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A
OF SUCH
As are ufualiy put into a PORTABLE CASE,
Containing their various Ufes in
Arithmetic, I I Architecture, Geometry,nbsp;nbsp;nbsp;nbsp;Surveying,
Trigonometry,   Gunnery, ijc.
With a fliort Account
Of the Authors who have treated on the
PROPORTIONAL COMPASSES And SECTOR.
To which js now added
An APPENDIX;
Containing, the Defcription and Ufe of the
The fecond Edition, with many Additions.
By J. ROBERTSON, F. R. S. Mafter of the RoyalAcademy at Portfmouth.
‘LONDON:
Printed for T. Heath and J Nourse in the Strand; J. H o d g e s on LondonBridge, andnbsp;J Fuller in AvemaryLane. M bcc lvh.
ocr page 8Page xvi. read March 5,1757 p10. 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.
Page 128. p. 12. for Ac=z—— N read Ar=:—;—N, •
° nbsp;nbsp;nbsp;^nbsp;nbsp;nbsp;nbsp;«fi
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.
ocr page 9Secretary to the Royal Society.
S I R,
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;muchefteemed and learned friend James Burrow, Efq; Viceprefident 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 publickly thanking him : Although I am confcious,
that
ocr page 10DEDICATION, that you have the higheft regard for the two refpedtable 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.
S I R,
Your mofl obedient Humble Servant,
•5oo5ocjfoc®ocjoC?Os$x$Dc;5De^«5oji^c^o53ojDC^cJxamp;6Soc5Soc^
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E R.
T is needlefs to enumerate the many pur* ^ I ^ pofes, to which mathematical inftrumentsnbsp;ierve their ufe feems quite neceffary to ferfons 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 abfolutely 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,
ocr page 12pieces, 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 conftruction 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 Rodolphusll 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 meafuringof 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.
fj In the colleflion of the late IfUliam Jones, Efq;
II.
ocr page 13II. 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 C07npajjes.
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 feóloral lines, of equal parts, ofnbsp;polygons, and of fuperficies that is, lines fo difpofed, 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.
but
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.
ocr page 14but leaves them for the exercife of the reader to difcover.
V. nbsp;nbsp;nbsp;Levinus Huljius, in his book of mechanical inJlruments, 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, furfaces, 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
been
ocr page 15been 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 againft. 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 conjideratione 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 compafs, under the title of Ufus amp; Fahrica circini cujufdam proportionis, ifc.” Galilaus begins with annbsp;addrefs to the reader, wherein he concludes, that a perfon 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
A 4 nbsp;nbsp;nbsp;lor d.
ocr page 16lord. Of which /aid injirument the abovenamed 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 confiderable 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 univerfity 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;Cö/rö to appear before them on the 18th of April,nbsp;the next day the caufe was heard and the parties difmilTed : But on the 4th of May following, their lordfhips 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 Eunpe, amp;c.”
IX. Anno
ocr page 17IX. 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 publilhed 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;chequerwork made by lines drawn thereon) whichnbsp;operates the problems performed by the french fedlorsnbsp;very accurately.
XL Anno 1615, Stephen MichaelSpackers, publifhed 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
ocr page 18in quarto at Marpurg, Anno 1617; fays, his ruler is applicable to the fame ufes as Jujlus Burgius'% inftrument. 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 pinhole, whereby he can apply the edge of another ruler, and fo conftitute a feófor 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 occafioned 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 fectov, 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 fixteen 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 deicription of the crofs ftaff.
XV. Mutio Oddi of Urhino printed at Milan, An.nbsp;2nbsp;nbsp;nbsp;nbsp;1633,
ocr page 191633, a quarto book, called the conJiruBion ünd ufe of the compaje polimetro, (or feftor.) The lines on this inftrument, were fuch as were common at that time: Henbsp;fays in the dedication to his friend Peter Linder of Nurenherg, 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 occafioned 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.
The following authors have alfo wrote on the fedtor, and fedloral lines,
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
a third
ocr page 20a 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 feöïor. 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
vol.
ocr page 21Vol. p. 1158, he treats on the fedtor, relates the conteft 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 ajlronomicum, 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.
XXXVI. J, Good, in his treatife on the fedicr, in 12mo. London, 1713.
( xiv )
XXXVII. Chrifiian WolfiuSt in his math, lexicon,
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
XXXIX. Mr. Belidor, in his new courfe of math. in 4to. p. 364, Paris, 1725.
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 arithmeticogeometricum, in fol. Leipfic, 1727. p. 86, gives a detail of the inventors of the proportional compaffes and feélor, 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; Horfcher, 1605; Horen, 1605; Hulfius, 1604, Claxius,nbsp;Lockmann, 1626, Metius, 1623; Patridge—;nbsp;deSaxonica, 1619; Scheffelts, 16975 Steymann, 16245nbsp;Uttenhoffers, 1626.
XLIII. Samuel Cunn, in his new treatife on the felior, 8vo. London, 1729.
( XV )
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 perfons 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 feétor: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,
ocr page 24Burgius, 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 fector: 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 perfons 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.
Royal Academy Port/moutU
March 5, 1755*
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THE
Page.
I. nbsp;nbsp;nbsp;^ F the common portable injlruments and
II. nbsp;nbsp;nbsp;Of con7paffes.nbsp;nbsp;nbsp;nbsp;3
III. nbsp;nbsp;nbsp;Of the b lacklead pencil, feeder, and tracingpoint. 7
T0 trace or copy a drawing. nbsp;nbsp;nbsp;ibid.
IV. nbsp;nbsp;nbsp;Of the drawingpen andprotra5lingpen.nbsp;nbsp;nbsp;nbsp;g
V. nbsp;nbsp;nbsp;Of theparailelruler, andits ufe.nbsp;nbsp;nbsp;nbsp;9
i ft. In drawing of parallel right lines. nbsp;nbsp;nbsp;i o
2d. In the dividing of right lines into equal parts.
3d. In the reduilion of rightlined figures to right
lined triangles of equal area. nbsp;nbsp;nbsp;11
VI. nbsp;nbsp;nbsp;Of the protratJor, and its ufe.nbsp;nbsp;nbsp;nbsp;13
I ft. In plotting and meafuring of rightlined angles ;
14
2d. In drawing of right lines perpendicular to each other ;
3d. /« infcribing of regular polygons in a circle, 15 4th. In defcribing of regular polygons on given rightnbsp;lines.nbsp;nbsp;nbsp;nbsp;16
VII. nbsp;nbsp;nbsp;Of the plane fcale, and its fever al lines.nbsp;nbsp;nbsp;nbsp;18.
Conflruliion of the fa'es cf equal parts. nbsp;nbsp;nbsp;ibid.
'Their ufe, joined with the protralior, in plotting of
rightlined figures. nbsp;nbsp;nbsp;2 2
ConfiruSiion of the other lines of the plane fcale, viz. I ft. Chords, ?.d. Rhumbs, 3d. Sinés, 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
Seft. nbsp;nbsp;nbsp;Page.
VUL ^he ufes of fome of the lines on theplane fcale. nbsp;nbsp;nbsp;27
y/ table, Jhewing the miles in one degree of longi~ tude to every degree of latitudenbsp;nbsp;nbsp;nbsp;29
IX. nbsp;nbsp;nbsp;Of the feSicr and its lines.nbsp;nbsp;nbsp;nbsp;30
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.
37
XII. nbsp;nbsp;nbsp;Of the ufes of the double fcales.nbsp;nbsp;nbsp;nbsp;40
^he ufe of the lines of lines.
I ft. ÏÖ two right lines given, to find a proportional, nbsp;nbsp;nbsp;41
2d. To three right lines given, to find a 4th proportional. nbsp;nbsp;nbsp;42
3d. To fet the fcales of lines at right angles to one another.nbsp;nbsp;nbsp;nbsp;43
4th. Betvieen two right lines to find a mean proportional. nbsp;nbsp;nbsp;ibid.
5 th. To divide a right line into equal parts. 44 6th. To delineate the orders of architeBure.nbsp;nbsp;nbsp;nbsp;45
Some terms in architeBure explained. nbsp;nbsp;nbsp;ibid.
Of the general proportions in each order. nbsp;nbsp;nbsp;47
To draw the mouldings in architeBure. nbsp;nbsp;nbsp;55
Table for defcribling the Ionic nbsp;nbsp;nbsp;volute.nbsp;nbsp;nbsp;nbsp;59
Ufes of fome tables for drawing the orders. nbsp;nbsp;nbsp;60
To delineate any order by the tables. nbsp;nbsp;nbsp;62
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.
XIII. nbsp;nbsp;nbsp;Some ufes of the fcales of polygons.nbsp;nbsp;nbsp;nbsp;72
XIV. nbsp;nbsp;nbsp;Some ufes of the fcales of chords.nbsp;nbsp;nbsp;nbsp;73
To delineate the ftation lines nbsp;nbsp;nbsp;of a furvey.nbsp;nbsp;nbsp;nbsp;75
XV. nbsp;nbsp;nbsp;Some ufes of the logarithmic fcales of numbers. 79
XVI. nbsp;nbsp;nbsp;Some ufes of the fcales of logarithrnic fines, and lo
garithmic tangents. nbsp;nbsp;nbsp;84
XVII. nbsp;nbsp;nbsp;Some ufes of the double fcales of fines, tangents, and
Xl3f
Sed.
'To find the length ofi the radius to a given fme^ tangent or Jecant.nbsp;nbsp;nbsp;nbsp;S8
.To find the degrees correfponding to a given ftne^ tangent orfiecant.nbsp;nbsp;nbsp;nbsp;8^
To a given number of degrees, to find the length ofi the verfiedfine.nbsp;nbsp;nbsp;nbsp;ibid.
To fiet the double lines to any given angle. ibid. To deficrihe an Ellipfis.nbsp;nbsp;nbsp;nbsp;go
To deficrihe a Parabola. nbsp;nbsp;nbsp;gi
To deficrihe an hyperbola. nbsp;nbsp;nbsp;g2
To find the diftance ofi places on the terrejlrial globe.nbsp;nbsp;nbsp;nbsp;93
XVIII. The ufie ofifiome ofi thefmgle and double ficales on the fieSlor, applied in the fiolution ofi all the cafiesnbsp;ofi plane trigonometry.nbsp;nbsp;nbsp;nbsp;9 ^
Case I. When among the things given, there he a fide and its oppofite angle.nbsp;nbsp;nbsp;nbsp;96
Case II. When two fides and the included angle are known.nbsp;nbsp;nbsp;nbsp;99
XIX.
Case III. When the threefides are known. 103 The confiruWion ofi the fieveral cafies ofi fiphericalnbsp;triangles, by the ficales on thefieSior.nbsp;nbsp;nbsp;nbsp;107
Case I. Given two fides, and an angle oppofite to one ofi them.nbsp;nbsp;nbsp;nbsp;108
Cale II. Given two angles, and a fide oppofite to one ofi them. nbsp;nbsp;nbsp;112
Case III. Given two fides, and the included angle.nbsp;nbsp;nbsp;nbsp;115
Case IV. Given two angles, and the included fide.nbsp;nbsp;nbsp;nbsp;1,18
Case V. Given the three fides. nbsp;nbsp;nbsp;122
XX.
Case VI. Given the three angles. nbsp;nbsp;nbsp;124
Ofi the proportional compafifies. nbsp;nbsp;nbsp;125
The figures referred to, are contained in fieven copperplates.
I
ocr page 28XX nbsp;nbsp;nbsp;CONTENTS.
Page.
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
II. nbsp;nbsp;nbsp;Of the weights of iron fhot.nbsp;nbsp;nbsp;nbsp;135
III. nbsp;nbsp;nbsp;Of the meafures of concave diameters. 136
IV. nbsp;nbsp;nbsp;Of the weights of fhot to given gun bores. 137
V. nbsp;nbsp;nbsp;Of the degrees in the circular head. 138
VI. nbsp;nbsp;nbsp;of the proportion of troy and averd. weights.
VII Of the proportion of Englijh and French feet and pounds.nbsp;nbsp;nbsp;nbsp;141
V\\\.Fa£lors ufeful in circular and fpherical figures.
142
IX. nbsp;nbsp;nbsp;Of the fpecific gravities and weights of bodies.nbsp;nbsp;nbsp;nbsp;147
Some ufes of the table. nbsp;nbsp;nbsp;151
X. nbsp;nbsp;nbsp;Of the quantity of powder ufed in firing of cannon.nbsp;nbsp;nbsp;nbsp;_nbsp;nbsp;nbsp;nbsp;154
XI. nbsp;nbsp;nbsp;Of the number of fhot or fhells in a finijhed
XII. Concerning the fall of heavy bodies. nbsp;nbsp;nbsp;161
XIII. nbsp;nbsp;nbsp;Rules for the raifing of water.nbsp;nbsp;nbsp;nbsp;164
XIV. nbsp;nbsp;nbsp;Of the fhooting in cannon and mortars. 167
XV. nbsp;nbsp;nbsp;Of the line of Inches.nbsp;nbsp;nbsp;nbsp;174
XVI. nbsp;nbsp;nbsp;Of the logarithmic fcales of numbers, fines,
verfedfines and tangents. nbsp;nbsp;nbsp;ibid.
XVII. nbsp;nbsp;nbsp;Of the line of lines.nbsp;nbsp;nbsp;nbsp;175
XVIII. Of the lines of plans or fuperficies. ibid. XIX. Of the line of folids.nbsp;nbsp;nbsp;nbsp;180
Nine Plates.
To the Binder.
The plates are all to ftand upright in the book, and no part to be folded upwards or downwards.
ocr page 29THE
Description and Use
Of the moft Neceflary
Sect, I.
A S E S of Mathematical Inflruments are lgt;v C m various forts and fizes ; and are comjjW monly adapted to the fancy or occafion ofnbsp;the perfons who buy them.
The fmalleft colledlion put into a cafe, commonly confifts of,
I. A fair of compaffes, one of whofe points may be taken off, and its place fupplied with,
B nbsp;nbsp;nbsp;A crayon
ocr page 302 nbsp;nbsp;nbsp;7he Defcription a?id Ufc
A crayon for lead or chalks.
A drawingfen for ink.
II. nbsp;nbsp;nbsp;A plane fcale.
With thefe inftruments only a tolerable Ihift may be made to draw moft mathematical figures.
But in fets, called complete pocketcafes, befide the inftruments above, are the following.
III. nbsp;nbsp;nbsp;A fmallcr pair of compaffes.
IV. nbsp;nbsp;nbsp;A pair of bows.
V. nbsp;nbsp;nbsp;A blacklead pencil, with a cap and feeder.
VI. nbsp;nbsp;nbsp;A drawingpen with a protra5iing~pin.
VII. nbsp;nbsp;nbsp;h. protractor.
VIII. nbsp;nbsp;nbsp;A parallelruler.
IX. nbsp;nbsp;nbsp;KfeClor.
In fome cafes, the plane fcale, protradlor, and parallelruler, 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.
Note, the fizc of a cafe is named from the length of the fcale or feftor.
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, drawingpen, 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
X. nbsp;nbsp;nbsp;A tracingpoint.
XI. nbsp;nbsp;nbsp;A pair of proportional compaffes.
XII. nbsp;nbsp;nbsp;A gunner’ll callipers.
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, defignmg, amp;c. and as the greateft part of thefe inftruments
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.
S E C T. II.
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 oppofition of the metals makes them wear more equable :nbsp;and by means of the ferew axle, with the help of anbsp;turnferew^ (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.
Tö ufe the fpring point.
Hold the compaffes in the left hand w'ith the ferew turned towards the right turn the ferew towards you,
B 2 nbsp;nbsp;nbsp;'nbsp;nbsp;nbsp;nbsp;or
ocr page 324 nbsp;nbsp;nbsp;^he Dejcription and Vfi
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 blacklead or chalks, by thenbsp;crayon; and with dots, by the dottingwheel; fornbsp;cither of them may be fixed in the fiiank in the placenbsp;of the fteel point.
As the dottingwheel 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 drawingpen.
The drawingpen 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
2 nbsp;nbsp;nbsp;the
ocr page 33cf Mathematical Lijiriiments. nbsp;nbsp;nbsp;5
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 praélice brings on eafy habits in the ufc ofnbsp;things, however a caution or two may be ferviceablcnbsp;to beginners.
To open and work the compajfes.
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.
To take a diftance between the points of the compajfes.
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 otherwife than nearly upright.
6 nbsp;nbsp;nbsp;'^he Defcription and JJfe
Po defcribe circles or arcs with the compajfes.
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.
Of the
B o V/ s.
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.
Of the Blackkad Pencil, Feeder^ and Tracing
Point.
TH E Blacklead Pencil is ufefu! to defcribe the firft draught of a drawing, before it is markednbsp;with ink becaufe any falfe ftrokes, or fuperfluous
lines,
ocr page 35of Mathematical Inflriiments. 7
lines, may be rubb’d out with a handkerchief or piece of bread.
The Feeder is a thin flat piece of metal, and is fometimes fixed to a cap that flips on the top of the pencil, and fcrves either to put ink between the blades of thenbsp;drawingpen, 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, ö’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.
Notc^ The coloured paper will ferve a great many times.
concerning this excellent mineral.
There is not perhaps, a more ufeful inftrument in being for ready fervice in making of Iketches or finiflacd plans; whether of architefture, fortification, machines, landlkips, ornaments, i£c than a blackleadnbsp;pencil; and therefore it may be proper to give a fewnbsp;hints
Blacklead 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
B 4 nbsp;nbsp;nbsp;into
ocr page 368 nbsp;nbsp;nbsp;l^he T)efcription and Ufe
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 blacklead ; 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 blacklead for ufe, is firft to fiuv it into long flips about the fize of a crowquill,nbsp;and then fix it in a cafe of foft wood, generally cedar,nbsp;of about the fize of a goofequill, or.larger; and thisnbsp;cafe is cut away with the lead as it is ufed.
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Of the DraivingPen^ and ProtraBingPin.
TI'I E Brawingpen 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,
ocr page 37clined, 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 drawingpen 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 drawingpen., will ferve for the drawingpen point, ufed with the compalTes. 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 Protradiingpin is a piece of pointed fteel (like the point of a needle) fixed into one end of a part ofnbsp;the handle of the drawingpen •, 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 drawingpen is a focket, into which a piece of blacklead pencil maynbsp;be put.
Sect. V.
Of the P A R A L L E LR U L E R .
This inflrument confifts of two Rulers, conneded 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
lo nbsp;nbsp;nbsp;'T’he JDefcription and Ufe
to each Ruler ; whereby they will be Parallel at every diftance, to which the bars will fufFer them to receed.
But the beft ParallelRulers are thofe, whofe bars crofs each other, and turn on a joint at their interfeflion •, 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.
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 parallelruler 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 neareft 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.
The manner of nbsp;nbsp;nbsp;parallelruler hert ^i
redted, is underftood to be the fame in the folution of the following Problems.
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
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ocr page 39any 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 parallelruler by thenbsp;points 5 and a, and parallel thereto, through thenbsp;points 4, 3, 2, I, draw lines , then ab, by the intcrleiftion of thofe lines will be divided into 5 equal parts.
PROBLEM III.
Any right lined quadrangle or polygon being given^ to make a right lin'd triangle of equal area.
Exam. I. To make a triangle of equal area to the quadrilateral abdc. (Fig. 3.)
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.
Exam. III. To make a triangle equal in area to the Hexagon, abcdef. (Fig. 5.)
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.
ocr page 4012 nbsp;nbsp;nbsp;T!he Defcription and Ufe
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 aüch.
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.
2d, Draw KC, and through b draw bl parallel to
KF, nbsp;nbsp;nbsp;meeting CD iiiL; draw kl, and the three Tidesnbsp;DC, CE, BK, are reduced to the two Tides dl, lk.
3d, Draw kg ; through h, draw hm, parallel to
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
MF.
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,
ocr page 41of Mathematical hjirumenfs. 13
The Protractor, is an inftrument of a femicircular 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 femicircumference 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 femicircumfercnce, 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
edae.
ocr page 4214 nbsp;nbsp;nbsp;^'he Dcfcriptlon and Vfe
edge, whereby the divifions on the limb will be much eafier pointed off,
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 protraftingpin, 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.
Exam. Ik To find the tiumber of degrees which meafure the angle abc. Fig. 9.
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.
IS
of Mathematical Injirumenfs.
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.
In a circle given to inferiheany regular Polygon, fuppofe an olfagon. Fig. 11.
Construction. Apply the blank edge of the protraBor 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 expreffing 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 in2nbsp;nbsp;nbsp;nbsp;terfeft
ocr page 44i6 nbsp;nbsp;nbsp;l’he Defcrlption and Ufe
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 compaffes 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.
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 lórt 
Dodecagon 
12 
•^0 00 
150 00 
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.
PROBLEM VIL
Upon a given right line ab, to deferihe any regular polygon. Fig. 12.
Con
ocr page 45of Mathematical Infruments. \j
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.
Exam. I. Upon the line ab defcrihe an hexagon. Fig. 12.
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 direéled; 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.
Exam. II. Upon the line ab to defcrihe a pentagon. Fig. 13.
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 direéled; 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
ocr page 46ig nbsp;nbsp;nbsp;1’he Defcription and Ufe
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.
Of the Plain Scale.
Marked EP.nbsp;Cho.
Ru.
Sin.
Tan.
Sec.
S.T.
Lon. Lat.
Ho.
In. Mer.
H E lines generally drawn on the plane fcale, are thcfe following:
I. Lines of equal parts.
II.
III.
IV.
V.
VI. V1Ï.nbsp;VIII.
IX.
X.
XI.
Chords. Rhumbs.
Sines.
Tangents.
Secants.
Half Tangents.
Longitude.
Latitude.
Hours.
Inclinations.
Of the Lines of equal Parts.
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; 4fgt; 0’c. which divifions diftinguifh bynbsp;Inbsp;nbsp;nbsp;nbsp;lines
ocr page 47of Mathematical Infruments. 19
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 lefthand 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.
Therefore to lay off a line 8 nbsp;nbsp;nbsp;87, or 870
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.
ocr page 48But if a length of feet and inches was to be exprefled, 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 fubdivifions 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 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 jV or 3 ï^ö^ whether leagues,nbsp;miles, chains, i^c.
Osi the diagonal line, joined to the 4th of the firft fubdivifions, count 7 downwards, reckoning the diftance 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 purpofes than any fet of fimply divided fcales hithertonbsp;made public, on one ruler.
One Side ) The divifions j 10, ii, 12,131,15, nbsp;nbsp;nbsp;18, 20,22, 27.
OtherSideJ to an inch ( 28, 32, 36,40, 45,50,60,70,85,100.
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.
23 nbsp;nbsp;nbsp;'^he Defcription and Vfé
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 protraétor 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.
If two angles and the fide contained between them were given.
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.
, five adjacent things., fides and angles, in a right lin'd quadrilateral, being given, to lay down the plannbsp;thereof. Fig. 1 7.
ExAfct.
ocr page 51Exam. 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.
If 3 of the things were fides, the plan might be formed with equal eafe.
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.
Plain Scale.
Preparation. Fig. 18. Pi. VI.
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
to AB, ED.
This mark or charafter A, fignifies the angle.
This mark — fignifies equal to.
By links, is meant the nbsp;nbsp;nbsp;part of a chain of four noles
or of 66 yards long.
ocr page 52Divide the arc ad into 90 equal parts; mark thé loth divifions with the figures 10, 20, 30, 40, 50^nbsp;60, 70, 80, 90 ; on D, as a center, with the coinpafles, 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.
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.
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.
1 nbsp;nbsp;nbsp;from
ocr page 53bf Mdthèmatlcal ïtiJlruniè'ntÈ. 25
frorti c to D for the right fines and from d tö 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.
IV. nbsp;nbsp;nbsp;Tloe Line of Tangents *.
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.
V. nbsp;nbsp;nbsp;The Line of Secants f.
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.
VI. The Line of HalfTangents (or the Tangentsnbsp;of half the Arcs).
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 correfponding figures of the arc ad.
The femitangents 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
ocr page 54amp;6 nbsp;nbsp;nbsp;^he Defcription and JJfe
fions of the femitangents above 90 degrees 'will be obtained on the line ca continued.
VII. nbsp;nbsp;nbsp;The Line of Longitude.
Divide ah, into 60 equal parts; through each of thefe divifions, parallels to the radius ac, will interfedl 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.
VIII. nbsp;nbsp;nbsp;The Line of Latitude.
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 divifions from b to d, with lo, 20, 30, idc. to 90 •, andnbsp;BD will be a line of latitude.
IX. The Line of Hours.
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.
X. The Line of Inclinations of Meridians.
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.
Sect.
ocr page 5527
of Mathematical Infruments.
Sect. VIII.
fhe ufes of fome of the Lines on the Plain Scale.
I. Of the Line of Chords. Pi. VI.
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.
Note, Degrees are generally denoted by a fmall “ put
over them.
II. Of the Line of Rhumbs^
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
ocr page 562 8 nbsp;nbsp;nbsp;^he Defcription and Vfe
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 bc ; 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 interfedtion b to c ; draw ac, and the angle bag lhall reprefent the Ihip’s courfe.
III. TTje ufe of the Line of Longitude.
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
ocr page 57of thè treatifes of navigation, and fomc other books, have thcfe tables , but the moft ufeful and efteemednbsp;are Sher‘win'% mathematical tables.
D. L, 
Miles. 
jU. L. 
Miles. 
D. L 
Miles 
I 
59^99 
31 
5Ï.43 
él 
29,09 
2 
5996 
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 
ÏO 
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'ïO 
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
30 nbsp;nbsp;nbsp;“The Defcription and Ufe
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.
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 diftinguifiied into fingle, and double.
The fingle fcales are fuch as are commonly put on plain fcales, and from whence dimenfions or diftances 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,
Inches, each Inch divided into 8 and lO Decimals, containing an loo parts.nbsp;Chords,
Sines,
T angents.
Rhumbs,
Latitude,
parts.
Cho,
Sin.
Tang.
Rum.
Lat.
Hou.
Lon.
In.Me
Num.
Sin.
V.Sin.
Tan.
Single lt;
1 nbsp;nbsp;nbsp;i
y mark’d»^
^^5 j Hours,
° 1 Longitude,
Inclin. Merid.
the Numbers,
Loga {sines, rithms fVerfed Sines,nbsp;of J T angents,
I ¦) CLines, or of equal parts, Chords.
Sines.
Secants,
Tangents to above 45° Polygons,
p Ian.
I Cho. 1 Sin.nbsp;mark’d lt;( Tan.nbsp;Sec.nbsp;Tan.nbsp;LPol.
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
ocr page 602 2 nbsp;nbsp;nbsp;Defcription and Ufe
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 fometimes 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.
Sect,
ocr page 61of Mathematical Injlruments. 33
Sect. X.
Of the Confruttiion of the Single Scales.
I. 'The Scale of Inches.
This fcale, which is laid clofe to the edge of the feélor, 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.
II. The Decimal Scale.
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 fubdivifions 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
Are fuch as have been already deferibed in the account of the plane fcale.
IV. The
ocr page 62This 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.
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 diftanccs exprefled by the logs, of the numbers, 2, g,nbsp;4, 5, 6, 7, 8, 9, refpeétively, (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
myProfeifor in GreJbam College, Anno 1624
From Mr. Edmund Gunter, the Inventor : Aftronpdown
ocr page 63down 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 éc 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,
I. From the fcalc of equal parts, take the diftances expreflèd 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 refpeftively ; 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.
D 2 nbsp;nbsp;nbsp;laying
ocr page 64laying 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.
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.
1. nbsp;nbsp;nbsp;From the fcale of equal parts, take the arithmetical complements of the logarithm cofines, (ornbsp;the fecants of the complements) of 5, 10, 15, 20, 25,nbsp;30, 35, 40, iSc. degrees; (rejeóting 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.
amp;'c. as the intervals will admit.
The
2. nbsp;nbsp;nbsp;Between every diftance of lo degrees, introduce as many degrees, ~ degrees ; 4 degrees; i de
grees,
ocr page 65The 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 reprefenting 35 minutes on the fines and tangents, andnbsp;168^ degrees, on the verfed fines.
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 this 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.
3S
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 feverally, the parts exprefifed by the numbers in thenbsp;tables (fuppofe Sherwin's) of the natural fines, correfponding 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.
Exam. To lay down 35° 15'; •whofe natural fine found in the tables is 14,
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.
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,
IV. 0/
ocr page 67of Mathematical Injlruments',
IV. Of the Scale of upper Tajigenh.
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.
V. Of the Scale of Secants.
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.
VI. Of the Scale of Chords.
1. nbsp;nbsp;nbsp;Make the length of this fcale, equal to that ofnbsp;the fines; and let the divifions to be laid down, exprefs 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.
VII. Of the Scale of Polygons.
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
D 4 nbsp;nbsp;nbsp;tJie
ocr page 68the 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 deferibed, there are found other fcales on the old feéfors, 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 feéfors, as now conftruéfed, 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 traél on the ufe of the gunnersnbsp;callipers.
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 feéfor.
Franfverfe dijiance^ is the diftance taken between any two correfponding divifions of the fcales of thenbsp;fame name, the legs of the feéfor 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 feéfor,nbsp;and the other foot is extended to the like divifion in
the
ocr page 69the fcak of the fame name on the other leg of the feftor.
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 diftances, 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.
Some TJfes of the Line oj Lines.
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 correfponding 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 exprefsing 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
take
ocr page 70take 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.
CD = 10 ; Plate VI. Fig. 21.
T0 three given lines AB = 3^ BC = 7 to find a fourth proportional.
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 ; 235
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 2rV: For 10 : 7nbsp;nbsp;nbsp;nbsp;3 : 2^5.
From this problem is readily deduced, how to increafe or diminifh a given line, in any affigned proportion.
Exam. To diminifio a line of 4 inches.^ in the pro' portion of S to 7.
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;
2 nbsp;nbsp;nbsp;and
ocr page 71of Mathematical Infrumeiits. 4i
and the correfponding tranfverfe diftance, taken twice, or thrice, or four times, amp;e. will be the line required.
7*0 open the fetlor fo, that the two fcales of lines Jhall make a right angle.
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.
To two right lines given, to find a mean proportional Suppofe the lines 40 and 90.
Operation, ift. Set the two fcales of lines at right angles to one another.
ed. Find the half fum of the given lines (= —
= 25)
= 65); alfo find the half difference of thofe lines 90—40
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.
44
^he Defcriptlon and Ufe
‘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 inftrument.
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.
Exam,
ocr page 7345
Exam. II. make a fcale of 7 inches long contain l8o fathoms.
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).
afligned
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
PROBLEM XV.
‘The ufe of the line of lines in drawing the orders of Civil Architecture.
In this place it is intended to give fo much of Architefture 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.
Definitions.
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;2. Or
ocr page 7446 nbsp;nbsp;nbsp;T^he Defcripüon and Ufe
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.
4. nbsp;nbsp;nbsp;The Column, that which fupports any weight.
5. nbsp;nbsp;nbsp;The Pedestal or foot whereon the Column isnbsp;fet for its better fecurity.
Each of thefe parts confifts alfo of three parts.
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 convexoconcave.
Plane members of different magnitude have different names.
9. nbsp;nbsp;nbsp;A Fillet or lift is the leaft plane or flat member.
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.
12 An Abacus, that at the top of the capital.
13. nbsp;nbsp;nbsp;The FacIjE or faces are flat members in thenbsp;Architrave.
14. nbsp;nbsp;nbsp;The Corona is a large flat member in thenbsp;Cornice.
The Convex members are,
15. nbsp;nbsp;nbsp;A.N A.STRAGAL of a ffuall femicircular convexity. ,
16. The
ocr page 7516. nbsp;nbsp;nbsp;A Fusarole when an Aftragal is cut into partsnbsp;like beads.
17. nbsp;nbsp;nbsp;A Torus a large femicircular convexity.
18. nbsp;nbsp;nbsp;An Ovola nearly of a quadrantal convexity.
The Concave members are,
19. nbsp;nbsp;nbsp;A Cavetto nearly of a quadrantal concavity.
20. nbsp;nbsp;nbsp;A ScoTiE of a concavity nearly femicircular.
The ConvexoConcave members are a Cymaife
and a Cima.
21. nbsp;nbsp;nbsp;A CvMAise or Oue, that v/hofe convex partnbsp;projects moft and by workmen is ufually called annbsp;Ogee.
22. nbsp;nbsp;nbsp;A Cima that whofe concave part projedts moft.
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.
Of the Tuscan Order.
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 farmhoufes,
ftables.
ocr page 76^8 nbsp;nbsp;nbsp;Tthe Defcription and XJfe
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.
1. nbsp;nbsp;nbsp;Height of the column equal to feven diameters, or modules.
2. nbsp;nbsp;nbsp;Height of the entablature equal to one fourthnbsp;of the column, wanting half a minute.
3. nbsp;nbsp;nbsp;Height of the pedeftal equal to one module.
4. nbsp;nbsp;nbsp;The capital and bafe, each half a module.
5. nbsp;nbsp;nbsp;Breadth of the bafe on a level is iJ module.
6. nbsp;nbsp;nbsp;Breadth of the capital equal to one module.
7. nbsp;nbsp;nbsp;Diminishing of the column is ^ module.
IS i:j modules.
8. nbsp;nbsp;nbsp;Projection of the beams fupporting the eaves
g. In colonades, the diftance of the columns in the clear is 4 modules.
10. In arches, and the columns fet on pedeftals. The diftance of the columns from middle tonbsp;middle is modules.
Height of the arch is modules.
Breadth of the pilafter between the column and paflage is 26 minutes.
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.
Of the Doric Order.
This order, fuppofed to be invented by Oorus a king oïAchaia, 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
Doric
ocr page 77Doric order ; notwithftanding which, there are many fine buildings of this order in other fituations, wherenbsp;they have a very pleafing efFedt.
The following general proportions are given by Palladio,
1. nbsp;nbsp;nbsp;Height of the column from yf to 8, and 84nbsp;modules.
2. nbsp;nbsp;nbsp;Height of the entablature is one fourth of thenbsp;column.
3. nbsp;nbsp;nbsp;Height of the pedeftal equal to zf modules.
4. nbsp;nbsp;nbsp;The Attic bafe is ufed vyith this order, it isnbsp;half a module in height, and fo is the capital.
5. nbsp;nbsp;nbsp;Breadth of the column’s bafe is 1 \ module.
6. nbsp;nbsp;nbsp;Breadth of the capital is i module mirnbsp;nutes.
7. nbsp;nbsp;nbsp;Diminishing of the column is 8 minutes.
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 midr
die is y~ modules.
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.
ocr page 7850 nbsp;nbsp;nbsp;^he Defcription and Ufe
Here follows a table for the particular conftrudlion of the ornaments with which the architrave and freezenbsp;are enriched.
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 
The column figned altitude gives the heights of the particular parts.
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 correfponding 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.
Of the Ionic Order.
This order, which is taller and flenderer than the Doric, does not appear with fuch a mafculine ftrengthi
and
ocr page 79and 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.
Palladio gives to the Ionic order the following general proportions.
1. nbsp;nbsp;nbsp;Height of the column to be 9 modules.
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.
6. nbsp;nbsp;nbsp;Diminution of the column is yf minutes.
7. nbsp;nbsp;nbsp;In colonades, the diftance of the columns innbsp;the clear is 2f modules,
8. nbsp;nbsp;nbsp;In arches, and the columns fet on pedeftals,
Diftance of the columns from middle to middle is yfr modules.
Height of the arch to its fofiit is 11 modules.
Breadth of the pilafters is 264 minutes, between the column and arch.
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
E 2 nbsp;nbsp;nbsp;onsi
ocr page 8052 nbsp;nbsp;nbsp;‘The Defcription and life
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 compaffes, obferving the limits of their altitude and projedion : 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.
Of the Corinthian Order.
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.
The general proportions alfigned by Palladio are ;
1. nbsp;nbsp;nbsp;The height of the column to be 94. modules.
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.
3. Height
ocr page 81of Mathematical Infirument^. 53
of the co
3. nbsp;nbsp;nbsp;Height of the pedeftal equal tonbsp;lumn.
4. nbsp;nbsp;nbsp;The height of the capital to be iJ module; ofnbsp;¦which the abaco is 4 of ^ module ; its horns projedling 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.
6. nbsp;nbsp;nbsp;The diminution of the column to be 8 minutes.
7. nbsp;nbsp;nbsp;In colonades, the intercolumniation is 2 modules.
8. nbsp;nbsp;nbsp;In arches, and the columns fet on pedeftals.
The diftance of the columns, from middle to
middle, to be 64 modules.
Height of the arch equal to 11^ modules.
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 fufaroles, an ogee, and a plateband , the firft, or lower facis projedts the fame as the top ot the fliaft.
The freeze, which projeéls 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 plateband of the architrave.
The breadths of the dentels are 34 minutes, and their diftance 14 minutes.
The breadths of the modilions are 11^ minutes, and their diftance in the clear 23 minutes.
E 3 nbsp;nbsp;nbsp;'nbsp;nbsp;nbsp;nbsp;The
ocr page 82The 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.
Of the Composite Order.
This order (the poor invention of the Romans^ and therefore frequently called the Roman order), is ufually compofed of the Cciinthian and Ionic; the Ionicnbsp;capital being fet over the two lower rows of leaves innbsp;the Corinthian capital.
Palladio gives us the following general proportions.
1. nbsp;nbsp;nbsp;The height of the column to be lo modules.
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.
3. nbsp;nbsp;nbsp;Height of the pcdeftal to be f of the column.
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 greateft breadth if modules.
6. nbsp;nbsp;nbsp;Diminution of the column equal to 8 minutes,
7. nbsp;nbsp;nbsp;In colonades, the intercolumniation is if modules.
8. nbsp;nbsp;nbsp;In arches, and the columns fet on pedeftals,
Diftance of the columns from middle to middle
is 7f 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,
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 modilion.
To draw the Mouldings in ArchiteTure.
' The terminations or ends of flat members, are right lines.
The aftragal, fufarole, and torus, are terminated by a femicircle.
T0 defcribe the Torus. Fig. i. Plate I.
On ab, its breadth, defcribe a femicircle.
To make an Ovolo, whofe breadth is ab. Fig. 2.
Make ac = 4 or of ab, and draw cb.
Make the angle cbd equal to the angle bod.
Then the interfeftion of bd with ca will give d the center of the are bc.
Or, Defcribe on bc an equilateral triangle; and make the vertex the center.
The former of thefe methods is the moft graceful.
To make a Cavetto, whofe breadth is Ai. Fig. 3.
Make ac = .^ or .i of ab ; draw bc, and produce the bottom line towards d.
Make an angle bcd equal to the angle cbd.
Then d, the interfedlion of cd with bd, is the center fought.
Or, On bc defcribe an equilateral triangle, and the vertex will be the center.
Fig. 4.
On
T0 make a Scotia^ whofe breadth is ab. Make af equal to f of ab.
On af defcribe the fquare ac, and on bf defcribé the fquare bd;
Then g is the center of the are ef, and d the center of the are fg.
make a Cima^ whoje breadth is ab. Fig. 5.
Make ac equal to about ^ of ab.
Draw the right line cb, which bifecl in d.
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
CD, DB.
Ho make a Cymaife^ or Ogee, whofe breadth is ab.
Fig. 6.
Make ac equal to about ^ of ab.
Draw the right line cB, which bifedh in d.
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.
Then e is the center of the arc cd, and f the center of the arc db.
Ho defcribe the curve joining the fhaft óf a column with its upper or lower fillet, the pro] edlion 0/ ab being given.nbsp;Fig. 7.
Make ac equal to twice ab.
Draw cd parallel to ab, and equal to of ac.
Then d is the center of the arc cb.
Ho draw the gradual dhninution of a Column. Fig. 8.
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.
Make
ocr page 85Make 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,
Then a curved line drawn through the extremities h, b^ will limit the gradual diminution required.
Palladio defcribes another method, which is more ready in pradbce.
Lay a thin ruler by the points D, H) e, and the bending of the ruler will give the gradual diminutionnbsp;required.
’To dejfcnibe the Volute of the tonic order. Figs. 9, 10.
The altitude ab, which is i of a module, or 26minutes, 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.
Divide
ocr page 86Divide 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 conftruftion. 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.
In either method, the centers of the inner curve may be thus found.
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 exaéfly 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 t43 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 Goldpj^n’s method.
p nbsp;nbsp;nbsp;^Ta
ocr page 87of Mathematical Injlruments. 59
A Table of the lengths^ in minutes.^ of the feveral radii of the outward and inner volutes.
N“ Rad. 
Outward Curve, 
Inward 
In parts of 
I ft rad. 
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 ïïï= 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 
To ufe this table, a fcale of of ^ module fhould be made, and divided into x 5 minutes, and the extream divifion decimally divided, whereby the lengthsnbsp;of the feveral radii may be taken : But as the feóbornbsp;is an univerfal fcale, there are two other columns added.
ocr page 886ö nbsp;nbsp;nbsp;^he Defcriptton and JJfe
ed, applicable to the feélor ; 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 feélor, and the curves more accurately defcribednbsp;than by any other method.
TV dcfcribe the Flutings and Fillets in channelled columns Fig. II.
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 prelènt, are more fond of copying his produdions, 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,
Doric
ocr page 89 ocr page 90S â– Â»
ocr page 91Boric, 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 thé 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.
I nbsp;nbsp;nbsp;Exa.m,
62
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“^^ = Ï314875 modules: And 18 f.
18
6 in. = 18,5 feet. Therefore nbsp;nbsp;nbsp;= *gt;37°9
= I f 49: inches nearly : And the fcftor may be fitted to this, as before diredled, according to the intendednbsp;fize of the draught.
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 exprefs 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
be
ocr page 93be beft to proceed as in the following example of the Ionic order.
To conJlmSl the Pedefial. Plate II.
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.
Min.
724
44
40
394
34t
53'ï'
30»
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 fubtrafted 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,
Bafe with 30 minutes........
This lefs by the plinth, 284» remains This lefs by the torus, 4, remains .
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
com
ocr page 9464 nbsp;nbsp;nbsp;^he Defcription and TJfe
compafies, which is, to bring them defer and clofer every diftance laid down.
And in the fame manner, for the cornice of the pedeftal, take a point C, 30 minutes below the cornice ; and tabulate as before.
Cornice with 30 min..........524 = CD
Ditto . . 
• 3i  
Ditto . . 
• 44  
Ditto . . 
• 14  
Ditto . • 
• 54  
Ditto . . 
• i4  
Ditto . . 
• 34 
This lefs by the fillet or cap, 24, leaves 504 = 01
ditto 464 =: C2
• nbsp;nbsp;nbsp;424 = 03nbsp;. 404 = 04
• nbsp;nbsp;nbsp;334 = c6nbsp;. 30 = cd.
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,
^hus for the Bafe.
The bale 334 min. with 30 added = 634 = so This lefs by the plinth lO min. leaves 534 =: si
Ditto.....torus nbsp;nbsp;nbsp;7quot;.....46 nbsp;nbsp;nbsp;= S2
Ditto.....fillet nbsp;nbsp;nbsp;14..... 444 nbsp;nbsp;nbsp;= S3
Ditto.....fcotia nbsp;nbsp;nbsp;44.....4044 nbsp;nbsp;nbsp;= S4
Ditto.....fillet nbsp;nbsp;nbsp;14 . • • ¦ . 384nbsp;nbsp;nbsp;nbsp;= S5
Ditto.....torus nbsp;nbsp;nbsp;54.....334 nbsp;nbsp;nbsp;= s6
Ditto.....aftragal24.....314 nbsp;nbsp;nbsp;=£7
Ditto.....fillet nbsp;nbsp;nbsp;14.....30 nbsp;nbsp;nbsp;= s8,
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.
For the Capital.
The capital 244 with 30 added, gives 544 = eg This lefs by the plateband 145 leaves 524 = finbsp;Ditto ...... ogee . . 34 • • • 49v =
This
ocr page 95 ocr page 96â– nbsp;nbsp;nbsp;v .f'
* gt;â€¢â€œ â– â€¢ fi. j â€˜v . â€˜ 
.......i
..t
':f:
r'
1'
f
*
.;K
â– :.i
*'â–
'Pi'
i â– nbsp;nbsp;nbsp;,
tf Mathematical Injirumentc. t This lefs by the rim of volutenbsp;nbsp;nbsp;nbsp;leaves 474 — ^3
Ditto......hollow
Ditto...... . ovolo . .
Ditto......aftragal
Ditto......fillet . .
To conftruSl the Cornice.
In the axis take gh = 36 for the architrave, hi = 27 for the freeze, and ik = 46 for the cornice. Then,
For the farts of the Architrave.
To the freeze 27 add hg 36, gives 63 =ig This Icfs by the firft face 64, leaves 564
Ditto......fufarole
Ditto......2d face .
Ditto......fufarole
Ditto......3d face
Ditto Ditto
ogee
fillet
5t
3f
It
lot ' 4t
24
42t = i'4 35 = ^5
3l4r=F6 30nbsp;nbsp;nbsp;nbsp;= F/.
II
= 12 4^tt— 13nbsp;44Tï = i4nbsp;34tt= 15nbsp;294 =16nbsp;27nbsp;nbsp;nbsp;nbsp;= IH.
For the Cornice.
To the freeze 27 add the cornice 46, gives 73 = hk
leaves 704=:hi 63i = H2nbsp;62t = H3
59 =H4
51 =H5 48 =:h6nbsp;404= H7nbsp;20 = hS
33 «9 32 =:hionbsp;27 =:HI.
7
1
3i
8
3
7i
6
This lefs by the fillet Ditto ..... cima
Ditto.....fillet .
Ditto.....ogee .
Ditto.....corona
Ditto.....ogee . nbsp;nbsp;nbsp;.
Ditto.....modilion
Ditto.....fillet . nbsp;nbsp;nbsp;.
Ditto.....ovolo . nbsp;nbsp;nbsp;.
Ditto.....fillet . nbsp;nbsp;nbsp;.
Ditto.....cavetto nbsp;nbsp;nbsp;.
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.
F nbsp;nbsp;nbsp;yfTABLS
ocr page 98
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  ,  ,  
— 
— 
— 
— 
—  
The Cornice  nbsp;nbsp;nbsp; 
 
 
— 
... 
0 
z6g 
— 
—  
THE DIE   
1 
C 
42 
1 
20 
0 
40  
The Bale    
— 
— 
— 
— 
0 
40 
— 
—  
Fillet .    
_ 
_ 
___ 
. 
_  
Cavetto ... 
— 
— 
— 
— 
0 
5 
0 
4ii  
Ogee     
— 
— 
— 
_ 
... 
_ 
——  
w 
Allragal ... 
— 
— 
___ 
_ 
_  
Fillet  ...  
— 
~ 
— 
— 
0 
0 
(46 l47i  
cq 
Cima    . 
— 
— 
— 
— 
_ 
_ 
_ 
_¦ 
Fillet    . 
_ 
__ 
\  
Torus    ^Plinth   . , 
— 
— 
— 
— 
— 
9 27v 
0 0 
50 5’ 
I he Order  The Entablature The Column The Pedeftal  

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 55ï i 53l 
o 
3f 
0 
js6 t S4i 
0 
3f 
0 
556 { 54f 
4l 
o 
S^i 
o 
4ï 
0 
S3i 
0 
5f 
0 
S3f 
li 
0 
S‘4 
— 
— 
0 
1 
0 
52?  
Si 
— 
— 
o 
4t 
0 
S 49: 146 
0 
H 
— 
— 
0 
44ï 
o 
0 
46 
— 
— 
— 
—  
5 43 
0 
3 
p 
46I  
_ 
 
.. 
o 
3l 
0 
— 
—— 
.— 
—  
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 
45Ï 
3f 
o 
4‘l 
— 
— 
— 
— 
— 
—  
— 
— 
— 
o 
4 
0 
5 43 146 
— 
—  
— 
— 
— 
— 
— 
—— 
— 
0 
3 
0 
47 
°i 
o 
47ï 
o 
01 
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 
rói 
o 
23' 
0 
C7 
0 
'} ^ 
0 
'7 
I 9 
 
F 2 
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 
38f 5 37l ?36inbsp;35:1 34i 29I 28i 27i 26 
0 0 5 0 
4 if 54f 
0 0 0 0 
27 24? $ 22i ( 30 33l 
0 0 6 0 
3f if 531 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 
— 
“Aiigulor v olutes 
f Ovolo Abacus I Fillet (, Cavetto B (ket Rim   
Ogee     ..
Abacus......
VoC fillet or rim J lute ^ chan, or hollow
Ovolo
Aftragal    . . Fillet 
Collarino   Middle Volute Courf.ofleaves, T 3d folding hal I zdnbsp;their height J ilinbsp;fJ fAHragal     I Fillet  
lt;; .lt; Body of the Column
Fillet 
Aftragal    
Fillet......
Scotia     _ Fillet......
Scotia    . Fillet    Torus  . _nbsp;.Plinth — . .nbsp;Safe    .
Shaft
Capital   
ocr page 101 ocr page 102
Order ; according to the Pro  

every Moulding and Part in the Entablature of each portio?is given by Palladio.
Ionic.
Roman.
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  
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 
— 
— 
—  
— 
— 
— 
_ 
_ 
_ 
_ 
0 
c 
0 
5 35f  
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72 nbsp;nbsp;nbsp;Jjefcription and XJfe
Some XJfes of the Scales of Polygons. PI. VI. PROBLEM XVJ.
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.
On a given line ab, to deferibe a regular pentavon. Fig. 23.
Solution, ift. Make ab a tranfverlè diftance to 5 and 5.
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 tranfverlè diftance to 60 and 60, on the fcales of chords.
i nbsp;nbsp;nbsp;2d.
i , yÂ» â€™ ' ' : . â– ' Â»? ; â€¢  lt;
v â€™'7
â€™â– 5?:....
' â–
â– ' .fei
â– =Jt
â€˜â– VA
ad. Divide 360 by 24; the quotient gives 15.
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.
With the chord of 60 degrees, divide the circumference into fix equal parts.
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.
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.
74 nbsp;nbsp;nbsp;'^he T)efcrtption and XJfe
_ ^0 protralfy or lay down, a right lined angle, bac, which Jhall contain a given number of degrees, PJ. VI.
Case I. When the degrees given are under 6o : Suppofe 46. Fig. 25.
ift. At any opening of the fe£lor, take the tranfverfe 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.
Case II. When the degrees given are more than 60 : Suppofe 148.
I ft. Describe the arc bc as before.
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.
ift. Describe the arch dg with the chord of 60 degrees.
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.
If
ocr page 109If 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 direéted 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.
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 fieldbook of a furvey. the bearings and diftances of the ftations were expreffednbsp;as follows:
O fignifies .Station.
B Bearing.
D —Diftance.
G i.B 7o°5o'DT080links.
O 2.Bi28 iu D 580.
G 3B 32 t5 Fgt; 605.
O 4.B287 30 D 70'D.
G 5 B 5045 nbsp;nbsp;nbsp;S'40
G 7.B 183 7.5 D yooi
ReturntoD314in 07. O 8,B 133 30 D 5rotoG5*
09.B18630D 39010^2.
ReturntoDyooin07. O10.B209 20 D 668 ccuting
Returntooro. Gir.B275 3oD 800.
G12.B 171500 7841001.
I nbsp;nbsp;nbsp;The
ocr page 11076
T^he Defcription and Ufe
The bearings are counted from the North, Eaftward. 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 tranfverfe 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 fecond 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.
agreeable to the
5th.
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.
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 pofuion 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
parts
ocr page 1127 8 nbsp;nbsp;nbsp;^he Defcription and life
parts 390, the length meafured in the field from the 9th ftation.
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 conftruótion 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.
Sect.
ocr page 113
Therefore, the Sedor being quite opened,  
orlt; 

If the I at the beginning ofnbsp;the fcale,nbsp;of the 1 ft interval, be taken for
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3
O o 3
g Cl,
So  quot;ïhe Defcripfion and XJfe
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, exprefs 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.
Exam. I. What is the prodiiSi of ^ hy 4 ?
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
principal
cf Maihematical Inf rumenfs. 8i
principar parts, that is on 2 units beyond 10 ; thereiore this pbint is to be cfteemed in this Exam, as x2.
Exam. II. What h the produS of ly 3 ?
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 diVifions 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.
Exam. III. What is theproduSl of 35 ly 24 ?
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
reach f rom (i. e. 35) to nbsp;nbsp;nbsp;(i. e. 840.)
10,0 nbsp;nbsp;nbsp;'
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.
Exam. IV. PFbat is theproduSi lt;ƒ 375 6o ?
Solution. The extent from i to 6, (or 60) in the
firft interval will reach from nbsp;nbsp;nbsp;or 375) in
10 nbsp;nbsp;nbsp;100
the firft interval, to 2 Pds 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 confidered, it will not be difficult to apply others to the fcale, and readily to affign the value of the refult.
Exam. V. What will be the quotient of ^6 divided by \\
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.
Exam,
ocr page 117Exam. VI. If 144. k divided by ^ 5 what mil be the quotient ?
Solution. The extent from 9 to i, will reach from 144 to 36.
Exam. VII. If lyzZ be divided by 12 •, what will be the quotient ?
Solution. The extent from 12 to i, will reach from 172b to 144.
Exam. VIII. To the numbers 3, 8, 15 ; find a fourth proportional.
Solution. The extent from 3 to 8 ; will reach from
15 nbsp;nbsp;nbsp;to 40.
Exam. IX. To the numbers 5, 12, 38 ; find a 4th pro^ portional.
Solution. The extent from 5 to 12, will reach from 38 to 91 4*
Exam. X. To the numbers l8, 4, 364J find a 4tb proportional.
Solution. The extent from 18 to 4j will reach from 364 to 801.
Exam. XI. T0 two Numbers l and 2 ; to find a feries of continued proportionals.
Solution. The extent from i to 2, will reach from
2 nbsp;nbsp;nbsp;to 4 j from 4 to 8 in the firft interval; from 8 to
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 Öf. 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.
Thii
ocr page 118$4 nbsp;nbsp;nbsp;Defcription and Ufe
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.
Some ufes of the Scales of Log. Sines and Log.
“Tangents.
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;
ocr page 119of Mathematical Infruments. 85
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.
Someufes of the double Scales of Sines^ ‘Tangents, and Secants.
¦m
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. '
G 2 nbsp;nbsp;nbsp;But
ocr page 120But if the Jecant of 28“ 30' was required ?
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'.
yilangeit greater taan 45 degrees (fuppofe 60 degrees) IS found thus.
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.
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 
of Mathematical Infruments, 87
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 ?
tangent.
fecant.
Againft 80 degrees ftands 5,671
The radius is
47.5
269,3725 nbsp;nbsp;nbsp;273,5525
So the length of the tangent on the twentyfifth 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.
Exam. T'o find the tangent of 78 degrees to a radius of 2 inches.
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
G 4 nbsp;nbsp;nbsp;tangen
ocr page 12288 nbsp;nbsp;nbsp;‘ïke Defcription and TJfe
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 tranfverle 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 tranfverfe diflance is the cocangent ot the degrees anfwering to the given line.
And rbr a fecant. Make the given line a tranfverfe 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 cofmenbsp;of the degrees anfwering to the given fecant line.
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 films. The tranfverfe diftance of 90 and 90 will be tire radius fought.
' 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.
a
ocr page 123of Mathematical Infruments'. S9
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 clivifions on both legs ; then will thofe divifions fhewnbsp;the degrees and parts correfponding to the givennbsp;line.
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.
Take the tranfverfe diftance of the fine complement of the given degrees.
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.
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
from
ocr page 124from 30 to 60 ; or from the fine of any degrees, to their complement.
Or on the fines, make the lateral diftance of 45 a tranfverfe diftance between 30 and 30.
0 deficribe an Elliffits,' having given ab equal to the longejl diameter •, and cd equal to the JhorteJi diameter.
thofe diftances to ae from e towards a, as at the points I, 2, 3, 4gt; 5gt;nbsp;nbsp;nbsp;nbsp;7} 8 i and thro’ thofe points
fines; and take he tranfverfe diftances of 10°, 20% 30% 40°, 50“, 6o“, 70“, 8o“, fucceflively, and apply
draw lines parallel to ec.
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 conftruftion of folar t'clipfes; but inftead of ufing the fines to every ten drgeces, the fines belonging to thenbsp;degrees and minutes c 'rrelpondiiig to the hours, andnbsp;quarter hours are to be uled.
91
of Mathematical Injlruments.
“T0 defcribe a Parabola wbofe parameter Jhall be equal to a given line.
Solution ift. ___
Draw a line to reprefent the axis, nbsp;nbsp;nbsp;i
in which make nbsp;nbsp;nbsp;A
AB equal to half nbsp;nbsp;nbsp;6
the given para nbsp;nbsp;nbsp;2^.r  ...... v
meter; divide AB nbsp;nbsp;nbsp;........................................
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
axis AB.
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^,
92
T^he Défcrjption and Ufe
To uefcrihiHyperho^i, the vertex A md a£y0ptotes ELI, BI, heinT given.
Solution ift. The affymptotes 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;cotangents 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, ƒ,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.
ocr page 127tf Mathematical Injlrumcnts.
Tl? find the difiance of places, on the terrefirial glohe ly having given their latitudes and longitudes.
This problem confifts of fix cafes.
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.
Exam, nbsp;nbsp;nbsp;ö/Bermudas, lat. 32“ ifit^longlL
68° 38' W. Ifiand of St. Thomas, lat o o, longit.
1* o E.
Required their diftance.
E.
’.iC
^LUTiox I ft. With the chord of 6ü* defcribe a circlenbsp;xeprefenting the equator, wherein take a point c to reprcfentnbsp;the beginning of longitude.
2d. From c apply the chord \ nbsp;nbsp;nbsp;A!
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.
Case IV. When the given places are in the fame parallel of latitude.
3 nbsp;nbsp;nbsp;Exam.
ocr page 12894
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;AÖ, 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, London’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 AÖ, 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
ocr page 129of Mathematical Infnmenfs, 9^
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 CapeHorn in latitude 55® 42' S. longitude 66° 00' W.
vB
:E
Solution ift. From a, Loni/on’s longitude, apply the chord of CapeHorn’^ 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.
To meafure bk on the chords , apply bh from b to i, and meafure the arc st i.
Sect. XVII.
T'he JJfe of fame of the fingle and double Scales, applied in the Solution of the Cafes of plain ’Trigonometry.
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.
This
ocr page 13096 nbsp;nbsp;nbsp;Defcription and Ufé
This problem confifts of three cafes.
Case I. when among the things givenj there be 3 fide and its oppofite angle.
Case II. When there is given two fides and the included angle,
Case III. When the three fides arc given.
Solution lt;ƒ C A S E L
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.
Example I.
In the triangle ABC : Given AB=rc6 7 nbsp;nbsp;nbsp;,
AC=64
AB=46° 30'
Required nbsp;nbsp;nbsp;A, amp; BC,
The proportions are as follow,
As fide AC : fide ab ;: fine A. b : fine L c.
Then the fum of the angles b and c being taken from 180°, will leave the angle A.
And as fine Ab : fine Aa ; ; fide ac : fide cb.
Firjl hy the logarithm fcales.
To find the angle c.
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..
And the fum of 46° 30' and 39° 24'is 85° 54'
Then 85“ 54' taken trom 180% leaves 94“ 6' for the angle a.
Ta
ocr page 131To find the fide bc.
The extent from 46“ 30' (=:^b) to 85” 54' the fupplement of 94» 6'nbsp;nbsp;nbsp;nbsp;on the fcale of log fines,
will reach from 64 (=ac), to 88 (=:bc), on the fcale of logarithm numbers.
Secondly by the double Scales.
To find the Angle c.
1. nbsp;nbsp;nbsp;Take the lateral diftance of 64 (=ac) fro.mnbsp;the lines.
2. nbsp;nbsp;nbsp;Make this a tranfverfe diftance of 46» 30' i=jLB)nbsp;on the fines.
3. nbsp;nbsp;nbsp;Take the lateral diftance of 56 (=ab) on the
lines. nbsp;nbsp;nbsp;¦nbsp;nbsp;nbsp;nbsp;_
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.
To find the ftde bc.
1. Take the lateral diftance of 64 (=ac) from the lines.
^ 2. Make this a tranfverfe diftance of 46“ 30' (=:Zb) on the fines.
3. nbsp;nbsp;nbsp;fake the tranfverfe diftance of 85'’ 54' (the fuppleinent 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.
Ex. n. In'the triangle ABC PI. VI. Fig. 27.
Given bc — nbsp;nbsp;nbsp;74
Ab = 104° 0'
4.C = 28 o Required ab amp; ac.
Now the fum of 104° 0' and 28° 0' is 132 ° 0'.
And 132° 0' taken from 180, leaves 48“ 0' for
the angle a.
The proportions are.
As fine Aa : fine Lc :: fide bc : fide ab.
And as fine La : fine Ab :: fide bc ; fide ac.
Firftt hy the Logarithm Scales.
Fo find AB.
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.
Lo find AC.
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.
Secondly by the double Scales.
‘To find AB.
1. nbsp;nbsp;nbsp;Take the lateral diftance 74 (= bc) on thenbsp;lines.
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.
To find AC.
1. nbsp;nbsp;nbsp;Take the lateral diftance 74 (= bc) on thenbsp;lines.
2. nbsp;nbsp;nbsp;Make this extent a tranfverfe diftance to 48° 0'nbsp;(= la) on the fines.
ocr page 133of Mathematical Injlrurnents. gg
3. nbsp;nbsp;nbsp;Take the tranfverfe diftance to 76® 0' the fupplement 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.
Solution lt;ƒ C A S E II.
The folution of this cafe depends on a well known theorem, viz.
As the fum of the given fides
Is to the difference of thofe fides,
So is the tangent of the half fum of the unknown angles
To the tangent of the half difference of thofe angles.
And the angles are readily found by their half fum and half difference being known.
Ex, III. In the triangle abc, PI. VI. Fig 28.
Given bc = 74 BA = 52nbsp;= 68“ 0'
Required C.A4.0 j amp; ACi!
Preparation.
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.
Also find the given fum of the fides, viz. EcjEA = 126,
And take the difference of thofe fides, viz. bc—
BA=22.
Then the proportions are
As bc ba : EC—BA : : tan. z : tan. x.
The
ocr page 134The difference of z and x gives the leflèr angle c,
And as fine z.c : fine z.b :: fine ba : fide ac.
Firjl by the Logarithm Scales.
Lo find the tangent o/n.
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.
Then the fum of 56° 0'and 14“ 31' or 70* 31' is the angle a.
And 14° 31' taken from 56° 0' leaves 41quot; 29' for he angle c.
To fiind AC.
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;thé 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
ocr page 135of Mathematical Infruments. tot
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, willreach 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.
Secondly by the double Scales.
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
1. nbsp;nbsp;nbsp;Take the lateral diftance 63 on the fcales ofnbsp;lines.
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.
4. TAKEXhe lateral diftance 11, onthelines;
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.
H 3 nbsp;nbsp;nbsp;But
ocr page 136But when m c greater i than 45 amp; N are each I lefs 5 degrees.
Then the third article of the foregoing operation is omitted.
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.
Ex. IV. In the triangle aec : Fig. 29.
Given AB = 45 BC =: 65nbsp;/B — 90
Required Aa ; Ac ; amp; ac.
The proportions arc.
For the Angle a.
As fide AB : fide ec :: radius ; tan A a.
And
of Mathematical hifruments. 103
And the a.a taken from 90“ leaves the /C, then ac may be found as direfted in the laft example.
Firfi by the logarithmic Scales.
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', becaufe 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.
Secondly by the double Scales.
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.
Solution o/ C A S E III. Fig. 30.
In the triangle abc :
Given bc = 926.
BA 558.
AC = 702.
Requir’d tB, ZC. ^a.
H 4 nbsp;nbsp;nbsp;There
ocr page 138There 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.
First. When all the angles are •wanted.
Suppose a perpendicular ad (PI. VI. Fig. 30.) drawn 10 the greateft fide bc, from the angle a oppolite 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.
T AKE the fum of the ftdes ac and ab, which is 1260.
Also their difference, which is 144.
Then on the fcale of numbers, the extent from 526 ( — Bc) to 1260, will reach from 14410 196.
And the half fum of 926 and 196, is 561 = dc.
And the half difference of 926 andi96 is 365 = 08.
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.)
Then 40“ 52' taken from 90°, leaves 49° 8' for 2Lb.
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.
Then 53“ 4' taken from 90°, leaves 36° 56' for the Lc.
Also the fum of 40° 52' and 53“ 4' gives 93° 56' for the ZCAB,
Seconrly,
ocr page 139of Mathematical Inftruments. 105
Seconplv, ‘Tofind either angle', fuppofe b.
Preparation.
Take the difference between eg and ba, the fides including the angles fought, and call it o == 368.
P'lND the half fum of ac and d, call it z = 5?^
And the half diff. of ac and d, call it x = 167
Then as I : nbsp;nbsp;nbsp;:: radius: fine i Z.B.
ABXBC
1. nbsp;nbsp;nbsp;The extent on the log. nünibers 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.
io6
'J’he Defcription and Ufe
By the douhky or fe5foral. Scales.
Tc find 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 tranfverfe 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 Ae 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 rightangled 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 fupply : However here will be fubjoined a few in fpherical trigonometry, as they will include fome operationsnbsp;not only curious, but perhaps not to be met withnbsp;elfcwhere.
Sec t,'
ocr page 141of Mathematical Injiruments. 107 Sect. XVIII.
T})e CoriftruBion of the feveral cafes of Spherical ’Triangles by the Scales on the SeSior.
rj^ H E cafes of fpherical triangles are fix.
Case I. Given two fides, and an angle oppofite to one of them.
Case II. Given two angles, and a fide oppofite to one of them.
Case III. Given two fides, and the included angle.
Case IV. Given two angles, and the included fide.
Case V. Given the three fides.
Case VI. Given the three angles.
Thefe fix cafes include all the variety that can arifc in fpherical triangles.
In the following folutions, are given three conftruftions 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.
So
ocr page 142io8
Solution c/ CASE I.
Exam. In the fpherical triangle ABD.
Given ab=: 29° 50'
DE— 63 nbsp;nbsp;nbsp;59
25 55
Required the triangle.
I. To put DB ott the primitive circle. Fig. i. i. Pi. VII.
ift. Make db = chord of 63* 59', and draw the diameter be.
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'.
3d. Make amp;d equal to ab, with the chord of 29“ 5°'*
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.
To meafure da.
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'.
To meafure gt;tB.
7th. Draw the diameter fg at right angles to be, cutting the circumference bae in r •, a ruler by bnbsp;amp; ^ gives ƒ j make fg equal to the chord of 90 deg,
a
ocr page 143a ruler on g and b, gives p in the diameter fg. Then on the chords gives the angle b = 36° 9'.
To me a fur e aa.
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'.
II.
I.
'¦10 put DA on the primitive circle. Fig. 2.
, 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'.
2d. Make O p, equal to the tangent of half the angle d ; viz. 12° 57'4
3d. On the primitive circle, make D d equal to the given fide db, with the chord of 63“ 59'.
4th. A ruler on b and d, gives b j then will bdz=; 63° 59
5th. Draw o B r, cutting the primitive circle in r.
6th. Maker.'v = the chord of 90“ j j^or twice the chord of 45°.
7th. A ruler on x and b, gives r,i on the primitive circle.
8th. Make mq — mf ~ chord of 29° 50'.
9th. A right line through x amp;c p, x Sc q, gives/amp; e\n G) r.
10th. On/e as a diameter, defcribe a circumference, cutting the primitive circle in a, lt;2.
lith. xA. ruleron A amp; O, gives F.
12th. Through a, b, f, defcribe a circumference, and the triangle abd is conftrudted with da on thenbsp;primitive circle as required.
III. To put ab on the primitive circle. Fig. 3. i.
ift. Make ab= the chord of 29° 50'; and draw the diameter bf.
2d. In
ocr page 144110 nbsp;nbsp;nbsp;T’he Defcription and Ufe
2d. In A b drawn perpendicular to ag, take A b — fine of AB 29° 50''.
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 ƒ 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.
Computation by the logarithmic fcales.
‘T0 find the angle a.
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;
To find the angle b.
Say, as radius, to the cofine of cb.
So tang. /Lc, to the cotang of a fourth arc.
And as tang, ab, to the tang, of cb.
So cofine of the 4th arc, to the cofine of a 5th arc.