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This item is filmed at the reduction ratio checked below/ Ce document est film* au taux de reduction indiquA ci-dessous. 10X 14X 18X. 22X 2SX 30X 12X 16X 20X 24X 28X 32X The copy filmad hara has baan raproduead thanks to tha ganarosity of: Seminary of Quebec Library L'axamplaira f ilmA fut raproduit grAca A la gAnifositA da: S (moaning "CON- TINUED"!, or tha symbol V (moaning "END"), whichavar applias. Maps, piatas, charts, ate, may ba fllmad at diffarant raductfon ratios. Thosa too larga to ba antiraly includad in ona axposura ara fllmad baginning in tha uppar laft hand cornar, laft to right and top to bottom, as many framas as raquirad. Tha following diagrams iliustrata tha mathod: La* imagas sulvantas ont 4tA raproduitas avac la plus grand soin, compta tanu da la condition at da la nattatt da l'axamplaira fiimA, at an conformity avac las conditions du contrat da filmaga. Laa axamplairaa originaux dont la eouvartura an papiar ast imprimte sont filmte an commandant par la pramiar plat at an tarminant soit par la darniira paga qui comporta una amprainta d'imprassion ou d'illustration, soit par la sacond plat, salon la caa. Tous laa autraa axamplairaa originaux sont filmis an commandant par la pramlAra paga qui comporta una amprainta d'impraaaion ou d'illustration at mn tarminant par la darni4ra paga qui comporta una taila amprainta. Un daa symbolaa suivanta apparattra sur la darniira imaga da chaqua microficha, salon la caa: la symbols — ^ signifia "A SUIVRE", la aymbola V signifia "FIN". Laa cartaa, planchas, tabiaaux, ate, pauvant Atra filmte h das taux da rMuction diff^rants. Lorsqua la documant aat trop grand pour Atra raproduit an un saul ciichA, 11 aat fiimi A partir da I'angia supAriaur gaucha, da gaucha h droita, at da haut an baa, an pranant la nombra d'imagas nAcaaaaira. Las diagrammas suivants illustrant la mAthoda. 32X 1 2 3 4 5 6 li KEY ill^i§liolni' § TO albmaliral mfrBanical Sralf: AN INSTRUMENT FOR SOLVING ALL PROBLEMS L\ AlilTHMETlC, GEOMETRY, AND TRlGOiNOMETRY, RIGHT-ANGLED AND OBLIQUE, PLANE AND SPHERICAL. ^VITIIOUT THE AID OF TABLES, EXCEPT THOSE OF LATITUDE AND LONGITUDE. By a. M. CHISHOLM, Esq. % l^ROVINCE OF NOVA SCOTIA. \\y. it iiKsiKMiiKKKi) tliiit nil lliis, tlic scvcntct'iitli (liiy nf April, A. 1). one tlumsanil iMglit huiulrod iiml sixty-one, Alexander M. Chisliolni, of Antigonishe. in tlio County of .Syilni'V, ill tliu saiil I'loviiu-t', has tk'|)o.site(l in this oflice the title of a hook or work, with a st'ale, the copyright whereof he claims in the words following : " Key to Chisholm's Matheiiiatical +>eale : a ([uadransnlar eiigraviMl Diagram, liy A. M. Chisholm, 1801," in conformity with Chapter one hundred and nineteen of the Revised Statutes. I'riiiinciiil Si'rrt'tii>'i/'s Ojjire, Ihlifax, April 17, 1801. JosEini Howe, Provincial Secretary. HALIFAX, N. S. PKINTKD JiY JAMES BOWKS & SONS. HOLLIS STUEKT. *• RECOM M EN DATIONS. Antkioni.ii, August Jth, IHiil. Havinc had an opi)ortunity tor some time past of testing the power and accuracy ol'C'hisholm's Mathematical Scale, I am happy to be able to state that it far exceeded my expectations. As a labor-saving instrument, particularly in Trigonometry and Navigation, I believe it has no equal. It should be taught in every school, and no navigator should be without a copy of it Tem'htr uf Mtitheinatics UoDK. McDonald, SI, FraiicLi Xavier^s VoUeije Alkxanuku CuisMol.M, Ksq., of Antigonish, has just ^hown me a very ingenious and, I believe, novel instrument, which he has invented, and which he calls •' A Mechanical and Mathematical Scale." From the brief examination of it which I have had the op- portunity of making, I am satisfied that it will jirove a valuable ac- quisition to Surveyors, Mariners, Kngineers, ;ind businc'-i men in jjeneral. If accurately graduated it must give correct result. Though exceedingly siuple, the sphere of its application is very extensive. The more thoroughly it is known and understood, the more fully it will be appreciated. Its introduction into Schools and higher Seminaries of Education will greatly facilitate the study of Mathematical Science, and |)rol)ably increase the number of its students. I sincerely wish the inventor much success. I'lTsbyterian College, Truro, August (ith, 1K61. .Famks Ross. TliiRO, 7th August, ISUl. I llAVK examined Mr. Cbisholm's Scale with I'"x]ilanations, and have no hesitation in stating that I believe it will be of great utility in our 8cliO(ils, provided it can lie printed at a moderate price. It fnrnislies ai Impressive and admirable illustration of the various departments of Practical Mathematics, and, 1. IIavini. had an iipportunity of examining a Scale invented l)v Mr. C'hisliulni, I feel eonvineed from the satisfactory results whicli it gave after some severe tests, that it is calculated to be of great service in scIkioIs. and perticularly to those engaged in navigation, who require to have a correct result, in a short space of timej in other words it is a labour-saving invention, and as such is deserv- ing of notice. 1 do not hesitate in giving it my unqualified approval. Jamks Woods. Prinrijial. 1[ai.11-a.\, X. S.. August aith, ISin. I llAVi: inspected with very great pleasure the " Mathematical Scale" invented by Mr. Chisholm. Thougn mnrked liy striking simplicity in its construction, it possesses a range and a precision much superior to any other scale with which I am acquainted. It is not encumbe'-ed with tables or a large array of figures, and yet it lays the results of Mathematical processes, which would invn've great labour and much time if wrought out in the ordinary way, before one at a glance. The most intricate problems which I ])ro])osed were solved with a rapidity for which I was not iircpared ; the question was hardly prcqiosed before the result lay t^uU and clear u|)on the scale, ami almost as self-evident as a simple axiom. In its simplicity' lies its astonishing power; since it is equally applica- ble to the solution of the most difficult prolilems in the various de- partments of Practical Mathematics, and to the elucidation of tlie abstruscr truths of pure science. It is a scale which will be more appreciated by any one in proportion to bis Mathematical skill; and those most advanced will see more clearly into its uiu'ivalled powers and be more readv to acknowledge its higli ca|)acity. I consider it as a powerful addition to the cause of Science, as abridging in a complete manner the toil of study, and the laborious calculations of professional men ; and have no doubt that the talented inventor will meet that reaily recognition from scientific men which bis Scale really deserves. Wll.I.IAM CJahvik, Tencticr nalhtiiii'if Ciilli'ijP^ iinii Spri-ptiinj i/tW. S. Litenirif unit Sriintitir Sttiifltf, St. Mauy's ('oi.i,i;(,k. Ilalitax, X. S., 21st August, iwil. I llAVi: testi'd '' I'lie Chisholm Scale" in the resoluticni of several Mathematical ])robleins, and I found it to he sufficiently accurate for all practical numerical results in the art of navigation, survey- ing, engineering, \c. I have no doub' that when •' The Chisholm Scale" is sufficiently brought bef(U-e the scientific world, that it will he at once iidoptcd instead of other s. At the distaneo of GO on A. a riuadrantal are is drawn, terminating at (ill on D. On this quadrant the degrees are marked and numbered at every tenth from A. to D. Within th(^ ((uadrant are marked the points, half and i|uarter points of the compass. 7. These divisions or lines together with a moveable index, graduated like the scale, and attached to it by a pivot at the angular point (jf .\. and !>., form the whcde ap[i:n'atus. X. Although the four sides of the scale are nninbcred in ex- actly the same manner, j-et it seldom liecomes ncce.-isary to have recourse to more than one side and the imlex, in the process of solving any one problem. The sides A. and C, being opposite and ]iarallel, are, in every respi'ct cijual ; as also are IJ. and D. ; but, in practice, the operat(jr will find the sides A. and 15. more convenient than C and 1). !•. The few figures marked on the scale, combined with the simplicity of its construction, render a more detailed descri|ition uiniece.ssary. It will suffice, therefore, to make a few remarks on its powers, comprehensiveness, and the labor it saves in cal- culation. It will reailily s(dve any pndilem in Arithmetic, (le- onietry. Trigonometry, and Navigatiuji, without the aid of any tables whal.soever, except those of latitudes and longitudes. In Trigonometry and Navigation especially, the branches in which its u.scs are jiarticularly important, the despatch with which the most difficult problems can be .solved by an expert operator, is, to say the lea.st, iiuTcdihle uidcss witne.s.sed. Another advan- tage in using it is, that, in any problem in the four branches al- ready referred to, it is in no Citse necessary to deviate from the rules now in use in the .schools. 10. The rules for solving Arithmetical problems by the scale will now he given, iiremisiiig, however, that in the two first ele- mentary rides, .Vdilition and Subtraction, the scale, like Loga- rithms, is n(Jt av.iilable. Attention to the following rules will obviate every difficulty. 5irLTIPLTCATI0N. 11. CASK T. — To nudtiply by any numbei from 1 to 10. lU'i.K. — Set 100 on index to the ])erpenilicidar of the nmlti- plicr, taken on A., then op|K)site the multiplicand on index will be found the product on A. Krum/ile 1. — Multiply 80 by 0. Set 100 on iiulex to !) on A., then opposite 80 On index will be found the product 720 on A. Although the scale .shews 72 instead of 720 in the product, it will be seen liy reference to Article ;!, that this number imiy be 720, 7200, 72,000, ^*cc. A little consideration will, there- fore, enable the operator to arrive at the correct result. AV. 2.— Multiply 1;") by 8. Set 100 on index to 8 on A., then opposite 15 on indcs is the priMluct 120 on A. O.VSK II. — To multiply by any number cxcecd'ng ten. Kl'I.k. — Set nudtiplier on index op[H).site 10 on A., then op- posite multi|)licand on .V, is the product on index. AV.— Multiply 10 by 12. Set 12 on index opposite ten on A., then opposite 40 on A. will be found the product 480 on index. • Unavailable, like Logarithms, for Addition or Subtniotion. A'ofe. — If side 15. instead of A. were used, the result would be the .s.'une. The operator can use that which he finds most convenient. CASK III. — To nudtiply by a Vulgar Fraction. Uli.K. — Set dcniiminator on index to |)erpendicular of nume- rator on .'i,. then opposite the multiplicand on index will be found the product on .V. AV— Multiply 8 by |. Set 4 on index to :( on A., then opposite 8 on index will be found the proiluct (i on A. Without moving the index the product of any other nuudicr by the .same fraction may be found. This method, although apjiarently difTerent from Oa.ses I. and II., is yet identical with them ; for when the index is set for a vulgar fracticm according to the directions given, 100 on index will be opposite to the corresponding value of the given vub'ar fraction, in decimals, on A. or IJ., according as A. or H. is used : thus, if the .scale besot for f, it will be found that 75 on A. is opposite to loo on index, and hence J is ei[ual to 75-100 or .75, which, being multiplied by 8, gives 0. Divisrox. 12. CASE T. — To divide by any number not exceeding ten. Hulk. — Set 100 on imlex to the perpendicular "f divisor on A., then iipposito the dividend on A. will be found the quotient on index. A'r.— Divide 48 by 0. Set 100 on index to the perpendicular of on A., then oppo site 48 on A. will be found the quotient 8 on index. C.\SK II. — To divide by any number not less than ten.' Ilii.K. — Set divisor on index to perpendicular of 100 on A., then opposite the ilividend on index is the quotient on A. A>.— Divide (iO by 12. Set 12 or 120 (Art. .'!) on index to 100 on A., then opposite 00 on index is the quotient 5 on A. CASH III.— To divide by a Vulgar Fraction. liuLh;. — Set the dciKuninator on index to the perpendicular of numerator on A., then opposite to the dividend on A. is the quotient on index. AV— Divide (iO by 5-0. Set on index to 5 on A., then opposite CO on A. is the quo- tient 72 on index. As in multiplying, .so in dividing, by a vulgar fraction, the rule is identical with that above given ; for, when the index ig set fiu' a regular fraction, the perpendicular from 100 on index to side A., will shew on A. the value of that vulgar fraction in decimals. From this also may be seen how well adajited the scale is for converting vidgar fractions into decimals, and vice rcrsii. RKDUCTTO>f. 10. As this rule depends altogether on Multiplication and Division, enough has been s;iid in Articles 11 and 12 to enable the learner to work without further instructions. KEY TO CIIISIIOLM'S MATHEMATICAL MECITANIOAL SCALE. I'ltOl'OltTION 14. Hrr.K 1. — Sot llu' lu-»l ti'iiii (ni index tn tlic sccnnil torni on A. (]!• 1!., ihcn llii' lliinl tcr n imlcx will slicw llio fniiilli tonn or luisweT on A. or B., lu'conliiij; as A. or H. lias Ijoen UBC'll. liui.K 'J. — Sot till' sci'oiiil Icnii on iinlc.x to tlic fii>l term on A. or ]{., llu'ii llio tliinl tunn on \. or I!, will .'•licw tin' luurlli torni or iin.^wi'r on index. Ante. — Tlio lii>l and lliird nro grncridlv taki'ii dn tlio sinic sido. aH are also I'lo socoiid and liMirlli. K.r. 1. — If :j y:mU of ulolli l■u.^l 4 ^liillinf;s. wliat will !t yard.'* co.-it .■ Tlii.- niaj- bo stated either of the two follow! Yils. w.iys : Sot 1(10 on indo.v to (1 on .\. nr 15., Ihon o])|Kwit(! SO (in indox i;-i Xi ICw. on A. or ]{., that I.h fmir largo divi.sion.s, each XI, and oi!j;lit small divi.iions, oadi two .sliiHinir.s. An inlorost, iiarlnor>lii|i, |iiiilit and |ii,>ioount, coniniis.sI(in anil linikor;i;;o, ki\, arc »ini])'v variatioii.s uf tho Unlo of Throo, the loarnor will liavo nu dill'iinlty in .^mK In'; ,iny |irolilom.s in llioni, with (ho aid uf tho nilo.s i;ivon in Arti'lo Id. Diioduui- nial.i lan ho iioiforiiiod by llio rnle given in Artiolo U. KXTKACTIUN OF SljrAUK ROOT. i; Till Yil» »«ll. d Yds. u Y'.ln. !l 8li. 4 .«h. Sh. Tlion, hy tho Hi>t rnlo, sot 3 on in 1) on indox will show l:i on A. or ]{. Or, sot 3 on index to !l iin \. or show I'J on A. or H, J!y Itulo '2. .sot 4 on indox to fi on .side A side A. or W. .show.s 1^ on indox. Or, set 'J on indox to '.'> uu A. or B., and 4 on A. or li. shcw.s I'J on index. /■'.'•. "J. — If 14 men perforin a iiiooe of work in ., then (i on hies may In Set 24 on indox to 1^ on A. or 15. Or, lpy liulo 2. sot 14 on index (o A. or ]i. shows oj on index. lo. "When remainders or fraeti'ms occur, their read on the scale, hy an exiiert oporatnr, with aliini>t p 'rfeet ac- ciiraey. By per.Hins nnaoiinainted with llie soalo, however, ro- conrso must he had, either to the diagonal on side li. for deei- inals. or in tho follnwing niannor for regular fractions : Sot tho index one division down on the perpondieuhir of the divisor (the lirst term in proportii.n ) on A. ; talio tho roiiiaindor on a pair of dividers, move tlii,' dividers along the index from the pivot towards side B. till they oxaody ciiineide witli flie space Letween the indox and side A., then will side A. shew the nu- merator of tho regular fraction. Kr. 1.— Divide TOO liy !l. Sot lOO on indox to !) on A., then TOO on A. will shew on indox TT with a remainder. Tu liiid the value of the remainder, take tho remainder on di- viders and set the index one division below on A. ; then if tho dividers ho moved ahmg the index, it will be found to ooincide with the space between the index and side A. at T, which i.s therefore the numerator of the fraction, and hence the tjuotient h TT T-0. If the same extent on the dividers bo applied to the space be- tween side li. and the diagonal, it will be found to coincide at 7T. and (ho whole lengdi of B. being 100, tliis number will be 77-100 or .TT, and in this ca.so, therefore, the iiuotionl is 77. TT 1-. The small diagon.il between the third and fourth divisions on D. may bo employed in the same nuinner. SIMPLE INTKHEST. 10. To calculate the interest of any princiiu |i'iire roof of a nuiiihcr is that number whioh, mul- tiplied by i(.'iolf, gives the propo.sod iiiimbor. liii.K, — Let (ho nnmhor, wliiwe root is re(|uirod, bo taken on .\. or |{.. and let ilOO un index bo sot to a trial divi.sor on A. or B. ; then, if the trial divisor or index show the given niini- hor on .V. or B., (he trial divi.^or is the rout roi|uirod : if not, vary the (rial divisor by moving (he index either way, accipniiiig as tho trial divi.sor shows a ro.Milt gre.itor or loss (lian (he civon nnmhor; and uinuo this uiidl (ho trial divi.sor (akeii on index show the givi'ii number on .\. or B. AV. — Itoipiirod (ho sipiaio root of (J(l() v If 20 be assumed as a (rial divi.Mir, .set 100 on indox to 20 on A., then 20 on index shows only 400 on A., which is h>ss than (he given number (KiO : hence the trial divi.sor 20 is le.ss than the root roiiuirod. If MO bo assumed, A. will be found to bo greater than the root roipiiiod : hence tho root inu.st lie botwoon 20 and /iO. By moving the index, the operator will iiiid (hat, when 100 on index is set to 21.5 nearly, or 24.4!) on A., 24.49 on indox will .shew (JOO on .\. A'life. — The sipiare root of any nnmhor, not cxeecdin"' 1000, is extracted niiiio convoiiiendy on .side B. than A. EXTRACTION OF THE CIBE BOOT. IS. Ri'LE.— Set 100 on index to trial divisor, or a.ssunioil root on A. or B.. then opposite trial divi-or on index is its s.piare on A. or B., and o]iposito (his sipi.-iro (akoii on index is the given nnmhor on A. or B., if the a.ssumod root be the corioot one : if othorwiso, the index must be moved, as in square root until the corioot root be found. When tho indox is set fiir any trial root it is not necessary to move it until the corroctiie.s.s oV iiieoiTeetness of (he (rial root is dotorminod. A'j'. — Itenuired the cube root of 40,0110. Here the given iiuiiilier eaii be divided only into two period.'', lionco there can bo only two figures and a deeinial fraction in the root. The cube root of the first jioriod 4(i is ;i +. l(K) on index, therefore, must be si't (o a nuniher bo(woon .'! and 4. Lot it bo .sol on index to ;!."i on A., and ;i.") on index will show 1225 on A., and 122.) on indox will shew nearly 4:1,000 on A., which is lo.-s than the given nunibor. Ilenoe ;).') is loss than (ho roijuired root. l!y a similar process lili will bo found to be greater. The correct root must therefore lie between Mi'i and ■'!ti, and by sotting 100 on index to :i.'i.S, and proceeding as be- fore, the result is found to be 40, 000 nearly. Ilenco";i,'i.S i,, nearly the root rcnuircd. PART II. PLANE TIUCONOMETRY. Ke.mauks, &c. 19. The linos joining the oorresponding divisions on the op- posite sides, 1). aii'l B., are calli'l jinrdllcls, in order to dis(iii- guish (hem from (ho.so joining A. and C, whioh are called yxv- _al. at any rate per piinliciihirs. cent., for one year. | -^o. The perjiendioular on (he fiOth division, being a tangent lUi.K.— -Set loo on indox to the rate per cent, on A. or 1!., i to (he arc, is called a line of tangents; and whenever tho word then opposite the iirineiiial on index is the an.. — What is the interost of X.siJ for one year, at ti per side A., is the sine of that degree, anil its niiiiiorical value to "0"' • I radius 00 Ls reckoned on B. ; the parallel on side D. is the co- Hino, and its numerical value is reckoned on A. ; and if the in- dex be set to any dogroo on the ipiadrant its intor.seetion with (ho lino of (angenis will show, on (he index, (he numerical value of (he -iocaiit, and on the lino of tangontw, tho numerical value of (ho (angent to the same radius tiO. The.se values being di- vided by tiO give tho natural sines, co.sinos, tte. 22. If 100 or 1 on side .V. or index be considered as rndiui', and a i|iiadrant coneeivod to lie do*rilied from 100 on- .\. to loo on I)., the side B, heconies (he tangent to the are which was coneeivod to bo (lins tbrinoil, and by placing the indox to any degree on the i|iiiidiaiit, (he perpendioular from 1(1(1 or I on index to side A. will he the natiinil sine of that degree; (he parallel on side I), (he na(ural eo,-ino, and (ho intoiseetion of in- dex with side B. will show on index tho natural .so,ant, and on side B. the natural (angont. 2:i. If (lie natural tangent of any degree above 59 bo ro- iiuirod, it will be neee.ssary to use the soini or quarter tangent as foniid on the perpendiculars of liO anil 15 respeelivolv. _ 21. In the solution of piobloins by (he scale, when the words sines, cosines, tangents, cntangonts, ^c, are used, they nm.-t be understood to moan their numerieal values to radius OO. 2.1. Tho division of radius into sixty equal parts agrees wldi the division of a degioo of huigitude on (ho equator into sixty minut.is; and (lius allords an oa.sy way of finding (ho lengdi of a degree of lungitiide in any )iarallol of latitude. Tho parallel from any degree on (ho quadrant to side l>. will be the long(h of a dogroo of longidide in the ]iarallol of that degree. For ex- ample, (ho lengili u\i a degree of luiigidido in (ho parallel of ;!0 is (he moasuro from ,'10 on qiiailran( (o side I)., which being reckoned on side \. shows 52, which is tho length of a degree of longiludo in the jiarallol of .'iO'. 2ti, The meridional difleronce of latitude can be readily found widioiit tho aid of any tables. Thus, set the index to "middle latitude on quadrant, and (he inter.-"c(:on of the (angont with the index sliows on (he index (ho length of a nioridional degree in (hat iiaialiol (assuiiiiug the iiiiddle of tho degree as tho"pa- rallel) ; and if (his bo iiiiildpliod by (he irfforeneo of latitude, in degrees, the jiroduct is (ho meridional difli'reni'o of ladtudo. 2T. Tho principles on which the calculations in Trigonometry are founded, are certain relations or ]iroportions oxi.stiiig between tie sides of triangles and certain linos cunnoetod with (he angles, called liiijiiiiDiiiflrifiil liiu'S or ra/ios, anil the principles on which the use of tho .scale, in Trigonometry, is ba.-iod, may be thus ex|ilaineil :— Lot A li (' be a triangle, and let I) E or any other ^..^ line bo drawn parallel (o I! C, one of (ho sides'of (ho flKur" '• triangle ; (hen A (' :, A B ; A 1> or A E : A = A U : A B. Now. by moans of (he index, an indolinito nnniber of triangles, with lino.'; iiarallol to some of the sides, can be formed ; ^aud hence uu indefinite number of proimrtions. RIGHT ANGLED TRIGONOMKTHV. Dkfinitio.ns anu Pkixciplfs. 2S. Every triangle consists of six parts, viz., throe sides and throe angles ; and when any (hroe of dioso are given, unless it bo the throo angles, the odier (hroe can be found. 29. The "um of the throe angles of any plane triangle is equal to two right angles or ISO". ;!0. The greatest side of every triangle is oopositc to tho greatest angle. !!1. The coiniileinont of an arc is its difference from a quad- rant. .'!2. Tho sup]ilement of an arc is its difference from a semi- oirole. ;i;5. The complement of an angle is its difference from a right angle. ^ lit. The supplement of an angle is its differonco from two right angles. ■i5. The .sine of an arc is a straight lino drawn from one ex- tremity of the arc, perpendioular, to the radius pa.ssing through the other oxdomily. 'M. The tangent of an arc is a straight lino touching tho arc 1 n A. ; mill if thu in- t it.s iiitcr.sciiiiiri with [, tliii iiiinirri'ul viiliiu I, till! iiiiiiii'i'ii'al viiluu riiuNU values being di- CDiiMiilcicd IIS rniliui', >l tVdiii 100 uii'A. to Lilt to Ihr an: wliiih • plai-ilifj; llir iiiilcx to liar tViiiii loo or I on of that ilc^rc'C ; tliu llii- inli;i>oc'tiiiii of in- itiiral Hcaiit, anil on •V alinvo TiO lio ro- (ir quni'ti'r tunj;(;iit an ;i'>|ice(ivi'K-. I'alo, wliiii till' wiii'iIh o iisL'il, tiny inu.-t Lu to railiiiM (iO. lal parlfi !i;;rocs with I' (.'((iiator into .sixty finiliiig till' lcii<;lli iif iliiilc. 'I'iio parallel will lie tiic li'nglh of at ilri;rci.'. For fx- e in tlio painlUil of ^i(K.' I)., wliicli Ijeing ) Iwi^lli of a ilc'grce can 1)0 roailily fimnil ;lio inili'x to iiiitlfUo of tliL' laiic;i'nt with a niL'i'iilioMal dogroe ' ilcgivu MM till' [la- ■fliTi'iKo of latitiiilo, It'i-oniM' of latituilo. oils ill 'rrifi;iinciint'try oils existing liotwooii L'toil with lliu angles, il the |iriiii'i|)li's on ry, is liaMil, may be ir any olhcr Hfo ^il||•s^lf iho nt,'"!-" 1. A C = A D : A [i. iiiiiiibi'i- nf irianglcs, au bo furiiicil ; and MKTRV. KS. iz., tliroo sides and arc given, unless it iiind. ly piano triangle is is opposite to the rcnce from a quad- roronce from a scnii- [fercnce from a right lifference from two triiwii from one cx- us pa. I'' is the seeant of A li ; so li Vi is the sine of li (', or the eosinu of A li ; (' II is the tangent of li (', or the eotangeiit of A H; and O II is the .seeant of li i\ or the eiseeant of A Ji. iii). The sine, tangent, and .secant of an me, are the .sine, tan- gent, and seeaiil of the angle niea.-nred by the are. Tims, the are A 15 measures A O |{ and li K ; A V, O V ; the sine, tangent, and .secant of the are .V ii are also the sine, tangent, and scciint of the angle A O 15. 40. The sine, tangent, and secant of an angle, are the cosine, cot.'.ngent and co,.{'caiit of the coniplcmcnt of that angle. Thus, ]i ti or its cipial H the sine of angle li () C, is the cosine of A O l> ; (' II and O II the tiingiiit and secant of li O (', are the eo(an;^cnt and cosecant of A O li. 41. The sine, tangent, and secant of an me, are eipial to the sine, tangent, and secant of its supplement. Thus, 15 ]•]. the siiii' of A li, is also the sine of 1! M 1> ; A F, the tangent of A 15, is eipial to the tangent of A 1 L, wliicli Ls eipial to li yi 1). Hi'iice, wlicii an angle is obtuse, its pupplonient iiin.st be u-sed. I'llOI'OSlTlONS. 4"J. When llie hypothennso of a right-angled triangle is made radius, its sides liccoirn> the sines of the opposite angles, or the eusines of the iidjaccnt angles. Thus, if A (' be considered as radius, it is evident, ,n,,^, by completing the figure (Art. HH), that li C is sine "h:"'-'' ii. angle A or cosine angle C, mid that A 15 is siiio angle C or co- sine angle A. 4.'i, When (he base is made radius, the perjiendieular beronies the tangent of its opposite angle, and the hypolhemise the se- cant of the .same ; or the perpendicular becomes the cotangent of the .adjacent angle, aiel the hypothcniist' the cosecant of the same. Thus, when .V 15 is made radius. 15 (' ln'coiiies tan- j.;,.,. gent angle A or eotangeiit angle (', and A (' becomes Bumc i. secant angle A or cosecant angle ('. 44. AV'ben the pcvpcndiciilar is made radius, the base is fan- gent of its opposite angle, and the hypnthenusc secant of the .same ; or the b.i.se is cotangent of its adjacent angle, and the liy- |Kithcnuse the co.secaiit of the same. Thus, when lit' is made radius, .\ 15 becomes tan- >i|,(. gent angle (' or cotangent angle A, and A C becomes Ui,'ui> •'i. .secant angle V or cosecant angle A. Itl I.KS Foil CoMl'UT.VTIOX. 45. CASE I. — When a side and one of the oblifjue angles an' given to tiiid a side. l{iii,K. — Make any side r.adins : then As the name of tlie given side Is to the given side, So is the ;aiiie of the side rci|uirod To the side reipiired. 40. CASi; II. — When two sides are given to find an ani»le. Ki.i.E. — Make one of the given sides radius : then As the side made radius Is ti' radius. So is the other given .side To .sine, tangent, or secant of the angle by it represented. Nute. — In working by the scale, let it be remeinbercd that iiir sin(>. tangent, and secant, their numerical values to radius CO must be used. The exaiii|ilcs in Trigonometry, N'uvigation, &c., arc taken out of Noric's Navigation. 47. Ex. — Qivon the liv|K)tlicnuse ii70 miles, the angle A .'■)l)" ."iiy and conscfpienlly the angle C 3!1 iJO' required the base .V li and the pcr|ieiidiciilar li ('. Making A (,' radius, A li will be sine angle C or en- s,,p sine angle A, and 15 C will be sine angle A or cosine "8""* •'■ angle C. Kadiiis : Side A C :!70 -= Sine angle A M',' .'JO : Side 15 C. — — = Sine angle C !5ij' 'M : Side A 15, ]5y the scale : First railius is ci(ual to (JO, and sine .'if')' 80 is c((ual to i>». Then by .\it. 14, .set radius or 00 on index to .'{70 on side A, then oppositi' sine angle .V TiO on index is found li (', liOS,,') on A, .\iiil if the sine of angle V Xl :iO which is iio.:!, bo taken on index, by the one setting we find opposite it on A the side A I5i;04.-2. Or, set the index to iy.\' !J0' on quadrant, then ojiiiositc Ji70 on index is found on A the side U C iJUS.5, aud on ii the side .V 15 liol.l'. 4S. (.liven the base A 15 Oli.'i and angle A 48' 4:), to find the hy|iotlieiiiise A (' and the perpeinliciilar I! ('; miikimj sw A II /■(k/i'iih, ji V heciimes tniiijeiU iiiKjIe A nr Cfifiiu- i'k"'"' "• i/ciil (iiKjIv C, (1)1(1 jl (J hccuimn sccdiU uiiyle A ur cusecant (iiiyh) (/. Then radius : A 15 (1'25 = Tangent angle A 48 4u' : 15 C. — — = Secant angle A 48^ 45' : A (J. liy the scale : Set (i'jr) on index to railins (U) on A, then opposite tangent 48 4.)' or (18.4 on .\ is li (' 71;! nearly on indix, anil op|iositc .seeant 48 4iV or HI on A is found A (.; !I4S nearly on index. Or, set inde.x to 4x 4.V on (iiiadrant, and opposite Oli.') on .\ is!l4Siin index; and if the parallel from HIS on index be traced to li, it shows on li the perpendicular 71;!. 4!l. (iivcii the hypotlieiui.se A (' MM, and the base li A 230; re(piiicd the angles A and C, and the perpeinlicular li 0. iMaking the hypothenu.se radius, li (] becomes sine sec angle A or cosine angle C, anil .V 15 sine angle C or eo- 'iKuros. sine angle A. Then, A C 400 : Radius = A 15 -I'M -. Sine angle C. liy the .scale : Set radius (iO on index to 4oO on A, then opposite 230 on A will be found sine angle C :i'>A on index : if the [larallcl of ;!.'>.4 on li be traced to quadrant, it will show on quadrant angle C •'IO !•'. wliich, being subtracted from W) , giy.'s angle A iVS iiV. Or, set 400 on index to 231) on side li, then the ]icrpendicu- lar from loO on index to . , shows on A the perpendicular 15 V 323, and the index cuts the quadrant in 30 H', which is the angle required. Ao/f.— The a of the sides about the right angle radius, when po;:sible,'as will be seen by next problem. .'■)0. (iivcn the base 15 A 35. ."i and the ]ierpcndicnlar 15 C 41.0 ; required the angles A and C and the liy|«ithenuse A C flaking .V 15 radius, 15 (' will be the tangent of angle seo .V or cotangent of angle (!, and A C will bo seeant of figure u. angle A or cosecant of angle V. Then .V 15 35.5 : lladius 00 = 15 C 41.0 : Tangent angle A. liy the scale : Set (iO on index to 35.5 on A, then opposite 41. (i on A is found on index tangent angle A 70.2 ; then, if index be set to 70.2 on the line of tangents, it will cut the i{uadrant at 4'J 31', which is the number of degrees on angle A ; aud if the pcrjien- dicular from 35.5 on A be traced to index, it will show ou index the liypothenuse A C 54.7, nearly. Or. trace the |icrpcn(licnlar of 41.0 on side A 'till it will in- tersect the parallel of 35.5 on side I! ; .set the index to the jioint of intersection : at this point will be found the bypotlienu.se 54.7 nil inilex, and on the qiiadiant will be fuuiid the angle C 41- 2"J', which, being subtracteil from 00 ', leaves angle A i'J 31'. E.XAMl'LKS KOll ExKUCISE. 1. (liven the hyiiotheiui.se lOX and the angle opposite the per- pendicular 25 3() ; required the base and the perpendieuhu-. Alls. — The ba.se is 07.4 and the perpciuliculur 40.00. 2. Given the base 00 and its op|)o8ito angle 71' 46'; required the lierpcndieular mid the liypothenuse. /I/(.||. — The perpendicular is 31. (iO and the hyixithenuso 101.1. 3. (iivcn the base 300 and the lierpendiculur 480 ; required the angles ami the liyiiolhenn.se. Aks. — The angles arc 53 »' and 30' 52', and the liy|M)tho- iiuse 000, OliLKifK-ANCLFn Tl!I(iO.V()Mi;TKY. 51. CA.SK I. — When two angle's and u side opposite to one of them are given. liiil-:. — .\s the sine of the angle opposite to the given side is to the given side, so is the sine of the angle oplxisito to the re- (piired side, to tin.' reqnireil side. 52. C.VSE 1 1. —When two sides and one of them are given. litl.R. — .Vs the side opposite to the given angle is to the given angle, so is the side ojiposite to the required angle, to the re- ipiired angle. A'o/c. — When two of the angles are known, the third is found ly siditracting their sum from 180 . Sec fii;uru H). uii angle opposite to {''■''■ 1. — (iivcn angle A 30 15', and tlio angle 15 05 30', and the side .V 15 53 ; required the sides A 1 (! and 15 C. As sine angle liS" l.V Is to its oppoHite side So is sine angle 10.5" 30' To its opposite side And so is sine of suj). ac 1.3' To its opposite side 37.2 on n. o3 on F. 57.8 on n. 82.5 on F. can be found more easily hy making cither 35.5 on B, 50.0 on F. Set 53 taken on index to 37.1 nn 15; tlien opposite .57.8 on 15 is found on index A C 82.5, and opposite 35.5 on 15 is found on iiiilex .side Ii (' 50.0. Kx. 2.— (liven the side A 15 330, the side M V 355, and the angle A 40 20'; required the angles 15 and C and the side .V C. As side given 3.')5 := Is to sine of its opposite angle 49" 20' " So is the other given side " To sine of its opposite angle 45' 5.S' " And without a mnve so is Sine of suiiiilemcnl 84^ ;)(>' >< To its opposite side " Si'o flLMiro 11. 355 on B. 4.1.5 on F. 3;i6 on n. 4H.1 on F. 5n.8 on F. " 4G(> + on It. Set 45.5 on index to 355 on side 15, (hen opposite 330 on 15 is foiiiKl sine angle C 43.1 nn index : trace the parallel of 43.1 on li. till it intersect the (piadrant. and at the point of intersec- tion is found on the quadrant 45 5S'..the angle at C. If the angles A and C be now added, and the sum .subtracted from ls(i , the remaimler is angle I! 84" 30': then, by the first case, A C can be found. 53. ('.\SE III. — Wlien two sides and the angle contained between them are given. liLLi:. — As the sum of the two given sides is to their differ- euce, so is tangent of half the sum of the unknown angles to the tangent of half their (litrerence. This half ditt'erenee,''added to half their sum, gives the greater angle, and subtracted, leaves the less. The angles being thus all known, the remaining side is found by Rule to Case 1. A'.i'.— ("iivcn the side A H 85, the side A C 47, and s^c the angle A 52 40' ; required the angles C and B and "«">■'' '-■ the side 1! C. I.SO' Anglo A = 52-' 40 B + C 127^ 20' i(n-|-C= 6.3 40' (A 15 + A C) 132 : ( A 15 . A C) 38 = Tang. 4 (C + B) 03 40' : Tang, i (0 ^ B). = - v "r ; Here, we musi use the .semi-tangent, found (Art. 23) on the perpendicular of 30 on A. to be 00.0 ; and on trial the operator will find it necessary to employ 00 and 10 in the first two terms of the proportion, in.-tead of 132 and 38. KKY TO CIIISIIOLM'S MATHEMATICAL MECHANICAL SCALE, Thus, set (>•) oil index to ID (in H, llu'ii ii|ipiinilc (10. (i on iuJex is lomiil soiiii-lniijjoiit of liiill' tin- clilt'cri'iico of llio nn- kiiown iin^li's 17..'> on II : if l"..') Ixi now liikcii on llio line t'f siMiii-liiii;;i'iiln, viz., llie |H'rpt'ii(liinliii- ol .'!() mul llic iiidrx set to it, the (|im(lr nt will lii^ <'nt by tin' inilcx lit ;!0 11', Imif the dill'iri'iice of llic iiiikiiowii iiiijiliM. Tlii'ii, O;) ID -j- ;t() II' — !»;i ."il greater iinjjlo C. (■„•! 10 _ ;(()' I r =. ;);t -'S losa angle JJ. 11 r is I'eiidily found liy lirst eane. (.)r, set the iiiilex to the given angle 52' 40 on quadrant | liiko 17 on index mid H.") on .side A, iinagiiie a right line drawn from t? on index to Hi) on nide A, and the n,,„ triangle is complete. The perpeiiilieiihir from 17 Hijuru i;i. on index to .side A is I)"."), and divide.s the hasj into two sog- tiients, 2«..'» and /id..'), and the triangle into two riglit-iiiigled triangles. If the perpeiidienlar ol M.,'), Kegineiit 1) It, ad- jiieent to the required angle, taken on A, and the parallel of 37.'> taken on 15, he traced till they intersect, and the index set to the point of iiilerseclion, this point shows on index the .side 11 (' (17.7, and the intersection ol index with the are of the quadrant, shows on the qiiiidriuit angle 1! ;!;J 2'J. 'J lie siniir ii-orhcd u-ith (lio nid of t lie asxisliiiy index. Kll.r.. — .Set tlie attached index F to the given angle .')2 10' on the qiiiwlrMiit, anil while ill this position, set the centre of the assisting index 11 to 17 0111'": bring the graduated edge in contact with the other given side H't ou A : then the circular ])art will indicate the angle at iircetiiig ol indices to bo I'-i 'il'. and the side sought to be 07.7. Two angles be- ing known, the third can easily be limiid by note to Art. •>■>, or thus : reverse the assisting index by placing its centre on H'> taken on A, and its graduated edge ou 17 taken on at- tached ini'.ex, the circular part will iL'iicate the angle to bo 'M 2'J', uiiil the side 07.7. M. CASK IV. — (iiveu the tlirco sides to find the angles. KuLK. — Draw a perpendicular from one of the angles upon tlie o|>{)osile side or this side prodnccil ; then calling this side ba.se, say iis ba.se is to the sum of the other two sides, so is the ditl'ciciicc of these sides to the ditl'erence of the segiiiouls of the ba.se. Then half this dilTerence added to half the sum gives the preater segment, and subtracted from lialf the sum — that is half the base — gives the le.-s. Then the triangle will be di- vided into two right-angled triangles, the angles of which cau be found by Art. l(i. £•.<-.— Ciivcii the side A H 107, the side R C 110 niid the side A C XH, to find the angles A li and C. A 15 : A C+C 15 = A C J. C 11 : A 1) ^ D 11 liu : i;i8 = 22 : 27.71 C or 7H.r) : II'J = 22 : 27.74 8ct !);i on index to 7S..") ou B ; then opposite 22 ou 11 is found on index 27.71 difference of the segments of the base Tlieu 27.74 Sec figure H. l;i.M7 half difTerence of the scginciits, 'iH.a half the sum or half the base. 92.37 sum gives greater segment D 11, 04.03 diff. gives less segment A D. Set side A C 88 taken ou index to its adjacent segment A 1) (M.O.'l taken ou A ; the index will shew ou qiiadraut the angle A 42' 44'. Again set side C 11 110 taken on index, to its ndjucent segment 1) 15 112.37 taken on A; the index will show ou quadrant, in like manner the angle 15 32 .03'. The same worked with the aid of the detached index II. First take the halves of the sides, namely ~>i.,'i, 44 and <')i>. On the attached index V take 44, and to it set the c;!ilre of the detached index. Move tlie indices till you get .O.J on de- tached index iu contact with 78.0 on side A ; the attached index will be found to intersect the ipiadrant at 42 41 angle A, and the circular part will indicate the angle at the meeting of the indicts to be 104' 23' angle C. Kx,VMfl.K.4 lOU E.XKKritlE, 1. Ciiven one side I2'.l, an adjacent angle oO'^ 30' and the ' opposite angle 81 .'iO : re(|iiire(l the third angle and the re- I maining siiles. I ^iiis. — The third onglo is 41 .'il', and llio rcnmiuitig sides are 108.7 and 87.08. 2. (liveii one side 110, nnothor side 102, and the con- j taiiied angle 1 13 30': rec(nireil the remaining angles and the I third side. Aim. — The reiiinining angles aro 34 37' and 31 47', and the third side is 1 77..'). 3. (iiveii the three sides rospectivoly 120.0, 12.')..'), nud 140.7 : required the angles. .I)i.«. — The angles are ol M', ^>l' M', and 73' 9', I'LANE SAILING. 5.j. In plane sailing, the enrtli is supposed to be an ex- tended iilaiie, 1111(1 the nieridiaiis are, llierelbre, consideii'd as being parallel to each other, the parallels of latitude ai right angles to the meridians, and the length of a degree ou the meridian, equator, and parallels ol latiliide e\ery where eipial. i')0. The coiirite is the angle which the ship's track iiiakes with the meridian.. The dliildiicv is the number of miles, &c., between any two places, reckoned ou the rhumb line of the course. o7. The diji'i i-rnce nf httiliide is the distance which a ship makes North or .South of the place sailed from, and is reckon- ed on a meridian. i>H, The dcjitirturc is the distance which a ship makes East or West, and is reckoned on a )ianillel of latitude. A'()(<'. — As the course is generally taken on the arc of the (|iia(lrant, the o|ierator will liiid it more coiivenieut to take ' the ditlcrence of latitude ou side A and the depiirture on side 11. I Ex. 1.— A ship from latitude 48 40 N., sails N ; K. by N. 2'.l(; miles, required her present latitude j and the departiin^ made good. I Then, by Trigonometry : Radius : Dist. 2it0 = Cosine coii. 3 pts. Sine (joiirao 3 pis. lly the Scale : Set radius 00 on index to 21)0 on 15, then opposite cosine 3 pts llt.l) on index is dill', hit. 2lf!.l on 15 and opposite sine 3 pts. 33.2 on index is dep. 104.4 on 15. Or, set the index to the course .'i pts, then the distance 200 on index will cut lie lierpendicular of the diU'ercnce of latitude 240.1 on side A, and at the same time will cut the parallel of the departure 104.4 on side 15. Then the prop(M-lioii will be — As rmlius OO on F is to cosine 3 pts 41*. 1* on A, .so is dis- tance 290 or 29.0 on K to 24.04 or 240.4 on A ; and so is distance 290 or 29. (i on F to departure 10.44 or li;i.4 ou A. .")9. The operator cannot fail to see that all tlie exercises in Navigation can be .solved by the scale in various ways ; but as a work of this kind must uecessarily be short, we will alter this confine ourselves to the easiest methods; and for this piiipose we must reserve the usual position of the ligiirc, by drawing the (liH'erence of latitude across the page, and the departure in a direittion from to]) to bottom. Ex. 2. — A ship sails .S. E. | 10. from .St. Helena, scc iu latitude l.')'o5'.S., until by ob.servation she is flijiiri' m. in latitude 18 49' S., require lier distance ruu and departure made good. Latitude .St. lleleua 15' 55' Latitude come to 18' 49 Sop ligurc l.V difl'lat. dep. DifTerence latitude 2 54 GO In miles 174 Kl.l.K. — .Set the index to the course 4^ pts., then opposite the dillerence of latitude 174 on A will appcui" the distance 274.3 on index, and the paralhd trace(l (roiii this point on index to side II, will show on 15 tlii^ departure 212. F.x. 3. — A ship friaii liilituile 3 10' .\., sails S. him. W. by W. .| \\ . until she has made .'loll milesofde- 'Ix""' "• parliire : reipiiied lu'r present laliliide and distance sailed. Uil.l-:. — Set the index to the course .'ij points, then oppo- site tlKMlepartnre .'l.'iO on 15 will appear the distiinco 415.1 on index, and the perpendicular traced from this point Iu sidu .\, will show on A the ditreronce of latitude 213.1. Lat. left 3' lli' N. Diff. lut. 213 milcf or 3 33' S. Lat in 17' H. Ex. A. — A ship from Capo St. Viiiceiil in lali- s„ tilde 37 3' N,, sails between the North ami West "liurf is. 430 miles, until her diUcrence of latitude is 214 miles: re- (piired her coiir.se st1 13' N. Dill'ereuco of lat. 1 25 GO In miles 85 Rtl.t;. — Trace the perpendicular of the latitude 85 taken on A, till it will intersect the parallel of the departure 72 taken on 11; set the index to the point of iiileiseclioii, and this point will show on index the dislaiice 1 I 1.4, and the iu- dex will show on quadrant the course 4i) 10'. E.\AMI"I.ES I'OIt EXKliCISK. 1. A ship from latitude 3G 30 N. sails SW. by W. 420 miles : what is her preseut latitude, and what departure haa she made ':* Anx. — Latituile in 32 37 N., and departure 319.3 miles. 2. A ship from latitude 3 54 S. has sailed N\V. ^ W. till slii^ arrives at latitude 2 14 N. : lYMpiiic I her distance run, and departure made good '< Alls. — Distance 017.8, and departure 190.2 miles. 3. — A ship sails between the north and west 170 leagues, from a port in latitude 38' 42 N., until her departure is 98 leagues: re(|iiircd her course and latitude in? Ana. — Course N. 35 12 W., and latitude in 45 31 N. TRAVKRSE SAILlNt!. 00. Rtr.i:. — Find by tlio scale the (lillercnco of latitude and departure corresponding to each course and distance, as in plane sailing; .set these down opposite the distance iu the proper column, observing that the diirerence of latitude must be placed iu the north column, if the course be northerly, and 1 I w. s. w, W. by N. S. by E. S. W. by \\ S. S. E. I KRY TO CIIISIIOLM'8 MATIIEBiATlCAI. MECIIANICAT, SCAr,K. (rniii tliiit puinl on •lure ili. , Nllils S. n,,„ ill's (if do- "«""• "• I 'littiiiu'C saili'il, |)iiiMls, iIkmi oppo- llir ilisliilM'o 1 1. 1. 1 III tliis piiiut lu aiilu 1(3 -JUiA. t ill lati- Hfp ml Wi'st "«"'■'•• l». is "J I I miles : re- iiilr ;;(ii)cl. () llu' pprpiMidiciiInr iO nil index is dc- IikK'x willi tliu urc SI' CO 9'. siiils bo- si'o limls she "K"!'' '"• I lie cuiirse stocrcd ) (he depnrlnrc 126 )mid I lie ditroreni'ii I'l'si'ilioii of index lie me the course 1 n-2 S. ;! ;j(; N. 2' l N. deini, in See belw ■en fiffurc 10. ai i;rx , 1)3' ob- lure ; ruipiired licr 2^ 38' X. 1' i;i' N. 1 25 GO liitiliide S."> lakcn I' tlic (leparliiru 72 if inlersecliiiii, nnd 111.1, and the iu- 1 tr. ils SW. by W. 120 .hat departure haa rtnre HI9..'5 miles. lil.'il N\V. ^ W. tin I her distance ruu, in;. 2 miles. I west 1 70 leagues, her departure is 1*8 in? ide in Ij 31 N. Il'ei'ciiec of latitude nc and distance, as the distance in the lee (if latitude must !0 be northerly, nnd 1 I I ill the Hiintli c(diinin, if iIk^ coiirHe lie Hiiiilhrrly ; and llint llii^ depiirtiiie must bi^ phieeil in the east ccdunm if the course be eiiHlcrly, and in the west ndiiiiin it it be westerly. Add up the columns of noiihin;.'. Munlliin;;, easlinx nnd we.-.lin;;, and net down the siiiii of eaih at llii^ liollniii ; Ihcii llii! dilt'ereiice between the siiiiis of the nurtli ami soulh coliiiiins will be the whole diU'ereiire of lalilnde made S. and hjiigiliide 31" 53' W. from Capu Verd, in latitude IT 45' N. and longitude 17° 32' W. Lat. IVrnambiico, 8° 4' .S. Lat. Cape W'rd, II 45 N. 8° It Long. 31" .53 W, Long. 1 7 32 W, Diir. latitude. 22' 41»' 00 G'4l' but. ('.i|ii' (io.«l Hope, 111" 22 but. St. IKkiiu l.') .05 SI" 22' la r..-) S. Lnng. IH'^r 8. Long. <") li'i Ulirorenco of latitude, IH= 'J7 m In miles, 11117 00" 17 Diff. long. -H" 1) .Mid. lilt. 25^ 8' la iallc'8, 1 1 Vj llfi.K.. — Set the index na directed (Art. 20) to middle lat- iliide 25" 8', then the intersection of tangent (.Vrt. 20) with the index shows on the index the length of a meridional de- gree in that parallel to be 00..'! ; and to find the meridional ditlerence of lalilnde, lirst, for degrees, multiply 00. ,3 by \H nnd the product is ir.l,!.4 ; then, for the minutes, set the in- dex as above directed, and the perpendicular ot the 27 min- utes on A will cut the index in 30, the proportional jiart for 27 miiiiiles. The former result 1H»3.1 added to the latter 30, gives the meridional did'erence of latitude 1223.1 nearly. Take half the meridional dillerence of latiiude (!1 1.7 seo on A, and trace its periiendicular till it intersect Hbuic ■,'j. the parallel of half the ditlerence of longitude 721.5 on IJ ; set the index to the point of intersection and it will show on the quadrant the course 111" 50, and the perpendicular of half the proper ditlerence of latitude 553.5 on A, traced to index, will show on iudex half the distance 858. Hence the distonee is 1710 miles, and the course 111' 50'. 01. When the ditlerence of hititude is large, especially in high latitudes, Ihe above method, like middle latitude sailing, is not strictly accurate. A correct result may, however, be obtained by taking the meridian ditTcrenco of latitude in parts not e.xceediug four decrees ; thus : — 15" 55' to 50" 10 18 18 20 20 22 22 24 24 20 20 28 28 30 30 32 32 34 34 34 22 Diir. \M. 0^5' 2 2 2 2 2 2 2 2 2 22 Mcr. Dlir. I.at. 5.2 125.0 120.0 128.4 130.4 132.4 134.4 137.4 140. 143. 26.8 D. L. 18" 27' Mcr. D. L. 1230.0 The meridional difTerencc of latitude thus found agrees ex- actly with the tables, and if operated with as in the preced- ing example, it will give the same result as thot found by Mereutor Sailing. Mid. lat. 3' 20' III miles, 1301) Dill', hnig. 1011 Set the index to middle latilinle 3° 220 leagues, or rather DOO miles, nsjuruji. on index, will appear on A the diU'erence of latitude 823 miles, and on H the departure 4D3 miles. Lat. left 2f)" 47' N. Diir. lat. ,S23 m = 13 43 S. Lat. in 1G° 4' Sum . = 45 51 Mid. I.it. Set the index to middle latitude 22" 55', then opposite the departure 411.3 ou A will appear on index difVereuce of loo"i- liide 537 miles. Long, left 24° 30' W. Difl'. long. 537 ni = 8 57 AV. Long 33" 33' Hence latitude iu is 10" 4' N., and longitude 33' 33' W. K.V. 4. — Sup|)ose a ship trom latitude 0' 10' N. and longi- tude ID^ 32' W., sails in the south-east quarter till she litis made 1 15 miles departure, and is by observation iu hititude 2" ID'S.: required her course stearcd, aistauce ruu, and longitude in. Lat. left 9°10'N. Lat. iu 2 19 S. 0" 10' N. 2 19 S. Diir. hit 11" 29' 60 C"51' Mid. lat. 3" 25' In miles 689 RfLE. — Find the point in which the perpendicu- lar of 089 on A, and the parallel of 415 on B inter- sect each other; set the index to this point, andon the quadrant will be indicated the course 31" 4', and the ab.ne mentioned point will show on index the distance 804.2 : if the iudex be Bee fiirure ;;.'» KKY TO CHISIIOLM'S MATHEMATICAL MECHANICAL SCALE. ■et to llitt miilillc laliliiilv .'I' '2'i', ninl llio ili'pnrliint II on A. o(i|HiKil<> it (111 inilux will be scimi llie ilillcii'inu Kiliidu III) iiiili'.^. I..MI-. Ml Jit^niMV. Dill", long. 4 1 (1 miloH — (i .■»ii K. '> tftkiMi 111' liiri- Long. in IS" iltl' W. IIiMK'i' lipr dinrsc ix S, .'tl" 4' K,, dliitaiioe run 804,2, anil lorpon(li(iilar Inieed to index, llioii opposite this point on index will bo found on 15 liulf llie dilloronec of loii<;iliide (i' 72 (U' Ii" 4 I'. Iloiico tli(! dillereneo of longitiido is 13' 28', mid longitude in llJ!) JO' K. The olijeet of this work beiii-' to tench tlio application of the sciilo to N'livigiilional purposes, and not to throw any ad- dilioiiiil Ii;;ht on Navigiilion, it is not tlioii^rht necessary to treat on objiiiuo and current sailiii;:s here. If the operator thoroughly nndcrstaiids Trigonometry and its application to Traverse .Sailiii;,', any cases that may occur in these, how- ever, will not eo.sl him a nioinent's thought when iu posse.s- hioa of the scale. sniKRiCAL trioo\omi:tuy. (I.'i. In treating on .Spherical Trigonometry at all, our ob- ject is merely to show that the scale is adapted as well to Spherieal as to Plane Trigonometry. W'e shall therefore give only a few examples. £x. 1. — In the spherical triangle ABC, right- s„ angled at 11, the hv|)Otlicnuso A C is 04', and the Afurciir. angle C 4(r : tiud li C. To find U C : Cot. A C 04= : IJad. = Cosine C 40° : Tang. B C. The cot. of 040 (Art. 21) is 29.3, radius is OO, and cosine 40' is 41.0. .Set radius 00 011 index to 211.3 on A, then op- posite 41.et the liypotlienuse A C and side B (" of the figiir(s A B (Kx. 1) be given, e(inal to 70' 24' and 05' 10' re- spectively : find angle C. • Because the pcrpemlicukr of the meriilional difference of latitude will not iutersect the iuJei, its liulf is usixl. Ilad. i Cot. A C 70' 21' - Tang. B C (l.V 10' : Coi. C. Itiidiu,'4 is 00, cotangent 70' 21' is 21.2, and the fiemi-tan- gent (d' O.'i' 10' iit I'l.'i ; llion, sot liO on index to 21.2 on B, and opposite O.'t on index is half the cosine (' 23.1 on II; Ihorolorc cosine angle (' is 40.2. The perpendicular of 40.2 taken 011 A, being Irncod to the arc nf the (piadrant, will iu- diciilo ou it llio iiinnber of degrees ill)' 42'. ASTUON'OMICAI, TUOBLEMS. 00. To find the sun's longitude on a given day. Ut'l.F. — Count the iiuinbcr ol days from the nearest equi- noctial point ; and if the sun is on the soiilh side of the e(pia- 1(U', their number will very nearly agree with the sun's longi- tude taken i degrees on the ipiadrmit of Iho scale. If the (h'clinalioii be north, count the ninnber of days as before, and suhlraci one day (or every thirty, and in proportion for a loss number, and the reniaindor will agree with the sun's longitude in dogreos and iiiinutcs on the (|iia(lraul. S'dlf. — Tlio sun's lotigituil(( is dftoii useful to discover data for the solution ol problems in Astronomy. Hx. 1. — K.'ipiiroil the sun's longiludo ou the 2olli day of November, |H(;o. The number of days from the 22(1 .September (the day ou which the sun was on the e(piator) to Iho 2'>th day of .No- veniber, is 04 ; hence the sun's longitude on that day was or. Kx. 2. — Ileipiircd the sun's longitude on the 25lli day of May, IMOO. From the 20tli JIarch (the day on which the sun was on the equator) to the 2rub May, arc 00 days; and subtracling a (lay for every 30, that is 2 !-.'» or 2.2 days, loaves 03.M or 03' ■\H', the sun's longitude. 07. To tiud llir sun's declination on a given day. ii'x. — Uo(piired the sun's declination on the 2Jlh day of November. The sun's longitude by (Art. (!0) is 01'. Then, as radius = 00 on F Is to sine 1 ' (the sun's di-cl.) ■'» I on B So is sine of 23" 28' (greatest decl.) = 24 on K To sine present decl. 20' S.")' 21.30 on B OH. The greatest declination and the present declinalion given to tiud the sun's longitude. L'x. — (liven the greatest declination 23" 2M', and the pre- sent declination 20' ,')5' : to find the sun's lo'igilude. Bi'i.K. — As sine of 23' 28' (greatest decl.) 21 on V Is to sine 20 55' (present decl.) 21.30 on B .So i* radius 00 on K To sine of sun's longitude 04 ' 54 on It 01). The latitude nnd docdination given to find the sun's amplitude, or the distance in degrees Iho snn is from the east or west at its rising or setting. Ex. — (iivon the latitude 40 N. and the declination 22 30' N. : reiiuired the sun's amplitude at rising, Bii.K. — As cosine hit. 40-' 40 on F Is to sine decl. 22 ' 80' 23 on I! So is radius 00 on V To sine ainplitudc 29'^ 50' nearly 20.8 on B 70. To find the time of the sun's rising and setting on a given day in any latitude. A''(c. — If the declinalion is not given, find it by Art. 07. Ex. 1. — Uequired the time of the sun's rising and selling in latitude 50- N., declination being 23' 88' N. As radius CO ou B Is to tang. hit. W 71 .2 on F •So is tang. decl. 23° 28' 20 on B The nsceiisional difrerenco ronverled into lime (allowing l.V to hour and I )o t minules (d' tiim^), |{i\es the lime that the sun rises before, or sets itfier, (i'(4iH'k in snminor, and the reverse in winlgr, in niirlli liitiliide. The above usceli- sioiml dilI'eroiie(( 31 ', converted Into lime, gives 2 bourn 4 ininnlos, which being added to o'clock, gives the time of the nun's selling H hours I minules. and being siiblriioted friHii (»■( lock gives .3 hours 50 minules ; therefore the sun sets at 4 minutes past H mid rises at 50 niinules past 3, Ex. 2. — Keipiii'od Iho lime of the sun's rising iu lut. 4U'N., the deeliiiatiou being 15'^ N. As r. ding fiO on F Is to tang. hit. 40^ flO.2 ou B bo is tang. decl. \!i 10.2 on V To sine ascensional difference 13" 13.0 on H 1.3 degrees eonverlod into lime gives 52 minutes, which, being siiblriicleil Ironi o'idock, gi\i's 5 hours 8 minutes ; hiuice the sun rises at H minules past 5 o'clock, 71. To find the length of the longest day in any latitude under 00" 32'. The hingest day will happen when Iho sun is in Ihc sol- stice, at which liiiio lhi> declination is 23 2H', Ex, — Itoijuired the longest day iu hililude 58^, As radius CO nn B Is III laiig. hit. .'J8'' !)5 on F So is Ian. of decl. 23" 28' 20. on B To sine ascensional difference 31" = 31.1 onF To sine nsceiisional difference 43' = 41 on F 43 degrees converted into lime is oipial to 2 hours 52 min- ules, and this addi d to o'chick (Art. 70) gives the time of the sun's Hotting M hours 52 minules, which, being doubled, gives till! length of the day 17 hours II miniilos. 72. To find the length of the longest day in any latitude above Ol!" 32'. Ex. — What is the leiiclh of the loii'.'esi day at tln^ North Cape, in the Island of Alaygeroc, in laliludc 71 30' N.'i" 1{| 1,1;. — Sol the index to lat. 71 3(f on (piadrant, and on the pei'peiidicular ol .'JO taken on A will bo loiiiid the semi- tan^^ont of the latitnih' H!).l. Then take HIM on index and sol it to the parallel of half radius 30 on B, and (ipposil(^ 00 ("inc of ascensional difl'er- cnce for hours) on index will bo foiiiiil on II 20.2, the lan- gont of dccliiialioii ou tln^ day on which llie sun ceases |o set in the given lalitiide. ,Scl the iinh^x lo 20,2 on lliu line of tjiigenis, and the doi linalion will iippeiir on the arc lo be 18 35', mid its sine will be found on side'B to bo IIM. Sot 2 1 (sine of gioalost declinalion 2.'1 28') on index lo 11). I (siiii? of aforesaid decl.) on I!, and on the arc of the (luadrant will iippoar the sun's hiiiiriliido when it censes to s t 51' ;!5'. Subtract 51" ;!5' from DO , and the remainder 38'' 25' doubled gives 70 ' 50', which, being taken iu time, is equal to 70 days 20 hours. The operation may bo more easily uiidoi'stood by the fol- low iiig pro|i(irlions : — As semi-tangent lat. 71" 30' AH.l on index Is to ball radius SO on B So is sine ascensioual diff. for hours. . 00 on index To tangent deid. when Ihe .«uii ceases (0 sii in the given latitude IH" 35' 20.2 on B The sine ol 18 35' is eipial lo IH.I Then, as siiK^ of greatest decl. 2.! 28'. 21 ou F Is Id sine ol above decl. 18 35 11). 1 on B So is radius 00 on F Sine of sun's long. 51' 35' 47 -|- ou B I into tinio (iillciwing ), |{i\i<)4 the* tirno ttiitt iirk in Niiiiiiiu-r, iind 'I'llC llllOVU UHI'CII- IMC, ^ivi'M 2 lioiirit 4 k, ^ivi'H llio tiiiiu of ml lirin;.' Hillitriu'lod i ; llirri'l'dru tliu Him iniiiMirM piiMt il. I'iHtllg ill lut. lU'N., «10 on F fio.l' oil H \t>.-2 on K la.lioiiH Hi iiiiiiiitPR, wliicli, Ti liiiiirH H miiuilos ; /(•lork. I liny ill liny latiliiilo 10 niiii is) ill liic Hul- 2H'. Hull' M^. CO nn B n.'. nil F :;M i|iiii>lraiit, and on l)u luiiiul llio <*vnii- llic parallel of liiilf >!' asrcMsional difl'er- I on II -M.-J, llu! taii- ihi' sun cuaMes lo set I \HK'2 on lliu line uf ar on tliu arc to be i-'H to Lo ll».l. "•\ 2H') on indox to il on llio arc of the wlii'ii it cua»c9 to 9 t 1 the I'eiuuinder 38" akcii ill time, is cquu) iderslood by the fol- ftil.l on index . . 'M) on n 00 on index (o . . -JO.a on U . . I'.M i'. 21 ouF 11). 1 on 11 . . 00 on F .. 17-foun / 6'- / / / /; />■ / ,,Vi./ 8 / S /,s> -t' i- \ '«l ^-*.; hr N rj;(,.>rt. , the hut lire the •ec- (cr- ««• mil on liii- rni- 6. ! I or it in he h0 on A m. ' Il fi- B 30 »1 a sp. ie an it, )g le re li. er 19 e- « le f /)/-- \ \ r' »^ T^-inn / K.I «,' '■^s^ ^ ^ -■?■- K F / r / /1 -7 „/ .:v^ / fi.iii'nu Q / / / / / 10 II .T,,-,y • JS / / / V 16 n /)i//'L,ii 17 \ \ \ • \ \ J Ar \l * .4 jT^il \ ,K 'Ihi-fl'f /,nl ^\ \ \, \ ■^ 2^ ■■<: »i \ / "? ./ .';^ '^ >"/ /f.-iii'nu / 5 % _;./ _>, A" / / i A2-, A' B ■N. / II z •'".^'^^ .13 — i >? '-■ \ \ '^' /^--. / : ^ A •:j,T '"■^ /.itK .^ /V a«-j ; SU ■' ^f I) ^ ^v B /'.•/ ■: fi '. ^: S vy l-l- .r ■7/. // i/LfiUt 'v- -s \ ■^ •f.-?. II :>V 95' be 8iil doubled ns above, tbe to be 70 day." 20 lioiii 73. To (iiul tlie Hill 1st. — When the br lln.i:.— Set l-> on leilgtb ill feet on I wi E.V. — Mow iiiiiiiY s long and '.) iiiclien wit Het \-> on V to 9 o F will .'"how 18 feet o AV. 2. — I{e in. I 12onF:8onl!: *uired. 3 Xote. — When the ^ions arc performed 1 4 E.C. — Ueiiuired tli >|uches the side of tin 4 12 ill. on K ; (1 in. ■ i 12 on F : (i in. on :ae.]iiired. * Er. 2. — Reiiiiired |nd 1.") iuelies the sii! 12 on .\ : !•') on 1' 12 on .\ : 1 j on 1' Kipiired. 7l'i. Hound and la] fJDg the rules in any I Ij Ex. — llow many s lud tbe '/ui 12 iiu'lie The rule is, to eon (le s(|uare. Applie ws : — •1 or 40 on A : 12 s. on F. 12 ou F : lO.ii on KEY TO CIIISIIOLM'S MATHEMATICAL MECHANICAL SCALE. II Ar 35' be subtrnctod from 90, iiml tlic remainder doubled ns above, tbe leiiglli of the longest day will be found to be 70 days 20 lioura. PART III. MKXSl.'UATION, 7,'J. To fnul tlic Huperfu'ial contents of n board or iiUink. Ist. — AVIion llie lireadlli is less tluin 12 indies. Rlii;, Set 12 on I to breadth in inclics on A or 15, then length in feet on I wiU show content in fe('t on A or H. £j.._IIi,«- many siiunre feet are there in a board 21 feet long and '.) inclies wide ? Ket 12 on V U> on 15, or 120 on V to 90 on B, then 2t on F will show 18 feet on B. iV. 2.— Ke on H == answer. /;,,.. ;). — In a plank 7 ft. in. long and 8 in. broad, how innuy square feet ? 12 ft. on B. 12 on F : 8 on 15 : ;'>7..") on F : : 2.") on 15 : the content rc- luired. Xote. — When the timber or stone is siiuare, both opera- lions arc performed by one move of the index. Ex. — Ucipiired tlie content ot a log 72 feet long, and (i iclies the side of the square. 12 ill. on F : (1 in. on 15 : : 72 ft. on F : : ■')(! ft. on B. 12 on F : (! in. on 15 : : 'M on F : : 18 on 15 : the content luired. K.r. -i. — Hciinired the solid content of a tree 18 feet long, ilid 1.') inches the side of the sipiare. ■ 12 on .V : I.') on F : 18 on A : : 321 or 2.'i..j on F. 12 on .\ : IJ on F : 22.0 on A : : 28.1-(-on F : content teqiiired. 70. Hoimd and tapering timber can bo measured by npply- jng the rules in any text book to tin' scale. i E.t. — IIow many solid leet in a iiiund tree .10 feet long, lud the girt 12 inches ? ■ The rule is, to con^•ider (-.narler ol the girt as the side (if <(Hnire. A|i|ilicd to the scale the oiieratiou is as liil- ws : — 4 or 40 on A : 12 on F : 10 on A : : lO'r, U)..') or 10 in. s. on F. 12 ou F : lO.ii on A : 'M on 20 ft. 3 in, on A. 12 on F : 10. .5 on A : 20 ft. .'3 in. on F : : 22 ft. 11 in. + or 2.'1 ft. nearly on A, vliieh is the content retpdred. A shorter rule is to assume the diameter as if it were the side of the square — say the diameter is 1 5 inches and log 20 feet long. Then, as 12 ou A is to 1.5 on F, so is 20 on A to 2."! on F, and (without a move) so is 20 on A to 31 on F. Then, as 100 on F to 7H,')4 ou A, so is 31 on F to 24.4 ou A : the rc- •piired solidity iu feet. 77. To find the area of a parallelograin ; whether it bo a square, a rectangle, a rhombus, or a rhoniboifl. Kii.i;. — MiiUijily the length by the perpendicular height, according to the directions given lor .Multiplication. Ex. 1. — Keipiired the area of a s(pmre whose side is 8 feet inches. As 100 on F : 8 ft. 6 in. or 8.5 on B : : 8.5 on F : : 72.25 or 72 ft. 3 in., the area required ou B. Ex. 2. — Kcciuired the area of a rhombus, whoso length is 12, and breadth or height 0.5. 100 on F : 0.5 ou A : : 12 on F : : 78 on A. Answer. 78. To find the area of a triangle, when the base and per- pendicular are given. Kt I.K. — Set half the base on F to 10 on A, then perpen- dicular on A will show area on F. Ex. I. — Keiinired the area of a triangle, whose base is CO and perpendicular height 20. \» 30 on F : 10 ou A or 15 : ; 20 on A or B : : 000 on F =- the area. Or, set the base on F to 20 on A or B, then perpendicular ou A or 15 will show area on F. Ex. 2. — Ucipiired the arei of a triangle, whose base is 80 and perpendicular height 0. As 80 on F : 20 on B : : on B : : 210 on F == area re- ipiired. In some cases the operation can be performed the more readily by taking the base on A or 15 to 20 on F, then oppo- site the perpendicular on F is the area on A or B. Ex. .'5. — What is the area ol a triangle, whose base is 120 and height 40 ? As 20 on F : 120 on A or B ; : 40 on F : : 210 on A or B = the area required. 79. (iiven any two sides of a right-angled triangle, to liiid the third side. CASH 1.— When the base and perpendicular are given, to find the hypothelinse. 1{,-, i-.J-JIovo the index so that the same point or number on F will at the same time be opposite one ol the sides on A, and opposite the other side on B, then the said number ou F is the hypothenuse reipiired. Ex. i. — In a right-angled triangle the base is 42, and the perpendicular 50 ; what is the length ol the hypothenuse ? Move the index until the working edge is at the point of intersection of the lines I'rom 50 on A and 42 on B, which shows 70 ou F = the length id' the liypotheuusc. C.VSK II. — When the liypolheniise and one of the sides are given, to find the remaining siiie. l{ii,,,:. — Set livpothenuse on F to tlie given side on A, then hypothenuse on F will show the remaining side on B. " Or, set hypotheiiiise on F to the given side on B, then hy- pothenuse on F will show the remaining side on A. Ex. — The hypolhenuse of a right-angled triangle is 53, and the base 45 : reipiired the perpendicular. As 53 ou F : 45 on A : : 5;5 on F : : 28 ou 15. Or, as 53 on F : 45 on B : : 53 on F : : 28 on X = the length of the perpendicular. 80. To lind the area ol a trapezium, the diagonal and the two perpendiculars being given. Kll.K. — Set IIH) on F lo the sum ol the iierpendicnlars on A or 15, then opposite half the diagouiil ou F is the reipiired area on A or 15. /.;,■. — Uequired the area of a trapezium, whose diagonal is 00, the perpeudieulars being ;itl and 44 respectively. As IfiO on F : 80 on A or B : : 30 on F : : 2400 on A or B = the .equired area. The area ol a trapezoid can be determined in nearly the same manner, the only variation in the operation being that the sum ot the parallel sides and half the perpendicular are used, instead of the sum of the perpendiculars and half the diagonal, as in the preceding article. The area of a regular polygon can be found by the dircc- liinis given for triangles, that is when the side aud the per- peudicular drawn to it from the centre are given ; for a reg- ular jiolygon can always be divided into as many equal triangles as it has sides. 81. To find the circumference of a circle, when the diam- eter is given. Hn.i;.— Set 100 on F to 3.1410 on A or B ; or, set 70 on F to 22 on A or 15, tlien opposite diameter on F is circum- ference on A or B. Ex. — What is the circumference of a circle, whose diam- eter is 8 ? As 7 on F ; 22 on B : : 8 on F : : 25.13 on B = circum- ference. Or, as 100 on F : 3.1410 on B : : 8 on F : : 25.13 on B. Another method : — As 100 on F : : diameter on A or B : : 3.1416 ou F : : circumference on A or 15. Or. as diameter on F : 100 ou A or B : : 3.1410 on A or 15 : circumference ou F. Niitv. — The diameter of a circle, whose circumference is given, may be found by reversing the operation described in either ot ilie preceding methods. 8!l. To find the area of a idrde. 1st. — When the diameter is given. l{ii.K. — Set 100 on F to 7854 or 78i on A or B, then the sipiare of the diameter ou F will show the are^ en A or B. Ex. — What is the are of a circle, whose di iineter is 9 ? As 100 ou F ; 7854 on A : 81 on F : C.30 -\- on A the area. 2d. — When the circumference is given. Bit.K.— Set 100 on F to .07958 or 79 0-10 on A or B, then the square of the circumference ou F will show the area on A or 15. Ex. — Ueipiired the area of a circle, whose circumference is 8. As 100 ou F : .07958 ou B ; : 04 on F : : 5. on B = area. 90. To find the area of a regular polygon, when only a side is given. Bti.i;. — Set the index to half the angle at the centre, con- tained by the two equal sides of any one of the equal tri- angles into which the jiolygon can be divided ; then 25 on B traced to the index, and thence to A, will show a iinautity on A, which, if multiplied by the number of sides the polygon contains, will give a constant mnlliplier. The product ot the sipiare of the side and this multiplier is the area of the poly- gon. Half the angle at the centre is always determined by dividing 180 degrees by the number of sides. Thus, for a nonagon it is 20 , tor an octagon 22 F, fiir a hexagon 30", &c. Ex. — Keipiircd the area of a regular pentagon, whose side is 10. Here, evidently, half the angle at the centre is 3l'i'. Then set the working edge of the index to 30° on the quadrant, and 25 on 15 traced to F will cut .344 -\- on A, wliiidi, being mnltiplied by (the number of sides) 5, gives 1.720 -|- 'be conslani multiplier for pentagons, ('onscqiiently the square of the side or 100 X 1.720 + = 172. + the area required. In computing the areas of regular polygons, the learner can also find the constant multipliers on the scale by means ol cotangents ; but this [iroperly belongs to Trigonometry, aud requires no explanation here. The method already de- scribed will be tbund to answer all purposes without having recourse to any other, so that the learner can at any time form a tabic ol multipliers for polygons in the space ot a few ./■ KEY TO CmsnOLM'S MATHEMATICAL MECnANICAL SCALE. tlty !»1. To find the side of a polygon, to contain a given ([uan- liuLK —Find tho multiplier for the regular polygon by tlie last article. Set 100 on F to the multiplier on A or li, then ODposite the area on A or B is the ncpiare of tlie xide on 1' . Ex.— What IB the side of a regular noiiagou, whose area is 395 feet '! , , ,. i o The multiplier for nonagons will he found to be ().IS + . Then, as 100 on F : 0.18 on B : : 395 on ]$ : : 04 on !• : the square root of 04 = 8 feet the length of liie side rc.iuired. 92 The area of a otrcle given to find tiie diameter. KULE.— Set 100 on F to .78.-)4 on \ or U, then opposite the area on A or B is the square of the diameter on F. _ £,c ^Vllllt is the diameter of a circle, whose area is oH.i) . As'lOO on F : 7854 on B : : 38.5 on B : : 49 on F : the squiiro root of 49 = 7 = the diameter. When tlie ciicumfereiice is required it may be detoriiiinea lu the same manner, using .07958 instead of .7854 ; or it may be found from the diameter. In the example given the circumter- eiioe may be thus found : — ,, , , , ti .i As 100 on F : .07958 on B : : 38.;) on B : ; 484 on F : the square root of 484 = ii2, which is tlie circumference. 93. To find the area of a sector of a circle, the chord and di- ameter being given. BuLE.— Find the area of the circle by Art. 89. Set the di- ameter on F to the chord on B, or set the radius on I" to halt the chonl on B. and the index will show half the number of de- grees in the sector on the ((Uadrant. Then the area of the sector can be determined by the toilow- ing proportion : — . , As 180 : area of circle : : half the number of degrees m the sector :: area of .sector. , ■ io i Ex. What is the area of a sector, whose diameter is IS, ana the chord of whcse arc is V . „ . .c r^ The area of the circle by Art. 89 is 2.)4. Setting 18 on F to G on B, or 9 on F to 3 on B, the index cuts 19' 4.i on the quadrant, which is half the number of degrees in tlie sector. Then, as 180 on B : 254 on F : : 19^ cm B : : 2, .u on !• = the area of the sector. 94. To find the area of a segment of a circle, the chord and diameter being given. . , , ■ i i Hui.K. — Find the area of tho sector as in tli^ la.st article, ami the area of the triangle as in Art. 78, and liie sum ov difference of these areas, aeeording as the segment is less or greater than a semi circle, shall be tiie area of the segment. Examples are unnecessary. 11.1 In calculating the area of a sector of a circle, when the chora and versed sine arc given, the diameter is easily found by divid- ing the .sum of the square of half the chord and of the versed sine by the vei-sed sine. OF SOLIDS. 95. To find tho solid content of a cube. Rui.E.— Set 100 on F to the side on A or B ; opjiosite the side on F is a certain quantity on A or B ; and opposite tins last quantity on F is the .solid content on .V or B. Or, imiltiiily the given side by it.self, and that product again l)y the side. ., . rvo A>. — Whiit is the solidity of a cube, whose side is 9 ( By setting H'O on F to 90 on A or B, 90 on F shows 81 on A or B, 81 on F shows 729 on A or B, = the solidity required. The content of a parallelopiped is found on the scale by mul- tiplying the length by the breadth, and that product by the alti- tude. I 96. To find the solidity of a cylinder. llLLE.— Find the area'of the bast by Art. 89. Jlultiiily the I area of the ba.so by the perpendicular heigiit of the cylinder. i jiV, What is the solidity of a cylinder, whose diameter is 3 and height 8 inches '! . . „ x • Tho area of the ba.sc (Art, 89, mensuration of surfiices) is 28 2 As 100 on F : 28.2 on B : : 8 on F : : 225 on B = solidity required. 97. To find the convex surface of a sphere. r,;lk.— Set 100 on F to the diameter on A or B, then oppo- site 3.1410 on F is a certain quantity on A or B, and opposite this (luantily on F is the convex surface of the sphere on A or B. ,. AV. AVhiit is tho convex surface of a sphere, whoso dianio- ter is 9 inches 1 Set 100 on F to 9 on A or B. 3.1410 on F is opimsile 28.2 + on A B. 28.2 on F is opposite 254 + on A or B, which is the con- vex siirfiici? required. Xole.—lu this, as in many co.«es, it may be sometimes more convent to have tlie quotient on the index, 98. To find the solidity of a sphere. H, ,,K._Tlie cube of tho diameter niultiiilicd by .5230 will be the solidity. ,.,-., Ej- —What is tho solidity of a sphere, whoso diameter is 2 inches 't 2 -1 = 8, and 8 X -5^30 = 4.1 88S. ( )n the scale the operation is performed by the direetiuus given for iiiuUiplieatiim. 99. To find tlie convex surface of a right cone. l{L;i,K._Set 100 on V to the ciicumfereiice of the base on A or B, ami the slant height on F will show double the convex surface on A or B. , , , . 1 , />._Wliat is the convex surface of a cone whose slant heiglit is 5 and the eireumference of whose base is 9.42. As 100 on F : 9.42 on B : : 5 on F : : 47 on B .. 4( : - == 23.5 the convex surface. 100. To find the sididity of a cone or pyramid. R,;,,K._M,iltiply tiie area of the base by the altitude, anil oiic-tiiird of tlie product will bo the .solidity. />,_\Viuit is tlie solidity of a cone, tho diameter of whoso base is 2 and altitude 50 feetV 2 X - X .7854 = 3.1410 = area of the base. ^.1410 X 50 = 52,03 -1- = solidity. 3 On the si'alo the operation is performed thus : — As 100 on F : .50 on B : : 3.1410 on F : : 1.57 -1- on B . 100 on V : 3 on B : : 157 + on F : : 52,03 on B = .solidity required. of tlie siiliere on A splierc, wlioso dinnie- r 15, wlikli is the coii- tf 111! i-oiiiotiiiK'H more )licd by .5230 will be ^ whoiso diameter is 2 % ly tlio dircctiuus given 1 it cone. nee of the base on A ow double tbc convex one wliose sUmt lieigbt is 0.42. s : : 47 on 15 .-. 47 : 2 " lyraniid. by the altitude , anil 1 the diameter of wlioso the base. I thus : — n F : : 157 + on 15 2.03 on 15 = .solidity TABLE OF LATITUDES AND LONGITUDES. Nnnie of I'laef. Coast of Great Britain and adjacent Islands. London, St. I'mil's, var. 24' 0'. 1H24 (ireenwieh Observatory Dual Ciistle l)over Castle l)un};cne.''s Light I'ortsniiiuth ( 'hurch Needles Liglit. Isle of Wight Weymouth Kddvstone Liglit Plymouth. New Church Ohl Church Kidniouth, St. Anthony's Head Land's Knd Stone Cape (\)rnwall Bristol Cathedral Holyhead Signal Tower Liverpool, St. Paul's W. and N, Coasts of Scotland, (ireenoek (ihisgow l»unnet Head 8tronmess. ( )rkiiey Islands E. Coast of Scotland, Inverness Dundee Kdiuburgh, Coll Lat. N.Lon.W.. Name of Place. Lat. N.Lon.W. Name of Place. E. Coast of England, 15erwi<'k Tyneniouth Light Y;inuoiuli W, Coast of Ireland. Cape (!lear Light l'.i.e Catoebe 8 51 Vera Cruz Mexico 53 17 54 10 54 59 55 14 20 20 31 West India Islands. llernmda, St. (Joorge's Town Kingston, Jam Port Royal l'«i'>t 54 :!(> 54 15, 53 21 ; 53 23 ' Cuba. 5 30 , 21 ! t ' irn de (^uba . . . ■ 4 i >'.iiva ";aH Ca,stle 20 Havana Light l55 58,07 12 .. .' 34 53 5() 10 27 51 48 41 22 54 43 15 10 18 39 1 12 55 38 30 5 28 35 17 . . .1 1 28 48 30 1 N. 20 57 11 1 7 32 58 49 ' 1 ILat. N. 10 27 04 15 10 13 ti4 48 .... 12 11 70 8 10 2ti 75 37 9 34 79 43 ... 17 29 8S 11 . . . 21 34 • 80 .>7 19 12 I 9(i 7 19 25 1 99 5 32 22 04 37 17 49 7