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Tous les autres exemplaires originaux sont film6s en commenpant par la premidre page qui comporte une empreinte d'impression ou d'illustration et en terminant par la dernidre page qui comporte une telle empreinte. Un des symboles suivants apparaitra sur la dernidre image de cheque microfiche, selon le cas: le symbols — ► signifie "A SUIVRE ", le symbols V signifie "FIN". Les cartes, planches, tableaux, etc., peuvent dtre film6s d des taux de reduction diff^rents. Lorsque le document est trop grand pour dtre reproduit en un seul clich6, il est film6 d partir de Tangle sup6rieur gauche, de gauche d droite, et de haut en bas, en prenant le nombre d'images n6cessaire. Les diagrammes suivants illustrent la mdthode. 1 2 3 32X 1 2 3 4 5 6 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 .T m !*• 1840 11^ 2.0 1.8 1.25 1.4 1 1.6 ^ 6" — ► m ^ /a /J ■dW ^># > '/ M Photographic Sciences Corporation s. -b '^s 4 V ,v \\ "% V 6^ %^ '% 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 X *< •< ■< + + + + N w 01 + + + 4- + S3 + + + H- + <M + 3 S^ + ^3 \\ II li II 1^ ^^^ 5^ <M CO I' I xn s •■■4 o t- c ■•J -as + to -E a 3 0) CO ■y 9 )<■ c o Cm O S a; p" C^l 5^1 II II -1^ •< cc -^ fi & 0) ctf c Wi a 72 N X 4; M BOOK III. o t- 00 05 ^ ^^^ n lil-J lil'O -«J 15 S' ^'~ 3 II 11 r-»-« tU 1 i -tJ -^ ■1 u 'i 1 1 X rs b :c 10 a £ v^ ^0 .X! M ^5 iO C 03 P -4^ to a «! o 9i a 2 »-i W) pC ■to^ l-H M -t-l -•^ ^ >% >» a a ...A ^ ^ X Ui 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 4- «? 1^ li ©1 05 CO "s "S + is M r-i Kl ^ Jl fC rO h^ (M -t '>?, + + 1^ «•: :i H P + ^ e CO JO 11 (I 'I! t' M ^' -I 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 IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I »ii||||M IIIII2.5 - lill|3-2 :| IIIIM * 2.0 1.8 1.25 1.4 1.6 < 6" — ► pm <^/ A c^ A O '/ 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 TEST TARGET (MT-3) 1.0 I.I ■f ilM ■■■■ Itt .^' m ^ m 111^ IM 1.8 1.25 1.4 1 6 ■* 6" ► y <^ A 'c^. e. el 4^ '* o / Photographic Sdences Corporation ,\ « <r :\ \ «\^ #>!.' ^ 23 WEST MAIN STREET WEBSTER, NY. 14580 (716) 872-4503 '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 , 1724 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 2-:T ]Sl 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 •4sl •(,''31 •90(5 IMS 1330 15:2 212 205 311754 19u6 2177 23:i9 2(")9 2S12; 3023 3l'31 3415 365(5 211 20f; 3807 4078 4289 4499 471 n 4929' 5130 5340 5551 57. ■1 21') 207 5970 6180 6390 <;599 (580!) 1 7018; 7227 7-13(5 7(546i 7^54 2ii!) 2(18 80(53 8272 848^ 868;) .88' .8 1 910(5' 9314 9522 97301 9:;3.s 2<.s 209 320146 0354 05.i2 0769 0977 11S4' ]3,)1 15i)8 l,so5' 2012 2'. 7 210 322219 2426 2633 283!) 3046 3252: 3458 3(5(55 3871 4077 20(5 211 4282 4^188 ■iVM 4899 5105 5310 5516 5721 5:)26 6131 205 212 6336 6.5-il 6745 6950 7155 7359; 7563 77()7 7972 817(5! 204 213 8380 8583 8787 89!)1 91!)4 9398; 9601 !)805 •••8l •211 203 214 330114 0617 0819 1022 1225 1427 1630 1832 2034; 223(5 202 215 2438 2640 2S42 30-14 3216 3447! 3(;49 3850 4051 ! 4253 2o2 216 4-154 4655 4H56 5957 5257 5458j 5(558 585!) eo59| 6260 201 217 6460 6(:afl 68(10 7(!(;0 7260 7459; 7(i59 7.S58 80581 8:i57 200 218 8456 8(i5(; 88.55 9054 9253 9-|51i 9650 9K4'»| ••47! •21(5 199 219 X. »t044-l (»(14S 0841 2 103: )| 3 1237 1435^ lt::k; l>-3,l| 'jiy>H', 2:: 5 1 ■■^ i 5 1 c> 1 7 8 1 9 If! A tABI.E OV UOr.AIMTHMS FIIOM 1 TO 10.000. I . h; I !j>i 1 2 3 4 321 IJ 1 5 6 7 8 1 9 D. 2-.) 3424-J3 )Hi2U •J.ili 3014 340iJ ; 3000, 3802! 3.»99 i 41;«) 197 221 43)2i 45,")) 4785 4JHl 51781 5374; 5570 57001 5.»02 (il.57 190 22J (i353l (554!) ' 0744' 0939 7135 73301 7525 7720i 7915 i 8II0: 195 223 8305! H500 8094! 8889 9083 9278, 9472 9000 9800 i«^54 194 221 3502 IS 0442 0030i 0829 1023, 1210, 1410 100;] 1796 1989 193 225 2183 2375 25()H 2701 2954 31471 3339 3532 3724 i 3916' 193 22(1 4108 4301 4493 4085 487(i 5008 520(J 5452 5643 ! 58:M| 192 227 0020 0217 0408 0599 0790 0981 7172 73031 7554 7744j 191 22,S 7035 8125 1 8310 8500 809f 888f 9070 9200 9450 9040l 190 221) !)H35 ••25 •215 •404 •59a •783 •972i 1161 13.50 1539 189 2;J') 36172(J 1917 2105 2294 2482 2071 2859 3048 3230 3424 188 2;}i 3012 3S00 39.S8 4170 4303 4551 4739 49201 5113 5301 188 23J 54 SH 5075 58()2 0049 0230 r;423| 0010! G790i 0983 71(;9 187 233 7350 7542 7719 7915 8101 8287 8473 8659 8845 9030 180 234 0210 9401 9587 9772 9958 •US •328 •513 •698 •883 1,H5 235 37100H 1253 1437 1022 1800 1991 2175! 2300 2544 2728 184 23G 2012 3090 3280 3404 3047 3831 4015 4198 4382 4505 184 237 4748 4932 5115 5298 5481 50(i4 5;i40 0029 6212 0394 1^3 23M 0577 6759 0942 7124 7300 74 S8 7070; 7852 8034 H210 1,82 23;) 8398 8580 8701 8943 9124; 93001 9ii7 9608 9849 ••30 181 240 380211 0392 0573 07.54 0934 1115 1290 1470 16,5(; 1837 181 241 2017 2197 2377 2557 2737 2917 3097 3277 345() 3()30 180 242 3815 3995 4174 4353 4533 4712 48911 5070 5249 5428 179 243 5000 5785 5904 0142 0321 6499 0077 G850 7o;m 7212 178 244 7390 7508 7740 7923 8101 8279 8450; 8()34 8811 8989 178 2-'J 9100 9343 9520; 9098 9875 ••51 •2281 ^405 •582 •759 177 24ti 390935 1112 1288 1404 1041 1817 I993I 2109! 2.345 2521 170 247 2097 2873 304S 3224 3400 35751 3751, 392() 4101 4277 170 24S 4452 4027 4',02' 4977 5152 5320 5.501! 5070 5850 0025 175 24!) 0199 0374 05481 0722 0890 70711 7245; 7419 7592 7706 174 250 3979 40 8114 82S7' 8401 8034 iWH 8981] 9154 9328 9.501 173 .^51 9074 9847 •-»20| •1;)2 •3ij5! ^538 •711 ^883 1050 1228 173 2.:2 401401 1573 1745 11117 20S;t 2201 2433 2OO5I 2777 29491 172 253 3121 3292 34(;4 3i)35 3.SO7 3978 4149; 4320! 4492 4')r.3i 171 2.54 4834 5005 5170 5340 5517 5088 5,S5^; 0029; 0199 03? • n 255 G540 0710 0^i8l 7051 7221 73)1 7.5olj 77311 7901 8070! 170 25J 8240 8410 8579 874.) 8918 9087 9257, 9420| 9595 97o4j 109 , 257 9933 •102 •271 •440 •009 •777 •940, 1114j 1283 1451 HY) 25 i 4ii:;20 1788 1950 2124 22)3 24'il 2029j 27901 2904 31.32 108 25) 3390 3407 3035 3803 3970 4137 4305, 44721 4039 4800, 10/ 2')0 414)73 5140 5;}07 5474 5041 580:i 5974! 0141 0308 64741 107 2'-l 0041 0807 0973 7139 73)0 7472 703S 7804 7970 8135, l«»i 2.;2 8301 8i07 8(333 8798 8904 9129 9295 9400 9025 9791 105 2G3 9950 •121 •2S0 •451 •010 •781 •945 lllOl 1275 1439 105 204 421004 n.is! 1933 20)7i 2201 2420 2.59()| 2754' 2918 3082 104 2o5 3240 3410 3574 3737 3,)01 4005 4228 4392 4555 4718; 104 200 4882 5045 52)8 5371 5534 5097 5800 6023 6180 031;): 103 207 0511 0074 OS'XJ 09,r) 71(;i 7324 7480 7ii4S 7811 7973: 102 20S 8135 82971 8459 H021 8783 8944 9100 9208 9429 9.591 102 20!) 9752 9914' ••75 •230 •3)8 •559 •720 •8.U 1042 1203 161 270 431334 1525 1085 1840 2007 2107 2328 2i.-iH 2649 28091 101 271 2909 3130 3290 3450 3010 3770 31,130 4090 4249 44091 160 272 4509 4729 4888 5048 5207 5307 5526 5085 5844 6004' 159 273 G103 G322 0481 0040 G798 6957 7110 7275 7433 7592| 1.59 274 7751 7909 8007 8220 83,84 8.542 8701 8859 9017 9175 1-38 275 9333 9491 9048 9800 9904 •122 •279 •437 •594 •752 1.58 270 440'.)09 1000 1224 13S1 1538 1095 1852 2009 2106 2323 1.57 277 2480 2037 2793 2950 3ior, 3203 3419 3570; 37.32 3889! 157 1 278 4045 4201 4357 4513 40(59 4825 4981 51.37 52* »3 5449 1.36 279 X. 5004 5700 5915 0071 0220 03 -iS 0537 6092 0S4Sj 7003 1.55 D. 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' 7453' 754(! ! 7(;4() 1 77331 7820! 792( 8013' 8100 ! 8199 8293 93 4(i(i S3s<l, K47.) , 8572 i 8005 1 8759' 8852 8945i 9038 9131 9224 93 4(17 1i317 0410 : 0503 J 050(i : y(;8o: 97-i2| 9875i 9907 ••oo •153 93 1 4.;s 0702 hi 0:!3) (1431 i 0524 \ 0017 0710 0802; 0895 ' 0988 1080 93 4;;') ' 117;j 12-;.) 13>S 1451 1 1.543' !('3( 1728 1.S21 1 1913 1 2005 93 47') C72<l.H 2100 22H3 2375 : 2407 2500 2052, 2744 ' 283(5 ! 2929 92 471 3021 I 3113 3205 ; 32)7 j 33,)0i 3182 3574: 3(!(>(; 375>^ 3S50 92 i 472 3012 j 4u3i 4120 : 4218 43101 4402 44941 45H(; 4077 ; 4709 92 ^ 1 . > 4S01 4)53 5015 5137 52281 5320' 5412! 5503 5505 ;)087 ! 92 474 : 577h 5S70 ; 5j(i2 i 0053 1 01451 02301 0328 WIO 0511 1 OOOi, 92 •IT.-i Oi!04 ' 07S5 ' 0S7O (i')lo : 7059i 7151; 7242 7333 7424 ' 7510 91 47fi 7007 ' 7(!08: 7781) i 78S,.: 7;)72 1 80<)3 81541 8245 ' 8330 1 84:i7 91 477 1 851 ■< HOOO 8700 1 8701 8882 i 8973 90041 9155 ; 9240 9337 91 47S 1 i)42'^ 0510 0010 • 0700 9791 : 98821 9973 ••03 •154 •245 91 47!t ' G8033,; 0420 0517 0007 0098! 0789! 0879 0970' 1000 1151 91 4H(» GS1241 1332 1422 1513 1003: 1093' 1784! 1874 i 1904 liOoo 90 481 2145 2235 2320 241G 2500 1 2590i 20801 2777 , 28(57 2,»57 90 4S2 3047 3137 3227 3317 3407 ! 34!)7 ! 3587! 3677 3707 3--)5/ 90 4S3 3047 4037 4127 4217 4307 , 4390 4*80' 4576 4066 4750 90 4Hi 4S45 i'.m 5025 5114 5204 ' 5294 5383] 5473 5503 5052 90 4sr> 5742 5831 5)21 6010 6100 6189 627!) 6368 6458 0547 89 4-i(; 0C3o G72G 0815 6',M)4 6994 70831 7172 7201 7351 74 iu 89 4S7 752'J 7018 77t)7 7700 7880 7975| 8004 8153 8242 8331 89 4S8 8420 8500 85)8 80,i7 877G 88G5j 89531 9042 9131 \)2-M 89 4S}> 'J3»J 03!)8 04 S6 9575 9;)04 9753 98411 9930 ••19 •107 89 4'»() G'JOIJO 0285 0373 0402 0550 0039! 0728! 081G 0905 0993 89 4»l 1081 1170 125S 1317 1435 1524 1012; 1700 1789 1877 88 412 I'JOo 2053 2142 2230 2318 2400 2494! 2583 2(571 2759 88 4' 13 2847 2;»35 3023 3111 3199 3287 3375 3403 3551 3039 88 4)4 37::7 3S15 3,)()3 3J91 4078 41601 4254 4342 4430 4517 88 4'.)J 4005 4003 4781 4808 4)50 5044i 5131 5219 5307 5394 88 i)C, 54S2 55G0 5G57 5744 5832 5919! 6007 (5094 6182 0209 87 4')7 0350 0444 0531 GG18 0700 0703; 68801 0908 7055 7142 87 4)S 722) 7317 7401 7401 7578 7605 77521 7K34 7926 8014 87 4'.M) 8101 8188 8275 8302 8449 8535 80221 8709 8796 8883 87 500 G38'J70 0057 9144 0231 9317 9404 9491' 9578 9664 9751 87 501 ' {)S3.S 0024 ••11 ••:)8 •184 •271 •;«8 •444 •531 •017 87 502 700704 0700 08771 0003 1050 1136 1222 1309 1395 1482 86 503 ioi)8 1054 1741, 1S27 1913 199;) 2086 2172 2258 2344 86 504 2431 2517 2003' 20S9 2775 2S(il 2947 3033 3119 3205 86 505 3201 ;,377 3403 : 3510 3035 3721 3.S07 3893 3979 4005 86 50(5 4151 4230 4322; 4408 4494 4579 4005 4751 4837 4922 8u 507 500S- 50;m 5170! 5205 5350 543j 5522 5007 5693 5778 86 508 5804! 5040 1 0035: 0120 (1206 62i)l 0370 (5402: 6547 G032 85 501) 0718 0803' 08'H! 0074 7059 7144 7229 7315 7400 7485 85 510 707570' 7<)55 7740 1 7826: 7911 7996 8081 81(50i 8251 8336 85 511 84211 85001 85011 8070 8701 8846 8931 90151 i)100 9185 85 512 0270 0355 0440 0524 9000 i)(i!)4 9779 9803 9948 ••33 85 513 710117 0202 0287 0371 0450 0540, 0(;25 0710 0794 0879 85 514 0;)03: 104HJ 1132 1217 1301 1385 1470 1554 1639! 1723 84 515 1807 1 1802| 1070 20(i0 2144 2229 2313 2397 2481 2566 94 516 2G50 2734 2818 2002 :j8o 3070 3154 3238 3323 3407 84 517 3491 3575 3C59 3742 3826 3910 3994 4078 41(52 4246 84 518 4330 4414 4497 4581 40(i5 474'yl 4833 4910 50(50 5084 84 510 N. 51(l7j 5251 1 5335 2 5418 •J 55021 558G 50(59 5753 7 5836 8 5920 9 84 i 4i 5 6 D. A TABLK OK L(K;AKITUMS KliOM 1 TO 10,000, m\ 94 m\ 94 W7 94 1241 94 3lHtl 293 224 153 080 005 1929 ;-i50 17()'J 94 93 93 93 93 93 92 92 92 )i>S7 1 92 J510 J337 ►245 1151 :i055 3S5/ 476oi 0517 1 74 iu «331 92-iU •107 0993 1«77 2759 3o39 ■1517 5394 10209 7142 8014 18883 J751 »017 11482 12344 1005 1922 778 i)032 7485 ii33G il85 1)879 l723 [!5CG 107 92 91 91 91 91 91 90 90 90 90 90 89 89 89 89 89 89 88 88 88 88 88 87 87 87 87 87 87 86 86 86 86 8G 86 85 85 85 85 85 85 84 94 84 246 84 084 84 920 84 9 D. 520 521 522 523 524 525 520 527 528 529 530 531 532 533 53i 535 530 537 538 539 5i0 541 iA2 543 544 545 546 547 548 519 550 551 552 553 554 555 550 557 55!} 559 5u0 5G1 5G2 503 501 565 506 rv;:7 508 509 570 571 572 573 574 575 576 577 578 579 X. n 710003 683S 7671 8502 9331 720159 09S6 1811 2634 3456 7242761 5095 5912 0727 7541 8354 9165 9974 730782 1589 732394 3197 3999 4800 5599 6397 7193 7987 8781 9572 740303 1152 1939 2725 3510 4293 5075 5855 0034 7412 748188 8903 9730 7505081 1279 2018 2816 3583 4348 5112 755875 (J636 7396 8155 8912 9668 760422 1176 1928 ^2679 1 ! 2 I 3 0087 0170 /'(i2.">4 69211 7004i 7088 7754! 78371 7920 8585 9414 0242 1008 80081 8751 9497 95s() 0325 1151 1893 1975 2716 2798 3538! 3020 4358; 4440 5170 5258 5993 6809 7023 8435 9246 •©55 0803 16G9 2474 3278 4079 4880 5079 6470 7272 8007 8860 9651 0442 1230 2C18 2804 3588 4371 5153 5933 0712 7439 0407 1233 2058 28S1 3702 5310 6075| 0150 6890 6972 7704 7785 8510 8597 9327 9408 •136 •217 0944 1024 1750 1830 2555 2035 3358 3438 4100 4240 4900 5040 5759 5838 6556 60:i5 7352 7431 8146 8225 8939 9018 9731 9810 0521 0600 1309 1388 209G 2175 2882 2901 3007 3745 4449 4528 5231 5309 GOll 6089 G790 6868 7507 7045 8266 1 8313 8421 90401 '.'118 919: 9814 '.i^'M 9908 05S0 0003! 0740 13')0i 1433 2125J 2502 2893' 2970 3000' 3736 44251 4501 5189 5951 6712 7472 8230 8988 9743 0498 1251 2003 5205 6027 6788 7548 8306 9063 9819 0573 1320 2078 2754| 2829 1 i 2 1510 2279 3047 3813 *578 5341 6103 6864 7624 83S2 9139 9894 0049 1402 2153 2904 tj33V 7171 8003 8834 9003 0490 1310 2140 2903 3784 4004 5422 6238 7033 7800 8<)7a 9489 •2.)8 1105 1911 2715 3518 4320 5120 5918 6715 7511 8305 9097 9889 0078 1407 2254 3039 3823 4000 5387 61G7 6945 7722 84i)8 9272 ••45 0817 1587 2356 3123 3889 4054 5^il7 G180 6940 770C 8458 9214 9970 0724 1477 222H 29'; 6421 7254 80M() 8917 9745 0573 1398 2222 3045 3800 4085 5503 632(t 7134 7948 8759 9570 •378 118G 1991 2790 3598 4400 5200 5998 6795 7590 8384 9177 990« 0757 1546 2332 3118 3902 4084 5405 6245 7023 7800 857< 9350 •123 0894 ir>04 2433 3200 3900 4730 5494 6256 7010 7775 8533 92'.tO ••45 0799 1552 230 3f*n 6 0501 7331 810',) 9.')0(» 9828 0055 14S1 23(15 3127 3948 4707 5585 0401 721l> 8029 8841 9051 •459 1266 2072 2870 3079 4480 5279 6078 6874 7670 84G3 9256 •«>47 08;>(; 1624 2411 319(( 3980 4702 5543 6323 7101 7878 8053 9427 •200 0971 1741 2509 3277 4042 4S07 5570 6332 05S,S 74-1 8253 90 S3 9911 0738 1503 2387 3209 4030 484; > 50(i7 (;483 7297 8110 8i)22 9732 •540 1347 2152 2950 3759 4500 5359 6157 6954 7749 8543 9835 •120 0915 1703 2489 3275 4058 4840 5021 6401 7179 7955 8731 9604 •277 1048 18181 258<' 3353 4119 4883 504G 6408 70921 7108 8 9 007llli754i 7504! 7587 8330 84191 91()5 92181 9i).)4i ^^77 082 l' 0903 ]040; 1728 2409: 2552 3291! 3374 41121 4194 49311 50131 57481 5830 6()40! 7400 7851 8009 93(;6 •121 OS75 ic:/ 23'u. 31^. 7927 80S5 9441 *»l9(ii il!»50 702 2453 32oa 6504 7379 8191 9003 9813 •021 1428 2233 3037 3839 4640 5439 6237 7034 7829 8022 9414 •205 09<)4 1782 2508 3353 4136 4919 56'.)9 6479 72r.6 8033 8808 9582 •354 1125 ■'95 .■..3 3» ■■) 4195! 4900 5722 6484 7244 8003 8701 9517 •272 1025 1778 2529 3278 8 H2731 9084! 98931 •7021 1508 2313 3117 3919 4720 "519 6317 7113 7908 8701 9493 •284 1073 1860 2647 M31i 4215i 4997' 5777 0556 7334 81.10 8885 96591 •431 1202i 1972 2740 ! 350(5 Us 83 83 83 83 83 82 82 82 82 82 82 82 81 81 81 81 81 81 81 80 80 80 80 80 80 79 79 79 79 79 79 79 78 78 78 78 78 78 78 77 77 77 77 77 77 77 4272 V7 5036 76 5799 76 6560 76 7320 76 8079 76 8836 76 9592 76 •347 75 1101 75 1853 75 2604 75 3353 75 9 D. 'It.. ]: ■■■Si iki MW •i i 1 'I' si 10 A TAHI.K ol-" I.O.SAli'.T.l.M.S Kl{( .'I 1 TO \'l.'1()Ci. !!l X. 1 i" ! 2 i 3 1 4 I 5 1 \7 \ 8 1 9 .)M) 1 7G3;(:28; 3503; 3578i 3053 3727 38(»2 3877; 3052: 4027; 4101 1 ryn 1 417G 1251 43211 Aim 4175 4550 4024 10')!! 4774 4848' 5S'J 4023 4008, .50721 5147 5221 52'.K) 5370 5445 5520i 55J)4; 5>s;} 5GG0 5743 5818 5802 50;i6 G041 G115 (ilOO G264i 6338 584 0413 G487 G5G2 6636 6710; G785: G850 G033 7007! 7082! 5S5 715G 7230 7304 7370! 7453 7527' 76011 7675 7740i 7823, 58(1 7808 7072 8046 81201 8104 8268! 8342 84161 84001 85G4, 587 1 Hfl:{8 8712' 878(ll 88G0I 803 1 0008: 0082 01561 0230 0303' 588 0377 04511 0525J 0500! 0G73 0746' 0820' OSO4! 0068! •»42 5H.) 770115 01801 0263' 0336 0410 0484 0557 0631 0705 0778 5;)() 770852 00261 000!) 1073 1146 1220 12031 13(]7! 1440i 1514! 5i>l 1587 1(5611 1734 1808 1881 li»55l 2028! 2102 2175! 2248! rm 23'^2 23!>5i 2468 2542^ 2(115 26881 2762 2835' 2008' 2081' rm 305-) 3328 3201; 3274' 3348i 3421! 3404i 3567! 3640 3713' 5'Jt 37.s(; 3^60 30331 4006 40701 4152! 4225 4208' 43';i; 4444 5!)') 4517 45':tO'4603j 4736: 4800! 4882 4055| 5028: 5100' 51731 5;>(( 524(i 5310 53021 5465' 5538 5610i 5683; 5756 5820| 5002 5!)7 5074 6(147 6J20. G103 G265; G338i G411I G483 G556i GG20 5')8 6701 (>774 G84G1 iY.n'.r. 6OO2! 70641 7137, 7200i '7282 7354' 5;);) 7427 74:»0 75:2: 7644' 77171 77S0' 7862! 7034' 8006' 807!)! Gitn 778151 8224 8211(1^ 83681 8441 8513i 8585' 8(158i 8730 8802i GUI 8874 801:7 '.>l)10 0001 0163, 02361 0308} 0380 0452, 0524; (m 050(1 iXKlO 0741 0813 0885: 0057 ••20 •lOl' •173' •245i «j();« 780:)17' ()3S0| 0461 0533 0605; 0677 0740 0.S21 0803 00G5 mi 1037 1100' 1181 1253! 1324; 1306 1468 1540' 1612 1684' 005 1755 1827 180!) 1071! 2(J42; 2114 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 2G 61(5010 4.63 959253 .90 657304 5.59 34263G 34 27 61G894 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 .90 658369 5.58 341G31 31 30 617727 4.62 959023 .90 658704 6.58 341290 30 31 9.618004 4.61 9.958905 .00 9.050039 5.58 10.340961 29 32 618201 4.61 953908 .90 650373 6,57 340027 28 33 618558 4 61 958850 .90 659708 6.57 340292 27 34 618834 4.60 958792 .90 6G0042 5.57 339958 20 35 619110 4.60 958734 .90 600370 5.57 339624 25 3G G10386 4.60 958077 .96 6G0710 6.56 339290 24 37 G196G2 4.59 958019 .96 661043 5.56 338957 23 38 619938 4.59 958501 .96 6G1377 6.56 338623 22 30 620213 4.59 953503 .97 601710 .6.65 838290 21 40 620488 4.58 953445 .97 602043 5.55 337957 20 41 0.6207G3 4.58 9.958387 .97 9.GG237G 6.65 10.337624 19 42 621038 4.57 958329 .97 662709 5.54 837291 18 43 621313 4.57 958271 .97 603042 6.64 830958 17 44 621587 4.57 958213 .97 663375 6.54 336025 16 45 621861 4.50 953154 .97 6G3707 5.54 333293 15 4G 622135 4.5G 95300G .07 6G4039 5 53 335901 14 47 622409 4.50 953038 .97 604371 5.53 335629 13 48 622682 4.55 957979 .97 664703 6.53 835297 12 49 G'2205G 4.55 957921 .07 605035 6.53 3349G5 11 60 62322i] 4.55 9578G3 .07 605300 6.52 334034 10 51 9.623502 4.54 9.957804 .07 9.005097 5.52 10.334303 9 52 623771 4.54 95774G .08 600029 6 52 333071 8 53 624047 4.54 9570S7 .98 6G03G0 5.51 833G40 7 64 62431! 4.53 957028 .98 600601 5.51 333300 ; 6 55 624591 4.53 957570 .98 667021 5,51 332070 5 5G 624863 4.53 957511 .98 607352 5.51 332G48 ,- 4 57 625135 4.52 957452 .98 6G7G82 6.50 332318 : 3 68 62540i 4.52 95739J] .98 Gmnfi 5.50 3:31087 i 2 69 62567'; 4.52 957335 .98 608319 6.50 331057 1 GO 62594; i Uosino. 4.51 957270; .9^ 60Sl)72 5.50 D. 331328 1 Siuo. jJL^l Cotang. Tang. IM. 23 22 21 i20 119 18 17 IG 15 14 113 Il2 h^ ho SINKS AND TANGENTS. (25 DEGREES.) 43 M. Sine. D. 1 4.51 Cosine. 9.957276 D. .98 Tang. I D. 5.50 Cotang. 1 1 10.331327 60 a. 625948 9.668673; 1 626219 4.51 957217 .98 669002: 5.49 330998 59 2 626490 4.51 95715S .98 609332 5.49 330668, 58 3 626760 4.50 957099 .98 6uJ661 5.49 330339, 57 4 627030 4.50 957040 .98 669991 5.48 330009i 56 5 627300 4.50 956981 .98 670320 5.48 329680' 55 6 627570 4.49 956921 .99 670649 5.48 329351! 54 7 6278401 4.49 956862 .99 670977 5.48 329023 53 8 628109; 4.49 956S03 .99 071306 5.47 328694 52 9 62837H| 4.48 956744 .99 671634 5.47 3283()») 51 10 628647 4.48 956684 .99 671963 5.47 328037 1 50 11 9.628916 4.47 9.956625 .99: 9,672291 5.47 10. 327709 i 49 12 629185 4.47 956566 .99 672619 5.46 327381, 48 13 629453 4.87 956500 .99 672947 5.46 327053 47 U 629721 4.46 956447 .99 673274 5.46 326726: 46 15 62998!) 4.46 956387 .99 673602 5.46 326398 45 16 630257 4.46 956327 .99 673929 5.45 326071 44 17 630524 4.46 95626S .99 674257 5.45 325743 43 18 630792 4.45 956208 1.00 674584 5.45 325416 42 19 631059 4.45 956148 1.00 674910 5.44 325090 41 20 631326 4.45 956089 1.00 675237 5.44 324763 40 21 9.631593 4.44 9.956029 1.00 9.675564 5.44 10.324436 39 22 631859 4.44 955969 1.00 675890 5.44 324110 38 1 23 632125 4.44 955909 1.00 676216 5.43 323784 37 21 632392 4.43 955849 1.00 676543 5.43 323457 36 25 632658 4.43 95578911.00 67(i869 5.43 323131 35 26 632923 4.43 955729,1.00 677194 5.43 322806 34 27 633189 4.42 955669 1.00 677520 5.42 322480 33 •-^8 633454 4.42 955009 1.00 677840 5.42 322154 32 29 633719 4.42 953548 1.00 678171 5.42 321829 31 30 633984 i.41 955488 1.00 678496 5.42 321504 30 31 9.634249 4.41 9.055428 1.01 9.678821 5.41 10.321179 29 32 634514 4.40 955368 1.01 679146 5.41 320854 28 33 634778 4.40 955307 1.01 679471 5.41 320529 27 34 635042 4.40 955247 1.01 679795 5.41 320205 26 35 635306 4.39 955186 1.01 680120 5.40 319880 25 36 635570 4.39 955126 1.01 680444 5.40 819556 24 37 635834 4.39 955065 1.01 680768 5.40 319232 23 38 636097 4.38 955005 1.01 631092 5.40 318908 22 39 636360 4.38 954944 1.01 681416 5.39 318584 21 40 636623 4.38 954883 1.01 681740 5.39 318260 20 41 9.636886 4.37 9.954823 1.01 9.682063 5.39 10.317937 19 42 637148 4.37 954762 1.01 682387 5.39 317613 18 43 637411 4.37 954701 1.01 682710 5.38 317290 17 44 637673 4.37 954640 1.01 683033 5.38 316iH)7 16 45 637935 4.36 954579 1.01 683356 5.38 8] 6644 15 46 638197 4.36' 954518 1.02 683679 5.38 816321 14 47 638458 4.36 954457 1.02 684001 5.37 815999; 13 48 638720 4.35 95439(5 1.02 684324 5.37 315076 12 49 638981 4.35 95433)11.02 684646 5.37 315354 11 50 639242 4.35 954274 1.02 684968 5.37 315032 10 51 9.639503 4.34 9.954213 1.02 9.685290 5.36 10.314710 9 52 639764 4.34 954152 1.02 685612 5,36 314388 8 53 640024 4.34 954090 1.02 685934 5.36 314066 7 54 64U284 4.33 9;)4029 1.02 686255 5.36 313745 6 55 640544 4.33 953968 1.02 686577 5.35 313423 5 56 640804 4.33 953900 1.02 686898 5.35 313102 4 57 641004 4.32 953S45 1.02 687219 5.35 312781 3 53 641324 4.32 953783 1,02 687540 5.35 312460 2 59 641584 4 32 953722 1.03 687H61 5.34 3121391 1 60 641842 Cosine. 4.31 D. 953060 1.03 688182 5.34 311818 Sine. 64^ 1 Cotang. D. Taug. M. ii -I ,t w 23 1 1 44 (2G DEGREES.) A TAIJLB OP LOGARITHMIC M.| Sine. D. Cosine. D. \ Taiii,-. D. Cotang. 1 ' J.r)41.S42 4.31 9.953GGU i.t.3 9.688182 5.34 10.311818 GO 1 G42101 4.31 953599 1.03 688502 5.34 311498 59 ! 2 042300 4.31 953537 1.03 688823 5.34 311177 58 1 3 G42»;i8 4.30 953475 1.03 G89113 5.33 310857 57 i 4 G42H77 4.30 953413 1.03 6894(13 5.33 310537 56 5 G43135 4.30 953352 1.03 689783 5.33 310217 55 ^ G G43393 4.30 953290 1.03 690103 5.33 3098!)7 54 7 G43G50 4.29 953228 1.03 690423 5.33 309577 53 j 8 G43908 4.29 9531G(; 1.03 690742 5.32 30925H 52 i ■■ 9 G441G5 4.29 953104 1.03 691062 5.32 308C3S 51 1 10 644423 4. 28 953042 1.03 691381 5.32 308019 50 11 9.644GH0 4.28 9.952980 1.04 9.691700 5.31 10.308300 49 1 12 644930 4.28 952918 1.04 692019 5.31 307981 48 i' 13 645193 4.27 . 952855 1.04 C92338 5.31 3076G2 47 14 G45450 4.27 9.52793 1.04 G9265G 5.31 307344 46 15 G4570G 4.27 952731 1 04 692975 5.31 307025 45 16 G459G2 4.26 952GC):) 1 04 693293 5.30 300707 44 17 64G21« 4.2G 95260G 1 04 693612 5.30 300388 43 18 G4G474 4. 26 952544 1 04 G93930 5.30 300070 42 IS) 64G729 4.25 952481 1.04 694248 5.30 305752 41 20 64G984 4.25 952419 1.04 G945GG 5.29 305434 40 21 9.G47240 4.25 9.95235G 1.04 9.694883 5.29 10.305117 30 22 G47494 4.24 952294 1.04 695201 5.29 304799 38 23 647749 4.24 952231 1.04 695518 5.29 304482 37 24 648004 4.24 952168 1.05 695836 5.29 304164 3G 25 648258 4.24 952106 1 05 696153 5.28 303847 35 26 648512 4.23 952043 1.05 696170 5.28 803530 34 27 6487GG 4.23 951980 1.05 696787 5.28 303213 33 28 649020 4.23 951917 1 05 697103 5.28 302897 82 1 29 649274 4.22 951854 1.05 697420 5.27 302580 31 1 30 649527 4.22 951791 1 05 697736 5.27 302264 30 t, 31 9.649781 4.22 9.9.51728 1.05 9.698053 5.27 10.301947 29 11 ' 32 650034 4.22 951665 1.05 698369 5.27 301631 28 1'; 33 650287 4.21 951G02 1.05 698685 5.26 301315 27 i ' 34 650539 4.21 951539 1.05 699001 5.26 800999 26 i 35 650792 4.21 951476 1.05 699316 5.26 800684 25 36 651044 4.20 951412 1.05 699632 5.26 300368 24 87 651297 4.20 951349 1.06 , 699947 5.26 800053 23 38 651.549 4.20 951286 1.06 700263 5.25 299737 22 39 651800 4.19 951222 1.06 700578 5.25 299422 21 ^1 40 652052 4.19 951159 1.06 700893 5.25 299107 20 i 41 9.652304 4.19 9.951096 1.06 9.701208 5.24 10.298792 19 1 42 652555 4.18 951032 1.06 701523 5.24 298477 18 I t 43 652806 4.18 950968 1.06 701837 5.24 298163 17 1 44 653057 4.18 950905 1.06 702152 5.24 297848 16 ' 45 653308 4.18 950841 1.06 702466 5.24 297534 15 46 653558 4.17 950778 1.06 702780 5.23 297220 14 . 47 653808 4.17 950714 1.06 703095 5.23 296905 13 48 654059 4.17 950650 1.06 703409 5.23 296591 12 49 654309 4.16 950586 1.06 703723 6.23 296277 11 50 654558 4.16 950522 1.07 704036 6.22 295964 10 51 9.654808 4.16 9.950458 1.07 9.70435C 6.22 10.295650 9 62 655058 4.16 950394 1.07 70466S 5.22 295337 8 53 655307 4.15 950330 1.07 704977 5.22 29502S 7 ; 54 65555(! 4.15 950266 1.07 70529C 5.22 294710 6 55 655805 4.15 950202 1.07 70560S 5.21 294397 5 56 656054 i 4.14 95013tJ 1.07 70591f 5.21 294084 4 67 656302 ! 4.14 950074 ,1.07 ' 70622e 5.21 293772 3 68 65655] 4.14 95001C |1.07 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