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I" I |""v<»^aifwpi|i^ii^ppMimp)pp(i^H|Bpi|M|||a|B 
 
MONOFORMULA 
 
 TKXT-IU)()K 
 
 OK 
 
 
 EOMETRY 
 
 FOfiLOWKI) UV TKKATISKS 
 
 nh' 
 
 PLANE AND SPHERICAL TRIIIONOMETRY 
 
 AVI) 
 
 AlTKi )XL\[AT1:: MEN.SIIUATK 
 
 I'.V 
 
 J. L. SEGUIN, A. M. 
 
 -* * •*•- 
 
 ^\ 
 
 Montreal 
 
 >KAl' AND DUMB INSTITUTION PRINT. OFFICK, 
 »i:i]i.E:-H:]sro, r=». ca. 
 1893 
 
ppi* 
 
 ■ ■I ^ll^.. I — „ ^., !>. > ■*■■ 
 
 n^-^^^^^pi 
 
 Sf 
 
 KntcrcMl uccording to Act of Parliament of Canada in the year one 
 thousand eight hundred and ninety three, by J. L. Sbgdin, at the Depart- 
 ment of Agriculture, 
 
—jWiiWIliii . i,i,'wpi'|ii^iii . v 
 
 ',-r' » m i ni »"lr*i 
 
 •mvufmw^*^" 
 
 ERRATA. 
 
 Page 92, Prob. 12, last line, instead of «', read s. 
 
 Page 97, Art. 8, instead of -fcxA^2, read ±2^^. 
 Page 165, 4th line, instead of ^^^yz (&c.), read if — ^^z 
 Page 167 Prob. 26, instead of 
 
 (Ac.). 
 
 1" ^ =S( -^^'^c); 
 
 I760n- 
 
 2- F= -s— , 
 
 read 
 
 1" V= 
 
 TTX' 
 
 2^(3x-&c) ; 
 
 2' F= 
 
 3 
 
 790r 
 
 8 
 
 Page 226, art. 139, 9th line, instead of x=p, read a =.-/>. 
 
 Page 239, Prob. 4, instead of mn read imn, and converseJy. 
 
 In fig. of page 242. the great circle ZCN' must pass through 
 
 P instead of Z. 
 
 Page 265, line before last, instead of Art 21, read App. D, II. 
 
 Page 266, 14th line, instead of Art. 27, read Art. 25. 
 
 Page 282, Note, 5th line, instead of Art 26, read Art. 22 ; 
 
 7th line, instead of Art. 25 (proceeding as in Art. 22), 
 
 read : Art. 21 (proceeding as in Art. 23); 
 
 10th line, instead of Art. 26 (proceeding as in Art. 23), 
 
 read : Art. 22 (proceeding as in Art. 24). 
 
 Page 283, Prob. 2, instead of (Art. 5), read (Art. 9). 
 
 Page 294, II, Uth i<k 17th line, instead of Art. 20, 
 
 read : paragraph I. 
 
 Table V, page 65. column of Pages, 7th, 19th and 20th line, 
 
 instead of 273, Art 27, read : 269, Art. 25 
 
 Page 227, Art 1 42, for the line beKiniiinj? with or (.34, /), substitute : 
 
 but, letting B^C=p and A = q^ in {I) of Art. 34, we have : 
 
 co8(fi+ C) -f cos^^2 cos^(.4+ «+ (7)cos^(i?-f C— ^) ; 
 
 he 
 
 nee. 
 
 sin 
 
 '■ho^ 
 
 - 2co8^(.4 4-i5 + C)qo&\{B-\- C-A) 
 
 sin/? sin (7 
 
 (t) 
 
SySi .'..UJM 
 
 
 r 
 
 pi 
 
 lu 
 
 3 
 
 y 
 
 pre 
 doc 
 
..^.....^NMaHNMi 
 
 PREFACE. 
 
 •• The iiifinitesimal method applied throughout mathematics is light in- 
 troduced into the mass, speed substituted for slowness. Then I do not, 
 for a .single moment, doubt but that the problem of mathematical teach- 
 ing lies principally in this point." 
 
 >■ The .'^pced, the clearness, the fecundity of mathematical teaching may 
 be doubled, more than doubled, by the decided introduction of the infini- 
 tesimal method." 
 
 ( f RATKY. 
 
 (Jratry is right. — The infinitesiDial method is a powerful 
 instrument in the uiatheinatical sciences in general, and it is 
 especially so well adapted to the study of geometry that it 
 enables us to sum up, in a single formula, the entire part of 
 this science which has for its object the mon.'^uration oi' plane 
 or curved surfaces, and of volumes. 
 
 In fact, this formula may serve for mea.»uring : 
 
 1" exuctlff, the area of all re<rtilinear s\irfaces, and like- 
 wise of any curvilinear surface whose curve is {'ontained in the 
 equation y = ax' -}- bx' -}- ex + / (Book III, Art. 54 ) ; 
 
 2'> approximately, the area of any curvilinear surface 
 whose curve is not contained in the preceding equation (Ap- 
 proximate Mensuration, Art. 10) ; 
 
 3 > t^artly, the volume of any solid generated by the revo- 
 lution of a plane whose curve is contained in the equation 
 f » ojt^ 4- hx -+ ex -j- / (Book III, Art. GO) ; 
 
 4" exactly, the volume of any solid which bears, to the 
 preceding solid of revolution, the same relation as a pyramid 
 does to a cone (Book III, Art. GO, Cor. 2) ; 
 
wmr 
 
 3SBI 
 
 PKKKArK. 
 
 r>" npproximiitcf//, thi* vo]uIlle^4 (•(' all regular Holidh not 
 conipriHed in the a}»ove clasHes IJ" and 4- ( A|>pn>xiniHtf> .Men- 
 suration. Art. 18 and Cor.) ; 
 
 i» ' i'X(ict/t/, tin' <!onvex surrace of any solid of* revolution^ 
 when the formula of rectification «»f' its generatinj; curve (solved 
 with respect to its ordinate//) is contained in the equation 
 // » </.r' -I- />-.- -\- /•;. + ./■ ( Book TFT. .\rt. 58); 
 
 7" exartfi/. tlie convex surface of any solid which bears, 
 to the preceding solid of revolution, the same relation as a 
 pyramid does to its inscribed cone (Book TTT. Art. 58 Cor. 2) ; 
 
 8" approximately, the surfaces of all regular solids not 
 comprised in the classes ♦»' and 7' ( Api)roxiniate Mensuration, 
 Art. 19). 
 
 We perceive, by the preceding: tableau, that oftentimes 
 the formula «;ives only an opproxinuition : but. as it will be 
 ascertained in the course of this treatise, this approximation 
 is practically equivalent to mathematical accurac}- (App. Men- 
 ,su ration. Art. 20). 
 
 It must be then acknowledged that this formula sums up, 
 with respect to mensuration, the whole plane and solid geome- 
 try, and ihat it thereby fully justifies the name of motinformuhi 
 which v.t' have iriven it. 
 
 This formula is nt» less Vhinutcrizetl by its simplicity than 
 by its extension. In fact, in its application to the nieasuring 
 of the area (.4) of a plane tijrnre. of the volume ( V) and of 
 the convex surface (S) of a suHd. its algebraic expressions are 
 
 V = '• (/f -f- 4 8+ />)*, 
 S = »• (/y + 4 /?'+ //). 
 
 • Any one who has read the StereometricoD of Sir Knight Haillarge will 
 recognize, in ecmation \" = |* (/i 1--*^''+ ft), the prismuidul tbrmiila that 
 tho Sir Knight applies to the cubing of the solids of the Island 2nd order . 
 
I'RKKAOE, 5 
 
 That i« to Hay, t«» ohtaiii tin- incaHuriMiieiit of the^e varioiirt 
 jreoiiietrieul iiia^^nitudt's (.1. I*. .S*). it suffices to J«l>|»ly hut one 
 and the sumo 
 
 FoRMILA. 
 
 •' To the parallel huM's [H ami h). add four times the median 
 section (/^'), parallel to the hases. and multiply the sum hy 
 th«* sixth part of the altitude (//)." 
 
 The application of the apprnximnte mtmotornntla. to aiialy- 
 tieal problems, may sometimes uecessitate lenirthy romputa- 
 tiouH, if we wish to obtain an approximation e«|uivailent to 
 mathematical aecura( y. But the monoformula, like iill other 
 geometrical formulae, has not for its immediate obje<t the sol- 
 ving of analytical problems. Pass from this province ol' abs- 
 tractions to that of arts, mechanics, etc.. where we have to 
 measure surfaces and volumes assuming so great a variety of 
 forms, no longer mere abstractions, but real magnitudes. In 
 this domain of really practical pr(d)lems, contrast the several 
 formulae and conipare their respective merits. Here. f<u- any 
 form and in any hypothesis, the only data re<(uired by the mo- 
 noformula are lines capable of being mechanically mea.sured. 
 and the only operations to be perft)rmed on the.se data are the 
 first four operations of arithmetic. Hence, by means of the 
 monoformula, any surface and any volume may be measured 
 by any one who knows the first lour rules of arithmetic and 
 who can make use of a scale of measiure. 
 
 Let us now cast a glance on the Table of formuhe and on 
 the Appendix. These fornuihe so numerous ami oftentimes so 
 complex can defy the most tenacious memory. Yet, this sum 
 of formulae is but a trifle when compared to tlmt which it 
 would be necessary to possess in order to solve all the problems 
 
.. JUL-insfseKSfv^jMim vn^wwmtmmmHi 
 
 6 
 
 I'RKKArK. 
 
 I I 
 
 that are within the province of tliu iiionotorniulu. BesideM, it' 
 the nature of the^4e forniultu be conHidered, we will at onoe 
 recognise that, in general, they are not within the reach of the 
 .student in elementary geometry. Moreover, they requ::^;, in 
 many ca8e«. the use of htgarithmic or trigonometrical tublen, 
 and u laborious calculation. Still more, any reader familiiir 
 with higher analysiH know8 that it in eany to find the differen- 
 tial of a geometrical magnitude, but he is likewiHc aware that 
 every differential is not integrable. A great number of geome- 
 trical forms have no formulae, nor can they have any. Now, if 
 we had to calculate some one of these forms, whether a sur 
 face or a volume, what would we do ? Should we divide this 
 .surface into approximate trapezoids and apply the formula of 
 the trapezoid ? Should we cut the solid into approximate frus- 
 ta of a cone or of a pyramid and apply their respective formu- 
 lae ? A reply to these (juestions will be found in Approximate 
 Mensuration, art. 20. 
 
 Owing to the efficiency of the analytical method, a single theo- 
 rem (under three different forms : Book III." art. 54, 58, 60) is 
 sufficient to put us in possession : 
 
 1° Of the formulae relating to volumes, which synthetical 
 geometry can impart to us only at the expense of so many 
 theorems and with so much slowness (Book li' art. 61) ; 
 
 2° Of many special formulae far above the sphere of syn- 
 thetical geometry (Book III, art. 55, 59, 61); 
 
 3° Of the monoformula ,vith its marvelous extension and 
 its incomparable simplicity. 
 
 In what an ample measure then is here realized the word 
 of Gratry : " The infinitesimal method applied throughout 
 mathematics is light introduced into the mass, and speed subs- 
 tituted for slowness." 
 
IMtKKACK 
 
 Anybody, iit*tt»r haviiifr taken notieo of tW extension, tlie 
 Miniplicity and the innnenHe praetieal advantages of the mono 
 forinuhi, will he convinced that it forces itself in practice and 
 thereby in its theory. But, tliis theory wiiich serves to prove the 
 exactness and the precise extension of the formula re<|uires the 
 use of the analytical method. The author is then amply justified 
 for having introduced analysis in the last Book of this treatise. 
 
 Let not the reader he discouraged at hearing that the infi- 
 nitesimal method is applied in this treatise. Infinitesimal cail- 
 culus, in its high spheres, is without doubt a science of diffi- 
 cult attainment ; but here, there is no (juestion ; treading 
 through the intricate path of higher analysis : f iiere is need 
 of only a few principles altogethir elemeiiti..v and fn'!\ as 
 easy to gn .s those of synthetic geometry. Hesiiles, if its 
 iiota*''^n be overlooked, this method i.n nothing new, even in 
 elementary geometry. Compare the two demonstrations of th<' 
 theorem in art. 12(), Book I, and 51. Book 111. and it wiP be 
 easily ascertained that the two methods, analytical and .syn- 
 thetical, applied in this case, as it may be done in many other 
 instances, differ only in their notation. 
 
 The monoformula, even when it is merely approximate, can 
 render immense services, since it may be applied whenever tl»e 
 special formula could not be made use of, either on account of 
 its not being known or of its necessitating too laborious a cal- 
 culation. In these hypotheses, the monoformula becomes a 
 convenient and precious substitute, as, by its use, a result prac- 
 tically equivalent to mathematical exactitude can be obtained. 
 
 But, let us hasten to say that it would not be rational to 
 apply the approximate monoformula to problems as convenien- 
 tly solved by exact formulae which are easy to memorize and 
 employ. 
 
« I'llKK.ArK. 
 
 W«* tlu'ii pi'oteHt, ill uiiti<-i))atioii, apiiii^t tho iiu]»itatioii of 
 aiming to subHtitnte. seaHoiiably and unseasonably, a t'orniula 
 freqiUMitly but approxiniat«\ for exact forinuho in ir«Muiietry, 
 the science exact pnr t'rrrfft'urr. 
 
 From the stand )H>int of the exact science, let us i'ei>eat. 
 with Hoileau : 
 
 " Rien n'est bean <jue le vrai. le vrai seul est ainiable' . 
 
 And to arrive at the counizance of truth in its fulness, to 
 otttain the niatheniatically exact f'onnnla, let us not shrink 
 from th«» labors ini|»osed u|Mm us by study. 
 
 T ackngwledge iny indebtedness to ]*rofessor S. Saindon for 
 SI collection of jtroblenis and liis excellent suujiestions in tlie 
 distribution of the matter in this work. 
 
 J. L. Seguin. 
 Montreal, February 189:>. 
 
 r 
 
 i. 
 
TO TlIK TKACIlKIl (F (IKO.MKTRY. 
 
 The Toju'lier of (jeonietiy is it's]>ect fully ie<juested to takt^ 
 the followinji; remarks in earnest consideration : 
 
 1° The part headed Appn>.rinuit< Mtusut'iitinn has been rle- 
 t ached from the treatise of ji;eometrv und postponed to the end 
 of the work, not indeed because it was rejrarded as an unim- 
 portant part of geometry or an element foreign to it, but 
 simply because this matter being outside the scope of common 
 geometry is not required by any programme of studies. 
 
 In the hypothesis that the pupil, for want of time, could 
 not niake a thorough study oi' this part, let liim at least make 
 so earnest a perusal of it as to ascertain the high approximation 
 of tlic uionoformula, whenever it is not nutfhcinttfiraf/i/ exact, 
 and know liow to jirocct'd in its practical applications. 
 
 2* Discarding, from this Treatise of (Jeometry, the Approx 
 imate Mensuration and the solutions of problems, which it is 
 not customary to give. wli;it will he left ? llanlly 120 pages of 
 theory and of practical applications. .\nd what will be the 
 re.sults arrived at by this theory so short, so elementary and 
 thereby accessible even to young pupils ? 
 
 (/. All the results, all the formuhe given by tlu» tnati.ses of 
 geometry in common use. 
 
 b. The formulae of the volumes of the : 
 prolate and oblate ellip.soids, 
 
 common paraboloids and hyf>erboloids, 
 
 tive diverging paraboloids, 
 
 frusta and segments of all tlu above sojjds. wIhi.m' variou** 
 
 forms are so frequently met with in arts, nieclianics, etc. 
 
 c. The formuh*; of the volumes of all polyedroids (segments 
 included) cirt. ims<'ribed abojit the preceding solids of rev(»- 
 lution. , 
 
I' 
 
 
 i 
 
 !• ! 
 
 10 
 
 1() THE TEACHER OP GEOMETRY. 
 
 </. The formulae of the areas (negmentK included) of the 
 ellipse, the commort parabola and the cubic parabola with its 
 multiplicity of forms. 
 
 e. The formulae of the convex .surfaces (segments included) 
 <>f all polyedroids whose curves are either a common or a cubic 
 parabola ; and likewi.se of all solids of revolution inscribed in 
 these polyedroids. 
 
 3^ Now. a theory wliich gives such results, whilst it spares 
 half the time devoted to the study (►f geometry, already affords 
 immense advantages. What then is the value of a theory which, 
 in addition to that, has the incomparably greater advantage of 
 giving us a monoformula which, though most simple, yet in- 
 cludes, in its wonderful extension, the special formulae of all 
 the above mentioned geometrical magnitudes {a to c included), 
 except three, viz : the area of the circle, the area of the ellipse 
 and the surface of the sphere (ra^, rah, 4Tra^) ; three for- 
 mulae both simple and easy to memorize. 
 
INTRODUCTION 
 
 (^KNERAL DEFINITION'S 
 
 1, Space is divisible int<» definite portions whifh are so 
 many definite extensions. 
 
 !<J, A point is that which has position, but no extension. 
 
 3. A line is extension in length, without breadth or thick- 
 ness. 
 
 Lines are straight or nirvcJ. 
 
 4. A straight line is one in which all the points lie in the 
 same direction. 
 
 5. A curved line is one whose direction changes at every 
 
 point. 
 
 The words line, curve, used alone, mean straight line, curved line, res- 
 pectively. 
 
 6. A surface is extension in length and breadth, without 
 thickness. 
 
 Surfaces are plane or curved. 
 
 7« A plane surface or a plane is a surface such that a 
 straight line joining any two of its points lies wholly in tlie 
 surface. 
 
 8. A curved surftuse is a surface no part of which is a 
 plane. 
 
 9. A solid, volume, or capacity is extension in Nnigth. 
 breadth and thickness. ^>^ 
 
 10. Parallel lines and parallel planes are those which 
 have the same direction, and thereby cannot meet however far 
 they may be produced. 
 
 * Thickness is also called allilnde, height, or depth ; the choice of the 
 word IS determined by usage. 
 
S^iBi«IIMM«uBMBitiiiipMW 
 
 1 
 
 12 
 
 INTIU)1H< TIOX. 
 
 i ! 
 
 11. An angle is tlu^ anuiiuit of divt'ijiciift' of two Yuwh, or 
 of two or iii(n<» plant's, wliieli meet or ti'inl to niot't at u cuiu- 
 nion poinf. 
 
 1.*^. A quantity i.s any tiling that can he incrt'a.sed. dimi- 
 nished and measured. 
 
 Then! are four specie:) of geometriciU quantities or niagiiitml s. viz : 
 iiiie. aur/iiff. t'o/uina ami ungle. 
 
 l«i. The operations to Ik> performed on jieometrieal majini- 
 tudes and the relations tliey hear to one another are expressed 
 hy ofyrhra.tr. st/mhoh. 
 
 14-. Equal magnitudes are those which, when applied 
 the one to the otlier. will coincide in all their parts. 
 
 15. Similar magnitudes are those wliich have the 
 same fm-m but not the same extent. 
 
 l(>t Equivalent magnitudes are those which liave the 
 sarjie extent, hut different forms. 
 
 17. To measure a quantity consists of finding hf)W 
 many tinu's it cojitains a (juantity of the same kind taken as a 
 unit of measure. 
 
 IS, As the linear unit is an arbitrary measure in leugtii 
 (foot, inch, meter, decimeter, etc.). 
 
 So the unit of surface is an arbitrary stjuare (s(|uare foot. 
 s<|uare incli, square meter, etc.), 
 
 And tlu' unit of volume is an arbitrary cu))e. (cubic foot, 
 cubic ijich, cubic nutter, etc.). 
 
 ID, The area of a jdane or convex surface is the ((uantity 
 of this surface expressed in terms of its unit of measure. 
 
 20. Solidity is the (juantity of a solid expressed in terms 
 of its unit of measure. 
 
 21. The length, breadth, and thickness of a geometrical 
 magnitude are its dimensions. 
 
 *ri*^. Geometry is the science whicli has for its object the 
 properties, relations and measurement of geometrical magni- 
 tudes. 
 
 *^ll. Plane geometry or geometry of two dimensions, 
 is that whi<'!» treats of plane surfaces. 
 
 I I 
 
(iKNKRAI, l»KKlXlTIONS. 
 
 13 
 
 indiny; how 
 
 *44:» Solid geometry <ir geometry of three dimensions, 
 
 is that which trentH <»f convex surfaces and voiuuies. 
 
 ^5. An axiom i.s a self-evident proposition. 
 
 !26* A theorem is a proposition to be denionstnited. 
 
 A theorem consists of two parts : the hjntntheniH or tliat 
 which is assumed, and the iisHertion or that winch is ass':rted 
 to follow from the' hypothesis. 
 
 !3T. The demonstration of a theorem is the c«»urse of 
 reasonin«r by which the truth of the assertion becomes evi- 
 dent. 
 
 2iH» A problem is a <(ue8tit»n to be solved! 
 
 39. A postulate is a self-evident problem. 
 
 30* A lemma is an auxiliary proposition. 
 
 •$1. A corollary is an obvious conse((uence of otie or more 
 propositions. 
 
 33* A scholium is a remark made on one or more propo- 
 sitions to point out their relations, use or extension. 
 
 Mi. 
 
 Axioms. 
 
 eonietrical 
 
 1. The whole is greater than any of its ]»urts. 
 
 2. The whole is eijual to the sum of a.l ii parts. 
 
 3. Things equal t<» the same thing jire e(jual to one aji- 
 other. 
 
 4. If equals be added to. or subtracted from, etjuals, the 
 results are equal. 
 
 5. If equals be multiplied or divided by e(|uals, the results 
 are equal. 
 
 ti. From one point to anotlier. but o»ie straight line <aii be 
 drawn. 
 
 7. A straight line is the shortest distance between two 
 points. 
 
 aensions, 
 
:■ 
 
 14 
 
 INTRODlirTION. 
 
 34, 
 
 Postulates. 
 
 1. A straiglit line can be tlrawn connecting any two points. 
 
 2. A 8traij>ht line can be prolonged indefinitely. 
 
 3. An angle, as well as a line, can be bisected. 
 
 AinmEVIATIONS. 
 
 ' 
 
 i ! 
 
 Ax. 
 
 for 
 
 axiom, 
 
 Ms. 
 
 for 
 
 measure. 
 
 Th. 
 
 a 
 
 theorem, 
 
 Int. 
 
 ii 
 
 introduction, 
 
 Prob. 
 
 n 
 
 problem, 
 
 Trig. 
 
 '• 
 
 trigonometry. 
 
 nyi». 
 
 u 
 
 hypothesis, 
 
 Supp. 
 
 " 
 
 supplement, 
 
 Ast. 
 
 a 
 
 assertion, 
 
 App. 
 
 fi 
 
 appendix. 
 
 Deni. 
 
 a 
 
 demonstration, 
 
 Fig. 
 
 u 
 
 figure. 
 
 Cor. 
 
 a 
 
 corollary, 
 
 L 
 
 •' 
 
 angle, 
 
 Sob. 
 
 u 
 
 Scholium, 
 
 A 
 
 a 
 
 triangle 
 
 Art. 
 
 u 
 
 article, 
 
 L R. 
 
 a 
 
 right angle, 
 
 Def. 
 
 u 
 
 definition, 
 
 L Adj. 
 
 it 
 
 adjacent angles. 
 
 Const. 
 
 u 
 
 construction, 
 
 L Opp 
 
 ii 
 
 opposite angles. 
 
 l_ L Corp. for corresponding angles formed by two parallels 
 and a secant. « 
 
 l_ LAI. int. for alternate-interior angles between parallels. 
 Q. E. 1). I'or which was to bo proved. 
 
 ' To abbreviate and simplify the demonstration of theorems, 
 an angle will often be represented by a single letter placed bet- 
 ween its sides and near its vertex ; also, frequent use will be 
 made of expressions similar to the following : • 
 
 !<• L AOD -f L BOD = 2L i2; 2o L a + L 6 = L m+L n; 
 P>'>Lrt = L6,asLopp ; 4" La-|-L ft ( = L c +L <i) = 2Ui2,&c ; 
 which expressions are read : 
 
 l'> The sum of the angles AOD and BOD is equal to two 
 right angles. 
 
 2" The sum of the angles a and b is equal to the sum of the 
 angles m and n. 
 
 3' The angles a and h are equal, . s opposite angles. 
 
UENERAL UEFIN1TION8, 
 
 15 
 
 40 The sum of the angles' a and b, which is equal to the 
 sum of the angles c and d, is thereby equal to two right angles. 
 
 A reference like, I, 47, 2, 3° is read : Book I, article 47, 
 corollary 2, 3°. 
 
 For a reference to an article in the same Book, no mention 
 is made of that Book. 
 
J 
 
 t 
 
 \l 
 
 
 • 
 
 
 
 
 
 
 
 
 i 
 
 IU)OK I. 
 
 PLANE GEOMETRY. 
 
 I >E FIX IT ION'S. 
 
 I, A plane figure is :i \t\nuv hounded hy (dtlun- Mtvaijrlit 
 1)1- curved linos. 
 
 !S, A roctilinear plane figure or polygon is a ]>lanr 
 fiiiurc )>onnded V»y straight lines. ealU*d siiffs. 
 
 3« Tlic perimiter is the broken line whi<-h hounds tin- 
 }M)lygon. 
 
 4. The vertices of a polygon are the pcnnts in wliieh the 
 sides meet. 
 
 .Tl, A diagonal of a polygon is any straight line joinin<> 
 tw<» vertices not eonseeutive. 
 
 (J, The base is the side on which the polygon is suj>po.sed 
 to rest. 
 
 7. The altitude of a polygon is the perpendicular dis- 
 tance between 'the ]>ase and the op[»osit«> side or angle. 
 
 8. Mutually e<niianiiular i)olvu<<iis are those whose anules 
 Sire e(|ual.each to each. 
 
 O. The sides similarly jtluced in these polygons are called 
 correspinHfiug auft's. 
 
 lO. Equal polygons are those which are mutually 
 equiangular and whose corresponding sides are equal. • 
 
 II. Similar polygons are those which are mutually 
 equiangular and whose corresponding sides are proportional. 
 
 155. The parts (sides, diagonals, angles, &e. ) similarly placed 
 in similar ))olygo:is are called homologous parts. 
 
 The name ot a i<o .vg >ii i-'< derived from the number of its sides or angles; thu:< : 
 
 13. A triangle is a polygon composed of three angles. 
 
 14. A quadrilateral is a polygon composed of f<nir sides. 
 
 15. A polygon is called pentagon, hexagon, hoptagon, 
 octagon, nonagon, decagon, undecagon, dodecagon, &c, accord- 
 ing as it has T), (1, 7, H, 9. 10, 11, 12, &c sides respectively. 
 
 10. lu general we say ; a polygon of 11, 12, &c sides. 
 
ARTICLE 26. 
 
 IT 
 
 PLANE ANdLES. 
 
 DEFINITIONS. 
 
 4, A plane angle or simply an angle is 
 the amount of divergence of two iitraiglit 
 lines which meet, or tend to meet, at a com- 
 mon point. 
 
 l^ 
 
 IH. The two lines (A/iand AC) are called 
 
 the sides, and the point (.4) in which the sides meet is called 
 
 the vertex, of the angle. 
 
 All angle is designated by naming its vertex. If many angles have 
 their vertices at the same point, they are respectively designated by three 
 letters, the middle one indicating the vertex and the others the two 
 sides. 
 
 19* Adjacent angles are two angles 
 formed by a straight line meeting another 
 straight line ; v. g. ACD and DCS. 
 
 20» A right angle is formed by a 
 straight line meeting another straight line 
 so as to make equal adjacent angles ; v. g. 
 ACD and DCB ' ^ 
 
 c 
 
 o 
 
 31. The former {CD) is then said to be perpendicular to 
 the latter {AB). 
 
 22» An oblique angle is one which is not a right angle. 
 
 Oblique angles are acute or obtuse angles. 
 
 33* An acute angle is one which is less than a right 
 angle. 
 
 S4* An obtuse angle is one which is greater than a right 
 angle. 
 
 !35. Complementary angles are those whose sum is equal 
 to a right angle. 
 
 26. Supplementary angles are those whose sum is equal 
 to two right angles. 
 
18 
 
 BOOK I. 
 
 H 
 
 ^7* Opposite angles am those whiuh lie 
 oil the opposite sides of two lines intersecting 
 each other ; v. g. « and c, or h and </. 
 
 Two straight lines and their secant ( line hy 
 which they are cut) form eight angles which 
 assume the following names : 
 
 !<28. Alternate-interior angles; that is, 
 two angles not adjacent, lying on opposite 
 sides of the secant and between the other 
 two lines ; v. g. «• and «, or d and m. 
 
 *29* Alternate-exterior angles ; that is, two angles not 
 adjacent, lying on opposite sides of the secant and without 
 ' the other two lines ; v. g. a and />, or h and o. 
 
 30. Interior angles on the same side ; that is, two angles 
 lying on the same side of the secant, and between the other 
 two lines ; v. g, d and «, or c and m. 
 
 li\. Corresponding angles; that is, two angles not adja- 
 cent, lying on the same side of the secant, the one within and 
 the other without the other two lines ; v. g. b and a, or d and 
 
 
 TllKORK.M. 
 
 The sum of adjacent nngh's is equal to 
 two right angles. 
 
 Hyp. Let the line CO meet the line 
 AB At the point O; ^ 
 
 AST. t\ienyf'i]\\-A0C-\-\-C0B='2i.K 
 
 ])EM. Assume BO to be perpendicular to AB. 
 
 Considering the sum of the angles AOl) and DOB as a 
 whole, the angles AOV and COB are the parts of this whole ; 
 hence, l.AOD-\-[^DOB=[^AOC+\-COB. Ax. 2 
 
 But, L.A0D-\-L.D0B^2\^R; Def. 20 
 
 therefore, L.A0C+1-C0B=='Z\„H. ' Q. E. D. 
 
ARTICLE 34. 
 
 19 
 
 Cor* 1*^ The sum of all the consecutive 
 angles (a -\- b -\- c), on the same side of a 
 tine {AB) is equal to two right angles. 
 
 2p The sunt of all the angles about the 
 same point {0) is equal to four right 
 angles. 
 
 For, l'> LaH-Lt+l-c(=:LvlOC+L6'05)=:2L/2; 
 
 likewise, Lw-|-L»-|-Lo =2L^; 
 
 therefore, 2° L.a+Lfe+ Lc -f L w + L?t + Lo =4L/2. 
 
 33* Theorem. 
 
 When two lines intersect each other, the opposite angles are 
 I'qual. 
 
 Hyp. Let the line AB intersect the line CD ; 
 
 AsT. then will 
 
 Dem. For (32), 
 
 likewise, 
 
 hence, 
 
 therefore, 
 
 For a similar reason, 
 
 34. 
 
 La=Lc,and i-b=sL.d, 
 La+L6=2Li2; 
 L6 + Lc=2L/f; 
 La+Lfc=LZ.H-Lc; 
 LasssLc. 
 
 Theorem. 
 
 ax. 2 
 
 ax. 4 
 
 Q. E. A 
 
 If a secant intersect two parallel lines, the sum of the inte- 
 rior angles on the same side is equal to two right angles ; con," 
 versely^ two lines are parallel, if the sum, of the interior angles 
 on the same side is equal to two right angles. 
 
 lo Hyp. Let ab and mn be two parallel 
 lines intersected by the secant am ; 
 
 AsT. then will Lrf+L«= 21. R. 
 
 Dem. Since, both lines ab and mn are pa- 
 rallel, their secant am has the same inclina- 
 tion to both of them ; hence, the sides of 
 both angles b and n have the same amount of divergence ; 
 
 that is, L6=L«/ 
 
 but, L6+Lti=2L/2; 32 
 
 therefore, by substitution, L.n-\-L.dss2L.R. Q. £. D» 
 
1^^ 
 
 { 
 
 9H 
 
 20 
 
 BOOK I. 
 
 2° Hyp. Let the Hecant am intersect the lines ah and mn 
 §0 that Lff+Ln^ 2L/e; 
 
 AsT. then vrill ab and mn be parallel. 
 
 Dbm. For, L<i-|-Ln-.2L7?. iy Ay;>- 
 
 and, L6-I-LJ-1 2 L/2; 32 
 
 hence, Lft + LrfaEaLd+Lw; 
 
 whence, L6=*Lw. 
 
 But, since these angles h and n are equal, their sides must 
 have the same amount uf divergence ; that is, both lines ah 
 and mn must have the same inclination to the secant am, and 
 thereby be parallel. 
 
 Cor.' 1. If the lines ab and nm are parallel : 
 1° Hie alternate-interior angles are equal, 
 2° The alternate-exterior angles are equal, 
 3° The corre9nonding angles are equal. 
 
 Cor. 2. Two lines ab and mn are parallel : 
 1° If the alternate-interior angles are equal, 
 2° If the alternate-exterior angles are equal, 
 3" //* the corresponding angles are equal. 
 
 35. 
 
 Theorem. 
 
 Ttoo angles are equal, if their sides he parellel and lie in 
 the same direction. 
 
 Hyp. Let5^/'and2)Ci;betwo 
 angles whose sides AB and CD, as 
 well as AF and CE, are parallel 
 and lie in the same direction ; ^ 
 
 AsT. then will La=Lc. 
 
 Dem. Prolong EC till it meets AB; 
 
 then, La=Li, 
 
 and Lc=L<6; 
 
 therefore, La=Lc. 
 
 Cor. Two angles are equal, if their sides are parallel and 
 in opposite directions. 
 
 For, angle a', formed by the prolongations of -4.6 and AF, 
 
 as LLoorp. 
 
 as LLcorp. 
 
 Q. E. D. 
 
ARTirLK 41 
 
 21 
 
 and angle c both have their »iiles parallel and l^iug iii oppouite 
 directions ; 
 
 besides, L»«'=b:L«, as Lojpp. 
 
 and Lc ass La; ht/ l)tm* 
 
 therefore, La'» Lc 
 
 TlilANilLES. 
 
 DEFINITIONS. 
 
 36. A triangle is a polygon fonipoaed of three angles and 
 three sides. 
 
 The triangle c?a««//i« J with respect (1°) to its sides and 
 (2°) to its angles . ..mprises the following classes and species : 
 
 H 
 
 CO 
 OQ 
 
 scalene triangle. 
 
 isosceles triangle, a varieti/ of which is 
 the equilateral triangle. 
 
 2° ^ 
 
 u 
 
 oblique-angled x , acute-angled triangle, do 
 gg I triangle ; S ) obtuse-angled 
 
 ^ ) triangle. 
 
 
 ^^ \ right-angled triangle. 
 
 37* A scalene triangle is one in which nil the sides are 
 unequal. 
 
 38. An isosceles triangle is one which has two equal sides. 
 
 39« An equilateral triangle is one in which all the sides 
 are equal. 
 
 40* An oblique-angled triangle is one in which the three 
 angles are oblique. 
 
 41* An acute-angled triangle is one iu which the three 
 angles are acute. 
 
w 
 
 ? I 
 
 \ 
 
 :r 
 • il 
 
 'n 
 
 111 
 
 iii 
 
 22 BOOK I. 
 
 4!3« An obtuse-angled triangle \» one which containH un 
 obtuse angle. 
 
 43. A right-angled triangle is one which contains a right 
 angle. 
 
 The wide opposite t,» the right angle of aright-angled triangle 
 is called hf/pothenusi. 
 
 44. 
 
 Theorem. 
 
 Two triangles are equal when two sides and the included 
 angle of the one are respectively equal to two sides and the in- 
 cluded angle of the other. 
 
 Hyp. Let ABC and ahc be two triangles, 
 in which Lvl=L a, AB=ab, and AC—ac; 
 
 AsT. then will ^ABC=^£\ahc. 
 
 Dem. Place A ABC on A «^><^ so that L^ 
 shall fall on Lf/, AB o\\ ah and AC on ac ; 
 then, since the vertices B and (7 will fall on h 
 and c respectively, /JC will coincide with he; therefore, the 
 two triangles will coincide in all their parts ; 
 
 that is, /^ABC= A^hc. Q. E. J). 
 
 45. 
 
 Theorem. 
 
 Two triangles are equal when two angles and the included 
 side of the one are respectively equal to two angles and the in- 
 cluded side of the other. 
 
 Hyp. Let ABC &nd ahc be two triangles (preceding figure) 
 in which L.l =La, L5 = L.b and AB s= ab ; 
 
 AsT. the Si will /\ABC= Aabc. 
 
 Dem. Place ^ ABC on Aahc so that L^ shall fall on La, 
 L5 on L?i, and AB on ah ; then will AC take the same direc- 
 tion as ac, and BC the same direction as be; hence, ^6* will 
 coincide with «c, BC with be and LC with L.c; therefore, 
 the two triangles will coincide in all their parts ; 
 
 that is, aABC = Aftbc. Q. E. D. 
 
ARTrrLK 40. 
 
 23 
 
 46. Thkork.m. 
 
 If. /rem a point without a xt might fixe, a pcrpevdirvhir he 
 let fall on that line., and two oh/iqne lines he drairn to that 
 same line, at pointa eqiiidistanf front the f(tot of the perpendi- 
 evlar : 
 
 1" the perpendirnlai' nHl he shorter than the oblique linen, 
 
 2 ' the tvo ohlique linen uill he equal. 
 
 Hyp. From the point ^, let a perpendi- 
 cular CD be drawn to the line A B. and 
 also the oblique lines CA and CB t<» the 
 points A and B equidistant from the foot 
 D of the perpendicular ; 
 
 AST. then will \CI)<CA,'1^^CA = CB. 
 
 Dem. On CD prolonged, lay off DF=D(K and draw AF. 
 
 1" By construction. \^ADC=\-ADF, asL. R. 
 
 and DC=DF; 
 
 besides, the side AD is common to both these right angles ; 
 
 hence, ^ADC=^^ADF. 44 
 
 and thereby, AC=^AF. 
 
 But, the straight line CF is shorter than the broken line 
 CAF ; therefore, the straight line 6*/), the one half of CF, is 
 shorter than CA, the one half of CAF. Ax. 7 
 
 2° By const., \^ADC=z\^BDC, r^L/?. 
 
 and AD=DB: 
 
 besides, the side CD is commo.i to both these right angles ; 
 
 hence, £sADC==£^BDC ; 
 
 therefore, AC=BC Q.E.D- 
 
 Cor. 1. The perpendicular is the shortest distance from a 
 point to a line. 
 
 Cot. 2. If a perpendic\dar CD he let fall n a line AB, at 
 its middle D, any point of that perpendicular is equidistant 
 from the extremities of that line. 
 
 Cor. 3. A line C F is perptendicula r to another line AB. at its 
 middle D, if each of two points (C and F, or C and D, or 
 etc.) of the former is equidistant from the extremities A and B 
 of the latter. 
 
t 
 
 I mi 
 
 I 1 
 
 Hi 
 
 i' \ 
 
 iiii • 
 
 :, I 
 
 24 
 
 WH)K I. 
 
 Theorem. 
 
 47» 3Vo triangles are equal when the sides of the one an 
 equal to the sides of the other ^ each to each. 
 
 Hyp. Let the sides of the triangles ABC 
 and ahc (iBg. of art. 44) be equal, each to 
 each ; 
 
 AsT. then will ^ABC—£\ahc. 
 
 Dem. Place ahc under ABC so that AB 
 and ah shall coincide, as shown in the annexed 
 figure, and draw Cc. 
 
 Since he sides AC and (JB are respectively equal to the 
 sides Ac and c7i, each of the points ^1 and B of the line AB is 
 equidistant from the extremities Cand c of the line Cc; hence 
 AB is perpendicular to Cc which is bisected at the point m 
 (46, 3). 
 
 Now, if the triangle ABC hfi revolved about the axis ^iJ 
 go that it shall fall on ABc^ Cm will coincide with cm, the 
 vertex Cwith the vertex c, and the sides J^'and (7B with the 
 sides Ac and cB respectively ; therefore, the two triangles 
 will coincide in all their parts ; 
 
 that is, AABC=Aahc. Q.E.D. 
 
 Sch. In two equal triangles, the equal sides are opposite to 
 the equal angles, and conversely. 
 
 48. Theorem. 
 
 In an isosrrIcK triangle, the angles opposite to the equal 
 aides are equal. 
 
 Hyp, Let ABC be an isosceles triangle, in 
 which AC=CB; 
 
 AsT. then will L^l = L/i. 
 
 Dem. Assume the angle (^ to be divided 
 into two equal angles <t and c by the lino (^/) 
 (called bisect, ix). 
 
 Since, AC=CB, hi/ hyp. 
 
 and CD is common to both triangles ACD and BCD, 
 
 then, aACD=aBCD; 44 
 
 therefore, L.1=L^. Q. E. D. 
 
ARTICLK 49. 
 
 25 
 
 the one, are 
 
 Cor. 1. An equilateral triangle ix also equiangular. 
 
 Cor. 2. .t triangle which has two equal angles is isosceles. 
 
 Cor. 3. A perpendicular, drawn from the vertex (C) to the 
 base {AB) of an isosceles triangle, bisects both the vertical 
 angle and the base. 
 
 Cor. 4. The bisectrix (CD) is perpendicxdar to the base 
 and divides the isosceles triangle into two equal right-angled 
 triangles. 
 
 49. 
 
 TilEOREJI. 
 
 to the equal 
 
 The sum of the three angles of a triangle is equal to two 
 right angles. 
 
 Hyp. Let ABC be any triangle ; 
 
 AsT. then will L«-|-L_/>-}-Lc=:2LR. j/X^ 
 
 Dem. Prolong AC to /), and assume 
 CF to be parallel to AB. "^ ^ 
 
 By this const., Lo=:La', as LLcorp. 
 
 and L6=Lft'; as LLal.int. 
 
 hence. La +L6 4-Lc=Lo'-i-L6'4-Lc; (1) 
 
 but, La'-fL6'-|-Lc=2Li2; 32, mr. 
 
 therefore. La -|-L/> -|-l_c=2Li?. Q.E.D. 
 
 Cor. 1. The angle BCD exterior ^> the triangle ABC is 
 equal to the sum of the opposite-interior angles a and b. 
 
 For, if angle c be eliminated from equation (1), the remain- 
 ders are equal; therefore, La-j-L6 (=Lrt'-|-L6') = L56^/>. 
 
 Cor. 2. Any two angles of a triangle being given, the 
 remaining one can be found by subtracting the sum of the 
 given angles from two right angles. 
 
 Cor 3. All the angles of a triangle may be acute, hut a 
 triangle can have only one right or only one obtuse angle. 
 
 Cor. 4. In a right-angled triangle, the two acute angles are 
 compltmentary. 
 
IWMIIIll ■ 
 
 m smaam 
 
 26 
 
 BOOK I. 
 
 QUADRILATERALS. 
 
 DEFINITIO.VS. 
 
 50. A quadrilateral is a polygon composed of four sides. 
 
 i>l. The quadrilateral comprises the following classes and 
 species : 
 
 trapezium, 
 
 trapezoid, 
 
 / 
 
 CD 
 < 
 
 parallelogram ; 
 
 rectangle, 
 
 g ' stjuare, 
 
 rhomboid. 
 
 7 
 
 rhombus. 
 
 ii2* A trapezium is iKiuiulrilateral which has no parallel 
 sides. 
 
 53. A trapezoid is a quadrilateral iu which but two sides 
 are parallel. 
 
 54. A parallelogram is a quadrilateral iu which the oppo- 
 site sides are parallel. 
 
 iy5» A rectangle li a parallelogram in which the four 
 angles are right angles. 
 
 50. A square is a rectangle in which the four sides are 
 equal. 
 
ARTICLK GO. 
 
 27 
 
 ,57. A rhomboid is a paralloloirram in wliicli tho four 
 angles are oblique, 
 
 58. A rhombus is a rh(>ni)»ni(l in which the four sides are 
 (Hjual. 
 
 59. 
 
 TllK(HlKM. 
 
 Till' oppotiiti' sulf's of o piriuillcliHjrniii arc equal. 
 
 Hyp. Let AIU'D ))\ anv narallelo- 
 jiraui : ^ 
 
 AsT. then will AB=DCxn\A AD=B(J. 
 
 I)EM. The sides yli5 and y>>C, likewise ^, 
 AD and BC, being parallel (Def. 54). 
 
 the diagonal will divide the parallelogram so that L« = La', 
 and Lc=Lc', as LLrt?. in<. ; hence, the two triangles ABD 
 and BCD are mutually equiangular, and sinee tliey have a 
 common side BD^ they are equal (45) : 
 
 therefore. AB= DC and A /)= BC Q. E. D, 
 
 Cor. 1. Two parallels hetwccn two other paralleh are equal. 
 Cor. 2. The opposite angles of a parallelogram are equal. 
 Cer, 3, A parallelognnn is ditrided iufo t\ro equal triangles 
 hy each of its diagonals. 
 
 60. 
 
 Thkorem. 
 
 The t no diagonals of a parallelogram bisect each other. 
 
 Hyp. Let ABCD be any parallelogram jol , ^^-rj^ 
 
 whose diagonals are AC and BD ; / Z,^;*'' / 
 AST. then will .40= Orand/*(9=0/>. A ^'"*^ ^^ '4 
 
 Bem. For. L<f = L^<', and Lr=Lr' .• as l^i^al. inf. 
 
 also AB=CD, . 51). 
 
 hence, AAB0=. z^CDO ; . 45. 
 
 therefore, J[0= OC and /?0= OD. Q.K. I). 
 
mam 
 
 imSmm 
 
 
 II 
 
 
 ;1 I 
 
 ' I I 
 
 ir;=x ------ JiKi"^ 
 
 'I 1 
 
 ill 
 
 t f 
 
 i . 
 
 i f 
 
 I I 
 
 ! ! 
 
 2H \UHtK 1. 
 
 AKKAS AXI> PROPORTIONS IX POLY(;ONS.* 
 
 TlIKiUlK.M. 
 
 01. 77/« ff/'fjf/ i>f iiHif paniUflotjnnn is equal to the product 
 of its hose and tiltitudc. 
 
 Hyi>. Assume ACH1) to ht» iiiiv miniUplo- 
 
 I tf""~J 
 
 ! it—.-- / 
 
 and «M|ual to the base .1/^ of ABCl) ; and '^ 
 
 let //= altitude J)F. /i= base .1/^, .l = iuva of ABCD ; 
 
 AsT. then will Ar=JixII. 
 
 Dem. Sint'o the // i)aralleloirranis o are e(|ual, they have 
 equal altitudes (/«.). Besides, the a^ea .1 of ABCD is equal 
 to the sum (n -\-o-{-{i-r=ii X") <>f its parts; whence, the 
 equation . I = » x " • {I) 
 
 Likewise, the altitude // iseqvuil to tlie sum(/«,-|- /*.-}-(('•«;=», x h) 
 of its parts ; whence the equation H=nxh. (2) 
 
 If the altitude h be taken as a unit of linear measure, equa- 
 tion (2) reduces to //=» ; iuid // substituted for ?i, in (1), 
 gives A = IIxu. (3) 
 
 Now, this equation, A = IIx", is always true, whatever 
 may be the altitude of the parallelogram a ; hence, it remains 
 true when this altitude assumes my of the values through 
 which it passes in progressively decreasing till it is reduced to 
 a nnithematical point. But, at this limit (since the parallelo- 
 gram (t is reduced to a mere line which is its base B), a=sB ; 
 therefore, by substitution of Ji for o, in (8), there obtains 
 
 finally : 
 
 A=IIxB. Q.E.D. 
 
 Cor. 1. Two parallelograms of equal bases and equal alti- 
 tudes have equal areas. 
 
 Cor. 2. Two 2>nv(ill*''ograms which have equal bases are 
 proportional to their altitudes ; and two parallelograms which 
 have equal altitudes arc proportional to their bases. 
 
 • In tlu> demonstration of the following theorems, a constant applica- 
 tj,)u will be made of the theory of proportions (Aljrebra) which th^^ 
 student must review if iieoes.^arv. 
 
ARTIC'LK r»2. 
 
 20 
 
 a 
 
 h X h 
 
 
 A n 
 
 
 A^H 
 
 a h 
 
 For, 1. A=BxH, and ossshxh reprcHent the respective 
 arens of two parallelograms, and divide the first by the second . 
 
 then. A JixH 
 
 therefore, if H=h, 
 and if Ji=h. 
 
 ii2* Thkorkm. 
 
 Tico triangles tvhich have thr sfUne base and the same atti- 
 tude are equivalent. 
 
 Hyp. Let .i^Cand ABD be two 
 triangles which have the same base 
 and the same altitude ; 
 
 AsT. then will they be equiva- 
 lent. 
 
 Dem. Through the vertices C and D, draw FG which will 
 be parallel to Afi ; also draw A (7 parallel to B(\ and BF 
 parallel to AD. 
 
 Since the parallelograms ABCG and ABFI) have the same 
 base and the same altitude, they are equivalent (61. 1). 
 
 Besides, each one of them is bisected by its diagonal ; 
 
 ABFD ^ .J. J. 
 hence, — ^ = A ABJ), 
 
 and 
 
 2 
 ABCG 
 
 ^ABC; 
 
 but, ABFD is equivalent to ABCG ; 
 
 therefore, /\ABC\» equivalent to £\ABD. Q. E. D. 
 
 Cor. 1. A triangle is equal to one half of a par(dlelograni 
 lohich has the same base and the .'tame altitude as the triangle. 
 
 Cor. 2. // the area, base and altitude of a triangle be de- 
 noted hy A, B, and IT, respectively, the formula of its area loill 
 
m, 
 
 'i\ 
 
 If I 
 
 I 
 
 I 
 
 j 
 
 ! 
 
 
 ' 
 
 ' \ 
 
 
 i 
 
 
 
 
 
 
 ^ . : 
 
 
 
 30 
 
 BOOK I. 
 
 Cor. 3. Two trianglea of equal bases and equal altitudes 
 have equal areas ; two triangles v^hich have equal bases are 
 proportional to their altitudes, anct two triangles which have 
 equal altitudes are proportional to their bases. 
 
 . For, let the areas of two t ngles be respectively repre- 
 sented by A= — — — and a=__:,and divide the one by the 
 
 other ; 
 then, 
 
 a bxh 
 
 hence, 1", if iJ=fe, and /i=:/t, A=ria, 
 
 A U 
 ' ' ah 
 
 A B 
 ' ' " b 
 
 a. 
 
 Theorem. 
 
 G3. The area, of a trapezoid is equal to the product of its 
 altitude and one half the sum of its parallel bases. 
 
 Hyp. Assume ABCD to be any trapezoid. 
 
 Draw the diagonal AC, the perpendicular 
 CF; and let B^AB, lower base ; b=CD, S*- 
 upper base ; I1=^CF, altitude ; and xt=area, 
 o^ ABCD; 
 
 AsT. then will ^4= ~^ *xH. 
 
 Dem. By this notation, a ^i^^^= 
 
 likewise, 
 
 A^CZ>= 
 
 2 
 BxH\ 
 
 bxJf 
 
 but, the area of ABCD, A= /^ABC-^ ^ACD ; Ax. 2 
 
 therefore, A = + = _L_ x H. Q. K . D. 
 
 2 2 2 
 
 64. 
 
 Theorem. 
 
 In any triangle, a line drawn parallel to one of the sides 
 divides the other two sides into proportional parts. 
 
ARTICLE 05. 
 
 31 
 
 Hyp. Let ABC be a trian;:l«', in whicli DB 
 is drawn ]t:irallel to ^17^/ 
 
 AsT. tluu \<\\\Al).n('^BK:E<\ 
 
 Dkm. Draw AE vlwA BI). 
 
 Since, the triaagles ADE and CDE have 
 their bases in the same line .4.^, and their ver- 
 tices at the same point £*, they have a common altitude ; 
 hence, they are proportional to their bases (62, 3) ; 
 
 that is, aADE:^(WE=AD:DC; (1) 
 
 likewise, aBED:/^CBE=BE:EC. (2) 
 
 But, the triangles ADE and BED have a common base DE 
 and the same altitude (their vertices being in the line AB pa- 
 rallel to DE) ; 
 
 hence, £^ADE=aBED, 62,3 
 
 therefore, in proportions (1) and (2), the first couplets being 
 equal, the second couplets are also equal ; 
 
 that is, AD:DC=BE:EC. Q. E. D. 
 
 Cor. The proportional parts into which two sides of a 
 triangle are divided are <dso proportional to the whole sides of 
 which they are respective parts. 
 
 For, by composition, we have from the preceding propor- 
 tion : 
 
 AD-\-DC:AD=BE-^EC:BE ; 
 
 that is, AC: AD= BC : BE. 
 
 Likewise, AC:DC=BC: EC. 
 
 65. 
 
 Theorem. 
 
 Anfj straight line dividing two sides of a, triangle into 
 proportional parts is parallel to the third side. 
 
 Hyp. Let ABC be any triangle in which the line DE divi- 
 des AC and BC so that AD:DC=BE:EC; 
 AsT. then will DE be parallel to AB. 
 
 Dem. Draw AE and BD ; then, the perpendicular distance 
 from E to AC is the altitude common to both triangles ADE 
 liudCDE; 
 
1 
 
 ii 
 
 ^^SSS^mS^^ShS^S. 
 
 I r ; ' 
 
 32 B(K>K I. 
 
 and th(neby{(>2,H),A^l/>A': A^//»;=.l/>: W; 
 
 likewi«e. ^nE1)'.^CI)E==fiE',EC; 
 
 but, b.vhyp, AD: DC = HE. EC ; ^ 
 
 hence. £\AI)E:A('l)E= /^HED: /\('I)E ; 
 
 that is, AADE=ABEJ). ^ 
 
 But these equal triangles have a uummon base DE ; hence, 
 their altitudes are equal, and thereby the two vertices yl and B 
 are equidistant from DE prolouired : 
 
 therefore. AH is parallel to DE. Q. E. D. 
 
 66. 
 
 Tfikokkm. 
 
 Two triangles are sinn'lar, if their corresponding ttideg are 
 proportionul. 
 
 Hyp. Let ABC and abc be two 
 triangles in which 
 
 AB:ab = BClbc=AC;ac; S^ »r 4i^ 
 
 AsT. then will these two triangles 
 be similar. 
 
 Dem. On AB mid AC, lay off Ah' and Ac equal to nh and 
 ac, respectively ; and draw h'c' , 
 
 By this const. AB\Ah' z= iVcxAc ; 
 
 hence. h'c is parallel to BC, (i5 
 
 and thereby. ^ABC is similar to A^'>'«' ; 
 
 consequently. AB:BC=Ab';h'c' ; 
 
 but, by hyp. AB\BC=ab\bc ; 
 
 whence. Ab''.h'c'=z ab'.bc. 
 
 Now, since the antecedents Ab' and ab are equal, the con- 
 sequents b'c' and be are also equal ; hence, all the sides of abc 
 and Ab'c' are equal, each to each. 
 
 that is, A<fb<-=AAb'c'. 
 
 But, AABC is similar to AAb'c ; 
 
 therefore, AABC is also similar to Aabc. Q. E. D. 
 
 6-7. 
 
 Theorem. 
 
 Two triangles which have an angle in each equal, and the 
 including sides proportional ar^e similar. 
 
ARTICI.R 68. 
 
 33 
 
 Hvr. Let ABC and ubc be two 
 triangles in which 
 
 L-4=L_tt, Hiid AB'.AC^ahiac; ^' 
 
 AsT. then will these two triangles _ 
 
 l)e similar. 
 
 I)EM. On AB and Ai\ layoff .1// and ^c' equal to at and ttc 
 respectively, and draw />'«•' ; 
 
 then, since Li4=La, 
 
 will /^aln-=^Ah'r. 44 
 
 By hyp. ^15Mr'= <</>:</»-, 
 
 b'c' is parallel to iJ6'; 65 
 
 {^ABC is .similar to/\Ah'c'. 
 AAf/v' = Aahc; 
 A ABC is also sixuilar to A"/"". <?• A'- D. 
 
 or 
 
 hence, 
 
 and thereby. 
 
 But. 
 
 therefore, 
 
 ' eoual to ah and 
 
 08. 
 
 Theorem. 
 
 Two mntualhj iqn'mngnlar triangles are similar. 
 Hyp. Let ABC and ahc be two triangles in which 
 L.4 = Lrx,L5=L6, andL_6'=Lc; 
 AsT. then will these two triangles be similar. 
 Dem. On A P- and AC, lay off Ah' and Ac' equal to «/> and 
 ttc respectively, and draw h'c' ; 
 
 then, AAb'c' = Aahc; 44 
 
 and thereby, L.Ab'c'z=z\_b ; 
 
 Li?=L6; hyhyj). 
 
 \^Ah'c' = L.B, 
 
 h'c' is parallel to 7J6^• 34, 2, 3 • 
 
 .45:.46'=^6Vl<'=^C'://c', 64 
 
 AB\ah=^AC\ac=BC\hc. 
 
 Therefore, if two triangles are mutually equiangular, their 
 
 corresponding sides are proportional, and thereby these 
 
 triangles are similar (11). Q. E. D. 
 
 but, 
 
 hence, 
 
 and thereby, 
 
 consequently, 
 
 or 
 
 8 
 
f! ifa.i.:N ' Buii 
 
 'M 
 
 BOOK I. 
 
 lii 
 
 I.I I ii 
 
 1 
 
 lit 
 
 Cor. ^Tico triatigles are ftunllar, if two angJex of the one 
 are equal to two aiigteH of the other, eaeh to each. 
 
 09. 
 
 TlIKOIlKM. 
 
 Two triangles are similar, if their sides are parallel, each 
 to eaeh. 
 
 Hyp. Let ABC and ahche two 
 triangles in which A Bund ah, AC 
 and ac, BC and 4c are parallel ; 
 
 AsT. then will these triangles 
 be ^(iinilar. 
 
 Dem. Since, by hypothesis, all the sides uf these triangles 
 are parallel, each to each, all the angles are equal, each to 
 eaeh (47). 
 
 Therefore, these triangles are mutually equiangular and 
 thereby similar. 
 
 70. Theorem. 
 
 Two triangles are similar, if their sides arc jt^^'P^ndicidar, 
 each to each. 
 
 Hyp. Let ABC and ahc be two 
 triangles in which ah is perpendicular to 
 AB, ac to AC, and he to BC ; 
 
 AsT. then will these two triangles be 
 similar. 
 
 Dkm. Prolong ba, ar, and rb till they meet AB, AC and BC, 
 respectively ; also join the points b and B by the line bB. 
 
 Then, the triangles Bub and Bob urtj right-angled at n 
 and o, respectively ; 
 
 whence, L Bhn -[■\_bBii=aL.R, 49, 4 
 
 and l-obB-\-[^oBb=al-R. 
 
 Adding, V^obn -\-\^oBh = '21-K. 
 
 But, L.obii -{-L.abr =2i_/?; as Ladj. 
 
 whence, L.obii -{'L.<>BH = L.obn-\-L.abc, 
 
 or • L_o/?M = L<f^o. 
 
 Likewise, L.tiiAo=[-bac, 
 
 and L.inCit = [_arh. 
 
AliTU'LK 71. 
 
 3<J 
 
 Tlu'ioforc, both triangles aro mutually equiangular and 
 there))}- .similar. Q. K. D. 
 
 Sch. In the preceding liypotliesis, we have considered hut 
 one ease, that in which the triangle abc lies within the triangle 
 ABC ; nevertheles.s, the demonstration is general. 
 
 For, whatever may be the relative position ol'these triangles, 
 it will always be possible to draw, within the greater, a 
 triangle who.se sides are parallel to the sides ot* the smaller, 
 t ach to each. 
 
 Cor. 1. In the triangles ABC and abc, any tiro sufen per- 
 Ijendiculct, to each other are honioloyons. 
 
 Cor. 2. Two angles whose sides arc pcrpendicnlar, each to 
 each, are equal. 
 
 71. 
 
 TlIKORKM. 
 
 In any triangle^ the line bisecting an angle die ida the oppo- 
 site side into segments proportional to the other two sides. 
 
 Hyp. Let ABC be a triangle, in ^E 
 
 which the angle 6' is divided, by CV, ^ •«?/ 
 
 into two equal angles a and b ; 
 
 AST. then will ^lZ>:Z>i^=^6";(7i?. 
 
 Dem. Draw BE parallel to DC, and 
 prolong AC till it meets BE. 
 
 By this con.st. A I):I)B=. A C:CE, 
 
 and 
 
 U 
 by hyp. 
 as LL. al. int. 
 as LL Corp. 
 
 hence, La = Li=Lc=Lc?, 
 
 and thereby, BC and CE are etiual as sides opposite to the 
 equal angles c and d. 
 
 Now, substituting i^Cfov its equal CE., in (1), there obtains : 
 
 A D : /)7i= .1 C : CB. Q. E. D. 
 
Ill; 
 
 mm 
 
 ii' 
 
 :llii 
 
 'i 
 
 I I.) 
 
 ! 
 
 f ■! 
 
 f_ 
 
 ': ! 
 
 ' i 
 
 / 7: 
 
 1 
 
 ; 1 
 
 
 • 
 
 1 
 
 i 
 
 36 
 
 73. 
 
 BOOK I. 
 
 Theorem. 
 
 If, from the i ertex of the right angle of a right-angled 
 triangle, a perpendicular be let fall on the hi/pofhenuse : 
 
 1" the perpendicular divides the given triangle into two 
 similar triangles; 
 
 2^ the perpendicular is a mean proportional between the 
 two segments of the hypothenuse ; 
 
 3'' each side of the right angle is a mean proportional bet- 
 ween the hypothenuse and the adjacent segments ; 
 
 4'' the segments of the hypothenuse are proportional to the 
 squares of the adjacent sides ; 
 
 50 the square of the hypothenuse is equal to the sum 
 of the squares of the other two sides. 
 
 Hyp. Assume ABC to be a right-angled 
 triangle ; and from the vertex C of the 
 right angle, let fall the perpendicular CD 
 on the hypothenuse AB ; 
 
 1" AsT. then will £^ACD be similar to ^BCD. 
 
 Dem. The triangles ABC and ACD are right-angled at C 
 and D, respectively, and they have a common angle A ; 
 
 hence, £^ACD is similar to aABC. 68 Cor. 
 
 Likewise, ^BCD is similar to aABC ; 
 
 therefore, ^ACD is similar to aBCD. 
 
 2^ AsT. then will CD^=ADxDB. 
 
 Dem. In the similar triangles ACD and DCB, 
 AD:CD=^CD:DB, or CD'^ADxDB. 
 3'> AST. thenyf'iW AC^^^ABx AD, and BC'=ABxBD. 
 Dem. In the similar triangles ABC and ACD, 
 AB:AC=AC:AD, or AC'-=ABxAD. 
 Likewise, in the similar triangles vl 56^ and BCD, 
 AB:BC=^BC:BV, or BC'=ABxDB, 
 AC^^AD 
 BC' BD' 
 
 (1) 
 (2) 
 
 40 AsT. then will 
 
ARTICLE 73. 
 
 87 
 
 t-angled 
 
 I * 
 
 nto two 
 ween the 
 onal let- 
 ,al to the 
 the sum 
 
 C 
 
 igled at G 
 68 Cor. 
 
 Dem. Dividing (1) by (2), there obtains : 
 
 BC' ABxBD BD' 
 5> AsT. then will AB'=A(P-\-BC\ 
 
 Bem. Adding (1) and (2), there obtains : 
 AC'+BCP=^AB {AD-\-DB) = ABK 
 
 Q.ED, 
 
 73. 
 
 Theorem. 
 
 Tf one of the sides of any acute angle of a triangle he made 
 the base of the triangle^ the square of the side opposite this 
 acute angle is equal to the sum of the squares of the other two 
 sides, minuj twice the jyroduct of the base and the distance 
 from the vertex of this acute angle to the foot of the perpendi- 
 cular let fall from the vertex of the triangle, upon the base or 
 the base produced. 
 
 Hyp. Assume ABC to 
 be a triangle in which 
 the angle A is acute, and y^ p 
 let a, b, c= sides opposite /#- 
 to angles A, B, C. respec- 
 tively, fig. (1) 
 
 ;>= perpendicular or altitude CD, 
 x=AD, distance from the vertex of the acute angle A 
 to the foot D of the perpendicular CD. ; 
 
 AsT. then will a-=6--f c'-^— 2 c x. 
 
 I)EM. In A BCD (fig. 1), according to art. 72, 5'», 
 
 a:'=f+BD'=p'{-{c—x)'=f-{-6^—2cx,-^x'. (1) 
 
 In A ACJ), &2=jo2-(-J Fr =/j2 -f X-, . ;^) 
 
 Subtracting (2) from (1), a'—b^= c-— 2 ex, 
 
 or <r=62-|- 0"— 2 ex. 
 
 InA^t'/>(fig. 2), 
 
 In A ACDJr^ij'-^- AD' =r.p'^x\ (4) 
 
 Subtracting (0 ^rom (3), n'—lr= —2rx-^c\ 
 
 Qf a-= l/-\-c"- -2 c^. 
 
38 
 
 BOOK I. 
 
 74. 
 
 Theorem. 
 
 If one of the sides of the obtuse angle of an ohtune-anjled 
 triangle he made the base of the triangle, the square of the side 
 opposite this obtuse angle is equal to the sum of the squares of 
 the other tuo sides^plus twice the product of the base and the 
 distance from the vertex of the obtuse angle to the foot of the 
 perpendicular let fall from the vertex of the triangle upon the 
 base produced. 
 
 Hyp. Assume ABC, to be a triangle in 
 which LJ5 is obtuse, and let a, b, c=side8 
 opposite to angles A, B, C, respectively ; 
 ^== perpendicular or altitude CD; x=sBD. 
 distance from the vertex of the obtuse angle 
 B to the foot D of the perpendicular CD ; 
 
 AsT. then will b'^=a^-^c^-\-2cx. 
 
 Dem. In the right-angled triangle ACD, 
 
 b^=p--\-AD^=p^-^ (c+xy=p^-^c--\-2cx + X'. (1) 
 
 InA5Ci>y=p=^+^/>- =p- +x-.(2) 
 
 ^Subtracting (2) from (1), b-— a^ =■. c^+2cx, 
 
 or b'^ ^a^^c^+2cx.Q.E.D. 
 
 75. 
 
 Theorem. 
 
 'iUl 
 
 '4 
 
 The sum of the interior angles of a polygon is equal to twice 
 as many right angles, less four, as the polygon has sides. 
 
 Hyp. Let ABCDE be any polygon. 
 
 From an interior point 0. draw the lines 
 OA, OB, &c, which will divide the polygon ^v.._j...---^r 
 into as many triangles as it has sides ; and 
 let w=number of sides of the polygon, 
 <S=sum of the angles of the polygon, 
 s=8um of the angles of the n triangle-^ AOB, BOC, rfr ; 
 
 AST. then will ^=2 «Li2— 4 L.R. 
 
 Dem. The sum of the three angles of a triangle = 2 L.R, 49 
 
 and the sum of the angles of n triangles =as2 n L7?; 
 
 that is, the sum ft=:2n, L.R. 
 
ARTICLE ii. 
 
 39 
 
 But, S=s niiiins tlu' sum of the angles about tlie point O; 
 that is, ininuH 4 \^R. 
 
 Therefi)re, *S' (=«— 4 L7?) = 2« L/2~4 Li?. (2. /:. 7>. 
 
 76. Cor. The sum of the inferior angles of a qimdrilaterdl 
 is equal t<t four right angles ; and, if two of these angles are 
 right angles, the remaining two are sujiplenientary. 
 
 77. 
 
 Theorem. 
 
 Tito similar j^oli/gons are divisible info homologous 
 triangles. 
 
 Hyp. Let ABCDE aud 
 ahcde be two similar poly- 
 irons, in which Lyl=Lf/, 
 L..B^-[-b, LC=Lc, d'c; 
 
 AST. then will these poly- 
 gons be divisible 4nto homologous triangles. 
 
 Dem. Divide these two polygons into triangles by diagonals 
 drawn from the homologous vertices .1 and a. 
 
 Since the homologous sides of similar polygons are propor- 
 tional, the equal angles B and h are included between propor- 
 tional sides ; 
 
 that is, /\,ABC is similar to /\ahe ; 67 
 
 consequently 
 
 [.ACB=:.'^arh 
 
 (1) 
 (2) 
 
 and, br ] ty p . L BCD=\^bcd. 
 
 8ubt. uctinu; (1) from (2), [-ACD=L.acd; 
 
 hence, aim.i the equal angles ACD and ard are included 
 between proportional sides, the triangles ACD and acd are 
 similar. 
 
 Likewise, the triangles ADE and ade are similar ; there- 
 fore, the two T/olygons ABODE and abcde are divisible into 
 homologous triangles. 
 
 Cor -on^'^rsely : Two polygons are similar, [f they are 
 comp's^iiJ ■' iih.lar triangles, similarly jtlaeed. 
 
Ill' 
 
 I 
 
 I i 
 
 t 
 
 '' I 
 
 I 
 
 BOOK T. 
 
 Theorem. 
 
 Tivo similar triangles are proportional to the squares of 
 their homologous sides. 
 
 Hyp. Let ABC and abc be two 
 similar triangles in which Lyi=L«, 
 
 AsT. then will these two 
 triangles be proportional to the 
 squares of their homologous sides. 
 
 Dem. From the homologous vertices C and e, let fall the 
 perpendiculars CV a I cd on the bases AB and ab, respecti- 
 vely. 
 
 Then , L Al/ (J= L adc ; 
 
 but, L.A=L.a; 
 
 hence, /\ACD is similar to A«ct/^ 
 
 andthereby, CD:cd=AC:ac=sAB:ab=BC:hc. 
 
 Again, 1 — : — =AC:ac=AB:ab:-=BC:bc. 
 
 " ' 2 2 
 
 Multiplying these two series of ratios, antecedent by ante- 
 cedents, and consequent by consequent, there obtains : 
 
 :'^A^L£P^:''l2^^=AC':ac^:^AB':ab-'=BC':br. 
 2 2 
 
 But ^— — and — - — - are the respective areas of the 
 
 triangles ^4^Cand abc ; and therefore, the areas of these two 
 triangles are proportional to the squares of their homologous 
 sides. Q. E. D. 
 
 as L.R 
 
 b\j hyp. 
 
 68, Cor. 
 
 79. 
 
 Theorem. 
 
 In two similar polygons, the perimeters are proportional to 
 their homologous sides, and the areas are proportional to the 
 squares of their homologous sides. 
 
ARTICLE 84. 
 
 41 
 
 let 
 
 fall the 
 
 h, 
 
 respocti- 
 
 
 as L.R 
 
 
 hy hyp. 
 
 
 68, Cor. 
 
 Hyp. Assume A B ODE an d 
 abcde to be similar polygons, 
 and let S and P=area and/li 
 perimeter of ABCDE, respec- 
 tively ; sandp=area and peri- 
 meter of abcde, respectively ; 
 .1 and «= respective areas of the homologous triangles ABC 
 and ahc ; 
 
 AST. then will P-.p^AB :ah =zBC :hc =:CD :cd ==d;c, 
 
 and S:s =^AK".a¥^BC-'.h6'^CD'\cd'^&c. 
 
 Dem. Any two similar miignitudes are proportional to their 
 similar parts ; but, in the polygons ABCDE and abcde, any 
 two homologous sides, as AB and ab, are similar parts of the 
 perimeters P and p ; and likewise, any two homologous 
 triangles, as A and a, are similar parts of the areas S and s ; 
 
 hence, P:p=sAB:ab=BC :bc=(i-c. 
 
 Likewise, S: s=. A :a ; 
 
 but. A: a=zAB-\ah-=&c; 78 
 
 therefore, *S': s^AB-'.ah-—dc. Q. E. D. 
 
 Cor. In two simihtr polygons, the homologous lines are 
 proportional, and the areas are proportional to the squares 
 of the homologous lines, 
 
 CIRCLE AND REGULAR POLYGONS. 
 
 DEFINITIONS. 
 
 80. A curvilinear plane figure is a plane figure bounded 
 by a curved line. 
 
 81. A circle is a plane figure bounded 
 by a curve in which all the points are equi- 
 distant from a point within, called the 
 csnter. 
 
 83. The circumference of a circle is 
 the curve which bounds the circle. 
 
 83. A radius of a circle is a straight line drawn from the 
 center to any point of the circumference. 
 
 84. A diameter oi a circle is a straight line passing 
 through the center and limited by the circumference ; v. g. AB. 
 
I « 
 
 ir 
 
 I" 
 
 42 
 
 BOOK I. 
 
 85. An arc of a circle is any part of its circumference; 
 V. g. AD. 
 
 86. A chord of an an^ is the straight 
 line joining the extremities of that arc ; 
 \. g. AB is a chord which subtends both ^ 
 the arcs AFB and AOB ; but unless ^ 
 otherwise stated, the smaller one is 
 meant. 
 
 87. A segment of a circle is a part 
 of the circle included between an arc and its chord ; v. g. 
 AFBA. 
 
 88. A sector of a circle is a part of the circle included 
 within an arc and the radii drawn to its extremities ; v. g. 
 ^CZ>(fig. of 81). 
 
 80. A secant is a straight line which cuts the circum- 
 ference at two points ; v. g. CD. 
 
 OO. A tangen. to ^ curve is a straight line whicli touches 
 the curve at only one point. 
 
 91. This point is called the point of contact or the point 
 of tangency; v. g. MX is a tangent, and is a point of 
 tangency. 
 
 93. An inscribed angle is one whose ^, — —-■—«—-— i^ 
 
 vertex is on the circumference, and whose /l^ <^ 
 
 sides are chords ; v. g. ahc. 
 
 93. A polygon is inscribed in a circle 
 ■when all its vertices are on the circum- 
 ference ; V. g. ahcd. "^ 
 
 The circle is then said to be cu'cumscribed about the 
 polygon. 
 
 94. A polygon is circumscribed about a circle when all 
 its sides are tangent to the circumference ; v. g. ABCD. 
 
 The circle is then said to be inscribed in the jwlt/gon. 
 
 95. An equilateral polygon is one in which all the sides 
 are equal 
 
 96. An equiangular polygon is one in which all the angles 
 are equal. 
 
ARTICLE 104. 
 
 43 
 
 uinference ; 
 
 or the point 
 
 li all the angles 
 
 OY* A regular polygon is one whicli i.s both tMiiiilaterul 
 and equiangular. 
 
 08» Tiie center <»f a regular polygon is the coninion (tenter 
 of the inscribed aJid circumscribed circles. 
 
 DO. The radius of a regular polygon is tlu' radius of the 
 circumscribed circle. 
 
 lOO. The apothem of a regular polygon is the radius 
 of the inscribed circle. 
 
 lOl* The angle at the center of a regular polygon is tlu* 
 angle formed by radii drawn to the extremities of any side of 
 the polygon. 
 
 HO'/d* The circumference of a circle is divided into 300 
 equal parts, called dvgncm ; the degree, into (10 e(|ual parts 
 'Called minutes ; and the minute, into GO e(|ual parts called 
 seconds. 
 
 The fractions of seconds are expressed in decimals, 
 
 103* Degrees, minutes and seconds are denoted by the 
 symbols °, '," ; thus 30° 23' 15" is read: thirty degrees, 
 twenty-three minutes and fifteen seconds. 
 
 104. 
 
 TlIEORKM. 
 
 In the same circle or in etjnal circles, 1" equal angles at the 
 center intercept equal arcs on the circum- 
 ference, and conversely ; 2' any two angles 
 at the center are proportional to their inter- 
 cepted arcs. 
 
 In the equal circles ABD and M\I* : 
 
 IHyv. If L.46'ii=Lil/0A', 
 then will are ^ii=arc J/.V ; 
 
 conversely, if arc .iZi=arc MX, 
 
 then will L.ACB=L.2M0X. 
 
 Dem. Appl^' the first circle to the second, so that L-ACH 
 shall coincide with LiUfOxV; then the point ^ will fall on M, 
 the point H on N, and the arc .4 'i will coincide with the arc 
 
44 
 
 H(K)lv I. 
 
 ■);!■!!! 
 
 J/iV, Conversely, ai>^)ly tlie arc AB on its equal MN, so that 
 they shall coincide ; then, the radius AC wUl fall on J/0, and 
 Ji(^ on iW, 
 
 Therefore, I'if L.16'/i=L JAVO, then &c. 
 
 2» Hyp. Let .l(7>aiid MOFhe two angles ;it the center, 
 intercepting the arcs AJJuud MF ; then will i-ACJ):[-MOF= 
 arc AD '.lira Ml*. 
 
 Dem. Assuming the .same unit of meaijure to be contained 
 'i\ times in the arc AD and 5 times in the arc .1//^, then will 
 
 AD.MP=)\:'y (1) 
 
 Now, the angles formed by the radii drawn to the points of 
 division will be equal, since they intercept equal arcs ; 
 
 hence, L_J6'/>:LJ/0/^=3:5, (2) 
 
 and, from (1) & (2), L.16V>:L_J70P=arc ^Drarc MP. 
 
 But, this relation is true whatever may be the unit of 
 Pleasure ; hence, it is true when the unit becomes infinitely 
 fimall, as is the case when the two arcs are incommensurable. 
 
 Therefore, 1^ any two angles at the center (fee. 
 
 Cor. An angle is measured by the arc included between its sides and 
 <ii'Kcribedft'om its vertex, as a center. 
 
 105. 
 
 Theorem. 
 
 !li 
 
 ^1 nuUas ptrjietulicalar to a chord hlsexts lluit chord and its 
 subtended arc. 
 
 Hyp. Let Cmn be a radius perpendicular ^^ ^ 
 
 to the chord AB, in the circle ABT; 
 
 AsT. then will Ani=mB, 
 and arc An= arc n B. 
 
 Dem. Draw the radii AC and BC 4 
 
 By this construction, ABC is an isosceles 
 triangle ; 
 
 hence, Ani=mB, 
 
 and \-ACm,=l^inCB ; 
 
 therefore. arc ^w=aro nB. Q. E. D. 
 
 Cor. 1. A perpendicular bisecting a chord passes through 
 the center of the circle. 
 
 Cor. 2. A diameter bisects the circle and its circumference. 
 For, l.ACn + \^ACT=i.BCn-{-\^BCT. as Ladj. 
 
 Cor. 3. A diameter (nT) perpendicular to a chord (^AB) 
 bisects both arcs subtended by that chord. 
 
 48,3 
 48,3 
 
ARTICLE 107. 
 
 
 45 
 
 For, nAT==nBT, 
 
 (1) 
 
 Cor. 2. 
 
 and nAsstiB. 
 
 (2) 
 
 Vem, 
 
 Subtracting (2) from (1),^^=^ 7'. 
 
 
 
 106. 
 
 Theorem. 
 
 Two equal angles at the center of a circle arc mhtended by 
 equal chords. 
 
 Hyp. In the circle ABT, let ACn and 
 BCn be two equal angles at the center 
 subtended by the chords An and ni?, res- 
 pectively ; 
 
 AsT. then will An=nB. ^ 
 
 Bem. In the isosceles triangles ACn and 
 BCn, 
 
 \—ACnz=. \^BCn ; hy hyj)' 
 
 hence, £^ACnz=z^BCn \ 44 
 
 therefore, An=z nB. Q. K. D. 
 
 Cor. In the same circle, or in equal circles, equal arcs are 
 subtended by equal chords. 
 
 107. 
 
 Theorem. 
 
 The tangent to a circle and the radius drawn to the point of 
 tangency are perpendicular to each other. 
 
 Hyp. Let DG be a tangent to the circle ABT, and draw a 
 radius CT to the point of tangency T; 
 
 AsT. then will DG and CT be perpendicular to each other. 
 
 Dem. By the definition of a tangent (90), the point of tan- 
 gency T is the only point common to both the circle and the 
 tangent ; hence, any other line than the radius CT, drawn 
 from the center to the tangent, will meet the latter without 
 the circumference, and thereby be longer than the radius CT ; 
 cousequently, the radius is the shortest distance between the 
 center and the tangent. 
 
 Therefore, CT is perpendicular to DG ; 46, 1 
 
 and conversely, DG is perpendicular to CT. Q. E. D. 
 
46 
 
 HOOK 
 
 I 
 
 Cor. -.1 perpendicular to a tangent, <it the point of tangeiicy^ 
 pasiseK through the center of the circle . ^ 
 
 108. 
 
 Theorem. 
 
 Tlirough am/ three points^ not in the same gtraight line, one 
 circumference, and only one, can he described. 
 
 Hyp. Let A , B and C be any three points, 
 not in the same straight line ; 
 
 AsT. then can one cireuniferenee, and//, 
 only one, be described through them. 
 
 Dem. Join the points .1, B and C by the 
 lines AB and BC, and let those latter be 
 bisected by the perpendiculars Em and Dn. 
 
 The perpendicular Em contains all the points which are 
 equidistant from .1 and B (art. 40, 2), and the perpendicular 
 Dn contains all the points equidistant from B and C ; hence, 
 the point of intersection Oof these perpendiculars is the only 
 point equidistant from A, B and ^7; therefore, one circum- 
 ference, and only one, can be drawn through these three 
 points. Q. E. D. 
 
 Cor. Tico circumferences cannot intersect at more than two 
 points. 
 
 For, if two circumferences have three points in common, 
 they will have the same center and become but one and the 
 same circumference. 
 
 109. 
 
 Theorem. 
 
 : 
 
 Two parallels lohich meet the circumference of a circle in- 
 tercept equal arcs. 
 
 Hyp. Let APBO be a circle in which 
 the two chords DF and OP are parallel to 
 the tangent MN ; 
 
 AsT. then will DO=FP, and AO=AP. 
 
 Dem. From the point of tan^ency J, 
 draw the diameter AB. 
 
ARTICLE 110, 
 
 47 
 
 tdugeiiajf 
 
 t line, one 
 
 which are 
 rpendicular 
 
 C ; hence, 
 
 is the only 
 
 ne circum- 
 
 these three 
 
 Q. E. D. 
 
 V. than two 
 
 In common, 
 ne and the 
 
 circle in- 
 
 IP 
 A/ 
 
 This diameter, beinj; perpendicular to the tangent ^l/!iV(107), 
 is also perpendicular to the chords OP and DF which, Ity 
 liyppthesiH, are parallel to this tan«;ent ; hence, each of the 
 arcs subtended by these chords is bisected by AB (105, H ); 
 that is, AO=AP, (1) 
 
 and AD^AF. (2) 
 
 Subtractinf<(l)from{2), DO^FP. Q. E. D. 
 
 110. 
 
 Theorem. 
 
 .In inscrihed angle in measured hy one hal/of it$ intercepted 
 
 arc. 
 
 This theorem admits of three hypotheses : 
 
 !•> The center of the circle may be in one of the sides of 
 the inscribed angle ; 
 
 2'^ it may be within the angle ; 
 
 3^ it may be without the angle. 
 
 lo Hyp. Let the center O of the circle 
 ABC be on the side AB of the inscribed 
 angle B ; 
 
 Ast. then willthe measureof L-S=^arc ^46^ 
 
 Dem. Draw the radius OC, and let ma 
 denote measure. 
 
 Then, L5=L(7, 
 
 and V.A0C=i.B^l.C=2 L5/ 
 
 whence, \~B=\ \^AOC. 
 
 But, the WIS of ^ \^AOC=^ arc AC ; 
 
 therefore, the msof L^=^ arc AC 
 
 2" Hyp. Let the center be within the 
 angle ABC ; 
 
 Ast. then will the ms of L-4J5C=: Jarc AC. 
 
 Dem. Draw the diameter BOD. 
 
 Then them«ofL^-Si)=i arc ^Z)(Dem.lo), 
 
 and the ms ofL.CBD=^ arc CD; 
 
 therefore, the measure of 
 
 l.ABD-\-l-CBD=^&rG (AD^DC); 
 
 that is, the measure of 
 
 L.ABC=^&ro AC, Q.E.D, 
 
 47, Sch. 
 49,1 
 
 104, t 
 Q. E. D, 
 
rmjim 
 
 48 
 
 BOOK T. 
 
 ■ ill 
 
 3' Hvp. Let the center be without the 
 angle ABC ; 
 
 AsT. then will the measure of 
 
 i.ABC==\nrGA(\ 
 Dem. Draw the diameter BOD. 
 Then, the measure of 
 
 L. Z)iiC=i arc Z>6' (l)em.l'>), 
 and the ms of Li>5^l=^ arc DA ; 
 therefore, the measure of 
 
 L.DBC-\^DBA = \ arc {DC— DA) ; 
 that is, the measure of 
 
 L.li?C=^arc AC. 
 Cot. 1. Any angle inscribed in a semi circle in a right 
 angle. 
 
 Cor. 2. All angles inscribed in the same segment are equal. 
 
 Q. E. D. 
 
 111. 
 
 Theorem. 
 
 An ang^.e formed by a tangent and t chord is measured hy 
 one half the arc aubtended by that chord. 
 
 Hyp. Let MX he a tangent to the circle^ /i 
 
 ABC, and AB a chord drawn from the 
 point of tangency A ; 
 
 AsT. then the ms of L5yliV"=^arc AB. 
 
 Dem. Draw the chord BC parallel to 
 MK 
 
 Then, Lfi=LJ5^iV, 
 
 and arc AC =Sirc AB ; 
 
 but, themsof LiJ=^arc AC =^ arc AB ; 
 
 therefore, the ms of L5^iV=^ arc AB. 
 
 as LL al. int. 
 
 lOU 
 
 110 
 
 Q. E, D. 
 
 113. 
 
 Theorem. 
 
 An angle formed hy two chords which intersect is measured 
 hy one half the sum of the intercepted arcs, the one hetwenn its 
 sides and the other between the sides of its opposite angle. 
 
ARTICLK 111. 
 
 49 
 
 Q. E. v. 
 
 ■U a right 
 
 Hyi*. Lt't AJi Hiul CJJ be two cliord^ 
 whi<li intorsect ; ^ 
 
 AsT. then will the iiieaisure ot* 
 
 L.AO/J=^ (are .l/>-f arc CB). 
 
 Dbm. J)raw the chord BF parallel to >f 
 CJK 
 
 Then, arc CJS«arc />F. 
 
 and L.AOJ)={.AJiF. 
 
 But. the measure ot* 
 
 l.ABF=^virc (AD-i- DF) =^{Arc AD+arc rii) ; 110. 
 
 therefore, the im of {^AOD=z\{a.vG yli^+arc CB). Q.E.J). 
 
 109 
 as LL. Corp. 
 
 IVi. 
 
 Theorem. 
 
 t are eq 
 
 udl. 
 
 neasurcd hy 
 
 1_L al. int. 
 
 110 
 Q. E. D. 
 
 is measured 
 e betwenn its 
 ^e angle. 
 
 Ail angle formed hy two secantt vhich meet without the 
 rlrdc is measured by one half the difference of the intercepted 
 arcs. 
 
 Hyp. Let AB and BV be two secants 
 neeting at the point B without the cir- 
 cumference A (W ; 
 
 Aht. then will the measure of ^^ 
 
 L^=^ (arc ^C— arc DE). 
 Dem. Draw the chord AE. 
 Then, {^AEC=L.A-\-[^B; 
 
 whence, \^B=[^AEC—[^A ; 
 
 but, the ms of L^J57(7=^arc AC ; 
 likewise, the ws of \^A=^axQ DE. 
 Subtracting (2) from (1), 
 
 the ms of L.AEC—l.A=\ (arc ^IC— arc DE) : 
 therefore, the ms of L5=^ (arc ^C— arc DE). Q E. D' 
 
 (1) 
 (2) 
 
 114. 
 
 Theorem. 
 
 If two chords intersect in a circle, their segments are red' 
 procally proportional. 
 
50 
 
 BOOK I. 
 
 J-^ 
 
 
 :jili 
 
 '■:■ : ;■. ■'■"':■■ 
 
 
 ii' i 
 
 #' 
 
 ■i .1 
 
 ■!'■' 
 
 1 , 
 
 1^ 
 
 
 ,1;i 
 
 11 
 
 Hyp. Let AB and C/> be two chords inter- 
 secting at F ; 
 
 AsT. then will AF'.CF=DF:BF. 
 Dkm. ])raw the chords AD and CB. 
 Then, (110,2),' L^=LC; 
 
 likewise, {^B=[^D; 
 
 hence, A-l/^^^ if* similar to ^CBF ; 
 
 therefore, AF : CF:^DF i BF. 
 
 68, Cor. 
 <^. JS'. J?. 
 
 115. 
 
 Theorem. 
 
 Tn:o secants meeting without a circle ami tcrminatinq in the 
 concave arc are reciprocaUy j)roportional to their external 
 segments. 
 
 Hyi». Let AB and AC be two secants 
 meeting at the point A without the circle 
 ajid terminating in the concave arc BC ; 
 
 AsT. then will AB. AO^AN -.AM. 
 
 ])Ejr. Draw the chords i^^V^and CM. 
 
 Then, the angle A is common to both 
 triangles ABNrxnA ACM, 
 
 and Lj5=LC'; 
 
 that is, A-1-S^Vis similar toA-1^-^; 
 
 therefore, . AB'.AC^^A:^ .AM, 
 
 110, 2 
 68. Cor. 
 Q. E, D. 
 
 116. 
 
 Theorem. 
 
 A tangent forming (tn angle with a secant terminating in 
 the eoneare arc. is a mean projjortional between the secant and 
 its ^'eternal aegmeiif. 
 
 Hyp. Let the angle A be formed by a tan-y/^ 
 gent ^1(7 and a secant -I /^ terminating at the 
 point B in tlie concave arc ; 
 
 AsT. then will AB'.AC^ACiAD. 
 
 Dem. Draw the chords BC and CD. 
 
 Thea, the angle A is comuion to both triangles ^liJCand ACDf 
 
 and LB=[^ACI); 110,111 
 
 the-efore, Zwl/?^' is sijuilar toA^l67>, 68,6V. 
 
 and thereby, AB.AC-^ACiAP. Q. F. D, 
 
ARTICLK 118. 
 
 51 
 
 117. 
 
 Theorem. 
 
 J perpendicular let fall^ from any point of the circum- 
 jVrf'Hcr, on a diameter, in a mean proportiotial hetwren the two 
 segmenfit of that diameter. 
 
 Hyp. AM.sume CI) to be a perpendicular let G^ 
 fall, from the point C of the circumference, 
 on the diameter AB ; A\ 
 
 Aht. then will AD:CD=(JD:J)B, 
 
 or CD'=AJ)\I)B. 
 
 Dem. Draw the ;hords AC and BC. 
 
 By this construction, A ^1^^^ is right-angled at C ; 110, 1 
 
 therefore, the perpendicular (W is a mean proportional bet- 
 ween the two segments of th^> hypotheiiuse AB ; 72, 2* 
 
 that is, CI}^=ADxDB. Q. E. D. 
 
 Sch. Let .4^=2 7? An=x, CD^y^^nd f)B=2 R—x. 
 
 By this notation, the algebraic expression of the theorem is 
 
 y-=a;(2 R~.r) = 2Rx—x\ 
 
 118. 
 
 Theorem, 
 
 A cireje nmi/ he inscribed, in, or rlrenmHerlhed about, a regU' 
 Jar polygon. 
 
 Hyp. Let ABCDEF be any legular 
 polygon; 
 
 AsT. then may a circle be inscribed in, 
 or circumscribed about, this polygon. /=^^ ffk.-^ -^^ 
 
 Dem. 1 ' J)raw the lines AO and BO 
 bisecting the angles A and B, and from jf 
 
 their point of intersection 0, draw lines 
 to the other vertices f', D, E, F of the polygon. 
 
 Then, the side OB is common to both triangles AOB 
 and BOC; 
 
 besides, L.ABO=\-OBC, by const, 
 
 and AB=BC; by hyp. 
 
 that is, A ABO= A OBC; 
 
 hence, AO=BO=CO. 
 
 Lik 
 
 ewise. 
 
 CO^DO=dc, 
 
l)i.4lpiU4!|i^j!f.j,^JlgBai>j^ 
 
 
 li''. '■'^i 
 
 wWi'm 
 
 52 
 
 BOOK r. 
 
 Therefore, .a circumference described, from the point 0, as u 
 center, andwitli a radius AO, passes through all the vertices 
 of the polygon, and is thereby circumscribed about the latter. 
 
 2^ The perpendiculars OM, OX, d'c, let fall on the sides 
 AB, BC, &c of the polygon, are equal altitudes of equal 
 triangles ; therefore, a circumference, described frojn the point 
 as a center, and with the radius OM, is tangent to the sides 
 of the polygon, and is thereby inscribed in the latter. Q. E. D. 
 
 Sch. The difference between the circumscribing ircumfe- 
 rence and the perimeter of the inscribed polyj^ 'n obviously be- 
 comes less as the number of sides of the polygon increas , 
 hence, if this number of sides becomes greater than any assign- 
 able quantity, the diflference between the circle and the ins- 
 cribed polygon will become smaller than any assignable quan- 
 tity ; therefore, a circle may be considered as a regular i>oly- 
 gon composed of an infinite number of sides. 
 
 111). 
 
 Thkorem. 
 
 The area of a reguldr jwli/yrni la equal tn half iln' j>rfKfiicf 
 of its perimeter and apothem. 
 
 Hx'P. Assume .4i?(7i)i5;'i^'tobeauy poly- 
 
 gon, and let 7l=area, P= perimeter, 
 
 //= apothem, ii= length of 
 
 one side, and 7j = number of sides, of the 
 
 polygon; 
 
 AsT. then will .1= — -- 
 
 2 
 
 Dem. Any regular polygon is divided, ])y its radii, into a." 
 many equal triangles as it has sides ; but, the area of each of 
 these equal triangles is equal to half the product of its base 
 (i^ or side of the polygon) and altitude (7/ or apothem of the 
 polygon) ; therefore, the area of the polygon is 
 
 A=nx — - — 
 2 
 
 That is (since ii x B=P, perimeter of the polygon). 
 
1120, 
 
 ARTICLE 121. 
 
 Theorem. 
 
 53 
 
 It is customary to represent the ratio of the diameter (2 Ry 
 to the circumterence ((7) of the circle, by the greek letter r, 
 
 - ; whence, C=2 tR.* This beingstated, prove that : 
 
 or 
 
 The area of a circle is equal to the square of its radius mul- 
 ti plied hy -. 
 
 Hyp. Assume A(Jlil) to l)e a circle, and 
 let ^= circumference, 7^=. radius, and 
 .l = area, of.l6'i?/;; 
 AsT. then will A = -1T-. ^' 
 J)em. a circle may be repirded as a regu- 
 lar pol^'gou composed of an infinite number 
 »»f small and e(|ual triangles whose altitude is 
 the radius R and whose common vertex is the center of the 
 circle (118, Sch). 
 
 Representing the infinitesimal bases of these triangles by 
 <r, 6, Cydy etc, which bases maybe considered as so many infini- 
 tely small straight lines, the area of each infinitesimal triangle 
 will be equal to half the product of its altitude R and base 
 (f, or b, or etc. But, the sum of the areas of these 
 triangles constitutes the area A of the circle ; that is 
 
 , axR , hxR , rxR , . R^ , j , , ^ \ i xi, 
 
 A=~— — 4- 4-_i: 4-etG =—(a4-h4-c4'etc) and the 
 
 sum (a-{-h-{-c-\-etc) constitutes the circumference (/ of the 
 
 circle ; 
 
 n f t Rr , 1 I , ^ \ OxR ^ttRxR p2 
 t herefore, ^1 = - (« -f- 6> -f c -|- etc) = — _ — = — = zR. 
 
 131. 
 
 Theorem. 
 
 Tu tico similar curvilinear plane figures, the curves are pro- 
 portional to any tv:o homologous lines, and the areas are pro- 
 portional to the squares of an}/ two homologous lines. 
 
 * It will be uroved in art. 49, Book III, that :r= 3. 1416, very n;arly. 
 
5i 
 
 BOOK r. 
 
 ■ o- 
 
 ■ii\ 
 
 m 
 
 Hyp. Assume ACEF and 
 acef to be two similar plane 
 figures ; A\ 
 
 and let A = area of ACEF, 
 a = area of ace/, 
 C= curve of A CEF ; c = curve of ace/ ; 
 
 L and /=any two homologous lines ; 
 AsT. then will C :c=L :l, and A :a — U:T-. 
 Dem. Suppose the inscribed polygons ABODE, and abcde 
 to be similar and divided into homologous triangles ; 
 and let /S^and /'=area and perimeter of polygon ABODE, 
 8 and ^= area and perimeter of polygon abcde. 
 Then, P : p=AB:ab =AO:ao =0F : c/= du; 79, Cor. 
 and S : s=AB^:nlr=AO'^:(io'=.&c. 
 But, these relations are always true, whatever may be the 
 number of sides of the inscribed polygons ; hence, they are 
 true when this number of sides becomes infinitely great ; that 
 is, when the perimeters P and p of the polygons become the 
 very curves O and c ; therefore, at the limit, there obtains : 
 C:c=:L: I, and A'.n^L'^i P. Q. E. D. 
 
 Cor. In two circles, the circumferences are proportional to 
 any two homologous lines {radii, diameters, chords, arc^^, 
 and the areas are proportional to the squares of any tico homo- 
 logous lines. 
 
 133. 
 
 PRACTICAL APPLICATIONS. 
 
 GRAPHICAL PROBLEMS. 
 
 A graphical problem is one whose solution requires a 
 geometrical construction. 
 
 Prob. 1. To bisect a straight line (AB). 
 
 Solution. From the extremities A and B, 
 as centers, with the same radius, draw arcs 
 intersecting at O and D, and join these points ^""^ 
 of intersection by a line ; then will this line 
 CDhxmvXAB. 
 
 ^sC 
 
 o 
 
 For, the points and D are equidistant 
 
 :*:^ 
 
ARTICLE 122. 
 
 55 
 
 from the extreinitie« A and Ji of AB ; therefore, the perpen- 
 dicular CD bisects AB. 46, 8 
 
 Prob. 2. To draw a perpend iaihtr to a line (^AB), at a 
 given point (0) c/ that line. 
 
 Solution. From the given point O lay off Oh=zOa, on tlie 
 line AB ; and from the points a and 6, with the same radius, 
 describe arcs intersecting at f\ and join by a line the point () 
 with the point of intersection C ; then will CO be perpendi- 
 cular to AB, at the given point O. 
 
 For, each of the points C and is equidistant from the 
 points a and h; therefore, CO is the required perpen- 
 dicular. 46, ii 
 
 I*rob. 3. To draw a perpendicular to a line {AB), from a 
 point (C) without that line. 
 
 Solution. From the given point C, as a center, with the same 
 radius, describe two small arcs intersecting AB at a and b ; 
 from these points a and ft, with the same radius, describe arcs 
 intersecting at D. Then, joining Cand />by the edge of a ruler 
 the line CO drawn along the ruler, from C to AB, will be the 
 perpendicular required. 
 
 For, each of the points C and D is equidistant from the 
 points a and b ; therefore, CD and thereby CO is perpendicular 
 to AB. 46, 3 
 
 Prob. 4. At a given point (A), in a straight line {AB), to 
 construct an angle equal to a given angle Qi). 
 
 Solution. From the ver- 
 tex a as a center, with any 
 radius, describe the arc be ; 
 from tht point J. as a center, 
 with the same radius ab, 
 describe an indefinite aro ^(7; again, from the point ^ as a 
 center, with a radius equal to the chord be, describe an arc 
 intersecting BC, and join, by a line, the point -1 with the 
 point of intersection C ; then will LJ.= La. 
 
 For, by construction, the arcs BC and be are equal ; hence, 
 the angles -t and a, being measured by equal arcs, are 
 equal. 
 
56 
 
 BOOK I. 
 
 ■:w 
 
 !! Hi 
 
 ::^^ 
 
 /7^ 
 
 IProb. 6. To bisect an angle or an ar«\ 
 
 SoiJiTiON. Let -t be the given angle 
 subtended by the arc BC. Draw the 
 chord £C, and a perpendicular from tlie 
 vertex A to that chord (Prob. 3). This ^ 
 perpendicular bisects the arc BC 
 (art. 105) and thereby the subtended angle .1. 
 
 Prob. 6. TJirough a given point ((7), to draw a llneparallet 
 to a given line (^AB). 
 
 Solution. Join the points A and C by a 
 line, and construct LC=LA (Prob. 4)-' 
 Since by construction, the alternate-interior 
 angles ^4. and C are equal, DO will be the 
 required parallel to AB. 
 
 Prob. 7. To construct a triangle^ xvhen two sides (m and « ) 
 and the included angle (a) aregiven. (* 
 
 Solution. Draw AB^-m, and construct y 
 L_.4=La (Prob. 4) ; also draw AC^=n, and 
 join the points B and C, by a line ; then will 
 ABC be the required triangle. 
 
 Prob. 8. To construct a triangle, when the three sides 
 (m, n and o) are given. 
 
 Solution. Draw AB=m; from the point vl 
 as a center, with aradius=7i, describe an arc ; 
 from the point B, as a center, with a radius 
 =0, describe a second arc intersecting the 
 first at C; finally, draw AC and BC. Then 
 will ABC be the required triangle. . — 
 
 Prob. 9. To Jlnd the center of a circle (ABC). 
 
 Solution. Draw the two chorda ABaud 
 BC, and bisect them by the perpendiculars 
 Dn and Em ; then will the point of inter- ^ \ 
 section of these perpendiculars be the 
 required center (Art. 105, l^ 
 
ARTICLE 122. 
 
 57 
 
 Prob. 10. On a given straight line (AB), to construct a 
 segment that shall contain an angle eqnal to a given angle (/>). 
 
 Solution. Construct \^ABF=L.E(VYoh. 
 \ ; draw a.i indefinite perpendicular CO to 
 the line AB, at its middle point O, and (I 
 also a perpendicular DB to the lino BF, at 
 
 the point /^ ; and from the point of inter- / ^"^7^/ 
 section C, as a center, with tlie ridius CB,^ 
 tlcscrihe a circumference ; then will any angle as ADB, in the- 
 sesinient APBA, be e<{ual to the angle ABF ( =L^). 
 
 F<n', hoth the angles AJ>I> and ABF iiry?, measured by half 
 the arc AGB (art 110 and 111) ; therefore, ABBA is ih& 
 required segment. 
 
 Prob. 11. To divide a straight line (^AB) into parts propor^ 
 fionul to any nmnhrr of girvn lints (»i, n, o, cf'c). 
 
 Solution. ])raw an indefinite line AC, rn 
 forming with AB, any angle ; on .1^', lay -'^-~ cC 
 
 (»flF ^bji=7», nia=:n, 'dnd no=-o ; join the »»v "1 1 
 
 points iJand o by the line Bo, and draw ^^'■'' \ \ \ 
 En and JJrn parallel to Bo ; then wiil the J) £ O 
 
 line AB be divided into the segments AD, DE and EB which 
 are proportional to the given lines wi, n and o (art. 64). 
 
 Prob. 12. To construct a fourth pt'oj)ortional to three given 
 lines (rn, n and o). 
 
 Sc^LUTiON. Construct any angle as BAC; • 
 on AB, lay oiF Am=ni, An=n, and on^ 
 
 rrt 
 
 ^ 
 
 ^^^ 
 
 AC, lay off Ao=^o ; join the points m 
 and o by the line mo, and draw Dn parallel 
 to o»i / 
 
 then will Am : An^=^Ao \ AD ; 
 
 or m : n— o :AJ) ; 
 
 therefore, AD will be the fourth proportional required. 
 
 Prob. 13. To construct a mean projwrtional between two 
 given lines (?/fc and u). 
 
 C 
 
 64, Cor, 
 
m 
 
 ih .1 ■ • 
 
 ;i*i!i 
 
 MM 
 
 58 
 
 liOOK I. 
 
 /77 
 
 -a 
 
 117 
 
 Solution. On an indefinite line AGj lay 
 oflF AD=:n, and DB=m ; on ^/? as a diame- 
 ter, describe the semi circle ACB, and draw / i. 
 CD perpendicular to AB at the point D ; 
 
 then will .4/> : DC=DCx T)B, 
 
 or n\DC=DC\ m; 
 
 therefore, the jwrpendicular 7>f' will be the mean propor- 
 tional required. 
 
 Prob. 14, Ti) divide a given Hue (AB) into extreme and 
 mean ratio (jhit is, into tiro such jtarts that the y renter part 
 shall hea mean jiroporticmal between the whole Hue aud the 
 smaller part). 
 
 Solution. Draw a perpendicular ^ 
 
 BC= 
 
 AB 
 
 I 
 to the line AB, at its ex- ' 
 
 <. 
 
 ''.^ 
 
 
 tremity B; from the point f\ as a 
 center, with a radius=^C, describe a S ** 
 
 circumference ; through the points A and 6^ draw the straight 
 line ACD terminating in the concave are and intersecting the 
 circumference at m ; from the point A as a center, with a 
 radius=ylMJ, describe the arc mn ; 
 
 then win AB\ An =A,i : Bn. 
 
 For, ADxAB=ABi Am, 
 
 or, by division, AD — ABlAB^AB — Avi : Am 
 
 116 
 
 (1) 
 
 But (by equation BC= ^\ , AB= 2 BC= Dm ; 
 
 hence, AD — AB=:^AD — Dm ^=Am=An. 
 
 Substituting An for itsequal AD — .1/^ and also for its equal 
 Aw, in proportion (1), there obtains : 
 
 An : AB=AB—Au : An ; 
 that is, AnlAB= Bn: Au, 
 
 or, bv inversion, ABi An= Am Bu. 
 
 Prob. 16. To construct a square equivalent to the sum of 
 two given squares Qehose sides arem and «, respectively^. 
 
ARTICLE 122. 
 
 59 
 
 m 
 
 '. 
 
 
 A 
 
 ■r ,/ 
 
 Solution. Construct a right angle DAF ; 
 on AF, lay off AB^=m, and on Al>, lay off 
 AC=-n ; join the points B and C by the line 
 />V; then, in the right-angled triangle ABC. 
 
 will BC'=AB'-VAC\ 
 
 or BC-=m^ -j-n'. 
 
 Therefore, the square whose side is BC will be the square 
 required. 
 
 Prob. 16. To construct a square equivalent to the difference 
 of two given squares (^whose sides are m and n, respectively.) 
 
 Solution. Construct a right-angle DAF; on AD, lay off 
 AC~h; from the point C, as a center, with a radius=wi, 
 describe an arc intersecting AF at B ; and join the points 
 (7andB, bythe line CB ; then, in the right-angled tiiangle 
 ABC. 
 
 will AB'=BC'-AC-, 
 
 or u{B'= m- — ;/-. 
 
 Therefore, the square whose side is AB will be the s(|uare 
 required. 
 
 Prob. 17. To inscribe a square in a circle. 
 
 Solution. Draw two diameters perpendicular io each other, 
 and join their extremities by chords ; then will these four 
 chords be the sides of an inscribed square. 
 
 Prob. 18. To iuscrihe a regular hexagon 
 
 {ACE). 
 
 Solution. Beginning at any point .1 of 
 the circumference, apply the radius six 
 times, as a chord ; then will ABCDEF be 
 a regular inscribed hexagon. 
 
 For, if the radii AO and BO be drawn, 
 the triangle ABO being equilateral is also 
 equiangular, and thereby each of its angles 
 is equal to 60^ ; hence, the angle at the center AOB, or its 
 subtended atcAB, comprises 60°, that is the sixth part of the 
 circumference ; therefore, ABCDEF will be the required ins- 
 cribed hexauon. 
 
 in a 
 
 nrcle 
 
60 
 
 BOOK 1. 
 
 1^ "M^^r 
 
 Prob 19. To inscnhf. tm vqulhttcntl tr'uingle in a circle 
 {ACE). 
 
 Join, by ehorils. the alternate vertices (.1 and T, C and E, 
 ^ and A) ot'the inscribed re}j;ular hexairon ; then will the ins- 
 cribed polygon be the required e«|uilateral triangle. 
 
 Prob. 20. 7% inscribe n ngnhtr decagon in a circle 
 (AD/I). 
 
 Solution. Divide the radius AO into 
 extreme and mean ratio (Prob. 14) ; then 
 will the greater segment OJ* he the required 
 length of each side A H, BC, CD, <fcc of the /tf\ 
 inscribed regular decagon. 
 
 For, assume the isosceles triangle AOB to 
 be constructed on the side AH of the decagon; then, the 
 
 angle at the center AOB comprises-l-j__or ?,Q° • hence, each 
 
 10 
 
 of the equal angles .150 and /^^O comprises 
 
 180°— 36° 
 
 or 72° 
 
 consequently, if the angle ABO be bisected l>y BP, the 
 triangles BPO and .1 HP will be isosceles, and thereby 
 OP=PH=BA; 
 
 that is, * AB=OP; 
 
 therefore, as the triangles AOh and ABP are mutually 
 equiangular and similar, AO : AH=:AH : AP, 
 
 or (substituting OP for itsei^ual .17i).iO: OP—OP:AP. 
 
 Prob. 21. To inscribe n regular pentagon in a circle 
 {ADII). 
 
 Join, by chords, the alternate vertices A and C, C and E d'c. 
 of the inscribed regular decagon ; then will the inscribed poly- 
 gon be the inscribed regular pentagon required. 
 
 Prob. 22. To inscribe a regular penttulecagon {fifteen 
 sides) in a circle (ADH). 
 
 From the vertex A of an inscribed regular decagon, apply 
 the radius AO. as a chord, on the circumference, which it will 
 meet at the point a ; then will the arc Ba be the fifteenth 
 
ARTICLE 123. 
 
 in 
 
 part of the circumference, and its chord Jin the side of the 
 inscribed regular pentadecagon required. 
 For, arc -5.1 = arc ^a— arc AJi, 
 
 or arc ^.1= 60° - 3i;° = 24° = _— . 
 
 15 
 
 Prob. 23. T(t hiscrihe <i polygon of tioiibh- the nmnber of 
 sides rtf (nnf given regnUir poh/gnn. 
 
 Bisect the ares subtended by the sides of the inscribed regu- 
 lar polygon, and subtend those semi ires by chords; then will 
 the inscribed regular polygon be tlie one required. 
 
 By this method, we may pass successively from an inscribed 
 square to inscribed regular polygons of 8, IG, 32. &c sides ; 
 from a regular decagon, to regular polygons of 20, 40, 80, &c 
 ijidei ; and from a regular pentadecagon. to regular polygons 
 of 30, GO, 120, i^c. sides. 
 
 24. To transform any given frinngh' iitfn an equivnlent 
 triangle double in tdtitudc. 
 
 25. To transform a trape-:oid into an eAjnirnlent triangle. 
 
 26. To divide any quadrilateral into tim equiralent parts. 
 
 27. To divide a straight line iiift> ports proportional to 1. ;; 
 and 5. 
 
 13;^. NUMKRKWL PIJOHLKMS. 
 
 A numerical problem is one whose solution recjuires nume- 
 rical operations. 
 
 Prob. 1. To find the area A of a jjaralklograni whose base 
 B is 15 and whose altitude H is 8. 
 Solution. A=BxH=15x 8=120. Art. 61. 
 
 Prob. 2. To find the altitude H of a parallelogram tohose 
 area A is IbO and whose base B is 15. 
 
 Solution. A=BxH ; whence, H=-= -=:12. 
 
 B 15 
 
 Prob. 3. To find the area A of a triangle whose base B is 
 
 12 and whose altitude H is 5. 
 
 Solution. A= 
 
 BxH 12x5 
 
 2 2 
 
 Art. 62, 2. 
 
02 
 
 HOOK I. 
 
 Prob. 4. To Jind tht! ham' B nj' n tn'nihjir. irhoMV iiren A i» 
 ♦10 iind vhoHt' altitude II is H. 
 
 Solution. A:^.*^^."; whence, B-- -^--^^^Ul^. 
 
 I*rob. 5. To find fill- arva A of a traprzoid ichosc altitude 
 \l in n aiidw/iosc parallel Ij(IH4:h li and h an' 12 and S, rrg- 
 pectiiu'hj. 
 
 Solution. A^^ + ^x IIrz:^^±5 x 5=50. Art. 63. 
 
 Prob. 6. The area A it/ a trapezoid i.s 100, t7» altitude II 
 t« 8, au(^ its han" B /.s 15 ; to jind its side h parallel to B. 
 
 Solution. .W^-T^xH : whunce, b=l^— B=^^!l-1 5=10. 
 '.' Jr 8 
 
 Prob. 7. To jiiid the diagonal \) of a square whose side C> 
 is ii. 
 
 Solution. The diairorml I) i.s the hypotheniise of the two 
 e(|ual triangles into which thesijuaTe is divided by the diagonal ; 
 
 hence (72,5'),D^C^+C-; whence,D=v/2C-= C \^2— 'Sx/I. 
 
 Prob. 8. To find the hf/pothenuse II of a right-angled 
 triangle whose sides A and B are 3 and 4 respectively . 
 Solution. H^=A2-|-B2; whence, II=%/a^-|-B^=\/P+P=\/25=5. 
 
 Prob. 9.ToJind the altitude CD and area C^ 
 
 A of an isosceles triangle ABC, in vdiich 
 AB=12, and AC=BCr=10. 
 
 Solution. The altitude or perpendicular 
 CD divides the triangle ABC into two equal right-angled 
 triangles ; Art. 48, 4. 
 
 hence, AC=^ AD^+DC^and AD=DB=6 ; 
 
 whence V\ DC =v/AC=*- AD^x/l00_36=8. 
 
 9() 
 
 A=- 
 
 ABxDC 
 
 12x8 
 
 ^48. 
 
 2 2 
 
 Prob. 10. To find the altitude H and the area A of an 
 equilateral triangle whose sidQ is 6. 
 
 Solution. Let ABC he this equilateral triangle in which 
 
ARTIOLA . 
 
 » I'rrrii, iiii«»-vi» — i m>-=.1 ( Irnh. '.» i. 
 
 III tli*^ ri^ht anj;Uul tiianj.'li* A('l>. 
 
 Ar-=AI>- , CI)-:^AI)--i-ll-: 
 
 whence. l| = v/A(;-- A P-— v/.'JCi— 1>=\/27=:3%/3. 
 
 anil 
 
 A = 
 
 Uxll 
 
 «ix;v" 
 
 ='•>%/;{. 
 
 Prob. 11. To jiinl th*ii r»<i \ nf n ni/n/nr Ikr.inqitti irliosr 
 si (If is J). 
 
 S(H-i;tion. This hexa<:;oii is ilivisible into .si.v iMjiial «Mjnihite- 
 ral trianfjjles whose coninion verf^x is at the center, and each 
 side of which is <». Hut ( I'roi) 1(»). the altitude of each of 
 these six trian<;les is the apotheiii (H) of the liexaLjon. and 
 thereby H=r{\/lJ ; therefore (since perimeter P=:(»x <»=•{<>), 
 
 , HxP :^y/:^x'M .^ .., 
 
 Art. 1 in. 
 
 NoTK. The area of any rcf^alar polygon may be computed by moans of 
 Table IV. 
 
 I'se will be made of Ibis Table TV, by applyintr tli'> sidjjoiued rule 
 which is fountl as follow.s ; 
 Lut ff=area of a legidar po'ygon whose side is c, 
 
 A=tabular area of a similar polygon whose side is unity: then, 
 .since the area> of Iwo similar polygons are proportional to the squares of 
 their homologous sides, there obtains : 
 
 A : rt=l- : f2, or a=-lXc- ; whence, the 
 
 Rule to find the area of a regular polygon whose side is given : 
 
 " Multiply the (a/tnlnr area of the similar poli/ffon hi/ the square of the 
 
 ifiven side.'' 
 
 Prob. 12. To find the anm of o tniptzinm ABCD nhost 
 iliiigoHid AC is 12, and in which fhr per/Zendicnfars Bni tnid 
 Olid [)n are 4 and 7 respect ivrli/. 
 
 Solution, a ABC=:2x 12=r24, -<» 
 
 ..,.d aAI)C=(;x 7=42; 
 
 whence, area of ABCD n^Glj. ^.x jl ^ ^^ 
 
 S I. To find the area of any irrep:u]ar 
 |ii)lygon, proceed as in Prob. 12. ^ 
 
 Prob. 13. To find the side of a square equivalent to a, 
 triangle, a pu> ^illelogram and a trapezoid, irhnnc (irfoa are S^ 
 12 and Hi, rrspecticely. 
 
waammm 
 
 mn 
 
 i 
 
 .,i;l:''I 
 
 HOOK I. 
 
 Soh;tion. If the isidt' of tli" squiirc bo deiiott'd by ./•. its 
 area will })e x- ; 
 
 he nee . .t-= 8 -f 1 - ■ f 1< •= 'i ' » . w 1 le iko , .ir= (5. 
 
 Prob. 14. To find the clirnm/eirmr V, of o circle irhoav 
 radius R is 10. 
 
 Solution. 0=:2rrR=:2()r. Art. 12(». 
 
 Frob. 15. To find the radius R of a circle inhose cirrmn- 
 jerence C is 100. 
 
 {^OLiiTJON. C=2tR ; whence, B^^'—-. 
 
 2;7 r 
 
 Prob. 10. To find the length J* of an arc o/40°, in a circle 
 whose radius Rz=15. 
 
 Solution. 360°:40^^=^2-R:L : whence, L= 
 
 lOr 
 
 Prob. 17. The area of a circle ACBl) /*• ;U4.U;, and the 
 <mgle MON <it the center in equal to 40° ; required 1" the 
 radius R of the circle^ 2'^ the area A itf the sector MONO. 
 
 Solution. !•> Since the area of the circle 
 is rR-^314.1G, and t=H.1410 (art. 120), ^ 
 
 then R= J^Ltii'=v/rO"0=10 ; 
 
 2<>, 360° : 40°=-R- : A ; wlience, 
 . rR2 lOOr 
 
 Prob. 18. To fiiid the sum S of the interior angles of a. re- 
 gular pentadccanon. 
 
 Solution. S=2nLR-4LR=30x 00° — 4x 00°=2340-\ 
 
 Art. 70. 
 Prob. 10. 7h find the number ii of sides of a regular jtoli/- 
 gon in which the sinnS of the interior angles is equal to 1440°. 
 Solution. S=2ijLR— 4LR ; whence, 
 S + 4 L R_ 1440° + 300° 
 ' 2LR ~" 18(1^ 
 
 n- 
 
 = 10. 
 
 Art. 70. 
 
 Prob. 20. When a vertical rod, ii ft high, casts a shadow 
 whose horizontal base is 2^/t long, uhat is the height h of it 
 
ARTrnr.K 1:^;^ 
 
 65 
 
 Htffli'i- that njstti^ at fhr sninr finit.a slimloir tr/insr hoiu'jtutal 
 hiW ix o() ft'ci long ? 
 
 Solution. The Hluulttws olbotli stwplo ami indure .similar 
 triangles in which 2^: ;')(>=(} : h ; wheiuc hm:! 20 ft. 
 
 21. To fi»d the area A of a ti'uiiKjh' vlmsr sl<lrs art 12, lU 
 am) 20. An8. A=:l)(i. 
 
 22. To find the area A of a sqmtre irhnsr illafjnnal is 
 equal to 8. Ans. A=32. 
 
 23. To find the area A of a rhomhoid vihof^e tlicyonals. (J 
 and 8. intersect at right angles. AN8. A:=-24. 
 
 24. To find the area A of a rectangle whose diagonal is 12, 
 
 the base beino 10. Ans. A=20 \/11. 
 
 25. What is the side S of a regular hexagon^ and the side 
 i^' of a regidar dodecagon, lohen these poli/gons an inscribed 
 ill a circle whose radius is unity. 
 
 Ans. S=l,and S'=v/2— v/:^ 
 
 26. When the three sid^s of a right-angled triangle are 
 15, 20, 25, what are the lengths of the tiro segments, s and s\ 
 into which the hypothenuse is <liride.d bi/ a perpendicularly 
 let fall from the vertex of the right angle, and what is the 
 length of p ? Ans. .s=^ I (l s'=9, and p=l 2. 
 
 27. If the length of the minute-hand of a clock is iJ, 
 irhat is the arc A described bi/ its revolving extremity, in 20 
 minutes. Axs. A=2 ;r. 
 
 28. Required the radius K of a rircle in which the 
 urea is numerically equal to thr circumference. Ans. R=::2. 
 
 --Kt 
 
BOOK II. 
 
 SOLID GEOMETRY. 
 
 DEFINITIONS AND PRINCIPLKS. 
 
 1. Two planes, or a .straight line and a plane, are parallel 
 when both indefinitely produced can never meet. 
 
 2. A plane and a straight line are perpendicular to eacli 
 other, when the latter forms right angles with the straight 
 lines of the plane passing through its foot. 
 
 •^. A diedral angle is the amoiuit t>f divergence of two 
 planes. ^ 
 
 These two planes are called /'(irts of the. augle^ and the line 
 in which the faces meet is called edge of the angle. 
 
 4*. The angle corresponding to a diedral angle is a 
 
 plane angle formed by two straight lines lying in the faces, 
 each in each, and perpendicular to the edge at the same point. 
 
 Thus, the lines oP^ op^ lyi"o in the respec- 
 tive faces 7^>, /A), and j)er})endicular to the 
 edge Din, at the same point y, form a plane 
 angle Po/> corresponding to the diedral angle 
 BoT). 
 
 A diedral angle has the same measure as 
 its corresponding angle and may be an acute, 
 an obtuse or a right angle. 
 
 5* A solid angle or polyedral angle is the amount of 
 divergence of several }>lanes meeting, or tending to meet, at a 
 common point. 
 
 This point is called the vertex of the angle; the lines in 
 wliich these jilants meet are called the edgen of the iingle ; and 
 tlie }>(»rtions of the ])lanes lying between the edges are called 
 the fteea oj' the angle. 
 
ARTICLE 8. 
 
 07 
 
 Thus, the three faces ABC, DC, FC, form ^ 
 :i .solid angle wliose vertex is the point C, and 
 wfioso odsres are AC BC, EC. The solid re- 
 j»resented by this figure has six polyedral 
 angles whose respective vertices are the points 
 A.B, CD, E, F. 
 
 i\, A solid angle is called triedral, tetraedral, 
 pentaedral, etc. angle, a<*cording as it has 3, 4, 5, etc, faces. 
 
 7. 
 
 Theorem. 
 
 ^ 
 
 Three points, not in a atraight line, determine the po»iti<ii- 
 of a jilane, and only one. 
 
 Hyp. Let A, B, C be the three points ; 
 
 AsT. then ^4, 7i, (^Metermine the posi- 
 tion of a plane, and only one. ^ 
 
 JJem. Assume a j)lane passing through 
 the 'points .1 and B to rotate about the straight line AB. 
 then one of the various positions, which it will occupy in this 
 motion of rotation, will bo determined by the meeting of the 
 third point C ; but, it is also obvious that, in any other posi- 
 tion, this plane cannot contain tlie three points .1, B, C ; 
 therefore, the three points ^l, B, 6' determine the position of 
 a })lane, and only one. 
 
 Cor. 1. ^1 straight line and a point without that line drter- 
 III i lie the position of a plane ; for, only one jdane can contain 
 flic III both. 
 
 Cor. 2. Two straight linen intersecting inch other determine 
 fill' position of a plane. 
 
 For, if two lines .16^ and BC intersect at (\ one of them 
 (as ^16') and any point (7i) of the other will determine the 
 position of a plane. 
 
 S. 
 
 TllKOREM. 
 
 The intersection if two planea is a straight line. 
 
68 
 
 nooK ir. 
 
 'W P 
 
 .;:i? •'■ 
 
 ^ iu 
 
 I'M : 
 ii.l'l 
 
 llvi*. Lvt AH nud CD be two planes 
 intersecting each other, 
 
 AsT. then will their intersection be a 
 straight line. 
 
 Dem. Through any two points m and n, 
 comnion to both planes, draw a straight 
 line niti. 
 
 Since this line nm has two points m and u, in each of the 
 two planes, it lies wholly in each of tiie planes. Besides, 
 no point without this line mn can be common to both 
 planes ; for, then two planes could pass through a line and a 
 point lying without it, which is impossible (7) ; therefore, the 
 intersection vni of the planes AB and CD is a straight 
 line. Q. E. D. 
 
 Cor. If two juiiiillel planoi arc intersected hy a third phinc^ 
 the lines nif iiite meet ion loill he two parallel straight lines. 
 
 For, both lines of intersection \^ lie wholly in the secant 
 plane, and 2 ' they cannot meet, since they lie in two parallel 
 planes, each in each ; but, two lines which lie in the same 
 plane, and w4iich caniu)t meet, follow the same direction and 
 thereby are parallel. , 
 
 9. 
 
 Thkorem. 
 
 Three parallel planes which intersect two straight linrs 
 divide them into proportional parts. 
 
 Hyp. Let HK, MN and OP be three /y, 
 parallel planes intersecting the straight lines 
 AC 'AuA BD tit the points .1, jP, C, and 
 B, E, I) ; Af, 
 
 AsT. thenwillJ/'':/''6'=i^A':A7>. \ ^ 
 
 Dem. Draw the line 7^6", and assume it to 
 pierce the plane MN" at G ; also draw AB^ q^ 
 EG, GF and CD. 
 
 The two parallel planes OF and JAV afe 
 intersected by a third plane A BC ; hence, the 
 Inies of intersection AB and FG are parallel (8, Cor) ; 
 
ARTICLK 21. 
 
 r»9 
 
 and thereby, 
 
 Likewise, 
 
 therefore. 
 
 m::ED=BG:GC; 
 JFiFr^JiEiUD. 
 
 Q. K. I). 
 
 FACETED SOLIDS. 
 
 DEFLNITIOXS. 
 
 10. A faceted solid i.s a solid }H»undod by either plane or 
 single curved faces.* 
 
 11. The edges of a faceted .solid are the lines in which the 
 faces meet. 
 
 1/i, The vertices of a faceted solid are the points in which 
 the edges meet. 
 
 13. The diagonal of a faceted solid is a straight line 
 joining tlie vertices of two solid angles not in the same face. 
 
 14. The convex surface of a faceted solid is the surface 
 composed of all its lateral faces. 
 
 15. The lower base of a faceted solid is the face on whidi 
 the solid is supposed to rest. 
 
 16. The upper base of a faceted solid is the face opposite 
 and parallel to the lower base. 
 
 A base still retains the name ot'baaa wii n it \i reduced to a point. 
 
 17. The altitude of a faceted solid is the perpendicular 
 distance between the planes of its bases. 
 
 18. A cross-section of a I'aceted solid i.s a section parallel 
 to a base of tVie solid. 
 
 19. A frustum of a faceted suiid is the portion included 
 between two of its cross-sections, or between a cross-section 
 and abase, the lateral edges admitting of no broken li:K - 
 
 ^O. A segment of a faceted solid is a frustum in which 
 one base is reduced to a point. 
 
 4 
 
 31. Similar faceted solids arc tlutse which are bounded 
 by the same number of similar faces, similarly placed. 
 
 
 * A dnnlti curved sur/ac; is one which will lie wholly iu the plane upon 
 which it is developed. 
 
70 
 
 liOOK II. 
 
 *^*4* The parts (faces, edges, angles, tSic) similarly pla('e<l in 
 similar faceted solids are called homologous parts. 
 
 There are two classes of faceted solids, vi/ : polyedrons and 
 polyedroids. 
 
 !S3Ji, A polyedron is a faceted solid bounded }»y polygons. 
 
 Ji4. A polyedroid is a faceted solid whose convex surfa(!t> 
 is composed of single curved faces ; v. g. a polygonal dome. 
 
 35» A wedge of a polyedroid is the portion included Ix't- 
 ween two secant planes which pass through two consecutive 
 lateral edges. 
 
 26. The curve of a polyedroid is the plane curve of any of 
 its lateral faces developed upon a plane. 
 
 A polyedroid is spicifivd by its curve and the polygon of its 
 cross-section. 
 
 !<5'7« The specific mime of a polyedron is derived from the 
 number of its faces, as follows : 
 
 Tetraedron, 
 
 or 
 
 pol 
 
 yedrcn 
 
 of 4 
 
 faces, 
 
 Fentaedron, 
 
 u 
 
 
 u 
 
 " 5 
 
 '• 
 
 Hexaedron, 
 
 (( 
 
 
 (( 
 
 " G 
 
 li 
 
 Octaedron, 
 
 a 
 
 
 n 
 
 '' 8 
 
 
 Dodecaedron, 
 
 u 
 
 
 I, 
 
 •' 12 
 
 
 leosaedron, 
 
 u 
 
 
 ii 
 
 " 20 
 
 
 Prism. 
 
 *^H* A prism is a polyedron uniform throughout itslen;:th. 
 
 A prism may be conceived as a solid gen- zl ,/f 
 
 erated by any polygon (^ABC) moving paral- 
 lel to itself, along a straight line (AD) ; and 
 tlie prism is either right or ohl\q\ie, according 
 ;js the straight line i^AD) is perpendicular or 
 oblique to the generating polygon (AB(^). 
 
 JliO. Cor. From this mode of generating the prism, it fol- 
 lows that, in a right or an obfique prism : 
 
 1" The upper base (^DEF) is equal and pa nd I el to its lower 
 base {ABC). ^ 
 
ARTin.K 40. 
 
 71 
 
 2" All sections jHtmftrl fo thr fiiisrs <irt' ci/iKtl to the latter 
 and equal t(t one tmother. 
 
 H' The eqntil ami paralli'l siilrs of the (jmerat'ing polf/(jnii 
 describe equal ami parallel iiarallelograms. 
 
 4'' If the gencmtuig polygon is (I jmrallelogram. the ojtjut- 
 sife faces of the prism are imrallel and equal parallelograms. 
 
 *AO, A parallelopipedon is a prism whoso opposite tares 
 are parallel and equal parallelograms. 
 
 lit. A rectangular parallelopipedon is one in whieli all 
 the faces arc rectangles. 
 
 3^. A cube is a rectangular parallelopipedon in which all 
 tlie faces are squares. 
 
 33* A right section of a prism is a section perpendicular 
 to its lateral edges. 
 
 34. When a prism is intersected l»y a plane oblique to its 
 bases, each part it^ called a truncated prism. 
 
 Wedoe and Prismoii). 
 
 35. A wedge is a polyedron bounded ^ 
 by a parallelogram, two triangles and two 
 trapezoids, which trapezoids may become 
 parallelograms ; v. g. A BCDEF. 
 
 36. The back of the wedge is the 
 parallelogram {AC). 4 
 
 37. The ends of the wedge are the triangles {ADE and 
 BCF). 
 
 38. The faces of the wedge are the trapezoids, or paralle' 
 lograms (.-IF and J)F). 
 
 39. The edge of the wedge is the line (fJF) parallel to tliB 
 back and in which the faces meet. 
 
 40. A prismoid is the portion of the wedge included bet- 
 ween its back and a section parallel to its back. 
 
72 
 
 BOOK H. 
 
 Pyramid. 
 
 Is 
 
 41 • A pyramid is a polyodron whose base 
 iH a polygon and whose lateral faces are 
 triangles. 
 
 4:*i. The vertex of the pyramid is the 
 common point (*S') at which the lateral faces 
 meet. 
 
 43. A pyramid is triangular, quadran- 
 gular, pentagonal, &c. according as its base is a triangle, a 
 <|uadrilateral, a pentagon, &c. 
 
 44. A regular or right pyramid is one whose base is a 
 regular polygon and whose lateral faces are equal triangles. 
 
 45. The slant height of a right pyramid is the perpendi- 
 cular distance between the vertex and any side of the base. 
 
 4C5. If a pyramid be cut by a plane oblique to its base, the 
 solid included between the section and the base is called * 
 truncated pyramid. 
 
 SOLIDS OF REVOLUTION. 
 
 DEFINlTfOXS. 
 
 47. A .»<(ilid of revolution is a solid generated by the com- 
 plete revolution of a plane figure about a fixed axis. 
 
 48. The convex surface of a solid of revolution is the 
 surface described by the curve of the generating plane. 
 
 49. The generating curve, whatever may be its position in 
 the convex surface, is called element of the surface. 
 
 50. The lower base of a solid of revolution is the face on 
 which the solid is supposed to rest. 
 
 51. The upper basa of a solid of revolution is the- face 
 opposite and parallel to the lower base. 
 
 5*4 » A cross-section of a solid of revolution is a section 
 perpendicular to the axis of the solid. 
 
 51^. A frustum <»f a s< lid of revolution is the portion in- 
 
AllTlCLK ()4. 
 
 7a 
 
 clnded between two i>t' its cross sections, or between a cross- 
 hcction and a l)ase. 
 
 54. A segment of a solid oi' revolution is a trustiun ii» 
 which one base is reduced to a point. 
 
 55. A wedge of a solid of revolution is the portion 
 included between two secant planes whose line of intersection 
 is the axis. 
 
 56. A lune of a solid of revolution is the convex surface 
 of a wedge. 
 
 5*7. A zone of two bases of a solid of revolution is the 
 jiortion of the convex surface included between two cross- 
 sections or between a cross-section and a base. 
 
 58. A zone of one base of a solid of revolution is the por- 
 tion of the convex .surface included between a cross section and 
 the vertex of the solid. 
 
 59. The altitude of a solid of revolution is the distance 
 between the planes of its bases. 
 
 (50. A solid of revolution is specified by its curve wliich is 
 the curve of its generating plane. 
 
 61. Similar solids of revolution are those whose curves are- 
 ^iniila. . 
 
 63. A polyedroid is circumscribed about a solid of 
 revolution when the upper and lower bases of the poly- 
 edroid are circumscribed about the upper and lower bases of 
 the solid of revolution, respectively ; and when each lateral 
 face of the polyedroid is tangent to the convex surface of 
 the solid of revolution, along a line called element of contact. 
 
 6J:5. The solid of revolution is then said to be inscribed in 
 the polyedroid. 
 
 64. 
 
 Right Cylinder. 
 
 In a revolution about ^IZ?, the rectangle AC ^ 
 will generate a right cylinder ; the side /)C 
 will generate tl»e convex surface of the 
 cvlinder, and the e<|ual sides AD. AT'wil/ ^ 
 
 -a:; 
 
 ■vM 
 
 i ■■■' t»i 
 
>i:vi. 
 
 74 
 
 liOOK II. 
 
 generate equal and parallel circles which are the bases of the 
 cylinder. The generating line CI), whatever may beitspcwi- 
 tion in the convex surface, is called element of the surface* 
 
 65. 
 
 l{i(*iiT Conk. 
 
 In a revolution about AB, the triangle 
 AB(\ right-angled tit B, will generate a 
 right cone whose convex surface and circu- 
 lar base will be described by the sides Ai^ / 
 and BC. respectively. 
 
 The straight line A(\ whatever may be its position in the 
 convex surface, is called the slant height of the cone. 
 
 66. Hpiieuk. 
 
 See Introduction to Spherical Triff. froiii 71 !(•!»<;. 
 
 6*7. Theorem, 
 
 The convex surf ace of (I right lir ism is equal to the 2)roduct 
 o/ (I Intend edge ofthisjtrism and the perimeter of its base. 
 
 Hyp. A.ssume ABC-E to be a right prism ; 
 let E='A lateral edge ; P= perimeter ABC 
 ;ind >V= lateral surface of ABC-E ; 
 
 AsT. then will S=ExP. 
 
 Dem. By the definition of a prism, all the '^ 
 rectangles AE, CF, BD, have equal altitudes 
 I'J ; hence, the sum of their areas is 
 {AB-hBC-^ CA)E=Px E ; that is, S=Px E. Q. E. D. 
 
 Cor. 1. The convex surface (*S*) of an oblique jjrism is 
 ^■quid to the product of its lateral edge- (^E) and perimeter {P) 
 ^f its right section. 
 
 For, let any right prism be cut by a plane oblique to its 
 bases, and connect the two truncated prisms so that their bases 
 jierpendicular to the lateral edges shall coincide in all their 
 parts ; then, the right prism "ill be changed into an oblique 
 
AHTH'I.K (»>.. 
 
 '5 
 
 nrisin. and tljo periini'trr /* n[' the Ikisc in tlie t'ornicr will 
 liccKiiM' the pcriniotcr /*' itfa riulit section in tho latter : tliat 
 i>. 7* is identical to /*'. Hut. in l»otli jirisins. the areas .V, a"* 
 well as the lateral ed<:es h\ are also identical : hence, in an 
 ul»li<(ue prism, S—Kx I*'. 
 
 Cor. 2. Thi' confer siir/dcr nf' II n'lj/it cijlindt'r is fqnal to 
 tin prod net of nn eirment of tin sin/iuc of this cf/finder and the 
 rirriun/errnce of its htisf. 
 
 Vov. a riirht cylinder is a riiiiit prism whose base is a circle, 
 iuid in which a lateral edge is an element of the ►^vrface. 
 
 Cor. 3. The roiiri'X surftn of mi ohliqiif ct/linder is equal 
 III tlif pi'odiirt of an chnifnt of fin surf arc of this riflindfr 
 mid, thr circnmfn'CHCi' of its right srction. 
 
 For. an obli«|ue cylinder is an obli<iue i»rism in which a 
 ri«:l»t section is a circle. 
 
 CJ8. 
 
 TlIEORK.M. 
 
 The surfaces, as veil as tin homologous faces, of two similar 
 /loli/edrons arc proportional to the squares of the hitmologoiis 
 edqes. 
 
 Hyi». Let <S' and .s = surfaces of two similar polyedrons, res- 
 pectively ; A and « = two homolo<i(>us faces. 
 
 L and / = two homologous lines of A and a ; 
 
 AsT. then will *S' : s = vl : a = L- : /-. 
 
 I )E.M. In the hypothesis, A\a=^L-\i-. 1,79, (Jitv. 
 
 Besides, similar parts of similar Uiagnitudesare proportional 
 to these magnitudes ; therefore (since viand (rare similar parts 
 of,S'ands), S\ s = Aia = L' -. P. Q. A'. D. 
 
 Cor. In am/ two similar solids, the honadogons faces, the 
 lumwlogous sections, as icell as the whole surfaces, are propor- 
 tional to the squares of ang tioo homologovs lines. 
 
 For, the surfaces of two similar solids may be divided into 
 infinitely small triangles, homologous each to each ; hence, in 
 t wo similar solids, S : s = A : a = L- : P. 
 
 m 
 
 Ml 
 
 ; { 'f 
 
7iJ 
 
 V.niilv" I. 
 
 iiU, 
 
 TllKoHK.M. 
 
 ':1 I.. 
 
 * 
 
 tP 
 
 til' ia. • 
 
 i 
 
 I ^^': 
 
 
 77*r rohuuf of n I'ttjlit i>r mi ith/i«fKr pn'sni in t^ifiiiil to the 
 proifiirf nf its hiist'uml a/fifiiifi'. 
 
 Hvi'. Assume any pi'if'Jn to Itc (lividiMl into iMjual se<;ments 
 by planes parallel to its Inise. and let 
 
 |'=voliune. y/=altitii(le. y/ = nuiiil)er of sefrments, of tin • 
 prism ; (;=:volunn'. /t=altitu(le. of each segment ; 
 
 /i=base ol' the prism and of each segment (these buses are 
 ecjual. 2\)) ; 
 
 Asr. then will V=J>x//. 
 
 Dkm. The volume ( l) of the prism is e«jual to the sum 
 (y-)-y-f-<l''' = " X I') of its parts ; whence, the e(juati«>n, 
 
 V=:.IX>K (1) 
 
 Likewise, the altitude ( // ) of the prism is e(|ual to thesum 
 (li,-{-lt-{-d-r=ii x/>) of its parts ; whence, the ecjuation, 
 
 J/=,ix/t. (2) 
 
 If the altitude // be taken as a unit of linear measure, equa- 
 tion (2) reduces to //=yt , then, substituting // for n, in (I), 
 there obtains V=Uxo. (H) 
 
 Now, this equation (8) is always true, whatever may be the 
 thickness of the segment v ; hence, it remains true when this 
 thickness assumes any of the values through which it passes in 
 progressively decreasing till it is reduced to a mathematical 
 point ; but, at this limit (since the segment v is reduced to a 
 mere surface which is its base 7i), v=B ; therefore, by subs- 
 tituting B for V in (3), there obtains finally : 
 
 V=IlxB. Q.E.D. 
 
 Cor. 1. llw volume of <i tight or an oblique cylinder in 
 equal to the jirodmt of its base und (dtitude. 
 
 For, the preceding demonstration applies to the cylinder as 
 well as to the jtrism. 
 
 Cor. 2. 1" T'lfo right nr oblique prisms or cylinders o/ 
 equal altitudess are to each other as their bases, 
 
 2 • 'fu'o right or obliqur prisms or cylinders of equivalent 
 bases are to each otlu r us tin ir altitudes. 
 
ARTICLK 7(». 
 
 4 I 
 
 3'J Tii'O right m- nhltqurjyriHmit or cifliiidtrn of tiiinil affifinlts 
 and equivalent huHen are equal ht vohimeH, 
 
 Sch. Denoting by Tanil r, the respective volumes of a prism 
 and a pyramid of eijual altitudes (//) andetiual bases (/i),aiid 
 
 jetting ",= ", there obtains ; ^'==« V=>iBxJf. 
 
 DOses are 
 
 7(>. 
 
 THEORKM. 
 
 If a pyramid he cut hi/ a ^^hnie p<iralhl to Ifs hasr, the seC' 
 tioii and the hase will he similar jwli/gans. 
 
 Hyp. Assume ahcd to be a section 
 parallel to the base ABC1> <»t' a 
 pyramid ABCD-S ; 
 
 AsT. then will both polygons be 
 similar. 
 
 J)em. Since the sides ah, he, dr. of 
 the section are respectively parallel 
 to the sides AB, BC, &c. of the 
 base (8, Cor) ; 
 
 then, L-4=La, L5=L/>, dr. 
 
 In the similar triangles ABS and ahS, AB : ah — BS : hS. 
 
 In the similar triangles J5C*S'and heS, BC : hc= BS : hS ; 
 
 whence, ABiah = BC\hc=rDx ai=(f'c. 
 
 Hence, the polygons ABCD and ahcd are mutually equian- 
 gular and their corresponding sides are proportional ; therefore? 
 these polygons are similar. Q. E. I). 
 
 Cor. In two similar pyramids, the altitudes are proportional 
 to the homologous edges ; and the homologous faces are propor- 
 tional to the squares of the altitudes and of any two homolo- 
 gous edges. 
 
 For, in the similar pyramids ABCD-S and ahcd-S, 
 lo AB:ah=BO:hc=CS:cS=PS:pS=dhc. 
 2" ABCS : AhcS=BC' : he' .- 
 
 whence, ABCS : AhcS=: PS' : pS-^= CS' : cS'= dw. 
 
 n 
 
 ^:4 
 
 ■■■■,:•[; 
 
 
t 
 ; 
 
 I 
 I 
 
 i 
 
 :i M 
 
 78 
 
 7K 
 
 IJiitiK II. 
 
 Tmkokk.m. 
 
 Tf two pifnimhh of fhr smin ttlflfitdr hv cut hif planes j"^''"^ 
 h! til thiir lntsf.t^ fltr sn-ffoiis rrjii i(/is/ii ii f f'min. tin' host's vi/l 
 he pi'ojxirfioiin/ l"/o tlir hnsrs, 2' /n flir mfumcs of flirsi 
 pi/niniufs. 
 
 \> Hyp. A.ssumo AIU'DS and S 
 
 FdH-SU) )»' two pyrajuids i>\' the 
 suinc altitiulo }*S, ami lot 
 
 .1 :=aroaofM/>V7>, 
 /^=art'aof'7'Y;//, 
 J' = aroa<)fa/><Y/, 
 />"=:art'a o{'/<jh ; 
 
 A' and i?'=.secti<»ns efjuidistant ^ 'S ^ 
 
 tVoni. and parallel to. the bases .land B. 
 a and A^.section.s e<|uidistant from, and iiarallel to. tin- 
 bi'se.*^ .1 and B, 
 &e. =!>e*ition.s ecfuidistant from, and parallel tii. the 
 bases .i and B ; 
 
 . .1 -n 'I -I' "' I' 
 
 A.^T. then Will = - --—zzzSiQ. 
 B H' b 
 
 Dk.m. In the similar poly<>;ons .1 BCD, ahcil^ A : A'~BC-l he-. 
 
 In the .similar triandes BCS and hcS, CS:rS=BC : bv. 
 
 In tlie .similar triangles (^ PS und cpS, 
 
 hence, 
 
 Likewise, 
 
 therefore. 
 
 rS:cS=PS :pS : 
 A:A'=PS'-,pS-. 
 B:B'=PS':pS'; 
 A^A' 
 B B' 
 
 A similar demonstration would prove that 
 
 A a „ 
 
 2" Hyp. Furthermore, let y=:»A ; Fand F' = the volumes 
 
 B 
 
 of the pyramids ABCD-S and F(tH-S, respectively ; 
 
 AsT. tlien will _ = w = ' ="- = z=&c. 
 V li B' h 
 
 Dem. The polygons A. A\ a, &c'. and B, B\ h, &c may b.' 
 
ARTICLK 72. 
 
 7a 
 
 regarded as .so niiiny infinitely tliin frusta wliitli constitute the 
 respective elements of the volumes I' and T"; but, tiie whole 
 is equal to the sum of all its parts ; 
 
 hence, V ^ (7J+ /r + 6-i &c.),(l) 
 
 and V = (A-^ A' -\-a-\-kc.) =r mi^J}-\-B' ■i-b-\-&i:.).(2) 
 
 |)ivid.(2)by(l), =wt; h"t, m =' =^ = —kc. \ 
 
 ■herefore, 
 
 V 
 
 A A 
 
 «i = = — - =&c. (3) 
 
 Q. E. 1), 
 
 Cor. 1. Two pyramids of equal altitiuh.n and iqiiivalent 
 hast's are equal iu volume. 
 
 For, if the areas -.1 and J^ are e(jual in (i ;,tlie volumes 
 Fand V are also equal. 
 
 Cor. 2. Tti-o right or oblique cones of equal altitudes are 
 proportional 1" to their bases, 2" to their sections parallel fo, 
 and equidistant J'rom, their bases. Besides, they are equal In 
 colume, ij^ their bases are equivalent. 
 
 For, both cones may be regarded as pyramids whose bases 
 are polygons having an infinite number of sides. 
 
 Thus, let Fand r/]f-'=volume and base of u right cone, res- 
 pectively, V and B =volunie and base of an oblique cone, oi 
 of a right or an oblique pyramid, of the same ultitude a^ the 
 right cone. 
 
 zR''^ and J5'.=: sect ions parallel to, and equidistant from, the 
 bases ttR' and B, 
 
 - r^ and 6 = sections parallel to, and ('(juidistant from, the 
 bases -i2- and B, 
 
 &c. =sections p.'rallel to, and equidistant from, the 
 bases -R- and B. 
 
 In this hyp., 1" and 2", —■- — z='- =-— =:«!it'^ 
 
 ^^ ' 'F B B' h 
 
 And,if r/2-and /^areequal, F= V',r:K'=B,::R' = B',7:r'^b,&(i 
 
 '72, Theorem. 
 
 Tf a solid of rivolutlon and tin- i-l rridnsfrlhrd pnlyedrold 
 are cut by planes perpendicular to the o.i ,,s nf revolt* tiai : 
 
 44 
 
 
 ! 
 ■ 1 
 
 ■ i 
 
 • i 
 
 ) >^ i 
 
80 
 
 BOOK H. 
 
 Mi 
 
 ':m 
 
 1 ' The sections of the j)(>lfje*f.roid are so many similar 
 polygons ; 
 
 2'* The volume of the solid of revolution <ind that of the 
 polyedroid are to each other as the sections determined by the 
 same secant jtlane in both solids ; 
 
 H> The convex surface of the solid of revolution and that of 
 the jwlyedroid are to each other as the sections determined 
 by the same secant j>la.ne in both surfaces. 
 
 l'» Hyp. Assume ABC J) to be a por- 
 tion of a polyedroid circumscribed / 
 about the solid generated by the revo- 
 lution of ^'17?C about the fixed axis^lZ? ; 
 the curve ANC being the element of ^ 
 contact, and the curve AOD, a lateral 
 
 edge. In this hypothesis, planes perpendicular to the axis AB, 
 at any number of points B, E^ 3f, <fec.. will determine polygonal 
 sections in the polyedroid, and circular sections (whose radii 
 will be BC(= T), 'EF{=: F), il!fA^(=7/),&c. j, iu the inscribed 
 solid of revolution. 
 
 Let r}"', ttF'-, -y-, c!£c. = these circular sections, 
 
 andi^j yi', 6, cfc'c=the polygonal sections circumscribed about 
 these circles - Y\ r 1"'^, ;r^-, &c., respectively ; 
 
 AsT. then will B, B', b, d'e. be similar polygons, 
 
 Dem. The planes ABC and ABDvfiW cutout of the parallel 
 polygons B, B', b, tfce, similar triangles BCD, EFG, MNO, ii-c. 
 
 For, these triangles are right-angled at C, F,^, &c., respec- 
 tively, uince the sides CD, FG, MO, d'c. of the polygons B, B', 
 b,d'c. are tangent to the inscribed circles rF"-*, ttF'^, Try-, &c. 
 and perpendicular to the radii BC, EF, MJV, &c. at the points 
 of tangency C, F, iV", &c. (I, 107) ; besides, their angles at 
 B, E, M, &c. are equal, as angles corresponding to the same 
 diedral angle ABCD ; hence, these triangles 5Ci), EFG,MNO, 
 &c. are mutually equiangular and thereby similar. 
 
 A similar demonstration will prove that the other planes 
 intersecting one another alon Mie axis AB and passing 
 through the elements of contaf d the lateral edges of the 
 
ARTICLE 72. 
 
 81 
 
 polyedroid, respectively, will divide the remaining parts of 
 the polyjrtms B, B' , b, A:c. into an equal number of similar 
 (riantrles similarly placed ; therefore, B, B\ h, «!tc. are similar 
 lM»ly<^(»ns. / Q, E. I), 
 
 2' Hvi'. Furthermore ; let - — ='i; Tand V" = volunK's of 
 the solid <»f revolution and of the polyedroid, respectively ; 
 
 :&C. 
 
 AsT. then will __>,_" — " _."•>' . 
 
 Dkm. The similar polygons /i, H' />, &c are proportional to 
 
 the s«juares of their homolouous lines BC(=:Y),EF{ = Y')^ 
 
 M.Vi =»/),&€.( 1,79, Cor) ; whence, B : Y-=B': Y-=h ; y-=&c., 
 
 zY' -Y' -,/• / 
 or — -•'- — tvc =n. 
 
 B~ B' ~ h~ 
 These various circles -Y-^ -Y'-, rry-, &c. and the polygons 
 />', B\ 6, &c. may be considered as so many infinitely thin 
 fruHt vhich constitute the respective elements of the volumes 
 laii'! ^ ; but, the whole is equal to the sum of all its parts ; 
 lience, r =(B-\-B'-]-b-\-kv.). (1) 
 
 ind F=(rP+sr- + -y- + &c.) = H(/J-fi?' + /> + &c). (^) 
 
 I)ivid.(2)by(l).l =n ; but, n -ZXl='lll'=l!!i:=.&ii ■ 
 
 V .Y r:Y'^__r:y-_^ 
 
 therefore, — --» — - — z=z 
 
 'F B B' 
 
 
 (H) (I K. />. 
 
 3" Hyp. Furthermore, 
 
 let C. r', r, &c=circuniferences of circles -F", r.Y"-. -y-. <Scp ; 
 
 P, P', ;>, &c=:perimeters of polygons B, B\ />, tScc. ; 
 
 »S' and J=oonvex surfaces of the solid of revolution 
 
 C 
 md of the polyedroid, respectively; and -y=zni; 
 
 AsT. then will -—m = rj=jj,=-=k('. 
 
 A r F ji 
 
 Dkm. In the similar polygons B, B\ b, &(;., the perimeters 
 
 /*, P',j>, &c. are proportional to the homologous lines BC(=Y)f 
 
 EF{^=:Y'),MiV{=i/).kc. ; 
 
 
 M 
 
82 
 
 BOOK II. 
 
 
 41 
 
 . 1 ji 
 
 whence, 
 or 
 
 or 
 
 2-r_2-y' 2-,/ 
 
 r _ c _ r 
 
 n — ^ 17' — :=&c=-.m. 
 
 P ~ P' p 
 
 The circiimfereiu'es t\ C\ r,,&c. and the perimeters P, P\p, 
 k(-\ may be considered as so many convex surfaces, in each one 
 ((f which the altitude is infinitely small, and which constitute 
 the respective elements of the surfaces .S^ and .t ; wheiu^e, 
 A = (/'+/^'-|-;,-f-&c.). (I) 
 
 and *V=,C+r"-f-.--f-&c.)=/». /^+/^'-fp + &c.), (2) 
 
 I>ivid.(2)by (1 ),'==: w; hut, m =y,^.~=^z=ScG.- 
 
 therefore, ' —ni=:z == . =z =&c. 
 
 A P r p 
 
 (1) V- -'>'• />• 
 
 Cor. 1. Letting Y, Y', y, Sn.\=the homologom lines CD, 
 FG, NO, &c, respectively, A'=(j.reii of the portloti ACD of the 
 
 c 
 
 convex surface oj the poli/edrold, and :rz=m' ; there obtains : 
 
 For, the homologous lines Y, Y', y, &c. are similar parts of 
 
 P P' i) 
 the similar perimeters 7^, P',/),&c; hence,— =;—-=■' =:ikc. (5) 
 
 C C r 
 Multiplying (5) by the above (4), ~=~= =^fkG=m\ 
 
 ByT)em.3", A' = (Y+ Y' + y+i^c), (6) 
 
 S={C+C'-]-c-{..^c.\:=ni'{Y-\-Y'-^y-\-ikc.). (7) 
 
 S 
 
 c c 
 
 T 
 
 Divid. 7)by((Ji,_ — ;H'=Y=;r^,= -^^c. 
 
 ^8) 
 
 73. Cor. 2. (riven the volume V and the radius Y of an}/ 
 cross-section of a solid of resolution, then {since the area of the 
 circle ~Y' and of the poli/gan B, circumscribed ahont -)'^ can 
 he amiputed) irill the cohime V of the circumscribed polijedroid 
 he found bij th' forninla. 
 
ARTICLE 75. 
 
 V'~V 
 
 B 
 
 83 
 
 {'^'\ 00 ) 
 
 Conversely. Given V and li, thr. rolume V v^iU Ix found 
 h}i tilt' formula V = V'"^ — • 
 
 14:. Cor. 3. Given the convex surface S of a solid of revo- 
 hifion and the radius Y of any of its cross-sections, then {since 
 the circumference ''Ir.Y or C^andthe.jierimeter P of any polj,- 
 tfini circumscribed about C can be computed) will the concur, 
 surface A of the circumscribed pofyedroid be found by the for- 
 
 m 
 
 ula 
 
 A =S 
 
 yP 
 
 c 
 
 (^\ (4) ) 
 
 Conversely. Given A and P, the convex surface S will be 
 
 (J 
 found by the formula ^^^=A — 
 
 75. 
 
 Theorem. 
 
 The volumes of tuo similar polyedrons are proportional to 
 the cubes of their homologous lines. 
 
 Hyp. Assume both polyedrons to be divided into hoiiiolo- 
 uous pyramids whose bases are the faces of the polyedrons ; 
 and let F=volume, 7i=z:base, //zr:altitude, of one of these py- 
 ramids, ?;=voliime, />;:=base. /i=altitude, of the homologous 
 ))yramid, 
 
 V and F' ^volumes of the polyedrons, respectively, 
 I J and /=:two homologous lines of the liomologous pyramids ; 
 
 AsT. then will V''.V"~[J:t\ 
 
 Dem. In this hypothesis, V-::=inB x H-, 
 
 and v=nb x /* ; 
 
 whence, V : v=zB x //: bxh. 
 
 Besides, B -. b=^ Jl'ih', 
 
 or h X IP=B X h'. 
 
 Multiplying, by (2), the second couplet of (1), term by term, 
 there obtains, V\ v==BIIx bll-.bh x Bh'=]/':P. 
 
 But, the altitudes, //and h. of two similar pyramids are pro- 
 portional to their homologous lines L and / lArt. 70. cor.); 
 
 69, Sch. 
 
 (1) 
 
 70, Cor. 
 
 (2> 
 
 tm 
 
 11 
 
 
 11 
 
 II 
 
 
It 
 
 
 
 ■ 'a' ', 
 
 ft-. 
 
 111?*! 
 
 84 
 
 BOOK II. 
 
 hence, Vw^I?'.!^. Again, similar parts of similar magni- 
 tudes are proportional to these magnitudes ; therefore (since 
 the volumes, Fand r, of the homologous pyramids are similar 
 parts of the whole volumes V and V" of the similar polyedrons), 
 
 F: V"—V:v=L^'.K Q. E. I). 
 
 Cor. The volumes (V and V") of any two similar solids are 
 to each other as the cubes of their homologous lines. 
 
 For, two similar solids may be divided into infinitely small 
 triangular pyramids, homologous each to each. 
 
 76. 
 
 PRACTICAL APPLICATIONS 
 
 Prob. 1. The altitude 11 of a right cylinder is 20 and the 
 radius R of its base is 8 ; required its volume \. 
 
 Solution. V=zB x H=^;rR2 x Hz=180r. Art. 69, 1 . 
 
 Prob. 2. The radius R of an oblique cylinder is 3, audits 
 element E of surface is 20 ; required its lateral surface S. 
 Solution. S=E x C=20x 2-R=120- Art. G7, 8. 
 
 Prob. 3. The altitude II of a right prism is 10, and its 
 base B is a triangle whose sides are 3, 4 and 5 ; required its 
 lateral surface S and its volume V. 
 
 Solution. S=:ExP=10x 12=120; 
 
 V=B X H=-. 0x10= GO. Art. 67 cV 69. 
 
 Prob. 4. Given the length 1. breadth b and thickness t of a 
 rectangular paralleloplpedon ; required the lateral surface S, 
 volume V and diagonal D. 
 
 Solution. S=P x K=2(b+t) x 1 : V=B x H=rbt x 1 ; 
 
 D=v/p+b--(-tl 
 
 Prob. 5. Given the volume \ of a solid of revolution and 
 the radius R of any of its rrosssections ; required the volume 
 V of the circumscribed right quadrangular j)olyedroid. 
 
 Solution. The area B of the square circumscribed about the 
 
 circle tR- is equal to 4R-. hence, V'=:T 
 
 . B .Y4R-_y4 
 
 -W 
 
 tR- 
 
 Art. 73. 
 
ARTICLE 76. 
 
 85 
 
 Prob. 6. Given the convex surface S of a solid of revolution, 
 and the radius 11 of any of its cross-sections ; required the 
 convex surface A o/' the circumscribed right hexagonal 
 pohjedroid. 
 
 Solution. The perimeter P of the regular hexaj'on circum- 
 scribed about the circumference 2-R is ('((ual to 4R\/H ; 
 
 hence, Az=S ^=S ^ ^y^=S h^. Art. 74. 
 
 C 2rrR - 
 
 Prob. 7. Given the radii a, b and c of three cylinders and 
 their common altitude H ; required the radius R of an equiva- 
 lent cylinder ivhose altitude is also H. 
 
 Solution. rR-'H=z.-a-H + rb-H + rc-H ; 
 
 whence, R=\/a--|-b"+c-. 
 Prob. 8. Required the edge X of <i cube equivalent to a 
 prism P, a cone C and a sphere S. 
 
 Solution. X-^P + C + S; wheence, X=i^P+CH-S. 
 
 Prob. 9. B=/o?rc/"iasR ABC, h^=upperhaseo\>c, A and2L= 
 two homologous sides o/*B and b, in the frustum ABC-c of a 
 pyramid; required b in function of A, :i and B. S 
 
 Solution In the similar tranglesB and b 
 
 (art. 68), X 
 
 A-:a='=B:b; ^^ 
 
 whence, b=B (-r ) • 
 
 Prob. 10. Required the median section B' (jyarallel to B) 
 of the preceding frustum. 
 
 Solution. In this median section B', the side A' homolo- 
 
 A + a 
 gous to A and a is equal to ' : hence (68), 
 
 A-:(^Lt^y=B:B'; whence, B' = B(^^+''y. 
 
 11. Required the altitude IT of a cylinder whose radius is 5 
 feet and which shall contain 250;r cubic feet. Ans. H= 10 ft. 
 
 12. Required the volume V of the rectangular parallelopipe- 
 don circumscribed about the preceding ci/linder. 
 
 If 
 
 
 f 
 
 'A. 
 
 fel 
 
86 
 
 BOOK II. 
 
 
 Ans. V = 1000 cubic ft. 
 
 13. A cylindric tub, whose, interior radius is 10 inches, is 
 half filled with water; an irrejiular «nrf impermeable body 
 plunged in that water makes it rise 5 inches; required the 
 vohimc V of the bodt/. Ans. V = 500r cubic inches. 
 
 14. A right cylinder to radius H is circumscribed about a 
 regidar hexagonal j)7' ism ; required the ratio of their couvex 
 surfaces (S and S') • Ans. S : S' = t : 1]. 
 
 16. A right cylinder to radius R is circumscribed about a 
 prism whose base is an equilateral triangle ; required the ratio 
 of th cir volumes V and V ' . Ans. V : V ' = r : f v/-^- 
 
 16. Required the edge x of a cube which is equivalent to a 
 rectangular parallelopipedou whose dimensions are 4, and9. 
 
 Ans. x=6. 
 
 17. Required the edge x of a cube whose entire surftce is 
 numerically equal to its volume. Ans. x=G. 
 
 18. The radii of two spheres are to each other as 1 is to 3 ; 
 required the ratio of their volumes V and V. 
 
 Ans. V:A^' = 1:27. 
 
 19. A wants to dig a circular pond so that the difference of 
 level between its horizontal bottom and the horizontal embank 
 mcnt shall be 10 feet, the interior radius 90 feet and the exte- 
 rior radius 100 feet ; what must be the thickness x of the hori- 
 zontal layer of soil taken out for the embankment ? 
 
 Ans. x=1.9 foot. 
 
 20. A, B and C, having bought a sugar loaf, want to divide 
 it equally among them by sections parallel to the base ; required 
 the altitudes, x, y and 7. of A, B and C's shares, supposing the 
 loaf to be a cone whose altitude is 10. 
 
 Ans. x=6.934; y=1.8; z=1.266 (art. 75, Cor). 
 
 Mi 
 
ARTICLE 7<, 
 
 87 
 
 77. APPLICATION' OP ALGEBRA TO GEOMETRICAL 
 
 PROBLEMS.* 
 
 Prob. 1. Given the Ju/pothennxe li tnid the sum s of the 
 other two sides of a right angled triangle ABC ; required these 
 two sides. q 
 
 Solution. Let x=AB, andy=BC. 
 
 Then, x-|-y = s. 
 
 and (I, 72, 50 x^+y-'^h^ A> 
 
 whence, 
 and 
 
 s±%/2h--s- 
 ^ 2 
 
 ,_srF%/2h-— s^ 
 
 2 
 
 
 Prob. 2. Given the base b and the alt itudeh of <iny triangle 
 ABC; required the sides of the inscribed rectangle whose alti- 
 tude is to its length as 1 is to c. 
 
 Solution. Assume amnc to he the 
 required rectangle. 
 
 JLetting 
 
 x = nin, y=am ; 
 
 then, Co = CD — nD=h— y. 
 
 In the similar triangles ABC and mnC, 
 
 AB:CD=mn:Co, 
 or b: h— X : h— y ; whence, hx = bh— by. 
 
 By the data, y: x= 1 :c ; whence, x=cy ; 
 
 hence. 
 
 x=. 
 
 cbh 
 
 andy=. 
 
 bh 
 
 eh -r b' *' ch-f-b 
 
 Prob. 3. Given the base and the attitude of any triangle 
 
 ABC ; to divide the triangle into three equivalent parts, by 
 
 lines parallel to the base. C 
 
 Solution. Let base AB = 2b, 
 
 altitude Co=h, 
 
 Cm=x, and Cn=y. 
 Then, AABC=bh,andeach ofthe ^ 
 
 three equivalent parts = - ; 
 
 ♦ These prob'ems belong to Book I ; yet they are placed here, on ac- 
 count of their being partly intended as a proximate preparation to the 
 Analytical geometry of the following Book. 
 
 
 ::il 
 
 
 \ i 
 
 M ^ i 
 
 
 
 Ha 
 
88 
 
 HOOK II. 
 
 ■1 
 
 heiico (I, 7S>, Cor), hh : — =li- : x-; wliencc'!,x = — =^ 
 
 ni. 
 
 and 
 
 2bli 
 
 bh ;:i__ = h- : y-; whence, y = hv^i| = Cn. 
 3 
 
 Throujih the })oiiitM m and n, draw lines parallel to AB, and 
 the triangle A13(' will be divided into the equivalent parts 
 re(iuired. 
 
 Prob. 4. frivcu the. haxr. h^diul the sum a offhrjii/pofhcriuse 
 and (lit i tilde, of anij I'ight-anghd triangle AHC ; required the 
 hf/pothenuse and the altitude. 
 
 Soi.iTTioN. Let hypotheniise AC = x, 
 
 and 
 
 Then, 
 
 and (I, 72, 5") 
 
 whence, 
 
 altitude BC=y. 
 
 X +.y =«, 
 
 X-— y-=b-; 
 
 x = 
 
 b- + «" , s--b- 
 jinH v = 
 
 2s 
 
 -, and y = 
 
 2 s 
 
 Prob. 5. Given the three perpend icularsii, b and c draion^ 
 
 J'rom anjj interior point o, to the three sides of an equilateral 
 
 triangle ABC ; required the side AB. 
 
 Solution. Letting altitude CD=h, 
 
 and 
 then, 
 and 
 or 
 
 Besides, 
 
 AB=AC =BC = 2x; 
 AD=DB=x (1,48, 3), 
 AC-=CD-'-f AD-(I,72, 5 ; ^^ 
 
 a r> 
 
 4x"= h--|- X-; whence, h=:x ^3. 
 AABC = CDxAD= hx=xV3. (1) 
 
 Joining, by lines, the interior point o and the vertices A, B 
 and C, the triangle ABC will be divided into three triangles 
 wliose bases are each equal to 2 x and whose respective alti- 
 tudes are the perpendiculars ao(=a), bo( = b), and co(=c) • 
 hence, the areas of these three triangles are ax, bx and ex, 
 respectively. 
 
 Therefore, AABC=ax4 bx + cx=(a+b+c)x. (2) 
 
 From (1) and (2), h= x^3 = a-fb-f c ; (3) 
 
 whence, AB= 2x = 51^+^1'. 
 
 \/3 
 
AKTK'LK 77. 
 
 89 
 
 Prob. 6. Given the base h undthr ultitudi \\>)j' uny triangle 
 AB(.\ and the area a of the inscribed rcctangb ■ required the 
 sides of the rectangle, c 
 
 {S(>h:T[()N. Jjet x=Do = ain. li — x = (^^. 
 
 Tlie area a = iiinxx; /nA 
 
 whence. 
 
 inn: 
 
 a 
 
 X 
 
 In the similar trianiiles ABC and nin('. CD: \li = (^t; nin 
 or 
 
 whence 
 
 x = 
 
 : I. =h-x:**; 
 
 . and nin= = 
 
 X 
 
 
 hrt |h-'_i!*''' 
 \ b 
 
 Prob. 7. Given, in, <( right-angled triangle ABC, the sum 
 s of the three .'i ides and the tdtitnde li or perpendicular let fall, 
 from the vertex C of the right angle, on the hypotJienuse ; 
 required the sides. C 
 
 Solution. Lettin<i ACzrrx. BC=ry ; 
 
 then AB=::H — (x-f-y). 
 
 In triangle ABC, ACM-BC-— AB-, ^^ TT^S 
 
 or xH y~J.s-(x-fy)|-; 
 
 whence, -xy= ^ •'*(x + y ) — '*'' 
 
 and 4 A ABC =2 AB x C I), 
 
 or 2xy=:2hjs-(x+y)( 
 
 !^(2h + s) 
 '27h+«)' 
 h8=* 
 
 ^ 2(h+s) 
 
 Combining ( 1 j and (2), 
 and 
 
 x+y: 
 
 (0 
 
 (2) 
 
 (3) 
 
 (4) 
 
 The solution of (3) and (4) will give the value of x and y, 
 and thereby the three sides. 
 
 Prob. 8. In any triangle ABC, the sides 'opposite to the 
 angles A, B and C are a, b, and c, respectively ; required, in 
 function of a, b and c, the median line m drawn from the ver- 
 tex of any angle C to the middle B' of the opposite side AB. 
 
 111 
 
 
 
IMAGE EVALUATION 
 TEST TARGET (MT-3) 
 
 1.0 
 
 IIM 
 
 I.I 
 
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 !*• 1840 
 
 11^ 
 
 2.0 
 
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 1.6 
 
 
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 /J 
 
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 Photographic 
 
 Sciences 
 Corporation 
 
 s. 
 
 -b 
 
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 V 
 
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 V 
 
 
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 23 WEST MAIN STREET 
 
 WEBSTER, N.Y. 14580 
 
 (716) 872-4503 
 

 i/.A 
 
 <fA 
 
90 
 
 BOOK II. 
 
 Solution. Letting x=DB', then 
 
 ■c 
 
 n— x=DB, and letting p=altitude CD; 
 
 then, in AACD, AC-=CJ>-fAD-, A 
 
 or 
 
 4 
 
 and, in ABCD, BC-=CD2, + BD^ 
 
 or 
 
 * 4 
 
 Subt. (2) from ( 1 1, b-'— a-=:2 ex ; whence, x= — . . 
 
 (1) 
 
 (2) 
 (3) 
 
 Again, in AB'CD,CD2=B'C--B'D^ or p=*=in='~x='. 
 
 Now, introducing, in (1), this value of p-, then in the result, 
 introducing the value of x taken from (3) and reducing, there 
 
 b- . c^' a^ 
 
 obtains: ;y=m^-|- -- — ,. ; whence, ni=^ v''2(a^-|-b^) — c^. 
 
 Prob. 9. Given the sum 2 a of the three sides of a right' 
 angled triangle ABC and the radius R of the inscribed circle ; 
 required the three sides. jp 
 
 Solution. Letting x=AF=AE, 
 and y=CE=CD; thenAC=x + y, 
 ABz=x+R,andBC=y+R. 
 
 By the data, ' 4* 
 
 2s=:2x-f2y-|-2R,or x-|-y+R=8 (1) 
 
 Besides, AB=*+BC-=AC-, or (x + R)-+(y+R)'=(x+y)=*;, 
 whence, xy=R(R + x+y)=R8. (2) 
 
 The solution of (1) and (2) will give the required sides. 
 
 Prob. 10. Given the three sides a, b, c of a trinngle; 
 required its urea A in terms of its sides. 
 
 Solution, From the 
 vertex C, let fall the per- 
 pendicular CD(=p),on 
 
 AB or AB prolonged, and ^ X \ \ ^ A/Cc^^ 
 
 ]etx = AD. C ^ " 
 
 — ^ 
 
 ji^j^^n., 
 
tifmmmi'mmmyi 
 
 '^mt 
 
 • 
 
 ARTICLE 7l. 
 
 
 III AACI), 
 
 p-=b2-X-. 
 
 (I) 
 
 In A ABC, 
 
 a=*=b-+c-— 2cx : 
 
 (2) 
 
 whence. 
 
 b2^c2_a2 
 x= ^ 
 
 2c 
 
 
 01 
 
 I, 7:{ 
 
 Itroducing this value of x in (1), 
 
 - u2 (t) -+c=-a^)^ 4bV-(b->c=^~a=)=-' 
 F=b^ 4^i = le^' 
 
 Now, bearing in mind that the diflFerence of two s<|uures is 
 equal to the product of the sum and diflFerence of their roots, 
 and factoring accordingly, 
 
 ', (2bc+b2+c--a-)(2bc-b--c2-fa-) 
 p.=_ _. , 
 
 or 
 
 .. |(b+c)=-'-a=-'(ja=^-(b-c)^'j 
 F 4c- 
 
 Factoring again, in conformity with tlie above principle, 
 
 (a + b+c)(b+c-a)(a + c-b)(a+b~c) 
 ^ ^ 
 
 m 
 
 Letting a-j-b-f c = 2s ; 
 
 then, b+c— a=:2(s— a), 
 
 a+c— b=2(H— b), 
 
 a+b— c=2(«— c). 
 Introducing the second members of these equations, in (3), 
 
 ., 28x2(8-a)x2(8— b)x2(s-c) 
 P= 5-: 
 
 whence, 
 
 p= \/s(s— a)(,s— b;(s— c.). 
 
 / cxp\ 
 
 Hence, the area A ( = — ;j- y = \/s(8— •a)(s--b)(8 — c). 
 
 l^J 
 
 Prob. 11. Given the three sides a, b, c of a triangle ABC ; 
 required the radius R of the circumscribed circle. 
 
92 
 
 BOOK ir. 
 
 Solution. From the vertex C of the ins- 
 cribed trian«5le ABC, let fall the perpendi- 
 cular CD, on AB ; draw the diameter CK. 
 the chord AE ; and let Cr)=p, CE = 2ir' 
 and a-f-b-f c = 28. 
 
 The inscribed angles B and K, having the same measure 
 
 ^' ' 'V are equal, and thereby the ri,L'ht-an,i;led triangles 
 
 BCD and ACE are j^imilar ; 
 
 hence, BC:CE = CD:CA, 
 
 or • a: 2R= p : b; whence, ab = 2Rp. 
 
 2 
 
 But (Prob. 10), 
 
 tlierefore, 
 
 whence. 
 
 P- ^%/s(.s-a)(s-b)(8-c); 
 
 iib ( = 2Rp) = ^ v/h(8— a)(8— b)(8~c) ; 
 13 abc 
 
 4V's(8— a) (s— b) (s— c) 
 Prob. 12. Given the three sides a, b and c of a triangle ; 
 required the radius R of the inscribed circle. 
 
 Solution. The given triangle may be divided into three 
 triangles whose bases will be a, b and c and whose equal alti- 
 tudes will be the radii R of the inscribed circle ; hence, letting 
 A=areaof the given triangle, and 2s = a-j-b-f-c ; 
 
 then , 
 
 But (Prob. 10), 
 therefore. 
 
 A=^(a+b + c) = Rs. 
 
 A=:: v/s(8— a) (s— b) (s— c) 
 )(8-b)(s-e.) 
 
 R = Ji«-»^. 
 
 Prob. 13. Given the base b and the altitude h of a paralle- 
 logram ; required, in /unction o/h and h, the base B and the 
 altitude H o/a similar triangle which has four times the area 
 of the given triangle. Ans. B=2b, andH=2h. 
 
 Prob. 14. Given the base b and the altitude h of any 
 triangle ABC (Jig. of Prob. 2) ; required the side x of the 
 
 inscribed square amnc. Ans. x = , -. 
 
 ^ b-f h 
 
ARTICLE 77. 
 
 1»3 
 
 Prob. 15* Gfiven the hase b and the diffcrt-nrt' d hrfiKccn 
 the hf/jiothenuse x and the altitude y of the trianyle ABC (Jig. 
 of Proh. 4) ; required x and y. 
 
 Ans. x=-^^, andy = _^. 
 
 Prob. 16. Given AC4-BC=s, an*^ perpendieular Cl)=p, 
 in triangle ABC, right-ang'ed at C (Jig. of Proh, 7) ; required 
 the hypotheniine h inj'unction ofi> and s. 
 
 Ans. h= — p±\/p--{-8-. 
 
 Prob. 17. Given the radius R of the circle inscribed in an 
 equilateral triangle; required, the side H of this triangle. 
 Ans. S = 2Rv/3. 
 
 Prob. 18. Given the diagonal d and the sum s of the hase 
 and altitude of a rectangle ; required the area A of the rec- 
 tangle. Ans. A='— ~ . ' 
 
 Prob. 19. Given the three perpendiculars a, b andc^ drawn 
 from a point vithin an equilateral trianyle to the three sides ; 
 required the relation which this sum hears to the altitude h of 
 the triangle. Ans. h = a-fb-f-t'. 
 
 Prob. 20. Given the three sides a, b and c of <t triangle 
 
 whose hase is c ; required its altitude p. 
 
 2 i . 
 
 Ans. Letting a+b+c=28, then p="v/s s— a)(s— b)(s— c). 
 
 Prob. 21. Given the hypothenuse \\ of a right-angled 
 triangle, and the quotient n of fheother two sides ; required n 
 
 h- 
 
 1 -T-n- 
 
 and b. Ans, 
 
 ''=''>|l7^^'^"*^^=s|l 
 
 Prob. 22. Given the hase b, and the altitude p, and the 
 
 ratio ~ of the other two sides of any triangle ; required a 
 n c 
 
 1 
 
 and c. 
 
 Ans. 
 
 arr^p-' 
 
 jb- + a-(l-n)|^ 
 
 :na. 
 
i 
 
 94 
 
 HOOK (I. 
 
 Prob. 23. Given the hi/pothenuse li of a tight-onrled 
 triangle, and the radius K of the inncrihedeirelr ; required the 
 other two itides a <ind b of the triangle. 
 
 Ans. a = r» ; b= 
 
 »> 
 
 2 
 
 Prob. 24. (riven thi radiun li ami a chord it of a circle; 
 required the distance d from the chord to the center of the 
 eirch: Ans. d = ^x/.|l^rZ^T 
 
 Prob. 25. Given the radius R of a circle and tvio parallel 
 chords}) and c on the same side (tf the center; required, the 
 distance d between these chords. 
 
 Ans. i\^W4K'-h'-W4lV-c^. 
 
 
 ii ' 
 
BOOK III. 
 
 ANALYTICAL GEOMETRY. 
 
 DEFINITIONS. 
 
 1. Analysis and Synthesis. Thetse two methods, applied 
 to geometry, have not, in modern times, the name signification 
 as they had in ancient times. 
 
 According to the AncienfH, geometrical analysis is the mode 
 of reasoning in which the advanced proposition is considered 
 a? pfranted and is used as a starting point for logical deduc- 
 tions which lead to some known truth; geometrical synthesis 
 is the mode of reasoning in which the truth of an advanced 
 proposition is deduced from definitions and from either self 
 evident or previously proved principles. 
 
 According to the Moderns, geometrical analysis consits in the 
 application of the algebruio or the infinitesimal method (art. 
 29, further on) to the demonstration of a geometrical truth ; 
 such is the geometrical analysis of this Book III. Geometrical 
 synthesis is the art of demonstrating a geometrical truth by 
 the aid of geometrical figures without the application of alge- 
 bra ; that is to say, in applying either the analytic or the 
 synthetic method of the Ancients ; such is the geometrical 
 synthesis of Books I and II. 
 
 3, A plane curve or simply a curve is a curved line lying 
 wholly in a plane. 
 
 To apply analysis to curves, usu will bu made of a system of rectan- 
 gular axes. 
 
 3* A system of rectangular axes consists of two straight 
 lines which intersect at right angles, and which are made use 
 of to determine the position of any point lying in their plane. 
 
 ■ ■' t 
 
 J 
 
96 
 
 B(K>K III. 
 
 %' 
 
 y i 
 
 ♦-♦^^■•-»- 
 
 \ 
 
 -♦-e^. 
 
 To determine this relative position of 
 a point in a plane, draw a horizontal 
 line Ji'X and a vertical line YY, inter- ^ 
 secting at the point 0. K 
 
 Divide these lines into equal parts- 
 l)eginnin^ at the point O (the unit of 
 length is arbitrary). From the point e, ' 
 
 draw r6 parallel to XA" and ca parallel to I'l"; the distances 
 ca and ch determine the position of the jmint c, with respect 
 to this system of rectangular axes. 
 
 The distance he ox its equal Oa is called the abscissa of 
 the point c, and tlie distance ac or its e(jual Oh is called the 
 ordinate of the point c; the abscissa and the ordinate are 
 collectively designated by the name of co-ordinates. 
 
 The axis XX\» the axis of abscissas or axis of X, and the 
 axis yy is the axisof ordinates or axis of Y; the axes of 
 X and Y are collectively designated by the name of co-ordinate 
 axes. 
 
 The point of intersection O of the co-ordinate axes is called 
 zero point of the axes, or, origin of co-ordinates, or simply 
 origin. 
 
 The abscissa is positise or negative according as it is reckoned 
 from to-f A'or from O to — A"; the ordinate is positive or 
 ?ie^a<ii;e according as it is reckoned from to -|- 1' or from 
 Oto-Y. 
 
 It is customary to represent the abscissa by x, and the 
 ordinate by y. 
 
 4» The equation of a line is the equation expressing the 
 relation between the co-ordinates of every point of that line. 
 
 5. A line is said to be of the first, second, third, &;c 
 order, according as its equation is of the first, second, third, 
 &c degree. 
 
 6. A plane figure is of the same order as its curve. 
 
 The tracing of a line by means op its equation. 
 7» Any line may be traced by means of its equation. 
 
ARTICLE 8. 
 
 97 
 
 <i 
 
 For. let XX and I'V be two rec- 
 tangular axes and eAe' any curve. 
 
 The various points A, b, c, d, <f'C 
 of this curve determine, with respect - f^ «, ... 
 to the axes, the various co-ordinates •*''^ ^•^ ^i"^ ^ 
 shown in the annexed figure. Then, [ >L M *« ^ *« 
 conversely, if the co-ordinates of the - 
 various points ^1, ft, c, d, d:c. be drawn 
 by means of a given equation, the line passing through the 
 origin A and the vertices h, c, d, &c. of the ordinates will be 
 the curve represented by the given equation. 
 
 8. Prob. 1. Trace the curve whose equation is y^=4s;. 
 Solution. Attribute to x the successive values : 0, 1,2, 3, 4&c, 
 and find the corresponding value of //, as fnfllows : 
 
 ;c=0, then y=: 
 
 In the given equat., //=rt2\/x, when 
 
 If the negative values — 1, — 2, — 3, &c, be successively attri- 
 buted to ic, the quantity y successively assumes the imaginary 
 
 values ±2v/"-^,±\/^, &c. 
 
 Then, draw two rectangular axes XX and Fl', and divide 
 them into equal parts, beginning at the origin A ; through 
 these points of division wi, w, o, &c of AX, draw the inde- 
 finite straight lines BB , CC\ BD', &c, parallel to YY. 
 
 Now, it ig to be observed that the first pair of co-ordinates 
 (a;=0,y=0), computed above, indicate the zero point of the 
 axes ; hence, the origin A is one of the points of the curve. 
 The second pair of co-ordinates (x=l, y=zb2) shows that, 
 when the abscissa x becomes equal to 1 (= J.m), the ordinate 
 y assumes the two equal and opposite values-|-2 and — 2 ; 
 accordingly, lay off, on the parallel BB\ the ordinates mh 
 (=-)-2) and mh' ( = — 2), and the position of two other points 
 (6 and //) of the curve will thereby be determined. 
 
 x=l. 
 
 " y=±2 
 
 x=2, 
 
 '^ y = rh2.8 
 
 rr,= .3, 
 
 '^ y = ±3.46 
 
 x = A, 
 
 •' 2/=±4 
 
 &c, 
 
 &c, 
 
98 
 
 BOOK III. 
 
 In order to find the position of the points represented by 
 the other pairs of co-ordinates, lay off, on the parallels 
 CC\ DD\ &c, the symmet.ical ordinatcs {nc and nr'), 
 (otZand oti'), <fcr, corresponding to the abscissas An, -4o, tf'c, 
 respectively. 
 
 But, the origin -i and the vertices (5, 6'), (c, c'), {d, tZ'), &c. 
 of these ordinates are so many points which belong to the curve 
 represented by the equation y^=4a;; therefore, if a line eAe' 
 be made to pass through these various points, it will be the 
 curve required. 
 
 Sch. In the given curve, y=db2\/x, any negative abscissa 
 ( — x) gives an imaginary value for its corresponding ordinate ; 
 but, imaginary ordinates imply an imaginary line ; hence, the 
 curve expressed by \he equation y'=4x cannot pass to the left 
 of the axis YY. 
 
 Prob. 2. Trace the curve whose equation is x^ — 4y^=0. 
 Solution. / a,_ o, then y=i 
 
 X 
 
 x=l, 
 
 , x=2, 
 In the given equat. y=--:tL ^y/^, when < _ ^ 
 
 ■4, 
 
 X- 
 
 (( 
 
 u 
 
 u 
 
 &C. 
 
 y = ±0.5 
 y = rfcl.4 
 y=±2.6 
 y=±4 
 
 &c. 
 
 Any negative abscissa (— x)give8 an imaginary value for its 
 corresponding ordinate ; hence, the curve, x^ — 4^^= 0, cannot 
 pass to the left of the axis YY. 
 
 Proceeding as in Prob. 1, the various 
 co-ordinates, and thereby the curve re- 
 presented by the equation gc'— 4y^=0, 
 will be drawn as shown in the annexed . 
 figure. 
 
 Sch. If the form of a curve was not 
 suflSciently defined by the vertices of 
 the ordinates corresponding to the 
 abscissas 0, 1, 2, 3, &c, it would be 
 necessary to determine the position of intermediate points, by 
 
ARTICLE 1). 
 
 !>9 
 
 iraeing the ordinates c«)rre.spoiiding to tho abscisftas > 
 UA, 1, U,2, ike. 
 
 Exercises. Trace the curv«'.s repreweiitetl by the following 
 e<{uatioii». 
 
 1 - y=^ X-'. 5- // =^\(x'-Ub','^\00). 
 
 
 i— VI 
 
 2.r-. 
 
 t>' y'z=x' 
 
 7" y"=^.r'- 10x^-f26a;). 
 
 3' .y=i (Sx-x-). 
 
 1), Equations of a .straight line. 
 
 Assume Ji'X and Kl'to be tworectan- ♦ 
 
 1 ^ 
 
 •'iilar axes. 
 
 Draw two indefinite parallel lines AC ^ 
 
 and A'a\ the former passing through 
 
 the origin; from the points C and </'- 
 
 let fall, on -\-XA^ the perpendiculars 
 
 rViand a7>; draw AD' parallel to .!/>. >" 
 
 and let 
 
 BC 
 
 = AA' = DD',m= 
 
 AB' 
 
 x=A/), and i/=Da'. 
 
 In thee«|ual triangles ^lJ9<t and A'D'a', 
 
 f)„=D'a' = Da'—DD' = y—c. 
 In the similar triangles vl At and AHC^ 
 
 AB:BC=AD:Da; 
 
 whence. 
 
 rtri 
 
 Da:=ADy. =mx. 
 
 AB 
 
 Substituting ij — c for Da, there obtains y — cs:smx, or 
 U=mx-\-c. 
 
 But, X and y are the co-ordinates of any point (a') of the 
 straight line A' a', and the equation y=mx-\-c expresses the 
 relation between x andy ; therefore (Def, 4), the 
 
 l'^ Equation of a gtraight line intersecting the axis ofY at 
 the distance c from the origin, is 
 
 y = nix-^c. 
 
 When c reduces to zero, the straight line A' a' becomes the 
 straight line Aa, and there obtains : 
 
 2' Equation of a straight line passing through the origin, 
 
 yz=.mx. 
 
100 
 
 BOOK Hi. 
 
 n, 
 
 ||;, 
 
 ■ lO. Kqi'ations ok thk 4;ir(?lk. 
 
 Let AD/i be a circlt*. Draw the ♦ -5. 
 horizontal (lianictor J /^. the ordiimtoN 
 DF and 67/, tlie chords M) and DM, 
 and the radiuj^ (YH. A 
 
 1 ' ^ssunie the origin of the co- 
 ordinates to be at the center T' of t lie > 
 circle, and let radius C(i=lt, aiiscissa*^ 
 VII— .r^ and ordinate GII=i/. 
 
 In the right-angled triangU^ CGJf, CG^=CII--\- ll(!\ 
 or /?-=.T^ -}->/-; but, this etjualion ii'=R' — x' expresses the 
 relation between the co-ordinates (x and y) of any point (^(1) 
 of the circle ; whence, the 
 
 Equation of ihr circle (^Origin at the center). 
 
 y'=zli'—:r'. 
 2' Assume the origin to be at the vertex A of the diameter, 
 and let AJi=2/i. AF=.i\ DF=y, and fB=2R-r. 
 
 By this notation, the right-angled triangle ADB gives 
 (I, il7, Sch.) for 
 
 Equation of the circle \Origin at the vertex of the ilianiefer). 
 
 y' = 2Iix — :r. 
 
 ELLIPSE. 
 
 • 11. The ellipse is a plane curve 
 (ADBE) in which the sum of two 
 lines (Fm and F'm) drawn, from 
 any one (m) of its points, to two^ 
 fixed points (F and F') is constant- 
 ly equal to a given line (AB). 
 
 The fixed points {F and F') are 
 the foci; the point {€) midway between the foci is the center; 
 the distance (Fm or F'm) of any point (m) of the curve to 
 either focus is a focal distance or radius- vector ; any straight 
 line passing through the center and limited by the curve is a 
 
ARTKLi; 1l'. 
 
 101 
 
 diameter; the diaiin'ttT (AH) which (•(intuiii!^ the foci i.t the 
 major axis or transverse axis ; the diaiiK'tiT ( />A') which 
 intersects the major aixis at ri}.'ht angles i» the minor axis ur 
 conjugate axis, of the ellipse. 
 
 Tf) (hsiribr. an rffipHt: mtrhnuicdlli/, fasten the extremities 
 of a threuil, whose length is ecjual to the major axis (A/i) to 
 two fixed points (/^md F') ; then, a pencil made to keep the 
 thread constantly stretched will deHcri})e an ellipse (.l/>/i A' j 
 in performing a complcite rev<»lution about the foci. 
 
 For, in every position of the tracing point (»«), the sum of 
 its focal distances ( Fin-\-F'm) will constantly be e(|Ual to the 
 lenjrth of the thread (.1/0- 
 
 It is customary to represent the major axis A I' by 'ln^ the 
 minor axis I)E by 2/^, and F(^ or F'C by r. 
 
 The ratio ^=t is called the eccentricity of the elLpse. 
 a 
 
 Vi. 
 
 Equations (»f th'^. ellipsk. 
 
 1" Assume the major axis and the minor axis to ue the axis 
 of abscissas and the axis of ordinates respectively, and the 
 center C to be the oriyiin. 
 
 Let Cn=iX, 
 
 mi 
 
 n = i/, and F'm — Fni='h 
 
 then (since FC=c), 
 
 Fn = FC—Cn = c—x, and F'li = FC-j- Cn ^c-\-x 
 Bydef. of the ellipse, Fm-\- F'm ='2a, 
 and thereby, Fni = a — ;:, 
 
 In the right-angled triangles i^mu and F'mu, 
 niu' -f Fn-= Fmr, or //- -f (f — .*• )-=(« — s )-, 
 mH'-\-F'n'=Fm', or i/'-^ (<--\-xy-== (a-^-z)'. 
 
 Subt. (1) from (2), and reducing, ex = az. or zzzz^"^ 
 
 a 
 
 Adding (1) and 1 2), and reducing, ^-'_)-r-|-a- =a-+2-. 
 Substituting . :.^ for z- in equatioji (4), there obtains : 
 
 (1) 
 
 (2) 
 
 (3) 
 1 4) 
 
 a 
 
 •I 
 
 C'.L 
 
 a 
 
 y'+c->ir-=«^+'^Y'Or ay +("'-<>') *' = a\a--<^). 
 
 (5) 
 
m 
 
 HOOK lit. 
 
 T1»oivf\>iv. l\v i««lM»lliuti»g /»M\»r »»*—«•*, in |ft), thorn ohtninH : 
 
 ,»«y^ t /)*.! ^ rr: O'h" ; whoiUM>, 
 
 A,'^Wf»f»f)M of the •7/»'/)«»\ 
 (Oviiiinrtt iWnMj^ov io»«l iiI>m(mm»h vooKoiumI otiiln* iiiiijitnixiN). 
 
 Ill) 
 
 f»' 
 
 wx^i*. tl\o »u>tit»!Uo •»i»» ( I'i of llu> )»olnl m will tiot lu' 
 
 ohrtMgtsi. hut tlio «hHoi».Ht\ ol' thl^ point will hoconic 
 An .<''f f'n —-.M •»•• 
 
 iVnotinvi Mm hy A', thojo ohtninn ; A' ::ri:ti-f •'' ; w'h'IU'''. 
 
 IntroUuvinji this viiluo ol' .rMn oquutiou ill), thoro ohtuina 
 
 fov \ ho 
 
 Ht^Hotiou of thr f'ifif}i>i', 
 
 ^Onjiin ti\ vortON ot ntixjov rtxii«. rtM«l alMU'innnw ivokoM*'*! y\\\ tliinrtxis) 
 
 3'^ In o«\urttivM»N <^(>) suni (T>. whon /» hooonios «m|uu1 t«» «». 
 thow obtains: i^'^r «r — ,j^ rtn»i v^ 2t?.r—.<^ which uro tho 
 o^«alio»^j< of tho oirolo to Viulius r». 
 
 AiiT^in, whon h Ihvh^uuvs uvoatov than n, tho vortioul nxi.s )lf> 
 Kvinnos tho m.^ijor «\is. and tho horixontal axis 2ci hooonnvHtlu' 
 minor axis ; hut. in this hy|H>thosis. *i is to ho ohangod into It, 
 and ^^nto<l. in ordorto comply withtlio usual notation which 
 invariahlv denotes tho nuijor axis hv 2(» and tho minor axis hv 
 
 21 ; \vhonoi\ 
 
 f!<fm}tton of* thr rUtf^sr. 
 
 ((Vicin at tho oontor. and ahscisvsas reckoned on the minor axis* 
 
 II' 
 
 Kqtuttiofi of thr fVipar. 
 <^ Origin «t vertex of minor «xi*. and abscissas r\^cko;io I on this axis), 
 
 i8i 
 
 fi* 
 
 r=^,i-^»'»— ''')• 
 
 (Ol 
 
 
IP" 
 
 AUTUJiiK 1*. 
 
 PAUAIIOfiA. 
 
 108 
 
 ( 
 
 IJI. Tlio pnrnbola U n pliino «Mir o f I— - 
 iiAff), Hiiy jMiliit. (nn orwliidi Ih imjmI- 
 (liFtdiiii IVmiii a DximI poJMl ( /''; niid allxcd 
 
 Tln» IIximI |m»1ii«, (A'i im «Iio fbmiH} <!•♦• 
 Itx«««t lino iVV) \h Him dlreotrix; (lie ^ 
 lino (-f A7') lliroii^li tlm CtmiiH, pnrjioridi 
 (Miliii* i(» tlio (liroch-ix, in tlio Axin of the y 
 pnrAboln. Tho vertex of the nxln tM t lio 
 |ioiiit {A) in wliinh tliH lu'm iiitMrMuttH tliiw'iiryn. 
 
 To ilfHi't'iln' ti pinahnfn. invvhunivnUff^ (aNii«ri orift ^nd 'if »i 
 ilir»Mnl, wlioHo Innglli in (m|iiiiI Ut tliM fti«lo (111 of a triaMj<ul)if 
 nilor (Illy, or wjtiiirp, to tlio v«'rf«'n // of HiiH H*jii«r«', «fi«l th« 
 oth»»i*«'ini to (bn fo«'ii« A'. Wliilo tlio Wfjimro \n nm'lo i»» Hli«l« 
 nioiig a nilor wlniHo od^o cnifiridoR willi tlio rlin-ctrix fT, a 
 juMH'il {H ) iiiioIm lo kr««|» llio flirofid roriHinrif ly »tr»>trlu«fl (i'rniu 
 A' to M, iitiil IVdiii » (o // (HI tin' 8ido fif jIm^ M^nnrf) will d<i»- 
 (•ril»o a luiniliola ( II A h). 
 
 K(»i', ill cvory poHit ioti of tlM« «i|uUn*, 
 
 Fh-^nll~(,'n-^nll, 
 
 or Fn!==:(i'ii. 
 
 14. 
 
 KiQIIATItiNH (»K THK I'AKAMor.A, 
 
 1" Let llio to'i^iii 1mi at iho point T', From iiny [cMfii (m) 
 of tlio curvo, draw tlio ordiiiuto 717;, atid hi 
 a= A('ta.AI'\ 2if~('F, j'~(']), tiiuli/==mj> 
 
 By dof. of tlio paniliolu, Fm—rn >'- ^ /// = ;/•. 
 
 In tho riglit-iiiij^iod Iriaiij^lo A/;m, Ffy'-{-mp*=i Fm-. 
 
 or {(Jp^Crf^mp'-Frn'^iy. 
 
 or, by the ubovo notation, (,/; — '^'i)^-\- y^ ■■./^ ; whence, 
 Equation of the jxirahola. (Origin at point (^), 
 
 2" If the origin be tranHforred to tho vertex A of tb** axis, 
 the ordinate />m (= V) of tho point m will not \}e changed, 
 
104 
 
 BOOK III. 
 
 but the abscissa of that point w will become 
 
 Ap—Cp—CA=x—a 
 Letting Ap=x—a=Xj and substituting JT for (x — a) in 
 equation (1), there obtains for 
 
 Equation of the jjarabola. 
 (Origin at the vertex of the axis). 
 
 r=4aX ' (2) 
 
 Hyperbola. 
 
 15. The hyperbola is a 
 plane curve conipdsed of two 
 opposite branches (GAH 
 and G'A'H), in which the 
 difference {F'ln—Fni) of 
 the distances from any point 
 (m) of the curve to two 
 fixed points {F & F') is^ 
 constantly equal to a given 
 line (^.l')- 
 
 The fixed points (F li' F') are the foci; the point ((7) mid- 
 way between the foci is the center; the distance (^Fmor F'm) 
 from any point (m) of the curve to either focus is a focal dis- 
 tance or radius-vector ; the straight line passing through the 
 center and limited by the curve is a diameter ; the diameter 
 which contains the foci is the transverse axis, of the hyper- 
 bila. 
 
 The vertices of the transverse axis are the two points 
 {A and A') in which this axis is intersected by the curve. 
 
 If, from the vertex A or A', as a center, with a radius equal 
 to CF, a circle be described, it will determine the vertices 
 {B & D) of the conjugate axis BD. 
 
 To describe an hyperbola mechanically, one extremity of a 
 ruler is fastened at one (jP') of the foci, so that the ruler may 
 be free to rotate about that focus. The difference in length 
 between the ruler and the transverse axis A A' is the length to 
 be given a thread, one end of which is fastened at one extrer 
 
m 
 
 ARTICLE 16. 
 
 ^0» 
 
 mity i-E) of the ruler, and the other end at the other focus (/^;. 
 A pencil (m) made to keep the thread constantly stretched 
 (from i^to wj, and from m to E) will describe the hyperbolic 
 branch GAH, when the ruler is made to rotate about the 
 focus F'. 
 
 For, in any position of the tracing point (m) , • 
 ^wi+m/'= length of the threads (^/»+wF') — ^L4' ; 
 
 whence, AA'=mF' — mP ; 
 
 therefore (def. of hyperbola) , the point ?h is always a point of 
 the curve. 
 
 If the focus / be made the center of rotation of the ruler, 
 und the above process be applied, the second branch (i'A'H' of 
 the hyperbola will be described. 
 
 It is customary to represent the transverse axis (AA' ) by 
 'la, the conjugate axis (BD) by 26, and FC or F'Chyc. 
 
 The ratio - =e is called excjentricity of the hyperbola. 
 
 16. Equations of the hyperbola. 
 
 1" Assume the origin to be at the center C ; draw the ordi- 
 nate mn of any point m of the hyperbola GAH, and let 
 Fm=r, F'm=:r', Cn=x, mn=i/. 
 
 By def. of the hyperbola, F'm—Fm= AA', 
 
 or r' — r =2a. 
 
 In the right-angled £^F'mny F'ni'=F'ii'-^mn-, 
 nr.smce F'n'={FC-hCnf=\c-^xy, r'-=(c-f :c)=^-|-/. 
 
 In the right-angled A^^">i, FTn'=Fn'-{-mn^, 
 or,iimce Fn'={FC—C}i}'=^{c—x)-, r'={c—x)'^-^if. 
 
 Subtracting (3) from (2), r'-^r^=4c.x. 
 
 Dividing (4) by (1), 
 
 , , 2cx 
 r -f r = 
 
 a 
 
 0) 
 
 (2) 
 
 (3) 
 (^) 
 
 (5) 
 
 Subtracting (1) from (5), 
 
 ex 
 ras a. 
 
 a. 
 
 /cor \ / cnt \ 
 Subst./ a ifor /•, in (3), / a\ =(c — x)--f-// ; 
 
 whence, ay3=(c-— a-)a;-— (c- — a-)a-. (6^ 
 
 In the right-angled l^ACD, CD'^AD-^ A(7\ or 6-=;«c-~a- -^ 
 
 \ I 
 
 nl 
 
 ' I 
 
106 
 
 BOOK III. 
 
 Then, by substituting V^ for {(?—a') in (6), there obtains for 
 Equation of the hyperbola (Origin at the center C) : 
 
 a' 
 
 ■a\ 
 
 (7) 
 
 2" If the origin be removed to the vertex ^4 of the axis, 
 the ordinate mn (= Y] of any point m of the curve will not be 
 changed, but the abscissa of the point m will become 
 
 Ah = Cn — CA=x — (I. 
 
 Letting An:=x — a= A", there obtains : x'^=X'-\-2aX-\-a^ ; 
 then, introducing this value of «- in (7^, there obtains for 
 
 Equation of the hyperhola (Origin at the vertex of the axis): 
 
 T-=^X'^aX-\-X\ 
 
 w 
 
 (8) 
 
 3" If the co-ordinates and the 
 axes be mutually changed in the 
 hyperbola DAD\ its equation, 
 
 f'=^{x^—ah = ~x'—V\yi\\\ be- 
 a^ d^ 
 
 a* 
 
 comea;"=-y*' 
 
 ■a^ ] hence, the 
 
 Equation of the hyjterhola FBF' conjugate to DAD' is 
 
 y2=_(x2-f.a2!. 
 
 (9) 
 
 17. DIVERGING PARABOLiE. 
 
 Under the generic name of diverging parabolee are com- 
 prised five plane curves of the third order, represented by the 
 cubic equation y'^=ax^-{-bx^-^cx-^f, in which the coeflScients 
 may assume any values, except those which are imaginary. 
 
 18. Equations of the diverging PARABOLiE. 
 
 Multiply y'^ =ax^-\-hoo^-i- ex -i-f hy p and let the second mem- 
 ber of the resulting equation be represented by the symbol 
 /*(xi; then, y^=sax^-{-hoo^-^cx-\-f hecomes py"=f{x)^ which is 
 read " jay^ is a function of a;." But this equation py'=/ (x) is 
 true for any value of y ; hence, it is true when y=0, and then 
 f{x)=0. 
 
I) 
 
 ARTICLE 17, 
 
 107 
 
 Since, /(x) is a function oT the tlilvd degree, it has only 
 three roots, two of which may be imaginary (App. A, III S. IV). 
 Denoting these three roots by /, m, n, and assuming the curve 
 to be to the right of the origin, we shall have but the following 
 five combinations and equations :^' 
 
 1" The three roots, /, m, n, may be real and u.iequal ; 
 
 o 
 
 whence, the equation of the 
 Parabola with Oval, 
 
 y'=it{.r—l)(x—m){x—n). ib) 
 P 
 
 2" Of these three real roots, the last two may be equal ; 
 whence, the equation of the 
 
 Crunodal Parabola, 
 
 f'=^.{x—l)(x—n]\ 
 
 ('•) 
 
 -Of. 
 
 3° Of these three real roots, the first two may be equal; 
 whence, the equation of the 
 
 Acnodal Parabola, 
 
 y-=-(.r — m)-(x — n). 
 P 
 
 (d 
 
 * In this hypothesis, the abscissa of the curve la invariably to thi' right 
 of the origin ; hence, its various values, and thereby its roots I, m, n, are 
 always positive; that is, x=l, x = m, x=sn ; whence, x — 1=0. x — ??» = 0, 
 
 z— n=:0, and 1/ {x)= ±(z—l)(x—m)(x—n)= if- (I) 
 
 P P 
 
 If the curve is assumed to perform half a revolution about the axis of 1'. 
 then any abscissa of the curve, in this new position, is iiegativo and there • 
 
 by 2=—/, x——m, 3=—n, whence. -i (z -}-/)(/ -f-m)(a:-f-«) = v- (2) 
 
 P 
 Therefore,(l)and(2)ropre3ent the same curve in two symmetrical posi- 
 tions with respect to the axis of 1"; that is, (1) and (2) have ccjual and 
 opposite abscissas, each to each 
 
I 31 
 
 m 
 
 i! 
 
 lOH IJOOK IIL 
 
 4 ' The tbwAe roots iii:iy be real and equal ; whence, the 
 equation of the \ 
 
 1 
 
 I/-— -(.<■— II )'\ 
 
 {e\ 
 
 rv'-Lasily. (wo roots may bo iniaj^Inary ; whence, the 
 e(juatioii of the 
 
 Pure Campomt^ 
 
 or 
 
 y'=L(x-—2/j-\-/'-\-iii-){x—ii}. (/) - 
 
 j: 
 
 10. Sch. 1. Each of the equations (A), (c), (d), (c), (/*), 
 assumes a more simple form, when the oriuin of the co-ordinate , 
 is removed to the vertex of the axis. 
 
 Thus, ill removing the origin from the point a to the vertex 
 I of the parabola with oval (fig. h)^ the abscissa / or ol reduces 
 to zero ; then, substituting zero for Z, in c<(uation (6), there 
 obtains for : 
 
 '>., 
 
 1" EtjHiifioii. of the ]}(U'aboh(, with oval, 
 
 "/— -.x'(.r— Mi) (.,•_„) = ] -j .7='— ()»,+ >/.) ;ir-j-m«,r 
 ? y> ( 
 
 By the same prot-ess. there obtains : 
 
 Eqimtioii of the crKnodal purahola, 
 
 /y-~ : .rl^.r — /<)-=:_ (;j;'' — 27i.r-\-n-x). • 
 
 p p 
 
 Equation of the iwnodal parabola^ 
 
 y'=^- {x -f- '«i)";f = - (;»r' -f 2mx- -f m-.r.).* 
 P P 
 
 ■}■ 
 
 ih) 
 
 («) 
 
 8" 
 
 {d} 
 
 * When the on^j,Iii is ri'inoveil frojn tliL' jwint o 1o the point n (fig. </) : 
 1" Tlio abscissa n or on r-'li.ccvi i„ zero ; 
 
AHTICLK in. 
 
 ton 
 
 .)' 
 
 .. 1 .. 
 
 Kqudtuni Iff the pun- i-niiipnnn. 
 ,p~^ S .,.3_2/.r;- + (/-+wr .1; ]■ . ' 
 
 (.) 
 
 (/•• 
 
 Sch. II. The (liverirititr panibct'ii) may })p tiau.s formed into 
 one another, when their equations are modified to this eifect. 
 
 Thus, by attributing, to the roots m and n of equation (6), 
 successive values tending to a common limit, the vertices ni 
 and n of the curve (/>) will thereby progressively approach each 
 other, since a change in the equation implies an identical 
 change in the curve which it represents, Wiien the abscissa m 
 becomes equal to the abscissa ?i, both in the equation and in 
 the corresponding curve, the two legs {crura) of the parabola 
 leach the oval and form the wo^e (aodus) n of the crunodal 
 parabola specified by equation (c). 
 
 Likewise, when I becomes equal to m, both in equation (h) 
 and in its curve, the oval reduces to a mere point m (fig. d.) ; 
 whence, the acnodal parabola specified by equation (tZ). 
 . Again, when I becomes equal to %, both in equation (t) and 
 in its curve, the oval vanishes at the point u (fig. e)^ and the 
 crunodal parabola (c) reduces to a cusp (cuspis) ; whence, the 
 cuspidal parabola specified by equation (c). 
 
 Finally, if the roots I and m become imaginary iu the 
 equation (Z>), the oval vanishes, leaving no trace, not even of 
 a point, and the curve (h) is thpreby reduced to the only cam- 
 paniform parabola which assumes the name of pure campana 
 represented by equation (/). 
 
 "M 
 
 ■J" The abscissa in or vm becomes negative, since mn stands on tlie left 
 of the origin n ; and then. 7-= — in, or a:-j-;M=0 : 
 
 hence, equation ?/^=-(x— »?»)2(z— ?i) (Origin at tlie point o) 
 
 Iwcomes 
 
 1l^:sz^(X-\-1 1)'^ X 
 
 (Origin at the point n). 
 
\ 
 
 f! 
 
 I 
 
 110 
 
 BOOK III. 
 
 CUBIC PARABOLA. 
 
 The cubic parabola is a plane curve of the third order, 
 consisting of one branch, represented by the cubic equation 
 l/=ax^-\-hxr-\-c£-{-f, in which the coefficients may assume any 
 values, except those which are imaginary, 
 
 21» Equations of the cubic parabola. 
 
 Since the second member of y=ax^+fex--f t-c-j-/', is 
 identical to the second member of i/'=ax^-\-bx'-\-cx-\-/,i\ie 
 equation of the cubic parabola admits of the same combina- 
 tions of roots as the collective equation of the diverging para- 
 bolse. Hence, the following equations express the various posi- 
 tions of the cubic parabola with respect to the co-ordinates 
 axes (10) : 
 
 First position, 
 '=i- < x^ — [m-\-n)xr-{-mnx V . (6') 
 
 Second position, 
 
 Jt 
 
 P 
 
 Third position, 
 
 y=-{x;'+2mx'-\-m^x). {d') 
 P 
 
 
 y «! 1^ \_/-^ 
 
 Fourth position. 
 
 iff) 
 
 Fifth position, 
 y=l I x3~2/a;--f i^-+m-)x | . (f) 
 
 m jL-l 
 
ARTICLE 22. 
 
 Ill 
 
 2!3. Curves REPRESENTED BY Y^^AX^^H- Bx2+ ex -I- p. (a) 
 1" As a cubic eijuation, the above equation represents but 
 the following five curves (art. 18) : 
 
 Parabola v)ith oval, i/^= < oi^—{m-\-ti)j^-^-innx >, 
 
 Crunodal jHirabola, y^= (x^ — 2nx^-{-n^x). 
 
 P 
 
 Acnodal JHirabola, y-= (x^-{-2nix'-{-m'x). 
 
 P 
 
 Cuspidal parabola, y'=z .tA 
 
 P 
 
 Pure Campana, y'=- ] ^^ — 2Jx^-\-{lr-{-m']x [ . 
 
 Each of these five curves is referred to rectangular axes 
 whose origin is at the vertex of the curve. 
 
 2" Letting a=0, in y^=saM^-{-ba?-\-cx-\-f, there obtains 
 i/^=bx'-}-cx-\-f, which, as a quadratic equation, represents but 
 the following four curves (App. C. I): 
 
 y'=iB? — x^, referred to the center, 
 
 referred to vertex of a diameter ; 
 
 (f=E?-x\ 
 Circle \ „ 
 
 1 y--2Rx-~x\ 
 
 62 
 
 y2— _(flj2 — 3.2, referred to center and major axis, 
 
 62 
 
 y^= ~{2ax—x^), referred to vertex of major axis, 
 
 Ellipse 
 
 a 
 
 \ 
 
 y'=z _(&2 — X'), referred to center and minor axis, 
 y^=.%{2bx—x^), referred to vertex of minor axis ; 
 
 (y-'_4a(x—a), referred to axis and directrix, 
 y^=4ax, referred to vertex 01 axis ; 
 
 62 
 
 y'=i-[x?'—a') referred to center & transverse axis, 
 
 a 
 52 
 
 Hyperbola y^=:^-{2ax-\-x^), referred to vertex of transverse axis 
 y2— r. (x2-J-a2), referred to center & covijugateaxis. 
 
 •i1 
 
 ■r 
 
112 
 
 BOOK in. 
 
 , 3''Lettinga=0,6=:n»^,c=r2?nn,/=n-,iny^s=saa7^-f ftx-' + cx + Z" 
 there obtains i/'^ni-x^-^-^imnx-^-n'^, or i/ = dtz(rnx-\-n) ; and 
 when ?t=:0, there obtains y = ±wix ; which represent the 
 
 Str 
 
 /y=±wja5. passing through the origin, 
 vttjf/f^ZiHc I y=rL(mx-|-n), cutting the axis of ]', at dis- 
 ( tance n from the origin (9, 1"). 
 
 */iti. Curves represented by the equation 
 
 Y = AX"4-BX-4-CX+F. Qt') 
 
 1" As a cubic equation, the above equation represents but 
 one curve (App.B.), the 
 
 Cubic U =\ d^—(m-{-n)x'-\-mnx'r, (h') 
 
 parabola [^^^^^ ^^^^^ .y^^^ 
 
 2' Letting a=0, in y=^di^-^bx?-\-cx-\-f, there obtains 
 y=zbx'-\-cx-{-f^ which, as a quadratic equa*^ion, represents but 
 one curve (App. C, II), the 
 
 1 , 
 y— .r-, 
 
 P 
 
 Common ) 1, . \ , on 
 
 / //= (.X- — wx), or t/ = -(7j.x — x^,) 
 parabola \' p ' P 
 
 y— - (x^— 2mx+m^+«-), 
 P 
 
 (1) 
 
 (2) 
 
 (3. 
 
 the axis of which is perpendicular to the axis of the parabola 
 represented hy y'=px=-iax. 
 
 3> Letting a=0, i=0, in?/=«x^H-&x^4-cx-|-/, there obtains 
 y = cx-^f; and when/ also reduces tozero, there obtains y = cx', 
 which represent the 
 
 fy=cx, passing through the origin, 
 y=cx-\-t\ cutting the axis of Y, at distance/ 
 s. from the origin (9, lo). 
 
ARTICLE 24. 
 
 113 
 
 24. 
 
 Solids op Revolution. 
 
 In a revolution about their axes of X, the plane figures spe- 
 cified by the following curves, will generate the following solids 
 of revolution : 
 
 FIGURE, 
 
 Specified by,* Solid of revolution. 
 
 R. angled A, y =mx. 
 
 Circle^ 
 
 Horizontal 
 ellipse, 
 
 Vertical 
 ellipse^ 
 
 Common 
 parabola, 
 
 Common 
 hyperbola, 
 
 Parabola 
 with oval, 
 
 Crunodal 
 parabola, 
 
 Acnodal 
 parabola, 
 
 Cuspidal 
 parCtbola, 
 
 Pure 
 Campana 
 
 y=«=2Rx-x^ 
 
 y^=^'(2ax-x2), 
 a' 
 
 r=^'(2bx-x=), 
 
 y-=4ax, 
 
 r=^'(2ax+x=), 
 a'' 
 
 y-= -j x^— (m-f-nix--fmnx I , 
 
 y-=-(x' — 2nx24-n-x), 
 P 
 
 y==-^(x3+2mx2+m2x), 
 y^=L^ 
 
 y,2=^. I x3-21x2+(l'+m2)x 1 , 
 
 Cone. 
 
 Sphere. 
 
 Prolate 
 ellipsoid. 
 
 Ol^late 
 ellipsoid. 
 
 Ck>miudii 
 paraboloid. 
 
 Common 
 hsrperboloid. 
 
 Paraboloid 
 with ovoid. 
 
 Crunodal 
 paraboloid. 
 
 Acnodal 
 paraboloid. 
 
 Cuspidal 
 paraboloid. 
 
 Pure 
 Ganipanoid. 
 
 J 
 
 I 
 
 I 
 
 * The origin, in these curves, is at {he vertex of the axis. 
 
114 
 
 BOOK III. 
 
 25. INFINITESIMAI. ANALYSIS. 
 
 The theory of infinitesimal analyairi, given in the following ragca, is 
 limited to a few elementary principles which the plan of this work 
 requires. 
 
 26. Variable Magnitudes, and Functions. 
 
 Let ABD be any curve passing through 
 the origin A. Draw the ordinates Bm, 
 Ciiy Do to the points B^ C, D of the _ 
 curve, and Bp parallel to XX. X- 
 
 When the arc AB is increased by BC y\ 
 and becomes AC, the abscissa Am is 
 increased by mn and becomes -4n ; the ordinate Bm(=pn) is 
 increased bypCand becomes Cn ; the plane ABm is increased 
 by C-B/n» and becomes ACn. If this plane ACn performs a 
 revolution about the fixed axis XX, the solid generated in the 
 revolution of the plane ABm is increased by the solid generated 
 in the revolution of the plane CBmn \ the convex surface 
 generated in the revolution of the curve AB is increased by 
 the convex surface generated, in the same revolution, by the 
 increment BC of the curve. 
 
 Hence, all these magnitudes are variable and depexuieiit on 
 one another. 
 
 If the relation between any two of these variables be ex- 
 pressed by an equation, and a value be attributed to one of them^ 
 the corresponding value of the other variable will thereby be 
 determined ; whence, the former is an independent variablOi 
 and the latter a dependent variable. 
 
 Thus, if the curve ABD be circular, or elliptic, or parabolic, 
 or &c, the relation between the co-ordinates cf each of its 
 points will be expressed by the equation y-=2flKB — x^, 
 
 or y2^-(2ax-~ic2), or &c (10 and 12). But, if a value be 
 
 attributed to the abscissa x, in these equations, the value of 
 the corresponding ordinate will thereby be determined ; hence, 
 X is the independent variable, and y is the dependent variable. 
 
ARTICLE 27. 
 
 115 
 
 In these equations nnri in .any similar expressions, the depen- 
 lent variable i» said to be a function of the independent 
 »ariable. 
 
 Tiie expiession of u function is generalized by the notation : 
 '/=/(x), which is read " y is a function of x." 
 
 The symbol /(x) represents any expression in terms of a?. 
 
 Thus, the function ya=/(.t) represents any one of the 
 equations we have already seen and, in general, any equation 
 between two variables. 
 
 A function may contain constants, besides variables; v. g. 
 the coefficients u, h, c, and the absolute term/, in 
 if=iaj^-\-bx--\-cx-\-f. 
 
 Constants are such quantities as retain, throughout the 
 same discussi'^n, the values once assigned to them. 
 
 Constants are represented by the loading letters, and 
 variables by the final letters of the alphabet. 
 
 27. 
 
 Infinites and Infinitesimals. 
 
 Let X be a variable, and n a constant, in the series : 
 H-\-nx-{-nx'-\-nx^-^&c^ (1) 
 
 • (2) 
 
 n n n a 
 n — — — . — (Kc. 
 
 X X' 
 
 When X increases, all the terms following the constant term 
 n increase in (1) and decrease in (2). 
 
 Ifx becomes infinitely great (=oo),that is greater than 
 any assignable quantity of the same kind, the series (1) will 
 assume the form : 
 
 ?i-}-w Qo-f n Qo^-|-w oo'-|-<fec, (3) 
 
 in which any consequent is infinitly greater than its antecedent 
 (then this antecedent, compared with its consequent, is equiva- 
 lent to zero) J and the series (2) will assume the form : 
 
 n n n J, 
 
 00 Co' CC) 
 
 (4) 
 
 in which any consequent is infinitely smaller than its antece- 
 dent (then this consequent, compared with its antecedent, is 
 equivalent to zero) . 
 
 
p 
 
 $ 
 
 I i 
 
 ill 
 
 
 ■ r 
 
 111 
 
 m "I 
 
 ill* 
 
 ii'i' 
 I' 
 
 110 
 
 liooK IK. 
 
 The quantities noo , na ^, noo ', &c. are called infinites of 
 the first, second, third, &c. order ; 
 
 and the quantities !L, -**.,, — , tf-f. are called infinitesimals 
 
 X C» ' GO •* 
 
 of the first, second, third &c. order. 
 
 28. 
 
 DiFPEIlENTIAL OP A FUNCTION. 
 
 In the hypothesis of Art. 20, the 
 curve ACD may receive any increment, 
 either finite or infinitesimal. 
 
 ZM 
 
 Let ijs now consider the particular**" 
 case in which the curve receives but in- «X 
 finitesimal increments. 
 
 Assume ACD to be a curve represented by any equation, 
 yz=f(x). When the finite arc AB becomes AC by receiving 
 the infinitesimal increment BC, the abscissa Am becofno i An 
 by receiving the inff'nitesimal increment mn ; the ordinate 
 Bm (=^w) becomes Cn(^=n2^-\-pC) by receiving the infinitely 
 small increment ^^(7 ; the plane ABm becomes ACn by receiv- 
 ing the infinitesimal increment CBmn (the plane CBnin is 
 infinitely small, since its altitude wi?i is infinitesimal) ; the solid 
 generated in the revolution of the plane ABni^ about the axis 
 AX, becomes the solid generated in the revolution of the plane 
 ACp^ by receiving the infinitesimal increment generated by 
 CBmn ; the convex surface generated in the revolution of the 
 curve AB becomes the convex surface generated in the revolu- 
 tion of the curve AC, by receiving the infinitesimal increment 
 generated by BC. 
 
 Each of these infinitely small increments is the differential 
 of the variable magnitude which receives this increment. 
 
 Notation. 
 
 Let Z=length of the finite curve AB, represented byy=/'.c), 
 c?/=/i6', differential of ?; 
 x=^m, abscissa of the point B of the curve, 
 dx=mn{=Bp), differential of a; ; 
 
ARTICLE 32. 
 
 117 
 
 of 
 
 3ntial 
 
 :/'.Xt, 
 
 ij=Bm, ordinate of the point B, 
 dy=zCp, (lifiFerential of y ; 
 
 .'l=areaof the finite plane figure ABm, 
 dA=CBmn, diiferential of ^1 ; 
 
 F=: volume generated by the revolution of ABm, 
 rfF= differential of F; 
 
 «S^= convex surface of the solid of revolution F, 
 (£^= differential of S. 
 
 29. The infinitesimal analysis comprises two operations 
 called differentiation and integration. 
 
 30. DIFFERENTIATION. 
 
 Differentiation is the operation of finding the differen- 
 tial of a quantity. 
 
 As shown in the above notation, this operation is indicated 
 by the symbol d prefixed to the quantity to be differentiated. 
 
 It must be borne in mind that, in such a case, the letter d 
 is the symbol of an operation and not a quantity. 
 
 31. 
 
 Method op differentiation. 
 
 To find the differential of an equation between two variables^ 
 give an infinitely small increment to the independent variable 
 and find the corresponding value of the dependent variable. 
 Then, subtract the original function from the function thus 
 modified ; the result reduced to its simplest form will be the 
 differential required. 
 
 This method, being too long in practice, i- o ily used in deducing rules 
 for diflerentia'ion. 
 
 32. Differentiation OF the equation /y=ax-f- 6. (1) 
 
 By Art. 28, when the independent -.ariable x receives the 
 infinitesimal increment (?.r, the dependent variable i/ also 
 receives the infinitesimal increment di/ ; 
 
 hence, j/-\-di/ = a{x-\-dx)-{-b = ax-{-adx-i-b. (2) 
 
 Subtracting (1) from (2), dy = adx. (3) 
 
 Comparing the original equation (1) with its differential (3), 
 it will be perceived that 
 
 W6\\ 
 
i a, 
 
 I 
 
 i!|! 
 
 118 
 
 BOOK III. 
 
 By (Hfferentiation, 1" a constant factor (a) is not changed ; 
 2o a constant term (h) is dropped. 
 
 From the same comparison, we may also deduce the 
 Bule (A) to differentiate an equation whose form is 
 
 yz=ax-\-b. 
 
 " For the variables x and y, substitute their respective diffe' 
 rentials dx and dy, and drop the constant term." 
 
 33. Differentiation of the equation y=x'^. (1) 
 
 In this equation, as in the equation of Art. 32, when the 
 independent v.ariable x receives the infinitesimal increment dx, 
 the dependent variable y also receives the infinitesimal incre- 
 ment dy ; 
 
 hence, y-\-dy=(x-\-dxy^. 
 
 Expanding (x-\-dx)^^ by the binomial formula (Algebra) , then 
 
 n(n-l)^n-2^^^y,_^^^ .9, 
 
 y-\-dy=x -^-nx dx-\-' 
 
 1.2 
 
 -Jx (dxy'-^Sc. 
 
 Subtracting (1) from (2), there obtains : 
 
 dy^m^'-'dx+'^il^x"-' (dxy+dcc. 
 
 But, the infinitesimals {dx)'^, (dxy, dx, of a higher order 
 than dx, reduce to zero when added to dx (Art. 27) ; hence, 
 all the terms of the above series, the first term excepted, 
 reduce to zero ; 
 
 whence, 
 
 n 
 
 n- 
 
 dy OT d(x )=nx dx. 
 
 (3) 
 
 Comparing the two members of equation (3), we deduce the 
 Rule (B) to differentiate a variable (x") raised to any con- 
 stant power : 
 
 " Diminish the exponent of thf variable by 1, and multipfi/ 
 the result (j'" ') by the primitive exponent (ii) and by fht 
 differential (rfx) of the variable.^' 
 
 34. Application of Rules (A) and (B). 
 
 Ex 1. Differentiate y=mx+c, equation of the straight line. 
 
 By rule (A), dy=mdx. 
 
 Ex 2. Diff. y''=K^—x', equation of the circle. 
 
 By rule (B), 2ydy=—2xdx, or dy= 
 
 xdx 
 
 xdx 
 
 s/R'- 
 
 X' 
 
ARTICLE 36. 
 
 119 
 
 Ex. 3. Diff. y=qx^-^bx^-\-cx-\-/, equat. of cubic parabola. 
 
 By rule (B), dy=3ax'clx-\-2bxd.r-{-cdx, 
 
 or dy=(3axr-\-2bx-\-c)dx. 
 
 Ex. 4. Diff. y^=zax^-j-hx^-{-cx-\-/, equat. of diverg. parabolae. 
 
 By rule (B), 2ydy=Sa^xd.v-\-2bxdx-\-cdx, 
 
 ^ _ ( 3a.r-' -\-2h x -^ c) dx __ {^ax'±2hx-[-c)dx 
 
 or 
 
 2y ''>/ax^-\-hx'-{-cx-\-f 
 
 Diffei'entiate the following equations : 
 y^=4ax, equation of the common parabola. 
 ^2adx^ l^ax=Jx^^dx. Ans. 
 
 y \» 
 
 dy 
 
 Ir 
 
 y'^-=. _(2rta;— cC-), equation of the ellipse. 
 
 a- 
 
 dy= 
 
 h{a — x^dx 
 
 a\/2ox — X' 
 
 Ans. 
 
 y-= — (2ax+J7^), equation of the hyperbola. 
 
 a' 
 
 Ans. 
 
 a\/2ax-\-x^ 
 
 y2=- I x^ — 2lx^-\-[l'-{-7n?)x \ , equat. of the campana. 
 pi ) 
 
 ^ _(3x'*— 4Zj7+;-+m2)cZx- 
 
 35. 
 
 DIFFERENTIAL.S 
 
 Ans. 
 
 OF GEOMETRICAL MAGNITUDES. 
 
 Adopting the hypothesis and notation of Art. 28, we will 
 find, as fellows, the 
 
 36. Differential of a plane curve. 
 
 Since, in the hypothesis of Art. 28, the 
 increment BC of the curve AB is infini- 
 tely smalljit may be regarded as a straight _ 
 
 line whose length is infinitely small \X 
 
 hence, in the right-angled triangle BCp, y 
 
120 
 
 BOOK III. 
 
 II In 
 
 '.r, 
 
 or, by the notation, {dl)^=.(dx)^-{-(di/f. 
 But, dl or BC is the differential of the plane curve I or AB ; 
 hence, the differential of a plane curve is 
 
 dl=:\/dx^-i-dy\ (I) 
 
 Ex. 1. Find the differential of the parabola y^=4ax. 
 
 Solution. The differential of f= 4ax is di/ = ?^. 
 
 y 
 
 Solving with respect to dx, and squaring the result, there 
 
 obtains 
 
 dx": 
 
 4a2 
 
 yW 
 
 Substituting ^-~ for dx^, in formula (I), there obtains for 
 
 4a^ 
 
 differential of common parabol 
 
 ^'--M 
 
 Ex. 2. Find the differential of the circumference of a circle. 
 
 Solution. The differential equation of the circumference 
 
 ofa circle is rfy = (Art. 34, Ex 2); hence, by sub- 
 
 for dl/', in formula (I), there obtains for 
 
 stituting 
 
 x^dx'^ 
 
 R'-x^ 
 differential of the circumference of a circle : 
 
 ,, r, , x^dx^ VW—x^]dx'-\-x^dy? 
 
 Jidx Rdx 
 
 or- 
 
 Vk'-x' y 
 
 37. Differential of the area op a plane figure. 
 
 
 In the hypothesis of Art. 28, the alti- 
 tude inn of the plane CBmn is infinitely 
 small ; that is, wn=i?/)=<ix, and -Bwi=y, _ . 
 Cn=np-\-pC=y-\-dy=y (since the in-*^ — 7 
 finitesimal dy, when added to the finite y 
 
 \ 
 
 m n .0 
 
 
 quantity y, is equal to zero ; Art. 27) ; hence, the infinitesimal 
 plane CBmn is a rectangle in which, the length =y, the alti- 
 tude— £?«, and the area dAssydx. (Book I, 61). 
 
for 
 
 IE. 
 
 nmal 
 alti- 
 
 ARTICLE 38. 
 
 121 
 
 But, this infinitesimal area dA of CBmn is the diflFerential 
 of the area^4 of the finite plane figure -^i ^wt ; hence, the 
 differential of the area of a plane figure is 
 
 dA = ydx. (II) 
 
 Ex. 1. Find the diflferential of the area of a circle. 
 
 Solution. Since, the equation of a circle is yz=iy/K' — x^, 
 if y/R^—xr be substituted for y, in formula (II), there obtains 
 for diflFerential of the area of a circle, dA=:y/ R' — x' dx. 
 
 Ex. 2. Find the diflFerential of the area of the plane figure 
 whose curve is represented by the equation y=ax^-f- ^aj-'-f ex -f/. 
 
 Solution. Introducing this value of y, in formula (II), 
 there obtains dA=(ax^-\-hx'-\-cx-\-f)dx, for the diflFerential 
 required. 
 
 38. Differential of the convex surface 
 of a solid of revolution. 
 
 In the infinitesimal plane CBmn, BC=dl=\^dx^-\-dy'", and 
 Bm=Cn=i/ (Art. 36 and 37); hence, the solid generated by 
 the revolution of this infinitesimal rectangle is a right cylinder, 
 the convex surface of which is generated by the side BC or d', 
 and in which the circumference of each base is 2^^ ; hence, this 
 convex surface is dS= 2-i/X BC= 2r.y x dl= 2ry x \/dx^-\-dy2 
 (product of its base 2-y and the element BC of the 
 surface ; II, 67, 2). 
 
 But, this infinitesimal surface of revolution dS is the diflFe- 
 rential of the convex surface *S' generated by the revolution of 
 the curve AB ; therefore, the differential of the convex sur- 
 face of a solid of revolution is 
 
 dS= 2-ydl= 2zijK/dx'-j-d,/. ■ (III) 
 
 Ex. 1. Find the diflFerential of the surface of the sphere. 
 
 Solution. Since the generating curve of this surface of revo- 
 lution is the circumference of a circle, whose diflFerential is 
 
 dl=— -(Art. 36, Ex. 2), the substitution of ^ for dl in 
 
 y y 
 
 the formula (III) will give, for the diflFerential of the surface 
 of a sphere, dS=^2zy — '-~2zRdx. 
 
 y 
 
f 
 
 '! m 
 
 : t 
 
 I I ! 
 
 ItlJ 
 
 \i^' 
 
 122 
 
 BOOK III. 
 
 Ex. 2. Find the differential of the convex surface of a com- 
 mon paraboloid. 
 
 Solution. Since the generating curve of this surface of 
 revolution is the parabola i/''=4ax, whose differential is 
 
 dl=il ^4^2-)- y2 1 Art. 30, Ex I), the substitution of^^l^:. ,,:■ 
 -"' 2« ' 
 
 for (U in (III) will give, for differential of the convex surface 
 
 of a common paraboloid, 
 
 Jill ' a 
 
 y'^ydy- 
 
 39. 
 
 Differential of the volume 
 
 OF A solid of revolution. 
 
 In the infinitesimal plane CBmn^ 
 Bni=Cn=::f/, and mnz=dx (Art. 37); 
 hence, the solid generated by the revolu- - 
 tion of this infinitesimal rectangle CBmn"^' 
 is a right cylinder, the two bases of which -Xj^ 
 are equal circles (-y") generated by the revolution of the equal 
 radii y or Bm and i/ ov Cn, and whose volume dV=±-::y^(.L\ 
 (product of the base -y- and altitude dx'). 
 
 But, this infinitely small volume dV^ generated by the revo- 
 lution of the infinitesimal rectangle CBmn, is the differential 
 of the volume F generated by the revolution of the finite plane 
 ABm\ hence, the differential of the vo'ume of a solid of 
 
 revolution is 
 
 dY=-irdx. (IV) 
 
 Ex, 1. Find the differential of the volume of a sphere. 
 
 Solution. Since the generating plane of a sphere (circle ) 
 is specified by the equation //-=/^V-^', the substitution oi 
 R^ — x'^ ioT y- in the general formula (IV) will give, for diffe 
 rential of the volume of a sphere, dV=^-{R' — x-)dx. 
 
 Ex. 2. Find the differential of the volume generated by the 
 revolution of the plane whose curve is represented by 
 ?/*'= ax^ + ^x' -f ex -\- f. 
 
 Solution. This value of,y- introduced in formula (IV 
 will give dV=:-(ax''^-\-bx--\-cx-\-f)dx, for the differential 
 required. 
 
ARTICLE 48. 
 
 128 
 
 40. INTEGRATION. 
 
 Integration, the reverse of diiFerentiation, is the operation 
 to be performed upon a given differential in order to find tht* 
 function from which the given differential may have been 
 derived. 
 
 This operation is indicated by the symbol | which if< called 
 integral sign. 
 
 The function which is found in performing the operation 
 indicated by the integral sign is called integral. 
 
 Differentiation and integration being two inverse operations 
 which neutralize each other, their symbols prefixed tt) a quan- 
 tity also neutralize each other ; v.g. j d{ax ) =ax '. 
 
 41. Constant term. 
 
 Since, a constant term disappears by differentiation (32), it 
 is necessary, in performing an integration, to add a constant 
 term to the integral. It is customary to denote this constant 
 term by C. 
 
 We will see, further on, how the value of C may be determined. 
 
 43. Constant factors. 
 
 Since a constant factor is not altered by differentiation (32), 
 it may be written without the symbol of differentiation ; 
 v. g. d{ax'^)=ad(x^^) ; hence, a constant factor may also be 
 written without the integral sign ; 
 
 V. g- 
 
 / 
 
 nax dx=na 
 
 s 
 
 X dx. 
 
 43. Integration op the differential equation dy = adx 
 
 To indicate the integration of the differential equation 
 
 dy = adx, 
 
 write I dy=:a | dx. 
 
 But, by def. 40, the symbols fand d prefixed to ti;'^ same 
 quantity neutralize each other ; hence, adding the consta.^t 6'^ 
 
 we shall have the integral 
 
 y=aX'^C. 
 
 I -A 
 
 I 
 
 
IT 
 m 
 
 !i 
 
 ! :: I 
 
 ii'': 
 
 I 
 
 1 11: 
 
 124 
 
 BOOK III. 
 
 44, Integration ok the diff. equation c/y^s nx'^-^dx. 
 By art. 33, nxn— i dx=d(x^^) . 
 
 To indicate integration, yurite j^nx^'''-^dx=fd(x^). 
 But, by def. 40, the symboit, « and d neutralize each other, 
 in the expression Cd^x*^) ; hence, adding 6*, there obtains : 
 
 fnx 
 
 w—i 
 
 d£=x''-^C. 
 
 Comparing the two members of this equation, we deduce the 
 Bule (N) to integrate a diflferential whose form is nx'"' — ^dx. 
 
 " Drop the differential {dx) of the variable, add 1 to the 
 exponent (**— i ), divide the result (nx**) 6y thenew exponent (**) 
 and add the constant C" 
 
 Application of rule (N). 
 Ex. 1. Integrate dys=SaxMx. 
 
 To indicate integ. write Cdi/=8aCx^dx. 
 
 X* 
 
 y=Sa- + G=^2ax*-\- C. 
 
 By rule (N), 
 
 Ex. 2. Integrate di/=(ax^—cx'-\-5x)dx. 
 
 To indicate integ. write fdy=na\x^dx—cCx^dx-{-6ixdx, 
 
 By rule (N), 
 
 /v»0 (Y*3 /v*A 
 
 ii=a—c — +5— + 6, 
 6 3 2 
 
 or 
 
 X 
 
 y=%{ax'-2,cx^ 15) + a 
 6 
 
 Integrate the folloioing differential equations : 
 
 Ex. 3. (Zy=15a;2£fa;. 
 
 y:=bx^^C\ 
 Ex. 4. rZu=(4x3+3a:2— 6sc)tfa;. 
 
 i<— x<+»^— Sx^+C 
 
 Ex. 5. (?J.=- -I X* — (m+»)x^-j-mnx^ ldx-\-fdx. 
 
 Ans. 
 Ans. 
 
 .4= -^ Si2x--15;mi f /i)a:+20mni +fx+C. Ans. 
 60;? i i 
 
 and 
 
ARTICLE 4b'. 
 
 125 
 
 INFINITESIMAL ANALYSIS APPLIED. 
 
 45. Problem. 
 
 To find the area of the 'parabolic figure whose curvcis repre- 
 sented by the equation y=s-x^. 
 
 Solution. Assume Oa'n' to be a para- 
 bola represented by y=:-x^, 
 
 P 
 and let x=On, abscissa of any point w' of Oa'n\ 
 y = MH', ordinate of any point n' of Oa'n\ 
 j4=area of any segment Onn'. 
 
 Substituting -x^ for y in the formula dA = i/dx (Art. 37), 
 > 
 
 there obtains, for diiFerential of the area Onn', dAz=i-xMx. 
 
 P 
 
 To indicate integration, write 
 Integrating by rule N of Art. 4-1, 
 
 I dA = - j xMx. 
 
 A=\^'+C. 
 p 4 
 
 The value of the constant C may be determined as follows : 
 It is obvious that both the area A and its altitude or ab- 
 scissa X reduc6 to zero at the origin 0. Substituting zero for A 
 and X, in the integral, there obtains = 0-|-C ; 
 
 but, since the con&tant C is equal to zero at the origin, it must 
 also, as a constant, be equal to zero, at any other point of the 
 parabolic area ; hence, the entire integral, or area of Onn% is 
 
 A= 
 
 46. 
 
 -. - = -X'*X ,=-/. 
 
 p ^ p 4 4 
 Probleji. 
 
 To find the area of the 2>icine figure whose curve is repre- 
 sented by the equation y=--(2ax — x-). 
 
1 
 
 1 
 
 '( M 
 
 HI 
 
 I 
 
 126 BOOK III, 
 
 Solution. Asssume Ofd to be a curve 
 
 whose equation isy= (2<tx—x-); 
 
 a 
 
 and let x=Oh, abscissa of any point /*of the curve, 
 i/=h/] ordinate of any point /of the curve, 
 -.l = area of any segment Ob/. 
 
 Substituting -(2ax—»-) fory, in formula dA=ydx (Art. 37): 
 
 a 
 
 -1/9 
 
 there obtains, for diff. of area 0/>/, dA=z-{2ax — X')dx. 
 
 a 
 
 To indicate integration, write | dA=^ i{2taxdx — x'dx). 
 
 Integrating by rule A^of Art. 44, A= ^ (^-f\ -f C. 
 
 In this integral, as in the integral of the preceding Prob. 
 and for the same reason, the constant C is equal to zero ; hence, 
 ^he entire integral, or area of any segment Ohf, is 
 
 a 3a 3a3 333 
 
 Sch. When x=2a (or Od)^ in the given equatiouj 
 y=-(2ttx— a;-), there obtains 2/=zero (or the point d); hence, 
 
 a?/ 
 
 substituting 2a for x, and zero fory, in formula-' (as -|-y)j there 
 
 o 
 
 obtains for area of OfdO, A=\(2ay=\{0dy. 
 
 4*7* . t*ROBLEM. 
 
 To find the area of the plane figure tohose curve is represented 
 by theeqiiationy=- '. x* — (ni-\-n)jL^-\-mnx- I -f-f. (a) 
 
 Solution. If this value of y be introduced in formula 
 dA^=ydx (Art. 37), there obtains, for differential of the area 
 of the figure specified by equation (a) : 
 
 dA^= \ X* — (wi-f-M)a;^+ m?ix- ^dx+fidx. 
 
AKTICl.K 48. 127 
 
 To indicate integration, write : 
 
 I dA=m. I < x*dx^{m-\-n)xMx-\-miixrd:c i -\-f Cdx. 
 By rule iVof Art. 44 : 
 
 J) Li) 4 .J ) 
 
 Since, area ^1=0, when its altitude .'r=0, then 0=(>-fC; 
 hence, the entire intcjrrul, or area of the plane figure specified 
 
 by (a), is A = ^\ I2x'—l^(m+n)x-^2(hnu \ +/x. (b) 
 
 48. * Problem. 
 
 1 
 
 To find thi' area of the cvspidal parabola y — x-. 
 
 Solution. Assume aOc to be the para- jr 
 
 bola y 
 
 1 A 
 __r2 
 
 i-* 
 
 and let x=Ob, abscissa of any point a of the curve, 
 
 yz=ab, ordinate of any point a of the curve, 
 
 ^4 = area of any segment Oab. 
 
 1 ^ 
 Substituting — .x^fory, in formula dA=ydx of Art. 37, 
 
 /P 
 
 there obtains for diflFerential of the area of any segment Oab : 
 
 1 * 
 dA = x^dx. 
 
 s/p 
 
 By rule ATof Art.44, ^=_L.x3eI+(7. 
 
 \^P f 
 
 Since, area .4=0, when its altitude a;=0, then 0=0-f C; 
 
 hence, the entire integral, or area of any segment Oab^ is 
 
 1 x^ 2x \ ^ 2 '^ 
 
 J = — — .' — = v X'^ = -xy=~xObxab, 
 
 k/P % ^ k/P 5 "^ 5 
 
 the double of which is 
 
 4 '^ 
 
 J = -a;y="'x Obxac. 
 
 5 5 
 
 i*'^ 
 
 ili 
 
!• 
 
 128 
 
 49. 
 
 BOOK III. 
 
 Problem. 
 
 Tojind the numerical value of tt. 
 (Ratio of the diameter to the circumference of the circle). 
 
 Solution. The expression, in terms of 
 TT, of a circular arc a of any number n'^ of 
 degrees, is obtained by the proportion : 
 360°:n° = 2-R:a ; whence, the are 
 
 a =2-/2 — . 
 
 This beinu: stated, assume the diameters AB and CD, in the 
 circle ACBI), to be rectangular axes whose origin is the 
 center 0. 
 
 Draw the chord J/^V equal to the radius AO. find parallel to 
 the axis ^/i ; also draw the radii ()M,ON, and the ordi- 
 nate NP. 
 
 Let C=circumference /16ViZ>, /?=l=radius OB, x = OP, 
 
 y--PN. 
 
 By cohstruetion, the triangle MNO being e{{uilateral and 
 thereby equiangular, each of its angles=60°; 
 
 whence, arc J/CiV^=GO°, 
 
 30 TT 
 
 MC= CiV=arc of 30° = 27ri2x 
 3fF=FiV=0P 
 
 360 6' 
 and 3rF=FiV=0P =i 
 
 In AOyP. NP'=Oir'-OP\ or :y'=l-x^ 
 
 (equation of the circle whose raiJiusis unity). 
 
 The diflFerential equation of the circle isc?^= 
 
 («) 
 
 (0 
 
 RcLx 
 
 y/R'- 
 
 X' 
 
 (art. 36; Ex. 2); and, when A'=l, then dl^ f^ =(l-x-)~^rfx. 
 
 v/l-x- 
 
 ,,2\— i 
 
 Expanding (1— a-) ^ by the binomial formula (Algebra): 
 
ARTICLE 49. 
 
 129 
 
 OP. 
 
 and 
 
 (a) 
 
 ^2 2.4 2.4.« 2.4.t)..S ^ '' 
 
 and I (<'= ( ((/x-f ^+ + 4- - 4-vxc.); 
 
 . .' .' ^ 2 2.4 2.4.G 2.4.(5.8 ^ ^' 
 
 hence, /= ■ x + j^-f 'JJL + ^•^•^' + 5:^1?'. +&c.(<?) 
 
 2.3 2.4.5 2.4.().7 2.4.0.8.9 
 
 From etjuations (a) and (/>), OP or 2'=^, when CN or 
 
 ?=arc of 30°=B^ ; therefore, by substituting ^ for x in (d), 
 i) 
 
 we shall have, for the numerical value of an arc (/=-.) of 30°, 
 
 b 
 
 in a circle to radius unity : 
 
 Z^arc3(r='=i+_i_4. _^ +_jj_.4. '^•^'^ -f-^c 
 
 - 2.3.2-^ ^2.4.5.2'^ 2.4.6.7.2' ^2.4.«).8.9.2'-' 
 
 Denoting the terms of this series by a, b, c, t?, ttr. then : 
 
 l='~ = a^- 7> -f 4- (? -f e +&C., 
 
 1 . 1 17 1 o, \- 3 ^' 3" , o 
 
 in which, a = A, h = = x — , c= = X — ,d=&c: 
 
 2.3.2^ 4 2.3' 2.4.5.2' 4 4.5' ' 
 
 , 7 " , a V . h 3" , c^, 52 , <i 7- , J 
 
 whence. ?= ■ — a+- X — -+- - X h-X — + -X - 4-cvC. 
 
 6 ^4 2.3 4 4.5^4 0.7^4 8.9 
 
 Since this series can be prolonged to infinity, the quantity r 
 is incommensurable ; but, the larger the number of terms 
 summed up in the series, the higher will be the approxi- 
 mation of TT. 
 
 
 A value of r, containing 8 decimal places of figures, may be 
 obtained by an easy and rapid computation, if logarithms 
 be made use of to compute the terms of the following 
 series. 
 
 
 
!■' ! 
 
 130 
 
 BOOK III. 
 
 5 i 
 
 
 a = 
 
 b = 
 
 d = 
 
 / = 
 
 9 = 
 
 h = 
 
 ; = 
 
 /c = 
 
 i = 0.50 000 000 000 000 
 
 = 0.02 083 333 333 333 
 
 = 0.00 234 375 000 000 
 
 = r.'M) 034 877 232 143 
 
 (.00 005 933 1)73 500 
 
 = 0.00 001 002 390 570 
 
 0.00 000 211 841 7J>(> 
 
 0.00 000 042 G20 544 
 
 0.00 000 008 814 000 
 
 0.00 000 001 802 000 
 
 0.00 000 000 400 108 
 
 a 
 4>< 
 
 1' 
 
 2:.3 
 
 h 
 4^ 
 
 3- 
 4.5 
 
 |x 
 
 5- 
 »;.7 
 
 ^ 
 
 7-' 
 8.9 
 
 -s>^ 
 
 9- 
 10.11 
 
 (x 
 
 (11 r' ,_ 
 
 12.13 
 
 !x 
 
 '13 - 
 14.15 
 
 ^ 
 
 (15r _ 
 10.17 
 
 1- 
 
 (17,,- _ 
 18.19 
 
 ix 
 
 (iDr _ 
 
 ' 20.21 
 
 Hence,'=a + 6-|-c + r^+tfcT = 0.52 359 877 407 958 
 6 
 
 Multiplying by 0, - = 3.14 159 205 
 
 Sen. I. When great accuracy is not required, writer — 3.1416 
 
 50. Sch. II. In a circle to radius R (the circumference being 
 equal to 2-R), there obtains for the length of an arc of : 
 
 1° 2rg^gx3.14_r59 205 ^/^.^^oi 745 329 244 
 
 ' 360 180 
 
 r, /g xOOl 7^'^ 329 244 _^ ^QQ ^^,^^^ ^gg ggj 
 
 ' 60 
 
 igxOjMM)29 088 821^^^,,^^ ^,, ^.^ .^^ 
 ' 60 
 
ARTICLE 51. 
 
 131 
 
 To find the length of an rav, use will be made of the above 
 table, by proceeding as in the following 
 
 Ex. Find the length of an arc of 23° 34' AV'.b, in a circle 
 to radius (i?) = 10. 
 
 SoL.Arcof23°=J?x23 XO.Ol 745 329 =A?X0.4 014 257, 
 
 ircof 34'=^X34 XO.OO 029 0888 =/?X0.0 098 902. 
 
 arcof 4rV5z=iBx41.5x0.00 000 4848 ^Ry^O.O 002 Oil; 
 
 hence, the required arc of 23° 34' 4r\5 =^X0.t 115 17=4.11 517 
 
 51* Theorem. 
 
 The area of a circle is equal to the square of its radius 
 muUipUed hy t:. 
 
 Hyp. Asi=iume AB and CD to be two rec- c 
 
 tangular dmmeters in the circle ACBD, 
 and let ^=area, and jB=radius, of -4Ci?Z) ; 
 AsT. then will A=-K^. 
 Dem. Let ?= length of any arc ; v. g. BN", 
 ti^=: differential of /, 
 a=area of the sector BON', 
 rfa=diflFerential of a. 
 Since, the differential dl may be regarded as an infinitely 
 small straight line, then the area, (^dii') of the infinitely small 
 triangle whose base is dl and whose altitude is the radius Ry 
 
 will be equal to — , — ■; 
 
 whence, the differential eoiiation da= 
 
 By integration, 
 
 a 
 
 Ryl 
 
 fC. 
 
 But, the constant Cis equal to zero, sipoe the area a and 
 
 the arc I of the sector ^O^V vanish at the same time ; hence 
 
 Rx I 
 the entire integral, or area of /?OiV, ia ((== 
 
 Therefore, when the variable arc I becomes the entire cir- 
 cumference 2,-:R. thoro ubtuiiis for the area of Im circle : 
 
 '1-Ry.R 
 
 
 : = -/^ 
 
 q. E. D. 
 
III! 
 
 1^ 
 
 132 
 
 BOOK III. 
 
 W- 
 
 Cor. 1. The area of any sector MONCM is equcd to half 
 the product of its arc MCN and radius OC. 
 
 Cor. 2. The area of any segment MCNM is equal to the 
 difference between the sector MONCM and the triangle MONM. 
 
 Cor. 3. The area of the portion of a circle included betiveen 
 any two chords AB and MN is equal to the difference betvjeen 
 the two segments whose respective bases are AB and J/iY. 
 
 53. Theorem. 
 
 The formula of the area (E) of the ellipse is E=-ab. 
 
 Hyp, Assume ADBD' to be a ♦ 
 circle circumscribed about the ellipse 
 AdBd' (origin at the vertex A of the 
 axis), ^1 
 
 and let 2a= major axis AB, 
 26= minor axis dd', 
 and J57= area of the ellipse AdBd': «/! 
 
 AsT. then will E=-ab. 
 
 Dem. Draw the ordinate MF, and let AF=x,MF= Y,niF=y, 
 
 J.=area of the elliptic segment AFm, t4.'=area of the circular 
 
 segment AFM. 
 
 By this notation, the equation of the circumscribed circle 
 
 , ^^ b :, 
 
 IS i = \/2a.i'-— .i-and the equation of the ellipse is y — -^y/lax-X' 
 
 (Art. 10 and 12). 
 
 Substitutinu' successively -\/2ax— .r' and\/2a.r— t'^ for y, 
 
 in formula dA~ydj: (art. 87), then? obtains for : 
 
 h 
 
 difFerential of the elliptic area AFm. dA= '</2a.v — x^dx 
 
 ^ a 
 
 differential of the circular area AFM. dA'— s^'lax—.r^dx. 
 
 Integrating both differentials : 
 
 4=- fx/2ax-.i^dx. (1) 
 a .' 
 
 and 
 
 ])ivid. (1) by (2), there obtains " ,= - ; Avhence, A~..^ J (3) 
 
 A'^ i ^/2ax-ird.v. (2) 
 
 A _b 
 
 ~ a 
 
 a 
 
ARTICLE 52. 
 
 133 
 
 Since this equality is true whenever both segments A and A' 
 Lave a common altitude (common abscissa x), then it is true 
 when this common altitude becomes the major axis 2a. But, at 
 
 this limit, A becomes the semi ellipse AdB{ss-\ and A' be* 
 
 comes the semicircle ^Z)i?/=a— 1. Therefore, by substitu- 
 
 IP — ' 
 tion of —and — for A and A' in (3), there obtains E=sT:ab. 
 2 2 
 
 Cor. 1. TJi3 area of an elliptic segment of one base or of two 
 hases perpendicular to the major axis is equal to the product of 
 
 the ratio - and the segment of the same altitude (same ab&- 
 
 cissa) in the circumscribed circle. 
 
 ¥ov {ahoxe Vein.), AFm=-AFM {4),sind Afn=~ AfK (5> 
 
 a a 
 
 circle 
 
 Subtracting (4) from (5), Afn^AFm=-{AfN--AFM), 
 
 a 
 
 or Fmnf=z^-FMNf. (6) 
 
 a 
 
 Cor. 2. Tha area (J.) of an elliptic segment of one base or 
 of tico bases perpendicular to the minor axis is equal w th^ 
 
 .oduczofthc ratio - and the area (A') of the segment of the 
 
 b 
 
 sirrn" 'ilritiufe (same abscissa) in the inscribed circle ; that is, 
 
 A^^A'. 
 
 For, the above equation (3), or u4r=^ ^4 '. is true whatever 
 
 a 
 
 'i\ybe the ratio- ; hence, it is true whe" b becomes greater 
 a 
 
 tic ' '.I ; t>at, ii. this hypothesis, the vertical axis 26 becomes the 
 major axis, and the horizontal axis 2a becomes th«^ minor axis 
 
 :ii 
 
 ■»35j(] 
 
I 
 
 II 
 
 11! I 
 
 'ill 
 
 3 'ir 
 
 m <! 
 
 
 ill 
 II, 
 
 l!!ll!l 
 
 
 ■it 
 
 il I' 
 
 134 
 
 BOOK III. 
 
 of the ellipse circumscribed about the circle ADBD' ; then, to 
 comply with the usual notation which invariably represents the 
 major axis by 'la and the minor axis by 26, a is to be changed 
 
 into h, and b into a, in the formula A=:-A' : 
 
 a 
 
 whence, 
 
 53. 
 
 A=W. 
 
 Theorem. 
 
 The area of a ')n7n.on parabola is equal to two thirds of the 
 circumscribed rectM 
 
 Hyp. Assume CAD to be a common parabola 
 expressed by y=2y^ax (origin at the vertex yl of »^ ^ 
 the axis, Art. 14). ^/l 
 
 Draw the rectangle CDEF ; and \et x=AB, ^ " 
 y^BC, .1 =area of ABC, and ^' =arer. of CADC ; \ j 
 
 AsT. then will A'^%ABx CD. £-^ 
 
 Dem. Substituting 1y/ax for y, in formula X 
 dA=f/d.r (Art. 37), there obtains for differential of the 
 area of ABC, 
 
 dA=2 y/axdx= 2a*x^dx. 
 
 To indicate integration, write | rfvi = 2a- | x^dx. 
 
 By rule N of Art. 44, 
 
 .h..j% 
 
 A = 2a^X'Li+C. 
 ■t 
 
 ■ I 
 
 Since .4=0 when x=0, then 0=0+ (7; 
 
 hence, the entire integral, or formula of the area of ABC, is 
 
 A= 
 
 2rt2j.Tr 4a^x'^ 2x 
 
 3 
 
 — ^X 2v/ax=|a?y. 
 
 When the abscissa x becomes AB, the ordinate y becomes 
 BC, and the area of ABC is A=^xy=:%ABxBC, 
 the double of which, or area of CADC, is A=i^ABx CD. 
 
ARTICLE 54 
 
 135 
 
 54. 
 
 THEOREM. 
 
 idx. 
 
 icomes 
 
 In the plane figure whose curve is represented hy an equation 
 contained in y=ax*-fb3t*+cx4-f, the area of any segment of 
 two parallel bases may be measured by the formula 
 
 the ordinates Y and y representing the parallel bases, the 
 ordinate Y' the median section^ and H the perpendicular dis- 
 tance between the bases, of the segment. 
 
 Hyp. Assume ABD to be a curye whose 
 equation is contained in y=aic'4-5x'+cr+/. 
 
 On the axis H-X-4, lay ofiF mn=:no ; draw 
 
 tlie ordinates Bm^ Cn, Do ; 
 
 and let -4m =x \ Ao^X' 
 
 Bm=v ) , Do: 
 
 
 An 
 
 Cn= 
 
 Ao-^Am_Jr-\-x —, 
 
 X « ^-A 
 
 2 
 
 1 
 
 (1) 
 
 ^'=:area of ABm, ^"=:area of ^Z)o, and ^=^"—^'. 
 
 TT 
 
 AsT. then will the area of BDom or ^=_(!r+4F'+y). 
 
 6 
 
 Dem. When the abscissa of the curve ABD successively 
 assumes the values Am, An, Ao, or x, X', X, its ordinate 
 becomes Bn, Cm, Do or y, Y', Y, respectively. 
 
 By this notation, y=ax' +6x' -^-cx +/j (*) 
 
 T=^aX^-\-bX'^-\-cX' -\-f, (2) 
 
 Y—aY^ -f & J:2 ^cX +/• (3) 
 
 If the value of X, in (1), be introduced in (2), there obtains: 
 
 r=a(£+£)'+j(£+r)'+c(£+f)+/. (4) 
 
 Expanding the binomials of (4) and multiplying the result 
 
 by 8, 
 
fl 
 
 HP! 
 
 ! 
 
 !i 
 
 ri 
 
 
 i; » 
 
 1 1 ? 
 
 ! r 
 
 liiM 
 
 136 
 
 :>. :^ :r^ :J^ cs 
 
 a* N^ c<i M 
 
 
 + + + +.S 
 
 1^ 
 
 + 
 
 
 
 o 
 2 
 
 + + 4-+^ 
 
 (M 
 
 CO 
 
 5^ 
 
 + 
 5^ 
 
 + + + + « 
 
 < Q 
 II + 
 
 1 
 
 «8 
 
 -r (M 
 
 + 
 
 4^ 
 
 8 
 
 + 
 
 + 
 
 a 
 
 II 
 
 Q 
 4- 
 
 
 
 C^ M 
 
 11 11 
 
 X 
 
 5<i 
 
 (M 
 
 + 
 
 
 + 
 
 QQ 
 
 c 
 
 ei 
 O 
 
 a> 
 
 a> , CO 
 
 
 
 « 
 o 
 
 a. -»^ 
 
 T3 a 
 e8 a 
 
 . •» S3 
 
 ^ Cm 
 
 92 
 
 BOOK III. 
 
 .-I 
 
 
 ! 
 
 I 
 
 
 8^M 
 
 J 
 
 8 
 
 S 
 
 .o 
 
 o 
 
 fcC 
 
 c 
 
 «0 t* OC 05 
 
 8 ^ 
 
 
 ^ 
 
 ^ + 
 
 (>V 8 
 
 + + + i 
 
 J + 
 
 4- •- 
 
 + 
 
 CO 
 
 + 
 
 C0 
 
 8 
 
 CO I 
 
 8 IfO 
 
 + 
 
 
 li 
 
 8 '-* ^ 
 
 ft 
 
 o 
 
 •♦J r 
 
 S 
 
 
 e 
 
 O 
 
 a) 
 
 -_ s 
 
 St 
 
 8 
 
 CO 
 
 
 rO + 
 
 T -^ 
 
 + "8 
 
 8 ^T 
 
 I ^ 
 
 '^-^ 8 
 
 «s aj 
 
 -f ■* 
 
 ■*a 
 
 ■ + a 
 fcN — 
 
 •—1 oa 
 
 r/3 ^^ 
 
 ^ + 
 
 O rN 
 m 
 
 V 
 es 
 
 
 + 
 
 
 ^ 
 
 8 
 
 o 
 a" 
 
 U 
 
 to 
 
 a> 
 
 a 
 
 o 
 
 a 
 
 o 
 
 :^ 
 
 O 
 
 
 8 ^ 
 f + 
 
 ^ a 
 la '^ 
 
 II II 
 
 -fist 
 II 
 
 J 
 
 CO 
 
 8 
 I 
 
 O 
 
 c- I 
 
 0) :; 
 
 •5 ^ 
 
 i ^ 
 
 a II 
 
 a-^ 
 
 r 5 
 
 .2 « a 
 
 O 
 
 f4 tn 
 
 2 CO ^ 
 
 o 
 
 a 
 
 , 
 
 ^ 
 
 
 ij. 
 
 e«-i 
 
 
 
 o 
 
 (1 
 
 n 
 
 «• 
 
 
 
 2 
 
 
 £ 
 
 fcC 
 
 B 
 
 ^^ to 
 C -^ 
 
 .2 « 
 
 a 0) 
 
 tc 
 
 <a 
 
 >» 
 
 a 
 
 c3 
 
 Cm 
 O 
 
 «e 
 
 :3 
 O 
 
 Cm 
 
 o 
 
 
 
 a 
 
 c2 
 
 c2 
 
 IS 
 
 S 
 
 • M 
 
 eS 
 O 
 
ARTICLE 54. 
 
 137 
 
 Cor. 1. In the plane figure whose curve is contained in 
 y=ax^H-bx"+cx + f, the area of <f segment of one ?)asc (Y) 
 
 is measured lii/ the formula A=:— (Y-f 4Y'-+-0), and when 
 both bases(Y and y)reduce to zero, formu!({(\0)(d8o reduces to 
 
 A=5(0+4Y'4-0). 
 o 
 
 Sell. I. The formula ^4=— (y-|-4y-f y) applies exactly 
 
 (') 
 
 but to the f\ane figures whose curves are represented by the 
 
 first, second and third degree in ij=ax'^-\-hx'-\-cx-^f . [a) 
 
 (Art. 23, 1' , 2>, 3'0- 
 
 For, assuming («) to become F= a A' +bX^'~ -fcA' ~"^+&c, 
 
 the above equations (5) and (9) will become : 
 
 2"-\Y+4r-\-y)=a(f-'' +l)(X'' + X''-'x+&c)+lkc, (5') 
 
 and 
 
 .4=(A'-.c) j Ji-(A" + A""\t+.Vc)+^c. I . (9') 
 Cw+1 j 
 
 Since the variables A, x, and the coefficients a, b, c, dx. may 
 
 assume any values, these equations remain true when x and 
 
 each of the coefficients, i, c, d, i^c. simultaneously reduce to 
 
 zero; but, then equation (5 j, divided by 2 ~", reduces to 
 
 (r+4r+y) 
 
 2 +1 v^t 
 
 =z a 1 — A , 
 
 and (9) reduces to ^1=1 x A . 
 
 ^ ^ 7i-f-l 
 
 Now,whenbysubstit.,.4=-(r-f4r'-f y) or E{Y+4Y'-\-y). 
 
 6 6 
 
 tl 
 
 len 
 
 ^'rx"=a^Ll±lv"' 
 
 u-j-l 
 
 o 
 
 n—'. 
 
 whence. 
 
 o"-.i 
 
 O^'t 
 
 (;x2"-^=:(H+l)(2''--^-+-l) 
 
 Therefore, the above Sch. is true, since this last equation is 
 satisfied by n=.l, ?i=2, m=3, and fails for any other value of»i. 
 
 
 j'jSf 
 
 ^ff 
 
^ip 
 
 m 
 
 138 
 
 BOOK III. 
 
 
 Sch. II. Because formula (10) is independent of the coeffic- 
 ients, a, h, c, f of the function (a), these coefficients may, with- 
 out aflFecting the formula (10), assume any value : positive or 
 negative, entire or fractional, rational or irrational ; they may 
 even vanish and thereby reduce the cubic equation (a) to an 
 equation of the first or second degree. 
 
 5t>. Sch. III. By the analysis in the above theorem, we also 
 
 obtain a formula ((>, 8 or 9) in function of the abscissa x (altitude of the 
 
 figure) aftVcted by coefficients, from which special formuhe • may be 
 
 deduced for mcasi;ring the area of any jjlano figure whose curve is 
 
 contained in 7/=ax'^ -\-bx^ -\-cx-{-/; as follows : 
 
 1 ^ 1 
 
 1" LLttin'Trt=-. l>^= — -(m-\-n), c=~vvi. /=0, in th.'iunction 
 
 P 1> 1> 
 
 ?/=a.r'i-|-6a^-+ca:-f:^. this function reduces to ?/=-■( x^--{m-\-n)x'^-\-:nnx I • 
 
 which i J an equation of the cubic parabola (Art. 23, 1"). Hence, if theje 
 values of the coefficients a, b, c,f, be introduced in the formula (6) of 
 Art 54, there obtains : 
 
 A=i— < -x^ — -(in-\-n)x-^-\-~mnx- v, 
 12 ^ p p p J 
 
 or A=—^< 3x- — 4(vi4-n)x -i-Omn V, 
 
 for the area of the cubic parabola whose equation u 
 y= ~ < a;3 — Qn-\-n)x'^ -\-mnx I. 
 
 By the same process, we will find the special formulas of the areas of 
 the other parabolic figures whose equations are contained in (a), and 
 which are tabulated below 
 
 2" Letting a=0, b=0,c==m, and/=0, in y=ax^-\-bx^-\-cx-\-/,ihis{\inc- 
 tion reduces to y=mx^ which is the equation of ABD reduced to a straight 
 line passmg through the origin A, (Art. 23, 3'>). Hence, introducing these 
 values cf o, b, c,/, in formula (9) of Art. 54, there obtains : 
 
 * There is no necessity of finding these special formulae, as well as the spec- 
 ial formulae of the convex surfaces and volumes in art. 59,61, since the formula 
 
 // 
 
 ~[B-[-'^D'-\-b) includes them all and replaces them all toa great advantage. 
 
 But, wc must not lose sight of the soiontifiQ interest which la attached to 
 those Rpccial formulae ; besides, these deductions are an excellent algebraic 
 
 cxcrci»c. 
 
ARTICLE 55. 
 
 139 
 
 A ' '-A'=^ "'^ X em{X-\.x)—:^~'^(,mX-\-mx)—^^ "~^( F-f ;/) , since mx=y 
 
 \ 1 it a 
 
 and mX-=Y. Now, to comply with the common notation, letting 
 A"—A''=A ; the altitude mn or X^x^H\ and the parallel bases JSmand 
 Cn or y and F=i' and iS', respectively ; there obtains for the area of the 
 
 trapezoid CBmn : 
 
 A=--{B'-\-b'). Again, assuming a nega- 
 
 z 
 
 tire straight line, y=^nx. to pass through the same origin A, the segment 
 whose altitude is mn and which is limited by the positive ordinatesy=s7nx, 
 Y-=mX, and the negative ordinates y'=—nx, Y'=—nX will give a tra- 
 pezoid of any form, the area of which will be 
 
 .4=^{(r4.r)-f-(y+y') \ovE(B^b). (1) 
 
 When the base b reduces to zero, the trapezoid reduces to a triangle 
 whose formula then is A=\By,H. (2) 
 
 When b becomes equal to ^, in (I ), the trapezoid becomes a parallelo- 
 gram whose formula then is A=By,II. (3) 
 
 RESUMi:. 
 
 Special formuliB of the areas of the plane figures whose cufves are contained 
 hi y=&x^^-\-h\^^-^cx-\-f•. 
 
 Cubic parabola, y= _ \x^—irti^n)x'--\-mnx I. 
 A= < \'x- — A(m-\-n)x-\-*jmn \. 
 
 Art 23 
 
 Alt. 2i,ng. (i') 
 
 Area, 
 
 Cubic parabola, ?/= - (z'' — 2nx'^-\-n'^x). 
 
 P 
 
 Area, 
 
 .4=-_(3a:2 — 8nz-|-6n2 ). 
 Cubic parabola. y= - (x'-^-\-2mx^-\-m'^z). 
 
 Area, 
 
 Cubic parabola, y= _ x^. 
 
 P 
 
 .4=-?i(3z24.8ma:4.6m2) . 
 12p 
 
 1 
 
 Area, 
 
 A=z 
 
 4p 
 
 Cubic parabola, y= ]. ix^—2lx^-\-{la-\-nfi}x\. 
 Area, A=z.^ / 3x^—8lx+6{i:i-^m') I . 
 
 Art 21, fig. (c') 
 
 Art 21, lig. {d') 
 
 Art 21, fig. («') 
 
 Art 21, fig. (/) 
 
140 
 
 IJOOK III. 
 
 ill 
 
 m 
 iiii 
 
 . ] 
 
 Cuiumon parabola, f/s=: _ z^. 
 
 Area, 
 
 1 
 
 Common parabola, ?/= - («/ — z"*). 
 
 Area. 
 
 X- 
 
 A= _ (3n— 2z). 
 
 1 
 
 Common parabola, ?/= - (2- — 2mx-|-w--|-w2)^ 
 
 App. C, fig. {h) 
 
 Area, 
 
 1= if_ j a;-— :^wja:4-:;(w-^+«2) j . 
 
 Area of a trapezoid : A:=.\H{li-\-b). 
 
 Area of a triangle : A=i\By,]L 
 
 Area of a parallelogram : A^=IiX.lI. 
 
 App. 0, fig. (/) 
 
 App. C, fig. {<l) 
 
 56. 
 
 Theorem. 
 
 I 
 
 The area of the surface of the sphere is equal to four of it& 
 great circles. 
 
 Hyp. Assume the surftice of a sphere to be 
 generated by the revolution of the souii cir- 
 cumference ADB about its fixed diameter 
 AB ; and let ^=radius AC, :;nd aS'= surface " ^ 
 of the sphere ; 
 
 AsT. then will S=4r:Ji\ 
 
 Dem. When the origin is at the vertex A, the equation of 
 the circle is y''^2Rx — xK and the diiferential of S is 
 ds=2-Rdx (Art. 38, Ex. 1). 
 
 Integrating, there obtains : s=2-Rx-\-C. 
 
 It is obvious that the surface s reduces to zero, when its 
 altitude or abscissa x reduces to zero, at the origin A ; at this 
 point then, the above integral becomes 0=04- (7; that is, the 
 entire integral, or area of a zone of one base (altitude =x) is 
 s=2rMx. ■ (1,, 
 
ARTICLE 57. 
 
 141 
 
 When the altitude x becomes the diameter AB or 2R, tho 
 generating arc becomes the semi circumference ADB ; then, 
 substituting 2R for x in (1), there obtains for the surface or 
 the sphere : 
 
 S=27:Rx2R=47:R'. (2) Q. E. D. 
 
 Cor. 1" In a whole sphere^ 2* in a spherical zone of one bane 
 3'> in a spherical zone of two bases, the area of the surface is 
 invariably equal to the product, of the circumference of a great 
 circle of the sphere and the altitude of this surface. 
 
 For, lo area {S) of a whole sphere : S-27:Rx 2R ; (2) 
 2'^area(ASf')ofazoneofonebase(altitude=X;:*S'=2-i2xX; (3) 
 3" subtracting (1) from (3), there obtains for area {S' — «) of 
 a zone of two bases (altitude= JT— x): S'—s—2t:R(^X—x). (4) 
 
 57. Relation op the curve op a solid op revolution 
 and op the circumscribed polyedroid. 
 
 V^ The curve of a solid of revolution, the abscissa of the 
 curve of the circumscribed poU/edroid and the element of contact 
 betvjeen the convex surfaces of both these solids are but one 
 and the same line {ANC). 
 
 For, assume ABCD to be a portion ^ 
 of a polyedroid circumscribed about the y 
 solid of revolution generated by the 
 plane ABC revolving about the axis 
 AB ; further assume the curve ANC ^ M £ B 
 
 to be the element of contact, and the curve ADD to be a lateral 
 edge. Then, if the lateral face ACD of the polyedroid be 
 developed upon a plane, the curve of the solid of revolut" >.. 
 which is also the element of contact ANC, becomes a straigM 
 line which is the abscissa (perpendicular distance from the 
 origin A to the ordinate CD) of the plane curve ADD. 
 
 2^ The equation of the curve of ariT/ right quadrangular 
 polyedroid is also the formula of rectification of the curve of 
 the inscribed solid of revolution, solved with respect to its 
 ordinate. 
 
 I 
 
 
 . ^1 
 
HI 
 
 HOOK 111. 
 
 For, when the right polyedroid is quadranj^ular, its cross- 
 section at any altitude (as .M) is a square circumscribed about 
 the circular cross-section (to radius MN) of the inscribed solid 
 of revolution ; and then, MJV=iyO, EF=FG,BC=CI),dc 
 
 Letting 2;=ab8cissa {AN), and y=ordinate {NO=:NM) 
 of any point (0) in the curve AOD of a right quadrangular 
 polyedroid ; then any equation, y =/(%),&» i/=az^-\-hz^-\-cz-\-f, 
 which expresses the relation between the co-ordinates ij and z 
 of any point (0) of the curve AOD, also expresses the length 
 of the ordinate y (as MN) of any point (iV) of the curve z 
 (^ANC) in terms of this same curve z. 
 
 Conversely, if the equation y=f(z) (1) 
 
 be solved with respect to z, the result z—f{y) (2) 
 
 will express the length of the curve ANC, in function of its 
 ordinate y. 
 
 This equation (2) is called formula of rectification of the 
 curve 2, and the equation (1) is the formula of rectification of 
 the curve z, solved with respect to it ordinate y. Hence, the 
 equation of the curve of any right quadrangular polyedroid is 
 also the formula of rectification of the curve of the inscribed 
 solid of revolution, solved with respect to its ordinate y. 
 
 Note. An example of both formula; (I) aad (2) will be found in equa- 
 tions (3) and (4) of App. N. 
 
 58. 
 
 THEOKEM. 
 
 If the equation of the curve of a right quadrangular polye- 
 droid he contained in the collective equation y=az**-f-bz--f-cz-f f 
 (z denoting the abscissa, and y denoting the ordinate of any 
 point in the curve of the polyedroid), the lateral area of any 
 frustum of the inscribed solid of revolution will he measured 
 hy the formula S = iH (C+4C'-fc). 
 
 Hyp. Assume the wedge ABCD to ^ 
 be a portion of a right quadrangular / 
 polyedroid circumscribed about the solid 
 generated by the revolution of the plane 
 ABC about the fixed axis AX ; the ^ 
 
ARTICLE 58. 
 
 143 
 
 curve ANC being the element of contact, and the curve AOD^ 
 whoso equation is contained in y = az'^-\-bz'^-\-cz-\-f^ being a 
 lateral ed^o of the polyedroid. 
 
 AsT. then will the area of the zone generated by the revolu- 
 tion of anyarc CNha measured by the formula <S'=^/i(Cf-4C"+c); 
 // denoting the element of contact, the circumferences C and c 
 denoting the bases, and the circumference C denoting the 
 median section of the zone. 
 
 Dem. On ANC, lay of£ NF=FC: draw the ofdinates BC, 
 EF, MN; 
 
 AN=z') 
 =NO)=yU 
 
 let 
 MNi=NO)=y 
 
 AC=Z\ 
 = CD)=Yh 
 
 jj,_ AC+AAr 
 2 
 
 EF{^=^FG) 
 
 BC{=CD) 
 
 =Z\ 
 
 (1) 
 
 =y\ 
 
 and «(!&«'= areas of the zones generated by the revolution of 
 the arcs -.liVand J.C, respectively. 
 
 When the generating curve ANC successively assumes the 
 values AN, AF,AC or z, Z\ Z, its ordinate becomes MN, 
 EF, BC ox y, Y', F, respectively. 
 
 By this notation, y=az^ -{-hz^ -j-cz -\-f, 
 
 (a) 
 
 r=aZ'^+bZ''--i.cZ'-hf, (2) 
 
 Y=aZ^-^bZ'+cZ +f. (3) 
 
 Introducing, in (2), the value of Z' from (1), we have : 
 
 Expanding the binomials of (4), and multiplying tne result by 8, 
 
 I 
 
 ! 
 
 i 
 
 M 
 
144 
 
 
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ARTICLE 59. 
 
 145 
 
 But, equation (5) proves that the quantity inclosed within 
 the braces of (9) is equal to 2 {V+4:Y' + y)\ therefore, by 
 substitution, there obtains for the area of the ^one, gonerated 
 by the revolution of any arc (Z — s) or CiV : 
 
 8'-ii-.^:zI{27tY+4x 27rr + 2Ty), 
 or, letting «'— «=>S, Z— z=:ir,and2;rr==C,27rr==iC",2rry=c, 
 
 O 
 
 (10) Q. E. D. 
 
 Cor. 1. If one base (c) or both bases (c and C) reduce to 
 zerOf in formula(10), the area (S) of the zone of one base or of 
 the whole surface of revolution will be measured by 
 |H(C+4C'4-0) or iH(0+4C'+0), respectively 
 
 Cor. 1. The area (A) of the convex surface of any polye- 
 droidy whose curve is contained, in the collective equation 
 y=az^-^-bz^H-cz4-f, ts measured 6v/o>'wiw?« A=^H(P+4P'+p) 
 in which H is the element of contact^ and P, P', p are the 
 perimeters of the polygons circumscribed about the above cir- 
 cumferences Q , C, c, respectively . 
 
 For, it has been proved (II, 72, S*^), that S=mA^ and 
 (C-l-4C"4-c)=:w(/'+4/''+j[>) J hence, since the element of 
 contact H is common to both surfaces *S' and -4, there obtains ; 
 
 ,Sr=iZr{C+4C"4-c)=m^l=m|-//(P4-4P'4-i>). 
 Therefore, A=^ ^H{P-\-^P'-^p). 
 
 Sch. I. Formula ^= iir(C+4C"4-^)and A=:\H{P-k-^P' -\-p) 
 apply exactly but to the convex surfaces specified in the above 
 theorem and Cor. 2. (Art. 54, Sch. I). 
 
 Sch. II. Since the formula (10) is independent of the coeffic- 
 ients a, 6, c,/' of the function (a), these coefficients may with- 
 out affijcting the formula (10) assume any value : poiiitive or 
 negative, entire or fractional, rational or irrational ; they may 
 even vanish and thereby reduce the cubic equation {a) to an 
 equation of the 1st or 2nd degree. i 
 
 59* Sch. III. By the preceding analysis (58), we also obtaia a 
 formula (6, H or 9), in function of the curve z (element of contact of the 
 convex surfaces of both the polyedroid and the inscribed solid of revolo* 
 
 10 
 
146 
 
 BOOK III. 
 
 tion) affected by coefficients, from which special formulae, in function of z, 
 may be deduced for measuring th ^ convex sujface of any solid of revolu- 
 tion injcribed in a right quadran>(u ar pol\ edro.d whose curve is con- 
 tained in y=az3-\-bz^-\-ez-{-/, as fellows : 
 
 1° Lettingc=-,6=s=— -(m-fn), c=-mn, and/=Oj in y=:az^-\-bz2-\-cz-y, 
 P P P 
 
 this function reduces to i/=-< z^-(m-\-n)z^-\-mm V which is the equation 
 
 (h') of the cubic parabola (23, 1"). Then, introducing these values of 
 a, b, c.f, in formula (6) of Art. 58, there obtains : 
 
 5«s- ^ ^ 
 
 or 
 
 p p p } 
 
 (1) 
 
 for the convex surface of the solid of revolution inscribed in the right 
 quadrangular polyedroid whose curve is represected by 
 
 pi 
 
 (wi-|-n)z2-j-ninz 
 
 !• 
 
 p p 
 
 Xow, introducing, in formula ^=5 — =5 of Art. 74, Book II, the 
 
 value of S given in (1), and letting F=«, there obtains : 
 
 Pz 
 
 A=— { ^z^—i(m-\-n)z-\-6mn \ , 
 
 .12p K i 
 
 for the convex surface of any polyedroid c rcumscribed about the prece- 
 ding solid of revolution. 
 
 By the same process, we will find the special fcrmtilae of the convex sur- 
 faces ofthesolidsof revolution and of the circumscribed polyedroids whose 
 curves are represented by the other eq nations contained in the 2nd and 
 3rd degree ofy^«2-^^ \-bz^-{-cz-\-/. 
 
 2° Letting a=0, i=0, c=7«, and/=0, in y=az^-\-bz^-\-cz-\-/, this func- 
 tion reduces to T/=m2, which is th. equation of th3 straight line passing 
 thro gh the origin (Art, 23, 3"). Tlieii, introducing these values of a, b, c,/, 
 in formula (9) of Art. 58, there obtains : 
 
 s'—8s:s^{Z—z)xGm[Z~z)=^T{Z—zXmZ-\-mz). 
 
 Substituting y for its valuj m^, and F for its value mZ, and letting 
 S=a' — s, there obtains, for convex surface of a frustum of a cone (slant 
 heightss:^— 2) : 
 
 5=:7r(^— «)( r-f y), or >S'=^,T(/2-f r), (1) 
 
 to comply with the common notation, in which the t«lant height (Z — z) is 
 denoted by JH, and the radii V and y of the bases are denoted by R and r 
 respectifely. 
 
ARTICLE 59. 
 
 147 
 
 When r reduces to z.?io, in (1), the frustum becotn'js a whole cono 
 whose convex surface then is : S=E-jR. (2) 
 
 Now, lettint^ 2nr ami 2n-^:=upper and lower circumferences of a frus- 
 tum of a cone ; j9 and 7*=ui)per and lower perimeters of a pyramid cir- 
 cumscrib. d about the f.ustum of a cone ; S and .l=convex surfaces of 
 these frusta, respectively, there obtains (II, 72, 3') : 
 
 2,T7e-}-2-r=2-(i?-|-r)=ni(P-|-/>), and S—mA, 
 Ih^n, substituting inA for S, and \m{I'-\-p) for :T{R-\-r), in (1), there 
 obtains for the convex surface of a frustum of a pyramid : 
 
 -r.^ P^P 
 
 A=Ex 
 
 2 
 
 (3) 
 
 When p vanishes, the frustum becomes a whole pyramid whose convex 
 
 surface then is 
 
 ._ ExP 
 
 When r becomes equal to R, in (1), and^ equal to P, in (3), the frus- 
 tum of a conu becomes a right cylinder, and the frustum of a pyramid 
 becomes a right prism whose convex surfaces then are 
 
 S=27rRxE, and A=PxE, respectively. 
 
 RtSUM^. 
 
 Special/ormulm of the convex surfaces of solids of revolution inscribed in right 
 
 quadrangular poly edroids whose curves are conlainedin ys^az^-j-bz^^-j-cz-l-f, 
 
 and of any polyedroid circumscribed about any of the preceding solids of 
 
 revolution. 
 
 Notation. In this B4@um^, the curve, specified by its annexed equation , is 
 assumed to be the curve of a right qmidrangular polyedroid ; .? denotes the 
 convex surfao;j of the solid of'revoladon in.-cnbed in this right quudraiigular 
 polyedroid, and A denotes the convex surlace otanu polyedroid circumscribed, 
 about this solid of revolution. 
 
 Cubic parabola, y= --I z^—(m-\-n)z^-\-mnz I. Art. 21, fig. {h'y 
 
 S=—^3^^^—^m-{^n)z^{-Gmn\■,A=—I \ 3z^—i(m-\-n)z-\-Gmn I . 
 6/) l ) I2p { I 
 
 Art.2i,fig. (c') 
 A= l-*(32!2— 8«2-f Gn-). 
 
 1 'In 
 
 Art. 21, fig. ((i') 
 
 Cubic parabola. y=~.{z^—2nz'i-^n'-z) 
 
 P 
 
 5= ^-(322— 8n2-j-6»2); 
 
 Gp 
 
 Cubic parabo a. y^~{z'^-{-2mz'^-\-m'^z). 
 
 P 
 
 Vlp 
 
 s^ 
 
 
 (3«''4.8?7»z-|-6m2); 
 
 .4=-^(3z^-f-8n«-f-G»n2). 
 \2p 
 
 
148 
 
 BOOK III. 
 
 Cubic parabola, yss-z'^, 
 
 V 
 
 2p 
 
 Art. 21, fig. («') 
 
 ^= 
 
 Pz» 
 
 4p 
 
 Cubic parabola, y=l | z^—2lz^-\-il^-^m^)z \ . Art. 21, fig. (f) 
 
 py. ) 
 
 Gp { J 12/) I J 
 
 App. C, fig. (6) 
 
 Pz' 
 
 A=^—(Zn—2z). 
 Qp 
 
 App. C, fig. (/) 
 
 App. C, fig. (i) 
 
 Commnn parabola. y=i-z'^. 
 
 P 
 
 c_ 27r28 ^ 
 *' ., — > 
 
 Common parabola, y=_(n2— «2), 
 
 5= l!!(:]n— 22 ) ; 
 
 Common parabola, y=^(z'-' — 2mz-\-m'^-^n^), 
 
 V 
 
 Tlio above P donotes th: perimeter of a polygonal cro.-s-soction ciroum- 
 pcribed iibout a circle to radius «. 
 Straight line, ?/r^wc, passing through the origin. 
 
 Right cone, S=:ttRx^; 
 
 Frustum of a cone, S==i.EiT(^R-\-r). 
 
 ■Right pyramid, -4=i \P')<,E ; 
 
 Frustum of a pyramid, A=\E(^P-\-p). 
 
 Right cylinder, 5=i 2TiJx P; 
 
 Right prism, A=^PxE. 
 
 60. 
 
 THEOREM. 
 
 In a solid of rcvoluHoriy whose curve is exjtressed hrj an 
 equation co»i(uritt^ in y"=ux^-)- bx"-f- ex -f-f, the volume of any 
 frustum may he measuredhy the formula 
 
 
 
jutn- 
 
 igin. 
 
 ARTICLE 60. 
 
 149 
 
 V representing the volume, rY- and ::y- the bases perpendi- 
 cular to the axis, rY'^ the median section parallel to the Lases^ 
 and H the altitude, of the frustum. 
 
 Hyp. Assume DAd to be any curve j. 
 whose equation is contained in ^ 
 
 y''—ax^-\-hx'-{-cx-\-f{ovigm at the vertex 
 A of the axis). jii~ 
 
 L_J — I U 
 
 On -\-XA, lay off ?rtH=no ; draw the 
 ordinates Bm, Cn, Do j «/ 
 
 andlet ^m=a;) ^o= 
 
 Bm = y ) } Do 
 
 m n o 
 
 
 
 "" 2 ~ 2 ' 
 
 Cu=: r j , mo= (X-x) = IT', 
 
 y and r"= volumes of the solids generated by the revolution 
 of the pliinos ABui and ADo, respectively ; 
 V" — T'' = V, volume of the frustum generated by the revo- 
 lution of BDoni ; 
 
 AsT. then will 
 
 v= ?!i(r2+4F'Hr)- 
 
 Dem. When the abscissa of the curve ABD successively 
 assumes the values Am, An, Ao, or x, A', A', its ordinate 
 becomes Bm, Cn, Do, or y, Y', Y, respectively. 
 
 By this notation, y'=ax^ -j-i.-c- -j-cx +/', (a) 
 
 r3=aX'H6-Y'=^+cA'-f/, (2) 
 
 r-= aX' 4- h.v+ cx + r. (3) 
 
 Introducing, in (2), the value of X' from (1), we have : 
 r-'=a(^y+b(^^y+,: (Ip) +f. (4) 
 
 Expanding the binomials of (4), and multiplying; the result by 8> 
 
 'i ■ 
 
 H I 
 
 n 
 
 i 
 
 
 !! 
 
 I 
 

 111 
 !il 
 
 1 
 
 !pM 
 
 'i'l 
 
 ;|| 
 
 ir' 
 
 >' jv 
 
 ii; 
 
 m 
 
 
 180 
 
 BOOK III. 
 
 
 58 
 
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 :i 
 
 
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ARTICLE 61. 
 
 151 
 
 o 
 
 Cor. 1. If one base (ttj^) or both bases (rY^ and :ry^ 
 reduce to zero, in formula (10), the volume (V) of the segment 
 or of the whole solid of revolution will be measured by 
 
 iHr(Y2+4Y'H0) or ;H-(0-r4Y'-^-^0), respectively. 
 
 Cor. 2. The volume (V) of the frustum of any polyedroid 
 circumscribed about the solid of revolution specified by 
 y'=ax'+bx^+cx-|-f, is measuredby the formula 
 
 in which H is the altitude, and B, B', b are the polygonal 
 cross- sections of the polyedroid, respectively circumscribed about 
 the circular cross-sections ttY", -Y'-, -y* of the solid of revo- 
 lution (tohose volume is V). 
 
 For, it has been proved (II, 72, 2'>) that V=nV, 
 nnd (-P-f4-r-+-3^2^=w(J5+45'H-6)j 
 
 hence, since both solids have the same altitude II, then : 
 
 F=t^iT(7ry2+4rF=» + ry2) = nF=rw^/i(^ + 4i;'-f6); 
 therefore, , V'= \h{B-^-^B' -\-b). 
 
 Cor. 3. If one base (b) or both bases {B and b) reduce to 
 zero, the volume (V ) of the segment or of the whole polyedroid 
 will be measured by ^H(BH-4B' + 0) or ^H(0 + 4B'4-0), 
 respectively. 
 
 Sch. I. The formula ^H-{Y'+AY"^-\-y-) applies exactly 
 but to the solids of revolution whose curves are included in the 
 collective equation y''=ax^-\-bx^-\-cx-\-f (Art. 22, and 54, 
 Sch I). 
 
 Sch. II. Since the formula (10) is independent of the coefl&c- 
 ientsa, b, c,/ of the function (a), these coefficients may, with- 
 out affecting the formula (10), assume any value : positive or 
 negative, entire or fractional, rational or irrational ; they may 
 even vanish and thereby reduce the cubic equation (a) to an 
 equation of the 1st or 2nd degree. 
 
 01* Sch. III. By the preceding analysis (60), we also obtain a 
 formula (6, 8 or 9), in function of the abaois^a x (altitude of the solid) 
 affected by coefficients, from which special formulse, in function of z, may 
 be deduced for mtasuring the volume of nry solid of revolution whose 
 curve is contained in y^:=ax'^-\-bx^-\-cx-\-f, aa follows : 
 
 'i I 
 
152 
 
 BOOK III. 
 
 1" Letting «=s-, 6=—- (m-f-n), <?=-»»», and/=iO, in i/^=sax^-\-bx^-\-cx4-/, 
 P P P 
 
 this functioa reduces ioy^^-l x^-(Tn-\-n)x'^-\-mnx v, 
 
 (1) 
 
 which represents the parabola with oval, or curve of the plane generating 
 the paraboloid with ovoid (24). Hence, if these values of a, b, c, /, be 
 introduced in formula (G) of Art. 60, there obtains : 
 
 or 
 
 r=^ !L \ ?24— I(m4-n)z3+-ffina;2 "I, 
 12 I /t p pi 
 
 12^1 
 
 •A(jn'\-n)x + Qmn \ , 
 
 (2) 
 
 for the volume of the paraboloid with ovoid. 
 
 Since the limits of the ovoid are z=U and z^:nt(Art. 19, !•), if «t be 
 substituted for x in (2), we shall have : 
 
 irnr 
 
 for the volume of the ovoid, ice. 
 
 F= -_^(2n— m), 
 I2p 
 
 (3) 
 
 B 
 
 Now, introducing in formula F=r-±1- (II, 73), the value of F given in 
 
 7ry2 
 
 (2) and (3), and letting Y^^x^, we shall have : 
 
 F=^ j 3z2— 4(ni4-n)x+6n»» | , 
 
 for the volume of the polyedroid circumscribed about the paraboloid with 
 
 ovoid, and 1"= -J^(2n— m), for the volume of the polyedroid circum- 
 12p 
 
 scribed about tlie entire ovoid. 
 
 By the same process, wo may find the special formula} of the volumes of 
 the other diverging paraboloids and of their circumscribed polyedroids. 
 
 2" Letting a=0, 6=— 1, c=2R, and /=0, in y^.=ax'^ -\-bx^-\-cx-\-/, this 
 function becomes y'^=.2Rx — x-^ which represents the circle. Then, intro- 
 ducing these values of a. b, cj, in formula (6) of Art. 60, there obtains : 
 
 F=-rV7^(— 4a;3_|_i2/i'i;2)=l.;r(3i2a;2— a;3). (1) 
 
 ■\Yl;cn a; becomes equal to the diameter 2 ii, equation (1) n duces to 
 lT(12yV'— 8^3); whence, the formula of the volume of a sphere is 
 
 4-^ffi3 
 
 if. m 
 
 The eqi ation r=jT(37?a;2 — x^) may be written 
 
 F=|;rz(2iea;— 2;2-fiS!a;). (3) 
 
 From the equation, 2Rx—x'iz=zy-, there obtains : Rx^=i\(x'^-\-y-)\ then 
 substituting y^ for 'iRx—x^, and ^{x^-\-y') for Rx, in (3), there obtains, 
 from (1) and (3), for the volume of a spherical segment of one base : 
 
 F=J,Ti2(3/2— X) or J7rar(y2-f Jx2). (4> 
 
TTTT- 
 
 ARTICLE 61, 
 
 153 
 
 or 
 or 
 
 In a circle to radius 
 
 
 Assuming the curve ABD to be circular, and -^JTA a diameter, the 
 solid generated by the revolution of the plane BDom will be a spherical 
 segment of two parallel bases, and the formula of its volume (V) will be 
 obtained by introducing the above values of ^, 6, c, /, in formula (9) of 
 Art. 60, as follows : 
 
 r= J 7r(J:— z) J (2RZ-X^-)-\-(2Rx—x^)-{'RX-\-Rx^Xz | . (B> 
 
 ^2^Y2), "^^ I Rx=\(7?^y^). 
 Introducing the second members of these equations in (5), there obtains : 
 
 or F=47r(X-a:)j|r2+|y24.i(X2-2Xa;+x2)}, 
 
 or r=i7r(X-a;)J3r2-|-3y2^_(X~i)2J. 
 
 Letting ^=altitude (X— i), then V^^lEn^Y^^y^Jf^m), (ft) 
 
 for the formula of the volume of a spherical segment of two bases. 
 
 Now, introducing, in formula V'snV — (II, 73), the value of Fgiven 
 
 in (2), (4), aud (6), and reducing, we will obtain the formulae of the 
 volumes of the corresponding circumscribed polyedroids. 
 
 By the same process, wc may find the special formulae of the volumes 
 of the other solids of the second order. ' 
 
 3" Letting a=0, b=m^, c=0 a.nd /=0, my^=a3^-{-bx^-\-cx-\-/,th\3 
 function reduces to y^=m^x'^, or y=vix, which is the equation of the 
 hypothenuse of the right-angled triangle generating a right cone. 
 
 Introducing these values of a, b, c,/, in formula (9) of Art. 60, then : 
 (F"— F) or V=^2H^—x) \ im2{X2-{-Xx-\-x2) J 
 
 =}7r{X—x)(m^X2-{-m^Xx+m2x^). (1) 
 
 From the above equation m^x'^^—yZ, 
 
 and nviX2^Y^, 
 
 th#re obtains : s^rn^X^ X w^.r- = m2A'jr=^ Yy. 
 
 Introducing these values in (1), and (to comply with the usual notation) 
 letting .ff=aUitude {X—x), R and r=radii Y and y, respectively ; thore 
 obtains V=lIlTr{R'^-^Rr-\-r^), for the formula of the frustum of a right 
 cone. 
 
 ?.i)^i;i 
 
t64 
 
 BOOK. UX, 
 
 When the upper base (n-r') vanishes, the fnutam of a cone become! an 
 entire cone; and there obtains ras^frXT-fi^) for the special formal* of 
 the volume of a right cone. 
 
 When r becomes equal to R, in formula ^ff7r(Ifl-\-Rr-{-r'), the frustim 
 of a cone becomes a cylinder ; hence, there obtains F=7ri2'x-^i for the 
 formula of the volume of a right cylinder. 
 
 Letting wlfl and Trr ^ bases of the frustum of a right cone, 
 
 7riZ"= median section parallel to irlfi, 
 
 B and bsa bases of the frustom of an oblique cone, and of a 
 right or an oblique pyramid, 
 
 B'^=i median section parallel to B \ 
 
 find asBuming these various frusta to have eqoal altitudes and equivalent 
 bases, there obtains (II, 69, 2): irR^=Bf irJt*ssB', and »rA«i ; 
 
 ■whence, v^tt/^x Trr^s=Tr Rr=\/^/i^ . But, a frustnm of a right cone may 
 be measured by the following two formulae : 
 
 F=Jir7r(i?2+i?r-fr2), above 
 
 and V=lffK{R^+4R'i-{.r'2)', Art. 60, (10) 
 
 therefore, by substituting B, B', b, for ttR^, ttR'^, ttt^, in these equations, 
 there obtains for the formulas of the volume of a frustum of an oblique 
 cone, and of a right or an oblique pyramid ; 
 
 V=IH(B-^4B' + b) or IH{B + >/B7b + b). 
 When b reduces to a mere point, or zero, in these formulae, the frustum 
 of a cone or of a pyramid becomes P" entire cone or pyramid ; hence, by 
 Substituting zero for b, there obtains F=|J7(i?+ 4J?' + 0) or ^B.IT, for 
 *he volume of an oblique cone, and of a right or an oblique pyramid. 
 
 When the upper base (i) becomes equal to the lower base (B), in for- 
 mula 5^(J?+>/if. 6+ i), the frustum of an oblique cone becomes an 
 oblique cylinder, the frustum of a right or an oblique pyramid becomes a 
 right or an oblique prism, and there obtains V=BX-fff for the formula 
 of the volume of an oblique cylinder, and of a right or an oblique prism* 
 
 ::1|| 
 
 RiSSUMI^. 
 
 Speeial/ormulte qfthe volumes qfthe tolida qf revolution whote curves are 
 contained in y5t=rfix3+ bx3+ cx+f, and qf their circunucribed polyedroidt. 
 
 Notation. In this Il69uni^, etich given curve speoifies the generating plane 
 of the subjoined solid of revolution, the volume of which is denoted by V, and 
 the volume of the polyedroid ciroumsoribed about this solid of revolution is 
 denoted by V. 
 
ARTICLE ni. 
 
 155 
 
 Parabola with oval,y2--:_ J x^^(m-\-n)x^-\-mnx \. Art. 19, 1* 
 
 r= !!Z! I z2— 4(m-j-n)24-6m« \ ; V'=^ -f x^— 4(m-|-n)ac+6mn "l . 
 12p i i 12^ l i 
 
 Entire ovoid, r= ![^(2n-»i); r= £!2(2n— m). 
 
 12/> 12;j 
 
 Crunodal parabola, i/^z=-(ifl—2nx^-{-^nx). 
 
 P 
 
 Art. 19, 2» 
 
 TTX 
 
 F= ^lz_(3a52— 8wz4-6n?); 
 
 12p 
 
 rz=JL(3x^—8nx-{.6n^). 
 I2p 
 
 1 
 
 Acnodal parabola, j/'^=~(x'i-\-2nix'i-\-mrx). 
 
 P 
 
 Art. 19, 3" 
 
 12/) 1 2/> 
 
 ^=^5^ 
 
 Cuspidal parabola, y'^^-x^. Art. 19, 4» 
 
 r— '^**- 
 
 ^p ^p 
 
 Pure campana, y2=l | rS— 2Za:2-|-(Z2-f-wj2)a; [ . Art. 19, 5" 
 
 p ( I 
 
 F=!^|3j;-'— 8Za:4-G(/24.m2) l ; 1^=^^ i 3a;2— 8/a;4-«(/24.7w2) l" 
 1 2^> I J 1 2^> I J 
 
 The above 1? is a cro8?-scction circumscribed about a circle to radius x. 
 
 Circle, 
 
 F=4;ri23i 
 
 7/2=2^a;— z-. 
 
 Art. 10, 2» 
 
 I"=^i?7;. 
 
 Frusta : 
 
 F=j7rr3i2(X2-x2)-r.l'3-z3)]. /; f i^,_^, ,_. y,_^. \ 
 
 Segments: 
 
 TTX 
 
 Bx^.., 
 
 F=7ra:2(3A— x) or_(y24.1_); F'= ll^{3/2— a;). 
 2 3 i2- 
 
 Tbe above ^ is a cross-section circumscribe J about u circle to radius R. 
 Horizontal ellipse, 7/2= l!(2ax— x2). Art. 12, 2' 
 
 a^ 
 
 F=|?ra62 ; 
 
 V'=*,Ba. 
 
 Frusta : 
 
 F=s !!^ j 3a{X2-z2)-(^-x3) j ; r=j^^ / 3a(X2-z2)-(.r3-z3) I 
 
 Segments : 
 
 F= -_(3a— z) ; 
 
 r^,^^X3a-x). 
 
 3o2 3o2 
 
 The above J? is a crosB-sectioQ circumscribed about a circle to radius h» 
 
 ; 
 
 ^ 
 
166 
 
 BOOK III. 
 
 I 
 
 
 I 
 
 ! I 
 
 «i 
 
 2 
 
 Vertical ellipse, y^= "L{;ibx—x2). 
 
 b^ 
 
 , Art 1 2, 3" 
 
 V=z\7:a^b\ 
 
 V=i\Bb. 
 
 Frusta : 
 
 Segments : 
 
 363 ^ " 
 
 Common parabola, y^^=4ax. 
 
 V=z2anx' , 
 
 r=2arr(.r2— a;Z); 
 
 3/.3^ 
 
 Art. 14, 2« 
 
 V= 
 
 2Bx' 
 
 a 
 
 Frusta 
 
 v'= diiX^-x^). 
 
 a 
 
 The abovo i^ isa croHs-section circumscribed about a circle to radius a. 
 
 72 
 
 Common hyperbola, y2_ __^{2ax-\-x^), 
 
 a- 
 
 Art. 16, 2« 
 
 V- .^(3a4-z); 
 
 r= g(3«+.). 
 
 Frusta : 
 
 . r=!I^| 3a(X2— a;2)>-(X3-;«r3) j ; V"=,— ^ | 3rt(.r2— ;c2)_(X3_;«:3) | 
 
 62, 
 
 Conjugate hyperbola, 2/2= __(a2+jr-). 
 
 Art. 16, 3» 
 
 ■7rA2f2 
 
 V= ifl(3a'^-{.x^'). 
 
 ■•' I. 
 
 3a- ^ ■ ' 3a^ 
 
 The above 5 is a cross-section circumscribed about a circle to radius 5. 
 Straightline, y=mx, passing through t! 
 
 Right cone, V=iIlTrR'^ ; its frustum, V=^IIrr{R r'"')* 
 
 Right pyramid, V'=:^IIXJB] its frustum, V'=^II(B-\-^ ' h-\-b' 
 Oblique cone or oblique pyramid, V=z^HyiB. 
 
 Frusta of these latter, ^^^^//(^-{-v/i^Ti+i). 
 
 Right cylinder, V^HnR"^] oblique cylinder, V'—By,H. 
 Right or oblique prism, V=By,H. 
 
AirncM' «12. 
 
 Vu 
 
 18 5. 
 
 -1' ,1. 
 
 7b-\-b)- 
 
 M ,tf 
 
 02. Sch. IV. Till! voluino «»f a woJ-o or ofa prisiuuid may 
 be lueusurecl by tbo tbriuula \l\l> t i/i' -t h). 
 1" Let / = len.t:th of the cdjic of any weJire, 
 X = length of the buck of any wedge ; 
 then will the formula f,//(/^ + 4f. +0) measure the volume.s 
 of the three wedges specified by the relations : L = l^ L^l, 
 and L <^l. 
 
 Assume ABCD-Iftohe aright 
 or an oblique parallelopipedon, 
 and observe that the plane 
 passing through the vertices 
 C, D, E, t\ divides the paralle- 
 lopipedon into two equal wedges, 
 and bisects the median section 
 an' parallel to the bases M 
 and m. 
 
 Further assume ADOPE to be a pyramid whose base is 
 the parallelogram AJJOF, a prolongation of ABdD. 
 
 Let ^=altitude common to the pyramid, parallelopipedon, 
 and wedge ABCDEF, 
 M=aYeM of the parallelogram ABCD, 
 a=-^J/, half the parallelogram art' and median section 
 
 "(parallel to J/) of the wedge ABCDEF, 
 iV=area of ADOP, base of the pyramid, 
 c= median section (parallel to iNT) of the pyramid, a 
 prolongation of a. 
 
 a). The volume (V) of the pyramid ADOP-E may be 
 
 measured(Art. 61,3'') by the formula V=^H{N-\-Ac). (1) 
 
 b). The volume (F') of the parallelopipedon ABC D-Il is 
 
 r'=//x.l/— iffxCJ/; but, the volume (F") of the wedge 
 
 ABCDEF is equal to one half of F' ; hence (since i/=2a), 
 
 tliure obtains for 
 
 Volume of the wedge ABCDEF, in which iy = ^: 
 
 V\^-hr or ^//x3i/) = ^/i(J/+4«). (2) 
 
 C). Adding (l)and(2),and letting(ilf-fiV)=^, (a-Hc) = ^' 
 there obtains for 
 
158 
 
 BOOK III. 
 
 ':ir':il' 
 
 ...•r?i; 
 
 li '!:i 
 
 I i '-i. 
 
 ,'■:^ 
 
 Volume of the wedge BCOPEF, in which i>>?: 
 
 V-\.V"=^H{A + '^A'). 
 
 d). If a pyramid equal to ADOP-E is cut off from one end 
 jf the wedge ABODE F, a wedge in which L-cCj will be left ; 
 hence, subtracting (1) from (2), and letting (J[f— iV)s=5, 
 {<i~k')=.B\ there obtains for 
 
 Volume of a wedge, in which L<J> '• 
 V"-V=yi{Bi-4B'). 
 2' Assume ABGD-II to be a prismoid, then will its volume 
 be nieaaured by the formula ^II(B-\-4:B'-^b). 
 
 For, the phne C'l^JS/F which divides this prismoid. into two 
 wedges, also divides the median section ad' {parallel to M 
 and m) of the prismoid into two parts, a and a', which are the 
 medinn sections of the two wedges. 
 
 From what precedes, the volume (F") of the wedge 
 ABCDEF is 
 
 F"=i//(Jf+4a), (2) 
 
 and the volume (F") of the wedge CDEFGH is 
 
 F"'=^^Zr(m+4a'). (3) 
 
 Adding (2) and(3), and letting (F" + F"')= F, {a+a')=M', 
 there obtains for the volume of the prismoid : 
 
 V=^H{M+4M'-\-m). 
 
 63. 
 
 MONOFORMULA. 
 
 It has just been proved that the formula ^ff{B-\-4B'-\-h) 
 r.iiiy be used for measuring : 
 
 1" the area of any plane figure, the curve of which is con- 
 tiiir.nd in y = ai:'^-\-bx'-\-cx-\-/ ; 
 
 2' the convex surface of any solid of revolution inscrihedm 
 ji rijjfht quadrangular polyedroid, the curve of which is con- 
 tained in y=:-ax'^-\-hx'-{-cx-{-f \ 
 
 3 ' the convex surface of any polyedroid circumscribed about 
 any of the preceding solids of revolution ; 
 
 4' the volume of a solid of revolution, the curve of whose 
 generating plane is contained in y'^^^ao^-^-bx^-^-cx-^-f \ 
 
ARTICLE 64. 
 
 159^ 
 
 Is con- 
 
 1b con- 
 about 
 1 whose 
 
 5 ' the volume of any polyedroid circumscribed about any 
 of tho i)ret'eding solids of revolution. 
 
 It will be proved, further on, that it may also be used 
 for measuring very approximately the area of any regular 
 plane figure, as well as the convex surface and the volume of 
 any regular solid. 
 
 For this reason, it is called monoformula {monos-formula), 
 name by which it will henceforth be designated. 
 
 64. PRACTICAL APPLICATIONS. 
 
 Prob. 1. Given the equation y=r^o'*^ to represent the para- 
 bola Oa'n'; required 
 
 1" the area of the segment tchose limits arc x = and X=o, 
 2" " " " x=4an<iX=8. 
 
 Solution. On(9A'', lay off Oa=a}i=4, + 
 and ac=cn=2 ; draw the ordinates aa\ * 
 cc', nn'] and let -4'=area of Oaa', 
 ^=area of Onn', and ^4—^1'= area 
 of ann'a'. O' 
 
 l'i In the Equation y=xou^'> 
 
 r rr=0, then ,v = 0, or 6, upper baseat the point 6>, 
 hen -j A" = 4, or Oa, '' F =0.64, or B', median section aa\ 
 A'=8,or On, •' y=5.12, or B, lower base nn', 
 X=S~II, altitude 0)1 of the segment Onn' ; 
 
 0.00 
 hence, area of Onn' or .4=^//^ 4/J' [ =% ^ 2.56 
 
 w 
 
 ofa«;i'or .4=^//) 4/r i=\ 
 
 5.12 
 
 =1x7.68=10.24. 
 2". lnjj = j^-^x\ 
 
 C rc = 4 or Oa, then ?/=0.64 or aa' = h, 
 when^ A'=:6or Oc, " y' = 2.16 or cc'=^', 
 (X=8orO;i, '• 7=5.12 or ??«' = i^, 
 
 X — x'=8 — 4i = 4=Jl, altitude an of the segment 
 
 i ) i 0.64 
 hence, area of ann'a' or A'-A'= III^ 4ii' >■ =| ■< 8.64 
 
 B ) I 5.12 
 
 = §X 14.4=9.6 
 
 ann'a' 
 
 m 
 
 ^*5ii';S'-,'; • • 
 
mrfTT^ 
 
 160 
 
 book! III. 
 
 Prob. 2. Given y=^\(x'-lQ^'+GSx) to rejiresent the 
 curve Anl ; required the area (A) of ^^ 
 AnpdA, between the limits x = and 
 
 X=8. 
 
 Solution. In y=^yi'(a5--16x+68), 
 
 when 
 
 C ;c=0, then ?/=0, or />. upper base at the point Aj 
 } A" — 4, " F' = 4, or B', median section i/i, 
 
 ( X=S, " r=1.6,or B, lower base <?^9, 
 X=S=H, altitude Ad of AnpdA ; 
 
 hence, ^=J/^j 47r >=f 
 
 = 4 X 17.(5= '^^-^i. 
 
 Prob. 3. 6^iye/iy-=4x=px to represent the common px^i'a- 
 tola BOC ; required the area (A) of the ^ 
 
 segment BOCB, hetween the limits x = and j/"'^^ 
 
 x=Ooror>. w"T 
 
 Solution. 1". By the special formula 
 
 O 
 
 -/f 
 
 In the given equation, 1/-= 4.1;, when .jr=0 
 or 0/>, then y=di:(5(or -1 Dll and —DC)] hence, by introdu- 
 cing these values of .r and// in .l = |^.ry. tliere obtains J. = 72. 
 
 2". By the monoformula, J=^//(i5-f 47^' + ^;). 
 
 The monoformula measures the area of this parabola extACthj, 
 by taking the bases and the median section parallel to the axis 
 of the curve. But, in the segment BOCB, the two buses 
 (i^ and i) are reduced to zero at the points /i and 0, tho 
 median section OJ) or B' = \K and the altitude BC or lf=]'2 ; 
 
 hence. .4 = ^/) 47^ =VS "»<> I ■='-• ' * 
 
 ( 7/ j ( 0) 
 
ARTICLE 64. 161 
 
 Prob. 4. Required the volume (V) of the frustum ABC -co f 
 a pyramid, in which AB= 12. 
 BC=AC (or A) = 10, ac(ora)=r), 
 and altitude op (or H) = 10. 
 
 Solution. Lower base B=l X 12 x 8=48, I, 123, Prob. 9. 
 
 median section JS'=^/^4^)''=30.72, II, 75,ProblO. 
 
 upper base b=B^^y =17.28 ; " Prob. 9. 
 
 hence, F=|Zr^ W f=V 
 
 17.28 
 
 122.88 
 48.00 
 
 = 1 X 188.1(5=312.6 
 Prob. 5. Given 10=R, radius of a sphere (origin of the 
 axes at the center); required : 
 lo the volume (V) of the sphere. 
 
 2'^ thevohime(V')of a segment hetioeen the limits x = 0c:&x=6, 
 
 3*^ the volume(y")of a frustum between the limits x — 5(frx=-3- 
 
 Sol. 1'* In the equation y'^=R'—jr of the generating circle, 
 
 C x =—R, then ?/-=0 : whence, 7r?/-=0 or h, upper base, 
 
 when-! X'= 0,'- Y'-'^E'; " rr-=-i?- or ii',med. sect. 
 
 ( Z= + /?, " r-=0 ; " rP=0 or £, lower base, 
 
 Jf—j:;=-ff -1-72=//, altitude o'* the sphere ; 
 
 hence 
 
 , V=^Il] iB' y=^^R] 4-7?- f=fri?»=i%0-«-7r. 
 
 I ^n (. ) 
 
 - o-w the equation 2/^=7?^— x-)=l 00— j;-, 
 r .r=0, then ^=1 00 ; whence, 7ry-=1 
 
 whenKY'=3, *' r2= 91 ; '• rl'-= 
 ( A'r=G, " i'-= 64; " rr-= 
 
 2" In the equation y^^^:=R^—x-y=100—x-, 
 
 .r=0, then ^=1 00 ; whence, ry-=100;r or h, 
 
 91ror5', 
 64rr or By 
 
 A — .i'=6=//, altitude nf the segment ; 
 
 lience, F=»//^ 4/?' ,' =«^ 304- ,^=528;:. 
 I B ) I 04.) 
 
 11 
 
i^i' 
 
 t< w 
 
 mi 
 
 * ! 
 
 » : 
 
 
 'I 
 
 -i 
 
 'A 
 
 lU 
 
 '^i\'! 
 
 id?' 
 
 162 BOOK in. 
 
 3" In the equation y^=:100 — or*, 
 
 whei 
 
 -V— x'=54-3=8=//, altitude of the frustum ; 
 
 hence, 
 
 C a:=— 3, then ^^=91 ; whence, Tzy^^Qln or 6, 
 jn ^ .Y'= + 1, " r-=99 ; " - r-=997r or £\ 
 ( A=+5, " y-=75; " ::r==75-or^, 
 .V— x=5+3=8=//, altitude of the 
 
 ;, r'-=ill} 4B' [-^^l 396:: ^ =^^ 
 
 I U ) H 75-3 
 
 =r^^' 80- V----9C-. 
 
 Preb. 6. 6rii;e?i m=4, n=8, p=^0, 
 in y''= ~ \ x"' — (m + n)x^+mnx |- , equation of the parabola 
 
 with oval ; required l'» the volume (V) of the paraboloid, bet- 
 uieen the liniifs xz=zH </HrZX=12; 2' //(« volume (V'^ of the 
 ovoid, between the limits x=iO and X —4. 
 
 Solution. 1' Performing the sub.stitution indicated, there 
 obtains : i/-=.^x(jr — 12x4-32) ; hut, in this e(|uation, 
 C :v= 8, then ?/-= ; whence, ry-= or h, 
 when- A"=10, " y'-r=20 ; " r.r'-=20,T or i^ , 
 ( A'=12, " r-=04 ; " rY-:=G4- or 7i, 
 
 ,V— x=:l 2 — 8=4=//. altitude of the required paraboloid j 
 
 ) Atr C_4 1 so,. 
 
 (64- 
 
 2'In v-=itx-(x^-12x+32), 
 
 ( x=zO, then y'=^0 ; whenco, r:t/-=zO or A, 
 when^ A'=:r2, "l'-=4; '• -r'-rrr4- or 7i', 
 ( A'=4. '^ r-=:0; " -¥'={) or M, 
 
 X — j-=z\—'Jr, aUituJe of the ovoid ; 
 III 4/i' '-—■»-■ l(i- ^ — i^2_ 
 
 c /O ( » 3 ; 
 
 Prob. 7. Reqitir<d, the c(tp(uiti/ (V) f>/"(f t/o/nc' affecting the 
 form of a, riqht Inxaijnnal volijcdroid inscribed in a semi 
 prolate ellipsoid ivhose ncmi-a.rrs are ni=l() and n=:6. 
 
 SoLUrroN. Tlio basn of" tliiy polyedroid is a regular hexagon 
 whose HI do a:, d ladius avo e;ich (Mjual to n or 6; hence, by 
 Table IVJo\v(-i- ])ase, /^— i;:}.5;>u 74:; ; 
 upper base, />=:0 or vertex. 
 
 hence, 1 
 
ARTICLE 64. 
 
 163 
 
 uig the 
 a semi 
 
 liexugoii 
 
 •I 
 
 In the equation y-z= — -(m- — x-)=-f^J^(l(^(^ — jr) of the 
 
 ellipse ; when x=5, one half the distance from 7Mo fe, then 
 y-=3r)X 0.75, which is the s(juare uftlie radius of the median 
 section B' (parallel to B). 
 
 But, the similar hexagons B and B' are to each other as the 
 s«iu;trcs of their radii ; hence, ir'.y-z=r>:B\ 
 or 3C::i;;x0.75=03..j:5074:i:/i'; whence, ii'= 70.1 -48056 ; 
 therefore, since m=\0^izrn^ altitude of the dome, then : 
 
 ( h-) (0 
 
 V=lll\ AB' ^ = V-; 280.5!>222-J 
 ( B ) ( 03.53074I 
 
 224 
 
 ^ Y X 374. 1 22007= 023.53828. 
 
 Prob. 8. Given \0 for the radinn (R) of a sphere ; required 
 1" the convex surface (S) of (he sphere ; 2' the zone (S') of one 
 base whosY. altitude (x) t.s 2 ; 3' the zone {^") of two bases 
 whose altitude (X — x) is 5. 
 
 Solution, lo 8=2- R x 27^1^400- ; 2" S'=^2:.Rr=40- ; 
 30 S"=:2-:R:\—j;)=W()r:. Art. 5d 
 
 Prob. 9. Given // = Jt;.- fo rcprrsnif f/ic. currr of a.- rifjht 
 quadranciularpolijedroid; niiulrrd 1 • tJic Huiits of this jxdye- 
 ilroid and of the i ir.-rrilicd Kolid <f rt'rofn/lon ; 2' the convex, 
 surf arts of both suiids. 
 
 Solution. 1 The limit.s of the curve y=;L--\ and thereby 
 ofbothrfMjnired'Surfaces. ;ive ;^= -J-4 a!HU = — 4( Appendix D, 
 TL Ex. 4). 
 
 1" III the iMjiiJiliu;; i/ — ^-z'-. 
 
 C z — — I. th'ii '/■=2; wlu'iice, 2-:y — 4- or c, 
 \\hv,u]z=^ 0, •• Y'-i': •• 2-:y'---rJ) or r", 
 
 ( Z---i4, •• )-.2: 
 
 Z 
 
 hence, 7A))wS—lIf -\ '. 
 
 2. )'--[: or r 
 //. ciciicrat ii!^ iirc of the zone j 
 
 iv) (4- 
 
 {, ._..._ 
 
 'I'^ 
 
);! 
 
 ii . 
 
 1 
 
 H 
 
 
 
 
 
 
 
 
 1 
 
 
 
 164 
 
 BOOK III. 
 
 Now, to find the convex surface of the polyedroid between 
 the same limits, 2 =±4, it must be borne in mind that the 
 generating arc (//) of the inscribed solid of revohition is iden- 
 tical with the element of contact (//) of the convex surface of 
 the polyedroid (57) ; 
 hence, element of contact //=Z— 2=4-j-4=8, 
 
 upper base &= perimeter p= 81/ =16, 
 
 median section B'= " P' = SY'= 0, 
 
 lower base 5= " P=SY=W; 
 
 r /> ) (IG 
 convexsurfaceof polyedroid, J. =^//-( 4B' f =M ^ 
 
 Prob. 10. Given yz= — (4az — z^) to represent the curve of a 
 
 right quadrangular jyolyedroid ; required the limits and the 
 convex surface of the inscribed solid of revolution. 
 
 Solution. 1" The limits of the given curve of the polye- 
 droid, and thereby of the inscribed solid of revolution, are 
 »=0 and Z=4rt (Appendix D, II, Ex. 5). 
 
 2'^ In the equation, y = —.(4az — 2-), 
 
 hence. 
 
 1 28 
 
 4a 
 
 I z = U, 
 when-{ Z'=2a, 
 lZ=4a, 
 
 2=0, then y=0 
 " Y=0 
 
 .1 
 
 it 
 
 whence, 2Ty=0 or c, 
 2rr' = 27raorC", 
 2-r=0 or (7; 
 
 Z — 2=4a=//, generating arc of the convex surface (S); 
 
 c ^ CO 
 
 hence, S=^ff-{ 4C 
 
 C 
 
 \aJ. 4x'l-a ^ = !/-«='. 
 
 Compare the monoformula with the common formula (5) in App, 0. 
 
 Prob. 11. Given y=^^(T? — 12z--f 48z) to represent the 
 curve of a right quadrangular j^oli/edroid ; required 1^ the 
 limits of this polijedroid, 2*^ the convex snrfure (S) of thr 
 inscribed solid of revolution, 3* the equation representing the 
 curve of a right hexagonal polyedroid circumscribed about this 
 solid of revolution, 4'^ the convex surface (A) of the latter 
 polyedroid. 
 
^,0. 
 ent th' 
 
 of thr 
 ling tlx' 
 lotitth>>< 
 latter 
 
 ARTICLE 04. 
 
 165 
 
 Solution. 1" The limits of the curve y-^-^^(z^ — 122^-1-482), 
 ind tliereby of the polyedroiJ a;id inscribed solid of revolution 
 ire,-=:0,andZ=8 (Appendix D, II, Ex. 7). 
 
 2' In the given equation -^^„-2(z- — 122-f 48), 
 
 sv 
 
 hen 
 
 ( ^ = 0, then y = 0; whence, 2,-y= or c, 
 
 r Z' 1 " Y' •* • '' '>-Y' — 8- nr r" 
 
 - X/ — 4, J^ — if J -- J — ^- or o , 
 
 ( Z=H, '^ r=«; " 2Ty=yi- or C; 
 Z^z=S=JI, generating arc of the convex surface S ; 
 
 hence, S=lll\ 46; l =« | -^.t | =-V- 
 
 3' Denoting, by y', the ordinate of the curve of the right 
 hexagonal polyedroid, there obtains y=v^3 y' (Appendix D, 
 I, Prob. 2'>). then substituting ^3 y' for 
 
 y in the given equation, there obtains y'=y^v^3(2'-122-4-482)^ 
 in which 
 
 C 2; = 0, then ?/ = ; whence, 12?/= 0, or hj 
 
 4", when-^ ;^'=4, •' y'=iV'> i " UY'=''{'y3 or B', 
 
 iZ=H, " i=^v/3; '' 12r=^y?-^3 or ii, 
 
 Z — z = ^=^II, element of contact of the surface A ; 
 
 30,^=1//] 4/;' [ = :^] %w?> [ = ^Pv/3. 
 i. J> ) ( ¥v/3 3 
 
 Note* Probloms as above may be varied at will by makiag use of the 
 various tqua'.ious coutained 1" in y=zax^-\-bx^-\-cx-\-/ (Art. 23) to repre- 
 sent the curves of curvilinear plane figures, 2" in y''=ax^-^bx--\-cx-\-/ 
 (Art. 22), to represent the curves of solids of revolution, 
 S" in y=az-^-\-bz--\-cz-\-f (Art. 13). to represent thj curves of circura, 
 scribed polyedroids. 
 
 Prob. 12. Given 2(\=:maj<)r (ccis, 16=:mi»or axis of an 
 t'.nipse; required 1" its area (K), 2" its cxcentricity (e). 
 
 Ans r'E = 80^, 2'e=5-. 
 
 Prob. 13. Required the later<d surface (S) of a right hexa- 
 goiial jyyramid vhose sbtut height is 20, the radius of the base 
 heing 5. Ans. S = 300. 
 
 hence 
 
 'Hi 
 
 ri 
 
 ; 'i 
 
 ni 
 
 •71 
 
 ■\A: 
 
I 
 
 
 !^ 
 
 ICC 
 
 BOOK III. 
 
 Prob. 1 1. A section parallel to the base and midway bet- 
 wee 11 the vertex and the base of the preceding pyramid deter- 
 mines a frustum tchose lateral surface (S) is required, 
 km. S=225. 
 
 Prob, 15. Required the convex surface (S) of a right cone 
 whose axis i-« 8, the radius of the base being 6. Ans. 60"", 
 
 Prob. 16. A section paralhl to the base and midway bet- 
 ween the vertex and the base of the preceding cone determines 
 a frustum whose entire surface (S) is required. 
 
 Ans. S = 907. 
 
 Prob. 17. Required the axis (A) of a cone similar to that 
 of Prob. 15, and lohose convex surface is 240". 
 
 Ans. A=1G. 
 
 Prob. 18. Gioen the notation : S:=convex surface, Il=alti- 
 tudc, a7id IX ^= radius of the hirse of a cone ; required !» the 
 formula ofH, 2" the .solution of tlii» formula with respect to H. 
 
 Ans. 1' S = -Rx/H-4-R'; whence, 2' H 
 
 I — - ' 
 
 Prob. 10. G^/i-'c^ S=GO,T, convex surf ace ; anc?R=6, radius 
 of tlie base of a cone ; required the altitude H and the slant 
 height \\ of the cone. Ans. H = 8, and h=10. 
 
 Prob. 20. Given B = 48, loioer base ; b=: 17.28, uj^pcr base 
 of a frustum of a 2)yramid ; and A = 10, one side of B; 
 required the side (a) o/b ; A and a being homologous. 
 
 Ans. a=6. 
 
 Prob. 21. Required the radius K of a sphere whose volume 
 is numerically equal to its surface. Ans. R=3. 
 
 Prob. 22. Giv€nn = -p = 8,inequation j= -(nx-x-) of Art. 
 
 P 
 23, 2'' ; required 
 lo the area (A) betv^een the limits x = and x = 8, 
 
 2o Ihe area (A') between the limits x=2 and x=6. 
 
 * 
 
 2" 
 
 Ans. 1" A=%2- ; 
 
 A'=V. 
 
ARTICLE &1. 
 
 167 
 
 Prob. 23. Given m=6, n=10, p=5, in equation (b') of 
 Art. 23, 1" ; required 1" the special formula of the area (A) 
 of the surface specified htj the resulting equation, 2o this area 
 between the limits x=0 and x=10. 
 
 Ans. lo A=3-\3x2-64x + 360)(Art.55,l<0,2"A=:io». 
 60 
 
 Prob. 24. Given H=10, altitude ; R=10, radius of the 
 lower bate ; r = 5, radius of the upper base of a frustum of a 
 cone; required its volume (V). 
 
 Ans. V=iip^. 
 
 Prob. 25. Required the volume (V) of a prolate ellipsoid 
 in which a=10 and b = 6. Ans. V=480-. 
 
 Prob. 26. Given 'p=:l=10, and Jii = C}, in the equation of 
 the 2iure cnnpaua (^Art. 22, 1"); required !<> the special for- 
 mula of the volume (V) of the campanoid, 2'^ the volume bet- 
 toeen thelimits x = and x=ilO, hi/ the nionoformnla. 
 
 Ans. 1- V=^\x2-80x + 816) ; 2. V=ilf*>-. 
 
 Prob. 27. Given m = 6, n = 10 <ind p = 9, in the equation of 
 the par ahola with ovnl (^Art. 22, 1"); required the volume (V) 
 <f the ovoid, (\oliose altitude is x=m = 6). 
 
 Ans. V=28r. 
 
 ^ - 
 Prob. 28. Given a.= 10, and h=Q, in y'= -,,(a--f-x-),eg'«a- 
 
 tion of a conjugate hyperbola ; required the volume (V) of the 
 solid of rcToiution generated by the plane CBgh (Jig. (f Art. 
 16, 3"), between the limits x=0 and x = 10. 
 
 Ans. V=480r. 
 
 Prob. 29. (rtre?i n = p = 16, inequation y= - (nz— z-) 
 
 (Art. 23, 2') of the curve of a right quadrangular jtolyedroid ; 
 required 1" the convex surface (S) of the inscribe solid of revo- 
 lution, between the limits x=0 and x = 8 ; 2" the area (s) of 
 the zone whose limits are z=2 and z=6. 
 
 Ans. 1" S=^|-^ ; 2" s=A4s. 
 
 ! i 
 
 I 
 
PLANE TRIGONOMETRY. 
 
 \,ri 
 
 DEFINITIONS AND EXPLANATIONS. 
 
 1, Plane trigonometry is that branch of mathematics 
 which has for its object the solution of plane triangles. 
 
 2* The six parts (sides and angles) of a plane triangle are 
 so related that when three of them are given, one being a side, 
 the other three may be found by computation. This computa- 
 tion is called the solution of the triangle. 
 
 3. The units of measure of the sides of a triangle are the 
 linear units (Int. to Geometry, 18). 
 
 The units of measure of the angles of a triangle, or of the 
 arcs by which those aiijrles are measured, are the daf/nrs, 
 minutes and seconds defined in Book I. Art. 102 i^ 10.^. 
 
 4, A quadrant is the fourth part of the circumference, and 
 thereby comprises 90°. 
 
 5, The complement of an arc is the difference between 
 that'aro and 90° ; thus, an arc of 40° 30' 20" is the comple- 
 ment of an arc of 49° 29' 40". 
 
 6. The supplement of an arc is the difference between 
 that arc and 180° ; thus, an arc of 60°, 30', 20" is the supple- 
 ment of an arc of 119°29'40". 
 
 "7, To define the trigonometrical 
 lines, use is made of a circle ACDF 
 whose radius is unity and which is 
 divided into four quadrants by two 
 diameters AD and CF, of which the />( 
 former is assumed to be horizontal 
 and the latter vertical. 
 
 8. The quadrants AC, CD, DF 
 and FA are respectively the first 
 second, third and fourth quadrant. 
 
ARTICLE IP. 
 
 ICOr 
 
 the 
 
 O. The right hand extremity .1 of the horizontal diameter, 
 from which arcs are usually reckoned is called the origin 
 of arcs. 
 
 10, An arc, one extremity of which coincides with the 
 origin of arcs, is said to be in that quadrant which contains the- 
 other extremity of the arc ; thus, AB is in the first quadrant, 
 and AM is in the second quadrant. 
 
 11. A line is positive when it is estimated upw<tr<l from the- 
 horizontal diameter, or rightward from the vertical diameter ;. 
 thus, PB and CP are positive. 
 
 13. A line is negative when it is estimated dnwnioard from 
 the horizontal diameter or leftward from the vertical diameter ; 
 thus, PG and O^Vare negative. 
 
 13. The cine of an arc (AB) is the perpendicular distance 
 (PB) from one extremity (5) of the arc, tothe diame:er (AD) 
 passing through the othor extremity (A; thus, LB perpen- 
 dicular to CF is the sine of the arc BC. 
 
 14. The cosine (complement-sine) of an arc (AB) is the 
 sine of the complement of that arc ; thus, LB is the cosine 
 of AB ; conversely, PB is the cosine of BC. 
 
 15. The tangent of an arc (-1^) is the perpendicular (AK) 
 to the diameter (AD) at one extremity of that are, limited by 
 the diameter prolonged (EK) through the other extremity 
 [B); thus, C>S^is the tangent of the arc BC. 
 
 16. The cotangent (complement -tanuent) of an arc {AB) 
 is the tangent of the complement (BC) of that arc ; thus, CS" 
 is the cotangent of AB ; conversely. ^lA" is the cotangent 
 of BC. 
 
 17. The secant of an arc [ABi i; the radius produced 
 (OK) through one extremity (B) of that arc. and liiuited by 
 the tangent (AK) to the arc at the other extremity ; thus, 
 OS IS tlie secant of the arc BC. 
 
 18. The cosecant of an arc (AB) is tiie secant (OS) of the 
 complement (BC) of that arc ; thus, OK is the cosecant of 
 the arc BC. 
 
 
170 
 
 PLANE TRIGONOMETRY 
 
 ill I arc 
 
 li). The verjQd-3ino of 
 
 ?, (AB) is the distance {f*A) from thu 
 
 foot of the sine to the origin of tho 
 
 are; thus, LC la the versed-sine of 
 
 ^ the arc BC. 
 
 *ZO, The covorsed-sine of an arc 
 (^1^) is the versed-sine (LC) of the 
 complement (BC) of that arc ; thus. 
 PA is the coversed-sine of the arc B( '. 
 J31« The eight trigonometrical lines : sine, cosine, tangent, 
 Ac, just defined are called circular functions. 
 
 The circular functions of an angle are identical to those of 
 the arc by which tlie angle is measured. 
 
 Circular functions referred to radius-unity, as above, are 
 natural cir( ular functions. 
 
 22* Relations of the natural circular functions. 
 
 Letting x=arc AB or angle AOB, and writing sin x, cos ;r, 
 tan .T, cot .T, sec x, cosec .v, versin .r, eoversin x, for sine x, 
 cosine a', &c., we have, from the above definitions : 
 
 1 
 
 sin .V. 
 
 cos .»'. 
 
 versin x. 
 
 eoversin x. 
 
 From the figure we have : 
 
 lo PA=OA-OP,ov 
 
 LC=OC-OL, or 
 
 AO 
 
 = 
 
 OB -r- 
 
 PB 
 
 :=. 
 
 OL = 
 
 LB 
 
 = 
 
 OP = 
 
 PA 
 
 = 
 
 
 LC 
 
 =: 
 
 ( 
 
 CS 
 OK 
 OS 
 
 versin x=l — cos x. 
 eoversin x=l— sin as. 
 
 tan .r. 
 
 cot ;r, 
 
 sec x. 
 
 cosec x. 
 
 2 " In the right-angled triangle OPB, 
 
 PB'+ OP'=l, or sin-.'B-}-cos-.x=l ; 
 whence, sin-x = l — cos^.r, 
 
 and cos^x = I — sin^x. 
 
 The symbols sin^a;, cos-a;, are read : sine square of z, cosine square of 
 8 111 the similar triangles OPB and OAK, 
 
 OP: PB=:OA:AK, or cos j.':sin x = l:tan x ; 
 
 sin X 
 
 (1) 
 
 w. 
 
 (2) 
 
 •> 
 
 (3) 
 
 4. 
 
 (4) 
 
 
 (5) 
 
 T). 
 
 whence. 
 
 tan X-. 
 
 cos X 
 
 i^>) 
 
ARTICLE 2'il 
 
 A^ain OP: OIi=:OA : OK, or (mis x : 1 =r.\ :sec ac ; 
 
 , 1 
 
 wluMicc, .see .rr:= 
 
 cos ./ 
 
 I \:\ the nimilnr trianixles OLli iiiid O^^S^. 
 
 OL: LB=OC:CtS, or sin ;i:co.s a;=l:cot x ; 
 
 cos .r 
 
 171 
 
 (7) 
 
 whence, 
 
 cot X— - 
 
 sin x 
 
 Aguiii OZ : OB=OC: OS, or sin .r: Iml :co.sec x ; 
 whence, cosec xz= : — . 
 
 Hin r 
 
 (8) 
 
 (5») 
 
 Multiplying (G) by (8), member Ay member, there obtains : 
 
 tan xcot x=:l ; (10) 
 
 whence, 
 
 and 
 
 tan .r=r= 
 
 cut .I'ZZZ 
 
 1 
 
 cot X 
 tan .1' 
 
 (12) 
 
 5' In the riLdit-an<:l(Ml triangles 0.1 A' and 0(^S, 
 
 OK'—OA'+AK-. or sec-./— 1-f tan-V. (Hi) 
 and OS'=^MC- -h ( 'S\ or vx)^eii\r=:\ | cot-.r. ( 1 4) 
 
 The preceding formulae arc t-ollected in the following 
 
 I. versin .r 
 
 Table A. 
 
 1 — ^.COS X 
 
 2. coversin x- = 1 — sin ./■ 
 
 ^5. sin-x + cos^x = 1 
 
 4. mxi'X = 1 — cos-x 
 
 T). cos-.f 
 (1. tan X 
 
 7. cot X 
 
 = 1 — sin"-'x 
 sin .r 
 
 cos .1' 
 cos .V 
 sin X 
 
 S. sec J' 
 
 1*. COSL'C X 
 
 10. tan X cot./" 
 
 11. tun ./• 
 
 1 2. cot .;• 
 IH. sec-.r 
 14. cosec^.?; 
 
 cos ..'• 
 
 
 1 
 
 
 1 . 
 
 1 
 
 
 cot ./• 
 
 
 1 
 
 
 tan .r 
 
 
 1-1- tan-./ 
 
 
 1 + cotv 
 
172 
 
 PLANE TRIGONOMETRY 
 
 PZrpa 
 
 34. Relations of the circular functions referred 
 to radius-unity and of those referred 
 to any radius k. 
 
 From the vertex of the anfrle aOh, des- 
 cribe the arc AB, with a radius 0A=1 ; and 
 the arc ah, with any radius Oa=zR ; draw the 
 sines PB,ph and let x=arc AB. 
 
 Then, in the similar triangles OFB and Opb, 
 OB:PB=Oh:jyh, or l:sin x=R:pb- 
 
 whence, 1" sin x='' 2' 2)h=R mm x. 
 
 And so on, for each of the other circular functions ; that is : 
 1" Any function of an arc to radius 1 is equal to the corres- 
 ponding function of an arc to radius R divided by E, ; 
 
 2^> Conversely, Miy function of an arc to radius R is equal to 
 the corresponding function of an arc to radius 1 multiplied 
 by R. 
 
 25, Limiting values and algebraic signs 
 
 OF THE circular FUNCTIONS. 
 
 The limiting valueo of a circular function are its values at 
 the beginning and at the end of each of the four quadrants. 
 
 The algebraic signs of these values are determined by the 
 principle that the sign of a variable quantity, up to the limit, 
 is the sign at that limit. 
 
 But (2o), the tangent, cotangent, secant, cosecant, versed-sine 
 and coversod-sine of an arc are all expressed in function of the 
 sine and cosine of that arc ; hence, to find the limiting values 
 and the algebraic signs of all circular functions, it suffices to 
 determine the limiting values and signs of the sine and cosine 
 at the beginning and at the end of each quadrant. 
 
 For that purpose, let us observe how the sine and cosine 
 increase or decrease from (P to 360°, 
 
 At the origin A, the arc is zero; then, the sine is zero, and 
 the cosine is the radius (0.'1= + 1)« Fig- of page 170 
 
ARTICLE 25. 
 
 173 
 
 As the arc increases from zero to 1)0^, the sine increases from 
 zero to its mcixlinum positive valuo (Of^^^ + l); l^^t, the 
 cosine decreases from its maximum positive value (Ovl=-f-l) 
 to zero (at the center C) ; hence, in the 
 
 1st quadrant. I «;;;^L + ^; 
 
 sin 90^=z-f 1. 
 cos 00°= ; 0. 
 
 As the arc increases from 90^ to 180°, the sine decreases 
 and remains positive ; the cosine beoiotes negative, but 
 increases until the arc reaches 180^, when the sine becomes 
 -f 0, and the cosine reaches its maximum negative value 
 (^ODz=. — 1); hence at the end of the 
 
 2nd 
 
 quadrant, \ 
 
 sin lSO°=:-|-0. 
 cos 180°=- 1. 
 
 When the arc increases from 180° to 270°, the sine passes 
 from zero to — 1 {OF), and the cosine passes from — 1, back 
 to — ; hence, at the end of the 
 
 3rd quadrant, I ^'J'^ ^1,'^ 
 
 mo. 
 
 o. 
 
 :— 1. 
 
 :— 0. 
 
 / :* 
 
 ■• i 
 i t 
 
 ^ i 
 ■I i 
 
 
 
 id-bine 
 I of the 
 
 values 
 ices to 
 
 cosine 
 
 cosine 
 
 Finally, when the arc increases from 270° to 360°, the sine 
 passes from — 1 to — 0, and the cosine passes from to+1 ; 
 hence, at the end of Ov^ 
 
 ,., J , f f^-n l}''0-=— 0. 
 
 4th quadrant, I ^.^^ 360^=+!. 
 
 Now, if the limiting values of the sine and cosine with their 
 proper signs, as just determined for each (juiidrant, be succes- 
 sively introduced in tho fornmla (ii) of Table A, there obtains : 
 
 tan 0- 
 
 r=0; tan W 
 
 tun 180' 
 
 
 
 = — 0; tan 270'^= J=cc ; tan 3^50° = ~^=-0. 
 — 1 — W ' 1 
 
 » 
 
 ';^1 
 
 Iro, and 
 
 k<''e 
 
 170 
 
 Porforming the same substitutions* in the ot'aoi forinuke of 
 Tahle A, the results will be those collected in the following 
 
174 
 
 
 
 PLANE TRIGONOMETRY. 
 
 
 
 25\ 
 
 
 Table B. 
 
 
 
 ArcrsO" 
 
 Arc: 
 
 =90° 
 
 Arc=.180° 
 
 Arcs 
 
 =270° 
 
 Arc=:360'' 
 
 sill = 
 
 
 
 sin 
 
 — 1 
 
 1 
 
 1 
 sin = sin 
 
 — I 
 
 sin = — 
 
 oo;; == 
 
 1 
 
 cos 
 
 =: 
 
 COS =— 1 
 
 COS 
 
 =— 
 
 cos = 1 
 
 V s'.n = 
 
 1) 
 
 V sin 
 
 r= 1 
 
 v.-.in = 2 
 
 v.sin 
 
 — 1 
 
 v.sin = 
 
 (■o.v.sin= 
 
 1 
 
 c().v.sin-= "' 
 
 co.v.sin= ] 
 
 co.v.si:i= 2 
 
 co.v.sin= 1 
 
 tan = 
 
 
 
 tan 
 
 = a 
 
 tan r=~0 tan 
 
 = ao 
 
 ;an — — (> 
 
 cot = 
 
 00 
 
 cot 
 
 = (; 'rot =— oc 
 
 c:;t 
 
 =: 
 
 cot = — cc 
 
 sec = 
 
 1 'sec 
 
 1 ^ _^ , 
 
 S(M! 
 
 ^^^ cr 
 
 sec = ] 
 
 cose<* = 
 
 X 
 
 (o.-ec 
 
 ^^^^ i ''OS C 1 oc 
 
 :'og c 
 1 
 
 rr^^— J 
 
 cosec = — cc 
 
 2i 
 
 Ft'nctions of Arcs r<):MrRisiN(} v^ x OO^dr.r. 
 
 (« clenoti-.ig^iiny number, and ,' any arc loss than 90°). 
 I' Fruintlu- (l(>:initi():-.s of the eirtailar functions, 
 
 q . i of . I /; = Pr>= OL=ccs (J 
 
 ( • s ( > 
 
 tan (; 
 
 ? AB=or=LP= 
 
 sin o 
 
 f7;r, 
 f7?r. 
 
 AB=AK-^ 
 
 CO 
 
 t of EC 
 
 :t . f 
 
 r,s'= 
 
 tan of BC, 
 
 SC'C «-> 
 
 f.i/;=oA'= 
 
 CUSt'C o 
 
 f/>'r. 
 
 cnsoc' oi 
 
 ,l/?=r 0X= 
 
 <ec of r>c 
 
 T>.notin- BG by ;r : tlien J />'— .IT— BC=\n)°—.v ; hence, 
 tiic Mibstitution of 00 ' — r f<!r ^! //, .uid .r i'ur />^ '. iiitheabove 
 
 (•((nations. \vi 
 
 11 
 
 'in 
 
 'MY 
 
 :\\' 
 
 ') 
 
 :CUS ,(• 
 
 cot (_00^ — .;•)=: tan ./■ 
 
 ^-o.s (inr— .r)r^sin 
 
 tan 
 
 r'^O' 
 
 -)--:= I'Ot 
 
 S.'C 
 
 l-OSCC 
 
 (!)0°— ;;■; 
 
 fosec ./: 
 
 (00- 
 
 ■)=.s. 
 
 sec .1- 
 
 sni o 
 
 tMJ/=A'J/-^Cy/: 
 
 -^rCOS O 
 
 f ('7J/. 
 
 cos of AM^OX'^LM^^r.x of CM. 
 Let t in- .r— Of, then .1 .V _ fMI^ + ,r ; and, niiice 
 negative ( Ai't. 25), there obtains, by snbstitution : 
 
 (KV 
 
 IS 
 
 <in (^90°-|-a')=co.sx. and cos (90°-f;c)=: — sin .«•., 
 
ARTICLE 27. 
 
 175 
 
 (>y is 
 
 Introducing these values in the formulae of Table A, there 
 
 obtains: tan (00°4-x)= — !^ 1 — ^ = _= — cot x ; 
 
 ^ ^ cos(90°+.r) —sin a; 
 
 and so on, as fullows : 
 
 tan (00°4-x)=: — cot a;. sec (nO''-T--i)^^^ — cosec re. 
 
 cot (00°-f xJTrr— tan 05. cosec (90'^+a.')= + sec x. 
 
 3 ' sin of A^f=yJl=sm of MD. 
 
 cos of AMz=OX=cos of MD. 
 
 Letting x=J/A then AM=l^i\° — x ; mid, since O^Y is 
 negative, there obtains by substitution : 
 
 sin (180° — a;)=sin j:, and cos (180° — x)= — cos .r. 
 
 Introducing these values iu the formula' of Table A, there 
 obtains : 
 
 tan (180°— x) = — tan ,r. sec (180°— ./) = - sec .f. 
 
 cot (180° — x) = — cot X. eosec (180° — ,/■) = +cosec a;. 
 
 Hy the same process, wo may find the functions of 180°-f-jL', 
 liTO^- ■•• 270°-(-a' and 3(30° — .r, iis shown in the following 
 
 37. 
 
 Table C. 
 
 Arc=90°+x 
 
 Arc=:270° — .r 
 
 >inTr= COS X COS =:r: — ,S1H ./• 
 t;;i!rr: — cot J' jCot — :; — t;in X 
 
 sin= — COS./' cos = — sin jc. 
 
 Arc=lSO^- 
 
 X 
 
 sin =: sm ,r |Cos = — cos ./■ 
 tan= — tan x cot r=r — cot x 
 Kec = — sec ./: icosec—- cosec.x 
 
 tan = 
 
 cot ./; cot = 
 
 1 
 
 tan X. 
 
 sec -- 
 
 — cosec X cusec = 
 Arc = 2703+3; 
 
 — sec X. 
 
 sin = 
 
 — cos X 'cos = 
 
 sin X. 
 
 1 t:in=: 
 
 — cot ;'• cot = ■ 
 
 — tan ./•. 
 
 Arc=180'^ + 
 
 c 
 
 S\U^=. — sin .-V 'COS rrr — COS .T 
 
 tan^^- tan x cot r~- cot .'• 
 secrrr — S(^C ./; IcoseC:=r:— cosec ./• 
 
 isoc -— c(^.sec./:cosec= — sec x. 
 
 Arc = 31)0^ -.f 
 
 sin = — sin .r [cos = cos J'. 
 tun = — tan x cot = — cot./;, 
 sec ;= sec X ;cosec = -cosecj". 
 
 If * 
 
 il 
 
 ■ 
 
 .*■• 
 
 :^! 
 
■TTTT^^^^^PT' 
 
 176 
 
 PLANE TRIGONOMETRY 
 
 By means of Table C, the functions of any arc, can he 
 expressed in term^ of the functions of an arc less than 90^. 
 Thus, sin 150^=:sin ( 90^ + 60°)= cos 60^ 
 sin300°=sin (270° + 30°) = — cos 30^ 
 tan 400° = tan (360° + 40°)= tan 40^ 
 
 28. 
 
 Fl^vctions op particular Arcs. 
 
 l'> Assuming BG to be equal to the radius 1, then the 
 triangle BGO is equilateral and thereby equiangular , further 
 assuming the angle at the center BOG to be bisected by the 
 radius OA, then .4G^=.4iJ=arc of SO"" ; 
 hence, sin 30° =z:P^=^56^=^. ^ (1) 
 
 By the formulae (5) and (6) of Table A, there obtains ; 
 
 ;OP=v/l-sin^30^= v/niJ=i^/3; 
 sin 30° h 1 
 
 cos 30°: 
 
 tan 30° = 
 
 2" 
 
 cos 30° |v3~v/3' 
 BC=AC-AB=viXG of 60° ; 
 cos m°z=OL^PB=\; 
 
 sin 60°=X5=v/l-cos-60°= x/r^=^^3 ; 
 
 (3) 
 
 tan 60° = 
 
 _ sin 60°_iv/3_ 
 
 
 (?) 
 
 (6) 
 (7) 
 
 cos 60° ^ 
 
 3> If .45=45°, then sin 45° = cos 45°, 
 
 , ^ ,-0 shi 45° , 
 and tan 4o^= - = 1. 
 
 cos 45° 
 &c.= c^e. 
 
 29. Circular functions of negative arcs. 
 
 Let A be the origin of urcs, and AB=.AG. 
 denoting .1/:? ))y a-, \\\^\\ AB=^x and A{r= — .»;, 
 But,- " sin of yt6J=i'C?=— sin of yl5, 
 
 and COS of AG=zOP.— cos o^ AB ; 
 
 hence, substituting x for AB, and — ./; for AG^ there obtains 
 sin ( — .c) = — sin .r, and cos ( — a')=:cos ./•. 
 
 whei 
 
he 
 
 )tains 
 
 ARTICLE 31. 
 
 17T 
 
 Introducing these values in the formulae of Table A, there 
 obtains : 
 
 tan ( — x) = — tan cc cosec ( — x)=. — cosec Ji7. 
 
 cot ( — .'£) = — cot X versin ( — a;) = +versin X. 
 
 sec ( — T) = +seca; co.v.sin( — .r)=l+8in x. 
 
 30. Inverse circular functions. 
 
 The inverse circular functions are those in which an arc 
 is regarded as a function of its trigonometrical linos, contrary 
 to direct circular functions in which the trigonometrical lines 
 are functions of arcs. 
 
 Thus, letting .r = arc AB, y—PB, and z:=OF : 
 then, ?/=«sin ;r, and ;; = cos or. 
 
 These direct circular functions become inverse circular func- 
 
 z, which are 
 
 sec 
 
 &c, 
 
 tions, by writing a'=si:i y, ar.d r=cos ^ 
 
 read ''a-is the arc whu>se sine is y, and x is the arc whose 
 
 cosine is 2." 
 
 It must l)e borae in mind that, in tlio i xprcssions sin 
 the symbol ^ ' i.s not an exponent. 
 
 iil. Relations of the sides and functions 
 OF a right-angled plane triangle. 
 
 Assume ABC to be a triangle right- 
 angled at B. f± 
 
 From J, as a center, with a radius 
 AEz=\^ de-^cribe th-j arc EF , let full the^x 
 perpendiculars ED^ GP^ on AB ; and lot 
 (f, 6, c = sides opposite to angles .1, B, C, respectively. 
 
 Then, in the similar triandos .l/iY7and ADE. 
 
 (2) 
 
 DF 
 
 1" 
 
 AE:ED=AC:CB, 
 
 or 1 :sin A = b:a ; 
 
 whence, 
 
 sin.i==''. rn, 
 
 and a = h sin .t 
 
 , 
 
 AE:AD=Ar:An. 
 
 III- 1 ; <•(!« A = b ; <• ; 
 
 whence. 
 
 cos.l-^ (.3). 
 
 and — /> fos A. 
 
 H' 
 
 AP:PG-AIi:B(\ 
 
 or 1 : tnti .l=c: n ; 
 
 \vlien<.'e. 
 
 tun vlr=: ' , (.')), 
 
 and^ '/==(■ tan .1. 
 
 (4) 
 
 (6) 
 
 ii 
 
 ■1] 
 
 '^ ■■'■ ■ :■ !<*?J 
 
 * ^ . il 
 
 '>., 
 
 12 
 
178 
 
 PLANE TRIGONOMETRY 
 
 Since either side about the right angle may be considered as 
 the base of the triangle, then the angles A and C may be 
 interchanged, provided their opposite sides, a and c, are also 
 interchanged ; thus, by this mutual change, the equations (5) 
 and (6) will become : 
 
 tan C= 
 
 a 
 
 (7), iind c = a tan C: 
 
 (8) 
 
 or (because tan 6'=cot A), c=^a cot A. (9) 
 
 From the preceding remark, and from the equations (2), 
 (4), (G) and (9), we may deduce the following principles : 
 
 Frin. 1. A formula ., in terms of tlie circular functions of a 
 plane right-angled triangle^ remains true when the acute angles 
 are interchanged^ provided their opposite sides are also inter- 
 changed. 
 
 Frin. 2. In a plane right-angled triangle, each side of the 
 right angle is equal to the sine of the opposite angle into the 
 hypothenuse. 
 
 Frin. 3. In a plane right-angled triangle, each side uf the 
 right angle is equal to the cosine of the adjacent acute angle 
 into the hypothenuse. 
 
 Frin. 4. In aplane right-angled triangle, each side of the 
 right angle is equal lo the tangent of the opposite angle into 
 the other side. 
 
 Frin. 5. In a plane right-angled triangle, each side of the 
 right angle is equal to the rotangent of the adjacent acute angle 
 into the other side. 
 
 33. Functions of the sum and difference 
 
 OF two arcs. 
 
 Assume AB and BC to be two arcs to 
 radius ()B=\. 
 
 Let fall the perpendiculars CP and ED 
 on AO, the perpendicular (.7E on BO ; and /^ 
 draw EF parallel to AO ; ulso let x = AB, ^ 
 and // = BC 
 
id as 
 'J be 
 ! also 
 
 » (5) 
 
 (9) 
 
 \8 of a 
 
 angles 
 
 inter - 
 
 of the 
 le into 
 
 of the 
 \te angle 
 
 ARTICLE 32. 
 
 179 
 
 Then, CP=ain (.v+//), CE =sin y, and OE=coh i/. 
 
 I. From the figure, we have : 
 
 Cr=PF-\-FC, or sin (:c-^fj) = PF-\-I'C=DE-^FC. (1) 
 The sides of the angles A0J3 and FCE being perpendicular 
 to each other, then [^AOB= L.FCE=zx. 
 In the right-angled triangle ODE (31, Prin, 2), 
 
 DE= OE sin .v = sin x cos y. 
 In the right-angled triangle FCE (31, Prin. 3), 
 
 FC— CE cos jc = cos X sin y. 
 Introducing these values o^ DE and FC, in (1), then : 
 
 sin (^ + ?/)=Hin x cos y-(-cos :v sin y. (2) 
 
 This formula (2) is true for all values of x and y ; hence, it 
 is true when y becomes — y, and then 
 
 sin (A:~y)=sin x cos ( — y)+co8 x sin ( — y) ; 
 but (29), cos ( — y)=co3 y, and sin ( — y)= — sin y ; 
 hence, sin (.v — y) = sin.rco8y — cos x sin y. (3) 
 
 Substituting 90°— jc for x, in (3), there obtains : 
 sin(90°— X— y)=sin (00 ^— ,v)cos y— cos (90o— jr)sin y ; 
 but(26), sin(90O— :r— y) or sin [90o_(A'+y) ]=cos {x-\-y)^ 
 
 and sin (90° — .r)=cos x^ and cos (90o-— .r)=sin x : 
 hence, cos (x-i-y)=cos x cos y — sin x sin y. (4) 
 
 Substituting — y for y, in (4), there obtains : 
 
 cos {x — y) = cos X cos ( — y) — sin x sin ( — y), 
 or cos {x — y)=cos x cos y-j-sin x sin y. (5^ 
 
 Collecting, (2), (4), (3) and (5), we have : 
 
 sin (x-f//)=sin x cos y-f-cos x sin y. (a) 
 
 cos (x'4-y) = cos X cosy — sin x sin y. (b) 
 
 sin {x — y)=sin x cosy — cos x sin y. (c) 
 
 cos (« — y) = cosx' cos y-f-sin x sin y. (d) 
 
 II. Dividing 1", (a) by (b); and 2 ', (b) by (a - there obtains : 
 ^^^ sin (x-f-y)_^i>» -c cos y-f-cus .r sin y 
 
 cos {x-\-y) cos X' cos y — sin x sin y 
 
 .^^^ cos (x-f-y) cos X cos y — sin x sin y _ 
 
 Sin (x-J-y) sm x cos y -f- cos .;; sin y 
 
 then dividing each term of both the numerator and the deno- 
 
 H 
 \ 
 I 
 
 i 
 
 > 
 1 
 
 y^^ 
 
 
 H^''''^^' 
 
 
 ^Hl''!;''^' 
 
 
 
 
 
 
 
 
 
 
 ^K,<'^ 
 
 
 
 
 ^^K,^.''»v 
 
 1^^ 
 
 
 .,; ; mU 
 
180 
 
 PLANE TRiaONOMETRY. 
 
 minator of the second member, in 1", by cos jc cos y ; and in 
 2", by sin x sin y, and reducing by the principle that 
 
 lo 
 
 2o 
 
 sin 
 
 cos 
 
 =tan, and 2' , =oot, there obtains : 
 cos sin 
 
 tanfx+y) 
 
 _ tan x-\-t'Any 
 1 — tanx" tan y 
 
 ^1 , V cot X cot ?/ — 1 
 
 cot(.r-f7/)= 'L 
 
 cot cc-f-cot y 
 
 III. Substituting — y, for y, in (e) & (f), and reducing 
 
 , N tan X — tan ?/ 
 
 tan(x— y)— '' 
 
 cot{x—y) 
 
 1 -f- tan X tan y 
 cot X cot ?/ + ! 
 
 33. 
 
 cot y — cot X 
 Functions of double and half arcs. 
 
 (e) 
 (t) 
 
 (li) 
 
 I. Letting x=zy, in formula) (a), (b), (e) and (t'),we havo 
 
 Ui') 
 
 a>') 
 
 . o 2 tan X , ,. 
 
 tan2j;=:., . ((>') 
 
 sin 2.r=2.sin ./■ cos x 
 cos 2xrfzCosv — sin-a;. 
 2 tan X 
 
 cot2a;= 
 
 1 — tan-./' 
 
 cot V' — 1 
 2cot X 
 
 (f) 
 
 II, Introducing successively, in (b'), the values of sin-a; and 
 cos^a', taken from the formulie sin-x=l — cos^o;, and 
 cos-a;=l — sin-.f of Table A, there obtains 
 
 (A) 
 (B) 
 
 cos2x=l — 2sin"X' ; whence, sin.r==^N/'J-(l— cos2j ). 
 and cos2.T=2cos'-'aj — 1 ; whence, cos.r=v/|.(l4-cos2a;). 
 
 Dividing 1", (A) by (B) ; 2', (B) by (A) ; then multiplying 
 both numerator and denominator by the denominator, and 
 reducing, there obtains : 
 
 sin 2a. • (E) 
 
 lo. 
 
 2o 
 
 tana;= — ^ 
 
 l-|-cos2x 
 
 sin2x 
 
 cota:= 
 
 1 — cos2u;" 
 
 (F) 
 
34. 
 
 ARTICLE 35. 
 
 Additional FoR.MULiE. 
 
 181 
 
 1 ' Adding' (c) to (a), then (d) to (h) : 2' subtracting (C) 
 from (a), then (d) from (b), there obtiiins : 
 lo sh>(x-|-^)-f-sin(.f — i/)^=2Hin X cosyi/, (1) 
 
 and C08(a;-|-y)-f-cos(x — i/)=z2cosx cosy. (2) 
 
 2' sin(.r4-''') — sin(.z; — j/) = 2cos x" sin y, (3) 
 
 jind coii(x— fj)— cos (x-\-y)== 28m X s'mi/. (4) 
 
 Letting j;+//=^), and x—i/=q, 
 then ic— ^(p-f(/), andy = ^(p — j). 
 
 Introducing tliese values of x and y, in (1), (2), (3) and (4), 
 there obtains : 
 
 sinp-j-sin 2^r=2 sin ^(jp -\- q) cos^(^p — g^). 
 cosj3-i-cos^=:2 CO.S ^(/>+g') cos^(j) — q). 
 sin/)— sin q=2 con ^(p-\-q) &[n^(p — q). 
 cosfy — cos/>=:2 sin l{j>-\-q) sin ^(j) — 5^). 
 Dividing (k) by (m), there obtains : 
 
 sin ;) ; sin q 2 sin h(p-\-q)cos ^(p—q) tan ^(p + q) 
 
 sin/;— sin q 2 cos ^(p^q)sin ^(p—q) tan ■h(j)—q) 
 
 (k) 
 
 ay 
 
 (m) 
 
 (n) 
 
 (o) 
 
 The other quotients obtained by dividing (k), (1), (m) and 
 (n). the one by the other, member by member, will give several 
 other useful formulae. 
 
 35. 
 
 Theorem. 
 
 In any plane triangle^ the sides are proportianal to the sines 
 of fJie opposite angles. 
 
 Hyp. Let ABC be any triangle ; 
 AsT. then will 
 
 sin A sin B sin C 
 
 IW'^ 'A(T^' ^~AB' 
 Dem. From the vertices .1 and B.A 
 with a radius-unity, draw the arcs ms 
 and of^ and let fall the jjcrpendiculars CD^ mn and o/>, on. the 
 base AB. 
 
 m 
 
 
 1 
 
 iU 
 
IS2 
 
 PLANE TRIOONOMETRV 
 
 In till' .similiii' triannjlcss A(W si\u\ Amu. 
 
 AC: rD= Am:mn, or AC: CD= 1 :sin .1 ; 
 vvheiice. CD=ACfi\\\A 
 
 In tho siniilur trianirles BCD and Hop, 
 
 BC: CD=Bo : op, or HC: CI)=^] :.sin li ■ 
 whence, 
 
 Hence, from (1) and (2), 
 or 
 
 Likewise, 
 Therefore, 
 
 or 
 
 sin^ 
 sin A 
 
 CD^BCsui B. 
 BCHmB:=ACii'mA. 
 BCiHinA^ACiH'mB. 
 AB:sinC=AC:HmB. 
 sin B sin C 
 
 H) 
 
 (2| 
 
 AC 
 
 sin B 
 
 " AB 
 sin C 
 
 36. 
 
 a b 
 
 Theorem. 
 
 (3) 
 
 
 tan ^(.1— y^)" '* 
 
 rtliizzsin ^lisin i?, 
 (/4-^>-^='*i»-'l-|-sin 7i:sin 7i, 
 « — 6:i = sin^l — sini?:sin B\ 
 
 a-^h sin ^l-(-sin B 
 
 a — b sin A — sin B 
 
 sin^+sin B tan \{A-\-B') ^ 
 
 sin .1— sin B~ tan^(.l— i^j' 
 a+6_ tan|(.l4-7?) 
 ~a—l> ^ tan ^{A—B) ' 
 
 TlIEdREM. 
 
 In any plane triangle, the sum of two sides is to their diffe- 
 rence, as the tangent of htdf the sum of their opposite angles. 
 is to the tangent of half the difference of those angles. 
 
 Hyp. Let A BC be any triangle. 
 
 AsT. tlien will 5> 
 
 a-f 6 tan 1{A + B) 
 T^b 
 De.m. From 
 by composition, 
 by division, 
 
 wlience, 
 
 But(:u,(o)), 
 
 therefore. 
 
 we deduce • 
 
 (1) 
 
 I f\ in (Oiij ptlanc triangle, the longest side he taken as the 
 base, and a perpendicular be let fall on this base from thr 
 Vertex of the op^msite angle, then the sum of the two segments 
 
 m 
 
p 
 
 a its th> 
 from th' 
 iegmcnts 
 
 ARTICLE 39. 
 
 183 
 
 or tlw vjholc Ixise, is to the samoftlw other tico sides, <is the 
 (Vitference, of these siiJes, is to the difference of the segments. 
 
 II VP. A.ssuine Alif to hv any plriiie trian<il(' From tlic^ 
 vertex C of the j,'reiitest aii^lo, let fall tho perpendicular 
 ( '/> on AB, and let s=A I), and s'= DB ; 
 
 AsT. then will AI)-\- I)/i:AC-\-CB=:AC-CB:AI)-I)/K 
 or c I h -jf- (I = b — a : s — s'. 
 
 i>KM. In the right-angled triangles ACD and DOB, 
 
 AC'=AD--tDC'', 
 
 CB-=DB'-^DC\ 
 Subtracting, AC-- CB'= AD'- DB', 
 or (AC-^CB)(AC-CB) = (AD+DB)(AD-J)Ii). 
 hence, AD+J)B:AC\CB=AC-CIi:AD-DB, 
 or c I h \- (t =^ h — <i ', s — s'. 
 
 (1) 
 
 Solution of rigiit-anoled plane triangles. 
 
 The solution of right-angled plane triangles admits uf 
 four cases : 
 
 WluMi the data for the solutiidi of a tria ij^le are pointed out. it \v".]l :il 
 Wiiys be understood that tho otlier parts are required. 
 
 3S. Case I. 
 
 Data. B=9()°, hypothenuse &, 
 and either side, as a, about the right angle. 
 
 Solution. By the equations (1) and (4) 
 of Art. 31, we have : 
 
 sinvl = 
 
 a 
 
 :i9. 
 
 c=icosil. 
 C=9()°-A, 
 
 Case II. 
 
 J5=9()^, and both sides, (i and c, about the right 
 
 Data. 
 
 angle. 
 
 Solution. From .4. + (7=90'^, and the equations (5j and 
 {2) of Art. 31, we have : 
 
 *' 
 
 k: 
 
 T 1 ! 
 
 m 
 
 .'■It 
 
 V4 
 
 N L 
 
 I 11 
 
 i 
 
 .^fl 
 
 m n 
 
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 TEST TARGET (MT-3) 
 
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 1.8 
 
 
 1.25 1.4 
 
 1.6 
 
 
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 pm 
 
 <^/ A 
 
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 Photographic 
 
 Sciences 
 Corporation 
 
 23 WEi^ 4AIN STREET 
 
 WEBSTER, NY. 14580 
 
 (716) 872-4503 
 
 s. 
 
 ip 
 
 V 
 
 :\ 
 
 iV 
 
 \ 
 
 
 iS^ 
 
 
 > 
 
 'ii" 
 
 

 
 CA 
 
!l 
 
 f 
 
 II 
 
 184 
 
 PLANE TRIGONOMETRY. 
 
 tan^=cotC=:-. 
 c 
 
 ^ = 90°—^, and 6'= 90°—^. 
 
 J a c * 
 
 40, 
 
 sinil sin 6' 
 
 Case III. 
 
 Data. B= 90°, hypothenuse b, and either acute an<;le. 
 
 Solution. From ^-j-C=: 90°, and the equations (2) and (4) 
 of Art. 31, we have : 
 
 a = b ainA. 
 c=b cosJ.= b sinC. 
 
 41. Case IV. 
 
 Data. ^=90°, either acute angle, as J., and lo the 
 opposite side a, or 2o the adjacent side c. 
 
 Solution. From A-\-C=90°, and the equations (9), (2), 
 (6) and (4) of Art. 31, we have : 
 
 C=9Q°-A. 
 
 c= a cot A. 
 
 1'^ 
 
 2- 
 
 a 
 
 sinA 
 a= c tan^. 
 
 b= -' 
 
 cos^ 
 
 Solution of onLiQUK-AxcjLKi) plane triangles. 
 
 The solution of oblique-anuled plane trianules admits of 
 four cases : 
 
 4:2. Cum I. 
 
 Data. Two antrles, as li and C, and the included side n. 
 
ARTICLE 44. 
 
 ir.5 
 
 Solution. From .4 + 5+ (7= 180°, and the equation (3), of 
 Art. 35, we have : 
 
 A — 180°-(i^4.C). Fig, Art. 36. 
 
 , sin li 
 sin.l 
 
 sinC , sin 6' 
 sin^l sin^ 
 
 Case II. 
 
 4:j. 
 
 Data. Two sides, as a and 6, and angle A opposite to a. 
 
 Solution. From A-^B-\-C= 180°, and the equation (3) of 
 
 Art. 35, we have : 
 
 b 
 
 sinjB; 
 
 sin ^4. 
 
 a 
 
 C=180° — (.4 + 5). A 
 sin C 
 
 c=:a 
 
 sin^l 
 
 Note. Since the sine of Ji is also the sine of its supplement B', the case 
 would be am6i//M0M«, if the (lata of the problem wc:e .«uoh as not to in- 
 dicate, in any way, which of the two un.u:le-! is to be taken. 
 
 In practical problems, howv-ver, there is no anibii^iiity. 
 
 44. Cfise III. 
 
 Data. Two sides, as a and b. and their included angle C. 
 Sol. Since ^4-^^+^"=- 180°, then \{A-^B)-\-^C: 
 whence, i(.4-f7^)— J)0°— ^6\ 
 
 90' 
 
 From (1) of Art. 3G,tani {A—By. 
 
 (I- 
 
 a-\-b 
 
 tan^(.4 + i^); 
 
 but, tan^(.4 + i?)=tan(90°— ^C) = cot^C; 20,1" 
 
 hence, letting M =1 h {A-\- B)-, and N:=. h {A—B), we have : 
 
 tan A = cotiC. 
 
 ,i-\-b 
 
 A=M-j-X, B=M-N. 
 
 c=a -. — - z=b- 
 
 m\A 
 
 m\B 
 
186 
 
 PLANE TRIGONOMETRY 
 
 
 I? 
 
 I':': 
 
 m^ij 
 
 45. Case IV. 
 
 Data The three sides a, />, c. Fig. Art. 36- 
 
 Solution. Assuiiiin^ CD to be perpendicular to AB^ and 
 letting »=r:.47>, and s'=Dn ; then, from (1) of Art. 37, 
 
 c 
 But, s:=^{s-\-s')-\-^{s—s')-=^{c-{-s—s'), and s'=c—s. 
 Hence, by these relations, and the equation (3) of Art. 31, 
 we have : 
 
 j_{b-\-a)(b—a). 
 
 8 — S 
 
 s'=: c — s. 
 s 
 
 C0Sv4r=: 
 
 COS-ff =: 
 
 a 
 
 46. 
 
 Logarithms. 
 
 The logarithm of a given quantity is the exponejithy which 
 a /?xefi quantity, called the base, must be affected to produce 
 the given quantity. 
 
 47. Tho common system of logarithms, to which alone 
 reference is made in this work, is that in which the base is 10. 
 Hence, in this system, any number is considered as some power 
 of 10, and the exponent of the power is the logarithm of the 
 number. 
 
 Thus assuming n = 10 , 
 
 then X is the logarithm of n, which is expressed by 
 
 los; u = X. 
 
 48. If nis an exact power of 10, its logarithm x is an entire 
 number ; thus, if we successively assume 
 
 n==lO, «=^100, w=lOO0, ^c, 
 then, will x=l, x=2, .r=3, «&c. 
 
ARTICLE 54. 
 
 187 
 
 Wlicn n is not an exact power of 10, its logarithm is composed 
 of two parts : an eutirt' part called the characteristic, and a 
 derimttl part called thj mautissa. 
 
 Thus, in 1.60S!>7<> —log 50, the entire part 1 is the charac- 
 teristic, and the decimal part. (598970 is the mantissa. 
 
 XoTE. The above oquation log n = x may be rcrerml by wrifiriff 
 ]0g x = n. which is read : •• th: number whose loparithiu is z equals n." 
 
 The following principles are demonstrated in Algebra. 
 
 4:i}» Prin. 1. The logarithm of the piodnct of two or morr 
 ^lumbers is equal to the snin of the Icgarlthms of these nmnliers 
 Thus, log (ahc.)=: log «-f log />-f logc. 
 
 50, Prin. 2. The logarithm of a quotient ix equal to the 
 logarithm of the dividend ininus the logarithm of the dicisor. 
 
 Thus, log/^— j;= loga-f log & — logo— log d. 
 
 51. Prin. 3. The logarithm of <tny power of a numljer is 
 equal to the logarithm of the niimher multiplied h\j the exponent 
 of the power. 
 
 Thus, log a^' = n log a. 
 
 5*Z» Prin. 4. llie logarithm of any root of a number is equal 
 to the logarithm, of the number divided by the index of the 
 roof. 
 
 Thus, log«»= - loira. 
 ' ° n " 
 
 53. Prin. 5. llie characteristic of the logarithm of an 
 entire or mixed number is positive^ and one less than the num- 
 ber of integral places in the number. 
 
 Thus, 3.G47 774-- log 4 444, 
 
 4.047 774= log 44 440, 
 0.G47 774= log 4.444 . 
 
 54. Prin. 6. The characteristic of the logarirhm of a 
 pure decimal is always negative and greater by 1 than the 
 number of zeros which immediately follow the decimal point ; 
 but, the mantissa, or decimal part of the logarithm, is always 
 positive. 
 
188 
 
 PLANE TRIGONOMETRY 
 
 This fact is indicated by ]>laeiiiir; thoi negative sign over tlie 
 characteristic, instead of before it. 
 
 Thus, H.550 788 is equivalent to— I]-;-. 550 7S8. 
 
 i5o« Prm.7. 1 • Adding 1 to, 2» suhtnictingl from, the cha- 
 racteristic of (I logarithm is equivalent to l'» multiplying, 
 2' dividing, hi/ 10, the numher corresjwnding tothat lagnrithin. 
 
 Thus, since 3.(547 774= log 4444 ; 
 
 then, 1" 4.(i47 774= log 44440, 
 
 and 2" 2.H47 774^.102: 444.4 . 
 
 5G« Prin. 8. A change of the decimal j)oint of a numher 
 implies a change in the characteristic, hut no change in the 
 mantissa of the corresponding logarithm. 
 
 Thus, log 33:J= 2.522 444 log 0.333=1.522 444. 
 
 log 33.3=1.522444 log 0.0333=2.522444. 
 
 56. Table of Logarithms. 
 
 In table I appended to this work, complete logarithms are 
 given for all numbers from 1 up to 100. For the others, the 
 mantissas alone are given, as the characteristic, in any case, 
 may be found by the principles of Art. 53 and 54. 
 
 Use of Table I. 
 
 •XT. Tojind the logarithm of a numher from 1 to 100. 
 
 Look on the first page of the ta'»le, in the column headen X, 
 for the given number ; the number opposite, in the colnniii 
 liead(Ml li»g., is the logarithm required. 
 
 Thus, ' log 26=1.414 073. 
 
 i>S. To find the logarithm ofani/ number. 
 
 Find the characteristic by the principles of Art. 53 and 54 
 iluMi. to find the mantissa, drop the decimal point, if there b(^ 
 jiny ; and with the entire number, proceed as follows : 
 
 1' It' this entire number contains four figures, keep it as it 
 is ; if it contains less than four figures, annex zeros enough to 
 make it a number of four fiuuros ; then find the left-liand 
 three fitrures of this number, in the column headed X. and 
 pass along the horizontal line, which begins with thes(> three 
 
ARTICLE 58. 
 
 ISO 
 
 fi inures, to the column liendeU 1»y the fourth fipire ; if the 
 number there found contaii.s si',i- fiiiures it is the niantisHti 
 required ; if ndt, tlie four fi.ures. there found ure the right- 
 hand ibur of the nianti.s.sa recjuired. Now, to find the other 
 two, foll»>w the same horizontal line back to the column headed 
 "0", droppinir to tlie line immediately below when any (fats 
 are passed orrr ; if tbe nuniber found in the " " column 
 contains six figures, prefix the left -hand two to tlie four already 
 found ; if not, follow up the •' " column till a number is 
 found containinu six fij^ures, the left-hand two of which are to 
 be prefixed to the four already found. 
 
 The result, to wliich a decimal point must be prefixed, wiU 
 be the mantissa required. 
 
 DotSritiiUfl tor ttiid ire to be cp'a'cd liy, zeros in the mantissa. 
 
 2' If ///lA' entire number c^jntains more than four figures 
 place a decinial point after the four left-hand figures, and find 
 the mantissa of the entire part, as above (1") ; then multiply 
 the decimal part by the corresponding tabular difference in the 
 column headed J), and add the product to the mantissa already 
 found ; the result will be the mantis.sa required. 
 
 It must DO borne in mind tiiat the tabular difference /) is millionths, and 
 tiiat ih,' i>roduc^ just mentioned is iml/ionUi-i, 
 
 Thus, to obtain the logarithm of 12;^.4r)G ; find first the cha- 
 racteristic, which is 2 : then, place the decimal point after 4 
 and find the mantissa of the entire part 1284, wliich is .0!>4 S20 ; 
 then, multiply the decimal part .5(1 by tbe tabular difference 
 351, and the product will be 107 which being added to.0l)4!-;20 
 will give .0'J5 017 for the numtissa re(|uired ; that is, 
 log 12:i.45(i=2.0l)5ttl7.'i= 
 
 Let tlie student verify the following eijuations : 
 
 1" loK :{():u=;j.5G0u2«) 2" io« :u2;;4=:4.5:}4 45S, 
 
 log 2:145= 1.1570 14;; 
 
 lo- .0()7s=2.s;u 2:50 
 
 lou .0li78V> = 2.83l 80li 
 
 log 24:i5.<i=:!.;{S(n;oH, 
 
 log 0.24155=: 0.7U5 428, 
 lou .50781)^1.754 21)4. 
 
 • NoTK. To (ind the logarithm of a fraction, roduci- it to a decimal, and 
 proceed us above. 
 
190 
 
 PLANE TRIGONOMETRY. 
 
 ■I 
 
 11 
 
 Ill' 
 
 59. To find the nnmhcr corresponding to a given logarithm. 
 Tliis problem admits of two cases : 
 
 Case I. Whtn the inantlasd can he found exactly in the 
 tabic, look in the t'oluiun headed " 0", for the left-hand two 
 figures of the given mantissii ; and, in the same or some other 
 column, for tlio other four figures ; then in the column headed 
 JV, opposite these four figures, will be found the left-hand 
 three figures, and at l<fp the other figure of the number 
 recjuired. 
 
 Thus, log— 1 2.370143 = 2845. 
 
 Case II. When the mantissa cannot he found exactly in the 
 table, take out tiie next less mantissa and the corresponding 
 tabular difference ; subtract tliis mantissa from the given one, 
 divide the remainder by the tabular difference, and annex the 
 €|uotient to the four figures already found . 
 
 In both cases, place the decinml point in the number thus 
 found, according to the given characteristic. 
 
 Ex. Find the number corresponding to log 2.654 321. 
 
 Solution. Given mantissa.654 321. 
 
 Next less in table .654 273 of 4511. 
 
 Tabular J9 96) 48.00 (5; 
 
 hence, log"* 2.654 321 = 451.15. 
 
 Let the student verify the following equations : 
 
 jog- 1 3.845 532=7007 log— * T.166 666=0.464 157, 
 
 ;jg~' 0.985 346=9.6682 log" ^2.865 344=0.07 334, 
 
 60. Multiplication by Logarithms. 
 
 To multiply by meanr of logarithms (40), find the alge- 
 braic sum of the logarithms of the factors ; the number corres- 
 ponding to this new logarithm icill he (lie product required. 
 Fx. Find the product of 234.5, 0.2345, 2.345 and 0.02345. 
 Solution. Add log 234.5=2.370 143, 
 
 to log .2345=1.370143, 
 to log 2.345=0.370 143, 
 to log .02345= 2.370143 ; 
 
 then will the product = log" ^ 0.480 572=3.02393. 
 
ARTICLE 63. 
 
 191 
 
 61. Division bv Logarithms. 
 
 To find a quotient h/ mrnns of logarithms (fiO), subtract 
 the. logarithm (>f the divisor from the logarithm of the divid- 
 end ; the nnmher corresponding to this neio logarithm will he 
 the quotient required. 
 
 Ex. Divide 450.7 by 7.G54 . 
 
 Solution. From log 450.7=2.651)631 
 take log 7.654=0.883 888; 
 
 O 7 / 
 
 then will the quotient = log~"^ 1.773 743=59.6682. 
 
 62. COLOGARITIIM OF A NUMBER. 
 
 The cologarithm (complenicnt-logarithm) of a number is the 
 remainder obtained by subtracting the logarithm of that num- 
 ber from 10 ; thus, the cologarithm of 5659 (whose logarithm 
 is 3.752 740) is 10-3.752 740=6.247 260. 
 
 The logarithm of a number being found, its cologarithm 
 (colog) may readily be obtained, by beginning at the left of 
 the logarithm and subtracting each figure from 9, to the last 
 significant figure which must be subtracted from 10. 
 
 By Prin. of 50, log =log a— log Z» ; (1) 
 
 and by definition, colog 6=. 10— log h ; 
 
 whence, log i = 10— colog 6. 
 
 Now, substituting 10 — colog 6 for log t, in (1), we have : 
 
 log y =log a + colog 6—10 ; hence, 
 h 
 
 63. To find a quotient h\j means of a cologarithm, add the 
 cologarithm of the divisor to the logarithm of the dividend, 
 subtract 10, andfind the numher corresponding to the resulting 
 logarithm. 
 
 The useof thf" cologarithm id very convenient in the operation of com- 
 bined multiplication and division, by making all mantissas additive. 
 
 Ex. Compute jc, in the proportion : 
 
 0.325x25.66 
 
 2.547 X 0.4563: 0.325= 25.66 :.r, or a;= 
 
 2.547x0.4563' 
 
If 
 
 ■I;;' 
 
 192 
 
 PLANE TRKiONOMETRY 
 
 Solution. Add log 0.325= 1.511 883 
 to log25.G6= 1.409 257, 
 to colog 2.547= 0.503971-10, 
 tocolog 0.45G3=10.3i0 750-10 ; 
 
 then will the quotient x=log~~ ^ 
 
 G4. 
 
 0.855 801 = 7.17 504. 
 Involution by Logarithms. 
 
 To raise a niunher to any power hif means of logarillims 
 (51), find the logarithm of the number, multipli/ it hy the 
 'exponent of the power, and find the number corresponding to 
 the resulting logarithm. 
 
 Ex. Find the 10th power of ^=0.75. 
 
 Solution. iMultiply log 0.75 =1.875 001, 
 by 10; 
 
 then will 
 
 _ 1 
 
 (0.75) = 
 
 log 
 
 — I 
 
 65. 
 
 2.750 610=0.056 313. 
 Evolution by Logarithms. 
 
 To extract any root of a number by means of logarithms 
 (52), //«(/ the logarithm of the number, diuidc it by the inde.i' 
 of the root, and find the number corresponding to the resulting 
 logarithm. 
 
 E.L\ Find the 10th root of f = 0.75. 
 
 Solution. Divide log 0.75= 1.875 061 + 9— 9. 
 by 10) =1 1) + 9.875 061 ; 
 
 then will (0.75)TO - log-^ 1.087 506=0.971045. 
 
 NoTK. When, as in tliis Ex., a neeative ohanicforistic is not divisible by 
 the index of til" root, adtl to it the .smaHest nefrativ.' niinihor which will 
 make it divisiiile, and prefix, to the mantis:::a, the same niiral)er with a 
 plus sign. 
 
 Table of Natural Sines, Cosines, Tangents 
 AND Cotangents. 
 
 A table of natural sines, cosines, tangents and cotan- 
 gents is u tablo in which the natural sine, cosine, cS:c., of an 
 arc or angle may be found. 
 
ARTICLE G5. 
 
 193 
 
 ble by 
 :h will 
 with a 
 
 kotan- 
 
 ot* an 
 
 Construction of a table of iKitu ml circular functions. 
 
 The equation (d) of Art .40, Book III, is a serial function of 
 X which may be reverted into the following serial function of 
 I (App. n): 
 
 X C=8inO= 4- - —- — + &C. 
 
 ^ 1 1.2.ir 1.2.3.4.5 l.2.3.4.5.«.7 
 
 Hence, 1", if the terms of this series be summed up alge- 
 braically, the numerical value of x, or sin I, will bo determined; 
 and the numerical value of cos / may then be found by the for- 
 mula: cos 1= \^l—x' ; 2', the numerical values of tan I and 
 cot I may also be found by the formulae : 
 
 . , sin / 1.7 cos I 
 
 tan /= , and cotf=-; 
 
 cos/ sin I 
 
 We perceive the possibility of constructing, by this process, 
 a table of natural circular functions for every minute of a 
 quadrant. 
 
 Trigonometrical computations might be performed by means 
 of these natural functions, but it is more convenient to make 
 use of logarithmic functions, as explained in the following 
 articles. 
 
 LoGARiTii.Mic Tables of Circular Functions. 
 
 66* A table of logarithmic sines, cosines, tangents, &c. 
 (Table II appended) is a table containing the logarithms of 
 natural sines, cosines, ^c, increased by 10. 
 
 Thus, the natural sine of an arc or angle of 30° being equal 
 
 to ^ or 0.5 (Art. 28), its logarithm is 1698 970 ; then, by 
 adding 10, there obtains 9. 098 970 which is the tabular loga- 
 rithm of sin 30°- 
 
 This addition of 10 to the logarithm of a natural function is 
 made in order to avoid negative characteristics. 
 
 Now, since addidg 10 to log 1 (logarithm of the radius to 
 which the natural circular functions are referred) is equivalent 
 to multiplying 1 by 10.000.000 (Art. 55). it follows that the 
 circular functions of Table II, are referred to radius 10,000.000. 
 
 liMf 
 
 /l3 
 
^iSSi 
 
 ^i-a...:, 17.J — !.« — ut.i... r I.I -rXj wJtum 
 
 194 
 
 PLANE TRIOONOMETRT 
 
 ConverHely. By sul>tiaciiii^ 1<> from tlio fnhufar lo^iirithinic 
 functions, we sliall have tlir natural iMLaritliniic functions, 
 the correspondiuL' nuinlirrs ofwhich will lie the natural func- 
 tions themselves. 
 
 Thus, the tabular lo-aritliniic sine 0.(108 070— 10=the 
 
 natural loiiaritlnnic sine i. (JOS 070, the citrresponding number 
 of whicli is the natural sine, 0.5, of an arc or an^le of 30°. 
 
 07* In table II, the logarithmic sines, cosines, tanjxents and 
 cotanjrer^ts are ^iven for each minute of a quadrant. 
 
 From 0° to 45°, these logarithmic functions read down- 
 wards^ their dej^rees beinj;; at the top of the paj^^e, and their 
 minutes in the left-hand column headed J/; from 45° to 00°, 
 they read My;i««»v/«, their degrees being at the bottom of the 
 page, and their minutes in the right-hand column. 
 
 The increase of logarithmic sines and the decrease of loga- 
 rithmic cosines, for 1 second, are respectively found on the 
 right, in the column headed I) ; the differences of logarithmic 
 tangents and cotangents, for 1 second, are placed between the 
 columns of these circular functions and are common to both ; 
 but, it must be borne in mind that these differences are addi- 
 tive in the former and suhtnictive in the latter. 
 
 Note. The syrahols : log sin , log cos , \of tun ,&c. are read ; 
 the arc (or angle) whose lo<?arithmic sine is. the arc whose logarithmic 
 
 cosine is, Ac. Thus, logsin""' 9.705 576=30° W 30" is read : the arc 
 (or angle) whose logarithmic sine is 9.705 576 e(|ual8 3u° 30' 30". 
 
 Use op Table It. 
 08. To find the logarithm of an arc or oitijle. 
 
 If the arc or angle is less than 45°, look for the degrees at 
 the top of the page, and for the minute.s in the left-hand 
 column headed il/; then follow the horizontal line, which begins 
 with the given nnnutes, till you reaeh the column designated 
 at top by the name of the function ; or, ii* the arc or angle 
 exceeds 45°, look for the degrees at the bottom of the page, 
 and for the minutes in the right-hand column of J/; then 
 again, follow the horizontal line till you reaeh the column 
 
AinicLK r.!). 
 
 195 
 
 inic 
 ons, 
 iinc- 
 
 =the 
 mber 
 
 s and 
 
 Imn- 
 their 
 
 of the 
 
 >f loga- 
 on the 
 rithmic 
 jen the 
 1 both; 
 addi- 
 
 are read ., 
 larithmic 
 
 the arc 
 
 ^rees at 
 ;ft-hantl 
 |h begins 
 Isi-'iiated 
 )!• angle 
 [he page, 
 \]I; then 
 column 
 
 iU'Mij.nal(Ml at ftnttimt \)y ihc name «»r tin- t'mietioii ; the logu- 
 ritlnnthorp found is the l()L;:;rithni r(M|uirtHl, if the given uro 
 <ir aiiple doos not contain seconds. 
 
 If tiu' arc or angle rontains seconds, set aside the logarithm 
 just found for degrees and, minutes ; then multiply the corres- 
 ponding number i;i tlie column /> (whicdi i.s milliouths) by the 
 i.um])er of given seconds, and (ufd this product to the logarithm 
 set asid(!, in the case of a siiu^ lU' a tangent, and suhtnict it 
 from tiiat logarithm, in the caseof a cosine or a cotangent. 
 A'.r. 1. Find the logaritlimic sim? of 40° :iO' 40". 
 SoLirno.N. Jiogsin 4(r^;{0'i= 9.P1M544 
 
 Tabular /). = 2.M'} 
 Cjiven seconds = 40 
 
 Product :^9ST47r mlditive 98 
 
 Log sin 40° :{0' 40" — 9.812 TT42 
 
 A'a,'. 2. Find inc logarithmic cosine of 55° 25' 20". 
 
 Solution. Log cos 55° 25— 9.7541^46 
 
 Tabular I). ~ 3.05 
 (liven seconds = 20 
 
 Product =6r.06 unhtractive 61 
 
 Log cos 55° 25' 20" -^ - 9.753 985. 
 
 69« To find the ai'c or tiinjlt corrc.spoiidiui^ to a logarithmic 
 /unction. 
 
 Look in the table for the given logarithm ; if found there^ 
 take the degrees from the top of the page, and the uiiuutes 
 from the left-hand column of M, in the case of a sine or a tun- 
 gent ; but, take the degrees from the bottom of the page, and 
 the minutes from the right-hand column of M, in the case of a 
 cosine or a cotangent. 
 
 If the given logarithm is not found in the table, look for the 
 next less^ set it aside, and take from the table, the correspond- 
 ing degrees and jninutes, as just e.\plained. ami ;-et them aside. 
 Then subtract the louarithm set aside from the given logarithm, 
 and divide the remainder by the corresj>onding tabular dif- 
 ference ; the ([uotient '.vill siive seconds to be addid to the 
 
 ':■ 1 " ■ 
 
 ''■■ I'i 
 
 I'M 
 
 ■^'1 
 
 »•!.' 
 
 
 ,yii 
 
196 
 
 PLANE TRIGONOMETRY 
 
 
 degrees and minutes set aside, for a sine or a tangent, and to 
 be snhtnictcd. for a cosine or :. cotaiiuent. 
 
 f y. 
 
 Ex. 1. Find the arc whose lo.>arithrai(; sine is 9.705 5T(». 
 
 Solution. Given log 9.705 57i». 
 
 Next less in table, 9.705 4()9 of :J0° 30'. 
 
 Tabular 2>, :}.57) " l07.00( 30" ofhlif in-. 
 
 Hence(«7, Note/, log sin~ ' !).705 :u(\ -= 'MP 80' 80". 
 / Ex. 2. Find the angle whose logarithmic cosine is 9.802 087 
 
 Solution. Given log 9.802 087 
 
 Next less in table, 9. 801 978 of 50^ 40'. 
 
 Tabular D, 2.57) ()4.00 ( 2^^' subtract ioe. 
 
 Hence, log cos" ^ 9,S02 087 =. 50o 89' 35". 
 
 70. 
 
 PRACTICAL APPLICATIONS. 
 
 Prob. 1. Given the hi/pothenuse b 1=8.55, and the siife 
 a =4.84, of the triangle ABC ; to find the other parts. 
 
 Solution. Applying logarithms to the for- 
 mulae : l'',sinyl=: ; 2, r=/>cos.4. of Art. 38, 
 
 there obtains : 1", lou; sin.l = V)^i<( — losr?> ; " 
 2', logt'=log/jH-log cos^. 
 
 But l",log a(4.34)=0.G37490 i 2", log 6(8.55) =0.981901; 
 
 colog^»(8.55)==9.0G8084 
 
 hence, J.=log sin" 9.705524 
 =30° 80' 15". 
 
 log cos.4(30O80'15" i = 9A)85801 
 
 hence. c = log 0.8t!72()7 
 = 7.8066 . 
 
 3", 6'=90o_,l = 59^ 29' 45". 
 
 Prob. 2. Given the two sides, a = 40.05, and c = 50.25, o/ 
 the right angle J\ ; to solve the triangle ABC. 
 
 Solution. Applying logarithms to the ftn-muhe : 
 
 l<',tanyl= - ; 2'\h— , of Art. 89, there obtains : 
 
 c sin ^4 
 
 lo, log tan^ = log(e — log c ; 2', log6 = loga — logsin.4. 
 
ARTICLE 70. 
 
 197 
 
 to 
 
 .2r>, of 
 
 But l",loga(40.05) = l.r)026():i 2",lojr<;(40.0':) =1.602603 
 colog c (5 i.jj) = h.2'js 8<!1 coloKsiu .l(:!H":{:n<)")=o.i2or> :j24 
 
 hence, .4 = log tan" ^ i).001467 hence, b = lo.ir~* 1.807927 
 = 38° 33' 19". =64.258. 
 
 3', Cz=: 90° -A =')\o 26' 41". 
 
 Prob. 3. Given the hf/pofhennfKt l)=:300, and the acute 
 ' angle A = 40° 30' ; to solve the triangle ABC. 
 
 Solution. Applying logarithms to the formulae : 
 
 1' . a=^ />sinvl ; 2', <•= l cos ^4, of Art. 40, there obtains : 
 1 , log a -.=z. log Z*+log sin .4 ; 2 ', log c = log 6+ log cos A. 
 
 But 1', log i(300) = 2.4771 21 
 logsin/i(40° 30') = 9.812r)44 
 
 hence, (t=log""^ 2.289605 
 zrzl94.x:U . 
 
 2«',log/>(3(K)) =2.477121 
 logcos.4(40° 30')= 9.88104G 
 
 hence, (•=log~^ 2.358167 
 =:228.123 . 
 3'>,6^^90° — .4^49° 30'. 
 
 Prob. 4. Given a side a =25.5, and the angles B =r30"25 
 25"<tn(^ C =80° 50' 35", in the oblique-angled triangle ABC; 
 to solve the triangle. Next fig. 
 
 Solution. Applying logarithms to the formulae : 
 
 , , sin/i 
 
 1", bz=: a- — r ; 
 
 siuyl 
 
 2-, C: 
 
 a .^. , of Art. 42, we have : 
 sin 7^ 
 
 1", l()g?> =. logft-flog sin B — log sin A ; 
 2 ', log c = log (f-f logsinC^ — log sin.4. 
 
 But, .4=180°- (/i+ 6^)= 68^44'; 
 
 and 1", log a (25.5) = 1.406540 
 
 Io!v.smZ/i30°25'25")=9.704485 
 colog sin.4{68°44') = 0.(i;}0630 
 
 hoKce, /> = iog"~^ I.I'M (inn 
 
 = 13.5312. 
 
 and 2', log a (25.5) = 1.406540 
 
 loR 8i.i(7(80°50'35") = 9.9941 20 
 colog sn.4(68°44') = 0.080630 
 
 lience. c ■■■■-- log~ ^ 1 . 48 1 290 
 = 30.29 . 
 
 rrob. o. Given the siihs a =4(M!. I)::j=:!l5ll. aitd the inrliuhd- 
 iiigle C -r.50° 50', in the oblique-angled triangle ABC ; ta 
 (ind the other po 'ts. 
 
 if' 
 
M 
 
 hi 
 
 ^i i! 
 
 
 w > 
 
 III 
 
 iri?> 
 
 PLANE TRKJOXOMETRY 
 
 S<iM t;o.\. Applying logurithiiis to the formulae 
 
 tan .\ = _ — - cot 
 
 u 
 
 al — ^,, of Art. 44, we have: a/ jnl \r 
 sin, 4 ^ ^^ ' ^ 
 
 1 , loji tan .V=lop:(a—i) -flog cot ^ (7— log (a-f- 6) ; 
 
 8". loir c -=z loga4-lo<r sin C — loir sin.l. 
 
 But 1 '\]o'^(a-h r- 50) = 1 .('.08970 
 
 log cot^ C(25° 25')=io.:j2:n:n 
 
 colog(«4-/.=750) = 7.124{)an 
 
 H",loga(400) =2.r)02or)n 
 log sin C( 50°50' ) = 9.S.S1)477 
 col()gsin.t(72':!-no")=.020 414: 
 
 hence,iV==:logtan~^ 0.147040 hence, <- = log~' 2.511051 
 
 =:7'50'10'. =r:j25.0511. 
 
 2". Now (44), J/=00°— |C=00° — 25° 25' = (;4''85': 
 lience, A=:M+ X^72° 34' 10", and i?=rJf— .V=:5G°35' 50". 
 
 ]?rob. 6. (rliH'ii the three sides a=30, b=:=40. c=5(), in the 
 triaiifjle AB(' : to find rhe other parts. 
 
 >^(iLUTi(>N. Applying logarithms to the formula) : 
 
 1". .V — .■•• =:J:^ ' '\ -L • 2". cos^i— : 3 I, COS/i = 
 
 e t> a 
 
 of Alt. 45, then : 1", log(.s— s')=log(Z> + </.)-f log,/< — ^/) — logr; 
 
 2 . 'iiU' COS. I :=::loU' .S — loU' h ; 3'. loU' COs/> = log s' — loU" <i. 
 
 j.'utl'.'o: C^ , a_T70i— 1.845008 
 
 log(A — o--l(); =1.000000 
 
 colog <:(50) =8.301030 
 
 henco,s— .s'^log""^ 1.140128 
 
 -^14. 
 Now (Art. 45, Formulae), 
 
 whence, 2" log .s(32) = 1.5051 50 
 colog 6(40, ^..S.30704O 
 
 hence, .4 = logcos~^ !).00:}000 
 
 = 30^ 52' 12". 
 3", log.s'(18) =1.255l'73 
 
 (•oloirrfi30) =8.522870 
 
 .— 1 
 
 , — 'V^ 
 
 •'<—2(^-r «—*•■) = •!-, ;herice,/> -logio.s * 0.77.S152 
 and .s' = «•_.>•= 18; ■ --53° 7' 48". 
 
 4 r=180^^M-l+/')=00^ 
 
 Rkmahk. Aiijfk' C 1;^ found to b • u right angle : and it will always he 
 io. wlu'tievor (/. h nnd c are multiple.^ of 3, 4 and 5, respectivoly. 
 
 HI ! 
 
ARTICLE 70. 
 
 199 
 
 Trob. 7. To find the height BC of a vertical object. 
 
 Data. The horizontal distance 
 /I /^— 105, and tlie angle of elevation 
 .l-=05^ 18' 15". 
 
 Golution. From Art. 41, 
 <f ~c tan ^zzr. 263. 1843. 
 
 Ans. ^ 
 
 XoTK. Angles are measured either by a Compass or a Theodolih'. 
 
 Prcb. 8. To find the height BC of an inaccessihle ohjeat. 
 
 Data. In the vertical 
 triiingle ADC, were mea- 
 sured : the horizontal base 
 A D= 500, and the angles 
 
 A~ 40^ 35', 
 and BI)C—{M)~' 25'. 
 
 Solution. 1' . In A A DC, J^, 
 
 DC=ADJh^ — 500 '!^'!i'l!15!,_-Joo - 1 2.933 243 . 
 sin.l67>~ sin 11)^' 50' 
 
 2'. In ABDC, CB. DChxu IWC =7Ab.7H\ . Ans. 
 
 Prob. 9. To find the dlstnncc of an inaccessible object A, 
 fmm the points B and C 
 
 Data. Were measured the base 
 iiC=\m, and the angles Ii=z72'^' 35', 
 an.] ' r=75^'15'. 
 
 F;)LUTi()N. By the formulae of Art. 42 : 
 .4-180^'- (ii+ 6M=:30° 10' ; 
 
 h--=a '- — ^r=750.445G ; 
 
 sin^l 
 
 sin 6' 
 sinJ. 
 
 c-=ia - — -=^7i i.SiSZ 
 
 Prob. 10. To find the distance AB hettoeen tvwindrcrssih/f. 
 objects. 
 
 'v<. 
 
 4 
 
 m 
 
il-il 
 
 ■ 1 ! \ ^ 
 
 tiSii 
 
 hi 
 
 ■,,-i; 
 t'i' 
 
 
 111 13. . 
 
 200 
 
 PLANE TRIOOXO.MET:!/ 
 
 Data. Were measured the base CZ?, 
 and the angles ADC, BDC, BCD and 
 ACD ; whence, BCA=BCD-ACD, 
 
 The solution oi AADC will giveylC, 
 and the solution of /\BCD will give 
 BC. 
 
 Assuming these solutions to give 
 ^(7=400, and BC=Hb(H and the com- 
 pass to give 50^' 50', for the included angle ACB ; then 
 (Prob. 5), JZ?=325.0511. Am. 
 
 11. Given hyjtothtnuse b=10, and altitude a=4 (fig. of 
 Prob. 2) ; requiredV^K. Ans. LA=23° 34'41".5. 
 
 12. Given hi/pothenuse b=10, and LAz=40° 30' 15"; 
 required a and c. Ans. a=6.49503 ; c=7. 60359. 
 
 13. jfivenH)=zradiu8 of a regular octagon ; required V> one 
 of the sides, 2 • the apothem, 3'> the area, of the octagon. 
 
 Ans. 1", 7.60366 ; 2", 9.2388 ; 3 •, 282.8425. 
 
 14. Required the arc A whose chord is 8, in a circle to 
 radius 10. Ans. A=47° 9' 23". 
 
 15. Required the area A of a segment whose arc is 40^, in u 
 ci'>-de to radius 10. Ans. A==2. 76666. 
 
 16. If the Peak of Tenerife is 12350 ft. high, and the 
 depression of the horizon from its summit 1" 58' 10", ichat 
 is the radius of the earth , supjmsed to he a perfect sphere. 
 
 Ans. 3957.41 miles * 
 
 17. // 3fount jEtna is 10963/if. high, what is the distance 
 from its summit to the apparent horizon, the radius of the 
 earth being 3956wu7c.s? Axs. 139.86 miles. 
 
 18. If the distance at rchich a mountain is visible at sea be 
 142 miles, tchat is its height f Ans. 2.138 miles. 
 
 19. At what distance can the top of a light-house 216 /V. 
 high he seen at sea ? Ans. 19.7 miles. 
 
 * NoTK. The t'ftVct of refraction, in iho normal state of the atmosphere, 
 must be corrected as follows : 
 
 When the apparent distance of the horizon is known, the re^i distance 
 is found by subtracting ^V of the former ; conversely, when the leal dis- 
 tant is known, the apparent distaiicj is found by add ng ^^j- of itself 
 to it. 
 

 •'V 
 
 INTRODUCTION TO SPHERICAL 
 TRIGONOMETRY. 
 
 DEFINITIONS AND GENERAL PRINCIPLES. 
 
 71. A sphere is a solid which may 
 be conceived to be generated by the 
 revolution of a semi-circle (ACB) 
 about its diameter (AB). 
 
 Tl*^* In this revolution, the ordinate- A 
 radius (CO) will generate a great 
 circle of a sphere ; and any other ordi- 
 nate (MP) will generate a small circle, 
 parallel to the great circle. 
 
 73. The axis of a great circle (CcD) of the sphere, or of 
 any of its parallels {MmN) is the diameter {AB) perpendicular 
 to that circle. 
 
 74. The poles of a circle of the sphere are the extremities 
 of its axis. 
 
 75. A spherical radius of a given circle (as MmN') of the 
 sphere is an arc (Z?m) of a great circle joining the pole (JS) and 
 any point {m) in the circumference of the given circle. 
 
 76. When the spherical radius is a quadrant, it is called a 
 quadrantal radius. 
 
 In a revolution about a pole (as B) : 
 
 77. A spherical radius (as Bm) will describe a zone of one 
 base ; two spherical radii of different lenghts will describe two 
 zones whose difference will be a zone of two bases ; a qua- 
 drantal radius (as Be) will describe the surface of a hemi- 
 sphere ; 
 
 78. The revolving extremity (m) of a spherical radius 
 {Bm) v/ill describe a circumference (MmN^ on the sphere ; 
 the revolving extremity (c) of a quadrantal radius {Be) will 
 describe a great circle (CcD). 
 
 
 :.•:^s■ 
 
it! 
 
 202 
 
 SPHERICAIi TRIGONOMETRY 
 
 >^;-'!i 
 
 79. A circular sector (as BO}f) less than 90°, revolving 
 about one of its radii {OB), will generate a spherical sector, 
 or spherical cone, whose vertex is at the center of the sphere 
 and whose b(tse is a zone of one base. 
 
 80« In half a revolution about its diameter (AB), a semi- 
 circle (ACB), whatever position (AcB) it may occupy, will 
 form, with iitiformrr trace (ACB), a diedral antrle (ABCc), 
 called a spherical wedge, or ungula, whose convex surface, 
 called a lune, is described by the revolving semi-circum- 
 ference (ACB). 
 
 81. The ungula, as well as its lune, is measured in degrees 
 by the arc (Cc) described by the revolving extremity (c) of 
 its ordinate-radius {Oc). 
 
 83. A spherical angle is the amount of divergence of two 
 arcs of great circles meeting at a point. 
 
 The arcs (as BM and 7^r) are the sides, and the point (J5) 
 at which the sides matt is the vertex, of the angle (MBc). 
 
 8ii. The measure of a spherical angle is the measure of 
 the diedral .angle formed by the planes of its sides. 
 
 84. A spherical polygon is a p:;rt of the surface of a 
 sphere bounded by arcs of great circles. 
 
 The bounding arcs are the sides, and the points at which 
 the sides meet are the vertices of the polygon. 
 
 A diagonal of a spherical polygon is an arc of a great circle 
 Joining two vertices of the polygon which are not consecutive. 
 
 85. Spherical polygons are classified in the same manner as 
 plane polygons. 
 
 86. A spherical pyramid is a pyramid whose vertex is at 
 the center of the sphere, and whose base is any spherical 
 polygon. 
 
 I'rom these definitions, it follows that : 
 
 87. If a diameter of the sphere be perpendicular to a circle 
 of that sphere, its extremities are the poles of the circle (74). 
 
 88. Each pole (^l or ^) of a circle (MmN) is equidis- 
 tant from every point of its c .rcumference. 
 
ARTICLE 05. 
 
 203 
 
 For. the spherical radii to all th • points (.V, m, &e.) of the circum- 
 ference are equal (75). 
 
 89. Every point in the circumference i»f a great circle is at 
 a quadrant's distance from each of its poles (75). 
 
 DO. A point (B) on tlie surface of a sphere, at a qua- 
 drant'.s distance from each of two points (as C and c) in the 
 arc of a groat circle (^CcD) is a pole of that arc (75). 
 
 01. If any point (:i.s o) in the circumferenco {(^I'D) of a 
 great circle be joined with either pole (as H) by a <(uadrant 
 (cB)^ the quadrant will be perpendicular to that circuni 
 fere nee. 
 
 For. the plane parsing tli.ongh cli also pa sck th!oii;ili the a\is AB of 
 CcD. 
 
 93. If two great circles interisecl each other, their axes 
 also intersect, and the spherical angles, at the intersection of 
 the circles, are equal to the plane angles at the intersection of 
 their axes, each to each, and the sum of any two adjacent 
 angles, either plane or spherical, is e([ual to two right angles. 
 For instanci', if/' and G are the poles of the circles ACB and 
 AcB^ respectively, the sides of the angles COc and FOG are 
 perpendicular each to each ; and the.se angles are tbereby e([ual 
 and subtended by equal arcs Cc and Hr (Bo(/k. I, Art. 70, 2). 
 
 93. Hence, a spherical angle (;is (^Am or niBM) is mea- 
 s(/re^n)y the arc (C<) included between its sides (produced if 
 necessary) and described from its vertex (.1 or B) by a (jua- 
 drantal radius (.ICor BC). 
 
 94. Since tho angle (as MBni) included between two arcs 
 of great circles is measured by another arc (6^') <^>f '^ ni'^^'^t 
 circle on the same sphere, the angles of a spherical triangle 
 may be compared by means of the arcs of great circles inter- 
 cepted by their sides, and spherical angles may be constructed 
 equal to given angles. 
 
 95. The volume of an ungula is to the volume of the 
 sphere, as the angular measure of the ungula is to the circum- 
 ference ; and likewise, the area of a lune is to the area of the 
 
)!> 1 
 
 l=! 
 
 
 204 
 
 SPHERICAL TRIGONOJIETRY 
 
 surface of the sphere, as the auirular measure (»f the lune is to 
 the circumference (81).* 
 
 Hence, letting /^=radiu8 of the sphere, 
 
 ri° = angular measiire of ungula or lune, 
 F= volume of unuula, 
 vl=area of lune ; 
 then, since ^ -/ii' and 4 t/^- = volume and surface of the sphere 
 respectively, we sliall have the following relations : 
 
 and 
 
 V n" , .1 n° 
 
 
 :j6u 
 
 or 
 
 whence, 
 
 V = , and ..1 = 
 
 270" 
 
 V=Ax ;^ and^= 
 
 47ye- 300^' 
 
 -JPu° 
 
 90" 
 8F 
 R' 
 
 (m) 
 (n) 
 
 In what follows, the arcs treated of, uuless otherwise indicated, aie 
 assumed to be a cs of gr-'at circles ; the distance, between any two points 
 on tlie surface of a sph.re is assumed to be ilie arc of a great oirclf which 
 they intercept ; each side of a polyj^on is as.4umed to be less than a senii- 
 circuuifjrence ; and the following propositions a.e regarded as self 
 evident : 
 
 1". The s.ni of any two sides of a plane triangle is gr ater than the 
 third side ; 
 
 2'. The sum of any two lat ral angles of a triedral angle is greater 
 than the third lateral angle. 
 
 9G. 
 
 Theorem. 
 
 Jill/ si'ile of a spherical triangle is less than the sum of the 
 <tth< r tiro sides. 
 
 IIyi'. Let .1/^(7 be a spherical triangle 
 whose lonjrest side is AC, on a sphere 
 whose center is ; 
 
 AsT. then will AC<{AB+BC). 
 
 Dk.m. For, in the triedral angle ABC-0, 
 whose lateral angles are measured by the 
 sides of A ABC, 
 
 therefore, AC<\An+BC). Q. E. D. 
 
 * Th:i same pr.nciplo will also apply to any o. her solid of revolution, 
 when thv; name of the solid is substituted for the word " sphere". 
 
ARTICLE 98. 205 
 
 J)"!. THKUllEM. 
 
 The mm of the sides of n sphirical triangle is less than the 
 r'n'caniference of II gmit rii'ch- (^olJU'^). 
 
 II VP. Let J/>'6'be a spherical A ; 
 
 i\.sT. then will ,l/iijri />Y'<:{«)0°. 
 
 De.m, Produce .l/i and .IT' till they 
 : loet at the pole />, opposite to the 
 pole .1 ; 
 ; hen, AB-\-BD+ AC+ (l)=)\m^ ■ 
 
 !.ut, liC<HD-{-CD ; 96 
 
 therefore, AB-j- AC-rBr<AB-}-BD-^AC-hC£> ; 
 
 that is, AB-^AC-{-BC<:Si\Oo. Q. E. D. 
 
 98. 
 
 Theorem. 
 
 //*, froini the vertices of a spheriad triangle, as poles, arcs be 
 (hscribed forming a sphericid triangle, the vertices of the 
 (ingles of the second triangle will be respectiveli/ poles of the 
 sides of thejirst triangle. 
 
 Spherical triangles thus related aro called polar triangles ; each 
 lu'iiig the j.olar t. iangle ofth • otlier. 
 
 Hyp. Let ABC be the polar triangle of 
 DEF; 
 
 AsT. then will DEF be the* polar triangle 
 of ABC. 
 
 Dem. Draw the ares BD and CD. 
 
 Since, 
 
 B is the pole of DF, 
 
 then, 
 
 BD is a quadrant ; 
 
 and since, 
 
 C is the pole of I)E, 
 
 then, 
 
 CD is a (juadrant ; 
 
 iieiice, 
 
 D is the pole of BC. 
 
 liikewise. 
 
 E is the pole of AC, 
 
 -.Mil 
 
 F is the pole of AB. 
 
 •■iff 
 
 ;■ .■■*;(") 
 
 i'SSi 
 
 '0 •:''''' '. 
 
 Therefore, DEF is the polar triangle of ABC. Q. E. D. 
 
n 
 
 *\ 
 
 
 
 
 
 206 
 
 »9. 
 
 sniERICAl. TRKIONOMETRY. 
 
 TllKOKKM. 
 
 /// tvi> poltii' trinvfjh x, 111)1/ iiiKj/i' of one triangle is me<i- 
 siirct/ In/ tht; sttpplniicnf of the side fi/incf opposite to it in flu 
 
 of In r. 
 
 Ilvi>. A.s.«iniie A IW and tf/JF to be 
 jxilar tiian^los ; and l«^t a. h. r.. tl. c. /'zrr sides 
 'j)>|M»^it(' to an<>les J, H, (\ JJ. A\ /'; y. h/\gw\l 
 
 AsT. then will <;+/= 1H0°, .1 +r/=lSOD, f ^/LJXb\ 
 
 Dem. Produce CA and TV? till the meet DE at 6^ and //. 
 Then; D is the pole of 67/. and DH=90° ; 
 
 likewise, A' is the pole of CO, and (?A;=90" ; " 
 Addinjr, DH +GE=\SO°, 
 
 or GH-\-DG+ GE =1S0^, 
 
 or 6?// I Z)A;=180°. 
 
 By notation, DE=f; 
 
 besides, arc6r//=LC; 
 
 hence, C+/=180°. 
 
 Similarly, A + dz=z 180°, n-^a-'[m°, &c.=&c. ^. A;. A 
 
 Cor. In two jwlar triangles, any side of one triangle is the 
 snpplement of the opposite angle in its polar triangle. 
 
 Art. 93 
 
 Art. 93 
 
 lOO. 
 
 Theorem. 
 
 The sum of the angles of a spherical triangle is greater tJiKU 
 fn-o right angles (180°) and less than six right angles (540°). 
 
 Hyi>. Assume ABC and BEF to be polar triangles ; and 
 let «, h, <\ d, e,/=sides opposite to angles ^4, B, C, J), E^ F : 
 AsT. then will J-f /i+C>180° and<540 -. 
 
 Dem. By hyp. A-l/^C' ^md A^^ A/'/'' are polar triangles ; 
 hence, .4 + (Z=180^\ /i+< = 180°, C+/=180° ; 
 w hence , A-\-B -i- C-\- d -f e -f /= 540 ' , 
 
 or .44-i?+6'=540° — ((/4-e+/). 
 

 
 AIITICLK 102. 
 
 207 
 
 
 But, 
 
 d+.Mf^:H\iV ; 
 
 97 
 
 
 licnco, 
 
 J -f- /i-f r=r>40''— (<:j«;o°); 
 
 
 nu'd- 
 
 t!iat is. 
 
 .i + /,>^.r:^i8o^ 
 
 
 ill tin 
 
 A,Li;'.iii, 
 
 '^4 ''-f./>0^; 
 
 
 
 tiicrordH'. 
 
 .1-f /i+r=r)4(> -oi^*^); 
 
 
 
 tliat is. 
 
 .i-^/?_i-r<r,4(i'. 
 
 Q. E. D. 
 
 Art. 93 
 
 p. E. D. 
 
 is the 
 
 terthun 
 
 (540^). 
 
 !S ; and 
 E,F: 
 
 >les ; 
 
 Cor. All thv nntjU's of <i s/)/urinU triiuujk may be right 
 ini(/hs, or all )nai/ he ohtiisn, 
 
 lOl. Definition. 
 
 Two spherical triangles are symmetrical, when all tlieir parts 
 are ei^ual. each ti) each, and placed in a reverse order in the 
 two trianirles. fr p 
 
 Thus, ABC, aMd DEF are 
 symmetrical triangles, 
 
 \ 
 
 
 if 
 and 
 
 lO*^. 
 
 L.4 = LZ>, L7^=LA\ L.C=l.F, 
 AB=BE, BC=EF, AC=DF. 
 
 Problem. 
 
 To construct a spherical triangle that shall he symmetrical 
 II- 1th res^icct to a given sj^herical triangle. 
 
 S(H-rTiox. Let ABC be the given 
 
 spherical triangle. From A, as a pole and 
 
 with the spherical radiua AB, draw an arc ; 
 
 from C, as a pole and with tlie spherical 
 
 radius CB, draw a second arc intersecting 
 
 the first at I) ; then join D and ^4, also D 
 
 and C, by a:cs of trreat circles ; and draw the radii AO, BO, 
 CO and DO. 
 
 ]' Byeonst. AD=.AB, CD=CB, 
 
 and ^1^,'is common to both triangles ABC and ADC ; 
 hence, these triangles are mutually ecjuilateral. 
 
 2' The chords of these e<jual arcs are equal ; consequently, 
 the two plane triangles formed by these chords, and whose ver- 
 tices are ^4, B, C, and .4, I), C, respectively, are mutually 
 
 
 m 
 
 m 
 
208 
 
 HIMIERICAL TRIUONOMBTRY 
 
 l'!:> I 
 
 i;-' 
 
 I ■'- 
 
 m 
 
 iy: 
 
 M 
 
 equilaterul, and tliereby equal ; bonidcs, tliey are equidistant 
 from the center O of the sphere. Now, the two pyramids 
 ABC (} and AD(>-0. ^vllo^e bases are these equal plane trianirles 
 and whose common vertex is the point 0, have equal bases and 
 equal altitudes ; hence, they areeijual. and tlieir lateral diodra] 
 anjrles are mutually e(|uian<iular. But, the spherical trianules 
 ^7^6' and ADC whose anules are respectively equal to the 
 'ateral diedral anules of these pyramids are also mutually 
 e(|uiangular. Therefore, since these trian<;les are mutually 
 equilateral, mutually equiangular and placed in a reverse order* 
 they are symmetrical. 
 
 Sch. If A]i=BC, then AJ)=DC, and these symmetrical 
 triangles are also isosceles. Henre, if /\ABC is placed on 
 ^ADC, so thatL^li^r shall coincide with [^ADC, Afiwith 
 CD, and CB with AD, then the two triangles will coincide in 
 all their parts. 
 
 Symmetrical triangles are not mperposable, if they are not isosce'.es. 
 
 103. Theorem. 
 
 Two spherical triangles, on the same sphere or on equal 
 spheres, are equal : 
 
 1" If they are mutually equilateral ; 
 
 2' if two angles and the included side of the one are equal 
 to two angles and the included side of the other, each to 
 each. 
 
 3> if two sides and the included angle of the one are equal 
 ^otioo sides and the included angle of the other, each to each. 
 
 The theorem, in each of these three hypoheses, maybe 
 demonstrated as in the analogous cases of plane triangles ; that 
 is, by applying the first triangle to the second, or «to the 
 triangle symmetrical with respect to the second. 
 
 104. Theorem. 
 
 1 ' If two sides of a spheriad triangle are equal, their oppo- 
 site angles are equal ; 2" converrely, if two angles are equ(d. 
 their opposite sides are equal. 
 
ARTirr,E 08. 
 
 1" IIyp. In the .spherical triangle DKF, 
 let J)F=EF; 
 
 AsT. then will LY^— LA'. 
 
 Dem, Join the middle f! of the base I)E 
 :ind the opposite vertex F. hy the arc FG ; '^ 
 then, ^DFCl and A^'FF nn- mutually equi- 
 lateral, and thereby, they are njiitually equiangular 
 t here fore . L Z> r=:l_ F. 
 
 209 
 
 l(i:}, 1 
 
 2" livr. In the .spherical triangle AJIC, 
 let L.i = L/i; 
 
 AsT. then will A( '=/}('. 
 
 Dem. Let ^DFF be the polar triangle 
 of AABC. 
 
 By hyp. L.l.^a_7i; 
 
 lience, DF^ EF, 
 
 and thereby, . L.l>-=z\^E \ 
 
 therefore, AC ^-- HC. 
 
 Cor. 1. If (ill the sldrs of n splierical triangle arc equal, all 
 its angles are also equal, and corirersely. 
 
 Cor. 2. Tlic arc diaion from flic vertex of an isosceles 
 triangle, to the middle of the base, is perpendicular to the base 
 and bisects the opposite angle. 
 
 99, Cor. 
 I)eni. 1" 
 99. Cor. 
 
 105. 
 
 Theorem. 
 
 The areas of two symmetrical triangles are equal. 
 
 Hyp. Let ABC and J)FF be C 
 
 two symmetrical triangles, // \ 
 
 inwnich AB=DE, A C = I)F b([\ ^"-^ N^'l^ 
 and BC=EF; ^Iji }^ 
 
 AST. then will AAnC=: ADEE. 
 
 Df.m. From a point M. on the surface of the sphere, equi- 
 distant from J. />' and C. let iin arc be drawn to cacli of these 
 vertices; and, from a point ^V iM^uidistaiit from ./>. A' and Fj 
 let also an arc lie drawn to oacli oi'tlic latter vertices. 
 
 14 
 
 #1 
 I* 
 
 
 
^!^ 
 
 210 
 
 SPHERICAL TRIGONOMETRY. 
 
 I r^' 
 
 ■ (■ 
 
 u ;[■ 
 
 But, a point J/" equidistant from .1, B and C is symmetrical 
 
 Avith respect to a point JV^ equidistant from />, E and F \ 
 
 c()nse(|uently, MA='MB=Mr=XD=y/'J=,VF; 
 
 that is, the trianjjles ABM and DhW, as well as J^CW and 
 
 EFX, are mutually equilateral and isosceles ; 
 
 hence, they are superposable. and thereby equal. 102 
 
 Therefore, AABM^ABFN; 
 
 ABCM=AFFJV; 
 whence, n\ 
 
 /\ABM+/\BCM-AACM = ADEN+AEFN—ADFN, 
 or AABC=:ADEF. 
 
 If M and N lie within the triangles ABC and DEF, then, 
 the minus signs of ACM and DFN, in (1), become plus signs ; 
 therefore, in both cases, AABC=ADEF. Q. E. D. 
 
 106. Theorem. 
 
 If the semi-circumferences of tico great circles intersect on 
 the surface of a hemisj)here, the sum of the opposite triangles 
 at the point of intersection is equal to a lune xchose angular 
 measure is that of the opposite angles in those triangles. 
 
 Hyp. Let ABC and EBF be two 
 semi-circumferences intersecting at B, 
 on the hemisphere whose base is the 
 great circle AECF ; 
 
 AST. then will ABAE+ABCF= 
 lune measured by \—CBF. 
 
 Dem. Produce the arcs BC and BF 
 till they meet at D. 
 
 Since, ABC and BCD are semi-circumferences, 
 then AB^CD. 
 
 Likewise, EB = FD, 
 
 and AE=CF; 
 
 hence, ABAE and A(^T)F are mutually equilateral and 
 equiangular ; consequently, they are symmetrical and equal 
 in area ; 
 
 whence, ABAE— ACRE. 
 
 But, ABCF+ ACI)F=]xiue BCDFB measured by i^CBF; 
 hence, ABCF+ABAE= lune measured by I.CBF. 
 
w^m 
 
 107. 
 
 ARTICLE 107. 
 
 Problem. 
 
 211 
 
 To find the formula of the urea of a spherical triangle. 
 
 SonTioN, Let ABC bo a spherical [^ .^^g; 
 
 triangle, on a heniispUere whose base is 
 the great circle DEFG. 
 
 If the sides of the triangle ABC be 
 produced till they meet the circle DEFG, 
 they will form three pairs of opposite 
 spherical triangles : 
 .», ADEk , Adll: 2 , BFO k HDK , :}», /\CHKk /^CEF; 
 that is, three lunes measured by the angles ^,ji5, C oi /S.ABC. 
 
 By Art. 95, equation (w), we have : 
 
 area of 1st \\xnQ='^'^^^^ = /\ADE-\- /\AGH, 
 area of 2nd lune = '^'^ ^ ^' = ABFG + ABDK, 
 
 (a) 
 
 90" 
 
 area of 3rd lune = '^'^-^=/\CIIK-^ ACEF. 
 
 90^ 
 
 But, the second members of (a), (/>) and (c) containing 
 
 three times the triangle ABC. their sum is equal to the area^ 
 
 2-K\ of the hemisphere ^?Ms 2 A ABC; 
 
 hence, adding (a), (6), (c), we shall have : 
 
 urea of the three Innes = r E' A±A±^= 27: K'-^ 2 A ABC, 
 
 90^ 
 
 or 
 
 whence, letting 
 
 AABC=^I^.^^-r:R^i 
 J = area of A^i^C, 
 
 180" O) 
 
 The difference (; .-l-|-l /i-fi C— 180") is called spherical excess of 
 
 Cor. </ie area of a spherical polygon is equal to its spherical 
 
 excess multiplied by '-- 
 
 zK' 
 
 180^ 
 
 ' t \'t 
 
212 
 
 SPHERICAL TRKJONOxMETRY 
 
 II' 
 
 li i '■ 
 
 \l i! 
 
 i!| 
 
 I ::i 
 
 I 
 
 For assume the .spherical polygon 
 ABODE to be divided into triangles, by 
 diagonals drawn from .1 ; 
 and let A = area of ABODE, 
 
 a', a", &c. = areas of /S.ABO, /\AOD, kc, respectively, 
 
 S=smn of the an.lcs of J BODE, 
 5', «", i^c. = sunisof the angles in /\ABO, /\AOD, &c. resp. 
 
 ?i = number of sides of ABODE. 
 Then ^l=:«' + a"-t-&c., 
 
 S=s' -f .s"4-cS:c., 
 knd(?»-2) = number of triangles in ABODE. 
 
 Bytheequiktion (p), a' = J:^(s'-mn, 
 
 <i 
 
 ' = ii^(.s"_180°), 
 
 i\lC. = <^C 
 
 Adding, a'+a"4-i^c= 
 
 "^•" .5 s'-\-s"+kc-in-2)lS0° X 
 
 or 
 
 A=^ \ S-(n-2) 180-= I 
 180^ ( ^ J 
 
 (2) 
 
 The difference " S — (n — 2)180° ' is the spherical excess of the polygon. 
 Sch. I. Letting 'S9— (w-2)180° " = r, in formula (^), 
 
 r 
 
 there obtains : 
 
 .l=.-72-X 
 
 180- 
 
 That is to say, on a sphere to radius /t, the area of a poly- 
 ^on whose spherical excess is : 
 
 r~ willbe :?^==i?^_^^lJl^?J^i''*===72^x 0.01 745 329, 
 180 180 
 
 1 will be 
 \ ■ (\ili be 
 
 72- X 0.01 745:120 
 
 = i?-X 0.00 029 089, 
 
 GO 
 72^X0.00 029j)89^^^0^,,,,,^^.^ 
 
 60 
 
 To find the area of a spherical polygon whose spherical 
 excess is given, u.se will be made of the above table, by pro 
 
 ceediiia: as in the followinir 
 
1 
 
 ARTICLE 115. 
 
 213 
 
 . resp. ; 
 
 2)180^ 
 
 \ 
 
 le poivgon. 
 mula (g), 
 
 )f a poly- 
 
 329, 
 
 h 089, 
 
 485 
 
 spherical 
 
 '1 
 
 by pv 
 
 u- 
 
 Problem. 
 
 Jo find the area of the polijgou vliose spherical excess ih^ 
 23° 34' 41".5, on a sphere to radius (R) =-- 10. 
 
 Sol. The area of a polygon whose spherical excess is equal • 
 to 23^' = i?-x23 X 0.01 745 329:= 7e-x0.40142r)7, 
 to 34' = 7^-x34 X 0.00 029 089=7^-x 0.0098902, 
 to 41".5= K' X 41.5 X 0.00 000 485=/?-x 0.0002011 ; 
 whence, the area required =i?-x 0.411517 = 41.1517 
 
 Sell. II. Since a spherical cone or pyramid may be divided 
 into infinitely small triangular pyramids of equal altitudes (^)^ 
 the sum of who.se infinitely small bases constitutes the 
 base (.1) of the cone or pyramid, it follows that the formula 
 of the volume ( V) of each of the latter is F= ^ AxR. 
 
 Stereograph ic Projection op the Sphere. 
 
 108. The projection of an object is the representation, on 
 a plane, of the main points and lines of that object. 
 
 109. The point of sight is the point where the eye is 
 situated. 
 
 1 lO. The primitive is the plane on which the delineation 
 of the object is made. 
 
 111. The axis of the primitive is the straight line drawn 
 from the point of sight perpendicular to the primitive. 
 
 112. The center of the primitive is the point common to 
 both the primitive and its axis. 
 
 113. The original is the object (point, line, o.c; to be 
 projected. 
 
 114:, A projecting line is a straiglit line joining any ori- 
 ginal point and the point of sight. 
 
 115- A projecting surface is a surface which contains the 
 jirojecting lines of all the points of an original line. If the 
 urigimil line is straight, the projecting surface is a projecting 
 plane. • 
 
 
214 
 
 SPHERICAL TRIGONOMETRY 
 
 !f •• 
 
 116. The stereographic projection of the sphere is tliat 
 in which a great circle is taken as a primitive, one pole of 
 Avhicli is the point of siglit. 
 
 117* Cor. TJw projection of any 2^olut in the intersection 
 of its projecting linevrith the jtrimitive. 
 
 Thus, if the point of sight be the pole 
 of tlie primitive AGBJf, on which a circle 
 UF IS to be projected, then CE, CF are 
 the projecting lines, and e,/ are the projec- 
 tions, of the points E^ F, respectively ; 
 EC F is the projecting plane, and cf'ia the 
 projection, of the diameter EF. 
 
 118. Prob. 1. To describe the jn'ojcction of a great circle 
 through any tioo jwints in the plane of the primitive. 
 
 Solution. Let ACBD be the primitive. 
 1'^ If one of the points is in the center of 
 the primitive, the diameter drawn through 
 the other point will be the projections^ 
 required. For, the great circle passes through 
 the pole of the primitive. 
 
 2' If one 6' of the points is in the circumfe- 
 rence, and the other F is neither in the circumference nor in 
 the center of the primitive ACBD ; draw the diameter CIK 
 then will the circle described througli the three points C.FJ). 
 be the reijuired projection. 
 
 111). Prob. 2. To describe the projection of a great circ.< 
 about some given point, as a jiole. 
 
 Solution. 1" If the given point is the center of the pri.ni 
 tive, the required projection is obviously tlie primitive itself 
 
 2' If tlie given point is in the circumference of the prim"; 
 tive ; draw two diameters perpendicular to each other, one ot 
 them I'nssing tiirouuh the given point ; then will the otiu r 
 diameter be tlie required projection. 
 
 lor, as tlie primitive (ACliD) passes through the pole (as C > 
 ot the ie([Uired projection (^l^), then conversely, the original 
 
ARTICLE 120. 
 
 :;i.> 
 
 is tliut 
 
 )()le ot 
 
 •section 
 
 at cii'ch' 
 
 »ce nor m 
 
 loter 
 
 CJ). 
 
 c.F.n. 
 
 te< 
 
 if iire 
 
 the pv 
 
 ive 
 
 unl- 
 
 it self- 
 
 ho pnai. 
 lor, one "1 
 the othii 
 
 C 
 
 lole (as 
 l\e origini 
 
 circle shall pass through the pole (0) of the primitive, and its 
 projection will be a diameter {AB). 
 
 3" If the given point /is neither in the 
 center nor in the circumference of the pri- 
 mitive ACBD ; draw two perpendicular 
 diameters (^Z? and CD) one of them, AB/^^ 
 passing tlirough f ; draw the chord CfF ; 
 and from F, lay off a quadrant FE ; also ■*' ^ 
 draw the chord CE^ cutting AB in e ; then 
 
 will the circle described through C, c, />, be the required pro- 
 jection. 
 
 For, assuming yl/7i to be the primitive and C its pole, tlion 
 r/ is the projection of a quadrant ^i^ (117), Now, assuining 
 ACBD tohe the primitive, since AfB passes through tlio 
 pole /of the required circle (CeD)^ the latter will pass through 
 the poles C and D of AB ; that is, the circle required will 
 pass through the three points (/, e and D. 
 
 Cor. Hence, the following method of finding the polo of a 
 projected great circle. 
 
 1' If the projection of <i gredt circle is a diaracter of tlir 
 primitive^ the poles of this projection are the extremities <>/ fhr 
 diameter 2^erpendicnlar to it. 
 
 2" If the given projection., Cel), is inclined to the primi- 
 tive ; draw the perpendlcnlar diameters CI) and AB ; throitqh 
 e, draw the chord QiV^^ laij o^ EF cqwd to a quadrant, dniv 
 the chord ¥C Intersecting AB in f ; then will the point {" hr 
 the pole of the given circle. 
 
 130. Prob. 3. To describe the projection of a small circle 
 about some given point as a pole. 
 
 Solution. 1" If the original small circle 
 is parallel to the primitive, whose center 
 is the pole of the original ; then draw 
 two perpendicular diameters AB mxd CD 
 in the primitive ACBD; lay off CF 
 ofjual to the spherical radius of the small 
 circle (v. g. 30°); draw DF intersecting 
 AB in JV; from 0, as a center, with the 
 
 
 Hi 
 
 
 n 
 
 m 
 
 LMm 
 
2U\ 
 
 SPHERICAL TllKJONO.MKTRY 
 
 li 
 
 •t 
 il 
 
 radius OxV. describe the circle J/.VP,and itwiil be the required 
 projection. For, C^iVis the projection of 6'/ (117). 
 
 2" If the original small circle is perpendicular to tlie primi- 
 tive, in whose circumference then is the pole (us C) of the 
 original ; lay off C/*^ equal to the spherical radius of the small 
 circle ; draw FG, tangent to (JF at F, and limited at G by 
 7-^^' produced ; then will the circle //T/i'^ described from G, as 
 a center, with the radius OF, be the projection required. 
 
 1^1. Cor. Hence, the following method of finding the pole 
 of a given projected small circle : 
 
 1" 1/ the small circle is concentric with the primitive, the 
 projected pole of the former is the center of the latter. 
 
 2" If the small circle, as EHF, is perjjendicalar to the pri- 
 mitive, its pole C IS the middle of the arc EC F. 
 
 IJiJi. Prob. 4. To describe the projection of a great circle 
 cutting the primitive at a given angle and through a given 
 point in its circnmference. 
 
 Solution. Let ACBD be the primitive, 
 and C the given angle and the given point ; 
 then draw the diameters 6'i>and .17/ perpen- 
 dicular to each other, and make L.(JCG~C^ 
 (v. g. 30o); and the circle CiS'/) described 
 from the center G, with the radius GC, will 
 be the required projection. 
 
 For, the distance between the poles and G is equal to the 
 measure of LA 6'^'= 6"'= .30-) (Art. 02). 
 
 1^*1. Prob. 5. To describe the projection of a great circle 
 that shall intersect the primitive and a given great circle at a 
 given angle. 
 
 Solution. Let ABDE be the primi- 
 t/'ve, BCE tlio given circle, and .1, C, 
 'le given angles (v. g. 70^ and 50'^ res- 
 pectively). About the pole of tb.e pri- 
 mitive, with a spherical radius 
 0^r=: A =70'^, describe the small circle 
 Imn, and about the pole F of BCE, with 
 
ARTICLE 123. 
 
 217 
 
 a spherical radius Fn=iC=^0°, describe the small circle 
 ?u 11 ; from either of the points of intersection, as n fur a pole, 
 describe the great circle ACD, and it will be the projection 
 required. 
 
 For, the distances of the pole n from the poles and i^^are 
 equal to the measures of L5.1C=:^1=70° 
 and L^C^=C= 50°, respectively (Art. 92). 
 
 mi 
 
 i.i' 
 
 mi 
 
 ■>-■'.' 
 
 
 m 
 
 i 5 K! 
 
 i.i 
 
SPHERICAL TRIGON /lETRY. 
 
 134. Spherical trigonometry is that branch of mathe- 
 matics which has for Hs object the solution of spherical 
 triangles,. 
 
 135. The six parts (sides and angles) of a spherical 
 triangle are so related that three of them being given, the 
 other three may be found by computation ; this computation 
 is called the solution of the triangle. 
 
 Note. In Avhat follows, the triangles treated of are assumed to lie oa 
 the surface of a sphere whose radius is unity, and each of their parts to be 
 less than 180°. 
 
 13G. IIelations between the functions 
 OF the sides and angles of a 
 
 RIGnT-ANGLED SPHERICAL TRIANGLE. 
 
 Let ABC be a spherical triangle right- 
 angled at B. Draw the radii OA, OB, OC ; 
 also draw the perpendiculars CF on OB, 
 FD on OJ. ; and join C and D by the' 
 straight line CD. 
 
 By this const., CDF, COF k OFD are right-angled As 
 The plane triangle CDO is also right-angled at D. 
 For, in ACDF, ^ CD'= CF^+FD', 
 
 and iiiAOFD, OF'z=0^'+FI)'. 
 
 Subtracting (6) from (a) , CD'- OF' = CF'- OD' ; 
 whence, CU'+OD' = CF'-\-OF\ 
 
 or CD'+OD'= CO'; 
 
 that is, CO' is the square of the liypothenuse in AC DO. 
 
 Now, let a, h, c z= sides opposite to angles ^4, B, C, resp. : 
 then, by the definitions of the circular functions, 
 
 CF =8111 a, OF=i cos a, CD = sin h, OD =: cos h 
 i.CDF=A, LCFD::=B,'uDCF=C, AO=BO=CO=l. 
 
 (a) 
 
f ■;; 
 
 ARTICLE 120. 
 
 1". In A67)F, sin CFD:C/)=smC/)F:CF. 
 or (above notation; 1 :sin/>=8inyl :siji k 
 
 whence, 
 
 and (:U, Frin. I) 
 
 2'. In AODF, 
 or 
 
 3'. In AODF, 
 or 
 
 and in ACBF, 
 or 
 
 whence, 
 
 or (since tan^I 
 
 i sm a 
 tana= -— ^.- = 
 
 sin fr = sin l> sin. I. 
 
 sin ('=sin h sin^\ 
 OD-OFcosDOF. 
 
 cos /> = cos <f ci»s r. 
 JJF=OFsm/)OF, 
 DF:=i'i)s ii sin c, 
 CF=DFUiuCDF, 
 
 sin a=DF Uu\A ; 
 sill r^ ta!w/ 
 
 210 
 Art. :{5 
 
 (1) 
 
 (2) 
 
 :}i. Prill. :j 
 
 :il, Vriii. 2 
 ;5l.r»riii.4 
 
 cot^l-' 
 and (31, Prin. 1) 
 
 40. In AODF, 
 
 or 
 
 and in ACDF, 
 
 or 
 
 DF cos <i sin <• sin r 
 
 sin r=tan « cot^-l ; 
 
 sin («=tan r cot/'. 
 DF=I)OtanJ)OF, 
 Z>Fz=ci>s A tun <•, 
 J)F= DC cosCDF, 
 
 cos i tan c^sin h eosA ; 
 
 (4) 
 31. J^riii.4 
 31. Prin. 3 
 
 whence, 
 
 / • cos 6 , ,. 
 or (since =cot 6) 
 
 cos.l = 
 
 cos l> tan (• 
 
 sin h 
 cos /I = cot /; tan c ; 
 
 'Cos6'=cot h tan a. 
 
 
 sin c 
 and (31, Prin. 1). 
 
 5'^. Equating the values ofi>/Mn 3' and 4'. there obtains : 
 cos a sin c= DC GOs(JDF=Hm h C()S.4 : 
 
 cos (( sin c 
 
 whence, 
 
 cos.l = . 
 
 sin b 
 
 Substituting, for sin c, its value from (2), there obtains : 
 
 cos.l=-.cos (t sin 6'; (8) 
 
 and (31, Prin. 1), cosr'rrcosf sin J. (<>) 
 
 G". Mult. (4) by (5), sin a sin r = tan (t tan r cotA o,otf\ 
 Dividing both members by tan a tan r. and reducing by the 
 
 principle that ' — (=8in _.l)=cos, there obtains : 
 
 tan sin 
 
 cos a cos c=cotil cotC. 
 
 iivH 
 
 wm 
 
 Vf] 
 
 '48 
 
220 
 
 SPHEKKWL TRKjOXOMETRY. 
 
 Finally, .suljstitutiiig, for ro.sr^ (mssc, its value jiiven in (3): 
 
 c().s/> = eot.l cotC. (lO) 
 
 The above formulae, from (1) to (10), cover every possible 
 case that can occur iii a riuht-anded spherical trianule. 
 
 Their combinations, wliich are too various to be easily 
 remembered, may be reduced to two simple rules by ineans of 
 
 coC 
 
 NaI'IEU's riRCULAK PARTS. 
 
 1^7. If, in the annexed right-angled 
 triangle ABC, the right-angled B be over- 
 looked, the sides a and r, with the comple- 
 ments of the hypothenuse (co. h) and of i^r"""""^ ^JB 
 the oblique angles (co. ^1, co. C), are called "^"^""^^ 
 
 Napier's circular parts. 
 
 138. When any one of these five circular parts is taken as 
 a middle part, the two parts which are immediately adjacent 
 to it, on the right and the left, are called adjacent parts, and 
 the other two (each of ^yhich is separated from the middle by 
 an adjacent part) are called opposite parts. 
 
 Now, bearing in mind that sin.l = cos(co. ^l), co8^1 = sin(co.vl) 
 tan-4 = cot(co. ^-1), sin h = eos(co. h), cos h = sini co. b), &.c. 
 (Art. 2()), we will eas jly ascertain that 
 
 1" the above formula) (1), (2), (8), (3) and (9), in which 
 the first members are the sines of the successive circular parts 
 (a, r, CO. .1, CO. h, co. C) of ABC, may respectively assume the 
 following forms : 
 
 sin a = cos(co. h) cos (co. A), (a) 
 
 sin c =: cos(co. />) cos (co. C), (b) 
 
 sin(co. ^4)=: cos a cos(co.C), (c) 
 
 sin(co. h)=. cos a cos c, (d) 
 
 sin(co. C)= cose cos(co. ^4). (e) 
 
 Comparing each of these formuhx; with the figure; we will 
 
 also ascertain the correctness of 
 
 1L*/iO* Napier's 1st Rule. The sine of the middle partis 
 equal to the product of the cosines of the opposite parts. 
 
 wl 
 
ARTICLE 131. 
 
 221 
 
 3): 
 (lO) 
 lossible 
 
 easily 
 eans of 
 
 taken as 
 adjacent 
 irts, and 
 liddle by 
 
 in (CO. -I) 
 6), &c. 
 
 in which 
 lar parts 
 lume the 
 
 (a) 
 
 (c) 
 
 (d) 
 
 (e) 
 
 we will 
 
 part is 
 
 is. 
 
 2' The above formula} (5).(4),(r)).(10) and (7), in which the 
 first members aro the sines of the successive circular parts 
 (a, c, CO. -.1, CO. b, CO. C) of AH(\ may respectively a.ssume the 
 following fornis : 
 
 .sin <i =z t\\\ (' tau(co. //), (f*) 
 
 sin r = tan a. tun(oo. A), {g) 
 
 sin(co. .1) = tan(co. A) tan c, (li) 
 
 sin(co. J>)-^ tan(co. .1) tan(co. C), (k) 
 
 sin(co. ^')= tan(co. b) tan tt. (1) 
 
 Comparing each of those forniulio with the figure, we will 
 
 again ascertain the correctness of 
 
 lliO. Napier's 2nd Rule. TIk sinr nf thr middle part 
 is vquiil to the product of tJir tangents of tlir adjacent parts. 
 
 Sch. These circular functions of arcs or auirles. to radius- 
 unity, may be converted into functions of arcs or angles to any 
 radius /iJ, by multiplying by 7^. the middle part, that is the 
 first member of each equation, from (tl) to d) inclusively 
 (Art. 24). 
 
 131. 
 
 Solution of rigiit-anoled 
 Spherical Triangles. 
 
 To solve, by means of Napier's Rules, any question that 
 may be proposed In right-angled spherical trigonometry, pro- 
 ceed as follows : 
 
 Out of the three circular parts concerned in the ((uestiou 
 (the given two and the required one), take such a middle part 
 that the other two .shall be equidistant from it ; that is. .shall 
 be either both adjacent or both opposite parts. The })art 
 required will then be found by one of the above Napier's 
 Rules. 
 
 Ex. Given <i and xl : to find />, c and <\ 
 
 Solution. 1". To find />, the middle part must be a ; be- 
 cause, it is the onlv combination between a, ^l and l< that 
 admits of a middle part a and ecpiidi.stant parts co. A andco. b, 
 which, in this case, are opdosite ; 
 
 I y 
 
 
 : ''■''^! 
 

 1 
 
 ooo 
 
 i 
 
 SPHERICAL TRIOONOMETRY 
 
 tluMi ( 1st Rule), sin u. ■-= co8(co. h) eon (co. .1) ; 
 
 sill ft 
 
 wlience, 
 
 sin /> 
 
 sin J 
 
 2', To Hiid c, the uiid<lU> jmrt i« obviously <•, and the equidis. 
 tiint parts aro a and vo.A ; 
 then (2nd llule), sin c = tan a tan (co. .1), 
 or sill c -- tan a cot.l. 
 
 ;{ . To find C, the middle part will be co. .1, and the ecjuidi.- 
 tant pirts, a and co. (■ ; 
 
 then (1st Rule), sin((;o. J)= cosft co8(co. C); 
 
 COS.! 
 
 whence, 
 
 sin 6' = 
 
 cos (I 
 
 13^* The solution of a right-angled spherical triangk 
 admits of six cases. 
 
 Case I. 
 
 Data. B= 90^, side a and opposite angle A. 
 Solution. By Napier's liules (131, Ex.), or the formulae 
 (1), (4) and (8) of Art. 126, we have : 
 
 . , sin a 
 sm = , 
 
 sinJ. 
 
 sin c = tan a coiA. 
 
 . ^ (iosA 
 
 sin 6= . 
 
 cos a 
 
 Case II. 
 
 Data. J5=90°, side a and adjacent angle C. 
 
 Solution. By Napier's liules, or the foruiulae (7), (5) and 
 (8) of Art. 12«i, wo have : 
 
 eosC 
 
 cot b : 
 
 tan (( 
 
 tan c=sin (t tan (7, 
 eosyl = cos <i sinC 
 
 Casy III. 
 
 Data. 7i=90°, hypothenuse h and either adjacent angle, 
 as A. 
 
[ui<liH- 
 
 [uidiij 
 
 ,rianglf 
 
 'ormulae 
 
 (5) and 
 
 angle. 
 
 ARTKLK 132. 
 
 223 
 
 Solution. By NapuM-'n Uulus. (.r the formula) (1), (C) and 
 (10) of Art. 12G, wo have : 
 
 .sin(t = sin h sinvl, 
 tan entail h con A, 
 c«»tC'=cos h inn A. 
 
 Case IV. 
 
 Data. ii=90^, hypotlionuse h and either side (as a) of ^. 
 Solution. By Napier's llules, or the formulae (3), (1) and 
 
 (7) of Art. 126, we have : 
 
 cos h 
 
 cos c=. 
 
 sinyl = 
 
 cos a 
 sin a 
 
 sin 6 
 cosC=tana cot b. 
 
 Ciise V. 
 
 Data. ^=90°, and both sides, a and c, of B. 
 
 Solution. By Napier's Rules, or the formulae (3), (4) and 
 (8) of Art. 120, we have : 
 
 cos b = cos a cos c, 
 
 cot.l = 
 
 sin c 
 
 sinC= 
 
 tan a 
 cos^l 
 
 cos a 
 
 Case VI. 
 
 Dat; . 7i=f)0 , and both oblique angles A and C. 
 
 Solution. By Napier's llules, or the formulae (8), (10) and 
 (9) uf Art. 120, we have : 
 
 cos .4 
 
 cos a = , 
 
 sin 6' 
 
 cos i= cot J. cot C, • 
 
 cos 6^ 
 cosc = 
 
 sinA 
 
 
 i'> 
 
 ^",3 
 
 tU 
 
224 
 
 SPHERICAL TRIGONOMETRY 
 
 ip 
 
 POLFTION OP QUADRANTAL SPHERICAL TRIANGLES. 
 
 1J^*>. A quaurantal spherical triangle is a spherical 
 triangle in which one side is a quadrant. 
 
 134. To solve a quadrantal triangle, proceed as follows : 
 1" Subtract each side a,id ande of the quadrantal triangle 
 from 180° ; the result will be its polar triangle (99). 
 2" Solve this right-angled polar triangle as above (1/^2). 
 3' Subtract each part of the riecht-angled trianu'e thus solved 
 from 180^ ; the result will be the required solution of the qua- 
 drantal trianirle. 
 
 lUii* An isosceles spherical triangle may be solved by 
 solving one of the e(jual right-angled triangles into which it 
 is divided by the arc drawn, from the vertex, perpendicular to 
 the base. 
 
 130. Theorem. 
 
 The sines of the sides of n spherical triayiglc are propor- 
 tional to the sines of their opposite angles. 
 
 Hyp. Assume ABC to be any spherical 
 triantrle, and let 
 
 a, Z/, c=: sides opposite to A, B, C ] 
 AsT. then will 
 
 sin A sin/i sin 6^ 
 sin (/. sin b sin c 
 T)em. From any vertex, as C, draw an i>i*c 
 CD perpendicular to the opposite side AB or AB produced, 
 and let p — CD. 
 
 By equation (1) of Art. 12(5, we have in the first figure : 
 
 sinji) =rrsin h sin^l, 
 sin^) = sin a ^mB ; 
 siii^jr=:sin b sin..4, 
 sinp = sin a smCBD. 
 sin CBD = sin ( 1 80 ' — />') = sin B 
 sin /) - y\n a sini)' ; 
 sin b sin. I ::= sin a mnB, 
 sin /> sinr=::sin<' auxB. 
 
 sin.l sin/> sin (7 
 
 (1) 
 
 and in 2nd fig, 
 
 But, 
 hence, 
 that is. 
 Likewise, 
 
 Therefore, 
 
 1 26. :i 
 
 in both ciiso.- 
 
 sm a 
 
 sin b 
 
 sin c 
 
ARTICLE 138. 
 
 225 
 
 137. 
 
 Theorem. 
 
 In a spherical triangle, the cosine of any side is eqnal to 
 the product of the cosines of the other two sides, plus theproditct 
 of the sines of these sides and the cosine of tha included adyle. 
 
 H\ P. Let ABC be any spherical triangle ; 
 AsT. then will 
 
 cos a = cos b cos c4- sin b sin c cosvl. 
 Dem. From any vertex, as C, draw an arc 
 CD perpendicular to the opposite side AB, (jr 
 Ali produced, and let 
 
 p — CD. m = AD, and n — l)B. 
 
 ill /\^UCD. COStt =:C0S^; COS i< =C(>S^) cos(c — 
 
 But. cos(c — v?i):zzCOsr cosyM-(-sinc sin???- ; 
 lu'iice, cosft := cos^>(cosc cos7?i-|-'^^nc sin7?i)' 
 
 \n^ADC, cos^ tan??i msin/j cos/1, (6) 
 
 uul cos/v = cos^) cos?ji ; 
 
 cos/> 
 
 ■ m). 
 
 1-2G, (3) 
 32. (>/) 
 
 i«) 
 
 12G. (3) 
 
 V 
 
 whence oos^ 
 
 Ci)S)?l 
 
 Introducing this vahu' ()t"c<)sy) in (rt), and reduchig, 
 we have eosa :=: eash cosc-psinc cos/> tan??i. 
 
 'riieret'ore(6), cosffc =^ cos/> cosc-f^''i'' ^^i"^' cos.l, 
 Likewise, .;os6 = cos^' cosf-j-=^'"" siiw eos/i. 
 
 •ciiid cost; :zr cosa cos^j-fsiiuf sinA votiC. 
 
 (1) 
 (2) 
 
 (J5) 
 
 4 
 
 ■}'\i 
 
 "m 
 
 i:icS. 
 
 TllKOnKM. 
 
 /// II .s/)lit'ric<iJ trianyh'. I'hc '^o.'-ittf of mnj angle is equal to 
 thr pradiirf of the sines of the otJicr tu-n angles and the cost; ■ 
 oi /lii'ir included side, minus '//< product of the cosines of these 
 anqles. 
 
 \\\\\ Let .l/>'(' he any spliorical triangle ; 
 
 As'i'. then will cos J sin /»' sin < ' cos f/ — ('os/?eosC 
 
 l>E.M. I.ct a'. I)'. (•' sides opposite tit angl(;s .1'. />'. (■', in 
 the polar ti'iaiiglc of MiC : then (Art, '.lit). 
 
 ,, .^. 1 su° — . I , />r- ISO'— />". r = \m' — r\ 
 
 
 15 
 
il; 
 
 226 
 
 SPHERICAL TRIGONOMETRY 
 
 Introducing these values of ^-1, a, kc. in (1) of Art. 137, 
 we have — cos .1'= cos/?' cosC — sin /J' sin<7' cos a'. 
 
 But, this equation remains true when .1' — .1, a' = a, &c. 
 Therefore, omitting tlie primes, and changing th 
 we have : cos^l=sinL' sin (7 cos a — cosi> cosC. 
 Likewise, cosZ? = .si!i.l sinC'cosZ* — cos.4 cosC, 
 and cos(7=sinyl sini? cos c — cos A cos/?. 
 
 The formulae (I), (2) and (3) of Art. 137 and 138 are not 
 suited for the use of logarithms : they may however be trans- 
 formed into others well adapted for that use, by the following 
 process : 
 
 Formulae of half arcs and half angles op 
 
 SPHERICAL triangles. 
 
 COS a — cos b cos c 
 
 signs, 
 
 (1) 
 (3) 
 
 139. From (1) of Art. 137,cos^l 
 Subtracting each member from 1, 
 1 
 
 sin b sin c 
 
 . sin b sin c-fcos b cos c — cos a 
 cosJ.= Z , . 
 
 sin b sin c 
 
 in} 
 
 (e) 
 
 (f) 
 (g) 
 
 Letting 2x=A, in (^)of Art. 33, 
 
 t hen will 1 — cos^ = 2 sin--|.4. 
 
 From (d) of Art. 32, sin b sinc+cos& cose— cos(fe— c,) 
 
 Introducing the second n;embers of (t) and (g) in (e): 
 
 sin b sin c 
 Again, letting x~p, and {b—c)= q, in (n) of Art, 34, 
 we have cos (b—c) — con «=2 sin-i((t + 6— c) sin^(a + c— 6); 
 
 hence, 2 .in^q^-^ «"'K^< + ^;-Osin^(a + c-?y) ^. 
 
 sin b sin c 
 Finally, letting a-\-b-l-c-^2s, then will a + b—c = 2{s—c), 
 and a-\-c—b-^2{s—b), hence, we shall have from (ki: 
 
 8m'^A = ^"'('^•--^) si" js — e) ^ 
 sin b sin c 
 
 Likewise, sin^^^i^zr: ^^"(^-^O ^] !l(lrz^, 
 
 sin « sin c 
 
 _sin(s — (f ) sin(s— 6) 
 
 ain'^C 
 
 sin a sin b 
 
 (1) 
 
 ^3i 
 
AUTKi.r, 142. 
 
 227 
 
 140. If we add 1 to ouqIi member of [n) of Art. 139, and 
 proceed as in that article, we shall obtain : 
 
 cos 
 
 .jj J sin *• sin(.s'— <i) 
 
 sin I) sin c 
 
 sin s sin(.s — b) 
 
 sin a sin c 
 
 sin s sin(.s — c) 
 
 cos-^(7: 
 
 (1> 
 
 (3> 
 
 sin a sin h 
 
 141. Dividing (1), (2) and (8) of Art. 139 by (1), (2) and 
 (3) of Art. 140, respectively, we shall obtain : 
 
 sin(s — h)&m{s — c) 
 
 tan^l^ 
 
 sin s sin(s — a) 
 _ sin(s — a)8in(s — c) 
 
 sin« sin(s — 6) 
 
 sin s sin(s — c) 
 
 142. From(l)of Art.138, cosa=5^!^«-^+^^-'i. 
 
 sinii sin 6 
 
 Subtracting each member from 1, we have : 
 
 - sinjKsinC — cosZ^ cosC— cos^l 
 
 1— cosa= . . . . ^ 
 
 sin/j sin 6 
 
 Since (139, /), 1 - cos a= 2 sin^^ii, 
 
 and (32 o).^ h'vuB smC — cosi? cosC=: — cos(5 4-C), 
 
 (1> 
 (2) 
 (3). 
 
 (P) 
 
 if we introda^' tlie second members of (r) and (s), in (p),. 
 
 , o • I cos(7>'-|-6')+cosvi 
 we have : 2 sin* ha= — — -^ — -^ , 
 
 s'mB sin(7 
 
 or(34,Z) 2sin^'ia=-^Mt:?±_9_«««^:^^t?:=li). (t) 
 
 sin/i sin 6' 
 
 Lettin,' ^ + 2?4-<7=2>Sr, then will B+C-A=2(S-A); 
 
 , . .,1 — oosaS^ cos(*V — A) 
 
 whence sin--*a= ^ 
 
 Likewis<f, 
 
 sin/i sinf 
 
 . .,,, — cos»S*cos(/? — B) 
 
 .qn-^6 = ' i ' n 
 
 sin^l sin 6 
 
 ,jin^4 sin.fi 
 
 (1) 
 (3) 
 C3) 
 
 '■4 
 
 I i - - Isi? 
 
228 
 
 SPHERICAL TRKiONOMFTRV 
 
 14:^5. By a process analotrous to the preceding one, we may 
 
 obtain the following: formula? 
 
 cos4rt= — — .- . - '-. 
 
 sinB s'lnC 
 
 eos(.S-r)cos(>S'-^) 
 
 cos-^o — - 
 
 sin 6' sin xl 
 
 cos-^c 
 
 _cos(S—A)iio»(S~B) 
 sinA sinii 
 
 (1) 
 
 (2) 
 
 taii-T," = 
 
 os,S'co.s(>S'— .1) 
 
 144. Dividin- (1), i2) and (li) of 142 by (1), (2) and (3) 
 of 143, respectively, we sliall have : 
 
 (1) 
 
 tan-'J,/; 
 
 taii-vlr 
 
 eoh , ')cos (S—C) 
 
 .(•OS (*S'— .l7co.s('^'— <')" 
 
 — (•i>sXcos(/S'— T) 
 
 CDS ( X .1 1 COS (,S' />) 
 
 Sch. These t'oviimliv. frcui Art. i:>!lto Ai't. 1 U iiielu.^ively, 
 irive sines, c(isiii:'.> Miiil r:iiiii'eiits rcfcn-cd to ;iii<i'les ar.d ai'cs 
 wliose radii art' iii!it\ . i'u re icr i hcsc riiiictioiis to arcs whose 
 rndii are any (jiiantity A', tlics'^ 'oiid iiicmlx'r of each formula 
 must he multi|ilie(l hv A'-. 
 
 I4.">. 
 
 Xaimki'.'s .Vnai, 
 
 Substitutii 
 
 !'• t lie >eco 
 
 f 
 
 !l(l liieiul 
 th 
 
 H'T o| ei 
 
 luat ioii ( ;; ) of 1 
 
 or cos r. Ill equation ( i ) oi the same article; ami. in the 
 
 I'or cos-o. we shall have, after 
 
 result, suhstitiitin ^ I — sin-/> 
 
 transposin;^ the tir-t memher to iii" second 
 
 (l= — cos 
 
 7 >iir/< -- sin 1/ sm A m: 
 
 or. dividini:' ]>\ — >\\i h sin r. ;ind t 
 
 cos J 
 
 h~ 
 
 I, cosr 
 
 ,i!isiMi.-in'::' co.- 
 h cos^ 
 
 «in h sill <■ CO,- 
 
 cos II sill It — Sill " co> l> ( 
 
 sill (• 
 cos /> sin r/ — sin /> cosf/ cosT 
 
 sin r 
 
 Jiikewise. cos/V= — 
 
 Addin-' (1 ) and (2). 
 and sin(je sin '^ cos />-f-co.s ^/ sin /* = siii(^/ -j-''' ) 
 
 then will cos.l -f-cos/^=( 1 — cosT) 
 
 sin 
 
 ( // + /«) 
 
 sm c 
 
 (3 
 
 »» 
 
 ^- 
 
 .VI, \n 
 
 • » 
 
 (•>) 
 
From 
 
 ARTICI,F, 145. 
 .sin.t:sin/)' = sin («:sin A, we deduce 
 
 by composition, sin /I +sin7i= ''.■-' (sin a -f sin 6), 
 
 sin a 
 
 by division, sin^#— sinZ?=:l. -(sin fi — sin h) 
 
 sin (I 
 
 229 
 
 (4) 
 (5) 
 
 Dividing (4) and (5), in succession by (3), 
 
 and writing 
 
 sin^"* n sin.i 
 for . 
 
 sin c 
 
 sin a 
 
 1 sinyl-4-sini5 sinf. sin a + sin j ,^v 
 
 we have 1 — = ^- x — ; ■ (o) 
 
 cos J. -f cos /i 1 — cos6' sin (a -i- L) 
 
 , sin^i — sinZi sinC sin « — sin i .«,v 
 
 and — — = y-X (7) 
 
 cosvl + cos/^ 1— cosC/ sin(a-f-i) 
 
 Dividing successively (k) and (ill) by (1) of Art. 34, and 
 then (k) by (ll); and in the quotients, letting ^ = ^1, and 
 q~B, we shall have respectively : 
 
 &\\\A-\-i^\r\B 
 cosJ.-|-cos^ 
 
 sin.l — sin/? 
 cos.4^-cos/i 
 
 sin.4-f«ii^-/> ^\ / A L>\ 
 
 = tan?;(.4 + i?). 
 
 (8) 
 (9) 
 (10) 
 
 cosy? — cos^4 
 But, equation (lO) remains true when A=C, and^=Oj 
 
 hence, 
 
 sinT 
 
 = cot^6''. 
 
 (11) 
 
 1 — cosC' 
 
 Introducing the second members of (8), (D) and (ll), in 
 (6) and (7), respectively : 
 
 tiin^{A-^B) sin ft 4- sin /> • /1o^ 
 
 - . .— ,r , (A/w> 
 
 sin(a-|-o) 
 
 and 
 
 Now, 
 and, 
 
 Dividini:, 
 
 cot^6' 
 taiivU J — /?)_sin (I — sin A 
 
 (13) 
 
 cot^C sin(ff--f/') 
 
 sina-j-hl -. 0-— 2siiiA('«-f/v) cos^((? — A), ?A, (k) 
 
 .siii(ff-f />) = L'siiiJ.(a + />) cos^(ce + /|). 33, (a') 
 
 sin a-fsin 1/ c()s-^-(a — />) 
 
 sin(f/-|-/^) cosT>(a-t-/-') 
 
 7 'Hi 
 
 
 1 
 
 
 m 
 
Ill 
 
 w. 
 
 m 
 
 230 
 
 SPHERICAL TRIGONOMETRY 
 
 Introducing the second membe" of this equation in (12), 
 
 tan^(.l -f B) _ cos^(a— 6) 
 
 (14) 
 
 cot^C cos^(« + 6) 
 
 By a similar process, equation (13) will be reduced to 
 
 tan^(^— .g) _ sin|(a— h) .^^. 
 
 Finally, if the formula) (14) and (15) be applied to the 
 polar triangle, by the process followed in 138, we shall have : 
 tan^(a-|-&)_cos^(^4.— iJ) /"Ifi^ 
 
 tan^c cos^{A-\-B) 
 tan^{a—h)_8{nl{A—B) 
 
 (n) 
 
 tan^c sin^(^l4-if) 
 
 These formulae (14), (15), (16) and (17) are called the 
 four Napier's analogle ). 
 
 146. Solution of op'.ique-anqled spherical triangles. 
 
 The solution of oblique-angled spherical triangles admits of 
 six cases. 
 
 Case I. 
 
 Data. The thrte sides a, b, and c. 
 
 Solution. Letting ^((t-\-b-j-c)t=.H, we have by Art. 140 : 
 
 .,-, , sin s a'mfs — d) 
 
 cos- 
 
 sin h sin c 
 
 .,■, !■> sin s sin/s — h) 
 
 cos'W= 
 
 sin a sin c 
 sin s sin(s — c) 
 
 siu a sin b 
 
 Case II. 
 
 Data. Two ndes, as a, b, and LA opposite to a. 
 
 Solution. From (1) of 136 and (17) of 145, we have : 
 
 sin b . 
 
 sini?=sinYl 
 
 sin a 
 
 ta„Jc=ta„4(<,-6)-e'414+^. 
 
 sin^(^4 — B) 
 
 sin C= sin J. 
 
 sin c 
 sina 
 
ARTICLE 146. 
 
 Case III. 
 
 Data. Two sides^ as a, b. and the included angle C. 
 Solution. From (14) and (15 of Art. 145, we have 
 
 ' cos^(a-|-6) 
 
 tani(.l-i?)=tanA^=cotiC-^^^^l 
 
 A = J/4- iV; and B= M— .W 
 sinC 
 
 231 
 
 sin (;=sin u- 
 
 s'mA 
 
 Case IV. 
 
 Data. The three angles A, B and C. 
 
 Solution. Letting ^( A i-B -{-€) — S, we have by Art. 14o 
 
 ^ siu/j sint' 
 
 sm.l sinC. 
 
 cos 
 
 a— cos(*S'— .4) cos (S—B) 
 
 \^C : j ; 
 
 s\uA sin/j 
 
 Case Y. 
 
 Data. Two angles, as A, B. and side a opposite to A. 
 Solution. From (1) of l?A> and (17) of 145, we have : 
 
 sin /> = sinfi 
 
 8in /,' 
 sin.l 
 
 ^ 1 ^1 ;,sin-\f.4 + Z^) 
 
 tan*c=tanA(a — 6;-—= _'. 
 
 "^ ^^ 'sin^u4 — 7i) 
 
 sinC=sinxl 
 
 sin c 
 
 sina 
 
 Case \ I. 
 
 Data. Two angles, as A, B, and the included side c. 
 Solution. From (16) and (17) of Art. 145, we have : 
 
 t;ll 
 
 
 
 ii 
 
 ■,H''-1- 
 
\r 
 
 i; 
 
 f' 
 
 232 
 
 147. 
 
 SI'IIKUICAL TllRiOxNOMETRY 
 
 tani(a + />) = tanm = tanJr^'Mir-_?l. 
 
 tanl(« - 6 ) = tan n = tan ir '"^MiZj^l. 
 (t=wj-j-?;, and h = m — n. 
 
 win6 = .sin.i -;- . 
 aina 
 
 PRACTICAL APPLICATIONS. 
 Areas of spherical polygons. 
 
 Prob. 1. To find the area (A) of a triangle^ in which 
 LA = ()2°, LB = 100° and LC = 58^ on a sphere to radius 3. 
 A=-R'^ A+ B+ C -]8(i" _^^ 62°-fino°-f.58°-lR0° _^_ 
 
 18(1° 
 
 lSi-° 
 
 (107, p) 
 
 Prob. 2. To find the area (A) of a, regular octagon^ in 
 which each angle is 140°, on a sphere to radius 3. 
 
 A=Iii!i|S— (n-2)18()' 1=9^(8X140— Gx180)=2t. (107, q) 
 
 148. Construction and Solution 
 
 OF right-angled spherical triangles. 
 
 Prob. 1. Given the side a = 30" 1.5', and the opposite angU 
 A zr: 48'' 30' ; to construct and i^olre the triangle. 
 
 Construction. Let .l/l/^/Y/be tlie pvi- ^ 
 
 niitive. Describe the circle J CA', making „ 
 L llM'zzz A — 4S° 30' (Art. i -l-l)] about 
 0^ a.s a pole, with a spherical radius €r\ 
 0/7 = 50'' 45' (CO. <?), describe the sn;all 
 circle CII (Art. 120, 1"); tlirough the 
 intersection C\ draw the dianu'ter BF; 
 then will ..47>Cbe the triangle r':'((uired. 
 
 Solution. Applying logarithms to the formulas : 
 
 -, • 7 sin " o • ^ i 1 M • /f cos. I 
 Jo gin l) z=z ; J" Sin c = tan a cot.! ; .) ' sinf = 
 
 sin ^4 ensrt 
 
 of 
 
, which 
 adius 3. 
 
 (107,11) 
 agon, in 
 
 1 (Ki^q) 
 
 ite anglt 
 
 s.l 
 
 ARTICLE US. 
 
 
 2'\logtaim(30°15') = 9.7r.r)S0r) 
 log cut J (48^30') = 0.04080^^ 
 
 hence.c- =log sin~ ^ 9.712013 
 — 31° 3' 41'' or 148° 50' 31 '. 
 
 )S 
 
 lt^ (I 
 
 of 
 
 Art. 132, Case I, wo shall have 
 
 l<',lo<;siiia(3(P15') = !).7()223() 
 colo;,' sin .4(48^30 ') = 0. 12r)544 
 
 hence, />=log sin ~ ' 9.827780 
 =:42»1G' 15" or 137° 43' 45 ". 
 
 3', logcos.4(48° 30')=: 9.821205, 
 cologcos rt(30° 15')= 0.003509 ; 
 
 hence, C = logsin" ' 0.884834 = 50-5'29" or 129054'31". 
 
 Sch. 1" The solution of the preceding problem gives a double 
 value to each of the quantities h, c and C, because the sine 
 of an arc or angle is also the sine of its supplement ; hence, 
 either of the triangles ABC or EBC fulfills the conditions of 
 the problem. In general, there will be a double solution of the 
 problem, whenever a sought quantity is determined by its sine. 
 Cases, like the above, in which any quantity is determined l^y 
 its sine, and which thereby admit of a double solution, are 
 called ambiguous cases. 
 
 But, if the triangle proposed for solution has been cons- 
 tructed by the rules given in Art. 118-123. it will always bo 
 easy to know which of the two triangles is to be taken. Foi 
 instance, if it were known that anyone of the three <ju;n,- 
 tities : side AB, side .IT or l.ACB is less than 9(P, then it 
 would be known that the triangle .1Z?C' alone could satisfy tlie 
 conditions of tlie problem. 
 
 2' Two parts of a spherical triangle are said to be of the 
 same species when they are ciicli greater than i>i)' or each !e>s 
 than 90°. Thus, in A^UIC\ L ^.' and its adjacent .^ide C/iareof 
 the same species ; in l\EBC. L. ^'and its a<ljacent side CBaxh 
 of different species. 
 
 Prob. 2. (iire)i the sidr ;i =:r 1 :',»h' 1 <»'. ,nnl (he aifjaeml 
 LC = 130° 10' ; tu coHStruet and solec the triamjle. 
 
m 
 
 234 
 
 SPHERICAL TRIGONOMETRY 
 
 Construction. Draw the diameter BD 
 and lay off B(W=a = UO'' 10'; make 
 l_^6'.l=49^ 50' (supp. ofLC"; and des- 
 cribe the circle CAF {Art. 122); then will 
 ABC be the triangle required. 
 
 Solution. Applying logarithms to the 
 formuloD : 
 
 1<', coti = ; 2", tanc=sinrt tanC; 3", cos^l=cosasin(7, 
 
 tana 
 
 of Art. 132, Case II ; and since, 2" and 3'» give negative 
 results, we have : 
 
 l",logcosC(130nO') = 9.809569 
 cologtana(130°10') = 1.926378 
 hence, h=z\os cot ^ 
 
 = 61° 26' 4". 
 
 9.735947 
 
 2'',logsina(130°10') = 9.883191 
 log tan (7il30°10')=l 0.073622 
 
 ,-1 
 
 hence, c=180°-log tan '9.956 713 
 
 = 137° 50' 39". 
 
 3", log cos «(130°10')= 9.809 569 
 log sin(7(130nO')= 9.883 191 ; 
 
 hence, ^ == 180'' — logcos"^ 9.692760 = 119° 28' 7". 
 
 Prob. 3. Given hypnthmnse b = SO^' 30', and LC= 50°50"; 
 to construct and solve the triangle. 
 
 Construction. Describe the great circle 
 CAF, making LyiCi5=(7 = 50" 50' 
 (Art. 122); about (7, as a pole, with a sphe- 
 rical radius Ci/=80° 30', describe the 
 small circle 7/^5(120, 2'); through the 
 intersection A, draw the diameter BE ; 
 then will ABC be the triangle required. 
 
 Solution. Applying logarithms to the formulae : 
 V\ sin c = sin 6 sin (7 ; 2'», tan a z= cosC tan b ; 
 3o, cot J.=cos b tanC, of Art. 132, Case III, we shall have : 
 
 l»,logsin 6(80°30') =9.994003 
 log sin 6'( 50° 50') = 9.889447 
 
 hence, c=logsin~~^ 9.883450 
 = 49° 52' 26" 
 
 2'>,logcosC(50°50')= 9.800427 
 log tan 6(80°30) = 10.776394 
 
 hence, rt=logtan"^ 10.576821 
 = 75"-^ 9' 36". 
 
 
ARTICLE 148. 
 
 
 ■ Vi- 
 
 3", log COS /^ (80° 30'i= 0.217000 
 log tun 6' (SO'' 5O')=:ln.0S0O40 ; 
 
 hence, Az^ log cot "" ^ 0.;}00r)5S= 78" 32' 47". 
 
 Prob. 4. Given the hi/potheuiisr b = US'' 25', and the 
 side a= G0° 40' ; to construct and solve the trianijle. 
 
 CoNSTRUCTio.N. Lay off Cli=ii — {\[)^\[y: and draw the 
 diameter BE, about r', as a pole, with a spherical radius 
 t'//=^*=G5° 2.')', describe the small circle //.4/i il20, 2 M; 
 through the intersection .1, describe the great circle CAF \ then 
 will ABC be the triangle required. 
 
 Solution. Applying logarithms to the formuhe : 
 
 ,„ cost ,1 . , sin a 
 
 l",cos c= : !•', sin.4== ; 3'., cosC=:tan a cot h, of 
 
 cosa sm h 
 
 Art. 132, Case I\ , we shall have : 
 
 l",lQgcos 6i65°2r)') = 0.G1011O 2'.!ogsin</i00°40'i =0.040400 
 cologcosa((»0"40') =:0.300902 | cologsin />,0rj^'2.j', =O.0412(J0 
 
 hence, c=logcos~^ 0.020012 hence,.! = log sin ~' 0.0S1G75 
 = 31*-^ 52' 30". = 73^' 28' 10". 
 
 3", log tan a (GO" 40')^ 10.250311 
 
 log cot h (05° 25): 
 
 hence, C = log cos 
 
 — 1 
 
 0.GG037G; 
 0.010087== 35^ 30'. 
 
 Prob. 5. Given both sides about the ri(jht angle : 
 a=: 40° 35', and C:=GO° 35'; to construct and solve the 
 triangle. 
 
 Const. Lay off.4J5=c = 60° 35', and 
 draw the diameter BCF ; about 0, as a 
 pole, with a spherical radius O//=z:50°25' 
 (CO. a), describe the small circle CII ^1 
 120, 1"), and through the intersection C, 
 describe the great circle ACE, then 
 will ABC be the triangle required. 
 
 
 
 
 :^!' 1 
 
 
 
 .. 
 
 1 1 
 
 '^ 
 
 
 '' 
 
 1 
 
 
 m 
 
 ._.,: iai 
 
i 
 
 tit', 
 
 I 
 
 
 2 ',Iog sin (• I ()0<?35')=0.040()r)4 
 colog tan ai4(»°;ir)'i=0.(M;7222 
 
 230 Sl'lIEUICA?. TUIGONOMETIIY 
 
 Solution. Aj»plyiutrl<)garithin«i to the fonnuhc : 
 
 1", cos o z=c<>s a cuHc; 2", cot.l — ; 3", sin6 = , of 
 
 tan.l cos a 
 
 Art. 132, Case V, we shall have : 
 
 l",log cos a 40^:^5' i:::^0.<SSor)(ir) 
 
 log cos c,«10'3r)'i=!).<;<U22(> 
 
 hence, Z>=log COS ~ 9.571725 j hence,. l=:log cot"' 10.0()727() 
 =()S" 5' 54". I =44^' 31' 12". 
 
 :{',logcos.l (44'-' 31' 12")=:1).S53002, 
 colog cos a (4(1"^ 35'; — 0.119405 ; 
 
 hence, C = log sin""' 0.072587=:<)9« 51' 22". 
 
 Prob. 6. Oicoi both oblique angles : A=110° 20' 
 arul C = 30° 30'; to construct and solve the triangle. 
 
 Construction. About the pole E of 
 the diameter i^//, with a spherical radius 
 EF=- ■-— 30'' 30', describe a small circle 
 Fnm ; about the pole 0, with a spherical k[ 
 radius 01= (>!)' 40' (supplement of Lvl), 
 describe the small circle Imn ; about the 
 intersection n of these small circles, des- 
 cribe the great circle ..4(76^ ; then will ABC be the triangle 
 required (Art. 123). 
 
 Solution. Applying logarithms to the formula} : 
 
 1 cos. I ., , i. ( ./7 o cos ^7 
 
 1", c;>sa=-^ — ^^ ; 2", cos b ■= cot A cotC ; 3", cos c =^ 
 
 sin 6 
 
 >' 
 
 sin.l' 
 
 of Art. 132, Case YI, and since 1" and 2" give ncgatice results, 
 we shall have : 
 
 l",logcos.4ill0=20' 1=9.540931 2>,logcot^ilUr20' 1 = 3.508873 
 
 loy;cotC(30°30' 1 = 10.229852 
 
 colog sin C ( 30° 30' )— 0. 2945:n 
 
 hence,i=180''-lo,f?co.s 'J. 798 T25 
 
 = 128" 59 3" 
 
 be n ce .( t= 1 .'^ n °-l ogo os ! > . 8 : ', ,") 4 (J J 
 
 = 133° 12' 20". 
 
 3\ log cos6'(30° 30') ^9.935320, 
 
 colog sin .4 f 1 1 ' 2 ) = 0.027942 ; 
 
 ~' 9.903202=23^ 14' 10 
 
 hen 
 
 ce. 
 
 c=log cos 
 
ARTICLE 149, 
 
 2:J7 
 
 111) CoN.STRrrTloN .\NI> NKMKRICAL SoLKTIoN 
 (iV OULIQIF, AN(U.KI) SIMIKUICAI- TRIANGLES. 
 
 Prob. 1. Girnt f/ir tliirc sidrs : a=l:j(l^' 10', b = 70«> 20' 
 (•=:14()'' I>0'; fit (•(iiiatriict oikI anfrr flif triumjlc. 
 
 Const. Jiuy utf .1 /;=(;== IP)'- :{()'. 
 Mild tlraw the (liiinu'tcrs HI) mikI J A': 
 ubdiit />, as a pole, with a s|)lu'ricu! 
 iMilius 1J(m := Vy^ \)^)' (su]»i»l('m('iit ut' (1), 
 d('scri})0 the small cirelt> (i(JA, ami about 
 
 .1, 
 
 18 a imle. witli a .^ 
 
 th 
 
 Mjhcri 
 
 cal radius 
 
 AE--h 70^' 20', dcsi-vilic th»' siiialhire'c 
 I'JCF (\n. 120, 2) cutting r;."J in (' : thn.uuli />. (\ /;, and 
 J. r. A', describe the great cinKs DC/i and ACK ; then will 
 AHC be the triangle re((uii'ed. 
 
 S(»lj;Tl<tN. Apjtlyiiig logai'itliuis to the fVimiuUc : 
 
 COS'-'i^l: 
 
 itin .s sin(.s — <( ) 
 
 i 
 
 '1 ■. cos 
 
 -J,/^= 
 
 sin .s sin 
 
 (.s— /,) 
 
 Sin h sm c 
 
 sin (I sin (• 
 
 .) '. cos 
 
 :i r' 
 
 .sin .s sin(.s — c) 
 
 r= 
 
 sin It sin h 
 
 )l' Ait. 1 n;, Case I, 
 
 we S 
 
 hall 1 
 
 lave 
 
 1 ', 'ogsinsfCi":;!!') = u.'JlTon;) i l' , loj,' sin .•< (!.>^:!0') = g,:ji:')'';) 
 U.g,sin(.<— fr)(ln='.>i)') = 
 
 .811(101 1 lo-r sill is—h 
 
 r!)«r.o') = M.'-o 
 
 colog sin h (T(i''2i)') = 0.020! o:-< 
 fuluw ,-inc (3y°30') = o.l'J64S9 
 
 loii' t'u.- 
 
 
 
 
 H).LT)12i;2 
 
 
 lieiice. J=21()gcos 'J.02riG 
 
 =130 
 
 '■> .■,! On" 
 
 coloo; sin </ (40^50') = 'i.nr.Hog 
 colo.u- sin c (?.;)''20') = •». i:i»UH9 
 
 I't. ."24034 
 
 ;'.7G20ir 
 
 lO"' COS.', B: 
 
 ■!; 
 
 h('nc='. fi=2 loa: 
 
 co.= 
 
 = 109" 21' 48' 
 
 3<>, log sin,s(9 • 30';= !).217t;00 
 log sin(.s — c)(;>;>^ 31') = <».S03:)l 1 
 colog sin «(-!!)= 50' ) = 0.1 1 (;s09 
 coloK sin bdi)"-' 20' ) = 0.02(;103 
 
 lieiirc 
 
 loscos^(7= 
 
 ^"=21 
 
 (,'_>' ens 
 
 
 li>. hiK)32; 
 l).;>'.2!n(;:ir l:!.")' .")■ 24- ' 
 
 •■^i 
 
 L-^t 
 
 b,ti 
 
 
 - '' ''■^' i^^lMl 
 
 
 '• ;;tlral| 
 
 
 ■' ''li 
 
 
 
 ■'"11 
 
 
 , ".->M 
 
 
 
 
 
 
 "^ 
 
 
 ...-ii 
 
 ,.i|liW& 
 
 ' ^ 11 
 
 HBIH 
 
 ^ra 
 
 ^HppR 
 
 ' n 
 
 Bii 
 
 
 ^^In^V^^ 
 
 
 
 1 
 
 H^ 
 
 l|r''''l 
 
 1 ||g^ 'If 
 
 ' Ml 
 ui u 
 
 K'^1 
 
 i H 
 
 VmMi'-v^ 
 
 Prob. 2. ( 
 
 J I mi SI 
 
 di' a -00 
 
 O . SI I 
 
 Ir ] 
 
 1 ---- ( .)^ .».) . fO/'< 
 
 niiijli A-:=^50° 51'; lo canstract and snlrc tliv fruuujli-. 
 
238 
 
 Sni ERICA L TRIGONOMETRY 
 
 CONSTRCCTION. Lay off A C ='h=:7 b° ^d' , J^A 
 
 and describe the great circle ABD, making ^, 
 
 LC^1^=.1=:50° 51' (Art. 122); about C, 
 
 as a polC; with a spherical radius 
 
 r/^— (f -----( iO^' 25', describe the small circle G> 
 
 FB(t : throuiilr f'and the intersection jB 
 
 or B' . describe either the great circle CBE 
 
 or CB'E ; then will ACB or ACB' be the required circh). 
 
 Solution. Applying logarithms to the formulae : 
 
 V\ umB^smA ; 2 , tanAc=r=tani(a— 6) -— 4->— ' / ; 
 
 sin(t " " &m-^(A—B) 
 
 3°, sinC=sin.'l '- — ' of Art= 146, we shall have : 
 
 sin a 
 
 10, logsin.4(5n«'5r) =9.839r.r9 3\log siaJ(50«'5r) 
 
 log siti bCi^^'i5') =9.086104 
 colog sin a(60°25') —0.060661 
 
 bence, i>=logsin~^ 9.936344 
 =59<'4.3U9''. 
 
 logsinc(T0'^33M2") 
 colog sin 0(60^25') 
 
 =9.889579- 
 =9.974510 
 =0.080661 
 
 hence, C=logsin 9.924750 
 
 =122»45M9". 
 
 2'S log tan^(&_rt){7^ 35')= 9.124284 
 log sini,.4 -}- .^)(55'' 17' 25")= 9.914860 
 colog sin^(Z^-.4)(4° 26' 24")= 1. 111175 ; 
 
 hence, c=2logta^j~' 10.150319=109° 26' 38". 
 
 Sch. ^=59° 43' 49", and (7=122'' 45' 49" fulfill the con- 
 ditions of /^ABC ; the supplements of B and C would give 
 AAB'C. 
 
 Prob. 3, Gicen a=45^, b=75^ 20', and the included 
 
 LC=--40° 30' ; to construct and solve the triangle. 
 
 Construction. Lay off 6'i^=:=iizrr45'',and 
 describe tlie great circle CAD, making 
 l.BC A— Cr ■.¥)•' 30' (Art. 122); about (7, 
 as a pole, with a spherical radius CF=zh 
 =75^^ 20', describe tlie small circle FAG 
 (120, M'»); through B, E, and the intersec- 
 tion A, describe the great circle BAE ; then 
 will ABC be the trianglv; required. 
 

 4-B). 
 
 =0.889579 
 =9.974510 
 =0.080661 
 
 9.924750 
 9". 
 
 28' 38". 
 
 the con- 
 (uld give 
 
 included 
 
 ARTICLE 149. 
 
 239 
 
 Solution. Applying logarithms to the formulae : 
 1 0, tani( J. + iJ) = tan A/:- coti C '''l^f j''-Ilt- ; 
 
 9.0 
 
 , tan4(^— ^)=tanA"=eot^( 
 
 sini(a-f6) 
 
 whence, A=M-{-A\ and B=M—N, 
 
 3" sin c=sina "-"-?, of Art. 140, Case III, we shall have 
 sin. 4 
 
 hence, 
 
 2",log cot^C(20"15) =10.433068 
 log3ini(6— a)(15''10*)= 9.417G84 
 cologsini(64-a)(60°10^)=n.06l742 
 
 1 
 
 hence, i\'=logtan ' 0.012494 
 =39' 1 5' 59". 
 
 1", logCOliC(20°15^) =10.433068 
 Iogco3^(6— a)(15<'10')= 9.984003 
 cologco3^(&+rt)(60^10')=0. 303225 
 
 l/=logtan""' 10.720896 
 =V9«'14' 1". 
 
 Then, 5=jli'+iV=118^ 30', and .l=J/-iV=39° 58' 2" 
 
 3o, log sin a(45°)=9.849485, 
 log sin«7(40o 30')=9.812544, 
 eolog sin^(39°58' 2")=0.192229 ; 
 
 hence. 
 
 9.854258=45^38' 12". 
 
 c=log sin ^ 
 
 Prob. 4. Given the three angles : A=70" 30', E=110^ 20', 
 C=50*' 10'; to construct and solve the triangle. 
 
 Construction. Descrbe the great 
 civGhBCE, making LCi>'Z)=()9^ 40' 
 (supp. of 7?) ; about O as a pole, with a 
 spherical radius 0^=70'^ 30', describe the 
 snuall circle mn ; about F pole of BCE, 
 with a spherical radius = (7=50° 10', des- ^ 
 cribe the small circle linn ; about cither /> 
 
 of the intersections m, w, as n, describe the great circle ACD ;. 
 then will-4j56'be the triangle required (Art. 123). 
 
 m 
 
 ¥] 
 '\\\ 
 
 
 ''il 
 
 '' H 
 
 m 
 
J,«.t>i,i"l,,' .i/jH-i " 
 
 240 
 
 SPHERICAL TRIGONOMETRV 
 
 SoLUTlON. Applying loL'arithms to the formula) 
 
 COS^^rt 
 
 co»(S—B)cos{S—C), 
 ti'mJj s'mC 
 
 cos 
 
 'U= 
 
 cos(S—A)coii(S—C), 
 ■ ,-- .7-7 ) 
 
 ■sin.l sill 6 
 
 
 COS-77C= 
 
 cos(,S'-.l)cos(*V-/?) 
 
 sin. I .siii/> 
 " Art. 146, Case lA', we shall liave : 
 
 log cos(.S— 6')(fi.")<'2t)' ) ='J. 020488 
 
 co]ogsiii/y(r,;»«'4o') =n.(i27;tr.' 
 
 cologsinC(r>'i«l(i') =0.1140s9 
 log cosla= 
 
 ■2) 
 
 rj.7<;i. ■;."); 
 
 'J '.'og cos(.S'— .1^45") =9.84948:. 
 
 lngcos(.S'— 6')(Gr><="Jn') ='j.f]2' 
 
 c:Aogs'inA(l()^-W') =o.O'.>.. ,:,;! 
 
 colog sin(?f5r."io') =o.n4(3.S:i 
 
 iiig 
 
 C()S.yy= 
 
 ■) 
 
 he 
 
 noe. 
 
 0=2 log cos 
 
 
 :Sr'( 
 
 (.'!•' - ..)' ' 
 
 9,ssoG7:, I hence. /y=:2log co 
 
 I =1()0''38"26' 
 
 19.GI0;',i; 
 9.8051;"): 
 
 log cos(*S'— J) (4.')^^ ) = !>.S404.Sr>, 
 
 log cos(,S'— /.' ) ( .")' 1 0' ) = <».I»1)S2:52, 
 
 ci)log sill .1 ( TO^'lio' j = (».()2r)(;-);], 
 
 eolog >in B({\^°A{y) - ().027!I42, 
 
 lou- 
 
 C(JS 
 
 
 lU.noiHTJ 
 
 hence, 
 
 — •2 lo'z 
 
 ('( >s 
 
 I). !).-)( (().")( ;=:)•{" ;}.")' 4S' 
 
 Prob. 5. ( 
 
 1 1 ri'ii 
 
 .I:=::!(i^ ;!(!'. LU=2:»^2(r 
 
 .s(( 
 
 /r a = ")()■ 
 
 10'; fn t'oiisffiirf mid sd/n flir fri'inir//r, 
 
 CoNSTliiriiiiN. Iia> ott" /i( 
 
 "ill Mir 
 
 and draw tin- diamcicr />^»' perpendicular 
 to tlie diauit'tcr ( ' h' : draw the ureal cirdi' 
 
 BAF 
 
 liiakwi'. 
 
 i_.l /»'('.-= /.'^L'.')^ _'0 : ahniil 
 
 the pule y) nt 
 
 HAF 
 
 W'.t 11 a siiiicncal railius 
 
 m}> 
 
 i-.ir=:}()^ -{ic. d 
 
 i'>rV\ 
 
 he t 
 
 le small circle 
 
 / 
 
 iy}n 
 
 intersecting IXt in ht and ;/ : ahum in. as a center, des 
 cribe the ureal circh^ <WF: then will A IW lie the ti'iauuh 
 
 refpu 
 
 ired. 
 
 For. the distanee hctwccn the pi)le> in and/) is 
 
 tl 
 
 le are 
 
 mil 
 
 = .l.^:i(r :;o' 
 
:9.84948r> 
 
 =n.()'2-. ."i:". 
 =0.ll463l> 
 
 fj.S05ir>7 
 
 
 
 1,1. iriiiugl^' 
 Is tUi' an: 
 
 ARTICLE 149. 241 
 
 Solution. Applying logarithms to the formulae : 
 
 lo, sin &=sin a -^— ; 2", tan|r=zztan| a— 6) - ,--A -Z_ .Z ; 
 sin.4 sin^(.4~7i) 
 
 3", sin(7=sinyl ~^ of Art. 146, Case V, we shall have : 
 
 3",logsin^(30°30')=9.7054G9 
 log 8inc(83^29'40") = 9.9971 95 
 colog sin a(50^10')— 0.1 14t;89 
 
 hence, C=log sin ~' 9.817353 
 =138'> 5710". 
 
 sina 
 l",logsinrt(50"10'):r=9.885311 
 log sini?(25^20')=9.631326 
 colog sin/l(30^30') — 0.294531 
 
 hence, 6= log sin — ^ 9.8111G8 
 =-40° 20' 43". 
 
 2", logtan^(^-6)(4°54'38") = 8.934073, 
 log sin^(^ + B) (27^55') =9.070419, 
 colog sini(yl-B) (2^35') — 1. 340089; 
 
 hence, c=21og tun""^ 9.950581=83'' 29' 40". 
 
 Prob. 6. Given LAz=40° 10', LB=120° 30', and the 
 included side c=50° 50', to construct and solve the tridugle. 
 
 Construction. Lay off 7i.4=ci=50°50'; ^' 
 
 describe the great circles ACF and BCD^ 
 
 making L5JCr=.l=:40° 10' and L.CBF= 
 
 59^ 30' (supplement of B) ; then will ABC 
 
 be the triangle required. 
 
 Solution. Applying logarithms to the formulae : 
 
 l'>, tan^(a + 6) = tanm = tan^c 
 
 cos-\(.l-/i). 
 
 cosi(^ + .ZJ)' 
 
 o X 1/ 7\ i J. ^ sini(/l — B) 
 2", tan-Ka— 6)=tanw = tanJrc — ^ -/; 
 
 " " sm^(^l + ^) 
 
 whence, a=w4-w, and b=ni — n ; 
 
 sin 6^= sin ^ 
 
 sin c 
 
 of Art. 146, Case VI, we shall have 
 
 sin« 
 
 1 '.'o<jlan.\d2r)°?5^) 
 
 lO.GTGSGy 
 
 l,,,,.os^(/?— J)(4(>"l(r) =!J.S!-'311t! 
 
 «ologcosi(.C-f^4)(8()°20')=0.T74908 
 licuce, m=lo":tau~'' 10,.334968 
 
 ■) 
 
 =05" ir r 
 
 •1". lop:tanic(2r)^2;r) =!), 076809 
 lo.cc sill ' (/> — . I)(4()" 1 (V ) =9. 8095(59 
 colug3iii>{/?-|-.-l)(8()'2()')=r().00(12H 
 
 hence, 
 
 
 «r=l()ur tan 
 =l7Mtri7' 
 
 9.492649 
 
 fji }Ja 
 
 I M 
 
 fi. 
 
 [I Jn 
 
 4 
 
 mi 
 
 Ah 
 
 % 
 
 16 
 
i I. 
 
 
 
 242 
 
 SI'IIKIUCAl. Ti;i!ioNu.\!Kr;iV 
 
 whorice. /y=rm+// = 82^^27'18". mid </=-- ,y,_//=-17''54'44". 
 ;i'. lo-' i<in.4(40^" 10') = !).'^0!K")(;!>, 
 log sill rfoO^ ,')()' ) = !>..SS!>47T. 
 coldg sin .^(47'-'r)4'44")rr0.12itr)2'. ; 
 
 hence, 
 
 150. 
 
 — 1 
 
 6'--Io-siii ' Ii.<2s572=:42^ 21'57". 
 
 ASTUOXOMTCAL VKOBLEMS. 
 
 Prob. 1. Giiu'u uriij tvit (,f fh.r hnir (jumififies ■ latitude of 
 the phi ci ; (Iccliitati'Hi. nm m''' ■!!/(■ (hkI asa usujuuI difference 
 of a celestid/ hodij : tojiiid tin- otiur tim. 
 
 NoTK. Tlie tecknical terms used in these p. oblems are defined in treatises 
 o:i Asti'uMiiiiiy, 
 
 Solution. Let PEj) be the meridian 
 of the place : Illi the liorizon and Z 
 thezenitli; EQ the e(|uator. and /■* its 
 north pole ; 6' the celestial bddy in the JI[ 
 horizon, when rising. 
 
 Then, [^PAh is the latitude- of the 
 place ; and in isABt\ right-angied at /> ; 
 CB is tlie north declination ; CA the amplitude : AB the 
 ascensional difference ; L.4 the co. latitude ; and 
 EA-\-AB{ = ^{)'^'-\-AB=^Jh^AB) is the semi-diurnal arc. 
 
 When the body C is in the southern hemisphere ; C B' is trie 
 south declination ; C A the amplitude ; AB' the ascensional 
 difference ; and AE — AB' the semi-diurnal arc. 
 
 It is obvious that the same fi:^ure may represent similar cir- 
 cumstances of the celestial body, when setting. 
 
 Now, let [-BAC=A, co. latitude, 
 
 CB=:a, declination, 
 
 AC=:h, amplitude, 
 
 AB=c, ascensional difference. 
 
 If the latitude of the place and the declination of a celestial 
 body be given, its amplitude /', and a.scensional difference c will 
 be found by tiie formulae of 132. Case I : 
 
A' 
 
 hide "/ 
 fcrencc 
 
 I trcalisi':-' 
 
 lI arc. 
 \B'\ii tlie 
 ;eensioniil 
 
 Imilar cir- 
 
 celestiul 
 Ince c will 
 
 1 ■ . sill A=: 
 
 Mil (f 
 si I!. I 
 
 AliTU'LK 150. 
 
 2 '. sill <' = ,;in a cot .1. 
 
 243 
 
 ISO 
 
 be 
 
 m 
 
 Likewise, uiveu a and <\ the jiarts A and h may al 
 t'dimd l»_v (lie I'ui-iiiulio nf Art. l.'L'. iVc 
 
 Prob. 2. (I'imi dm/ tim (if' till j'diii- ijitii litilics : suits Ion- 
 ijitiiilr. riijht (ixi-ciisimi. iJt c/ iiia/ ioji ; niiil nbJiquitij of the 
 cr/iptic : to ji 11(1 (lir i)th< r tirii. 
 
 Siiij ikln. Iji't E\> 1)1' tluM^(|uat<)r ; Mm 
 ilic ecliptic; l*l'it a meridian |tassin^' 
 tlirouiili the center Col' the sun ; PEfd^ 
 I he solstitial eohire : then. .)/ is tlu; summer 
 and HI the winter .solstice : J is the vernal *^ 
 (•(piinox, and the ])oint diametrically ojtpo 
 site to vl is the autumnal e(|uinox. 
 , Besides, AB^^c is tlie suns right ascension, 
 J}C=:a is the suns declination, 
 AC=^/> is the suns longitude, 
 and L.BA(J= A is the obliquity (jf th<' ecliptic. 
 
 Hence, since the triangle ABC is riglit-angled at /J, any two 
 of the four parts a, h, c and J being given, the other two may 
 be foutid by the f'ormuh« of Art. 11^2. 
 
 Prob. 3. Given thr riylit (isceusiun and ilerlination of a 
 cihistud body, to find lis lomjltiidt' and latitude, and conver- 
 scfi/ ; the ohllquitij of the ecliptic bcimj yiven In both cases. 
 
 Soiii'TiON. Let JJ be any celestial body and XDii a meridian 
 of the ecliptic, (i)receding fig.). 
 
 Tlien, AB=R is the right ascension of the body, 
 
 JiD=d is its declination, 
 A/j=/j is its longitude, 
 LJ)=/ is its latitude. 
 
 .!/>== 79 is its distaiu-e from the vernal point J, 
 L.liA(J=() is the obli<|uity of tlie ecliptic ; 
 also let \^/JAn=M: L/>J/. = A'; then, M=0±:X, accorS- 
 ing as iVis witliout or witViin the angle O ; and X= M ;^0. 'i^ 
 
 * Tlic (liHcrciue of iwoi a ii it cs, uiifii it is not, known which is the 
 g!'('at<':. is tj.\i)rt'S.-('il }iy tlit; .-yiiil)ui ^. 
 
 ■Jk 
 
 •i i'.;.*fl 
 
 r- s,iF^ 
 
 ;fm 
 
I 
 
 ii! 
 
 11 
 
 
 244 
 
 SPHERICAL TRIGONOMETRY 
 
 Now, assuming 0, R and d to be given ; then, L and I will be 
 found as follows : 
 
 1'^ In AABD, we have (Art. i:i2, Case V) : 
 
 sini? 
 
 cosZ)=cos(Z Q08R ; and cotJ/= 
 
 tan d 
 
 9o 
 
 Since N=M—0, in /\ADL, we have (132, Case III): 
 sin /=sinZ> sinA^; and tan7> = tanZ> cosiV. 
 When 0, L and I are given, then 72 and (?may be found by 
 a similar process. 
 
 Prob. 4. Given the latitude of the place, the sun's decliiia- 
 tiou and altitude; to fnd the sun^ azimuth and the hour of 
 the day. 
 
 Solution. Let NMnm be the meridian 
 of the place ; EQ the equator, P its pole ; 
 Mm the horizon, oN-^ the zenith, and D the 
 center of the sun ; then, DL is its altitude, "^ 
 DB its declination, and PJ/the latitude of "^^ 
 the place ; hence, in /\NPD, 
 
 \—XrD=-A is the required azimuth from north, 
 [^DXP=B is the required horary angle, 
 DN=^a is tlie uiven co. altitude. 
 DP=h is the given co. declination, 
 NP=c is the given co. latitude. 
 Now (Art. 146, Case I), ^((f + 6+c) = s, 
 
 and lo. 
 2o. 
 
 cos^i.t=: ^"'f ^"^^:^-^^> . 
 sin h sin c 
 
 cos-i/j: 
 
 •sin s sin(,s — h) 
 
 sin a sin c 
 
 Prob. 5. Given the how of the day, and the sun's decli na- 
 tion and altitude; to find the latitude of the place. 
 
 Solution. Let the notation of /\XPI) (preced. tig.) be the 
 game as in Prob. 4 ; then, since the given parts are DX=:" 
 (CO. altitude DL), DP:=:ih (co. declination D l>}, and angle 
 ylnz:supplement of the given horary angle EPB, we shall 
 have (Art. 14G, Case II): 
 
t '1 
 
 be the 
 angle 
 
 sin /> 
 
 245 
 
 siniy=siM-l 
 
 Mil II 
 
 and then, eo. latitude c=tan-^( '< — />)' -V JIL- J; 
 
 whence, the reijnired latitude /'J/^riMI'-' — c. 
 NoTK. It must 1).' borne in iiiiiid lliiil oiu' liou:- =ir)^. 
 
 Prob. 6. Giroi the longituihs and latitudes, or the right 
 inrensions and declinations of two celestial bodies ; to find 
 their distance. Q^ 
 
 SttLT'Tiox, Let ED be the ecliptic ; (7, /*, 
 its poles ; T^tlie vernal efjuinox ; ^1 and B 
 two celestial bodies wiiose loniiitudes VR, 
 r//, and latitudes RA, LB are uiven ; 
 
 then, in A^li^C': 
 
 CH = a is the given co. latitude BL^ 
 CA = b is the given co. latitude .1/^, 
 
 and [-ACB-=^c is the given difference VL — VR of the 
 
 longitudes ; hence, the required distance, AB=c, may be 
 
 found as follows (Art. 14G, Case III): 
 
 tani(.l-fi>')=tan3/=cot*6'-^t^'^^^ 
 
 cos^(a-}-6) 
 
 tan^(.4-i?)r=:taniV:=cotK^ ^^"aO^-^j. 
 
 whence, 
 and then, 
 
 sin-^(a-|-6) 
 i>'=J/+.Y, and A=M-Xl 
 sinC 
 
 sin c=sin a 
 
 sin^l 
 
 Sch. I. The distance between the two bodies may be found 
 by tlie same process, when their right ascensions and declina- 
 tions are given. 
 
 Sch. II. Assuming VR. VL, and ^.1, LB to represent the 
 longitudes and latitudes of two points .1 and B on the surf-ice 
 of the earth, their distance Al> may also be found by the same 
 process. 
 
 NoTK The stndi-nl i.s auvis d to tind tlie numerical solutions of these 
 iistrononiical problems, ))v making use o; the data registered in the t'ollow- 
 ni^' h'fiheiiieris. ' 
 
 Sec Explanation of the Eji/iemeris, Ait. IT/i, «&c. 
 
S: 
 
 i 
 
 I 
 
 240 
 
 SI>HF.1{|:\I. 'll!I(i(tN().MKTHV 
 
 J50 AsTlUiNu.MICAl, Kl'IIK.Mi;iU> hull I'liK Vi:.\lt ISIKJ, 
 
 (tuil for flic Miiwi/iiiii 1)/' fin- lioi/nl Oh.sirriiftirij 
 (i/ ( I ri I II II' I rli . 
 Al>|)aiTiit ()l)li(|iiit V c)l thr ccliptie. 
 Aui;u,><t l^t : L»;^' 27' ID' .1 ; AuLu.^t 12tli : 2:;^' 27' l:>".(;2. 
 I. Sun. Aiiii s'l' lSit;5. at virii)i vnini. 
 
 
 Longitude. 
 
 1 
 
 liOligitudf, r: 
 
 1 
 
 2 
 
 12!r2;!'4i)".7 
 
 i:)i) 21 <; .0 
 i:ii IS :;2 .<; 
 
 l:'.2 !G' n,"4 r 
 1.;:; i:; 2i> .d ; s 
 .:;4 1 I (» .2 ; ;• 
 
 I-(iiigitu(le. 
 
 ID 
 1 1 
 
 12 
 
 Longitud*'. 
 
 1 :;.•.' s'3-".2 
 I :;(j <; ."■» .o 
 !:;7 ;i 4ii .4 
 
 L'^.S-" \"J"A 
 
 i:;8 58 ,-.:! .7 
 1 ;;;» r.t; :',2 .2 
 
 Day 
 
 R. Ascension 
 
 ■4-Dccli]j:ilion: 
 
 l>ayJ 
 
 i!. 
 
 Asmision. 
 
 ■f DccliiMition. 
 
 
 h III > 
 
 o / ; ' 
 
 h 
 
 ni s 
 
 ' 1 1 
 
 1 
 
 8 47 2<i.:ir, 
 
 1 7 54 52 :; 
 
 7 
 
 ;i 
 
 in 2';. 215 
 
 1<; 18 l;i.-j 
 
 ') 
 
 S 51 12.: I ; 
 
 IT :?;> 2'.t.2 
 
 s 
 
 U 
 
 1 1 1 5 '1 1 
 
 ii; 1 1(1.4 
 
 :{ 
 
 S A") 4.78 
 
 IT 2;; 4s 7 
 
 '.) 
 
 '.1 
 
 IS :', 5'.» 
 
 15 4;'. 58.0 
 
 4 
 
 s oS 5<;.ui 
 
 IT 7 5; 2 
 
 10 
 
 ;> 
 
 21 51.40 
 
 15 2(! 24.4 
 
 r. 
 
 ;» 2 4'; DC 
 
 I (J 51 :',i; :• 
 
 1 1 
 
 It 
 
 25 ;!s (;:, 
 
 ] 5 8 35.'.t 
 
 <) 
 
 It G ;!i; 75 
 
 1 li 35 G I 
 
 12 
 
 y 
 
 2!> 25?.4 
 
 14 50 32.8 
 
 II. Jupiter. Ar(irsT IS!),'}, -At menu midnlijhf. 
 
 D.'iy.ii; 
 
 . A.^ftMinion 
 
 4- T)t'(']i:iiiti(in 
 
 D.iy. 
 
 U.Ascension. 
 
 -f Hec 
 
 
 1 n\ >s 
 
 u r t 1 
 
 h 111 s 
 
 ' .' 
 
 1 
 
 ! 42 42 10 
 
 IS 44 5G. 
 
 » 
 
 ;; 4G 2 !)G 
 
 18 r,i 
 
 2 
 
 ! 43 IT.l',) 
 
 is 4 3 11.2 
 
 8 
 
 3 4G 34.42 
 
 IS :,: 
 
 3 
 
 '. 43 51 14 
 
 IS 4 4 5(;.(; 
 
 <i 
 
 3 47 5 30 
 
 18 5 1 
 
 4 
 
 ! -14 25 15 
 
 IS 4G 30.7 
 
 10 
 
 3 47 35 58 
 
 18 :,« 
 
 ; ) 
 
 : 44 5S 32 
 
 IS 48 20 i; 
 
 1 1 
 
 3 48 5.20 
 
 IS r.T 
 
 G 
 
 ! 45 30 O'J 
 
 18 4<i 5! 1.1 
 
 1 2 
 
 3 48 34.; 14 
 
 IS 5; 
 
 iuatiou 
 
 0.1 
 
 ]o. 
 
 in. Moon Ai^firsT 1S!»:5 at mean mu.hiujhf. 
 
 I); 
 
 IV 
 
 onii'i 
 
 Hull 
 
 40 22 
 5 1 25 
 
 ij. 1 
 
 45,8 
 
 10 
 
 Ti 
 
 12 
 
 GS 42 2T. 
 83. 1(1 20 
 y7 45 5:; 
 
 .'2 ri,.i 
 
 -.4 4 ■■2 
 
 ] ^I 14 
 
 O.i 
 
 15 1 : 
 
 -j-LatitiKlc. 
 
 I{. Ascension. 
 
 -f-Doclii 
 
 n'io 
 
 'J / ( ' 
 
 h m s ■ 
 
 J / 
 
 1 1 
 
 1 34 41.7 
 
 2 2'3 4.3.01 
 
 1 G 2G 
 
 14 : 
 
 2 42 G.2 
 
 3 25 22.72 
 
 21 30 
 
 2I.( 
 
 3 40 2S 
 
 1 25 22.44 
 
 25 23 
 
 4i; > 
 
 4 25 :;4.] 
 
 5 23 12 (;5 
 
 27 -tl 
 
 55.' 
 
 4 5.3 50.2 
 
 G 3.5 5 ;17 
 
 28 G 
 
 50. 
 
 5 .". 2 5 
 
 7 40 21 21 
 
 2(; 34 
 
 3G.; 
 
 4 52 ;;s.5 
 
 8 42 ;m.' g 
 
 23. IG 
 
 25. ( 
 
 4 23 53.S 
 
 '.I 40 24.7 7 
 
 IS 35 
 
 -10.^ 
 
 :', '.','.> :;s*", 
 
 !M ;!:; 4S i3 
 
 ] 2 50 
 
 47.' 
 
 Xot:;. Tiie iiIkivl' ],()n;:■itnde^^ I.irttitiitii's. 11. Ascensions iiinl Declina- 
 
 tioiiS a 
 
 s are npiiarrnt, an 
 
 d tl 
 
 s;:j:ii (-L) moan.- 
 
 Xni-li 
 
ARTIfLK 1."!. 
 
 247 
 
 EXPL 
 
 tVNATlMN (»!•' •rilK. KlMIF.MERlS. 
 
 li>'^. Bct'orc luakiirz u>*' <»f the Kjtliciiicri.x. it i- iiccojiiMiry 
 
 to a.scertain. in ovcrv iii>taiit('. tlic ( 
 
 li.>t 
 
 (in:e 
 
 "\ tlic .>uii 
 
 ' /// timi!) f'n>in tln' iiu'riilian ol' (ircciiwirli , or what i.>(ii)ii- 
 iihuily cailcd. th(^ citrrojiniKUnM' ( in'ciiwicli liiiK.' ; ai.tl tlii> 
 is evidently equal to the '.ivcii time tukU'V the a.s.siuncil incri 
 
 <ii in. illCfnlSrd 01' '/ 
 
 I III I il ,.^ 
 
 liiil \)\ tlic (liffrrcnce ('// tl 
 
 mi i m: t In 
 
 iwo meridians, acfordiiiu' ;is liic assumoil 
 W ishi-ii rd (U' E-mhrti nl uf ( Irci'iiwiidi. 
 
 ncndian i> tu tlio 
 
 'V\ 
 
 Ills : since 
 
 24 1 
 
 lour; 
 
 !(;(! 
 
 1 1 
 
 I : i .)' L 
 
 when it is <!1 
 
 I a 
 
 fter 
 
 mean iitMin at a ])laee 1 5 • \Xv>x ot' ( Jri-emvich. it is 7h after 
 
 ircan noon at (Ireenwieli ; and it is for tliis Ureenwitdi t;n 
 
 that Nve must deduce the re(|uired ([uantitie 
 I'ipiiemcris. 
 
 fi 
 
 le 
 om the 
 
 It 
 "'Sep 
 
 
 15 
 
 « > rn 
 
 7'r 
 
 I) I'oiivei't ."htn<' into tlir mr respond iiicj di:rfirr.< af 
 
 Vi 
 
 d<: 
 
 Iii(/kt Asi-fiision or nf Tirn.'itrial Jjongifiide : 2" rini rir^i h/. fn 
 convert drtji'ces of Riiiht Axmifiion or of Tcrrcstrioi Lon.fir in!, 
 
 oirs 
 
 into tinir. j)rorrcd as fofl. 
 
 1" Reduce the given hours to minutes, and divide th( 
 result by 4 ; tlicn, in the (|Uotient. th«' minutes of time wi!' Ix; 
 (lesirees. and the seconds of time will he minutes o 
 
 f sp: 
 
 ICS 
 
 (1 
 
 md so on. 
 
 Tl 
 
 lU? 
 
 lours 20m l(ls=^20^^ 2' 80 
 
 2' (.'onversely. luultiply the i;iven depress. minute>, iY-,-,. hy 
 4 : then, in tiie i>roduct. the deurees will he minutes of time, 
 tlie minutes of space will be seconds of time. Xc. 
 
 Tl 
 
 lUS. 
 
 S7" :n'42" = r)h 50m 2t;s. 8. 
 
 II 
 
 J H'i 
 
 w 
 
 urn 
 
 1,>4:. The day of the imtnth. in the Kjdiemeris. i^ as>ii:iicd 
 to beiiiu at nwini 710011 (if the corresnondiiiLi' riril d !•/, an 1 is 
 continued to the t'ollowin:^ mean noon ; it may thfrtr'tr'' 'u- 
 called the mean (itttrniKniilrnl '/•'//. 
 
 Venice the f//-// (A/// commences at the pncrdiiKj mMi,!.;iit. 
 
 the civil reckoninuis always 
 iiomicul reckoning. Thus : 
 
 12 h 
 
 ours lu auvunce o 
 
 hi 
 
 .f tl 
 
 u- 
 
 •9[ 
 
!':!^ 
 
 248 
 
 SPHERICAL TRIOONOMETRY 
 
 < 14 
 
 Ex. 1. August 1st at ()h HOni A.M., civil time, 
 
 iis July 31st ISh l>()in, astronoinical time. 
 
 Ex. 2. August lOtli lOli 2()iu A.M., civil time, 
 
 is August Otli '11\\ 20ui, tisfronomicdl time. 
 
 Ex. '1 August 10 (lays 4h lOin, <istronomical time, 
 
 is AuLTUst Kith 4h lOni P.M., civil time. 
 
 Ex. 4. August 14 days 21h 25m, antroiiomic(d time^ 
 
 is August 15th 91 1 25ni A. 31., civil time. 
 
 !«>(>• 1" To reduce the time under any given meridian to 
 the conrspondincj astronomical time of Greenwich ; 2", conver- 
 sely, to reduce the time at Greemvich to the corresponding 
 astronomical time under any given meridian, proceed as 
 follows : 
 
 1" To the given time (at the given place) expressed astrono- 
 mically, add the longitude in time, when it is ^y ; and subtract 
 it, when it is i/. 2" To the given time (at Greenwich) expressed 
 astronomically, add the longitude in time, when it is E ; and 
 subtract it, when it is ^V. 
 
 Ex. 1. Find the time at Greenwich corresponding to 
 August 10th at lOh 20m A.M., at Chicago, in longitude 
 87°3C'42" IF. 
 
 Sol. Given time(astron.), Aug. Od 22h 20m (Art. 154, Ex. 2) 
 Plus longitude in time 5h 50m 2(Js.8(Art. 153,2') 
 
 Corresponding Greenwich time lOd 4hl0m2l)s.8. 
 
 Ex. 2. Find the time at Montreal, in longitude 73^34'37".5]r, 
 corresponding to August 15th at J)h 25m A M, at Greenwich. 
 
 Sol. Given time(astron.)August 14d 21h 25m Os (Art. 154), 
 J/ta?«s loniritude in time 4h 54m 18s.5 ; 
 
 Astronomical time at Montreal 14d 17h 30m 41s.5. 
 
 156. In the above Ephemeris, the Suns ajiparoit Right 
 Ascension [R. A) and Declination {Dec), at mean noon, denote 
 the apparent position of the Sun reckoned from the true ver- 
 nal equinox, at the instant the Greenwich mean time clock 
 indicates noon (Oh Om s). 
 
T 
 
 ARTICLE 15n. 
 
 24f> 
 
 Now. to Jind the K.A. tintf \k'c. /or nny othtrmmn tlmcund 
 /)/(ic<\ proceed as in the followintr 
 
 />',*•. Find ti.o Sun"s jqijiaront Dec. and li.A., for lontritudo 
 SI" 30' 42" W, August KHh, at ]<ili 2()ni A.M. 
 
 Solution. (Jroonwich tinie(15r).Ex.l),Aug.lod 41i Klni^lls.S 
 Sun's Dec. at mean noon, on Kitli =15^^2»)'24".4, 
 
 Suii"s Dec. at mean noon, on lltli. =1 a'' 8'H5".1) ; 
 
 Variation of Dec, in 24 hours, = 0^17'4S".5. 
 
 The variation (11 for the above 4h 10 m 2r.s.Swillbe 
 found by the proportion : 24 h : 4 h 10 m 2(i s.S— 17'4S".5: V; 
 whence, F=3'5".8. 
 
 Hence, required Dee.=Dec. of preceding //ooji— ]'rr:ir)''23' 
 18".G. 
 
 By the same process, the required A'..4. will be found=: 
 Oh 22m 30s. 94. 
 
 The proportional part V \s additive ov sublractivc, in tlie case of a Dec, 
 according as the latter is increasing or decreasing ; hut. I" is always addi^ 
 tive, in the case of a H.A, since the latter is always increasing. 
 
 ill 
 
 ilj 
 
 lit 
 
 i. 
 
 ■•(If 
 
 !-. ■ ' f 1 
 
 './K 
 
 ^::*^ 
 
 ■'?;] 
 
 i% 
 
 n 
 
 
i i 
 
 APPROXIMATE MENSURATION. 
 
 (Siip/'l'iii' iif h> ( II niin ! ni . Ili'iilc III. Arti'r/is ')[, 'tX. ♦id) 
 
 Dki'i 
 
 MIMiN.s. 
 
 •w 
 
 1. A pointo*" 'uttexion 
 hit! 
 
 l> :i iMtilit ;it 
 
 1 ;i curve (..•-.iiuc^ frmii in'iii>_: cuii 
 
 vex t(» neiiii: C'Mic.is e. i»r t lie reveiX". tn- 
 
 Avai'Us ;i (jo-on 
 
 litiat 
 
 e axis. 
 
 rn 
 
 n 
 
 Ills. 1 
 
 f ()/, 
 
 II i> cMiicave. alK 
 
 1 >n>/ 
 
 ciiiivex tnwanls tin- axi> O.V. the iMi'm 
 v is a ))i>iiit of intlt'xioii. 
 
 V 
 
 :^, 
 
 
 1 
 
 'fir 
 
 \f 
 
 - ' 
 
 1 
 1 
 1 
 
 O ^ 6 c c/ 
 
 *> 
 
 'i. .V variai>:e iiuaiitiiy reaclie.- i inaximun or a minimum 
 
 value, acconlinu' as tlii> value is urraternr less tliaii the v 
 
 line.- 
 
 ■sv 
 
 hiel 
 
 1 iinuH't 
 
 liatcl 
 
 y preecile and I'olluw it 
 
 Tints. su|)ii(»>ini: (ii>-r=zl)r^-'-(lx (dirt'ereul ial oC.c), 
 
 an 
 
 a 1 
 
 II in- 
 
 hi. 
 
 II. l' ' ivi'^hli' ^^'j-r' . tlieii the ordiiuito hh 
 
 w 
 
 ill bo :i iiiaxitiuiin in the i-urve Oh' I and a iniuiuiiiia in O'h'^l' 
 
 !)lHK<'ri(»N {)V riUVAI'l KK, 
 
 In what follows, the (Mii-\atiire at 
 
 any point of a eiiive 
 
 \v 
 
 he retiardi'd 
 
 as heiiiii; directed towards the co 
 ordinate axis which forms, wit h the 
 tangent to llie curve, at that jinint. an 
 an<ile not ureatei' tiuin (i(^\ 
 
 Thus, if eacli (d' the two tanuents <tf\.\\\i[fil to tlio cuvvi 
 Ovod, at the points n ;ind if. fitrins witli the axis 0A\ or it' 
 parallel '<</, an aisLile of (><h . tiie points a and d are the limit.- 
 
 d' th 
 
 e concave arc ((nd towan 
 
 ds tl 
 
 le axis 
 
 f A' 
 
 md then 0.( i- 
 
 a convex arc towaro 
 
 Is tl 
 
 U' axis o 
 
 f r 
 
rainimuin 
 
 tlic value,- 
 
 rdiiiato hli 
 1 ill 0'b''I. 
 
 AKTH'M; I. 
 
 251 
 
 1. I.KNdTIIS n|.- Tlir, l'\H.\l! 
 
 il,.r, 11 .z=.ii.r' -^ li.r- -^r.r -\- f 
 
 .\M» <il' .\\> OTIIKI! t'l lt\ KS COM I' \1{KI> 
 
 A.' 
 
 • nine (III 
 
 ./ lo lie ;i |»aralMiIa ir^=<t. '-{-f' 
 
 t.i-~f- i-r -j- /, i jirocct 
 
 I. 
 
 ttr 
 
 111:. I ami the ai'c un t<> lie iiitinitely small. 
 
 hraw the ordinate nni. the unliiiate iprulnirzcil y>/'. aii<l 
 purailej tn 0,\'. 
 
 Let "/' ilitVercnt ial (/.r. A;/irr<riff('r<'lH ial <///. ami l„»vf/rn:<l. 
 
 I r t he line '//'he tangent ti» t he i-iir\ >•. at the iniint n. it will 
 
 ('(»i lie Hie With t he iiiiinitf.-.iiii' 1 art; -//r am 
 
 I th.' t 
 
 imt hiiii will he >iiiii!.ii- ; w it 
 
 f;iee 
 
 (ir ' I 
 
 /" tlli'Jiii:' .il,i\(l 
 
 y. or 
 
 II 
 
 ,1 
 
 il.i' lie (•!)> ii 
 
 1/ if Mil 
 
 tan K I I n.. '1'1\ 
 
 naii'^uvs ciif 
 
 }->, 
 
 Nuw. Jiti'ci'tMit iat iiiL! //—(<./•'-(-/>.'•■ 4- '''-r/" there ohrain.^ 
 
 (I'l = ( .'J(M---f- ■_'/>,' +(• )./.(■ ; 
 
 Wliejice. .nil-- 
 
 if,r ('US ;i 
 
 i\\. ". 
 
 le 
 
 A\ hen ihe taimeiil ^at the |mmiiI "I Id'coint's ])ara]lel tu tl 
 axis (J.W the aiiulo '' vanishes; lieiiei'. when tin- ji »int of 
 tan^cney is ; 
 
 at (I . then 
 
 w lie nee 
 
 <r>/ 
 
 IIX 
 
 :taii.O^'— Orzi;5'/.(;--fl.'6./~ 
 
 — h±s//r. 
 
 i)ilC 
 
 .ill 
 
 (U 
 
 '('/ 
 
 It <it)", then -:Z3=tan.(i(r=:^/:5--^;]'/x- + 26x + c 
 
 (if. 
 
 whence 
 
 a.i- 
 
 x_— 
 
 . A± \/A-'+ ',\n^/ :;"—:]'/.( 
 
 Wa 
 
 at IMh'. thoi 
 
 '('/. 
 
 (I.r 
 
 w . nee 
 
 .V 
 
 -tan iMi^'r^: X —;;" 
 
 v: 
 
 X-^L'/.T-i-. 
 
 SubtiMetiiig' (1 ) iVoni (2j, tliei'e obtain.- 
 
 {^) 
 
 (:-5) 
 
 X — ./• = 
 
 or X — ,/•= 
 
 _ — /y ±: \/A- +>y(/ ^/H—}\itc — /> =t= \/ A- — 
 
 )'/<■ 
 
 .><? 
 
 .)" 
 
 ± s/fr-^'A,, ./;j — Haf-qr >//r- 
 
 .j(>r 
 
 .Wr 
 
 for the ])v ection ol' the eiirve; from O"-' to±(iO^, on the 
 
 a.\ 
 
 is of X 
 
 
 
 m ^^' 
 
 *it 
 

 - \ Y 1 
 
 
 liV: 
 
 I: 
 
 
 
 it-.-, •;' 
 
 
 Al'PUOXIMATE MENSURATION. 
 
 Subtractiiic: (?) from (3), there obtains A'— x = x, for tho 
 projt^ciiuK of the curve, from ±(()0^ to 90°), on the axis of .V. 
 
 Since tlie coefficients >i,h, c can take any value, t'le projec- 
 tion of the paraboli\3 i/ ^^a.r^-'rf >•'>'■'-{- cx-^-f, from O*-' todrtJO*^ 
 up^ni tlie axis of A. can also assume any value ; hence, the 
 parabolic arc, from 0'^ tortliO"^, can assume any length, since 
 the arc necessarily varies in length with its projection. 
 
 On the contrary, the parabolic arc, from±(()0" to 0<l°), is 
 always infinite, since irs projection is always infinite. 
 
 5. Therefore, 1" an arc rcpvesented by 
 
 ,j - r<rc-'-f Lv- + ex +/ (a) 
 
 can assume the length of any other plane curve y=f{x). (I)) 
 from 0" to.±-(»0" (portion directed towards the axi:^ of A). 
 
 2" The parabolic arc (jl), fromdr(()0° to {X)"^), cannot 
 assume the length of any otiier plane curve (I)), when this 
 curve (b) is finite within the latter limits (portion directed 
 towards the axis of Y). 
 
 The relations between the curves (a) and (b), specified in 
 2"^, become the relations specified in 1", when the portion of the 
 curve (b) directed towards the axis of Y has been turned to- 
 wards the axis of A', by a mutual change of its co-ordinates. 
 
 Note. For i he moth' of proceeding iii this mutual cliange ofco-orili- 
 no.tcs, see App. C. II : and Art. 27. Proh. 2, 3, further on. 
 
 C« Sch. I. Maximum and minimum oniinates of the 
 paraholxv y = ax-'' -f bx" -f ex + f . 
 
 In the above equation (1), the two roots : x: 
 
 
 represent tlie abscissas corresponding to the maximum and 
 minimum ordinates of the curve. 
 
 Thus.ditYerentiating y— 4?^(.rM8,r--)-0t!.r) ^ p 
 equation of the parabola Onl, there ob- ^ 
 
 tains ' '^:zrfV)('^'*''~3^'^" + ^^"^)' 00 
 
 But. when the tangent to the curve 
 becomes parallel to the axis of A' at the 
 points h' and^). it reduces to zero ; that is 
 
ARTICLE 
 
 
 253 
 
 ^.t 
 
 *JLz=0. and equation (?<) becomes -i.l = X.(3.r- — 30x4-06) = 
 «/.r d.r 
 
 0--.T-— 12.C+32 ; whence, .7'=4 or S. 
 
 The niaxinuim and niininiuni ordinates corresponding to 
 thest abscissas x—z-V & .>'=:S, may no.w be found as follows ; 
 
 In the given equation y=i^-^x'p[y — 18.x4-i^6), 
 
 , ( .x = 4, or 0/>, then ?/ = 4, or maximum hh' , 
 
 when ■; J /-w . •> o • • j 
 
 [ .»=», or (/(t, " // = .{. 2, or minimum dp. 
 
 Sch. II. The curve reaches 0° at the vertex of its maximum 
 
 and of its minimum ordinate. 
 
 7. Elementary segments. 
 
 Any plane figure whose curve is represented by (b)(Art. 5) 
 can be divided into such segments that each of them : 
 
 1" shall contain an arc either exclusively convex or exclu- 
 sively concave towards the axis of X. by tracingevery maximum 
 and every minimum ordinate, and an ordinate to every point of 
 inflexion of the curve ; 
 
 2" shall be diviM))le. by its median section parallel to its 
 bases, into two api)roximate trapezoids.* 
 
 A segment fulfilling both these conditions will be called 
 elementary segment. 
 
 Thu!:', let Anl be any curve(b) and 
 assume the arc Am to be its only 
 portion directed towards the axis 
 
 of r. 
 
 Draw the maximum ordinate h))^ 
 tile minimum ordinate dp^ the ordinate co to the point of 
 iiitiexion o, the ordinate am se[)arating the arc ;;/// (which may 
 vary from .30° to 00"^, Art. 3) towards the axis of X fnmi tlic 
 arc J);i towards the axis of 1', and torn this arc Avi towards 
 the axis of A' by taking am for its axis of A', and A(i for its 
 axis of !'( Art. 5, 2). 
 
 • As til" trapezoids into wliicli ii curviliiioaf plane Hfrii'.v is ilividod, in 
 offlor to obtain ,1 rough ai)proxiniatlou of its area by the foiiaitlaol' the 
 trapezoid 
 
 1 
 
 *1li 
 
 
254 
 
 AI'lM'.dXIMA IK MKNSLKATION. 
 
 ?j! ! 
 
 i<_V I his r()ii>triU'n(i!i. the liuui'c .\ii/)(/\< dividcil into sc^- 
 inenis ncc-.^sai'lly rnltilliiiu' t lie Hi>t iiltuvc cuiKlitinn (1"). iiud 
 ■!'i!i«' M'.-ninl our is iiov yet (lUiijirKMl with, tlicyiiiiiv be suli- 
 (iividiHl into sinullcr scuuu'iits whifh then will fulfill hotli coii- 
 <l!(i(i!is ol' ilif ('!('ni('iii;n-y scuniciit. 
 
 Ii i> olivioiis tluit, ill liuy |i:'i'!tliiili(' liuur*' sjiccKicd by (Ji), 
 : lio. jionion wlioj-c arc is towards tlio axis ot' .\' is likewise 
 (livi^iii'o into olcijuMitiiry scuinonts. 
 
 Now. any cU'niriiiary sc :in('nt sjicciiit'd ])y (}>/ is vory noai'ly 
 «MH',al to an clcnicnlary >('!.:nK';it s|)(M'iH('d ))y (JJ). :is is [H'ovcd 
 in Aft. 10. t'urtlu'v on and illustraKMl in tlu' follow in t;;- 
 
 Problem. //'. ;■/(' jiIkhi' ti'l''i'< >i'h<isr nirrc is crjirrssnl hi/ 
 y=::: ,j/5-gX'(X* — 4Sx~ r)7r> ) ■ 1(1. an r/cuinifa n/ xrijinciif is (jirni, 
 hrtirct'ii (he limits x S mtd X- 12 ; /<> </i i/iicr. frorii flir pai'U- 
 lio'tr rqiiatioa y = ax''-;-bx--|- ex -j- f. d niniv ricaJ rqiuifioit 
 specifi/iiKi a. sf(fii}< lit ''irij iif(i)'li/^< qiK(/ III f/ic qirai <>it<'. 
 
 S()i,rTi«sN. In ('(juation '/== 4 oVy*''('" — l!^.''+.')7(i') -f 1 (I, {m,) 
 I- (I. then 1/ ^ 10, 
 
 , \'.=- .^, •■ v^l4.0!>»;.\ , . .., .., 
 
 when ,^._i(, u 1-1 <)()() ) '"-t approxnnate trapezoid.-'- 
 
 CZu, ■■ ;v~i5:i84:}^'hi 
 
 If tlie values ol' these four pairs id' co-ordinates be sticces- 
 sively introduced in 1/ = ax-^ -\- bx'- -}- <\c -{-/. there obtains : 
 
 10, 10=04-/, 
 
 2". 14.0'.ii;=((S'' -I-//S- 4-r8 -fin, or i\A<i-{- 8i4-c=0..")! 2. (1) 
 
 :!', ]4.',;(i()=,((10/'4-/;(li))-;-{-rli)-f-ln. or 1 oii(/-|-l oi-|-r=0.4;Hi. cj) 
 
 -1", l,-),!S'4=(/(l'J)-4-//(12V-'-|-rl'J-f 1(1, or MW/-f 1 2^y-fr=().4?.2. (;i) 
 
 Solv li.n" (\). (■-') it (:U. w ' liavc : <;=z'K"'^-\'k />='.'. (•=.!. •J4 
 
 Tntrodnein<j; these valuesof 'f. />. c /'. i:i i/=z<ix'-{-/t,r-Mr.i -f/ 
 there obtains : ,/ — ,__ J^-.j,.,r( — !),*•-+ 14(.l.r4-4S0) + 10, („) 
 
 for tlie re({uired e<iuatioii. 
 
 / x= s. tlK'ii //--=I4f)!i'; ill {III), uiid //=:14.(i',m; ill (II). 
 { .r= !>, •• //=14.r),"»G2r) •• //=14.:i4H:) 
 
 For. wiu'U ' u-=10, •• )/=i4.'J •• //=J4.!) 
 
 j/=ii, •• y=ir).n22r. •• //=ir).r.'n.-, 
 
 ( 3-=12, '• V=ir).184 " 'v=l').18+ "J 
 
 • .\ii appro.viniate trapezoid, one ottbe bases of which reduce:- to zerc> 
 ;ti'.l n'tair.s the name of trapezoid. 
 
AKTK 1,1-; U. 
 
 255 
 
 into !S('^- 
 
 (1-"). !IU(1 
 Ix.tli COll- 
 
 hI by (;i), 
 < likewise 
 
 erv ucjirlv 
 is yji'iiVCMl 
 
 j/v.x-.sTf/ Ay 
 
 /// /.s (jirt II- 
 
 fhr juira- 
 
 ctjiiatioit 
 
 nil , 
 
 -1(1, (/M) 
 
 .rapozoid.'"' 
 
 be succes- 
 iiis : 
 
 ! 2, 
 
 
 (>) 
 
 .III, 
 
 
 (^) 
 
 ■-> *> 
 
 
 (■•) 
 
 =.i.'- 
 
 4 
 
 
 >j--^(\t 
 
 + / 
 
 1(1, 
 
 
 ('0 
 
 ',m; 
 
 iti 
 
 (a). 
 
 A'^:^ 
 
 
 li 
 
 ■J ( 1 r. 
 
 
 .• 
 
 84 
 
 
 " * 
 
 llCtr 
 
 to zero 
 
 Add iiu tlic unliHute.- n!' ( /// ) and tliox' <•!:'(/() seitaictrly, 
 we will lind ().(MH.") ior tli*- dift'ciciice of their Miin> : therefore, 
 these seiinieiits arc \vy\ nearly e(|iuil.'=^ 
 
 <S. Sell. I r t he s(vj;nient> \ ni ) and (u) l>e traced.it will he 
 found tit s'njiit. tliat each ol' i licni cijniiiriscs ali'mt .'](l°. 
 
 This value niav !jc I'nund mat hematieallv as follows : 
 
 Miffcrentiatini:' //== I ,,'•,, ^,(.r= — [s.,--|- oTCt,/- ) -j- 1 '•• we have : 
 
 '^^-^ ^L(4./''— 144./-+li:>L'.r,^ ■'■ ,.,-_;u;.;;.;-i>sS), 
 ilx 4(MMr 1(10(1 " 
 
 for tlie tan<^eiit of ihe arc \vhu.<c abscissa is ./■ (Art, 4). 
 T?ut, in the ('(luation ' ■'^zi:^ ' (x- — l:{(U'+ 2.SS), 
 
 1 
 
 wlien 
 
 [./•=: 8. then ''^ = (f.r)12=tan27^^ nearly (Table HI). 
 
 
 Therefore, the i^iven are nii) comprises about '11'-'. l)etween 
 its limits ,/r::::S and .iz=z\'l. \\\ the same process, it will betound 
 tinit the s(\uinent {ii) also comprises about '11'. between the 
 same limits. 
 
 t), Kr(»m what jirecedes (7), we nniy dtuluct' the followin*.;; 
 Rule (M), to divide, into elementary seiinients. any plane 
 figure whose curve, // = /"(,(). i,< not contained in 
 // = ( >.r' -f /y.j;'-' 4- rx -f ./". 
 
 •• 1" J)irlde the fujiiri' /rom ()■' /'<d=(M(('^ cr (i(l°) liiti) si>(j- 
 iiiriif.s ii'hnsr fjiisis sli'i/l III III j-^ii ui/ii'iiinr fo flf (i.iisoj'}^ 
 
 I ii.n's of thi' cum:), 
 
 2* JJiviih' f/ir rimni III ii'l [I trt nf tin JiiJKi'' info scymcnts 
 ir/iosa hilsrs sllii/l f"- iiiii'illlil til tlir n.ns iif tin mm'. 
 
 .')" Krt'fij S)'<jnii'iit must III tmi /imil to siirU hiulls o.i to allow 
 its mnliiin si'iiimi. oornlhl tn its Ixisi^s., t<i <Iii*iili it Into tico 
 liiir/sii/fii-tiini. lit host rii'iijlili/. fin funu nl <t t r<i ju ianl. 
 
 4' Xo iiVifnii'llt hiiixf rniitiii n i! jini lit III' i ii ]h .liiiii irifhilt itn 
 
 II nuts. 
 
 * The pxaot ilcgiec of apprn.Kiiii itu)ii of the segmt'iita (ni)aiid (h) will 
 
 li'- '. ivca ill Prol). ."i. tnillier on. 
 
 fln^^i 
 
 .^1 
 
 -•?T 
 
 
25!; 
 
 APPIIOXI.MATE r.iExsnr.'iTiox. 
 
 lO. 
 
 THEOKE3I. 
 
 If a 2ylo.ne figure, whose curve is represented hi/ (inij equation 
 (l>) (Art. 5, 1"), he divided into elementary segments (hi/ rule 
 M, Art. 9), the area of a segment of two parallel hases mat/ he 
 measured very approximately hy the formula 
 A = ^A(H4-4B'-|-b); A r<prcscntinq the <irea, B and b the 
 parallel hascs, B' the median section parallel to the hases, and 
 IT the altitude of the segment. 
 
 Hyp. ^Issume j::=l and .r=H to bo tlie limits of an elemen- 
 tary segment ; 
 
 xVsT. then will A = ^II ( B -j- iB' -\- h ) very approximately. 
 
 Dem. In the equation y=f(.v') which represents the curve 
 of the given segment, 
 
 x=0, let ^ = 0, 
 
 , . x=l, let y=:m — 2Gj). or h, upper ])ase, 
 
 x=2. let y=m — 4C«, or B , median section, 
 
 x = 8, let y = m — 54??. or B, lower base. 
 
 Introducing successively the values of these four pairs of 
 
 co-ordinates in y^^ajJ-^-^-hx'-^cx-i-J] and then proceeding as in 
 
 Prob. of Art. 7, there obtains y=zn(x''^ — 27.r)-\-vi, (c)--- 
 
 In this equation (c). as in the given curve y=zf(x), 
 
 r.r = l, then /y = 7?i — 2t)». or /;, upper base. 
 
 when -| ./•:=2, •• // = w? Vhi, or 7^'. median section. 
 
 (^,/'=.'^, •' y=ni — 54?i, or 7i. lower base. 
 
 That is, between the limits .rr=l anda::=r:3, the twoseginuiit • 
 compared have equal altitudes (I>—1 --2), equal median s(M'' 
 tions, and e<[ual bases each to each. Besides, these sognicnt.-^ 
 containing neither any point of inflexion, nor any maximum m 
 minimum ordinate, within their limits, all the ordinates that 
 may be drawn between tlieir parallel bases regularly increase 
 from the lesser to the greater base. But, since tlie ordinates 
 regularly increase and are etjual each to eacli when they 
 
 * la \\\\^ ciit'vo (''). tlie abscis.->a .i-.-=i) corre.'jpomis to tlie point nt 
 intiexion. and the abscissas .c=rt^ oorrespoud to the luaximum and mini- 
 mum ordmates. 
 
quntion 
 
 (hij rule 
 
 mat/ he 
 
 nd h thr 
 'ses, and 
 
 [ elemen- 
 
 ately. 
 iie curve 
 
 pairs of 
 Yuvi as ill 
 
 11. 
 
 -I'U'uieiit • 
 iaii s(M'' 
 
 sov:ni(Mit> 
 iinuiu or 
 
 ates that, 
 increase 
 ordinate^ 
 \en thev 
 
 loiut 
 
 ARTICLE 11. 
 
 2:u 
 
 and iiiiiii- 
 
 become the bases and the median sections of these segments, it 
 is obvious that any two ordinates corresponding to the same 
 altitude (same abscissa) in the latter are very nearly equal ; 
 hence, these two segments compared are very nearly ecjual. 
 
 By a similar demonstration, it may be proved that any other 
 elementary segment whose curve is represented by (b) is very 
 nearly equal to an elementary segment whose curve is repre- 
 sented by (a); but, the area of the latter is measured exactly 
 by the formula ^//(i^4-45' + />)(Book III, 54): therefore 
 the area of the former may be measured very approximately 
 by the same formula. Q. E. D. 
 
 Cor. Ill the pl-ane figure whose curve is represented bij (l>), 
 the area of an elementary segment of one 6a«e (B) may he 
 measured very approximately hy the formula 
 
 A=pi(B + 4B'-fO). 
 
 Sch. I. The smaller the number of degrees comprised in the 
 arc of an elementary segment, the higher will be the approxima- 
 tion obtained by ^//(i?4-4ZJ'-f 6). 
 
 Sch. II. The formula }Jl{B-\-'iB' -\-h), applied to the 
 elementary segments into v/hich may be divided any plane 
 figure whose curve is represented by //-=/'(x), will always give, 
 for the area of this figure, an approximation practically equi- 
 valent to mathematical accuracy. 
 
 This formula then is a convenient substitute for the exact 
 formula^ wlignever the latter cannot be employed. 
 
 11. Lengths of the rriiVEs y-=(ix''-\-],x--\-cx-\-f 
 
 AND OF ANY OTHER CURVE COMPARED. 
 
 Placing the origin at the poixit in which the co-ordinates of 
 the curves y''=.ax'* -{-hx^ -\- ex -{• f simultaneously vanish, and 
 substituting zero for .land y, there obtains 0=^0 + /". 
 
 In this hypothesis then, this collective expression reduces to 
 
 y'z=ax^-\-h.i^-\-cx^ of y=dt:\^ax^-\-bj~-\-cx. (1) 
 
 Diflferentiating y'^=^ax'^-\-hx'-\-cx^ there obtains : 
 
 2ydyz={3ax'-^2bx + c)dx\ whence, _a-. "" ^ 
 
 ,y f 
 
 I It 
 
 dx 
 
 ^y 
 
 17 
 
■;i '' 
 
 iii 
 
 Jfli 
 
 ■■'M li 
 
 258 
 
 APPROXIMATE ."iIEXSrUATION. 
 
 wliicli represents the tangent ti» tlio curve. ;'.t the point whose 
 tibsci'-s:i is .;• (Art. 4). 
 
 When the tanjent reehices to zere in heconiinu' jiavallel t(v 
 
 the axis of A', there obtains 
 
 
 -.() 
 
 that is. Vi(ix--\- 'lJ>x-\-C' \) ; whence. 
 
 — ]>diz\^lr — )\n<' 
 
 ?M 
 
 ■ (-'' 
 
 When the point of tangency is at IXT'^, tlie tangent is infinite 
 
 .m • ... , (''// ?>:lX--\-'lJ)X-\-C 
 
 (irig. _,) j ; tlien — 'rz; x = — 
 
 Ix 
 
 21' 
 
 ; whence i'nzO • 
 
 but (1;, r==tN/a.V''+6A-+cA ; lience, zhv/a.A-'+/;.\-+c.\'=0, 
 or, squaring and factoring A'(<«A'--4-/>A-f-<") = 0; 
 
 (3) 
 
 whence, 
 
 A'=0, or 
 
 7;±v/6-— 4 
 
 ac 
 
 'la 
 
 Subtracting (3) from (2), we liave : {x—\)- 
 
 ■ hzh\//r—:i,ic 
 
 3<t 
 
 or 
 
 {X— A ; = 
 
 0((r 
 
 ■hdz\/b' — 4<ic 
 
 9, 
 
 a 
 
 hzt2^0'—Sac =f- 8n/6-— 4ac 
 
 <6a 
 
 for the projection of the curve (from 0° todb90°), upon the 
 axis of A'. 
 
 yine-e. tlie coefficients o. h. c. can assttme any value, the pro- 
 jection ix — X) of any curve containeil in //- — :.'/. /■''-j-iaj'4-<^-'-'. 
 (from 0° tortitO'-^), up<in the axis ot' .\. can also assume any 
 value ; hence, tlie arc of these curves, from U ' torhOO", can 
 assume any length, as tiiis arc necctSsarily varies in length with 
 its projection. 
 
 12. Tlierefore, the curves /y'=a.r''-}-Z;./--i-c,y (n) 
 can a.ssume the lengtii of any other curve Y--=F{X), {\}) 
 between the same limits (>'-' and !tO-'. 
 
 13. Sch. I. . To make the curves (a) and (1)) coincide,, 
 both in decrrees and leni:th. tliere i^- never need of takinirthe 
 
i fi 
 
 ARTICLE 15. 
 
 •Ibd 
 
 diroction of curvature into .account ; there is never need either 
 of invertiiruthe co-ordinates of the curve, contrary u'what has 
 Ik'Ch <loiic' ill Art. 5. 
 
 .14. Sch. II. Micxlrtiiiiu. and minimimi fifdinutr.s of the 
 
 '■I' rrcs y"=ax'' -r bx--(- ex. 
 
 In eii. union {'I) of Art. 11, the two roots a'rrr 
 
 —hdtz\/h--'6ac 
 
 reprt'sent the abscissas correspondinir to the maximum and 
 niiniuuim ordinates of the curves. 
 
 Thus, differentiating ^' J^ 
 
 //- = -j-V(-''''^ — 18.t-+9C.r), equation of 
 the pure campana, one half of which is 
 Onl, there obtains : 
 
 ill/ _rU-— 3G.i'4-0() . . 
 
 dx lihj o7f 6 c c/ "^ 
 
 But, when the tangent to tliat curve, at the points h' and » 
 
 becomes parallel to the axis of A", there obtains 1-^= 0, and 
 
 dx 
 
 dy _ 3r-— 36x'+9() 
 
 ^0, 
 
 e(|uation (n) reduces to 
 
 ^ dx 20.y 
 
 or ,r- — l 2j--|- 32-^0 ; whence, ;c = 4or8. 
 
 The maximum and minimum ordinates correspondin"' to 
 these abscissas. a:=:4and .rn=8, may now bo found as foMows : 
 
 Tn the iriven ecjuation ?/-=-jipr(.r-— 18x-f9G), 
 
 ( ./•=:4, or Ob, then y=±4, or drmaximum hh\ 
 when I • 
 
 (^ ,i.;=S, (.r Of/. '• // = rtv/12..S. oriminimum fi;). 
 
 Sch. III. The curve reaches 0'-' at tiic vortex of its max- 
 imum and of its minimum ordinate. 
 
 15. PlLT'.MENTAllY FIU'STUAr OF A SoF.lD 
 
 (IF IlEVoHTlo.V AM) OF A l'ol,YEI)RolI>. 
 
 Any }»lane fiLnirc whose curve is represented by the equation 
 (a) or (b) of xVrt. 12. can be divided into 'such segments that 
 each oi' them 
 
 
 .1 
 
260 
 
 APPROXIMATE MENSURATION. 
 
 ''■}■ 
 
 1" shall contain, within its limits, neither any point of 
 inflexion, nor any maximum or minimum ordinate ; 
 
 2 " shall he divisible, hy its median section, parallel to its 
 hases, into two approximate trapezoids.^ 
 
 The solids irenerated by the revolution of these segment t* 
 about their axis of A' will be called elementary frusta of a 
 solid of revolution, and the frusta of a polyedroid circum- 
 scribed about these elementary frusta of a solid of revolution 
 will be called elementary frusta of a polyedroid (an elemen- 
 tary frustum, in which one base is reduced to zero, becomes an 
 elementary segment). 
 
 When the curvilinear figures specified by the equations (a) 
 and (b), of Art. 12, are divided by their maxima and minima 
 ordinates^and by the ordinates corresponding to their points of 
 inflexion, it is obvious that their segments fulfill the first above 
 condition (1"), and if the second one is not yet complied with, 
 they may be subdivided into smaller segments which then will 
 fulfill both conditions (1" & 2') of an elementary segment. 
 
 Now, any elementary segment specified by (b) is very nearly 
 equal to an elementary segment specified by (a), as is proved in 
 Art. 18, further on, and illustrated in the following 
 
 IG. Problem. In the jilctne figure whose curve is expressed by 
 the equation y-=:-2-^-^x^(x^ — -lOx + 'lOO), an elementary segment 
 is given between the limits x = (> and x = 10; to deduce, from 
 y^=ax^-|-bx"-i-cx-f-f, a numerical equation representing the 
 curve of a segment very nearly equal to the given one. 
 
 Solution^ In the given equation 
 
 y'=TiTr^^'(^'-40x+400), (m) 
 
 '»= 0, then y =0 
 x= 6, '' y —^^-^o Mat approximate trapezoid, 
 
 ^rc=10', 
 
 when 
 
 
 y2= 50.00 S 2nd 
 
 * As the trapezoids into which a curvilinear plane figure is ^ided in 
 order to obtain a rough approximation of its area by the formula of the 
 trapezoid. 
 
ARTICLE 17. 
 
 261 
 
 1 
 
 If these four pairs of numerical values be successively 
 introduced in y-=ajc^+6x-+cu-f-/, there obtains : 
 
 1", 0=0+/, 
 
 2", 35.'28=a6» +66'^ -f-c 6, or 36<x+ C6-|-c=5.88 (1> 
 
 3 , 46.08=o8» +682 +c 8, or 64a+ 86+c=5.76 (2) 
 
 4", 50 =a(10)» + 6(10)2+clO. or 100a+106+o=s6.00 (3) 
 
 Solving (1), (2) and (3), we have : a=— 0.08, fc=1.06, c=2.40 . 
 
 Introducing these values of «, h, c,/, in i/'^=aoc^ + bxr + cx-\-ft 
 there obtains, for the required equation : 
 
 f x= C, then 7/^ 35.28 in (w), and y^= 35.28 in (n) 
 
 »/ =±6.434675 " " »/ =±6.4265 ' 
 
 z= i, 
 
 when ■{ x= 8, ' 
 
 »/-'= 46 (IH 
 
 •' i/^= 46.08 
 
 >( 
 
 1 ^= », ' 
 
 • // =±7. )(!(t357 " 
 
 •' »/=±7j01 
 
 .( 
 
 U=io, ' 
 
 _*/-= 5;) •• 
 
 " y-'= 50 
 
 i( 
 
 For. 
 
 Adding the ordinates of(»i)and tho.se of (?t) separately, we shall 
 tind ().0014<)8, for the difference of their sums ; therefore, 
 these segments are very nearly equal. 
 
 ITt Sch. Ifthe.se segments (m) and (n) be traced, it will 
 be found at sight, that each of them comprises about 30°. 
 Thi.s value may be found mathematically as follows : 
 Differentiating the equation, i/ = -^^jj{x*—40x^-\-^0Qx^), of 
 segment (m), there obtains : 2i/di/=^^-^{4x'^—120x--\-800x)dx ; 
 
 whence ^ <^ — 
 
 = ±^ X ==fc: (x— 10), for 
 
 dx 10 .r-20 10^ '' 
 
 the 
 
 tangent to the curve (m), at the point whose abscissa is x 
 
 (Art. 4). 
 
 dx 
 
 But, in this equation 
 
 :j'^^2{x-U)), 
 
 when J 
 
 \x= 6, then ^i(=0.56568=tan.30°, nearly. (Table III) 
 
 x=10. 
 
 dx 
 
 Hence, the given arc (m) comprises about 30'', between the 
 limits x=:G and x=10. 
 
 By the same process, it will be found that the arc (n) also 
 comprises obout 30°, between the same limits. 
 
 ■ 
 
 I 
 
 I. 
 
 i; 
 
I'-l 
 
 » 
 
 'It 
 
 ! 
 
 • , I 
 
 
 « I, 
 
 ' '1. 
 
 1 ll 
 
 1«. 
 
 i*:i-.tl 
 
 2f;2 
 
 AlTMltiXIMATE ,M KXSrHATrON'. 
 
 IS. TUKOKEM. 
 
 //' ll sdHiI- of I'irniiitiiiti. spici/ict/ />i/(iii tqii<(/i<ni Y = F(X) 
 (>[' Art. 12. hi' (lir'nlcil Intn ilniiriihirn friistn (l.'>), tlie rohimc 
 of iiiili frnatnm min/ hr mntsiir'dvryi/ upproximnti'lii In/ tin 
 formidii V = ,\HriY--f4Y'--f y-): V rejirrficntiinj flir nnhnm . 
 -Y' 'Did -y- fhrh'tsrs pn'/>niilicii/>i r f" fhr 'i.ris. -Y'' the rm- 
 diitn srrtinn jxinilli/ to tin /kisis, (iiid H thr dltiti'ih \ of tlw 
 fnisfiiii;. 
 
 Hvi'. Assuiiio .i'=z\. ;ui(l x=:zn to lie tlu' limits of any 
 eleiiu'iitaiy fVustuiu ; 
 
 AsT. tluMi will ]^=i//,T(r- + -n"- + //-) veryiipproxiinately. 
 
 Dem. Iti th'.' equati(Mi Y=F{X) of the curve ol' tlii> 
 frustiun, 
 
 i x=0. let //- = <». 
 
 when 
 
 [ ;r=l, " ij-=^m — 2()», 
 
 
 
 y'-z=')n — {Cm, 
 r, = .\ •' >f'=^>yi — •>-!•''• 
 Introduciii : -ucce.ssivi'iy these I'our pair.< ofnunierical values, 
 in y-=zax^-\-lKrr-{-cx-\-J\-d\\'\ then iiroeeoding as in Art. !(!, 
 there obtuNis i/-r=:nQi''' — 21.c) -j-m. (c)^' 
 
 In this curve {r). as in the given curve I'm:/' (A'), 
 r.rr^l, there obtains y-=m — 2(>/^ 
 when -■ ,/==2, " ip = m—4C)n. 
 
 (^./•=:8, " j/-=m — T)!)/. 
 
 That is, between the limits x=l and .»=;]. the two gene- 
 rating segments of the frusta have equal altitudes (3 — 1=2), 
 equal median sections and equal bases, each to each. 
 
 Besides, tlie segments containing neither any point of 
 inflexion, nor any maximum or minimum ordinate, within their 
 limits, all the ordinates that may be drawn between their 
 parallel bases regularly increase, from the lesser to the greater 
 base. 
 
 * The abscissa a:=0 conespondrf to the point of inttexio!!, in the carve 
 (c), and the abscissas x=-\- 3 correspond to its niaxinmm and minimum 
 ordinates. 
 
ARTICLE IS. 
 
 203 
 
 
 But. wince the ordhiiitos reirulurly iiicre-;se and arc efjiial each 
 to C'lic'li when tht'V beooiiu' tlic ]);iscs ami tlio niciliaii sections 
 of these se;^ineiits. it is nlivimi^. that any two oidinates eorres 
 j)oiHlintr to tiie same altitude (>au»e absci>sa). in tlie hitter, are 
 very netirly equal ; lieiicc. the two si'unients conipared ai-e very 
 iiearlv e'Hiiil : tiierel'oie. tlic two i'ruMa generated 1)V the revo- 
 hition of tiiesi' s( i:iiH'iit^. al'out tiic fixed ajiis ol' ^A'. are also 
 very nearly ('i|ua!. 
 
 By a similar ilem')n>trat ion. it niav l)e ]>rovcdinat anvotlier 
 elementary Irustuni of a solid of revolution sitt'eilted hv 
 l'=F(X) is vci-y nearly iMjual to an elementary frustum of a 
 solid of I'evolution >iieeitied hy /y -'=:* /.v- ' -j- A. ;-'-(-( •./•-f- /" : hut, 
 the Volume of the latt'-i [.< measured c.cdctli/ by the i'ormula 
 /;//-( J'- rd] '"-{-,//-;( l^ook JIT. (lOi: therefoic. the volume of 
 tlu! former is measureil very apjiroximately by the sanie fi>rmula. 
 
 Co". 1. Tlir I'riiatiim "f (Ik- pi)/i/ii/rni(f cirfnuisrri/jff^ dhmif 
 flit: I Idui'iifdi'ii t I'li'^fii Hi '>t' fli'' •"''"/»"' I'/' rrrn/iifiiiji iijifi'l^inJ In/ 
 Y=F(X) initij III iiH iisn /■/ il II rji (I iipyiKriindtcIijhij the formuhi 
 
 V'.4i{(nj-4B'-fb). 
 
 For. rr=j\//(r}'- + 4r:r--r-/r)=>/rzz^/,^j//f/,' + .l/?' + b) 
 (Book in. (■.(),l^; but. it has just liecn proved that the volume 
 ( V) of an elementary frustum of a >olid of revolution specified 
 by y^=-F(X) may be measured ai)proximitely by the formula 
 1'=:j\//-( J'-'-f4}"--|-//"'); therefore, the volume (T'^ofan 
 elementtiry frustum of tlir polyedroid cireumscribed about this 
 frustum of a solid of revolution may a!>o be measured very 
 appr(.)Xjnato]y by the formula V'=l///t 11 \/>'-j-lj). 
 
 Cor. 2, riif vojinni' {\ ) of miii rh mtntarij segmmit of a 
 solid of n CO ut'ion spectfitd hij \z=V X), mid the volume (V) 
 of the elementury segment of fhi- rirnimscrllied poli/edroid arc 
 measured very approximate!// h>/ the formula: 
 y=:^J'HT(Y- + 4Y'-4-0), and Y':=:z}.,ll[B + AB -rO), respcc- 
 tivehj. 
 
 Sell. I. The smaller the number of degrees comprised in 
 the element of contact of an elementary frustum or segment, 
 the higher will be the approximation obtained by the mono- 
 formula. 
 
 f* 
 
 tg 
 
264 
 
 APPROXIMATE MENRTTRATION. 
 
 Sch. II. The rnonoformula applied to the elementary frusta 
 and segments of a solid of revolution or of a polyedroid whose 
 curve is represented by y=F(X), will always give, for the 
 volumes of these solids, an approximation practically o(jui- 
 valent to mathematical accuracy. This formula then is a con- 
 venient substitute for the exact formula, whenever the latter 
 cannot be employed. 
 
 19. Assume the function y=f{z) (6) 
 
 to represent the equation of any plane curve, except the 
 
 equations contained in y^=-az^-\-h'^-\-cz-\-f, (a) 
 and let it be required to prove the following 
 
 THEOREM. 
 
 If the, curve of a right polyedroid he represented by (b), the 
 convex surfaces of the elementary frusta of this polyedroid and 
 of the inscribed solid of revolution may be measured very 
 approximately by the monoformula. 
 
 Hyp. Assume (/>) to be the e({uation ^ 
 of the curve AOD of aright polyedroid / 
 circumscribed about the solid generated 
 by the revolution of the plane ABC 
 about the fixed axis AB ; *" M £ S 
 
 AsT. then may the convex surfaces of the elementary frusta 
 of both solids be measured very approximately by the mono- 
 formula. 
 
 Dem. Let the portion ACD of the convex surface of the 
 polyedroid be bounded by the lateral edge AOD^ the ordinate 
 CD of the curve AOD, and the element of contact ANC. 
 
 Now, if this convex surface ACD be develope^l upon a plane 
 it will become the curvilinear plane figure specified by the 
 function y=/(x) of Art, 10. But, it has been proved, in that 
 article, that the elementary segments into which this plane 
 figure can be divided (by rule J/, Art. 9) may be measured 
 very approximately by the monoformula : hence, if the portion 
 ACD of the lateral surface of the polyedroid be divided into 
 
 ' -i^ \ < 
 
ARTICLE 19. 
 
 2C.5 
 
 elementary sef^ments, each one (a** DCNO) of these segments 
 may be measured very approximately by the formula 
 
 .4':=:i7(Y4-4Y' + y).* 
 By Art. 72, 3" and Cor. 1, Book III, 
 
 hence, since a lateral face {DCNO) of any elementary frustum 
 of the rijrht polyedroid may be measured very approximately 
 by the monoformula .4'=^//'(Y4-4Y -f y), the whole lateral 
 surface of this frustum may be measured very approximately 
 by the formula .l=J//(7*-}- 4/^' ijj), and the convex surface 
 of the elementary frustum of the inscribed solid of revolution 
 may also be measured very approximately by the formula 
 
 .V=^//(6'.+ 4(7'-f c). Q. E. D. 
 
 Cor. 7'Ae convex surfaces iS and A of any elementary segment^ 
 of the solid of revolution and of the circnmscrihed polyedroid 
 specified by (b) may be measured very approximately by the 
 formulw S = ^H(C + 4C -fO)«n(/ A=«:^H(P + 4P' + 0)', res- 
 pectively. 
 
 Sch. I. The smaller the number of degrees comprised in the 
 element of contact of these elementary segments and frusta, 
 the higher will be the approximation obtained by the mono- 
 formula. 
 
 Sch. II. The monoformula applied to the convex surface of 
 the elementary frustum or segment of a solid of revolution or 
 of a .polyedroid specified by the above equation (6), will al- 
 ways give, for the area of this convex surface, an approxima- 
 tion practically equivalent to mathematical accuracy. 
 
 This formula then is a convenient substitute for the exact 
 formula, whenever the latter cannot be employed. 
 
 I 
 
 • Since the curve AOD of this surface ACD cannot transcend the 
 limitsH-60''(a maximum value wliich it can reach in rto other solid than 
 
 41 
 
 in the right triangular polyedroid, Art. 21), there is never need of invert- 
 ing its co-ordinates. 
 
 
ii 
 
 1" 
 
 
 1- i{i 
 
 
 '< : ! 
 
 11 
 
 I 
 
 2yk) 
 
 . *^0. 
 
 A I'l'lKtX r:\I.\TE M CNSURATION. 
 
 General .Scholium. 
 
 Till' jtrocess of" ai)]»r()xiniution indicated in Sch. IT of 
 Art. 1(1. 1 S and !!•. must not lie mistaken for tlie common pro- 
 cess which consists of dividi)!}.': a <-urvilinear plane figure into 
 approximate tr:ij)ez()ids that are measured by tin; formula of 
 the trapt.'zoid ; and of dividing a solid into approximate fru;:ta 
 of a cone or of a pyramid, the V(jlumes and convex surfaces of 
 which are measured by their respective formula). 
 
 By applying these various formultx) in the some circum- 
 stances as the monoformula, it is easily ascertain that the 
 approximation obtained by the latter has such a superiority 
 over the result obtained by the former that tlio monoformula 
 forces itself, to tlie exclusion of the other ai)proxinuite for- 
 mulae (Art. 27. ]^-ob. :J. 4. 5. Ck 7, 12, IH. M, 15, 17, 18). 
 
 31. Equations contain k\ » in ij'- — hx'' -\- b.r'-rcx' -f ex -f-/'. 
 
 The above e((U:itio!! h;is four ]o.)ts which niay ])e either 
 
 real or imaginarv. Ik'notiuii these four roots by A', I, tn. ii.Wi.' 
 shall \\\\o. the lollowiug combinations : 
 1" Four real and e<(ual roots : 
 
 1 
 
 P 
 2f' Four real roots, three of whicli are equal : 
 
 i/--= (,)■ — l)if(X — li). 
 
 l> 
 3" Four real roots, two* of whicdi are equal : 
 
 '/'"= ix—lr^x — m ){x — «). 
 
 4" Four real and unequal roo.ts : 
 
 y-~ ix'—k}IX — I)i.v—m)(.i 
 V 
 kc. &c. 
 
 ■n). 
 
 (a) 
 
 ih 
 
 (0) 
 
 (d) 
 
 Proceeding, as in J3ook III, U', to sim))lify, v. g. the equa- 
 tion {€), there obtains : 
 

 •MITir-M.; 
 
 r-' \ 
 
 x'(j: — rii)(x—n)=' •, .r-—(vi-{-)i).f-\- 
 
 r 
 
 Wll 
 
 ] 
 
 207 
 
 (0 
 
 Tliis curve (e) is raised, parallel to itself, the heiulit /' 
 nbuve its axis of A. by addiii'j, the constant /to its ordinate, as 
 
 foil 
 
 ows 
 
 r= 
 
 I' 
 
 1 
 
 '■- — (^iit-\-i) ).i -^nin ■ -^/ 
 
 2*-^. ivji:ATI(»NS CONTAINKD FN y 
 
 ) 
 
 = a.r'4-L 
 
 'j.r-f-<'x 
 
 iiJ) 
 
 •-\->'X-{-t. 
 
 The above equation admits of the same < "'.(binations of rttots 
 the e(iuation of Art. '27) : heneo. it contains the folhjwinu' 
 
 equat 
 
 ions 
 
 
 //= 
 
 ■«)' 
 
 (a') 
 
 
 2o 
 
 ?/== .{.r—mVi.r—n^ 
 
 i^n 
 
 KC. 
 
 :t."ve 
 
 Proceediniz' as in Art. 2."). theri ibtains 
 
 X- \ 
 
 ) 
 
 y = '^- - .>•■—(/» + ;/ ).x + m« • +/ 
 
 P 
 
 i 
 
 J 
 
 *>:a 
 
 rilolJLKM. 
 
 (9') 
 
 To find tJie fornviJii oi h< ri>Jum>' >>/ " ^oliil of rrvahttum 
 
 whose curve is rcjin^o ii.'i 
 
 ,/ /- 
 
 1 
 
 ) 
 
 ( 
 
 (?n4-n).r'4- '""■'" . -r./ 
 
 (^) 
 
 Solution. Introducing' this value of //-. in formula 
 
 dV= -if (U\Book 
 
 III. H!)'!. then" obtains f-.r differential of 
 
 the volume of a solid of revolution specified by {(j): 
 
 dV 
 
 - \ 
 
 I 
 
 \ 
 
 .(•'—(/»* + « ).'"'4- 
 
 vin.r 
 
 ) 
 
 ,]x^-fd.v. 
 
 To indicate intetiratitm. write 
 
 S'^U 
 
 I x^d.r—{m+n)jMjL-^-mn.rdj: -f 
 
 I 
 
 ^f 
 
 dx. 
 
f 
 
 
 
 if ;: 
 
 r^ 
 
 I 'if • 
 
 11 :■ 
 
 •Si- 
 
 208 APPROXIMATE MENSURATION. 
 
 Integrating (Book III, 44), there obtains : 
 
 p ( 5 4 o J 
 
 Since, this volume V and its altitude x simultaneously 
 vanish, then C=0 ; therefore, the entire integral, or volume of 
 the solid of revolution specified by {g),is 
 
 V=^ \ 12x'^lb(m-^n)x+20mn \ +7:fx. 
 6 dp ( ) 
 
 (A> 
 
 24. Problem. 
 
 Assume the curve of a right quadrangular poly edro id to be 
 represmted hy y= ^ z* — (m+njz^+mnz- |- +f (g) 
 
 andjind the foniiula of the convex surface of the inscribed 
 solid of refolution. 
 
 Solution. Introducing this value of // in dS=''27rydz 
 (Book III, o8), there obtains, for differential of the convex 
 surface generated by the revolution of the curve s : 
 
 ci>S': 
 
 •?- 
 
 /' 
 
 5* — (m + n^s^-i-mnz' > dz-j-2T:fdz. 
 
 ■} 
 
 Tc indicate integration, write : 
 
 ■] z*ds — (m+n)z^dz+mnz^dz i -\-2rf I dz 
 
 Integrating (Book III, 44), there obtains : 
 
 -3 1 _ 
 
 'fz + C. 
 
 .(■--7/ 
 
 p ( 5 4 6 ) 
 
 The constant C is equal to zero, since the surface >S* and its 
 generating curve z simultaneously vanish ; therefore, the 
 entire integral, or formula of the convex surface generated by 
 the revolution of the curve 2;, is 
 
 S=— I 122--15(m+«)z4-20mn \ ^2i:fz, (K) 
 
 30/) ( ) 
 
 h ' 
 
AUTICLE 
 
 200 
 
 KM 
 
 J55. ANALYTICAL PROBLEMS 
 
 WITH SOLUTIONS COMPARED. 
 
 (Approximate Monoformula applied). 
 Prob. 1 . Let DK be a chord parallel to the diameter AB inthe circle AP6 
 to radius (R)=10, and find the area (A) of the s^egment 
 ABDE whose altitude ("u=-4. 
 
 Solution. In equation, yssv^i?-— .c-=v^lOO— z- 
 (origin at the center C) 
 
 I X =0. then Ji =zy/R^—x' =\/l (To=l 0.000000 ) 
 
 I Xj = l, <• ?/i=v/lOO— \=z\^~Wi=z 9.949875 |- Ist segment 
 
 wat^'i < a:^=2, '• y.,=\^lm— 4=\/ 96= 9.797066 | CBom 
 
 I ^3=3, - !/,=x/ liH)- 9==v/ 91= 9.539381 I 2nd segment 
 
 \ x^=4, '• ?/^=VlO()— 1G=V 84= 9.165160 j moDn. 
 
 1" By the exact formula. If a radius be drawn to tiic point Z). the 
 .segment CBDn will bi- divided into a triangle CDn (who.se area is 
 \CnXL>n) and a sector BCD (whose anca is "IRy^ATC BD). hence, area 
 (A) of ABBE (the double of CBJJn}=snx BDxIi+ Cnx^n. 
 
 But, P-1 or A=0-4=sia 'J3^ 34' 41".5. . 
 
 and the length of arc BD (of 23^ 34'4I ".r))=4. 1 1 .". 1 :•^(;iook [11. 50. Ex.); 
 
 h?nce. i?Xarc 7>'yv=:i)X4.1 151 7=41. 15170 ; 
 
 i),'sides, CnXDn=x^x>/l= 4X9.16516=36.436064; 
 
 therefore, by exact formula, iJxarci¥/>-|-^'"X Dn, or ^4=:77. 81234 . 
 
 2" By thp monoformula, J=J.//(/^-f-4/i' + h). 
 
 f Dn=>/^= 9.165160. or A, upper base, 
 In segment CBDn\ mo:=7/.,=: 9.797966, ovB'. mvdian section, 
 
 ( CB=^y=\i) 
 Cn=*4=4= 
 
 hence, area of CBDn or .4'= I// { 4^' 
 
 ( ^ 
 that is, by monoformula, .4 or tA' 
 By exact formula, J 
 
 Deficit 
 
 , or B, lower base ; 
 
 //, altitude: 
 
 ( 9.165160 
 ■=i { 39.191864 
 
 ( 10. 
 
 i|X58,357024=77.809365 . 
 3*77.812340 . 
 = 0.002975. 
 
 m 
 
 K, ■ '*■ 
 
 K a 
 
h' 
 
 
 V ■ 
 
 |1' 
 
 W 
 
 hi 
 
 270 
 
 AI'I'IMXI.MA I K MENSlKATJdN. 
 
 If CU1)h 1k' ilividcd iutci two SL'^uu-nts ot fquiLi altitudes 
 
 
 thoi 
 
 r r 7; 
 
 1, aro.'i oi ( />';«<. (I 
 
 uri'ii ()) innJJii. (ii 
 
 .r=.'A 
 
 '- ■'.'/, 
 
 in. 0(1(1 0(10, 
 ;')!». "Hit 5o(», 
 
 .1"='/. 
 
 ( ,r 1 _= f 
 
 '.'.TOT 'JG<.; 
 
 V •, i'/ 
 
 i';;:ri 
 
 !S.l. 
 
 4. 
 
 iG.") KJO; 
 
 .■iU'i-. hy llioiior.. l'(.r-|-.l")=: -•! :=r;X I I'J.TIS J l(j 
 
 ,S120T7, 
 
 iJy tlie '.'Xiiot tormulii. 
 Oeficit 
 
 .-1 
 
 =7 7. SI 2340 
 = O.0O(i20:5 
 
 Prob. 2. Find the arcd of the ni't/mviif DIIFI* iti I he inrcvdini) circle 
 asHiniiinij t/tc chord \)V to he eciuiil to 8, 
 
 SoLiTioN". Ill contbrmity with rule J/ of Ait. '.'.the bases of IjBFD 
 must be parallel to tlieaxi- All. Now. the area of the raixtiliuear triangle 
 Bi)a will be compuied :ii conformity with this rule, if the rectangle 
 CalJii be subtracted from the segment C DDn measured as in I'rob. 1 
 and the result doubled will be the recjuired area. 
 I]y Prob. I. {-"). twice the area of 
 Subtracting twic:- 
 tliere obtains by mouof. area of 
 
 (:;7>'/>«=77.8r2oi 
 
 CaDn or J('«X />«=73.321280, 
 Dl'FI}= 4.41)0797 . 
 lly exact forml. (Trig. Ait. To. Proli, 14). area of DBFD= 4.491060 . 
 Deficit = 0.0002G3 . 
 
 ' Prob. 3. Assumin;/ a=b=G, in equation .?/-= ^{2ax-\-z-) of the hyper- 
 
 a- 
 
 bola (Book IIFH), -l). there obtains y-'=!2.f-fx2 ; 
 
 }on\ let thisei/i/(iti(in rejiresent the hijpcrbola 'HOC. ard 
 
 jiiiil the ■trea (A) tf tin fn'/ine/it !'>(>(' 
 (ID— 4. 
 
 '•hose lib 
 
 iscii-sa 
 
 SoLrTioN. 1" iv the exact formul 
 
 I : 
 
 J- 
 
 -// 
 
 A=i/(a-t-x)— 
 
 ''' ^r..J "."-^-''("-h->-^ } 
 
 M 
 
 10 Lf 
 
 1 
 
 '/// 
 
 In e(iiiiiliou //-=:1 2.c -)-.<-. wiicu .(=1 
 
 \ 
 
 I in. 
 
 (App. E' 
 
 then 
 
 lience. 
 
 !=s,G4-.o-ii>i^ h.gC!>^!!±^'yr^)U4o.4 
 
 (I. -CM 
 
 \ 
 
 ''X*' 
 
 S 
 
 2" Toapidythe monof. tw the area ut\/J<'('fj. this .segment must be turned 
 towards the axis of A', iy nintually changing it; i^o-or'linates (proc edin v 
 
 •s i,n ,\pi'. '". 11. 
 
 aim 
 
 ■h). 
 
AIlTiriiK Zi). 
 
 TA 
 
 
 } - 
 
 ft). I'y imituuily (■li;iM<_''infr I'H-od-onliii.'ites iti .'/"= ('_'//.(■-!- \-'). and 
 
 ?(ilv 11 Li" tlu' result with f'Speot to //. tluM'e oliliiins //=: _ (y/7/'-|— '■' — ^')- 
 
 I) 
 
 //). .S ibtniciiiig ill • coii.stiuil c, 
 
 -). Clianfriuj;' the sijXM of ihc onlinatt 
 
 /) 
 
 ft 
 
 Xiiw, letting' </=//=.-r(;, ^=4 or (>/). in ihisciiuatioii. there ohtiiiii-^ : 
 
 v=l'i — v ;;i;-|-/'. or cquiitiou of the liyporliola J>(JC\ in which />V' is the 
 iiorizotitii! axi^ot'.r, and O/J the v. rtical axi.-? of -|- J- 
 ((*;ijiii ul tiic point />). IJiit, in ('(i-jation ,'/=:l" — v :>(J4--£'". 
 ; r<:=0. or ijoint />. then ;/ =4, or <>/J, 
 
 ^ x=2 •> }/,=;>,. (;Tr,4?.. 
 
 ^^.]„,n x=4. or 7V», '• ?/,^=2.788.S!», or mu, 
 
 I x=i\ " ^3=1.51471. 
 
 ( 2;=8, or TJIj, " ^4=0, ' orpoiiit /> : 
 
 hence, since the altitude (Z??i=/j/j>=77) of each segment is equal to 4, 
 then 
 
 area of DO inn 
 
 ( '> ) { y ) ( 400000, 
 
 ,or.4=i//]4/>^ =^ 4//J 1=5^ 14.70172i 
 i B ] ( //, j ( 2.78889 ; 
 
 f l> ) i 1/,) f 2.78881), 
 
 , or -rr=}Jr] 4// l=.^ ] 4,v, ■ =5 J (3.0o884, 
 ( 7; J ( //, ) ( 0,00000 ; 
 
 area of Bynii 
 
 therefore, 'by monoformula. A=2(A'-\-A'") =1 X 30.338:;4=40.4.- 1 1 2 . 
 Dy the exact formula, A ^40,43. 
 
 Fxcc.^fj = 0.00112. 
 
 Prob. 4. W.'uil would I/O the approximation ohtaincd bij npplning the /or- 
 miila of the trapezoid to I'rob, 3, in the same circumstanees as the inonoj. ? 
 
 Solution. Mea:-!uring two parallel base.?, // a;id ?/p and three inter- 
 mediate sections (//, ,//,,//,,) as in I'roli. ;;. t]i • altitude // of eaeli trape- 
 zoid Avi'l bo 2, and there will obtain for the .area of /fO/i : 
 
 ^7/0/4-//i)4-i//(.'/,+.'/,)+^^0/.4-/':)+V%,i4-,v,>. 
 
 .'/ 1 
 
 4,00000, 
 7.;',")(t8ti. 
 
 Doublin^^ there obtains A=lf-{ 2//j \ =J <{ ,"i.57 77s. 
 
 I 2}/, I I :!.02lt42, 
 
 [ 11 \ i [ 0.00 000 ; 
 
 = 2 X TuTys'b w===3i> .'J 1 fi 1 3 . 
 
 lly the exact, fornuda. .1 =40,4,'). 
 
 Deficit by the formula of the trapezoid = o.r)3.!X7 . 
 
 Excess by the monofoimula. as (k.«t»U2. 
 
 'm. 
 
 .\ 
 
 1! %■ 
 
 II 
 
 m 
 
 •if;! 
 
 m 
 
 f 
 
 I '■'■ '<' I 
 
 li 
 
lomimpiiaa 
 
 070 
 
 APPROXIMATE MENSURATIOX. 
 
 
 n 
 
 m 
 
 
 If' 
 
 5»1 
 
 Prob. 5. Given m=n=24 p=tOuO, f=10, m equations (a ) anrf (b) 0/ 
 Art. ^1 ^ Book III ; rrqiiirfid the area (A) nf a seffmen!. between the limits 
 x=8. and X=l 2 ; 1 " bi/ the exact formula. 2" i// //*« monofonnula, 3" % </t,, 
 formula of the trapezoid. 
 
 Solution. Perfonning tlie substitution indicated, tlu-re obtains, for 
 equation of the curve : //=jJj^2;'^(z-'— 482:-f-o7G)4-10, („) 
 
 and for formula of tlie area of the plane figiiie specified by (a); 
 
 1" By the exact formula (i): 
 
 ^=J7kff'^'(-l''-«0A'4-960)4-loX-,-ij-^5y^r<(j:-'-G0x4-9G())-l0x=59-2512 
 2" In 2/=:;j^y^z2(3;2_4835_j_576)_{_io, (equation (?n) of Art. 7, Prob.) 
 
 x= 8, there obtains y =14.096, 
 
 x= 9, " 2/^ = 14.55625, 
 
 when <; a:=10, '• y, = 14.9, 
 
 a—ll, " 3/3 = 15.11225, 
 
 x=12. 
 
 3/4 = 15.184.. 
 
 Between the limits i=:8, and a;=12. the altitud ■ of the seg. i3(12— 8)=:4 
 hence, A=IH\ 4B' 
 
 1/ ) r 14.096. 
 
 ^^4^2 Hfi '^S-^'O^, 
 y,) (15.184; 
 
 = §X 88.76=59.2533. 
 
 By the exact formula, , A =59.2512 . 
 
 Excess = 0.0021 . 
 
 If this surface bi' divid"d into two segments of equal altitudes, //= 
 
 ( h ) C ,y ") C 14.096 
 ( ^ } 
 
 ( M 
 
 whence, A'-{-A"= A 
 
 A' of 1st seg. 
 
 ^"of2ndseg. = ^//-J 4/i' "^ = 
 
 j-^ 58.225, 
 
 By the exact formula, 
 Excess 
 
 ( 3/2 3 (14.900, 
 
 yA C 14.900, 
 
 4y.; U~H -60.449, 
 
 .V4 ) ( ifii^^-t ; 
 
 =-^X 177.754=59.2513:5 
 
 = 59.2512. 
 
 = 0.00013, 
 
 7>" By measuring two parallel bases (?/and y^) and a median section(//^) 
 the formula of the trapezoid will give : 
 
 ( V •) ( 14.096 ) 
 
 A=U] 2y, > = 'ii 29,800 }■ =59.08. 
 *" ( 3/;) "(15.184) 
 
 Deficit by the formula of the trapezoid 
 Excess by the monoformula 
 
 = 0.1712 
 = 0,0021 
 
ARTICLE 25. 
 
 27:; 
 
 and (b) of 
 
 the limits 
 
 ', 3" by the 
 
 (tains, for 
 
 (") 
 
 ;=r)9-2512 
 t. 7, Prob.) 
 
 12— a)=4 
 
 2533 . 
 ^512. 
 U21 . 
 es, 11=2, 
 
 :r)0.25i:i;5 
 
 :59.2512. 
 0.00013. 
 
 seotionO/^) 
 
 :5i).08. 
 
 0.1712 
 0.0021 
 
 By measuring two parallel t)ased (y &»/j) and .'{ intermediate section? 
 
 C.Vi.y2,.V3)> 
 
 f y ^ r 14.0900^ 
 
 I 2y J 20.1125 
 
 there obtains :^=yi'-{ 2y., )' = h\ 20.8000 J- =50.2085. 
 
 2//3 
 
 I 1/^ 
 
 \ 30.2245 
 (.15.1840 J 
 
 Deficit by the formula of the trapezoid 
 Excess by the monoforinula 
 
 = 0.0427. 
 = 0.00013 
 
 Prob, 6. Assmning GAD15M to be a square terminated by a semi circle 
 and letting AO=2R=20, AG or Ct=a=20, Co=x=5 ; //nf/ the volume 
 (W) generated by the revolution of the plane DPSt about the fixed axis GH, 
 ] ' /(// tlte exact formula. 2 " by the monqfonnula. 
 
 Solution. 1" 13y the exact formula : . j 
 
 V7=-[d'x + R-x—^T^-\-a{xy-\-All)] '' 
 (App. K). 
 
 The arc Z>Por J=2rr/ex ^^^^ = 
 
 360" 
 
 .1 
 
 T 
 
 
 V 
 
 
 D 
 
 {DP=30'', since it.s sine Co=5=r,lii!). 
 
 In the equation, y=:y/ji-—x-=^l ( H)—x-, ^ 
 
 of the circle, when x or Co=i>, there obtains y or oP=8.66025. 
 
 Introducing the numerical valuer of J, li, a, z and y in the exact 
 formula, there obtains : r=4o71 5583r. 
 
 2" By the monoformula P=,v//(Z?4-4^'-f A), 
 
 the equation y=y/R'—x-=\/loo — x-, 
 
 f x=:0, then y=R: v,-hence. -{n + Ri^=B. to radius Z)/, 
 when J .r=2..'J. •• i'^=<».(i8244 : •• rr(a4-l'')-=Z?\ to radius nl, 
 l..V=ri, •• l'=8.r)6025: " ':r(a^r)-=zb, to radius P5; 
 
 A'— 2:=5=//, altitude Co of the frustum ; hence, 
 i b ) ( T(a4- }')■-) ( 8-1.40044, 
 
 V=]Ji\ 47/ \=^};Jl\ 4XT(«-f J''V- \='-\ :>o24.lor)'.2, 
 IB) { T:{a-\-R j- j ( 000.00000 ; 
 
 ==;iT X 524K6()536~=7r43 7 1.338. 
 By the exact formula. V =t4371..'S58 . 
 
 Deficit = tO,220. 
 
 Prob. 7. IVhat would be the approximation obtained by applying the for- 
 iioda of the volume of a frustum of a cone to /'rob. (I, in the same circumstances 
 ■IS 'he monoformula ? 
 
 Soi.i'TioN. WIicii two piini lei Oimes, T(</-f-A')'-', n-(a-f- }')-', and a median 
 section. -:((/+ )')-'. are nr asu'ed, as in I'rnl;. 6, the altitude (//) of each 
 frustum is Cn=mo='\, and there obtains : 
 
 18 
 
m 
 
 u 
 
 I 
 
 n 
 
 
 I i! 
 
 27^ 
 
 T 
 
 ArPROXIMATE MEXSURATIOxV. 
 
 iy/rr[o<+A')--f(«+A')(./+r')4-(.^+i'')-l 
 
 f (a-{-R)'^ \ t '.lOo.oofiuuo, 
 
 I (a + Ii)(a-\-y') I I «;m). 173200, 
 
 I (;/-}- J" )(,/ 4-1') I I .«:>(!. /on mi. 
 
 [ (rt-fi')- j [ H'J1,(» 101)30; 
 
 =;iTX iJ^-~l^TijT7=,T4 -153.898 
 liy tiu' exact i'onnala. V =t437],M8. 
 
 Deiirit by the formula of a fru<tiiiii of a cone = -1 7.G6 . 
 
 Deficit by the iiionofommla = to.'_>2. 
 
 Gauging of casks by the monoformula. 
 
 A segment ADJJA. either circular or elliptic, -^^ 
 
 generates a spindle, either circular or elliptic, in 
 its revolution, about the fixed axis AB. \n this y^ 
 revolution, the arc FJJE and the two symme- 
 trical ordinates Fm and Eh generate the surface 
 of a frustum of a .spindle which may obviously be 
 
 considered as the interior surface of a cask whose curvature is either 
 circular or elliptic, according as FDE is an arc of a circle or of an 
 ellipse. 
 
 This being stated, solve the following 
 
 Prob. 8. Find the capacity (V) of a circular cask whose dimensions art 
 the/allowing : 
 
 mh=l=5i), length of the cask, 
 CD=R=26.1, bung radius, 
 Fm=Fh=r='10, head radius, 
 
 JJo=a:=50, radius of the arc FDE, 
 FI)E=A=52.36, circular arc Avho,^e radius and chord arc a and I. 
 Solution, l". Hy the exact formula (Apj). 3''): 
 
 F=t| an{ 1 --;^,) —(<i—Ii)(rl + aA) I 
 
 Tntroducing. in this foniiula. the numerical values of A, a, I, R, r, 
 there obtains : ]''=r73028-l. 
 
 2". Bv the monoformula 
 
 
 B \ 
 
 .r^\ 
 
 By the exact formula, 
 Excess 
 
 _.„_! ] 425.78, 
 "■•'■' 1 400.00; 
 
 =-Y-7rX 1825.78=7730429.66 . 
 =t30284. 
 == T145.66. 
 
.. 1.1. *Ja. 
 
 2.0 
 
 
 i+rn 
 
 :5—/ 
 
 e is cither 
 I or of an 
 
 tensions art 
 
 larc a and l- 
 
 L4, a, i, -R; /"j 
 
 1^30429.66 . 
 I-3028-4. 
 rU5.66. 
 
 vciy iiiiii:(tx'niiit(''y : linic ■, siititiiH'tiiii,' tliis excess ^/.tX^'o^-'^ — '*)'» fi^ODi 
 tilt' iiii)iio:'oiiiiula ;l/.T('2/V-'-|-/-'). l\\>-vc olitiiiiis : 
 
 nnniiproxiii'.iitc forimila. iinictii'iiliy (Miiiivulfiit lu imithematical accuracy, 
 ill till.-; ]i:ir;ic' ila:' case. 
 
 ^Vc re (1. (Ill iKiLi'i- 'J;V2 of the Traitu de r'oduieirie, \>\ Evsseric and 
 I'as'al : •• In praetie ■, usi^ nia :c; (or j^'aiij^fiiif^ a cask) ofa process 
 e-taliii. lietl liy tlu! niiiiisti'i'ial in -t ii ' ion, in pluvio-e, year VII. Tlii.-< 
 ]»:d 'es.5 coMsits o!' coiiveitiiig a ca. k iiit ) a cylinder liaviiij^ the interior 
 1 ii<j.'th of the cask for its altitude and iiaviuLC, I'o:' its base, a ciiclo whose 
 dianie'.or is the buu^ diameUr diiniiiished by one third the ditference bet- 
 ween this diameter and the head diamete:-'. 
 
 Applying to Prob, 8, this ru e whose algebraic form is 
 
 there obtains by this commercial formula 7=^29931. 
 By the exact formula. r=7r30284 . 
 
 Deficit by the commercial formula, t353 . 
 
 Excess by the mouoformula, t145.666. 
 
 Hence, the monofonnula, even without correction, gives a more exact 
 result than the commercial foniuila. 
 
 Prob. 9. Find the capacitij (V) of an elliptic jcask whose dimensions are 
 thejollowiny : 
 
 I =34, length of the cask, 
 
 ^=15, bung radius, 
 
 r ^12, head radius, 
 
 a =40, serai major axis of the ellipse, 
 
 b =34.64 semi minor '• 
 
 ^1=35,116, circular arc whose radius and chord are a and ^, 
 
 Solution. 1" By the exact formula, 
 
 ] =T I hV ( 1 -JL. ) -(b-R)(rl + bA) I . Ai.p. i*/ 
 
 Int.'oducing, in tlii? formula, the numerical values ol .1, a, b, I, R, r, 
 there obtains ; I''=::(;71(). 
 2" By the nionoforniula. 
 
 v=iiiUir\^^},ih^^=li.h^^^ 
 
 144; 
 
 l^y the exact foriiiubi, 
 Kxcess by the monofonnula. 
 
 ■,'-rXii'J4=Tt;732. 
 =-G7Ji», 
 
 _.)0 
 
'! t 
 
 276 
 
 APPROXIMATE MF,N'ST-Il.MH "N. 
 
 ) 
 
 
 m 
 
 mm ij 
 
 mm h 
 
 i. ! 
 
 
 Letting ll7rx=22-rT, there obtains a:=l .'J= };;=,-„ X. '5-= ,-,T(/^—r)2.very 
 approximate]}- ; iimce, siilitrnctiiig tlii.s excess J/-Xi'o('^ — '"^' *''*^'" ^^''' 
 monoformula J(T(2A'2-|-r-), there obtains : 
 
 r=J/T{2/;24-r2_^,(i2-r)'^}, 
 
 an approximate forrauUi, practically equivalent to mathematical accuracy^ 
 
 in this particular case. 
 
 3" By tiic commercial formula, there obtains : V=-CiCi6A. 
 
 By the exact formula, V =t(J7 I o . 
 
 Deficit by the commercial formula = t46 . 
 
 Excess by the monoformula = Tr22 . 
 
 Hence, the Uionoformula, even without correction, gives a higher 
 approximation than the commercial formula. 
 
 Note. If thi.-< investigation be continued, it will invariably be ascer- 
 tained that the nioMolbnnula \l-('lR'-\-r-)<^\\'csan appioxiniaiioii more than 
 twice higher than that obtained by thi' commercial formula /rr[jt(2/^-4-r)]'-; 
 and that the formula Ihl'^R^-^-r^ — i^j(^— '')"]» applied to usual forms, 
 gives a result which may bn regarded as exact. 
 
 In Prob. 8 and D, the maximum section (bung section) of the cask has 
 been taken for the median section between the two bases (the heads)' 
 contrary to the rule of Art. 1.5 ( 1" & 2 "). 
 
 For this reason, the munofonnula has given but a very rough approx- 
 imation in comparison to that which it would give, if the rule were 
 observed (by dividing the cask into two synnnetiical frusta, by its bung 
 section, then computing one of them and doubling the result). 
 
 Prob. 10. Assuming the shn ft of a Corinthian column to he divided into 
 three frusta of equal alti/wics, anil the loicer base of th'. middle frustum to bf 
 two feet in diameter ; find the volume of each frustum. 
 
 Solution. By App. (i2), //=15.68 
 feet, altitude of the shaft, and 
 a=0.494<).'}37, length cf arc Bo of 
 28''2r27'\4. 
 
 In the equation, y=co3 
 
 sax 
 2lt 
 
 of the conchoid, 
 
 X = 0, then?/'' =cos20 =co32 0"=' =1.000 000, 
 
 x^—\n, " 2/2j=cos2>a=co32 7' .5^2 r\85=0.984 763, 
 
 when / a;,=;i//, " ?/2.=cos2k=cosn4°10*43*\7 =0.940 000, 
 
 ^2 = 5^. 
 
 y/2,=cos23a=cos22l«'16' 5".55=0.868 392, 
 7/-'^=cos-a =cos228«2r27'\4 =0.774 400. 
 
AHTKLK 25. 
 
 27; 
 
 1" Hy the exact formula, V= ' (:iit-^s'ma-{-iin2a), 
 
 da 
 
 App. R. 
 
 (in which //=15.68, a=0.49493;n , sin rt=8iti 2H«2r 27". 4=0.474973, 
 sin 2rt=8in 5(]«42'54".8=0,835950), there obtains, for the 
 volume of the whole shaft, r=Tl4.76ia94 cubic feet. 
 2" V>y the monofonnula. 
 
 Since the shaft is divided into three parts of equal altitudes, 
 J//(or/t=iXl5,G8). the lower and middle frusta are equal, and in each 
 
 of the latter 
 
 {b ) I m/^ ) f 1.000 000, 
 
 AB' [=:lh\4z,,\ l^lhrr} r..939 0.'-,2, 
 U ) ( rrj/% ) ( I). 940 000 ; 
 
 both lower frusta. 
 
 whence, 
 
 2r=~i^"^-X5,879052i=;rl0.242G16 
 3X3 
 
 Upper frustum I 
 
 "■=H'i}= 
 
 ■At 
 
 whence. 
 
 T' 
 
 T15.G8 
 
 3X6 
 
 0.940 000, 
 3.47;! .'Jt38, 
 ( (1.774 400; 
 
 X5.187 968=T 4.519297 
 
 Therefore, by the monoformula, 
 By the exact formula. 
 Excess 
 
 2r+F'=r=Tl4.761913 cubic ft. 
 r=7rl4.761894 '• 
 = 7r 0.000019 " 
 
 Prob. II. Assuviing the sliaft of P rob. 10 to be in marble of density 2.8 ; 
 find its weight. 
 
 Solution. By formula P=1000rZ)ozs, A pp. 5 
 
 there obtains, for the required weight P : 
 1", by the monof. 1000FZ>oz8=rrl4761.913x2.8=129852.37ozs; 
 
 2°, by the exact formula, 1000 FZ)ozs=t14761. 894x2.8=1 29852. 20ozs ; 
 hence, the monoformula is in error, by an excess of 0. l7oz5, 
 
 in a weight greater than 8000 pounds. 
 
 Prob. 12. Letting f=0, m=n=20, and p=200, in equations (g) and(h) 
 of Art. 22 ; find the volume (V) of the solid of revolution specified by (g), bet- 
 ween the limits x=s=6, and X=10, 1" by the exact formula (h), 2" by the 
 monoformula. 
 
 Solution. Performing the substitution indicated, there obtains : 
 equation of the curve of this solid, y2z=:^yj^x^(x^ — iOz-\-iOO), 
 exact formula of the volume •' F=3^'(y^Ta;3(3z^— 150a;-f--00). 
 
 1 » By the exact formula : 
 
 («7) 
 (A) 
 
 ^=3^VTr^'l''(3^'2-150.r+200)-^^i,^7ra:3(322-150i-|-200), 
 
 we have: 7=^179.69. 
 
IMAGE EVALUATION 
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 ► 
 
 y 
 
 <^ 
 
 A 
 
 'c^. 
 
 e. 
 
 el 
 
 
 4^ 
 
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 o 
 
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 Photographic 
 
 Sdences 
 Corporation 
 
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 'X 
 
 k 
 
 > 
 
c W&S\ D 
 
 '^^O 
 
!l 
 
 '! 
 
 278 
 
 APPROXIMATE ME.\S[:RATH)N. 
 
 2" In equation //•'='- y-'(.i-—'r'r4-4<i()). (equation (w) of Art. ]t>) 
 
 / .;• = C. ilicro .)])taiiis ,//- =;}5.280, 
 
 y- J =11.405, 
 
 whei 
 
 
 
 y-3=4n.oor), 
 
 7/2, =5(1.000 
 
 x^ — .i'.— l\) — i]:^ [ = 11. altitudi' (*r tlu' frujstuin ; lioiice, 
 By tho exiict i'onutilu, V =rl70.nOOOO . 
 
 Excess = -o.o4:]:}:i. 
 
 Ifliiis frustum )■(.' divided into two fr-iistfi of cqwal altitudes (//'=2). 
 theiv obtitius fo • the volume of the : 
 
 1st frustum. T''=v//' ■' -;-//', \- ='^ tt } 4 1;- . 1=}.-J li\:>J]2, 
 
 ( -irj ■ ( )r\) ' { 4.;.;!S; 
 r -'/• , ] 1 '/-, ] ( -ill. OS. 
 
 2nd fru5tum.r"='//"-' I-'-' -=-- ' 4//-' - = 'rrJ i;;().02. 
 
 j;>.of) 
 
 lience, r'-f-r"= 
 ]\v the exact formula. 
 
 r 
 
 T' 
 
 = : T X ^o'.'. ! i,S=T 1 70.0033 . 
 
 ==rlT9.0t:i()0, 
 
 V = r().()U33 . 
 
 Prob. 13. What voiiH In; 1)11: approximation ohtained bii 'ippli/iii'i /he 
 formula of a/niMum n/ti coiif to I'rob. 1 2. in the Kinm: circinns/ances ./s the 
 monoformrda ? 
 
 SoLUTiox. ]" Taking tw.) pa.allel bases, rr?/- and r//- j. and a median 
 section, -/z'-^, the iiltiiiide of eaeli cjf the two ajiproximate frusta of a cone 
 will be equal tn 2 ; lio>ic ', the volume < V) of this sum Avill be 
 T=^//;rO/-4-7/.y,4-//- J + ' //-(//^4-//o»/,+y-,). 
 r r 1 f 35. 2H, 
 
 i .'/'/.> I 40.32, 
 
 or r= y H- { '2>/'.. )• = ^r -{ 92. 1 » k 
 
 I .V.^y":., I ■' 4,^-00, 
 
 I y-4 J I 5< ••<><> ; 
 
 = ;i-:X 205.70 =:rl 77. 17333. ^ 
 
 By the exat formula, T' =r]71K0H. 
 
 Deficit by foniil. of a fru.stum of a cone = -2.51007 . 
 
 Excess by- the inonofonimla = -0.04333 . 
 
ARTin.F. 25. 
 
 '27.) 
 
 2" Taking l\v» parallel liases. -»/-' and ~>/'- ^, and llirco intormodiate 
 BfCtioiis. rr//-',. -//-^. r//- . til" altitude (//') of each of the approximate 
 frusta of a CO. le will be 1 : hence, the volume ( I') of their sum will h.' : 
 
 F==i,//'T(y-+»/»/,4-2//^,4-//,//,4-2.v:',4-//,?/.4-LV/:;,4-//,//,+/^ 
 
 =lT(3r).284-:58.214-82.8l4-4Li.G84-0'J.lG4-t7.r)2-|-U8.Ol4-4!».50-l-r)O.()()). 
 
 =i-x5:]7.it=-i'::hi(] . 
 
 By the exact formula. T' =t1 T'j.i;'j . 
 
 Deficit liy the fo.niula of a frustum of a cone = xu.G3 . 
 
 Excess by tlK- monoibrmala = ru.oo:?;; . 
 
 Prob. 14. Assuntinr/ ("lADlJlI (u />r a nctanylc tcrmitutkd bij a semi 
 circle, and h-Hing A!;=:2l»~rJ, Ad or ^, 
 
 Ct=](), Apn= arc of 4i"- — A : jlncl the .'""""^^ 
 convex i^urfare C^) ijcwrittrd h]i the rcvolu- I ^<Z/. ^- 
 
 t/on njthearc A. alotil the axis Gil. ^. 
 
 Soi-rriox. 1" \\w the exact formula. 
 S^-l-IU-{-'l-:,tA. Api>. .M 
 
 (in which /,'=.. //=1 d, /j^"- 
 
 a'=.-lm=.-ir— ?»C=1— ens 4O'^=0. 233956, and 
 A=Avn=i\n' of \'<'^—'lnIl.^l .— ".~, tlieiv ohtiiins .9=45.335064. 
 
 2" lly the inonMlormula, >'—'// (''-|-4r"-|-c) (in which 
 
 Jl=ixvc ^lp>i=. :,~. 
 
 C"=2rr (^4-0/') ='2--(^/-i-,sin 'J<;'')=2-(li)-|-0.:;42'>'.>(i), 
 
 C =2-7 (lmA-mn)=2~{ii-\-An 40^)=l>-(1()J-0.(; i'JT:-i.-^). tlu-ie ohlains : 
 
 >S'= 
 
 ( '.' ") » ) 2,tx 10.000000") ., ,.r lu.oodooo 
 ■}jf]Ar'[= ^~ Ux2-x i<>.;i4-*i»L'(. ^----■".^ n.:.5(;s(i,vo 
 
 ' (' t'X'.U l>tx10.(U-J7S^ -M H).(1427n;.S 
 
 ) --n 
 
 
 
 ""27 
 
 log 2 
 
 =0.;}01080, 
 
 
 I02 -- 
 
 = u.094;;(i(), 
 
 
 loo()2.0108G8 
 
 = 1.7024().S, 
 
 
 colog 27 
 
 = 2 oC^diii; ; 
 
 
 hence, .S^=log ^ 1. (')5(U:U= 45.335000 . 
 By the exact i'onnula, *S^ =45.3!>5or,4 . 
 
 Deficit = O.OOOOiU. 
 
 -'M 
 
280 
 
 APPROXIMATE MENSURATION. 
 
 tf!| 
 
 I" 
 
 i! 
 
 i 
 
 11; 
 
 or 
 
 ( C ) ^ f 27rXlO 
 
 S=in\ 2C' |-=-^-^ 4n-XlO. 
 
 \ c f 2X9|2-X10. 
 
 ,342020 
 ,642788 
 
 Prob. 15. What would be the approximation obtained by applying the 
 formula of the convex surface of a frustum of a cone to Proh. 14, in the same 
 circumstances as the monoformula ? 
 
 Solution'. Taking two parallel bases, C and c, and a median section C\ 
 as in Prob. 14, the generating arc (II) of each of the two zones will be 
 equal to -J-, and the sum (S) of thc'r areas will be 
 
 S=^II(C+C')+^H(C'-^c), 
 
 (10 
 =^7r2 \ 20.684040, 
 ( 10.642788 ; 
 
 =^7r2 X 41 .3l.'6838=45.320 . 
 
 By the exact formula, S =45.335 . 
 
 Deficit by the formula of a frustum of a cone ^ 0,015 . 
 
 Deficit by the monoformula = 0.0000G4 
 
 Prob. 16. Drato the semi-circle AcB (fiff. of Prob. 14), on th6 left of AB; 
 and, the other conditions being the same as in Prob. 14, find the surface 
 generated by tlie revolution of the arc Ap'n' of Ad". 
 
 Solution. I» By the exact formula, Sz^2TTaA — 27rJZz, App. M 
 
 there obtains : &=s42.394936 . 
 
 2» By the monoformula, S=i\H{C-\-AC-\-c), (in which 
 
 ir=arc ^p'n'=27ri2X7V?j=§-^» 
 c=27r X -4(7=27r X a=27r X 1 0, 
 
 C7'=2n-X (so— op') =27r(a— sin 20^)=2-(10— 0.342020), 
 (7=27rX(^n»—TOn')=27r(a— sin 40*')=27r(10— 0.642788), ) we have : 
 
 5=iir 
 
 v\ _ 2rr 
 
 But, log 2 
 log -2 
 log 57.989132 
 
 colog 27 
 
 hence, >Si=log 
 By the exact formula, 
 Excess 
 
 2n-XlO ) ( 10.000000, 
 
 4X27rX 9.657980 \=i^^tt^\ 38.631920, 
 
 27rX 9 357212) ( 9.35 7212; 
 
 zj^yir^X 57-989132. 
 
 = 0.301030, 
 = 0.994300, 
 = 1.7633465, 
 
 = 2.568636 ; 
 
 — 1 
 
 1.6273125=42.394960. 
 
 8 =42.394936 . 
 
 = 0.000024. 
 
 Prob. 17. Oiven f=0, m:=ns=24 and ps=s4000, inthe equations (g) and 
 (h) of Art. 23 ; required, between the limits zss8 and zs=:12, the convex sur- 
 face (S) of the solid qf revolution inscribed in the polyedroid whose curve is 
 (g)> I' by the exact Jormula (h), 2" by the monoformula. 
 
ARTICLE 25. 
 
 2Sl 
 
 SoLDTiON, Performing the substitution indicated, there obtains : 
 Equation of tlie curve of the polyedroid, 
 
 .'/=TFo52'(^^-482+5T«)+10, (y) 
 
 and exact formula of the convix surface of the inscribed solid of revo- 
 
 lution, 
 
 '^=TIjk7^2='(2'-60z+960)+20n-z. 
 
 (A) 
 
 1» By the exact formula (/t), &=7rll8.5024 . 
 
 20 In equation y=j^\jjz\2-^—48z-\-5'!6) + 10, (m) of Art 7, Prob. 
 
 a = 8, there obtains y =14.096, 
 2i= 9, " yi = 14.55625, 
 
 when 
 
 11, 
 
 it 
 
 ^3-15.11225, 
 24=12, " ^4 = 15.184; 
 
 24 — 2=4=//, generating arc of required zone >S^ ; hence, 
 
 ( c ) C 2^y ) (14.096, 
 
 S=IH\ 4C' [=U 4x2ry, f =|:r^ 59.600, 
 l O) ( 2ry;3 (15.184; 
 
 By the exact formula, 
 
 Excess 
 
 =|xX 88.880=7rll8.5067 . 
 
 S =xll8.5024. 
 
 = rrO.0043 . 
 
 If the generating arc ^(or 24— z) be dividea into two halves {ff'=as2) 
 there obtains, for the area of the : 
 
 1st zone 
 
 ( c ) i 2rry ) ( 14.096, 
 
 , S'=:^n' \ iC [ =§ \ 4X 27ry, \ =f7r ] 58.225, 
 
 i (^ ) ( 2ry2 ) ( 14.900; 
 
 i'c ) ( STry^l f 14.900, 
 
 2nd zone, S^'^^IT \iCr\=U ^y.'i^y, \^^t^\ 60.449, 
 
 [ C ) ( 27ryJ ( 15.184; 
 
 hence, 8'+^'=. 
 
 By the exact formula, 
 Excess 
 
 S =i^7rXl77.754=s7rll8.50266 
 iS =7rll 8.5024. 
 
 = 7r0.00026. 
 
 Prob> 18. What would be the approximation obtained by applying, t<t 
 Prob, 17, the formula of the convex turface ofajruttum of a cone, in the same 
 circumstances as the monoformula f 
 
 Solution. Taking two parallel bases, 2n-y and 27ry4, and three inter- 
 mediate sections, 2n-yi, 27ryj, 2:7y3, as in Prob. 17, the generating arc 
 (A) of each zone will be equal to 1 , and their sum (S) will be : 
 
 .^1 
 
 Vl 
 
 i 
 
 J\ 
 
 '\i\ 
 
 If 
 
 4^1 
 
 M 
 
 n 
 
 
 ! 4 
 
V. 
 
 I I 
 
 f i 
 
 282 
 
 APPROXIMATE MENSURATIOX. 
 
 5=== V/('^r7/4-2-v,)+^/'(--.'/; -f --V2) + i^('.'TV,+2r,V3)+U(2Ti/3 + 2^7/ J, 
 
 r 
 
 
 or S=lh-\ -Ix-l-ii 
 
 2.y,, j 
 
 ( ;'/ 
 i -^/^ 
 
 f 14.0000, 
 I 2i).1125, 
 
 I 30.2245, 
 1 ir).lS40; 
 
 = rxri874r70\ 
 
 ^rxn.^.5024. 
 
 By the exact fornmbi. >*s' 
 
 Deficit )»y formula of a IV. <ii';i cone.— rrO.OSr); . 
 
 Excess by the inojiot'oriiiula = -0.()002(j. 
 
 VoTK. Til Student w'll ho sin'j)iif(l uitii an u'. umlaut matter for vaiy- 
 ing tlicse jiroblcms wi;li -oliitioiij ('oiiipaivd. hy fiiidiuir the exaci 
 formula : 
 
 1° of ihe areas of the pane fi;j.'U'AS whcse curves are represented by the 
 various ei|uati')n.> of Art. 'JG r prooeeding as i.i Art. 47. nook III); 
 
 '2" of the vohmies of the solids of revolution whose curves are represented 
 by the various iqiiations of A.-t. '.'5 (piocoedin;^ us in Art. "22;: 
 
 .3" of thi-' Convex surfac-ps of the :-oIids of n'volution im^criUd in the 
 polyedroids whose curves ae represented 1 y ihe various eijuatious ot 
 Art.'20 (proceeding as :n Art. '23) 
 
 26. 
 
 MECHANIC A]. l^KOlMJvMS. 
 
 J" 
 
 -^ 
 
 A mechanical problem is one in vvhicii the lines required for its solu- 
 tion are measured mec'ianlcally. 
 
 Prob. I. To comjnite. in ifiiniir'j meters. Jnj the monoformula. tJi>.' area (A.) 
 of an elliptic segment v:hose arc H(>r=no". 
 
 Solution. Draw the chord DC, and bLsect it by 
 the perpendicula ' 0/f. 
 
 In conformity with ml." J/ (Art. 9 ). the elementary 
 segment <^BD must Ije given bases and a m.diau 
 section parallel to the a.\i.- OH: accordingly, 
 througli the middle n of Bl>. draw mn parallel 
 to OD. 
 
 Kow, by a mechanical measurement, 
 
 let BD=zH meters, altitude of segment OBD, 
 
 OD=B '• , lower b:i.- (/,') •■ , 
 
 vm=B' •• , median .'ciion i^A") '• , 
 
 pointi^=zero . upi)er base (/>) « ; 
 
f 2TJ/J, 
 
 ■ for vary- 
 tlio exaci 
 
 ed by tlie 
 
 ■epresontec] 
 
 •ibcd ill the 
 (Illations ol 
 
 for its solu- 
 |/('' area (A) 
 
 AllTICLE 2(». 
 
 283 
 
 Avlu'iice. ar.'ii of O/ID. A'=!Jf\ 4/?' >=|//-', 4//' : 
 
 therefore. 
 
 2.r=.l 
 
 =L\lf(H-\- \/>'i -i|U.'ir> meters. 
 
 Prob. 2. To conqnile, in square meters, hy tif mou'tformuln. the arm (X) 
 of '1)1, hi/jierholic seffiiict}' wfiose arc II()C='')'i'''. 
 
 Som:tiox. Draw tlif clinvd /!(' (pn-cciMliiiir lis:-.), ami Itisci't it ))\ the 
 ]icr]ir .(lini'ar O//. 
 
 In couforniity willi nil ■ .)/ (An. '.). tlir r/cine/Wiri/ s^yinei't O/i/) must 
 1)6 given liases nml a in ilian section paralli'i to OJf; uccorUnyli/, through 
 the niidil'c n of /1/K draw >:i)i paidlrl to (//>. 
 
 By a mechanical niea.surein 'nt. 
 let /y/>=// meteiR. altitude of scfrmeiit ')/,'/). 
 ()1)=iN " . lower ha -^e (A') " , 
 
 mn—N' •' , nu'd. .^.'Ctlon (N^) 
 ]»oint j^=zero , upper l)a.>e (A) " : 
 
 whence, area of 
 
 • OBD, A'=].If\ 4.V' I : and ■lA'=A=^^JI{X-{- kV) sq. w* 
 { ^' ) 
 
 ' Prob. 3. To rompiifr, h;/ thu'inonoj. thf itri'<t(.\)nf (he se;pnent OfdO.Wtose 
 r/ion7()d=2lI mt/cr.s, it)ul n-ltosf. arc Ofd {sjKci's not ;ji-er>)* comprises, at 
 aiijht. (ihout .")•'". 
 
 Solution. Divide the uivtn altitude Od into 
 four ecjual parts, by tin points a, b, c, and draw 
 ordinates to these iioiiitri. 
 
 By a mechanical measurement, 
 let >i, b, c meters be the respective lenj^ths of 
 these ordinates. 
 Hince the ordinate // divides 0/dO into two O ^ 6 '^ c/ 
 
 * Since the monoformula is independent of the axes, la and 1h, of the 
 curve there is no need of them ; on the eoinrary. were it required lo apply 
 the special formula. 
 
 A=.H(a^N)-± log { "B±^±[±lL} I . 
 .)/ V lib ) 
 
 it would be necessary to determine them: but. hoir by the elementary 
 geometry ? 
 
 • The monoformula. b ing applicable to any plane figure, is indepen- 
 k'nt of the species of the curve. On the contrary, ^vere it required to apply 
 the special formula, it would be necessary to determine this species; but, 
 
 how? 
 
 I 
 
 W 'M 
 
 
' 
 
 i ! 
 
 284 
 
 APPROXIMATE MENSURATION. 
 
 elementary segments, of equal altitudes //, iu each of which a base (B) 
 reduces to zero, the one at the point O and the other at the point d, 
 then 
 
 area of Obf, 
 
 area of bcif, A 
 
 
 =A=^II(2a-\-b-\-2c) sq. meters 
 
 i * 
 
 ( 0, 
 
 Prob. 4. The curve of a solid of revolution (species not given) comprises 
 about 30'^,from the vertex to the circumference of the base ; required, in 
 square meters, by the inonoformula, the convex surface (S) of this segment. 
 Solution. A mechanical measurement has given : 
 //meters=length (//) of the arc of 30<', 
 
 2R " ^diameter of lower circumference (C=2T^)of the segment, 
 2^' •' = •• of median •' (C"=2Ti2'), bisecting arcSO*' 
 
 zero " := " of upper " (c=0)) vertex; 
 
 then, -S=4/r-! 4(7' (==*//■! 4x2rri2' } =J^7r(i2+4i2') square meters. 
 { C S I 2^R ) 
 
 Prob. 5. Find, by the monoformula, the convex surface(^)of a regular ttta- 
 gonal dome circumscribed about a semi ellipsoid. 
 
 Solution. Divide the element of contact into three ai 38 of 30*, at tight, 
 and bisect these arcs. Then, by a mechanical measurement commencing 
 at the vertex, 
 
 let //, W, H" meters^lengths of the three arcs of 30°, respec- 
 tively, and a, b, c, d, e,f meters^the respective sides of the octagons 
 passing through the successive points of division ; 
 then, since the upper base (vertex) is zero, there obtains : 
 
 A=^iff\ 4rt I: 
 
 upper zone. 
 
 -, 4rt 
 
 =*//(4a-f6), 
 
 ^H'{b-\-4c-{-d) \ :=5 square meters. 
 
 median zone, A'ss^H' -; 4c >■ =1 
 
 ( d ) 
 
 id) 
 lower zone, ^"=|//" ■{ 4c i=*H"{d-\-ie-y) 
 
 Prob. 6. Find, by the monoformula, the convex surface (S) of an oblate 
 elliptic dome (axes not given)* whose curve comprises about 60", from vertex 
 to base. 
 
 * Since the monoformula is independent of the axes, 2a and 2b, of i.be 
 curve, there is no need of knowing them; on the contrary, were it 
 required to apply the special formula (App. F), it would be necessary to 
 determine them : but, how, by means of elementary geometry ? 
 
ARTICLE 26. 
 
 285 
 
 Solution. Hisect the curve, at sight, and again bisect each arc of 30*. 
 
 Hy a mechanical measurement, 
 let Hkir meters=the respective lengths of the arcs of 30", 
 and rt, b, c, d meter3=:the respective diameters of the circumferences 
 passing through the successive points of division ; 
 
 tlien, since the base at the vertex is zero, there obtains for the elementary 
 zones of 30°, viz ; 
 
 upper zone, A=i\U \ AC V = '. // 
 -4'=^//'* AC (• = '//' 
 
 lower zone, 
 
 =:5 square meters. 
 
 4 
 
 ■'Hi 
 
 ^%']^: 
 
 ( 
 
 lii- 
 
2R{] 
 
 APPENDIX. 
 A. 
 
 J. 
 
 (;exkual foi.m ok an i^qfation. 
 
 
 Any equation witli oiio unknown quiintity of the degree n (n heint^ 
 an entire and positive number) can assume thiform 
 
 x"-\-Ax^~ ' -j_y;x"~--f-CV"~''+ JIx-\-K=<). ( 1 ) 
 
 For. if all the terms of an ((luation with o.e unknown (piantity r. of 
 
 the degree n, lie divid.d by the coefUcient of a:'*, and if all the tcrnn of 
 
 the second member l)e transposed to the fiist member, the equatioji will 
 
 assume the above form (1). 
 
 To simplify thejollowing discussion let the first member of cquatinii (1) be 
 
 represented by the symbol f(.\), caZZef/ function ofr. 
 
 II. 
 
 Roots of the Equation f{x) = 0. 
 
 Any quantity which, being substituted for x, inf(x), reduces this func- 
 tion to ztro, is a root of the equation/(z)=0. 
 
 Theorem. 
 
 If f(\) reduces to zero when x=a, 1" this function will be divisible by 
 (X — a), 2" the quantity a tvill be a root of the equation f(x)=0. 
 
 For, 1" divide f{z) by {x — (/) till a remainder independent of x is 
 obtained, and let 
 
 C^=quotient ; and /?=remainder (if there be any); 
 then. f{x)=Q(x-a)-{.K. (2) 
 
 l)Ut. since/(2;) reduces to zero wh-'u x=a, the equation (2) reduces to 
 ()=t)-\-R ; that is, the equation/(2:)=0 is divi:?ible exactly by {x—<i); and 
 thereby, f(x)=(J(«-a) . {?,) 
 
 2" Conversely. if/(a:) be divisible exactly by (x — a), the quantity x 
 will be a root of the e(iuation/(a;)=0 ; since, this quantity a. when subs- 
 tituted for X in (3), reduce3/^a:) to zero. 
 
 III. Theorem. 
 
 If the equation f(x)=:0 be of the degree n, it will have n roots, neither more 
 nor less. 
 
APPENDIX 11 
 
 2.^7 
 
 I (tt heinu 
 
 (I) 
 
 ;itity r. of 
 
 term:? of 
 
 itio'.i will 
 
 lion (1) l/c 
 
 this func- 
 
 ivisible by 
 It of X is 
 
 (2) 
 
 lioduces to 
 c—(t): and 
 
 [[uantity x 
 turn subs- 
 
 lilher more 
 
 l" AsPiinu' r? to bo a rnot of /"(j)='">. 
 
 Ry this liyj)otlu'3i-',y'(r) is divisible exactly )iy {x — a), 
 
 -fii 
 and tlie deffrt't' of ihf iinoiicnt will be {n — 1), siii'M- — = 
 
 
 u—\ 
 
 Kepregeiiting this quotient by/](j:). tlier • obtains : f{x)-:=(x—n)j\(x)\ 
 wlieiic '. e(juatioii (4) may be written (x — a\t\[x)=^). (j) 
 
 Ajjain.assiunt' i to be a root of./^(j):=0. (e<|uatiuii of degree n — i ). 
 
 Hy this hypothesis. /(/; is divisible by (x — i). and the degree of the 
 f)iiotienl will be {« — 2). 
 
 Representing thii (piotient by/^ (/). the ((piation (5) may be written : 
 
 Likewise, equation (0) may assume the form 
 
 {x-a){x-i){x~c\f^,X-^)=u. 
 and so on, uiitill/(.Ti is resolved into n binosnial factors; 
 that is, /(x)=(x—f/)(x—A)(x—c)(A-c, to M factors). (7), 
 
 But,/(a;) reduces to zero when x is equal to any one of the n quantities 
 a, b, c, &c, ; therefore, /{x) has n roots. 
 
 2" This equation/(.v)=0 cannot have more than n roots. 
 
 For, if .r bj given any value (p) which is not one of the « values 
 a, b, c, &c., the second member of ecjuation (7) will become 
 (p — a){p — b)(p — c)(&c. ), an expressio i which cannot reduce tozero,sincr 
 each factor is different from zero ; therefore, no quantity different from 
 the n quantities a, b, c, &c. can be a root ofy(jr)=0. 
 
 IV. 
 
 Theorem. 
 
 If the coefficients of an tguation be real, the equation admits of but an even 
 number of imaginary roots. 
 
 For, \fa-\-b^y—l be a root of/(.r)=0, the qtiantity a— iy^ — 1 will also 
 
 be a root of this equation ; because, the radical factor ^ — 1 cannot be; 
 eliminated unless an even number of these imaginary roots be multiplied 
 
 by one another ; otherwise. \/— I would be a factor in the coefficients of 
 /"(.r), contrary to the hypothesis. 
 
 B. 
 
 LINES OF THE THIRD ORDER. 
 
 In his Treatise " De Enumeratione Linearum Tertii Ordinis," 
 Newton demonstrates that the lines of the third order. 
 
 ax^ + bx'^y-{-cxy^-{-J)/^-l-ex''--{-fxt/-\-ffi/-\-hx-\-ky-\-l=sO, 
 are divided into four classes, of which the : 
 
 
 
 8 1' ., 
 It' ,'■ 
 
 
 
I>! 
 
 288 
 
 APPENDIX C. 
 
 J 
 
 Ist Class 
 
 comprises 69 Specioa of hyperbolic curvcg repreatMited by the equation 
 
 2nd Class 
 
 comprises but one Species (a curve composed of two branches, tiie one 
 hyperbolic and the other parabolic) representeil by the equation 
 
 xy^ax'^-\-bx'^-\-cx-\-d ; 
 
 3rd Class 
 
 comprises 5 Species of diverging parabola; represented by the equation 
 
 y'^=iax-^-\-bx'^-{-cx-{-d ; 
 
 4th Class 
 
 comprises but one Species, called cubic parabola, represented by the 
 equation y=ax^-\-bx^-\-cx-\-d. 
 
 c. 
 
 I. Curves represented by the General Equation 
 OF the Second J^egree. 
 
 Since the general equation of the second degree, 
 
 Ax'-^Bxy-{.Cf-\-DK^Ey-\.F=(), (a) 
 
 is true for any value of x and y, it remains true when x and y alternately 
 vanish ; 
 
 .4x24./)x-|-F=0 ; 
 Cy^+i?y-f F=0 ; 
 
 Cf-{-Ey=Ax'Jf-Dx. 
 
 thus, when y^O, in (a), then 
 and, when x=::0, in (a), then 
 whence, 
 
 (1) 
 (2) 
 
 Solving ( 1 ), there obtains : 
 
 }■ 
 
 _ —E±:^E-—ACF 
 2C 
 
 i 1 
 t I 
 
 Multiplying this value of y by ^, in (2), and letting the product equal—G'. 
 then equation (2) reduces to Cy'-—(T^A±^-\-IH, 
 
 which may take the form : \'=zb\^-\-c\-\-f, ^3) 
 
 .or pf=b'\'-\.c'x-\-f. (4) 
 
 Now, I" assuming i=0, in (.;), thor.' obtains y-=c\-4-/, 
 w'hich represents the common parabola referred to its axis and dir>"Ctrix 
 (Rook III, 14, 1"'); then, assuming />=() aii(l/=0. in (;^), we have y-=cx, 
 which represents the common parabola refen-fd to the vertex of its axis. 
 
 2 " Li'tting the second member of i quatlon (4) be represented by the 
 symbol/(x); then (4) becomes py-=/(x). 
 
'W: 
 
 "(fi^'T 
 
 ation 
 
 the one 
 
 ^uation 
 
 ted by the 
 
 JATION 
 
 (a) 
 1 alternately 
 
 (1) 
 (2) 
 
 equal — G. 
 (3) 
 
 diroctrix 
 |ve y'=cx, 
 
 its axi^. 
 (ed by the 
 
 APPENDIX C 
 
 2^:;) 
 
 Denote the two n)Otaof/(x) by m and «. Thesi* roots, which maybe 
 either both real or both Imaginary ( App. .4. Ill A IV), admit of only th»' 
 following thn-e combinations and equations : 
 
 Ist, two real and equal roots : whence, the equation : 
 
 y''=i-(x±"»V^: 
 2nd, two real and unequal roots ; whence, the equation : 
 
 y2=±-(x±m)(x±M); 
 
 3rd, two imaginary root:< ; whence, the eijuation : 
 
 y2=-i-l| x=b("'+/(>/-l) } j x±i;,:- V-1) j 
 1 
 
 (ft) 
 
 ('') 
 
 P 
 Extractinj; the square roftt of ( '), and disca. .ag tli** itr;\rinary value 
 
 (/) 
 
 — ti\ -my, there obtains : v=± — .(xiJ- m). 
 ^hich represents the straight lini' cutting the axis ot I' ic the distance 
 db — from thi- orit,'in (I?ook III, 1», 1"). 
 
 When /rt=0, in (/), then y=± x. 
 
 \^P 
 which represents the straight Hue passing through the origin. 
 
 If, in the curve represented by (c). the o.igin b^ removi'd to the point 
 
 in which m=0, equation (c) will be reduced to y-=dr-(x±n)- (y) 
 
 P 
 
 Discarding — -(x4-») w-hos.' square root is ima'zinarv. and l^x — n\ 
 P ' P 
 
 V 
 
 which represents the same curve as L(x-|-«) with opposite abscissas 
 
 /' 
 (Book III, IS,*), then(//) contains only ilio following t%vo ctjuations : 
 
 1st. y'-=-(«x — \-), which represents the i llipse referred to the vertex 
 P 
 of eilh r axis ; or the circle, a variety of the ellipse in which /)=:l 
 
 (Hook III. 10 and 12): 
 
 'Jnd. v'-=-(;jx + x-'), which represents the c uniiion hvperboia, 
 P 
 referred to the vertex of its transverse axis (Rook III, 1'). J")- 
 
 If, in the curve represented by (rf), the origin be removed to the point 
 
 in which 7?ii=0, the equation (./) will become v*=-(x--|-«^)» 
 
 P 
 
 (l) 
 
 
 ■I'l 
 
 ) K 
 
 
 ,w% 
 
 3\ 
 
 
 ■■■m\ 
 
 m 
 
 19 
 
290 
 
 APPENDIX C. 
 
 |1 
 
 t ! 
 
 i. 
 
 which IV presf'ts the coninion hypc.bohi 
 FBF conjii£?a;e to DAD', and referred 
 toils c nter C. (I'ook III. 10. 3°V 
 
 Now. if tlio o.-igin of the cd-ordi.uites 
 of /'i>7'" be transferred from C to any 
 point — T, the ordinate gh of any point 
 g in FBF' will not be changed, but the E^ 
 abscissa of this point g will become 
 
 — Xh= — XC-\-Ch. L'.'tting this new abscissa — Xh-=X, and the con- 
 stant — A'C=m. then X=zm-\-x : whenc:; x'x=X'-—2mX-\-m- . 
 
 Introducing this vahu of a:- in (/i, w • have : 
 
 
 The name process will giv- V-=~{X'--\-2mX-\-jtfi-\-n-), for the 
 
 P 
 equation of FBF', when the origin is removed from C to any point -|-X 
 
 Therefore, the equation {a) reduced to an equation of the first degree 
 represents but the straight line ; and, as a quadratic equation, it only 
 represents the circle, the ellipse, the common parabola and the common 
 hyperbola. 
 
 II. 
 
 Common Parabola represefted by 
 
 y=b.\- + cx-f f. (a) 
 
 Multiply the equation (a) by^). and represent the second member of the 
 resulting equation by the symbol/ (x); then (a) becomes />y==/(x). 
 
 Denote the two roots of/(x) , by m and n. These two roots, -which may 
 be either both real or both imaginary, admit only of the following three 
 combinations and equations, when the curve is assumed to be to the right 
 of the origin (Book III, 18,*): 
 
 1 " Two real and equal roots ; whence, the equation : 
 
 y=-(x— m)-i 
 P 
 
 2» two real and unequal roots ; whence, the equation : 
 
 y=_(x— nt)(x— n); 
 P 
 Z" two imaginary roots ; whence, the equation: 
 
 y=i| x-(m+«v/-l) } { x-(m-nv/-l) } 
 
 =i(X' — 27»x-|-rw24-n2) . 
 P 
 
 (6) 
 
 C^) 
 
 (3) 
 
 It 1 1 
 
APPENDIX C. 
 
 291 
 
 It 
 
 (a) 
 mber of the 
 
 , which may- 
 owing three 
 to the right 
 
 (&) 
 
 (^) 
 
 (3) 
 
 If, ill the curvi's repre.-entvii by (A) .>(ul {c), ih oiigiii be rcniovod to 
 the point in wiiicli m=zO. tlioir eqr.iitioiis (h) and (r) w'lV be reduced to 
 
 
 1 .. 
 
 (.), and v=-(x- — nx). 
 P 
 
 (2) 
 
 The.e eqii;iliuns (1), (2) and (:>) represent tlie common parabola, 
 y2=r=;/x. as it may b ■ ascertained by comparing the following curves and 
 their ('(juat'ons : 
 
 (a) 
 
 the con- 
 
 
 4-m-. 
 X-\-m'-\-n') 
 
 y-=4ax^=px 
 
 (origin at 0, Book III. 1 4. 'J"). 
 
 For the 
 
 / 
 
 r point -\-X, 
 first degree 
 on, it only 
 
 1 ., 
 
 y=- X- 
 
 P 
 
 the common I 
 
 (origin at 0). 
 
 (*) 
 
 
 v=-(x^— nx) 
 P 
 
 (origin at D). 
 (origin at B). 
 
 (C) 
 
 (c) 
 
 ^fc. c 
 
 V=i(x2— 2mx-fm24-n2) 
 
 P 
 (origin at^). 
 
 (rf) 
 
 1" If the co-ordinates be mutually changed in the eommo:; jiarabola 
 ^OC, fij,'. («)) its e(|uation y^=p\ will become x'^=py ; but, this 
 
 result solved with respect to ?/ will givf v=- x-. (i) 
 
 P 
 
 which is the equation of the common parabola liOC, fig. (i); honce, this 
 
 mutual change of co-ordinates has no other effect than to change the 
 
 iaorizontality of fig. (a) into the verticality of *ig. (i). 
 
 
 
 
 
I' 
 
 292 
 
 APPENDIX 
 
 w»* 
 
 2" Let D0:= — or c, in the commoa parabola (6>, and subtract this 
 P 
 
 constant c from the length _ z^ of its ordiaates y ; then will the curve 
 
 (6) be lowered, parallel to itself, the distance c or DO below the hori- 
 zontal axis (FG) of ^, and thereby become the curve BOC, fig. (c'), 
 
 represented by y=- x^— c= -(x'^— m*) (Origin at D). (c') 
 
 P P 
 
 3" If the origin be removed from £> to B, in fig. (c'). the ordinate 
 mn[=Y) of liny point m of the curve will not be changed, but the ab- 
 scissa of thlj point m will become Bit=BD-\-Dn=BD-\-x. Denoting this 
 constant BD by m, and the abscissa Bn by X, we have Bn or X=to-|-x i 
 •whence, x=X — m, and x^=X^ — 2mX-f-wi'-. 
 
 Introducing this value of zs, iu ^^c'), we have 
 
 or (letting 2m:=n), 
 
 Y=I(X2-2mX), (c") 
 
 Y=:(x^— «X). 
 p 
 
 4" Assume AB to be perpendicular to FG, flg. (c), and equal to 
 
 ^^ "^ "'; then, if this constant "^^"^"''be added to the length !:(X^— 27raX) 
 
 P P P 
 
 of the ordinate F, in the common parabola BOC,. fig. (c), the latter will 
 
 be raised, parallel to itself, the height — Z__ or ^^ above the axis (/''(?) 
 
 P 
 of -l', us in fig. (c?) represented by the equation 
 
 Y= i (X3-2mX)-f ^ji:!l!=l(X2— 27wX-|-7n24-n2). Id) 
 
 P P P 
 
 But, these equations (6), (c), {d), just obtained are identical to the above 
 
 equations (!). (2), (3), contained m the collective expression 
 y=^x2-|-cx-f-/; therefore, y=bx^-^cx-\-f represents but the common 
 2)(>rabola whose axis is perpendicular to the axis of the common parabola 
 y-=px. 
 
 Sch. If the sign of the ordinate be changed in BOC. fi;?. (c). the 
 
 I 
 
 1 
 
 equation y=- x- — c, of this curve will become v=c — - x'' 
 
 which represents the common parabola BOC of tli,' following fig. (/). 
 This change of sign of the ordinate has no other effect than to malie the 
 curve perform half a revolution about the axis (FG) of X 
 
 1 2 
 y=c— -x2 
 
 V 
 (origin at D), 
 
 (/) 
 
 Jtr 
 

 ract this 
 ihe curve 
 
 the hori- 
 l fig- {<=')> 
 0). {C) 
 
 ; ordinate 
 It the ab- 
 Qoting this 
 
 TlX), (c") 
 
 al to 
 (X2-2toX) 
 
 e latter will 
 the axis (FG) 
 
 to the above 
 
 the comvion 
 \ion parabola 
 
 fiS- (c)- the 
 
 ring fig- (/)• 
 I to make the 
 
 AITENDIX D. 
 
 I. Equation op the curve 
 
 OF A CIRCUMSCRIBED POLYEDROID.* 
 
 Assume vlOX> to be the curve of a ^ 
 right polyedroid circumscribed about / 
 the soHd of revolution whose curve is 
 ANC or z • 
 and let x=zAM^ altitude of both solids, ^ 
 
 i/ = MN', ordinate to point A^.in curve ANC, 
 
 293 
 
 u 
 
 0. 
 
 In AMNO, MN: JVO=cos a : sin a, 
 or y: y' = cos a:sin a ; 
 
 NO y' sin a 
 
 MN 
 
 AOD, 
 
 whence, = ^ 
 
 = tana. 
 
 y cos a 
 
 The cross-section of a right polyedroid is a regular polygon 
 which may be divided, by its radii, into as many equal triangles 
 as it has sides ; and the apothem of the polygon (radius of the 
 inscribed circle) bisects the angle at the center, in each of these 
 equal triangles. 
 
 Thus, the cross-section of a triangular polyedroid is an equi- 
 lateral triangle divided, by its radii, into three equal triangles, 
 in each of which the angle at the center is equal to 120^ ; 
 hence, one half (NMO) of this angle at the center is equal 
 to 60". 
 
 Therefore, in a right polyedroid whose cross-section is : 
 
 1" an equilateral triangle, ^=vi_ = tan.a— tan.60° = v/3 ; 
 
 MN y 
 
 ='~ = tan.tt=tan.30'^: 
 
 MN y 
 
 NO_y' 
 
 2" a regular hexagon, 
 
 3" a square, 
 iVc. 
 
 MN 
 
 &c. 
 
 y 
 
 :tan.a=tan.45° = l ; 
 
 This being stated, solve the following 
 
 • When tlie curve --l(>/> of a itolyodroid is represented liy an equiition 
 cont lined ill y=nv-\-ltz'-^cz-\-f, the curve AN'C of the inscribed soliil of 
 levolutic n is rcpr'-sented by ii transcendental e(|uation (App. P) 
 
 
 
 
 '1? 
 
 
 'M!Ul 
 
 Hi 
 
 1'', 
 
M 
 
 f 
 
 1:11 
 
 iiili 
 
 ill 
 
 
 li 
 
 294 .......:' r: ;v 
 
 Problem. AsHnmivij thr cnm of a right qnddrdngnlar 
 poli/edi'oitl^ circumsrri/if'd f(J>oiif n solid of rrro'utlon, tohc repre- 
 sented bij the eqiiiitioii yrrrifSz — ::'-): Ji u d th e equation of the 
 Curve V^ of a ri<jht trimujiilur, and 2' <>/' " right Itexogonali 
 poli/edroid rirei(mscrif)('(l <dii>iit fhr tihore solid of n vohttioii. 
 
 .Solution. From tlie a!)ov(MM|u;iti(>:is d" and 2'). we liave : 
 1' 1/= 'J.^. 2" //=^/.'5//'; tluMi. introducing ^successively -these 
 
 values of y, in the given ('((uation )/-=^^(^z — ,:-), we have : 
 
 1" )/'zj=:j^^'S(Sz — ;:'). 2' . — -{^z — :,-'). for t lie two equations 
 
 required. 
 
 II. Limits of tup; (Mrvk <ti' a i'olvkdkoid 
 
 OR of the INSCI11I5E1> SOLID OF REVOLUTION. 
 
 Denotini; the curve ( 'J-^') ""f the generating plane {AMy) 
 of a solid of revolution, by -. the differential of this curve vrill 
 
 he dz=^s^d.i'--\-(h/' (Book III. 'Ji'/); whence, dx=:\^<h'' — dij'-. 
 But. the quantity under this radical sign becomes negative, 
 and thereby imaginary. \\\\vn the increment (/:; of the curve is 
 less than the increment di/ of the ordinate : hence, when 
 
 Air=-d;S-. or dii = ^dz. or •-':=il, the curve roaches the 
 -' dz 
 
 limits (positive and negative), byond which it is imaginary. 
 Now, by Art. 20, i!^^ '^ - 
 
 = tan.«(a constant (juantity). 
 
 or 
 
 MN y 
 Differentiating ?/'=// tan.'/, tliere obtains dij'^=<ifj tana 
 
 dg: 
 
 tan. a 
 
 7 / 1 
 
 Substituting — ~ for <^//. in the above equation _ i=±:l, 
 tana dz 
 
 there obtains : ' ' =d=tan.a. for both limits byond which the 
 dz 
 
 curve ^0/>of any polyedroid beconies imaginary. Therefore, 
 
 from the equations (1". 2'. 3, ^*kc) of Art. 20. there obtains 
 
 for the limits of the curve of a right polyedroid which is : 
 
AT'PFNDTX T), 
 
 295 
 
 he rep re. - 
 m of the 
 \,exngonah 
 evolution. 
 
 svo liave : 
 •ely 'these 
 
 have : 
 equations 
 
 ) 
 
 ON. 
 
 .e (AMN) 
 curve will 
 
 s uoi:ative, 
 10 curve is 
 nco. when 
 
 ches the 
 
 liinary. 
 antity). 
 
 = (hj tana 
 
 which the 
 
 Therefore, 
 Ire obtains 
 \\\ is : 
 
 d]l' 
 
 I' triangular, "•-=:±tan.a=±tan.r)0°=±v/3, 
 dz 
 
 2 (luadranuular, '' -= 
 o ' hexaironal. " = 
 
 " =±tan.4{>°=±l, 
 
 =±tan.30^^: 
 
 1 
 
 7h' 
 
 cS:c. = i<CC. 
 
 That is, 1" the curve xWD (the limits of wliieh are those of 
 
 •L) cannot transcend ±00 ', in a riuht trianuular polyerlmid ; 
 dz 
 
 2" ADD connot transcendd=:45'-\ in a riirht quadraniiular 
 
 polyedroid ; l> ' AOI) cannot transcend ±30*^'. in a viuht 
 
 hexagonal polyedroid ; &c.. v^c. 
 
 The ]>rinciple wo have just laid down enables us to doterniine 
 
 the limits of a polyedroid and of a solid of revolution, by 
 
 determinint:; the limits of their curves. 
 
 E.c. 1. Assuming tJie curve X{)\) of (t right (pindrangnlor 
 polyi^droid to he represented htj the equdtion y = ^(8z — z-) 
 ichatu'ill be its limits ? 
 
 Solution. IHiFerontiating )/=^^(^9>p: — z'). there obtains 
 dy=^(S — 2z)d;,, or 'y^=:^(-l:— ,t); but (above equation 2'), 
 
 the limits of the curve, in this case, are -i=r±l ; 
 
 hence, ]:(4— ^O^r-bl ; 
 
 whence, the required limits of ADD are ,i— (), and ,-=8. 
 
 Ex. 2. Assuming the curve ADD of <i, right triangular 
 
 2)olyedroid to he rep)resented by the equation y= (.3z- — z-'), 
 
 what %cill he its limits 1 
 
 Solution. Diiferentiatins; ?/= (Zz-—z^'), 
 
 '' 8x/3^ ^' 
 
 there obtains --= — — -(62— 3r); but (above equation 1"), the 
 dz 8\/3 
 
 limits of the curve, in this case, are ■^=zfc\/3 : 
 
 dz 
 
 X 
 
 
IF 
 
 I 
 ill, 
 
 !' it 
 
 ■ i 
 
 296 
 lience, 
 
 APPENDIX D. 
 
 1 
 
 8x/8 
 
 -(62-322) = d=%/3; 
 
 .Solution. When -^y = K then —^ = ^H (Art. 21). 
 
 whence, the required limits are z = 4, and z= — 2. 
 
 Ex, 3. In ;in ohUqne jwh/edroid whose curve AOD is 
 
 YO 
 
 represented hy y-=^i''-<^\{)^th< r<(fi()~ — ^ As equal to 8; Ji7id 
 
 the limits q/"AOI>. 
 
 XO 
 
 MN dz 
 
 Now, diiFerentiatinjr ?/ = ^2^+10, 
 
 there obtains '-'^ =^8;:-=^-- ; henee, ^r-=±8 ; and the limits 
 dz 
 
 oi AOD are :;=-f-4 and :; = — 4. 
 
 Find the limits of a right quadrangular polyedroid whose 
 
 curve AOD is successively represented by the equations of the 
 
 following examples.* 
 
 Ex. 4. ii=\z^. Differentiating, there obtains : -_:/=^2z ; 
 
 dz 
 
 hence, ^2r===fcl ; whence, the limits are r=+4 and - = — 4. 
 
 Ex. 5. //= — (4a;—:;')- Diff.there obtains : _i= — (4a— 2:r) 
 •^ 4a^ ^ dz Aa> ' 
 
 hence, — (2a — :j) = ±1 ; whence, the limits are c=0and-=4a. 
 2a 
 
 Ex. (). ii=l{z^-Q>z-+'^z). Diff.wehave : ^=l(3:;2_i2;+8); 
 
 dz 
 
 hence, ^(3:5' — 122 + 8)==fcl ; whence, the limits are 
 
 r=0 and :;=4. 
 
 E.r. 7. i/= — (:;^—12s-'+48s). Differentiating we have : 
 
 '^I—Jl (322—24; 4- 48); 
 rf2 48 ^ " 
 
 hence, -q(32" — 242-|-48)=:±l ; whence, the limits are 
 
 2:=0 and 2 = 8. 
 
 • Itmustbe borne in ir.'nd that tliL' point O*' of the curve is invariably 
 the vertex of its uoxinium and of its minimum ordinate (4, Son. It). 
 
 
:m 
 
 AOD is 
 8; Jind 
 
 the limits 
 
 id whose 
 ms of the 
 
 d- = -4. 
 
 L(4a— 2:j) 
 
 tn 
 
 d;=4a. 
 
 122+8); 
 
 lare 
 
 lave 
 
 are 
 
 invariably 
 II). 
 
 appendix d' 
 Problem. 
 
 297 
 
 Given l=x-4- — -|-- 
 
 o.v5 
 
 3 5x7 
 
 -f-&c. (Book III. 49, equation d); 
 
 2.3 ' 2.4.5 ' 2.4.6.7 
 'ofind x[=isin 1) in function qf 1, by the inverse method of v, 
 
 .1 A_ 3 . 3 5 
 
 Solution. Let 
 
 i=«c. 
 
 2.4.5 2.4.G.7 
 
 By this notation, the given equation becomes 
 
 ^=2-f-ar''+Aa:- + CA"+(7a:''-f cj-c. (1) 
 
 Raising equation (1) to the 3nl, 5th, 7th and yth power (and, to avoid 
 a useless work, discontinuing the multiplication as soon as the product 
 shall contain x with an exponent higher than 9), there obtains : 
 
 l^=T^-\--iax^-^2{d'-irh)x'-\-[a^-\-Qab-\-2c)x^. (2) 
 
 /»=a:S4-5a^'+5(2a'^-f6)z«. (3) 
 
 r'=x''-\-lax^. (4) 
 
 P=x^. ^ (5) 
 
 Now, letting x=iAl-irBl?-\-Cl=-\-Dl''-\-EP-\-^c, (6) 
 
 and substituting the second members of (1), (2), (3), &c. for I, Z^, V>, &c. 
 
 in (6), we have ; x=:Ax -\-Aaj(^-\-Ab]^-\- Acx''-\- Ad^ 
 
 4- Bx^^ZB(v>-\-ZB{a^-^h)x^-\-B(^a^-\-%ab-\.Zc(7? 
 4- Cx^-\- 5 Cax^^ 5C(2a2-fA) x» 
 
 + Dx'-\- iDai^ 
 
 4- Ex^. 
 
 Transposing x to the second member, and factoring, there obtains : 
 
 0=^ 
 — 1 
 
 x-\-Aa 
 
 23_|_ Ab 
 
 -\-3Ba 
 
 x^+ 
 
 Ac 
 
 +3j5(a 
 
 ^4-/>) 
 
 4- 
 
 5C« 
 
 + 
 
 D 
 
 .Dividing by at, there obtains : 0=.4 — \-\-Aa 
 
 4- B 
 
 z'-f- Ad a:9+&c. 
 
 -f-5(a34-6rt64-3c) 
 + bCi2d^-\-b) 
 4- '/>« 
 
 -j-3i?« ■ ;.' 
 
 + C 
 
 Since, this equation (7) is true for any value of z, it must be true, 
 when a;=0, and then ,4=1. (8) 
 
 Substituting 1 for A, in (7), and dividing by x'^\ 
 
 1, 
 
 
 I 
 
 ;?■'"'( 
 I^M 
 
 0=a + ^4- A 
 4- 3i?rt 
 
 + c- 
 
 a:2-|-&c. 
 
 • 
 
 
 When a;=0, there obtains B=—a. 
 
 (9) ^ 
 
 By the same process, there obtains : 
 
 
 C=—3Ba—b=[W^—b. 
 
 (10) 
 
 D=r>— c — 3/;(a-'4-i) — 5Ca=8a6— 12a3— c. 
 
 (11) 
 
 J?=— f^ — &c. 
 
 
 ■i 
 

 APPENDIX E. 
 
 lilt oiiuciti},'' these vnlucs of J. /?. C, «kc.. in (0). tliore obtains : 
 
 x=:l—ul^4-(:;,i:'—/>)r'4-iHah—\2ir^—r)l-Jf-&c. (12) 
 
 rtiijilly, inir()!iu(;i:ij(. i.i (\2). t!ie .'imiKTiciil viilucs of a. h, c, &c : 
 
 ,v =siii /)=_ — ___ 4- — + .__ 
 
 1 I. 'J.:: i.2.:{.4.r) ^ 2.:> a J> .(',.: i .2.:;.4.r).(;.7.8.y 
 Tliis number ut terms is suflicienl to sbow the l.aw of tlie series. 
 
 Arp:.\ of the Common Hyperbola. 
 
 Assume DAJ to he :i:i bv]ieil)olic cur.o, wiiose 
 
 r> 
 
 equation is 7/=^y/'2ax-\-x'' (urij^iii iil \er;ex A). 
 a 
 
 % 
 
 In '/.l=-('.></.r-fy-)iiv, (ii!(erciit:;il of.l/^. J(A 
 
 It 
 
 ]iit a-\-x=t ; th.-:i. 'L\=/t. k /-— (/-'=li'/.{-]-.i-. 
 
 Introilucinu' tiiese viilucs of .//;i mI of 'I'lx-f-x-, 
 
 li } 
 
 \\\ tliediirercntial. there obtains : ./J= ■i-—ii-)~dt. 
 
 the intetrral of whieli is 
 
 .1==- S //J _.,:_— In ; »./, 
 
 ^^^)]+C^ 
 
 Eestoiinj^, to t. its value (^a-\-x), tliere obtains : 
 
 But. .-I=r0 when a:=o : lieuc', Ui= — l,il,-^l,i-i.C. or C—\ahXl'i- 
 Therefore, the entire integral, or area of any sediment Alh>. is 
 
 h . 
 
 Doublingthis result, and substituting y for- v 2.o-|-x-, there obtains : 
 
 A=xj{a-\-x)—<i},y^l\ -d. V-^- r- 
 
 <- lib ) 
 
 In this formula, the symbol I represents tlic naperian logarithm ; hence, 
 letting log=lo;;arithni of the common system, 
 
 and (Algebra), J/=0.4:543, modulus of the common system, 
 
 there obtains : il/X^=log; whence, Z=:_2. 
 
 ' M 
 
 '' "'•'ejb»'.-bv'substitutini'- the common _JZ for L there obtains for 
 
 .)/ 
 
 Area of the hyperbola (altittide=jt): 
 
 A=y(a-{-x)— log^ ^' '^ \ ^ ' \. 
 M y. lib i 
 
(12) 
 
 h 
 -A 
 
 ')i 
 
 ; hence, 
 
 for 
 
 APPKN'DIX F 
 
 F. 
 
 Convex surface of a Prolate Ellipsoid. 
 
 29£ 
 
 DifTereatiatiiig i/-=^('2(ix — z-), einiatioii of tlu.' curvt! ;ienerating the 
 
 (I- ' 
 
 convex sii;fa(.'C' of a pnilate ellii>soiil (origin at the vortcv of the axis), 
 
 there nbtiuus : t///=_-l '- — ; whence, ///-=_ L — i 
 
 (I- 1/ ' a^ jf- 
 
 Introducing tiie.-e vahu's of// a!;d ihi'- in thi' <fiMi,ral ditleT'iitial. 
 
 dS=^2~yy/(h'-\-Aii-. of a suiface of levolntion. aid iviiuciuLf. there 
 
 obtains for dKfv'i'entia! of tii- couvcv .^iufarc of a [)foLit(; ellipsoid : 
 
 I' . 
 
 dS=2~- /r,-',,:i ^.,.t- — /;^jriax~-x-)dx. 
 a- 
 
 Letting a — x=u : then .//= — du, and 'lu.x — .f-='<- — u-. 
 
 Again, in the ellipse, c'-=d- — f/-, and 
 
 fHook in. I 1 Sc 12). 
 
 Introducing these vaUu's nt' (<i " — //'). (•J'/.r— r-;. and d.i:. in the ditferea- 
 tial, and reducing, tliere obtains : 
 
 a' sj c- 
 
 2- _ / — H-d>i. liio integral 
 
 C, 
 
 ofwhichis S=—2-—<~ /-— >/--4- ^m ~l 
 
 ul-lsl e- 2<- a S 
 
 or, restoring to u its value {(i—x). 
 
 a ) 
 
 *=-?{<— >j:-»'+ 
 
 i'lx — x--i- ^m 
 
 ■ 1 
 
 Since, - , =-_, and 5=:0 when x= i ; then, C^=l,-(l,-l-ii ); 
 
 whence, the entire integral, or area of any zone (altitude=x): 
 
 S=hT ^ i-f-a + s//;iA.e2. 2>ix-x-) - - ^^» [ • 
 
 When x=2a, there obtains, for the surface of the wiiole prolate 
 ellipsoid : 
 
 S=2b-(b-\-a 
 
 sin 'e 
 
 Cor. Area of a zone of two bases ('altitude:=X — x): 
 
 A' — a 
 
 '—brr j ^^-^\/6-^-f.e-^^2(/..l-A'-)-f'^-^v//.24-e-Y2ax— r^) 
 
 a/ . —ie(a—A') . —leia — x)\ 1 
 — -(sin J^ i— sm J^ 1) V. 
 
 e\ a a / i 
 
 mi'm 
 
 .^5 
 
 f4 
 
 'm 
 
 i J 
 
J3 
 
 I 
 
 i 
 
 300 
 
 APPENDIX H 
 
 Sch. Since the symbol sin 'c represents the circular are whose sine 
 is equal to the excentricity (e) of the generating ellipse, the quantities e 
 
 and sin 'f may be eleminatod from the expression a- 
 
 when the 
 
 ellipsoid has but little excentricity ; for both quantities are then sensibly 
 equal. 
 
 G. 
 
 Convex Surface of an Oblate Ellipsoid. 
 
 Difl'erentiating »/-=_(2/y/ — x-), and s'lnarin:^' the result, we have : 
 
 //- 
 
 dii^=:- ' ' '^ !lf-: hence, ihe differential of the convex surface of an 
 
 oblate ellipsoid is d^=-!J-y,/^fii,ij^i^fji_^i)^2bx—z^)dx. 
 
 Letting b—x=u ; then. dx=—du, and 2bz—x^=:b'^—u'^. 
 Introducing the values of dx,(2bx—x-), and (b2—a'^)(Ai>\K F), in the diflFe- 
 
 ^rtac lb* 
 rential, aird reducing, there obtains : dS= — Z — _ L--\-u'^ du. 
 
 b^ >/ o"* 
 
 Byintegration .9=-!f5| _^ j^^+il/(«,4- j?+^^ 
 
 Hence, the entire integral, or area of any zone (altitude=a;), is 
 
 „ f b—x . b"^ b{a-X-c) 1 
 
 S=^a-l a— ^ixj^^\b—xf\~rr log -T - l. 
 
 When x=2ft, there obtains, for the surface of the whole oblate ellipsoid : 
 
 c. ., i I h- 1 rt-i-c 1 
 
 «=..«, a+_log-^}. 
 Cor. Area of a zone of two bases {altitude=X —\): 
 S=a:T j ^^^bi^ci{b-~Ay-^^^b*-}-cHb—x)2 
 
 4- Z X <^(b-^)^^f'*-^c-{b-x)i > _ 
 
 H. 
 
 Convex Surface of a Common Paraboloid 
 
 The differential of the convex surface of a common paraboloid is 
 
 dS=~H4a'i-^>/-)-ydt/. the integral of which is S=—(ia'^-\-y^)'^-\-C» 
 a ' ^ 3a 
 
 Since y-z=Aax. and C=— i^-rt!* there obtains for 
 
 Convex surface of a comiaon paraboloid (altitude=:a;): 
 
 S=J- { (a-'-f a^) ' — aS I . 
 3a V } 
 
APPENDIX K 
 
 301 
 
 K. 
 
 Convex Surface of an Hyperboloid. 
 
 The differential of the convex surface of an hyperboloid is 
 
 Let a4--v=M ; then, dx=idu, and 2ax-^x^=u- — a**. 
 Introducing dii and m- — a^, in tlie differential, there obtains : 
 
 o-fj 2ri - 
 
 dS='^^- s/a-/j.iJt-i^a'-\-/j')(U'—u-) iu=. ''-^'^■i\^(ar'-\-b^ui—a*du. 
 
 Again, c-=a2-|-i-, and C'=a'c- ] whence. b'=:a'e'—d^. (IJook III, 15) 
 Substituting d-e' for {d'-\-b'), in the differential, and reducing : 
 
 dS=^^ \ «=*— - du, the integral of which is 
 
 or, restoring to u, its value (a-i^x)\ 
 
 S=bK i ^\/e2(a4-a:)2-a2 — ~l fa-\-x-\-- \/?J(7fI)^rZ^ \ 1 4.(7. 
 
 When x=0. 5=0, and 0=6x1 \/a'e^—a^—~daJf--\^d^e^-d^ \ I +(7. 
 Hence, the integral, or convex surface of an hyperboloid (altitiide::^?;), H 
 
 „ , f a-\-z , a , ae-\-h \ 
 
 S:=^brA -^-k/ eHa-\.zf—d^^b-V--rr^^% ~ \ 
 
 ^ a \^> ^Me e(^a-{-x)-^K^>\a-\-x-)—dn 
 
 Cor. Area of a zone of two bases {altitude=X — x): 
 
 f a-\-X . a\x . 
 
 5=67r ——>/ e\a^Xf—d- l—V e^{a^x)-'—a^ 
 
 e(a-f-x)4-\/«-(a-f-A-^^— a^ " ' 
 
 4— 1 log g(a+a^)+^ e.-{a-\-x ) i—a^ "l 
 ^^^ e{a + X) J^\/e\a-\-X)^—ni ' 
 
 
 3 
 
 I 
 
 
 'K( 
 
 i 
 
 M 
 
302 
 
 Al'PENlUX 1, 
 
 , 1 
 
 !• 
 
 S 
 
 L. 
 
 PltdHLE.M. 
 
 1" Assnmini/ (rADHH lo !»■ a rcctun^li. tfrminated by a semi-rircli' ADI!, 
 to lotatr (tliiiut l/w Jixed axis ijll, Jinil Iht 
 formula oj titt volume of the solid of revo. «l ^v 
 
 lulion 
 
 Solution. Let R=JW=CD, 
 
 az= (H z=ml, A=:arcJJu, 
 x=Ciii, 
 
 y=\^J{^ — x^=mn=i>iii'.^ 
 The differential of the volume {V) of this solid of revolution is 
 dV=7Tilny<lz=-(a + y)-idx=-(a'i-{.R^—x^-\-2a^U'-x-)dx ; 
 
 whence, F=t j d'x-\-R'x-iiX?-\--la (u^/W^^-^-^R^bxvT^j^ j -f C7. 
 
 Since F=0, when jc=0, then C=0. 
 
 Hence, substituting A for its value Rs\n 
 
 — I X 
 
 - , there obtains : 
 
 Volume generated by the revolution of Dnlt about GH ; 
 
 F=:r[z^a2_}_i^•■')— Jr»-|-rt(z.v-fye-4)]. (a) 
 
 When x=i2 or CA^ then 2/=u, and yl=,\/fT or quadrant DA. 
 Introducing these values of A, x, y, in (a), and doubling the result* 
 there obtains : 
 
 Volume generated by the revolution of GAD B II ahoMt Gil : 
 
 Sch. If the volume, 2Rza^. of the cylinder generated by the revolution 
 ofABIIG, be subtracted from the volume Fin (b), there obtains : 
 Volume generated by the revolution of the semi-circle -.4/?^-4 about Gil. 
 V=pRJ^-\--aX'^R''. {&) 
 
 2" Draw the semi-circumference AcB, and find the formula of the volume 
 generated by the revolution of the plane AcHHCl about Gil {above notation). 
 Solution. The differential of the retjuired volume is 
 
 dV=-{ln'fdx=-(a—y')-dx. 
 Proceeding as above (I"), there obtains : 
 
 Volume generated by the revolutioti oi cn'lt, about GH : 
 F=-[,v(a^+i?2)-ir''-«(xi/-fi2J)], 
 (in which A represents the arc en' whosL' sine is x or Cm). 
 
 Volume generated by the revolution of AcBHG, about GH i 
 F=T(27?a^+AA^— rtTA2). 
 Sch. I. If the vol' mo F, in {/), be subtracted from the volume. 2Rna^, 
 of the cylinder generated by the revolution of the rectangle ABllG, 
 about GH there obtains : 
 
 id) 
 
 if) 
 
yr^^W' 
 
 AI'l't:.M)lX M. 
 
 3U3 
 
 I,: ADI5, 
 
 )!+.. 
 
 (o) 
 DA. 
 
 he result) 
 
 (ft) 
 tevolutioa 
 
 Is : 
 
 lljout GIf- 
 
 the volume 
 lotation). 
 
 id) 
 
 (/) 
 itG, 
 
 Volume (fi'ncnitt'd by iln" nvniution of iu> «oti)i-circle AcHA, »l»oiit Clf 
 
 Sch. II. AdiliMjr (<')nii(l (-/). tlicn- ol'tains : 
 \'cilutiie },a'iiL'ia'o(l hy ilii- ri'Vdliiiio.'i of llie circle AJJBc, about GJl : 
 
 V=2-aX-R- 
 
 (AJ 
 
 M. 
 
 I'kohle.m. 
 
 ! " Find t/icfurtiiu/a of tin: cnnrrx Kiir/acr i/'-timittd L>/ tin: reioliition of the 
 genu circuni/trcnce .\\)\>,(tli(iiit (III (iiifci-fditi;? lij^j. 
 Solution. Let /i=.IC'=67>. a=AG=Cl. .l=fin" Ajm, xr^Am, 
 
 y=z^'lKx—x''=inn=vin' (ori^^in ftt the vertex .1 of Al>). 
 
 The diflertntial of the surface ^i-'iierotid by the revolution oi Apn id 
 
 dS=z2-(lm + mn)s/dx-^di/'=2::(a-lf7j)^'dx^ + dtf. 
 
 ( Jl x\'^d^^ 
 
 Substituting ^ 1 for its value dy"^, in (a), there obtains 
 
 (0) 
 
 rf5=2T(fl4-v) \^^:==2^Rdx-\- 
 
 2anRdx 
 
 , 
 
 y 
 
 the integral of which is 
 
 S=2-:TRx-\-2-!raR versed sine 
 
 — 1 X 
 R' 
 
 Substituting A for its value, R versed sine — , there obtains : 
 
 R 
 
 Surface generated by the revolution oi Apn^oXtOMl Gil : 
 
 S=:27TRx+2-aA. (R 
 
 When 2;=2i?, then A=-R or semi-circumference ADB ; whence, tk 
 
 surface generated by the revolution of the semi-circumference ADB : 
 
 S=i-R'-{-2-aX-R- (c) 
 
 2" Find the formula of the surface generated by the revolution of the semi- 
 
 circuv'/erence AcB, aljout Gil {above notation, 1"). 
 
 Solution. The ditteiential of the surface generated by the revolution of 
 
 the arc Ap'n' (or A) is 
 
 dS=2-{lm—r,}n')^/iA'+d!i'=2-{a—rj)^dx--^dy\ 
 
 Proceeding (is i.bove ( 1 "), there obtains : 
 
 Surface generated by the revolution oi Ap'n', about GH : 
 
 Sz=2raA—2-Rx. (d) 
 
 When a;=2/»', then A=7:R or senii-cireuraference AcB: whence, the 
 
 Surface generated by the revolution of the semi-circumference AcB : 
 
 S=z2-ax-lf-'i R'. (/) 
 
 Sch. Adding (c) and (/). there obtains : 
 
 Surface (f the ring generatt d by the revolution of the circle ADBc, 
 about the axis Gil : 
 
 S=2T:ax2^R. {g) 
 
 " 5 
 
 ^'i 
 
 11 
 
 I , 'I 
 
 HI" 
 
i 
 
 304 appendix 
 
 Volume of a prustuji op an Elliptic Spindle, 
 
 WITH BYMMETRICAL BASES. 
 
 When an elliptic s jfment ADD A revolves 
 abo;. I its fixed choid AB,\hQ solid of revolution is 
 called elliptic spindle. ^ 
 
 Th'i formula of the vohim.) of a f.ustum of 
 this spindle may be obtainad as foliotvs . 
 
 Let a=9i'mi-naajoraxisG^O, 6=s8emi-miaoraxisi)0, R=DC,r=Fm=:Eh^ 
 
 b , 
 
 x=iFn=pO, y=~>/ d^—x'^-=.Fp (origin at the centero); 
 
 then, mp—CO=DO—DC=(b—R)=m, Fm=Fp—mp=7j—m. 
 
 The differential of the volume generated by the revglution of any seg- 
 ment CDFm is 
 
 f 62 b . -, 
 
 dV=r.{Fnifdx=TT{;y—mfdz=:nrX i2_^2;2_>2m-V a-— x'^-fni^ lir, 
 
 the integral of which is 
 
 Since, F=0 when a:=0, then C=.'^. 
 
 Doubling the iiitcgril, and letting «j=(ft—i2). 2x=l or EF, and 
 
 2osin ^-=iA(ov arc EDF whose radius and chord are a and I), 
 a 
 
 here obtains, by introducing A, I and (b — R) in the double integral : 
 
 Volume of a frustum of an elliptic s[>iadle, with symmetrical bases : 
 
 V=:T^bn{l-JL.\^(b^R){rl-{-bA) \. (a) 
 
 Sch. When b=a, the elliptic spindle is changed into a circular 
 spiniiif, and there obtains : 
 
 Volume of a frustum of a circular spindle, with symmetrical bases : 
 
 V::=~LiH(l-Jl-\^(a-R)(rl-{-aA) \. (6) 
 
 o. 
 
 Cycloid. 
 
 When a circle FnE rolls on a stiai.lit 
 line AC, any point n of its ci'-Cum'er(Mici' 
 describes a curve .l/yf' called a cycloid. 
 
 The circle FuE is the generating c rf.l A 
 lU'.d the voint n is the generating point. 
 
.E, 
 
 APPENDIX (> 
 
 EylATIOX (»K TIIK ('VCI.OID. 
 
 Lit a=fJF, diameter of the jfonerntiniir circlf ; 
 
 x=Ain, iiiul y:=.,iin:=.Fo, M-ioCil .-iiie of tlu' ir.c F>i 
 
 then. .'l/''=/''/)=:ver3ed sine" \y=''vei sod sine"'"''. 
 
 !>;> 
 
 
 771. 
 
 any seg- 
 
 } 
 
 tic, 
 
 and 
 
 i), 
 jral : 
 
 (a) 
 circular 
 
 sea ; 
 
 (&) 
 
 and /''//( or on=^y/<ii/ — i/-. 
 
 a —1-'/ / 
 
 Hut, a; or .h/;=J/''— A'm=., versed sine ' —\/^„u—,/'-; whenc . the 
 
 Equation of the cyehiid ^urii^in at the i.oiut A): 
 u __ , L' // 
 
 a 
 
 -V mi — '/-. 
 
 (I) 
 
 FOKML-I.A OK IfKl'TIKICATION OK T!;;: '^' V;T,.ili). 
 
 Lettiiii,' ^=len.t:tli of a plnu' eurvc, its dilfe-.ent'al will l.e 
 
 ihz=-:\^^'dx'-^.l)f. (-21 
 
 The differintial of (1) is ./.f7=-l- ■" ^■' : \vlie:.<v. ./.■-= ■'■':/;''_!.. 
 
 s^ ail — II- <n; — ii- 
 
 Introducin-- tiie value of <//-' in {■l). \\v' (ii!iereri;ial of ;i e.-\i)id \vi;i iie 
 
 dz=.-^ j.' ' ',. .-4-1^^'/"=^ / — <1 11=^11 - i a — 7 ', - ,'y, 
 siay—y- <" — >/ 
 
 the integral of which is := '^2(i-i 'i—y) \ -\- (^=z 3z ■j>/,/- _. ,, .7 4- C. 
 
 Since 2r:=:n when »/='X th 11 C--=^'l<i- 
 Therefore, th.' intejirral. o,' formula of rociticatiou of acvcloid. is 
 
 This equation, solved witli respect to //, liecomes 
 
 (3) 
 (4) 
 
 FORMl'I.A OF T!IK CnNVnX S'TI^K ' K\i:!;.\'! '' i !;V TIK R'! V 01' Tio.N 
 OK A CvCLOin i.lA^'t A1.0( !■ ITS liASI-: (.il'). 
 
 Ilv tlie iiliove notation, the dinercnr'nl of ilic siir''.ic> <^ rn'nUfd li\ tlie 
 
 revolution nf tiic rNcioid A ,' -< '. aimut ■< I ast- . I ' '. u i 1 !jc : 
 
 1 : : • 
 
 the i.;t('j>Tn! ct' v.hi- h is >' — r:;r ' - ... -..,rvi ■(''.'//—)/) — '' '• 
 
 Since .V;:^ I wiien y=:'K \\u-\i '-.- -'t-. 
 
 Tlence, tlie i iitire iiitei:-:-:*!, oi- f iviur.'a of li.e snr hi'e ueiicrat'-d liy tlic 
 revolution of the i-vclo'.;' A/iC. a- out its 1 :is" .1''. i^ 
 
 S=\-[•:<r^'-:=s::!'l^^ — ///• 'J-/ ^-y\]. (f)) 
 
 NoTK, Coraiiare (5) witli tlie niono'c.'.i'ila (I'ook III. (U.I'ruh. 10). 
 
 . i-M 
 
 I 
 
 m 
 
 ■■:SI 
 
 :rm 
 
 
 m 
 
 
 jwj 
 
 ".il 
 
 20 
 
306 
 
 Al'l'F.NDiX i 
 
 P. 
 
 i"QrATioN OK T)iK ciRVi': . L\7 ' OF A Solid ok i»Kvoi,fTio.\, 
 inscriJidl iu <i /Djramuloid ir/ta.sf curvf AH!) /.,• cuiiUiimd in y=m:'.--\-n7.-\-p. 
 
 If the curve AOh of fi pvramiduid he lepre- 
 seiited by one of tlie ecjuations ( 1 ),(2) and (,;), / 
 in Hook III. '_'.'!. 2". the equation of the fiirve 
 ^4J\'C of the in.scril)ed solid of revolution may 
 be obtained as follows : 
 
 Vi \! \' 
 
 \\ \\ \\ + 
 
 Let the equution ii=-[nz — Z') represent the curve AODofa right 
 
 P 
 quadrangular pyramidoid. 
 
 Differentiating ?/=~(«2— 2-). there obtains di/=-{fi~2z)dz ; then 
 P P 
 
 introducing this value of rf?/. in dx=y/dz'—(h/'^ (Hook III. 36), 
 
 J 
 
 we shall have : dx= / ({^-----i " — -zY'dz- ==. ~ y/p'—i n— 2z)^ dz. ( 1 > 
 
 >,' p- V 
 
 Let n — 2z-r=.u ; then, J;= — hiu. 
 
 Substituting— ^du for dz, and ti- for (n — 2z)'', in (1). there obtains : 
 
 the integral of which is 
 
 1 / 
 
 dx=~ ~S^ It- — ?/- du, 
 -/' '' 
 
 I f U y 1)'- , « 1 
 
 ^=-•274 2^^'-"' +2^'" p|+^' 
 
 or, restoring to u its value (n—2z), 
 
 "'"""i'i ^«-"-)x/r-(«-2^~)'^+i'=sin~'^ } + C. 
 Letting c=(). when .v=(i, the entire integral wijl be : 
 
 Tlie e(iualion of the curve JA'(7(that is, the equation which cxprcssi'S 
 the relation between the co-ordinates x and // of every point of that 
 curve z) Aviil l)e obtained by substituting, for z. its value in function of y, 
 in the equation (2). 
 
 For this purpose, solvi' '/=:(«- — z'-]. as follow.s ; 
 
 2z — »=-f-\/»' — -ij'l/- or n — 2z=:.-^y/ii^ — ipy. 
 Introducing this value of (« — 22», in (2), tht.3 obtains : 
 
 
 ApyS^ji- — n'-\-4ity -\-p'hi\ 
 
 — 1 iw — 
 
 j"-^') }• 
 
^l! _'J> 
 
 (l> 
 
 ATM'KNDIX R 
 
 Equation of the curve AOl) of a ii<ilit tiuadrangular {)yraini(loid : 
 
 307 
 
 {A) 
 
 /' 
 Formula of rectification nf the curve .LVCoftbe iuscribed soldi of rev o" 
 
 lutioii : 
 
 Kquatiou of tlie curve AXC of the insciiiied solid of revolution: 
 
 1 . _!« 
 
 4p { f> 
 
 Sch. Letting /<=n=4a. in equations (--1), (-4'), (.4''), there obtains: 
 Equation of the curve AOIJ of a right quadrangular pyramidoid : 
 
 ^=_(4a2— z2), which is (4) of App. O. 
 
 4a 
 
 Formulii of rectification of th.- curve -47V6' of the inscribed solid of 
 
 revolution : 
 
 2==:2(a=f:v/a-— ay), which is (3) of App. O. 
 
 Equation of th- curve ANC of the inscribed solid of revolution : 
 
 x=-Y^( >^«y— 2/^-fasin~^ l^^ni^V which is (I) of App. O. 
 
 } ;^) 
 
 jpresSL'S 
 lof that 
 lou of y, 
 
 [)}• 
 
 B. 
 
 Proportions of the column in the five 
 Orders of Architecture : 
 
 Toscan, Doric, Ionic, Corinthian and Composite. 
 
 If the columns of the five orders be given _a common altitude, the 
 seventh, eight, ninth and t'nth part of that altitude will determine the 
 diameter of the lower base of the shaft in the toscan, doric, ionic, and 
 Corinthian or composite order, respectively. 
 
 The radius of this base is a unit of measure called module. 
 
 Some architects gradually swell the shiift of the column, in the five 
 orders, in such a manner as to give it the following proportions : two 
 modules to th.' diam.'ter of the lower base, two modules and one twelvth 
 to the diani'jter of the cross-section between the two lower third parts) 
 and one module and ten twelvths to the diameter of the upper base. 
 
 J 
 
308 
 
 APPENDIX R 
 
 CoNt fHH!) 
 
 Thfc lonorittKUiial median s rtioii of tin- sliiif't of a column is a plan 
 figun; whose curve, cullod r<>»rlit///. is di'-cfiljcd as foliowri : 
 
 JJm. 
 
 no. 
 
 If tilt' cc'.iter (' lie tal< 
 
 cil :is 
 
 oriiiiii of tli(.' roctanj^ular axes AH and />/■' 
 
 tiie cosines HCinsi 
 
 hi aii't iti> 
 
 if tlic respeotivi' arcs n*-', /, 
 
 >tii. 
 
 itil 
 
 and />'«. wilt deUMiiiiiic til- ordiiiatcs TA', Ji....l/.. FK <if tl.e poin'.- 
 
 Ji. i k. E ol the coiicUu^d. tlio abscissa of which wiT tic o.r./ (1. 
 
 CF. respectively. 
 
 rf ti 
 
 le lower third part, who.: ' .'ixiji of abscissas \i('l> pralimijeL 
 
 t^iven co-ordinate.s syiiunetrical to ilios.- kX tii )nidd!e thiid part, the 
 curve passing through the vertici's of the ofdiiiates will be the conchoid 
 of the column. 
 
 Now. taking th»' radius (JH as a unit of leuiith. and denotin.;' the whole 
 altitude of the' shaft bv //; then. iTi-e.itisi' of Architecture) 
 
 //=:! 
 
 \)'l. Ill the tos;'au onl 
 
 //■=! :;.;;(;. in th ■ doric orde:'. 
 //=!,"). G."<, in the other ihr «• orders. 
 From the abov proportions of the coluinii. !he:e .alsn obtains 
 
 FF_\ 
 Cl~l 
 
 cosine t)t no 
 
 )f Hu 
 
 wbence. arc /lo^r-.'lH'' 
 
 ".4=(i.4'.i4!':;:'.T in leu'jrtii. 
 
 Kv'AlMN (IF TUK ('ONCUOlll 
 
 Let .i(=i>.4itltt;'.;!7, length of the arc .//'>, 
 
 2-= CJ, abscissa of any p nut (/i of tlie conchoid, 
 
 (/= /; (=:ms=roA' /ini). ordinate of the smne point (?). 
 
I plain' 
 
 '/" 
 
 ml i»ar'--i 
 ii . . . . ''»'" 
 
 part, tlu' 
 oiu'hoi'l 
 
 APPKNMirX S. 
 
 :i<)9 
 
 r.ut. by ('onstnictioti. tlic I'lisitli <'F(o': -.H\ ot' tlic upi'tT two lliirds 
 of'tlie shaft is to the iihsci.s.sa C'l (oi' - !. :i< the length of the arc Bo (or a) 
 
 IS 
 
 to it> iKH'iion l>)>i lu;' rus 
 
 
 '/)•• 
 
 •J// 
 
 iit ),- 
 
 ,- : j=(/ : (:0> // ; Wlicliri-. ('u~ .'/=. 
 
 H 
 
 Therct'oro tho ('iiiiatinii nf the iiiiovc i-otiflmid is 
 
 Vj=rr()S 
 
 ■111 
 
 \'0MMK OF Tin; SlIAKT OK A COLl'MN. 
 
 (1> 
 
 Substitntini.'" 
 
 Mix 
 ■111 
 
 t'. ir //-'. ill </r=T//-7x. tliere obtains t'or (lift'erential 
 
 )t til" voliiir.e ot llif dlifit't ot" a coliinui ; 
 
 .,- ..War., 
 
 \ll 
 
 Mil' III 
 
 tciiTiil ot wiiicii is r= ' I '" "4- -ill '"' |4-C. 
 
 .'-^iin'e l—ii wlicii .r=o. tlieii ('=.'K 
 
 Wlicn .» = .',//, tiicre obtains foi' lie volunie ( T'; of the middle third 
 part, or of tlie lower tliird pail whirh is i(|ual to the middh- one : 
 
 I = (r/-4-siii (/.). 
 
 (2) 
 
 When x='^J/. tluic obtains, fo v<i iinic (1'') of the upper two tiiinls : 
 
 (3) 
 
 1'"= ■■( L''/-(-sin2(^). 
 
 A(lduig(2) and (I!), and leitin.i; ( l''-|- 1'")= >'. then: obtains; 
 Volume (if the shaft of a column. 
 
 \'= ' ,^:;'^-j-rii!w/-l-siirJ«/) 
 
 (4) 
 
 
 i 
 
 ."M'Kill'i.- Wr.!(:i:i", 
 
 Lettinjr l'rr:\yi\ ■'•: i.i'a lindy wlic/< v > iiiii ■ is I' .it llie tctiipi'nunre ot 
 • zeio de).Te cunti;j,' a,!'. 
 /<=wcr2i!; it an e([ a! V 'iiinif ( I' i of distilii'd watci', at tln' 
 temp iature ol Univ ilt'!^-;' cs >"i'iit:.n;ad'' ; 
 
 then \\:11 th" raMo be ili«' sprrijsi- in i;iiil t\\' \\\v hoily \\'iio-e weii^iit is/'. 
 /■ 
 
 !f tkis speeitii- wci^^'it (rei itive density j be d note ! i.y /.', there nbtaiiLS : 
 
 1 
 
 
 :/-' 
 
 (I) 
 
310 
 
 APPENDIX S 
 
 1 
 
 
 Since the weight of a cubic foot of water, at four degrees oeiu. is equal 
 to 1000 ounces, the weight (;j) of any volume ( V) of water, measured in 
 cubic feet, will be equal to 1000 T ounces; whence, the equation: 
 
 P=pl)=l 000 VJ> ozs. ( 2 ) 
 
 In the metric system, th,' weight of a cubic meter, decinieter, ceuti- 
 
 nieter, of distilled water, at four d grecs ceut. being equal to a !o:i. a 
 
 kili)grara, a gram, resjtectively, it follows that ;i volume ( T) of water is 
 
 iilways numerically equal to its weight (/>): whence, the equation : 
 
 P=p£t=iVD. (:;) 
 
 BND. 
 
■ht] 
 
 A TABLE 
 
 f;,\ 
 
 m 
 
 OF 
 
 LOGARITHMS OF XTLMnKRS 
 
 From 1 TO lO.oiMi. 
 
 (3) 
 
 X. 
 
 1 
 
 Log. 
 
 t X. 
 i 2<'. 
 
 L,:--. : 
 
 X 
 
 ' 51 
 
 Log. 
 
 X 
 
 L:., 
 
 nooooo 
 
 1.4 1 4073 ; 
 
 
 707570 
 
 i 70 
 
 1.8S0S14 
 
 2 
 
 o.:}i)i(»;]i) 
 
 27 
 
 1.431304 ; 
 
 52 
 
 .f 
 
 7I0i'ii:) 1 
 
 ' 77 
 
 i.8,soi'a 
 
 3 
 
 0.477121 
 
 1 28 
 
 ].447l.'.S ! 
 
 53 
 
 
 7--'4270 ; 
 
 ' 7s 
 
 l.s:i 2(1^5 
 
 4 
 
 0.(;n2(i(in 
 
 ■ 20 
 
 1 . 4023 18 
 
 54 
 
 
 7323.(4 '■ 
 
 ' 70 
 
 1 ;<,»7027 
 
 5 
 
 . (i!)8:>70 
 
 i 30 
 
 1.477121 
 
 55 
 
 
 74(1303 ! 
 
 SO 
 
 l.o03r);)O 
 
 (i 
 
 0.778151 
 
 ! 31 
 
 1 101302 
 
 50 
 
 
 7481SS 
 
 Hi 
 
 1.00 sis.-) 
 
 7 
 
 0.8150!)8 
 
 32 
 
 1. '-05 150 
 
 57 
 
 
 755875 : 
 
 H2 
 
 1.013S14 
 
 8 
 
 0.U0800O 
 
 i 33 
 
 1.518514 
 
 1 58 
 
 
 7(;312S 
 
 83 
 
 1.011)078 
 
 
 
 0.05124:1 
 
 ; 34 
 
 1.531471) 
 
 I 50 
 
 
 77i)s52 
 
 84 
 
 1.024270 
 
 10 
 
 1. 000001) 
 
 I 35 
 
 1.544008 i 
 
 1 00 
 
 
 778151 
 
 85 
 
 1.020410 
 
 u 
 
 l.oii;!;):! 
 
 ' 30 
 
 1.55(13(13 
 
 ' 01 
 
 
 7S.-,330 
 
 8() 
 
 1.0311'.»8 
 
 12 
 
 1 0701S1 
 
 1 *" 
 
 1 508202 
 
 02 
 
 
 7:)23'.t2 
 
 i 87 
 
 1.03;>510 
 
 i:) 
 
 i.n;wi;{ 
 
 3S 
 
 J 57.>78t i 
 
 03 
 
 
 7'.);)341 
 
 1 88 
 
 1.0444^3 
 
 ]l 
 
 1 ] 101 28 
 
 ' 3!) 
 
 1 .5'.no(;5 ' 
 
 i <'•* 
 
 * 
 
 S0G181 
 
 89 
 
 1.040.3:'.) 
 
 ir, 
 
 i.i7()0:»i 
 
 ' 40 
 
 1 (;o20(;o i 
 
 1 (;5 
 
 
 8 12!) 13 
 
 00 
 
 1.054243 
 
 ii> 
 
 1.201120 
 
 ' 41 
 
 1.012781 
 
 0(; 
 
 
 810544 
 
 01 
 
 ].0.5;)((11 
 
 17 
 
 1.2:iOU'.> 
 
 1 42 
 
 1 . (;23211) 
 
 07 
 
 
 420075 
 
 02 
 
 1 
 
 1.0(;:]7>i 
 
 IS 
 
 1.255273 
 
 1 43 
 
 1. 033408 
 
 08 
 
 
 832500 
 
 03 
 
 I.0(;s4.s3 
 
 I'J 
 
 1.278751 
 
 44 
 
 ].()43453 
 
 00 
 
 
 838S4'.) 
 
 1)4 
 
 1.073128 
 
 20 
 
 1.301030 
 
 45 
 
 1.053213 
 
 70 
 
 
 S450!)8 
 
 05 
 
 1.077724 
 
 21 
 
 1.322211) 
 
 ' 4f; 
 
 1 .002758 
 
 71 
 
 
 S51258 1 
 
 00 
 
 1.0Svi)71 
 
 1^2 
 
 1.312423 
 
 \ 47 
 
 l.(;720'.)8 
 
 72 
 
 
 S57n33 
 
 i»7 
 
 1.0S(;772 
 
 L;? 
 
 1.3C,1728 
 
 i 4s 
 
 l.()S1241 
 
 73 
 
 
 ^03323 : 
 
 ;)8 
 
 1 0(1220 
 
 21 
 
 1.3811211 
 
 ■ 4!) 
 
 1 ('.001 '.k; 
 
 ' P 
 
 
 Si;!t2:!2 ! 
 
 '.10 
 
 1 '.)0.5.;:!5 
 
 25 
 
 1.3',)7i»40 ' 
 
 1 
 
 50 
 
 1 .(;',(^o70 
 
 75 
 
 1 
 
 
 s750i;i ' 
 
 : Kt't 
 
 2 OmIOOO 
 
 '■■ B: 1 
 
 '*1 
 
 
 m 
 
 IIemauk. — 111 the folluwinu- table, tlic two leudinu' H:.:u:x>.'^ in 
 the first column of ioirarithins must ))e prefixed ti> all the nimi- 
 be.'\s of the same liorizontal line in the next nine ndunins ; l»ut, 
 when a point (*) occurs, its place is to be supplied by a cypher, 
 and the two leading tigures are to be taken from the next lower 
 line. 
 
 :fP;i 
 
 Mm 
 
A TABLE OF LO(iaVKITHMH FKOM 1 TO lO.OOO. 
 
 N. 
 
 (J 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 1 8 
 
 9 
 
 1). 
 
 I'M) 
 
 OOtK.'IO 
 
 04;ii 
 
 ()i()H 
 
 13(il 
 
 2106 
 
 25,i8 
 
 :50'J9 ;M«ii 
 
 :l89ll 4;J2 
 
 1(11 
 
 1:521 
 
 4751 
 
 51S1 
 
 5()09: ()038 
 
 6466 
 
 6894 
 
 7321 1 77 IS 
 
 81 7! 1 428 
 
 KI'J 
 
 .S(i()(l 
 
 !)02(; 
 
 !)451 
 
 98-'' •300 
 
 •724 
 
 1147 
 
 1570 19!)3 
 
 2415 
 
 424 
 
 lu;} 
 
 0128: i7 
 
 325!) 
 
 30S0 
 
 41 ±521 
 
 4940 
 
 5300 
 
 5779! (5197 
 
 (5(516 
 
 419 
 
 KU 
 
 7(i:i;j 
 
 7451 
 
 78i;8 
 
 828i! H700 
 
 9116 
 
 9532 
 
 9947! •:501 
 
 •775 
 
 410 
 
 1(15 
 
 0211HJ) 
 
 1003 
 
 2010 
 
 24281 2841 
 
 8252 
 
 3(;(;4 
 
 4075 
 
 448(5 
 
 489(5 
 
 412 
 
 Kx; 
 
 5;{0() 
 
 5715 
 
 0125 
 
 «533j 6!)42 
 
 7;i50 
 
 7757 
 
 81(54 
 
 8571 
 
 8978 
 
 408 
 
 1 .7 
 
 !);5,s-l 
 
 !»7m;) 
 
 •195 
 
 •OOOi 1001 
 
 1408 
 
 1812 
 
 2210' 201!) 
 
 3021 
 
 404 
 
 lOS 
 
 03;,i2i 
 
 3H20 
 
 4227 
 
 4()28' 5029 
 
 54:50 
 
 58:50 
 
 (52:501 602!) 
 
 7028 
 
 400 
 
 l(J') 
 
 7l2(i 
 
 7S25 
 
 8223 
 
 8()20, 9017 
 
 !I114 
 
 !)MI! 
 
 •207| ^(502 
 
 •!)!)8 
 
 396 
 
 110 
 
 04l;5,'3 
 
 17H7 
 
 2182 
 
 2570 2909 
 
 mi, 2 
 
 3755 
 
 41481 4540 
 
 4!)32 
 
 3!)3 
 
 111 
 
 5;{2:{ 
 
 5714 
 
 0105 
 
 64951 ('.s,s5 
 
 T21-} 
 
 7(504 
 
 8(»53! 8442 
 
 8s;50 
 
 389 
 
 112 
 
 !»21H 
 
 !)!i(K; 
 
 99: .'3 
 
 •:5,S0; •700 
 
 1153 
 
 15:58 
 
 19241 2:50!) 
 
 2094 
 
 386 
 
 11 ;) 
 
 OS.'iOTS 
 
 3f(;3 
 
 3«!(! 
 
 42;}0i 4(il3 
 
 4!)!)0 
 
 5378 
 
 5700 [ 6142 
 
 6524 
 
 382 
 
 lU 
 
 (i'05 
 
 72-'(i 
 
 70<.() 
 
 8OI0! H\-J.{> 
 
 8S05 
 
 91. S5 
 
 9503 !)'»52 
 
 •:5i'o 
 
 379 
 
 115 
 
 0(KI(;!)S 
 
 1075 
 
 1452 
 
 1829 
 
 2200 
 
 25^^2 
 
 2!)58 
 
 3:5:531 3709 
 
 40H3 
 
 ;576 
 
 ]!'; 
 
 4i5H 
 
 4^::52 
 
 5200 
 
 55S0 
 
 5953 
 
 <;:!2() 
 
 <5(i!ti) 
 
 7071^ 74-i3 
 
 7815 
 
 372 
 
 117 
 
 Ml SO 
 
 8557 
 
 8! 128 
 
 !)298 
 
 9608 
 
 •«:58 
 
 •40 7 
 
 •770 1145 
 
 1514 
 
 ;«59 
 
 lis 
 
 071.SS2 
 
 2250 
 
 2017 
 
 2985 
 
 3:552 
 
 3718 
 
 4085 
 
 4451 4816 
 
 5182 
 
 me 
 
 llf) 
 
 5517 
 
 5;)12 
 
 (5270 
 
 6(i40i 7004 
 
 7308 
 
 7731 
 
 H094| 8457 
 
 Hbl!) 363 
 
 12(1 
 
 071>]'<1 
 
 !>5I3 
 
 9904 
 
 •200 
 
 •0-.'(i 
 
 •!)87 
 
 1347 
 
 1707: 2007 
 
 242(; 360 
 
 121 
 
 0H27S5 
 
 3144 
 
 3503 
 
 3 m 
 
 42J!) 
 
 457(i 
 
 4931 
 
 52;)1, 5(547 
 
 6004 Hal 
 
 l±i 
 
 (iilOO 
 
 ()71(i 
 
 7071 
 
 7420 
 
 7781 
 
 8130 
 
 8490 
 
 8845] 9198 
 
 9552 ;555 
 
 12.! 
 
 !»!)05 
 
 •258 
 
 •01 1 
 
 •903: 1315 
 
 1(507 
 
 2018 
 
 2370! 2721 
 
 3071 351 
 
 IL'I 
 
 093122 
 
 3772 
 
 4122 
 
 4471 i 4820 
 
 51(5!) 
 
 5518 
 
 58(56 
 
 6215 
 
 65(52 349 
 
 V^', 
 
 (iillO 
 
 7257 
 
 7004 
 
 l!)51i 8298 
 
 8644 
 
 8'):)0 
 
 93:55 
 
 OO.'-'l 
 
 ••2(5 :340 
 
 1 i-.k; 
 
 100:171 
 
 0715 
 
 105!) 
 
 1403 
 
 1747 
 
 2091 
 
 2434 
 
 2777 
 
 311!) 
 
 34(52 
 
 343 
 
 1_7 
 
 8S(I4 
 
 41 Mi 
 
 44:7 
 
 4S28 
 
 5109 
 
 5510 
 
 5851 
 
 6191 
 
 65:51 
 
 6871 
 
 340 
 
 1-S 
 
 7210 
 
 751!) 
 
 7888 
 
 8227 
 
 8505 
 
 89o:5 
 
 !)241 
 
 957!) 
 
 9!)16 
 
 •253 
 
 338 
 
 i V>\\ 
 
 1105':0 
 
 0)20 
 
 1203 
 
 15!)9i 19:54 
 
 2270 
 
 2005 
 
 2! (40 
 
 3275 
 
 3(509 
 
 335 
 
 VM) 
 
 ll:iM:5 
 
 4277 
 
 4<)11 
 
 4944 1 5278 
 
 5(511 
 
 5!)43 
 
 (5270 
 
 (5008 
 
 (5!)40 
 
 333 
 
 l;il 
 
 7271 
 
 7003 
 
 li»:54 
 
 82(55 
 
 85:)5 
 
 8! (20 
 
 !)250 
 
 !)58(5 
 
 9i)15 
 
 •245 
 
 330 
 
 l.":2 
 
 120574 
 
 o:!;\3 
 
 1231 
 
 1500 
 
 I808 
 
 2210 
 
 25)1 
 
 2871 
 
 3198 
 
 3525 
 
 328 
 
 Jij;} 
 
 3h52 
 
 4178 
 
 4504 
 
 48:50 
 
 5156 
 
 54H1 
 
 5800 
 
 6131 
 
 645(5 
 
 6781 
 
 325 
 
 i;ji 
 
 7105 
 
 742!) 
 
 7753 
 
 8070 
 
 83!)9 
 
 8722 
 
 !)015 
 
 9;508i 9(590 
 
 ••12 
 
 323 
 
 i;35 
 
 130334 
 
 0055 
 
 0!)77 
 
 1298 
 
 1019 
 
 1!:;59 
 
 2200 
 
 25801 2900 
 
 321!) 
 
 321 
 
 i;«j 
 
 353!) 
 
 3858 
 
 4177 
 
 44!)(5 
 
 4«14 
 
 5133 
 
 5451 
 
 57(5!) (5080 
 
 (i4o:s 
 
 318 
 
 i;^7 
 
 G721 
 
 7037 
 
 7:ir>4 
 
 7071 
 
 7!)H7 
 
 8:503 
 
 801H 
 
 8934 !»24!) 
 
 95(54 
 
 315 
 
 1:]S 
 
 087!) 
 
 •194 
 
 •50S 
 
 •822 i 11:50 
 
 1450 
 
 1703 
 
 207()i 23^i» 
 
 2702 
 
 314 
 
 1;]!) 
 
 14:^015 
 
 3:527 
 
 3039 
 
 3951 
 
 4203 
 
 4571. 
 
 48s5 
 
 5li)(5 5507 
 
 581 K' 
 
 311 
 
 110 
 
 1401 2S 
 
 ()l:58 
 
 6748 
 
 7058 
 
 7307 
 
 7(57(5 
 
 7!)85 
 
 82! )4; 8003 
 
 8!»11 
 
 md 
 
 Ul 
 
 !I211) 
 
 9527 
 
 !)8;55 
 
 •142 
 
 •44!) 
 
 •750 
 
 10(53 
 
 i;-570 1(57(5 
 
 1!)82 
 
 307 
 
 1:2 
 
 1522SS 
 
 2594 
 
 2!)00 
 
 3205 
 
 3510 
 
 3S15 
 
 4120 
 
 4424 
 
 4728 
 
 50:52 
 
 305 
 
 M:) 
 
 533(; 
 
 5(;40 
 
 5!)43 
 
 0210 <5549 
 
 6852 
 
 7154 
 
 7457 
 
 775!) 
 
 80(i] 
 
 303 
 
 1!,1 
 
 s;](i2 
 
 8004 
 
 8905 
 
 9200 9507 
 
 !)8(;8 
 
 •1(^8 
 
 •40!) 
 
 •7(5!) 
 
 lOOH 
 
 301 
 
 145 
 
 16l;}(iM 
 
 l()(i7 
 
 1!)(;7 
 
 220(5 ^m.4 
 
 2863 
 
 3101 
 
 3460 
 
 3758 
 
 4055 
 
 2!)!) 
 
 i'r< 
 
 435:5 
 
 4050 
 
 4!)47 
 
 52U 5541 
 
 5838 
 
 6134 
 
 6430 
 
 6720 
 
 7022 
 
 2!)7 
 
 117 
 
 7317 
 
 7013 
 
 7!)08 
 
 82o;5 8497 
 
 8792 
 
 !)08(; 
 
 9380 
 
 9074 
 
 99(58 
 
 295 
 
 l!s 
 
 170202 
 
 0555 
 
 0848 
 
 1141 14:54 
 
 172(5 
 
 201!) 
 
 2311 2(50:5 
 
 2895 
 
 293 
 
 n) 
 
 31 SO 
 
 ;3478 
 
 3709 
 
 40{;o 4:351 
 
 4(541 
 
 4!):52 
 
 5222 5512 
 
 5802 
 
 2!)1 
 
 150 
 
 17(ii):)l 
 
 6381 
 
 <)070 
 
 695!) 
 
 7248 
 
 753(5 
 
 7825 
 
 8113 8401 
 
 868!) 
 
 289 
 
 151 
 
 s:>77 
 
 9204 
 
 9552 
 
 9839 
 
 •120 
 
 •413 
 
 •09!) 
 
 •!)85 1272 
 
 1558 
 
 287 
 
 152 
 
 181M4 
 
 2129 
 
 2415 
 
 2700 
 
 2!)85 
 
 3270 
 
 3555 
 
 38:5!) 
 
 4123 
 
 4^107 
 
 285 
 
 15:5 
 
 4()01 
 
 4!)75 
 
 5259 
 
 5r)42 
 
 5825 
 
 6108, 
 
 0391 
 
 (i()74 
 
 695(5 
 
 72:59 
 
 283 
 
 151 
 
 7521 
 
 7803 
 
 8084 
 
 8300 
 
 8647 
 
 89281 
 
 9209 
 
 !)4!K) 9771 
 
 ••51 
 
 281 
 
 155 
 
 190332 
 
 0(;i2 
 
 0892 
 
 1171 
 
 1451 
 
 1730: 
 
 2010 
 
 2289 25(57 
 
 2840 
 
 279 
 
 ] A) 
 
 3125 
 
 3403 
 
 3081 
 
 3!)59 
 
 4237 
 
 4514; 
 
 47!)2 
 
 506!) 534(5 
 
 502:5 
 
 278 
 
 157 
 
 5K!)!) 
 
 0170 
 
 6453 
 
 6729 
 
 7005 
 
 7281 1 
 
 755(i 
 
 7832; 8107 
 
 8382 
 
 276 
 
 15S 
 
 8057 
 
 8932 
 
 J)20(! 
 
 !>481 
 
 9755 
 
 ••90 1 
 
 •:5n3 
 
 •577! •.S5() 
 
 1121 
 
 274 
 
 159 
 
 201397 
 
 1('70 
 
 1943 2210 
 
 2488 
 4 
 
 _27("5i: 
 
 3o;5.5 
 
 3305 3577 
 
 3848 
 
 272 
 I). 
 
 \. 
 
 II 
 
 1 
 
 ^2^ 
 
 3 
 
 5 
 
 G 
 
 7 8 9 
 
424 
 41!) 
 41G 
 412 
 
 4()S 
 404 
 400 
 31)0 
 31)3 
 
 389 
 
 386 
 
 382 
 
 37!) 
 
 370 
 
 372 
 
 301) 
 
 306 
 
 303 
 
 300 
 
 357 
 
 355 
 
 351 
 
 34!) 
 
 :340 
 
 343 
 
 340 
 
 338 
 
 335 
 
 333 
 
 330 
 
 328 
 
 325 
 
 323 
 
 321 
 
 318 
 
 315 
 
 314 
 
 311 
 
 301) 
 
 307 
 
 305 
 
 303 
 
 301 
 
 21)1) 
 
 2!>7 
 
 2!i5 
 
 203 
 
 2!)1 
 
 28!) 
 
 287 
 
 285 
 
 283 
 
 281 
 
 279 
 
 278 
 
 276 
 
 274 
 
 272 
 
 \1 
 
 Id 
 |;i 
 
 1- 
 
 1). 
 
 
 
 A TAHLE OK I,0(;AK1THM.S KHOM 1 
 
 TO 10,000. 
 
 
 
 X. 
 
 
 
 I 
 
 2 
 4'^t,3 
 
 3 
 
 4 
 "5204 
 
 5 6 
 ~5...i'5746 
 
 7 
 6016 
 
 8 y 
 
 ~62.'-(i ~65.")("i 
 
 D. 
 
 IC'I 
 
 20412' 
 
 4934 
 
 271 
 
 ICl 
 
 6.S2( 
 
 70 /( 
 
 7;;.i5 
 
 7031 
 
 7904 
 
 8173 
 
 8441 
 
 871 ( 
 
 8'.)7i) 
 
 !iut7 
 
 2(59 
 
 IV'l 
 
 951.-, 
 
 97' ;3 
 
 •e51 
 
 •319 
 
 •586 
 
 •853 
 
 1121 
 
 13.SS 
 
 1654 
 
 ; 1921 
 
 267 
 
 103 
 
 212l8>i 
 
 2454 
 
 2';2( 
 
 2986 
 
 3252 
 
 i 3518 
 
 37S3 
 
 40'! 
 
 4314 
 
 i 4571 
 
 2(;(; 
 
 ICl 
 
 I'M 4 
 
 510! 
 
 5373 
 
 5638 
 
 5902 
 
 61(56 
 
 (5430 
 
 (5(5" t 
 
 (5957 
 
 7':'i 
 
 2'! 
 
 105 
 
 7 1. SI 
 
 7747 
 
 8()]( 
 
 8273 
 
 853() 
 
 1 8798 
 
 9060 
 
 !)323 
 
 !)5,s5 
 
 : 9Nl( 
 
 262 
 
 KJO 
 
 220108 
 
 037( 
 
 0(;3i 
 
 08: )2 
 
 i 1153 
 
 1414 
 
 1(575 
 
 r.)3( 
 
 219(5 
 
 ' 2-:5( 
 
 2hl 
 
 ICT 
 
 271(1 
 
 2976 
 
 3236 
 
 3196 
 
 1 tf^-" 
 
 4015 
 
 4274 
 
 45:53 
 
 47! '2 
 
 5051 
 
 259 
 
 lOS 
 
 5391) 
 
 55(;8 
 
 5S26 
 
 6()M4 
 
 (5342 
 
 1 6600 
 
 6858 
 
 7115 
 
 7372 
 
 7(i3( 
 
 258 
 
 lOK 
 
 7887 
 
 8111 
 
 8400 
 
 8657 
 
 8913 
 
 ; 9170 
 
 942(5 
 
 902 
 
 91*38 
 
 •193 
 
 25(5 
 
 170 
 
 230419 
 
 0704 
 
 096(1 
 
 1215 
 
 1470 
 
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 1979 
 
 2234 
 
 2488 
 
 2742 
 
 2.54 
 
 171 
 
 29;)0 
 
 3-50 
 
 3501 
 
 3757 
 
 4011 
 
 ! 42(54 
 
 4517 
 
 4770 
 
 5023 
 
 527r 
 
 2.-,3 
 
 172 
 
 5528 
 
 57.S1 
 
 6033 
 
 6285 
 
 6537 
 
 678!) 
 
 7041 
 
 72! )2 
 
 7r)44 
 
 7795 
 
 252 
 
 173 
 
 8()|(i 
 
 81:97 
 
 8548 
 
 8799 
 
 9019 
 
 9299 
 
 9550 
 
 9H09 
 
 ••50 
 
 •300 
 
 2r;o 
 
 171 
 
 240549 
 
 07"!) 
 
 1048 
 
 1297 
 
 ir>46 
 
 17!»5 
 
 2044 
 
 2293 
 
 2541 
 
 27:10 
 
 21') 
 
 175 
 
 3038 
 
 32S6 
 
 3534 
 
 3782 
 
 4030 
 
 4277 
 
 4525 
 
 4772 
 
 5019 
 
 52(;() 
 
 248 
 
 170 
 
 5513 
 
 5759 
 
 oooi; 
 
 6252 
 
 6499 
 
 <)745 
 
 6991 
 
 7237 
 
 7482 
 
 77l:.-s 
 
 24(5 
 
 177 
 
 7973 
 
 8:^19 
 
 8-164 
 
 8709 
 
 8954 
 
 9198 
 
 9443 
 
 9(5t7 
 
 9!)32 
 
 •17(5 
 
 245 
 
 178 
 
 250:20 
 
 0604 
 
 09i)8 
 
 1151 
 
 1395 
 
 163S 
 
 1881 
 
 2125 
 
 23(58 
 
 2610 
 
 2!;! 
 
 17!' 
 
 2(53 
 
 3096 
 
 333S 
 
 3580 
 
 3S22 
 
 -10(54 
 
 430(5 
 
 454m 
 
 4790 
 
 5031 
 
 2.2 
 
 180 
 
 255i73 
 
 5514 
 
 5755 
 
 5996 
 
 6237 
 
 (5177 
 
 (5718 
 
 6958 
 
 7198 
 
 7439 
 
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 7079 
 
 7918 
 
 8158 
 
 8398 
 
 8t!37 
 
 8877 
 
 9116 
 
 9355 
 
 9594 
 
 9833 
 
 23') 
 
 1k2 
 
 260071 
 
 0310 
 
 0518 
 
 0787 
 
 1025 
 
 12<53 
 
 1501 
 
 1739 
 
 1976 
 
 2214 
 
 2;-, > 
 
 im 
 
 2451 
 
 2(iS8 
 
 2!)25 
 
 3162 
 
 3399 
 
 3(j3(5 
 
 3h73 
 
 4109 
 
 4316 
 
 45,^2 
 
 237 
 
 irM 
 
 4818 
 
 5054 
 
 52; )0 
 
 5525 
 
 5761 
 
 5996 
 
 6232 
 
 61(57 
 
 (5702 
 
 (5937 
 
 2.35 
 
 185 
 
 7172 
 
 7196 
 
 7641 
 
 7875 
 
 8110 
 
 8344 
 
 8578 
 
 8,si2 
 
 9046 
 
 9::79 
 
 2M 
 
 186 
 
 i)513 
 
 9746 
 
 99S0 
 
 •il:i 
 
 •416 
 
 •(57!) 
 
 •912 
 
 1111 
 
 1377 
 
 l(;o9 
 
 2:.:] 
 
 1!j7 
 
 271842 
 
 2071 
 
 2396 
 
 2538 
 
 2770 
 
 3001 
 
 3233 
 
 34(54 
 
 3(596 
 
 3!/li7 
 
 2;>i 
 
 188 
 
 415S 
 
 43Si) 
 
 4()20 
 
 4S50 
 
 5(N1 
 
 5311 
 
 5542 
 
 5772 
 
 6002 
 
 (5232 
 
 2:-,o 
 
 180 
 
 6402 
 
 661)2 
 
 6921 
 
 7 151 
 
 73S)) 
 
 7(i09 
 
 7S38 
 
 8067 
 
 h296 
 
 8.j_5 
 
 2l;» 
 
 100 
 
 2787(54 
 
 89S2 
 
 9211 
 
 9439 
 
 9667 
 
 9.S95 
 
 •123 
 
 •351 
 
 •578 
 
 •,'•'■■.6 
 
 22. S 
 
 1!>1 
 
 281033 
 
 12;:i 
 
 1488 
 
 1715 
 
 1942 
 
 21(59 
 
 2396 
 
 2(5-2 
 
 2849 
 
 3075 
 
 227 
 
 102 
 
 3301 
 
 3527 
 
 3753 
 
 3979 
 
 4205 
 
 4431 4(55(5 
 
 48h2 
 
 5107 
 
 5332 
 
 216 
 
 103 
 
 5557 
 
 57s2 
 
 6007 
 
 6232 
 
 61 5() 
 
 «5(5S1 (5905 
 
 7130 
 
 7354 
 
 75/8 
 
 2-5 
 
 1!M 
 
 7802 
 
 80::(; 
 
 8:>}9 
 
 8173 
 
 8(;;)6 
 
 .s;)20 9143 
 
 93;56 
 
 95s9 
 
 9.M2 
 
 2:-3 
 
 105 
 
 290035 
 
 0:^57 
 
 01 so 
 
 oro2 
 
 0925 
 
 1147 13(.9 
 
 1591 
 
 1813 
 
 20;; 1 
 
 2' " 
 
 If);; 
 
 2256 
 
 247S 
 
 26:-» 
 
 2920 
 
 3111 
 
 33o3 35S4 
 
 S.'^fll 
 
 4025 
 
 42 i 6 
 
 22? 
 
 197 
 
 4466 
 
 4(«S7 
 
 4':)07 
 
 5127 
 
 5317 
 
 5567' 57M7 
 
 (5007 
 
 6226 
 
 (544!i 
 
 220 
 
 1!)8 
 
 ()(;65 
 
 6«S4 
 
 7101 
 
 7323 
 
 7512 
 
 7761! 7979 
 
 81;»,S 
 
 841(5 
 
 8t 35 
 
 21;) 
 
 199 
 
 8853 
 
 9071 
 
 9289 
 
 95(»7 
 
 9725 
 
 99431 ^161 
 
 •378 
 
 •5: (5 
 
 •M3i 
 
 •2V^ 
 
 200 
 
 301030 
 
 12±7 
 
 1404 
 
 1(581 
 
 I'-'iH 
 
 21141 2331 
 
 2517 
 
 27(.4 
 
 29^0' 
 
 217 
 
 201 
 
 3196 
 
 3112 
 
 3028 
 
 3814 
 
 405:) 
 
 4275! 4491 
 
 470(5 
 
 4' 21; 
 
 51;. 6 
 
 216 
 
 202 
 
 5351 
 
 55;;6 
 
 57;a 
 
 59!iii 
 
 61-11 
 
 (54251 (5(539 
 
 6S54 
 
 7o(i8' 
 
 7:':i2 
 
 215 
 
 203 
 
 7496 
 
 7710 
 
 7924 
 
 8137 
 
 8351 
 
 851)4' 8778 
 
 89! (1 
 
 9204' 
 
 9417 
 
 213 
 
 204 
 
 9630 
 
 9843 
 
 ••5(; 
 
 •2(58 
 
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 1330 
 
 15:2 
 
 212 
 
 205 
 
 311754 
 
 19u6 
 
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 2180 2258 2320 
 
 2401 
 
 i'M 
 
 2473 
 
 2544, 
 
 2616 
 
 26H8i 
 
 2750, 2831 
 
 2002, 2074 3046 
 
 3117 
 
 ()07 
 
 3180 
 
 3260' 
 
 3332 
 
 3403 
 
 3475; 3516 
 
 3618 3680! 3761 
 
 3832 
 
 G08 
 
 3004 
 
 3075 
 
 4046 
 
 4118 
 
 4H0 42!,1 4332, 4403; 4475 
 
 4546 
 
 GOO 
 
 4617 
 
 4680 
 
 4760 
 
 4831 
 
 40021 4074 5045 5110; 5187 
 
 5250' 
 
 GIO 
 
 785330 
 
 5401 
 
 5472 
 
 5543 
 
 5615; 5686; 5757; 5828! 5800 
 
 5070 
 
 Gil 
 
 G041 
 
 6112 
 
 6183 
 
 G254 
 
 G325 G30n! 0467 G538I GGOO 
 
 6680 
 
 G12 
 
 G751 
 
 6822 68.13 
 
 6064 
 
 7035 7106: 7177 '.248' 7310 
 
 7300 
 
 fil!{ 
 
 74G0 
 
 7531 7602 
 
 7673 
 
 7744' 7815' 7885' 7056: 8027 
 
 8008' 
 
 GU 
 
 81G8 
 
 8230 8310 
 
 8381 
 
 8451, <S522i «5''^ *^!J<>ai ^T^i 
 
 8804' 
 
 G15 
 
 8875 
 
 8046 0016 
 
 0087 
 
 0157' 0228 0200! 0360; 0440 
 
 0510 
 
 (jlC. 
 
 0581 
 
 0651 0722 
 
 0702 
 
 08631 0033 
 
 •<»*4 ••74' ^144 
 
 •215; 
 
 G17 
 
 700285 
 
 0356 0426 
 
 0406 
 
 0567 0637 
 
 0707' 0778' 0848 
 
 00181 
 
 (il:i 
 
 oo.ss 
 
 1050 1120 
 
 1100 
 
 1260, 1340 
 
 1410 1480! 1550 
 
 1620; 
 
 Gill 
 
 1601 
 
 1761' 1831 
 
 1001 
 
 1071! 2041 
 
 2111 2181' 2252 
 
 2322! 
 
 (;2ii 
 
 702302 
 
 2462 2532 
 
 2602 
 
 2672' 2742 
 
 2812 2882' 2052 
 
 3022! 
 
 O.'l 
 
 3002 
 
 1 3162 
 
 3231 
 
 330] 3371, 3441 
 
 3511 35811 3651 
 
 3721! 
 
 GL'J 
 
 37'.)(> 
 
 1 3.S<.0 
 
 3030 
 
 4000 4070 4130 
 
 4200 4270 
 
 4340 
 
 4418, 
 
 cm 
 
 44.S8 
 
 i 4558 
 
 4627 
 
 4607 4767 483(1 
 
 4006' 4076 
 
 5045 
 
 5115 
 
 G-'l 
 
 5i\5 
 
 ' 5254 
 
 5324 
 
 53; )3 
 
 5463 5532 
 
 5602 5672 5741 
 
 5811 
 
 (YA~) 
 
 58 SO 
 
 5040 
 
 6010 
 
 (loss 
 
 61 5S (1227 
 
 ; 6207 6366' 6436 
 
 6505 
 
 G2(l 
 
 G574 
 
 1 (i(;!4 
 
 6713 
 
 1 6782 
 
 6852 0021 
 
 1 GOiiOi 7060 7120 
 
 7108 
 
 G'27 
 
 72(18 
 
 7337 
 
 7406 
 
 i 7475 
 
 7545 7614 
 
 1 7683i 7752! 7821 
 
 7800 
 
 G28 
 
 i 7000 
 
 8020 
 
 80;I8 
 
 ■ 8167 
 
 8236: 831 (5 
 
 ' 8374' 8443i 8513 
 
 8582 
 
 G20 
 
 8(151 
 
 8720 
 
 8780 
 
 1 8858 
 
 8027 8006 0065' 01341 0203 
 
 0272 
 
 G?0 
 
 1 700341 
 
 1 0400 
 
 0478 
 
 1 0547 
 
 0616 0(is5 0754 0823 0802 
 
 0061 
 
 G'll 
 
 i SOOO-J!) 
 
 ' Ofi:)s 
 
 01671 023(1 
 
 0305 (.373 0442' 0511 0580 
 
 ' 0648 
 
 ■(i;W 
 
 1 (1717! 07S(1 
 
 ' 0854; 0023 
 
 00.12 10(11 
 
 i 1120' 1108 
 
 1266 
 
 1 1335 
 
 m:) 
 
 1 1404 
 
 1 1472 
 
 1541 
 
 1600 
 
 ' 16781 1747 
 
 i 1815; 1884 
 
 1052 
 
 2021 
 
 g;!1 
 
 20;s! 
 
 ! 2158 
 
 : 2226 
 
 2205 
 
 ' 2363! 2132 
 
 ' 2500 2568 
 
 2637 
 
 ' 2705 
 
 r.;{-) 
 
 > '2774 
 
 1 2842 
 
 ! 2010 
 
 1 2070 
 
 3017' 3116 3184' 3252 
 
 3321 
 
 1 3380 
 
 j G:iti 
 
 3457 
 
 1 3525 35114 
 
 3662 
 
 ' 3730 37')8 3867 3035' 4003 
 
 ' 4071 
 
 i g;37 
 
 4130J I2ns 427G' 43 !4 4tl2 41s(r 4548 4G1G; 4(1S5 4753 
 
 g:<s 
 
 j 4821 i 4 .s't 4:)57 5tl25 5i)'»3 5101 5220 5207: 53(15 5433 
 
 (;;vi 
 
 5501 
 
 55; lO 
 
 t 56:17 
 
 5705 
 
 ! 57'"3 5X41 
 
 5;t08 507G 
 
 604 f 
 
 , 0112 
 
 D. 
 
 "75 
 
 75 
 
 75 
 
 74 
 
 74 
 
 74 
 
 74 
 
 74 
 
 74 
 
 74 
 
 74 
 
 73 
 
 73 
 
 73 
 
 73 
 
 73 
 
 73 
 
 73 
 
 73 
 
 72 
 
 72 
 
 72 
 
 72 
 
 72 
 
 72 
 
 72 
 
 72 
 
 71 
 
 71 
 
 71 
 
 71 
 
 71 
 
 71 
 
 7] 
 
 71 
 
 71 
 
 70 
 
 70 
 
 70 
 
 70 
 
 70 
 
 70 
 
 70 
 
 70 
 
 70 
 
 G'» 
 
 61) 
 
 60 
 
 60 
 
 GO 
 
 GO 
 
 60 
 
 60 
 
 60 
 
 60 
 
 68 
 
 68 
 
 68 
 
 68 
 
 (58 
 
 .\. 
 
 ;). 
 
A TABLE OF LOGARITHMS FROM 1 TO 10,000. 
 
 11 
 
 D. 
 
 ~75 
 
 75 
 75 
 74 
 
 I 74 
 i 74 
 i 74 1 
 
 74 
 ;! 74 ' 
 i' 74 : 
 ■\ 74 
 ! 73 , 
 
 II 73 i 
 }! 73 ! 
 i: 73 i 
 
 73 
 73 ; 
 73 ! 
 73 i 
 72 ' 
 72 I 
 72 
 
 72 
 72 
 72 
 72 
 71 
 
 «; 
 
 71 
 
 7(1 
 7(1 
 70 
 70 
 70 
 
 5i 
 
 70 
 
 1 
 
 70 
 
 15 
 
 (>!» 
 
 JS- 
 
 ()'.» 
 
 ^\ 
 
 ()<.) 
 
 2 
 
 (ill 
 
 2' 
 
 (V.) . 
 
 1 
 
 G'.l 
 
 H! 
 
 (iU 
 
 5' 
 
 (I'.l 
 
 1 
 
 li',) 
 
 5 
 
 Gil 
 
 
 
 08 ' 
 
 1 
 
 GS 
 
 ;»; 
 
 68 i 
 
 3 
 
 G8 
 
 ) 
 
 GH 
 
 1). 
 
 N. 
 
 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 D. 
 
 G40 
 
 806180 
 
 6248 
 
 631G (j?.^ 
 
 6451 
 
 6519 
 
 "6587 
 
 6655 
 
 6723 
 
 67iMi' 68 
 
 G41 
 
 6858 
 
 6926 
 
 6994 70i;l 
 
 7129 
 
 7197 
 
 72G4 
 
 7332 
 
 7400 
 
 7467' 68 
 
 642 
 
 7535 
 
 7603 
 
 7670; 77:58 
 
 7806 
 
 7873 
 
 7941 
 
 8008 
 
 8076 
 
 8143 68 
 
 643 
 
 8211 
 
 8279 
 
 8346; 8.V14 
 
 8481 
 
 8549 
 
 8616 
 
 8684 
 
 8751 
 
 8818 67 
 
 644 
 
 8886 
 
 8953 
 
 9021 9088 
 
 9156 
 
 9223 
 
 9290 
 
 9358 
 
 9425 
 
 9492 
 
 67 
 
 645 
 
 9560 
 
 9627 
 
 9694 9762 
 
 9829 
 
 9896 
 
 9964 
 
 ••31 
 
 ••98 
 
 •165 
 
 67 
 
 646 
 
 810233 
 
 0300 
 
 0367 
 
 0434 
 
 0501 
 
 0569 
 
 0636 
 
 0703 
 
 0770 
 
 0837' 67 
 
 647 
 
 0904 
 
 0971 
 
 1039 
 
 1106 
 
 1173 
 
 1240 
 
 1307 
 
 1374 
 
 1441 
 
 15081 67 
 
 648 
 
 1575 
 
 1642 
 
 1709 
 
 1776 
 
 1843 
 
 1910 
 
 1977 
 
 2044 
 
 2111 
 
 2178 67 
 
 649 
 
 2245 
 
 2312 
 
 2379 
 
 2445 
 
 2512 
 
 257" 
 
 2646 
 
 2713 
 
 2780 
 
 2847, 67 
 
 650 
 
 812913 
 
 2980 
 
 3047 
 
 3114 
 
 3181 
 
 3247 
 
 3314 
 
 3381 
 
 3448 
 
 3514; 67 
 
 651 
 
 3581 
 
 3G48 
 
 3714 
 
 3781 
 
 3848 
 
 3914 
 
 3981 
 
 4048 
 
 4114 
 
 4181 67 
 
 652 
 
 4248 
 
 4314 
 
 4381 
 
 4447 
 
 4514 
 
 4581 
 
 4647 
 
 4714 
 
 4780 
 
 4847 67 
 
 Q53 
 
 4913 
 
 4980 
 
 5046 
 
 5113 
 
 5179 
 
 5246 
 
 5312 
 
 5378 
 
 5^145 
 
 5511! 66 
 
 654 
 
 5578 
 
 5644 
 
 5711 
 
 5777 
 
 5843 
 
 5910 
 
 5976 
 
 6042 
 
 6109 
 
 6175' 66 
 
 655 
 
 0241 
 
 6308 
 
 6374 
 
 6440 
 
 G500 
 
 6573 
 
 6G39 
 
 6705 
 
 6771 
 
 6838 66 
 
 656 
 
 6904 
 
 6970 
 
 7036 
 
 7102 
 
 71G9 
 
 7235 
 
 7301 
 
 7367 
 
 7433 
 
 7499 
 
 66 
 
 657 
 
 7565 
 
 7631 
 
 7698 
 
 7704 
 
 7830 
 
 7896 
 
 79G2 
 
 8028 
 
 8094 
 
 8160 
 
 66 
 
 658 
 
 8226 
 
 8292 
 
 8358 
 
 8424 
 
 841)0 
 
 855() 
 
 8G22 
 
 8688 
 
 8754 
 
 8820; 66 
 
 659 
 
 8885 
 
 8951 
 
 9017 
 
 9083 
 
 9149 
 
 9215 
 
 9281 
 
 9:U6 
 
 9412 
 
 9478| 66 
 
 660 
 
 819544 
 
 9610 
 
 9G76 
 
 9741 
 
 9807 
 
 9873 
 
 993l» 
 
 •i»*4 
 
 ••70 
 
 •136! 66 
 
 661 
 
 820201 
 
 02G7 
 
 0333 
 
 0399 
 
 0464 
 
 0530 
 
 0595 
 
 OGGl 
 
 0727 
 
 0792! 66 
 
 662 
 
 0858 
 
 0924 
 
 0989 
 
 1055 
 
 1120 
 
 11 •<G 
 
 1251 
 
 1317 
 
 13S2 
 
 1448j 66 
 
 663 
 
 1514 
 
 1579 
 
 1645 
 
 1710 
 
 1775 
 
 1841 
 
 1906 
 
 1972 
 
 2037 
 
 2103' 65 
 
 664 
 
 2168 
 
 2233 
 
 2299 
 
 2364 
 
 24.)0 
 
 2i:)5 
 
 2560 
 
 2G26 
 
 2G91 
 
 275(i! 65 
 
 665 
 
 2822 
 
 2887 
 
 2952 
 
 3018 
 
 3083 
 
 3148 
 
 3213 
 
 3279 
 
 3344 
 
 3109 65 
 
 666 
 
 3474 
 
 3539 
 
 3G05 
 
 3()70 
 
 3735 
 
 3800 
 
 3865 
 
 3930 
 
 3996 
 
 4061 
 
 65 
 
 667 
 
 4126 
 
 4191 
 
 4*256 
 
 4321 
 
 4386 
 
 4451 
 
 4516 
 
 4581 
 
 4646 
 
 4711 
 
 65 
 
 668 
 
 4776 
 
 4841 
 
 4906 
 
 4971 
 
 503G 
 
 5101 
 
 51G6 
 
 5231 
 
 5296 
 
 5361 
 
 65 
 
 669 
 
 5426 
 
 5491 
 
 5556 
 
 5G21 
 
 5G86 
 
 5751 
 
 5815 
 
 5880 
 
 5945 
 
 6010 
 
 65 
 
 670 
 
 826075 
 
 6140 
 
 6204 
 
 62G9 
 
 6334 
 
 6399 
 
 6464 
 
 6528 
 
 6593 
 
 6658 
 
 65 
 
 671 
 
 6723 
 
 6787 
 
 6852 
 
 6917 
 
 6981 
 
 704».' 
 
 7111 
 
 7175 
 
 7240 
 
 7305 
 
 65 
 
 672 
 
 7369 
 
 7434 
 
 7499 
 
 75()3 
 
 7()2S 
 
 7()92 
 
 7757 
 
 7821 
 
 7886 
 
 7951 
 
 65 
 
 673 
 
 8015 
 
 8080 
 
 8144 
 
 8209 
 
 8273 
 
 8338 
 
 8402 
 
 8467 
 
 8531 
 
 8595 
 
 64 
 
 674 
 
 8660 
 
 8724 
 
 8789 
 
 8853 
 
 8918 
 
 8'.)h2 
 
 9046 
 
 9111 
 
 9175 
 
 9239 
 
 64 
 
 675 
 
 9304 
 
 93G8 
 
 9432 
 
 9497 
 
 95G1 
 
 9G25 
 
 9690 
 
 9754 
 
 9818 
 
 9882 
 
 64 
 
 07G 
 
 S347 
 
 ••11 
 
 ••75 
 
 •139 
 
 •204 
 
 •2G8 
 
 •332 
 
 •396 
 
 •4G0 
 
 •525 
 
 64 
 
 677 
 
 830589 
 
 0053 
 
 0717 
 
 0781 
 
 0845 
 
 9909 
 
 0973 
 
 1037 
 
 1102 
 
 1160 
 
 64 
 
 6Td 
 
 1230 
 
 1294 
 
 1358 
 
 1422 
 
 1486 
 
 1550 
 
 1614 
 
 1678 
 
 1742 
 
 1806 
 
 64 
 
 ar-) 
 
 1 io'/O 
 
 1934 
 
 1998 
 
 20G2 
 
 2126 
 
 2189 
 
 2253 
 
 2317 
 
 2381 
 
 2445 
 
 64 
 
 <bO 
 
 832509 
 
 2573 
 
 2637 
 
 2700 
 
 27G4 
 
 2828 
 
 2892 
 
 2956 
 
 3020 
 
 3083 
 
 64 
 
 1 tsi 
 
 3147 
 
 3211 
 
 3275 
 
 333S 
 
 3402 
 
 34()6 
 
 3530 
 
 3593 
 
 3G57 
 
 37211 64 
 
 1 jW2 
 
 3784 
 
 3848 
 
 3912 
 
 3975 
 
 4039 
 
 4103 
 
 41 66 
 
 4230 
 
 4294 
 
 4357 
 
 64 
 
 1 li'-.'. 
 
 .4421 
 
 4484 
 
 4548 
 
 4G11 
 
 4675 
 
 4739 
 
 4802 
 
 4866 
 
 4929 
 
 4993 
 
 64 
 
 68 
 
 5056 
 
 5120 
 
 5183 
 
 5247 
 
 5310 
 
 5373 
 
 5437 
 
 5500 
 
 55G4 
 
 5627 
 
 63 
 
 6H5 
 
 5(191 
 
 5754 
 
 5817 
 
 5s-il 
 
 5-,»44 
 
 6007 
 
 6071 
 
 6134 
 
 6197 
 
 (52G1 
 
 63 
 
 686 
 
 6321 
 
 6387 
 
 6451 
 
 G514 
 
 6577 
 
 6G41 
 
 6704 
 
 6767 
 
 6830 
 
 6894 
 
 63 
 
 687 
 
 6«o7 
 
 7020 
 
 7083 
 
 7146 
 
 7210 
 
 7273 
 
 7336 
 
 7399 
 
 7462 
 
 7525 
 
 63 
 
 688 
 
 7588 
 
 7(J52 
 
 7715 
 
 7778 
 
 7841 
 
 7!>04 
 
 7967 
 
 8030 
 
 8093 
 
 815G 
 
 63 
 
 689 
 
 8219 
 
 8282 
 
 8345 
 
 8408 
 
 8471 
 
 8534 
 
 8597 
 
 8660 
 
 8723 
 
 8786 
 
 63 
 
 690 
 
 838849 
 
 8912 
 
 8975 
 
 9038 
 
 9101 
 
 9164 
 
 9227 
 
 9289 
 
 9352 
 
 9415 
 
 63 
 
 691 
 
 9478 
 
 9541 
 
 9G04 
 
 9GG7 
 
 9729 
 
 9792 
 
 9855 
 
 9918 
 
 99.S1 
 
 ••43 
 
 63 
 
 692 
 
 840106 
 
 0109 
 
 0232 
 
 0294 
 
 0357 
 
 0420 
 
 0482 
 
 0545 
 
 0608 
 
 0671 
 
 63 
 
 693 
 
 0738 
 
 07!)6 
 
 0S59 
 
 0921 
 
 0984 
 
 1(>1( 
 
 la ..; 
 
 1172 
 
 12U 
 
 1297 
 
 63 
 
 dH 
 
 1359 
 
 1422 
 
 1485 
 
 1547 
 
 1()10 
 
 1672 
 
 1735 
 
 1797 
 
 18(iO 
 
 1922 
 
 63 
 
 j ()0o 
 
 1985 
 
 2047 
 
 2110 
 
 2172 
 
 2235 
 
 2297 
 
 23()0 
 
 2422 
 
 2484 
 
 2547 
 
 62 
 
 696 
 
 2609 
 
 2G72 
 
 2734 
 
 2796 
 
 2859 
 
 2921 
 
 2983 
 
 3()4ti 
 
 3108 
 
 8170 
 
 62 
 
 697 
 
 8233 
 
 3295 
 
 3357 
 
 3120 
 
 3482 
 
 3514 
 
 3G0G 
 
 3GG9 
 
 3731 
 
 37931 62 
 
 »!98 
 
 3855 
 
 3018 
 
 39801 4042 
 
 4104 
 
 4166 
 
 4229 
 
 4291 
 
 4353 
 
 4415 
 
 62 
 
 
 4477 
 
 
 
 453^ 
 
 1 
 
 4'"/l 
 
 j 4(iG4 
 
 4726 
 
 4 
 
 47d8 
 5 
 
 4850 
 
 4912 
 
 7 
 
 4974 
 
 5036 
 
 62 
 I). 
 
 2 
 
 3 
 
 G 
 
 8 
 
 9 
 
 ■: t: 
 
 M 
 
 i M 
 
 m' 
 
 20 
 
|1' : 
 ll 
 
 III! 
 
 ii 
 
 12 
 
 
 A TABLB OP LOGARITHMS FROM 1 
 
 TO 10,000. 
 
 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 5 
 
 6 
 
 7 
 55:J2 
 
 8 
 
 9 
 
 D. 
 
 700 
 
 845098 
 
 5160 
 
 5222 
 
 5284 
 
 534G 
 
 54U8 
 
 5470 
 
 5594 
 
 5G56 
 
 62 
 
 701 
 
 5718 
 
 57.S0 
 
 5«42 
 
 59U4 
 
 59uU 
 
 6028 
 
 6090 
 
 G151 
 
 6213 
 
 6275 
 
 62 
 
 702 
 
 0337 
 
 63' »9 
 
 64(51 
 
 6523 
 
 6585 
 
 6G4G 
 
 6708 
 
 G770 
 
 6832 
 
 6894 
 
 62 
 
 703 
 
 6955 
 
 7U17 
 
 7U79 
 
 7141 
 
 7202 
 
 72G4 
 
 7326 
 
 7388 
 
 7449 
 
 7511 
 
 62 
 
 704 
 
 7573 
 
 76:34 
 
 7696 
 
 7758 
 
 7819 
 
 7881 
 
 7943 
 
 8004 
 
 80GG 
 
 8128 
 
 62 
 
 705 
 
 8189 
 
 8251 
 
 8312 
 
 8374 
 
 8435 
 
 8497 
 
 8559 
 
 8G20 
 
 8G82 
 
 8743 
 
 62 
 
 70G 
 
 8805 
 
 8866 
 
 8928 
 
 8989 
 
 9051 
 
 9112 
 
 9174 
 
 9235 
 
 9297 
 
 9358 
 
 61 
 
 707 
 
 9419 
 
 9481 
 
 9542 
 
 9G04 
 
 9GG5 
 
 9726 
 
 9788 
 
 9849 
 
 9911 
 
 9972 
 
 61 
 
 708 
 
 850033 
 
 0095 
 
 015G 
 
 0217 
 
 0279 
 
 0340 
 
 0401 
 
 04G2 
 
 0524 
 
 0585 
 
 61 
 
 709 
 
 0646 
 
 0707 
 
 0769 
 
 0S30 
 
 0891 
 
 0952 
 
 1014 
 
 1075 
 
 113G 
 
 1197 
 
 61 
 
 710 
 
 851258 
 
 13-20 
 
 1381 
 
 1442 
 
 1503 
 
 15G4 
 
 1625 
 
 1686 
 
 1747 
 
 1809 
 
 61 
 
 711 
 
 1870 
 
 1931 
 
 1992 
 
 2053 
 
 2114 
 
 2175 
 
 2236 
 
 2297 
 
 2358 
 
 2419 
 
 61 
 
 712 
 
 2480 
 
 2541 
 
 2()02 
 
 2GG3 
 
 2724 
 
 2785 
 
 2846 
 
 2907 
 
 29GH 
 
 3029 
 
 61 
 
 713 
 
 3090 
 
 3150 
 
 3211 
 
 3272 
 
 3333 
 
 3394 
 
 3455 
 
 3516 
 
 3577 
 
 3637 
 
 61 
 
 714 
 
 3698 
 
 3759 
 
 3820 
 
 3881 
 
 3941 
 
 4002 
 
 40G3 
 
 4124 
 
 4185 
 
 4245 
 
 61 
 
 715 
 
 4306 
 
 4367 
 
 4428 
 
 4488 
 
 4549 
 
 4G10 
 
 4670 
 
 4731 
 
 4792 
 
 4852 
 
 61 
 
 71G 
 
 49 1'^ 
 
 4974 
 
 5034 
 
 5095 
 
 515G 
 
 5216 
 
 5277 
 
 5337 
 
 5398 
 
 5459 
 
 61 
 
 717 
 
 E. ' >. 5W0 
 
 5G40 
 
 5701 
 
 5761 
 
 5822 
 
 5882 
 
 5943 
 
 6003 
 
 6064 
 
 61 
 
 718 
 
 61. .; ; 6245 
 
 6306 
 
 63G6 
 
 6427 
 
 6487 
 
 6548 
 
 6G08 
 
 6668 
 
 60 
 
 719 
 
 6729 
 
 , , 6850 
 
 6910 
 
 6970 
 
 7031 
 
 7091 
 
 7152 
 
 7212 
 
 7272 
 
 60 
 
 720 
 
 857332 
 
 Vc Vi53 
 
 7513 
 
 7574 
 
 7G34 
 
 7694 
 
 7755 
 
 7815 
 
 7875 
 
 60 
 
 721 
 
 7935 
 
 7995 
 
 8056 
 
 8116 
 
 8176 
 
 8236 
 
 8297 
 
 8357 
 
 8417 
 
 8477 
 
 60 
 
 722 
 
 8537 
 
 8597 
 
 8657 
 
 8718 
 
 8778 
 
 8838 
 
 8898 
 
 8958 
 
 9018 
 
 9078 
 
 60 
 
 723 
 
 9138 
 
 9198 
 
 9258 
 
 9318 
 
 9379 
 
 9439 
 
 949$ 
 
 9559 
 
 9619 
 
 9679 
 
 60 
 
 724 
 
 9739 
 
 9799 
 
 9859 
 
 9918 
 
 9978 
 
 ••38 
 
 ••98 
 
 •158 
 
 •218 
 
 •278 
 
 60 
 
 725 
 
 860338 
 
 0398 
 
 a458 
 
 0518 
 
 0578 
 
 0G37 
 
 0G97 
 
 0757 
 
 0817 
 
 0877 
 
 60 
 
 726 
 
 0937 
 
 0996 
 
 1056 
 
 1116 
 
 1176 
 
 1236 
 
 1295 
 
 1355 
 
 1415 
 
 1475 
 
 60 
 
 727 
 
 1534 
 
 1594 
 
 1654 
 
 1714 
 
 1773 
 
 1833 
 
 1893 
 
 1952 
 
 2012 
 
 2072 
 
 60 
 
 728 
 
 2131 
 
 2191 
 
 2251 
 
 2310 
 
 2370 
 
 2430 
 
 2489 
 
 2549 
 
 2608 
 
 2668 
 
 60 
 
 729 
 
 2728 
 
 2787 
 
 2847 
 
 2906 
 
 2966 
 
 3025 
 
 3085 
 
 3144 
 
 8204 
 
 3263 
 
 60 
 
 730 
 
 863323 
 
 3382 
 
 3442 
 
 3501 
 
 3561 
 
 3G20 
 
 3680 
 
 3739 
 
 8799 
 
 3868 
 
 59 
 
 731 
 
 3917 
 
 3977 
 
 4036 
 
 4096 
 
 4155 
 
 4214 
 
 4274 
 
 4333 
 
 4392 
 
 4452 
 
 59 
 
 732 
 
 4511 
 
 4570 
 
 4630 
 
 4689 
 
 4748 
 
 4808 
 
 4867 
 
 4926 
 
 4985 
 
 5045 
 
 59 
 
 733 
 
 5104 
 
 3163 
 
 5222 
 
 5282 
 
 5341 
 
 5400 
 
 5459 
 
 6519 
 
 6578 
 
 5637 
 
 59 
 
 734 
 
 5696 
 
 5755 
 
 5814 
 
 5874 
 
 5933 
 
 5992 
 
 6051 
 
 6110 
 
 6169 
 
 6228 
 
 69 
 
 735 
 
 6287 
 
 6346 
 
 6405 
 
 6465 
 
 6524 
 
 6583 
 
 6642 
 
 6701 
 
 6760 
 
 6819 
 
 69 
 
 736 
 
 6878 
 
 6937 
 
 6996 
 
 7055 
 
 7114 
 
 7173 
 
 7232 
 
 7291 
 
 7350 
 
 7409 
 
 59 
 
 737 
 
 7467 
 
 7526 
 
 7585 
 
 7644 
 
 7703 
 
 7762 
 
 7821 
 
 7880 
 
 7939 
 
 7998 
 
 59 
 
 738 
 
 8056 
 
 8115 
 
 8174 
 
 8233 
 
 8292 
 
 8850 
 
 8409 
 
 8468 
 
 8527 
 
 8586 
 
 59 
 
 739 
 
 8644 
 
 8703 
 
 8762 
 
 8821 
 
 8879 
 
 8938 
 
 8997 
 
 9056 
 
 9114 
 
 9173 
 
 59 
 
 740 
 
 869232 
 
 9290 
 
 9349 
 
 9408 
 
 9466 
 
 9525 
 
 9584 
 
 9642 
 
 9701 
 
 9760 
 
 69 
 
 741 
 
 9818 
 
 9877 
 
 9935 
 
 9994 
 
 ••53 
 
 •111 
 
 •170 
 
 •228 
 
 •287 
 
 •845 
 
 59 
 
 742 
 
 870404 
 
 0462 
 
 0521 
 
 0579 
 
 0G38 
 
 0696 
 
 0755 
 
 0818 
 
 0872 
 
 0930 
 
 58 
 
 743 
 
 0989 
 
 1047 
 
 llu6 
 
 1164 
 
 1223 
 
 1281 
 
 1339 
 
 1898 
 
 1456 
 
 1515 
 
 68 
 
 744 
 
 1573 
 
 1631 
 
 1690 
 
 1748 
 
 1806 
 
 .1865 
 
 1923 
 
 1981 
 
 2040 
 
 2098 
 
 58 
 
 745 
 
 2156 
 
 2215 
 
 2273 
 
 2331 
 
 2389 
 
 '^448 
 
 2506 
 
 2564 
 
 2622 
 
 2681 
 
 63 
 
 746 
 
 2739 
 
 2797 
 
 2855 
 
 2913 
 
 2972 
 
 3030 
 
 8088 
 
 8146 
 
 8204 
 
 8262 
 
 68 
 
 747 
 
 8321 
 
 3379 
 
 3437 
 
 3495 
 
 3553 
 
 3611 
 
 8669 
 
 2727 
 
 8785 
 
 3844 
 
 58 
 
 748 
 
 8902 
 
 3960 
 
 4018 
 
 4076 
 
 4134 
 
 4192 
 
 4250 
 
 4308 
 
 4366 
 
 4424 
 
 68 
 
 749 
 
 4482 
 
 4540 
 
 4598 
 
 4656 
 
 4714 
 
 4772 
 
 4830 
 
 4888 
 
 4945 
 
 5003 
 
 68 
 
 750 
 
 875061 
 
 6119 
 
 6177 
 
 5235 
 
 6293 
 
 6351 
 
 6409 
 
 5466 
 
 6524 
 
 6582 
 
 68 
 
 751 
 
 5640 
 
 6698 
 
 6756 
 
 5813 
 
 6871 
 
 5929 
 
 6987 
 
 6045 
 
 6102 
 
 6160 
 
 68 
 
 752 
 
 6218 
 
 6276 
 
 6338 
 
 6391 
 
 6449 
 
 6507 
 
 6564 
 
 6622 
 
 6680 
 
 6737 
 
 68 
 
 768 
 
 6795 
 
 6853 
 
 6910 
 
 6968 
 
 7026 
 
 7088 
 
 7141 
 
 7199 
 
 12S6 
 
 7814 
 
 58 
 
 754 
 
 7871 
 
 7429 
 
 7487 
 
 7644 
 
 7602 
 
 7659 
 
 7717 
 
 7774 
 
 7832 
 
 7889 
 
 68 
 
 755 
 
 7947 
 
 8004 
 
 8062 
 
 8119 
 
 8177 
 
 82ai 
 
 8292 
 
 8849 
 
 8407 
 
 8464 
 
 57 
 
 756 
 
 8522 
 
 8579 
 
 8637 
 
 8694 
 
 8752 
 
 8809 
 
 8866 
 
 8924 
 
 8981 
 
 9089 
 
 57 
 
 767 
 
 9096 
 
 9158 
 
 9211 
 
 9268 
 
 9325 
 
 9383 
 
 9440 
 
 9497 
 
 9555 
 
 9612 
 
 57 
 
 i7S6 
 
 9669 
 
 9726 
 
 9784 
 
 9841 
 
 9898 
 
 9956 
 
 ••13 
 
 ••70 
 
 •127 
 
 •185 
 
 57 
 
 759 
 
 880242 
 
 0299 
 
 03.56 
 2 
 
 0413 
 3 
 
 0471 
 
 0528 
 
 0585 
 
 0642 
 
 0699 
 8 
 
 ot:6 
 
 9 
 
 67 
 
 
 
 1 
 
 4 
 
 6 
 
 7 
 
 D. 
 
A TABLE OP LOGARITHMS FROM 1 TO 10,000, 
 
 13 
 
 D. 1 
 
 
 N. 
 
 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D 
 
 ~62 1 
 
 7o0 
 
 880H14 
 
 0S71 
 
 0i)2H 
 
 0985 
 
 1042 
 
 TU99 
 
 115G 
 
 1213 
 
 T271 
 
 1328 
 
 57 
 
 62 1 
 
 761 
 
 1385 
 
 Mrl2 
 
 149!) 
 
 1556 
 
 1613 
 
 1G70 
 
 1727 
 
 1784 
 
 1841 
 
 1898 
 
 57 
 
 62 I 
 
 7G2 
 
 1955 
 
 2012 
 
 20G9 
 
 212G 
 
 2183 
 
 2240 
 
 2297 
 
 2354 
 
 2411 
 
 2468 
 
 67 
 
 62 1 
 
 763 
 
 2525 
 
 2581 
 
 2G38 
 
 2G95 
 
 2752 
 
 2809 
 
 2866 
 
 2923 
 
 2980 
 
 3037 
 
 57 
 
 62 1 
 
 764 
 
 3093 
 
 3150 
 
 3207 
 
 32G4 
 
 3321 
 
 3377 
 
 MM 
 
 34!)1 
 
 3548 
 
 3605 
 
 57 
 
 62 1 
 
 7G5 
 
 3G61 
 
 3718 
 
 3775 
 
 3832 
 
 3888 
 
 3945 
 
 4002 
 
 4059 
 
 4115 
 
 4172 
 
 57 
 
 61 1 
 
 7GG 
 
 4229 
 
 4285 
 
 4342 
 
 4399 
 
 4455 
 
 4512 
 
 4569 
 
 4025 
 
 4G82 
 
 4739 
 
 57 
 
 61 1 
 
 767 
 
 4795 
 
 4852 
 
 4909 
 
 49G5 
 
 5022 
 
 5078 
 
 5135 
 
 5192 
 
 5248 
 
 5305 
 
 57 
 
 61 I 
 
 7G8 
 
 53G1 
 
 5418 
 
 5474 
 
 5531 
 
 5587 
 
 5G44 
 
 5700 
 
 5757 
 
 5813 
 
 5870 
 
 67 
 
 61 I 
 
 7G9 
 
 5926 
 
 59S3 
 
 6039 
 
 6096 
 
 6152 
 
 G209 
 
 6265 
 
 6321 
 
 6378 
 
 6434 
 
 56 
 
 > 61 1 
 
 770 
 
 886491 
 
 G547 
 
 ()G04 
 
 GGGO 
 
 671G 
 
 6773 
 
 6829 
 
 6885 
 
 6942 
 
 6998 
 
 56 
 
 ) 61 1 
 
 771 
 
 7054 
 
 7111 
 
 71G7 
 
 7223 
 
 7280 
 
 7336 
 
 7392 
 
 7449 
 
 7505 
 
 7561 
 
 56 
 
 ) 61 1 
 
 772 
 
 7617 
 
 7G74 
 
 7730 
 
 7786 
 
 7842 
 
 7898 
 
 7955 
 
 8011 
 
 8067 
 
 8123 
 
 56 
 
 7 61 
 
 773 
 
 8179 
 
 823G 
 
 8292 
 
 8348 
 
 8404 
 
 8460 
 
 P516 
 
 8573 
 
 8629 
 
 8685 
 
 56 
 
 5 61 
 
 774 
 
 8741 
 
 8797 
 
 8853 
 
 8909 
 
 8965 
 
 9021 
 
 t)077 
 
 9134 
 
 9190 
 
 9246 
 
 56 
 
 2 61 ; 
 
 775 
 
 9302 
 
 9358 
 
 9414 
 
 9470 
 
 9526 
 
 95S2 
 
 9638 
 
 9694 
 
 9750 
 
 9806 
 
 56 
 
 9 61 ' 
 
 776 
 
 9862 
 
 9918 
 
 9974 
 
 ••30 
 
 ••8B 
 
 •141 
 
 •197 
 
 •253 
 
 •309 
 
 •365 
 
 56 
 
 4 61 
 
 777 
 
 890421 
 
 0477 
 
 0533 
 
 0589 
 
 0G45 
 
 0700 
 
 0756 
 
 0812 
 
 0868 
 
 0924 
 
 56 
 
 8 60 ^ 
 
 778 
 
 0980 
 
 1035 
 
 1091 
 
 1147 
 
 1203 
 
 1259 
 
 1314 
 
 1370 
 
 1426 
 
 1482 
 
 56 
 
 r2 60 ' 
 
 779 
 
 1537 
 
 1593 
 
 1649 
 
 1705 
 
 17G0 
 
 1816 
 
 1872 
 
 1928 
 
 1983 
 
 2039 
 
 66 
 
 f5 60 
 
 780 
 
 892095 
 
 2150 
 
 2206 
 
 2262 
 
 2317 
 
 2373 
 
 2429 
 
 2484 
 
 2540 
 
 2595 
 
 56 
 
 n 60 
 
 781 
 
 2651 
 
 2707 
 
 2762 
 
 2818 
 
 2873 
 
 2929 
 
 2985 
 
 3040 
 
 3096 
 
 8151 
 
 66 
 
 78 60 
 
 782 
 
 3207 
 
 3262 
 
 3318 
 
 3373 
 
 3429 
 
 3484 
 
 3540 
 
 3595 
 
 3651 
 
 3706 
 
 66 
 
 79 60 
 
 783 
 
 3762 
 
 3817 
 
 3873 
 
 3928 
 
 3,)84 
 
 4039 
 
 4094 
 
 4150 
 
 4205 
 
 4261 
 
 55 
 
 78 60 1 
 
 784 
 
 4316 
 
 4371 
 
 4427 
 
 4482 
 
 4538 
 
 4593 
 
 4648 
 
 4704 
 
 4759 
 
 4814 
 
 65 
 
 77 60 1 
 
 785 
 
 4870 
 
 4925 
 
 4980 
 
 503G 
 
 5091 
 
 514G 
 
 5201 
 
 5257 
 
 5312 
 
 53G7 
 
 55 
 
 75 60 2 
 
 786 
 
 5423 
 
 5478 
 
 5533 
 
 5588 
 
 5644 
 
 5699 
 
 5754 
 
 5809 
 
 5864 
 
 5920 
 
 55 
 
 72 60 1 
 
 787 
 
 5975 
 
 6030 
 
 6085 
 
 6140 
 
 6195 
 
 6251 
 
 6306 
 
 6361 
 
 6416 
 
 6471 
 
 55 
 
 68 60 ■ 
 
 788 
 
 6526 
 
 6581 
 
 6636 
 
 6G92 
 
 6747 
 
 6802 
 
 6857 
 
 6912 
 
 6967 
 
 7022 
 
 65 
 
 !63 60 < I 
 
 789 
 
 7077 
 
 7132 
 
 7187 
 
 7242 
 
 7297 
 
 7352 
 
 7407 
 
 7462 
 
 7517 
 
 7572 
 
 55 
 
 58 59 ; 1 
 
 790 
 
 897627 
 
 7G82 
 
 7737 
 
 7792 
 
 7847 
 
 7902 
 
 7957 
 
 8012 
 
 8067 
 
 8122 
 
 55 
 
 52 59 1 
 
 791 
 
 8176 
 
 8231 
 
 8280 
 
 8341 
 
 8396 
 
 8451 
 
 8506 
 
 8561 
 
 8615 
 
 8670 
 
 55 
 
 45 59 1; 1 
 
 792 
 
 8725 
 
 8780 
 
 8835 
 
 8890 
 
 8944 
 
 8999 
 
 9054 
 
 9109 
 
 9164 
 
 9218 
 
 55 
 
 37 59 1 I 
 
 793 
 
 9273 
 
 9328 
 
 9383 
 
 9437 
 
 9492 
 
 9547 
 
 9602 
 
 9656 
 
 9711 
 
 9766 
 
 55 
 
 28 59 ;: 
 
 794 
 
 9821 
 
 9875 
 
 9930 
 
 9985 
 
 ••39 
 
 ••94 
 
 •149 
 
 •203 
 
 •258 
 
 •312 
 
 65 
 
 19 59 f 
 
 795 
 
 900367 
 
 0422 
 
 0476 
 
 0531 
 
 0586 
 
 0640 
 
 0695 
 
 0749 
 
 0804 
 
 0859 
 
 65 
 
 09 59 f 
 
 796 
 
 0913 
 
 0968 
 
 1022 
 
 1077 
 
 1131 
 
 1186 
 
 1240 
 
 1295 
 
 1349 
 
 1404 
 
 65 
 
 98 59 1 
 
 797 
 
 1458 
 
 1513 
 
 1567 
 
 1G22 
 
 1676 
 
 1731 
 
 1785 
 
 1840 
 
 1894 
 
 1948 
 
 64 
 
 86 69 
 
 798 
 
 2003 
 
 2057 
 
 2112 
 
 2166 
 
 2221 
 
 2275 
 
 2329 
 
 2384 
 
 2438 
 
 2492 
 
 64 
 
 73 59 
 
 799 
 
 2547 
 
 2601 
 
 2655 
 
 2710 
 
 2754 
 
 2818 
 
 2873 
 
 2927 
 
 2981 
 
 3036 
 
 64 
 
 60 59 1 
 
 800 
 
 903090 
 
 3144 
 
 3199 
 
 3253 
 
 3307 
 
 3361 
 
 3416 
 
 3470 
 
 3524 
 
 3578 
 
 64 
 
 h5 59 
 
 801 
 
 3638 
 
 3687 
 
 3741 
 
 8795 
 
 3849 
 
 3904 
 
 3958 
 
 4012 
 
 4066 
 
 4120 
 
 64 
 
 kO 58 
 
 802 
 
 4174 
 
 4229 
 
 4283 
 
 4337 
 
 4391 
 
 4445 
 
 4499 
 
 4553 
 
 4607 
 
 4661 
 
 64 
 
 Es 58 
 
 803 
 
 4716 
 
 4770 
 
 4824 
 
 4878 
 
 4932 
 
 4986 
 
 6040 
 
 6094 
 
 5148 
 
 5202 
 
 &; 
 
 Ks 58 ; 
 
 804 
 
 5256 
 
 5310 
 
 5364 
 
 5418 
 
 5472 
 
 5526 
 
 5580 
 
 5634 
 
 5688 
 
 5742 
 
 64 
 
 Bl 58 
 
 905 
 
 5796 
 
 5850 
 
 5904 
 
 5958 
 
 6012 
 
 6066 
 
 6119 
 
 6173 
 
 6227 
 
 6281 
 
 54 
 
 ^p2 58 
 
 806 
 
 6335 
 
 6389 
 
 6443 
 
 6497 
 
 6551 
 
 6G04 
 
 6658 
 
 6712 
 
 6766 
 
 6820 
 
 54 
 
 ^b4 58 j 
 
 807 
 
 6874 
 
 6927 
 
 6981 
 
 7035 
 
 7089 
 
 7143 
 
 7196 
 
 7250 
 
 7304 
 
 7358 
 
 64 
 
 ^B4 58 
 
 808 
 
 7411 
 
 7465 
 
 7519 
 
 7573 
 
 7626 
 
 7680 
 
 7734 
 
 7787 
 
 7841 
 
 7895 
 
 54 
 
 ^Es 68 
 
 809 
 
 7949 
 
 8002 
 
 8056 
 
 8110 
 
 8163 
 
 8217 
 
 8270 
 
 8324 
 
 8878 
 
 8431 
 
 tn 
 
 ^K2 68 
 
 810 
 
 908485 
 
 8539 
 
 8592 
 
 8646 
 
 8699 
 
 8753 
 
 8807 
 
 8860 
 
 8914 
 
 8967 
 
 64 
 
 ^BO 68 
 
 811 
 
 9021 
 
 9074 
 
 9128 
 
 918J 
 
 9235 
 
 8239 
 
 9342 
 
 9396 
 
 9449 
 
 9503 
 
 64 
 
 ^B7 68 ■ 
 
 812 
 
 9556 
 
 9610 
 
 9G63 
 
 9716 
 
 9770 
 
 9823 
 
 9877 
 
 9930 
 
 9984 
 
 ••37 
 
 63 
 
 ^B4 68 1 
 
 813 
 
 910091 
 
 0144 
 
 0197 
 
 0251 
 
 0304 
 
 0858 
 
 0411 
 
 0464 
 
 0518 
 
 0571 
 
 63 
 
 ^V9 68 1 1 
 
 814 
 
 062' 
 
 0678 
 
 0731 
 
 0784 
 
 0838 
 
 0891 
 
 0944 
 
 0998 
 
 1051 
 
 1104 
 
 53 
 
 ^ft4 67 ■ ' 
 
 815 
 
 1158 
 
 1211 
 
 1264 
 
 1317 
 
 1371 
 
 1424 
 
 1477 
 
 1530 
 
 1584 
 
 1637 
 
 63 
 
 ^B9 67 ; ■ 
 
 816 
 
 1690 
 
 1743 
 
 1797 
 
 1850 
 
 1903 
 
 1956 
 
 2009 
 
 2063 
 
 2116 
 
 2169 
 
 53 
 
 ^^^^v ^M 
 
 817 
 
 2222 
 
 2275 
 
 2328 
 
 2381 
 
 2435 
 
 2488 
 
 2541 
 
 2594 
 
 2647 
 
 2700 
 
 53 
 
 
 
 818 
 
 2753 
 
 2r06 
 
 2859 
 
 2913 
 
 2966 
 
 3019 
 
 3072 
 
 8125 
 
 3178 
 
 3231 
 
 63 
 
 V D 1 
 
 
 819 
 N. 
 
 3284 
 
 3337 
 
 1 
 
 3390 
 2 
 
 nun 
 
 3 
 
 3406 
 4 
 
 8549 
 5 
 
 3602 
 6 
 
 J655 
 
 7 
 
 3708 
 8 
 
 3701 
 9 
 
 59 
 
 D 
 
 (J 
 
 ''' i. 
 
 ill 
 
 < 'I 
 
14 
 
 A TABLE OF LOOAUITHMS FROM 1 TO 10,000. 
 
 N. 
 
 
 
 1 
 
 2 
 "3!)20 
 
 3 
 
 4 
 
 5 
 
 6 
 4132 
 
 7 
 
 8 9 
 
 D. 
 
 820 
 
 913814 
 
 3867 
 
 3973 
 
 4026 
 
 l079 
 
 4184 
 
 4237 4290! 
 
 53 
 
 821 
 
 4343 
 
 4396 
 
 4449 
 
 4502 
 
 4555 
 
 4608 
 
 46(50 
 
 4713 
 
 4700, 4819 
 
 53 
 
 822 
 
 4872 
 
 4925 
 
 4977 
 
 5030 
 
 5083 
 
 5136 
 
 51 S9 
 
 5241 
 
 5294; 5347 
 
 53 
 
 823 
 
 5400 
 
 5453 
 
 5505 
 
 5558 
 
 5611 
 
 5664 
 
 5716 
 
 57(59 
 
 5S22I 5875 53 1 
 
 824 
 
 5927 
 
 5980 
 
 6033 
 
 6085 
 
 6138 
 
 6191 
 
 6243 
 
 6296 
 
 634!) 6401 
 
 53 
 
 825 
 
 6454 
 
 6507 
 
 6559 
 
 6612] 
 
 6664 
 
 6717 
 
 6770 
 
 6H22 
 
 6875 6927 
 
 53 
 
 82G 
 
 6980 
 
 7033 
 
 7085 
 
 71381 
 
 7190 
 
 7243 7295 
 
 7348 
 
 7400| 7453 
 
 53 
 
 827 
 
 7506 
 
 7558 
 
 7611 
 
 7663 77161 
 
 7768 7820 
 
 7873 
 
 7!)25j 7978 52 
 
 828 
 
 8030 
 
 8083 
 
 8135 
 
 8188 
 
 8240i 82931 8345| 
 
 8397 
 
 8450' 8502 52 
 
 82!) 
 
 8555 
 
 8607 
 
 865!) 
 
 8712 
 
 8764 
 
 8Sl6i 8S69 
 
 8!)21 
 
 8973' 9026 52 
 
 830 
 
 919078 
 
 9130 
 
 9183 
 
 9235 
 
 9287 
 
 9340! 5)392 
 
 9444 
 
 9496 !)549l 52 
 
 831 
 
 9601 
 
 9653 
 
 9706 
 
 9758 
 
 9810 9802 9914 
 
 9!»6" 
 
 ••19 ••71 52 
 
 832 
 
 920123 
 
 0176 
 
 0228 
 
 0280 
 
 0332 03S4 0436 
 
 0489 
 
 0541, 05!)3: 52 
 
 .S33 
 
 0645 
 
 0697 
 
 0749 
 
 0801 
 
 0853 0;J06 0!)58 
 
 1010 
 
 10.,2^ 1114' 52 
 
 834 
 
 1166 
 
 1218 
 
 1270 
 
 1322 
 
 1374 1426 1478 
 
 1530 
 
 1582: l';J4' 52 
 
 8:55 
 
 1686 
 
 1738 
 
 1790 
 
 1842 1S!)4; l:;4li| 1998; 
 
 2050 
 
 2102; 2154 52 
 
 m\ 
 
 2206 
 
 2258 
 
 2310 2362| 2414 24()6l 25181 
 
 2570 
 
 20221 2674! 52 
 
 837 
 
 2725 
 
 2777 
 
 28291 2K.-.1 2933 2l)S5, 3037; 
 
 30 i) 
 
 31 iO: 3192: 52 
 
 83H 
 
 3;>44 
 
 3296 
 
 3348' 3399 
 
 3151 3503 3555 
 
 3ti07 
 
 3658! 3710| 52 
 
 83i) 
 
 3762 
 
 3-114 
 
 38651 3917 
 
 3969 4(t2l| 4072 
 
 4124 
 
 41761 4228; 52 
 
 810 
 
 92427!) 
 
 4331 43S3: 
 
 4434 
 
 44«(3 
 
 4538 45S9| 
 
 4011 
 
 46'.>3; 4744, 52 
 
 841 
 
 4'"96 
 
 4848 
 
 489!) 
 
 4951 
 
 5003 
 
 5054 
 
 5106 
 
 5157 
 
 5209! 5201 1 52 
 
 842 
 
 5;ii2 
 
 53o4 
 
 5415 
 
 5467 
 
 5518 
 
 5570 
 
 5621 
 
 5073 
 
 5725 5770! 52 
 
 843 
 
 5828i 5S79' 
 
 5931 
 
 5982 
 
 60341 6085; 
 
 6137 
 
 6188 
 
 6240' 6291 51 
 
 844 
 
 or 12 
 
 O;! )4! 6445 
 
 6497 
 
 654-i 
 
 6600 
 
 6651 
 
 6702 
 
 6754 6805 51 
 
 845 
 
 *.. ;7 
 
 Oiii)8 6959 
 
 7011 
 
 7002 
 
 7114 
 
 7105 
 
 7216 
 
 7268; 7319 51 
 
 840 
 
 7370 
 
 7i'22 
 
 7473 
 
 7524 
 
 7576 
 
 7627 
 
 7078 
 
 7730 
 
 77811 7832! 51 
 
 847 
 
 7833 
 
 7935 
 
 7986 
 
 8037 
 
 80.S8 
 
 8140 9191 
 
 8242 
 
 82!)3' 8345 51 
 
 S48 
 
 8396 
 
 8447 
 
 8498 
 
 8549 
 
 8601 
 
 8652 
 
 8703 
 
 8754 
 
 8S051 8857 51 
 
 «4<) 
 
 8908 
 
 8959 
 
 9010 
 
 9061 
 
 9112 
 
 9163 
 
 9215 
 
 92(56 
 
 9317; 93(58 51 
 
 850 
 
 929419 
 
 9470 
 
 9521 
 
 9572 
 
 9623 
 
 9674 
 
 9725 
 
 9776 
 
 9827 
 
 9879 51 
 
 851 
 
 9!)30 
 
 9!)S1 ••32 
 
 ••83 
 
 •134 
 
 •185 
 
 •236 
 
 •2S7 
 
 •338 
 
 •389 51 
 
 852 
 
 930410 
 
 0491 0;'42 
 
 05;)2 
 
 0643 
 
 0694 
 
 0745 
 
 07.)0 
 
 0847 
 
 0898 51 
 
 853 
 
 0949 
 
 1000 1051 
 
 1102 
 
 1153 
 
 1204 
 
 1254 
 
 1305 
 
 1356 
 
 1407 51 
 
 854 
 
 1458 
 
 1509 1560 
 
 1010 
 
 1661 
 
 1712 
 
 1703 
 
 1814 
 
 1865 
 
 1915 51 
 
 855 
 
 1966 
 
 2017 
 
 2068 
 
 2118 
 
 2169 
 
 2220 
 
 2271 
 
 2322 
 
 2372 
 
 2423' 51 
 
 85() 
 
 2474 
 
 25:^4 
 
 2575 
 
 2626 
 
 207/ 
 
 2727 
 
 2778 
 
 2829 
 
 2879 
 
 2!)30 51 
 
 857 
 
 2981 
 
 3031 
 
 3082 
 
 3133 
 
 3183 
 
 3234 
 
 3285 
 
 3335 
 
 3386' 3437 51 
 
 858 
 
 3487 
 
 3538 
 
 3589 
 
 3039 
 
 3690 
 
 3740 
 
 3791 
 
 3841 
 
 38!)2! 3943! 51 
 
 850 
 
 3!»93 
 
 4044 
 
 40:)4 
 
 4145 
 
 4195 
 
 4246 
 
 42'.)6 
 
 4347 
 
 4397; 414S 51 
 
 860 
 
 9344!)8 
 
 454!) 
 
 45!)9 
 
 4050 
 
 4700 
 
 4751 
 
 4S01 
 
 4852 
 
 4902 
 
 4953 50 
 
 8()1 
 
 5003 
 
 5054 
 
 5104 
 
 5154 
 
 5205 
 
 5255 
 
 5306 
 
 5356 
 
 5406 
 
 5-457! 50 
 
 862 
 
 5507 
 
 5558 
 
 5()08 
 
 5658 
 
 5709 
 
 5759 
 
 6809 
 
 5860 
 
 5910 
 
 59(J0' 50 
 
 863 
 
 6011 
 
 6061 
 
 6111 
 
 6162 
 
 6212 
 
 6262 
 
 6313 
 
 6363 
 
 6413 6403 50 
 
 864 
 
 6514 
 
 6564 
 
 6(il4 
 
 6065 
 
 6715 
 
 6765 
 
 6815 
 
 6865 
 
 091(5; 69r>6 50 
 
 805 
 
 7016 
 
 7066 
 
 7117 
 
 7167 
 
 7217 
 
 7267 
 
 7317 
 
 7367 
 
 7418 
 
 74<;s 50 
 
 866 
 
 7518 
 
 7508 
 
 7618 
 
 7668 
 
 7718 
 
 7769 
 
 7819 
 
 78(59 
 
 7919 
 
 7!^69 50 
 
 867 
 
 8019 
 
 8069 
 
 8119 
 
 8169 
 
 8219 
 
 82()9 
 
 8320 
 
 8370 
 
 8420 
 
 8470 50 
 
 868 
 
 8520 
 
 8570 
 
 8020 
 
 8670 
 
 8720 
 
 8770 
 
 8820 
 
 8870 
 
 8!)20 8970 50 
 
 860 
 
 9020 
 
 907C 
 
 9120 
 
 917(] 
 
 9220 
 
 9270 
 
 9320 
 
 9369 
 
 9419 94159; 50 
 
 870 
 
 939519 
 
 956!) 
 
 9619 
 
 966!] 
 
 971!J 
 
 !)7()9 
 
 9819 
 
 9869 
 
 9918 9!)08 50 
 
 871 
 
 940018 
 
 0061^ 
 
 ! 0118 
 
 016e 
 
 1 021H 
 
 0267 
 
 0317 
 
 0367 
 
 0417 
 
 0467! 80 
 
 872 
 
 0516 
 
 056f 
 
 ) 061(] 
 
 066t 
 
 . 071( 
 
 0765 
 
 0815 
 
 0865 
 
 0915 
 
 0904 50 
 
 873 
 
 1014 
 
 1064 
 
 [ 1114 
 
 116E 
 
 \ 1213 
 
 1263 
 
 1313 
 
 1362 
 
 1412 
 
 1402 50 
 
 874 
 
 1511 
 
 1561 
 
 1611 
 
 166( 
 
 ) ma 
 
 176C 
 
 180!) 
 
 185!] 
 
 1909 
 
 1958 50 
 
 875 
 
 200!^ 
 
 205!' 
 
 i 2107 
 
 215- 
 
 J 220"; 
 
 225(1 
 
 230G 
 
 2355 
 
 2405 
 
 2455 50 
 
 876 
 
 2504 
 
 2554 
 
 [ 260S 
 
 1 205f 
 
 J 270S 
 
 ! 2752 
 
 2801 
 
 2851 
 
 2901 
 
 295( 
 
 } 50 
 
 877 
 
 300C 
 
 304t 
 
 ) 3091 
 
 314^ 
 
 ] 319^ 
 
 ! 3247 
 
 3297 
 
 334C 
 
 33!)6 
 
 344. 
 
 ) 49 
 
 878 
 
 349E 
 
 3544 
 
 [ 359E 
 
 1 364t 
 
 ) 869S 
 
 ! 3742 
 
 3791 
 
 3841 
 
 3890 
 
 393' 
 
 ) 49 
 
 87!) 
 
 398il 
 
 4m 
 
 J 
 
 < 408S 
 2 
 
 i 413' 
 
 J 418( 
 
 ) 423( 
 
 . 428.1 
 
 433J 
 
 . 4384 
 8 
 
 443; 
 
 3^ 49 
 
 N. 
 
 
 
 3 
 
 4 
 
 5 
 
 6 
 
 1 "7 
 
 9 
 
 1 f>- 
 
A TABLE OP LOGARITHMS KKOM 1 TO 10,000, 
 
 15. 
 
 52 
 
 52 
 
 52 
 
 52 
 
 52 
 
 52 
 
 52 
 
 52 
 
 52 
 
 52 
 
 52 
 
 52 
 
 51 
 51 
 
 51 
 51 
 51 
 61 
 51 
 51 
 51 
 
 51 
 51 
 51 
 
 50 
 
 50 
 
 50 
 
 50 
 
 50 
 
 50 
 
 50 
 
 50 
 
 50 
 
 50 
 50 
 50 
 50 
 50 
 50 
 50 
 i 50 
 49 
 i 49 
 '_49_ 
 
 1 D. 
 
 N. 
 "hho~ 
 
 
 944483 
 
 I 
 4532 
 
 2 
 
 3 
 
 4 
 4680 
 
 5 
 
 6 
 4779 
 
 7 
 
 8 
 4877 
 
 9 
 4927 
 
 D. 
 
 49 
 
 4581 
 
 4631 
 
 4729 
 
 4M28 
 
 881 
 
 497U 
 
 5U25 
 
 5074 
 
 5124 
 
 5173 
 
 5222 
 
 5272 
 
 5321 
 
 5370 
 
 5419 
 
 49 
 
 882 
 
 5469 
 
 5518 
 
 5567 
 
 5616 
 
 5665 
 
 5715 
 
 5764 
 
 5813 
 
 6862 
 
 5912 
 
 49 
 
 88!) 
 
 5961 
 
 6010 
 
 6059 
 
 6108 
 
 6157 
 
 6207 
 
 6256 
 
 6305 
 
 6354 
 
 6403 
 
 49 
 
 884 
 
 6452 
 
 6501 
 
 6551 
 
 6600 
 
 6649 
 
 6698 
 
 6747 
 
 6796 
 
 6845 
 
 6894 
 
 49 
 
 885 
 
 6943 
 
 6992 
 
 7041 
 
 7090 
 
 7140 
 
 7189 
 
 7238 
 
 7287 
 
 7336 
 
 7385 
 
 49 
 
 88G 
 
 7434 
 
 7483 
 
 7532 
 
 7581 
 
 7630 
 
 7679 
 
 7728 
 
 7777 
 
 7826 
 
 7875 
 
 49 
 
 887 
 
 7924 
 
 7973 
 
 8022 
 
 8070 
 
 8119 
 
 8168 
 
 8217 
 
 8266 
 
 8315 
 
 8364 
 
 49 
 
 888 
 
 8413 
 
 8462 
 
 8511 
 
 8560 
 
 8609 
 
 8657 
 
 8706 
 
 8755 
 
 8804 
 
 8853 
 
 49 
 
 88i) 
 
 8902 
 
 8951 
 
 8999 
 
 9048 
 
 9097 
 
 9146 
 
 9195 
 
 9244 
 
 9292 
 
 9341 
 
 49 
 
 •8i)() 
 
 949390 
 
 9439 
 
 9488 
 
 9536 
 
 9585 
 
 9634 
 
 9683 
 
 9731 
 
 9780 
 
 9829 
 
 49 
 
 8!)1 
 
 9878 
 
 9926 
 
 9975 
 
 ••24 
 
 ••73 
 
 •121 
 
 •170 
 
 •219 
 
 •267 
 
 •316 
 
 49 
 
 892 
 
 950365 
 
 0414 
 
 0462 
 
 0511 
 
 05(50 
 
 0608 
 
 0657 
 
 0706 
 
 0754 
 
 0803 
 
 49 
 
 89?} 
 
 0851 
 
 0900 
 
 0949 
 
 0997 
 
 1046 
 
 1095 
 
 1143 
 
 1192 
 
 1240 
 
 1289 
 
 49 
 
 894 
 
 1338 
 
 1386 
 
 1435 
 
 1483 
 
 1532 
 
 1580 
 
 1629 
 
 1677 
 
 1726 
 
 1775 
 
 49 
 
 895 
 
 1823 
 
 1872 
 
 1920 
 
 1969 
 
 2017 
 
 2066 
 
 2114 
 
 2163 
 
 2211 
 
 2260 
 
 48 
 
 89G 
 
 2308 
 
 2356 
 
 2405 
 
 2453 
 
 2502 
 
 2550 
 
 2599 
 
 2647 
 
 2696 
 
 2744 
 
 48 
 
 897 
 
 2792 
 
 2841 
 
 2889 
 
 2938 
 
 2986 
 
 8034 
 
 3083 
 
 3131 
 
 3180 
 
 3228 
 
 48 
 
 898 
 
 3276 
 
 3325 
 
 3373 
 
 3421 
 
 3470 
 
 3518 
 
 3566 
 
 3615 
 
 3663 
 
 3711 
 
 48 
 
 899 
 
 3760 
 
 3808 
 
 3856 
 
 3905 
 
 3953 
 
 4001 
 
 4049 
 
 4098 
 
 4146 
 
 4194 
 
 48 
 
 900 
 
 954243 
 
 4291 
 
 4339 
 
 4387 
 
 4435 
 
 4484 
 
 4532 
 
 4580 
 
 4628 
 
 4677 
 
 48 
 
 901 
 
 4725 
 
 4773 
 
 4821 
 
 4869 
 
 4918 
 
 4966 
 
 5014 
 
 6002 
 
 6110 
 
 6158 
 
 48 
 
 902 
 
 5207 
 
 5255 
 
 6303 
 
 5C51 
 
 5399 
 
 5447 
 
 5495 
 
 6i:43 
 
 6592 
 
 5640 
 
 48 
 
 903 
 
 5688 
 
 5736 
 
 5784 
 
 5832 
 
 5880 
 
 5928 
 
 6976 
 
 6024 
 
 6072 
 
 6120 
 
 48 
 
 904 
 
 6168 
 
 6216 
 
 6265 
 
 6313 
 
 6361 
 
 6409 
 
 6457 
 
 6505 
 
 6553 
 
 6601 
 
 48 
 
 905 
 
 6649 
 
 6697 
 
 6745 
 
 6793 
 
 6840 
 
 68L- 
 
 6936 
 
 6984 
 
 7032 
 
 7080 
 
 48 
 
 90() 
 
 7128 
 
 7176 
 
 7224 
 
 7272 
 
 7320 
 
 7368 
 
 7416 
 
 7464 
 
 7512 
 
 7559 
 
 48 
 
 907 
 
 7607 
 
 7655 
 
 7703 
 
 7751 
 
 7799 
 
 7847 
 
 7894 
 
 7942 
 
 7990 
 
 8038 
 
 48 
 
 908 
 
 8086 
 
 8134 
 
 8181 
 
 8229 
 
 8277 
 
 8325 
 
 8373 
 
 8421 
 
 8468 
 
 8516 
 
 48 
 
 909 
 
 8564 
 
 8612 
 
 8659 
 
 8707 
 
 8755 
 
 8803 
 
 8850 
 
 8898 
 
 8946 
 
 8994 
 
 48 
 
 910 
 
 959041 
 
 9089 
 
 9137 
 
 9185 
 
 9232 
 
 9280 
 
 9328 
 
 9375 
 
 9423 
 
 9471 
 
 48 
 
 911 
 
 9518 
 
 9566 
 
 9614 
 
 9661 
 
 9709 
 
 9757 
 
 9804 
 
 9852 
 
 9900 
 
 9947 
 
 48 
 
 912 
 
 9995 
 
 ••42 
 
 ••90 
 
 •138 
 
 •185 
 
 •233 
 
 •280 
 
 •328 
 
 •376 
 
 •423 
 
 48 
 
 913 
 
 960471 
 
 0518 
 
 0566 
 
 0613 
 
 0661 
 
 0709 
 
 0756 
 
 0804 
 
 0851 
 
 0899 
 
 48 
 
 914 
 
 0946 
 
 0994 
 
 1041 
 
 1089 
 
 1136 
 
 1184 
 
 1231 
 
 1279 
 
 1326 
 
 1374 
 
 47 
 
 915 
 
 1421 
 
 1469 
 
 1516 
 
 1563 
 
 1611 
 
 1658 
 
 1706 
 
 1753 
 
 1801 
 
 1848 
 
 47 
 
 91G 
 
 1895 
 
 1943 
 
 1990 
 
 2038 
 
 2085 
 
 2132 
 
 2180 
 
 2227 
 
 2275 
 
 2322 
 
 47 
 
 917 
 
 2369 
 
 2417 
 
 2464 
 
 2511 
 
 2559 
 
 2606 
 
 2653 
 
 2701 
 
 2748 
 
 2795 
 
 47 
 
 918 
 
 2843 
 
 2890 
 
 2937 
 
 2985 
 
 3032 
 
 3079 
 
 3126 
 
 3174 
 
 3221 
 
 3268 
 
 47 
 
 919 
 
 3316 
 
 3363 
 
 3410 
 
 3457 
 
 3504 
 
 3552 
 
 3599 
 
 3646 
 
 3693 
 
 3741 
 
 47 
 
 920 
 
 963788 
 
 3S35 
 
 3H82 
 
 3929 
 
 3977 
 
 4024 
 
 4071 
 
 4118 
 
 4165 
 
 4212 
 
 47 
 
 921 
 
 4260 
 
 4307 
 
 4354 
 
 4401 
 
 4448 
 
 4495 
 
 4542 
 
 4590 
 
 4637 
 
 4684 
 
 47 
 
 922 
 
 4731 
 
 4778 
 
 4825 
 
 4872 
 
 4919 
 
 4966 
 
 5013 
 
 5061 
 
 5108 
 
 6155 
 
 47 
 
 923 
 
 5202 
 
 5249 
 
 5296 
 
 5343 
 
 5390 
 
 5437 
 
 6484 
 
 5531 
 
 5578 
 
 6625 
 
 47 
 
 924 
 
 5672 
 
 5719 
 
 5766 
 
 5813 
 
 5860 
 
 5907 
 
 5954 
 
 6001 
 
 6048 
 
 6095 
 
 47 
 
 925 
 
 6142 
 
 6189 
 
 6236 
 
 6283 
 
 6329 
 
 6376 
 
 6423 
 
 6470 
 
 6517 
 
 6564 
 
 47 
 
 920 
 
 6611 
 
 6658 
 
 6705 
 
 6752 
 
 67-99 
 
 6845 
 
 6892 
 
 6939 
 
 6986 
 
 7033 
 
 47 
 
 927 
 
 7080 
 
 7127 
 
 7173 
 
 7220 
 
 7267 
 
 7314 
 
 7361 
 
 7408 
 
 7454 
 
 7501 
 
 47 
 
 928 
 
 7548 
 
 7595 
 
 7642 
 
 7(i88 
 
 7735 
 
 7782 
 
 7829 
 
 7875 
 
 7922 
 
 7969 
 
 47 
 
 929 
 
 8016 
 
 8062 
 
 8109 
 
 8156 
 
 8203 
 
 8249 
 
 8296 
 
 8343 
 
 8390 
 
 8436 
 
 47 
 
 930 
 
 968483 
 
 8530 
 
 8576 
 
 8623 
 
 8670 
 
 8716 
 
 8763 
 
 8810 
 
 8856 
 
 8i)08 
 
 47 
 
 931 
 
 8950 
 
 8996 
 
 9043 
 
 9090 
 
 9136 
 
 9183 
 
 9229 
 
 9276 
 
 9323 
 
 9369 
 
 47 
 
 932 
 
 9416 
 
 9463 
 
 9509 
 
 9556 
 
 9602 
 
 9649 
 
 9()95 
 
 9742 
 
 9789 
 
 9835 
 
 47 
 
 933 
 
 9882 
 
 9928 
 
 9975 
 
 ••21 
 
 ••()8 
 
 •114 
 
 •161 
 
 •207 
 
 •254 
 
 •300 
 
 47 
 
 934 
 
 970347 
 
 0393 
 
 0440 
 
 0486 
 
 0533 
 
 057!) 
 
 0626 
 
 0672 
 
 0719 
 
 0765 
 
 46 
 
 935 
 
 0812 
 
 0858 
 
 0904 
 
 0951 
 
 0997 
 
 1044 
 
 1090 
 
 1137 
 
 1183 
 
 1229 
 
 46 
 
 936 
 
 1276 
 
 1322 
 
 1369 
 
 1415 
 
 1461 
 
 1508 
 
 1554 
 
 1601 
 
 1647 
 
 1693 
 
 46 
 
 937 
 
 1740 
 
 1786 
 
 1832 
 
 1879 
 
 1925 
 
 1971 
 
 2018 
 
 2064 
 
 2110 
 
 2157 
 
 46 
 
 938 
 
 2203 
 
 2249 
 
 2295 
 
 2342 
 
 2388 
 
 2434 
 
 2481 
 
 2527 
 
 2573 
 
 2619 
 
 46 
 
 989 
 X. 
 
 26()6 
 
 
 
 2712' 2758 
 
 2H04 
 
 2851 
 
 2897 
 
 2943 
 
 2989 
 ~7 
 
 3035 
 8 
 
 3082 
 
 46 
 D. 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 9 
 
 m 
 
 ii'J 
 
 ' i 
 
 ^I'P 
 
 m 
 
 
''' 
 
 i 
 
 16 
 
 
 A TABLE OP LOOAUITHMS FROM 1 
 
 TO 10,000. 
 
 
 
 X. 
 
 940 
 
 
 
 1 
 "3174 
 
 2 
 
 3220 
 
 3 
 
 4 1 5 
 
 6 
 
 7 
 
 8 
 3497 
 
 9 
 3543 
 
 D- 
 
 973128 
 
 3266 
 
 3313 3359 
 
 3405 
 
 3451 
 
 40 
 
 941 
 
 3590 
 
 3636 
 
 3682 
 
 3728 
 
 3774 3820 
 
 3.S00 
 
 3913 
 
 3959 
 
 4005 
 
 46 
 
 912 
 
 4051 
 
 40;)7 
 
 4143 
 
 4189 
 
 4235 4281 
 
 4327 
 
 4374 
 
 4420 
 
 4466 
 
 46 
 
 943 
 
 4512 
 
 45-,8 
 
 4604 
 
 4650 
 
 4(596 4742 
 
 478H 
 
 48.34 
 
 4880 
 
 4926 
 
 46 
 
 91-t 
 
 41)72 
 
 5018 
 
 5064 
 
 5110 
 
 5156 5202 
 
 5248 
 
 5294! S.'MO 
 
 5386 
 
 46 
 
 945 
 
 5132 
 
 5178 
 
 5524 
 
 5570 
 
 5616 5662 
 
 6707 
 
 57.53 
 
 5799 
 
 5845 
 
 46 
 
 94(5 
 
 58;)1 
 
 5:)37 
 
 5983 
 
 0029 
 
 6075 6121 
 
 01(57 
 
 6212 
 
 0258 
 
 0304 
 
 46 
 
 947 
 
 63.")0 
 
 6396 
 
 6442 
 
 6488 
 
 6533 
 
 0579 
 
 (5625 
 
 (5671 
 
 0717 
 
 0703 
 
 46 
 
 948 
 
 6808 
 
 6854 
 
 6900 
 
 6946 
 
 6992 
 
 7037 
 
 7083 
 
 7129 
 
 7175 
 
 7220 
 
 46 
 
 949 
 
 7266 
 
 7312 
 
 7358 
 
 7403 
 
 7449 
 
 7495 
 
 irAl 7586 
 
 7632 
 
 7(578 
 
 46 
 
 950 
 
 977724 
 
 7769 
 
 7815 
 
 7861 
 
 7906 
 
 7952 
 
 7998 8043 
 
 8089 
 
 8135 
 
 46 
 
 951 
 
 8181 
 
 8226 
 
 8272 
 
 8317 
 
 8363 
 
 8409 
 
 8454 
 
 8500 
 
 8.546 
 
 8591 
 
 46 
 
 952 
 
 8637 
 
 8683 
 
 8728 
 
 8774 
 
 8819 
 
 8865 
 
 8911 
 
 8956 
 
 9002 
 
 9047 
 
 46 
 
 953 
 
 9093 
 
 9138 
 
 9184 
 
 9230 
 
 9275 
 
 9321 
 
 9366 
 
 9412 
 
 9457 
 
 9503 
 
 46 
 
 954 
 
 9548 
 
 95i)4 
 
 9639 
 
 9685 
 
 9730 
 
 9776 
 
 9821 
 
 9867 
 
 9912 
 
 9958 
 
 46 
 
 955 
 
 980003 
 
 0049 
 
 0094 
 
 0140 
 
 0185 
 
 0231 
 
 027(5 
 
 0322 
 
 0367 
 
 0412 
 
 45 
 
 950 
 
 0458 
 
 0503 
 
 0;J49 
 
 0594 
 
 0640 
 
 0685 
 
 0730 
 
 0776 
 
 0821 
 
 0867 
 
 45 
 
 957 
 
 0912 
 
 0957 
 
 1003 
 
 1048 
 
 10!)3 
 
 1139 
 
 1184 
 
 1229 
 
 1275 
 
 1320 
 
 45 
 
 958 
 
 1366 
 
 1411 
 
 1456 
 
 1501 
 
 1.547 
 
 15!r2 
 
 1637 
 
 1683 
 
 1728 
 
 1773 
 
 45 
 
 959 
 
 1819 
 
 18(14 
 
 i:)o;) 
 
 1954 
 
 2000 
 
 2045 
 
 2o:>o 
 
 2135 
 
 21S1 
 
 2226 
 
 45 
 
 960 
 
 982271 
 
 2316 
 
 2362 
 
 2407 
 
 2452 
 
 2497 
 
 2.543 
 
 25;-l8 
 
 2(533 
 
 2(578 
 
 45 
 
 901 
 
 2723 
 
 27(;;» 
 
 2814 
 
 2859 
 
 2904 
 
 2949 
 
 2994 
 
 3ti40 
 
 3oe5 
 
 3130 
 
 45 
 
 962 
 
 3175 
 
 3220 
 
 32(;5 
 
 3310 
 
 3356 
 
 3401 
 
 »14(5 
 
 3491 
 
 3536 
 
 3581 
 
 45 
 
 963 
 
 8626 
 
 3671 
 
 3716 
 
 3762 
 
 3,S07 
 
 8852 
 
 8897 
 
 8942 
 
 3987 
 
 4032 
 
 45 
 
 964 
 
 4077 
 
 4122 
 
 4167 
 
 4212 
 
 42.57 
 
 4302 
 
 iUI 
 
 4392 
 
 4437 
 
 4482 
 
 45 
 
 965 
 
 4527 
 
 4572 
 
 4617 
 
 4662 
 
 4707 
 
 4752 
 
 4797 
 
 4842 
 
 4887 
 
 4932 
 
 45 
 
 966 
 
 4977 
 
 5022 
 
 5067 
 
 5112 
 
 5157 
 
 5202 
 
 5247 
 
 5292 
 
 5337 
 
 5382 
 
 45 
 
 967 
 
 5426 
 
 5471 
 
 5516 
 
 5561 
 
 5(50(5 
 
 5651 
 
 5(596 
 
 5741 
 
 5786 
 
 5830 
 
 45 
 
 968 
 
 5875 
 
 5920 
 
 5965 
 
 6010 
 
 6055 
 
 6100 
 
 0144 
 
 618!) 
 
 6234 
 
 6279 
 
 45 
 
 969 
 
 6324 
 
 6369 
 
 0413 
 
 6458 
 
 6.503 
 
 6548 
 
 0593 
 
 6G37 
 
 6(582 
 
 6727 
 
 45 
 
 970 
 
 986772 
 
 6H17 
 
 6861 
 
 6906 
 
 0951 
 
 6996 
 
 7040 
 
 7085 
 
 7130 
 
 7175 
 
 45 
 
 971 
 
 7219 
 
 7'2()4 
 
 7309 
 
 7353 
 
 7398 
 
 7443 
 
 7488 
 
 7532 
 
 7577 
 
 7622 
 
 45 
 
 972 
 
 7r,66 
 
 7711 
 
 7756 
 
 7800 
 
 7845 
 
 7;-i90 
 
 7;)34 
 
 7979 
 
 8024 
 
 8068 
 
 45 
 
 97-; 
 
 H113 
 
 8157 
 
 8202 
 
 8247 
 
 8:391 
 
 8336 
 
 83S1 
 
 84^5 
 
 8470 
 
 8514 
 
 45 
 
 974 
 
 8559 
 
 8604 
 
 8648 
 
 8693 
 
 8737 
 
 8782 
 
 8820 
 
 8871 
 
 8910 
 
 89(50 
 
 45 
 
 975 
 
 9005 
 
 904.) 
 
 9094 
 
 9138 
 
 9183 
 
 9227 
 
 9272 
 
 9316 
 
 9361 
 
 9405 
 
 45 
 
 976 
 
 9450 
 
 94!)4 
 
 9539 
 
 95S3 
 
 9028 
 
 9672 
 
 9717 
 
 97(51 
 
 9806 
 
 9850 
 
 44 
 
 977 
 
 9895 
 
 993.I 
 
 9983 
 
 ••28 
 
 ••72 
 
 •117 
 
 •1(51 
 
 •206 
 
 •250 
 
 •294 
 
 44 
 
 978 
 
 99033!) 
 
 03 S3 
 
 0428 
 
 0472 
 
 0516 
 
 0561 
 
 0605 
 
 0650 
 
 0694 
 
 0738 
 
 44 
 
 979 
 
 0783 
 
 0827 
 
 0871 
 
 0916 
 
 0960 
 
 1004 
 
 1049 
 
 1093 
 
 1137 
 
 1182 
 
 44 
 
 980 
 
 991226 
 
 1270 
 
 1315 
 
 1359 
 
 1403 
 
 1448 
 
 1492 
 
 1536 
 
 1.580 
 
 1625 
 
 44 
 
 981 
 
 1669 
 
 17] 3 
 
 1758 
 
 1802 
 
 1846 
 
 1890 
 
 1!)35 
 
 1979 
 
 2023 
 
 2067 
 
 44 
 
 982 
 
 2111 
 
 2156 
 
 2200 
 
 2244 
 
 22S8 
 
 2333 
 
 2377 
 
 2421 
 
 2405 
 
 2509 
 
 44 
 
 983 
 
 2551 
 
 2598 
 
 2642 
 
 2686 
 
 2730 
 
 2774 
 
 2819 
 
 2863 
 
 2907 
 
 2951 
 
 44 
 
 984 
 
 2995 
 
 303;) 
 
 3083 
 
 3127 
 
 8172 
 
 321()j 3260 
 
 3304 
 
 3348 
 
 339k 
 
 44 
 
 985 
 
 3436 
 
 3480 
 
 3524 
 
 3568 
 
 3613 
 
 3657 
 
 3701 
 
 3745 
 
 3789 
 
 3^33 
 
 44 
 
 9CS(5 
 
 3877 
 
 3921 
 
 3965 
 
 4009 
 
 4053 
 
 4097 
 
 4141 
 
 41h5 
 
 4229 
 
 4273 
 
 44 
 
 987 
 
 4317 
 
 4361 
 
 4405 
 
 4449 
 
 4493 
 
 4537 
 
 458] 
 
 4(525 
 
 4669 
 
 4713 
 
 44 
 
 938 
 
 4757 
 
 4801 
 
 4845 
 
 48H9 
 
 4!)33 
 
 4977 
 
 5021 
 
 50( 5 
 
 5108 
 
 51.52 
 
 44 
 
 i)8!) 
 
 5196 
 
 5240 
 
 5284 
 
 5328 
 
 5372 
 
 5416 
 
 5460 
 
 5504 
 
 5547 
 
 .5591 
 
 44 
 
 990 
 
 995J35 
 
 5(;7i) 
 
 5723 
 
 5767 
 
 5811 
 
 5854 
 
 5898 
 
 5942 
 
 5986 
 
 6030 
 
 44 
 
 991 
 
 6074 
 
 6117 
 
 6161 
 
 6205 
 
 0249 
 
 6293 
 
 6337 
 
 03^(1 
 
 6424 
 
 6468 
 
 44 
 
 992 
 
 6512 
 
 6555 
 
 6599 
 
 6643 
 
 0(587 
 
 6731 
 
 6774 
 
 6818 
 
 68(52 
 
 0906 
 
 44 
 
 993 
 
 6949 
 
 6993 
 
 7037 
 
 7080 
 
 7124 
 
 7168 
 
 7212 
 
 7255 
 
 7299 
 
 7343 
 
 44 
 
 994 
 
 7386 
 
 7430 
 
 7474 
 
 7517 
 
 7501 
 
 7605 
 
 7(548 
 
 7692 
 
 7736 
 
 7779 
 
 44 
 
 995 
 
 7823 
 
 7867 
 
 7910 
 
 7954 
 
 7998 
 
 8041 
 
 8085 
 
 8129 
 
 8172 
 
 8216 
 
 44 
 
 996 
 
 8259 
 
 8303 
 
 8347 
 
 8390 
 
 8434 
 
 8477 
 
 8521 
 
 8564 
 
 8608 
 
 8652 
 
 44 
 
 097 
 
 8695 
 
 8739 
 
 8782 
 
 8826 
 
 8809 
 
 8913 
 
 8956 
 
 9000 
 
 9043 
 
 9087 
 
 44 
 
 998 
 
 9131 
 
 9174 
 
 9218 
 
 9261 
 
 9305 
 
 9348 
 
 9392 
 
 9435 
 
 9479 
 
 9522 
 
 44 
 
 999 
 
 9565 
 
 9<;09 
 
 9652 
 
 9(596 
 
 97.39 
 
 9783 
 
 9826 
 
 9870 
 
 9913 
 
 9957 
 
 43 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
D- 
 
 40 
 46 
 46 
 46 
 46 
 46 
 46 
 46 
 46 
 46 
 46 
 46 
 46 
 46 
 46 
 45 
 45 
 45 
 45 
 45 
 45 
 45 
 45 
 45 
 45 
 45 
 45 
 45 
 45 
 45 
 45 
 i22; 45 
 
 44 
 44 
 44 
 
 44 
 44 
 44 
 44 
 
 iSj 44 
 
 r^'< 44 
 
 44 
 44 
 44 
 44 
 44 
 44 
 44 
 44 
 44 
 44 
 44 
 43 
 
 [9 
 \^ 
 
 |7 
 
 D. 
 
 Uil 
 
 A TABLE 
 
 OK 
 
 LOGARITHMIC 
 
 :.;:! 
 
 SINES AA'D TANGENTS 
 
 van EVERY 
 
 
 ••; i 
 
 
 || 
 
 '! . 
 
 1 
 
 '1 
 
 SI 
 
 1^' 
 
 Wm 
 
 I 
 
 SI 
 
 \ 
 
 WK 
 
 DEGREE AND MINUTE 
 
 OP THE QUADRANT. 
 
I 
 
 18 
 
 (0 DEGREES.) A TABLE OF LORARITHMIC 
 
 Bine. 
 
 
 
 1 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 10 
 11 
 12 
 13 
 14 
 15 
 1(5 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 23 
 26 
 27 
 28 
 29 
 30 
 31 
 32 
 33 
 34 
 35 
 36 
 37 
 38 
 39 
 40 
 41 
 42 
 43 
 44 
 45 
 46 
 47 
 48 
 49 
 50 
 51 
 52 
 53 
 54 
 
 50 
 57 
 
 '>H 
 59 
 60 
 
 0.000000 
 6.463726 
 764756 
 940847 
 7.065780 
 162696 
 241877 
 308824 
 366816 
 417968 
 463725 
 7.505118 
 542906 
 577668 
 609853 
 639816 
 667845 
 694173 
 718997 
 742477 
 764754 
 7.785943 
 806146 
 825451 
 843934 
 861062 
 878695 
 895085 
 910879 
 926119 
 940842 
 7.955082 
 968870 
 982233 
 995198 
 8.007787 
 020021 
 031919 
 043501 
 054781 
 065776 
 8.076500 
 086965 
 097183 
 107167 
 110926 
 126471 
 135810 
 144953 
 153907 
 162681 
 8.171280 
 179713 
 1879H5 
 196102 
 204070 
 211895 
 219581 
 227134 
 234557 
 241855 
 
 10 
 
 Coiiiie. 
 
 5017.17 
 2934.85 
 2082.31 
 1615.17 
 1319.68 
 1115.75 
 966.53 
 852.54 
 702.63 
 689.88 
 629.81 
 579.36 
 536.41 
 499.38 
 467 
 438.81 
 413.72 
 391.35 
 371.27 
 353.15 
 336.72 
 321.75 
 308.05 
 295.47 
 283.88 
 273.17 
 263.23 
 253.99 
 245.38 
 237.33 
 229.80 
 222.73 
 216.08 
 209.81 
 203.90 
 198.31 
 193.02 
 188.01 
 183.25 
 178.72 
 174.41 
 170.31 
 166,39 
 162.65 
 159.08 
 155.66 
 152.38 
 149.24 
 146.22 
 143.33 
 140.54 
 137.86 
 135.29 
 132.80 
 130.41 
 128,10 
 125.87 
 123.72 
 121.64 
 11!) 63 
 
 9 
 
 9. 
 
 D. 
 
 9. 
 
 9 
 
 9 
 
 Cosine, i D. 
 
 000000 
 000000 
 000000 
 000000 
 000000 
 000000 
 ,999999 
 999999 
 999999 
 999999 
 999998 
 999998 
 999997 
 999997 
 999996 
 999996 
 999995 
 999995 
 999994 
 999993 
 999993 
 999992 
 999991 
 999990 
 999989 
 999988 
 999988 
 999987 
 999986 
 999985 
 999983 
 .999982 
 999981 
 999980 
 999979 
 99997 
 999976 
 999975 
 999973 
 999972 
 999971 
 999969 
 999968 
 999966 
 999964 
 9i)9;)63 
 999961 
 999959 
 999958 
 999956 
 999954 
 .999952 
 999950 
 999948 
 999946 
 999944 
 999942 
 999940 
 999938 
 999936 
 999934 
 
 9 
 
 .00 
 
 .00 
 
 .00 
 
 .00 
 
 .00 
 
 .01 
 
 .01 
 
 .01 
 
 .01 
 
 .01 
 
 .01 
 
 .01 
 
 .01 
 
 .01 
 
 .01 
 
 .01 
 
 .01 
 
 .01 
 
 .01 
 
 .01 
 
 .01 
 
 .01 
 
 .01 
 
 .02 
 
 .02 
 
 .02 
 
 .02 
 
 .02 
 
 .02 
 
 .02 
 
 .02 
 
 .02 
 
 .02 
 
 .02 
 
 .02 
 
 .02 
 
 .02 
 
 .02 
 
 .02 
 
 .02 
 
 .02 
 
 .02 
 
 .02 
 
 .03 
 
 .03 
 
 .03 
 
 .03 
 
 .03 
 
 .03 
 
 .03 
 
 .03 
 
 .03 
 
 .03 
 
 .03 
 
 .03 
 
 .04 
 
 .04 
 
 .04 
 
 .04 
 
 .01 
 
 JPang. 
 
 0.000()(MJ 
 6.463726 
 764756 
 94(»H47 
 7.065786 
 162696 
 241878 
 808825 
 866817 
 417970 
 463727 
 7.505120 
 542909 
 577672 
 609857 
 639820 
 667849 
 694179 
 719004 
 742484 
 764761 
 7.785951 
 800155 
 825460 
 843944 
 861674 
 878708 
 895099 
 910894 
 926134 
 940858 
 7.955100 
 968889 
 982253 
 995219 
 8.007809 
 020045 
 031945 
 043527 
 054809 
 065806 
 8.076531 
 086997 
 097217 
 107202 
 116963 
 126510 
 135851 
 144996 
 153952 
 162727 
 8.17i;5'2S 
 1797(i3 
 188030 
 19615(1 
 204126 
 211953 
 219641 
 227195 
 234()21 
 2Un!?l 
 
 ^ I Cotan g. I 
 
 5017.17 
 29.34.83 
 2082.31 
 
 Iiitinite 
 13.536274: 
 23.5244 
 0591531 
 
 1615.17:12.9.342141 
 
 1319.69 
 1115.78 
 996.53 
 852.54 
 762.63 
 689.88 
 629.81 
 579.33 
 536.42 
 499.39 
 467.15 
 438.82 
 413.73 
 391.36 
 371.28 
 851.36 
 336.73 
 821.76 
 808.06 
 295.49 
 283.90 
 273.18 
 263.25 
 254.01 
 245.40 
 2.37.35 
 229.81 
 222.75 
 216.10 
 209.83 
 
 837.304 
 758122 
 691175 
 6.33183 
 582030; 
 5.36273 
 12.4948801 
 457091. 
 422328! 
 890143! 
 360180 
 332151 
 305821 
 280997 
 257516 
 235239 
 12.214049 
 193845 
 174.540 
 15605(: 
 138326 
 121292 
 104901 
 089106 
 073866 
 0.59142 
 12.044900 
 031111 
 017747 
 
 (;o 
 
 59 
 
 58 
 
 57 
 
 5<5 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 40 
 
 39 
 
 88 
 
 87 
 
 36 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 29 
 
 28 
 
 27 
 
 004781 26 
 
 203.92 11.992191; 
 
 198.33 
 193.05 
 188.03 
 183.27 
 178.74 
 174.44 
 170.34 
 166.42 
 162.68 
 159.10 
 155.68 
 152.41 
 149.27 
 146.27 
 143 36 
 
 979955 
 968055 
 956473 
 945191 
 934194 
 11.923469 
 913003 
 902783 
 892797 
 883037 
 873490 
 864149 
 855004 
 8460481 
 837273' 
 
 110..57,11.82H672i 
 
 137.90 
 135.32 
 1.32.84 
 130,44 
 128.14 
 125.90 
 123.76 
 121.68 
 
 ii;t.n7 
 
 820237 
 811964 
 803844 
 
 7958741 
 788047 
 7803.59 
 772805 
 765379 
 758079 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 
 
 Sine. 1 89 | Cotang-. D. i Targ. i M. 
 
n 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 85 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 29 
 
 28 
 
 27 
 
 20 
 
 ;)l) 
 
 24 
 
 55 23 1 
 
 73 
 
 22 
 
 )1 
 
 21 
 
 )4 
 
 20 
 
 )9 19 
 
 )3 18 
 
 VS 17 
 
 )7 
 
 16 1 
 
 
 
 14 
 
 9 
 
 13 
 
 4 
 
 12 
 
 :H 
 
 11 
 
 3 
 
 10 
 
 2 
 
 9 
 
 ' 
 
 8 
 
 . 
 
 7 
 
 1 
 
 6 
 
 1 
 
 i 5 
 
 7 
 
 4 
 
 ) 
 
 I 3 
 
 ) 
 
 2 
 
 
 1 
 
 
 
 
 M. 
 
 
 
 BINES 
 
 AND TANGFNTS 
 
 . (1 DEOKKE ) 
 
 
 19 
 
 |M. 
 
 Sine. 
 
 D. 
 
 Cosine. 
 
 D. 
 
 Tang. 
 
 D. 
 
 Coung. 1 
 
 
 
 8.241855 
 
 119,63 
 
 9^999934 ' 
 
 .04 
 
 8.241921 
 
 119,67 
 
 11.758079' 60 
 
 1 
 
 249033 
 
 117.68 
 
 999932 
 
 .04 
 
 249102 
 
 117.72 
 
 750898 
 
 59 
 
 2 
 
 2500!)4 
 
 115.80 
 
 999929 
 
 .04 
 
 256165 
 
 115.84 
 
 743835 
 
 68 
 
 3 
 
 263042 
 
 113,98 
 
 999927 
 
 .04 
 
 263115 
 
 114.02 
 
 736885 
 
 57 
 
 4 
 
 2698H1 
 
 112,21 
 
 999925 
 
 .04 
 
 269956 
 
 112.25 
 
 730044 
 
 56 
 
 5 
 
 276614 
 
 110,50 
 
 999922 
 
 .04 
 
 276691 
 
 110.-54 
 
 723309 
 
 55 
 
 6 
 
 283243 
 
 108,83 
 
 999920 
 
 .04 
 
 283323 
 
 108.87 
 
 716677 
 
 64 
 
 7 
 
 289773 
 
 107.21 
 
 999918 
 
 .04 
 
 289856 
 
 107.26 
 
 710144 
 
 53 
 
 8 
 
 296207 
 
 105.65 
 
 999915 
 
 .04 
 
 296292 
 
 105.70 
 
 703708 
 
 52 
 
 9 
 
 302546 
 
 104,13 
 
 999913 
 
 .04 
 
 3C2634 
 
 104.18 
 
 697366 
 
 51 
 
 10 
 
 308794 
 
 102,66 
 
 999910 
 
 .04 
 
 308884 
 
 102.70 
 
 691116 
 
 50 
 
 11 
 
 8.314904 
 
 101,22 
 
 9.999907 
 
 .04 
 
 8.315046 
 
 101.26 
 
 11.684954 
 
 49 
 
 12 
 
 321027 
 
 99, S2 
 
 999905 
 
 .04 
 
 321122 
 
 99.87 
 
 678878 
 
 48 
 
 13 
 
 327016 
 
 98,47 
 
 999902 
 
 .04 
 
 327114 
 
 98.51 
 
 672886 
 
 47 
 
 14 
 
 332924 
 
 97.14 
 
 999899 
 
 .05 
 
 333025 
 
 97.19 
 
 666975 
 
 46 
 
 15 
 
 338753 
 
 95,86 
 
 999897 
 
 .05 
 
 338856 
 
 95.90 
 
 661144 
 
 45 
 
 16 
 
 344504 
 
 94,60 
 
 999894 
 
 .05 
 
 844610 
 
 94.65 
 
 055300 
 
 44 
 
 17 
 
 850181 
 
 93,38 
 
 999891 
 
 .05 
 
 350289 
 
 93.43 
 
 649711 
 
 43 
 
 18 
 
 3557H3 
 
 92.19 
 
 999888 
 
 .05 
 
 855895 
 
 92.24 
 
 644105 
 
 42 
 
 19 
 
 361315 
 
 91,03 
 
 999885 
 
 .05 
 
 361430 
 
 91.08 
 
 638570 
 
 41 
 
 20 
 
 36(1777 
 
 89.90 
 
 999882 
 
 .05 
 
 866895 
 
 89.95 
 
 633105 
 
 40 
 
 21 
 
 8.372171 
 
 88.80 
 
 9.999879 
 
 .05 
 
 8.372292 
 
 88.85 
 
 11.627708 
 
 39 
 
 22 
 
 377499 
 
 87.72 
 
 999876 
 
 .05 
 
 377622 
 
 87.77 
 
 622378 
 
 38 
 
 23 
 
 382762 
 
 86.67 
 
 999873 
 
 .05 
 
 882889 
 
 86.72 
 
 617111 
 
 87 
 
 24 
 
 387962 
 
 85,64 
 
 999870 
 
 .05 
 
 888092 
 
 85.70 
 
 611908 
 
 36 
 
 25 
 
 393101 
 
 84.64 
 
 999867 
 
 .05 
 
 893234 
 
 84.70 
 
 006766 
 
 35 
 
 26 
 
 398179 
 
 83.66 
 
 999864 
 
 .05 
 
 398315 
 
 83.71 
 
 601685 
 
 34 
 
 27 
 
 403199 
 
 82,71 
 
 999861 
 
 .05 
 
 403338 
 
 82.76 
 
 596662 
 
 33 
 
 28 
 
 408161 
 
 81.77 
 
 999858 
 
 .05 
 
 408304 
 
 81.82 
 
 591(;96 
 
 32 
 
 29 
 
 413068 
 
 80,86 
 
 999854 
 
 .05 
 
 413213 
 
 80.91 
 
 586787 
 
 31 
 
 30 
 
 417919 
 
 79.96 
 
 999851 
 
 .06 
 
 418068 
 
 80.02 
 
 581932 
 
 30 
 
 31 
 
 8.422717 
 
 79,09 
 
 9.999848 
 
 .06 
 
 8.422869 
 
 79,14 
 
 11.577131 
 
 29 
 
 32 
 
 427462 
 
 78,23 
 
 999844 
 
 .06 
 
 427618 
 
 78.30 
 
 572382 
 
 28 
 
 33 
 
 432156 
 
 77,40 
 
 999841 
 
 .06 
 
 432315 
 
 77.45 
 
 567685 
 
 27 
 
 34 
 
 436800 
 
 70.57 
 
 999838 
 
 .06 
 
 436962 
 
 76.63 
 
 563038 
 
 26 
 
 35 
 
 441394 
 
 75.77 
 
 999834 
 
 .06 
 
 441560 
 
 75.83 
 
 558440 
 
 25 
 
 36 
 
 445941 
 
 74,99 
 
 999831 
 
 .06 
 
 446110 
 
 75.05 
 
 553890 
 
 24 
 
 37 
 
 450440 
 
 74,22 
 
 999827 
 
 .06 
 
 450613 
 
 74.28 
 
 549387 
 
 23 
 
 38 
 
 454893 
 
 73,46 
 
 999823 
 
 .00 
 
 455070 
 
 73.52 
 
 544930 
 
 22 
 
 39 
 
 459301 
 
 72,73 
 
 999820 
 
 .00 
 
 459481 
 
 72,79 
 
 540519 
 
 21 
 
 40 
 
 463665 
 
 72,00 
 
 99981(i 
 
 .06 
 
 463849 
 
 72,06 
 
 536151 
 
 20 
 
 41 
 
 8.467985 
 
 71 . 29 
 
 9.999812 
 
 .06 
 
 8.468172 
 
 71,35 
 
 11.531828 
 
 19 
 
 42 
 
 472263 
 
 70.60 
 
 999H0!) 
 
 .06 
 
 472454 
 
 70.66 
 
 527546 
 
 18 
 
 43 
 
 476498 
 
 69,91 
 
 999805 
 
 .06 
 
 476693 
 
 69.98 
 
 523307 
 
 17 
 
 44 
 
 480693 
 
 69,24 
 
 999801 
 
 .06 
 
 4S0892 
 
 69.31 
 
 519108 
 
 16 
 
 45 
 
 484848 
 
 68.59 
 
 999797 
 
 .07 
 
 485050 
 
 68.65 
 
 5H:)50 
 
 15 
 
 1 46 
 
 488963 
 
 67.94 
 
 999793 
 
 .07 
 
 489170 
 
 68.01 
 
 ',JM.'1.0 
 
 14 
 
 !47 
 
 493040 
 
 67.31 
 
 999790 
 
 .07 
 
 493250 
 
 67.38 
 
 .1 7V» 
 
 13 
 
 148 
 
 497078 
 
 66.69 
 
 999780 
 
 .07 
 
 4!)7293 
 
 66.76 
 
 50i707 
 
 12 
 
 ! 49 
 
 501080 
 
 66.08 
 
 999782 
 
 .07 
 
 501298 
 
 66.15 
 
 498702 
 
 11 
 
 ' 50 
 
 505045 
 
 65.48 
 
 999778 
 
 .07 
 
 505207 
 
 65,55 
 
 494733 
 
 10 
 
 1 51 
 
 8.508974 
 
 64.89 
 
 9.999774 
 
 .07 
 
 8.509200 
 
 64,96 
 
 11.490SO(» 
 
 9 
 
 52 
 
 512867 
 
 64.31 
 
 999769 
 
 .07 
 
 513098 
 
 64.39 
 
 48(;9()2 
 
 8 
 
 53 
 
 616726 
 
 63.75 
 
 999765 
 
 .07 
 
 516961 
 
 63,82 
 
 483f»39 
 
 7 
 
 54 
 
 620551 
 
 63.19 
 
 999761 
 
 .07 
 
 520790 
 
 63,26 
 
 479210 
 
 6 
 
 55 
 
 624343 
 
 62.64 
 
 ' 999757 
 
 .07 
 
 524586 
 
 62,72 
 
 475414 
 
 5 
 
 56 
 
 628102 
 
 62.11 
 
 999753 
 
 .07 
 
 528349 
 
 62.18 
 
 471651 
 
 4 
 
 57 
 
 631828 
 
 61.58 
 
 999748 
 
 .07 
 
 532080 
 
 61.65 
 
 467920 
 
 3 
 
 58 
 
 636523 
 
 61.06 
 
 999744 
 
 .07 
 
 535779 
 
 61.13 
 
 404221 
 
 2 
 
 59 
 
 639186 
 
 60.55 
 
 999740 
 
 .07 
 
 531447 
 
 60.62 
 
 460553 
 
 1 
 
 60 
 
 542019 
 
 . 00 04 
 D. 
 
 999735 
 Sine. 
 
 ,07 
 
 543(!84 
 Cotanr. 
 
 <i0 12 
 
 456916 
 Tang. 
 
 
 M. 
 
 Cosine. 
 
 • 1 
 
 V t 
 
 ' ' ill 
 
 :|i 
 
 mm 
 
20 
 
 
 (2 DEC 
 
 UKK8.; A 
 
 TAIII.K OK LOOAItlTHMIO 
 
 
 
 
 Sine. 
 
 60.04 
 
 Cosine. 
 
 9.999735 
 
 D. 
 
 .07 
 
 Tang. 
 
 8.5430H4 
 
 60.12 
 
 Cotang. ' 1 
 
 11.45G916 GO 
 
 8.{>t2819 
 
 1 
 
 M(m2 
 
 59.55 
 
 999731 
 
 .07 
 
 646G!'l 
 
 59.62 
 
 453309 
 
 59 
 
 2 
 
 64!)995 
 
 59, OG 
 
 99972G 
 
 .07 
 
 5502(;S 
 
 59.14 
 
 449732 
 
 68 
 
 3 
 
 55353!) 
 
 58.58 
 
 999722 
 
 .08 
 
 65.3817 
 
 68. GG 
 
 446183 
 
 57 
 
 4 
 
 557054 
 
 58.11 
 
 999717 
 
 .08 
 
 55733G 
 
 68.19 
 
 4426G4 
 
 56 
 
 6 
 
 500540 
 
 57.65 
 
 999713 
 
 .08 
 
 5G0828 
 
 57.73 
 
 4.39172 
 
 55 
 
 6 
 
 5G3999 
 
 57.19 
 
 999708 
 
 .08 
 
 664291 
 
 57.27 
 
 435709 
 
 54 
 
 7 
 
 567431 
 
 56.74 
 
 999704 
 
 .08 
 
 6G7727 
 
 56.82 
 
 432273 
 
 53 
 
 8 
 
 570H3(> 
 
 56.30 
 
 999699 
 
 .08 
 
 571187 
 
 56.38 
 
 4288G3 52 
 
 9 
 
 574214 
 
 55.87 
 
 999694 
 
 .08 
 
 674520 
 
 55.95 
 
 42.5480 ' 
 
 10 
 
 5775G(5 
 
 55.44 
 
 999G89 
 
 .08 
 
 577877 
 
 .55.52 
 
 422123 
 
 11 
 
 8.580892 
 
 55.02 
 
 9.999685 
 
 .08 
 
 8.581208 
 
 .55.10 
 
 11.418792 -xo 
 
 12 
 
 5841.93 
 
 54.60 
 
 999G80 
 
 .08 
 
 584514 
 
 54. G8 
 
 41-5480 48 
 
 13 
 
 5874G9 
 
 54.19 
 
 999G75 
 
 .08 
 
 587795 
 
 54.27 
 
 412205 47 
 
 U 
 
 590721 
 
 53.79 
 
 999G70 
 
 .08 
 
 591051 
 
 53,87 
 
 408949 46 
 
 15 
 
 59394H 
 
 53.39 
 
 999GG5 
 
 .08 
 
 594283 
 
 53.47 
 
 405717 
 
 45 
 
 IG 
 
 597152 
 
 53.00 
 
 999GG0 
 
 .08 
 
 597492 
 
 .53,08 
 
 402508 
 
 44 
 
 17 
 
 G00332 
 
 52.61 
 
 999655 
 
 .08 
 
 600677 
 
 52.70 
 
 3!»9323 
 
 43 
 
 18 
 
 G03489 
 
 52.23 
 
 999G50 
 
 .08 
 
 603839 
 
 52.32 
 
 39Gl(ir 42 
 
 19 
 
 606623 
 
 51.86 
 
 999645 
 
 .09 
 
 606978 
 
 51.94 
 
 8:);ju22' 41 
 
 20 
 
 G09734 
 
 51.49 
 
 999640 
 
 .09 
 
 J610094 
 
 51.58 
 
 88990G 40 
 
 21 
 
 8.G12823 
 
 51.12 
 
 9.999635 
 
 .09 
 
 8.613189 
 
 51.21 
 
 11.386811 39 
 
 22 
 
 615891 
 
 50.76 
 
 999629 
 
 .09 
 
 616262 
 
 50.85 
 
 383738 38 
 
 23 
 
 618937 
 
 50.41 
 
 999624 
 
 .09 
 
 619313 
 
 50.50 
 
 380687 37 
 
 2i 
 
 621962 
 
 50.06 
 
 999619 
 
 .09 
 
 622343 
 
 50.15 
 
 377G57 36 
 
 25 
 
 624965 
 
 49.72 
 
 999614 
 
 .09 
 
 625352 
 
 49.81 
 
 374648 35 
 
 26 
 
 627948 
 
 49.38 
 
 999608 
 
 .09 
 
 628340 
 
 49.47 
 
 371GG0 34 
 
 27 
 
 630911 
 
 49.04 
 
 999603 
 
 .09 
 
 631308 
 
 49.13 
 
 3G8G92 33 
 
 28 
 
 633854 
 
 48.71 
 
 999597 
 
 .09 
 
 63425G 
 
 48.80 
 
 3G5744' 32 
 
 29 
 
 63677G 
 
 48.39 
 
 999592 
 
 . ^9 
 
 637184 
 
 48.48 
 
 3G281G' ?' 
 
 30 
 
 639680 
 
 48.06 
 
 99958G 
 
 .09 
 
 640093 
 
 48. IG 
 
 359907 { 
 
 31 
 
 8.642563 
 
 47.75 
 
 9.999581 
 
 .09 
 
 8.642982 
 
 47.84 
 
 11.357018 !,„ 
 
 32 
 
 645428 
 
 47.43 
 
 999575 
 
 .09 
 
 645853 
 
 47.53 
 
 3ij4147 28 
 
 83 
 
 648274 
 
 47.12 
 
 999570 
 
 .09 
 
 648704 
 
 47.22 
 
 35129G 27 
 
 34 
 
 651102 
 
 40.82 
 
 999564 
 
 .09 
 
 651537 
 
 46.91 
 
 3484G3 26 
 
 86 
 
 653911 
 
 46.52 
 
 999558 
 
 .10 
 
 654352 
 
 46.61 
 
 345G4S 25 
 
 36 
 
 656702 
 
 46.22 
 
 999553 
 
 .10 
 
 657149 
 
 46.31 
 
 342851 24 
 
 37 
 
 659475 
 
 45.92 
 
 999547 
 
 .10 
 
 659928 
 
 46.02 
 
 340072 23 
 
 38 
 
 662230 
 
 45.63 
 
 999541 
 
 .10 
 
 662689 
 
 45.73 
 
 337311 
 
 22 
 
 39 
 
 664968 
 
 45.35 
 
 999535 
 
 .10 
 
 665433 
 
 45 44 
 
 834567 
 
 21 
 
 40 
 
 667689 
 
 45. OG 
 
 999529 
 
 .10 
 
 6G8160 
 
 45.26 
 
 331840 
 
 20 
 
 41 
 
 8.670393 
 
 44.79 
 
 9.999524 
 
 .10 
 
 8.G70870 
 
 44.88 
 
 11.3291.30 
 
 19 
 
 42 
 
 673080 
 
 44.51 
 
 999518 
 
 .10 
 
 673563 
 
 44.61 
 
 326437 
 
 18 
 
 43 
 
 675751 
 
 44.24 
 
 999512 
 
 .10 
 
 676239 
 
 44.34 
 
 3237611 17 1 
 
 44 
 
 678405 
 
 43.97 
 
 99950G 
 
 .10 
 
 678900 
 
 44.17 
 
 321100 
 
 16 
 
 45 
 
 681043 
 
 43.70 
 
 999500 
 
 .10 
 
 681544 
 
 43.80 
 
 318456 
 
 15 
 
 46 
 
 683665 
 
 43.44 
 
 999493 
 
 .10 
 
 684172 
 
 43.54 
 
 315828' 14 1 
 
 47 
 
 686272 
 
 43.18 
 
 999487 
 
 .10 
 
 686784 
 
 43.28 
 
 313216 
 
 13 
 
 48 
 
 688863 
 
 42.92 
 
 999481 
 
 .10 
 
 689381 
 
 43.03 
 
 810619 
 
 12 
 
 49 
 
 691438 
 
 42.67 
 
 999475 
 
 .10 
 
 691963 
 
 42.77 
 
 308087 
 
 11 
 
 50 
 
 693998 
 
 42.42 
 
 999469 
 
 .10 
 
 694529 
 
 42.52 
 
 305471 
 
 10 
 
 51 
 
 8.696543 
 
 42.17 
 
 9.999463 
 
 .11 
 
 8.697081 
 
 42.28 
 
 11.302919 
 
 9 
 
 52 
 
 699073 
 
 41.92 
 
 99945G 
 
 .11 
 
 699617 
 
 42.03 
 
 800383 
 
 8 
 
 53 
 
 701589 
 
 41.68 
 
 999450 
 
 .11 
 
 702139 
 
 41.79 
 
 297861 
 
 7 
 
 54 
 
 704090 
 
 41.44 
 
 999443 
 
 .11 
 
 704646 
 
 41.55 
 
 295354 
 
 6 
 
 55 
 
 706577 
 
 41.21 
 
 999437 
 
 .11 
 
 707140 
 
 41.32 
 
 292660 
 
 , 6 
 
 56 
 
 709049 
 
 40.97 
 
 999431 
 
 .11 
 
 709618 
 
 41.08 
 
 290362 
 2879][7 
 
 4 
 
 57 
 
 711507 
 
 40.74 
 
 999424 
 
 .11 
 
 712083 
 
 40.85 
 
 8 
 
 58 
 
 713952 
 
 40.51 
 
 999418 
 
 .11 
 
 714534 
 
 40.62 
 
 285465 
 
 2 
 
 59 
 
 716383 
 
 40.29 
 
 999411 
 
 .11 
 
 716972 
 
 40.40 
 
 2830$8 
 
 , 1 
 
 60 
 
 718800 
 
 40.06 
 
 999404 
 
 .11 
 
 719396 
 
 40.17 
 J). 
 
 280604 
 
 
 H. 
 
 Cosine. 
 
 D. 
 
 Sine, 
 
 A^l 
 
 Cotang. 
 
 Tang. 
 
28 
 27 
 26 
 
 ?2 
 
 23 
 
 ^ 
 
 22 
 
 >7 
 
 21 
 
 
 
 20 
 
 !0 
 
 19 
 
 J7 
 
 18 
 
 11 17 
 
 
 
 16 
 
 6 
 
 15 
 
 8 
 
 U 
 
 6 
 
 13 
 
 9 
 
 12 
 
 7 
 
 11 
 
 1 
 
 10 
 
 £ 
 
 9 
 
 a 
 
 8 
 
 7 
 G 
 5 
 4 
 
 i 1 
 _ 
 
 M. 
 
 
 
 8INES i 
 
 KSn TANGPNT8 
 
 (3 DEGRKES.) 
 
 
 21 
 
 M. 
 
 
 
 Sine. 
 
 40.06 
 
 Cosine. 
 
 9.999404 
 
 D. 1 Tang. 
 
 D. 
 
 40.17 
 
 Cotang. 
 
 11.280604; 60 1 
 
 8.718H()0 
 
 .11 
 
 8.7193iN) 
 
 1 
 
 721204 
 
 39.84 
 
 99i)398 
 
 .11 
 
 721806 
 
 89.95 
 
 278194 
 
 59 
 
 2 
 
 723:)! 15 
 
 39.62 
 
 999391 
 
 .11 
 
 72420 J 
 
 89.74 
 
 2757!)(i 
 
 58 
 
 3 
 
 725972 
 
 39.41 
 
 9!)93H4 
 
 .11 
 
 7265HM 
 
 89.52 
 
 273412 
 
 57 
 
 4 
 
 728337 
 
 39.19 
 
 9!)9378 
 
 .11 
 
 728951 » 
 
 39.30 
 
 271041 
 
 56 
 
 5 
 
 7306HH 
 
 38.98 
 
 999371 
 
 .11 
 
 731317 
 
 89.09 
 
 2686S3 
 
 55 
 
 (] 
 
 733027 
 
 38.77 
 
 999364 
 
 .12 
 
 733663 
 
 88.89 
 
 266337 
 
 54 
 
 7 
 
 73r)3r)l 
 
 38.57 
 
 9!)i»357 
 
 .12 
 
 735996 
 
 38,68 
 
 264004 
 
 53 
 
 8 
 
 737607 
 
 38.30 
 
 999350 
 
 .12 
 
 73H317 
 
 38.48 
 
 2616S3 
 
 52 
 
 9 
 
 73990!) 
 
 38.16 
 
 99!)343 
 
 .12 
 
 740626 
 
 88.27 
 
 259374 
 
 51 
 
 10 
 
 742251) 
 
 37.!)6 
 
 999336 
 
 .12 
 
 742922 
 
 38.07 
 
 257078 
 
 50 
 
 11 
 
 8.744536 
 
 37.76 
 
 9.999329 
 
 .12 
 
 8.745207 
 
 37.87 
 
 11.254793 
 
 49 
 
 12 
 
 746802 
 
 37.56 
 
 999322 
 
 .12 
 
 747479 
 
 37.68 
 
 252521 
 
 48 
 
 13 
 
 749055 
 
 37. 37 
 
 999315 
 
 .12 
 
 749740 
 
 87.49 
 
 250260 
 
 47 
 
 14 
 
 751297 
 
 37.17 
 
 99930!i 
 
 .12 
 
 751989 
 
 87.29 
 
 248011 
 
 4(> 
 
 15 
 
 75352H 
 
 36.!»8 
 
 999301 
 
 .12 
 
 754227 
 
 37.10 
 
 245773 
 
 45 
 
 16 
 
 755747 
 
 36.79 
 
 9992!)4 
 
 .12 
 
 756453 
 
 36.92 
 
 243517 
 
 44 
 
 17 
 
 757955 
 
 86.61 
 
 999286 
 
 .12 
 
 758668 
 
 36.73 
 
 241332 
 
 43 
 
 18 
 
 760151 
 
 36.42 
 
 999279 
 
 .12 
 
 760872 
 
 86.55 
 
 239128 
 
 42 
 
 19 
 
 762337 
 
 36.24 
 
 999272 
 
 .12 
 
 7630(55 
 
 86.36 
 
 236935 
 
 41 
 
 20 
 
 764511 
 
 36.06 
 
 9!)9265 
 
 .12 
 
 765240 
 
 86.18 
 
 234754 
 
 40 
 
 21 
 
 8.766675 
 
 35.88 
 
 9.9!)9257 
 
 .I2i 8.767417 
 
 86.00 
 
 11.232583 
 
 39 
 
 22 
 
 768828 
 
 35.70 
 
 9!)9250 
 
 .13 
 
 76957H 
 
 a5.83 
 
 230422 
 
 88 
 
 23 
 
 770970 
 
 35.53 
 
 999242 
 
 .13 
 
 771727 
 
 35.65 
 
 228273 
 
 87 
 
 24 
 
 773101 
 
 35.35 
 
 9!)9235 
 
 .13 
 
 77;!S(!6 
 
 85.48 
 
 226134 
 
 36 
 
 25 
 
 775223 
 
 35.18 
 
 999227 
 
 .13 
 
 77^-,,)!)5 
 
 35 31 
 
 224005 
 
 85 
 
 26 
 
 777333 
 
 35.01 
 
 9!)9220 
 
 .13 
 
 77H114 
 
 35.14 
 
 221886 
 
 a4 
 
 27 
 
 775)4:J4 
 
 34.84 
 
 999212 
 
 .13 
 
 780222 
 
 34.97 
 
 21977W 
 
 33 
 
 28 
 
 781524 
 
 34.67 
 
 999205 
 
 .13 
 
 782320 
 
 34.80 
 
 217680 
 
 32 
 
 29 
 
 783605 
 
 34.51 
 
 999197 
 
 .13 
 
 78440.S 
 
 34.64 
 
 215592 
 
 31 
 
 30 
 
 785675 
 
 34.31 
 
 999189 
 
 .13 
 
 78648G 
 
 84.47 
 
 213514 
 
 30 
 
 31 
 
 8.787736 
 
 34.18 
 
 9.999181 
 
 .131 8 7885,5-i 
 
 34.31 
 
 11.211446 
 
 29 
 
 32 
 
 789787 
 
 34.02 
 
 999174 
 
 .13 
 
 790613 
 
 34.15 
 
 209387 
 
 28 
 
 33 
 
 791 82S 
 
 33.86 
 
 99916() 
 
 .13 
 
 792662 
 
 33.99 
 
 207338 
 
 27 
 
 34 
 
 79385!) 
 
 33.70 
 
 99915H 
 
 .13 
 
 794701 
 
 33.83 
 
 20526!) 
 
 26 
 
 35 
 
 795881 
 
 33.54 
 
 99S)150 
 
 .13 
 
 796731 
 
 33.68 
 
 203299 
 
 25 
 
 36 
 
 797894 
 
 33.39 
 
 999142 
 
 .13 
 
 798752 
 
 33.52 
 
 201248 
 
 24 
 
 37 
 
 799897 
 
 33.23 
 
 999134 
 
 .13 
 
 800763 
 
 33 37 
 
 199237 
 
 23 
 
 38 
 
 801892 
 
 33.08 
 
 999126 
 
 .13 
 
 802765 
 
 33.22 
 
 197235 
 
 22 
 
 39 
 
 803876 
 
 32.93 
 
 9!i9118 
 
 .13 
 
 804758 
 
 33.07 
 
 195242 
 
 21 
 
 40 
 
 805852 
 
 32.78 
 
 999110 
 
 .13 
 
 806742 
 
 32.92 
 
 193258 
 
 20 
 
 41 
 
 8.807819 
 
 32.63 
 
 9.999102 
 
 .13 
 
 8.808717 
 
 32.78 
 
 11.191283 
 
 19 
 
 42 
 
 809777 
 
 32.49 
 
 999094 
 
 .14 
 
 810683 
 
 32.62 
 
 189317 
 
 18 
 
 43 
 
 811726 
 
 32.34 
 
 999086 
 
 .14 
 
 812641 
 
 82.48 
 
 187359 
 
 17 
 
 44 
 
 813667 
 
 32.19 
 
 999077 
 
 .14 
 
 814589 
 
 82.33 
 
 185411 
 
 16 
 
 45 
 
 815599 
 
 32.05 
 
 999069 
 
 .14 
 
 816529 
 
 82.19 
 
 183471 
 
 15 
 
 46 
 
 817522 
 
 31.91 
 
 999061 
 
 .14 
 
 818461 
 
 32.05 
 
 181.53;) 
 
 14 
 
 47 
 
 819436 
 
 31.77 
 
 999053 
 
 .14 
 
 820384 
 
 31.91 
 
 17961(i 
 
 13 
 
 48 
 
 821343 
 
 31.63 
 
 999044 
 
 .14 
 
 82229H 
 
 31.77 
 
 177702 
 
 12 
 
 49 
 
 823240 
 
 31.49 
 
 999036 
 
 .14 
 
 824205 
 
 31.63 
 
 175795 
 
 11 
 
 50 
 
 825130 
 
 31.35 
 
 999027 
 
 .14 
 
 826103 
 
 31.50 
 
 173897 
 
 10 
 
 51 
 
 8.827011 
 
 31.22 
 
 9.999019 
 
 .14 
 
 8.827992 
 
 31.36 
 
 11.172008 
 
 9 
 
 52 
 
 828884 
 
 31.08 
 
 999010 
 
 .14 
 
 829874 
 
 31.23 
 
 170126 
 
 8 
 
 53 
 
 830749 
 
 30.95 
 
 999002 
 
 .14 
 
 831748 
 
 31.10 
 
 168252 
 
 7 
 
 54 
 
 832607 
 
 30.82 
 
 998993 
 
 .14 
 
 833613 
 
 30.96 
 
 166387 
 
 6 
 
 55 
 
 834456 
 
 30.69 
 
 998984 
 
 .14 
 
 835471 
 
 30.83 
 
 164529 
 
 5 
 
 66 
 
 836297 
 
 80.56 
 
 9989^6 
 
 .14 
 
 837321 
 
 80.70 
 
 162079 
 
 4 
 
 57 
 
 838130 
 
 30.43 
 
 998967 
 
 .15 
 
 839163 
 
 80.57 
 
 160837 
 
 3 
 
 68 
 
 839956 
 
 30.30 
 
 998958 
 
 .15 
 
 840998 
 
 80.45 
 
 159002 
 
 2 
 
 59 
 
 841774 
 
 30.17 
 
 998950 
 
 .15 
 
 842825 
 
 30.32 
 
 157175 
 
 1 
 
 60 
 
 843585 
 
 80.00 
 
 998941 
 
 .15 
 
 844644 
 
 80.19 
 
 155356 
 
 
 M. 
 
 
 Cosine. 
 
 D. 
 
 Sine. 
 
 86' 
 
 Gotang. 
 
 D. 
 
 Tang. 
 
 
 m 
 
 i 
 
 I'. 
 
 m 
 
 mm 
 
 {III 
 
 ,iS 
 
p^ 
 
 SH 
 
 22 
 
 
 (4 DBGREE8.) A TABLE OF LOGARITHMIC 
 
 
 U. 
 
 Sine. 
 
 3). 
 
 Cosine. 
 
 ID. 
 
 1 Tang. 
 
 D. 
 
 Cotang. 
 
 1 "" 
 
 
 
 8.848585 
 
 80.05 
 
 9.998941 
 
 .15 
 
 8.844644 
 
 80.19 
 
 11.155356 
 
 60 
 
 1 
 
 845387 
 
 29.92 
 
 998932 
 
 .15 
 
 846455 
 
 80.07 
 
 153545 
 
 59 
 
 2 
 
 847183 
 
 29.80 
 
 998923 
 
 .15 
 
 848260 
 
 29.95 
 
 151740 
 
 68 
 
 3 
 
 848971 
 
 29.67 
 
 998914 
 
 .15 
 
 850057 
 
 29.82 
 
 149943 
 
 57 
 
 4 
 
 850751 
 
 29.55 
 
 998905 
 
 .15 
 
 851846 
 
 29.70 
 
 148154 
 
 56 
 
 5 
 
 852525 
 
 29.43 
 
 998896 
 
 .15 
 
 853628 
 
 29.58 
 
 146372 
 
 55 
 
 6 
 
 854291 
 
 29.81 
 
 998887 
 
 .15 
 
 855403 
 
 29.46 
 
 144597 
 
 64 
 
 7 
 
 856049 
 
 29.19 
 
 998878 
 
 .15 
 
 857171 
 
 29.35 
 
 142829 
 
 53 
 
 8 
 
 857801 
 
 29.07 
 
 998869 
 
 .15 
 
 858932 
 
 29.23 
 
 141008 
 
 52 
 
 9 
 
 859546 
 
 28.96 
 
 998860 
 
 .15 
 
 86068G 
 
 29.11 
 
 139314 
 
 61 
 
 10 
 
 861283 
 
 28.84 
 
 998851 
 
 .15 
 
 862433 
 
 23 00 
 
 137567 
 
 60 
 
 11 
 
 8.863014 
 
 28.73 
 
 9.998841 
 
 .15 
 
 8.864173 
 
 28.88 
 
 11.135827 
 
 43 
 
 12 
 
 864738 
 
 28.61 
 
 998832 
 
 .15 
 
 865906 
 
 28.77 
 
 134094 
 
 48 
 
 13 
 
 866455 
 
 28.50 
 
 998823 
 
 .16 
 
 867632 
 
 28,66 
 
 132308 
 
 47 
 
 U 
 
 868165 
 
 28.89 
 
 998813 
 
 .16 
 
 869*51 
 
 28.54 
 
 130649 
 
 46 
 
 15 
 
 869868 
 
 28.28 
 
 998804 
 
 .16 
 
 871064 
 
 38.43 
 
 128936 
 
 45 
 
 16 
 
 871565 
 
 28.17 
 
 998795 
 
 .16 
 
 872770 
 
 28.82 
 
 127230 
 
 44 
 
 17 
 
 873255 
 
 28.06 
 
 998785 
 
 .16 
 
 874469 
 
 28. 2i 
 
 125531 
 
 43 
 
 18 
 
 874938 
 
 27.95 
 
 998776 
 
 .16 
 
 876162 
 
 28.11 
 
 123838 
 
 42 
 
 19 
 
 876615 
 
 27.86 
 
 998766 
 
 .16 
 
 877849 
 
 28.00 
 
 122151 
 
 41 
 
 20 
 
 878285 
 
 27.73 
 
 998757 
 
 .16 
 
 879529 
 
 27.89 
 
 120471 
 
 40 
 
 21 
 
 8.879949 
 
 27.63 
 
 9.998747 
 
 .16 
 
 8.881202 
 
 27.79 
 
 11.118798 
 
 39 
 
 22 
 
 881607 
 
 n7.52 
 
 998738 
 
 .16 
 
 882869 
 
 27.68 
 
 117131 
 
 38 
 
 23 
 
 883258 
 
 27.42 
 
 998728 
 
 .16 
 
 884580 
 
 27.58 
 
 115470 
 
 37 
 
 24 
 
 884903 
 
 27.81 
 
 998718 
 
 .16 
 
 886185 
 
 37.4,7 
 
 113815 
 
 33 
 
 25 
 
 886542 
 
 27.21 
 
 998708 
 
 .16 
 
 887833 
 
 27.37 
 
 11.2107 
 
 35 
 
 26 
 
 888174 
 
 27.11 
 
 998699 
 
 .16 
 
 889476 
 
 27.27 
 
 110524 
 
 34 
 
 27 
 
 889801 
 
 27.00 
 
 998689 
 
 .16 
 
 891112 
 
 27.17 
 
 108888 
 
 33 
 
 23 
 
 6&1421 
 
 26.90 
 
 998679 
 
 .16 
 
 892742 
 
 27.07 
 
 107258 
 
 32 
 
 29 
 
 893C35 
 
 26.80 
 
 998669 
 
 .17 
 
 894366 
 
 20.97 
 
 105634 
 
 31 
 
 30 
 
 894643 
 
 26.70 
 
 998659 
 
 .17 
 
 895984 
 
 26.87 
 
 104016) 30 t 
 
 31 
 
 3.896246 
 
 26.60 
 
 9.998649 
 
 .17 
 
 8.897596 
 
 26.77 
 
 11.102404 
 
 29 
 
 32 
 
 897842 
 
 26.51 
 
 998639 
 
 ,17 
 
 899203 
 
 26.67 
 
 100797 
 
 28 
 
 33 
 
 899432 
 
 26 41 
 
 998029 
 
 17 
 
 900803 
 
 26.58 
 
 099197 
 
 27 
 
 34 
 
 901017 
 
 26.31 
 
 998619 
 
 .17 
 
 902398 
 
 26.48 
 
 097602 
 
 26 
 
 35 
 
 902596 
 
 26.22 
 
 998609 
 
 .17 
 
 903987 
 
 26.38 
 
 096013 
 
 25 
 
 36 
 
 904169 
 
 26.12 
 
 998599 
 
 .17 
 
 905570 
 
 26.29 
 
 094430 
 
 24 
 
 37 
 
 90573(; 
 
 26.03 
 
 998589 
 
 .17 
 
 907147 
 
 26.20 
 
 092853 
 
 23 
 
 38 
 
 907297 
 
 25.93 
 
 998578 
 
 .17 
 
 908719 
 
 26,10 
 
 091281 
 
 22 
 
 39 
 
 908853 
 
 25.84 
 
 998568 
 
 .17 
 
 910285 
 
 26.01 
 
 089715 
 
 21 
 
 40 
 
 910404 
 
 25.75 
 
 998558 
 
 .17 
 
 911846 
 
 25,92 
 
 088154 
 
 20 
 
 41 
 
 8.911949 
 
 25.66 
 
 9 998548 
 
 .17 
 
 8.913401 
 
 25.83 
 
 11.080599 
 
 19 
 
 42 
 
 913488 
 
 25.56 
 
 998537 
 
 .17 
 
 914951 
 
 25.74 
 
 085049 
 
 18 
 
 43 
 
 915022 
 
 25.47 
 
 99852/ 
 
 .17 
 
 916495 
 
 25.65 
 
 083505 
 
 17 
 
 44 
 
 916550 
 
 25.38 
 
 998516 
 
 .18 
 
 918034 
 
 25.56 
 
 081966 
 
 16 
 
 45 
 
 918073 
 
 25.29 
 
 998506 
 
 .18 
 
 919568 
 
 25.47 
 
 080432 
 
 15 
 
 46 
 
 919591 
 
 25 20 
 
 998495 
 
 .18 
 
 921096 
 
 25.38 
 
 078904 
 
 14 
 
 47 
 
 921103 
 
 25.12 
 
 998485 
 
 .18 
 
 922619 
 
 25.30 
 
 077381 
 
 13 
 
 48 
 
 922610 
 
 25.93 
 
 998474 
 
 .18 
 
 924136 
 
 25.21 
 
 075864 
 
 12 
 
 49 
 
 924112 
 
 24.94 
 
 998464 
 
 .18 
 
 925649 
 
 25.12 
 
 074351 
 
 11 
 
 50 
 
 92o60!> 
 
 24.86 
 
 998453 
 
 .18 
 
 927156 
 
 25.03 
 
 072844 
 
 10 
 
 51 
 
 S. 927100 
 
 24.77 
 
 9.998442 
 
 .18 
 
 S. 928658 
 
 24.95 
 
 11.071342 
 
 9 
 
 52 
 
 928587 
 
 24.69 
 
 998131 
 
 .38 
 
 930155 
 
 24.86 
 
 069845 
 
 8 
 
 53 
 
 93006H 
 
 24.60 
 
 998421 
 
 .18 
 
 931647 
 
 24.78 
 
 068353 
 
 7 
 
 54 
 
 931544 
 
 24.52 
 
 998410 
 
 .18 
 
 933134 
 
 24.70 
 
 066866 
 
 6 
 
 65 
 
 933015 
 
 24.4? 
 
 998399 
 
 .18 
 
 634616 
 
 24.61 
 
 065384 
 
 5 
 
 66 
 
 934481 
 
 24.8^^ 
 
 998388 
 
 .18 
 
 930093 
 
 24.53 
 
 063907 
 
 4 
 
 57 
 
 935942 
 
 24.27 
 
 998377 
 
 .18 
 
 937565 
 
 24.45 
 
 062435 
 
 3 
 
 58 
 
 937398 
 
 94.19 
 
 998366 
 
 .18 
 
 939032 
 
 24.37 
 
 060968 
 
 2 
 
 59 
 
 938850 
 
 24.11 
 
 998335 
 
 .18 
 
 940494 
 
 24.30 
 
 059506 
 
 1 
 
 80 
 
 94C296 
 Cosine, i 
 
 21.03 
 
 99834-i 
 Sine. 
 
 .18 
 85 
 
 941952 
 Cotang. 
 
 24,21 
 D. 1 
 
 058048 
 
 
 
 Tang. M. j 
 
3 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 50 
 
 13 
 
 48 
 
 47 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 40 
 39 
 3d 
 37 
 36 
 35 
 34 
 33 
 32 
 31 
 l6i 30 
 29 
 
 28 
 27 
 26 
 25 
 24 
 23 I 
 22 
 21 
 20 
 19 
 18 
 17 
 16 
 15 
 14 
 13 
 12 
 11 
 10 
 9 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 
 M. 
 
 
 
 SINKS . 
 
 V.Vi) TANGENTS. 
 
 (5 DEGREES.) 
 
 
 23 
 
 ;m. 
 
 Sine. 
 
 8.!»402;HJ 
 
 D. 
 
 Cosine. 
 
 9.998344 
 
 P. 
 
 .19 
 
 Tang. 
 
 D. 
 
 Cocang. 
 
 11.05S048| 60 
 
 
 
 24.03 
 
 8.941952 
 
 24.21 
 
 1 
 
 94173S 
 
 23.94 
 
 998333 
 
 .19 
 
 943404 
 
 24.13 
 
 0.56.596! .59 
 
 2 
 
 943174 
 
 23.87 
 
 998322 
 
 .19 
 
 944852 
 
 24.05 
 
 0551481 58 
 
 3 
 
 944606 
 
 23.79 
 
 998311 
 
 .19 
 
 946295 
 
 23.97 
 
 0.>3705 
 
 57 
 
 4 
 
 946034 
 
 23.71 
 
 998300 
 
 .19 
 
 947734 
 
 23.90 
 
 052266 
 
 56 
 
 5 
 
 947456 
 
 23.63 
 
 998289 
 
 .19 
 
 949168 
 
 23.82 
 
 050832 
 
 55 
 
 6 
 
 948874 
 
 23.55 
 
 998277 
 
 .19 
 
 950597 
 
 23.74 
 
 049403 
 
 54 
 
 7 
 
 950287 
 
 23.48 
 
 9982()() 
 
 .19 
 
 952021 
 
 23.66 
 
 047979 
 
 53 
 
 8 
 
 951696 
 
 23.40 
 
 998255 
 
 .19 
 
 953441 
 
 23,60 
 
 046559 
 
 52 
 
 9 
 
 953100 
 
 23 32 
 
 998243 
 
 .19 
 
 9.')48.)6 
 
 23.51 
 
 045144 51 1 
 
 10 
 
 954499 
 
 23.25 
 
 998232 
 
 .19 
 
 9562(i7 
 
 23.44 
 
 043733' 50 
 
 11 
 
 8.955Si)4 
 
 23.17 
 
 9 998220 
 
 .19 
 
 8.957674 
 
 23.37 
 
 11.042826 49 
 
 12 
 
 957284 
 
 23,10 
 
 998209 
 
 .19 
 
 959075 
 
 23,29 
 
 040925 
 
 48 
 
 13 
 
 958670 
 
 23.02 
 
 998197 
 
 .19 
 
 960473 
 
 23.23 
 
 039.527 
 
 47 
 
 14 
 
 SC0052 
 
 22.95 
 
 998186 
 
 .19 
 
 961866 
 
 23.14 
 
 038134! 46 
 
 15 
 
 961429 
 
 22 88 
 
 998174 
 
 .19 
 
 9632.55 
 
 23.07 
 
 036745! 45 
 
 16 
 
 962801 
 
 22.80 
 
 998163 
 
 .19 
 
 964639 
 
 23 00 
 
 03.5361! 44 
 
 17 
 
 964170 
 
 22.73 
 
 998151 
 
 .19 
 
 966019 
 
 22.93 
 
 033981 i 43 
 
 18 
 
 9(55534 
 
 22.66 
 
 998139 
 
 .20 
 
 967394 
 
 22,86 
 
 03260(5 42 
 
 19 
 
 9()68i»3 
 
 22.59 
 
 998128 
 
 .20 
 
 968766 
 
 22.79 
 
 031234 41 
 
 20 
 
 968249 
 
 22.52 
 
 998116 
 
 .20 
 
 970133 
 
 22.71 
 
 0298(57 
 
 40 1 
 
 21 
 
 8.969600 
 
 22,44 
 
 9.998104 
 
 .20 
 
 8.971496 
 
 22.65 
 
 11.028504 
 
 39 
 
 2? 
 
 960947 
 
 22.38 
 
 998092 
 
 .20 
 
 972855 
 
 22. 57 
 
 027145 
 
 38 
 
 23 
 
 972289 
 
 22.31 
 
 998080 
 
 .20 
 
 974209 
 
 22.. 51 
 
 025791 
 
 37 
 
 24 
 
 973028 
 
 22.24 
 
 998068 
 
 .20 
 
 975560 
 
 22.44 
 
 024440 
 
 36 
 
 25 
 
 974962 
 
 22.17 
 
 998056 
 
 .20 
 
 976906 
 
 22,37 
 
 023094 
 
 35 
 
 26 
 
 976293 
 
 22,10 
 
 998044 
 
 .20 
 
 978248 
 
 22.30 
 
 021752 
 
 34 
 
 27 
 
 977619 
 
 22.03 
 
 998032 
 
 .20 
 
 979586 
 
 22 23 
 
 020414 
 
 33 
 
 28 
 
 978941 
 
 21.97 
 
 998020 
 
 .20 
 
 980921 
 
 22.17 
 
 019079 
 
 32 
 
 29 
 
 980259 
 
 21.90 
 
 998008 
 
 .20 
 
 982251 
 
 22.10 
 
 017749 
 
 31 
 
 30 
 
 981573 
 
 21.83 
 
 997996 
 
 .20 
 
 983.577 
 
 22.04 
 
 01(5423 
 
 30 
 
 31 
 
 8.982883 
 
 21.77 
 
 9.997985 
 
 .20 
 
 8.984899 
 
 21.97 
 
 11.015101 
 
 29 
 
 32 
 
 984189 
 
 31.70 
 
 997972 
 
 .20 
 
 986217 
 
 21 , 91 
 
 013783 
 
 28 
 
 33 
 
 985491 
 
 21.63 
 
 997959 
 
 .20 
 
 987532 
 
 21,84 
 
 012468 
 
 27 
 
 34 
 
 986789 
 
 21.57 
 
 997947 
 
 .20 
 
 988842 
 
 21,78 
 
 011158 
 
 26 
 
 35 
 
 988083 
 
 21.50 
 
 997935 
 
 .21 
 
 990149 
 
 21.71 
 
 009851 
 
 25 
 
 3fi 
 
 985)374 
 
 21.44 
 
 997922 
 
 .21 
 
 991451 
 
 21.65 
 
 008549 
 
 24 
 
 37 
 
 990()60 
 
 21.33 
 
 997910 
 
 .21 
 
 9i>2750 
 
 21.58 
 
 0072.50 
 
 23 
 
 38 
 
 991943 
 
 21.31 
 
 997897 
 
 .21 
 
 9J4045 
 
 21.. 52 
 
 005955 
 
 22 
 
 39 
 
 993222 
 
 31.25 
 
 997885 
 
 .21 
 
 995337 
 
 21.46 
 
 004663 
 
 21 
 
 40 
 
 994497 
 
 21.19 
 
 997872 
 
 .21 
 
 99(J624 
 
 21.10 
 
 003376; 20 
 
 41 
 
 8.9957()8 
 
 21.12 
 
 9.997860 
 
 .21 
 
 8.W)7908 
 
 21 , 34 
 
 11.0020921 19 
 
 42 
 
 997v.;]6 
 
 21.06 
 
 997847 
 
 .21 
 
 999188 
 
 21.27 
 
 0()0«12: i8 
 
 43 
 
 998299 
 
 21.00 
 
 997835 
 
 .21 
 
 9.000465 
 
 21.21 
 
 10.999.535 17 
 
 44 
 
 999560 
 
 20.94 
 
 997822 
 
 .21 
 
 001738 
 
 21.15 
 
 9982621 16 
 
 i5 
 
 9.000816 
 
 20.87 
 
 997809 
 
 .21 
 
 00.3007 
 
 21 09 
 
 99(5993 i 15 1 
 
 46 
 
 002069 
 
 20.82 
 
 997797 
 
 .21 
 
 004272 
 
 21 03 
 
 995728 14 
 
 47 
 
 003318 
 
 20.76 
 
 997784 
 
 .21 
 
 005534 
 
 20.97 
 
 9944(36' 13 
 
 48 
 
 0045()3 
 
 20.70 
 
 997771 
 
 .21 
 
 006792 
 
 20.91 
 
 9932081 12 
 
 49 
 
 005805 
 
 20.64 
 
 997758 
 
 .21 
 
 008047 
 
 20.85 
 
 9919,53 11 
 
 50 
 
 007044 
 
 20.58 
 
 997745 
 
 .21 
 
 009298 
 
 20.80 
 
 990702 10 
 
 51 
 
 9.008278 
 
 20.52 
 
 9.997732 
 
 .21 
 
 9.010546 
 
 20.74 
 
 10.9894:54 9 
 
 52 
 
 009510 
 
 20.46 
 
 997719 
 
 .21 
 
 011790 
 
 20.68 
 
 988210 8 
 
 53 
 
 010737 
 
 20.40 
 
 997706 
 
 .21 
 
 01.'5031 
 
 20,62 
 
 986969 7 
 
 54 
 
 011962 
 
 20.34 
 
 997693 
 
 .22 
 
 014268 
 
 20.. 56 
 
 985732 6 
 
 55 
 
 013182 
 
 20.29 
 
 997680 
 
 .22 
 
 015r)02 
 
 20.51 
 
 984498 5 
 
 56 
 
 014400 
 
 20.23 
 
 997667 
 
 .22 
 
 016732 
 
 20.45 
 
 983268 4 
 
 57 
 
 015613 
 
 20.17 
 
 997654 
 
 .22 
 
 017959 
 
 20.40 
 
 982041 3 
 
 58 
 
 016824 
 
 20.12 
 
 997641 
 
 .22 
 
 019183 
 
 20.33 
 
 980817! 2 
 
 59 
 
 018031 
 
 20.06 
 
 997628 
 
 .22 
 
 020403 
 
 20 28 
 
 979597! 1 
 
 60 
 
 019235 
 
 20,00 
 
 997614 
 
 ,22 
 
 021(>'J0 
 
 20 ':s\ 
 
 978380! 
 
 
 Coaine. 
 
 D. 
 
 Sine. 
 
 84^ 
 
 Cotang. 
 
 _ D.~ 
 
 Tang. 
 
 M. 
 
 % 
 
 
 ■ I 
 
 m' 
 
l! 
 
 k 
 
 24 
 
 
 (G DEGREES.) A 
 
 TAULB OP LOGARITHMIC 
 
 
 
 1 Sine. 
 
 J.U1'J;;35 
 
 D. 
 
 20.00 
 
 Cotiine. | D. 
 
 Taug. 
 
 9.021C20 
 
 D. 
 
 Cotang. 
 
 
 y. 997014 
 
 .22 
 
 20.33 
 
 10.978380 
 
 60 
 
 1 
 
 020-135 
 
 19.95 
 
 997601 
 
 .22 
 
 022834 
 
 20.17 
 
 97716G 
 
 59 
 
 2 
 
 021(132 
 
 19.89 
 
 0975SS 
 
 .22 
 
 024044 
 
 20.11 
 
 975956 
 
 58 
 
 3 
 
 022825 
 
 19.8-4 
 
 99757-4 
 
 .22 
 
 025251 
 
 20.06 
 
 974749 
 
 57 
 
 4 
 
 02401G 
 
 19.7! 
 
 9;)75G1 
 
 .22 
 
 026455 
 
 20.00 
 
 973545 
 
 5G 
 
 ■ 5 
 
 025203 
 
 19.73 
 
 997547 
 
 .22 
 
 027655 
 
 19.95 
 
 972345 
 
 55 
 
 G 
 
 026386 
 
 19.67 
 
 997534 
 
 .23 
 
 028852 
 
 19.90 
 
 971148 
 
 54 
 
 7 
 
 027507 
 
 19.62 
 
 997520 I 
 
 .23 
 
 03004G 
 
 19.85 
 
 9G9954 
 
 53 
 
 i « 
 
 028744 
 
 19.57 
 
 997507 
 
 .23 
 
 031237 
 
 19.79 
 
 968763 
 
 52 
 
 ! 9 
 
 029918 
 
 19 51 
 
 907493 
 
 .23 
 
 032425 
 
 19.74 
 
 967575 
 
 51 
 
 10 
 
 031089 
 
 19.47 
 
 997480 
 
 .23 
 
 033609 
 
 19.69 
 
 966391 
 
 50 
 
 11 
 
 9.032257 
 
 19.41 
 
 9.997460 
 
 .23 
 
 9 034791 
 
 19.64 
 
 10.9G5209 
 
 49 
 
 12 
 
 033421 
 
 19. 3 J 
 
 997452 
 
 .23 
 
 03590^ 
 
 19.58 
 
 9G4031 
 
 48 
 
 13 
 
 034582 
 
 19.30 
 
 99743!) 
 
 .23 
 
 037144 
 
 19.53 
 
 9G285G 
 
 47 
 
 14 
 
 035741 
 
 19.23 
 
 997425 
 
 .23 
 
 038316 
 
 19.48 
 
 961684 
 
 46 
 
 15 
 
 03689G 
 
 19 20 
 
 997411 
 
 .23 
 
 039485 
 
 19.43 
 
 9G0515 
 
 45 
 
 IG 
 
 03-i048 
 
 19.1 J 
 
 9973J7 
 
 .23 
 
 040G51 
 
 19.38 
 
 959349 
 
 44 
 
 17 
 
 033197 
 
 19 10 
 
 997383 
 
 .23 
 
 041813 
 
 19.33 
 
 958187 
 
 43 
 
 18 
 
 040342 
 
 19.05 
 
 9973GLt 
 
 .23 
 
 042973 
 
 19.28 
 
 95702V 
 
 42 
 
 10 
 
 0414S5 
 
 18 99 
 
 997355 
 
 .23 
 
 0:14130 
 
 19.23 
 
 955870 
 
 41 
 
 20 
 
 042625 
 
 18.94 
 
 997341 
 
 .23 
 
 045284 
 
 19.18 
 
 954716 
 
 40 
 
 21 
 
 9.043762 
 
 18.89 
 
 9.997327 
 
 .24 
 
 9.046434 
 
 19.13 
 
 10.9535GG 
 
 39 
 
 22 
 
 044895 
 
 18.84 
 
 997313 
 
 .24 
 
 047582 
 
 19.08 
 
 952418 
 
 38 
 
 23 
 
 046026 
 
 18.79 
 
 997299 
 
 .24 
 
 048727 
 
 19.03 
 
 951273 
 
 87 
 
 24 
 
 047154 
 
 18.75 
 
 997285 
 
 .24 
 
 049869 
 
 18.98 
 
 950131 
 
 36 
 
 25 
 
 048279 
 
 18 70 
 
 997271 
 
 .24 
 
 051008 
 
 18.93 
 
 948992 
 
 35 
 
 2G 
 
 049400 
 
 18.65 
 
 997257 
 
 .24 
 
 052144 
 
 18.89 
 
 947856 
 
 34 
 
 27 
 
 050519 
 
 18 60 
 
 997242 
 
 .24 
 
 05S277 
 
 18.84 
 
 946723 
 
 33 
 
 28 
 
 051635 
 
 18.55 
 
 997228 
 
 .24 
 
 054407 
 
 18.79 
 
 945593 
 
 32 
 
 29 
 
 052749 
 
 18.50 
 
 997214 
 
 .24 
 
 055535 
 
 18.74 
 
 944465 
 
 31 
 
 30 
 
 053859 
 
 18.45 
 
 997199 
 
 .24 
 
 056659 
 
 18.70 
 
 943341 
 
 30 
 
 31 
 
 9.054966 
 
 18.41 
 
 9.997185 
 
 .24 
 
 9.057781 
 
 18.65 
 
 10.942219 
 
 29 
 
 32 
 
 056071 
 
 18.36 
 
 997170 
 
 .?4 
 
 058900 
 
 18.69 
 
 941100 
 
 28 
 
 83 
 
 057172 
 
 18.31 
 
 997156 
 
 .f4 
 
 060016 
 
 18.55 
 
 939984 
 
 27 
 
 31 
 
 058271 
 
 18.27 
 
 997141 
 
 .24 
 
 061130 
 
 18.51 
 
 938870 
 
 26 
 
 35 
 
 059367 
 
 18.22 
 
 997127 
 
 .24 
 
 062240 
 
 18.46 
 
 937760 
 
 25 
 
 3() 
 
 060460 
 
 18.17 
 
 997112 
 
 .24 
 
 063348 
 
 18.42 
 
 936652 
 
 24 
 
 37 
 
 001551 
 
 18.13 
 
 997098 
 
 .24 
 
 U64453 
 
 18.37 
 
 935547 
 
 23 
 
 38 
 
 062639 
 
 18.08 
 
 997083 
 
 .25 
 
 065556 
 
 18.33 
 
 934444 
 
 22 
 
 39 
 
 063724 
 
 18.04 
 
 9970G8 
 
 .25 
 
 066655 
 
 18.28 
 
 933345 
 
 21 
 
 40 
 
 064806 
 
 17.99 
 
 997053 
 
 .25 
 
 067752 
 
 18.24 
 
 932248 
 
 20 
 
 41 
 
 9.065.-^^ 
 
 17.94 
 
 9.997039 
 
 .25 
 
 9.068846 
 
 18.19 
 
 10.931154 
 
 ly 
 
 42 
 
 066962 
 
 17.90 
 
 997024 
 
 .25 
 
 069938 
 
 18.15 
 
 930062 
 
 18 
 
 43 
 
 0680S6 
 
 17.80 
 
 997009 
 
 .25 
 
 071027 
 
 18.10 
 
 928973 
 
 17 
 
 44 
 
 069107 
 
 17.81 
 
 996994 
 
 .25 
 
 072113 
 
 18.06 
 
 927887 
 
 16 
 
 45 
 
 070176 
 
 17.77 
 
 996979 
 
 .25 
 
 073197 
 
 18.02 
 
 926803 
 
 15 
 
 46 
 
 071242 
 
 17.72 
 
 996964 
 
 .25 
 
 074278 
 
 17.97 
 
 925722 
 
 14 
 
 47 
 
 072306 
 
 17.68 
 
 996949 
 
 .25 
 
 075356 
 
 17.93 
 
 924644 
 
 13 
 
 48 
 
 073366 
 
 17.63 
 
 996934 
 
 .25 
 
 076432 
 
 17.89 
 
 923568 
 
 12 
 
 49 
 
 074424 
 
 17.59 
 
 996919 
 
 .25 
 
 077505 
 
 17.84 
 
 922495 
 
 11 
 
 50 
 
 075480 
 
 17.55 
 
 996904 
 
 .25 
 
 . 078576 
 
 17.80 
 
 921424 
 
 10 
 
 51 
 
 9.076583 
 
 17.50 
 
 9.996889 
 
 .25 
 
 9.079644 
 
 17.76 
 
 10.920356 
 
 9 
 
 52 
 
 077583 
 
 17.46 
 
 996874 
 
 .25 
 
 080710 
 
 17.72 
 
 919290 
 
 8 
 
 53 
 
 078631 
 
 17.42 
 
 996858 
 
 .25 
 
 081773 
 
 17.67 
 
 918227 
 
 7 
 
 54 
 
 079676 
 
 17.38 
 
 996843 
 
 .26 
 
 082833 
 
 17.63 
 
 917167 
 
 6 
 
 55 
 
 080719 
 
 17.38 
 
 996828 
 
 .25 
 
 083891 
 
 17.59 
 
 916109 
 
 6 
 
 56 
 
 081759 
 
 17.29 
 
 996812 
 
 .26 
 
 084947 
 
 17.55 
 
 915053 
 
 4 
 
 57 
 
 082797 
 
 17.25 
 
 996797 
 
 .26 
 
 086000 
 
 17.51 
 
 914000 
 
 8 
 
 58 
 
 083692 
 
 17.21 
 
 996782 
 
 .26 
 
 087050 
 
 17.47 
 
 912950 
 
 2 
 
 59 
 
 084864 
 
 17.17 
 
 996766 
 
 .26 
 
 088098 
 
 17.43 
 
 911902 
 
 1 
 
 60 
 
 085894 
 
 17.13 
 
 990751 
 
 .26 
 
 089144 
 
 17.38 
 
 910856 
 
 
 
 
 Cosine. 
 
 D. 
 
 Bine. 
 
 a- 
 
 Cotang. 
 
 ' D. 
 
 Tang. 
 
 ^^ 
 
56 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 41 
 1 43 
 
 42 
 
 41 
 
 40 
 
 89 
 
 38 
 
 87 
 
 86 
 
 85 
 
 84 
 
 33 
 
 32 
 
 81 
 
 80 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 24 
 
 23 
 
 22 
 
 211 
 
 20 
 
 ly 
 
 18 
 17 
 16 
 15 
 14 
 13 
 12 
 11 
 10 
 9 
 8 
 7 
 6 
 5 
 4 
 8 
 2 
 1 
 
 
 
 
 SINES AND TANGENTS. 
 
 (7 DEGREES.) 
 
 
 25 
 
 M. 1 
 
 
 
 Sine. 
 
 D. 
 
 Cosine. 
 
 D. 
 
 .26 
 
 ^ Tang. 
 9.089144 
 
 D. 
 
 17.38 
 
 Cotang. 
 
 60 
 
 J.U85894 
 
 17.13 
 
 9 996751 
 
 10.910856 
 
 1 
 
 086922 
 
 17.09 
 
 996735 
 
 .26 
 
 090187 
 
 17.34 
 
 909813 
 
 59 
 
 2 
 
 087947 
 
 17.04 
 
 996720 
 
 .26 
 
 091228 
 
 17.30 
 
 908772 
 
 58 
 
 3 
 
 088970 
 
 17.00 
 
 996704 
 
 .26 
 
 092266 
 
 17.27 
 
 907734 
 
 57 
 
 4 
 
 0899i)0 
 
 16.96 
 
 996688 
 
 ,26 
 
 093302 
 
 17.22 
 
 906698 
 
 56 
 
 5 
 
 091008 
 
 16.92 
 
 996673 
 
 .26 
 
 094336 
 
 17.19 
 
 905664 
 
 55 
 
 6 
 
 092024 
 
 16.88 
 
 996657 
 
 .26 
 
 095307 
 
 17.15 
 
 904633 
 
 54 
 
 7 
 
 093037 
 
 16.84 
 
 996G41 
 
 .26 
 
 096395 
 
 17.11 
 
 903605 
 
 53 
 
 8 
 
 094047 
 
 16.80 
 
 996625 
 
 .26 
 
 097422 
 
 17.07 
 
 902578 
 
 52 
 
 9 
 
 09.')056 
 
 16.76 
 
 996610 
 
 .26 
 
 098446 
 
 17.03 
 
 901554 
 
 •51 
 
 10 
 
 09(;062 
 
 16.73 
 
 996594 
 
 .26 
 
 099468 
 
 16.99 
 
 900532 
 
 50 
 
 n 
 
 9.01)7005 
 
 16.08 
 
 9 996578 
 
 .27 
 
 9.100487 
 
 16.95 
 
 10.899513 
 
 49 
 
 ]2 
 
 098')(;0 
 
 16.65 
 
 996562 
 
 .27 
 
 101504 
 
 16.91 
 
 898496 
 
 48 
 
 l:} 
 
 09.)065 
 
 16.61 
 
 996546 
 
 .27 
 
 102519 
 
 16.87 
 
 897481 
 
 47 
 
 n 
 
 100062 
 
 16.57 
 
 996530 
 
 .27 
 
 103532 
 
 16.84 
 
 896468 46 | 
 
 ]> 
 
 101056 
 
 16. 53 
 
 996.514 
 
 .27 
 
 104.-)42 
 
 16.80 
 
 895458 
 
 45 
 
 ](] 
 
 102048 
 
 16.49 
 
 996408 
 
 .27 
 
 1U5550 
 
 16.76 
 
 894450 
 
 44 
 
 17 
 
 103037 
 
 16.45 
 
 996482 
 
 .27 
 
 106556 
 
 16.72 
 
 893444 
 
 43 
 
 18 
 
 104025 
 
 16.41 
 
 996465 
 
 .27 
 
 107559 
 
 16.69 
 
 892441 
 
 42 
 
 19 
 
 105010 
 
 16.38 
 
 996449 
 
 .27 
 
 108560 
 
 16.65 
 
 891440 
 
 41 
 
 20 
 
 105992 
 
 16.34 
 
 996433 
 
 .27 
 
 109559 
 
 16.61 
 
 890441 
 
 40 
 
 21 
 
 9.106973 
 
 16.30 
 
 9.996417 
 
 .27 
 
 9.110556 
 
 16.58 
 
 10.889444 
 
 39 
 
 22 
 
 107951 
 
 16.27 
 
 996400 
 
 .27 
 
 111551 
 
 16.54 
 
 888449 
 
 38 
 
 23 
 
 108927 
 
 16.23 
 
 996384 
 
 .27 
 
 112543 
 
 16.50 
 
 887457 
 
 37 
 
 24 
 
 109901 
 
 16.19 
 
 996368 
 
 .27 
 
 113533 
 
 16.46 
 
 886-467 
 
 36 
 
 25 
 
 110873 
 
 16.16 
 
 996351 
 
 .27 
 
 114521 
 
 16.43 
 
 885479 
 
 35 
 
 26 
 
 111842 
 
 16.12 
 
 996335 
 
 .27 
 
 115507 
 
 16.39 
 
 884493 
 
 34 
 
 27 
 
 112809 
 
 16.08 
 
 996318 
 
 .27 
 
 116491 
 
 16.36 
 
 883509 
 
 33 
 
 28 
 
 113774 
 
 16.05 
 
 996302 
 
 .28 
 
 117472 
 
 16.32 
 
 882528 
 
 32 
 
 29 
 
 114737 
 
 16.01 
 
 996285 
 
 .28 
 
 118452 
 
 16.29 
 
 881548 
 
 31 
 
 30 
 
 115098 
 
 15.97 
 
 996269 
 
 .28 
 
 119429 
 
 16.25 
 
 880571! 30 [ 
 
 31 
 
 9.116650 
 
 15.94 
 
 9.996252 
 
 .28 
 
 9.1204(J4 
 
 16.22 
 
 10.879596 
 
 29 
 
 32 
 
 117613 
 
 15.90 
 
 996235 
 
 .28 
 
 121377 
 
 16.18 
 
 878623 
 
 28 
 
 33 
 
 118567 
 
 15.87 
 
 996219 
 
 .28 
 
 122348 
 
 16.15 
 
 877652 
 
 27 
 
 34 
 
 119519 
 
 15.83 
 
 996202 
 
 .28 
 
 123317 
 
 16.11 
 
 876683 
 
 26 
 
 35 
 
 120469 
 
 15,80 
 
 996185 
 
 .28 
 
 124284 
 
 16.07 
 
 875716 
 
 25 
 
 3:5 
 
 121417 
 
 15.76 
 
 996168 
 
 .28 
 
 125249 
 
 16.04 
 
 874751 
 
 24 
 
 37 
 
 122;)62 
 
 15 73 
 
 996151 
 
 .28 
 
 126211 
 
 16.01 
 
 873789 
 
 23 
 
 38 
 
 123306 
 
 15.69 
 
 996134 
 
 .28 
 
 127172 
 
 15.97 
 
 872828 
 
 22 
 
 39 
 
 124248 
 
 15.6" 
 
 996117 
 
 .28 
 
 128130 
 
 15.94 
 
 871870 
 
 21 
 
 40 
 
 125187 
 
 15.62 
 
 996100 
 
 .28 
 
 129087 
 
 15.91 
 
 870913 
 
 20 
 
 41 
 
 9.126125 
 
 15. 59 
 
 9.990083 
 
 .29 
 
 9.130041 
 
 15.87 
 
 10.869959 
 
 19 
 
 42 
 
 127(H)0 
 
 15. .56 
 
 996066 
 
 .29 
 
 130994 
 
 15.84 
 
 869006 
 
 18 
 
 43 
 
 127993 
 
 15.52 
 
 996049 
 
 .29 
 
 131944 
 
 15.81 
 
 868056 
 
 17 
 
 44 
 
 128925 
 
 15.49 
 
 990032 
 
 .29 
 
 132893 
 
 15.77 
 
 867107 
 
 16 
 
 45 
 
 129854 
 
 15.45 
 
 996015 
 
 .29 
 
 133839 
 
 15.74 
 
 866161 
 
 15 
 
 46 
 
 130781 
 
 15.42 
 
 995998 
 
 .29 
 
 134784 
 
 15.71 
 
 865216 
 
 14 
 
 47 
 
 131 70G 
 
 15.39 
 
 995980 
 
 .29 
 
 135726 
 
 15.67 
 
 864274 
 
 13 
 
 48 
 
 132630 
 
 15.35 
 
 995963 
 
 .29 
 
 136667 
 
 15.64 
 
 863333 
 
 12 
 
 49 
 
 133551 
 
 15.32 
 
 995946 
 
 .29 
 
 137605 
 
 15.61 
 
 862395 
 
 11 
 
 50 
 
 13t470 
 
 15.29 
 
 995928 
 
 .29 
 
 138i542 
 
 15.68 
 
 861458 
 
 10 
 
 51 
 
 9.ia-)})87 
 
 15.25 
 
 9.995911 
 
 .29 
 
 9.139476 
 
 15.55 
 
 10.860524 
 
 9 
 
 52 
 
 136303 
 
 15.22 
 
 995894 
 
 .29 
 
 140409 
 
 15.51 
 
 859601 
 
 8 
 
 58 
 
 137216 
 
 15.19 
 
 995876 
 
 .29 
 
 141340 
 
 15.48 
 
 858660 
 
 7 
 
 54 
 
 138128 
 
 15.16 
 
 995859 
 
 .29 
 
 142269 
 
 15.45 
 
 867731 
 
 6 
 
 55 
 
 139037 
 
 16.12 
 
 995841 
 
 .29 
 
 143196 
 
 16.42 
 
 866804 
 
 5 
 
 56 
 
 139944 
 
 15.09 
 
 995823 
 
 .29 
 
 144121 
 
 15.39 
 
 865879 
 
 4 
 
 67 
 
 140850 
 
 15.00 
 
 B95806 
 
 .29 
 
 145044 
 
 16.36 
 
 864956 
 
 8 
 
 58 
 
 141754 
 
 15.03 
 
 995788 
 
 .29 
 
 .146966 
 
 16 33 
 
 864034 
 
 2 
 
 &9 
 
 142655 
 
 15.00 
 
 995771 
 
 .29 
 
 146885 
 
 15.20 
 
 863116 
 
 1 
 
 60 
 
 143555 
 Cosine. 
 
 14.96 
 B. 
 
 995753 
 
 .29 
 82° 
 
 147808 
 CoUng. 
 
 15.26 
 
 852197 
 Tang. 
 
 
 M. 
 
 Sine. 
 
 D. 
 
 Ilk 
 
 iili! 
 
 i'ji. 
 
16 
 
 
 (8 DEGREES.) A 
 
 TABLK or L06ARITUMIO 
 
 
 M. 
 
 
 Bine. 
 
 "D. 
 
 Codne. 
 
 D. 
 
 .30 
 
 Tang. 
 
 ^. 
 
 Cotang. 
 
 60 
 
 9.143555 
 
 14.96 
 
 9.995753 
 
 9.147803 
 
 15.26 
 
 10.852197 
 
 1 
 
 144453 
 
 14.93 
 
 995735 
 
 .80 
 
 148718 
 
 15.23 
 
 851282 
 
 59 
 
 2 
 
 145349 
 
 14.90 
 
 995717 
 
 .80 
 
 149632 
 
 15.20 
 
 850368 
 
 68 
 
 8 
 
 146243 
 
 14.87 
 
 995699 
 
 .30 
 
 150544 
 
 15.17 
 
 849456 
 
 67 
 
 4 
 
 147186 
 
 14.84 
 
 995681 
 
 .30 
 
 151454 
 
 15.14 
 
 848546 
 
 66 
 
 5 
 
 148026 
 
 14.81 
 
 995664 
 
 .30 
 
 152868 
 
 15.11 
 
 847637 
 
 65 
 
 6 
 
 148915 
 
 14.78 
 
 995646 
 
 .80 
 
 153269 
 
 15.08 
 
 846731 
 
 64 
 
 7 
 
 149802 
 
 14.75 
 
 995628 
 
 .80 
 
 154174 
 
 15.05 
 
 845826 
 
 63 
 
 8 
 
 150686 
 
 14.72 
 
 995610 
 
 .80 
 
 155077 
 
 15.02 
 
 844923 
 
 52 
 
 9 
 
 151569 
 
 14.69 
 
 995591 
 
 ,30 
 
 155978 
 
 14.99 
 
 844022 
 
 51 
 
 10 
 
 152451 
 
 14.66 
 
 995573 
 
 .80 
 
 156877 
 
 14.96 
 
 848123 
 
 50 
 
 11 
 
 9.153330 
 
 14.63 
 
 9.995555 
 
 .80 
 
 9.157775 
 
 14.93 
 
 10.842225 
 
 49 
 
 12 
 
 154208 
 
 14.60 
 
 995537 
 
 .80 
 
 158671 
 
 14.90 
 
 841329 
 
 48 
 
 13 
 
 155083 
 
 14.57 
 
 995519 
 
 .30 
 
 159565 
 
 14.87 
 
 840435 
 
 47 
 
 11 
 
 155957 
 
 14.54 
 
 995501 
 
 .81 
 
 160457 
 
 14.84 
 
 839543 
 
 46 
 
 15 
 
 156830 
 
 14.51 
 
 995482 
 
 .31 
 
 161347 
 
 14.81 
 
 838653 
 
 45 
 
 16 
 
 157700 
 
 14.48 
 
 995464 
 
 .81 
 
 162236 
 
 14.79 
 
 837764 
 
 44 
 
 17 
 
 158569 
 
 14.45 
 
 995446 
 
 .81 
 
 163123 
 
 14.76 
 
 836877 
 
 43 
 
 18 
 
 159435 
 
 14.42 
 
 995427 
 
 .31 
 
 164008 
 
 14.73 
 
 835992 
 
 42 
 
 19 
 
 160301 
 
 14.39 
 
 995409 
 
 .31 
 
 164892 
 
 14.70 
 
 835108 
 
 41 
 
 20 
 
 161164 
 
 14.36 
 
 995390 
 
 .31 
 
 165774 
 
 14.67 
 
 834226 
 
 40 
 
 21 
 
 9.162025 
 
 14.33 
 
 9.995372 
 
 .31 
 
 9.166654 
 
 14.64 
 
 10.833346 
 
 89 
 
 22 
 
 162885 
 
 14.30 
 
 995353 
 
 .31 
 
 167532 
 
 14.61 
 
 832468 
 
 88 
 
 23 
 
 163743 
 
 14.27 
 
 995334 
 
 .31 
 
 168409 
 
 14.68 
 
 831591 
 
 87 
 
 24 
 
 164600 
 
 14.24 
 
 995316 
 
 .81 
 
 169284 
 
 14.55 
 
 830716 
 
 86 
 
 25 
 
 165454 
 
 14.22 
 
 995297 
 
 .31 
 
 170157 
 
 14.53 
 
 829843 
 
 85 
 
 20 
 
 166307 
 
 14.19 
 
 995278 
 
 .31 
 
 171029 
 
 14.50 
 
 828971 
 
 84 
 
 27 
 
 167159 
 
 14.16 
 
 995260 
 
 .31 
 
 171899 
 
 14.47 
 
 828101 
 
 33 
 
 28 
 
 168008 
 
 14.13 
 
 995241 
 
 .32 
 
 172767 
 
 14.44 
 
 827233 
 
 82 
 
 29 
 
 168856 
 
 14.10 
 
 995222 
 
 .82 
 
 173634 
 
 14.42 
 
 826366 
 
 81 
 
 80 
 
 169702 
 
 14.07 
 
 995203 
 
 .32 
 
 174499 
 
 14.89 
 
 825501 
 
 80 
 
 31 
 
 9.170547 
 
 14.05 
 
 9.995184 
 
 .82 
 
 9.175362 
 
 14.86 
 
 10.824638 
 
 29 
 
 32 
 
 171389 
 
 14.02 
 
 995165 
 
 .32 
 
 176224 
 
 14.33 
 
 823776 
 
 28 
 
 83 
 
 172230 
 
 13.99 
 
 995146 
 
 .32 
 
 177084 
 
 14.31 
 
 822916 
 
 27 
 
 84 
 
 173070 
 
 13.96 
 
 995127 
 
 .32 
 
 177942 
 
 14.28 
 
 822058 
 
 26 
 
 85 
 
 173908 
 
 13.94 
 
 995108 
 
 .32 
 
 178799 
 
 14.25 
 
 821201 
 
 25 
 
 86 
 
 174744 
 
 13.91 
 
 995089 
 
 .32 
 
 179655 
 
 14.23 
 
 820345 
 
 24 
 
 87 
 
 175578 
 
 13.88 
 
 995070 
 
 .32 
 
 180508 
 
 14.20 
 
 819492 
 
 23 
 
 88 
 
 176411 
 
 13.86 
 
 995051 
 
 .32 
 
 181360 
 
 14.17 
 
 818640 
 
 22 
 
 89 
 
 177242 
 
 13.83 
 
 995032 
 
 .32 
 
 182211 
 
 14.15 
 
 817789 
 
 21 
 
 40 
 
 178072 
 
 13.80 
 
 995013 
 
 .32 
 
 183059 
 
 14.12 
 
 81694. 
 
 20 
 
 41 
 
 9.178900 
 
 13.77 
 
 9.994993 
 
 .82 
 
 9.183907 
 
 14.09 
 
 10.816093 
 
 19 
 
 42 
 
 179726 
 
 13.74 
 
 994974 
 
 .32 
 
 184752 
 
 14.07 
 
 815248 
 
 18 
 
 43 
 
 180551 
 
 13.72 
 
 994955 
 
 .32 
 
 185597 
 
 14.04 
 
 814403 
 
 17 
 
 44 
 
 181374 
 
 13.69 
 
 994935 
 
 .32 
 
 186439 
 
 14.02 
 
 813561 
 
 16 
 
 45 
 
 182196 
 
 13.66 
 
 994916 
 
 .33 
 
 187280 
 
 13.99 
 
 812720 
 
 15 
 
 46 
 
 18301G 
 
 13.64 
 
 994896 
 
 .33 
 
 188120 
 
 13.96 
 
 811880 
 
 14 
 
 47 
 
 183834 
 
 13.61 
 
 994877 
 
 .33 
 
 188958 
 
 13.93 
 
 811042 
 
 13 
 
 48 
 
 184051 
 
 13.59 
 
 994857 
 
 .33 
 
 189794 
 
 13.91 
 
 810206 
 
 12 
 
 49 
 
 185466 
 
 13.56 
 
 994838 
 
 .33 
 
 190629 
 
 13.89 
 
 809371 
 
 11 
 
 50 
 
 186280 
 
 13.53 
 
 994818 
 
 .33 
 
 191462 
 
 13.86 
 
 808538 
 
 V) 
 
 51 
 
 9.187092 
 
 1«.51 
 
 9.994798 
 
 .83 
 
 9.192294 
 
 13.84 
 
 10.807706 
 
 9 
 
 52 
 
 187903 
 
 13.48 
 
 994779 
 
 .33 
 
 193124 
 
 13.81 
 
 806876 
 
 8 
 
 53 
 
 188712 
 
 13.46 
 
 994759 
 
 .33 
 
 193953 
 
 13.79 
 
 806047 
 
 7 
 
 54 
 
 189519 
 
 13.43 
 
 994739 
 
 .33 
 
 194780 
 
 13.76 
 
 805220 
 
 6 
 
 55 
 
 190325 
 
 13.41 
 
 994719 
 
 .33 
 
 195606 
 
 13.74 
 
 804394 
 
 5 
 
 56 
 
 191130 
 
 13.88 
 
 994700 
 
 .33 
 
 196430 
 
 13.71 
 
 803570 
 
 4 
 
 57 
 
 191933 
 
 13.36 
 
 994680 
 
 .38 
 
 197253 
 
 13.69 
 
 802747 
 
 3 
 
 58 
 
 192734 
 
 13.83 
 
 994660 
 
 .33 
 
 198074 
 
 13.66 
 
 80192G 
 
 2 
 
 59 
 
 193534 
 
 13.30 
 
 994640 
 
 .33 
 
 198894 
 
 13.64 
 
 80110C 
 
 1 
 
 CO 
 
 194332 
 , Cosine. 
 
 13.28 
 D. 
 
 994620 
 
 .83 
 
 199713 
 
 13.61 
 D. 
 
 800287 
 
 
 
 Sine. 
 
 81= 
 
 Cotang. 
 
 Tang. 
 
 M. 
 
SIXES AND TANGEXT8. (9 nE(JREE3.) 
 
 27 
 
 60 
 59 
 58 
 
 57 
 
 66 
 
 55 
 
 54 
 
 63 
 
 62 
 
 61 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 441 
 
 43 1 
 
 42 
 
 41 
 
 40 
 
 89 
 
 38 
 
 87 
 
 86 
 
 85 
 
 84 
 
 33 
 
 32 
 
 81 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 IP 
 9 
 8 
 7 
 6 
 5 
 
 M. 
 
 M. 
 
 Sine. 
 
 D. 
 
 Cdtine. { 
 
 D. 
 
 .33 
 
 Tang. 
 
 9.199713 
 
 D. 
 
 Cotang. 
 
 10.809287 
 
 60 
 
 
 
 i). 194332 
 
 13.28 
 
 9.994020 
 
 13.61 
 
 1 
 
 195129 
 
 13.26 
 
 994600 
 
 .33 
 
 200529 
 
 13.59 
 
 799471 
 
 59 
 
 2 
 
 195925 
 
 13.23 
 
 994580 
 
 .33 
 
 201345 
 
 13.56 
 
 798655 
 
 58 
 
 3 
 
 196719 
 
 13.21 
 
 994560 
 
 .34 
 
 202159 
 
 13.54 
 
 797841 
 
 57 
 
 4 
 
 197511 
 
 13.18 
 
 994540 
 
 .34 
 
 •»02971 
 
 13.52 
 
 797029 
 
 56 
 
 5 
 
 198302 
 
 13.16 
 
 994519 
 
 .34 
 
 203782 
 
 13.49 
 
 796218 
 
 55 
 
 6 
 
 199091 
 
 13.13 
 
 994499 
 
 .34 
 
 204592 
 
 13 47 
 
 795408 
 
 54 
 
 7 
 
 199879 
 
 13.11 
 
 994479 
 
 .34 
 
 205400 
 
 13.45 
 
 794600 
 
 53 
 
 8 
 
 200GG6 
 
 13.08 
 
 994459 
 
 .34 
 
 206207 
 
 13 42 
 
 793793 
 
 52 
 
 9 
 
 201451 
 
 13. OG 
 
 994438 
 
 .34 
 
 207013 
 
 13 40 
 
 792987 
 
 51 
 
 10 
 
 202234 
 
 13.04 
 
 994418 
 
 .34 
 
 207817 
 
 13,38 
 
 792183 
 
 50 
 
 11 
 
 9.20:5017 
 
 13.01 
 
 9.994397 
 
 .34 
 
 9,208019 
 
 13 35 
 
 10.791381 
 
 49 
 
 12 
 
 203797 
 
 12.99 
 
 994377 
 
 .34 
 
 209420 
 
 13.33 
 
 790580 
 
 48 
 
 13 
 
 201577 
 
 12. 9G 
 
 994357 
 
 .34 
 
 210220 
 
 13 31 
 
 789780 
 
 47 
 
 14 
 
 2053)4 
 
 12.94 
 
 994330 
 
 .34 
 
 211018 
 
 13. 2S 
 
 788982 
 
 46 
 
 15 
 
 200131 
 
 12.92 
 
 99431G 
 
 .34 
 
 211815 
 
 13,20 
 
 788185 
 
 45 
 
 IG 
 
 20.1!)()G 
 
 12.8;) 
 
 994295 
 
 .34 
 
 212>;il 
 
 13.24 
 
 787389 
 
 44 
 
 17 
 
 207079 
 
 12.87 
 
 994274 
 
 .35 
 
 213405 
 
 13,21 
 
 786595 
 
 43 
 
 18 
 
 208452 
 
 12.85 
 
 994254 
 
 .35 
 
 214198 
 
 13 19 
 
 785802 
 
 42 
 
 lU 
 
 20 y222 
 
 12.82 
 
 994233 
 
 .35 
 
 214989 
 
 13.17 
 
 785011 
 
 41 
 
 20 
 
 20.)'J32 
 
 12.80 
 
 9;)4212 
 
 .35 
 
 2157S0 
 
 13 15 
 
 784220 40 
 
 21 
 
 9.210700 
 
 12.78 
 
 9.994191 
 
 .35 
 
 9.21()5G8 
 
 13. 12 
 
 10.783432i 39 
 
 .22 
 
 211526 
 
 12.75 
 
 994171 
 
 .35 
 
 21735G 
 
 13.10 
 
 782644; 38 
 
 L3 
 
 212201 
 
 12.73 
 
 99415J 
 
 .35 
 
 218142 
 
 13.08 
 
 781858 37 
 
 21 
 
 213055 
 
 12.71 
 
 994129 
 
 .35 
 
 218926 
 
 13.05 
 
 781074! 3G 
 
 25 
 
 213 S18 
 
 12.08 
 
 994108 
 
 .35 
 
 219710 
 
 13.03 
 
 780290 35 
 
 2G 
 
 214579 
 
 12.06 
 
 994087 
 
 .35 
 
 220492 
 
 13.01 
 
 779508 34 
 
 27 
 
 215338 
 
 12. G4 
 
 994060 
 
 .35 
 
 221272 
 
 12.99 
 
 778728 33 
 
 2S 
 
 210097 
 
 12.01 
 
 994045 
 
 .35 
 
 222052 
 
 12.97 
 
 7779481 32 
 
 2!) 
 
 210854 
 
 12.59 
 
 994024 
 
 .35 
 
 222830 
 
 12.94 
 
 7771701 31 
 
 30 
 
 217009 
 
 12.57 
 
 994003 
 
 .35 
 
 223006 
 
 12.92 
 
 7 ••^94 30 
 
 31 
 
 9.21 S3u3 
 
 12.55 
 
 9.993981 
 
 .35 
 
 9.2243i2 
 
 12.90 
 
 10.77)618 29 
 
 32 
 
 210116 
 
 12.53 
 
 9939(50 
 
 .35 
 
 225156 
 
 12.88 
 
 774844' 28 
 
 33 
 
 210SG8 
 
 12.50 
 
 99393'.) 
 
 .35 
 
 225929 
 
 12.86 
 
 774071^ 27 
 
 3i 
 
 220G18 
 
 12.48 
 
 993918 
 
 .35 
 
 22(J700 
 
 12.84 
 
 773300, 26 
 
 3.) 
 
 221307 
 
 12.40 
 
 993890 
 
 .36 
 
 227471 
 
 12.81 
 
 772529 23 
 
 30 
 
 222115 
 
 12.44 
 
 993375 
 
 .36 
 
 2282.39 
 
 12.79 
 
 771761 24 
 
 37 
 
 222 SGI 
 
 12.42 
 
 993854 
 
 .30 
 
 229007 
 
 12.77 
 
 770993 23 
 
 3S 
 
 223o06 
 
 12.39 
 
 993832 
 
 .36 
 
 229''73 
 
 12.75 
 
 770227 
 
 22 
 
 33 
 
 224349 
 
 12 37 
 
 993811 
 
 .36 
 
 230)39 
 
 12,73 
 
 769461 
 
 21 
 
 40 
 
 225092 
 
 12.35 
 
 993789 
 
 .30 
 
 231302 
 
 12.71 
 
 7G8G98' 20 
 
 41 
 
 9.225833 
 
 12.33 
 
 9.993708 
 
 .36 
 
 9.232065 
 
 12.09 
 
 10.7679331 19 
 
 42 
 
 22G573 
 
 12.31 
 
 993746 
 
 .36 
 
 232826 
 
 12 07 
 
 767174! 18 
 
 43 
 
 227311 
 
 12.28 
 
 993725 
 
 .36 
 
 233586 
 
 12.05 
 
 766414 
 
 17 
 
 44 
 
 228048 
 
 12.26 
 
 993703 
 
 .36 
 
 234345 
 
 12.62 
 
 765653 
 
 16 
 
 45 
 
 228784 
 
 12.24 
 
 993681 
 
 .86 
 
 235103 
 
 12.60 
 
 764897: 15 
 
 46 
 
 220518 
 
 12.22 
 
 993000 
 
 .36 
 
 235859 
 
 12.53 
 
 764141 14 
 
 47 
 
 230252 
 
 12.20 
 
 993638 
 
 .36 
 
 23(1014 
 
 12.56 
 
 763386 13 
 
 48 
 
 230984 
 
 12.18 
 
 993610 
 
 .86 
 
 237368 
 
 12 54 
 
 762G32 
 
 12 
 
 49 
 
 231714 
 
 12. IG 
 
 993594 
 
 .37 
 
 238120 
 
 12 52 
 
 761880 
 
 11 
 
 50 
 
 232444 
 
 12.14 
 
 99.3572 
 
 .37 
 
 23)872 
 
 12.30 
 
 761128 
 
 10 
 
 51 
 
 9.233172 
 
 12.12 
 
 9.993550 
 
 .37 
 
 9.23:)(;22 
 
 12.48 
 
 10.760378 
 
 9 
 
 52 
 
 233S99 
 
 12 09 
 
 993528 
 
 .87 
 
 240371 
 
 12.46 
 
 759629 
 
 8 
 
 63 
 
 234G25 
 
 12.07 
 
 993500 
 
 .87 
 
 241118 
 
 12.44 
 
 758882 
 
 7 
 
 54 
 
 235349 
 
 12.05 
 
 9934S4 
 
 .37 
 
 241805 
 
 12.42 
 
 758133 
 
 6 
 
 55 
 
 23r,073 
 
 12.03 
 
 9934G2 
 
 .37 
 
 242010 
 
 12,40 
 
 757390 
 
 5 
 
 56 
 
 23ii7:)5 
 
 12.01 
 
 993440 
 
 .37 
 
 243354 
 
 12.38 
 
 75G646 
 
 4 
 
 57 
 
 2375J5 
 
 11 99 
 
 993418 
 
 .87 
 
 241097 
 
 12.36 
 
 755903 
 
 3 
 
 cs 
 
 238235 
 
 11.97 
 
 99339ii 
 
 .37 
 
 2418.39 
 
 12.34 
 
 755161 
 
 2 
 
 59 
 
 23M953 
 
 11 95 
 
 993374 
 
 .37 
 
 245579 
 
 12,32 
 
 754421 
 
 1 
 
 00 
 
 23i)(;70 
 
 11 93 
 
 99.33:51 
 
 ; .87 
 
 2403 !!» 
 
 12.30 
 
 753681 
 
 
 
 
 Gosiiie. 
 
 J). 
 
 ! iHue. 
 
 80 
 
 Cotang. 
 
 D. 
 
 Tang. M. 
 
 Bi 
 
 ■1 i*^! 
 
 22 
 
28 
 
 
 (10 DEGREES.) A 
 
 TAULE OF L0GAH;TU.MI0 
 
 
 H. 
 
 Sine. 
 
 D. 1 
 
 Cosine. 
 
 D. 
 
 Tang-. 
 
 D. 
 
 Cotang. 
 
 60 
 
 
 
 U. 239070 
 
 11.93 
 
 9.993351 
 
 .87 
 
 9.24G319 
 
 12.30 
 
 10.753081 
 
 1 
 
 2-103HG 
 
 11.91 
 
 993329 
 
 .37 
 
 247057 
 
 12.28 
 
 752943 
 
 59 
 
 2 
 
 241101 
 
 11.89 
 
 993307 
 
 .37 
 
 247794 
 
 12.26 
 
 752200 
 
 58 
 
 .3 
 
 241814 
 
 11.87 
 
 993285 
 
 .37 
 
 248530 
 
 12.24 
 
 751470 
 
 57 
 
 4 
 
 242520 
 
 11.85 
 
 993202 
 
 .37 
 
 2492G4 
 
 12.22 
 
 750736 
 
 56 
 
 5 
 
 243237 
 
 11.83 
 
 993240 
 
 .37 
 
 249998 
 
 12.20 
 
 750002 
 
 55 
 
 6 
 
 243947 
 
 11.81 
 
 993217 
 
 .38 
 
 250730 
 
 12.18 
 
 749270 
 
 54 
 
 7 
 
 244G5f, 
 
 11.79 
 
 993195 
 
 .38 
 
 251401 
 
 12.17 
 
 748539 
 
 53 
 
 8 
 
 2453G3 
 
 11.77 
 
 993172 
 
 .38 
 
 252191 
 
 12.15 
 
 747809 
 
 52 
 
 9 
 
 2460G9 
 
 11.75 
 
 993149 
 
 .38 
 
 252920 
 
 12.13 
 
 747080 
 
 51 
 
 10 
 
 246775 
 
 11.73 
 
 993127 
 
 .38 
 
 253648 
 
 12.11 
 
 746352 
 
 50 
 
 11 
 
 9.247478 
 
 11.71 
 
 9.993104 
 
 .38 
 
 9.254374 
 
 12.09 
 
 10.745626 
 
 49 
 
 12 
 
 248181 
 
 11.69 
 
 993081 
 
 .38 
 
 255100 
 
 12.07 
 
 744900 
 
 48 
 
 13 
 
 248883 
 
 11.67 
 
 993059 
 
 .38 
 
 255824 
 
 12.05 
 
 744176 
 
 47 
 
 14 
 
 249583 
 
 11.65 
 
 993036 
 
 .38 
 
 256547 
 
 12.03 
 
 743453 
 
 46 
 
 15 
 
 250282 
 
 11.63 
 
 993013 
 
 .38 
 
 257269 
 
 12.01 
 
 742731 
 
 45 
 
 16 
 
 250980 
 
 11.61 
 
 992990 
 
 .38 
 
 257990 
 
 12.00 
 
 742010 
 
 44 
 
 17 
 
 251677 
 
 11.59 
 
 9929G7 
 
 .38 
 
 258710 
 
 11.98 
 
 741290 
 
 43 
 
 18 
 
 252373 
 
 11.58 
 
 992944 
 
 .38 
 
 259429 
 
 11.96 
 
 740571 
 
 42 
 
 19 
 
 253067 
 
 11.56 
 
 992921 
 
 .38 
 
 2G0146 
 
 11.94 
 
 739854 
 
 41 
 
 20 
 
 253761 
 
 11.54 
 
 992898 
 
 .38 
 
 260803 
 
 11.92 
 
 739137 
 
 40 
 
 21 
 
 9.254453 
 
 11.52 
 
 9.992875 
 
 .88 
 
 9.261578 
 
 11.90 
 
 10.738422 
 
 39 
 
 22 
 
 255144 
 
 11.50 
 
 992852 
 
 .38 
 
 262292 
 
 11.89 
 
 737708 
 
 38 
 
 23 
 
 255834 
 
 11.48 
 
 992829 
 
 .39 
 
 263005 
 
 11.87 
 
 736995 
 
 37 
 
 24 
 
 256523 
 
 11.46 
 
 992806 
 
 .39 
 
 263717 
 
 11.85 
 
 736283 
 
 36 
 
 25 
 
 257211 
 
 11.44 
 
 992783 
 
 .89 
 
 264428 
 
 11.83 
 
 735572 
 
 35 
 
 26 
 
 257898 
 
 11.42 
 
 992759 
 
 .39 
 
 265138 
 
 11.81 
 
 734802 
 
 84 
 
 27 
 
 258583 
 
 11.41 
 
 992736 
 
 .39 
 
 265847 
 
 11.79 
 
 734153 
 
 83 
 
 28 
 
 259268 
 
 11.89 
 
 992713 
 
 .39 
 
 266555 
 
 11.78 
 
 733445 
 
 32 
 
 29 
 
 259951 
 
 11.87 
 
 992690 
 
 .39 
 
 267261 
 
 11.76 
 
 732739 
 
 31 
 
 80 
 
 260633 
 
 11.35 
 
 992666 
 
 .39 
 
 267907 
 
 11.74 
 
 732033 
 
 30 
 
 81 
 
 9.261314 
 
 11.33 
 
 9.992648 
 
 .39 
 
 9.268071 
 
 11.72 
 
 10.731329 
 
 29 
 
 82 
 
 261994 
 
 11.31 
 
 992619 
 
 .39 
 
 269375 
 
 11.70 
 
 730625 
 
 28 
 
 83 
 
 262673 
 
 11.30 
 
 992596 
 
 .39 
 
 270077 
 
 11.69 
 
 729923 
 
 27 
 
 84 
 
 263351 
 
 11.28 
 
 992572 
 
 .39 
 
 270779 
 
 11.67 
 
 729221 
 
 26 
 
 85 
 
 264027 
 
 11.26 
 
 992549 
 
 .39 
 
 271479 
 
 11.65 
 
 728521 
 
 25 
 
 86 
 
 264703 
 
 11.24 
 
 992525 
 
 .39 
 
 272178 
 
 11.64 
 
 727822 
 
 24 
 
 87 
 
 265877 
 
 11.22 
 
 992501 
 
 .39 
 
 272870 
 
 11.62 
 
 727124 
 
 23 
 
 88 
 
 266051 
 
 11.20 
 
 992478 
 
 .40 
 
 273573 
 
 11.60 
 
 726427 
 
 22 
 
 89 
 
 266723 
 
 11.19 
 
 992454 
 
 .40 
 
 274269 
 
 11.58 
 
 725731 
 
 21 
 
 40 
 
 267395 
 
 11.17 
 
 992430 
 
 .40 
 
 274964 
 
 11.57 
 
 725036 
 
 20 
 
 41 
 
 9.268065 
 
 11.15 
 
 9.992406 
 
 .40 
 
 9.275658 
 
 11.55 
 
 10.724342 
 
 19 
 
 42 
 
 268734 
 
 11.13 
 
 992382 
 
 .40 
 
 276351 
 
 11.58 
 
 723649 
 
 18 
 
 43 
 
 269402 
 
 11.11 
 
 992359 
 
 .40 
 
 277043 
 
 11.51 
 
 722957 
 
 17 
 
 44 
 
 270069 
 
 11.10 
 
 992335 
 
 .40 
 
 277734 
 
 11.50 
 
 722266 
 
 16 
 
 45 
 
 270735 
 
 11.08 
 
 992811 
 
 .40 
 
 278424 
 
 11.48 
 
 721576 
 
 15 
 
 46 
 
 271400 
 
 11.06 
 
 992287 
 
 .40 
 
 279113 
 
 11.47 
 
 720887 
 
 14 
 
 47 
 
 272064 
 
 11.05 
 
 992263 
 
 .40 
 
 279801 
 
 11.45 
 
 720199 
 
 13 
 
 48 
 
 272726 
 
 11.03 
 
 992239 
 
 .40 
 
 280488 
 
 11.43 
 
 719512 
 
 12 
 
 49 
 
 278388 
 
 11.01 
 
 992214 
 
 .40 
 
 281174 
 
 11.41 
 
 718826 
 
 11 
 
 60 
 
 274049 
 
 10.99 
 
 992190 
 
 .40 
 
 281858 
 
 11.40 
 
 718142 
 
 10 
 
 51 
 
 9.274708 
 
 10.98 
 
 9.992166 
 
 .40 
 
 9.282542 
 
 11.88 
 
 10.717458 
 
 9 
 
 62 
 
 276367 
 
 10.96 
 
 992142 
 
 .40 
 
 283225 
 
 11.86 
 
 716775 
 
 8 
 
 63 
 
 276024 
 
 10.94 
 
 992117 
 
 .41 
 
 283907 
 
 11.85 
 
 716093 
 
 7 
 
 64 
 
 276681 
 
 10.92 
 
 992093 
 
 .41 
 
 284588 
 
 11.83 
 
 716412 
 
 6 
 
 65 
 
 377887 
 
 10.91 
 
 992069 
 
 .41 
 
 285268 
 
 11.81 
 
 714782 
 
 6 
 
 66 
 
 277991 
 
 10.89 
 
 992044 
 
 .41 
 
 285947 
 
 11.80 
 
 714058 
 
 4 
 
 67 
 
 278644 
 
 10.87 
 
 992020 
 
 .41 
 
 286624 
 
 11.28 
 
 718376 
 
 8 
 
 68 
 
 279297 
 
 10.86 
 
 991996 
 
 .41 
 
 287801 
 
 11.26 
 
 712699 
 
 2 
 
 69 
 
 279948 
 
 10.84 
 
 991971 
 
 .41 
 
 287977 
 
 11.25 
 
 712023 
 
 1 
 
 60 
 
 280599 
 
 10.82 
 
 991947 
 
 .41 
 
 288652 
 
 11.28 
 
 711848 
 
 
 
 M. 
 
 Cosine. 
 
 D. 
 
 Sine. 
 
 79- 
 
 Cotang. 
 
 D. 
 
 Tang. 
 
28 
 
 27 
 
 26 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 1 17 
 16 
 
 1 15 
 14 
 18 
 12 
 11 
 10 
 9 
 8 
 7 
 
 5 
 
 2 
 1 
 
 
 
 
 
 SINES AND TANGENTS. 
 
 (11 DEGREES.) 
 
 
 29 
 
 M. • 
 
 
 Sine. 
 
 J.2«05!»9 
 
 D. 
 
 10.82 
 
 Cosine. ' 
 
 9.991947 
 
 D. 1 
 
 .41 
 
 Tang. 
 
 9. 28^052 
 
 D. 
 
 Cotang. 
 
 11.23 
 
 10.711348 
 
 60 
 
 1 
 
 28124s' 
 
 10.81 
 
 991922 
 
 .41 
 
 289320 
 
 11.22 
 
 710074 
 
 59 
 
 2 
 
 2HlH\)t\ 
 
 10.79 
 
 991H97 
 
 .41 
 
 289.)99 
 
 11.20 
 
 710001 
 
 58 
 
 3 
 
 282344 
 
 10.77 
 
 991873 
 
 .41 
 
 2900711 
 
 11.18 
 
 709329 
 
 57 
 
 4 
 
 2H:5iftO, 
 
 10.70 
 
 9918-48 
 
 .41 
 
 291342 
 
 11.17 
 
 708058 
 
 50 
 
 5 
 
 2S:) >m 
 
 10.74 
 
 991823 
 
 .41 
 
 292013 
 
 11.15 
 
 707987 
 
 55 
 
 G 
 
 2HUS0i 
 
 10.72 
 
 991799 
 
 .41 
 
 2920S2 
 
 11,14 
 
 707318 
 
 54 
 
 7 
 
 2s")i24 
 
 10.71 
 
 991774 
 
 .42 
 
 293.}r)0 
 
 11.12 
 
 70(5050 
 
 53 
 
 8 
 
 2^.')7(;0 
 
 10. Gil 
 
 991719 
 
 .42 
 
 294017 
 
 11.11 
 
 705983 
 
 52 
 
 9 
 
 28(U08 
 
 10.07 
 
 991724 
 
 .42 
 
 294(!84 
 
 11.09 
 
 705316 
 
 51 
 
 10 
 
 287048 
 
 10.00 
 
 991(599 
 
 .42 
 
 295349 
 
 11,07 
 
 704051 
 
 50 
 
 11 
 
 9.2.-i7G87 
 
 10.04 
 
 9.991074 
 
 .42 
 
 9.290013 
 
 11.00 
 
 10.703987 
 
 49 
 
 12 
 
 28S326 
 
 10.03 
 
 991G49 
 
 .42 
 
 2!)0()77 
 
 11.01 
 
 703323 
 
 48 
 
 13 
 
 2889G4 
 
 10.01 
 
 991G24 
 
 .42 
 
 297339 
 
 11.03 
 
 702G01 
 
 47 
 
 14 
 
 289G00 
 
 10.59 
 
 991599 
 
 .42 
 
 298001 
 
 11.01 
 
 701999 
 
 46 
 
 15 
 
 290236 
 
 10.58 
 
 991574 
 
 .42 
 
 298002 
 
 11.00 
 
 701338 
 
 45 
 
 16 
 
 2'J0870 
 
 10.50 
 
 991549 
 
 .42 
 
 299322 
 
 10.98 
 
 700G78 
 
 44 
 
 17 
 
 291504 
 
 10.54 
 
 991524 
 
 .42 
 
 299980 
 
 10.96 
 
 700020 
 
 43 
 
 18 
 
 292137 
 
 10.53 
 
 991498 
 
 .42 
 
 300038 
 
 10.95 
 
 6993G2 
 
 42 
 
 19 
 
 2927G8 
 
 10.51 
 
 991473 
 
 .42 
 
 301295 
 
 10.93 
 
 698705 
 
 41 
 
 20 
 
 293399 
 
 10.50 
 
 991448 
 
 .42 
 
 301951 
 
 10.92 
 
 698049 
 
 40 
 
 21 
 
 9.294029 
 
 10.48 
 
 9.991422 
 
 .42 
 
 9.302007 
 
 10.90 
 
 10.69739'i 
 
 39 
 
 22 
 
 294G58 
 
 10.40 
 
 991897 
 
 .42 
 
 303261 
 
 10.89 
 
 696739 
 
 38 
 
 23 
 
 29.5286 
 
 10.45 
 
 991372 
 
 .43 
 
 303914 
 
 10.87 
 
 690080 
 
 37 
 
 24 
 
 295913 
 
 10.43 
 
 991346 
 
 .43 
 
 301567 
 
 10.86 
 
 695433 
 
 36 
 
 25 
 
 29G539 
 
 10.42 
 
 991321 
 
 .43 
 
 305218 
 
 10.84 
 
 694782 
 
 35 
 
 20 
 
 2971 04 
 
 10.40 
 
 9912!)5 
 
 .43 
 
 3058G9 
 
 10.83 
 
 694131 
 
 34 
 
 27 
 
 297788 
 
 10.39 
 
 991270 
 
 .43 
 
 30C519 
 
 10.81 
 
 693481 
 
 33 
 
 28 
 
 298412 
 
 10.37 
 
 991244 
 
 .43 
 
 307108 
 
 10.80 
 
 692832 
 
 32 
 
 29 
 
 299034 
 
 10.30 
 
 991218 
 
 .43 
 
 307815 
 
 10.78 
 
 692185 
 
 31 
 
 30 
 
 299055 
 
 10.34 
 
 9911!)3 
 
 .43 
 
 308403 
 
 10.77 
 
 691537 
 
 30 
 
 31 
 
 9.3i)i)27G 
 
 10.32 
 
 9.991107 
 
 .43 
 
 9.309109 
 
 10 75 
 
 10.690891 
 
 29 
 
 32 
 
 300895 
 
 10.31 
 
 991141 
 
 .43 
 
 3097.54 
 
 10.74 
 
 690240 
 
 28 
 
 33 
 
 301514 
 
 10.29 
 
 991115 
 
 .43 
 
 310398 
 
 10.73 
 
 689602 
 
 27 
 
 34 
 
 302132 
 
 10.28 
 
 991090 
 
 .43 
 
 311042 
 
 10,71 
 
 688958 
 
 26 
 
 35 
 
 302748 
 
 10.20 
 
 991004 
 
 .43 
 
 311085 
 
 10.70 
 
 688315 
 
 25 
 
 30 
 
 303304 
 
 10.25 
 
 991038 
 
 .43 
 
 312327 
 
 10.08 
 
 687673 
 
 24 
 
 37 
 
 303979 
 
 10.23 
 
 991012 
 
 .43 
 
 312967 
 
 10.07 
 
 687033 
 
 23 
 
 38 
 
 304593 
 
 10.22 
 
 990986 
 
 .43 
 
 313608 
 
 10.65 
 
 686392 
 
 22 
 
 39 
 
 305207 
 
 10.20 
 
 . 9909G0 
 
 .43 
 
 314247 
 
 10.64 
 
 685753 
 
 21 
 
 40 
 
 305S19 
 
 10.19 
 
 990934 
 
 .44 
 
 314885 
 
 10.62 
 
 685115 
 
 20 
 
 41 
 
 9.300430 
 
 10.17 
 
 9.990908 
 
 .44 
 
 9.315523 
 
 10.01 
 
 10.684477 
 
 19 
 
 42 
 
 307041 
 
 10.10 
 
 990882 
 
 .44 
 
 310159 
 
 10.60 
 
 683841 
 
 18 
 
 43 
 
 307G50 
 
 10.14 
 
 990855 
 
 .44 
 
 316795 
 
 10.58 
 
 683205 
 
 17 
 
 44 
 
 308259 
 
 10.13 
 
 990829 
 
 .44 
 
 3174.30 
 
 10.57 
 
 682570 
 
 16 
 
 45 
 
 30H867 
 
 10.11 
 
 990S03 
 
 .4^4 
 
 318064 
 
 10.55 
 
 681936 
 
 15 
 
 46 
 
 309474 
 
 10 10 
 
 990777 
 
 .44 
 
 318697 
 
 10.54 
 
 681303 
 
 14 
 
 47 
 
 310080 
 
 10.08 
 
 9907i}0 
 
 .44 
 
 319329 
 
 10.53 
 
 680671 
 
 13 
 
 48 
 
 310085 
 
 10.07 
 
 990724 
 
 .44 
 
 319961 
 
 10.51 
 
 680039 
 
 12 
 
 49 
 
 311289 
 
 10.05 
 
 990697 
 
 .44 
 
 320592 
 
 10.50 
 
 679408 
 
 11 
 
 50 
 
 811893 
 
 10.04 
 
 990671 
 
 .44 
 
 321222 
 
 10.48 
 
 678778 
 
 10 
 
 51 
 
 9.312495 
 
 10.03 
 
 9.990044 
 
 .44 
 
 9.321851 
 
 10.47 
 
 10.678149 
 
 9 
 
 62 
 
 3130>)7 
 
 10.01 
 
 990618 
 
 .44 
 
 322479 
 
 10.45 
 
 677521 
 
 8 
 
 53 
 
 313698 
 
 10.00 
 
 990591 
 
 .44 
 
 323106 
 
 10.44 
 
 676894 
 
 7 
 
 64 
 
 314297 
 
 9.98 
 
 990565 
 
 .44 
 
 323733 
 
 10.43 
 
 676267 
 
 6 
 
 55 
 
 314897 
 
 9.97 
 
 990538 
 
 .44 
 
 324358 
 
 10.41 
 
 675642 
 
 5 
 
 66 
 
 315495 
 
 9.96 
 
 990511 
 
 .45 
 
 324983 
 
 10.40 
 
 675017 
 
 4 
 
 57 
 
 316092 
 
 9.94 
 
 990485 
 
 .45 
 
 325607 
 
 10.39 
 
 674393 
 
 3 
 
 58 
 
 316689 
 
 9,93 
 
 990458 
 
 .45 
 
 326231 
 
 10.87 
 
 673769 
 
 2 
 
 59 
 
 317284 
 
 9.91 
 
 990431 
 
 .45 
 
 326853 
 
 10.36 
 
 673147 
 
 1 
 
 60 
 
 317879 
 
 9.90 
 
 990404 
 
 .45 
 
 327475 
 
 10.35 
 
 672525 
 
 
 
 
 Cosine. 
 
 D. 
 
 F'ne. 
 
 78^^ 
 
 Cotanif. 
 
 D. 
 
 Tarv. 
 
 M. 
 
 !>! 
 
 1 *' 
 
 Jl 
 
 mi 
 
30 
 
 
 (12 DEGREES.) A 
 
 TABLB or LOOARITHMIC 
 
 1 
 
 
 ■fiT" fline. 1 
 
 ^- 1 
 
 Cotine. 
 
 D. 
 
 Tang. 
 
 b. 
 
 Cotang. ! 
 
 
 
 J. 317879 
 
 9.90 
 
 9.990404 
 
 .45 
 
 9.327474 
 
 10.85 
 
 10.672526 
 
 60 
 
 1 
 
 818473 
 
 9.88 
 
 990378 
 
 .45 
 
 828095 
 
 10.83 
 
 671905 
 
 59 
 
 2 
 
 819066 
 
 9.87 
 
 990351 
 
 .45 
 
 828715 
 
 10.82 
 
 671285 
 
 68 
 
 8 
 
 819658 
 
 9.86 
 
 990324 
 
 .45 
 
 829384 
 
 10.80 
 
 670666 
 
 67 
 
 4 
 
 820249 
 
 9.84 
 
 990207 
 
 .45 
 
 829953 
 
 10.29 
 
 670047 
 
 56 
 
 6 
 
 820840 
 
 9.83 
 
 990270 
 
 .45 
 
 830570 
 
 10.28 
 
 669430 
 
 55 
 
 6 
 
 821430 
 
 9.82 
 
 990243 
 
 .45 
 
 831187 
 
 10.26 
 
 668813 
 
 54 
 
 7 
 
 822019 
 
 9.80 
 
 990215 
 
 .45 
 
 831803 
 
 10.25 
 
 668197 
 
 63 
 
 8 
 
 822607 
 
 9.79 
 
 990188 
 
 .45 
 
 832418 
 
 10.24 
 
 667582 
 
 62 
 
 9 
 
 823194 
 
 9.77 
 
 990161 
 
 .45 
 
 333033 
 
 10.23 
 
 666967 
 
 61 
 
 10 
 
 823780 
 
 9,76 
 
 990134 
 
 .45 
 
 333646 
 
 10.21 
 
 666354 
 
 60 
 
 11 
 
 9.324366 
 
 9 75 
 
 9.990107 
 
 .46 
 
 9.334259 
 
 10.20 
 
 10.665741 
 
 49 
 
 12 
 
 324950 
 
 9 73 
 
 990079 
 
 .46 
 
 334871 
 
 10.19 
 
 665129 
 
 48 
 
 13 
 
 825534 
 
 9.72 
 
 990052 
 
 .46 
 
 335482 
 
 10.17 
 
 664518 
 
 47 
 
 1:1 
 
 820117 
 
 9 70 
 
 990025 
 
 .46 
 
 330093 
 
 10.16 
 
 663907 
 
 46 
 
 15 
 
 826700 
 
 9 69 
 
 989997 
 
 .46 
 
 336702 
 
 10.15 
 
 663298 
 
 45 
 
 16 
 
 827281 
 
 9 68 
 
 989970 
 
 .46 
 
 837311 
 
 10.13 
 
 662689 
 
 44 
 
 17 
 
 827862 
 
 9 66 
 
 989942 
 
 .46 
 
 337919 
 
 10.12 
 
 662081 
 
 43 
 
 18 
 
 828442 
 
 9 65 
 
 989915 
 
 .46 
 
 338527 
 
 10.11 
 
 661473 
 
 42 
 
 19 
 
 829021 
 
 9 "64 
 
 989887 
 
 .46 
 
 339133 
 
 10.10 
 
 660867 
 
 41 
 
 20 
 
 329599 
 
 9'62 
 
 989860 
 
 .46 
 
 339739 
 
 10.08 
 
 660261 
 
 40 
 
 21 
 
 0.330176 
 
 9 61 
 
 9.989832 
 
 .46 
 
 9.340344 
 
 10.07 
 
 10.659656 
 
 89 
 
 22 
 
 830753 
 
 960 
 
 989804 
 
 .46 
 
 340948 
 
 10.06 
 
 659052 
 
 38 
 
 23 
 
 831329 
 
 9 58 
 
 989777 
 
 .46 
 
 341552 
 
 10.04 
 
 658448 
 
 87 
 
 24 
 
 831903 
 
 9 57 
 
 989749 
 
 .47 
 
 3t2155 
 
 10.03 
 
 657845 
 
 36 
 
 25 
 
 832478 
 
 9 56 
 
 989721 
 
 .47 
 
 842757 
 
 10.02 
 
 657243 
 
 35 
 
 26 
 
 333051 
 
 954 
 
 989693 
 
 .47 
 
 843358 
 
 10.00 
 
 656642 
 
 34 
 
 27 
 
 833624 
 
 9 "53 
 
 989665 
 
 .47 
 
 843958 
 
 9.99 
 
 650042 
 
 33 
 
 28 
 
 834195 
 
 9 52 
 
 989637 
 
 .47 
 
 844558 
 
 9.98 
 
 655442 
 
 82 
 
 29 
 
 834766 
 
 9 50 
 
 989609 
 
 .47 
 
 845157 
 
 9.97 
 
 654843 
 
 81 
 
 SO 
 
 335337 
 
 9 '49 
 
 989582 
 
 .47 
 
 345755 
 
 9.96 
 
 654245 
 
 80 
 
 81 
 
 9.335906 
 
 9 48 
 
 9.989553 
 
 .47 
 
 9.346353 
 
 9.94 
 
 10.653647 
 
 29 
 
 82 
 
 836475 
 
 946 
 
 989525 
 
 .47 
 
 346949 
 
 9.93 
 
 653051 
 
 28 
 
 83 
 
 837043 
 
 9 45 
 
 989497 
 
 .47 
 
 847545 
 
 9.92 
 
 652455 
 
 27 
 
 34 
 
 837610 
 
 9' 44 
 
 989469 
 
 .47 
 
 848141 
 
 9.91 
 
 651859 
 
 26 
 
 85 
 
 838176 
 
 9 43 
 
 989441 
 
 .47 
 
 348735 
 
 9.90 
 
 651265 
 
 25 
 
 36 
 
 838742 
 
 9 41 
 
 989413 
 
 .47 
 
 849329 
 
 9.88 
 
 650671 
 
 24 
 
 87 
 
 839306 
 
 9 40 
 
 989384 
 
 .47 
 
 349922 
 
 9.87 
 
 650078 
 
 23 
 
 38 
 
 839871 
 
 9' 39 
 
 989356 
 
 .47 
 
 350514 
 
 9.86 
 
 649486 
 
 22 
 
 39 
 
 840434 
 
 9 37 
 
 989328 
 
 .47 
 
 851 IOC 
 
 9.85 
 
 648894 
 
 21 
 
 40 
 
 340996 
 
 9" 38 
 
 989300 
 
 .47 
 
 351697 
 
 9.83 
 
 648303 
 
 20 
 
 41 
 
 9.341558 
 
 9' 35 
 
 9.989271 
 
 .47 
 
 9.352287 
 
 9.82 
 
 10.647713 
 
 19 
 
 42 
 
 842119 
 
 9 34 
 
 989243 
 
 .47 
 
 352876 
 
 9.81 
 
 647124 
 
 18 
 
 43 
 
 842679 
 
 932 
 
 989214 
 
 .47 
 
 QgQJflij 
 
 9.80 
 
 646535 
 
 17 
 
 44 
 
 843239 
 
 9 31 
 
 989186 
 
 .47 
 
 ^^viiufl 
 
 9.79 
 
 645947 
 
 16 
 
 45 
 
 343797 
 
 6 30 
 
 989157 
 
 .47 
 
 85^ 
 
 9.77 
 
 646860 
 
 15 
 
 46 
 
 844355 
 
 9 29 
 
 989128 
 
 .48 
 
 855227 
 
 9.76 
 
 644773 
 
 14 
 
 47 
 
 844912 
 
 9.27 
 
 989100 
 
 .48 
 
 855813 
 
 9.75 
 
 644187 
 
 13 
 
 48 
 
 84546<J 
 
 9.26 
 
 989071 
 
 .48 
 
 856398 
 
 9.74 
 
 643602 
 
 12 
 
 49 
 
 846021 
 
 9.25 
 
 989042 
 
 .48 
 
 856982 
 
 9.73 
 
 643018 
 
 11 
 
 60 
 
 84657a 
 
 9.24 
 
 989014 
 
 .48 
 
 857566 
 
 9.71 
 
 642434 
 
 10 
 
 51 
 
 9.347134 
 
 9.22 
 
 9.988985 
 
 .48 
 
 9.358149 
 
 9.70 
 
 10.641851 
 
 9 
 
 52 
 
 847681 
 
 9 21 
 
 988956 
 
 .48 
 
 858781 
 
 9.69 
 
 641269 
 
 8 
 
 53 
 
 84824C 
 
 1 9.20 
 
 988927 
 
 .48 
 
 35931S 
 
 9.68 
 
 640687 
 
 7 
 
 54 
 
 848795 
 
 ! 9.19 
 
 988898 
 
 .48 
 
 859893 
 
 9.67 
 
 640107 
 
 6 
 
 55 
 
 84934£ 
 
 \ 9.17 
 
 988869 
 
 .48 
 
 860474 
 
 9.66 
 
 639526 
 
 6 
 
 66 
 
 84989^ 
 
 ( 9.16 
 
 98884(] 
 
 .48 
 
 86105S 
 
 9.65 
 
 638947 
 
 4 
 
 57 
 
 85044J 
 
 ) 9.15 
 
 988811 
 
 .49 
 
 1 861632 
 
 9.63 
 
 638368 
 
 8 
 
 68 
 
 85099^ 
 
 I 9.14 
 
 988782 
 
 .4£ 
 
 1 86221C 
 
 1 9.62 
 
 63779C 
 
 2 
 
 69 
 
 85154( 
 
 ) 9.13 
 
 98875J 
 
 .49 
 
 1 862787 
 
 ' 9.61 
 
 637213 
 
 1 1 
 
 60 
 
 85208J 
 
 i 9.11 
 D. 
 
 988724 
 
 .4? 
 
 ) 863364 
 
 t 9.60 
 
 636636 
 
 
 
 
 Cosine. 
 
 Sine. 
 
 77" 
 
 Cotang. 
 
 D. 
 
 Tang. 
 
 M. 
 
Ti T"?; — : . ' 
 
 [CosineT 
 
 D. 
 
 ^- ' .Cosine^ j D 
 
 9.!>,iS721~4r) 
 988695 ■ 
 988666 
 988636 
 988607 
 988578 
 98854S 
 98851!) 
 98848,J 
 988460 
 9884301 
 9.988401 
 988371 
 988342 
 988312 
 988282' 
 9882521 
 988223 
 9881»3 
 988163 
 98S133I 
 9.9S8103 
 988073 
 988043 
 988013 
 987983 
 987953 
 987922 
 987892 
 987862, 
 987832 
 1.987801 
 987771 
 987740, 
 987710 
 987679 
 987649 
 987618 
 98758V 
 987557 
 987520 
 9.987490 
 987405 
 
 987434 
 
 987403, 
 
 987372 
 
 987341 
 
 987310 
 
 9872791 
 
 987248 
 
 987217, 
 9.987180' 
 987155 
 987124 
 987092 
 987061 
 987030 
 98G99H| 
 986967 
 986930 
 _98^04 
 
 Sine. 
 
 9T363364 
 363940 
 364515 
 365090, 
 365664 
 366237 
 366810 
 367382 
 8079531 
 368524 
 3690941 
 9.369663 
 370232 
 370799 
 871367 
 8719331 
 372499, 
 3730641 
 373629 
 8741931 
 874756 
 9.37.1319 
 375881 
 876442] 
 877003, 
 377563 
 878122 
 378681 
 879239, 
 379797 
 3803541 
 ,380910 
 3814661 
 382020 
 3825751 
 883129 
 88;J682| 
 384234 
 8847861 
 88.1337 
 38.1888 
 38i;438 
 88(;987 
 387.536 
 3880S4 
 888631 
 389178 
 889724 
 3i;0270 
 3908151 
 891360' 
 9.3919031 
 392447 
 892989 
 
 I, 
 
 I ill. 
 
32 
 
 
 (14 DEGRBES.) A 
 
 TABLE OF LOGAUITHMIC 
 
 
 
 M. 
 
 Sine. 
 
 D. 
 
 Cosine. 
 
 D. 
 
 Tang. 
 
 D. 1 
 
 Cotang. 1 
 
 ' 
 
 J.383G75 
 
 8.44 
 
 97986904 ' 
 
 .52" 
 
 9.39G771 
 
 8.96 ] 
 
 LO. 0032291 00 
 
 1 
 
 3H4182 
 
 8.13 
 
 98G873 
 
 .53 
 
 8973011, 
 
 8.96 
 
 602091, 59 
 
 2 
 
 384GH7 
 
 8 42 
 
 98G841 
 
 .53 
 
 39784(;i 
 
 8.95 
 
 602154 53 
 
 3 
 
 38511)2 
 
 8.11 
 
 98G809 
 
 .53 
 
 3983831 
 
 8.94 
 
 (')01017 57 
 
 4 
 
 385G!)7 
 
 8.40 
 
 986778 
 
 .53 
 
 398919 
 
 8.93 
 
 6010H1 56 
 
 5 
 
 38G201 
 
 8.39 
 
 98674G 
 
 .53 
 
 39i)455 
 
 8.92 
 
 600545 55 
 
 G 
 
 38G704 
 
 8.38 
 
 986714 
 
 .53 
 
 3;»99!)0 
 
 8.91 
 
 000010 54 
 
 7 
 
 387207 
 
 8.37 
 
 986683 
 
 .53 
 
 400524 
 
 8.90 
 
 5994761 53 
 
 8 
 
 387709 
 
 8.3G 
 
 98GG51 
 
 .53 
 
 401058 
 
 8.89 
 
 698942 52 
 
 9 
 
 388210 
 
 8.35 
 
 98GG19 
 
 .53 
 
 401591 
 
 8.88 
 
 5!)8409! 51 
 
 10 
 
 388711 
 
 8.34 
 
 986587 
 
 .53 
 
 402124 
 
 8.87 
 
 597870' 50 
 
 11 
 
 9.389211 
 
 8 33 
 
 9.986555 
 
 .53 
 
 9.402G5G' 
 
 8,86 
 
 10.5973441 49 
 
 12 
 
 389711 
 
 8 32 
 
 986523 
 
 ,53 
 
 403187 
 
 8.85 
 
 596813: 48 
 
 13 
 
 890210 
 
 8 31 
 
 986491 
 
 .53 
 
 403718 
 
 8.84 
 
 . 596282 47 
 
 14 
 
 890708 
 
 8 30 
 
 986459 
 
 .53 
 
 404249 
 
 8.83 
 
 595751 
 
 46 
 
 15 
 
 39120G 
 
 8 28 
 
 986427 
 
 .53 
 
 404778 
 
 8.82 
 
 595222 
 
 45 
 
 16 
 
 391703 
 
 8 27 
 
 986395 
 
 .53 
 
 405308 
 
 8.81 
 
 594692 
 
 44 
 
 17 
 
 892199 
 
 8 26 
 
 98C3G3 
 
 .54 
 
 405836 
 
 8.80 
 
 594164 43 
 
 18 
 
 392695 
 
 8 25 
 
 986331 
 
 .54 
 
 4(JG3G4 
 
 8.79 
 
 593036 42 
 
 19 
 
 893191 
 
 8 24 
 
 986299 
 
 .54 
 
 406892 
 
 8.78 
 
 593108 41 
 
 20 
 
 393685 
 
 8.23 
 
 98626G 
 
 .54 
 
 4074191 
 
 8,77 
 
 592581 
 
 40 
 
 21 
 
 9.394179 
 
 8 22 
 
 9.986234 
 
 .54 
 
 9.407945 
 
 8,76 
 
 10.592055 
 
 39 
 
 22 
 
 394G73 
 
 8.21 
 
 986202 
 
 .54 
 
 408471 
 
 8.75 
 
 591529 
 
 38 
 
 23 
 
 89516G 
 
 8.20 
 
 986169 
 
 .54 
 
 408997 
 
 8,74 
 
 591003 
 
 87 
 
 24 
 
 395658 
 
 8.19 
 
 986137 
 
 .54 
 
 409521 
 
 8,74 
 
 590479 
 
 36 
 
 25 
 
 396150 
 
 a. 18 
 
 986104 
 
 .54 
 
 410045 
 
 8,73 
 
 589955 35 | 
 
 26 
 
 396641 
 
 8.17 
 
 986072 
 
 .54 
 
 410569 
 
 8.72 
 
 589431 
 
 34 
 
 27 
 
 397132 
 
 8.17 
 
 986039 
 
 .54 
 
 411092 
 
 8.71 
 
 588908 
 
 33 
 
 28 
 
 897621 
 
 8.16 
 
 986007 
 
 .54 
 
 411G15 
 
 8.70 
 
 588385 
 
 32 
 
 ; 29 
 
 398111 
 
 8 15 
 
 985974 
 
 .54 
 
 412137 
 
 8.69 
 
 587863 
 
 31 
 
 80 
 
 398G00 
 
 8.14 
 
 985942 
 
 .54 
 
 412658 
 
 8.68 
 
 587312 
 
 30 
 
 31 
 
 9.399088 
 
 8.13 
 
 9.985909 
 
 .55 
 
 9.413179 
 
 8.67 
 
 10.586821 
 
 29 
 
 32 
 
 399575 
 
 8.12 
 
 98587G 
 
 .55 
 
 41 3699 
 
 8.66 
 
 586301 
 
 28 
 
 33 
 
 4000G2 
 
 8 11 
 
 985843 
 
 .55 
 
 414219 
 
 8.65 
 
 585781 27 1 
 
 34 
 
 400549 
 
 8.10 
 
 985811 
 
 .55 
 
 414738 
 
 8.64 
 
 585262 
 
 26 
 
 35 
 
 401035 
 
 8.09 
 
 985778 
 
 .55 
 
 415257 
 
 8.64 
 
 584743 
 
 25 
 
 36 
 
 401520 
 
 8.08 
 
 985' • 
 
 .55 
 
 415775 
 
 8.63 
 
 584225 
 
 24 
 
 . 37 
 
 402005 
 
 8.07 
 
 985712 
 
 .55 
 
 416293 
 
 8.62 
 
 583707 
 
 23 
 
 38 
 
 402489 
 
 8.06 
 
 985679 
 
 .55 
 
 416810 
 
 8.61 
 
 583190 
 
 22 
 
 39 
 
 402972 
 
 8.05 
 
 985646 
 
 .55 
 
 417320 
 
 8.60 
 
 582674 
 
 21 
 
 40 
 
 403455 
 
 8.04 
 
 985613 
 
 .55 
 
 417842 
 
 8.59 
 
 582158 
 
 20 
 
 41 
 
 9.403938 
 
 8.03 
 
 9.985580 
 
 .55 
 
 9.418358 
 
 8.58 
 
 10.581642 
 
 19 
 
 42 
 
 404420 
 
 8.02 
 
 985547 
 
 .55 
 
 418873 
 
 8.57 
 
 581127 
 
 18 
 
 . 43 
 
 404901 
 
 8.01 
 
 985514 
 
 .55 
 
 419387 
 
 8.56 
 
 580613 
 
 17 
 
 44 
 
 405382 
 
 8.00 
 
 985480 
 
 .55 
 
 419901 
 
 8.55 
 
 5800991 16 
 
 45 
 
 405862 
 
 7.99 
 
 985447 
 
 .55 
 
 420415 
 
 8.55 
 
 579585 15 
 
 46 
 
 406341 
 
 7.98 
 
 985414 
 
 .5G 
 
 420927 
 
 8.54 
 
 579073' 14 
 
 47 
 
 40682C 
 
 7.97 
 
 985380 
 
 .5C 
 
 421440 
 
 8.53 
 
 578560 
 
 13 
 
 48 
 
 f0729L 
 
 > 7.96 
 
 985347 
 
 .5G 
 
 421952 
 
 8.52 
 
 578048 
 
 12 
 
 : 49 
 
 49777^ 
 
 r 7.95 
 
 985314 
 
 .50 
 
 422463 
 
 8.51 
 
 577537 
 
 11 
 
 50 
 
 40825^ 
 
 [ 7.94 
 
 985280 
 
 .56 
 
 422974 
 
 8.50 
 
 577026 
 
 10 
 
 51 
 
 9.408731 
 
 7.94 
 
 9.985247 
 
 .50 
 
 9.423484 
 
 8.49 
 
 10.576516 
 
 9 
 
 52 
 
 409201 
 
 r 7.93 
 
 985213 
 
 .56 
 
 423993 
 
 8.48 
 
 576007 
 
 8 
 
 53 
 
 40908S 
 
 ! 7.92 
 
 985180 
 
 .56 
 
 424503 
 
 8.48 
 
 575497 
 
 7 
 
 54 
 
 41015'i 
 
 r 7.91 
 
 985148 
 
 .56 
 
 425011 
 
 8.47 
 
 574989 
 
 6 
 
 ., 55 
 
 410635 
 
 ! 7.90 
 
 985119 
 
 .56 
 
 425519 
 
 8.46 
 
 574481 
 
 5 
 
 50 
 
 41110f 
 
 ; 7.89 
 
 98507! 
 
 .56 
 
 426027 
 
 8.45 
 
 573973 
 
 4 
 
 57 
 
 41157e 
 
 » 7.88 
 
 98504C 
 
 .56 
 
 426534 
 
 8.44 
 
 573466 
 
 3 
 
 53 
 
 412055 
 
 ! 7.87 
 
 985011 
 
 .56 
 
 427041 
 
 8.43 
 
 572959 
 
 2 
 
 5!) 
 
 41252:1 
 
 t 7.86 
 
 98497s 
 
 .56 
 
 427547 
 
 8.43 
 
 572453 
 
 1 
 
 GO 
 
 41299C 
 1 Cosine. 
 
 } 7.85 
 
 984944 
 
 .51 
 
 428052 
 
 8.42 
 
 571948 
 
 
 
 Sine. 
 
 75» 
 
 1 Cotang. 
 
 D. 
 
 1 Tan^r. 
 
 M. 
 
SINES AND TANfiENTS. (15 DEGREES.) 
 
 33 
 
 M. I Sine. 
 
 
 
 1 
 
 2 
 3 
 4 
 
 5 
 
 G 
 
 7 
 
 8 
 
 •J 
 10 
 11 
 12 
 13 
 U 
 15 
 Iti 
 17 
 18 
 11) 
 20 
 
 21 t) 
 
 22 : 
 23 
 24 
 25 
 2G I 
 
 27 I 
 
 28 I 
 29 
 30 
 31 9 
 32 
 33 
 34 
 35 
 36 
 37 
 38 
 39 
 40 
 41 
 42 
 43 
 44 
 45 
 
 I 
 
 41'21»!H) 
 41:MG7 
 41:J',)3H 
 41440H 
 414S78 
 415347 
 415.S15 
 41G-283 
 41()751 
 417217 
 417(181 
 .418150 
 418(il5 
 419079 
 419544 
 420007 
 420470 
 420933 
 421395 
 421857 
 422318 
 .4227781 
 42.7i38 
 423G97 
 42415G 
 424G15 
 425073 
 425530 
 425987 
 4'2G443 
 411(5899 
 ,427354 
 427809 
 4282G3 
 428717 
 429170 
 429G23 
 430075 
 430527 
 430978 
 431429 
 .431879 
 432329 
 432778 
 43322G 
 433G75 
 
 46 
 
 434122 
 
 47 
 
 434569 
 
 48 
 
 435016 
 
 49 
 
 43i')4G2 
 
 50 
 
 435908 
 
 51 
 
 9.436353 
 
 52 
 
 43()798 
 
 53 
 
 437242 
 
 54 
 
 437G86 
 
 55 
 
 438129 
 
 56 
 
 438572 
 
 57 
 
 439014 
 
 58 
 
 439456 
 
 59 
 
 439897 
 
 GO 
 
 440338 
 
 
 1 Cosiue. 
 
 7.85 
 7. 84 
 7.83 
 7.83 
 7.82 
 
 7. 
 7. 
 7. 
 7, 
 7, 
 7 
 7 
 
 7.73 
 7.73 
 7.72 
 7,71 
 7.70 
 
 81 
 
 M) 
 79 
 78 
 77 
 7(J 
 75 
 
 69 
 68 
 67 
 67 
 6() 
 65 
 64 
 63 
 62 
 61 
 60 
 7.60 
 7.59 
 7.58 
 7.57 
 7.56 
 7.55 
 7.54 
 7.53 
 7.52 
 7.52 
 7.51 
 7.50 
 7.49 
 7.49 
 7.48 
 7.47 
 
 7. 
 7. 
 7. 
 7. 
 7. 
 7. 
 7. 
 7. 
 7. 
 7. 
 
 .46 
 .45 
 .44 
 .44 
 .43 
 .42 
 .41 
 .40 
 .40 
 .39 
 7.38 
 7.37 
 7.36 
 7 36 
 7.35 
 J7.3£ 
 
 D. 
 
 9 
 
 
 
 98 19 14 
 9S4910 
 981S76 
 984842 
 984808 
 984774 
 9 < 17 40 
 984706 
 984672 
 984637 
 981(;03 
 984569 
 9S4535 
 981500 
 98li(;(; 
 984432 
 984397 
 981363 
 984328 
 984294 
 9842^9] 
 
 9.984224 
 9ill90 
 9841551 
 984120] 
 984085 
 9840 ■)() 
 984015 
 983981 
 983946 
 983911 
 
 9.983875 
 983840 
 983805 
 983770 
 983735 
 983700 
 983664 
 983629 
 983594 
 983558 
 983523 
 983487 
 983452 
 983416 
 983381 
 983345 
 983309 
 983273 
 983238 
 983202 
 983166 
 983130 
 983094 
 983058 
 983022 
 982986 
 982!>50 
 982914 
 982878 
 982842 
 
 Cosine. I D^ 
 
 .01 
 
 .57 
 .57 
 .57 
 .57 
 .57 
 
 9 
 
 9. 
 
 .57 
 
 .•" 
 
 .."i7 
 
 .57 
 
 .57 
 
 .58 
 
 .58 
 
 .58 
 
 .58 
 
 .58 
 
 .58 
 
 .58 
 
 .58 
 
 .58 
 
 .58 
 
 .58 
 
 .58 
 
 .58 
 
 .58 
 
 .58 
 
 .58 
 
 .58 
 
 .59 
 
 .59 
 
 .59 
 
 .59 
 
 .59 
 
 .59 
 
 .59 
 
 . 59 
 
 .59 
 
 .59 
 
 .59 
 
 .59 
 
 .59 
 
 . 59 
 
 .59 
 
 .59 
 
 .60 
 
 .60 
 
 .60 
 
 .60 
 
 .60 
 
 .60 
 
 .60 
 
 .60 
 
 .60 
 
 .CO 
 
 .en 
 
 .(,i; 
 ,()() 
 
 Sine. 1 74 Cotang. 
 
 Tang. 
 
 9.42H(t52 
 428557 
 4291)62 
 4295(;() 
 4301)70 
 430573 
 431075 
 431577 
 4321(7!) 
 4325.SII 
 4;i';i»S0 
 
 9.43.3580 
 434080 
 434579 
 435078 
 43557(i 
 436073 
 43(;570 
 437067 
 437563 
 43S05!) 
 
 9.4385,54 
 439018 
 439543 
 440036 
 440529 
 441022 
 441514 
 44200(i 
 442497 
 442988 
 
 9.44.3479 
 443968 
 444458 
 444947 
 445435 
 445)23 
 44(5411 
 446898 
 447384 
 447870 
 
 9.448356 
 448841 
 449326 
 449810 
 450294 
 450777 
 451260 
 451743 
 452225 
 452706 
 
 9.453187 
 453668 
 454148 
 4.54628 
 4.55107 
 45.5586 
 456064 
 456.542 
 4.57019 
 4.")74!)() 
 
 8.42 
 
 8 41 
 
 8 40 
 
 8. 39 
 
 8,3S 
 
 8 .38 
 
 8. .37 
 
 8 36 
 
 8 , 35 
 
 8,34 
 
 8 33 
 
 8.32 
 
 8.:;2 
 
 8.31 
 
 8.30 
 
 8.29 
 
 8 28 
 
 8.28 
 
 8,27 
 
 8.26 
 
 8.-25 
 
 8.24 
 
 8.23 
 
 8.23 
 
 8.22 
 
 8.21 
 
 8, '20 
 
 8,19 
 
 8.19 
 
 8.1H 
 
 8.17 
 
 8.16 
 
 8.16 
 
 8.15 
 
 8.14 
 
 8.13 
 
 H.12 
 
 8.12 
 
 8.11 
 
 8.10 
 
 8.09 
 
 8.09 
 
 8,08 
 
 8,07 
 
 8.06 
 
 8,00 
 
 8,05 
 
 8.04 
 
 8,03 
 
 8.02 
 
 8.02 
 
 8.01 
 
 8.00 
 
 7.i)9 
 
 7.99 
 
 7,98 
 
 7.97 
 
 7,96 
 
 7,96 
 
 7,!)5 
 
 7,94 
 
 10 
 
 Cotang. 
 
 ru.571i)4s 
 571413 
 
 570938 
 5704311 
 56'.)1)301 
 .569! 27: 
 5(18925 
 .568123 
 5i)7i)21' 
 .51)7 120 : 
 
 5i;,;:;2o, 
 5(;(;420 
 .")(;.">:)2ii 
 
 5(;51!?1 
 rn'A'.>22 
 .5(U424 
 5(;3!)27 
 .")63l;!0 
 5()2933 
 .5(<2i;!7 
 561 I'll 
 5(51146 
 5li()952 
 560457 
 .5.")1)9(54 
 5.59471 
 .558!»78 
 55^486 
 .5.57994 
 55750:ij 
 5570121 
 10.. 556.521! 
 55(5<*32: 
 "42 
 
 10. 5( 
 
 10 
 
 10 
 
 55r)0.-).3i 
 554.565' 
 .5-)4077 
 5.53589; 
 553102! 
 55261(5! 
 552130; 
 ..5.51(5441 
 5.511.591 
 550(574! 
 .5.50191)' 
 549706; 
 .549223 
 548740 
 548257 
 547775 
 547291 
 ,5468131 
 54(53321 
 .545852 
 .545372 
 544S93 
 .544414! 
 54.393(5, 
 54.n458 
 5429SI' 
 542.¥t4 
 
 (50 
 
 59 
 
 58 
 
 57 
 
 56 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 50 
 
 4!) 
 
 48 
 
 47 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 40 
 
 39 
 
 38 
 
 37 
 
 3(5 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 O 
 
 2 
 1 
 
 
 
 
 Taug. M. 
 
H 
 
 
 (16 DEGREES.) A 
 
 TABLI OP LOQARITBMIO 
 
 
 M. 
 
 Sin*. 
 
 D. 
 
 Coiine. 
 
 D. 
 
 TaniL 
 
 D. 
 
 CoUnf. 
 
 
 
 
 J.44033H 
 
 7.34 
 
 9.982842 
 
 .w» 
 
 9.457496 
 
 7.94 
 
 10.542504 
 
 60 
 
 1 
 
 440778 
 
 7.83 
 
 982805 
 
 .GO 
 
 457973 
 
 7.93 
 
 542027 
 
 69 
 
 2 
 
 441218 
 
 7.32 
 
 9827G9 
 
 .61 
 
 458449 
 
 7.93 
 
 541551 
 
 68 
 
 3 
 
 441058 
 
 7.31 
 
 982733 
 
 .61 
 
 458025 
 
 7.92 
 
 511075 
 
 57 
 
 4 
 
 44209G 
 
 7.81 
 
 982(596 
 
 .61 
 
 459400 
 
 7.91 
 
 540600 
 
 66 
 
 5 
 
 44253,5 
 
 7.30 
 
 982660 
 
 .61 
 
 469875 
 
 7.90 
 
 640125 
 
 65 
 
 G 
 
 442973 
 
 7.29 
 
 982624 
 
 .01 
 
 460349 
 
 7.90 
 
 539651 
 
 64 
 
 7 
 
 443410 
 
 7.28 
 
 982587 
 
 .61 
 
 460823 
 
 7.89 
 
 539177 
 
 53 
 
 8 
 
 443847 
 
 7.27 
 
 982551 
 
 .61 
 
 461297 
 
 7.88 
 
 638703 
 
 62 
 
 9 
 
 444284 
 
 7.27 
 
 982514 
 
 .61 
 
 461770 
 
 7.88 
 
 638230 
 
 51 
 
 10 
 
 444720 
 
 7.26 
 
 982477 
 
 .61 
 
 462242 
 
 7.87 
 
 537758 
 
 60 
 
 11 
 
 9.445155 
 
 7.25 
 
 9.982441 
 
 .61 
 
 9.462714 
 
 7. 86 
 
 10.537286 
 
 49 
 
 12 
 
 445590 
 
 7.24 
 
 982404 
 
 .61 
 
 463186 
 
 7.85 
 
 536814 
 
 48 
 
 13 
 
 44G025 
 
 7.23 
 
 982367 
 
 .61 
 
 463(558 
 
 7.85 
 
 536342 
 
 47 
 
 14 
 
 44G459 
 
 7.23 
 
 982331 
 
 .61 
 
 464129 
 
 7.84 
 
 635871 
 
 46 
 
 15 
 
 44(5893 
 
 7.22 
 
 982294 
 
 .61 
 
 464599 
 
 7.83 
 
 535401 
 
 45 
 
 IG 
 
 44732G 
 
 7.21 
 
 982257 
 
 .61 
 
 465069 
 
 7.83 
 
 534931 
 
 44 
 
 17 
 
 447759 
 
 7.20 
 
 982220 
 
 .62 
 
 465539 
 
 7.82 
 
 534461 
 
 43 
 
 18 
 
 448191 
 
 7.20 
 
 982183 
 
 .62 
 
 466008 
 
 7.81 
 
 533992 
 
 42 
 
 16 i 
 
 448G23 
 
 7.19 
 
 982146 
 
 .62 
 
 466476 
 
 7.80 
 
 533524 
 
 41 
 
 20 
 
 449054 
 
 7.18 
 
 982109 
 
 .62 
 
 4G6945 
 
 7.80 
 
 533055 
 
 40 
 
 21 
 
 9.449485 
 
 7.17 
 
 9.982072 
 
 .62 
 
 9.467413 
 
 7.79 
 
 10.532587 
 
 39 
 
 22 
 
 449915 
 
 7.16 
 
 982035 
 
 .62 
 
 467880 
 
 7.78 
 
 632120 
 
 38 
 
 23 
 
 450345 
 
 7.16 
 
 981998 
 
 .62 
 
 468347 
 
 7.78 
 
 631653 
 
 37 
 
 24 
 
 450775 
 
 7.15 
 
 981961 
 
 .62 
 
 408814 
 
 7.77 
 
 531186 
 
 36 
 
 25 
 
 451204 
 
 7.14 
 
 981924 
 
 .62 
 
 4(59280 
 
 7.76 
 
 630720 
 
 35 
 
 26 
 
 451632 
 
 7.13 
 
 981886 
 
 .62 
 
 469746 
 
 7.75 
 
 530254 
 
 34 
 
 27 
 
 4520G0 
 
 7.13 
 
 981849 
 
 .62 
 
 470211 
 
 7.75 
 
 629789 
 
 33 
 
 28 
 
 452488 
 
 7.12 
 
 981812 
 
 .62 
 
 470076 
 
 7.74 
 
 629324 
 
 32 
 
 29 
 
 452915 
 
 7.11 
 
 981774 
 
 .62 
 
 471141 
 
 7.73 
 
 628859 
 
 31 
 
 30 
 
 453342 
 
 7.10 
 
 981737 
 
 .62 
 
 471605 
 
 7.73 
 
 628395 
 
 30 
 
 31 
 
 9.453768 
 
 7.10 
 
 9.981699 
 
 .63 
 
 9.472068 
 
 7.72 
 
 10.527932 
 
 29 
 
 32 
 
 454194 
 
 7.09 
 
 981662 
 
 .63 
 
 472532 
 
 7.71 
 
 627463 
 
 28 
 
 33 
 
 454G19 
 
 7.08 
 
 981625 
 
 .63 
 
 472995 
 
 7.71 
 
 624005 
 
 27 
 
 34 
 
 455044 
 
 7.07 
 
 981587 
 
 .63 
 
 473457 
 
 7.70 
 
 526543 
 
 26 
 
 35 
 
 455469 
 
 7.07 
 
 981549 
 
 .63 
 
 473919 
 
 7.69 
 
 526081 
 
 25 
 
 3G 
 
 455893 
 
 7.06 
 
 981512 
 
 .63 
 
 474381 
 
 7.69 
 
 625619 
 
 24 
 
 37 
 
 45G316 
 
 7.05 
 
 981474 
 
 .63 
 
 474842 
 
 7.G8 
 
 625168 
 
 23 
 
 38 
 
 456739 
 
 7.04 
 
 981436 
 
 .63 
 
 475303 
 
 7.67 
 
 624697 
 
 22 
 
 39 
 
 457162 
 
 7.04 
 
 981399 
 
 .63 
 
 475763 
 
 7.67 
 
 624237 
 
 21 
 
 40 
 
 457584 
 
 7.03 
 
 981361 
 
 .63 
 
 476223 
 
 7.66 
 
 623777 
 
 20 
 
 41 
 
 9.458006 
 
 7.02 
 
 9.981323 
 
 .63 
 
 9.476683 
 
 7.65 
 
 10.523317 
 
 19 
 
 42 
 
 458427 
 
 7.01 
 
 9812,i5 
 
 .63 
 
 477142 
 
 7.65 
 
 522858 
 
 18 
 
 43 
 
 458848 
 
 7.01 
 
 981247 
 
 .63 
 
 477601 
 
 7.64 
 
 522.399 
 
 17 
 
 44 
 
 459268 
 
 7.00 
 
 981209 
 
 .63 
 
 478059 
 
 7.6;; 
 
 521941 
 
 16 
 
 45 
 
 459688 
 
 6.99 
 
 981171 
 
 .63 
 
 478517 
 
 7.63 
 
 521483 
 
 15 
 
 40 
 
 4G0108 
 
 6.98 
 
 981133 
 
 .64 
 
 478975 
 
 7.62 
 
 521025 
 
 14 
 
 47 
 
 460527 
 
 6,98 
 
 981095 
 
 .64 
 
 479432 
 
 7.61 
 
 520568 
 
 13 
 
 48 
 
 460916 
 
 G.97 
 
 981057 
 
 .64 
 
 479889 
 
 7.fel 
 
 r"M]i i2 
 
 49 
 
 461304 
 
 6.96 
 
 981019 
 
 .64 
 
 480345 
 
 7.60 
 
 11 
 
 50 
 
 461782 
 
 6.95 
 
 980981 
 
 .64 
 
 480801 
 
 7.59 
 
 ,1 
 
 10 
 
 61 
 
 9.462199 
 
 G.95 
 
 9.980942 
 
 .64 
 
 9.481257 
 
 7.59 
 
 .,•43 
 
 9 
 
 52 
 
 462616 
 
 6 94 
 
 980904 
 
 .64 
 
 481712 
 
 7.58 
 
 IS288 
 
 R 
 
 53 
 
 463032 
 
 6.93 
 
 98086G 
 
 .64 
 
 482167 
 
 7.57 
 
 .'il,S33 
 
 1 
 
 64 
 
 463418 
 
 6.93 
 
 980827 
 
 .64 
 
 482021 
 
 7.57 
 
 517379 
 
 6 
 
 65 
 
 463864 
 
 6.92 
 
 980789 
 
 .64 
 
 483075 
 
 7 50 
 
 516925 
 
 6 
 
 6G 
 
 464279 
 
 6.91 
 
 980750 
 
 .64 
 
 483529 
 
 7.55 
 
 510471 
 
 4 
 
 57 
 
 464694 
 
 6.90 
 
 980712 
 
 .G4 
 
 48.3982 
 
 7.55 
 
 510018 
 
 3 
 
 58 
 
 465108 
 
 6.90 
 
 980673 
 
 .64 
 
 484435 
 
 7.. '54 
 
 515005 
 
 2 
 
 69 
 
 465522 
 
 6.89 
 
 980635 
 
 .64 
 
 484887 
 
 7. 53 
 
 515113 
 
 1 
 
 CO 
 
 465935 
 
 6.88 
 i D. 
 
 98059(5 
 
 .64 
 
 485339 
 
 7.53 
 P. 
 
 514001 
 
 
 
 
 Cosine. 
 
 Sine. 
 
 73^ 
 
 Cotang. 
 
 1 Tang. 
 
 M. 
 
30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 2 
 
 a 
 
 10 
 9 
 9 
 
 7 
 
 6 I 
 
 5 
 
 4 
 
 8 
 
 2 
 
 1 
 
 
 
 
 
 BINES AND TANGENTS. 
 
 (17 PEGRKES ) 
 
 
 85 
 
 X. 1 
 
 Sine. 
 
 D. 
 
 Cotine. 
 
 D. 1 
 
 Tany. 1 D. 
 
 CoUng. 1 
 
 
 
 3.465935 
 
 6.88 
 
 9.980596 
 
 .64 
 
 9 485339 7.52 
 
 10.514(!61i 60 
 
 1 
 
 466348 
 
 6.88 
 
 980558 
 
 .64 
 
 "485791 
 
 7.52 
 
 614209, 59 
 
 2 
 
 466761 
 
 6.87 
 
 980519 
 
 .65 
 
 486242 
 
 7.51 
 
 513758 
 
 58 
 
 3 
 
 467173 
 
 6.86 
 
 980480 
 
 .66 
 
 486693 
 
 7.51 
 
 513,307 
 
 57 
 
 4 
 
 4675H5 
 
 6.8,5 
 
 980442 
 
 .65 
 
 487143 
 
 7.50 
 
 512857 
 
 56 
 
 5 
 
 467996 
 
 0.85 
 
 980403 
 
 .65 
 
 487593 
 
 7.49 
 
 512407 
 
 55 
 
 G 
 
 4684ft7i 
 
 6.84 
 
 980364 
 
 .(» 
 
 488043 
 
 7.49 
 
 611957 
 
 54 
 
 7 
 
 468817 
 
 6.83 
 
 980325 
 
 .65 
 
 488492 
 
 7.48 
 
 511.508 
 
 53 
 
 8 
 
 469227 
 
 6.83 
 
 980286 
 
 .65 
 
 488941 
 
 7.47 
 
 511059 
 
 52 
 
 9 
 
 469637 
 
 6.82 
 
 980247 
 
 .<» 
 
 489390 
 
 7.47 
 
 510610 
 
 51 
 
 10 
 
 470046 
 
 6.81 
 
 980208 
 
 .65 
 
 489838 
 
 7.46 
 
 510162 
 
 50 
 
 11 
 
 D. 470455 
 
 6 80 
 
 9.980169 
 
 .65 
 
 9 490286 
 
 7.46 
 
 10. .509714 
 
 49 
 
 12 
 
 470803 
 
 6', 80 
 
 980130 
 
 .65 
 
 490733 
 
 7.45 
 
 .509267 
 
 48 
 
 13 
 
 471271 
 
 6.79 
 
 980091 
 
 .65 
 
 491180 
 
 7.44 
 
 508820 
 
 47 
 
 14 
 
 471679 
 
 6.78 
 
 980052 
 
 .65 
 
 491627 
 
 7.44 
 
 508373 
 
 46 
 
 15 
 
 472086 
 
 6.78 
 
 980012 
 
 .65 
 
 492073 
 
 7.43 
 
 507927 
 
 45 
 
 16 
 
 472499 
 
 6.77 
 
 979973 
 
 .65 
 
 492519 
 
 7.43 
 
 507481 
 
 44 
 
 17 
 
 472898 
 
 6.76 
 
 979934 
 
 .66 
 
 492965 
 
 7.42 
 
 ,507035 
 
 43 
 
 18 
 
 473304 
 
 6.76 
 
 979895 
 
 .66 
 
 493410 
 
 7.41 
 
 506,590 
 
 42 
 
 19 
 
 473710 
 
 6,75 
 
 979855 
 
 .66 
 
 493854 
 
 7.41 
 
 ,506146 
 
 41 
 
 20 
 
 474115 
 
 6 74 
 
 979816 
 
 .66 
 
 494299 
 
 7.40 
 
 505701 
 
 40 
 
 21 
 
 9.474519 
 
 6 74 
 
 9.979776 
 
 .66 
 
 9 494743 
 
 7.40 
 
 10.505257 
 
 39 
 
 22 
 
 474923 
 
 6.73 
 
 979737 
 
 .66 
 
 495186 
 
 7.39 
 
 504814 
 
 38 
 
 28 
 
 475327 
 
 6.72 
 
 979697 
 
 .66 
 
 495630 
 
 7.. 38 
 
 504370 
 
 37 
 
 24 
 
 475730 
 
 6.72 
 
 979658 
 
 .66 
 
 496073 
 
 7.37 
 
 ,503927 
 
 36 
 
 25 
 
 476133 
 
 6.71 
 
 979618 
 
 .66 
 
 496515 
 
 7.37 
 
 503485 
 
 35 
 
 26 
 
 476530 
 
 6.70 
 
 979579 
 
 .66 
 
 496957 
 
 7.36 
 
 50.3043 
 
 34 
 
 27 
 
 476938 
 
 6.69 
 
 979539 
 
 .66 
 
 497399 
 
 7.36 
 
 502601 
 
 33 
 
 28 
 
 477340 
 
 6.69 
 
 979499 
 
 .66 
 
 497?4i 
 
 7.35 
 
 5021.59 
 
 32 
 
 29 
 
 477741 
 
 6.68 
 
 979459 
 
 .66 
 
 498282 
 
 7.34 
 
 501718 
 
 31 
 
 30 
 
 478142 
 
 6.67 
 
 979420 
 
 .66 
 
 49872*? 
 
 7.34 
 
 501278 
 
 30 
 
 31 
 
 9.478542 
 
 6.67 
 
 9.979380 
 
 .66 
 
 9 499163 
 
 7.33 
 
 10.5008.37 
 
 2.i 
 
 32 
 
 478942 
 
 6.66 
 
 979340 
 
 .66 
 
 499603 
 
 7.33 
 
 500397 
 
 28 
 
 83 
 
 479342 
 
 6.65 
 
 979300 
 
 .67 
 
 500042 
 
 7.32 
 
 499958 
 
 27 
 
 84 
 
 479741 
 
 6.65 
 
 979260 
 
 .67 
 
 500481 
 
 7.31 
 
 499519 
 
 26 
 
 85 
 
 480140 
 
 6.64 
 
 979220 
 
 .67 
 
 500920 
 
 7.31 
 
 499080 
 
 25 
 
 86 
 
 480539 
 
 6.63 
 
 979180 
 
 .67 
 
 5013.59 
 
 7.30 
 
 498641 
 
 24 
 
 37 
 
 480937 
 
 6.63 
 
 979140 
 
 .67 
 
 501797 
 
 7.30 
 
 498203 
 
 23 
 
 88 
 
 481331 
 
 6,62 
 
 979100 
 
 .67 
 
 502235 
 
 7.29 
 
 497765 
 
 22 
 
 39 
 
 481731 
 
 6.61 
 
 979059 
 
 .67 
 
 502672 
 
 7.28 
 
 497328 
 
 21 
 
 40 
 
 482128 
 
 6.61 
 
 979019 
 
 .67 
 
 .503109 
 
 7.28 
 
 496891 
 
 20 
 
 41 
 
 9.482525 
 
 6 60 
 
 9.978979 
 
 .67 
 
 9.503546 
 
 7.27 
 
 10.496454 
 
 19 
 
 42 
 
 482921 
 
 6.59 
 
 978939 
 
 .67 
 
 503982 
 
 7.27 
 
 496018 
 
 18 
 
 43 
 
 483316 
 
 6.59 
 
 978898 
 
 .67 
 
 504118 
 
 7.26 
 
 495582 
 
 17 
 
 44 
 
 483712 
 
 6.58 
 
 978858 
 
 .67 
 
 .504854 
 
 7.25 
 
 495146 
 
 16 
 
 45 
 
 484107 
 
 6.57 
 
 978817 
 
 .67 
 
 505289 
 
 7.25 
 
 494711 
 
 15 
 
 46 
 
 484501 
 
 6.. 57 
 
 978777 
 
 .67 
 
 505724 
 
 7.24 
 
 4,)4276 
 
 14 
 
 47 
 
 4S.1395 
 
 6.56 
 
 978736 
 
 .67 
 
 506159 
 
 7.24 
 
 493841 
 
 13 
 
 48 
 
 485289 
 
 6.55 
 
 97869(5 
 
 .68 
 
 500593 
 
 7.23 
 
 49.3407 
 
 12 
 
 49 
 
 485682 
 
 6.55 
 
 97865.) 
 
 .68 
 
 507027 
 
 7.22 
 
 492973 
 
 11 
 
 50 
 
 486075 
 
 6.54 
 
 978615 
 
 .68 
 
 50746(1 
 
 7.22 
 
 492540 
 
 10 
 
 51 
 
 9.486467 
 
 6.53 
 
 9.978574 
 
 .68 
 
 9.. 507893 
 
 7.21 
 
 10.402107 
 
 9 
 
 52 
 
 486800 
 
 6.53 
 
 978533 
 
 .68 
 
 50S32('> 
 
 7.21 
 
 491(i74 
 
 8 
 
 53 
 
 487251 
 
 6.52 
 
 978493 
 
 .68 
 
 5087.5'.> 
 
 7.20 
 
 491241 
 
 7 
 
 54 
 
 487()13 
 
 6.51 
 
 978452 
 
 .68 
 
 500191 
 
 7.19 
 
 490Sni) 
 
 6 
 
 55 
 
 4880.11 
 
 6.51 
 
 978411 
 
 .68 
 
 509622 
 
 7.19 
 
 490378 
 
 5 
 
 56 
 
 4SS424 
 
 6.50 
 
 978371 
 
 .68 
 
 5100.>4 
 
 7.18 
 
 48:)946 
 
 4 
 
 57 
 
 4-!^l«14 
 
 6.50 
 
 978320 
 
 .68 
 
 510485 
 
 7.18 
 
 489515 
 
 8 
 
 58 
 
 4Hi)204 
 
 6.49 
 
 9782S,H 
 
 .6a 
 
 510916 
 
 7.17 
 
 489084 
 
 2 
 
 59 
 
 48S»5;I3 
 
 6,48 
 
 978217 
 
 .68 
 
 511346 
 
 7.16 
 
 48H654 
 
 1 
 
 60 
 
 1 
 
 489982 
 , Cosine. 
 
 6.4a 
 D. 
 
 97820r 
 
 .68 
 
 511770 
 Cotang. 
 
 7.16 
 D. 
 
 488224 
 
 
 
 Sine. 
 
 11^ 
 
 1 Tang. 
 
 H. 
 
 I 
 
 ■ 
 ;1 
 
 fi:; 
 
\r. 
 
 3ij 
 
 
 (18 OEGUEE.S.) A 
 
 TAULli OF LOGAKITHMIC 
 
 
 
 
 Sine. 
 
 'j.4H;)i»;s2 
 
 D. 
 
 6.18 
 
 Cosine. < 
 9.978206 
 
 .68 
 
 Tang. 1 
 
 D. 
 
 Cotang. 1 
 
 9.. 511776 
 
 7.16 
 
 10.48822* m 
 
 1 
 
 4!)0371 
 
 6.48 
 
 978165 
 
 .68 
 
 512206 
 
 7.16 
 
 4877941 .59 
 
 2 
 
 4!K)75!) 
 
 G.47 
 
 978124 
 
 .68 
 
 512635 
 
 7.15 
 
 487305 
 
 58 
 
 3 
 
 4'.ill47 
 
 6.46 
 
 978083 
 
 .69 
 
 5130()4 
 
 7.14 
 
 48693(j 
 
 57 
 
 i 
 
 491535 
 
 6.46 
 
 978042 
 
 .69 
 
 513493 
 
 7.14 
 
 486507 
 
 56 
 
 5 
 
 491922 
 
 6.45 
 
 978001 
 
 .69 
 
 513921 
 
 7.13 
 
 486079 
 
 55 
 
 6 
 
 492308 
 
 6.44 
 
 977959 
 
 .69 
 
 514349 
 
 7.13 
 
 485051 
 
 54 
 
 7 
 
 492C!)5 
 
 6.44 
 
 977918 
 
 .69 
 
 514777 
 
 7.12 
 
 485223 
 
 53 
 
 8 
 
 493081 
 
 6.43 
 
 977877 
 
 .69 
 
 515204 
 
 7.12 
 
 484796 
 
 52 
 
 9 
 
 4y34{iG 
 
 6.42 
 
 977835 
 
 .69 
 
 515631 
 
 7.11 
 
 4843li9 
 
 51 
 
 10 
 
 493851 
 
 6.42 
 
 977794 
 
 .69 
 
 516057 
 
 7.10 
 
 483943 
 
 50 
 
 11 
 
 9.494236 
 
 6 41 
 
 9.977752 
 
 .69 
 
 9.516484 
 
 7.10 
 
 10.48.3516 
 
 49 
 
 12 
 
 494621 
 
 6.41 
 
 977711 
 
 .69 
 
 516910 
 
 7.09 
 
 483090 48 
 
 13 
 
 495005 
 
 6.40 
 
 977609 
 
 .69 
 
 517335 
 
 7.09 
 
 4826(;5j 47 
 
 14 
 
 49.5.'?88 
 
 6.39 
 
 977628 
 
 .69 
 
 517761 
 
 7.08 
 
 482239 46 
 
 ^r^ 
 
 495772 
 
 6.39 
 
 977586 
 
 .69 
 
 518185 
 
 7.08 
 
 481815 45 
 
 lii 
 
 4!>;>154 
 
 6.38 
 
 977544 
 
 .70 
 
 518610 
 
 7.07 
 
 4813901 44 
 
 17 
 
 490537 
 
 6 37 
 
 977503 
 
 .70 
 
 519034 
 
 7.06 
 
 4809661 43 
 
 18 
 
 490919 
 
 6.37 
 
 977461 
 
 .70 
 
 519458 
 
 7.06 
 
 480542! 42 
 
 li) 
 
 497.301 
 
 6.36 
 
 977419 
 
 .70 
 
 51!)882 
 
 7.05 
 
 480118; 41 
 
 2i) 
 
 497682 
 
 6.36 
 
 977377 
 
 .70 
 
 520305 
 
 7.05 
 
 4790'J5 40 
 
 21 
 
 ;}. 498064 
 
 6 .35 
 
 9 977335 
 
 .70 
 
 9.. 520728 
 
 7.04 
 
 10.479272! 39 
 
 '21 
 
 49S414 
 
 6.34 
 
 9772s)3 
 
 .70 
 
 521ir>l 
 
 7.03 
 
 478849 38 
 
 2:} 
 
 49H825 
 
 6.31 
 
 977251 
 
 .70 
 
 521573 
 
 7.03 
 
 478127 .37 
 
 24 
 
 499204 
 
 6.33 
 
 97720'.> 
 
 .70 
 
 521995 
 
 7.03 
 
 478005, 36 
 
 2.1 
 
 499584 
 
 6.32 
 
 977167 
 
 .70 
 
 522417 
 
 7.02 
 
 477583' 35 
 
 2») 
 
 49;)9(}3 
 
 6.32 
 
 977125 
 
 .70 
 
 522838 
 
 7.02 
 
 477162 34 
 
 27 
 
 50(1.342 
 
 6.31 
 
 977083 
 
 .70 
 
 523259 
 
 7 01 
 
 476741' 33 
 
 28 
 
 500721 
 
 6.31 
 
 977011 
 
 .70 
 
 523680 
 
 7.01 
 
 476320 32 
 
 2'.) 
 
 .501099 
 
 6.30 
 
 976999 
 
 .70 
 
 524100 
 
 7.00 
 
 475900 31 
 
 30 
 
 501476 
 
 6.29 
 
 976957 
 
 .70 
 
 524520 
 
 6.99 
 
 475480 30 
 
 31 
 
 9.501851 
 
 6.29 
 
 9.976914 
 
 .70 
 
 9.524939 
 
 6.99 
 
 10.475001 29 
 
 32 
 
 502231 
 
 6.28 
 
 976872 
 
 .71 
 
 5253.59 
 
 6.98 
 
 474641^ 28 
 
 33 
 
 502(507 
 
 6.28 
 
 976830 
 
 .71 
 
 525778 
 
 6.98 
 
 474222; 27 
 
 34 
 
 502984 
 
 6.27 
 
 976787 
 
 .71 
 
 526197 
 
 6.97 
 
 473S03 26 
 
 35 
 
 503360 
 
 6.26 
 
 976745 
 
 .71 
 
 526615 
 
 6.97 
 
 4733S5| 25 
 
 3»i 
 
 5()3735 
 
 6.26 
 
 976702 
 
 .71 
 
 527033 
 
 6.96 
 
 4729671 24 
 
 37 
 
 504110 
 
 6.25 
 
 976660 
 
 .71 
 
 627451 
 
 6.96 
 
 47254:)i 23 
 
 38 
 
 504485 
 
 6.25 
 
 976617 
 
 .71 
 
 527868 
 
 6.95 
 
 472132J 22 
 
 3!) 
 
 504860 
 
 6.24 
 
 976574 
 
 .71 
 
 528285 
 
 6 . i)5 
 
 4717151 21 
 
 40 
 
 505234 
 
 6.23 
 
 9765321 .71 
 
 528702 
 
 6.94 
 
 471298' 20 
 
 41 
 
 9.50.5(;08 
 
 6.23 
 
 9.976489 
 
 .71 
 
 9.529119 
 
 6.93 
 
 10.470881: 19 
 
 42 
 
 505981 
 
 6.22 
 
 976446 
 
 .71 
 
 529535 
 
 6.93 
 
 470465' 18 
 
 1 i^ 
 
 506354 
 
 6.22 
 
 976404 
 
 .71 
 
 529950 
 
 6.93 
 
 470050 17 
 
 44 
 
 506727 
 
 6.21 
 
 976361 
 
 .71 
 
 530366 
 
 6.92 
 
 4696.34, 16 
 
 45 
 
 507099 
 
 6.20 
 
 976318 
 
 .71 
 
 530781 
 
 6.91 
 
 4692191 15 
 
 46 
 
 507471 
 
 6.20 
 
 976275 
 
 .71 
 
 531196 
 
 6.91 
 
 468804: 14 
 
 47 
 
 507843 
 
 6.19 
 
 976232 
 
 .72 
 
 531611 
 
 6.90 
 
 468389 
 
 13 
 
 48 
 
 508214 
 
 6.19 
 
 976189 
 
 .72 
 
 532025 
 
 6.90 
 
 467975 
 
 12 
 
 49 
 
 508585 
 
 6.18 
 
 976146 
 
 .72 
 
 532439 
 
 6.89 
 
 4«75(;i! 11 1 
 
 50 
 
 508956 
 
 6.18 
 
 976103 
 
 .72 
 
 532853 
 
 6,89 
 
 467147 
 
 10 
 
 51 
 
 9.50932G 
 
 6.i7 
 
 9.976060 
 
 .72 
 
 9.5.33266 
 
 6.88 
 
 10.466734 
 
 9 
 
 52 
 
 509696 
 
 6.16 
 
 976017 
 
 .72 
 
 533679 
 
 6.88 
 
 466321 
 
 8 
 
 53 
 
 510065 
 
 6.16 
 
 975974 
 
 .72 
 
 534092 
 
 6.87 
 
 465908 
 
 7 
 
 54 
 
 510434 
 
 6.15 
 
 975930 
 
 .72 
 
 534504 
 
 6.87 
 
 465496 
 
 3 
 
 55 
 
 510803 
 
 6 15 
 
 975887 
 
 .72 
 
 534916 
 
 6.86 
 
 465084' 5 
 
 5<) 
 
 511172 
 
 6.14 
 
 975844 
 
 .72 
 
 535328 
 
 6.86 
 
 464672 4 
 
 57 
 
 511.540 
 
 6.13 
 
 975800 
 
 .72 
 
 535739 
 
 6.85 
 
 464261 3 
 
 58 
 
 511907 
 
 6.13 
 
 975757 
 
 .72 
 
 5.36150 
 
 6.85 
 
 463850 2 
 
 5i) 
 
 512275 
 
 6.12 
 
 975714 
 
 .72 
 
 5.36r>6] 
 
 6.84 
 
 463439 1 
 
 00 
 
 512642 
 
 6.12 
 
 975670! .72 
 
 53()972 
 
 6.84 
 
 463028 
 
 
 1 Cosine. 
 
 1 D. 
 
 I 8in<3. (71 
 
 Cotang. 
 
 D. 
 
 Taiig. ; M. 1 
 
m 
 
 51) 
 58 
 57 
 56 
 
 54 
 53 
 52 
 51 
 
 50 
 49 
 
 48 
 I 47 
 
 i 44 
 43 
 
 i 42 
 41 
 40 
 39 
 38 
 37 
 3() 
 35 
 34 
 33 
 32 
 31 
 30 
 29 
 28 
 27 
 26 
 25 
 24 
 I 23 
 
 21321 22 
 
 51 21 
 .298 20 
 li 19 
 
 i5: 18 
 
 |)0;)0l 17 
 J)G34, 16 
 1)219 15 
 lt;804: 14 
 k389i 13 
 b75 12 
 
 1147 
 |734 
 l321 
 
 1908 
 
 |49G 
 
 )84 
 
 i! 11 
 
 10 
 
 5 
 
 372 4 
 201 3 
 
 150 2 
 
 :39i 1 
 
 )28 
 M. 
 
 
 
 SINES AND TANGENTS. 
 
 (IL> DEOilEES.) 
 
 
 37 
 
 M. 
 
 Sine. 1 
 
 D. 
 
 6.12 
 
 Cosino. 1 D. ! 
 9.97.')t;7(l .73i 
 
 Taag. ' 
 
 <) 5:!.i.>(2i 
 
 D. 
 
 (> si 
 
 Cotang. 1 
 
 10.403I)2H, 
 
 00 
 
 1 
 
 9.512042 
 
 1 
 
 513009 
 
 6.11 
 
 975027 
 
 .73 
 
 5373S2 
 
 6.83 
 
 4(52018: 
 
 .59 
 
 2 
 
 513375 
 
 6.11 
 
 9755S3 
 
 .73 
 
 537792 
 
 0.83 
 
 40220S 
 
 58 
 
 3 
 
 513741 
 
 6.10 
 
 975539 
 
 .73 
 
 538202 
 
 6.82 
 
 461798 
 
 57 
 
 4 
 
 514107 
 
 6.09 
 
 975190 
 
 .73 
 
 538611 
 
 6.82 
 
 4(51389 
 
 50 
 
 5 
 
 514472 
 
 6.09 
 
 975452 
 
 .73 
 
 539020 
 
 6.81 
 
 4009.SO 
 
 55 
 
 6 
 
 514837 
 
 6.08 
 
 975408 
 
 .73 
 
 539429 
 
 6.81 
 
 4(505711 
 
 54 
 
 7 
 
 515202 
 
 6.08 
 
 975365 
 
 .73 
 
 53!)837 
 
 6.80 
 
 4001(531 
 
 53 
 
 8 
 
 5155()6 
 
 6.07 
 
 975321 
 
 .73 
 
 510245 
 
 6 80 
 
 459755 
 
 52 
 
 9 
 
 515930 
 
 0.07 
 
 975277 
 
 .73 
 
 510053 
 
 79 
 
 459347 
 
 51 
 
 10 
 
 516294 
 
 6.00 
 
 975233 
 
 .73 
 
 541061 
 
 79 
 
 458939 50 
 
 11 
 
 9.516057 
 
 0.05 
 
 9.975189 
 
 .73 
 
 9 541468 
 
 6 78 
 
 10.458532 49 
 
 12 
 
 517020 
 
 6.05 
 
 975145 
 
 .73 
 
 541875 
 
 78 
 
 45.S125 48 
 
 13 
 
 517382 
 
 6.94 
 
 975101 
 
 .73 
 
 542281 
 
 6 77 
 
 4.5771iH 47 
 
 14 
 
 517745 
 
 6.04 
 
 975058 
 
 .73 
 
 542088 
 
 77 
 
 457312 40 
 
 15 
 
 518107 
 
 6.03 
 
 975013 
 
 .73 
 
 .543094 
 
 6.76 
 
 45(5906 45 
 
 16 
 
 518408 
 
 6.03 
 
 974909 
 
 .74 
 
 543499 
 
 6 70 
 
 456501 44 
 
 17 
 
 518829 
 
 6.02 
 
 974925 
 
 .74 
 
 543905 
 
 G 75 
 
 4.50095 43 
 
 18 
 
 519190 
 
 6.01 
 
 974880 
 
 .74 
 
 544310 
 
 6' 75 
 
 455090 
 
 42 
 
 19 
 
 519551 
 
 6.01 
 
 974836 
 
 .74 
 
 544715 
 
 6.74 
 
 4.55285 
 
 41 
 
 20 
 
 519911 
 
 6.00 
 
 974792 
 
 .74 
 
 545119 
 
 6 74 
 
 4548H1 
 
 40 
 
 21 
 
 9.520271 
 
 6.00 
 
 9.974748 
 
 .74 
 
 9 545524 
 
 6 73 
 
 10.454476 
 
 39 
 
 22 
 
 520031 
 
 5.99 
 
 974703 
 
 .74 
 
 545928 
 
 6 73 
 
 454072 
 
 38 
 
 23 
 
 520990 
 
 5.99 
 
 974659 
 
 .74 
 
 54<)331 
 
 672 
 
 4536(59 
 
 37 
 
 24 
 
 521349 
 
 5.98 
 
 974614 
 
 .74 
 
 54()735 
 
 6 72 
 
 453265 
 
 36 
 
 25 
 
 521707 
 
 5.98 
 
 974570 
 
 .74 
 
 547138 
 
 6 71 
 
 452862 
 
 35 
 
 26 
 
 522060 
 
 5.97 
 
 974525 
 
 .74 
 
 547540 
 
 0.71 
 
 4524(50 
 
 34 
 
 27 
 
 522424 
 
 5.96 
 
 974481 
 
 .74 
 
 517943 
 
 6 70 
 
 4520.57 
 
 33 
 
 28 
 
 522781 
 
 5.96 
 
 974436 
 
 .74 
 
 54H.345 
 
 6 70 
 
 4516.55 
 
 32 
 
 29 
 
 523138 
 
 5.95 
 
 974391 
 
 .74 
 
 548747 
 
 6.(59 
 
 451253 
 
 31 
 
 30 
 
 523495 
 
 5.95 
 
 974347 
 
 .75 
 
 549149 
 
 6.69 
 
 4.50851 
 
 30 
 
 31 
 
 9.523852 
 
 5.94 
 
 9.974302 
 
 .75 
 
 9 51!)550 
 
 6 (58 
 
 10.4.5(1450 
 
 29 
 
 32 
 
 524208 
 
 5.04 
 
 974257 
 
 .75 
 
 549951 
 
 (58 
 
 4.50949 
 
 28 
 
 33 
 
 524564 
 
 5.93 
 
 974212 
 
 .75 
 
 550352 
 
 67 
 
 449648 
 
 27 
 
 34 
 
 524920 
 
 5.93 
 
 974107 
 
 .75 
 
 550752 
 
 0.67 
 
 44924S 
 
 26 
 
 35 
 
 525275 
 
 5.92 
 
 974122 
 
 .75 
 
 551152 
 
 6 66 
 
 44SS4S 
 
 25 
 
 30 
 
 525030 
 
 5.91 
 
 974077 
 
 .75 
 
 551552 
 
 6 (56 
 
 448448 
 
 24 
 
 37 
 
 525984 
 
 5.91 
 
 974032 
 
 .75 
 
 551952 
 
 6 65 
 
 44sn48 
 
 23 
 
 38 
 
 526339 
 
 5.90 
 
 973987 
 
 .75 
 
 552351 
 
 65 
 
 447(54!) 
 
 22 
 
 39 
 
 526693 
 
 5.90 
 
 973942 
 
 .75 
 
 552750 
 
 6 05 
 
 447250 
 
 21 
 
 40 
 
 527046 
 
 5.89 
 
 9738S)7 
 
 .75 
 
 553149 
 
 64 
 
 440851 
 
 20 
 
 41 
 
 9.527400 
 
 5.89 
 
 9.973852 
 
 .75 
 
 9.553548 
 
 6 64 
 
 10.4101.)2 
 
 19 
 
 42 
 
 527753 
 
 5.88 
 
 973307 
 
 .75 
 
 553946 
 
 (5 63 
 
 44(5051 
 
 18 
 
 43 
 
 528105 
 
 5.88 
 
 973761 
 
 .75 
 
 551314 
 
 (53 
 
 445(550 
 
 17 
 
 44 
 
 528458 
 
 5.87 
 
 973716 
 
 .76 
 
 554711 
 
 6 62 
 
 445259 
 
 16 
 
 45 
 
 528810 
 
 5.87 
 
 973671 
 
 .70 
 
 555139 
 
 0.(52 
 
 444801 
 
 15 
 
 46 
 
 529101 
 
 5.86 
 
 973625 
 
 .70 
 
 555530 
 
 6.(51 
 
 4414(54 
 
 14 
 
 47 
 
 529513 
 
 5.86 
 
 973580 
 
 .76 
 
 555933 
 
 6 61 
 
 444007 
 
 13 
 
 48 
 
 5298(54 
 
 5.85 
 
 973535 
 
 .76 
 
 556329 
 
 (5 00 
 
 413071 
 
 12 
 
 49 
 
 530215 
 
 5.85 
 
 973489 
 
 .76 
 
 550725 
 
 <50 
 
 443275 
 
 11 
 
 50 
 
 530505 
 
 5.84 
 
 973444 
 
 .76 
 
 5.57121 
 
 6 59 
 
 442fH7:» 
 
 10 
 
 51 
 
 9.530915 
 
 5.84 
 
 9.973398 
 
 .70 
 
 9 557517 
 
 0.59 
 
 10 442483 
 
 9 
 
 52 
 
 531265 
 
 5.83 
 
 973352 
 
 .76 
 
 557913 
 
 0.59 
 
 442087 
 
 8 
 
 53 
 
 531614 
 
 9.82 
 
 973307 
 
 .76 
 
 55S.308 
 
 6.58 
 
 441(592 
 
 7 
 
 54 
 
 531963 
 
 5.82 
 
 973261 
 
 .70 
 
 558702 
 
 (5 58 
 
 44129H 
 
 
 
 55 
 
 532312 
 
 5.81 
 
 973215 
 
 .76 
 
 559097 
 
 0.57 
 
 440903 
 
 5 
 
 56 
 
 532661 
 
 5.81 
 
 973169 
 
 .76 
 
 559491 
 
 6.57 
 
 440.509 
 
 4 
 
 57 
 
 533009 
 
 5.80 
 
 973124 
 
 .70 
 
 559885 
 
 6 50 
 
 440115 
 
 3 
 
 68 
 
 533357 
 
 5.80 
 
 973078 
 
 .76 
 
 5(50279 
 
 6 56 
 
 439721 
 
 2 
 
 59 
 
 533704 
 
 5.79 
 
 973032 
 
 .77 
 
 5(50673 
 
 ' 0.55 
 
 439327 
 
 1 
 
 60 
 
 534052 
 
 5.78 
 
 972!t86| .77 
 
 5010ii() 
 Cotang. 
 
 5.-. 
 
 438931 
 Tang. 
 
 (» 
 
 Cotino. 
 
 D. 
 
 Sine. 
 
 i70 
 
 D. 
 
 TW. 
 
 
38 
 
 
 (20 DEGREES.) A 
 
 TABLE OF LOGARITHMIC 
 
 
 M. 
 
 
 
 Sine. 
 
 D. 
 
 5.78 
 
 Coiine. 
 
 D. 1 
 
 .77 
 
 Tanjj. 
 
 D. 
 
 Cotang. 
 
 10.4389;M 
 
 (50 
 
 9.534052 
 
 9.972986 
 
 9.5010(50 
 
 6.55 
 
 1 
 
 534399 
 
 5.77 
 
 972940 
 
 .77 
 
 501459 
 
 6.54 
 
 438541 
 
 59 
 
 2 
 
 534745 
 
 5.77 
 
 972894 
 
 .77 
 
 561851 
 
 6.54 
 
 438149 
 
 58 
 
 3 
 
 53r)092 
 
 5.77 
 
 972848 
 
 .77 
 
 562244 
 
 6.53 
 
 437750 
 
 57 
 
 4 
 
 535438 
 
 5.70 
 
 972802 
 
 .77 
 
 56263(5 
 
 0.53 
 
 437304 
 
 56 
 
 5 
 
 535783 
 
 5.70 
 
 972755 
 
 .77 
 
 563028 
 
 6.53 
 
 430972 
 
 55 
 
 6 
 
 53(5129 
 
 5.75 
 
 972709 
 
 .77 
 
 563419 
 
 6.52 
 
 430581 
 
 54 
 
 7 
 
 530474 
 
 5.74 
 
 972663 
 
 .77 
 
 563811 
 
 6.52 
 
 430189 
 
 53 
 
 8 
 
 530818 
 
 5.74 
 
 972017 
 
 .77 
 
 564202 
 
 6.51 
 
 435798 
 
 52 
 
 9 
 
 537103 
 
 5.73 
 
 972570 
 
 .77 
 
 564592 
 
 0.51 
 
 435408 
 
 51 
 
 10 
 
 537507 
 
 5.73 
 
 972524 
 
 .77 
 
 564983 
 
 6.50 
 
 43.5017 
 
 50 
 
 11 
 
 9.537851 
 
 5.72 
 
 9.972478 
 
 .77 
 
 9.505373 
 
 6,50 
 
 10.434(527 
 
 49 
 
 12 
 
 538194 
 
 5.72 
 
 972431 
 
 .78 
 
 565703 
 
 49 
 
 434237 
 
 48 
 
 13 
 
 538538 
 
 5 71 
 
 972385 
 
 .78 
 
 566153 
 
 6.49 
 
 438847 
 
 47 
 
 14 
 
 538880 
 
 5.71 
 
 972338 
 
 .78 
 
 566542 
 
 6,49 
 
 433458 
 
 46 
 
 15 
 
 539223 
 
 5.70 
 
 972291 
 
 .78 
 
 5(56932 
 
 6.48 
 
 4330(58 
 
 45 
 
 lU 
 
 539505 
 
 5.70 
 
 972245 
 
 .78 
 
 507.320 
 
 6,48 
 
 432080 
 
 44 
 
 17 
 
 539907 
 
 5.69 
 
 972198 
 
 .78 
 
 507709 
 
 -•.47 
 
 432291 
 
 43 
 
 18 
 
 540249 
 
 5.69 
 
 972151 
 
 .78 
 
 508098 
 
 6,47 
 
 431902 
 
 42 
 
 VJ 
 
 540590 
 
 5.68 
 
 972105 
 
 .78 
 
 568486 
 
 6.46 
 
 431514 
 
 41 
 
 20 
 
 540931 
 
 5.68 
 
 972058 
 
 .78 
 
 568873 
 
 6.46 
 
 431127 
 
 40 
 
 21 
 
 9 541272 
 
 5.67 
 
 9.972011 
 
 .78 
 
 9.5092(51 
 
 6.45 
 
 10.430739 
 
 39 
 
 22 
 
 541013 
 
 5.67 
 
 971964 
 
 .78 
 
 509648 
 
 6,45 
 
 430352 
 
 38 
 
 23 
 
 541953 
 
 5.60 
 
 971917 
 
 .78 
 
 570035 
 
 6,45 
 
 4299(55 
 
 37 
 
 M 
 
 542293 
 
 5.66 
 
 971870 
 
 .78 
 
 570422 
 
 6.44 
 
 429578 
 
 3(5 
 
 25 
 
 542032 
 
 5.65 
 
 971823 
 
 .78 
 
 570809 
 
 6,44 
 
 429191 
 
 35 
 
 2G 
 
 542971 
 
 5.65 
 
 971770 
 
 .78 
 
 571195 
 
 6,43 
 
 428805 
 
 34 
 
 27 
 
 543310 
 
 5.64 
 
 971729 
 
 .79 
 
 571581 
 
 6,43 
 
 428419 
 
 33 
 
 28 
 
 543049 
 
 5.64 
 
 971682 
 
 .79 
 
 571907 
 
 6.42 
 
 428033 
 
 32 
 
 2d 
 
 543987 
 
 5,63 
 
 971635 
 
 .79 
 
 572352 
 
 6.42 
 
 427048 
 
 31 
 
 30 
 
 544325 
 
 5.63 
 
 971588 
 
 .79 
 
 572738 
 
 0,42 
 
 427202 
 
 30 
 
 31 
 
 9.544063 
 
 5 62 
 
 9.971540 
 
 .79 
 
 9.57.3123 
 
 6,41 
 
 10.42(5877 
 
 29 
 
 32 
 
 545000 
 
 5.62 
 
 971 193 
 
 .79 
 
 573507 
 
 6.41 
 
 420493 
 
 28 
 
 33 
 
 545338 
 
 5,01 
 
 971446 
 
 .79 
 
 573892 
 
 6.40 
 
 426108 
 
 27 
 
 34 
 
 545074 
 
 5.01 
 
 971398 
 
 .79 
 
 574276 
 
 6.40 
 
 425724 
 
 20 
 
 35 
 
 540011 
 
 5,60 
 
 971351 
 
 .79 
 
 574000 
 
 6,39 
 
 425340 
 
 25 
 
 30 
 
 540347 
 
 5 (10 
 
 971303 
 
 .79 
 
 575044 
 
 6.39 
 
 424950 
 
 24 
 
 37 
 
 54(5083 
 
 5 59 
 
 971250 
 
 .79 
 
 575427 
 
 6,39 
 
 424573 
 
 23 
 
 38 
 
 517019 
 
 5.59 
 
 971208 
 
 .79 
 
 575810 
 
 6,38 
 
 424190 
 
 22 
 
 3!) 
 
 547354 
 
 5.58 
 
 971101 
 
 .79 
 
 570193 
 
 6,38 
 
 423807 
 
 21 
 
 40 
 
 547089 
 
 5 5S 
 
 971113 
 
 .79 
 
 570570 
 
 6,37 
 
 423424 
 
 20 
 
 41 
 
 9.54S024 
 
 5,57 
 
 9.971000 
 
 .80 
 
 9.570958 
 
 6,37 
 
 10.423041 
 
 19 
 
 42 
 
 548359 
 
 5,57 
 
 971018 
 
 .80 
 
 577:^41 
 
 6.36 
 
 422059 
 
 18 
 
 43 
 
 548093 
 
 5,50 
 
 970970 
 
 .80 
 
 577723 
 
 6.36 
 
 422277 
 
 17 
 
 44 
 
 549027 
 
 5,50 
 
 970922 
 
 .80 
 
 578104 
 
 6,30 
 
 421890 
 
 16 
 
 45 
 
 549300 
 
 5,55 
 
 970874 
 
 .80 
 
 578480 
 
 6.35 
 
 421.514 
 
 15 
 
 4U 
 
 549093 
 
 5,55 
 
 970827 
 
 .80 
 
 5788(57 
 
 6,35 
 
 421133 
 
 14 
 
 47 
 
 550020 
 
 5,54 
 
 970779 
 
 .80 
 
 579248 
 
 6,34 
 
 420752 
 
 13 
 
 48 
 
 550859 
 
 5,54 
 
 970731 
 
 .80 
 
 579029 
 
 0.34 
 
 420.371 
 
 12 
 
 4'J 
 
 550092 
 
 5,53 
 
 970083 
 
 .80 
 
 580009 
 
 6,34 
 
 419991 
 
 11 
 
 50 
 
 551024 
 
 5.53 
 
 970035 
 
 .80 
 
 .580389 
 
 6,33 
 
 419(511 
 
 10 
 
 51 
 
 9.551350 
 
 5.52 
 
 9.970580 
 
 .80 
 
 9.. 58070!) 
 
 0,33 
 
 10.419231 
 
 9 
 
 52 
 
 551087 
 
 5.52 
 
 970538 
 
 .80 
 
 .581149 
 
 6,32 
 
 4188.51 
 
 8 
 
 53 
 
 552018 
 
 5.52 
 
 970490 
 
 .80 
 
 581.528 
 
 6.32 
 
 418472 
 
 7 
 
 54 
 
 552349 
 
 5,51 
 
 970442 
 
 .80 
 
 581907 
 
 6.32 
 
 418093 
 
 6 
 
 55 
 
 552(580 
 
 5.51 
 
 970394 
 
 .80 
 
 582286 
 
 6,31 
 
 417714 
 
 5 
 
 56 
 
 553010 
 
 5.50 
 
 970345 
 
 .81 
 
 582065 
 
 6.31 
 
 4173.35 
 
 4 
 
 57 
 
 553341 
 
 5.50 
 
 970297 
 
 .81 
 
 583043 
 
 6,30 
 
 4109.57 
 
 3 
 
 58 
 
 553070 
 
 5.49 
 
 970249 
 
 .81 
 
 583422 
 
 6.30 
 
 410578 
 
 2 
 
 59 
 
 554000 
 
 5.49 
 
 970200 
 
 .81 
 
 583800 
 
 6,29 
 
 410200 
 
 1 
 
 CO 
 
 554329 
 Oosino. 
 
 5.48 
 D. 
 
 970152 
 
 .81 
 69° 
 
 584177 
 
 6,29 
 
 415823 
 Taug._ 
 
 
 
 Sln'>. 
 
 i Cotang. 
 
 D. 
 
 M, 
 
38 
 
 37 
 
 3(5 
 
 35 
 
 Si 
 
 33 
 
 32 
 
 31 
 
 30 
 
 25) 
 
 28 
 
 27 
 
 2(5 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 Ul 10 
 18 
 17 
 16 
 15 
 U 
 13 
 12 
 11 
 10 
 
 M 
 
 
 
 SINKS AND TANGENTS. 
 
 (21 DEGREES.) 
 
 
 39 
 
 
 
 Sine. 
 
 D. 
 
 5.48 
 
 Cosine. { 
 
 D. 
 
 Tang. 
 
 D. 
 
 Cotang. 
 
 
 J. 554329 
 
 9.970152 
 
 .81 
 
 9.584177 
 
 6.29 
 
 10.415823 
 
 60 
 
 1 
 
 554658 
 
 5.48 
 
 970103 
 
 .81 
 
 684555 
 
 6.29 
 
 41.5445 
 
 69 
 
 2 
 
 554987 
 
 5.47 
 
 970055 
 
 .81 
 
 684932 
 
 6.28 
 
 415068 
 
 68 
 
 8 
 
 555315 
 
 5.47 
 
 970006 
 
 .81 
 
 585309 
 
 6.28 
 
 414691 
 
 57 
 
 4 
 
 655843 
 
 6.46 
 
 909957 
 
 .81 
 
 585686 
 
 6 27 
 
 414314 
 
 56 
 
 5 
 
 555971 
 
 5.46 
 
 969909 
 
 .81 
 
 586062 
 
 6.27 
 
 413938 
 
 65 
 
 6 
 
 550299 
 
 5.45 
 
 969360 .811 
 
 586439 
 
 6.27 
 
 413561 
 
 64 
 
 7 
 
 556626 
 
 5.45 
 
 969811 
 
 .81 
 
 686815 
 
 6.26 
 
 413185 
 
 63 
 
 8 
 
 556953 
 
 5 44 
 
 969762 
 
 .81 
 
 687190 
 
 6.26 
 
 412810 
 
 52 
 
 9 
 
 557280 
 
 5.44 
 
 969714 
 
 .81 
 
 587566 
 
 6.25 
 
 412434 
 
 51 
 
 10 
 
 557606 
 
 5.43 
 
 969665 
 
 .81 
 
 587941 
 
 6.25 
 
 412059 60 
 
 11 
 
 9.557932 
 
 5.43 
 
 9.969616 
 
 .82 
 
 9.588316 
 
 6.25 
 
 10.411684 49 
 
 12 
 
 658258 
 
 5.43 
 
 969567 
 
 .82 
 
 588691 
 
 6.24 
 
 411309 48 
 
 18 
 
 658583 
 
 5.42 
 
 969518 
 
 .82 
 
 589066 
 
 6.24 
 
 410934 47 
 
 U 
 
 558901) 
 
 5.42 
 
 969469 
 
 .82 
 
 589440 
 
 6.23 
 
 410.500 40 
 
 15 
 
 659234 
 
 5.41 
 
 969420 
 
 .82 
 
 589814 
 
 6.23 
 
 410186 
 
 45 
 
 16 
 
 559558 
 
 5.41 
 
 969370 
 
 .82 
 
 590188 
 
 6.23 
 
 409812 
 
 44 
 
 17 
 
 559883 
 
 5.40 
 
 969321 
 
 .82 
 
 590562 
 
 6.22 
 
 405)438 
 
 43 
 
 18 
 
 660207 
 
 5.40 
 
 909272 
 
 .82 
 
 590935 
 
 6.22 
 
 409065 
 
 42 
 
 19 
 
 660531 
 
 5.39 
 
 969223 
 
 .82 
 
 5!)1308 
 
 6.22 
 
 408692 
 
 41 
 
 20 
 
 560855 
 
 5.39 
 
 969173 
 
 .82 
 
 591681 
 
 6.21 
 
 408319 
 
 40 
 
 21 
 
 9.561178 
 
 5.38 
 
 9.969124 
 
 .82 
 
 9.592054 
 
 6.21 
 
 10.407940 
 
 39 
 
 22 
 
 561501 
 
 5.38 
 
 90!)075 
 
 .82 
 
 592426 
 
 6.20 
 
 407^74 
 
 38 
 
 23 
 
 561824 
 
 5.37 
 
 969025 
 
 .82 
 
 55)2798 
 
 6.20 
 
 407202 37 ! 
 
 24 
 
 562146 
 
 5.37 
 
 968976 
 
 .82 
 
 593170 
 
 6.19 
 
 406829 
 
 36 
 
 25 
 
 662468 
 
 5.36 
 
 968926 
 
 .83 
 
 593542 
 
 6.19 
 
 406458 
 
 35 
 
 26 
 
 562790 
 
 5.36 
 
 96So77 
 
 .83 
 
 593914 
 
 6.18 
 
 406086 
 
 84 
 
 27 
 
 563112 
 
 5.36 
 
 968827 
 
 .83 
 
 594285 
 
 6.18 
 
 405715 
 
 33 
 
 28 
 
 563433 
 
 5.35 
 
 968777 
 
 .83 
 
 594656 
 
 6,18 
 
 405344 
 
 32 
 
 29 
 
 563755 
 
 5 35 
 
 968728 
 
 .83 
 
 595027 
 
 6.17 
 
 404973 
 
 31 
 
 80 
 
 564075 
 
 5.34 
 
 968678 
 
 .83 
 
 59535)8 
 
 6.17 
 
 404602 
 
 80 
 
 31 
 
 9.564396 
 
 5.34 
 
 9.968628 
 
 .83 
 
 9.55)5768 
 
 6.17 
 
 10.404232 
 
 29 
 
 32 
 
 564716 
 
 5.33 
 
 968578 
 
 .83 
 
 596138 
 
 6.16 
 
 403862 
 
 28 
 
 33 
 
 565036 
 
 5.33 
 
 968528 
 
 .88 
 
 596508 
 
 6.16 
 
 403492 
 
 27 
 
 84 
 
 565356 
 
 5.32 
 
 968479 
 
 .83 
 
 596878 
 
 6.16 
 
 403122 
 
 26 
 
 35 
 
 665676 
 
 5.32 
 
 968429 
 
 .83 
 
 597247 
 
 6.15 
 
 402753 
 
 25 
 
 36 
 
 565995 
 
 5.31 
 
 968379 
 
 .83 
 
 597616 
 
 6.15 
 
 402384 
 
 24 
 
 87 
 
 566314 
 
 5.31 
 
 968329 
 
 .83 
 
 597985 
 
 6.15 
 
 402015 
 
 23 
 
 38 
 
 566632 
 
 5.31 
 
 968278 
 
 .83 
 
 598354 
 
 6.14 
 
 401646 
 
 22 
 
 39 
 
 566951 
 
 5.30 
 
 968228 
 
 .84 
 
 598722 
 
 6.14 
 
 401278 
 
 21 
 
 40 
 
 567269 
 
 5.30 
 
 968178 
 
 .84 
 
 599091 
 
 6.13 
 
 400909 
 
 20 
 
 41 
 
 9.567587 
 
 5.29 
 
 9.968128 
 
 .84 
 
 9.599459 
 
 6.13 
 
 10.400541 
 
 19 
 
 42 
 
 567904 
 
 5.29 
 
 968078 
 
 .84 
 
 599827 
 
 6.13 
 
 400173 
 
 18 
 
 43 
 
 568222 
 
 6.28 
 
 968027 
 
 .84 
 
 600194 
 
 6.12 
 
 899806 
 
 17 
 
 44 
 
 568539 
 
 5.28 
 
 967977 
 
 .84 
 
 600562 
 
 6.12 
 
 399438 
 
 16 
 
 45 
 
 568856 
 
 5.28 
 
 967927 
 
 .84 
 
 600929 
 
 6.11 
 
 899071 
 
 15 
 
 46 
 
 669172 
 
 5.27 
 
 967876 
 
 .84 
 
 001296 
 
 6.11 
 
 398704 
 
 14 
 
 47 
 
 569488 
 
 5.27 
 
 967826 
 
 .84 
 
 601062 
 
 6.11 
 
 398338 
 
 13 
 
 48 
 
 569804 
 
 5.26 
 
 967775 
 
 .84 
 
 002029 
 
 6.10 
 
 397971 
 
 12 
 
 49 
 
 570120 
 
 5.26 
 
 967725 
 
 .84 
 
 602395 
 
 6.10 
 
 397605 
 
 11 
 
 50 
 
 570435 
 
 5.25 
 
 967674 
 
 .84 
 
 602761 
 
 6 10 
 
 897239 
 
 10 
 
 51 
 
 9.570751 
 
 5.25 
 
 9.967624 
 
 .84 
 
 9.603127 
 
 6.09 
 
 10.396873 
 
 9 
 
 52 
 
 571066 
 
 5.24 
 
 967573 
 
 .84 
 
 003493 
 
 6.09 
 
 896507 
 
 8 
 
 53 
 
 57138C 
 
 5.24 
 
 967522 
 
 .85 
 
 003858 
 
 6.09 
 
 896142 
 
 7 
 
 64 
 
 671695 
 
 5.23 
 
 967471 
 
 .85 
 
 604223 
 
 6.08 
 
 395777 
 
 6 
 
 65 
 
 67200g 
 
 5.23 
 
 967421 
 
 .85 
 
 60158H 
 
 6.08 
 
 895412 
 
 6 
 
 66 
 
 572322 
 
 1 5.23 
 
 967370 
 
 .85 
 
 604953 
 
 6 07 
 
 895047 
 
 4 
 
 57 
 
 57263C 
 
 » 5.22 
 
 967315) 
 
 .85 
 
 605317 
 
 0.07 
 
 8940H3 
 
 3 
 
 68 
 
 67295C 
 
 1 5.22 
 
 96726H 
 
 .85 
 
 605682 
 
 6.07 
 
 894318 
 
 2 
 
 69 
 
 57326f 
 
 \ 5 21 
 
 967211 
 
 .85 
 
 60604G 
 
 6.06 
 
 893954 
 
 1 
 
 CO 
 
 57357f 
 
 ) 5.21 
 D. 
 
 96716f 
 
 .85 
 
 600410 
 i Cotang. 
 
 6.06 
 
 893590 
 
 
 M. 
 
 
 Cosine. 
 
 Sine. 
 
 1 68^ 
 
 Tang. 
 
 J 
 
10 
 
 
 (22 DEGREES.) A 
 
 TABLE OP LOGARITHMIC 
 
 
 
 "mTT 
 
 Sine. 
 
 D. 1 
 
 Cosinfi. 1 D. i 
 
 lanti'. 
 
 D. 
 
 Cotang. 1 
 
 ' 
 
 J . 573.375 
 
 5.21 
 
 9.907100 
 
 .85 
 
 9.000410 
 
 0.00 
 
 10.393590: (SO 
 
 1 
 
 57:5^48 
 
 5.20 
 
 907115 
 
 .85 
 
 000773 
 
 0.06 
 
 3932271 59 
 
 2 
 
 571200 
 
 5.20 
 
 9070(;4 
 
 .85 
 
 607137 
 
 6.05 
 
 392803 58 
 
 3 
 
 571512 
 
 5.19 
 
 •(07013 
 
 .85 
 
 607500 
 
 6.05 
 
 392500; 57 
 
 4 
 
 57H24 
 
 5.19 
 
 9009.;l 
 
 .85 
 
 007803 
 
 6.04 
 
 302137' 56 
 
 5 
 
 57513(5 
 
 5.19 
 
 900910 
 
 .85 
 
 608225 
 
 6.04 
 
 301775' 55 
 
 G 
 
 575417 
 
 5.18 
 
 900850 
 
 .85 
 
 608588 
 
 0.04 
 
 391412 54 
 
 7 
 
 575758 
 
 5.18 
 
 90G80S 
 
 .85 
 
 608950 
 
 6.03 
 
 391 050 1 53 
 
 8 
 
 5700GO 
 
 5.17 
 
 900750 
 
 .86 
 
 009312 
 
 6.03 
 
 390088; 52 
 
 9 
 
 57(3379 
 
 5.17 
 
 900705 
 
 .86 
 
 (509074 
 
 6.03 
 
 390320 
 
 51 
 
 lu 
 
 57(;689 
 
 5.16 
 
 9(50053 
 
 .86 
 
 610036 
 
 6.02 
 
 10.38(5904 
 
 50 
 
 11 
 
 ) 57()9:t9 
 
 5.]'o 
 
 9.'.)(5(5002 
 
 .80 
 
 9.610397 
 
 0.02 
 
 389003 
 
 49 
 
 12 
 
 577309 
 
 5.1'J 
 
 000550 
 
 .80 
 
 610759 
 
 6.02 
 
 3S0241 
 
 48 
 
 13 
 
 577018 
 
 5.15 
 
 90041);) 
 
 .86 
 
 611120 
 
 6.01 
 
 388880 
 
 47 
 
 U 
 
 577927 
 
 5.15 
 
 9(50417 
 
 .86 
 
 611480 
 
 0.01 
 
 388520: 40 t 
 
 15 
 
 57H230 
 
 5.14 
 
 900395 
 
 .86 
 
 611S41 
 
 0.01 
 
 388150 
 
 45 
 
 10 
 
 578545 
 
 5.14 
 
 900344 -80 
 
 612201 
 
 6.00 
 
 387799 
 
 44 
 
 17 
 
 578853 
 
 5.13 
 
 900292 ,86 
 
 612501 
 
 6.00 
 
 387439 
 
 43 
 
 18 
 
 579102 
 
 5.13 
 
 900240 
 
 .80 
 
 612921 
 
 6.00 
 
 387079 
 
 42 
 
 19 
 
 579470 
 
 5.13 
 
 90(5188 
 
 .80 
 
 613281 
 
 5.99 
 
 38(5719' 41 
 
 20 
 
 579777 
 
 5.12 
 
 9(50130 
 
 .86 
 
 613041 
 
 5.99 
 
 10.380359; 40 
 
 21 
 
 ,■). 580085 
 
 5.12 
 
 9 900085 
 
 .87 
 
 9.614000 
 
 5.98 
 
 3800001 39 
 
 22 
 
 580392 
 
 5.11 
 
 900)33 
 
 .87 
 
 6f4359 
 
 5.98 
 
 385(541, 38 
 
 23 
 
 580099 
 
 5.11 
 
 905981 
 
 .87 
 
 614718 
 
 5,98 
 
 385282 
 
 37 
 
 24 
 
 581005 
 
 5.11 
 
 905928 
 
 .87 
 
 615077 
 
 5.97 
 
 384923 
 
 3(5 
 
 23 
 
 581312 
 
 5.10 
 
 90587(5 
 
 .87 
 
 615435 
 
 5.97 
 
 384505 
 
 35 
 
 26 
 
 581018 
 
 5.10 
 
 905824 
 
 .87 
 
 615703 
 
 5.97 
 
 384207 
 
 34 
 
 27 
 
 581924 
 
 5.09 
 
 965772 
 
 .87 
 
 61(5151 
 
 5.96 
 
 383849 
 
 33 
 
 28 
 
 582229 
 
 5.09 
 
 905720 
 
 .87 
 
 616509 
 
 5.90 
 
 383491 
 
 32 
 
 29 
 
 582535 
 
 5 09 
 
 965608 
 
 .87 
 
 616807 
 
 5.96 
 
 383133 
 
 31 
 
 30 
 
 5S2840 
 
 5.08 
 
 965015 
 
 .87 
 
 617224 
 
 5.95 
 
 10.382776 
 
 30 
 
 31 
 
 9.583145 
 
 5.08 
 
 9.905.503 
 
 .87 
 
 9.617582 
 
 5.95 
 
 382418 
 
 29 
 
 32 
 
 583449 
 
 5.07 
 
 965511 
 
 .87 
 
 61793!> 
 
 5.95 
 
 382061 
 
 28 
 
 33 
 
 583754 
 
 5.07 
 
 965458 
 
 .87 
 
 618295 
 
 5.94 
 
 381705 
 
 27 
 
 34 
 
 584058 
 
 5.06 
 
 965400 
 
 .87 
 
 618052 
 
 5.94 
 
 381348 
 
 26 
 
 35 
 
 584361 
 
 5.06 
 
 965353 
 
 .88 
 
 019008 
 
 5.94 
 
 380992 
 
 25 
 
 36 
 
 584605 
 
 5.06 
 
 965301 
 
 .88 
 
 619304 
 
 5.93 
 
 3800301 24 I 
 
 37 
 
 584908 
 
 5.05 
 
 965248 
 
 .88 
 
 619721 
 
 5.93 
 
 380279 
 
 23 
 
 38 
 
 585272 
 
 5.05 
 
 965195 
 
 .88 
 
 62007(5 
 
 5.93 
 
 379924 
 
 22 
 
 39 
 
 585574 
 
 5.04 
 
 905143 
 
 .88 
 
 620432 
 
 5.92 
 
 379568 
 
 21 
 
 40 
 
 585877 
 
 5.04 
 
 965090 
 
 .88 
 
 620787 
 
 5.92 
 
 10.379213 
 
 20 
 
 41 
 
 9.580179 
 
 5.03 
 
 9.9(5.5037 
 
 .88 
 
 9.621142 
 
 5.92 
 
 378858 
 
 19 
 
 42 
 
 586482 
 
 5.03 
 
 904984 
 
 .88 
 
 621497 
 
 5.91 
 
 378503 
 
 18 
 
 43 
 
 580783 
 
 5.03 
 
 964931 
 
 .88 
 
 621852 
 
 5.91 
 
 378148 
 
 17 
 
 44 
 
 587085 
 
 5.02 
 
 964879 
 
 .88 
 
 622207 
 
 5.90 
 
 377793 
 
 16 
 
 45 
 
 587386 
 
 5.02 
 
 964320 
 
 .88 
 
 022561 
 
 5.90 
 
 377439 
 
 15 
 
 46 
 
 587688 
 
 5.01 
 
 964773 
 
 .88 
 
 622915 
 
 5.90 
 
 377085 
 
 14 
 
 47 
 
 587989 
 
 5.01 
 
 964719 
 
 .88 
 
 623269 
 
 5.89 
 
 376731 
 
 13 
 
 48 
 
 588289 
 
 5.01 
 
 904(500 
 
 .89 
 
 623623 
 
 5.89 
 
 376377 
 
 12 
 
 49 
 
 588590 
 
 5.00 
 
 964613 
 
 .89 
 
 623976 
 
 5.89 
 
 376024 
 
 11 
 
 50 
 
 588890 
 
 5.00 
 
 9645(50 
 
 .89 
 
 624330 
 
 5.88 
 
 10.375670 
 
 10 
 
 61 
 
 9.589190 
 
 4.99 
 
 9.9(54507 
 
 .89 
 
 9.624689 
 
 5.88 
 
 375317 
 
 9 
 
 52 
 
 589489 
 
 4.99 
 
 964454 
 
 .89 
 
 62503C 
 
 5.88 
 
 374964 
 
 8 
 
 53 
 
 589789 
 
 4.99 
 
 964400 
 
 .89 
 
 625388 
 
 5.87 
 
 374612 
 
 7 
 
 54 
 
 59008P 
 
 4.98 
 
 964347 
 
 .89 
 
 625741 
 
 5.87 
 
 374259 
 
 6 
 
 55 
 
 590387 
 
 4.98 
 
 9642i)4 
 
 .89 
 
 626099 
 
 5.87 
 
 373907 
 
 5 
 
 56 
 
 590686 
 
 . 4.97 
 
 964240 
 
 .89 
 
 62644S 
 
 5.86 
 
 373555 
 
 4 
 
 67 
 
 590984 
 
 4.97 
 
 964187 
 
 .8G 
 
 626797 
 
 5.86 
 
 373203 
 
 3 
 
 58 
 
 59128S 
 
 ! 4.97 
 
 96413.T 
 
 .80 
 
 627148 
 
 5.86 
 
 872851 
 
 2 
 
 69 
 
 59158C 
 
 1 4.96 
 
 96408( 
 
 .83 
 
 627501 
 
 5.85 
 
 372499 
 
 1 
 
 GO 
 
 59187(' 
 1 Cosine. 
 
 ) 4.96 
 
 96402( 
 
 .8'J 
 67" 
 
 627852 
 
 ! 6.85 
 
 872148 
 
 
 M. 
 
 1 
 
 D. 
 
 Sine. 
 
 Cotang. 
 
 D. 
 
 Tang. 
 
jU.5yi87^ 
 592170 
 592473 
 592770 
 593007 
 5933(;;j| 
 59365;) 
 , 593955 
 
 8 594251 
 
 9 594547 
 iO 594842, 
 il 9.595137 
 ?2 595432 
 13 595727' 
 It 596021 
 
 15 596315, 
 
 16 696609 
 
 17 596903 
 
 18 597196 
 
 19 597490 
 
 20 597783 
 
 21 9.598073 
 
 22 59836S 
 
 23 598660 
 21 598952 
 2'J 599244 
 2<; 599536 
 
 27 599827 
 
 28 600118 
 
 29 600409 
 
 30 600700 
 
 31 9.600990 
 
 32 601280 
 
 33 601570 
 ^ 601860 
 S ^02150 
 86 602439] 
 
 37 602728 
 
 38 603017 
 
 89 603305 
 
 f ? 603594 
 
 fl 9.603882 
 *2 604170 
 f3 604457 
 *» 604745, 
 *? 605032 
 *6 605319 
 *J 605606 
 *f 605892 
 *^ I 606179 
 
 606465 
 [9.606751 
 
 607036 
 
 607322 
 
 607607 
 
 607892 
 
 608177 
 
 608461 
 
 608745 
 
 609029 
 
 609313 
 
 "^i>iiS. (_'3 DEGREES ■» 
 
 Ti i~?r — ^ •' 
 
 50 
 51 
 52 
 63 
 54 
 
 66 
 
 56 
 
 67 
 
 58 
 
 ^- '_Cosine^ 
 
 ~97964026 
 
 963972 
 
 963919 
 
 963S05 
 
 963811 
 
 963757 
 
 963704 
 
 963650 
 
 963596 
 
 963542 
 
 963488 
 
 ). 963434 
 
 963379 
 
 9633251 
 
 9632711 
 
 963217 
 
 963163, 
 
 963108 
 
 963054 
 
 962999, 
 
 962945 
 
 9.962890 
 
 9628361 
 
 962781 
 
 962727 
 
 962672, 
 
 962617 
 
 962562 
 
 962508' 
 
 962453 
 
 962398 
 
 9.962343, 
 
 962288 
 
 962233 
 
 962178, 
 
 962123 
 
 962067 
 
 962012 
 
 961957 
 
 961902 
 
 „ 961846 
 
 9.961791 
 
 961735 
 
 961680 
 
 961624 
 
 961569 
 
 961513 
 
 961458 
 
 961402 
 
 961346 
 
 ^ 961290 
 
 9-961235, 
 
 9611791 
 
 961123 
 
 961067] 
 
 961011 
 
 960955| 
 
 9608991 
 
 9608431 
 
 960786 
 
 960730J 
 
 Cotang. 
 
 ■3 =^ 
 
 •^0 G28905 
 
 •90 C29255 
 
 •^P 629G0G' 
 
 ■ 629956,1 
 
 6303001 
 
 630056! 
 
 631005 
 
 •90 9.631704.' 
 
 •90, 6320.53J 
 
 •90 632401 
 
 •90 C32750! 
 
 •90 633098! 
 
 •90 633447 
 
 •91 633795 
 
 •91 C34143, 
 
 •91f 634490 
 
 634838 
 
 ,10. 372148! lo" 
 ' 371797 59 
 371446 58 
 371095 57 
 370745 56 
 370394 55 
 3700-44 .54 
 3096941 53 
 3693441 52 
 368995 51 
 , 368645! 50 
 ,iO. 368296) 49 
 ' 307947 Is 
 367599 47 
 3072.50 46 
 366902 45 
 306553 44 
 306205 43 
 305857 
 3G5510 
 , 365102 
 10.364815 
 304468 
 364121 
 363774 
 363428 
 363081 
 362735 
 362389 
 362044 
 , 861698 
 10.361353 
 861008 
 860663 ^, 
 360318 26 
 359973 25 
 359629 24 
 359284 23 
 858940 22 
 858596 21 
 , ^ 358253 20 
 10.357909 19 
 857566 is 
 857223 17 
 856880 ifi 
 856537 15 
 356194 14 
 355852 13 
 355510 12 
 855168 11 
 . 354826 10 
 10.354484 
 853143 8 
 353801 
 353*60 6 
 853119 5 
 a'',2778 4 
 862438 8 
 862097 
 351767 
 851417 
 
 42 
 41 
 iO 
 39 
 38 
 37 
 36 
 
 36 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 29 
 
 28 
 
 n 
 
 il k 
 
12 
 
 
 (24 DKOREKS.) A 
 
 TABLE OF LOQAUITHMIC 
 
 
 
 M. 1 
 
 
 Sine. 
 
 D. 
 
 Cosino. 
 
 D. 
 
 Tang. 
 
 D. 
 
 5.06 
 
 Cotang. 1 
 
 9.609313 
 
 4.73 
 
 9.930730 
 
 .94 
 
 9.648583 
 
 LO. 351417 
 
 60 
 
 1 
 
 609507 
 
 4.72 
 
 9G0G74 
 
 .94 
 
 648923 
 
 5.06 
 
 351077 
 
 59 
 
 2 
 
 G098S0 
 
 4.72 
 
 960618 
 
 .94 
 
 649203 
 
 5.66 
 
 350737 
 
 58 
 
 3 
 
 6101G4 
 
 4.72 
 
 960501 
 
 .94 
 
 649002 
 
 5.66 
 
 350398 
 
 57 
 
 4 
 
 610447 
 
 4.71 
 
 960505 
 
 .94 
 
 649942 
 
 6.65 
 
 350058 
 
 56 
 
 5 
 
 610729 
 
 4.71 
 
 960448 
 
 .91 
 
 650281 
 
 5.65 
 
 349719 
 
 55 
 
 G 
 
 611012 
 
 4.70 
 
 960392 
 
 .94 
 
 650620 
 
 6.05 
 
 349380 
 
 54 
 
 7 
 
 Gil 204 
 
 4.70 
 
 960335 
 
 .94 
 
 650959 
 
 6.G4 
 
 849041 
 
 53 
 
 8 
 
 61157G 
 
 4.70 
 
 9G0270 
 
 .94 
 
 651297 
 
 6.04 
 
 818703 52 
 
 y 
 
 G1185H 
 
 4.09 
 
 9G0222 
 
 .94 
 
 651030 
 
 6.64 
 
 848304; 51 
 
 10 
 
 G12110 
 
 4.69 
 
 900105 
 
 .94 
 
 651974 
 
 6.63 
 
 348026; 50 
 
 11 
 
 9.G12121 
 
 4.69 
 
 9.900109 
 
 .05 
 
 9.652312 
 
 6.63 
 
 10.347688 49 
 
 12 
 
 G12702 
 
 4,68 
 
 900052 
 
 .05 
 
 652050 
 
 5.03 
 
 347350 48 
 
 13 
 
 012983 
 
 4.68 
 
 959995 
 
 .95 
 
 652088 
 
 5.03 
 
 347012 47 
 
 11 
 
 G132()l 
 
 4.67 
 
 95993:S 
 
 .05 
 
 653326 
 
 5. 02 
 
 340G74i 46 
 
 15 
 
 613145 
 
 4.67 
 
 959382 
 
 .95 
 
 653GG3 
 
 5.62 
 
 346337 
 
 45 
 
 10 
 
 613825 
 
 4.67 
 
 959825 
 
 .95 
 
 651000 
 
 6.62 
 
 346000 
 
 44 
 
 17 
 
 G14105 
 
 4.00 
 
 959708 
 
 .95 
 
 654337 
 
 5.61 
 
 345003 
 
 43 
 
 18 
 
 G14385 
 
 4.00 
 
 959711 
 
 .05 
 
 654G74 
 
 6.61 
 
 34532G 
 
 42 
 
 10 
 
 G14G(J5 
 
 4.00 
 
 959054 
 
 .95 
 
 655011 
 
 6 61 
 
 344989 
 
 41 
 
 20 
 
 614044 
 
 4.05 
 
 959590 
 
 .95 
 
 655348 
 
 5.01 
 
 344052 
 
 40 
 
 21 
 
 9.615223 
 
 4.65 
 
 9.959539 
 
 .05 
 
 9.655084 
 
 6.00 
 
 10.344310 
 
 39 
 
 22 
 
 615502 
 
 4.05 
 
 959482 
 
 .05 656020 
 
 6,60 
 
 343980 
 
 38 
 
 23 
 
 G15781 
 
 4.04 
 
 950425 
 
 .95 
 
 650350 
 
 5.00 
 
 343044 
 
 87 
 
 24 
 
 GIGOCO 
 
 4. 64 
 
 959308 
 
 .95 
 
 650092 
 
 5.59 
 
 343308 
 
 36 
 
 25 
 
 61G338 
 
 4.G4 
 
 959310 
 
 .90 
 
 657028 
 
 5.59 
 
 342972 
 
 85 
 
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 61(5010 
 
 4.63 
 
 959253 
 
 .90 
 
 657304 
 
 5.59 
 
 34263G 
 
 34 
 
 27 
 
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 4.03 
 
 959195 
 
 .90 
 
 657099 
 
 6.59 
 
 342301 
 
 33 
 
 28 
 
 617172 
 
 4.62 
 
 959138 
 
 .90 
 
 658034 
 
 5.58 
 
 341900 
 
 32 
 
 20 
 
 617450 
 
 4.62 
 
 959081 
 
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 658369 
 
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 31 
 
 30 
 
 617727 
 
 4.62 
 
 959023 
 
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 658704 
 
 6.58 
 
 341290 
 
 30 
 
 31 
 
 9.618004 
 
 4.61 
 
 9.958905 
 
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 9.050039 
 
 5.58 
 
 10.340961 
 
 29 
 
 32 
 
 618201 
 
 4.61 
 
 953908 
 
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 650373 
 
 6,57 
 
 340027 
 
 28 
 
 33 
 
 618558 
 
 4 61 
 
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 .90 
 
 659708 
 
 6.57 
 
 340292 
 
 27 
 
 34 
 
 618834 
 
 4.60 
 
 958792 
 
 .90 
 
 6G0042 
 
 5.57 
 
 339958 
 
 20 
 
 35 
 
 619110 
 
 4.60 
 
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 600370 
 
 5.57 
 
 339624 
 
 25 
 
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 4.60 
 
 958077 
 
 .96 
 
 6G0710 
 
 6.56 
 
 339290 
 
 24 
 
 37 
 
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 4.59 
 
 958019 
 
 .96 
 
 661043 
 
 5.56 
 
 338957 
 
 23 
 
 38 
 
 619938 
 
 4.59 
 
 958501 
 
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 6.56 
 
 338623 
 
 22 
 
 30 
 
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 21 
 
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 41 
 
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 670320 
 
 5.48 
 
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 6 
 
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 670649 
 
 5.48 
 
 329351! 54 
 
 7 
 
 6278401 
 
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 8 
 
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 671963 
 
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 328037 1 50 
 
 11 
 
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 5.47 
 
 10. 327709 i 49 
 
 12 
 
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 672619 
 
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 13 
 
 629453 
 
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 956500 
 
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 672947 
 
 5.46 
 
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 629721 
 
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 673602 
 
 5.46 
 
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 16 
 
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 326071 
 
 44 
 
 17 
 
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 5.45 
 
 325743 
 
 43 
 
 18 
 
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 4.45 
 
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 19 
 
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 23 
 
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 37 
 
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 25 
 
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 322480 
 
 33 
 
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 5.42 
 
 322154 
 
 32 
 
 29 
 
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 321829 
 
 31 
 
 30 
 
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 321504 
 
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 29 
 
 32 
 
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 5.41 
 
 320854 
 
 28 
 
 33 
 
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 5.41 
 
 320529 
 
 27 
 
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 5.41 
 
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 26 
 
 35 
 
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 5.40 
 
 319880 
 
 25 
 
 36 
 
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 24 
 
 37 
 
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 319232 
 
 23 
 
 38 
 
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 4.38 
 
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 631092 
 
 5.40 
 
 318908 
 
 22 
 
 39 
 
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 4.38 
 
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 681416 
 
 5.39 
 
 318584 
 
 21 
 
 40 
 
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 20 
 
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 311498 
 
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 5.33 
 
 310857 
 
 57 
 
 
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 310537 
 
 56 
 
 
 
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 691381 
 
 5.32 
 
 308019 
 
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 9.952980 
 
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 f 70654] 
 
 5.21 
 
 29345S 
 
 2 
 
 
 
 69 
 
 66679S 
 
 1 4.13 
 
 94994J 
 
 ll.O'i 
 
 f 7068^1 
 
 \ 6.21 
 
 293146 
 
 1 
 
 
 
 60 
 
 657041 
 
 f 4.13 
 
 949881 
 
 1.01 
 
 f 70716( 
 
 5 5.20 
 
 292834 
 
 
 
 
 
 1 Coiine. 
 
 D. 
 
 Sine. 
 
 68» 
 
 Cotang. 
 
 D. 
 
 Tang. 
 
 M. 
 
 - 
 
f)0 
 51) 
 58 
 57 
 5(3 
 55 
 
 y.'!570-t7 
 
 I <i.575i2 
 
 I 0577:10 
 
 ()5,Sfi;j7 
 
 65S5;J1 
 (J5<S77H 
 C51W25 
 651*271 
 0*5!J517 
 |a.«5;)7G3 
 6U0009 
 (j(J0255 
 600501 
 C607iG 
 C(iO!)yi 
 
 6G1236 
 G«H81 
 6G1726 
 6G1070 
 >.C622U 
 662459 
 662703 
 662y4G 
 6631 i)0 
 663433 
 6G3677 
 663920 
 664163 
 , 664406 
 9.664648 
 664891 
 665133, 
 665375 
 665617 
 665859 
 666100 
 666342 
 666583 
 , 666824 
 9.667065 
 667305 
 667546 
 667786 
 668027, 
 668267 
 668506 
 668746 
 668986 
 669225 
 .669464 
 669703 
 669942, 
 670181 
 O70419 
 670658 
 670896 
 671134 
 671372 
 _671609 
 Cosine 
 
 ._ ^^'-''^^ ^-^^ l-ANGENTS. C27 
 
 ^Cosine. D. 
 
 y.iMii^s.si iTo7, 
 
 919810 1,07 
 949752,1.07 
 9496881.08 
 949G23I 1.08,1 
 94955811.08 
 949494,1.08 
 949429 1.08 
 9493G4 1.08 
 94930011.08 
 9492351.08 
 9.9491701.08 
 9491051.08 
 9490401.08 
 9489751.08 
 948910 1.081 
 9488451.08 
 9487801.09 
 9487151.09 
 948650 1.09 
 948584 1.09 
 9.9485191.09 
 9484541.09 
 94838811.09' 
 948323J1.09 
 948257(1.09 
 948192|1.09 
 94812G,'1.09 
 9480601.09, 
 
 DEGRt 
 
 9479951.--, 
 9479291.10 
 9.9478631.10 
 9477971.10 
 947731(1.10 
 947665|1.10 
 947600(1.10 
 9475331.10 
 9474671.10 
 9474011.10 
 947335 1.10 
 „ 9472691.10 
 9.94720311.10 
 947136(1.11 
 947070 |l. 11 
 947004(1.11 
 946937(1.11 
 9468711.11 
 9468041.11 
 9467381.11 
 9466711.11 
 ^ 94660411.11 
 9- 946538(1. 11 
 946471 1.11 
 9464041.11 
 9463371.11 
 946270,(1.12 
 9462031.12 
 9461361.12 
 9460691.12 
 946002(1.12 
 _945935(1J2 
 
 9.7071Gg( 
 
 707478( 
 
 707790 
 
 708102 
 
 708414 
 
 70a72G 
 
 709349 
 
 709GG0 
 
 709971 
 
 710282 
 
 9.710593 
 
 710904 
 
 711215 
 
 711525 
 
 711830 
 
 712140 
 
 712450 
 
 712700 
 
 713076 
 
 713386 
 
 9.713090 
 
 714005 
 
 714314 
 
 714024 
 
 714933 
 
 715242 
 
 715551 
 
 715860 
 
 716168 
 
 716477 
 
 >• 716785 
 
 717093 
 
 717401 
 
 717709 
 
 718017 
 
 718325 
 
 718633 
 
 718940 
 
 719248 
 
 „ 719555 
 
 9.719862 
 
 720169 
 
 720476 
 
 720783 
 
 721089 
 
 721396 
 
 721702 
 
 722009 
 
 722315 
 
 722621 
 
 722927 
 
 723232 
 
 723538 
 
 723844 
 
 724149 
 
 724454 
 
 724759 
 
 725065 
 
 725369 
 
 _725674| 
 
 4& 
 
 !lU. 292831 
 292522 
 292210 
 291898 
 2915X0 
 291274 
 290903 
 290(i51 
 290ai0 
 290029 
 , 289718 
 f J 0.289407 
 289096 
 288785 
 288475 
 288104 
 287854 
 287544 
 287234 
 286924 
 i.« 280G14 
 (10.280304 
 285995 
 285680 
 285370 
 285067 
 284758 
 284449 
 284140 
 283832 
 1,^ 283523 
 (10.283215 
 282907 
 282599 
 282291 
 281983 
 281670 
 281367 
 281060 
 280752 
 i,n 280445 
 (10.280138 
 279831 
 279524 
 279217 
 278911 
 278604 
 278298 
 277991 
 277685, 
 ,,„ 277379 
 P0.277073| 
 276768, 
 276462 
 276156] 
 275851 
 275546, 
 275241 
 2749a5| 
 274631 
 _274326| 
 
 00 
 59 
 58 
 57 
 56 
 55 
 54 
 53 
 52 
 51 
 50 
 49 
 48 
 47 
 46 
 45 
 44 
 43 
 42 
 41 
 40 
 39 
 38 
 37 
 36 
 35 
 34 
 33 
 32 
 31 
 30 
 29 
 28 
 27 
 26 
 25 
 24 
 23 
 22 
 21 
 20 
 19 
 18 
 17 
 16 
 15 
 14 
 13 
 12 
 11 
 10 
 9 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 
 ' 'I 
 
 r 
 

 4G 
 
 
 (28 DKOnSES.) A TABLE 07 LOGARITHMIC 
 
 
 M. 
 
 U 
 
 1 Sine. 
 
 D. 
 
 Codne. 
 
 D 
 
 l.lli 
 
 Tang. 
 
 9. '725674 
 
 D. 
 
 Cotang. 
 
 60 
 
 y.U71009 
 
 3.96 
 
 9.94593S 
 
 5.UM 
 
 10.27432(j 
 
 1 
 
 671847 
 
 3.95 
 
 945868 
 
 1.12 
 
 725979 
 
 1 5.08 
 
 274021 
 
 59 
 
 2 
 
 672084 
 
 3.95 
 
 945800 
 
 1.12 
 
 726284 
 
 ' 5.07 
 
 27371( 
 
 58 
 
 3 
 
 672321 
 
 3.95 
 
 945733 
 
 1.12 
 
 726588 
 
 5.07 
 
 27341'^ 
 
 67 
 
 4 
 
 672558 
 
 3.95 
 
 945666 
 
 1.12 
 
 72G892 
 
 5.07 
 
 273108 
 
 66 
 
 5 
 
 672795 
 
 3.94 
 
 945598 
 
 1.12 
 
 727197 
 
 5.07 
 
 27280.<] 
 
 55 
 
 6 
 
 673032 
 
 3.94 
 
 945531 
 
 1.12 
 
 727501 
 
 5.07 
 
 272499 
 
 54 
 
 7 
 
 673268 
 
 3.94 
 
 94546411.13 
 
 727805 
 
 5.06 
 
 272195 
 
 53 
 
 8 
 
 673505 
 
 3.94 
 
 945396 
 
 1.13 
 
 728109 
 
 5.06 
 
 271891 
 
 52 
 
 9 
 
 673741 
 
 3.93 
 
 945328 
 
 1.13 
 
 728412 
 
 5.06 
 
 271588 
 
 51 
 
 10 
 
 673977 
 
 3.93 
 
 945261 
 
 1.13 
 
 728716 
 
 5.06 
 
 271284 
 
 60 
 
 U 
 
 9.674213 
 
 3.93 
 
 9.945193 
 
 1.13 
 
 9.729020 
 
 5.06 
 
 10.270980 
 
 49 
 
 12 
 
 074448 
 
 3.92 
 
 945125 
 
 1.13 
 
 729323 
 
 5.05 
 
 270677 
 
 48 
 
 1:) 
 
 674684 
 
 3.92 
 
 945058 
 
 1.13 
 
 725)626 
 
 5.05 
 
 270374 
 
 47 
 
 li 
 
 074919 
 
 3.92 
 
 944990 
 
 1.13 
 
 729929 
 
 5.05 
 
 270071 46 
 
 15 
 
 675155 
 
 3.92 
 
 944922 
 
 1.13 
 
 730233 
 
 5.05 
 
 2697G7 45 
 
 i<; 
 
 075390 
 
 3.91 
 
 944854 
 
 1.13 
 
 730535 
 
 5.05 
 
 2G9405 44 
 
 17 
 
 075024 
 
 3.91 
 
 944786 
 
 1.13 
 
 730838 
 
 5.04 
 
 269102 
 
 43 
 
 Is 
 
 075859 
 
 3.91 
 
 944718 
 
 1.13 
 
 731141 
 
 5.04 
 
 268859 
 
 42 
 
 I'J 
 
 070094 
 
 3.91 
 
 94465011.13 
 
 731444 
 
 5.04 
 
 268550 
 
 41 
 
 20 
 
 070328 
 
 3.90 
 
 944582 
 
 1.14 
 
 731746 
 
 5.04 
 
 268254 
 
 40 
 
 21 
 
 9.670502 
 
 3.90 
 
 9.944514 
 
 1.14 
 
 9.732048 
 
 5.04 
 
 10.267952 
 
 39 
 
 22 
 
 G70790 
 
 3.90 
 
 94444(i 
 
 1.14 
 
 732351 
 
 5.03 
 
 267649 
 
 38 
 
 2;j 
 
 077030 
 
 3.90 
 
 944377 
 
 1.14 
 
 732G53 
 
 5.03 
 
 267347 
 
 37 
 
 21 
 
 077204 
 
 3.89 
 
 944309 1.14 
 
 732955 
 
 5.03 
 
 267045 
 
 36 ■ 
 
 25 
 
 077498 
 
 3.89 
 
 9442411.14 
 
 733257 
 
 5.03 
 
 260743! 35 | 
 
 20 
 
 G77731 
 
 3.89 
 
 944172 1.14 
 
 733558 
 
 5.03 
 
 266442 
 
 34 
 
 27 
 
 077904 
 
 3. 88 
 
 94410411.14 
 
 733S00 
 
 5.02 
 
 2GG140 
 
 33 
 
 2;j 
 
 078197 
 
 3.8S 
 
 944030|1.14 
 
 734102 
 
 5.02 
 
 205838 32 
 
 2;) 
 
 G78130 
 
 3.88 
 
 943907 
 
 1.14 
 
 734403 
 
 5.02 
 
 265537 31 
 
 30 
 
 078003 
 
 3.88 
 
 943899 
 
 1.14 
 
 7347(J4 
 
 5.02 
 
 205230 30 
 
 31 
 
 9.678895 
 
 3.87 
 
 9.943S30 
 
 1.14 
 
 9.735000 
 
 5.02 
 
 10.2049341 29 
 
 32 
 
 079128 
 
 3.87 
 
 943701 
 
 1.14 
 
 735307 
 
 5.02 
 
 204033! 28 
 
 03 
 
 679300 
 
 3.87 
 
 943;i93 
 
 1.1.-) 
 
 735008 
 
 5.01 
 
 2(;4332i 27 
 
 34 
 
 079592 
 
 3,87 
 
 943024 
 
 1.15 
 
 735909 
 
 5.01 
 
 204031 20 
 
 35 
 
 079824 
 
 3.80 
 
 943-)5.) 
 
 1.15 
 
 730209 
 
 5.01 
 
 203731, 25 
 
 3(] 
 
 080050 
 
 3.80 
 
 943180 
 
 1.15 
 
 7.30570 
 
 5.01 
 
 203430; 24 
 
 37 
 
 080288 
 
 3.8i» 
 
 943417 
 
 1.15 
 
 730M71 
 
 5.01 
 
 203129 23 
 
 3S 
 
 080519 
 
 3.85 
 
 94334S 
 
 1.15 
 
 7;;7i7i 
 
 5.00 
 
 202829 22 
 
 39 
 
 080750 
 
 3.85 
 
 943279 
 
 1.15 
 
 737471 
 
 5.00 
 
 202529 21 i 
 
 40 
 
 680982 
 
 3.85 
 
 943210 
 
 1.15 
 
 737771 
 
 5.00 
 
 202229 
 
 20 
 
 41 
 
 9.081213 
 
 3.85 
 
 9.943141 
 
 1.15 
 
 9.73S()71 
 
 5.00 
 
 10.201929 
 
 19 
 
 42 
 
 081443 
 
 3.84 
 
 943072 
 
 1.15 
 
 738371 
 
 5.00 
 
 2(il029 
 
 18 
 
 43 
 
 681074 
 
 3.84 
 
 943003 
 
 1.15 
 
 738071 
 
 4.99 
 
 201329 
 
 17 
 
 44 
 
 681905 
 
 3.84 
 
 942934! 
 
 1.15 
 
 738971 
 
 4,99 
 
 201029! 
 
 Ifl 
 
 45 
 
 682135 
 
 3.84 
 
 942804 
 
 1.15 
 
 739271 
 
 4,99 
 
 200729 
 
 15 
 
 46 
 
 682365 
 
 3.83 
 
 942795 
 
 1.10 
 
 73,).570| 
 
 4,99 
 
 200430 
 
 14 
 
 47 
 
 682595 
 
 3.83 
 
 942720 
 
 1.10 
 
 739870 
 
 4,99 
 
 200130 
 
 13 
 
 48 
 
 682825 
 
 3.83 
 
 942650' 
 
 1.10 
 
 7401G9! 
 
 4,99 
 
 259831! 
 
 12 
 
 49 
 
 683055 
 
 3.83 
 
 942587 i 
 
 I.IG 
 
 7404G8! 
 
 4.98 
 
 259532 
 
 11 
 
 50 
 
 683284 
 
 3.82 
 
 942517! 
 
 I.IG 
 
 740007! 
 
 4,98 
 
 259233' 10 j 
 
 51 
 
 9.683514 
 
 3.82 
 
 9. 942448 1 
 
 1.10 
 
 9.741000' 
 
 4.98 
 
 10.2.58934 
 
 9 
 
 52 
 
 683743 
 
 3.82 
 
 942378! 
 
 1.10 
 
 741305! 
 
 4.98 
 
 258035 
 
 8 
 
 53 
 
 683972 
 
 3.82 
 
 942308 i 
 
 1.10 
 
 741004 
 
 4 98 
 
 258330 
 
 7 
 
 54 
 
 684201 
 
 3.81 
 
 942239 1 
 
 1.10 
 
 741!)02i 
 
 4.97 
 
 258038 
 
 6 
 
 55 
 
 684430 
 
 3.81 
 
 942169, 
 
 I.IG 
 
 742201' 
 
 4.97 
 
 2577.39 
 
 5 
 
 5G 
 
 684658 
 
 3.81 
 
 942099, 
 
 I.IG 
 
 74:1,5.59! 
 
 4.97 
 
 257441! 
 
 4 
 
 57 
 
 084887 
 
 3.80 
 
 942029 
 
 1.16 
 
 742858 
 
 4,97 
 
 2.57142 
 
 3 
 
 58 
 
 685115 
 
 3.80 
 
 941959 
 
 1.16 
 
 743156: 
 
 4.9T 
 
 250844 
 
 2 
 
 59 
 
 685343 
 
 3.80 
 
 941889 
 
 1.17 
 
 743454| 
 
 4,97 
 
 25:;510 
 
 1 
 
 GO 
 
 0S5571 
 
 3.80 
 
 d: 
 
 941819 
 
 1.17 
 
 7437.52 
 
 4 , 90 
 D. i 
 
 250248! 
 
 
 
 1 
 
 Cosine. 1 
 
 Sine. 
 
 Ui" 
 
 tJotang. 1 
 
 Tang. 1 
 
 M. 
 
?«• 
 
 1 
 
 i;)2(i 
 
 (>0 
 
 1021 
 
 59 
 
 mn 
 
 58 
 
 3412 
 
 57 
 
 JIOS 
 
 56 
 
 280;} 
 
 55 
 
 H'M 
 
 54 
 
 2195 
 
 53 
 
 52 
 
 51 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 8 
 7 
 G 
 5 
 4 
 3 
 2 
 1 
 
 
 M. 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 60 
 
 It 
 
 
 1 
 2 
 8 
 4 
 5 
 6 
 7 
 8 
 9 
 10 
 11 
 12 
 13 
 14 
 15 
 IG 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 26 
 
 '9.685571 
 685799, 
 686027 
 686254 
 686482| 
 686709 
 686936 
 687163 
 687389 
 687016 
 , 687843 
 9.688069' 
 088295 
 088521 
 688747 
 088972 
 689198 
 689423 
 089648 
 689873 
 , 690098 
 '9.690323 
 690548 
 690772 
 690996 
 691220 
 691444 
 691668 
 691892 
 692115 
 , 692339 
 [9.692562 
 692785 
 693008 
 693231 
 693453 
 693676 
 693898 
 694120 
 694342 
 , 694564 
 19.694786 
 695007 
 695229 
 695450 
 695671 
 695892 
 696113 
 696334 
 696554 
 ,„ 696775 
 I9.69C995 
 697215 
 697435 
 697654 
 697874 
 698094 
 098313 
 698532 
 698751 
 _698970 
 Cosine 
 
 T» ~t ri . . ■/ 
 
 Cosine^ I D. 
 
 9^9418l9iri7 
 941749 1.17 
 9416791.17 
 941609 1 17 
 94153911.17 
 941469)1.17 
 941398 1.17 
 9413281.17 
 941258 1 17 
 941187,1.17 
 „ »: 11171.17 
 9.941c ''-'l 18 
 
 940076 ' 18 
 9400051 18 
 940831 1.1« 
 9407631.18 
 9406931.18 
 9406221 18 
 9405511.18 
 94048()'l 18 
 940109,1 18 
 9.94033S1 18 
 9402671.18 
 9401961 18 
 940125,1.19 
 940051 '1.19 
 9399821 19 
 939911 1 19 
 9398401.19 
 939768 1.19 
 9396971.19 
 .939625,1.19 
 939554 1 19 
 9394821.19, 
 9394lo;i.l9 
 9393391 1.19 
 
 9392671.20 
 9391 95 ll 20 
 9391231.20 
 939052 1 20 
 938980 ;1. 20 
 9.93890811.20 
 938836 1.20' 
 938763:1.20, 
 938691 11.20 
 938619:1.20 
 9385471.20 
 938475 1.20 
 938402 1 2] 
 938330 1.21 
 0.7 : 9382581.21 
 if J 9.93S185 1.21 
 ^•66 938113 1.21 
 
 9.743752I 
 744050 
 744348 
 744645 
 744943 
 745240 
 745538 
 745835 
 746132 
 746429 
 746726 
 9.747023 
 747319 
 747616 
 747913 
 748209 
 748505 
 748801 
 749097 
 749393 
 7496891 
 9.749985, 
 750281 
 750576 
 750872 
 751167 
 751462 
 751757 
 752052 
 752347 
 752642 
 9.752937 
 753231 
 753526 
 753820 
 754115 
 754409 
 754703 
 754997 
 755291 
 755585 
 9.755878 
 756172 
 756465 
 756759 
 757052 
 757345 
 7576381 
 7579311 
 758224 
 758517 
 
 9.7.T; 
 
 938040 1.21 
 
 9379071.21 
 
 937895 1.21 
 
 9378221.21 
 
 9377491.21 
 
 937076 1.21 
 
 937604 I 1.21 
 
 _.937531jl^l 
 
 SiHer~l60 
 
 7591021 
 7593951 
 759687 
 759979 
 760272 
 760504 
 760856 
 761148 
 ___761439| 
 Cotanp-. 
 
 Cotang. 
 [10.256248:' 
 255950' 
 255652 
 255355 
 255057 
 254760 
 254462 
 254165 
 253868 
 253571 
 , 253274 
 '10.252977 
 252681 
 252384 
 252087 
 251791 
 251495 
 2511«>9 
 
 250903 
 250607 
 250311 
 '10.250015 
 249719 
 249424 
 249128 
 248833 
 248538 
 248243 
 247948 
 247653 
 , 2473,18 
 [10.24700.}' 
 246709;' 
 246474 
 246180 
 245885 
 245591 
 245297 
 245003 
 244709 
 , 244415 
 (10.244122 
 243828 
 243535 
 243241 
 242948 
 242655 
 2423621 
 242009; 
 241776 
 , 241483 
 '10.241190' 
 240898' 
 240005! 
 240313- 
 240021 1 
 2397281 
 239430 
 2391441 ; 
 238852,' ; 
 238561' 
 
 Tan^: 
 
 
 ^ 
 
■!i 
 
 II 
 
 1 
 
 i 
 
 48 
 
 
 (30 DtOKEES.) A TABLE OF LOOAUITHMIO 
 
 
 1 
 
 1 
 
 M. 
 
 Sine. 
 
 J_»._ 
 
 ! Cosine. 
 
 D. 
 
 'i'any. 
 
 1 D. 
 
 Cotang. 
 
 1 
 
 
 
 
 9.09H970 8.04 
 
 9.937531 
 
 1.21 9.70143! 
 
 4.80 
 
 10.238501 
 
 60 
 
 
 1 
 
 699189 S.Gl 
 
 93745H 
 
 1.2i 
 
 !i 701731 
 
 4.86 
 
 23H2G'J 
 
 69 
 
 
 2 
 
 69940'; 
 
 1 a.iVi 
 
 937383 
 
 1.22i 702(12.' 
 
 4, HO 
 
 237977 
 
 58 
 
 
 3 
 
 699G2( 
 
 ; 3.04 
 
 937312 
 
 1.2i 
 
 ! 702314 
 
 4. HO 
 
 2370S(j 
 
 57 
 
 
 4 
 
 G99H1J 
 
 [^ 3.03 
 
 93723H 
 
 1.2'. 
 
 ! 7020lt( 
 
 4 . H5 
 
 237391 
 
 ' 5(5 
 
 
 5 
 
 70000i 
 
 ! 3.03 
 
 937105 
 
 1.22 702H97 
 
 4.H5 
 
 237103 
 
 55 
 
 
 G 
 
 700280 i 3.03 
 
 937092 
 
 1.22 7031 HH 
 
 4.85 
 
 23081': 
 
 ; 54 
 
 
 7 
 
 70049)" 
 
 1 3.03 
 
 937019 
 
 1.22 703179 
 
 4.85 
 
 230521 
 
 53 
 
 
 8 
 
 70071( 
 
 . 3,03 
 
 93(5940 
 
 1 . 22| 70377( 
 
 4.85 
 
 23(5230 
 
 ! 52 
 
 
 9 
 
 70093;^ 
 
 1 3.02 
 
 930H72 
 
 1.22 
 
 701001 
 
 4.85 
 
 235939 
 
 i 51 
 
 
 10 
 
 701151 
 
 ' 3.02 
 
 930799 
 
 1.22 
 
 701352 
 
 4,84 
 
 235048 
 
 ; 50 
 
 
 11 
 
 9.701301- 
 
 ' 3.02 
 
 9.936725 
 
 1.22 
 
 9.704043 
 
 4.84 
 
 10.23.5357 
 
 j 49 
 
 
 12 
 
 701585 
 
 ; 3.02 
 
 930052 
 
 1.23 7(M933 
 
 4.84 
 
 235007 
 
 48 
 
 . 
 
 13 
 
 701802 
 
 ! 3.01 
 
 030578 
 
 1.23 70.5224 
 
 4. 84 
 
 234770 
 
 47 
 
 
 14 
 
 70201<J 
 
 1 3.01 
 
 930505 
 
 1.23 705514 
 
 4.84 
 
 234480 
 
 46 
 
 
 15 
 
 70223(j 
 
 ' 3,01 
 
 930431 
 
 1.23 705805 
 
 4.84 
 
 234195 
 
 45 
 
 
 16 
 
 702452 
 
 3.61 
 
 93G357 
 
 i.2;i 
 
 7(!(;0!)5 
 
 4.84 
 
 233905 
 
 44 
 
 
 17 
 
 702009 
 
 3.00 
 
 030284 
 
 1.23 
 
 70(5385 
 
 4.83 
 
 233015] 43 1 
 
 
 18 
 
 702885 
 
 3.00 
 
 930210 
 
 1.23' 700G75 
 
 4.83 
 
 233325 
 
 42 
 
 
 19 
 
 703101 
 
 3.60 
 
 930130 
 
 1,23 700905 
 
 4.83 
 
 233035 
 
 41 
 
 • 1 
 
 20 
 
 703317 
 
 3.00 
 
 930002 
 
 1.23 707255 
 
 4.83 
 
 232745 
 
 40 
 
 
 21 
 
 9.703533 
 
 3.59 
 
 9.935988 
 
 1.23 9.707545 
 
 4.83 
 
 10.232455] 39 
 
 
 22 
 
 703749 
 
 ; 3.59 
 
 935914 
 
 1.23, 707834 
 
 4.83 
 
 2321(56] 38 
 
 
 23 
 
 7039G4 
 
 ' 3.59 
 
 935840 
 
 1.23| 708124 
 
 4.82 
 
 231870 37 
 
 
 24 
 
 704179 
 
 i 3.59 
 
 935700 
 
 1.21 
 
 708413 
 
 4.82 
 
 231587 36 
 
 
 25 
 
 704395 
 
 , 3.59 
 
 935092 
 
 1.24 
 
 708703 
 
 4.82 
 
 231297] 35 
 
 
 26 
 
 704610 
 
 1 3.58 
 
 935018 
 
 1.24 
 
 7(58992 
 
 4.82 
 
 231008 34 
 
 • 
 1 
 
 27 
 
 704825 
 
 3.58 
 
 935543 
 
 1.24 
 
 709281 
 
 4.82 
 
 230719 33 
 
 
 28 
 
 705040 
 
 3.58 
 
 935409 
 
 1.24 
 
 709570 
 
 4.82 
 
 230430 
 
 82 
 
 
 29 
 
 705254 
 
 8.58 
 
 935395 
 
 1.24 
 
 769800 
 
 4.81 
 
 230140 
 
 31 
 
 
 30 
 
 7054G9 
 
 3.57 
 
 935320 
 
 1.24] 770148 
 
 4.81 
 
 229852 
 
 80 
 
 , 
 
 31 
 
 9.705683 
 
 3.57 
 
 9.93524G 
 
 1.24; 9.770437 
 
 4.81 
 
 10.229503 
 
 29 
 
 , 
 
 32 
 
 705898 
 
 3.57 
 
 935171 
 
 1.24 
 
 770720 
 
 4.81 
 
 229274 
 
 28 
 
 
 33 
 
 706112 
 
 3.57 
 
 935097 
 
 1.24 
 
 771015 
 
 4.81 
 
 228985 
 
 27 
 
 
 34 
 
 70G32G 
 
 3.56 
 
 935022 
 
 1.24 
 
 771303 
 
 4.81 
 
 228697 
 
 26 
 
 
 35 
 
 70G539 
 
 3. 50 
 
 934948 
 
 1.24 
 
 771592 
 
 4.81 
 
 228408 
 
 25 
 
 ) 
 
 36 
 
 70G753 
 
 3.50 
 
 934873 
 
 1.24 
 
 771880 
 
 4.80 
 
 228120 
 
 24 
 
 
 37 
 
 70G9G7 
 
 3.56 
 
 934798 
 
 1.25 
 
 772168 
 
 4.80 
 
 227832 
 
 23 
 
 
 38 
 
 707180 
 
 3.55 
 
 934723 
 
 1.25 
 
 772457 
 
 4.80 
 
 227543 
 
 22 
 
 
 39 
 
 707393 
 
 3.55 
 
 934649 
 
 1.25 
 
 772745 
 
 4.80 
 
 227255 
 
 21 
 
 
 40 
 
 707600 
 
 3.55 
 
 934574 
 
 1.25 
 
 773033 
 
 4.80 
 
 226967 
 
 20 
 
 
 41 
 
 0.707819 
 
 3.55 
 
 9.934499 
 
 1.25 
 
 9.773321 
 
 4.80 
 
 10.226679 
 
 19 
 
 
 42 
 
 708032 
 
 3.54 
 
 934424 
 
 1.25 
 
 773608 
 
 4.79 
 
 226392 
 
 18 
 
 
 43 
 
 708245 
 
 8.54 
 
 934349 
 
 1.25 
 
 773890 
 
 4.79 
 
 22610. t 
 
 17 
 
 
 44 
 
 708458 
 
 3.54 
 
 934274 
 
 1.25 
 
 774184 
 
 4.79 
 
 225810 16 1 
 
 
 45 
 
 708670 
 
 3.54 
 
 934199 
 
 1.25 
 
 774471 
 
 4.79 
 
 225529 
 
 15 
 
 1 
 
 46 
 
 708882 
 
 3.53 
 
 934123 
 
 1.25 
 
 774759 
 
 4.79 
 
 225241 
 
 14 
 
 
 47 
 
 709094 
 
 3.53 
 
 934048 
 
 1.25 
 
 775040 
 
 4.79 
 
 224954 
 
 13 
 
 
 48 
 
 709306 
 
 3.53 
 
 933973 
 
 1.25 
 
 775333 
 
 4.79 
 
 224667 
 
 12 
 
 ( 
 
 49 
 
 709518 
 
 3.53 
 
 933898 
 
 1.20 
 
 775621 
 
 4.78 
 
 224379 
 
 11 
 
 
 50 
 
 709730 
 
 3.53 
 
 933822 
 
 1.20 
 
 775908 
 
 4.78 
 
 22-4092 
 
 10 
 
 
 51 
 
 •). 709941 
 
 3.52 
 
 9.933747 
 
 1.20 9.770195! 
 
 4.78 
 
 10.2238051 
 
 9 
 
 
 52 
 
 710153 
 
 3.52 
 
 933671 
 
 1.20 
 
 770482 
 
 4.78 
 
 223518 8 
 
 
 53 
 
 710304 
 
 3.52 
 
 933590 
 
 1.20 
 
 776769 
 
 4.78 
 
 223231 7 
 
 
 54 
 
 710575 
 
 8.52 
 
 933520 
 
 1.20 
 
 777055 
 
 4.78 
 
 222945 6 
 
 
 55 
 
 710786 
 
 3.51 
 
 933445 
 
 1.26 
 
 777342 
 
 4.78 
 
 222658 
 
 5 
 
 
 56 
 
 710997 
 
 8.51 
 
 933309 
 
 1.26 
 
 777028 
 
 4.77 
 
 222372 
 
 4 
 
 
 57 
 
 711208 
 
 8.51 
 
 933293 
 
 1.26 
 
 777915 
 
 4,77 
 
 222085 
 
 3 
 
 ' 
 
 53 
 
 711419 
 
 3,51 
 
 933217 
 
 1.26 
 
 778201 
 
 4.77 
 
 221799 
 
 2 
 
 
 59 
 
 711029 
 
 3.50 
 
 933141 
 
 1.20 
 
 778487 
 
 4.77 
 
 2'21512 
 
 1 
 
 
 60 
 
 711839 
 
 8.50 
 D. 1 
 
 933000 
 Sine, 
 
 1.2G 778774] 
 
 4.77 
 
 221220 
 
 
 H. 
 
 
 
 Cosine. 
 
 69 i 
 
 Cotang. i 
 
 1>. 1 
 
 Tang. 
 
60 
 59 
 
 58 
 57 
 
 712050 
 
 7i22(;r» 
 7l2Ki!> 
 712«7!> 
 
 7128H!>J 
 
 7l»)!),sf 
 
 71.'J.'J0S| 
 
 713517, 
 
 71372(1 
 
 , 71393.J 
 
 9.7HMJ 
 
 7143521 
 714501 
 7M7(io] 
 7lt!)7Hi 
 71518(; 
 7153!)4I 
 7l5(J02 
 715809 
 I 7l(J017| 
 ^•71(52241 
 71C432 
 710C39 
 7l(;S4Gi 
 7170531 
 717259] 
 7l74(;c 
 7l7(i73 
 717879 
 , 7180;^5 
 (''•718291 
 718497 
 718703 
 7189091 
 7191141 
 719320, 
 719525 
 719730 
 719935) 
 720140 
 '■720345 
 7205-49 
 7207541 
 720958 
 7211G2 
 7213(JG 
 721570 
 721774] 
 721978 
 u ^22181 
 9.722385] 
 722588 
 722791 
 722994, 
 723197 
 723400, 
 723603 
 723805 
 7240071 
 
 iCosiHeT 
 
 
 '•''osiae.j D 
 
 '^•^•••i.'ioGfi r7,]j 
 
 932990 1 '>] 
 {'•■!2;»14i;^- 
 !'32S3S 1 -j;, 
 
 !'327(;2 1 27 
 !'3:.>(;85 1 27 
 932(Jt.i,«'l 'j; 
 
 J'.i2533r27 
 M2157 1 27 
 y323S0 1 27 
 932304 1 27 
 
 ^'32228 1 27 
 
 •'^J-'loli; 27 
 932075 1 2n 
 93l!);j,s 1 2.S 
 931921 1 28 
 931815 I '28 
 9317G8 1 28 
 931(;91 1 2,s 
 931(514 1 28 
 931537 1 '28 
 5. 9314(50 1 28 
 9-31383 i;2« 
 931300 1 28 
 
 9^12291:29 
 931152 1 29 
 931075 I '2nl 
 930998 r 29 
 930921 1 '29 
 93(3843 1' 29 
 9307GCr2'> 
 9.930G88 1'29 
 930011129 
 930533 1 29 
 930450 1' 29 
 930378 1 '29 
 930300 1 30 
 930223 1 30 
 030145 1 30 
 9300671 "30 
 929989 1 30 
 9.9299111 '30 
 929833 1 '30 
 9297551 30 
 9290771' 30 
 929599 1 30 
 9295211.30 
 929442 1 30 
 929364 1.31 
 929280 1 31 
 „ 9292071.31 
 9.9291291 31 
 929050 1 31 
 9289721 31 
 928893^1 31 
 
 928815 1.31 
 
 928736 1.31 
 
 9280571.31 
 
 928578 1 31 
 
 928499 1 31 
 _928420 1.3l| 
 Sine. 58 
 
 Tang. 
 
 0.778774,' 
 77!>()(;() 
 
 7793 K; 
 779(i.'jj 
 779918 
 
 780203 
 7804.'^!) 
 780775 
 
 4D 
 
 ^•^•221220 1^0" 
 220910 59 
 220054; 68 
 220.3081 67 
 2200,S2 60 
 219797 55 
 21 9511 I 54 
 219225 53 
 218940 52 
 218(554' 51 
 , 2 1 8301) 50 
 'O.2I8O8I' 49 
 21779!)' 48 
 21751.4 47 
 217229 40 
 21 G944 45 
 21(5059 44 , 
 21(5374' 43 
 21(5090' 42 ' 
 215805 41 
 , 21,3521 40 
 "^•215230; 89 
 214952 38 
 214608 87 
 214384 80 
 214100 35 
 213810 34 
 213532 33 
 213248 82 
 2129041 81 
 -,„ 212G81I 30 
 ^0-212397 29 
 212114 28 
 211830 27 
 211547 26 
 211264 25 
 210981 24 
 210698 23 
 210415 22 
 210132 21 
 ,„ 209849 20 
 10.209567 19 
 209284 18 
 209001 17 
 208719,' U 
 2084371 15 
 208154 14 
 207872 13 
 207590 12 
 207308 11 
 ,,> -'07020 10 
 1 10. 206744 ' " 
 ' 206462' 
 208181' 
 205899, 
 205617, 
 205336 
 205055 
 204773 
 204492 
 204211 
 
 m 
 
IM 
 
 50 
 
 
 (.S2 DEGREES.) A 
 
 TABLE OF LOOARITHMIO 
 
 
 M. 
 
 
 
 Sine. 
 
 9.724210 
 
 D. 
 
 Goiine. 
 
 9.928420 
 
 1.32 
 
 lang. 
 
 9.79.5789 
 
 D. 
 
 Cotang. 
 
 -60- 
 
 3 37 
 
 4,68 
 
 10.204211 
 
 1 
 
 724412 
 
 3.37 
 
 928342 
 
 1.32 
 
 79G070 
 
 4. 68 
 
 203930 
 
 59 
 
 2 
 
 724G14 
 
 3.36 
 
 928263 
 
 1.32 
 
 796351 
 
 4.68 
 
 203649 
 
 58 
 
 3 
 
 724S1G 
 
 3.3G 
 
 928183 
 
 1.32 
 
 798632 
 
 4.68 
 
 203368 
 
 57 
 
 4 
 
 725017 
 
 3.36 
 
 928104 
 
 1.32 
 
 796913 
 
 4.68 
 
 203087 
 
 56 
 
 5 
 
 725219 
 
 3.36 
 
 928025 
 
 1.32 
 
 797194 
 
 4.68 
 
 202806 
 
 55 
 
 6 
 
 725420 
 
 3.35 
 
 927946 
 
 1.82 
 
 797475 
 
 4.68 
 
 202525 
 
 54 
 
 7 
 
 725G22 
 
 3.35 
 
 927867 
 
 1.32 
 
 797755 
 
 4.68 
 
 202245 
 
 53 
 
 8 
 
 725823 
 
 3.35 
 
 927787 
 
 1.82 
 
 798036 
 
 4.67 
 
 201964 
 
 52 
 
 d 
 
 726024 
 
 3.35 
 
 927708 
 
 1.82 
 
 798316 
 
 4.67 
 
 201684 
 
 51 
 
 10 
 
 726225 
 
 3.35 
 
 927629 
 
 1..S2 
 
 798596 
 
 4.67 
 
 201404 
 
 50 
 
 11 
 
 9.726426 
 
 3.34 
 
 9.927549 
 
 1.32 
 
 9.798877 
 
 4.67 
 
 10.201123 
 
 49 
 
 12 
 
 726326 
 
 3.34 
 
 927470 
 
 1.33 
 
 7991.57 
 
 4.67 
 
 200843 
 
 48 
 
 13 
 
 726827 
 
 3.34 
 
 927390 
 
 1.33 
 
 799437 
 
 4.67 
 
 200563 
 
 47 
 
 U 
 
 727027 
 
 3.34 
 
 927310 
 
 1.33 
 
 799717 
 
 4.67 
 
 2002831 46 
 
 15 
 
 727228 
 
 3.34 
 
 927231 
 
 1.33 
 
 799997 
 
 4.6G 
 
 2000031 45 
 
 IG 
 
 727428 
 
 3.33 
 
 927151 
 
 1.33 
 
 800277 
 
 4.66 
 
 199723 
 
 44 
 
 17 
 
 727628 
 
 3.33 
 
 927071 
 
 1.83 
 
 800557 
 
 4.66 
 
 199443 
 
 43 
 
 18 
 
 727828 
 
 3.33 
 
 926991 
 
 1.83 
 
 800836 
 
 4.66 
 
 199164 
 
 42 
 
 19 
 
 728027 
 
 3.33 
 
 926911 
 
 1.83 
 
 801116 
 
 4. 66 
 
 198884 
 
 41 
 
 20 
 
 728227 
 
 3.33 
 
 926831 
 
 1.33 
 
 801396 
 
 4. 66 
 
 198604 
 
 40 
 
 21 
 
 9.728427 
 
 3.32 
 
 9.926751 
 
 1.33 
 
 9.801675 
 
 4.66 
 
 10.198325 
 
 39 
 
 22 
 
 728626 
 
 3.32 
 
 926671 
 
 1.33 
 
 801955 
 
 4.66 
 
 198045 
 
 38 
 
 23 
 
 728825 
 
 3.32 
 
 926591 
 
 1,33 
 
 802234 
 
 4.65 
 
 197766 
 
 37 
 
 24 
 
 729024 
 
 3.32 
 
 926511 
 
 1.34 
 
 802513 
 
 4.65 
 
 197487 
 
 36 
 
 25 
 
 7'i9?93 
 
 3.31 
 
 926431 
 
 1.34 
 
 802792 
 
 4.65 
 
 197208 35 
 
 26 
 
 ^j9422 
 
 3.31 
 
 926351 
 
 1.34 
 
 803072 
 
 4.65 
 
 196928: 34 
 
 27 
 
 729G21 
 
 3.31 
 
 926270 
 
 1.34 
 
 803351 
 
 4.65 
 
 196649 
 
 33 
 
 28 
 
 729820 
 
 3.31 
 
 926190 
 
 1.34 
 
 803<)80 
 
 4.65 
 
 196370 
 
 82 
 
 29 
 
 730018 
 
 3.30 
 
 926110 
 
 1.34 
 
 803908 
 
 4.65 
 
 196092 
 
 81 
 
 30 
 
 730216 
 
 3.30 
 
 926029 
 
 1.84 
 
 804187 
 
 4.65 
 
 195813 
 
 80 
 
 31 
 
 9.730415 
 
 3.30 
 
 9.925949 
 
 1.84 
 
 9.804466 
 
 4.64 
 
 10.19.5534 
 
 29 
 
 32 
 
 730613 
 
 3.30 
 
 925868 
 
 1.84 
 
 804745 
 
 4.64 
 
 195255 
 
 28 
 
 33 
 
 730811 
 
 3.30 
 
 925788 
 
 1.34 
 
 805023 
 
 4.64 
 
 194977 
 
 27 
 
 34 
 
 731009 
 
 3 29 
 
 925707 
 
 1.34 
 
 805302 
 
 4.64 
 
 194698 
 
 26 
 
 35 
 
 731206 
 
 3.29 
 
 925626 
 
 1.34 
 
 805580 
 
 4.64 
 
 194420 
 
 25 
 
 86 
 
 731404 
 
 3.29 
 
 925545 
 
 1.35 
 
 805859 
 
 4.64 
 
 194141 
 
 24 
 
 37 
 
 731G02 
 
 3.29 
 
 925465 
 
 1.35 
 
 806137 
 
 4.64 
 
 198863 
 
 23 
 
 38 
 
 731799 
 
 3.29 
 
 925384 
 
 1.35 
 
 806415 
 
 4.63 
 
 193585 
 
 22 
 
 39 
 
 731996 
 
 3.28 
 
 925393 
 
 1.35 
 
 806693 
 
 4.63 
 
 193307 
 
 21 
 
 40 
 
 732193 
 
 3.28 
 
 925222 
 
 1.35 
 
 806971 
 
 4.63 
 
 193029 
 
 20 
 
 41 
 
 9.732390 
 
 3.28 
 
 9.925141 
 
 1.35 
 
 9.807249 
 
 4.63 
 
 10.192751 
 
 19 
 
 42 
 
 732587 
 
 3.28 
 
 925060 
 
 1.35 
 
 807527 
 
 4.63 
 
 192473 
 
 18 
 
 43 
 
 732784 
 
 3.28 
 
 924979 
 
 1.35 
 
 807805 
 
 4.63 
 
 192195 
 
 17 
 
 44 
 
 732980 
 
 3.27 
 
 924897 
 
 1.35 
 
 808083 
 
 4.63 
 
 191917 
 
 16 
 
 45 
 
 733177 
 
 3.27 
 
 924816 
 
 1.35 
 
 8083G1 
 
 4.63 
 
 191639 
 
 15 
 
 4G 
 
 733373 
 
 3.27 
 
 924735 
 
 1.36 
 
 808638 
 
 4.62 
 
 191362 
 
 14 
 
 47 
 
 7335G9 
 
 3.27 
 
 924654 
 
 1.36 
 
 808916 
 
 4.62 
 
 191084 
 
 1? 
 
 48 
 
 733765 
 
 3.27 
 
 924572 
 
 1.30 
 
 809193 
 
 4.62 
 
 190807 
 
 12 
 
 49 
 
 733961 
 
 3.26 
 
 924491 
 
 1.3G 
 
 809471 
 
 4.62 
 
 190529 11 
 
 50 
 
 734157 
 
 3.26 
 
 924409 
 
 1.36 
 
 809748 
 
 4.62 
 
 190252 10 
 
 51 
 
 9.734353 
 
 3.26 
 
 9.924328 
 
 1.36 
 
 9.810025 
 
 4.62 
 
 10.189975 9 
 
 52 
 
 731549 
 
 3.26 
 
 92424G 
 
 1.36 
 
 810302 
 
 4.62 
 
 189698 8 
 
 53 
 
 734744 
 
 3.25 
 
 924164 
 
 1.36 
 
 810580 
 
 4.62 
 
 189420 7 
 
 54 
 
 734i)39 
 
 3.25 
 
 924083 
 
 1.36 
 
 810857 
 
 4.62 
 
 189143 6 
 
 55 
 
 735135 
 
 3.25 
 
 924001 
 
 1..3G 
 
 811134 
 
 4.61 
 
 188866 
 
 6 
 
 50 
 
 735330 
 
 3.25 
 
 923919 
 
 1.3G 
 
 H11410 
 
 4.61 
 
 188590 
 
 4 
 
 57 
 
 735525 
 
 3.25 
 
 923837 
 
 1.36 
 
 811687 
 
 4 61 
 
 188313: 3 
 
 58 
 
 735719 
 
 3.24 
 
 92.3755 1.37 
 
 811964 
 
 4.61 
 
 188036' 2 
 
 5!) 
 
 735914 
 
 3.24 
 
 923673 1.37 
 
 812241 
 
 4 61 
 
 187759! 1 
 
 (iO 
 
 736109 
 
 3 24 
 
 923.591 1.37 
 
 HI 251 7 
 Cotang'. 
 
 4.(;i 
 
 D. 
 
 1874H3 
 
 
 
 1 
 
 Cosine. 
 
 Sine. 
 
 07j_ 
 
 Tann-. 
 
 >'. 
 
 I 
 
114 
 
 lir 
 
 12 
 11 
 10 I 
 
 9 
 
 8 
 i7 
 
 6 
 |6 
 
 1^ 
 3 
 
 1 
 
 
 N». 
 
 
 
 8IKES ASD T.ANGENTS. 
 
 (33 DEGKEES.) 
 
 
 SI 
 
 M. 
 
 
 Sine. 
 
 D. 
 
 3.24 
 
 Cosine. | 
 
 D. 
 
 1.37 
 
 Tang. 
 
 D. 
 
 4.61 
 
 Cotang. 
 
 60 
 
 9.736109 
 
 9.923591 
 
 9.812517 
 
 10.187482 
 
 1 
 
 73G303 
 
 3.24 
 
 923509 
 
 1.37 
 
 812794 
 
 4.61 
 
 187206 
 
 59 
 
 2 
 
 736498 
 
 8.24 
 
 923427 
 
 1.37 
 
 813070 
 
 4.61 
 
 186930 
 
 58 
 
 3 
 
 736692 
 
 8.23 
 
 923345 
 
 1.37 
 
 813347 
 
 4.(;o 
 
 186653 
 
 57 
 
 4 
 
 736880 
 
 8.23 
 
 923263 
 
 1.37 
 
 813623 
 
 4.60 
 
 186377 
 
 56 
 
 5 
 
 737080 
 
 8.23 
 
 923181 
 
 1.37 
 
 813899 
 
 4.60 
 
 186101 
 
 55 
 
 G 
 
 737274 
 
 8.23 
 
 923093 
 
 1.37 
 
 814175 
 
 4.60 
 
 185825 
 
 54 
 
 7 
 
 737467 
 
 3.23 
 
 923016 
 
 1.37 
 
 814452 
 
 4.(.0 
 
 185548 
 
 53 
 
 8 
 
 737661 
 
 8.22 
 
 922933 
 
 1.37 
 
 814728 
 
 4 . 60 
 
 185272 
 
 52 
 
 
 
 737855 
 
 3.22 
 
 922851 
 
 1.37 
 
 815004 
 
 4.60 
 
 184996 
 
 51 
 
 10 
 
 738048 
 
 8.22 
 
 922768 
 
 1.3S 
 
 815279 
 
 4,60 
 
 184721 
 
 50 
 
 11 
 
 9.738241 
 
 8.22 
 
 9.9226S6 
 
 1 38 
 
 9 815555 
 
 4.59 
 
 10.181445 
 
 49 
 
 12 
 
 738434 
 
 3.22 
 
 922603 
 
 1 . 38 
 
 815831 
 
 4.59 
 
 184169 
 
 48 
 
 13 
 
 73S627 
 
 3.21 
 
 922520 
 
 1 . ;'.8 
 
 816107 
 
 4.59 
 
 183893 
 
 47 
 
 14 
 
 73 iS20 
 
 3.21 
 
 9224P- 
 
 ». 38 
 
 816382 
 
 4.59 
 
 183618 
 
 46 
 
 15 
 
 73!)013 
 
 3.21 
 
 9223.>.:; 
 
 '..38 
 
 81(!658 
 
 4.59 
 
 183312 
 
 45 
 
 10 
 
 73!)206 
 
 8.21 
 
 922272: 
 
 1.3S 
 
 816933 
 
 4.59 
 
 1830; i7 
 
 44 
 
 17 
 
 7393,)8 
 
 3.21 
 
 922189 
 
 1.38 
 
 817209 
 
 4.59 
 
 182791 
 
 43 
 
 18 
 
 7395:i0 
 
 3.20 
 
 922106 
 
 1.38 
 
 8174S4 
 
 4.59 
 
 182516 
 
 42 
 
 11) 
 
 739783 
 
 3.20 
 
 922023 
 
 1.38 
 
 817759 
 
 4.59 
 
 182:111 
 
 41 
 
 20 
 
 739975 
 
 8.20 
 
 921940 
 
 1.38 
 
 818035 
 
 4.58 
 
 181965 
 
 40 
 
 21 
 
 J. 740167 
 
 3.20 
 
 9.921857 
 
 1 . 39 
 
 9.818310 
 
 4.58 
 
 10.181690 
 
 39 , 
 
 22 
 
 740359 
 
 8.20 
 
 921774 
 
 1.39 
 
 818585 
 
 4,58 
 
 181415 
 
 38 
 
 23 
 
 740550 
 
 3.19 
 
 921691 
 
 1.39 
 
 818860 
 
 4.58 
 
 181140 
 
 37 
 
 24 
 
 740742 
 
 3.19 
 
 921607 
 
 1.39 
 
 819135 
 
 4.58 
 
 180865 
 
 86 . 
 
 25 
 
 740934 
 
 3.19 
 
 921524 
 
 1.39 
 
 819410 
 
 4.5S 
 
 180590 
 
 35 
 
 2« 
 
 741125 
 
 3.19 
 
 9214-fl 
 
 1 . 39 
 
 819684 
 
 4.58 
 
 180316 
 
 34 . 
 
 27 
 
 741316 
 
 3.19 
 
 921357 
 
 1.3!) 
 
 819959 
 
 4.58 
 
 1S0041 
 
 33 
 
 28 
 
 741508 
 
 3.18 
 
 921274; 
 
 1 . 39 
 
 820234 
 
 4.58 
 
 179766 
 
 32 
 
 29 
 
 741699 
 
 3.18 
 
 921190 
 
 1.3.) 
 
 820508 
 
 4.57 
 
 170492 
 
 31 
 
 30 
 
 741889 
 
 3 18 
 
 921107 
 
 1.39 
 
 820783 
 
 4.57 
 
 170217 
 
 30 
 
 31 
 
 9.742080 
 
 3.18 
 
 9. '.(21023 
 
 1,39 
 
 9,821057 
 
 4.57 
 
 10 178913 
 
 29 
 
 32 
 
 742271 
 
 3 18 
 
 920939 
 
 1.40 
 
 821332 
 
 4 57 
 
 178668 
 
 28 
 
 33 
 
 742462 
 
 3.17 
 
 920856 
 
 1.10 
 
 821606 
 
 4 57 
 
 1783 11 
 
 27 
 
 34 
 
 742652 
 
 3 17 
 
 920772 
 
 1.40 
 
 8218.S0 
 
 4 57 
 
 178120 
 
 20 
 
 35 
 
 742842 
 
 3.J7 
 
 920688: 
 
 1.40 
 
 822154 
 
 4.57 
 
 177810 
 
 25 
 
 36 
 
 743033 
 
 3.17 
 
 920604' 
 
 1 40 
 
 822429 
 
 4.57 
 
 177571 
 
 24 , 
 
 37 
 
 743223 
 
 3.17 
 
 9205201 
 
 1.40 
 
 822703 
 
 4 .-J 
 
 177297 
 
 23 ■ 
 
 38 
 
 743ii:> 
 
 3.16 
 
 920 4361 
 
 1.40 
 
 822977 
 
 4 56 
 
 177023 
 
 22 i 
 
 39 
 
 7i'^;0: 
 
 3 16 
 
 9203521 
 
 1,40 
 
 8232.j(( 
 
 4.56 
 
 17.. 750 
 
 21 , 
 
 40 
 
 7^V<32 
 
 3 16 
 
 920268: 
 
 1,4(1 
 
 823524 
 
 4 56 
 
 176476 
 
 20 
 
 41 
 
 9.7-;-n-^ 
 
 3.16 
 
 9.920184' 
 
 1,4(1 
 
 9.H237;)8 
 
 4 56 
 
 10.176202 
 
 19 
 
 42 
 
 "i417l 
 
 3.16 
 
 920090 
 
 1.40 
 
 824072 
 
 4.56 
 
 175:)28 
 
 18 
 
 43 
 
 •..4l;i61 
 
 3.15 
 
 920015 
 
 1,40 
 
 824345 
 
 4 56 
 
 175655 
 
 17 
 
 44 
 
 714550 
 
 3 15 
 
 919931 
 
 1.41 
 
 824619 
 
 4.56 
 
 1753S1 
 
 16 
 
 45 
 
 744739 
 
 3 15 
 
 919846 
 
 1.41 
 
 824893 
 
 4.56 
 
 175107 
 
 15 
 
 40 
 
 744928 
 
 3.15 
 
 9197()2 
 
 1,41 
 
 825166 
 
 4.56 
 
 17H31 
 
 14 
 
 47 
 
 745117 
 
 3.15 
 
 ;»19,)77 
 
 1,41 
 
 825139 
 
 4.55 
 
 17 1.-).. 1 
 
 13 
 
 48 
 
 745306 
 
 3.14 
 
 919593 
 
 in 
 
 825713 
 
 4.55 
 
 1712(7 
 
 12 
 
 49 
 
 745i;»l 
 
 3.14 
 
 91950.S 
 
 1.41 
 
 8259:S(; 
 
 4 . 55 
 
 1711)11 
 
 11 
 
 50 
 
 7453(3 
 
 3 14 
 
 919124 
 
 1.11 
 
 r,2(;25;i 
 
 4 . 55 
 
 173711 
 
 10 
 
 51 
 
 1.745871 
 
 3 14 
 
 9.919339 
 
 1.41 
 
 9,,S2,i532 
 
 4.55 
 
 :o.i73t(;s, y 
 
 52 
 
 74605i» 
 
 3.14 
 
 919254 
 
 1.41 
 
 h-2r,805 
 
 4.55 
 
 173195 8 
 
 53 
 
 7462 IS 
 
 3 13 
 
 919169 
 
 1.41 
 
 82707H 
 
 4.55 
 
 172922 
 
 7 
 
 04 
 
 746436 
 
 8 13 
 
 919085 
 
 1,41 
 
 8:.73-.l 
 
 4.55 
 
 1720 !9 
 
 6 
 
 55 
 
 746624 
 
 3 13 
 
 919000 
 
 1.41 
 
 827624 
 
 4.55 
 
 172376 
 
 5 
 
 5(5 
 
 746812 
 
 3.13 
 
 918915 
 
 1.42 
 
 827807 
 
 4.5-i 
 
 172103 
 
 4 
 
 57 
 
 7469tt;) 
 
 3.13 
 
 91 8830 
 
 1 42 
 
 82-' 170 
 
 4 54 
 
 171830 
 
 3 
 
 5S 
 
 747187 
 
 3.12 
 
 918715 
 
 1 42 
 
 82s;t2 
 
 4 54 
 
 1715.58 
 
 2 
 
 59 
 
 747374 
 
 3.12 
 
 91865U 
 
 1.42 
 
 828715 
 
 4 . 54 
 
 171285 
 
 1 
 
 CO 
 
 747562 
 
 8.12 
 
 918571 
 
 1.42 
 
 82S987 
 
 4.54 
 
 171013 
 
 
 
 
 Cosine. 
 
 D. 
 
 Sine. { 
 
 te 
 
 Cotang. 
 
 D. 
 
 tang. 1 
 
 J^ 
 

 
 «2 
 
 
 (34 DEGREKS.) A 
 
 TABLE OF LOGARITHMIC 
 
 
 M. 
 
 
 1 Sine. 
 9.7475G2 
 
 D. 
 
 Cosine. 
 
 'J.>.)18574 
 
 1.42 
 
 ! Tang. 
 
 ' 9.828987 
 
 D. 
 
 1 Cotang. ; 
 
 10.171013 00 
 
 3,lii 
 
 4.54 
 
 1 
 
 74774S) 
 
 3.12 
 
 91848!) 
 
 1.42 
 
 ' «292C(] 
 
 4.54 
 
 17(171(1 59 
 
 2 
 
 74793( 
 
 3 12 
 
 91S404 
 
 1.42 
 
 829532 
 
 4.54 
 
 1701(iS 58 
 
 3 
 
 71^12:] 
 
 3.11 
 
 ' 918318 1.42 
 
 82!)805 
 
 4.54 
 
 17019;! 
 
 i 57 
 
 4 
 
 7-is.'51i 
 
 3.11 
 
 918233 
 
 1.42 
 
 83U077 
 
 4.54 
 
 16!)923 
 
 50 
 
 5 
 
 748197 
 
 3.11 
 
 918147 
 
 11.42 
 
 83034! 
 
 4.53 
 
 109051 
 
 55 
 
 
 
 74HGS3 
 
 3.11 
 
 !tlS0(;2 
 
 1.42 
 
 83()(;21 
 
 4.53 
 
 10937!) 
 
 54 
 
 7 
 
 748870 
 
 3.11 
 
 9171170 1.13 
 
 «3(l.sy3 
 
 4.53 
 
 109107 
 
 53 
 
 8 
 
 74905(j 
 
 3.10 
 
 , •I17s:)11.13 
 
 831100 
 
 4.53 
 
 108835! 52 
 
 9 
 
 749243 
 
 3.10 
 
 917.S05 1. 13 
 
 831437 
 
 4.53 
 
 108503' 51 
 
 10 
 
 74942!) 
 
 3.10 
 
 917719,1.13 
 
 83170!) 
 
 4.53 
 
 108291! 50 
 
 11 
 
 9.749G15 
 
 3.10 
 
 9.917031 
 
 1.43 
 
 9.83l!)81 
 
 4.53 
 
 lO.KJHOl! 
 
 , 49 
 
 12 
 
 749801 
 
 3.10 
 
 91754s 
 
 1.43 
 
 832253 
 
 4.53 
 
 107747 
 
 48 
 
 13 
 
 749987 
 
 3.09 
 
 9174G2 
 
 1.43 
 
 832525 
 
 4.53 
 
 107475 
 
 47 
 
 14 
 
 750172 
 
 3.09 
 
 91737GI1.43 
 
 8327!)6 
 
 4.53 
 
 107204 
 
 ! 40 
 
 15 
 
 750358 
 
 8.09 
 
 917290 
 
 1.43 
 
 i 833008 
 
 4.52 
 
 100932 
 
 45 
 
 IG 
 
 7r)().-',43 
 
 8.09 
 
 917204 
 
 1.43 
 
 833339 
 
 4.52 
 
 IGOOOl 
 
 44 
 
 17 
 
 750729 
 
 3 09 
 
 917118,1.44 
 
 833011 
 
 4.52 
 
 100389 
 
 43 
 
 18 
 
 750914 
 
 3.08 
 
 917032 1.44 
 
 833882 
 
 ! 4.52 
 
 106118 
 
 42 
 
 li) 
 
 751099 
 
 3.08 
 
 ; 910940,1.44 
 
 i 834154 
 
 i 4.52 
 
 105840 
 
 41 
 
 20 
 
 751284 
 
 3.08 
 
 1 9108591.44 
 
 ' 834425 
 
 4.52 
 
 105575 40 
 
 21 
 
 9.7514G9 
 
 8.08 
 
 ! 9.9107731.44 
 
 9.831096 
 
 4.52 
 
 10.105304' 39 
 
 22 
 
 7)1054 
 
 3.08 
 
 91GG871.44 
 
 834907 
 
 4.52 
 
 105034' 38 
 
 , 23 
 
 751839 
 
 3.08 
 
 91G0001.44 
 
 835238 
 
 , 4.52 
 
 104702 
 
 37 
 
 24 
 
 752023 
 
 3.07 
 
 9105141.44 
 
 835509 
 
 4.52 
 
 104491 
 
 30 
 
 25 
 
 752208 
 
 3.07 
 
 ' 9104271.44 
 
 835780 
 
 4.51 
 
 104220 
 
 35 
 
 . 20 
 
 752392 
 
 8,07 
 
 91034111.44 
 
 83G051 
 
 4.31 
 
 103949 
 
 34 
 
 27 
 
 752576 
 
 3.07 
 
 9102541.44 
 
 830322 
 
 4.51 
 
 103078 
 
 33 
 
 28 
 
 7527G0 
 
 8.07 
 
 910107 
 
 1.45 
 
 830593 
 
 4.51 
 
 103407 
 
 32 
 
 21) 
 
 752944 
 
 8.06 
 
 910081 
 
 1.45 
 
 830864 
 
 4.51 
 
 103130 
 
 31 
 
 30 
 
 753128 
 
 3 06 
 
 915994 
 
 1.45 
 
 837134 
 
 4.51 
 
 102800 
 
 30 
 
 31 
 
 9.753312 
 
 3. 00 
 
 9.915907 
 
 1.45 
 
 9.837405 
 
 4.51 
 
 10.102595 
 
 29 
 
 , 32 
 
 753495 
 
 3.00 
 
 915820 
 
 1.45 
 
 837075 
 
 4.51 
 
 102325 
 
 28 
 
 33 
 
 753079 
 
 3.06 
 
 915733 
 
 1.45 
 
 837940 
 
 4. 51 
 
 102054 
 
 27 
 
 ' 34 
 
 7538G2 
 
 3.05 
 
 915040 
 
 1.45 
 
 838210 
 
 4.51 
 
 101784 
 
 26 
 
 35 
 
 754040 
 
 3.05 
 
 915559 
 
 1.45 
 
 838487 
 
 4.50 
 
 101513 
 
 25 
 
 36 
 
 754229 
 
 8 05 
 
 915472 
 
 1.45 
 
 838757 
 
 4.50 
 
 101243 
 
 24 
 
 37 
 
 754412 
 
 3.05 
 
 915385 
 
 1.45 
 
 83!)027 
 
 4.50 
 
 100973 
 
 23 
 
 38 
 
 75451^5 
 
 3.05 
 
 915297 
 
 1.45 
 
 839297 
 
 4.50 
 
 100703 
 
 00 
 
 39 
 
 754778 
 
 3.04 
 
 915210 
 
 1.45 
 
 8395G8 
 
 4.60 
 
 100432 
 
 21 
 
 40 
 
 754900 
 
 3.04 
 
 915123 
 
 1.40 
 
 839838 
 
 4.50 
 
 160162 
 
 20 
 
 41 
 
 9.755143 3.04 
 
 9.915035 
 
 1.40 
 
 9.:U0108 
 
 4.50 
 
 10.159892 
 
 19 
 
 42 
 
 755320 
 
 3.04 
 
 914948 
 
 1.40 
 
 840378 
 
 4.50 
 
 159022, 
 
 18 
 
 43 
 
 755508 
 
 3.04 
 
 914800 
 
 1.46 
 
 840(!47 
 
 4.50 
 
 159353 
 
 17 
 
 44 
 
 755090 
 
 3.04 
 
 914773 
 
 1.46 
 
 840917 
 
 4.49 
 
 159083, 
 
 10 
 
 45 
 
 755872 
 
 3.03 
 
 914085 
 
 1.46 
 
 841187 
 
 4.49 
 
 158813 
 
 15 
 
 4(5 
 
 750054 
 
 3.03 
 
 914598 
 
 1.46 
 
 841457 
 
 4.49 
 
 158543 
 
 14 
 
 47 
 
 75023G 
 
 8.03 
 
 914510 
 
 1.46 
 
 841720 
 
 4.49 
 
 158274' 
 
 13 
 
 48 
 
 750418 
 
 3.03 
 
 914422 
 
 1.46! 
 
 841990 
 
 4.49 
 
 158004 
 
 12 
 
 49 
 
 750000 
 
 3.03 
 
 9143341 
 
 1.46J 
 
 812200 
 
 4.49 
 
 157734 
 
 11 
 
 50 
 
 750782 
 
 8.02 
 
 914240 ! 
 
 1.47! 
 
 842535 
 
 4.49 
 
 157405 
 
 10 
 
 51 
 
 9.75G9G3 
 
 8.02 1 
 
 9.914158 
 
 1.47 
 
 9.842805 
 
 4 49 
 
 10.157195 
 
 9 
 
 52 
 
 757144 
 
 3.02 
 
 914070 
 
 1.47 
 
 843074J 
 
 4.49 
 
 156926 
 
 8 
 
 53 
 
 757326 
 
 8.02 
 
 913982 
 
 1.47 
 
 843343 
 
 4.49 
 
 156657 
 
 7 
 
 54 
 
 757507 
 
 3.02 
 
 913894! 
 
 1.47 
 
 843012 
 
 4.49 
 
 1563881 
 
 6 
 
 55 
 
 757088 
 
 3.01 
 
 913S00 
 
 1.47 
 
 8438S2 
 
 4.48 
 
 156118 
 
 5 
 
 56 
 
 757809 
 
 3.01 
 
 913718 
 
 1.471 
 
 8441511 
 
 4.48 
 
 155849 
 
 4 
 
 57 
 
 758050 
 
 3.01 
 
 913030 [ 
 
 1.47 
 
 844420 
 
 4.48 
 
 155580 
 
 3 
 
 58 
 
 758230 
 
 3.01 
 
 913541 1.47! 
 
 84408S) 
 
 4.48 
 
 155311! 
 
 2 
 
 59 
 
 758411 
 
 8.01 
 
 913453 1.47 
 
 844958 
 
 4.48 
 
 155012 
 
 1 
 
 (>0 
 
 758591 
 
 3.01 
 
 913305 1 47 
 
 8152271 
 Cotang. 1 
 
 4.48 
 
 154773 
 
 
 
 1 i 
 
 Cosine, i 
 
 D. 1 
 
 Sine. 
 
 od 1 
 
 D. 
 
 Tang, i 
 
 M. 
 

 
 SINES AND TANGENTS. 
 
 (35 DEGREES.) 
 
 5:^ 
 
 M. 1 
 U 
 
 Sine. 1 
 9.758591 
 
 D. 
 
 Cosine. 
 
 9.',ti;i;!ii: 
 
 )) 1 
 1.47 
 
 Tang. 
 
 9.84.5227 
 
 D. 
 
 4.4S 
 
 Cotang. : 
 10.154773 (iO 
 
 3.01 
 
 1 
 
 758772 
 
 3.00 
 
 913270 
 
 1.47 
 
 84.5490 
 
 4.48 
 
 154501 59 
 
 2 
 
 758952 
 
 3 on 
 
 9131 87 
 
 1.48 
 
 845701 
 
 4.48 
 
 1542.30 58 
 
 o 
 
 759132 
 
 3.00 
 
 913099 
 
 1.48 
 
 H4r)033 
 
 4.48 
 
 1531K57; 57 
 
 4 
 
 759312 
 
 3.00 
 
 913010 
 
 1.48 
 
 840302 
 
 4.48 
 
 15309.S' 50 
 
 5 
 
 759 192 
 
 3.00 
 
 912922 
 
 1.48 
 
 840570 
 
 4.47 
 
 1534.30' 55 
 
 6 
 
 759072 
 
 2.99 
 
 912833 
 
 1.4H 
 
 840S.",9 
 
 4.47 
 
 1.531(51' 54 
 
 7 
 
 759852 
 
 2.99 
 
 912744 
 
 1.48 
 
 847107 
 
 4.47 
 
 1.52893 53 
 
 8 
 
 700031 
 
 2.99 
 
 912055 
 
 1.48 
 
 847370 
 
 4.47 
 
 152(524 52 
 
 9 
 
 700211 
 
 2.99 
 
 9125(;o 
 
 1.48 
 
 847044 
 
 4.47 
 
 152350 .51 
 
 10 
 
 700390 
 
 2.99 
 
 912477 
 
 1.48 
 
 847913 
 
 4.47 
 
 1520S7; 50 
 
 11 
 
 9 700509 
 
 2.98 
 
 9.912388 
 
 1.48 
 
 9.848181 
 
 4.47 
 
 10.151S19 49 
 
 12 
 
 '700748 
 
 2.98 
 
 912299 
 
 1.49 
 
 848449 
 
 4.47 
 
 151551' 48 
 
 13 
 
 700927 
 
 2.98 
 
 912210 
 
 1.49 
 
 848717 
 
 4.47 
 
 1512S3 47 
 
 U 
 
 701100 
 
 2.98 
 
 912121 
 
 1.49 
 
 848980 
 
 4.47 
 
 151014 46 
 
 15 
 
 7G1285 
 
 2.98 
 
 912031 
 
 1.49 
 
 849254 
 
 4.47 
 
 150740 45 
 
 IG 
 
 7G14G4 
 
 2.98 
 
 911942 
 
 1.49 
 
 849.522 
 
 4.47 
 
 15017.S 44 
 
 17 
 
 7G1G42 
 
 2.97 
 
 911853 
 
 1.49 
 
 849790 
 
 4.40 
 
 150210 43 
 
 IH 
 
 701821 
 
 2.97 
 
 911703 
 
 1.49 
 
 8.50058 
 
 4.40 
 
 149942' 42 
 
 19 
 
 701999 
 
 2.97 
 
 911074 
 
 1.49 
 
 850325 
 
 4.40 
 
 149(575 41 
 
 20 
 
 702177 
 
 2.97 
 
 911584 
 
 1.49 
 
 850593 
 
 4.40 
 
 149407; 40 
 
 21 
 
 9 702350 
 
 2.97 
 
 9.911495 
 
 1.49 
 
 9.8.50801 
 
 4.40 
 
 10.149i;!9 39 
 
 22 
 
 702531 
 
 2.90 
 
 911405 
 
 1.49 
 
 851129 
 
 4. 46 
 
 148871 38 
 
 23 
 
 702712 
 
 2.90 
 
 911315 
 
 1.50 
 
 851390 
 
 4.46 
 
 148001 37 
 
 24 
 
 702889 
 
 2.90 
 
 911220 
 
 1.50 
 
 851004 
 
 4.46 
 
 148330 36 
 
 25 
 
 703037 
 
 2.90 
 
 911130 
 
 1.50 
 
 851931 
 
 4.46 
 
 148009, 35 
 
 20 
 
 703245 
 
 2,90 
 
 91104G 
 
 1.50 
 
 8.52199 
 
 4.40 
 
 147801 34 
 
 27 
 
 703122 
 
 2.90 
 
 910950 
 
 1.50 
 
 8524CG 
 
 4.4(5 
 
 147534 33 
 
 28 
 
 703()00 
 
 2.95 
 
 910800 
 
 1.50 
 
 852733 
 
 4.45 
 
 1472(57, 32 
 
 2!) 
 
 703777 
 
 2.95 
 
 910770 
 
 1.50 
 
 853001 
 
 4.45 
 
 14(9)9 31 
 
 30 
 
 703!)54 
 
 2.95 
 
 910080 
 
 1.50 
 
 8532(;8 
 
 4,45 
 
 1407:52 30 
 
 31 
 
 9.704131 
 
 2.95 
 
 9.010590 
 
 1.50 
 
 9.853535 
 
 4.45 
 
 10 140405! 29 
 
 32 
 
 704308 
 
 2.95 
 
 91050(1 
 
 1.50 
 
 853802 
 
 4.45 
 
 140198 28 
 
 33 
 
 701485 
 
 2.94 
 
 910415 
 
 1.50 
 
 8540G9 
 
 4.45 
 
 145! )3 1' 27 
 
 34 
 
 704602 
 
 2.94 
 
 910325 
 
 1.51 
 
 854336 
 
 4.45 
 
 14.5(5(541 26 
 
 35 
 
 7(i4838 
 
 2.94 
 
 910235 
 
 1.51 
 
 854(503 
 
 4.45 
 
 145397 25 
 
 30 
 
 705015 
 
 2.94 
 
 910144 
 
 1.51 
 
 854^70 
 
 4.45 
 
 145130 24 
 
 37 
 
 705191 
 
 2.94 
 
 910054 
 
 1.51 
 
 855137 
 
 4,45 
 
 144s(53 23 
 
 38 
 
 7()53(')7 
 
 2.94 
 
 9099G3 
 
 1.51 
 
 85.5404 
 
 4.45 
 
 144590 22 
 
 31) 
 
 705544 
 
 2.93 
 
 909873 
 
 1.51 
 
 855071 
 
 4.44 
 
 144329 21 
 
 40 
 
 705720 
 
 2.93 
 
 909782 
 
 1.51 
 
 855938 
 
 4.44 
 
 1440(52 20 
 
 41 
 
 9.705890 
 
 2.93 
 
 9.909G91 
 
 1.51 
 
 9.85(5204 
 
 4.44 
 
 lfl.14.37110 19 
 
 42 
 
 700072 
 
 2.93 
 
 909G01 
 
 1.51 
 
 85i;47J 
 
 4,44 
 
 143.521) 18 
 
 43 
 
 700247 
 
 2.93 
 
 909510 
 
 1.51 
 
 85(57,37 
 
 4.44 
 
 14.3203' 17 
 
 4-t 
 
 700423 
 
 2.!)3 
 
 909419 
 
 1.51 
 
 857004 
 
 4,44 
 
 1429:)6 10 
 
 45 
 
 700598 
 
 2 92 
 
 909328 
 
 1.52 
 
 857270 
 
 4.44 
 
 1427.30 15 
 
 id 
 
 70()774 
 
 2.92 
 
 !K)9237 
 
 1.52 
 
 8575:{7 
 
 4.41 
 
 1421;,;! 14 
 
 47 
 
 70(J949 
 
 2,92 
 
 909140 
 
 1.52 
 
 85"M(I3 
 
 4,44 
 
 1421. (7 13 
 
 48 
 
 7(i7r24 
 
 2 . 92 
 
 909055 
 
 1.52 
 
 8.580(59 
 
 4,41 
 
 141!;31 12 
 
 49 
 
 7(57300 
 
 2,92 
 
 908904 
 
 1.52 
 
 85833(5 
 
 4,44 
 
 i4io;;i 11 
 
 50 
 
 707475 
 
 2.91 
 
 908873 
 
 1.52 
 
 858(502 
 
 4,43 
 
 UV.VM 10 
 
 51 
 
 9.707019 
 
 2,91 
 
 9.908781 
 
 1.52 
 
 9.8.58S(;8 
 
 4,43 
 
 10.141132' 9 
 
 52 
 
 707824 
 
 2,91 
 
 908090 
 
 1 52 
 
 8.59134 
 
 4,43 
 
 14(M00 8 
 
 53 
 
 7079!>9 
 
 2 91 
 
 908599 
 
 1 , 52 
 
 859400 
 
 4,43 
 
 140000' 7 
 
 54 
 
 708173 
 
 2,91 
 
 908507 
 
 1.52 
 
 8.591 1(5(5 
 
 4,43 
 
 140334 6 
 
 55 
 
 708318 
 
 2.90 
 
 908410 
 
 1.53 
 
 85!»932 
 
 4 43 
 
 1400(5S 5 
 
 50 
 
 70S522 
 
 2,90 
 
 908321 
 
 1.53 
 
 8.5()I'.I8 
 
 4 43 
 
 119S0'.' 4 
 
 57 
 
 708097 
 
 2,90 
 
 908233 
 
 1 .53 
 
 8504(51 
 
 4 43 
 
 149530 3 
 
 58 
 
 708871 
 
 2,90 
 
 908141 
 
 ,1.53 
 
 8.5(l7;)0 
 
 4 43 
 
 149270; 2 
 
 59 
 
 709015 
 
 2 90 
 
 908049 
 
 ll 53 
 
 850995 
 
 4 43 
 
 1 4!)005 1 
 
 00 
 
 709219 
 
 2.90 
 
 907958 
 
 4,53 
 
 8512(51 
 
 4 43 
 
 1487.39 
 
 
 Cosine. 
 
 D. 
 
 Sine; 
 
 Ml 
 
 Cota^g. 
 
 D. 
 
 Tang. M. 
 
54 
 
 
 (36 DEGREES.) A TABLE OF LOOARITHUIO 
 
 
 M. 
 
 
 1 Sine. 
 
 D. 
 
 2.90 
 
 Cosine. 
 
 y.90795h 
 
 ! D. 
 1.53 
 
 Taii^-. 
 
 D. 
 
 Cotang^. 
 
 60 
 
 9.7092HJ 
 
 y. 801261 
 
 4.43 
 
 10.13873'J 
 
 1 
 
 709399 
 
 2.89 
 
 907800 
 
 1.53 
 
 801527 
 
 4.43 
 
 138473 
 
 59 
 
 2 
 
 7095G(] 
 
 2.89 
 
 907774 
 
 1.53 
 
 861792 
 
 4.42 
 
 13820H 
 
 58 
 
 3 
 
 709740 
 
 2.89 
 
 907082 1.53 
 
 86205>'- 
 
 4.42 
 
 137942 
 
 57 
 
 4 
 
 709913 
 
 2.89 
 
 907590 1.53 
 
 862323 
 
 4.42 
 
 137077 
 
 56 
 
 5 
 
 7700S7 
 
 2.89 
 
 90749-. 
 
 1.53 
 
 862589 
 
 4.42 
 
 137411 
 
 55 
 
 6 
 
 77fl20( 
 
 2.88 
 
 90740() 
 
 1.53 
 
 862854 
 
 4.42 
 
 13714G 
 
 54 
 
 7 
 
 770433 
 
 2.88 
 
 907314:1.54 
 
 803119 
 
 4.42 
 
 130881 
 
 53 
 
 8 
 
 77(li;OG 
 
 2.88 
 
 907J22 
 
 i.rA 
 
 863385 
 
 4.42 
 
 130015 
 
 52 
 
 9 
 
 77U779 
 
 2.88 
 
 907129 
 
 1.54 
 
 803050 
 
 4.42 
 
 130350 
 
 51 
 
 10 
 
 77.1952 
 
 2.88 
 
 9070:J7 
 
 1.54 
 
 803915 
 
 4.42 
 
 13G085 
 
 50 
 
 11 
 
 9.771125 
 
 2.88 
 
 9.900945 
 
 1.54 
 
 9.804180 
 
 4.42 
 
 10.135820 
 
 49 
 
 12 
 
 77129H 
 
 2.87 
 
 900852 
 
 1.54 
 
 804445 
 
 4.42 
 
 135555 
 
 48 
 
 i;{ 
 
 771470 
 
 2.87 
 
 900700 
 
 1.54 
 
 864710 
 
 4.42 
 
 135290 
 
 47 
 
 14 
 
 771043 
 
 2.87 
 
 90G0G7 
 
 1.54 
 
 864975 
 
 4.41 
 
 135025 
 
 46 
 
 15 
 
 771815 
 
 2.87 
 
 906575 
 
 1.54 
 
 865240 
 
 4.41 
 
 134700 
 
 45 
 
 Hi 
 
 771987 
 
 2.87 
 
 906482 
 
 1.54 
 
 805505 
 
 4.41 
 
 134495 
 
 44 
 
 17 
 
 772159 
 
 2.87 
 
 906389 
 
 1.55 
 
 865770 
 
 4.41 
 
 134230 
 
 43 
 
 18 
 
 772331 
 
 2. 80 
 
 906290 
 
 1.55 
 
 860035 
 
 4.41 
 
 133905 
 
 42 
 
 I'J 
 
 772503 
 
 2.80 
 
 900204 
 
 1.55 
 
 800300 
 
 4.41 
 
 133700 
 
 41 
 
 20 
 
 772075 
 
 2.80 
 
 906111 
 
 1.55 
 
 800504 
 
 4.41 
 
 133436 
 
 40 
 
 21 
 
 9.772S47 
 
 2.80 
 
 9.900018 
 
 1.55 
 
 9.800829 
 
 4.41 
 
 10.133171 
 
 39 
 
 22 
 
 773018 
 
 2.86 
 
 90592511.55 
 
 867094 
 
 4.41 
 
 132906 
 
 38 
 
 23 
 
 773190 
 
 2.86 
 
 9058321.55 
 
 867358 
 
 4.41 
 
 132642 
 
 37 
 
 24 
 
 7733;;i 
 
 2.85 
 
 90573911.55 
 
 867623 
 
 4.41 
 
 132377 
 
 36 
 
 25 
 
 773533 
 
 2.85 
 
 90564511.55 
 
 867887 
 
 4.41 
 
 132113 
 
 35 
 
 20 
 
 773704 
 
 2.85 
 
 905552 '1.55 
 
 868152 
 
 4.40 
 
 131848 
 
 34 
 
 27 
 
 773875 
 
 2.85 
 
 905459 
 
 1.55 
 
 808416 
 
 4.40 
 
 131584 
 
 33 
 
 28 
 
 774040 
 
 2.85 
 
 905366 
 
 1.56 
 
 868680 
 
 4.40 
 
 131320 
 
 32 
 
 29 
 
 774217 
 
 2.85 
 
 905272 
 
 1.50 
 
 868945 
 
 4.40 
 
 131055 
 
 31 
 
 30 
 
 774388 
 
 2.84 
 
 905179 
 
 1.56 
 
 86!t209 
 
 4.40 
 
 130794 
 
 30 
 
 31 
 
 9.774558 
 
 2.84 
 
 9.905085:1.56 
 
 9.869473 
 
 ■4.40 
 
 10.130.-27 29 1' 
 
 32 
 
 774729 
 
 2.84 
 
 904992 '1.56 
 
 809737 
 
 4.40 
 
 130203 
 
 28 
 
 33 
 
 774899 
 
 2.84 
 
 904898'!. 56 
 
 870001 
 
 4.40 
 
 129999 
 
 27 
 
 34 
 
 775070 
 
 2.84 
 
 904804 1.56 
 
 870265 
 
 4.40 
 
 129735 
 
 26 
 
 35 
 
 775240 
 
 2.84 
 
 9047111.56 
 
 870529 
 
 4.40 
 
 129471 
 
 25 
 
 30 
 
 775410 
 
 2.83 
 
 9046171.56 
 
 870793 
 
 4.40 
 
 129207 
 
 24 
 
 37 
 
 775580 
 
 2.83 
 
 9045231.56 
 
 871057 
 
 4.40 
 
 128943 
 
 23 
 
 38 
 
 775750 
 
 2.83 
 
 904429 1.57 
 
 871321 
 
 4.40 
 
 128679 
 
 22 
 
 39 
 
 775920 
 
 2.83 
 
 9043351.57 
 
 871585 
 
 4.40 
 
 128415 
 
 21 
 
 40 
 
 77G090 
 
 2.83 
 
 9042411 
 
 ^57 
 
 871849 
 
 4.39 
 
 128151 
 
 20 
 
 41 
 
 9.770259 
 
 2.83 
 
 9.904147, 
 
 ^.57 
 
 9.872112 
 
 4.39 
 
 10.127888 
 
 19 
 
 42 
 
 77(5*29 
 
 2.82 
 
 904053 
 
 1.57 
 
 872370 
 
 4. 39 
 
 127624 
 
 18 
 
 43 
 
 770598 
 
 2.82 
 
 9039591 
 
 1.57 
 
 872640 
 
 4.39 
 
 127360 
 
 17 
 
 44 
 
 770708 
 
 2.82 
 
 903864' 
 
 1.57 
 
 872903 
 
 4.39 
 
 127097 
 
 16 
 
 45 
 
 770937 
 
 2.82 
 
 9037701 
 
 1.57 
 
 873167 
 
 4.39 
 
 126833 
 
 15 
 
 i 40 
 
 777100 
 
 2.82 
 
 903076 
 
 1.57 
 
 873430 
 
 4.39 
 
 126.570 
 
 14 
 
 1 47 
 
 777275 
 
 2.81 
 
 903581 
 
 1.57 
 
 873694 
 
 4.39 
 
 126306 
 
 13 
 
 48 
 
 777444 
 
 2.81 
 
 903487 
 
 1.57 
 
 873957 
 
 4.39 
 
 126043 
 
 12 
 
 49 
 
 777<;i3 
 
 2.81 
 
 903392; 
 
 1.58 
 
 874220 
 
 4.39 . 
 
 125780 
 
 11 
 
 no 
 
 777781 
 
 2.81 
 
 903298 
 
 1.58 
 
 874484 
 
 4.39 
 
 125516 
 
 10 
 
 51 
 
 9.7779.50 
 
 2 81 
 
 9.903203 
 
 1.58 
 
 9.874747 
 
 4.39 
 
 10.125253 
 
 9 
 
 52 
 
 778119 
 
 2.81 
 
 903108 
 
 1.58 
 
 875010 
 
 4.39 
 
 124990 
 
 8 
 
 53 
 
 778287 
 
 2.80 
 
 903014' 
 
 1.58 
 
 875273 
 
 4.38 
 
 124727 
 
 7 
 
 54 
 
 778455 
 
 2.80 
 
 902919 
 
 1.58 
 
 875536 
 
 4.38 
 
 124404 
 
 6 
 
 55 
 
 77S024 
 
 2.80 
 
 902824 
 
 1.58 
 
 875800 
 
 4.38 
 
 124200 
 
 5 
 
 50 
 
 778792 
 
 2.80 
 
 902729 
 
 1.58 
 
 8760G3 
 
 4.38 
 
 123937 
 
 4 
 
 57 
 
 778900 
 
 2.80 
 
 902034 1.58 
 
 870326 
 
 4.38 
 
 123G74 3 
 
 58 
 
 77912M 
 
 2.80 
 
 902539 1.59 
 
 876589 
 
 4.38 
 
 1234111 2 
 
 5;) 
 
 7792 15 
 
 2.79 
 
 !t02444 1.59 
 
 870851 
 
 4.38 
 
 123149 1 
 
 (0 
 
 i 
 
 77.) 403 
 
 2.79 
 1). 
 
 90234it 1.59 
 
 87'7114 
 Cotr .2. 
 
 4.38 
 
 122886 : 
 
 Sine. 1 
 
 03 1 
 
 D. 1 
 
 Tan^, i 
 
 M. 
 
s- 
 
 
 '30 
 
 60 
 
 73 
 
 59 
 
 08 
 
 58 
 
 i2 
 
 57 
 
 77 
 
 56 
 
 11 
 
 55 
 
 4G 
 
 54 
 
 53 
 
 52 
 
 51 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 29 
 
 28 
 
 27 
 
 2G 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 
 
 779631 
 
 779798 
 779966 
 780133 
 780300 
 780467 
 780634 
 780801 
 780368 
 7811341 
 .781301 
 781468 
 781634 
 7 SI 800 
 781966 
 7821321 
 7822981 
 7824611 
 782630 
 782796 
 .782961 
 783127 
 783292 
 7fj3458 
 783623 ' 
 783788 i 
 783953 J 
 7841.1,8 5 
 , 784282 2 
 L 784147 2 
 1 9. 78461 2 
 784776 
 784941 
 785105 
 78.5269 
 785433 
 785597 
 785761 
 785925 
 ,„ 786089 
 |9. 786252 
 786416 
 786579 
 786742 
 786906 
 787069 
 787232 
 7873f>5 
 787557 
 , 787720 
 i 9- 787883 ' 
 788045 i 
 788208 i 
 788370 U 
 788532 2 
 788694 2 
 788856 2 
 789018 
 7891 sol 
 
 _78fm<;^ 
 
 iCiMilne. D. 
 
 AND T,,,,^^^^ (37 DKOREES.) 
 
 iM)02349r59 
 9022531:59 
 902158 1.59 
 9020631.59 
 901967 1.59 
 901872 1 59 
 
 901776 1.59 
 901681 |l. 59 
 90158511.59 
 901490 1 59 
 
 „ 901394 1.60 
 9.9012981 60 
 901202 1 60 
 901106 1 60 
 90101011 60 
 90091ijl'60 
 
 9008181. 'go; 
 
 9007221 60 
 900626 1 60 
 9005291 60 
 900433 1 61 
 9. 9003371 '61 
 900240 1 '61 
 900144 1. 61 
 900047 1 61 
 8999511' 61 
 8998.;4l'ci 
 8997571 ■611 
 899o60l'6ll 
 899.364 1 '611 
 899467 1 621 
 .899370 1 62-' 
 8992731 '621 
 899170 1. '621 
 899078 1 62 
 8989811 '62 
 898884 1 '62 
 8987871 '62' 
 8986891' 69 
 898.5921 '62^ 
 ^ 898494 1. '631 
 9.8983971 63 
 898299 1 '63 
 8982021 '63 
 898104 1 '63 
 898006 1 63 
 897908 1 '63 
 897810 1 '63 
 8977121.63 
 8976141.63 
 897516 1.63, 
 .897-418 1 64 
 897320il 64 
 897222'! 64 
 897l2.'Vr64 
 897025,1. 64 
 896926 1 64 
 896828 1 64 
 896729 1' 64 
 8966311.641 
 _«96,'>32il 641 
 
 Tang-: 
 
 97877114 
 877377 
 877640J 
 877903 
 878165 
 878428 
 878691 
 878953 
 879216 
 879478 
 879741 
 9.880003] 
 880265 
 88052S 
 j 880790 
 881052 
 881314, 
 881576 
 881839 
 882101 
 882363 
 9.882G25 
 882887 
 883148 
 883410 
 8830721 
 8839341 
 8841961 
 884457 
 I 884719 
 ! 384980 
 ' 9.885242 
 885503 
 885765 
 886026' 
 88G288 
 886549 
 886810 
 887072 
 88733SI 
 887594 
 9.887855 
 888116; 
 888377 
 888G39 
 888900 
 8891G0 
 889421 
 889682] 
 889943 
 890204 
 9.890465 
 890725, 
 89098g' 
 891247, 
 891507 
 891 76n,' 
 89202s 
 892289 
 8925J9 
 «92810 
 
 4.35 
 
 Cotang. 
 
 10.122886|-eo" 
 122623) 59 
 122360 58 
 122097 57 
 121835 56 
 121572 55 
 121309 54 
 121047 53 
 1207841 52 
 120522 51 
 120259 50 
 1^.119997 49 
 119735 48 
 119472 47 
 119210 46 
 1189481 45 
 118686 44 
 118424 43 
 1181G1 42 
 117899 41 
 ,„ 117637| 40 
 10.117375 39 
 117113 38 
 1168521 37 
 116590,1 3G 
 110328 35 
 116066! 34 
 115804' 33 
 115543 32 
 115281 31 
 115020 30 
 10 114758 29 
 114497) 28 
 1142351 27 
 113974) 26 
 113712 25 
 113451 24 
 113190 23 
 112928 22 
 , 112667 21 
 I 112406 20 
 10.11214.5 19 
 111884 18 
 111623 17 
 I 111361 16 
 I 111100 15 
 110840' U 
 
 norm; 13 
 
 110318 U 
 
 110057! 11 
 
 ,' 109796,' 10 
 
 j 10. 109535; 9 
 
 I 109275 
 I 1090141 . 
 1087531 6 
 
 I084<>3; 
 108232 
 107972( 
 107711 
 107451' 
 I-,J£II90, 
 
56 
 
 
 (:!S DEOUKES.) ' 
 
 L TAni,E OK LOOAniTJI.MIC 
 
 
 M. 
 
 U 
 
 ' Sine. 
 
 9.7«'J312 
 
 D. 
 
 Cosine. 
 
 l)T89(i.j;t.i 
 
 D. 
 
 1.04 
 
 Tang, 
 
 9!892MU 
 
 ' 4.34 
 
 1 Cotang. I 
 
 2.09 
 
 10.10719(1 CO 
 
 1 
 
 7S:).-)04 
 
 2.09 
 
 89043:! 
 
 1.05 
 
 89;l{»70 
 
 ' 4.34 
 
 100930, 59 
 
 2 
 
 78!JC()5 
 
 2.09 
 
 «9(;3:i-) 
 
 1.05 
 
 893;i;u 
 
 ' 4.34 
 
 10000! 
 
 1 58 
 
 3 
 
 78!»S27 
 
 2.09 
 
 8902;!il 
 
 1.05 
 
 8i»:{5;)i 
 
 ! 4.34 
 
 100 lOD 
 
 57 
 
 4 
 
 7s;)Ji« 
 
 2.09 
 
 89i;i:J7 
 
 1.05 
 
 89.3S51 
 
 i 4 . 34 
 
 100H9 
 
 50 
 
 5 
 
 7i)0119 
 
 2.09 
 
 89(;o:!-; 
 
 1.05 
 
 894111 
 
 4.34 
 
 105S8!I 
 
 55 
 
 G 
 
 7!>(J;310 
 
 2.08 
 
 «9.VJ3;t 
 
 jl.05 
 
 894;{7] 
 
 4.34 
 
 105(;29 
 
 54 
 
 7 
 
 790171 
 
 2. OS 
 
 S9.">S40 
 
 1 . <i5 
 
 894032 
 
 j 4.33 
 
 105308 
 
 ' 53 
 
 8 
 
 79(),;32 
 
 2. OH 
 
 895741 
 
 1.05 
 
 S94S92 
 
 1 4.33 
 
 105108 
 
 : 52 
 
 
 
 790793 
 
 2. OS 
 
 «!».",'; 1 1 
 
 1.05 
 
 895152 
 
 4,83 
 
 104848 
 
 51 
 
 10 
 
 790!).j4 
 
 2. OS 
 
 «;(5512 
 
 1.05 
 
 895 H 2 
 
 4,33 
 
 104588 
 
 50 
 
 11 
 
 9 791115 
 
 2 (iS 
 
 9.s)51l:3 
 
 1,00 
 
 9. S'. (5072 
 
 4.33 
 
 10.10432^ 
 
 49 
 
 12 
 
 791275 
 
 2.07 
 
 .S!).131;J 
 
 1.00 
 
 .S95!)32 
 
 4.33 
 
 101008 
 
 : 4s 
 
 13 
 
 791 13G 
 
 2.(i7 
 
 895244 
 
 1.00 
 
 890192 
 
 4.33 
 
 103808 
 
 ' 47 
 
 14 
 
 791590 
 
 2.07 
 
 S'.)51i5 
 
 1.00 
 
 890452 
 
 4.33 
 
 10354S 
 
 40 
 
 15 
 
 791757 
 
 2.07 
 
 8!».5()45 
 
 1. 00 
 
 890712 
 
 4.;!3 
 
 103288: 45 
 
 1(5 
 
 7iU917 
 
 2.07 
 
 894:)45 
 
 1.00 
 
 890971 
 
 4.33 
 
 103029 
 
 44 
 
 17 
 
 792077 
 
 2.07 
 
 894810 
 
 1.00 
 
 897231 
 
 4.33 
 
 102709 
 
 43 
 
 18 
 
 792237 
 
 2.00 
 
 894740 
 
 l.OC 
 
 897491 
 
 4.33 
 
 102509 
 
 42 
 
 19 
 
 7923:17 
 
 2.00 
 
 894040 
 
 1.00 
 
 897751 
 
 4.33 
 
 102249 
 
 41 
 
 20 
 
 792557 
 
 2.00 
 
 8!)4540 
 
 1.00 
 
 898010 
 
 4.33 
 
 101990 
 
 40 
 
 21 
 
 9.792710 
 
 2.00 
 
 9.894440 
 
 1.07 
 
 9.898270 
 
 4.. 33 
 
 10.101730 
 
 39 
 
 22 
 
 792870 
 
 2.00 
 
 894:540 
 
 1.07 
 
 898530 
 
 4.33 
 
 101470 
 
 38 
 
 23 
 
 793035 
 
 2.00 
 
 89^240 
 
 1.07 
 
 898789 
 
 4,33 
 
 101211 
 
 37 
 
 24 
 
 793195 
 
 2.05 
 
 894140 
 
 1.07 
 
 899049 
 
 4.32 
 
 100951 
 
 3(i 
 
 25 
 
 793354 
 
 2.05 
 
 894040 
 
 1.07 
 
 899308 
 
 4.32 
 
 100092 
 
 35 
 
 20 
 
 793514 
 
 2.05 
 
 893940 
 
 1.07 
 
 899508 
 
 4.32 
 
 100432 
 
 34 
 
 27 
 
 793G73 
 
 2.05 
 
 893440 
 
 1.07 
 
 899827 
 
 4.32 
 
 100173 
 
 33 
 
 28 
 
 793!^32 
 
 2.05 
 
 893745 
 
 1.07 
 
 900080 
 
 4.32 
 
 099914 
 
 32 
 
 2'J 
 
 793991 
 
 2.05 
 
 893045 
 
 1.67 
 
 900346 
 
 4.32 
 
 099054 
 
 31 
 
 30 
 
 794150 
 
 2.04 
 
 893544 
 
 1.07 
 
 900005 
 
 4,32 
 
 099395 
 
 30 
 
 31 
 
 9.794308 
 
 2.04 
 
 9.8ii3444 
 
 1.08 
 
 0.900804 
 
 4.32 
 
 10.099130 
 
 29 
 
 32 
 
 791107 
 
 2.04 
 
 893343 
 
 1.08 
 
 901124 
 
 4.32 
 
 098876 
 
 28 
 
 33 
 
 794020 
 
 2.04 
 
 893243 
 
 1.08 
 
 901383 
 
 4.32 
 
 098017 
 
 27 
 
 34 
 
 794784 
 
 2.04 
 
 893142 
 
 1.08 
 
 901042 
 
 4.32 
 
 098358 
 
 20 
 
 35 
 
 794942 
 
 2.04 
 
 893041 
 
 1.08 
 
 901901 
 
 4 32 
 
 098099 
 
 25 
 
 3(5 
 
 795101 
 
 2.04 
 
 892940 
 
 1 08 
 
 902100 
 
 4.32 
 
 097840 
 
 24 
 
 37 
 
 79525!> 
 
 2.03 
 
 892839 
 
 1.08 
 
 902419 
 
 4 32 
 
 097581 
 
 23 
 
 38 
 
 795417 
 
 2.03 
 
 892739 
 
 1.68 
 
 902079 
 
 4.32 
 
 097321 
 
 22 
 
 39 
 
 795575 
 
 2.03 
 
 892038 
 
 1.08 
 
 902938 
 
 4.32 
 
 0970(;2 
 
 21 
 
 40 
 
 795733 
 
 2.03 
 
 892530 
 
 1.08 
 
 90:J197 
 
 4.31 
 
 090803 
 
 20 
 
 41 
 
 9.795S91 
 
 2.03 
 
 9.8924:]5 
 
 1 . (19 
 
 9.90:U55 
 
 4,31 
 
 10.090545 
 
 19 
 
 42 
 
 790049 
 
 2.03 
 
 892334 
 
 1.09 
 
 90.'?714 
 
 4.31 
 
 090280 
 
 18 
 
 43 
 
 790200 
 
 2 03 
 
 892233 
 
 1 . ()9 
 
 903973 
 
 4 31 
 
 090027 
 
 17 
 
 44 
 
 796304 
 
 2.02 
 
 892132 
 
 1.69 
 
 904232 
 
 4.31 
 
 095768 
 
 10 
 
 45 
 
 V90521 
 
 2.02 
 
 892030 
 
 1.69 
 
 904491 
 
 4.31 
 
 095509 
 
 15 
 
 46 
 
 790()79 
 
 2.02 
 
 891929 
 
 1,69 
 
 904750 
 
 4,31 
 
 095250 
 
 14 
 
 47 
 
 790836 
 
 2.02 
 
 891827 
 
 1.09 
 
 905008 
 
 4.31 
 
 094992 
 
 13 
 
 48 
 
 790993 
 
 2.02 
 
 891720 
 
 1 09 
 
 9052(57 
 
 4.31 
 
 094733 
 
 12 
 
 49 
 
 797150 
 
 2.61 
 
 891024 
 
 1.09 
 
 905526 
 
 4.31 
 
 094474 
 
 31 
 
 50 
 
 797307 
 
 2.61 
 
 891523 
 
 1 70 
 
 905784 
 
 4.31 
 
 094210 
 
 10 
 
 51 
 
 9.797404 
 
 2.61 
 
 9.891421 
 
 1,70 
 
 9.9O0O43 
 
 4.31 
 
 l0.o;)r,957 
 
 9 
 
 52 
 
 797G21 
 
 2.61 
 
 891319 
 
 1,70 
 
 900302 
 
 4.31 
 
 093098 
 
 8 
 
 53 
 
 797777 
 
 2 61 
 
 891217 
 
 1,70 
 
 900500 
 
 4.31 
 
 093440 
 
 7 
 
 54 
 
 797934 
 
 2.61 
 
 891115 
 
 1.70 
 
 900819 
 
 4 31 
 
 093181 
 
 6 
 
 55 
 
 798091 
 
 2.61 
 
 891013 
 
 1 70 
 
 907077 
 
 4 31 
 
 092923 
 
 5 
 
 S6 
 
 798247 
 
 2.61 
 
 890911 
 
 1.70 
 
 907336 
 
 4.31 
 
 092004 
 
 i 
 
 57 
 
 798403 
 
 2.60 
 
 890809 
 
 1.70 
 
 907594 
 
 4 31 
 
 092400 
 
 3 
 
 58 
 
 798560 
 
 2 60 
 
 890707 
 
 1 70 
 
 907852 
 
 4 31 
 
 092148 
 
 2 
 
 59 
 
 798716 
 
 2.60 
 
 890005 
 
 1.70 
 
 908111 
 
 4.30 
 
 091889 
 
 1 
 
 60 
 
 798872 
 
 2.60 
 D. 
 
 890503 
 Sine. 
 
 1.70 
 51" 
 
 908309 
 Cotarg. 
 
 4.30 
 
 091031 
 
 
 
 Cosine. 
 
 D. 
 
 Tang. 
 
 Jl 
 
Hj 52 
 H ol 
 B 50 
 S 4!) 
 ^, 4.S 
 47 
 40 
 45 
 44 
 
 in 
 
 42 
 
 37 
 30 
 85 
 34 
 33 
 32 
 31 
 30 
 2!) 
 28 
 27 
 2G 
 25 
 24 
 23 
 22 
 21 
 20 
 19 
 18 
 17 
 1(J 
 15 
 14 
 13 
 12 
 31 
 10 
 9 
 8 
 7 
 6 
 5 
 4 
 i 
 2 
 1 
 
 
 i_' 
 
 iy.7y«,s72 
 79902s, 
 799.184 
 799339 
 799495, 
 799051 
 799806 
 7999G2 
 «00I17, 
 
 800272 
 800427 
 '.80058'^' 
 800737 
 800892, 
 801047 
 801201 
 80135G 
 801511 
 8016-05, 
 80LSJ9 
 
 in '^^^•m 
 
 ^•802128 
 80228*5 ' 
 802436/ 
 802589' 
 802743/ 
 8028971 
 8030501 
 803204 
 803357 
 ,„ 803511' ■ 
 i^-803G(U 
 : i 80;isi7 
 ■ 1 803970, 
 804123 
 804276,' 
 804428, 
 80458J 
 804734 
 
 mm; 
 
 I 805039 
 
 805:M;j' 
 8054!j5 
 805;i47, 
 8057911 
 mjr,il 
 80,;i0;j) 
 80.;254' 
 80640( 
 , 8(;(;.r,7 
 "^•80670;^, 
 
 80.;S60 
 807011 
 
 807163, 
 
 807314 
 
 807I(;5 
 
 807(;i5, 
 8077(;(; 
 807917 
 808067 
 
 jCosiiie. I iji 
 
 i>. 890503 r70 
 890400 ;i 71 i 
 890298 1 '71- 
 890195,1 .'71,' 
 890093 1.711 
 88999()'l 71 1 
 889888 1:71 
 889785,^1.71 
 8896S21 71 
 889579:1. '71 1 
 88947711 71! 
 9.889374;1 72' 
 889271 |l 72i 
 88916811,72,' 
 
 C'*!J DEORBEs.) 
 
 *^ o«ijie 
 
 889064 1 '79' 
 
 888961 ;i.' 72 
 
 SS8858,U.72 
 
 888755|1.72l 
 
 8886511.721 
 
 888548 1.72 
 
 „ 88844411.73 
 
 9-888341I1.73 . 
 
 888237 1.73' ' 
 
 888134 1.73' 
 
 • 888030 1.73i 
 
 887926/1.73 
 
 8878221.73' 
 
 887718 1.73' 
 
 887614 1.73,' 
 
 8875101.73 
 
 „ 88740611.74 
 
 9-8873021 74I 
 
 887198 1. "74' 
 
 88709311 74 
 
 88698911.74, 
 
 886885 1.74 
 
 8867801 74 
 
 8S6G7G1 74, 
 
 8865711. '74 
 
 886^66 1.74 
 
 886362 1 75 
 
 3. 886257 1*75' 
 
 886152 1 75 
 
 88G047l'75l 
 
 8859421 "75 
 
 8858371.75' 
 
 885732 1 75 
 
 885627 1 75 
 
 8855221.75' 
 
 885416 1.75 
 
 8853111,76 
 
 -.885205 1.76' 
 885100 1 76 
 8H4994 1 76i' 
 
 88^889 I 76 
 884783 1 76 
 
 8846771.76,' 
 
 884572 1.76 
 884466 1 76 
 884360 1.76, 
 
 484254|l__77,' 
 
 Sine. 150 
 
 909144! 
 , 909402' 
 f 9096(;o; 
 9099181 
 910I77J 
 910435 
 ' 910693' 
 9109511 
 9.911209 
 911467 
 911724! 
 911982: 
 912240! , 
 912498 < 
 912756' ■ 
 913014 
 913271 
 913529] 
 
 9.913787, 
 914044 
 9143021 
 914560 
 914817 
 915075! 
 915332, 
 915590, 
 915847 
 916104, 
 9.916362 
 OlGGlOi 
 916877 
 917134 
 917391 
 917648, 
 917905 
 918163, 
 918420 
 „ 918677 
 9.918934 
 919191' 
 919448 
 919705 
 919962, 
 920219 
 920476, 
 920733 
 920990 
 n 921247, 
 9.9215031 
 921760,' 
 922017, 
 922274 
 922530,' 
 922787 
 923044 
 923300,' 
 923557 
 _J)23813,' 
 Cotanf- 
 
 "~' — "— — ^^ ' 
 
 l'>.091(;;;i m 
 091372 59 
 091114 58 
 090856,' 57 
 090598 56 
 090310! 55 
 0900S2' 54 
 089823' 53 
 0895(;5' 5'> 
 089307 51 
 08f)019 5o 
 10.088791! 4y 
 088533, 48 
 08S:.76,' 47 
 08S018 4G 
 
 os77(;o: 45 
 087,:(;2 44 I 
 087214 43 
 I 0869' 6 4'^ I 
 I 086729 41 
 ( 086471; 40 
 iO. 086213 30 
 OSoDoG: 38 
 085698, 37 
 085440 3G 
 08ol83i 35 
 084925! 3i 
 084668; 33 
 084410, 32 
 084153 31 
 ,,„ 083896' 30 
 10.083638,' 29 
 
 i 083381 1 28, 
 083123,' 27 
 082866 26 
 082609 25 
 082352 24 
 082095 23 
 081837 22 
 081580 21 
 , 081323 20 
 '.081066/ 19 
 080809| 18 
 080552 17 
 080295 16 
 080038 15 
 079781 14 
 079524 13 
 079267 12 
 079010 u 
 Itn ^^8753| 10 
 'JO. 078497 
 078240 
 077983 
 077726 
 077470 
 077213 
 076956 
 0767001 
 0764431 
 076187 
 
I ! 
 
 I''' 
 p:: 
 
 58 
 
 
 (40 DEGREES.) A. 
 
 TABLE OF LOGARITHMIC 
 
 
 M. 
 
 
 
 Sine. 
 
 D. 
 
 Cosine. 
 
 1.77 
 
 Tang. 
 
 9.923813 
 
 D. 
 
 Cotang. 
 
 Li) 
 
 9.80S0C7 
 
 2.51 
 
 9.881254 
 
 4.27 
 
 1U.U70187 
 
 1 
 
 808218 
 
 2.51 
 
 884148 
 
 1.77 
 
 924070 
 
 4.27 
 
 075930 
 
 59 
 
 2 
 
 808308 
 
 2.51 
 
 881042 
 
 1.77 
 
 924327 
 
 4.27 
 
 075673 
 
 58 
 
 S 
 
 808519 
 
 2.50 
 
 883930 
 
 1.77 
 
 924583 
 
 4.27 
 
 075417 
 
 57 
 
 4 
 
 808009 
 
 2.50 
 
 883829 
 
 1.77 
 
 924840 
 
 4.27 
 
 075160 
 
 56 
 
 5 
 
 808819 
 
 2.50 
 
 883723 
 
 1.77 
 
 925096 
 
 4.27 
 
 074904 
 
 55 
 
 6 
 
 808909 
 
 2.50 
 
 883617 
 
 1.77 
 
 925352 
 
 4.27 
 
 074648 
 
 54 
 
 7 
 
 809119 
 
 2.50 
 
 883510 
 
 1.77 
 
 925009 
 
 4.27 
 
 074391 
 
 63 
 
 8 
 
 809209 
 
 2.50 
 
 883404 
 
 1.77 
 
 925805 
 
 4.27 
 
 074135 
 
 52 
 
 9 
 
 809419 
 
 2.49 
 
 883297 
 
 1.78 
 
 920122 
 
 4.27 
 
 073878 
 
 51 
 
 10 
 
 809509 
 
 2.49 
 
 883191 
 
 1.78 
 
 920378 
 
 4.27 
 
 073622 
 
 50 
 
 11 
 
 9.809718 
 
 2.49 
 
 9.883084 
 
 1.78 
 
 9.920034 
 
 4.27 
 
 10.073366 
 
 49 
 
 12 
 
 809808 
 
 2.49 
 
 882977 
 
 1.78 
 
 920890 
 
 4.27 
 
 073110 
 
 48 
 
 13 
 
 810017 
 
 2.49 
 
 882871 
 
 1.78 
 
 927147 
 
 4.27 
 
 072853 
 
 47 
 
 14 
 
 810107 
 
 2.49 
 
 882704 
 
 1.78 
 
 927403 
 
 4.27 
 
 072597 
 
 46 
 
 15 
 
 810310 
 
 2.48 
 
 882657 
 
 1.78 
 
 927059 
 
 4.27 
 
 072341 
 
 45 
 
 16 
 
 810405 
 
 2.48 
 
 882550 
 
 1.78 
 
 927915 
 
 4.27 
 
 072085 
 
 44 
 
 17 
 
 810014 
 
 2.48 
 
 882443 
 
 1.78 
 
 928171 
 
 4.27 
 
 071329 
 
 43 
 
 18 
 
 810703 
 
 2.48 
 
 882330 j 
 
 1.79 
 
 928427 
 
 4.27 
 
 07.1573 
 
 42 
 
 19 
 
 810912 
 
 2.48 
 
 882229 
 
 1.79 
 
 928083 
 
 4.27 
 
 07i317 
 
 41 
 
 20 
 
 811001 
 
 2.48 
 
 882121 
 
 1.79 
 
 928940 
 
 4.27 
 
 071060 
 
 40 
 
 21 
 
 9,811210 
 
 2.48 
 
 9.882014 
 
 1.79 
 
 9.929190 
 
 4.27 
 
 10.070804 
 
 39 
 
 22 
 
 811358 
 
 2.47 
 
 881907 
 
 1.79 
 
 929452 
 
 4.27 
 
 070548 
 
 38 
 
 23 
 
 811507 
 
 2.47 
 
 881799 
 
 1.79 
 
 929708 
 
 4.27 
 
 070292 
 
 37 
 
 24 
 
 811055 
 
 2.47 
 
 881092 
 
 1.79 
 
 929904 
 930220 
 
 4.26 
 
 070036 
 
 36 
 
 25 
 
 811804 
 
 2.47 
 
 881584 
 
 1.79 
 
 4.26 
 
 009780 
 
 35 
 
 20 
 
 811952 
 
 2.47 
 
 881477 
 
 1.79 
 
 930475 
 
 4.26 
 
 069525 
 
 34 
 
 27 
 
 812100 
 
 2.47 
 
 881309! 
 
 1.79 
 
 930731 
 
 4.26 
 
 069269 
 
 33 
 
 28 
 
 812r48 
 
 2.47 
 
 881201; 
 
 1.80 
 
 930987 
 
 4.26 
 
 009013 
 
 32 
 
 29 
 
 812390 
 
 2.40 
 
 881153' 
 
 1.80 
 
 931243 
 
 4.26 
 
 008757 
 
 31 
 
 30 
 
 812544 
 
 2.40 
 
 881040, 
 
 1.80 
 
 931499 
 
 4.26 
 
 008501 
 
 30 
 
 31 
 
 9.812092 
 
 2.46 
 
 9.880938; 
 
 1.80 
 
 9.931755 
 
 4.26 
 
 10.0()8245 
 
 29 
 
 32 
 
 812840 
 
 2.40 
 
 880830 
 
 1 . 80 
 
 932010 
 
 4.26 
 
 007990 
 
 28 
 
 33 
 
 812988 
 
 2.40 
 
 880722 
 
 1.80 
 
 932200 
 
 4.26 
 
 007734 
 
 27 
 
 34 
 
 813135 
 
 2.40 
 
 880013 
 
 1.80 
 
 932522 
 
 4.26 
 
 007478 
 
 26 
 
 35 
 
 813383 
 
 2.40 
 
 880505 
 
 1.80 
 
 932778 
 
 4.26 
 
 007222 
 
 25 
 
 30 
 
 813430 
 
 2.45 
 
 880397 
 
 1.80 
 
 933033 
 
 4.26 
 
 000907 
 
 24 
 
 37 
 
 813578 
 
 2.45 
 
 8802891 
 
 1.81 
 
 933289 
 
 4.26 
 
 oo;i7ii 
 
 23 
 
 38 
 
 813725 
 
 2.45" 
 
 880180 1 
 
 1.81 
 
 933515 
 
 4.26 
 
 0G0455 
 
 22 
 
 39 
 
 813872 
 
 2.45 
 
 880072; 
 
 1.81 
 
 933800 
 
 4.26 
 
 000200 
 
 21 
 
 40 
 
 814019 
 
 2.45 
 
 879963, 
 
 1.81 
 
 934050 
 
 4.26 
 
 005944 
 
 20 
 
 41 
 
 9.814100 
 
 2.45 
 
 9.879855; 
 
 1.81 
 
 9.93J311 
 
 4.26 
 
 10.005089 
 
 19 
 
 42 
 
 814313 
 
 2.45 
 
 87974<) 
 
 1.81 
 
 934507 
 
 4.26 
 
 005433 
 
 18 
 
 43 
 
 814400 
 
 2.44 
 
 879037 
 
 1.81 
 
 934823 
 
 4.26 
 
 005177 
 
 17 
 
 44 
 
 814007 
 
 2.44 
 
 879529 
 
 1.81 
 
 935078 
 
 4.26 
 
 004922 
 
 16 
 
 45 
 
 814753 
 
 2 44 
 
 879420 
 
 1.81 
 
 935333 
 
 4.26 
 
 004007 
 
 15 
 
 40 
 
 814900 
 
 2.44 
 
 879311 
 
 1.81 
 
 935589 
 
 4.26 
 
 004411 
 
 14 
 
 47 
 
 815040 
 
 2.44 
 
 879202 
 
 1.82 
 
 935844 
 
 4.26 
 
 004150 
 
 13 
 
 48 
 
 815193 
 
 2.44 
 
 879093 
 
 1.82 
 
 930100 
 
 4.26 
 
 003900 
 
 12 
 
 49 
 
 815339 
 
 2.44 
 
 878984 
 
 1.82 
 
 930355 
 
 4.26 
 
 003045 
 
 11 
 
 50 
 
 815485 
 
 ' 2.43 
 
 878875 
 
 1.82 
 
 930010 
 
 4.26 
 
 003390 
 
 10 
 
 51 
 
 9.815031 
 
 2 43 
 
 9.878700 
 
 1.82 
 
 9.!)3(J80« 
 
 4.25 
 
 10.003134 
 
 9 
 
 52 
 
 815778 
 
 2.43 
 
 878050 
 
 1.82 
 
 937121 
 
 4.25 
 
 002879 
 
 8 
 
 53 
 
 815924 
 
 2.43 
 
 878547 
 
 1.82 
 
 937370 
 
 4.25 
 
 002024 
 
 7 
 
 54 
 
 81G009 
 
 2.43 
 
 878438 
 
 1.82 
 
 937032 
 
 4.25 
 
 002308 
 
 6 
 
 55 
 
 810215 
 
 2.43 
 
 878328 
 
 1.82 
 
 937887 
 
 4.25 
 
 002113 
 
 5 
 
 50 
 
 810301 
 
 2.43 
 
 878219 
 
 1.83 
 
 938142 
 
 4.25 
 
 001858 
 
 4 
 
 57 
 
 810507 
 
 2 42 
 
 878109 
 
 1.83 
 
 938398 
 
 4.25 
 
 001002 
 
 3 
 
 58 
 
 810052 
 
 2.42 
 
 877999 
 
 1.83 
 
 938053 
 
 4.25 
 
 0G1347 
 
 2 
 
 59 
 
 810798 
 
 2.42 
 
 877890 
 
 1.83 
 
 938908 
 
 4 25 
 
 001092 1 
 
 ()0 
 
 810913 
 Cosine. 
 
 2.42 
 D. 
 
 877780 j 
 Sine. "\ 
 
 I 83 
 4^. 
 
 03'.) 103 
 
 4 25 
 
 000837: 
 
 Cotan^^. 
 
 Tang. 
 
 M. 1 
 
28 
 27 
 20 
 25 
 24 
 23 
 22 
 21 
 20 
 I'J 
 18 
 17 
 IG 
 15 
 14 
 13 
 12 
 11 
 10 
 9 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 
 
 
 SINKS AND TANGENTS. 
 
 (41 DE0REK8.) 
 
 
 89 
 
 M. 
 
 
 
 MHO. 
 
 I). 
 
 Cosine. 1 
 
 D. 
 
 1.83 
 
 Tang. 
 9.!»391«3 
 
 D. 
 
 Cotafis. 
 
 
 9.816943 
 
 2.42 
 
 9.877780 
 
 4.25 
 
 10.060837 
 
 60 
 
 1 
 
 817088 
 
 2.42 
 
 877670 1 
 
 83 
 
 939418 
 
 4.25 
 
 060582 
 
 59 
 
 2 
 
 817233 
 
 2.42 
 
 877560,1 
 
 83 
 
 939673 
 
 4.25 
 
 060827 
 
 58 
 
 S 
 
 817d79 
 
 2.42 
 
 877450 1 
 
 83 
 
 939928 
 
 4.23 
 
 060072 
 
 57 
 
 4 
 
 817524 
 
 2.41 
 
 877840 1 
 
 83 
 
 940183 
 
 4.25 
 
 059817 
 
 56 
 
 5 
 
 817668 
 
 2.41 
 
 877230 
 
 
 84 
 
 940438 
 
 4 25 
 
 059562 
 
 55 
 
 6 
 
 817813 
 
 2.41 
 
 877120 
 
 
 84 
 
 940694 
 
 4.25 
 
 059306 
 
 54 
 
 7 
 
 817958 
 
 2.41 
 
 877010 
 
 
 84 
 
 940949 
 
 4.25 
 
 059051 
 
 53 
 
 8 
 
 818103 
 
 2.41 
 
 876899 
 
 
 84 
 
 941204 
 
 4.25 
 
 058796 
 
 52 
 
 9 
 
 818247 
 
 2.41 
 
 876789 
 
 
 84 
 
 941458 
 
 4.25 
 
 058542 
 
 51 
 
 10 
 
 818392 
 
 2.41 
 
 876678 
 
 
 84 
 
 941714 
 
 4.25 
 
 058288 50 
 
 U 
 
 9.818536 
 
 2.40 
 
 9 8765(58 
 
 
 84 
 
 9.941968 
 
 4.25 
 
 10.058032' 49 
 
 12 
 
 818681 
 
 2.40 
 
 876457 
 
 
 84 
 
 942223 
 
 4.25 
 
 057777, 48 
 
 18 
 
 818825 
 
 2.40 
 
 876347 1 
 
 84 
 
 942478 
 
 4.25 
 
 057:)22 
 
 47 
 
 14 
 
 818969 
 
 2.40 
 
 876236 1 
 
 85 
 
 942733 
 
 4.25 
 
 057267 
 
 46 
 
 is- 
 
 819113 
 
 2.40 
 
 876125 1 
 
 85 
 
 942988 
 
 4.23 
 
 057012 
 
 45 
 
 le 
 
 819257 
 
 2.40 
 
 87601411 
 
 85 
 
 943243 
 
 4.23 
 
 056757 
 
 44 
 
 17 
 
 819401 
 
 2.40 
 
 87590^1 1 
 
 85 
 
 943498 
 
 4.25 
 
 056502 
 
 43 
 
 18 
 
 819545 
 
 2.39 
 
 875793 
 
 1 
 
 85 
 
 943752 
 
 4.25 
 
 056248 
 
 42 
 
 19 
 
 819689 
 
 2.39 
 
 875682 
 
 1 
 
 85 
 
 944007 
 
 4.25 
 
 055993 
 
 41 
 
 20 
 
 819832 
 
 2.39 
 
 875571 1 
 
 85 
 
 944262 
 
 4.25 
 
 055738 
 
 40 
 
 21 
 
 9.819976 
 
 2.39 
 
 9.8754^91 
 
 85 
 
 9.944517 
 
 4.25 
 
 10.055483 
 
 39 
 
 22 
 
 820120 
 
 2.39 
 
 875348 
 
 1 
 
 85 
 
 J^H77] 
 
 4.24 
 
 055229 
 
 38 
 
 23 
 
 820263 
 
 2.39 
 
 875237 
 
 1 
 
 85 
 
 945026 
 
 4.24 
 
 054974 
 
 37 
 
 24 
 
 820406 
 
 2.39 
 
 875126 
 
 1 
 
 86 
 
 945281 
 
 4.24 
 
 054719 
 
 36 
 
 25 
 
 820550 
 
 2.38 
 
 875014 1 
 
 86 
 
 945535 
 
 4.24 
 
 054465 
 
 35 
 
 26 
 
 820693 
 
 2.38 
 
 874903 1 
 
 86 
 
 945790 
 
 4.24 
 
 054210 
 
 34 
 
 27 
 
 820836 
 
 2.38 
 
 8747911 1 
 
 86 
 
 946045 
 
 4.24 
 
 053955 
 
 33 
 
 28 
 
 820979 
 
 2.38 
 
 87468011 
 
 86 
 
 94G299 
 
 4.24 
 
 053701 
 
 32 
 
 29 
 
 821122 
 
 2.88 
 
 874568 
 
 
 86 
 
 946554 
 
 4.24 
 
 053446 
 
 31 
 
 30 
 
 821265 
 
 2.88 
 
 874456 
 
 
 86 
 
 946808 
 
 4.24 
 
 053192 
 
 30 
 
 31 
 
 9.821407 
 
 2.88 
 
 9 874344 
 
 
 86 
 
 9.947063 
 
 4.24 
 
 10.052937 
 
 29 
 
 82 
 
 821550 
 
 2.88 
 
 874232 
 
 
 87 
 
 947318 
 
 4.24 
 
 052682 
 
 28 
 
 33 
 
 821693 
 
 2.87 
 
 874121 
 
 
 87 
 
 947572 
 
 4.24 
 
 052428 
 
 27 
 
 34 
 
 821835 
 
 2.87 
 
 874009 
 
 
 87 
 
 947826 
 
 4.24 
 
 052174 
 
 26 
 
 35 
 
 821977 
 
 2.87 
 
 873896 
 
 
 87 
 
 948081 
 
 4.24 
 
 051919 
 
 25 
 
 36 
 
 822120 
 
 2.87 
 
 873784 
 
 
 87 
 
 948336 
 
 4.24 
 
 051664 
 
 24 
 
 87 
 
 822262 
 
 2.87 
 
 873672 
 
 
 87 
 
 948590 
 
 4.24 
 
 051410 
 
 28 
 
 88 
 
 822404 
 
 2.87 
 
 873560 
 
 
 87 
 
 948844 
 
 4.24 
 
 051156 
 
 2.2 
 
 89 
 
 822546 
 
 2.^"^ 
 
 873448 
 
 
 87 
 
 949099 
 
 4.24 
 
 030901 
 
 2(1 
 
 40 
 
 822688 
 
 2!t>o 
 
 873335 
 
 
 87 
 
 949353 
 
 4.24 
 
 030647 
 
 20 
 
 41 
 
 9.822830 
 
 2.86 
 
 9.873223 
 
 
 87 
 
 9.949607 
 
 4.24 
 
 10.050393 
 
 19 
 
 42 
 
 822972 
 
 2.86 
 
 873110 
 
 
 88 
 
 949802 
 
 4.24 
 
 050138 
 
 18 
 
 43 
 
 823114 
 
 2.86 
 
 872998 
 
 
 88 
 
 950116 
 
 4.24 
 
 049884 
 
 17 
 
 44 
 
 823255 
 
 2.36 
 
 872885 
 
 
 88 
 
 950370 
 
 4.24 
 
 049630 
 
 16 
 
 45 
 
 823397 
 
 2.36 
 
 872772 
 
 
 88 
 
 950GJ5 
 
 4.24 
 
 049375 
 
 15 
 
 46 
 
 823539 
 
 2.36 
 
 872659 
 
 
 88 
 
 950879 
 
 4.24 
 
 049121 
 
 14 
 
 47 
 
 823680 
 
 2.85 
 
 872547 
 
 
 88 
 
 951133 
 
 4.24 
 
 048867 
 
 13 
 
 48 
 
 823821 
 
 2.85 
 
 872434 
 
 
 88 
 
 951388 
 
 4.24 
 
 048612 
 
 12 
 
 49 
 
 823963 
 
 2.35 
 
 872321 
 
 
 88 
 
 951642 
 
 4.24 
 
 048358 
 
 11 
 
 50 
 
 824104 
 
 2.85 
 
 872208 
 
 
 88 
 
 951896 
 
 4.24 
 
 048104 
 
 10 
 
 51 
 
 9.824245 
 
 2.35 
 
 9.872095 
 
 
 89 
 
 9.952150 
 
 4.24 
 
 10.047850 
 
 9 
 
 62 
 
 824386 
 
 2.35 
 
 871981 
 
 
 89 
 
 952405 
 
 4.24 
 
 047595 
 
 8 
 
 53 
 
 824527 
 
 2.35 
 
 871868 
 
 
 89 
 
 952059 
 
 4.24 
 
 047341 
 
 7 
 
 54 
 
 824668 
 
 2.34 
 
 871755 
 
 
 89 
 
 952913 
 
 4.24 
 
 047087 
 
 6 
 
 55 
 
 824808 
 
 2.34 
 
 871641 
 
 
 89 
 
 953167 
 
 4.23 
 
 01G833 
 
 5 
 
 56 
 
 824949 
 
 2.34 
 
 871528 
 
 
 89 
 
 953421 
 
 4.23 
 
 046579 
 
 4 
 
 57 
 
 825090 
 
 2.34 
 
 871414 
 
 
 89 
 
 953675 
 
 4.23 
 
 046325 
 
 8 
 
 58 
 
 825230 
 
 2.34 
 
 871301 
 
 *■ 
 
 89 
 
 953929 
 
 4.23 
 
 046071 
 
 2 
 
 59 
 
 825371 
 
 2.34 
 
 871187 
 
 1 
 
 89 
 
 954183 
 
 4.23 
 
 045817 
 
 1 
 
 60 
 
 825511 
 Cosine. 
 
 2.84 
 D. 
 
 871073 
 Sine. 
 
 ■1 
 
 90 
 
 954437 
 
 4.23 
 
 045563 
 
 
 M. 
 
 4 
 
 8^ 
 
 Cotang. 
 
 D. 
 
 Tang. 
 
 24 
 
60 
 
 
 (42 DEOnEES.) A TABLH OP LOOARITHMIO 
 
 
 M. 
 
 Sine. 
 
 D. 
 
 Cosine. | p._ 
 
 _lang. 
 
 D. 
 
 Cotang. 
 
 
 
 
 a.H25511 
 
 2.34 
 
 U.871()73il.'.Ht 
 
 9,954137 
 
 4.23 
 
 10.015563 
 
 60 
 
 1 
 
 825051 
 
 2.33 
 
 870960 ! 1.90 
 
 954(i!>l 
 
 4.23 
 
 045309 
 
 59 
 
 2 
 
 825791 
 
 2.33 
 
 87084G 
 
 1.90 
 
 954915 
 
 4.23 
 
 045055 
 
 58 
 
 3 
 
 825t)31 
 
 2.33 
 
 870732 
 
 1.90 
 
 95520(1 
 
 4.23 
 
 044800 
 
 57 
 
 4 
 
 826071 
 
 2.33 
 
 870618 
 
 1 . 90 
 
 955154 
 
 4.23 
 
 04454(1 
 
 56 
 
 5 
 
 826211 
 
 2.33 
 
 870504 '1.90 
 
 955707 
 
 4.23 
 
 0442i»:! 
 
 55 
 
 G 
 
 82G351 
 
 2.33 
 
 8703!)(l 1.90 
 
 955!)(>1 
 
 4.23 
 
 044039 
 
 54 
 
 7 
 
 826491 
 
 2.33 
 
 87027G 
 
 1.90 
 
 956215 
 
 4.23 
 
 043785 
 
 53 
 
 8 
 
 826631 
 
 2.33 
 
 870161 
 
 1 . 90 
 
 9564(i9 
 
 4.23 
 
 04353J 
 
 52 
 
 y 
 
 826770 
 
 2.32 
 
 870U47 
 
 1.91 
 
 956723 
 
 4.23 
 
 043277 
 
 51 
 
 10 
 
 826010 
 
 2.32 
 
 869933 
 
 1.91 
 
 956977 
 
 4.23 
 
 04301*3 
 
 50 
 
 11 
 
 9.827019 
 
 2.32 
 
 9. 80981 H 
 
 1.91 
 
 9.957231 
 
 4.23 
 
 10.0427(;9 
 
 49 
 
 12 
 
 827189 
 
 2.32 
 
 86970-1 
 
 1.91 
 
 957485 
 
 4.23 
 
 042515 
 
 48 
 
 13 
 
 827328 
 
 2.32 
 
 869580:1.91 
 
 957739 
 
 4.23 
 
 042261 
 
 47 
 
 11 
 
 827407 
 
 2.32 
 
 869474:1.91 
 
 957993 
 
 4.23 
 
 042007 
 
 46 
 
 15 
 
 827606 
 
 2.32 
 
 869300 1.91 
 
 95824(i 
 
 4.23 
 
 041754 
 
 ^45 
 
 16 
 
 827715 
 
 2.32 
 
 869245 '1.01 
 
 958500 
 
 4.23 
 
 041500 
 
 44 
 
 17 
 
 827884 
 
 2.31 
 
 869130 1.91 
 
 958754 
 
 4.23 
 
 041246 
 
 43 
 
 18 
 
 828023 
 
 2.31 
 
 869015 1.92 
 
 959008 
 
 4.23 
 
 040992 
 
 42 
 
 19 
 
 828162 
 
 2.31 
 
 868900 1.92 
 
 959262 
 
 4.23 
 
 04073H 
 
 41 
 
 20 
 
 828301 
 
 2.31 
 
 868785 1.92 
 
 95951G 
 
 4.23 
 
 040484 
 
 40 
 
 21 
 
 9.828439 
 
 2.31 
 
 9. 868670 1.92 
 
 9.9597G9 
 
 4.23 
 
 10.040231 
 
 39 
 
 22 
 
 82S578 
 
 2.31 
 
 868555 1.92 
 
 960023 
 
 4.23 
 
 0.39977 
 
 38 
 
 23 
 
 828716 
 
 2.31 
 
 868440 
 
 1.92 
 
 960277 
 
 4.23 
 
 0.39723 
 
 37 
 
 24 
 
 828855 
 
 2.30 
 
 868324 
 
 1.92 
 
 960531 
 
 4.23 
 
 039469 
 
 36 
 
 25 
 
 828993 
 
 2.30 
 
 868209 
 
 1.92 
 
 960784 
 
 4.23 
 
 039216 
 
 85 
 
 26 
 
 829131 
 
 2.30 
 
 868093 
 
 1.92 
 
 961038 
 
 4.23 
 
 038962 
 
 34 
 
 27 
 
 829269 
 
 2.30 
 
 867978 
 
 1.93 
 
 961291 
 
 4.23 
 
 038709 
 
 33 
 
 28 
 
 829407 
 
 2.30 
 
 867862 
 
 1.93 
 
 961545 
 
 4.23 
 
 038455 
 
 32 
 
 29 
 
 829545 
 
 2.30 
 
 867747 
 
 1.93 
 
 961799 
 
 4.23 
 
 038201 
 
 31 
 
 30 
 
 829683 
 
 2.30 
 
 867631 
 
 1.93 
 
 962052 
 
 4.23 
 
 037948 
 
 30 
 
 31 
 
 9.829821 
 
 2.29 
 
 9.867515 
 
 1.93 
 
 9.962306 
 
 4.23 
 
 10.037694 
 
 29 
 
 32 
 
 829959 
 
 2.29 
 
 867399 
 
 1.93 
 
 962560 
 
 4.23 
 
 037440 
 
 28 
 
 33 
 
 830097 
 
 2.29 
 
 867283 
 
 1.93 
 
 962813 
 
 4.23 
 
 037187 
 
 27 
 
 34 
 
 830234 
 
 2.29 
 
 867167 
 
 1.93 
 
 963067 
 
 4.23 
 
 036933 
 
 26 
 
 35 
 
 830372 
 
 2.29 
 
 867051 
 
 1.93 
 
 963320 
 
 4.23 
 
 036680 
 
 25 
 
 36 
 
 830509 
 
 2.29 
 
 866935 
 
 1.94 
 
 963574 
 
 4.23 
 
 036426 
 
 24 
 
 37 
 
 830646 
 
 2.29 
 
 866819 
 
 1.94 
 
 963827 
 
 4.23 
 
 036173 
 
 23 
 
 38 
 
 830784 
 
 2.29 
 
 866703 
 
 1.94 
 
 964081 
 
 4.23 
 
 035919 
 
 22 
 
 39 
 
 830921 
 
 2.28 
 
 866586 
 
 1.94 
 
 964335 
 
 4.23 
 
 035665 
 
 21 
 
 40 
 
 831058 
 
 2.28 
 
 866470 
 
 1.94 
 
 964588 
 
 4.22 
 
 035412 
 
 20 
 
 41 
 
 9.831195 
 
 2.28 
 
 9.866353 
 
 1.94 
 
 9.964842 
 
 4.22 
 
 10.035158 
 
 19 
 
 42 
 
 831332 
 
 2.28 
 
 866237 
 
 1.94 
 
 965095 
 
 4.22 
 
 034905 
 
 18 
 
 43 
 
 831469 
 
 2.28 
 
 866120 
 
 1.94 
 
 965349 
 
 4.22 
 
 034651 
 
 17 
 
 44 
 
 831606 
 
 2.28 
 
 866004 
 
 1.95 
 
 965602 
 
 4.22 
 
 034398 
 
 16 
 
 46 
 
 831742 
 
 2.28 
 
 865887 
 
 1.95 
 
 965855 
 
 4.22 
 
 034145 
 
 15 
 
 46 
 
 831879 
 
 2.28 
 
 865770 
 
 1.95 
 
 966105 
 
 4.22 
 
 033891 
 
 14 
 
 47 
 
 832015 
 
 2.27 
 
 865653 
 
 1.95 
 
 966362 
 
 4.22 
 
 033638 
 
 13 
 
 48 
 
 832152 
 
 2.27 
 
 865536 
 
 1.95 
 
 966616 
 
 4.22 
 
 033384 
 
 12 
 
 49 
 
 832288 
 
 2.27 
 
 865419 
 
 1.95 
 
 966869 
 
 4.22 
 
 033131 
 
 11 
 
 50 
 
 832425 
 
 2.27 
 
 865302 
 
 1.95 
 
 967123 
 
 4.22 
 
 032877 
 
 10 
 
 51 
 
 9.832561 
 
 2.27 
 
 9.865185 
 
 1.95 
 
 9.967376 
 
 4.22 
 
 10.032624 
 
 9 
 
 52 
 
 832697 
 
 2.27 
 
 865068 
 
 1.95 
 
 967629 
 
 4.22 
 
 032371 
 
 8 
 
 53 
 
 832833 
 
 2.27 
 
 864950 
 
 1.95 
 
 967883 
 
 4.22 
 
 032117 
 
 7 
 
 54 
 
 832969 
 
 2.26 
 
 864833 
 
 1.96 
 
 968136 
 
 4.22 
 
 031864 
 
 6 
 
 55 
 
 833105 
 
 2.26 
 
 864716 
 
 1.96 
 
 968389 
 
 4.22 
 
 031611 
 
 6 
 
 56 
 
 833241 
 
 2.26 
 
 864598 
 
 1.96 
 
 968643 
 
 4.22 
 
 031357 
 
 4 
 
 57 
 
 833377 
 
 2.26 
 
 864481 
 
 1.96 
 
 968896 
 
 4.22 
 
 031104 
 
 3 
 
 58 
 
 833512 
 
 2.26 
 
 864363 
 
 1.96 
 
 969149 
 
 4.22 
 
 030851 
 
 2 
 
 69 
 
 833648 
 
 2.26 
 
 864245 
 
 1.96 
 
 969403 
 
 4.22 
 
 030597 
 
 1 
 
 60 
 
 833783 
 
 2.26 
 
 864127 
 
 1.96 
 47^ 
 
 969656 
 
 4.22 
 
 030344 
 
 
 
 
 Cosine. 
 
 D. 
 
 Sine. 
 
 Cotang. 
 
 D. 
 
 Tang. 
 
BINES AND TANCiENTS. (43 DEGREES.) 
 
 CI 
 
 (iO 
 59 
 58 
 57 
 50 
 55 
 54 
 53 
 52 
 51 
 50 
 49 
 48 
 47 
 46 
 45 
 44 
 43 
 42 
 41 
 40 
 39 
 38 
 37 
 30 
 35 
 34 
 33 
 32 
 31 
 30 
 29 
 28 
 27 
 26 
 25 
 24 
 23 
 22 
 21 
 20 
 19 
 18 
 17 
 16 
 15 
 14 
 13 
 12 
 11 
 10 
 9 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 
 M. 
 
 M. < Sine. 
 
 D. 
 
 1 Cosine. 
 
 JD. 
 
 1 Tang. 
 
 1, P. 
 
 Cotang. 1 
 
 U 9.833783 
 
 2.20 
 
 9.804127 
 
 1 . 90 
 
 9.909050 
 
 4.22 
 
 10.030314! 60 
 
 1 
 
 833919 
 
 2.25 
 
 804010 
 
 1 . !)0 
 
 9(;!):io!) 
 
 4.22 
 
 0300!)1 
 
 59 
 
 2 
 
 8:iK).-)l 
 
 2.25 
 
 803892 
 
 l.!)7 
 
 970102 
 
 4.22 
 
 02!)«;!8 
 
 58 
 
 8 
 
 83418!) 
 
 2.25 
 
 8(i3774 
 
 1 . 97 
 
 970410 
 
 4.22 
 
 029584 
 
 57 
 
 4 
 
 83 1;^.") 
 
 2.2.-. 
 
 8)i;!(;50 
 
 1.97 
 
 97O0(i!) 
 
 4 . 22 
 
 0293.31 
 
 50 
 
 5 
 
 831k;(i 
 
 2.25 
 
 8();r)38 
 
 1.97 
 
 970!)22 
 
 4 22 
 
 029078 
 
 55 
 
 C 
 
 8315!ir. 
 
 2.25 
 
 803 U9 
 
 1.97 
 
 971175 
 
 4.22 
 
 028825 
 
 54 
 
 7 
 
 834730 
 
 2.25 
 
 H03301 
 
 1.97 
 
 971429 
 
 4.22 
 
 028571 
 
 53 
 
 8 
 
 83181!,") 
 
 2.25 
 
 803183 
 
 1.97 
 
 971 0S2 
 
 4.22 
 
 028318 
 
 52 
 
 9 
 
 8319!)!) 
 
 2.24 
 
 803004 
 
 1.97 
 
 971935 
 
 4.22 
 
 028005 
 
 51 
 
 10 
 
 8351 ;ii 
 
 2.24 
 
 802940 
 
 1.98 
 
 972188 
 
 4.22 
 
 027812 
 
 50 
 
 u 
 
 9.83520!) 
 
 2.24 
 
 9.802827 
 
 1.98 
 
 9.972441 
 
 4.22 
 
 10.027559 
 
 49 
 
 12 
 
 8354(13 
 
 2.24 
 
 80270!) 
 
 1.98 
 
 972094 
 
 4.22 
 
 027300 
 
 48 
 
 13 
 
 83.->538 
 
 2.24 
 
 8025!)() 
 
 1 . 98 
 
 972948 
 
 4.22 
 
 027052 
 
 47 
 
 14 
 
 835072 
 
 2.24 
 
 802471 
 
 1.98 
 
 973201 
 
 4.22 
 
 020799 
 
 40 
 
 15 
 
 835807 
 
 2.21 
 
 802353 
 
 1.98 
 
 973454 
 
 4.22 
 
 020540 
 
 45 
 
 10 
 
 835941 
 
 2.24 
 
 8(;223t 
 
 1.9H 
 
 973707 
 
 4.22 
 
 0202!)3 
 
 44 
 
 17 
 
 830075 
 
 2.23 
 
 802115 
 
 1 . 98 
 
 97.3900 
 
 4.22 
 
 ()2(i040 
 
 43 
 
 18 
 
 830209 
 
 2.23 
 
 80r,)90 
 
 1.98 
 
 974213 
 
 4.22 
 
 025787 
 
 42 
 
 19 
 
 830343 
 
 2.23 
 
 801877 
 
 1.98 
 
 974400 
 
 4.22 
 
 025534 
 
 41 
 
 20 
 
 830477 
 
 2.23 
 
 801758 
 
 1 . 99 
 
 974719 
 
 4.22 
 
 025281 
 
 40 
 
 21 
 
 9.830011 
 
 2.23 
 
 9.8(;1038 
 
 1.99 
 
 9.974973 
 
 4.22 
 
 10.025027 
 
 39 
 
 22 
 
 • 830745 
 
 2.23 
 
 86151!) 
 
 1.99 
 
 975220 
 
 4.22 
 
 024774 
 
 88 
 
 23 
 
 830878 
 
 2.23 
 
 801400 
 
 1.99 
 
 975479 
 
 4.22 
 
 024521 
 
 87 
 
 24 
 
 8.37012 
 
 2.22 
 
 801280 
 
 1.99 
 
 975732 
 
 4.22 
 
 024208 
 
 3(5 
 
 25 
 
 837140 
 
 2.22 
 
 801101 
 
 1 . 99 
 
 975985 
 
 4.22 
 
 024015 
 
 35 
 
 20 
 
 837279 
 
 2.22 
 
 801041 
 
 1.99 
 
 970238 
 
 4.22 
 
 023702 
 
 34 
 
 27 
 
 837412 
 
 2.22 
 
 800922 
 
 1.99 
 
 976491 
 
 4.22 
 
 023509 
 
 83 
 
 28 
 
 837540 
 
 2.22 
 
 800802 
 
 1.99 
 
 970744 
 
 4.22 
 
 023256 
 
 32 
 
 29 
 
 837079 
 
 2.22 
 
 800082 
 
 2.00 
 
 970997 
 
 4.22 
 
 023003 
 
 81 
 
 30 
 
 837812 
 
 2.22 
 
 860502 
 
 2.00 
 
 977250 
 
 4.22 
 
 022750 
 
 30 
 
 81 
 
 9.837945 
 
 2.22 
 
 9.800442 
 
 2.00 
 
 9.977503 
 
 4.22 
 
 10.022497 
 
 29 
 
 32 
 
 838078 
 
 2.21 
 
 800322 
 
 2.00 
 
 977750 
 
 4 22 
 
 022244 
 
 28 
 
 33 
 
 838211 
 
 2.21 
 
 860202 
 
 2.00 
 
 978009 
 
 4.22 
 
 021991 
 
 27 
 
 34 
 
 838344 
 
 2.21 
 
 860082 
 
 2 00 
 
 978202 
 
 4.22 
 
 021738 
 
 26 
 
 85 
 
 838477 
 
 2.21 
 
 859902 
 
 2.00 
 
 978515 
 
 4.22 
 
 021485 
 
 25 
 
 30 
 
 838610 
 
 2.21 
 
 859842 
 
 2.00 
 
 978708 
 
 4.22 
 
 021232 
 
 24 
 
 37 
 
 838742 
 
 2.21 
 
 859721 
 
 2.01 
 
 979021 
 
 4.22 
 
 020979 
 
 23 
 
 38 
 
 838875 
 
 2.21 
 
 859001 
 
 2.01 
 
 979274 
 
 4.22 
 
 020726 
 
 22 
 
 39 
 
 839007 
 
 2.21 
 
 859480 
 
 2.01 
 
 979527 
 
 4.22 
 
 020473 
 
 21 
 
 40 
 
 839140 
 
 2.20 
 
 859300 
 
 2.01 
 
 979780 
 
 4.22 
 
 020220 
 
 20 
 
 41 
 
 9.839272 
 
 2.20 
 
 9.859239 
 
 2.01 
 
 9.980033 
 
 4.22 
 
 10.019967 
 
 19 
 
 42 
 
 839404 
 
 2.20 
 
 859119 
 
 2.01 
 
 980286 
 
 4.22 
 
 019714 
 
 18 
 
 43 
 
 839530 
 
 2.20 
 
 858998 
 
 2.01 
 
 980538 
 
 4.22 
 
 .19462 
 
 17 
 
 44 
 
 839668 
 
 2.20 
 
 858877 
 
 2.01 
 
 980791 
 
 4.21 
 
 019209 
 
 16 
 
 45 
 
 830800 
 
 2.20 
 
 858750 
 
 2.02 
 
 981044 
 
 4.21 
 
 018956 
 
 15 
 
 46 
 
 839932 
 
 2.20 
 
 858035 
 
 2.02 
 
 981297 
 
 4.21 
 
 018703 
 
 14 
 
 47 
 
 840004 
 
 2.19 
 
 858514 
 
 2.02 
 
 981550 
 
 4.21 
 
 018450 
 
 13 
 
 48 
 
 840190 
 
 2.19 
 
 858393 
 
 2.02 
 
 981803 
 
 4.21 
 
 018197 
 
 12 
 
 49 
 
 840328 
 
 2.19 
 
 858272 
 
 2.02 
 
 982056 
 
 4.21 
 
 017944 
 
 11 
 
 50 
 
 840459 
 
 2.19 
 
 858151 
 
 2.02 
 
 982309 
 
 4.21 
 
 017691 
 
 10 
 
 51 
 
 9.840591 
 
 2.19 
 
 9.858029 
 
 2.02 
 
 9.982562 
 
 4.21 
 
 10.017438 
 
 9 
 
 52 
 
 840722 
 
 2.19 
 
 857908 
 
 2.02 
 
 982814 
 
 4.21 
 
 017186 
 
 8 
 
 53 
 
 840854 
 
 2.19 
 
 857780 
 
 2.02 
 
 983067 
 
 4.21 
 
 016933 
 
 7 
 
 54 
 
 840985 
 
 2.19 
 
 857665 
 
 2.03 
 
 983320 
 
 4.21 
 
 016680 
 
 6 
 
 55 
 
 841116 
 
 2.18 
 
 857543 
 
 2.03 
 
 983573 
 
 4.21 
 
 016427 
 
 5 
 
 56 
 
 841247 
 
 2.18 
 
 857422 
 
 2.03 
 
 983826 
 
 4.21 
 
 016174 
 
 4 
 
 57 
 
 841378 
 
 2.18 
 
 857300 
 
 2.03 
 
 984079 
 
 4.21 
 
 015921 
 
 3 
 
 58 
 
 841509 
 
 2.18 
 
 857178 
 
 2.03 
 
 984331 
 
 4.21 
 
 015669 
 
 2 
 
 59 
 
 841640 
 
 2.18 
 
 857050 
 
 2.03 
 
 984584 
 
 4.21 
 
 015416 
 
 1 
 
 60 
 
 841771 
 
 2.18 
 
 850934 
 
 2.03 
 46° 
 
 984837 
 
 4.21 
 
 015163 
 
 ; 
 
 M. 
 
 Cosine. 
 
 D. 
 
 Sine. 
 
 Cotang. 1 
 
 D. 
 
 Tang. 
 
IMAGE EVALUATION 
 TEST TARGET (MT-3) 
 
 V 
 
 /. 
 
 {•/ 
 
 .^^. 
 
 .<? 
 
 €?j 
 
 '•^' % .. 
 
 WJ'... 
 
 V 
 
 i/i 
 
 
 1.0 
 
 I.I 
 
 1.25 
 
 '■>■ 112 
 
 112,5 
 
 2.0 
 
 U ill 1.6 
 
 
 VI 
 
 e. 
 
 e), 
 
 
 
 Photographic 
 
 Sciences 
 Corporation 
 
 ^^ 
 
 A^ 
 
 <^ 
 
 ^v 
 
 
 
 ^1} 
 
 >, 
 
 \ 
 
 %^' .^v '««^ 
 
 ?3 WEST MAIN STREET 
 
 VSBSTER.N.Y 14S80 
 
 (716) 872-4503 
 
 
.v^ 4i 
 
 L<? 
 
 
 C 
 
 i/x 
 
 il 
 
 1^* 
 
■ss 
 
 n ' - . . .. 
 
 t 
 1 
 
 i 
 
 !■ 
 1 ■ 
 
 S2 
 
 
 (44 DKGUUE8.) ▲ TABLE OF LOOABITHSnO 
 
 
 1 
 
 M. 
 
 
 Sin*. 
 
 0. 
 
 Coiine. 
 
 2.03 
 
 Tan^. 
 
 0. 
 
 1 Cotang. 
 
 60 
 
 9.84J77I 
 
 2.18 
 
 9.85G934 
 
 9.984937 
 
 4.21 
 
 10.015163 
 
 
 \ 
 
 84t90a 
 
 2.18 
 
 ^ 856812 
 
 2.03 
 
 985090 
 
 4.21 
 
 014910 
 
 59 
 
 V- 
 
 812033 
 
 2.18 
 
 8.50690 
 
 2.04 
 
 985843 
 
 4.21 
 
 014657 
 
 58 
 
 
 3 
 
 843163 
 
 2.17 
 
 856d68 
 
 2.04 
 
 985596 
 
 4.21 
 
 014404 
 
 87 
 
 
 4 
 
 842294 
 
 2.17 
 
 856446 
 
 2.04 
 
 985S48 
 
 4.21 
 
 014152 
 
 56 
 
 
 5 
 
 842424 
 
 2.17 
 
 856323 
 
 2.04 
 
 986101 
 
 4.21 
 
 013899 
 
 65 
 
 
 6 
 
 842555 
 
 2.17 
 
 856201 
 
 2.04 
 
 986854 
 
 4.21 
 
 013646 
 
 54 
 
 
 7 
 
 842685 
 
 2.17 
 
 856078 
 
 2.04 
 
 986607 
 
 4.21 
 
 013393 
 
 53 
 
 
 d 
 
 842815 
 
 2.17 
 
 855956 
 
 2.04 
 
 986860 
 
 4.21 
 
 018140 
 
 52 
 
 1 
 
 9 
 
 842946 
 
 2.17 
 
 855833 
 
 2.04 
 
 987112 
 
 4.21 
 
 012888 
 
 51 
 
 
 10 
 
 843076 
 
 2.17 
 
 855711 
 
 2.05 
 
 987365 
 
 4.21 
 
 012635 
 
 50 
 
 ■ 
 
 n 
 
 9.843206 
 
 2.16 
 
 9.855588 
 
 2.05 
 
 9.987618 
 
 4.21 
 
 10.0123S2 
 
 49 
 
 
 12 
 
 843338 
 
 2 16 
 
 855465 
 
 2.05 
 
 987871 
 
 4.21 
 
 012129 
 
 48 
 
 
 IS 
 
 843466 
 
 2.16 
 
 853342 
 
 2.05 
 
 988123 
 
 4.21 
 
 011877 
 
 47 
 
 
 14 
 
 843595 
 
 2.16 
 
 855319 
 
 2.05 
 
 988376 
 
 4.21 
 
 011024 
 
 46 
 
 
 15 
 
 843725 
 
 2.1(i 
 
 855096 
 
 2.05 
 
 988629 
 
 4.21 
 
 011371 
 
 45 
 
 
 16 
 
 843855 
 
 2.16 
 
 854973 
 
 2.05 
 
 988882 
 
 4.21 
 
 011118 
 
 44 
 
 
 17 
 
 843984 
 
 2.16 
 
 854S50 
 
 2.05 
 
 989134 
 
 4.21 
 
 010866 
 
 48 
 
 
 18 
 
 844114 
 
 2.15 
 
 854727 
 
 2.06 
 
 989387 
 
 4.21 
 
 010013 
 
 42 
 
 ■ 
 
 P 
 
 844243 
 
 2.15 
 
 854603 
 
 2.06 
 
 989640 
 
 4 21 
 
 010360 
 
 41 
 
 
 ^0 
 
 844372 
 
 2.15 
 
 854480 
 
 2.06 
 
 989893 
 
 4.21 
 
 010107 
 
 40 
 
 
 21 
 
 9.841502 
 
 2,15 
 
 9.854356 
 
 2.06 
 
 9.990145 
 
 4.21 
 
 10.009355 
 
 39 
 
 
 22 
 
 841(;31 
 
 2.15 
 
 854233 
 
 2.06 
 
 990398 
 
 4.21 
 
 009602 
 
 88 
 
 
 23 
 
 844760 
 
 2.15 
 
 854109 
 
 2.06 
 
 990(>51 
 
 4.21 
 
 00934i) 
 
 37 
 
 \'\ 
 
 24 
 
 841889 
 
 2.15 
 
 853986 
 
 2.06 
 
 990903 
 
 4.21 
 
 009097 
 
 36 
 
 '! ■ 
 
 23 
 
 845018 
 
 2.15 
 
 853862 
 
 2.06 
 
 991150 
 
 4.21 
 
 008844 
 
 35 
 
 ■ , 
 
 26 
 
 845147 
 
 2.15 
 
 853738 
 
 2.06 
 
 991409 
 
 4.21 
 
 008591 
 
 34 
 
 1 i ■ 
 
 2V 
 
 845276 
 
 2.14 
 
 853614 
 
 2.07 
 
 991602 
 
 4.21 
 
 008338 
 
 33 
 
 ;I 1 
 
 28 
 
 845405 
 
 2.14 
 
 853490 
 
 2.07 
 
 991914 
 
 4.21 
 
 008086 
 
 82 
 
 
 29 
 
 845533 
 
 2.14 
 
 853366 
 
 2.07 
 
 992167 
 
 4.21 
 
 007833 
 
 31 
 
 ; 1 
 
 30 
 
 84'>662 
 
 2.14 
 
 833242 
 
 2.07 
 
 992420 
 
 4.21 
 
 007580 
 
 SO 
 
 ( i 
 
 31 
 
 9.845790 
 
 2.14 
 
 9.853118 
 
 2.07 
 
 9.992672 
 
 4.21 
 
 10.007328 
 
 29 
 
 M 
 
 32 
 
 845919 
 
 2.14 
 
 852994 
 
 2.07 
 
 992925 
 
 4.21 
 
 007075 
 
 28 
 
 V 
 
 33 
 
 846047 
 
 2.14 
 
 852869 
 
 2.07 
 
 993178 
 
 4.21 
 
 006822 
 
 27 
 
 \ 1 
 
 III ' 
 
 34 
 
 846175 
 
 2.14 
 
 852745 
 
 2.07 
 
 993430 
 
 4.21 
 
 006570 
 
 26 
 
 T I 
 
 35 
 
 846304 
 
 2.14 
 
 852620 
 
 2.07 
 
 993683 
 
 4.21 
 
 006317 
 
 25 
 
 1 ' i 
 
 36 
 
 846432 
 
 2.13 
 
 852496 
 
 2.08 
 
 993936 
 
 4.21 
 
 006064 
 
 24 
 
 
 37 
 
 846560 
 
 2.13 
 
 852371 
 
 2.08 
 
 994189 
 
 4.21 
 
 005811 
 
 23 
 
 h ; 
 
 88 
 
 846688 
 
 2.13 
 
 852247 
 
 2.08 
 
 994441 
 
 4.21 
 
 005559 
 
 22 
 
 ' . ! 
 
 89 
 
 846816 
 
 2.13 
 
 852122 
 
 2.08 
 
 994694 
 
 4.21 
 
 005300 
 
 21 
 
 
 40 
 
 846944 
 
 2.13 
 
 851997 
 
 2.08 
 
 994947 
 
 4.21 
 
 003053 
 
 20 
 
 1 
 
 41 
 
 9.847071 
 
 2.13 
 
 9.851872 
 
 2.08 
 
 9.995199 
 
 4.21 
 
 10.004801 
 
 19 
 
 
 42 
 
 847199 
 
 2.13 
 
 851747 
 
 2.08 
 
 995452 
 
 4.21 
 
 004548 
 
 18 
 
 
 4S 
 
 847327 
 
 2.13 
 
 851622 
 
 2.08 
 
 995705 
 
 4.21 
 
 004295 
 
 17 
 
 i : 
 
 44 
 
 847454 
 
 2.12 
 
 851497 
 
 2.09 
 
 995957 
 
 4.21 
 
 004043 
 
 16 
 
 
 45 
 
 847582 
 
 2.12 
 
 851372 
 
 2.09 
 
 996210 
 
 4.21 
 
 003700 15 
 
 
 46 
 
 847709 
 
 2.12 
 
 851246 
 
 2.09 
 
 996463 
 
 4 21 
 
 003537 14 
 
 ■ 
 
 47 
 
 847836 
 
 2.12 
 
 851121 
 
 2.09 
 
 996715 
 
 4.21 
 
 0032851 13 
 
 
 48 
 
 847964 
 
 2.12 
 
 85099(5 
 
 2.09 
 
 996968 
 
 4.21 
 
 003032 12 
 
 1 
 
 49 
 
 848091 
 
 2.12 
 
 850870 
 
 2.09 
 
 997221 
 
 4.21 
 
 002779 11 
 
 J 
 
 50 
 
 848218 
 
 2.12 
 
 850745 
 
 2.09 
 
 997473 
 
 4.21 
 
 002527 10 
 
 
 51 
 
 9.848345 
 
 2.12 
 
 9. 850019 
 
 2.09 
 
 9.097726 
 
 4.21 
 
 10.002274 
 
 9 
 
 
 52 
 
 848472 
 
 2.11 
 
 850493 
 
 2.10 
 
 997979 
 
 4.21 
 
 002021 
 
 8 
 
 
 63 
 
 848599 
 
 2.11 
 
 850368 
 
 2.10 
 
 998231 
 
 4,21 
 
 001709 
 
 7 
 
 ' 
 
 64 
 
 848726 
 
 2.11 
 
 850212 
 
 2.10 
 
 993484 
 
 4 21 
 
 001510 
 
 6 
 
 1 
 
 55 
 
 848852 
 
 2.11 
 
 850110 
 
 2.10 
 
 998737 
 
 4.21 
 
 001203 
 
 5 
 
 . 
 
 56 
 
 848979 
 
 2.11 
 
 849990 
 
 2.10 
 
 998989 
 
 4.21 
 
 oomi 
 
 4 
 
 : 
 
 57 
 
 849106 
 
 2.11 
 
 849864 
 
 2.10 
 
 999242 
 
 4.21 
 
 000758 
 
 3 
 
 
 58 
 
 849232 
 
 2.11 
 
 849738 
 
 2.10 
 
 999495 
 
 4.21 
 
 000505 
 
 2 
 
 t 
 
 59 
 
 849359 
 
 2.11 
 
 849611 
 
 2.10 
 
 999748 
 
 4 21 
 
 000253 
 
 1 
 
 
 (50 
 
 849485 
 
 2.11 
 
 849485 
 Sine. 
 
 2.10 
 
 10.000000 
 
 4 21 
 
 10,000000 
 
 
 
 > 
 
 Cosine. 
 
 D. 
 
 45° 
 
 Cotang. 
 
 Tang. 
 
 M. 
 
TABLE IV. 
 
 6a 
 
 TABLE III. 
 
 NATURAL TANGENTS. 
 
 1° 
 
 Lengths 
 
 D. 
 
 19" 
 
 Lengths 
 
 Id. 
 
 .37" 
 
 Lengths 
 
 1 
 
 1 ^• 
 1 55" 
 
 Lengths 
 
 1.42815 
 
 D. 
 
 73" 
 
 Lengths 
 
 .0174.50 
 
 .344328 
 
 .75.3554 
 
 3.27085 
 
 2 
 
 .034921 
 
 20 
 
 .363970 
 
 38 
 
 .7812.'»0 
 
 50 
 
 1.48256 
 
 74 
 
 3.48741 
 3.73M5 
 
 3 
 
 .054408 
 
 21 
 
 .3838(54 
 
 39 
 
 .809784 
 
 57 
 
 1.53986 
 
 75 
 
 4 
 
 . 0(59927 
 
 22 
 
 .404026 
 
 40 
 
 .839100 
 
 58 
 
 1.60033 
 
 76 
 
 4.01078 
 
 5 
 
 .087189 
 
 23 
 
 .424475 
 
 41 
 
 .869287 
 
 59 
 
 1.66428 
 
 77 
 
 4.3.3148 
 
 6 
 
 . 10)1 04 
 
 24 
 
 . 445229 
 
 42 
 
 .900404 
 
 60 
 
 1.73205 
 
 78 
 
 4.70463 
 
 7 
 
 .122785 
 
 25 
 
 .466308 
 
 43 
 
 .932515 
 
 61 
 
 1.80405 
 
 79 
 
 5.14455 
 
 8 
 
 .140541 
 
 26 
 
 .487783 
 
 44 
 
 .965689 
 
 68 
 
 1.88073 
 
 80 
 
 5.67128 
 
 9 
 
 .158381 
 
 27 
 
 ., 509525 
 
 45 
 
 1.00000 
 
 63 
 
 l..)6261 
 
 81 
 
 6.31375 
 7.11587 
 
 10 
 
 .170327 
 
 28 
 
 .531709 
 
 46 
 
 1.035;53 
 
 64 
 
 2.05030 
 
 82 
 
 11 
 
 .194380 
 
 29 
 
 .554809 
 
 47 
 
 1.07237 
 
 65 
 
 2.14451 
 
 83 
 
 8.14435 
 
 12 
 
 .212.557 
 
 30 
 
 .577350 
 
 48 
 
 1.11061 
 
 66 
 
 2.24604 
 
 84 
 
 9.51436 
 
 18 
 
 .230808 
 
 31 
 
 .600861 
 
 49 
 
 1.15037 
 
 67 
 
 2.35585 
 
 85 
 
 11.4301 
 
 14 
 
 .249328 
 
 32 
 
 .624869 
 
 .50 
 
 1.10175 
 
 68 
 
 2.47509 
 
 86 
 
 14.8007 
 
 15 
 
 .207949 
 
 33 
 
 .649408 
 
 51 
 
 1.23490 
 
 09 
 
 2.60.509 
 
 87 
 
 19.0811 
 
 16 
 
 .286745 
 
 34 
 
 .074509 
 
 52 
 
 1.27994 
 
 07 
 
 2.74748 
 
 88 
 
 28.63(>J 
 
 17 
 
 .305781 
 
 35 
 
 .700208 
 
 53 
 
 1.32704 
 
 71 
 
 2.90121 
 
 89 
 
 57.2900 
 
 18 
 
 .324920 
 
 36 
 
 .720.543 
 
 .54 
 
 1. 37038 
 
 72 
 
 8 07768 
 
 190 
 
 « 
 
 lo 
 
 TABLE IV. 
 Areas (A) of some Regular Polygons, 
 
 EACH SIDE OP WHICH IS UNITY. 
 
 Names. 
 
 Sides. 
 
 Abeas. 
 
 Names. 
 
 BiDBS. 
 
 Areas. 
 
 Triangle, 
 
 Square, 
 
 Pentagon. 
 
 Hexagon, 
 
 Heptagon, 
 
 3 
 4 
 5 
 6 
 
 7 
 
 0.4330127 
 l.OOOOQOO 
 1.7204774 
 2.5980762 
 3.6339124 
 
 Octagon, 
 
 Nonagon, 
 
 Decagon, 
 
 Undecagon, 
 
 Dodecagon. 
 
 8 
 
 9 
 10 
 
 11 
 
 12 
 
 4.8284271 
 1.1818242 
 7.6942088 
 4.3G.56399 
 11.1961524 
 
 The use of this part 1" will be found in Book I, art. 123, Prob. 11. 
 
 2« Lengths op Circular Arcs op I'*, or 1', or 1"^ 
 
 TO R.\DIUS R. 
 
 Length of an arc of \^=RxO-0\ 1 453 292, 
 
 « " " r==/ex 0.000 290 888, 
 
 " " « 1"=/2X 0.000 004 848. 
 
 The use of this part 2" will be found in art. 50 of Book III. 
 
r- 
 
 64 
 
 TABLE IV 
 
 3" Area op a Spherical Polygon whose spherical 
 excess is 1", or 1', or 1", on a sphere to radius r. 
 
 Area of a polygon to spherical excess of 1*'=^^X'*-'^1 745 329 244, 
 '< , <; « l'=^^XO-00 029 088 821, 
 
 « « »« l"=r/23x 0.00 000 484 814. 
 
 The use of this part 3" will be found in Trig. Art. 107, Sch. I. 
 
 Numbers often used in Calculations. 
 
 C/ircumference of a circle to diameter 1 1 
 
 Surface of a sphere to diameter 1 [ ==t ; 
 
 Area of a circle to radius 1 ] 
 
 Area of a circle to diameter 1 : 
 
 Tolume of a sphere to diameter 1 : 
 
 Tolume of a sphere to radius 1 : 
 
 Area of an equilateral triangle, whose side isl . = 
 
 Area of a s luare, " " . : 
 
 Area of a "regular pentagon, " " . i 
 
 Area of a regular hexagon, " '• . : 
 
 Area of a regular heptagon, ** *< . : 
 
 Area of a regular octagon, " " . ; 
 
 Area of a regular nonagon, " " . ; 
 
 Area of a regular decagon, " " . : 
 
 Area of a regular undecagon, " ' . : 
 
 Area of a regular dodecagon. " " . ; 
 
 Length of an arc of I*', to radius 1 "I 
 
 Area of a polygon, to spherical excess 1", >■ : 
 
 on a sphere to radius 1 J 
 
 Length of an arc of 1 ', to radius 1 ...... . "j 
 
 Area of a |)olygon, to spherical excess !'» J- = 
 
 on a sphere to radius 1 j 
 
 Length of an arc of I ", to radius 1 "i 
 
 Area of a polygon, to spherical excess 1", >■ = 
 
 ou a sphere to radius 1 j 
 
 3.141592G 
 
 0.7853082 
 
 0.5235988 
 4.1887902 
 
 0.4330127 
 1.000000 ' 
 1.7204774 
 2.5980762 
 3.63.39124 
 4.8284271 
 6.1818242 
 7.6942088 
 9.3656399 
 11.1961524 
 
 0.0174533 
 0.0002909 
 0.0000048 
 
 Log. 
 
 0.497150 
 
 1.895090 
 
 r. 7 1 8999 
 0.622089 
 
 1.636501 
 0.000000 
 . 235648 
 0.414652 
 0.560374 
 0.683805 
 0.791133 
 0.886164 
 0.971537 
 1.044069 
 
 2.241877 
 4.463726 
 6.685575 
 
TABLE V. 
 
 05 
 
 GEOMETRICAL FORMULA. 
 
 Areas (A) op Plane Figures. 
 
 1241877 
 
 163726 
 
 585575 
 
 Names. 
 
 Parallelog. 
 Triangle. 
 
 Trapezoid. 
 
 Regular 
 Polygon. 
 
 <3ircle. 
 
 Circular 
 Segments. 
 
 Horizontal 
 Ellipse. 
 
 Tertical 
 Ellipse. 
 
 Elliptic 
 Segments. 
 
 Parabola. 
 
 Hyperbola. 
 
 Hyperbolic 
 segment of 
 "two bases. 
 
 Parabolas. 
 
 Equations. 
 
 5=2^'. 
 nJi=2nB'=P. 
 
 do 
 
 y 
 
 ^ a 
 
 b . 
 
 " a 
 
 y=T\/26i-z^ 
 
 do 
 
 {yr=.2y/ax. 
 2yz=H, k x=zB' 
 
 y=i~\^2ax-\-x^. 
 
 do 
 
 y=ai»4.6z«4-cx4-y 
 
 FonMUL/K. 
 
 A=\H{B-\-AB'-\-0)=:\BxH- 
 A=\H{B+\B'^b)^\H{B-\-b) 
 A=^Hn(B+AB' + 0)=K^X ^• 
 
 Reference, 
 
 Assnab. 
 
 Reference, 
 A=^H{O^AB'+0)^^xy. 
 
 ^=y(a+x)-^logfy±^^ 
 la ab 
 
 .4=a(r-y)4.Xr-a:y 
 
 —— log a^'-t-6(«4--^) 
 Jf ay-}-6(a+a;) 
 
 ^=j//(r4-4r+y). 
 
 Elementary segments of any plane figure whose formula is 
 either wanting or requires too laborious a computation ; t. g. 
 ihe above hyperbolic segment : 
 
 Approximate A:s\H{r-\.^r-^y). 
 
 Paoks. 
 
 28,61. 
 29.61,62 
 30,62. 
 52,63. 
 
 100,131. 
 
 100,132, 
 273. 
 
 102,133, 
 165. 
 
 102,133, 
 
 104,134, 
 160. 
 
 106,274, 
 App. E. 
 
 106, 
 App. E. 
 
 112,136, 
 159. 
 
 256.273, 
 Art. 27. 
 
wm 
 
 i6. 
 
 TABLt V. 
 
 Volumes (V) and Convex Surfaces (^S) of Solids. 
 
 Names. 
 
 Right or 
 Oblique 
 
 Higbt or 
 
 Oblique 
 
 Cylind'jr. 
 
 Right or 
 
 Oblique 
 
 Pyramid. 
 
 Right or 
 
 Oblique 
 
 Gone. 
 
 Ftustum 
 of Right or 
 Oblique 
 Pyramid. 
 
 Frustum 
 o|l Right or 
 Oblique 
 Cone. 
 
 Wedges 
 Prismoid. 
 
 Sphere ; 
 
 Spherical 
 Segment; 
 
 Spherical 
 Frustum. 
 
 Spherical 
 W'^dge. 
 Spheric-il 
 Lune. 
 
 Spkberical 
 Pyramid 
 
 Spherical 
 Polygon. 
 
 Kqi'atio.ns. 
 
 
 B=B'=b 
 
 B=4B' 
 P=2P' 
 
 B=z4B' 
 C=2C' 
 
 \{P^P)^P' 
 
 i((74-c)=C' 
 
 FORMULiK. 
 
 /=\/2/fcc-vt» 
 
 do 
 
 x=iH. 
 
 do 
 
 X-x=U. 
 
 n'^sangul^r 
 measure. 
 
 H=2i;B^4B' 
 
 5-(n-2)180'= 
 
 spherical ex- 
 cess. 
 
 r=-j/i(Z?4-4i?'4-6)=/?xi/'. 
 
 V=:lH(B-^4B'-{-b)^BX H. 
 5=i//(C+4C"-|-c)«.Cx E, 
 
 S=^H(P-\-4P'-\.0)MAffKP. (right 
 pyrauiid). 
 
 F=i//( B + 4i?'-|-0)«i^X B. 
 S^\U{C^4C'^0)may,r:R. (right 
 cone, 
 
 V=^\ir{BJ^-4B'-\-b). 
 
 5=J^(P+4i>'4.^)«JJ5r(i>-f;,)(right 
 
 pyramid). 
 
 S=.\n{Bj,4B'^b), 
 cone). 
 
 V^\HlB-if.4B''\-Q). 
 V^\H(B^.4B'Jrby 
 
 r=^A7r(0 + 4F2+0)a»j7rJP, 
 5=47rif. 
 
 r=i57r(r2+4r»4-o). 
 
 Siss.2nIU. 
 
 r=^irT(r*-l-4r2+y«). 
 
 r= Tiz^x 
 
 «• 
 
 270" 
 
 9U» 
 
 V=zlJf(B-\-4B' + 0)=xiffxB. 
 
 5=5_^| 5--(n~2)180« "I. 
 I8O0 l ^ ' J 
 
 Paoe9. 
 
 74,76,84 
 
 75,76,84 
 
 150,154,. 
 145,161. 
 
 150,154, 
 146,106. 
 
 150,154, 
 145,106. 
 
 150,154,. 
 145,661.. 
 
 157,158; 
 
 100,151,. 
 161,140, 
 163. 
 
 151,161, 
 141,1631 
 
 I50,161> 
 141,163: 
 
 204. 
 
 212,233; 
 
LIDS, 
 PA0E9. 
 
 74,76,84 
 
 75,76,84 
 
 150,154,. 
 145,161. 
 
 150,154, 
 145,106.. 
 
 150,154r 
 145,1G6. 
 
 150,154, 
 145,661.. 
 
 157,158; 
 
 100,151,. 
 161,140, 
 
 163. 
 
 151,161, 
 Il41,163l 
 
 |i50,161k 
 L41,163i 
 
 204. 
 
 TABLI V. 6T 
 
 Volumes ( V) and Convex Surfaces (S) op Solids. 
 
 Names. 
 
 Prolate 
 Ellipsoid ; 
 
 Segment ; 
 Frustum. 
 
 a 
 
 do. 
 do. 
 
 Oblate 
 Ellipsoid ; 
 
 Segmipnt ; 
 
 Frustum. 
 
 Common 
 Paraboloid ; 
 
 Frustum. 
 
 12,232;. 
 
 Hyperboloid ; 
 
 Frustum. 
 
 Conjugate 
 Hyperboloid. 
 
 Eqcatioms. 
 
 
 3 
 
 do. 
 
 '.H. 
 
 do. 
 
 (X-jr)=i?; 
 
 y==2i/«>. 
 
 FoRMULi*. 
 
 r=J//r(0+4r'-+0)=«JrraA». 
 
 do. 
 
 iX-x)=H. 
 
 a ' 
 
 do. 
 
 (X^x)^//. 
 
 y=± -Va^-f *' 
 
 r=j//-(F24-4r»-fo). 
 
 (S=formula of App. F. 
 ^ssformula of App. (/'), Cor. 
 
 r= Ji7rr(0-f 4 r'+0)=$Ta»6. 
 
 r=i//;r(r-'4-4r2-fo). 
 
 &asformula of App. Q. 
 
 r=iiyn-(r24.4r'«+y2). 
 
 &BU)rmula of App. (tf), Cor. 
 
 r=i//rr(r2_^4F2 + y3). 
 
 3a ( J 
 
 r=i//>r( 1-3-1-4 }"»-fO). 
 (S==formula of App. K. 
 
 S=formula of App, (if). Cor, 
 
 r=i/7T( 1-24.4 1"»+y2). 
 
 Paqeb. 
 
 102,155, 
 167. 
 App. F. 
 
 151. 
 App. F. 
 
 150. 
 
 102,151,. 
 156, 
 App. Q^ 
 
 151. 
 150. 
 
 104,151,. 
 156. 
 App. //. 
 
 150. 
 App. H. 
 
 106,151, 
 
 150. 
 
 106.150, 
 167. 
 
«8 
 
 TABLE V. 
 
 Volumes (For F) and Convex Surfaces (S or A) 
 
 OF Solids. 
 
 Xamks. 
 
 KyUATlONS. 
 
 Diverging 
 Paraboloids. 
 
 Frusta. 
 
 Polyedroids 
 •Segments. 
 
 Jrusta. 
 
 Polyedroids 
 
 Segments, 
 
 Frusta. 
 
 ■Solids of 
 revolution. 
 
 Zone of 
 one base. 
 
 Zone of 
 two bases. 
 
 Formula:. 
 
 specified 
 
 in Cor. 2 & 3, 
 
 on page 151. 
 
 y=sa2^-\-br^-\-cz-\-J 
 
 z^H. 
 
 {Z-z)=H. 
 
 specified in art. 58, 
 on page 142. 
 
 Z-z=iff. 
 
 r=j//7r(r2-f.4j"24-o). 
 
 r=J^T(P-f4r2+y»). 
 
 r'=i/y(04.4^'+0). 
 
 V'=lII(B^iB'-\-b). 
 ^=i^(0-f4/>'4.0). 
 
 ^=J^(/^4.4i"4.0), 
 
 ^=i^(P+4i"4-p). 
 
 5=i^(0-|-4C^'4.0). 
 5=i^(C-|-2(7'4.0). 
 
 A=:SX 
 
 p 
 
 2x1' 
 
 V^^//(B-^iB'^b). 
 
 Pages. 
 
 111.150, 
 151, 
 
 162.167. 
 
 151. 
 
 151,162, 
 
 151. 
 
 112,145, 
 163,164, 
 167. 
 
 145,163, 
 164. 
 
 82,83, 
 84,85. 
 
 Volumes (F) and convex sur- 
 faces (A) of polyedroids circum 
 scribed about any solid of revo- 
 'lution whose volume {V), con- 
 vex surface {S) and radius (F) 
 are given. 
 
 The Elementary Frustum or 
 Segment of any solid whose 
 formula is either wanting or 
 requires too laborious a compu- 
 tation will be measured very 
 
 ;approximately by : 
 
 Note. Special formulae will be found in Riiumf, on pages 130, 147 
 «nd 154. and in the Appendix. 
 
 262.264, 
 
 273 
 to 289. 
 
.167. 
 
 264, 
 
 
 TABLE V. 69 
 
 TRIGONOMETRICAL FORMULAE. 
 
 Right-Anglei) Plane 
 Triangles. 
 
 Case I. 
 
 Data. J?=s90«, hypothentise b. 
 and either side, as a, about thu 
 right angle. 
 
 Solution. 
 
 sin.4=!-. 
 b 
 
 c^b coaA. 
 
 Case II. 
 
 Data. ^==90'', and both sideg, 
 ■a and c, about thi' right angle. 
 
 Solution. 
 
 tan.4=cotC^-. 
 c 
 
 J— 90«— C7, or C^90*'—A. 
 
 . a c 
 
 sinA sinC 
 
 Case III. 
 
 Data. B=90^, hypothenuse b, 
 and either acute angle, as A. 
 i 
 
 Solution. 
 
 C=909— A 
 a=sb sinA. 
 <ssb coaAssb ainC. 
 
 147 
 
 Obuque-Angled Plane 
 Triangles. 
 
 Case I. 
 
 Data. Two angles, as Ji and (7, 
 and the included side a. 
 
 Solution. 
 
 A=18<)''—{B+C). 
 
 , sin/i 
 0=a 
 
 8iiK4 
 
 sinC . sinC 
 
 Bin.<4 s'niB 
 
 Case II. 
 
 Data. Two sides, as a and b, 
 and angle A opposite to a. 
 
 Solution. 
 
 s'mB=~ ainA. 
 a 
 
 C=180«—(A'\-B). 
 
 sinC 
 
 c^a. 
 
 sin.4 
 
 Case III. 
 
 Data. Two sides, as a and b 
 and their included angle C. 
 
 Solution. 
 
 tan A" 
 
 cotiC. 
 
 a-\-b 
 
 A=M-\-N ; B^M^N. 
 
 sinC ,s:n(7 
 Csssu . =T=0 . 
 
 sin.^ sini^ 
 
70 TABLE V. 
 
 RiOHT- Angled Triangles. Oblique- Angled Triangles. 
 
 Case IV. 
 
 Data, -fiss'jno, either acut- 
 angic, as A, and 1" the oppoaitu 
 side a, or 2" the adjacent side e. 
 
 Solution. 
 
 C=90°—A. 
 c^a coiA. 
 
 fc=a cot^ 
 sin^ 
 
 20 
 
 a=sc iATiA, 
 
 cosA 
 
 Hight-Anolbd Spherical 
 Triangles. 
 
 Case I. 
 
 Data. ifasDO", side a and opposite 
 Solution. 
 
 siniss 
 
 Bin a 
 
 aiaA 
 gin csstan a cotA. 
 
 sinCa 
 
 cos a 
 
 C08--1 
 
 Case II. 
 
 Data. .£=90^, side a and adja- 
 cent [_^C. 
 
 Solution. 
 cosC 
 
 cot b^ 
 
 tana 
 tancsssina tanC 
 cos.4=co8a sinC. 
 
 Case IV. 
 
 Data. The three sides a, b, e. 
 Solution. 
 , ,.^(a-\-b)(a^b) 
 
 € 
 
 *=J(c+«-«'). 
 
 C0S.^xb1. 
 b 
 
 a 
 C=180<'-(^+5). 
 
 Oblique-Angled Spherical 
 Triangles. 
 
 Case I. 
 
 Da^m The three sides a, b, e.. 
 Solution. 
 
 sin 6 sin c 
 
 siu a sin e 
 
 ^Qgj^^^8m«_8in(«-c)^ 
 sin a sill b 
 
 Case II. 
 
 Data. Two sides, as a, b, and 
 \__A opposite to a. 
 
 Solution. 
 
 sin5«=8in.4j!!L?. 
 am a 
 
 tanJc=tanUa-6)i!!lMh?):. 
 "*' 8ini(.4-^) 
 
 siin e 
 
 sin(7=ssin.4 
 
 eina 
 
b, and 
 
 Right-angled 
 Spherical Triangles. 
 
 Case III. 
 
 Data. /?=90<', hypolhc- 
 nuse b, ami either adjacent 
 angle, an A. 
 
 SOLUTIOH. 
 
 sin a=sin b sin A. 
 tan a=:tan b coaA. 
 cotCsscos b iAnA. 
 
 Case IV. 
 
 Dilta. J?==90<', hypothenuse 
 bf and either side of Ji as a. 
 
 Solution. 
 
 cos ft 
 coia 
 
 _sinff 
 siu& 
 
 cosC=:tan a cot &. 
 
 Case V. 
 
 Data. i?=90«, and both 
 sides a and e of i?. 
 
 Solution. 
 cos6=co3a 9OS0. 
 
 TABLE V. 71 
 
 Oblique-Angled Spherical 
 Triangles. 
 Case III. 
 
 Data. Two sides, as a, b, nnd the 
 included Lt. 
 
 Solution, 
 
 tan \ (^4-/0s=tan.V=cot » C "^o^Ua-ft) 
 
 cosi(a-|-6) 
 
 tnni (.4-2?)=tanA=cot» C !i!L'i^r±> 
 
 sini(a-f6) 
 
 A=3f^X; n=M^X. 
 
 sinC 
 
 COSCsss- 
 
 8inj4=- 
 
 C0t.4ss 
 
 smc 
 
 tana 
 
 C03-4 
 
 COS a 
 
 Case VI. 
 
 smC/sss- 
 
 Data. TJ^OO", and both 
 oblique aiiglos -1, C. 
 Solution, 
 cos .4 
 siaC 
 cos b=co\.A cotC. 
 c sC 
 
 cos rt=- 
 
 COSC=- 
 
 sin^l 
 
 C03^ic= 
 
 sm c=3ina. 
 
 sin.4 
 
 Case IV. 
 
 Data. The three angles A, B, C. 
 Solution. 
 1{A-\.B^£)^S. 
 
 sin// sinC 
 
 cos^i6=£££j!=l11!:2!:^=£). 
 
 sin.4 sinC 
 », __co9(S—A)roMS—B) 
 
 ' ' '■ ^*^ — I ■ I.I — .. ill .^ • 
 
 sin^4 Ejin// 
 
 Case V. 
 
 Data. Two angles, as A, B, and side 
 a opposite to [_ .4. 
 
 Solution, 
 
 . . . sinff 
 sin &=3m a 
 
 sin^ 
 tanic=tanK«-6)^-^-±^. 
 
 sinC=8in^ ^llli. 
 sin a 
 
 Case VI. 
 
 Data. Two angles, as A, B, and the 
 included side c. 
 
 Solution. 
 
 tani(a4-6)=tanm=tan'c S^!ildll^^ 
 
 ^o^\{A-\-B) 
 
 tani(a— 6)=tann=tanic s'^U^--^). 
 
 ' sini(^+if) 
 a=m-j-«; 6=w» — «• 
 
 ■sinC=:sin^!llL^ 
 sin a 
 
CONTENTS. 
 
 GKOMKTllY. 
 
 INTRODUCTION. 
 
 Paoe 
 
 General Definitions 11 
 
 Axioms 13 
 
 Abbreviations 14 
 
 BOOK I. 
 
 PLANE GEOMETRY. 
 
 Definitions 16 
 
 Plane Angles IT 
 
 Triangles 21 
 
 Quadrilaterals 26 
 
 Areas and Proportions in Poly- 
 gons 28 
 
 Circle and Regular Polygons.. 41 
 
 Graphical Problems 54 
 
 Numerical Problems 61 
 
 BOOK IJ. 
 
 SOLID GEOMETRY. 
 
 Definitions 66 
 
 Polyedrons and Polyedroids. . . 69 
 
 Solids of Revolution 72 
 
 Convex Surfaces, Volumes and 
 
 Proportions in Solids 74 
 
 Practical Applications 84 
 
 PxfiB 
 
 Application of Algebra to 
 Geometrical Problems 87 
 
 BOOK III. 
 
 ANALYTICAL GEOMETRY. 
 
 Definitions 95 
 
 Tracing of Curves 96 
 
 Equations of the Straight Line. 99 
 
 Equations of the Circle 100 
 
 Equations of the Ellipse 101 
 
 Equations of the Common Para- 
 bola 103 
 
 Equations of the Common 
 
 Hyperbola. 105 
 
 Equat* "s of the Diverging 
 
 Parabolas , 106 
 
 Equations of the Cubic Para- 
 bola 110 
 
 Solids of Revolution specified 
 by y2=aa:»-f 6z'+cz4-/. .... 113 
 
 Infinitesimal Analysis 114 
 
 Infinitesimal Analysis applied 
 
 to Mensuration 125 
 
 Areas of Plane Figures 131 
 
 Areas of Convex Surfaces.... 140 
 
 Volumes of Solids 148 
 
 Practical Applications 150- 
 
CONTENTS. 
 
 75 
 
 TKKIONOMETIIY. 
 
 PLANE TRIGOXOMETRV. 
 
 Vaue 
 Definitiono ins 
 
 Rclation.s of the Natural Circu- 
 lar Functions 1 To 
 
 Limiting Values and Algebraic 
 Signs oC Circular Functions. 1 T'J 
 
 Functions of Arcs comprising 
 
 Functions of Particular Arcs.. ITC 
 Circular Functions of Negative 
 
 Arcs IT'". 
 
 Inverse Circular Functions. ... ITT 
 Relations of the Sid;s andFunc- 
 tions of Right-Angled Plane 
 
 "" ■'. gles 177 
 
 Functions of the Sura and Diffe- 
 rence of two Arcs 178 
 
 Functions of Double and Half 
 
 Arcs 180 
 
 Additional FormuUe 181 
 
 Theorems 181 
 
 Solution of Right-angled plane 
 
 Triangles 183 
 
 Solution of Oblique-anglcdplane 
 
 Trirngles 184 
 
 Logarithms 
 
 Table of Logarithms 186 
 
 Paoi 
 Tiible of natural Circular Func- 
 tions? 192 
 
 Loguriihiiiic Table of Circular 
 
 Fuiictioiia 103 
 
 Practical Applications 106 
 
 INTRODUCTION TO SPHERICAL 
 TIUGONOMKTRY. 
 
 Definitions and General Prin- 
 
 ciplci, ^ 2'tl 
 
 Thoorenis 204 
 
 Stprcograri^iic Proj; ctloa of the 
 Sph<', ■ 213 
 
 SPHERICAL TIIIGONOMETRY. 
 
 Definitions, . 21? 
 
 Pelations of the Functions of the 
 Sides and Angles of a Pvicrht- 
 angh d spherical Triangle, . . 218" 
 
 Solution of Riglit-anghd sphe- 
 rical Triantrl:.'S 221 
 
 Solution of Quadnntal sphe- 
 rical Triangles 224 
 
 Formulae of Hait Area and Half 
 
 Angles of Spherical Triangles. 220 
 
 Solution of Oblique-angled 
 spherical Triangles 23' 
 
 Practical Applications 23.' 
 
 APPROXIMATE MENSURATION. 
 
 Definitions 250 
 
 Areas of Plane Figures 256 
 
 Volumes of Solids 262 
 
 Areas of Convex Surfaces 204 
 
 Equation of the curve of a 
 
 Polyedroid 20] 
 
 Limits of the Curve of a Polye- 
 droid and of the Inscribed 
 
 Solid of Revolution 267 
 
 Practical Applications., 273 
 
 APPENDIX. 
 
 290 
 
"^""^^"mijm 
 
 •74 
 
 CONTENTS. 
 
 TABLES. 
 
 Page 
 
 1 
 
 TABLE I. 
 
 Logarithms of Numbers .... 
 
 tABLEn. 
 Logarithmic Circular Functions 1 7 
 
 TABLE in. 
 Natural Tangents C3 
 
 TABLE. IV. 
 
 Paob 
 
 I* Areas of Regular Polygons 
 2» Lengths of Circular Arcs 
 3" Areas of Spherical Polygons. 63 
 
 TABIJBV. 
 
 Ooomctricnl k Tiigonometrical 
 Formulre 65 
 
 ■*-' 
 
 \ 
 
Paob 
 ygona 
 Arcs 
 irgona. 63 
 
 Btrical 
 
 65