the needle the flower-de-lucc in the box;' then draw a circlenbsp;O P Q. R upon the paper, and as large as thenbsp;per will hold.

Through the center of this circle draw the line NS parallel to that side of the table whichnbsp;parallel to the meridian line in the box, and thisnbsp;will be the meridian of the plan.

Move the chamfered edge of the index on O� till you observe through the sights the severalnbsp;marks A, B, C, and the edge thereof will cut thenbsp;circle in the points 1,2,3, amp;c. Then havin�nbsp;taken and protracted the several bearings, the dis- 'nbsp;tances must be measured as shewn before.

To draw the plan. Through the center �and the points 1, 2, 3, amp;c. draw lines 0 A, 0 B, 0 C�nbsp;� D, � E, 0 F, and make each of them equalnbsp;to its respective measure in the field; join thenbsp;points A, B, C, amp;c. and the plan is finished.

Example 2. To take the plot of a field AB C P; ^c.from several stations, fig. 14, plate 13.

Having chosen the necessary stations in the field, and drawn the circle 0 F QR, which yor*nbsp;must ever observe to do in every case, set npnbsp;your instrument at the first station, and bring thenbsp;needle to the meridian, which is called adjustingnbsp;the instrument; move the index on the ceiit^nbsp;�, and take an observation at yl, B, C, 0 2, R�nbsp;and the fiducial edge thereof will intersect thenbsp;circle in 1,2, 3, amp;c. Then remove your instrument to the second station in the field, and applying the edge of the index to the center 0 annnbsp;the mark 0 2 in the circle, take a back sight tj�nbsp;the first station, and fasten the table in thisnbsp;tion; then move the index on the center Q,nbsp;direct the sights to the remaining angular mark^nbsp;so will the fiducial edge thereof cut the circle

points 4, 5, 6, amp;c. The several distances being measured with a chain, the work in thenbsp;b^ld is finished; and entered in the book thus:

Having chosen O 1 upon paper to represent Ihe first station in the field, lay the edge of a pa-I'allel ruler to � and the mark 1, and extend thenbsp;other edge till it touch or lay upon � 1, andnbsp;olose by its edge draw a line 1,1 =520. Then laynbsp;the ruler as before to � and the mark 2, and, extending the other edge to � 1, draw thereby thenbsp;jine 1,2=344, which gives the corner B, as thenbsp;hue 1,1 does the corner ji. After the same man-^^or project � 2, together with the corners C, H.nbsp;�^gain, apply the edge of the ruler to � and thenbsp;point 4, and extend the other edge till it touch 0nbsp;and draw the line 2,4=370, which will givenbsp;l-be point or corner D, Thus project the reraain-^^g corners �, F, G, and the plan is ready fornbsp;olosing.

Adjust your plain table at the first station aiBCD, and draw the protracting circles and meridians NS; then by proposition i, project the angles or corners o�AB CD, B E FC, and also thenbsp;second station, into the points 1,2, � 2, 3, 4, 5, 6-Again, erect your instrument at the first station,nbsp;and lay the index on the center 0 and the marknbsp;0 2, and direct the sights by turning the tablenbsp;to the second station, then move the index till yonnbsp;observe the third station, and the edge thereofnbsp;will cut the circle in 0 3. Then remove the instrument to the third station, lay the index on thenbsp;center 0 and the mark 0 3, and take a backnbsp;observation to the first station; after which by thenbsp;last proposition find the points 7, 8, � 4, 9, in thenbsp;circle 0 P QR. As to the measuring of distances,nbsp;bothinthis andthetwosucceedingpropositions thatnbsp;shall be passed over in silence, having sufficientlynbsp;displayed the same heretofore; what I intend tonbsp;treat of hereafter, is the method of taking andnbsp;protracting the bearings in the field, with the manner of deducing a plan therefrom.

Choose any point � 1 for your first station; apply the edge of a parallel ruler to the center 0nbsp;and the point 1, and having extended the othernbsp;edge to 0 1, draw the line l,lz=370, which winnbsp;give the corner A. In like manner find the othernbsp;corners B, C, D, together with 0 2; which beingnbsp;joined, finishes the field AB CD. After the san^nbsp;method construct the other fields BE^C,nbsp;DIHK, IFGH, and you have done.

Example 4. To take the plot of a field AB C D ^ going round the same, fig. IF, plate 13.

Set up your plain table at the first station in the field; move the fidueial edge of the index pn thenbsp;'tenter �, and take an observation at the mark,nbsp;Placed at the second station, then will the samenbsp;fiducial edge cut the circle OP QR in the point 1.nbsp;*hen remove your instrument to the second sta-and placing the edge of the index on � andnbsp;^he point 1, take a back sight to the first, or last

station; then directing the index on the center 0 to the third, or next station, the edge thereof willnbsp;cross the circle in the point 2. In like manner thenbsp;instrument being planted at every station, a backnbsp;sight taken to the last preceding one, and the index directed forward to the next succeeding station, will give the protracted points 3, 4, 5, 6.

Choose any point � l, to denote the first sta-Lay the edge of a parallel ruler on the cen-0 and the point 1, and extend the other edge it touch 0 1, and draw by the side thereof thenbsp;1,2=530; then apply the ruler to 0 and thenbsp;^^'ark 2, and extend the other edge to 0 2, andnbsp;thereby the line 2,3=440; again, lay thenbsp;^fige of the ruler to � and the point 3, and thenbsp;^ther edge being extended to 0 3, draw the linenbsp;3,4=405; after the same method lay down thenbsp;�'crnaining stations, and the traverse is delineated.

Example 5. To take the plot of several fields B, C, D, hy circidat'ton,fig. \Q, plate 13.

Prom the projecting point 0 by last example, project the stations in A, into the points 1, 2, 3,nbsp;4; then the instrument being planted at the second station, frofn the same projecting point Onbsp;project that station the second in A into the pointnbsp;2^ denoting the instrument being planted anbsp;Second time at that station, which is done thus :nbsp;W the index to 0 and the point 2, and take anbsp;l^ack sight to the first station, that being the station immediately preceding that you are at in thenbsp;lleld book ; them on the center 0 take a fore observation at the next succeeding station, and thenbsp;ir^dcx will cut the circle in the point 2^. Thusnbsp;P^�oject every other remaining station.

BEMARKS.	0.	�A.	0.	REMARKS.
		�1 �		In Field A.
	53	O
	80	290
Corner	85	610
	Go	605
		�2		In Ditto.
	70	0
	65	560
		�3		In Ditto.
	60	0
	75	36s
	55	680
-		�4		In Ditto.
	50	0
	48	440	to�l	Close here A,
Return to		�2		In A.
		0
Corner	60	80	into	B.
	50	403
		�5		In B.
	50	0
	60	340
	50	650
		�6		In Ditto.
	50	0
Ag. Hedge	51	500
		560	by�	for closing C,
Closes at Cor. of A	50	620	=	the HedgCj
		682	to� 3	in A.
Return to		�4		In A.
		0
		60	70
		into C.
		380	60
		682	68

far more accurate than the old method of going round the field, and measuringnbsp;all the angles. It was first introduced into practice by Mr. Talhoi.

If the ticld contain four sides. Jig. 23, plate 0? begin at one of the corners A. 1. Measure thenbsp;Measure the side AB. 3. Tak^nbsp;4. Measure the side B C, andnbsp;angle BCD. 5. Measure the side CD, and thenbsp;angle CD A, and side DA, the dimensions arenbsp;finished; add the four angles together, and if thenbsp;sum makes 3()0�, you may conclude that yen'quot;nbsp;operations are correct; the above figure may henbsp;measured by any other method as taught before?nbsp;by measuring the diagonal, See.

If the field contain more than four sides, fS' 24, 25, 26, plate 9, having set up your marks, cu-dcavour to get an idea of the largest four-sidennbsp;figure, tliat can be formed in the field you ni'enbsp;going to measure; this figure is representednbsp;tJie figures by the dotted lines.nbsp;nbsp;nbsp;nbsp;'

Then beginning at A, take the angle BAk'� measure in a right line towards B, till you com^nbsp;against the angle/, there with your sextant,nbsp;cross, let fall the perpendicular � e, as taught in tn ^nbsp;method of the triangle, observing at hownbsp;chains and links this offset or perpendicular ^nbsp;from the beginning or point X, which note in y^^�^nbsp;field book, and measure ef, noting it also innbsp;field book; then continue the measure of the nnnbsp;AB to B; take the angle ABC, and measure 1nbsp;line B C; take the angle B C ID, and measure tnbsp;line CD; take the angle CDA, and measure tnbsp;line DA; observing as you go round, to let

B. Set the offsets to the right or left of your ^olumn of angles and chains, according as theynbsp;to the right and left of your chain line in the

, have already observed that this instrument Quid never be used where much accuracy is re-1 ired, for it is scarcely possible to obtain with anynbsp;t^^^^^^nty the measure of an angle nearer than tonbsp;Pq degrees, and often not so near; it has there-�quot;c long been rejected by aceurate surveyors.¹

This instrument takes the bearing of objects from station to station, by moving ti'.c index tiHnbsp;the line of the sights coincides vvith the next station mark; then counting the degrees betweennbsp;the point of the compass box marked N, and thenbsp;point of the needle in the circle of quadrants.-

Thus let it be required to survey a large woofb fig. 51,plate Q, by going round it, and observingnbsp;the bearing of the several station lines which cn-comjiass that wood.¹

I'lie station marks being set up, plant the cH' cumferentor at some convenient station as at a, thenbsp;tlowcr-de-lucc in the compass box being from you;nbsp;direct the sights to the next station rod h, and setnbsp;down the division indicated by the north end oinbsp;the needle, namely 200� 30', for the bearing of thenbsp;needle.

Remove the station rod h to c, and place the circumferentor exactly over the hole where thenbsp;rod b was placed, measuring the station lines, andnbsp;the oflsets from them to the boundaries; noquot;'nbsp;move the instrument, and place the center thereofnbsp;exactly over the hole from whence the rod bnbsp;taken. The flower-de-luce being from you, turUnbsp;the instrument till the hair in the sights coincide^nbsp;with the object at the station c, then will the northnbsp;end of the needle point to 292� 12', the bearingnbsp;of Z'c; the instrument being planted at c, andthonbsp;sights direeted io d, the bearing of cd, will bonbsp;33R45'. In the same manner proceed to takonbsp;the bearing of other lines round the wood, obscr'-ing carefully the following general rule;

Keep the flower-de-luce from you, and take tn^ bearing of each line from the north end ot thonbsp;needle.

Instead of planting the circumferentor at every station in the field, the bearings of the several linesnbsp;may be token if it be planted only at every other

So if the instrument had been planted at h, and the flovvcr-de-hieein thebox kept towards you whennbsp;you look back to the station a, and from you whennbsp;you look forwards to the station r, the bearings ofnbsp;the lines a h, and be, would be the same as beforenbsp;observed; also the bearings of the lines cJ, andnbsp;e, might be observed at lt;r/, and ef, and � a, at �;nbsp;^'O that instead of planting the instrument sixnbsp;times, you need in this case plant it but threenbsp;times, which saves some labour.

But, since you must go along every station line, measure it, or see it measured, the trouble ofnbsp;Setting down the instrument is not very great, andnbsp;then also you may examine the bearing of eachnbsp;hoe as you go along; and, if you suspect an errornbsp;the work, by the needles being acted upon bynbsp;^oine hidden magnetic power, or from your ownnbsp;Mistake, in observing the degrees that the needlenbsp;Pomts to, you'mny correct such error at the nextnbsp;station before you proceed.

As when the instrument was planted at a, and he sights directed at b, the flower-de-luce fromnbsp;the north end of the needle pointed to 260''nbsp;^0 ; now being come to direct the sights backnbsp;^ ^ mark at u, keeping the flowcr-dc-lucc towarda

SO shall the north end of the needle point to 30', as before at a, and then you may benbsp;sure the bearing of the line ab\% truly observed.

But if the needle doth not point to the same number of degrees, See. there hath been some errornbsp;in that observation, which must be corrected.

OF THE IMPROVED CIB.CUMFERENT0R, WITS MR. gale�s method OP USING IT; A METHOD THAT IS APPLICABLE WITH EaUAl¹nbsp;ADVANTAGE TO THE THEODOLITE, amp;C.

For the sake of perspicuity, it will be necessary to give again the example before used in page 222,nbsp;and that not only because it will exhibit morenbsp;clearly the advantages of Mr. Gales method,¹ butnbsp;because we shall have occasion to refer to it whennbsp;we come to his improved method of plotting; andnbsp;further, because I have thought this mode so advantageous, and the tables so conducive to accuracy and expedition, that I have caused occasionally the traversing quadrant to be engraved innbsp;smaller figures under the usual one of the limb ofnbsp;the theodolite.

Set the circumferentor up at ^,fg. 1, plate 18, take the course and bearing of B C, and measurenbsp;the length thereof, and so proceed with the sidesnbsp;CD, D E, EF, F G, GA, all the way round tonbsp;the place of beginning, noting the several coursesnbsp;or bearings, and the lengths of the several sides in 3-field book, which let us suppose to be as follow:

B. By north 7� west, is meant seven degrees to the westward, or left hand of the north, asnbsp;shewn by the needle ; by north 55� 15' cast, fifty-five degrees fifteen minutes to the eastward, ornbsp;*�ight hand of the north, as shewn by the needle.

In like manner, by south 62� 30' east, is meant sixty~two degrees and thirty minutes to the east-t^'Rrd, or left-hand of the south; and bv southnbsp;40 west, forty degrees to the westward, or rightnbsp;oand of the south.

The 21 chains, 18 chains 20 links, amp;c. arc the ^oiigths or distances of the respective sides, asnbsp;*lleasured by the chain.

When the boundaries of a survey have crooks bends in them, it is by no means necessary tonbsp;take a new course for every small bend; the bestnbsp;most usual way, is to proceed in a strait linenbsp;**�001 one principal corner to another, and whennbsp;are opposite to any bend in the boundary, tonbsp;^*easure the rectangular distance; termed the ofF-from the sti-ait line to the bend, noting thenbsp;�3^nie in the field book, together with the distancenbsp;the strait line from whence such offsets werenbsp;^ade. The offsets, as already observed, arc gc-**erally measured with an offset staff.

Tor the purpose of noting these offsets, it is ne-'^Cssary that the field book should be ruled into

the courses and distances; the adjoining column^ on the right and left hand to contain the measurenbsp;of the offsets, made to the right or left hand respectively; and the outside columns on the rigtgt;tnbsp;and left hand, to contain remarks made on th^nbsp;right and left hand, respectively, such as thenbsp;names of the adjoining fields, or the bearingnbsp;any remarkable object, amp;c.

Example, Let fig. 3, plate 18, represent a field to be surveyed, whose boundaries are crooked.

Set up the instrument at, or near any convenient corner, as at 1, and take the course, afid bearing, as before dii'ected, north 7� west, notenbsp;this down in the middle column of the field book;nbsp;and measure with the chain as before directed, tillnbsp;you come opposite to the first bend, so that thenbsp;bend be at right angles to the station line; notenbsp;the distance thus measured, 3 chains 6o links, ii^nbsp;the middle column, and measure the offset froinnbsp;thence to the bend 40 links, noting the samenbsp;the adjoining left hand column, because thenbsp;boundary is on the left hand of the station lin*^�nbsp;and note in the outside column, the name or ownernbsp;of the adjoining field, proceeding in the samenbsp;manner all round the field, noting the courses;nbsp;distances, offsets, and remarks, as in the fol'nbsp;lowing:

FIELD BOOK.

To survey a tract of Jatid, consisting of any nuni-her of fields lying together.

1. Take the outside boundaries of the whole tract as before directed, noting in your field booknbsp;where the particular fields intersect the outsidenbsp;boundaries; and then take the internal boundaries of the several fields from the place where theynbsp;so butt on the outside bounds.

Let fig. 4, plate 18, represent a tract of land to be surveyed, consisting of three fields.

First, begin at any convenient corner, as at A) and proceed taking the courses, distances, oft'setsgt;nbsp;and remarks, as in the following field book.

FIELD BOOK.

i^emarks.

'^Rterdonfarm.

acob Williams,�

�Rerton farm.

Off sets.	Station lines.	Off sets.	REMARKS.
	2d. N. 18�30.W 0nbsp;nbsp;nbsp;nbsp;58 9 �0	0 16 0 (Xl	The above corner.
	3d. N.12�15.E. O 00 4 nbsp;nbsp;nbsp;10 8 QO	0 00 0 60nbsp;0 00	West field.
0 00	4th. N. 37� E. 5nbsp;nbsp;nbsp;nbsp;40 17 00	0 40 0 00	To the corner of west field and north field.
0 70 0 00	5th. S. 76� 45. E. 6 nbsp;nbsp;nbsp;50 16 00	0 00	North field.
	6th. S.5�15.W. 9 60 14 nbsp;nbsp;nbsp;30 19 00 25 nbsp;nbsp;nbsp;00	0 56 0 10nbsp;0 70nbsp;0 00	To the corner of north and east fields. N. B.nbsp;A gate into each fieldnbsp;20 links from thenbsp;corner.
	7 th. S.74'�30,W. 2nbsp;nbsp;nbsp;nbsp;30 7 nbsp;nbsp;nbsp;65	0 86 0 00	East field.
To the place of beginning.
8th. At the corner of east and west field,nbsp;viz. The first offset onnbsp;the first station line.
	North. 8 nbsp;nbsp;nbsp;30 16 61	3 75 3 00	East field.

REMARKS.	Off sets.	Station lines.	Off sets.
		9 th.
		East.
	0 00	0	0 (X)
North field.		5 40	0 6o
		13 77	0 �)
		lOlh.
West field.	Back to the last station.
Corner of west		N.62'�03.\V
and north fields	0 00	0	0 00
against Iluntcn	0 70	3 40
don farm.	0 (Xl	7 m	0 00

REMARKS.

North field.

To take a survey of an estate, manor, and lordquot; ship. An estate, manor, or lordship, is in realh)nbsp;a tract of land, consisting of a number of held��nbsp;it differs in no respect from the last article, cS'nbsp;cepting in the number of fields it may contain, aU�nbsp;the roads, lanes, or waters that may run throng^'nbsp;it, and is of course surveyed in the same manner-It is best in the first place to take the whole ejnbsp;the outside boundaries, noting as above directednbsp;the several offsets, the several places wherenbsp;boundaries are intersected by roads, lanes, ornbsp;ters, the places where the boundaries of the re�'nbsp;pective fields butt on the outside bounds, ^nbsp;where the gates lead into the respective field�?nbsp;whatever other objects, as windmills, houses,nbsp;that may happen to be worthy of being taken nO'nbsp;tice of. If, however, there should be a large streaiRnbsp;of water running through the estate, therebynbsp;viding it into two parts, and no bridge nearnbsp;boundary, then it will be best to survey thatnbsp;Avhich lies on one side of the stream first, and f quot;nbsp;terwards that part which lies on the other

tlicreof: if the stream be of an irregular brearlih, both its banks forming boundaries to several fieldsnbsp;should be surveyed, and its breadih, where it en-and where it leaves the estate, be determinednbsp;Ihe rides of trigonometry.

In the next place, take the lanes or roads, (if such there be) that go through the estate,nbsp;rioting in the same manner as betore, where thenbsp;^^ivisions between the several fields butt on thosenbsp;lanes or roads, and where the gates enter intonbsp;Iliosc fields, and what other objects there may benbsp;^''orth noticing. Where a lane runs through annbsp;estate, it is best to survey in the lane, because innbsp;doing, you can take tlic offsets and remarksnbsp;on the right hand and the left, and therebynbsp;earry on the boundaries on each side at once. Ifnbsp;^ large stream run through and separate the estate,nbsp;d. should be surveyed as abovc-mcnlionid; butnbsp;^�nall brooks running through a meadow, requirenbsp;'(nly a few offsets to be taken from the nearest station line, to the principal bends or turning innbsp;Ihc brook.

In the last place take the internal divisions or boundaries between the several fields, beginningnbsp;any convenient place, before noted in the fieldnbsp;book, w'herc the internal divisions butt on the ont-fi/lo of the grounds, or on the lanes, amp;c. notingnbsp;always every remarkable object in the field

book.

Example. Let fg. 5, plate 18, represent an, estate to be surveyed; begin at any convenientnbsp;place as at A, w here the twm lanes meet, proceednbsp;Noting the courses, distances, amp;c. as before di-from A to B, from B to C, and so on tonbsp;li, B, G, H, I, K, L, M and A, quite round

Then proceed along the lane from A to N, ^ and I, setting the courses, distances, offsets,nbsp;remarks, as before.

This done, proceed to the interna! division^ beginning at any convenient place, as at O, ai^fnbsp;proceed, always taking your notes as before directed, from O to P, Q and R, so will you have withnbsp;the notes previously taken, the dimensions of thenbsp;north held; go back to Q, and ])rocced from �nbsp;to H, and you obtain the dimensions of the copf^e.nbsp;Take E S and S P, and you will have the dinieti'nbsp;s-ions of the home field; go back to S, and takenbsp;S T, and you will have the dimensions of the land?nbsp;applied to domestic purposes of buildings, yards?nbsp;gardens, and orchards, the particulars and sep*'!'nbsp;rate divisions of which being small, had better bonbsp;taken last of all; go down to N, and take NiD?nbsp;noting the offsets as well to the brook, as to thenbsp;fences, which divide the meadow from the southnbsp;atid west fields, so will vou have the dimension^nbsp;of the long meadow, together with the minute�nbsp;for laying down the brook therein; go back tonbsp;and take U B, and you obtain the dimensionsnbsp;the west field, and also of the south field; g�7nbsp;back again to N, and take N W,, noting the off'nbsp;sets as well to the brook aamp; to the fence whichnbsp;divides the meadow' from the cast field, thus wn'nbsp;you have the dimensions of the east field, and thenbsp;minutes for laying down the bi'ooks in the inc�'nbsp;lt;low; then go to L, and take L X, whichnbsp;you the dimensions of the cast meadow, and 9nbsp;the great field; and lastly, take the internal dtv^'nbsp;sions of the land, appropriated to the dome^f^*^nbsp;purposes of buildings, yards, orchards, gardens,nbsp;The method of taking the field notes is so lO'nbsp;tirely similar to the examples already given,that theynbsp;would be altogether unnecessary to repeat here.

To the surveyor there can need no apology for introducing, in this place, the method used bynbsp;Mihie, one of the most able and expert sur-^�yyors of the present day; and I think he will consider himself obliged to Mr. Milne, for communicating, with so much liberality, his deviations fromnbsp;common practice, as those who have hithertonbsp;iiiade any improvements in the practical part ofnbsp;��nrveying, have kept them as profound secrets, tonbsp;detriment of science and the young practitioner. As every man can describe, his own me-ttiods, in the clearest and most intelligible manner,nbsp;t have left Mr. Milne'?, in his own language.

'' The method I take to keep my field notes ia ^nrveying land, differing materially from those yetnbsp;Pnblishcd; if, upon examination, you shall think itnbsp;niay be useful or worthy of adoption, you may, ifnbsp;you please, give it a place in your treatise on sur-''eying.

What I, as well as all surveyors aim at, in going ^^-'out a survey, is accuracy and dispatch; the firstnbsp;Only to be acquired by care and good instru-^�^Cnts, the latter by diligence and long practice.

f'rom twenty years experience, in the course of y^�kich I have tried various methods, the followingnbsp;tvhat I at last adopted, as the most eligible fornbsp;Carrying on an extensive survey, cither in England^nbsp;Scotland, or any other cleared country,nbsp;j Slaving taken a cursory view of the ground thatnbsp;am first to proceed upon, and observed thenbsp;P aiiicst and clearest tract, upon which I can mea-j^�re a circuit of three or four miles, I begin at anbsp;1 convenient for placing the theodolite upon,nbsp;making a small hole in the ground, as at A,

flute 20, one assistant leading the chain, the othci* having a spade for making rnarks in the ground, Inbsp;measure in a direct line from the hole at A tonbsp;noting down, as I go along, on the field sketchnbsp;the several distances from place to place, anfinbsp;sketching in the figure of the road, as 75 linksnbsp;the east side of the avenue, 400 links, touching thenbsp;south side of the road, with an oflset of 40 links tonbsp;the other side, 500 links to the corner of a wood?nbsp;and 817 to the corner of another wood; whichnbsp;last not allowing me to carry the line farther, Inbsp;make a mark at B, where I mean to plant the iH'nbsp;strument, and beginning a new line, measure alongquot;nbsp;the side of the road 885 links to C, where I makenbsp;a mark in the ground, writing the same in my fieldnbsp;sketch, taking care always to stop at a place fromnbsp;which the last station can be seen, when a polenbsp;placed at it. Beginning a new line, and measuring in a direct line towards D, I have 345 links tonbsp;a line of trees, 390 links opposite a corner of palednbsp;inclosurcs, where I make a mark to have recoursenbsp;to, and 893 links to the end of the line, where inbsp;make a inai'k in the ground at D; from thenccnbsp;measuring in a direct line to E, I here have fiOOnbsp;links of steep ground, therefore make a mark mnbsp;the ground at the bottom of the steepness, so thatnbsp;when I come to take the angles with the theodolite, I may take the depression thereof; continumgnbsp;out the line, I have 1050 links to a bridge, 'vithnbsp;an offset of 20 links to the bridge, and 80 links tonbsp;the paling; at 1247 there is a line of trees on thenbsp;left, and here also the line comes to the noi't ^nbsp;side of the road, and 1388 links to the end of tjmnbsp;line at E, and so proceed in like manner round tnonbsp;circuit to F, G, H, I, K, L, M, N; and from thencenbsp;to A where I began. The holes, oi� marks madenbsp;in the ground, are represented by round dots

the field sfcctch; steepness of ground is expressed Oil the sketch by faint strokes of the pen, as atnbsp;1^5 F, amp;c. Figures ending a station line are written larger than the intermediate ones; oiFsets arenbsp;'Written down opposite the jfiaccs they were takennbsp;and arc marked either to the riglit, or left ofnbsp;the line on the field sketch, just as they happen tonbsp;on the ground. If the field sketch here givennbsp;quot;^'�is enlarged, so as to fill a sheet of paper, therenbsp;quot;^ould then be room for inserting the figures of allnbsp;^hc offsets, which the smallness of this does notnbsp;^hniit of.

In measuring these circuits, or station lines, too 'lUieh care cannot be taken by the surveyor tonbsp;tiieasure exact; I therefore, in doing them, alwaysnbsp;choose to hold the hindermost end of the chainnbsp;^�'yself

The next thing to be done is to take the angles bearings of the above described circuit, and alsonbsp;^hc altitude or depression of the different dccli-^'ties that have been measured up or down.

Having previously prepared a sheet of Dutch paper with meridian lines drawn upon it, as innbsp;2], also a horn protractor with a scale ofnbsp;^ins upon the edge thereof, and a small ruler

is � settle steady at the magnetic north, which ^j^^^^'^Pspensably necessary at first setting off, atnbsp;Same time taking care that no iron is so nearnbsp;� place as to attract the needle.

The best theodolite for this purpose is the large one, sec jig. 2, plate l6, the manner of using elnbsp;which I shall here describe.

The spirit Iceel C, having been previously adjusted to the telescope A, and the two telescopes ])ointing to the same object, I begin by levellingnbsp;the instrument, by means of the four screws ISfnbsp;acting between the two parallel plates N, first ii^nbsp;a line with the magnetic north, and then at rightnbsp;angles thereto. This accomplished, I turn thenbsp;moveable index, by means of the screw G, till dnbsp;coincides w�ith 180� and 300�, and with the magnetic north nearly, screwnng it fast by means of thenbsp;nut H, and also the head of the instrument bvnbsp;means of the pin L; I make the north point i''�nbsp;the compass box coincide with the needle very exactly, by turning the screw K; both telescopesnbsp;being then in Tie magnetic meridian, I looknbsp;through the lower one I, and notice what distaidnbsp;object it points to.

Then unscrewing the moveable, index by the aforesaid nut H, the first assistant having hee'gt;nbsp;previously sent to the station point marked N?nbsp;I turn about the telescope A, or moveable index�nbsp;by means of the nut G, till it takes up the polenbsp;now placed at N, raising or depressing the telescope by means of the nut E, till the cross hairs ofnbsp;wires cut the pole near the ground. This done�nbsp;I look through the lower telescope to see that hnbsp;points to the same object it did at first; if so, thenbsp;bearing or angle is truly taken, and reading it upo�jnbsp;the limb of the instrument F, find it to be 40nbsp;S. W, I then take my sheet of paper, and placingnbsp;the horn protractor upon the point A, pla^^nbsp;along the meridian line, passing through it, I pf*^'nbsp;off the same angle, and with the first ruler drawnbsp;faint line with the pen, and by the scale set o

length of the line, which I find to be, by my fii'st sketch,yi/rz/ɒ�20, 7690 links; and writing bothnbsp;heaving and distance down, as in plate 21, andnbsp;^g:iin reading oft' the angle to compare it withnbsp;''hat I have wrote down, I then make a signal tonbsp;fhc first assistant to come forwai'd with his pole;

the mean time I turn the moveable index about, tdl the hair or wire cut the pole which the secondnbsp;^i^sistant holds up at B, and looking through thenbsp;lower telescope, to see that it points on the samenbsp;''hject as at hrst, I read the angle upon the limbnbsp;'^fthc theodolite, 48� 55' S. E. and plotting it oft'nbsp;''Pon the sketch with the horn protractor, draw anbsp;^rait line, and prick off by the scale the length tonbsp;817 links; writing the same down on the fieldnbsp;^ketch, and again reading the angle, to sec that Inbsp;h^tve wrote it down right, I screw fast the move-^ble index to the limb, by means of the screw H,nbsp;�^quot;d making a signal to the second assistant to pro-''Ced with his pole, and plant it at the third sta-C, while I with the theodolite proceed to B,nbsp;�''living the first assistant with his pole where thenbsp;'�istruinent stood at A.

j. I^lanting the instrument by means of a plumb-'iic over the hole, which the pole made in the S^ound at the second station, and holding thenbsp;j^ovcable index at 48� 55' as before, and the limbnbsp;.f'^^lled, I turn the limb till the vertical wire innbsp;telescope cuts the pole at A nearly, and therenbsp;^cicw it fast. I then make the vertical wire cutnbsp;j pole very nicely by means of the screw K; thennbsp;'^king through the lower telescope to see and re-th^ � quot;'�hat distant object it points on, I loosennbsp;th^nbsp;nbsp;nbsp;nbsp;screw, and turn the moveable index till

Vertical hair in the telescope cuts the pole atC Ver*nbsp;nbsp;nbsp;nbsp;making the limb fast, I make it do so

the index. Reading the degrees and minutes M'hich the index now points to on the limb, I plotnbsp;it in my field sketch as before, the bearing or anglonbsp;being 48� 30', and the length 885 links. Sendingnbsp;the theodolite to the next station C, the same operation is repeated; coming to D, besides taking thenbsp;bearing of the line D E, I take the depression froiUnbsp;to the foot of the declivity 600 links below ibnbsp;which I perform thus: Havin'^ the instrumentnbsp;and the dt)able quadrant level, I try what part ofnbsp;the body of the assistant, who accompanies monbsp;for carrying the instrument, the telescope isnbsp;against, and then send him to stand at the marknbsp;at the bottom of the declivity, and making thenbsp;cross wires of the telescope cut the same part of hi^nbsp;body equal the height of the instrument, I findnbsp;the depression pointed out upon the quadrant tonbsp;be f of a link upon each chain, which upon sixnbsp;chains is four links to be subtracted from 1363nbsp;links, the measured length of the line, leavingnbsp;1384 links for the horizontal length, which Inbsp;mark down in my field sketch, plate 2, in thenbsp;manner there written. Also from the high groundnbsp;at D, seeing the temple O, I take a bearing to dnbsp;and plotting off the same in my field sketch, dra^vnbsp;a strait line; the same temple being seen fromnbsp;station L, and station M, I take bearings to it?nbsp;from each; and from the extension of these threenbsp;bearings intersecting each other in the point O,nbsp;a [iroof that the lines have been truly measured^nbsp;and the angles right taken.

Again, from station E to station F, the ground rises considerably, therefore, besides the bearing?nbsp;1 take the altitude in the same manner I did thonbsp;depression; and do the same with every conamp;i'nbsp;derablc rise and fall round tlae cii�cuit.

Coming to station N, and having the moveable ji^dex at �l� 15' N. W. taking up the pole at M,nbsp;making fast the limb, I loosen the index serew,nbsp;�Hid tarn it till I take up the pole at the first station A. This bearing ought to be 40� 55' N. E.nbsp;being the same number of degrees and minutes itnbsp;bore from A south-west, and if it turn out so, ornbsp;tvithin a minute or two, we eall it a good closure ofnbsp;the circuit, and is a proof that the angles have beennbsp;�'iccuratcly taken.

But if a greater error appears, I rectify the moveable index to 40� 55', and taking up the pole at A, Hiake fast the limb, and take the bearing to M,nbsp;^nd so return back upon the the circuit again,nbsp;till I find out the error. But an error will seldomnbsp;Hr never happen with such an instrument as herenbsp;described, if attention is paid to the lov/er telescope; and besides, the needle will settle at thenbsp;Same degree (minutes by it cannot be counted)

the box, as the moveable index will point out Hn the limb, provided it is a calm day, and nonbsp;extraneous matter to attract it. If it docs not, Inbsp;then suspect some error has been committed, andnbsp;Return to the last station to prove it, before I gonbsp;farther.

With regard to plotting off the angles with the born protractor in the field, much accuracy is notnbsp;�Accessary; the use of it being only to keep thenbsp;held sketch regular, and to preserve the figure ofnbsp;the ground nearly in its just proportion.

Coming home, I transfer from the first field sketch, plate 20, all the intermediate distances,nbsp;offsets, and objects, into the second, flute'll,nbsp;nearly to their just proportions; and then I amnbsp;^eady to proceed upon surveying the interior partsnbsp;it ^be circuit. For doing this, the little lightnbsp;beodolite represented at Jig. 4, plute 15, or thenbsp;Hiore complete one, fg. 7, plate 14, will be sutu-

cicntly accurate for comiuon surveying. It is unnecessary for me to describe farther the taking the angles and mensurations of the interior subdivisions of the circuit, but in doing of which, thenbsp;young surveyor had best make use of the hornnbsp;protractor and scale; because, if he mistakes anbsp;chain, or takes an angle wrong, he will be soonnbsp;sensible thereof by the lines not closing as he goesnbsp;along; the more experienced will be able to fillnbsp;in the work sufficiently clear and distinct withoutnbsp;them. The field sketch will then be such as isnbsp;given in plate 22, which indeed appears rathernbsp;confused, owing to the smalliiess of the scalenbsp;made use of to bring it within compass; but ifnbsp;this figure was enlarged four times, so as to fill anbsp;sheet of paper, there would then be room for entering all the figures and lines very distinctly.

Having finished this first circuit; before I begin to measure another, I examine the chain I havenbsp;been using, by another that has not been in use,nbsp;and find that it has lengthened more or less, asnbsp;the ground it has gone through was rugged ornbsp;smooth, or as the wire of which it is made is thicknbsp;or small; the thick or great wire drawing ornbsp;lengthening more than the small.

The careful surveyor, if be has more than two or three da3's chain work to do, will take care tonbsp;have a spare chain, so that he may every now andnbsp;then correct the one by the other. The oflsetnbsp;staff is inadequate for this purpose, being toonbsp;short. For field surveying, when no uncommonnbsp;accuracy is required, cutting a bit off any link onnbsp;one side, 50 links, and as much on the other side,nbsp;answers the purpose better than taking away thenbsp;rings.

P'ieusuration becomes very sensible, notwithstand-the utmost care; therefore, surveying larger tracts of country, as counties or kingdoms, requiresnbsp;^ different process. Surveys of this kind arc madenbsp;ti'oin a judicious series of triangles, proceedingnbsp;from a base line, in lengtli not less than threenbsp;^�^iles, measured upon an horizontal plane with thenbsp;greatest possible accuracy. Thomas Milne.�

t'he following Tables are inserted for the occasional use of the young Surveyor.

�Ut-* Tables, each perpendicular ciiliiinn is of one denon'ination tlitough-^ all the Lateral ones are equal, hut o( diiferent denominations.

PLOTTING,

By plotting, we mean the making a draught of the land from the field notes. As the instrumentsnbsp;necessary to be used by the surveyor in taking thenbsp;dimensions of land, are such wherewith he may_nbsp;measure the length of a side, and the quantity ofnbsp;an angle in the field; so the instruments com-inonly used in making a plot or draught thereof,nbsp;are such wherewith he may lay down the lengthnbsp;of a side, and the quantity of an angle on paper.nbsp;They therefore consist in scales of equal parts fornbsp;laying down the lengths or distances, and protrac'nbsp;tors for laying down the angles.

Scales of equal parts are of different lengths and differently divided; the scales commonly used bynbsp;surveyors, are called feather-edge scales; these arOnbsp;made of brass, ivory, or box; in length about lOnbsp;or 12 inches, but may be iDade longer or shorternbsp;fit pleasure. Each scale is decimally divided, thenbsp;whole length, close by the edges, which are madenbsp;sloping in order to lay close to the paper, and num-bered 0, 1, 2, 3, 4, amp;c. which are called chains,nbsp;and every one of the intermediate divisions is tennbsp;links, the numbers are so placed as to reckonnbsp;backwards and forwards; the comnienccment of

scale is about two or three of the larger divisions from the fore end of the scale; these are numbered backwards from o towards the left iiandnbsp;Wilh the numbers or figures 1, 2, 3, amp;c. thesenbsp;scales are often sold in sets. Secj^^. '2, flate 22.

The application and use of this scale is easy and. *^xpcdiLiouSj for to lay down any number of chai nsnbsp;^I'oin a given point in a given line; place the edge.

the scale in such a manner that the o of the scale may coincide with the given point, and the edgenbsp;the scale wdth the given line; then wdth the pio-^meting pin, point off from the scale the givennbsp;�^^stance in chains, or chains and links.

F G H, flate 2, represents a new scale or ^'ithcr scales of equal parts, as several may be laidnbsp;uowu on the same instrument; each scale is cii-t'lded to every ten links, or tenths of a chain,nbsp;'''hieh are again subdivided by their respective no-l^ius divisions into single links; the protracting pinnbsp;moved with the nonius by means of the screw d,nbsp;that the distances may be set off with suchnbsp;St'Cat accuracy, as not to err a single link in set-off any extent, Avhich in the scale of four

, I K L, fJate 2, answers the same purpose as foregoing instrument, and may be used as anbsp;Protractor also.

tor Mr. Gale\ method of plotting, a method tiieh will recommend itself to every attentivenbsp;^tirveyor, two scales should be used, one of aboutnbsp;inches long, the other about 10 inches, eachnbsp;J^'ided on the edge from one end to the other; anbsp;.^risp should be fixed on tbe shorter scale, wherebynbsp;,^^'^ray at pleasure be so fixed to the other scale,nbsp;fo rnove along the edge thereof at right angles,nbsp;� purpose of laying off perpendicular lines.

Fig. F G H, plale 1, represents an instrument for this purpose, with different scales on thenbsp;F G; both this, and that represented at Jig. IK F�nbsp;form excellent parallel rules. They are also madenbsp;only with the clasp, if desired.

The protractor is a circle or semicircle of thif* brass, divided into degrees, and parts of a degree;nbsp;on the outer edge.

The common semicircular protractor, jig. 2; Jgt;late 3, is of six or eight inches diameter, the lim'^nbsp;divided into 180 degrees, and numbered bodinbsp;ways, 10, 20, 30, amp;c, to 180; each degree is sid^'nbsp;divided into two parts. In the middle of the dia'nbsp;meter is a small mark, to indicate the center of th*^nbsp;protractor; this mark must be always placed onnbsp;the given angular point.

The common circular protractor is more usefn* then the semicircular one; the outside edge is tl�'nbsp;vided into 300 degrees, and numbered 10, 20, 3Gnbsp;to 3�0; each degree is subdivided into halve�*nbsp;In the middle of the diameter is a small mark;nbsp;which is to be placed on the angular point, whennbsp;an angle, is to be protracted; 'the diameter, reprO'nbsp;senting a meridian, must be placed on the meO'nbsp;dian of any plan, where the bearing of any objectnbsp;is to be laid down. The application of this instrn'nbsp;ment is so easy and simple, that examples ofnbsp;use arc unnecessary here.

Fig. Topiate 11, and 4 and 5,plate 17, represent' the three best circular protractors; the princip'^nbsp;is the same in all; the difference consists in sup*^'nbsp;rioiTty of execution, and the conveniences thnnbsp;arise from the construction. Fig. 4, plaie l7;nbsp;a round protractor, the limb accurately divided t�nbsp;360�, each degree is divided into two parts, whicnbsp;are again subdivided to every minute by thenbsp;nius, which moves round the limb of the protrac*

oil a conical center; that part of the index *^eyond the limb has a steel point fixed at the end,nbsp;'^'hosc use is to prick off the angles; it is in a di-^ ^e't hiiG with the center of the protractor, and thenbsp;index di vdsion of the nonius, all which is evidentnbsp;�I'oin an inspection of the ligurc.

The common method of plotting is this: take a �'Jii'-ct of paper of convenient size, draw a linenbsp;^nercon, to represent the magnetic meridian, andnbsp;Ussigii any convenient point therein to representnbsp;place where the survey commenced; lay thenbsp;^%e of the protractor on this meridian line, andnbsp;ringing the center thereof to the point so assignednbsp;represent the plaee of beginning, mark off thenbsp;^^grees and minutes of the first course or bearingnbsp;the limb of the protractor, and draw a linenbsp;^oni the place of beginning through the point so

any convenient point therein, as A, to represert^ the place of beginning the surv^cy; lay the edge f�¹nbsp;the protractor on the line NAS, with the ccntc''nbsp;thereof at the point A, and mark off seven deg''^^^nbsp;071 the limb to the westward, or left hand otnbsp;north, and draw the line AB through the poi^^nbsp;so marked oft', making the length thereotnbsp;chains by the scale of equal parts. �Znly.nbsp;another meridian line N B S, through the pointnbsp;parallel to the former.¹ Lay the edge of thenbsp;tractor on this second meridian line N B S, vvjmnbsp;the center thereof at the point B, and marknbsp;55� 15' on the limb to the right hand, or east'nbsp;ward of the north; and draw the line B C throng^nbsp;the point so marked off, making the length thereo¹nbsp;18 chains 20 links by the scale of equal paft^nbsp;Sdly. Draw another meridian line N C S, throng¹�nbsp;the point C, parallel to the former; lay the edg^nbsp;of the protractor to this third meridian line N C b;nbsp;with the center thereof at the point C, andnbsp;down the third course and distance C D, innbsp;same manner as before; and so proceed withnbsp;the other lines, DE, E F, EG, GA; and ifnbsp;last line shall terminate in the place of beginning?nbsp;the work eloses, as it is called, and all is right. Bnfnbsp;if the last line do not terminate, in the place n¹nbsp;beginning, there must have been a mistake eith^^nbsp;in taking the notes, or in the protraction of thein �nbsp;in such case, therefore, it will be necessary fn g�jnbsp;over the protraction again, and .if it be not fonn'nbsp;then, it must of course be in the field notes,nbsp;correct which, if the error is material, theynbsp;be taken again.

The foregoing method of plotting is liable to inaccuracies of practice, on account of hav-a new meridian for every particular line of tlicnbsp;purvey, and on account of laying oft every newnbsp;'��le/cow the pohit of termhuUion of the preceedingnbsp;whereby any little inaccuracy that may hap-in laying down of one line is communicatednbsp;the rest. But there is a second method of plot-by which these inconveniences are avoided,nbsp;''lul by which also the accuracy or inaccuracy ofnbsp;field work is decided with precision and ccr-^�linty; I would, therefore, recommend this secondnbsp;�'Method to the practitioner, as far preferable to anynbsp;^Ther I have seen.

2lt;r/ Method. Take out from the first table in the '^Ppe7tdix to this Work the northings, southings, east-and westings, made on each of the several linesnbsp;*^^^thc survey, placing them in a kind of table innbsp;^heir respective columns; and, if the sum of thenbsp;l^'^rthings be equal to the sum of the southings, andnbsp;sum of the castings equal to the sum of thenbsp;quot;'estings, the work is right, otherwise not.*

f hen in an additional column put the w'hole Quantity of northing or of southing made at thenbsp;'^^�inination of each of the several lines of the sur-

^ * The truth of this obsei-vation cannot but appear self-evi-the reader. For the meridians within the limits of an �quot;'�ary survey having no sensible difference from parallelism,nbsp;^'^cessarily follow, that if a person travel any way soe-hlanbsp;nbsp;nbsp;nbsp;small limits, and at length come round to the

''quot;here he sat out, he must have travelled as for to the West as to the southward, and to the eastward as to thenbsp;�liffinbsp;nbsp;nbsp;nbsp;though the practical surt'eyor will always find it

vey; which will be detennincd by adding or siih-tracting the northing or southing made oil each particular line, to or from the northing or southingnbsp;made on the preceding line or lines. And ii^nbsp;another additional column, put the whole quantity^nbsp;of easting or westing made at the termination olnbsp;each of �ie several lines of the sur\'ey, which wiUnbsp;be determined in like manner by adding or sub'nbsp;tracting the casting or westing made on each pat'nbsp;ticuhir line, to or fro�i the casting or westing madenbsp;on the preceding line or lines.

The whole quantity of the northings or southings, and of the eastings or westings, made at the terminations of each of the several lines, being thusnbsp;contained in these two additional columns, thenbsp;plot may be easily laid down from thence, by ^nbsp;scale of equal parts, without the help of a protractor.

It is best, however, to use a pair of scales with u clasp,¹ whereby the one may at pleasure be so fastened to the other, as to move along the edgenbsp;thereof at right angles, so that the one scale maynbsp;represent the meridian, or north and south liue^nbsp;the other, an east and west line. It may also benbsp;observed, that in order to avoid taking the scalesnbsp;apart during the work, it will be necessary thatnbsp;the whole of the plot lay on one side of that scalenbsp;which repecsenls the meridian; or, in other words?nbsp;the stationary scale should represent an assumednbsp;meridiem, laying wholly on one side of the survey-On this account it will be necessary to note iu ^nbsp;third additional column of the preparatory table?nbsp;the distances of each of the corners or terminations of the lines of the survc}g east or west fromnbsp;such assumed meridian. In practice, it is rather

WiDfe convenient that this assumed meridian should on the west side, than on the cast side of thenbsp;^Urvey^

�1 he above-mentioned additional column, as has already observed, contains the distance ofnbsp;^:ich'of the corners or terminations of the lines ofnbsp;survey, eastward or westward, li'om that mag--l^'^hc meridian which passes through llic place ofnbsp;�*^giuning the survey. In order, therefore, thatnbsp;assumed meridian should lay entirely withoutnbsp;survey, and on the west side thereof, it will onlynbsp;necessary that its distance from that meridian,nbsp;Mheh passes through the place of beginning,nbsp;��nould be somewhat greater than the greatestnbsp;^'*antity of wmsting contained in the said secondnbsp;'Additional column. Let then an assumed numbernbsp;^oiiicwhat greater than the greatest quantity ofnbsp;|''csting contained in the second additional co-be placed at the top of the third additionalnbsp;I^A^Uimn, to represent the distance of the place ofnbsp;beginning of the survey from the assumed meri-d'an. Let the several castings, contained in thenbsp;^AAcond additional column, be added to this as-^AAined number, and the several w estings subtractednbsp;it; and these sums and remainders beingnbsp;|cspectively placed in the third column, will shewnbsp;distance of the several corners or t erminationsnbsp;the lines of the survey from the assumed meridian.

^ ^63 in case a single sheet should not be large ^Atough; and on the left hand of the intended plotnbsp;a pencil line to represent the assumed meri-A)n which lay the stationary scale. Place thenbsp;A^eablc scale to any convenient point ou the

edge of the former, and point off by tlie edge of the latter, according to any desired number ofnbsp;chains to an inch, the assumed distance or nuin-her of chains contained at the top of the above-mentioned third column of the preparatory table-Move the moveable scale along the edge of tbt�nbsp;stationary one, to the several north or south distances contained in the first of the above-mcnti'nbsp;oned additional columns; so will the points tbe^nbsp;marked off represent the several corners or ternii'nbsp;nations of the lines of the survey; and lines bci�^nbsp;drawn from one point to the other, will of cours*^nbsp;represent the several lines of the survey.

Let it be required to make a plot of draught of the field notes, p. 222, accordingnbsp;Mr. Gales method.

T. he northings and southings, eastings and ^vest' ings in the above tabic, arc'takcn from thenbsp;table in the Appendix; thus, first find the.emirs'-'nbsp;7 degrees in the table, and over against 21 chains,nbsp;in the column marked dist. you nave 20.843 in th^nbsp;column marked N. S. which, rejecting the rightnbsp;hand figure 3 for its insignificancy, is 20 chain'

links for the quantity of 7iorlhing, and in the 'Column marked E. W. you have 2.559, very nearnbsp;ohains and 50 links for the westing made on thatnbsp;vourse and distanee. 2. Find the next coursenbsp;^^�15 in the table, and over against 18 chains innbsp;I'he column dist. you have 10 chains 26 links innbsp;Ihc column marked N. S. and 14 chains 79 linksnbsp;the column marked E. W. and over againstnbsp;links in the column dist. you have 11 links innbsp;column N. S. and 16 links in the columnnbsp;�parked E. W. which, put together, make 10nbsp;Chains 37 links for the northing, and 14 chainsnbsp;�a links for the easting made on the second course;nbsp;�*^d so of the rest;

The north and south distances made from the Face of beginning to the end of each line, confined in the next column, are determined thus.

the first course, 20 chains and 84 links of l^iaithing was made; on the second course, 10chainsnbsp;links of northing, which, added to the pre-makes 31 chains 21 links of northing; onnbsp;third course was made 6 chains 65 links ofnbsp;^Fithing, which subtracted from the precedingnbsp;ri 'trains 21 links of northing, makes 24 chainsnbsp;flinks of northing; and so of the rest.

-The east and west distances made from the r ace of beginning to the end of each line, con-in the next right hand column, are deter-^ in the same manner; thus, on the firstnbsp;Was made 2 chains 56 links of westing;' onnbsp;second course 14 chains 25 links of easting,nbsp;56^rnbsp;nbsp;nbsp;nbsp;subtracting the preceding 2 chains

Ij , of westing, there remains 12 chains 39 � of easting; on the third course was madenbsp;Ppf^^^ios 77 links of easting, which added to thenbsp;,.,^v4iog i'2 chains 39 links, makes 25 chains

The distances of the end of each line from the assumed meridian, contained in the next rightnbsp;hand column, are thus determined. The first assumed number may be taken at pleasure, providednbsp;only that it exceeds the greatest quantity of westing contained in the preceding column, whereb)nbsp;the assumed meridian shall be entirely out of thenbsp;survey. In the foregoing example, the greatestnbsp;quantity of westing contained in the precedingnbsp;column is 2 chains 50 links; the nearest wholenbsp;number greater than this is 3 chains, which is accordingly taken and placed at the top, to repm-sent the distance between the assumed incridiaonbsp;and the place of beginning of the survey; fromnbsp;this 3 chains subtract 2 chains 56 links of westing, there remains 44 links for the distance between the terminations of the first line and thenbsp;assumed meridian. The 12 chains SQ links mnbsp;casting in the next step, is added to the assumednbsp;3 chains, which make 15 chains 39 links for thenbsp;distance of the termination, by the second brgt;e-from the assumed meridian. The 25 chains mnbsp;links of casting in the next step, being added mnbsp;like manner to the assumed 3 chains make 2^nbsp;chains l6 links for the distance of the terminatiddnbsp;of the third line from the assumed meridian;nbsp;so on, always adding the eastings and subtractidSnbsp;the westings from the first assumed number. ^nbsp;The preparatory table being completed, tak^nbsp;sheet of paper or more, joined together if neces^nbsp;sary, and near the left hand edge thereof rulenbsp;line as N. S. jig. 2, plate 18, to represent thenbsp;sumed meridian; on this line lay the stationdrgt;nbsp;.scale, and assuming � as a convenient point thejenbsp;to represent the point directly west from the P \nbsp;of beginning, bring the moveable scale tonbsp;point a, and lay off the first number contained

last-mentioned column of the preparatory table, viz. 3 chains, from ato A, and A will represent the place of beginning of the survey. Move thenbsp;loveable scale along the stationary one to the firstnbsp;^orth and south distance from the place of beginning, viz. 20.84 from a to b, and lay off the cor-^�esponding distance from the assumed meridian,nbsp;0-44, from b to B, and draw AB, so will AB represent the first line of the survey; again, movenbsp;the moveable scale along the stationary one, to thenbsp;third north and south distance from the place ofnbsp;beginning, viz. N. 31.21 from a to c; and lay offnbsp;the corresponding distance from the assumed meridian, viz. 15.39, from c to C, and draw BC, sonbsp;t'^ill B C represent the second line of the survey,nbsp;�^gain move the moveable scale along the station-^^ry one to the third north and south distancenbsp;troni the place of beginning, viz. 24.56, from anbsp;d, and lay off the corresponding distance fromnbsp;the assumed meridian, viz. 28.16, and draw CD,nbsp;trhich will be the third line of the survey. Pro-r^'^cd in the same manner till the whole be laid,nbsp;'^oivn, and AB C D E F G will be the requirednbsp;plot.nbsp;nbsp;nbsp;nbsp;^

'I'fiis method of plotting is hy far the most perfect, the least liable to error of any that has been con-*f lved. It may appear to some to require morenbsp;abour than the common method, on account ofnbsp;h� computations required to be made for thenbsp;Preparatory table. These computations arc how-made with so much ease and expedition, bynbsp;be help of the table in the appendix, that this ob-J^etion would vanish, even if the computation werenbsp;^0 other use but merely for plotthig-, but it must henbsp;^^^ed that these computations are of mucJi furthernbsp;10 determining the area or quantity of land

taincd with equal accuracy in any other way* When tins is considered, it will be found that thisnbsp;method is not only preferable on account of h�nbsp;superior accuracy, but is attended with less labomnbsp;on the whole than the common method.

If a pair of scales, such as are above recom' mended, be not at hand, the work may be laidnbsp;down from a single scale, by first marking off th*^nbsp;N. and S. distance on the line N. S. and aftci'nbsp;wards laying off the corresponding east distancednbsp;at right angles thereto.

7h plot the field notes, p. 313. I. Lay down ah the station lines, viz. 7� W. 21 chains, N. 55. i5jnbsp;E. 18. 20, amp;c. contained in the middle column ofnbsp;the field book, as before directed, without payio?nbsp;any regard to the offsets, until all the station lines?nbsp;represented in fig. 3, pl.\S, by dotted lines, be laidnbsp;down; then lay off the respective offsets at rigWnbsp;angles from the station line, at the respective diS'nbsp;tances at which they were taken, as is done afnbsp;fig. 3, a bare inspection of which will makenbsp;work perfectly plain.

whole of the outside boundaries, the station lio^^ first, the offsets afterwards, as directed in the pf^'nbsp;ceding articles; and then lay down the intern^nbsp;divisions or boundaries of the respective fields,nbsp;the same manner as is done 'n\fig. 4.

The protractor, whether a whole or a sciwci' cle, ought not to be less in diameter thannbsp;or eight inches, to insure the necessarynbsp;accuracy in plotting the angles of a survey. ^nbsp;degrees on the limb arc numbered various Jnbsp;but most commonly from 10� 20quot;, amp;c. to 30

�fc would be right to repeat the numbers the contrary way, when the breadth of the limb wnll ad-^�iit of it. Others are numbered 10� 20�, amp;c. to ^ 80�, and the contrary. Surveyors will suit them-slt;?lves with that kind best adapted to their modenbsp;'Of taking angles in the field; so that in whichnbsp;over way the limb of the instrument they surveynbsp;'^'ith is graduated, the protractor had best be the

same.

It is true, a surv�cyor of much practice will read off an angle mentally very readily, in which evernbsp;'�'�ay the instrument happens to be graduated.

For instance, I have in the field taken the angles made with the magnetic meridian, in which Oase I count no angle above 80� 59^; if the anglenbsp;ooiues to g0� 0', I call it due �. or due W. if it isnbsp;^9� 59', I write to it N. E. or S. E. or else N. W.nbsp;Or S. W. as it happens to turn out.

Now 89� 59' N. E. upon a protractor numbered lo 360� 0', and placing 300 to the north, reads thenbsp;Same; but 89� 59'S. E. reads 90^1', and 89� 59'nbsp;S. W. reads 269� 59'; and lastly, 89� 59' N. W.nbsp;*�eads 270� 1'.

Again, if the angle or bearing is due south, it ^oads on this protractor 180�; if 0� 1'' S. E. it readsnbsp;^^9� 5^'; and if 0� l' S. W. it is 180� 1'; and sonbsp;of any other intermediate angle or bearing.

The great inducement to surveyors for taking angles by the bearings the lines make withnbsp;I'I'e magnetic meridian, is the having the needle as

check in the course of the survey; and when the oircuit comes to be plotted, having it in his powernbsp;^0 prick off all the angles, or bearings of the cir-*^oit, at once planting of the protractor.

is furnished with a clamp for making it fast to the limb, while it is carrying from one station tonbsp;another.

The protractor for plotting this way of surveying had best be graduated 10, 20, amp;c. to 180, on the right hand, and the same repeated on the left?nbsp;and again repeated contrary to the former; but asnbsp;protractors for general use are graduated 10, 20,nbsp;amp;c. to 360, I shall here describe a very useful andnbsp;convenient one of this latter kind, and then proceed to give an example of plotting all the anglesnbsp;of a circuit by it from one station.

This protractor is represented at fig. 5, fJate 17-Its diameter, from pointer to pointer, is inches; the center point is formed by two lines crossingnbsp;each other at right angles, which are cut on anbsp;piece of glass. The limb is divided into degreesnbsp;and half degrees, having an index with a noninSnbsp;graduated to count to a single minute, and is fur-nished with teeth and pinion, by means of whichnbsp;the index is moved round, by turning a small nut.nbsp;It has two pointers, one at each end of the index,nbsp;furnished with springs for keeping them suspendednbsp;while they are bringing to any angle; and beingnbsp;brought, applying a finger to the top of the pointer, and pressing it down, pricks off the angle¹nbsp;There is this advantage in having two pointers,nbsp;that all the bearings round a circuit may be laid,nbsp;or pricked off, although the index traverses butnbsp;one half of the protractor.¹

The protractor, represented Jig. plate 11, is circular, of the same diameter as the foregoing; the center is also formenbsp;by the intersection of two lines at right angles to each other,nbsp;which are cut on glass, that all parallax may thereby be avoided Jnbsp;the index is moved round by teeth and pinion. The limbnbsp;divided into degrees and half degrees, and subdivided tonbsp;minute by the nonius; the pointer may be set at any convenieanbsp;distance from the center, as the socket which carries it

Let jf^. 5, plate 13, be the circuit to be plotted. � draw the magnetic meridian N. S.fig.Q, plateis,nbsp;^nd assigning a point therein for station 1, placenbsp;l^be center of the above described protractor uponnbsp;with 360� exactly to the north, and 180� to thenbsp;�outh of the magnetic meridian line N. S. Fornbsp;^he conveniency of more easily reading off de-Srees and minutes, I bring the nonius to the southnbsp;Side of the protractor, or side next to me; andnbsp;Seeing by the field sketch, or eye draught, 5,nbsp;^hut the bearing from station 1 to station 2 is 87�nbsp;^0quot; S. E. I readily conceive it wanting 2� 30' ofnbsp;being due east, or g0� 0'; therefore, adding mentally 2� 30' to 90� 0', I make 92� 30' on this protractor, to which I bring the nonius, and thennbsp;^ake a prick by pressing down the pointer of thenbsp;Protractor, and with my lead pencil mark it 1.nbsp;�^he bearing from station 2 to station 3 being 10�nbsp;15' N. E. seej?^. 5, I bring the nonius, for thenbsp;�^ke of expedition and ease of reading off, to 10�nbsp;I5' S. W. which is upon this protractor 190� 15',nbsp;�iitd pressing down the north pointer, it will pricknbsp;10� 15' N.E. which I mark with my pencil, 2.nbsp;The bearing from station 3 to station 4 beingnbsp;13' N, E. I readily see this is wanting 4� 47'nbsp;being due east; therefore I bring the nonius tonbsp;'''^thin that of due west, which upon this protrac-I'^r is 265� 13', and pressing down the N.E.nbsp;Pointer, it will prick off 85� 13' N.E. which Inbsp;mark 3.

� to the bar B C, and moveable with it, there is another bar hy �nbsp;nbsp;nbsp;nbsp;different scales of equal parts are placed, so that

gjj^^^nsferred to any distance from the center, containing the , nyniber of degrees marked out by the index; the u.se ofnbsp;be evident from Mr. Milne�s obsen'ations..

The beai'ing from station 4 to station 5 bcin^ 11� 5' N. W. I bring the nonius to 108'� 55',nbsp;with the opposite, or north-west pointer, prick offnbsp;11� 5' N. W. which I mark 4.

In like manner I prick off the bearings froii^ station 5 to 6, from 6 to 7, and from 7 to 1, whichnbsp;closes the circuit; marking them severally withnbsp;my pencil 5, 6, 7.

Then laying aside the protractor, and castin.? my eye about the tract traced by the pointer oinbsp;the protractor for the bearing from station hnbsp;marked 1, I apply the ruler to it and station 1, an^nbsp;drawing a line, mark off its length by compassesnbsp;and scale, eight chains, which fixes station 2.

Again, casting my eye about the tract traced by the pointer of the protractor, for the 2 bearing�nbsp;marked 2, I apply the parallel ruler to it arulnbsp;station 1, and moving the ruler parallel eastward�nbsp;till its edge touches station 2, I from thence drawnbsp;a line northward, and by scale and compasses marknbsp;its length, five chains, to station 3. Thus thenbsp;bearing lO� 15' N. E. at station 1, is transferred tonbsp;station 2, as will easily be conceived by supposingnbsp;the points a, b, moved eastward, preserving thegt;'^nbsp;parallelism till they fall into the points of stationnbsp;2 and 3, forming in Jig. 6, the parallelogramnbsp;a, b, 3, 2.

It is evident this bearing of 10� 15' N. E. is hy this means as truly plotted as if a second meridiannbsp;line had been drawn parallel to the first from station 2, and the bearing 10� 15' N. E. laid off fromnbsp;it, as was the common method of plotting previous to the ingenious Mr. William Gardiner com-^nbsp;municating this more expeditious and more accurate mode to the public. However, its accuracynbsp;depends entirely upon using a parallel ruler thatnbsp;moves truly parallel, which the artist will do

Station 3 being thus fixed, to find station 4, ^ Cast my eye about the tract traced by the pointernbsp;the protractor for the bearing marked 3, and applying the parallel ruler to it and station i , movenbsp;^Ite ruler parallel thereto northward, till it touchnbsp;^bition 3; and from thence draw a line north-cast-'''ard, and by scale and compasses marking itsnbsp;length, three chains, station 4 is found.

Now conceive the points c, d, moved north-'^�lt;ird, preserving their parallelism till they fill into the points of stations 3 and 4; and thus siation 4nbsp;has the same bearing from station 3, as c or d hasnbsp;h�oin station ].

. In the same manner I proceed to the next bcar-'�ig 4; and placing the ruler to the point made by the pointer of the protractor at 4 and station 1,nbsp;moving the ruler parallel till its edge touchnbsp;station 4; from thence draw a line, and mark offnbsp;*ts length, 5 chains, to station 5. Here e, f, isnbsp;transferred to 4th and 5th stations, and the. 5thnbsp;station makes the same angle with the meridiannbsp;rom the 4th station, as e, or f, does with the rne-^�fiian from the first station.

In like manner, applying the parallel ruler to station 1, and the several other bearings of the cir-^^�it 5, 6, and ^ ; the points g, h, will be transferrednbsp;� 5th and 6th, and i, k, to �th and 7th; and lastly,nbsp;.he bca ring from the 7th station will fall exactlynbsp;^'^to first, which closes the plot of the circuit.

, It is almost unnecessary to observe, that the 5gt;tted lines marked a, b, c, amp;c. fig. 6, are drawnnbsp;. only for illustrating the operation: all thatnbsp;^ necessary to mark in practice are the figuresnbsp;�nbsp;nbsp;nbsp;nbsp;3, 8ic. round the tract traced by the pointer

1. nbsp;nbsp;nbsp;The area or content of land is denominatednbsp;by acres, roods, and perches.

2. nbsp;nbsp;nbsp;The most simple surfaee for admensuratioonbsp;is the reetangular parallelogram, which is a plan^nbsp;four-sided figure, having its opposite sides equ^^*nbsp;and parallel, and the adjaeent sides reetangular tonbsp;one another; if the length and breadth be equahnbsp;it is called a square-, but, if it be longer one waynbsp;than the other, it is commonly called an ohlong-

It has been already observed, that the best mS' thod of taking the lengths is in chains and ligt;^^^ �nbsp;hence, the above-mentioned product will give th�nbsp;area in ehains and decimal parts of a chain. Annbsp;as 10 square chains make an acre, the area innbsp;will be shewn by pointing off one place of the decimals more in the product than there are bothnbsp;the multiplicand and multiplier-, which answers tn ^nbsp;same purpose as dividing the area iqto chains byl^^

and the roods and perches will be determined by ^multiplying the decimal parts by 4 and by 40.

Example. Suppose the side of the foregoing �f]uare be 6 chains and 75 links; Avhat is the area?

, Suppose the length of the foregoing oblong be 9 chains and 45 links, and the breadth 5 chains andnbsp;links; how many acres doth it contain?

A. R. P.

If the parallelogram be not rectangular, but oblique angled, the length must be multiplied by the perpendicular breadth, in order to give the area.¹

3. After the parallelogram, the next most simp^� surface for admensuration is the triangle.

perpendicular, let fall thereon from the opposite angle, and that product will be the area of the tri'nbsp;angle.-j-

ABC, right angled at B, let the base AB be 8 chains andnbsp;25 links, and the perpendicular BC 6 chains and 40 links;nbsp;how many acres doth it contain ?

noposite angle F, be 7 chains and 30 links; how ��iany acres doth it contain ?

It would be the same thing, if the perpendicu-l^t Were to be multiplied by half the base; or, if the base and perpendicular were to be multipliednbsp;together, and half the product taken for the area.

f � The next most simple surface for admensu-'�ation is a four-sided figure, broader at one end than at the other, but having its ends parallel tonbsp;?�te another, and rectangular to the base. Its areanbsp;fletermincd by the following

Add the breadths at each end together, ^�^d multiply the base by half their sum, and thatnbsp;Pi�oduct will be the area. Or otherwise, multiplynbsp;base by the whole sum of the breadths at each

end, and that product will be double the area, the half of which will be the area required.

Example. Let the base be to chains and links, the breadth at the one end 4 chainsnbsp;75 links, and the breadth at the other end 7 chainsnbsp;and 35 links; what is the area?

5. Irregular four-sided figures are called trap^' �zia. The area of a trapezium may be deterinin^^nbsp;by the following

Rule. Take the diagonal length from on� extreme corner to the other as a base, and multiplynbsp;it by half the sum of the perpendiculars, fallingnbsp;thereon from the other two corners, and that prO'nbsp;duct will be the area.

Example. \n the trapezium ABCD, let the diagonal AC be 11nbsp;chains and 90 links; the

If the diagonal AC were multiplied by the ^hole sum of the perpendiculars, that productnbsp;j'ould be double the area; the half of which wouldnbsp;the area required.

6. The area of all the other figures, whether re-�ular¹ or irregular, of how many sides soever the |}�ure may consist, may be determined either bynbsp;ividingthe given figure into triangles, or trapezia,nbsp;^^d measuring those triangles, or trapezia, sepa-^htely^ the sum total of which will be the area re-9^ired; or otherwise, the area of any figure maynbsp;^ determined by a computation made from the

The regular figures, polygons and elides, do not occur in surveying. On this head, therefore, I shall only ob-~ that the method of determining the area of a regular poly-gd¹� 'quot;^iiich is a figure containing any number of equal sides andnbsp;iiu k ingles, is, to multiply the length of one of the sides by thenbsp;pi of sides which the polygon contains, and then to multi-to^ product by half the perpendicular, let fall from the centernbsp;one of the sides, and this last product will be the area.nbsp;ji)5 length of the perpendicular, if not given, may be deter-hy trigoiiometiy, thus;

courses and distances of the boundary lines, ac^ cording to an universal theorem, which will benbsp;mentioned presently.nbsp;nbsp;nbsp;nbsp;,

7. The method of determining the area by dividing the given figure into trapezia and triangles? and measuring those trapezia and triangles separately, as in article 5 and article 3 of this part?nbsp;is generally practised by those land measurers,nbsp;who arc einployed to ascertain the number ofnbsp;acres in any piece of land, when a regular landnbsp;surveyor is not at hand; and for this purpose,nbsp;they measure with the chain the bases and perpendiculars of the several trapezia and triangles in �nbsp;the field.

Thus, for example, suppose the annexed figure be a field to be measurednbsp;according to thisnbsp;method, the landnbsp;measurer would m ea-

pezium A B C G; the base G D, anti thenbsp;pcrpcndicnlars K F,nbsp;and L C, of the tra-nbsp;nbsp;nbsp;nbsp;A.

Divide 360 degrees by double the mimber of sides contained n� the polygon, and that quotient will be half the angle at the ceH'nbsp;ter; then say, as the tangent, or half the angle at the center,nbsp;to halt the length of one of the sides; so is radius, to the p^^'nbsp;pendicular sought.

The area of a cikcle is thus determined; if the diameter be given, say, as 1 : is to 3.141592 :: so is the diameter : to the cir-cumlerence; or, if the circumference be given, say, as 3.l4J-5.f-is to 1 so is the circumference : to the diameter. Thennbsp;tiply the circumference by ^ of the diameter, and the producnbsp;will be the area.

Pt'ziiim GCDF; and the base D F, and perpen-�^^icular Al E, of the triangle D E F.

El order to determine where on the base lines perpendiculars shall fall, the land incasurctnbsp;^pminonly makes use of a square, consisting of anbsp;piece of wood about three inches square, \^-ithnbsp;�Apertures therein, made with a tine saw, from cor-to corner, crossing each other at right anglesnbsp;^11 the center; which, when in use, is tixed on anbsp;�Eatf, or the crosss; see jigs. 2, 3, and �, flate 14.

File bases and perpendiculars being thus mca-?^i'ed, the area of each trapezium is determined as article 5, and the area of the triangle as in arti-3 of this part; which, being added together,nbsp;amp;i''es the area of the whole figure required.

Ellis method of determining the area, where the ''�liole of each trapezium and triangle can be seennbsp;One view, is equal in point of accuracy to anynbsp;l^^etliod whatever. It may likewise be observed,nbsp;l^at a plot or map may be very accurately laidnbsp;^o\vn from the bases and perpendiculars thus nica-?^^i'ed; this method, however, is practicable onlynbsp;Open or cleared countries; in countries coverednbsp;l^bli wood, it would be altogether impractic.'ible,nbsp;'^�^ause it would be impossible to see from onenbsp;EU't of the tract to another,-so as to divide it intonbsp;le necessary trapezia and triangles. But in opennbsp;i^ountries, the only objection to this method, isnbsp;great labour and tediousness of the field work;nbsp;Ignbsp;nbsp;nbsp;nbsp;survey, the number of the trapezia and

.^^'anglcs would be so very great, that the nieasur-of all the several bases and perpendiculars be a very tedious business. The same endsnbsp;effectually answered, with infinitely less la-by taking the courses and distances of thenbsp;^'^odary hues, in the manner mentioned p. 310-

8. nbsp;nbsp;nbsp;Many surveyors who take their held notes ir*nbsp;courses and distances, make a plot or mapnbsp;thereof, and then determine the arm from the plo'^nbsp;or map so made, by dividing such plot or map intonbsp;trapezia and triangles, as in the above figure, andnbsp;measuring the several bases and perpendiculars tynbsp;the scale of equal parts, from which the plot otnbsp;map was laid down. �

By this method, the area will be determined something nearer the truth, but it falls short of thatnbsp;accuracy which might be wished, because theionbsp;is no determining the lengths of the bases and pot'nbsp;pendiculars by the scale, within many links; andnbsp;the smaller the scale shall be by which the plotnbsp;laid down, the greater, of course, must be the i�'nbsp;accuracy of this method of determining the area-

This method, however, notwithstanding its ni' accuraev, will always be made use of where tabl^'nbsp;of the northing, southing, easting and westing,nbsp;not at hand.

9. nbsp;nbsp;nbsp;Proper tables of the northing, southing, cast'nbsp;ing and westing, fitted purposely for the surveyor ^nbsp;use, such as are contained in the Appendix tonbsp;Tract, ought to be in the hands of every survey o'''nbsp;By these tables the area of any sun^ey is detet'nbsp;mined with great accuracy and expedition, bynbsp;easy computation from the following

ff the sum of the distances, on an east �vVest line, of the two ends of each line of thenbsp;^rom any assumed 77ieridian laying entirely out op ^nbsp;survey, betnultiplied by the respective northi^^^nbsp;SOUTHING made on each respective line\ thenbsp;UEB,�NCE between the sum of the north

�^ucTS, and the sum of the south products, �w// double the area of the survey.*

Por example. Let it be required to determine the area of the survey mentioned in p. 310, fromnbsp;field notes there given.

The northings, southings,castings and westings, also the north and south distances, and the eastnbsp;and voest distances made from the place of hegin-^^iiigto the end of each line; and likewise the distances of the ends of each line from the assumec}

Demonstration. Let AB C D E F be any survey,nbsp;let the east and westnbsp;��nes Aa, B b, C c, Dd, E e,nbsp;F f, be drawn from thenbsp;�gt;vds of each of the lines ofnbsp;survey to any assumed'nbsp;�'��eridian, as N. S. layingnbsp;''direly out of the survey.

I say, the difference between the sum of the products i, ^ff'Bbxab, Bb-hCcxbc, Cc Ddxed; and the sum ofnbsp;a * Products Dd Eexde, Ee Ffxef, Ff Aaxfa; will benbsp;the area of AB C D E FA.

douK^' by article 4, the former products will be, respectively, the areas of the spaces ABba, BCcb, and CDdc, andnbsp;will of course be double the area of the space Aadnbsp;A. In like manner, the latter products will be, respec-aijjj double the areas of the spaces DEed, EFfe, and FAafjnbsp;course be double the area of the spacenbsp;thenbsp;nbsp;nbsp;nbsp;But it is evident, that the difference between

of the space ABCDEFA; consequently, the dif-their doubles must be double the area of ABC �

meridian, have been already placed in their respective columns, and explained in the preparatory to the explanation of the method of plottingquot;nbsp;there recommended; but it will not be impropernbsp;to give them a place here again, together with thenbsp;further columns for determining the area; in ordernbsp;tliat the whole, together with the respective usesnbsp;of the several parts, may appear under one viequot;';nbsp;as follows.

Again, the diflerence between the sum of the products Mm-h Nn X mn, Nn Ooxnbsp;no, tiq-l-ltrx qr; and the sumnbsp;of the products Oo-|-PpXop, ^nbsp;Pp Clqxpq, ilr-|-MmXrm;nbsp;will be double the area of M N ^nbsp;O P Q R M.

For, reasoning as before, the sum of the first mentionedj^ ducts will be double the area of QqoUNMRQ MSriv^ignbsp;and the sum of the last mentioned products will be dounbsp;the area of �qoOPCl-1-MRrmM. Rut it is evident,nbsp;QqoOPQ�QqoONMRQ-MNOPgRM: conse�iuequot;^'^�

NOPQRM; and, consequently, the dift'erence of theiry-must be equal to double MN�P�RM. And the like m other possible figure.

N.	s.	E.	W.
20 84			2 56
10 37	6 65	H 45 12 77
	s 43		7 07
	'3 96	I 04
3 47			II 90
	5 64		7 23
34 lt;gt;8	34 68	28 76	28 76
The agreement between the sum of the northings, and the sum of thenbsp;southings, and between the sum ofnbsp;the eastings, and the sum of the
westings shews that the survey has
been truly taken.

/ nbsp;nbsp;nbsp;For the Plotting-.			/	For the Arc
North and	East and	Distances of	.Sums of the
south dis-	west distan-	the ends of	distances of
tances made	CCS made on	each line	the two ends
on the whole	the whole	from an as-	of each line	North pro-
from the	from the	sun;ed mcri-	from the as-	ducts, in
place of be-	place of be-	dian laying	sumed meri-	acres.
ginning to	ginning to	out of the	dian, for
the end of	the end of	survey.	multipliers.
each line.	each line.
		E. 3 00
N. 20 84	W. 2 56	E. 0 44	3 44	7 i68g6
N. 31 21	E. 12 39	�5 39	15 S3	16 41571
N. 24 56	E. 25 16	28 16	43 55
N. 16 13	E. 18 09	21 09	49 25
N. 2 17	E. 19 13	22 13	43 22
N. 5 64	E. 7 23	10 23	32 36	II 22892
0 00	0 00	3 00	13 23
				34 81359
			North prod. Subtract
The first and t^ird of these columns				2
only are used in plotting j		the middle
column is 0	nlv a preparatory step tor		Area	Acres
the distance	gt; contained in the third
column.
				Hoods
				Perches

South products, in acres.

7 46172

The three last columns relating to the area, arc the only ones that remain to be explained.

The first of these three columns contains the sums of the distances of the two ends of each lin�nbsp;of the survey from the assumed meridian; which,nbsp;according to the foregoing universal theorem, arcnbsp;multipliers for determining the area. These arenbsp;formed by adding each two of the successive numbers in the preceding left hand column together.nbsp;Thus 3.00 is the distance of the place of beginningnbsp;from the assumed meridian, and 0.44 is the dis'nbsp;tance of the end of the first line from the assumednbsp;meridian; and their sum is 3.44 for the first multiplier, Again 0.44 added to 15.39 rnakes 15.83nbsp;for the second multiplier; and 15.39 added tnnbsp;*28.16 makes 43.55 for the third multiplier; andnbsp;so of the rest.

The north products contained in the next column, are the products of the multiplication of the several northings contained in the column�nbsp;marked N, by their corresponding multipliers innbsp;the last mentioned column. Thus 7.16896 is thenbsp;product of the multiplication of 20.84 by 3.44;nbsp;an additional place of decimals being cut off,nbsp;give the product in acres.

The south products contained in the next right hand column are, in like manner, the products ofnbsp;the multiplication of the several southings in th^^nbsp;column marked S, by their corresponding multi'nbsp;pliers. Thus 28.9607 5 is the product of the multiplication of 6.65 by 43.55; an additional placenbsp;of decimals being cut off, as before, to give thenbsp;product in acres; and so of the rest.

The north products being then added together into one sum, and the south products into anothernbsp;sum, and the lesser of these sums being subtractei^nbsp;from the greater, the remainder, by the aho)'^

Universal theorem, is double the area of the survey: ^hich being divided by 2, gives the area required:nbsp;''iz. 51.73087, equal, when reduced, to 51 acres,nbsp;^ roods, and 30 perches Q-IO.*

Nothing can be more simple or easy in theope-^^tion, or more accurate in answering the desired ^nds, both with respect to the proof of the work,nbsp;plotting, and the computation of the area,nbsp;fhan this process. The northings, southings, east-'ugs and westings, which give the most decisivenbsp;proof of the accuracy or inaccuracy of the fieldnbsp;'Vork, are shewn by inspection in table 1. Fromnbsp;^*'ese the numbers or distances for the plotting arenbsp;formed by a simple addition or subtraction. Andnbsp;multipliers, for determining the area, arenbsp;formed from thence, by a simple addition, withnbsp;^Uuch less labour, than by the common method ofnbsp;flividing the plot into trapezia, and taking the mul-f�pl�ers by the scale of equal parts.

The superior accuracy and ease wdth w'hich uvery part of the process is performed, cannot, itnbsp;imagined, fail to recommend it to every practitioner, into whose hands this tract shall fall.

to. If the boundary lines of the survey shall uave crooks and bends in them, which is generallynbsp;1^0 case in old settled countries, those crooks andnbsp;l^onds are taken by making offsets from the stationnbsp;*oes; as explained in page 311. In these cases,nbsp;too area comprehended within the station lines isnbsp;determined as in the last example; and the areas

, * It must be observed, that, in order to determine the area y this method of computation, it is indispensibly necessary thatnbsp;6 Several lines of the survey be arranged regularly, one afternbsp;^aothgr, all the way round the survey, in case they should notnbsp;^ve been so taken in the field, because, without such arrange.-the northings, or southings made on each line, and thenbsp;ances from the assumed meridian, would not correspond with

comprehended between the station lines and the¹ boundaries arc determined separately, as in thenbsp;third and fourth articles of tliis part, from the ojf-sets and base lines, noted in the field book; andnbsp;are added or subtracted respectively, according asnbsp;the station lines shall happen to be within ornbsp;out the Held surveyed; which, of course, gives thenbsp;area required.

For example. Let it be required to determiaa the area of the survey mentioned in the exampk'�nbsp;page 312, from the ncld book there given.¹

The bearings and lengths of the station lines ai'C the same as in the last example; and, consc-qucntly, the area comprehended within those station lines must be the same; viz. 51.73087.

Then, for the area comprehended between the first station line and the boundary of the field,.nbsp;have (sec the field book, page 313) first, a basenbsp;line 3.6o, and a perpendicular offset 0.40; which?nbsp;according to article 3 of this part, contains an areanbsp;of 0.07�200. Secondly, a base line 4.85, (being th�nbsp;difference between 3.00 and 8.45 on the stationnbsp;line, at which points the offsets w�ere taken,) andnbsp;an offset or breadth at the one end 0.40, and atnbsp;the other end 0.10; w'hich, according to article ^nbsp;of this part, contains an area of 0.12125. Thir^b�nbsp;a base line 7-155 (being the diflercncc betweennbsp;8.45 and 15.f)0 on the station line,) and an off�^dnbsp;or breadth at the one end 0.10, and at the othernbsp;end 0.�5; wdiich according to the 4th article enbsp;this part, contains an area of 0.2f)812. FouriW�nbsp;a base line 5.40, (being the difference betweennbsp;15.6o and 21.00 on the station line,) and a pe^'nbsp;pendicular offset 0.65; which, according to l^henbsp;3d article of this part, contains an area of �. 17550-These added together make 0.03087 for the area

the offsets on the first station line; and, as this station line is vo�tl�n the survey, this area must, ofnbsp;�bourse, be added to the above-mentioned area comprehended within the station lines.

Next, for the area comprehended between the second station line and the boundary of the field,nbsp;'^'�c have a base line IS.tiO, and a perpendicular ornbsp;^ttset 0.60; whieh, according to article 3 of thisnbsp;part, contains an area of O.54�00. And, as thisnbsp;�''lation line is without the survey, this area must ofnbsp;bourse \iQ.sidHracteddim\x\ the above-mentioned area.

On the fir� station line On the second -1 )u the thirdnbsp;On tht fourthnbsp;On fifthnbsp;t)n ihejLrtbnbsp;On the seventh -

11. When the survey consists of a number of fields lying together, it is best to determine thenbsp;area of the whole tirst; and afterwards, the areanbsp;each of the fields separately; the sum of which, hnbsp;the work be true, will of course agree with the areanbsp;of the whole; which forms a check on the truthnbsp;of the computations.

But here it must be observed, that the bouo-daries of each field must be arranged for the coiH' putation, so as to proceed regularly one after anO'nbsp;ther all the way round the field; because, as wa�nbsp;observed in the note to the Qth article of this part,nbsp;without such arrangement, the northing or southiti^nbsp;made on each line, and the distances from thenbsp;assumed meridian, would not correspond with on�nbsp;another.

For example. Suppose it were required to ascertain the area of the survey mentioned in pag^^ 314, of the several fields therein contained,nbsp;the field book there given.¹

The bearings and lengths of the station lines on the outside boundaries are as follow; viz.

Then, for determining the area of the east field, quot;�e have the following lines, viz.

The area comprehended within these lines, com-* ted accordins to the foregoing universal theo-

^^T-here are no offsets to be added in this field, the ofi the offsets to be subtracted, computed as innbsp;'��e last example,nbsp;nbsp;nbsp;nbsp;are ........................ 2.22052

1st. The offset from the south-east-�erly corner to the first station line nbsp;nbsp;nbsp;ch. !�

th.e field book,................ S. nbsp;nbsp;nbsp;8��nbsp;nbsp;nbsp;nbsp;W.nbsp;nbsp;nbsp;nbsp;Q.�P

The area comprehended within these lines, computed according to the foregoing universal theorem, is....................................A. 22.lSOf8

The areas of the offsets to be added, computed as in the last example,.......................... 0.62257

[ 306 ]

MARITIME SURVEYING.

It was not my intention at first to say any thii^S concerning maritime surveying, as that subjectnbsp;had been already very well digested by Mr,nbsp;doch Mackenzie, whose treatise on maritime survey'nbsp;ing ought to be in every person�s hands whonbsp;engaged in this branch of surveying, a branchnbsp;which has been hitherto too much neglected,nbsp;few general principles only can be laid downnbsp;this work; these, however, it is presumed, will benbsp;found sufficient for most purposes; when the pra^quot;'nbsp;tice is seen to be easy, and the knowledge thereofnbsp;readily attained, it is to be hoped, that it will constitute a part of every seaman�s education, and thenbsp;more so, as it is a subject in which the safety ofnbsp;shipping and sailors is very much concerned.

1. nbsp;nbsp;nbsp;Make a rough sketch of the coast or harboubnbsp;and mark every point of land, or particular variation of the coast, with some letter of the alphabet�nbsp;either walk or sail round the coast, and fix a staffnbsp;with a white rag at the toj) at each of the place;�nbsp;marked with the letters of the alphabet. If therenbsp;he a tree, house, white cliff, or other remarkablenbsp;object at any of these places, it may serve insteadnbsp;of a station staff�.*

2. nbsp;nbsp;nbsp;Choose some level spot of ground upon whicnnbsp;a right line, called a fimdarnental base, may benbsp;measured cither by a chain, a measuring

a piece of log-line marked into feet; generally speaking, the longer this line is, the better; ds

^'tuation must be such, that the quot;whole, or most pRrt of the station-staves may be seen from bothnbsp;pnds thereof; and its length and direction must,nbsp;possible, be such, that the bearing of any station-�tafF, taken from one of its ends, may differ atnbsp;^cast ten degrees from the bearing of the same staffnbsp;^'T-ken from the other end; station-staves must benbsp;at each end of the fundamental base,nbsp;if a eonvenient right line cannot be had, twonbsp;^kies and the interjacent angles may be measured,nbsp;�i-iid the distance of their extremes found by con-�huction, may be taken as the fundamental base.

If the sand measured has a sensible and gradual ^^cclivity, as from high-water mark to low-water,nbsp;then the length measured may be reduced to thenbsp;horizontal distance, which is the proper distance,nbsp;hy making the perpendicular rise of the tide onenbsp;of a right-angled triangle; the distance mca-^Ored along the sand, the hypothenuse; and fromnbsp;thence finding the other side trigonometrically, ornbsp;hy protraction, on paper; which will be the truenbsp;^Ogth of the base line. If the plane measured benbsp;the dry land, and there is a sensible declivitynbsp;there, the height of the descent must be taken bynbsp;�I Spirit-level, or by a quadrant, and that made thenbsp;forpeaidicul ar side of the triangle.

If in a bay one strait line of a sufficient length P^ruiot be measured, let two or three lines, form-angles with each other, like the sides of a po-�ygon, be measured on the sand along the circuitnbsp;'7 the bay; these, angles carefully taken with anbsp;hyodolitc, and exactly protracted and calculated,nbsp;'^ill giyg strait distance betwcecn the two far-test extremities of the first and last line.

.. Find the bearing of the fundamental base by I Compasses, as aceurately as possible, with Had-� quadrant, or any other instrument equally

exact; take the angles formed at one end of base, bctwL�cn the base line and lines drawn t*�nbsp;meh of the station-staves; take likewise thenbsp;gles formed between the base line and lines drawnnbsp;to every remarkable object near the shore, ��snbsp;houses, trees, windmills, churches, amp;c. which nngt;ynbsp;be supposed useful as pilot marks; from the otb^'*nbsp;end of the base, take the angles formed betwcci*nbsp;the base and lines drawn to every one of the stationnbsp;staves and objects; if any angle be greater thannbsp;the arc ol� the cpiadrant, measure it at twice, bynbsp;taking the angular distance of some intermediatenbsp;object from each extreme object; enter all thesenbsp;angles in a book as they arc taken.

4. nbsp;nbsp;nbsp;Draw' the fundamental base upon paper froinnbsp;a scale of equal parts, and from its ends respectivelynbsp;draw' unlimittcd lines, forming with it the anglesnbsp;taken in the survev, and mark the extreme of caehnbsp;line with the letter of the station to which its an-gle corresponds. The intersection of every tquot;^nbsp;lines, whose extremes are marked wdth the sanionbsp;letter, will denote the situation of the station ofnbsp;object to which, in the rough draught, that lett^'*nbsp;belongs; tlirough, or near all the points of infc''quot;nbsp;.section w'hich represent station staves draw awa''nbsp;ing line w'ith a pencil to represent the coast. ^

5. nbsp;nbsp;nbsp;At low waiter sail about the harbour, aiH^nbsp;take the soundings, observing whether the

be rocky, sandy, shelly, amp;c. These soundmn^ may be entered by small numeral figures in

while at anchoi, and denote the same in the chart by small darts. The time of hig'h water isnbsp;denoted by Roman numeral letters; rocks arenbsp;denoted by small crosses; sands, by dotted shad-the figures upon which usually shew thenbsp;^ cpth at low water in feet; good anchoringnbsp;places are marked by a small anchor. Upon corn-

near the shore, eare must be taken to ex-ainiiie and correct the outline of the chart, by coservina; the inflections, creeks, amp;c. more mi-dutely.

6- In a small sailing vessel go out to sea, and . ^ke drawings of the appearance of the land, withnbsp;ds bearings; sail into the harbour, observe thenbsp;^Ppearance of its entrance, and particularly whe-there be any false resemblance of an entrancenbsp;y which ships may be deceived into danger; re-*dark the signs or objects, by attending to which,nbsp;harbour may be entered in safety; morenbsp;^^Pecially where it can be done, let the ship steernbsp;. the anchoring place, keeping two remarkablenbsp;�jeets in one, or on a line.

^ '^dseg, churches, trees, amp;c. on shore, are drawn in their proper figures; in a proper place ofnbsp;g-^ chart, draw a mariner�s compass, to denote thenbsp;Elation of the rhumbs, amp;c.; and on one side ofnbsp;flower-de-luce, draw a faint half flower-dc-lucenbsp;P��d of north by compass ; draw also a

or the pen and common ink, and are as as any others, but they are frequently donenbsp;B b

in water colours; for which purpose the beginnet will derive more advantage from viewing a pro[)etnbsp;drawing, or from overlooking a proficient at wotk^nbsp;than from a multitude of written instructions.

yin example to illustrate the foregoing precepl^� Let AB C D E F G H, f,g. 11, plate '23, representnbsp;a coast to be surveyed, and IK L, an adjacei'

'Vo make the actual ohservations on shore. BysailnA or walking along the shore, a rough sketch is inat^eanbsp;and station staves set up at the points of land. A,

Ci, D, E, E, G, H, and also at I, K, L, on the island �, precept 1. During this operation, it is observt^t*jnbsp;that no proper place offers itself at which a fn'^Jnbsp;damcntal base can be drawm, so as to comnaannbsp;at once a view of all the stations; it wdll, thet^quot;nbsp;lore, be necessary to survey the coast in two sC'nbsp;parate parts, by making use of two base lines; 'nbsp;the coast had been more extended, or irrcgulat?nbsp;greater number of base lines might havenbsp;necessary. Between the points B and C,nbsp;ground is level, and a considerable number offPnbsp;southernmost points may be seen from both poiP^quot;^^nbsp;conformable to precept 2; make B C, therefoi'^�nbsp;the first funftaincntal base, its length bynbsp;surement is found to be 812 fathoms,nbsp;bearing by compass from B to C, is N. 11�nbsp;nbsp;nbsp;nbsp;.'g

from each end of this base, measure with y quadrant the angles formed between it, and hinbsp;tlrawn to each station in sight, precept 3, ent*^*'�nbsp;tabulate them as follows:nbsp;nbsp;nbsp;nbsp;�

I After having made these observations, it will requisite to proceed to the northern part of thenbsp;^oast; in all cases where a coast is surveyed in separate parts, it is best to measure a new funda-^^^C'ltal base for each part, when it can be eonve-^�*cutly Jojig. aline from the station E drawn to-Wds is well adapted to our purpose; let E P,nbsp;^ereforc, be the second base line, its length bynbsp;� ^�Measurement is found to be 778 fathoms, andnbsp;tgt;earing by compass from E to P, N. 38� 20' E.nbsp;icasure the angles formed by lines drawn fromnbsp;,end of this base, as before directed, and tabu-them as follows:

1 Second fundamental base E P=:778 fathoms, ^��ring of P, from E. N. 38� 20' E.

parts of this survey is preserved the ^^.*^^ond fundamental base being drawn fromnbsp;hy Pij whose situation was before determinednbsp;��^rvations from the first base line; if thisnbsp;position of the second base had not beennbsp;and it had been taken at a distancenbsp;every point determined in situation from thenbsp;E b 2

first base, the comiection would have required observation of the bearing of one of the said point�nbsp;from each end of the second base: thus, suppo-�^^:nbsp;the line I P to be the second base line, instead otnbsp;E P; the position of I P with respect to the giveifnbsp;point E, may be known by taking the bearings ntnbsp;E, from I and P.

All the observations which arc required to made on shore, being now eom])lctcd, it will benbsp;adviscable to construct the chart before we pr^'nbsp;cecd to the other observations on the water, fnt?nbsp;by tliis method, an opportunity is otfered of draquot;'nbsp;ing tlic waving line of the coast with more coiTcrt''nbsp;ness than could other^^'ise be obtained.

On the point B as a center, describe the n c s w, the diameters n s and w c being at rigb*^nbsp;angles to each other, and representing the ine!quot;�^'nbsp;dian and parallel; from n set off the arc nnbsp;equal to the quantity of the variation of thenbsp;pass; draw the diameter N S, and the diam^l-f.!nbsp;WE at right angles to the same; these lines jnbsp;represent the magnetie meridian, and its paraU^nbsp;of latitude.

From the center B, draw the first base h'^ BC=812 fathoms, N. 11� 14'E. by coiupl'^��nbsp;from one extremity B, draw the unlimited hp ,nbsp;Ba, Bl, Bk, Be, Bd, forming, respectivelynbsp;the base, the angles observed from thence ;nbsp;the other extremity C, draw the unlimited hp^.nbsp;Ca, Cl, Ck, Ce, Cd, forming, respectivelynbsp;the base, the angles observed from thence;nbsp;intersections of lines terminated by the samenbsp;ter, will give the stations A, L, K, E, f'quot;nbsp;cept 4,

I^i-aw the line BQ, N. 38� 20'E. by compass, �ind from E, draw EP=778 fathoms in the samenbsp;^^d-ection or parallel to it, EP will be the secondnbsp;dase line; from one extremity E, draw the un-d.mited lines E h, E i, E g, E f, forming, respcc-tivcly with the base, the angles observed fromnbsp;^dencc; from the other extremity P, draw theiin-dmited lines Ph, Pi, Pg, Pf; the intersectionsnbsp;lines terminated by the same letter will givenbsp;stations H, I, G, F, pi'ccept 4.

Near the points A, B, C, D, E, F, G, H, draw ^ Waving or irregular line to represent the coast;nbsp;draw also by the help of the points I, K, L, a linenbsp;d) represent the coast of the island.

i^rccept 5, sufficiently explains the business to ^j^d^ch it is necessary to attend on the water; no-^^'rig more, therefore, need be added here, exceptnbsp;elucidation of the method, by the place atnbsp;y^^Wh any observation is made, may be found onnbsp;? ehaj.^ . place of an observer may be dctcr-dy the bearings of two known objects; the

preferring each of these methods; but, distance in the second method is knownnbsp;to estimation, we have given the preference

in the present example; all the sound-m' L arc supposed to be laid down by these the, r drrt, to prevent confusion and prolixity,nbsp;3Vlnbsp;nbsp;nbsp;nbsp;bearing arc drawn only from the points

From the southern extremity M, of the reef of rocks, which runs off the point L, the bearing oinbsp;the stations C and B arc taken by the compass, asnbsp;follows, viz. C, N. 62� E. and B, N. 85� E.

On the chart, therefore, from C, parallel to the rhumb B R, draw the opposite rhumb C m, S. 62nbsp;W. and from B, draw the opposite rhumb Bnbsp;S. 85� W. their intersection M, will be the extremity of the reef of rocks.

Again, from the sand head N, the bearings oi the stations L and K are taken by the compasSjnbsp;as follows, viz. L, N. 17� W. and K, N. 5�E.

On the chart draw the rhumb B n, N. 17� ^ � and from L, parallel to the same, draw the op'nbsp;posite rhumb L n; draw likewdse the rhumb Bnbsp;N. 5� E. and from K, parallel to the same, druquot;^nbsp;the oppose rhumb K n; the intersection of thesenbsp;two last lines will be the sand head, and the sitn^'nbsp;tion of the point of land O, which lies so that Jtnbsp;cannot be seen from cither of these base lines,nbsp;determined in like manner from the bearings of b*nbsp;and K, though in practice it may have been mot^nbsp;convenient to settle its place from a single beat'nbsp;ing and distance of the station A.

The position of the compass is given from tn^^ of the circle N E S W, made use of in the coO'nbsp;struction, and the scale of geographical mih�!�nbsp;found from the consideration, that 1021 fathoO�j�nbsp;make one geographical mile; take therefore

'd ' willnbsp;:alo

SO that it may float strait on the surface of the Water; if the line has been well stretched, ornbsp;�^�ueh used before, it is the better; also preparenbsp;two ropes somewhat longer than the gi'calestnbsp;of the water to be measured, with a pig ofnbsp;tead or iron ballast, which we call an anchor, 50nbsp;6o pounds weight, tied by the middle to onenbsp;of each rope, that when it is at the bottom itnbsp;be able to anchor a boat, and bear to benbsp;^*^i'etchcd strait without shifting the place of thenbsp;�^'chor. Let the measuring line be thoroughivnbsp;immediately before you begin to use it, andnbsp;then stretched on the water close by the shore,nbsp;?^*(1 its length measured there with a pole. Then,nbsp;the direction intended to be measured, take twonbsp;^�cniarkable sharp objects on the land in a line,nbsp;near the shore, the other as far up in the conn-as you can; if such are not to be had, plact'nbsp;^Woys on the water at proper distances in thatnbsp;^h'ection.

2. Take the objects, or buoys, in a line, and folding one end of the measuring line fast on thenbsp;Ore, carry out the other in a boat, in thatdirec-till it is stretched strait at its full length bynbsp;man in the boat, and exactly at the end of thenbsp;let another man drop the anchor, ubich willnbsp;one length of it. There keep the boat, andnbsp;end of the measuring line, close to the anchornbsp;^ope, drawn tight up and down, till another boatnbsp;in the other end which w�as on the shore, andnbsp;i^yg it strait in the directionnbsp;^he land marks, or buoys, and there lt;lrops ano-anchor, which will mark the second lengthnbsp;the measuring line. Go on thus till the wholenbsp;� P posed distance is measured; and immediatelynbsp;^ *^'1� let the measuring line be again measured withnbsp;l^pole on the water near the shore, as at first, andnbsp;� ^he lengths difter, take the mean between them

for the true length. It is obvious, that to measure with any exactness this way, the sea must not onlynbsp;be smooth, but void of a swell, and of all stream oinbsp;tide; either of which will hinder the line from 1}'nbsp;ing strait. This method of measuring a strait lin*^nbsp;may be convenient on some occasions; and if camnbsp;is taken to keep the anchor rope right up andnbsp;down when the measuring line is applied to ibnbsp;will be found sufficiently exact for many purpose?�nbsp;but not for a fundamental base line from whichnbsp;other distances are to be deduced.

There is another way of measuring a strait line� mechanically, on the sea, which is so well knova^nbsp;to seamen, that it is needless to describe it parti'nbsp;cularly here; and that is, by heaving the log overnbsp;a ship�s stern while she is under sail, and observing how many knots of the log line run out in hannbsp;a minute; for the line is so divided that the shipnbsp;will run, or is supposed to run, so many miles mnbsp;an hour, in a strait eourse; and twice as much mnbsp;two hours, and so on. But this conclusion i*nbsp;founded on three suppositions, neither of w�hich i*nbsp;certain, viz. that the log remains in the samenbsp;place during the wdiolc half minute that the linenbsp;running out from the ship�s stern; that the phipnbsp;continues to sail with the same velocity, and als^nbsp;in the same direction, during an hour, or two,nbsp;that she did during the half minute; the contrarynbsp;of which is more likely in most cases. For thenbsp;log line may shrink, or stretch, while it is runningnbsp;out; or may drag after the vessel by the w�cigi^^nbsp;of the line, or by not running easily and readi^Xnbsp;off the reel; the swell of the sea may alter the ph^enbsp;of the log; and currents, or streams ofnbsp;stronger orweakcr below the surface than on it,nbsp;unsteady helm, lec-rvay, and varying winds, maynbsp;change the direction, or celerity of the ship�s nio-tion; for neither of which can any certain allcWquot;

Q-Hee be made. This way, therefore, of measuring ^ strait line, or distance, is not to be depended onnbsp;exact; but is mentioned here, because rocks,nbsp;shoals, or islands, sometimes lie so far from thenbsp;^'Oast, that there is no other way of forming anynbsp;^^otion of their distance. If any such distance isnbsp;be measured after this manner, let the log-linenbsp;he thoroughly wet when it is measured; let thenbsp;�Ciigth between each knot be 51 feet, which is thenbsp;} 2oth part of a geometrical mile, as half a minutenbsp;TYwth part of an hour. Choose neap-tide, asnbsp;j^och slack water as can be got, and a moderatenbsp;hrcczc of following wind; let the line be run offnbsp;the reel so as never to be stretched quite strait;

if the half minute is measured hy a watch that �hc\vs seconds, rather than by a glass, it will ge-j^crally be more exact. Perhaps one second shouldnbsp;he allowed for the loss of time in calling out atnbsp;^he beginning, and stopping it at the end of thenbsp;except the person who holds the watch cannbsp;heutrivc to observe the going out of the red rag atnbsp;^he beginning, and also to stop the line himself atnbsp;he end of the time; which does not seem a diffi-h^h^t matter.

J o find the distance of two places hy the flash and^ J�port rf a gun. Sound moves 114*2 English feetnbsp;h' One second of time, or �120 feet, which makesnbsp;�^geographical mile, in nearly; therefore, letnbsp;^ be fired at one place at an appointed time,nbsp;�h'd observe the time that elapses between thenbsp;�quot;ish and report, and so many seconds as you ob-^hi'Ve, so many times 1142 feet are you distantnbsp;the place; the operation should be repeatednbsp;^ '0 or three times for greater certainty. The dis-to be measured in this way should never benbsp;hhan two miles, on accouvit of errors that maynbsp;^^^se in taking the time.

Method 1. By points. Lay the rough plan upon the clean paper, on wliich yon intend to drawnbsp;fair copy, press them close together by weights?nbsp;and keep them as flat as possible; then with itnbsp;pointrel, or needle point, prick through all thenbsp;corners of the plan to be copied; separate thenbsp;papers, and join by lines the points on the. cleaitnbsp;paper. This method can only be used in plans?nbsp;whose figures are small, regular, and bounded bynbsp;strait lines.

Method 2. By tracing paper. Rub the back of the rough plan with black-lead powder, and havingnbsp;wiped off the superfluous lead, lay the blackednbsp;part upon the clean paper, or place a sheet otnbsp;black tracing pajwr between the rough plan ninlnbsp;the clean paper; weights arc to be placed as ntnbsp;the former method, to maintain the papers in thonbsp;same position.

Then, with a blunt point of brass, steel, ivory, trace exactly the lines of the plan, pressingnbsp;the paper so much, that the black lead under thenbsp;lines may be transferred to the clean paper; whefnbsp;the whole of the plan has been thus delineated, pPnbsp;over the black-lead marks with common, or India'^nbsp;ink.

Methods. �y a copying glass. This is a hng� square, or rectangular piece of looking glass,nbsp;in a frame of wood, which can be raised to anynbsp;angle, like a desk, the lower end resting upon In^nbsp;table; a screen of blue paper fits to the upper edgt?nbsp;and stands at right angles to it.

Place this frame at a convenient angle against a strong light: fix the old plan and clean papernbsp;fii'inly together by pins, the clean paper upper-^^lost, and on the face of the plan to be copied;

them with the back of the old plan next the Rlass, namely, that part which you intend to copynbsp;lirst. 'The fight through the glass will enable younbsp;perceive distinctly every line of the plan uponnbsp;^he. clean paper, and you can easily trace over themnbsp;''ith a pencil; and having finished that part whichnbsp;'Covers the glass, slide another part over it, andnbsp;'^opy this, and thus continue till' the whole benbsp;Copied.

Method 5. By the assistance of proportionable and ^Biangular compasses, fg. A and N, plate 1, andnbsp;fig- 12, plate?,. These will, in many instanstan-cos, assist the draughtsman very much, and lessennbsp;^l^e labour of copying.

Method 6. By the pantograplier. There is no �Method so easy, so expeditious, nor even so accu-^^te, as the pantograplier. It is an instrument asnbsp;Useful to the experienced draughtsman, as to thosenbsp;'I'ho have made, but little progress in the art. It

targing^or copying of the same size, giving the Cotliues of any drawing, however crooked or complex, with the utmost exactness; nor is it confinednbsp;any particular kind, but may with equal facilitynbsp;cc used for copying figures, plans, sea charts,nbsp;^iiaps, profiles, landscapes, amp;c.

, � have not been able to ascertain who was the ^'cntor of this useful instrument. The earliestnbsp;ccount I find, is that of the Jesuit Scheiner, about

the year 1031, in a small tract entitled, Fanta^rd-^hice, sive ^rs nova DeUneancli. The principles are self-evident to every geometrician; the mechanical construction was tirst improved by my hi'nbsp;ther, about the year i750. It is one, amongnbsp;other scientific improvements completed by him,nbsp;that others have many years after, assumed tonbsp;themselves.

The pantographer is usually made of wood, ot brass, from 12 inches to two feet in length, ami ,nbsp;consists of four flat rules. Jig. IQ, flute Wl, twonbsp;of them long and two short. The two longernbsp;are joined at the end A by a double jiivot,nbsp;which is fixed to one of the rules, and works innbsp;two small holes placed at the end of the other.nbsp;Under the joint is an ivory castor to support thisnbsp;end of the instrument. The two smaller rules arenbsp;fixed by pivots at E and H, near the middle of thenbsp;larger rules, and are also joined together at theirnbsp;other end, G.

By the construction of this instrument, the four rules always form a parallelogram. There is 3nbsp;sliding box on the longer arm, and another on thenbsp;shorter arm. These boxes may be fixed at anynbsp;part of the rules by means of their milled screws;nbsp;each of these boxes arc furnished with a cylindrienbsp;tube, to carry either the tracing point, crayon,nbsp;or fulcrum.

The fulcrum, or support K, is a leaden weight � on this the whole instrument moves when innbsp;To the longer instruments are sometimes placednbsp;two moveable rollers, to support and facilitate thenbsp;motions of the pantographer; their situation maynbsp;be varied as occasion requires.

The pencil holder, tracer, and fulcrum, must in cases be in a right line, so that when they arcnbsp;^ct to any number, if a string be stretched overnbsp;^hem, and they do not coincide with it, there isnbsp;error cither in the setting or graduations.

The long tube which carries the pencil, or ^^rayon, moves easily up or down in another tube;nbsp;^herc is a string affixed to the long, or inner tube,nbsp;passing afterwards through the holes in the threenbsp;^rnall knobs to the tracing point, where it may, ifnbsp;^^acessary, be fastened. By pulling this string, thenbsp;pencil Ts lifted up occasionally, and thus pre-''ented from making false or improper marks uponnbsp;^�he copy.

Vo reduce hi any of the proportions i, t, i, amp;c, marked on the two bars B, and D. Suppose, fornbsp;^^ample, i is required. Place the two sockets at tnbsp;the bars B and D. Place the fulcrum, or leadnbsp;'Veiglit at B, the pencil socket, with pencil at D,nbsp;'*�h1 the tracing point at C. Fasten down upon anbsp;�'anooth board, or table, a sheet of white paper un-the pencil D, and the original map, amp;c. undernbsp;tracing point C, allowing yourself room enoughnbsp;the various openings of the instrument. Thennbsp;quot;^dh a steady hand carefully move-the tracing pointnbsp;^ over the outlines of the map, and the pencil at Dnbsp;dill describe exactly the same figure as the original, but i the size. In the same manner for anynbsp;nfher proportion, by only setting the two socketsnbsp;the number of the required proi)oilion.

. The pencil-holder moves easily in the socket, to ^U'e -way to any irregularity in the paper. Therenbsp;a cup at the top for receiving an additionalnbsp;^'�iight, cither to keep down the pencil to the pa-or to increase the strength of its mark.

, There is a silken string fastened to the pencil-n'der, in order that the pencil may be drawn up

oft' the paper, to prevent false marks when crossing the original in the operation.

If the original should be so large, that the iti-strument will not extend over it at any one operation, two or three points must be marked on the original, and the same to correspond upon thenbsp;copy. The fulcrum and copy may then be removednbsp;into such situations, as to admit the copying of thenbsp;remaining part of the original; first ohserving?nbsp;that when the tracing point is applied to the threenbsp;points marked on the original, the pencil falls onnbsp;the three corresponding points upon the copy. Innbsp;this manner, by repeated shiftings, a pentagraphnbsp;may be made to copy an original of ever so largenbsp;dimensions.

Suppose I. You set the two sockets at i, as before, and have only to change places between the pencilnbsp;and tracing point, viz. to place the tracing pointnbsp;at D, and the pencil at C.

7b copy of the same size, hut reversed. Place the two sockets at 5, the fulcrum at D, and thenbsp;pencil at B.

There arc sometimes divisions of 100 unequal parts laid down on tlie bars B and D, to give anynbsp;intermediate proportion, not shewn by the fractional numbers commonly placed.

Pentagraphs of a greater length than two feet are best made of hard wood, mounted in bras'^jnbsp;with steel centers, iqjon the truth of which dc'nbsp;pends entirely the ecpiable action of this usefnlnbsp;instrument.

LEVELLING.

The necessity of tinding a proper channel for ^^onveying water occurs so often to the surveyor,nbsp;^hat any work on that subject, which neglected tonbsp;j^'�eat on the art of levelling, would be manifestlynbsp;iiriperfect; I shall therefore endeavour to give thenbsp;'�eadcr a satisfactoiy account of the instrumentsnbsp;^*scd, and the mode of using them.

3, plate 17, represents one of the best con-^Ifucted spirit levels, mounted on the most eom-plete staves, similar to those affixed to a best theodolite.

The achromatic telescope. A, B, C, is inovea-either in the plane of the horizon, or with a ^Oiall inclination thereto, so as to cut any objectnbsp;'^d^ose elevation, or depression, from that plane,nbsp;does not exceed 1 '2 degrees; the telescope is aboutnbsp;t'Vo feet long, is furnished with tine cross wires,nbsp;ddc screws to adjust which are shewn at a, for de-dorinining the axis of the tube, and forming a justnbsp;doe of sight. By turning the milled screw B,nbsp;the side of the telescope, the object glass isnbsp;ffioved outwards, and thus the telescope suited tonbsp;different eyes.

. The tube c a with the spirit bubble is fixed to pde telescope by a joint at one end, and a capstannbsp;'oaded screw at the other, to raise or depress it fornbsp;^0 adjustment. The two supporters D, E, on

�which (he telescope is placed, are nearly in the shape of the letter Y, the innCr sides of the Y�s arcnbsp;tangents to tlic cylindric ring of the telescope*nbsp;The lower ends of these supporters arc let perpen'nbsp;dicularly into a strong bar; to the lower end ofnbsp;the support E, is a milled nut F, to bring the io-strument accurately to a level; and at the othernbsp;end of the bar at FI, is a screw for tightening tlicnbsp;support D at any height. Between the two sup'nbsp;ports is a compass box G, divided into four quarters of go� each, and also into 300�, with a mag'nbsp;netical needle, and a contrivance to throw thenbsp;needle oft' the center when it is not used; au^�nbsp;thus constituting a perfect circumferentor.

The compass is in one piece with the bar, or is sometimes made to take on and oft' by two serc^v?�nbsp;To the under part of the compass is fixed a coiU'nbsp;cal brass ferril K, which is fitted to the bell-metalnbsp;frustum of a cone at the top of the brass head oinbsp;tlic staves, having at its bottom a ball, moving n�nbsp;a socket, in the jdate fixed at the top of the threenbsp;metal joints for the legs. L, L, are two strongnbsp;brass parallel plates, with four adjusting screW^?nbsp;1^, h, h, h, \vhich are used for adjusting the hoi'i'nbsp;zontal motion. The screw at M is for regulatingnbsp;this motion, and the screw N for making fast th^nbsp;ferril, or whole instrument, when necessary.nbsp;these two screws the instrument is either movcunbsp;through a small space, or fixed in any positie'^nbsp;�with the utmost accuracy. The staves being exactly the same as those applied to the best theO'nbsp;dolites, render any further description of then)nbsp;here unnecessary.

It is evident from the nature of this instrurnen that three adjustments are necessary. 1. To plac�nbsp;the intersection of the wires in the telescope?nbsp;that it shall coincide with the axis of the cylin'^'�*'

till by trial the bubble comes to the middle i't both positions of the telescope. This verynbsp;mode of adjustment of the level is one greatnbsp;provement in the instrument; for, in all the olonbsp;and common constructed levels, which didnbsp;admit of a reversion of the telescope, the spi^'�^nbsp;level could only be well adjusted by carryingnbsp;instrument into the field, and at two distant stR'nbsp;tious observing both forwards and backwai�^l�nbsp;the deviation upon the station staves, and coi'nbsp;rcctiug accordingly; which occasioned no sni!'�nbsp;degree of t ime and trouble not necessary bynbsp;improved level.

7o adjust for the horizontal motion. The Ic^'^ is said to be completely adjusted, when, afternbsp;two previous ones, it may be moved entirely rounfnbsp;upon. its staff-head, without the bubble changi^�onbsp;materially its place. To perform which, bringnbsp;telescope over two of the parallel plate screws h,nbsp;and make it level by unscrewing one of the^^nbsp;screws while you arc screwing up the opposite oO^gt;nbsp;till the air bubble is in the middle, and the serequot;'�nbsp;up firm. Then turn the instrument a quart�s'�nbsp;round on its staff, till the telescope is directlynbsp;the other two screws h, h, unscrewing one serequot;'�nbsp;and screwing up at the same time the oppo-^'^nbsp;one, as before, till the bubble settles again innbsp;middle. The adjustment of the staff phltcsnbsp;thus made for the horizontal motion of the instmnbsp;ment, and the telescope may be moved roundnbsp;its staff' without any material change of thenbsp;of the bubble, and the observer enabled tonbsp;range of level points. The level tube is ^nbsp;made to adjust in the horizontal direction bynbsp;opjiosite screws at its joint e, so that itsnbsp;be brought into perfect parallelism to that of ^nbsp;telescope.

there is a sliding vane A, fg. Q, plate 17, with ^ brass wire across a square hole made in the vane;nbsp;when this wire coincides with the horizontal wii'Cnbsp;of the telescope, it shews the height of the app�'nbsp;rent Icvx'l above the ground at that place.

Levelling is the art which instructs us in lindiit? how much higher or lower any given point on thenbsp;surface of the earth is, than another given pointnbsp;the same surface; or in other words, the differencenbsp;in their distance from, the center of the earth. .

Those points are said to be level, which atc equi-distant from the center of the earth. Thenbsp;art of levelling consists, therefore, 1st. In findingnbsp;and marking two, or more level points that shahnbsp;be in the circumference of a circle, whose centetnbsp;is that of the eailh. idly. In comparing the pointsnbsp;thus found, with other points, in order to ascertai_�|nbsp;the difference in their distances from the earth ��nbsp;center.

Letj^.^. 1, plate�IZ, represent the earth; A? center; the points B, C, D, E, P, upon the cit'nbsp;cumference thereof are level, because theynbsp;equally distant from the center; such are thenbsp;ters of the sea, lakes, amp;c.

the bubble keeps to the middle, it is adjusted; if not, by of a screw-driver turn one of the screws at the pnjper end, t*nbsp;be 30 raised or dcpi'essed as to cause the bubble to stand tb�nbsp;versing, at the same time altering the inclination of thenbsp;which the level is tried. If a bubble stand the reversion o

are inclined, and both require to be corrected by half the angle of the deviation shewn by reversion. Edit.

To know how much higher the po 'mt B, jig. 2, piate 23, is than C, and C lower than D. We mustnbsp;and mark the level points, E, F, G, nj)on thenbsp;�'iidii AB, AC, AD, thereby comparing B with E,nbsp;C with F, and D with G; we shall discover hownbsp;^uch B is nearer the circumference of the circlenbsp;than C, and, consequently, how much furthernbsp;from the eentcr of the earth, and so of the othernbsp;points.

The first, which is the most simple and independent, is by the tangent of a circle, for the two ^^tremities of the tangent give the true levelnbsp;points, when the point of contact is precisely innbsp;middle of the line. But if the point of con-^^�ct with the circumference be at one of the extre-^�iities of the line, or in any other part except thenbsp;Middle, it will then only shew the apparent level,nbsp;one of its extremities is further from the circumference than the other. Thus the tangentnbsp;^ 9� fis- 3? plate 23, marks out two true levelnbsp;at B and C, because the point of contactnbsp;^ is exactly in the middle of the line B C, and itsnbsp;fr'^o extremities are equally distant from the circumference and the center A.

�ut the tangent B D, jig. 7, plate 23, marks two P^mts of apparent level, because the point B,nbsp;^''here it touches the circumference, is not thenbsp;J^iddle of the line; and, therefore, one of its ex-emities B, is nearer to the center than the othernbsp;� I � is further from the center, in proportion asnbsp;_ more distant from the point of contact B;nbsp;�'ftich constitutes the diflference between the truenbsp;^�^d apparent level, of which \ve shall speak pre-

sently. Every point of the apparent level, except the point of contact, is higher than the true level.

As the tangent to a circle is perpendicular to the radius, we may make use of the radius to determine the tangent, and thus mark the leve)nbsp;points. Thus let A, Jig. 5, j)lafe 23, represent thenbsp;center of the earth, AB the radius, and C, B, Pjnbsp;the tangent; the two extremities C, D, are equallynbsp;distant from the point of contact B, consequentlynbsp;the angles B CA, B DA, will be equal; the anglednbsp;at the tangent point arc right angles, and the ra-dius common to both triangles, and the sides CAgt;nbsp;DA, arc equal, and the points CD, are two levelnbsp;points, because equally distant from the center.

It is evident from this, that if from any point of the radius two lines be drawn, one on each side,nbsp;making equal angles with it, and being of an eqnalnbsp;length, the extreme points of these lines will benbsp;level points. Thus, if from B, of the radius BA,nbsp;two equal lines B C,. B C, Jig. 6, plate 23, benbsp;drawn, making equal angles, CBA, DBA, thennbsp;will C and D be equally distant from the center;nbsp;though the level may be obtained by these obliqu�nbsp;lines, yet it is far easier to obtain it by a line ps��'nbsp;pendicular to the radius.

When the level line is ])erpendicular to the r^' dius, and touches it at one of its extremes,nbsp;other extremity will mark tlie apparent level,nbsp;the true level is found by knowing how muchnbsp;apparent one exceeds it in height.

To find the height of the apparent above tb� true level for a cct�tain distance, square that distance, and then divide the product by the dianaCquot;nbsp;ter of the earth, and the quotient will be thenbsp;quired difFerenee; it follows clearly, that thnbsp;heights of the apparent level at different distancesnbsp;are as the squares of those distances; and, consequently, that the difiercncc is greater, or smalls i

proportion to the extent of the line; for the cx-^Pinity of this line separates more from the eircnm-ffrence of the circle, in proportion as it recedes from the point of contact. Thus, let A, jig, 7,nbsp;plate 23, be the center of the earth, BC the arcnbsp;'^'hich marks the true level, and B, E, D, the tan-Sont that marks the apparent level; it is evident,nbsp;that the secant A D exceeds the radius AB, by C D,nbsp;''��^�hich is the difterence between the apparent andnbsp;frue level; and it is equally evident, that if the linenbsp;Extended no failher than E, this difference wouldnbsp;*'ot be so great as when it is extended to D, andnbsp;that increases as the line is lengthened.

When the distance docs not exceed 25 yards, the difference between the two levels may be neg-freted; but if it be 50, 100, amp;c. yards, then thenbsp;error resulting from the difference will becomenbsp;Sensible and require to be noticed.

The j^rst method of finding and marking t\vo level points, is by a tangent whose point of contact is exactly in the middle of the level line; thisnbsp;method may be practised without regarding thenbsp;ditference between the apparent and true level;nbsp;but, to be used with success, it would be wellnbsp;place your instrument as often as possible at ai^nbsp;equal distance from the stations; for it is clear,nbsp;that if from one and the same station, the instrument remaining at the same height, and used nrnbsp;the same manner, two or more points of sight benbsp;observed, equally distant from the eye of the observer, they will also be equi-distant from the center of the earth.

Thus, let the instrument, fig.S, flute 13, be placed at equal distances from C, D;' E, F, thenbsp;two points of sight marked upon the station-staveSnbsp;C G, D H, will be the level points, and the difte-rence in the height af the points E, F, will sherr^nbsp;how much one place is higher than the other.

Second method. This consists in levelling froua one point immediately to another, placing the n^'nbsp;strument at the stations where the staves werenbsp;fixed. This may also be performed without noticing the difference between the true and apparentnbsp;level; but then it requires a double levelling�nbsp;made from the first to the second station, and reciprocally from the second to the first.

13, the station staves C B, D E, which in the pr*'-^' tice of levelling may be considered as parts ofnbsp;radii AB, AE, though they be really two perpeo^nbsp;diculars parallel to each other, without the risk onbsp;any error, In order to level by this method, p*^�nbsp;the instrument at B, let the height of the eye,nbsp;the first observation, be at F, and the point of sigLnbsp;found be G; then remove the instrument to j

fix it so that the eye may be at G; then, if ^he line of sight cuts F, the points F and G arenbsp;i'^vcl, being equally distant from the center of thenbsp;'^^rth, as is evident from the figure.

But if the situation of the two stations be such, ^hat the height of the eye at the second stationnbsp;^'ould not be made to coincide with G, but onlynbsp;'^'ith H, jig. 10, j^ldten?,, jet if the line of sightnbsp;J?ives I as far distant from F as G is from H, thenbsp;two lines F G, H I, will be parallel, and their ex-t^'emities level points; but if that is not the case^nbsp;the lines of sight arc not pai'allel, and do not give,nbsp;^cvel points, which however might still be obtainednbsp;t'y further observations; but, as this mode is notnbsp;pi'aetised, w'e need not dw�cll further upon it.

The velocity of running water is proportional to the fall; w'hcre the fall is only three or four inchesnbsp;a mile, the velocity is very small; some cuts havenbsp;hfien made with a fall from four to six inches; a fournbsp;*�'ch fall in a strait line is said to answ'er as w^ell asnbsp;^gt;ic of six inches wdth many windings.

The distance from the telescope to the staff ^�hould not, if possible, exceed 100 yards, or fivenbsp;�chains; but 50 or 60yards arc to be preferred.

term that simple levelling, when level Points arc determined from one station, whethernbsp;he level be fixed at one of the points, or betweennbsp;fhein.

.1'hus, let AB, fig. 1, plate be the station Points of the level, C, D, the two points ascer-ihcd, and let the height

In this example, the station points of the level are below the line of sight and level points, as if�nbsp;generally the ease; but if they were above, as n*nbsp;3%'nbsp;nbsp;nbsp;nbsp;24, and the distance of A to C be

feet, and from B to D nine feet, the difference wiH be still three feet, that B is higher than A.

Compound levelling is piothing more than ^ collection of many single or simple operation^nbsp;compared together. To render this subject clearer�nbsp;we shall suppose, that for a particular purpose itnbsp;were necessary to know the difference in the levelnbsp;of the two points A and N, fg. 3, plate 24; Anbsp;the river Zome, N on the river Belann.

As I could find no satisfactory examples in anf English writer that I was acquainted with, an�nbsp;not being myself in the habits of making actualnbsp;surveys, or conducting water from one placenbsp;another, I was under the nexessity of using thosunbsp;given by Mr. Le Fehvre.

Stakes should be driven down at A and N, actly level with the surface of the water, and tbus^nbsp;should be so fixed, that they may not be chang^i^nbsp;until the whole operation is finished; the groundnbsp;between the two rivers should then be surveY^j �nbsp;and a plan thereof made, which will greatlynbsp;the operator in the conducting of his business;nbsp;this he will discover that the shortest waynbsp;to N, is by the dotted line AC, GH, HNj

tie will from hcncc also dctcnninc the necessary iiuinbcr of stations, and distribute them properly,nbsp;^oine longer than the others, acconling to the nature of the ease, and the situation of the ground,nbsp;lu this instance 12 stations arc used; stakes shouldnbsp;then be driven at the limits of each station, as atnbsp;A, B, C, D, E, F, amp;c. they should be drivennbsp;^hout 18 inches into the ground, and be about twonbsp;three inches above its surface; stakes should alsonbsp;driven at each station of the instrument, as atnbsp;2, 3, 4, he.

Things being thus prepared, he may begin his quot;'ork; the first station will be at 1,. eqni-distantnbsp;h�oin the two limits A, B; the distance from A tonbsp;^5 166 yards; and, consequently, the distance onnbsp;^^leh side of the instrument, or from the stationnbsp;�take, will be 83 yards.

Write down in the first column, the first limit A; in the second, the number of feet, inches, andnbsp;tenths, the points of sight, indicated on the stationnbsp;^hiffat A, namely, 7-d.O. In the third column,nbsp;^he second limit B; in the fourth, the height indicated at the station staff B, namely, 0'. 0.0.nbsp;Eastly, in the fifth column, the distance from onenbsp;Edition staff to the other, in this instance 166 yards.

Now remove the level to the point marked 2, quot;quot;hich is in the middle between B and C, the twonbsp;pieces where the station staves ai-c to be licld, ob-^'�'��ving that B, which was the second limit in thenbsp;|4'mcr operation, is the first in this; then writenbsp;uown the observed heights as before, in the firstnbsp;folumn B; in the second 4. 6. 0; in the third, C;

.As, from the inequality of the ground at the station, it is not possible to place the instru-in the middle between the two station staves.

find the most convenient point for your station, aS at 3; then measure exactly how far this is fromnbsp;each, station staff, and you will find from 3 to C,nbsp;l6oyards; from 3 to D, 80yards: the remaindernbsp;of the operations will be as in the preceding station-In the fourth operation, it will be necessary tonbsp;fix the station, so as to compensate for any errornbsp;that might arise from the inequality of the last;nbsp;therefore, mark out 80 yards from the station staffnbsp;D, to the point 4; and l6o yards from 4 to E;nbsp;and this must be carefully attended to, as by suchnbsp;compensations the work may be much facilitated-Proceed in the same manner with the eight remaining stations, as in the four former, observingnbsp;to enter every thing in its respective column;nbsp;when the whole is finished, add the sums of eachnbsp;column together, and then subtract the less fromnbsp;the greater, thus from 82:2; 5, take 76 ; 0 : 7�nbsp;and the remainder 5 : 4 : 8, is what tfie groundnbsp;N is lower than the ground at A.

[ 397 ]

The next thing to be obtained is a section of this level; for this jiurpose draw a dotted line, asnbsp;o o, fig. 4, flate 24, either above or below thenbsp;plan, which line may be taken for the level or ho-J'izontal line; then let fall perpendiculars uponnbsp;this line from all the station points and places,nbsp;quot;�here the station staves were fixed.

Beginning at A, set off seven feet six inches quot;pon this line from A to a; for the height of thenbsp;li�vel point determined on the staff at this place,nbsp;draw a line through a, parallel to the dotted linenbsp;o o, whieb will cut the third perpendicular at b,nbsp;fhe second station staff; set off from this pointnbsp;downwards six feet to B, which shews the secondnbsp;limit of the first operation, and that the ground atnbsp;�H, is one foot six inches higher than at A; placenbsp;your instrument between these two lines at thenbsp;height of the level line, and trace the ground ac-�^Ording to its different heights.

Now set off on the second station staff B, four heet six inches to C, the height determined by thenbsp;level at the second station; and from c, draw a linenbsp;parallel to o o, which will cut the fifth perpendicular at rf, the third station staff from this pointnbsp;�et off 5 feet 6 inches A downwards to C, whichnbsp;^ill be our second limit with respect to the preceding one, and the third with respect to thenbsp;then draw your instrument in the middlenbsp;oetween B and C, and delineate the ground withnbsp;'fs different inequalities. Proceed in the samenbsp;^^aiiner, from station to station, to the last N, andnbsp;} ou will have the profile of the ground over whichnbsp;ffie level was taken.

'Fo trace more particularly the profile of each station. It is necessary to observe in this place,nbsp;that if the object of the ojicratiouhe only to knovvnbsp;the reciprocal height of the two extreme terms, a*'�nbsp;in the preceding example, then the method therenbsp;laid down for the profile or section will be suft-cient; but if it be necessary to have an exact detail of the ground between the said limits, thenbsp;foregoing method is too general; we shall, therefore, institute another example, in which we shallnbsp;suppose the level to have been taken from A to Nnbsp;by another route, but on more uniform ground?nbsp;and less elevated above the level of the two rivers?nbsp;in order to form a canal, marked O, P, Q, R, S, 1?nbsp;U, X, Y, for a communication from one river tonbsp;the other.

Draw at pleasure a line Z X, fig. 5, plate 24, represent the level line and rcgidatc the rest; theiinbsp;let fall on this line, perpendiculars to represent thenbsp;staves at the limits of each station, taking carenbsp;tliat they be fixed accurately at their respectivenbsp;distances from each other.

As the difference between the extreme limits ought to be the same as in the former example,

5 feet 4 inches A, set oft� this measure upon the perpendicular at o, the first limit, and from o, prolonging the perpendicular, mark off at a, the heightnbsp;dctcrrnfned at the first station staff, as in the pi'O'nbsp;ceding example; then do the same with the second and third, and so on with the following, td*nbsp;this part of the work is finished; there then remains only to delineate in detail the ground between the station staves; the distances arc 3*^'nbsp;Slimed larger in this instance on account ofnbsp;detail.nbsp;nbsp;nbsp;nbsp;,

I�o get the section of the ground between O an P, place your instrument at one of the limits as i-gt;.

fixing it, so that the cross hairs may answer to the l^int b; then look towards the first limit o, rais-i^ig or depressing the vane, till it coincides withnbsp;fhe intersection of the cross hairs, and the line ofnbsp;sight, from one point to the other, will mark thenbsp;level or horizontal line.

Now to set off the height of the brink of the fiver above the first limit, drive a stake dowmnbsp;vlose to the ground at a, upon which place yournbsp;station staff, and observe at what height the hairsnbsp;intersect the vane, it will be at 4. 10; then lavingnbsp;off upon the line o z, the distance from the firstnbsp;ftfrin to the first stake, from whcnco let fall a perpendicular, and set off thereon 4. 10 to a, whichnbsp;gives the height at the first stake, or, which is thenbsp;*arne,thc height from the edge of theriverabove thenbsp;surface of the water, as is plain from the section.

Drive a second stake at b, in a line between the limits; place the station staff upon this stake, andnbsp;observe the height 4. �. intersected by the crossnbsp;luiirs, the instrument still I'emaining in the samenbsp;situation; set off on the level line the distancenbsp;from the first stake a to the second h, and thennbsp;frt fall a perpendicular, and mark upon it 4. 6. tonbsp;which gives us the height of the ground at thisnbsp;place.

To mark out the small hollow c, drive down a fhird stake even with the ground, in the middlenbsp;quot;^l^it at c; but as before in the station line, marknbsp;^pon the level line the exact distance from thenbsp;�ocond stake b, to the third c\ then let fidl a pei--Pondicular from c, and set off thereon 6:8:0nbsp;� pointed out by the cross hairs on the staffj whichnbsp;ootermincs the depth of the hollow, as may benbsp;�oen by the section.

^ith respect to the ground between each stake, the distances arc now very short, it will be

easily expressed by the operator, whose iudgmetit will settle the small inequalities by a comparisonnbsp;with those already ascertained.

Proceed thus with the other stations, until you arrive at the last, and you will obtain an accuratenbsp;section in detail of 3quot;our work; by such a section,nbsp;it is easy to form a just estimation of the land tonbsp;be dug away, in order to form the canal, by addingnbsp;thereto the depth to be given to the canal.

Another example of compound levellhig. In thC levelling exhibited in plate 25, we have an exani'nbsp;pie taken where the situation w'as so steep andnbsp;mountainous, that it was impossible to place thnnbsp;staves at equal distances from the instrument, otnbsp;even to make a reciprocal levelling from one station to another.

Such is the case between the first point A, taken from the surface of a piece of water, w'hich fall^nbsp;from the mountains, and the last point K, at thenbsp;bottom of a bason, where it is proposed to makenbsp;fountain, and the height is required, at which ^nbsp;jet d�eau will play by conducting the water froinnbsp;the reservoir A, to the point K of the bason, bynbsp;tubes or pipes properly made, and disposed withnbsp;all the usual precaution.

From the manner in which the operator proceed in this instance, it is evident, that the in'nbsp;Strumc-nt shoidd be adjusted with more than nt'nbsp;dinary care, as the true cirstance from one monn-tain to the other cannot be attained without inucl*nbsp;trouble.

The height is here so great, that it w'ill be nC' cessary to go by small ascents from A to D, ^n ^nbsp;it will probably be commodious in some pnA ^nbsp;the w'ork to use a smaller instrument; underneatnbsp;is a table of the different level points as ascertain'nbsp;cd, which, with the profile and plan, render tlus

Only two IcveHings are made here between A '�^'d D, though it is evident from the plan, thatnbsp;would have been necessary; but as our design is only to shew the manner of proceeding innbsp;particular case, and as more would have con-the plan and section, they arc here neglected.

, -111 the fourth, the height found was l6. 8; but as distance E, 350 yards, was considerable, it wa*nbsp;'^cessary to reduce the apparent level to the truenbsp;the same was also done in all other casesnbsp;^I'e it was necessary; at the last limit we firstnbsp;the height from N to o, then from o to I, thennbsp;I to K, which added together, and then cor-for the cun�ature, gives 47. 3. 0. Now,nbsp;y adding each column together, and then sub-,^ctin�

The general section o{ \\\\^ operation is delineate'^ tinder the plan at Jig. 2, and is sufficiently pla'*'nbsp;from what has been ah'eady said in the precedni?;nbsp;instances. But an exact profile of the mounta'i'nbsp;is not so easy, as it would require many opcrf^'nbsp;lions, as may be seen in the section; some of thesi�nbsp;might however be obtained, by measuring frof�nbsp;the level line already mentioned, without moviii!?nbsp;the instrument, as at Jig. 3.

Example 3, Jig. 1, plate 20, Mr. Le gives us in that of a river, being one part of fh^jnbsp;river Haynox, from Lignebruk to Villelmrg,nbsp;the inode he obseiwed in taking this level.

The first operation was that of having stak^*^ driven at several parts of the river, even withnbsp;water�s edge; the first stake A, a little above th^nbsp;mills at Lignebruk, shews the upper waternbsp;when the water is highest, and is our first limd�nbsp;the stake h shews the low water mark at the saii�*^nbsp;mills, the stake B is the second limit.nbsp;nbsp;nbsp;nbsp;.

The stakes C and D, above and below the iTi''' of Mazurance, shew the height of the waty'�fnbsp;when at the highest and lowest, and their 'nbsp;ference; these stakes form our third andnbsp;limits. Lastly, the stakes at E and F, abovenbsp;below the mills atnbsp;nbsp;nbsp;nbsp;mark as before f*�.

difference between the highest and lowest of the water, and are also the last limits of ^nbsp;operation.nbsp;nbsp;nbsp;nbsp;. t

Particular care was taken, that the marks sho^ all be made exactly even with the edgenbsp;water, and they were all made at the difh^'�^nbsp;parts of the river, as nearly as possible at thenbsp;instant of time.

The principal limits of the levelling determined and fixed, it only remains to binftnbsp;fevel between the limits, according to the wet

already pointed out, using every advantage that may contribute to the success of the work, and atnbsp;the same time avoiding all obstacles and difficulties that might retard, or injure the operations.

The lirst rule is always to take the shortest pos-�liblc way from one limit to the other.

However, this rule must not be followed if there are considerable obstacles in the way, asnbsp;liills, woods, marshy ground, Sea or, if by goingnbsp;^sidc, any advantage can be obtained; thus, in thenbsp;present instance, it was found most convenient tonbsp;go from A 2 to B, by the dotted line A c d e f g hnbsp;J k B, which, although it appears the longest, wasnbsp;�ii effect the shortest, as you have only to levelnbsp;from one pond to the other, at Ac, de, fg, hi,nbsp;^ B (at the top of the plate,) the distances c d, e 1,nbsp;g h, i k, the surfiiccs of the several ponds beingnbsp;assumed as level lines, thereby abridging the worknbsp;'without rendering it less exact; more so, as it wasnbsp;�^ot tlie length of the river that was required, butnbsp;�^oly the dcclivityL

Having levelled from A to B, proceed from B C, following the dotted line B1 m n o C, whencenbsp;^''0 obtain the difference in height between thenbsp;**^irface of the water at A, and that at C.

The next step was to level between C and D, above and below the mills, to find the differencenbsp;oetween the water when at its highest and lowestnbsp;��'diiations. From D, levels were taken across thenbsp;Country to p-, leaving on account of the pond ornbsp;ake which was assumed as level, we began at q,nbsp;mm thence to r, where we left off; beginningnbsp;'�gain at s, then levelling from thence to /, and so'nbsp;to L, above the mills at Villehotirg, and finishednbsp;n� ^�^^ow them.

By these operations we obtained the knowledge how much the waters above and below the millsnbsp;n d 2

of lAnehruk arc liighcr than those of MazurancCf and these, than tliosc of V'lilebown. with all thonbsp;necessary consequences. From this example, thi?nbsp;importance of a thorough knowleilgc of the grouiK-in order to carry on such a work is very evident;nbsp;this picec of levelling was near live German iniF'^nbsp;in length, in a strait line, and nine or ten with thenbsp;bendings of tile river. For the protilc or sectionnbsp;of the foregoing operation, fig. 2, plate 20, tii'^^nbsp;draw the dotted line AG, on which let fall pet'nbsp;pendiculars from the principal limits ABCJ^nbsp;produced; then beginning at the highest watet'nbsp;mark at Lignebruk, set off three feet to six, for thnnbsp;difference between high and low water; from ^nbsp;draw the dotted line b c, parallel to AG. Fromnbsp;the point set off on the perpendicular four feet tnnbsp;B, the dilfci'ence found between b and B; from ^nbsp;draw B d, parallel also to AG; then set off thi't^nbsp;feet from d to E downwards, for the diffcrciK'*^'nbsp;found between B and C, and 4^ feet from C tonbsp;for the diflerenee in height of the mills atnbsp;ranee. P'rojn D draw the line D e parallel to A O��nbsp;and from the point c to L, set off three feet for th^*nbsp;difference of the level between D and F. Ai^nbsp;lastly, from E to F, set off one foot six inchesnbsp;the difference between the higher and lowernbsp;ters at the mills of Villehourg, shewing that tlmmnbsp;arc ly feet difference which the upper waters �Anbsp;Lingebruk arc higher than the lower waters ^nbsp;Villebourg.

The pocket measuring tape of 100 feet length, with the centesimals of a yard, isnbsp;be a useful article in the practice of levelling*

fined, the glass is parallel. Or a glass may be accurately examined by laying it on a piece of papeo and viewing the top of a wall or chimney, See*nbsp;about 15 or 20 yards distant; for, if the two surfaces be not parallel, the object will appear doublenbsp;and surrounded with a light fringe, and thus maynbsp;every part of the glass be examined, and its defects discovered; the examination will be morenbsp;perfect if a small telescope be used.

To examine whether a surface be a perfect plane, take the sun�s diameter very accuratelynbsp;with your sextant, when its altitude is considerable; then examine in the surface y'ou wish to trynbsp;the two images, without altering the index; ifnbsp;be concave, the two images will lap over, if coU'nbsp;vex, they will separate, and the quantity of thisnbsp;error may be found by the sextant. If, therefore^nbsp;you use glass, amp;c. as a reflecting surface for aunbsp;artificial horizon, you must either allow for thenbsp;error, which makes the given altitude too great, dnbsp;the glass be convex, and too small, if concave, ofnbsp;you must make both your adjustments and observations from the same reflecting surface. But thi�nbsp;will not entirely obviate the difticulty, as the surface is apt to vary from the sun�s heat during ^nbsp;long course of observations.

The angle, observed by means of the artifici^ horizon, is always double the altitude of the star?nbsp;amp;c. above the horizon; consequently, you caiiuonbsp;take an altitude of more than 45� with an octapbnbsp;or of 6o� by the sextant.

Every thing being ready, and the instrumct' properly adjusted, move backward, till you see thnbsp;reflected image of the sun in the water.nbsp;image be bright, turn one or more of the drunbsp;glasses behind the horizon glass,

Hold now the sextant in a vertical plane, and direct the sight to the sun�s image in the artificialnbsp;horizon. Then move tlie index tell you see thenbsp;*^thcr image reflected from the mirrors come downnbsp;the sun�s image seen in the horizon, so as tonbsp;touch, hut not pass it; thcTi bring the edges innbsp;Contact in the middle, between the wires of thenbsp;telescope, as before directed, and the divisions onnbsp;the arc will shew the double altitude.

Correct the double altitude for the index error, before you halve it. Then to this half altitude addnbsp;the sun�s semi-diameter, and subtract the correction for refraction, and you will have the true altitude of the sun�s center above the real horizon.

The altitude of a star must betaken in the same lUanncr as that of the sun; the double altitudenbsp;'^ust be corrected for the index error, if any, thennbsp;halved, and this half corrected for refraction givesnbsp;the true altitude above the real horizon.

In taking tJie sun�s altitude, whether for the Purpose of calculating time, or for double alti-Pidcs, it is best to fix the index to some particularnbsp;�^hvision of the instrument with great nicety, andnbsp;^hcn wait till the sun is risen or fallen to that al-^hude.

-Phis is much better than observing its altitude und moving the screw to it, as the screw when thusnbsp;j�Uddcnly moved is very apt to alter a small triflenbsp;Y the inequality and pressure of the threads, afternbsp;he hand is removed from it; whereas, when it isnbsp;.*^cd to some division previous to the observation,nbsp;h may repeatedly tried and examined bet'orenbsp;he observation is taken.

. tku accurate observer will find that the error of ^'xtant will vary according as Im takes it, by

excess; this is owing to some spring in the index, or inequality in tlie adjusting screw, which itnbsp;very difficult totally to obviate.

The best way to correct this, is always to move the index the same way in making your observa-tions, as you did in taking the error of adjustment; though, where a great number of observations are taken, it were best both to settle the adjustment, and take the observations alternately hvnbsp;moving the index baekwards and forwards, or h}'nbsp;setting the objeets open, and making them kTnbsp;over alternately. A mean of all these will certainly be the most free from error, as the errorsnbsp;will counteract each other.

This may also correct a faulty habit which an observer may have contracted, in forming the contact between the two objects; and though therenbsp;may seem to be some impropriety in the mode,nbsp;yet a mean of them will be much nearer the truthnbsp;than any single observation, where a person pm'nbsp;fers seeming to real accuracy.

The lower limb of the � or ([ always come� first into contact, when you move the index fo^'nbsp;ward, and the index shews the double altitude otnbsp;the 0 �s upper limb, if the moveable sun is uppe^'nbsp;most, but of the lower limb when the moveabwnbsp;siin is lowermost.

At sea, they generally take the altitude of 0�s lower limb, because it is most natural to briHonbsp;that to sweep the horizon. But, by land, ^nbsp;most correct to take the altitude of the 0's upp*-limb; t. Because it is highest, and less liable Tnbsp;be affected by refraction. '2.' Because the jnbsp;diameter and refraction are both sulffractivc, nunbsp;the operation is more direct than when one isnbsp;the other minus.nbsp;nbsp;nbsp;nbsp;'

Observations taken by means of tlie fore horizon Rlass are called fore observations, because in themnbsp;toth objects are before the observer.

Previous to every observation, the instrument should be examined, in order to see whether thenbsp;index or horizon g-lasses be firm, or whether anynbsp;nfthe screws be loose; the horizon glass must alsonbsp;adjusted.

One or two of the dark glasses should be plaeed before the horizon glass, always projxirtioning thenbsp;stren gth of the shades to the brightness of the sun�snbsp;i'nys, that the image may be looked at w'ithout injuring the eye.

Hold the quadrant in a vertical position, the arc i^ownwards, either by the braces or the radii, asnbsp;niay be most convenient, or still bettor accordingnbsp;the foregoing direetions. Let the eye be at thenbsp;dipper hole in the sight vane, and the lower part ofnbsp;^^ie limb against the breast.

Turn yourself towards the sun, and direct the �ight to that part of the horizon that lies directlynbsp;^^uder it, keeping the quadrant, as near as you cannbsp;Judge, in a plane passing through the sun�s centernbsp;��'ud the nearest part of the horizon, moving at thenbsp;Same time the index with the left hand, so. as tonbsp;bring the imag-c of the sun down towards the ho-iizon; then swing the quadrant round in a linenbsp;Purallel to the line of sight; by tliis means thenbsp;Ullage of the sun maybe made to describe tlie arcnbsp;a cirele, with the convex .side downwards,nbsp;if that edge of tlie sun which is observed,nbsp;just grazes upon the horizon, or if the horizon justnbsp;uuches it like a tangenty'without cutting it, the

observation is rightly made, and the degrees minutes pointed out by the nonius on the arcnbsp;shew the apparent altitude of the sun. But if thcnbsp;Sun�s edge dip below, or cut the horizon, the i�'nbsp;dex must be moved backward; if, on the contrary�nbsp;it falls short of it, the index must be moved forward, until it just grazes the horizon.

Dr. Maskelyne gives the following advice: that in taking the sun�s altitude, the observer shoul^^nbsp;turn his quadrant round upon the axis of vision�nbsp;and at the same time turn himself upon his heel�nbsp;so as to keep the sun always in that part of thenbsp;horizon glass whieh is /at the same distanee as thenbsp;eye from the plane of the quadrant; and that uO'nbsp;less care be taken to observe the objects in thenbsp;proper part of the horizon glass, the measured angles cannot be true. Iii this method the reflectednbsp;sun will describe an arc of a parallel circle roundnbsp;the true sun, whose convex side will be downwards, and consequently when by moving the inquot;nbsp;dex, the lowest point of the arc is made to touchnbsp;the horizon, the quadrant will stand in a verticalnbsp;plane, and the altitude above the visible horizonnbsp;will be properly observed.

Great care should be taken that the situation ol the index be not altered, before the quantitynbsp;makes is read off.

The observed or apparent altitude of the sun requires three corrections, in order to obtain thenbsp;true altitude of the sun�s center above the ho

1. The first correction is to obtain the obscr'�^ altitude of the sun�s center.nbsp;nbsp;nbsp;nbsp;. �

All astronomical calculations respecting't�.� heavenly bodies, arc adapted to their centers; y ^nbsp;in taking altitudes of the sun, it is usual to hroypnbsp;his lower limb in appai-cnt contact with the hori

*^011, In this case it is evident, that a quantity equal to the seniidiameter of the sun must be addled to the observed altitude, to give the altitudenbsp;ef his center. But if on any occasion, as fromnbsp;elouds, the altitude of the upper liinl) betaken, thenbsp;Seniidiameter of the sun must be siditractcd.

The mean semidiameter of the sun is id minutes, which may be taken as a constant quantity 111 common observations, as the greatest variationnbsp;this quantity scarcely exceeds one quarter ofnbsp;^ minute.

�i. The second coiTcction is, to rectify the cr-*�oi's arising from refraction.

One of the principal objects of astronom}^ is to the situation of the several heavenly bodies.nbsp;It is necessary, as a first step, to understand thenbsp;*^auscs which occasion a variation in the appear-uiice of the place of those objects, and make usnbsp;Suppose them to be in a different situation fromnbsp;'^�hat they really arc: among these causes is to benbsp;�quot;'^ukoned the following.*

The rays of light, in their passage from the ce-�^stial luminaries to our eyes, are bent from their true direction by the atmosphere; this bending isnbsp;^^lled refraction, and they are more or less re-�quot;quot;ictcd, according to the degree of obliquity withnbsp;}'^hich they enter the atmosphere, that is accord�ll? to the altitude of the object; from this cause,nbsp;^�'cir apparent altitude is always too great; thenbsp;^^^^mtity to be subtracted from tlic observed alti-^ide may be found in any treatise on navigation.

.*1 he following pleasing and easy experiment quot;'dl giyg reader an idea of what is meant by

propert)', to which wc are indebted for all advantages of vision, and the assistance we receivenbsp;from telescopes, amp;c.

Experiment. - Into any shallow vessel, a basoi^j put a shilling, and retire to such a distance,nbsp;that you can just sec the farther edge of the sh�'nbsp;ling, but no more ; let the vessel, the shilling,nbsp;your eye, remain in the same situation, whilenbsp;assistant fills up the vessel with water, and th¹^�nbsp;whole shilling will become visible, the rays coni'nbsp;ing from the shilling being lifted or bent upward¹nbsp;in their passage through the water. For the sain^nbsp;reason, a strait stick put partly into water appearsnbsp;bent.

The dip of the horizon is the quantity that thr' apparent horizon appears below the true horizninnbsp;and is principally occasioned by the height ofnbsp;observer�s eye above the water; for, as he is eJe'nbsp;vated above the level of the sea, the horizonnbsp;views is below the true one, and the observed alti'nbsp;tilde is too great, by a quantity proportioned to fd^'nbsp;height of the eye above the sea: the quantity to donbsp;subtracted from the altitude will be found in an)nbsp;Treatise upon Navigation.¹

Mr. Nicholson says, that observers at sea genO' rally choose to stand in the ship�s waist when theynbsp;take altitudes, because the height of the eye abo'¹-.nbsp;the water is not so much altered by the motion enbsp;the ship; but this is of no consequence, foi' ��nbsp;rough weather the edge of the sea beheld fm^^ ^nbsp;small elevation is made uneven by waves, who^^nbsp;altitudes amount to two or three minutes, or uioio¹

^vliicli circumstance produces as great an unccr-^'dnty as the rise and fall of the object seen from 'he poop, when the sliip pitches. These arc minute causes of error, but not to be disregarded bynbsp;fhosc who wish to obtain habits ot accuracy andnbsp;^sactness.¹

The meridian altitude of the sun'|' is found hy ^Itending a few minutes before noon, and takingnbsp;his altitude from time to time; wdicu the sun�snbsp;^dtitude remains for some time without any con-^^idcrahlc increase, the observer must be attentivenbsp;mark the coincidence of the limb of the sunnbsp;quot;�'th the horizon, till it perceptibly dips below thenbsp;^dge of the sea. The quantity thus observed isnbsp;'he .meridional altitude,

TO TAKE THE ALTITUDE OF A STAR. Before an observer attempts to take the altitudenbsp;star, it will be proper for him to exercise him-f'-df by viewing a star with the quadrant, and learn-to follow the motion of the reflected imagenbsp;^''ithout losing it, lest he should take the image ofnbsp;^oine other star, instead of that whose altitude henbsp;desirous to obtain. His quadrant being properlynbsp;�Adjusted, let him turn the dark glasses out oi thenbsp;and then,

'2-. Hold the quadrant in a vertical position, Agreeable to the foregoing directions.

3- Look, through the sight vane and the transparent part of the horizon glass, strait up to the which will coincide with the image seen in

the silvered part, and form one star; but, as sooo as ) on move the index forward, the reflected imagenbsp;will descend below the real star; you must folloquot;^nbsp;this image, by moving the whole body of the qua�nbsp;th'ant downwards, so as to keep it in the silverednbsp;part of the horizon glass', as the motion of the iu'nbsp;dex depresses it, until it comes down exactly to thenbsp;edge of the horizon.

It is reckoned better to observe close than opcTi; that is, to be well assured that the objects touchnbsp;each other; and this opinion is well founded,nbsp;many }x*rsons are near-sighted Avithout knowiui?nbsp;it, and see distant objects a little enlarged, by thenbsp;adtlition of a kind of penumbra, or indistinct shading oftquot; into the adjacent air.

There arc but two corrections to be made to the olAscrvcd altitude of a star, the one for the dipnbsp;of the horizon, the other for refraction.

The first subject for consideration is, wlicthe�' the sun�s declination be north or south; and,nbsp;eondly, whether the required latitude be north utnbsp;south. If the latitude, or deeliitation, be hotunbsp;north, or both south, they are said to be of th*^-same denomination; but if one be north, and ih*'nbsp;other south, they arc said to be of different deuo-ininations. Thirdly, take the given altitui^nbsp;from 90� to obtain the zenith distance.

Rule 1. If the zenith distance and dcclinutip|| be of the same name, then their difference ^ ^nbsp;give the latitude, whose demomination is the samenbsp;with the declination, if it be greater than the 2e^nbsp;nith distance; but the latitude is of a contmr)

Rule �1. If the zenith distance and declination have contrary names, their sum shews the requirednbsp;^^titude, whose name will be the same as the destination.

Or, by the two following rules you may find the latitude of the place from the altitude and dedication given.

Rule 1. If the altitude and declination are of difFcrent names, that is, the one north, the othernbsp;South, add 00 degrees to the declination; fromnbsp;that sum subtract the meridian altitude, the rc-ciainder is the latitude, and is of the sanie name asnbsp;the deel illation.

Rule 2. If the meridian altitude and declination are of the same denomination, that is, both Corth or both south, then add the declination andnbsp;quot;Ititude together, and subtract that sum from ponbsp;degrees, if it be less, and the remainder will be thenbsp;hititude, but of a contrary name. But if the sumnbsp;exceeds po degrees, the excess will be the latitude,nbsp;'¹f the same name with the declination and altitude.

Rxample 1. Being at sea, July 2g, 1779, the �Meridian altitude of the sun�s lower limb was observed to be 34� 10' N. the eye of the observernbsp;being 25 feet above the sea. The latitude of thenbsp;place is required.

The sun�s declination for the third year ¹ aftci¹ leap year on July 29, is found in Table IILnbsp;18� 46' N. the dip for 25 feet elevation is five nii-nutes, the refraction for 34� is one minute; therefore,

Example Q.. October 20, 1780, sun�s meridiquot; onal altitude, lower limb 02� 09' S. required thenbsp;latitude. Height of the eye 30 feet.

178O is leap year, and,the sun�s declination i¹� October 20, is 12� 45' S. The dip for 30 foetnbsp;elevation is six minutes/ the refraction for 02�

The annual course of the seasons, or the natural year, coH' sisting of nearly 3()5 days six hours, and the current year beii'Snbsp;reckoned 305 days, it is evident that one whole day wouldnbsp;lost in four years if the six hours were constantly rejected.nbsp;avoid this inconvenience, which, if not attended to, would causenbsp;the seasons to shift in process of time through all the months onbsp;the year, an additional day is added to the month of Februa¹/nbsp;every fourth year; this fourth year is termed leap year, andnbsp;found by dividing the year of our Lord by 4; leap year leaves 1'¹^nbsp;remainder; other years arc called the tirst, second, or thirdnbsp;after leap year, according as the remainder is 1, 2, or 3.

The I^duiical yllmanack, and liohcrts.mis Treatise on contain the best tables of the sun�s declination, amp;c. amp;c.

Example 3. Jan. 7, 1776, altitude of sun�s lower limb at noon 87� 10' S. height of the eyenbsp;30 feet, required the latitude.

Example 4. In the year 1778, July 30, the meridian altitude lower limb was 84� lo' N.nbsp;�quot;'Squired the latitude, the height of the eye beingnbsp;^0 feet.

Example 5. Being at sea in the year 1777, ^^ose weather prevented the meridian observationnbsp;thnbsp;nbsp;nbsp;nbsp;night proving clear, the nor-

the Great Bear was observed to come to its least altitude 30� 10'. Required the latitude; the-height of the eye being 20 feet.

Example Q. June 11, 17/0, the sun�s rneO' diau altitude of the upper limb below the poi^nbsp;was observed to be 2� 03'. The height of the oh-server�s eye being 16 feet; required the latitude-

The enlightened edge of the moon, or th^^ edge which is round and w�cll defined, must 0nbsp;brought in contact with the horizon, whether ^nbsp;be the upper or under edge; in other respects, to ^nbsp;same method is to be used in taking the altitude 0nbsp;the moon as was directed for the sun.nbsp;nbsp;nbsp;nbsp;j

Between nc�W and full moons the enlighten^ limb is turned towards the west; and during t ^nbsp;time from the full to the new moon, the enhginbsp;tened limb is turned towards the cast.nbsp;nbsp;nbsp;nbsp;^

If that telescope, which shews objects inverte 7 be used, then the upper edge or limb of thenbsp;will appear the lower, the left side will appealnbsp;right, and the contrary.

methods of det ermining the longitude by observation, consists in discovering the differencenbsp;of apparent time between the two places undernbsp;consideration.

The angular motion of the moon being much greater than that of any other celestial body, tl'Cnbsp;observation of its place is much better adapted tonbsp;discover small differences of time, than similarnbsp;observations made with any other instrument'nbsp;The only practical method of observing its placenbsp;at sea, is that of measuring the angular distancenbsp;between it and the sun and a tixed star.

The most obscure, or rather, the least lumiiion^ of the two objects must be viewed directly,nbsp;the othdr must be brought by reflection in appa'nbsp;rent contact with it.

The well-defined image of the moon must always made use of for the contact, even though dnbsp;should be necessary for that purpose to make tb^nbsp;reflected image pass beyond the other.

In the night-time it is necessary to turn doquot;^'' one or more of the green screens, to take oflnbsp;glare of the moon, wdiich would otherwise prevc'i^nbsp;the star from being seen.

The central distances of the sun and moon every three hours of time, at Greenwich, on suchnbsp;as this method is practicable, are set down innbsp;Nautical Almanac. From these distancesnbsp;are to compute roughly the distance between theifnbsp;nearest limbs at the time of observation.

Mr. Lucllam says, the moon should be viewed directly� through the unsilvered part of the horizonnbsp;glass, but the sun by reflection; and, if it be verynbsp;bright, from the unsilvered part of the glass.

If the sun be to the left hand of the moon, the sextant must be held with its face downwards; butnbsp;'vith the face upwards, if the sun be to the rightnbsp;hand of the moon.

Set the index to the distance of the nearest limbs �f the sun and moon, computed roughly as before;nbsp;and, placing the face of the sextant agreeable to thenbsp;foregoing rules, direct the telescope to the moon,nbsp;putting the sextant into such a position, that if younbsp;look edge-ways against it, it may seem to form anbsp;line passing through the sun and moon, a circumstance that can be only obtained by practice, thenbsp;parent of aptness; then give the sextant a sweepnbsp;Or swing round a hue parallel to the axis of thenbsp;telescope, and the reflected image of'the sun willnbsp;pass by the moon to and fro, so near that you cannot fail of seeing it.

The nearest edges, or limbs, may now be brought into exact contact, by moving the index,nbsp;^ud then using the adjusting screw; observing,nbsp;tlrst, that on giving the sextant a motion roundnbsp;the axis of the telescope, the images of the sun andnbsp;^�'oon only touch at their external edges, and thatnbsp;the body of the sun must not pass over, or be uponnbsp;the body of the moon. And, secondly, that thenbsp;�dge of the sun touch the round or enlightenednbsp;^%e of the moon. Then will the index point outnbsp;the observed or apparent distance of the nearestnbsp;^�Iges of the sun and moon.

But the observed distance requires several corrections, before the true distance of the centers of he objects, as seen from tire center of the earth,nbsp;^�an be found.

Correction 1. Is the sum, or their semi-diameters, which is to be added, to give the apparent distance of the centers of the sun and moon.

The semi-diameter of the sun for every sixth day, and of the moon for every noon and midnight, atnbsp;Greenwich, are to be found in the Nautical Almanac; from these their semi-diameters are to benbsp;computed at the time of observation, by the rulesnbsp;to be found in the same work.

Correction 2. Is to free the apparent distance of the effect of refraction and parallax, which willnbsp;then be the true distance of the centers of the sunnbsp;and moon, as seen from the earth.

For this purpose, two sets of tables, with directions how to use them, are to be found among the Requisite Tables to the Nautical Almanac; beingnbsp;a set of tables published for that purpose by thenbsp;Board of Longitude. 8 vo. 1781.

TO TAKE THE DISTANCE BETWEEN THE MOON AND SUCH STARS AS ARE SELECTED IN THEnbsp;NAUTICAL ALMANAC, FOR THE PURPOSE Ofnbsp;FINDING THE LONGITUDE AT SEA.

The distance of these stars from the moon�s center for every three hours at Greenwich, is givennbsp;in the Nautical Almanac, from whence their distance from the enlightened edge may be roughlynbsp;computed as before.

The star must be viewed directly; the moon is generally bright enough to be seen by reflectionnbsp;from the unsilvered part of the glass; the propernbsp;.shade to take off the glare of the moon is soonnbsp;found. When the star is to the left hand of thenbsp;rnoon, the sextant must be held with its face up-\vards; but if the star be to the right hand of tlifJnbsp;niopn, with its face downwards,

Set the index to the distance roughly computed, iind placing the face of the octant by the foregoingnbsp;rules, direct the telescope to the star. Then placenbsp;the sextant so that, if seen edge-ways, it may seemnbsp;to form a line passing through the moon and star,nbsp;and give it a sweep round a line parallel to the axisnbsp;of the telescope, and the reflected image of thenbsp;moon will pass so near by the star, that you willnbsp;see it in the field of the telescope; a proof that thenbsp;sight is directed to the right star.

The enlightened edge of the moon, whether cast or west, must then be brought into contact withnbsp;the star, by moving the index. To know whethernbsp;the contact is perfect, let the quadrant gently vibrate in a line parallel to the axis of vision, for thenbsp;star should just graze the edge, without enteringnbsp;at all within the body of the moon; when this isnbsp;the case, the index will shew the apparent distancenbsp;of the moon from the star, which, when corrected,nbsp;gives the true one.

Correction 1, For the semi-diameter of the moon. This may be found in the Nautical Almanac for every noon and midnight, at Greenwich;nbsp;and from thence computed, by the rules therenbsp;given, for the time of observation. If the observednbsp;Or enlightened limb be nearest the star, the semi-diameter thus found is to be added; if the enlightened edge be the furthest from the star, then thenbsp;semi-diameter is to be subtracted.

Correction 2. Is for refraction and parallax, to he found from the table as directed before lor thenbsp;sun and moon.

These corrections being properly made, you have the true distance of the moon�s center fromnbsp;the star, as seen from the center of the earth.nbsp;�Aom this distance, and the time of observation^nbsp;die longitude may be found.

The star to be observed is always one of the brightest, and lies in a line nearly perpendicularnbsp;to the horns of the moon, or her longer axis; butnbsp;if you have any doubt whether the sight be directed to the proper star, set the index to the supposed distance as before, hold the sextant as nearnbsp;as you can judge, so that its plane, seen edgeways, may coincide with the line of the moon�snbsp;shorter axis, and moving it in that plane, seek thenbsp;reflected image of the moon through the telescope.nbsp;Having found the reflected image of the moon,nbsp;turn the sextant round the incident ray, that is, anbsp;line passing from the moon to the instrument, andnbsp;you will perceive through the telescope all thosenbsp;stars which have the distance shewn by the index;nbsp;but the star to be observed lies in a line nearlynbsp;perpendicular to the horns of the moon, therenbsp;will, therefore, be no fear of mistaking it.

The basis of all astronomical observation is the determination of the exact time of any appearancenbsp;in the heavens. By corresponding altitudes thisnbsp;time may be determined, without the apparatus ofnbsp;a fixed observatory; they are also useful in findingnbsp;a meridian line, and may be easily and accuratelynbsp;made by a sextant.

For these observations it is necessary to be provided with a clock. These altitudes should be observed, in our latitude, at least two hours distant from noon. The best time is when the sun isnbsp;on or near the prime vertical, that is, the cast ornbsp;west points of the compass.

About these times in the forenoon, take several double altitudes of the sun, write down tfle dp-

grees, minutes, and seconds shewn on the arc, and nlso the exact time shewn by the clock at each observation; and let the difFerent observations madenbsp;in the forenoon be written one below the other innbsp;the order they are made.

In the afternoon, set the index to the same degree and minute as the last morning observation; note very exactly the time shewn by the clocknbsp;.when the sun is come down to the same altitude,nbsp;lt;and write down the time on the right hand of thenbsp;last morning observation; proceed in the samenbsp;manner to find the time by the clock of all the altitudes corresponding to those taken in the morning, and write down each opposite to that morning one to which it corresponds.

Take now the first pair of corresponding altitudes, add them together, and to half their sum add six hours; this being corrected for the changenbsp;of the sun�s declination between the morning andnbsp;evening observations, you will have the time ofnbsp;solar noon derived from this pair of observations.nbsp;Do the same for each pair, and take the mean ofnbsp;the times thus found from each pair, and younbsp;will have the exact time shewn by the clock atnbsp;solar noon.

The time by the clock of solar or apparent noon being thus obtained, the time of mean noon maynbsp;be had by applying the pi-oper equation of time.

Or ihis. Add 12 hours to the time of the. afternoon observation, from which subtract thenbsp;time of the forenoon one,¹ and add half the difFc-rence to the time of the forenoon or morning ob-

This gives the, whole inter^'al between the observations, to Which, if your clock is known to vary much [p tiiat time, younbsp;�Rust add the clock�s loss, or subtract its gain, d\iring that interval,nbsp;3ad half this corrected (if necessary) interval must be added to thenbsp;tirne of forenoon, �cc.

Having this time nearly, it must be corrected by the table of equations for equal altitudes, on ac-count of the sun�s change in declination, in thenbsp;interval between the observations; and you mustnbsp;also apply the equation of time foupd in the Nautical Almanac with contrary signs, subtractingnbsp;when it is -j-, and adding when it is �.

As the seconds differ add them together, and divide the sum by 3, by which younbsp;obtain a mean.nbsp;nbsp;nbsp;nbsp;.

For accurate Tables of Equations to Equal Altitudes, and the Method of finding tie Longitude at Sea by Time-keepers, I refer thenbsp;reader to Mr. Wales's pamphlet, 8vo. 1704,

MERI�)I0XAL problems by the stars, AX'D VARIATION OF THE MAGNETIC NEEDLE BYnbsp;THE SUN.¹

7o fix a inerulian line by a star, when it can he seen at its greatest elongaimi on each side of thenbsp;foie. '

� Provide two plummets. Let one hang from a fixed point; let the other hang over a rod, supported horizontally about six or eight feet abovenbsp;the ground, or floor in a house, and so as to slidenbsp;Occasionally along the rod: let the moveablenbsp;plummet hang four or five feet northward of thenbsp;fixed one. Sometime before the star is at itsnbsp;greatest elongation, follow it in its motion with thenbsp;moveable plummet, so as a person a little behindnbsp;the fixed plummet may always see both in one, andnbsp;just touching the star. When the star becomesnbsp;stationary, or moves not beyond the moveablenbsp;plummet, set up a light on a staff, by signals, aboutnbsp;half a mile off, precisely in the direction of bothnbsp;the plummets. Near 12 hours afterwards, whennbsp;the star comes towards its greatest elongation onnbsp;the other side of the pole, with your eye a foot ornbsp;two behind the fixed plummet, follow it with thenbsp;moveable plummet, till you perceive it stationarynbsp;as before; mark the direction of the plummetsnbsp;then by a light put up precisely in that line: takenbsp;the angle between the places of the two lights withnbsp;Hadley s quadrant, or a good theodolite; the center of the instrument at the fixed plummet, and anbsp;lgt;ole set up in day-light, bisecting that angle, willnbsp;exactly in the meridian seen from the fixednbsp;plummet.

This observation will be more accurate if thf-eye is steadied by viewing the plumb-lines and star through a small slit in a plate of brass stuck upright on a stool, or on the top of a chair-back.

This problem depends on no former observations whatever; and, as there is nothing in the operation, or instruments, to affect its accuracyinbsp;but what any one may easily guard against, it maynbsp;be reckoned the surest foundation for all subsequent celestial observations that require an exactnbsp;meridian line. The only disadvantage is, that itnbsp;cannot be performed but in winter, and when thenbsp;stars maybe seen for 12 hours together; whichnbsp;requires the night to be about 15 hours long.

To fix a yneridtan line by two drcunifoJaf stars that have the same right ascension, or dijfefnbsp;precisely ISO degrees.

Fix on two stars that do not set, and whose right ascensions are the same, or exactly 180 degrees different: take them in the same verticalnbsp;circle by a plumb-line, and at the same time Jet anbsp;light be set up in that direction, half a mile, or anbsp;mile off, and the light and plummet will be exactly in the meridian.

In order to place a distant light exactly in the direction of the plumb-line and stars, proceed innbsp;the following manner. Any night before the observation is to be made, when the two stars appear to the eye to be in a vertical position, set upnbsp;a'staff in the place near which you would have thenbsp;plummet to bang; and placing your eye at thatnbsp;staff, direct an assistant to set another staff upright in the ground, 30 or 40 yards off, as near innbsp;aline with the stam as you can. Next day setnbsp;the two staffs in the same places; and in theirnbsp;direction at the distance of about a mile, or half anbsp;ptile northward^ cut a small hole m the ground.

for the place where the light is to stand at night; and mark the hole so, that a person sent there atnbsp;night may find it.

Then choose a calm night, if the observation is to be made without doors; if it is moon-light, sonbsp;much the better, and half an hour before the starsnbsp;appear near the same vertical, have a lighted lan-thorn ready tied to the top of a pole, and set upright in the distant hole marked for it the nightnbsp;before; and the light will then be very near thenbsp;nieridian, seen from the place marked for thenbsp;plummet. At the same time, let another pole, ornbsp;rod, six or eight feet long, be supported horizontally where the plummet is to hang, six ornbsp;�seven feet from the ground, and hang the plumb �nbsp;line over it, so as to slip easily along it, either tonbsp;one hand or the other, as there may be occasion,nbsp;or tie a staff firmly across the top of a pole, six ornbsp;seven feet long; fix the pole in the ground, andnbsp;make the plumb-line hang over the cross staff.nbsp;Let the weight at the end of the line be prettynbsp;heavy, and swing in a tub of water, so that it maynbsp;not shake by a small motion of the air. Thennbsp;�shift the plumb-line to one hand or the other, tillnbsp;one side of the line, when at rest, cut the starnbsp;which is nearest the pole of the world, and thenbsp;middle of the light together: as that star moves,nbsp;continue moving the plumb-line along the rod,nbsp;so as to keep it always on the light and star, tillnbsp;the other star comes to the same side of the plumbnbsp;line also; and then the plummet and light willnbsp;be exactly in the meridian.

There are not two remarkable stars near the north pole, with the same right ascension pre-lt;^isely, or just a semi-circle�s diffei�ence: but therenbsp;�i^e three stars that are very nearly so, viz. the polenbsp;star, e in Ursa Major, and y in Casiopeia. If

cither of the two last, particularly e, arc talen irt the same vertical with the pole star, they wiUnbsp;then be so very near the meridian, that no greaternbsp;exactness need be desired for any purpose in surveying. At London y is about l' west of thenbsp;meridian then; and e much nearer it, on the westnbsp;side likewise. Stars towards the S. pOle propernbsp;for this observation arc, y, in the head of thenbsp;cross; in the foot of the cross; and a, in thenbsp;head of the phenix.

7o find a meridian line hy a circumpolar star^ when it is at its greatest azimuth from the pole. Bynbsp;a circumpolar star is here meant, a star whosenbsp;distance from the pole is less than either the latitude, or co-latitude of the place of observation.

1. nbsp;nbsp;nbsp;Find the latitude of the place in which younbsp;are to observe the star.

2. nbsp;nbsp;nbsp;Fix on a star whose declination is known,nbsp;and calculate its greatest azimuth from the north,nbsp;or elevated pole.

3. nbsp;nbsp;nbsp;Find at Avhat time it will be on the meridiannbsp;the afternoon you are to observe, so as to judgenbsp;about what time to begin the observation.

4. nbsp;nbsp;nbsp;Prepare tw'o plummets, and follow the starnbsp;with one of them, as directed in page 427, untilnbsp;the star is stationary; then set up a light hal anbsp;mile, or a mile off, in the direction of the plumb-line and star, and mark the place of the light innbsp;the ground, and also of the plumb-line.

5. nbsp;nbsp;nbsp;Next day set up a pole where the light stood,nbsp;W'itb a flag flying at it: then with a theodolite,nbsp;or Hadley lt;1, cjuadrant, set the index to the degreenbsp;and minute of the star�s azimuth from the north,nbsp;found before; direct, by waving to one hand ornbsp;the other, an assistant to set up a staff on thenbsp;same side of the pole Avitb a flag, as the pole ofnbsp;the world was from it, so as at the plumb-line these

two lines may make an angle equal to the azimuth of the star, and the plumb-line and staff will thennbsp;be in the meridian.

When any star is descending, it is on the west-side of the pole of the world; while it ascends, it is on the east-side of it.

The pole of the world is always between the pole star and Ursa-Major: so that when Ursa-Major is W. or E. of the pole star, the pole of thenbsp;World is W. or E. of it likewise.

To find the greatest azimuth of a circumpolar star from the meridian, use the following proportion.

As co-sine of the lat. : R :: S. of the star�s distance from the pole ; S. of its greatest azimuth.

The greatest azimuth is when the star is above the horizontal diameter of its diurnal circle.

On the N. side of the equator, the pole star is. the most convenient for this observation; for thenbsp;time when it is at its greatest elongation from thenbsp;pole, may be knawn sufficiently near by the eye,nbsp;by observing when e, in Ursa-Major, and 7, innbsp;Casiopeia, appear to be in a horizontal line, ornbsp;parallel to the horizon; for that is nearly the time.nbsp;Or, the time may be found more precisely bynbsp;making fast a small piece of w'ood along the plumb-line when .extended, with a cross piece at rightnbsp;angles to the top of the upright piece, like the letter T; when the plummet is at rest, and bothnbsp;stars are seen touching the upper edge of the crossnbsp;piece, then they are both horizontal. It is another conveniency in making use of this star, thatnbsp;it changes its azimuth much slower than othernbsp;stars, and therefore affords more time to take itsnbsp;direction exact.

On the south side of the equator, the head of the cross is the most convenient for this observa-

tion, being nearest to the S. pole; and the time its coining to the horizontal diameter of the diurnal circle, when it appears in a horizontal line withnbsp;the foot of the cross, or the head of the phenix;nbsp;and at the equator its greatest azimuth is one hournbsp;l6 minutes before or after that, as it is E. or W.nbsp;of the meridian. The pole of the world is between the head of the cross and the last of thesenbsp;stars.

y'17 Jrml vi'hen any siar �UJill come to the merldiagt;t either in the south or north. Find the star�s rightnbsp;ascension, in time, from the most correct tables,nbsp;also the sun�s right ascension for the day andnbsp;place proposed; their ditFerence will shew the dit-fcrence between their times of coming on the meridian in the south, or between the pole andnbsp;zenith; which will be after noon if the sun�s rightnbsp;ascension is least, but before noon if greatest.nbsp;Eleven hours 58 minutes after the star has beennbsp;on the meridian above the pole, it will come to thenbsp;meridian north of it, or below the pole.

Given the latitude of the flace, and the decimations and rio/it ascensions of Hvo stars in the same verticalnbsp;line, to find the horizontal distance of that verticalnbsp;from the meridian, the tune one of the stars will takenbsp;to come from the vertical to the meridian, and thenbsp;frecise time of the observation.

Let Z PO be a meridian, Z the place of observation, H O its horizon, and Z P a vertical circle passing through two known stars, s and S; P thenbsp;()olc, E Q. the equator, P s D and P S d circles ofnbsp;declination, or right ascension, passing throughnbsp;these stars: then is ZP the co-latitude of thenbsp;place, P s and P S the co-declination of the tw'Onbsp;stars respectively, and the angle s P S the nearest

distance of their circles of right ascension; V O the arc of the horizon between the vertical circle andnbsp;the meridian; and d Q the arc of the equatot between the star S and the meridian. When, in thenbsp;triangle P Z s, the angles P Z s and s P Z, measured by the arcs VO and D E, arc found, thenbsp;problem is solved.

Solution 1. Begin with the oblique-angled triangle P s S, in which are given two sides P s and P S, the co-declinations of the two stars respectively, and the included angle SPs; which anglenbsp;IS the difference of their right ascensions when itnbsp;is less than 180 degrees; but if their difference isnbsp;more than 180 degrees, then the angle SPs isnbsp;equal to the lesser right ascension added to whatnbsp;the greater wants of 300 degrees. From hencenbsp;find the angle SsP by the following proportions;

As radius is to the co-sine of the given angle S P s, so is the tan. of the side opposite the re-'quired angle P s, to the tangent of an angle, whichnbsp;call M.

M is like the side opposite the angle sought, if the given angle is acute; but unlike that side, ifnbsp;the given angle is obtuse.*

Take the difference between the side P S, adjacent the required angle, and M; call it N. Then,

Sine N : sine M :: tan. of the given angle SPs '� tan. of the required angle S s P, which is like thenbsp;given angle, if M is less than the side P S adjacentnbsp;to the required angle; but unlike the given angle,nbsp;if M is greater than P S.

Two arcs or angles are said to be like, or of tlie same kind, 'vhen both are less than 90�, or both more than 90�; but are saidnbsp;to be unlike, when one is greater, and the other less than 90�gt;nbsp;^nd are made like, or unlike to another, by taking the supple-�Rent to 180� degrees of the arc, or angle, produced in like proportion, in place of what tlie proportion brings out.

1. Next in the oblique-angled triangle P s there�are given two sides, PZ, the co-latitude ofnbsp;the place of observation; PS, the co-dcclinationnbsp;of the stars; and the angle PaZ opposite to onenbsp;of them, which is the supplement of PsS, lastnbsp;found to 180�: from thence find the angle PZsnbsp;opposite the other side, by the following proportion :

This angle PZs, measured by the arc of the horizonVO, is therefore equal to the horizontalnbsp;distance of the vertical of the two stars from thenbsp;meridian. Let the direction of the vertical benbsp;taken by a plumb-line and distant light, as beforenbsp;directed, or by two plumb-lines, and marked oiinbsp;the ground. Next day, the degrees and minutesnbsp;in the arc V O, may be added to it by Hadley^nbsp;quadrant, and a pole set up there, which will benbsp;in the direction of the meridian from the plumb-line.

3. Last, in the same triangle PsZ find the angle s P Z between the given sides, by the followingnbsp;proportions:

As radius is to the tan. of the given angle PsZ, so is the co-sine of the adjacent side P S, to thenbsp;co-tan. of M.

M is acute, if the given angle and its adjacent sides arc like; but obtuse, if the given angle andnbsp;the adjacent sides are unlike.

As the co-t. of the side adjacent to the given angle Ps, is to the co-t. of the other side PZ;nbsp;is the co-s. M, to the co-s. of an angle, whichnbsp;call N.

N is like the side opposite the given angle, n that angle is acute; but unlike the side opposit^^nbsp;the given angle, if that angle is obtuse.

Then the required angle s P Z is either equal to the sum or difihrcncc of M and N, as the givennbsp;sides arc like or unlike.

The angle s P Z, thus found, added to s P S, and their sum subtracted from 180� will leave thenbsp;angle d P Q, or the arc d Q that measures it,nbsp;which is the arc of the equator the star S mustnbsp;pass over in coming from the vertical, ZV, to thenbsp;�meridian. Which converted into time, and measured by a clock or watch, beginning to reckonnbsp;the precise moment that a plumb-line cuts bothnbsp;stars, will shew the hour, minute, and second, thatnbsp;S is on the meridian.

Find, by the right ascension of the star and sun, at wdiat time that star should come to the meridiannbsp;in the north, the night of the observation; subtractnbsp;from it the time the star takes from the vertical tonbsp;the meridian, and the remainder, corrected by thenbsp;sun�s equation, will be the time when the two starsnbsp;Were in the same vertical.

The lowest of the two stars comes soonest to the meridian below the pole, the highest of themnbsp;comes soonest to that part of the meridian whichnbsp;�S above the pole.

The nearer in time one of the stars is to the me-ndian when the observation is made, it is the' bet-ter;,for then aji ordinary watch will serve to measure the time sufficiently exact, It is still more advantageous if one of them is above the pole,nbsp;when the other is below it.

The nearer one of the stars is to the pole, and the farther the other is from it, the more exact willnbsp;this observation be, because the change of the vertical will be the sooner perceived. For this reason,

north latitudes, stars northward of the jcenith are preferable to those tliat ate southward of it-

If the two stars arc past the meridian when they are observed in the same vertical, then the arc d �nbsp;gives the time S took to come from the meridiannbsp;to the vertical, and must be added to the timenbsp;when that star was on the meridian, to give thenbsp;time of the observation; and the arc of the horizonnbsp;V O must be marked on the ground on the side oinbsp;the vertical, contrary to what it would have beennbsp;in the foregoing supposition; that is, eastwardnbsp;hvu the pole, and westward above it.

When two stars come to the vertical line near the meridian, it may be difficult to judge on whichnbsp;side of it they are at that time; for determiningnbsp;this, the following rules may serve.

The right ascension of two stars may be either each/m, or each than 180 degrees, or onenbsp;more and the other less.

When the right aseension of each of the stars is either less, or more, than 180 degrees, they willnbsp;come to the same vertical on the east-side of thenbsp;meridian when the star, ^vith the greatest rightnbsp;ascension is the lowest-, but on the ^vest-side oi thenbsp;meridian when it is highest.

JVhen the right ascension of one of the stars is more than 180 degrees, and that of the other less.

If the right ascension of the highest is less than 180�, but greater than the excess of the other�snbsp;right ascension above 180�, then they come to the-veilical on the east-side of the meridian. But Hnbsp;the right ascension of the higher star is less thannbsp;that excess, they come to the vertical on the west-side of the meridian.

If the right ascension of the higher star is more than ISO degrees, and that excess is less than thenbsp;right ascension of the lower star, then they come tonbsp;the vertical on the west-side of the meridian; but

tKe excess of the higher star�s right ascension above 180� is more that the right ascension of thenbsp;lower star, then they come to the same vertical onnbsp;the east-side of the meridian.

Case a. IVJmi the Hvo stars are in the same �vertical, soutinvard of the zenith, plate l6, fig. 3.

When two stars ai'c observed in the same vertical line southward of the zenith, or toward the depressed pole, the operations and solutions arenbsp;nearly the same as in Case 1. For let P Z O be anbsp;meridian, Z the place of observation, OH its horizon, ZV a vertical circle passing through thenbsp;two stars S and s, P the pole, QE the equator,nbsp;PD and Pd two circles of declination, or rightnbsp;ascension, passing through the two stars respectively; then is ZP the co-latitude of the place,nbsp;P S and P s the co-declination of the two stars respectively, the angle S P s the difference of theirnbsp;right ascensions, the angle SZQ, or OV the arcnbsp;of the horizon which measures it, the distance ofnbsp;the vertical from the meridian; the angle sPZ,nbsp;or d Q which measures it, the arc of the equator,nbsp;which s must pass over from the vertical to thenbsp;meridian, which converted into time, and measured by a watch, will shew when s is on the meridian; and subtracted from the calculated timenbsp;that the star should come to the meridian, will,nbsp;M'hen corrected by the sun�s equation, give thenbsp;true time of the observation. S P s is the trianglenbsp;to begin the solution with; and in the trianglenbsp;sPZ, the angles sPZ and sZP, when found,nbsp;Will give the solution of the problem, as in Case 1.nbsp;Por the supplement sZB to 180� degrees is thenbsp;angle VZO, or its measure OV, the horizontalnbsp;distance of the vertical from the meridian; thenbsp;angle sPZ, or its measure dO,, is the equatorialnbsp;distance of the vertical from the meridian; for all

On the south-side of t he zenith, when the highest of the two stars has the least right ascension, theynbsp;come to the same vertical on the east-side of thenbsp;meridian; but when the highest star has thenbsp;greatest right ascension, they come to the samenbsp;vertical on the west-side of the meridian.

Problem. To find the suns amplitude at rising, or setting, and from thence the quot;variation nfi the tnagnetic needle, with an azimuth compass.

Make the needle level with the graduated circle in the box. Then, when the sun�s lower edge is a semi-diameter above the horizon, take thenbsp;bearing of its center, from the IN. or S. whichevernbsp;is nearer, through the sights making the threadnbsp;bisect the sun�s disk, and that subtracted from 90�,nbsp;will be the sun�s magnetic amplitude, or distance,nbsp;from the E. or W. points by the needle

Next, calculate the sun�s true amplitude for that day, by the following proportion:

As the co-sine of the latitude is to radius, so is the sine of the sun�s declination at setting or rising to the sine of bis amplitude from the W. or E.

Which will be N. or S. as the sun�s declination is N. orS: and the distance in degrees and minutes between the true E. or W. and the magnetic, is the variation of the needle.

An easy and sure way to prevent mistakes, which the unexperienced arc liable to in this calculation, is, to draw a circle by hand, representingnbsp;the visible boundary of the horizon, and on it tonbsp;mark the several data by guess; then, by inspecting the figure, it will easily appear how the variation is to be found, whether by addition or sub-trgetion, and on which side of the north it lic�lt;nbsp;For example;

V!aie 16, fig. 5. Suppose- the variation was sought at sun-setting. Draw by hand a circlenbsp;N W S E, to represent your visible horizon; in thenbsp;middle of it mark the point C, for your station;nbsp;from C, draw the line C W, to represent the truenbsp;gt;vest; then on the north or south side of that line,nbsp;according as the sun sets northward or southwardnbsp;of the true west, draw the line C O, representingnbsp;.the direction of the sun�s center at setting, andnbsp;another line C w, for the magnetic west, either onnbsp;the north or south side of �, as it was observed tonbsp;be, and at its judged distance. Then by observing the situation of these lines, it will easily occurnbsp;whether the magnetic ajnplitudc and true amplitude are to be added, or subtracted, to give thenbsp;variation; and on which side of the true north thenbsp;variation lies. In the present sujjposition, w �nbsp;is the magnetic amplitude, and �W the truenbsp;amplitude; �W therefore must be subtractednbsp;from � w, to give w W the distance of the onenbsp;from the other; and n, the magnetic N. QOquot; fromnbsp;w, must be westward of N, the true north.

This method is sufficiently exact for finding the variation, but it is not exact enough for fixing anbsp;precise meridian line; because of the uncertaintynbsp;of the refraction, and of the sun�s center: but ifnbsp;the sun ascends, or descends, with little obliquity,nbsp;the error then will be very little.

First, let the latitude of the place be exactly found. Next, let the quadrant be carefully adjusted for observation. Then, two or three hoursnbsp;before or after mid-day, take the altitude of thenbsp;sun�s center as exactly as possible, making the vertical wire of the telescope bisect the sun�s disk;nbsp;*md, without altering the plane of the quadrant in

the least, move the telescope vertically till you sec some distant sharp object on the land, exactly atnbsp;the vertical wire; and that object will be in thenbsp;direction of the sun�s azimuth when the altitudenbsp;of its center was taken. If no such object is to benbsp;let a pole be set up in that direction, about

Add the complement of the latitude, the complement of the altitude, and the complement of the sun�s declination to go� together, and take thenbsp;half of that sum, and note it down; subtract thenbsp;complement of the declination from the half sum,nbsp;and take the remainder; then take the complement arithmetical ¹ of the sines of the complement of the altitude, and of the complement of thenbsp;latitude, and add them together, and to them addnbsp;the sines of the fore-mentioned half sum and remainder; half the sum of these four logarithms i9nbsp;the co-sine of half the azimuth required. Therefore, find by the tables what angle that co-sine answers to, double that angle, and that will be thenbsp;sun�s azimuth from the north.

If the sun�s declination is S. in north latitude, or N. south latitude, in place of taking the complement of the declination to 90�, add go� to it, andnbsp;proceed as before.

In south latitudes the azimuth is found in the same manner; only, the sun�s azimuth is foundnbsp;frorn the S,

The complement arithmetical of a logarithm is foimd thus: begin at the left hand of the logarithm,, and subtract each figurenbsp;from 9, and the last figure from 10, setting down the several remainders in aline; and that number will be the arithnictica}nbsp;iComplement reijuired.

Then, to find the variation, place your needle helow the center of the quadrant, set it level, andnbsp;find how many degrees the pole, or object, in thenbsp;sun�s azimuth, bears from the north by the needle;nbsp;and the difference between that and the azimuthnbsp;found by calculation, is the variation of the needlenbsp;sought.

If the sun ascends, or descends with little obliquity, a meridian line may be fixed pretty exactly this way, because a small inaccuracy in the altitudenbsp;of the sun�s center will not be sensible in the azimuth. But, when the sun docs not rise high onnbsp;the meridian, this method is not to be relied on,nbsp;when great exactness is necessary; for then everynbsp;inaccuracy in latitude, altitude, and refraction,nbsp;occasions severally a greater error in the azimuth.nbsp;To mark the meridian line on the ground, placenbsp;the center of a theodolite, or Hadley's quadrant,nbsp;where the center of the quadrant was when thenbsp;sun�s altitude was taken, and putting the index tonbsp;the degree and minute of the azimuth, direct, bynbsp;waving your hat towards one side, or the other, anbsp;pole to be set up, making an angle with the formernbsp;pole placed in the azimuth, equal to the sun�snbsp;azimuth found; and that last-placed pole will benbsp;in the meridian, seen from the center of thenbsp;quadrant.

N. B. The sun�s declination in the tables must be corrected by the variation arising from the difference of the time between your meridian andnbsp;that of the tables; and also for the variation ofnbsp;declination for the hours before, or after noon, atnbsp;which the sun�s altitude was taken.

The two last problems are constructed and fully explained in Roherison s, Moore's, and other treatises on navigation; it is, therefore needless to benbsp;niorc particular here.

PRACTICAL GEOMETRY^

It is as easy to trace geometrical figures on the ground, as to describe them on paper; there is,^nbsp;however, some small difference in the mode ofnbsp;operation, because the instruments arc diflerent.nbsp;A rod, or chain, is here used instead of a scale;nbsp;the spade, instead of a pencil; a cord fastened tonbsp;two staves, and stretched between them, instead ofnbsp;a rule; the same cord, by fixing one of the stavesnbsp;in the ground, and keeping the other moveable,nbsp;answers the purpose of a pair of compasses; andnbsp;with these few instruments cvciy geometrical figure necessary in practice, maybe easily traced onnbsp;the ground.

Problem 1. To draw upon the ground a strait line through two given points. A, B, fig. 1, plate 27-

Plant a picket, or staff, at each of the given points A, B, then fix another, C, between them,nbsp;in such manner, that when the eye is placed so as

* I have been obliged to omit in this Course, so liberally' communicated by Mr. I.andman, the calculations that illustratenbsp;the examples; this, I hope, will not in the least lessen its use, asnbsp;this work will fall into the hands of very few who are ignorantnbsp;of the nature and application of logarithms.

to sec the edge of the staff' A, it may coincide with the edges of the staves B and C.

The line may be prolonged by taking out the staff A, and planting it at D, in the direction ofnbsp;B and C, and so on to any required length. Thenbsp;accuracy of this operation depends greatly uponnbsp;fixing the staves upright, and not letting the eyenbsp;be too near the staff, from whence the observationnbsp;is made.

We have already observed, that there is no operation more difficult thtm that of measuring anbsp;strait line accurately; when the line is short, it isnbsp;generally' measured with a ten feet rod; for thisnbsp;purpose, let two men be furnished each with a tennbsp;feet rod, let the first man lay his rod down on thenbsp;line, but not take it up till the second has laid hisnbsp;down on the line, so that the end may' exactly' coincide with that of the first rod; now let the fii'stnbsp;man lift up his, and count one, and then lay itnbsp;down at the end'of the second rod; the secondnbsp;man is now to lift up his, and count two; and thusnbsp;continue till the whole line is measured. Stavesnbsp;should be placed in a line at proper distances fromnbsp;each other by Problem 1, to prevent the operatorsnbsp;from going out of the given line.

When the line is very long, a chain is generally used; the manner of using the chain has beennbsp;already described, page 204.

Problem 3. To measure distances by pacing, and to make a scale of paces, vehich shall agree zvithnbsp;another, containing fathoms, yards, or feet.

In military concerns, it is often necessary to take plans, form maps, or procure the sketch of anbsp;field of battle, and villages of cantonment, or to reconnoitre fortified places, ^vhcl�e great accuracy isnbsp;BPt required, or where circumstances will not ql-

low the use of instruments; in this ease it is necessary to be well accustomed to measuring by the common pace, which is easily effected by a littlenbsp;practice; to this end measure on the groundnbsp;300 feet, and as the common pace is 2.1 feet, ornbsp;120 paces in 300 feet, walk over the measurednbsp;space till you can finish it in 120 paces, within a-pace or two.

Example. Let us suppose that we have the map of a country containing the principal objects; asnbsp;the villages, towns, and rivers, and it be necessarynbsp;to finish it more minutely, by laying down thenbsp;roads, single houses, hills rocks, marshes, amp;c. bynbsp;measuring with the common pace; take the scalenbsp;belonging to the map, and make another relativenbsp;to it, in the following manner, whose parts arenbsp;paces. Let the scale of the map be 200 fathoms,nbsp;draw a line AB, j?^.3, plate 27, equal to this scale,nbsp;and divide it into four equal parts, AC, CD, amp;c.nbsp;each of which will represent 50, fathoms; bisectnbsp;AC at E, divide AE into five equal parts, A e, e f,nbsp;f g, amp;c. each of which will be five fathoms; drawnbsp;GH parallel to AB, and at any distance therefrom; then through the points of division A, ef,nbsp;gh, E, C, See. draw lines perpendicular to AB,nbsp;and cutting GH, which will be thereby dividednbsp;into as many equal parts as A B. Two hundrednbsp;fathoms, at 2h feet per pace, is 480 paces; therefore write at H, the last division of G H, 480 paces,nbsp;at I 300, at K 240, and so on. To lay down onnbsp;the plan any distance measured in paces, take thenbsp;number of fathoms from the line A B, corresponding to the number of paces in GH, which will bcnbsp;the distance to belaid down on the plan.

When plans, or maps, are taken by the plain table, or surveying compass, this is an expeditiousnbsp;method of throwing in the detail, and a littlenbsp;practice soon renders it easy.

Distances of a certain extent may be measured by the time employed in pacing them; to do this,nbsp;a person must accustom himself to pace a givennbsp;extent in a given time, as 600 paces in five minutes, or 120 in one minute; being perfect in thisnbsp;exercise, let it be required to know how manynbsp;paces it is fi-om one place to another, which took,nbsp;up in pacing an hour and a quarter, or 75 minutes.nbsp;Multiply 75 by 120, and you obtain 120, the measure required.

This method is very useful in military operations; as for instance, when it is required to know the itinerary of a country for the inai'ch of annbsp;army, or to find the extent of a field of battle, annbsp;encampment, amp;c. It may be performed very wellnbsp;egt;n horseback, having first exercised the horse SQ as

to make him pace a given space in a determinate time, and this may be eticcted in more or lessnbsp;time, according as he is trained to walk, trot, ornbsp;gallop the original space.

Problem 4. To walk in a strait line from proposed point to a given object. Jig. �1, plate�1�J.

Suppose the point A to be the proposed point from whence you are to set out, B the given object; fix upon another point C, as a bush, a stone,nbsp;or any mark that you can find in a line with B;nbsp;then step on in the direction of the two objectsnbsp;B, C; when you are come within 10 or 15 pacesnbsp;of C, find another object D, between C and B, butnbsp;in a line with them; it is always necessary to havenbsp;two points constantly in view, in order to walk innbsp;a strait line.

This problem is of great use in measuring distances, and surveying by the pace; particularly in reconnoitring either on foot or on horseback. Itnbsp;may also be used when a battalion is ordered tonbsp;take a position at the distance of 3 or 400 paces,nbsp;and parallel to that in which it is standing; to effect this, let two non-commissioned officers, whonbsp;are well exercised in the practice of this problem,nbsp;step out from the extremities, or wings of thenbsp;battalion, place themselves square with the frontnbsp;of the battalion, then fix upon two objects straitnbsp;before them, and on a signal given, set out andnbsp;step the required number of paces, then halt, thusnbsp;becoming two guides for placing the battalion.

Problem 5. To place a troop E F, beiwee?2 two given points A, B, fig. 4, plate 27.

To perform this, let two persons, each with a pole in his hand, separate about 50 or 6o pacesnbsp;from each other; and then move on till the polesnbsp;are situated in the direction A^B, i. e. so that thenbsp;pefson at C may see the point D in a direction

with B, at the same time that the person at D sees C in a direction with tlic object A.

Method 2, fi^. 5. To perform the same with one person. Take a staff C F, fig. 5, plate 27,nbsp;tlirec or four feet long, slit the top thereof, andnbsp;place a strait rule, six or seven inches long, in thisnbsp;slit; place this instrument between the two points,nbsp;and see if A and B coincide with E, then view Dnbsp;, from B, and if A coincides therewith, all is right;nbsp;if not, move the rule forward or backward till it isnbsp;in the direction of D, E.

Problem 6. To raise a perpe^idicular fironi n point C, to the given line AB, fig. �, plate�1'].

Set off two points E, D, in AB, equally distant from C; fold a cord in two equal parts, place anbsp;staff in the middle at F, fasten the two ends to thenbsp;staves E and D; then stretch the cord tight, andnbsp;the point F will be the required point, and the.nbsp;line C F wall be perpendicular to AB.

Problem 7., Frojn a given point F, out ofi the Jitie KB, to let faU a perpendicular QY, fig.T,pi .�V].

Fold your cord into two equal parts, and fix the middle thereof to the point F, stretch the twonbsp;halves to A, B, and where they meet that line plantnbsp;two staves, as at E, D, divide the distance E Dnbsp;into two equal parts at C, and the line C F will benbsp;perpendicular to AB.

Problem 8. To raise a perpendicular BQ, at the end B of the line B F, fig. 8, plate 27.

From D taken at pleasure, and with the length D B, plant a staff at the point A, in the directionnbsp;F B; with the same length set off from D towardsnbsp;C, plant a staff C in the direction DA, and BGnbsp;will be perpendicular to F B.

Method 2, fig. Q, plate 27. By the numbers 3, 4, 5, or any multiple thereof. Set off from C to

plant a staff at D, take a length of three feet anti five feet, or yards, according as the former measure was feet or yards; fasten one end of the cordnbsp;at C, and the other at D; then stretch the cord sonbsp;that three of these parts may be next to the pointnbsp;C, and five next to D; plant a staff' at H, and C Hnbsp;�will be perpendicular to CA.¹

Problem g. To clrazv a line C F parallel to the line AB, and at a given distance from it, Jig. 10,nbsp;plate �27-

At A taken at jdeasuse in AB, raise a perpendicular AC, then from B taken also at pleasure in AB, draw BF perpendicular to AB; make AC,nbsp;B F, each eepud to the given distance; plant stavesnbsp;at C and F, and in the direction of these twonbsp;plant a third, E, and the line C E F wall be thenbsp;required line.

Pkoreem 10. To make an angle Ah c on the ground equal to a given angle ABC, fg. 11 and 12,nbsp;plate�IT.

Set off any number of equal parts from B to C, and from B to A, and with the same parts measure AC; describe on the ground with these threenbsp;lengths a triangle a b c, and the angle a b c thereofnbsp;will be equal to the angle ABC.

Problem 11. To prolong the line AB, notwithstanding the obstacle G, and to obtain the length thereof, fig. 13, plate 2/.

At B raise the perpendicular B C, draw C D perpendicular to B C, and E D to C D, make E Dnbsp;equal to BC; then raise a perpendicular EFtonbsp;E D, and E F will be in a strait line with AB.

The measures of the several lines AB, CD, EF, added together, give the measure of the line AF.

]\Iuch trouble and time maybe saved by using a cross-staff, before described; seep. 211. Edit.

Problem 12. To draw a line B C, parallel to the inaccessible face b c the bastion a, b, c, d, e, innbsp;order to place a battery at C, where it will producenbsp;the greatest effect, jig. 14, plate 27.

Place a staff at the point A, out of the reaeh of musket shot, and in a line with b e; from A drawnbsp;AG, perpendicular to bA; set oft from A to Gnbsp;about 375 paces, and at G raise GB perpendicularnbsp;to AG, and produce this line as far as is requisitenbsp;for placing the battery, i, e. so that the directionnbsp;of the fire may be nearly perpendicular to one thirdnbsp;of the fiice b c.

Problem 13. To make an angle upon paper equal to the given angle AB C,jig. \b and 16, pl.�l^.

Set off 36 feet from B to C, and plant a staff at C; set off also 36 feet from B to A, and plant anbsp;staff at A, and measure the distance AC, whichnbsp;We will suppose to be equal to 52 feet. Let D Gnbsp;be a given line on the paper; then from G withnbsp;36 equal parts taken from any scale, describe thenbsp;arc D H, then with 52 parts from the same scale,nbsp;mark off the distance DH; draw GH, and thenbsp;angle DGH is equal to the angle ABC.

Problem 14, To measure from the outside an angle AB Q, formed by two walls AB, CB, fg. 17,nbsp;plate IT.

Lay off 30 feet from B to E in the direction AB, and plant a staff at E; set off the same measure from B to F in the direction B C, and measurenbsp;FE, and you may obtain the measure of yournbsp;angle, either by laying it down on paper, as in thenbsp;last problem, or by calculation.

Problem 15. To ascertain the opening of an inaccessible flanked angle BCD q/quot;the bastion ABnbsp;C D E, fig. 18, plate 27 �

Place a staff at F, in the direction of the face ^ C, but out of the reach of musket shot; make

F G perpendicular to F B, and equal to about 25(T paces; at G make GH perpendicular to GF?nbsp;and produce it till it meets, at H, the direction otnbsp;the face D C; place a staff at I in a line with DH?nbsp;and the angle I H K will be equal to the requirednbsp;angle B C D, and the value of IH K may be obtained liy Problem 13.

Problem if). 7b ascertain the length of the line AB, accessible only at the tv)o extremities A, Bjnbsp;Jig. \g, plate 17.

Choose a point O accessible to A and B, draw AO, O B, prolong AO to D, and make OD equalnbsp;to AO; prolong quot;BO to C, and make OC equalnbsp;to B O, and C D will be equal to AB, the requirednbsp;line.

Problem 17- To let fall a perpendicular froW the inaccessible point D, upon the right line AB, Jig-11, plate 17.

From A and B draw AD, B D, let fall the perpendiculars AH, BF, upon the sides AD, BDj plant a staff at their intersection C, and another E?nbsp;in the direction of the points C, D, and E C Dnbsp;will be the i-cquired perpendicular.

Plant a staff at C in a line with AB, so that B C may be about I of the length of AB, draw anbsp;line C E in any direction, the longer the better.nbsp;Set up a staff at D, the middle of C E, find thenbsp;intersection F of EB, DA; draw DG parallel tonbsp;BC; measure BF, FE; and you will find ABnbsp;by the following proportion; as FE � BP'': vBCnbsp;orDG::BF:AB.

Inaccessible distances may be obtained in other modes, when circumstances and the ground willnbsp;permit, as in the following problem.

Problem 19. Let it he required to measure the distance of the produced capital of the ravelin,nbsp;from the head of the trenches B, to the angle A,nbsp;fg. �12, plate 2^.

Draw B E perpendicular to AB, and set ofF From B to E 100 feet, produce B E to C, and makenbsp;E C equal one eighth or tenth part of B E; at Cnbsp;raise the perpendicular C D, plant stages at E andnbsp;C, then move with another staff on CD, till thisnbsp;staff is in a line with E and A. Measure C D, andnbsp;you will obtain the length of AB by the followingnbsp;proportion; as EC : CD :: BE : AB; thus ifnbsp;B E = 100 feet, E C = 10 feet, and C D = 38 feet,nbsp;then as 10 : 100 :: 38 : 380 feet, the distance of

AB.

Plant a staff at C, a point from whence you can readily see both A and B. By the preceding problems, find the length of CA, C B, make C D asnbsp;many parts of CA, as you do C E of C B, and joinnbsp;DE; then as CD : DE :: CA : AB.¹

Problem 21. To determine the direction of the capital of a bastion produced, fig. 24, plate 27.

Upon the produced lines B D, B E, of the two faces C B, AB, place the staves E and D; find thenbsp;lengths of E B and D B, by Problem ] 9, andnbsp;measure D E. Then as the capital G B divides

An easy method of finding the distance ofifiorts or other objects.� ft you go off at any angle, as pO, and continue in that directionnbsp;Until you bring the object and your first station under an anglenbsp;uf 63, the distance measured from the first station is equal to halfnbsp;the distance the object is from your first station. But should thenbsp;ground not admit of going off at an angle of pO, go off at an angle of 45, and whatever distance in that direction you find thenbsp;object and the first station under an angle of 106^, is half thenbsp;�iistance of the object and first station. See also pages 21Snbsp;and;2S3 of these Essays.

the angle ABC, and its opposite EB D, into tivo equal pai'ts, we have as DB ; BE ;; DF ; EE,nbsp;and as D B B E : B E D E �- E F, and consequently the point F, which \idll be in the directionnbsp;of the capital G B produced.

Problem 12. Through the pohit C to draw ^ Vine I K parallel to the Inaccessible line AB, Jig. 25,nbsp;plate 17.

Assume any point D at pleasure, find a point E in the line AID, which shall be at the same time innbsp;a line witli C B; from E draw E G parallel tonbsp;DB, and through C draw GCF parallel to AD,nbsp;meeting B D in F. Plant a staff at H in the lino�nbsp;E G, so that H may be in a line with F A, and anbsp;line I, H, C, K, drawn through the points B C,nbsp;will be parallel to AB.

Problem 23. To take a plan of a place A, B, C, D, E, by similar triangles, fig.llS arid 17, pl.17�

1. Make a sketch of the proposed place, as^f/-27- nbsp;nbsp;nbsp;2. Measure the lines AB, BC, CD, DE,

EA, and write the measures obtained upon the corresponding lines a b, be, c d, dc, ea, of thenbsp;.sketch. 3. Measure the diagonal lines AC, AD,nbsp;and write the length thereof on a c, ad, and thenbsp;figure will be reduced to triangles whose sides arenbsp;known. 4. To obtain a plan of the buildings,nbsp;measure B G, B F, G H, I K, F K, amp;c. and writenbsp;down the measures on the sketch. 5. Proceed ionbsp;the same manner with the other buildings, amp;c.nbsp;6. Draw the figure neatly from a scale of equalnbsp;parts.

Problem 24. To take a plan of a wood, �marshy ground, by measuring round about it, fig- 2�nbsp;and IQ,, plate 17�

Make a rough sketch,29, of the wood, set tip staves at the angular points, so as to form thenbsp;circumscribing lines AB, BC, CD, and measure

these lines, and set down the measures on the corresponding lines in the sketch, then find the value of the angles by Problems 13 and 14.

1. Draw a sketch of the river. 2. Mark out a line AB,_;7^.30. 3. At C, upon AB, raise the perpendicular CH. 4. Measure AC, CH. 5. Measurenbsp;� from C to D, and at D make DI perpendicular tonbsp;AB, and measure D I. 6. Do the same the wholenbsp;length of A B, till you have obtained the principalnbsp;bendings of the river, writing down every measurenbsp;when taken on its corresponding line in the sketch,nbsp;and you will thus obtain sufficient data for drawing the river according to any proportion.

Pr o B L E M 2�. To take a plan of the neck of land A, B, C, D, E, F, G,fg. 32 and 33, plate 27.

Take a sketch of the proposed spot, divide the figure into triangles ABC, ACD, ADE, AEG,nbsp;by staves or poles placed at the points A, B, C, D,nbsp;E, P', G, measure the sides AB, B C, AC, CD,nbsp;AD, AE, Ah', F G, A G of the triangles, writingnbsp;down these measures upon the corresponding linesnbsp;ab, be, ac, amp;c. then measure AH on AB, andnbsp;at H raise the perpendicular H I, and measure itsnbsp;length; do the same at K, M, amp;c. writing downnbsp;the measures obtained on their corresponding linesnbsp;ah, hi, ik, kl, amp;c. Proceeding thus, you willnbsp;ascertain a sufficient number of points fornbsp;down your plan by a scale of equal pails.

Measure a line F E, from the foot of the build-ifig, so that the angle CDA may be neither too

acute nor obtuse; thus suppose EF=130 feet, place your theodolite at D, and measure the anglenbsp;ADC = 34� 50'.

Which, by working the proportion, you will find to be 80 feet 66 parts, or 89 feet 7-9'2 inches, tonbsp;which adding four feet for D E, or its equal C F,nbsp;you obtain the whole height 03 feet 7-0^^ inches.

Problem 28. To find the angle formed by th^ line of aim, and the axis of a piece of ordnance pro-duced, the caliber and dimensions of the piece beingnbsp;knoxmi, fig. 35, plate 28..

Suppose the line BI to be drawn through B, the summit of the swelling of the muzzle, and parallel to C D, the axis of the piece; the angle ABInbsp;will be equal to the angle AE C, formed by thenbsp;line of aim AF, and the axis CD; then in thenbsp;I�ight-angled triangle AIB, we have the sides Alnbsp;and BI to find the angle ABI, xvhich we obtainnbsp;by this proportion; as IB : AI ;; radius, tangentnbsp;of angle AB I equal to AE C required.

Problem 29. The elevatmi, three degrees, of a light tveelve-powider being given, to find the heightnbsp;to which the line of aim rises at the distance of 1200nbsp;yards, which is about the range of a twelve-pounder,nbsp;with an elevation of three degrees, fig. 35, plate 38.

The line of aim, which we suppose to be found by the preceding problem, forming with the axisnbsp;of a light twelve-pounder 1� 24', will make withnbsp;the horizon an angle of 1� 36'; thus the height atnbsp;the horizontal distance of 1200 yards will be th�nbsp;second side of a right-angled triangle, where thenbsp;angle adiacent to the side of 1200 yards is 1� 36 ,nbsp;and giay therefore be obtained by the followingnbsp;proportion; as radius to the tangent of 1� 36', so isnbsp;1200 to 33 yards 1 foot 6 inchcs=F G,

Problem 30. The Jirst embrasure A of a ri-rochee battery being direct, to find the incTmatwn of the seventh embrasure i. c. the angle formed bynbsp;the line of direction B C, and the hreast-veork A B,nbsp;at the seventh embrasure A B; it is supposed that allnbsp;the pieces are directed towards a point Q at the distance of 1500 feet, Jig. 3�, plate 28.

The line of direction A C of the first embrasure is supposed to be perjiendicular to the breast-worknbsp;AB; therefore we Iiave to find the angle AB C ofnbsp;the right-angled triangle B AC, in which the rightnbsp;angle is known AC=:1500 feet, and AB is determined by the size, the distance, and number ofnbsp;embrasures: thus, suppose the distance from thenbsp;middle ofoone embrasure to another be 20 feet,nbsp;this multiplied by �, will be 120 feet, and equal tonbsp;A B; then as A B to A C, so is radius to the tangentnbsp;of angle ABC, 85� 20'.

Problem 31. As the hurler DE, fig. 37, plate 28, is always perpendicular to the directingnbsp;line of the gun, and as at least one end of it oughtnbsp;to be laid against the breast-work, it will make annbsp;angle AFD, which was found by the precedingnbsp;problem to be 85� 20'; therefore, knowing thenbsp;length DE of the hnrler, and consequently itsnbsp;half D F, it will be easy to calculate the distancenbsp;BF from the breast-work where the middle F ofnbsp;the hurler ought to be placed, upon the line ofnbsp;direction of the gun.

Problem 32. To ascertain the height of a huild-ing from a given point, from vchence it is impossible to measure a base in any direction, the points A and P'nbsp;being supposed to he in the same horizontal line, fig,nbsp;38, plate 28.

Aleasure the angles C E D, AE D, let C E D be equal 43� 12', the angle AED equal 2� 2G', andnbsp;the height E F from the center of the instrument

to its foot five feet; then in the right-angled triangle AD E we have AD=E F=;5 feet, the angle D EA=2'^ 26' to find D E^ which may be obtained by the following proportion; as radius ? co-tangent of angle DEA, so is AD to DE II8nbsp;feet; then in the right-angled triangle CDE wenbsp;have DE 118 feet, angle CED33�l2'to findnbsp;CD, which is found by the following proportion;nbsp;as radius to tangent of angle C E D, join D E tonbsp;D C 77 feet, which added to E F 5 feet, is 82 feet,nbsp;the height of the tower.

Problem 33. The distance AC, 135 foises, from the 'point C, to the funked angle of the bastionnbsp;being given, and also AB, 186 toises,the extetior sidenbsp;of the polygon, to find B C, fig. 3g, plate 28.

1. Find the angle B by the following proportion; as AB is to the sine of angle C, so is AC tonbsp;the sine of angle B, 39� 8'; because it is plainnbsp;from the circumstances of the gase that B must benbsp;acute, and therefore angle BAG is also known.

2. nbsp;nbsp;nbsp;BC is found by the following proportion; asnbsp;sine angle C is to AB, so is sine of angle BAC :nbsp;BC, 213 toises, three feet.

1. At B measure the angle P B C in the direction F B. 2. Set off any distance B D as a base.

3. nbsp;nbsp;nbsp;Measure the angle B D C, C B D is the supplement of F B C, and BCD is the supplement of

CBD-fBDC.

Then as sin. ZBCD : BD ;: CDB : BC, B C being found, we have in the right-angled triangle F B C, the side B C and angle F B C, to findnbsp;F C, which is found by this proportion; as radiusnbsp;to sin. angle F B C, so is B C : F C, F C added tonbsp;AF, tlie height of the instrument, gives the heightnbsp;of the tower,

Pr o B LEM 3 5. To ascertain the d'lstance hetween f ivo inaccessible objects, C D, fig. 41, jdate 28.

1. Measure abase AB, from wliosc extremities you can see the two obiects C, D. 2. Aleasurenbsp;the angles CAB, DAB, DBA, CBA, CBD.nbsp;3. In the triangle BAG we have the side AB andnbsp;angles ABC, BAC, to find B C, which is foundnbsp;by the following proportion ; as sine angle ACE:

� AB :: sine BAC : BC. 4. The angles CAB, CBA, added together and subtracted from 180,nbsp;gives the angle AC B. 5. In the triangle AB Dnbsp;we have the angles DAB, ABD, and eonse-quently ADB, and the side AB to find B D,nbsp;which is found by the following proportion; as thenbsp;sine of angle D B is to AB, so is the sine of anglenbsp;DAB : B D. (). In the triangle CBD we havenbsp;the two sides B C, BD, and the angle CBD, tonbsp;find the angle C DB, and the side CD; to findnbsp;C D B we use this proportion; as the sum of thenbsp;two given sides is to their difierence, so is the tangent of half the sum of the two unknown anglesnbsp;to the tangent of half their diftercnce; the anglenbsp;C D B being found, the following proportion willnbsp;give CD; as sine angle C DB to B C, so is sinenbsp;angle C B D to C D.

Problem 36. To draiv a line through the point B, parallel to the inaccessible line C D, fig.41, pi. 28.

Find the angle B C D by the preceding problem, and then plaee your instrument at B, and withnbsp;BC make an angle CBE equal BCD, and EBnbsp;w ill be parallel to C D.

same direction, though there are obstacles which prevent one extremity of the line being seen from the other, fig. 42, plate 28.

1. Assume a point C at pleasure, from which the tw'O extremities of the line AB may be seen,

3. Measure the distances CB, CA, and the angle ACB, and lind the value of the angle CAB-3. Measure the angle A CD, and then in the triangle ACD \vc have the side AC, and the twonbsp;angles DAC, ACD, and therefore the third tonbsp;find C D, and as sine angle AD C to AC, so is sinenbsp;angle CAD to CD. 4. Set oif CD, makingnbsp;with AC an angle equal the angle ACD, equalnbsp;the measure thus found, and the point D will benbsp;in a line with AB; and thus as many more points,nbsp;as G, may be found as you please; in this rnannei'nbsp;a mortar battery may be placed behind an obstacle,nbsp;so as to be in the direction of the line AB.

Thus also you may fix the position of a richo-chec battery nbsp;nbsp;nbsp;43, so as to be upon the cur

Problem 38. To measure the height of a hill, �whose foot is inaccessible^ fg.AA, flatelB.

1. Measure a base F G, from whose extremities the point A is visible. 2. Measure the anglesnbsp;ABC, ACB, ACD. 3. In the triangle ABCnbsp;we have BC, and the angles ABC, ACB, to findnbsp;AC; but as sine angle BAG is to EC, so is sinenbsp;angle AB C to AC. 4. In the right-angled triangle AD C we have the side AC, and the anglenbsp;ACD, to find AD; but as radius is to sine anglenbsp;AC D, so is AC to AD, the height required.

First, choose two places so remote from each other, that their distance may serve as a commonnbsp;base for the triangle to be observed, in order tonbsp;form the map.

Make a rough sketch of these objects, according to their positions in regard to each other; on thisnbsp;sketch, the different measures taken in the coursenbsp;of the observations are to be set down.

Measure the base AB, whose length should be proportionate to the distance of the extreme objects from A and B; from A, the extremity of thenbsp;base, measure the angles EAB, FAB, GAB,nbsp;CAB, DAB, formed at A with the base AB.

From B, the other extremity of the base, observe the angles EBA, FBA, GBA, CBA, DBA.

If any object cannot be seen from the points A and B, another point must be found, or the basenbsp;changed, so that it may be seen, it being necessarynbsp;for the same object to be seen at both stations, because its position can only be ascertained by thenbsp;intersection of the lines from the ends of the base,nbsp;with which they form a triangle.

It is evident from what lias been already said, that having the base AB given, and the anglesnbsp;observed, it'tvill be easy to tind tlic sides, and fromnbsp;them lay down, with a scale of equal parts, thenbsp;several triangles on your map, and thus fix withnbsp;accuracy the position of the different places.

In forming maps or plans, where the chief points are at a great distance from each other, trigonometrical calculations arc absolutely necessary.

But where the distance is moderate, attcr having measured a base and observed the angles, insteadnbsp;of calculating the sides, the situation of the pointsnbsp;may be found by laying down the angles with anbsp;protractor; this 'method though not so exact asnbsp;the preceding, answers sufficiently for most military operations.

Problem 40. The nse of the surveymg cojnpass, 3, plate Iff, in determhiiug the particulars of -va-

The first station being at A, plant staves at the requisite places, and then place the compaSs at A,nbsp;directing the telescope to C; observe the numbernbsp;of degrees north AC, made by the needle and thenbsp;telescope, or its parallel line drawn through thenbsp;center of the compass, and mark this angle in yournbsp;sketch, see Jig. 46; observe and mark in the samenbsp;manner the other angles, north AO, north AP,nbsp;north AQ,; then measure AC, and at the secondnbsp;station C, observe the angles north C M, northnbsp;C Q, north C O, north C P, north C D, and sonbsp;on at the other stations; in observing the angles,nbsp;attention must be paid, when the degrees pass 180,nbsp;to mark them properly in the sketch, in order tonbsp;avoid mistakes in protracting.

Problem 41. To raise perpendiculars, andform angles equal to givc?i angles by the surveying compass.nbsp;Jig. 47, plate 28.

The direction of the capital F E being given, describe the square AB C D in the following manner; place your compass at F, and as AB is to benbsp;perpendicular to F E, direct the telescope to E,nbsp;and observe, when the needle is at rest, the number of degrees it points to; then turn the compassnbsp;box upon its center, till the needle has describednbsp;an arc of 00�, and place a staff in the direction ofnbsp;the telescope towards A, and AF will be perpendicular to F E; continue AF towards B, and makenbsp;AF, B F, each equal 30 toises at E; in the samenbsp;manner raise the perpendicular D C, and makenbsp;ED, EC, each equal 30 toises; join DA, CB,nbsp;and you have the square AB C D.

Make C G, DK, each equal to ' of AB, through G and K draw the line G K, set off G H,nbsp;K I, each equal to i of AB or CD, draw the linesnbsp;of defence 1C, H D, place the compass at I, andnbsp;the telescope in the direction IH; observe wherenbsp;the needle points when at rest, and turn the compass till the needle has described an arc of 100, thenbsp;value of the angle LIH of the flank, place a staffnbsp;at L in the direction of the telescope, and at thenbsp;same time in the line IH, which gives the lengthnbsp;of the face D L and the flank I L; the face CMnbsp;and flank H M arc ascertained in the same manner.

of AD, draw OZ, make RT equal to t AB; at T and with T D, form an angle D T U, equal tonbsp;105�; plant a staff at U, so that it may be in thenbsp;line Z O, and at the same time in the directionnbsp;T U, which gives the flank T U and face O U.

Make QS equal f AB, at S form an angle CSP equal to 120�, draw the line S P, meeting the river,nbsp;and the lines O, U, T, D, L, I, H, will be the linenbsp;of the tete dc pont required.

Place the table at A, where you can conveniently seethe greater part of the field, and having madenbsp;a scale on it, fix a fine needle perpendicular to thenbsp;table at the place that you fix upon to representnbsp;the point A; the fiducial edge of the index is always to be applied against the needle.

Turn the j)lalu table so that the index rna}' point to the object B, and be so situated as to take in thenbsp;field; then ])lant a staff' at C, in a line with B,nbsp;point the index to the windmill F^, and draw thenbsp;indeiinite line AF; then point it to K, the rightnbsp;wing of the cavalry KL, and draw AK, then tonbsp;L, and draw AL, afterwards point the index tonbsp;the steeple I, and then to the points AI, O, P, N,nbsp;FI, E, G, then draw a line on the table parallel tonbsp;the north and south of your compass, to representnbsp;the magnetic meridian.

Remove the plain table to C, planting a staff� at C, measure AC, and set off' that measure by yournbsp;scale from A upon the line AC, and fix the needlenbsp;at C; then set the index upon the line AC, andnbsp;turn the table till the line of sight coincides withnbsp;A, fasten the table, point the index to F, and drawnbsp;C F, intersecting AF in F, and determining thenbsp;position of the windmill F; from C draw the indefinite lines CM, CK, CL, amp;c. which will determine the points MKL, amp;c. draw KL andnbsp;m n parallel thereto, to represent the line of cavalry.

Remove the table from C to D, setting up a staff' at C, measure C D, and set off'the distance onnbsp;C B from your scale, place the needle at D, thenbsp;index on C D, and turn the table till C coincides with the sights, and take the remarkablenbsp;�objects which could not be seen from the othernbsp;stations.

Staves should be placed at the sinuosities of the river, and lines drawn at the stations A, C, D, B,nbsp;to these staves, which will give the windings of thenbsp;river.

Flaving thus determined the main objects of the field, sketch on it tha roads, hills, amp;c.

The plan of trenches, taken with accuracy, gives a just idea of the objects, and shews how you maynbsp;close more and more upon the enemy, and be covered from their enfilade fire, and also hoiv to pro-cebd in the attack without multiplying uselessnbsp;works, which increase expense, augment the labour, and occasion a great loss of men.

Measure a long line AB, parallel to the front of the attack D F, place the table at A, and set up anbsp;staifiat B, point the index to B, and draw a line tonbsp;represent AB, fasten the table, and fix a needlenbsp;at the point A, direct the line of sights to thenbsp;flanked angle of the ravelin C, and draw AC;nbsp;])rocced in the same manner with the flanked angles D, E, F, G, to draw the lines at the openingnbsp;of the trenches, plant staves at H and R, from Anbsp;draw a line on the table in the direction AH, measure AH, and set ofl'that measure from your scalenbsp;upon thfe line AH on the table.

Remove the plain table from A to B, Set up a staff at A, lay the fiducial edge of the index againstnbsp;the line Ab, and turn the table about till the staffnbsp;at A coincides with the line of sight, then fastennbsp;the table, direct the sights to C, and draw B C,nbsp;intersecting A at C; in the same manner ascertainnbsp;the flanked angles D, E, F, G, draw a line in thenbsp;directiQii B R, and set off the measure thereofnbsp;fi'om your scale.

Remove the table from B to R, and set up a staff at B where the plain table stood, lay the indexnbsp;upon the line corresponding with RB, then turnnbsp;the table about till the line of sight is in the direction BR, screw the table fast, direct the sights towards P, and draw on the table the line R P, andnbsp;by thp scale lay off its measure on that line.

Remove the plain table from R to P, lay the index upon the line P R, and turn the table aboutnbsp;till the line of sight coincides with R, screw thenbsp;table fast, and draw a line upon it in the directionnbsp;PQ,; measure PQ, and take the same number ofnbsp;parts from your scale, and set it off on the line, andnbsp;so on with the other station Q.

Having removed the table to the station S, and duly placed it with regard to Q, from the point S,nbsp;draw the lines S V, S T, S U, setting oft' from yournbsp;scale their lengths, corresponding to their measuresnbsp;on the ground.

Plaving thus taken the zigzags R P Q, S TW, and the parts U U WZ of the parallels, you removenbsp;the plain table to H, and proceed in like mannernbsp;to take the zigzags HI K, amp;c. as above, whichnbsp;will represent on your plain table the plan of thenbsp;attack required.

Levelling is an operation that shew�s the height ofone place in respect to another; one place is saidnbsp;to be higher than another, when it is more distantnbsp;from the center of the earth than the other; whennbsp;a line has all its points equally distant from thenbsp;center, it is called the line of true level; whence,nbsp;because the caxlh is round, that line must be anbsp;curve, and make a part of the earth�s circumference, as the line AB E D, all the points of whichnbsp;are equally distant from the center C of the earth;nbsp;but the line of sight AG, which the operation ofnbsp;levelling gives, is a right line perpendicular to thenbsp;semi-diameter of the earth CA raised above thenbsp;true level, denoted by the curvature of the earth,nbsp;and this in proportion as it is more extended; fornbsp;which reason, the operations which we shall give.

arc only of an apparent level, which must be corrected to have the true level, when the line of sight exceeds 300 feet.

Suppose, for example, that AB was measured Upon the surface of the earth to be 6000 feet, asnbsp;the diameter of the earth is near 42018240 feet,nbsp;you will find BF by the following proportions.

* 42018240 : 6000 :: 6000 : BF equal to 0,85677 f. which is 10,28124 in. that is to say, between two objects A and F, 6000 feet distant fromnbsp;each other, and in the same horizontal line,nbsp;the difFcrcnce B F of the true level, or that ofnbsp;their distance from the center of the earth, �snbsp;10,28124 in.

When the difference between the true and apparent level, as of B F, has been calculated, it will be easy to calculate those which answer to a lessnbsp;distance; for we may consider the distances B F,nbsp;h f, as almost equal to the lines A I, A i, which arenbsp;to each other as the squares of the chords, or ofnbsp;the arcs AB, ab, because in this case the chordsnbsp;and the arcs may be taken one for the other.

Thus to find the difference f b of level, which answers to 5000 feet, make the following proportion; 6000 f. : 5000 f. :: 0,85077 : fb, which willnbsp;be equal to 0,71399 f. or 8,50788 in.

The point F, which is in the same horizontal line with A, is said to be in the apparent level of A,nbsp;and the point B is the true level of F; so that B Fnbsp;is the difference of the true level from the apparent.

* As the arc AB=:6(XX) feet is but veiy small, it may be considered equal to the tangent AF, and in this respect, AF is a Riean proportional between the whole diameter, or twice the radius B C, and the exterior part B F.

level between points B and A, which are not in the same horizontal line; then at A make use of annbsp;instrument proper to take the angle BCD, andnbsp;having measured the distance C D, or C I, by anbsp;chain which must be kept horizontal in differentnbsp;parts of it on the ground ALVB, you may in thenbsp;triangle CDB, considered as rectangular in D,nbsp;calculate B D, to which add the height C A of thenbsp;instrument, and calculate the difference of levelnbsp;DI, as we liave shewn above.

But as this method requires great accuracy in measuring the angle BCD, and an instrumentnbsp;vfcry exactj it is often better to get at the same endnbsp;with a little more trouble, which is shewn by thenbsp;following method.

Place the level at E, at equal distances from B and G, fix one station staff at B, the other at G;nbsp;your instrument being adjusted, look at B, and letnbsp;the vane be moved till it coincides with the line ofnbsp;sight; then look at the staff G, and let the vanenbsp;be moved till it coincides with the line of sight H,nbsp;and the difference in height shewn by the vanesnbsp;on the two staves, will be the difference in the level between the two points B and G. Thus, suppose the vane at G was at 4 f. 8 in. and at B 3 f.nbsp;9 in. subtract one from the other, and the remainder 11 inches, will be the difference in thenbsp;level between the two points B and G; you maynbsp;proceed in the same manner with the other points;nbsp;but more need not be said on this head, as I havenbsp;already treated this subject very fully in the foregoing part of this woi'k.

[ 467 3

ESSAY ON PERSPECTIVE;

Definition 1. Perspective is the art of delineating the representations of bodies upon a plane, andnbsp;has two distinct branches, linear and aerial.

Definition 2. Linear perspective shews the method of drawing the visible boundary lines of objects upon the plane of tlie picture, exactly where those lines would appear if the picture were transparent; this drawing is called the outline of thosenbsp;objects it represents.

Definition 3. Aerial perspective gives rules to fill up this drawing with colours, lights, andnbsp;shades, such as the objects themselves appear tonbsp;have, when viewed at that point where the eye ofnbsp;the spectator is placed.

To illustrate these definitions, suppose the picture to be a plate of glass inclosed in a frame PLN, fig. \, plate ZQ, through which let thenbsp;eye of the spectator, placed at E, view the objectnbsp;QRS T; from the given point E, let the visualnbsp;lines EQ, ER, E S, amp;c. be drawn, cutting thenbsp;glass, or picture, in q, r, s, amp;c. these points ofnbsp;intersection will be the perspective representations

of tlie original Q R S T, amp;c. and if they are joinedf qrst will be the true perspective delineation ofnbsp;the original figure. And lastly, if this out-line be_nbsp;so coloured in every part, as to deceive the eye ofnbsp;a spectator, viewing the same at E, in such a manner, that lie cannot tell whether he views the realnbsp;object itself, or its representation, it may be trulynbsp;called the picture of the object it is designed for.

Definition A. When the eye, or projecting point, is supposed at an indefinitely great distance, compared with the distance of the picture and the object to be represented, the projecting lines beingnbsp;then supposed parallel, the delineation is called anbsp;parallel one, and by the corps of engineers, military perspective.

Definition 5. If this system of parallel rays be perpendicidar to the horizon and to the picture,nbsp;the projection is called a plane.

Definition 6. If the parallel rays be horizontal, and the picture upright, it is called an elevation.

Definition 7- A right line E e, fig. 1, plate 30, from the eye, E, cutting the plane of the picturenbsp;at right angles, and terminating therein, at e, isnbsp;called the distance of the picture.

Definition 8. The point e, where this central ray cuts the picture, is called the center of thenbsp;picture.

Definition 9. The point X, fig. 1, plate 30, where any original line cuts the picture, is called the intersection of that line.

Definition 10. The scat F, fig. 2, plate 30, of any point E upon a plane, is where a perpendicularnbsp;from that point cuts the planej thus, if a perpendicular E F, be drawn from any elevated point E,nbsp;to the ground plane FGN at F, this point F isnbsp;called the scat of the point E upon the groundnbsp;plane.

T)efinition 11. Apparent magnitude is measured by the degree of opening of two radials passingnbsp;through the extremes of bodies whose apparentnbsp;magnitudes are compared; thus, the apparentnbsp;magnitude of Q R, fig. 3, plate 30, to an eye at E,nbsp;is measured by the optic angle QER. Hence itnbsp;is evident, that all objects viewed under the samenbsp;angle, have the same apparent magnitude.

Definition 12. The intersection G N, fig-1, plate 30, of the picture with the ground plane, isnbsp;called the ground line.

Definition 13. A plane passing through the eye, and every where parallel to the ground {or groundnbsp;plane, as it is commoidy called, because it is supposed every v/here flat and level) is called the horizontal plane, as E h n, fig. 2, plate 30.

Definition 14. The intersection of the horizontal plane with the picture is called the horizontal line-, thus h e n, fig. 1, plate 30, is called the horizontal line, being the intersection of the horizontalnbsp;plane E h e n, parallel to the ground plane F G N.

Definition 15. The center of any line is where a perpendicular from the eye cuts it.

Definition 1(). A plane i D Fgh, fig.l, no. 2, plate 30, passing through the eye E, perpendicular to the ground, is called Xho. vertical plane-, andnbsp;that part of the ground plan gfN which lies tonbsp;the left hand, is called amplitudes to the left, andnbsp;all that lying on the other side, amplitudes to thenbsp;right, their measures being taken on the base linenbsp;G F N, or its parallels, as depths arc by f g, or itsnbsp;parallels.

Propositioii 1. Parallel and equal strait lines appear less, as they arc farther removed from the eye.

Let O P, Q R, fig. 3, plate 30, be two equal and parallel strait lines, viewed at the point E; QRnbsp;fjeing the farthpst off, ^yill appear the least.

For, draw E Q and E R, cutting O P in r; then Q, R and O r have the same apparent magnitude,nbsp;(definition 11,^ but O r is only a part of O P.

Corollary. Similar figures parallelly situated, appear less, the farther they are removed from thenbsp;eye.

Proposition Q,. All original parallel strait lines, P R, fig. 3, plate 30, which cut the picture at P,nbsp;appear to converge to the same point O therein;nbsp;viz. that point which is the intersection with thenbsp;picture, and a line passing through the eye parallel to the original parallel lines.

For, since P R is parallel to E O Q, let R Q be parallel to O P, and therefore equal to it. By thenbsp;last proposition, the farther R Q is taken fromnbsp;O P, the nearer it (the representation of the pointnbsp;R) approaches to the fixed point O, and the casenbsp;is the same with any other line parallel to PR;nbsp;and, therefore, if all the parallels be indefinitelynbsp;produced, that is, at least till the strait lines measuring their distance become invisible to the eye,nbsp;they will all appear to vanish together in this point,nbsp;which, in consequence thereof, is called the vanishing point of all those parallels.

to converge to a right line therein, viz. that line O o, which is the intersection of the picture, withnbsp;a plane QOEoq, parallel to them all passingnbsp;through the eye.

For, since p S, fig. plate 30, is parallel to o q, therefore p S will appear to converge to its vanishing point o, by the foregoing proposition; andnbsp;since both the points R, S, equally appear to tendnbsp;to the respective points Oo, therefore the line R Snbsp;also appears to approach to the line Oo; and thisnbsp;is the case with all planes parallel to P R S p, and

tlicrcforc if they arc produced till they become invisible, they will appear to meet in Oo, whence this line is ealled the vanishing of all those parallelnbsp;planes.

Corollary 2. The representations of original lines pass through their intersections and vanishingnbsp;points; and of original planes, through their intersections and vanisiiing lines.

For, in this case the line E O, which should produce the vanishing point, nevmr cuts the picture.

Corollary y. Hence the representations of plane figures parallel to the picture, are similar to,theirnbsp;originals.

Let q r s t u, fig. 4, phpe 30, be the representation of the original figure Q. R S T U, to an eye at E; then all the lines q r, rs, st, amp;c. being respectively parallel to their originals Q R, R S, S T,nbsp;amp;:c. as well as the diagonals, t q, T Q, amp;c. therefore the inscribed triangle qtu, and QTU, arcnbsp;similar to each other, and so of all the other triangles ; and, therefore, the whole figure q r s t tonbsp;all ST.

Corolla7y 5. The length of any line in the representation is to that of its respective original, as the distance of the picture to that of the originalnbsp;plane; for all the triangles E Q U, E q u, E Q R,nbsp;E q r, amp;c. are similar; therefore, as E q is to q u,nbsp;so is E a to aU; that is, as q u is to Q U, so isnbsp;the distance of the picture to the distance fromnbsp;the plane QU.

Problem 1. Having the c�7iter and distance of the picture given, to fi.nd the representation of anynbsp;given point thereon, fig. b, plate 30.

From the eye E, to the center of the picture e^ draw E e, and parallel to it Q d from the given

point Q, cutting the picture in cl; join e d, and draw E Q.J intersecting e d in q, the perspectivenbsp;place of the original point Q.

For, e d is the representation of the original line d Q, indefinitely produced from its intersection q,nbsp;till it appears to vanish in e; therefore q must benbsp;somewhere in this line, pr. �2, ror. 2; it must alsonbsp;be somewhere in EQ, and therefore in the point q,nbsp;where they intersect.

Problem 2. The center and distance of the �picture being given, to fnd the representation of anbsp;given line Q. R, fg. 6, plate 30.

Produce RQ, to intersect the picture in X, draw EV parallel to RQX, cutting the picticrenbsp;in V, the vanishing point of the line RX; drawnbsp;ER and EQ, cutting VX the indefinite representation of R X, in r and q, and r q is the representation of R Q.

Problem 3. Having the center and distance of the picture, to fnd the representation of an originalnbsp;plane, -whose position with respect to the picture is

Draw any two lines, except parallel ones, upon the plane, and find their vanishing points by thenbsp;last problem; through these points a line beingnbsp;drawn will be the vanishing line required.

Problem 4. Having the center and distance of the picture given, to find the vanishing point of linesnbsp;perpendicular to a plane whose representation isnbsp;given, fig. 7, plate 30.

Let P L be the vanishing line of the given representation, X its center, c that of the picture, and eE its distance; join EX, and perpendicularnbsp;tP it draw E f, cutting X e in f,

For, E P L is the original plane, producing the vanishing line P X L, arul E f being a visual ray,nbsp;parallel to -aril the original lines that arc perpendicular to the plane E P L, or its parallels; f isnbsp;therefore the vanishing point of all of them.

Corollary 1. When the vanishing line passes through the center of the picture; that is, Avhennbsp;the parallel planes are perpendicular to that of thenbsp;picture, the points X and e coinciding with Efnbsp;become perpendicular to Ee, or parallel to thenbsp;pieturc, and therefore the lines will have parallelnbsp;representations.

Corollary 2. If the original lines were desired to make any other angle with the original plane thannbsp;a right one, it is only making x Ef equal to it.

Problem 5. The center e, and dista7ice Ee, ^ the picture being given, and the vanishing point f, ofnbsp;u line parallel to E f, to fnd the vanishing line ofnbsp;planes perpendicular to that line whose parallel E fnbsp;produces the vanishing point f, fg. 7, plate ZO.

Join fe, and make EX perpendicular to, cutting f c, produced in X, draw P L perpendicular to ex, and PL will be the vanishing line required.

' The planes may form any angle instead of a right one, if that f E X be made equal to the same.

Pro b lem �. The cetiter and distance of the picture heug given, and the vanishing point of a line, to find the vanishing line of planes, perpendicidar tonbsp;the line whose vanishing point is given, fig. 7, pi. 30.

From f, the given vanishing point,quot; through c, the center of the picture, draw feX, and throughnbsp;the eye E draw E f, perpendicular to which drawnbsp;Ex, cutting fe in X; make PXL perpendicularnbsp;to Xef, and PL will be the vanishing line.

Problem 7- Having fiven the center and dis-(ance of the picture, the inequation of two planes, the

�van'islnng luie of one of inent, and the vanishing gomt of their common intersection, to find the vanishing ttnlnbsp;of the other glane.

Case 1. Let the inclination of the planes be a right angle, and let PL be the vanishing line, Pnbsp;the vanishing point of their common intersection.nbsp;By Problem 5 find f, the vanishing point of linesnbsp;perpendicular to the plane, whose vanishing linenbsp;is PL; join Pf, which is the vanishing line required.

For, since this last plane is pei�pendicular to the former, f will be the vanishing point of one line innbsp;it, (prop. 2) and P being the vanishing point ofnbsp;another, therefore P f is the vanishing line.

Case 2. When the inclination n e m, fg. 8, plate 30, is greater or less than a right angle.nbsp;Let N X be the vanishing line of one plane, x thenbsp;vanishing point of its interseetion with the othernbsp;plane; from the eye E draw Ex, and perpendicular thereto, the plane N E M intersecting x Nnbsp;in N, and the pieture in N f M, make the anglenbsp;N E M equal to n e m the given inclination, joinnbsp;X M, which will be the required vanishing line.

For, the planes being parallel to the original planes by construction, M x, N x are their vanishing lines, cor. 2, pr. 2.

Pdnd the representation of any one of its faces by Problem 3, and of the others by the last; if anynbsp;side be convex or concave, a number of pointsnbsp;may be found therein by Problem 1, and curvesnbsp;drawn evenly through them will represent thenbsp;curved superficies required; or squares, or othernbsp;regular figures may be inscribed or circumscribednbsp;about the original figures, and these plain figuresnbsp;being projected by the foregoing methods, together

with a few points therein, by which means the carves may be similarly drawn about or withinnbsp;these projected squares, amp;e. by this means also maynbsp;tangents be drawn to the representations of allnbsp;kinds of curve lines, in all kinds of situations.

So far I have endeavoured to render the principles of perspective obvious by a mere inspection of the figures. For the young artist, whose mindnbsp;seldom conforms to mathematical reasoning,nbsp;may, in all the foregoing problems, suppose thenbsp;plane of the picture placed upright upon that ofnbsp;the paper, which paper he may consider as thenbsp;ground plane, the operator�s eye E, being al-v\�ays in its proper situation with respect tonbsp;the picture, as well as to the planes of the originalnbsp;objects. The data, or things required to be knownnbsp;before objects can be put into perspective, arenbsp;their plans and elevations, see Defin. 5 and 6,nbsp;which must be actually laid down by a convenientnbsp;scale adapted to the size you mean your picturenbsp;should be, which may be very easily accomplished;nbsp;for you must remember, that the two extreme visual rays, that is, those which pass from the eye tonbsp;the two opposite borders of the picture, must notnbsp;make an ajigle less than two-thirds, or greater thannbsp;three-fourths of a right one. With respect to thenbsp;distance of the picture, it must be remembered,nbsp;that objects cannot be seen distinctly nearer to anbsp;common eye than six inches, and therefore in thenbsp;sinallcst miniature pieces the distance must exceednbsp;that quantity; the height of the eye should benbsp;Jibout half the distance of the picture, and aboutnbsp;the third part of the picture�s whole height. Fornbsp;example, if the whole height of the picture be

three, the distance from it should be two, and the height of the eye one. If the plan, when laidnbsp;down by its proper scale, be bounded by a quadrangle G N c d. Jig. 9, no. 2, plate 30, then thenbsp;ground line of the picture is supposed to be placednbsp;upon the shortest side of it, and upright to itsnbsp;plane; therefore this side of the figure should benbsp;made exactly equal to the breadth of the picture,nbsp;and the other two adjacent sides should, if produced, meet at the distance of the picture, as at Fnbsp;the foot of the observer, as is represented in Jig. Q,nbsp;no. 3. The problems already given are sufficientnbsp;for all cases that can happen in putting objects innbsp;whatsoever position into perspective; and thoughnbsp;they are perhaps solved in such a manner, as tonbsp;give the clearest ideas of the genuine practice ofnbsp;perspective, yet others may possibly prefer some

Example 1. Having given the center e of the picture P L N f. Jig. 2, no. 2, plate 30, its distancenbsp;eE 12 inches, height of EF six inches, to findnbsp;the representation of a point T, whose depth innbsp;the plan is eight inches, and amplitude to the leftnbsp;five inches.

Method 1. Form f in fN, take f n=five inches, and having drawn T perpendicular to fN, make itnbsp;equal to eight inches, then will T be placed in itsnbsp;proper situation, and n will be its seat on the picture; join T F, (F as usual being the foot of thenbsp;observer�s eye upon the ground plane, that is, innbsp;the present example, six inches perpendicularlynbsp;below the eye E) cutting fN in t, draw to perpendicular to fN, and join T E, intersecting t onbsp;in o, the representation of the point T required.

Otherwise, if the point T be situated on the ground plane, ch�aw F T, cutting f N in t, and t s

If the point T has any elevation perpendicular over the same point, as at S, make T S equal to thenbsp;height it should have, and draw E S, cutting thenbsp;perpendicular ots in s, the perspective point of Snbsp;required.

Remark. This method is very convenient in some cases, as wlicn there arc many windows, amp;c.nbsp;perpendicularly over one another, amp;cc.

Method �I. In the horizontal line cE,//V. 2, 710. 3, plate 30, from the center of the picture e,nbsp;take e E equal to the distance of the picture; fromnbsp;the given point Q draw Qd perpendicular to thenbsp;ground line G N, make d Q ^ equal to d Q, andnbsp;join Q ^ E and e d, intersecting each other in q,nbsp;the representation of the point Q.

Methods. Perpendicularly over the center e, fig. 2, 710. 4, plate 30, of the picture, take E e,nbsp;equal to the distance of the picture, draw Q.d perpendicular to G N, join E Q and e d, intersectingnbsp;each other in q, the point as before.

Remark. Both these methods arc in fact the same as that in Problem 1, as may be seen by comparing the figures with each other, being markednbsp;with the same letters for that purpose. It may benbsp;farther remarked, that both may be alternatelynbsp;used in the same piece, remembering to use thatnbsp;which you judge will give the bluntest intersection, that is, whichever makes the angle e q E thenbsp;greatest.

And now supposing a clear knowledge both of the theory and practice to be obtained, it may notnbsp;be amiss to shew how naturally a practice morenbsp;elegant and simple may be deduced, viz. by supposing all the planes to coincide with the paper^nbsp;or plane of the picture.

Thus, fig. 5, fJafe 30, if the triangles q e K# q (] Q revolve round d q e till they fall into thenbsp;plane P L N, no change takes place in any of thenbsp;comparative distances E e, E Q, amp;c. and thenbsp;point q preserves its situation on the picture, as in

Method 4. Let m N f s r, fig. 2, no. 5, flate 30, be the ground line, f e E perpendicular to it, passing through the center of the picture e, and letnbsp;fE ^ equal to fN, be the distance of the picture,nbsp;N P being perpendicular to N Ph and the representation of the point Q, whose distance from N rnbsp;is rQ.

Draw E Q, cutting Nr in s, taking Nm equal to Q r, and draw e m, cutting N P in o, draw o q

Revuirk. The lines E Q c m need not be drawn, but a dot made at s, where the ruler crosses n rnbsp;and e m; a T square ¹ may also be applied to thenbsp;line N s r, and a dot made on its fiducial edge at s;nbsp;then if the square be slid up till the fiducial edgenbsp;crosses the mark at o, the point s will be transferred to q, the representation of the given pointnbsp;obtained without drawing any lines over the picture. (This method was first discovered by Mr.nbsp;Beck, an ingenious artist, well known for manynbsp;tiscful contrivances.) It is also more accurate

See fg. 20, plateel. This is a very useful article in dra-w-ing; a niler a, is fixed as a square to h-, there li alsQ a moveable piece c. This ruler applied close to the side of a true drawingnbsp;board, will admit of parallel lines being drawn, as.well as obliquenbsp;ones, with more ease and expedition than by the common parallelnbsp;rulers. Edit.

But nevertheless, when the two points in representation arc projectccl very near together, it is the best way to work by the intersections and vanishing points of the original lines; for then having the full extent of the representation, its truenbsp;directions may be ascertained very correctly. The.nbsp;vanishing points by this method are thus found;nbsp;suppose for instance of the line I Q intersectingnbsp;the ground line in I, in f e E ^ take e E equal tonbsp;E � f, that is, equal to the distance of the picture;nbsp;draw E V parallel to I Q, intersecting the horizontal line h o in V the vanishing point of I Q. and allnbsp;its parallels.

Example 2. To put any plane figure, as QRST, or M N O P, into perspective, plate 30, fig. g.

This is, in faet, only a repetition of the last; for by finding the projections of the several points, asnbsp;before, nothing remains but to join them properly,nbsp;and the thing is done; thus the points m, n, o, p,nbsp;projected from M, N, O, P, being joined, givenbsp;m p o n, the representation of the quadranglenbsp;M P O N, amp;:c. But if any line, as p o, happensnbsp;to be very short, so that a small error in point onbsp;would considerably alter the direction of it, findnbsp;the intersection and vanishing point of its original; or, if the vanishing point fills at too great anbsp;distance, as it very frequently does in practice,nbsp;find the intersection E *, and vanishing point Z ofnbsp;any other line passing through the point O, as thenbsp;diagonal OM, and the truth of the projectionnbsp;niay be depended upon. All the three last methods may be made use of in projecting the same,nbsp;figure; thus the point p, by the second method,nbsp;that is, make e � in the horizontal line equal tonbsp;E ^ k, the distance of the picture, draw P Y perpendicular to S X, and take Y k equal to Y P, join

Ek and c Y, intersecting in p, which is a bcttei* intersection than the third method would give^nbsp;viz. by drawing EP; but yet inferior to that innbsp;the fourth method.

Example .3. To put any solid body into per-spcctivc. Find the scats of all the ];oints upon the ground plane, and project them as before;nbsp;let G be the seat of one of the points, g its projection, produce eg to the ground line aE�, andnbsp;make E ^ H perpendicular, and ecjual to the heightnbsp;of the given point, from its seat join H c, and itnbsp;will cut the perpendicular g g in g the representation of the required point.

Make G G ^ perpendicular and equal to the given height, draw G c, cutting S U in c % thennbsp;e ^ g ^ parallel to S X, will cut the perpendicularnbsp;gg^ in the point'gquot;quot; required; or the point gnbsp;may be transferred to g by the T square, without drawing g g- or c g

Proceed in this manner, till you have obtained all the requisite points in the figure, marking or numbering them as you proceed, or else correct them bynbsp;strait or curved lines according to your original, bynbsp;which means you may instantly see the connectionnbsp;of your work at all times without contusion.

Remarks. 1. Sometimes when there is a great number of small parts in a body to be perfectlynbsp;made out, it may not be amiss to draw squaresnbsp;over the ground plane of the object, and find thenbsp;seats of its several elevated points; then by turning these squares into perspective, the positions ofnbsp;the several points will likewise be found by inspection, and the horizontal row of squares willnbsp;serve as a scale for the altitudes of bodies, whosenbsp;seats lie in that row, or both plans and elevationsnbsp;may be used, as in Jig. 10, plate 30.

2. Perspective may also be practised without having any recourse to ground plans; for, by taking the horizontal angles, the amplitudes of objectsnbsp;are to be ascertained, and by vertical ones theirnbsp;heights and depths in the picture; the angles thusnbsp;taken may be entered in a table of this form.

In protracting which angles you must make th� distance of the picture radius and lay down thenbsp;angles by a line of tangents adapted to that radius,nbsp;which is therefore best done by means of the sector; or more expeditiously by the fourth method,nbsp;fiage 478; for, if a protractor be fixed at Enbsp;^g. 9, plate 30, horizontal angles will cut thenbsp;ground line S X in the same places, as if the linesnbsp;forming them passed over the original points innbsp;the plan; and if the. protractor be fixed at e, thenbsp;same may be said of S U with respect to elevations.

Solid bodies may also be put into perspective, -by drawing lines in particular directions, as fromnbsp;the center of a circle, or of concentric ones, andnbsp;finding the representations of the same, raise perpendiculars of the proper heights, always supposing the bodies to be transparent, fg. 10, plate 30.

Luminous bodies, as the sun, moon, lamps, amp;r. arc generallv considered as points; but the artistnbsp;takes the advantage of their not being perfectlynbsp;so, by softening the extremities of his shadows,nbsp;which are so softened in nature, and for this reason, winch may be thus explained.

Let R S, fg. 11, no. 2, plateel, be the radius of a luminous body, whose seat is rs, and center is S:nbsp;from the opaque body O, draw x r, touching thenbsp;extremities of both the bodies on both sides, bynbsp;which means a penumbra, or semi-shade, qxn,nbsp;will be formed on each side of the main shadow,nbsp;which becomes extremely tender tow^ards the outernbsp;extremity, and from thence gradually strengthensnbsp;till it blends with the uniform shadow, which will,nbsp;if the diameter of the luminous be greater thannbsp;that of the opaque one, measured in the directionnbsp;of their centers, converge to a point, as �/i, fig. 11,nbsp;flaf-e 31.

N. B. The opaque body which casts the shadow is called the shading body; and those that are immersed in the shadow, are called shadowednbsp;bodies.

Through the luminous body draw planes touching all the illumined planes of the object, and the intersections of these planes with the given plane,nbsp;wall give the boundary of the shadow required.

Example 1. Let the luminous point be the sun, the plane of projection ARSB, fig. 12, plate 3hnbsp;and the object the parallelogram ABCD.

The sun�s rays, on account of his distance, may be supposed parallel, therefore the planes 0 RAj

O S B, ave parallel; and, therefore, since C D is parallel to BA, RS is parallel to it also, and thenbsp;shadow a parallelogram, whose length is to thenbsp;heiglit of the object, supposed upright, as radiusnbsp;to the tangent of the sun�s altitude^ and itsnbsp;breadth as radius to the sine of the inclination ofnbsp;its rays, with BA the base of the parallelogram.

Therefore find the seat of one of the rays, as OAR, and make the angle ADR equal to thenbsp;complement of the sun�s altitude, and make thenbsp;parallelogram B R, and the thing is done.

Example 2. Let the luminous body be a lamp placed at O, fig. 13, plate 31; the object, a parallelogram ABCD, standing upon the planenbsp;ARSD.

Having drawn O G perpendicular to the jdane ARSD, and the rays O B R, O C S; from G, thenbsp;seat of the lamp on the plane AS, draw GAR,nbsp;G D S, intersecting the rays OB, OC, in R and S,nbsp;and ARSD will be the shade required.

Continue the sides CADB to A', Aquot;, B', B'quot;, 8cc. fig, 14, plate 31, and where AG, B H, cut thenbsp;body GHIK, draw KGA'^, IHB, making thenbsp;angles A'GAquot;, BHB', equal to the inclination ofnbsp;the plane GI to GB, proceed in the same manner with every new plane M N; or if the object isnbsp;curvilinear, tangents will always pass through the �nbsp;lines C A' and D B respectively, except when theynbsp;are perpendicular.

Example 4. Let a lamp O, fig. 15, plate 31, throw a shadow on the body RSUT, and letnbsp;Q. T U be the central line of the shadow from thenbsp;parallelogram, AB, upon the ground; draw OB,nbsp;GP, parallel to Q.T, and from T to OP drawnbsp;t P, touching the plane U s of the body; from P

draw P S u, P w U, cutting the extremities of the shadow in U and u, and w U u S will be the shadow of AB upon the face US; proceed in likenbsp;manner with all the other illumined faces.

N. B. If t P never meets O P, it denotes the fiicc of the body to be parallel to AB, and the shadow on that face to be a parallelogram.

The shadows, being thus ascertained, may be put into perspective by the foregoing rules.

Let fall a perpendicular upon the reflecting plane, to which draw a radial from the eye, asnbsp;much below the horizontal line, as the real objectnbsp;appears to be above it.

Example 1. Let AB, fig. 18, plate �il, be any' object placed on the water; from B draw Bb,nbsp;perpendicular to the surface Ab, which continuenbsp;till the angles BEb and CEb are equal; that is,nbsp;(Eb being a horizontal line) till b c is equal tonbsp;b B, and b C will be the reflection of AB.

Example 2. When objects are upright, the lines may be produced below the horizontal line, asnbsp;much as the real ones are above it.

In this kind of projections, the eye is supposed to be placed at an indefinite distance from the object in the diagonal, and looking down upon it innbsp;an angle of 45�, so that the top, one side, and onenbsp;end, are seen under the same angle, and thereforenbsp;appear in their true proportions with respect tonbsp;each other; ahd therefore heights, lengths, andnbsp;breadths must be laid down by the same scale,nbsp;and all parallel lines made parallel, see fig. xi,nbsp;plate 30.

Before we can give rules for regulating the force of lights and shades in a picture, we must consider what degree of it the bodies themselves arcnbsp;endued with, according to their several positionsnbsp;with respect to the illuminating body.

Proponl�on 3. The intensity of light upon any plane is reciprocally as the square of the distancenbsp;of that plane from the illuminating body.

Let AB C D, jig. l6, glate 31, be the shadow of the square abed upon a plane parallel to it, whichnbsp;projection will therefore be a square, and in proportion to abed as the square upon OA to thatnbsp;upon O a; therefore since the real quantity of lightnbsp;is the same as would be received upon ABCD,nbsp;the intensity of it is reciprocally as the squarenbsp;upon AB to the square upon a b, or as square O Anbsp;to square O a.

For example, if parallel planes arc at the distance of one, two, and three feet from a luminous point, the intensity of light upon them would benbsp;one, one-fourth, one-ninth, amp;c.

Corollary. All parallel planes are equally illuminated by the sun at the same moment.

Proposition 4. If the sun�s beams fall perpendicular upon one face AB, fg. 17, plate'�\, of an object, and inclined upon another AC, the intensity of light on the faces, is as radius to the sine ofnbsp;the angle of incidence.

Produce AB to b, the quantity of light Ab receives is the same as would be received on AC if bA were away; therefore the brightness is as ACnbsp;to Ab, that is, the brightness of Ab, ox A B is to

Proposition 5. A plane uniformly enlightened, does not appear so to an eye in different situations.

For, as all bodies are porous, the little exuberances will have their light and dark sides, and the eye will view more of the former, as it is morenbsp;nearly situated in a line with the rays of light, andnbsp;more of the latter, the more it faces them.

The subject of this proposition is one great cause of the graduation of light upon the faces ofnbsp;buildings and other planes, and not altogethernbsp;owing to a greater teint of air, as the artists call it,nbsp;on that part which is the farthest off.

Remark. It is very necessary to observe, that transparent and polished bodies are not includednbsp;among those mentioned in this proposition, fornbsp;they seem most illuminated in that part whichnbsp;makes the angle of reflection equal to that of incidence; but if bodies of this kind are not flat, asnbsp;water when just broken by small rippling waves,nbsp;then the light is reflected from some part of almost every wave, and so is extended to a greatnbsp;space, but is strongest perpendicular under the luminary, and gradually decreases on each side.

The case is the same in the sky, which is brightest near the sun�s apparent place, and graduates into a deeper azure as it retires farther off,nbsp;and for a reason nearly the same; for the pellucidnbsp;particles floating above us, having large intersticesnbsp;between them, act in the same manner as the rippling waves in disturbed water; and, therefore,nbsp;the more obliquely the light strikes upon them,nbsp;the more united their force will be to an eye situated in the proper angle of reflection.

Proposition 6. All shades and shadowing objects would be equally dark and indistinguishable, ifnbsp;they received no secondary or reflected light.

For, light is not visible of itself, but by striking upon other bodies renders them so, and these en-Jightened bodies serve as lights to bodies otherwise in shade, and such lights are called secondarynbsp;or reflected ones, the chief of which is the sky.

Proposition 7. Every body participates of the colour of the light by which it is illumined; for,nbsp;blue rays thrown upon a yellow body will producenbsp;a green; red rays, purple; and purple rays, thatnbsp;is, blue and red, black.

Corollary. Hence shadows arc often observed green in the morning or evening, for the sky isnbsp;always very green at those times compared withnbsp;other times of the day, owing to the warm raysnbsp;being more copiously reflected downwards by thenbsp;sun�s beams striking more obliquely on the atmosphere, which partly acts as a prism, andnbsp;the shadows become more blue, as the sky becomes so; but clouds are of all colours, and asnbsp;they are denser than the blue part of the sky, theynbsp;throw stronger reflections, and cause many accidental teints in the shadows of bodies; therefore,nbsp;as the shadow of every body is partially enlightened by all the bodies surrounding it, it must partake of the colours of all of them; and this is thenbsp;grand source of harmony in painting, of wdflehnbsp;system the colour of the original light serves as anbsp;key, and is to be attended as nicely to in painting,nbsp;as in music.

Proposition 8. Bodies partake more of the colour of the sky, as they arc farther off�.

For, the sky being only a body of air every where surrounding us, its natural colour supposednbsp;to be blue, the farther off any body is, the morenbsp;of this blue air is intercepted between us and thenbsp;body, and therefore the bluer it is, and that innbsp;projxirtion to its distance.

Various have been the methods used to facilitate the practice of perspective, as well for those whonbsp;understand, as those who are ignorant of that art;nbsp;and, though some have supposed that the warmthnbsp;of imagination and luxuriance of fancy, which impels the mind to the cultivation of the fine arts, isnbsp;not to be confined to mechanical modes, yet uponnbsp;enquiry they will find, that the most able and accomplished artists arc often obliged to have recourse to some rules, and to use some mechanicalnbsp;contrivances to guide and correct their pencil. Sonbsp;great is the difficulty, and so tedious the operationnbsp;of putting objects in true perspective, that theynbsp;trust mostly to their eye and habit for success;nbsp;how well they succeed, w'c may decide from thenbsp;portraits drawn by the best artists, and the different judgments formed concerning them. Mr.nbsp;Eckhardt has well observed, that there is no artistnbsp;who will be hardy enough to say, that he can delineate by the eye the same object twice with exactness, and prcvserve a just and similar proportionnbsp;of parts in each. In one of the figures, we shal)nbsp;find some of the parts larger than in the other�nbsp;both cannot be right: yet, supposing them perfectly the same, neither may be conformable tonbsp;nature. Add to this, many situations of an objectnbsp;occur, which no eye, however habituated, can represent with accuracy.

On this account, I have a long time endeavoured to complete an instrument that shoidd give the out-linc of an object with accuracy. Thesenbsp;Essays have now swelled so far beyond my intentions, that I must be as concise as possible. I must,nbsp;however, acknowledge the valuable hints commurnbsp;nicated by AIr.i%�wooi/, and other ingenieus men,

The methods most generally in use are, 1. The camera obscura. 2. The glass medium or plane.nbsp;3. A frame of squares. The inconveniences andnbsp;inaccuracies which attend these expedients, induced Sir Chrisfopher Wren, Mr. Fergiisoyi, Mr.nbsp;Hirst, My Father, Mr. Watt, Mr. Fckhardi, Ferenbsp;Toussaint, and others, to have recourse to differentnbsp;contrivances to remedy their defects; of whichnbsp;those by the Rev. Mr. Hirst, My Father, Mr. JVatt,nbsp;and Mr. Echhardt, arc undoubtedly the best;nbsp;Mr. Eckhardt's, and Mr. Hirst's vary but littlenbsp;from each other.

Those represented at fig. 1 and 2, plate 32, appear to me far superior to any that have been hitherto contrived; the object is delineated on annbsp;horizontal plane, the pencil B,may be moved in anynbsp;direction, whether curved or strait, with the utmost freedom. By either, the artist may be surenbsp;of obtaining the measure of every part of the object with exactness; and this is performed withoutnbsp;any loss of time. The instrument may be mov-cdnbsp;from any place, and brought back to the same withnbsp;great exactness; and the outline may be formednbsp;cither of a number of points, or one continuednbsp;line, at the pleasure of the draftsman.

That represented at fig. 2, is the simplest of the two instruments: fig. 1, though more complex,nbsp;merits, for its contrivances and motions, the attention of the mechanic, as well as the draftsman.nbsp;They both move with facility in every dircctioy,nbsp;and the whole operation consists in lookingnbsp;through the sight C, which may be placed in anynbsp;convenient situation, and moving the pencil B, sonbsp;that the apex A, of the triangle may go over thenbsp;object, whose outline will be delineated at the samenbsp;tunc by the pencil B.

To lessen the expense, and render the instru-Rtcat more portable, I have constructed an instru-

ment somewhat similar to that represented at jig. 2, plate'62, but which moves only in a verticalnbsp;plane, the board on which the drawing is madenbsp;being in the same plane with the triangle.

To these may be added the parallel rule, and the perspective compasses. The distance of thenbsp;rule from the eye, as it has no sights, must be regulated by a piece of thread tied to it, and heldnbsp;between the teeth.

Fig. 3, plate 32, is a pair of pocket brass perspective compasses, by Mr. Jones, that have been found very useful and convenient for taking readily the relative proportions of buildings landscapes, amp;c. andnbsp;protracting them on the drawing. A, A, are thenbsp;two legs, made of small tubes about six inches innbsp;length; B, B, are two sliding legs moving to different distances out of the tubes A, A; D is anbsp;sight piece with a small hole; E, E, are two steelnbsp;points to take the sights by; F, F, are two morenbsp;small steel points at the ends of the sliders to marknbsp;down the distance on the paper, after an observation ; C is an arc with teeth fixed on one leg, bynbsp;which, and the pinion G, the other leg is moved tonbsp;the proper angle while observing. This arc isnbsp;.sometimes divided into degrees, and otherways subdivided by a set of figures, so as to give distances bynbsp;inspection, amp;c. according to the pleasure of thenbsp;purchaser. The sight C, turns down; the slidersnbsp;B, B, go inwards; the arc takes off, and the wholdnbsp;packs into a smtill narrow case.

FINIS.

ADDENDA,

As this treatise is designed to comprehend a general collection of the most approved methods of surveying, I think the following method of surveying a large estate by Mr. Emerson, and thenbsp;new method of surveying and keeping a fieldnbsp;book by Mr. Rodhaiii, as published in Dr. IJultorUsnbsp;Mathematical Dictionary, 2 vols. 4to. 1706, will benbsp;of real information to many surveyors; and, in mynbsp;opinion, as deserving of practice as any other method I am acquainted with.

If the estate be very large, and contains a great number of fields, it cannot be done by surveying ajl the fields singly, and then putting themnbsp;together; nor can it be done by taking all the angles and boundaries that inclose it. For in thesenbsp;cases, any small errors will be so multiplied as tonbsp;render it very much distorted.

1. Walk over the lordship two or three times, in order to get a perfect idea of it, and till you cannbsp;carry the map of it in your head. And to help

your memory, draw an eye draught of' it on paper, to guide you, or at least of the principal partsnbsp;of it.

2. nbsp;nbsp;nbsp;Choose two or more eminent places in thenbsp;estate for your stations, from whence you can seenbsp;all the pi�incipal parts of it; and the fewer stationsnbsp;you have to command the whole, the more exactnbsp;your work will be; and let these stations be as farnbsp;distant from one another as possible; and they willnbsp;be fitter for your purpose, if these stationary linesnbsp;be in or near the boundaries of the ground, andnbsp;especially if two lines or more proceed from onenbsp;station.

3. nbsp;nbsp;nbsp;Take what angles, between the stations, younbsp;think necessary, and measure the distances fromnbsp;station to station, always in a right line; thesenbsp;things must be done, till you get as many anglesnbsp;and lines as are sufficient for determining all yournbsp;points of station. And in measuring any of thesenbsp;stationary distances, mark accurately where thesenbsp;lines meet with any hedges, ditches, roads, lanes,nbsp;paths, rivulets, amp;c. and where any remarkable object is placed, by measuring its distance from thenbsp;Stationary line; and where a perpendicular from itnbsp;cuts that line. And always mind, in any of thesenbsp;observations, that you be in a right line, which younbsp;will know by taking backsight and foresight, alongnbsp;your stationary line; which you must never omit.nbsp;And thus as you go along any main stationary line,nbsp;take offsets to the ends of all hedges, and to anynbsp;pond, house, ruill, bridge, amp;c. orwitting nothingnbsp;that is remarkable, and all these things must benbsp;noted down, for these are youv data, by which thenbsp;places of such objects are to be determined uponnbsp;yorrr plan. And be sure to set marks irp at thenbsp;intersections of all hedges with the stationary line,nbsp;that you may know rvherc to measure from, when

you come to sun-ey these particular fields, wliich must immediately be done, as soon as you havenbsp;measured that stationary line, whilst they are freshnbsp;in memory. By these means all your stationarynbsp;lines are to be measured, and the situation of allnbsp;places adjoining to them determined, which is thenbsp;tirst grand point to be obtained. I would havenbsp;you lay down your work upon paper every night,nbsp;when you go home, that you may see how younbsp;go on.

4. As to the inner parts of the estate, they must be determined in like manner by new stationarynbsp;lines. For, after the main stations are determined,nbsp;and every thing adjoining to them; then the estatenbsp;must be subdivided into two or three parts by newnbsp;stationary lines; taking inner stations at propernbsp;places, where you can have the best view; andnbsp;measure these stationary lines as you did the first,nbsp;and all their intersections with hedges, and all offsets to such objects as appear; then you may proceed to survey the adjoining fields, by taking thenbsp;angles that the sides make with the stationary line,nbsp;at the intersections, and measuring the distances tonbsp;each corner, from the intersections. For everynbsp;stationary line will be a hash to all the future operations; the situation of all parts being entirely dependent thereon; and tlierefore they should benbsp;taken as long as possible; and are best to runnbsp;along some of the hedges or boundaries of one ornbsp;more fields, or to pass through some of their angles.nbsp;All things being determined for these stations, younbsp;must take more inner stations, and continue to divide and subdivide; till at last you come to singlenbsp;fields; repeating the same work for the inner stations, as for the outer ones, till all be done. Andnbsp;close the work as oft as you can, and in as fewnbsp;lines as possible. And as it may require some

judgment to choose stations the most convenJ-ently, so as to cjuise the least labour; let the stationary lines run as far as you can along some hedges, through as many corners of the fields, andnbsp;other ronarkable points, as you can. And takenbsp;notice how one, field lies by another; that you maynbsp;not misplace them in the draught.

5. nbsp;nbsp;nbsp;An estate may be so situated, that the wholenbsp;cannot be surveyed together; bscausc one part ofnbsp;the estate cannot be seen from another. In thisnbsp;case, y'ou may divide it into three or four parts, andnbsp;surveys the j)arts separately, as if they were landsnbsp;belonging to different persons; and at last joinnbsp;them together.

6. nbsp;nbsp;nbsp;As it is necessary to protract or lay downnbsp;your work as you proceed in it, you must have anbsp;scale of a due length to do it by. To get such anbsp;scale, you must measure the whole length of thenbsp;estate in chains; then you must consider how' manynbsp;inches long the map is to be; and from these younbsp;will know how many chains you must have in annbsp;inch, and make your scale, or choose one alreadynbsp;made accordingly'.

7. nbsp;nbsp;nbsp;The trees in every hedge row must be placednbsp;in their pro[)er situation, which is soon done by thenbsp;plain table; but may be done, by the eye withoutnbsp;an instrument; and being thus taken by guess, innbsp;a rough draught, they will be exact enough, beingnbsp;only to look at; except it be such as are at anynbsp;remarkable places, as at the ends of hedges, atnbsp;stiles, gates, amp;c. and these must be measured.nbsp;But all this need not be done till the draught benbsp;finished. And observe in all the hedges, what sidenbsp;the gutter is on, and to whom the fences belong.

8. nbsp;nbsp;nbsp;When you have long stations, you ought tonbsp;have a good instrument to take angles with, whichnbsp;should be exact to a quarter of a degree at least;

and hardly any common surveying instrument will come nearer. And though the plain table is not atnbsp;all a proper instrument to survey a whole lordshipnbsp;with, yet it may very properly be made use of tonbsp;take the several small internal parts; and such asnbsp;cannot be taken from the main stations; and is anbsp;very quick and ready instrument.¹

Example. Walking over the lordship, I pitch upon the four stations A, B, C, D, fig. 1, plate 33,nbsp;from wdiich I can command the greatest part of it,nbsp;there I set up marks. Then I measure along AB,nbsp;which is a right line, and the boundary on one sidenbsp;of the land. In measuring along, I set down thenbsp;distances measured, when I come at the corners ofnbsp;the fields a, a, a, a, where the hedges come in, andnbsp;likewise where I cross the brook b b. Then having got to B, I set down the whole length of AB.

Next I measure from B to C, and in my way, I set down how far I have measured when I crossnbsp;the hedges at c, c,c,c\ and likewise where I crossnbsp;the brook again. Thus I measure forward tillnbsp;I come at C, and then I set down the length of thenbsp;stationary line B C.

After the same manner I measure along the stationary line CA, observing to set down the intersections with the hedges, as before; till I come at A, where I set down the length of C A. Then thenbsp;three points. A, B, and C, are determined; andnbsp;may be laid down in the plan; and all the foresaidnbsp;points.

The angles by this method require to be taken very' correctly; and, as instnrments are now constructed with an extraordinarynbsp;degree of perfection, to a skillful observer the angles, howevernbsp;numerous, can be of no reasonable objection; for, the chain itself,nbsp;and oftentimes the manner of using it, may create as many errorsnbsp;as might be found by taking a multitude of angles- by the Iheornbsp;dolite, tat.

Being come to A again, I go from A towards B) and in my way I survey every single field adjoin-*nbsp;ing to the stationary line AB. To do which thenbsp;shortest way, I take the angles at every intersection a, that the sides of each field make with thenbsp;stationary line AB; and then I measure theirnbsp;leugtlis; by which every field is easily laid down.nbsp;In the same manner I proceed from B to C, andnbsp;measure every field adjoining to BC. And then Inbsp;go to A, and measure every field in my way thither.

Next I go from A towards D, and set down, as before, all my crossing of the hedges; and thenbsp;length AD, when I eome at D. And in likenbsp;manner I measure along D C, setting down all thenbsp;crossings of the hedges as before, with whatevernbsp;else is remarkable, as Avherc a highway crosses at d.

Having finished all the main stations, we must begin to make inner stations. Therefore I take Pnbsp;and G for two stations, being in the lines AB andnbsp;B C, the hedges from F to G running almostnbsp;straight; then I measure from F towards G, andnbsp;at/, I find a hedge going to the left, and going onnbsp;to^, I find another hedge going to the right; andnbsp;at h, I eross the burn. At i, there is an angle, tonbsp;which I make an offset. Going on further, I comenbsp;at a cross hedge /, going to the right; and thennbsp;measure on to G, the end of the station. Now innbsp;going from F to G, we can take all the angles thatnbsp;the sides of the fields make with the stationarynbsp;line FG, and then measure their lengths; bynbsp;which these fields may be laid down on paper.

Then I take another inner station at I, and measuring from A to 0, I come to the opposite corner of the field; then measuring on to p, I eross anbsp;hedge; then I proceed to my station 1. Then Tnbsp;measure from I to F, and take an offset to n, wherenbsp;the hedge crosses the brook. Then I come to the

Corner of the last field at m; and then measure to the opposite eorner at F;, the other station. Innbsp;your going from A to I, you may take the anglesnbsp;that the hedges make with your stationary linenbsp;A I, and measure these hedges, and then they maynbsp;be laid down. And the like in going from fnbsp;to F.

All this being done, take a new station H, and measuring from B towards H, all the hedges lie almost in a right line. So going along we come atnbsp;a cross hedge, and going further we come at a tree,nbsp;in the hedge we measure along; going further wenbsp;comeat two other cross hedges; and a piece further we cross the brook; going on we come at anbsp;cross hedge; going on still we come to anothernbsp;cross hedge; all these hedges are to the left.nbsp;Then going on still further, we have a windmill tonbsp;the right; and afterwards a cross hedge to the left,nbsp;and then we measure on to the station H. Thennbsp;measuring from H towards C, we have a house onnbsp;the left; and then go on to C. And the fields maynbsp;be all surveyed as you go along B H and H C, andnbsp;then laid down. And after this manner you mustnbsp;proceed through the whole, taking new stations,nbsp;till all be done.�

Mr. John Rodham�s new method of surveying, WITH THE PLAN OP THE FIELD-BOOK, plate 34.

� The field book is ruled into three columns. In the middle one are set down the distances onnbsp;the chain line at which any mark, offset, or othernbsp;observation is made; and in the right and leftnbsp;hand columns are entered the offsets and observations made on the right and left hand respectively of the chain line.

It is of great advantage, both for brevity ant^ perspicuity, to begin at the bottom of the leaf andnbsp;write upwards; denoting the crossing of fences,nbsp;by lines drawn across the middle column, or onlynbsp;a part of such a line on the right and left oppositenbsp;the hgures, to avoid confusion; aud the cornersnbsp;ot fields, and other remarkable turns in the fencesnbsp;where otfsets arc taken to, by lines joining in thenbsp;manner the fences do, as will be best seen by comparing the book with the plan annexed to thenbsp;field-book, as shewn in plate 34.

The marks called, 0, h, c, amp;c. are best made-in the fields, by making a small hole with a spade, and a chip or small bit of wood, with the particularnbsp;letter upon it, may be put in, to prevent one marknbsp;being taken for another, on any return to it. Butnbsp;in general, the name of a mark is very easily hadnbsp;by referring in the book to the line it was made in.nbsp;After the small alphabet is gone through, the capitals may be next, the print letters afterwards, andnbsp;so on, which answer the purpose of so many different letters; or the marks may be numbered.

The letter in the left hand corner at the beginning of every line, is the mark or place measuredand, that at the right hand corner at the end, is the mark measured to: but when it isnbsp;not' convenient to go exactly from a mark, thenbsp;place measured from, is described such a distancenbsp;from one mark towarA?, another-, and where a marknbsp;is not measured to, the exact place is ascertainednbsp;by saying, turn to the right or left hand, such a dis^nbsp;tame to such a mark, it being always understoodnbsp;that those distances arc taken in the chain line.

The characters used, arc [~ for turn to the Tight hand,~\ for turn to the left hand, and A placednbsp;over an offset, to shew that it is not taken at rightnbsp;angles with the chain line, but in the line with

some strait fence; being chiefly used when crossing their dircetions, and it is a better way of obtaining their true places than by offsets at right angles.

When a line is measured whose position is determined either by former work (as in the casenbsp;of producing a given line, or measuring from onenbsp;knowm place or mark to another) or by ,itself (asnbsp;in the third side of a triangle) it is called a fastnbsp;line, and a double line across the book is drawn atnbsp;the conclusion of it; but if its position is not determined (as in the second side of a triangle) it isnbsp;called a loose line, and a single line is drawn acrossnbsp;the book. When a line becomes determined innbsp;position, and is afterwards continued, a doublenbsp;line half through the book is drawn.

When a loose line is measured, it becomes absolutely necessary to measure some line thatnbsp;will determine its position. Thus, the first linenbsp;a h, being the base of a triangle, is always determined; but the position of the second side hj,nbsp;does not become determined, till the third side j bnbsp;is measured; then the triangle may be constructed,nbsp;and the position of both is lt;letermincd.

At the beginning of a line, to fix a loose line to the mark or place measured from, the sign ofnbsp;turning to the right or left hand must be addednbsp;(as at / in the third line;) otherwise a stranger,nbsp;when laying down the work, may as easily construct the triangle Itjb on the wrong side of thenbsp;line a h, as on the right one; but this error cannotnbsp;be fallen into, if the sign above named be carefullynbsp;observed.

In choosing a line to fix a loose one, cate must be taken that it does not make a v^ery acutenbsp;or obtusc angle; as in the triangle � B r, by thenbsp;angle at B being very obtuse, a small deviation

from trutli, even the breadth of a point at p or r, would make the error at B, when constructed,nbsp;very considerable ; but by constructing the triangle y)By, such a deviation is of no cojisequence,nbsp;Where the words, leave-off, are written in thenbsp;field-book, it is to signify that the taking of offsets is from thence discontinued; and of coursenbsp;somethins: is wanting; between that and the nextnbsp;offset.�

Mr. Thomas Keith, Teacher of the mathematics, has considerably improved the German parallelnbsp;ruler, or that of Mr. Marquois, see pages 27 andnbsp;28. By making the hypothenuse, and the perpendicular line to it from the opposite angle, in thenbsp;ratio of 4 to 1, and adding several scales, amp;c. itsnbsp;uses are considerably extended for drawing plansnbsp;of fortifications, and other branches of the mathematics.

Fig. 2, 3, 4, and 5, plate 33, represent the two faces of the ruler, and the triangle of half thenbsp;real dimensions.

The slider is a right-angled triangle, the perpendicular is divided into inches and tenths, and the base has three indices, and other divisions requisite to be used with the scale.

The ruler contains l6 different scales, which, by the help of the slider may be increased to 2Q,nbsp;without guessing at halves and quarters. The figures on the one end shew the number of divisionsnbsp;to an inch, and the letters on the other end arenbsp;necessary to exemplify its use. It likewise contains the names of the polygons, the angles at theirnbsp;centers, and a scale of chords by which all polygons may be readily constructed.

To draw a line parallel to the slider. Lay the hypothenuse, or sloped edge of the triangle in thenbsp;position you intend to have your line, place thenbsp;scale against the base of the triangle, and draw anbsp;line along the slope edge, keep the scale fixed,nbsp;and move the slider to the left or right hand, according as you want a parallel line above or belownbsp;the other.

To draw a line parallel to the scale. Lay the scale in the position you intend to have your lines,nbsp;and draw a line along the edge of it. Place thenbsp;base of the triangle against this edge, the middlenbsp;index standing at O on. the scale, and make anbsp;mark against any one division on the perpendicular; turn the triangle the other side uppermost,nbsp;and make a mark against the same division; joinnbsp;these marks, and this line will be parallel to thenbsp;former. The same may be done by sliding thenbsp;triangle, without turning it, if the lines arc notnbsp;required to be very long.

To draw a line parallel to the slider at a great distance. Draw a line along the slope edge of thenbsp;triangle, and fix the scale as before, make a marknbsp;against 2 on the perpendicular; the index beingnbsp;at O, turn the triangle the other side uppermost,nbsp;and make a mark against the same division, takenbsp;away the triangle, and move the scale to thesenbsp;marks, apply the triangle again; proceed thus tillnbsp;you have got the proper distance, then draw a linenbsp;along the slope edge, and it will be parallel to thenbsp;former.

To draw a line parallel to the scale at a great distance. Draw a line along the edge of the scale,nbsp;place the base of the triangle ngainst that edge,

the middle index being at O, and make a mark against 2 on the perpendicular; turn the trianglenbsp;the other side uppermost, and make a mark againstnbsp;the same division; take away the triangle, andnbsp;move the scale to these marks; proceed thus tillnbsp;you have got the proper distance, then draw anbsp;line along the scale, and it will be parallel as required,

If a line is wanted perpendicular to the scale, apply the base of the slider to it, and draw a linenbsp;from the scale along the perpendicular of the triangle; should a longer perpendicular be wanted,nbsp;apply the perpendicular of the triangle to the scale,nbsp;and draw a line along the base.

If a perpendicular is wanted to a line, which has been drawn along the hypothenuse of the triangle,nbsp;keep the scale fixed, and apply the hypothenusenbsp;of the triangle to it, then draw a line along thenbsp;perpendicular of the triangle.

Havnng made choice of any one scale, take the side of your polygon from it. Take the degreesnbsp;under the name of your polygon, and subtractnbsp;them from 180; at each end of the above side ofnbsp;your polygon, make angles equal to half the remainder, and the distance from the intersection ofnbsp;the lines, which form the angles, to either end ofnbsp;the side of your polygon, will give the radius of itsnbsp;circumscribing circle.

Or, the three angles, and one side being given of a triangle, the radius, or remaining sides, maynbsp;he found by trigonometry,

If the side of your polygon zvas taken from the scale C. Move the slider from O to 1, 2, 3, he.nbsp;on the scale D, and you will draw a parallel line ofnbsp;the width �f 1, 2, 3, he. divisions on the scale C.nbsp;This scale is 20 fathoms to an inch.

If the side of your polygon was taken from the scale C, calling the dhnsions two each. Move thenbsp;slider from O, on the scale A, to 1, 2, 3, amp;c. onnbsp;the same scale, and yon will draw a parallel line ofnbsp;half the width of 1,2, 3, he. divisions on the scalenbsp;C. This scale is 40 fathoms to an inch.

If the side of your polygon was taken from the scale C, calling the divisions three each. When thenbsp;middle index stands at O, the divisions marked Cnbsp;on the slider will make straight lines with 5 and 6nbsp;on the scale D. By moving the slider to thenbsp;right or left till the other divisions thereon makenbsp;straight lines successively with 5, 6, he. you willnbsp;draw a parallel line of one-third of the width ofnbsp;1, 2, 3, he. divisions on the scale C. This scalenbsp;is 6o fathoms to an inch.

If the side of your polygon was taken from the scale A. Move the slider from O, to 1, 2, 5, he.nbsp;on the scale B, and you will draw a parallel line ofnbsp;the width of 1,2, 3, amp;c. divisons on the scale A.nbsp;This scale is 10 fathoms to an inch.

If the side of your polygon was taken from the scale D. Move the slider from O, to 2, 4, 6, amp;C'nbsp;on the scale B, and you will draw a parallel line ofnbsp;the width of 1, 2, 3, amp;c. divisions on the scale P.nbsp;This scale is five fathoms to an inch.

Note. There is no absolute necessity for always moving the slider from O, and it may be used either side uppermost. Any one of the three indices may likewise be made use of, amp;c.

To render the scale still more perfect, Mr. Keith has made the following additions.

1. nbsp;nbsp;nbsp;The divisions on the scale A, and the fourthnbsp;scale from A, have been subdivided for the purposenbsp;of constructing sections, amp;c. Ten of the divisions on the scale A make one inch; if, therefore,nbsp;you call these divisions six each, the small divisionsnbsp;on the same scale, nearest to the left hand, will benbsp;one each; if you call the large divisions on thenbsp;scale A, five, four, or three each, then the sets ofnbsp;smaller divisions in a successive order from thenbsp;left hand division above mentioned, tow'ards thenbsp;right, will be one each.

2. nbsp;nbsp;nbsp;Five of the divisions on the fourth scale fromnbsp;the edge make an inch; if, thei�efore, you callnbsp;these divisions 3, 5, 7? 9? or 11 each, then eachnbsp;of the sets of small divisions, from the left handnbsp;towards the right, will be subdivided into parts ofnbsp;one each. Thus you have a scale divided intonbsp;2|, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, andnbsp;6o equal parts to an inch, exclusive of the scales

3. The line P on the slider stands for polygons;nbsp;JR, radius; the figures in this line being the radii

of the several polygons under which they stand, the side being 180 toises in each. If the side ofnbsp;your polygon be different from 180 toises, say, asnbsp;180 toises are to the radius under the number ofnbsp;sides your polygon contains, so is the side of yournbsp;polygon to its radius. Ex. What is the radius ofnbsp;an octagon, the exterior sides being 120 toises?nbsp;as 180 : 235 ' 18nbsp;nbsp;nbsp;nbsp;120 : 150 ' 78 Ans. The

remaining lines are to be read thus�The English foot is to the French foot as 107 is to 114, or as lnbsp;is to 1.005, a toise six French feet, a fathom sixnbsp;English feet. The English foot is to the Rhyn-land foot as 0715 is to 10000; the llhynland footnbsp;is to the French foot as 1033 is to 1008; the Rhyn-land rod is 12 Rhynland feet.

It may not be amiss to remark, that either the scale or the slider will erect a perpendicular instantaneously, for the long divisions across thenbsp;scale make, right angles with the edges.

This article is generally included in the magazine case of instruments for the military of-ffeer, or engineer; is a very useful mathematical instrument in the artillery service, and, as its description was omitted by our late author, I havenbsp;taken the opportunity of inserting some account ofnbsp;it here.

The principal uses of this instrument are to take the diameters of common shot, the bore or calibernbsp;of a piece of ordnance, estimate the weight of shot,nbsp;quantity of powder, amp;c. for guns of given dimensions, and other particulars in practical gunnery.

Fig. 6 and 7? are the representations of the two faces of the callipers marked A, B, C, D. Theynbsp;consist of two thin flat brass rulers moving on a

joint, curved internally to admit the convex figure of a ball, whose diameter is to be taken by the pointsnbsp;at the end A; these points are of steel, to preventnbsp;much wear. The rulers are made from six to 12nbsp;inches in lcnp:th from the center, accordinp- to thenbsp;number of lines and tables to be engraved uponnbsp;them. The usual length for pocket cases is sixnbsp;inches, the scales upon which I shall now describe.

On Fig. 6, ruler A, is contained, l. A scale of inches, divided into tenths, and continued to 12nbsp;inches on the ruler B.

2. nbsp;nbsp;nbsp;A table shewing the quantity of powder necessary for charging the chambers of brass mortarsnbsp;and howitzers.

3. nbsp;nbsp;nbsp;On the ruler B, is a line marked Inches, beginning from the steel point, for giving the diameters of the ealibers of guns in inches.

4. nbsp;nbsp;nbsp;A line marked Guns, contiguous to the preceding, shewing the nominal pounders, or weightnbsp;of shot for the respective bores of the guns innbsp;inches.,

5. nbsp;nbsp;nbsp;On the semicircular head of the rule is anbsp;semicircle divided into degrees, figured in contrarynbsp;directions, to measure angles by, and give the elevation of cannon, amp;c.

6. nbsp;nbsp;nbsp;Next to the preceding, is a circular scalenbsp;marked Shot Diameters, which, with the chamfered edge marked Index, shews the convex diameter in inches of a shot, or other object placednbsp;between the points. The quadrant part of thenbsp;joint prevents this being represented in thenbsp;figure.

7. nbsp;nbsp;nbsp;on the ruler C, a table marked Brass Guns,nbsp;shewing the quantity of powder necessary for tfit^nbsp;proof and service charges of brass guns.

y. Two circular scales on the head of the ruler, marked Shot, shewing the weights of iron shot, asnbsp;taken by the points of the callipers.

10. nbsp;nbsp;nbsp;A table shewing the quantity of powdernbsp;necessary for proof and service chtnges of ironnbsp;guns, from h to 42 pounders.

11. nbsp;nbsp;nbsp;To these are sometimes added various figures of a circle, cube, amp;c. with numbers.

12. nbsp;nbsp;nbsp;A table of the weights and�pecific gravities of a cubic foot of various metals, ivory, wood,nbsp;waters, amp;c.

1. nbsp;nbsp;nbsp;The line of inches, graduated contiguousnbsp;to the exterior edges of the sides A and B,nbsp;Jig. 6, when opened to a straight line, makes anbsp;measure of 12 inches and tenths for the purposenbsp;of a common rule.

2. nbsp;nbsp;nbsp;The table shewing the quantity of powdernbsp;necessary for mortars and howitzers. This tablenbsp;is adapted for both sea and land. Thus, by inspection merely is shewn that a 13 inch brassnbsp;mortar at sea I'cquires 30 pounds of powder; bynbsp;land only 10 pounds. Mortars and howitzersnbsp;under 10 inches have only the land quantitiesnbsp;inserted.

3. nbsp;nbsp;nbsp;The line of inches for concave diameters.nbsp;It commences from the steel point at 2 inches, andnbsp;continues on to 10 inches, and is subdivided tonbsp;halves and quarters. When used, the legs arcnbsp;placed across each other, and the steel pointsnbsp;brought to a contact with the internal concavenbsp;�grfacc of the gun, at the diainctrica!, or greatest

possible distance; the callipers then being taken out, and inspected where in the scale one external edge of the rule is upon the other scale, thatnbsp;division will give in inches and quarters the caliber of the gun required.

4. nbsp;nbsp;nbsp;The line shewing the weight of shot for thenbsp;bores. Adjoining to the preceding line is the onenbsp;marked Guns, proceeding from 11 to 42 pounds.

It is therefore evident, that at the same time the caliber in inches is given, the weight of the shotnbsp;is also given by inspection on the scale, by thenbsp;side of ihe ruler crossing both; and, from thenbsp;weight of the shot, the caliber in inches. Thusnbsp;4? inches shews a g-inch shot, and vice versa.

5. nbsp;nbsp;nbsp;The semicircular graduation on the joint-he-ad. The degrees are figured to 180 contrary-ways, so as one set to be the eomplement of 180 tonbsp;the other; and they are used to lay down or measure angles, find the elevation of cannon, amp;c.

First. To lay down any angle, suppose 30 degrees. Open the legs till the chamfered edges cut 30,nbsp;and two lines drawn against the outside edgesnbsp;will be at the inclination, or angle of 30 degrees,nbsp;tending to a proper center. If a center point benbsp;desired, cross the legs till the other chamferednbsp;edge cuts 30; the outside edges will then be atnbsp;the angle of 30 degrees, and the plaee of their intersection the angular point.

Secondly. To measure an entering or internal angle. So apply the outward edges of the rulers, that they may exactly coincide with the legs ornbsp;sides of the given angle. The chamfered edgenbsp;on the semicircle on the outward set of figuresnbsp;will point out the degrees contained in the givennbsp;angle.

Thirdly. To measure a salient, or external angle. Place the legs of the callipers over each other.

and make their outside edges coineide with the legs or sides of the angle to be measured; thennbsp;will the chamfered edge cut the degree requirednbsp;on the inner semicircle, equal to the angle measured.

Fourthly. To determine the elevation of a cannon or mortar. A rod or stick must be placed intonbsp;the bore of a cannon, so as to project beyond itsnbsp;mouth. To the outward end of this stick, anbsp;string with a plummet must be suspended; thennbsp;with the legs of the callipers extended, so as tonbsp;touch both the string and rod, the chamferednbsp;edge will cut the degrees equal to the complement of 90� of the true angle of the elevation ofnbsp;the cannon, figured on the outer semicircle. Sonbsp;that�if the edge had cut 30 degrees, the truenbsp;elevation would have been 6o� the complementnbsp;to 90.

6. nbsp;nbsp;nbsp;The circular line of inches on the head nextnbsp;to the preceding, marked Shot Diani. This scalenbsp;extends from O to 10 inches, and is subdividednbsp;into quarters. The chamfered edge marked Inches,nbsp;is the index to the divisions.

To measure hy this scale the convex diameter of shot, amp;c. Place the shot between the steel pointsnbsp;of the callipers, so as to shew the greatest possiblenbsp;extent; the chamfered edge, or index, will thennbsp;cut the division shewing the diameter of the shotnbsp;in inches and quarters.

N. B. The diameter given by this scale gives the shot rather less than by the scale No. 4, onnbsp;account of a customary allowance of what is callednbsp;�Vmdage.

7. nbsp;nbsp;nbsp;The table marked Brass Guns, to shew thenbsp;quantity of powder necessary for proof and servicenbsp;charges of brass guns. This table is. arranged foi*nbsp;the size of the piece, called either Heavy, Middle,

or Lights into their rcs])cctivc columns. Ex. SUj)-pose it is required to know the quantity of powder for a proof and service charge of a licavy 4*2 pounder. Under 42 arc 31 lb. 8 oz. for the proofnbsp;charge, and 21 lb. for the service charge.

8. The line of lines, marked Lin. This is line of equal jiarts, and for various occasions maynbsp;be so used, brnr other problems, being the samenbsp;as usually laid down upon the sector, I must refer the reader to the description of that instrumentnbsp;already given in this work.

g. Two circular scales on the joint head for giving tl\c weight of iron shot, marked Slioi..nbsp;These scales arc engraved upon the chamferednbsp;part oftlic head of the callipers; they arc, in fact,nbsp;but one continued scale, only separated for thenbsp;advantage of being more conspicuous. Theynbsp;shew the weights of iron shot from I to 42 pounds.nbsp;The diameter of the shot is to lie taken by thenbsp;points of the callipers, and then by an index engraved on the ruler is pointed out the propernbsp;weight.

10. nbsp;nbsp;nbsp;The table shewing the quantity of powdernbsp;necessary for proof and service charges of ironnbsp;guns, marked Iron Guns. An inspection of thenbsp;figure, or the callipers themselves, shew the tablenbsp;formed into three columns. The first, markednbsp;Iron Guns, shewing the weight of the shot; thenbsp;second. Proof, shewing the quantity of powdernbsp;for the proof charge; the third, Service, the quantity requisite for a service charge; and all thesenbsp;adjusted from \ to 42 pounders.

11. nbsp;nbsp;nbsp;The mathematical figures. The most vacant part of the callipers are sometimes filled upnbsp;�by six matlicmatical figures, with numbers annexed to each to assist the learner�s memory, andnbsp;indicating as follows.

1. nbsp;nbsp;nbsp;A circle with two diametrical lines, aboutnbsp;which are the numbers 7- 22,, and 113. 355; theynbsp;both arc the proportion of the diameter to the circumference, the latter being nearer the truth thannbsp;the former. The following examples may benbsp;therefore readily worked by them.

2. nbsp;nbsp;nbsp;A circle with an inscribed and circumscribednbsp;square, and inscribed circle to this smaller square.nbsp;To this figure arc annexed the numbers 28. 22.nbsp;14. 11. denoting that the larger square is 28, thenbsp;inscribed circle is 22. The area inscribed at thenbsp;square in that circle 14, and the area of thenbsp;smaller inscribed circle 11.

Both the proportion of the squares and circles are in the proportion of 2 to 1; and from them thenbsp;area of any circle may be found, having its diameter given. For example.

3. nbsp;nbsp;nbsp;Represents a cube inscribed in a sphere; thenbsp;affixed number SQt shews that a cube of iron inscribed in a sphere of 12 inches in diameter, weighsnbsp;89j pounds.

4, nbsp;nbsp;nbsp;Represents a sphere inscribed in a cube, withnbsp;the numbers 243 affixed; it is to shew the wT�ightnbsp;in pounds of an iron globe 12 inches in diameter,nbsp;or a globe inscribed in a cube, whose side is 12nbsp;inches.

5, nbsp;nbsp;nbsp;Represents a cylinder and cone, the diameter and height of which are one foot. To the cylinder is affixed the number 304,5, the weight innbsp;pounds of an iron cylinder of the above dimensions. The cone shewn by the number 121,5nbsp;that it is the weight of the same base and height,nbsp;and is onc-third of that of the cylinder. Conesnbsp;and cylinders of equal weight and bases are tonbsp;one another as 1 to 3.

6, nbsp;nbsp;nbsp;Represents an iron cube, whose side is 12nbsp;inches, to weigh 404,5 pounds. The figure of thenbsp;pyramid annexed to the cube, the base and heightnbsp;of wdiich, if each 12 inches, denotes the height tonbsp;be one-third of the cube�s weight, viz. 154f.

Remark. Globes to globes are as the cubes of their diameters; cubes to cubes, as their cubesnbsp;of the length of their sides; of similar bodies, theirnbsp;weights arc as their solidities. Hence the dimensions and weight of any body being given, thenbsp;weight or dimensions of any other similar bodynbsp;may be found.

The sixth figure shews that a cube of iron, whose sides are each 1 foot, weighs 404,5. Therefore,

As 1 (the cube of 1 foot) ; 404,5 ;; 8 (the cube of 2 feet) : 3716, the weight of an iron cube whosenbsp;sides are 2 feet.

2. nbsp;nbsp;nbsp;The diameter of an iron shot being six inches,nbsp;required the iveight.

By the fourth figure a one foot iron ball weighs 243 pounds, and 6 inches=,5 thenbsp;nbsp;nbsp;nbsp;of a foot.

Therefore, as 1 (the cube of 1 foot) : 243 (the weight) ,125 (cube of ,5 or iV ) : 30,375 pounds,nbsp;the weight required.

Another ride. Take i the cube of the diameter in inches,and 4 of that eighth, and their sum will benbsp;the weight required in pounds exactly. Or, thenbsp;weight of a four inch shot being nine pounds, thenbsp;proportions 64 ; 9 may be used with equal exactness.

3. nbsp;nbsp;nbsp;The veeight of an iron ball being given, to findnbsp;its diameter.

This rule is the converse of the preceding, the same numbers being used; as 243 (the weight) ; 1nbsp;(eube of 1 foot;) or, as Q pounds : 64 (cube of 4.)

Another rule. Multiply the weight by 7, and to the product add i of the weight, and the cubenbsp;root of the sum will be the diameter in inches.nbsp;Thus, an iron ball of 12 pounds weight will benbsp;found to be 4,403 inches. The methods of thisnbsp;rule are too evident to require worked examples.

4. nbsp;nbsp;nbsp;The following example will makeup a sufficient number, by which the learner may knownbsp;how to apply readily any number from the figuresnbsp;just desbribed.

A parcel of shot and cannon weighing fve tons, or 11200 pounds is to be melted, and cast into shot ofnbsp;three and five inches diameter, and weight of eachnbsp;sort to be the same-, required the number there willnbsp;be of each.

2d. Cube of 3 inches, or cub Then, as 1 (cube of 1 foot)nbsp;of a 1 foot ball) ::nbsp;nbsp;nbsp;nbsp;: 3,79, weight

Now 5 inches =:tV of a foot, its cube tVW. Therefore, as 1 : 243 ::nbsp;nbsp;nbsp;nbsp;: 17,67, weight of

A table of specific gravities and weights of bodies is added, or not, at the pleasure of the purchaser. It is not essential to the general usesnbsp;of the callipers, although many curious and usefulnbsp;problems relative to the weights and dimensionsnbsp;of bodies may be obtained from it in the most accurate manner.¹

The quantity of lines placed upon the callipers may be increased or arranged at pleasure. Thenbsp;following 19 w'ere the greatest number that I evernbsp;knew of being placed upon them.

4. nbsp;nbsp;nbsp;The weight of iron shot proper to given gunnbsp;bores. 5. The degrees of a semicircle. 6. Thenbsp;proportionofTroyandAverdupoiseW'eight. 7-Thenbsp;proportion of English and French feet and pounds.

9. nbsp;nbsp;nbsp;Tables of the specific gravity and weights 01nbsp;bodies. 10. Tables of the quantity of poivder necessary for proof and service of brass and iron gunS.nbsp;11. Rules for computing the number of shot 01nbsp;shells in a finished pile. 12. Rules concerning thenbsp;fall of heavy bodies. 13. Rules for the raising 01nbsp;water. 14. The rules for shooting with cannon

See Mr, Adams�s Lectures, five vols. 8vo. a new and proved edition of which is now in the press, and under nry correction and augmentation.

or mortars. 15. A line of inches. 16. Logarithmic scales of n Limbers, sines, versed sines, and tangents. 17. A sectoral line of equal parts, or the line of lines. 18. A sectoral line of plans or superficies. ig. A sectoral line of solids.

Fig. 8, plate 33, is a representation of a quadrant used for elevating a cannon, or mortar, in the most expeditious manner. The bar A, isnbsp;placed in at the mouth; the index B, brought to.nbsp;the are till the bubble of tbe spirit level settics innbsp;the middle. The angle is then read ofi' to minutes upon the are by the nonius at C.

Fig. g, plate 33, is a representation of a small level and perpendicular. It is used to find thenbsp;center line of a piece in the operation of pointingnbsp;it to an object, or to mark the point for a breechnbsp;hole, amp;c. The spirit level A, determines the position on the gun, and the spring index point B,nbsp;serves to mark the necessary points upon the surface to obtain the line by.

Are a set of brass rings, all connected to one center, with holes suitable to the diameters, ornbsp;pounders of iron shot, from four to 4�2 pounders;nbsp;being all respectively marked, and are too evidentnbsp;to need, a description here.

� 500, � 14, for By making, amp;c. read By making the base of the triangle and a perpendicular line to it, drawn from the opposite angle, in thenbsp;ratio of 4 to i.

A LIST

Of the principal Imtmmcnts descrihei in ibis IVorh, and tleir Prices, as made and sold by W. and S. Jones, Holborn, London.

Partial, or complete magazine cases of instruments, as represented in the plate, from ll. 18s. to �nbsp;PLATE II.

Parallel rulers, fg. A, B, C, D, and E, according to the length and mounting, from 2s, 6d. each, tonbsp;N. B. Oi figs. A, or B, or C, or D, or E, one of themnbsp;is included in the cases above, to order.

-4. Sectors, according to the length and materials they are made of, from 2s. 6d. to �

Bevel, amp;c. rulers, fgs. 4, 12, 15, 16, 17, and 13, various prices from 10s. 6d, to � � � � PLATE XI.

Brass stand and counterpoise for a ditto � 4 4 Second best ditto 8l. 8s. to � � � � 10 10

'[Mgt;iiw,Rmtedtiramp;PuhJishediyGffsgt;:^XAsmsN?(ioFketSlreet.asifieAct(lirecfy.Jumxi^ffi.

CATALOGUE

^n�tvmient�,

Best doubk-jolnted standard gold spectacles, with pebbles, and fish-skin gold-mounted case................ l6 l6 O

Totoishell spectacles, silver-jointed, with pointed and other shaped sides, peculiar for their lightness and uninterrup-

[ 2 j

Achromatic stick telescopes, of various lengths, from 18s. to 4 0 0 The new-improved ditto, with three sliding brass tubes, bynbsp;which an instantaneous view of the object is obtained,nbsp;and shuts up to a short length for the pocket, of one foot

The new-improved 2| feet achromatic refractor, on a brass stand, mahogany tube, with two sets of eye-glasses, onenbsp;magnifying about forty times for terrestrial objects, andnbsp;the other about seventy-five times for astronomical purposes, packed in a mahogany box................... g nbsp;nbsp;nbsp;gnbsp;nbsp;nbsp;nbsp;O

New-improved ditto, answering the purpose of an opera-nlass, with a compass, and helioscope for viewinc; the sun,

New-improved achromatic pocket telescope, which, by a small apparatus within its tubes, is readily converted into

\n improved portable seven-inch achromatic telescope in brass, with a stand that packs up into the tube of the

telescope, adapted for astronomical uses............ '3 nbsp;nbsp;nbsp;13nbsp;nbsp;nbsp;nbsp;6

Reflecting Telescopes, fitted up either upon the Grr-gor'uin, Ntnutonian, or Herscbdiari principles, with improved wood, or metal stands, and other apparatus lor making celestial observations in the most commodious andnbsp;accurate manner.�The general prices are as follow:

[3]

X�oose reflectors that are constructed upon the principles o/'.-K�ewton or Herschel are about twice the above lengths in the tubes. The reflectorsnbsp;upon the usual Gregorian construction arc made with the vertical motionnbsp;upon a new principle, so as to render them more firm and steady while innbsp;use, than any reflectors mounted in the old manner.

A four feet, seven inch aperture, Gregorian reflector, with the vertical motion upon a new invented principle, asnbsp;well as apparatus to render the tube more steady in observation; according to the additional apparatus of small fl. s. d.nbsp;speculums, eye-pieces, micrometers, amp;c. from 70I. to .. JOO 0 0nbsp;Three feet long, mounted on a brass stand, common mounting ........................................... 23 nbsp;nbsp;nbsp;2nbsp;nbsp;nbsp;nbsp;O

Telescopes, both refracting and reflecting,fitted up with equa; torial,amp;c. motions, micrometers, adjusting, compensating,

The new opake and transparent solar microscopes, with improved apparatus, from lOl. 10s. to................. 16 16 O

Ditto of a larger size, with additional megalascopic apparatus, from 141. 14s. to.......................... I9 19 Q

mahogany box..................................21 nbsp;nbsp;nbsp;Qnbsp;nbsp;nbsp;nbsp;O

The Lucernal Microscope, as improved by W. Joves, exhibiting images of opake and transparent objects bynbsp;night or day, in a manner singularly pleasing, brilliantnbsp;and distinct, with upwards of 100 objects, proper apparatus, patent lamp, amp;c........................... 16 16 O

A portable optical apparatus, consisting of a scioptic ball and socket, a solar microscope, Wilson�s microscope, anbsp;pocket compound microscope, a pocket telescope, and

A new set of moveable painted sliders, shewing the fundamental principles of astronomy, with the real and appa*

�.

For a descripthrt of this instrument, as axiell as of spectacles, reading-glasses, amp;'c, see the late Mr. G. Adams�s Essay on Vision, 8-yo. price $s. now sold by W. andnbsp;S. Jones.

Concave and convex glass mirrors, in plain black frames, four, five, six, and seven inches diameter, each ps. 123.

Concave mirrors, ground cylindrically, possessing several curious properties in the deformation of objects j according

A curious set of optical models, where the rays of light are represented by silken strings, and illustrating the principles of vision, telescopes, prisms, amp;c. packed in five cases 6 l6 6

A new very portable theodolite, by rack-work, measuring angles with equal accuracy as those of the common largenbsp;sort, is at the same time applicable for taking altitudes,

A 4-inch further improved ditto, by which the vertical and horizontal angles are shewn at the same time, with rack-work motions and portable parallel plate staves, amp;c.,.,. 10 10

�. s. d.

Circumferentors, much used in wood lands, from 2l. 2s. to 4 4 0 An improved ditto, contrived to answer the purposes of a

Improved ditto, with rack-work and pinion, and moveable divided limb, making a very portable cross-staff, compass,

Measuring tapes, one, two, three,and fourpoles,5s. 7s.6d. Qs. 0 10 6 Pedeometers for ascertaining distances in walking or riding,nbsp;of a watch size for the pocket, and also to apply to carriages, from 31.3s. to............................ 12 12 O

Sets of protracting and plotting scales; instruments fordivid-ing lines or transferring divisions on paper. An instrument for describing circles from four to six inches radius,

�Gunners levels or perpendiculars�Shot gauges�Shell ditto�Gunners quadrants, with a plummet or level, ornbsp;adjusting screw, amp;c. and all other instruments for mili-' tary poirposes.

Ebony and brass mounted best sextants, from 4l. 4s. to .... 8 18 � Metal ditto, all brass, framed on a principle the least liablenbsp;to be warped or strained, with adjusting screws, telescopes,nbsp;and other auxiliary apparatus, the most proper for takingnbsp;distances accurately, to determine the longitude at sea, amp;c. 12 12 Q

Horizontal sun-dials, in brass, made for any latitude, of four, five, or six inches diameter, divided into five minutes

Tor a general description and representation of the instruments used in surveyings level~ lin^ts nrid other branches of practical geometry, see the late Mr. G. Adams's Geometrical and Graplrical Essays, an improved edition iji \V. Jones, in two vols.nbsp;8w. 1797, with thirty-five folio copper-plates. Trice 14s.

A portaWe Transit Instrument, with a cast-iron stand, to ascertain the rate of chronometers, the longitude, amp;c.nbsp;the axis is twelve Inches in length, and the telescope about

Ditto, with a brass framed stand, and other additions .... 20 nbsp;nbsp;nbsp;0 O

and Copernican systems, from 7I. 7*- to.............50 nbsp;nbsp;nbsp;0nbsp;nbsp;nbsp;nbsp;0

Manualorreriesofthecommoncon3truction,2l. 12s.6d. to.. nbsp;nbsp;nbsp;6nbsp;nbsp;nbsp;nbsp;6 0

Tellurian and planetarium together, making the New Portable Orrery, packed in a neat mahogany box, according

An orrery shewing the motions of Alercury, Venus, theEarth and Moon, by wheel-work, theEarth is a Ijinch globe,

Other planetariums and orreries in great variety, the motion.? by wheel-work, exemplifying all the motions and phrenomena of all the planets, the Georgium Sidus included, tfom 401. to............................1000 nbsp;nbsp;nbsp;0nbsp;nbsp;nbsp;nbsp;0

Twelve inch ditto, improved by Ferguson, with the new discoveries of Capt. Cooke, .^c. and the horary circles,

The Terrestrial, containing all the latest discoveries and communications, from the most correct and authenticnbsp;observations and surveys to the year l/.OS, engraved fromnbsp;an accurate drawing by d/r. Arrowsmitb.�The Celestialnbsp;containing the po.sltions of nearly �000 stars, clusters,nbsp;nebnhe, planetary nebulae, amp;c. correctly computed andnbsp;laid down, by IF. Jones, for the year 1800, from thenbsp;latest observations and discoveries by Dr. Maskdyne, ,

N.B. These are the only modern English 18-inch Globes extant, the [.dates being engraved from entire new drawings, and are dedicated to the Bight Eton. Sir Josephnbsp;Banks, Bart. P. B. S. and the Bev. Dr. Maskelyne, Astronomer BtyaJ �,

[8]

a general description cf orreries and other astronomical instruments^ see the late Mr. G. Adams�s Astronomical Essays, ^vo, with sixteen plates^ price los. (gt;d.nbsp;sto w sold by W. and S. Jones.

With a great variety of receivers, and other apparatus, described by various authors.

�. s. d

Exhausting and condensing s3frlnges, from lOs. -6d. to nbsp;nbsp;nbsp;1 H �' 6,

Electrical machines and complete apparatus, for medical purposes, packed in boxes, the cylinder from seven to

An electrical cannon, to be discharged by inflammable air. . nbsp;nbsp;nbsp;0 16 O

Acurious collection of working moilels, to be set in motion by the electrical fluid, consisting of a corn-mill and a three-barrelled water-pump, worked by one crank only; an orrery,nbsp;shewing the diurnal motion of the earth, age and phas'es ofnbsp;the moon, 8ec. and astronomical clock, shewing the aspectsnbsp;of the sun and moon, age, phases, amp;:c. all delicately made ofnbsp;card paper, cork, and wire only, p.acked in a deal case ....nbsp;nbsp;nbsp;nbsp;212nbsp;nbsp;nbsp;nbsp;6

Kinnersley�s electrical air nbsp;nbsp;nbsp;thermometer................. I nbsp;nbsp;nbsp;1nbsp;nbsp;nbsp;nbsp;O

C' !� d.

A variety of other apparatus, too numerous to be inserted here, which, as well as the machines, are mounted from the most approved, eligiblenbsp;methods, so as to render them in action both powerful and jiermanent.

For rt cam-pleie df-icriptinn of electrical apparatus, see the late Mr. G. Adams�s Ess.iy on Electricity, �A-price-js. inboards,now soldby\i .and S. Jomes.

Barometers, plain mounted, from ll. lls. 6d. to....... 2 nbsp;nbsp;nbsp;12nbsp;nbsp;nbsp;nbsp;()

Thermometers for all the various purposes, from gs. to nbsp;nbsp;nbsp;.nbsp;nbsp;nbsp;nbsp;.nbsp;nbsp;nbsp;nbsp;. .nbsp;nbsp;nbsp;nbsp;3nbsp;nbsp;nbsp;nbsp;3nbsp;nbsp;nbsp;nbsp;O

and cold, in the absence of the observer, from ll. lls.6d. to 2 12 � An hygrometer, shewing the moisture and dryness of the air 0 10 .6nbsp;Barometers, thermometers, and hygrometers, all in one neat

A box of magnetical apparatus, illustrating a variety of cu-yions and entertaining properties in magnetism, consisting chiefly of the following articles: a set of six artificial barnbsp;magnets; two horse-shoe magnets; six small iron balls;nbsp;a magnetometer; two magnetical spinners; a small dipping needle; a gimbal compass; two brass magnet tablc.s;nbsp;an armed combined magnet; six magnetic needles, withnbsp;six pointed stands; and sundry' other illustrative and entertaining articles, all packed in a mahogany case 5l. 5s. to 7nbsp;nbsp;nbsp;nbsp;7 O

Dipping needles, variation amp; other compasses, in great variety. I�yrometers, shewing the expansion of metals, fronj 3l. 3s. to 10 10 O

[�]

The mechanical powers, for illustrating and demonstrating the laws of motion, gravity, amp;c. a set neatly made innbsp;brass, consisting of the balance, the puliies, the ditferentnbsp;kinds of levers, the inclined plane, the wheel and axle, thenbsp;screw, a compound engine, a compound lever, a doublenbsp;cone to move up an inclined plane, friction wheels,

A small carriage with inclined plane, and wheels of diftcrent sizes, amp;e. experimtiitallv proving the friction, resistance.

A whirling table, for explaining and demonstrating the laws of the planetary motion, the demonstrations of the doctrine of the tides, and other properties of gravity and centrifugal force, from 101. lOs. to..................... l6 l6

Ditto, ditto, with a variety of other necessar)'- apparatus, jiacked in a fish-skin case, forming Cronstedfs complete

Magellan�s new portable lamp furnace, with the blow-pipe, small glass retorts, amp;c. ike. for chemical as well as mine-

E 12}

/�. 5. d.-

phials with glass.stoppers, with Gottlmg�s printed description of ditto .................................... 4 8 O

A mahogany case, containing, in phials, a variety of prepa-tions for young persons to perform amusive and instructive

Instruments of Recreation and Amusement.

The sensitive fishes, that have the property of swimming to a piece of bread placed at the end of a stick ; and, whennbsp;the other end is presented, of retreating and going back,nbsp;sensible, as it were, of no substance for them to eat ....nbsp;nbsp;nbsp;nbsp;0nbsp;nbsp;nbsp;nbsp;6nbsp;nbsp;nbsp;nbsp;6

The sagacious swan, that with a machine makes three kinds of amusements�1st. the swan will point out the secrets ofnbsp;the cards; 2d. it will point answers to l6 humourous enigmas; and 3d. disclose any particular hour that was thought

A box containing four numbers and four letters, the order of which may be discovered, if ever so secretly placed,

Ditto with five numbers, no perspective, but another very similar box, made in neat mahogany boxes, and more

A magic painter, exhibiting a copy of any one of eight different paintings secretly chosen.................... 0 10 6

foursecretly chosen; an elegant and curious instrument .. nbsp;nbsp;nbsp;2 12nbsp;nbsp;nbsp;nbsp;6

A box containing five pieces of different metals, which may anyway be secretly placed, and their situation be told by

the magical perspective........................... 1 nbsp;nbsp;nbsp;8nbsp;nbsp;nbsp;nbsp;0

An optical paradox, containing two perspectives, between which a board may be placed, and the object will be seennbsp;through them just as well as if the board was not there,

An optical deception, containing from six to twelve ditl'erent paintings, and. which are looked down upon through anbsp;perspective, and immediately there appears another verynbsp;different object, without any alteration of the instru-' ment whatsoever,.or concern of the person using it, from

ll. 11s. 6d. to................................ 3 nbsp;nbsp;nbsp;3nbsp;nbsp;nbsp;nbsp;O

A diagonal opera glass, that shew.? persons on one side, when the glass is presented to the object directly before

A mathematical recreation, containing near seventy figures on a card ; any one figure being thought of, is readilynbsp;pointed o.ut by any one usingnbsp;nbsp;nbsp;nbsp;it , ...... 0 nbsp;nbsp;nbsp;1nbsp;nbsp;nbsp;nbsp;0

[ 13 ]

The two curious mathematical cubes, one of which is gauged � � so as to prove it to be larger than the other, yet the largernbsp;one will actually pass through the smaller one, and not in

The mathematical paradox, a piece of wood of one figure, fits, exactly, and passes through a triangular, a square,

A mechanical instrument, consisting of a cube and two wooden handles, that supports itself on a point, althoughnbsp;the entire form and weight appear evidently all on one side 0 12 Onbsp;A cylindrical mirror that produces two or three curious

optical effects................................... 1 nbsp;nbsp;nbsp;1nbsp;nbsp;nbsp;nbsp;O

A magic or electrical botile, that is charged by the rubbing of a ribband only, and will give a shock to five or six persons, with apparatus, in a pocket case.............. 0 10 6'

A set of the artificial fire-works imitated, containing a series of brilliant and entertaining scenes of fire-works, cascadesnbsp;of fire, amp;c. producing altogether a pleasing effect, and notnbsp;attended with any trouble, noise, or danger, when using;

Besides the preceding, a great variety of other articles too numerous to be included in this catalogue, as well as any instrumental articlenbsp;made from particular drawings, or as described by the different writersnbsp;upon mathematics, philosophy, chemistry, amp;c. amp;c.

Merchants, shopkeepers, schoolmasters, and others that sell again, are supiplled with the best articles, and with good allowance.

Letters from the country or abroad, containing orders or previous inquiries, explicitly and piunctually attended to.

Les acad�mies, ebser-vatoires et ecoles des pays ctrajigcrs ainsi que Ics ncgociants, mirebands et aiitres fersennes peu-vent se procurer toutes seriesnbsp;dinstrunents de la meilleure qualii�, tant pour les ma�rtaux, qstc la mainnbsp;i oeuvre, avec la plus grande expedition, et ati plus juste prix.

A Description and Use of the New Portable Orrery, to wfiich is prefixed a short account of the so|ar system, includingnbsp;the new planet, the burning mountain ip the moonnbsp;lately discovered by Dr. Herschel, and the probable reasons why the comet did not appear, as lately expected,nbsp;with two copper-plates, 4th edition ................. 0 1 Q

r 14]

�, s. d.

A Description and Use of the Hadley's Quadrant, with an account of all the new apparatus added to it, for takingnbsp;observations accurately, in order to determine the longitude at sea, illustrated by copuer-plate figures, id edit.. . O 1 Onbsp;A Description and Use of the Fockel-C-'ase of Mathematical

Cowley�s lllu.stration of Sol.iD Grometry, contaiulng 42 copper-plates of moveable figures; a work very usefulnbsp;and conveiiiont for teachers and young students of geometry, as the figures, when folded up, form exactly thenbsp;solid figures of the Platonic bodies, conic sections, andnbsp;Several portions of Euclid�s Elements, amp;c. amp;c. board.s . . O IS fi

'Die New Encyclopocdia Britanriica (printed at Elt;linburgh) a new edition, quarto, in 13 vols. or 3� parts, now

This Dictionary, of Arts and.Sciences is upon a newand enlarged plan, and contains the systems of the different arts and sciences, under thenbsp;diff erent heads, as well as the explanations of the various detachednbsp;terms.

Hutton�s (Dr.) Mathematical Dictionary, 2 vols. boards .. nbsp;nbsp;nbsp;2 12 O

The Philosophical 'I�ransactions of the Iloyal Society, containing 11 vols. of the Abridgement; and from thence, the Continuation at large to the present time; the Index,nbsp;with Birch�s and Sprat's history, 5 vob, all in uniform

W. and S. Jokte s take this Opportunity of informing the Public that they have purchased the Stock and Copyright of the several Philosophical Essays by the latenbsp;Mr. George Ad a^is, deceased, of Fleet Street-, and thatnbsp;they are novo sold at their Shop in Holborn. The follovjingnbsp;are those novo mprint, and to be had as above.

I. nbsp;nbsp;nbsp;AN essay on electricity, explaining^clearly and fulljnbsp;the Principles of that useful Science, describing the various Instrumentsnbsp;that have been contrived, either to illustrate the 'llieory, or render thenbsp;Practice of it entertaining. The difi'er�nt Modes in which the Electri-lt;'al Fluid may be apjdled to the human F'rame for medical Purposes,nbsp;are distinctly and clearly pointed out, and the nece.s.sary Apparatusnbsp;explained. To which is now added, A Letter to the Author,nbsp;from Mr. John Birch, Surgeon, on the Subject of Medic.vlnbsp;Elf.ctuici'Cy. Tcurli Edit'wn, 8vo. Price /s- in boards, illustratednbsp;with six Plates.

II. nbsp;nbsp;nbsp;AN ESSAY ON VISION, briefly explaining the Fabric ofnbsp;the Eye, and the Nature of Vision; intended for the Service of thosenbsp;whose E}'es are weak and impaired, enabling them to form an accuratenbsp;Idea of the State of their Sight, the Means of preserving it, togethernbsp;with proper Rules for ascertaining when Spectacles arc necessary, andnbsp;how to choose them without injuring the Sight. 8vo. Second Edition.nbsp;Illustrated with Figures. Price 3s. in Boards.

containing, 1. A full and comprehensive View, on a new Plan, of the. general Prlncijiles of Astronomy, with a large Account of the Discoveries of Dr. Herschel, kc. 'I. The U.se of the Celestial and Terrestrialnbsp;Globes, exemplified in a greater Variety of Problems than are to benbsp;found in any other Work: they are arranged under distinct Heads,nbsp;and interspersed with much curious but relative Information. 3. Thenbsp;De.scription and U.se of Orreries and Planetaria, amp;c.nbsp;nbsp;nbsp;nbsp;-i. An Intro

duction to Practical Astronomy, by a Set of easy and entertaining Problenus. Third Edition, 8vo. Price 10s. 6d. in Boirds, illustratednbsp;with sixteen Plates.

IV. nbsp;nbsp;nbsp;AN INTRODUCl'ION TO PRACTICAL ASTRONOMY,nbsp;or the Use of the Quadrant and Equatorial, being extracted from thenbsp;preceding Work. Sewed, with two Plates, 2s. O'J.

V. nbsp;nbsp;nbsp;GEOMETRICAL AND GRAPFIICAL ESSAYS. Thisnbsp;Work contains, 1, A .select Set of Geometrical Problems, many ofnbsp;which are new, and not contained in any other Work. 2. The Description and Use of tho.se Mathematical Instruments that are usually putnbsp;into a Case of Drawing IiRstniments, Besides these, there are alsonbsp;described several New and Useful Instruments for Geometrical Purposes.nbsp;3._ A complete and concise System of Surveying, with an Accountnbsp;of some very essential Improvement.s in that useful Art. To whichnbsp;is, added,, a Description of the mo.st improved Theodolites, Plane.

[2l

Tables^ and other Instruments used in Surveying; and most accurate Methods of adjusting them. 4. The Methods of Levelling, for thenbsp;Purpose of conveying Water from one Place to another; with anbsp;Description of the most improved Spirit Levels. 5. A Course ofnbsp;Practical Military Geometry, as taught at the Royal Academy,nbsp;Woolwich. 6. A short Essay on Perspective. The Second Edition,nbsp;corrected, and enlarged with the Descriptions of several Instrumentsnbsp;unnoticed in the former Edition, By W. Jones, Mathematical Instrument Maker; illustrated by thirty-live Copper-plates, in 2 vols.nbsp;8vo. Price 14s. in Boards.

VI. AN APPENDIX to the GEOMETRICAL AND GRAPHICAL ESSAYS, containing the following Table by Mr. John Gale, viz. a Table of the Northings, Southings, Eastings, and Westings, to every Degree and fifteenth Minute of the Quadrant, Radiusnbsp;from 1 to 100, with all the intermediate Numbers, computed to thenbsp;three Places of Decimals. Price 2s.

ESSAYS ON THE MICROSCOPE,

BY THE LATE AUTHOK, 111 Quarto, with thirty-four folio Plates, .separate. The Second Edition, with manynbsp;Corrections, Additions, and Improvements, by

LECTURES

In Five Volumes Svo. The Second JEdition, with upwards of Forty large Plates, considerable Alterations and Improvements; containingnbsp;more complete Explanations of the Instruments, Machines, amp;c. andnbsp;the Description of many others not inserted in the former Edition.

Degrees.	Nat. Tangent.	Nat. Secant.
76	4,011	4,133
77	4,331	4,445
78	4,701	4,810
79	5,144	5,241
60	5,671	5,759
81	6,314	6,392
82	7,115	7,185
83	8,144	8,205
84	9,514	9,567
85	11,430	11,474
86	14,301	14,335
87	19,081	19,107
88	28,636	28,654
69	57,290	57,300

No. sides	Names.	Areas.
3	Triangle	0,433
4	Square	1,
5	Pentagon	1,7*2
6	Elexagon	2,598 3,634
7	Heptagon	2,598 3,634
8	Octagon	4,828
9	Nonagon	6,182
10	Decagon	7,694
11	Undecagon	9,365
12	Uuodecagon	11,196

UEMAllKS.	Oti- sets.	Station lines.	Ofl'- sets.	REMARKS.
	1st. At the corner.
	against Win. Humphrey's, and H. Der-
	man�s land.
		N. 7� W.
Dernian�s land	0 0	0 0
Dernian�s land	0 40	3 nbsp;nbsp;nbsp;6'0
	0 10	8 45
	0 65	15 nbsp;nbsp;nbsp;60
A corner	0 O	21 00
Higgin�s land		2d. N.55�15.E.
Higgin�s land	0 0	0 0	0 0
		6 10	0 �0	To the bound-
A corner	0 0	18 20	0 0	ary of the held.
		3d.
Horne�s land		S. 62� 30. E.
Horne�s land	0 0	O 0	0 0
		7 40	0 go	'I'o the bound-
A corner	0 0	14 40	0 0	ary of the field.
		4th.
'� 'Vard�s land		S. 40� W.
	0 0	0 O
10 do		8 00
3A a corner	0 gj	11 00
		5th.
aBove corner � Ward�s land		S. 4� 15. E.
	0 68	0 66
	1 25	7 50
A corner	0 0	14 OO
'^luniphrey�sland	j�th. N.7:H45.N.
'^luniphrey�sland	0 0	0 0	0 0
		3 20	0 30	To the bound-
		7 50	1 25	ary of the field.
		12 nbsp;nbsp;nbsp;40	0 o	A corner.

Areas of the Otisets.
To be added. A. 0,63687	I'o be subtracted.
	0.540(X) p.֒lStX)

0.36350 I.O8875

	0.68750 0.37510


2.08812	2.15660

Mill.	Angle.		Angle.		Div.
1	4,-5�	00'	45�	00'	1
�2	63	26	26	34	2
3	71	34	18	26	3
4	75	58	14	02	4
5	78	41	11	1.0	5
6	so	32	9	28	6
8	82	52	7	08	8
10	84	17	5	43	10

Paces	Feet	Inc.	r.	Feet	Inc.	P,	Feet	Inc.		Feet Inc.		p.	-^ Feet Inc.
I	2	6	16	40	0	31	77	6	46	�5	0	61	�52	6
2	5	0	f?	4^	6	32	80	0	47	II7	�	62	�55	0
3	7	6	18	45		33	82	6	48	�20	0	63	�57	6
4	10	0	19	47	6 !	34	85	0	49	�22	6	(14	160	0
5	12	6	20	so	0 i	35	87	6	;5o	I2S	0	6s	r6e	6
6	15	0	21	52	6:	56	90	0	�51	J27	6	66f 165		0
7	gt;7	6	ZZ	SS	0	37	92	6	5�	130	0	67	167	6
s	20	0	^3	57	6	3�	95	0	'53	132	6	68	170	0
9	22	6	'14	60	0	39	97	6	44	135	0	69	172	6
10	^5	0		62	6	40	100	0	is 5	137	6	70	175	0
1 r	27	6	26	65	0	41	102	6	.59	140	0	71	177	6
IZ	30	0	^7	67	6	42	105	0	�57	�42	6	72	180	0
13	32	6	.8	70	0	43	107	6	�58	14s	0	73	182	6
'4	35	0	.9	72	6	44	no	0	.59	147	6	74	185	0
15	37	6	30	75	0	45	112	6	60	150	0	75	187	6

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ir V

GRAPHICAL ESSAYS,

A GENERAL DESCRIPTION

GEORGE ADAMS,

THE SECOND EDITION,

CHARLES, DUKE of RICHMOND, AND LENNOX,

THESE ESSJYS

PREFACE.

advertisement

THE EDITOR.

iry;.

LIST

OF AUTHORS,

WORKS

In the Press,

MICROSCOPE,

PHILOSOPHY,

CONTENTS.

Psgf

GEOMETRICAL

ESSAY

�3, the double barred ditto.

E.xatnple 2. To take off the nuniher 76,4. Observe the points where the sixth horizontal cuts

twentieths.

parts.

39697

38795

cerned.

required.

*quot;lrcle.

�creaftcr.

NO , �^ NO , .

4. nbsp;nbsp;nbsp;4

AHb.

and EF.

L A,

I B = nbsp;nbsp;nbsp;10,9918

Jinks.

r 175 ]

GEOMETRICAL PROBLEMS.

SURVEYING.

C 19s 1

ground.1

DESCRIPTION, USE, and method of adjustingnbsp;HADLEY�S QUADRANT.^

t269]

DESCRIPTION, USE, and method of adjustingnbsp;HADLEY�S SEXTANT.

[ 279 ]

VA

FIELD BOOK.

FIELD BOOK.

�VV

book.

1^878400

PLOTTING,

iB

link

same.

and the roods and perches will be determined by ^multiplying the decimal parts by 4 and by 40.

Example. Suppose the side of the foregoing �f]uare be 6 chains and 75 links; Avhat is the area?

, Suppose the length of the foregoing oblong be 9 chains and 45 links, and the breadth 5 chains andnbsp;links; how many acres doth it contain?

A. R. P.

�^ucTS, and the sum of the south products, �w// double the area of the survey.*

d\

. nbsp;nbsp;nbsp;7 W.

[ 306 ]

MARITIME SURVEYING.

'd ' willnbsp;:alo

LEVELLING.

[ 397 ]

PRACTICAL GEOMETRY^

AB.

CBD-fBDC.

[ 467 3

ESSAY ON PERSPECTIVE;

FINIS.

ground.¹