Printed for T. Heath and J Nourse in the Strand; J. H o d g e s on London-Bridge, andnbsp;J Fuller in Ave-mary-Lane. M bcc lvh.

ERRATA.

Page xvi. read March 5,1757- p-10. Line 16. for Plate iii read Plate iv. p. i2. 1. 15, for KF r. KC. p. 26. 1. 26. fornbsp;into r. in. p. 40.!. 19. for Scale r. Scales, p. 63. 1. 6. delenbsp;K. 1. 29. for B3r. B5. p. 80. 1. 9. r. As the divifor, is tonbsp;unity; fo is the dividend, to the quotient. And as the divifor, is to the dividend j fois unity, to the quotient, p. 8r.nbsp;1. 19. for 25 r. 35. p. 95 1.23. for xvii r. xviii. p. 96.nbsp;1. 12. read Ex. i. PI. vi. Fig. 26. p. too. 1. 4. for N. r. x.nbsp;1. 20. forN. r. X. p. loi. 1. 34. for N r. x. 1. 35. for Mnbsp;r. z. 1.36. forNr. X. p. 107. 1. i. forxviii r. xix. p. 109.nbsp;1. 17. for on B read on P. p. no. In the computation fornbsp;the letter C read D. p. 125. I. 24. for xix read xx.

Line 16 for � read^. p. 145.1. 19. for 4. r. p.149.1.24. for balks r. bulks, p. 155.1. 19 for i. 8. r. i. o.

PETER DAVALL, Efq;

IT is no new thing for a lover of Science to addrefs his produd:ions to a friend eminentlynbsp;diftinguilhed for his general knowledge, asnbsp;well as particular fkill in the parts whereon thenbsp;Author writes : On this account I heartily wilh,nbsp;that inflead of the fubjects contained in the following fheets, I had a work of a more elevatednbsp;kind wherewith to do greater honour to the namenbsp;of my friend ; however, fuch as they are, I hopenbsp;they will, with your ufual franknefs and goodnature, be accepted. Indeed I muft obferve,nbsp;that the late Prefident of the Royal Society, Martin Folkes, Efq; honoured the firft Editionnbsp;of this book with his Patronage ; and alfo, ournbsp;much-efteemed and learned friend James Burrow, Efq; Vice-prefident of the Royal Society,nbsp;thought the book fo worthy his perufal, as to remark all the typographical and other errors, andnbsp;to make fome ufeful obfervations, a lift whereofnbsp;he favoured me with, and for which I truft younbsp;will permit me to take this opportunity of pub-lickly thanking him : Although I am confcious,

DEDICATION, that you have the higheft regard for the two re-fpedtable names, which I here mention out ofnbsp;gratitude; yet I would not be underflood thatnbsp;you are to accept hereof in this public manner,nbsp;merely becaufe thofe confiderable perfonages havenbsp;already favoured the Work ; I offer this as a tribute for your acquaintance and friendfhip, andnbsp;flatter myfelf that you will find in this impreflionnbsp;fome things, which if they have not difficulty tonbsp;recom.mend them, have at leaft, I apprehend, fonbsp;much utility accompanying them, as to rendernbsp;the whole in fome degree interefting, and perhaps not unworthy the notice of the mofl fkilfulnbsp;in the Mathematical Sciences. I am.

REA

T is needlefs to enumerate the many pur* ^ I ^ pofes, to which mathematical inftrumentsnbsp;ierve their ufe feems quite neceffary to fer-fons employed in moft of the adive ftationsnbsp;in life.

The Architehl, whether civile military, or naval, never offers to effed any undertaking, before he hasnbsp;firft made ufe of his rule and compajfes ; and fixed uponnbsp;a fcheme or drawing, which unavoidably requiresnbsp;thofe inftruments, and others equally neceffary.

The Engineer, cannot well attempt to put in execution any defign, whether for defence, oftence, ornament, pkafure, amp;c. without firft laying before his view, thenbsp;plan of the whole ; which is not to be convenientlynbsp;performed, but by rulers, compajfes, amp;c.

There are indeed, very few, if any good Artificers, who have not in fome meafure, occafion for the ufenbsp;of one or more mathematical inftruments; and whenever there is required, an accurate drawing of a thingnbsp;to be executed, or reprefented -, that colledion of inftruments, ufually put in portable cafes, is then abfo-lutely neceffary: And of thefe, the moft common ones,nbsp;or others applicable to like fervice, muft have beennbsp;in ufe, ever fince mankind have had occafion to provide for the neceffary conveniencies of life : But thenbsp;parallel ruler, the proportional compajfes, and the feSor,nbsp;are not of any great antiquity.

However, by means of the opportunity, which the author had of confulting moft, if not all the principalnbsp;A 2nbsp;nbsp;nbsp;nbsp;pieces,

( iv )

pieces, that have been wrote on this fubjefb f � he thinks it will fufficiently appear from what follows,nbsp;who were the inventors of thefe latter inftruments jnbsp;and when they were firft known and made ufe of.

I. Gafpar Mordente^ in his book on the compa^eSy printed in folio at Antwerp, 1584 �, gives the conftruc-tion and ulc of an inftrument, invented by his brothernbsp;Fabricius Mordente, in 1554; and by him prefentednbsp;to the emperor Maximilian II. in 1572 : Fabricius prefented it afterwards, with fome improvements, to Ro-dolphusll- thefon of Maximilian: In 1578, Gafparnbsp;ftudied to apply the inftrument to various ules by thenbsp;command of the then governor of the Netherlands.nbsp;The inftrument confifts of two flat legs, moveablenbsp;round a joint like a common pair of compaffes ; butnbsp;the ends or points are turned down at right angles tonbsp;the legs, fo as to meet in one point when the legs arenbsp;clofed. In each leg there is a groove, with a Aidernbsp;fitted to it, carrying a perpendicular point 5 fo thatnbsp;thefe alfo appear like one point when the legs arenbsp;clofed, and the Aiders are oppofite. This compafs isnbsp;jointly ufed with a rod, containing a fcale of equal parts;nbsp;whereof 30 are equal to the length of each leg. Asnbsp;the operations with this compals, depend on the properties of fimilar triangles, therefore its principles arcnbsp;the fame with thofe of the fedtor : And mioft, or allnbsp;the problems that are performed by the line of linesnbsp;only, can with almuft the fame eafe, be performed bynbsp;thefe; the tranfition from this inftrument to the fedlornbsp;is very natural and ealy.

The ufe of this inftrument, is exemplified in problems concerning lines, fuperficies, folids, and mea-furingof inacceffible diftances.

The author, p. 22, fays, he invented an inftrument there deferibed -, which is our parallel ruler with parallel bars : The parallel ruler with crofs bars, is anbsp;more modern contrivance.

( V )

II. Daniel Speckle, in the year 15S9, publiflied innbsp;folio, his mditary architeHlure, at Strajhiirg \ where henbsp;was architedl. In his fecond chapter, he takes notice of compafies then in ufe of a curious invention,nbsp;whcfe center could be moved forwards or backwards,nbsp;fo that by the figures and divifions mark�d thereon,nbsp;a tight line could be readily and correftly divided intonbsp;aoy number of equal parts, not exceeding 20. Thisnbsp;inllrument has been fince called the proportional C07n-pajjes.

In the fame chapter he mendons another compafles, with an immoveable center, and broad legs, whereonnbsp;were drawn lines proceeding from the center, andnbsp;divided into equal parts; whereby a right line couldnbsp;be divided into equal parts not exceeding 20 ; becaufenbsp;the divifions on the lines ftill kept the fame proportion, to whatever diftance the legs were opened.nbsp;This inftrument was afterwards call�d the fePior.

III. nbsp;nbsp;nbsp;Dr. Thomas Hood, printed at London, Annonbsp;1598, a quarto book, intituled, The making and ufenbsp;of a Geometrical Inftrument called a Sedior. This inftrument confifts of two flat legs, moveable about anbsp;joint; on thefe are fe�loral lines, of equal parts, ofnbsp;polygons, and of fuperficies that is, lines fo dif-pofed, as to make ail the operations that depend onnbsp;limilar triangles quite eafy, and that without the laying down of any figure. To the legs is fitted a circular arc, an index moveable on a joint, and fights,nbsp;whereby it is made fit to take angles.

IV. nbsp;nbsp;nbsp;Chriftopher Clavius, in his pratlical geometry,nbsp;printed in quarto at Rome, Anno 1604, in page 4,nbsp;fhews the conftruftion and ufe of an inftrument,nbsp;which he calls the inftrument of parts �, it confifts ofnbsp;two flat rulers moveable on a joint; on one fide ofnbsp;thefe legs, are the feftoral lines of equal parts-, onnbsp;the other fide, are thole of the chords : After fire wingnbsp;fome of their ufes, he concludes with faying, he is 'nbsp;fenfible of many others to which it may be applied.

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V. nbsp;nbsp;nbsp;Levinus Huljius, in his book of mechanical in-Jlruments, printed in quarto at Frankfort^ ^nno 1605nbsp;gives, in the third part, the defeription and ufe of annbsp;inftrument, which JuJius Burgius call�d iht proportionalnbsp;Compafs., Hulfius fays, the ufe of it had not beennbsp;pubiilhed before, although the inhrument had beennbsp;long known,

VI. nbsp;nbsp;nbsp;Anno 1605, PFHp Horfeher, M. D. pubiilhed at Mentz, a quarto book, containing the ulenbsp;and conftrudtion of the proportional compajjes. Thisnbsp;author does not pretend to be the inventor; but thatnbsp;feeing fuch an inftrument, he thought he could,nbsp;from Euclid, fhew its conftrudtion and the groundsnbsp;of its operations'.

VII. nbsp;nbsp;nbsp;Anno 1606, Galilieus pubiilhed in Italian, anbsp;treatife of the ufe of an inftrument which he calls,nbsp;Fbe geometrical and military compafs. On this inftrument are deferibed feftoral lines of equal parts, fur-faces, folids, metals, inferibed polygons, polygons ofnbsp;given areas, and fegments of circles. In the prefacenbsp;to an edition of this book, printed at Padua, Annonbsp;1640, by Paola Frambotti, Galila^us fays, that onnbsp;account of the opportunity he had of teaching mathematics at Padua, he thought it proper to feek outnbsp;a method of Ihortening thole ftudies. In anothernbsp;part of the preface he fays, that he Ihould not havenbsp;pubiilhed this traft, but in vindication of his own reputation ; for he was informed that a perfon had bynbsp;fome means or other, got one of his inftruments,nbsp;and pretended to be the inventor, although himfelfnbsp;had taught it ever fince the year /59/.

VIII. Anno 1607, Baldejfar Capra, publiftied anbsp;treatife of the conftrudtion and ufe of the compajs ofnbsp;proportion, (or fedlor.) He claims the invention ofnbsp;this inftrument; and hence arofe a difpute betweennbsp;QaliUus and Capra; forne particulars of which have

( vii )

been mentioned by feveral, and particularly by �Thomas Salujbury, Efq; in his life of Galilaus, publifhed atnbsp;the end of the fecond volume of Yin mathematical collections and tranjlations^ at London, in fol. Anno 1664.

It appears from thefe accounts that one Simon Marius a German who was in Padua about the year 1607, tranflated into latin, the book publifhed the year before by GaliUus, and caufed his difciple Capra to printnbsp;it as his own: Marius dreading a profecution, retired,nbsp;and left Capra in the lurch, who was proceeded a-gainft. At that time GaliLeus publifhed an apology,nbsp;intitled, � The defence of Galilaeus Galilasi, a Florentine gentleman, reader in the univerfity of Padoua,nbsp;againji the calumnies and impojiures of BaldefTar Capranbsp;a Milanefe, divulged againji him as well in his conjide-ratione ajtronomica upon the new fiar of 1604, as (andnbsp;more notorioufly) in lately publifhing for his invention thenbsp;conflruCiion and ufes of the geometrical and military com-pafs, under the title of Ufus amp; Fahrica circini cu-jufdam proportionis, ifc.� Galilaus begins with annbsp;addrefs to the reader, wherein he concludes, that a per-fon robbed of his inventions, fuffers the greateft lofsnbsp;that can be fuftained, becaufe it defpoileth him ofnbsp;honour, fame and deferved glory:quot; He proceeds, andnbsp;fays, � into this ultimate of miferies and unhappinefs ofnbsp;condition, BaldefTar Capra, a Milanefe, with unheardnbsp;of fraud, and unparalleled impudence hath endeavoured tonbsp;reduce me, by lately publifhing, and committing to thenbsp;prefs my geometrical and military compafs, as his propernbsp;invention, and as a production of his own wit, (forfonbsp;he calls it in the work itfelf) when it was 1 alone, thatnbsp;ten years fince {viz. Anno 1597) thought of, found andnbsp;compleated the fame, fo as that no one elfe hath any fh arenbsp;in it-, and I alone from that time forward imparted, dif~nbsp;covered and prefented it unto many great princes, and othernbsp;noble lords; and in fine, only that I a year fince caufednbsp;the operations thereof to be printed, and confecrated to thenbsp;glorious name of the mojl ferene prince of Tufeany, my

( v�i )

lord. Of which /aid injirument the above-named Capra^ hath not only made himfelf the author, but reports menbsp;for its fhamelefs ufurper, {thefe are his very words) andnbsp;confeqiiently bound to blujh within my felf with etctrearnnbsp;confufton, as unworthy to appear in fight of learned andnbsp;ingenuous men.� Galilceus then proceeds, among othernbsp;things, ro produce the atteftations of four confider-able perfons, Ihewing that ten years before that time,nbsp;he had taught the ufe of the inftrument, and thatnbsp;Capra who had for four years paft feen them makingnbsp;at the workman�s houfe, had never challenged thenbsp;invention, as his own.

Galilreus after this, fays that he went to Venice, and laid the affair before the lords reformers of the uni-verfity of Padoua, on the 8th of rlpril 1607,nbsp;fame time {hewing them his own book, publilhednbsp;June the loth i6c6; and that of Capra\, publilhednbsp;March the 7th 1607. The lords thereupon citednbsp;C�/r� to appear before them on the 18th of April-,nbsp;the next day the caufe was heard and the parties dif-milTed : But on the 4th of May following, their lord-fhips pronounced fentence, and fent it to Padoua tonbsp;be put in execution ; the amount of their fentencenbsp;was, that having fully confidered the affair, it appeared to them that GaliUus had been abufed, andnbsp;that all the remaining copies of Capra's book fhouldnbsp;be � brought before their lordfhips to he fuppreffed innbsp;fuch fafhton as they fhall think fit, refervPg to ihemfelvesnbsp;to proceed againji the printer and bookfeller, fcr thenbsp;tranfgreffions they may have committed againji the lawsnbsp;of printing; ordering the fame to he made known accordingly.

'The fame day all the copies of the faid book were fent to Venice unto the lords reformers �, there being foundnbsp;440 ith the hooljeller, and 13 with the author, henbsp;having dijiributed 10 of them into ftindry parts of Eu-npe, amp;c.�

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IX. nbsp;nbsp;nbsp;Anno i6io, John Remmelin, M. D. publifliednbsp;at Frankfort, a quarto edition of two tradis of Johnnbsp;Faiilhaber *, one of thefe contains the ufe of the feiior,nbsp;on which are lines of equal parts, fuperficies, folids,nbsp;metals, chords, l^c. He fays, that G- Brendel, anbsp;painter, ufed this inftrument in perfpedlive painting.

X. nbsp;nbsp;nbsp;Z). Henrion, in his mathematical memoirs. Annonbsp;i6i2, gave a fhort tradl of the ufe of the compafs ofnbsp;proportion (or feSior.) In 1616 he printed a book ofnbsp;the ufe of the fedlor �, and a fifth edition, in the yearnbsp;1637, the preface to which, fecms to be wrote innbsp;the year 1626, wherein he fays, that about the yearnbsp;1608, he had feen in the hands of M. Alleaume, engineer to the king of France, one of thefe fedtors;nbsp;whereupon he wrote fome ufes of it, which he pub-lilhed in his memoirs, as above. He alfo declares,nbsp;that before his firft publication, he had not feen anynbsp;book on the ufe of a fedlor, and therefore calls whatnbsp;he publilhes his own. He charges Mr. Gunter withnbsp;having ufed many of his propofuions. This authornbsp;printed at Paris 1626, an odlavo book of logarithms,nbsp;at the end of which is a tradl call�d logocanon, or thenbsp;proportional ruler ; which is a defcription and ufe ofnbsp;an inftrument, he calls a lattice, (perhaps from thenbsp;chequer-work made by lines drawn thereon) whichnbsp;operates the problems performed by the french fedlorsnbsp;very accurately.

XL Anno 1615, Stephen Michael-Spackers, pub-lifhed in quarto at Uhn, a treatife of the proportional rule and compafs of G. Galgemeyer, revifed by G.nbsp;Brendel, a painter at Laugingen. On thefe proportional compaffes, are lines of equal parts, of polygons, fuperficies, folids, ratio of the diameter to thenbsp;circumterence �, rcdudlion of planes, and redudlion ofnbsp;folids. The ufe and conftrudlion of thefe lines, arcnbsp;flrewn by a great variety of examples.

XII. Benjamin Bramer, in his book of the defcription of the prcppcrtional ruler and parallelogram, printed

in quarto at Marpurg, Anno 1617; fays, his ruler is applicable to the fame ufes as Jujlus Burgius'% in-ftrument. Bramerh inftrument confifts of a ruler,nbsp;on which are lines of equal parts, of fuperficies, ofnbsp;folids, of regular folids, of circles, of chords, andnbsp;of equal polygons �, at the beginning of each fcale,nbsp;is a pin-hole, whereby he can apply the edge of another ruler, and fo conftitute a fe�for for each fcale.

XIII. nbsp;nbsp;nbsp;Anno 1623, Adriano Metio Almariano, printed at Amfterdam a quarto book, fhewing the ufc ofnbsp;an inftrument called the ru/e of proportion. In hisnbsp;dedication, he fays, that whilft he was reviewingnbsp;fome things relating to praftical geometry, he metnbsp;�with Galileo's book of the ufe of the fcdor, whichnbsp;gave him opportunity to improve on it, and occa-fioned the publifhing of this book.

XIV. nbsp;nbsp;nbsp;Mr. Edmund Gunter, profeflbr of aftronomynbsp;in Grejham college, printed at London, Anno 1624, anbsp;quarto book, called the defcription and ufe of the fec-tov, on which are federal lines, jft. of equal parts jnbsp;2d. fuperficies; 3d. folids �, 4th. fines and chords;nbsp;5th. tangents 6th. rhumbs ; 7th. fecants: Alfo lateral lines of, 8th. quadratures; 9th, fegmentsnbsp;loth, inferibed bodies; nth. equated bodies-, 12th.nbsp;metals ; On the edges are a line of inches and a linenbsp;of tangents.

Mr. Gunter does not fay any thing concerning the invention, and has no preface but at the end of thenbsp;trad, in a conclufion to the reader, he fays, that thenbsp;fedor was thus contrived, meft part of the booknbsp;written, and many copies dilperfed, more than fix-teen years before, this article being written Maynbsp;1, 1623, brings the time he fpeaks of to about thenbsp;year 1607, which was before the time Henrion fnysnbsp;he firft faw the fedor.

The fcales of logarithm numbers, fines, and tangents, were firft publifired in J 624, in Gunter s de-icription of the crofs ftaff.

( xi )

1633, a quarto book, called the conJiruBion �nd ufe of the compaje polimetro, (or feftor.) The lines on this in-ftrument, were fuch as were common at that time: Henbsp;fays in the dedication to his friend Peter Linder of Nu-renherg, he firft taught the ufe of it

In the preface he fays, that about the year 1568, Comraandine, who then taught at Urbino, did contrive a pair of compaffes with a moveable centre,nbsp;to divide right lines into equal parts; which was donenbsp;at the requeft of a gentleman named Bartholomewnbsp;Euftachio, who wifhed to avoid the trouble of thenbsp;common methods, or of being obliged to have manynbsp;compaffes for fuch divifions of right lines.

He farther fays, that about that time, Guidibaldoy tnarquefs of Monte^ who lived at XJrbino for the fakenbsp;of Commandineh company, being frequently at thenbsp;houfe of Simone Boraccio, who made Commandine�snbsp;proportional compaffes, did contrive, and caufe to benbsp;made, an inftrument with flat legs, (like the fedtor)nbsp;which performed the operations of the compafs morenbsp;eafily. Oddi fays alfo, that great numbers were made,nbsp;and in few years, had many ufeful and curious additions, with treatifes written on its ufe in diverfe languages, and called by different names, which oc-cafioned the doubt of who was the true author, everynbsp;one having found means to fupport his caufe; Butnbsp;Oddi fays, he not intending to decide the difpute,nbsp;leaves it to time to difcover and feems contentednbsp;to have pointed out who was the firft inventor; hisnbsp;chief intention being that of making the ufe public,nbsp;and the conflrudlion eafy to workmen.

XVI. nbsp;nbsp;nbsp;Anno 1634, P. Petity printed in 8vo. atnbsp;Paris, a treatife on the fedfor. He thinks GaliUusnbsp;was the inventor.

XVII. nbsp;nbsp;nbsp;An. 1635, Matthias Berneggertis �pnnx.tA. ztnbsp;Strajhurg a 4to. edition of Galilaus�^ book on thenbsp;fedtor, which confifts of two parts: To this is added

( xii )

a third part, fhewing the conftrudion of GaliUus*i lines, and fome additional ufes and tables.

XVIII. An. 16 Nicholas For eft Duchefne at Pern, in i2mo. a book of the feftor. He feemsnbsp;to be little more than a copier of Henrion.

XIX. nbsp;nbsp;nbsp;An. 1645, Bettinus in his Apiaria tmiverfa.,nbsp;amp;c. apiar. 3d. p. 95, and apiar. 12, p. 4. In hisnbsp;Mrariwn philo. math. 4to. an. 1648, vol. I. p. 262.nbsp;In his Rccreationum math. appiarilt;�. See. i2mo. an.nbsp;1658, p. 75, applies the fedor to mufic.

XX. nbsp;nbsp;nbsp;John Chatfield printed at London, in 12mo.nbsp;his trigonal ftdlor, anno 1650.

XXI. nbsp;nbsp;nbsp;An. 1656, Nicholas Goldman ypxmvtd 2.x. Leyden, in folio, his treatife on the fe��or. He fays thatnbsp;Galilaus was the firft who publilhed the deferiptionnbsp;of the feftor, an invention ufeful in all parts ol thenbsp;mathematics, and other affairs of life.

XXII. nbsp;nbsp;nbsp;John Collins printed at London, in 4to. hisnbsp;book of the feclor on a quadrant, an. 1659.

XXIII. Pietro Ruggiero, inFx's military architecture, in 4to. printed 2.xl^lan, an. 1661, p. 230, appliesnbsp;the fedor to the pradice of fortification.

XXIV. nbsp;nbsp;nbsp;An. 1662, Gafpar Schottus printed atnbsp;Strajburgh his mathefts c.ffartea, in 4to. in which henbsp;gives a defeription and ufe of the fedor ; In the preface he mentions GaliUo as the inventor of the fedor.

XXV. nbsp;nbsp;nbsp;J. Templar printed in 12mo. at London,nbsp;an. i66j, 3. '000k CdiWed the femicircle on a feclor. Henbsp;fays, the applying of Mr. Fcrftcr�s line of verfednbsp;fines to the fedor, was firft publifiied an. 1660, bynbsp;John Brosvn, mathematical inftrument maker innbsp;London.

XXVI. nbsp;nbsp;nbsp;Daniel Schwenter in his practical geometry,nbsp;reviled and augmented by George Andrevo Bocklein,nbsp;printed in 410. at Nuremberg, an. 1667, treats onnbsp;the defeription and ufe of the fedor.

XXVIl. John Caramuel printed at Campania, an. 1670, his mathefts nova, in 2 vols-, folio. In the zd

( xiii )

Vol. p. 1158, he treats on the fedtor, relates the con-teft between Galilaus and Capra, and thinks the fame might have been objefted againft others, as well asnbsp;againft Capra: He alfb fays, that Clavius had fuchnbsp;an inftrument before that of Galilaus appeared; andnbsp;Clavius having taught lor a long time at Rome, hadnbsp;many fcholars, fome of whom might have carriednbsp;his inftruments to feveral countries. Caramuel mentions a ftory of a Hollander {hewing to Galilaus annbsp;inftrument of this fort, that he had brought from hisnbsp;country, and of which Galilaus took a copy.

XXVIII. John Brown, in his book on the triangular quadrant, printed in 8vo. at Z.oW(?K, an. 1671.

XXIX. nbsp;nbsp;nbsp;John Chijiopher Rohlhans, in his math,nbsp;and optical curiofities, printed in 410. at Leipfic, an.nbsp;1677, P- 216.

XXX. nbsp;nbsp;nbsp;An. 1683, Stanijlawa Soljkiego printed atnbsp;Kracow, his geometria et architedlura Poljki, in folio,nbsp;p. 69, treats on fome fectoral lines.

XXXI. nbsp;nbsp;nbsp;Henrick JaJper Nuis, printed at Tezwolktnbsp;in 4to. his ReSlanguliim catholicum geometrico ajlrono-micum, an. 1686.

XXXII. Be Chales, in his curfus mathem. printed at Leyden, in 2 vo!s. fol. an. i6go. Vol. 2d. p. 58,nbsp;relates the conteft between GaliUus and Capra, andnbsp;afcribes the invention of the proportional compafs tonbsp;Dr. Horfcher, or Jtijius Biirgius.

XXXIII. An. tboi, an edition in 8vo. of Mr. Ozanands treatife of the fedtor, was printed at thenbsp;Hague.

XXXIV. P. Hojle printed at Paris his courfc of mathematics, in 3 vols. 8vo. an. 1692. In vol. 2d.nbsp;p- 27. he gives a tradf on the fedlor.

XXXV. Thomas Allingham in his Jhort treatife on the feSicr, in 4to. London, 1698.

2 XXXVI r.

8 VO. printed at Leipjic, an. 1716, under the word circinus proportionum, relates, that Levinus Hulfius,nbsp;in his treatife on the proportional compaflTes, printednbsp;at Frankfort the 10th of May, 1603, fays, that henbsp;firft faw the faid inftrument at Ratijbon, on the daynbsp;of the imperial dyet: That he had fold them far andnbsp;near before 1603 ; and that it had been inaccuratelynbsp;copied in feveral places: Wolfius fays farther, thatnbsp;Jttfius Burgius was certainly the inventor, but ufed tonbsp;let his inventions lye unpublifhed.

He then relates the conteft between GaliUns and Capra, and ends with fhewing the difference betweennbsp;the inftruments of Burgius and Galilaus.

XXXVIII. M. Bion, in his conftruftion of mathematical inftruments, tranflated by Edmund Stone, fol. London, 1723-

XL. Roger Rea, in his fe5lor and plane fcale combed, 8VO. London, 1727, 2d edition.

XLI. Vincent Tofco, in his compendium of the math. in 9 vols. 8VO. Madrid, 1727, vol. I. p. 359,

XLII. Jacob Leupold, in his theatrum arithmetico-geometricum, in fol. Leipfic, 1727. p. 86, gives a detail of the inventors of the proportional compaffes and fe�lor, which goes on to p. 121, and then henbsp;gives a lift of the authors who have wrote on proportional inftruments, %iz. Bramer, 1617; Capra, 1607;nbsp;Cafati, 1664-, Conette, 1626-, Bechales, 1690; Bolz,nbsp;1618; Faulbaber, i6io; Galgemeyer, 1615; Bren~nbsp;dell, 1611 : Galilaus, 1612-, Goldman, 1656; Horf-cher, 1605; Horen, 1605; Hulfius, 1604-, Claxius,nbsp;Lockmann, 1626-, Metius, 1623; Patridge�;nbsp;deSaxonica, 1619; Scheffelts, 16975 Steymann, 16245nbsp;Uttenhoffers, 1626.

XLIV.

XLIV. William Webjler, in his appendix to tranfiation of P, Hoji�s mathematics, 8vo. 2 vols,nbsp;London, fj^o.

There may be feveral other authors who have wrote on the conftrudtion and ufe of the fedtor, or on fomcnbsp;of the fedloral lines; but thofe above, arc all thatnbsp;have come to hand ; and indeed thefe are many morenbsp;than are wanted to determine this enquiry ; whichnbsp;may be colledted chiefly, from Mordente, Speckle, Hood,nbsp;Clavius, Hulftus, Galiheus, Oddi, Sallufbury, Caramuel,nbsp;Dechales, Wolfius, and Leupold-, the others fervingnbsp;only to inform the reader what works are extant onnbsp;this fubjedt. From the whole he may obferve, thatnbsp;there are few countries in Europe, but have one ornbsp;more treatifes on the proportional compafles andnbsp;fedlor, in their own language; and this is fufficientnbsp;to fliew, that thefe inftruments have been in univerfainbsp;efteem.

As the publication of Mordente�s, book was in 1584, it is not improbable, as Caramuel relates, that a Hollander (or one from the neighbourhood of Antwerp')nbsp;might fhevv one of Mordente�s, inftruments to Gdi~nbsp;licus : Neither is it improbable that Galilaus had feennbsp;both Mordents�amp; and Speckle�s books, the former having been publilhed thirteen years, and the latternbsp;eight years, before GaliUus, by his own accounts,nbsp;thought of his inftrument.

As Mutio Oddi, was a native of Urhino, and from what he fays in his dedication, it is not improbablenbsp;but he was acquainted with one or more of the per-fons he mentions in his preface, or at leaft with fomcnbsp;of their acquaintance, from whom he might gathernbsp;the particulars he relates; to which, if any creditnbsp;may be given, Commandine was the inventor of thenbsp;proportional compaflTes, and Guidobaldo of the fe�tor:nbsp;And in the intercourfe between Italy and Germany^nbsp;fome of Simone Borachio�s work might get into thenbsp;hands of many ingenious Germans, and give Jufius

( )

Burgius, to whom the proportional compafs is ufually afcribed, opportunity of getting an early copy andnbsp;alfo put into Speckle�s way, the inftrument he mentions to have feen: His defcription pretty nearly agreeing with what Oddi fays was contrived by Guidobaldo.

But while we are fearching among foreigners for the inventor of the feftor, what are we to think of ournbsp;countryman Dr. Hood? who in 159^ publifhed hisnbsp;account of an inftrument which he really calls a fec-tor: And though we fhould allow that Hood as wellnbsp;as Galilaus might have leen Mordente�s and Speckle�snbsp;books; and both of them might have feen fome ofnbsp;Borrachio's work, yet it is not very probable thatnbsp;Hood could have got the form of his inftrument fromnbsp;GaliUus the year after he thought of it; and as Hoodnbsp;publifhed eight years before GaliUus, Hood certainlynbsp;has an equal right with Galilieus-, if not a greater, tonbsp;the honour of the invention of the feflor.

After all, it may be faid, that it is not impofilblc for the fame thing to be difeovered by different per-fons who have no connexion with one another; examples of a like coincidence of thoughts being knownnbsp;on other fubjefts.

To the prefent edition, there is added an appendix on the gunners callipers, which was promifed to thenbsp;public in the former impreffion, publifhed at the beginning of the year 1747 ; and befide this, the bodynbsp;of the book has been augmented by more than threenbsp;Iheets of additional illuftrations and problems, andnbsp;another plate : By all thefe additions, it is conceivednbsp;the book is now rendered more generally ufeful.

What is done in the foregoing eflay, and in the following work, is fubmitted to the reader�s judgment; the author intending no more than to have thenbsp;honour of invention afcribed to whom it is due; andnbsp;alfo to give fome aftiftance to beginners in the mathematical ftudies.

CON-

CONTENTS.

Seftion.

ConfiruSiion of the other lines of the pla-ne fcale, viz. I ft. Chords-, ?.d. Rhumbs-, 3d. Sin�s-, 4Th.nbsp;Tangents-, 5ch. Secants-, 6th. Half Tangents-,nbsp;7th. Longitude-, 8th. Latitude-, ^t\\. Hours-,nbsp;1 oth, Inclination of Meridians.nbsp;nbsp;nbsp;nbsp;2 3

a nbsp;nbsp;nbsp;VIII.

txviii CONTENTS.

X, nbsp;nbsp;nbsp;Of the (onflruSiion of the ftngle fcales on the feBor. 33nbsp;XL Of the confiruSiion of the double fcoles on the feBor.

Three tables, fhewing the altitudes and projeBions, of every moulding and part in the pedeftals, columns, and entablatures of each orderaccording to the proportions given by Palladio.

CONTENTS.

AP-

appendix.

Of the callipers, and what they contain. 132 Art.nbsp;nbsp;nbsp;nbsp;I.nbsp;nbsp;nbsp;nbsp;Of themeafures of convex diameters.nbsp;nbsp;nbsp;nbsp;134

The plates are all to ftand upright in the book, and no part to be folded upwards or downwards.

CASE,

PORTABLE COLLECTION,

Mathematical Inftruments.

A S E S of Mathematical Inflruments are lgt;v C m various forts and fizes ; and are com-jjW monly adapted to the fancy or occafion ofnbsp;the perfons who buy them.

I. A fair of compaffes, one of whofe points may be taken off, and its place fupplied with,

With thefe inftruments only a tolerable Ihift may be made to draw moft mathematical figures.

But in fets, called complete pocket-cafes, befide the inftruments above, are the following.

In fome cafes, the plane fcale, protradlor, and parallel-ruler, are included in one inftrument.

The common, and moft efieemed fizeof thefe inftruments, is fix inches -, though they are fometimes made of other fizes, and particularly of four inchesnbsp;and a half.

Some artifts have contrived a very commodious flat cafe, or box, where the infide of the lid or topnbsp;contains the rulers and fcales: The compafies, drawing-pen, fy�c. lie in the partitions of a drawer, thatnbsp;drops into the bottom part of the cafe, but not quitenbsp;to the bottom �, leaving room under it for black leadnbsp;pencils, hair pencils, Indian ink, colour cells, amp;c. andnbsp;befide the inftruments already enumerated, in boxesnbsp;or cafes of this fort are put

But the cafe of inftruments called the magazine. Is the moft complete colledtion ; for this contains whatever can be of ufe in the pradlice of drawing, defign-mg, amp;c. and as the greateft part of thefe inftruments

are

of Mathematical Inftruments. nbsp;nbsp;nbsp;^

are fcarcely ever ufed but in the ftudies or chambers of thofc who have occafion for them ; therefore itnbsp;will be ufelefs to infift on pocket cafes; for fewnbsp;perfons care to load themfelves with the carriage ofnbsp;what is called a complete fet.

Of the Compasses attd Bows.

COMPASSES arc ufually made of filver or brafs, and thofc are reckoned the beft, part ofnbsp;whofe joint is fteel; and where the pin or axle onnbsp;which the joint turns, is a fteel ferew �, for the oppofi-tion of the metals makes them wear more equable :nbsp;and by means of the ferew axle, with the help of anbsp;turn-ferew^ (which ftiould have a place in the cafe) thenbsp;compaffes can be made to move in the joint, ftiffer ornbsp;eafier, at pleafure. If this motion is not uniformlynbsp;fmooth, it renders the inftrument lefs accurate in ufe.nbsp;Their points flrould be of fteel, and pretty wellnbsp;hardened, elfe in taking meafures off the fcales, theynbsp;will bend, or be foon blunted. They alfo Ihould benbsp;well polifhed, whereby they will be preferved freenbsp;from ruft a long time.

To one point of the fmallercompafles, it is common to fix in the ftrank a fpring, which by means of anbsp;ferew, moves the point �, fo that when the compafs isnbsp;opened nearly to a required diftance, by the help ofnbsp;the ferew the points may be fet exadly to that diftance v which cannot be done fo well by the motion innbsp;the joint.

Hold the compaffes in the left hand w'ith the ferew turned towards the right turn the ferew towards you,

or flackcn it, and the fpring point will be brought nearer to the other point: On the contrary, by turning the fcrew from you, or tightning it, the fpringnbsp;point will be fet farther from the other point.

The ufe of thefe lelTcr compaffes, is to transfer the mcafures of diltances from one place to another �, or,nbsp;to defcribe obfcure arcs.

Of the large fizcd compafies, thofe are eftecmed the beft, whole moveable points are locked in by anbsp;fpring and catch fixed in the llrank ; for if this fpringnbsp;be well effedted, the point is thereby kept tight andnbsp;fteady ; the contrary of which frequently happens,nbsp;when the point is kept in by a fcrew in the fhank.

The ufe of thefe compafies is to defcribe arcs or circumferences with given radius�s; and it is eafy tonbsp;conceive, that thefe arcs or circumferences can benbsp;defcribed, either obfcurely by the fteel point; in ink,nbsp;by the ink point *, in black-lead or chalks, by thenbsp;crayon; and with dots, by the dotting-wheel; fornbsp;cither of them may be fixed in the fiiank in the placenbsp;of the fteel point.

As the dotting-wheel has not hitherto been effected, fo as to defcribe dotted lines or arcs, with any tolerable degree of accuracy, it feems therefore to benbsp;ufelefs: and, indeed, dotted lines of any kind arcnbsp;much better made by the drawing-pen.

The drawing-pen point, and crayon, have generally (in the bell fort of cafes) a locket fitted to them: fo that they occupy but one of the holes, or partitions, in the cafe.

The ink, and crayon points, have a joint in them, [uft under that part which locks into the fhank of thenbsp;compaffes; becaufe the part below the joint fhouldnbsp;Hand perpendicular to the plane on which the linesnbsp;are defcribed. when the compafs is opened.

If inftead of the larger cornpafs being made with fhifting points, there were two pair put into thenbsp;cafe} to one of which the ink point was fixed, and to

the other the crayon point; this would fave the trouble of changing the points in the compafs at every time they were ufed ; and would increafe the expence,nbsp;or bulk of the cafe, but a trifle.

Most perfons at firft, handle a pair of compafles very aukwardly, whether in the taking of diftancesnbsp;between the points, or deferibing of circles. To benbsp;furc long pra�lice brings on eafy habits in the ufc ofnbsp;things, however a caution or two may be ferviceablcnbsp;to beginners.

With the thumb and rriiddle finger of the right hand pinch the compafles in the hollow part of thenbsp;fhank, and it will open a little way ; then the thirdnbsp;finger being applied to the infide of the ncareft leg,nbsp;and the nail of the middle finger a6ling againft thenbsp;fartheflr, will open the compalTcs far enough to introduce the fingers between the legs: then the hithernbsp;one being held by the thumb and third finger, thenbsp;farther leg may be moved forwards and backwardsnbsp;very eafily by the fore and middle fingers, the forenbsp;finger preffing on the outfide to Ihut, and the middlenbsp;one a61ing on the infide to open, the compalTes to anynbsp;defired extent. In this manner the compalTes arenbsp;manageable with one hand, which is convenient whennbsp;the other hand is holding a ruler or other inftrument.

Hold the compalTes upright, fet one point on one end of the diftance to be taken, there let it reft ; andnbsp;'^as before fhewn) extend the other point to the othernbsp;end.

Always take care to avoid working the compafles with both hands at once; and never ufe them other-wife than nearly upright.

Set one foot of the compafles on the point defigned for the centre, hold the head between the thumb andnbsp;middle finger, and let the fore finger reft on thenbsp;head, but not to prefs it: then by rolling the headnbsp;between the finger and thumb, and at the fame timenbsp;touching the paper with the other point, a circle ornbsp;arc may be defcribed with great eafe, either in lead ornbsp;ink.

In defcribing of arcs it ftiould be obferved, that the paper be not prefl'ed at the centre, or under thenbsp;foot, with more weight than that of the compafles �,nbsp;for thereby the great holes and blots may be avoided,nbsp;which too frequently deface figures when they arenbsp;made by thofe who arc aukward or carelefs in the ufenbsp;of their inftruments.

The bows arc a fmall fort of compafTes, that com� monly fhut into a hoop, which ferves as a handle tonbsp;them. Their ufe is to defcribe arcs, or the circumferences of circles, whofe radius�s are very fmall, andnbsp;could not be done near fo well by larger compafles.

Sect. III.

TH E Black-lead Pencil is ufefu! to defcribe the firft draught of a drawing, before it is markednbsp;with ink becaufe any falfe ftrokes, or fuperfluous

The Feeder is a thin flat piece of metal, and is fome-times fixed to a cap that flips on the top of the pencil, and fcrves either to put ink between the blades of thenbsp;drawing-pen, or to pafs it between the points, whennbsp;the ink by drying, does not flow freely.

The �Tracing Point is a pointed piece of fteel; and commonly has the feeder fixed to the other end ofnbsp;the handle. Its ufe, is to mark out the outlines of anbsp;drawing or print when an exalt;5l copy thereof is wanted, which may be done as follows.

On a piece of paper, large enough to cover the thing to be copied, let there be flrewn the ferapingsnbsp;of red chalky or of black chalky or of black lead; rubnbsp;thefe on the paper, fo that it be uniformly covered ;nbsp;and wipe off, with a piece of muflin, as much as v^illnbsp;come away with gentle rubbing. Lay the colourednbsp;fide of this paper, next to the vellum, paper, onnbsp;which the drawing is to be made ; on the back ofnbsp;the colour�d paper, lay the drawing, ��r. to be copied.nbsp;Secure all the corners with weights, or pins, that thenbsp;papers may not flip : trace the lines ot the thing to benbsp;copied, with the tracing point; and the lines fo tracednbsp;will be imprefs�d on the clean paper.

And thus, with care, may a drawing or print, be copied without being much damaged.

There is not perhaps, a more ufeful inftrument in being for ready fervice in making of Iketches or finifla-cd plans; whether of architefture, fortification, machines, landlkips, ornaments, i�c- than a black-leadnbsp;pencil; and therefore it may be proper to give a fewnbsp;hints

Black-lead is produced in many countries, but the bell yet difeovered is found in the north of England :nbsp;it is dug out of the ground in lumps, and fawed out

into fcantlings proper for ufe; the kinds moft proper to ufe on paper muft be of an uniform texture, whichnbsp;is difcoverable by paring a piece to a point with anbsp;penknife; for if it cuts fmooth and free from hardnbsp;flinty particles, and will bear a fine point, it may benbsp;pronounced good.

There are three forts of good black-lead ; the foft, the midling, and the hard ; the foft is fitted: fornbsp;taking of rough fketches, the midling for drawing ofnbsp;landflcip and ornaments, and the hard for drawing ofnbsp;lines in m.athematical figures, fortification, architecture, fife. The indifferent kinds, or thofe which innbsp;cutting are found flinty, are ufeful enough to carpenters or fuch artificers who draw lines on wood, amp;c.

The befl: way of fitting black-lead for ufe, is firft to fiuv it into long flips about the fize of a crow-quill,nbsp;and then fix it in a cafe of foft wood, generally cedar,nbsp;of about the fize of a goofe-quill, or.larger; and thisnbsp;cafe is cut away with the lead as it is ufed.

Sect. IV.

TI'I E Brawing-pen is an inftrument ufed only for drawing of right lines �, and confifls of twonbsp;blades, with fteel points, fix�d to a handle. The bladesnbsp;by being a little bent, caufe the fteel points to comenbsp;nearly together; but by means of a ferew pafling thro�nbsp;both of them, they are brought clofer at pleafure,nbsp;as the line to be drawn flrould be ftronger or finer.

In ufing this inftrument, put the ink between the blades with a common pen, or with the feeder ; andnbsp;f by the frrew) bring them to a proper diftance fornbsp;drawing the intended line: hold the pen a little inclined,

cf Mathematical Infiruments. nbsp;nbsp;nbsp;g

clined, but fo that both blades touch the paper then may a line be drawn very fmooth, and of equalnbsp;breadth, which could not be done fo well with a common pm.

Note., Before tht drawing-pen is put into the cale, the ink fhould be wiped trom between the blades;nbsp;otherwife they will foon ruft and fpoil, efpecially withnbsp;common ink. And that they may be clean�d eafily,nbsp;one of the blades ftrould move on a joint.

The diredions given about this drawing-pen., will ferve for the drawing-pen point, ufed with the com-palTes. But it muft be obferved, that when any arcnbsp;is defcribed of more than an inch radius, then the inknbsp;point fhould be bent in the joint fo that both thenbsp;blades of the pen touch the paper, otherwife the arcnbsp;defcribed will not be fmooth.

The Protradiing-pin is a piece of pointed fteel (like the point of a needle) fixed into one end of a part ofnbsp;the handle of the drawing-pen �, into which, the piecenbsp;with the pin in it, generally fcrews. Its ufe is tonbsp;point out the interfcdions of lines; and to mark offnbsp;the divifions of the protrador, as hereafter direded.

Sometimes on the top of the drawing-pen is a focket, into which a piece of black-lead pencil maynbsp;be put.

This inflrument confifts of two Rulers, con-neded together by two metal bars, moving cafiiy round the rivets which faften their ends ; thelenbsp;bars are fo placed that both have the fame inclination

to each Ruler ; whereby they will be Parallel at every diftance, to which the bars will fufFer them to receed.

But the beft Parallel-Rulers are thofe, whofe bars crofs each other, and turn on a joint at their inter-feflion �, one end of each bar moving on a centre,nbsp;and the other ends Aiding in grooves as the Rulersnbsp;receed.

This inftrument is very ufeful in delineating civil and military architedfure, where there are many Parallel lines to be drawn ; and alfo in the folution ofnbsp;feveral geometrical Problems ; fome of which are asnbsp;follows.

PROBLEM I.

A right line ab being given, to draw a line parallel thereto, that JJjall pafs through a given point c (Fig. i.nbsp;PI. III.)

Construction. Apply one edge of parallel-ruler to the given line ab j prefs one ruler tight againft the paper, and move the other untill its edge cuts thenbsp;point c; there ftay that ruler, and by its edge drawnbsp;a line through c, then this line will ht parallel to ab.

If the point c happens to be farther from the line AB, than the rulers will open to ; ftay that ruler near-eft to c, and bring the other clofe to it, where let itnbsp;reft, and move forward the ruler neareft to c, andnbsp;fo continue till one ruler is brought to the point intended.

PROBLEM II.

A right line ab being given, to divide it into any propos�d nun.ber of equal parts-, fuppofe 5. (Fig^ 2.)

Construction. Draw the indefinite right line bc, fo as to make with ab, any angle at plcafuie j with

of Mathematical Infnments. n

any convenient opening of the compafles, lay off on Be, the required number of equal parts, viz. i, 2,nbsp;3, 4, 5 lay the edge of the parallel-ruler by thenbsp;points 5 and a, and parallel thereto, through thenbsp;points 4, 3, 2, I, draw lines -, then ab, by the intcr-leiftion of thofe lines will be divided into 5 equal parts.

Any right lined quadrangle or polygon being given^ to make a right lin'd triangle of equal area.

Construction. Prolong ab; draw CB ; and through D, draw de parallel to cd, cutting ae in e ;nbsp;then a line drawn from c to e forms the triangle ace,nbsp;of equal area to the quadrangle abdc.

Exam. II. Given the pentagon abcde ; requir'd to make a triangle of equal area. (Fig. 4.)

Construction. Produce dc towards f; draw ac; through B, and parallel to ac draw bf cutting dc innbsp;F; and draw af. Then the area of the trapeziumnbsp;AFDE will be equal to the area of ihapentagon abcde.

Again. Produce ed towards g �, draw ad gt; through F, draw FG parallel to ad, and draw ag. Then thenbsp;area of the triangle age, will be equal to that of thenbsp;trapezium afde ; and confequently, to that of thenbsp;pentagon abcde.

Construction. Draw fd, and parallel thereto, through E, draw eg meeting cd produced in o, andnbsp;draw gf. Then the triangle fgd is equal to the tri*nbsp;angle fed, and the given Hexagon is reduced toUienbsp;Pentagon abcgf equal in area.

Again. Draw ag ; through f, draw fh parallel to AG, meeting cg produced in h ; draw ah, and thenbsp;pentagon is reduced to the trapezium a�ch.

Laftly, Draw ac, faralkl thereto, through h, draw Hl, meeting bc produced in i, and draw ai.nbsp;Then the trapezium is reduced to the triangle abi,nbsp;which is equal in area to the given Hexagon abcdef.

Exam. IV. Given the nineJidedfigure abcdefghi, to make a triangle of equal area. (Fig 6 )

Construction, ill, Draw ib, and through a draw AK. parallel to ib, meeting hi produced in k,nbsp;and draw bk ; fo the three fides hi, ia, ab, arc reduced to the two Tides hk, kb.

KF, nbsp;nbsp;nbsp;meeting CD iiiL; draw kl, and the three Tidesnbsp;DC, CE, BK, are reduced to the two Tides dl, lk.

KG, nbsp;nbsp;nbsp;meeting gf in m, and draw km-. To the threenbsp;Tides kh, hg, gf, are reduced to the Tides km, and

4th, Draw KF; through m, draw mn, parallel to ke, meeting fe in n, and draw kn ; To the threenbsp;Tides KM, MF, FE, arc reduced to two Tides kn, ne.

5th, Draw LN, and through k, draw ko, parallel to LN, meeting ef produced in o, and draw lo ; Tonbsp;the three Tides en, nk, kl, are reduced to the twonbsp;Tides Eo, OL.

Laftly, Draw le, and through D, draw nv parallel to LE, meeting oe produced in p, and draw lp ; Tonbsp;Thall the triangle olp bc equal in area to the givennbsp;nine Tided figure.

Proceeding in the Tame mannera figure of any number of Tides may be reduced to a triangle of equalnbsp;area.

Sect. VI.

Of the Protractor.

The Protractor, is an inftrument of a femicir-cular form; being terminated by a right line reprefenting the diameter of a circle, and a curve linenbsp;of haU the circumference of the fame circle. As atnbsp;Fig. 7. The point c, (the middle of ab) is thenbsp;centre of the femicircumference adb, which femicir-cumference is divided into 180 equal parts call�dnbsp;degrees and for the convenience of reckoning bothnbsp;ways, is numbered from the left hand towards thenbsp;right, and from the right hand towards the left, withnbsp;10, 20, 30, 40, amp;c. to 180, being the half of 360,nbsp;the degrees in a whole circumference. The ufe ofnbsp;this inftrument is to protra^, or lay down an anglenbsp;of any number of degrees, and to find the numbernbsp;of degrees contained in any given angle.

But this inftrument is made much more commodious, by transferring the divifions on the femicircum-fercnce, to the edge of a ruler, whofe fide ef h parallel to AB j (fee Fig. 7.) which is done by laying a nr/ernbsp;on the centre c, and the feveral divifions on the femicircumference ADB, and marking the interfedions ofnbsp;that ru'er on the line ef, which may eafily be conceiv�d by obferving the lines drawn from the centrenbsp;c to the divifions 90, 60, 30 ; fo that a ruler withnbsp;thefe divifions mark�d on 3 of its fides and numbered both ways, as in the Protractor, (the fourthnbsp;or blank fide reprefenting the diameter of the circle)nbsp;is of the fame ufe as a Protractor, and is much betternbsp;ada,,ted to a cafe.

That fide of the inftrument on which the divifions are mark�d, is call�d the graduated fide, or limb of the inftrument, v. hich fhould be Hoped away to an

PROBLEM IV.

A number of degrees being given to protrabl, or lay down an angle whofe meafure Jhall be equal thereto.nbsp;And an angle being frotraSied., or laid down, to find whatnbsp;number of degrees meafures that angle.

Exam. I. fio draiv a line from the point a, that fioall make an angle with the line ab of 48 deg. Fig. 8.

Apply the blank edge of the protraftor to the line ab, fo that the middle or centre thereof (which isnbsp;always mark�d) may fall on the point a ; then withnbsp;the protrafting-pin, make a mark on the paper againftnbsp;the divifion on the limb of the inftrument numberednbsp;with the degrees given ; {viz. 48.) counting from thenbsp;right hand towards the left; a line drawn from a,nbsp;through the faid mark, as ac, lhall with ab, form thenbsp;angle required, viz. 48 degrees.

If the line had been to make an angle with ab, at the point b then the centre muff have been laid onnbsp;B, and the divifions counted from the left hand towards the riglit.

Apply the blank edge of the protradlor to the line AB, fo that the centre fliall fall on the point b ; thennbsp;will the line bc cut the limb, of the inftrument in thenbsp;number exprefting the degrees that meafure the givennbsp;angle ; which in this example is 125 degrees, countingnbsp;from the left hand towards the right.

PROBLEM

PROBLEM V.

From any given point a, in a line ab, /0 draw a line perpendicular /o ab. Fig. 10.

Lay the protradtor acrofs the line ab in fuch a manner that the centre on the blank edge, and thenbsp;divifion numbered with 90, on the limb, may both benbsp;cut by the given line ; then keeping the ruler in thisnbsp;pofition, Aide it along the line, till one of thefe pointsnbsp;touch the given point a, draw the line ca, and it willnbsp;be perpendicular to ab.

In the fame manner, a line may be drawn, perpendicular to a given line, from a given point out of that line.

PROBLEM VI.

Construction. Apply the blank edge of the pro-traBor to ab the diameter of the Circle, fo that their centres fhall coincide j fet off a number of degreesnbsp;from B to D equal to an angle at the centre of thatnbsp;polygon, (viz. 45.) and through that mark draw anbsp;radius CD �, then fhall bd the chord of the arc ex-preffing thofe degrees, be the fide of the intendednbsp;polygon; which chord taken between the compaffes,nbsp;and applied to the circumference will divide it intonbsp;as many equal parts as the polygon has Tides, viz, 8 ;nbsp;and the feveral chords being drawn will form thenbsp;polygon required.

It will rarely happen that this operation, though true in theory, will give the fide of the polygonnbsp;exaft ; tor when the chord of the arc prickt offnbsp;from the protraftor, is taken with the compaflesnbsp;and applied to the circle, it generally falls beyond, or Ihort, of the point fet out from : for itnbsp;mull: be obferved that the point where two lines in-2nbsp;nbsp;nbsp;nbsp;terfeft

terfeft one another is not to be readily determined in a pradical manner; and a very fmail error in thenbsp;taking the length of the chord, being feveral timesnbsp;repeated becomes confiderable at laft. Here the com-paffes with the fpring point will be found of great ufe.

A TABLE, Jloewing the Angles at the Centres and Circumferences of regidar Polygons fromnbsp;three to twehe Sides inclifroe.

This table is conftruded, by dividing ^560, the degrees in a circumference, by the number of Tides innbsp;each polygon ; and the quotients are the angles at thenbsp;centers the angle at the center fubftrafted from 180nbsp;degrees, leaves the angle at the circumference.

Construction. From the ends of the given line, draw the lines ad, bc ; fo that the angles bad, abg,nbsp;may each be equal to the angle at the circumferencenbsp;in that polygon �, make ad, bc, each equal to ab \ fromnbsp;the points d and c, draw lines that fhall make withnbsp;DA, CB, angles equal to the former �, make thefe linesnbsp;each equal to ab ; and fo continue, till a polygon isnbsp;form�d of as many fides as required.

Draw ad, bc, fo that the angles bad, abc, may be each 120 degrees; make ad, bc, each equal tonbsp;AB : alfo, make the angles adf, bce, each equal tonbsp;120 degrees, and make df, ce, each equal to ab ;nbsp;draw FE and �tis done.

Or it may be done by the help of the parallel ruler, when the polygon has ^n even number of fides.nbsp;Thus,

Having form�d the three fides ad, ae, bc, as be-, fore dire�led; through d, draw df parallel to bc ;nbsp;make df equal to ab ; through f draw fe parallel tqnbsp;AB : make fe equal to ab and join ce.

Draw ac, bd, that each may make with ab, an angle of 108 degrees. Make ac, bd, each equal tonbsp;AB ; on the points c and d, with the com paffes openednbsp;to the diftance ab, deferibe arcs to crofs each other innbsp;E ; draw ec and ed, and �tis done.

In any regular polygon, having found all the fides but two, as above dire�led; thofe may be found asnbsp;the laft two in the pentagon were.

But a regular polygon deferibed upon a given line ab may be conftrutbed with more accuracy, thus.nbsp;See Fig. 12, 13.

Make an angle bap, and another abp, each equal to half the angle of the required polygon �, on thenbsp;point p, where the lines ap, bp, cut one another, andnbsp;�with the radius pa defcribe a circle, in which if thenbsp;given line ab be applied, the polygon fought will benbsp;formed.

SECT. VII.

Lines of equal parts are of two forts, viz. fimply divided, and diagonally divided. PI. V.

1. Simply divided. Draw 3 lines parallel to one another, at unequal diftances, (Fig. 14 ) and of anynbsp;convenient length; divide this length into what number of equal parts is thought neceffary, allowing fomenbsp;certain number of thefe parts to an inch, fuch as 2,nbsp;2i� 3gt; 3tgt; 4gt; 4-fgt; 0�c. which divifions diftinguifh bynbsp;Inbsp;nbsp;nbsp;nbsp;lines

lines drawn acrofs the three parallels. Divide the left hand divifion into 10 equal parts, which diftinguilhnbsp;by lines drawn acrofs the lower parallels only ; but,nbsp;for diftinftion fake, let the 5th divifion be fomewhatnbsp;longer than the others: and it may not be inconvenient to divide the fame left-hand divifion into 12 equalnbsp;parts, which are laid down on the upper parallel line,nbsp;having the 3d, 6th, and 9th divifions diftinguilhed bynbsp;longer ftrokes than the reff, whereof that at the 6thnbsp;divifion make the longcft.

There are, for the moft part, feveral of thefe fimply divided fcales put on rulers one above thenbsp;other, with numbers on the left hand, fliewing innbsp;each fcale, how many equal parts an inch is dividednbsp;into; fuch as 20, 25, 30, 35, 4�� 45inbsp;feverally ufed, as the plan to be exprefled fhould benbsp;larger or fmaller.

The ufc of thefe lines of equal parts, is to lay down any line exprelTed by a number of two places or denominations, whether decimally, or duodecimally divided as leagues, miles, chains, poles, yards, feet, inches,

c. and their tenth parts, or twelfth parts: thus, if each of the divifions be reckoned i, as i league, mile, chain,nbsp;^c. then each of the fubdivifions will exprefsnbsp;thereof; and if each of the large divifions be callednbsp;10, then each fmall one will be 1 and if the largenbsp;divifions be 100, then each fmall one will be 10, ftfe.

parts, let them be leagues, miles, chains, fet one point of the compaffes on the 8th of the large divifions, counting from the left hand towards the right,nbsp;and open the compafics, till the other point falls onnbsp;the 7th of the fmall divifions, counting from the rightnbsp;hand towards the left, then are the compaffes openednbsp;to exprefs a line of 8nbsp;nbsp;nbsp;nbsp;87 or 870 leagues, miles,

chains, ^c. and bears fuch proportion in the plan, as the line meafured does to the thing reprelented.

20 nbsp;nbsp;nbsp;�The Defcription and life

But if a length of feet and inches was to be ex-prefled, the fame large divifions may reprefent the teet, but the inches muft be taken from the uppernbsp;part of the firft divifion, which (as before noted) isnbsp;divided into 12 equal parts.

Thus, if a line of 7 feet 5 inches was to be laid down ; fet one point of the compafles on the 5thnbsp;divifion among the 12, counting from the right handnbsp;towards the left, and extend the other to 7, amongnbsp;the large divifions, and that diftance laid down in thenbsp;plan, fhall exprefs a line of 7 feet 5 inches : and thenbsp;like is to be underftood of any other dimenfions.

II. Diagonally divided. Draw eleven lines parallel to each other, and at equal diftances; divide thenbsp;upper of thefe lines into fuch a number of equalnbsp;parts, as the fcale to be exprefled is intended to contain, and from each of thefe divifions draw perpendiculars through the eleven parallels, (Fig. 15.) lubdividenbsp;the firft ol thefe divifions into 10 equal parts, bothnbsp;in the upper and lower lines ; then each of thefe fub-divifions may be alfo fubdivided into 10 equal parts,nbsp;by drawing diagonal lines j viz. from the loth below,nbsp;to the 9th above �, from the 9th below, to the 8 thnbsp;above-, from the 8th below, to the 7th above,nbsp;amp;c. till from the ift below to the oth above, fonbsp;that by thefe means one of the primary divifions onnbsp;the fcale, will be divided into 100 equal parts.

There are generally two diagonal fcales laid on the fame plane or face of the ruler, one being commonly half the other. (Fig. 15.)

The life of the diagonal fcale is much the fame with the iimple fcale; all the difference is, that a plannbsp;may be laid down more accurately by it : becaufe innbsp;this, a line may be taken ot three denominations jnbsp;whereas Irom the former, only two could be taken.

Now from this conftrudfion it is plain, if each of the primary divifions reprefent i, each of the firftnbsp;fubdivifions will exprefs -rV of i j and each of the

fecond

of Mathematical Infriiments. ai

fecond fubfiivilions, (which are taken on the diagonal lines, counting from the top downwards) will exprefsnbsp;�j^ of the former fubdivifions, ora loothoi the primary divifions ; and if each of the primary divifionsnbsp;exprefs lo, then each of the firft fubdivifions will exprefs I, and each of the ad, f', and if each of thenbsp;primary divifions reprefent too, then each of the firftnbsp;fubdivifions will be lo ; and each of the ad will be i,nbsp;^c.

Therefore to lay down a line, whofe length is exprefs�d by 347, 34 -j-V or 3 -�^�^- whether leagues,nbsp;miles, chains, i^c.

Osi the diagonal line, joined to the 4th of the firft fubdivifions, count 7 downwards, reckoning the dif-tance of each parallel i ; there fee one point of thenbsp;compaffes, and extend the other, till it falls on thenbsp;interfedion of the third primary divifion with thenbsp;fame parallel in which the other foot refts, and thenbsp;compafles will then be opened to exprefs a line of 347,nbsp;34nbsp;nbsp;nbsp;nbsp;; or 3nbsp;nbsp;nbsp;nbsp;amp;c.

Those who have frequent occafion to ufe fcales, perhaps will find, that a ruler with the 20 followingnbsp;fcales on it, viz. lo on each face, will fuit more pur-pofes than any fet of fimply divided fcales hithertonbsp;made public, on one ruler.

The left hand primary divifion, to be divided into 10 and 12 and 8 parts-, for thefe fubdivifions are ofnbsp;great ufe in drawing the parts of a fortrefs, and of anbsp;piece of cannon.

It will here be convenient to fhew, how any plan exprefled by right lines and angles, may be delineatednbsp;by the fcales of equal parts, and the protradtor.

PROBLEM

PROBLEM VIII.

Three adjacent things in any right lined triangle icing given^ to form the plan thereof.

Exam. Suppofe a triangular field., abc, (Fig. i6.) the fide ab=327 yards; ac=;2o8 yards ; and thenbsp;angle at A=44i degrees.

Construction. Draw a line ab at pleafure ; then from the diagonal fcale take 327 between the pointsnbsp;of the compaffes, and lay it from a to b ; fet thenbsp;center of the protra�tor to the point a, lay off 44i'nbsp;degrees, and by that mark draw ac : take with thenbsp;compaffes from the fame fcale 208, lay it from a tonbsp;c, and join cb ; fo fhall the parts of the trianglenbsp;AECj in the plan, bear the fame proportion to eachnbsp;other, as the real parts in the field do.

The fide cb may be meafured on the fame fcale from which the fides ab, ac, were taken : and thenbsp;angles at b and c may be meafured by applying thenbsp;protraftor to them as fhewn at problem IV.

Draw a line to exprefs the fide ; (as before) at the ends of that line, point off the angles, as obfervednbsp;in the field ; lines drawn from the ends of the givennbsp;line through thofe marks, fhall form a triangle fimilarnbsp;to that of the field.

PROBLEM IX.

, five adjacent things., fides and angles, in a right lin'd quadrilateral, being given, to lay down the plannbsp;thereof. Fig. 1 7.

of Mathematical Infruments. 2^

Exam. Give^ Z. ¹ a = 70�; ab = 215 links ; Z_ B = 115�; Bc = 596 links; Z_ c = 114�,

Construction. Draw ad at pleafure; from a draw ab, fo as to make with ad an angle of 70�;nbsp;make ab=2 i 5 (taken from the fcales) i from b, drawnbsp;BC, to make with ab an angle of 115�: make bc =nbsp;596; from c, draw cd, to make with cb an angle ofnbsp;114�, and by the interfeftion of cp with ad, a quadrilateral will be form�d fimilar to the figure in whichnbsp;fuch meafures could be taken as are exprcfled in thenbsp;example.

Following the fame method, a figure of many more fides may be delineated ; and in this manner, ornbsp;fome other like to it, do fome furveyors make theirnbsp;plans of furveys.

1�he ConfruBion of the remaining Lines of the

Describe a circumference with any convenient radius, and draw the diameters ab, de, at right angles to each other ; continue ba at pleafure towards f gt;nbsp;through d, draw bg parallel to bf ; and draw thenbsp;chords BD, BE, AD, AE. Citcumfcribe the circle withnbsp;the fquare hmn, whofc fides hm, mn, fhall be parallel

54 nbsp;nbsp;nbsp;�he Dejcription and �/�

I. To conJiruB the Line of Chords *.

Divide the arc ad into 90 equal parts; mark th� loth divifions with the figures 10, 20, 30, 40, 50^nbsp;60, 70, 80, 90 ; on D, as a center, with the coin-pafles, transfer the feveral divifions of the quadrantalnbsp;arc, to the chord ad, which marked with the figuresnbsp;correfponding, will become a line of chords.

Note, In the conftrudtion of this, and the following fcales, only the primary divifions are drawn ; the in^*nbsp;termediate ones are omitted, that the figure may notnbsp;appear too much crouded.

* The chord of an arc, is a right line drawn from one end of the arc to the other end.

il. The Line of Rhumbs^.

Divide the arc be into 8 equal parts, which mark Vvith the figures 1, 2, 3, 4, 5, 6, 7, 8 ; and dividenbsp;each of thofe parts into quarters; on b, as a center,nbsp;transfer the divifions of the arc to the chord be,nbsp;Ivhich marked with the correfponding figures, will benbsp;a line of rhumbs.

f The rhumbs here, are the chords anfwering to the points of the mariners compafs, which are 32 in the wholenbsp;circle, or 8 in the quarter circle.

III. The Line of Sines *

Through each of the divifions of the arc ad, draw fight lines parallel to the radius ac ; and cd will benbsp;divided into a line of fines which are to be numbered

The fine ofi an arc, is a right line drawn from one end of an arc perpendicular to a radius drawn to thenbsp;Other end.

And the verfied fime, is the part of the radius lying be^ tUveen the arc and its right fine.

frorti c to D for the right fines and from d t� c for the verfed fines. The verfed fines may be continuednbsp;to 180 degrees by laying the divifions of the radiusnbsp;CD, from c to E.

A Ruler on c, and the feveral divifions of the arc AD, v/ill interfefl; the line dg, which will become anbsp;line of tangents, and is to be figured from d to g withnbsp;10, 20, 30, 40,

* The tangent of an arc, is a right line touching that arc at one end, and terminated by a fecant drawn throughnbsp;the other end.

The diftances from the center c to the divifions on the line of tangents being transferred to the line afnbsp;from the centre c, will give the divifions of the linenbsp;of fecants; which muft be numbered from a towardsnbsp;F, with lo, 20, 30, amp;c.

f The fecant of an arc, is a right line drawn from the centre through one end of an arc, and limited by the tangent of that arc.

A RULER on E, and the feveral divifions of the arc ad, will interfedl the radius ca, in the divifions ofnbsp;the femi, or half tangents ; mark thefe with the cor-refponding figures of the arc ad.

The femi-tangents on the plane fcales are generally continued as far as the length of the ruler they are laidnbsp;on will admit; the divifions beyond 90� are found bynbsp;dividing the arc ae like the arc ad, then laying anbsp;ruler by e and thefe divifions of the arc ae, the divifions

fions of the femi-tangents above 90 degrees 'will be obtained on the line ca continued.

Divide ah, into 60 equal parts; through each of thefe divifions, parallels to the radius ac, will inter-fedl the arc ae, in as many points; from e as a centre,nbsp;the divifions of the arc ea, being transferred to thenbsp;chord EA, will give the divifions of the line of longitude.

A RULER on A, and the feveral divifions of the fines on cd, will interfeft the arc bd, in as manynbsp;points i on E as a centre, transfer the interfeftionsnbsp;of the arc bd, to the right line bd ; number the di-vifions from b to d, with lo, 20, 30, idc. to 90 �, andnbsp;BD will be a line of latitude.

Bisect the quadrantal arcs bd, be, in 0, b ; divide the quadrantal arc ab into 6 equal parts, (which givesnbsp;15 degrees for each hour) and each of thefe into 4nbsp;others; (which will give the quarters.) A ruler on c,nbsp;and the feveral divifions of the arc ab, will interleftnbsp;the line mn in the hour, points, which are to benbsp;marked as in the figure.

Bisect the arc ea intoc-, divide the quadrantal arc be into 90 equal parts lay a ruler on c and thenbsp;feveral divifions of the arc be, and the interfeftions ofnbsp;the line hm will be the divifions of a line of inclinations of meridians.

On 2 of the ufes of the line of chords is to lay down a propofed angle, or to meafure an angle already laidnbsp;down. Thus, to draw a line ac, that fhall make withnbsp;the line ab an angle containing a given number ofnbsp;degrees, (fuppofe 36.) Figure 19.

On a, as a centre, with a radius equal to the chord of 60 degrees, deferibe the arc bc ; on this arc, laynbsp;the chord of the given number of degrees from thenbsp;interfedlion b, to c �, draw ac, and the angle bac willnbsp;contain the given number of degrees.

Note, Degrees taken from the chords are always to be counted from the beginning of the fcale.

Lhe degrees contained in an angle already laid down, may he nieafured thus. Fig. 19.

On a as a centre, deferibe an arc bc with the chord of 60 degrees �, the diftance bc, meafured on thenbsp;chords, will give the number of degrees contained innbsp;the angle bac.

If the number of degrees are more than 90 -, they muft be taken from, or meafured by the chords, acnbsp;twice j thus if 140 degrees were to be protradled,nbsp;70� may be taken from the chords, and thofe degreesnbsp;laid off twice upon the arc deferibed with a chord ofnbsp;60 degrees.

Their ufe is to delineate or meafure a fhip�s courfe % which is the angle made by a fhip�s way and the meridian.nbsp;nbsp;nbsp;nbsp;Now

Now having the points and ^ points of the compafs contained in any courfe ; draw a line ab (fig- 19.) fornbsp;the meridian on a as a centre, with a chord of 60*nbsp;defcribe an arc b-c ; take the number of points andnbsp;points from the fcale of rhumbs, counting from o,nbsp;and lay this diftance on the arc bc, from the inter-fedtion b to c ; draw ac, and the angle bag lhall re-prefent the Ihip�s courfe.

If any two meridians be diftant one degree or 60 geographical miles, under the equator, their diftancenbsp;will be Icfs than 60 miles in any latitude between thenbsp;equator and the pole.

Now let the line of longitude be put on the fcale clofe to the line of chords, but inverted ; that is, letnbsp;60quot; in the fcale of longitude be againft 0� in thenbsp;chords, and o* degrees longitude againft 90� chords.nbsp;Then mark any degree of latitude counted on thenbsp;chords; and oppofite thereto, on the line of longitude,nbsp;will be the miles contain�d in one degree of longitude,nbsp;in that latitude.

Thus 57,95 miles, make 1 degree of longitude in the latitude of 15 degrees ; 45,97 n^hes, in latitudenbsp;40 degrees; 36,94 miles, in latitude 52 degrees; 30nbsp;miles, in latitude 60 degrees, i�c.

But as the fraftional parts are not very obvious on fcales, here follows a table fhewing the miles in onenbsp;degree of longitude to every degree of latitude.

This table is computed, upon the fuppofition of the earth being fpherical, by the following proportion.

As the radius is to the cofine of any latitude, fo is the miles of longitude under the equator to the milesnbsp;of longitude in that latitude.

Every perfon who is defirous of acquiring mathematical knowledge, ftiould have a table of the logarithms ol numbers, fines, tangents, and fccants; mod

of Mathematical Injlruments. 29

of th� treatifes of navigation, and fomc other books, have thcfe tables -, but the moft ufeful and efteemednbsp;are Sher�win'% mathematical tables.

A TABLE, Jheiving the Miles in one Degree of Longitude to eneery Degree of Latitude.

D. L,	Miles.	jU. L.	Miles.	D. L	Miles
I	59^99	31	5�.43	�l	29,09
2	59-96	32	50,88	62	28,17
3	59.92	33	50,37.	63	27.24
4	59.�5	34	49,74	64	26,30
5	59.77	35	49,'5	65	25.36
6	59.67	36	48,54	66	24,41
7	59.56	37	lt;17,92	67	23.44
8	59.42	38	47,28	(8	22,48
9	59.26	39	46,63	69	21,50
�O	59.C9	40	45,97	70	20,52
11	58,89	41	lt;15,28	71	'9�53
12	58,69	42	44,59	72	18.-4
gt;3	58,46	43	43,88	73	^7.54
	58,22	44	43.16	74	16,54
gt;5	57.95	45	42.43	75	15.53
16	57,67	46	4'.68	76	'4-, 52
17	57,38	47	40,92	72	I 'Ij'�O
18	5 ��,lt;=6	48	40.15	78	12,48
'9	56,73	49	39.36	79	11.45
20	56,38	5�	38,57	So	10,42
21	56,02	51	37,76	8i	0,38
22	55,63	52	36.94	82 ,	8.35
23	55.23	53	36,11	83	7,32
24	54.81	54	35.27	84	6,28
25	54,3^!	55	34.41	85	S.23
26	53,93	56	33.55	86	4.18
27	53.46	57	32,68	87	3,14
z8	52.96	58	31,79	88	2,C^
29	52,47	59	3.6.90	89	1,05
30	5 i,t,'6	60	30,00	90	c,oo

The ufcs of the fcales of fines, tangents, fccants, and half tangents, are to find the poles and centersnbsp;of the feveral circles reprefented in the orthographicalnbsp;and ftereographical projeftion of the fphere; whichnbsp;are referved until the explanation and ufe of the linesnbsp;of the fame name on the feftor are Ihewn.

The lines of latitudes, hours, and inclinations of meridians, arc applicable to the pradice of dialing ; on which there are feveral treatifes extant,nbsp;which may be confulced.

Sect. IX,

Of the S ^ c r o r.

ASedfor is a figure form�d by two radius�s of a circle, and that part of the circumference comprehended between the two radius�s.

The inftrument called a feftor, confifts of two rulers moveable round an axis or joint, from whencenbsp;feveral fcales are drawn on the faces of the rulers.

The two rulers are called legs, and reprefent the radii, and the middle of the joint expreffes thenbsp;center.

The fcales generally put on. feftors, may be diftin-guifiied into fingle, and double.

The fingle fcales are fuch as are commonly put on plain fcales, and from whence dimenfions or di-ftances are taken as have been already direfted.

The double fcales are thofe which proceed from the center ; each fcale is laid twice on the fame facenbsp;of the inftrument, viz. once on each leg: Fromnbsp;thefe fcales, dimenfions or diftances are to be taken,nbsp;when the legs of the inftrum.ent are in an angularnbsp;pofition, as will be (hewn hereafter,

The

of Mathematical Inflruments, 31 , Tihe Scales commonly put on the beji Seniors, are

The manner in which thcfe fcales are difpofed of on the fedfor, is beft feen in the plate fronting thenbsp;title page.

The fcales of lines, chords, fines, tangents, rhumbs, latitudes, hours, longitude, inch merid.nbsp;may be ufed, whether the inftrument is ftiut ornbsp;open, each of thefe fcales being contained on one ofnbsp;the legs only. The fcales of inches, decimals, log.nbsp;numbers, log. fines, log. verfed fines and log.^tangents, are to be ufed with the fedor quite opened,nbsp;part of each fcale lying on both legs.

The double fcales of lines, chords, fines, and lower tangents, or tangents under 45 degrees, are allnbsp;of the fame radius or length; they begin at thenbsp;center of the inftrument, and are terminated near thenbsp;other extremity of each leg �, viz. the lines at the

divifion lo, the chords at 6o, the fines at 90, and the tangents at 45 ; the remainder of the tangents, ornbsp;thofe above 45�, are on other fcales beginning at ^ ofnbsp;the length of the former, counted from the center,nbsp;where they are marked with 45, and run to about 76nbsp;degrees.

The fecants alfo begin at the iame diftance from the center, where they are marked with 10, and arenbsp;from thence continued to as many degrees as thenbsp;length ol the fector will allow, which is about 75�.

The angles made by the double fcales of lines, of chords, of fines, and of tangents to 45 degrees,nbsp;are always equal.

And the angles made by the fcales of upper tangents, and of fecants, are alfo equal ; and fome-times thefe angles are made equal to thofe made bynbsp;the other double fcales.

The fcales of polygons are put near the inner edge of the legs, their beginning is not fo far removed from the center, as the 60 on the chords is ;nbsp;Where thefe fcales begin, they are mark�d with 4,nbsp;and from thence are figured backwards, or towardsnbsp;the center, to 12.

From this difpofition of the double fcales, it i? plain, that thofe angles which were equal to eachnbsp;other, while the legs of the fedtor were clofe, willnbsp;ftill continue to be equal, although the fedor b?nbsp;opened to any diftance it will admit of.

This fcale, which is laid clofe to the edge of the fe�lor, and fometimes on the edge, con*nbsp;tains as many inches as the inftrument will receivenbsp;when opened ; Each inch is ufually divided into 8nbsp;equal parts, and alfo into 10 equal parts.

This fcale lies next to the fcale of inches ; it is of the fame length of the fedlor when opened, and isnbsp;divided into 10 equal parts, or primary divifions jnbsp;and each of thefe into 10 other equal parts ; fo thatnbsp;the whole is divided into 100 equal parts. Andnbsp;where the fedtor is long enough, each of the fubdivi-fions is divided into two, four, or five parts ; andnbsp;by this decimal fcale, all the other fcales, that arenbsp;taken from tables, may be laid down.

The length of a fedor is ufually underftood when it is Ihut, or the legs clofed together. Thus a fedornbsp;of fix inches when fiiut, makes a ruler of twelvenbsp;inches when opened, and a foot fedor, is two feetnbsp;long when quite opened.

III. The Scales of Chords, Rhumbs, Sines, Tdh~nbsp;gents. Hours, Latitudes, Longitudes, and Inclination of Meridians j

Dcfcription and Ufe

]V. l�he Scale of Logarithmic Numbers.

This fcale, commonly called the artificial numbers, and by fome the Gunter's fcale, or Gunter'^ ¹ line, is a fcale exprefTing the logarithms of commonnbsp;numbers, taken in their natural order. To lay downnbsp;the divifions in the beft manner, there is neceflary anbsp;good table of logarithms, (fuppofe Sherwins,) andnbsp;a fcale of equal parts, accurately divided, and ofnbsp;fuch a length, that 20 of the primary divifions ftiallnbsp;make the whole length of the intended fcale of numbers, or logarithm fcale.

L'he ConJiriiBion.

1. nbsp;nbsp;nbsp;From the fcale of equal parts, take the firftnbsp;10 of the primary divifions, and lay this diftancenbsp;down twice on the log. fcale, making two equal intervals ; marking the firft point i, the fecond i, (ornbsp;rather 10) and the third 10, (or rather 100.)

2. nbsp;nbsp;nbsp;From the kale of equal parts, take the di-ftanccs exprefled by the logs, of the numbers, 2, g,nbsp;4, 5, 6, 7, 8, 9, refpe�tively, (rejefting the indices :) lay thefe diftances on each interval of the log.nbsp;fcale, between the marks 1 amp; 10, jo amp; 100, reckoning each diftance from the beginning of its interval,nbsp;viz,, from I, and from 10, and mark thefe diftancesnbsp;with the figures 2, 3, 4, 5, 6, 7, 8, 9, in order.

Thus the firft three figures of the logarithms of 2�, 3, 4, 5, 6, 7, 8, 9, are, 301, 477, 602, 699,nbsp;778, 845, 903, 954 ; thefe are the numbers thatnbsp;are to be taken from the fcale of equal parts, and laid

of Mathematical Injlruments.

down in each interval, obferving that the extent for each is to be applied from the beginning of the intervals.

3. nbsp;nbsp;nbsp;The diftances exprelTing the logs, of thenbsp;numbers between 10 amp; 20, 20 amp; 30, 30 amp; 40,nbsp;40 amp; 50, 50 amp; 60, 60 amp; 70, 70 amp; 80, 80 amp; 90,nbsp;90 amp; 100, (rejedting the indices) are alfo to be takennbsp;from the fcale of equal parts, and laid on the log.nbsp;fcale, in each of the primary intervals, between thenbsp;marks I amp; 2, 2 amp; 3, 3 amp; 4, 4 amp; 5, 5 amp; 6, 6 amp;nbsp;7, 7 amp; 8, 8 amp; 9, g �c 10, refpeftively -, reckoningnbsp;each diftance from the beginning of its relpe(ftivenbsp;primary interval.

4. nbsp;nbsp;nbsp;The lafl: fubdivifions of the fecond primarynbsp;interval are to be divided into others, as many as thenbsp;Icale will admit of, which is done by laying down thenbsp;logarithms of fuch intermediate divifions, as it ihallnbsp;be thought proper to introduce,

V. The Scale of Logarithm Sines,

I. From the fcalc of equal parts, take the diftances exprefl�d by the arith.metical complements ¹ of thenbsp;logarithmic fines, (or the fecants of the complements)nbsp;of 80, 70, 60, 50, 40, 30, 20, 10, degrees re-fpeftively ; rejedling the indices ; and thefe diftances,nbsp;lay on the fcale of log. fines, reckoning each from thenbsp;mark intended to exprefs 90 degrees.

Thus. To the fines of 80% 70�, 60�, 50�, 40�, 30�, 2o�, 10�, the three firft figures of the arithmeticalnbsp;complements of their logarithms, are, 007, 026,nbsp;063, jg2, 301, 466, 760; thefe are the numbers to be taken from the fcale of equal parts, ufed lor

By the arithmetical complement of any fine, tangent, is�e. is meant the remainder, when that fine, tangent, isnbsp;fubftradted from radius, or 10,000000, isfe.

36 nbsp;nbsp;nbsp;T!he De faquot; ip tl on and life

laying down the logarithms of numbers, and every extent of the compafles is to be laid from the rightnbsp;hand towards the left, beginning at the point cholenbsp;for 90% which ufually ftands diredly under the endnbsp;of the line of numbers.

2. nbsp;nbsp;nbsp;In the fame manner, lay off the degrees undernbsp;10 j and alfo, the degrees intermediate to thofe ofnbsp;10, 20, 30,

3. nbsp;nbsp;nbsp;Lay down as many of the multiples of 5 minutes, as may conveniently fall within the limits ofnbsp;thofe degrees which will admit of fuch fubdivifions ofnbsp;minutes.

VI. fhe Scale of Logarithmic fangents.

j. This fcale, as far as 45 degrees, is conftrudled, in every particular, like that of the log. fines j ufingnbsp;the arithmetical complements of the log. tangents.

2. The degrees above 45, are to be counted backwards on the fcale: Thus 40 on the fcale, reprefents both 40 degrees, and 50 degrees �, 30 on the fcale, reprefents both 30 degrees, and 60 degrees; and thenbsp;like of the other mark�d degrees, and alfo of theirnbsp;intermediate ones.

VII. fhe Logarithmic verfed Sines.

1. nbsp;nbsp;nbsp;From the fcale of equal parts, take the arithmetical complements of the logarithm co-fines, (ornbsp;the fecants of the complements) of 5, 10, 15, 20, 25,nbsp;30, 35, 40, iSc. degrees; (reje�ting the indices,) andnbsp;the double of thefe diftances, refpeftively, laid on thenbsp;fcale (intended) for the log. verfed fines, will give thenbsp;divifions exprelfing 10, 20, 30, 40, 50, 60, 70, 80,nbsp;L?r. degrees; to as many as the length of the fcalenbsp;will take in.

2. nbsp;nbsp;nbsp;Between every diftance of lo degrees, introduce as many degrees, ~ degrees ; 4 degrees; i de

of Mathematical Injlruments. 37

The fcales of the logarithms of numbers, fines, vcrfed fines, and tangents, fhould have one commonnbsp;termination to one end of each fcale; that is, the 10nbsp;on the numbers, the 90 on the fines, the o on thenbsp;verfed fines, and the 45 on the tangents, fhould benbsp;oppofite to each other : The other end of each of thenbsp;fcales of fines, verfed fines, and tangents, will run outnbsp;beyond the beginning (mark�d i) of the numbers ;nbsp;nearly oppofite to which, will be the divifions repre-fenting 35 minutes on the fines and tangents, andnbsp;168^ degrees, on the verfed fines.

Sect. XI.

Of the ConfruBion of the Doubl� Scales.

I. Of the Line of Lines,

This is only a fcale of equal parts, whofe length is adapted to that of the legs of the fedor :nbsp;Thus in the fix inch fedor, the length is about 5^-inches.

The length of thi-s fcale is divided into 10 primary divifions j each of thefe into 10 equal fecondary parts;nbsp;and each fecondary divifion, into 4 equal parts.

Hence on any fedor it will be eafy to try if this line is accurately divided : Thus. Take between thenbsp;compafies any number of equal parts from this line,nbsp;and apply that diftance to all the parts of the line inbsp;and if the fame number of divifions are contained between the points of the compafies in every application, the fcale may be received as perfed.

II. Of

The Defeription and life

II. Of the Line of Sines.

1. nbsp;nbsp;nbsp;Make the whole length of this fcale, equal tonbsp;that of the line of lines.

2. nbsp;nbsp;nbsp;From the fcale of the line of lines, take off fe-verally, the parts exprefifed by the numbers in thenbsp;tables (fuppofe Sherwin's) of the natural fines, corre-fponding to the degrees, or to the degrees and minutes, intended to be laid on the fcale.

3. nbsp;nbsp;nbsp;Lay down thefe diftances feverally on the fcale,nbsp;beginning from the center-, and this will exprefs anbsp;fcale of natural fines.

Take this number as accurately as may be, from the line of lines, counting from the center and thisnbsp;diftance will reach from the beginning of the fines, atnbsp;the center of the inftrument, to the divifion expreffingnbsp;35� 15'; and fo of the reft.

In fcales of this length, it is cuftomary to lay down divifions, expreffing every 15 minutes, from o degrees to 60 degrees; between 60 and 80 degrees,nbsp;every half degree is exprefled -, then every degree tonbsp;85 ; and the next, is go degrees.

III. Of the Scale of Langents.

The length of this fcale is equal to that of the line of lines, and the feveral divifions thereon (to 45 degrees) are laid down from the tables and line of lines,nbsp;in the fame manner as has been deferibed in thenbsp;fines; obferving to ufe the natural tangents in thenbsp;tables,

This fcale is to be laid down, by taking ^ of fuch of the natural tabular tangents above 45 degrees, asnbsp;are intended to be put on the fcale.

Although the pofition of this fcale on the fedtor refpedls the center of the inftrument, yet its beginning, at 45 degrees, is diftant from the center, ^ ofnbsp;the length or radius of the lower tangents.

The diftance of the beginning of this fcale, from the center, and the manner of laying it down, is juftnbsp;the fame as that of the upper tangents ; only in this,nbsp;the tabular fecants are to be ufed.

1. nbsp;nbsp;nbsp;Make the length of this fcale, equal to that ofnbsp;the fines; and let the divifions to be laid down, ex-prefs every 15 minutes from o degrees to 60 degrees.

2. nbsp;nbsp;nbsp;Take the length of the fine of half the degreesnbsp;and minutes, for every divifion to be laid down, (asnbsp;before diredted in the icale of fines ;) and twice thisnbsp;length, counted from tlie center, will give the divifions required.

Thus, twice the length of the fine iS� 15', will give the chord of 36� 30quot;; and in the fame mannernbsp;lor the reft.

This fcale ufually takes in the fides of the polygons from 6 to 12 fides inclufive: The divifions arc laid down, by taking the lengths of the chords of

40 nbsp;nbsp;nbsp;l�be Defcriptmi and Ufe

the angles at the center of each polygon; and thefe diftances are laid from the center of the inftrument.

But it is beft to have the polygons of 4 and 5 fides alfo introduced ; and then this line is conftrufted fromnbsp;a fcale of chords, where the length of 90 degrees isnbsp;equal to that of 60 degrees of the double fcale ofnbsp;chords on the fedlor.

In the place of fome of the double fcales here de-feribed, there are found other fcales on the old fe�fors, and alfo on fome of the modern French ones, fuchnbsp;as, fcales of fuperficies, of folids, of inferibed bodies,nbsp;of metals, amp;c. But thefe feem to be juftly left outnbsp;on the fe�fors, as now conftru�fed, to make room fornbsp;others of more general ufe : However, thefe fcales,nbsp;and fome others, of ufe in gunnery, lliall hereafternbsp;be deferibed in a tra�l on the ufe of the gunnersnbsp;callipers.

Sect. XII.

Of the Ufes of the Double Scale.

IN the following account of the ufes, as there will frequently occur the terms lateral diftance, andnbsp;tranfvefe diftance ; it will be proper to explain whatnbsp;is meant by thofe terms.

Lateral dijiance, is a diftance taken by the compalTes on one of the fcales only, beginning at the center ofnbsp;the fe�for.

Franfverfe dijiance^ is the diftance taken between any two correfponding divifions of the fcales of thenbsp;fame name, the legs of the fe�for being in an angular pofition : That is, one foot of the compafles isnbsp;let on a divifion in a fcale on one leg of the fe�for,nbsp;and the other foot is extended to the like divifion in

of Mathematical Injlruments. 41

It muft be obferved, that each of the feftoral fcales have three parallel lines, acrofs which the divifions ofnbsp;the fcale are marked : Now in taking tranfverfe dif-tances, the points of the compaffes muft be always feCnbsp;on the infide line, or that line next the inner edge ofnbsp;the leg; for this line only in each fcale runs to thenbsp;center.

PROBLEM X.

To two given lines AB = 2, BC � (gt;�, to find a third proportional. Plate VI. Fig. 20.

Operation, i. Take between the compafles, the lateral diftance of the fecond term, {viz. 6.)

2. nbsp;nbsp;nbsp;Set one point on the divifion expreffing thenbsp;firft term {viz. 2.) on one leg, and open the legs ofnbsp;the fedlor till the other point will fall on the corre-fponding divifion on the other leg.

3. nbsp;nbsp;nbsp;Keep the legs of the fedtor in this pofition ;nbsp;take the tranfverfe diftance of the fecond term, {viz.nbsp;6.) and this diftance is the third term required.

4. nbsp;nbsp;nbsp;This diftance meafured laterally, beginningnbsp;from the center, will give (18) the number exprefs-ing the meafure of the third term: For 2 : 6 :: 6 : 18.

Or, Take the diftance 2 laterally, and apply it tranfverfely to 6 and 6 (the fedor being properlynbsp;opened), then the tranfverfe diftance at 2 and 2 beingnbsp;taken with the compafles and applied laterally fromnbsp;the center of the fedor on the fcale of lines, will givenbsp;,66 3= 4, the third term when the proportion isnbsp;decreafing: for 6 : 2 ;: 2 ; 4.

Note., If the legs of the fedorwill not open fo far as to let the lateral diftance of the fecond term fallnbsp;between the divifions expreffing the firft term; then

42 nbsp;nbsp;nbsp;�The Defeription and life

take i, f, -i, or any aliquot part of the fecond term, (fuch as will conveniently fall within the opening of thenbsp;fefSIor) and make fuch part, the tranfvcrle diftance ofnbsp;the firft term; then if the traniVerfe diftance of thenbsp;fecond term be multiplied by the denominator of thenbsp;part taken of the fecond term, the produft will givenbsp;the third term.

PROBLEM XL

Operation. Open the legs of the fedlor, until the tranfvcrfe diftance of the firft term, (3) be equal tonbsp;the lateral diftance of the fecond term, (7) or to Ibmenbsp;part thereof; then wdll the tranfverfe diftance of thenbsp;third term, (10) give the fourth term, (237) required ;nbsp;or, fuch a fubmultiple thereof as was taken of thenbsp;fecond term : For 3 ; 7 :: 10 ; 23-5-

Or, Set the lateral diftance 7 tranfverfely from 10 to 10 (opening the fedlor properly); then the tranfverfe diftance at 3 and 3 taken and applied laterally,nbsp;will give 2-rV: For 10 : 7nbsp;nbsp;nbsp;nbsp;3 : 2^-5.

From this problem is readily deduced, how to in-creafe or diminifh a given line, in any affigned proportion.

1. nbsp;nbsp;nbsp;Open the feftor until the tranfverfe difiance ofnbsp;8 and 8, be equal to the lateral difiance of 7.

2. nbsp;nbsp;nbsp;Mark the point to where 4 inches will reach,nbsp;as a lateral diftance taken from the center.

3. nbsp;nbsp;nbsp;The tranfverfe diftance, taken at that point, willnbsp;be the line required.

If the given line, fuppofe 12 inches, fhould be too long for the legs of the fedlor, take or or 4,nbsp;Cdc, part of the given line tor the lateral diftance;

and the correfponding tranfverfe diftance, taken twice, or thrice, or four times, amp;e. will be the line required.

PROBLEM XII.

Operation. Take the lateral diftance from the center to the divifion marked 5 between the points ofnbsp;the cornpaffes, and fet one foot on the divifion marked 4 on one of the fcales of lines, and open the legsnbsp;of the feblor till the other foot falls on the divifionnbsp;marked 3 on the other fcale of lines, and then willnbsp;thofe fcales ftand at right angles to one another.

For the lines 3, 4, 5, or any of their multiples, conftitute a right angle triangle.

PROBLEM XIIL

To two right lines given, to find a mean proportional Suppofe the lines 40 and 90.

3d. Take, with the compafTes, the lateral diftance of the half fum (65), and apply one foot to the halfnbsp;difference (25), the other foot tranfverfely will reachnbsp;to (60) the mean proportional required : For 40 : 60nbsp;;; 60 ; 90.

PROBLEM

PROBLEM XIV.

�fo divide a given line into any propofed number of equal parts: (fiippcfe 9).

Make the length of the given line, or fome known part thereof, a tranfverfe diftance to 9 and 9 : Thennbsp;will the tranfverfe diftance of 1 and i, be the partnbsp;thereof ; or fuch a fubmultiple of the f part, as wasnbsp;taken of the given line.

Or the f part, will be the difference between the given line, and the tranfverle diftance of 8 and 8.

The latter of thefe methods is to be preferred when the part required falls near the center of the inftru-ment.

To this problem may he referred the method of making a fcale of a given length, to contain a given number ofnbsp;equal parts.

The praiftice of this is very ufeful to thofe who have occafion to take copies of furveys of lands �, draughtsnbsp;of buildings, whether civil or military; and in everynbsp;other cafe, where drawings are to be made to bearnbsp;a given proportion to the things they reprefent.

Exam. Suppofe the fcale to the map of a furvey is 6 inches long, and contains 140 poles-, required to opennbsp;the fedlor fo, that a carrefponding fcale may be takennbsp;from the line of lines.

Solution. Make the tranfverfe diftance 7 and 7 (or 70 and 70, viz. equal to three inches (=nbsp;and this pofidon of the line of lines will produce thenbsp;given fcale.

If it was required to make a fcale of 140 poles, and to be only two inches long.

Solution. Make the tranfferfe diftance of 7 and 7 equal to one inch, and the fcale is made.

of Mathematical Infiruments.

Solution. Make the tranfverfc diftance of 9 and 9 equal to 3^ inches, and the fcale is made.

Exam. III. �To make a fcale which fhall exprefs 286 yards, and be \ % inches long.

Solution. Make the f of 18 inches (or 6 inches) a tranfverfe diftance to the f of 286 (= 957) andnbsp;the fcale is made.

Or, Make the of 18 inches (= 4i- inches) a tranfverfe diftance to f of 286 (= 717), and the fcalenbsp;is made.

Exam. IV. To divide a given line (fuppofe of 5 inches') into arty affignedproportion (as of \to 5).

Solution. Take (5 inches) the length of the given line, between the compafles, and make this a tranfverfe diftance to (9 and 9) the fum of the propofednbsp;parts; then the tranfverfe diftances of thenbsp;numbers (4 and 5) will be the parts required

In this place it is intended to give fo much of Ar-chitefture as may enable a beginner to draw any one of the orders; but that the following precepts maynbsp;be rightly underftood, it will be proper to explain anbsp;few of the terms.

I. Architecture is the art of building well; and has for its objeft the Convenience, Strength, andnbsp;Beauty of the building.

2. nbsp;nbsp;nbsp;Order in Architefture, is generally underftcodnbsp;as Ornament, and confifts of three grand parts,nbsp;namely;

3. nbsp;nbsp;nbsp;The Entablature, which reprefents, or is,nbsp;the weight to be fupported.

5. nbsp;nbsp;nbsp;The Pedestal or foot whereon the Column isnbsp;fet for its better fecurity.

6. nbsp;nbsp;nbsp;The Pedefial is compofed of a Base, or lowernbsp;part, a Die, and a Cornice, or upper part.

7. nbsp;nbsp;nbsp;The Column is made up of a Base, a Shaft,nbsp;which is a middle part, and a Capital, the uppernbsp;part.

8. nbsp;nbsp;nbsp;The Entablature confifts of an Architrave,nbsp;or lower part, a Freeze, the middle part, and anbsp;Cornice, the upper part.

So that an Order may be faid to confift of nine large parts, each of which is made up of fmaller partsnbsp;called Members; whereof fome are Plane, fome Curved, either convex or concave, or convexo-concave.

10. nbsp;nbsp;nbsp;A Plinth is that flat member at the bottomnbsp;of the Pedeftal, or of the bafe of the Column.

11. nbsp;nbsp;nbsp;A Plateband, that at the top of the Pedeftal,nbsp;or the upper member of the Architrave in the Entablature.

of Mathematical Injlruments. 47

16. nbsp;nbsp;nbsp;A Fusarole when an Aftragal is cut into partsnbsp;like beads.

21. nbsp;nbsp;nbsp;A CvMAise or Oue, that v/hofe convex partnbsp;projects moft and by workmen is ufually called annbsp;Ogee.

23. nbsp;nbsp;nbsp;Soffit is the under part of the Crown of annbsp;Arch, or of the Corona of an Entablature.

24. nbsp;nbsp;nbsp;Trigliphs (i. e. three channels) is an Ornament in the Freeze ot the Doric Order.

25. nbsp;nbsp;nbsp;Metops {i. e. between three�s) is the fpace ofnbsp;the Freeze between two Trigliphs.

26. nbsp;nbsp;nbsp;M0DI1.10NS, or Mutules, are the bracketsnbsp;or ends of beams fupporting the Corona. In thenbsp;Corinthian Order they are generally carved into anbsp;kind ot Scrol.

27. nbsp;nbsp;nbsp;Dentels are an Ornament looking fomewhatnbsp;like a row of teeth and are placed in the Cornice ofnbsp;the Entablature.

It is cuflomary among Architefts to eftimate the heights and projedlions of all the parts of every ordernbsp;by the diameter of the colu.mn at the bottom ol thenbsp;fhaft, which they call a module ; and fuppofe it tonbsp;confift of 60 equal parts, which are called minutes.

This order, which fome writers liken to a ftrong robuft labouring man, is the moft fimple and unadorned of any of the orders: The places moft recommended to ufe it in, are country farm-houfes,

ftables, gateways to inns, and places where plainnefs and ftrength are reckoned moft neceflary : Thoughnbsp;there are inftances where this order has been appliednbsp;to buildings of a more public and elegant nature.nbsp;The general proportions affigned by Palladio.

2. nbsp;nbsp;nbsp;Height of the entablature equal to one fourthnbsp;of the column, wanting half a minute.

10. In arches, and the columns fet on pedeftals. The diftance of the columns from middle tonbsp;middle is modules.

The ovolo under the corona, in the cornice of the entablature, is commonly continued within the corona, giving it a reverfe bending in the foffit, fomethingnbsp;like a cyma.

This order, fuppofed to be invented by Oorus a king o�Achaia, may be likened to a well limbed genteel man ; and although of a bold afpedt, yet not fonbsp;fturdy and rufticly clad as the Tufean. Architedtsnbsp;place this order indifferently in towns: But when theynbsp;would decorate a country feat with it, the open champaign fituation feems beft for the reception of the

of Mathematical Infruments. 49

Doric order ; notwithftanding which, there are many fine buildings of this order in other fituations, wherenbsp;they have a very pleafing efFedt.

4. nbsp;nbsp;nbsp;The Attic bafe is ufed vyith this order, it isnbsp;half a module in height, and fo is the capital.

8. nbsp;nbsp;nbsp;In colonadcs, the diftance of the columns in thenbsp;clear is 24 modules.

9. nbsp;nbsp;nbsp;In arches, and the column fet on pedeftals,nbsp;Diftance of the columns from middle to mid-r

Height of the arch to its foffit is lof- modules, Breadth of the pilafters is 26 minutes.

In the Doric order the architrave has two faces and a plinth; the upper face is ornamented with rows ofnbsp;fix drips or bells, covered with a plain cap ; Thenbsp;freeze is divided into trigliphs and metops : Thenbsp;breadths of the drips, cap and trigliphs are eachnbsp;module ; The trigliphs confift of two channels, twonbsp;half channels, and three voids; the breadths of thenbsp;channels and voids are each 5 minutes: The axis ofnbsp;the column continued, runs through the middle void,nbsp;leaving the drips three on each fide: The metops, ornbsp;diftances between the trigliphs, are equal to the heightnbsp;of the freeze, and are commonly ornamented wfthnbsp;trophies, arms, rpfes, (^c.

Here follows a table for the particular conftrudlion of the ornaments with which the architrave and freezenbsp;are enriched.

That figned projcftion fhews the breadths of thofe parts on each fide of the middle line of thenbsp;column continued.

And under the word profile ftand the numbers {hewing how far the feveral parts projedt beyond thenbsp;planes or faces of the members on which they arenbsp;made.

The foffit of the corona in the cornice of the entablature, is ufually ornamented with drips correfpond-ing to the trigliphs, and rofes, arms, fsfc. over the metops.

The (haft of the column is fometimes fluted ; that is, cut into channels from top to bottom, the channels meeting one another in an edge, and are in number twenty.

This order, which is taller and flenderer than the Doric, does not appear with fuch a mafculine ftrengthi

of Mathematical InflrUments.

and is by fome writers compared to the figure of a grave matron. The lonians who invented this order,nbsp;applied it chiefly to decorate their temples ; Butnbsp;when applied to the ornamenting a country palace,nbsp;the rich and extended vale feems a proper fite :nbsp;Workmen indeed ufe it indifferently in every place.

2. nbsp;nbsp;nbsp;The altitude of the entablature is equal to 4nbsp;that of the column, and divided for the architrave,nbsp;freeze, and cornice, in the proportion of 4, 3, 5.

3. nbsp;nbsp;nbsp;The height of the pedeftal equal to 2 modulesnbsp;374 minutes; or f - of the column.

4. nbsp;nbsp;nbsp;Height of the bafe fr module; its breadth i module 22i minutes.

5. nbsp;nbsp;nbsp;.Height of the capital and volute is 314 minutes, and the breadth of its abaco is 1 module 34nbsp;minutes.

7. nbsp;nbsp;nbsp;In colonades, the diftance of the columns innbsp;the clear is 2f modules,

The diftance of the modilions in the entablature is 22 minutes, and the breadth of each modilion isnbsp;1 o minutes �, the axis of the column produced alwaysnbsp;paffes through the middle of a modilion, which in thisnbsp;order is a plain block reprefenting the end of a beam.nbsp;The three moft elegant remains of the ancient Ionic ordernbsp;in Rome have their cornice ornamented with dentelsnbsp;inftead of modilions; and it is the opinion of fome,nbsp;eminent for their tafte in Architefture, that in thisnbsp;order dentels would have a better effedf than modili-

ons; the heights of thefe dentels were ufually twice their breadth, and their diftances half their breadth.

The freeze of this order is ufually made fwelling, and is formed by the fegment of a circle, whofe chordnbsp;is parallel to the axis of the column, and the fwelling projedting as far as the plateband of the architrave.

The volutes of the capital are now made to projeft in the diredlions of the diagonals of the fquare cap,nbsp;or abaco, over the volutes, lb that their drawingnbsp;lliould be expreffed like the volutes in the Romannbsp;order : They are much better drawn by an eafy hand,nbsp;than by any rules for defcribing them with the com-paffes, obferving the limits of their altitude and pro-jedion : But the volutes in the ancient examples ofnbsp;this order were curled in a plane parallel to the architrave. Thefe volutes are fuppofed to reprefent thenbsp;plaited trelTes in which the Grecian women ufed tonbsp;drefs their hair.

The fhaft of the column is fometimes fluted, leaving a fillet or lift between each channel : In this order there are 24 flutes and fillets.

This order, the moft elegant of all, is by fome compared to a very fine woman clad in a wantonnbsp;fumptiious habit: It was invented at Corinth, andnbsp;foon fpread into other places to adorn their publicnbsp;buildings. A proper rural fituation for this order,nbsp;feems to be a fpot commanding a rich and beautifulnbsp;profped in a fine watered vale.

2. nbsp;nbsp;nbsp;Height of the entablature equal to 4 that ofnbsp;the column ; the architrave, freeze and cornice to benbsp;in the proportion of 4, 3, 5; and the projection ofnbsp;the cornice equal to its height.

4. nbsp;nbsp;nbsp;The height of the capital to be i-J- module; ofnbsp;�which the abaco is 4 of ^ module ; its horns projedl-ing over the bottom of the column ^ of a module.

5. nbsp;nbsp;nbsp;The height of the bafe equal to i module ; andnbsp;its greateft breadth to be one module and a fifth.

Breadth of the pilafter, between the column and fides of the paffage, to be 27 minutes.

In this order, the fhaft is frequently cut into 24 flutes, which are feparated from one another by as .nbsp;many fillets.

The capital is compofed of three tiers of leaves, eight leaves in a tier, with their ftalks or fcrols, encircling the body of the capital, which reprefents anbsp;bafket, whofe bottom is juft as broad as the diameternbsp;of the top of the column within the channels: Thenbsp;ornaments of this capital are beft done by hand, without rule or compafs, obferving the proper altitudesnbsp;and projedlions of the parts.

The architrave confifts of three facias, three fu-faroles, an ogee, and a plateband -, the firft, or lower facis projedts the fame as the top ot the fliaft.

The freeze, which proje�ls the fame as the top of the fhaft, has its lower part turned into a kind ofnbsp;cav'etto, terminating with the extremity of the plate-band of the architrave.

The breadths of the modilions are 11^- minutes, and their diftance in the clear 23- minutes.

^4 nbsp;nbsp;nbsp;The Defcripilon and Ufe

The middle of a dentel fhould be under the middle of a modilion, and the axis of the column pafles through the middles of both dentel and modilion.

This order (the poor invention of the Romans^ and therefore frequently called the Roman order), is ufu-ally compofed of the Cciinthian and Ionic; the Ionicnbsp;capital being fet over the two lower rows of leaves innbsp;the Corinthian capital.

2. nbsp;nbsp;nbsp;The height of the entablature equal to f of thenbsp;column �, the architrave, freeze, and cornice, in thenbsp;proportion of 4, 3, 5 ; the freeze fwelling like that ofnbsp;the Ionic.

4. nbsp;nbsp;nbsp;Height of the capital equal to if module ; ofnbsp;which the abaco is f module, its horns projedtingnbsp;from the center of the column i module.

5. nbsp;nbsp;nbsp;Height of the bafe gif minutes, and its great-eft breadth if modules.

Height of the arch equal to I2f modules: In the clear, the height is to the fpan as 5 to 2.

The breadth of the pilafters between the column and arch is modules, or 42 minutes.

In this order the fliaft, if fluted, is to have 24 channels and 24 fillets, one between each two flutes.

The volutes of the capital are angular, to have the fame appearances on every fide, and they are drawnnbsp;like thofe in the Ionic,

Th?

of Mathematical Injlrumenfs.

The modilions in this order are worked into two faces, with an ogee between them; the breadth of thenbsp;lower face 9^ minutes, that of the uppernbsp;nbsp;nbsp;nbsp;the

diftance of two modilions at the upper faces is 20 minutes, and at the lower faces 23 minutes; the axis of the column palling through the middle of a modi-lion.

E 4 nbsp;nbsp;nbsp;'

56 nbsp;nbsp;nbsp;��he 'Defcription and �fe

On CD and db, make ifofceles triangles, whofe legs t)E, DF, may be each -� of the bafe cd, db ; and thenbsp;Vertexes e and f will be the centers of the arcs cDjnbsp;DB.

Or, The centers of the arcs cd, db, may be found by dt feribing equilateral triangles on the right lines

Through d draw the right line ef, fo, that the angle CDE may be equal to the angle dce j meeting the upper and lower lines in e and f.

Ho defcribe the curve joining the fhaft �f a column with its upper or lower fillet, the pro] edlion 0/ ab being given.nbsp;Fig. 7.

Draw the axis ab of the intended length of the fhaft; and parallel thereto, at half a module diftance,nbsp;draw CD ; make ce equal to half the proper diminution, and draw ef parallel to ba.

of Mathematical Injlrumcnts. 57

Make ag equal to one third of ab ; and fo high is the fhaft to be parallel to its axis; through g draw hinbsp;at right angles to ab.

On hi defcribe a femicircumference cutting the line EF in the point 4 divide the arc H4 into equal partsnbsp;at pleafure, fuppole 4 ; and through thofe points drawnbsp;the lines 11, 22, 33, 44.

Divide the line gb into a like number of equal parts, as at the points a,b, c-, and through thefe pointsnbsp;draw lines parallel to ih; making aa � ii, bb = 22,

^^ = 33-

Then a curved line drawn through the extremities h, b^ will limit the gradual diminution required.

Lay a thin ruler by the points D, H) e, and the bending of the ruler will give the gradual diminutionnbsp;required.

The altitude ab, which is -i of a module, or 26|-minutes, is divided into 8 equal parts, viz. 4 from c to A, and 4 from c to b ; upon cd = 3f, one of thefenbsp;parts, a circle is defcribed, and called the eye of thenbsp;volute, which correfponds with the aftragal of thenbsp;column.

Palladio gives the following manner of finding the 12 centers of the volute, which he difcovered onnbsp;an old unfiniflied capital. Fig. 9.

Within the eye of the volute infcribe a fquare, whofe diagonal is cd -, in this fquare draw the two diameters 13, 24, and thefe four points i, 2, 3, 4, arenbsp;the centers of the arcs ai, ib, B3, 34, which formsnbsp;the firfl; revolution.

The centers of the arcs forming the fecond and third revolutions are thus found ; fee the eye of thenbsp;Volute drawn at large. Fig. 9.

58 nbsp;nbsp;nbsp;^he Deferiptim and life

Divide the radii op, 02, 03, 04, each into 3 equal parts, as at the points 5, 6, 7, 8, 9, 10, u, 12, andnbsp;thefe will be the centers of the remaining arcs, thenbsp;laft of which is to coincide with the point c, in thenbsp;eye.

Goldman ebferving that in this conftrudtion the ends and beginnings ot the arcs were not at right angles to the fame radii, contrived the following con-ftruftion. See Fig. 10. and its eye drawn at large.

Upon one half of cd, deferibe the fquare 1, 2, 3, 4; and draw the lines 02, 03 ; divide 01, 04, eachnbsp;into 3 equal parts; then lines drawn through thofenbsp;points parallel to i, 2, their interfeflions with 14,nbsp;02, 03, will be centers of the volute.

So the points, i, 2, 3, 4, 5, 6, 7, 8, 9, 10, ii, 12, will be the centers of the twelve arcs which togethernbsp;form the outward curve of the volute.

Take oa equal to I of 01 ; divide oa into three equal parts, and thefe divifions will give centers of thenbsp;inner curve �, the two eyes drawn at large will fliewnbsp;how the 12 inner centers are found, where they arenbsp;diftinguilhed by large points; the 12 centers of thenbsp;outward curve being marked by the figures.

In the deferibing of thefe volutes, it will frequently happen., that the laft quadrant will not fall on its truenbsp;termination, occafioned by the radii of the feveralnbsp;quadrants not being exa�fly taken by the compaffes ;nbsp;In order to avoid this inaccuracy, at leaft in fomc degree, here is fubjoined a table lltcwing the length ofnbsp;each radius, computed from Goldman�s method ; Butnbsp;it may alfo be applied to Palladio�s, the radius ofnbsp;the largeft quadrant not differing t4-3- of a minute,

�ffW of a module from the truth ; and excepting the arc deferibed from the firft center, the reft may benbsp;made quadrants in the fame manner as fliewn in Gold-pj^n�s method.

A Table of the lengths^ in minutes.^ of the feveral radii of the outward and inner volutes.

To ufe this table, a fcale of of ^ module fhould be made, and divided into x 5 minutes, and the ex-tream divifion decimally divided, whereby the lengthsnbsp;of the feveral radii may be taken : But as the fe�bornbsp;is an univerfal fcale, there are two other columns added.

ed, applicable to the fe�lor ; where the Jonger radiu5 14,166 is made a tranfverfe diftancc to 10 and 19,nbsp;or 100 and 100, on the line of lines, and all the othernbsp;radii of both curv^es are proportioned thereto : Nownbsp;the centers of the corves being found as fhewn in thenbsp;eyes of the volute, the feveral radii may be taken fromnbsp;the fe�lor, and the curves more accurately defcribednbsp;than by any other method.

In the Doric, the circumference of the column being divided into 20 equal parts (here the f circumference is divided into 5), of which ab is one; on ab deferibe a fquare, and the center c of that fquare is thenbsp;center of the channel or ilute required.

In the Ionic, and Corinthian, divide the circumference of the column into 24 equal parts (here the 4 circumference is divided into 6), of which 'ad is one ;nbsp;divide ad into 4 equal parts; then ae � -ad is thenbsp;breadth of the flute, and ed � ^ad is the breadth ofnbsp;the fillet.

The flutes are femicircles defcribed on the chords of tlreir arcs in the column.

In the three following tables are contained the heights and projedions of the parts of each order, according to the proportions given by Palladio; thenbsp;orders of this architect were chofen, becaufe thenbsp;� Engli/h, at prel�nt, are more fond of copying his pro-dudions, than thofe of any other archited.

The firft table ferves for the pedeftal, the fecond for the column, and the third for the entablature, ofnbsp;each order. Each table is divided into feven principalnbsp;columns: In the flrft, beginning at the left hand, isnbsp;contained the names of the primary divifions; in thenbsp;fecond thofe ot the feveral divifions and members innbsp;the orders; and the other, five, titled with 'Tufean,

of Mathematical Infruments. 6i

Boric, Ionic, Corinthian, Roman, contain the numbers cxpreffing the altitudes, and projeftions taken fromnbsp;the axis, or middle of the column, of the feveralnbsp;members belonging to their correfponding orders.

The column containing each order, is divided, firft into two other columns, one fliewing the altitudes,nbsp;and figned Alt. and the other, the projeftions, andnbsp;figned Proj. Each of thefe is alfo divided into twonbsp;other columns, one containing modules, and markednbsp;Mo. and the other, the minutes and parts, and marked Mi.

Under the table of the pedeftal there is another table, fhewing the general proportions for the heightsnbsp;of the orders.

In each of the orders of architedture, the height of the order, and the diameter of the column, have anbsp;conftant relation to one another.

Therefore, if the diameter of the column be given, the height of the order is given alfo: And having determined by what fcale the order is to be drawn,nbsp;fuch as i inch, i inch, 2 inches, to a foot ornbsp;yard, ^c. Take from fuch fcale, the part or partsnbsp;exprefling the diameter of the column, and make thisnbsp;extent a tranfverfe diftance to 6 and 6 {i. e. 60 and 60)nbsp;on the fcales of lines, and the fedlor will be openednbsp;fo, that the feveral proportions of th� order may benbsp;taken from it.

Exam. Suppofe the diameter of a column ii to he I'i inches; and the drawing of the order is to he delineatednbsp;from a fcale of an inch to a foot: that is, the diameternbsp;of the column in the drawing is to he an inch and half.

Make the tranfverfe diftance of 6 and 6, on the fcales of lines, equal to if inch, and the fedlor isnbsp;fitted for the fcale.

If the height of the order is given, divide this height, by the height of the order in the table ; andnbsp;the quotient will be the diameter of the column.

^he Defcription and XJfe

Exam. JVhat mujl be the diameter of the column in the Ionic order, when the whole height of the order isnbsp;fixed at feet 6 inches.

The height of the order in the table is 13 mo, 297 mi. = *3�^^ = �314875 modules: And 18 f.

= I f 4-9: inches nearly : And the fcftor may be fitted to this, as before diredled, according to the intendednbsp;fize of the draught.

To delineate an Order by thefe Tables.

Having determined the diameter of the column at bottom, and fet the fedtor to the intended fcale, drawnbsp;a line to reprefent the axis or middle of the order.

On this line, lay the parts for the heights of the pcdeftal, column, and entablature, taken from thenbsp;table of general proportions.

Within each of thefe parts refpedlively, lay the feveral altitudes taken from the tables of particulars,nbsp;under the word Alt. Through each of the pointsnbsp;marked on the axis, draw lines perpendicular to thenbsp;axis, or draw one line perpendicular, and the othersnbsp;parallel thereto.

On the lines drawn perpendicular to the axis, lay the projedlions correfponding to the refpedtive altitudes thefe projeftions are to be laid on both fidesnbsp;of the axis, for the pedeftal and column ; and onlynbsp;on one fide, for the entablature, join the extremitiesnbsp;of the projeftions with fuch lines as are proper to ex-prefs the refpedfive mouldings and parts: And thenbsp;order, exclufive of its ornaments, will be delineated.

As the altitudes of many of the parts are very fmall, it will not be convenient, if pofiible, to take from thenbsp;fcale of lines, fuch fmall parts alone �, therefore it may

of Mathematical Injlruments. 63

In the line ad, which reprefents the axis of the order, take the bafe a a =: 424- min., the die ad ==nbsp;I mod. K �5 min.-, and the capital do = 224 min.nbsp;Then to draw the fmall members in the bafe andnbsp;cornice, proceed thus.

To the minutes in the bafe, 424, add fome even number of minutes, fuppofe 30 = ab, and the fumnbsp;724 is equal to ab; then compofe a table, fuch as thenbsp;following one, wherein the alt. of the plinth is fub-trafted out of the No. 724 ; then the torus out of thisnbsp;remainder; then the cyma out of this remainder 9nbsp;then the fillet out of this; and laftly, the cavetto outnbsp;of this remainder. Thus,

This lefs by the fillet, o|, remains This lefs by the cyma, 5, remainsnbsp;This lefs by the fillet, 04� remainsnbsp;This lefs by the cavetto, 34, remainsnbsp;the minutes firft added.

Then the feveral numbers in the table may be taken from the line of lines on the fedbor, and applied from B towards a. Thus, ,

Make Bi = 44, B2 = 40, B3 = 394� B4 = 344� B3 � 334; draw lines through thtfe points at rightnbsp;angles to ad, and on thefe lines lay the refpeftivenbsp;prqjcdlions, as fhewn in the general table j then thenbsp;proper curvature or figure being drawn at the extremities of the numbers, the bafe of the pedeftalnbsp;will be made.

It will be found moft convenient to lay off the numbers from the greater to the lelTer ones 9 for thennbsp;there is only one motion required in the joints of the

And in the fame manner, for the cornice of the pedeftal, take a point C, 30 minutes below the cornice ; and tabulate as before.

� nbsp;nbsp;nbsp;354 = C5

These numbers laid from c towards d, gives the altitudes of the members of the cornice.

In like manner the mouldings about the bafe and capital are laid down, by taking 30 minutes in. thenbsp;fhaft both above the bafe and below the capital; having firft fet on the axis, the refpedlive heights of thenbsp;bafe, fhaft, and capital,

The bale 334 min. with 30 added = 634 = so This lefs by the plinth lO min. leaves 534 =: si

s8 is here fuppofed to be 30, though the plate is not high enough to admit 30 minutes to be laid innbsp;the fhaft of the column.

The capital 244 with 30 added, gives 544 = eg This lefs by the plateband 145 leaves 524 = finbsp;Ditto ...... ogee . . 34 � � � 49v =

Tables may be made in like manner for either of the orders, to be taken from the feftor; The projections from the axis being all of them large numbers,nbsp;they may be taken from the feftor eafily enough afternbsp;it is fee to the diameter of the column, as before Ihewn.

A LITTLE refleftion will make this very clear, and perhaps more fo, than by beftowing more words thereon.

TABLE

A Table Jhewing the Altitudes and Projedlions of

A Table of general

R S T._____

every Moulding and Part in the Pedejlals of each

Proportions for the Orders.

__ _TABLE

A 'I'able, Jhewing the Altitudes and Proje�liom of

	Tufcan				Doric.
Alt.		Proj.		Alt.		Proj.
Mo.	Mi.	Mo.	Mi.	Mo.	Mi.	Mo.	Mi.
-	�	�	�	�	�		�
0 c 0 0	10 10	0 0 0 c	30 29 245 2;\|	0 0 0 c n 0	I � 2f 6f li I.I, 10	0 0 0 0 0 0 0 0	38-f 5 37l ?36inbsp;35:1 34i 29I 28i 27-i 26
0 0 5 0	4 if 54f	0 0 0 0	27 24? $ 22i ( 30 33l	0 0 6 0	3f if 53-1 if	0 0 0 0	3'! f 26 ( 30 33f
�	�	�	�	0	_ 1 gt;2	0	36�
		�	�	�	�	�
�	�	�	�	0	If	0	3^
		�	�	0	4I	0	333
	�	�	�	0	, I * 4	0	36?
�	__				�		��
�	�	�	�	�	�	�	�
�.	�	�	�	�	�	�	�,
0	I2f	0	40	0	,1 �� 2	0	40
0	ilt;r	0	40	0	10	0	40
0	2' i	�	�	0	3-
b	2f	�	�	7	0	�	�
0	30	�	�	0	30		�

Names of the Members.

Collarino - - -Middle Volute -Courf.ofleaves, T 3d folding hal I zdnbsp;their height J ilinbsp;f-J fAHragal - - - - -I Fillet---- - -

^ [.Aftragal.....

(quot;Torus - - - - -

Aftragal - -

Fillet----

_ _ nbsp;nbsp;nbsp;_T A B L E

A Table, Jhewing the Altitudes and Projections of

Third.

Corinthian.

Air.		Proj.		Alt.		Proj.		Alt.		Proj.
Mo.	Mi.	Mo.	Mi.	Mo.	Mi.	Mo	Mi.	iVio.	Mi.	Mo.	Mi.
o		i	1 2	0		1	'4	0	2 ^	1	i8i
o	7	�	�	0	6\|	�	�	0	8
o	i	1	4	0		I	6\|	0	1	1	10
o	3f		3 of	0	3	l;	sf 4	0	3l	l;	9 6
o	8	0	S9;	0	^ I / 3	1	3	0	9f	I	5
�			��	�	�	�	�	0	H	0	55
		��	'�	0		I	2	0	ri	0	54
o	3	c		0	. t	1 *	1
	3		1 33			{ 0	59
		�	-'	�	�	�	�	0		0	53
V	7l		rz	0	7gt;	0	40,	0	If	0
1	�	�	�	�	�	�	�	0	:?4	0	51
o	1?	rgt;	37	0	1	0	40	0	I	0	51
o	6	0	36		4f	0	39	�	�	�	51
�	�	�	_	_	_		_	0	c	0	5 35f
							36				1 29
��		�	�	0	I	0	36	�	�	�	1 29
	��	�	T-	0	51	0	35	�	�	�.	�
			�	�	--	�			2	0	30
0	I	0	3'i	0	1	0	32	0	2	0	287-
	�		�	0	4f	0		�	�	�	�
o	?	0	27	�	�	�		�	�	�	�
o	46	�	�	0	3 7f	�	�		30	�	�
	27	0	H	C'	28i	0	2/		'0	0	35
0	3	�	�	0	3*^		�	0	40
0	2-t	0	.�4	0	2	�	34:	0	H	0	35
	�	�	�	�	�	�	�	0	4	0	32
o	4i	c	5 33 ( 30	0	5	0	5 33f 1 30	0	3f	0	5 3- i 29
	�	�	�-	0	2	0	2g�	�	�	�
o		0	29	0	loi	0	28	�	�	�	--
0	2	0	29	0	ri	0	28	0	lA	�	7.9
0	8-	0	27I	0	8i	0	27	0	15	0	28
.�	�	�	�		-	-T	__	0			5 27f
									3	r	t 207
0	I;	0	27f	0		0	27	�	�-	__
0	6\	0	26\|]	0		0	2�	0	I 1	0	26

Sect. XIIL

In a given circle., whofe diameter is ab, to inferibe a regular otiagon. Fig. 22,

Solution. Oi�Etir the legs of the fedlor, till the traofverfe diftance of 6 and 6, be equal to ab :nbsp;Then will the tranfverfe diftance of 8 and 8, benbsp;the fide of an oeftagon which will be inferibed innbsp;the given circle.

In like manner may any other polygon not exceeding 12 Tides, be inferibed in a given circle.

PROBLEM XVII.

2d. At that opening of the fedlor, take the tranfverfe diftance of 6 and 6; and with tliis radius, on the points a, e, as centers, deferibe arcs cutting in c.

3d. On c as a center, with the fame radius, deferibe a circumference palfing through the points a, B ; and in this circle may the pentagon, wliofe fide isnbsp;AB, be inferibed.

By a like procefs may any other polygon, of not more than 12 fides, be deferibed on a given iine.

The fcales of chords will folve thefe two problems, or any other of the like kind : Thus,

In a circle whofe diameter is ab, to deferibe a regular polygon of 24 fides. F ig. 24.

Solution, tft. Make the diameter ab, a tranfverl� diftance to 60 and 60, on the fcales of chords.

of Mathematical Injlruments. 73

3d. Take the tranfverfediftanceof 15 and 15, and this will be the chord of the 24th part of the circumference.

As there are great difficulties attending the taking of divifions accurately from fcales; therefore in thisnbsp;problem, where a diftance is to be repeated feveralnbsp;times, it will be beft to proceed thus.

In every divifion of 60 degrees, lay down, ift.The chord of 15 degrees. 2d. The chord of 30 degrees.nbsp;3d. The chord of 45 degrees, beginning always atnbsp;the fame point.

If methods like this be purfued in all fimilar cafes, the error in taking diftances, will not be multipliednbsp;into any of the divifions following the firft.

Sect. XIV.

Some TJfes of the Scales oj Chords.

These double fcales of chords, are more convenient than the fingle fcales, fuch as deferibed on the plain fcale ; for on the feftor, the radius withnbsp;which the arc is to be deferibed, may be of any lengthnbsp;between the tranfverfe diftance of 60 and 60, whennbsp;the legs are clofe, and that of the tranfverfe diftancenbsp;of 60 and 60, when the legs are opened as far as thenbsp;inftrument will admit of. But with the chords on thenbsp;plain fcale, the arc deferibed, muft be always of thenbsp;fame radius.

PROBLEM

PROBLEM XVIIL

_ ^0 protralfy or lay down, a right lined angle, bac, which Jhall contain a given number of degrees, PJ. VI.

ift. At any opening of the fe�lor, take the tranf-verfe diftance of 60 and 60, (on the chords;) and with this opening, deferibe an arc bc.

2d. Take the tranfverfe diftance of the given degrees 46, and lay this diftance on the arc from any point B, to c ; marking the extremities b, c, of thenbsp;laid diftance.

3d, From the center a of the arc- draw two lines AC, ab, each pafling through one extremity of thenbsp;diftance bc, laid on the arc ; and thefe two lines willnbsp;contain the angle required.

2d Take the tranfverfe diftance of 4 or 4, of the given degrees 148 ; fuppofe 4 = 494 degrees ; laynbsp;this diftance on the arc thrice; viz. from b to a, fromnbsp;a to h, from b to d.

3d From the center a, draw two lines ab, ad; and the angle bao will contain the degrees required.

When an angle containing lefs than 5 degrees, fuppofe 345 be made, it is mofl convenient to proceed thus.

2d. From fome point d, lay the chord of 60 degrees to G ; and the chord of 564 degrees (= 6o* � 34�) from D to E.

3d. Lines drawn from the center a, through o and E, will form the angle age, of 34 degrees.

of Mathemattcal Injlruments, 7^

If the radius of the arc or circle is to be of a given length j then make the tranfverfe diftance of 60 andnbsp;60, equal to that affigned length.

Either of thefe fcales of chords, may be ufcd fingly in the manner dire�ted in the ufe of chords onnbsp;the plane fcale.

From what has been faid about the protrafting of an angle to contain a given number of degrees, it willnbsp;be eafy to fee how to find the degrees which are contained in a given angle already laid down.

problem XIX.

To delineate the vifual lines of a furvey ly having given^ the bearings and diftances from each other^ of thenbsp;Jtations terminating thofe vifual lines.

Exam. Suppofe in the field-book of a furvey. the bearings and diftances of the ftations were expreffednbsp;as follows:

O 6.B273 55 P 5;

[ft

The bearings are counted from the North, Eaft-ward. 1 heretore all the bearings under 90 degrees, fall between the N. and E. or in the ift quadrant.

Bearings between 90� and 180�, fall between the E. and S. or in the ?d quadrant.

Those between 180� and 2 /0�, fall between the S. and W. or in the 3d quadrant.

And thofe between 270� and 360% fall between the W. and N. or in the 4th quadrant.

Solution, ill. Take from the chords the tranf-verfe diftance of 60 and 60, (the fedtor being opened at pleafure,) with this radius deferibe a circumference,nbsp;and draw the diameters NS. WE. at right angles.nbsp;Pl. VI. Fig. 31.

2d. The firft bearing 70�. 50'is in the firft quadrant, but being more than 6c*, take the tranfverfe diftance of the half of 70� 50', and apply this extentnbsp;in the circumference twice from N. towards E, andnbsp;the point correfponding to the ift bearing will be obtained, which mark with the figure 1.

3d. The fecond bearing 128� 10', falls in the fe-cond quadrant; its fupplemenc to 180quot; is 51� 50', that is 50' from the S. point. Now take the tranfverfenbsp;diftance of 51� 50', and apply it in the circumferencenbsp;from S. towards E, and the point correfponding to thenbsp;fecond bearing will be found, which mark with thenbsp;figure 2.

4th. The 3d bearing 32� 13', is to be applied from N. to 3 ; The 4th bearing �87*? 3o',is in the 4th quadrant ; therefore take it from 360�, and the remaindernbsp;72� 30', is to be applied from N. towards W. andnbsp;the point 4 reprefenting the 4th bearing will benbsp;known.

In this manner proceed wdth all the other bearings, and mark the correfponding points in the circumference with the numbers 5, 6, 7, (iff.nbsp;nuUiber of the bearing or ftation.

of Mathematical Injlruments. 77

5th. Chufe fome convenient point on the paper to begin at, as at the place markt O i. Lay a parallelnbsp;ruler by c the centre of the circle, and the point in itsnbsp;circumference marked i, and ( by the help of thenbsp;ruler; draw a parallel line thro� O i, the point chofenbsp;for the firft ftation, in the direftion of the (fuppofed)nbsp;radius C i �, and on this line lay the firft diftance �, thatnbsp;is, take from a convenient fized fcahquot; of equal partsnbsp;the extent of 1080, and transfer this extent from G inbsp;to G 2 j and this line will reprefent the firft diftancenbsp;meafured, laid down according to its true pofilion innbsp;refpefl; to the circle firft deferibed.

6ch. Lay the ruler by the centre C, and the point in the circumference noted by the figure 2, and parallelnbsp;to this pofition of the ruler, draw thro� the point � 2nbsp;aline O 2 O 3, in the diredlion of the (fuppofed)nbsp;radius C 2, and on this line lay from O 2 to 0 3 thenbsp;extent 580 taken from the fame fcale of equal partsnbsp;the io8o was taken from, and this line fhall reprefentnbsp;the fecond meafured diftance laid down in its true po-fuion relative to the firft diftance.nbsp;nbsp;nbsp;nbsp;,

Proceed in this manner from ftation to ftation until the line O 7 O 10 is drawn.

7th. Take from the fcale of equal parts 314, and apply this extent in the line G 7 o lo from O 7 tonbsp;O 8, and the relative point, where the eighth ftationnbsp;was taken, will be reprefented by the point O 8 ; thennbsp;by the parallel ruler draw the line O 8 G 5, in thenbsp;diredion of, and parallel to, the (fuppoledj radiusnbsp;C 8 and if the preceding work is accurately performed, this line will not only pafs thro� the pointnbsp;� 5, but the length of the line G 8 O 5 will be equalnbsp;to 510, as the ftation line was meafured in the field.

8rh. Now as the 9th ftation falls on the fame point as the 5th ftation did. draw the line O 9 O 2, andnbsp;this line will not only be parallel to the (fuppofed)nbsp;radius C 9, but will alfo meafure on the fcale of equal

9th. The 10th ftation is taken from the end of the line 700 meafured from the 7th ftation; thereforenbsp;drawing from O 10 a line parallel to the (fuppofed)nbsp;radius C 10, this line will concur with the firft meafured line at the diftance of 668 from the point o lO.

loth Returning to� 10 again, the fame point is taken for the 1 ith ftation, and the line O 11 � 12 isnbsp;to be drawn parallel to the (fuppofed) radius C 11, andnbsp;to be made of the length of 800 from the fcale ofnbsp;equal parts; aiid this will give the point O 12 for thenbsp;12th ftation : Then drawing the line O 12 O i, ifnbsp;the operation is every where truly done, this line willnbsp;not only be parallel to the (fuppofed) radius C 12, butnbsp;will alfo meafure on the fcale of equal parts 784, thenbsp;fame as was meafured in the field in proceeding fromnbsp;� 12 to � I.

By fuch methods as thefe, the furveyor obtains a cheque on his work, and can make his furvey clofe (asnbsp;�tis called) as he proceeds.

The drawing of the vifual lines of a furvey is, tho� an eflential part, but a fmall ftep towards the makingnbsp;a plan ; for the remaining work the reader is refer�dnbsp;to the treatifes already extant on that fubjed.

What has been faid about the delineating of the vifual lines of a furvey, may be applied to navigationnbsp;in the conftru�tion of a figure to reprefent the variousnbsp;courfes and diftances a fhip has failed in a given time,nbsp;called traverfe failing ; for the courfes are the bearingsnbsp;from the Meridian, and the diftances failed are of thenbsp;fame kind as the diftance between ftation and ftationnbsp;in a furvey.

If the I at the beginning ofnbsp;the fcale,nbsp;of the 1 ft interval, be taken for

And the primary and intermediate divifions in each interval, muft be eftimated according to the valuesnbsp;fet on their extremities, wz, at the beginning, middle and end of the fcale.

In arithmetical multiplication, or divifion the parts may be confidered as proportional terms; for innbsp;fimple multiplication ; as unity or r, is to one faftor;nbsp;fo is the other faftor, to the produft ; And in divifion ; as the divifor, is to unityj (or to the dividend,)nbsp;fo is the dividend, (or unity,) to the quotient.

Now as the common logarithms of numbers, ex-prefs how far the ratios of their correfponding numbers are diftant from unity; it follows, that of thofe numbers which are proportional, that is, have equalnbsp;ratios j their correfponding logarithms will have equalnbsp;intervals, or diftances : and hence arifes the rule fornbsp;working proportionals on the logarithmic fcale.

Rule. Set one foot of the compafles on the point or divifion reprefenting thefirft term, and extend thenbsp;other foot to the point reprefenting the fecond term:nbsp;Keep the compafiTes thus opened fet one foot on thenbsp;point exprefiing the third term, and the other footnbsp;will fall on the fourth term, or number fought.

Solution. Set one foot on the i atthe beginning, and extend the other to 3, in the firft interval ; withnbsp;this opening, fet one foot on 4, in the firft interval,nbsp;and the other foot will reach to 12, found in the fecond interval.

Obferve. In this Exam, the i, 3, and 4, are valued as units in the firft interval; and the one in the middle is 10; the diftance between this i or 10, andnbsp;the 2 or 20, in the fecond interval, is divided into 10nbsp;principal parts, exprefs�d by the longer ftrokes; everynbsp;one in this Exam, is taken as an unit -, now as thenbsp;point of the compaffes falls on the fecond of thefe

principar parts, that is on 2 units beyond 10 ; there-iore this pbint is to be cfteemed in this Exam, as x2.

SoLUi ION. In the firlt interval, take the diftance between i and 3; and thi.s diftance will reach fromnbsp;(4 or) 40 in the firft interval to (12 or) 120 in thenbsp;fecond interval.

Obferve. T he i and 3 in the firft interval, are taken ' as units : but as the values given to the civifions innbsp;either interval, may as well be call�d 40, as 4; andnbsp;being taken as 40, the i at the beginning of thenbsp;fecond interval will be jeo ; and the 2 in the fecondnbsp;interval will be 200 : confequently the principal di-Vifions between this i and 2 will each exprefs 10; andnbsp;fo the fecond of them will be 20, which with the 100,nbsp;exprefs�d by the i, makes 120.

Solution. The diftance from i in the firft interval, to 24 in the fecond, will reach from 25 in the firft interval, to 84010 the fecond.

Obferve. In the firft application of the compafies, the primary divifions in the firft interval are taker asnbsp;units, and thefe in the fecond interval, as tens: Eutinnbsp;the fecond application, the primary divifici.s in thenbsp;firft interval are reckon�d as tens ; and thole in thenbsp;fecond, as hundreds.

As the extent out of one interval into the other, may fometimes be inconvenient, it will be proper tonbsp;fee in fuch cafes, how the example may be folved innbsp;one interval. Thus,

In either interval, take the extent from i to 2 (i. e. 24) and this extent, (in either interval) will

82 nbsp;nbsp;nbsp;^he Defcription and JJfe

In this operation ; the fecond term is reckoned a tenth higher than the firft term ; therefore, as it fallsnbsp;in the fame interval, the fourth term muft be a tenthnbsp;higher than the third term.

the firft interval, to 2 -Pd-s in the fecond interval j which divifion muft be reckoned 22500 : For had thenbsp;point fell in the firft interval, it would have been onenbsp;place more than the 375, bccaufe 60 is one placenbsp;more than i �, but as it falls in the fecond interval,nbsp;every of whofe divifions is one place higher than thofenbsp;in the firft interval, therelbre, it muft have twonbsp;places more than 375, which is taken in the firft interval.

If the operations in thefe examples be well con-fidered, it will not be difficult to apply others to the fcale, and readily to affign the value of the refult.

Solution. The extent from 4 to i, in the firft interval, will reach from 36 in the fecond interval to nine in the firft.

It is to be obferved, that when the fecond term is greater than the firft term; the extents are reckonednbsp;from the left hand towards the right : and when thenbsp;fecond term is lefs than the firft, the extents are takennbsp;from the right hand towards the left: that is, the extents are always counted the fame way towards whichnbsp;the terms proceed.

of Mathematical Infruments. 8^

16 nbsp;nbsp;nbsp;in the fecond interval; from 16 to 32; from 32 tonbsp;64 ; ^c. Alfo the fame extent will reach from i 4- to

3 nbsp;nbsp;nbsp;; from 3 to 6 ; from 6 to 12 ; from 12 to 24 1nbsp;from 24 to 48 i �f. And the fame extent will reachnbsp;from 2 f to 1; �, from 5 to 10; from 1 o to 20 �, fromnbsp;20 to 40 ; i^c. And in a like manner proceed, if anynbsp;other ratio was given befides that of i to 2.

This Example is of ufe, to find if the divifions of the line of numbers, are accurately laid down on thenbsp;fcale.

There are many other ufcs to which this fcale of log. numbers are applicable, and on which feveralnbsp;large treatifes have been wrote ; but the defign here,nbsp;is not to enter into all the ufes of the fcales on thenbsp;feftor, only to give a few examples thereof; but afternbsp;all that has been faid, when examples are to be wroughtnbsp;whofe refult exceeds three places, �tis beft to do it bynbsp;the pen, for on inftruments, altho� they be very largenbsp;ones, the loweft places of the anfwers, at'beft, are butnbsp;guefs�d at.

Sect. XVI.

These fcalesare chiefly ufed in the folution of the cafes of plain and fpherical trigonometry, which will be fully exemplified hereafter : But innbsp;this place, it will be proper to fhew, how proportional terms are applied to the fcales.

In plane trigonometrical proportions, there are always four terms under confideration ; fuppofe two fides and two angles, whereof, only three of the termsnbsp;are given, and the fourth is required: Now the fidesnbsp;in plane trigonometry, are always applied to the fcalenbsp;of log numbers ; and the angles are either appliednbsp;to the log. fines, or to the log. tangents; according asnbsp;the fines or tangents are concerned in the proportion.nbsp;Therefore, when among the three things given, ifnbsp;two of them be fides, and the other an angle ; or ifnbsp;two terms be angles, and the other a fide.

Rule. On the fcale of log. numbers, take the extent between the divilions expreffing the fides ; and this extent applied from the divifion expreffing the angle given, will reach to the divifion ftewing the anglenbsp;required.

Or, the extent of the angles taken, will reach from the fide given to the fide required, on the line ofnbsp;numbers.

So in fpherical trigonometry, where fome of the cafes arc worked wholly on the fines, others partly onnbsp;fines, and partly on tangents ; the extent taken withnbsp;the compalTes, between the firft and fecond terms,nbsp;when thofe terms are of the fame kind, will reachnbsp;from the third term to the fourth.

Or, the extent from the firft term to the third, when they are of the fame kind, will reach from thenbsp;fecond term to the fourth.

Sect. XVII.

PROBLEM XX.

Given the radius of a circle ( fuppofe equal to 2 inches) required the fine, and tangent of q,o' to that radius.

Solution. Open the fedor fo that the tranfverfe diftanceof 90 and 90, on the fines; or of 45 and 45nbsp;on the tangents j may be equal to the given radius ;nbsp;viz. two inches : Then will the tranfverfe diftance ofnbsp;28�� 30', taken from the fines, be the length of thatnbsp;fine to the given radius; or if taken from the tangents,nbsp;will be the length of that tangent to the given radius. '

S6 nbsp;nbsp;nbsp;�fhe Defcription and XJfe

Make the given radius two inches, a tranlVerfe diftance to o and o, at the beginning, of the linenbsp;ot' fecants j and then take the tranfvcrfe diftance ofnbsp;the degrees wanted, viz. 28� 30'.

Make the given radius, fuppofe 2 inches, a tranfverfe diftance to 45 and 45 at the beginning of the fcale ofnbsp;upper tangents; and then the required degrees 60� 00'nbsp;may be taken Iroiii this fcale.

1 he fcales ol upper tangents and fecants do not run quite to 7 6 degrees } and as the tangent and fecantnbsp;may be fometimes wanted to a greater number of degrees than can be introduced on the fedor, they maynbsp;be readily found by the help of the annexed table ofnbsp;the natui al tangents and fecants of the degrees abovenbsp;75} the radius of the circle being unity.

Meafurc

Meafure the radius of the circle ufed, upon any fcale of equal parts. Multiply the tabular number by thenbsp;parts in the radius, and the produft will give the lengthnbsp;of the tangent or fecant fought, to be taken from thenbsp;fame fcale of equal parts.

Exam Requ red the length of the tangent and fecant cf 80 degrees to a circle whofe radiui-, meaftired on a fcalenbsp;of 2^ parts to an inch, is of thoje parts ?

5.759 47.5

So the length of the tangent on the twenty-fifth fcale will be 2694- nearly. And that of the fecantnbsp;about 2734-

Or thus. The tangent of any number of degrees may be taken from the feftor at once ; if the radiusnbsp;of the circle can be made a tranfverfe diflance to thenbsp;complement of thofe degrees on the lower tangent.

Make two inches a tranfverfe diftance to 12 degrees on the lower tangents ; then the tranfverfe diftance ofnbsp;45 degrees will be the tangent of 78 degrees.

In like manner the fecant of any number of degrees may be taken from the fines, if the radius of the circle can be made a tranfverfe diftance to the cofine ofnbsp;thofe degrees. Thus making two inches a tranfverfe diftance to the fine of 12 degrees; then the tranfverfe diftance of 90 and 90 will be the fecant of 78nbsp;degrees.

From hence it will be eafy to find the degrees anfwering to a given line, exprefling the length of a

tangent or fecant, which is too long to be meafured on thoih fcales, when the feftor is fet to the givennbsp;radius.

Thus. For a tangent, make the given line a tranf-verle diltance to 45 and 45 on the jower tan.eents ; then take the given radius and apply it to the lowernbsp;tan -ents ; and the degrees where it becomes a tranf-verfe diflance is the cocangent ot the degrees anfwer-ing to the given line.

And rbr a fecant. Make the given line a tranf-verfe diftance to 90 and 90 on the fines. Then the degrees anfwering to the given radius applied as ^nbsp;tranfvcrle diftance on the lines, will be the co-fmenbsp;of the degrees anfwering to the given fecant line.

PROBLEM XXL

Given the length of the fine, tangent, or fiecant, of a-'y degrees ; to find the length of the radius to that fine,nbsp;tangent, or fecant.

Make the given length, a tranfverfe diftance to its given degrees on its refpedtive fcale ; Then,

' In the ioiver tangents. The tranfveife diftance of 45 and 45 near the end, of the fedor will be the radiusnbsp;iought.

In the upper tangents. The tranfverfe diftance of 45 and 45 taken towards the centre of the fedlor on thenbsp;line of upper tangents, will be the centre fought.

In the fecants. The tranfverfe diftance of o and o, or the beginning of the fecants, near the centre of thenbsp;fedor, will be the radius fought.

PR OB-

PROBLEM XXII.

Given the radius and any line reprefer ting a fi^e, tangent or fecant to find the degrees correfponding to that line.

Solution. Set the feftor to the given radius, according as a fine, or tangent, or fecant is concerned.

Take the given line between the compaffes; apply the two feet tranfverfly to the fcale concerned, andnbsp;Aide the feet along till they both teft on like clivi-fions on both legs ; then will thofe divifions fhewnbsp;the degrees and parts correfponding to the givennbsp;line.

PROBLEM XXIII.

fofind the length of a verfed fine to a given number of degrees, and a given radius.

Make the tranfverfe diftanceof 90 and 90 on the fines, equal to the given radius.

If the given degrees are lefs than 90, the difference between the fine complement and the radius, givesnbsp;the verfed fine.

If the given degrees are more than 90, the fum of the fine complement and the radius, gives the verfednbsp;fine.

PROBLEM XXIV.

Ho open the legs of the feElor, fo that the correfponding double fcoles of lines, chords, fines, tangents, may make,nbsp;each, a right angle.

On the lines, make the lateral diftance 10, a diftance between 8 on one leg, and 6 on the other leg.

On the fines, make the lateral diftance 90, a tranfverfe diftance from 45 to 45 j or from 40 to 50 ; or

5 o nbsp;nbsp;nbsp;T'he Defcriptkn and Ufe

Or on the fines, make the lateral diftance of 45 a tranfverfe diftance between 30 and 30.

PROBLEM XXV.

0 deficribe an Elliffits,' having given ab equal to the longejl diameter �, and cd equal to the JhorteJi diameter.

fines; and take he tranfverfe diftances of 10�, 20% 30% 40�, 50�, 6o�, 70�, 8o�, fucceflively, and apply

3d Make EC a tranfverfe diftance to 90 and 90 on the fines; take the tranfverfe diftances of 80�, 70�,nbsp;60�, 50�, 40�, 3amp;�, 20�, 10�, fucceflively, and applynbsp;thofe diftances to the parallel lines from i to i, 2 to 2,nbsp;3 to 3, 4 to 4, 5 to 5, 6 to 6, 7 to 7, 8 to 8, andnbsp;fo many points will be obtained thro� which the curvenbsp;of the ellipfis is to pafs,

The fame work being done in all the four quadrants, the elliptical curve may be compleated.

1 ills Problem is of confiderable ufe in the con-ftruftion of folar t'clipfes; but inftead of ufing the fines to every ten d-rgeces, the fines belonging to thenbsp;degrees and minutes c 'rrelpondiiig to the hours, andnbsp;quarter hours are to be uled.

PROBLEM XXVI.

like a line of fines to every ten degrees, as at the points 10, 20, 30, 40, 50, fcfc. andnbsp;thro� thefe points draw lines at right angles to the

2d. Make the lines Aa, lob, 20c, 30J, 40^, refpedively equal to the chords of 90� 80�, 70�, 6o�,nbsp;509, to the radius ab, and the points a, b, c, d, e,nbsp;{3c. will be in the curve of a parabola.

Therefore a fmooth curve line drawn thro� thofe points and the vertex b, will reprefcnt the parabolicnbsp;curve required.

The like work may be done on both Tides of the axis when the whole curve is wanted.

As the chords on the feftor run no farther than 60, thofe of 70, 80 and 90 may be found by taking thenbsp;tranfverfe diftance of the fines of 33�, 40�, 45� to thenbsp;radius ab, and applying thofe diftances twice alongnbsp;the lines 2Gf, 10^,

PRO-

problem XXVII.

Solution ift. The af-fymptotes bh, bi, being drawn in any poficiorr, rhe line ba,nbsp;bifcfting the angle ibh, andnbsp;the vertex a taken, draw ai,nbsp;AC, parallel to bh, ei.

2d. Make AC a tranfverfe diftance to 45 and 45 on thenbsp;upper tangents, and applynbsp;to the affymptotes from b,nbsp;fo many of the upper tangents taken tranfverfly as maynbsp;^ be thought convenient, asnbsp;ED 50�, BE 55\ BF 60� BG 63�, BH 70% and drawnbsp;Bii, �^, parallel to ac.

3d. Make ac a tranfverfe diftance to 45 and 45 on the lower tangents, take the tranfverfe diftance of thenbsp;co-tangents before ufed, and lay them on thofe parallel lines; thus make Dr/=4o�, �^=35�, p'f=:30%nbsp;c^=i:25�, H/?)i=:2o�, i�c. and thro� the points a, e, �,nbsp;(dc. If a curve line be drawn it will be the hyperbola required.

There are many other methods of conftruding the curves in the three laft problems, and a multitude ofnbsp;entertaining and ufeful properties which fubfift amongnbsp;the lines drawn within and about thefe curves, whichnbsp;the inquifitive reader w'ill find in the treatifes on conicnbsp;feftions.

PROBLEM XXVin.

Tl? find the difiance of places, on the terrefirial glohe ly having given their latitudes and longitudes.

Case I. If both the places are under the equator. Then the difference oi longitudeqs their diftance.nbsp;Case II. When both places are under the famenbsp;meridian.

Then the difference of latitude is their difl:ance. Case III. When only one of the places has latitude,nbsp;but both have different longitudes.

^LUTiox I ft. With the chord of 6�* defcribe a circlenbsp;xeprefenting the equator, wherein take a point c to reprcfentnbsp;the beginning of longitude.

of Bermudas longitude 68� 38' to E, and that �f St. 'Thomas^nbsp;longitude to a, the arc ab, itnbsp;being the difference of longitude.

3d. From B, the place having latitude, draw the diameter bd, apply the chord of the latitude' 32� 25,nbsp;from E to E, and draw ef at right angles to bd.

4th. Draw Fc, make fg, equal to fc, and draw EG ; then EG meafured on the chords will give thenbsp;diftance fought, about 73 degrees.

Hhe Defcriptlon and life

Exam. Required the dijiance between the Lizard and Pengwin IJland, both in latitude 49�, 56' N. the longitude of the Lizard being 5� 14' W. and that lt;?/Pengwinnbsp;IJland i'd W.-

Solution ift. From c, the commencement of the longitude, apply the chord of thenbsp;Lizard�s longitude to a, and ofnbsp;Penguin�s longitude to b, andnbsp;draw the diameters Aa, 'Rb.

Apply the chord of the common latitude 49� 56' fromnbsp;A to D, and from b to e ; drawnbsp;DF and EG at right angles tonbsp;A�, and join gf ; then gf meafured on the chordsnbsp;will give the diftance fought, about 29 degrees.

Case V. When the given places are on the fame fide of the equator, but differ both in latitude andnbsp;longitude.

Exam. What is the dijiance between London in latitude 32' N. longitude., Q� 0' and Bengal in latitude 22� 0' N. longitude 92� 45' E.

Solution. From a, Lon-don�s longitude, apply Bengal�s longitude 92� 45' to c, taken from the chords ; alfonbsp;apply the chord of London�snbsp;latitude from a to b, and ofnbsp;Bengal�s latitude from c to d.

2d. Draw the diameters A�, cc, and be, df, at right anglesnbsp;to Aa, cc, and join fe.

3d. Make bg equal to df, and eh equal to ef, join cH ; 1 bus gh meafured on the chords will givenbsp;the diftance required, which is about 72 degrees.

Case VI. When the places are on contrary fides of the equator, and differ both in latitude and longitude.

Exam^ ft^hat is the dijlance between London in la~ titude 51* 32' N. longitude 0� 0' and Cape-Horn in latitude 55� 42' S. longitude 66� 00' W.

Solution ift. From a, Lon-i/on�s longitude, apply the chord of Cape-Horn�^ longitude to c,nbsp;draw the diameters Aa, cc-, alfonbsp;apply the chords of London'?, latitude from A to B, and ofnbsp;/forw�s latitude from c to d.

2d. Draw BE and d f at right angles to Aa, cc join ef andnbsp;m^e EG equal to ef,

3d. At right angles to Aa, draw gh, and make It equal to df ; join bh, which meafured on the chordsnbsp;will give the diftance required, which is about 123nbsp;degrees.

T'he JJfe of fame of the fingle and double Scales, applied in the Solution of the Cafes of plain �Trigonometry.

PROBLEM XXIX.

IN any right lin'd plane triangle, an? three of the fix terms, viz. fides and angles, (provided one of themnbsp;he a fide) being given, to find the other three.

The Solution of the examples falling under this cafe depend on the proportionality there is betweennbsp;the fides of plane triangles, and the fines of theirnbsp;oppofite angles.

Then the fum of the angles b and c being taken from 180�, will leave the angle A.

The extent from 64 (z=ac) to 56 (=ab) on the fcales pf logarithm numbers, will reach from 46� 30'nbsp;(= Ab) to 39� 24', (=Ac.) on the fcale of logarithmnbsp;fines..

of Mathematical Infnimnts,

The extent from 46� 30' (=:^-b) to 85� 54' the fupplement of 94� 6'nbsp;nbsp;nbsp;nbsp;on the fcale of log fines,

2. nbsp;nbsp;nbsp;Make this a tranfverfe diftance of 46� 30' i=jLB)nbsp;on the fines.

4. nbsp;nbsp;nbsp;Find the degrees to which this extent is a tranfverfe diftance on the fines, wz. 39� 24'; and this isnbsp;the angle fought.

3. nbsp;nbsp;nbsp;fake the tranfverfe diftance of 85'� 54' (the fup-pleinent of 94� 6' = Z_ a) on the fines.

4. nbsp;nbsp;nbsp;Find the lateral diftance this extent is equal to,nbsp;on the lines; and this diftance, viz, 88, will be thenbsp;fide required.

The

98 nbsp;nbsp;nbsp;'The Defcription and Ufe

The extent from 48� o' (� La) to 28� o' (=Lc) on the fcale ot logarithm fines, will reach from 74nbsp;(=Bc) to 46, 75, (=:AB,) on the fcale of logarithmnbsp;numbers.

The extent from 48* 0' to 76� 0'(= fupplement of 104� 0') on the fcale of log. fines, will reach fromnbsp;74 to 96, 6 (=;Ac)on the fcale of logarithm numbers.

2. nbsp;nbsp;nbsp;Make this extent a tranfverfe diftance to 48� 0'nbsp;(= Aa) on the fines.

3. nbsp;nbsp;nbsp;Take the tranfverfe diftance of 28� 0' (=Ac)nbsp;on the fines.

4. nbsp;nbsp;nbsp;To this extent find the lateral diftance on thenbsp;lines, vise,. 46,75 and this will be the length of ab.

2. nbsp;nbsp;nbsp;Make this extent a tranfverfe diftance to 48� 0'nbsp;(= la) on the fines.

3. nbsp;nbsp;nbsp;Take the tranfverfe diftance to 76� 0' the fup-plement of 104.� 0' (= Z.b) on the fines.

4. nbsp;nbsp;nbsp;To this extent, find the lateral diftance on thenbsp;lines, viz. 96, 6, and this will be the length of AC.

And the angles are readily found by their half fum and half difference being known.

Take the given angle 68� 0'from 180�, and half the remainder, viz. 56� o' is the half fum of the unknown angles which call z; and let x ftand for thenbsp;half difference of thofe angles.

Then the fum of z and x gives the greater angle a.

loo 1�he Defcription and Ufe

Take the extent from 126 {� fum of the given fides) to 22 (�diff- of thofe fides) on the fcale ofnbsp;logarithm numbers; lay this extent from 45� d tonbsp;the left on the logarithm tangents �, ftay the loweftnbsp;point, and bring that which relfed on 45 degrees, tonbsp;56� 0'; remove the compafs, and this extent laid fromnbsp;45� d towards the left, gives 14 . 31'equal n.

The extent from 41� 29'(= 4.c) to 68� 0' (= z. b) on the logarithm fines, will reach from 52 (� ba) tonbsp;7 2, 75 (� ac) on the fcale of logarithm numbers.

In finding the tangent of (n, or) the half difference of the unknown angles, there were two applications of the compafles to the fcale of tangents: Now this happens becaufe the upper tangents whichnbsp;Ihould have been continued beyond 45�, or to thenbsp;right hand, are laid down backwards, or to the leftnbsp;hand, among the lower tangents (the logarithmicnbsp;tangents afeending and defeending by like fpaces atnbsp;equal diftances on both fides of 45^), and therebynbsp;the length of the fcale is kept within half the lengthnbsp;neceflliry to ky down all the tangents in order, fromnbsp;th� left towards the right. Hut fuppofing they werenbsp;fo laid down, then the point of 56� 0' will reach asnbsp;far to the right of 45� as it does now to the left, and

the extent on the numbers from 126 to 22 would reach from the point 56� taken on the right of 45�, to 14� gi'nbsp;at one application; the faid extent being appliednbsp;from 45� downwards, will-reach as far beyond 14� 31',nbsp;as is the diftance from 45� to 56�; therefore the legsnbsp;of the compafles being brought as much clofer as isnbsp;that interval, will reach trom 45� to the degreesnbsp;wanted.

IftDEED when the half fum is lefs than 45�, then the extent from the fum of the fides to their difference,nbsp;will reach from the tangent of the half fum, downward, to the tangent of the half difference, at once.

And when the half fum of the unknown angles, and their half difference, are both greater than 45�,nbsp;then the extent from the fum of the fides to their difference, will reach from the tangent of the half fumnbsp;of the angles, upwards (or to the right) to the tangent of the half difference of thofe angles, at once.

Becaufe 126 the fum of the fides will be longer than the fcales of fines, therefore take 63, the half ofnbsp;126, and II, the half ot 22, the difference of thenbsp;fides �, for the ratio of 63 to 11, is the fame as that ofnbsp;1261022. Then

2. nbsp;nbsp;nbsp;Make this extent a tranfverfe diftance to 56nbsp;degrees, on the upper tangents.

3. Take the tranfverfe diftance of 45� on thenbsp;upper tangents, and make this extent a tranfverfe diftance to 45� on the other tangents.

To this extent, find the tranfverfe diftance on the tangents, and this will be 14� 3 \ � n.

And this is the manner of operation, when m is greater than 45 degrees, and n is lefs.

102 nbsp;nbsp;nbsp;�the. Deferiftlon and life

Now having found the angles a and c, the fide AC may be found as in the firft or fecond examples.

But in this cafe, the third fide ac may be found without knowing the angles. Thus,

1. nbsp;nbsp;nbsp;Take the lateral diftance of (34 deg.) the halfnbsp;of (68,) the given angle, from the fines,

2. nbsp;nbsp;nbsp;Make this extent a tranfverfe diftance, to 30nbsp;on the fines.

3. nbsp;nbsp;nbsp;With the fedlor thus opened, take the diftancenbsp;from 74 on one leg, to 52 on the other leg, eachnbsp;reckon�d on the lines.

4. nbsp;nbsp;nbsp;The lateral diftance, on the lines, of this extent,nbsp;gives the fide ac = 72, 75.

From the two firft articles of this operation, is learn�d how to fet the double fcales to any givennbsp;angle.

When the included angle b is 90 degrees, the angles a and c are more readily found, as in the following example, whofe foluiion depends on this principle. That one of the given fides has the fame proportion to radius, as the other given fide has to thenbsp;tangent of its oppofite angle.

And the a.a taken from 90� leaves the /-C, then ac may be found as direfted in the laft example.

The extent from 45 (� ab) to 65 (=bc) on the numbers, will reach from 45 degrees to 55� iS'nbsp;(= /.a) on the tangents.

Here the angle a is taken equal to 5;,� 18', be-caufe the fecond term bc is greater than the firft term AB ; But if the terms were changed, and it was madenbsp;BC to AB, then the degrees found would be 34� 4Tnbsp;= Z.C.

1. nbsp;nbsp;nbsp;Take the lateral diftance of the firfl: term,nbsp;from the lines.

2. nbsp;nbsp;nbsp;Make this a tranfverfe diftance to 45 deg. onnbsp;the tangents.

3. nbsp;nbsp;nbsp;Take the lateral diftance of the fecond term,nbsp;from the lines.

4. nbsp;nbsp;nbsp;The tranfverfe diftance of this extent, foundnbsp;on the tangents, gives the degrees in the angle fought.

If the firft term is greater than the fecond, then the lateral diftance of the firft term, muft be fet to 45 degrees on the lower tangents, and the lateral diftancenbsp;of the fecond term, muft be reckon�d on the famenbsp;tangents.

But if the firft term is lefs than the fecond, then the lateral extent of the firft term muft be fet to 45� onnbsp;the upper tangents, and the lateral extent of the fecondnbsp;term muft be reckon�d on the fame tangents.

104 nbsp;nbsp;nbsp;�nbsp;nbsp;nbsp;nbsp;Defcription end JJfe

There are ulually given for the folution of this cafe by the logarithtnu: fcales two methods j the onenbsp;beft when ah the angles are to be found, the othernbsp;beft wlien one angle only is wanted ; both methodsnbsp;will be^here delivered.

Suppose a perpendicular ad (PI. VI. Fig. 30.) drawn 10 the greateft fide bc, from the angle a op-polite thereto ; then ad divides the triangle abc intonbsp;two right angled triangles bda, cda; in which ifnbsp;CD and DB were known, the angles would be found,nbsp;as in the folution or Cafe I.

Then on the fcale of numbers, the extent from 526 ( � Bc) to 1260, will reach from 14410 196.

The extent from 558 (= ba) to 365 (=bd) on the numbers, will reach on the log. fines from 90�nbsp;(= Z.BDA) to 40� 52' (=2LBAD.)

And the extent from 702 (=ca) to 561 (= cn) on the numbers, will reach from 90� (� Acda) tonbsp;53� 04' (= /-CAD) on the fcale of log. fines.

Take the difference between eg and ba, the fides including the angles fought, and call it o == 368.

1. nbsp;nbsp;nbsp;The extent on the log. n�nibers from i to 535nbsp;(=: z), will reach from 167 (= x ) to a 4th point;nbsp;mark it and call it g.

2. nbsp;nbsp;nbsp;The extent from i to 558 (= ab), will reachnbsp;from 9�6 (= Bc) to a 4th point 5 mark it and callnbsp;itH.

3. nbsp;nbsp;nbsp;The extent from the point h to the point g,nbsp;will reach from i, downward to a 4th point, mark itnbsp;and call it K.

4. nbsp;nbsp;nbsp;The extent from k, to the middle point betweennbsp;it and the 1 next above k, taken on the log. numbers,nbsp;will reach on the log. fines from 90quot; to 24� 34', whichnbsp;doubled gives 49� 8' for the angle b.

But the fcale of log. verfed fines being ufed, the work will be confiderably fhortened. Thus,

1. nbsp;nbsp;nbsp;On the log. numbers take the extent from 535nbsp;(=z.) to 926 (=: bc), this will reach from 558 ( = ba)nbsp;to a 4th point, where let the foot of the compaffesnbsp;reft.

2. nbsp;nbsp;nbsp;Then the extent from that 4th point 10167nbsp;(=x), will reach on the line of verfed fines fromnbsp;o degrees (at the end) to 130� 52', which taken fromnbsp;j 80� leaves 49� 8' for the angle b.

1. nbsp;nbsp;nbsp;Take the lateral diftance 702 (�ac, the fidenbsp;eppofite to the angle b) from the lines,

2. nbsp;nbsp;nbsp;Open the legs of the feftor until this extent willnbsp;reach from 926 (= cb) on one fcale of lines, to 558nbsp;(= ab) on the other fcale of lines.

3. nbsp;nbsp;nbsp;The feftor being thus opened, take the tranf-verfe diftance between 30� and 30� on the fines, thisnbsp;diftance meafured laterally on the fines, one footnbsp;being on the Centre, will give 24� 34' for half the angle B.

The other angles may be found as A-e was, or according to the ditedions in fome of the precedingnbsp;cafes.

Although in thefe examples, oblique triangles were taken as being the moft general; yet it may benbsp;readily feen, that thofe concerning right-angled triangles are only particular cafes, and may be, for thenbsp;general, more eafily folved.

Variety of other examples, {hewing the ufes of thefe fcales, might be given in various parts of thenbsp;mathematics, which the reader may of himfelf fup-ply : However here will be fubjoined a few in fphe-rical trigonometry, as they will include fome operationsnbsp;not only curious, but perhaps not to be met withnbsp;elfcwhere.

T})e CoriftruBion of the feveral cafes of Spherical �Triangles by the Scales on the SeSior.

In the following folutions, are given three con-ftruftions to every cafe, whereby each fide is laid on the plane of projedion, or (as it is commonly called,nbsp;the) primitive circle.

To abbreviate the diredions given in the following conftrudions, it is to be underftood, that the primitive circle is always firft defcribed, and two diametersnbsp;drawn at right angles.

The fedorisalfo fuppofed to be fet to the radius wanted, on the fcale tiled ; and the tranverfe diftanccnbsp;of the degrees propofed is to be taken from the chords,nbsp;or fecants, or tangents, amp;c. according to the namenbsp;mentioned in the conftrudion.

'The JDefcription and Ufe

2d From D, with the fecant of the l. d� 25� 55', cut the diameter O i in c ; on c as a center, withnbsp;that radius, defcribe the circumference da, and thenbsp;angle bda will be 25* 55'.

4th. With the tangent of ab, 29� 30', from J, cut � B produced in h and from h, with that radius, cutnbsp;Da in A or a.

5th, Through b, a, e, defcribe a circumference, and the triangle bda will be that required �, whofcnbsp;parts DA, /.B, and jLa may be thus meafured.

6th. Make o p equal to the tangent of half the angle bda, wz. 12� 57V ; then a ruler on p and A,nbsp;gives e ; and d e meafured on the chords, gives thenbsp;degrees in da, viz. 42� 9'.

7th. Draw the diameter fg at right angles to be, cutting the circumference bae in r �, a ruler by bnbsp;amp; ^ gives � j make fg equal to the chord of 90 deg,

of Mathematical Infruments. 109

a ruler on g and b, gives p in the diameter fg. Then on the chords gives the angle b = 36� 9'.

8 th. A ruler on a and p, gives n ; and on a and p, gives m; and nm meafured on the chords, givesnbsp;52� 9, for the fupplement of the angle dab, which isnbsp;127� 51'.

, ill. With the fecant of theangle d, 25�, 55', from D, cut the diameter in c ; and on c, with the famenbsp;radius, defcribe the arc db, and the angle bda willnbsp;be 25�, 55'.

3d. On the primitive circle, make D d equal to the given fide db, with the chord of 63� 59'.

10th. On/e as a diameter, defcribe a circumference, cutting the primitive circle in a, lt;2.

12th. Through a, b, f, defcribe a circumference, and the triangle abd is conftrudted with da on thenbsp;primitive circle as required.

3d. Make the angle b a g, � L. n 25� 55'; from A draw a ^ at right angles to a g^ and fromnbsp;d, the middle of a b^ draw de perpendicular to a b,nbsp;cutting A^, in ^ ; from with the radius e a, deferibenbsp;a circumference Aph.

4th. Fromi^', with the fine of BD, 63� 59', cut the circumference Apb inf-, and draw a/.

5th. From a, draw ac at right angles to � a, meeting e � (perpendicular to a O,) continued, in cnbsp;and on c, with the radius ca, deferibe a circumference adg.

6th. Make Bm= bd, with the chord of 63� 59'} from m, with the tangent of 63� 59'' cut o b produced, in � j on K, with the fame radius, cut adonbsp;in D.

7th. Through b, d, f, deferibe a circumference, and the triangle abd will be that which was required.

The fines of the angles of fpheric triangles are as th^ fines of their oppofite Tides.

Then the extent of the compafles on the line of fines from 29� 50' (= ab) to 25� 55' (= 4. c) ;nbsp;will reach from the fine of 63'� 59'' (= cb) to the finenbsp;of 52� 9^ (= l~ a).

But by conftruftion the 4. a is obtufe therefore 127� 5�'' (the fupplement of ^2.� 0') is to be takennbsp;lor the angle quot;

Then

of Mathematical Infriments. m

The extent from the fine of 90� to the fine of 26� i' (= comp, of 63� 59^), will, on the tangents,, reachnbsp;from 25� 55' to 12� 2' : But the 4th arc is to be anbsp;cotangent; therefore 77� 58'' (the comp. 12quot; 2') is ,nbsp;the 4th arc.

The extent from the tangent of 29� 50' to the tangent of 63� 59', will reach on the line of fines from . 12� 2' (= comp, of 77� 58O to 48� f.

But the 5th arc is to be a cofine; therefore 41� 51' (the comp, of 48� f) is the fifth arc.

The extent from the tangent of 29� 50' to the tangent 63� 59' is thus taken. Set one foot on thenbsp;tangent 29� 50', and extend the other to the tangentnbsp;of 45�: Apply this extent on the tangents from 63�nbsp;59' towards the left; reft the left hand foot, and extend the other to 45�, and the compaffes will then havenbsp;the required extent.

The extent on the fines from 90� to 64� f comp, of 25� 55') will reach on the tangents fromnbsp;63� 59' towards tlte right to 61� 31' the 4th arc,

Alfo the extent on thefines from 26� \' (= comp, of 63� 59^) to 60� \o' (= comp, of 29� 50') willnbsp;reach from the fine of 28� 29'' (the complement of 61�nbsp;31') to the fine of 70� 37'.

But

Tiefcription and life

ift. FromB , with the fecant of L b, 36� 8', cut the diameter O e in c ; on c, with the fame radius.

2d. Make the angle naq equal to 37* 50', the difference between 127� 50' and 90�.

3d. Make aq = tangent of da, 42� 9'; on 0 with the fecant of 42� 9' deferibe an arc qqj. on c withnbsp;c q, cut the arc q q^in cn

4th. Draw 0^0 G cutting the primitive circle in D, and bd will be a fide of the triangle.

5th. From with on, cut b^f in a j and through 0, A, G, deferibe a circumference, and thenbsp;triangle bad is that required. Whole parts bd, banbsp;and i. D are thus meafured.

7th. Make� p= tangent of halOz. b, u/z.i8�4'; ' a ruler on p and a gives x; then ba? meafured onnbsp;the chords gives 29� 50', for ba.

8th. Draw a diameter perpendicular to gd, cutting the circumference dag in j ; a ruler on d and s gives OT ; make mn 90 degrees, then g n meafurednbsp;on the chords, gives 25� 55' for the L.-D.

of Mathematical Infrumcnts. ir^

I ft. From A, with the fecant of the fupplement of the Z_A, viz. 10'', cut the diameter Of continued in c; on c, v;ith the fame radius, defcribe anbsp;circumference a^e.

2d. Make O p = the tangent of half the fupplement of Z_ A, viz. 26� 5'; and make a x = chord of AD, 42� 9^: a ruler on p and x, gives dj then isnbsp;AD equal to 42� 9^.

3d. On G, with the tangent of the angle b, 36� defcribe an arc me-, on d, with the fecant ofnbsp;i- B, 36� 8^, cut the arc rnc in c; on r, with thenbsp;fame radius, defcribe a circumference db, then thenbsp;triangle adU, will be that required.

I ft. Lay down ad with the chord of 42� 9': Draw the diameter dfi and another Oh, perpendicular to DF.

2d. On a, with the fecant of the fupplement of /L A, viz. 52� 10^, cut the diameter e 0 in c;nbsp;and on c, with the fame radius, defcribe the circumference abg.

3d. Make O p equal to the tangent of half the fupplement 4. a, viz. 26� f a ruler by o and pnbsp;gives X.

4th. Make xm = xn with the chord of 4.B, 36� 8''*, a ruler by g and n gives r, by G and m gives J �, on tgt;nbsp;the middle of rs, with the radius hs, cut O h in p.

5th. A ruler on f and p, gives h % make hk � ^d ; a ruler or f and k gives c, with the radius c d, defcribenbsp;the circumference def; and the triangle abd, is thatnbsp;fought.

Then the extent from the fine of 36� 8'' to the fine of 42� will reach from the fine of 52� 10^ fdienbsp;fupplement of 127� 50') to the fine of 63� 59^ =nbsp;fide BD.

The extent from the radius, or the fine of 90� to the fine of 37� 50' (the complement of 52� io), willnbsp;reach on the tangents, from 42� 9' to 29� 02'' =: 4thnbsp;arc.

And the extent from the tangent of 36� 8' to the tangent of 52� 10'', will reach on the fines from 29�nbsp;0%' ro 58� 54' 5th arc.

Then the difference between 58� 54'' and 29� o%' gives 29� 5�/ for the fide ab.

Ihe extent from the tangent of 36� 8' to 'he tangent of 52� jo' is taken as lliewed in the fecund operation of the firfl cafe.

of Mathematical Injlruments.

Now the extent from the fine of 9 � to the fine of 47� 5�^ complement ot 42� 09'), will reachnbsp;from the tangent of 52� c' to the tangent of 44� 40'.nbsp;But the 4th arc being a cotangent will be 46� 20, thenbsp;complement of 43� 40'.

Alfo the extent trom the fine of 37� 50' (the complement of 52� 10^) to :he fine of 53� 52' ^the compliment of 36� 08^), will reach from the fine of 46� 20' to the fine of 72� 14' the 5th arc.

Then the difference be ween 72� 15' and 46� 20^ viz. 25� 54'' will be the angle d,

In applying the firft extent, viz. from the fine of 90� to line ot 47�nbsp;nbsp;nbsp;nbsp;1', to the tangents; fet one foot

on the tangent of 45� and let the o�her too reft where it falls; move the foot from 45� to 52� lo'j thenthijnbsp;extent will reach from 45� to 43� 40'.

I ft. Make ab = chord of 29� 50'', draw the diameter b�f, and another G e perpendicular thereto.

3d. From O, with the tangent of half A e, viz '. 18� 4�^5 cut O E in p, the pole of bdf.

4th. Make B A' = BD, 63� 59'; a ruler on p and a, gives D. Through a, d, g, defer.be a ciicumference,nbsp;and the triangle adb is that required, whofe parts ad,nbsp;A A, and A D may be thus meafured.

ii6 nbsp;nbsp;nbsp;Defcrlptlon and Ufe

gth. A ruler on a and s gives z, make z y � chord of 90�; a ruler on a and y gives p the pole of a j o;nbsp;a ruler on p and d, gives and a n meafured on thenbsp;chords gives 42� 8' for ad.

6th. Gjy meafured on the chords, gives 52� ii'' for the fupplement of a. a ; therefore Z. a = 127� 49''.

7th. A ruler on d and p gives r, on d and p, gives m; and rm, m.^afured on the chords gives 2 5* 56^ fornbsp;the angle bda.

iff. Make db = chord of 63� 59': draw the diameter BF and perpendicular thereto, the diameter O o.

ad. From b, with the fecant of Z-b, 36� 8', cut G G in cj on c with cb, defcribe the circumferencenbsp;baf.

3d Make G p � tangent of half Z. b, 18� 4'', and D X = chord of AB 29� 50^, a ruler on p and tt givesnbsp;a; through d, a, e, defcribe a circumference, and thenbsp;triangle abd is that required.

1. nbsp;nbsp;nbsp;In a right line ri, touching the primitive circlenbsp;in any point b, take hd = tangent of bd, 63� 59^;nbsp;and be � tangent of ab, 29� 5ok

3. nbsp;nbsp;nbsp;Fromi, �, with 0(2, �f, defcribe arcs croffingnbsp;in from x, d, draw the diameters ae, df; andnbsp;others og, oh, perpendicular to ae, fd.

4. nbsp;nbsp;nbsp;From d, a, with bd., eb, defcribe arcs croffingnbsp;in E; and draw Jb, ab.

5. nbsp;nbsp;nbsp;From b draw bc, perpendicular to ab, andnbsp;meeting �g. produced in calfo draw bc perpendicular to (/b, and meeting Gh in c; then c is the centernbsp;of a circumference through a, b, e ; and c the centernbsp;�of that through d, b, f; and the triangle abd is thatnbsp;required.

of Mathematical JnjlrumenU.

Then the half difference of the required angles added to their half fum v/ill give the greater angle a.

And rhe half difference of thofe angles taken from their half fum will give the leffer angle d.

Now the fum of the given fides 63� 59' and 29� is 93� 49', their difference is 34quot; 09^; the halfnbsp;fum = 46� 54^, and the half difference is 17� 04.fi.

Then the extent from the fine of 46� 54^ to the fine of 17� will reach from the tangent of 71� 56^ (thenbsp;complement of 18� fi) to the tangent of 50� 57^ thenbsp;half difference of the required angles.

Here the extent on the fines is from right to left or decreafing; fo,the extent on the tangents muff benbsp;from left to right, which in this cafe is decreafing.

Alfo the extent from the fine of 43� 6'' (the complement of 46� 540 to the fine of 72� 56'' fthe complement of 17� 04''), will on the fcale of tangents reach from^ 71� 56' (the complement of 18� fi) tonbsp;76� 53'' the half fum of the required angles.

The angles being known, the other fide may be found by oppofite fides and angles, and is 42� 08'.

Which 4th arc will be like the fide ufed when the given angle is acute, otherwileic will be of a contrarynbsp;kind with the fide ufed.

Then take the difference between the 4th arc and the other given fide, call the remainder a 5th aic.

Now the extent from the fign of 90� to the fine of 43� 5 ' ( � complement of 36� 08') will reachnbsp;from the tangent of 29� 50^ to the tangent of 24� 51'nbsp;the 4th arc.

Then the extent from the fine of 65� 09' (the complement of .;4�nbsp;nbsp;nbsp;nbsp;) to the fine of 50� 52' (the

complement of ;.cp 08'') will reach from the fine of 60� 10' (the complement of 29� 50'') to the fine ofnbsp;47� 5T; whofc complement, viz, 42� 09' is the fidenbsp;required.

I. Make db = chord of 63� 59'; draw the diameter BF, and dr.aw Go perpendicular to bf.

of Mathematical Infrumenfs. 119

2 From B, with the fecant of /I b, 36� 8', cut �g in c; and c will be the center of baf.

3. nbsp;nbsp;nbsp;From D, with the lecant of Ad, 25� 55'j cutnbsp;�H in r, and c will be the center of dae; and thenbsp;triangle dab is that which w'as requiredj whofe partsnbsp;DA, BA, and Aa, are thus meaiured.

ruler on p and a gives x; then dx meafured on the chords gives 42� 10'' for ad.

5. nbsp;nbsp;nbsp;Make Gp == tangent of 4- Ab 18� 4', a ruler onnbsp;p and A, gives 2; then b2 meafured on the chords,nbsp;gives 29� 54^ for ab.

6. nbsp;nbsp;nbsp;A ruler on a and^, gives �, on a and p, gives w,nbsp;and nm meafured on the chords gives 52� 10'' the fup-plement of the angle a. Therefore Aa = 127� 50''.

2d. Make Gp = tangent of 4r Ad, 12� 574; and make dx ~ chord of bd, 6^� 59'; a ruler on p, v,nbsp;gives B; and db is 6^� 59''.

3d. Make the angle cbc = A B, 36� 8''v through c, draw me perpendicular to bG, cutting nr in r; onnbsp;r, with the radius rB, deferibe the circumference abgjnbsp;and the triangle abd, is that which was required.

ill. From B, with the fecant of Ab, 36� 8^ cut r�F in C', and c is the center of the circumference of

2d. Make bv =: chord of bd, 63� 59''; and Op = tangent of 4 A b, 18=4'; a ruler on p and a' gives d ;nbsp;then is bd = 63� 59''.

' 3d. Make the angle cor = /.d, 25� 55'; then rnc drawn perpendicular to G d, meeting or in r, givesnbsp;c the center of the circumlerence adgj and the triangle ABD will be that required.

J20 nbsp;nbsp;nbsp;^he Dcfcriptioji and Ufe

The 4 th arc will be of the fame kind with the angle firft ufed if the given fide is lefs than 90�; but of anbsp;contrary kind if that fide is greater than 90�.

Arcs are laid to be of the fame kind, when both are lefs, or both greater, than 90 degrees.

The required angle will be of the fame kind with the angle ufed in the proportions, if the 4th arc is lefsnbsp;than the other angle-, but of an unlike kind when thenbsp;4th arc is greater than the other angle.

Now the extent from the fine of 90* to the fine of 26� 01' (the complement of 63� 59'J will reach fromnbsp;the tangent of 25� to the tangent of 12� 02': Butnbsp;this is the complement of the 4th arc, which is

77� 5^'-

Then the extent from the fine of 77� 4S'* to the fine of 41� 50^ will reach from the fine of 64� 5' (thenbsp;complement of 25� 55') to the fine of 37� 50,nbsp;which is the complement to 52� lok

But as the 4th arc v/as greater than 36� 08', the angle fought is to be of a contrary kind to 25� 55'nbsp;(~ /_n), chat is, that a is to be obtufe; fo 127� 50'

of Mathematical Injlrumenfs. 121

Now all the angles and one fide being known, the other Tides may be found by the proportion fubfiftingnbsp;between the fines of angles, and the fines of theixnbsp;oppofite fides.

�Then the half difference added to the half fum gives the greater of the fought Tides.

And the half difference fubtradled from the half fum gives the leffer of the fought fides.

And the extent from the fine of 58� 59 (= complement of 31� 01') to the fine of 84� 54' (the complement of 5� 6'), will reach from the tangent of 31� 59' to the tangent of 35� 58'.

3d. From n, with the tangent of ad, 42quot; 9', cut EA produced in c; and from c, with that radius, de-fcnbe the are from with the tangent of bd,nbsp;63� 59', cut FB produced in c-, and from c, with thenbsp;radius cmy cut tuc re in d.

4th. 1 hrough A, D, E j B, D, F, defcribe circumferences, and the triang�e adb is that which was required ; whofe angles a, b, d, are thus meafured.

5th. A ruler on a and 0, gives x; on b and gives 2; make xy, zv, each 90�; a ruler on a and ynbsp;gives p, in a radius perpendicular to ae; and a rulernbsp;on B and v gives p, in a radius perpendicular to

6rh. Ej meafured on the chords, gives 52� 12' for the fupplement of the 4.a; thereiore 4.a = 127� 48'.

8th A ruler on d and p gives t, and on d and p gives S-, then is meafured on the chords, gives 25�nbsp;58' for the angle D.

The Tides ad, de, are put on the primitive circle, by a conttruftion lb like the foregoing one, that it isnbsp;neediefs to repeat it. See figures 2. 5. and 3. 5.

of Mathematical Injiruments.

And the difference of bd and x is 51quot; 40'; the half difference is 25� 50'.

Now take the extent on the line of fines, from the half fum 38� 9' to either of the containing fides, as tonbsp;29� 50^ j apply this extent from the other containingnbsp;fide 42� 09' towards the left, there let the foot reft,nbsp;and extend the other point {viz. that which was fet onnbsp;42� 09'') to the half difference 25quot; 50''; then this extent applied to the line of verfed fines, will reach fiomnbsp;o degrees (at the beginning) to 52� 12'; the lupple-ment of which, or 127� 48' will be the degrees innbsp;the angle a.

And ad = 42 09 Their difference call x =21 50nbsp;The fide oppofite to the is ab = 29 50

Then the extent on the fines from 25� 50' to 63 �^9' will reach from the fine ot 42� 09'' to fome pointnbsp;beyond 90�; therefore apply the extent between 25*nbsp;50' and 63� 59' from the fine of 90� downwards, letnbsp;the point reft where it falls, and bring that pointnbsp;which w'as fet on 90� to 42� of-, then will the difiance

between the feet fhew how far the firft extent would reach beyond 90�: Now apply this extent on the finesnbsp;from the point oppofite to the middle i on the linenbsp;of numbers, the other foot falling upwards to thenbsp;right, let it reft there, and extend the other foot tonbsp;the half difference 4� 0'; Then this extent applied tonbsp;the verfed fines, one foot being fet on the point oppofite the middle i on the line ot numbers, the othernbsp;foot will fall on 154� 5''; the fupplement whereof,nbsp;viz. 25� 55' will be the angle d.

ift. From B, with the fecant of 4b, 36� 8^, cut Gf in c, and c will be the center of the circumferencenbsp;through B, D, E.

2d. From G, with the tangent of 52� 10^ the fupplement of 4a, deferibe an arc xc.

3d. Make an angle ca^ = 4d, 25� 55'; make aq equal ex. (= fecant of 52� 10'.)

4th. From c, with the radius cq, cut xc in c-. From c, with the radius qa, deferibe a circumferencenbsp;ADO; and the triangle abd, is that which was required: whofe fides ab, bd, da, are meafured as follows.

5th. A ruler on b and a gives d, and on a and h, gives ��, make dg, fh, each 90 degrees; a ruler on_^nbsp;and B gives p, and on h and a, givesp, in Of, Gh,,nbsp;drawn perpendicular to be, ag.

7th.

7th. Then b�, ew, aw, mcafured on the chords, gives 29� 50^ i 63� 59'; 42� 9'; for the relpeftivenbsp;meafures of ba, ed, ad.

The diredlions for this conftrudtion, may be eafily applied to the putting either of the other fides on thenbsp;primitive circle. Fig. 2. 6. and 3. 6. PI. VII.

The fum of the fupplement of Z.a and x is 62� 23''; the half fum is 31� ii4r'-

The difference of the fupplement of Z.A and x is 41� 57^; the half difference is 20� 58V,

Now on the fines, the extent from the half fum 31�- \ \f to 25� 55'' will reach from 36� 08' to anbsp;fourth fine; and the extent from that fourth fine to thenbsp;fine of the half difference 20� will reach on thenbsp;verfed fines from the beginning to about 64� the fidenbsp;fought.

Sect. XIX.

Those compaffes are called proportional, whofe joint lies between the points terminatingnbsp;each leg; in fuch a manner, that when the compaffesnbsp;are opened, the legs form a crofs.

120 nbsp;nbsp;nbsp;DeferiptioJi and life

Simple proportional compafles, are fuch, wbofe center is fixed: One pair of thefe, ferve only for onenbsp;proportion.

Thus, if a right line is to be divided into 2, 3, 4, 5, amp;c. equal parrs; or the cord of -fj t tgt;nbsp;of a circumference is to be taken j there muft be asnbsp;many of fuch compaffes, as there are diftindt operations to be performed.

In each cafe, take, the length of the right line, or of the radius of the circle, between the longer pointsnbsp;of the legs; and the diltance of the fliorter points willnbsp;be the part required.

Compound proportional compaffes, are thofe �wherein the center is moveable; fo that one pair ofnbsp;thefe will perform the office of feveral pairs of thenbsp;fimple fort.

In the ffianks of thefe compaffes are grooves, wherein Aides the center, which is made faff by a nutnbsp;and ferew.

On each fide of thefe grooves, fcales are placed �, which may be of various forts, according to the fancynbsp;of the buyer; But the fcales which the inftrument-makers commonly put on thefe compaffes, are onlynbsp;two, viz. lines and circles.

By the fcale of lines, a right line may be divided into a number of equal parts, not exceeding thenbsp;greateft number on the fcale; which is generally 12.

Exam. I. To divide a given right line, (fuppofe of y.p inches long,) into a propofed number of equalnbsp;parts, (as II.)

Operation. Shut the compaffes; unferew the button; move the Aider until the ,ine arrofs it, coincides with the I�th ciivihon on the fcale of lines;nbsp;ferew the button fail:; open the compaffes, until thenbsp;given line can he received between the longer pointsnbsp;cf the legs; then will the diftance of the fhorter

of Mathematical Infruments.

By the fcale of circles, a regular polygon may be inferibed in a given circle; provided the number ofnbsp;Tides in the polygon, do not exceed the numbers onnbsp;the fcale, which commonly proceed to 24.

Exam. II. To inferibe in a circle of a known radius, (ftippofe 6 Inches) a regular polygon of 12 fides?

Operation. Shut the compaffes; unferew the button; Aide the center until its mark coincides withnbsp;the 12th divifion on the fcale of circles; ferew thenbsp;button faft; take the given radius between the longernbsp;points of the legs ; then will the diftance of thenbsp;fliorter points, be the fide of the polygon required.

'These fcales are applicable to fcveral other ufes befide the fore ming ones, in the fame manner, as thenbsp;like lines on the feftor are.

From thefe operations it is evident, that the lengths of the longer and fhorter legs, (reckonednbsp;from the center,) muft always be proportional to thenbsp;diftance of their extremities.

Therefore, to divide a right line into 2, 3, 4, 5, 6, 7, 8, ISc. equal parts; the lengths of each leg,nbsp;from the center, will be expreffed by the foHhwingnbsp;feries, the whole length of the inftrument being takennbsp;for unity.

These divifions may be very accurately laid on the legs of the compdlles by the help of a good feftor.nbsp;(See Prob. 14,)

O R, the diviftoHS of this fcale of lines may be found by the following conftruefion.

Draw the indefinite right line de; and from any point A, without de, draw Aa, equal to the lliank

of the compafl�s, making any angle at a, with de.nbsp;Through a diaw the rightnbsp;P line AB, parallel to de, andnbsp;I ^ -f,nbsp;nbsp;nbsp;nbsp;� equal to the given line from

then the point c is the center of the fcrew pin. And through c, drawing bc, meeting de in b-, then is

If the center of the fcrew-pin be diflant from the mark in the Aider, the � part of N.

Then ac � 5000, or 3333, or 2500, Cfc. when the divifions on the lhank refpedl the center pin.

And nc = 5 5�quot;5 ^ 335s or 2525, (Sc. 1 I 4975 or 3308 or 2475, amp;c. Snbsp;when the divifions refpedt a mark in the flider, diftantnbsp;ft om the center pin, of the length of the Inftrument.

The fcale for dividing of the circle, or the divifions for regular polygons may be found thus.

Find the angles at the center, of as many regular polygons as are to be defcribed on the compafl�s.

Seek the fines belonging to the half of each angle, to the radius I.nbsp;nbsp;nbsp;nbsp;To

of Mathematical Infty'uments. 129

Then will the reciprocal of thefe numbers, be the lengths of the polygonal divifions, on the legs of thenbsp;compafles, reckoned from the longer point j the lengthnbsp;of the inftrument being accounted unity.

For the longer and Ihorter legs, for points) are in the fame ratio, as are the radius and chord of thenbsp;angle at the center.

And as the fum of the radius and chord, is to the radius; fo is the fum of the longer and Ihorter legs,nbsp;(or points) to the length of the longer point.

And hence was the following table compofed, which Ihews the decimal parts on the leg, from tirenbsp;longer point to the center.

These divifions may be truly laid off by the help of a good fedtor -, making the whole length of thenbsp;proportional compaflTes, a tranfverfe diftance to 10nbsp;and lOj on the line of lines.

The complements, to unity, of the numbers in the table, will give the diftances of the divifions fromnbsp;the olher point of the inftrument.

130 nbsp;nbsp;nbsp;�T�he Defcription and Ufe

If the mark in the Aider, is at fome diftance from the center, as it commonly is, then this diftance,nbsp;which is always known, muft be added to, or fub-tradled from, the foregoing numbers, according tonbsp;that fide of the center the mark is on -, and the fums,nbsp;or remainders, will give the diftances of the divifionsnbsp;from one of the points.

Abovt A4ichaelmas, 1746, was finiflied a pair of proportional compafTes, with the addition of a very curious and ufeful contrivance ; (fee the plate frontingnbsp;the title page) viz. into one of the legs (A) at anbsp;fmall diftance from the end of the groove, was ferew-ed a little pillar of about 4 of an inch high, andnbsp;perpendicular to the faid leg ; through this pillar, andnbsp;parallel to the leg, went a ferew pin (pb) ; to onenbsp;end of this ferew, was foldered a fmall beam (cc) nearly of the length of the groove in the compafles thenbsp;beam was Ait down the middle lengthwife, which received a nut (�) that Aid along the Ait {dd); thisnbsp;nut could be ferewed to the beam, fall enough to prevent Aiding; one end {e] of the ferew of the nut (�)nbsp;falls into a hole in the bottom of the ferew to thenbsp;great nut (g) of the compafles j the ferew pin {bb)nbsp;pafled through an adjufter {h): To ufe this inftru-ment, Ihut the legs clofe, flacken the ferews of thenbsp;nuts ^ and �; move the nuts and Aider k to the divi-fion wanted, as near as can be readily done by thenbsp;hand j ferew fall the nut/; then by turning the adjufter h, the mark on the Aider k, may be broughtnbsp;exadlly to the divifion ; ferew faft the nut g ; opennbsp;the compafles ; gently lift the end e, of the ferew ofnbsp;the nut/, cut of the hole in the bottom of the nut^ ;nbsp;move the beam round its pillar a, and flip the pointnbsp;e, into the hole in the pin�; flacken the ferew of thenbsp;nut /; take the given line between the longer pointsnbsp;of the compafles, and ferew faft the nut /: Thennbsp;may the Aaorter points of the compafles be ufed without any danger of the legs changing their pofition ;

this

this being one of the inconveniendes that attended the proportional compafies before this ingenious contrivance which was made by Mr. ��h'.mas Heath,nbsp;Mathematical Inftrument-maker in the Strand, London.

The proportional compaffes had not been long invented before there were feveral learned and ingenious perfons who contrived a great variety of fcales to benbsp;put thereon j but thefe are here omitted, becaufe thenbsp;credit of the proportional compafTes is greatly fallen,nbsp;fince the invention of the fedlor, the latter being anbsp;much more ufeful inftrument than the former, and notnbsp;fo fubjed to be put out of order for if one of thenbsp;points of thefe compaffes Ihiould be blunted or broke,nbsp;the inftrument cannot be ufed, until the damagednbsp;point be replaced by a new one. However, thofenbsp;who are defirous of knowing the conftruftion and ufenbsp;of fuch fcales on the proportional compafTes, may benbsp;amply fatisfied in confulting Hulfivs, Horfcher, Gal-gemaire Eton, and others mentioned in the prefacenbsp;to this book.

AP-

APPENDIX.

GUNNERS CALLIPERS.

Pair of Callipers is an inftrument ufed to w wnbsp;nbsp;nbsp;nbsp;diameters of convex and concave

The inftrument called the Gunners Callipers, con-lifts ot two thin rulers or plates, which are moveable quite round a joint, by the plates folding one over thenbsp;other.

The length of each ruler or plate is ufually between the limits of fix inches and a foot, reckoned from thenbsp;centre of the joint; and from one to two inchesnbsp;broad ; But the moft convenient ufeful fize is aboutnbsp;nine inches long. The figure is beft feen in thenbsp;plate.

On thefe rulers are a variety of fcales, tables, proportions, amp;c fuch as areefteemed ufeful to be known by gunners and other perfons employed about artillery : But except the taking of the calibre of Ihot andnbsp;cannon, and the meafuring of the magnitude offalimtnbsp;and entring angles^ there are none of the articles withnbsp;which the callipers are ufual'y filled, elTential to thisnbsp;inftrument; the fcales are, or may be, put on thenbsp;fedtor ; and the tables, precepts, amp;c. may be putnbsp;into a pocket-book, where they will not need fo muchnbsp;contradlion: However, for the fake of thofe who are

APPENDIX.

defirous of having a fingle indrument perform many things, the following articles and their difpofition onnbsp;the callipers are here offered : Some of which werenbsp;propofed many years ago by my muchefteemed friendnbsp;Mr. Charles Labelye, engineer to the works ofnbsp;minjler-bridge.

X. nbsp;nbsp;nbsp;Tables of the quantity of powder neceffary fornbsp;proof and fervice of brafs and iron guns.

XI. nbsp;nbsp;nbsp;Rules for computing the number of fhot ornbsp;Ihells in a finifhed pile.

XVI. nbsp;nbsp;nbsp;Logarithmic fcales of numbers, fines, ver-fed fines and tangents.

134 APPENDIX.

The Callipers propofed for the reception of the foregoing articles is nine inches long, and each leg twonbsp;inches bi oad at the head, and at the points ; part ofnbsp;the bread h between the ends is hollowed away in anbsp;curve, in order to contain the curvature of the ball,nbsp;whofe diameter is taken between the points �, one ofnbsp;ten inches diameter is the largeft that can convenientlynbsp;be taken with a nine inch Calliper; thofe of fix inchesnbsp;cannot well be applied to a iliot of more than fevennbsp;inches diameter.

For the eafe of reference *, it will be convenient to diftinguiOa the four faces of the Callipers by the letters A, B, c, D : Each of the faces a and d confift of anbsp;circular head and a leg ; the other faces b and c con-lift only of a leg.

ARTICLE I.

On part of the circular head joining to the leg of the face a, are divifions diftinguiftied by the title ofnbsp;(Ijot diameters : Thefe are to Ihevv the difiance in inches, and tenth parts of an inch, of the points of thenbsp;Callipers when they are opened.

Open the points of the Callipers fo, that they may take in the greatefl: diameter of the ball; then willnbsp;the bevil edge marked e fhew among the forefaid divifions, the diameter of that ball in inches and tenthnbsp;parts, not exceeding ten inches.

Open the feftor until the radius of the circle, whereon is iHarked the fcale of divifions on the headnbsp;of the Callipers, taken with the compaffes, falls tranf-verfely in the fcales of lines, on the divifions fhewingnbsp;the diftance between the centre of the Callipers andnbsp;its points: Then the tranfvetfe diftances of the feveral

APPEND

divifions on the fcalcs of lines, chords to the circle of divifions on the head of thenbsp;Callipers will give the divifions required.

Thus in the nine inch Callipers, the radius of the head, or circle of divifions being one inch, and thenbsp;breadth at the points two inches; the diftance betweennbsp;the centre and points will be (lt;/ 82 �) 9,055385 :nbsp;Then one inch being made atranfverfe diftance on thenbsp;fcales of lines to 9nbsp;nbsp;nbsp;nbsp;; the tranfverfe diftances of 10,

9, 8, 7, 6, amp;c. being applied to the circle on the head of the Callipers appropriated for the fcale, fromnbsp;the mark where the divifions commence, will give thenbsp;feveral points, which being cut by the bevil edge Enbsp;will fliew how far the points of the Callipers are di-ftant.

ARTICLE II.

On the circular bevil part e of theface b, is a fcale of divifions denominated by weight of fhot. Thefenbsp;are to fhew the weights of iron fliot when the. diameter is taken between the points of the Callipers:nbsp;For then the number cut by the inner edge of theleg a,nbsp;fhews the weight of that iron fliot in pounds averdu-poife, when the weight is among the following ones,nbsp;niz.

Observing that the figures neareft the bevil edge anfwer to the fhort lines; and thofe figures behind tliemnbsp;anfwer to the divifions marked with the longer ftrokesnbsp;or lines.

These divifions are to be laid down from a table fhewing the diameters of iron fliots to given weights.nbsp;Such a table is computed by knowing the weight of

four inches diameter is found to weigh nine pounds: Then the weights of fpheres being to one another asnbsp;the cubes of their diameters. Say, As 9 itgt; is to 64,nbsp;the cube of 4.

Nowfetting the points of the Callipers to touch one another, make a mark on the bevil edge e oppofitenbsp;to the inner edge of the leg a and this mark willnbsp;be the beginning of this fcale of weights : The othernbsp;divifions will be obtained by opening the points ofnbsp;the Callipers to the diftances relpedling the weights tonbsp;be introduced, as fhcv n by the table, and marking thenbsp;divifion oppofite to the inner edge of the leg a,

ARTICLE III.

On the lower part of the circular head of the face A, and to the right hand of the divifions for the diameters of fhot, is another fcale of divifions, againftnbsp;which ftands the words Bores of Guns.

Slip the legs of the Callipers acrofs each other, until the fteel points touch the concave furface of thenbsp;gun in its greateft breadth ; then will the bevil edgenbsp;F, of the face b, cut a divifion in the fcale Ihewingnbsp;the diameter of that bore in inches and tenth parts.

In the nine inch Callipers thefe divifions may be extended to 18 inches diameter , but 14 inches is fuffi-cient for both cannon and mortars ; And in the fixnbsp;inch Callipers a diameter greater than 10 inches cannot be Cjvuveniently introduced.

APPENDIX.

Set one inch the radius of the circle on which the divifions are to be put, as a tranfverfe diftance to thenbsp;divifion 9-rB-^ on the fcale of lines on the feftor ; Setnbsp;the points of the Callipers together, and make a marknbsp;on the circular head where it is then cut by the bevilnbsp;ege F : Then the feveral tranfverfe diftances takennbsp;from the feftor, and applied on the circumference ofnbsp;the circular head of the Callipers, from the faid mark,nbsp;the feveral divifions ihewing the diftance of the pointsnbsp;of the Calliper are thereby obtained.

Within the fcales of fliot and bore diameters on the circular part of the face a are divifions markednbsp;Pounders.

When the bore of a gun is taken between the points of the Callipers, the bevil edge f will cut one of thefenbsp;divifions, or be very near one of them : Then thenbsp;number ftanding againft it will fhew the weight ofnbsp;iron (hot proper for that gun', not exceeding 42nbsp;pounds.

belong to the longeft ftrokes or lines -, and the figures I. 2. 4. 6. 9. 16. 24. 32. 42 belong to the fhortnbsp;ftrokes.

J HE diameters given by thefe pounders are larger than thole given for the fame weights of fhot ; be-caufe there is an allowance made, called PFindage, thatnbsp;the fhot may roll eafily along the chace.

AR-

13S APPENDIX.

ARTICLE V.

These degrees are placed on the upper half of the circular head of the face a, where are three concentric fcales of degrees : The outward fcale has 180 degrees numbered from the right to lett, with 10. 20.nbsp;^o. 40. amp;c. to 180 : The middle fcale is numberednbsp;in the fame manner, but the contrary way : And thenbsp;inmoft fcale begins in the middle with o, and is numbered from thence both ways with 10. 20. 30. amp;c. tonbsp;90 degrees.

Apply the legs of the Callipers fo that its outfide edges coincide with the legs of the given angle ; thennbsp;will the bevil edge f cut the degrees fhewing the meafure of that angle in the outfide fcale.

Slip the legs of the Callipers acrofs each other, fo as their outfide edges may coincide with the legs ofnbsp;the given angle ; then will the bevil edge e cut thenbsp;degrees (hewing the meafure of that angle : Thefenbsp;degrees are to be counted on the middle fcale.

Hence an angle of any number of degrees may be readily laid down by the Callipers, either on paper,nbsp;or in the field.

Thus. Open the Callipers, the legs being croffed, until the edge e cuts the degrees on the middle fcale;nbsp;the croffing edges of the inftrument will then form thenbsp;fides of that required angle : The Callipers then laidnbsp;fiat on the paper or ground, lines drawn by the ftraicnbsp;fides will exprefs that angle.

�39

Thirdly. To find the elevation of cannon and mortar or of any other oblique plane or line.

Pass one end of a fine thread into the notch on the plate B, and to the other end tie a bullet, or othernbsp;weight: Apply the ftrait fide of the plate a to thenbsp;fide of the body whofe inclination is wanted ; hold thenbsp;plate A in this pofition, and move the plate b untilnbsp;the thread falls upon the line near the centre markednbsp;Perp. Then will the bevil edge f cut the degrees,nbsp;counted on the inner fcale, fliewing the inclinationnbsp;which that body makes with the horizon.

Note. When the edge f cuts o on the inner fcale ; and the firing cuts the Perp. mark, then the ftrait fidenbsp;of the leg A is horizontal: If the head of the Callipersnbsp;is elevated above the other end, then the edge f muftnbsp;Aide downwards towards the ftrait fide of the leg a :nbsp;But if the head of the Callipers is held lower than thenbsp;other end, then muft the edge f Aide the contrarynbsp;way.

As the outfide of a cannon or mortar is not parallel to its chafe ; therefore a ftrait ftick muft be applied to the bottom or top of the bore, touching the chafe ; and the fide of the Callipers be laid on thatnbsp;ftick.

ARTICLE VI.

On the face c near the point of the Callipers is a little table fhewing the number of pounds that arenbsp;contained in an equal weight expreffed in poundsnbsp;Troyj and the contrary.

These numbers are taken from very accurate experiments made in the year 1744 by the late Martin polkes., Efq-, Prelident of the Royal Society, afiiftednbsp;by feveral other gentlemen of that learned Body.

APPENDIX.

Now 18 C wt. is equal to 2016 ib (=18x112). Then 1 : 1,21545nbsp;nbsp;nbsp;nbsp;:nbsp;nbsp;nbsp;nbsp;:nbsp;nbsp;nbsp;nbsp;2016nbsp;nbsp;nbsp;nbsp;:nbsp;nbsp;nbsp;nbsp;2450nbsp;nbsp;nbsp;nbsp;IfeTroy.

Or, 0,82274: i nbsp;nbsp;nbsp;:nbsp;nbsp;nbsp;nbsp;:nbsp;nbsp;nbsp;nbsp;2016nbsp;nbsp;nbsp;nbsp;:nbsp;nbsp;nbsp;nbsp;2450nbsp;nbsp;nbsp;nbsp;lb Troy.

Or, 576: 700 nbsp;nbsp;nbsp;:nbsp;nbsp;nbsp;nbsp;:nbsp;nbsp;nbsp;nbsp;2016nbsp;nbsp;nbsp;nbsp;:nbsp;nbsp;nbsp;nbsp;2450nbsp;nbsp;nbsp;nbsp;ib Troy.

Either of thefe methods may nbsp;nbsp;nbsp;be tilednbsp;nbsp;nbsp;nbsp;as the ope

Example II. What is the worth of a ton of gold ; fup~ pofing I ib froy makes guineas.

And I : 1,21545 : : 2240 : 2722,6 lb Troy. Also 444^ Guineas, makesnbsp;nbsp;nbsp;nbsp;fteriing.

But if Troy pounds were given to be converted into Averdupoile pounds, then the numbers in the Troynbsp;column muft be the firft terms of the proportions.

Example III. If a brafs gun weighs 2450 lb Troy j What is its weight in Averdupoife ?

Then i : 0,82274 : : 2450 : 2015,716 Avc. Or, 1,21545 : 1,00000 : : 2450 : 2015,7nbsp;Or, 700 :nbsp;nbsp;nbsp;nbsp;576nbsp;nbsp;nbsp;nbsp;: : 2450 : 2016.

Although the Averdupoife pound is heavier than the Troy pound, yet the Troy ounce is heavier thannbsp;the Averdupoife ounce, nearly in the proportion of 90nbsp;to *^^ 2.nbsp;nbsp;nbsp;nbsp;Ex-

APPENDIX.

Example IV. In a chejl of filver containing 4380, pieces of eighty each piece weighing 4 of an ounce Troy:nbsp;How many ounces Averdupoife.

Close to the former table is another, fhewing the number of cubic inches in a gallon, both in wine andnbsp;beer meafures and confequently their proportions :nbsp;One ufe is fhewn by the following Example.

How long will 33 butts of beer ferve a crew of ^24. men., allowing to each man 3 wine quarts a day ?

Now 33 butts contain 3564 beer gall. (=108x33) And 231 : 282nbsp;nbsp;nbsp;nbsp;3564 : 43504 wine gallons.

Then 17403 divided by 324 gives very near 54. Consequently 4 of 54, or 18 days, is the timenbsp;that the beer will ferve.

Say 282 : 231 : : wine gallons : beer gallons. Or 94 : 77 : : W. G.nbsp;nbsp;nbsp;nbsp;: B. G.

Of the proportion of the Englijh and French feet and Vnbsp;nbsp;nbsp;nbsp;pounds.

Near the point of the face d of the Callipers are two tables fhewing the proportion between the poundnbsp;weights of London and Paris, and alfo between thenbsp;lengths of the foot meafurc of England and France.nbsp;Thefe are according to the accurate ftandards fettlednbsp;between the Royal Societies oiLondon and Paris aboutnbsp;the year 1743.

APPENDIX,

Example I. Suppofe a crew of 54 Englijh failors were to attack a French fort, and carry off 6 pieces of hrafsnbsp;cannon weighing one with another 980 French :nbsp;Hew much wotdd each John's fhare come to,fuppoJingnbsp;they could fell the cannon at i L a hundred weightnbsp;Englijh ?

Near the point of the Callipers on the face a is a table containing four rules of the circle and fphere.

APPENDIX.

Diam. X3,I4i6 zrcircumf. ( Sq. Diatn. xo,7854 r=area lt;nbsp;Sq. Diam. X3,i4i6 =furfacenbsp;CubeDiam. xoJ5^36 =folidity

Then (0,5236x1728=) 904,7808 is the folidity. Upon the circular heads of Callipers are ufuallynbsp;placed certain mathematical figures with numbers fetnbsp;to them ; which figures and their numbers may benbsp;placed near the points of the Callipers here deferibed,nbsp;the circular head being appropriated for another ufe.

144 APPENDIX.

The numbers in figure i, are ufcful for finding the circumference of a circle by knowing its diar.ieter jnbsp;or to find the diameter by knowing the circumference. Thus

Fig. 2. There is a circle inferibed in a fquare ; a fquare within that circle, and a circle within the innernbsp;fquare : To this figure are fet the numbers 28. 22.nbsp;14. II. Thefe numbers fignify, that if the area of thenbsp;outward fquare is 28, the area of the inferibed circlenbsp;is 22 ; the area of the fquare inferibed in that circlenbsp;is 14, and the area of its inferibed circle is 11.

It may be obferved, that the fquares are in proportion to one another as 2 to i ; and the two circles are alfo in the fame proportion.

Figure 3. Reprefents a cube inferibed in a fphere j the number 904 fixed to it fhews, that a cube of iron,

APPENDIX. nbsp;nbsp;nbsp;145

Figure 4. Is to exprefs a fphere infcribed in a cube ; Now this figure with its number 2464 is to fhew thenbsp;weight in pounds of an iron fphere of 12 inches diameter ; or of a fphere infcribed in a cube whofe fide isnbsp;12 inches.

Figure 5. Reprefents a cylinder and cone, whofe diameters and heights are each one foot; To the cylinder is annexed the number g69-rT fhewing thenbsp;weight in pounds of an iron cylinder of 12 inchesnbsp;diameter and 12 inches in height: And the numbernbsp;121 -rhr joined to the cone fhews that an iron conenbsp;the diameter of whofe bafe is 12 inches, and the heightnbsp;12 inches, weighs 121 -j-ls- pounds.

Figure 6. Shews that an iron cube, whofe fide is 12 inches, weighs 470 pounds �, and that a fquarenbsp;pyramid of iron, whofe bafe is a fquare foot, and itsnbsp;height 12 inches, weighs 1564 pounds.

These numbers which have hitherto been fixed to the four laft figures are not ftriftly true.

And the weights of fpheres being to one another as th ecubes of their diameters :

Therefore 64 (=4x4X4): 9 :: 1728 (=i2X 12x12) ; 243 pounds, for the weight of a fphere ofnbsp;iron which is 12 inches in diameter .� Confequentlynbsp;the number 243 fliould be ufed inftead of 2464 in thenbsp;4th figure.

Again. The folidity of a cube infcribed in a fphere of 12 inches in diameter, is 332,55 cubic inches.

And the weights of bodies of a like matter being in the proportion of their folidities.

APPENDIX.

So 8 54^ tb is the weight of an iron cube infcribed in a fphere of iz inches in diameter.

Also 3504 tb is the weight of an iron cylinder of a foot in diameter and height.

And 1484 tb is the weight of an iron pyramid, having its bafe a fquare foot, and its height equal tonbsp;12 inches.

On the leg b of the callipers is a table fhewing the weights of a cubic inch or foot of various bodies innbsp;pounds averdupoife. To the table here annexed isnbsp;joined the fpecific gravities of thofe bodies, which arenbsp;omitted on the callipers for want of roorq.

Arable /hewing the weights of bodies and their fpecijic gravities.

APPENDIX.

In the Foregoing table is cojitained fuch bodies as praftical engineers and others may have occafion tonbsp;know their refpedlive weights; there are indeed anbsp;great number of other bodies whole Ipedfic gravitynbsp;have been determined by ingenious men : But tholenbsp;only which were apprehended to be the moll ufefulnbsp;were feledtcd for this fubjeff.

Every one will readily conceive how the column of weights may be obtained, namely by procuringnbsp;maffes of a cubic inch or foot of the folids, and carefully weighing them in nice fcales to the fmalleft degree ol averdupoife weight: And for the fluids, theirnbsp;weights may be determined by having cubical or cylindrical veffels made to hold a known quantity of cubical inches, and in them to weigh the fluids.

The fpecific gravity of a body being the relation which that body has to fome other body fixed uponnbsp;as a ftandaid to compare by ; and rain water beingnbsp;found to be alike, or very nearly fo, in all places ;nbsp;and therefore chofen by philofophers as the propernbsp;ffandard ; confequently by the word fpecific gravitynbsp;of a body is meant no more, than that it is fo manynbsp;times heavier or lighter than water, when comparednbsp;together in equal balks.

Thus fine filver is fomething more than ii ; that is, a mafs of fine filver will weigh fomething abovenbsp;eleven times the weight of an equal mafs of water:nbsp;And, fo a common brick weighs twice as much asnbsp;the rain water that would fill a mould fitted to the.nbsp;brick.

Now the weights of equal maffes of feveral bodies being determined, their fpecific gravities may be readily found, they being in the fame proportion to onenbsp;another as their weights: 1 And as the comparifonnbsp;is made to rain water, of which, by repeated experiments, it has been found that a cubic foot weighednbsp;624; pounds averdupoife', therefore dividing the weightnbsp;of a cubic foot of anybody-, by 62-^ *, the quotient will be

L 3 nbsp;nbsp;nbsp;the

APPENDIX.

the fpecific gravity of that body, relative to rain watef whole fpecific gravity is reprefented by unity.

The difficulty of procuring maffes of metals and other bodies in all parts homogenous, and of havingnbsp;both them and the vel��ls of capacity conftruded tonbsp;a mathematical exaftnefs, has rendered this methodnbsp;of eftimating the fpecific gravities from the weights ofnbsp;equal bulks, liable to exception ; And therefore another method has been contrived to come at thefe fpecific gravities, hydroftatically.

It is a well known thing that any body will weigh lefs when it is immerfed in v^rater than when it isnbsp;weighed in the open air ; and from a very little reflection, it will be feen that the difference between thenbsp;weights of any body when weighed in air and in water, will be equal to the weight of fo much water asnbsp;is equal in bulk to the body immerfed : But the difference between the weights of a body in air and in water, will fliew the weight of a bulk of water equal tonbsp;the body fo weighed ; 'Therefore to find the fpecific gravity of any body, find its weight in air and in rain water, and take the difference of thofe weights ; then thenbsp;weight in air divided^ by that difference, will give thenbsp;fpecific gravity required.

If the folid whole fpecific gravity is wanted, be lighter than water, fo that it cannot fink by its ownnbsp;weight, let it be joined to another fo weighty that thenbsp;compound may fink : But firft let the lofs of weightnbsp;which the heavy body alone fuffains in water be foundnbsp;as before ; and then let the lofs of weight which thenbsp;compound body fuffains be difeovered ; from whichnbsp;take the lofs of weight of the heavier, and the remainder is the lofs of weight fuftained by the lighter j bynbsp;which dividing the weight in air of the fighter body,nbsp;and the quotient will fliew the fpecific gravity.

When the fpecific gravity of fluids are to be compared to each other ; take a folid of any m.atter and fhapCj fuppofe a glafs balk hting by a horfe hair, and

APPENDIX. nbsp;nbsp;nbsp;151

Immerfe this folld in each fluid, and find the lofs of weight of the folid in each fluid, the weight of thenbsp;body in air being firft known then will thefe iofTesnbsp;exprefs the fpecific gravities of thole fluids : P'or fincenbsp;the lofs of weight in each liquor is equal to the weightnbsp;of as much of the liquor as is equal in bulk to thenbsp;body weighed; therefore by taking the iofles ofnbsp;weight fuftained by the fame body in the feveralnbsp;liquors, the abfolute weights are obtained of fuch portions as are equal in bulk, and coniequently the Ipe-cific gravities of thofe liquors.

In this method of finding the fpecific gravity of fo-lids, it is not necelfary that they ftiould be reduced to any regular fliape ; neither is there wanted a velTel ofnbsp;a known figure and capacity to contain the fluids; andnbsp;coniequently the fpecific gravities of bodies, whethernbsp;foiids or fluids, may be very eafily come at : But fromnbsp;the fpecific gravities to find the abfolute weights ofnbsp;any alTigned mafs of feveral bodies, there muft benbsp;another experiment made, which is to find the lofs ofnbsp;weight in water, of a body of a known magnitudenbsp;fuppofe of a cylinder of a homogenous metal, the fo-lidity of that cylinder being moft accurately calculated j then will the abfolute weight of an equal mafsnbsp;of water be known *, and confequently the weight of anbsp;cubit foot of water may be accurately obtained, fromnbsp;whence the abfolute weight of a cubic foot of any othernbsp;body whofe fpecific gravity is known^ may be found bynbsp;multiplying the fpecific gravity of that body by the weightnbsp;of a cubic foot of water.

APPENDIX.

Exam. I. What is the weight of a block of marble 7 feet long, 3 feet broad, and 2 feet thick ?

Exam. II. What is the weight of a inch iron bomb fiell, the metal being two inches thick on a mean ?

Here the folidity of two fpheres muft be found, one of 13 inches diameter, and the other of 9 inchesnbsp;diameter; then their difference being taken will givenbsp;the folidity of the fhell.

Exam. III. How many pigs each of 12 inches long, 6 wide and 4 thick, may be cajl out of 10 ton of meltednbsp;lead ?

And 22400 divided by 707,0458, gives 31,681 cubic feet, which the 10 ton will make.

*53

From feveral experiments it appears that middling fized men, or thofe between 5 feet 6 inches and 5 feetnbsp;9 inches in height, weigh about 150 pounds, and arenbsp;in bulk equal to about 2^ folid feet; and the fmallnbsp;fized men, or thofe between 5 feet 3 inches, and 5nbsp;feet 6 inches in height, weigh about 135 pounds, andnbsp;are in bulk equal to about 2^ folid feet ; And fromnbsp;thofe expriments it alfo appears, that moft men arenbsp;fpecifically lighter than common water, and muchnbsp;more fo than fea water. Confequently could perfonsnbsp;who fall into water, have prefence of mind enough tonbsp;avoid the fright ufual on fuch occafions, many mightnbsp;be preferved from drowning : And a very fmall piecenbsp;of wood, fuch as an oar, would buoy a man abovenbsp;water while he had fpirits to keep his hold.

A GENTLEMAN who had been on board of a Mal-tefe Ihip of war, obferved hanging to the tafarel, a block of wood almoft like a buoy, and fo ballancednbsp;that one end fwam upright, carrying a little flagftafFnbsp;with a fmall vane �, the perfon who was on duty onnbsp;the poop had orders to cut the rope by which the buoynbsp;hung, upon any cry of a perfon�s falling over board jnbsp;and as the block would be in the (hip�s wake by thenbsp;time the perfon floated therein, he was fure of havingnbsp;fomething at hand to fuftain him, till the boat couldnbsp;come to his alTiftance; and Ihould that take fo longnbsp;time to do, as that the diftance from the fhip to thenbsp;man rendered him invifible, yet the boat would havenbsp;a mark to row towards, Ihewn them by the vane.

Exam. IV. How many /pars of �white fir, each of 20 feet long and a foot fquare, are to be lafhed together^ tillnbsp;the raft is fufficient to floaty in common �water, 100nbsp;barrels of gunpowder condubted by four middling fizednbsp;men, fo as to keep the barrels three inches clear of thenbsp;water ?

154

Now the deal will of it felf fink in the water, until the weight of the water difplaced is equal to the weightnbsp;of the wood.

Then 62,5 : 1 : : 711,248 ; 11,38 the number of cubic feet which each fpar will have immerfed bynbsp;its own weight.

As the barrels are to be 3 inches clear of the water, therefore the fpar muft be funk 9 inches �, and confe-quently 15 feet folid of each fpar muft be immerfed :

Then 15�11,38=3,62 the additional cubic feet of water to be difplaced by each fpar, by its incumbent weight.

Consequently 56 fuch fpars lafhed together will make a float fufficient for to fuftain the given weightnbsp;in the manner propofed.

On the circular head of the callipers, on the face D is a table contained between five concentric feg-ments of circular rings -, the inner one markt Guns,nbsp;fhews the nature of the gun, or the weight of ball it

APPENDIX. 'iss

carries: The two next rings contain the quantity of powder ufed for proof and fervice to brafs guns ; andnbsp;the two outermoft rings fliew the quantity for proofnbsp;and fervice, ufed in iron cannon.

The numbers in this table exprefs the Englifh ufage, which for the moft part, allows the weight ofnbsp;the flrot for proof, half its weight in fervice, and onenbsp;fourth of its weight of fhot for falutes.

The French allowance of powder, for the charge of the piece for fervice, ufed to be two thirds of thenbsp;weight of the Ihot ; twice as much for proof, andnbsp;one fourth of the weight of Ihot for falutes.

APPENDIX.

Guns carrying fhot of the weight 1 lb. lb. 2 lb. 4 lb. 5 4 lb. 8 lb. 26 lb. 36 lb. are now out of ufe innbsp;the Britifh navy.

The ufe of this table is obvious: For feek the name of the gun in the inner ring, and the weights ofnbsp;powder for proof and fervicc will be found betweennbsp;the fame two ftrait lines, like radii j and in one ofnbsp;the other rings, according as it is tituled at the end.

Thus to a brafs 9 pounder, there is allowed 91b. of powder for to prove, or try the goodnefs of thenbsp;gun when it is firft call �, and 4 lb. 8 oz. of powdernbsp;for each charge in common fervice ; But an iron 9nbsp;pounder has 9 lb. for proof, and 6 lb. for fervice.

On fhip board, after there are five or fix rounds fired on warm fervice, the allowance of powder is tonbsp;be proportionally leflened each time the gun is loaded,nbsp;until the charge is reduced to one third of the weightnbsp;of the fhot ; And the guns as they grow warm innbsp;firing, are not to be wetted left the gun be in dangernbsp;of fplitting by checking the metal with cold water.

The ingenious Mr. Robins, from fome hints he gathered from a manufcript lent him by the Rightnbsp;Honourable Lord Anfon, advifes to leflen confiderablynbsp;the common charges allowed to cannon in fervice ;nbsp;For from thofe papers it appeared that in fervice,nbsp;where 24 pounders have been ufed to batter in breach,nbsp;the charge was only 8 pounds of powder: Indeed thenbsp;velocity of the ball could not be quite lb great withnbsp;8 pounds of powder as with 12, and confequently thenbsp;fhot would not be drove fo far into the rampart, andnbsp;the breach not made altogether fo loon; notwith-ftanding which, the advantages attending the fmallernbsp;charges, greatly overbalanced the difference of a fewnbsp;hours in making a fufheient breach.

In fea fervice it would perhaps be found of greater ufe to begin with one third of the weight of fhot in

APPENDIX.

powder, and to diminilh that to one fourth or one fifth as the gun waxed warm �, for by fome experiments it has appeared, that fuch fmall charges ofnbsp;powder has produced greater ravage in timber, thannbsp;has been found with the ufual charges : From whencenbsp;it may be reafonably concluded, that if a fhot hasnbsp;juft force enough to go through one fide of a fhip,nbsp;there will be a greater quantity of fplinters rent outnbsp;of the plank, andconfequently do more mifchief, thannbsp;if the Ihot went with a velocity fufficient to drive itnbsp;through both fides of the Ihip.

ARTICLE XI.

Iron fhot and fhells are ufually piled'up by horizontal courfes into a pyramidal form, the bafe being either an equilateral triangle, or a fquare ; or a redlan-gle j in the triangle and fquare, the pile finifhes in a fingle ball; but in the reftangle, the finifhing is anbsp;fingle row of balls.

In the triangular and fquare piles, the number of horizontal rows, or the number counted on one ofnbsp;the angles from the bottom to the top, is always equalnbsp;to the number counted on one fide, in the bottom row.

In triangular piles, each horizontal courfe is a triangular number, produced by taking the fucceffive films of the numbers i and 2 ;nbsp;nbsp;nbsp;nbsp;1,2 and 3 ; i, 2, 3

Numbers in order 1.2. 3.4 .5 .6 . 7.8 .9 .io.n,amp;c. Triangular numb, i . 3.6.10.15 . 21 . 28.36.45.55.66, See.

And the number of fhot in a triangular pile is the fum of all the triangular numbers taken as far, or tonbsp;as many terms, as the n�mber in one fide of the bottom courfe.

Count the number in the bottom row, and multiply

158 APPENDIX.

that number more two by that number more one ; Then the produft multiplied by one fixth of the faidnbsp;number, the produft will be the fum of all the Ihotnbsp;in the pile.

Exam. I. How many jhot are in a finijhed triangular piky in one fide of whofe bottom courfe are 20 Jhot ?

Exam. II. Required the number of jhot in a finijhed pile; there being in one fide of the triangular bafe 40nbsp;Jhot?

In fquare piles, each horizontal courfe is a fquare number,produced by taking the fquare of the numbernbsp;in its fide.

Number in the fide 1.2.3. 4* 5*6. 7. 8.9. 10, amp;c. Squares, orhoriz. courfes 1.4.9. 16.25.36.49.64.81.100, amp;c.

And the number of Ihot in a fquare pile is the fum of all the Iquares, taken from one, as far as the number in the fides of the bottom courfe.

Count the number in one fide of the bottom courfe j to that humber add one, and to its double add one jnbsp;multiply the two fums together �, then their produdtnbsp;being multiplied by one fixth of the faid number, thenbsp;produd will give the number of Ihot contained in thatnbsp;pile.

Exam. III. How manyJJoot are in a fqiiare fmijhed �pile., one fide of its bafe containing 20 foot ?

Exam. IV. Required the number of fisot in a fquare finifhed pile, one fide of the lower courfe, or tier, havingnbsp;40 fhot in it ?

From thefe examples it may be obferved, that where room is wanted, �tis moft convenient to havenbsp;the Ihot flowed in triangular piles; For on the equilateral triangle, which is lefs than half the area of anbsp;fquare on one of its fides, there can be piled a greaternbsp;number than half of what can be raifed on the fquare :nbsp;Indeed the heightof a fquare pile is fomewhat lefs thannbsp;a triangular one, as a fhot will fink lower in the fpacenbsp;between 4 others, than in the fpace between 3 others,nbsp;all the fhot being of equal diameter ; they being fonbsp;reckoned in every pile.

In redtangular piles, each horizontal courfe is a rectangle, the upper one being one row of balls: Now every fuch oblong pile may be confidered as confift-ing of two parts, one a fquare pyramid, and the othernbsp;a triangular prifm.

To find the number of fhot in a rectangular pile. ift. Take the difference between the number innbsp;length and breadth in the bottom courfe.

APPENDIX.

2d. Multiply the number in breadth, more one, by half the breadth the produdt multiplied by thenbsp;faid difference, will give the number in the prifmaticnbsp;pile.

gd. Upon the fquare of the breadth, find (by the laft rule) the number in a pyramidal pile.

4th. Then the fum of thefe two piles will fhew the number in the reftangular pile.

N. B. The number of horizontal courfes, or rows, is equal to the number in breadth at bottom ; Andnbsp;the number lefs one, in the top row, is the difference between the number in length and breadth atnbsp;bottom.

Exam. V. How many Jhot are in a finijloed pile of io courfes, the number in the top row being 40.?

Consequently the fum of 2870 and 8190, or 11060 fhot will be the number contained in that rectangular pile.

If any of thefe piles are broken, by having the upper part taken off, and the remaining number of fhot are required ; it may be obtained by computing whatnbsp;the whole finifhed pile would contain and alfo whatnbsp;the pile wanting, or taken away contained �, for thennbsp;their difference will fhew the number remaining.

APPENDIX.

In Examples I amp; HI. The letter n ftands for 20. And Examples II amp; IV. The letter n ftands for 40.nbsp;In Example V.nbsp;nbsp;nbsp;nbsp;The letter n ftands for 20.

When heavy bodies are fuffered to fall, it is well known they fall in lines perpendicular to the furfacenbsp;of the earth.

The force with which any body in motion ftrikes an obftacle, depends on the weight of that body, andnbsp;on the velocity or fwiftnefs with which it moves.

Thus a man by throwing, with the fame ftrength, a pound of iron and a pound of cork, will hit a muchnbsp;harder ftroke with the iron than with the cork.

Also a man and a boy each throwing a pound of iron againft the fame objedt, the ftroke given by thenbsp;man will be ftronger than that given by the boy, onnbsp;account of the man�s weight flying the fwifteft.

I62 nbsp;nbsp;nbsp;APPENDIX.

The fame heavy body by falling from different heights, will ftrike blows of different ftrengih, thatnbsp;being the ftrongefi:' where the height is greateft. Con-fequently heavy bodies by falling acquire velocitiesnbsp;greater and greater according to the length of theirnbsp;tall.

2d. That the /paces fallen through are as the fquares of the timesor as the fquares of the velocities.

3d. That a body moving uniformly with the velocity obtained by falling through any height., will fall twicenbsp;as far in the fame time it was paffing through thatnbsp;height.

Experiments ffiew that heavy bodies fall about 16 feet in one fecond of time : Confequently at the endnbsp;ofthefirft fecond of time, a falling body has acquirednbsp;a velocity that would carry it down 32 feet in the nextnbsp;fecond of time.

�ft. That the fquare root of the feet in the fpace fallen through, will ever be equal to one eighth of thenbsp;velocity acquired at the end of the fall.

id. 'Fhat the fquare root of the feet in the fpace fallen through, will ever be equal to four times thenbsp;number ot fcconds of time the body has been falling.

qd. And that four times the number of feconds of time in which the body has been faffing, is equal tonbsp;one eighthof the velocity in feet per fecond, acquirednbsp;at the end of the fall.

From thefe three rules moft of the queftions relative to the fall of bodies may be readily folved.

As thef� rules cannot, for want of room, be put in words at length on the callipers, they are, on the facenbsp;A ot one of the legs, exprefled in an algebraic manner.nbsp;Thus,

Exam. II. From what height muji a bullet fall to acquire a velocity of 160 feet fer fecond ?

Exam. III. How long muJi a bullet he in falling to acquire a velocity of 160 feet per fecond.

Thent= ('4�=) 5 feconds of time.

APPENDIX.

Exam. IV, How many feconds will it require for a hea^ vy body to fall through a fpace equal tonbsp;nbsp;nbsp;nbsp;�

So that it will require 'if'. lt;fquot; of time for the body to fall through 3375 yards.

ARTICLE XIII.

Experiments have fliewn, that taking horfes and men of a moderate ftrength,one horfe will do as muchnbsp;w'ork in raifing of water, and fuch like labour, asnbsp;five men can.

I r has been alfo found, that one man in a minute, can raife a hogfhead of water 12 feet high upon a mean :nbsp;For a flout man, well plied with ftrong liquor, willnbsp;raife a hogfhead of water 15 feet high in a minute :nbsp;Now as the quantity of liquor equal to a hogfhead wasnbsp;raifed to thefe heights only by way of experiment foranbsp;few minutes, fuch numbers ought not to be efteemednbsp;as the common labour of a man who is to work 4 ornbsp;5 hours on a flrecch : But it may be reckoned, thatnbsp;of common labouring men, taken one with another, one of them will raife a hogfhead of water to Snbsp;feet in height in one minute, and work at that rate fornbsp;fome hours.

It is quite indifi�rent in what manner the man is fuppofed to apply his force ; whether by carrying thenbsp;water in manageable parcels up a flair cafe, or raifing it by means of fome machine ; For the advantagesnbsp;gained by ufing of engines arifes chiefly from the eafcnbsp;with which the power can be applied.

APPENDIX. nbsp;nbsp;nbsp;165

Here hogfheads are reckoned at 60 gallons, this eftimate being nice enough for any computations onnbsp;water engines.

Examp. I. How many hogsheads can fix horfes raife, by an engine, to 2^ feet high in 3 hours ?

Hence this ride. Mui-tiply eight times the power by the time, the produdl divided by the height, givesnbsp;the hogfheads.

Ex amp. ir. It is propofed to throw out of a pond, by an engine, 432 tuns of water in 3 hours by fix ho fesnbsp;to what heighth can the water be raifed ?

i66 APPENDIX.

Hence this rule. Multiply eight times the power by the time; the produdl divided by the hogfheads,nbsp;gives the height in leet.

Examp. III. How long will it require fix horfes to raife with an engine 1728 hogfheads of water to the height

of 2 nbsp;nbsp;nbsp;^

Hence this rule. Multiply the hogflieads, by the height in feet �, the produft divided by 8 times, thenbsp;power will give the time in minutes,

Ex AMP. IV. How many hor fes will it require to work an engine, to raife 1728 hogjheads to the height of 2^nbsp;feet, in 3 hours ?

The time to be done in is 3 hours, or 180 minutes. The general rule is hxf=px8xt.

Hence this rule. Multiply the number of hogf-heads, by the height in feet gt; the produdf divided by 8 times the number of minutes, gives the number onbsp;men.

It has been proved by many writers, that the flight of fh�t, or the track they defcribe in the air, is a curvenbsp;line called a parabola : But then they fuppofe thatnbsp;the refiftance made by the air is fo inconfiderable asnbsp;fcarcely to atFedl the motion of heavy bodies.

Upon this fuppofition then, which is very far from being true �, there have been collebled the followingnbsp;obfervations and rules.

I. All bodies proje�fed by any force-, are urged with two motions, viz. one in the direction of the powernbsp;exerted by the engine, and the other in a perpendicular diredion to the earth, by the force of gravity and the track or path defcribed by the bodynbsp;with thefe two forces is a curve called the parabola.

APPENDIX.

horizon and the part in which the body de-Icends will be alike to that in which it afcended.

fcent, be on the fame level with the point from which it was projedted, thofe points are equallynbsp;diftant from the vertex, or higheft point of thenbsp;curve.

will fall there again in the fame obliquiiygt; and with the fame velocity it was projeded withal.

jeded with the fame velocity, at different elevations, will be in proportion to one another ; as the right fines of twice the angles of elevation.

cities, the heights to which they will rife in the air, are in the fame proportion to one another asnbsp;the verfed fines of twice the angles of elevation.nbsp;Vn. When equal bodies are projeded with equal velocities, the times of their continuance in the airnbsp;will be in proportion to one another as the rightnbsp;finesof the angles of elevation.

Vjll. In the fame piece, different charges of equally good gunpowder will produce velocities, nearlynbsp;in the fame proportion as the fquare roots of thenbsp;weights of the charges.

but with different velocities, the horizontal ranges will be in proportion to one another, as the fquaresnbsp;of the velocities given to the fliot, or as thenbsp;weights of the charges of powder nearly.

APPENDIX.

XII. The greateft height to which a projedied body can rife, at an elevation of 45 degrees, is equalnbsp;to one fourth part of its horizontal range.

To hit an obje�l that lies above or below the horizon of the piece, the beft elevation, is equalnbsp;to the complement of half the angular diftancenbsp;between the obje�l and the zenith.

At elevations equally diftant from 45 degrees, both above and below it, the horizontal rangesnbsp;will be equal.

XV. The time which a heavy body, projefted at an elevadon of 45 degrees, will continue in the air,nbsp;before it arrives at the horizon, will be equal tonbsp;the time that body would take to defeend, by thenbsp;force of gravity, through a fpace equal to thenbsp;horizontal range.

It has been found that a 24 pounder at an elevation of 45 degrees., and charged with 16 pound of powder,nbsp;has ranted its Ihot upon the horizontal plane aboutnbsp;6750 yards.

Therefore 3375 is the impetus, or perpendicular fpace which a 24 pounder muft tall through to acquirenbsp;fuch a velocity, as, at an elevation of 45 degrees,nbsp;would prqjedt or throw that fhot on the horizon to thenbsp;diftance of 6750 yards.

Now a heavy body falling by the force of gravity through a fpace equal to 3375 yards or 10125 feet,nbsp;will, at the end of the fall, acquire a velocity of gt;' 04,8nbsp;or about 805 feet per fecond (as fhewn at Art. XII )

The chief of the above principles are fiiortly ex-prelTed on the face e of the manner.

APPENDIX.

To apply thefe rules to the praftice of Ihooting, it is to be underftood that the gunner fhould make an experiment with every gun he has the care of at fome elevation, fuppofe at 45 degrees, and with the ufualnbsp;charge of powder, and then knowing how far the piecenbsp;has ranged the lliot on the horizontal plane ; he maynbsp;apply the refult of thofe experiments to other elevations and quantities of powder.

Examp. I. Suppofe the greatefl horizontal range to be S'] yards: How far will thejame piece., and with annbsp;equal charge of powder, range a foot at an elevation ofnbsp;25 degrees ?

With equal charges the horizontal ranges are as the right fines of twice the angles of elevation.

Is to the fine of 50�, or the fine of twice 25% So is the greateft horizontal range 6750 ydsnbsp;To the horizontal range required. 5170 yds.

Fxamp. II. The greatefi horizontal range of a 24 pounder being Sj^o yards : To what height -tvill that footnbsp;rife at an elevation lt;?� 25 degrees?

APPENDIX.

The logarithm verfed fines on the callipers are the fupplements of the real verfed fines ; therefore in thenbsp;ufing of this line the fupplements of the double anglesnbsp;are to be ufed.

Then the extent from the verfed fine of 90� to the verfed fine of 130� (the fupj lement of 50�) will onnbsp;the line of numbers reach from 16877 to 603.

Or thus. Take 12927-�^ of 5170, the horizontal range on an elevation of 25�.

Then. The extent on the log, tangents from radius to 25�, vvill on the line of num.bers reach from j 2927-to about 603 yards.

Examp. III. At an elevation of'2^ degrees, how many feconds will a 2 y pounder continue in the air before it arrives at the horizon ?

Then As the fine of the elevation 45 degrees Is to the fine of the elevation 25 degreesnbsp;So is the time ill air at 45�, viz. 357 fecondsnbsp;To the time in air at 25�, viz. 21. feconds.

On the line of log. fines take the extent from 45 degrees to 25 degrees; then will this extent, applied

And hence may be eftimated the lengths of fules proper for (hells to be fired at given elevations and

Examp. IV. Required the elevation necpjfary to firike an ohjc�l on the horizon at 5170 yards dijlance^ the great-ejl random of that piece beirg 6j 1^0 yards ?

So is radius, or twice the fine of 45 degrees. To double the elevation required, viz. 50 deg.

The half of which, or 25 degrees, is the elevation neceflary to be given to the piece.

Examp. V. At an elevation of degrees, 16 of powder will threw a 2^ pounder 6j^oyards: How muchnbsp;powder will throw the fame fhot 51 qo yards at the famenbsp;elevation ?

Examp. VI. At an elevation of 25 degrees, a 24 pounder was ranged on the horizon 5170 yards : Required the impetus that would have given an equal velo-c:ly to that Jhot ?

APPENDIX.

of elevation, as was ufed at 2 5 degrees, the fliot would have the greateft horizontal range.

And with equal charges in the fame piece, the impetus is the fame at any elevation.

Consequently, to folve this queftion nothing more is required than to find the greateft horizontalnbsp;range, which is double to the impetus.

That is, thcextenton the line of fines from 5o�to go* Will on thelineof numb reach from 5170 to 6750.

Examp. VII. Suppofe the horizontal range of a piece to he yards : Required the atigle of elevation propernbsp;to Jirike an object 11� above the level of the piece, the horizontal diflance of that objeSi being 4.6^0yards ?

Say, As the greateft horizontal range 6750 Is to the given horizontal diftance 4680nbsp;So is the cofine of the objccl�s elevation 78� 00^.

APPENDIX.

And the diflerence of thefe halves (51��13�. 48=} 37�. 12'. is the lefler elevation.

So that the piece pointed at either of thefe elevations, �with the charge of powder that gave the horizontalnbsp;range, the objeft will be ftruck.

But in ail (hooting on afcents or defcents, it is beft to take the angle between the objedt and zenith, andnbsp;get the complement of the half of that angle ; thennbsp;the piece being elevated to that complement, find bynbsp;trials what charge will reach the objedl; For on thisnbsp;elevation, a lefs charge of powder will do the bufinefsnbsp;than on any other elevation.

So in the foregoing example the diftance of the objedl from the zenith is 78�,

The half ol 78 is 39, and the complement of 39 is 51�, for the bell elevation.

This line, the ufe of which is well known, is placed on the edge of the callipers, or on the ftrait borders ofnbsp;the faces c, d.

These fcales are placed along the faces c, d of the callipers, near the ftrait edges, and are marked andnbsp;numbered as is fhewn in fedtion X ; fome of the ufesnbsp;of thefe fcales are alfo fliewn in the XV and followingnbsp;fedlions.

The line of lines is placed on the callipers on the faces c, D, in an angular pofition, tending towardsnbsp;the centre of the inftrunaent *, its conftrudlion and ufesnbsp;are the fame as defcribed in treating of the fedor ; the

These lines lie on the faces c,d, of the callipers, and like the line of lines tend towards the centre of the in-ftrument ; They are marked near the ends of the callipers with the word Plan, and have the figures lo, 9,nbsp;8,71 6, 5, 4, 3, 2, I, I, running towards the centre;nbsp;Each of thefe primary divifions is fubdiyided into tennbsp;parts ; and each of the fubdivifions is alfo dividednbsp;into two, or more parts, according to the length ofnbsp;the callipers.

These divifions reckoned from the centre along either leg, are as the fquare roots of all the whole numbers under 100 ; and alfo, of the half numbers : That is, the diftance from the centre to thefirft 1, is as thenbsp;fquare root of i : From the centre to the next divifionnbsp;is as the fquare root of : To the next as 2, the nextnbsp;as 24 �gt; the next as 3, amp;c.

And the diftance from the centre to the fecond i, is as the fquare root of 10 ; from the centre to the nextnbsp;divifion is as the fquare root of i o4-; to the next asnbsp;II ; to the next as 114-, amp;c. So that the diftancesnbsp;from the centre to 2, to 3, to 4, and fo on to 10, are asnbsp;the fquare roots of 20, 30, 40, and fo on to 100 �, andnbsp;the intermediate divifions and fubdivifions are efti-mated as before fhewn between i and lo.

This line is eafily conftrufted from a table of the Iquare roots of ail the units and half units under 100;nbsp;together with a fcale of the intended length of the linenbsp;of plans, divided into 500 or 1000 equal parts; andnbsp;fuch a fcale is the line of lines.

In the followng folutions, the lengths of lines are fuppofed to be taken between the points of a pair ofnbsp;compafTes : And when the callipers are faid to be opened to any line; it means, to the diftance of the poiatsnbsp;of the compafs between which that line was taken ;nbsp;the points being applied tranfverfely to the legs of thenbsp;callipers, as fhewn lor the fedfor at fedfion Xll.

ift. On the line of plans leek the divifion reprefent-ing the given number : Obferving, that numbers of an odd number of places are beft lound between thenbsp;divifions i and i ; and thofe of an even number ofnbsp;places, between the 2d i and the 10 at the end.

2d. Take, with the compafles, the diftance between that divifion and the centre of the callipers ; and thisnbsp;extent being applied, from the centre laterally alongnbsp;the line oflines.) will give the fquare root of the number propofed.

If the integer places in the given number are even, the root will confift of half as many places : But if the

number

APPENDIX.

number of integers be odd, increafe it by one, and the integer places in the root will be half that number of places.

Thus numbers of two, four, fix, eight integer places, will have roots confifting of one, two, three, four, amp;c.nbsp;places : And numbers confifting of one, three, five,nbsp;amp;c. places, have roots of one, two, three, amp;c. places.

Examp. II. Between two given numbers (fuppofe and 9 j to find a mean proportional.

ift. Take the greater of the given numbers (9) laterally from the line of lines, and make this extent a tranfverfe diftance to ( 9 and 9) the fame number onnbsp;the lines of plans.

2d. Take the tranfverfe diftance between (4 and 4) the lefler given number on the lines of plans, and thisnbsp;extent applied laterally on the line of lines, will givenbsp;(6 for) the mean proportional fought.

Bv this example it is eafy to fee how to find the fide of a fquare equal to a fuperficies whofe length andnbsp;breadth are given.

Examp. III. 'itwo JirnilaTy or like, fuperficies beings given ; to find what proportion they have to one another.

i/. Take one fide of the greater fuperficies between the points of the compafles, and make this extent anbsp;tranfverfe diftance on the line of plans betw'een lonbsp;and 10 ; or loo and ico : or on any other number.

2d. Apply a like fide of the lefs fuperficies tranf-verfely to the line of plans, and the divifions it falls on will fttew the number, that to the former numbernbsp;(taken tranfverfely for the fide of the greater fuperficies) bears the fame proportion of the lefler fuperficies to the greater.

This propofition may be wrought laterally on either of the legs, reckoning from the centre : For like Tidesnbsp;of limilar p!ans being laid froim the centre on either

Examp. IV. To find the ftdes, or other lines, of a fu-ferficies a, which Jhall be fmihr to a given fuperficies b, and in a given proportion to b, fuppofe as ^ to j ?

ijl. Tothefcales of plans, apply tranfverfely, any given line of b to the confequent of the given ratio, asnbsp;from 7 to 7.

2d. Take the tranfverfe diftance, on the plans, of the given antecedent, as from 3 to 3, and this �xtentnbsp;will be a like line of the figure a.

3J. As many lines being thus found as is neceirarygt; the figure a may be conftruded.

Ex AMP. V. To find the fides, or other lines, of a fuperficies D, which Jhall be like to either of two given plane figures a and b and alfo be equal to the fum or difference of A and B.

jfi. Find (by Ex. 3.) two numbers expreffing the proportion of the given figures a and B j and take thenbsp;fum and difference of thofe numbers.

2d. Find (by Ex. 4.) the fides of a fuperficies d, fimilar to a, fo that a may be to d as 3 to 10 for thenbsp;fum, or as 3 to four for the difference �, or if like to b.,nbsp;fo that B may be to d as 7 to 10 for the fum, or as 7nbsp;to 4 for the difference.

And thus, a fufficient number of lines being found the figure d may be conftruded.

179

Examp.VI. Three numbers bemg given to find a fourth in a duplicate proportion : Or, the like fides a, b, of twonbsp;fimilar figures a, b, being known, and alfo the area a, ofnbsp;one, to find the area of the other.

On the fcale of plans, take the given fuperficies a laterally ; and on the fcale of lines, apply this diftancenbsp;tranfverfely to the given fide ^ofrhat fuperficies: Takenbsp;the tranfverfe diftance ot the given fide b of the othernbsp;fuperficies, from the fcale of lines ; then this diftancenbsp;applied laterally on the fcale of plans, will ftiew thenbsp;area of b.

Thus. If 40 poles be the fide of a fquare whofe area is 10 acres ; what is the area of that fquare whofe fide isnbsp;Go poles ?

Take the lateral diftance 10 on the fcale of plans; apply this diftance tranfverfely to 40 and 40 on thenbsp;line of lines : Then the tranfverfe diftance of 60 andnbsp;Go on the lines, applied laterally to the fcale of plans,nbsp;will give 2 24 acres the area required.

Again. How many acres of woodland mamp;afure, of x�et to the pole, is in that field which contains 288nbsp;acres, at 164 feet to the pole ?

Apply the lateral diftance of 288, taken from the fcale of plans, to the line of lines, tranfverfely from 18nbsp;to 18 ; then the tranfverfe diftance of 164 and 16400nbsp;the lines, will, on the fcale of plans, give 242 the areanbsp;in woodland acres.

Examp. VH. To open the callipers, fo that the lines of plans make with one another a right angle ?

Then fet the callipers fo, that this extent fhall be a tranfverfe diftance to the halves of the lormer numbe.--,

APPENDIX.

Thus : The lateral extent of 6o on the plans, put tranfverfely to 30 and 30 on the plans, will fet thofenbsp;lines at right angles to one another.

These lines are laid on the faces of d, c, of callipers, like feftoral lines tending to the centre, and are diftinguifhed by the letters sol placed at their ends.

There are twelve primary divifions on thefe lines marked 1,1,1,2, 3,4,5,6,758,9, lOjeach of the elevennbsp;Ipaces or intervals is divided into ten other parts �, andnbsp;each of thefe parts is divided into two or more parts,nbsp;according to the length of the inftrument.

These divifions are beft taken from a fcale of equal parts, fuch as the line of lines, and thence transferrednbsp;to the fcales of folids, reckoning from the centre jnbsp;from whence the feveral diftances of the divifions are,nbsp;as the cube roots of fuch numbers under 100 as are in-fended to be introduced.

Thus, the diftance of the firft i from the centre is as the cube root of ^nd the greater divifionsnbsp;following to the fecond i, exprefs the cube roots ofnbsp;.A� to the number i, which the fecond inbsp;ftands for ; and if thefe fpaces are fubdivided, theirnbsp;diftances from the centre are as the cube roots of^Vr�

centre to the fecond i is as the cube root of i, and the greater divifions between thenbsp;fecond i and the third i, are as the cube roots of thenbsp;whole numbers 2, 3, 4, 5, 6, 7, 8, 9 ; the intermediate fmaller divifions are as the cube roots of thenbsp;mixed numbers to which they belong; Thus if thenbsp;fpace between the divifions reprefenting the roots of inbsp;and 2 is parted into 4 ; then thofe fubdivifions will be

The diftance between the centre and the third i is as the cube root of lo ; and fo the following divifionsnbsp;marked with 2, 3,4, amp;c, to 10, are as the cube rootsnbsp;of 20, 30, 40, amp;c. to 100 : each of thefe fpaces arenbsp;divided into 10 parts, which are as the cube roots ofnbsp;the intermediate whole numbers �, and if thefe lubdi-vifions are again divided, thefe latter divifions will benbsp;as the cube roots of the mixed numbers to which theynbsp;belong.

On the French inftruments, the divifions of this line is ufually extended to 64 ; and confequently onlynbsp;the cube roots of all the integer numbers under 64 arenbsp;thereon exprefifed : Now whether the divifions proceednbsp;only to 64 or to 100, the beft way of laying themnbsp;down is from a table of cube roots ready computed,nbsp;reckoning the length of the greateft root, or the lengthnbsp;of the fcale of folids, to be equal to the length of thenbsp;line of lines, taken from the centre.

The following table contains the cube roots of all the whole numbers from i to 100.

i82 appendix.

Table of cube numbers and their roots.

The numbers in the foregoing table may be laid on the line of folids in the following manner.

Make the length of the line of folids equal to the length of the line of lines, apply this extent tranf-verfely to 4,642 on the line of lines -, then the othernbsp;numbers in the table taken tranlVerfely from the linenbsp;of lines, are to be laid laterally, from the centre, onnbsp;the line of folids.

Some ufes of the lines of folids.

Examp. I. I�o find the cube root of a given number.

Seek the given number on the line of folids, and take its extent from the centre.

Then this extent applied laterally to the line of lines will give the cube root fought.nbsp;nbsp;nbsp;nbsp;It

APPENDIX.

' It fhould be remarked, that a given number of of 15nbsp;nbsp;nbsp;nbsp;2nbsp;nbsp;nbsp;nbsp;ornbsp;nbsp;nbsp;nbsp;3nbsp;nbsp;nbsp;nbsp;places,nbsp;nbsp;nbsp;nbsp;has a root of one place,

of .4� nbsp;nbsp;nbsp;5nbsp;nbsp;nbsp;nbsp;ornbsp;nbsp;nbsp;nbsp;6nbsp;nbsp;nbsp;nbsp;places,nbsp;nbsp;nbsp;nbsp;has a root of two places,

of 7, nbsp;nbsp;nbsp;Snbsp;nbsp;nbsp;nbsp;ornbsp;nbsp;nbsp;nbsp;9nbsp;nbsp;nbsp;nbsp;places,nbsp;nbsp;nbsp;nbsp;has a root of three places.

The primary divifions from i to 10 may be reckoned either as fo many hundreds, or as fo many hundred thoufands, or as fo many hundred millions.

And hence the divifions between the centre and the firft of the primary ones, are to be eftimated for numbers of one, two, four, five, feven and eight places.

Examp. II. To a number given, to find another in a triplicate ratio of two given numbers.

Thus. Suppofe afhot of 4 inches diameter to weigh ptfe; required the weight of that flrot which is 8 inchesnbsp;in diameter ?

Here a number is to be found, that to 9 fliall be in the triplicate ratio of 4 to 8.

Now from any fcale of equal parts, fuppofe inches, take 4 ; and make it a tranfverfe diftance to 9 and 9nbsp;on the line of folids (reckoning the 10 at the end, asnbsp;100^ : Then will the extent of 8 inches, appliednbsp;tranfverfely to the line of folids, give 72 for the number fought, which is the pounds weight of a fhot of 8nbsp;inches diameter.

appendix.

Again. Suppofe a Jhip of 2000 tons burthen is 144-feet 6 inches on the keel, and 51 feet by- the beam: Required the length and breadth of another fimilar fhip that flsall be of 14.1^ tons burthen ?

From any fcale of equal parts take 1444 and make this extent a tranfverfe diftance to 2000 on thenbsp;line of folids; then will the tranfverfe diftance ofnbsp;1415 taken on the line of folids give the length of thenbsp;keel, which applied to the faid fcale of equal partsnbsp;will give about I28-| feet.

Also the extent in equal parts of 51 being made a tranfverfe diftance to 2000 on the lines of folids; thennbsp;the tranfverfe diftance on the folids of 1415 will givenbsp;in equal parts 464 feet for the breadth by the beam.

ift. From any fcale of equal parts take the meafure of the greateft of the given lines or numbers, and apply this extent tranfverfely to that number on the linenbsp;of folids ; then the tranfverfe extent on the folids, ofnbsp;the leaft of the given numbers, being taken, will benbsp;the greater of the required means, whofe meafure willnbsp;be found on the faid fcale of equal parts.

id. Make the extent of the greater mean, a tranfverfe diftance to the greater of the given numbers, on the line of folids; then the tranfverfe diftance of thenbsp;lefler of the given numbers, taken from the line ofnbsp;folids, will give the leffer of the required means.

The lateral extent of 41-3-, talcen from the line of lines, apply tranfverfely to 4i|- and 414 on tlje line ofnbsp;folids; then the tranfverfe extent of 9 and 9 taken onnbsp;the folids, and applied laterally to the line of lines willnbsp;give 25 for the greater of the two means.

Apply the faid extent of 25 tranfverfely to 41-^ and 414 on the line of folids; then the tranfverfc extent onnbsp;the folids from 9 to 9 applied laterally to the line ofnbsp;lines, will give 15 for the leffer mean.

Examp. IV. V0 find the fide of a cube equal to a farallelopi-pedon vehofe lengthy breadth and depth arenbsp;given.

2d. Find the meafure of the mean proportional on the line of lines, and apply it to the lines of folidsnbsp;tranfverfely, at the numbers exprefling that meafure:nbsp;Then the tranfverfe extent of the length being takennbsp;from the line of folids and applied laterally to thenbsp;line of lines, will give the fide of a cube equal to thatnbsp;parallelopipedon.

Thus, Suppofe a parallelopipedon, whofe length is ']% , breadth 64, and depth 24,

The number 64 taken laterally from the line of lines and applied tranfverfely to 64 and 64 on thenbsp;line of plans ; then the tranfverfe diftance of 24 andnbsp;24 on the plans meafured laterally on the line of linesnbsp;gives about 39,2 for the mean proportional.

Apply the extent of the mean proportional, to 39,2 tranfverfely on the line of folids ; then the tranfverfenbsp;extent of 72 and 72 on the folids, being applied to thenbsp;line of lines laterally, will give 48 for the fide of thenbsp;cube equal in folidity to the given parallelopipedon-

APPENDIX.

tranfverfely on the line of folids from i o to i o, or from any other number to its oppofite.

id. Apply the like fide of the folid b tranfverfely to the lines of folids, and obferve the number it fallsnbsp;on : Then will the numbers on which thofe tranfverfenbsp;extents fall, fhew the ratio of the folids a and 3.

Examp. VL A folid a being given to find the dimen-fions of a fimilcir folid b, that to a fhall have any ajfigned ratio.

ift. On the line of folids feek two numbers ex-preiling the terms of the given ratio.

2d. Take the extent of one fide of the given folid A, and apply it tranfverfely on the lines of folids to thenbsp;antecedent of that given ratio ; then the tranfverfe extent of the confequent taken on the lines of folids willnbsp;be a like fide of the folid b.

Apply the fide of the cube a to the lines of folids tranfverfely from i to i ; that is from 10 to 10 -, thennbsp;will the tranfverfe diftance of the numbers 2 and 2 ornbsp;20 and 20 fliew the fide of the cube b.

Again. To find the diameter ofa fphere'�, that to the fphere a , whofe diameter is given, fhall he in the ratio of 0, to 2.

Make the diameter of the fphere a a tranfverfe diftance to 2 and 2 on the lines of folids then will the tranfverfe diftance of 3 and 3 on the line of folids benbsp;the diameter of the fphere b.

Examp. VII. Any number of unequal fimilar folids being given; to find the fide of a fimilar folid equal in magnitude to the film of the magnitudes of the given folids.

of one of the given folids, and apply this extent to the lineoffolids tranfverfely, to any number (fuppofe lonbsp;at the 3d i).

Also take in the fame equal parts, the numbers (hewing the fimilar fides of the other folids, and applynbsp;thefe extents to the lines of folids tranfverfely, notingnbsp;the numbers they fall on.

Then will the tranfverfe extent on the line of folids of a number equal to the fum of the noted numbers,nbsp;be the like fide of the fimilar folid required, whichnbsp;applied to the fame fcale of equal parts the others werenbsp;taken from will give the meafure of that fide.

Thus. What �will be the diameter of that iron Jhot caft from 3 other Jhot �whofe diameters �were 4 inches, 4, 4nbsp;inches, and 5 inches ; fuppojing no wajle in melting ?

Make 4 inches a tranfverfe extent on the line of folids, to any number fuppofe 10. Then 4,4 inchesnbsp;applied tranfverfely to the folids will give about 134 ;nbsp;and 5 inches alfo applied tranfverfely to the folids willnbsp;give about : Now the fum of the noted numbersnbsp;10 and 13� and i9x will be 43; then the tranfverfenbsp;extent of 43 on the line of folids will give 6i inchesnbsp;for the diameter of the new fhot.

Examp. VIII. To find the dinierfions of a folid�which Jhall be equal to the difference of t�wo given fimilar folids^nbsp;and alfo fimilar to them.

Apply a dimenfion of one folid tranfverfely to the line of folids at any number; and alfo note what number on the line of folids, the like dimenfion of thenbsp;other folid falls tranfverfely on take the differencenbsp;of thofe noted numbers �, and on the line of folids takenbsp;tranfverfely the extent of the remainder, and that willnbsp;be alike dimenfion of the fimilar folid required.

Thus. With the powder out of a Jhell of lo inches concave diameter is filled a fhell of 7 inches : What ftzednbsp;fhell will the remaining powder fill ?

The extent of 10 inches being applied tranfverlely to the lines of folids, at any number fuppofe 100 ; thenbsp;extent of 7 inches will fall tranfverfely on the lines ofnbsp;folids, about the number 34^; The difference between 100 and 34-^ is 65^: Then the tranfverfe extentnbsp;at 65-I. on the line of folids, will give 8,7 inches fornbsp;the concavity of that fhell which the remaining powdernbsp;will fill.

?	, ,,, 1 1,,,' ' Tl' ‘i' '	I . . V 1 nbsp;nbsp;nbsp;.	. 1 ■......!■	-■■ .■■.1 -.-	,■■■......,.»ig.,-...^-^-^-1	OOT
		' ‘ I8‘ ' ' ' '	■ .........	ok	f nbsp;nbsp;nbsp;ilr .	' nbsp;nbsp;nbsp;71T

	.............		‘ ff ‘ ‘ nbsp;nbsp;nbsp;‘ '	ie		—tr-
	n?:rr,,, iy, „,,k., ,,quot;	1 nbsp;nbsp;nbsp;, TC ,	..... \’-. . . . . 1		Ir . nbsp;nbsp;nbsp;—^-

Names.	aj rt c/gt;	Angles at Center	Angles at Cir.
Trigon	3	120� 00'	60� 00
Square	4	90 00	90 00
Pentagon	5	72 00	108 00
Hexagon	6	60 00	120 00
Heptagon	7	51 25f	128 34,^
Oftagon	8	45 00	135 00
Nonagon	9	40 00	140 00
Decagon	10	36 00	144 00
Endecagon	II	32 43Tr	147 l�rt-
Dodecagon	12	�^0 00	150 00

N� Rad.	Outward Curve,	Inward	In parts of	I ft rad.
N� Rad.	Outward Curve,	Curve.	Outward.	nward.
1 i *	^ = 14,166,	^1=12,604 40	100,000	88,969
2	^ = 12,500	^ = 11,146 4�	88,235	78,677
3	^ = 10,833	f	76,468	6S,379
4	�= 9,.66	8,229	64,705	58,087
	70	1010
5	7.777	7.0x4	54,901	49,510
6	6,666 9	^=6,04. 144nbsp;nbsp;nbsp;nbsp;� 'X	47.05S	42,642
7	5.555		39.2x5	35.7SX
8	12= 4,444	590 ��= 4.097	3X.372	28,920
9	g ^= 3,611	3.368 144	25.490	23.774
10	ff- 3.055	111= 2,882 144	21,56s	20,343
11	�= a.soc	777= ^�395 144	17,647	16,906
12	35 nbsp;nbsp;nbsp;P75 tI= ^�94417^= ^.909		1 13.725	13.475

Ditto . .		� 3i
Ditto . .		� 44
Ditto . .		� 14
Ditto . �		� 54
Ditto . .		� i4
Ditto . .		� 34

Degrees.	Nat. Tangent.	Nat. Secant.
76	4,011	4,133
77	4^331	4,�'45
78	4,701	4,810
79	5 gt;44	5,241
80		5,759
81	6,314	6,392
82	7�gt;gt;5	7,185
83	8,144	8,205
84	9 514	9,567
85	11,430	11,474
86	14,301	gt;4,335
87	19,081	19,107
88	28,636	28,654
89	.57, 90	57,300

	Altitude.	Projeftion	Profile.
	Min.	Min.	Min.
Capital	5	i6	3
Freeze	45	�	�
Trigliphs	40	15	F -F zF
Plinth	4f	i6	3
Cap	ix	*5	2
Drips	3f	15	2

		Tufcan.				Doric.
[Namesor tne iviembers.		Alt.		Proj.		Alt.		Proj.
		Mo.	Mi.	Mo.	Mi.	Mo.	Mi.	Mo.	Mi.
	��FiUct . nbsp;nbsp;nbsp;.nbsp;nbsp;nbsp;nbsp;-nbsp;nbsp;nbsp;nbsp;-	�	�	�	�	0	3i	0	56
	Ogee - nbsp;nbsp;nbsp;- -nbsp;nbsp;nbsp;nbsp;-	�	�	�	�	�	�	�
	Corona - nbsp;nbsp;nbsp;-nbsp;nbsp;nbsp;nbsp;-nbsp;nbsp;nbsp;nbsp;-	�	�	_	_____
	Fillet . - - nbsp;nbsp;nbsp;-	�	�	�	�	�	�	�	�
	Cima . nbsp;nbsp;nbsp;-nbsp;nbsp;nbsp;nbsp;-nbsp;nbsp;nbsp;nbsp;-		_	�		0	q

qC	Fillet - nbsp;nbsp;nbsp;-nbsp;nbsp;nbsp;nbsp;-nbsp;nbsp;nbsp;nbsp;-	�	_	_		0		0
o O	Aftragal - - - �	�	�	�	�	�	\ ri	�	t45i
	Cavetto . nbsp;nbsp;nbsp;-nbsp;nbsp;nbsp;nbsp;-	_	� .	_	_	0	5	0	41 T
	L Fillet - , - , -	�	�		�	�	5	�
	The Cornice - nbsp;nbsp;nbsp;-	-	-	�	...	0	z6-g	�	�
	THE DIE - -	1	C		42	1	20	0	40
	The Bale - - -	�	�	�	�	0	40	�	�
	Fillet . - - -		_	_	___	-.	_
	Cavetto ...	�	�	�	�	0	5	0	4i-i
	Ogee - - - -		�	�	�	_	...	_	��
w	Allragal ...	�	�		___			_	_
	Fillet - ... -	�	~	�	�	0		0	(46 l47i
cq	Cima - - - .	�	�	�	�	_	_	_	_�
	Fillet - - - .	_		__			\
	Torus - - - -^Plinth - - . ,	�	�	�	�	�	9 27v	0 0	50 5�

Ionic.			Corinthian.					Roman
I Proj.			Alt.		Pro].		Alt.		Proj.
Mi,	Mo.	Mi.	Mo.	Mi.	Mo.	Mi.	Mo.	Mi.		Mi.
	o	sH	O		0	37	0	2f	0	57
3f	o	i 55� i 53l	o	3f	0	js6 t S4i	0	3f	0	556 { 54f
4l	o	S^i	o	4�	0	S3i	0	5f	0	S3f
li	0	S�4		�	�		0	1	0	52?
Si	�	�	o	4t	0	S 49: 146	0	H	�	�
	0	44�	o		0	46	�	�	�	�
						5 43	0	3	p	46I
_	-	..	o	3l	0	5 43	�	��	.�	�
						1 43
3r	o	4ii	�	�	�	�		�	�	�
�	�	�	�	�	�	�	0		0
	�	�	o	�9	__	�	0	2lt;;\|
3?	o	4�i	I	36	0	' 2	2		0	J 2'
42i	�	�	o	3�	�	�	0		-
�	�	�	_	�.	�	�	0	I	0	45�
3f	o	4�l	�	�	�		�	�	�
�	�	�	o	4	0	5 43 146	�	�
�	�	�	�	�	��	�	0	3	0	47
�i	o	47�	o	0-1	0	47	�	�	�	�
5	�	�	o	5	�	�	0	7i	0	5 45l 1 S4l
oi	o	S3l	o		0	ss	0	I	0	54i
4	0		o	4	0	57	0	4f	0	57
28i	o	r�i	o	23'	0	C7	0	'} ^	0	'7

2: 0 CT) 51 agt; . l/i	Length on the Leg.	Z\| 0 51 ngt; c/3	Length on the Leg.	0 Cn ol	Length on the Leg.
3	ogt;3333	11	0,6596	19	0,7523
4	0,4142	12	0,6589	20	0,7617
S	0^4597	�3	0,6765	21	0,7706
6	0,5000	14	0,6921	22	0,7785
7	o,53.H	'5	0,7063	23	o,/86o
8	0,5665	16	0,7193	24	0,7931
9	0,5940	17	o,73�3
10	o,6r8o	i8	0,7423

Nature	Brals				Iron
of guns	Proof		Service		Proof		Service		Salutes	Scaling
Pounders	ib.	OZ.	ife.	OZ.	ife.	OZ.	fe.	oz.	ife.oz.	Ife.	oz.
I	I	.0	0.	8	1	0	0.	8	0.8	0	IF
iF	I	.8	0	12	I	.8	0	12	0.12	0	.2
2	2	0	i	0	2	0	t	8	1.0	0	.3
3	3	0	I	8	2	.0	T	8	1.8	0	.4
4	4	0	2	0	4	�0	2	.0	2.0	0	.6
5t	5	4	2	10	5	�4	2	. 10	2.10	0	.8
6	6	.0	3	.0	6	.0	3	.0	3-0	0	.8
8	8	.0	4	0	8	0	4	0	3-12	0	. 10
9	9	0	4	8	9	.0	4	.8	4-0	0	. 12
12	12	0	6	0	12	. 0	6	.0	4.12	I	.0
i8	ji8	0	9	.0		. 0	9	.0	6.0	I	.8
24	(21	0	12	.0	18	.0	11	.0	7.0	2	.0
I 26	,22	. 0	*3	. 0	19	. 0	12	. 0	7.12	2	�4
! 32	26	. 12	16	.0	2 t	.8	14	. 0	9.4	2	.12
j 36	128	. 0	18	.0	22	. 0	15	. 0	10.0	3	. 0
1 42	131	.8	21	. 0	25	.0	17	.0	n .4	3	.4

Cubes i	lt;oots (	Jubes t	lt;.oots	Cubes ]	^oots	Cubes	lt;oots lt;	Jubes	R.oots
I	1,000	21	V59	41	5.448	61	3.936	81	4.327
2	1,26c	22	2,802	42	3.476	62	3.958	82	4.344
3	1,442	23	2,844	43	3.503	63	3.979	83	5.362
4		24	2,884	44	�.530	64	4-,000	84	4.379
5	i,7icj	25	2,924	45	3.557	65	4,021	%	4.397
6	1,817	2�	2,962	46	3.583	66	4,041	86	4.414
7	igt;9'3	27	3,00c	47	3.609	67	4,061	87	4.43*
8	2,000	28	3,036	48	3.634	68	4,082	88	4.448
9	2,080	29	3,072	49	3.659		4,102	89	4,465
10	2,154	30	5.107	50	3.684	70	4.I2I	90	4,481
u	2,224	31	3.H1	S'*	3,708	7*	4.141	9*	4.498
12	2,285	32	3.175	52	3.732	72	4,160	92	4.5*4
13	2^35'	33	3.207	S3	3.756	73	4.179	93	4.53*
gt;4	2,410	34	3.240	54	3.78c	74	4,198	94	4,547
15	2,465	35	3.271	55	3.803	75	4.217	95	4,563
16	2,52c	36	3.302	56	3,826	70	4.236	96	4.579
17	2A7gt;	37	3.332	57	3.848	77	4.254	97	4.595
18	2,621	38	3.362	58	3.871	78	4.27:	98	4,610
gt;9	2,66f	39	3.39	1 59	3.893	79	4.291	99	4,626
20	2,714! 40		3,420! 60		3.91'	80	4.30c	lOO	4,642

(T 2a

T R E A T I S E

Mathematical Inftruments

GUNNERS CALLIPERS.

ERRATA.

PETER DAVALL, Efq;

REA

( iv )

( V )

( vi )

( vii )

( v�i )

( ix )

( xi )

( xii )

( xiii )

2 XXXVI r.

XLIV.

( )

CON-

CONTENTS.

Seftion.

a nbsp;nbsp;nbsp;VIII.

txviii CONTENTS.

To

CONTENTS.

AP-

appendix.

CASE,

PORTABLE COLLECTION,

Mathematical Inftruments.

are

of Mathematical Inftruments. nbsp;nbsp;nbsp;^

Of the Compasses attd Bows.

To

Sect. III.

Sect. IV.

cf Mathematical Infiruments. nbsp;nbsp;nbsp;g

to

PROBLEM I.

PROBLEM II.

of Mathematical Infnments. n

Sect. VI.

Of the Protractor.

PROBLEM IV.

PROBLEM

PROBLEM V.

PROBLEM VI.

SECT. VII.

20 nbsp;nbsp;nbsp;�The Defcription and life

fecond

of Mathematical Infriiments. ai

PROBLEM

PROBLEM VIII.

PROBLEM IX.

of Mathematical Infruments. 2^

1�he ConfruBion of the remaining Lines of the

54 nbsp;nbsp;nbsp;�he Dejcription and �/�

I. To conJiruB the Line of Chords *.

il. The Line of Rhumbs^.

III. The Line of Sines *

of Mathematical Injlruments. 29

A TABLE, Jheiving the Miles in one Degree of Longitude to eneery Degree of Latitude.

Sect. IX,

Of the S ^ c r o r.

The

of Mathematical Inflruments, 31 , Tihe Scales commonly put on the beji Seniors, are

Tangents to 45�

Double

Dcfcription and Ufe

]V. l�he Scale of Logarithmic Numbers.

L'he ConJiriiBion.

of Mathematical Injlruments.

V. The Scale of Logarithm Sines,

36 nbsp;nbsp;nbsp;T!he De faquot; ip tl on and life

VI. fhe Scale of Logarithmic fangents.

VII. fhe Logarithmic verfed Sines.

of Mathematical Injlruments. 37

Sect. XI.

Of the ConfruBion of the Doubl� Scales.