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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http: //books .google .com/I mr^ .! »t.'. 7f .• . ,.-.1 :.. *f ,< '*> >- ' ." »i w Wf^m s ^^ iffi^' Ep- '^^ ^-^ «Sl^'S^H*^ r,"5^"^v%wi iS '•^*li iiSrp^'7> ^ ^^^in^bdEwS^ ^W WJ>3-- iiiiiiiiiniinniiimiiiiiniyiimnn|ff THE GIFT OF Prof .William H.Eutta niiiiiiii!i:iitiiiiiiniiiiiiii: fA -- ■ - ! 7 i -* t» v' - ' 1 ^ A ,i • r* I H i ... if AN EASY INTRODUCTION TO THE MATHEMATICS; IN WHICH THE THEORY AND PRACTICE ARE LAID DOWN AND FAMILIARLY EXPLAINED. To each subject are prefixed, A BRIEF POPULAR HISTORY OF ITS RISE AND PROGRESS, CONCISE MEMOIRS OF NOTED MATHEMATICAL AUTHORS ANCIENT AND MODERN, AND SOME ACCOUNT OF THEIR WORKS. The whole forming A COMPLETE AND EASY SYSTEM or ELEMENTARY INSTRUCTION IN THE LEADING BRANCHES OF THE MATHEMATICS; DESIGNED TO FURNISH STUDENTS WITH THE MEANS OF ACQUIRING CONSIDERABLE PROFICIENCY^ WITHOUT THE NECESSITY OF VERBAL ASSISTANCE. Adapted to the use of SCHOOLS, JUNIOR STUDENTS AT THE UNIVERSITIES, AND PRIVATE LEARNERS, B8FECIALLT THOSE WHO STUDY WITHOUT A TUTOR. IN TWO VOLUMES. BY CHARLES BUTLER. n^PT irh S|f t^m, Uv ftii r«y ^m fiuyy ns, h ir^rt r»t$ ^nrwfumt. «»IARMnN. Shake off your ease, and send your name rJfA tv To immortality and fame, / "^^CLx^l^ By ev'ry hour that flies. Watts. I O^st^j^y, ^^ VOL. II. OXFORD: PRINTED BY BARTLETT AND NEWMAN; AND SOLD BT LONGMAN, HURST, RE£S, ORME, AND BROWN, PATERNOSTER ROWj LONDON; PARKER, OXFORD; AND DEIOHTON, CAMBRIDGE, 18H. ^, o (a^^u^. i*/-^^-- /4./:iint?- CONTENTS. ALGEBRA. PAQB GsNERAL Problems. Their Nature and Properties explained 1 Method of registering the Steps of an Operation . . 17 AaiTUMETICAL PROGRESSION. Its Rules A^ebraically investigated 36 and applied . . 39 Problems exercising Arithmetical Progression . . 40 Permutations 49 Combinations 43 Simple Interest, its Rules invest^ted and applied • . 45 '^ Discount^ its Rules investigated and applied 48 ^i Tbe Doctrine of Ratios ' • * . ^ ^ Continued Fractions .••«•• 58 Proportion^ Direct • • . . 62 /V3 Inverse^ or Reciprocal Proportion 69 \ Harmonical Proportion . r 70 ^ Contra-harmpnical Proportion 73 t Comparison of variable and dbfenobnt Quantities . 74 ^ Geometrical Progression. Its Rules investigated 89 and applied 87 Problems in GeometricaLProgression ..... 89 Compound Interest^ its Rules investigated and applied . 91 Properties of Numbers^ an Investigation of those ivhich * are most generally useful . 93 SauATioNs of several Dimension?* • A general View of the Nature, Formation^ Roots, &c. of Equations Ill Generation of tbe higher Equations . . , « .113 Depression of Equations 117 Transformation of Equations • 118 To find the Limits of the Roots ^ i^e aS iv CONTENTS. « PAOB To find the possible Roots of an Equation . • . 129 By Newton*8 Method of Divisors 132 Recurring Equations 134 Cubic EctUATioNS, Cardan's Rule 138 BiauADRATic EauATioNs, Des Cartes' Rule 143 Euler'sRule 146 Simpson's Rule 147 Afpboximation. To revolve Equations by the simplest Method . .150 By Simpson's Rule 153 By Bernoulli's Rule 155 Exponential Equations 159 • Dr. Button's Rule for extracting the Roots of Num- bers by Approximation 162 Problems producing Equations of three or morb Dimensions » . . . 163 Indeterminate Analysis 165 Solution of Indeterminate Problems 173 Diophantine Problems 176 Infinite Series^ their Nature, &c 181 To reduce Fractions to Infinite Series 182 To reduce compound quadratic Surds to Infinite Series 184 Newton's Binomial Theorem 185 To find the Orders of Diffiirences 190 To find any Term of a Series 191 To interpolate a Series 199 To revert a Series 195 To find the Sum of a Series 197 The Investigation and Construction of Logarithms, both hyperbolical and common 1204 GEOMETRt. • Historical Introduction .'211 On the Usefulness of Geometry . . . .- . . .241 Description of Mathematical Instruments . . . 242 Of Geometry considered as the Science of Demon- stration, "with some Account of the Principles of Reasoning, as introductory to the Study of Budid 250 Observations on some Farts of the first Book of £uclid*s Elements %S9 CONTENTS. V PAOB On Euclid's second Book 291 On Euclid's third Book 297 On Euclid's fourth Book 301 On Euclid's fifth Book 304 On Euclid's sixth Book 308 An Appendix to the above six Books of Euclid . .314 Pbactical Geometry^ exemplifying and applying Euclid's Theory; the Use of the Mathematical Instru- ments, &c 327 Methods of constructing Scales of equal Parts . . 343 To construct Scales of Chords, Sines^ Tangents, Se- cants, &c 344 The Mensuration of a great variety of plane and solid Figures, Land, Planks, Timber, Stone, &c. 346 TfilGONOMBTRY. Historical Introduction 359 On the new (French) Division of the Quadrant (note) 367 Definitions and Principles of Plane Trigonometry 371 Variation of the Algebraic Signs 375 Introductory Propositions 380 Investigation of Formula 389 Method of constructing Tables of natural Sines, Tangents, &c 394 Method of constructing Tables of Logarithmical Sines, Tangents, &c 399 The fundamental Theorems of Plane Trigonometry 400 Solution of right angled Triangles 407 Solution of oblique angled Triangles 413 Mensuration of inaccessible Heights and Distances 423 Description of the Quadrant ibid. Theodolite 426 Mariner^s Compass ibid. Perambulator 429 Guntei's Chain ibid. Measuring Tapes, Rod, &c. . . 430 Problems . . . ' ibid. Conic Sections. ' Historical Introduction * . . .441 The Parabola . 417 The Ellipse 458 The Hyperbola 478 List of Subscribers 501 ERRATA. 7 To the note at the bottom of the page add> '' The sign *.* denotes therefore** 18 Last line^ for ss^Ae di fferenc e, read d^the difference. 44 Line 8^ for n— 1 read n— 1. Line 20, for n.n — l.n— 2.n— 3.n— 4^ read n.n— l.n»9^— 3.fi— 4. 64 Line 3, for Fo since, read For etnce ad, ^6 Art. 68, after the word convbrtxndo^ add, Euclid pr. £. . Books, 71 The note at the bottom is useless here> as it occurs in the latter part of T?ie Froperties of Numbers, pp, 108, 109* 97 Last line, for«6=1.9> read &-f l.g. 123 Dele the third and five following lines. ^52 Art. 15. line 9, dele *< or simple:* 320 The three lines JG, BD, and EC in the figure, should intersect in the point Fon the circumference. Two or three of the figures in Part X. are very indifferently * cut^ but it is hoped that there is nothing which can possibly mislead> or affect the demonstrations. AN EASY INTRODUCTION TO TKX MATHEMATICS, &c- PART IV. ALGEBRA. OENEfeAL PROBLEMS^ ART. 1. «/jlLGEBRA is divided into two kinds^ numeral and literal, both depending on the same principles and employing the same operations. ^» Numeral algebra ' is that chiefly used in the solution of numeral problems, in which all the given quantities are ex- pressed by numbers^ the unknown quantities only bei^g de- noted by letters or other convenient symbols. This kind of fdgebra has been largely treated of in the preceding volume. 3. Literal or specious algebra ^ is that in which all the quan- • Numeial algebra is that part of the science, which thcc Earafeaos received from the Arabs, about the siddie of the 15th cfoHiry. It doe* oot appear thai the latter people, or even Diophaotns, (who is the only Oitek writer oa the subject at present known,) nnderstood any thing of the general methods' now in use ; accordingly we find but little attempted bcyoad the solution <^ nuaie* ijcal problems, in the writibgs of liucas de Bnrgo, Cardan, Drophantus, Tar- talea, BombeUi, f^eletarios, Stevinus, Reoorde, or any other of 'the early au- thors who treated on algebra. >> Vieta, the great hnpiover of ahnMt every branch^ of the M a thiwa atics YOIi. II. B S ALOSBRA. Fakt IV. titksj both kaown and unknoim, are lepreaented by letteiB and other general ebaracten. This general mode of designation is of the greaitest use ; as efery conclusion, and indeed evety step by which it IS' obtained, becomes an universal rule Ibr performing' every possible operation of tite kind* 4. In literal algebra, the initial letten a, 6, c, d, &c. are usuaBy employed to represent known or ^ven quantities, and the final letters x, y, z, to, v, &c. to represent unloiown quantities, whose values are required to be found. 5. A general algebraic problem is that in which all the quan- tities concerned^ both known and unknown, are represented by letters or other general characters. Not only such problems as have their conditions pn^osed- in general terms, are here im* plied, every particular numeral problem may be made general, by substituting letters for the known quantities concerned in it : when this is done, the problem which was originally proposed in a particular form, is now become a general problem. 6. Every problem consists of two parts, the data, and the qtuBsita'; the data Include all the conditions and quantities given, and the qusesita the quantities sought. 7* The process by which the quaesita are obtained by means of the data, that is, by which the values of the unknown quan- tities are found, is called the analysis \ or the. analytical ■rr- known in his time, is considered as the first who introdaced the literal aota* tion of given quantities into genera! practice, about the year 1600. Cardan had indeed given specimens of such an improvement, in his algebra, as early as 1545 ; but as the advantages of a general mode of notation were thea in all probability not sulBcienUy understood, the method was not adopted wtil about the time we h«fe mentioned. The impioTement of Viet* was forthor i^vanced and applied by Thomas Harriot, the fathcar of modern algebra, abont 1620; likewise by Onghtred in 1631, Des Cartes in 1637, and afterwards by Wallis, Newton, Leibnits, the Bemoallis, Baker, Raphson, Sterling, £uler,&ie. and is Justly peilierred by all modem algebraists, on account of the universality of its application. The letters of. the alphabet are called by Vieta, tpeciesf- whence algefara has been named oritAmeiicu spedata: reasoning in species, as applied to the solotion of mathe m a t ica l problem% appears to have been bor* rowed from the Civiliaiis, who determine cases at law between imaginary per- sons, representing them abstractedly by A and 9; these tliey call •cases te a^ecUi! this is the more probable, as^^^ta hunself was a lawyer. « The MTord data means tbtngs given, and puuiia things sought. * Thawofd analysis^ (from the Greek mmOiw* c^m^i} i» lU geneval sense. Pakt IV. GENERAL VROELEMB. t iNTSSTIGATtDBTl it 18 alsO lUUned the SOI.UTI<IN>Or KIBoiUTlON of the proyem. S. When the values of the unknown qdanlhies are fbund and express^ in known termsj the subetituttng these values^ each for its respective unknown quantity in the given equations; that 18^ by reasoning in an aider the convesse of anal)sis9 and there- by ultimately proving that the quantities thus assumed have the properties described in the problem^ is called the synthesis % or SYNTHETICAL OEMOMsxaATioKof theprobliMayandfiequentlf the coirposiTioN. 9. When the value of any quantity^ which was at fifBt un« knowa> is found and expressed in known terms, the translate ittg of this value out of algebraic into oommon language, whece« In the relation of the quantities- concerned is simply declared, is called deducing a theorem ^5 but if the tianslation be exhibited in the form oi9i precept, it is called a canon <j or rulb. implies the reaolvii^ of any thing which is compounded, into its constituent si9<* pie elements : thus in algvbra, several quantities, known and unknown, being tomponnded together, analysis is the disentangling of them; by its opera- tion, each of the quantities included in the composition is disengaged from the rest, and its value found in terms of the kitown quantities concerned. This being the proper business of algebra, the science itself on that account is frequently termed analysis, which name however implies other brandies besides algetoi. ^ Synthesis (from the Greek rvy^irif, compotUia) is the converse of analysis. By analysis, as we hate shewn, compound quantUies are decompounded ; hj synfliesis, the quantities disentangled and brought out by the analysis, are again compounded, by which op^iM^oo the original compoijnd quantity it re- produced ; hence synthesis is colkd. (Ae method of dgmtmniraiunt^ mni analgia the metifid ef investtgifUiQH,. <* A theorem (from the Greek ^t^fftifMh a epecukuioni) .is a proposittoa ter* minatittg in theory, in which something is simply itiSrmed or denied. Theorems, as we have observed before, are. initestigated or discovered by anaJ^sis^ and their truth demonstrated by syntbesi«« ^ s A caaoa (froin the Greek »mmf) cf role (from the Latia nguim) is •• system of precepts difectiog^wiiat operations mu^ be perfoimed^ in ordea ta, produce any pr<^osed result^ such^ are the rules of eonmon arithmetiq* U is . noticed f^bove, that a theorem, ^d a canon, are of nearly the same iiaport, . differing only in the form-of words in which tl^ey are laid down ; the distiae* tion may appear trifling, but it is observed by writer|> whose skiU and judg- ment are nn^estiviii^i^^x and on that iiccoant we tb$nght pioper aot tirdy to omit it. b2 4 ALQESSA. Part IV VO. A coKoirLABT ** 10 a truth obtwrtfd intonawBiitriy, umI by the bye; an addiUooaltnith, over aod above wbat the prahl^a yipopteed to aeareh out, or prore. 11. A ftCHouvM ia a remark or eaplaDatory ofcaorvalioiiy io^ tended to illuatnite 80inetbui§^ preoediiig'. 19. To make what ha» been delivered perlbctly pfattn» to the analytical investigation of several of the following proUems^ is added the synthetical demonstration ; instances are given of de* dooir^ theorems and of deriving canons or rules from the analy-* sis ; examples are likewise proposed, where necessary, to shew the method Of applying the gehend condnsions to particular cases ; and finally, tbe manner of converting any porticukir numerical problem into a general form, and of substitttting and deriving expressions for the unknown quantities, in a great variety of ways, are shewn and explained. PROBLBM 1 '. Given the sum and difference of two magni- tudes, to find the magnitudes* Analysis. Lei x=:the greater magnitudey y^the less, i= the given sunh d=stke given difference. Then by the problem ^r-f yas*. And x— ysrrf. ^ • »-fd Whence by addition 2j;sx«4-<f, or xs 2 ^ Tbe t«nii cofollaiy ir derived from the Latin oonMty.^ometkimg' given over etnd above f and teiiolinm fiKim rx*yjm9 a ekvrt comment, ■ Sereral of the problems here given, with others of the kind, may be found in Sannderifoa'* Elemento of A4j^bm» 2 vot 4to. 1740. in the Abri%ment of the Mme, and in' Ludhun't Rudiments of Mathematics. ^ In the lechnieal bmgnage^ the mathematicians, Q. E. f. denotes, quod erai investigandom* which woe to be imiettigaied ; Q. £. D. quod erat de- Bionstimmlum, iViAicA wot to he demmatraUd ; and Q. E. F. quod emt facir toAwa^^'Wkichwa$tohed$ne* Tbe iirst is subjoined to analytical investiga- tioni, the seeottd to synthetical demonstrations, and the third to the proof t)^at a proposed ptaetical operation is actuaUy performed and done. We hare adapted the distinctions of anafyeU, tynihesis, thmremy camm, &c. and like- wise tbe above abbrtfviations in* a few instances, to assist this learner in a knowr ledge of their use, wheb any boeh eontaining the» may happen te flOl into his hands^ Pabt IV. GENERAL PROBLEMS. .5 ' -STNTHifiB. Bemwte hf ihe prMem x^^fttis, «nd iX'^t^zad, if the valuet ftmnd 6jr efte analysis he really equwalent to x ami f reepecthely, then those values being euhetituted for x and p m the gwen equations^ and the latter value added to the former in ihe fipst equation, and subtracted from it in the secomdj the results will be s and d. Let us make the expemnent ^ s-^d «— d 2^ First — - — I — -— xs— a^, .tMch atuwers the firet ixmtftfion, namely that x-^ysxs^ Seax&dUf — == — ssd^ which answers the second con^ ^2 « 3 . diijum^ namely that x^^y^es^d; wherefore the values of x and y J<mnd by the <malysis, jure those which the problem requires. TiifBQftigt^ 1. If the differenoe of any two magnitudes be «dded to their sum, half the result will be the greater magni- .titde; bnt if the difference ht Miiatracted from thQ spn, half the-reeuH will be the less. . Scholium, llie form of any general algebraic expression may be changed at pleasure, provided its value be not altered thereby : by this means ^ theorem may sometimes be laid down in a more convenient form than thai derived immediately from s-\-d the analysis. The value of x found idx>ve, viz. -——may he thue s d f^— d s d expre0ed^7;4-— j and the value of y, viz.-—-— ,thi|s, —— — : , hence we obtain the above theorem in a pioife convenient form* ■ viz. Theorem %. Half tlie differenoe of two magnitudes being added to half their sum, the result will be the greater 3 and half the difierente being subtracted "from half the sum; the re- sult will be the less. Corollary* Hence it appears, that theorenis ^^ip4 canons may be derived from uny general algebi^ic investigadQn, which will solve every perticular c£»e subject to the same conditions with the general problem^ to which that investigation belongs. Cam ON I. (From theqran 1.) Add the difference of any tinfo mUgnitudes to their sum, and divide the result Vy ^» ^^^ ^lotieat 93 6 ALGSMLL Past W. ^vffl be the greater magnitude. SuMraet the diftffwme from the mm, and divide the result by % the quotient wiH be the kas. Canon 3. (from theorem 2.) Add half the differenoe of anj two magnitudes to half their 8um« and the Tegult will be the greater magnitude. Subtract half the difference from half the sum^ and the Ksult will be the leas. SxAuPLEs.-^l. Giv^i the sum of two numbeiB 20>and their difference 12, to find the numbers. -^ 30-f 12 32 By canon 1. — - — ^ ---=16 =z the greater number, 20-12 8 ^'-— - — =r«~s43xlfte leu nmmber. 2 2 ^ 20 12 By cowan 2. ~+— =:10+6=16=«Ae greater number. 20 12 "5 — =ia— 6=r4=/^ k8B number, as before. 2. If the sum of two numbers be dl> and theur difference 14^ what are the numbers ? ^ 31 + 14 45 ^^. ,, By canon 1. — - — ^—=9fl^= the greater. 31—14 17 « ,, , — - — i=z-^=S\^the Ubs, 2 < 2> ' 31 14 By ctmon 2. -3-+— sxl5i+ 7=92^8=*^ freoesr. 2 2 y — Y=16*— 7=«8f satte Isst, «» 6^efe. S« The sum of two numbers is 16^ and their difference 6, to find the numbers ? Am. 11 and 5. 4. Given the sum 109> and the difference 51, to find the numbers ? Jne. 754^ and 244-. 5. Given the sum of two numbers 44., and their diflference I4., to find the numbers ? Jns. 244- and l^. . 6. Given the sum 123> and difference 104> to find the numbers ? Problem 2. What magnitude is that, to which a given mag- nitude being added, and from it the same given magnitude b^ng subtracted, the sum shall be to the remainder in a ^iven ratio? • Paut ly. GENERAL PROBLEMS. 7 A«rAi.7Sis. Let xssihe magmiude reqmred, a^zthe gk)€n magnitude to be added and subtracted; r and s the tern^i of the gwen nafta; then by thefirohkm, x+a ; f — a :: r : s,\' rx-^ar ar-^tig r+f 5=sj74-a*, •.• rx-^sxzszarA-aSy (tnd x= = a, the mag- nitude required \ Q. E. I. _. ar+ai ar-^as+ar^as ^ar Synthesis. First, \-ass = , r— « r — s r—s ^ „ ar-^-as ar'\'as-^ar'\-as %ae Secondly, — a= ■ " = r—s r—s r — * 2ar ^ag 2a 2a ^ ^ ^ Xr : X « : : r : f. Q, E. D. m • r— « r—s r—s Examples. — 1. What number is tbat^ which with 3 added to it^ and also subtracted from it, the sum is to t|ie remainder as 9 to 7 ? Here a^S, r=9, «=7. and a?=-i^x3=---x 3=8x3 =24. - 2. Required a number^ which being increased and 4eQreased 1 by 'T^, the sum is tQ the remainder as 3 to 1 ? tiere as=-— , r=s3, s^sl, \' x^^- — r X T-r=-:r X t:::— :r: 12 3—1 12^ 2 12 24 3. If 10 be added to, and subtracted from, a certain number, the sum will be to the remainder as 11 to 9 } what is the num* ber? Ans, 100. 4. If -^ be added to, and subtracted from^ a required number^ the results will be as 15 to 13 ; what is the number ? 1 Here it is plain if r==s, then s + a^x-^a, consequently a=o, whence any iBagnitode taken at pleasure for x will satisfy the conditions of the problem. li r y* (the qoantity —-^ a, or) the Talne of x will be affirmatii^e ; bat if r— # r^ #9 the Talne of x will be negative : in the former case the ratio is that of the greater tfi«gva/tfy, but in the latter, it is the ratio of the ietser inequmliiy, and the given problem is changed into the following ; ** To find a magnitude, from and to which a given magnitude being subtracted and added, the remainder •ball be to the sum as r to s*' B 4 S ALOBBRA* Past IV. DMgakude into two psHi in m giTeniatio. Analysis. Let aatihe given magttUude, x^aneoftheparU, then will a^Tszthe other part; also, let r and s represent the terms of the given ratio. Then 6y the problem x : a-^x :: r : s,\' sx^ar^rx, and rx , ^^ J «*" ar^as—ar as -^sxssar, •.! x= , and a-^x^za' Q. £. /. c ^ or as ar-^-as r-^-sa Synthesis. First, 1 = — - — ^= ~ — =a. r+» r+j r+# r-^-s Secondly, —— : -~— :: ar : as :i r : s. Q. E. D. ExAMFLM. — 1. Divide the number 32 into two parts> in the ratio of 9 to 7. JEferea==32, r=9,«==7, a»dx==--— -=2x9s:l«, and a 9 + 7 -JC=(-^=) 32-- 18=14. 3 2 4 2. Dinde — into two parts, in the ratio of — to — . 7 "^ 5 9 oi, ^^ ^_ 3 2 4 3 2 2 4 6 Here «=:---, ^=-^-. *="r'* «»rf J^= — X — i = — 7 5 9' 75 6^9 35 38 6 45 3 9 27 ^ 3 27 '*-4T-35^ 38 =y^ 15=153"' ''"^ "-"^^^ 7^133"= 399—189 210 30 931 931 133' 3. Divide 60 into two parts, in the ratio of 1 to 3. Ans, 15 and 45. 4. Divid€f 5 into two parts, in the ratio of 20 to 19, Problem 4. To divide a given number into two parts, such, that certain proposed multiples of the parts being taken, their sum shall equal another given number ? Analysis. Let ais^the given number to be divided, x and »=» the parts respectively, r=zthe multiplier of x, sz=the multiplier of y, and bv^the sum of the multiples of ts and y; then by the pro*- blem, x^y=za, and rx-^sy=zb. From the first of these equations, Vie ftave y^a-^x; and from the lalt^r^ ysz ; •.• a— xa; Pahi^ IV. GENERAL PROBLEMS. 9 , . b'-as ar'^aS'-b-\'as ar—b -. _ , = (a— X=:) a = a:- :. <?. E. I. r-^s r — 8 r— -« ^ ^ 6—05 ar—b ar-^as r-^$.a Synthesis. First, 1 = = =«. r— « r— # r— * r— » ^ „ 6— a» ar— 6 6r — a$r asr — bs Secondly, x r-\ X5=( 1 ■ r—s r— « r — s r— « br—bs .r—s.b , ^ ^ ^ = =) s6. 0. £. D. r— « r— 5 Examples. — 1. Let 100 be divided into^two parts> so that foor times one part beilig added to three times the other^ the sum will be 355. 6— <i# Here a=ziOO, r=4, 5=3, and 6=355: •.• x= ao r— 5 355—100x3 355—300 ^ , ar— 6 100x4—365 — = =55, and »= = =ss 4—3 1 * ^ r— » 4—3 400—355 =45. 1 2. To divide 13 into two p^rts, so that three times one part, added to five times the other^ will make 47. 47—13x5 Here a=13, r=3, 5=5, and 6=47? '.• 3-5 472:6B_ — 18_ 13x8— 4739— 47 -8_ —2 ""—2"" '^" 3—5 ■" -2 "^—2"" * 3. To divide 23 into two parts, so that the Bum of 9 times the first part, added to 7 times the second, may make 199. ^ROBLEM 5. Given the sum and quotient of two numbers, to findt them. Analysis. Let s=:the given sum, qszthe given quotienty x and yzs: the numbers required; then by the problem, x^ysss, and X ' — =9. From thejirst x^zs—y, and from the second x=^qy, •.• 5 _ , . 05 ^ „ - qy^s-^yyorqy-^-y^s, •.• y=— --,a«dx=(9y=)--^. Q,EJ. q-r^ 9 + 1 ^ 05 5 qs-\-s 0+1.5 Synthesis. First, -^ 1 =•= =2 =«. 9+1 9+1 9+1 9+1 Secondly, -?i*-^— i~=-l=n. Q, E. D. 9+1 9+1 1 10 ALGKBRA. PaktIY. ExAMPLss.— 1. The sum of two numbers is 54, and tlieif pa- tient 8, to find tbe numbers ? rr .-. « 9' 8x54 433 ^„ J ' ^ 9 + 1 8+1 9 * — -H— '^^-fi 9+1 8+1 9 2. Given the sum 3, and quotient 11, of two numbers, to find them? 33 3 3 1 Heres^S, g=ll, •.'x=j-=2-~.a»<iy= - =— . • 3. If the sum be 144, and quotient %^, what ase the nlim' bers ? Ans. 100 and 44. 4. Let the sum be 91, and quotient 65 required the numbers ? Problem 6. The sum of two numbers and the difference of their squares being g^ven, to find the numbers ? AifALTsis. JjCt sssthe given sum, b^the given difference of their squares^ x and y^the required numbers : then bjf the problem, j?+y=s, and a^'-y'^sszb. From the first, x^ss^^y; this value being substituted for x in the second, it becomes («— yl* — y«=«*— ^2 5 2«y+y*— y*=) »*— 25y=6, v 2«y=«*-.6, and ws= ■; ^s whence x:=:^(s^y=:)s — —=z — ="2^. «. E I. Synthesis. First, ^-^ — : — = — =». ' ^s ^ ^s 2s Secondly, — JL— 2 ««— fe> 54^.255^4.5. 2« 45 a 54_25a5^fc« 4,95 ^ ^ _ 4 «* 4 5* Examples* — 1. Given the sum 14, and the difierence of the squares 28, of two numbers, to find them ? "■ When Tfi "^ b, j^ will be negative, and the first given equatiMi it changed into s—y^s, bat the second remains the same ; for the sign of y* is not altered by changing the sign of y. Tbe problem by this change becomes the following ; Given the difference, and the difference of tlw sqnares, to find the numbers. See Ludlom, p. 150. FA»nr. GENERAL PROBLEMS. U Here «s=14, 6=28, •.• x= ^ ^ - =■*— =8, a«(i ysa 2x14 28 '^ 14^— 28 168_ 2 X 14 "■ 28 "" ' 2. If the sum be 4, and the difference of the squares likewise 4f what are the numbers ? Here »3x4, 6=4, •.* xx=24, yas I4.. 3. The sum is 101, and the difference of the squares 100, what are the numbers ? Problem 7* Ghren the product and quotieot of two numbers, to find the numbers ? Analysis. Let psz the given product, qssthe given quotient, X and y^the required numbers respectively ; then by the problem, X xy=p, and — =59 ;from the latter^ x=:qy ; tJus substituted for x y P P in the former, gives qy'^:=p \' y*=-^, and yc=:^-=--; •/ x=qys: 9Vj-V~ = VP9' Q'E.I. p p*<7 Synthesis. First, VP9 X v'-^-s y'i—i = ^p«=ap. Secondly, ^pq-^ v^— = ^pq X v"— == V— = q P P ^^^ssq. Q. E.D. Examples. — 1. Given the prodoct 196, and quotient 4, to find the numbers ? Here p=:196, 9=4, '.v^^^ X 4=^784=28=1:; and 196 V-4-=V49=7=y. 2. The in*oduct is — , and the qi^otient I4 ; required the numbers ? „ 55 26 5^ 42 flerep=-, 9=-, •/ x= ^3^=-^, and V^Vj^^' 3. If the product be 605, and the quotient 5, what are the anmbers? IS ALGEKU. Paxt IV. Problbm 8. Given the imn and |»nodiict of two nuaiben. to find them? AvALTHs. Let iz^the gwen tum,pssihe given product, x amd ff=zthe numbers required. Then by the problem, x-^-yszs, and xy^p;from the first jr^t— x; this vabie substituted for y in the second, it becomes sx—a/^ssp, •/ j* - sx= —p ; complete the square* and x«-»+_=--^p=^-_r, .,. x__.=: -|, ^__^=: + _ ^, ..j.= ^and jf=(«^^5=)« — =^ T 2 ; 2 2 Secondly, ^ x T"^^' Q. E. D. Examples. — 1. Given the sum 17, and product 72> to find tlie numben ? K^-. 1^ iro 17+^/289-288 17+1 ^ J5ferei=17,p=72, v j:= — — ^ = — =^^0 or 8, a»4f jf= 3 — - — =:8 or 9; whence, y 31^9, thenysiSf but ifs^szS, thenyssg. 2. If the sum be — , and product -—, what are the nurabeis ? i« o w 11 1 2 1 Here #= — ^, p= — , *= — ^. v= — . 12"^ 6 ' 3 '^ 4 3. liet the sum be 21, and product 90, required the numbers ? PaoBLEif 9. The sum of two numbers, and the sum of their squares being given, to find the numbers ? Analysis. Let s::sthe sum, a:sithe sum of the squares, x and yssthe numbers sought. Then hy the problem, x+y^s, and aj»+ y^=za; now from thefcrsty:ss^x, vy*=*a^2*r+JC«; thisvalue substituted for y« in the second equation, it becomes j«+««— 9«r+ «• stf ; that is, 2x*— 2 Jtrasa— «*, v x^-^sx^ — — , v jr«— «r+ £ _ a^l» js _ 3a--j« 4*"^ 2 ■*'4"^^ T"' Past IV. GENERAL PROBLEMS. 13 fVheri ike nquare u completed, the procea ma^ he simpiyied by substituting a more convenient expression for the known side of the equation ; thus, in the above equation, instead of , let R^ *« JR« * — ^ be substituted, and it will become a^-^sx-\ — = — : whence by evolution, x---^^±^—=z±j; v ar=(-^±~=)-|-, 2 2 2 Synthesis. First, -^t=- 4 = — =». #»HP2^«+/J« 2*«+2B« *«+il« . . r , ^ = — _ SB — — SI {by restoring the value of ^ 2 Examples. — I. Let the sum 9> andj|;i^ sum of the s^iyaret 45, be proposed, to find the nimibere ? Here sss9, a=45, then B= J^aZIil^sx ^90—81= ^9=3^ - 9-f-3 12 ^ . 9—3 6 2. Let the sum 2.25, snd tke sum of the squares 2.5625» he given. Hct-c *=:2.25, a=2.5<525^ 1?S5.25, ar=l.25, yss:!. 3. Given the sum 15, and sum of the sqoaies 137> to find the numbers ? Problem 10. Given the product, and the sum^of the squares of two numbers, to find them ? ANALYSIS. Let p^the product, asathe sum of the squares, X and yiathe required numbers •; then by the problem, xy=:p, and • Since 9a^fi»Iti, it follow*, that if ^ ^ 2a, the probtem wiU be impoisU ble ; because R^ will be negativQ inlbat case,<aiid consequentlj will btve od square root. • Let x=the greater of two numbers, y^the less, «=: their sum, S3 jdM difference, p *stke product, q «& tbe quotient, a =» the sum tf 14 ALGEBRA. Past IV. jt*+y'=a. Prow the first, y=— , z y*=^; iub^iUuie thii value for y* in <&« second^ and a*+^=:a, •.• i:*4-p*s=ar", or a?* tf* a* a* "^4 p* '-^ajfizs—p^^ •.• X*— ax*H — =( — — p«=s i-, lo/iicA 6y «<6- , a+R , p J a+U . and *=+ ^-=- ; ai$o y^{^sz) p-«.-h ^-=- i but, m order ^0 o6<ai]i the value of y in terms of a and R, toe must tuiatituie for p Us equal ^ , {which is derived from the above equa-- tf«— 4p« |i« a + R iian ss-j-,) wherefore f/^p^±^/-^» be€omes=z± Sywthesis. First, ± ^ -=^ X + V -3— = v^ — 7 — » {yoKvch by restoring the value of B^, viz, «•— 4p^)=s ^ ^ 4p« Q. £. D. £xAMrLE8.-^L If the prtiduct be ^4, and the sum of the squares 52, what are the numbers ? Here p=24, a=52, 11=^^34^^-:) 2o,*=r ^51±^=3 o^ ^ 52-20 82 2. Given the product I.32« and the siim of the sqiiarie^ 2.65, to find the numbers ? £ferep=1.32, 11=2.65, li=.23> a?=:1.2, y=:l.l. ibe sqnaresy ft =3 the differenee of tb« squares ; any UK> of thesie eight (jty y, s, dtp*qt^ 9XiA h) beiDg giTeOy the remiunios six may thence be foDod, as was lint ifaewn by Dr. Pell, in his Additions to Rhonius's Algebra, 1688. Tbefe pro- blemt ma/ be fouod wrought oat at length in ff^ar^M Yoi^ng Mathematician's Guide, 8th cdHioay London, 1724. Past IV. GENERAL FBOBLEMS. 15 3. Given tke product Uf, and the sum of the squaits ^50, to find the numbers ? Problem 11. A vintner makes a mixture of 100 gallons, with wine at 6 shillings a gallon, and wine at 10 shillings a gallon : what quantity of each sort must he put in^ so as to afford to sell the compound at 7 shillings a gallon without loss ? Analysis. Lei a=i6, 6s: 10^ sszlOO, ib=75 xssthe quajt" iitff at 6 shillings J y^the quantity at 10 shilMngs* Then by the problem, x-i-y=«, and ax'^by^ms;from ike first, x=zs^y',from ^, , ms — by ms—by the second, jt= , •.• »— 1^= , or as'^ay:=fns^by, or a-^m ay-^byz^as-^ms; that is, a — b.y:=ia^m,s, \* yt:z -.*, '.'xsi a — 6 . a— OT as—bs as-^ms ms — bs m—b a—b a^b a—b a— 6 a-^b Q. E. I. tit— 6 a— m a*— & Stnthbsis. First, r^H t-.«= ^.s=«. Likewise a^b a—b a^b m — b , a— w am—ah ab'—bm am — bm a -^b ax i-<+6x — i-«= ^-'^^ r •*= i— •*== — i a—b a—b a—b a—h a-^b a — b V . ms=ims, Q, E. D. m.'—h The above problem resolved in numbers, gives j?= 7.*= ^ XlOO= — Xl00= — X 100=75 gallons at 6 shillings s^ 6 — 10 —4 4 and tf=^5^^.*=-^ X 100=^ X 100=---x 100=25 gaZ^ons ^ a— 6 6—10 —4 4 ** al 10 sfullings a gallon, Pboblem 12. Towards the expense of building a bridge, A paid 1000/. more than B, and 2000/. more than C, and the square of A*s payment equalled the sum of the squares of the other two 3 what sum did each contribute > Analysis. Let aszlOM, then 2a=2000, also let x^zCs payment, then willx-^a=:B*s pay ment, and x4-2 a=zA*s payment; whence by the problem x-h2a)*=x+a)*4-x®; that m, x*■f4xa-f- 4a*=a?*+2aa^-a*4-x*, or 3a*=x*— 2j:a; that w, x*— 2ar= 3 a*, •/ a?*— 2flaf+a*=4a*, •/ «— a=-jh^4a*=+^^' ^""^ *=^ 3a=3000=C« share, \' jr4-a3r4a=s40b0s=:Fj share, and x-J- ^ a=5 fl=5000=if # share, Q, E, L X6 AIX3£BBA. Part IV. Stutubsis. ra\'^ss{$quare^ jripafmentzs) 4al*-fs3*=s (sum of the squares of Bs and C«=) 25 a«. Moreover At pof- ment (Sa) exceeded Es (4 a) hy a, and Cs (3 a) Ay 2 a. Q. E. D. Problbm 13. It 18 required to divide 11 into two such parts, tbat the product of their squares may be 784. Analysis, ^t aszllybzs7S4,xa^dysa Sports required s then by the problem, x-^ysza, and $f^^^b;fr(m the first, y=«— X; the square of this value substituted for y* in the second, gives a — ^^ X x*=6, whence by evtdution a— xucs ^b; that is, or— x*= a' a^ a9_^ /A ^b,orx^^ax=s^ ^b, •/ j?«-ajc-».— =(--— y'^^Z — 1^=) R^ a R^ R ^ a+R 4' • • • X- a ==—=:--_. Q. E, I. a-^R a-^-R 2a ^ a4-R Synthesis. Fwst, -^=— -f._s= — =a. rftat — « + /? 2 • 3 2 ' 2 X 2 "1 ^ 4 == 16 =(*^ restoring the value of R*=za*^4^b=s) a4_2«*-f8a»^fe+a*— 8a»v^6+16 6 165 Te ^16=*- <?^0- The solution of the problem in numbers, is x=z^-—^=: 2 a-h ^/a' —4 ^/b ll+^121-4v'784 ^ ^ a-^R ^ = -^ =7, and y=— -=4. Phoblem 14. Given the sum of two numbers 24, and the product equal thirty-five times their diflference, to find ihe numbers ? Analysis. Let x and y be the numbers required, #=s24, mi= 85 J tl^n by the problem, x+y=s, and xy==(i».i^=:) mx^my. From the first, y=s—x; this value substituted in the second, git7e«*x--r» = (wix— »w4-»ix=)2 wix— »w, or x' +2 m— *.x3asw5,- whence (putting a=2wi— *) x»+ax=m5, •/ x^+ax-f ?l=(m*+ a' Ams^' R' a R ^-R^a T""^ 4 — ^ T' '*' ^"^T"^- 2"' ""^ ^"^ " 2 * **^^"^ TAkr IV. GENERAL PROBLEMS. ¥7 Synthesis* First, 1 = — =s, 2 2 2 ^ ' , R-a 2ff--Jl+a 2«JR— JJ»4-2aft— 2m-o« Secondly. __X — — -=x ^ 4 . , ■ ■ • — — = (since a+s=z2m) 4 ■■■ ; (to AtcA> because 4nw+a"=it*,)=3 4mA— '4ma — Ams ' R—a 2*— -R+a 4 - 2 2 Q. E. D. R— a The answer to this problem in numbers is^ xs — — =s 74—46 28 , ^ 2i-R + a 48—74+46 20 , = — =14i and »=———= — ss — ^10. 22'^ 2 2 2 TO REGISTER THE STEPS OF AN ALGEBRAIC OPERATION. The register p is a method whereby the place from whence any «tep is derived, and the operation by which it is produced, are dearly pointed out^ by means of symbols placed opposite the said stept in the margin. The symbols employed are + for addition, — for subtraction, X for multiplication, h- for division, ^ for involution, *m for jevolution, a for completing the square, = for equality, and ir. fbr transposition. When the regbter is used in the solution of any problem, it reqiures three columns ; the right hand column contains the alg^- 9 The re^ster will be fonnd to be a rery coDTenieiit mode of reference, where mn ample detail of the work U required ; bat at modern algebraists prefer noting down results, and omit as much as possible particularising those intermediate steps which are in a great degree evident, the register is now less in use than formerly. We are indebted to Dr. John Pell, an eminent English mathematician, for the invention : it was first published in Rhonius's Algebra, translated out of the High Dutch into English by Thomas Brancker, altered and augmented by Dr. Pell, 4to. London, 1688. The learner will be enabled, by the specimen here given, to apply the method to other cases if he thinks proper ; at least he •bonld understand its use, as it is employed in the writings of Emerson, Ward, Carr, and some other books which are still read^ VOL. II. Q la ALGEDSA. Pam iy« braic operation, in the n^ct the steps ate numbered, and in the left hand column opposite to eaoh step are placed, first the num« ber of Hie $t^ from whence it is derived, aiul then the symbcd denoting the operation by which it is obtained. And here it must be noted, that the numbers 1, 2« 3, &e. in the register column, always denote -the numbers of the steps, as fli'st, secotd; third, &c. but when a figure has a dash over it, as 3, it denoUft a number concerned in the operation. In the following estample an additional column is placed Cfx tbe left, fOff th^ purpose of exphdnl^ the process, 15, Given ---+~ss7, and ^-—=±3, to find x and y. 6 2 ^16 • ^ j^t ^2dS, dxx^, mitt7^ caslS, £j?p/aisatioii. /» equatum 1. $uhtr acting ^. Multiplying eq. 3. tn/o 6. Multiplying eq. 5. iiilo ir. V 2>tvu2ti^ eqiMiott 6. ^ ^. E^ua^m^ ^Ae4M and 7th steps, Multiplffkng kq. S, h^ d. Multiplying e^. 9. info y. Transposing in eqt^tttkm 10. Dividi$ig equation 11.^6. Camp, the square, 4rc. ineqA2. Register. Giveh 1 U d 3x6 2^ Sxc 4=57 8xd 10 tr. 12to*^. Evolving the root of eq, 13. " 13 *m ' ^ - -. dm ijni Jddvng — to eq. 14. 14-^ -^ * , 2 /^rom //te 7^A and 15^^ e^. 7 . • . 15 By restitution in the \^th eq. By restitution in the I6th eq. ibrestit. \6restit. —=18, or 24, Wherefore if y=8, then isslS} but if y-se, then »s=24. No, 1 2 > 3 4 5 •6 ^pjjr^cn*. 4 y cSh* 7J 8 9 10 frdj»iy^ijr*'5=ciitt», 11 I^T— Wmyas— cd»V. cdn* y*»-dmy=s r-. . ^ d«w« y*— dmy-f ss 12 131 14 15 16 17 18 Operation, X y ^ c 07= dm- *f i' sy ( d'm' cdn' «) 4 h bd*m*'^4cdn' R' 4b • dm+JR 2 xsscn' X 2cn' dnt±|{ dm+£' 2x7+2 ^ ^ x=5l6?<9x-— r— • 2x7+2 2 cl44x— , or 144X 2 80 ALGMfiRA. PAIT IV, 16. Giyen the difference 9, and quotieat 4, of two nmnbers, to find them ? Let x^ the greater, y^theUu, d=s9, 9=4. Register Given I 2xy 1+y 3s4 5-y 6-1-9—1 1+7 I 3 4 5 6 7. 8 OpercUion. x^ysid, X —=9. y x=ct+y. 9ysd+y. 9y-y=d. d_ 17. Given the sum of the squares of two numbere 61, and the difference of their squares 11^ to find the numbers ? Let xssihe greater, yisthe le$s, a=:6l, (ssU. { .Roister Given l+« 3-h2 4 *M 1-2 6-»-2 7 No, 1 2 3 5 6 7 OperaHan. 2ar»=a+6. T' • « X'sz- 2 =6. 2y»=:a—6. y=sv- =5. PittW. GENERAL FBDttLEMB. fl 18. The ^fierefeoe of two nunabere eaneeds their quotient bj Sj'and their product exceeds their suih by dO: what lare Hul' numbers? Let xs=itfie greater, y^the lea, a=S, h^^O. Registers Given J Ixy 3 if. 4ss5 6-y 70and iuhst No. Operation* 8 AAI* 9<r. 10. 1 — =sx-^^— (t, 4 xy-^xrsy*+ay. y'-fa'-lyssk 3 6 7 8 9 10 11 12 y'+a— i.y-f «*+ r::? y+ 4 ""4 g— 1 jft fi— a+l y-^1 il— a+l 2 =r8. 19. The square of the greater of two numbers oniltiplied into the less, produces 75 i and the square of the less tnultiplied into the greater, 45: what are the numbers? Let xszthe greater, y:=sithe less, az=75, 6=45 3 <^^^ x'yss c« and xy'^=b, Inf the problem j divide the first by the second, and — = — •.•a?=-T-; substitute this value for x in the second,, y b b ■ ■ an* b' b' ' av md^ssb, oray*:£zb% vy^ss— , cmd jr=V^— 3, V2r:t:(-^ =)5. 90, To divide 100 into two piirts, sueh, that their product may equal the difference of their squares^ Let x=ithe greater part, yssthe less, atnlOO; then by the fTohlem, T-f y=a, and xytrt* ^y^ ;from the first x^a—y $ this iubstituted in- the $,econd,:ii becomes ay— ft*sst(a?»— y»a=) a«-«- c3 9t AlCBBHA. IktolV. 38.1966011^5 -/ x=(a^y= a-^^±^-^*^)^^±*-/*=. 61.80339888. 21. What tv«ro nunrfMri tti& thoee, whose diflfereoce is 4, and the product of their cubes ^f^\ ? Lei d=4, p=9261, txi'=zthe less, then^ x ^d:=t ke greater ^ fohence by the problem, (op* xx+^f 5cp, •.' »x«-hd='v'P» '^^ J • • d* <J' . M, x» + dar=3 ^p, •/ x'-^dx-i — -=« ^pH — . j iftu resohed, gives 4 4 28. The greater of two numbers is to the leas as 3 to 2« and the sum of their squares is' 906 5 required the i^umbers } Lei assS, bzs^ c=:208, xszihe greater number, then (a : b bx b*x^ a'^ : : OP : ) — ^thelesS', v by theprab,x*4- sac. vxas^/ a ^ r ^ a' ' ^ a* -ft* bx ^x =sl2, and — = — s=8. a 3 2&. t>ivide thentwiber 35 into two safib paits^ that the sum of their square roots may be 7. Let azs^S, 6=7> x=z(me part, then a-^xssthe oihpr, \\by theprob, >/x4- ^a— «^6; square both sides, and x-^-2^ax— x* 6*— a +a-r«=r6s •/ ^aJp— a?»a-— -J ag^tin square both sides, oife^ «^-:t.=:^i^'. „hich resolved, gives x^^^ ^a^^fc^^^ 16, or 9, and a— »=9, 0^ 16. 214. What number is t^t^ to which its biqi^viiiate bei;^ ailded, and from the sum twice its cube subtracted^ the remaqi- der will be 1722 ? 9 Here we, moat endently take the negative valve of + y^S, otherwise y would come oat greater than 100, and consequently s would be negative j which is contrary to what was proposed. » Ilcielhe<«ffirm^tiv« Kslve of 4*^« V5Wil«t (etdten. Fi»Y m GENfiilAL l^dfiLEAtS. «§ d;+.5==a^ 6y the ftroh. wA!l?Aoe^-^l.Sir> + .(li95s±d^ •/ ^ cem I 1 1 ■ ■■ pleting the square, and Tedxlction, *=a ^^.75+ ^a-j-.^Sar'^.S, •.* 25. To fiad two idumbere/such^ that their sUtn^, ptdduct, ^d the difference of their squares, may he etfiaX to each other i Letx^s. the greater, y=i the less, then by theprob. x-^y=sxyjand m-^yxssi^^ffi; divide thelatterhyX'\'y,undl^f;M*^iftPrstxeii'^yt iubstiiute this value for x in the first, and l-k-Stys^y^y*, whence y^—^ V^^=ai^^^=:L6180339887, «fC. cOid *=*(! +|f=«)-|- 4 Tb 26. The product of two numbers Is 1944? and the sUftoUd iroot of the greater is to the cube it)ot of the 1^^ as li to 1 1 %rhfit at« the numbets ? X ^tx %x \^ Let x^^.the ^eatet, then, I4. 1 1 ; : ;r : (77^)—-, *.' -^ = 83^ - 8 — :szthe less. Let <^=5ri p=«1944i then hy the problem (r*X W:s&) C3^^p^',*X:=M^ j^^^^ ^^h6\:^^^\' x^ii^tA^^thigfeatet^ o 8 <md csr^isi(— ><3»te)8S5*^ [est. 27 ^. . tto fufii^ 4nd product ^f two imp^fs ^H ^ml« nod if to either sum or product the 9mp of the square besdded^ th? |P9$ult will he \% y what aire the numbers ? Let ^ £tad' y repr^emi the numkertf, d^t^, ih^ ^+yM4|^ Md x-^y^x^ -fy'a'WK iy theprob. Take twice $he fifHj r^m ih0 • ^ ^u^ \. \ 4Ut4.1 hy completing iht s^strei»^f ^-^st^y^-^^^-^-^^t)—^ j 1 4 fl +1 ■4- ^4 (1 4- 1 *•• fry eHolutvon x-^^y^—^ssst^ ^ — — »== -^ i--^- — —f •»• i+yad ■ T-' -^ ' =x4, iffhence also xy±± •^ ■ ^ ' — r- =a4* From ^/^ i^uafe of the last i0i onig, take fim Hmtiif fh^ tagiytmM if*-^^ xy 4.y*si^5 %• hy «o«wfkm,i^y±i:3(J, 0nd x^y, \* :fy=***«fc|r*aDi4/ *t*xi^^/andft±% c4 ALOSBIU. r^tTm 98. G^ven the product p(^li^) and Ibe $um of the ftxirtb {K^were f (=»337>) of two mimbers^ to find them ? Le< ar=s:^^ greater, if^the leu, then xjfvp, OMd ir^+y^stf add twice the square of the first to, and subtract itfrdm, the second, and extract the squa re root of the sum and di fference^ and there will arise x* +y* = ^«+2p*, and x»— y* = ^«— 2p* : taketh^ sum and difference of these two equations, and extract the square root from each, and t^ ^-j-^*-h2p* + V«— *P* =*> «'»^ y=* * « - ' -• I 29. The sum of two numbers is &, ftnd if they he divided alternately by each other> the sum ^ the -quotients w91 be 4{-i required the numbers ? * Let a=:25^ b^z4^, xzsthe greater number, then a-^x^^the X fl*'"*X less, and by the problem 1 =6 ; whence :t' + d» —2 aX a— d? X 4 x»=flto— &r«, or 2j?* + 6ct'— 2<ur— fl&xss— a» ; <toit»> 2+6 .a?* —2+6.00?=: -a* ; divide this eguation 6y 2+6i and x»— ittsae -— r — rj w^6»( ' by completing the square, and extracting the roof, 2 + 6 tr a» a* ^ a i- a* a* ^c. .:=^+ ^-. -^^=:20, or 5 j and a-x^-T ^/•5~2+ 6 =5, or20. " . 30. 6ivetf th6 sum of two numbers 9, and the sum of their e^jibes 189> to find the numbers ? Let 2«=:9> 6=sl89, ^xtnthe difference of the required numbers f then {by proh, 1.) «+2rs=t^ greater, and s-^xssthe less, and by the proh, (*+j1^+«— j?]'s=^+3t»ir+S«P*+a:*+*»— > 3«'ar+3air»— x'rs) 2«»+6«r'=s6, or 6*e«ss6— 2*»; \- »*» 6*— £«* 6*-2«' 1 '91 g^ > g»d *±S;/— - — ac— , tp^nce t+«=K~+— =p5ya»d 9 1' *^j:s=;-;-— .--=4. '22 > ^ 31. Given the simy e, and the sum of the biquadrat^ 272» of two numbek^; to find them ? Let ^s^S, 2xs£^e difference of the required numbers, 6.s«.. 27.2 ; th$n,^as in the preceding problem, s-^xssthe greater, and s^ xsithe less, whence *+xl*+«— a7*=6; ic^ic^ tnvo^r^d and re* . ; IfARTiV; G£N£BAIi PR09L£MS. « duced, we Aarc2«*+12«*x'+2x*=6^ or j:* + 6«*x»=— 6— «*j V 6y completing the «^tkir«, jc*+6«*x«+9f*32---fc+8**|'.' 6y evolution, x' +3*» = + v'"S"*+8**, x* =s— 3 «• + V^ 6-I-8**, 4mc2ar3=;^^— 3**Hh V''^^+8«*=1; whence «-fa:=4, aiwf *— 39. Given the sum \0, and the sum of the fifth powers 17050» of two numbers, to find them ? ^ Let ^ssslO^ 6=17050^ 2 j=f^e difference of the numbers required ; then S'\'X::=ithe greater^ and s^x-^^the less, and by pro* eeeding as in problems 30 and 31> we have^s^-^-Ws'x* -h 10 sx* acA, whence xs:t ^ ^ 1 s* s=2, •.' «-f-x=s7> ond s — «s=3. 33. Given the product p, and the sum of the nth powers s, of two numbers, to find them ? Let X and y represent the numbers, then by the problem ar"H- y^sss, and xyszp ; from the second equation yssJ^f thisvalue^ub^ X t stituted for y in the first, gives x"-fi— sat, or a?**+p"as««", or s^ s* X*"— «r"s:— p") hence, completing the square, x^'-'Sac^-] s=— 4 4 5 7^ s s* —p" J whence ^— -g-= ± >/-^-/>'> ^—'^± V^ — P'> ««<i *== ,t « •' - « «4-a/«'— 4p' . p V-5-± 'v^X'^*^'* V"^ ' ami ya-^*=p-^• * « * 34. Given the product p, and the difference of the nth powers d^ of two numbers, to find them ? Let X and y be the numbers, then xy=sp, and x*— -y"=d ; when ce by pro ceeding ai in th e foregoing p roblem, xsw ^2 « ALOKBRA. ^AiT lV^ 36. Required the values of x and y in the lis. V**x ^y'atsSyS and 12V*— ^y^s^^ te« ass* V*, »=x V'y, lAca it'sBX, aad ifi^f; v fAe gteeM equaiumt become u*^szZ t*, and 19 u^zss^ ; divuie tAe last bui one by^t*, and zae — ; ^Aif equatiim added to the preceding, gvees a* 12 uaB«+— > or »' -«-24 axs— 44 j Ito tgaa^ioa retobed, girel tt=:2, ••• «=(^») 2, «« (tt'=) 8, CMd f:xM{z' =. >4. 36. If 18 oxen in 5 weeks can eat 6 acres of grass, and 4d oxen in 9 weeks eat 21 acres of the same^ how many must there be to eat 38 acres in 19 weeks, the grass being allowed to grow uniformly ? Let ais:lS, 6=s5, c=56, ds45, msa2U ^^9, rs38, trrlf, Irs the quantity eaten by an ox in a week, w=:the quantity on an acre at first, xszthe weekly increase on an acre after the first 5 weeks, x^sthe number of oxen required, p25(«— 6^)4, l=i (t— -fcsrr) 14 ; then will rw=:the grass on r acres ai first, and riz=5 the inermae on r acres m t weeks ; the mm of these, by theproblemg equcUs the qu€Mtity x oxen ate in s weeks, that is, ixs=fir+r<2; again, mwt^the grass on m acres at first, and mpx^sthe increase of the safne in p weeks ; the sum of these two equals what d oxen ate in n weeks, that is, mw'\*mpx=idn; also cw^(the grass on C acres at first y^ the quantity a oxen can eat in b weeks, thai is, cw=sabf whence wssz — • to mp times the first equation, add rt times the second, and mpsX'\*mriw-\-mprtzszdnrt'^mprw^n^tz, or mp^ szdnrt^mprw-^mrtw ; for w in this equation, substitute its equal a6 , - ^. , _ _ abntpr abmrt — , and the equation becomes mpsxxdnrt+ - — ^ — , of c c c , . - , , cdnt-^abmp-^abmt cmpsxsz cdnrt -f- abmpT'^abmrt ; whence «3C ^ -« X cmps cdnt-j-^abmxp-^t 34020+ 1890 x —10 „^ ^^ ,. rss ■^ ^ xrrr ■ . ' ^" ^ XS8aB:0Oy the cmps 9576 answer. 37* A waterman, who can row 11 miles an hour wifh the tide^ and 2 miles an hour i^nst it, rows 5 miles op a nver atodbacli Paut m OENIiKAIi PROBLEMS. «P ag^ iii 3 hmr^^ now 8M;i^0Bi9g the Ucto t» wii uttiflnteilx tlie Let mssll^ »=s2, ps5j rs3» v:=ithe velocittf refiiifed* oiul irs&fAe /im^ Ae rowed with the tide, then will r^x^the time he towed agaimt it; whence {x : p :: I hour : )—=:hi8 velocity with X the tide, and (r—x : p i: 1 hour : ) =sAtf velocity againtt (ifje; now .since the tide assists himssv when he goes with it, it Tilust evidently retard himszv when he goes against it^ whence P ^vsAthe difference of his velocity with, and against tide, •/ — — • X P P ■^va^v, ov t7S£X-.^p^«.,ii^ . 9I4M9 because his-vehn^ty with, i$ r— a? 2a? f2r— 2a? to his velocity against, tide, as m to n -, so his time of rowing with, if to hie time Of towing c^in^, tide, as n to m, since, the time if nr inversely as the velocity ; wherefore x ; r^x ; : n : m, •/ x= M-f-fl 6 7 isj^ofdn hour ss the time he rowed with tide, and r— jr=s2 — hours:=:the time he rowed against it } for x substitute its value — P P in the equation above derived. And it becomes v=s (-C — — i- — 3=) ^ ^2x 2r-^Six ^ n ^ 12 05^ (Rl 3510 , 19 ., . ^ pH---—pH-2r— ---;=: --■--~3Es----=s4 —- miles per hour=sthe '^ 13 ^ 13 12 66 793 44 ^ lulocity of the tide. 38. The ages of five persona, A» B, C, D« and B, Bve mi^» that the sum of the first four is 95^ that of the three first and l^st 97> that of the two first and t\vo last 103, that of the first and three last 106> and that of the four last 107 ) required the age of each ? Let a=z95, 6=97, c=103, (f=sl06, e==l07, s=the sum of all their ages, and let x, y, z, v, w, be put for their ages retpec- tively ; then wiU s—wsza, «— «=6, »— zapc, s — y:std,.and «— * ' Velocity ^fmm tlM hmtk mh^, ft«ift»> it ttM aftntiMi^f mHiieD, wbatcbf » BlMii% hsdy- fuam <wwr • cntoisk tpM^ ia a ontaifi time; pr in tammoa btngtiage, it it tbe degree of twiftaet^ with wbicb a body moves : it is liUcwiiM* mMMd eekvit9s (mi^tlie l«Ciii ««Ai^ Mft orvfin^lAt. tt AtX^EBRA; pARtlV. tse; aid the»e fine eqwUUms together, md the sum is (5«— X— y— 2— »— io=:5«— »=) 4*=a+6+c+d+e; whenee ssaf ', now if this value he substituted for s in the five preceding equations, we shall thence obtain the required numbers, viz. 10=32, ©=30, 2=24, y=s21, and x=20, being the ages of E, D, C, B, and A, respectively. \ 39. To find a point in the straight line which joins two lumi- naries, or in the line produced, which is equally enlightened by^ both •. Let asstheir distance apart, x^the distance of the least of them from the required point, then a^x^zrzthe distance of the other : lei the quantity of Ught emitted by the first in a given time be to that emitted by the second in the same time, asm ton; then fgjiU — : be the ratio of the effects they produce, supposing. ^' a±x\ i»=», and -J : will be the ratio, supposing m and n un^ tn tqual: but these effects are by hypothesis equal; whence — = X n — • ■ . -, •/ iiia'-j-2a;n4:4'WW?*=siw:*,orm— n«r*+2awu:= a'+2aj:-f a? 2am ma' 2am am I* -^ma', '/ X* A a:= — , •/ar'H x-^- 1 = ■ III —— ^^-^— i am \' ma* am am — ^, '/ x± =»± v/ TO— » m — n " m-^n — ' m — n * ma' m — n and x: ^-- am , am 1* ma^ -f am+ -/m»a* ^-{-a.m-^ Vmna* (-f +>/ =3 -=*-2r 3») =» ^ m—n^ m—w m—n wi— » m — n s= the distance required. 40. The weight w, and the specific gravity of a mixture, and the specific gravities a and b, of the two simples which compose it, being given, to find the qua,ntity of each « ? • A'lnmiaary, (from the Latin hiinen, light,) is a body that gires light, as the fan, moon, a plan^, star, &c. * The double sign serves both cases, tIx. a^x when the point tvqoired is-, beyQud the smaller luminary, and a-^x when it is beti9«;eii them ; als9 in the answer, the upper sign — applies to the ficst case, and the lower sign -^ iO'^lke > second. ■ The gravity of a body, (from the Latin gramSf heavy,) is tta weighty < Part IV. GENSRAL PROBLEMS. 129 JUt xm^ iDe^fhi of the simple, whose «peci/Sc gravHy is the great€»t, then w—xazthe weight of the other. X a v)—x I the magnitude of the I ' ' ' "" hody, whose weight is w s w x- w— J? w Whence — h — ; — = — , or bsx-^-asw^asxssabw,'.' bsx-^ a b s , abw—asw b — smw mxssabw — asw, or a?= — ; = — . bs--as f,^a^ ' 41. Suppose two bodies, A and B, to move in c^ipoeite direc- tions towards the same point with given velocities, the distance of the places from whence they set out, and the'difierence of the times in which they beghi to move, being likewise given, thence to determine the point where they meet } Let d^sithe distance from A to B at the time of setting out, SO-srAs distance from the point of meeting, then d — x=Rs distance from the point of meeting; let t=:the difference between the times of their beginning to move, and suppose A moves through the space a in the time n, and B through the space b in the time m, then nx (a:n::x :) — sithe time of As motion, and (6 : m : : d— x : ) d^"" x^m nx . — r — ^ithe time of B*s motion; whence by the problemj^ — — d — x.m bt-\'dm — T — =*, VX5=-- .a. o on-^am ^juid the specific grarity is its weight compared with that of a body of equal hulk, hot of a difflereDt kind : thus, a cabic £D0t of oommon water weighs 1000 oances avoirdapois, and a cubic inch of each of the following substances weighs •8 follows ; Tix. fine gold, ]9640os. fine silver, 11091 ox. cork, 240 ox. new falXen snow, 86 ox. oommon air, 1.232 ox. &c. &c. these numbers, then, repre- sent the specific gravities of the aboTe-mentioned substances respectiTcly, com- .fared with*co«imoa water. — ^Tables of the spedfie grarity of a great variety of bodies, both solid and fluid, may be found in the writings of Mersenne, Muf- chenbroeck, Ward, Cotes, Emerson, Hntton, Vyse, Martin, &c. and are useftil ^or computing the weight of such bodies as are too large and unwieldy to be inoved ;' by means of their kind and dimensions, which must be prerionsl^ known. N W AL&SAIIA. PAftf tV. SxAicnms^*^!. A tett out from London to>M«Rk Diiriiam dteUnt 257 miles, md lAweli II tsfleB iA4liOtfi$ B«ils<nit from Durham 8 boon later, and travels towards London at the rate of 10 miles in S hours : whereabouts on the road wHl they meet? Here d==257> t^B, #sfell, msl4, 6s^10» ms3.* Then x=— ^J!±?^?^ X ll = m ^ lailef from Xo«A«. 10X4+11x3 73 -^ . 2* Supposing Africa to be 9QfiOO miles round, and a. ship to iail from the Isthmus of Suez down the Bed Siea, with int^at to coast it round that vast conthient, sailing on an average ^ miles an hour } — a week after anethief ship sails from the opposite side of the same Isthmus with the santt intent* and p il ing the traits of Gibmltar, sails at the mte of 3^ miles an Ixniff;*** Aear what place* on tlie coast will they raedt ? 4$. If two bodies, A and B, move in (he same direction and in the same straight line, their velocities, distance at setting out, land the interval between the times of their beginning to move, being given, thenCe to determuie the point where they will come together. Let A ie the farthest from the required point, ct=<^ distattce froth A to B, xzsAts distance from the point, then wiU x — d^B's distoHde; tiko let tmihe Mtfvmlof tsmt Aehmeoi ihtir wetting xmt, and let A move through the space a in the time r, and B thret^ tx the space h in the time <; then U>iU {a :r n x : ) — ^the time of As motion, and{h:szi x^d : ) -Il-ie^Ae time of Bs motion; nohmee bf the prohlem, !!i--iZLjft±f, *.• x^J^^:a, when A « o hT'^as sets out first; and — r =^ •/ x= — ^-.o^ when B ^ets b a as^hr ^ cut first. EKAMPLBS.---1. A ship sbiIb from the D«wnS) east, toii^aitfe Petersburg, at the rate of 54 mites in «8 htmt ; «4 hwurs aftelr another ship saib frbm Lisbon, distant ttom the l>owns 660 miles west, in pursuit of her, and goes at the rate of 8 miles aa hour : whereabouts will the latter ship overtake the former^ PlkT IV. GENERAL PROBLEMS. 91 Heted^&SO, ^ssM, assS, rzl, b^M, Jrcstt; 0nd4i^au$e B sets out first, therefore xsz 54x344-23x560 xSss 8x23—54x1 858.21538, 8(c. miles from Lisbon, or (858.21538, ^.--560^) 308.21538, SiC miles from the Downs. 2. Suppose the skip from Lisbon sets sail 24 hourii before tbe other ? -,. 54 X 24--23 X 550 ■ ^^«w,«^ , zt a^ Then x=i -—■ — ; — 7r''7z;r- x 8=<>98.triS8, ?rc. mites from. ! . 54x1—8x23 Lishm, or .(698.7138, ^c--550=d) 14$i7138» ic. miles fron^ the Downs, 3. A is trOO tniles south of London, and sets otit on a journey north^^Fard, travelling 37 miles etery ^4 hours ; B from London pursues the same roiit, selling out 49 hours lafter A, and tra- velling at the rate of ll miles every 8 hours : where will they be together ? 43. Qiven the forcfs of several agents « separately, to deter- mine their Joint force ? Let A, B, C, D, ^c. be the agents, and suppose A -1 B "I b \n can produce an effect, ^ c > times, in the time ^ r Uc CaU the gtMn ^eet 1, aitkd hi ^stihe iime in whidi theif can- produce ity all operating together : Thsn will ax'\ m.{time) : a,{effect) : : x A time : — * m bx n X ^c. d *c. n 'BX dx s The effect produced in the time x, b^ A B C D > An agent, (in Latin agent, from »ym to drire,) is that by which any thing - 18 done or effected, niilosophert call that the agent, which is the iinmediate cause of any effect, and that on which the effect is produced they 38 ALGEBRA. PaatIT. . Bui the ^m of thne effects it equtd to the gkfen efeei I, pro- dttced by thejomt apemttum of aU the agents^ m th^ time x ; whenct ax hx ex dx a b , c , d . —+ h h— *c,=l, or X.--+ h— +— *c.5=l,vx= ffififf m n r $ a . b c d ^ m n r s Examples. — 1. A can reap 5 acres of wheat in 8 days, B caa reap 4 acres in 7 days, and C 6 acres in 9 days ; how lon^ will they require to reap a field of SO acres, all working together > Here m=8, a=s5, n=s7> 6=4, r=9, c=s6. 1 1 168 Thenx=z __X30=-- j x30=-yt xSOss a b c 5 4 6 dlo m n r 8 7 ^ 32 2. A vessel has three cocks. A, B. and C -, A can fill it twice in 3 hoiurs, B 3 times in 4 hours, and C 4 times in 5 hours -, in what time will it be filled with the three tocks all open together ? 44. If two agents, A and B, can jointly produce an effect ia the time m, A and C in the time n, and B ^nd C in the time r ; in what time will each alone produce the same effect ? Let Jyy^sthe time < B> would require to produce the given effect; and let the effect be called 1. call the patient ; the effect, as communicated by the agent, they call an eeticn ; but as reeeived by the jiatient, a pauiou : a smith striking oa an anvil has been frequently proposed as a proper example ; thus the smith is the superior agent^ the hammer with which he strikes is the it^itrufr agents the blow he strikes is the adtofi, the anvil is the patient^ and the blow it receives, the pasnon. Part IV. GENERAL PROBLEMS. ds 0? {time) : 1 {effect) Then is TO Umie) : — X X 1 1 1 1 1 TO TO TO n ft TO 7 n I z r y r II J A in the time m B TO A n C n B r C r fflience—+'^=:l, or (1) —4—=—. a? y ^ :p y TO — +— =1, or (2) —+_=—. r r ^ X z n *• *• , ,ov 1 1 1 — 1---=1, or (3) —+—=--., y z ^ y z r Add equations I, % and 3 together, and the sum will be 1 1 I — 4— -f — X2=— +— +— , or (4) —+—+—=— -4 j:yz TOMr x y z 2to 2n 4-3- ; /roTO eq. 4' subtract eq, \, 2, an^ 3 severally, and the re^ mainders are JL— ^ J ^ 1_ z ^m 2n ^r to 1 _ 1 ^ JL JL y'~2w «n 2r n 1 _ 1 JL J, 1 a? ""2 TO 2n 2r r ^ 2TOnr ► whence < Xrs- TOf-f-mn — nr 2TOnr nr+mn-^mr- 2TOnr »r4-rar — win * Examples. — 1. A and B can unload a waggon in 3 hours^ B and C in 2^^ hours, and A and C in 2^^ hours j how long will each be in doing the same by himself? Here m:Kz3, 71=24, »*=^t> *=a 7i ^ ^^^ ^ = ' ^ 24x2iH-3x2i— 3x2^ 37.126 4.6875 VOL. II. =7.92 /lour^. 34 ALG£BRA. Past IV. 2x3x24X^ 37.125 24 X 2^+3 X 24-3 x2i 7.6875 =:4.82926889 hours. 2 X 3 X 24 x^i 37125 ^ ^ , «^^.^e L 2=:-: — — -^ ^ = =4.21276595 hours. 3 x2i+3x 24-^x24 8.8125 2. A quantity of provfeions will serve A and B 8 mcmtlis, A and C 9 months^ and B and C 10 months ; how long would the same quantity serve each person singly ? . Ans. A 14 fit. 20$4 days, B 17 m. 16|f days, C 33 m. ^ff daySf reckoning 30 days to a month. 45. It is required to divide the number 22 into three such parts^ that once the first, twice the second, and thrice the third being added together, the sum will be 47» and the sum of the squares of the parts 166 ? Let X, y, and z, denote the three parts respectively, a=22j h =47> c=166j thenify the problem x+y-^-zssa, x+2y-|-3z=6> and j?*+y*-f-z*=c; subtract the first from the second, and y-\-2z szb^a, whence y:=^b — a-^2z; subtract double the first from the second, and z— a:=5— 2fl, whence xssz+2a— 6^ let f^b^a, ^=6— 2 a; these values being substituted in the two latter equa- tions, they become yszf-^^z, and xzs^z^g; svhstitute these values for y and x in the third given equation, and it will become z*-^2g9 +g*+/*-4/z+4z«+z«=:c, or z^^^I±Mz J'''^''^ .^ puth^ ^ o 2f4-jT c f* fi* -^^^j and the latter equation becomes z^^hzss — "^ ^ ,m which by completis^ the square, 8fC. it becomes zss — + V g H-j (f/ohich^ by restoring the values of c,f, g, and k, viz, c=166, /=r6-a=47-22=26, gs=5>-gtfs=47— 44= J 1. 2/-f 5f 50-1-3 63^ 53 166—625-9 . 2809 3, and A=-=^^ — S= = — )= \- ^ \ 3 -3 3^6^^ 6 ^36 =9, whence x=z{z^g:=i) 6, and yas (/— 2«=s) 7. 46. Required the values of x and y in the following equations, viz. a^+3!«y-ha?*y«+a:y'+y*=211=ii,- and a«+«*'9*+«*y*+ t?y-hy«=U605=6? Divide the second by the first, add the quotient to, and sub* tract it from the first, and the results will be (2:r^+2xV+2^:s Paat IV. GENERAL PROSLEMS. i5 =— fl— — , i^^ ttJiW fAe two ei]u<xiion9, o^t^tf derit^rf, 6^oin« spxn, •/ p= — i iAis being squared, and the nquare added to «•— s w' in 111* p*=w, gi«c5 «*=mH — ^, or «♦— iiw*=«% ••• 5=^—4- ^ f-n* n =13, andp=( — =) 6. ^ow since (*=?) a?*+y*=:13, and (p=) xy:=:6, if the square root of the sum and difference, of the former and double the latter be taken, we shall thence obtain x=i3, and ' 47. Given the sum =5, and the product =p, of any two num- bers, to find the sum of their nth powers ? Let X and y represent the ttco numbers, then will x4-y=5, and xpssp. First, {x+yl'^sa) j^-f^ipy-f y«32#«, and ^xyrm^p; subtract the latter from the former, and ^fi-^-y^ss^-^QpnstkesHm (ff the squares. Secondly, x'^ + y^jx -j- y:=^s^^%pjSy or x^ +xy.x + y -f-y'=s'— 2sp, which (by substituting sp for its equal ^y^x-^-y) becomes a:^-fsp+y^=«^— 2fip, •/ 3^-\-y^z=::,^'Ssprzthe sum of the cubes. Thirdly, a?3+y'.a?4-y=«3— 3«p.s, or a?*-ha?y.a:*+y«-|- y*=:s*-^35*p, which {ky a^stiiuUng p^'^—^p for its equal xy.x^'^y^) becomes a:*4-p,«*— 2p4-y*=:«*— 3«^, •/ :t*+y*=(**— 3s*p— p.«*— 2p=) *^— 4s''p-f 2p'=/^e sum of the biquadrates. In like manner it may be shewn, that s^'^5^p'\-bsp''ssthe sum of the Jifth powers; s^— 6y*p+9s'p*— 2p^=*/»e sum of the sixth powers, ^c. By comparing together these several results, the law of con- tinuation will be manifest; for it appears from the foregoing pro- cess, that The sum of any powers is found by multiplying the sum of the next preceding powers by s, and from this product subti*acting the sum of the powers next preceding those multi« plied hyp. D 2 36 AL6BBRA. Part IV. Thus, the sum of the 4th fHwensss x sum of the cuhes^p x sum of the squares. The sum of the ^th powers^s x sttm of the 4th powers^p x sum of the cubes. The sum of the 6th powerszss X sum of the ^th powers—p x sum of the 4th powers, Stc ^c. Hence the sum of the nth powers of x and y will be as follows ;. n— 2 n— 3 n— 4 n— 4«— 5 n— 6 . ft— 5n— 6n— 7 w— 8 . 13. To investigate the rules of arithmetical progressioa. Let a^ihe Ua»t term I ^^^^^^ ^^^^ ^^ ^^^^^ z=itJie greatest -> n=zthe number of terms d=zthe common difference of the terms 8=: the sum of all the terms, ' . Then will a+a+<i+a+2(2+a+3d+^ SfC. io a-f ft— l.cE he an increasing series of terms in arithmetical progression;. And 24-z— d+«— 2rf-f-« — 3rf+, 8(C. to z-^n — l,d will he a decreasing series in arithmetical progression. 14. Now since in the increasing series a -^n^ I, d=: the greatest term, and z:=: the greatest term by the notation, therefore z^:^ a •\' n— l.d (theorem 1.) JVhence by transposition, 8sc, assz — n— l.d z—a Z'^^a (theor, 2.) d= (theor. 3.) and nzx — ; — 1-1 (theor.4.) ^ • n— 1 ^ ' d ^ Whence, of the first ttrm, last term, number of terms, and difference, any three being given, the fourth may be found by one of these four theorems, 15. Next^ in order to find «, and to introduce it into the fore- going theorems^ let either of the above series, and the same series inverted be added together ; and since the sum of each series is:=: . s by the above notation, the sunt of both added together, will 6t'i- ^ dently be 2 s. Thus, The series a-|-a4-d+a+2d+a-i-3d+^c.=«^ The series inverted a-^-S d+a-\'2 d-\'a'\'d'{-a , , , =«. i.t ■« ■ I- 1 Their sum 2a+3d-h2a + 3d-f 2a-f 3d+2a-f 3<f=^.2« Part IV. ARITHMETICAL PROGRESSION. 37 That is (2 a-f 3(2.11, or a-i-a+3d.», or^ since a+3 rfsrz) ' ' ■' ft JL, T ft ««— — » n • a+z.n=s2«, whence *=( — ^— =) <»+«--^ (theok. 5.) From this equatumare deriveda=: z (theok. 6.) z=^^ — ra (t^eor. 7.) and n= (theor. 8.) Also by equating the vaiues of z in 2« . ... s theorems 1 and 7> (»w^ o+w — 1.4=-- — -a.) we obtain a=r n ' n ——.a (theor. 9.) fl=( ■ ;=s) — . -(theor. 10.) 2 n.»— jl «— I » «— 1 5=—n.2a+7i—l.d (theor. ll.)andn=- (theor. 12.) 16. In like mannevy by equating the values of a in theorems 3 and 6, (viz, z^n — l.d= z,) loc derive z= 1 .d 9 n 2 2 wz — ^^ (theor. 13.) d= — . (theor. 14.) sr= — n.Sz— n-'l.d ^ ' n n— 1 ^ 2 id4~z^ A/ld-l-z^— 2 d* (theor. 15.) andn= ^^~i (theor. 16.) and a z-'-a equating the values of nin theorems 4 and 8, we have — — J- 1= 2* —-7, whence z= ./a— 4^d)*4-2 d«— i^ (theor. I7.) a= . z-f-a.z— a ^z+T^*— 2d5+4.d (theor. 18.) d=r-—:;; (theor. 19.) z— a+ci z-j-a , #= — -^ — •—^^ (theor. 20.) 17. Hence any three of the five quantities a, z, d, n, s, being given, the other two may be found : also if the first term a=zo, any theorem containing it may be expressed in a simpler manner. IS. The following is a synopsis of the whole doctrine of arithmetical progression, wherein all the theorems above de- rived are brought into one view* d3 58 ALGEBRA. Pait] PTheor. I Given | Req.| Solutiog when a^o. I. XI. m. V. IV. XX. VII. X. VIII. XIX. XVII. XII. II. XV. IX. xm. VI. XIV. XVIII. XVI. a, d, n-l 2=:a-i-n— l.d Theor. a, d,z I. a, n, 8 2, d. i: z, d, s 5=4..n.2a-|-n— I.d d= n ■ I .. ■ 2— <t 71=-— -+i 5= z — a+d a-f« ^8 'zz=z a n d= ^ s^na n n — I W=" a+2 XXI. 2=11— I. d XXII. f=4-n.n— l.d XXIII. d=- XXiV. n 5 = 2.-- 2 XXV. a XXVI. XXVII. 2^ XXVIII. ^at— XXIX. d= 2+a.z— a 2«— a — 2 2= ^a— 4J*+2<f*— 4^ n= ^— «4- >v/-W— 3«4.2(i« a=2— n — l.d ai^Bi^vMriiH 5=4-n.?2— »— -l.d ^aa^^aMte* « w— 1 , a= :r— .d w 2 « 2 2s a= 2 n 2 n2-*» "" n *n— I «= >v/2-R31*— 2 ds+4-d n XXX. XXXI. XXXII. When a:=zQ. n— I *=• 2 + cZ 2 T*"2 2* n 2 » n— 1 ff= 2 d=- 2«— 2 2=v'id«+2dM «= id+vI^M^ S 8 8 When d=iOj then azsz^:—^ } 8:=znazznz ; n= — = — n a Pabt IV. ARITHMETICAL PROGRESSION. d9 ExAMPLBfl. — I. la an arithmetical ftogretsioa, the first term is 3, the number of terms 60, and the common difference S : what is the last term^ and the sum of the series ? Here a^S, n^bO, d=x^. Whence, tJum. 1. z=3+50— 1x^=2 101 =si^ last term. And, theor. 2. «=4-x 50x3 x 3 + 50- lx2=2600=f^ sum. 2. Given the first term 3^ the last term 101^ and the number of terms 50 5 to find the common di£ference and the sum of the series ? Here ascS, 2=101, n=50. Whence, theor. 3. ^=*r --=2=^^e common difference. 50 And theor. 5. «=3 + iOl x --=3600=s the sum. 3. The first term is S, the common difference % and the last term 101 5 required the number of terms, and the sum ? Here 0=3, d=:2, z=101. • 101—3 Wherefore, theor, 4. »= ^.l^zzBO^ithe number of 'terms. ^ J ^L ^^ 101-3+2 101+3 ^^^ - And, theor. 20. «= ^ X ^ =2600==th£ sum. 4. The first term is 3, the number of terms 50, and the sum of the series 2600, to find the last term, and £fference ? Here a=3, 11= 50, s=^600. 2 X 2600 Then, theor. 7. 2= — 3=101=*Ae last term. '50 .. ^ .X , , ^ 2600—50x3 ^ ^. j#wf, <*ew. 10. <l3s:--x ; 3=3=3: iAe cmmuw 50 ^ 50—1 difference. b: Given the first term 5, the last term 41, and the sum of the series 299, to find the number of terms, and the common differenced Ans. 6y theor* 9. »3=13, and by theor. 19. ds^iS. 6. Given the first term 4, the common deference 7> and the turn 355, to find the last terra, and number of terms? Ans. by theor. 17. zsxejf and by theor. 12. fi=10. 7. Tte last terra is 67> the difierence 7, md tht number of D 4 40 ALGEBRA. Paet IV. terms 10> being given, to find the first term and sum ? Jng. by thear. 2. asz4, and hy theor, 15. f =5355. 8. Let the common di£ference 3, the number of terms 13> and the sum 299 be given, to find the first and last terms ? Ans. by theor. 9. a=5, and by theor, 13. z=41. 9. Let the last term 67, the number of terms 10, and the sum 355, be given, to find the first term and difference ? Am. by theor. 6. a =4, and by theor. 14. dsT. 10. If the last term be 9>'the difiference 1, and the sum 44, required the first term, and number of terms ? Ans, by theor, 18. a=5, and by theor. 16. n=8. 11. The first term O, the last term 15, and the number of terms 6, being given, to determine the di£ference and sum ? Ans. by theor. 23. d=3, and by theor, 24. «=45. 12. Bought 100 rabbits, and gave for the first 6d. and for the last 34d. what did they cost ? Ans. SL 6s. Sd. 13. A labourer earned 3d. the first day, 8d. the second, ISd. the third, and so on, till on the last day he earned 4s. lOd. how long didHie work ? Ans. 1*2 days, 14. There are 8 eqdidifierent numbers, the least is 4, and the greatest 32 -, tvhat are the numbers ? Ans, 4, 8, 12, 16, 20, 24, 28, and 32. 15. A man paid 1000^. at 12 equidifiercnt payments, the first was 10/. — ^what was the second, and the last ? Ans. the second 23;. 6s. 8d. the last 1661. I3s. 4d. 16. A trader cleared 502. the first year, and for 20 years he cleared regularly every year bl. more than he did the preceding; •what did he gain in the last year, and what was the sum of his gains? 17. The sum of a series, consisting gf lOQ terms, and be- ginning with a cipher, is 120 5 required the conunon difference^ and last term ? 19. PROBLEMS EXERCISING ARITHMETICAL PROGRESSION. 1. To 6nd three numbers in Arithmetical Progression, the common difference of which is 6, and product 35 } Let the three numbers be x---6, x, and j;+6 respectwehf. Then by the problem, (x— 6.x4;+6=) «'-.-36a?5=35, orx^— 36x —35=^05 this equation divided by Xrf 1^ give» (x^-^o;-- 35=0, or) Pabx IV. AMTHMETICAL PROGRESSION. 4i «*— jp=:35 5 whiah resolved, we h(me a?=^35.25+.5, whence a:--6=:^35.25-f 5.5, c/nd .a:+ 6=^^^5.25 +6.5 : the numbers therefore are .43717, 6AS7l7,(ind 12.43717, nearly. 2. An artist proposed to work as many days at 3 shillings per day, as he had shillings in his pocket; at the end of the time having received his hire, and spent nothing, he finds himself worth 44 shillings j what sum did he begin with ?. Let x=his number of shillings at first, whence also x=:the number of days he worked : we Jiave therefore here given the first term x, the common difference 3, and the number of terms x-^-l, in an arithmetical progression, to find the last term} now by theor. 1. (z=a+n--l.d, or) 44=af-f a:+l — 1 x3, that is, 4x= 44, whence a:=ll shillings =z the sum he began with, 3. To find three numbers in arithmetical progression, such« that their sum may be 12, and the sum of their squares 56 ? Let x^zthe common difference, 3 5=(12) the s^um, then wUl s=^the middle number, s — x=^the less extreme, and s+x=:the greater extreme, also let fl=56j then by the problem, («— x)*-fr 5«-|-7+il«i=) 3««-|-2a;«=a, whence 2:r2=a— 3«S and xss a— 3«« 56-48 , , ^ ", V — 5 — = V ; — =^ 3 therefore s=i4, s— x=2, and s+xrz6, % 2 that is, 2, 4, and 6, are the numbers required. 4. To find four numbers in arithmetical progression, whereof the product of the extremes is 52, and that of the means 70 ? Let xzzithe less extreme^ y=the common difference ; then will X, x-i-y, x-^^y, and x-^-Sy, represent the progression. Let a= 52, Z>=70, then by the problem (a?.^+3y=) a?*+3xy=:a, and* (j;-|-y.a?+2y=) a?*+3a:y+2y*=6;/roTO<^ latter equation sub' b — a tract the former, and 2y*=6— a, whence y=^— — =35 suo- stitute this value for y in the first equation, and it becomes a^+dx 81 9 =a ; completing the square, Sfc. we obtain a:= a/^H =4 : 4 2 wherefore 4, 7> 10, and 13, are the numbers required. 5^ The sum of six numbers in arithmetical progression is 48, and if the common difference d be multiplied into the less ex- treme, the product equals the number of terms -, required tbQ terms of the progression ? ALGSMU. Past IV. Let a^s-the first term, then da=6, and a^s^—i also, since s^s (•i^fi.da+n — l.ds) mH — '--^—-d by theor, 11. we have bff sub* stitutum, 48=6 a+---— .d^ <^< is, 6 a+ 15 d=:48 -, whence 2 a-f 2 5d=:16^or (ptt/^iii^— /or a) 5 <?+ 12=16 d, or <P— — d= — 18 •7- i whence fry completing the square, 4rc. if =?^ therefore azc o 6 (— =) 3; coiuegiMiUZy the numbers are S, B, 7,9,11, and 13. 6. The continual product of four numbers in arithmeticil progression is 880^ and the sum of their squares 214 ; what are the numbers ? Let p=:880> «=214, 2x=<fte common difference^ y^Sxav the less extreme; then will y— 3x, y— x, y-h^Pt ond y4-3x=*^ <eni» of th e progression s wherefore by the problem, y— 3x.y — x. y+x.y"+3x=p,afirf y— 3xl*-f y— xp+y+xl«-f y+3xl*=«; these equations reduced, become y*— 10y*x*+9x*=p, and 4y*-h20x* s=#; /rom tAe 2a^^€r of these y*= 5x*, therefore y** — — 4 16 Ssjfi '-^+25x*; tf <toe values Be substituted for their equals in the s^ 5 sx^ 5 <x^ former, we have — ~ — I- 25 x* h 50x* + 9x*=:p, whence 16 2 2 -T — =-! — ! -. or (mUtttue ass —. and — =-^ — — --')x*— «x*= — 5 then by completing the square, ^c, x^ J±^=ily rutariag the values of a and R) H, »A.«ce y= (^-- — 5x*=i:) 64-: therefore y— 3x=2, y— x=:5, y-f x=:8, and y -|- 3 x= 1 1 ^ /A6 numbers required, 20. To find the number of permutattons^ which can be made with any number of given quantities. Defs The permutations of quantities are the diflerent orders in which they can be arranged. Pabt 1Y. PERMDTATIONS. 43 Let a arid b he two quantitiisj thete wUl evidently admii of two permutations, viz. ab and bo, whkh number of pemmtaiUme may be thus expressed, 1x2. Let a, by and c, be three quantities j these admit of six pemm* tations, abc, bac, cah acb, 6ca, and cba, viz. 1x2x3. Let a, b, c, and d, be four quantities) these admit of 24 per* mutations; thus, abed bacd cabd dabc abdc bade cadb dacb aebd bccid cbad dbac acdb bcda cbda dbca adbc bdac cdab dcab adcb bdca cdba dcha That is, 4 things admit o/ 1x2x3x4 permutations. In like manner, 5 tilings admit qflx2x3x4x5 ^ 1x2x3x4x5x6 >permuiations. 7 Ix2x3x4x5x6x7j S;c 5fc. jind therefore n things admit of 1 x 2 x 3j 8sc. to n^ permutations. Examples. — 1. How maay ways can the musical notes uty re, mi, fa, sol, la^ be sung ? Ans. Ix2x3x4x5x 6:s720 ways. 2. How many changes can be rung on 12 bells J Answer, 479001000. 3. How many permutations can be made with the 24 letters of the alphabet ? . 21. To find the number of combinations that can be mad« out of any given number of quantities. Def. The comUnations of quantities^ or things^ is the takii^ a leas collection out oi a greater as often as it .can be done> without regarding the order in which the quantities so taken are surranged. Thus, if a, b, and c, be three quantities, then ab, ac, and be, are the combinations of these quantities, taken two and two : and here it is necessary to remark, that although ab and ba form two different permutations, yet they form but one combination; in the same manner ac and ca make but one combination, as also be and cb. Let there be n things given, namely a, b, c, d, S;c. (to n terms,) then if a be placed before each of the rest, n— 1 permutatUmf ^ 44 ALGEBRA. Pabt IV. ioiU be formed; if h be placed before each of the rest, n — 1 pet' mutations will in like tnanner be formed; and if c, d, e, 8;c. be placed respectively before each of the rest, n— 1 permutations in each case will arise; consequently, if each of the n things be placed before all the resty there will be formed in the whole n.n — 1 permutations; that is, there can ».n— 1 permutations be formed of n things taken two at a time. Hence, if instead of nwe suppose n — 1 things, b, c, d, e, 8(C, the number of permutations which these afford of the quantities taken twS and two, will (by what has been shewn) be n — l.n — 2 } now if a he prefixed to each of these permutations^ there will be n— 1^ — 2 permutations in which a stands first; in the same man^ ner it appears, that there will be fi--l.n— 3 permutations in each case when b, c, d, e, dtc respectively stand first ; and therefore when each of the n things have stood first, there wUl be formed in the whole n.n — l.n— 2 permutations of n things taken three and three. . By similar reasoning it appears that n things taken 4 at a time afford n.n— l.n— 2.n— 3 •% 5 at a time tt.n— l.n— 2.n— 3.n— 4 ^'^ r at a time . . . „. n-l.n-g.n-3.n--4 (tations. n.n— l.n-r2.n— 3.n— 4 . . . n— r + 1-' This being premised, we may readily obtain the number of combinations, each consisting of ^, 3y 4, B, 8fC. to r things, which can be made out of any given number n ; for it appears by the pre- ceding problem, that 2 things admit of 2 permutations, but by the definition they admit of but 1 combination ; and therefore any^ number of things taken .2 at a time, admit of half as many combinations as there are permutations; but the number of permutations in n things, taken two and two, has been shewn to be n.n — l-, therefore the number of combinations in n thirds, taken two and two, will be — ^ . or which is the same — ^ — -— . 2 1.2 If three things be taken at a time, then 6 permutations will arise from every 3 things so taken, and but 1 combination ; and therefore any number of things taken 3 at a time, admit of one sixth as many combinations, as there are permutations ; but the number of permutations in n things taken 3 at a time, has been shewn to be n.n— l.n— 2 5 and therefore the number of combina^ Pakt IV. COMBINATIONS. 45 tions in n things, taken ^ at a time, will be n.n — l.n — 2 or n.n— In— 2 1.2.3 JBjf similar reasoning it mat^ be shewn, that the number of combinations in n things, taken 4 5 n.n— l.n — ^2.n— 3 1.2.3.4 > at a time will be < n.n— l.n— 2.n— '3.n— 4 1.2.3.4.5 n.n-^ l.n— 2.n— 3.n— 4.n — 5 1.2.3.4.5.6 n.n— l.n— 2.n— 3, 5rc. to n— r+l 1.2.3.4, 4c. to r Examples. — 1. How many combinations can be made of 2. letters, out of 10 ? rr ,^1. n.n^ 10X9 ^^ . Here n=10. whence = — - — =45. Ans» 1.2 2 2. How many combinations of 5 letters can be made out of the 24 letters of the alphabet ? Here n=24, then njn^ 1 .n— 2.n— 3.n— 4 = 10626. Ans, 1.23 4.5 3. In a ship of war there are 40 officers, and the captain in- tends to invite 6 of them to dine with him every day ; how many parties is it -possible to make, so that the same 6 persons may not meet at his table twice ? 22. To investigate the rules of simple interest. Def. 1 . The sum lent is called the prvnci'pal, 2. The money paid by the borrower to the lender for the use of the principal, is called interest. 3, The interest (or quantity of money to be paid) is previ- ously agreed upon ; that is, at a certain sum for the use of lOOZ. for a year : this is called the rate per cent, per annum '. y Per cent, means by the hundrefi, and per annam, by the year ; the term 5 per cent, per annum , means 5 pounds paid for the use of 100/. lent daring the space of a year, &c. VarioDS rates of interest have been i^iven in this country for the use of 46 ALGrEBEA. Part IV. 4. The principal and interest being added together^ the sum is called the amount. Let pxzthe principal lent, r=ithe interest of I pound for a year, t=zthe time during which the principal has been lent, i^ the interest of p pounds for t years, a=^the amount; then toiU 1 (pound) : r {interest) : : p (pounds) : pr^the interest of p pounds for a year: and 1 (year) : pr (interest) : : t (years) ; ptr=zi (thbob. l.)zzthe interest of p pounds for t years, or t parts of a • m • year: hence p^ — , *= — ,andr=z — . If to this interest the ^ "^ tr pr P^ principal be added, we shall have ptr^^pssa (thbor. 2.) hence by transposition, ^c. p= (theor. 3.) t=z ^ (theor. 4.) '^ If-fl ^ ^ pr ^ and ras — -± (theor. 5.) The following is a synopsis of the whole doctrine of simple interest. Theor. Given. {Req.j Solution. .,«,r.{* irzzptr. aszptr-^-p, a mency, at different periods, from 5 to 50 per cent, but the law at present is, that not more than 5 per cent, per annum can be taken here, although the legal rate of interest is much higher in some of our colonies. The interest of money is oompntod as follows ; In the courts of law in years, quarters, and days. On South Sea and India bonds calendar months and days. On Exchequer bills .... quarters of a year and days. Brokerage, or commission, is an allowance made to brokers and agents in foreign, or other distant pfaioes, for buying and selling goods, and perform- ing other money transactions, on my account ; it is reckoned at so much per cent, on the money which passes through their hands, and is calculated hj the rules of simpU interest, the time being always considered as 1. The same Yules senre for finding the value of any quantity of stock to be bought or Mid, and likewise iot finding the price of insurance on hovses, ships, goods, Ac PjiitIV. simplb intbbjest. 4r £xAMPi.B8.*-l. Required the simple interest of 7^/. lOf. for 4 years^ at 5 per cent, per annum ? Herep^{76Bl 10*.=) 765.5. t=:4. r=(— =).05. Tbm i=zptr (tkeor, I .) =^765.6 x 4 x .05= 153.1 =: 1631. 2*. Anst^er. 2. What is the amount of 752. 10«. 6d. for S^ jetm, at 44 per cent, per annum ? Here p=(75i. 10«. 6(f.=) 75.525, ^=(84.=) 8.5, r=(ii=) .0475: whence {theor. 2.) prr+p=:75.525X&5Xi0475 4-7^.585 5=106.01821875= 106i. 0«. 4d:^.49=a, (he amount. S. What sum of money being put out at 3 per oent. simple interest, will amount to 4022. 10s. in 5 years ? Here a=(4022. 10y.=) 402.5, f=5, r=(— =).03: vjhere- r i.x. ox « 402.5 402.5 „, , 4. In what time will 3502. amount to 4022. IO9. at 3 per cent, per annum ? f/cr« p=350, a=402.5, r=.03. nn. /.I. ^x«— P 402.5—3.50 52.5 ^ ^ ^ Then (theor, 4.) i-= = — -3:5 years:=ti. the ^ ' pr 350X.O3 10.5 ^ answer. 5. At what rate per cent, will 752. amount to 772. Ss. l^xL in 1^ 3'ear, ? Her€j)=75, fl=:(772. Ss. Hd.=) 77.40625, 2= (14-=) 1.5. r.^ .^ .X «— P 77.40625—75 ^,.^ ^ T?iew (theor.B.) rar i.=i^^ r-^ =.021 38 s=2-iV per pt 75x1.5 '^ otsmp. neathfyssir, the answer. 6. What is the interest of 2542. 17*. 3d. for 24- years, at 4 per cent, per annum ? Ans. 252. 9s. S^d. 7. What is the amount of 2502. in 7 years, at 3 per oent per annum ? Ans, 3022. lOs. Od. K 8. What sum being lent for 4 of a year, will amount to 15«. C^d- at 5 per cent ? Ans* 15 shillings. 9. In what time will 252. amount to 252. 1 Is. 3(2. at 4^- per cent, per annum ? Ans. half a year. 10. At what rate per cent, fer annum will 7962. 1^ ■ttotii^ to 9762. Os« 4^(2. in 5 years ? Ans. 44- per cent. 48 ALGEBRA. Part IV. • 11. Required the interest of 140L lOf. 6d. for ^^ yeais^at 5 per cent, per annum ? 1^. To find the amount of 2002. in 8 years^ at 44 per cent, per annum ? 13. Suppose a sum^ which has been lent for 120 days at 4 per cent, per annum, amounts to 243/. 3^. l-^d, what is the sum ? 14. In what time will 7252. 15«. amount to 7312. 25. 8^. at 4 per cent, per annum } 15. At what rate per cent, per annum will 5592. 45. Od. amount to 7352. 7*. Od. in 7 years ? 23. To investigate the rules of discount. Def. 1 . When a debt which by agreement between debtor and creditor should be paid some time hence, is paid imni^diately, it is usual and just to make an allowance for the early payment 3 this allowance is called the discount. 2. The sum actually paid (that is^ the remainder, after the discount has been subtracted from the debt,) is called the present worth. 3. The debt is considered as the amount of the present worth, put out at simple interest, at the given rate^ and for the given time *. Let p::=:the given debt, r=zthe interest of 1 pound for a year, tzzztlie time the debt is paid before it is due, in years or parts of a year; then will l-{-tr.:=^the amount of 1 pound at the rate r, and for the time t: {Art. 22. theor. 2.) then also will the amount of 1 pound be to 1 pound, {or its present worth,) as the given debt, to its present worth ; also the amount of 1 pound, is to the interest of 1 pound, as the given debt, to the discount ; that 1*5, 1 + ^r : 1 : : p : P l + ^r :=.the present worth of p pounds paid t time before d«e> at r Tptr per cent, interest: also l-\-tr ; tr :: p : -^ — =2/tc discount aU ^ ^ H-2r lowed on p pounds, at the said rate, and for the said time. Examples. — 1. What is the discount^ and present worth of 2502. paid 2 years and 75 days before it falls due, at 5 per cent. per annum simple interest ? ■ In Smart's Tables of Interest, there is inserted a table of discounts, by wbich tb« diaooant of aoy snm of money may be calculated with ease and cz|>edition. Part IV. DISCOUNT. 49 Here p==950^ r=s:.05, <s=(« y.75d=) 2.^548 years, ^ ^ ^«50X8.80548X .05^87^685 ^3,33^5^ l + tr H-2.20548x.05 1.110274 242. ld«. 7d^=the discount P 250 250 1 + *r 1 + 2.20548 X .05 1.1 10274 2252. 35. A\d.:=the present worth, 2. Required the present worthy and discount, of 4872. I2s. due 6 months hence^ at 3 per cent, per annum ? Ans. pr. worth 480/. 7*. lO^d. disc. 7l 4*. l^d. 3. Sold goods for 8752. 5s. 6d. to be paid for 5 months hence } ivhat are the present worth and discount at 44- per cent, per annum ? Ans. pr. worth 8592. Ss. Z\d. disc. 162. 2«. 2^d. 4. What is the present worth of 1502. payable as follows ; viz. one third at 4 months^ one third at 8 months^ and one third at 12 months ^ at 5 per cent, per annum discount ? 5. How much present money can I have for a note of 352* 15s. 8(2. due 13 months hence, at 4^ per cent, per annum discount ? OF RATIOS. 24. Ratio • is the relation which one quantity bears to another in magnitude, the comparison being made by considering how often one of the quantities contains, or is contimQed in, the other. Thus, if l^ be compared with 3, we observe that it has a certain relative magnitude with respect to 3, it is 4 times as great % as S, or contains 3/otfr times; but in comparing it with 6, ire discover that it has a different relative magnitude with respect to 6, for it contains 6 but twice. ■ Ratio is a Latin word implying comparison. The stodent must be carefiil not to oonfoond the idea of ratio with that of proportion, as some thronf^ inattention have done : he mnst bear in mind, that ratio is simplj the C9mp€iri8on ^ one quantity to another, both being quantities of the san^e kind ; whereas proportion is the equality of two ratios : the former requires two qaaotittes of the same kind to express it, the latter requires at least three quantities, which must be all of the same kind ; or four quantities, whereof the two first must be of a kind, and the two last likewise of a kind. See the note on Art: 53, and the note on Art. 137' Part I . Vol. 1. VOL. 11. £ 50 AX^BBRA. Paxt !▼. 25. The ratio of iw6 quantifies il usuaDf expre»«d by inter- podng two dots^ placed vertically^ betw^n them. Thus the rating of a i^h^ cuhI o^ 5 ^d 4, ore vntxtieni a : h, and 5 : 4. 96. The former quiuntity is cdkd the mUeedentj and tiiie kit* ter the consequent. Thus in the above ratios, a and 5 are the antecedents, and h and 4 the consequents. The antecedent and consequent are Called terms of the ratio. 37* To determine what multiple^^ part^ or parts the antece* dent is of the consequent, (that is» to find how often it eontains or is contained in the consequent,) the former must be divided by the latter j and this division is expressed by placing the con- sequent below the antecedent like a fraction. Thus the ratio of a to h, or a i h, is likewise prcfAr^ ex- pressed thus — , and 5 : 4 fhus -~. o 4 28. Hence^ two ratios are equal, when the antecedent of the first ratio is the same multiple, part> or parts of its consequent, that the antecedent of the other ratio is df its consequent 5 or in other words, when the fttM^iofi made by the terms of the former ratio (Art. 27«) is equal to the iractioa made by the tenm of the latter. Thus the ratio of 6 :S is equal to the ratio of 3 : 4>/^ 6 3^ 8 ~4' 29. Hence, if both terms of any ratio be ttiultifdi^ or di** tided by the same quantity, the ratio h Hot alfefed. 3 Thus if the terms ofS:4or — be both multiplied by any number, suppose 6, the result = — , which fractv&n is e©i* dmtly equal to the givt/n frattitm 4 5 that u^t x 4 k tke as 18 ** 24 > in like manner ^ if the terms of the taiia a i b, or xp be both multipUed by my qtiantzty n, the resuhing fatio an t b bn, or -j^ is the same as a : b, or -fr- ; and the same in general, on b FitfrlV. lUflOK 61 30. Ilettee> oaei rMo i§ g^^after than another, when the tinte- cedept of the fbriOQer ratio is a greater multiple) part> or parts ii$ its-eonseqiient^ thati the antecedent of the latter ratio is of its consequent ; or> when the fraction constituted by the teroit mi the fin»t rtttto> is ^reatcnr thsin that conttitnted by the termi of the latter. J!^ tf : S » greater than 8 : 4^ for 6 contains^ thrte Hme$, p ft whereas 8 contains 4 but twice, or --- U greater than — . 31. Having two or more ratios given^ to determine which is the greater. ^ Rule. Having expressed the given ratios in the form of frac- tions, (Art. S7-) reduce these fractions to other equivalent ones having a common denominator, (Vol. 1. P. 1. Art. 180.) The lat- ter will expr^ the given ratios h^mag a common c6nse(pieiiti wherefore the numerators will express the relative magnitudes 6t the ratios respectively. £xAMFi.£S.-*-l. ^Vhich is the greater ratio, 7 : 4^ or 8 : 5 ? 7 d These raths expressed «» fortn qf fra^ti^n^i^ar^ — ^ ---^ ' " '4 5 whence 7x5=85, and 8x4=32, these are the new numerators; tik^4xB=:M,^ common denominator. «« /. 7 35 ' 8 32 I««r^arc —•=—;„ and -—=--- 3 and the former q£ these 4' 2Cr 5 20 ^ . ^ heing the greater, shews that the raiiq of 7 i4,is greater thim ihe roHo ofSiB. .12. Whkb is tiie greaier rMicb tteat of 6: Iti of ^t df 23:fe> 8 2^ '"' These raH^ epDpreseed Wee fraetiens^ are -- and — , wfiich reduced to other equivalent fractions with a common denofninator, 256 253 become ~~, and --3 retpeetwely s ^^ former^of these being ^ the greater, shews thatidJie ratuy S : 11, is greater than the ratio 23:32. . 3. Which is greatest/ th/^ »(ip of 18 : 25> or that of 19 : 27 ^ 4n»s the format, 4. Whiehi is \h» greatest, and whieh «h^ least, of the ration 9 : 10, 37 : 41, and 75 : 83 1 59 ALGEBBA. Pabt IV. 39. When the antecedent of a ratio is greaiter than its poase- quent, the ratio is called a ratio of thegreaUr inequaliUf. Thus b : 3, II : 7, and 2 : I, are ratios of the greater m^ equaUty. 33 \ When the antecedent is less than its consequent, tkt ratio Is called a ratio of the lesser inequality. Thus 3 : 5, 7 : 11> ^d I :% are ratios of the lesser in* equality. 34. And when the antecedent is equal to its consequent, the ratio is called a ratio of equality, ' Thus 5 : 5, 1 : 1, and a : a, are ratios of equality, 35. A ratio of the greater inequality is diminished by adding a common quantity to both its terms. ThuSi if I be added to both terms of the ratio 5:3, it 6e- 5 90 ,6 18 , , ^ ..... . comes 6:4} out -—=:—, and — =--, the latter of which {hemg the ratio arising from the addition of 1 to the terms of the given ratio) is the least, and therefore the given ratio is diminished : and in general, if x be added to both terms of the ratio 3:9, it 3 • • 3+JC ^comef 3+« : 9-f-r, that is -—■ becomes ; these fractions re* ducti to m common denominator a$ before, become ^±^ and , 4+2x ^ ■■ I tH II ».,»■. ^ Wketk tbe aotecedent is a mitltiple of its coiMeqnent, the ratio \g named a multiple ratio ; but when the antecedent is an aliqnot part of its conse^nenty tiie ffalia is naned a tubmuiUple ntio. U tha antecedent aoataias the aonseqnent ^ twice, as- V2 : 61 fdnple, 1 thrice, asF 12 :'4 >it is eaU^ a|:trit>le»> }k ration fbnr times, as 13 : 3 J (^ quadruple, J &c. &C. U the antecedent be contained in^ tbe consequent twice, as- S-i \9\ Tsubdapl^ T thrice, as 4 : 13 >it is called a^ subtriple, ^ration- four times^ as 3 : 12 J (,sabqnadn\ple,cj Sec. &c. There is a great variety of denominations applied to different ratios by tfie early writers, whUsb is Mcessary to be^ nUdei^tood by those who read the works either of the ancient mathematicians, or of their commentators, and nmy ba- seen in Chambers' and Hatton*» Dictionary : at present it ia uioal ta nalne ratios by tbe least numbers that will express thea». Part IV. RATIOS. 63 ■ ■ • respectively ; and since the latter is evidentlif the least, it ^ r* * Sf follows that the given ratio is Mmintshed by the addition of sif to ' each of its terms. 36. A ratio of the lesser inequality is increased by the addt- •tion of a common quantit}i to each of its terms. Thus if I he added to both termsofthe ratio 3 it}, it becomes 4 : 6, but -7-=rr> and —•=—-, the latter of which being the 6 30 6 30 *^ greater, shews that the given ratio is increased : in general, let 2 : 3 have any quantity x added to both its terms, then the ratio becomes 2-|-x : S+x, that is — becomes : these reduced to a 3 3+x' 6+2a? 64-3jc common denominator, become ,and , of which the 9+3 Jp 2+3 a?' -^ latter being the greater, it shews that the given ratio is' increased^ 37* Hence, a ratio of the greater inequality is increased by taking fr«m each of its terms a common quantity less than either. • Thus by takvi^ 1 from the terms of 4:3, it beeomes 3 : 2, A Q ^ O but — — ~, and -—=—-, the latter being the greater, shews that the given ratio is increased, 38. And a ratio of the lesser inequaUty is diminished by tak* iDg from each of its lerms a common quantity less than either. Thus by taking 2 from the terms of 3 1 4, it becomes I : % hut —=---, and ---=---, the latter being the leasts shews that the given roUio is dimunshed, 39. Hence, a ratio of equality is not altered by adding tOi or subtracting from, both its terms any common quantity. 40. If the terms of one ratio be multiplied by the terms of another respectively, namely antecedent by antecedent, and con- sequent by consequent, the products will constitute a new ratio, which is said to be compounded of the two fonnerj this compo- sition is sometimes called addition of ratios. Thus, if the ratio 3i4 be compounded with the ratio 2 : 3, the resulting ratio (3x2:4x3, or) 6 : 12 is the ratio com^ pounded of the two given ratios 3 : 4 and 2 : 3, or the sum of the ratios 3 ; 4 and 2 ; 3. e3 54 AWEBHA. Tmt it. 41. If the ratio aihhe compounded with itself^ the refultlog ratio a^ib^w the ratio of the squares of a and b, and is said to )^ double the i^^tio a : bf and the mtio a : 6 is mi to be ha^ the ratio cfiib^; in like manner the ratio a' : 6^ is spill to.be triple .the ratio a : b, and a : 6 one third the ratio a^ :V} also «* : £i* is said to be n ^tmef the ratio of a : (^ ^d ai : bi om ii*^of ihe ra- tio o£ a:b, 41. B. Let a : 1 be a given ratio^ then ^r, l,a^ :l, €^ : I, a^:l, are twice, thrice, four times, n times the giv^n r^tio, where n shews what multiple or pail of the ratio |t" : 1 ihe |^en xdti» <i : 1 is ; hence tbie indices I, 2^ 3, 4« , . . «, are caUed ik» mea* sures of the ratios of a, a^, a^, o^, ... a* to 1 r^p^ctive]y> of the logarithms of the quantities a, o^ c^, aS • • . a** 49. If there be several ratios, so that the consequent of thf first ratio be the antecedent of the second 3 the consequent of the second, the antecedent of the third $ the consequent of the third, the antecedent of the fourth, &e. then wfll the ratio compounded of all these ratios, be that of the fiiBt antecedent to th% last con- sequent. For letaih,b:€,e:4,die,'8fe»he any number of given ra* tios ; the$e compounded by Art, 4Q. pUl be Qixb:f^cxd:hxc^4 dxbxcXd a X e, or) , — = — ,oraie, the ratio of ihfifir^mtecedmt bv^cxdxe e ^ • ^ . -r a to the last consequent e. 4S. Hence, in any series of quantities of the same Und, 4ht tot wfll have to the last, the^ratio compounded of the ratios of the fim to the.second^ of the secoQd to the thir4» of the thJini to the fourth, &c. to the last quantity. 44. If two ratios of the greater inequafity be tompoundei together, each ratio is increased. Thus,, let 4:S be compounded with B i^,ihe resulting ratio ^ 4 5 (4 X 5 ; 3 X 2 or) —- w greater thga either -^ytr -^,m ^Vf^f^^kn reducing thesefractions to a common denominator. Art, 31. 45. If two ratios of the lesser inequality be compounded to- gether, each ratio is diminished. Thus, let S: 4 be compounded unth 2 : 5, the resulting ratio (3 X 2 : 4 X 5 or) — , w less than either of the givi^ r^w— or — , as appears by reducing thes^ fractions as before* 40. If a ratfa) of the greater inequality be compounded with a ratio of the leaa, the former will be diminished^ and the latter increased. Thus, let 4:3 be compaunied mth ^:&, the r^nMng r9tio 3 4 (4xS:3x5 4>r) :^,ii Usg than the nUw--, but greater than the 15 S ratio -r. 6 47. From the composition of ratios, the method of their de- coBoposition evideatly £d11owb; for since ratios may be repre- sented like fractions, and the sum of two ratios is found by mul- tiplying these fractions representing them together, it is plain that in order to take one ratio from another, we have only to divide Ihe fraetion r^resenting the formerly that representhig o the latter. Hence^ if the ratio of (3:4 or) — be compounded 5 with the ratio of (5 : 7 or) —^ w^ obtaux the ratio of (15 : 28 or) — i DOW if 'from' this raObfo^ve decompound the fahatr of tl^e givea nHoB, naibely ---, the tvsult wfflbe (— x --as—a) — ^, which is the latter of the given ratios -, and if from the com- 15 5 poimded ratio — , we decompound the latter given Mtfo — , tine »8 7 15 7 105 3 result will be (55X— =r-^=)-7=the fbrmelr given ratio: wbeoQe «to subtiact one ratio Cram another, thU is the riile. Auxs. I^t the ratios be represented like fractions. (Art. 27.) Invert the termos of the ratio to be subtracted, and then multi- *ply the correspondent terms of both fractions tpgether ; the pro- duct reduced to its lowest terms will exhibit the remaining ratio, or that which heing compounded with the ratio subtracted, will give the ratio fit)m which it was subtracted. ExAiifrFLBs. — 1. S^rom 5 -.T^ let 9 : 8 be subtracted. '5 9 l%jgie raiiot reprsnented like fractifms^ are — and — . 7 " b4 66 ALGEBRA. Past IT. 8 5 8 40 The latter inverted, becomes — j wherefore — ^ "5"=^* ^ 40 : 63,. the difference required. 9, ¥tam 6 : 5^ decompound 7 ' 10. Thug — X — =--= — > or 12 : 7» <ft« difference required, o 7 oo / 3. From the ratio compounded of the ratios 8 : 7i 3 : 4^ and 5 : 9^ subtract the mtio compounded of the ratios 1:2, 8:3, 9:7, and 20: 21. Thus _x-X-x-X-^X-^X-=.~=7:24, the difference. 4. From a : b decompound x:y. Ans, asf : bx. 5. From 11 : 12 dk»mpound 12 : 11. Ans. 121 : 144. 6. From 3 : 4 take 3 : 4. Ans. 1 : 1, 7* From a : x take 3 a : 5 x, and from ax : y^ take y :9ax. S. From the ratio compounded of a : b, x : z, and 5 : 4, take the ratio compounded of 5 fr : x, and 2 a : 3 z. 48. If the terms of a ratio be nearly equaU or their diffisrence when compared with either of the terms very small, then if this difference be doubled, the result win express double the given ratio ', that is> the ratio of the squares of its terms, nearly. Let the given ratio be a+x:a, the quantity x being very sv^l tshen compared with a, and consequently stiU smaller when compared with a+x; then wiU (a+x]*, or) a*+2aae+x* : a* be ih^ ratio qf th^ squares of the terms a+x and a : and because x is small when compared with a, xjs (or x^) is small when compctred with ^a.x, and much smaller than a.a;^ wher^ore if on aecoumt of the ejpceeding smallness of ofi, compared with the other quantities, it be rejected^ then {insteqd of a* +2 ax + a?* : a*) we shall haoe a» 4-2 ox : a^ ; that is, {by dividing the whole by a) a+2x : a, for the ratio of the squares of a+x : a, which was to be shewn. .Examples. — 1. Re(juired the ratio of the square of 19 to the square of 20 ? Here a=s 19, x= 1, and ■ ■ =g^, ther^ore by the preceding a "y* X ^ V a 19 article, — ^ ==2p5 ^^^ •*> *^ **^**^ ^f ^^e square of 19 to the a -^ 2x '2 1 *ART IV. RATIOS. 6T /■o/^ ,.. «, , T, 1^' ,361 ,7681 .19 square of 20 is 19 : 21, nearly. For — =( — =^)' , and -- •^ * «0« MOO '8400 «1 7600 19 ^^ AAivk ' ^^^^^^^^^^y the ratio — is somewhat too great, but it 19 exceeds the truth by only ; which is inconsiderable. ^ ^ 8400 2. Let the ratio of 8o|* : 79l* be required? „ a-f-x 80 «4-2x Here as=79, jps=l, ctmsequently =z:-, ond — r— = a 79 79 81 --, or 81 : 79=*^ ratio of 86l« : 79lS nearly. _ 80» ,6400 .505600 ^ 81 505521 ^. ^ ,, For -—=(--— -=) — -— --, and — 5= • lomc/i t/iere- 79* ^6241 ^493039 79 493039 79 fore differs from the truth by only 493039' 3. Let the ratio loS* : ill)* be required ? jins. ^. 4. Required the ratio iooil « : 1000 1« ? Ans. — . 6. What are the ratios 3009)* : 3oIo]S and lOOOOl* : 100051*? 49. Hence it appears^ that in a ratio of the greater inequality, the above proposed ratio of the squares is somewhat too small ^ but in a ratio of the less inequality, it is too great. 50. Hence also^ because the ratio of the square root of a+ 2x:al8 a-^x :a nearly^ it follows that if the difierence of two quantities be small with respect to either of them, the ratio of their square roots is obtained very nearly by halving the said di£Perence. Examples. — 1. Given the ratio 120 : 122> required the ratio • 1201t:122]x? ^ 120 a a 120 Here a=120, 2 j:=2, 755= -t^* •' —rz=7^> ^^ 130 : 122 a+2a? a-^-x 121 121 ss the ratio of ISSIt : I22I i, nearly. 2. Given the ratio 10014: 10013, to find the ratio of their square roots ? Ans. 20027 : 20026. 4. Given 9990 : 9996 and 10000 : 10000.5, to find the ratios of their square roots respectively ? J» AUmnUL Paw IV. 51. Bjr fifaiiiUar nasonb^ it may be shewn^ that the ratio of (Hie cubes, pf the fimrth powers^ of the nth powers, is obtained \jiy taking 3^ 4, n tiroes the difbrence respectivety, provided S, 4j or n times the difference is afxtaSl with respect to either of the terms. And likewise, that the ratioof the 3rd, 4th, or nth roots are obtained nearly by taking ^, -^^ i part of the difference respectively. S%. When the terms of a ratio are large numbers, and prime to eadi otlier, a ratio may be found in smelter numben nearly equivalent to the former, by means of what are called continuied firactions <. h Thug, let ^git7€n raiiQ he esftetrnd bf — , cmd let b contain a, c times, with a remainder d; let a contain d, e times, with a) 6 (c a remainder/; again, iet d eon" d) a (e tain f, g times, with a remainder f) ^ (JS h, and so on ; then by multiplying h) f (k each divisor by its quotient, and I) h (m adding the remainder to the pro* n)JJp ■duct, there arises f, Ac b=ac+d, a^de^f, d-fg^ h, h^s^lm-^n, l^np+q, BfC. b jac^d \ d , Hence the given fraction — ac ( ■ ■ =) c-\ — , but aszde-k- • "^ a ^ a a fi thU value substituUd for a in the preceding equation, ise shall have — =(c-f-r — ■=) c+ ji but since d^szfg-^h, by a substituting this value for d in the preceding equation, we shall ■ ■■ ■' ■ l lll H llll | | ,1 III IWI »—— 1M I I f , « Th0 acMiod ef 4ii4iQ^ tbe appronmate vahit of a ratio in small munben, has been treated of bj Dr. Walltg,iD his TteaHse tf Jlgebra, c 10, 11. and in a tract at tbe end of Horrox's Works ; hj Huygens, in Descript, Autom, Planet. Op. ReUq, p, 174^ 1. 1 ; by Mr. Cotes in his ffarmonia JUensiSramm, .and by several others. have — =<c+ — i-y- <=) c^- . ^ j &ft< oMte/sM+l, 4y «if&^£iftf ^ifl^ /^t5 {Vo/Ke /or f m the precedii^ equation, we thaU , ^ 1 1 have-'=z(c+ =).c+ 5 6tt* As|m+n> ^^ J . ■ ■■■ g-^ '■ '■ ■■' A 1 therefore ly substituting as Iff are, — =(cH j ss) . ' ^ l^-H — ' r^ gi- m » i» n *+ ' 1 - * cH ' r ,•> 5«« Z=«p+g, therefore e+ : ^+ *+-^ ■ n / tc + J =) C4--r- c+ ■ j t ■■ ■ e-{- fir+ 5 «+ ft+ — i.^ *+- "^ ,,1 J ^^ Sfc, a continued fraction, P Now in this continued fraction, if one term onhg (viz. c or y)6« h taken, it wiU be an approximation to the ratio — in small numr heri: if twe^fns,viz.c^^i:=^~') be taken, it wUl be a ^ ^ ^ 1 40 ALGEBRA. Part IV. .« nearer approjAmaium than the farmer, to the ratio — ; but neces- sarily expressed hy a greater number of figures: if three terms be taken, viz. c+i 1 =(c-h -4.=c+^=) S?!±£±£,a c-f— ^ gg-H ge+l g«+l ^ g nearer approxitnation to the ratio — expressed by stUl more figures; if four terms be taken in, we shall have c-) 1 = *+T (c+f 1 ^c^^l k = k 1 CI ^^-^'^ ' egk^e+k gk+l -^^egk+e+k ' "^ egk-^e-^k • 1 ExAMFLSB.— 1. Required a aeries of ratios in smaller num- bers, continually approximating to the ratio of 12345 to 67891 ? 12345) 67891 (5 61725 6166)12345(2 12332 13)6166(474 52 "96 91 56 52 4) 13 (3 12 1 Here 6=67891, a= 12345, c=5, d:=:6\66, «=2, /=rl3, g=:474, ^=4, /c=3, /=1. Then ---=-—, an approximation to the given ratio, in the least whole numbers possible. ^ „ cc+1 5x24-1 vll . ,. J Secondly, »( ^/"o"' ^ ^^(^^^r approximation. e % % Pav IV. BATIOS. m -PL* i; ^«+<^+^ ,5x474x2 + 5+474 ,5919 Thirdly, — ~^( tzz — :: — : =) > ^ ^' ge+l ^ 474x2+1 ' 949 nearer approximation than the former, cejr*+ce+r*+|rAf+l Fourthly, «f^+e+ip 6x2x474x3+5x2+6x3+474x3 + 1 ^ 15668 5— ( 2s) , a still ^ 2x474x3+2+3 ' 2849 nearer approximation than the last, 2 . Required approximate values for the ratio 763 1 7 1 ; 3 101000 in more convenient numbers ? Operation. 753171) 3101000<4 3012684 • 88316)753171(8 706528 46643) 88316 (I 46643 41673) 46643 (1 41673 4970)41673(8 39760 1913 *c. Here 0=753171, 5=3101000, c=4, d=88316, 6=8,/=; 46643, g=l, A=4ie73, kszl, 1=^4970, ot=8, i»s1913. e 4 Therefore — =— , thefint approximation, ce+l 4x8+1 ,33 ^, , , ^, s:( — -de) — , the $econd <qtpfoxtmaium, e ■ o 8 cge-^C'\'g ,4x1X8+4+1 .37 ,. ,,. , — : — = ( =x) — , the third approxi-. ge^l ^ 1x8+1 ^9 if motion. C'6'gAf + (?€+ c/f-^grAf + 1 €gk'\-e-i-k 4x8x1x1 + 4x8 + 4x1 + 1x1 + 1 V 70 ^. . . -s(- ■ — . =3) •-—, the fourth ap- ^ 8x1x1+8+1 ' if ^ ^ proximaiion, dsc jjrc 3. The ratio of the diameter of a circle to its circumference is nearly as 1000000000 to 3141692653 } required approximating vjihies of tbw ratio in smaller numbers } 5 ® 383 Ah8. TlmfirH —, <Ae «ecoft<l — , the third 7-^, **« fourth 1 . 7 '"^ 355 . m ' *'• 4. Required approximate expr^^Oflff in small numbers for the ratio 78539811635 : 10CX)O00000O, being tbxt df the area of a circle^ to the square of its diameter, neady ? ^ 1 3 4 7 11 17« 355 „ . ^ 1' 4' 6' 9' 14'«19'452' 5. IF the side of a square be 1234000, its diagonal will be 1745139, nearly ; required approximatioDs to this ratio in smaller numbers ? OF PROPORTION \ 53. Four quantities are said to be proportionals, when the first has to the second the same rtttio which the third has to the fourth; that is, when the first is the same multiple, part, oc parts of the second that the third is of the fourth. ' Ratio is the comparison of magnitudes or quantities ; proportion is the equality of ratios ; hence there mast be two ratios to constitute that equality which is called proportion ; that is, there must be three terms at least to expresf the two ratios necessary to a comparison. Some authors have, with the most unaeeounlable nejfligeaee, eeafonnded and perplexed t)i«ir inexpe** rienced readers with the definitions they liave given of ratio and proportitm. Dr. Hntton; to whose useftil labours almosteriery branch of the mathematics is indebted for elucidation or improvement, in his system of Elementary Mathematics for the use of the Boyal Military Academy, thus defines them : ** Ratio is the proportion which one magnitude bears to another magnitude of the same kind, with respect to quantity ;" and immediately after, ** Proportion is the epuilUy of rattog" Now it has always been held «s a necessary maxim in logic, that <* in every definition the ideas implied by the tenna oi the definition, should be more obvious to the mind than the idea of the thing defined/* otherwise the definition fails of its- purpose ; it leaves us just as wise as it found us. Wherefore, supposing the above definitions of ratio and proportion to be adequate and perspicuous, as they ought to be, if we appfy this doctrine to them^ it will follow from the fonb^r, that the idea of proportion is more obvious than that of ratio ; and from the latter^ that theiden^nf laitip is more obvious than that of proportion ; but the supposition that both these conclusions are tttie, implies a idanifest absurdity,- and consequently, that one or both of these definitioDs must be fimlty. It iB but jastioe to suppose, tiiat^ the learned Doctor must have used the tenn frvjpoHwn^ in the foriaier ditff^iitiiMi^ 64. 'ttis prdpoftiOd>ar equalitjof ratkiB, Is taBuftfly eiipi^ssed by four dots, thus : : interposed between the tiro-iMios. Thus, d:b::c:d, shews that a has to h the same ratio that c has to d, or that the four quantities, a, b, c, and d, are propor* tionals, and are usually readj a is to b, as c to d. 55. Tht first and last terms of the proportion (viz. a and d) are called the extremes, and the two middle terms (6 and c) the means. 56. Sinte it has been shewn^ (Art. 97.) that any ratio is truly taqirened by piwii^ its terms in tlie form of a f^ntioD ; therein fare, when four quantities are propostionakir that is, whto tte first has to the second the same ratio whdck the third has to tko fourth, it follows, that the firaction constituted by the terms of the first ratio, will be equal to»the fraction constituted by the terms of the other ratio placed in the same order. a c b d Thus, if ai b::c:d, then will -p-=---, or — = — . h di a c 57. If fow fuaiiities are proportioiials> the priKiiiot of tka extremes isieN|Ml t^ the product of the means. a e Let a \\ lie id, then by the preceding article, -t-=s-t; muU d • ^ c tiply the terms of this equation by bd, and (-r- x bdsz— x bd, or) tk d ad=zb€. Euclid 16,6. 58. Hence, if three quantities are proportionals, the product of the extremes is equal to the square of the mean. a c Let a:c:ic;dj then — s=-y, by what has been shewns mul" e a a c tiply botk sides by cd, and ( — xc<f=a— xcfll, or) ad:=z(^. Eudid 17> 6. ■coonting to its vuigar acceptation, (natetly, the oo «yw ao>«f oat thii^ wiU» aaother,) and in the latter, according to its mathematical import. The leaxncv «n§^ to bo eau tio a e d to ttndy not to be hnpoeed on by tbe double meaning of words, and especially to scorn the mean artifice of araiiing himself on any occasion of the aJhbi^^ify of language. A wrangler may confound bis opponent by using the siUne word in two or three different senses ; btit triitii (whkfi is- tfate gtttnd object of science) is discovered only when our rt^ttoniog inidceeds by meafks of t^rms which are strictly limited in their signification. 64 AXiGSRRA. PaUt IV. 59. Hence, if three temis of any proportion be given> the fourth may be found : Fo'^ since I z=hc, if a, d, and h, are given, then ---zrzc; if a, o d, and c, are given, -— =& ; if a, b, and c, are given, — =d ^ and be ■ if d, b, and c, are given, then -r=^ '• 60. if the product of two quantities be equal to the product of two others, then if tlie terms of one product be made the means, and the terms of the other product the extremes, the four quantities will be proportionals. ad be Thus, if ad=bc, divide both sides by bd, and (-rz^-n* ^^) •^ • ^ bd bd a c --= --, that i8,a:b:;cid. Euclid If, 6. o a 61. If the first term be to the second, as the third to the fourth, and the third to the fourth as the fifth to the sixth, then will the first be to the second as the fifth to the sixth. Let a:b:: c: d, and c: dii e if, then will aibi: e :f; for <^ c . c e ^ . a e . , , . b d d f of 69. Hence, if the same ratio subsists between every two ad- jacent terms of any rank of quantities, that id, if the terms are in continued proportion, the first term will be to the second as the last but one to the last. Or b For, let a, b, c, d, e,f, g, h, k, I, Ssc. be such, then '-r-= — 6 c c d e f g h k a k d e f g h k I b I b:: k:l. 63. If four quantities are proportionals, they are also pro- portionals when taken inversely. « This article furnishes a demonstration of the Rale of Three, except tint part of it which respects the reducing of the terms : but the latter is obvioas ; since in order to compare quantities, it is plain we must bring them to a sim- ple form, and likewise the quantities compared must be of the saqie deaomi- na^ion, otherwise a comparison cannot be made. Paet IV. FftOFORTlON. «} a r Let a:b::c:d, then will b:a::d:c; for since ---sr-j-, let a unity be divided by each of these equal fractions, and the qMotients (1-+— r-=) — , and (l-H-v=) — ^ill be equal, wherefore biaxi o a a . c d:c; this operation and property is usually cited under the name iNVBRTfiNDo. Euclid pr. B. Book 5. 64. If four quantities be proportionals^ they are also propor- txDnals when 'taken alternately. ^ a c Let a: b::c: d, then will aicwbid; for ----r=---, where- a fore multiplying each of tkbse equals by — , we have (—- x — =» c b a b . — r X — , or) — ss-r> Ihat is, a:€::b:d; this is named altbe- d e c d NANDO, or PERMUTANDO. Euclld 16, 5. $5. If four quantities be proportionals, the sum of the first and second is to the second^ as the sum of th« third and fourth to the fourth. a Let a:b::cid, then will n+b: b::c-^d:d. Because — = b -—, let unity be added to each, and (--+ ls=--;-f I, that is) — r— a b d o = , wherefore a-|-&: 6 ::c-f d: d; this is named comfo- NBNDO. Euclid 1^,5. 66. In like manner, the first is to the sum of the first and second, as the third to the sum of the third and fourth. _ a-^b c+d , ,6 d , ,^ . >.^v For since -^r^ = — -~, mvertendo r= > also (Art, 62.) b d a+b c-^a ^ • . , J ^ . b , d , ^ abd bed adh=zbc: wherefore ( ^xad^- ,xbc, on ^-2=- — -^ m- a c vide these eouaU by bd, and r=: -, or ar a-f 6 : : c : c4- d. a^b . c-^d 67. If four quantities be proportionals, (he excess of the first above the second is to the second, as the excess of the third above the fourth is to the fourtli. a Let a:b::c:d, then v?v[l a-^b ; h r : c— d : d. Because —ss VOL. IX. F M ALGSfiRA. T^n If^ C M C -—, let unity be subtracted from each, and •(-r^l=-;r'~l* ^) a^^b c-^d — -—=—-—, that M, a— 6: 6::c— d:d; this is called divi- b a DBNDo. Euclid 17> 5. 68. In like manner/ the first fe to its excess above the second,' as the third to its excess abo%'e the fourth. a — b c — d b d Because —j— = , by the preceding^ and siMX — = — , b d a c ^ a^b b c— rf d a— ^ c— d , therefore — ; — -x — = — r— X — = — • — r= --, or a — b : a :: b a d c a c e — d : e, and invertenda (Art. 63.) a: a^^h : : c : c— d; this is CONVBRTENOO. 69. Hence^ because a-^b : at: «^d : c, the excess of the €i«t above the second is to the first, as the excess of the third above the fourth to the fourth. 7X). If four quantities be proportionals, the sum of the first and second is to their difference, as the sum of the third and fourth to their difference. Let a: b :; c : d, then mil 4+ 6 : a— 6 : : c+d : c— d; for a-^b c-\-d a— 6 c — d since —■ — =—--—, (Art, 65.) and — r— -as , {Art, 67.) divide, b d d ' , ^ , , , • , fl+ft ^—b c+d c— i the former equcus by the latter, and (— ; — i — ; — = — ; — •- — 7—1. •^ . b h d a or) ^ — r= r, that is, a-^-b : a — b : : c4-d : c— ^. a— 6 c^d 71 . Hence, the difference Of the first atid second is to thtelr sum, as the difference of the thi)*d and fourth to theh* Sum. Far since a-j-b : a — b :: c-fd : c-^d, therefore imicrtendo a— 6 : a + b : ; c—d : c+d. 73' If several quantities be pfoportlonals, «s any one of ike antecedents is to its consequent, so is the sum of any number of the anteoedents, to the sum of their respectiv« consequents. Let a : b :: c : d :: e :f :: g : h .: k i l : : m ; n, 8(c. then ioiU a: b :: a+c-|-e+g^+>-f wi : 6-f-d+/+*-f f-hn. Because a : b :: c : dt therefore ad^^bc, and abszba; also, because a : b ::■ e :f, therefore afssbe; in like manner^ ah^ssbg, alzsibk, and anss bm: wherefore {ad-\-af+ah-\ral+ans=^bc^be'\-bg+bk'j^bm,or) flxdH-/+^+/+ii=6xc-f e+^-f/f-Hw, wherefore a:6::c4> Part !V. PROPORTION. 07 H-f+*+»» J <i+/+^+^-f*»; ond the like may be proved, whatever number of antecedent^ and their respective consequents be taken. 73. If fonr qaantities be proportitoals^ and if eqaimultiples or «qiuil ^arls of the first and aecand, and equimultiples or #qu$l pait9 fk the third and fourth, be tdceD> the resiidting quaatities will likewise be proportionals. Thus, if a : b i: c : d. Tbm will 1. ma mb : me : md • 2. ma ; mb lie : nd t. ma : mb :: r n r ^ n 4. r — a : ft It :: mc : md 5. f» -^tf : m — b :: m r < — c : s s For in each case, (by multiplying extremes and i^eans,) ad=:bc, or -7-=--r-> or a : b :: c: d. o a 74' HeMe^ if two quantities be prime to each dther, they Vt^ the le«iit in that proportion. 75. If four quantities be proportlonais^ and the first aad third be multiplied or divided by any quantity^ and also if the second and fourth be multiplied by the same or any other quantity^ the results will be proportionals. Xtf t a : b :: c : d. Then will 1. ma • • nb : : mc nd • 8. ^ m • • b . c . mm 1 y • n m d n ft 3. ma • b "^ : : me : n 1 , 4. ma • • mb :; mc : nd t 5. • m ft ft nb ; : — : m nd, 8ic. k 1 Bar in eOfdh case, ^rmdtipiyingmftremesahd [ means,} ad:a^bc,0t a c ...... -rrsz—-', or a:b : : c : d. b d 76. Hence, if four quantities be proportionals, their e^ui-' multiplefl^ as also their like parts, are proportionals. F 2 68 ALGBWIA. . Pait IV. 77. Heoce also, if instead of the first and second tenns, or of the first and third, or of the second and fourth^ or of the. third and fourth, other quantities proportional to them be sub- stituted, the results in each ca^e will be proportionals. 78. In several ranks o€ proportional quantities, if the cor- respcHiding terms be multiplied together, the product will be proportionals. Thus, let a : h :: c : d^ And e :f '.'. g ". h\ then tc'ill aek : hfl :: cgm : dhn. And k : I :: m: nj, aek : bfl :: cgm : dhn, and the like may he sheum of any number of ranks. 79. Hence it follows^ that the likQ powers of proportional quantities (viz. their squares, cubes, &c.) are proportionals. For, let a : b :: c : d And a : h :: c : d Also a: b :: c : d, 8fC. then by multiplying two of these tanks together, as m tfie former article, we have «* : 6*': : c* : d^, and by multiplying all the three, a^ : 6^ : : c^ : d' ; and the like nun^ he shewn of all higher powers whateder. 60. Hence also the like roots of proportional quantities are proportionals. For, let a: b :: c : d, then will or : br n cr: dr^ for -t'= ^.. /. tt c ffT cr III » -3-, therefore ^-r'=- \/--r» ^"^^ *** r~~3~» ^ or : frr : : c»- : a o a b^- d^ dr, and the same may he shewn of any other roots. The c^ration described in the three foreg;oing articles. Is called COMPOUNDING THE PROPORTIONS. 81. If there be any number of quantities, and also as many others, which take^n two and two in order are proportionals, namely, the first to the second of the ^t rank, as the first to the second of the other rank ; the secotid to the third of the first rank, as the second to the third of the oth^r rank, and so on to tlie last quantity in each f then will the first be to the last of the first rank, as the first to the last of the other rank. PabtIV.. PROPQRTION. 69 » . ... d: e :: k : I Then will a : e ::f: I; for if the above four proportions hfi compounded^ {Art, 78.) we shall have abed : bcde : ifghk : ghkl, .abed fghk . €t f , ^ ^ . , ^^ ^^ ghkl' ^^ Tl' ^*^^'** a:e::f:l,and the like may be demonstrated of any number of ranks. This IB called sx jeolvkli in fropostion£ ordinata, or simply BX mwjo ordinato. Euclid 22, 5. 82. If there be any number of quantities^ and as many others^ which taken two and two in cross order are proportionals* namely^ the first to the second of the first rank, as the lost but one to the last of the other rank ; the second to the third of the first rank^ as the last biit two to the last but one of the other rank, and so on in cross order ; tben will the first be to the last of the first rank, as the first to the last of the other rank. ra : b :: k : I Let a : b : c : d : el , ^. i, ^i a) f> - ^ '» h : k Andf:g '.h'.k'.lS^*^'' ^^^\ c:d::g:h \d:e::f:g Then wiU a : e ::f : I; for compounding the above four pro* portions, (Art, 78.) there arises abed : bcde : : khgf : Ikhg, or (t-t-= ., t j that is,) — sr-^-* wherefore a: e ::f: I, which was ifcde Ikhg 'el -^ '' to he shewn ; and the like may be proved ef any numher of ranks. llib is called ex jaayALi in proportiokb pbbturbata, or siniply, BX mq,uo pbrtukbato ^ Euclid 23> 5* INVERSE, OR R£GIPRCX:AL PROPORTION, 83. The foregoing artides treat of the pn^rties of what Ib called DiBBCT Pbopo&tion, where the first is to the second as the third is to the fourth ; but when the terms are so arranged. ^ It must be undentood, that what we bate delirered on proportion, refers to eommenturabU magnUude* only : it is io sobstaDce tbe tame as the Slih book of Euclid's £iemeiita, except that- the doctrine there deliverid iocludes both eommenmrabU and meommensurabie nagnitndet ; Eaclid has effectod this double object by means of his fifth definition, which although strictly feneraly has been justly complained of for its ambiguity and clumsiness. F3 '';fc 70 . hUSSmti^ y«w IV. that the first is to the second, as the fourth to the third, it is then oamed Ivybbsb PaopoKTioH» and the fovri^iuMBtlties in the order thev stand, are said to be rnvtasKLY paoPonTioNAL. Thusy 2 : 4 : : 12 : 6^ and 9 : 5 : : 10 : 18> *c. are inverseUf proportional. 84. Hence« an inverse prpportion may be made direct, by chaining the otder of the terms in either of the ratios which constitute the proportion. 85. The reciprocals of any two quantities will be inversely proportional to the quantities. Let a and b be two quojitities, then vfiU a : ( : ; -r* : — , for muHipl^ing both terms of the latter ratio by aby tee shaH have a : b:: (-r- :-:-::) a : b, therefore a: bz: -r-i — ; inlikeinanr o a o a 11 ... ner b : a :: — : ^r-, that is, the direct ratifi^ o^ tfte qui9fi^tiB» i^ a V » the same as the inverse ratio of theit reciprocals ; and the inverse ratio of the quantities^ the same as the direct of their reciprocals. Hence, inverse proportkn i* Ukt^i^ frequently chilled reci- rfiOCAL FROPQ&TIQN. HAKMONICAL PROPORTION. 86. Three quantities are said to be in harmonical or mueieal pro[>oriion,' tvhen the first is to the t^iird, as the difierenee of ike fii-dtaAd second, toi ihe di^«aAe««:eiof the seeond WMt thirds fUid fouii t^nm are mi to be in h^H^mwical proportipnA i?f hen the lirst is to t\^ fyvLTiium the. dtflFwenoettf tlw &^ and seoeAd m to the difference of the third and fourth. TAds, tf A: e:: a^-^b : b^c,> then an the (htee quantities, 4> bw^dsy hafimimkaUit. pfoppr^ipnoL A^d \fia,:dr.:0r^bi^'^d,.tkm!air^'th$fQufyai,b,c,audd, Mrmim^flUy proportional; &7. Hence^ if all the terms of any harmonical proportion be either multiplied or divided by any quantity whatever^ the ropults .w'iU still be in hai^oiopiqal proportion. 88. If. double the product of anjf two quantities be divided •by their s«di> the ^otient will be a bann(»mcai mean betn^eeti the tw'o qtiantilies. 9iw IV. VRpgmXW^ « duct, and 04-1= their sum, wherefore r is the harmonical a-^-b mean required, for (Art. 86.) a . 6 : : a : ( — xa r=: a^f-A a tt'+'O ', , = — ; =) T^^; that is, the first is to the third, as ike d^fttence between the first and second to the difference be- tween the second and third. Examples. — 1. To find a harmonical mean between 9 and 6. «T ^ . « , ^ab ^ , . , ^ Here a=2, 6=6, a»d ^-=---=3, the mean required; for e:©:: (3— «:6— 3 ::) 1 :3. % |l«quired a harmonicat mean between 24 and 12? Jns. 16. 3. Heqttired the harmonieal mean between 5 and 20? Ahs.S. 4. Required tbe harmonieal mean between 10 and 30 ? « 89. If the product ei two given qaaatitiM be divided bf the difierence between double the greater and the less^ or double tfete le$s Mod the greater, the quotient will be the third harnMni* cal proportional to the two given quantities. Let a ijmd b be twogi^sen quasUities, whereof ais the greater $ 4he» tnU be the iln/rd harmoni eal proportional to a and b : I ■ ■ ■ . A . ab , Mka — ^ oh—h^ b^—ah , ab fora--: 1 1: a— 5 : ( — =_- r=- — ---=so^ - ■ ' -^j •^ 2a-.6 a.2a^ 2 a— 6 6— 2 a 6— 2a ^ ab > the difference between ihe secgnd and third '• 2 a— i I I I I III I PI t ■ . t ■- T ^ **?' — t Td wlmt bas been safd on this subject, the following pftrtiealan rclttinf to the comparison, &c. of the three Ikinds of pro po rtionals, my be a^Asd; viz, I, The reciprocals of an arithmetical progression are in harfnooical pro- gression, and the reciprocals of a harmonif^ai pr<^ression, are in arithmetical pragifMioD. 7%us, a, a'\'d, a'\-Zd, a-^-Sidt are arUhmeticuUy proportional, ^*^ T' HM ' 5+2^ ^+34' '***^ reciprocaUy are hartMrncaUg preporHMol^ and tibe contforse* F 4 <f •# 73 ALOEXRA. Paw IV. ExAMFLBs.-*!. To find a tldrd bsmiomcal proportiDnal to 48 and 39. rr .« r «« J «* 48x33 1536 ^^ • Here a=:48, 6=32, and r=- — -- — -.=s-----=54, 2 a- 6 2x48—32 64 the number required; far 48 : 24 : : (48—32 : 32—24 : : ) 16 : 8. 2. Required a third hannoaical proportional to 2 and d,> Ans» 6. 3. Required the third harmonical proportional to 20 and 8 1 Ans, 6. 4. Required the third harmonical proportional to 10 and 100 > 90. Of four harmopical proportionals any three being given^ the fourth may be found as follows. , Let a, h, c, cmd d, he four quantitkn ia harmmical propor* tion, then since a : d:: a— 6 : c— d, (Art» 86.) by multiplying extremes and means, ac — ad=ad — 6d; from this equation OMg three of the qwmtities being given, the remaining one may be found. ac Thus, a, 6, and c, being given, we have ds= - — r one of the bd extremes i if b, c, and d, be given, then azpr-^ — the other ear* treme; if a, b, and d, be given, then ess——* om ^f the a mean* ; ./ a. c. and d. be given, then b^^-^tl^ theother mean. 2. If there be taken an arithmetical mean and a harmonteal mean between any two quantities, then the fonr quantities will be geMnetrically propoctiooal* Thutf between a and h the harmonical mean is — rT> and the arithme' a'\'b ^ab a-^h Heal mean — - — , and a : — r-r : : — - — : b. 2 a-^b 2 3. The following simple and beautiful comparison of the three Unda of pro«> poTtionals, is given by pappus, in his third book pf Mathematical CoUeotiopt* Let a, bf and c, be thejirst, second, and third tertnt ^ thent C JrUhmetieals a\a' < Geometrieals a l^Harmomcals a 4. There is this remarkable difference between the three kinds of proportion ^ namely/ from any given term there can be raised A continued arithmetical series, increasing but not decreasing, '\ . A continued harmonical series, decreasing but not increasing, > A continued geometrical series, both increasing and decreasing, J ^^*^/* In the< Geometrieals a: b>:: a-^b : 6— tf. Tart IV. PROPORTION. 73 Examples. — 1. Let there be given 3, 4, and 6, being the first, second, and third terms of a harmonical proportion, to find the fourth ? Here fl=3, 6=4, c=6, and -_^=(-__=-==) 9, the fourth term required; far 3:9:: (4—3 : 9—6 : : ) 1 : 3. 2. Given the second, thirds and fourth terms, viz. 4, 6, and D, to find the first ? Here £>=r4, €:^6, thud, vtherefore a^s — f— -=(- -= 36 — =) 3, the first term required. 3. Given 3, 6, and 9, being the first, third, and fourth terms^ to find the second ? rr. « ^ J ^ J . 2fld— flrc 54—18 . ^ ^ .rore «=s3, c=6, d=:9, and 6=—; — s=:{— — as) 4, • d 9 f/i€ second term required. 4. Given 3, 4, and 9, being the first, second, and fourth, to find the third ? tr o 1 . J « J 2ad-M 54-36 . ^ Acre aa=3, o=4, d=9, and c= — — =s( — - — =) 6, a o the third term, as was required. 5. Let the first, second, and third' terms in harmonical pro- portion, viz. 36, 48, and 7^> be given to find the fourth ? . ^ 6. Given d4, 36» and 54, or the second, third» and fourth terms, to find the first ? 7. Given 97% 36, and 81, being the first, second, and fourth tanauB, to find the third ? 8. Let 48, 96, and 144, being tbe first* third, and fourth, be ^ven, to find the second ? 91. Three quantities are said to be in contra-harmonical PROPORTION, when the third is to the first, as the difference of the first and second to the difference of the second and third. Thus, let a, b, and c,l)e three quantities in contra-harmonv^ cal proportion t then will c : a : : acssb : &CV)c. 98. Tbe following is a syDopsis of the whole doctrine of pro- portion, as contained in the preceding articles. 74 AUUtSBJL PabtIV. Let fiiur qinmtities a, 6» c, aod d, be pr^portionaU^ tben are ttiey also proportionals ia all the foUowkig fprms -, viz. 1. Directly • . . a : 6 : : c : d. 8. Inversely b : a :: d: c. 3. Alternately , n : e : : b : d, 4. Alternately and inversely . . . . c : a : : d : 6. 6. Compoundedly a : a+6 : : c : c+d. 6. Compoundedly and inverstcly a-f 6 : a : : c-|-d : c. 7. Compoundedly and alternately a : c :: a-i-b : c^d. 8. CampouadedJy. alternately, \^,^.,^^^, «+j. and inversely J 9. Dividedly a : a — b : : c : c — d. or, a : b — a : : c : d— c. 10. IXvidecHy and alternately . . . . a : c : : a-^b : c— d. or, a : c :i 6— a : d— c. 11. Mixedly a+6: a— 6 : : c+d : c— d. 1*. Mixedly and inversely a— ^ : a-^-h :t c— d : c+d. 13. Mixedly and alternately a-f 6 : c-f d : : a*-& : c— d. 14. By multiplication ra : r6 : : «c : sd, 15. By division : — : — : : — : — , r r $ s IS. By invidution a* : *■ : : c* : d". 17. By evolution av : ^r : : c*^ : dy. 18. They are inversely proportional when a : b :: d: c. 19. They are in harmonical proportion when a : d : t tf wo 6 : €^d. Sa Three numbers are in contra^hsnaon]^ proportion when c : a : : a c/) 6 : c c/) d. The 14th> 15th, leih, and 17tb partieidaiB admk of inver- sion, alternation, composition, division, &c. in the same mnncr with the foregoing ones, m is evident from the niBtare of proportion. The comparison of VARIABLE and DEPENDANT QUANTITIES \ 93. A quantity is said to be variable, when from its nature and coDstitution it admits of increase or decrease. _ — ' ■ ^ TM doetrine of Tariable aofl depeadBiit qinntitieB, «» laid doMm in the fuUowiu{; articles, sbo«M bo v«ll iui4«nt90cl hyaU tki^te vho intwA i^ntd PaktIV» variable ahp DJ^SNPAMT QUANTinES. n 94. A (juaatity is sajki to be hmatitMe or eMittoiil»iidien its Ofiture is such that it do«s not cbaoge its value. 95. Two variaUe q^Hiatitisft are aaid to be depend€mi, whett ent of tbi^iii being increased or decreased, the other k Increased (BT decureafi^d reepectiveljF, in the same ratio. Thus, let A and B be two variabU qumiiUM, mtch, tM when A i« changed into any other value u, B u necessarUf ch^niged mtQ a ^corresponding value b, (in which oast A : a :: B : b,) ihm A and B are said to be mutually d^^itndant.r d6r. To every proportion four terms are necessary, but in af^lyijlg the dnrfiiiie td pvaetice, although four quantitks are always understood, two only are emplc^ed. This concMe mode of expression is found to possess some advantages above the common method, as it saves trouble, and likewise assists the inind, by enabling it to conceive more readily the relations which the variable and depeadaol quantities under coinsideratioA bear to each other. 97. Of two variable and dependant quantities, each is aaid to vary directly as the other, or to vary as the other, or simply to be as the other, when one being increased, the other is neces- sarily increased in the same ratio, or when one is decreased, the other also is decreased in the same ratio. Thus, if r be any number whatever, and if when A is in^' creased to rA, B is Tiecessarily increased to rB, (that is, when A A\r4'.vB\ rB,) <?r p^hm 4 is docreoMd to—, B is necessarii^ r B A B decreased to -r-, (iluit is, when A : ■*-:: B : — ,) then A %s said r r r to vary directly as B: or we say simply, A is directly as B. Example. A labpor^r agrees tp work a week for a certain sum ; now if he work 2 weeks, he receives twice that sum, if ke work' trtit'half a week, he receives but half that sum, and *o on ; in tWs cstse, the sum he receives is directly as the time he works. tUti Isaac I^ewton's Principla, or any other scientific treatise 00 Natoral Philotopby or AstroDomy. See on this subject, JUtdlamfs Rudiments, hth M'lt, p. S3.*>— 250. and If^ocMTs Algebra^ 3d Edit. p« 103 — 109* 1 76 ALGEBRA. Part IV. 98. Ohe (piantity is said to vary inversely as another, when the former cannot be increa8ed> but the other is decreased in the same ratio ', or the former cannot be decreased, but the other must nccessprily be increased in the same ratio ; that is, the former cannot be changed, but the reciprocal of the latter is changed in the same ratio. Example. A man wallu a certain distance in an hour; now if he walk twice as hst, he will go the given distance in half an hour -, but if h& walk only half as fast, he will evidently require two hours to complete his journey i in this case his rate of walking is inversely as the time he takes to pei*fiirm it. 99. The sign ec placed between two quantities, signifies that they vary as each other. Thus A K B implies that A varies as B, or that A is as B; ulso A K -^ skews tlmt A varies as the redprocal of B, or that ■ A is inversely as B, 100. One quantity is said to \'ary as two others jointly, when the former being changed, the product of the two latter must necessarily be changed in the same i*atio. Thva A varies as B and C jointly, that is, A 9^ BC, when A cannot be changed mto a, hut the product BC must be changed into be, or that A : a :: BC : be. 101. In like manner one quantity varies as three others jointly, when the former being changed, the product of the three latter is changed in the same ratio. Thus Ak BCD, and the like, when more quantities are concerned. Example. The interest of money varies as the product of the principal, rate per cent, and time, or I ic PRT. « loss. One quantity is said to vary directly as a second, and inversely as a third, when the first cannot be changed, but the second multiplied by the reciprocal of the third, (that is, the second divided by the third,) is changed in the same ratio. B Thus A varies directly as B, and inversely as C, or, A tc -t7# B h when A : a:: -^ : — . C c • ' Part IV. VARIABLE and DEPENDANT QUANTITIES. 77 Example. A fermcr must einploy as many reapers, as are Erectly as the number of acres to be reaped, and inversely as the number of days he alV;>ts for the work, or B jc — . 103. U JtQ B, and ^ oc C, then wiU ^ * BC For smce B:b::A: -^=ra, and C : c :: -r^- i -57.=v<»= <*<? i* jj BC final value of A arising from iU successive changes in the ratios of v4hr Bil^andC: c; wherefore smce'^;r;:sza, or Abc^aBC, A :a:: BC : be, or A fKi BC. 104. in like manner it may be shewn, that if ^ oc B, A u: C, s and A9i D, then A oe BCD -, also if ^ «c B, and ^ ec ~, then B 1 -^ * "^i and likewise li A tt B,A ec C,and-4 cc -yr. then A « BC -gj-, the proof of all which is the same as in the former article. 104. B. If ^ cc BC and B be constant, then ^ oc C5 if Cbe B constant, then A k B-, if -rf «c -tt and C be constant, then A « B^ if B be eonstant, then^^ tc -r;. For since the product BC varies by the increase or decrease of C only, when B is constant, and A varies aJs that product, there* fore when B is invariable, A must evidently vary as C,- and when B alone is variable, and C constant, A {varying as the product AB) must in like manner vary as B: after the same manner it may- he shewn, that when A ee BCD, if B€ be constant, then A ^ D i if D be constant, then A k BC; if C he constant, then A ee BD ; and if B be constant, then A cc CD ; and in general^ if A be as any product or quotient^ and if any of the factors be given, A will be as the product or quotient (as the case tfiay be) of all tfie rest, 105. If the first quantity vary as the second, the second as the third, the third as the fourth, and so on, then will the first vary as the last. Let A, B| C, a«d D, he any number of variable quantities. m AtOSBRA. PaktIT. and a,b,t<md d, torfespondit^ mlues of them ; and let A ^ B, Bit C,andCtt D; then teiU Ate D. Because A:a:: B :b. And Bib:: C:c. And C:c:: D-: d, therefore ex (cquo (Art, 81.) A: an D : d, that is, A k D ; and the same may he shewn to be true of any nufiAer of variable quantUies, 106. If the first be as the second^ and the second inversely as the thirds then is the first inversely as the third. 1 I • l4et A n By and B ti -—, then is A ^t -^> For since A:a::B:b, And B : 6 : : — : — , therefore ex aquo A: a:: — : — , o c \^ c 1 • that is, A 96 -j;, 167. If eadi of two quantifies Vary as a thiiti, then will both their sum and difference^ and also the square root of their pro* dnct, vary as the third. Let A 9c C, and B 9^ C, then will A;j^B K C, and^AB n C. Because A :a:t€;e,^ i , ., . AndC:c::B:bJ^^y^^^''' Therefore ex aqucUi Aia::fiib^qnd aUemmdH AnBi: a:b, wherefore componendo et dividendo A±B : B i:a-^b:b, whence altemando A±B : a±b ::B:b; but B:bi:C:c, where- fore ex aquali A±B : a±b i:C:c, that is, A^ 9fi C, or the sum and the difference of A and B will each be as C. Again, because A And B Therefore (Art. 78.) AB Whence (Art, 80:) ^AB a :: C : c, b :: C : c, aJb:\ O :'c^, ^ab : : C : c, that is, ^AB cc C. 108. If one quantity vary as another, it will likewise vaiy aa any multiple or part of the other. Let m be any constant quantity, and let A 9^ B, then, wUl A ee taS, and A ec — . m ' Because A : a :: B : b, by hypothesis, and B : b :: mB : mb. Art. 73. Ther^ote A : a :; mB : mb, that u, A tn mB, PabtIV. variable ani> DBPSKBANT OUANTITIES. 19^ And B : b :: — : — . m m Therefore A \ a :\ — : — . mm Thai'ts,A9^ ~. m Since A^ B,AiM tquml to B imdHfi^ed €t ^vkM tf $ofm' R h constant quantity j for A : a :: mB :mb :: — : — , whence alter-* m tn nando A : mB : : a : mb :i B b And A '. — : : a : — ^ if m b% uummd, ao thai Av^mB, ar m m . B b A= — , then will a^smb, 4tr a= — reipectively, m ffi ^ 110. If the corresponding values of A and B be known^ then will the value of the constant quantity m be likewise known. For if a and b be the known corresponding values of A and B, then since A^mB, or A=^ — j by substUuting a and b for A m ■ cmd Bi we shall hate a^s^mbf or a=; — ; whence m=-;-^ or «!« m b ' b H /I — .• wherefore dUo (since As^mB, ot Aa-^) -rfat-r M t, ^)r« a ' ^ m 6 a 111. If the product of two quantities be coBttaot/ iNn will the fiietOTs be inversely as each other. 1 1 Let AB be a constant quantity, then is A t^ ~ and B m -^ /or AB being coMPant, it mm/ be OMsider^ ae 1 5 iha$ is, AB « 1, whence A « -^, and JB oc ~ . B A 119. ileiiQs, ia the cefMtant product ABC, A m -^^^ B « 1 1 I 1 1 AC ^ * 'jW S€ t^ -*j, AC 9c -^, 4md AB n -^i 9tw^ U>e Uk« may be shewn wh^n the product consists of any number of fectors. ^ ^ 8a ALQBBKA. PAsrlt. 113. If the quotient c^ two quantitks be oooatMt^ tbeo %xe those quantities directly as each other. Let— ec 1« then, (multiplying both sides by B,) wiUA ce B, and B K Af and the like may be shewn wJien the quottent is com^ posed of any number of quantities, 1 14. If two quaotities vary as each other^ their like multiples and also their like parts will vary ^$ each other respectively. Let A K B, and let m be any quantity constant or variable, A B * then will mA ec niB, and — aq — . m m . For since by hypothesis A : a :: B : b, therefore mA : ma:: mB : mb {Art, 73.) that is, mA « mB, Also — : — : : — : — , therefore — « — . m m m m mm 1 15. If two quantities vary as each other^ their like powers and like roots will vary as each other respectively. Let A%B, then since A:a:: B: b {Art, 95.) A^ : a"" : : \B" : b\ and A^ :a^ ::B~-: 6v, {Art. 79.) that is. A' k B\ Iff v« >n ec B^ 116. If one quantity vary as two others jointly^ then will each of the latter vary as the first directly, and as the other inversely, A A Let A fic BC, then £ « 77, and C «c — . For since BC oe A, divide both by C, and B «e 77 ; divide both by B^ and C cc -^ . B 117* If the iirst of four quantities vary as the second* pind the third as the fourth^ then will the product of the first an^ third vary as the product of the third and fourth. Let A ti B.andCK D, then is AC k BD. For A: a:: B:b. And C : c :: p : d, " Therefore {Art 79.) AC: ac:: BD: bd: or AC ce BD, - 118* If four quantities be proportionals^ and one or two of them be constant, to determine how the others vary. Let A i B :: C : D, then will AD== BC, and therefore AD ce BC, Let A be constant^ then D ce BC, {Art. 104.) let D ^ART IV. VARIABLE AND DEPENDANT QUANTITIES. 81 ^ coiiBtani, then A oe BCx lei B be constant, then C ee AD; let C be constant, then B k AD, Next, let A and B be both constant, then D k C; let A and C be constant, then D oc B; let D and B heconstant, then A «e C; lei D and C be constant, then A % B> let A and D be constant^ then B and C will be both constant, or %. vary inversely as each other, that is, B k -^» and C te -^ ; (Art. 111.) in like manner, if B and C be constant, then A and D vUl both be constant, or vary inversely as each other, nam A « ~, and D «e -j. lastly, if three of the quantities be con- stant, the fourth will evidently be constant. 119. To shew the use and great convenience of the conclu- sions deiived in the preeediog artides, the following examples are subjoined. Examples. — 1. Let Pssany principal or sum of money lent out at interest^ i{=the ratio of the rate per cent. T=the time it has been lent at interest^ and J=the interest; to determine the relative value of each. First, supposing all the quantities variable. Then Ice PRT {Art. 22.) whence Pss-—-, R m :—^ and T «6 — , (Art. 114.) Let I be given, then P «c ^, R «c p^, and T te •^^, (Art. 104.) let P be given, then I 9^ RT, R k -=, «id I I T ic -=-, (Art. 111.) let R be given, then I tc PT, P «c -^^ and \ I I r Bc -5", (^rt. 111.) let T be given, then I ec PR, P « -5-, ond P jK R «e -^; let I and P be given, then R «c -=r>^ui(£ T ce s"; let I P T H and R be given, then P cc -=;, and 3* oc -5-; let I and Tbe given, then P flc ~> and R tn —; let P and R be given, then I k T; R V let P and T be given, then 1 9t R» Lastly, let R and T be given, then I 9c Pi and if any three of the quantities be given, tbe fmurth wiU be given. VOL. II. O % SuppcMf the qiiuidtie$ of inotioii in taro monipg htOm tfft be in the ratio comppunded of the qqantitie^ of 0i9Uer« «nd tin veloekiesj to determine the other dicgnvtances. Brst, let Msithe qumtUy Qfmotitm, Q^zfuamtUiy ^ mUter, Vzsvelociiy; then M 9^ QV by hypothesis, wherefore Qm-prs and if Mbe given, Q « j^ ;, also ^ « -g-* ««d M being given, yK^;ifQbe given, then M 9c V; and if Vbe given, M k Q* Secondly, suppose the quanUty of matter Q to be in the com* pound ratio of the magnitude m, and density D, or Q % mD; by substituting mD for Q in the abov^ expre$sions where Q is M 1 found, we shall have M ce mDV, mD « j^, mD st rp-, M bang- M 1 given: Fee ^--r^rfWVm —^^ M bnng giutsifrom ibete ii is mil mU plain that a great variety of other expressions may be obtained, qni still more, by considering one or more of the quantities invariable* Lastly, since the magnitudfss qf bodies are as the cubes of their homologous lines, {or d^,) that is, (P k m; if d^ be substi" iut^dfor m, by proceeding as before, toe $bull obtain at length aU the possible relations of the above quantities : but the prosecution of this is left as an exercise for the learper. GEOMETRICAL PROGRESSION. 120. To investigate the rules and theorems of Geometrical Progression, Let aszthe least term, 1 u j i *i ^ - z^ihe greateH term, T "^ '^ *** «**'«^- n=^the number of terms, r=the common ratio, s=zthe sum of all the terms, * Then will a4-ar+or*H-ar*> ^c. to ar'^'^^^ be m increasing geo* metrical progression. * A progression, consisting^ of three or four terms only, is nsually Galle4 geometrical proportion, or %im^\f proportion. One important property of s gttomftrieal progression is tbis, namely, the product oC the tw« extreme tern* is equal to that of any two terms equally distant 6om tlw cadrHDea : hmos^ ia U» IV. GSOMETRrCAL PROGRESSION. 8S ' K Z Z Z And z-\ 1 — 5-I--JJ *c. to-—-^ will be a decreasing geO' From the farmer of these we have ar^'-'^szthe last term of the series, hut z^ the last term by the notation, wher^^e ar"— *=c2 ; from this equation we obtain a=-j~-j, (theor. 1.) zs:iaf'-^ (theok. 2.) r=~ a r (theob. 3.) and since l: riia+ar-^' «r* : ar-^-ar^+ar^, (Art 72.) that i*, 1 : r : r i—x : s-^a, therefore 9-'-aszr,s~'Z.whencer= (theoe.4.) a:=s—r^'~z (theor. 5.) _ ^ — z^ ^ — l.#+a- - rz— a . , xs (THEOR. 6.) and s= (theor. 7.) out smce r ^ ' r — 1 «=rar"— > by th. 2. substitute this value for z in th. 7, and szs 7- (theor. 8.) whence a= (theor. 9.) and since rr= ^i3* ... ^ . , rz — a (th, 3.) and sss-. (th, 7.) if for r in the latter ^ its 1^^ he «.±|--i-a value —1"—' be substituted, we shall have *= a a (theor. 10.) and because (th. 4.) s—az^sr-^zr, and (th, 1.) z z . . «= r, therefore (s — a=) » --^isr-^zr. or sr-~s=i (zr^ z zr» — z . T*— 1.Z , r* — l.z ^ "r-r=s „ . =) — r-T- * whMmce s=z (theor. 11.) con* r^ 1 !*"-»' J sequenHy ztsz ^^ (thbor. 12.) The dhove theorems are all that can be deduced in a general manner^ without the aid of logarithms in some cases^ and of equatioDs of several dimensions in others. The theorems want- ing are four for finding n, two for r, one for a, and one for z t the fout theorems for finding the value of n, may be expressed four proportioDals, ihe product of the two extremes is equal to the product of the' two means';, and in three proportionate^ the product of the extremes if etpttl to the ^tputt ef tile liteall. 6 9i ^ 84 ALG£BRA. Pakt IV. logarithmically; the remaiidiig four cannot be g^ven in a general manner, but their relation to the other quantities maj be expressed in an equation, by means of which any particular value will be readily known. 121. We proceed then, first, to deduce the equations from whence the remaining values of r, a, and z, may be found in any paiticular case ; next, we shew how the theorems found are to be turned into their equivalent logarithmic expressions; and lastly^ we shall deduce logarithmic theorems for the four expressions of the value of n. Firsts because 2=ar»— * (th. 2.) and z= {th. 6.) sr^8'{-a therefore ar^'-^^i , whence ar"=fr— t +a, or ar»— sr=s rs a—s , ,^ . ,. , . a— «, w r* = (theor. 13.) which u as near as we can * a a get to the value of r, and which (supposing a, s, and n, given in numbers) if n be greater than 2, will require the solution of a high equation to find its value, Secof{dly, because «— a=«r — zr^ {th, 4.) and (fh. 1.) a=5 z z 7, therefore (<— a=) s -=ssr— zr, and zf^x=:sr* — fP^vaal V \ f |-TT 1 z sr^-^^, or 2— <.r»— «r*"-'s=— «r to^ccr*— r"— *=— , z^s z—s (theor. 14.) this equation being solved, the value of r wiU be known, ^^ TUrtUy, since s—a=tr—xr, (th. 4.) and r=— |*~S («A.3.) a 71. _ . z \ z therefore s-^a^s — ■■— >— «. — a 1 a »— 1 (theor. 15.) by the solution of which equation («, fi, and z, beisig given) a will be found. Fourthly, by the same equation, viz, a,s — il"— '=2.4 — 2''— ', (theor. 16.) s, n, and a, being given, 2 will likewise be known. 1^^. It remains now to put the above theorems into a loga- rilhmical form> to place the whole in one point of view, and to deduce the four theorems for finding the value of n : observing that to multiply two factors together, we add their logarithms together 3 to divide, we subtract the logarithm of the divisor from that of the dividend ; to involve or evolve^ we multiply Pa«t IV. GEOMETRICAL PROGRESSION. 85 or divide respectively the logarithm of the root or power by its index^ as directed in Vol. I. Fart 2. Let A- represent the logarithm of And L the logarithm of the (juantity to which it is prefixed; then will the following synopsis exhibit the whole doctrine of geometrical progression^ as investigated in the preceding arti- cles i^. k Some of the foUowidg logarithmic expresftkmt are extremely inconTenieiity particularly theor. 10. Th« batt method of computing the ?aloe of t in that theorem, will be, first to find the log. of z, subtract the log. of a from it, add this remainder to the log. of z, and divide the sum by Hf— ] ; find the natural number corresponding to the quotient, from which subtract a, and find the log, of the remainder. Secondly, from the log. of 2, subtract the log. of a, divide the remainder by n-^ ], find the natural number corresponding to the quotient, subtract I from it, aad subtract the log, of this ^emahider from that of tho former; and thellM ill other oases. QS ■ V 86 Theor. II. VIII. VU. XVII. VI. XIX. w. xvm. Given. a,r,n a, r, z a,8^r Req. s n ALGEBB4. Solution by Numbers. Pav a,z,s XIIL XVI. III. X. I. XI. a,n, 8 a,n,z r, n, z IX. XII. V. XX- XV. XIV. I n n 8 z^zar^-^ r— 1 rz — a '~r-l r «■ *— a • 5—2 . r8 a'-^8 a a 2.«-2l"-»=ai- ra|"-i I i^^"-»i>»<^^i^«^P^»«^^^pW Solution by Logarithms. Z=^A+R.n^l S^sA-k- i.^— I— X.n— l.ttA«ra JBi iSs^s L.TZ ^a'^Ls'^ 1 + 1 Z=sl».r— 1^+a— fi ni l I «— ^«». n=? JL-f— i.<»+a— -4 As: L.f'-^a -^X.«*— « n=:- Z-A L.S — i— £.5— z + 1 ■I—* R= g.—l'-'-a <=' 1^- a=- r»— 1 r,w, * r, 2, « »,«,< n «•— ; r"— 1.Z «: r-l.r»-» Z^A n— 1 5=:I..2.i]a-»-.a-.L.3*-*-.l -4=Z-.fi.n— 1 r— 1.« a= .B.^1 z= a=« — r.»— 2 .1 I a.,— a^"-i=2iZ^"-. *— 2 «--2 5= t.?*— 1 + Z— I..r— I + JB.n-] ^=I..r— 1+5— R.r"— 1 Zrsli.r-. 1+ jR.«— 1 +S-.X.r"-i -4=£.«— r.#— .« RUt IV. GEOMETBJiGAL FBOGKBSSION. m L To sliew how tie 17^, IBth, lOfhi snd 9(Hh th^rana are derived. Z— -el . «=— ^ + 1 (TfliOK. 17;) and because R=zLa^a^Lj^t (th. 4.) suhsiituie this value for R in theor. I7. and ns 2 ^ 7 = + 1 (thbok, 18.) again, for Zin theor. 17. sub' »■ « ifillifi^ ii« raZtte /row theor, 6. aHct »!=: (— ^^^^ ' « ^ 1 *=) ^ (thkor. 19.) Lastly, for A xa theor. 17. 2 A substitute its value from theor. 5. and n=(--^ — 1-1=) — ^t h 1. (theob. 20.) £tAMPLE8.-^l. Given the ratio % the number of terms 6, and the last term 96> of a geometrical progression^ to find the first term> and the sum of the term^ ? Bete rss^, ttss^, zae96> whence (theor. 1.) as:— ^=: By IiOgarithnH). Z: =... 1 .9822712 g.n— lasO^SOlOSOOx 6at l.6051500 «-*-f*-*=aa3 0.4771212 pfience as=3« I.1*— lssX.2«-l=£.6S=1.799S406 +2ai;N9gg 1.9822712 X.f-l+Zs:.,: 3.7816117 ^ f Lrr-lsLass 0.0000000 1 +RM^lsiL.^ X 5= ... . 1.5051500 <« 2.2764617 fshence ssslB9t 04 88 . ALQEBRA. Part IV. 2. Given the ratio 2> the number of terms 6, and the sum of the terms 189> to find the first and last terms ? Here rsz% nss6, J33^89^ and (theor. 9.) «=-; — r = 1x189 V 1Q9 ^ • r— l.f--'^ , . (-—r— —=) -^=3 J alsQ «= — - — : — (Umr. 12.)= ^2«— 1 ' 63 r"— 1 ^ ' 1 X 2* X 189 ,32 X 189 ^ ^ 2^-1 =)nS3-=^^' By Logarithms. L.r— 1= 0.0000000 4-^ = 2.2764617 — L.r'— 1= 1.7993405 -^= 0.4771212 whence a=3. X.r— 1 + 5.11—1= ... 1.5051500 4-iS= 2.2764617 I..r-1 + fi.n— 1 + 5= 3.78161 17 -Lr*— 1= 1.79984(^ Z= 1.9822712 whence zss96. 3. Given the first term 3^ the ratio 2> and the last term 96» to find the number, and sum of the terms ? vr ^ ^ ^ , , V r«— « ,2x96—3 Here a=3, r=2, 2=96, and (theor. 7.) — -r = { ; =) 189=». By Logarithms. Z= 1.9822712 —-4= 0.4771212 -♦-iJ=0.3010300) 1.5O515Cl0(5 L.rz— a=I..189= 2.2764617 — L.r— IssJL.ls Q.000000O 5= 2.2764617 whence «=189. therefore n= 5 + 1 = 6, theor 17. 4. Given the first term 4, the ratio 3, and the sum of the terms 484, to find the last term, and number of terms ? Here a=4, r=3, «=484, and {theor. 6.) 2= ^"" ' ■ = ^ 3 'a Logarithi L.r— l.«+a=L.972=2.9876663 — ie= 0.4771212 Z= . . . fl.5 105461 whence z=324. L.r— 1.5+a=L.972=2.9876663 --4= 06020600 -i-i2=. . . . 0.4771212) 2.3856063 (5 whence n=5, ^Aeor. 19. PiBT IV. GEOMETRICAL PROGRESSION. 89 5. Given the first term % last term 2048, and sum of the terms 2730, to find the ratio, and number of terms ? vHere a=:2, 2=2048, «=2730, and Uheor, 4.) r=-^^— ss 2730-2 _ 2728 _ ^730-2048""^ 682 "" ' By Logarithms. 2=3.3113300 —-4=3.3010300 Z— -4=3.0103000 I.«-a=L.2728=3,4358444 -L.*-.z= 1.682= 2.8337844 B= 0.6020600 whence r=4. L.Jira=L.2728=3.4358444 -.L.»^= X.682= 2.8337844 X, .THi— i,.«-.z=a602oeoo therefore .6020600)3.0103000(5 whence n=5 + l=6, theor. 18. 6. Given r=4, n=:6, and ^=2730, to find a and z. iliu. a= 2, z=2048. 7. Given rsx2, n=6, and z=96, to find a and «. .4n«. a=3, «=189. 8. Given the ratio 5, last term 12500, and sum of the terms 15624, to find the first term, and number of terms. Ans. a =4, n::ze. 9. Given a=:4, n:=:6, and z=: 12500, to find r and t. Answer r=5,«= 15624. • 10. Given r=3, n=4, and z=81, to find a and *. ll.^Given r=i6, w=5, and «=1555, to find a and z. 12. Given a=3, r=10, and n=20, to find « andz. 124. PROBLEMS IN GEOMETRICAL PROGRESSION. 1. Of three numbers in geometrical progression, the difference of the first and second is 4, and of the second and third 12 j required the numbers ? Let X, y, and z, be the numbers. Then y— «=4, or xz=zy—4', z—y= 12, or z=y4-12. Wherefore since by the problem x :y::y:z,by substituting the values of xaadz in this analogy, we shall have y — 4 : y : :y: y+ 12 5 wherefore, (by multiplying extremes and means,) y— 4 .y+12=) y*+8y— 48=y*, or 8y=48; wherefore y=6, ar=2, «=18. M ALGEBBA. Part IV. %. The product of three numbers in geptnetricfd ^rogfesftion is 1000^ and the sum of the first and last 25 5 required iht numbers ? « Let X, y, and x, be the numbers ; then since xiyiiyiZjwe have xz=iy^, {Art, 120. Note,) and . {xyzzsixz.yss) ^^ssiooa, whence ^=10; also xzTz(y^=) 100^ and by the problem X'\-z^ 25 : from the sqwire of this equation subtract four times the pre" ceding, and x*— 2x2+2*— 225: extract the square root of this, and X — 2=15 5 add this to, and subtract it^from, the equaHixm x-f 2=25, and 2x=40, or x=20, also 2 z=10, or 2=5 j whence 5^ 10^ and 20, are the numbers. 3. To find any number of mean proportionals between two given numbers a and b. Let n— 2=i/ie number of mean proportionals, then will n= the number of terms in the progression : also let r= the ratio, then (theor. 3. Geom, Prog.) r= — B-.1 5 and by logarithms, log. b — hg. a H-n— 1=20^*. r ; whence r being found, if the less extreme he coff- tbaudiy muUvpUed, or ike greater divided^ 6y r> ifte retsUU miU he the mean proportionals required, BxAMPLKs.— »1. To find two mean proportionate betw^n 12 and 4116. 4116)7 ^ V Here ac:12, 6=c4U6, ♦»=4, and r=r{J±J =3431t=)7 ; 12 ' whence 12x7=84> the first nXan, and 84x7=5S8> the secofid mean, 2. To find four mean proportionals between 2 and 48^. An». 6, 18, 54, and 162. 3. To fibd five mean proportionals between 1 and G4* 4» There are four numbers in geometrical progresBion'^ the ^uiB of the extremes is 9, and the suqqei of the cubes of 'th» means 72 } what are the numbers ? Let X, y, u, and z, be the, numhers. Then by thepix>blem, arH-2=9, or x.=9*— 2. X : y : : m.: 2, op xz^uy, whence xz= (9— 2.2=). 9 2—^2*. x'.ywyiu^or xu^y^ (J?«y=) xH^y^, y: u::u:z,or zy=zu'* ... {zyuss) X2*=«i %tt {xz.x+zsz) 92— *«.9=X*2+XZ«. >lnf the problem. l^AXTlF. COMPOUND INTEREST. 91 ^n4 (y^+fi'ss) T^sssfla^+xz^, and things that are egico/ to the same are equal; therefore 9«— «*.9=72, or 9z— z*=8, or 2>— .92=^8; iphence bff oowtpleting the square, 4rc. zss:S, xss (9-2=) 1, y=(V^*«) «* tt=(V^«*=) 4. 5. Of foiff numbers in geometrical progRtsion, tbe product of tlie two least k 8, and of the two greatest 1S8 j what are the numbers ? Let X, y, «, and x, be the numbers. ^ 8 Then xy^B, or xss^r- y 198 VflsiVie,or z:s. u 8 198 ORCsttif^ or — . sstttf y tt 2%6rc/are (8 x 128=) 1024=ttV, or uy=3% and «=—. f 8 39 J?a^ (x : y : : y : ttj that is,) — : y : : y : — , where miuUipUfing 956 extremes and means, y^s— j>, or y^=956i whence y=4^ a:= 8 39 198 ( — =)9, tfss ( — =) 8, z=( — =s) 16, (i^ntun^tf required^, 6. The sum of 3 numbers in geometrical progression is 14, and the greater extreme exceeds the less hj6; what are the numbers ? Ans. % 4> and 8. 195. Def. Compound Interest is that which is paid for the «se» not only of the principal or sum lent, but for both princi-^ pal and interest, as the latter becomes due at the end of the year, half-year, quarter, or other stated time. To investigate the rules of Compound Interest, Let p:=sthe principal, r^r:the rate per cent, t^the time, i2= (14-r=) the amountoflLfor a year, called the ratio of the rate ppr cent, a^the amount. Then since 1 pound : is to its amount for any given time and. rate : : so are any number of pounds : to their amount for the sam^ time and rate^ therefore as p ipRssthefost, pR I pR^:= second, | > year's amou^t,. p ipB^the first ipR I pB^:= second, I: R::2pE^: pR^szthird, \ pB? : pR^^fourth, 92 ALGEBBA. Part IV. Whence we have theorem 1. pR^=a, theor. 2. ~=p, theor. 3. V^=^. THEOR. 4. ^f^^^^^S'P ^^ ^j^ ^y^^ ^^^^ P log. R * of which follow immediately from the first; the fourth cannot be conveniently €xhU}ited in nutnbers without the aid of logarithms. By means of these four theorems, all questions of compound interest may be solved. Examples.— 1. What is the amount of 1250i. lOu. 6d, for 5 years, at 4 per cent, per annum, compound interest ? Here p:sz(UBOl. lOs. 6d.=) 1250.525, ^=5, J«=±1.04. Thentheor, 1. (p/J*=) 1250.525 x foS^s: 1250.525 x 1.2166 . =1521.388715=1521/. 7«.9^.=a. 2. What principal will amount to 200Z. in 3 years, at 4 per cent, per anniun ? Here ar=200, JR=1.04, teS, emd theor. 2. (^=) ?^ = 1.124864 =^7y«y^92=17y/. 155. U^d.^ip. 3. At what rate per cent, per annum will 500i. amount to 578/. I6s. 3d. in 3 years ? Here p=500, fl=(578/. 16*. 3d=) 578.8125, ^=3; and, ^r « /♦ fl V • 578.8125 1 theor. 3. (V-=)V gQQ " =(y V144.7031. 5ee FoZ. J. P. 3. ^r*.63.=)yx5.25=1.05=12.. te^^orc, (*ince jR-l =r,) we Aare fi— l=.05=r, «w. 5 per cent, per annum. 4. In how many years will 225Z. require to remain at interest, at 5 per cent, per annum, to amount to 260/. 9s. 3^d. ? Here p=225, -R=1.05, a=(260/. 9s. 34d.=) 260.465625; whence, theor. 4. (^t^-^^P^ ^S- 260.465625- fo^. 225 ^g' R log. 1.03 ■"' 2.4157506-^ 2. 3521825 0.0635681 0.021 1893 ""0.0211893 "^^ ^^"'"'^ ^• 5. What sum will 500/. amount to in 3 years, at 5 per cent. per annum ? Ans. 578/. 16«. 3d. 6. What principal wiU amount to 1521/..7*. 9id. in Syeare, at 4 per cent, ptr annum ? Ans. 1250/. lOs. 6d. Part IV. PROPERTIES OF NUMBERS. 93 7. At what rate per cent, will 7912. amount to 16421. I99.9id. in 21 years ? Jm. 4 per cent 8. In how many years will 7^11. be at interest at 4 per cent, to amount to 1642/. I9s, 9^d. Ans. 21 yean. If the interest be payable half-yearly, make ^ssthe number of half-^years, that isstwice the numbir of years, and r=:half the rate per cent, but if the interest be payable quarto*]?, let lasthe number of quarter-years^ viz. 4 times the number of years, and r=one-fourth of the rate per cent, and let JRsr-f- 1 in both cases, as before ^ 126. To determine some of the most useful properties of numbers. Def. 1. One number is said to be a multiple of another^ when the former contains the latter some number of times exactly, without remainder. Thus 12 t« a multiple of I, 2, 3, 4, and 6. CoR. Hence every whole number is either unity, or a multiple of unity. 2. One number is said to be an aliquot part of another, when the former is contained some number of times exactly in the latter. Thus 1, 2, 3, 4, and 6, are aliquot parts of 12, for 1 is tV, 2 is ^, 3 w ^, 4 M 4^, and 6 is ^ of 12. Cor. Hence no number which is greater than half of another number^ can be an aliquot part of the latter. 3. One number i» said to measure another number, when it will divide the latter without remainder. Thus each of the numbers 1 , 2, 4, 5, 10, and 20, measures 20. 4. One number is said to be measured by another, when the latter will divide the former without remainder. Thus 20 is measured by 1, 2> 4, 5, 10, fsnd 20. Cor. Hence every aliquot part of a number measures that number, and every number is measured by each of its aliquot parts, and by itself. ^ It was at first intended to investigate and apply every rule in aritbmeticy but want of room obliges us to omit Equation of Payments, Loss and Gain, Barter, Fellowship, and Exchange; these will be easily understood from the doctrine of proportion, of which we have amply treated. ^ M ALGEBRA. Past IV. 6. Any nttmbtr which lAesiftttret two or mor^ numbers^ is called their common measure; aM the greatest nuttiber tbftt will raeasttre theoi^ is cslM ih^it greatest conmion measure. Thus 1, 2, 3, and 6, are ihe common measures cf 12 and 18 i mtd 6 tf thevr greatest common measure. Cot. Heoce the greater common m«asmre of several num^ bers cannot be greater than the least of those numbers \ and when the least number is not a common measure, the g r eates t cdomoQ measure caiinot be greater than half the least. Def. ^. cor. 6. An even number is that which can be divided into two equal whole numbers. ThMs 6 is an even number, being divisible into two equal whole numbers, 3 and 3, 8se. 7. An odd number is that which cannot be divided into tw6 equal whole numben } or^ which differs from an even number by unity. Thus, 1» 3, 5, 7, &c. are odd numbers. Cor. Hence any even number may be represented by 2 a^ i^nd any odd number by 2 a+ 1, or 2 a— 1. S. A prime number is that which can b6 measured by itself and unity only \ Thus, I, 2, 3, 5, 7, 1 1, 13, 17, 19, 23, &c. are prime num- bers. 1 Hence it appears, that no even nniiiber except 3 can be a prime, or thai all primes except 3 are odd ttumben ; Imt it doea not fbttow that all the odd numbers are primes : every power of an odd nniibcr ia odd, odaseqiieBtly the powers of all odd kwmbers greater than 1, after the first power, will be composite numbers. Several eminent mathematicians, of both ancient and modem times, have made fruitless attempts to discover some general expression for finding the prime numbers : if n be made to represent any of tbe nambers 1, 2, 3, 4, &c. then will all the taDtes of 6 n + 1 •»! 6 n-^ I constitute a series, including all the primes above S; but this series will have some of its terms composite numbers: thus, let ns=I, then 6ii+l»7 and 6ft— l«B5y both primes; if n=2, then 6n 4- 1 = 13, and 6 n— 1 » 1 1 , both primes ; if iib3, then 6n+l = 1.9, and 69t— 1 » 17, both primes, Sec. Let »s6, then 6ft-|- 1 ssST a prime, but 6 }i— 1 s35 (::35 X 7) a composite number; also if irsg, then 6ii-{-> 1 »" 49 a composite number, and 6 n — 1 se47 a prime, Stc. For a talble of wB tbei prime numbers, and all the odd composite numbers, undcfT 10,000, see j^. HuttmCs MathemaHcal Dtctionafy, 1795. Vol. H. p. 276, 378. Sair 1% FROPfiRTlSS Q^ NUMBERS. 9h 9. Namben are said to be prime to each #dier, when unity IS their gi-eatest common roeasture ». Thus, 11 and 26 are prime to each other, fm' no uwmber greater than 1 will divide both without remainder, la A composite number is ^atwhkh is measured bf any ownber greater than unity. Thus,C i9 a composite mmber,for % and 3 wiU each meeh mreit. Cob. Hence every composite number will be measured by two numbers : if one oi these numb^B be known^ the oflMf wiU be. the quotient arising from the division of the eottiposite Dumber^ by the known measure. Thus, 6=3 X 2, and-^-z^^y also -^=2. 2- 3 11. The component parts of any number, are the numben (eacb greater than unity) which multiplied toget^er^ produce that number exactly. Thus, 2 and 3 are the component parts of 69 for 2x3cb6; 3, 4, and 5 are the component parts of 60, for 3 x 4 x 53=60, &c. 12. A perfect number'* is that M^iiefa is equal to the sum of all its aliquot parts. ■ Nombcn which are priaie to erne another, mre not aeceMarily pritme$ in the sense of def. 8. thus 4 and 15 are composite nnmbers according to def. 10. bnt they are prime to each ethers since unity only will divide both. Hence two even nujjabers cannot be prime to each other. In the Scholai's Guide to Arithmetic, 7th Ed. p, 104. 9. it is asserted, tiat " If a number cannot be divided by some nnmber less than the square root thereof, that nnmber is a pnmc." Now tbia cannot be troe ; for neitber of the sqaavs nnmbers &» 3&9 49> 4fe. fte. can be. neaturcd by any number Icaa than its square root, and yet these numbers are not primes : a slight alteration in tbe wording will however make it perfectly correct ; thus, *< If a number which is fM a Sfuair09 cannot be divided by some number less than the square root thereof, that nnmber is a prime.** This interpretation was undoubtedly in^ tended by the learned author, akhongh his words do not seem to warrant it. ■ The IbUowing table is said to coatain all the pex&ct namben at present 6 8589869056 88 IS7438691328 406 2305843008 1399^1^ 8128 S4178516398381.58837784576 33550336 9903530314283971830448816128 These nnmbers were extracted from the Ada of the Petersburg Academy, in several of the Tolnmes of which^ Tracts on the subject may be feond* 96 ALGEBRA. Past It. \ Tims, 6 is a perfect number, for its aliquot parts ute !(= — 6 of 6) 2 (=— of 6) andS (=-- of 6) and 1+2 + 3=6. 13. An imperfect number is that which is greater or less than the sum of its aliquot parts ; in the former case it is caUed jan abundant number, in the latter, a defectine nunU^er. Thus, 8 and 12 are imperfect numbers; the former (viz. 8) is an abundant number, its aliquot parts being 1, 2 and 4, the 9um of which l-h2+4=:7> is less than the given number 8. 7%e loiter (viz, 12) is a defective number, its aliquot parts beia^ I, % 3, 4, and 6, the sum of which, vix, 16, is greater than the given number 12. 14. A pronic number b that which is equal to the sum of a square number and its root Thus^ 6, 12, 20, 30^ 8sc, are pronic numbers; for 6=s(4+ ^4=) 4+2; 12=(9+^9=) 9+3 5 20=:(16+Vl6=) 16 + 4i 30s=(25+ V26=x) 25 + 5, *c. Property 1. The sum^ difiference^ or .product of any two whole numbers^ is a whole number. This evidently follows from the nature of whole numbers, for it is plam that fractions cannot enter in either case, ' CoK. Hence the product of any two proper fractions is a fraction. 2. The sum of any number of even numbers is an even number. Thia, let 2 a, 2 b, 2 c, 8fc, be even numbers, (See def, 7* cor.) Then 2a+2&+2c+, ^c,z:^their sum; but this sum is eoi- dently diioisihle by 2, it is therefore an even number; def, 6, CoR. H^[ice if an even number be multiplied by any number whatever, the product will be even. 3. The sum of any even number of odd numbers is an even number. Thus, (def 7. cor.) Iet2a+h 2 6+ 1, 2 c+ 1, and 2 d+ 1, be an even number of odd numbers. Then will their sum 2 a+2 6+2 c+2 d+ 1 + 1 + 1 + 1, 6e m even number; for the former part 2a+26+2c+2d is even, by def 6. and the latter consisting of an even number of units is like* wise even ; wherefore the mm of both will be even, by property 2. Con, Hence if an odd number be added to an eveo> the sum will be odd. fhRT ly. PROPERTIES OF NUBfBERS. 9f 4. The sum of any odd number of odd nuinben» is an odd number. For let ^a-^l, 2 6-4-1, Sc+1, be an odd number of odd numbersy then 2a+2 6H-2c+l+l + l==<A«ir 9um, the former part of which 2a+26+2c, being divisible by 2, {def 6.) a an even number, and the latter part 1 + 1 + 1, comisting of an odd number of units, is odd : now the sum of both, being that of an eten num- ber added to an odd, wiU, by the preceding corollary, be an odd number. 5. The di&rence of two eren numbers, will be an even number. For let 2 a and 2 6 6e two even numbers, then since 2 a->2 b and 2 6+2 a will each be divisible by 2, it is plain that the difftt- rence of ^ a and 2 6 wUl be even, whichever of them be the- greater, 6f The di£Eerence of two odd numbers is even. jFbr let 2a+l and 2 6+1 be two odd numbers, whereof the former is the greater; then stftc«2a+l— 2 6+ ls2a— 2 bis the proposed difference, which is divisible by 2, it is therefore an even number. 7. The difference of an even number and an odd one will be odd, whichever be the greater. Let 2 a be an even number, 2 6+ 1 an odd number greater than 2 a, and 2 c+1 on odd number less than 2a; wherefore (2 6 + 1—2 a=) 2 6^2 a+ 1 ss efte difference, supposing the odd num- bet to be the greater ; and (2— 2c+l=) 2 a— 2 c-^l=sthe diffe- rence, supposing the even number the greater. Now each of these differences differs from the even numbers 26— 2 a, or 2a— 2c 6y unity : the difference therefore in both cases is an odd number. 9. The product of two odd numbers is an odd number. For fel 2a+ 1 and 2 S+ 1 6e any two odd numbers, then wiU (2a+1.26+l = ) 4ab-^2b+2a-{-l=:iiheir product ; butthesum of the three first terms is evidently even, being divisible by 2, cmd the tohole product exceeds this sum by unity, the product is there' fore an odd number, (def. 7 .) 0. If an odd number measure an odd number, the quotient will be odd. For let a + 1 be measured 6y 6+ 1, and let the quotient be q ; (J J. \ *fcw, 7 — -=9 5 then will bssl.qssa+i ', and since 6=1, apd O "^ JL VOL. II. H 9S ALGEBRA. Fait it. d-f 1 are odd, it is plain that q must he odd, othervnse an odd number multiplied by an even number, would produce an odd num- ber, which is impossible, (proper. 2. cori) 10. If an odd number measure an even number^ the quotient will be even •. 2a fibrtet— — =g, then2b+l.q=i2a; and since 2fc+l is ^ Mr. Boanycastle, in treating on this subject, (Scbolar's Gaidey 5th Eifit. p> S03.) has committed a tiifting oYeniglit. Plop. 10. in hit book is as ioUnirc i " If an odd or even number measures an even one, the quotirat will be even." The fermeir p<Miit«on is here shewn to be true, but the latter is evidently £ilse« namely, " if an even number measure an even number, the quotient is even.** 2a In proof of his assertion be says, « let r-r* q ; then 2 (.9«t8 a ; and siaoe ftm and 2 b are even numbers,, q must likewise be an even number." This oenso^ qnence however does not necessarily follow ; q may be either even or odtf, for any even number (2 b) multiplying any odd number (q), will evidently pro- duce an even number. (See proper. 3.) Henoe the quotient of an «vea nwa- 8 ber by an even number, may be either even or odd ,• thus, ~=*4 an even num^ dfr; but -rr^S anoddnnmber. Mr. Keith has fidlen into the fame error, or (whicfi is more probable) has copied it from the above work. See his Cbm* plete Practical jirithmeticiany 3d Edition, p. 283. Cor. to Art. S2. The first named Author is likewise mistaken when he says, (Prop. II.) '' If nn odd or an even number meaaiires an even one, it will al«o measure the half of it." Now the half of any number will evidently measure the whole, and the half measures itself, that is, it is contained once in itself; wherefore it follows, according to the tenor of the reasoning there employed, that if one quantity be contained once in another, the former quantity measures the latter, but the whole is contained once in the whole, and therefore measures it : but what- ever measures the whole meastures its half, says Mr. B. whereiore the whoU must necessarily measure the half! Thi< nftiittfce seems to have arisen from »- circumstance which might easily have happened—that of confounding the idea of a measure with that of an aliqtcot part : bad it been said that every aliquot part of the whole measures the half, ^^^ assertion would have been perfectly accurate. Should the freedom of the above remarks require an apology, I feel it necessary to testify my unreserved admiration of the eminent talents of the teamed and respectable authors in qaestion, and to assure them tibat nothing invidious can possibly be intended : but truth is the grand object of the sciences, ^nd he who is engaged in the arduous and important office of instruction, forfeiti alt claim to fidelity and confidence, if he does not point out error wherever he may happen to find it ; and he is scarcely less blameable who omits to do it with becoming caqdour, and under a sense of his own fallibility. hmT IV. PROPERTIES. Of; NUMBERS. . » 4ui odd ftufnto-, wkd^mm eem m$, iifbU9w$ thai q rniUl b$ m>en ; f^henoise the product of two odd mumb^n mould 6t km*^ tDhich u impossible, (proper. 8.) 11. An even aumber caoxiot measure aa odd oamber. 2a+l Jf possible, let ^ ^ ■■=?; wherefore V{a-|-I=S6.g.* hut since 2 b is an even numbeff 2 b.q is also even, (proper. 2. cor.) that is, an odd number (9 a+ 1) » eguoZ to an even one, (8 b,q,) which is absurd ': wherefore an even number, 8fC^ I'd. If one nurmber measure another^ it will measure everj multiple of the latter. * fl na Let nssas^ idude number, and -r^qf ^^^ ^^ T^*^' But since ^ is by hypothesis a whole number^ nq must be a whofe number, (proper. 1 .) thai is, b measures n times a. 13. That number which measures the whole, and also a part of another number, will likewise measure the remainder. a-^-b a Fbr let asid --* be each a mhelU number. c c Then wiU (- — =) — be a whale number, (pnfper.l.} ^ C C .0 \r r ^ 14. If one number' measure two other numbers^ it will like* wise measure their siun and diffiifenCe. Let e measure bo^ a mtd k^ tibea wiU — and — be both c c a b ci-f-5 a b . whole numbers z wherefore ( — | — =) , and ( =) •^ ^ c c ' c ^cc ~I^, will also be whole numbers, (proper. I.) c CoR. Hence the commoti measure of two numbers will like* wise be a common measure of the sum and di&rence oi SBf multiple of the one, and the other. Thus, if , and <-, he whole numhers, then w%U c c c and "^ be whole numbere. c 15. If the greater of two numbers be divided by the leas, and if the divisor be divided by the remtdnder, nhd the last di« ▼isor by the last remainder continually, until nothing remain^ 100 AiLGBBRA. PaetIV, the last diVisor of aH will be the greatest common measure of the two given numbers. Let a and b. be two numbers, and let a be contained in b,f times with c remainder ; let c be contained a) b (p ina,q times with d remainder ; and let d be c) a (q contained in c, r tiines exactly ; then will d) c (r d be the greatest common measure of a and b. . o For since (6=:ap-\-c, or) b—ap^ss^c, and a-^qcszd, it follows {from proper* 12.) that every quantity which measures a askd b, will likewise measure ap, and also b-^ap or c, (proper, 13.) in like manner, whatever quantity measures a and c wiU also measure a und qc, and likewise (a^qc, or) d; wherefore any quantitff which measures <2, must likewise meeuure c and a and b, but d measures d, therefore it is a common measure of a and b. It Kkewve appears, that d is the greatest comnum measure of a and b; for since rd=sc and (c^-|-(J=) rdq+dz=:a, and (ap+c^) rdqp-i- dp+rd^tby that is, rq+l,d=ia, and r9p-fP+''-^=^> it follows that d is the greatest common measure of these two vtUues of a and b, or that it is a multiple of all the common measures, except the gres^est, of a and b. Otherwise, since it appears that every common measure of a and b measures d, and d itself measures a and b, it follows that d is the greatest common measure of a and 6'. 16. The sum and the diffiavnoe of two numbers will each measure the difiference of the squares of those numbers. For smce a+6.a— 6=a»— 6», it follows that 7— =a— 6> find -— =a+6. a— 6 17. The suni of any two numbers measures the sum of their cubesi and the difiference of any two numbers measures the dif- fepence of their cubes. _, a*4-6« , a* — 6» For ----=:a*— a6+ft*; ond — — r-.=aH«6+fcS asap- pears by actual d^ision. 9 See Wobd'r Ahrebrd^ tWrd Edition, p. «. The «boTe is a demonstratFon of the Tole ia page I48r of the ^t volume. Paxt IV. PROPBRTIBS OP NUMBERS. 101 ^ ^CoR. Hence if the (MPoduet of any two tuimben be tubtracted from the sum of their squares, the remainder mcafores the sam of their cubes ; and if the said product bte added to the stmi of the 8quares> the sum measures the difference of their cubes. . 1 8. If any power of one number, measure the same power of another, the former number measures the latter. JFor let — be a whole number produced by -r^T-'T^' *^« ^^ ^ tr bob o ^ term$; then will *t- ^ a whole number ; for if not, let it if pom' ble he a fraction, then thu fraction being multiplied continually a* into iteelf, wiU at length produce {-tA a whole number, which i$ tr C abewrd: wherefore ~is a whole number, or b meaeuree a. b Cob. Henoe if one number measure another, any root or power of the former will measure the like root or power of the latter respectively. 19. If the similar powers of two numbers be multiplied toge- ther, the product will be a power of the same kind with that of the &ctors. For if a^ be multiplied by 6', the product a*" b^ is. likewise an, ^ n^ power, the root of which is ab. Cor, Hence e?ery power of a square number is a square, every power of a cube number a cube, and in geneial eveiy. power of an »*^ power is an n^ power \ 20. If any power of one number be divided by the sama power of another number, the quotient will be a power of the same kind with that of the said numbans. 0^ Let (f and b* be the n^ powers of a and b ; then is -r^ also a an n* power, for its root is—. Cor. Hence the quotient of one square- by another, is a square ; the quotient of one 6ube by another is a cube, &c. « And it it obvieottluit all Hm powtrt •£ a piinc number (eacMpt the fin( power) will be eompottte. h3 103 AL6EBRA. P^^t IV. «l. If two Bvmbeis dMfer by unity, their lum if fpal to the difference of their Mjuares. Ltt a and a+l be any im numbm J^ffkrmg fty unity : thm toiU ««+! be tkeir mm, also (a+ lj*-.^«<^-f «a-|-l— 0^=) 5Ja-M«f*« <fcy«f«iceo/<Acir t^iMrtf « MipA u the $ame as ikeir sum. C0R..I. Henee the differences of 0*, !•, 9^, S«» 4*, &c/ » • . (ssO, 1, 4, 9, 1.6, &c.) are the odd numbers 1, 3, 5, 7, &c. Cor. 2. Hence the squares of all whole numbers may be found from the series of odd numbere 1, 3, 5,^, 9, &c. by addition only. Thus, 1=1«; l+3=(4=)?«j 1+3+5=^(9=) 3*; 1+3 + 5+7=(16=:) 4«5 l+3 + 5+7+9=(25=:) 5»i and so on at pleasure, 92. An odd number which is prime to another number, is nicewise prime to double the latter. For let a be an odd number, and b any other number ; then since a, being odd, cannot be measured by any even number^ (proper. 11.) it must be measured by an odd one: wherefore if a and 9 b have a common measure^ it must be an odd nunther ; but 9 bis eri- dently even, (def 6.) and if an even number be measured by an odd one, the quotient toiU be even^ (proper. 10.) and since this even quotient can be halved, it is plain that the foremeniloned odd num» her, which meaeures 9 b, mill be cteltriwati hi^f ess many tinms in h amitis4$^9b,, that at, it - meaeures' b^ whence a and b' have a com* men measure; but they, are pwimq to audi Mier^uiherefoTe a am^ %h have no cdmtiKMi meaiUre. ' Cob. Hence' if an odd nuniber be prime to any other num* ber> it is prime to twb^ ftnat, eight/ sltteen^ &c. tunes the latter. 23. If each of two numbers be prime to a third number^ their product is prime to it. Let a and b be each prime to c, then will ah be prime to c. Then, since neither a and c, nor b and c, have any common ' I|i the Scholar's Qai4e, p. 204. prx>|». 19. cpr, lite 0* UJbj RUitsbi but with<{at it, the eondasion doee not follow. , . , V4BT IV. PROPERTIES OF KUMBERS. 103 measure, it is pUdn that ah and c can haoe no eomuum measure; wherefore ab is prime to c. 34. If one number be prime to another, every power of the Ibnner will be prune to the latter. Let a be prime to b, then wHl a" be prime to 6, For since a and hhaoe no common measure^ a.a.a.a» SfC, and b Cjonnot hove a eommon measure; wherefore {a,a>a.aj SfC.z^) a" is prime to b. fid. The.8mn of two numbers wl^cli are prime to each other^ 18 prime to each of the numbers. Let a be prime to b, then wUl a^^b be prime to a and b. For if not, let e be their common measure; wherefi^re, since c measures a-^b a both a+b and a, that is, and — are whole numbers, by jub* c • c • b tracting the latter from the former, the remainder — is a whole c. nunUfcr, (proper. 1.) In like manner, because and — are whole CO a numbers^ by subtracting the latter from the former, — will be also V a ' b a whole number; wherefore — and — are both whole numbers, c c that is, thenismbersa and b, which by hypothesis are prime to each other, haoe a common measure c, which is absurd. , CoR. Hence if a part of any number be prime to the whole^ the remaining- part is prime to the whole. £6. In a series of continued geometrical proportionals begin- ning at linity, all the odd terms will be squares j the first, fourth, •eventh, tenth, &o.. terms will be. cubes s. and the seventh term will be both a square and a cube. Thus, letl,r, r^, r^, r*, r^, r®, r', r*, r^, 8fC. be an increasisig geometrical series, beginning at 1. Then wiU I, f^, r*, r^, r*, 4kc. {that is, all the odd terms) be squares ; I, r^, r^, r^, (or the 1st, 4tK 7th, and lOth,) wUl be cubes ; also r^, (or the Jth term,) is both a square and a cube: and the like may be shewn in a decreas* ing series, Sr. Every square number o^st end in either 1> 4> 5^ 6, 9, orO. The truth of this will appear by Sj^wisrii^ the first ten numr bers\,^,^,^iuto\D. * h4 104 ALGEBRA. Part IV. Cob. Hence no square can end in 9, 3^ 7> or 8. 28. A cube number may end in either of the ten digits. This voiU likewise appear by cubing those numbers, Coa. Hence 2> 3, 5, 6, 7, B, 10^ &c. can have no exact sqaars root, nor can 3> 3, 4, b, 6, 7, 9, 10, &c. have an exact cube root. 29. All the powers of numbers ending ih 0> I, S, and 6, vnXi end in the same figures respectively 5 and all powers ending in the above figures, will have their roots ending in the same figures respectively. Thus iol*=100, 10l'=1000, l9|«ssl<KXX)« SfC. ending in 0. ll]«=:121, in»=1331, m^= 14631, Sfc. ending in I. 5l*= 25, 5l*=125, 5]*=:625, SfC. ending in S. 6)*= 36, 6)«=216, 6?*= 1296, 8(C. ending in 6. and the like for the roots of powers ending as abovCf as is plain. SO. All numbers ending in 4 or 9, will have their even powers end in 6 and 1 respectively ; and their odd powers the same ss their roots, viz. 4 and 9, respectively. 7%M» il«^=;16,4?»=64, 4l*=266, *c. 9l«=81, §?'=729, 9l*=6561, «rc. 31. The powers of numbers ending in 2 will end in 4, 8, 6, and 2, alternately ; numbers ending in 3 will have their powers ending in 9, 7, 1> and 3, alternately; numbers ending in 7 will have their powers ending in 9, 3, 1, and 7^ alternately 5 and numbers ending in 8 will have their powers ending in 4, 2, 6, and 8, alternately. 77^19 will appear by involving such numbers. Cor. Hence numbers ending in 1 and 9 will have their even powers end in the same figure, viz. 1 -, numbers ending in 3 and 7 will end their like even powers with the same figure, vis. their squares with 9» their 4th powers. with 1, &c.; numbers end- ing in 2 and 3 will end their even powers alike, viz. their squares with 4> their 4th powers with 6 ; numbers ending in 4 and 6 will have their even powers end alike, viz. with 6 ', and in gene^ ral, the like even powers of any two numbers equally distant from 5, will end in the same figure. 32. The right hand places of any number being ciphers, if the right hand significant figure be odd, the number will be divi*- sible by unity, with as many ciphers subjoined as there are d- j>hers on the right of the saifj number -, if the right hand signi* Pabt IV. PROPERTIES OF NUMBERS. lOS ficant figure be even, it wiU be divkiUe b^ 2, with as many ci- phera -subjoined. Thvs 12S0 is dwiiible hy 10, 3100 hf 100, 7000 by 1000, «c: Also 1240 is divisible by 30, S£00 by 900, 8000 by 2000, 4c. Off d </ie ZiAre is true in all simUar cases. 33. Every number ending in 5, is divisible by 6 without i^mainder. This is plain, since all such numbers are either 5, &r multiples of 6. Cor. Hence, numbers ending in O or 5 are divisible by 5, 34. If the two right hand figures •£ any number be measured by 4, the whole is measured by 4 j and if the three right hand figures be measured by 8, the whole is measured by 8. Thus the two right hand figures of each of the numbers 184, 2148, 37128, 13716, 71104, *c. being divinble by 4, each of these numbers is measured by 4. jilso the three right hand figures of each of the numbers 13398, 97464, 9916, 100800, 9040, 4c. being measured by 6, each of the numbers is measured by 8 j and the same is true in all similar cases. 35. In any even number, if the sum of its figures be measured by 6, the number itself is measured by 6. Thus the sum of the figures in the eten number 738 t« I85 which b&ng measured by 6, the number 738 itself is likewise mea^ sured by 63 and the like of all other similar numbers, 36. If the sum of the figures in the first, third, fifth, &c. places in any number, be equal to the sum of those in the second, fourth, sixth, &c. places, the number itself is divisible by 11. Thus the number 4759 is divisible by 11, because 44*5 {the sum of the first and t^ird)s=7+9, {the sum of the second and fourth ;) in like manner 1934563 is divisible by II, for 1 +3-f 5-f 3=9+4+6 ; and the same is true of all similar numbers, 37* Any part of the sum or difierence of numbers is found by dividing each of the given numbers separately by the num^ ber denoting that part 3 and any part of their product is found by dividing one only of the numbers by the number denoting the part *. ■ '■ ill' I ■ — III II... ■ «i « 1 1 1.» • The properties 32 to 37 iaclasivt, with some others^ are iotrodaced in « 109 ALGSS&A. PAKTlir. TkH9 half the sum ^ ea-i-Ab^Scii Sa+S6^4e. Jnd half the product of 6ax4bxSc u Sax4hx8c,or 6ax26x8c, or6ax46x4Cj ^ach be'mgsa^S abe. 38. Every even square number is measured by 4, and erery odd square divided by 4 leaves 1 remainder. For nnce the root of an even square must be even, (proper. 8.) let 2n be its root; then ^^s4n^ the square, which is evidently divisible by 4. Again, since the root of an odd square must be odd, (proper. Il.)let2n + lbe such root, *ik€n^»+ll*=4n«-i-4n+ 1 thesqwxre; Ujhich being divided by 4, wj^l evidently leave 1 remaining. 39. If any number, and also the sum of its figures, be each divided by 9^ the remainders will be equal. Met n he any number composed of the digits a, h, c, and d; then, according to the establisfied principles of notation, 1000 a + 1006+ 10c+d=:n; but 1000a =(99 94-La=) 999a-i-fl; 1006= (994-1.6=;) 99 6+6; 10c=(9+l.c=) 9c+c: therrfore n=: (1000 a+ 1006+ 10 c+ds=)999a+996 + 9c+a+6+c+diCOtt- sequently — =111 a+11 6+c-| — — i—, or the number n being divided by 9 leaves 1- — remainder, which is the same as tlie remainder of the sum of its digits divided by 9', as was t0 be shewn, CoR. Hence the operations of addition^ either of whole num- bers or decimals, may be proved by casting out the nines; for it is plain that if the excess of nines in two or more numbers be taken, and likewise the excess of nines in these excesses, the last excess will equal the excess of nines in the sum of the given numbers j since the sum of the excesses of the parts (taken feparately) is evidently equal to the excess of the whole t. t • note on p. 155, 156. Vol. I. as usefal for readily finding the measures of nam* bcrs, and fpr redueiog fcactions to their lowest terms. > To shew the method of proving addition by casting oot the nines, the following examples are subjoined. £x. 1. Ex. 2. 357S .. • * 8 68.496 .... 6 6832 .. ..I I Excenes ^^^fj • • - • « | Ercestes 7654 ....4 V^ ^^ 4.7121 6 8323 563«T •••• > of 4.7121 6 V «;. •• -I \mnes, ^^« ^^^ • • • • JL \niL. ....2^ 8S7.S091 .,..8-^ Pabt IV. PROi»£RTIBS OF NUMBERS. lOf 40. If each of two nnmbeTB be ^fivMed by 9> and the product of , the semainden also divided by 9* this remaiiidei' shall equal the remainder ariaiiig from the product of thie two given num-* hers divided iof 9. F&r ifit 0^-). a and &B+b be tlm ttpo numben, whkk being dU ab vid^d by 0, toill evidently leave'a and bfor remainders^ and — ^ the product of the$e remMnden diaided by 9. ^+MX9B^b _ »\jiB^9aB'{'9Jh^ab 9 ""^ ' 9 ""^ aB+Ah-i ; wherefore — is the remainder of the product of the two given numbers divided by 9« and it equals the product of the remainders of the two given wam^beri.diindfd by 9^0$ found ahomf which was to be shewn *^. ' WII' M In Ex. I., the nines b«ing cast ovt of the top tine» the 8 placed opposite remains in excess; in like manner 1, 4, and 7> are respectively the excesses «l thel second, third , and jfourtb, lines: now these foar excisies heing added together, and the nines cast ont of the sum, the excess will be 9, and if the nines be cast oat of the sum of the numbers proposed, (263^1«) the excess ia Ulewise 2, which two excesses agreeing, the work is presumed to be right for the reasons gireti in property 39. and its corollary. But there are two cases in iM^ich Hiis mode of proof does not succeed ; the first is idien a mistake of 9y or any nultij^ of ^s lias been made- in tlie addiag ; and tbe second Is whea all, or any of the figures haine beei^ transposed: in each of tbese cases, al» thongfa the work is mantfiestly wrong, the proof will make it appear right. Subtraction may likewise be proved by the same method, but this will be con- aidered rather as a natter <d coyiosity than use : in subtradtog tfae ezcesees^ if the Viwes one be the greater^ 9 mqst be borrowed, as in Ex. 2. below. E«. 1. Ex. 3. From 237165 .... 6") 37.4& .... 11 Take 123428 ... ,2 >JBreiiwse. 3.12^4 • . . .^ > J!?e»t. 1 I37a7 . . . . 4 J . 34.326^. ...63 In Sx. 1. basving ea«t tbe nines ont of tbe t^o given numbers, the lower czeess 2 is subtracted from tbe upper excess 6 ; then the difference 4 being c^nal to the excess of nines in (1 13737) the remainder, shews the work to be jpgbt) «Db)e^ b^wever to tbeiexeeptions- stated above. In £x. 2. the 4 cannot be taken from 1, therefore 9 is borrowed ; the rest •V Mk tbe preeeAnif example. - i « Tbe pTttctieal plication of tbts property of the number 9, is fully exem< pHfiini in tlM.pM>o£i suls^iaed to tfae operations of nuUipKcation and divisioQ ef both whole numbers and decimals. See Vol. I. p. 34—38. 47—49^^15. 319. 10$ ALOERRA. Part IV. 41. Any ariUimetical pragretekm cui be increased m tfj^i- turn, bat not decreased; a barmonical prqgreauon can be de* creased in infimiium, but not increased; bat a geometrical pro- gression can be both increased and decreased in it^bniMm *. First; let a-|-a+r+a+2r-f-,^. be an arithoietical progres- sion ; this series can evidently be increased at pleasure by the constant addition of r : but if you take the series backwards, and decrease its terms suooessively by r, it will become <i4'r+ a-f a~r-f a-*-8r-h, 8fC. now when ei/ft«r of the quantities r, 2r» 3 r, becomes equal to a, that term is equal to O, and (he series evidently can proceed no further. Secondly, let — | 1 -f, 4rc.be a barmonical series, in •' a a-fr a+^ which the last term is the least ; this can evidently be decreased at pleasure by the constant addition of r to the denominator. Now taking this series backwards, and continually subtracting r from the denominator, it becomes H h h — tt'^* *^- ^^ Q'^r a a — r a**%r when r, 9 r, S r, or some multiple of r, becomes equal to a, it is plain tbe next term of the series will be negative, or the series terminates, without the possibility of further increase. Thirdly, let a+ar-^-ar^'^ar^, be a geometrical series; thi» series may be increased by constantly multiplying by r, or de- creased by constantly dividing by r, as is evident, without the possibility of its terms becoming negative. The nuiuber 3 poticMes tbe tame property, bat 9 is mwiUr prdemd, at being tbe moBt convenient for practice : we may add, that tbe tame incoDTenience attends the proving of multiplication and division by this method, as that men- tioned in the precediiig note. Tbe rate for proving addition by casting out the nines was, according to Mr. Bonnycastle, first pablishcd by Dr. Wallis in 1657 ; but the property of the number d» on which tbe rule is founded, was most piohably known to tbe Arabians long before that time : Lucas de Bmgo, who wrote in 1494, was well acquainted with this property, and shewed the method of proving the primary operations of arithmetic by it, as is witnessed by Dr. Uutton. Matf^. Diet. Vol. I. p. 66. X This property of the three kinds of progressioas was first noticed by Pappus, a Greek Mathematician of tbe Alexandrian School, who flourished in the latter part of the fourth century, in the third book of his Collections. Past IV. PROPERTIES OF NUMBERS. 109 49. If a harmonicai mean and an ariUunetical mean be taken between any two numljln^ the four terms will be pro- portionals. Let a and h he any two nutAers, then will — —- ^ ^a arith^ metical mean, and — tt <> harmankal mean between a and b: then wiw a : ■■* : : : by for the product of the meam (ab) it equal to the prodmct of the extremee {ab), which is the criterion of pro- porOonality. (Art. 56.) 43. The square root of a rational quantity cannot be partly rational, and partly a quadratic suri^ For if possible, let ^xssa+ jy/% of which jjb is an irredu- cible surd ; square both sides, and x^m^ +9 a ^^6+ 6^ or> 9 a^6 X— a* — & =*— a* — 6> V j^b^ — ~ , that is, an irreducible surd equal to a rational quantity, which is absurd; wherefore ^x cannot equal any quantity of the form ofa-^ ^b, 44. If each side of an equation contain rational quantities , and irreducible surds^ then will the rational parts be equal to the ra- tional^ and the surd parts to the surd. Lei 4?+ ^«=a+ ^b, then will x=a, and V'=* V^- For if x be not =a, let x^a-^m, then a+iii+ ^z:sa+ j^b, ^ ^ + iv/2= ^^b, that is, j^b is partly rational, and partly surd, which is proved to be impossible in proper. 43. 45. From the forgoing property we derive an easy method for extracting the square root of a binomial surd^ as follows. Example. To find the square root of m+ ^n. First assume ^x+ V*^ V^wH" V* **^ squaring both sides x4-2^«B-f «=:m+ v^n; wherefore {proper. 44.) x-^-z^m, and 9 ^xzx ^n; these equations squared gioe x' + 9 xz^z* =sm*, and 4xzszn; subtract the latter from the former, and x* -^2 xz-^-z* ssm*— «, V by wofoilioa*— xss^ia*— «; but X'\-z^m, v t= ^ ,andz^ ^ /.• vm+ ^n^{^x+ ^z=) ^ Z-- ^ V 21- , the root required. PART V. ALGEBRA. OF EQUATIONS OF SEVERAL lilMENSIONS. A GENERAL view of the nature^ fonnaticm, mnd roots of •qaations. 1 . A simple equation is that which contaiiii the unknown quantity in its first power ohly. Tku9 cur+ftssc. 2. A quadratic equation is that whick contains the second power of the unknown quantity^ and no power of it higher than the second. Thus ta^-^bx^c, 3. A cubic equation is that which contains the thirds and no higher power of the unknown quantity. Thus a3fi^bx*'\-cx=::d, or ax^ + bx^=::c, or wfi-^bx=sc. 4. A biquadratic equation is that which contains the fourth^ and no higher power of the unknown quantity. Thus ac^-h&a?*— cr®+(ir — c=o, 8fc. 5. In like manner^ an equation of the fifth degree is that which cooftains the fifth, and no higher power of the unknown quantity j an eqtiation of the sixth degree contains the iixth power J one of the seventh degree the seventh power of the unknown quantity^ &c. &c. i 6. All equations above simple^ which contain only one power of the unknown quantity^ are called pure. Thus ax^=b is a pure quadratic, a3?i=:h is a pure cuhie, ua^zsih a pure biquadratic, S(c. 7* All equations containing two or more different powers of the unknown quantity^ are called affected or adfected equations. Thus aot^-^hx^s^e is an adfected quadratic; ckc*— iBr*s3C, amd aa:' + &r=c are adfected cubics ; a^'^sf^-i-ax^sb, and a**-^to*aac;, and ax^-^bx^ + cx*^dx-^esso, are adfected biquadtFodct, 112 ALGEBRA. Part V. 8. An equation is said to be of as many dimensions, as there are units in the index of the highest power of the unknown quantity contained in it. - Thus a quadratic is said to be an equcUion of two dimensions ; a cubic of three ; a biquadratic of four, <rc. 9. A complete equation id that which contains all. the powers of the unknown quantity » from the highest (by which it is named) downwards. Thus ax^—bx+cszo, is a complete quadratic ; ax^—hs^-bcx — dsso, is a complete cubic ; a?*— Jf*— ac^+a?— a5=o, a complete ii- quadraiiCy ^c. 10. A deficient equation is that in which some of the inferior powen of the anknown quantity are wanting. As aa?*— 6a:*+c=so, a deficient cubic; aa:*— 6a;*-hca?— d=o, a deficient biquadratic, S;c, 11. An equation is said to be arrsMEiged according to its di- mensions, when the term containing the highest powet of the unknown quantity stQSids first (on the left) ; that which contains the next highest, second ; that which contains the next high^, third ; and so on. Thus the equation x*— ar♦4■6a^'— ca7®-fd|3P— ^«=o> m arranged according to its dimensions, Cos. Hence every complete equation of n dimensions will contain n-i-l terms. 12. The last term of any equation being always a known quantity, is usually called the absolute term : and note, this last or absolute term may be either simple, or compound, consisting of leveral known quantities connected by the sign + or — 5 ^which t€>gether are considered as but one term. 13. The roots of an equation are the values of the unknown quantity (expressed in known terms) contained in that equa- tion ', hence, to find the roots is the same thing as to resolve the equation. 14. The roots of equations are either possible, or imaginary. Possible roots are such as can be accurately determined, or their values approximated to, by the known principles of Algebra. Thus y^a, ^^a-^b, *^c, ^c. are possible roots. 15. Imaginary or impossible roots ar^ such as come under the form of an e»en root of a negative quantity, which cannot be determined by any known method, of analysis. Thus V**"** * V***^* * V"~^/ *^* ^^ impossible roots* Paut V. NATURE OF EQUATIONS. IW 16. The limits of the roots of an equation are two quantities, one of which is greater than the greatest root 3 and the other, less than the least. The greater of these quantities is called the iuperior limits and the less, the inferior limit. Also the limits of each particular root, are qutotities which &11 between it and the preceding and following roots. 17* The depression of an equation is the reducing it to another equation, of fewer dimensions than the given one possesses. 18. The transformation of an equation is the changing it into another^ differing in the form or magnitude of its roots from the given equation. OF THE GENERATION OF EQUATIONS OF SEVERAL DIMENSIONS. 19. If several simple equations involving the same unknown quantity be multiplied continually together, the product will form an equation of as many dimensions as there are simple equations employed '. Thtis, the product of tmo simple equation» is a quadratic ; the continued product of three simple equations is a cubic; that of four, a biquadratic; and so on to any number of dtmensUms, For^ let X be any variable unknown quantity, and let the given quantities a, b, c, d, Ssc be its several values, so that xs^a,, x^b, xssic, x^d, SfC. these by transposition become x-^as^o, x^b^o, X— csso, x-^d^o, 8(C. if t he continued product of these simple equations be taken, (viz. x^ajr— 6.x— cor— d. Ssc.) it will m^f f This metikod of gemsntmg roperiot tqiiations by the eontimul maltipli- catioo of inferior oaei , was the invention of Mr. Thomas Harriot^ a oelc« brated Xnglish mathematician and philoeopher, and was first pnbUsbed at JjondoQ in the year 163 1* beinf ten years after the antbor^s decease, by his friend, Walter Warner, in a folio woik, entiUed, Artis Jnafyiice Praxis^ ad /B^uatumes AlgebraiettM nova, expeHtay et generdU metkodo^ t^emh^emdas^ By this excellent contrivance the relations of the roots and coeiBcients, and the whole mptery of equations, are completely developed, and their rarions relations and properties discovered at a single glance. See on this subject iSitr Isaac Newton's Ariihmetica UmversaUt, p. 256, 257. Madaurin** jRgebra, p. 139. ^» Huiton't Mathematical Dictionary^ Vol. I. p. 90. ;^mpaon*9 Algebra, p. 131. &c. Dr. WaSHtU Algebra ; Pr^essor yilantU Elememis qf Matkematieal Ana^sit, p. 48. and various other writers. VOL. II. 1 114 ALGEBBA. Part V. m constitute an equation (=zo) of qs many ^mennons as there are factors, or simple equations, employed in it^ composition: for example. Let X — a=o Be multip, info x—b^^o The product U ^'-«|,+«t^^, „ quadratic. Multiplied into x—c=io The product is a?'— a"| +a6^ —6 >3i^+ac >x—abc=o, a cubic, — cj +bcj Multiplied into x — dsso The product is x*'~a'^ +a^T ^abc\ ,-i-flc J "Obd \x+abcd=zo, a -f-fld I -pft— acd f biquadratic. + 6r { — 6cdJ + bd\ -t-cdJ *c, S(C. From the inspection of these equations it appears^ that SO. The product of two simple equations b a quadratic. 91. The continual product of three simple equations^ or of one quadratic and one simple equation, is a cubic. 22. The continual product of four simple equations^ or of two quadratics^ or of one cubic and one simple equation^ b a biqua- dratic 5 and so on for higher equations '. ^. The coefficient of the first term or higher power in each equation b unity. 84. The coefficient of the second term in each, b the sum of the roots with their signs changed \ Thus, in th4( quadratic, whqse roots are-^-amnd'^b, the coefi" eientis.'^a'^b^in the cubic, whose roots aTe'\-a, + b, and-i-c, it ■ It M in like manner eTideot, that the roots of the componnded equatioot will have not only the same roots with its component simple e^ationsy but that its roots will hare the same signs as those of the latter. ■ Hence, if the sum of the affirmative roots be equal to the sum of the ne- fattve roots,' tlie coefficient of the second term will be ; that li, the icoQiid tenn will vanish : and conversely, if in an equatioa the second term be wantr ing, the sum of the jaffirmative roots and the sum of tl^e negatiYe loota ate equal. / Paet V. NATURE OP EQUATIONS. lis is — fl— fc— c; in the biquadratic, whose roots are+af + bt-^-Cj and+d, it is — a— fc— o — d, 8(C. ' 25. The coefficient of the third term in each^ is the sum of all the products that can possibly arise by combining the roots, with their prober signs, two and two. Thus, in the cubic, the coefficient of the third term M+a6-f ac-^be; in the biquadratic, it iS'{'ab+ac+ad+bc-{'bd-{-cd, SfC. 26. The coefficient of the fourth term in each, is the sum of all the products that can possibly arise by combining the roots, with their signs changed, three by three. Thus, in the biquadratic, the coefficient of the fourth term 18 — abc^ahd^acd-^bcd. In like manner, in higher equations, the coefficient of the fifth term will be the sum of all the products of the roots, having their proper signs, combined four by four \ that of the sixth term, the roots, with their signs changed, five by five, &c. 27. The last, or absolute term, is always the continued pro- duct of all the roots, 4^aving their signs changed. Thus, in the quadratic, whose roots are -^^ a and-^-b, the last term is-^ab (or—ax —b) ; in the cubic, the absolute term is —abc (=: — ax— fcx— c); in the biquadratic, ^e absolute term is-\- abed (=— a X — 5 x — c x — d), ^c. 28* The first term is always positive, and some pure power of X. 2S.B. The second term is some power of x multiplied into ^a, — b,—c, ifc. and since x is affirmative^ and each of these quantities negative, it follows that the second term itself is negative, since 4- X — produces — . 29* The third term wUl be positive, for its coefficient being the sum of the products of every two of the negative quantities- — a,— 6,— c, 4rc. and (since-*- X— produces +) therefore these sums, multiplied by any power of x, (which is always positive,) will always give a positive result. SO. For like reasons the fourth term will be negative, the fifth positive, the sixth negative, and so on i that is, when ,tbe roots are all positive, the signs of the terms of the- equation will be alternately positive and negative : and convei'sely, when the signs of the terms of the equation are alternately + and — , all the roots will be positive. 12 lie ALGEBRA. PaktV. Cor. Hence, if the signs of the even terms be changed, the signs of all the roots of the equation will be changed. 31. Let now the roots of the equations, above referred to* be' supposed negative 5 that is, x= — a, a?= — b, a?= -r c, x=: — d, 4rc. then by transposition, x-)-a=:o, j:+&=:o, x4-c=ao, x+d^o, 4rc. ^i^^^tm^mm ^m^t^-^n^^ fl^H^^^p* «^i^H^^^ the product of these, or x+a.x+b.x+cjB+d, Sfc, wiU bean equation, having all its terms affirmative; for since all the quantities composing the &ctors are +, it is plain that the pro^ ducts will all be -h . Cor. Hence, when the signs of all the roots (in the above simple equations, having both terms on one side) aj^e -<• , the signs of all the terms of the equation compounded of them will be-f ^ and conversely, when the signs of all the terms of an equation ^*e 4^, the signs Of all its roots will be — . 32. If equations similar to the foregoing be generated, having sotne of the toots +, others ^, it will appear, th^ there will be as many changes in the signs of the terms, (from + to — y or from — to+,)9s the equation has positive roots 3 and as inlany continuations of the same sign, (-hand+/or — and — ,) as the ^quatiom has negative roots : and conversely, the equation will have as lAanjr affirmative roots as it has changes of signs, and as many negative roots as it has continiiations of the same sign \ Cor. It follows from what has been said, that every equation has as many toots as its unknown quantity has dimensions. To be particular j a quadratic has two roots, which are either both affirmative, both negative> or one affinnatite and one i^Hb ^ ThU supposes the roots to be all possible. Ererj equation w3( have either an even number of impossible roots, or node : hence a quadratic wSl bare both its roots possible, or both impossible ; a etibfc one ot thYee possible roots^ and twof or none impossible ; a biqnadratie will have eHhet fdar^ two, or none of its roots possible, and none> two, or fouSr, impoisib^ *^ and the like of hig^her equations. An impossible root may be considered, either as affirmative or ne^tire. The di Acuities attending the doctrine of impoa^le or imaginary roots, have hitherto bid defiance to the skill and address of the ^rned : a great number of theories atid invesfigations have appeared, it is tfne ; bat our knowledge of the origin, nature, properties, &c. of imaginaiy roots i» sUU very imperfect. The following Authors, among others, have treated on the sttl^ect, via. Cardan, Bembelli, Albert Oirard, Wallis, Newton, Mao- laurin, James Bernoulli, Emerson, Euler^ D'Alembert, Waring, Hnttoo, Sterling, Playiair, &c. PahtV. depression of equations. lir i^egative. A cubic has three roots, which are either all afErma- tive, all negative; two affirmative, and one negative; ot one affirmative, and two negative : and the like of higher equations. 33. If one root of an equation be given, the equation may h^ depressed one dimension lower ; if two roots be given, it may be depressed two dimensions lower ^ and so on, by the following rule *. RuLB. When one root is given, transpose all the terms to one side> whereby the whole will=o; transpose in like manner .^e value of the root> then divide the former expression by the letter, and a new equation will arise=o^ of one dimension lower than the given equation. Examples. — 1. Let of*— 9x*-|-36x— 24=o be an equation, whereof one of the roots is known; namely, x=33. By transposition x— 3=o, divide the given equation by this quantity. Thus, jr— 3)a:'— 9 a:«+26a?— 24(x*— 6x+8=o, the resulting a:*— 3 a:* equation, which being re* -r.6x*4-26 « solved by the known rule for c-6x*+18j; quadratics j lis two remain^ " 8^—24 ing roots will be found, viz, 8 07^24 x^4, and xsS. 5. Letap*-h4«'+19a«— 160«=140p, whereof one root= — B, be ^ven, to depress the equation. Here by transposition, a?*+4x' + 19a:*— 160 a?— i400=o, /md ar+5=o; then, dividing the former by the latter, we have — I 1 rsif'— a?*-f 24 x— 380=0, the re- a?+5 ' . sultvf^ equation, 3. Given x=3 in tb^ equation x^-**5x 4-6=0, to depress it. 4. If jr— 4=sob« ft divisor of the equation a:*— 4 a:*— x-|-4=d, to de^ness the equation, and determine its two remaining roots. Ans, the resulting equation is jr*^l=:o, and its roots -^l \ ' When the_ absolute term of an equation so, it is plain that one of the roots is 0, and consequently the equation m^y be divided by the unknown quantity, and reduced one dimenslpn lower. In lika loanner, if the two last lerms be wanting, the equation may be reduced two dimensions lowe?) if ^hrec;, three dimensions, &c. »3 118 ALG£BRA. Part V. 5. To depress the equations a?*— 5a?®+2x+83=^, aod j:*— sis oi^+ 18 j?4-40=o, on^ root of the former beiog +4, and one of the latter —5. 34. If two of the roots be given, x-f ^=o, and xH h<= Oy the given equation being divided by the product of these^ x+rjc+*, will be depressed thereby two dimensions lower ; thus, 6. To depress the equation x'--5j?*+2x+8=o, two of its roots, —1 and -f-2, being given. Thus, x-f 1=0, and j:— 2=0| then x+lj:— 2=a^— a?— 2, o:^— 5a?*-|-2j:+8 the divigor ; wherefore •- =rx— 4, whence x— 4=o is the resulting equation, 7. Given jt'—S a:*— 46x— 72=o, having likewise two values of X, viz. —2 and —4, given, to depress the equation. An" swer, X — 9=0. B. Given a:*— 4x' — 19 jr» +46 x 4-120=0, two roots of which are +4 and — 3, to depress the equation. 35. To transform an equation into another, the roots of which u ill be greater, by some given quantity, than the roots of the prO' posed equation* Rule I. Connect the given quantity with any letter, different from that denoting the unknown quantity in the proposed equa* tion, by the sign — , and it wiU form a residual. II. Substitute this residual and its powers, for the unknown quantity and its powers in the proposed equation, and the result will be a new equation, having its robts greater, by the given qiiantity, than those of the equation given'. ' The truth of this rale is clear from the fivst example, where since y — 3 »!', it is plain that y^x-\- 3, or that the equation arising from the substitution of y— 3 for X will have its roots (or the Talue's of y) greater by 3, than the values of X in the proposed equation : this will be still more evident, if both the fiven and the resulting equation be solved ; the roots of the former will be found to be —7 and + 3, those of the latter —4 and -|> 6. Let it not be thought strange that the negative quantity ~7, by being increased by 3, becomes —4, or a less quan- tity than it was before ; for a negative quantity is said to be increased, in pro- portion as it approaches towards an affirmative value ; thus, — 3 is ssud to be greater than —4, —2 than ^3, —1 than —3, and than — 1 : in the pre- sent instance, it is plain that »7 added to + 3 will give —4 for the sum. Hencef, if the roots of an equation be increased by a quantity greater than tb)^ Part V, TRANSFORMATION OF EQUATIONS. 119 ExAifPLBS. — 1. Given a:*+4a?— 21=o, to ti'ansform it into Another equation, the roots of which are greater by 3 than those of the given equation. Operation. Explanation, Let y-3=x, then Having substituted y-3 for x^ I -r«— (iZIil » — ^i/«— 6 «/ -I- O substitute y-3)9 for jfi, y-3.4 for * — ^ y ^' ""^a' — ^y-f-y 4^^ and -.21 for itself; I then add 4- 4l^= (y — 3.4^) + 4 y — 12 ^1 the quantities arising from these J«2J -— ^ ^ ^ 2]^ substitutions together, and make the — * result y* — 2y — 24«»o, which equa* J?*4-4j?--21 =s y^— 2y— 24= tion wiU have its roots greater by 3 „„ . » ^ than the roots of the equation given Wherefore y*^^y-^U=zo, in the quegtion. is the equation required. 2. Given the equation a^'+a?*— JLOa?+4=o, to transform it into another, the roots of which ai*e greater, by 4 than the va- lues of X. Let y— 4=x, then X^z=(y^4\S=:) y3 — I2y«4.48y— 64 ■f x^=(f^^=^) ... 4.^2- 8 y4- 16 — lOx =:(y— 4.— 10=) .... — lOy+40 + 8= + 8 This transformed equation is evidently divisible byy(ory + o, ory^o)'y therefore is one of its roots: by this division U be" comes y*— 11 y+30=o, the two roots of which are +6 and +5 j hence the three roots of the equation y'— 11 y*+30y=o, being o . + 6, and + 5, those of the proposed equation x* + x**- 10a?+ 8=0 are known; for {stwe xsBy<^4) its roots ioill be 0-^4^ 6—4^ and 5— 4; or —4, +2, <MMi 4-1. CoR. Hence, when the roots of an equation are increased by a quantity equal to one of the negative roots, that root is taken away, or becomes in the transformed equation ^ and in this case, the transformed equation may be depressed one dimension lower. 3. To increase the roots of the equation x?-^6a?*4-12x-^8 s=o, by 1. ■ ' ■■ ■'■'■' I ■■ ■ 111 II I < I 1 1 1 I III III I ■ I' l greatetjfc negative root, the negative roots will be' changed into- affirmative ones. It may be likewise useful to remark, that a de&cicnt. equation may be made complete by this rule. 14 120 ALGEBRA. Pakt V. 4. To increase the roots ofa:*— 4 j:' -1-6 j:*— 13=0, by 5. 36. To transform aw equation into another ^ the roots of which will be less than tlwse of the proposed equation, by some gioen /quantity, KuLB. Connect the given quantity with some new letter bf the sign +, and proceed as directed in the preceding rule *. Examples. — 1. Transform the equation x*--2 j?— 24=:o into another, the roots of which will be less by 3 than those of the given equation. Operation. Let y-^S^X, then EspUmatum. — 2j:=(y-|-3.— 2=)— 2y— 6 and— 24foriUclf,thefumoftbe8c ■^24= -*24 i« y* + 4y— 21=0, the equation ia-.2x-24=r y.+4y-21 ^^^^^«*- I I.I ■■■■- — ■» Wherefore y*+4y^21=:o, is the equation required. This equation being solved, the roots wUl be found to he -^S and —7; wherefore those of the given equation are +3+3 and —7+3, or +6 and —4. 2. To transform the equation^^— a«p^+&r— c:^o to another^ the roots of which shall be less by e. Let y+eszx, then *Srr(^+;|s=:)y9 + Sy«C+3yc*+e' -j +fca: =a(y+c.6=) by+be I quired. — c= — c -^ 3. Duninish the roots of a^— 6«^+9ap— 12ssa, by 6. 4. IMminish the roots of a7^+5a^— 6x*+7x— Sso, by 10* 37. To exterminate the second term of an equation. RuLB I. Divide the coefficient of the second term, by the in* dex of the highest power of the unknown quantity in the given equation. II. Change the sign of. the quotient, and then eonnect it with some new letter ; tins will form a binomial. • The trath of this mle will be plain from ex. I . for y + 3 being made equal to JT, or ifssjr— 8, that is, y less than or, by 8 ; the roots or values of y in tba transformed equation, will be less by 3 than the corresponding values of x in |hf proposed f tj^uation, ^s is eridentf Pa«t V. TRANSFOBJilATlON OF EQUATIONS. l«l m. Substitute this binomial and its powers^ for the unknown quantity and its powers in the given equation^ and there will arise a new equation wanting its second term '• Examples. — 1. To transform the equation a^-|-12 *•— 8x— 9 =o> into an equation wanting its second term. Operation. 13 First — = +4. Let y— 4=:ir. Then, a^^(y^4)p^) y^— 12y« + 48y— 64* — Sx =(y— 4.— 8=) — 8y+ 32. — 9 = — 9. jr» + 12a«— 8x— 9= .... y *— 66y + 151=o. Explanation* I first divide tbe coefficient 12 of the second tenn by the index 3 ; the qno* tient 4 I annex to a new letter y, first changing its sign from + to —-a making 1^—4 ; this quantity and its powers are next substituted for x and ita powersy as in the two foregoing rules ; then adding the like quantities together, the sum b the equation y^ *— 56y -|- 15 1 no» wasting its second term, as was proposed. 2. To destroy the second term from the equation a:*— 0x^4- fcc»— ca?+d=o. a First, — — is the coefficient of the second term dimded by the index of the first. Let y be the new letter, then by the rule, y-{'—zsix, whence ■ 3y«o« 3ya» a* ^4 16 64 ca "~CX SBr ••«.••• ••" Cy "~ ■-—;* 4 4"d= ...,. -f-rf ^. 3y«fl« 3y»a» ya» ^ 6ya "S^ a* , ^*--i — r+'^*-T+T-^+^~64+ ' Thb rule is necessary to the solution of cubic and biquadratic equations ; and the truth of it will appear from an attentive examination of the process in ex, 1. Tbe third> fourth, and fifth> &c. terms may be exterminated from auf 1» ALGSBAA. Part v. [• dsso, which J properly contr acted », becomes jf* -f ^— -^ •y^ — S'+'S — ^-y ^?7S ™^> '^ c^tta^ion regKtrea. o " 256 3. Given ar* — 4af+8=o, to exterminate the second term. —4 Thus, = —2 ', then let y + 2=x, and proceed cw before. 4. Given a:*+ 10 x— 100=0, to destroy the second term. 10 ' , Tfitis, -jrss-l-Sj te^y--5=x, and proceed. 2 5. To exterminate the second term from x'—S x'-j*4x— 5=o. 3 Thus, — =*:— 1, let y + l:=x, and proceed. . 6. Let the second term be taken away from the equation x*4-24x5— 12a?*+4x— 30=0*. 7. To take away the second term from the equation x^— 50x*+40x^— 30x«+20x— 10=0; 38. To multiply the roots of an. equation by any given quantity, that is, to transform it into another, the roots of which will be any proposed multiple of those of the given equation. Rule I. Take some new letter as before^ and divide it by the given multiplier. II. Substitute the quotient and its powers^ for the unknown quantity and its powers^ in the given equation^ and an equation equation, but these transformations being less nseful and more difBcult than the above, we have in the text omitted the rales : in general, to take away the second term reqnires the solution of a sioif le equation ; to take away the third term, a quadratic ; the fourth term, a cubic ; and the n^ term requires the solution of an equation of n — 1 dimensions. See the note behw, f This contraction consists in the reducing of the fractional coefficients of the same powers of y to a common denominator, and then adding or subtracting, according to the signs; putting the coefficients of the same power ofy under the vinculum, &c. &c. i> In like manner, to take away the third term from the equation x^ — ax' -^hx^c=o, we assume y4~^=^> where e must be taken such that (suppos- »— I 7 ing ai=the index of the highest power of x) n, -3— « • .^»— I, ae+b=:o» In which case a quadratic is to be solved ; and in general, to take out the m*^ term, by this method, an equation of m— 1 dimensions must be solved, as was observed in a preceding note. See Wbod^s Algebra, p. 141. Past V. TRANSFORMATION OF £QUATIONS. 1S3 win thence arise^ whose roots are the proposed nnJtsple of those of the given equatioa. . . Rule I. Assume some new letter as before, and place the given quantity under it, for a denominator. II. Substitute this fraction and its powers, for the unknown quantity and its powers respectively, in the given equation, and a new equation will arise, having its roots respectively equal to the given equation multiplied by the given quantity*. Examples. — 1. To transform the equation x*+5j:— 3=o,. into another, the roots of which are 10 times as great as those of the given equation. o —2 = -2 Lei r-=ap. 10 Then j?« = 100 -f5x = + Whence a«+5x ^2=i!-+-^^ — 2=o, that is, y«+50y-200 100 2 =0, the equation required \ 2. liCt the roots of 3 0^—12 a?* + 15 X— 21=0, be multiplied by 3. 9 Thus, -|-=*' t Then3a^^C'p=^y^ 4y< + 15x = +5y -21 =: -gl 3%ere/are (^-^+5 y-21, or) y«-12y*+45y — 189:=o, the equation requured* < This nile reqairct neither pro«f Dor explaiMtion ; it it fometimet ufeful for freeing an equation from fractions and radical qnantities. k Hence it appearf, that to mnltiply tfae rooU of an equation hy any quan- tity, we have only to n^ultiply its terms respectively by those of a geometrical progression, the first term of which is 1, and the ratio the mikltiplying qoMi- 124 ALGEBRA. Part V^ 4. Let the rooU of x' -*3 x+4=so^ be doubled. 5. Let the roots of ar'-flSa:*— 20x-f 50=o, be multiplied by 100. 39. To transform any given equation into another j the roots of which are any parts of those of the given equation. Rule I. Assume a new letter as before^ and let it be multi-r plied by the nimiber denoting the proposed part. II. Substitute this quantity and its powers^ for the unknown quantity and its powers> in the given equation ', the result will be an equation, the roots of which are respectively the parts pro- posed of those of the given equation ^ ExAMPLBS. — 1. Let the roots of »•— x— 6=o, be divided toys. Assume 3 y=x ; then wiU x«:s 9y* —X = . . —3 y fFA«ice (9y*— 3y— 5=0, or) y* — ^ =o, is the equa^ • 3 9' tion required, 2. Let the roots of x»+7x*— 29x+2=:o, be divided by 5. 3. Given x*— 2x^—3 x+4=o, to divide its roots by 8. 40. To transform an equation into another, the roots of which are the reciprocals of those of the given equation. Rule I. Assume a new letter, and make it equal to the reci- procal of the unknown quantity in the given equation. tlty . thus, in ex. 1 . the roots of the equation are to be multiplied by 10 ; wherefore mnltiplying the given equation x* + 5 :r— Saso by the geometrical progreision 1 10 , 100 The product is x' + sOr— 200so, as above, where y in the above example answers to x in this ; and the like in other cases. 1 This rule is equally evident with the foregoing ; and in like manner, the roots of an equation are divided by any quantity, by dividing its terms by those of a geometrical progression, whose £rst term is 1, and ratio, the said quantity : Thus, ex, 1. to divide the roots of x' — ar — 5 bo by 3, pivide its terms respectively by I 3 9 X 5 The qnotientf are x • — "5"— "T""* <>> w above ; where y in that, answers to x in this. It is sometimes necessary to have rci codrse to this rule, to exterminate surds from an equation. Pabt v. transformation of equations, iss II, Substitute the reciprocal of this letter and its powers^ for the unknown quantity and its powers^ in the given equation j the result will be an equation, having its roots the reciprocals of those of the g^ven equation. Examples.— 1. Let the roots of «*— 2j;a-h3«— 4r=p, be transformed into their reciprocals. Assume y=:— , that is »=— , then will X y y y +3«= — +1 y -4 = . ; -4 ^'^^ (77— r7+— — 4a=o, or muUiplying by f, ehang-^ 9 tf if kig the signs, and dimding by 4,) yS-.i-y«+i. y^L -<,, the equation required, 2. Let the roots of a^+lOa?— 25=o, be changed into their reciprocals. 3. Change the roots of a?— ac«+fcxr-c=so, into their reci- procals. 4. Change the roots of «*-f at»-»— fca;"r-«+caf-«— d=ao, into their reciprocals. 41. To transform an equation into another, the roots of which are the squares of those of the gioen equation* RuLjs. Assume a new letter equal to the square of the un- known quantity in the given equation 5 then by substituting as in the preceding rules an equation will arise^ the roots of which are the squares of those of the ^ven equation. Examples.— 1. Let the roots of the equation x^+9:r— 17so^ be squared. Assume yj=^x^ Then x«=sy —17= —17 Whence y-^O^y'^lT^^o, the equation required ". II. 1 1 1 * The roots of the propoied equation fro 1.6 «d4 ^lOSi those of th« 126 ALGEBRA. Pakt V. 2. Let the roots of ar*— a?*+r— 7=o* be squared. Assume yssj^ Then a:^s=yi -7 = ....-7 Whence y^^y-^ sjy^T^o^ the equation required. 3. Square the roots of x^+Sx*— 3a?-.12=o. 4. Square the roots of x*— (mp*4-^— cx+d=:o. 5. Square the roots of xr — 7xt— 8=o. OF THE LIMITS OF THE ROOTS OF EQUATIONS. 42. Let x— a.x--6j7— c^-hd=o, be an equation^ having the root a greater than h, b than c, and c than d*; *'hk wfaich^ if a quantity greater than a be substituted for x, (as every factor i^^ on thb supposition, positive,) the rescdt will be positive; if a quantity less than a, but greater than b, be substituted, the re- sult will be negative, because the first factor will be negative, and the rest positive. If a quantity between b and c be sub- stituted, the result will again be positive, because the two first fsuctora are negative, and the rest positive ; and so on ^. Thus, transformed equatiuD are 2.56, and 113.36,' which are the squares of the for- mer respectively. * *' In this series the greater is <f, the less is — </ ; and whenever a, b, c, — <f, &c. are said to be t^e roots of an equation, taken in order, a is supposed to be the greatest* Aiso in speaking of the limits of the roots of an equation, we understand the limits of the possible roots." This note, and the article to which it refers, were taken .from Mr. Wood's Algebra ; see likewise, on this subject, Maclaurin* 8 Algebra y part % cb. 5. Pf^olfius's Algebra, part 1. sect. 2. ch. 5. Sir Isaac Newton* sArithmeiica Universalis, p. 258. &c. JCh\ J9^arwg*s AMUcUuma AlgebraictB, 8cc • To illustrate this, let the roots of the equation x* — /»x* + ?* ' — rx-^-s^o be a, b, e, and if,* then x— aso, x—b^o, x — cso, and x^^dsso ; and let g, which we will suppose less than a, but greater than 6, be substituted for x in the latter equations ; then will ^— a be negative, and the rest, viz. g—h, g'^c, and g — d, positive, and consequently their product will be positive ; and g'^Oy (a negative quantity,) multiplied into this positive result, will- therefore give a negative product: if h, which is less than 6, but greater than c, be sub*, stitttted for Xj we have A— a and h^-^b both negative, and their product posi- tive} but A"-»c and A»- (fare both. negative, therefore their product isitosi- Part V. LIMITS OF THE ROOTS. IftT quantities which are limits to the roots of an equation^ (or between which the roots lie^) if substituted for the unknown quantity^ give results alternately positive and negative.** 43. *' Conversely, if two magnitudes, when substituted for the unknown quantity, give results one positive and the other negative, an odd number of roots must lie between these mag- nitudes : and if as maoy quantities be found as the equation has dimensions, which give results alternately poiitive and ne- gative, an odd number of roots will lie between each two suc- ceeding quantities 5 and it is plain that this odd numb^ can- not exceed unity, since there are no more limiting terms than the equation has dimensions.** 44. If when two magnitudes are severally substituted for the unknown quantity, both results have the same sign, either an even number of roots, or no root, lies between the assumed magnitudes. Cor. Hence, any magnitude is greater than the greatest root of the equation, which, being substituted for the unknown quan- tity, gives a positive result. 45. To find a limU greater than the greatest root of an equation. Rule. Diminish the roots of this equation by the quantity 6, (Art. 36.) and if such a value of e can be found, as shall make every term of the transformed equation positive, all its roots will be negative, (Art. 31. Cor.) consequently e will be greater than the greatest root of the eqtuition. ExAMi>LE8. — 1. To find a limit greater than the greatest root rfa*— 5ar+6=o. Let a?=:y-fe TZien iriM jt«=:y«+2 ye+€* — 5a:= — 5y— 5e +6 = +6 Whence (y*4-2ye— 5y + ^— 5e+6=a, or) ys+ge— 5^ +e.e— 5+6=0, is the transformed equation ^ now it appears by trudsj that 4 being substituted for e in this equation, it will be* five ; and theie two products mnUipiied, give likewise a poMtirc product. In like manner it may be shewn, by substituting^ k, which is less than c, and great- er UiaB 1/, the result will be negative ; and substituting m, less than the least root, the result will be positive. 138 ALGEBRA. Paet V. come y^+3y+3=d, of which all the rooU are negative; where- fore 4 Is greater than the greatest root of the equation a^— 5x4- .6=:o, ' - 2. To find a limit greater than the greatest root of x'— l^x^ -f 41x«-43sse;o. Let xssjr-f 6, a$ before. Then ioii/x»=sy*-f 3y*e+Sy€«+«» -|-41x= 4-41y +41e —43 = -43 JfTAerc/bre (y«+3y«e— I2y*+3ye* — 24ye+41 y+c»— I2c«+41c —43=0, or)y'+3.c— 12.y«+3e»— 24c+41.y+e.c' — 12e-f41 •^43=^0, is the transformed equation j where (by trials) it isfoundp that if S be substituted for e, the terms will be allposUive; viz. ^ + 12y'+41y+29=o; whence S is greater than the greatest root of the given equation, 3. Required a limit greater than the greatest root of x^— 6 x* — 25 X— 12=0. Ans. 9. 4. find a limit greater than the greatest root of x*— 5x'+ 6x*— 7x+8=o. 5. To find a limit greater than the greatest root of x^+3 x'— 5x«+8x— 20=0. 46. To find a limit less than the least root of an equation. KuLB. Change the signs of the even terms, (the second, fbuirth* sixth, &c.) and proceed as before ^ then will the limit greater than the greatest root of the transformed equation, with, its sign changed, be less than the least root of the given equa- tion. See Cor. to Art. 30. and Art. 45. ExAMPLBs.— 1. Let X*— 7x+8=o, be given to find a limit less than the least of its roots. This equation, by changing the sign of its second term, becomes x'-f7x-f8=o. Let x=y-f c. Then x*=y»+2ye-|-c« -|-7x= +7y+7« + 8 = ...........+8 ^yhence {y • 4-2 ye4-r y+c' 4-7 e4-8=o, or) y ' +2e4-7.y +«+ 7»e4-8=o, is the transformed equation; and i/"— 1 be substi- / P^ET V. LIMITS OF THE ROOTS. 189 bUed for e, aU Us terms will be posiiive^ for the equatum he- €Oi»e«y^4-5y«f ftsco; whetefwre'^l ualimiU less ikan the leasi root of the equation s' — 7 J'-f 8=5«. 9. To find a limit less than the least root of x«-f-x'«»lOjr4^ €=so. Changing the signs of the second and fourth terms, the e^aHon becomes x' ^3f* ^lQx^6:=iQ» Lei x=cy 4- e, then voill — «*=s — y* — 2ye — €» — lOopss — lOy — lOe -6= -6 ** « — . I ■ I III ■ ■■ ■ — ff^hetue y'-fSe— l.y»-f 3€'— »e— 10.y+«'— e— IQ.c— 6 =0^ is the transformed 'equation, in which 4 being substituted for e, U becomes y«-f 11 y'-^S0y'\'^:s;o$ wherefore —4 is less than the least root of the equatiofi x^-^x' ^ 10 x -^6=^0, 3. To find a limit less than the least root of x'-f- 12«— 90 =0. Ans. —14. 4. To find a limit less than the least root ci x'-^Ax'-^Sx-^ 6=0. 5. To find a Hmit less than the least root of a?* —5 a?' —3=0. 6. To find the limits of the roots ot jr'+«»— 10« + 9=o. Ans.-^Z and'^6. 7. Baqnirfdthttttfliitoiof a»«-^4«»4*8a?'-14«+^=o? a What are the limtta of the fools c^jp* ---2a;' — 5 x+ 7^0 ? 9. What are the limits of the roots of a?«+ 3 a;'— 5x4- 10«o? RESOLUTION OF EQUATIONS OF SEVERAL DIMENSIONS. 47* When the po^ibte roots of an equation are integers, either positive or negatiuoe, they may be discoffered as follows, RuLB I. Find all the dt^^sors of the last term, and suhsdtnte them soceessively fyr the imkaown ^^uaatitj^ In the proposed equation. II. When by the substitution of either of these divisoiB for the rooty the rewilf ing equation becomes = o, that divisor is a root ei &e giifM eq^aadoA', otherwise it is not. HI. U none <tf the (Nfison^ suooeed, the rools are either fractional, irrational, or impossible. VOL. I. K ISO ALGEBKA. Pabt V. IV. When the last term admits of a great aumber of diilaors,! It will be convenient to transform the given eqiMttion into ano- ther, (Art. 35, 36.) the last term of whid» will haye femx divisoTB. Examples. — 1. Let x'— ^a:'--5x+6=sa, be given, to find its integral roots by this method. First, the divisors of the last term 6, ore-f 1,-1, +2,—^, -f3,— 3, + 6, and—6-y now + l being substituted for x in the given equation, it becomes + 1 —2—5 + 6=o ; wherefore -^1 is a root. Next J let — l be substituted, and the equation becomes — 1— S -J- 5 + 6s=8 ', wherefore — I is not a root, Thirdly y let -^-^ be substituted, and the equation becomes $ — 8— lO-^er:— 4; wherefore + ^ is not a root. Fourthly, let —2 be iubstituted, and the ^nation becomes —8 — 8 + 10 -f 6=0; wherefore —2 w a root* Fifthly, let+S be substituted, and the equation wiU then be- come+27— 18— H^ + 6=^0 *, wherefore + 3 i« likewise a root. Thus, the three roots of the given equation are'\- 1,-2, and + 3 3 and it is plain there can be no more than three roots, since the equation arises no higher than the third degree f consequently there is no necessity to try the remaining divisors, 2. Givenx*— 6 0?' — 16 a? + 21=0, to find the roots. The divisors of the last term 21, are+ 1,— 1, +3,— 3. + 7> — 7> + 2 1^ and —21 ; these beif^ successively substituted for x, we shall have SubstitmioDs. i^^to + 1 — I + 3 Results. + 1— 6— 16+21=0 + 1— 6+16+21=32 +81— 54— 48+21=0 —3 + 7 -7 +^1 —21 4-81- .54+ 48+21=96 +2401— 294—112+21=2016 + 2401— 294+li2+2!=2240 + 194481-2646—336+21=191520 + 194481—2646+336+21 = 19219^1 M'fterefore + 1 and +3 are the only roots which pan befdmnd by this method; the ttoo remaining roots are therefore impo^silde^ *ein^— 2+^^—3. .. Pakt V. RtoOLUTlON OF EQUATIONS. ISl 3. Given x*— 4 j!9*-19 x«+ 106 J?— l«Oajo, to find the roots; S'mce the last term 1^0 has a great number of divisors, it wiU be proper to transform the equatim into another, whose abso* lute term will have fewer divisors ; in order to which, let xsz^-f 2> then (Art. 36.) j^=:j(*+8y»+^4y«+ 3Sy+ 16 — 4x3=5: _4yS_^y«_ 48y^ 3«' — 19x«= — 19y*— 76y— 76 + 106j;=: +I06y+212 ' — 1% =? ..—120 y*+4y'— 19y«+ 14y=o Here ^Ae last term vanishing, the number assumed, viz, +2, is mi€ of the roots of the oiigiwU equation, (Art, 33. note,) and the transformed equation being divisible by y, will thereby be reduced one dimension lower : thus, y^ + 4 y^— 19 y + 14=o ; the divisors of <Acto^/crml4,arc+l,— 1,+2,-*2,+7,— 7.+ 14,— 14j each of these being substituted for y in the last equation, +1,4-2, and ^7 are found to succeed, they are therefore the roots of the transi* formed equation ^^4-4^*— 19y-hl4=o; wherefore, since x=y4- 2, three of the roots of the original equation will be (l-^-^sz) S, (2+2=) 4, and (—7+2=)— 5, which with the number 2 <w- sumed above, gioe + 2>+3^ + 4> and — 5> for the four roots re- quired. 4. Given x'— 3ax^— 4a^x+12a'sso, to find the roots. The numeral Visors of the last term are + 1,-1, 4-2,-2, +3,— 3,+4,—4, 4-6,-6,4- 19> antfi —IS ; and of ^toe, 4-2,-2, ojid— '3 are found to succeed ji wherefore the roots are 4- 2 a,— 2 a, and — 3 a* 5. Required the roots of x^4-a?— 12=a? Ans, 3, and —4. 6. What are the roots ^f a:»4-4x«4-a?— 6=o? Ans. 1,-2, and —3. 7. What are the roots of a!5 4-2jr*— 19x-20=o? Ans.-^l, —4, aitd+5. 8. Required the roots of a?>— 14 «« +51x4-126=0? Ans. —2^+7, and+9. 9. Whataretherdotsofx*— 15x^+10x+24=o? Ans,--!, +2»+3, and —4. 10. Required the roots of x'+4x'— 7x— 10=o? K 2 la ALOBBIU. Pa»t V* 4g. SIR ISAAC NBWTONS M£THQI> OF DISCOVI^mG THE ROOTS 0£ EQUATIONS BT MEANS OF DlViBOKS; Rule I. For the unknown qiiMitity In the given equatkni^ substitute three or more terms of the arithmetical progresiioil 2> 1> 0^— 1>— 2^ &c. and let these t«nni lie placed in a column one under the other. il. Substitute each number in this column successively fo^ the unknown quantity In the proposed equation ; collect all the terms of the equation arising from each substitution into one sum^ and let this sum stand opposite the number substituted from whence it arises : these sums wiH form a second cc^iinm* III. Find ail the divisors of the 8ums> and place th^ai ill lines opposite their respective sums : these will form a third co* lumn. IV. From among the divisors collect one or more aritlimeti- cal progressions^ the terms of which difieir either by unity> or by some divisor of the coefficieifit of the highest power of the unknown quantity, observing to take one term only (of each progressioh) out df each line of the divisors : eaeh of these pro* gressions will form an additional column. V. Divide that term of the progression thus found> (or of each progression, if there be more than one,) which stanifii against O in the assumed pitogreadon, by the conmion dift^rvnoe of the terms of the fortner } and if the ]progres6ion1te increas- ing, prefix the sign -|- to the quotient ; but if it be decreasifig;^ prefix the sign — : this quotienf will be a i^oot of the equatnM». Hence there will be as many roots found by this ma^iod) ii there are progressions obtained fl*om the divisMi. EixAMVLES^ — 1. Givenx'— 24:— 24s«^ to &d tte tdiies of x. Operation. I, 2, 3, 4, 6, 8, 12, 24 l>5,2g. 1, 2, 3,. 4, 6, 8, 12, 24 1. 3, 7, 21 1, 2, 4, 8, 16 Whence, the roots are +6 and —4. JExpUufotion, The left haod column is the assumed progreition» the tevms of rabilltnted. successively for x in the given equation: firsts by subslitatiiii^ 2 Substitutions, Results, 2 —24 I —25 -24 — 1 —21 —2 -1(5 Prog'i deritmd.\ 4 a ' & & 6 4r ' 7 3 8 2 sli hich txw V. N£WTON*S METHOD OF DIVISORS. 133 iir j% tiKB aqikftti0a atamiBlbt to —04, wbkh h tli« 'nmtit io tliit ossc ; this I put in the seoood colnmn, and itg divisors 1 > S, 3, 4, H^ &c. in the third. Secondly, I substitute 1 for s, and the whole equation amounts to —35, viitdi is the second retuit^ and it« divisors are J ^ 5, and 25. ThirdJl)r» bjr svbstitut- iug fpr Xy the result, is —34, and its divisors 1 , 3, 3, 4, 6, &«. as in the first case. Fourthly, by sabstitnting — 1 for x, the result is «-'21 , and its dansors aue I, 3, ?• and SI. Ftflhly, by svbstiUAior '-^^ ^ 'V**'^ is "-I6» the d^ vu#rf «f which are 1, S, 4, 8, and 16. Sixthly, I try t9.obtM a progreswm, hj taking one number out of each line of the (divisors : and first I tvy for an in- creasing one ; the only one that can be found is 4, 5, 6, 7t and 8, Tis. 4 out af Che first line, 5 out of the second, 6* out of the third, 7 out of the fourth, and % iMit of the fifth $ these numbera eonstitute the fourth column. SeTenthly, I tiy fisr a decreasing progression, and (proceeding as belbre) find that 6, 5, 4, Sf mild 2, which constitute the fifth column, is the only one that can be obtmined. Eighthly, the number 6 and 4, standing opposite the in the assumed progresik. sion, divided by the common difference I , gives 6 and 4 for the roots of the equa- tion. The former being a tenn of the incrcasmg progression,' must have 4> pre- fixed to it ; the latter being a term of the decreasing progression, must have — prefixed ; wherefore the roots are + 6 and —4. 2. Givea of^— 64?' -7 jp+60*=#, to find the roots. Opbration. Dwisions, I, 2, 3, 5, 6, 10, 15, 30 1,2,3,^,6,8, 12, 16, &c. 1,2,3,4,5,6, lO, 15, &c 1# 2, 3^ 4j 5, 6, 10, 15, &c. 1, 2, 3, 6, 7> 14, 21, 42 Substitutions, ResuUs. 2 30 1 48 60 — I 60 -2 42 Prog:d(rived.\ 2 3 5 3 4 4 4 5 3 5 6 2 6 7 I Roots 4, 5, and Expktnaiion, Proceeding as before, I obtain three progressions, two increasing, and one decreasing, and the numbers 4, 5, and 3, standing opposite the 0, bein^ dirided Vj 1 the common difference, the quotients are the soots, nz. 4- 4 and -f 5 in ihe increasing progressions, and —3 in the decreasing one. 3. Given «*— x* — 10a:+6=<>, to find the root$. Substitutions. Results, 2 -10 1 — 4 ^ + e —1 + 14 -« + 14 Divisors, 1, 2, 5, 10. 1,2,4 1, 2, 3, 6 1,2,7,14 Progressions. 5 4 I > I 1, 2, 7, 14 Here vfe can derive only one progression, and ikat a one; wherefore t/ie only root discovered l^ this s^ethod ja i^jS : but by means of this root the given equation may be depressed to a quadratic, (Art, 33.) and the two remaining roots found by the Jmown rule far quuiraties; thus, Mee x-f^seo, d^idmg the pror ^H-ir' •A^lOxcf^ posed equation by this, we obtain {^ ZTZ ss);^* — 4«+ s^^ k3 134 ALGEBRA. Paet V, 5=:o, the two roots of which are (2+ v^.=t) 3.4142135624 onrf .6857864376. 4. Required the roots of 6 x*— 20 x» — 12 x* — 1 1 x— 20=o > Dioison. \PTOg, 1,2,7, 11, 14,22,77, 154 2 I, 3, 19, 57 3 1, 2, 4, 5, 10, 20 4 1,5 5 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, &c. 6 Here we obtain only one progression, consequently -^^ A is the only Toot found, 5. Given j?*+a?'-- 29 x»— 9x4-180=0, to find the roots. Subsiit. Results. 2 — 154 1 -57 -20 -1 + 5 -2 +210 Subst Results. 2 70 1 144 180 -1 160 -2 90 Divisors, Progressions, \i 2 3 4 5 6 5 4 3 2 1 7 6 5 4 3 1, 2, 5, 7, 10, 14, &c. 1 1, 2, 3, 4, 6, 8, &c. 2 1, 2, 3, 4, 5, 6, &c. 3 1, 2, 4, 5, 8, 10, &c. 1, 2, 3, 5, 6, 9, &c. Here are four progressions, two increasing and two deereasi$ig, and the roots are 3, 4,-3, and — 5. 6. Required the roots of x« — x — 12a:o ? Ans, +4 and —3, 7. Required the roots of x* +2 x» —23 x— 60=0 ? Ans, +5, —4, and -^3. 8. What are the roots of 2x5— 5x*-|-4x— 10=o ? An^ swer, one root + 24-. 9. Required the roots of x' — 3 x» — 46 x— 72 = o ? Ans, + 9, —2, and -B-4. 10. Tofindtheroots ofx*— 6x* + 10x— 8=0? RECURRING EQUATIONS. 49. A recurring equation is one having the sign and coeffi- cient of any term, rec]coning from the banning of the equa- tioil, the same with those of the term equally distant from the end 5 and its roots are of the form a, — , b, -r-, or the recipro- a b .cals of one another. » • • .. 50. If the recurring equation be of an odd number of dimen^ sions, + 1 or — J is a root y and the equation may be depx^esed to one of an even number of dimensions. (Art. 33.) Tamt ▼. RECURRING SQUATIONS. 135 Thus, let x^^^x'-^-lcno', +i » evidenibf one rool; ihatefwre, (Art 32.) — «' + ! TAtf equation x'— x^l=o, 6et»^ resolved hy the rule for f iia<2rafuSf, it$ roots wiU be found to be -^ "^ . Cor. Hence, a cubic equation of the form Ji^±px'±px'^i may always be reduced to a quadratic* and its roots found. 51. If the given equation be of even dimensions above a quadratic, its roots may be found by means of an equation of half the number of dimensions. Thus, by supposing the equation to be the product of thefae* i i tors X— flj? , X— 6jr— T-, 4c. by actual multiplication, and a putting m=:a-l , n=r&-| — r-, 4c. we obtain x* — mx-h 1, x* — nx a + 1, 4c. wherefore by multiplying these quadratic factors toge^ ther, and eqiictting the coefficients of each term of the product, with that of the corresponding term of the given equation, the t)a- lues of m and n will be readily found : and since for every single value ofm there will be two values of x, it follows that the equc^ tion for finding m will be of but half the number of dimensions ne- cessary for finding the value of xby other methods. Examples.— 1. Let x*— 3x'+3x»— 3x +l=obe the pro- posed equation. Assume the product (x*— nix+ Ijf*— nx-f 1=) x* ^m-j-n,3i^-{' pm^^^jc'-^m'^n.X'^'l^the proposed equation: then making the coefficients of like powers of x in this product and the given equa^ tion equal, we shall have m-|-n=s3, and mit4-2=2^ or nussto-, wherefore, if n^:zo, then m:=z3, and the two equations x'— -mx •fl=o^ and X*— nx-flssoj become respectively x'— 3x+I 3+ a/5 =0, and x' + l^oj from the former of these x=g( -^^ -as) k4 IS6 ALdBBAA. FaktT. ^SieasaiSSSt, imd ^rt^MOllS -, whieh tw6 taluei of « mre the reciprocals of each other. From the latter, ifiz, a^ + la^o, we obtain *= + v^ — 1, or + ^ — 1, bud — ^-*1, /oi" the two re- maining values of x, 2. Let a:'— 1=0 be given^ to find the values v£x. Here it is plain that -f 1 1# a root, or x-^-lsto, wherefore di* viding the given equation by this, we have (- si)3fl-{'X+lsso, the two roofs of which are ■ ■ ■ ~" . cM ^ "" ' « . 1-4- ^•^ft 3. Given a:* -f- 1 =:o, to find the values of*. Arts. —1, ^ , 4. Let the equations or^—lsso, «•+ lwo>»*-— lr*o, andV-h Iso, be proposed^ to find the values of « in eaieli. Literal equations^ wherein the given quantitj^ and the lan- known one are alike afiected^ may be reduced to others of fewer dimensions^ by the following rules. 52. H^hen the given equation is bfevin ^mensUms, Rule L Divide the equation by the equal powers of its two quantlti^ in the middle tenh. II. Assinne a new equation, by putting some letter equal to the sum of the quotients arising fh)m the division of the given and unknown quantity, alternately, by each other. III. Substitute in the former equation the values of its terms ^ound by the latter, and an equation will arise of half the di- mensions of the given one^ from the solution of which the roots of the given equation may be detennined. Examples. — 1. Required the roots of ar*— 4al?^-5a«x•— irf'J;4■a*=o? Fir^j dividing the whole equation by the equal powers in the *P* 4j? 4fl fl' UtiddU term, it becomes ( |-.^-f 5— — ^--ssio: or^ which is the -a» a XX' 9mte,) -r^ 4. — f- 5s=o. Let — | b=«> thenhysqutfr- a X* a X ax x^ a' ing, -^.f ~«f ftiaap', and by sulistitutmg z' and^for th^r va&tes a?' a* in the equation —--{ —4. — | h5=o, it becomes «•— 4f+3 a' X' ax KwtT. HECUBttlNQ EOUATIONS. 137 =0, whence z=3, or 1 y but since — + — =x, if the former valae d X be taken, then — | — Ss3 ; whence «*-^3 axss — 1^» fMch eokei, a X a gives xss{-^3±^bB^) 3.618034a, or S61966 eu But if the X a \ _D latter value ofz, namely 1, be taken, then ( — | — =1, or) ar— •^ ax ax=:^a\ whence j=s — "^ are 1^ ^too remaining roots, . «. Gii«B 7*?*^— ^««*— 8fi«'*JP+7«^=o, to fiod the faliM of X. This divided by aV becomes 7—+ 26. h- t==<>' -^* ^ a* a?» a« x' *s a x^ k* X a z'sz 1 > then 2*— 2= — h^ — , which multiplied by z^--^^ — , ax a^ X' ^ "^ a X <p3 a «p £|» 3p3 |i|3 a* X a x^ a^ x^ X* a* 3z=s— +— . a^ X* }%e»e wttecj ^fe^ifwfed a* before, we obitnn 72'— 262*— 21 z +52=0, one root of whWh {by Art. 47) e< 4, and by means of this, the equation may bedepressed to the quadratic If-^^z-^ IS =0, {Art. 32.) the two roots of which are +1.2273804, and — 1.5130947. Wherefore, since «=— H — , or jc*— aarss— «% by a X 4^ M2t£tio» of this we obtain xs=: —"^ , i« which, if the three values ofzbe successively substituted, the six roots of the given equation will be obtained, S. To find the roots of «*+6aa?*-20a*«*+6a*a?-fa*=sp. 4. To find the roots of a;*-204ia:» + 1««^x*-20«»«+«*sb:o. 5. Hcqtnrcdtheroot8 6fa^-aa?*-fl*x+rf^=^ 63. When the given equation is of odd dinunsixms. Rule. Divide the equation by the sum of the known ^nd un- koown quantities, and proceed as before. aLAMPi.Bs.— 1. Given «?*-3 ax^+e.a's^-^e f^x' -3 a^x+a*, to find the roots. 1S» AliGfiBBA. Paut V. First, dividing by X'\'a,ihe quotient is x^ — 4x^a+10£'a' <* 4jra'+(r*=o; wherefore dividing this by x*a*, according to the fretting ruUt the quotient is — | 4* — I 4- 10=o ; let z a^ X* a X X a . X' a* - «= — I , then z*= — I \'2, and substituting these va- ax a* a?* lues as before^ 2*— 4z-|-6=o; whence 2=5:2+ v'— 5; but si since ar a aZ'\'a>/z' — ia z = — , we navex' '-azx^-^a* ; whence x= — = — ^ ■» ax 2 and substituting for z its values found above^ we obtain four of the roots, which together with —a, (since x+asso,) make up the five roots of the equation, 2. Given*'— or*— a'-'jj-Ha'sso, to find the roots. Ans, a, a, and —a. 3. Required the roots of Jc*H-4 OB*-- 12 a'r* — 12 «•*» + 4 a*« -^a^sio} 4. To find the roots of x^— or*— o^x+a'sso. CARDAN'S RULE FOR CUBIC EQUA- TIONS. 54. Let X* -}-cLP=s6 be any cubic equation* wanting its seeond term > it is requii*ed to find one of jfs roots, according to Car- dan's method ^ P This rule bean Cardan's name from the circumstance of bis baring been -the first who published it, namely at Milan in 1545» in aivork entitled. An Magna : but it was invented first, in or about the year 1505, by Scipio Ferreas, Professor of Mathematics at Bononia; and afterwards, v'is. in 1535, by Nicholas Tartalea, a respectable mathematician of Brescia; from the latter Cardan con- trived to extract the secret, which he afterwards published in violation of the most solemn protestations. The rules which Cardan thus obtained were for the three cases j^ + hx»€, X^^hx-^e, and afi -f c»»hx ; and it must be acknowledged m justice to him, that he greatly improved them, extending them to all forms and variatiM of cubic equations, in a manner highly creditable to his abilitiei as a mathematician. See 'nurtalea's QumHti H JmfeiUiam diverse, ch. 9« Boesut's Hist, of the Math. p. 907. Montucla's Hut, desMath, t, 1. p. 591. Pr. Hut- ton's Math, JDiet. vol. 1. p. 68—77. The root obtained by this method is always real, although not always the greatest root of the equation : and it is remarkable, that this rule always exbi* bits the root under an imaginary form, when all the ro«>ts of the equation are real ; and under a real form, when two of the roots are imaginary. See Dr. Button's Paper on Cubic Equations, in the Philotoph, Trans, for 17^. Part V. CUBIC8, CARDAN'S RULE. 139 Assume y-^-zszx, and 3 yz= —a; suUthute these values for X and a in th e proposed equation, it becomes (y^+^y'x+3pz* Hhf^+a.y+«=y» +2* +3 yz.y+z-|-a.y+z=y» +2»— a.^47+a .jf-f-z=r) ys-fz'=6; from the square of this tidce four times a 4 a' the cube of yz= — ~, and the result is y«— 2 y V + 1"=5» +— - , 4 d* the square root of which is y'— z»s=:^6«-j j buty'+z*=:6; *7 wherefore the sum and difference of these two equations being taken^ the former is 2tf*=i+ A/^^+-7ziry and the lattw^z'se x7 4 a' 111 *— V^+i^^^^ Whence is found y=» V— 6+ ^/-r^*+sr«'i and ^7 » 4 3/ «=^ Vy^— ^-4-^'+^«'> whence j?=(y+z=:) Vy^^ VT^'+^^' + ' 'v/-^^- -^^T^' +^«'» ^^^^ ^ ^*'- dan*s theorem : but the rule may be exhibited in a form rather more convenient for practice ; thus, because z=r — — -, we have x 3y 4« W'rrb-\' J — 6* +r-:rt' ; whcucc the rule is as follows. ^2 ^4 27 55. Rule I. If the given equation have all its terms, let the second term be taken away by Art. 37. II. Instead of a and h in either of the above general theorems^ substitute the coefficients of the corresponding terms^ with their proper signs» in the transformed equation; then, proceeding according to the theorem, the root will be obtained. If a be negative, and —a' greater than —h*» the root 37 4 cannot be found by this rule^. 4 This is called the Irreducible Case ; it exhibits the root, although real, under an impossible form : thus the root of the equation xs— 1 5x^:4 ii 4, but by Cardan's rule it is' >v/2+ ^ — 121 + » -/2— V* 121, an impossible form. 140 ALGEBRA. Paut V. ExAVFLSS.— 1. CSfwen x'-^-S rssd8> lo fifid the volue of x. Here the second term is wanting, wherefore a=6» &^88> and ^a •^^•"m^m aa^HiVawr^i^a .^W~b+V-^'+^'-W^+V^'+^'=- 88 88)' Si' 88 Sii' 6)' Let the cube root of each of these imaginary expressions be extracted, thej be- come 8-f v^— 1 + 2— V^— 1 9 which being added together, the impostiUepftrls destroy eadi other, and the Mm is 4, agreeably to what has been obsenred. It is remarkable, that this case never occurs except when the equation has three real roots, as we bsre before obsenad. The irreducible case has exercised the abilities of the greatest algebraists for these three hundred years past, but its solution still remains among the de< ^iderata in science. Dr. Wallis thought he had discovered a general rule, but it was afterwards found to apply only to particular cases. Baron Maseres gare a series, which he deduced by a laborious train of algebraic reasoning from Newton's BinomialTheorem» whereby this case is resolved without theintervea- tion of either negative or impossible quantities. Dr. Button has likewise disco- Tered several series applicable to the solution : (see Philoi. Traru, vol. 68. and 70.) other series for tlHs purpose may be seen in Ctmrmilfs Afy^bra, p. S. Art. 19. Soma's jflgebra. Art. 178-9. Landen's lAicuiratitms, Zm CaUU*t Le^ontde Math, Art. 399. &c. , Lorgna's Memoirs qfthe HaKan ^ctsdewy, t. i. p. 707. &c. The irreducible case may be easily solved by irigonometry ; as «arly as 1579) BombeUi shewed that angles are trisected by the resolution of a cubic equation. Vieta, in 161 5, shewed how to resolve cubics and higher equations by angular sections. In 1639, Albert Girard solved the irreducible case by a table of sines, giving a geometrical con&tniction of the problem, and exhibiting the roots by means of the hyperbola and circle, Halley, De ttotvre, Emerson, Siikipson, CrakeK, Cagnoli, Wales, Madielyne, Tbacker, :Sic. hawte employed the eaue method of sines : and lastly, Mr. Bonaycastif , Professor of the Makhanatkies at the Royal Ifilitary Academy, «Woolwicii, has communicated additional observations on the irreducible case, and an improved solution by a taUe of natural sines. See HuttorCs Math. DicU vol. 2. p. 743^. When one root is obtained by Cardan's rule, the two other roots may be de- rived not only by depressing the equation, as in ex. 1 . but likewise as follows : let r=> Cardan's root, and v and tr^stbe two other roots, then will v 4- w= -^r, N r 1 r3--46 r _ \ r"*— 4ft ami, vwrssiy whesoe »« — -^r H- -t-V ■" ' ' • , and w* — -^ + o v ' ' ' """ . PAn y. CUB1C& CASDANS RULE. itt 3 *V'^^M^^i4^^l6=s4<449--.449s4s^^ root required. If the two remaining' roots be required^ deprei^ the given e^iia^ fion, (-rfr^ 33.) thus {--^ 7—=) ^•+4ar4.22=o, of which |A« roote (/ottiul by the rule fitr quadratia. Vol. I. P. 3. Art, 97.) are —2+3^—2. 3. Given y' — 6 j^*'^- 3 y ** 4=0'> to find- the value of y. First, to take away the second term, {Art. 37') let y^{x^ Then. y5=:^+da?*+12«+ 8 —6 y*= . —6 a?«— 24 a? —24 -|-3y =....+ 3jc-h 6 -4 = - 4 Whence x^ * — 9 a:— 14= o, or j?*— 9 a:=s 14. -3 Here a=:^-*9,i»14, aiMix=V74- V49--27 -3 -^ :Vu.«90415--~^;^=2,269- 3^+4.690415 ^ V^l^^^l^ —3 =2.269+ 1.322»3.591^ the root ear vahie of x; mber^re 2.269 y^(x+2=) 5.591 = ^^ root of the proposed equation, 3. Let y'+3y«+95f=13begiven, tofindy. Here, putting y^x^ I, the equation is transformed (Art. SJ.Y into a;'H-6j:=20j whence asset, 6=20, and xss'^lO+^/WS ^T ;;=' V20.3923— r-:Tj--^=:2.732-.732=2 1 wherefore y=(jr— 1=)2— 1=1,*^ 'root required. 4. Given x*— 12a;=16^ to find x. Ans. a:=4. 5. Given j:^— 6j?=— 9, to find x. Ans. x=— 3. 6* Giveq y5+30y=117. to find y. Ans. y=3. 7. Given ^54.^^—350, to find y. Ans. yrs&OS. 8. Given y^^ 15 y«+81 9=s243, to find y. Ans. y=9. 9. Given y»-.6y«+10y— 8=0, to find y, AnK^y^^. 10. Given y« + 20 y ^ 100, to find y . " 14C ALGEBRA. Pau V. COliJPLETING THE CUBE. " 55. B. In eveiy complete cubic equation, haying its signs cither all -f > or alternately + and -— , if the coefl|cient of the third term be equal to three times the square of one third of the coefficient of the second term> the cube may be completed by adding the cube of one third the coefficient of the second term, with its proper sign, to both sides of the equation j and then, by extracting the cube root from both sides, the root of the equation will be'found '. ExAMPLBs.— 1. Given j^ + 6j^ + 12xs=56, to find the value of J7. Here i of 6sz2, and 12=3x2*; wherefore adding 2?^ io both sides, the given equation becomes jr'4-6i:*+12x+ 8= (56 + S=) 64. The cube root of this is :r4-2=4; wherefore J?s2. 2. Given a^— 12a:«+48jr=:61, to findx. Here i o/— 12=— 4, and 3.— 4l«=483 wherefore ^V^^ —64 is to he added, and the equation becomes x' — 12x*-f 48l:— 64=(189— 64=) 125. The cube root of which is j— 4=5; whence ai;=9. 3. Given 6x» — 90 jr» +450 ar= 729.75, to find x. First, dividing by 6, we have x* — 15x*-f76x=121..625. Also 4. 0^—15=^5, 3.-5|«=+75, ond-.5l«=— 125, to he added; wherefore x» — 15x«-|-75x— 125=(121.625— 125=)- 3.375; andx— 5=(v'— 3.375=) — l.5,t(7^cre/'or«x=(5— 1.5=) 3.5. 4. Given r» -f.3x'+3x=26, to find x. Ans. x=3, 5. Given x» — 18 x" -f 108 x= 189, to find x. Ans, x= —3. 6. Given x» +21 x* + 147^=400, to find x. 7. Given x^ — 2 1 x« + 147 x= —64, to find x. 2x 1 • 8. Given 2 x*—x» +--=—-, to find x. 27 2 » This rale is evident ; for let (r +«!*=*) x* +3aa:» + 3««x^tfa be a complete cube, it is plain that + a is 4. the coefficient of the second term, 3 .+aS»the coeiScient of the third tenn, and the cube of+a, w^a^ the third term ; wherefore if jr4 +3 ojt* + Za*x^h be given, it is plain that the cube is completed by adding the cnbe of one third the coefficient of the second term to both sides, ma king x * +*'* + rt^»x+<i* =6+rt«;then extracting the cube root xHtfl=^ ^fc-f-a*, and x« +a + 3 ^ft^i|3, which is the rule. The root of aj^mplete cnbe is found by taking the root of the first term and the root of the latt^ and ooaiMsctinf them by the s\gn of the last. PaitV. BIQUADRATIOS. DSS CARTES' ftULE. 145 56. DES CARTES' RULE FOR BIQUA- DRATIC EQUATIONS '. RuLV 1. Take away the second term from the given equa* tion, (Art. 37.) and it will be reduced to this form, x*-\-ax^ rbx -^-c^szo; wherein the coefficients a, b, and c, may represent any quantities whatever, either positive or negative. II. Assume the prodact x'-J-fxr+9.a:'+rx+« equal to the transformed equation j?*4-flu?*+6r+c=o, and let the two fac- tors be actually multiplied together ; then will the product • Lewis Ferrari, the friend and papil of the celebrated Cardan, was the firat who discovered a mle for the solutioD of biquadratics ; nsmelyi aboat tib« year 1540. His rule, which is called the liaHan method^ was first published bj Cardan with a demonstration, and likewise its application to a great va- riety of suitable examples : it proceeds on a very general principle, completing •oe side of the equation up to a square by the help of multiples, or parts of its own terms, and an assumed unknown quantity ; the other side is then made a square, by assuming the product of its first and third terms, equal to the square of half the second : then by means of a cubic equation, and other cir- cumstances, tlie management ot which greatly depends on the skill and judg- ment of the operator, the root is found. The mle we have given above was invented by that eminent French philo- sopher and mathematician, Ren^ Des Cartes, whose name it bears ; and was first published in his Geometry, lib. 3^in 1631 , but without any investigation : like Ferrari's method. It requires the intervention of a cubic and two qnadraticr; both methods are sufficiently Uborions, but that of Des Cartes has in some respects the preference. The reason ol the rule is extremely obvious ; for it is plain that any biqua- dratic may be eonsidered as the product of two quadratics ; and if the coeflB- cients of tte terms of these latter can be found in terms of «, fr, c, &c. the coefBcienti of the transformed biquadratic, (as we have shewn they can by maaiM «# a cubic, &c.) then those quadratics being solved, their roots wi|l evidently be those of the transfoxmed biquadratic, from whence the roots of the givett equation will be known. All the roots of a complete biquadratic equation will be real and unequal. ' l^t, when 4 of the square of the coefficient of the second term is greater than the product of the coefficients of the first and third terms. Secondly, wlien ^ the square of the coefficient of the fourth term is greater than the product of the coefficients df the third and fifth terms. Thirdly, when 4 the si|uar« of the coefficient of the third term is greater than the product of the coelBcients of the second and fourth tertns r in all other cases besides these three, the complete biquadratic equation will have imaginary roots^. 144 ALQEBRA. PaktT. **:?}.' +^|.-+j}x+^= X* * +aar» + &r + c. III. Make the coefficients of the 9«ne power of x on each ode this equation equal to each other, in order to find the vafaies of the aflsamed coeffidents p, q, r, and $; then will p4-rs=o, f ^.g^-jM-ssOy jm4-^=^ and qs^^c; from the first of these we get rs — p, from the second s+q=(a — ^pr =since r= b — p) a+p', and from the third «— 9s;— -. IV. From the square of the last hut one, subtract the square b* of the bflty and 4f«33a' +2 «|>' +p^— — , or (since ^azzc) 4tf 6» ^a* +S ap' +p* ^, which equation reduced, is p* +2 ap* + a' '-4 c.p* =z&', from the solution of which (by Cardan*s rule er otherwise) the vahie of p will be found. V. Having diseovered p, the value of *='X"+^+5~» *^ that of ^=5-^+^ — ^-, will likewise be thence determined; that % 3 2p &, (since r=:^p,) sdl the quantities in tiie two assumed Catctoi? j?» +pa?+9ur' +rx+*, excq>t the value of x, are known. VI. Next, liiid the roots of the two assumed quadratics x* + pX'\-q=o, and x» 4-rx+#=o, and we shall have, from the for- 9 P' T mer, 0?=—- ~+ ^"2 — 9* *"^ ^o™ ^^® latter, x= ( jh ^.^^Mt or since rs;:— ps5j)^4- V^*^*- Wheroftwe the fiwf 4 3 4 roots of the transformed biquadratic equation x*+ax' + bX'^c p p* P P* P p' and — ^ — v'^"" 9 ** *^® roots of the proposed equation. » 4 ' SKAimns.~l. To find the fonr roots of the biquadrstie FaktV. biquadratics. SBS CARTES' RULE. 146 rmi^ io iake uHBOf the setmi term, {AH. 37») Mt z^x^ z*x3a:*H-4x*-f 6«»-f 4«:+l ^4z*±s -'4««'^I2x'^13jr— 4 •— *8z » . — 8 J: — 8 + 3S = +32 «*— 6^' — 16j: 4-21 =o Here, putting a=—6, b^^l6, and c=+21j the assumed cubic (p°+2ap*+a*— 4c.p»=&') becomes by substitution p* — 1^|»* — 48 p' =256. ^om tAtf^ /e^ the second term be taken away, by putting p'=:y-f-4 3 then will p«=y'-|-12y«+48y+e4 — 12p*= — 12y'— 96y— 192 — 48|>»=s — 48y— 192 —256 s= . . —256 ""* y*— 96y=576 To find the root of tMs equation by 'Cardah*s rule, {Jfrt, h^ 55^ here «=— 96, 6=576, and ^.Z— 6+^—6*+— a' — 2 4 27 1 ■ i.iainpi. ".!■ Ill 'iPl lilt I . 1.1 ' '^"i**^ 'v^T^' +2r**^' Ar««+ v^2944-S2768 27 -32 ^^+V^9ii=32^=^'^=^' "^^>^ P=(^y+4:±:) ^ t ,«.P*.^ -^-6 16 —16 . /«.P' ^2^2^2p 2^2^ 8 ^'^^2^2 Wherefore the two quadratics to be solved s viz. x* -^px-^-q zso, and x' +rX'^.87so, (&y sub^ituting the abope values of p, q, r, and s,) become a?» +4 j:=: —7* ond x* — 4 x= —3 -, the two roots qftheformerofthesearex:=i'-'^^^'^3', andoftheiatter,x^S; •«-a>.*Mt<M*ai<MM«iifc>«««-«rfB.*«.^M«>MaM«a« * We have the solution of both these quadratics (or rather th^ ttttstrers) iajgMMul teniB» in the »ol« f tie. -^± v''4^«> "^ ^T4 '^^ — *' ^* which the valaes of p, q, and i, being subttitotedy the roots of the transformed eqaation will come out as before* vol.. JU L 146 ALGEBRA. Fast V. and 1. Wherefore the four roots of iht tT€ai»fofrmed eqfauOiwn X*— 6a:*— 16j:+21=o,arc —2+ v^— 3 3— ^—3 . . .Sand 1 5 hut iwce z=x+ 1^ by aiding unity, to each of these roots, we shall have the four roots of the gwen equation z«— 4 z* — 8 z-^SZ =0, as follows; 2= — 1-f- V^^t 2=— 1— V^* «=4, and zss 2, (M tea* required ". 2. Given z*— 42'— 3z*— 4z+l=o, to find the values of z- i^n«, z=-=~ — ana =^^2: . 2 2 3. To find the roots of x*— 3 a?*— 4«— 3=o. Jns. «= 2 « 57. EULER'S RULE FOR BIQUADRATIC EQUATIONS '. Rule I. Let x*-^ax' -^bx-^-cszo, be a general biquadratic ^ a a' c equation wanting its second term, and let J^-^j ^==75+ T* and h^-. II. With these values of/, g, and A, let the cubic equation z* — /z' -f-gz— A:=o be formed, and let its three roots (found hj any of the preceding methods) be p, q, and r. III. Then will the four roots of the proposed biquadratic be as fblloWSj viz. When -^^ is positive l-st root, ar=s ^p-k- ^q-k- ^r 2nd root, x= ^p-f ^9— ^r Srd root, a:=VP-" a/9'+ V 4th root, x» v'p*-^ V9~ v''" Wlien —h is nqgative^ a:= ^/V—A/q-V^r x^^ ^/P'^r ^q-¥ V^ x^^^p^^q^^r ii I > " This rale applies to that casa only in which two- of the roots-are potsible, and two impossible. ▼ The learned and renerable Leonard Euler, joint Professor of Mathematics alt the University of Petersbarg, was the inventor' of this method; which he first published in the 6th volume of the Petersburg Commentaries for the year 1738 ; and afterwards in bis Algebra, translated fifom the German iot» £reneh, in= 1774»<and lately into English^ Paet V. BIQUADRATICS. SIMPSONS RULE. 147 Examples.— 1. Given x«— 95;r'+60x— 36ss:o^ to find the four roots. a 25 Here a=i^, 6= — 60, and c=36 ; wherefore f:=z(—s:z) —, a' c 769 225 g= (—+--=) ----, and A= — 3 consequently by substituting 16 4 16 4 <Aese values in the cubic equation z'-^fz* -j-gz — hzso, it becomes 25 , . 769 225 a* z* -\ z szo, 2 16 4 The three roots of this equation being fotmd, foiU be z^ 9 25 ] ■-r=p* Jf=4=9, a»dz=---=r; and since -- b is negative, the four roots will be 9 25 9 ^ 9 25 9 25 «= ^^^g-.^r=-^— -^4-v'-4-=— 6 2. Given a?*— 6a?"4-4=o, to find the roots. Ans, x= + l, +2,-1, and —2. 3. Given a?«— 3 x'— 36 a?' +68x^*240=0, to find the roots, jfni. xss— 2,— 5,+4, and-^e. 4. Findtheroot8of««+x««-29x'—9x-f 180=0. Ans.x^ 3,4^—3,011(2—5. 6, Findtherootsofy«—4sr'— 19^^+46^+120=0. 58. SIMPSON'S RULE FOR BIQUADRA- TIC EQUATIONS \ This method supposes the given biquadratic to be equal to the difference of two assumed squares > thus, - ^^— ^— — ^— -— ^^^— — ^— — ^— ^— ^— — — ^— ^— — — — I X This rule was first giveii by Mr. Thomas Simpson, Professor of the MaftheiDatics at the Royal Military Academy, Woolwich ; and published in the second edition of his Algebra, about the year 1747 : it is in some instances pre- ferable to either of the preceding methods, and some trouble is saved by it, as here we are net under tha necessity of exterminating the second term from the complete biquadratic equatiooi which in the preceding rules is indispensable. L2 148 AWVnUL Faet V. BuLE I. Let X* +031' -hfar* '♦riar -f 4ag » ^ >e tt e propose d equation, and equal to the difEerence x* +— <v+<^ ' — Bx-i- C| '. II. Square the two latter quantities, making the the squares equal to the imposed equation, and jou will have j:*+aa*+2-4r' ^ — B*jr»— 2BCr— C» J in. Blake the confident cix in each terpi on one side of the equation, equal to the coeflkient of the same power of x on the other j then will 1 1 First, 2.*+— o«— J5«=fc, or 2^+— a«-6=:B«. 4 4 Secondly, aJ— 2 BC=c, or aA—c:si9 BC. Thirdly, A^»^0=zd, or i<*— d=C». IV. Multiply the first and last of these equations together, and the pioduct (B^O) will evidently h« «q^ to (—AB^O) 4 one fourth the square pf the «Moad \ that f9» 2iC>h— 4^— 6^ 4 1 1 ' ' — '-'— ^ «^ 4 4 •••ti V. Let ifc=s— oc— d, i=— c/-f"A-T-4*^^i and hy this sub- stitution, the preceding equation will become A^^^'^kA* '^kA '1=0. 2 VI. Find the root or value of ^ in Hm^ cubic equation, by any of the foregoing methods ; which being done, B and C will ■ I L - ■ ■ I . - > 2B likewbe be known, since ^=s v84+'T'<»'-^A» I»4 C=5* VII. And sinoe the proposed qunitityjp«-Hur* + to' -fcar+d is equal to nothing, its equ?a a;*+4'ax+-rfl*-ftc4-Cl' irtH 2 likewi se be e qual to nothing; wherefore it follows^ that 1 x'-^'—ax-^A «=&*♦- cj*. Tkn V. BIQUADRATICS. SIMPSONS RULE. 14» VIIL Ettfact the square root fi^m both sides of this equation, and or' +~-ax+^=r + Br +C, whence a?»H a+Bjp=4-C— i#; tvhich equation solved, gives xss'\ — B a-{- ""2 4 — VT^a'-f — aB-f--B«-|-C— -rf; wherein all the four roots of *o 4 4 — the given equation are exhibited* according to the variations of the signs ^. ExAMFLM.-^!. Giv^ ar<—Gi*—-58««—H4«— 11=50, to find the values of x. Hire asu-^e, &98-<-Sd, e3«-114, and d=s-ll, whence k 1 11 ==(^ ac-d=) 182, Z=(--c»+d.— a»— 6=) 2512 j whence by iubstituting these values in the cubic equation A' bA' + kA-^ -^1=0, it 6flcome« -rf* +29-4* + 182^^—1256=0, the root of y Dr. HttttoD remarkti that Mr. Simpson has subjoined aa observation to this rule, which has since been proved to be erroneous ; namely, that ** the ▼alue of A, in this equation, will be commensurate and rationai, (and' there- fore the easier to be disoovered,) not only when all the roots of the fiv«9 ofoattoo are eem me tm n riiief but when they are trrsrfMNO^and even impossible ; aa wiU appear from the ^camples wah^lptdm" This, oootinues the I>octor, is a strange rcaeon for Simpson to give in proof of a proposition : and it is wooderftd that he |sU on no examples that di^rove it, as the instances in which hia assertion holds true, are veiy few indeed in comparison with thosa ia which it feib. MttK JDki.yoh h f, m. When dthcr jS^e, jBa>a, or €>■•, the roots of the proposed biq^adratio win be obtaiUBd by the resohitiao of a quadratic only. Simfton'$ Alg. «l* tekt. p. 16$. Besides the rules by Ferrari, Des Cartes, Eulcr, and Simpson, two other rules ibr the solution of biquadratics have been discovered x one by La Foati^ne, of. the Royal Academy of Sciences at Paris, and inseipted in the Ifcamin of that leanud society for 1747 ; ud. the other by Dr. Edward Waring,. LuCMlftn Professor of Blathematies at Cambridge, iq a profowi4 waak, cntilled^AIMiiMiMiM jagehmemf published in the year 1770. AXt tempt* 'havu not- been wanting to diseover methods of resolving equations of the h%her orders^ but they have hitherto been unsuooessful; no general rule Urn the solution of adfeeted eqnolions above tho fsorth ponret^ has y«^ heiy discovered. 1.3 150 ALGEBRA. Past V. which (found hy Cubics) is A^A; whence B^(,^/2A'\"2<^*'^h =) 5^3, C=(?^=) 3^3, and t :=z ±-LB^l.a± 1 Zn i 5 3 21 = 11.761947, or 3.101693, or +2.830127+ ^—1.1865334798, for the four roots; the two latter, expressed by the doMe sign, are impossible, 2. Let the roots of j;«— 6x'+5f'+2x— 10=0, be fband. Jns. x=5, — 1, 1 + i/ — 1, and 1— ^—l. 3. Givenj;'*— 12 JT— 17=0, to find tbe values of X. Ans.T=i 2.0567, or .6425, or .7071+^—4 7426406. 4. Given x«— 25x' +60x= -36, to find the roots. Ansmer x=3, 2, 1, and —6. 5. Given x*— x»+2x«— 3x+20=o, to find the roots. RESOLUTION OF EQUATIONS BY APPROXIMATION •. 59. The foregoing rules require for the most part great labour and circumspection, and after all^ they are applicable » ■» ■■ ■ — - I. ■ I ■ , . ■ ■ Methods of apprmamatiiu^ to the roots of nunhen, were enplojed ss early as the time of Lacas de Bnrgo, who flourished in the ISth eenUiry; bat the first who are known to hare applied the doctrine to the resolotion of eqas* ttons, were Sterinns of Bruges, and Vieta, a cclehrated mathematician of Lower FiAtoa ; the former in bis Arithmetic, printed at Leyden, in 1585, and in his Algebra, pablished a little later ; and the latter in his Opera Math^ tnatiea, written about the year 1000, and pablished by Van Schooien, in 1646. . Their methods, although in some respectaimprored by Ooghtred in his Key t» the Mathematies, 1648, were still very tedious and imperfect: to remedy these defects. Sir Isaac Newton turned his attention to the subject, and it is to his successful application to this branch, that we are principally iadebted for a general, easy, and escpeditious method of approximating to the roots of all sorts of adfected equations, as may be sten in his' tract De Anafyn per EquaHenui ^umere terminerum infSMitat, 1711, and elsewhere. Dr. Halley inrented two roles for the same purpose, one called his rmHemal ikeerem, and the other, his irrational theorem, both of whkh are still justly esteemed for their utility. This necessary part of Algebra is likewise indebted to the labours of WaUis, Raphson, De Lagni, Thomas Simpson,. and others ; whose methods have been given by various writers on the subject. Fait V. APFROXniiOION. 151 onfy to eqoations of particular Idods^ all of which taken toge- ther^ form but a small part of the numerous kinds and endless variety of algebraic problems^ which may be proposed. But as we have no general rules whereby the roots of high equations can be founds we must be content to approximate as near to the required root as possible^ when it cannot be found exactly. 60. The methods of approximation are general, including equations of every kind and description, applying equally to the foregoing equations, and to all others which do not come under the preceding rules : hence approximation is the most general, easy, and useful method of discovering the possible roots of numeral equations, that can be proposed. 61. It must be observed, that one root only is found b^ these methods, and that not exactly, but nearly. We begin by making trials of several numbers, which we judge the most likely to answer the conditions of the proposed equation; then, (by a process to be described hereafter,) we find a number nearer than that obtained by trial ; we repeat the process, and thereby ob* tain a number nearer than the last 5 again we repeat the pro* cess, and obtain a number still nearer, and so on, to any assign^ able degree of exactness* 62. The simplest method of approximation, KuLB I. Find by trials a number nearly equal to the root of the proposed equaticm. If. Let r=the number thus found, and let zsthe diffierence between r and the root x of the equation : so that if r be less than X, then r'{-zssx; but if r be greater than x, then r^zzsx, III. Instead of x in the given equation, substitute its equal r+x, or r^z, (according as r is less or greater than x,) and a new equation will arise, including only z and known quantities. IV. Reject every term in this equation which ccmtains any power of z higher than the firsts and the value of z will be found by a simple equation. > V. If the sign of the value of 2 he -f, this value must be added to the value of r; but if—, it must be subtracted, and the result will be nearly equal to the root required. VI. If this root be not sufficiently near the truth, let the operation be repeated ; thus, instead of r in the equation jus^ paw resolved, substitute the corrected root, apd the secon4 l4 15S hJ/RSKBA fjtn* T mine of z being added or rabtiaettd accordtog toi^agft, a nearer af^roxnnatioa to the root wifl be haA, and if a still nearer appeoxiniation be required, the operation may be re- peated at pleasure^ observii^ alwa^ to sufaetltiite Ite last cor- reeted root for the new iraloe of r. Examples. — 1. Given x*+x=:14, to find x by approxi- mation. By trials it soon appears that x must he nearly equal to 3.; let therefore r=3^ oad r+2=x; wherefore substituting this value of X in the giten eqiuition, it becomes r + rl*+ r + z= 14, that is, r'+2r2: + r*+r+z=14} whence by transposition, and rejecting , . , 14— r»—r 14—9—3 «*, we ODinfii 2 rz -^xs: 14 — r» -?- r, oaa xs= — , ■ sb- ^ .■ ^ — ?.r-fl 6 + 1 2 SS-— 3S.28, and a:=:(r4.z=3+.28=) S.28> aeariy. /• For a nearer value of or^ let the operatimi be repeated. Thus, let r=3.28 *3 and substituting this valufi for rintht 14-r»— r .^ ^ 14— 10.7584— S.28: equatum ^=-^;rfr' ** **^^^^ "=( e^eTI ■== — .0384 . ^ ^_ . , , =s)— .00508, nearly; wherefore jr=(r+r=s3.28— 7.56 .00508=) S.27492» extremely near. 2* J»et «'^-*-2:x* +3ss:5 be giseo, t» find dr. /^ appears by trials, that x^S nearly, wherefbre lei f b=5, nad r+z3=dr as:before; then wiltf jp»s5:r«-h3r'z-f3faf»+ z?^ — 2jr?= — 2r* — 4rz —2a* >=>£n 4-30? = ,3r +32 J From which,, rejecting tUl the terms which contain z* or 9^, we obtain (r»+3r'z— 2r»— 4rz+3r+3z=5» or) 3r»z— 4rz+ ■ Sometimes it happens that the correction consists of several figures ; in that case, if a second operation be necessary, it will be convenient not to snb- alitute aU the %nrea for r, but ooIjl oneflgore, or two^ spdi as will nearly express the valneof the wbo)^ : thus, if x alter the first opeiatioa be 3.5^ for a second operalioal will, piit r»(not 3.68, bttt).3.^ if; «t the ooiiQliisioQ of this second process ;r^ 3.648917, and a third be deemed neeesswy, I will not employ all these agnres, but instead, of them put rs 3.65, and proceed. This method is to be attended to in all cases, as it saves miich trouble, and prodtices searcely any effect on the approximation. PXnT. APFROXIMAtlC»f. 15S 3z=5— r»+2r'— Sr : whence z= — -— ^ — : r— ^ t g7-.lg^3 "^ jQ=)-.7; »A«icea?={3— .y-)2^ne(irfy. For a nearer approximation. Let fs2.3, iAt« vff/ue mbetiiuted for r in the preceding 5— W.167+10.58— 6.9 -3.437 ' ecmatum, we heme z=:( as— ^ =s) — ^ ^ 15.87—9.2+3 9.67 ^ S6, iotoicex=3(^3— .36aB) 1;.94, f<tJ< neater Ann hefion; and i/* 1.94 6e substituted for r in i^ eguaiion above aUuded to, a third approximatkm wiU be had, wkerebf a nearer value of a wiUhe o6- tained, 3. Given x' — 5 xssSl, to find x. Jus, X78i603S77S» 4. Given x« + 2 a7--40:? 0> to. find x. Am. xis5.403135. 5. Given x* + j:' +x=:90, to find x. Ans. x=74.10283> 6. Given 2x' 4: 4 x' —245 x-*-70^o> tafind x. Jfi». x=s 10.265. 7. Given x*— 12x+7=o> to find x. ^)m. x=:2.0567- 8. G^een x' -4>10x^20a:9> to find ^le value of x. 63. The following method affords a motfter approximatum to - the unknown quantity than the former rule \ Rule I. Let a number be found by tmls nearly equfd to the required root, and let z=tbe diSerence of the assumed number and the true root, as before. <> This method is given by Miu Simpcon in p. 162. of his Algebra, where be has extended the dpctrine beyond what our limits wiU admit : the above rule is in its simplest, form, imd triples the number of figares tme in the root, at p every operation ; he calls it an approximation of the teeond degree^ (s« -^ p being the^rj*,-) and since g« ^^^^^_^, ^ j^, if the first value of z (vi<, -^] be substituted in the second term of the denomii^^tor, and the following. op terms be rejected, it will become z» -^ — ^, an approximation of the second de- grtty the same as the above rule. If for z its second value ^— — be substi- p toted, then gg - , an approximation of the third degree^ which h* 154 ALGKRRA. Pait V. II. Sidistibite the ttBumed quantity -jhz, in the given equa* tion, as directed in the preceding mle; and the given equation will be reduced to this fimn, iiz+6z'-|-cz' +, &c.=sp. o bz* cz* III. By transposition and division we have z=<^- , a u Q &c where, if aU the terms after the first be rejected, we shall P P have z= — ; and if 9 be put for -=--, and its square substituted bq' for z' in the seobnd term, we shall have zso— •-^. a EzAMPLBS. — 1. Given x'-*2jr« +3 xs5, to find ar. Hare x=:3 nearly; let 3+2=jr, then, «*=s 27+27 z+9z«+z»-| ^5x' = -18— 12z— 2z* . . V=5, that'u, +3j: = 9-f 3z J 18+18z+72'+«*=5, or 18z4-7«*+2' = -13. Here a=18/t=7, c=l, p=-13, 9=(^='ZH-:)-.72. 9 — ^'^^"'•'^^ 18 =)— -^SlCsz; wherefore x=s (3+z=3— .9216=) 2.0784. For a second approximation, Let2-^z=six; then a?»= 8+12z+6z»+«'^ -2 x^= --8^8 z-2 z« . . V =5, that is, +3x= 6+ 3z J 6-f7«-h4z«+z»=s5, Of 7«+4z«+z«= — l. c fcy making «»-^^rr^, araltiplying both tenns of the Craciion hj l + tq, and rejecting ht'q» (as very small) from the product, becomes — fll^r-* a* +b+as.f By similar methods* and by putting «-—"+ r; , the approximating mlt of thc/owfAdegreeis ap.a + w p_ ^^^^^ quintuples the nunH her of figures true at every operation. Bkit V. APPROXIMATION. 155 Here fl=7, 6=4, c=l, p=— l, and q:=(—ss^^ss)^ a Tf .14285; wherefore a— !!il=(— .14285— -~X— .I428a•=) — a ^ 7 .15451064=2. ^dj;=(2.0784-. 15451064=) 1.92388996, very nearly. ^. Given a7»4-20ar=100, to find the talue of x. Am. a?= 4.1421356. 3. Given a:*— 2 r=5, to find r. ^rw. x=2.094551. 4. Given a?'— 48 x«+200=o, to find x. Ans. i= 47.91287847478. 5. Given «♦— 38 af'+SlO a:' + 538 xH-289=o, to find x. An- swer, 07=30.5356537528527. 6. Given j?*+6a?*—10a?s-112ar«-207a?-110=o, to find x. Ans. a?=4.4641016151. 7. Given 2 a?" +3 x+4=50, to find the value of a:. 64. BERNOULLrS RULE Has been sometimes preferred on account of its great shnpli* city and general application : it is as follows. Rule I. Find by trials, two numbers as near the true root as possible ^ * This is perhaps the most easy and general metbod of re9olYixig equations of ererj kind, that has ever yet been proposed ^ it was invented by John Bernooliiy and published in the Leipsic Acts, 1697. The most intricate and difficult forms of equations, however embarrassed and entangled with radical, compound, and mixed quantities, readily submit to this rule without any previous reduction or preparation whatever ; and it may be (Conveniently employed for finding the roots of exponential equations. The rale is founded on this supposition, that the first error is to the second, as the difference between the true and first assumed number is to the diffe- rence between the true and second assumed number : and that it is true accord* ing to this supposition, may be thus demonstrated. liet a and 6 be the two suppositions ; A and B their results produced by si« wilar operations ; it is required to find the number from which N is produced by a like operation : in order to which. Let N—A^ r, N^ B^s^ and x » the number required ; then by hypothesis, r : * : : ar— « ; x^h, whence dividendo r— * : « : : i— a : jr— ft, that is, --• ^x^hf which is the rule when both the assumed quantities, a and 6, are (ett than the tme root ^. 159 AUEOmk. Vkm r. II. Substitute these assumed numbers for the unknown quan« tttjr m the ^ven* equatidn^ and mark the errdr which arises from each with the sign +> if it be loo greats and — ^ if too Itttle. III. Multiply the difference of the assumed numberf bjr fife least error, and divide ihe product bj the difimnee df the er- rors when they have like signs, but by their sum when they have unlike. IV. Add the <|uotient. to the assumed number beloii^n^ to the least error, when? that number is too littld*^ but subtiact 'when it is too great 5 the result will be the root^ nearly. V. The operation may be repeated, if necessary, as in Ihe Ibr- mer rules> either by taking two new assumed numbers^ or using one of die fiormer numbers^ and assumiog a new one. Examples. — 1. [Given 10jr*+9a:' + 8 j:«+7Jf=1234, to find iT. Here hy triah k appears to be greater than 3 ; wkerefof&let 3 and 4 be the two aswmedr numbers*. Next, let ji and B be eaeb greater Hiaa JIT, then wifl N'^A^ — r, Ari N^B^ —Si but — r :— * : : +r ; +*, wherefore r— * : # : : a— ^ : h — x^ nt a — 6'j < ss &•— Xy which is the nde when the assumed quantities^ a and Vy aie each greater than jr. Lastiyy M oviief result ^ be too little, and the other B too great ; then will rbe positfre and^ negative. Wherefore r-f-«': (— r, oi*, which is the samej a — .h9 41 •: : a'^h : h^as <>iM «*> <** i>^«V wfakh it the iQk^- wHefl«0ii« of tlM assumed quantities is too great, and the other toe small. Q. £. D. All qpes- tions in double position are resolved by this method. ^ The convenience of substituting two numbers which differ by unity is this, it saves the trouble of multiplying the least error by* that difference. If the numbers substituted have decimal ^aces, the same method is to be observed : thus, suppose they are 1 .34 and 1.35, and the least error 12<5794, in this case the diffbrence of the supposed numbert ia .01, and the multiplication is per- formed by simply removing the decimal mark two places to the left, makiag the product . 1 25794 ; and the like in other instances. Famt V. APPROXIMATION. 157 first 8mfp9ei^4m, Eque^ion. Second Supposition, or 9^$. Wff=4. 810 =10a:«= £560 243 = 9x^= 576 72 , . SK 8jp«s= 138 21 = 7a? = ^8 1146 :ssrmtltss 3292 — 88 ' ' ' = error= ...... 4-^58 , Difference of the assumed numbers 4+3=1. Least error 88. Sum of the errors {they Mug unlike) 88+ 1 X 88 88 2058=21463 wherefore "^7:^^=2777^= .041, the correction to be %i4o %14o added to 3 the number from whence the least error crises, 3 being too little; wherefore 3.0 41 is th e root or value ofx, nearly. 2. Given ^l+a?+ V2+«'+ V3+^=16, to find x. Firom a few trials it appears that x is somewhat greater than 8, rnhfirefm-e assuming S and 9 for the values ofxy the work uMl stand thus *. Erst Supp. Equation. Second Supp. <w *»8. or irasO. 3 = v ^l+j? = 3.16228 4.041^4 =»v ^+j» = 4.36207 4T6378 s=;V3+y'= 5.20149 11.80502 =zresult^ . . . .12.72584 —4.19498 z::error=sz —3.27416 • The logarithms are of excellent service in all cases of this rule, where rooU and powers are required to be foond, op where the terms are mixed and com- plicated: thus in the pf<«seat instaBce, supposing ar«8, then 1 +x=9, the square root of which (vl«. 8) imm^iately ooeivrs ; but let ar-9, then I +»- 10, to find the square root of which, by the conimon method, xeqoires rather a long process. I therefore take the logarithm of 10, divide it by 3, (the index of the square,) and the quotient is a logarithm, the natural number cprrespond- ing to wfai^b (s 3.16228, as above. Next, supposing ar«8, then » V'SHhrT^ V^66. I find the logarithm of $6, divide it by 3, and the natural number %reqiDg with the quotient is 4J04m, «5 above. Let ^«9> Mi«P * ^/U^ » 'i/83, which by a simila r process is found to be 4,36207, as aboye. Lastlp if jr*8, «han *V« + a:«-^V'5l«5 if ar«9, then ♦^5T*9«*>/7«2# «» roots of both which are found by a similar operation, and ar^ as above, viz. 4.76378 and 5.20149. 8ee VoL I. Part 2. Art. 38. 158 ALGEBRA. Pakt V. Diff. of assumed numherszsil, least error 3.^7416, diff. of the errors {having like signs) 4.19498— 3.274 16= .9^062; 3.^7416 wherefore =3.5309^ the correction to be added; com- quenthf 12.5309 is the value of x nearly. For a second approximation^ Let the numbers 11 and 12 be assumed, then First Supp. Equation. Second Supp, or 0?= 11. orx=12. 3.38525 = ^1+jp = S.60555 4.9732 =V 2+j;« = 5.26563 6.0435. =♦^3+^:'=: 6.4502 14.40195 =rc»ttZf= 15.32138 — 1.59805 :zzerTor= —.67862 Least error .67862, diff. of errors (1.59805— .67862=) 67862 491943; w^nce^--— -=.73809, the correction to be added l» » 12 5 to^cforc a?= 12.73809, rery nearZy. 3.. Given ai — -xr+a^^j^.^^x^'j^^ ^" ^ =45. to ^ x^x—l find the value of x. Here x will be found by trials to be nearly equal to 10» wherefore let 10 and II be two assumed numbers; then^ First Supp. Equation. Second Supp. or x=zlO. or ar= II. 7.74264 =:A=: 8.42718 4 g —4-14358 as — -^Ts= —4.43549 5 67.6616 3= +x 3 ^x' -H2 ar v'*' H*a? ss79.S363 x+l ^.seeee =5 — = —.34497 70.894 ^result=i 82.88302 4-25.894 ^errorsz .... +37*88302 Least error 25.894, dij^. of errors (37.88302—25.894=) 25 894 11.08902; «*'^^/^''« iY-^gQ^=2.1598, to be substracted fron 10 : consequently »= (10— 2.1598=) 7.8402 nearly ; and if Part V. APPROXIMATION. 15^ greatet exactitess be required, the operation may be repeated at pleasure, 08 in the second example. 4. Given d^+3a;s20^ to find the value of x. Am. x^ 3.13939. 5. If a:'+a;«+a?=20, what is the value of 4:? An$. xsa 2.3^174. 6. Let 2a:»+3x«+4a?=100 be given, to find Jl. Am. x^ 3.0696. 7« Given -—a?*— 12 a?*— 50=0, to find x. Answer, X3» 4 11.9782196186948. x^ 8. Given — +3x*— 5a^— 56a!«— 10S4.a:=55, to find x. Ans. ar=2.2320508075. 9. Given >v^l+a?' + v'2+a;*+^3+a:*=l0, to findx. Ans. a?= 3. 0209475. IOOjp « /5-4*.ir" EXPONENTIAL EQUATIONS. By the foregoing rule, the roots of Exponential IJquationB may be approximated to, with the assistaiice of logarithms* 65. An exponential equation is that in which the indices, as well as some of the quantities themselves^ are unknown qu8Q"> titles to be determined. Examples. — 1. Given x*=sl000, to find the value of x. li appears by trials that x is greater than 4, but less than 5. Let 4.4 and 4.5 be the numbers proposed. Then since x x log. ofx^log. of 1000, that is, Rrst, (4.4xlog. o/4.4s) 4.4 X0.6434527» 2.83 119188 But the log. of iq0Oaa 3.O000000O Error -- ^0.16880612 Secondly, (4.5 X tog. of 4.5=) 4.5x0.6532125=2.93945625 Log, of 1000= 3.00000000 Error —0.06054375 StAtract this error from thefonntr, and the dJiff. is 0.10826437 Then 4.5— 4.4s.l= di/f. of numbers found by trial, and 1 X 06054375 .06064375, kast error ^ therefore '- .naa^A^^ ' =.055922, the 160 AL6EBBA. PartV* correcUon; wherefore js:xz (4,5 +,0^^922:=) 4.559^2^ i^ cmswer, very nearly i for 4.SbO^^^'^^^^=^(Jby logarithms) 1009.315, which reeuU exceeds the truth by 9.315. To repeat the operation. Let 4.55 and 4.56 be the assumed numbers. Then (4.55 x%. 4.55) =4.55x0.65801 14 =2.99395 187 Log. of 1000 K 3^0000000 Error— 000604813 Also (4,56 X log. 4.56=) 4.56 X 0.6589648=3.00487948 Log. of 1000=3.00000000 Error (least) ^0X)04S7945 Then 0.00604S13 +0.00487945= .01093758, sum of the errors. Tkerefore :25iii^i^!?:5?=:2^;g^l=. 00234. cor- •^ .00487945 .00487945 rectum. Whe refore ^= 4.56-^.00224=4.55776, nearltf. For 4.5.5776l*-5*776= 1005.6, which is too great by 5.6; and for a still nearer approximation, the operation may again be re- peated; thus, let 4.556 and 4.557 be proposedy and proceed as brfore. 2. Given x^^lQO, tx> find x. Aks. ^7=3.597285. 3. Given a»=7837577897, to find z. AM. «= 11.295859. 4. Given x*as 123456789, to find x. Ans. d^c±8.6400268. 5. Given y'=3000, and a?y=5000, to find x and y. Ans. «*» 4.691445, and y=5*510132. a. Given a?*s=400, to find x, Ans. d?=2.32443i8. 66.^ Two or more equation^ imvohing «« many unknown quantities, may he resolved by a^itpreximaUen, as follows^ RvLM, I. Reduce M the equatiods to one, (by either of the methods for redncing equations containing two or more un- known quantities. Vol. I. Part 3. Art. 90 — ^95.) this equatien viU contain only one unknown quantity. II. Find the value of this unknown quantity by one of the preceding rules ^ from whence that of the others may be obtained. Examples. — 1. Given x-fy+z=^2, 2t— 3y+5z=40, and 3«4-4y**2«^afc— IQD, to find x, y^ and t. ftAMT T. APPROXIBIATION. let Erom eq» I. :p»n— y-^2; iukitUuie ikii value of s in the second and third, and (44-~2y— 9s--df +5c=:40, or} 6y-*3z =45 aba (M— 3y*-3z+4y-S<*s--100j «r)^j:»4-3«— fs= 166 ', let now the value of y (= — - — ) in the last but one he sub* d r4-4 stituted in the last, and it becontes {2z» +3z =166, or) 10z'H-12z=834. • Now it appears from trial, that z is greater than 4, but less tJian 5 ; fee tkes9 two numbers therefore be substituted for t, then by the last rule, \st Supp, Equation. Znd Supp, or 2=4. or 2=5. 640. =10«»= 1260 48 s» + J2«ss 60 688 r= result = 1310 — 146 = error z:^ +476 For a nearer approximation. Let 4.2 and 4.3 be put for z, and 1st Supp, Equation. ^nd Supp, 740.88 =102^= 795.07 50.4 s=-f 122= 51.6 ^91.28 =zresult=sz 846.67 —42.72 sserror:^ +12.67 ^ ^ 12.67 x.l 1.567 v^ooo-r^ ,. 'T^ ( • ■ 1 rL« =)X)22874, 1^ oorreciuis. ^42.72+12.67 65.39 ' Wherefore a= (4.3— .022874=) 4.277126, «ciy nearly. Whence y=(-^^ac) 3.366275, cwd x=(22— i^— 2=) 6 14366599^ Msr^. 2. Given «— x=10> x^+x2=900, and xyzvtzSOOO, to find x, y, and 2. Erom eq. 1. t=10+a?; ^«H<ttfe this value for 2 ill ^A« 900— IOjc— x« second, and it becomes xu + 10 x + x* =900, and y = ; X write this value for y, and 10 +x for z in the third, and it will become (9000+800*— 20 «*—x'= 3000, or) a^+20««— «00«= 6000. VOL. II. M I IGZ ALGEBRA. Pakt T. &re hf trials x isfimnd to he greater than 93, bnt leu than 24 3 then Mtmg these two numbers as snpposUians, and proceed- ing as before, x =23.923443456^ 9s3.69655893S, and zsl 33.923443456, nearly. 3. Given jc^+y=157> and y'— 2r:s6, to find x and y. Jnt, j:= 12.34, y =4^21. 4. Given x+xy=&0, and jr^— y*=495, to find x and y. Ans. xs=8> y=:9. 5. Given i^+3r'=12, and i'+y'sS, to find xmnd y. 6. Given ar+yzs20, y-|-2z=22, and x+xy=:28, to find x,f, andz. 67. Dr. BUTTON'S RULE for extracting the rooU of numbers by approximation. Rule f . Let N=the number of which any root is required to be extracted, — =the index of the proposed root, r=the number found by trials, which is nearly equal to the root, namely, r^=:N nearly, and let x=the root, or i^zs^N exactly. 11. Then will x= '*"^^'^^""" — V r, neariy '. n+l.r*+fi— l.A^ ' The rale is thnB demonstrated; let iVathe given ntimber, the root of I which it is proposed to evolve; — sthe index of the root, r as the nearest it- tional root, v= the difference hetween rand the exact root, x^r + v^the enct root; then since i^^a^r+v, we shall have i\r=r+t;J»=r"+iir*— »v+» •--^~r« - ' V' + ,&c. (Vol. I. P. 3. Art* 54.) and hy transposition and diTision, TV"— I* Ji— 1 «» fi^ltf> ■■ggp+-*'-— « — ^y&c. in which, rejectingr —r-' — on acconnfc of its Mr" -*2r '' ^ 2 r saiallness, v may be considered as « ^ . Bat from the first eqnatioB, ff-l n— 1 JV— r«=itr« — » t; + ».-g-»«- •»» + ,&«. = (iir"-» +11.-2"'* ""**') X^,* which, if the former value of v (vie. r r) be substituted, we shall have «— I N—r^^ 2nr^ + n— l.iV— »r« + r" iVr-r»=.j:»r«»-»+-^ —)Xv^ ^ Xf= — y y; consequently t;^ — :- — , and are(r+v*; ^»* «+l.f* + »— l.-Y l^ART V. APPfiOXIMATJON. 163 III. To find a nearer value, let this value of a? be subetituted for r in the above theorem^ and the result will approach nearer the root than the former. IV. In like manner, by continually substituting the last value of X for r, the root may be found to any degree of exactness. Examples. — 1. Let j:*=19 be given, to find the value of x. Here iV=rl9, ns=4, and the nearest whole number to the fourth root of 19 is ^', let therefore r =2, then iciW r"= 16, and xzs n+l.iV+^^l.r- 5 X 19 + 3x16* ^ 286 ^ — ^-- xr=(- — -__^>_.-_x2t=)-— =2.08, nearly. ;r4rr.r»-n-l.2\r '5X16+3X19 ^37 ^ To repeat the process for a nearer approximation. Let rst^,OS, then r" =5 (2^08/*=) 18.71773696 j these numbers being substituted in the theorem, we shall fiave xz=^ 5x19+3x18.71773696 ^ ^ ,151.15321088 ^ ^6x18.71773696+3x19 ^150.5886848 2.0677975, extremely near ; and if a nearer value of x be require 'ed, this number must be substituted for r, and repeat the operation. 2. Given rc'ssSlO, to find x, Ans. a?=7.999, ^c. 3. Given x*=790O, to find x, Ans. j:=6.019014897. 4. Extract the sixth root of 262140. Ans. j:=3.9999, ^c. 5. Required the sixth root of 21035.8 ? Ans. a?=5.254037. 6. Extract the sixth root of 272. es. PHOBLEMS PRODUCING EQUATIONS OF THREE OR MORE DIMENSIONS. 1. What number is that, which being subtracted from twice its cube, the remaipder is 679 ? Ans, 7. 2« What number is that, which if its square be subtracted from its cube, the remainder will exceed ten times the given number by 1100 ? Ans. 1 1. r+==. = — =: == — .»', which is the rule. This is the «+!.»* + «— I.A^ w+ l.r^+H— l.AT inTcatigation of the rule io Vol. I. page 260 : the theorem was first i^iven b^ Dr. Hntton, in the first Volatne of his Mathematical Tracts j it includes all the rational formulae of Halley and De Lagni, and is perhaps more convenient foi^ nemery and operation than any other rule that has been discovered. M 2 164 ALGEBRA. Part V. 5. What number is that^ whieh being added to its 8<]uare^ the sum will be less by 56 than — its cube ? Am. 8. 4. There is a number, thrice the square of which exceeds 9 twice the cube by .972 j required the number ? Am. —. 5. If to a number its square and cube be added, four times 43 the sum will equal —- of the fourth power ', required the num- 54 bet ? Ans, 6. 6. If the sum of the cube and square of a number be mt^i- plied by ten times that number, the product shall exceed twice the sum of the first, second, third, and fourth powers by 180; what is the number ? Ans, 2. 7. Required two numbers, of which the product multiplied by the greater produces 18, and their diffierence multiplied by the less, 2 ? Ans, 3 and S. 8. The di^s being 16 bouts long, a persM ntfao was asked the time of day, replied, *' If to the cube <tf the hours passed since sun-rise you add 40, and from the square oi the hours to come before sun-set you subtract 40, the results wrill be equal *' required the hour of the day ? Ans. Sin the Tiwming. 9. To find two mean proportionals between I and 2. Ans^ r. 25992, and 1.5874. 10. The ages of a man and his wife are such, that the sum ef theur square roots is 11, and the difference of their cubes 31031 f what are theif ages ? Ans, 36 and 25. 1^1. If the cube root of a lather's age be added to the square root of his son's, the sum will be 8 $ and if twi6e the cube root of half the son*s age be added to the square root of the fiitha^'s, the sum will &e IS 3 what is the age of each i Ans. thefaihefs e^, the son's 16. 13. There are in a statuary's shop three cubical blocks of marble, the side of the second exceeds that of the first by 3 inches ; and the side of the third exceeds that of the second by 2 inches 5 moreover, the solid content of all the three to- gether is 1136 cubic inches 3 required the side of each ? Afi^> 4, 7, and 9 inches. PART VI. ALGEBRA. THE INDETERMINATE ANALYSIS !• A PROBLEM 18 said to be indeterminate, or unlimited, when the number of unknown quantities to be found is greater than the number of conditions, or equations proposed ^ ■ For some accouat of the subject, see the note on Diopbanttne problems. ^ If the namber of putsita exoec4 the nvmber of datm, the problem is nn- limited. If the qtutrita be equal in number to the data, the pioblitm is limited. If the data exceed the quauita, the excess is either deducible from the other conditions, or inconsistent with them ; in the former case the excess is redaadant, and thnreibre unnecMsary ; in the latter it renders the problem absurd, and its solution impossible. To give an example of each. - 1, Lei x-i-y^S hegivtHtto/indtke wUmes^ X andy. Here we haye but one condition proposed, and two quantities required to ba fonndy the problem is therefDre unlimited; for (admitting whole numbers only) X may si, then ys5 ; if xs»9, then jf»4 ; if x^a, then y^S $ if xa>4» then jr»9 ; if jr^S, then jr^s 1. 3, Lei x+yssS, arndx^y^A, he given. Here we have iwa conditions proposed, and #100 quantities to be found, whence the problem 1ft UmUed; (see Vol. I. P. 3. Art. 89.) for «r»5, jf«l : and no other numbers can poasibly be found, that will lulil the eonditions. 3. Lei *+y«6, «—y«4, iwrf«y —5,4* ^w«i. Here is a redundancy, three conditions are laid down, and but two quantitiey to be found. By the preceding example x—h,y^\ \ wherefore Ay —5 X 1 — 5, or the latter condition {xy^h) is deducible from the two former. 4. Let x+y=6,x--y^4, and xyisli, he given. Here is not only a redundancy, but an inconsisteney ; for the grntest pro- duct that can possibly be made of any two parts of 6, is 9, that is, Ay »9 ; it cannot then be divided into two parts, x and y, so that «y— 18; wherefore the latter condition is inconsistent with the two former, and renders the pro- blem impossible. There is a mistake in the appendix to L adlam' e Rudimentif 5th edit. p. 338. Art. 107' by which the subject is altogether perverted. M 3 \ 166 ALGEBRA. Pabt VT. 2. An indeterminate problem will frequently admit of innu« merable answers^ if fractions, negative quantities^ and surds be admitted 3 but if the answers be restricted to positive whole numbers J the number of answers will in many cases be limited. 3. The indeterminate analysis is the method of resolving indeterminate problems 3 it depends on the following self-evident principles^ viz. '^ The sum, differences and product of two whole numbers^ are likewise whole numbers." '^ If a number measure the whole^ and likewise a part of another number^ it will measure the remaining part." 4. In the given equation ax^^by'\-c, to find the values of x and y in positive whole numbers. Rule I. Let W stand for the words whole number, then (since x and y are by hypothesis whole numbers) the above equation aj=s6y -|-c reduced^ will be a?s= =fr, II. If JZf be an improper fraction, reduce it to its equi^'a- a lent mixed quantity; (see Vol. I. p. 880. ex. 9, 10.) that is, let Jj^^ifn-^-^^ : from which rejecting m, we have = ^ a a ^ « by Art. 3. III. Take the difference of -^^ or any of its multiples, and a y or any of its multiples, viz. — , -^, — , &c. in order to re- •^ a a a ^ duce the coefficient of y to unity, or as near unity as possible, and the remainder will he^W. IV. Take the difference of this remainder and any of the foregoing fractions, or any other whole number nearly equal to it, then will the remainder ;= W, V. Proceed in this manner^ till the coefficient of y becomes unity, or ?^ = fr. VI. Let— ^=», then will yszap—g; and if any whole num- 'a Pabt VI. IND£TERA1INATE ANALYSIS. 167 ber whatever be substituted for p, the value of y wUl be known ^ whence x (= ) will likewise be known. a ^ Examples. — 1. Given 4x=5y— 10, to find the values of x and y in whole numbers. ^ , 5y — 10 „, , 5tf— 10 ^ y— 2 First, x=-^ =^i 6tt<-^ =sy— 2+ '■ 9 fohenc9 «— 2 (rejecting y— 2) ^ =sW^=p, therefore y— 2=54p, oiid y=4p 4 .« ,. ^ ^ , ,5y— 10 10-^10 ^ + 25 te* /)=o, then y =2> whence x= (— ^ = — - — =) O- Secondly, letp he taken=:ly then ^=(4^+2=;) 6, and x= 5y--aO_20_ 5v— 10 T^irdty, letp^% then y=(4y^2=) 10^ and j=( ? i =^=) 10. 4 ^ Fottr*%, fe^ p=S, ^/i£»y=(4p+2=t) 14, and x=(-=— — 4 =) 15- Fifthly, let ptB4, then y=i}8, and j;=20. Sixthly, let p=5, then y=22, and 07=25. 8sc. 4e. Hence it appears, that the values of x (viz. 0, 5, 10, 15, 2Q, 25, jS^.) di£fer by the coefficient (5) of y ; and the values of y (viz. 2, 6, 10, 14, 18, 22, &c.) by the coefficient (4) of x; and it is plain, that this will be the case universally in every equation of the form axzs,hy — c, viz. the successive values of X will di£fer by h, and those of y by a. 2. Given 17^=13^—14, to ^n4 ^ ^nd y in positive whole numbers. 13t/— "14 17 n First, x= — ^- — s=Fr, afap ^^W; wherefore (Art, 3.) 17J|_13y-14^4jH:14^ ^ 4jH^ lJy+56 17 17 17 '•17 ^ vT^ ^W, that i,,i£?ii+3==»r,»fce«ceH^ti=Fr,. and (i^^ 17 17 17 M 4 we led AJLGEBRA. PaetVI. ^ .-. — ?[Z— rs) lUssWszp, whence y=17p+55 lei p^o, then 13tf-14 65-14 ,„ y=5, and j?=( — j^ = --^^ =) 3. And by continually adding 13 to the value of x, and 17 to the ▼alue of y, we obtain the following values^ viz. x=3, 16, 29, 42, 55, 68, 81, 94, 107, &c. y=5, 22, 39, 56, 73, 90, 107, 124, 141, &c. 3. Let 4x+7y=s23, be given, to find x and y. ^ 23— 7y X ^ 3— 3tf , . .. K First, x=( ^=) 5— yH --^, whence rejecting 5— y, 4 4 have '-=^=ir, »A*re/Te (ll?+«-Ziii=) y±?=ir=p; 4 4 4 4 consequently y-\- 3=4 p, and y=Ap—Z; let p^l, then y=:(4jj— 3=) 1, andx=z{ ?= — =) 4-, which are the only affirvM' 4 4 ^ii7# answers the question admits of, 4. Given 19a?+14^=1000, to iind jrandj^. First, x=:.{ j= — ??=) 52+— ^^-^5 r6;«c«i?^ 52, ^ 12— 14y „^ ^, 19y 12r-14y ^5y+12 hxive —^^^W, cojisequently (^+ ^^ ■ =) -^5—= TMr » .5v+13 20y-f48 .20y-fl0 ^ „, , JT, a^o ( ^^ x4= ^J =).^-ZL^+g=:y, „fc«we — j_. = jrr; wherefore {—^ j^=) ^^= ^=P' ^'^ y=19p — 10^ ief p=l, theny=z9, and x=z{ "^ — ^=) 46. Let p=:2, /^«n ^=28, and ^s32. Ze^ p=3, t^en y=:47f andxsz 18. X,e< p=s4, then y=^G6, and x=:4. These are all the cffirmative values of x and y ; for if pbe ' taken :=:^, then u;tZ/y=85, and ^ =r — 10, a negative quantity, Th£ above values will be obtained by adding the coefficient (1 9) of X, to the preceding value of y ; and subtracting the coeffi- cient (14) of y, from the corresponding value of x; and the same is universally true of every equation of the form of <fcr+fey=<?. 6. Given 13 a?=21 y — 3, to find the least values of x and y in whole numbers. Ans. a?=3, y=2. Past VI. IND£T£BMINAT£ ANALYSIS. 109 6. Given 41jrs43y— 53> to find x and y. Jm. xalO, 7. Given 8a;+9y=25^ to find x and y. ^w. xs=2> ysl. 8. How many positive values of x and y in whole numbers can be found from the equation 9x=2000— 13y? Ans. 17 values of each, 9. Given 13jr=14y+36^ to find J? and y. 10. Given 101 x=s4331-.177y, to find j? and y ^ 5. To find a whole number, which being divided by given numbers, shaU leave given remainders. Rule I. Let x=the number required; a, b, c, ^rcsrthe given divisors; f, g, K ^c.=the given remainders; then will a b c 11. Make the first fraction =p, find the value of x from it^ and substitute this value for x in the second fnictioo. III. Find the least value of p in the second fraction, (Art. 4.) in terms of r, and thence x in terms of r. IV. Substitute this last value for x in the third fraction, whence find the least value of r in terms of s^ and thence the value of X in terms of s, V. Substitute this ^-alue in the fourth fraction, &c. and pro- ceed in this manner to the last fraction, from whence the value of X wiU be known. £xAMPL£s.-^l. What number is that, which being divided by 3^ will leave 9 remainder, and being divided by 2, will leave 1 remainder ? J— 2 X— 1 _^ Let xz=the number, then will — -— =IF, and — ---=rF| 3 2 let =p, then wiU x=3p+2 ; substitute this value forxin the frac^n ^^, and it becomes ^ ^W: but -^^^t wherefore (^-^^'■-'^=)^^^ tr^cep=2r-l; let % % % c By similar metbods indeterminate equations, involving three or mor« unknown quantities, may be resolred. 170 ALGEBRA. Paht VI. r be takenszl, then p=:(3r— ls=2— 1=) 1, and x=(3f)4-2=) 5, the number required, 2. What is the least number which can be divided by 2, 3, 5, 7, and 11, and leave 1, 2, 3, "4, and 5, for the respective remainders ? Let x:=zthe number, then fZLL=: fT, -— = ^, fZ? = ^, 3 '3 * 5 ' «C"'""4 X 5 T 1 —— = /f;; and __s=:^^ fcy //,g problem. Let -— -=:p, M01 «c— 2 a;=2pH-l 5 substitute this value for x in the fraction , and o it becomes^-l^:=zfV; but ^=W, wherefore (gP^^P-^^.,) -3 3 33 0+1 4— -=fF=r, and p=3r— 1, wherefore a?=(2p4-ls=) 6r— 13 $tt6£<i/tt^e this value for x in the third fraction , and it he* 5 f;omes ^Irl^fv but ^=:fV, wherefore (?Iri-.^=) !JZi S 5 "^ ^ 5 5 ^ 6 = ^='> a«d r5=:5»4-4, consequently a7=(6r— 1=:) 30<+23| /^i* value being substituted for x in the fourth fraction ^^, it - 30«+]9 2*4-5 becomes — =4<+2-f =:W, whence (rejecting 4»+2) - — #r; a«o ( — -; — X3= — - — =) — ' 1-2, wherefore (rejecting the 2) -Jl^zzztV; but y=^, consequently (— - —_ ss) -^— =fr=^, wAcncc 5=7^+1, and Jr=(30«-f23=) 210 * + 53 5 /Aw value substituted for x in the fifth fraction ^^, ., . 210/-h48 t4-4 ^ it becomes = 19 1 + 4+--Y-,/rom whence rejecting 19 1 t+4 + 4, we have —-z=zfV:=zu, whence /=n u— 45 let «=!, iAcii ^ =(11 tt— 4=) 7i and a:=(210 ^+63=) 1523. 3. Required the least whole number^ which being divided by Part VL INDETERMINATE ANALYSIS. 17I Sf will leave 2 remainder 3 but if divided by 4, will leave 3 re- mainder ? jins, 11. 4. Eequired the least whole number^ which being divided by 6, 5j and 4, will leave 5, % and 1^ for the respective remainders ? Am. 17. 5. To find the least whole number^ which being divided by 3, 5, 7> and % there shall remain 2^ 4^ 6, and O, respectively. Am, 104. 6. Required the least whole number^ which being divided by 16^ \7, IS, 19^ and 20^ will leave the remainders 6, 7> 8> 9^ and 10, respectively ? 6. Any equation involving two difierent powers only of the unknown quantity^ may be reduced by substitution to the form of an indeterminate equation, involving two variable quan- tities. Hence, all commensurate quadratic equations, commen- surate cubics wanting one term, commensurate biquadratics wanting two terms, &c. may be resolved by this method. It will be proper for the convenience of reference, to premise the fol- loviring table of roots and powers *'. Roots 1,2,3, 4, 6, 6, 7, 8, 9, 10, 11, 12. Squares 1,4, 9, 16,25, Se, 49, 64, 81, 100, 121, 144. Cubes 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728. Examples. — 1. Let aj*+4x=32 be given, to find x. 32 j;« ' 4 Ftrst, by tramposition and divmouj —T"' Secondly, X 1 it %$ plain, that whatever equimultiples of 4 and 1 be taken, the fractiom whose terms are constituted of these equimultiples re- 4 4 8 spectively will be equal to -- and to one another, that is, -T"— "o" 12 16 20 „ „„ . ^, . ., .. ,, ... 32— J* = — = — = — , ^c. Wherefore, thirdly^ if the quantity 3-45 ^ he made equal to aether of these fractiom, which (after transposing the known quantity 32) will give the resulting numerator equal to the square of the denominator, that denominator will be the value * See on this subject, Dodson's Mathematical Bepontory, Vol. I. Emerson's Algebroy Simpson's Algebra and Select Exercises, Vilaut's Elements ef Ma^ tkematieal Analysis, &c. IW ALG£BIIA. Pabt VI. of X in the proposed equation ; that is, — Z£. a= JL— Z.— 1?=:!5 X 13 3 4 5=-T-> *c. here it is plain, that if the fraction — be taken, we shall ^i?cS9-«*=sl6, or jr«=(S3— 16=) 16, whence xs=4. %. Given s^^Sxss40, to find ar. By transposition and division, as before, we have = 6 19 18 34 . r_„-r 40-24=16, '^''•' I and xss4, the answer. 3. Given «*+S arsSS, to find «. „^^ 8S>~j' _3_g_9 13 15 18 gl 24 * 1 S 3""4""5""6""7""8* And /^-«* =^*> \ whence x=s8> ^Ae answer. 4, Given a:^— 5^ q?El44 to find :p. Here fIllil=-l=i?=i5=:??-?5-£2-.?5 HfA ^ r »*r=(36+14=)49. Wherefore { j J ^i. ^ I oita ;r=7, ^A« answer. 6. Given «• — -Hr=118^, to find x. 4 Here il=i^=±=±=s±=l=l*=li-ii_l_!: « 12 3 4 5 6~7""8~9 10 ir "^ -^ i and xzsll, the imswer. 6. Given 4s^— 5 «--6ao, to find x. Her ^ilf =-i=— -15 fVkerefare ( ^7^'^'"^'' ->r I ^*7(^^7^=)^> •^ I ana x=2, I and ar=3. Consequently a:= +2, or +3. 7. Given y«4-4y'=96, to find y. 96'"C)' 4 8 ie< r=y^ tA«n will tj'+4r=96, and = — =— =s » 1 2 12 16 20 24 2& 32 , r 96— »«=(96— 32=)644 345678 \ and 9=8, ^^ oiMw^. Pabt Vf. INDETERMINATE PROBlJaiS. 17S But vzsf, whence jf s=» ^v^Q v^=) «. Orihui, «• 4 16 36 64 Because t^+4oss96, therefore _ _ _ _ . •^ 24-17 1 4 9 16 I 24— VBS16, or rsaS j wAence y=2, cw i^e. 8. Given jf*— 7sf=36, to find y. ^^^ y'-36 _7_^14^gl gg y 12 3""4* I ofid y=s4, tAe a$uwer. 3 9. Gi?cn z?— l^zs— -, to find 2. 4 — -SS-— «ik€ncez3al. X 1 Here ^i5^if!=s £.—??— ?1 ^^ ^^ JO. Given 9z*— z'slOO, to find t. 4 f*25- r .^=*(«25-100=)125, ^"^^ I and zsx (» ^25=) 5. 1 1. Given «>+2 «sS^ to find x. Am. xs2. 12. Given s^ — 5 xai6> to find x. Aw. xs6. 18. Given «*+30=9 x, to find x. Ans, x=iB, or 4. 14. Given y'+70s39y> to find y. Ans. y=:5. 16. Given 2^—21 z+20sko, to find z, Ans. zss4. 16. Given 60—^=11 x, to find x. Ans. x=3. 7. INDETERMINATE PROBLEMS*. 1. How must tea, at 7 shillingn per pounds be mixed with tea at 4 shillings per pound* so that the mixture may be worth 6 shillings per pound ? Let xisthe mmber of pounds at 7 slullings, then 7xsztheir vahie; yssthe number at 4 shMings, then 4y^their value. Whence by the problem 7 x+4yss{6*x+y:=2)6x+6y, or xsz 2y»or l:xx=2xy v 4? : y : : 2 : 1 */ there must be twice as much in the nuxture at 7 shUiiags, as there is at 4 shillings. • These problems «tc of the kiad which belong to the rale of Alligation. / 174 ALGEBRA. Part VI ^. Twenty poor persons received among them 20 pence ; the men had 4d. each^ the women i^d. each^ and the children -^cf. each ; what number of men> women, and children, were re- lieved ? Let x=the number ofmeuy y=zthe number of women, z=zthe number of children; then by the problem x-Hy-h2=20, and {4x-\- 4.y-|-4.z=20, or) 16x+2y + z=80: subtract the first equation V from this, and 15a:+y=60, or y=(60— 15a:=)4— ar.l5, or --2- 15 SO 45 =-—=—=—-, 3fc. but by the problem y -^ 20 */ y=15 j and since Ji M *J 4— j:=1 \' x=zS, a»dj;=:(20— x— y=)20— 18r=2. 3. How many ways can 1002. be paid in guineas and crown- pieces ? Let x=:the number of guineas, y^the number of crowns. Then by the problem 21x+5^=:(100x 20=)2000. 2000— 6 tf , ^ 5-5y 5— 5y „, Whence a7=( ^=)96 + ^, v - = W, v ^21 ' ^ 21 21 .5-5y ^ . 20-20y „ » 21y „_ 20-20y , 21 tf 20+y -5—=)— -2-=^r=p, vy=21p— 20; letpznl, then y^i crown, andxss( ?=) 95 guineas: and if {^\) the coefficient of 21 x, be continually added to the value of y, and (5) the coefficient of y, continually subtracted from that of x, the corresponding values ofx and y will be as follows, viz. ir=95, 90, 85, 80, 75, 70, 65, 60, 55, 50, 45, 40, 35, 30, 25, 20, 15, 10, 6, 0. ^ y=l, 22, 43, 64, &y, 106, 127, 148, 169, 190, 211, 232, 253, 274, 295, 316, 337, 358, 379, 400. Jns, 19 ways. 4. To divide the number 19 into three parts, such that seveo times the first part, four times the second, and twice the third, being added together, the sum wiB be 90. Let the parts be x, y, and z ; then by the problem x-hy-f z=: 19, a»d7ar-f-4y+2«=90;/rom*^/«*»=19 — y— «, thisvalue being substituted for x in the second^ it becomes (133— 7y— 7a:-h 4y+2«=)133-3y-5z=90j or (3y=r43-5z, or)y=^— Pabt VI. INDETERMINATE PROBLEMS. 175 1— Sz 1— Sz Sx 1— Sz Sz = 14-z+-^, ... -f^^W; also ^=ir, •/ (^-4-y 1+z =)"-^— ==^=P '•' x=3p— 1 J if /) 66 to/fc«=l, then z=2, yss 43—5* ( — - — =)11, and J?=(l9-y — 2=) 6; ifp^2, then will 2=5, o y=6, and ^=85 i/'p=3, ^/»«n 2=8, y=l, and a:=10: <Ae«e ar« a/2 the possible values in whole numbers. 5. How many ways is it possible to pay 100/. in guineas at 21 shillings each, and pistoles at 17 shiUings each ? Jns. 6. 6. If 27 times A/s age be added to 16 times B.*s, the sum will be 1600 5 what is the age of each ? Jns. J/s 48, B:s 19. 7. A Higler*s boy, sent on a market day With eggs, fell down and smash*d them by the way } The news reached home, and Master, in a rage, Vow*d him a whipping, bridewell, or the cage : *' 'Tis through your negligence the eggs are lost, '' So pay me if you please the sum they cost." The boy, since nought avail his tears and prayers. Fetches his leathern bag of cash down stairs ; The cash a year's hard earnings had put in. But much he wisb*d to sleep in a whole skin. *' How mai^ were there. Master ?*' In a doubt. The Higler calls his wife to help him out $ Says she, ** I counted them by twos, threes, fours, '' fives, sixes, sev*ns, befoi'e he left these doors ; *' And one, two, three, four, five, and nought, remained *' Respectively, nor more can be explain*d." At nine a groat, ingenious Tpros, say. What sum will for the sad disaster pay ? Ans, 4<. 4d^. 8. Is it possible to pay lOOZ. with guineas and moidores only > jins. It is impossible. 9. A, who owes B a shilling, has nothing but guineas about him, and B has nothing but louis d'ors at 17 shillings each -, how» under these circumstances, is the shilling to be }»aid ? Ans. 4 must give B 13 guineas, and receive 16 lonis d'ors change. 10. With guineas and moidores the fewest, which way Three hundred and fifty-one pounds can I pay ? 176 ALGEBRA- Part VI. And when puid ev'ry way *twi]l admit, the amount Of the whole is required ?— Take paper and count 8. DIOPHANTINE PROBLEMS. Unlimited problems relating to square and cube numbers, right angled triangles, &c. were first and chiefly treated of by Diophantus of Alexandria, and from that circumstance they are usually named Diophantine Problems '. These problems, if not duly ordered^ will firequently bring out answers in irrational quantities 5 but with proper management this inconvenience may in many cases be avoided, and the an- swers obtained in commensurable numbers. The intricate nature and almost endless variety of problems of this kind, render it impossible to lay down a general rule for their solution* or to give rules for an innum^able variety of particular cases which may occur. The following rules will, per- haps, be found among the best and most generally applicable of any that have been proposed. RuLB I. Substitute one or more letters fix* the req[aired root of the given square, cube, &c. so that, when involved, either the given number* or the highest power of the imknown quantity^ may be exterminated from the given equation. ' Diophaotnt lias been considered hf aoue writers m the mruAoK of Alge- bra; others have ascribed to him the inventioa of unUmited problenM : bat the difficult nature of the latter, and the masterly and elegant solutions he has given to most of them, plainly indicate that both opinions are erroneous. Diophantus flourished, according to some, before the Christian sra ; some place him in the seooul eantury after Christ, others in the fourth, and others in the eighth or ninth. His Arl^hmeticsp (out of which ba^e been extracted most of the curious problems of the kind at present extant,) consisted origi- nally of thirteen books, six of which, with the imperfect seventh, were pub- fished at Basil in 1575, by Xylander ; this fifagneot is the only work 00 Alge- btn, which hat detoended to us firomthv aneieiitst the TCBuuMog books luive ■ever been discovered. See f^ol, I. p.' 337. Of those who have written on, and MoocMl^y e«ltivated, the Diophantiae Algebra, the chief are» Bachet de Meseriacy Bxaacker, Bernoulli, BoonyGastle, De Billy» Euler, Fermat, Kersey, Ozanam, Frettet^ Saundenon, Vleta, and Wolfius. V^t VI. DIOPHANTINE PROBLEMS. 177 II. V, after this open^ooy the unknown quantity be of but one dimension^ reduce the equation^ and the answer will be found. III. But if the unknown quantity be still a square* cube^ ftc. substitute some new letter or letters for the root^ and proceed as before directed. IV. Repeat the operation until the unknown quantity is re* duced to one dimension ; its value will then readily be found, from whenoe the values of all the other quantities wiU likewise be known. 1. To divide a given square number into two parts^ so that each may be a square number. Analysis. Let a'^szthe ginen $quare number, «*3soiie oflhe parts f then wiU iifi'^:fiszthe other part, which, by the problem^ mutt likewise he a square* Let rx — assnthe side of the latter square, then wiU (rx— 0]*=:) r*a5*— 2 ara?+a*=tf*— x*, whence xa ^ar "3 at* -^ — -s:^^ side of the first square, and ra:— a=(-^— j— a=) as^'^a ^ dor ) ■ ^ 'szthe side of the second square; wherefore « and r«+l ^ f J r*4-ll GIMP'S ^^^_ ^V I '—j^'^ mre tiie parts re^ed ; where a and rmmf be any numbers taken at pleasure, provided rbe greater or less tthon unity '. Q,£,L ~ 4 a«i* Synthesis. First, -- — 1'*4 r*+V r*-hl =( r*-f2»*+i rr. ■ ..,1 ] _i_fl*_ lOAiCA if the first condition. Secondly, -5 — -.|« and -7— rf 0*"^ evidently both squares^ which is the second c^dtHon. Q, E, D, £xAMPLE8.— Let the square number 100 be proposed to be cMeM into two parts^ whkh will be squares. ii»4- f Mr. Bonnycastle, in his solution of the problem^ (Algebra, third Edit, p. 143.) has omitted this restriction, which is evidently necesMury ; for if r be 9i|fpoMd^ltiMici^«ttltlM;itni«falwr«ftiwfraat|Mi ::r-~r vaiusli, and tha sttetioo become nugatory. VOU II. ■$( 178 ALGJ^BRA. Past VI. Here a^sslOO, and aszio. First, a$tume rs2> then wiU ^ar 40 xsz-^ — !=( — sz)S=the sideofthejirttiquare, and ra?— a=6s=: the iide of the second ^jitare; for 8)*+d'=:(64r|-S6=)100, as was required. eo Secondly, assume r:s:3, then wiU «s(— c=)6, and rr— a=8> as htfore. 80 380 . Thirdly, assume r=:4, then a?= — , aifrfra?— a=(-— — 10=) \i55|» 6400+22500_2890O_ ^^ ^ 289 «89 ^ 150 „ 80 17 Divide 36 into two square numbers. Bere a*=36, a:^^'^ assume r=2, then xss—, and rx^a^ To divide 25 into two square numbers. Ans. 16 and 9. To divide 81 into two square numbers^ 2. To find two square numbers having a given diffieienee. Let dvsthe given difference, axbssd, whereof a y.b, and let x=iihe side of the less square, and x-^bs=the side of t?te greater ; thenwiU jr+Al*— 3?«=(a;«+2&F+6*— a?«=) 2 &r4-6«=a6; dwide this by b, and 2x+bsza, v xss ^i the side tf the less square; 2 a — If a-^b and a7-j-fcxi(— - — 1-&=)---— =*/ie side of the greater square: 2 2 , . i+I]' a'+2a5 4-&' ^. , . .' wiierefore J = =riAe greater square required. Synthesis. l*5r*<, — — I a«d i ore €tHdei>% MA ^guares^ Secondly, — I r=(— --=sa6=s)d ; itiicc 4, 4' ' 4 fy hypothesis abzs^d. Q, E. D. Example^.— To find two 3qi|«re numbeis^ whereof the greater exceeds the less by 11. PartVJ; DIOPHANTINE PBOBLEMS. W Here dsll(sll x 1)^ Ut asU> ^^l- Then — ^ — =( — — sA)6ssMe of ih^ greater square. jiPid -^I^s( — ^^=s)5=«td6 of frtHen square* Whence 6] * =36, and 5l * =25, are the squares required. To find two square numbers difiering by 6. Here d=6 (=3x2), a=3, 6=2. Then -i— =-—=«<ie of <Ae greater. Jlnd ^-^^= — =Mde of *^ less; •.• — and -— are the squares required. To find two squares, whose difference is 15. Ans. 64 and 4^. To find two squares differing by 24. 3. To find two numbers, whose sum and difference will be both squares. Let xzsaneofthe numbers, s'-^xssthe other ; then wiU their sum (x+jr*— x=) x', eoidentlff be a square number. And since (*/ — «— »jr» ) »* ^2 xs^their d^fisrence, mu st U^e - wise be a square; let itg side be aeswnedssx'^r, then wiU (x— r) »' =)x«— 2aT+r*=x»— 2x, or2xr— 2x=r», v xss , and r*}« r* . r* r» 2r-2 ^4r« — ** *"2r— 2| 2r-2 ^4r«— 8r+4. 2r-2 o» T^-.* .« _^j — +r=: — thenumbersrequiredfWhere 4r»— 12r» + 12r— 4 /4 f:ri)s.4 r ffM^ be any number greater than 2 K 36 X 45 Examples.— Let r=3, <fe«i wi/Z ir=--, and x«— x=-^ ^^c ID iO 46-1-36 81 J 9 - ,. niifwfcers sottg^e ; for ^ =— ^'^^ le' '^w^'^^*- * If 3 be snlntitated in this example for r, both numbers will come out»3 i that is, their sum will be 4, and difference ; wherefore r must not ooFy be greater thi&a 1, <a» is asserted in Bonnycastle*s A%tfbra, p. 146.) hut greatet : thanS. N 2 ISO ALGI&m. Fait VI. Let f 3s5, to find th* nvmbcn. 4. To divide a givcB nuooberj. which is the sum of two known squares, into two other squares. Let a' +b'ss the number given, rx-^aszihe tide of ihe first required square, sx—b^the side of the second, where r ^s. Then will rj;— al * 4- «i— 6^ * = (f*J?*— 2 arj:+ a*-f- *• x* —2 bsx — 2ar+*2 6#^=o, or r»+*».a?*=:2ar+2 6«.x; ••• dividing ijf x. 2.ar+6< tt7C ^at>c r* -|-«* j:=2 ar+2 6*. '/ x=— ^ : consequently r«— a= — riri a^stde of the first square, and sx-^bsz — j— ^b=zside of the second. 42 Examples. — ^Let a=6> 6=:4> rzsS, <=:3; (A^ loiU ^»7^> 108 « ^ 58 fx— a=--— , and w:— o=— -. 17 1/ Let asz4, b=sS, r=:2, and sszl, be given. 6. To find two aumbeiB, of whieh the sum is equal td the square of the least. Ans. 6 and S. 6. To divide the nnmber Sa into two partst^iudi that their product IwiU be a square Munfaer. Ans. 27 smd 3. 7. To (fivide the number 129 into two parts, the difference of which will be a square number. Ans, 105 and 24« 8. What two numbers are those, whose product added to the sum of their squares, will make a square ? Ans. 5 and 3. 9« To find two squares, such that their sum added to their \S 1 product may likewise make a square. Ans, — and ---. If 8f 10« To find two mimbeis, one of which being taken from their product, the remainder will be a cube. Ans, 3 and 108. 11. To find two numbers^ such that either of them being ad- ded to the square of the other, the sum will be a square. An-' 16 .43 ^er-and^. . 12. To find three numbens, such that their su^xif an4 likewise the aim of every two of them, mil eaeh be a J<piare numbinr. Ans, 42, 684, and 22. PART VII. ALGEBRA. INFINITE SERIES •. 1. A SERIfiS is a ntak of quantities, which usually proceed according to some given law, increasing or decreasing sucoea- sively; the sin|de quantities winch constitute the sories are caOad its terms. 9. An increasing or diverging series is that in which tha tanna suiicesaiTBly incraaae* €t$ I, 8, S, 4, isc a-f-3 a-f 7 a^ 3re. S. A decieasiiig or conveigii^' aeries is that in which tba ttnoa sttceeasiveljF decrease, as d> 3^ 1, 4c. lOa^^Ja^^ a, Use. * The doctrine and application of infinite series, justly considered as the greatest improvements in analysis which modern times can boast, were mtrodneed about the year 166»8, by Nicholas Mercator, who is supposed to have taken the first bint of such a method from Dr. Wallis^s Arithmetie of Inteitw; bat it waa tfce genius oi Ktntan that first gave it a body and fofm. The principal use of infinite serie% is to approximate to the valoet and sums of such fractional and radical quantities, as cannot be determined by any finite ezpreuions ; to find the fluents of fluxions, and thence the length and quadrature of curves, &c. Its application to astronomy and physics is very ex- tensive, and has supplied the means whereby the modem improvements in those sciences have been made. The intricacy of this branch of science has exercised the abilities of some of the most learned mathematicians of Europe, and its usefulness has induced many to direet their chief attenlioB to iti te- provement : among those authors who have written on the sulyject, the follow- ing are the principal ; D'Alembert, Barrow, Briggs, the BemonUis, Lord Bronncker, Bonnycastle, Des Cartes, Clairant, Colson, Cotes, Gfaaier, Cob* dorcct, Dodson, Euler, Emerson, Fermat, Fagnanus, Goldbacb, Oiavesande, Gregory, Haltey, De lUdpital, Harriot, Huddens, Huygens, Horsley, Hotton, Jones, Kepler, Keill, Kirkby,. Lan#ai, De Lsfns, Leibdita, Lorgna, ManfiredV^ Monmort, De Moivre, Maclaurin, Montano, Nichole, Newton, Oughtred, Ric- catl, RegnaM, ftranderson, Stusius, Sterling, Stuart, Simpson, Taylor, Varig- nbn^ VioUy WaUis, Waring, fto. &«• N 3 183 ALGEBKA. PaxtVII. 4. A neatnd serin is tliat in whidi the terms neither increase nor decrease^ as I, 1, 1, 1^ Sgc. a+a4»a+a« 4rc. . 5. An arithmetical series is that in which the terms^incveaae or decrease hy an equal difference, a$ I, S^ 5> 7» 4rc. 9, 6, 3, 0, 8(C, 11+2 a+3 a, lire. 6. A geometrical series is that in which the terms increase hy constant multiplication, or decrease by constant division, oi h 3 3, 9, 27, 3fc. 12, e, 3,~, *c. a+3tf+4«+8tf, *c. 7. An infinite series b that in which the terms are supposed to be continued without end ; or such a series, as from the nature of the law of increase or decrease of its terms requires an infi- nite number of terms to e^qiress it. 8. On the contrary, a series which can i>e completely ex« pressed by a finite number of terms, is called a finite or termi« nate series. 9. Infinite series usuaUy arise fitim the division of the name- rator by the denominator of such inctions as do not give a terminate quotient, or by extracting the rootof a surd quantity. 10. To reduce fractions to inJinUe series. Rule I. Divide the numerator by the 'denominator, until a sufiicient number of terms in the quotient be obtained to shew the law of the series. II. Having discovered the law of continuation, the series may be carried on to any length, without the necessity of forther division. 1, Reduce -—— - to an infinite series \ l+« ^ If » be aa integer, theo wiU 1. — j7-=sa»-- » + a»-- «*+«■-- S6« + ,&c. to ^. *■--•», which aerie* e¥i- dentiy termiiiatet. 2. "^^^ ^tf"- «—*■- 86+ «"—sft2-,&c which termiBttes in-4"- », when n is an even number, bat goes on inMnitelf when n is odd. 3. r ~«* — * "-"«■ -- »ft+ a" 7- **» — , Sbc, which series terminates tn +b'^ ^i, when n is an odd namber, bnt goes on indefinitely when n is n«i». PAtT Vn. INFINITE SERIES. 183 Opbbatiom'. I + x) 1 * (1 — *+ «*— ap* + , 4c. t}^ series required. ^•4"^ Expkttudion* .^x— j:* . This operation 18 similar to those in Art. 50. .^X—'X* ^'^ ^* ^^* ^ '^ ^* unnecessary to proceed . ■• farther in the work, since we can readily X discover the law by which the terms of the x'+Jc' quotient proceed, vis. by constantly mnlti- ^__ 5 plying by x, and making the terms alter* nately + and — ; knowing this, we may oon- — 3r — J* tinue the quotient to any length we please, X* ^c.^^^^^ troubling ourselves with the work 2. Reduce to an infinite series. X— « Operation. d CZ HZ* ■ QZ* ar— «) a * ( h— . + h— —+* *c. the <eri€| required. az a X ~ ExplmuaUm. , Here the law of continuation is mani- ^_f!f^ fest, the signs being all +, and each X ' X* term arises by multiplying the nume- • , • rator of the term immediately preceding ^ it by z^ and its denoipinator by «;. X' az* az* az* Id*" fu* az* X* ' X* az* X* 8fc* 4. The difference a* —6* is not measured by the sum ai-b, Hencey first, the difference of th§ nth powers of any two numbers is mea- sured by the difference of the numbers, whether f» be even or odd. Secondly, it is measured by the tmn of the numbers, when n is even, bu^ not when n is odd. Thirdly, the ntm of the nth powers is measured by ^he «mm of the numbers when n is odd, but not when n is even. In each of the quotients which <er- mmniCf the number of terms is equal to the index ji. See an ingenious appli- cation of these condnsioiu in the Rar. Mr. Bridga's Loetunt on Alg^a^ p. 248. n4 11. When any qaantity is common tommftmm, the seriM may be simplified by dividiiig;eYery term by that ijuantity^ putting the quotients under the vinculum, and placing that qoanti^ ^ before the vinculum, with the sign x between. Thus, in the above series — is canmum toaU the temu, mid dividing hif —, ihe qwtiemt tf 1+— +^-f— +,*c.«Aicfcmioti. I emi put under the vincuium and amnected mlh thedioiew— ha the a ^ z z* z' sign X, the series becomes — x l-f— H — + — ^, Sfc. wHxch is a X X X* X* simpler form than that in the example. 3. Reduce- to an infinite series. A*.l+x+««+*»-f ,*c. X "*" X 4. Reduce to an infinite series. Jns. zH 1- 1 — fl— z n a* a* 5. Let -— be converted into an infinite series. Jns.—-^ *+« X az az' az' ^ a z z» z» p+-:^-i;r+'*«-o^-xl--+---+,*c. See ex.9. - a' ijt "Jf+6 ^«roed into an infinite series. Ans. — x :r 6 6* 6» 7. Reduce — , and likewise its equal , to infinite series, 3 »+l 3 10' IpO^ 1000^ 10000 1111 10 iol* idp lot* 1111 1 111 Ana II i» -I- I t ' . Ac. ac— -v 1 m< 4...,——^ ■ I I II U II ■■ 11 » I. I 12. To reduce compoMfid quadratic surds tg infinite^ series. Ruu. Sxtraet the square root, (Art. 57. Fart 3. VoL 1.) attd continue the work until the law of the series be discoveied j after Fait ¥11. INFINITE SEBIES. 18$ ^hich the root may be carried to any lengthy as in tlie preceding rule^ and it will be the series required. Examples.—!. Convert a«-h««|+ to an infinite series. Opbration. ExpUauftum, 2 ' ^ The lawoCcontiniiation it not *^"q]/ * obviou» in this example, bn^ the f eries may be made tome* 2' -L ^ what more simple by dividtng 4«* all the tcrmi after the first b| .9 2* «* , z* -—, it win then become jf^ ^ — ) 2* I-t: 4x« ar* 64r« 8x^""64Jc« 3. Let ^««— jf* be converted mto an infinite series. Ans. a— ^""8a» 16a* ""' *^' b b' 3. Change v^* + ^ into an infinite series. Ans, a-f- 2a Sa* 4* SacjMress 1 +2e\-l- in an infinite series. IS. SIR ISAAC NEWTON'S BINOMIAL THEOREM *. For readily Jindir^ the pomert and roots of binomial quantities. Rule I. Let P=the first term of any given binomial^ <?= the quotient arising firom the second term being divided * This theorem was first discovered by Sir I. Newton in \&S9, and sent (in the above form) in a letter dated Jotte 13tb, 1G76, to Mr. Oldenbnigh, at that time Secretary of the Royal Society, In order that it might be comnmiHcated to M. Leibnitz. As early as the beginning of tho l6th centory, Stifelins and elbcn knew bow to determine ti^e integral powers of a biooisial» not menly by continued moltiplication of the root, but also by means of a table, which Stifellns bad formed by addition, wlierein were arranged the coefficients of the termtol any power within the limits of the table. Victa seems also to have 186 r AUSmSA. PamtVII. by the fixst; then will PQ=the second term. Let ^sthe in- n dex of the iNiwer CMT nxit ixqiiii^d to be found, viz. m Qiidentood the law of tlie coefidoits, but the method of gtoentiog them soc cessivelj one from another, was fixtt taught by Mr. Henry Briggs, Savitian ProfetMir of Geometry at Oxford, about the year 1000 : thns the theorem as far as it relates to powen, appears to hare been complete, wanth^ oaly the algebraic form ; this Newton gave it, and likewise extended its appUcation and use to the extractioa of roots of every description, by infinite series, which probably nerer was thonght of before his time. The theorem was obtained at first by induction, and for some time no demonstration of it appaan to hare been attempted $ several mathematicians have however since given denon* ftrations, of which the following is perhaps the most simple. Let I+d««l+«r+f««+«r*+«jr* + ,&c.l r — i- , . y each to II +1 terms. i+y)"«i+/»y+«y»+ry*-i-*r*+»*c J . Then by subtraction l + jr/« — 1+^ ■ — i».jr— y + ^.jr*— y'-f. rje» — ya + , &c to « terms ; wherefore 1+jr— l+jf x—\ > that is^ (by actual division ; see the preceding note,) I-f 4:]»~» + l+y.l+jr> - «+ ,&c (to « terms] 8j»-f- f!]r7ir+ rJTT^Ty* + #jr* +*'y-h*y2,+y* +9 9tcton terms. Let jr«y, then n,f+x\^ — * »p+ Sq*+3rx' + 4mx* + , Ac. t o n terms, whence j^r+3»« i»+2<jw+3r*» + 4*r*+, &cx l"+i ^p + 2qx+3rx» + 4sx»+, StCl >+2j+j»jr+3r+2fjr»+4#+3rjr»+,&c. (4). Butbe<sanse l+«|*«i^ p»-^qx' +<"** -h, &c. by the above assumption, therefore fi.l~^fjr)*ssis^ fl^ + mqx * +nn F* + , &c. (S) wherefore the two series ^ and j9 (being each equal to nA-^x) •) are equal to one another^ and consequently the coeflicients •f the same powers of x will be equal ; that is, 1. /»*», $. gj+^»jy, or 2f +»»»•, V 2t=n*->^»nM^, ind ,«l!iZi " « « « - — :r *— 3.y «•»— 1JI-.2 3. 3 r + 2 y =r wjr, or 3 r=«-2.y, •.• r= -j- = — ; &c, &c &c. -- r— *-s »4«— I 11.11.^ l.|t^3 Hence i^jr^-^Bl +jMr+-j— .*• +. ^ ^\+,&c(C) Now since «+»=.« x 1 + V' '•' «+^"=«» >< 1+-^! -(by subatitnting ^ *" a h b «.«— 16' ~ for jr in the series C) a'X l + ».— + -5^.-7+, &c. =««+»ui«»-'i+ Past Vlf . INFINITE SERIES. IBT Thtor, n=it8 denominator 5 then P+PQ]^ will expreM the^vea binomial with the index of the required power or root plaoei over it. II. Let each of the letters A, B, C, D, ^. represent theiraltte of the term in a series^ which immediately precedes the term in which that letter stands. III. Then will the root or power of the binomial P+ PQl? be expressed by the following series, viz. PIt + ^ ^Q-^ -5 — BQ TO— 2n ^^ III— 3n ^^ + -T— - CQ+ — — DQ+, *c. *>n 4a IV. If the terms and index of any binomial, with their proper signs, be substituted respectively for those in the above general form, then will {he series which arises express the power or root required. ExAMPLBs.— 1. To extract the square root of (fi-^g^ in an infinite series. z» 1 Here Pssa\ Q= , and (since -— %$ the index of the tquare root) i»=l, n=r2 j then P+PQtfssfl*— **1*> «wJ P|v=(^=) a^the first term A. . TO ^^ 1 ^ «« 1 2» fl2* . «• n ^2 a* 8 o» 2a* 2a <A« second term B. TO— « «^ .1—2 „ «• 1 z* 2* . + Tr*'^=<— ^*^-?=— ^-«5'^"-^=>- - — zsthe thkrd term C:. 8 a' ***"" ut* —•*• + , &o. in which, il?i.» be tabttitated for «% ^ for — , and i^, -B, C, &c. for the preceding teimt, the wries will become if »» + — .AQ-^ -^ — J?P+ — = — . CQ+ , Ike. at above. Jf the index - . be a positive whole number, the series will terminaie at the " + IM term ; bat if it be negative, or fractional, the series will not termi* nate : all which is maailest from the above cnmptes. Itt ALGXBKA. PauVII. + — ^ C(}s=:(-^-xCx =— S-><-.S-7X 7=)- — — -sslAe/oar^A ierm D. ^--^--^^^the fifth term £. 5n ^10 a* 10 ISSa? a' 72»o "" ^>^ ^ =^^ *"^^ *«»^ ^« S56ii» ifC. 8(Cm Wherefore the square root of the gieen binomial, or -^ — -^, z* X* z* hz* 7z^ fl«— z«|T=sa— .rrr-^t *c. as required^ ' 2a 8a» 16a* 128 a' 256 a» ^ 2. Find o+Hf in an infinite aeries. Here Pssa, Q=— , m=S, n=5, oiui P+JPQJv =«+3t. a P)^=s«t the first term of the series A. H ifQ=(---x-^X — ss—xof X — =^) — 7 *«e second «er» ^. n 5 a 5 a 5|jr + -- — BQ=(-— • xBx— =— — X— ^ X— =) ths ^ 2n Mo a 5 5<jf a ' 35^ third term .• C -f — — CQ=:(— — xCx— =--^X rX— =) Sn 15 a 15 25(fi' ^ ' rt« fourth term D. 125 a^ -^ w-3n^_ 3-15 _ 6 3 _ 76* 6 ^ 4» ^20 a 5 i25aiJL a < — — -- the fifth term E, €^ba^ ^ ., , 3 6 3 6* 76* 216* *c. 4c. Wherefore a+6»3-=tfr+ — --f - 5«* 26rf 12607' — + 6fl5av « / r. -, ^ V 3 36 3 6« 7 6» 216* +,4c,(icfcicA6yifr^9.)=fffxl+ — +-^-- — — — +, ^ ''^ / '^ "Tg^ 25a*^125a» 625a*^ 4c. P**» Vfl. INHNITB SERIES. 1S9 3. To find the value of ' in an Infinite series, and then multiptt **«. retuUing teria by y* ; wherefore in the pretent can P=y», Qa=fl, m=-l, nsS, and j5+p^^_, y <erifi , ^^ 'y y the 3rd *en» /• . C. the 4th term • , j}^ +—7-— i)Q=( — -— xDx— S3— ^X— - — X— =) 4» ^8 y» 8 16y» y' ^ ------ <^ 5<A <€rm R. «c. *!!. 2%a «ri« ffua^ipiied 4y y*, according to what was pre* Vy +^ '^ y 2y'^8y« 16y'^128y» 4. To invoh^ 1«, or hi equal ll+l, to theciibe. 1 " m Here Ps=Il, Q= j^, i»=3, n=l 5 f^, as te/ore, P+P(SF (+yXl3Slx^==)+863(+|.xS<J8x^=s)+3S(+lx r ^X— ta;)+l, where {nnee the oo^fit^m^of^ next term wdl heo) the eeries mu»t emdmtfy terminate. VFkerefore cotketmg the «tettt ierme, (1331*f*dfi8-f 13-hls) 17«8 iitbewkei^ 18, a$ wag required. 190 ALGEBRA.. :. FAiTVa 5. Find the value of x+p -r in an infinite series. Ans. xt+ Sjtt 9a4 81xT 6. To find r- — in an infinite series. Jns. -^x c c* c 7. Find ^a*+6 in an infinite series. Ans, 04-5——^^^ + 6' •, *c. 16a* 8. Ettract the 5th root of 2488SS by infinite series. Ans. 12/ 9. Find ==ra ^ infinite series. Ans.-^-^ — 2I4.-? -i jf-jTy)* "^ X' X* JC* OE* r» 10. Ilnd in an infinite series. 1 1 . Tb find * ^x* ^z* in an infinite series. \% Find y x y — 1;] ^ in an infinite series. 14. A series being given, to find the several orders of differences, RuLB I. Subtract the first term from the second^ the second from the thirds the third JErom the fourth, and so on-; the seve- ral remainders will constitute a new series, called the first order of differences* II. In this new series, take the first term from the second, the second from the third, &c. as before, and the remaindecs will form another new series, called the second order of differ^ ences. III. Proceed in the same manner for the third, fourth, fifth, Sfc. orders, until either the difierences become O, or the work be Carried as &r as is thought necessary *. * Let o, b, c, d, f, S^e, be the terms of a given series, then if JD^tbe first term of the «ith onlerof.diffiBr«iiGes, the foliowiiig theorem will Exhibit the vaJae 9i jD:riZf ±u-{-nb±n,'^A!+n.—n—M±n,'^.—,^.€'h,^c. (to n-f. 1 terms) »/>, where the upper tfifos aost.be tihiii when • it «a OTeii^ number, and the lower signs when ft is odd. Pax* Vn. INHNITE $BBI£S. 191 Examples.— 1. Given the aeries i, 4> 8, 13^ 19« 26^ &c. Xo find the several orders of differences. Tkui I, 4, S, 13^ 19j 26, ^c. the given seriei. Then . . . 3j 4, 6, 6, 7» ^c. the first differeneee. And \» I, 1> Ij ^c. the second differencee, AUo 0, 0, O, iicthe third differences. where the work evidently must termtnaie. 9. Given the series I, 4, S, 16, S% 64, 19B, &c. to find the several orders of di^renoes. J9ere 1, 4, 8, 16, 32, 64, 128, 4c. given series. And ... 3, 4, 8, 16, 39, 64, ftc. Ut diff. 1, 4, 8, 16, 3«, *c. gnddi/. 3, 4, 8, 16, 4rc.3rddj^. 1, 4, 8, SfC.4thdiff. 3, 4, 8(C.6thdif. 1, «c. 6MdtJ. «c. 3. Find the several orders of differences in the series li 3, 3, 4, &c. Ans. First differences 1, 1, 1^ 1, Sfc. Second diff. 0, 0, 0, *c. . 4. To find the several orders of differences in the series 1, 4, 9, 16, 26| &c. Ans. First differences 3, 5, 7> 9, 4rc. iSecond. 8, 2, 2, *c. I%trd 0, 0, *c. 5. Required the orders of difierences in the series 1, 8, 97, 64; 125, &c. 6. Given 1, 6, 20, 60, 105, &c. to find the several orders of dijflferences. 7. Given the series 1, 3, 7s 13» 21, &c. to find the third and fixnth orders of differences. 15. To find any term of a given $erie$. RvLS I. Let a, b, c, d, e, &c. be the given series ; d^d^SdV", ^, &c« respectively, the first term of the first, second, third, fourth, &c. order of differences, as found by the preceding arti* cle; nsthe number denoting the place of the term required. If the dilferenccf be rery gnat, the logarithms «f the qnantttict may b*. used, the dUTereiicefl of which will be much smaller than those of the quantities tlkmsehres; and at the close of the operation the natural number answeribf to the logazitbmical resnlt will be the auwtr. See JEmsTM*'* JDigtrtnixai Mttkod,pTop. 1. Wt AL6BBAA. PaktVIT. .«'"+— j—.-^,-j-.-—^+lrc.asto the «* tenn leqmred •. £xAMPLEs.-^l. To find the 10th term of the series 8, S, 9, U, 90, he. Here {Art. 12.) % 5, 9, 14, 20, *c. ««rief . 3, 4, 5, 6, ^c. IH d^. 1, ], 1, isc^rnddiff. O, 0, Ssc,3rdd^. Where <P=3, cP'=l, d"»=:0, olfo a=2, nsslOj vAerefore - . ""-^ ^ . »— ^ «— 2 ^. ^^ . 10—1 ^ 10—1 10—2 Iss) 2+27+36=:65slAe 10t& lerm reqiared. 2. To find the 20th term of the series % 6, 12, 20, 30, Bsc. Here a=2, »=20; and Art, 12. 2, 6, 12, 20, 30, «c. MTter. 4, 6, 8, 10, *c. Ill diff. % % 2, Ac 2iid di^. or d' =4, d" 3=2) wkemse +342=r 420= the 90th term required. S. Required the 5th term of the series 1, 3, 6, 10, &c. ^tu. 15. 4. To find the 10th term of the series 1, 4, 8, 13, 10, Ac Ans.6^ 5. To find the 14th term of the scries 3, 7, 1«> la 25, ftxu Ans. 133. 6. Required the 20th term of the series 1, 8, 27. 64, 125. r &c. ^i». sooa 7. To find the 60th term of 1, 4, 8, 13, 19, &c. 8. To find the 10th term of 3, f, 12, 18, 26, &e. 16. If the succeeding terms of a given series be at an wHts distance from each other, any intermetUate term may be found by mterpolaiUm, asfaUows. • For ths ioTMtifitioB <f this twkt, m JEmenot^* DjftnMlmf MttMtf Part VII. INFINITE SERIES. 193 RvLE I. Let y be the term to be interpolated^ x its distance from the beginning of the series, d*, d", d»", dS &c. the first terms of the several orders of d]£ferences. II. Then wina4-JdHj.^^.d"-har. ^T'V^'r^ .d"'-f*.^^> — ^- . -7- .d^+ *c,=y, the term required '. Examples. — 1. Given the logarithms of 105> 106, 107, 108, and IQ9, to find the logarithm of 107.5. Stries. Logarithms. XH diff, 2nd diff, Zrddiff. Mhdiff. 105 0211893 .„^^ 106 ... . 0253059 lii?^ -387 « lor . . . . 0293838 ^^ -379 ""^ -0. 108 ... . 0334238 ^^^ —373 ""^ 10& . . . . 0374265 **^^' 5 Here a?5= (107.5—105=2.5) -^=iAe distance of the term y, o=.0211893, d»=41166, d" = — 387, dM» = -.8, d«'=-.2. iP "~ 1 X""^! wT— 2 jC— 1 rA€» y=a-f«d*-f-a?.-— -.d^+x.— — -. .d"»+jp. . iS « 3 2 X— 2x— 3^5, , 5 _, 5 3 ,.. 5 3 1 5 3 1 1 -. B _,, 15^.. 6 _,,,, 5 ^ ■2 ^T^T>< -T^ ^'=^+-2 ^'+T^" + i6^"^-l2s^'== j0211893+|-x41166+~X-387+^X-8-^X.-2 = ) 0211893+102915-725-2.5 -|-.078=.031407128,*^eZo^arU^w required, 2. Given the logarithmic sines of 3® 4\ 3° 5', S^ 6^ 3° 7\ and 3<> 8S to find the sine of 3° 6» IS^*. Series. Logarithms, 1st diff'. 2nd diff. Zrddiff. 3M».... 8.7283366 g^.,^ 3 5 .... 8.7306882 '^J'J^ -126 3 6.... 8.7330272 q^^^" -127 t . 3 7 .... 8.7353535 ,^^*?r -123 "^^ 3 8.... 8.7376675 ^^^ Herexsz(S^ 6^ 15"— 30 4»=a2oi5»=)-j-=fAedi«fa«ceo/</ie terwiy, to be interpolated ; a=8.7283366, d'=23516, d»i = — 126, ' This rule is investigated in Eoierson's Differential Method, prop. 5, VOL. 11^ O 194 ALGEBBA. Pakt TIL ii«"=l, and y=fl+xJ« 4-^.^.4** -h*.^^.^^^"=(«+~ 3 2 3 4 45 15 ^' +^' +T^*"=)8 7«8a3W •♦-.O05W11-.O0OO1771W5 + .0000000117=8.73300999996^ the log. sme regvirvdL 3. GiTen the series —-, — p --, --, --, to find line term which 50 51 5» 53 54 stancb in the middle, between rr and --. .^nt. •-*-• 52 53 105 4. Given the Icgvithmic sines of V O', V V, 1» 2', and V S\ to find the logarithmic sine of 1^ i> 40>^ 4ns. 8.2537533. 6. Given the series — , —-, -—-, -—, -—, &c. to find the nuddk 23450 term between — and — . 5 6 17. If ihefcrit differences of a series of eqniMffkrent terms he snuUl, any intermediate term may h^fownd by interpolation, as follows. RuLK 1. Let a, b, c, d, e, &a repres^t th^ given series, and fissthe number of terms given. II. Then will a-^nb+n.—^^.c-^n.—-—. .d+n.— r— .— -— o 2 2 3 2 3 .——.«+, &C.SO, fipom whence, by transposition, &c. any re-. ijuired term may be obtained i. Examples.— 1. Given the square root of 10, 11, 12, 13, and 15, to find the square root of 14. Here ns5, and e is the term required. a=(Vl0=)3.1622776 fc=(^U=)3^166248 c=(^12=r)3.46410l6 d=(v^l3=)3.6055512. /=(Vl5=)3.8729833 And since n=s5, the series must be continued to 6 terms. ^, . , n— 1 w— In— 2 , n^ln— 2 Therefore a^nb-jrfi* .c-^n. ■ . M4-n.^ • — •^ ^ S323 «— 3 9—1 n— 2 n— 3n— 4 - 4 2 3 4 5 -^ f For the investigation of tbb rule, sec Emerton's Difftreniial Method^ prep* €• Pa£t VII. INFINITE SERIES. 195 Whence, hy trampositian, in order to find e, we thall have n— 1 »-2 n-3 , n— 1 , n-in— «^ «.-~^.-^.— j-.e=: — a + n6— n.-^.c + n.— - -3— •<* + «• —^ r — . -— — . —r-'fi t"^ t« numbers becomet 5 c= —3.1622776 S3 4 5 + 5 X S.3166S48— 10 X 3.46410164-10 x 3.6055513+3.6729833 = 56.5 1 16193 -37.8032936= 18.7083257, (wd c= i?^^5???5Z, 5 3.74166514=^^6 root, nearly, 2. Given the square roots of 37> S8> 39, 41, and 42, to find the square root of 40. Am, 6.32455532. 3. Given the cube roots of 45, 46, 47> 48, and 49> to find the cube root of 50. Ans, 3.684033. 4. Given the logarithms of 108, 109, 110, 111, 112, and 114, to find the logarithm of 1 13. Am. 2.0530784. 18. To revert a given series. When the powers of an unknown quantity are contained in the terms of a series, the finding the value of the unknown quantity in aootiier series, which involves the powers of the quantity to which the given series is equal, and known quanti- ties only, is ddled reverting the series ^. ' Rule I. Assume a series for the value of the unknown quan- tity, of the same form with the series which is required to be re- verted. II. Substitute this series and its powers, for the unknown quantity suid its powers, in the given series. III. Make the resulting terms equal to the corresponding terms of the given series, whence the values of the assumed co- efficients will be obtained. Examples.— I. Let aa?+fc:c*-|-ca?^ + da?* + , &c.=2 be given, to find the value of x in terms of z and known quantities. ^ Various methods of rerersion may be seen, as giren by Demoivre, io the Philosophical Transactions, No. 240. in Maclaorin's Algebra, p.263,&c. Col- ton's Comment on Newton's Fluxions, p. 219; Uorsley's Ed. of Newtoo's Worisa, vol. I. p. 291, &c. Stuart's ExpUaalion of Newton's Analysis, p. 455. Simpson's Fluxions, &c. &c. O 2 J96 ALGEBRA. Past VU. Lei ^^x, them U it piam tkai tf 3^ amd U9 pamten he 99hUi' iutedinthegwemteriafarxoMdUsfomen, the mOees rfzwnU he n,2n,Sn, 4m, isc. amd 1 -, whemee «s=l, amd the diferauxt ofihete imdkes are O, I, % 3, 4, 4rc. JFberefore the mdke^oftie serieg to he astmmed, must hace the tame differemces; let therrfare thisserie»heJz'^Bz*^&-^nz^'^,tse.=x. Jmd if tkit eeria be mvohed, amd substituted for the several powen of x,im thegivem series, U will become aJz+aB2!^-^aC3^-^aDt*+, tec. * -\'bJ^7^'\'^bAB7?'^^bACi^-\',ke. * * * + 6B«r*+, ftc >=rz. * * * + d^t*^,ke. Whence, by equating the terms which comtaim Uke powers ofz. tte obtain {aAzt=z, or)A=. — ; (aB;^-f 6.A;*so,wAaice)B=3( — bA* b =) ^,(aCz'+26JBz'+c^z»=o; whemee) C=(— a a' ' / \ ^bJB+cjP ^aP^ac ^ ^ ^bAC^rbB^-k-^cA^B-k-dJ^ , =)— ^r- J ^=(-— ;; =) habc^blP'-'C^d ^ ^kc. and consequently xsi^Az+Bsfi-^Cfi+ySse, ,2 bz^ ^b^^ac , Sfc'— 5a^+a'il =) T-H r — ^ jB*+, 9sc. the senes a (^ a* a^ required. This oDDclusion forms a general theorem for every similar se- ries^ involvings the like powers of the unknown quantity. 2. Let the scries x—af2_^jj3^jj*^^ ^.==z, be pfoposed for re- version. Her^ az=tl, 5=— I^ c=l^ d==— I, 4rc; tto^ rofoes 6dii^ substituted in the theorem derived from the preceding example, we thence obtain x=2^z* +a^'+z*-f•, 8(c, the answer required. X* X' X* 3, Xet X — ^'\'^ T-+> &c.=y, be given for reversion- 's o 4 Substituting as before, we have a^l, 6=:~~-j ^^T' ^'''^^ it 3 s — 7-> 4c. These values being substituted, we shall have x=: jf+ 4 y> «* «* ^+^+|--f, SfCfrom which if y be given, and sufficiently small for the series to approximate, the value of x wiU be known. Pa«t VII. INFINITE SERIES. 197 Let 2"=x, then, if z he transposed, the indicis will be I, nm^ nm-^np, nm-^^np, nm+Snp, *c. where, if the twe least, 1 and nm, be made equal to each other, we shall have fi= — : and the m differences are -C., -£, -X, -£, ^, The series therefore to be m m m m *' I l+p l+2p l+Sp assumed for xisAzln-i-Bz m ^Cz m ^Dz^^nT +^ ^c.=ztf Mi* series being involved, and the like terms of bath compared as before, we have ^=1, B=-.l, c^l-^m^^pMb^o.mc ^ ^^ ^"W^~"^"W »» 9mJ from, whence the pfllue of x being found, theorems for innumerable cases may thence be deduced. 5. Revert the series z+--;-H 1 1-. &c.=«. Ans zsix x^ x^ off 1.2.3 "^1.2^.4.5 1.2^.4.5.6.7^' *^* 6. Revert the /series aj? + &jp» + ca?»+ilr*.-f, Ac. sr^+A«»-f 19. To/jid t^ turn ofn terms of an infinite series. RuLB I.' I^t a, b, c, d, e, Slc. be the jgiven series^ .«s=the sum of « terms, and cf , d", d"', d^ &c. respectively the fi^t jterms of the several orders of differences, found by Art. 12. II. Th.u win na+n.'^.i+n.^!^.dr+n.!^.^, »— 3 ^„ n— 1 n— 2 n— 3 »— 4 ^ -_-.tf -|-».-_..-,_._«.«^.iP^^ &c.ss#, the sum of n terms of the series, as was required'. ' XliM.nil6 i$ inveitigated by Mr. Emenon, Ui bit D^ertnHai Meihod^ pmp. 3. The tOTettigations of this aod tome of the foregoing^ raies, aUhongh not ^iBcvtt^ are rather prolix, aod require too qiveh room to be admitted witbiii the compass of notes ; for this reason they are omitted. The follow- ing problems on Ihe siiB»mation of series, which afed bat a very imperfecj; specinea of timt upble biaodi, wei» taken mostly firoitti>M£ms'«Afo<A«ma<Ma/ RepMUcry^ voL I. where a great Q«mber of problems on the sabjept» with in>- O 3 19B ALGEBRA. Fart VII. Prob. 1. To find the sttm of n tenns of the series 1^ 2, d» 4, Firsts bff Art. 13. I, 2, S, 4, 5, isc. the given tenet. \, \, \, \, S(C, first differences. O, O, O, 4rc. second differences, Herea^l, d'sl, <«'»=oj thenwiUna^n!^Xd^:sz ( — : ,whichj (smce a ana d' eacA =1)= s:) 2 — - — =s«, iAe sum required. The sum of n terms of this series may likeivise be found as follows. Let 1+2+S+4+5+, Sf c. ... -f«=t Invert this series, ondii-f-ji— 1+»— 3+«— 3+»+4-h, S;c. • • • • T* 1 3Sa» ■ MiiMM ■■ M M^ta^M^aaaM •^^■MiM^^ ^M^wa^^n^ — ^dd bothseries together, and »-f 1 -|-]|+ 1-f i>-hl-ffi+ 1+n+l + ,SfC. . . . +w+l=r.2«; that is, n.n+l=:2 s, whence s=:^^ — , as before. Examples.—-!. Let the simi of 20 terms of the above series be required. Acre )is20> a»d #=3— —at — - — sllO« the answer. 2 2 2. Let the sum of 1000 terms be required. Ans. 500500. 3. Let the suiti of 12345 terms be required. Prob. 2. To find the sum of n terms of the series 1, 3, 5, 7, 9, &c. Here 1, 3, 5, 7> 9, Sfc. the given series, 2,%t2f %, Sec. . , first difference. O, 0, 0, &ic. . . second difference, 9t— 1 Wherefore a=xl, d'=2, i'z^o, and na+n.— — -.d'=(fifl+ .d'ss (since aal and dsrS) n-f-B^^^-ns) tt'szej^ lAesMm f«- quired, genlooB sdatiofis, may ^ feeen. I'be doctrine of iHArit* Series wUi probaWy nerer be comptete ; but it would reqsire a very large treatise to do anple Jtu* tke to tbe subject, evea ia Hs present ^ pAiT Vli. INHNITB SMlES. 1^ Or thus, Let 1+3+5+7+9+, *c.,. . . .+2n— 1=5. This inverted, m*»— l+8»-3+3»— 5+2«— 7+2n— 9+, *c. +1=«. TAe smi of both is 2n+2n-t-2n+2n+2n+^ ^c. . . +2ftr=2 1. Whence n terms of this sum is 2 n.n=2 «^ or <=sn% (u before. EXAM^LB^.-—!. To find the sum t)f 10 tenn^ of the above se- ries. Here nve:lO, and sts(n^va) 100, the answer. 52. To find the sum of 50 terms. Jns, 2500. 3. To find the sum of 1928 terms. PsoB. 3. Td find the sum of n terms of the series l>f squares I, 4, 9y 16, 25, &c. Here I, 4, 9, 18* 25, ftc. the series. 3, 5, 7, 9^ 9!t 1st ^. % % 2, *c 2nd diff. O, O, 4c 3rd diff. V n— 1 Whence a=:l> rfar3» d^=B2, d*'s=o, «»id na+n.— — ^+n. »— In— 2 ^, n— 1 . ^ n— In— 2 3n*— n^ -^.-—.rf ^(,+8 ».-^+g „.^._^..-_^+ - n'— 3n*H-2n .n.n+1.2n+l ,, . , -I -^ — : — ) : V — .ssf, the sum required. 3 ^ Examples. — 1. Let the sum of 30 terms of the above series be required. ^ ^ . n.n+ 1.2 n+1 30x31x61 Heren=305 wherefore — ^2— g — L-=: s9455> the answer, 2. Let the sum of 70 terms be required. Prob. 4. lb find tie sum of a tenns of the sories a+«+d+ a+2d+a+8d+, &c. «— 1 n.n— l.d Here, 6y «Ac rule, na+n.---.d=sna+ — ^ =:«, Wc n^m Or thus, Si nce #^>gfiCTtt-fa-fd+ a+2d+g-h3 d+, *c. ^ f + l-H4-l-f l + l+>4c. xai ^ *^ 1 -^0+1+2+3+4+, *c*>^d J "" ' irefcaw«Ae«umo/**c/rUo/«fte»e>+l + l + l + l+>*<^(^o o 4 800 ALGEBRA.. PiWRT VIL « terms) =n.- and the sum of (^ latter, -fO+ 1+^+3 + * 5fC, (to n terms) sz^^^^^^, (theor. 92. Jrithmetkal Progression,) -wherefore na+ -^ .(i=«j <w before. Or thus. Because a + a+d + c+9d +« + 3<'+' *c -f a4-»— l.d=«> if «d a + »d— d-f fl -f nd— 2d+ a 4-«d — 3<i+ a-f-«rf— 4d+, ^ c . . . +a=#, * ofbo^lT } ^ **+ nd— d+ 3 a+nd-^d+2 a+»d— d+2 a+?id— 4+, ^c +2a+nd— d=2«. ■ » — .— — •m^ , . ; — ; ^ 2a4-n— l.d.n ' That u, aa+nd— d.«i=?2*, or «s=( — : — •—. z=i)na-l^ ? n.n— 1 — T — Ay as before. Prob. 5. To find llie sum of n terms of the serte ]> x, 47% a?», &c. Let 1 +ar-j-j?* +a:* +, ^c. (to j:*— i)=s; mM^pfy <^w serks ky x, and x-f ar' +a?* +J?*4-, ^c. (*o 3?*)=;:^; subtracting the wp- ^" 1 per from the lower, we feaue— l+a;"=«a:— «; whence 5= -, <Ae 5ttm required. When JT is a proper fraction, the sum of the series in mfinitum may be found in the same manner. Thus l+x-\-x'-{'X^'\', ^c.=«. ^nd x+x' -\-x^ ■j-x'* +, 8iC,=zsx; whence, subtracting as be- fore, -^ Is^sx-^Si md s^ , the sum of the smes in mfinitum. Prob. 6. To find the sum of. an infinite number of terms of the circulating decimal .99*99/ &c» First, .99999, *c.=— + ,— +-^+ — ^ +, *c. 5=*, tha^ 10^ 100^ 1000^ 10000^ 1 X J • 1 1 i»ART VTL. INFINITE SERIES. SOU + 1 h , ^c. = — : subtrchct the last hut one from ike latt, 100 1000 9 andl=:( lOs S Q 5 — =) ~, or «= 1, the sum required. Hence, I' .1111, 8!C.or — 2 .2222, Ssc or — *f .3333, *c. or — .4444, fifc. or — Thesumof^ ^ .5555, fire, or — 9 2 .6666, fifc. or ~ 7 .7777, fifc. or ~ 8 .8888, 5rc. or ~ , 9 2^ 9 3" 4 >o/.9999, 5rc.=^ ^ 5 2^ 3 9 £ 9* Prob. 7* To find the sum of n terms of the series a^+er+c/V +a+2dl*+fl+3tf]«+, &c. i'trj^, 6y actually squaring the terms, we have a* =ra« o+27p==a«+2x2ad+ 4 d- a+3?l2=a*4-2x3ad-|- 9 cP a+4d]«=a*+2 x 4 ad+ 16 d^ S(C. fifc. Jff%€nce l + l + l-fl + ^c- {ton terms ) x a* -f O-hl-f^+3 + fifC. (^0 n egrm) X 2 fld 4ro-|-l4-4-f 9 + *c. (ton e6rww)xd* l + l + l + l + *c. But ^0+l+^+3 + ^c.U^„^^^^^ the sum of ] i .0+l+4+9+*c.J 1x2 «.n — 1.2n— I 1X2X3 108 AU^EBRA* pAkT vn. Whence (n.a^+n.^l.ad^'i^f'''^^^<^=) ^ 1x2x3 n— 1.2n— 1 n.a^+»— l.<ui-i .d'=x5, the ^m required. Pkob. 8. To find the sum of the infinite series H--^+-x+ « 10 First, let "7"+"«"+'x4-7r+> ^c. ad infinitumxzs^ or, which is the same, which, divided by ^, becomes or^ which is the same, that is, 1 1 1 1 p I T 3 3 4 5'^ Whence 1= ^ > a«rf therefore 5=3, <Ac «t»m required. "Prob. 9. To find the sum of n terms of the above series. 1111 1 Letz=:-+-+-+^^,8iC.to^. ^^1 1 1 1 1 1 • . ^ And z h =-;r + -:r4— r+-^+^ ^<^« *o l^«+l""2^3^4^5"^^"' n+1' Whence, subtracting the third from the first, 1 1 1111.^1 *=•7^"H"T^ 1 h > gfC. to " ^ } 1 n+1 2 6 ' 12 20 n,n+l rr,,^ . . ^ 1111 , ^ 1 That w,_-=i~.+--.+--+--+, %c. *o -=zr5 « + i 2 o 12 20 n.n+1 PAtTVH. INFINITE SiRIES. 90S This, multiplied by, 2> becomes 9fi 1 1 1 1 . . 2 =^+-Tr+— +:^+, «fc. to «+l 1 ' 3 6 '10 n^fTfl' 111 2n That is, the sum of i-\ — ■] 1 — +, S^c, to n terms = 3 6 10 ' n-fl pROB. 10. To find the sum s of the infinite series -r-+-r +-^ S 4 o 4-, &c. Let x=—, then toiZ/ x+a:*+a?*-|-x*+a;*-f , fifC.=«; Substitute =(5=)x+x'+j:^+x*+a:*+, ^c. 1— X hy actual multiplication, comes out =:x, that is, :t:=z; and there* fore, substituting x for z in the\second step, it becomes x+a:* -fx* X +r*-fx*ss— ^ — =»; in which, by restoring the value of x, we 1 — x quired. Pkob. 11. To find the sum of 1000 terms of the series 1 + 5+9+13+ 17 + , &c. Ans. 1999000. Pkob. 12. To find the sum of 20 terms of the series 1+3 + 9+27+81 +, &c. Jm. 174339220. Prob. 18. To find the smn of 12 terms of the series 4+9+ 16+25+, &c. i^TW. 1562. ; Prob. 14. To find the sum of n terms of the series c^ +a+3i^ +a+2d]'+a+35)3 + ,&c. -rfn*. «o»+ + 2 n.it-^ 1.2 «— 1.3 ad' n^^^n^+n\d^ ^6 "*" 4 • Prob. 15. To find the sum of n terms of the series 1+3+^ 7+15+31+,&c. -4iM. 2" + »— 2 + ». 1 1 Peob. 16. Required the sum of the infioite series i^^'^ "*" 8 16^* 3 804 ALGEBRA. Past VH. 13 3 Frob. 17. To find the sum of the infinite scries -• + t'^ 4 . ^ -f— +, &c. An$, 2. lo Pbob. 18. To find the sum of — f 1-~+ -- -f , &c. ad ia- 3 9 27 81 finitum. Jnt.lh Prob. 19. To find the sum of the infinite series I • 1 .2.3 «.o.4 pROB. 20. To find the sum of « tenns of the above series. , 11+1.11+2—2 Ans. — ■ - 4.n+l.n+2 1 Prob. 21. To find the sum of the infinite series , ^ ^ + 1.2.3.4 2.3.4.5^3.4.5.6^' 18 Prob. 22. To find the sum of n terms of the above series. 1 1 Am, ^^ 3.»+l.»+2.n+3 20. THE INVESTIGATION OF LOGA- RITHMS. Let there be given &^=iN, in which expression x is the loga- rithm of a'3 it is required to find the value of x^ that is^ the loga* rithm of (a"=) the number N. Let a=l+*, and ^=sl+n; then foill l + bY=s l+n, from tvhkh, extracting the y* root, we obtain 1 + 6)7=1 +«'y", v ^ * J? XX b ^ X X X b^ (Art. 11.) i+6l7=l+— .6+—. i._+_A-l. 2.- ' y y y ^ y y y 2^ T:nJl^=i+JL.„+l.i_i.^+l.i_i.±.2.^+,^e. y y y ^ y y y «^ X Here, if y be assumed indefinitely great, the quantities — , Part VII. INVESTIGATION OF LOGARITHMS. *5 — , may be considered asszo, since they will in that case be inde* y finitely^ small with respect to the numbers \, % 3, 4> ^c. '^ y ^ y y y -2, ^c. These values being substituted in the above series, we shall .i ^' X X b' X b' have {lHh6)y=r+»t'=) 1 + -.6 -^+---s~^ «fc.= l + ^ y y 2 y 3 1 1 n? 1 ft' X I ' . ' 1 y y 2 y 3 y y . n— 4-n*-f4^n*— ,*c. ,, ...... n-^« -|.4«3 -, ^c. or, 3^= ^_T^,^ ,^3_^ ^^^ =(^ st«6aWuti7ig for n and b, their equals JV— 1 and a— 1) 0-1— 4a^»+4a^'-,SfC. ci/to* o/ ^Ac iioo toiter fractions^ then the last but one will be- come X {or the log. o/ 1 + «) =-j^^«— i»' +i»' — i«* +* ^c. wAic^ imet^ w/^ n i« a tofto/e number, does not converge, and therefore is of no use; but we may obtain by means of it a series which will converge sufficiently fast for our purpose, as follows: I . I . 1.1 21. Since log. H-»=— .n-~n2+Y«'— j»*+yn'->*c. for n let — n be substituted , and the above expre ssion becomes kg. i«n=^.-n-^n«-^«'--«^--.«*-, *c. And if the lower equation be subtracted from the upper, ^ , l+« 1 the remainder is {log. 1+n — iog. 1— nss) log. YZI^^'^H- ^ N he substituted for n in this equation, and it will become j^^o ^ r I ^ — I — ■ — h, *c. that is, log. N—lof. **iyr-i^3.j?-il* 5.iv-i]» n V» ALGXBRA. Fait VII. Whence, by transposition. 2 1 which latter is a very convenient series for finding the logarithm of any whole number N^ provided N be greater than 2^ and the logarithm of N— 2 previously known. 22, Since a*=:N, it follows from the nature of logarithms, {see Vol I. P. 2. Art. 18, 37.) that x x log. a= togf. H; hui (AH. 20.) a x^log. N: wherefore log. «ssl 5 and log. — -^atlog. a^log. a^o. Wherefore, {since — =1,) log. 1=0. Having therefore the Iqga- a rithm of 1 given, we can thence find the logarithm of 3 ; for let N=iS, tJien N— 2=1, the logarithm of which is o, a» we have shewn ; wherefore, by svhstitutii^ 3 for N in the above expression, we shall have log. 3=]g^-"2-+32i+^5+> 8!e.^{log. 1=) 0. 23. Having found the logarithm of 3, we may thence find those of all the odd numbers in succession ; thus, 2 T i I Let ]\r=5i then, log. 5=^.---4-^-n[+r-:;T+> S^c + log.S. M 4 3.4' 5.4* 2 11 1 Ut N^7i then, log. y=^.~+_+_^^-|-,«rc.-f iogr.5, I* ■ ■■ Let JV=i9j tbm, log. 9=^—+ —+^+, iic.+log.7- Let N=n; then, log. li=Z.^+-J-+-i_4.. *c.+ log. 9. 24. The logarithm of the number 2 is thus found. Los. of 4 (by what has been shewn above) :=:^:rz. 1 1- — ^ -^ V ' ^ M 3 3.3^ +-, 8fc.+log.2. 5.3* ^ But log. 4=?og. 22=2xZo^. 2; therefore ^xlog, 2=-^. "^^ TTi ■*■ TTb + ' *^- + %• 2 5 whence, by transposition, . (2 X ^. Because Faut VII INVESTIGATION OP LOGARITHMS. 207 . 25. Having shewn die BieUiod of finding the kgaiithins of aU the prime numbers^ those of the composite numbeis will be rea^ dily obtained by addition only | thus, flog. 4=^0^.2+ tog. 2. log. 6:=zlog.S + log.2. log. S^log.4+log,2. log. 9=2og. 3 + /og. 3. log. lO=zlog. b-^hg. 2. hg. 12stog. 6+tof. 2. Ike. ^6. Bat before we can apply the above expressions to the ac- tual construction of logarithms, the value of the quantity M. most be determined 5 it is called the modulus^ of the system, and may be assumed equal to any number whatever : whence it is plain that (by varying the value of AT) innumerable systems of logarithms may be formed for the same scale of numbers, in eadk of which the magnitude of the logarithm- of any number tdli depend on the value of M; moreover ilf depends on the va« 1 1 loe of a, (since ikf=a— 1— —.a— ll«-f — .a— IF— , &c.) which • 3 therefore is called the hose of the system, and may be varied at pleasure. If jif=i, then win log. iv==Ar-I-4-l^-ri1*+4-^'^-^ « 3 &c. the logarithms of this system are denominated Napier's or hyperbolic logarithms. Lei N^~JP^\9^ I.^-Ol^-, 8fc.=p; iken if M he A 3 P ^ modidus, we shall have log. Nsz -^^ if Jlfssl, then toill hypu l^. Nszp ; and if this vdtue efphe suhstvtuied in the preceding etiaation, it becomes log, ^= — ■ ' — , whence also hyp. log. N^Mx log. N. 27. Hence hyperbolic logarithms are changed into others, k The name moduhu was lint gi^eo to this fiictor by Mr. Cotes, in a learned paper on the nature and constaniction of logarithms, printed in the Philoso- pkieal TramactioiUy No. 888, and afterwards in a tract entitled Logometria. The modulus is a fourth proportional to the fluxion of the number, the fluxion of the logarithm, and the number itself ; or it is the number which expresses the sQhftaageiit etf the l^gmrithmic ob iogiMtie (afenra. WB ALGEBRA. J^astVH. whote modoliB is ilf, by dividiiig the former bj M: and loga- rithms whose modulus is M^ are changed into hyperbolic loga- rithms, by multiidying the former of these by M. Lei N=za, then s'mce log. N= ^'!^ ' — , we shall have iy M htm Ijut J\r subsiitutum^ log, fl=r ^' ^' — ; hut it has heen shewn that kg. aszl, wherefore bif multiplication (aM:=l x M:=)M=^hyp. log. a. But since the value of a may he assumed at pleasure, fef a=lO 3 substitute this value for a in the above equation, and M^rzhyp. hg. 10. Logarithms derived from this assumption are usually called Briggs*s> or the Common Logarithms 9 and to construct a table of them, it is plain we must first find the hyperbolic logarithm of 10, which has been shewn to be the modulus of that system. Now log, lO=log,^xS=log, 2-hlog, by and the modulus of the system of hyperbolic logarithms is unity, or M^l. Therefore, {Art, 24.) hyp. log. 2=2x-j4-j^+g;p+» *c,= .69314718. Hyp. io^. 3=2 x—+ — -f-^+,*c.= 1.09861228. Hyp. log. 5=2x— +— 5+--+, SiC-hlog. 3=1.6094379^1. d8. Having found the hyperbolic logarithms of 2 and 5, we have from the nature of logarithms, hyp. log. I0=ihyp. log.^ -^hyp. log. 5=(.69314718+1.60943791=)2.S0258509=Af, the 2 modulus of tfie system of common logarithms; and since -rz^ = .868588964, thk quotient being substituted for its 2.30258509 equal —, will become a constant multiplier of the general series, that is, com. log. 2V^= .868588964 x-r= h _ -f +> ^-1 3.]^=1]' 5.i^=il* ^c.+to^. IST— 2j which is a general thcOTem for finding the common logarithms of all the prime number above 2; the theorem for finding the. logarithm of the number 2 being Paat VII. INVESTIGATION OF LOGARITHMS. 909 .868588964 X — H ; H l + , *c. (Art. «4.) and since theloga^ ^thms of the composite numbers are derived from those of the prime numbers by addition only^ we are now in possession of the means of constructing a complete table of these useful numbers. 29. To construct a table of common logarithms. Let A=r.86SB88964, then the above theoreni for finding the logarithm of^ wUl become --• + v^+"T&+* *'^* ^^^^^ ** ^ 3 3.3 5.3' rived the following practical rule for finding the logarithm of the number 2. Rule I. Divide the factor .868588964 by 3^ and reserve the quotient. II. Divide the reserved quotient by 9, and in like manner reserve the quoticftit | divide this last quotient by 9, and reserve the quotient ; and so on, continually dividing by 9> as long as division c^ be made. III. Set the reserved quotients in order> under one another^ and divide them respectively by the odd numbers 1, 3^ 5^ 7, 9, &c. placing the quotients one under another as before. IV. Add the last mentioned quotients together^ and the sum will be the logarithm of 2^ as was required. Examples. — 1. To find the logarithm of the number 2. Operation. 1 ) .289529654(.289529654 3) 32169962( 10723321 5) 3574440( 714888 7) 3971 60( 56737 9) 44129( 4903 11) 4903( 446 13) 546( 42 15) 60( 4 3) .868588964 9). 289529654 9) 32169962 9) 3574440 9) 397160 9) 44129 9) 4903 9) 545 9) 60 6 Ans. log, o/2 =.30l029D95 ExplanatioM. The firit (or left hand) oolomo cooUiiu the divitors 3, 9) 9, &o. the scoond contains the dividend^ and successive quotients, which arise by dividing each nnmber in it by the opposite divisor ; the third contains the divisors, Ij 3, 5, 7f Sec. In the fourth column the reserved quotients above mentioned are arranged under one another in order, each opposite its respective divisor. The fifth con- siitft of the quotients arising from the division of each of the reserved qnotieou by its proper divisor ; the sum of these latter, subjoined at the bottom^ is the logarithm required. Note. In some of the above divisions, where the reawinder is very large, the VOL, !!• P no AL6EBBA. Fabt VII. faft ifBotirat figure is afnmied gvcaCer bj out tkao it oOf hi ftfictly to be ; tim, w it iervrs «d1/ tio aake 19 for other nnaU remaiaden lett, will be fio- dnctire of 00 error of conseqaence in tbe icsalt. 2. To find the common logarithm of the number 3. Here, by assuming A as before^ ike general theorem for find- tag the common logarithms ofaU tmmbers greater than % will he- come -- — ^+ ■ -♦-■ 4-. 8fC.+log» JV— 2. In tltis ease , JVs=3, V 2^- 1^2, ^—T^ ^-,2 X 4, iV~T)*=:2 X 4 X 4, TT^^ssi 2x4x4x4, i?— D»=2x4x4x4x4, *c. SiC whence it '» plain, that the first column of divisors ^ust be 2, 4^ 4, 4, 4, &c. and the other column of divisors, in this and eveiy other case, .will be tlie odd numbers, 1, 3, 5, 7, &c. and proceeding as be- fore^ the work will stand thus : 2).8685SS964 l).434294482(.4342944a2 4) .434294482 3).108573620{ 36191207 4). 108573620 5) 27143405( 5428681 4) 27143405 7) 67S585I( 969407 4) 67a5851 9) 1696463( 188496 4) 1696463 H) 4241 16( 38556 4) 424116 13) 106029( 8156 4) 106029 15) 26507( 1767 4) 26507 ir) 6627( 389 4) 6627 19) 1657( 67 4) 1657 31) 414( 19 4) 414 23) 103 ( 4 4) 103 25) 25( 1 25 Sum .477121252 To which add (log. N'^2:si)log. l =.OO0OO0Qao The sum is tfte log, rf 3=. 477121252 In a similar manner the logarithms of the other prime num- bers are obtained, and by means of them those of the compo- site numbers, as has been already shewn. 3. To find the logarithm of 5. Ans. .698970004. 4. To find the logarithm of 7- ^^fns, .845098040. 5. To find the logarithm of 4. jhu. .602059991. 6. To find the logarithms of 8, 9, 10, 11, 12. PART VIIL GEOMETRY. HISTORICAL INTRODUCTION. fjrEOMETRY ' is the science of magnitude, or local ex- tension ; it teaches and demonstrates the properties of lines, surfaces^ solids, ratios, and proportions, in a general manner, and with the most unexceptionable strictness and preeision. Geometry, or measuring, must have been practised as an art at the commencement of society, or shortly after, when men began to build, and to mark out the limits of their respective territories. That thb art had reached a considerables degree of perfection at the time of the general deluge, can hardly be doubted from that stupendous nonumenl oi human folly, the Tower of Babel, which was begun about 115 years after* that period : Herodotus informs us, that this vast building had a squase base^ each side of which W9s a furlong in length ; Strabo affirms that its height was likewise a furlong; and Glycas says^ that the constant labour of fqr^y years was con* sumed in erectiog this unfiaished and useless fabric. The Pyramids,' Obelisks, Temples, and other public edifices with which Egypt abounded, existed prior to any authentic date of profane history : many of these had been in ruins probably * The name Gemtketry is derived from yn the earth, and fmr^ to measare. The iuYention of measaring if ascribed to the JEgyptialis by UerodotHs, Diodomsy Strabo,, and Proclas; to Mercury by others among the ancients.^ and to the Hebrews by Jo^ephns. 212 INTRODUCTION. PartVBI for ages before the earliest historians lived, who speakcrf thek magnificence as surpassing that of the most splendid struc tures in Greece ''. Can it be supposed possible, that buildings, whose magnificent remains alone were sufficient to excite the wonder and admiration of a learned and polished nation like the Greeks, could have been raised without the assistance of Geometry } The priest$ of Memphis informed Herodotus, that their king Sesostris divided the lands bordering on the Nile among his subjects, requiring that the possessor should pay an an- nual tribute proportionate to the dimensions of the land he occupied; and if the overflowing of that river occasioned any diminution, the king, on being applied to, caused the land to be measured, and claimed tribute in proportion only to what remained. " I believe," adds Herodotus, " that here Geometry took its birib, and hence it was transmitted to the Greeks." On the strength of this conjecture we frequently hear it affirmed, that ^^ Geometry derived its origin from the annual inundation of the Nile ^ but it is plain that this as- • , ^ Sevvrai inttaftces of lbi» lamf be given.' The tomb of OsymandyM, oife of their kings, is said to have been dnconunpnJy nuigni.ficc«t ; it was sumoad* ed by a circle of gold, 365 cubits in circumference, divided into as many eq^oal parts, which shewed the rising and setting of the sun for every day in the year : fhift circle was carried away by Caabyses, kis^ of Bmia, when he eoo^eMd Egypt, A.X. 525. Gognei Orig. des Loi», ^-c. T. 2. /tv. S. MoUm'* Anc HiH^ vol, /. p. 3. The fEunous Labyrinth contained 12 palaces surrounded, by 1500 rooms, adorned with innumerabk ornaments and statues of the finest parble, jind most exquisite woskamiitbip ;. tiierc were besides, 1500 tsfater> nuieous apartments, which Herodotus (who surveyed this nobla and beautifid structure) was not permitted to see, because the sepulchres of their king? were there, and likewise the sacred crocodiles and other annuals, which a nation so wise iu other reafMcts worshipped as gods : <' Who** (says the learned and pious Rollin) << can speak this without confusion, and without deploring the blindness of man !" Tbe magnificent city of Thebes, with its numerous and splendid palaces and other public edifices, which was ruined by Cambyses, is the last instance to be mentioned, although many more might be added. It extended above 23 miles, had an hundred gates; and could send oat at every gate 20,000 fighting men, and SOO chariots. PartVIIL geometry, 2l3 ^ertion deseives little credit ; for as a science, Oeometry never existed in Egypt before the time of Alexander, and as an art it must have been known there (as we have shewn above) long before the age of Sesostris; for according to tlie very pro- bable conclusions of our most accurate and best informed chronologers, Sesostris was the Egyptian king, who invaded Jerusalem, A. C. 971 ; on which occasion he is mentioned in a King^ ch« xiv. v. 25, under the name of Shishak.: now we have direct proofs, on the most unquestionable autliority, that measuring was understood by the Jews who came from Egypt, many centuries earlier than that date; see Genesis, ch. vl. V. 15, 16. Exodus, ch. xxv. xxvi. xxvii. and various other parts of the Mosaic 'history. Not to take up the reader's time with conjectures about the origin of Geometry, which at best must be vague and un-^ certltin, we hasten to inform him, that the Greeks, to whose taste and industry almost every science stands indebted, were the first people who collected the scattered principles and practices of Geometry, which .they found in JEgypt and other easte^ countries, and moulded them into a form and con- sistence. Until it passed through their masterly hands. Geometry could not by any accommodation of language be properly termed a science; but by their consummate skill and indefatigable labours, a few scanty and detached princi- ples and rules, heretofore chiefly applied to the measuring jof land, (as the name Geometry imports,) at length grew into a;Qd became the most complete and elegai^t science in the .WiOii4* We .^dore th^t benign Providence, who has repeatedly condescended to make even wicked and idolatrous nations useful instruments for promoting the execution of his merciful designs to man. Thales ^ ranks among the earliest of the Grecian philoso- « Tbalesy the &ther of the Greek philotopfayy and the first of the seTen wim ;#WB ^ GNpeece, was boxfi at Milctum, A. C. €40 ; alteT acquiriof the besrt leai^ p3 iU INTRODUCTION. PabtVUL fhttSy whe travelled into foreign comitries m quest of that knowledge which their own could not supply, A. C. 640* He became not only an able geometer, but was likewise very skilful in every branch of Mathematics and Physics, as these Sciences then stood. We are unacquainted with the parti* ieulars of his acquirements and discoveries in Geometry, but he is mentioned as bnng the first who measured the height of the pyramids at Memphis, by means of . their* shadows, and who applied the circumference of a circle to the Bieasur« ing of angles. Pythagoras'^ was-another eminent Grecian philosopher, who ing his own country aiforded, be trarelled 411 the £ast, aod returned with a mind enriched with the knowledge of Geometry, Astronomy, Natural Philoso- phy, &c. which he improved by his own skill and application. He divided the celestial sphere into five soncs ; be observed the apparent diametcT of the snn* making it half a degree ; he understood the cause and course of eclipsci » cal- culated them with accuracy, and divided the year into 365 days. He disliked taionarcby, because he considered it as little better than tyranny, to every spe- cies of which he was an avowed enemy. One evening as he walked out to contemplate the stars, be bad the misfortune to fall into a ditch, on which an old woman, who saw him, exclaimed, *< How can you possibly know what is doing in the heavens, when yuu cannot see what is even at your feet !'* He died at the Olympic Games, at the age of upwards of 90 years^ Thales was the founder of the Ionian tect, and had for his scholars some of the most eminent philosophers of antiquity, among whom are mentioned Anaxlroander, Anaxi- menes, and Pythagoras. It is uncertain whether he left any writings ; Augus- tine mentions some books on Natural Philosophy ascribed to bim ; Simplicins, some on Nautic Astrology ; Laertius, t^vo treatises on the Tropics and Equi- noxes j and Suidas, a work on Meteors, written in verse. ' P^tbagoFBB,a celebrated philosopher of Samos. He was early instructed i» music, poetry, astronomy, and gymnastic exercise, with whatever else might tend to enlighten his mind, and invigorate his body. At the age of eighteen he resolved to travel for that instruction, which the ablest philosophers oi Samos were incompetent to supply : be spent 25 years in Egypt, *here havtug ingratiated himself with the priests, he became acquainted with all the learn- ing of that country ; having travelled through Chaldea, and visited Babylon, he returned, passing through Crete, Sparta, and Peloponnesus, from whence he crossed over into Italy, and finally fixed hia residence at Crotona. Here be opened a school, which, by the fame of bis mental and personal accomplishments, was aoon crowded with popils, many of whom came from distant parts of Greece asd Ualy* Hit icbcdJurs^ wbo wwe called (be Jtaliim «crr« were fonned bj FAvrVah GEOMETRY. 815 was CBdowed with an equal tUrat for uaefol kaowledge, and employed the same means to gmtify it, A. C. 590* The 32nd and 47Ui paopeMtions of tbe fiist book of EocUd's Elements are ascribed to him ; from the latter of which be was led to determine, that the diagonal of a square is incomioensurable to its side : every person moderately acquainted with Geo- metry will adroowledge, thai the useful purposes to which these important proportions maybe applied are innumerable^ About this time, or shortly after, die following celebrated tlie fwlct Mfi/ud fnm tlie B^^fplian prieits ; •moog other aosteriticty he en- joined them a five years tUeiioe, during which they were only to hear ; after this they were allowed to propose doabts, ask questions, &c. in which they w«re permitted to say, not a lUtk in mam^ w*nU, but wuiek in «f few wardt MpcsMk. Qaery. Might not the prattling, self-sufficient young gentlemen in some of our academies, be admirably benefited by an institution of this kind? Besides the propositioos mentioned above, Pythagoras wa9 the author of the following, vie. only three rectilineal figures can fill up the space about a point ; namely, the equilateral triangle, the square, and tbe hexagon. He invented the multiplication table ; the obliquity of the ecliptic was first discovered by him ; he called the world tutfuty and asserted that it was made in musical proportion ; the ann he called tke fiertf globe of uttiiy, and maintained that the seven planets move round him in an harmonious motion at distances corresponding to the musical divisions or intervals of tbe monocbord : he taught the true aolar system, which had been asserted by Phildans of Croiona, hut being foiw gotten and lost during many ages after, was at length revived by Copernicus, and demonstrated by the illustrious Newton. The modesty of Pythagoras was not less conspicuous than his attainments ; on being addressed at a public assembly with the splendid appellation of r«f •(« wise ffMtis, he disclaimed tbe title, and requested that they would rather call him ^tXMra^$f, a lover of wisdom ; a circumstance which first gave rise to the terms phUotophy and philosopher. Some authors affirm, that Pythagoras offered 100 oxen as a sacrifice to Apollo, in gratitude for the discovery which that god enabled him to make of the 47th proposition of tbe first book of Euclid ; this is extremely improbable, as he was a firm believer in tbe doctrine of the transmigration of souls, which forbade taking away the life of any animal : nor is it much more credible that be sub- ptitnted little oxen made of flour, clay, or wax ; no, this would doubtless have been considered as an intolerable aiTront, which the meanest heathen god in tbe catalogue would disdain to put up with. The whole story is perhaps nothing better than a fiction, an ingenious sample of ancient priest-craft. p4 216 INTROireCTlON- Part VHI. problems tMfgah to be a^taled among the learned ; tmien^ff the rectification and quadcatnre of the chrde, the trisectioa of an angle, the findmg two mean proportiqnak, and the duplication of the tube ^. Some of the ancients mAveA these problems, but their solutions were either meehanical, hf approximatum, or* depended on the properties of certain curres njot considered as geonwtrical; consequendy their mcihods did not fulfil the necessary condition, requiring that these problems, which without d^)ute are elementary, should be solved by pure elementary Geometry. Some of the most eminent geometers of both ancient md modern times have engaged in this arduous undertaking, and not one among them all has succeeded : no solution of either of these famous problems, strictly and purely geometrical, has ever yet appeared. What a useful lesson does this address to the noisy advocates for the omnipotency of reason ! they may hence learn, that the reasoning powers of the human mind, although unquestionably great and excellent, have their limits, narrower perhaps than these philosophers have been accustomed or are willing to allow ; and consequently that reason, although the most noble, and distinguishing boon that Heaven has ever conferred on man, was not given him to be deified, L^t them contemplate with becoming attention the * The rectification of a'circle is the finding a right line equal to its circom* ference, and its quadrature is the finding a square equal to its area. !%€ find- ing two mean proportionals consists in this ; having two right lines given, thence to find two others, such, that the four lines will be continued proper* tionals. Tbe duplication of the cube consists in finding the side of another cubc^ which cube sl^all be in magnitude just double the former : the two latter pro- ' blems depend oq each other, ^nd form but one, known by the name of the De^ liah problem^ which \\. obtained from the following circumstance : a plague threatening to depopulate /Vthetis, the oracle of Apollo at Delpho9 was consulted^ and returned for answer, *' Double the altar and the plague shall cease." The geopieters immediately set to work to find the side of a cube double d this altar> vj^hich was likewise cubical ; but after much labour they found to their great iportification, that the solution could not be effected b^ auy of the method^ then ii> use, pAar VHI. GEOMETltT. ^17 numerous iiwannountable oUtacIes which oppose tbemselves at the very threshold of almost every department of know- le4gey and candour wiU oblige them to confess that the men- tal powers are still very imperfect, and consequemly that saperior attainments in any science ought ahvays to he accompanied with modesty, diffidence, and humility. Of those who engaged with ardour in theabove-^mentioned tlifiicult researches, Anaxagoras of Clazomene wa^ one of the eaicliest, A. C. 500; he was an excellent geometer, and com- posed a treatise expressly on the quadrature of the circle, which, according to Plutarch^ was written during hb im« prisonment at Athens. (Enopidus of Chios and Zenodorus flourished about A. C. 480; to the former are ascribed the 9tb, 11th, 1 2th, and 23d propositions of Euclid's first book of Elements. Zenodorus proved, that figures of equal areas are not necessarily contained by equal bpundacies, as some bad asserted; one only of his treatises has escaped the ravages f^ time; it has been preserved by Theon in his Commentaries, and is the earliest piece on Geometry at present extant. The school of Pythagoras produced a great number of learned geometricians : with the names of some of them we are acquainted, but scarcely any thing is known of their discoveries and improvements; as most of their writings, through the constant .mutability of human afiairs, during a long lapse of ages, have been destroyed or lost. One famous discovery in Geometry, however, remains to be noticed as originating among the disciples of Pythagoras, namely, the ingenious theory of the five regular bodies ^ f Tbey are Vikewise denominated the Platonic bodies, ^d are a« follow. 1. The THraidnfi, or regular triangular pyramid, contained by four equila- teral and equal triangular faces. 2. The Hexaedron, or cube, contained by six equal square faces. 3. The OetaSdron, contained by eight eqaal equilateral triangular facet. 4. The Dodecmidron, contained by twelve equal and regular 1 218 INTIOIIUCTION. P4«T Vin. Hffpocntes ' of Chios, A. C. 450. distiQgiusbed himaelf «8 the ficst who squared a curvilineal space ^; in hb attempts 4o aol^e tfae . oelebrated problein of doubliiig the cuhe^ he discovered^ that if two mean proportioiials between the side of a given cube and double tliat side be found, the. least of these means will be the side of the required cube ; the same IS demonstrated in Euclid 33. 1 1. but it w^s soon. discovered that tlte difficulty. Instead of being removed, was only a lit* tie disguised; for the two mean proportionals themselves could not be found by any pure geome^ical process, and the problem continues, to the present hour, to bid defiance to the mnited skill and labours of the ablest geometricians. Geometry was cultivated with the greatest attention by Plato * ; his school was a school of geometers, as appears from lientagonal £ices ; and 5« The IcosaSdren, eontained by twentf equal and equilateral triangular faces, These iKre, t<^geUier with the i^Aov^ wludi aiax be considered as a sixth, are all the regular solids that can possibly be made. The following are called mixed solids, each being compounded of two of tfte former: viz, 1. The JSsoctoSdrott^ contained by fourteen planes, Tix. six equal •quarety and eight equal and equilateral triangles. 3. The leowUdecmSdnih contained by thirty-two planes, viz. twelve equal and regular pentagons, and twenty equal and equilateral triangles. See a treatise on the Regular and Mixed Solids, by FInssas, subjoined to Bamiu^M EueUd, Ltmdu^ 1751. T%e five flCfular solids may be constructed with pasteboard, the method of dohiy which was first shewn by Albert Darer, an ingenious magistrate of Nuremberg, in his Imtitutumes Geomefrictt, Paris, 1533. See also Hawney's C&mpleie Meamrer, 9di Ed. p. 268. Bonnycastle's IfUroducHtm to Mauwtiiamg 4re. 4th Ed. p. 181. &c.' Hutton's Maik» IHciumary, vol. I. p. 215, and vol. U. p. 355. &c. r I am equally uncertain whether there be any further particulars of this geometrician in existence, and whether the above date be correct: he must not ht confounded with a learned physician of the same name, in the Island of Cos, who was much esteemed for skill and fidelity in his profession. * This curve is the lunula : if three semicircles be described on the three sides of a right angled triangle, their intersections will form two lunar spaces, the sum of which is equal to the area of Che triangle ; the proof of which de- pends on Euclid 47* 1 > 31 ! 6, and 2. 12. Proclus ascribes the lunula to (Eno- pidas. * The original name of this eminent philosopher was Aristocles, and he )feceived that of Plato from the broadness of his shoulders j be was bora at paiitvui. :^ GEcniBTinr. . 21s the fcllowifig mscriptioD which he caused to^be-fihoed oter the door; let no ohb pssschk to Bamut BBftKr WBo it UNSKiixBD IN- OBOMETRT. Likc hk {Nnedceessovs, Plato attempted the duplicatiaii of the cube ; for this purpose he contrived an 4B$tniineDt> comirting of straight roles, moving in grooves perpendicularijr to each other, by means of which he was enabled to find two mean proportionab : but the pro* Mhem about 430 |ttMB bclbn Cbrift» wd «daca|cd with Um gnmtert atteatiMi |K>th to his QU^vtol and corporeal improTemcnts ; having in his early years ac- quired considerable skill in music, painting, poetry, philosophy, gymnastic ezer^tMy Sec. he at SO jcarf old becaoie a disciple of SocntM, who stilcd him tJke Swtm 1/ the Academy, Plato» on the de^th of his beloved master, retired to Megara, where he was kindly entertained by Euclid the philosopher : from thence he passed over into Italy, where he perfected himself in natural philo- sophy oB^er Arehytas and Philolaus ; from Italy he went to Cyrene, where ho received kistmctions in geometry from Theodoras : he afterwards travelled into Egypt, where he acquired arithmetic, astronomy, and, as it is supposed, an ac- quaintance with the writings of Moses ; after visiting Persia, he returned to A- tbons* where he opened a school, and taught pbilotopby in the Academia, whonco his disciples were called Academic*, Plato afterwards made several excursions abroad, in one of which being at Syracuse, he had the misfortune to displease Dionysius, and uarrowly escaped with his life. The tyrant, however, delivered him into the hands of an envoy from liacedemon, which then was at war with Atben$, a^d he was sold for a slave to a Cyrenian merchant, who immediately liberated and sent him to Athens. The ancients thought more highly of Plato than of all their philosophers, calling him the divine Plato ; the mott wise ; Oemogtsaereds the Hmner ^ phUoBt^hersy Hfc, The orator Cicero was so en- thasiastic in his praise, that he one day exclaimed, <' err ate tnehercule malo cum Platone, quam cum istis vera sentire" The Platonic philosophy appears to be founded chiefly on the Mosaic account of the creation, &c. hence, in the early 9gcs of the .church, Platonism and Christiainity were incorporated and blended together by some of the fathers of the Eastern church ; but this union is severely and justly censured by Gisborne, Milner, and others, as extremely detrimental to the genuine spirit of Christianity. After the death of Plato, which happened A. C. 348, two of his disciples, Xenocrates and Aristotle, succeeded him : the former taught -in tlie Academy, and his disciples were called Academics ; tlie latter taught in the Lycseam, and his scholars obtained the name of Peripntetic*, from the circumstance of their receiving their instructions, not sitting, as is usual, but waUung. The works of PUto are numerous : they are all, except twelve letters, written in the form of dialogue ; the best editions are those nf Lyons, 1588. Frankfort,>/. 1602. and Deuxpontp^ 12 vol. 8to, 17 1«. 920 iNTtOBUenON. Part VIIL MSB was meekankalf and oonsequently ccmld not be admitted as a geomUrical sdntiQii of the probltiD. The circle was the only curve ifitberto admitted into Geometry, but Plato introduced into that science the theory of the conic sections, or those corves which are formed by a plane cutting a cone in various directions. The numerotn properties of these celebrated curves, and their usefulness in Geometry, soon became apparent, and excited the attention of mathematicians, who considered this branch of Geometry of a distinct and more exalted nature than that which treated ei the circle and rectilineal figures only ; and hence it obtained the name of the higher or sublime Geometry. By means of the properties of these curves, Archytas. of Tarentum *, the master of Plato, taught the method of finding two mean proportionals, and thence the duplication of the cube, A. C. 400. Menechmus accomplished the same thing about that period, or shortly after : they both effected the solution by means of the intersection of two conic sections ; a circum- stance which merits particular notice, as being the origin of the celebrated theory of geometrical locif of which so many important applications have been made by both ancient and modern geometrieians. Were it possible to describe the conic sections by one simple continued motkHi, like the circle, the above solutions would possess all the advantages of geometri* cal construction, according to the sense implied to the term by the ancients; but failing in that particular, they do not fulfil the necessary condition. The great problems we have so frequently mentioned, ^ Archytas is said to be tbe inventor of the crane and screw ; he contrived also a wooden pigeon, which could fly : the ten categories of Aristotle are a* scribed to him ; as are also several works, but none of them have docendcd to us. He was a wise legislator, and a skilful and valiant general, having o«bi- manded the army seven times without having been once defeated. He WR> at last shipwrecked and drowned in the Adriatic Sea. Part VIII. GEOMSIVr. 221 aMioogh now given up as impoiriUe to be ilolved by the proposed method, were stuped by the aneteots with iBoenant ardour; and the researdiea to which speculations of this kind gave birth^ proved a fruitful source of discoverMs in Geometry. The numerous and extensive applications of Greometry to other branches <rf knowledge, espedally to Astronomy, made a systematic arrangement of its principles and conclusions, according to their logical connexion and dependance, indis- pensable. Of those who undertook to compos Elements of Geometry, Hippocrates, Eudoxus, Leon, Thaetetus, Theu- dias, and Hermottnius, were the chief, and the usefulness of their labours in this respect was apparent ; but their treatises, of which scarcely any thing is known, were all super- seded by the Elements of Euclid ^, which have maintain- ed their supericvity ov» other systems of the kind through every succeediDg age to the present, and still hold their rank as the only classical standard of elementary Geometry. Eu- did^s Ekments, as we now have them, are comprised in fif- , teeo books, and the subjects they treat of may be arranged in three divi«<ms; of which the first includes the theory of superficies, the second that of numbers, and the third that of solids : the first four books explain and demonstrate the properties of lines, angles, and planes ; the fifth treats in a general manner of the ratios and proportions of magnitudes ; 1 Endid was one of the mott cclebratfd ipattieiiMtiGiaiisof tlie Ale«uulriaA ■chool ; be was bom at Alexandria, and taiight with great applause, A. C. 280. He wrote several works, as mentioned in the text, of which the Elements is tiie ddef. Ill -this work be availed himtelf of the labmun of those who bad gone before bim, collecting and properly arranging the principles and propositions which had already been given by others, supplying the deficiencies, and strength* ening and confirming the demonstrations. The particulars of his life, and time of liiadflBtb, are uakaolni : it it said that King PtolMiy Lagtts, on examto- l|ig tbe Elamcirts, asked htm if it was not .possible to arrive at the same oon- cluaions by a shorter method ; to which Bwclid replied, ** There is no rojra/road to -Geometry." 2Z2 INTKODUCnON. Pabt VIIl. tht rixth of the' propMrtmiB, &c; of plane figuries ; .tke seventh, e^th, and ntntb, explain and prove diefiundamenf tal properties of nmnben f the tenth contains the theory of commensurable and ineommensuraUe lines. and spaces ;. and the remaining five books unfold the doctrineof solids* The first six books^ with the eleventh and twelftli, are. all that are now usually studied ; the -modern improvements in analysis having furnished much shorter and more conveDienk methods of attaining to an adeqpute knowledge of the sub^ jects contained in the remaining books, than those given in the Elements. The Elements of Euclid furnish all thsu is necessary for determining the perimeters and areas tjf rectilineal figures^ the superficies and solid contents of bodieg contained by rectilineal planes, and for descrilHng them on paper: in them it is proved, &at a cone is equal to one*thirdof its cir^ cutnscriblng cylinder ; that the solid content of a cjplhkder is found by multiplying the area of its base into its altitudes •we are likewise taught, what ratio similar plane figures,, aid also similar solids, have to one another; that the periphertor of circles are as their diameters, and the areas as the squares of their diameters ; that angles are measured and compared by means of the intercepted circumferences, &e. These and several other properties of the circle are given in the Ele- ments, but it is no where directly sheivn how the circum-' ference (that is, its ratio to the given diameter) or how the area of a circle may be found : it is true, that a method of ^proximation both to the circumference and area seems to be implied in the sceoiid proposition of the twelfth boak,.bul. no further notice is taken of it in any of the subsequent propositions. In hia demonstiations^ Euclid has observed for the most part all that strictness, for which the ancients were so distin- guished : from a small number of definitions and self-evident PaetVIII. geometry. 223 priaciplefs, tie ha0 deduced with moontestiible evidence truth of all the proposilbns which he proposed for proof. ^n» rigorous strictness haS; however, sometimes led him ueoessArily into aa indirect and complicated chain of reasoning, which' makes hb demonstrations in a few instances tedions and dif-^ ficuk. To remedy this defect, several of the moderns have undertaken with suceess to simplify and render more direct and appropriate, such ot the demoDstrations as seemed fio^ require improvement ; but others, who have lessened the number of propositions by retrenching those which they deemed superfluous, have in general been less happy: by removing those links, which appeared to them unnecessary, the chain of demonstration has in many cases been broken and spoiled. The Elements have been translated into the language of evtry country where learning has been encouraged, and en^ riched with numerous and valuable commentaiies* The Arabs were the first people who engaged in tUi way : on the revival of learning ammig them, their grand eare was to obtain the mathematical works of the best Greek authors, and translate tlKminto the Ar^c language. There wtre probably several translations of Euclid ; one in particular is mentioned as made by Honain £bn Ishak al Ebadi, a learned physician^ who flounced in the reign of the KhaUf Al Motawakkef, A. D. 847. Adelard, a monk of Bath, in the twelfth cen- tury, appears to have been the first who made %. Latin trans-^ lation of the Elements, which he did firom 4ie Arabic, as no Greek copy of Euclid had then been discovered. Carapanus of Novaia translated and commented on the Elements in 1250, which work Was revised and further commented on by Lucas De Burgo, about 1470. Orontius Fln«us published the first six bodes with notes in 1530, which is said to have been the firirt edition that appeared in print. Pdetarius published the first six books in 1 557, and about the same time Tartalea gave a commentary on the whola of the iBftfloi^bdoks.* ail INTRODfUCnON. Part VIII. In 1670 BtlliDg8iey*s Eiiclid appeared, with a very plain and useful pw&ce and notes by the learned and eccentric Dr. John Dee. Candalla published the Elements, with addi- tions and improrements, in 157^> which work was itfterwards reprinted with a pnrfix commentary by Clavius the Jesuit. Many edilionsof the Elements have since appeared, the chief of which are those of De Cfaales, Tacquet^ Herigon, Barrow, Ozanam, Keill, Whiston, and Stone ; but Dr. Robert Sim- son's translation of the first six and the eleventh and twelfth books^ with the Data, first publi^ed in the year 1 7^6, is that now most generally used in the British Empire. Playfair's Euclid is an improvement on Samson's ; and In- gram's edition contains some particulars chiefly relating to practical Geometry, which are not to be found in either. Be- fore we conclude this enumeration^ it will be necessary to observe, that Dr. David Gregory », the Savitian Professor of Astronomy, published at Oxford, in 170^9 the whole of tlie worics^ of Euclid in Greek and Latin $ this he b said to have done in prosecution of a design of Dr. Bernard *, his prede- "* Dftvid Oregiory ww bora at Aberdeen in laSl ; here and at Sdinbrn^ be received bis maUiematical and classical education : in I6d4 he was elected Professor of Mathematics in the University of Edinburgh ;''and it deserves to be noticed, that he, in coi^nnction with bis brother James, first introduced the Newtonian phUoiophy into Scotland. Tbrouf h the Inentty inteifefenee of Newton and Flamstead, our author obtained the Saviliaa Profesaorship of Astronomy at Oxford, where he was honoured with the degree of M. D. His works are EjtefcitaH& Geometriea, Stc; 4to. Edinb. 1684. Chtoptriem et IHtp" irie^ Sphmiem' JEkmenimfOxmo, l^h* jiHrwMntim, P^fneaf^et Gemmtrite Mkmemta, and some others: be died in 1710, at Maidenhead in Berkshire. B Dr. Edward Bernard rendered himself fieimous by being the first who un- dertook to ec^ect the work» of the ancient mathematicians for puUicatioo ; he likewise tiioaght to, England the 5tb, ^h, and 7th books of the Cooicsof A{i(ottoniu8, being a c<9y of the Arabic Version which the celebrated Golios bad obtained in the East. He succeeded Dr. Wren in the Professorship in 1673, and resigned it in 1^1, on being presented to the Rectory of Brightwell m Berkshire. He died in I696> in the SJStb year of his age. His' work» on ma- thematical subjects are mostly inserted in the Philosophical Transactions: they consist of Observations on the Obliquity of the £cliptic, various \/istr0nowdeat •ad Cki»tt$gtgietd TabUs, ^^ PabtVIII. INTftODUCTION. ^25 cesser, and in obedience to a precept of Sir Henry Saville % the. founder of the Professorship, reqiiiring that those who fill die chairs of Geometry and Astronomy should publish the mathematical works of the ancients. Dr. Gregory's is the completest edition of Euclid extant. According to Pappus and Proclus, several mathematical treatises, brides the Elements, were written by Euclid : hts Data, a work still extant, is calculated to facilitate the method of resolution, or analysis, shewing from certain things givf n by hypothesis, what other things may thence be found. His three books of Porisms are said to have been a curious collec- tipa of important particulars relating to the analysis of the ibore diflScult and general problems ; but no part of this wof k, or of any other on the same subject written by the ancients, had been preserved, except a small specimen by Pappus; from whence several modern geometricians, particularly Fermat, BuUiald, Albert Girard, Halley, Simson, and Play- &ir, have attempted to restore either completely, or in part, what the ancients are supposed to have delivered on the sub- ject. Euclid wrote, besides these, a work on the Division of « Henry SaVille'was bom at Bradley in Torkshire, A^ D. 1549» and entered at Merton College, Oxford, in 1561, of which college he was chosen a fellow* and took his degiree of M. A. in 1570. In 1578 he trarelled through different parts of Siuope for improvement, and on his retnm was appointed Greek Tutor to Qaeen Elizabeth. In 1585 he was made Warden of Merton College* over whkh he presided 36 years, with eqaM credit to himself and advantage to that learned body. He was chosen Provost of Eton -College in 1596, and received tlie bononr of knighthood from Sing James I. in 1604, after declining the most flattering offers of preferment in either church or state. Sir Hanty Soiville was an accomplished gentleman, a profound scholar, and a munificent, patron of learning, to which (on the death of his mily son) he devoted his wholef fortune. In 1619 he foanded two professorships at Oxford, one for Geometry, apd one for Astronomy, each of which he endowed with estates. In addition to tfaie several legacies he left to the University, he bestowed on it a great i|«ABttty of mathematical books, rare and curious manuscripts, Greek types, &c. &c. He died at Eton College in 1722, leaving behind him several works» . of which the only one pertaining to our present subject is his CoUeciion rf Mathematical Lecturer on EucUd^t Elements, 4to. 1621. VOL. 11 • g 226. GEOMETOY. PartVIII. Superficies ; Loci ad Siiperficiem ; four books on Conic Sec- tions ; and treatises on other branches of the Mathematics. Archimedes ', one of the greatest geometricians of anti- quity, was the first who approximated to the ratio of the cir- P Archimedes was born at SyracQto, and related fo Hiero, King of Sicily: lie was remarlcable for bis extraordinary application to mathematteal studies, but more so for bis skill and surprising inventions in Mechanics. He excelled likewise in Hydrostatics, Astronomy, Optics, and almost every other science ; he exhibited the motions of the heavenly bodies in a« pleasing and instructivs manner, within a sphere of glass of his own contrivance and workmanship ; he likewise contrived corions and powerful machines and engines for raisiag weights, hurling stones, darts, &c. launching ships, and for exhausting the water out of them, draining marshes, &c. Whdn the Roman Consul, Mar- cellus, besieged Syracuse, the machines of Archimedes were employed t these showered upon the enemy a cloud of destructive darts, and stones of vast weight and in great quantities ; their ships were lifted into the air by his cranes, levers, hooks, &c. and dashed against the rocks, or precipitated to the bottom of the sea ; nor could they find safety in retreat : his powerful bnmiqg glasses reflected the condensed rays of the sun upon them with such effect^ that many of them were burned. Syracuse was however at last taken by stormy and Archimedes, too deeply engaged in some geometrical speculations to be conscious of what had happened, was slain by a Roman soldier. Maroellna wa« grieved at his death, which happened A. C. 210, and took care of his funeral. Cicero, when he was Questor of Sicily, discovered the tomb of Archimedes overgrown with bashes and w^eeds, having the sphere and cylinder engraved on it, with an inscription which time had rendered illegible. His reply to' Hiero, who was one day admiring and praising bis machines, can be regarded only as an empty boast. ^* Give me/' said the ezultij^ philosopher, " a place to stand on, and I will lift the eMrtV (A»« ^mi r« fw, mu rifi^ ynf *t9n^t*») This however may be easily proved to be impossible ; for, granting him a place, with the simplest machine, it would re4|aire a man to move swifter than a cannon shot during the space of 100 years, to lift the earth only &ne inek in all that tinie«**— Hiero ordered a golden crown to be made, but suspecting that the artists bad purloined some of the gi4d and substituted base metal in its stead, be employed our philosopher to detect the cheat ^ Archimedes tried for some time in vain, but one day as he went into the bath, he observed timt his body exdvded just as much water as was equal to its bulk ; the th«mght immediately struck htm that this discovery had furnished ampls data for solving his difliculty; upon which be leaped out of the bath, and ran through the streets homewards, crying ont^ <«^»« ! tv^%m ! / have found it i J have /mmd it /—The best edition of bis works is that of Torelli, edited at the Clarendon Press, Oxford, fol. ITS^y by Pr. Robertson, Suviltan Professor sf Astronomv. l^AnrVni. INTRODUCTION. 227 tumference of a circk to its diameter, A.C. 250: this he eflected by circumscribibg about, and inscribing in the circle Iregular polygons of 96 sides, and making a numerical calcu^ lation of their perimeters ; by means of this process he made the ratio as 22 to 7j which is a determination near enough the truth for common practical operations, where great exact- ness is not required, and has the advantage of being express- ed by small numbers. He was the next after Hippocrates, who squared a curvilineal space 3 he applied himself with ardour to the investigation of the measures, proportions, and properties of the conic sections, spirals, cylinders, cones, spheres, conoids, spheroids, &c. On these subjects the follow- ing works of his are still extant, viz. two books on the Sphere and Cylinder; and treatises on the Dimensions of the Circle ; on Spirals ; on Conoids and Spheroids ; and on the Centres of Gravity. The next geometer of note after Archimedes, was Apol- lonius Pergsdus, A. C» 230 : this great man studied for a long time in the schools of Alexandria under the disciples. of Buclid, and was the author of several valuable works on Geometry, which were so much esteemed, that they procured him the honourable title of the great Geometrician. His principal work, and the most perfect, of the kind among the ancients^is his treatise on the Conic Sections, in eight books ; seven only of these have been preserved, the four first in the original Greek, and the 5th, 6th, and 7th in an Aramc version \ 4 AceorStpg to l^ppUB abd Eutocitu, the following works were likewise -Written by A|»dlloniQs, viz. 1. The Section of a Spa^e. J2. The Section of a Ratio. ' 3. The Determinate Section. 4. The Inclinations. 5. The Tangen- cies, and 6. The Plane Loci ; each of these treatises consisting of two books. Pappus has left us some particulars of the abore works, which are all concern- ing them that now remain ; but from the^ scanty materials, many restorations liave been made^ ris. by Vieta, SnelUus» Ghetaldus, Fermat, Schooteu, Alex. Andefioii, HaUey» Simaon^ Horsley^ Lawson, Wales, and Barrow. The best edition of the C«ntci of ApolloAios is that by Dr. Halley, foi, Oxw. 1710* a 2' Xbeag^of Arohimedes ^d AvQlhouhiS hm with jusd«« th^ sci^i^ce oever acquired so* great a dtsgree oi brilliancy at aqy otber p^uod of the Greciao history. XbeduglicsuioQ of the oube,,qjuiadr«ture'Of the circloi tri*- section.of an^ aogle^ &c. were probleiss of which the ancients tu»ver lost.^igbt;. ijaaoy of the proposilioiiSHin tbe Elements^ payiticuliurly. piy^ 27^ 2S> wd- 29' of. th& sixth book, are inti- HM^tely connected with the aolution^. and probably originated in, the atlm»pt(» to. obtain it* Thj»^ application oft the conic sections tQ this purpose by M en^hmus^ has been, fdready noticed : about ihe aaoae time IKoostratus; invented: the-qna^ di^triiS) a iQ€K;hanical cuITC^po6sesl»()g the triple adiwitits^ of tfjsACting and. multiplying aa ang^e>. and squaring the Qit€li»i Tibet conchoid of Nicomede^^ who- flourished A. C« 250, has been applied by both ancient and*, modern, geome? tQra^ually.to the trisection, finding: two mean prc^rtiotials, 2ind tbeioonstruction of other solid probkni9^;.for which pur- poses, this. ourve has be^n preferred by Archimedes, Pappusi and Nfiwton» to any other. (See Newton- l^^rt^Ama^ca Uni^ t€nalisi,p. 288).2H90 The cissoid, another curve, heie^ an unprovament on the conchoid; was ini^nted by Diocles about laOyearebefcore Christ. Hero^ . DosithfittSy Eratosthe^eB^ . and ; Hypsides, ^ who : fbu^ liabed in^ the^ second century befdre Christ, and Geminius who flourished in the first, were all eminent for. their skill in* Geometry: indeed the science continued to be cultivated with il^rdour by. a numerousilistKotf geometricians, produced by the Alexandrian school,* until'thatfasnous seat^pf learning' fell a prey to the blind and merciless bigotry of the Arabs. The fiffst:who wrote on., the spberci and. its circles to any con»^ m Swiiloaof tb« GyUndeF and Cone, prinM- fiieni tht ov^inai GUseek) witl|ia> LbtiQ tvMMlatioQ, Tam VIII. INTRODUCTION. 2» derable extent, at lealt whose works have been preserve4^ was Tbeodosius^ A. C. 60 : this work, in which the Jproposi'- tions are demonstrated with equal strictness and el^^nce, forms the basis of spherical Trigononietryy as pntctiled by the BMxlems* About the same titne, or shortly after^ Mehelaui wrote lus treatise on Chords, which b lost ; but his wblck on Spherical Triatigles, containing the constmeiioti and tri« gonometrical method of resolving them, accorditig to the ancient practice, is still extabt. We are particulariy indebted to Pat^s, A. D. 380, and Procluis, A. D^ 4dO> fioT their kbo-^ Tious researches ; many particulars relating to the scienees df the Greeks would have been lost to pcBterity, but for their writings: the former was an etninent mathematiciaii a£ Alexandria, and author of several learned and useful wdrks^ particularly eight books of Mathematical Collections, of which the first and part of the second are wantit)^. These books een- twn a great variety of useful information relilting to Geome* try. Arithmetic, Mechanics, &c. with the sokitiori of proMcm^ oi different sorts. Proclus likewise studied at Alexandria^ and afterwards presided over the Platonic school at Athens } be wrote, besides many otber w<H*ks, Commentaries on the fint book of Euclid, on the Mathematics, on Phil€isophyi also a treatbe De Splwrra, wbieh Was published by Dr. Bftia« bridge, Savilian Professor of Geometry at Oxford, in 1690. The writings of the Greek geometFieians were trfeslate4 and commented on by several learned Arabians, but tfi^ improvements they introduced were chiefly of the practH eal kind ; among these may be meotioned the fundamenUd propofiitiofM of Trigonometry, in wht€b,by (be substitution of sines instead of the chords, and other conveaknt Abridge- ments, they greatly simpKfied the theory and solictiotis of plane and spherical triangles. These improvements are a-r jBieribed to MaiMMnet Ebn Mssa^ ft geometer of whom there ttill exists a work on Plane atid Splrerical Figtarres. We Bk«f«* as 2S0 <5EOMETRY. Part VIII. wise possess a work on Sarveying, written by Mahomet of Baghdad^ which some modern authors have ascribed to Euclid. A few learned men, famous for their skill in Geometry, flourished in the West during the fifteenth century. Of these the chief were the Cardinals Bessarionand Cusa ', Purbach, Nicholas Oresme, Bianchini, George of Trabezonde, Lucar de Burgo, Schonerus, Walther, and Regiomontanus; the latter wrote a treatise on Plane and Spherical Trigonometry, A. D, 1464 ; in which, among other improvements, he introduced the use of the tangents, and applied Algebra to the solution of geometrical problems : this, is the more surprising, as it occurred several years before the publication of any of the worka^f De Burgo, who is generally supposed to have be^a the introducer of Algebra into Europe. . 43il the revival of learning iit Europe about the beginning of the sixteenth c^tury, the study of Geometry began to be cultivated with great attention ; the works of the Greek geometricians were eagerly sought after and translated into Latin or Italian, and served as guides to those who had a taste for that correct reasoning, for which the ancient Geo- metry is so ji^tly famed, or were desirous of availing thenn selves of the knowledge of its application and use, as ctm^ nected with the necessary business of life. As early as 1522, John Wenier, a celebrated astronomer of Nuremberg, pub- lished some tracts on the Conic Sections, and on other geo^ metrical subjects. Tartalea composed a treatise on Arith- metic, Algebra, Geometry, Mensuration, &c. entitled, ^Tra^- tato di humeri et Misure, 155G, being the first modem work ' Nicolas De Cusa was bom of poor parents, A. D. 1401 ; bis application to learning and bis personal merit, boweyer, raised bim to tbe rank of bisbop and' cardinal", bis claim to tbe honour of baving squared tbe circle was ably re- futed by Begiomontanus i ne.veTtbelesiJ be was a man of very extraordinaiy pattsy and excelled in tbe knowlttdga of law, divinity, natural pbilosopby, aad feometiryi on wl(icb 8ob|eet9 he i> said to hare written some eycdlent trea^seif. He died in 1464. ^ 'v Part VlIL INTRODUCTION. 231 which teaches how to find the area of a triangle by means of its three sides, without the aid of a perpendicular. Mauro- licus was a respectable geometer, ^and wrote on various sub- jects ; his treatise on the Conic Sections is remarkable for Its perspicuity and elegance. Aurispa, Batecombe, Butes, Ramus, Xylander, Foilius, Cardan, Fregius, Bombelli, Ficinus, Durer, Zeigler, Fernel, Ubaldi, Clavius, Barbaro, Byrgius, Commandine, Pelletier, Dryander, Nonius, Lina- cre, Sturmius, Saville, Ghetaldus, R. Snellius, and many others who flourished at this period, were cultivators of Geometry; and if they made few discoveries, still their labours as translators, commentators, or teachers, were be- neficial in diffusing knowledge, and merit our grateful ac- knowledgments. Vyious approximations to the ratio of the circumference of a circle to its diameter, were given about the beginning of the 1 7th century, approaching much nearer the truth than any that had hitherto appeared ; viz. by Adrian Romanus, Willebrord Snellius, Peter Metius, and Ludolph Van Ceu-* len ; according to the conclusion of M etius, if the diameter be 113, thcs circumference will be 355, which is very near the truth, and has the advantage of being expressed hy small numbers. By continual bisection of the circumference. Van Ceulcn found, that if the diameter be 1, the circumference will be 3,14159, &c. to 3G places of decimals; which dis- covery was thought so curious, that the numbers were en- graved on his tomb in St. Peter's Church- yard, at Leyden •. * The simplest (and consequently least accarate) ratio of the diameter to the circumference is as 1 to 3 ; a ratio somewhat nearer tl^tn this, is as 6 to 19. We have noticed before that Archimedes determiqed the ratio to be as 7 td 22 nearfy, which is nearer than the above. A nearer approximation is as 106 to 333* That of Melius is still nearer, viz. as 113 to 355 ' A nearer approximation than the 1^ j^^2 ^^ g^^^ last is J * ' / still nearer is : ... as 1815 to 5702, &c. Q 4 m GEOMETRY. P^jit VIIL Geometrical problems had long before this period beeo solved algebraically^ by Cardan, Tartalea, Re^montaou^, and BombelU ; but a regular and general method of apply- ing Algebra to Geometry, was first given by Vieta, about the year )580$ as also the elements of angular sections. De^ Cartes improved the dbcovery of Vieta, by introducing a general method of representing the nature and circumstances of curve lines by algebraic equations, distributing curved into classes, corresponding to the different orders of equation^ by which they are expressed ; A. P. IG37- A method of Ixingents, and a method de maximis et minimis, nfUcb resembling that of fluxions or increments, owe their ori- gin to Fermat, a learned countryman and competitor of Des Cartes, with whom he disputed the honour of first ap- plying Algebra to curve lines, and to the geometrical con- • struction of equations, secrets of which he was in posstssion before Des Cartes' Geometry appeared. About this tim^, q^ a little earlier, Galileo invented the cycloid ; its properties were afterwards demonstrated by Torricellius. The improvement of Des Cartes, now called the nm Ceqi^etrtfj was cultivated with ardoiu* and success by math^- nqatieians in almost every part of Europe; his work w^ translated out of French into I^atin, and published by Fran- cis SchoQten, with a commentary by Schooten, and notes by M- de Beaune, 16*49. The Indivisibles of Cavalerius, pu^- lisbed in 1635, was a new and useful invention, applied to Van Cenlen's nomben, as mentioned above, were extended to 72 places of figures by Mr. Abrabam Sharp, about 1706 ; Mr. Macbin afterwards extended the same to 100 places, and M. De Lagni has carried them to the amaxinc length of 128 places: thus, if the diameter be 1000, &c. (to 128 places}' till circumference will be 31415, 92653, 58979, 32384, 62643, 38327, 95028, 84197, 16939, 93751, 05820,^494, 45923, 07816, 40628^ 62089, 98628, 03482,53421, 17067, 98214, 80865,' 13272, 30664, 70938,446+, or 7**. This number (which includes those of Vm Ceulen, Sharp, and Machtn) is sufficiently near the truth for any purpose, so that except the ratio could be completely found, we need not wish for a greater de^^ree of accuracy. PfBT VUl. INTRQDyCTiON. m ieteamne th^ ttrea^^ of cunre$, tti^ soUdides cf hodics n^f^r rated by their reiBrolutiqi) about 9, fixed lioej &c. Boberva)^ af /early as 1634^ had employed a ftimilar metho^^ wbi^b hf lipplied to the cycloid, a eurve at thi^t tin^e jmd^ cel^brfi^f^ for its numerous and singular properties; be likewise i^r vented a general method for tangents, applicable ^ike t^ geometrical and mechanical curves. The inverse method ^ tangents derived its or^in from a problem, which De Beaune proposed to bis friend Des Cartes, in 1647* In 1655 tl^ learned Dr. Wallis published his Arithmetica InfinitormQ ; being either a new method of reasonbg on quantities, or else a great improvement on the Indivisibles of Cav^eriiif ^bove mentioned ; peculations which led the wfiy to in^i)b| aeries, the binomiid theorem, and the method of fluxions : thb work treats of the quadrature of curves and many other pro- blems, and gives the first ei^pression known for thf area qf 4 circle by an infinite series* One of the greatest discoveries in modern Geometry was the theory of evolutes, the autluv of which was Christmn Huy^cns, an ingeniou$ Dutch mathematician, who pjublished it at the Hague in 1658, in a work entitled^ Horeksgiufla Oscillatorium, sive de Motu Pendulorum, &c. In 16G9 were published Dn Barrow's Optical and Geomcr trical Lectures, containing many v.ery ingenioMS and proibujid researches ojq the dimensions and properties of curves, and i^pecially a method pf tangents, by % mode of calculatioi^ differing firom that of fluxions or ioorements in scarcely any particular, except the notation. About this time the use of geometrical loci for the solution of eqviatioQs, was carried to a great degree of perfection by Slusiua» a canon of Liege^ in his Mesolabium ei Problemaia Solida: he likewise in* isejted in the Philosophical Tram^actions, a short and easy method of drawing tangents to all geometrical curves, with a demoqstratipq of the same \ and likewise a tract ovl the HSi GEOMETRY. FabtYDL Optic Angle of Alhazen. Besides those we hare mendoDed, maojf others of this period devoted their attention to the rectification and quadrature of curves, &c. of whom Van Heuraet, Rolle, Pascal, Briggs^ Halley, Lallou^re, Tor-^ riceHtus, Herigon, Niell,. Sir Christopher Wren, Faher^ Lord Brouncker, Nicholas Baker, G. St. Vincent, Mercator, Gregory, and Leibnitz, ware the principal. Tlie seventeenth century is famed for giving birth to two noble discoveries; namely, that of logarithms in Hi 14 by Lord Napier, whereby the practical applications of Geometry are greatly facilitated ; and that of fluxions, to which pro- blems relating to infinite series, the quadrature and properties of curves, and other geometrical subjects connected with Astronomy, Pliysics, &c. and which were formerly considered as beyond the reach of human sagacity, readily submit. For this subKme discovery, the learned are indebted either to the profound and penetrating genius of Sir Isaac Newton % or < Sir I»aac New^n, one of the greatest mathematieiaqB and pfailosophert that ever lived, was born in Lincolnshire, in 1643. -Having made some profi- ciency in the classics, &c. at the gi-ammar school at Gfantham, he (being an •nlf child) waa taken home hj bts mother (who was a widow) to be her com- panion, and to learn the management of his paternal estate : but the Iotc of books and stady occasioned his farming concerns to be neglected. In 1660 he was sent to Trinity College, Cambridge ; here he began with the study of Euclid, bat the propoeitions of that book being too easy to arrest his atteolUott long, he passed rapidly on to the Analysis of Pes Cartes, Kej^ler's Optics, &c. making occasional improvements on his author, and entering his observations, &c. on the maigin. His genius and attention soon attracted the favourable notice of Dr. Barrow, at that time one of the most eminent .mathematicians in England, who soon became his steady patron and friesf). In 1664 he took his de- gree of B. A. and employed himself in speculations and experiments on the na^ ture of light and colours, grinding and polishing optic gUwses, and opening the way for his new method of .fluxioqs and infinite series. ^ The next- year, the plague which raged at Cambridge obliged him to, retire into the country ; here he laid the foandatioii of his universal system of gravitation, the first hint of which be received from seeing an apple fall from a tree ; and subsequent reasoning induced him to conclude, that the same force which brought down the apple might possibly extend to the moon, and retain her in her orbit : he afterwards extended the doctrine to all the bodies which compose the solat system, and . P4 RT VIII, INTRODUCTION. ?85 to that of L^ibnite, or to both, for both laid claim to the in- vention. No sooner was the method made public, thail 9 d€monstra;tc4 the same in the mo^' ^dent manner, GeaGnniiif the laws iriiidl Kepler bad discovered^ by a laborious train of obseryation and reasoning |( namely, that ** the planets move in elliptical orbits ;" that " they describe equal areas in equal times ;" and that " the squares of their periodic times are as the cubes of their distances." Every part of natural philosophy not ooly T^eived improvement by his* inimitable tpach, ,bQt. became a new science nnder bis hands : his system of gravitation, as we have observed, confirmed the discoveries of Kepler, explained the immutable laws of nature, changed the system of Oopernicus from a probable hypothesis to a plaib and demon- strated truth, and eflpectually overturned the vortices and other imaginary machinery of Des Cartes, with all the improbable epicycles, deferents, and islamsy apparatus, with which the ancients and sdtaie of the moderns', had en- cumbered the universe. In fact, his PhUosophia Naturalia Principia Matker matica contains ap entirely new system of philosophy, built on the sol|d basis of experiment and observation, and demonstrated by the most sublime Geometry ; and bis treatises and papers on optics supply a new theory of lig^ht and colours. The invention of the reflecting telescope, which is due to Mr. James Greguryy would in all probability have been lost, had not Newton interposed, and by his great improvements brought it forward into public notice. In 1667 Newton was chosen fellow of his College, and took his degree of M. A. Two years after, his friend Dr. Barrow resigned to him the mathematical chair ; he became a Member of Parliament in 1688, and through the interest of Mr. Montagu, Chancellor of the Exchequer, who had been educated with him at Trinity C>o]lege, our author obtained in I696 the appointment of War- den, and three years after that of Master, of the Mii^ : he was elected in 1699 member of the Royal Academy of Sciences at Paris ; and in I7O8 President of the Royal Society, a situation which he filled during the remainder of his life, with no less honour t<^hiiuself than benefit to the interests of science. In 1705| in consideration of his superior merit, Queen Anne conferred on him the hoQou]: of knighthood: he died on March 20th, 17^7, in the 85th year of his age. Virtue is the brightest ornament of sciience : Newton is in- debted to this for the bett part of his fame ; he was 9k great man, and goodwm he was g^reat : to the most exemplary candour, moderation, and affability, he added every virtue necessary to constitute a truly moral character ; above all, he felt a firm conviction qf the truth of Revelation, and studied the Bible with the greatest application and diligence. But such is the folly bf man, that the tribute, which is due to the gaeat first cause alone, we trans- fer to the instrument; Newton, Marlborough, Nelson, Wellington, &c. have a// our praise, while the great soujrce of knowledge, stren^h, victory, and every benefit we enjoy, is foigotten. How would the modest Newton have reddened with shame and indignation, could he haive heard all the ex- travagant encomiums, little short of adoration, which have with foolish and 996 GBOUEm. Part VIII. sharp und virulent contest eosoed : at kngth the Rojni ^ociely was appealed to, and a Committee iqipoiiited to exa- mine letters, papers, and other documents, and thence to £onn a decision on the claim of each. The result of the inquiry was, *^ That Sir L Newton had invented hb method hefore the year 1669, and eoosequeotly fifteen years before M. LeibnitK had given any thing on the subject in the JLeipsic Acts :'* the same Report in another part says, ** that it did not appear that M. Leibnitz knew any thing of the difierential calculus, before his letter of the 21st of June, 1677-" It appears however that this decision, which con- firmed the claim of our illustrious countryman, did not give entire satisfaction to the continental mathematicians of that period, nor are their successors better disposed to yield the palm to Newton; they still contend that Leibnitz, ad- mitting that he was not theirs/ inventor, (and some refuse to concede this point,) borrowed nothing of his method from bis rival; a fact which some well informed Englishmen have much questioned. Other tracts containing improvements in Geometry were given by Newton; as, i. EnumeratiQ Idnearwn Tertii Ordinis. 2. Tract at us Duo de Speciebus et Magnitudine Figurarum Curvilinearum. 3. GenesU Curvitrum per Urn. bras: in these, as well as in bis Principia and other works, he has for the most part employed hb own new ^naly$ii% by which the doctrine of curves has been amaaongly extended and improved. Geometry had hitherto consisted of two kinds, JElemea* Wyt or that which treats of right lines, cectilineal figures, the circle, and solids terminated by these ; and Higher^ at Tramcendetit Geometryy which treats of all sorts of curves, * impious yrofusioa been lavislied on bU memory ! . His worHa» collected in S ▼oluraes 4to. with a TsUuable Commentary by Dr. Horsley^ were pubUsbed ia 1784. pABrr¥ia INTROmJGTR)N. 28^ except the circle, and the sdidir gfeocratcd by their revolu- tion : to these, as has been €A>serve€l, the diseoyeries of Sir Isaac Newton have added a third, viz. the Sublime Geom^y^ #r tke doctrine and application of fluxions ". Of those anthors, who have since applied themselves to the evkiva^on and improvement of the new calculus, (as the doctrine of fluxions was called,) and to the extension of its applications, the following are the names of some of the chief; vir, Agnesi, IVAlembert, Bossut, the Bernoulli's, Cheyne, Cotes, Craig, Clairaut, Colson, Caifooli, Condorcet, Emerson^ Euler, Fontaine, Fagnanus,.Guisnee, Le Grange, L'H<>pital, Hayes, Hinl^on, Harris, Htttton, Joites, Jack, Landen, Lorgna, D^e Lagni, Manfred!, Maseres, NIaclaurin, Nicole, Nieuwentyt, Reyneau, Riccati,^ Raphson, Rowe, Smith|» Sterling, Saunderson, Siuif»on, Tirj^lor, Vince, Walmsley, Waring, &c. The IbllQwittg inventions, which are either nearly allied to the method, of fluxions or- capable of similar application, have been already noticed in the Introduction to Part III. viz. Dr. &xx>k Taylor's Methodus Incrementorumy 17 15 ; Kirk- by^s Bdetriae of Ultimaton, l^iS] Landen's Residual' Analysis, 1764; and Major Glenie's Doctrine of Vniversat Comiparisa»ylJS9f and his Aftecedental Calculus, 179S. It' has been the error' and misfortune of some eminent' « <*On peat^tiser kiG^Mtt^ie de dMRSr^ntes flkaniires. £n ^I^mentair^, et «» tTHnseendant^^ La O^m^rie ^Mnientaire iw consididTe qtie les propri^s d^ ' tijgnes dtoSte*, det lignes elreukiires, et dt% sdltdes ternilD^s park;es fibres: Lq ' oeff«le«H & teiik fi^re carviligne doat on^'p&rle datis les'^l^meos de G^o* <'< lA O^oin4t#«e-traiM0endafite est proprem^nfe celle qoi a pour objet toutes l4|»€oitflibsidiffi^Bie8*da cercle, comme les sections coniqties, et Ics coixrbes' dNm genre pltii» ^iev4* «< Far \^ on aaroit trois divistem de la'G^dm^fie : G^ou^trie ^l^mentaire, •V de» ligtRsilK^itei, etda cerele ; G^m^rie traftsoendante, oa des conrbes ; tx O^ouk^trie sabUme, oa des nouveaaxcalcah/^ IfAlemherty EficpcUtpedie, 258 GEOMETRY. Part Vllt and otiierwke deserving characters, to direct their attentiod ahnost exclimvely to malhetnatical demonstration^ whereby they have been induced to, deny or undervalue the force and evidence of moral certamty; the celebrated Dr. Edmund Halley * was one of these* Revelation is a subject, ^hich among very many otlvers does not admit ci mathematical proof; and therefore he affirmed with equal rashness and impiety, that ^' the doctrines of Christianity are incompre-* liensible, and the religion itself is a cheat/* This hardy declaration roused the iodignation of Dr. Berkeley % the * Edmund Halley was born in London, A. D. 1656. After making coo-' iiderable progress in tfanicl^sica at St. Pani*s Sebool, and obtaining some knowledge of tbe mathematics, he was sent in 1673 to Oxford, where be Applied himself closely to mathematics and astronomy. Having conceived thf design of completing the catalogue of stars, by increasing it fi'om his own ob* servation by those in tbe southern hemisphere, he embarked for St. Helena ia November, 1676; he returned in 1678, having completed his catalogue, oa which occasion the University of Oxford honoured him witb the degree of M. A. and tbe Royal Society elected him one of their Fellows. In 1691 ktf applied for the appointment of Savilian Professor, but being charged witk infidelity and scepticism, and his pride scorning to disavow the charge, be did not succeed ; however in 1 703 h^ succeeded Dr. Wallis as Professor of Geo- metry at Oxford^ and had the degree of LL. D. conferred on him. Id 1713 be became Secretary to tbe Royal Society, an office which six years after he tt* signed, on being appointed Astronomer Royal : in prosecuting tbe duties of this office, he is said to have missed scarcely a single observation duridg eight tea years which he held it ; he died in 174;2* ^r. Halle/s numerous obsenrationft on the heavenly bodies, the winds and tides, the variation of the magnetic needlci and other valuable tracts on mathematical subjects, published separately or in the Philosophical Transactions, have rendered his name fomouS all over Europe^ - y Gewge Berkeley was born at ^ileriu in Ireland, in the year 1684: after receiving tbe first part of his education at Kilkenny school^ he became a Pen- sioner of Trinity College, Dublin, in 16999 "id a Fellow in 1707 : in 17S1 be took the degreesr of B. D. and D. D. and three years after was promoted- te tbe Deanery of Derry, and to the Bishopric of Cloyne in 1733 ; in 1753 he removed with his family to Oxford, where be died the following year. Besides tbe ri^plies and rejoinders to which the above dispute gaVB birth. Dr. B^rkel^ wrote Arithmetica absque Algebra^ out Eudide Demonstraiaf 1707 ; a Muike* matical MiaceHany^ inscribed to Mr. Molineux ; Theory qf Fitumt 1709 ; The Principles of Human Knowledge , 1710; Dialogues between Hylas and P/tUonus, 1713. In tbe two latter it is attempted to be proved, that the common notion of the existence of matter is false*; that we eannot be certain that P4RTVXIL INTRODUCTION. . tS9 learned and virtuous bishop of Cloyne^ who, to 'aaseirt the truth and honour «f injured religion, published in 1734 The Analy^. In this work, whi<:h is addressed to Ha!ley as an infidel mathematician, he shews that the mysteries in faith, &c, are unjustly objected to, especially by the mathe* maticians, who, be affirms, admit much greater mysteries, and even falsehoods, into science; of which, he says, the doctrine of fluxions furnishes an example. This avowed attack on a new branch of science, the principles of which had not then in every particular been established with sufficient firmness, called forth the zeal and abilities of its admirers; and produced, besides a direct answer, as it is supposed by Dr. Jurin, Robins'-s Discourse concerning the Method of Fluxions, &c. 1 735 ; V^lton's Vindication, &c« 1735 ; and Smith's 'New Treatise of Fluxions, with answers to the principal objections in the Analyst, 1737: but the most complete vindication of the method of fluxions to which this contest gave rise, together with a firm establishment of its principles, &c. are to be found in Maclaurin's Complete System of Fluxions, with their application to tlie most con-- nderable Problems in Geometry and Natural Philosophy, In 2 vol. 4to. published at Edinburgh, in 17^2 : this is indeed the most complete and comprehensive work on the science that has ever yet appeared. Of the modern elementary writers on Geometry, who have given systems of their own, and not strictly followed Euclid, the following are the principal; viz. Borelli, Pardies, Wolfius, there are any such things as external sensible objects ; and that they are, as far as we can know, nothing more than mere impressions made upon the mind by the immediate act of God, according to certain rules called laws of nature. He was a truly excellent man, and the line by which Pope has characterised him, by ascribing to him << every virtue under heaven," is said not to have for exceeded the truth. In addition to the above works, h^ wrote The Minute PhUoM/pher ; wn^ tracts on religious and political subjects % Siris, or the Tirtaes of Tar Water -, and another piece on the same subject. ^m GEOJifitRY. Part Vlli Stufrifttt% IttMrfiMn^ Mttrch^fti^ Hfamilton, Emerson, Sinip- sbii, Bonwycdstle, and Button^ those of the three last are valuable and useftH perferftitfn^s. Those who have writtcii dii the ^object 6( pratotic^l Geom«ry, are Bayer, Bonny- eilstle, CkVkid, Gantd^rus, Gregory, Herigoto, Hawneyy Hukius, Kapler, Ltgiitbody, Le Oerc, Ikfallet, Ozanam/ Ramutf, Reinhold, Scliwinterus^ Seheffelt^ Tacquet, Voigtel,- Wolfiiis, and many othei^. i ' PA^t niL. USEFULNESS OF GEOMETRY. S4J ON THE USEFULNESS OF GEOMETRY. W O question is more frequently asked by beginners in Geome- iryj, than the following: Of what use, u the study of EucluVs Elements ? The industrious, the idle, the sensible, and the dull, from different motives, are equally concerned in the inquiry : they almost daily agitate It with a 4egree of importunity, which sometimes proves troublesome to the Tutor* because he iSnds himself incapable of answering 'the question compktely to his own or their satisfaction. The difficulty hqpever lies not in the ignorance of the Tutor, or the want of usefulness in the science, but in the nature of things : for no art or science whatever can teach its own use ; how then can one, who is learning merel|F the principles of Geometry, expect to understand fully its use- ftilness, or that his Tutor, however learned he may be, can by any explanation do justice to a science, of which the various and useful applications will perhaps never be completely deter- mined ? To try to satisfy alUthe absurd and vexatious scru[)]es, which the idle, the querulous, or the captious, please to stajt against any braflRrh of learning, would perhaps be a vain attempt ; but it will be proper to advise the diligent and well-disposed stu^ dent, (and to sucli the advice can hardly be needful,) that it is his duty, and will be to his advantage, to study attentively and without scruple, any branch of learning which his friends may think proper to recommend to him as useful, and which the experience of wise and good men in every age has proved to be so. But in the present instance, an implicit reliance on authority is not at all necessary ; the obvious uses of Geometry are suffi- cient to recommend it to the. candid and impartial inquirer ^ some of these we shall briefly enumerate. Gecnnetry is useful* as it 4|)pliea to the businesses and concerns of society, and as fua€laroental^ to other sciences and arts connected with tKem. Whatever relates to the comparison, estimation, &c. of distances, spaces, and bodies, belongs to Geometry ; and consequently on its principles and conclusions immediately depend Mensuration, Surveying, Perspective, Architecture, Navigation, Fortification, with many other branches equally conducive to public benefits ia sfaort> it is difficult to acquire a tolerable degree of know- VOL. 11. R 242 GEOMETRY. Part VIIL ledge in philosophy^ or any art or science, \tithoat some ac- quaintajQce with Geometry. In addition to the direct and practical uses of the science, there is another, ivhich Lord Bacon calls " collateral and inter- venient." Geometry strengthens, corroborates, and otherwise improves the reasoning faculties, inuring the mind to patient labour, teaching it method, and supplying it with the means of contriving and adopting proper expedients for the prosecution of its researches. GeoAietry may then be justly con^dered as a highly valuable science, both with respect to its practical appli- cation, and as a complete model of strict demonstration : and in the latter view it recommends itself to the diligent attention ofevery lover of truth. In what follows, we shall treat of Geometry in the two-fold tiew abm'e explsdned, by briefly shewing the practical applica- tion of Euclid's doctrine, and likewise by considering it purely as a system of demonstration. The demonstration of a proposition does not depend on the correctness of the diagram, which therefore may be drawn by hand 5 but in the practical uses ot the propositions which we mean to exemplify, accurate figures should be made, and for this purpose instruments must be employed : we will therefore give a brief description of such instruments as are necessary for the construction of figures, and explain their fitrther uses hereafter, repeating, that the instruments are by no means neces* sary to the demonstration, DESiCRIPTION OF A CASE OF MATHEMATICAL IN- STRUMENTS. A common pocket case of Mathematical Instruments cod^ tains, 1. a pair of Plain Compasses 5 2. ajiair of Drawing Com" passes > to the latter belong 3. a Port Crayon, 4. a fiettiog Pen, and 5. a Steel Pen : 6 *a Drawing Pen, with 7. a Pointer 5 8. a Protractor 3 9. a Plain Scale 5 10. a Sector j 11. a Parallel Uul«r; and 12. a Black-lead PenciP. ' ' • ' • Cases of Mathematical Instrtimciits may be had at all prices, £roBi five shil- ling) tQ^six guineas ; a case that costs tweDty-fiveor thirty sbittiDgv will be suft Part VIII. MATHEMATICAL INSTRUMENTS. «43 The PLAIN COMPASSES are used for the following piu:- poses: - 1. To draw a blank or obscure line by the edge of a rulerj through any given point or points. 2. To take the distance between two points, and apply it to any line or scale $ or to take the length of one line, and apply it to another. 3. To measure any line by taking its length between the points of the compasses, and apply them to the divisions of a proper scale. 4. To set off any proposed distances on a given line. 5. To describe obscure circles, intersecting ai'cs, &c. G. To lay off any propoeN^d angle, and to measure a given angle, by means of a scale of chords, &c. The DRAWING COMPASSES ^ one of the legs is filled ficiently good to answer the leartier's purpose, and be should not go tnueh ttn« itt that price. ^ M^gmines or ooaiplete collection of every kind oi aseful drawing instrument, will cost from five to forty guineas. lo using the instruments, lines and figures should be drawn as fine, neat, Md exact as possible ; the paper on which the drawing is made should, if pos- sible, not be pricked through or deeply scratched with the compasses ; i% should be laid on a quire of blotting, or other paper, daring the operation \ sod the drawer should sit so that the light may be on his left, and not by any nwans in front. The drawing pen should not be dipped in tlie ink, but ink shonld be taken from the stand with a common pen, and put into it. The points of the instruments should be cleaned and wiped quite dry after they kave been used^ and every means employed to guard against rust, which will otherwise spoil the instruments. ^ In the best sort of compasses, the pin or axle is made of steel, as ako half the joint itself, as the opposite metals rnbbiug on each other are found to ivear more equally ; the points should be of bard well-'poliflbed steel, and thii joint work with a smooth, easy, and aaifoipii motion. In the dnMring eom- psases, the shifting point is sometimes made with a joint, and fusnished with a fine spring .and screw ; so that, having opened the compasses jaeaW^ to the re- qatrod extent, by turning the screw the point will be moved to the true eiLtciit within a AatV^ hreadthy for which reason they aie named Hair Cow^MiMn. There are various otlter kinds of compasses not appertaining to a common case of instruments, which are noi less nseful to Ae praatical geometrician than those we have described; vie. ' I. Bom CmnpaMiet^A imall sqrt whtdi sbat np in a hoop ; tbeir use is to it* scribe the circuniferenccs and arcs of very small circles. R ^ 244 GEOMETRY. Pam Vlll. with a triangular socket and eerew, to receive and fi»ten for use the following supplementaiy parts; viz. 1. a STEEL POINT j which being fixed in the toclDrt, makes the com- passes a plain pair^ having all the uses above described. S. A PORT CRAYON, with a short piece of blade-lead or slate pencil, finely p<nnted and fitted on it lor drawing circles and arcs on paper, or on a slate, 3. A STEEL P£N> for drawing lines or circles with ink; the small adjusting screw passing through the sides of the pen> serves to open or close them, for the purpose of drawing lines as thick or fine as may be thought necessary. 4. A DOTTER % whidi is a small indented wheel, fixed at the end of a common steel drawing pen ; from which it receives ink for the pui*pose of drawing dotted lines or cii'des. In the Port Crayon, Dotter, and Steel Pen, there is a joint for setting the lower part of the instrument perpendicular to the paper, which must be done in order to draw a line well. The PRAWING PEN is fixed in a iHrass handle, and its use b to draw straight ink lines by the edge of a ruler. The han- dle or shaft unscrews near the middle^ and in the end of' the 2. Spring CompasKs, or IXviderst made of hardened steel, haTjog an arched head, which by itf spring opens the legs ; the opening being directed by a cir' eatar screw, and worked with a not. 3. Proportional Covnptuses, both simple and compound; their nses are to di- ▼ide a given line Into any number of equal parts \ to find the sides of similar planes or solids in any given ratio ; to divide a circle into any number of equal parts, &c. 4. Trisecting Compasses, invented by H. Tarragon, for trisecting arcs and angles. 5. Trialtgular Compasses with three legs, for taking three points at Mioe. 6. Tharn'Up Compasses are the plain compasses, with two additional points fixed near the b«ttom of the legs, the one carrying a port crayon, and the other a drawing pen ; these are made with fc joint to torn op, so as to be oscd. or not, as occasion may require. 7. Beam Compassesfor describing very large circles. 8. BUiptieal Con^Muwes for describing ellipses. 9. Spiral Compasses, for describing spirals. 10. Cylindrical and SpkeHeal Compassts, or Calt/wHr, for mcasariDg the dia- meters of cylindrical andnpherical bodies, &o. && c The Btotting Pen, not being easily cleaned, soon bectancs rusty and use- less ; the best way to draw a dotted tine is fir^ to^nw 1^ Jiae^MLpeiK^^JUld then to dot it with the writing o^ dnvwiBg pen. P4itT Vlll MATHEMATICAL INSTRUMENTS. «46 upper part is fixed a fuie SUel Pint or POINTER, for making dots, small, neat, and with the greatest exactness. The PROTRACTOR « is a brass sesniciicle divided into ISO degrees* ^d nuQibered each way from end to end ; the exterv pal edge of the Protractor's diameter is called th^ fiducial edge^ and IS the diameter of the circle* the small notch in the mid^ die of the fiducial edge being the centre. The use of the Fro* tractor is to measure any angle, to make an angle of any pro* . posed qumber of degrees, to erect perpendiculars, 8w, The PLAIN SCALE is used for measuring and laying down distances : it contains on one side, a line of 6 inches, a line of &0 equal parts, and a diagonal scale. On the other side it lias a line of chords marked C, and seven decimal scales of various sizes. > The line or scale of inches has each inch divided into 10 equal parts, and is used for taking dimensions in inches and tenth parts of an inch. The line of 50 equal parts being 6 inches in length, is pro- perly a decimal scale of a foot> for by it the foot is divided into 10 and likewise 100 equal parts. By this line, and the line of inches above described, any given decimal of a foot may be re- duced into inches ', and likewise any given number of inches to the decimal of a foot. Examples. — 1. Reduce .^ of a foot into inches. 50 t Here, opposite 30 i» the second line (for M==T7^=o;:=-2) itmda 2tV inches, in the first: therefore ,^ foot =^2-^0- inches. 2. Reduce 5-^ inches to the decimal of a foot. Opposite 5-iV in the first line, stands 45 in the second ; tliere- , fore St^t inchess^ .45 foot. ^ A Protractor in the form of a right angled paraHe)ogniii,i9 not only more conveai<:ot for the case than the Mmicircular one, but likewise measures some angles with greater exactness, and is therefore to be preferred. The Protractor, Scales, and Sector, sbonld be made of either iroiy, steel, or silrer, rather than brass, for brass ii^ttres the sight when nstd long together^ especially by candle- light. The improTcd Protractor.lips an index moving about the centre, cutting the circumference, and wiU set off an angle tme to a single minute, b3 246 GEOMETRY. Pa»t VllT. 3, To find tlie value of 3 inches. Jtn. .95 foot. 4. To find t!ie value of .15 fixit. Ans. 1 ineft A- The Diagonal Scale is likewise a centesimal scale, for by it an unit is divided into 100 equal parts ; and any number of tho^e ]y.iTti may be taken in the compasses, and laid down <m pajier nilh sufficient exactness fbr most practical purposes. To explain the constnictinn and use of the Diagooal Scale, let ABCD be a section of the scale, which b equally divided (siip|>ose into inches) fiimi B (onards A in the paints E, 1, 2, ■ 3, &e. Let BC=.BE .- and let each of these be divided into 10 eijiial parts in the points marked by the small figures. I, 3, 3, 4 , &c. I, II, Til, IV, &c. also divide CF in the same manner in the points a, b, e, d, &c. and let the lines passing through B,' E, 1, % 3, be perpendicular to AB, and the lines kl, nil, mill, olV, &c. parallel to it, join 9 C, 8 a, 7 fc, 6 c, 5d, Src, Since 9 B=B/=aC, and 9 C by its inclination to 6C meets it inC, if the dislaiice of yCandifCat B, that is 9B, be called J, then will their distance on the next parallel marked / be -'-, and at the next parallel marked //, it will be — : at the lo *^ 10' next marked ///, it will be — ; at the next marked IV, it will be — ; and so Od, deeceasing successively by — , down to the point C, whers the lines meetj and consequently the dbtance is nothing. ]f 8Bbe called 3, then will the distance fh>m8ato£Coa the parulltfl marked /, be 1.^; on the parallel marked II, \-fji ■ on tbe paiallel markttti III, IrVi oa the parallet marked IV^ ) tV i and the like for other divisions. Pait Vllf . MATHEMATlCAIi ttWTRUMENTS. 24T ExAMrx.Bs.^1. Ikit it be i^uired to find 3.4. on thrscide. Here it wm be com^enieni to begm at £$ wkerefwre if the diiimee of itbe lines EFmdSfbe iakm in the compc^us on tftif jnrailel marked IF, U wUl be 3.4, the number required^ 2. To find 7.8 on the scale. Ea^tend the eompassesfrom ET to 7 h on the' parailel marked VIII, and it will be the distance. 3. To find 3.45 by the scale. In this ease we must take each of the primary divisions marked with the large figures, I, % 3, SfC.for unity, and then the smaller divisions, E I, SfC. will each represent one tenth, and the parallel differences each one hundredth; wherefore we must extend the compasses from 3 D to 4e on the parallel marked F, and it ioill be the distance required. fiooo"! r ^^ 100 I rp. .„ j 10 10 ^"^t!^ ^r 1 1 Aiid eachsuc- jUachsub- ! .1 leegsbe paral- •1 I f ""rL K I -01 flel difference .01 ^ ^^> ^ .001 &c. J L &c. The^Dlagonal Scale has the decimal and centesimal division At each end, the unit of one being double that ^f the other, for the convenience of drawing figures of different sizes *. The other side of the Plain Scale contains seven lines deci<- naally divided and subdivided ; these are called Plotting Scales, and serve to construct the same figure of seven different sizes : by the help of these we can accommodate the figure to the dimen- sions of the page or sheet on which it is required to be drawn. The number at the beginning of each of these lines shews bow many of its subdivisions make an inch. The line of chords marked C on this side of the Plain Scale^ is used for the same purposes as the Protractor^ viz. to meltsure ^'lay down angles^ ^c. The method of using both will be explained hereafter. * Tbe.laethod of diagonals was invented by Richard Chanseler, an Englisfi* Jaan, aad first published by Thomas Digges, Esq. in his Jl^, seu ScaUt Mtt^ thmatictt, London, 1573. R 4 k «4* GBOUETRV. Paht VIII. The SIiCTOR ' is nn tDfltniiDent coAftisltBg of two fl&t nilers or legs, moveabk on a joint or 9lxw, Hm rotddlc point of which 48 the centre *, it contaias all the Ikiet usually set on the Pkiii Scale, and several others, which the peculiar conetruetioB (if thlB useful instnirnent renders universal. The hoes on the Sector are distinguished into two klads, sin* gle and double. The single lines on the best Sectors are as follow i 1. A line of Indies decimally divided. 2. A line of a Foot centesiaially divided on the edge^ 3. Gunter*s line of the Logarithms of Numbers, marked n 4. Logarithmic Sines s 5. Logarithmic Tangents t 6. A line of Chords Cho. 7. . . . Sines Sin, 8. . . . Tangents Tang, 9. . . . Rhumbs Rhum. 10 I^Oitude Lat 11. . . . Hours > Ifoa. 13. . . . Longitude Lon. 13. . . . Inclination of Meridians .... In.Mer. 14. . . . Logarithmic Versed Sines ^ . . , V. Sin^"^ The doubk lines are, 1. A line of Lines^ or equal parts .... marked Lin. 2. . . . Chords Cko. 3. . . . Sines Sin. 4. . . . Tangents to 45 degrees Tan, 5. . . . Secants Sec. 6. . . . Tangents above 45 degrees Tang. 7. . . . Polygons Pol. f The first printed account of the Sector appeared at Antwerp in 1584, by Gasper Mordente, who says that bis brother TVtbrietus invented the Sector ia the year 1554. Soaic ascribe tha invention to Guide Uhal^Oy A. I>: 1568:. otbi^rs again to Jnstus Byrgias, a French matbemattcal initramcnt maker, who abo flonrisbed in the I6th centary. Daniel Speckle next treated of the Sector, TIC. at Strasbarg in 1 58P, and Dr. Thomas Hood wrote on the same mbject at London in 1^98, as did Samuel Foster, in a postbamous work pnj^lisbed at London by Leyboume, in 1661. Many others bare sioee explained the nature and uses of this instrnment ; but the most complete account of any will be found in Mr. Robert^'n's Treatise of Mathematical Instruments. PaktVih. mathematical instruments. «4® The scftles of Lines, Glierda* Skies, TangcntSi MmndMt, Lati- tudes, Longitufte»> Hoiifs, and Ind. Mend, being set on one leg oolkf, may be u«ed with the instnimaiit either ihut or Oftn, The scales of Inches, Decimals, Log. Numbers, Log. Sines, Log. Versed Sines, and Log. Taofeats» are on both Ic^ and must be used with the instrument open to its utmost extent. The double lines proceed from the centre or joint of the Sec- tor obliquely, and each is laid twice on the same face of the in- strument, viz. once on each leg. To perform operations pecu- liar to the Sector, or, as it is called, *' to resolve proUenis sector" wise,** its legs must be set in an angular position, and then dis- tances are taken with the compasses, not only " laterally," (or in the direction of its length,) but '^ transversely," or '* parallel- wise," viz. from one leg to the other. The PARALLEL RULER ' consists of two straight flat rules, connected by two equal brass bars, which turn freely on four pins or axes, fixed two on each rule at equal distances, so that the rules being opened, or separated to' any distance within the li- mits of the bars, they will always be parallel, and consequently the lines drawn by them will be parallel. The BLACK LEAD PENCIL should be made of the best black lead, and its point sci*aped very fine and smooth ; it is used for drawing lines by the edge of a saile or ruler where ink lines are not wanted. Plans and figures which require exactness, should be first drawn with the pencil, and then if they are not right, it will be easy to take out the faulty part with a piece of India rubber, and make the necessary correction -, after which the pencil lines may be drawn over with ink. The pencil is not less convenient as a substitute for the pen in writing, calculating, &c. A piece of good clean India nibber, of a mode- rate size and thickness, must always accompany a case of Mathe- matical Instruments. ff The FdraUel Ruler qauiiUy put. into a case of (jastrameatft is onLy six inches long, #Dd too small for most purposes ; the better sorts ar« from six ioches to two feet in length, and sold separate. Tbe Double Parallel Ruler consi^ of three rules, so connected that the two exterior rules move not only parallel, but likewise opposite to eadk other ; fur some account of its constructtoQ aad use see Martm'n Frincipie* ^ Per^ctive^ p. 2a. S50 * GBOUBTRY. Past VIII. The §angomg short deflcriptioo ww deemed necessaiy, tmt the uses of the InBtmaieDts must be deferredycSotil the learner has acquired suflbaent skill in Geometiy to understaiid them. OF GEOMETRY, CONSIDERED AS THE SCIENCE OF DEMONSTRATION. As the reader is supposed to be unacquainted with logic^ it will be proper in this place to introduce a few particulars taken from that ait^ which may serve as an introduction. 1. The uiind becomes conscious of the existence of external objects by the impressions it receives from them. There are five inlets or channels^ called the organs of sense, by which the mind receives all its original information 5 namely, the eye, the eaTj the nose^ the palate, and the touch : hence seeing, hearing, smelling, tasting, and feeling, are called the five senses. This great source of knowledge, comprehending all the notices con- veyed to the mind by impulses made by external objects on the organs of sense, is called sensation. ^. Pbrcbption is that whereby the mind becomes conscious of an imtpression -, thus, when I feel cold, I hear thunder, I see light, &c. and am conscious of these eifects on my mind, tbis consciousness is called perception, 3. An idea results from perception 3 it is the representation or impression of the thing perceived on the mind, and which it has the power of renewing at pleasure. 4. The power which the mind possesses of retaining its ideas, and renewing the perception of them, is called memory 3 and the act of calling them up, examining, and reviewing them, is called REFLECTION. 5. In addition to the numerous class of ideas derived by seu" sation wholly from without, the mind acquires others by refiec" twn ; thus by turning our thoughts inward, and observing what passes in our own minds, we gain the ideas of hope, fear, love, thought, reason, will, &c. The ideas derived by means of sen- sation are called sensible ideas, and those obtain^ by reflec* tion, INTELLECTUAL IDEAS. 6. Erom these, two sources alone (viz. sensation and reflec- tion) the mind is furnished with ample store of materials for its future operations; sensation supplies it with the original Pajt VIII. PRINCIPLES OP BEASOMING. «5l stock derived iVom Without^ and reflection increases that stock> deriving other ideas by means of it from within. ' 7. A SIMPLE IDEA is that which cannot be divided into two or more ideas y thus the ideas of green^ red^ hard, 96it, sweety &c. are simple. 8. A COMPLEX IDEA is that which arises from joining two or more simple ideas togettierj thus the ideas of beer, wine> false* hood^ a house, a square, are complex, being each made up of the ideas of the several ingredients or particulars which compose it^ together with that of their manner of combination. 9. In receiving its impressions, the mind is wholly passive ; it cannot create one "new simple idea : those from ^thout ob- trude themselves on it by means of the senses, and those from within, which arise from the mind's contemplating the im- pressions it has already received, are equally spontaneous and (with respect to the mind) involuntary. But although the mind cannot create one original simple impression, yet when it is stored with a number of simple ideas, it possesses a wonderful power over them : it can combine several simple ideas together, so as to form a complex one, and vary the combinations at plea- sure } it can compare its ideas, and readily determine in what particulars they agree, and in what they disagree. Having combined several simple ideas so as to form a complex one, the mind can again separate or resolve this complex Idea into its component simple ones : this it can do both completely, and in part ; it can retain just as many of the simple ideas in compo- sition (out of the number which forms the entire complex one) as it chooses, and reject the rest ; and if to this arbitrary com- bination a name be given, whenever we hear that name pro- nounced, the idea compounded of the whole of the parts pre- scribed, and no more, occurs immediately to the mind. 10. From the comparison of ideas arises what is called bela- I'lON ; and among other relations that which in mathematics is called RATIO, being a relation arising from the comparison of quantities in respect of their magnitude only. 11. In comparing several complex ideas together, we find, that although they differ with respect to some of the simple ideas of which they are compounded, yet they agree in sonw general character : thus, a triangle and a square differ with respeet to 262 GEOMCIT RY. PArr Ylil. tl^ir fonPy t)ie number of their sides* and the niimbeF aod mag- nitude of their angles ; but they agree in one general character, they are both Jigures, A lion and a sheep differ widely from each other in many particulars ; but in their general character they agree^ viz. they are both animals. IS. This most important power of the mind over its oomples ideas is called abstraction, and the general idea produced by its operation is called an abstract idra. 13. An abstract idea then comprehends in one general cl3ss> not only all the simple ideas, bi^t all the complex ones &om which it is abstracted : thus the idea of beast is a complex idea, and includes the ideas of lion, horse, bear, wolf, rabbit, &c. the idea of hnimal is likewise complex, including those of man, beast, birdj fish, insect, &c. 14. Hence an abstract or general idea is merely a creature of the mind» and can have no existing pattern or aixrhitype : we can form in the mind the abstract idea of a triangle, viz. one that shaU include the ideas of all particular triangles ; but we cannot describe on paper any figure capable of representing a triangle in general, via. all the varieties of triangles that can be made. 15. Hence also whatever is true of an abstract idea is likewise true of every particular complex or simple idea included under it i thu8» if it be pnn^ed generally tiiat two sides of a triangle are together greater than the third, it follows that the same thing is thereby pi'oved, and must be true of each and eveqf, individual triangle: in like manner whatever is proved of plaSe rectilineal figures in general, will necessarily be trUe (not only of every kind, but) of every particular rectilineal figure that can be made ; thus, since it follows from prop. 32. book 1. of foiclid, that all the interior angles (taken together) of every rectilineal figure are equal to twice as many right angles, wanting four, as the figure has sides, the same thing must be true of each parti- cular kind of such figure -, as of squares, triangles, trapeziums, polygons, &c. and likewise of every particular figure included iwder those kinds. 16. Upon an examination of our ideas of the objects that surround us, we shall find that several of them resemble each other, except in one, two, oar perhaps more circumstances > Jabt VUl. PRINCIPLES OP REASONING. 5253 now if We leave out frDio otir consideration the particulari ill whidi they disagree^ and retain those only in which they agree, we shall obtain the abstract idea of a tracias, which, as it id supposed to arise fit>m the lowest possible degree of abfitraction, is called tbk inferior species ; and the indi- viduals which compose it, being supposed capable of no subordi- nate arrangement, are called farticulars. If this idea of species be compared with our ideas of other species, we shall in lilce manner perceive that they disagree in sofne of their circum- stances only ; wherefore by leaving these out as before, we shall obtain tlie idea of a species superior to the former, viz. which in- cludes the former, and one, two, or more others. In like man^ ner by continual abstraction we pass through the sticcessive gradations of species, until at length we arrive at a point where no further abstraction is possible : the ultimate idea thus obtain- ed, .as including the ideas of all the several species, is called a GENUS. 17* Thus by successive acts of abstraction, a guinea is gold, metal, siitetanee, being ; a herring is fish, animal, sub- stance, being ; Tray is greyhound, dog, beast, animal, substance, being ; ah oak is tree, vegetable, substance, being ; James is scholar, man, anhmai, substance, being, &c. In the examples here proposed it mliy be observed, that aubgtance is common to them all J the^ Idea of substance includes therefore those of metal, imimal, and vegetable, and consequently the subordinate ideas of guinea, herring, Tray, oak, and James. Substance then is to be considered as the pRoXimatb ^envs of these, including them a\\', bring is the highest ch* superior oehus, and im-^ plies merely existence. 18. As a general knowledge of the operations Of the mind in componnding, compaiing, and abstracting its ideas, is necessary to those who would folly understand the plan and scope of Eu- elid, so it will be equ«dly profitable to shew, in as plain a manner as possible, how our abstract and other complex ideas are nnlbkled, so as to make them intelligible by words (expressed either by the voice or writing) to others. 19. And first, simple ideas are expressed by words arbitrarily assma^ as their repi^eseniatives ; so that whenever any word is read or proDoimced, the idea it stands for immediately occurs to the mind of the reader or hearer : but should it happen in any 454 GBOMETRY. Pabt VUI. iDstance otherwise, tbe object whkli jifoduoes the idea must be presented to him, and he muBt be informed that suck a word a the sign of that idea^ or should the idea have two or thiee different words to express it, these should all be.prooouDoed, and probably the idea will occur to him &om one of them : there is no other method of communicating a simple idea from one mind to another. I point a person to the object, I tdii him its name, and immediately his minil associates the latter with the idea of the former, making the name the constant reprc* sentative of the idea. 20. But although simple ideas cannot be conveyed to tlie mind by any verbal descriptioo, the case is di0erent with respect to complex ideas ; these may be communicated with great faci- lity : for since a complex idea is composed of several simple otiss, if the names of the latter be pronounced, together with thei)* mode of connection, the complex idea will immediately occur to the hearer; provided his mind be previously furnished with jts component simple ideas, together with a knowledge of the names or signs by which th^ are expressed* 21. It has been shewn, that if the difference betweea indivi- duals, agreeing in their general and noost r^nai'kable properties and circuoistances, (and which is called their nuuekal oipfeb- ENCB,) be rejected, we obtain the abstract idea of a species; if the di£ference between this species and another species (called the spec I FIG dipfrrence) be rejected, we get the ide^of a species, which includes and is superior to the former; and if in like manner we continually drop the successive specific difkf' ences, we shall at length arrive at the genus, or srunmit of oar research. 29. Hence an easy method fHresents itself of unfolding a. com- plex idea, or of communicating our con^lex ideas to other per- sons by means of definitions, namely by following a contrary order : we name the genus or kind, to this name we jpin that of the specific difference, and both together will convey to the mind of the hearer the complex idea we mean to describe. Agflia, if we consider this specie? as a genus, and join to it the next lower specific difference^ the result will give a precise idea of the nest inferior species 3 proceeding in this manner thrqiiigh all tbe suc- cessive ranks of species t« the lowest, to which jpining tbe numeral difference; We at leii^th obtain the idea of a particular PartVIII. PRINCIFLBS C»* BSASONING. S5& iDdKidual t thii proteaa U exeoapUfied in th^ defioUions prefixed to the Elements of Eudid. ' 23. It tmj be noticed^ that in Imyiog down a definitbn there 18- no necessity to- have recourse to the Kighesi genus, or even %o remote species ; the proximate superior spmiies may in ail casea be taken for the genwf, and as that is always su]^M»ed to be kuown» we have only to add to its name that of the specific differenee. 94. Thus, in defining a right an§^ed trangle^ I describe U to be a triable lisving a right €mgle : triangle ia the species or liind to which the figure belongs, and its having a right angle is tiie circumstaai^ by which it di^rs firom every other species of tri- angle. 1 do not say^ *' a right angled triangle is a being,** or " a figure,** or '^ a plain figure,** these species are too remote 3 but 1 Gall it a '* triangle,** which is the proximate speciea to right angled triangle : now the idea of triangle being previously known, that of aright angled triangle will likewise be. known by Rei- fying, that it has a right otsgle, ^5. The obvious use of definitions is to fix our ideas, so that wbenever a definition is repeated, Ibe precise idea intended by it, and no other, may immediately occur to the mind > and when- ever, an idea m present to the mind> its definition may as readily occur. S6. Adequate and precise definiti<M)s may then be considered as the true foundation of every sysl^em of instruction ; for when our ideas are fitly represented by words whose signification is fixed, there can be no danger of mistake either in communis eating or receiving knowledge. 27. There are some ideas of which the mind perceives their ^Igreement or disagreement immediately, without the necessity of t^Pgmn&at or jtt^ooff this necessary determination- of the mind IS' called a jupoment, and, the evidence or certainty with which it spontaneously acquiesces in tins determination^ is called INTUITION 3 also the. irresistible force with which the mind is impelled to its determination, is called intuitive evipenc£. 98. The feculty by which we pei'ceive the validity of self- evident truth, is Called common senae », which signifies " that instinctive persuasion of truth which arises from tHtuiiive evi- » ff See An Eiaay on th$ nature qnd iptmutahility of Truth, by James Beattie, Llf,f^^p, 1. c. 1. 856 GBOMfifiatr. pAiit vni. dente:** it is aoteeofolit to scienoe, and altko^gli no jMirt of it, yet *^ it is the foundation of all reaaonteg.** 39. There are some, ideas of which the mind cannot perceive the agreement or di8agi«emeat» withont the intervention .4if others, which the logicians call jaiddie terms ; the proper dioiee end management of these are the chief hosiness of science. 30. These midcye temiEr serve as a Chain to connect two re- tnole ideas, that is^ to connect the subject of our inqttlry with some self-evident truth : thus, suppose A aad D to be two ideas, of whicli the truth of ^ is self-evident^ but that of JD not so; and let it be admitted that J and D cannot be brought toigetheo so as to afibrd the requisite means of comparison fbr determiniag their relation ; In this case I must seek for some idtermediite ideas, the first of which is Connected with A, the last with I>, and the succeesive intervening ohes with each other :' let these be B and C; now if it be iHtuitivelif certain, that B agtces with Ay that C agrees with B, and that D agrees with C> it Mlows with no less certainty that D agrees with ^* this latter certainty is how- ever not intuitit>e, btit of the kind which is called denwnstrabk \ and the process by which the mind becomes conscious of this de- monstrable certainty is called itSAsoNiKO, or demj>nstkation. 31. Every well ordered system of science will therefore con- sist of DEFINITIONS and PROPOSITIONS : defifdtims are used to expfaiin dbtinctly the meaning of tb^ terms employed, and to limit and fix our ideas rMpecting them with absohite precision. That which affirms or denies any thing, is called a proposition : I am ).the sun shines ; vice will inevitably he punished 5 two and three are five, &c. are prc^po^ions. 39. Propositions are either self-evident, or- demon^ral4e; and since thdre cAn be no evidence tfupeHor to intukioni it !bi« lows that self-evident propositions not only requite no proof* as some have said, they admit of none '. -; I ' - ■ — -/ ^- - - .^ ^ « » I ■ « III I II III I H I I t^J—fciAjl ■ > ■ I ■»-»— i»— .<.^J»^>. ^ Every itep of a cteSaonstAtion mtist follow frtim tnitli* pmtvhuatp kn^wn -with inhtUive ctttAnty \ bnt the conclvuion or tbiof to ht proved, depeadiiig HA a ooDnected Mries of intvitioas, and no less cettein than each of the pre- ceding steps, is nevertheless not dignified with the name of intuitioH; It is obtained (as we have noticed above) by demonstration, i For every proposition is proved by means of others which are more evident than itself, but nothing can be more evident than that which is setf-evideni ; wherefore a self-evident proposition can admit of no proof. Fart WIL PRINC1FZ4CI <» RBASONINQ. 85» 33. DtmoaAahle propaaiHons fti% sudh to do not^&dttiR of « determination by any single efot of the niiad; to 9Ttiv€ lit< a consciousness, of ^Itveir^nilh^ We ate.pUiged fnequently. (as we kare obssrrcd akovo) to have recourse to several intermediate Btej^ the first«of which resta mVtk intuitive certainty on tome self-evident trutb^llfe r^ witli tie saiwe intuitive certainty 4^ pend on each other in succession, and the prifpsltiAi, or tpoth to be proved, depends with Mke intuitive certainty on the la^ of these ', so that the thing to be pijved must evident^ be true, since • it depends on a self-evident troth, which d^ndance is consti-* toted and shifnrn by a series oft- truths following or flowing from each other witk intuitive certainty. 34. Propositions are likewise dtvidyd iotO practical a«i tkeo^ T^ical. A practical proposition is that which pfoposes soma o|ieration^ or is immediately directed to, and^rminites in pfac^ tie«; thiS) to draw a. straight line, to describe a cirde» to con« |(nicta'triai^le, &c.*are pfac^ical profmtitiont^ 35. A theoretical propositioa is that in which some troth is poposed fbr consideration, and which terminates in theory: thus, the whole is greater than its fiart i contentmelit Is better tlian richcfs i two sides of a triangle are togel^r greater than the thirds 4tc. are.«^^oreitca2 proposi^isfis. ^6. ProposUioi^, both practical and the^|«ticai> ar6 either H^'^midentOY JtemfiMtrahle, 37. A »elf^€videni practical prapoiition is named by Euclid a I'OSTtJLAXE^^od a self-evident iimnreHcal proposition, an ax mac. 38. A demgHstrable practical propofition is called a PKOfiLBic^ ^KdemonstraUe theoretical propgaition^ a tdbosbm. 38. Hence, postulates and axioms being, intuitive truths or nsixims of common sense, admit of no demonstration ^ hut4>ro- hlems and theorems not being self>-evident5 therefore require to be deoiohs^ppttei^. ^ 4Q, Definitions, postulates, and axioms, m^ the sole principles on which demonstr^ion is founded ; this foundati^, narrow and sBght as it niayseem^is comiuually extended and strengthened by^^l^Ml eonstaot accession ol new materiab<j for every truth, as soon as it is demonstrated^^ hpooams a principle of equ«l force and validity with truths whieh are 8e)f*evident, and rear toniiig may^be btdlt on it with the same degree of certainty as OB Iheui: thus reasoning, fay its p^igress, continually inir VOL« II. s «& GBomerBY. pam vul creases its Iwb^iaid the powers of-tiM mind, ampie at tiiey are, must lieAce be ioadequate to the use of all that vaat aocunm- lation and mrietyr of means, provyed for tlieiK employment. 41. When ftrom the exanriaatmn and csmpaiison of two known troths a thuti follows as •& evident conseipieniee^-the known truths are called niHiiiafts^^ha-tnlth derived an ikfs* B«»CB> amk the«st of deriving it finun the. f»reiiitset is called ORAWIHG, or MAKING AN INPBRENCB. Thus» if 4«M^ and two be equ^to fopr, and three and one be equal to four, these being the premitm* it follows as an infereuoe that two and two, and ihr^ and «ae» are etpial to |)ie same ^vis, to four) : noW> since things that are equal to thi^same are equal to 0ne another, it follows as a further 'ifi^cnctf, that tW0 and two are equal to three and one, 42. This axamiAs will fiimish a general, although necessaoly an imperfect, notion of Euclid*s method of proving his propo* sitions : his demoikitratiows are nothing more than a regular a«$ well connected chain of suecessive intuitive inferences, the first of which is drawn from self-evident premises, and the last Hw thing which was proposed to be prored. 43. Hence, although demonstration is necessarily founded on self-evident truth, it is noi at all necessary in every case that Ave should have recoipse to first principles^ for this woukl make de- monstration a most unwieldy machine, requiring too mueh la- bour to be of extensive use : every inference fiedrly drawn from self-evident principles is of equal validity with inl^itive truth, and may be employed for the same purposes ; thus £nclid> in his demonstrations, makes U8# of the truths he has befose do* m<Mistrated with a confidence as weU foiinded as though Uiey were self-evident, and merely refers you to the proposition wheit the truth in question is proved. This saves a. great deal of trou* ble, for truths once established may with the stnolwst propriety be employed as principles for the proof and discovery of others. 44. It frequently happens in the course ;(if a demonstration, that an inference presents itself, which is useful in other cases^ although not imaawdiately so with respect to ther proptosillon snder conskleration ; when such 'an inlsrence is made, it is ealled a COROLLARY, and the act of making it naouciNG a cobollart. 45. A LEMMA b a proposition not immediately connected with the subject in hand, but is assume4 for the sake of shorteniflig Pabt WIL on EUCLID'S JPIRST BOOK. 86^ <]it draiom^Mien of ofte or nunrt of the isttoNiiig pn^osi^ tiOBB. 46. A SCBQUUM lA a note or oUeryaUon, aefving to coBfiroa» explahi, illustrate, or apply the subject to which it refers. 47. Euclid in his fifemeots ewplo^FS two methods for establish* iog^ the truth off what he intends to prove f namely, direct and imdirett, both proceeding hp a series of inferences in the manner explained above. Art. 41> 43« 48. A DiftECT DBMONSTKATioN is that wiiich proceeds from intuUive or demomtrated truths^ by a chain of successive infe- rences» the last of which is the thing to be proved. 49. An inoirbct or apologicai. d&monstration, or as it is frequently named, aanucTio ad . absukdum, consists in as- suming as true a proposition which directly contradicts the one we mean to prove -, and proceeding on this assumption by a train of reasoning in all respects like that employed in the direct method^ we at leogtk deduce an inference which contradicts seane self-evident or demonstrated truth, and is therefore absurd and Mse ; consequently the proposition assumed must be false, aoA therefore the proposition we intended to prove must by a necessary consequence be true, since two contrary propositions cannot be both true or both false at the same time ''. NOTES AND OBSETEIVATIONS ON SOME PARTS OF THE FIRST WyOK OF EUCLID'S ELEMENTS. 5<X The first book of Euclid's Elements contains the princir pies of all the following books ; it demonstrates some of the most general properties of straight lines, angles, triangles, parallel lines, parallelograms, and other rectilineal figures, and likewise the possibility and method of drawing those lines, angles, and figures. It begins with definiiiom, wherein the technical terms necessarily made use of in this book are explained, and our ideas k Mathematical demoostrations " are notbing more than series of entliy- meines; «Tery thing is concluded by force of syllo^sm, only omitting the {[reinises, which either occur of their own accord, or are recollected by means of quotations." This might easily be shewn^ by examples,' but the necessary «xplaoatioii8, &c« w<^iihi take up too much room. See on tfali subiect The Ble- menU tf Logic, hy. PTmsm Jhtneemy Professorvf PhOmoplaf in thM Maritchmi C^Utge t^ ,M9riecM, 9tb £4. a book wbiob ought to be recommended to the pemsal of- students in Geometry. s 2 «M *€»OMETRY. Past Vlli respecting tliein Mcertained and fixed; next are 1^ down tlie poitulates and axiomsy or those self-evident truths^ which consti- tute the basb of geometrical reasoning i and lastly, the propo- tUions (whether problems or theorems) are given in the order of their connexion and dependance^ the denionstcatkms of which depend solely on the definitions, postulates, and* axioms, previ. ously laid down 3 and from the demAnstratiQas uae^ corolla- ries ure occasionally derived. X)n the Dejinitions, * • 51. Definition 1. The first definition, as given by Euclid, and likewise in Dr. Simson's translation, has beerf justly com- plained of as defective 5 it includes no positive property of a point 5 we learn from it not what a point is, but what it is not 5 " it has no parts, nor magnitude :** now since every adequate definition admits of conversion, let us try the experiment on this ; when converted it will stand thus, " that which has no parts nor magnitude is a point;" but this is' evidently untrue, for although a point be without extension, that which is without extension is not necessarily a point, it may be nothing. It is therefore necessaiy to substitute another definition of a p6int, which shall include a positive property as well as the ne- gative one above described; this will help the student over a difiiculty, which (notwilhstanding Dr. Simson*s illustration in his note on this definition) might have discouraged him in his first attempt at Geometry. Instead then of Dr. Simson*s defini- tion and note, let the following be substituted : 52. Def. *' A point is that which has position, but not magnitude V* 53. The idea of a point (as above defined) is evidently an abstract idea : a mathematical point cannot therefore be made on paper or exhibited to the eye ,• we may indeed represent it by a dot, but this dot, make it as small as you possibly can^ will have lei)gth, breadth, and thickness too; still it may be used as a m>ark or representation of position or situation, shewing simply 1 This impnvettetit wat probably first sn^sested by Dr. Hooke, who say% that *' a point ba» pMitiou, and a relation to roa^aitade, bat has itMlf no magnitude \" his id«at on this snbject have>been adopted by both Plfyfiiir and Ingram. ^ PWT WII. ON EUCLID'S raaST BOOK. ^1 to where, ear ftom wheoee, lines a*e to be drawn, distanced esti- mated, &c. A point then, as made on paper, is to be considered as a mark indkaiing merely position } this mark must necessarilf have magnitude, but it is made the representative of that which has Dot. 54. Dtf.^. ** A line is length without breadth." The obser- vatidDs contained in the foregoing article may with equal pro- piiety be applied to this defioi^o,. To repre§mt a mathematical line, which is without breadth or thiekness, (or rather to repre- sent the idea of such^ line,) we are obliged td have recourse t« Inline which has ixith. The line w« draw on papejr is not there*- fore the line we have defined^ but merdly the mark by which the iito of such a line is represented. Th» abstract 4dea of length (without breadth and thickoess) is perfectly familiar to every one; thus, if it be asked, " what is the length (or di&tance) from hence to London ?** the answer is, " thirteen miles :" ^his would, as we might suppose, be satisfactory; but should the mquirer farther ask, how toide 9 or Aoto thick ? every one wqpld yiy or despise him for his stupidity. « 55. I>e/l 3. This is not properly a definitwn, but an inference from the two former, for '^ that which terminates a line can have no breadth, since the line in which it is has none ; and it can have no l^ngth) for in that case it would not be a termina* tien, but a part of that which is suppo^^d to be terminate," and would Gonsequeiatly itself have terminations or extremities : wbenee the termination of a line can have- no magnitude, and having necessarily position* it must therefore be a point, by Art. 53* b^.Def. 4. Wfi bave before remarked, (Art. 7, 19, 20.) that a simple idea -admits oi no definition ; .no definition can possibly be gtieR of stfoightnessi to lie ^' evenly between its extreme points** is a very awkward paraphrasis of the word straight, and will not perhaps be so well understood by a learner, as the defi- nition would be were it to run thus, " a straight line is that vhkh i|> not erookedf* 57. Hence it follows, that "z straight line is (iie shortcut dillance betwaen its extreme points/' this h^ bten proposed instead of £uclid*s defiaitioa b^i^some, but it haspheen objected to by othefii^ Professor Flayfi^ir has^iven the foUowing* l^hich is Q«tfainly an itiprovemeatj viz. '' lines whieh cannot poi^cide s3 202 GEOMETRY: Pakt VUI- in two points, witliout coinotding altogvAher, 9Stt caBed lines ;" but it msKf be added, tbaK neither of the two krfter defi- nitions is suffidently simple and perspicuous to stand at the be- ginning of a system of Elements. 58. All other lines besides straight Hnes are called curve Uim, or simply curvet ; and henoe we define curves to be '* those lines which do not lie evealy iietween their extreme points," or **iprhich are not the shot test distanee between thesr extreme points.** 50. Def. 5. We have shewn that the id«R of length only (or of what the mathematician* call a line) is perfectly fuiuliar to every one ; the idea of a superficies (or of length and breadUl without thickness) may be shewn to- be equdly so : in calculating the content of a field, it is well known thatt the superficial c0k* tent is always understood, in which length and breadth onjy are concerned 5 thickness does not enter at all into the consideration. eo. Oiir ideas of a geometrical solid, superficies, line, and point, are obtained by abstracti(Mi« (See Art; 1^ — 17.) Thus in ^ontemplatii^ any material body that first offers itself to our consideration, we shall find that liesides being made up of mUh ter, it has extension, or, length, breadth, and thickness ; now, if from the complex idea of this body, we exclude the idea of matter, there will remain the abstract idea of extension, or of length, breadth, and thickness only, namely, of that which in geometrical language is called a solid. If from the complex idea of this solid we exclude the idea of tbtfikness, we thence obtain the abstract idea of length and breadth only, or «f a geometrical superficies. Again, if from the complex idea of a superficies we exclude the idea of breadth, the result will fiu> nish us with the abstract idea of length only, or of a geometri- cal line. And lastly, if from the idea of line we exclude that of lefl^h, " we get the very abstract idea of a pmnt: though I confess,'* says Mr. Ludlam, '' the operation of the mind in this case is so very subtile, that it can hardly be distinctly and clearly traced out." 61. Def. 6. To this definition we may add, that if thft extre- mities or boundaries of the superficies be straight lines, it hi called a rectilineal superficies -, if curves, it is caBed a curviliimal superficies ; and if some of the boundaries are straight lines, and the rest curves, it is tailed m mvatilineal superficies. 6% The defiokioa of a plane superficies, a» originally i^tveB PiitVIIL ON £UCLm*S FIBIir BOOK. 363 by Euclid, is as faXtaw. *' A plaM tuperDcics is tliat which lie$ evenfy between its extreme lines ;*' the term '^ lies evenly" has already been objected to as obsoure. (Art. 56.> Or. tkoaon, convinced of its impropriety^ has subotiluted another definition, which has the advantage of indudiag^ the esMAtlaL property of a |»lane, and consequeatiy of distiiq^i^ing it fiom every other knd of superficies : for be«des a plane, theve are various kinds of superficies, as the spherical, cflmdricml, amical, and many others. Aooording totliis definilioB, a plane superficies " is lliat in which ami^ two points being taken, the straight line between them lies wholly in that superficies^" the term '^ plane," in popular language, means that which is perfecilf fiat, or kndh ■owiftwo points be taken in a sifierficies which is not perfectly flat, it is plain that the intermediate parts of the straifht line, which joins those points, will foll-either above or behw the super- ficies ^ we see moreover not only the propriety, bat the absolute necessity^ of the distinction ^' any «wo points," for two points may be taken (i» one partieolar direction) in tha. sucfcce of a •one or cylinder, which will agree with the definition, but not amf two pohuts. ^. Def, 8. To give the jj^arner an id^ of what is h^ire meant by the teni|^ '' angle," or '' indinatloii of two lines," it will not be aimiss to have recourse to a fiuniliar exampla : let a pair of com- passes be opened to several different extents, these will be so OKiny different angles 5 when the legs are opened to but a small distance^ this opeuii^, or (as it is here called) iocliQfltion of the legs, will be a small angle i when opened wider, the legs will form at their meeting a larger angle than before, and so on. 64. The two lines which fbrm>4ftff (as it is usually expressed) contain an angle, are sometimes called the legs. The m^nitude of any angle dpes not at all depend on the length of the legs, or lines which contain it 3 in the example above proposed, the legs of the eon^passes may |m^ an in<^, a foot, 4^ any other length, or one. may be longer than the other, and yet the o|{|»« i«g, vni^inatiotti m an^ contained by them may still remaui the same. 65. De/. d*." The object of the eighth definition is to define * <' Tiie fint nine defihitloiis might batir been gittd in HA form of an inin^ 4acyon,liir nowof then are 9cometrical|.«Ke(t the u§M0 » inended by S 4 264 ClK)MFraT. BartVIIL in general every anglB wfaieh caA be described on a f^nei ivhether such angle be contained by straight or oorve lines j but since o^vilineal angles-are not treated of in the EtementSi that definition might itaVd beep omitted. - Ii|- the ninth, where *' a plane reotillti«al€»gie'* is defined, the word *^ plane'* is a redun* dancy ; for the angular point, as well ar every point in the lines whitth contain any vectilin^l angle, must necessarily be all in one and the same plane, a» is -proved in the second propositicm of the eleventh book. The note subjoined to Def. 9. in th« Klements is merely to shew how we .are to read, write, or to determine the place oP an angle when it is read ta us : if an «ngle be expressed, by three letters, as is usual, the anguh^ point is alwfiys lihderstQod to be at the mt4d|? letter ; > thus, if JBC den^e an angle, 4liis angle is alwaye understood to be at the middle letter A and not at either Aor C, • ^ ^. Def, 10^11, 19. When a straight toe meets another straight line, (without crossing or tutting it,) two angka are ibrttied at the point where they meet; if the8» angles be equal to each other, they aris called r^ht angles: but if one be greater than the other, the forcoer (which is greater than a right angle) is eelled^an d)iu9e angle; and the fctter (which is less than a right angle, see prop. IS\ boi^l,) is called an acute mgle. '67. Def, 13. In <he sense of this definition, pdnts are 'the boundaries of a line, lines iof a superficies and superficies of a solid. 68. ii$f. 14. Hence, according to £u«elid> neither a line nor an angle can be called a figure, because they are not either of them ** tndosed by one or more boundaries." t)r. Simsoo ;'* this is Mf. Ingram's opinion, and be a^ds, <' The t«rms by which a line and a super^cies are defined, give some explanation of the meaning of the^e words, but give no geometrical criteria by which to "know them ; and the best way of accfiHring proper'ide^s of them, is by coostderMig their relation to a BoUd and -to one another, as Dr. Sims«n has done.*' See on this dnfaiject the note on Def. 1 , iSknvm*9 Eutiid, idth £il. p. $80. A defiatfion then may he said to be geometrical, when it furnishes some criterion t« which we may refer, and ]|y which the idea of the thing defined m^^ be completely arrived at and ob- tained, at the rMuIt of any demonstration where it is concerned: other defini- tions are usually called metaphysical; they are employed in all d^s where gieometrical 4l«fiintion8 cannot l)e ^l#kn, as necessary for explaining in the beet pnanner poUibtc Hie aature of tlpt^thii^ defined, the meaatog of terma, fto. PitT Villi ON EUCLOrS nSST BOOK. MS ^. B^. ISff^We have here a complete and Mkkctorf in« <tance of the method of defining a species by means of the genus imd special difference. (Art. 23, 24.) '' A cireU i$ a plane figurtt' it belongs to that class of figures, which have dSX tbeir parts in the same plane, a|id consequently agrees in this general character with a triangle, a square, a polygon, an ellip-^ sis, &c. it is *' contmed by one line caUed the cvrcumftrmee}* kere we have a limitation whereby all such figures as are con* taioed by more than one line, as the triangle, square, polygon, &c.are excluded; '^ and u such that all straight lines dtaam ftom a eertam point within the figure'* (called in the next follow- ing definition '^ the centre"*) to the circumference, are equal to Me another : this latter clause operates as an additional limita* tion, which excludes the ellipsis and all irregular curvilineal figures from the definition, because there is no point in either of those figures, from whence all the straight lines drawn to the circumference are equal. Here then we are informed, first, to what general class of figures a circle belongs, and secondly, by what it dififers from every other figure of that class; whence the definition furnishes us with an adequate and precise idea of the figure called a circle. 70. Another definition, in substance the same as Euclid's, is this ; f A circle is a figure generated (or formed) by a straight line revolving (or turning) in a plane about one of its extreme points, which remains fixed," the fixed point being the centre^ 9od the line described by the revolving point the circumference, 71. The circumference of a circle is likewise called the peri* phery : it is sometimes improperly named the circle ; a circle, in the proper acceptation of the term, means the space included within the circumference, and not the circumference exclusively, jl. To describe a circle with the compasses^ you have only to fix one foot at the point where the centre is intended to be, and (the compasses being opened to a proper extent) turn the other (sot quite round, and it will trace out the circumference. 73- After Def. IJ. add the following, which is in continual use, viz. 'f a radius, or semidiameter of a circle, is a straight-line drawn from the centre to the circumference." - 74. Def, 18, 19. Any part of a circle cut off by a straight In}^, is called a segment of a circle; if the straight line pass throu^. the cen|^, it is a diameter, (Deftpl7.) and divides the 9M GEOMKTKS. tA%r VllL cirde ioto two c^iia/ segaKiits, criled mmkr<Mk$: hot if liie ftnig^t line wbieh cuts tlie cirde docs not pass tfaroi^h tke centre, it will divide the drde into two um eq^ l segments, the greater of which is said to be'' a $egmemt grmtimr than a sema- circfe," and the less '' a MgfweiKl len than a mrnrdreUT Bfthe terms '' segment of a didr/* and *' sCTn-drde,** we are alwaqrs to undeistand the tpaee induded between a port of the circoni' ference and the stnight line by which that part is cot off, unless the contrary be expressed. 75. Any part of the drcumfeienoe is catted on are^ and the straight line which joins the extremities of an arc, (or which divides the drde into two segments,) is called a chord, Wz. it is the common chord of both the arcs into which it divides the whole drcumference. 76. Def. 23. We have nothing to do professedly with poty« gons in the first book, yet since the definition is introduoed, it may not be improper to observe, that a polygon, having all its sides and angles respectively equal, is called an equilateral, «^t- angular, or regular polygon. These figures are named according to their number of sides } thus, five, 1 pa Pentagon, a Hexagon, A polygon having ^ seven, y sides, is called ^ a Heptagon, an Octagon, &c. 77. Def. 24, 25, 26, 27, 28, and 29. Triangles are distin- guished into three varieties with respect to their sides, and three with respect to their angles : the three varieties denominated from their sides, (as laid down in Def. 24, 25, and 26.) are equi- lateral, isosceles, and scalene; the latter, although defined here, does not occur under that name in any other part of the Ele- ments, llie three varieties which respect their angles, are right* angled, obtuse-angled, and acute^angled, Def. 27, 28, and 29. 78. Def, SO. A square, which according to this definition '^ has all its sides equal, and all its angles right angles/* must evidently be just as wide as it is long ; hence there can be no such thing as a long square, although we read of such a figure in some books ^ ' ll.llll ■! Illll II 11^ I. Ill I I ■■.»! ■ 1 » Euclid's deSmtion of a tqaare may be coosidered as iaulty, for wHb ithe essential properties of a square be has incorporated ao iiifarcmce, wbicb is tbe Pmt VIIL on EUCLIVS HBST book. 967 79. Def. 31. Since the wOfd Mot^ does not cmce occur in mxf rab6ei]uent pmrt of tlw Etements, it ehoukl not bave found ft place here. The figure defined k ft species of that which Is called in the second bookt and elsewhere^ a rectangle. 80. Def. 35. In the definition of parallel lines as here bid down. Dr. Simson has iroprored on Euclid, and his definition is better adapted to the kamer's comprehension than either ttt those approved by Wolfius> BosooTich, Thomas Simpsony D*Alembert, or Newton ; the truth is, that no inferenoi can be dnmrn from any definition hitherto given, sufiicient to fix the doctrine of paralki lines on the firm basis of nniAjectiooabla evidence** tabject of the cor. to prop. 46« b. I. It would be more ttrictly scientific to doiiDc a tqvaro to be << a four-eMed ignre having all ite tWiet eqtuil) and eMt of it# anf let a dgfat Juagk ;" for that " an eqnilateral foar-eiiled figure ie a parallelogram," and that " erery parallelogram ha?iBg one right angle has all its angles right aiigles," are plainly inferences from the definition given in this note, and that of a parallelogram, prop. 34. b. I . the like observations extend to Def. 33. In both instances Euclid has abandoned his own plan, and transgressed a rale which od^ht never to be violated wHhont absolute nc- eesvity ; the d^g^rtnre is however juttifiable in the present instaaoc, as Euclid's definition wiU he more easily understood by a beginner than that which we have proposed. • Having explained the definitions as they stand in Euclid, we may bo allowed to remarh, that a more methodical arrangement of them would be a desirable improvement; should any future Editor think this hint worth his attention and adopt it, it will be conducive to ckganee, correctaess, clear- ness, and slmpKcity, which are undoubtedly points of importance, especially at the beginning of the Elements. The alteratioas 1 would propose are as follow: Def. 18. A segment of a circle is the figure contained by a straight line, and the circumference it cuts off. 19. If the straight line be a diameter, the segment is oalkd a semicircle. From the 20th to the 29th inclusive, may stand as at present. 30. Paraltel straight lines are such as are in the same plaae, and which, bein||L^ftfbdoced ever so far both ways, do not aoeet. 31. A parallelogram is a four-sided figure, of which the opposite sides aaa parallel. 39. The diameter or dkigonal of a parallelogram is a straight line wUflb joins any two of its opposite angles* 33. A rhombus is a parallelogram whfch has all iu sides eq««l^ but its anglea are not right angles. M8 OBOMKnur. . PaktVUI. On the P^tMatet. 81. A postulate, as we have befcx^ •bserv^ed, is a self-«Ttdent practical proposition : on this subject Mr. Ludlam very justly remarks^ that ** Euclid does not here require a practical dexterity in the management of a ruler and pencil^ but that the postulates are here set down that his readers may admit the pasminUiy of 'what he may hereafter require to be done.'* On this we remark, that our conviction of the possibility of any operation depends on our having actually performed it in some particular instance ourselves, or known that it bto been performed by others $ hav« ing thus satbfied itself of the possibility in particular instances, the mind immediately perceives that the possibility exteads to every instance^ or that the operation is true in general. On these considerations it has been affirmed, that " the mathenu- tical sciences are sciences of experiment and^i)servatioQ, founded solely upon the induction of particular fects, as much so as mechanics, astronomy, optics, or chemistry ^" T^s doctrine, to Its fullest extent, it would perhaps be unsafe to adopt. 82. In applying the postulates, we proceed in an order the converse of that laid down in the preceding article : we admit what is affirmed in the postulate to ha true in general, 1. e. in all cases $ and since it is true in all cases, it follows ai a necessary inference, that it is true in the particular case under considera- tion. We will now begin to exemplify the use of the mathe- 34. A rhomboid is a parallelogram of which all its sides are oot equal, nor any of its angles right angles. 35; A rectangle is a parallelogram which has all its angle# right angles (or ^icb has oae of its angles a right angle ; see the foregoing note,) 36. A square is a rectangle which has all its sides equal. >97. All other four>sided figures besides these are caUed trapesiums. Note. A trapezinm which has two of its sides parallel, is sometimes called a tiapesoid, and a straight line joining the opposite angles of a trapesium is ealled its diagonaL ThedefinitioM preoeding the 18th might stan4 as they do at present, if instead of the first definition, that which we fawe proposed (see Art. £)^)i»were adopted. P The postulates prefixed to the Elements are in number (as they ought to be) the fewest possible; for, as Sir Isaac NewtoQi observes, « postulates are principiss which Geometry borrows from the arts, and its excellence consists in the paucity of them." The postidates of £nctid are all problems derivted from the mechanics. Ingram. P4iT VUl ON EUCXID3 FIRST BOOK. S6S> matfeal instmiiients, to afford the student an oppoftunity of practteal as well as mental improvement. 83. Postulate 1. If it be granted, that '' a straight line may be drawn from any one point to any other point,** it follows as an evident consequence, that a straight line can be drawn from the point jti to the point B. Lay a straight scale or ruler, so that its edge nuiy touch the two proposed points A and B, then with a pen or pencil draw along the edge of the scale or ruler a line from A to B, and what was granted in general will in this par- ticular instance be performed. 84. Post. 9. To produce a line means to lengthen it. A straiglit line of two inches in length, may according Xm this postnkte be produced until it is three, four, five, or more inches m length. Lay the edge of your scale touching every point of the ^ven line, and with th& pencil or pen, as before, draw the line to the length proposed. 85. Post, 3. Bsttend the points of the compasses to the re- hired distance, then with one foot fixed on the given point as a centre, let the other be turned completely round on the paper, and it will describe the- circle required. On the Axioms. 86. An axiom is a self-evident theoretidfeil proposition, which neither admits of, nor requires proof. Axioms evidently depend in the firait instance oh particular observation, from whence the nund intuitively perceives their truth in general : Hke the pos* ^tes, these ^neral truths being previously laid down and ac- knowledged, are applied to the proof c^ the demonstrable pro- poftitioDs which follow. 87. Axioms I, 2, 3, 4, 5, 6, 7, 9, and 10, are too plain to Kquire illustration ^ the 10th is what is usually caUed an identi- cal proposition, amounting to no more than this, namely, that ''all right angles are right anglies.** 88. Ax, 8. i&oald the learner feel disposed to hesitate at this ^on, he may be informed, that every one readily admits its truth in practical matters ; a farmer who has two quantities of com, eadii of which exactly fills his bushel, would be surprised if any one should deny that these two quantities areequid to each other. 89. The Jf^th apdom, ios^t is called, Is not propeiiy an axiom, but a jMpQposition which requires proof ; the learner, if he can- 870 GEOMETRY. FastVUL not readily uiKler9Ca»d its import, may pass on until he has read the 2Sth proposition : it mmt then be resumed as necesBiny to the demonstration of the 99th. On the Propositions, 90. The propositions in Euclidj we have before shewn^ are either problems or theorems ; the problems shew how to per- form certain things proposed, and the theorems to estaUish and confirm proposed truths : both reipiire demonstration, and the process is nearly the same in both ) indeed proUuus may be changed into theorems, and theorems into pnoblems, by a slight alterai^ion in the wording. The demonstnition of the first ^irepo- sttion depends solely on the definitions, postulates, and axioms ; that of the second proposition on these and the first, and so on : the truths obtained by the proof of propositions being always employed, where necessary, in succeeding demonstrations. 91. Every geometrical [Ht)po6ition may be considered as com- prehending three particulars, viz. the enunciation, the construe* tion, and the demonstration. The enunciation declares in gene« ral terms what is intended to be done or proved. The con- struction teaches to draw the necessary lines, circles, &c. and applies the enunciation to the figure thus constructed. The demonstration is the system of reasoning which follows, whei^by what was enunciated is clearly and fully made out and proved, . 92. Tlie numbers and letters in the margin are references te the proposition, axiom, postulate, or definition, where the par^ ticular cited in the corresponding part of the demonstiutien is to be found, or is proved ; thus 1 post, means the firat postulate -, 15 def. the 15th definition; 3 ax. the third axiom; 2. 1. means the second proposition of the first book, &c. the first number always referring to the proposition, and the second to the book. 93. Before the student begins to learn the demonstration, he mu^t be able to define accurately all the teems of science which occur in the proposition, and to repeat the postulates, axioms, enunciations, &c. referred to in the margin -, next, the enunci- ation of the proposition must be well understood and learned by heart : ali this will, in a very short time, become perfectly eiat^y. The construction of the figure comes next^. th^ figure should be inade solely from (he directiona wJbinh immediately follow the ^mmcifitlpai i£ thia be, thmightdiffioidt at first, the figure in Paut VIIL on £UCUD*S FIB8T BOOK. S7I £uclld may be taken as a guide : every part of tbe figure may be drawn by hand, avd the more accurately thia b done, the better will it assist the recollection ; the instruments may be employed for this purpose, but they are not ahtolutekf necessary^ M the truth of any proposition does not in the least depend on the accuracy of the construction : letters must be made at the angles and Qthor prominent parts of the figure ; these liay (at first) be copied from the figure in Buclid. Lastly, in order to prepare the way for demonstrating the first proposition, as well as some of the following ones, in a complete and satisfectory manner, it will be necessary to premise the three following axioms : 94. Axiom 1. If a point be taken nearer the centre than the drcumferenoe is, that point is within the circle. 95. Axiom 2. If a point be taken more distant from the centre than the circumference is, that point is without the circle. 96. Axiom 3. If a point be taken within the circle, and ano* ther point without it, any line which joins these two points will cut the circumference. 97. Previous to attempting the first proposition, the student must be prepared (agreeably #o what has been said in Art. 93.) to answer the following questions : viz. what is a proposition ? (for the answer, see Art. 31.) what is a problem ? (see Art. 38.) what is a point ? (see Art. 52.) what is a line ? (see Def. 2.) mhat is a straight line ? (see Def. 4.) what is a triangle ? (see Def. 2L) what is an equilateral triangle ? (see Def. 24.) what is a circle ? (pte Def. 15.) what is the firftt postulate ? what is the third poalmlate ? what is Euclid's first axiom ? — We will now shew how the first proposition ought to be demonstrated. EnunciaMon, M. PkioposiTioN 1. Problem. To describe an equilateral triangle upon a given "^ straight line. {See the figure in Euclid,) JjbH AB be the given. straight line ; it is required to de* scribe an equilateral triangle upon it. 4 la Eactid it is << a given Jlnite $trst|;bt line $" here tbe word " finite*' is •Qpcrflaoos, for whatever is given must of necessity be^nite; a line is said to he « given," wMli wwtb«r lidi^eqQal to it can iM aetnallf dmwa ; (see EucUd's Basa, ^>iir. i .) hat who man diaw a line equal to aa tafioite line ? «W " GEOMETRY. ' PartVIIL Construction. Sroitt the centre J, at the distance JB, describe* the circle BCD, by the Sd postulate ' j and from the centre B, at the distance BA, describe the circle ACE by the 3d postulate; these circles mil cut one another, by Art. 94, 95> 9(y ; then from the point C, where they cut one another, draw thcr straight lines CA, CB to the points A and B, by the 1st postulate i ABC shall' be an equilateral triangle. ^ Demdnstfatiort. Because the point A is the centre of the circle BCD, AC is equal to AB, by the Ibth definition^ and becaiuse the point B is the centre of the circle ACE, BC is equal to BA, by the nth definition: therefore CA, CB are each of them equal to AB; but things which are equal to the same are equal to one iano- ther, by the 1st axiom; wherefore CA and CB are equal to one another, being each equal to AB; consequently the three straight lines CA, AB, and BC are equal to one another, and form a triangle ABC, by the ^Ist definition, which is therefore equilate- ral, by the 24th definition, and it is described upon the given straight line AB, because AB is one of its sides. Which was required to be done* 99. With* similar accuracy every proposition in the Elements ought to be demonstrated ; the difficulty of acquiring a habit of strict and close reasoning would by this practice very soon ht surmounted^ and the powers of the mind gradually strengthened aod enlarged. 100. Prop. 3. Having read over attentively the demonstra- tion, it may perhaps be objected, that in drawing the straight line from A, we are confined by Euclid's figure to ofie part icuW direction AL ; the proposition seems at first sight to be limited in this respect, but it is not so -, for if from ^ as a cen^, with the distance AL, a circle be described, straight lines may be drawn from the centre A to the circumference in everjf direc- tion by the 1st postulate, and each of these lines will be ^ual to , AL by the 15th definition. 101. Prop. 2. and 3. have Oeen objected to as sufficiently evt-^ ' Tbe Mbteaces ia- Italic «rc not in Euclid b«t thef^ife n^cesaaff, awl thonld be ioppUcd by tfaii'«t«de«t m m pwai that be^iMi4ctt(a«dt lis flMbjecw ' \ Part VIIT. ON EUCLID'S FIRST BOOK. S73 dent without proof; but it appears to have been the design <^ tlie aneient geometers to erect a oomplete system of science on as harrow a basiB as possible : hence E^cUd lays down aelf-evi* dent principles which admit of no demonstration^ and of these the fewest mimber possible that can be talcen to efifect his pur- poae; by means of which and the definitions he demonstrates ail such of his proposiHont a$ am $ittcepiible ofproqf, without re- gard to their being easy or difficult, or to the degree of ^videnoo inth wfaidi their truth may «c first sight appear. 108. The third propo^ion being mueh less difficult than either the first or second, iSL may be asked, why was it not pot first? The answer is, the pvopf ctf this proposition d^iends on tke Moond, and that of the second depends on the first, and iifccemt^ depmAence is the only ord^r that can possibly be at- tended to in any connected system of reasoning. lOS. The following lemma should be understood before the fourth pn^tosition is attempted. Lemma, Let LMN, PQR be two equal angles, and let them be applied to (laid upon) e^ch other, so that the paimi M may coincide with the point Q, and Uie straight fine q ML with the ^ ^ straight line QP; then ^ill MN fall upon Qft. For if LMN be appli- j^/ "N ^ J^ JSi «i XQ PQR as above, and MN do not fall upon QR, let it fell otherwise as QT, then the angle litf AT becomes PQT; but LMN is by hypo- thesis equal to PQR, therefore the angles PQ T and PQR are equal to each other, the greater equal to the less, which is akurd ; ther^oi*e MN cannot Ml otherwise than on QR, which Wfts to be shewn. 104. This kind of proof, we have already observed, is what is called " reductio ad absurdum." The method of proving the equality of two figures by laying them one on the other, and shewing that their conrespondlng parts ooipcide, is called supri^- poiUim, and has been ol^ected to, not from itsMrant of evidence^ but beeause it has been considei^ Ungeometrical, as depending OB BO poslisllatei indeed we are no more.boimd to admit the VOL. II. T * 274 GEOMETRY. Pakt VIII. INMsilMlity of appljin^ one figure to mntitber, tiMm we wart to admit the poasibtlity of joiamg tifo points, pt wh i rin g o stniglit line, or describiiig; a drde: henee a |M»fla te to that effect secoM o ccttoju y • ' 105. Prop. 4. This and the cightli are important propDritions, as on them depends the whole doctrine of tiian^es; thef aie both proved by sop^^lasit]ony which has becQ ex|daineQ sbove. '"^ it is worth while to reniarfc," sagfs Mr. Lndbm, " with what cantion and aoooracy all fiodid s prapositioos are wonkaL A earelws writer might say, tf two hima^tt kmoe two nde$ tmd an tmgU equal, then the third side of lAe one will ba eqmmi io the third tide of the other, Sfc, But Eiidid cantioDB yoa not oidy that tlie sides most lie eqoal each to each, but also that the angier spoken of must be tliat which Is eoalataed Ay itie respectively equal sides. We vnXi shew tint two triangfes maj have (as was mud) two sides respectively eqoal, and also one angle, yet neither their thmi sides nor the figures themselveB wiU be equal.*' 105. «< Let JBC be an isosceles triai^, J the vertex, BC fhe base« -Produce the base BC to A and join jiD; then we shall have two triangles formed^ viz. ABD and ACD, having two sides and an angle respectively equal ; that iB« the side JB in the tiiangle ABD, j. equal to the side AC in the triangle ^CD; also the side AD common to both triangles. The an- gle ADC is also common to both triangles j yet ^^ ^^- ^I^ ' « Evclid/' says Mr. Ingram, ** never supposed any thing to be possible which he hat not before shewn to be possible ; this was not merely to avoid . impottibilities, ai sone allege, bnt to secure evidence, and to nake hit reader as certain of bis coocliisions as he himself was." Miem. ^ JEttfRd^ p. S8K It mast be confessed Ihit it Euclid's general rule, to which the Instance ia ques- tion is undoubtedly an CMeption, notwithstanding the great difficulty Mr. Playfair finds in admitting the fact : to avoid it, the learned Professor hat sfiewn how the fourth and eighth propositions may be proved withoqt the aid of suprapotition ; but the postulate he requires for that purpote cannot coasts- tently with geometrical correctness be gsanted, becaate it is a de m aO iUa Me propositiott. CVmipare hit pot€«Ml* {Skm. of Gfm. p. 355:) with the 1 8th proposition of the 6th book of Euclid} Bn4^hey wiU be found to bn the Past VIII. ON EUCLID'S FWST BOCK. 275 tliie third side BD^ in the famter tHiMigte^ k not equal to the third side CD io the latter ; for CD by the construction is only a part of BD .* nor are the ftgures ABD and JCD equal, for the fiwner contains the lattiT, as appears from the Hg^ure '." 107. Prop. 5. Cor. Every eqnilateral triangle may be con- sidered as isosceles. Let ABC be such a triangle -, and since JB ^AC, the angle B=the angle C; and since BA:=BC, the angle A:^ the an^e C, both by the proposition ; wherefore, since BtsC and AstC, it follows (from axiom 1 ) thatBs^; wherefore the three angles A, B,and C. are equal to each other, that is, the equilateral triangle ABC is also equiangular. * 106. The enunciation of every theorem consists of two parls, viz. the SUBJECT and the predicate. The subject is that of which something is affirmed or denied, and the predicate is that which is affirmed or denied of the subject : thus, in prop. 4. two triangies having two sides of the one equal to two sides of the other, each'lfp each, and the included angles equal, is the subject ; and that such triangles will have their hoses equul, their other oi^les equal, and be equal in all respects, is the predicate. The subject of prop. 5. is, an isosceles triangle, and the predicate \<^ that the angles at its base are equal to each other, and likewise the angles under the base. 109. Two propositions are said to be. the con ve ass of each other, when the subject of one is made the predicate oi the other, and the subject of the latter the predicate of the former. Propo- sitions wherein the subject and predicate thus change places, are called CONVBBSB fkopositions *. > LodfauD^s RudSmtnit t^ Mtakematiet, 5th £dL p. 183, 184. " Two convene pfoporitiona, although in aiost c«tr« betk true, are Dot in ftU case* so ; one may be true, and the other feUe ; thtt», the proposition, '< If two triangles have the three sides ^f the one respectively equal to the three sides of the other, the three angles of the one will be respectively eqnal.to the three Uigles of the other," may be proved to be true ; but its converse, vis. '* li the three angles of one triangle be respectively eqaal to the three angles of ano- ther, then vill the sides of the first tfiai^e be respectively equal to those of the other," is not neceasarily true j there may be a million triangles ciri:uBi- T C 216 QEQMESRY. Fakt VHi 110. Prop. 6. 16 the oonverse of prop. 5. and its prpoC is lay reductio ad absurdum ; the words *' the base DC is equal to ib» base AB, aad" may be left out as. unnecessary, and instead cf '' therefore ^£ is not unequal to AC, &c;* it will be more proper to read, '' therefore DB is not equal to JC; and is. the same ttaaner it may be proved, that no stiaaght line, either greater or less than AS, can be equal to .^IC, wharefore AB k equal to AC, which was to be depaonstrated." 111. The corollary to prop. 6. may be thus pjrotved: (aee the fig. to Art. 107) because the angle B=the cmgle C^ ;.* the side ^Cacthe side AB, (by the prop.) aad because the angle A^ the angle C, •.- the aide 4C=;the side AB, v ACsiAB^BC, which was to be shewn. This and the corollary to prop. &. are the converse of each other. 112. Prop. 7. Many of the propositions in Euclid ape mccdf subsidiary, that is, they are in themselves of no other xme„ than as necessary to the (Mroof of otiier propositions that are uselul^ oi this kind are prop. 7, 16, and 17> of the first book The de^ monsUrati&B of this proposition i^ another instance of reductio ^d absurdumi we here suppose aa imposstbllity to be possible, in order to shew the absurdity of that supposition : a figure is hete mode to represent what no figure con represent* L a. an im* possibility -, Ibr we suppose not only that the lines AC and AD are equal to one another, but also that CB aad DB are ec^aal to one another, which the demonstration shews cannot be tiue« unless the points C and D coincide, and then the two triangles torib^d aibeat od« a a dhftr . , which have tbtiar mrsttpaadiiia aag^eg mM eq«Ml to each otkier, but it ia ^laia that the coixe^poDding sides of no two of the tri- angles can possibly be equal, since one of these triangles always coBtains the ether. Comferse and contrary propositions are not to be codfouoded, they are alto- gether dtiferent ; the former we have explained above : two propositions are contrary to one another, when one afiiiBia what the other denief^ or d«aies what it afirms ; th«s, if it be olfimed that '* two and three «iw five," the eonirmry propoaition is, that two and three ar^ not &ve. Again, *< two straight iinst cfitmot. incloee a ^paoe,? and ^* two straight linee can inolo«e a space,*' ara C0n/r«iy. propi9sition»k Two contrary propositions caaikot be both trne w false : thus, A is equal to JB, and A is not equal to B, are ooatrary^ pffof^tiona ^ now ^ it is evident, that if the fermer of these be true, the latter caoaot; and if the latter be true, the former, caoaot » in the same maaiier it.iAiiar he ahew^ that they cannot be both false* Part VUt ON EUCLID'S FIHST BOOK, ^7 will altogetiier coincide and form but one triangle. It is possible tbat AC and AD terminated at the extremity A may be equals but tben €B and DB terminated at the extremity B can^ not be equal : in like manner CB and DB may be equal, but if they are, AC and AD cannot ; and this is all that vm required to be proved. The enunciation of prop. 7- which in the oilgi- nftl is awkward and unintelligib]^ has been improved, by Dr. Simeon ; he has likewise added the second case, which is not- \i^ the Greek fext of Euclid, although it is found in the Arabic version -, this case is demonstrated by means of the latter past of prop. 5. which is cited in no ether part of the Elements. US. Prop, 8. The 7th propositidn is of no other use than as it serves to demonstrate this : we have here a second instance of a pro^ by suprapoGJition ; and eince it is shewn that the tri- angles so applied completely coincide, it fallows fh}m the 6th juciom, that the trian^es are equal ; that the s&ies of the one are respectively equal to the sides of the other 3 and the angles 6^ the one, to the angles of the other. 114. Cor. Hence, if the three sides of one tnangln be respec- tively equal to the three sides of another, the two triangles witt be both equal, antd equiangular to each other '. 115. Prop. 9. If the angles BAP, C^F be bisected, the whole aagle BAC will be divided into /our equal parts ; and if each of these parts be bisected, the angle BAC will be divided into eight equal parts ; again, if each of these parts be bisected^ the whole angle S^C will be<Kvided into sixteen equal psurts, and so on. Hence by this propoMtkHi, an angle way be divided into any number of equal jiarts^ provided thut number bfi some power of the number 2. 116. Cor, Hence, if a straiglxt line bisect an angle of a^ equilateral triangle^ or if it bisect the angle incluided by the * The terms equUtngular and eqmangular to ench taAer, ma^t oot ^e mis- understood or confounded ; a figure is said to be equiangular, when -iktt ite angles are equal ; and two figures are Mid to be equfangMiar 49 each other, whea <Mb of the angles in one Bgofe is equal to its correepondttig angle in tlte <4hei^ alttoygh neither of tbe^e figqres may be e<||iiai)giri«r m tbe former sense : a similw observation applies to the terms epUtaterai and MuUatenUfif ncftoCAer. The converse of vrop* 8* » 90^ necestafily true, as is shewn in the note 0% Art. 109. T» 578 GEOMETRY. Part VIIL equal sides of an isosceles triangle, it shall likewise bisect the base. (See the (ig. in Euclid.) For AC^BCy and CD is common 5 also the angle ACD^ \he angle BCD, therefore (prop. 4.) the base .^l>=the base BD. 117. It has been' shewn in prop. 9. and Art. 115. that any angle may be bisected geometncally^ but the geometrical trisec- tion of an angle (except in one particular case> see the note on Art. 140.) still remaibs among the desiderata in science; no tkiethod having yet been discovered whereby any section, except the bisection^ can be performed by the Elements of Geometry '. 118. Prop. 10. The word " finite," as used in this place, b , redundant. See the note on Art. 98. The method of bisecting a given straight line with instruments will be shewn hereafter. 119. Prop. 11. Drawing a straight line perpendicular to a given straight Ikle from a given point in the latter, is called '^erecting a perpendicular .*' 120. From the corolbry to this pro|)09ition it appears, that two straight lines can meet one another in only one point ; for if they meet each other in two points J and B, (see the figure in Euclid)) the parts inttircepted between A apd B must either coincide or inclose a space ; bat they cannot coincide, otherwise the two "straight lines would have a common segment, which by y A117 angle may be tritected oigcehraieaHy as follows : From tile angular pbink.^ as a centre, w^b ikity for radius, describe tbe •ore BC, draw the cb«rd BCmc, and let ar»tbe dMHrd of Br, one third the arc BCi then will jrS-^S ors -«c, which solv- ed by Cardan's rule, gives ~1 be turned into a number, (by restoring C^ 1^ value of c, &fc.) and chords be drawn from .9 and C to the points r and «, and ^r ^n be joined, these lines will trisect the given angle BAC, as wai rc%iiired. Several methods of trisecting anangle may be found in the works of thoee who have written of the higher Geometry, as Psappus, Vi«ta, Gnian^e, L'H6pi- tal, Simpson, Macla«rin, Emerson, ITOmerique, Waring, &c. Past VIU. ON EUCLID'S FIRST BOOK. «;» the coroUaiy is impossible ; neither can they inclose a space, (axiom 10.) therefore they cannot meet each <Hher in mure tlian lOne point. 121. Prop. 12, Drawing a perpendicular to a given straigbt line, from a given point wUh&ut it, is called ** letting fall a per- pendicular." We are told in the proposition to " take any point . D upon the other side of AB ;" by " other side," we are to under- Aland the side opposite to that on which C stands. 122. Prop, 13. Leamen are generally perplexed with de- jnoBstritions of which they cannot previously undewtand some- thing of the plan and scope,, and with none more frequently thsn that <if prop. 13. Let such as find it difficult observe, first, (hat CBE, EBJ) are by construction two right angles; secondly, that the three angles CBA, J BE, BBD, are equal to the above two, consequently to two right angles ; and thirdly^ that the two given angles DBA, ABC are equal to the last-men- tioned three, conseqtiently to the fore-mentioned two, and con- sequently to two right angles, which was proposed to be proved. 123. Cor, Hence, if the angles ABD, ABC be unequal, the greater is obtuse, and the less acute i the former being as much greater than a right angle^ as the latter is less, as is evident from the prc^position. 124. The 13th and 14th, the 18th and 19th, and the 24th and Vfttb, are converse propositians ; the 29th is the converse ci the 27th and 28th, and the 48th of the 47th. 125. The following is not completely the converse of prop. 15, but it is partly so. If two straight lines AE, EB, (see fiuclid*s fig. pr. 15.) on the opposite sides of CD, meet CD in any point £, so as to make the vertical angles A EC, DEB equal, .then will^£ and EB be in the same straight iine. For the four ao^es at E being equal to fbur right angles by^cor. 2, and the two CEA, AEDxth» two DEB,. BEC, each of these equals will be the half of four right angles, that is, equal to two right angles j whence (prop. 14.) AE and EB are in the same straight line. 126. Prop. 20. Dr. Simson remarks, (from Proclus,) ^^t ^' the Epicureans derided this proposition as being manifest Xq Asses i* some of tie modems have done the same, but equally without reason: according to Ji)uclid*s plan, a deipoqstjt^tioi^ was necessary, as will appear by referring to Art. 101. T 4 . . ' 280 GEOMETRY. Pakt Vllt. 127. Prop. 21. " It is essentkl to tlie tnitli of thk propni- tkm, that the straigfat lines drawn to the point within the urian- g;le, be drawn from the two extremities of the base " omitting this limitation^ there are cases in which the sUm of the two Unes drawn from the base to a point within the triangle, will exceed the sum of the two sides of the triangle, which may be shewn as follows : Let ABC be a triangle, right angled at A, D any point in JB, let CD be joined, and BA produced to G ; then since CAD is a right «Dg\e, CAG is also a right angle, (prop. 13.) but CAG is greater than CDA, (prop. 16.) .* CAD is likewise greater than CDA, / CD is ^Q greater than CA, (prop. 19.) From CD cut off DE^ AC, (prop. 3.) In« sect CE in F, (prop. 10.) and* join BF; then will the sum of the two straight lines BF*^ I> A O and FD be greater than the sum of BC anA CA, the sides df the triangle. Because CFsizFE by construction, •.• CP+FB^EF'\'FB, but CF+FB > BC, (prop. 17) •• EF+F» > BC; to these ub- equals, let there be added the equals . . . ED^AC and we shall have (by axiom 4.) EF+FB-^^ED ^ BC+ AC,, but EF^ED=:FD / BF^FD y BC^AC. Q. E. D. and the same may be proved if the angle CAB be obtuse. 128. Prop. 22. To invalidate the force of an objection which has been made»to the demonifetration c5f this proposition, it will be necessary to prove that the two circles (set Simson's figure) must cut each other : thus, because any tWo of the straight lines DF, FG, GH, are together greater than the third (by hypo-' thesis), •.• FD ^ (FG+ GH, or) FH, •.• the circle DKL must meet the line FE somewhere between JF and H, (see Art. 95.) for the like reason, the circle KHL must meet DG between D and G ; consequently these circumferences wilt pass both wiiliout and within each other, and therefore must cut leach other. SeQ Art. 96. IPakt VIII. ON EUCLID*S FIfiST BOOK. SSI l^. Pfjop. ^. It in«i«t be ol»erred» tkat the two equal (viz. ooe in each triangle) must be alike situated in the triangles $ both must be either between the given angles^ or oppoeite equal angles^ otherwise the triangles will not necessarily be equal. Let 4^0 be a triangle^ right angled at A, from whence let AD be drawn per- pendicular to the base BC, (19.1.) this will divide the triangle into two others, ^D^ and ADC, having a right a^gle in each^ (viz. at A) and the angles ABD, CAD equal % and also the side AD common $ these triangles therefore have two angles of the one equal to two an- gles of the other, each to each, but the common side AD not lyin^ either between given* or opposite equal angles, the triangles are therefore not necessarily equal. 129. Prop, 29. We have before remarked^ tha* this proposi- tioD is the converse of the 27th and 2Sth. It has given the gemneten of tK>th ancient and modem times more trouble than all the rest of Euclid's propositions put together^ to demon- strate it the 18th axiom was assumed -, but this axiom is by no means self-evident, and therefore the 29th, which depends on it, cannpt be said to be proved, unless the axiom itself be previously proved> which cannot easily be done, but by introducing aa axiom scarcely less exceptionable than that which was to be deoionstrated, " This defect in Euclid," says an ingenuous com- mentator,^' is therefore abundantly evident, but the manner of correcting it is by no means obvious -," the methods chiefly em* ployed for that purpose are the following three i I. '' A aew de*- fittition of parallel lines :*' 2. '* A new manner of reasoning on the properties of straight lines without a new axiom :'* and 3, '' The introduction of a new axiom less objectionable than Eu- * See the 8th prop. b. 6. al«o Ludlam's RodimeDts, p. 18^. "W^^re two nwnbers are placed, as (12. 1.) in the above artfcte, the. ant tiQiAber refers to the proposition, and the second to the book ia £actid ; alto If no fii^nre be mentioned, that belonging to the proposition in Euclid which $| under consideration, b always meant. 282 GEOIUST&Y. Pa&t VIIL clid*8 13th •.** Omitdng the two former methods, we shall qroil ourselves of the laller^ by introducing an axiom which Euclid himself seems to have tacitly admitted, (see prop. 35, 36, 37i and 38, book 1.) although he has not formally proposed it. The axiom is as follows : 130. Axiom. If two straight lines be drawn through the same point, they are not both parallel to the same straight line. By the help of this axiom (if it be admitted as such) we may demonstrate the 29th proposition in the following manner, without the aid of Euclid*s 12th axiom. 131. If AGH be not equal to GHD, one of them must be greater than the other i let AGH he the greater, 4md at the point G in ^ .. ^P^"^"?"^!^ . ? K the straight line GH make the angle A^GH ^GHD, (23. 1.) and produce KG to L ; then will KL be parallel to CD, (27. 1.) ... two ^^ straight lines passing through the same point are both puuDel to CD, which by our axiom is impossible. The alleles AGH and GHD are therefore not unequal^ that is, they are equal. The latter part of the demonstration may proceed as in Sipison, be- ginning at the words, but the angle AGH is equul to the fngle EGB, kc. 1^2. Cor. Hence, if two straight lines KL and CD make • Boscoricb, Thomas Simpson, Bezoiit, - Wolfius/ lyAlerobert, Sturmios, VarigQon, and several otben, are for adoptini^ a n^w definition of parallel lines ;, Ptolemy^FVanoescbiniSy&e. have endearoored to demoofltrate the prapertics of parallel lines without the help of either a new defiotUao or a new azi«a» bat bave fai^ : Professor Playfair introduces the axiom we have adopted above, which on the whole seems to be tlie best, and preferable in several respects to Euclid's. Clavius has be&towed greater attention on the subject than any modem geometer : whether he considered his demonstration as founded on a newaxioai or not, it is not quite certain, but it appears that bis reasoning dependa on a proposition which ought not to be admitted as selfrevideot. A further elucidation of this subject may be found in the notes on the 29th prop, jn Si$Mon*s Euclid^ Ingram** Euclid^ Pla^air'a JSlemetiU of Geometry^ Simpwn's Elements ^ Geometry ^ &c. Part Vllf. ON EUCLID'S FIBST BOOK. MS with another straight line EF the ai^gles KGH^ GHCtogether less than two right angles, KL and CD will meet towards IT and C, or on that side of EF on which are the angles which are less than two right angles. For if not, KL and CD are either parallel, or meet towards L and D; but they are not parallel, for if they were, the angles KQH, GHC would be equal to two right angles (by prop. 29.) which they are not: neither do KL and CD meet towards L and A for if they did, the angles LGti, GHD, being in that case two angles of a tiiangle, (17. 1.) would be less than two right angles; but this is impossible, for the four angles KGH, LGH, CHG, DHG, are together equal to four right an- gles, (IS. 1.) of which the two KGH, CHG ar^ by hypothesis less than two right angles j therefore the remattiing two LGH, J>HG are greater than two right angles. Therefore, since KL and CD are in the same plane and not parallel, they must meet* somewhere 3 but it has been shewn that they cannot meet to- wards L and D, wherefore they must meet towards K and C, or on that side of £Fon which are the angles KGH, GHC, which are together less thsin two right, angles. Q. £. D. Thus, by the assistance of our axiom, we have demonstral^pl £uclid*s 13th, which is neither self-evident, nor easily understood by a be* ginntib * 133. Prop, 32. This proposition, which is ascribed to l^ha- goras, is one of the most useful in the whole Elements, as will be evident in some sort frotti the following corollaries derived immediately from it, viz. 134. Cor. 1. The exterior angle is* the difference between the interior and adjacent angle and two right angles, and each of the inteiior angles is equal to the difference between the two remaining interior angles and two right angles. Thus, let R represent a right angle, J, B, and C the interior -angles of the triangle : (see£uclid*s figure:) then wUl the exterior angU JCDzrz^R-'C, also J^i^R-^B-^C, B=z^R'^A^C, and 135. Cor. 2. The difference between the exterior aagie and either of the two interior opposite angles, is eqaal to the other interior opposite angle. Th^is, ACD'^JszB, and ACD-BzszJ, 136. Cor, 3. If one angle of a triangle be "jl right angle, the M€ GBOMBTBT. Part Vin. other two ft^glcft taken togedier neke a right ang^ come- qoently each of them is acute: these acote angles aie calM comjdemmU of one another to a right angle. ThuSf if C be a right angle, thett will A be the compUmaUmf B, and B ^ compiemetU of ^. 137. Cor. 4. If one a^e be obtuse* tbe reBttiniiig two wiH be together less than a right ai^gbj and cooseqaently both acute. 138. Cor. 5. If the sum of two ang^ ia cme Iriangle be equal to the sum of two angles in another^ the leaiaioing angle In the one will be espial to the reaaiaining angle in the other ; and if one angle in one triai^le be eqfual to one angle in another, the sum of the two remaining aisles in the fimaer win be ei|ual tQ the sum of the fwo remainii^ angles in the latter. 139. Cor. 6. If one ai^le at the base of an isosc^es tnan^ be equal to one aii^le at the base of another isosceks triangtey the two remaining angles in the one will be e^al to the two remaining angles in the other, each to each ; and if the vertical angle of one isoscelies triangle be equal to the vertical a^gle of another, ^^eh of the angles at the base of the one will be equal to each of the a|(gles at the base of the other. 140. Cor. 7. Bach angle of an equilateral triangle is one- third of two right angles^ or two-thirds of oda right ang||p ^. 141. Cor. 8. " AH the interior angles,*' &c. as Cor. 1. in Simson. 14^. Cor. 9. All the interior angles of any rectilineal figure, are equal to twice as many right anglesj except four, as the figure has sides. Thus, let n^the number of sides, Si=ihesum of the interior engles in an^ rectUineal figure, then wiU Cor. 8. stand thrn^ <S+4RaB^.jR. and Cor. 9. thus^ Szs^n^4.R. -r^ * Hence, if the angle <tf an equilateral triangle be bisected, (9. I.) each of the^narts will be one-tbird of a right angi^ which is the only angle that can be geowetrifisUy trisected. Fa rt VIIL ON EUCLID'S VlRfiT BOOK. 143. Cor. la Hence, tbm interior angles of the kOowine^ rectilineal figures will^ be as below : if tbe figure kaTo Three Four Five Six Seven Eight Nine Ten Eleven Twelve L sides, the sum of its . interior angles wills 8— 4=s4 10—45=6 14-4=10 16-4=19 IS— 4s 14 20—4= 16 22—4=18 L 24- 4=20 J right ai^es. 144. The converse uf the former part of prop. 34. is as follows : " If the opposite sides of a quadrilateral figure be equal> the figure will be a parallelogram.*' Let ABCD be a quadrilateral figure, having its opposite sides equal, viz. AD^BC, and ^B=DC, ^then will AD be parallel to BC, and AB to DC, Join BD, ^ j^ then because ADssBC, and AB^xiDC, also BD common, •.• the angle /rDB= the angle DBC, and ABD=BDC, (8. 1. and Art. 113.) •.• AD is parallel to BC, B C and AB to DC (27. 1.) '.• ABCD is a parallelogram, according to the definition, prop. 34. 14.5V' CW. ilence, if the opposite sides of a quadrilateral figure be equal, its opposite angles will likewise be equal by prop. 34. 146. The converse of the second part of prop. 34. is this -. ^ If the opposite angles of a quadrilateral figure be equal, the figure will be a parallelogram.'* Let the angle BAD=iBCD, (see the above figure,) and ADOszABC; and since these fouf angles are the interior angles of a quadrilateral figure, they are toother equal to four right angles 3 (by Art. 143.) let now the above equals be added and the wholes will be equal, (Ax. 2.) that is, BAD+ADC^^BCD-^ABC, •.• the former two angles, as ip^ell as the latter two, will be (half of four right angles, or) two right angles, *.* (by prop. 29.) AD is parallel to BC, and AB to DC; that is, ABCD is a parallelogram. 146. In the right angled parallelogram ABCD, if the side AB be supposed to move along the line BC, and perpendicular 2sa GfiOMSTRY. Fart Vllf, ■— aMWM^H ■ ■ ■■■ ' " ■■■■ . ■ ■■■■ ■ ■ " — — - — — ■ -■ , .. . » — — — — ■ ■ «« to Hi wlien ^HarriTes at C, At ■ T ' i r— i { )I> it will coincide with DC and by its motion it wiH h:i%'e described or generated the parallelogram A BCD; let AB consist of suppose 4 equal parts, each of which we will call unity^ (or 1.) let |}in= one of those parts, and Br, rs, su, &c. each=J3iit; now it is plain, that when AB arrives at r, it will by its mcrtion have described the four rectangles between AB and jrr, each of which will be the square of {Bm, that is of) unity; in like manner, when AB arrives at s, u, v» z, C, it will have described 8, 12, 16, 20, 24 squares of {Bm, or) unity : whence it appears, that the area A BCD or 2^, is found by multiplying* the number of equal parts (calfed units) contained in AB, or 4, by the num- ber of like parts in BC, or 6. In like manner, if AB contaia n units, and BC m units, the area ABCD will contain n x m=:nm units : if »=m, the figure ABCD will be a square, and nm will become n' or m'. Hence the area of a rectangle is found by multiplying the two sides about one of its angles into each other, and the area of a square by multiplying the side into itself. 147. Prop, 35. fVom this proposition, and the jnreceding article, we derive a method of finding the area of any pai^e- logram whatever : for let ABCD (see Simson*s first figure) be supposed to be a right angled parallelogram, its area will be ABxBC, (by Art. 146.) or the perpendicular ^£?, drawn into (or multiplied by) the base BC; but DBCF^ABCD by the proposition, •/ Di?Cf=:perp. ^Bxbase BC. 148. Fience we have the following practical rule for finding c The terms muUipi^itt^ and dividing^ do ooi occur in geometrical laogoage ; thus, ia the expression AB X BC^ABCDy AB is said to be drawn iMio BC,. waA/iBCD is not called the product of AB and BC, but their rectangle; and AB in expressions like the foUowiog ~^^> AB is not said to be divided by C, but C is said to be applied to AB, The old writers are v^ry particidar in this rf spect, but the moderns are less so, as we frequently find arithmetical terms made use of in their geometric«U problems ; but this abuse should as mnch as possible be avoided. PaxtVUL on fiUCLTD'S first book. «87 the ai?a of a panllelogram. 1. Let &n a perpeodicuhr on the faose from any point in the o|]|K)site tide. 2. Multiply the base aod perpendicular together^ and the product will be the area required. 149. Prop. 37. Since every triangle b half of the palallelo- gram described upon the same base, and between the same parallels, (see abo prop. 41.) and the tOrea of the parallelogram isszperp. X base, (by the last article^) *.* the area of the triangle will be -^ J that is, half the perpendicular multiplied into the base, or half the base multiplied into the perpendicuHtTf will give the area of the triangle. 150. Prop. 38. Cor. Hence, if the base BC be greater than the base EF, the triangle JBC wiH be greater than the triangle £DF; and if BC be less than ER the triangle ABC will be Ian tlum the triangle EDF. Also, if ABC be greater than EDF, then IS BC greater than EF; and if less, less. 151. In prop. 42. we are taught how ** to describe a paralle- logram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.*' In prop. 44. we' are to describe a parallelogram with the two former conditions, ttd also one more : we are " to apply a parallelogram to a pvm straight line, which parallelogram shall be equal to a given triangle, and have one of ks angles equal to a given rectilineal angle;'* ta^' t^PPfy ^ parallelog^m to a straight line," means to make it on that straight line, or so that the said line may be one of its sides. 153. Prop 45. The enunciation of this proposition is general, if by <' a given rectilineal figure** we are to understand *< any given rectilineal figure :" but the demonstration applies to only a partu^ular case ; for it extends no further than to four-sided figures, and doe& not even hint at any thing beyond ; but the defect is easily supplied as follows ; sua GEOUETRY. Part VIII. Let ABCOND be any rectilineal figure $ join DB, DC, CNj then having made the parallelogram FKML equal to the quadrilateral figure ABCD^ as in the proposition. api>ly the pa* ndlelogram LS=^DCN to the straight line LM^ having an angk LMS^E, then it may be prored as before^ that ¥L and LP are in the aame straight line aa are KM and MS: also that fS is parallel to FK> and cpi^quently that FK8P is a parallelogram and equal to 4BCND j and applying as before a parallelogram PT^NCO, having the angle PST^E, to tiie straight line PS, FKTR may in like manner be proved ta be a parallelogram equal to JBCONIK and hax-ing an angle FKT=zEi and by a similar process a parallelogram may be made equal to any ^vea rectilineal figure whateverj and having an angle eqi^ to any given rectilineal angle. The foregoing illustration being under^ stood, the corollary to this proposition will be evident. Cor, Hence we have a method of determining the difference of any two rectilineal figures. Thus AUCOND exceeds BOON by the parallelogram FM, 153. Prop. 46. Cor. In a similar manner the rectangle con- tained by any two given straight lines may be described. 154. The squares of equal straight lines are equal to one another. Let the straight lines AB and CD be equals then will the squares ABEF, CDGH described on them be equal. For since AB= M CD by hypothesis, and HC^CD (Def. 30.) ••• HC^AB,hx3XFA^AB. (l>ef.-30.) / HC^FA; ^ ^ ^ wherefore if the square FB be applied to the square HD, so that A may be on Cj.and AB on CD, B shall coincide with D X L ■ Paht Vni. ON EUCLID'S FIRST BOOK. 2i59 lecause AB^CD-, and AB coinciding with CD, ^F shall coin- cide with CH because the angle BAF=:DBn, (Def. 30. and Ax. 11.) also ^coinciding with C, and -^Fwith CH, the point P shall coincide with H, because AF=zC[I; in the same manner it may be shewn, that FE and EB coincide respectively with HG and GD, therefore the two figures coincide, and consequently are equal by Ax. 8. Q. E. D/ Cor. 1. Hence two sqimres cannot be described on the same straight line and op the same side of it. Cor, 2. Hence two rectangles which are equilateral to one another will likewise be equal. 155. If two squares be equal, the straight lines on which they stand will also be equal. Let ABEF=zCDGH, (see the preceding figure) then will AB^CD', for if not, let AB be the greater, and from it cut off AK^CD (3.1) and on AK describe the square AKLM, (46.1) then since AK=z CD, the square ^L=the square CO, (Art, 154.) but AE:=CG M|jypQthesis, •/ AL^^AE the greater to the less which is impossmie, ••• AK is not equal to CDy and in like man- ner it may be shewn that no straight line, either greater or less than AB, can be equal to CD, ••• AB=CD. Q. E. D. 156. Prop, 47. This proposition, which is known by the nam<( of the PytJiagorean Theorem, because the philosopher Pythagoras was the inventor of it, is of very extensive application ; its pri- mary and obvious use is to find the sum and difference of given squares, th^ sides of right angled triangles, &c. as is shewn in the following articles ^, 157. To find a square equal to the sum of any number of given squares. Let A, B, C, D, &c. be any number of given straight lines ; it is required to find a square equal to the sum of the squares described on A, B, C, D, &c. Take any straight line EM, and from any point £ in it draw EP perpendicular tq EM (11.1) i take EFz==A, EG:=iB ' This proposition has been proved in a variety of ways by Ozanam, Tac- ^uet, Stunhias, Ludlaxn, Mole, and others ; it supplies the foundation for computing the tables of sines, tangents, &c. on which the practice of TrigoQo- metry chiefly depends, and was considered by Pythagoras of such prime im- portance, that (as we are told) he offered a hecatomb, or sacrifice of 100 oxen, to the gods for inspiring him with the discovery of so remarkable and useful a property. VOL. II. r 290 GEOMETRY. PAirVni (3.1), join FG, make EL=zFG, jEH^C, join HL, take EN:^ HL, EM=zD, and join MN; the square of MN win be equal to the mm of the squares of ^, B, Cy and D. Because EF^A, andEG=B, vFGl*=: (f!E)«-|-£G!«(47. 1.) =r) ^-f B*> and be- cause EL=^FG, and C«=)^ + ^4.C; and because EN-LH, and EJ*f= A v MiV)«=(EN|«+£itfl«=T5l«+D«=) ^+B«+C«+1>*. which was to be shewn , and in the same manner any number oC squares may be added together, that is, a square may be found equal to their sum. .^. 158. To find a square equal to the difiference of the squares of two given unequal straight lines. Let A and B be two unequal straight linesj whereof A is the greater; it is required to find a sqviare equal to the excess of the square of A above the square of B, In any straight Hne CH take CD =:A, DEz:zC, (3. 1.) from D as a centre with the distance DC describe the circle CKF, from E draw £F perpen- dicular to CH (11.1), and join DF; EF wiU be the side of the square required. Because FD=z (DC=i ) A, DE=: B, and DEFis a right angle, V (47. 1.) FB\''=(DEI^+EFi^=:) B^+Wi^ that is ^=^JB«+ EF\^', take B« from each of these equals, and ^-JB«=£J^^ that is, EF is the side of the square, which is the differenct required. A B Part VIII. ON EUCLID'S FIRST BOOK. 291 169. Hence* if any two sides of a right angled triangle b© given^ the third side may be found. (See the preceding figure.) For since S£l«+£?^«=:5y''a, v ^DS)*+EFf^:szDF. Examples.— 1. If the base DE of a right angled triangle be. « inches^ and the perpendicular EF 8 inches^ required the longest side, or hypothenuse DF • ? Here ^J5£)H£?1«= v^6«-h8«= ^36+64= ^100=10= DF. 2. Given the hypotl\enuse =20, and the base =11, to find the perpendicular ? Thus v^*— 111*= ^400—121= ^279= 16.703293= ^/^e perpendicular required. 3. Given the hypothenuse 13, and the perpendicular 10, to fiod the base ? Thus v^i3)2— To) 2= ^169—100= ^6D=S.3066239=</ic hose required, 4. Given the base 7» and the perpendicular 4, to find the hypothenuse ? Arts. 8.0622577. 5. Given the hypothenuse 12, and perpendicular 10, to find the base ? Ans. 6.6332496. 6. Given the hypothenuse 123, the base 99, to find the per- pendicular ? ON THE SECOND BOOK OF EUCLID'S ELEMENTS. . 160. The second Book of Euclid treats wholly of rectangles and squares, shevt^ing that the squares or rectangles of the parts of aline, divided in a specified manner, are equal to other rectan- gles or squares of the parts of the same line, differently divided : by what rectangle the square of any side of a triangle exceeds. * In a -right aogled triangle the longest side, (viz. that opposite the right angle) is called the hypothenuse, the other two sides are called legs, that on wbidi the figure stands is called the base, and the remaining leg tiie perfendicuiar, u 2 i92 eEOMETRT. Paet VUf . or fidk short ti tlie torn of the sqptures of the other two flides^ &c. 161. RecUn^es and squares may in every case he represented hy numbers or letters, as well as by gecmietrical figures* and frequently with greater convenience ; thus, one side of a rec- tangle may be called a, and its adjacent ade h, and then the rectangle itself will he expressed by ob ; if the side of a square be represented by a, the square itself will be represented by att or a* ', and since in this book, the magnitudes and comparisons only, of rectilineal figures are considered, its object may be at- tained by algebraic reasoning with no less certainty and with much greater &cility than by the geometrical method employed by Euclid -, we will therefore shew, how the propositions may be algebraically demonstrated. 162. Def, 1. Euclid tells us what '' every right angled parallelogram is said to be contained by*** but he has not in- formed us either here, or in any other part of the Elements, what we are to understand by the word rectangle, although this seems to be the sole object of the definition ; instead then of Euclid's definition, let the following be substituted. '* Every right angled parallelogram is called a rectangle } and this rectangle h said to be contained by any two of the straight lines which contain one of its angles V' 163. Prop 1. Let the divided line BCss.$, its paits BD^zot DE=^b, and EC=c; then will «=a+6-fc. Let tbe undivided line As^x, the n if the a bove equation be multiplied by x, we shall have M?=(a-h6+c.x=) ax+bX'\-cx', " that is, the rectan* gle sx contained by the entire lines s and x, is equal to the seve- ral rectangles ax, bx, and ex, contained by the undivided line x^ and the several parts a, h, and c« of the divided line s" Q. £. D. Cor, Hence, if two given straight lines be each divided into any number of parts, the rectangle contained by the two straight lines wUl be equal to the sum of the rectangles contained by each of the parts of the one, and each of the parts of thS other. Thus, let s^za + b-^-c, as before, Andx^y-\-z. Then sx=i{a-{-b-\'C.y-^z=,)ay'\'by-\-cy-^az-^bz'{-cz. * • m I ■ I . ■ . , .1 'f * The rectangle contained by two straight lines AB, BC, i» ft^fteiaUf ealled << the rectangle under JB, BCs" or simply " tbe rectangle uiB^ BC" P4M Vra . ON EUCLIDS SECOND BOOK. 29S 164. Prop. 9. Let ABsss, ACsza, and CBszb. Then a-^bszg, multiply these equals by s, and as-^-hs^ss; that is, the rectangle contained by the whole line s and the part a« together with that contained by the whole line « and the other part b, are equal to the square of the whole line s. Q. E. D. This proposition is merely a particular case of the former, m which if the line « be divided into the parts a and b, and the undivided line x^::8, we shall have 5J?=ax-f-&r, become sszsas-^ k, as in this proposition. 165. Prop. 3, Let ^B=*, AC:=a, and CB=6, then will s^ «+&«and 56=(a-|-6.6sr) a5+^; in like manner £a=r(a-).6.a=) aa+a6; that is^ in- either case the rectangle contained by the whole s, and either of the parts a or b, is equal to the rectangle (^ contained by the two parts a and b, together with the square of the aforesaid part a, or 6 as the case may be. Q. E. D. This proposition is likewise a particular case of the first, in which the undivided line is equal to one of the parts of the divided line. 166. Prop. 4. « Let AB=:s, AC:=za, and jBC=6, then will asa-^-b; square both sides^ and ««=s(a-|-6]*=) aa-^Siab-^bb; that is, the square of the whole line s, (viz. ss) is equal to the siun of the squa^res of the parts o^and b, (viz. aa-^bb) and twice the rectangle or product of the naid parts, <viz. 2 ab,) Q. E. D. 167. Prop. 6. Let^C=CB=a, CD^x, then will^2>=fl-h «, and DBi=^a^x, and their rectangle or product a-f -J^.a— g=s oa^xx; to each of these equals add xx, and a-^-xM—x+xxs^aa^ tbat is^ the rectangle contained by the unequal parts, together with the square of (x) the line between the points of section is e^ual ,to the square of (a) half the line. Q. £. D. In the corollary, it is evident that CMG=the difference or excess of CF above JLG, that is, of the square of ( Cg, or) A C above the square of CD; but CMG \&==iAa=z{AC+CDx ^C-CD=:) ADxDB, therefore (C5^*-Cl}l«, that is) A ^^^ cBi^^ADxDBjOT as we have shewn above au-^xxsaa-^-x. a— J?. n il " ' . P ■ I !■ Ml I I I II I I ■ ' !■ m,,,. * In Euclid's demonstration there is no neeessity to prove the figure CQKB rectangular in the manner he has done ; it may be jhewn thus, ** because COKB is a parallelogram, and the angle CBK (the angle of a square) a right ug^e, therefore all the angles of CQKB are right angles by Cor. 46. I. vs 294 GEOMETRY. Part VIIL 16S. Prop. 6. Let AC^CB^a, BD^x, tben will AB:=s^a, at^d AD^^ia-^-X', then the rectangle contained by AD and DB will be 2 a-f J7.x=9 ax-^-xx. to these equals let aa (the square of half AB) be added^ and 2 a-f-ar.j:+aa=(aa+3 «rx+xr=) a+J)*; that is, the rectangle contained by the line produced and part produced, together with the square of half the line bisected^ is equal to the square of the line made up of the half, and part produced. Q. E. D. Cor. Hence, if three lines x, a-^ Xy and 2a+x be arithmeti- cally proportional, the rectangle contained by the extremes (x.2tf -fx) together with the square of the common difference a, (or aa) is equal to (a •fx]*) the squai*e of the middle term. 169. Prop, 7. Let AB=s, AC=a, CB^b, then s=ra-|-6, and M=(a4-6l*=aa+2a6+66=) ^ab-^bb-^aa, to these equals add bb, and m+6&= (2a6+2 W+fla=2.a-f-6.ft+aa=) 2s6+a<l,• that is, the square of the whole line, (or ss) and the square of one part 6 (or bb,) is equal to twice the rectangle contained by the whole 5, and that part 5, (or ^sb,) together with (aa) the square of the other part. Q. E. D. Cor. Hence, becaifse 2«6+a«=5«+66, by taking 2«6 from both, we have aa= w— 2 sb-^- bb ; that is, the square of the differ- ence of two lines («) AB and (5) CB, is less than the sum of the squares of («) AB and (ft) CB, by twice the rectangle (2 sh) 2.AB.CB contained by those lines. 170. Prop. 8. Let A B=s, AC=:a, CB=b, then «=o-f-ft, or <t=s— ft, •.• aa=(s— ft]*=:=) w— 2*ft-|-ftft, to each of these equals add4sft, and 4s6+aa=w4-SJsft+ftft=5-hft'*5 that is, f4 5ft, or) four times the rectangle contained by the whole s, and one part ft, together with (aa) the square of the other part a, is equal to (5+TI| ■ or) the square of the straight line made up of the whole 5, and the part ft. Q. E. D. 171. Prop. 9. Let AC^CB^a, CD=x, then will the greater segment ^D=:a+a?, and the less segment DB^=:a — x. Then a-f ^l*=««+2aa?+a:r And a— x] * =rtfa— 2 ax'\-xx The sum of both=2cra4-2xj?=2.aa4-xx That is, aH-x)*-f fl— X '=2.aa4-xx, or the sum of the squares of the unequal parts (a+x and a— x) is equal to double the square of the half a, and of the part x between the points AHT Vlir. ON EUCLID'S SECX>ND BOOK. «95 of section; or, which is the same thing, *' the aggregate of the squares of the sum and difference of two straight lines a and x k equal to double the squares of those lines." Q. £. D. 172. Prop. 10. Let JC=iCB=sa, BDz=zx, then will AD^ Sfl+T, and CDzsza+x. Now *iJa-f-T)»=4aa+4ar+a:x Add XX to this, and the sum is 4aa^4ax-\-2xx Also a+x]*=aa+2aa:+xjr, add aa to this, and it becomes 2aa+^aa7-|-Tx; now the former of these sums is double of the latter, that is 4fla4-4ai:+2a?x=2.^aa+2ax+xx; or, the square of the produced line Za-{-x, together with the square of the part produced x, is double the square of a half the line, and the square of a4<d the line made up of the half and the part produced. Q. E. D. 173. Prop. 11. This proposition is impossible by numbers, for there is no number that can be so divided, that the product of tlie whole into one part, shall equal the square of the other part ,* the solution may however be approximated to as follows : Let ^£=2 a, AHzzzx, HB=:y, then by the problem x-^y^ 3.0, and ^exy^xx; from the first equation y=2a— x^ this value being substituted for y in the latter equation^ we shall have iaa^^€a=^xx, or aRr+2a«=:4 aa, this solved (by Art. 97. part. 3.) gives x= -j- ^5 aa~^a, and y=(2 a— j?=3 a— >/5 aa^a=z) 3a— j^baa, or which is the same «=1.236068> &c. xa, and ysz ^63931, &c. xa« 174. Prop. 12. Let ^jB=a, BC=6, CD=x, and AD=:z; Then (47. 1.) -i'B'l' = S5l»+S3)'=6TI)»+2z= bb-^^bx-^xx+zz And CS)' + AC\'=: bb * -^xx+zz (Subtract the latter from the former,) Therefore ^B> -7751'' +56]-= 2bx * * That is, the square of AB/ihe side subtending the obtuse angle^ is greater than the sum of the squares of CB and AC, the sides containing the obtuse angle, by (2 bx) twice the rectangle BC,CD. Q,E.D. 175. Prop. 13. Let ABsia, CB—b,AC^c, AD=zd, BD=zm, DC:=n; then the first case of this proposition is proved as follows : First, 66 -f »^m=26OT + nn (7. 2.) To each of these equals add V 4 9»e OEOHETRY. Part VUl. dd, and bb-jrtnm'^ddss^bm+dd'^'nn. But 4xas mm +c((2> and cc=dd-^nn (47. 1.) '•' if cw and cc be substituted for their equab in the preceding equation^ we shall have fc6-hart=2 6ra-f cc, or cc= 6& + <za — 2 ^171 . Second case. Because aa=cc+664*26n (19. 2.) add 66 to both sides, and aa+66=cc+2 664-2 6n, but 6m=6n+66 (3.2.) '.' 2 6m=2 6n4-2 66 ; substitute 2 6m for its equal i^ the preceding equation, and tfa + 66=cc-|-2 6m, or cc::=:<ia-h66«^2 6m. Third case. Here the points C and D coincide, *.* 6=m/ wherefore since cc+66=aa (47. 1.) to each of these equals add 66, and cc + 2 66 =±(za-p 66, or cc=aa+66— 266, which correfrr ponds with the former cases since 2 66 here answers to 2 bm there. Wherefore cc is less than aa4-66 by 2 6m, or 3C)»^ ifii)'4-5c)» by 2, Ca BD. Q. E. D. 176. Prop. 14. By help of this problem any pure quadratic equation may be geometrically constructed* To construct an equation is to exhibit it by means of a geometrical figure, m such a manner, that some of the lines may exjMress the cour ditions^ and others the roots of the given equation. Examples. — 1. Let x'ssab be given to find a? by a geome- trical construction. See Euclidts figure. Make BEi=ia^ EF^b, then if BFbe bisected in the point 6, (10. 1.) and from G^ as a centre, with the distance GF, a arcle he described, and EH be drawn perpendicular to BF from the point Ej (11. 1.) it is plain that EH will be the value of x^ For by the proposition EH]'szBExEF=iab, but by hypothesis x^zs ab, *.' JSH)»=ra?*, and ElJ^x; which was to be shewn. But the root of x^is either +J7 or —ar, now both these roots may be shewn by the figure, for if £H=: + J?, and EH be produced through D till it meet the circumference below BF, the line inter" cepted between E and the circumference will ^z^x, for in this case BE X £jF=— a; x — a:= H-r% as before. 2. Let x' =s:36 be given, to find the value of x. Here, because 36=9x4, 7nake JB£=:9, £F=4; then pro^ ceeding as before, eSI* =9X4=36, and EH^6. 3. Let a:» =120=12 x 10 be given. Make i5£=l2, JEF=J0, then JMB«=120, 0std EHsi (^120=) 10.95445=3?. 4. Let (r'=3 be given. Pakt Vra. ON EUCUD'S THIB© BOOK. «^ Here 3=3x1; make BEszS, EF=1, then EH)*=3, and £J7=5l.73205=u ON THE THIRD BOOK OF EUCLID'S ELEMENTS. 177* This book demonstrates the fimdamenta] properties of circles^ teaching many particulars relating to lines> angles, and figures inscribed ; lines cutting them ; how to draw tangents i describe or cut off proposed segments, &c. 178. Def. I, ** This," as Dr. Simson remarks, *' is not a de- finition, but a theorem 5" he has shewn how it may be proved : and it may be added, that the conv«*se of this theorem is proved In the same manner. 179. Def. 6 has been already ^ven in the first book, and might have been omitted here, (see Art. 74.) Def. 7 is of no use in the Elements, and might likewise have been omitted. Ia the figure to def. 10 there is a line drawn from one radius to the other, by which the figure intended to represent a sector of a circle is redundant : that line should be taken out. 180. Prop. 1. Cor. To this corollary we may add, that if the bisecting line itself be bisected, the point of bisection will be the centre of the circle. 181. Prop. 2. X^is proposition is proved by reductio ad ab« surdum. The figure intended to represent a circle is so very unlike one, that it will hardly be understood, the part AFB of the circumference being hent in, in order that the line which joins the points A and B may fall (where it is impossible for that line to fall) without the circle. The demonstration given by Euclid i^ by reductio ad absur- dutn. Commandine has proved the proposition directly ; his proof depends on the following axiom which we have already given, viz. '* If a point be taken nearer the centre than the circum- ference is, that point is within the circle." Thus, 182. Let AB be two points in the circumference ACB, joip AB, this line will fall wholly within the cirde. Find the centire £96 GEOMSTRV. pajit vni. D, (Art. 179.) m JB take any point E, and join DA, DE, and DB, Be- cause DA=:DB, ••• the angles DAB DBA are equal, (5. 1.) but DEB } than jD-<^B (16. 1.) consequently ^ than JDJB^j / DB > DE (19. 1.) / by the axiom the point £ is within the circle, and the same may be proved of every point in AB, •/ AB fells within the circle. Q. E. D. 183. Prop. 4. It is shewn in prop. 3. that one line passing through the centre may bisect another which does not pass through the centre ; but it b plain that the latter cannot bisect the former, since it does not pass through the centre, which is the only point in which the former can be bisected. 184. Prop. 16. A direct proof may here be given as in Art. 181. prop. 2. provided the corresponding axiom be ad- mitted, namely, '^ If a point be taken ferther fiom the centre than the circumference is, that point is without the circle/ Thus, Let BEA be a circle, D its centre, BA a diameter, and CAT a straight line at right angles to the diameter BA at the extremity A, the line C^r shall touch the circle in A. In CT take any point C, and join DC cutting the circle in £, then because DAC is a right angle, DCA is less than a nght angle (17.1.) '.-J^C^D^ (19.1.) *.* D is farther from the cen- tre than Ay consequently by the axiom C is without the cirdc, and the same may be shewn of every point in CT, -.- CT is without the circle. Q. E. D. Cor. Hence it appears that the shortest line that can be drawn from a given point to a given straight line, is that which is per- pendicular to the latter. 185. In the enunciation of this proposition we read, that " no straight line can be drawn between that straight line (i e, the touching line, or tangent) and the circumference irom the ej(« Part YIII. ON EUCLID'S THIRD BOOK. <299 tremity (of the diameter) so as not to cut the circle ;" this ap- pears to be an absurdity, for how can a line be said to be between'^ the tangent and circamference, if it cut the latter ? and how can a line which cuts the circumference be between it and tlie tangent ? The like may be observed of the sentence^ ^* therefore no straight line can be drawn from the point A between AE and the circumference^ which does not cut the circle/* It was for the sake of the latter part of the demonstration that the seventh definition of this book was introduced ^ both may be passed over, as they do not properly belong to the Elements. 186. Prop, 24. The demonstration of this proposition is manifestly imperfect j after the words " the segment AEB must coincide with the segment CFD,** let there be added, '^ for if AEB do not coincide with CFD, it must fall otherwise (as in the figure to prop. 23.) then upon the same base^ and on the same side of it^ there will be two similar segments of circles not coinciding with one another^ but this has been shewn (in prop. 23.) to be impossible > wherefore, &c." Without this addition, the proposition cannot be said to be fairly proved. 18/. Prop. 30. It is of importance to shew that DC falls without each of the segments AD and DB, and since the centre is somewhere in DC (cor. 1.3.) it must be likewise without each of those segments 3 . wherefore (by the latter part of 25. 3.) each of the segments ^D and DB is less than a semicircle. 188. By means of prop. 35. and 36. the . geometrical con- struction of the three forms of affected quadratic equations may be performed. The first and second forms are thus constructed \ * The geometrical construction of an equation is the redocing it to a geo- metrical figure, wherein the conditions of the pr»powd equation being ex- hibited by certain lines in the figure, the roots are determined by the inter- sections which necessarily take place in consequence of the construction. The ancients made great use of geometrical constructions, which is probably owing to the imperfect state of their analysis ; but the improvements of the rooderns, particularly of Mercator, Newton, Leibnitz, Wallis, Sterling, Demoivre, Taylor, Cramer, Euler, Maclanrin, and others, have in a great itteasure superseded the ancient methods. Simple equations are constructed by the intersection of right lines, quadra- lies by means of right lines and the circle^ but equations of higher dimensions require the copic sections^ or curves of superior kinds, for their construction ; Sm GEOlfBTRT. Part VUI. Tint fotm -xx+ax^he. * Second form xx^axtsbc. fhxm C as a c^Dtre with a dktanoe 2=4.4 describe the circle JGB, then (mippoBing ft ^ c,) with the dintanrfi 6— c ia the compasfies (taken firom any convenient scale) from any pcMiA £ in the ciicamferenoe» describe a small arc cutting the cireum- lerenoe GB in F, join EF, and produce it to D, making FD s=c, and from D draw DBCA passing through the centre C, then will DB and DA be the values of X in both the first and second forms, viz. x=s +DB or— D^ in the firet formt and x:b+DA or — DB*m. the second form. For since ABssa by construction^ if DB^x, DA will be «+«, but if DA szx, then DBssx^-a; but DA.DB=DE.DF (37. 3.) or (x+a.r=) xx+ox^ftc in the first form> and {xjc — a=) xx'^cuo^bc, in the second^ and since the two proposed equatiood differ only in the sign of the second term> it is plain that they will have the same roots with contrary signs^ (see Art. 30. part 5.) 189. If we suppose (=c, the construction will be still more simple^ for (6— c=) EF=:o, that is £F will vanish, and DFwill consequently touch the cir cle i n G, and become DG, and we shall then have DA,DB=zDG]9 ; wherefore if a right angled triangle DGC be constructed having GC^^, and DGszb, {BD=D C^CG in the fast form, and its negative value -JDCTCG. DA^DC-k-CG in the secondj and its negative value '^DC-'CG. 190. To construct the third -form of affected quadratic equa- tions, or arr— ax = — ab. From the centre C with the distance CB asj-a, describe tfae circle AEF as before, from any point £ draw EF^b-i-c, make TUrions methods of construction eqnations may be seen in tbe writings e# Slusios, Vieta, Albert Oirard, Schooten, Fermat, Des Cartes, Ghetaldos, De la Hire, Barrow, Robcrval, Halley, Newton, Gregory, Baker, Hyac, Star- mini, De I'Hdpitai, Sterliog, Madaarin, Simpson, Emerson, and others. Part VIII. ON EUCLID'S THIRD BOOK. 301 flDssb, then DF=c, join DC and produce it both ways to A and B. Since ABsza, if AD be called x, then wiU DJBsa— x, but JD.DBszED.DF (35.3.) that is, (x.a — x=) ox — xxssbc, or which is the sum xx — ax= — be as was proposed to be shewn. The like conclusion will follow by supposing DFssx, whence the two roots of the given equation are AD md DB. 191. If 6=c, then will EDssDF, and AB will be perpen- dicular to EF (3. 3.) and EC being joined^ we shall in that case have a right angled triangle^ the hypothenuse of which will =^a^ and one of its sides ssb, wherefore the sum and difference of the hypothenuse and the other side will be the two roots of the equation as is manifest. ON THE FOURTH BOOK OF EUCLID'S ELEMENTS. 192. This book will be found of great use to the practical geometrician^ it treats solely on the inscription of regular rec« tilineal figures in^ and their circumscription about a circle; and of the description of a circle in and about such rectilineal figures. 193. Prop. 1. The reason why the straight line required to be placed in the given cirate ifiust not be greater than the dia- meter, appears from the 1 5th proposition of the 3rd book, where it is proved, that the diameter is the greatest straight, line that can be placed in a circle. 194. Prop. 4. From this proposition it appears, that the three lines which bisect the three angles of a triangle^ will all meet in the same point within the triangle. Also the sides of any triangle being known, the segments intercepted between their extremes, and the points of contact, may be found K ■ ■-■i^ii iiiii II ■ ■ ■ ii« i Thra, Iet^fra»40, ^C»30, and BC^30, then will AB+ BC^SO; horn ibis lubtract AC^AE+FC^m, and the remainder is BB-hBF^SOi ther«foie JR£«1W'«15,FC«C0«(^C— ^Z**) 5, and AQ^AE^iAC'^ CO») 25, d04 GEOMETRY. ' PARxVIIt 195. Prop. 5. We hence learn that it is possible to describe a circle through any three given points, provided they are not placed in a straight line; for by joining every two points, a triangle v^ill be formed, and the proof will be the same as in the proposition. Also only one circle can pass through the same three points. (10. 3.) 196. " The line DF is called the locus of the centres of all the circles that will pass through A and jB. And the line EFis the locus of the centres of all the circles that will pass through A and C. And this method of solvit^ geometrical problems, by finding the locus of all those points that will answer the several conditions separately, is called constructing of problem by tlie intersection of geometric loci V* 197. Prop, 6. Hence the diameters of a square (being each the diameter of its circumscribing circle) are equal to each other 3 they also bisect the angles of the square, and divide it into four triangles, which are equal and alike in all respects : . and since the square of jBD=the sum of the squares of BA and AD (47. 1.) =2.^*, it follows that B5lH^'=3iJl*+ 198. Prop, 7. Because the side of a square is equal to the diameter of its inscribed circle (for GF==JBD,) and the square of the diameter is equal to twice the inscribed square> (see the preceding article) 3 therefore a square circumscribed about a circle is double the square inscribed in it. 199. Prop, 10. Since the interior angles of ABD:=2 right angles (32. 1.) and the angle B=iD=^9lA, *.• the angles at 4, £, and Dj are together equal to (A-^-^A+^Assz) bA, that k LudlcaxCs Rudiments, p. 207, Loci are expressed by algebraic equatioDS of different orders, according to the nature of the locus. If the equation be constructed by a right line, it is called locus ad rectum; if by a circle, loau ad circulum ; if by a parabala, locus ad paraholam / if by an ellipsis, locus ad eUipsim. Th« loci of such equations as are right lines or circles the ancients called plane loci; of those that are conic sections, solid loci; and of thos^ that are of curves of a higher order, sursolid loci. But the moderns distin- guish the loci into orders, according to the dimensions of the equations by which they are expressed.—- fTu/Zon. The following authors^ among many others, have treated of this subject, viz. Euclid, ApoUonius, Pappus, AristaeoSy Viviani, Fermat, Des Cartes, Slusius, Baker, De Witt^ Civg, L'Hdpital^ Sterling, Maclaunu^ Emerson^ and Euler. Pabt VIIL ON EUCLID'S FOURTH BOOK. 303 is 5^=:2 right angles^ and A=r^ of 2 right angles ; wherefore if ijf be bisected^ each of the parts will be -^ of one right angle. Hence by this proposition a right angle is divided into live equal parts, and if each of these parts be bisected, and the latter again bisected, and so on, the right angle will be divided into 10, 20, 40, 60, &c. equal parts 5 and since the whole circum- ^rence subtends four right angles (at its centre), the circum- ference will, by these sections, be divided into (4x5, 4 x 10, 4x20, &c. or) 20, 40, 80, &c. equal parts; and by joining the points of section, polygons of the same number of sides will be inscribed in the circle. 200. Prop. 11. Because by the preceding article, CAD^s^^ of two right angles, and the three angles at Ay which form the angle BAE of the pentagon, are equal to one another (being in equal segments 21. 3.) '•* BAE =f of two right angles or 4 of one right angle. 201. Prop. 13. It follows, that if any two angles of an equi- lateral and eqaiangular figmre be bisected, and straight lines be drawn from the point of bisection to the remaining angles, these •ball likewise be bisected 5 and if, from this point as a centre, with the distance from it to either of the angles, a circle be described, this circle shall pass through all the angles, and con- sequently circumscribe the given equilateral and equiangular fi^e. See prop. 14. tBb. Prop, 15. Hence the angle of an equilateral and equi- ^gular hexagon, will be double the angle of an equilateral tri- angle, that is, 4 of 2 right angles, or 4 of one right angle. This proposition b particularly useful in trigonometry. 203. Pr^yp*, 16. All the angles of a quindecagon (by cor. 1. pr.32.b.l.)areequalto(2x 15— 4r=) 26 right angles 5 wherefore 26 11 rr= 1 — right angle = one angle of an equilateral and equi- ps 15 angular quindecagon. If each of the circumferences be bisected, each of the halves bisected, and so on continually, the whole cir- cumference will be divided into 15, 30, 60, 120, &c. equal parts^ and these points of bisection being joined as before, equilateral uid equiangular polygons of the above numbers of 8ides> will be inscribed as is manifest. 204. Hence, by inscribing the following equilateral and equi- angular figures, and by continual bisection of the circumferences 304 GEOMETRY. PaktVIII, subtended by their sides^ the circle will be divided into the ffdiowing numbers of equal parts, viz. by the Triangle, into 3, 6, 12, 24, 4S, 96, 192,384, &c^ Square 4, S, 16, 32, 64, 128, 256> 512, &c. I equal Pentagon 5, 10, 20, 40, 80, 160, 320, 640, &c. | pots. Quindecagon 15, 30, 60, 120, 240, 480, 960, 1920* &c.^ The numbers arising from inscribing, bisecting, &c. an before, of the Hexagon, ^ ^Trian^e, -uigun, ■ ^^ included in those of the <Z?^' Decagon, | | Pentagon, Triaecmtagon,-^ ^Quindecagon, and so on continually : whence it appears that the cirde may be geometrically divided into 2, 3, 5, and 15, equal partSj and likewise into a number which is the product of any power of 2 into either of those numbers : but all other equal divisions of the circumference by Geometry, are impossible. ON THE FIFTH BOOK OF EUCLIIXS ELEl^CEMTS. 205. In the fifth book, the doctrine of ratio and proportion is treated of and demonstrated in the most general manner, preparatory to its application in the following books. Some of the leading propositions are of no other use, than merely to furnish the necessary means of proving those of whicMk use is obvious K 206. Def, 1. By the word part (as it is used here) we are not to understand any portion wJiatever of a magnitude less than I Students accustomed to algebra, will find Professor Playfair's method of demonstrating the propositions of the fifth book, much more convenient and easy, than that of Dr. Simson. There are those who would entirely omit the fifth book, and substitute in its place the doctrine of ratio and piopcntion as proved algebraically (p. 49 — 74. of this volume;) which might do very well, if no referenc& were made to the fifth book ; or if the sixth might be allowed to rest its evidence on algebraic, instead of geometrical demonstration ; but if this cannot be admitted, it will be advisable to read the fifth book at least once over, in order folly to understand the sixth, where it is Heferred to not less than 58 times ; in that book there are 17 references to the 1 1th piopo«tiaa» 10 to the 9th, 8 to the 7th, and 5 to the 2^d ; these four may therefore be considered as the most useful propositions in the fifth book. Pai^t niL ON EUCLID'S WITH BOOK. 805 the whole 5 it ioipliefi that part cnly, which in Arithmetic is called an aliquot part. The second deiinitioa is the converse of the first. 207. The third definiticm will be easily understood from what has been said on the subject in part 4. Art. 24. &c. 208. Def. 4. The import of th^p definition is to restrain the magnitudes^ which '* are said to have a ratio to one another,", to such as are of the same kind : now of any two magnitudes of the same kind, the less may evidently be multiplied, until the product exceed the greater : thu8> a minute may be multiplied till it exceeds a year, a pound weight until it exceeds a ton, a yard until it exceeds a mile, &c. these magnitudes then have r^pectively a ratio to one another "'. But since a shilling can- not be multiplied so as to exceed a day, nor a mile so as to exceed a ton weight, these magnitudes have not a ratio, to each other. 209. Def. 5. '* Ojie of the chief obstacles to the ready under- standing of the 5th book, is the difficulty most people find in reconciling the idea (^ proporticNoi^ which they have already acquired, with that given in the fifth definition j" this obstacle b increased by the unavoidable perplexity of diction, prodiiced by taking the equimultiples of the aitemaie magnitudes, and imifiediately after, transferring the attention to the multiples of those that are adjacent 5 operations, which cannot easily be de- scribed in a few words with sufficient clearness; besides, the de- finition is en<nimbered with some unnecessary repetitions, vi^ich aaight be left out, without endangering its perspicuity or preci- sion. On the subject of this definition, as it appears to me,, much more has been said than is necessary. Euclid here lays* down a criterion of proportionality, to which we are to appeal in all cases, whenever it is necessary to determine whether mag- * In onicr to make the comparUon implied here, it is bowetek- -neceteary that the magnitmlefl compared should be, net only of the same kind, but like- Wittj 0/ the same demmimtion: properly speaking, we cannot compare a minnte with a year, a pound weight with a ton, or a yard with a mile ; but we can compare a minute with the number of minutes in a year, a' pound with the number of pounds in a ton, and a yard with the number of yards in, a. mile 5 the ratio of a guinea to a pound can be determined only after they are both reduced to the same denomination ; then, and not before, we find that tbey have a ratio, viz. the former is to the latter as 21 to 20. VOL. II. . 2C 905 GSmSTBY. Paet YUI. mtndes are, nr «re not pssportioMils; ani k» Im ^vite « irt dik bm^ BB k» lliu tiMive pin Mii eiflkit cnnfln of ifei a^lkatkm; so that, admitCiii^EDfdid's criterion to be >it;te mode of itk r ea e a is, if I am not deceiicd, a» Mmple, aai tiie efidenee as aatisfiietflfy, as can be re^^md. 2M>. Bat bow ave we to kiyvr, whetkcr Sadidrs rtaadaidof pfoportiopality be just or not; tfiat is^ wbetber It doe» or daii not s^ieeiiltli our raeehred nolioBB of piopaftiMQlity,* aa didalail hj eommen sense? we wffi eo mp gf e Budiifs doctrine, m \mi down in tiie fiMi de&iition> wilb tbe notion wU^ alt ponons^ whefber learned or not, have of proportion, and they witt be foand to agree. 1211. Ask any man wbat be meaBw by ** two thiag» b^i^ m the saune p roportion to one nnother, that twootber thii^are ?** and he win immediatdy answer, *' when Hie first is as laige when compared with the aeeond, as the third is, when compared widi the ftfurtb." Now, the obriovs melbod of finding ko» i&rge one magnitode h, when c ompar ed with another, ia to find bMr often it eontains, or is contained in, tbe other; or inmnreeorrecf and scientific langm^, to find what nn^tl^le, part, or parts the former magnitnde is of tbe kitar; wbidi is ^ected,by dbridiiig the nmnbcr representing tbe one, by that r^iesentiog Ae ether. Wherefore, tbe common notion of proportkmaliily when aden- rately expressed, wiU be as foOows. 21S. '* Two megnitodes are proportional to two Others, wh^i the first is the same mnKi]^, pwt, or parts of tbe second, as the thhd is of the fbnrth 5" or, when tbe ifiiotient of tbe first di- i4ded by tlie second, equals the quotieift of tbe tlnrd dkided by^ thefottrth: under tbete eircumstanoes ^'tbe foor magnitudes aire said to be proportionals.'* This is in substance tiie snme as def. eo. of the 7th book of Sodld's Elements, and Mr. Ludkm has shewn that it Mjpees with BocMd's doc^ne as delivered, in his 6th booli, that is, if fonr magnitudes he proportionals ac- cording to def. 5. 5. they tre proporliOMds nceonii^g tn tbb article ; and if they be proportionals ac<!onling to this article, they are likewise proportionals according to def. 5. 5. first, if a:b :: c: d by 5. def. 5. book, then fiH31 ox (l»^X e, and-^-s PauVBI. ONEUCUD'aHITHBOOK- 307 And (15. 5.)k a : h :: ad : bd ^ Atad (II. 5.) ..-.•... c : 4. : : ad : bd And (15. 5.) ........ c : d :: 6c : 2kI Whei^elore (11.5.) . . ad ibd :: be i bd Consequently (9. 5.) . . . adz:zbc, and the ---• parts of these equab^ ' will likewise evidently be equals that is (<»^XTj=*^^Tj°'')*T" =— , so that if four magnitudes « : 6 : : c : d be proportionals ^ecQfding to £uclid*s 5th definition^ they are also, proportion's by Art. 211. Q.KD. See also Art. 56. p»rt 6. 214. It remains to be shewn that *« if ftmr magnkudes be profxnitionais according to Art. 211. they are afeo proportionab to def. 5. 5. dBudid." c . Let -7-=-j* then will adzzbc agreeably to Art. ?11, ana if ad=sbc, then will a: b :: e : d agreeably to def. 5. 5. Euclid. For let m and n be two multii^^iers, and let the first and tinisd, (yisi, a 9iid c) be multiplied by m, and the second and fourth (or b and d) by «; if ma be greater than nb, then will n^ be greater than nd, and if tqual equals and if leas less. For since a x ds6 x c, it follows that nut x nd^nb x ntc, *.- if ma be greater than 96, it is plain that mc must be greater than nidp if equal e^iaL and if less; wherefore Uy def. 5. 5. a, 6^ c^ fuod d^ are proporUonals. Q. £. S. 215. k will be readily seen that tlie d^finatium (Axt 21 K)^ which we derive from the popular notloa of proporHonalsi is restrained to magnitudes which can be expressed by cooHuen- fiurate ni|inbei»« Euclid^s 5th definitioa i^^plies eq^iaUy to cqpi- mensurate aqd incommensurate magnitudes ^ this capacity of universal a{^licalion gives it a d^dded pi^&renoe ta the defini* tkm in Art. 211. and we have ahewa that both 3gree as &r as th^j CM.be comp^r^d t^getber. 21^. JDrf* 6. and 3. properly form but oxwe definition^ which ma^ sUind ap fel(owB» viz. " magnitudes which have Ih^ saioe. ratio are .4»41ed proportionals^ and this identity of ratios >, called proportion.'* 217. Tb^ loth and lUkdefinitioms o^g^t to have b^en i^9<^d X 2 308 geometry: FartVUI. afbsr def. A, since duplicate, triplicate, quadnqdkate, &c. ntiot are particular species of compound ratio > thus^ let a, h, e, dy e, kc. be any quantities of the same kind, a has to e the ratilycGin- pounded of the ratios of a to* fr, of & to c, of c to d, and of d to e, (see Art. 40— 42. part 4.) and if these ratios be equal to one another, a will have to e the quadruplicate ratio- of a ta &> (or o^ : b*y that is, the ratio compounded of four ratios each of which is equal to that of a to 6 ; in like manner a will have to d the triplicate ratio (or a? : ¥) and to c the duplicate ratio (or €fi : b*) of a to b ; wherefore it is pkia that each is a parti- cular kind of compound ratio. 918. Def, i% The antecedents of several ratios are said to be homologous terms, or homoU^ous to one another, likewise the consequents are homologous terms^or homologous to one another -, but an* antecedent is not homologous to a consequent, nor a consequent to an aiUecedent ; the word homologous is unneces- sary, we may use instead of it the word similar or like, either of these sufficiently expresses its meaning. ON tri& SIXTH BOOK OF EUCLlD'S ELEMENTS. 219. The principal object of the sixth book is to apply the dlDctrine of ratio and proportion (as delivered iii the 5th) ta lines, angles, and rectilfncral figcfres 5 we are here taught ho^ to divide a straight line into its aliquot parts; to divide it simi- larly to another given divided straight line 3 to find a mean, third and fourth proportional to given straight lines ; to deter- mine the relative magnitude of angles by means of their inter* cepted arcsi and the converse ; to determine the ratio of similar xvctOineal figures; and to express that ratio by straight lines with many other useful and interesting particulars. 220. Def, 1. According tx> Euclid *' similar rectilineal figures are (first,) those which have their setefal angles^ equal, each to each, and (secondly,) the sides about the equal angles proportion nalsf now each of these conditions follows from the other, and therefin'e both are not necesssoy : any two equiangular rec- tiltjaeal figures wi& always have the sides about their equal angles proportionals 5 and if the sides about each of the angles of two rectlHnea] figures be proportionals, those figures will be equiangular, the one to the other. See prop. 18. book €, 221. De/; 2. Instead of this definition which is of no use^ PahtVIH. on EUCLID'S sixth book. ^ 309 Dr. Simsoa has substituted the following. '^ Two magnitudes are said to be reciprocally proportional to two others, when one of the first is to one of the other magnitudes, as the remaining one of the last two is to the remaining one of the first," (see Simson^s note on def. 2. b. 6.) this is perhaps the best definition that can be given for the purpose. 222. Def. 3. ITius in prop. 11. b. 2. the line ^B is cut in extreme and mean ratio in the point H, fyt BA : Mi : : 4ff • HB as will be shewn farther on. 223. Def, 4. In practical X^eometry and other branches depending on it, the Ui}ie or plane .on which a figure is supposed to stand is denomjinated the hose; Euclid ma^es either side indififerei\tly the base, and a perpendicular let fall from the op* posite ^ngle (called the vertex) to the base, or the base pro* dtK-ed, is called the altitude of the figure (for an example see ^e Jtture^ JSgures to prop. J3. b. 2.) 224. Prop. 1. Let ^==the altitude, JB^the base of oo^ parallelogram or triangle 3 a=the altitude, 6==the liase of anq- tfaeri then will .^jB=the first parallelogram, a^=the second; AB ah -r-=the fi[rst triangle, and —the second; andif^:Ma,tlienwlU * <& AB a6 > : : B : 6; and if B=:6, then will J ^B 06 > : : A^ 2 ' T J I ? ' 2 J c^ that is, parallelograms and triangles of equal altitudes are to QQe another as their bases ; and if they have equal bases^ ^ey are to oi^e another as their altitudes. Q. £. D. 225. Pr<yp, 2. Hence, because the angle ADE^ABC, and JED^ACB (29. 1.) and the angle at A common, the triangle ADE wiH l>e equiangular to the triangle ABCy (32 1.) And if there be drawn several lines parallel to one side of a triangl^ they nvill in like manner cut the other two sides into jHX>portio- nal segments 5 and conversely, if several straight lines cut twp sides of a triangle proportionally, they will be paraflel to Ae re- maining side, and to one another. Hence also if straight lines be ^wn parallel to one, two, or three sides of any triangle, another triangle will, in each case, be formed, wbiph i^ equiangular tp the ^ven one. 226. Prop, 5. Although in the enunciation it is expressly saud, that the equal angles of the two triangles ABC, DEF ai-e X 3 aio • GEOMETRY. Paet VIH. oppt)site to the homologous sides^ yet this circumstance is iiM bnce ndtieed in the demonstration -, and hence the learner will be iieady to conclude^ that the proposition is not completely proved; hut let him attentively examine the demonstration^ and he will find^ that although nothing is expressly affirmed about the equality of the angles which are opposite to the ho- mologous sides, yet the thing itself is incidentally made out ; thus A^ atnd DB bemg the antecedents^ it app^atrs by the de- monstration that the angle C opposite to AB is equad to die angle JT opposite to Dfi ; and BCund EF being the consequents, it is incidentally shewn that the angfe A opposite to J3C is equal to the angle D opposite to BP; also AC and DF being both aintecedents or both consequents, their opposite angles B and £ are in like manner shewn to be equal. These observations are li&ewise applicable to prop. 6. 227* Prop. 10. By this proposition a straight line may be divided into any number of equal parts as will be shewn when 'Wt treat of the practical part of Geometry. 22S. Prop. 11. A third proportional to two given straight lines may £^o be found by the following method, (see the figure to prop. IS.) Let AB and BD be the two given straight lines, draw BD perpendicular to AB (11. 1.) join ADs at the point D drav^BC at right angles to AD (11. f.), and produce AB tiQ i€ cut DC in C; then will BC be the third proportional to AB and BD. For since ADC is a triangle, right angled at !>, from whence DB is drawn perpendicular to the base, by cor. to prop. 8. A B :BD :: Bb: jBC, that is BC is a third prOpottibnaJ toABanSiBD. Q.£P. Let^B=:a, ADscb, then a: b :: b : — =:J3C which is the a fiame thii% performed algebrttieaVy. 2^9. Prop. 12. Uet o, h, and c, be the fhree given stral^t he linefl^hen will a: b :: c : — z=HF, the fourth pronortional re- • a quWed. ^30. Prop. 13. libt ABssa, BCssb, and the required meaoa tkx, thete simife «:«::%: 6, we facve (by moltiplyinjg exUemn and means) xx:=zah, and x=: ^^ab^szDB °.. » It has bfeien asserted in the introdcKstion to this part, Uutt there is no knonrn geometrical method of finding more than one mean proportional be- pAWr VOL ON EUCLUyS SBOOl BOOK. dll £sAiiFi*Y«.-» 1* To t»i a ineAB pe^^oitioiMi betweieii 1 Here aitzl, hsil€, mid d:ss^a&ssv^]#=34» the fneau f|r Hr To find a mean proporUooa! betweeft 15 and 11. Hare as=l$, frsll, and ^ss^a^a^/lSxllsV^^'" ll8453dS57S, <Ae rc9»trtf<2 man. 2^1. Prop. 19. By the help of this useful proposition we are enabled to construct similar triangles^ having any given ratio to each others thu8> let it be required to make two similar trian^ gles^ one of whieh sh2dl be to the other as m to n. Make BC s=m, BG=:n, and between BC and EG find a mean proportional EF (IS. 6.) upon BCand EJPmake simflar triangles ABC, DBF (18. 6.) then by the^present proposition mm:: ABC : DEF, SxAMPLBs.— 1. Let the side of a triangle ABQ viz. BC:=ze, It is required to make a similar triangle, which shall be only half as large as ABC. Bisect BC in G (10. 1.) and between BC and BGfiud a mea^ proportional EF (13. 6,); if a triangle be made on EF similar to ABC it will be hal f of A BC. Thus BC being=:8, EG wiU=4, and ^BCx BG= ^^8 x 4= ^32=5.656854= JEF. tK. Let £F=S» required the side of a trian gle fiv e times as hige as ni^y and simikr to itr Ans. v^x40srv'^20sa 17^8854383 the Me rehired. 232. Prop. 20. Hence, if the homologous sides of any two similar rectilineal figures be known^ the ratio of the figures to one another may be readily obcainedj nsunely, by finding a third {proportional to the two given sides : for then, the first line will be to the thirds as the figure on tihe first, to the similar and simi- larly described figure on the second^ as is manif<»t from the ^" tffttB tw^hnm ftraigibi ham a «od (/ tbitAajFhoWwrwlKiaoi^lfebnu* cally )tf the feU^wiv^ theorems. One mean proportional will be j^ab Two means » >/«* J» » >/tfi* Three means ,*^a»b, « v^<l»4», * j/ab* Four means , • ^a*b, • js/a*b*f iv^tf«6i, » /t/«rft* nmeans ■+ V«»*. '+ V«""-*-ft'.'+ V*"""^* x4 312 GEOMETRY. Fart VlU* tecond cor. to the profiosition. Hence also any rectilmeal figure may be geometrically increased, or decreased in any as^gned ntio. Thus, let it be required to find the side of a pentagon one fifth as large as ABCDE^ and similar to it 5 find a mean proportional between AB and ■!■ AB (13. 6.) let this be FG, and upon FG describe the pentagon FGHKL similar and similarly situated to ABCDE (IS. 6.) then will the former be i of the latter. Again, let it be required to find the Me of a polygon 3 times as large as ABCDE, and similar to it ? TTius j^ABxiAB=:th€ side required. 233. Prop, 22. By means of this proposition, the reason of the algebraic rule for multiplying surd quantities together, may be readily shewn. Thus, let it be required to prove that ^a x ^b:sz ^ahy first, since unity : the multiplier : : the muUipli-' cand : the product; therefore, in the present case, 1 : ^a : : jy/b : ^ox ,/6=the product, but by the proposition (1* : ^a* : ; ^6* : j^a^X ^h\ that is) 1 : o : : 6 : a6=the square of the product^ wherefore ^a5=the product. 234. Prop, 23. Hence, if two triangles have one angle of the one equal to one angle of the other, they will have to each other the ratio coitipounded of the ratios of the sides about their equal angles ; this will appear by joining DB and GE ; for the triangles DBC, GEC have the same ratio to one another, that the parallelograms DB and GE have (1.6.). Also it appears from hence, that parallelograms and triangles have to one ano- ther respectively, the ratio compounded of the ratios of theif bases and altitudes. 235. Prop. 30. This proposition has been introduced under a different form in another part of the Elements, (viz. 11.2.) there, we have merely to divide a straight line, so that the rectangle con- tained by the whole and the less segment^ may equal the square of the greater } we have to determine the properties of a figure, but the idea of ratio does not occur 3 here we are to divide a line, so that the whole may be to the greater segment, as the greater segment is to the less, and the idea of figure has no place } but our business is solely with the agreement of certain ratios. 1 do not recollect a single reference to this proposition in any subsequent part of the Elements, except in some of the books which are omitted. 236. Prop. 31. What was provied of squares in prop. 47. b. I. Part VITL ON EUCLID'S SIXTH BOOK. Sl$ is here shewn to be true of rectUineal figures in general ; and the same property belpngs likewise to the circle^ and to all similar carvilineal and similar mixed figures^ with respect to their dia- meters or similar chords ; but the six former books of Euclid s Elements do not furnish us with sufficient principles to extend the doctrine beyond what is proved in this proposition. We are here taught how to find the sum and difference of any two simi- lar rectilineal figures, that is, to find a similar figure ecjual to the said sum or difference. See the observations on 47. 1. 237* Prop, 33. This useful proposition is the foundation of Goniometry, or the method of measuring angles. If about the aagular point as a centre with any radius, a circle be described, it is here shewn, that the arc intercepted between the legs of the angle will vary as the angle it subtends varies 3 thus, if the angle be a right angle, the subtending arc will be a quadrant (or quarter of a circle) 3 if it be half a right angle, the sub- tending arc will be half a quadrant } if it be equal to two right aagles, the subtending arc will be a semi-circle 3 and if it equal- four right angles, the subtending arc will be the whole circum- ference. Now if two things vary directly as each other, it is plain that the magnitude of one, will always indicate the contemporary magnitude of the other) that is, it will be a proper m^isure of the other. Such then is the intercepted arc described about aa aagle, to that angle 3 and therefore if the whole circumference be divided into any number of eqiial parts, the number of those parts intercepted between the legs of the angle, will be the mea- sure of that angle. It is usual to divide the whdle circumference into 360 equal parts called degrees, to subdivide each degree into 60 equal parts called minutes, and each minute into 60 equal parts called seconds, &c. wherefore, if an angle at the centre be subtended by an arc which consists of suppose 30 degrees, that angle is said to be an angle of 30 degrees, or to meastire 30 degrees; if it be subtended by an arc of 45 deg. 54 min. the angle is said to measure 45 deg. 54 min, 8sc. 238. Hence the whole circumference which subtends ^o«r right angles at the centre (Cor. 1. 15. 1.) being divided into 360 degrees, a semicircle which subtends two right angles will con- tain 180 degrees, and a quadrant which subtends one right angle wiU contain 90 degrees, wherefore two right angles are said to measure 180 degrees, one right angle 90 degrees, &c. and note« 814 GBOMfiTRY. pAftT VID. degttm, mintsteB, «nd necordt, aM thus amked ^ ^ '^ tinif H degrees, 3 mintiles^ 4 seocttds, are ufiual^ wrintea 1^« 3^ 4^> &c. ^8. B. Hence, if eiKmt eny sngiiter fokti C sAaeiictm, eevenH eODcentric circles be dtesciibed, ccftti^ CA aiij C# ki tlie pdioU X, Z, A, By (he ore ^Bj will be f tlie ivlkole eiraxttifefettee ef which it is an mt, as the «fe KZ is to the Whole e(i«diirft»ettee of ithkh it is an wc. f^edhice BC to I>» aiid ^hfoog^ <:4sim HK at fight ttngles to DB <11. 1.) ; thm BA : tBB:\ ftligi^ J?Cil : ftttgle BCH (13. 6.) */ Bil : 4K BH : : angle BCif : 4K MUgle BCA, (13. 5.)i that is, BA is to the whole circumference BBDK, as the angle BCA, is to four right angles; in the same maimer it is ih6wn> that XZ is to the Whole chx<uniference ZXB as the.siime angle BC^f to four right angles ; wherefore AB : the whole eircamfereoce BHDK : : XZ : the whole drcamferefice ZXE, Q. £. D. 1399. Hence also, if the dreumferences tt Hiese two drcles he eteh dmded into 36D degrees, as above (Art. 936.) At wOl ^ontfain as many degreed of the c h f canife f e nce BSDK, as XZ does of the drcomilerence ZXE. AN APPENDIX TO THE FIRST SIX BOC«S OF EUCLID. Cmttainmg some useful propositions which are not in the Elements. 240. If one eide of a triang^ be ^iseoted» the sum of thb squares of the two remainii^ ndes is doable the square id hall tl^e side bisected^ and of the square of the line dcawA from the point of bisection to the opposite angle. Let ABC be a triangie, having BC bisected in D, and D^l dr awn from D to the opposite angled; then will BSl^-f ^Q^ss fi.BS)HSS|». Let AE be perpendicodar to BC, ihm foeciMtsc BEA k a ri^t angle, 2Zi|'aB^H S3^ Md ^es€£)«'f iO|«, (47- 1.) Part Vltl. APPENDIX TO EUCLID. +£C)H2.E3^. But since BC is divided in« to two equal parts in D, and into two un- equal parts in £, 5£|« 315 =s 2 . JBD> + .2cl^ =2.'55l«+^ .f£5l^ But Se|*+£51»=d31«, (47. i.) SE^P + E2*=2.W + 2.S31'=) 2.fiSl«+D7)«; and the same may be proved if the angle at C be obtuse, by using the 10th proposition of the second book instead of the 9th. Q. B. D. 241. In ai^ pandldogram, the sum of the squares of the diameters, is equal to the sum of the squares of the sides. Let JBCD be a parallelogram, ^C and BD its diameter?, then wm 2c1*+551^=^H5c]*-H'^'+S^'- Because the angle AED^ CEB (15. 1.) and EJD^ECB t59.L) the triangles AEDy CEB have two angles of the one =:two angles of tBie other each to each, B C and a side opposite to the equal angles in each, equal, viz. AD= Be (34. 1.) •.• -^E=ECand D£= =£B(g6. 1.); and because BD is bi sected in E, S 2[i^^AS]^^2. BEi^+E2\\ and 5cl«+c5?* =:(2.Bll^-h£Cl^ ( Art. 239.) =) 2.B£)«+E^^ v 55l*+23l* + 5(3^ e2J«= 4.S£)« +:B?«=(since4.1i9*=^BBlSand 4.e31» =:30«.by 4.2)B2S«+"5S|*. Q.E.D. Cor, Hence the diameters of a paraBfelc^nrai bisect each otheir. 242. If the sum of any two opposite angles of a quadrilateral figure be eipial to two right angles, its four angles will be in the circumference of a circle. Let ABCD be a quadrilateral figure, having the sum of any two of its opposite angles equal to two right angles, and let a circle be described passing through the tlu-ee points. A, B, 1>, (5. 4. afld A*t. 194.) I «ftf the ckcumfctence sbdill MkeWise ptos 316 GEOMETRY. pa»t vni. through the fourth point C; for if not^ let the fourth point fidl without the circumference at a, and join DC; then since -by hypothesis the sum of any two (^posite angles of the figure are equal to two right angles, •/ B^D+B6D=tworightang. les, but B^D-f BCl>=tworight angles (29.3.) v BAD-^-BGD^zBAD+BCD, take away the common angle BAD, and BGD^BCD^ the interior and oppo- site equal to the exterior which is impossible (16. 1.) %* the fourth point cannot &11 without the circle, in the same manner it may be ^ewn that it cannot fall within it, '.* it must fsdl on the cir- cumference at. C. Q. E. D. Cor. If one side BCof a quadrilateral figure inscribed in a circle be produced, the exterior angle DCGssthe interior and opposite BAD ; for DCG+DCB=two right angles (13. 1.) and B^D-f I>CB=two right angles (22.3.) •/ DCG+DCBs^BJD +DCB, take away DCB, and DCG:siBAD. 243. If the vertical angles of se%'eral triangles described on the same base, be equal to each other, and the circumference of a circle pass through the extremities of the base, and one of the vertical angles, it shall likewise pass through all the others. Let ACB, ADB, AEB, &c. be the several equal vertical angles of triangles described on the common base AB, if a cir- cle pass through A, B, and C, it shall likewise pass through the remaining points D, £, &c. Take any point IT in the circumference on the other side of AB, and join AK, KB, then wiU ACB-^ AKB:=:2 right angles, (22. 3.) 5 but ADB=AEB==ACB by hy- pothesis, '.* each of the angles AEB.ADB together with^JiTB =2 right angles, •.• (Art. 241.) the angles E and D are in the circimiference. Q. £. D. 243. If two straight lines cut one another, and the rectangle Part VIII. APPENDIX TO EUCLID. 317 contained by the segments of one of them^ be equal to the rec- tangle contained by the segments of the other, the circun^. fet^nce which passes through three of the extremities of the two given straight lines, shall likewise pass through the fourth. Let AB and CD cut each other in E, so that AE x £5= CExED, the circumference ACB, which passes through the three points A, C, and B, shall likewise pass through the fourth D. For if not, let the circumfe- rence, if possible, cut CD in some other point G; then since A, C, B, and G, are in the circumfe- rence^ the rectangle AE x EB=s CExEG (35,3.) but AExEBsz CExED by hypothesis j •/ CEx EG=:CExED, V EG^ED, the lesssathe greater, which is ab- surd j therefore G is not in the circumference ; and in the same tfiannei' it may be shewn, that no othft point in CD, except D, can be in the circumference. Q. E. D •. "Join CB, and through K draw KP parallel to Fd then since the ai^le ^EC^ABC+ DCB (S2. 1 .) if the angtllar point E were in the circumference. It if plain that it would be subtended by an arc equal to AC+ DB ; and con- seqaently, if E were ai the centre* it would be subtended by an arc etfual to ^ "^1 («0« 3.) Again, if JSrCbe joined, it may be proted (29. l. and 3fi.8.) that CP and HK are equal, but the arc BDP^^CPB—CP^) ^PB'-HKi and sin«e the angle BKP^BFCi and BKP is subtended by tfce arc BOP^ if BKP were in the clmimference, it would be subtended by an arc equal to BDP: but if it were at the centre, BKP would be subtended , BDP CPB—HK by an arc « (— — (20. 3.) that 48=) j by what has been shewn. And since an angle is measured by the subtending arc described about the angular point as a centre (Art. 262.) it follows, that if two straight lines JB» CO cut one another within m circle^ the angle AEC ie measured ^by half the *»tm gfthe subtending arcs AC and BDy and {hy similar reasoning) the angle ^ED is measured by half the sum of the arcs APD, CKB. But if two straight lines CF, FB cut one another without the circle, the angle BFC is measured by half the diference of the intercepted arcs CPB and HK; this is «oiuieeted with Art. 261. 262. 815 ovmemY. PaatVDJ. 5244. hU tkef^ bf tiro i^iiH^ lilies CP and i^& cii^^ drde in two fdaii$, mad ««ch <ilher ia a pcunt F whboiat the eirde; aodletCf'cuttbeciceiinfei^aoeiii CaiidH^ai^ it in Bi i^u iS 9^ ^9iBi Klmtakenm FB,9oibat CFxFHrsi BFxFK, 1 9t^ the point JT is in the drcumfeience. For if Dot^ kt tba ciieiuiifereiice HJB cut FB in X^ then CFxFBssBFxFL (3^. 3. car.) but CFxFH^BFxFK by hypcytbesis, v BFx FLszBFx FK md FL=:FK, the laos^cth^ greater^ which is absurd. *.' L is not in the cireum&reaace } and in like manner it may be shewn that no other point in £F, except B and K, can be in the cinnim£ereoce; K ia therefore in the circumference. Q. £. D. 945. If a straight line AB be drawn from the eoOanemil^ A of the diameter AC, meeting the perpenjlictilar ED in ^ then will the rectangle BA x AE:n CA X AD. Join BC, CE, then because the aiigle ABC in a aevuoird^ is a right angle (31. 3.) CBE is also a rig^ angle (13. 1.) and if a circle CDEB be described on C£ as a diamatev^tta dffcufldb* rence shall pass through the pcants C, B, £> and D; and 8uae& BE and CD meet in the point A, BAxAE^CAxAD by 35^ oreor. 36.3 Q. £. D. Henc» EA:AD::€A:ABQ3^ 16.6.) 246. If an arc of a circle he W^ctfid, and £rqm the egj^trf^mi* ties qf the arc and the point of i^isection^ straight Unes be dtasm to any pcnnt in the drcum&Kenee^ titon wlK the som of the two lines drawn from t^ extremities of the arc, have to the line drawn from the point of bisection^ the same ratio which the chord of the arc has to the chqrd of half the are. Let AB be an arc bisected in C, a)|d D Mff point ia the Part Vni. APPENPOX TO EUCLID. 319 caoamfeiwm, 'yAa Al>, CD, BJ^, 4C and BC, th^n will ifX>+ DB:DC::BJ: AC. Bee^me ACBX^Ib a quaAAiiteral fignre inscribed in a ctrck, AM.CD {^JD.CB^DB.AC (JD.6.) nAicli. htOHMc CBati#Q otAB.AC^BD. AC,^AC,jm^EI> (1.8.) «m1 be- ^soae the skies of eqiud reetan^es are reciprocally proportional (14. 6.) AD^BDiCDiiAB.AC. Q.£.l>. 247. If two points be taken in the semidiameter of a circle, sacb, that tlie rectangle cc^tained by the s^;inent8 between them and the centre, is equal to the square of the semidittneter ; the straight lines drawn fixnn these points to any point in the circum* ference, shall have the same ratio^ that the segments of the dia- meter between the two fore-mentioned points and the circum- ference* liave to one anotluer. Let I> be the centre of the drefe/ ABC and DF the semi- ^Bameter p r odtt eed, in whieh let ^aod Fbe taken, snefa, that mOFx^AM^s then if B» 5, •ad n be drawn irom E zxA ^ , ^ ^ _^ ^, to any point B in the dreum- F ^a1 E j> j ferenee ^B : FB : : EA : A¥. Join AB, ED ; the n sinc e *»f hypothesis £f>.l>Fs(:3S|« «) BS\*', M>F:DB: : DB: DE (17. 6.) j that is, the sides about the common angle D of the triangles FBDt EBD are propor- tionals, *.* these triangles are equiangular (6. 6,), and the angle FBD=iBEI>^EAB^ABE (32.1.) > but EAB=ABD (5.1.) •• {FBD^) PBA-^ABD^sABD-k-ABE, take away the common angle ABD, ajad FBA=:ABE, •.' B-4 bisects the angle FBE, •/ MB:BF::EA; AF(3.6.) Q. E. D. Cor. Hence, if FB lie produced to G, and fiC joined, the ex- terior angle jiBG will be bisected by ^c For since ABC is ^l right angle (di.S.) it is half the sum of the angl^ fbe ^nd JSBG (13. 1.) : but ABE^^ FBE, '.' ^BCsz^EBG, 248. If from the three angles of any triangle, perpendicularly 520 GEOMETRY. Part Vnt be drawn to tlie opposite sides^ these perpendiculars shall inter- sect one another in the same point. Rrst, In the acute angldd triangle ^BC^ let the perpendicu- lars BD and CE intersect one another in ^, join jfF, and pro* duce it to G, AG is perpendicular to BC. Join DE, and let a circle be described about the triangle AEF (5. 4.) then since by hypothesis AEF is a right angle, AF will be the diameter of the eirde (31.3.) ; and because ADFs, ABF, the circumference of the same circle shall pass through the point D (Art. 242.) and the points A, B, F, D, will be all in the cb- cumference. But because the angle EFBslDFC (15.1.) and BEFsz CDF (by hypothe- sis) '.* the triangles BEFoxid CDF are equiangular (32. 1.) V BFiEF:: CF: FD (4. 6.) -.' BF: CF:: EF: FD (16. 5.) and since the an^e BFCssEFD (15. 1.) and the sides about these equal angles are proportionals, the triangles BFC and EFD are equiangular (6. 6.) •.• the ai^le FCB^BDF^EAF (21. 3.) / EAP:=zFCGs and AFE^CFG (16. 1.) •.• AEF^FGC (32. 1.) j but AEF\& a right angle by hy- pothesb^ '/ FGC is also a right angle and AG is perpendicular to JJC. Secondly, In the right angled tiiangle AFD» draw DH perpen- dicular to AF, '.* AD, AD, and FD, are the three perpendiculars^ and it is plain that they all meet in D, Thirdly, In the obtuse angled triangle BFC, BE ]^ perpendi- cular to CF produced, CD perpendicular to BF produced, and GF perpendicular to ^C^ and it appears by the foregoing de- monstration, that these three perpendiculars BE, CF, and CD intersect each other in the same point J. Q. E. D. 249. If a straight line tpuch a circle, and from the point of contact two chords be drawn, and if from the extremity of one of them, a straight line be drawn parallel to the tangent meeting, the other chord (produced, if necessary)^ then wiU the two chords and the segment intercepted between the parallels^ be proportionals. Tam VSH. APPENDIX TO litCLljy. 921 T.- Let TA touch the circle m A, from whence let the chords AB and jiC be drawn^ and from C the extremity of one of them J let CD be drawn parallel to TJ (31.1.) meet* ing^B in D, then will BA : AC II AC', AD. Join BC, then because the angle .^CBs TAD (32 . 3.) = ADC (29. 1 .) and BAC common, the tri- angles ACB, ADC are equi- angnlar^ and AB : AC : : AC : AD (4. 6.) Q. K D. I Cor. 1. Hence BA.ADssAI!\2. 2. If ^B pass through the centre, then will ACS be a right angle (31.3.), and CD will be perpendicular to^^B (18.3. and 29. 1.) ; and since AB : AC:: AC: AD; the side AC of th& triangle ACB is a mean proportional between the hypbthenuse AB and the segment of it, AD adjacent to AC, as is shewn id cor. 8. 6. 250. If a perpendicular be diawn from the vertitol angle of any triangle to the base, (produced if necessary), then will the rectangle contuned by the sum and difference of the sides of the triangle^ be equal to the rectangle contained by the sum and difference of the segments of the base. Let ABC be a triangle, and CD a perpendicular drawn from the vertical angle C to the base AB, meeting it (pro* VOL. 1I« 1^ 3j» eEQlfESBY. PmitVIBw dttced if neces aary^ as' in % g.) in D, then w i U i^€-f CB .AC-^ CB as^Z> + DB.JD'^ DB. Krom C as a centre, with the akort€8t side CB for a distance, describe a circle, catting AC pro- duced in G and H, and ^B (pro* duced in fig. 2.) in £ ^nd B. Then since CGz^CH=CB, •/ AH^AC + C0=the sum of the sides, and AG^IaC-^CG^) AC^CB:=i their difference 3 and because I>B ssD£ (3. 3.) (AD+DB=z) AB is the sum of the segments, and {AD ^DB^AD'-DE::z) AE their difference in fig. 1 . sdso (AD -A. B^^^--.a — -^B +DBss^D+i>£=:^£s=:the sum of the segments in fig. S. and (^D—DB=)^Bstheir difference. Wl^ecefare, (cor. 36.?.) AH.AG^ABJiE: that is* the rectangle contained by the sum and difference of the sides AC and CJ6, is espial to the r^taogle^ contained by the sum and di&rence of the segments. Ap an4. BD, intercepted between the extremities A and jB of the base*. (or base produced,) and the perpendicular CD. Q. £. D« Cw. 1. Hence AB:AH::AG:AE (16. 6.) that is, tl|e base of a triangte. : is to the sum of the sides : : as the difference of sides to the sum : (fig. 2.), or difference (fig. 1.), of the s^ments of the bate, according as the perpendicular CD fsdis without, or within the triangle. This inference is particularly useful in trigonometry, when the three sides of a triangle are given to find the angles. 2. Becau se DBssPE , and B£=2 BD, '.' AB^E=^(AB,ABT B £=:) AB^Bt^ B PszASI^T^ABBD : v since JCl^^VS* zsAC+CB.AC'^CB (cor. 6. 2.) ^21b\^+^AB.BD, the rec- tangle contained by the sum and difierence of two sides of a triangle, is equal to the square of the base minw or plus twice the rectangle contained by the base, and its least segment. 3. If ABC be a right angle, the point B coincides with D, and the circle described from C with the distance CB will touch the base AB in A and (36. 6.) HA.A6si (^*; that is^ since B coincides with Dzs) aB^i *.* the rectangle contained rim mi: AFPENIXnC TO EUCLID. Stt kf the sum and difiercace o£ the hypotlieniise, and one of the •ides is equal to the square of the other nde. 4> Since by eor.e . ^C+ CB.^C— eA=3:^75l«+2.ilB.BJ>, and- M-h C3.dC^ CB^:^^^Cfff^ (5. 2.) •/ A^^^aS^^AS\^^ ^JB.BD, and ^«s±31bl«+e^«tf2.^B:BD. Or the square* of the side AC is less or greater than the sum of the squares of AB and CB, hj twice the rectangle contained by the base, and the s^ment Cft according as the angle ABC k acute or obtose. This is the same as 1^ and Id. 9 Euclid. 250. B. The chord of one sixth part of the circumference being given, to find the chord of half that arc, and thence to inscribe withm the circle a polygon of a great number of sides. Let ABD be a semicircle, C its centre^ dmw the chord DA^^AC (1. 4 ), bisect the m DA in E (30. 3.). and join EA; EA wiU be the side of a regular polygon of 12 sides. Bisect EA, and draw a straight line ixDm A to the point of section, and it will be thesideof a polygon of 24 equal sides; and by continually bisecting, we obtain the sides of po^fgons of 48, 96^ 192, 884^ &c. equal sides. 251. To find the circumftrence and area of a circle, having ^ diomeler given p. RtJLB. Eir»t. Since there is no geometrical method for deter- mining accurately, the length of the whole, w any part of the wo m fe r ence^ we must be content with an approximation; which however, may be obtained to such a d^pree of exactness, &9 to differ from the truth by a line less than any given line. Secondly, If two similar polygons of a great number of sides, be doe inscribed in, and the other circunascribed about « cirde. .' Hiii prdblenuwill serve to iliew by what laborion9> methodi Wq^llis, Romairas, Metins, SneUins, Vao Cenlen, and others, obtained appraximatiuiis to the citdes periphery ; the same HAnf^ may however be performed with madi nore ezpaiUtion and ease, by the method of fluions, infinite serie% &c. be« %^i<a ilscfrtMe and Aj^pHcatim n/ Flmeumi^ p«it. 1. sect. S, GSOHETBY. Pi«VHt the circumference wffl'be greater than the sum of the sides cf the former, but less than the sum of the sides of the latter 5 aiKl thtrefore^ if the numbers expressing these sums agree in a certain number of figures, those figures may be considered as' expressing (as far as they go) the length of the drqiaafereiice which lies between the two polygons > and if half the di£krenoa of the remaining figures be added to the less number, or sub* tracted from the greater, the result will afford a still more accurate expression for the length of the circumferenee. Draw any straight line AC, and ofi it describe the eqcfilateral trian^e ABC (1.1.) fi'om C ^ a centre, with the distance CA = CB describe the ore ^£jDB; then because ABa^AC^ihiQ side of an equi- lateral and equiangular hexa- gon inscribed in the circle (15. 4.) •/• AEDB Will be one sixth of the whole circum« ference. Let f=^C=l, c=^B=i, the arc AE=ED=iDB, and a:^AE=zihe chord of one third of the arc AB; then since the arc EB ia double the arc AE, the angle EAH=^ACE (20.3.) and AEC is common^ -.* the triangles AEC and AEH are equi- angulat (32. 1.) and CA:AE::AE: EH (4. 6,) ; that is, r ; ^EH; alsQ CEiAEi: AH : EH •.* AEssiAH, in XX like manner it is shewn that BD=zBK, ',- AH=^BK, •/ AH-^- BKzt9ix, and HK^iAB-^AH-^BK^) c— 2x; but CE : ED XX : : CH : HK (4. 6.)i or r : a: : : r— . — : c-^^-^ whence, multij^y- fly- ing extremes and means, cr'^irx^zrx--* — 5 which bytransposi- r tion, &c. (since c and r each =1,) becomes **— 3a?=— 1, the root of which is the chord of AE, or of xt part of the whole circumference. Next to trisect the arc AE, let 3 y— ^ss^r, the chord of AB, Part VHI. APPlENDIX TO EUCLID. 335 we shall hare ap*=^fy»—27y*-|.9yr—y9 ^ ' and — 3ar=— 9y+3y' and + 1 = 4-1 Their sum x* — 3 a:+ 1 = — 9y-h30y» — 27y« >9y^— y + 1 =o, <he root of which is the chord of ^V pa^ of the whole circurn* Terence. Again, to trisect the arc of which y is the chord 5 let 3 2— «»=y, and if this value be substituted for y in the last equation', we shall obtain an expression in which the Talue of z will be the chord of the -rW part of the whole circumference. Proceed- ing in this manner after sixteen trisections, the chord cff -nHulirsis part of the circumference (the radius being unity) will be found to be .Oo6oOOOZ4SQ6979^S9SS^OSS, which num- i)er being multiplied by 3582803^6 (or the number of sides of the polygon, of which the above number expresses the length of one side) the product will be 6.283 ia53d71795859684897'5l? =the perimeter of the inscribed polygon. "- ' 352. Next, we arc to find the length of the side of a circuoir ^bed polygon of thf$ same number of . sides, in order to which^ let AB:^the side D^ of the inscribed polygon ,as .found above, ^ DE the side of a similar circuinscribed polygon 5 bisect AB in H, join ' CH and produce it to F. Th en 'c7p— 2^ ^=1151^ or 1«— .O00O006l2163499644916|^= 1- .000000000000000147950723611871658 0846470516 =£ .99999999999999985204^! 76388128342, te.:?:CF|«, the square root of whicl^ number is .99999999999999999 &c.=CH; now CHiHA:: CF,I^^ ^DF (4. 6.) that is -0000QQ012163409644916016.X l _. .^9909999999999999 .000000012163499644916, &c. = DF, which number multiplied hy2give8 .O000OO0iJ432699929832, &c.= DE But .00000002432699928983, 8iC.:^AB and since: these two numbers agree as far as the 16th place of decimals, and the arc APB lies between DE and A By it follows, feat those 16 decimal placies will express the length of the ar^ T 3 ^FB very nearly; tbat is, tlie above number will difler from the troth by a very small decimal, whose highest |ilaoe is 17 placo below unity. Whence XXI0000(»4386999^9ssthe length of the arc JFB or of the m a tja i ri part of the whole circanh- ference extremely near. Now if ihe length q£ the arc JFB m above determined be multiplied into the denominator of tUi fraction, the product will be 6.^83185306ia9S4rS3sthe cir- cumference of a circle whose diameter is 3, yery nearfy. 253. Having found the circumference of a circle, we can readily find the area» if not with strict accuracy, at least suffi- ciently near the truth for any practical purpose, in order to which, let us suppose an indefinite numl^ of straight lines drawn from the centre to the circumference, thesjs will divide .the circle into as many sectojjps, the bases of which will he infde- Anitely small orc*^ and their common altitude the rsidius of the circle } now since these small urcs coiiicide indefinitely near with U&e sides of a circumscribed or inscribed polygon of the saaae number of sides as there are sectors, these sectors may evidently be considered a^ triangles, the bases of which are the above small arcs, and their oonmum altitude the radius ; but half the I)a8e of a triangle, multiplied into the altitude, will give the area (42. 1.) wherefore, (half the sum of the bases, that is) half the circumference of the circle, multiplied into the radius, will give the area of the triangles, that is, the area of the <»r^Ie j thus 6.2831853, &C.X1 ^ « , . . , . ;; ss3.1415<)26> &c.=sthe area of a circle> whose diameter is 2. 254. Having fbund:4lie cireon^bienoe of a cirde, whose dia- meter is 2, we are by means of it enabled to find the Gircnm* ference of any other circle, whatever its diameter may be ; for let the inseribed polygon (whose sides coincide indefinitely near with the circumference) have n sides, the length of each being r ; and let a similar polygon be inscribed in any other circle having the length of its side szs, then will wr=the periphery of the first polygon, and n«=that of the second. Let Issthe ra* dius of the former circle, ^ssthat of the latter ; then if linea be drawn from each centre to the point* of division, in the re- spective circumferences, we shall have 1 : r : : « : ^, (4. 6.) whence (16. 5.) 1 : * : • r : «, and consequently (15. 5.) l-.t:zm ft ne, that !«, the peripheries of the similar polygons are to »AHT VIB. PRAdWCAti PItOBLEMS. MT «tlttr as the raiii of tJbeir drctowcribed erfcteti but theie iH^pOtts ooineide inAttivSMy near with their ciremoferenM: wherefore the chmnifereneeB of ckties«ait «s tHeir radii. 255. The aite of one ctrde bein^ known^ that of anoUier fiitle having ;a given diameter, wulj be found ; let i>s:the dia- flMler of a circle, ^sits ^xesLj aind d»Uie diameter of another ^iirf^ whos^ area 4? is re^edj then .(«. 13.) D^iO^iiAi t^ Whence ^=-^5-, the area required. FIUCTIGAL OEOBIfiTRY. SS5. Practical iGreometry teaches the appli^tion .<^ theoKs t!(jal Geometry, as delivered by Euclid and other inters, to practical uses ^ 256. To draw a straight line from a given point if, to re p r e e e ni my length; in yards, feet, inches, Sfc. Rule. I. Let each of the divisions on any convenient scale of equal parts represent a yard, foot, inch, or other unit of the measure proposed. II. Extend the con^nsses on that scale untU the number of spits proposed be included exactly between the points. III. With this distance in the compasses, and one foot on A, describe a small arc Bit B s lay the edge of a straight scale or nder fhtfn ^ to B, and draw the Une AB with a pen or pencij^ ind it wUl be the line required. A — '—^- j B iBxAMPLBs. — 1. To draw a straight line ftom t)ie point 4 to ^present 13 inches. ^ Hie ^idlb|nii^ problems are intended as aa introdoetion to the practical ^plication of fome of the principal propotitions in the Elements of Euclid, ttd likewtte to assist the stodent in acquiring a knowledge of the use of a <tte of mathematical instrameots. From a great rariety of problems nsaally liven by writerf on Practical Greometry, we have selected such as appear most Becessary, and likewise such methods of solving them as appear most simple ttd obvious ; to a learner well acifuatnted with Buefid, other methods will occur, and he should be encouraged to exereise his ingenuity in disooTering fod applying them. The best Momentary treatises on Practical Geometry a.o^ HciUQiation^ are those of Mr. Bonnycastie and Dr. Hntton. MS gbombtrt: Faet vni. mOi 01^ foot en O txi€^ tke 44k^ io Ae l^ih ^mtim m ike •oaU yw thoote to adap<» and ^ppiff that d i ti m$e e from A m above direeied, md U mill give the loigth propoied, 3. Tb draw a line thai shall r epre ac nt 35 yards. Let each jniwumf dvMtm he cmuidtnd as 10 yards, thm miU each subdiv Mum represent 1 yard; appliy the compares frem 3 haekwardM (to the left) tothe&th eabdimsum, and 35 svbdieu sions will be included between the points: apply this from the given point and draw the line as before. 3. To draw a line equal to 9(S3. On the diagonal scale, lei each primary dkrision represent 100, then will each subdivision represent 10, and the distance which each diagonal slopes on the first parallel wiU be 1, on ths second 2, on the third 3, and soon; therefore for 363 extend from the number 2 backwards to the sixth subdivision, on the third partUlel, {viz. the 4th line downwards) and it wiU be (he distance required, S57' To measure any straight Une \ Rule. Extend the compasses from one extremity of the given line to the other, and apply this distance to any convenient scale of equal parts, the number of parts intercepted between the points, will be the length required. ' Note. If the sides of a rectilineal figure are to be measured* the sanie scale must be used for theto all 5 and one scale must be used for each of two or more lines, when their relative length is required to be ascertained \ . 268* To bisect a given strniiht line 49. # ' By the word meature it meant the relative measure of a line, thai is, tbe length of that liae compaied witi)^ tl^e ki^|[1^ of aootlier line, both bciof mft- tured from tbe same scale ; if we call tpe subdivisions of tbe scal/s feet oc yards, tbe l^^e will represent a line of as many feet or yards as it contains such •ubdiyisions ; to find tbe abso^utf ipeasare of a line in yards oj feet, w« must evidently apply ^ scale of actual yards or feet to it. • Any scale of equal "parts may be employed for tbis purpose, but it will b« pjoper to cbvoae one tbat will bring tbe proposed figure witbin the limits you intend it to occupy ; every part (f i«. every line) of tbe figure nmst be mea- ftured by one scale, and not one line of tbe figure by one sc^e, and aootbef line by anotber. tlMT VIII. FRACnCAL PROBLEMS. xm AvLS I. With any dittance in the compasses greater tiian lialf the given line, let arcs be de- scribed from the centres A and Bj cutting each other in C and D. A- II. Draw a straight line from C to X>, and it will bisect the given straight line, as was re- quired ». \°- / \ ■\ c/ ^ \ ./D l9 ^59. From a given point J0$ in ti gwen strtnghl hne JBi toereet a perpendicular FD. Rule I. From any point C (without AB) as a centre, with the distance CD, describe the circle £I>jP cutting AB (pro- duced if necessary) in E and D, and draw the diameter ECF. ^ II. Join FD, and it will be i ^ perpendicular to AB, as wias "~" required «. By the Peotractok. Lay the centre of the protractor on A and let the 90 on itii cii'cumference es^ctly eoincide with the given line 3 draw the Une FD along the radius, and it will be the perpendicular required. Si59.B. From a given point F, to let fall a perpendicular to a gwen straight line AB» See the preceding figure. Rule I. In 4B take any point £, join FE, and bisect it in C, (Art. 258.) II. From C as a centre with the distance CF or CE, describe * If the points jiC and J?C be joiaed, ibis rule may be proved by ^lid S. I.- « The proof of this role depends on Euclid 31. 3. Of tb9 Tariont methods fcr erecting a perpendicular^ ^ven by writers on Practical Geometry, this is tip mo^t simple and easy. ^so GEosoeneiT. ^aetviJ. Die circle EDFj jcin FD, tttd it w91 be die perpendictihr required*. 260. Through a given point B to draw a straight line paraUel to a given straight line AB. RuLB I. Take any point.Fin AS, and from £ and Fas ce«« tres> with the distance EF, describe the ares EG, FH. II. Take the distance £G in % the compasses^ and apply it fix)m 7L " ■ ^ — ^ Fto Hon the arc IV. / """•••^.^ / III. Through E and Hdmw [ -^^7 the stnught line CD, and it ^^1^ ^ ^ 5 uriB be parallel to ^B Bs was requh^d ^ Bt THB PABAI.LBL RVLBR. lAy the rukr 90, that the edge of one of its pamBeb tasy exactly coincide with the line AB. Heldiog it steady la Uitt position, move the other parallel up or down untfl it cut the point E, through which <hraw a line CED, and it will be panl* M to AB. If £ be too near, or too distant for the extent of the rukr, first draw a line parallel to, and at any convehieiit d^tsuace from AB, to which draw a parallel through £ u before^ and it will be parallel to AB. ^61. At a given point A, in a given stra^ht line AB, to make an angle BAC, which shall measure an^ gioen number ff degifes. RuLB I. JBxtend the aompaases from the beginning Of the scaleof chords (mark- ed C,) to the 60tb dcr ^s gtee, and from the given ppint A, with ,tbis distance, describe an arc cutting AB (produced if neces- jBaiy) in £. 11. Extend the compasses fro;m the beginning ct the scale of « This depends on Euclid 31.3. y Since the arcs EG, HFaie equal, the angles JEFG, F£lfai the ceot^ are equal, (Euclid 87. 8.) and therefore A£ is jMuraUel to CD^ EdcUd S7. 1. Past VI0. FRACHCAL PRCIBLElifS. ^l idionfo, to tbe aumber deacrting tlie meftsore of the |»D|Mfad angle, and from JE as a centre^ with this distaooe, cut 4m above arc in the point R III. Through F draw the straight line JB, jund the angle BACi^ill be the an^^le required ". fixAMPLas'-^l, Let (he angle pi^pqsed ooBBSMre .30 dqprees. Bm>i$^g described the arc RFwUh the radim W, extend the tmpaues from the b^mofg of the eeale to SOj la^^ tkkti^ tent from E^ and draw a line through the point mturked with the fiompasses, and the angle of 30P wiU be made. . S. At the point A in AB make an angle measuting 160 degrees*. Here the proposed angle bemg grmter than 90, ii mil be eon^^ient to take ii at twice $ ia^f 9/ BXP first, m E¥$ ihemfnm i^> % <^ 70^ tnore; drm a Uft^ through Jhe extwmHyofike 70, and it wiU make wOh AB M^^nngle ^ ISO d^rem, By tw« (J{iip7fiAc:TAi. Laj the central point on A, and the fiducial e4ge of tbe ndiua along AB, so that they exactly coincide 3 then with the pointer, make a fine dot^ opposite the proposed degree (reckoning from "tile line AB) on tbe ci/cumferenee ; through A and this dot, draw a straight line, and it will make with AB the angle required. iW«»4w«n^M*f*«saaMaiMMBi«fMi^ ■ If the cireumference of a circle be divided into 860 equal parts called iegreet, one sixth part of the circamference will ueasnre 60 degrees, and its chard wiH be equal to the radias of the circle (EtKlid 15. 4.) ; wherefore, if thfc 4hit 60 degrees on any scale of chorda be taken in tbe compasses, and a circle he deBcri)>ed with that distance as radius, the chords on the scale, wiU be tbe ffopcr measure for the chord of every arc ai that circnmference, as well as for the circmnfereBce itself; and since the arc intercepted between tbe legs of the ^ligle, (being described f^m the angular point as a centre,) is the measure of ^ angle it subtends, (Euclid 33.6. Art. 236.) the rule is manifest. By this yvoUem an ang^e inay be made,.eqaai to any given angle. * 1V> measure, or lay down, an angle greater than 90*, the arc must be takea hk tbe compasses at twice; thus for 100% take 60* first, and then 40* ; or 60* flrtt, and then the remaining fit)*, &c. For an mv of 170* take 90» and 80«, or 60*, 50*, and tO», vis. at three times, ftc. &c. If two straight tines cut Me another within a circle, their angle of inclination is measured by half the ^^A of tbe int^Mepted afeti but if they cut without the citde, their angle of hieUaaUon is meaMNd by half te Werenca of the inlercepted arcs. See thi Bote on Art* S48. ' . < I ". i3S QBOMBTRY. Pa«t VOT. X ExAMPLBs. lltke «t given poititf , ih, given straigfat lines, the /oUowing angles, via. of 20^, 35S 45^ 58«, 9a>, 160^, and iri°i. 262. To mefluure a given angle BAC. See the preceding figure. RuLB T. Frcmi the angular point S as a centre, with 60^ from the scale of chords as a radius, describe the arc EF, cutting the legs of the given angle (produced if necessary) in E and R IJ. Extend the compasses from £ to F, and apply the extent to the scale of chords, so that one point of the compasses be on <the beginning of the scale -, then the number to which the other point reaches will denote the measure of the given angle \ ExAMPLB.' To measure the angle BAC. Htming with the radius 60^ described the arc EF, extend the jcopipasses JrcmE ta-F; then ^ihis extent reaches from the 6«- ginning of theseale toSS^, the awgle BAC measures 35 degrees. Bt THE' F)|{)t«ACTOK. Lay the fiducial edge on ABl'so that the central notch may ^ The reasoa of th^ rule will be cvicknt from the preceding note. Axt iogenious method of measuring angles, by means of an undivided semicircle, and a pair of compasses, without the assistance of any scale wbateyer, wu pTOpwed by M. De Lagni, in the memoirs of the French Academy of Sciences ; some account of bis method may be found in Dr. Hutton's Mathematical Dictionary, under the word Goniometry. Thomas Fajitet De Lagsi was bom at Lyons in the 17th century, an4 died in 1734 % l^e ,was successively professor royal of Hydrography at Rochford, sub-4irector of .the Generfd Bank at Paris, and associate geometrician and pensioner in the Ancient Academy. De Lagni excelled in Arithmetic, Algebr^, s^nd Geometry, sciences which are indebted to him for improvements ; he invented a binary Arithmetic, re<|uiring only two ^gures for all its operations ; likewise some convenient approximating theorems for the solution of higher equations, particularly the irreducible case in cubics. He gave a general theorem for the tangents of tmUiiple;-areSf and determined the ratio of the circumference of a circle to its diameter to 120 places, which is the nearest approximation for the purpose, that has been made. Our author was particularly foj^d of calculating, and It may be truly said of jiiim, that ** He felt ^the ruling passion strong in death i" for on his death bed, when he was apparently insensUilcv one of his friends asked him, What is thf square of 12 ? to which he immediately replied, 144 i we regret, that the last foments of this ingenious man, were not emplo^d on subjects of iq$nitejy greater importance. pabt vnr; pRAcrrcAL problems. Sss be on Ay then will the degrees (on the circumference) inter- cepted between AB and AC^ be the measure of the angle. ' Example. I'd measure the angle BAG by the protractor. Lei the centre coincide with A, and the fiducial edge with AB; count the degrees {on the circumference) from AB to AC, and the number will he the measure of BAC. 263. To diofde a given angle ABC into any number of equal parts. Rule I. From the angular point B as a centre, with the radius 6(P (from the scale of chords,) de- scribe the arc EF as before, and find the measure of the angle' ABC. II. Divide the num- ber of degrees in this measure by the num^ ber denoting the ntmaber of parts in- to which the angle is to be divided, and the quotient will be the degrees each part will measure. III. Extend the compasses, from the beginning of the scale* of chords, to the degree denoted by the above quotient, and apply this extent successively along the arc EF. IV. Through B and each of these divisions, draw straight lines Ba, Bb, Be, Bd, &c. and the angle ABC will be divided, as was proposed <•. Example. To divide the angle ABC into 5 equal parts. Having described EF with the radius 60°, Ut EF measure • If either of the lines SC, BA be less than the proposed radius, (vi«. the chord of 60«») it must be produced to the circumference EF\ likewise BC, BA may be either, or both, so long, that EF cuts them ; in cither case the rule is the same as is plain. See the note on Art. 261 . So to measure an angle with the protractor, it will sometimes be necessary to produce the line* contain- ing the angle, until they meet the circumference of the instrument ; this may be done with a lead pencil, and the produced parts may be rubbed out, after the angle is measured. 3M. cfficof^miy. fABTVin. €Ufpoie 55 degreay them — s= 1 V^thtnliiwJber ofdegn^ m each of 5 the parUf take 11<» (Jram ihe $caU of ebard$) in ike eompasies, attd apptjf a from E to a, from a to b,jrom btoc, and from c to df and Unroagh the jwnit a» b, c, and d, draw Ba, Bb, Be, and Bd, and ABC wiU be dmded into S equal part9. 864. In like oiaiiner the whole drcumference muf he di?kled into any number of equal parts, and by joining the points of di¥i^on> polygonsof any number of sides may be inscribed in it } and if straight lines be drawn perpendicular to the several ladii which pass through the points of divbion» at their extremi- ties, polygons of the same number of sides will be drcumscribed about the circle, as is evident. Bt tbb Pbotractoh. Let the fiducial edge coincide with the diameter of the cirde^ and the oentral notch with the centre, and suppose a polygon of 36 equal sides be required to be inscribed in the drde, mark with the pointer opposite every 10th degree (on the protractor) ; draw straight lines from the centre to these points, and join the points where they cot the circumference $ and a po^rgon of S6' sides will be inscribed : and if at the extremities of these radii, and perpendicular to them, lines be drawn meeting each other, a polygon will be circumscribed about the circle, similar to the former} and by a sunilar method, any other regular polygon may be inscribed, or circumscribed. ExAMPLE8<»l. To inscribe in, and circumscribe about^ a given circle, an equilateral triangle, and a square. 2. To inscribe in, and circumscribe about, a circle, regular polygons of 10, 15, 30^ 24, and 30 sides, respectively. . S65. To divide a given straight Ime^AB into anj^ numher of equal parts. RvLB I. Draw the straight UmAD making any angle with AB; II. Beginning ati#, wi^aqr extent in^tbe-companeBy tdse at- many equal dirisions (al, 12; 23, 3c, &c.) in AD^BbAB is to lie divided into, let these terminate at C, and johi CB. \ 9 PiiitVBL JfRAXmCAL FKOBUSMS. 33& UI.TI«ougli S these divisions C^x draw 8tiii%]|t Hnes parallel to CB, and » *- cutting -4B in ^ ..•••*'' \ the points a, .-••"'t » ^ c, &c. these X-^'^t^ j jf- will divide AB into the number of equal parts required < ExAMpLss— '1. It is required to divide a given line AB into 4 equal parts. FtrsU draw an indefinite line AD, making m^, «f?gZe {DAB} with AB. Secondly, open the compasses to my convenient extent, (or A\) and with it lay off the equal distances A,!-, 1,2; S, 3 Old 3, C. Thirdly, join CB, and through 3, 2, and 1, draw 3 c,. 26, 1 1^ each paraUel to CB, (^rt. 260,) then wiUAB be divided ifUo 4 equal parts in a, h, and c. 2. To divide a line of 44- incliei in length into 10 ^equal parts. Note. By this proUem meif itraight line may be divided into parts which are proportiond to thoaeof a given <tivided straight line*. 266. Tojind a third^ffoportUmal to two given straight lines 4 andB. RvLB I. Draw two indefinite straight lines CD, CF, making anyan^eDCK IT. In these, trite CG ^ equal to A, CD and CJE ' each equal to B, and join -----------^--•---— — • GE. III. Through jD draw c l)#pettai^ ta G£ (Art 2W.) and CF wiii be the _ tkffd proportional re- quired; that \s, {CO : CE :: CD : CF, or) 4 I If :; B I CFt. ' The rcMoa of this rale will appear firom EacUd 10. S. it is pretcsahle t» the oompiez methodt propoied bj tome of the mgdern writers, * SeeSncUd to.9. ' Thi»U the saac with Eudidll.S. ZS6 GEOMETRY. Pak* Vffl. 267. To find a fourth proportional to three given. $tf0^[hi Ima A, B, and C. Rule I. Draw two indefinite Unes OD, OF, as before. II. Take OD equal to J, OF equal to B, j^ and OG equal to C III. Join DF, and through G draw GE parallel to DF (Art. 260.) > and 0£ will be the fourth proportio- nal required ^ for ( DO : OF :: GO i OE, that i$)A:B::C:OEK 26S. To find a mean proportional between two given straight lines A and B. Rule I. Draw the indefinite straight line HK, and in n take HD equal to A, and DK equal to B. II. Bisect HK in C (Art. 258.), and from C as a centre A.- .••" -•.,JE H^ Q -r -K with the distance CM {^CK) describe the semicircle HEK. III. Through D, draw DE perpendi* cular to HK, (Art. 259.) and it will be the mean proportional required; for (ED :DE:: DE: DK, that h) A : DE :: DE : BK 269. To find the centre of a given circle ABD. Rule I. Draw any straight line BD in the given circle, and bisect it in H, (Art. 268.) f This is tbe same with Eadid 1ft, G. k Tbit is fincUd's 1^. 6. BkUlp Vllf . PRACTICAL PROBLEMS. Sd7 IL Throtigli S dnck AS perpendicular to BD, (Art. 259.) umL produee it to E. m. Bi9Bct JE in C, (Art I 258.) the point C will be the i ceatle,of the ^vea circle *. ...•".«••»., 270. To draw a tangent to a circle from any given point, either in the circumference, or without the circle, RtfLB 1. Find the cehtre C, (Art. 269.) and fot T be a given point ^thout the circle^ from which the taogent is required to be drawn. II. Jdin CT, and on it describe the eetnicircle CAT. III. Join^r^ and it will touch the circle as was required. IV. If the tangent be required to be drawn ftom any point itf JM the eifiBunifNtn^e> join CAy ahd dtttw AT perpendicular to it (Art. dA9.) y AT mm touch th^ 271 . To describe a triangle, hatikg its Ihtee sides gibeki ItuLE 1. Let Ai B, and C, b6 the thli^.i»ide8 of the i«f|^i#d triangle, draw a straight line DE equal to one of A them, suppose A, (Art. 256.). II. Take the length of the line B in the compasses, and from D as a centre, with this distance, describe an arc. III. From E as a.cen- B C * This rule depends on Eadid 1. 3. Other methods xuay be derived from Euclid 19, 3; a\,3ySif8i and Tarious other parts of tlie Elements. ^ This depends on Euclid 31. 3. and 16. 3. VOL. IJ. Z S3§ GEOMETRY. Fabt VUh tre^ ^th the length of the line C in the compo38e8> describe an are, cutting the former arc in F. IV/ Join DR EFi and D£F wiU be a trian^e, having its ades respectively equal to A, B, and C ^ Examples. — 1. Desoibe a trisngle of w&ich the sides aj<6 4, 3, ftnd 2, respectively, and measure the angles. Jns. 10$^^ AT, Old 290^. 2. Describe a triangle, the sides of whiteh afe 25, 36, and 47, and find the measure of its angles. 272. To describe a triangle havit^ two sides and the i$icluded angle given. HuLB I. Draw a straight line AB equal to one of the given sides. II. At the point A, make the angle BAC equal to the proposed angle, (Art. 261.) 3 and make AC equal to the remaining given side. III. Join BC, and ^BC will be the triangle required ". Examples. — 1. Given ABssB, AC^6, and the angle BAC=^ SCP ', to describe the triangle, and measure the remainij^ side CB, and likewise each of the angles C and B. Am. side CB^ 4.25, ang. 0=100°, ang. B=^hO. 2. Given 2 sides equal to 210 and 230 Ftspectively, and (he uicluded angle \0p^ to find the rest. 273. To describe a triangle having two sides ABj, AC, and an smgle ABC, opposite to one of them, given. . I H' ■ «. I I ■ > Hm proof of thit rule may b« found in £uclid 32. T. * This rule and th* t^o next are sulSciently ob^ou*. i*ART VIII. PRACTIfcAt PROBLEMS. 339 Bulb I. Draw the side AB, and at its extremity B make an aogle^^Cequaltothepro- posed angle (Art 261.) 1 and produce the line BC. IJ. From ^ as a centre, with the given length of AC in the compasses, de- scribe an arc, cutting BC in C. Hi. Join AC, and ABC will be the required triangle. Note. If the given angle be (a right angle, or obtuse, viz.) opposite the greater given side (as in fig. 1.), the arc will cut BC (on the same side of B), in one point C only; but if the given angle be (acute, viz.) opposite the tew side (as in fig. 2.), the arc will cut BC in two points C, D-, and either of th© tri- angles ^JSC or ^BD will answer the proposed conditions; hence this case is ambiguous* Examples. — 1. Given -^B=195, -^C=291, and the angle ABC^i^OP (fig. 1.) ; to construct the triangle^ and deteymtne (instrumentally) the remaining side and angles. Ans. BCs^^iG, ang. ^=48^ C=499. 2. Given ^JB=136, ^C=53, and the angle Bss^^% (fig.^) to find the rest. Ans. BC^zUT. axig, BCA^s^^^, ang., BAC^ 58O4., or J9i>=1834^ ang. D=81o, ang. BAD^TG^^. 274. To descrU>e a triangle, having two angles, and the adjacent fide, given. Rule I. Draw a straight line AB, ^{ual \o the given side. H. At A and B respec- tively, make angles CAB, CBA eqticd to- the given angles (Art. ^61.); and pro- duce AC, BC, to meet in C; i/BC will be the triangle required. Examples.— 1. Given -4Bss:72, ang. B=322ot» a^- '^^^^0 to make the triangle, and find the rest An$, ^CsA9t/. CBsk Sef, ang. CaBlS7°^. z2 940 GEOftlETRV. PitRT VlII, 2. Given ^BsfclO, apg. -rf=s45^ aog* BssW, to canstroct the triangle, and fiad the rest. 275. To describe a triangle, having two tingles and a iide offo^ site one of them, given, KuLB I. Add the two given angles tc^ether, and subtraet their sum from 180° (see Art. 236.B). II. Draw AB equal to the given side^ and at the point A, make the angle BAC equal to the above remainder (Art. 261.). lii. At the point B, make the angle ABC equa], to one of the given angles ; then wiH ACB be the other, and the triangle will be described *. Note. If AB be opposite the less an^e, then ABC is the tn* apgle 3 but if AB be oppfosite the greater, then ABD will be the triangle required. Examples. — 1. Given -rfB=40, the angle -4BC= 80®^ and the angle -4CB=70^» to describe the triangle^ and find the rest. Ans. AC:=z86, jBC=45, ang. ^^r30». 2. Given ABss40, and two angles=100^, and 40°, to make the triangle^ and determine the rest. 876. To describe a rectangle, the sides of which are giten. RiTLs I. Let A be one side of the rectangle, and B the others draw CD equal to A, II. At the point C, draw CE perpendicular ta CD (Art. 259.) ; and make it equal to B. III. Through E draw EFpa- rallel to CD (Art. 260.), through D draw DF parallel to CE, and £CX>F will be the rectangle con- tained by A and By as^ wa^ required *• B -*•*- nr |.i )y ^ »i>»i ' >>' ■ 1 1'l , n The three angles of a triangle are together equal to two right angles (EiMlid dft. 1.) that M, to }80«>; wherefore if tiie sum of two angles of a tiMq(l9 1)0 ««fatfa#«ttd fiBont 1 80», the vcmahidsr ^Ml be tbe AM angle. • The proof wi this problem may be inferred from Ettdid 4«; t. Paut vnt PRACTICAL PROBLEMS. 341 In like manner a square may be described on a gifen line CD, by making CE equal to CD «*. ^7* To make a figure^ similar to a given rectilineal figure "^ having the sides of the former greater, or less, in any ratioy tJian those of the given figure. Rule. I. Let ABCDE be the given figure, draw the lines EB, EC, &c. from any one of the angles £, to the other angles B and C-, and first, let H: it be required to increase the figure, to another whose side is EF. II. Produce EJ, EB, ^ £C, and ED, to F, G, H, and K; and draw FG parallel to AB, GH tp BC, and HK to CD (Art. 260.); EFGHK will be similar to the given figure ABCDE, HI. In like manner, if it be required to lessen the figure, to another whose side is EL-, through L draw LM, MN, and W respectively parallel to AB, BC, and CD (Art. 260.) -, and LMNPE wiU be similar to ABCDE \ 27S. To make a regttlar polygon of any number of sides, on a given straight line AB, Rule I. Let n=the number of sides of the polygon to be ' SeeEadid 46. 1. « The trath of this oonstniction is evident, for the triangles ELM, EAB, BFG^ beiog efiaiangalar, EL : LM :iEAi AB : : £Ft FO (£noli<f 4. 6.) ia like maimer it may be shewn that the sides abdat the renuuaing equal angles of the figares are profoitionads, wherefore (Euclid def. 1 . 6.) the tbre^ figures are iimilar. Z^ Mt GEOMETRY. Part Vm i, then will the sum. of its interior angles be=:2n— 4 right an- 1 2n — 4 eles, and each of its angles =—- — ^^ right angles \ JI. At the points A and B make the angles BAC, ABC each equal to half the above angle> that is=— —^ (Art. 261 •.). III. From the point C where these lines intersect^ with the distance CA's^ CB, describe a circle. IV. Take the distance AB in the compasses^ and apply it to the circumference (as AF, PE, ED, &c.)> which will contain it as many times exactly, as the proposed polygon has sides ; draw the straight lines AF, FE, ED, &c. and the polygon will he described. Examples. — 1. To make a regular pentagon on AB. Here n=5, •/ ^^ =(-f. of a right angUz^^ of 90^=) 54^ n Make BAC, ABC each :^54^; from the centre C with the radvu CB or CA describe the circle ^AB, th^ AB taken in the com' passes, and applied to the circumference, will meet it in the points ABDEF and A ; which points J>eing joined, the pentagon will he described as proposed, % To make a hexagon, and a heptagon on AB. n — 2 For the hexagon, «ss6 j •.* =s(4 of a right angle =) W n z=:BAC. 71—2 For the heptagon, nss^ -, •.• — ^— = (j-ofa right angle =)64''y zszBAC; and proceed for both figures as before. ' This depends on cor. 1. 32. 1. of Eaclid. • That the lines Cj^, CB drawn from the centre t6 the angnlar points A and B bisect the angles FJB, AMD, appears from Eaclid book 4 ; \ix. in the equilateral triangle, prop, 6 ; in the square, j>rop. 6 j in the regular pentagon, prop. 14 ; ^"oA in the regular hexagon, prop. 15 ; and the same may be proved of any regular polygon whatever. Paht Vin. PRACTICAL PBOBLEMS, 343 n n (0 ™i^i / / 279. Tb construct a scale of eqluU parts. RfTLB I. Draw three lines A, B, and C, at convenient dis- tances, and parallel to one another (Art. 260.) -, and in C, take the pu-ts .Ca, ab, he, cd, &c. equal to one another. II. Through C, draw DCE perpendicular to Ca (Art. 259.) j and through a, c, d, &c. draw lines parallei to PCE, cutting the parallels J, B, and C; the distances w Ca, aby be, cd, &c. are called the / \^^, primary divisions of the scale. / /' III. Divide the left hand pri- / v^ ^/y^ mary divisions Ca, into 10 equal '^ parts (Art. 265.) 5 and draw lines through these points, parallel to DCE, across the parallels B and C; this primary division will he divided (• into 10 equal parts, called subdivi- sions of the scale. f \P IV. Number the primary divi- sions from left to right, viz. 1, 2, S> &c. and the scale will be com* plete. 280. To make a scale of which any number of its subdivisiofis will he equal to an inch. Rule I. Let one of the primary divisions Ca, of the scale C, be an inch ; and let it be divided into 10 equal parts, as above. IT. From any point D in AD, 4.<;p draw Da ; draw DS making any angle with DJ, and make DS= Ca. III. Take the number of sub- divisions (which are proposed tp mak^ an inch) in the compasses io» from the scale C, and ^pply this distance from D to E. IV. Draw ES, and through C draw CG parallel to El$, and make DH-sDO. z4 t<s - 1^ h SM tSBOUKTBY. Vau VHt v. Through J7, dnw AL iW9Ufll to C^ cittiiig I>^ will HK be one of the primary divisions, containing lO of the parts proposed ', VI. If lines be drawn thro^igh D to each of the subdivisioos in Ca, it will divide the line HK into 10 equal parts (Art. 3^1.)' which will be tlus subdivisions of the scale HL ; and if the suo cessive distances Kl, 12, 23, 34, &c. be taken in KL, each equal to HK, these will form the primary divisions^ and the scale HL will be constructed. £xAMPi.Es. — 1 . To construct a plane scale, having 20 of its subdivisions equal to an inch. Take the distance Cb (=2 inches =20 subjiivisions of Ca) in the compasses, make DE^Cb, DS==Ca, and proceed as before, 2. To construct a scale of which 35 subdivisions make an Inch. Extend the compasses from d backwards to the fith subdivision between C and a, this extent ( =35 subdicisions of the scale Cd) being applied from D in the straight line DE, proceed as before, 3. To make scales of which 15, 25 j 30f an4 40 resfective subdivisions will equal an inch* 2S1. To construct scales of chords, sines, tangents, secants, Sgc, Rule 1. With any convenient radius CA describe the circle ABDE, draw two diameters AD, BE, perpencticular to each other (Art. 259.), produce EB indetoitely towards F, draw DT parallel to EF (Art. 260.), and join AB, BD, DE, and EA. II. Divide the quadrant BD into 9 equal part6^ (Art. 263.), and from the centre C, through each of the dixisions^ ds%w, straight lines cutting DT in 10. 20i, 30« 40, &c. this will be tlw scale of tangents. III. From D as a centre, through each of the di^fisions of the quadrant, describe arcs cutting BD in }0^ 2(1, 30, 40> &c. thl^. will be the scale of chords. ■ ■ ■ >- " ■ ' ■ I . - I II « - ■ ■ . . 1 _ ■ I ■ . , , . 11.11 1. 1 1 I > ■« • » * To demonstrate the truth of this construction, let the number of subdivi- sions of HK contained in Ca=Ba be called n, also by construction Ca con- tains 10 subdivisions of itself; •.• I)£=n, T)S-10; bat DE : DS:i DC: {DG^) DH (4, 6.j and DC : DH : : Ca : HH; ••• DE> xDSxxCax HK, or lOCa C9 n : 10 : : Gi : HK, ',' HK^ ; let ««ao (as in Ex. 1.) than J^ilT^-tr ; 2Ca let «=35 (as in Ex. 2.) then HK-'-z-, &c. Q. E. D. Paet vni. PRACTICAL PB<»LEMS. 345 2V. Tbrough the dmuoMflof thequadna^ dnw Itaea parallel to BC, cutting CD m 80, 70, 60, 50, &e. khia wiJl be tke acale of SUMS and cosines. V. If. straight lines be dravm ftom A to the sewial divisiona (io, 20, 30, &c.) of DJ, cutting the radios in 10, 20k, 30, 40, &c CB will be a scale (^ semi-tangents. VI. If from the centre C, through the several divisi^M of I>r, arcs be described, cuttiog BF i» ]iO> Sp, 30, te« JHF wiU be a acale of secants. 346 GEOMETRY. Past VUI. Vn. Divide the radios AC into GO equal parts^ draw straight lines through each of these divisions parallel to CB» cutting the arc AB \ and from ^ as a centre, through the points where these parallels cut the quadrant AB^ describe arc* cutting ^0 in 10, 90, 30, 40, &c. AB will be a scale of longitudes. VIII. Divide the quadrant ^£ into 8 equal parts, and through these, from £ as a centre, describe arcs cutting AE in 1, 3, 3, 4, &c. A¥» will be a scale of rhumbs, IX. Draw straight lines from B, through the several divbions of the scale of sines (CU), these will cut the quadrant £D in as many points > from A as a centre, through each of these pointt, describe ara cutting £D in 10, SO, 30, &c. £D will be a scale of latitudes. X. If the above constructions be aocorately made, with a circle the radius of which is 3 inches, the several lines will exactly correspond with those on the common scales ^ wherefore to construct a scale, we have only to take the several lines re- spectively in the compasses, and apply them (with their respective divisions) to a flat ruler; and what was required will be done. 9m, To find the area of a parallelogram ACDE. Rule. Let a=the altitude AB, 6=the base CD: then will a6s=the area required '. Examples.— 1. To find the area of a square whose side is 12 inches. Here as=12, fe=12, and a&sl2x 12^144 square inckes=i ihe area required, 2. To find the area of a parallelogram, the base of which is 20 inches, and its altitude 25.109. Here a=25.109, &s:20, and a6s=25.109x 20=502.18 square inches = the area required, 3. To find the area of a rhombus, whose base is 42, and altitude 23. 4. To find the area of a rhomboid, whose base is 10, and altitude 7-^. " Every paraUdogram, is eqnal to the rectangle contained by its base and ^rpendicalar altitude (see Eaclid 85. 1 ; 1, 9,&c.) ; whence the rale is phuii. Part VIII. PRACTICAL PROBLEMS. 347 283. To find the area of a triangle ABC. Rule. Let &1I a perpendi- 3 cular BD from the vertical angle B to the base JC, and let a:=BD, b=AC, then will ab , ---=the ai-ea required ». Examples. — 1. The perpendicular height of a triangle is 2$ inches, and its base 16 inches ; what is the area ? fiere a^^, 6=16, and —=z—^ — =224«oMarcincftc*,tAtf 2 2 area required, j2. 1 ne base of a triangle is 1.03, and its perpendicular alti- tude ^,11, what is the area ? Ans. 1.08665. 3. The altitude 7A-> and the base 84. being given, to find the area of the triangle. 284. To find the area of a triangle, Itaving its three Hdes given, JluLE. Let a, h, and c, represent the three sides respectively, a4-6-4-c ■ " and let — ^: — =p> then will ^p.p— a.p— 6.p— c=the area of o the triangle/. ' This depends on Enclid 41. 1. y LetA Br ^ayA C^hy BC^^Cy AD^Xy then /)C=6-x, and (Euc 47. 1.) c» — 5— ;rl « ^BHi * =a.« — ar* , 01 c»— 6» + 26a:— ar»=a«—ar», whence a:= '""— . But BD)' -^aSI" -^ISi ' ^AB-^AD.AB-'AD:^ (a-\^ 26 rt3+6a— c* Sah+a''' + b' —c 2fl6— o^—fta + ca 2I ) >< (^^ '■ 26 ) *" 26 ^ 26 ^5^»— c» c»— a— 6|» 1 ■ ~ and thearea4.^CXfii>=iV("+^'— '^*)X(^''-o^O=-r _„____ ^ ■ a+b + ca+b — c c+O'—bC'^a+b V(«+ 6+ c,a-k- 6— <?.<?+ o — 6.C— «+ 6) = v ^ — • — 5 ^' 2 «+6+c this expression, by putting p^ ^ is the rule. Q. B. D. , becomes ^pp-^cp^b.p'-'ay which On-, If «»!!+ 6, and d^b c/> c, then will >/*«— fl« . a» — rf« be the rule. J^omtycastk^s MeMuratum,p, 47.^ 348 GEOMSniT. Faht VUL fixAMFLEs.— 1. To tmd die are 4, 5, and 6. gf a ixiaa^, wkoee aides Here a=4, fc=5. c=6, p=(— :: =—=) 7-5. <m(f 2 ^p.^Z^.^Il6.p^= V7.5 X 7^—4 X 7.5—5 x 75— C= • V7.5X 3.5x3.5x1.5= vd8.4375=9.S«15«l*e orw r«fwr«rf. 2. Required the area of a tiiangle^ of wluch the threeaides are 20, 30, and 40, respectively? Ans. 290.4737> &c. 3. The sides are 12, 20, and 25, required the area of the triai^le? 285. To find the area of a r^ular foUfgon, ha»mg and also the number of sides given. RujLB I. Let ABDEF be any regular polygon, bisect the angles FJB, JBD by the lines AC, EC, and from the point of intersection C let fall the perpendicular CH, II. Let n=:the number of eldea of the polygon, a^s-CH, and 6= nhn. AB, then will — -- =the area of the 2 polygon •. enenie, Tbis rule is g^ven, without a dem<m8tratioOy in the Geodrnt «f Hen> thfr yoQiii^er ; but the inventiiHi is snppoted to bekmg' to some piecedingy and non profound Geometer. Tartalea is the first among the modems who introdoocs the rule, viz. in his TraUaio di Numeri et Mkwe, foL Venice^ 1959. ha * Hiis rale is evident, for the area of each of the triangles wil( be ^e'*^ (Art. 283.] ; but there are n triangles, where£Mre the area of their sum, (m. ha nia of the giTea polygon,} will be « X -^^ "S"* if tiie side of each of the following figures be unity, then will the radias of the iQScribed and circamfcribed circles be as bdow : PlBT VJIf , PRACTICAL PROBLEMS. 349 £xAMn.E8.«-l. The aide of a pentagon is 4, and the perpen- dicular from the centre 2,61, required the area ? „ ^ , , ^ . nba 5x4x2.01 Here 11=6, 6=4, a=s2.0l, and — = =20.1, the area required, 2. The side of a hexagon is 7.3, and the perpendicular from the oentre 6^2 required the area ? Here «»6. 6=7.3, a=6.32, and !^^g X 7.3 X 6.39 ^ 2 2 138.408, lAe area required, 3. To find the area of an octagon, whose side is 9.941, and perpendicular 12. Ans, 477.168. 4. To find tlie area of a heptagon, whose side is 4.845, and perpendicular 5. Inscribed cirtfle, Ctremn. cir. Psrp. keighi. Equilateral triangle Square Pentagmi ••«.<«.. Hexagon Octag«n Decagon • • . . Dodec^gOQ 0.57735027 0.70710678 0. 8506508 1.00000000 1.30656296 1.61803398 1.98185165 0.86602540 1.53884176 0.28867513 0.50000000 0.68819096 0.86602540 1.80710678 1.53884176 1.8668201ff Hence the areas of thete figures may be readily found, and likewise those of siauUr figures, whateyer be the length o£ the given side ; since simi- lar polygons are to one another as the squares of their homologous sides, (£ttcfid 20.6.) or as tfa« squares of the diamet«r» of their eircumscribiog circles by 1. 12. If the square of the side of any regular polygon in the following table, be lAnHipUtd into the number ttandiflg agaiiitt its name, the produot will be the area. « Ao. qf sides. Names. Multipliers, 3 • . . • Trigon, or equilateral triangle 0.43301 3— 4 . . . . Tetragon, or square 1 .000000 5 . . . . Pentagon 1.720477 + 6 . . . . Hexagon 2.598076 + 7 ... . . Heptagon 3.633912 -f 8 . . . . Octagon 4.828427 + 9 ..^..Nonagon 6.181824 -i- 10 ....Decagon 7-694209— 11 .... Dodecagon 9.365640-- 12 .. ..Dodecagon 11.196152 + 550 GEOMETRY. Part VIII. S86. To find th0 area of any g'w^ rectilineal figure JBOUEE Rule I. Join the opposite angles of the figure, viz. AC, AD, FD, so that it may be divided into triangles ABC^ACD, ADF, FDE. II. Find the area of each of the tri- angles ABC, ACD, ^DF, ADE, (Art. 283.), and add these __ areas together, the sum will be the aoreaof the hgareABCDEF. ExAMPLBS.— 1. Let AC=zlO, BH^4, CL^S, AD^li, CL=z6, FD^S, EN^3, and FK=:S. ACBH 10x4 40 Then 2 2 AD.LC l^xe 2 ■ ~ ^D.FKVZx^ 2 *■ 2 = — =20=arca of ABC. 72 =^—=^S6:=area of ACD. eo =— =30=arca of AFD. FD.NE 8x3 24 .^ . ^^^ 2 2 2 *^ Their sum 98=:area of ABCDER 2. Let AC=z4t^, BH=^10, AD^bO, Ci-=20> fD=10a, £iNr=s2o, and FK^U, to find the aiea. Am. 2076. 287. The diameter of a circle being given, to find the drcutn- ference; or the circumference being given, to find the diameter. Rules I. As 7 : 22 -x ^ ^, or, as 113 : 355 \ ' '' '^^ ^^^^^^' '' ^^^ '"" or, as 1:3 1415927 /'"°'^""'^""\ • The first of these prgportions is that of Archi®edes, which is the easiest, although the least exact, of any of the ruUs Which have been proposed for this purpose ; the second proportion is that of Mctius ; the third is Van CeolenV * rule, and depends on Art. 252, where it is shewn, that if the diameter be «, the circumference will be 6.2831853, &c. wherefore, if the diameter be 1, the circumference will be 3.1415927 nearly, which is the same as therole. Pabt VIIL practical PROBLEMS. 351 ...i. ,,» 1 '. •• the circumference : the diame- or, as 355 : 113 1 :: the ci: or, as 3.1415937 Examples. — 1. The diameter of a circle is 12, required the circumference ? ^ 29 X 12 264 Tkiu, « 7 ; 22 : : 12 : — 5— =^:r=37.714285 th^ cir^ 7 7 (umference nearly. Or, as 113 : 355 ; : 12 : rri^==-_-=37.699ll5 the circumference mare nearly. Or, as I I 3.1416927 : : 12 : 31415927 X 12=37.6991124 the circumference very nearly. 2. The circumference is 30, required the diameter ? SO X 7 105 Thus, <w 22 : 7 : : 30 : -— =—=9.54545, &c. the dia^ iMier, 113x6 678* Or, 05 355 ; 113 :: 30 : —=-^=9.549295, &c. the 71 71 iiameter. 30 Or, as 3.1415927 : 1 : : 30 : ——_ =9.549296, &c. ^ diameter. 3. The diameter of a circle is 6, required the circumference ? Ans. 18.8495562, &c. 4. The circumference is 5, required the diameter? ^ns. 1.5915493, &€. 5. If the diameter be 100, what is the circumference ? And if the circumference be 100, what is the diameter ? 288. Tojind the area of a circle. Rule I. Let c=the circumference, d=the diameter, then Will -7-=the area of the circle. 4 Or, 2nd. .7854d«=the area. Or, 3rd. .07958 c»=the area. Examples.— 1. The diameter of a circle is 4, required the circumference and area ? These proportions are the conrerte of the fonaer. 3M GfiOMETRY. pAitr Vffl» Tfti» (JrL 25^.) 3.1415927 X4=]2.5d63706stibedrcian^ _,, cd 1^.5663706x4 , , Then — = = 12.5663708= <^ area, by rule 4 4 . 1. (Jrt, 253.) Or, .7854 <P=. 7854 X 16 =s 12.5664 = tft4? area, by rule 2. Or, .07958 c*= (.07958 x 12.566370b? '^= .07953 x 157.913675, &c.=) 12.566769= ^Ae area, by ruU^. 2. Required the area of a circle, whose diameter is 7, and its drcumference 22 ? Jns. 38^ 3. What is the area of a circle, whose diam^er k 1> and dr- camfeirence 3.1415927? 289. To find the area of any irregular mixed figure JBCDEF, Rule I. Inscribe the greatest possible rectilineal figure ACEF in the proposed figure, and let ASCy CDE be the remain- ing irregularly curved boundaries. II. From as many points JL^---^^^ S as possible in the curve ABC, let fall perpendiculars (Art. 259), to^C; and find their sum. III. Divide this sum by the number of perpendicu- lars taken, and multiply the quotient by the base AC, the product will be the area of the curved space ABC. IV. Proceed in like manner^ to find the area of tlie space CDE. V. Find the area of the rectilineal figure ACEF by Art. 286. then lastly, add the three aieas together, and the sum will be the area of the figure ABCDEF s « This method of approximatioa is used for measuriag fields and other endosates, which bsve very cfoolied and ifreg^^la^ bonndaries ; -the ^eatef the numbef of perpendiculars be, the nearer truth will th« approximation bc,.aa is evident. To find the area' of a regularly tapering board, measure across the two ends, add both measures together, and half tfie sum multiplied into the length of the board, will give the ar«a. iPikiVni. PRACTICAL t%6dL£MS« 363 Examples. — 1. Let AE^^O* the perpendicular FH=10^ the perpendicular CK=9, ACszl4, C£=L1, the sum of 9 perpen- diculars let fall on AC,^S7i ^^^ ^^^ ^^^ o^ 7 perpendiculars let fell on €E, = 25, to fi nd the area of the figure ABCDEF. ^ AExEH-^-KC ,20x10+9 20x19 380 • ,^ Brst, r—l =( p-J— = = =) 190 * » ^2 2 2 '^ sKthe area of the rectilineal space ACElFi 37 iSecowd/y, —=4.1111, &c. then -^Cx4.1111, &c.=(14x 4.1111^ &c. =) 57.5555, &c. =<Ac area of the curved space ABC. -*. 25 * J%irdiy, —=3.571428, &c. then CEx 3.571428, &c. = 7 (11 x 3.571428, &c.=) 39.285708, &c. =*/rc area of the cui^ed space CDE, Lastly^ these added together, viz, 190 =</ie area ACEP 67.555565= ABC 39 .28570^ = ...CDE The sum 286.841263= ABCDEF, as ivAs re* . quired. 3. het AE^lOl, fH=25, CJK:=21, -4C;i=87, CJE=79> the sum of 20 perpendiculars on ^C=103, and the sum of 17. on C£=72 5 to find the area of the figure ABCDEF . 290. To find the solid content of a prism. Rule. Find the area of its base by. some ot the preceding rules, and muUiply this area into the perpendicular height o( the prism, the product will be the solid content ^. Examples. — 1. The side of a cube is 13 inches, required its solidity ? Thus 13 X 13= 169=arca of the base {Art 282.) Then 169 X 13=2197 cw6ic inches z=xthe solidity of the cube. Or ^fciw, 13x13 X 13= (13) 3=) 2197 = */*« solidity, as before. If the board do not taper regularly, measure the breadth in several places^ «dd all the measures together, divide the sum by the number of breadths taken, and multiply the quotient by the length of the board, and it will give thtf area. ^ This rule depends on Euclid 2 cor. 7. IS. VOL, IJ. . A a 354 GEOMETRY. P^t VUI. 5^ The skies dbout one of the angles of the base of a rectan- gular prism are 7 and 5 respectively^ and the altitude of the prism 20; required the solidity ? Thus 7x5si35=area of the base; then 35x20ss700 ike solidity. 3. The sides of the base of a triangular prism are 2, S> and 4, respectively, and the perpeqdicular altitude 30; requited the soUdity? Q4.34.4 Thus {Art. 284.) p=s. ^ ^ =4.5, and ^415 X 4.5-2 X 4.5-3 X 4.5—4=3 v^.4S755s2.»47375=:anw of the base. Then 2.9047S75X 30^5 87. 1421250^ lAe solidUy. 4. The base of a prism is a regular hexagon, the side of which is 8 inches, and the altitude oi the prism is 4 feet ; re- quired the solidity ? Here {Art. 285.) 6=8, «= ^8«— 4«=( ^48=) 6.9282, - _ nba 6x8x6.9282 .^^^^^« • r .l ,n=s6, and -rr^ s =166.2768 square uicto=<fcc 2 2 ' area of the base: wherefore by the rule 166.2768x48 {inches) =7981.2864 cti6ic inches =4 cubic feet 1069.2864 cubic inches. 5. The length of a parallelopiped is 16 feet, its breadth 4^ feet, and thickness 6i feet ; required the solidity ? Ans. 486 cubic feet, 6. The length of a prism is 5 feet, and its base an equilatenl triangle, the side of which is 2^ feet; required the solidity? Aris. 13.5315 cubic feet. 7' The base is a tegular pentagon^ the side of which is 12 inches, and the length d feet 3 required the solidity of the prism ^ 291. To find the solid content of a pyramid. Rule. Find the solid content of a prism, having the same base and altitude as the pyramid, by the last rule ; one third part of this prism will be the solid content of the pyramid *. Examples. — 1. The altitude of a pyramid is 20 feet, and its base is a square, the side of which is 12 feet ; required the solidity ? * This depends on cor. U 7. 1?. Eiidid. Part Wir. PRACtflCAt PftOM^EMS. 3fci * I »28SO=:5o/tdify o/* the circuTUseribing prism, and — ^«d60 9 euhic feet :a: the solid content of the pyrcanid. 3. The altitude of a pyramid is 11 fytt, and iU bade a i«gu1ar hexagon, the side of which is 4 feet 5 what is the solidity ? Here (^rf. 285.) 5=4, a= v'4«-2«=: ^12=3.464101 6, «-/5 « ^'*^<» 6x4x3.4641016 «-6,fl«^.-5-«-7- *: — '-^ ^41.56^199»tfre<i of ike hase^ ako 41. 5692 Idftx lias 457.^6 141 12= wZidi^y 0/ the cir- M.r.o^'U' ,A^ /^rv^X 457.2614112 cumcnhmg prism {Art. 290.), •/ -3:162.4204704 cuhicfeet :sxthe solidity of the pyramid, 3. What is the solid content of a triangular pyramid, the height of which is 10, and each side of the base 3 ? Answer, 12.99039. 4. What is the solidity of a Square pyramid, each side of its base being IS, and the altitude 25 ? 292. njbtd the selvi i:(Ment of « cylinder. RuL£. Multiply the area of the base by the perpendicular altitude, and the product will be the solidity '. ■»<»— i»i*»*« I 'This ride depends on £ttctid 1 1 and 14 <tf book l^i The eoiivex »uper. ficies of a cylinder is found by mnltiplying the circumference of the base by the altitude ol the cylinder ; to which, if the areas of the two ends be added> the sum will be the whole external superficies. To find ths solidUy (f squared timber. 1. 1^ the stick be eiiualiy broad and thick throughout, find the area of a section any where taken, and multi- ply it into the length, the product will be the Solidity. S. If the stifck tapers regularly from one end to the other, find half the sum of the areas of the two cnds^ and mnltipT y it inter the lengtlr. 3. If the stick dorrnot taprr regutarly, fiad the areas of seveval different sections, add them together, and divide thi$ tain by the number of sections taken, this quotient multiplied into the leugUii will give the solidity. To find the solidity of rough or unsquared timber. Multiply the square of one fifth of the mean girt by twice the length, and the product will be the solidity. Or, multiply the square of the circumference by the length, take ^ of the product, and from this last number subtract ^ of itself, the remainder ^U be the solidity. See on this subject fftUton*s and BonnyeasHe'i Tr^tise$ on Mensuration, A a 2 VSt ©BOMETRt: pAETVni. £xAMPLB$*— 1. The altkude of a cylinder is 12 €eet> and the diameter of its base S feet ; required the solidity ? First, 3 X 3,lAlB997^^M4776l:=zciramferenceofthe hose. -aft. 387. Then, i2i£i?^ZSl=7.o685836=afetf of the hose. Art. 4 388. V 7«0685S36xi2s84.8230032 cubic feet ^the ioMUy fefUxrei. 9. The altitude is 90 feet^ and the drcumference of the base eo feet ; required the solid content of the cylinder ? Jns. 636.64 feet, 3. The diameter of the base is 4 feet, and the altitude 9 feet 5 required the solidity of the cylinder ? ^3. To find the solid content of a cone. AtfLE. Find the solidity of a cylinder of the same base and altitude with the ^ven cone^ by the last rule > one third of this will be the solid content of the cone K Examples 1. The circumference of the base of a cone is IS feet, and its altitude 10 feet ; requiml the solid content ? 12 ftf^^- ' > — 2=3.819718=: dtam. of the base. Art. 287. 3«14159«7 then^ — x-^-—^ — s6xl.909859s:11.459154s:area of 2 2 the base. Art 9SS. Whence 11.459154 x 10= I14^69l54=5o/idi^ of the civ- cumscribing cylinder. Art. 292. 114 59154 lastly, — 1-- — =2:33.19718 cubic feet =ithe solidity of the cone. ' I For the fouddatiou'of the rale ait EudM' 10. 13. Let fl«t fae axU of a rcone^ <f~the semidiameter of its base, then (Euclid 47. 1.) A/a^+^f^^rthe slant height of the cone ; and if the slant height be multiplied into the cir- cumference of the base, the ptoduct will be the oonyex superficies of the cone, to which^ adding the areaortbe base, the sum will be the whcfle extermd^ superficies. Rules for finding the superficies and solidities of the several sections of a prism, pyramid, cone, cylinder, sphere, &c. may befoudd in Mr. Bonnycastle^s excellent Introduction to Msnturatwn, a work which, cadadt b^ too highly commended. fWtrVBI. PRACTICAL PROBLBBCS. • d57 4 9. The altitude is 13> and the diameter of the base^i re- ,^ired , the solidity of the cone ? Am, 28.2743344. 3. The area pf the base is 30> and the altitude 14 } required the solid content (^ the cone ? S94. To find the solid content of a sphere. ■' RuLB. Find the solidity of a cylinder^ of which the altitude* and the diameter of its base^ are each equal to the diameter of the given sphere -, two thirds of this will be the solidity of the sphere K ^ EacUd has proved that " spheres are to each other in the ^plicate ratio of^ their diameters" (18. 18.) ; hat this Is the m^j property of the sphere to be found in the Elements. We are beholden to Archimedes for the most 9 part of onr orij^nal information on this salgeet ; the abenre rnle, which was ,taken from his treatise ** on the sphere and cylinjder/' may be easily denon- * ttrated by ** indivisibles,*' *f the metl^od ef mcremeots/' << SioioDi,'' aid wme other modem methods of computation j but I believe it cannot be effected by elementary Geometry. The superficies of a sphere is equal to the convex surface of its cirenmsoibin)^ •j^der s it is likewise equal to four times the area of a great circle of the sphere. If the diameter of a sphere be 2, then will tiia cifcumlereBce of a greal prcle be 6.S8318 llie superficies of a great ciyclis • • • . • 3.14159 The superficies of a sphere 1S.56637 The solidHy of the sp^re 4.18790 rite Mde ...• 1.62209 And of the inscribed tetraedroa i superficies • * . 4.6188 Vsolidity 0.15132 {its side .. •• 1.1547 superficies • . . 8.0000 solidity 1.5396 rits side •..• 1.41421 The inscribed octaSdroti J superficies ... 6.9382 Isolidity 1.33333 riU side ••». 0.71364 The iDMSribed dodecaSdroa < superficies . . . 10.51462 Isolidity 8.785l6r ^lU side •••• 1.05146 The inscribed icosa£droii < superficies... 9.57454 Isolidity 2.53615 Hence the superfiiiial and solid content of a soUd, similajr to any of the jfbofe, may be readily obtained, its side being given j the superficies being u^ the squares (Euclid 20. 6,), and the solidities as the cubes (cor. a» 12.) eC the homologous sides. 9%l . CanMHTRT. PuTVm. its soUdi^ ? ito^'i base. Art. 287. Sfcoiii% £2i£d^fZZ^87J9<»58S€84fce ^ylinderV tee. iln. eas. 7%tri^, 7.0685836x32=91.2057508=^ soU^ of the e^Under. Jrt 292. Lattly, * of 21.2057508=14.1371672 cu&ic/eee=<J^ sofi- dUy of the^here. 9' ThacUam^ter of a sphere u^ 17 incbes^s requiired its so|i» 4ity? jfM$.l.4»»e»qihicffieL 3. Jl ^aeartii be a pwfect aphfliv of 8000 mass diameter> Wr Mw my eabie nriloi of wattiBr^tocfr it contam? PART IX. TRIGONOMETRY. HISTORICAL INTRODUCTION. Trigonometry • is a sdejice which inches how tp determine the sides and angles of triangles, by means of the relations and .properties cff certain right lines drawn in and about the circle ; it Is divided into two kinds, plane and sphe^ rical, the former of which applies to the computation of plane rectilineal triangles, and the latter to triangles formed by the intersections of great circles, on the surface of a sphere. This science is justly considered as an important link con- necting theoretical Geometry with practical utility, and mak- ing the former conducive, and subservient to the latter. Geo- graphy, Astronomy, Dialling, Navigation, jSurveying, Men- suration, Fortification, &c. are indebted to It, if not for their existence, at least for their distinguishing perfections ; and there is scarcely any branch of Natural Philosophy, which can be successfully cultivated without, the assistance of Trigonometry. We are in possession of no documents that will warrant us even to guess at the period when Trigonometry took its rise ; but there can be do doubt that it must have been in- herited not very long after the flood. The earliest inhabitants of Chaldfea and I^pt were acquainted with Astronomy, which " The Dftme is derived from v(ut three, yn^s a comer, and fur^w to measure. The objects of Trigbnometry are the sides and angles only, whateyer respects the areas of triangles beloD|[8 to Geometry. A a4 S60 TRIGONOMETRY. Part IX. (admitting it to have been at that time merely an art, and iii its rudest state) would still require the aid of some method similar to Trigonometry to make it of any benefit to mankind* We may reasonably 8uppo9e that the anojent Greeks eultivated Trigonometry, in common with Geometry and Astronomy 5 but none of their writings on the subject have been preserved. Theon **, in his Commentary on Ptole- my's Almagest, mentions a work consisting of twelve books on the chords of circular arcs, written by Hipparchus, an Astronomer of Rhodes, A.C. ISO *. This work is believed by the learned to have* been a treatise on the ancient Trigo- . ^ TlieoD, a respectable mathematician and pbilosopliery and pr^ident of the Alexandrian school, ibnrished A. D. 370. He was not mbre famous for his acqnirements in science, -than for bis veneration of the DEriT, and his frm belief in' the constant ^aperintendence of divine providence; .he r^oom* inends meditation on the presenpe of .God^' af the most delightful and nseful 'employment, and proposed, tbaf in order to deter the profligate from committing crime*, therer should be written at the corner of every k'treet; Remember GoA 8E^s TBBE, O Sinner. Dr. Simson, in bis notes on the Elements of Euclid, has ascribed most of the faults in that book to Tbeoni without mentioning oi^ what authority he has done so. c HipP<trchns was bom at Nice, in Bithynia: here, and afterwards at Kbodes and Alexandria, bis astronomical observations were made. He dis- covered that the interval between the vernal and autumnal equinox is longer by 7 days than that between the autumnal and vernal ; he was the first who krranged the stars into -49 constellations, and determined their longitude* and apparent magnitudes ; and his labours in this respect were considered so valaabie, that Ptolemy has inserted his -catalogue of. the fixed stars in his Almagest, where it is still preserved! He also di^lcovered the precession of the equinoxes, and the parallax of the planets ; and, after the example of Thales, and Sulpicius Gallus, foretold the exact time of eplipses, of which be made a calculation for 600 years. He determined' the latitude and longitude, and fixed the first meridian at the 'F&rtuhatdf Ifuuke, or CetMfy Inlands; in which particular he has bee^ followed by most succeeding geographers. . Astronomy is particularly indebted to him , for collecting tbs[ detached and scattered principles and observations of his predecessors, arranging them in a system ; thereby laying that rational and solid foundation, upon which succeeding astronomers have built a most sublime and magaificeBl superstructure. Of the several works said to have been written by bim^ ^is Commentary on the Pbcsnomena of Aratus is the only qd^ tba| remslins. • . JP4»T IX, INTIIODUCTION. 3dl wvfietTy, and Is the most ancient on that subject of which. ^we baye any account. The Spherics of Tlieodosius * is the earliest work on Tri- gonometry at present known. It was written about 80 years before Christ, and consists of three books, " containing a variety of the most necessary and useful propositions relating to th§ sphere, arranged and demonstrated with great perspi- ^cuity and elegance, after the manner of Euclid's Elements/* We are in possession of three books on spherical triangles by Menelaus *. He is considered as the next Greek writer wjio tfeated expressly on the subject, and lived about a hun- dred years after Cb"st. This work of Menelaus was greatly ^ Theodosins was a native of Tripoli, in Bithynia ; and, according to Stral?e, excelled in mathematical knowledge. The work above-mentioned consists oC Ihree books ; the first oif whicli contains 23 propositions, the second 23, and the third 14. It was translated into Arabic, and afterwards from the Arabic into Latin, and pub)isftd at Venice; but the Arabic edition being very defective, a complete edition was obtained by Jean Pena, Regius Professor of Astronomy at Paris, and published there in Greek and Latin, A. D. 1658. Long bef«re this time, a good Latin translation of the work had been made by ViUltio, a respectable Polish mathematician of the 13th century, and the first of the moderns who wrote to good purpose on optics. The Spherics of Theodosius have been cnrichW with notes, commentaries, and illustrations, by Clavins, Hdegan^us, Gu^rinus, and De Chalcs ; but the best editions are those of Dr, Barrow, 8vo. London, 1675 ; and Hunt, 8vo. Oxon, 1707. There are still in existence in the National Library at Paris, two other pieces by Theodosius, one on The Ccel&tial Hwaes, and the other on Days and Nights: a Latin translation of which was published by Peter Dasypody, A.D. 1572. « Menelaus was a respectable mathematician and astronomer, probably of the Alexandrian school, but we have no particulars of his life or writings, except that he is said to have written six hooks on the chords of circular arcs, which is supposed to have been a treatise on the ancient method of construct- rag trigonometrical tables, but the work is lost. A Latin translation of the three books on spherical triangles was undertaken by RegiomonUnus, but wa» $rst published by Maurolycus, together with the Spherics <rf Theodosius, and his own, (Messanae, 1558, fol.) An edition of this work, corrected from a Hebrew manuscript, was prepared for the press by Dr. Halley, and published to Costard, the author of the History of Astronomy, in 8v6. 1768, iG2 TRIGONOMETRY. Pabt K. improved py Pcoletny, who, m the fint boc^ of his Almagest, has introduced a table of arcs and their chords, to every half degree of the semicircle ; he divides the radius, and also the ate equal to one sixth of the whole circuBiference (whose chord is the radius) each into 60 equal parts, and estimates all other ares by siirtieths of that arc, add their chords l^ siiltieths of that chord (or radius) ; which method he is sup posed to have derived from the writings of Hipparchus, and other authors of antiquity. No farther progress seems to have been made in the sci- ence, until some time after the revival of learning among the Arabians, namely, about the latter part of the eighth century; when the ancient method of computing by the chords of arcs was laid aside by that people, and the more convenient me- thod of coiAputing by the sines, substituted in its stead. This improvement has been ascribed by some to Mahomed Ebn Musa, and by others to Arzachel, a Moor, who had settled ifi Spain, about the year 1 100 : Arzachel is the nrst we read of who constructed a table of sines, which he employed in his numerous astronomical calculations instead of the chords, di- viding the diameter into 300 equal parts, and computing the magnitude of the sines in those parts. We are indebted to the Arabs for the introduction of those axioms and theorems into the science, which are considered as the foundation of modem Trigonometry, and likewise for other improve- ments. The sexagesimal division of the radius, according to the method of the Greeks, was still employed by the Arabian^ althoug'h they had long been in possession of the Indian, or decimal scale of notation. But shortly after the diffusion of science in the west, an alteration was made by George Purbach, Professor of Mathematics at Vienna, who wrote about the middle of the 15th century; he divided the radium into 600000 equal parts, and computed a table of sines io Paw IX. INTRODUCTION. d«9' these part% for emy ten imnutes of the quadmnt^ bjr 4it de-* cimal notation. This work was further prosecuted by Regio* montanus^ the disciple and friend of Purbaidi; but as the plan of his master was evidently defective, he afterwards changed it altogether, by computing anew the table of sines for every minute of the quadrant, to the radius 1000000. Regiomon- t&aus also introduced the use of tangents into Trigonometry, the table of which he named Canon Fecundusy on account of the numerous advantages arising from its use. He likewise enriched the science with many valuable theorems and pre- eejpts; so that, excepting the use of logarithms, the Trigo- nometry^^f Regiomontanus was little inferior to that of our own times. About this period the mathematical sciences^ began id be studied with ardour in several parts of Italy and Germany, and it can hardly be supposed that a science so obviousiy useful as Trigonometry, would be without its share oi admi-* rers and cultivators, although scarcely any of their writings on the subject . have been comniitt)ed to the press. John Werner of Nuremburg, (who was born in 146B, and died in 158B,) is said to have written five books on tiiangles; but whether the woric exists at present, or is last, we are not in- fbrmed. A brief treatise on plane and spherical Trigono- metry was written about the year 1500, by Nicholas Coper- nicus, the celebrated restorer of the true solar system. This tract contains the description and construction of the canon of chords, nearly in the manner of Ptolemy; ttf which is subjoined a table of sines to the radius lOOOOO with their differences,, for. every ten minutes of the qua- dranty the whole forming a part of the first book of his AMo&iunies Orbium CcBk^iumf first published at Nurem^- burg, fol. 1543. Ten years after, Erasmus Reinhold, Pror* fessor of Mathematics at Wirtemburg, published his Ca^ nan Facundus, ox table of tangents; and about the same 56^1 TRIGONOMETRY. Part IX. 4iiiie Fnmciscos Maandjco% Abbotof Bfanna^in ISdly^aiid one of the best Geometen of the age, published his Tabidm Benfficaj or canoo of secants. But a more complete work on the subject than any that had hitherto appeared^ was a treatise in two parts by Viet^ one qf the ablest mathematicians in Europe, published at Paris, in 157-^. The first part, entitled Canon Matkenmti^ cus seu ad triangula, cum appendicibus, contains a taUe of sines, tangents, and secants, with their difl&rences for every minute of the quadrant, to the radius 100000. The tangents and secants Ufw^fis the end of the quadrant are carried to 8 or g figures, sind tbe arrangement is simibir to that at present in use, each number and its compliment standing ip ^e same line, (^pposite one another. The second part of this volume, entitled Vniversalium Inspeethnum ad Cwonem Matkemati" cum liber singularisy contains the OHistnictioo of the fore^* going table, a complete treatise on plain and sphc^eal Tn-^ gonometry, with their application to various parts of A^ Mathematics; particulars relating to the quadrature of the circle, the duplication of the cube ; with a variety oi other curious and interesting problems and observations of a mis- cellaneous nature *• Besides the above masterly performance, Vieta w^ the author of several tracts on pli^ne an4 spherical Trigonometry, which may be fouiid in tbe cotlectioii of hisf works, published by Schooten, at Leydep, in 164& The triangular canon was next underta]cen by George Joachim Rheticus, a pupil of the great Cc^rnicus, and Pro^ fessor of Mathematics at Wirtemburg; ^' he computed the f For further particulars of this iDterestiog volume, see The History tf Trigonometrical Tablet, p. 4, 5, 6, 7, bj Dr. Hntton. It appears tkatt scafoely^ any copies of this ezcelleot work are now to be ioood ; for tbe Doctor utji, ia concluding his account of it, ** I never saw one (copy) besides that which is in my own possession, nor ever met with any other person at all aeqwuntecl with such a book," p, 7. t^iJiTlX. INTRODUCTION'. 36& i^on of sines and co-sines for every ten seconds of the ^quadrantj and for every single second of the first and last degree ;" he had proposed^ in obedience to the desire of his master, to complete the trigonometrical canon, and extend it ftirther than had hitherto been done; but, dying in iSjG, the completion of this vast design was at his re- quest consigned to his pupil and friend Valentine Otho, mathematician to the EHectoral Prince Palatine ; who, after several years of indefetigable labour and intense application, accomplished the wcnrk, and it wa& printed at Heidelberg, in 1596, under the title of 0pm Palatinum de Trianguiis,- We have here an entire table of sines, tangents, and secants, for every ten seconds of the quadrant to ten place? of figures^ with their differences, being the first complete eanon of these numbers that was ever published. But notwithstanding the pains th^ had been taken in the calculation, the tables in this valuable performance were afterwards, found to contain a considerable number of errors, particularly in the co-tangents and co-secants ; the correc- tion of these was undertaken by Bartholomew Pitiscus, a skilful mathematician of that time, who, having procured the original manuscript of Rheticus, added to it an au3d- liary table of sines to 21 places, for the purpose of supply- ing the defect of the former^ and published both in folio, at Frankfort, in 1613, under the title of Thesaurus Ma- them€Uicus^ &c. Pitiscus then re-calculated the co-tan- gents and co-secants to the end of the first six degrees in Otho^s worky which rendered it sufficiently exact for alstrono- mical purposes^ and published his corrections in separate sheets, making in the whole 86 pages in folio. The Geomeirica Triangulorum of Philip Lansbergius, in four books, was published in 1591 > a brief, but very elegant work, containing the canon of sines, tangents, and secants, with their construction and application in the solution pf SC6 TRIGONOMETRY. Paiit IX. plane and spherical triUDgles; the whofe betog fully aad dearly explained. This is the first work in which the tan- gents and secants are carried to 7 places of decimals to the last degree of the quadrant. * A comply and masterly work on Trigonometry by Pids- cus, was published at FVankfort, in 1500; the- triangalar canon is here given^ and its construction and use clearly described, together with the application of Trigonometry to problems of surveying, altimetry, architecture, geography, djalUng, and astronomy ; forming the most commodious and useful treatise on the sul]gect at that time extant. Several other writers on Trigonometry appeared towards the close of the 16tb, and at the beginning of the 17th century, of whom Christopher Clavius, a Jesuit of Bamberg, may be considered as one of the chief. In the first volume of his works, (which were printed at Mentz, in 5 volumes, folio, 16I2,) he has given an ample and circum- stantial treatise on Trigonometry. In this woric the caBon of sines, tangents, and i^ecants, is computed for every minute to 7 places of decimals, and carried forward to the end of the quadrant, the sines having their differences computed te every second, and construction of the tables being accom- panied with clear and satisfactory explanations, chiefly derived from the methods of Ptolemy, Purbach, and Regiomontanos. Van Ceulen, in his celebrated treatise De Circulo tt ad- scriptisy first published about the year 1600, treats of the chords, sines, and other lines connected with the circle; which work, with some other of Van Ceulen's pieces, wss afterwards translated into Latin, and published at Leydeu, in 16199 by Willebrord SneUhis, who has also himself given in li» Doctfinm Triangulorum CamniciBy the construction of sines^ tangents, and secants, together with a very usefel synopsis of the calculation of plane and spherical triangles. A eanoD of sines, taageats^ and secants^ to every mimte Paet IX. INTRODUCTIPN. 867 of the quadrant, was published in 1G27> at Aonsterdaiii, by Francis Van Schootea, the ingenioiis comm/eiilatar oh the Geometry of Des Cartes, His assariVn), that bis tiyi>le was without a single error, has been since found to h^ meonect ; some of his numbers have been discovered to err in the last %usej being hot always calculated to the nearest unit '. ' - ■ - — ^ — 9 In tho early ages of Geometry the circamfcfcftee of the circle igms divUled into 360 degrees^ each degree into 60 minutes, each minute into 60 flecoods^ Sec. ; this method was adopted by the moderat, and still prevails among the Bpglish, and most other nations in Bwofo } but the Frensfa aiathcmaticians have introduced an improvement, whkb, when it is generally Q|ider8tood and adopted, will be of the greatest advantage to Trigonometry. Towards the latter part of the eighteenth century, a new system of weights and measures was instituted iu France, in which they were decimfdly divided and saiMHvided; this was followed by another of eq^al importance, a new division of the qma- drant. By this new method, the whole circumference is divided into 400 equal parts called degvces, r^w^h degree into 100 minutes, each minute into 10<> seconds, &c. conseipiently the quadrant will contain 100 degrees. One aidvantage in tbhi method is its convenient identity with the common decimal scale of numbers, for !<>, 83', 45", in the new French scale will be expressed by the very same figures in common deciaials, viz. by 1.9345^ ; in like manner 91«, 3', 4% French, is expressed by S1.03O4» common decimals ; ITO**, 1', «", 84"' by 170.010234*; 5', O", 11'" by .05001 1«; 12', 18", 14"' by .121814% Sec. Among the works on this plan %t present in use, are I^es Tables Porta' iimt de Callet, 2 Edit. Paris, 1795 ; the Trigonometrical Tables of Borda, improved by Delambre j 4to. an IX. ; a^ thm taUes lately published by Hobert and Ideler, at Berlin. Likewise tables on the above plan, to an extent hitherto unknown, have b«an for ipaAy yesrs under the hands of M. Ptony, assisted by a Qimbef pf fibU mativKDUUticiaqt, a work which, ieaides its great usefulness, will be the most ample monument existing, of human industry, in the provini^ of calculation. To reduce degreeff mumUis, ^r. i^ I4« Ftvish 9wh> t«to degrees, minutes, Sfc. of the common scale, and vice versd» l^iacis Ui« qufkdfwt is 4ivi|l«4 by IhA FTtmh method into 1 00% and by the comvifin q^ei^ intq StQ% '.* \QQf> Frmpk ss»90^ csmman.} '.- To udme Freneh degrees, minutes, Sfc, into conunon. Rule. Express tjbe Fkua^ nne^piTi «(Mi|ii%, mhtract from this -rr of itself ; mark off the pr^p^ decim*^ iB the re^uii^dar, mtihipfy these by 60, xnark off the decimals ; multiply these agaia by 60, an^ mark off the decimals AS b^^ior^,- %^. ; the resulting ^ole i|umbers wiU \» the degrees, minutes, second^ &c. te«mired, a^oc^ing . to th^ ^t^gi^ $<^ak. £xAMPLE8.--ri. In %4% ^', SA" |t<«Mil^> Im»« mdBy 4^¥«ei9 niontas, sccondsi &c. common ? S6« TftlGONOHilETRt. Pa ir IX; The invention of logarithms by Lord Napier, in IGH^and their subsequent improvement by Mr. Henry Briggs, greatly facilitated the pmctical opei^tionk of Trigonometry. Besides the invention of logarithms, we are indebted to Napier for the method of computing spherical triangles by means of the five circular parts, and other valuable improvements in spherical Trigonometry. The docfrine of infinite series, introduced about tbe year 16^8, by Nicholas Mercator, and improved by Newton, Leibnitz, the Bemouflis, and others,' soos found its applica- tion to Trigonometry, by fun^ishinc; expressions for the sines, tangents, &c. for which purpose the exponential formute of Mr. Demoivre are extremely convenient.' But the gi-eatcst aiid most useful improvement of modem times In the analysis of sines, co-sines, tangents, &c. which Fint,/roffn d4S 56^, 32" »34.56dS» Subtract ^ of the same s 8.45639 The remaimder eai.lOSSS Multiply the decimals by fiO' 6.41280 Multiply the decimals by €0 24,76800' Multiply the decimals by 60 46.08000 Thereon 84S 56', 3S" French s^SlS 6', ^4"^ 46'"» 08 ctfMMum. S. In 8% 12', 8" French, how many degfcet^ miiiatM, &c. common ? i^> 7% 18', 88% 81'". 8. In 12*, I', 9!* French, how manyddgrees, &c. common? 4. In a*, 8', 7" F^'eoch, how many degrees, &c. eonmon I To reduce common degrees into French, RufcB. Turn the minntes, secomU, See. into decimals, to the whole add f of itself; then the integers of the sum will be degrees, the two left hand decim^ minutes, the two next d<^imal8 seconds, &c. ExAMnjBs.— 1. To redoce 34% 56^, St" commion, to French measme. First, to 34% 56', 32" e 34 .942222% 3ec. ^dd ^ of the same^ 8.882469 The sum is 38.82469 1» 38% 82', 46", Bl'^Freneb' 2. In 24% 44', 6" common, how many degrees French T Ans. 24% 15^.^ a. Turn 28% 27', 58" common into jFremrA. Am. 26% 17V35". 4. Turn 1% 2'^ 34" common into /^eiicA. Part IX. INTRODUCTION. sGd we owe to the penetrating, comprehensive, and indefatigable ttiind of the venerable Euler : by substituting the analytical mode of notation, in the room of the geometrical, which had hitherto been chiefly used, he simplified the methods of pre- ceding writers, investigated a great variety of formulae, ap- plicable to the most difficult cases, and made the trigonome- trical analysis assume the form of a new and interesting science* Admitting that the Continental mathematicians are out superiors in the theory of Trigonometry, as well as in their writings on the science *, still we have some very good and useful treatises on the subject; the chief of which arc those of Thomas Simpson, Emerson, Maseres, Horsley, Keith, Vince, and Woodhouse ; but Mr, Bonnycastle's Trea^ Use on Plane and Spherical Trigonometry^ is the most com- plete work on the subject of any that have hitherto appeared in this country* ■MM * See the Quarterly Review for Nov£mber^ 1810, page 40). VOL. II. B b / f J TikT iXi DEFINITIONS AND PRINCIPLES. 87l PLANE TRIGONOMETRY'. DEFINITIONS AND PRINCIPLES. i. JrLANE Trigonometry teaches how to determine^ ffooi proper data, the sided and angles of plane rectilineal triangles^ by means of the analogies of certain right Hnesj described ini and about a circle. 2. Every triangle contairm 6ix parts^ viz. three sides^ and three angles; any three of these^ whereof one (at least) is a side, being given> the remaining three may be fbtlnd. 3. The sides of place rectilineal triiSLngles are estimated in feet^ yards^ ^hon»9^ chains^ &c. or by abstract numbers : and each of the angles, by the arc of a circle, included between the two legs 3 the angular point being the centre. 4. It has already been observed (Art. 237. t>aft H.), that the whole circumference is supposed to be divided into 360 degrees, each degree into 60 minutes^ each minute into 60 seconds, &c. -, as many degrees^ minutes, and seconds therefore, as are con- tained ih the arc intercepted between the legs (^ an angle, so many degrees, minutes,* and seconds, that angle is said to mea- sure ', and, note, in the following definitions, whatever is affirm- ed of an arc, is likewise affirmed of the angle (at the centre,) which stands on that arc. 5. Draw any straight line JC^ from C as a centre With the distance CA, describe the circle JEN* produce AC to L, and through the centre Cdraw £CK perpendicular to AL; in the arc EA take any point By join BA, BE, and BCy and produce th6 latter to ^; through A and B draw AT^ BD each parallel to CEi, and produce them to S and G; join CG, and produce it to R and 5, produce CB to T, through E and B draw REM, MFB, each parallel to CA, and join J5L, MN; then since TA, J^D are both parallel to EC, they are parallel to one another (30. 1.), and both perpendicular to CA (39. 1.) } for a like reason EH and FjB * An easy tract on Plane Trigonometry maj^lie found in Lndtam's Rudt- nenl* of MathemtUks, Mr. Bridge's le<iHit«s on the same subject, publisbad^ ia 1810, is likewise a neat and useful work. B b 2 87d PLAKB TEaGONOHBTRT. PikT IX, are parallel, and both perpendicular to EC, and BD^FC, and FB ^CD (34. 1.) 6. Because the four right angles ACE, ECU LCK, KCA are sub- tended by the whole circumference, each of these angles will be sub* tended by one fourth part of the wIk^ cir- cumference, which is called a auADbAKT j the arc ABE is therefore a quadrant. 7. The difiference of any arc firom a quadrant, or 90^, or of any angle from a right angle, is called THE COMiaEMBNT of that arc or angle. Thusy the arc BE is the complement of the arc AB; and the angle BCE is the complement of the angle ACB K 8. The difference of any arc from a semicircle, or \S(P, or of any angle from two right angles, is called the supplement of that arc or angle. Thus, the arc BL is the supplement of the arc AB, and the angle BCL of the angle ACB ^ 0. The chord of an arc is a straight line drawn from one end of the arc to the other. y b Id li&e manner AB is the complement of BE^ and the angle ACB of the angle BCE, The name complemeni likewise applies to the excels of an dre Bboye a quAdrant, or of an angle aborc a rfght angle ; thus EB Is the cwkkp^ nent of the arc BML, and of the angle BCL ; but in most practical qveitiotis it is usoally restrained to what an arc or acute an|^]e wants of 90«. « The arc AB is likewise the supplement of the arc BML, and the angle ACB of the angle BCL, The term supplement means also the excess of air arc abote a semicircle, thus the arc AB is the supplement of the arc AMN., The difference of aa arc from the whole circumference i» tenned it» swfglc ment to a circle. f AKf CL DEFINITIONS AN© PBINCIPLES. 873 Thu^f % straight line JB U the ck9r4 of the wtc AB, or of thfi qngk ACSl, C^. The chord o( 9QP Is e^ual to the raitius (cor. 15. 4.) ; and the chord of 180^ is the diameter. 10. Ths co-chord of an arc, is the chord of the complement 4if that arc. Thus, the stra^ht line BE (or the chord of the arc BE) is iifl co-chord of the arc AB, or of the angle ACS. 11. Thb supplemental chord of an arc, is the chord of its supplement. Thus, BL {or the chord of the arc BML) is the supplemeri-' tal chord of the arc AB, or of the angle ACB: Cor. Hence it appears tluit the diord of any arc, is likewise the chord of its supplement to a whole circle i also that the chord can never exceed the diameter (15. 3.) Thus, BL is not onty the chord of the art BML, but also <\f the arc BKL. 12. The sine of an are^ is a straight line drawn from one end of ^he arc, perpendicular to the diameter which passes through the other end of the arc. Thus, BD is the sine of the arc AB, and of the angle ACB* Cor, Hence the sine of an arc, is the same as the sine of it^ silpplementj for BD is not only the sine of the arc AB, but also of the are BML ; for it is drawn from one extremity B, (of the arc BML\) perpendicular to the diameter AL, passing through the other extremity L, 13. The co-sine of an arc, la that part of the diam^te^ (passing through the beginning of the arc,) which is intercepted between the sine and the centre^ and is equ?d to the ji^e of th^ complement of that arc. Thus, CD is the co-sine of the arc AB, and of the anglf ACB ; and it is equal to BF (34. 1) the sine of BE, which is the jcmplevf^t of AB. Cor. Hence the sine of a quadrant^ or of a right angle Qa opt qxdj e^qual to, but) is the radius ^ and the co-sine of a quadr r^i^t or riglit angle is nothing. Thus, if the pqint B be supposed to move to E, the arc AB ^\ll beofJim^ 4Ej the, sine of which is EC; and thp point D coin^ dding with C$ the co^sine CD will vanish, BbS W4 PLANE TRIGONOHBTRr. pAmr IX. Hence also the sine or co-sine can never exceed the nuiias, 14. The vbrsbd siwb of an arc, is that part of the diameter which is intercepted between the beginning of the arc and its sine. Thus, DA if the vprsed sine of ifu arc AB, and of the angle ACB; and AP is the versed sine of the arc ABM, and of the pngle ACM. Cor, Hence the versed sine of an ore lets than a quadrant^ is equ^lto the difference; and of an arc ^eater than ^ qt^i^rant, to the sum of the co-sine and radius. Thw, 4D (the versed sine ofAB) ^CA—CD, and AP {the versed sine of ABJif) rpCA+ CP. Hence also the versed sine (being alwajrs within the qrcle,) can nerer e^^ceed the diameter, (15. S.) 15. The co-versed 91NB <vf an arc, is the ^ versed sine of its com- plement. . Thus, EP is the co- versed sine of the arc AB, and of the angle ACB. Cor, Hence the co- versed sine is equal to the excess of the Radius, above the sine. 16. The tangent of an arCf is a straight line at right angles to the dia- meter, passing through one end of the arc, and meeting a diameter pro. ^ duced through the other end of the arc. Thus, AT is the tangent of the arc AB, and of the angle ACB, Cor, Hence a tangent may be of any magnitude (according to the magnitude of its arc) from nothing to infinity. Hence also the tangent of 45^ is equal to the radius (6. 1.) 17. The co-tangent of an arc, is the^ tangent of the coow plement of that arc, ' I V. 4 (I PiBT IX. DEFINITIONS AND FRINCIFLE8. 378 Thus, EH (the tangent of EB) U the co-tangent of. the arc AB, and of the angle ACB. 18. Tub secant of an arc, is a straight liae diawn from the centre, through the end of the arc, and produced till it meet the tangeivt. Thus, €T %8 the secant of the arc AB, and of the angle ACB. Cor. Hence a secant can never be less than the radius> but it increases (as 4he are increases) from the ra^^us to infinity. 19. Thb co-^bbcant of an arc is the secant of its complex ment. Thus, CH {the secant of EB,) is the co-secani of the arc AB, and of the angle ACB ^ THE VARIATIONS, AND ALGEBRAIC SIGNS, OF THE TRIGONOMETRICAL LINES IN THE FOUR QUAD- RANTS. SO. If the sine, co^ine^ tangent, co-tangent, secant, co-secaiit, versed sine, and co- versed sine for every aix in the first quadrant AE be drawn, they will serve for the three remaining quadrants EL, LKy KAt that is, for the whole circle, as will be shewn forther on -, but previous to this, it will be necessary to suppose the point B to coincide with A, and to move ^om thence roun4 the whole circumference, and this will lead us to explain the manner of applying the algebraic signs tH smd — to the Unas peculiar to Trigonometry. 21. When the point B coincides with A, the arc AB wil) =a» and the points D and T wjU coincide with A-, wherefore AT=zo, BI>sso, DA=o, CB and CD each s radius ; that is, the tang^at, sine, and versed sine, (of o degfteea, .or) at the be- ginning of the quadrant will be nothing, and the secant and cck sine will be radius. * Some of the trigonometrical lines reoeived their nunct from^he parts of an archer's bowj to which they bear a similitade; thns, arc oomea froiti arcus, > bow} CHORD from chorda^ ihe string of a bow; saoitta (now generally called the versed sine) from sagitta, an arrow ; sine from sinus, the bosom, alliiding to that part of the chorda or string, which is held near the breast in the act of shooting, the sine being half the chord of double the are. The prefix CO is an abbreviation of the word complemeni; thns co-sine, eo^tamgent, ftc. imply con^lemeni sine, c^mplemeni tangent^ &c. or the sine, tangent, Sec. ef the coBplement of a given are, B b 4 5y« PLANE TBIC30N0METRY. Pabt is. 32. The sine BD increases (with the motioii of B) from o, during the first quadrant AE; when the point B coincides with E, the sine BD will evidenfly ciHndde with EC, and beeome radius f it then decreases during the second qoadcant, at tkeeod of which^ (when B is supposed to arrive at L,) it is iigain s9. Puring the progress of B^ through the third quadrant LK, the sine again increases from o, and on the arrival of B at the point K, it again becomes radius ; after which it graduafly decreases through the fourth quadrant KJ, at the end of whidi (where the arc is 360 di^gree^j) it is =;o, after which it again increases as before. 23. The sines are con- sidered 9s affirmative or oegative with respect to their direction from the diameter LA, to which they are referred 5 those on one side that diame- ter being eonsidered as affirmative/ those on the Other side, and in a con- ^ trary directionj will be negative 5 fbr instance, the sipes of the first and second quadrants which are on one side the dia« meter being reckoned -h, those of the third and fourth quadrants^ being on the other side will be — . ^4. The co-sine at the beginning of the first quadrant is radius, and decreases wi^h the motion of the point B through the arc AE to o ; when B arrives at E, D coincides with C; that is, the co-sine of a quadjaixt (or 90^) is =0. It afterwards increases from to. th^ ^nd Z- of the secon4 quadrant, where it ifi again radius j i^ the third* it co^i^tinually clecreases^ a( the ^ (K) of which it is again nothing ; (ifterwards, during the fourth quadrant KA, it again increases, at the end of which (viz. at the point A) it is again radius. ^5,. The co-sines originate at the centre C; consequently if » ^ •] F • • • • • • /c \^ •[* * • •• D N Pakt IX. ALGEBRAIC &lGm. 37T those in the direeticm CA be considered as affirmfttive, those in the opposite direction CL will be negative. The co-sines then of the first and fourth quadrants will be alike> viz. -f 3 those of the second and third will also be alike, but contrary to the former> viz.—. 26. At the beginning of the first quadrant (at A) the tangent is nothing; from o it increases continually^ until the point B coincides with E, when it becomes parallel to the secant^ (which will then coincide with CE) and is therefore infinite. When the point B has passed £. the tangent will change its direction^ and (with the motion of B} will continually decrease, until B arrives at L, or the end of the second quadrant, when the tangent will ag^n become nothing} from it changes its direction to AT, and increases until B arrives at K, the end of the third quadrant ^ when- it is again infinite, it decreases from infinite during the fourth quadrant, at the end of which it is again nothing, 9J. The tangent originates at the point A ; consequently, if tlie tangent in the direction of ^ The called affirmative, that in the direction of AS will be negative ) but we have shewn that the tangents of the first and third quadrants are in the direction of AT 9 wherefore they are both + ; whence the tangents of the second and fourth quadrants being in the direction of AS will, ibr the reason given above, be both — . 28. The secant at the point A is equal to radius, and it in- creases (by the motion of B) with the tangent^ and with it be- comes infinite at £, the end of the first quadrant. In the second quadrant £L, the secant changes its direction from CT to CS, and decreases from infinity to radius ; in the third qua- drant LKj it increases again in the direction CT, from radius to infinity : in the fourth quadrant KA, the secant once more change^ iU directioa to CS, a,pd decreases from infinity tp radius, 29. Theaeoaat has its origin at the centre C from whence its length is computed^ and it will change its aiga 09 often as the revolving radius CB passes the diameter ^K; having the same algebraic sign as the co-sine 5 whence it appears that the secants of the first and fourth quadrants will be +^ those of the second and third — . 30. The changes' which take place in the magnitudes and directions of the co-tangent EH, and the co-secant CH, may be S7S PLANE TRIGONOMETRY. Fakt IX. explained in the same manner; the co-tangent being computed from the point £, will change its direction^ and consequently its algebraic sign every quadrant^ the first and third being +>the second and fourth will be — . The co-secant at the point A is infinite^ at the point £ it is radius, at the point L T it is infinite, and at K it is again radius. In the first and second quad- rants its sign will be +> in the third and fourth — ^ being the same as the sine. 31. The versed sine at ^ is s= Of at £ it is radius ; at L it is the diameter; at K it has decreased to radiust and continues it;s decrease to A, where it is nothing. This line being computed from Ay will be always affirmative. 39. It may be remarked, in general, of the above lines» that as oft as they become ir^nite or nothings they change their direction, and consequently change their algebraic sign 3 these changesi may be exhibited in one point of view, as follows * : < It is Bometimes necessary in analytical oompatations to employ am l^reater than the whole circumference, which ara will faU in the 5th, eth, 7th, &c. quadrant (counting the quadrants again ronnd the circle) ; in these cases, the proper sign of the arc in question most be particilarly attended to; it may be readily found from the above table. Let a S3 any arc, its sine, tangent, &e. may be fonHd in tennt of the rat from the foregoing figure, by means of similar triangles : thus, r.cpsft co-sec a. tan a I. Sine of a s= /y/r'^—coa'ass r. tana r* co-tan a r r. tana cosa.seca V'r'+tan'a ^v^r* + co-tan » a tana, co-tan g r,y8ec»o—r> co-sec a sec a CO- sec a co-sec a sec a / Part iXp AL0JSBRA1C SIGNS. S7» 1st 2nd 3rd quad. quad. quad. Sine and co-secant + + — Cp-sine and secant -|- — — Tangent and co-tan. + — + Versed sine + + + 4th quad. + 9. Co- sine of a« V*"*— mo** ~ r. co-tana r' r. sin a sin a. co- tang ^ tan a r r' r. co-tana sin a. co-sec a ^r' + co-tan « a ^^a -f-tan'a tan a. co-tan o r^co-sec'a — r* sec a. 1 sec a co-sec a sec a co-sec a 3. Tangent of «=-;;j:j;;^- r>/r'»— cos^a j^nWa-^r» = r. sin a r^tona r. SID a cos a *sina.co-tan»a >/r»-8in«a r. sec a cos a, sec a cosa sia a. co-sec a CO- tan a co-sec a co-tan a ^co-sec^a— r« ra r. cos a 4. Co-tangent of a» rs.sin a r. cos a tan a ^r Vra—sin^ ^ ^co-sec^a — ra = sin a cos a. tan » a ^r* — cos«a r. co-sec a cos a, sec a sec a tana sin a sin a. co-sec a >v/sec»a— r' »',^r» 4- co-tan 'tf co-tana sin a. CO* sec a ' sin c. co-tan a r. CO- sec a r. tan a eo-tan a. tan a COS a sin a cos a r. co-sec a tana, caseca ^ co-tan a 3* cosa ^co-sec a a— r« 6. Cosecant of a= V**' + co-tan* « sin a tana. CO- tang __y \fr* -f tan«a ^'^ sin a tana cos a. tan a r. co-tan a __ COS a r. seca cos a. sec a tan a sin. a co>tan a. sec a r. sec a »• Vsecaa— r» And since the versed sine of a=r-cosa; the co-versed sin e ^r-sin fi the Mppleme ntal versed si ne ^r 4- cos a; thechord = V^ar.r-cosa^- thc co.chord = V2r.r-8ina; and the supplemental ehard « dao PLANS TBIOOMQHSTRY. FAmT]J^ At the hefftmifig aa4 end of each i|i|adrant^ the values of these lines will he as follow : (fi SCR 18a> 270P • Sine O + rad. • • . 7- rad. Co-sine + rod. O •5- rad. O rhrad. Tangent inf. inf. O Co-tangent inf. O inf. O wt/. Secant •+• rad. inf. -^rad. inf. -hrod. Co-secant inf -f rod. inf. -^rod, w/ Versed sine Q + rad. • 4- diam. 4- rad. O INTRODUCTORY PROPOSITIONS. 33. The sine^ co-sine, tangent, and secant of any arc^ are re9pecti?ely equal to the sine, co-sine, tangent, and secant of the supplement of that arc. Let the arcs AB and AM be supplements of each other, viz. AB less than a quadrant, and AM greater^ then will the sine BD of the arc AB, be equal to the sine MP of the are AM, and also the co-sine CD to the co-sine CP. For since AM-^- AB ^2r,r+coBa; ^|tb^r of these latter may be fowad in terms of any of the above by proper sabstitution, regard being bad in every case to the c^iapgc of signs, when the arc a is greater than a quadrant. From these expressions iiar the trigoDometrical lines belonging to a single wecy others may be derired which are applicable to a great variety of cases, viz. far the sums, diffcMDces, multiples, sub-multiples, &c. of given arcM ; but the pro^cution of this bssIJb) part of Trigonometry further than is necessary for constructing the sin^, tan- gepts, &c. would require piore room tha|i c^ conveniently be spared ; w^ must therefore refer the inquisitive student for the gratification of his wishes, to the writings of £u1er, Cagnoli, Vince, Woodhouse, BooDycgstle, and tome othei^ who have treated expressly on tte tubject. PaitIX. INTRODUCrrORY PROPOSITIONS. Ml ^ISOPssAM+ML', taking AM from both, the arc ABzsML, *.♦ the angle BCA^MCL (27.3.) j also tlie angles niPC, BDC are right angles, and the side MCzszBC, / (26. 1.) MP=tiD' and CPzsCDi that is, the sine and co-sine of any arc or angle, are Respectively equal to the sine and co-sine of the supplement of that arc or angle, observing that the sines MP and BD will be both -t-, but the co-sines will have different signs, viz. CD will be +, and CP — Likewise AS the tangent, and CS the secant of the arc AM are respectively equal to AT the tangent, and CT the secant of the arc AB. For the angle TCA^MCL (as shewn above)r=^CS (15. 1.), the angles at A right angles, and the side CA common, *.* (26. 1.) AS=zAT, and CS^CT; that is, the tangent and secant of any arc or angle, are respectively equal to the tangent and secant of the supplement of that arc or angle. In like manner the sine* co-sine, tangent, and secant of an arc terminating in the third qnadrant LK, will be thode of an arc which is the excess of the proposed arc above a semicircle. Thus the sine of the arc AMN is PN=:PM (3.3.) = BD, the sine of the arc AB) and the co-sine PC^CD, the co-sin6 of AB; only this ^e and co-sine (PN and PC) will be nega- tive. AT will likewise be the tangent, and CT the secant of the arc AMN» (as appears from Art. 16 and 18) j the former of whkh will be -f, and the latter — . The sine, co-sine, tangent, and secant of an arc terminating in the fourth quadrant KA wiM be respectively the same with those of an arc which is the supplement of the proposed arc to the whole chrcle. Thus the sine of the arc AMNG is GD, which is=:BX> (3.3.) the sine of the arc AB, only GD is negative -, the co-sine CD is the very same as the co-sine of the arc AB, AS is the tangent of AMNG, which is ^AT; and CS the secant, which is =Cr; AS will be — , CS+ -, see Art. 32. 'I^h6 Versed sine AP of any arc AM, terminating in the second quadrant, is fequal to the difference of the versed sine of Its supplement and the diameter, or to the sum of the co-sine and radius. Thus, (6ihce Ato=:LP) AP:=z{AL-^LP=z) AL-AD^ PC-f- CA. ^The versed sine of any arc, terminating in the third ^8^ PLANB TRIGOKOMETRY. pA%i nf/ and fourth quadrants, is the same with the versed sine of its supplement to the whole circle : thus AP is the versed sine oP the arc AMN, and also of the arc NGA ; also AD is the versed iine of the arc AMNG, and likewise of AG its supplement to the whole drde. It has been already observed that all the ver^ sines are affirmative or -f- . Thus we have shewn tliat the sine, co^sine, tangent, and secant of any arc AB^ will be respectively equal in magpaitude to the sine, co-sine, tangent, and secant of its supplement to either a semicircle, or to a whole circle, diffinring only in the algebrai<! signs; and therefore if the sine, co-sine, tangent, and secant for every degree and minute of the first quadrant be computed, and the whole arraoged in a table, this table wiH serve for tte whole circle. 34. The sine of any nrc is equal to half the chord of double that arc: and conversely, the chord is double the sine of half the arc. Because CA cuts B G 2k right angles BD=DG (3. 3.) V BDzzzxBG; also the arc B^=the arc AG (30. 3.) •.• the arc BA:=i^ the arc BG; that i$, BD the sine of the arc BA is half the chord BG of (the arc BAG, which is) double the arc BA. Q E. D. The con- verse is sufficiently evi- dent from the preceding demonstration. Car. Hence, because the chord of 60o=:the radius (Art. 9. cor.) •/ the sine of 30°= (4- the chord of 60°=:) ^ radius. Hence also the co-sine of 60o= (sine of 30°=) ^ radius; and the versed sine of 60°= (radius — co-sine =) 4- radius. 35. The sine or co-sine of any arc, together with the radius being given, we may thence determine the rest of the trigono* metrical lines belonging to that arc, as follows : l*ABt JX INTRODUCTORY PROPOSITIONS. 3S3 Rwt, Let CB the radius, and BD the sine of the arc BA, he given, to find the cosine CD; then ( 47. 1.) CB)« =gDl'-h eS:«, and C5)V^*=CSl«, •/ CD= ^CB|«-B5I« j that is, the ohsine of an arc is equal to the square root of the difference of the squares of the radius and sine. Secondly. Let CB the radius, and CD the co»s ine be given ,, to find BD the sinej thus, (as ^bove) -BDrav'CB)'— Cd)'; that is, the sine of an arc is equal to the square root pf the differ- ence of the squares of the radius and co-sitie. Thirdly. Since -rfD=C^— CD, and ^1*=^C4-CP; there- fore the tersed sine of any arc less than a quadrant, is equal to the difference of the radius and co^sine; but of any arc greater than a quadrant, it is equal to the sum of the radius and co-sine. Fourthly. Because -S?t«=55)»+"S5]« (47.1.) '.• B^= V5B)*+D3)*5 that is, the ehordof any arc is equal to the tquare root of the sum of the squares of the sine and versed tme,^. Fifthly. Because £5]«= (gf)*-!-'^^ (47. 1.) ;= DC^+ CErc3«=:)5C|«+C£^irftD]* ••• EB= ^Dtl^+ CE^BDY ; that is, the co^chord of an arc is equal to the square root of the sum of the squares of the co^sine and the excess of the radius ffbotfe the sine. Sixthly. Because the right angled triangles BCD, TCA^ fiCP, and HCE have the acute angle TCA which is common to the two former, equal to each of the acute angles CPF, CHE in the two latter (by '129. 1.) 5 these four triangles are equiangular (3S. 1.)^ and have the sides about their equal angles proportionals (4. 6.) 5 whence we have the following analogies. if CD iDB'.iCA : AT \ DB.CA *■ co'sine : sine : : radius : tangent ^ ' ~* CD ' sine X radius ^.- ,. ,. sine W TANGENT = : = (if radlUS = 1) ; — g. co-sme ^ co-sme 'InHkemanner it is shewn that XJ»= (>v/5S]« +^SZl • =) ^BD) a +Z^^ <v> The tupplemental chord is equal to the square root of the sum of the squares ff the sine and suppiemental versed sine. ( Hence it appears, that when the sine and co-sine have like algebraic signs, ^« tangenl will be + , tmt when they have unlike »igns, the tangent wiU 384 PLANE TRIGOMCMBrRY. PAftt iS« o f^^ '^^ ''<^ '^'^ 1 -Clf^^^l^s: i co-sine : radius : : radius : secant ^ ' CD CB]^ radiuslft 1 ■— , or SECANT as ■ ■ ' . =g(if r?id.=sl) : — ^ CD co-sine ^ ^cosine cDBiCB II EC :CH 1 . . r£f==:£M?= ^sine t radius : : radiua : co-secant-' DB jl -, , or co-sECAMTza: — , " S=(if rad.ssl) ' DB sme sine ^ cDB:DC ::EC i EB \ . . £^-.^^£1^ ^ sine t co-sine : : radius : co-tangent ^ ' DB * co-sine X radius ,.^ , ^.co-sine. or CO-TAHGEWT a- — ■ : — ■ it=(lf rad.ssl)-^^- *. sine sine r r^ I AC ::CE : EH \ ^_ '- tangent : radius : : radius : co-tangent ^ ' "" ACCE ^^ radius]* ,,^ . ,^ -;;l v >- =-=-;-, Or CO-TAHTBtNT = •.={if rai.sal) TJ TA ' tangent ^ ^ 1 . tangent* \. cTA :TC t:CE : CH \ ^,, TC.CE *- tangent : secant : : radius : to-sfecant ^ TA secant x radius ,,_ _ , . secant , or CO-SEC ANT=z ' — ' s(if rad.si.) — — ^ '. tangent tangent Cor. Hence the radius is a mean proportional between the co-sine and secant 5 between the sine a&d co-secant, and between the tangent and co-tatigent. 36. The secant of 60° is equal to the diameter. For since the co -sine of 60°=-i^ radius (cor. Art. 34*) =t CB, if this value be substituted for CD in the secoxid analogy (given above), we shall have Cr=:(— -L—) _-=-_ =-3 CB; that is^ the secant of 60^ is equal to the diameter. Q. E. D. *» Hence the secant will always have £he same algebraic sign with the co-sine. * Hence the co-secant will bare the same algebraic sign with the sine. k Hence the co- tangent will be + when the sine and co.sine have ]ike«^« and -*• when they have nnlike, viz. it will always have the same sign as tke tangent (see the 1st analogy.) 1 Hence, when the tangent and secant have like sign«, tfcie 00 secant will be -f > bnt when they have nolike, -«-. PaatIJL INTKMDUCTOaY PAOFOSITIONS. 985 Cor, Hence the tangent of 60^=s'twice the sine; for since. CBiCTiiBD: TA (4. 6. and 16. 5.) and Cr?=2 CB •/ TA:st 2 BD (cor. 4. 5.) Z7' From what has been ddiivered> we can readily determine the arithmetical values of the chords co-ehord^ supplemental chords sine, co-sine, tangent, co-tangent, .secant, -co-secant, versed sine;, co-versed sine, and supplemental versed sine of the arcs of 30°, 45®, 60°, and 90° to any given raAus ^ thus, let the radius =1, then Art 36. secant of 60° Art 19. co-secant It of 30° I jineof 180° j ] ^ . „, J, . ^,«^ ^-^thediameter=5:2.00Q000a Art, 31. versed sine Aft. 9. cor. chord of 180° Art 9. cor. chord of 60^ Art 10. co-chord of 30° Art\6, cor, tangent of 45° Art 17. co-tang, of 45° >'=the i-adius = 1.0000000. Art 13. cor, sine of 90** Art 31. versed sine of 90° Art 24. co-tine of 180° /-sine of 30° ^ -rfrf.34. cor. J co-sine of 60° i , « I J • e j^f'=^ i ^^ radrus=:0.500000p. ^versed sine of 60^ I Art 15. cor. co-versedsineof 30°^ Art 34. cor. Art 13. Art. 35. tangent of 30° 1 . sine 30°. -4re. 17. co-tang, of 60° -> co-s;ine 30° 5 ;^^ - , =g O.6773503. .8660254 ^r^ 35. versed sine of 30° 1 =rad.— co-sinp 30°= Art 15. co-versed sine of 60° J 1 - .8660254==: 0.1339746. Art 35. chord of 30P 1 ^ ^sili^^of30°+;^^«of30° At, 10. co-chord of 60P ^=:= ^.25 + . 0179492= 0.5176380. ArtZ^. secant of 30° x_ rad?]^ _ 1 _ Art 19. co-secatit of 60° / co-sine 30° .8660264. 31.154700$, VOL. 11. P C ■''• r^^'°'}=^'-f=^-=''»- 0.8660254. SM riuANS TRfGOSOUKTRY. pAiT 1X» Att 84. »i«» of 450 Art, 13. cO^^hM df 4&. Jrt. 35. versed sine of 45* 1 =srad.—co*8in^3s 1-^.7071066 ^ri. 25. convened sine of 45^ Jkt. 35. Meant of 45^ Art 19. co-secant of 45^ Art, 85. cfiord of 45* Art. 10. co-chord of 48*' Art. 35. tangent of (5d^ ^r^ 17. co-tangent of 30^ }==:t*^<»a erf SK>«=iv^.fai:i« -^ =s:*t V^«= O.7d710^a } . .a^928982« eo-tiwt .7©71068 1.4'14S18tf \ = y/sinel^ 4- V- sine^^s=^ 0.7653668' }8inc =rad. X— -r oo-si sine of §0^ ooHMneof 60^ .8660954 .5 1.7320508. In like manner (Art. 35.) tlie chor4 of the suippleoieni of . ©O^.. 90^ ^1.414^135^ ^^1 ,. A .ri20ol .■-^,. =^;=^ rW66366S 450 h^h-^^1 «f^ 1350 H ^slS;i^+«"P^^^^;^^=i 1.8477591 300 1500- 1.93l851j^ 38. The sine, co-sine, tangent, seoaiH^ &c* of any ore AB of a circle, vrha^ radius is Crf, is to the sine, co-sine, tangent, secant, &c. of a similar arc DE, whose radius is CD, as €A t^ CD. From the point B let fall JBF perpendicular to CD (12. l.)| and through A, £, and D, draw AK, EG, and DT, paral- lel to BF (31. 1.), then will BFhe the sine of the arc BA, CF its co-sine, AK its tangent ; EQ the sine of ED, CG its co-sine, and DT its tangent (Art. 12. 16'.) ; and since AB and DE each subtend the com- mon angle at the centre C, they are similar, that is, they contain each the same number of degrees (part 8. Art. 239.) 3 now siocc the angles at F, A, G, and D, are right angles, and the angle at CcommoD, the triangles BCF, KCA, ECG, and TCD, are similar fa^tix iNrR«Knn«Y pi»posith)ns. am <32. l.)> and liave the sides about tbeir eqiiftl aQglel proportioaab (4. 6.) J that is. First, FB : BC:: Gf : £C, and ^teroatdy (16.6.) FB : €E :iBC: ECi that is, «ne of arc BA i sine of arc ED : ; rod. (BC) of the former arc : rad. (EC) of the latter. Secondly, FC: CB:: GC: CE, and alternately FC i GC:: CB : C£; that is, cosine of arc BA : o^sine of arc ED : : rad. (CB) of the former : rad. (CE) of the latter. Thirdly, KA : AC : : TD : DC, and alternately KA:TD:: AC : DC; that is, tang, arc BA : tamg^. arc ED : : rod. of BA : rad. of ED. Fourthly, KC zCAz: TC : CD •/ altenirt^ ^C.TCix CA : QDi that is, aeoant of arc BA : secant arc fiD .: ; rod, o^ fiA : rad. of JSJJ. Fifthly, Because BC : CF:: EC-, CG / fcy conversion (prop. B.5.) BC'.FAiiECt GD, / inversely (prop. B.5.), fA I (BC^) AC :: GD: (ECz?:) DC.' alternately FA : GD : : AC : DC; that is, verud sine of arc BA : versed sine of are ED : : rad. of BA ; rod. of ED. Wherefore the sines, co-sine^ taogents, socants, and versed sines of ^ given angle in different circles, are respectively as the radii of those drdes. Q. E. D. Hence, if sines, co^siaes, tangents, &c. be computed tq a given radius, thej may be Ibiind to any other radius, by th^ above proportions. S9. The co-sine of any arc, is equal to half the chord of the Supplement of double that are. Let AE be an arc, C the leentre, join CE, and from 4 <^^ 4L perpendicular to CE (19. 1.), and produce it to S, join BD, ^nd froai the centre Cdraw CJf peipendicular to BD, '.* DF=9 CO? 88S HiANE TRIGONOMSTRY. Pa&t IX. FB (S,S.)', afsoCX is the co-sine of AE (Art. 13.) » BD the suppleaiental chord of (AEB^) double of AE (Art 11.), and FB=balf the said supplemental chord. Because DBA is a right angle (13. 3;), and VLB, CFB right angles (by construction), •.* FB is pji- rallel to CL, and BL to PC (58. 1.), •/ FBLC is a paral- lelogram, and CL:=xFB (34. l.)5 that i8» the cosine of them AE is equal to half the supplemental choid of (^H) double of AE. Q. E. D. 40. The chord of an arc Is a mean prdporlional between its Tersed sine and the diameter. Draw BK at right angles to DA (12 1 ), then because DBA is a right angle (31.3.), DA: AB:: AB: AK (cor. 8.6.); that is, the diameter is to the chord of the arc AEB, as the same chord is to the versed sine of AEB. Q. E. D. 41. The sum of the tangent and secant of any arc, is equal to the co-tangent of half the complement of that arc. Draw CH at right angles to DA (12. 1.), and let AEhe any arc, AS its tangent, CS its secant, and the arc EH its com- plement. Bisect EH in B (30. 3.), and di-aw CBT meeting AS produced in T. Then AT is the tangent of the arc AEB (Art. 16.) that is, the CO' tangent of HB (Art. 17.) which is half the conipleuient of AE. Because AT and CH are parallel, the angle HCB=CT4 (^9.1); but HCB=zBCE \' BCE^CTA \- 5C=6T (6. 1) AS+SC=AT ; that is, the sum of the tangent and secapt of the arc AE\s equal to (AT) the co-tangent of (HB) lialf the complement of AE. Q. E. D. 42. The radius is to the co-sine of an arc, as twice the sine to the sine of double that arc. Because the right angled triangles ALC, AKB have ihe apgle at A common, they are equiangular (32. l.)> '•* ^P- Partjx, investigation of formula. S89 CL : : AB : BK, that is radius : co-sine of JE : : twice the sine f^AE : sine of double of JE. Q. E. D. THE INVESTIGATION OF FORMULA, NECESSARY FOR THE CONSTRUCTION OF THE TRIGONOME- TRICAL CANON. 4S. The sines and co-sines of two unequal arcs being given to determine the sine and co-sine of their sum and difference. Let KFy FE be two unequal arcs of which the sines and co-sines are given^ and let KF be the greater^ from which cut M FD=:FE the less (34.«.)* Jo»n ED, and from the centre C draw CF perpendicular to ED (12. 1.) '.• EL^ID (3. 3.) ; draw DHt FG, LO, EM, each perpendicular to the diameter €K, and DS, LN each parallel to it (31. 1.) meeting LO, EM in the points S and N. Because EL^zLD EF=iFD, •/ (30.3)5 and because LN is parallel to DS, the angle ELN=^LDS (29. 1.), ••• the right angled triangles ELN, LDS having all their angles equal, and the homologous sides EL, LD equal, are equal and similar (26. 1, and def. 1.6.), •.* EN=lS and NL=:SD; also in the parallelograms NMOL, SOHD, we have NM=zLO, NL=zMO, DH=SO, and SD=^OH (34. 1.). / NL=zMOz=SD=iOH. Let the arc KF=A, the arc FEz=B, and the radius CF=:R', then will the arc /CJE=(^F+FE=) A-j-B, and the arc KDs^{KF^ K M. OG M O G H C FD^KF^FE:s:)A^B; sdso FG is the sine 1 ^^ ^ and EM is the sine *> « ^ - p CG , . . co-sine / CM . . . co-sine J EL .« . sine -1 « « DH,^, , sine Cli c c 3 CH . . . co-sme } SM PLANE tBiGOKOSfBTRT. pAAt^IX. BecaiM NL is pandM ta CO, and JPQ to LO md the angles at 6> O, and N rigbt angles, tke triangles CFG, CLO, and ELN are equiangular (29 and 32. 1.), consequently (4. 6.) ^.^ w,^ ^» »^ »^ PO.CL . anil, cos ^ CF: FGi.CLi LO, •.• 10= ( =r) CFi CG : : EL : E-N, •.* £2V=( — — ;— =) CF ' R CF ' R CFi FG::EL: LN, •/ X^=-(— >-— - =±) CK R But Eitf (=itfi?+ EN^LO-^EN), or sin ^+ J5=: sin utf. cos JJ-fcos ^. sin if CM {=zCO'-MO=:CO'-LN), or eos ^-ir£: toA J. COS jB— sin ^. sin £ B Dff (:=:SO=:L0^L8^L0^E^), or sia ^--5: sin u^. cos B— cos A. sin^ £ • CH (z=CO+OH^CO+LN)y or cos ^-.JRar eos A. cos 5+sin -rf. sin S ^ ■ ^ 44. These formula for the sines aod co>sines of the arcs A-JtB which are, it is plain, adapted to any radius B> may be simplified and rendered more convenient ^putting B=l > they will then become Formula 1. Sin ^+if=8in A, cos j8+cob A. sin B. %, Cos ^+ J3=cos A. cos J?-^6in A. sin B. 3. Sin ^— ^ sssiin A, cos J3«-cos A, sin J?. 4. Cos A'^B^CQ» A. eos ^-f-sin A. sin 5. 45. To find the sine and cocaine of multiple arcs, that is^ if i^ foe any arc^ to find the sine and co-sine of nA. Add the^r^^ and third of the aboye formul^B togetJ^er« and in the sum let ^ be substituted for B, and B for A, and w^ shal l have sin £4--^+sin B'-Asz^ cos A . sin B, that is^ • sin £ -I- -4=2 cos A. aia J3— sin B— ^. {Y), FiHT IX. INyESTIOATIOW OF tOBMVhM. Ml Add the second wadfwirlh together^ and substitute S for A, and ^for B as before : then, cos j g 4-^ 4. cos B— ^5s2 cos J. cos g ; that i?, cos ^-f^=2co8 ^: cos 5- cos JB— -rf (Z) Let n--i.A=B; this value beings substituted for B in the expressions Y and Z, we have the two following theorems for the sines and co-sines of multiple arcs, viz. Theor, 1. Sin nif=2 cos J. sin n-rl -<^— sin n— 2^. 2. Cos n//2=2 cos ^. cos n— 1 -^— cos n— 2-^. Ifi which general theorems, if n be expounded by 1, 2, 3, ^» ^, &c. we have the formulae for all particular mukiple orcif viz. if „--2. / *• ^"^ 3if =c2 cos A. sin ^ (from theor. 1.) * ^ 6. Cos 2A =2 cos A. cos /^— cos (= 1) (theor. 2.) fi=r3 / '^' ^^^ SJ=:2 cos J, Bin 2-<#— sin J (theor. 1.) ' ^ 8. Cos S.<f =2 cos A. cos 2^— cos A (theor. 2.) „_^ r 9. Sin 4-^=2 cos A, sin 3-4— sin 2-4 (theor. 1.) ^10. Cos 44 =:2 <;o8 A. cos 3-4— cos 2-4 (theor. 2.) „_5 f 1 1. Sin 5-4—2 cos A. sin 4^— sin 3-4 (theor. 1.) * 1 12. Cos 5^ =2 cos -4. cos 4-4— cos 3 A (theor. 2.) &£. &c. &c. 46. These formulae may be continued to any length, and by means of them* the sine and co-sine of evety degree and minute of the quadrant, may be computed, as will be shewn ; but, hav" log found the sines and co-sines to the end of the first 30 de« grees by this method, those from 30^ to 60^ may be obtained by an easier process, by means of the following formula. Add formula; 1 ami 3 (Art. 44.) together, and sine A^6 +sin -4—5=2 sin -4. cos JJ; let-45=30P, then will -sin. -4^2^ (cor. Art. 34) -, substitute these values of A and sin. A in thd above expression, and it will become •in 300+JB-l-sin 30— Jg— (2xtXCOS B^) co sJg; ••• Formula 13. sin 3G-iri3=co6 JB— sin 30— A 47. The tangents of two unequal arcs A and B being given, to find the tangents and co-tangents of their sum and difference^ It has been shewn (Art. 3^.)y that when radius =1, the sine tangent of any arc =: r— : wherefore, bv substituting for ° ^ co-^me ' ° c c 4 SK PLANE TRIGONOMETRY. Part IX the sine and eo^ine their respective values as given in the for- mulae. Art. 44. we shall have Formula 14. Tan ^+5=^--^±^= cos A-jrB sin A, cos jB+cos A. sin B cos J. cos B^&in A, sin B ' F.lS.Tan^— ^= sin A-^B sin A. cos B—eosA. sin 5 cos A-^B ^^ -^- c*^^ -B-f sin A, sin 5' If both terms of the right hand fractions be divided by €ot A, cos By they wiU become sin A sin B 4 F. 16. Tan -i+jB= cos A cos J? tacn ^+tan B sin -4. sin B 1— tan ^. tan B 1— • ^' ■ ■ • ■ cos -4. cos B (Art. 35.) sin ^. sin B F, 17. Tan A^B^ cos ^. cos JB tan -^— tan B sin ^. sin B ~ l-|-tari-4. tan B l-f ■ — cos ^. COS J? (Art. 35^.) ■ .^ cos A-^-B , .^ ^^ ^ 1— tan-4.tan^ F 18. Cotan^+B=-;r*5^ (Art.35.>= ^^^^^^. . cos -4— JS 1 H- tan A. tan B F. 19. Cotan^— J3=-T sm A—B t^^ -^— tan JB 48. To find the tangents and co-tangents of multiple arcs; that is, if A be any arct to find the tangent and co-tangent of nA. . tan -<^-f-tan B ^ Since tan ^+B^ ^_,^j^^ j^ ^ (Art. 47.) First, let B=Aj then F. 20. Taii 2^= (tan 2TB=) ^ *^^ ^ 1— tan2]« F. 21. Co-tan 2^= ( —- Art. 35.=) L_Arf «f.^Nl-^!^*_ 1 tmm^m^> tan 2^ * '^ 2 tan ^ 2 tan i< tan ^1* c^^P^ - (Art. 35. analogy 5.) 4. co-tan ^-4. tan A. Secondly. Let J9=s2-dr, then will Pabt IX. INVESTIGATION OF R>RMUL.£. 89S ^ 2 tan -4 tan A+ «^ « . tan -^+ tan 2^ 1— tan^ f. 28. Tan 3^= utu^-rtau^^ 1— tan -rf. tan 2-4 ,_2tanr^* 1— tan ^1« 3 tan -rf— taiT^l^ 1—3 taiH?)* P. 23. Co-tan 3^= (— Vr Art. 35.=) ^ ""^ ^^^ ^ * tan 3^ 3 tan ^-tan A]^ In like manner, 1— 6Tan A*+Uin~3\* F. 25. Co-taa 4^=±ll^^-!+^3l. 4 tan -4-4 tan ^1* &c. &c. 49. These formulae may be extended to every minute of the quadrant j but although it seemed necessary to shew how the tangents and co-tangents of multiple arcs are expressed in tcrms^ of the tangents of the component arcs themselves, yet we have shewn how to compute the tangents and co-tangents for the first 45° by means of the sines and co-sines, which is in many respects preferable to the above method. The tangents and co-tangents of arcs above 45°, may be found by a very easy process, the formula for which is deduced as follows : It appears from formulae 16 and 17> Art. 47. that Tan A+B -== J « > let ^=45°, then (Art. 16. cor.) — l+tan-<f.tan B ' ^ tan. As=l, •— — 14- tan B Uence^ tan 45°+J?=-r^^ =, and Un 45«--B= l—tan B l--tattB l+tanB' Subtract the latter from the former, and Tan 45^TI-tan i^B=I±^_J-±:ii= 1— tanB 1-ftanB i+tan JBt«-l -tan B)« 4 tan fi ^ , . , ^ ^ ■ : = — ==r— ; but smce tan 2JB= 1— tiOl* l-taniil« 2. tanB ^ , , , ^^ 4. tanB — ■ - (formula 20. Art. 4S) ; •/ 2 tan 2B= — =r , SM J^LANB TKIGOKOMETRT* Pavt tX. for thif fraction substitute iU equal (2 t an 9B) in the lastego a- tion but one, and we shall have tan 45"+B— tan 45®— 5= 5 tan. 2B; hence arises Formula 26. Tan 45M^=tan 45'*— B+2 tan 2B -. THE METHOD OF CONSTRUCTING A TABLE O? SINES, TANGENTS, SECANTS, AND VERSED SINES. 50. In the preceding articles the methods of deriviqg ex- pressions for the sines, co-sines, tangents, &c. of the sum, difference, and multiples of arcs in terms of the sines, co-sines, &c. of the arcs themselves, have been shewn ; but before we can employ these formulae in the actual eoBsthiction of the trigono- metrical canon, in which the numerical values of the sine, tan- gent, &c. of arcs for every minute of the quadrant are usually exhibited, it will be necessary to compute the sine and co-sine of 1 minute, and from these we shall be able^ by means of what has already been proved, to determine not only the numerical values of the rest of the sines and co-sines^ but likewise those 'of the tangents, co-tangents, secants, co-secants, versed sines^ and CO- versed sines, which constitute the entire canon. 51. To find the sine and cosine of an arc of 1', the radm being unity. It has been shewn (part 8. p. 231, 232.) that if the Iradius of a circle be unity, the semi-circumference will be 3.1415926535898 nearly -, this semi-circumference consists of ISO degrees, each degree being 60 minutes j that is, of (180x60=) 3.1415926535898 10800 minutes ; •.• -— ar. 0008906882086= the 10800 length of an ate of 1', the radius being unity. But in a very small arc, as that of V, the sine coincides indefinitely near with the arc ",* wherefore the above nombei ■^ The trigonometrical formuls, iatroducecl iato this work, 4re those odIj Which are necessary for the construction of a table of sines, tangents, &c. Several of tb« French and G^nuaa matbepiaticiaos hare excelled in this spedcs of investigation, and produced a great variety of theorems suited to eveiy ease in Trigonometry. The English reader will find a collection of fonnul8, applicable to the most delicate investigations in Mechanics, Astronomy, &c. in Mr. Boqinycflstle's Treatise on Plane and Spherical Trigonometrift London, 1806. B In SfaBptoqft Doctrine and application iff Fluxions, part 3. p. SOl* io^ Fa*t1X. CONStRUCnON OP S1N£S, &c. 99B XHM90B$SI%, &e. may be tftken Ibr.tke length of the sioc of l ^ Wherefore also (Art. S5.) the co-sine of V^^l-^sin 1')*= ( V-9^999991538405, &c.=) .99999996. 52. Construetum nf the mneg and ea-tmtsfram O U S€P. Since (Art. 51.) the sine of r» (.0008906888086, &c.s> X)0029O9, whieh is its nearest Tslue to seven places of decimals, and co-sine of l'=s .99999996. Let ^=an arc of 1', tlten the above numeral values being substituted respectively for sine and co-sine of 1' in formula 5. Art. 45. we shall have By Fmnula 6. sip ^'=2 cos 1'. sin V =2 x .99^9996 x .00O29O9=.OOO581S, here the sine cf 3' is found F. e. Cos 2'=2 cos Kcos r -*1 ^2 X. 99999996 X. 99999996 -.1=: ,9999998^ here the co^sine of 2' isfdund- F. 7. Sin 3'=2 cos T. sin 2'— sin 1' =? 2 X .99999996 x .0005818— .0002909=: .0008727* here the sine of 3' is found. F. 8. Cos 3'=2 cos 1'. cos 2"— cos 1' = 2 x .99999996 x .9999998— .99999996=. 9999996, here the eo^ineofS' is found. F 9. Sin 4=2 cos 1'. sin 3'-sin 2'= 2 x .99999996 x .0008727— . 0005818= .001 1 636 . F. \0. Cos 4'=9*C08 V. cos 3'— cos 2's2 x .99999996 x .9999996— .9999998= .9999993. F. 11. Sin 5'=« cos 1'. sin 4'— sin 3'= .0014544. F. 12. Cos 5'=2 cos 1'. cos 4'— cos 3'=.99999«9. And m this manner proceed to find the sine and co*sine of every nunute as fiir as 30**. 52. B. To find the sims and co^mes from 30* iQ 60* By formula 13. Art. 46. sin 30°-|-5=coe B— sin 30—^. i "f in Wrkce\ Pluxitnu, p. ««0. » w shewn tbat (radiat Uin^ 1,) the siae of aojT .b00290S88i086)^ .0008 9088890 86)^ .0002908893086)^ .oowms«208ir-.-r-— -^ + jXi T^:^ -I- he. a:.oeD3906Sfil676»&«. «tb9 WM f4 I'y wbifih 4if«r8 from ti»« al>oT« tjcpvctfiioo for tiie length of thf» cirv 9f l' hy w\f .(^000.009141 i tM !«« th» ar^ of 1' ABd its tine, ooiwtide W » dmrn^l plMSM indwsiv^f thMreisr^ thft liiieof I'to^plaMsof d«c»Mi9 (Hw wvidNr t<» vbtf^ln the t»h)«« w wwOly coKput«4) tXMit^ coiMidit wA its IW^ 396 PLANE TRIGONOMETRY. Part IX. ° Let J5=sl, then sin 30^ 1'=C08 I'-sin 29^ 59'= .99999990 -.4997481 = .5002519. jB=2' .Sin 30'' 2'=cos 2— sin 29« 68' c=. 9999998— .4994961 ==.5005037. J?=3' .Sin 80«» 3'= cos 3'— sin 29° 57'=: .5007556. 5=4' .Sin 30'> 4'=:cos 4— sin 29° 56= .6010073. 5=5' .Sin 30° 5 =cos 5'— sin 29° 55'=.501259l. &c.# &c. . .&c. 53. Having computed the sines in this manner as fas as 60"} the co-sines from 30^ to 60" will likewise be known 3 the co-sine of any arc above 30*^ being the same as the sine of an arc as much fceZoti; 60**. Thus, cos 30** l'=sin 59^' 59'=.8658799. cos 30° 2'=-sin 59° 58'=.S657S44. cos 30° 3'=sin 59° 57'=. 8655887- cos 30° 4'=:8in 59° 56'= .8654430. &c. &c. &c. COS 60° =sin 30° =.5000000. 54. To find ^e sines and co- sines from 60° to 90^. The sine of any arc above 60^ is the same as the co-sine of an arc at the same distance below 30° -, and in like manner, the co-sine of an arc above 60° is the same as the sine of an arc equally below 30° : thus. Sin 60° l'=cos 29" 59'= .8661708. cos 60° l'=sin 29° 59' =.4997481. Sin 60* 2'=cos 29o 58' Sin / 6(y 3'= cos 22' 5/ cos 60° 2'=sin 29° 58' cos 60° 3= sin 29° 57' &c. &c. 55. To find the versed sines and co-versed sines of the quadrant. Jn any arc less than 90*^ tlie versed sine is found by sub- tracting the co-sine from radius (cor. Art. 14.); and in ares greater than 90°, it is found by adding the co-sine to radius : thus. ' • The learner is supposed (in this and the following articles,) to bare com- pated all the preceding sines, co^sines, tangents, &c. ; if he has not, he mast, in order to work the examples, take them from a table. By means of th« fdrmnlse here given, any natural sine, tangent, secant. Sec. in the table, yfbifih. is supected to be wrong, may be examined, and if necessary, corrected. Part IX. CONSTRUCTION OF SINES, &c. 397 ver.sin r=l— cos l'=(l — .99999996=:) .00000004 ver. sin 2 = 1 —cos ^'= (1 — .9999998= ) .0000002 ver. sin ;V=rl— cos 3' =(1 — .9999996=) .0000004* ver. sin 4'= 1 —cos 4'=0000007 ver. sin 5'=il-.cos 5=.000001l &c. &c. ▼er. sin 90° l'=l +cos 89** 69'=1.0002909 ver. sin 90^ 2'rsl+cos 89^ 58'=1.0005818 &c. &c. Versed sines for arcs greater than 90, do not occur in the com*, mon tables. 56. The co-versed sine is found by subtracting the sine from the i-adius (cor. Art. 15,) ; thus, co-versed sin r=asl— sin l'a=(l — 0002909=) .9997091 CO- versed sin 2'=1— sin 2'= (1-. 0005818=) .9994182 co-versed sin 3 = 1— -sin 3= (1— .0008727=) 9991273 &c.» &c. 57. To find the tangents and co-tangents from 0' to 45". By Art. 35. anal. 1. it appears that the tangent of any arc ^= (radius being 1.) = vtanr I =illLL;=C^22!2£e-=) .0002909 co-tan 89° 59' J cos 1 \99999996 *-> 2' . I =ii!?i;= (:^22551i =) .0005818 co-tan 89° 58 J cos 2 \9999998 tans' ,_sin3l^ 0008727 ^3^ )o 57 J co-sin 3' \9999996 ^ tan 4' 1 sin 4' .0011636 sine co-sine co-tan 89^ 57' ^ co-sin 3' \9999996 ,l=—^=(^^^-=) 0011636 co-tan 89° 56 J co-sin 4' \9999993 &c. &c. And proceed in this manner to 45**. 58. To find the tangents and co-tangents from 45^ to 90°. Because (formula 26. Art. 49.) the t ngents of 45*^ + i? = (an. 45°— -B+2 tan. 2-B; therefore if 5=1', then 1*^^ ^^7.0 ^^ } =tan 44° 59+2 tan 2=5 t co-tan 44° 59 -^ (.9994184 + 2 X. 0005818=) 1.0C058^0. 368 PLANE TRIGOIfOllRRT. PitT IX. 1.0011642. n— Q' /tan 450 3' 1 , . , 1.0017469. £=4 .... I^"'/^''*' ,^ }=tan 440 56+2 tan S^^ *. co-tan 44. 56. -^ 1.0033298. &c. &c. And in this manner the tangent <if ev«rf suoceedii^ minute of ihe remainder of the quadrant^ must be found. 59. To find the seeants and c6'$ecaittB ^ the ^uadroftt. By the second analogy Art. 35. fire have seie /i:x r- ths •^ ^'^ cos Jt fadius being tmity; whence if _ fsec 1' 1 1 1 _ /T-r, then^^ g^ g^ 59' J "^ cos 1' "^ .99999996""^ 1.00000004. • if^3 . . • . < ^^ ^ . Vac ^^r ■* - ss) (^co-sec 89® ST'J cos 3 \9999996 ' ^0000004. r6ec5' 1 J , 1 _, "^-"^ • • ' • jco-sec 89® 55' |- cos 5'""\9999989'"^ 1.0000011. {sec T 1 1 orvo £.0' >= -^=1.0000021 so-sec 89® 63 J cos 7^ &c« &c. 60. By this method the sejcants and co-se<sants of every minute pf the quadnii^t may be computed^ but it is necessary to employ it only for the odd minutes -, the secants and co-secants of the even minutes may be obtained by a process which is somewhat more easy ; a^ follows By art. 41. tan 4+sec ^ssco-tan -^ 90—^. •.• sec ^=co-tan 4. 90— <^— tan 4. ("see 2' *) HQ ^fi' Let 4=b2', theni ^ ^ ^^ .^/ J.=^(co-tan -— taa «' ' 1 co-sec 89® 58 J ^ 2 =) co-tan 44» 59'-tan2'=s(| .0005819— .0005818=5)1.0000001. p«>T m cofiflniiucnoN or sikes, &e. p6» a(1.601l642— .0011636=) 1.0000006. =(l.p017469-. 0017455=) 1.0000016. (^co-sec 89° 52 J = 1XHX)0027. &c. &c. 61. The numbers thus computed are called natural sines, tangents> &c. they are computed for every degree and mimite of the quadrant, and arranged ia eight columns^ titled at the top and bottom 3 these together constitute the table of natural sines^ tangents^ &c. directiooB for the use of which are given in the introduction to every system of trigoaosietric^il tables >*. OF THE TABLE OF LOGARITHMIC SINES, TANGENTS, &c. 69. The logarithmic or artificial sines, tangents, &c. are the l^rkhms of the sines, tangents, &c. computed to the radius io) '®=ieO0000000Oj for since the sines, co-sines, and many of the versed stnes and tangents c<»Bputed to the radius 1 are proper fractions, their logarithms will have a negative indexj (v(4. 1. page 287.) but by assuming the above number for radius, these fractions become Whole numbers, their logarithms affir- Hoative, and the figures expressing any sine, tangent, &c. will be the same in both cases, as likewise their logarithms, excepting the indices^ which (as we have observed) will he frequently nega* tive in the former case, but always affirmative in the lattCTj therefore, in order to find the logarithm of the sine of an arc, ejaculated to the radius 10)'^, we most add 10 to the index of the logafrithm of the same sine to the radius 1 : for, let r= the radius, «=<fee sine of any arc to rad. r^ Ri=^a different radius, S=the sine of an arc (to rad. R) simitar to the former, then {Art. 38.) P For an accoont of the tables of sines, tangents, &c with ample directions to assist the learner in their use, see Dtt Hutton's Math^ftwticai Tables, iedit. p. 151,152. 400 PLANE TRIGONOMETRY. Fart IX. r:R::8zSi which if r=cl and Hs=10^**, becomes 1 : lo!'® : : s : S, V 5=io^'Ox5, •/ hg. S^lOxlog. lO+log. s:={8rnce log. 10=1) 10+ Zog.«. Q. E. D. Examples.— 1. To find the logarithmic sine of l'. To log. of .000^909 (s=«i« 1') =—4.46374^7 jidd 10 The sum is 6A6S7437=:thelog.sine of X to radius 10000000000. « ^. To find the logarithmic tangent of 2*. 35'=s To log. of .0451183 (=fa» 2^35') =—2.6543527 Add 10 The sum 8.6543527 m Ihelog. tangent of 2% 35'. 3. To find the logarithmic secant of 7*. 5'; The log. of 1.0076908 (=5cc 7* 5')=0.0033273 Add 10 The log. secant of 7* 5'.= 10.0033273 . 4. To find the logarithmic versed sine of 20" 12'. To log. of .0615070 (=»er. s. of 20' 12') ==-2.7889245 Add 10 The log. versed sine of 20«» 12'= 8.7889245 In this manner the logarithmic sines> co-sines^ tangents, &c. are computed -, viz. by adding 10 to the index of the logarithm of the nat-ural sine^ co-eine^ tangent, &c. respectively comespond- ing to the radius 1 '. Having shewn the method of computing the trigonometrical canon, both in natural numbers and fogarithajs, the next thing to be done is to demonstrate the propositions on which the practical part of trigonometry is founded. •the fundamental theorems of plane trigonometry. .63. In a right angled triangle the hypothenuse : is to either of the sides : : as radius : to the sine of the angle opposite to that side. 4 By the preceding rules any logarithmic sine, tangent, secant, &c. in the table, suspected to be inaccurate, may be examined, and the error (if aoy should be found) corrected. ' The log. sine of 1' (as here given) exceeds the truth by .0000176 becaasc ,tbe sine of T is only .000390888 and not .0002909. See Art. 51. PaztIX. tONDAMENTAL theorems. 401 Let JOB be a triangle, rigiit angled at Ai frOm € as a centre with any radius CD describe a circle I>£, and draw DF perpendicular to CA. Because DF is parallel to BJ (^8.1.) CBiBAtiCD: DF and! .^ ^v CB,CA.,CDx CF S^ '^ But 1>F ia the sine of the angle C (Art. 12.), Md CF is the co- D sine of the angle C (Art. 13), or the sine of the angle (CDJPs) jB; •• hyp. CB : side Bil:: radius (CD) ^ : sin ang. C (DF) oppoaite to BA: in like manner hyp. CB : side CA:: radius {CD) : sin. ang. B (CF) opposite to CA, Q. £. D. 64. If CD be the radius to which the trigonometrical canon 18 computed, then will DF be the sine of C, and CF the sine of B,2iS actually exhibited in the eanon; and therefore, having the hypothenuae CB, and one side BA^ of a right angled tri- angle given, the angle C (opposite BA) may be found, for CB : BA : : tabular radius : tabular sine of C, which sine being found in the table, the angle of which it ia the sine, will be known. Hence> the angle C being known, the angle £=90»— C is likewise known. 65. In a right ftngled trifloogle, one of the sides about the light angle : ia to the other : : as radius : to the tangent of the angle opposite the latter side. About the angular point C» of the triangle ABC, with any radius CE, describe the arc DE aa before, and draw £G at right angles to C^ (II. 1.) meeting CB in G, EG will be the tangent of the angle C (Art. 16.) •.• CA:AB:: CE: EG (4. 6.) ; that K side CA : side AB : : radius : tan. ang. C. In like manner, if from B as a centre with the radius BA a circle be described, AC will be the tangent of the angle B; and it may in like manner be ahewn, that BA : AC : : radius : taxu ang B. Q. £. t>. TOL. 11. p d 4M PLAN£ TRXGONaMETRT. Fait IX. 6(1. If C£ bft the radioa to which the canon la computed, £d will be the tabular tangent of C; wherefore^ shoe €A : AB :: CE: EG, we have only to find EG in the tangents, and its corresponding tngle C will be known } wharefore the two sides about the right angle of any right angled triangle being given, the angle C, and likewise the angle B (=:90*^€^) in»f be found. €7* The sides of any plane triangle are to each other as the tines 6f their opposite angles. Let ABC be a triangle, from B draw BD 'perpendicular to AC produced if necessary ; and CE perpendicular to AB, If a circle b^ described from B as a centre, with the radius BC, then it is evident that CE will be the sine of the angle ABC} and if from the centre C, with the same radius, a circle be described, BD wfll be the Mne of the angle BCA (Art. 12.) f wherefore, since the angle A is common to the right angled trL angles AEC, ADB, these triangles are equiangular (3^. 1.), and AB:BD::AC:CE {4.6) .' AB : AC :: BD : CE (16.5.); that is, side AJBisUie AC : : sin. ai^. ACB oppaii^Q AM : sin. ang. ABC opposite AC. Q. £. D. In the case in which the perpend&eular BD fiills without tbe triangle ABC, BD is actucUl^ the sine oi the exterior angle BCD i but BCA k the supplement of BCD (13. 1. asad Art. 8.) and since the sine (^ an. angle J^likewiBe tt^e sine of ite supple- ment (cor. Art. 12.) BD is therefore the sine of the angle BCA. 68. Hence, if we have two sides AB, AC oi saiy triangle ghen, and likewise an angle ACB opposite (AB) one of them ; the angle ABC opposite the other given side (AC) may be found i and thence the renmining angle A. For since AB : AC : : sin. ang. ACB : sin. ang. ABC, the three first terms beiog pAfet IX. FimDAMESTAh. THEOREMS. 40) given, the fourth, or sine of ABCy atid consequently the atigle j^BC is known 3 whence also the angle ^= 180^*^^8^480 is known. Lastly, from the two given sides AB, AC, and thd three angles which we have found, the third side BC will be obtained, for invcrtendo, sin. ang. ABC : sin. ang. BAC : : side AC : side BC, 69. If half the difierence of two quantities be added to half their sum, the result will be the gfeater of the two proposed quantities -, but if half the did^rence be taken from half their sum, the result will be the kss. Thus, let A and B be two quantities, of which A is the greater; S:^ their sum, i>=± their difference. And A^B^dS''^''^^'^' Their sum ^A=:S+D, •.• A=z~—ts—-^—. Their difference 2B=S-1>, •.' B=— — - =-5^— - Q.E. D. S D ,5 Cor, Hence, if from {A=) ■q-+-x- we take •^, the remain* der is — -, that is, *' if half the sum be subtracted from the 2 greater, the remainder id half the difference." 70. If within a triangle, a perpendicular be drawn from the opposite angle to the base, then will the base : be to the sum of the other two sides : : as the difference of these sides : to the. difference of the segments of the base. Liet ABC be a triangle, having the straight line CD drawa. from the angle C perpendicular to the base ABs then will A 8 : AC-^ CB : : AC-CB : AD—DB. From C as a centre with the distance CB the least of the t wa sides, describe the circle EBF, cutting CB in £, and AC pro^ duced in G and JP; then because CF:x: CB (15 def. 1.) AFzzAQ ]>d 2 404 PLANE TRIGONOMETRY. Part IX. + CB=s:the suip of the sides j and because CG=±€B, AC^ €B=x (AC-^CGzt^) ^G±=the di£fereDce of the sides. M&o, since DE=:DB (3.3.), AD--^ DB^iAD-^-DEz^) ^E=the differenqe of the segments -A.. E (AD and DB) of the base. Because from the point A without the circle, AB and AF are drawn cutting the circle, AB.AE^AF.AG (cor. 36.3,),-.* AB : AF:: AG : AE (16.6.) ; that is, the base : sum of the sides : : difference of the sides : difference of the segments of the base. Q. E. D. When the three sides of a triangle are given, the angles are found by this proposition. 71. In a plane' triangle, twice the rectangle contained by any two sides, is to the diffefrence of the sum of the squares of these two sides and the square of the base, as radius to the co-sine of the angle contained by the two sides. Let ABC be a triangle 2^B.BC: 31i?^4-Sc|«-:33« :: radius ; co-sine of ABC Draw AD per- ^ndicular to BC (produced if neces- sary), then 52) « ^-'icl* =^2 ^ 2C5. BI> (13.2 ), vZ5)«+Sc|*— 3C|^=2CB.BD; but ^CS.BA : ^CB.BD : : AB : BD (1. 6.) 5 that is, twice the rectangle con- tained by the sides : is to the difference of the sum of the squares of the sides, and the square of the base : : as AB : to BD; but B being the centre, and AB radius, BD will be the co-sine of the angle ABC (Art. 13.), •.• twice the rectangle contained by the sides, is to the difference of the sum of the squares of these two sides and the square of the base, as radius, to the co-sine of the angle contained by the two sides) and the same may in like manner be proved when the angle at B is obtuse, by using the I2th proposition of the second book of Euclid, instead of the 13th. Q. E. D. When the three sides only of a plane triangle are given, S Past IX. HJNDAMENTAL THEOREMS. 40S the angles may be found by means of this proposition^ withput letting fall a perpendicular^ as In the preceding article* 7^« In a plaice triangle^ tfj^^um of any two sides : is to their difference ; ; as the tangent of half the sum of the angles at the jbase s to the tangent of half the difference. Jjet ABC be a triangle^ from C as a centre with the .least side CB as radius^ describe the circle EBF-^ produce AC to F, join BE, BF, and draw ED perpendicular to EB. Because CE^CF^CB, AF=i{AC-\-CF^) AC+CBzsthe sum of the sides, and AE s (AC^CE=) AC-CB=z difference of the sides. Also C, tCB=iCB4+ CAB j(32. 1.) ^ s;sthe sum of the angles at the bwe, •.• FEB=i{^FCB ^ by 30. 3.=) half the sum of the angles at the base. And since CEzszCB, the angle CEBszCBE (6. i.) } but CEB:=zCAB -^EBA (3«. 1.) 5 •/ CBE=iCAB-^EBA; to each of these equals add EBA, %• {CBE-\'EBA=i) CBA^CAB-i-^EBA or CBA-^ CABz=:^EBA; that is, ^EBA^^the difference of the angles (CBA, CAB) at the base, •.• EBA^half the difference of the angles at the base. Now since EBFis a right angle (31. 3.)» and BED a right angle by construction, if from £ as a cenire with the radius EB a circle be described, it is evident that FB is the tan^ gent of FEB (Art. 16.) j that is, FB is tlie tangent of half the sum of the angles {CAB, CBA) at the base; and if from ^ as a centime with the same radius (EB) a circle be described, it will be equally plain that ED is the tangent of EBA; that is, ED is the tangent of half the difference of the angles {CAB, CBA) at the base. Again, becaui^e ED is parallel to FB (27* 1 -), and the angle A common, the two triangles AFB, AED are equi- angular (29. 1.), •.• AF: FB iiAEiED (4. 6.) and AF.AE: : FB : ED (16. 5.) ^ that is, the sum of the sides : is to their difference : : as the tangent of half the sum of the angles at the base : to the tangent of half their difference. Q. E. D. When two sides and the included angle are given, the re- maining angles may be fo^nd by this proposition with the help of Art. 69. Dd 3 4W PLANE TKIGOMOMETRT* Fast IL SOLUTION OF THE CASES OF PLANE TRJANGLB$. 73. There agre three ways of solving trigonometrical problems, V17. hy geometrical conMtruction, h^rithmetical computation, and hutrumentally, or bj the' scale and compasses. The first of these methods has been already explained in part 8. under the head of Practical Geometry ; the second consists in the application of the principles laid down in the foregoing theorems, by the help of either natural numbers, or logarithms 3 and by the third, the proportions are worked with a pair of compasses on the Ganters* scale 'j the method of doing which will be explained in the foHowing examples, where the conditions are exhibited in th^ form of a Rule of Three stating, having either thefirti and second terms^ or the^r^^ and third, always of the same Idnd. 74. iVhen the first axd second terms are of the same kind. Extend the compasses from the first term to the s^o^iid, on that line of the Gunter which is of the same name with Ihfise terms ; this extent will reach from the third term to the fourth, on the line which is of the same name with the third and fourtli. 75. When the first and third terms are of the seme hmd. Extend the compasses (on the proper line) fircHn the first to the third ; that extent will reach (en the proper line) from the second to the fourth -, observing in all eases, that when the proportion is increasing, the extent must be taken forwards oa * Tbk scale was inrented by the fUr. Edmimd Guattr, B.D. professor of Afttffoaomy at Grctbam College, probably about the year 1$$4 ; it it a bioad 4(1 f o)er tWQ feet in length, on which are laid down (besides all the lines com- mofl to the plape scale) logaritluuic lines of nvmbers, sines, versed sines, tan* ctntf, meridional parts, eqaal parts, sine rhumbs, and tangent rhombs ; that is, t^e actual lengths (taken on a scale of equal parts) are expressed by the figures constituting the Ic^arithms of the quantities in question. With these logar- Kbmic scales, all questions relating to proportion in numbers may be solved, fb»>tlie compasses being extended fmm the first term to the second or third, t(at extent will reach from the second, or from the third to the fourth, aocordiog as t^e ^rst and second, or first and third terms are of the same kind. For aii ample description of this scale, see Robertson's EUmentt of Navigation, vol. 1. p. 114. 4th. edit* likewise Mr. Donne*s directions usually sold with his improved scale ; and for an account of the improrements by Mr. Robertson, see a tract on the subject, published in 1778, by William Monntaine, Esq. F. R. S. FamtUC: of BKfHT AM0LI2> TBUNOLES. Mf the sqOfi^ but VirlMti the ptoptije^oA kdecreaib^Atiumi be tiken SOLUTION OF RIOHT ANGLED TRIANGLES. 76. Case 1. Given the bjpothenuse AB, and one side AC, of* i7gbt angled triangk -, to find tbe j^waining side BC, wd tlie angles A and £ ". ^^^^ Because^^5cl«4^«==351« (47. 1.) / BCJi «=r45;*-^Cl«> and SC=; v'^^'— ^** whmce BC is found . likewise (Art. 63.) hyp. AB : tade AC :i radius : sin. angle B; that is, ain B^ ^Cx radius , , ..• i . « ^p 5 or by logarithms <, log. sin B =log. AC+log. rod.-— log. .AB i whence the angle B is found, both by natural numbers and logarithms. Lastly^ since the three angles of any tri- aagje aire equal to two rig^t angles (32. 1.) ^ = 180», and the angle C (a right angle)=90% •.• B+A:=^ (180«>-.C=lScr-9(y=) 90^ but the angle B has been found, •/ .4=90— B is likewise known •. By a similar process AB and BC being given> AC and the angles B and A may be found. * Before yon begin to work any qne3tion in Trigonometry, you mast draw » sketch resembling, as nearly as you can guess, the figure intended ; pladny letters at the angles, and eacb number given in the question opposite the tide or angle to which it belongs ; some authors mark the given sides and angles by a small stroke, drawn across the given side, or issuing from the given angle ; t^e unknown paits they mark with a dphtr (o). * It must be remembered, that multiplication of natural numbers is per- fomed hy the addiiiM of their k>garithm8, division by subtraction, involution l»y wutHpHcaiiony and evolution by divvrioHf if these particulars be kept in mind, there will be no difficalty in solving tri^nometrical problems by logar- itlmis, see vol. l. part. 8. «- The angle A may be found in the same page of the table in which B is fomnd ; thus, if the degrees and minutes contained in B be foand at the top and on the iefi hand respectively, of the page, those contained in A will be fvoo^jkt theftoMom and on tint right; viz. the degrees at the bottom of the page, sutd the minutes on therrighPhand, in a fine with tlie minutes in B» n d 4 4M HAKE.TBlGONOMSrRY. Pa&t IX-^ • BzAKFtBft."-!. Qhtm the liypotlieimse ABssl9CK and tb6 perpendicular ^Cs95> to find the base BC and the angk& 4 andJB. B9 caiutruetion. Draw any straight line £C, at C draw Ci^ perpendicular tq BC, and make it eqiul to 95 taken from any convenient scale of equal partS} from ^ as a centre with the radius 190 takeiv from the same scale^ cross CB in J3, and join 4B. Take the length of CB in the compasses, and apply it to the abo?e« mentioned scale> and it wiU be found to measure 7B nearly; next measure the angles A and B by the scale of chords or the protractor, and they will be known, viz. ^ss38* and ^s=5f2*« nearly *. By cakulaiion. First, to find BC. We have BCs= v'SS)*— ^«= (^i5o)«-96>=^5S76=) 73.3143, &c. f Seeondhf, to find the angle B. We have sin £=; ^^2?=(^^^=).7916666 the natural sine of B, and the nearest angle in the table corresponding with this sine is 52^ 30' •) wherefore the angle B=52* 20', and^=(90"— J5=90'-» 62' 20'=) 37* 40'. > The sides and angles of triangles are yery ezpeditioosly determined both by the plane scale and the Ganter, but these methods are not to be depend^ on fn cases where accnracy is required ; they are neyertbeless nsefal where great exactness is no object, and as convenient checks on the method of calculation. y The side £C may likewise be ibund trigonometricallyi after the angle A AB.%m A has been found ; thus (Art. 63.) AB : BC : : rad : sin A^ •.• BC^ ;; — > rad this solution may be performed by the Gunter ; thus, extend on the sines from 900 to 37<>4, this extent will reach on the numbers from 120 to 7d-^sJSC nearly. > This, although it is the angle which has the nearest sine in the table to the above, is not perfectly exact ; the natural sine of 58* SO' being only .7916792. which is less than .7916666 by .0000874} now the sine of 52«8l' exceeds that of 52(> 20' by 1 777 > therefore our angle 52<> 20' is too small by -rrrr of a minute ; that is, by 29" -tttt ' whence, in strict exactness, aogl« 5=52« 20' 29" iVrV^J and angle-<rf=37*> 39' SO" -rf'^fj" FiBt IX. OP MGHT ANGLED TRIANGLES. 4W The same by loganthms. Since log. sin B^log, ifC+log. jfad.-log. ^iJ, V to log. ^C=log. 95= ....... 1.9777236 Add log. radius =log. 10000000000= 10.0000000 And from the sum = 11.9777336 Subtract log. JB—log. 120 = ... . 2.0791812 ■ Remains log. sin J?=52»20'= .... 9.89854^4 Whence angle -4= (90^—5=:) ST 40' as before. Jnstrumentally, by the Gunter, Extend the compasses from 120 to 95 on the line (of num- bers) marked Num. that extent will reach from (radius) 90* on the line (of sines) marked sin, to 52'*4.=52* 20'=the angle P. We cannot find the side BC by this method^ without anticipating case 4. 2.' In the right angled triangle JBC, given the hypothenuse i*B=I35, and the perpendicular ^C= 108, required the ba^e * An observation similar to that in the preceding note occurs iiere : the log. tine in the table which is the nearest to the above, is that of 52<* 20', vis. 9.8984944, bat this is less than the above, being too small by 480, wherefore 520 ir ig too ii^Q foff the an^le B; now the difference between the log. sine of 52<> 20', and that of 52« 21' is 975, whence the above value of B is -Stt- of a minute, or 29"-^ too small j that is, the angle J3=62« 20' 29"tt-, and -.^«37^ S9' ao'^T^ by this mode of calculation. It is worth .while to observe, that the difference of about -x-w of a second between this result, and that in the foregoing note, arises from the circum- stance of the logarithms, as well as the sines, being approximations, and not absolutely exact. When the sine, tangent, &c. found by operation is not in the table, 1. take the nearest from the table, and find the difference between that and the one found by operation; call this difference the numerator. 2. Find the difference of the next greater and next less than that found by operation, and call this difference the deneminator, 3. Multiply the numerator by 60 and divide the product by the denominator, the quotient will be seconds, which must be added to, or subtracted from the degrees and minutes corresponding to the nearest tabular number, according as that number is less or greater than the namber found by operation. This rale will serve both for natural and logarithmic sines, tangents, Sec. and Tikewise for the logarithms of numbers, observing in the latter case (instead of multiplying by 60) to subjoin a cipher to the numerator, and having divided by the denominator, the first quotient figure must occupy one place to the right ol the right hand figare in the nearest tabular number, and be added, or sub* tracted, according as that namber is too little, or too great. \ *4 410 Pl4/^£ 14EUOOM0a|BQntY. Pakt IX. BQ» vA tb« fliaglcs JkvfAB^ An$. BC^^h ang. A^=99^ hi, 5. Givi^ AB^Q9l, BCatl6, required th^ remaining side and angfef ? Am. AC^19^ ang. ^as43* 5'^ cii^, J3=47' 55'. 77, Case 2. Giuen the two sides AC and CB, to find the hypothenuse AB and the angles ^ and B, first, (47. 1.) ^-6== ^^/ACJl^+VSi"^} whence AB is found. Secondly, (Art. 65.) AC : CB : : radius : tangent ang. A: or tan A:sz """ •'•• ' - • " } and by logarithms, log. tan. -^=log. BC +log. rad.— log. AC, •/ the angle A is found, both by natural numbers and logarithms, and the angle B^^^Cf—A is likewise found. ExAMPLjEs. — 1. Given the side ^C=123, and the side CJ5s= 132, to find the hypothenuse AB and the angles A and B. By calculation \ F*r#<, ^5= V^* + CB)«=;r V 123l«+ 132l«= ^32553= 180.424. o J, t . , T ^ CBxrad. 132 Secondly, by natural munbers, tan ^o^ — >r " '^123^ 1.0T31707=natural tangent of 47* l'=ang. A, \' ang. -B= (9(r— ^^) {W-.47' 1 '=42« 59'. Thirdly, by logarithm, log. tan. ulalqg. CB-J-log. yad.- log. ^J? •.• to log. CB 13a= 2.1»5739 Add log. radius 10000000000 =10.0000000 And from the sum = 12.1205739 Subtract log. ^B 123 = 2.0899051 Remains lag. tan. ang. ^==47^ l'= 10.0306^88 And ang. B^W^^A:=i4aPbtf as before. Instrumenlally, Extend the compasses from 123 to 132 on the line (of numbers) marked Num, this extent will reach from (radius =) 45^ on the line (of tangents) marked Tan. to 47^ I'scthe angle A. T ^ In this and the foUowiaf e^Mc of rigU aa|^ triasf let, tibc ^onstracilian n purposely emitted, it beiog perfectly- «a«y and- obvioss, frani wittt kas liaea given on the subject in the PractUi^. G^ameti^,.wmu the end of part 9* fARt IX OF SIGHT ANEOJD TRIANGLES. 411 The side JB is not foaifd vutrumeaially for a reason simi- lar to that before given. 2. The perpendicular AC^^^tOO, and the base BC=110 of a right angled triangle ABC being given, required the hypothe* nuse AB, and the ^ngles A and B ? An9» i#B=;^8.254^ ang. ^=28° 49', ang. B=6V 11'. 3. Given AC=^4, and BC=S, to find AB, and the angles A and B. Ans. AB=zS, ang.A=zSe^ 52', ang. 5=53° 8'. 78. Ceue 3. The hypothenuse AB and the angle B being given^ to find the sides AC, CB* and the angle A. First, since the angle at JS is given, the angle A=z90P-^B. Secondly ^ (Art. 63.) AB : ^C : : radius : sin ang. B •.• AC^ ^ ^d " ' ^ •'^^ Ipg.^C =:log. sin B+log. AB—log. radiqs ; whence AC is found both by natural numbers and logarithms. Thirdly, 5F«=J?C)« + CffI« (47. 1.) '/ VSi^^ABi^-^-lT)^ and C5= ^AB'^ AC.AB-'A C (cor. 5.2.); al so log. C W^'^^^^^^^^- ^^-^C , . . ^j ^ ^^„„^^ boti^ by 2 natural numbers and logarithms. ExAMPi^Es.-^!. Given the hypothenuse AB^=^\6o, and the angle 5=35" 30', to find the sides AC, CB, and the angle A. By calculation. First, ang. ^=9Q»'-J?=«^ (90^-35^ 30'=) 54^30'. X Ar^ ^^^ B.AB , . Secondly, (by naitural numbers) AC=^— — ^ — =(smce raa=l, sin 35° 30' x ^-8=) .580703x16.^=5=95.815996 5 but the same may be done more r^dily by logarithms -, thus, be- cause Jog. -^C'=log. sin 5+ log. AB'-^oQ. rad. •.• To log. sin B. or 35° 30'=^ .... 9.7639540 Add log. AB. or 165= . . And from their suin= • • l^ubtract log. radius^^ . . . Remains log. AC 95.816= 2.2174839 U. 98 14379 10.0000000 1.9814379 T \ f 419 PLANB TEUGOMOM BTRT; ^ Fait IX TUrdlxj, CB^ ss/AB^-ACAB—AC^ <V:16&+95.816x 165-95.816S ^860.816x69.184= V18044.?94144=) 134^29. The same by logarithm s, log. C^^^ \i^:AB-\-A€^\og. AB—AC that is, to log. A J^-^AC, or 260.816= 2.4163348 Add log. AB-^AC, or 69.184=: . . . . 1.8400057 The sum divided by 2 2) 4.2568399 Gives the log. of CB=: 134.329 » . . . .^2.1281699 Instrumentally. 1. Extend from (radius or) 90* to 35* 30' (sang. B) on tbe }ine of sines; this extent will reach from 165 (backwards) to ^bout 95 ^ on the line of numbers^ for the side AC (opposite the ang. B.) 2. £xtend on the Une of sines^ from 90" to 54* 30' (comp. B.) 'y this extent will reach on the lines of numbers from 165 to fkbout 134 -iV for the side CB, Ex.— 2. Given the hypothenuse 4B=25, and the aogh £=49% to find the sides AC, CB, and the angle A} Ans. 4C= 18.893, CB= 16.4017, ang. A=z4l\ 3. Given ^BslOO, and the angle ^=45^ to find the lest? Ans. BC=-iC=70.7108, ang. B=45^ 79. Case 4. Ope side AC, and its adjacent angle A hemg given, to find the other sides AB^ BC, and the remaining angle B, Ftrsti angle 5=90**—^. Secondly, because (Art. 67.) AC : CB ^ . ^ ^» sin A.AC : : sin B : sm A, *.• CJ5=: — : — =-— 3 and log. CB=log. sin -4-l-log. .4C— log. sin B. Thirdly, because (Art. 63.) AB : AC:i radius : sin B, '.' AB^ — / ^ : also log. sm B AB=log. AC +log. rad. —log. sin B. j^ Examples. — 1. Given the perpendicular ^C= 1023, and the angle ^=12*» 45' 5 to find the angle B, and the remaining sides AB, BC. PiiBT IX. OF BIGHT At<6LBD TBIAMGLES. 4i^ B^ calculation. First, aog. B=i90»-^=:(90»-12r4b'=) 77^ 15'. c ^1 ^^ HnJ,AC .2206974X1083 ^^, .^lo Secondly, CB=z — . _ = ^.o.qq =231.4812; ^ sin jB .9753423 and by logarithms^ log. CBs=log. sin -^+log. -^C— log. sin B; that is, to log. sin J. 12^ 45'= 9.3437973 Add log. AC 1023= . 3.0098756 From the sum = 12.3536729 Subtract log. 8in,B 77** 15'= .... 99891571 Gives log. CB. 231.4812= 2.3645158 «,L. « ^„ -^C.rad. 1023x1 ,^,««^^ Thirdly, AB=: —. — -- = = 1048.862. ^ sin 5 .9753423 And by logarithms^ log. .-^J5=log. -^C+log. rad— log. sin B; that is, to log. AC 1023= 3.0098756 Add log. radius= 10.0000000 And frorn'the sum= 13.0098756 Subtww't log. sin J5 77^ 15'= . . . 9.9891571 Gives 1<^. AB 1048.862= 3.0207185 Instrumentally, 1. To find CB, extend from (sin B, to sin A, that is, from) sin 77^7 to sin 12K 3 this extent will reach on the line of num*« hers from (AC) 1023 to 2314-. 2. To find AB, extend from (sin B to radius, that is, from) 77^-i^ to 90^ on the sines; this extent will reach from 1023 to about 1049 on the numbers^ Ex. — 2. Given the perpendicular ifC=400, and the angle A=^4T^ S(f, to find the hypothenuse AB, the base BC, and the angle jB? Ans, \^B=592.072^ BC=436.52^ ang, J?= 42« Stf . 3. Given ^tf C=82, ang. ^1=33^ 13'^ to fikid the rest ? Ans. ABssi979^ CB=63.69, ang. B=^%69 Alf. SOLUTION OF THE CASES OF OBLIQUE ANGLED TRIANGLES. The foregoing calculations are efiected both by natural numbers and logarithms, serving as a useful exercise for the learner; but principally to shew, that both methods termiimte in the same result. • \ 414 PLAN& tRfCKlNOMBIllT. PaAyIK. Trigonometrical operartidtis are liowever seldom performed by the natitml aumherft, abd tkeneffere^ in the fottMvitag cases^ we ghall employ only the logarithmic ptoOMs. 80. Case. I. Let there be given the two angles B and C, and the side AC opposite to one of them j to find the angle J, and the sides JJ5 and JSC. First, the aisles ^ and C A being given, and ^ = ISO^— B+C, the angle A wUl be inown. Secondly, (Art. 6f.)' AC : AB : : sin i? : sin C *•' AB:^ ACAnC ^, .^ , »' ~' 'C ■ M , „ ■ ;-or by lo^itopms, l^g. AB=:\og, -4C+log. sin C— log. sin B; '.' ABisinown. Thirdly, (Art. 67.) AC i CB :: sin 5 : sin ^ •/ GB= AC, sin A ^ , . * * ^^ ^ i^-' — . p . By logArithms, log. CBaftlog.^+log. sin 4- log. sin B; '.' CB is known. Examples — 1. Given the angle 5=46®, the ai^Ie Car59^ and the side AC (opposite JB)r=i^O; to find the angjte-4 and the sides y^jS, BC. "^ . By construction. ... From any scakof eqnai pavta. take ACsslQO, at C ixttfae the a ngle AC B=i59y and at A make the angle GrfBsat(l8d'»- B+C=1800-46o + 59<>:?:) 75^5 then take the length of J^, and of BC respectively in the compasses^ and apply them to the above-mentioned scale, and AB will=143, £C=161^ By computation. 1. Log. ^B=slog. ifC-hlog, sin C'^log. tin B •/ To log. AC IWic 34>7P1813 Add log. sin C.59 9.9330656 And from the ^m:t: lB.0l3246a Subtract log. sin B 4€P=st 9.8569341 Remains log. AB 14^.9845= . , . 2.1653127 PaAt IX. OF OBLIQUE AN6L£D TRIANGLES. 41$ 2. Log. CBisilo^: A€+\og, sin .<— log. sin B. \' To log. AC 190= 2.079181« Add log. sin J 75o= ... 9.9849438 And from the snni= 18.0641250 Subtract log. sin B 46<>= 9.8569341 Ronaaina leg. CB 161.1354 . 4 . . 2.2071909 Instrumentallj^. 1. Extend on the sines from 46*" (ang. B), to 59^ (ang, C) ; this extent will reach on the numbers from 120 {AC), to about 143 (AB). 2. Extend from 46° to 75° on the sines -, this extent wilt reach froDd 120 {AC), to about 161 {CB), on the numbers. Bx. 2. Giveft the angle AstBSP 43', the angle Css7# Y, and the side ABta^eiO', to §nd the angle B, and the sides AC, CB } Am. ang. B^Af 10^^ jlCss46&.08» C£s542. 3. Given the side ^^=1075, the angle ^=34'^ 46^, and the angle 0=22"* 5' ; to find the r«st ? Am. BC=2394, ^C= 1630.5, any. -rf=123»9'. 81. Case 2. Let there be given the two sides AB, AC, and the angle B, opposite AC: to find the angle B^Cand C> and the remaining side BC, Mrst, {Art 67) AC : AB : : sin £ : sm C; -.* sin C= ■ ; ■■■' 5 which by logarithms is, log. sin C^ log. AB-^log, sin J&— log. AC; ••• angle C is known, ^ X /a x\ Secondly, angle l?.^C =180— jB+C, •/ angie B-iC w ifc«oio». •^ < This case will be always ambiguous when the given angle B is acute, &fld AB greater than AC, (a;i in the first example) ; for the above expression \^ the sine of both AsB^Axa, or of its supplement AzB (for the sine of an angle and the sine of its supplement are the same, by cor. Art. 1 S.) ; conse- quently the angle A will be either BAx or BAts, according as the angle AsB, 6r its stipplement AzB be taken ; and the correspondiqg value oi BC will be either Bx or Bz, But if the given angle be either obtuse, or a right ' 416 PLANE TRIGONOliETRT. Part £| Thirdbf, (Art. 67-) JC iBC-^nn Bisin BAC, .• BCz jiCsin BAC : — = — j that is, by logaritfams, log. 5C=Iog. -4C4-lo| sill Xy sin J?ufC— log. sin B: */ JSC if known. Examples. — 1. Given AB=204, ^C=145, and the angle J =35®; to find the side BC and the angles BJC and C. jBy coMtruciion. Draw .<^jB and make it =204 by any scale of equal parts and make the angle J?=35® 5 from .^ as a centre with the radio (AC=) 145 taken from the same scale, cross jBCin z and jf, join Az, Ax, either of which will be AC 3 then will Bz or Bx bi the value of BC, these being measured by the above scale> ivill be BzzsSl^ and Bx=252j. for the values of AC; also by the scale of chords, or protractor, BAx=z9V, BAz^l^^ for the corresponding values of BAG; likewise ^J5=:54^ AzB=s I26*i for those of C. By calculation. To find the angle C. Because log. sin Cslog. AB+lc^, sin J?— log. AC; V To log. AB 204= 2.3096302 Add log. sin B 35«= 9.7585913 From this sum= • . . . . 12.0682215 Subtract log. AC 145= 2.1613680 Remains log. sin C-l or its supp. > =9.9068535 I viz. 1260 12' J Next, to find the angle BAC. ^350+53048' ^ f Q»>^ First, B+C:=} or S=J or I350 + 1260 n'J 1 16l« 12' angle, each of the remaining angles will be acnte (32. 1.) ; therefore when the angle B is either obtuse, or a right angle, C muH be acute ; consequentijr when B it not less than a right angle, no ambiguity can possibly take place If the angle B (in any proposed example nnder this case) be either acute, obtuse, or a right angle, and AC greater than»^B, there is no ambiguity ; but it must be remarked, that if JiChe less than j^B X nat. sin B (or the peiptB* dicnlar drawn from A to the base BCt) the question is impossible. ^ eaBgfcf Ai^i»iiaTlX. OF OBLIQUE ANGLED TBIANGLfiS. 417 ^..^ /angleS-4C=180— £4-C=J or >=-J or Lastly, to find tfie side BC Since log. BC=log. ^C+log. sin B^C-log. sin B. If BACt=i9l^ 12' To log. AC 145= 2.1613680 /• Of 12' ^ nl pr* Add log. sin BAC'l or its sup. > 9*9999047 thenar t 88*48' J luiii And from tbe sums 12.1612727 orir' Subtract log. sin B SS^rs 9.7585913 de;H Reoiaiiis kg. ^Css262.744a . . . 2.4026814 ^^' If i9i#Ca 18* 48' , fcf To log. irfC 145SS 2.1613680 B=l^ Add kig.sio^i#€iy4tfg 9.5082141 And from the sums .... 11.6695821 Subtract log. sin B 35**= . . . 9.7585913 Remains log. £0=81.4687=: 1.9109906 ifMrumentallif, To find the angle C, Extend the compasses from 204 to 145 on the line of numbers^ that extent will reach, on the sines from 35* to 53' 48', the supplement of which is 126" 12', either of these is the angle C. To find the side BC. Extend on the sines from 35^ to 88* 48'>.that extent will reach on the numbers from 145 to 253 ; or extend on the sines from 35° to 18° 48', this will reach from 145 to Sli on the line of numbers. Ex.— 2. Given the side ^£=266, BC^ 179, and the angle C=107°40'$ to find ^C, and the angles A and B? Ans, i<C= 149. 8. ang. A=:S9^ 53', ang. B=32° 27'. 3. Giwn -rfC=236, ^C=350, and the angle B=38°40'j required the rest? Ans. AS==IS4A7, or S62.04, ang, -4= 67<> 54', or 112° 6', ang. C=73° 26' or 29° 14'. 82. Case 3. Let the two sides BA^ AC, and the included angle A, be given 5 to find the side BC and tlie angles B and C VOL. II. K e \ 418 PLANE TRIGONOMETRY. FftHi-IX. Let AB y AC, then (18. 1,) ^ the ang. C^B', and since B^ C = 180°- ^ (32. 1.) 4. C-fB =:ix 18Cy>— -4=90P— i^j V ^C-k-B it known. But (Art. 73.) ABj^AC : 'B^ ^C A B-^A C (:: tan ^C+B ; tan 4^ C-B) : : tan 90- 4- A : tan i C-B, / tan 4. C^ AB-^AatangO-^^ A . , . IfB+^C ' ^ loganthms, log. tan 4. C—B log. -^^— -4C+log. tan 90-4- ^— log. ^iB+^C •.• 4- C^B » JmotD». Whence (Art. 69.) the greater angle C=4^ C+Jg -fj. C— B^ and the leas^vix. 5=4- C-|-B— f C— B> '.' ^ ang2e« C and R are known. Lastly, (Art. 67.) AB : BC :: sin C : sin -^, ••• BC^ AB.sin A — ^j^-^ > oy logarithms, log. J5C=log. -^B+ Jog. sin -rf— log. sin C •.• BC is known. Examples — 1. Given AB:=:90, .rfC=30, and the angle A szSOPy t^find lihe aide BC and the angles JB and C. Bif construction. Make AB^20 by any scale of equal parts, at A (with the scale of chords or protractor) make the angle BACz=i8GP, and make AC^SO, by the above scale of equal parts, join BC; then, the angles B and C, and the side BC being measured, will be as foUowsi viz. ang. B=63o 24', ang. C=z3GP 36', side BC=33, nearly. By carculation. 4- B4.C=:9O<^-4. ^:=(9 O0-4(y>=:) 50°; this b eing k noway in ^rder to find 4 B~C, w e hav e log, ta n 4 B— C=(log. -rfC— -4B+log. tan W-^^A-^log.AC-^ABzs) log; 10+ log. tan 5(y>— log. 50. •.• To log. 10= , :i.ooooooo Add log. tan SO^rs 10.0761865 From the sums ........ 1 1.0761865 Subti-act log. 5 0a± . ...... 1.6989700 Remwns tott4 B-^C 13® 24'4. 9.377216& PjibtIX. of <»LfQtJ£ AKG1.BB triangles. 4i9 Also ^ ^+C;++ JB--C=i£60» + 13« 34'i=sdSo 24'4-=aiJgle B. And 4. B+ C— i B— 1'=»60«- 13*» 24'4^s=»6* 35'i.=cat)gle C, Lastly^ log. BCalog. ^B+log. sin ^— log. sin C; V To log. ^B 20= ........ 1.3010300 Add log. sin A 80«=s {».99335l5 From the sums 11.2943815 Subtract log. sin C 36^ 35^4- . . 9.7753250 Remains log. BC 33.0412stt . . 1.5190565 Irutrumentally, For the first proportion^ extend from 50 to 10 on the numbers) this extent wiU reach on the tangents from 50^ (the contrary way, because the tangents above 45^ are set back again f) to about S^'i, that is^ from 45<> to 13''4-. Extend, for the second proportion, from 36*^ 36' to 80^ on the sines ; tliis extent will reach from 20 to about 33 on the numbers. Ex.-^2. Given the' side ABsz^lB, the side ^C=478.d, and the included angle AzsSi9 AQ'i to find EC, and the Angles B and C? Am. BC=s:326.1, ang. Bsl23« 9', ang, Csa 220 6'. 3. Given ifB=116, AC=zB7, and the angle Jr=115^ 37' > required the rest ? Ansi BCs 172.5> ang. B^^V ^\ ang. C» 37« 20'. G3. Case 4. Let the three sides ABy BC, and CA, of the tri-^ angle ABC be given 5 to find the three angles A^ B, and C. ^»" i^iar ^ When the ratio to be niemiared is in the tangents, and one of the term» below, and the other above 45* ; ba¥in$ talwn the extent of the. two fbruer tenne on the nombers, &c. as the case may be, Kppiy this distance 00 the tan- gents, from 45» downwards (to the left) and let the foot of (be compasses rest on this point, which for distinction we will call s; with 00*6 foot on o^ bring the other foot from 45% to the given term of the ratio; apply the distance (of z from the given term) from 45^ downwards, then, one foot ol the compasses being on 45, the other will (with this extent) exactly readk the term re^^uired to be founds £e2r 4^0 l^LANE TRIGOTJOMEtlnr. Past % Bnt, By ieiUngfaU o perpendicvlar AD. Let BJ be thegreater side, AC j^ the less^ and BC the base; then (Art. 70.) BC : BA+AC : : BA^ AC : BD-^DC, -.- BD-DC:s BAJfACBA^AC BC and log. i^D-Dc' =Iog. 5-4+^C+log. BA^AC—log. BC '.' BD'-DC u Awwic». But BD+DC {szBC) M 5riw», ".' *A« AaZtre* of <^efc arc likewise known. Bin (Art. 69.) BD+DC . BD^DC , . ^ ^ and 2 2' JBD+DC BD-DC 2 2 •.' l/jc segments BDy DC are known. Now in the right angled triangle ABD we have AS, BD and the right angle ADB given. •/ (Art. 63.) AB : BD : : rad. : sin -BJfD, or sin BADss ^^. In logarithms, log. sin ^^D=log. ^JD+lo-log. AB • AB; •/ J?-4I> is known, •.• also its complement} viz. the mgle ABC is known. And in the right angled tntakgU ADC we h^ve AC, CD and the right angle -41>C given, •.• as above, CA : CD:: rad. : nn CAD, or sin CAD=s '^ ' . By logaritha», log. sin CAD =log. CD+10— log. CA\- CAh, and consequeHilly fts com- ptetnent, viz. the angle Cis known. Also BACz=^BAD+DAC is known. The solution without a perpendkulmr. By Art. 71 - 2 BA.AC ; b1)^ +A(\*'^Sc^ : : mditts : cos^T ^ rad.53l*+^Cl«-5r}» ,,cos^= -^ZaC s (since rad=: 1, see also cor. 5. 2.) ' 2"B^ ^C • ^ logarithms, lo g, co^ A=lO +]og' Sa"* + AC -^BC.ACSC -log, ^BA-^lo^.^AC '.' the angle A is known f and B+ CslSO^--^, to find the angles B and C. f^Kf DC. OF qi||4tfJS AN^1|D TBIA{4GLES. 4SI (Art. S %.) Log, ta n, i- B ^CzsJo g. AC-^A B -^-log. tan. M'-^^A -log. AC^ABj then 4- ^4- 0+4- ^- Cs=ang. Bi . ^^^ ^^ 4.B-HC-i5-C=ang. c' ^ -^ence ^&e three {mgle$.A, B, and C, are known*. Examples.— 1. Given the aMe -rfB=12, AC=^U, and JBC= 10^ to find the angles A, B, and C. £y comtructum, 1. Draw the straight line JSC=:10> taken from any convenient .scale of eqjual parts^ from £ as a centre with the radius 12 de- scribe an arc, and froijoi C with the radius 11 cross the above arc in A, (both the latter distances being taken from the same scale with JBC,) and join 4B, AC' 9. Measure the angles by means of the scale of diords^ or protractor^ and they will be nearly as follows) viz. A^hl^, JB== 59°^, and C= 6d°i. jPy ealcnilaHon. First, let Ajy he perpendiculqar to BC; .see the last figure Jjutpne. Jgf)+Z)C= JgC=10 BD-DC^ ^- ^ = -^^^2.3. ^^ BD+DC BD-'DC V jBP== ^^ r{- ^ ;^5+.1.15==^.1.5 the greater figment 3 and DC— ^ ?-- — — r-=^-^1.15=3.85 the less seg- meat ; Then log. sin. 54I>=log. -BD+ lO^log. ^5=0.7888751 + 10—1.0791812=9.7096939 5 v ang. BiiD=30' 50', and v its complement 5=f59*' 1(/. Jn li)£e manner^ log. sip. CAD^sslog. CD+lO—log. C-4=p 0.5854607+ 10—1.0413927=9.5440680= / ang. CAJQ^^O^ 29^ > the complement of which is 69* 3I'==the angle C. Also the ang. BJC=BJD+C4D=30 50'+?0» 29'« 61* W. « On haviog found the angle A, the remainiDg angles B and Cmay be fonnd (perhii|»8 more omyeiiieatly) by Art. 67. thvts BC: CA ; : tin ^ : sin B^ CA,%vbl a ■ ■ ^^ ; V B w ifcnott7»; whence also C^X^O'-^A^k- B \ '•' C is likewUe hnotvn, z e S 4» PLANE TRICSONOMETRY. Paet I£. ThetobiiummMoui aperpendictilari see tbe kst figure. Natural cos ^=5.__ 2 1^^.^C 264 .6250000 •/ angle AszbV Wy •/ C+B««18a»— 51« IS': 128» 41', and -.-±5ik64* 20'4. 2 • Ixjg. tan. = log. ^B— itfC+log. tan. 64» 2<y4^-log. -rfB+^C=0+10.31S4222— 1.3617279==«.9566944 / C-t- B C— B •. angle C«s-~-- +-— - «s64* 20'^ +5* l(f^=^S^ 31'. 2 2 angle B— -^ o"""^*" 20>-5' l(/4.=59* lO'. 2 2 ' Instrumenially, first method. 1. Extend from 10 to 23 on the line of numbers ; this extent will reach^ on the same line^ from 1 to 2iV> the difference of the segments of the base. 2. Extend from 12 to 6.15 on the numbers -, this extent will reach on the sines from 90" (radius) to SO* &0'=BJD, the complement of which is 59* 10'=ang. B. 3. Extend from 11 to 3.85 on the numbers > that extent will reach from 90* to 20"^ on the sines, the complement of which is 69i=^C, Second method. 1. Extend from 264 («a=2B^.-^C) to 165(=SZ|*+^C*— *B9*) on the numbers j that extent will reach from 90° to 384- on the sines^ the complement of which }s 51-;-==angle A. 2. Extend on the numbers from 23 to 1 3 that extent will reach from 64* j. to 45*" } and back again to 54. on the tangents, for half the difference of the angles B and C. Ex. 2. Given the three sides, viz. -^B=100, AC=s»40, and BC=s70.25 'y to find the three angles ? Ans. ang, A=33'' 35', ang. B=l&' 22^ ang. C=128« 3'. 3. Given ^B»:^68.95, JC=^7^, and BC^^OO, to find the Jingles? Ans. ang. -^=112** 6\ ang. B=3S^ 40'> ang. 0=» .^9^ J4'. I^BTIX. 1NA0CB»IBL£ ISUQfiTS & DISTANCES. 4Si3 THE APPLICATION OF PLANE TRIGONOMETRY TO THE FINDING OF THE HEIGHTS AND DISTANCES OF INACCESSIBLE OBJECTS. The uses to whick Hane Trigonometry may lie applied are 50 various and extensive, that merely to point them out would require a very large vc^ume 3 and to understand them> the stu- dent must be well acquainted with Geography, Astronomy^ and the numerous branches of Natural Philosophy^ of which this science fnrms a necessary part. At present we shall confine our* selves to one of its immediate and obvious applications^ namely^ that of determining the hdghts and distances 4)f inaccessible objects. The following instruments are used in this branch of men« suration> namely, a quadrant, a theodolite, a mariner's. compass, a perambulator, Gunter's chain, measuring tapes, a measuring rod, station staves, and arrows ^ the description and uses of which are as follow : 84. The uuadrant 'is an instrument for measuring angles in a vertical position 9 that is, to determine<the angular altitude ' Besides the common surveying qimdrant, of Which that described abore is the simplest form, there are yarioas other- kinds, as the astronomical quadrant, the sinical quadrant, the herodictical quadrant, i>ayis's, Gunter's, Hadlej'Sy Oole's, CoUins's, Adao^s's, 9fid some others. Quadrants may be bad at any f rice from one to twelTe guineas. The height of an object may be taken in two senses, viz. 1. its perpendicu- lar distance (in fathoms, yards, feet, &c.) from the ground ; 3. its angular height, or the number of degrees contained iti the angle St the eye of the ob* «enrer, ^hicb the perpendicular height subtends ; the former we have, for wfiitinction, denexnioated hHgkl, the latter tUtUude. I 1fiC4 4M HMfi fnoiO»<mMXKr. IPawIX. ♦♦ d BBCf pfOpOMQ OCQCCt. jtBC 18 a quadiant, to the centre C of which the weight IFisfredy Mttpended, b^ meaot of the string CW$ <« are two sights^ through which the eye of an observer at Jl sees the object O* The arc AB of the quadrant is divided Into degrees, which are subdivided into halves, quarters, or single minutes. In using this instrument, the obser- Wk^'-^X^^^t^-^.^^A'Z^''''^^^^^^^ D ver turns it about the centre C, until the oh* ject O is visible through the sights «# ; and as he turns it, the line CW, revolving freely about the centre C, moves along the circumference AB^ when he sees the object through the sights, the arc BWvinXL be the measure of its angular altitude, that is, of the angle OAD, Draw OD perpendicular, and AD parallel to the plane of the horizon \ then because the angles at E and D are rigM angles and the angle A common, the triangles CAE, OAD are equiangular (33. 1.), •/ the angle ACE=iAOD; but DOA-\- DAO» (a right angles) ACB, from these equals take the equals DOA^ECA, aad the remainder DA€hssECB. And since the arc BfV is the measure of the angle ECB (Part 8. Art. 237.) it is likewise the measure of DAO, or of the angular altitude of the object above the plane of the horizon. 85. The theodolite S in its simplest form, consists of a brass I Some of the best theodolites are adapted to measuring vertical as well as hwizonttJ aSgles to a single minute ; being fitted with vertical arch, lerel, telescopic sights, and rack-work motions. The prices of theodolites are from two to ibrty guineas. The circumferentor is an instrument for measuring horizontal angles, chiefly used in wood lands, and its price is from two to fire B Taut a. INACClBSIBIiB BBteSTS & DUCTANCES. 48( €irc^e of about a foot in diameter, having it« circumference divided into 360 degrees, and these subdivided into halves, quarters, or minutes; the index sCs turns About the centre C# and has fixed on it two sights s s; there are likewise fixed oo the cir^mierence two*sights » n ; this cirde i« fixed in {^ hori* zbntal position on three legs of a convenient height for making observations. The theodolite is used ^ ^ for measuring the angular distanciM of objects situated on the plane of the hori- zon j thus. Let A and B be two objects, place the instru- ment in such a position that one of them, as A, may be aeen through the fixied fights n and n by an eye atF. Turn the index 9 s about the centre C, until Che other ot^ect B appears trough the sights $ s to an eye situated at E ; then will the angle ACB^ which is meli«ared by the arc nr^ be the angular distance of the given objects A and^. 86. Thb mariner's compass ^ is an instrument used for find- ing the position or bearings of objects with respect to the meri- dian, and for determinii^ the counie of a ship : wliat principaHy requires explanation is the eard ; it is a round piece of stiff pasteboard, having its circumference divided into thirty-two c gnineas. The semicircle is a macb simpler and cheaper iustrument than the theodolite, and serves very well for measuring angles on the plane of tbe horizon where very great accuracy is not required. ^ The invention of the mariner's compass is usually afcribed to Flavio Gioia, an Italian, A.D. 1302; but it is stated by some authors that the Chinese had a knowledge of it as early as the year 1 1 30 beibre Christ. The price of this usefcrl instrument is from half- a- crown to twelve guineas. 4s< MANE TuieosraiiEniY. put is. eqml pMts, cdled poMb ; K Bteel wire, called (he needle, vrbMi hu beea rubbed with « kttd- ttoDe, it fixed acroM the und^ ude of the card from N to S, by which means (when the card is ezactl; balanced on id centre) the pMot N is directed to the north, and conse- quently the point S to the south, and each (rf the remaining pcnnts to its .reapective paaition in dw horizon j in the centre of the card URdemeath, b fixed a finely polished conical brass socket, aboat one third of an inch deep. The compass box is a basin of brass or wood, having a fine pointed steel needle fixed perpendicularly in its bottom : on the point of this, the above-meationed socket in the bottom <^ the, card being placed, the card is bidanced and turns freely as im- plied by the attractive force of the magnet. The box is sus- pended within a brats hoop or ring, by means oi two gimbdi placed on opposite sides, which serve as an axis, and admit &«e motion i and this hoop is in like manner suspended on the oppo- site sides of a square wooden box by gimbols, at 90" distance fixtm the former, a contrivance intended to secure the horizontil position of the inner box and card, wliatever may he the motion of the ship in which the compass is placed '. * Tboie wbo crou (oreita, deurti, and aDinbabited coantiiag, find thi> |D>tniiiient a nectunj compsDioa to direct themi tbej kerp tbe oompan alwBjt before tbtm, and Ibllaw tbe direction of tbat poiDt which indicate] tbe ■itnUion of tbe place tbe; wisb to arrive at. Tbe like metbod U emplojel id heerios a ibip, wbicb i( kept in •och « pmitioo, tbat the propoeed pm^ majp, of ill o«H accord, etand in a direction towanU tbe bead irf tbe ibip. Note,NbElDeB]UHrfiliy«Jl; tiJli E, itortknordMiuti HEhK. mrrlk^u* in iMrdt j N £, northeait, &c &c. Y^icb will be eaillj uaderahrad. NORTil^ Pts Degrees SOUTH 1 Nb£ Nb W 1 no 16' SbE Sb W Kne NNW 3 es 30 SSE SSW N£bN NWbN 3 33 45 SEbS SWbS NE NW 4 45 S£ SW NEbE NWbW 5 56 15 SEbE SWbW £N£ WN W 6 67 80 ESE WSW BbN WhN WEST 7 78 45 EbS WbS EAST 8 90 EAST WEST ^abtI^. inaccessible heights & distances. 4«r 87. A table shewing the degrees and minutes that every point of tlie compass makes with the meridian ^ Exjdawttion. In the preceding figure tbe line N S is called the 'mei^idia$iHne s the two first colamns of the table ex- tend from nxurth both ways to east and west, as tb« two last do from south ^ the two first points in the first and second columns make the same angle with the meridian line N S (ll« 15') reckoning frvm the north point, that the two first in the 5th and 6th columns do, reckoning from the south, and the like is CTidently true of the points in any horizontal line of the table. The angles made by the points in tbe first and second columns witii the meridian are therefore measured by the arc9 intercepted between them and the north point, viz. tbe first column, on the east side of north ; and the second on the W€9t : in like manner the angles made by the points in the 5th and 6th columns with the meridian are measured by the respective arc* intercepted between them and the touth point, those in the 5th column being on the east of south, and those in the sixth on the west: for example, N N E is 92<» 30' to the east ef north, N N W is the same distance W€9t of north ; SS E is the same distance east of smithy and S S W is tbe same distance west of south. In the third column each number denotes the distance from north or soUtb of the points agdinst which it stands ; and tbe numbers in the fourth column shew the degrees ai^d minutes of ihe arc intercepted between the north or souths and the points against which they stand. 88. The use of the above Table. When a question is proposed in which the conditions require that lines should be drawn in given positions with the meridian expressed in points of the compass, the construc- tion may be made with the greatest fiaicility, by means of this table 5 to eflfect which this is the KuLE. — 1. Describe a circle and draw the diameter NS for the meridian, N being the north point, S the south. 2. Take the degrees and minutes from the table which cor- respond with the points mentioned in the question, and mea- sure arcs from the meridian equal ta them. ^ The table is thus constructed : divide 360 (» the number of degrees in the circumference pf a circle) by 32 (=» the number of points in the compass,} and the quotient is ^ part of the circumference — 11<* 15', or 1 point of the compass ; this doubled is 23<* 30' for two points ; its triple is 3d<*45' for three ]H>ints, and so on. 4S8 fUNK TBiaO»rQMinEf . »iwIX 8. Ikiw liQei Uiraigli 4ie ecvtre |p t^ pmote Hups sured, and construd your figure by drawing its aides retpectiT^f parallel to these, and each of its proper length taken fipom a scale of equal parts. 4. If the position of one of the lines be required^ draw f line parallel to it through the centre of the circle^ measure tht angle this line malces with the meridian, then the point of the compass which stands opposite this measure will give the besyr* ings or position required '; and its length, taken in the eom* passes, and applied to a sqale of eqpal parts> will give it| measure. Examples. — 1. A man intends to travel from C to Z which lies N N W from C 6 miles, but he must arst call ^t D» whi4 lies N £ 3 miles, then at A N b W frt>m D 5 miles, and lastly at £', which is S Wfrom H 41- miles; at Hhowfar is he distant from Z, and what course must he travel to arrive there > Here I first draw cCZ through the point d, distant 2t® SO^ from N (answering to N N W) ; next I draw itb at 45^ distance from N (answering to N E) ; next 1 draw rn at II® 15' distance on the left of N (answering to N b W) } and since a8zsNbzat4&^, it is plain that ah will be the S W as well as the N£ line. I then take CDs3, draw DH parallel to rn and make it=5, whence I draw HK parallel to ab and make it= 4,, 1 then join KZ and find its measure to be 2^ miles nearly, and its bearings (shewn by the paraUel xv, the position of which is. measured by the arc Nv) ' Ttie pQtitiop, or heariji|pi ctf » Une nipy likewise he known by simply 4fAwiqg .a meridian from the g^Tep point, and measnrii^ the angle which that iiae mak^ witli it $ the d^reet cont^pfd in it .beiq| fo^nd in the table wiU shew the point of the compass required. PiRT IX. INACG^^lBLlft itfiiaairS k DISTANCES. 489 N Ifl^ fi"*, ^ilkt tt> )lbJ& f>^B> of 74^egraM to tbe eastward of north by east 2. B is 8 ihileB NW From C, and il 4 miles N from B-, requirad th« course uid distance frwn AtoCP An». €ourse S 3P4E. Distance 11 mites. 3. A ship sailed S E 12 leagues^ N N E 20 leagues, and NN W SO leagues $ required her distance from the point sailed from^ and hter course back ? 89. The perambulator », called also a pedometer, waywiser, and surveying wheely is an instrument for measuring large dis- tances on ground nearly level ; it consists of a wheel 8^ feet in circumference, which the noeasurer drives before him, by means of two handles, fixed at the end oi a hollow shaft, terminating in two cheeks to receive the wheel, and in which its axis turns. The wheel goes over one pole of ground in every two revolu- tions, and its motion is communicated by the intervention of various clock-work movements within the shaft, to a dial, fixed near the handles, the index of which points out the distance passed over. * The Gunter's chain "* is used to measure smaller distances tban those to which the perambulator is applied j its length is 66 feet=:22 yardsrs4 poles, and is divided into 100 links, each 7,92 inches in length. This is the most convenient instrument of any that has been contrived for measuring land, because 10 •*■ ttt! bearings «f two tAjectB from vackotber may be estimated otther in <fc^ee», or po^t ; degrees may be tarned into fiomts, or poiats into degve^ l»y referring to the table ; thtts, if an object bear 8«» 46' to the east of sooth* bytaming tethfefeblc I ftttd that the exact pof nit of bearing is SEbS; if it bear 25*»to the ^est of north, the bearing 'm ptim* n NNW«»a«'W; that !i,'j|«aO''Wt8tof NNW. Or the Teckonlng mf^bemadetotiie neareet ^rterpoiki, tfciis N14»4'W is N bW^W; S««»7<{-£ U SS£i£; ikVXe manner Nier4« 41' £ is N£ b£4 E, fto. fte. ■ Ttie prtce of Ifliis instrtiment Is Mit «v« to ten gnioeas. l^ie name hdm^&r is likewise appHed to «h instraifleiit of a watch tiae ^ the ipockct, tot Kscertidning distances, either walking or ridiag, and eoels from tfafiee to tfteen guftteas. Tbe ptra'ftrbiilator, CWmten^s «iate, and ta^es,wiU measofe iWth »uiB<irent txactn^s for most fmrfM^s whete <he gtomid is level, but where it is not, distances should be TofBd by trigonott^tikal ealevkttien. • theOnritfcr«sc1iain Will cdl* IHMn iws to foOTteen thlUiDgB, aooov^to^ to its strength, apdihe pie(le«ftidli ^ '430 PLANE TRIGONOMETRY. P«t IX, diains in lengthy aod oaeia breadlii^ (sslOOOOO squave links) make just an acre. 91. The measuring tapes ' are of cftxe, two^ thfee, or !b«r poles in length; they are applied to the same purposes as tbe chain, and, if kept dry, will measure with tolerable exactness. 92. The measuring rod may be of six> eight, or ten feet in length ; it is divided into single feet, which are subdivided inta halves and quarters, or into tenths of a foot, for the convenience of measuring small distances. 93. Station staves or prickets, are staves of about five or six feet in length, having a small flag fixed at one end, the other end being sharpened to a point for fixing in the ground; these staves are used in measuring, for marking stations, which are required to be seen and distinguished at a distance. 94. The arrows arc of wood or iron, pointed at one ehd, and their use is to stick in the ground as a mark, at the end of every chain or other measure. 95. Fboblems% Prob, 1. An observer at 113 feet distance from the foot of an obelisk, finds its angular altitude to be 40^ ; required its height, that of the observer's eye above the plane of the horizon beic^ 5 feet? p These tapes are sold at the sbops of the niathttniatica} instrament makert, and cost from five to twelve shUHngs> according to their length. The above instruoients^ at the prices we have mentioned^ will perhi^ be found too expensive for the student's pocket ; in that case his own ingeouitjE may supply him with all that is necessary for measuring vertical and horizon- tal angles and distances. A theodolite may be made with a circular piece of stiff pasteboard, gradnated and nailed (through its centre) on the top of a piece of mop^stick, the other end of the etiek being sharpened to a point for fixing it in the ground. A qoadraat likewise may be made of pasteboard,. ' in like manner graduated, and having a piece of lead, or a stone, hung frobi itt centre by a strnig. The chain or tapes may have their place supplied by a string previously measored, divided, and subdivided, according to the mind of the operator. The measuring rod may be made of any stick, of a proper length and thickness. The station staves may be made of sticks having one end pointed and the other split, for the purpose of holding a piece of white papcr^ and the arrows may be cut ou^ of any hedge. With apparaitus of this kind, I have frequently known altitudes and difitanff^ determined, with sufficient ezactii^ for any commoo purpose. PaatIX. inaccessible heights & distances. 431 Npte, In ftftdiag the height of ofatiects^ to the observed hngtit must be added, that of the obeerver*9 eye above the ylaoe of the horizon. Let AB be the obelisk, CB the dis- tance of the observer, and J?£ the height of his eye ; then JIE is the part re- quired to be found. In the tri- angle ACE, we have given C£=11S, the angle -rfC£=40%' consequently C.il£s:(90— 40ss) 60^, and the angle CEA a right angle; to find ^£. Now (Art. 67) CE: EA :: sin A : sin ACE, •.• EAss CE. Bin ACE " ^, 2 , and log. E^srlog. C£+log. sin ^C£— log. sin A 3s8.05SO784+9.a0S0675-9.884254O=:1.9768919> the natural Hittiber, corresponding to which is 94.8182s=il£, *.* ^£+££=a 94.81884- 5=:99.8182 feetss99 feet 9 inches iff4==the height lequired. Pro6. 2. The angular altitude of a spire, known to be 137 feet high^ is 51^ \ now supposing the height of the observer's eye to be 5-i- feet, how &r is^he distant from the foot of the spire ? l^oit. In questions of this kind, the height of the eye must be subtracted from the given height, previous to the operation. Here are given ^£sl37y £B=S^, •.• ^£=137^5^=s 131.5, AEiy a right angle, and angle ^JD£=:51^ ••• ang. DAE-zs: (900— 5P=) 390. . (Art. 67.) I>£ : E-df : : sin BAE : sin ADE •.• EA. sin BAE I>£s sin ADE 131.5 xsm 39^ , .._ , |^ r-zTT — > ••' IPB i>£=log n sm 51* ^ ^ ^^ 433 PLANS TRIGONOHKTRY. Fast IX. ISl.S+Iop. tin Sd-log. Oa 51"=«.ll»258+9.7»B«7lfl- 9.6906(n6=S.0Cr%95O .- i)£ so 106.487 feet=IOS §e^ fi inches -rfr- Prob. 3. Wanting to calculate the perpendicular height of & cliff, I took its angular altitude IS" 3<y, but after measuring 950 yards in a direct tine towards its base, I was unexpectedly Slopped by a river; here however T ag;ain took its altitude 69° SO'i required the height of the cliff, and my distance from the centre of its base P Let ^ be the first station, B the second, C the sum- mit of the cliff, and D its base; then ^6=950, the an- gle ^=ir 30', -A- angle /iBC= (1800 —690 3tf=) 110" 30' ■.■ ang^CB=(18O-12''3O' + ll0',30'=18O'-123'=) S7»i ■/ in the triang^ ABC we have Ote side AB and the three angles given, to find BC. Now (Art. 67.) AB : BC :: ua sin ACB ' sin .rf— log. sin ACB= (log 950+log. sin 12»30'— log. sin 57°=) 2.9777236+9.3353368-9.9235914=2.3894690, ■.■ BC =24S,17I; having ftinnd BC, there is given in the triangle . BCD the right angle BBC, the angle CBI>=69" Stf, the angle BCI>=(90*-69" 30'=) 80P3tf and the side BC=iM5.m,-.* (Art. 63.) BC : BD :: rad : sin BCD, ;■ BD= f^'^^" = 85.8608 yards. Also (Art. 63.) BC : CD :: rad : rin CBDj BC sin CUT} :■ CD= — '-—^ =249.645 yiuds. Prob. 4. Two persons, situated at jt and B, distant ^ miles, observed a bright spot in a thunder doud at the same instanlj its altitude at A was 46°, and at B 63° 30'; required its perpe»- dicular hei^t Irom the earth ? BtsI. Angle .<iCB={180"— 46«+6S'' SB's) 70* ,30", Aen (Art. 67.) AB -.BC:: sia ACB : sin BAC, w SC=^^^^^ '< 1 P^HT K. . INACCESSIKLB HSfOHTS ft DISTANCES. 433 = 2.1361^ miles. Wherefore in the, rj|ght apgled triapgle BCD, BC : CD : : rad : sin CBD (Art. 63), / CD^ ^^' ^'^^^^ ^ /-03/f/ rad 1.9117 mile=th6 height required. Prob. 5. Two towns, A and B, are invisible and inaccessible to each other, by reason of an impassible mountain, situated between them; but both of them are visible and accessible from the point C, viz. A bears N E from C distance 3 miles, and B bears N b W from C distance B-^ miles 3 required the bearings and distance of A and B from each other ? First, Since CJ lies N E, or 45® on the east of the meridian, and CB lies N b W or 11^ 15' on the west, V angle C= (45°+ 11° 15'=) 56o 15'; •.• (Art. 72.) CB-\-CA : CB— tA : : tan — - — : tan — - — : or 8.95 : 2.25 : : tan 61° 52'i : tan 27** V 57'' 3 then (Art. 69.) angle 4= (6F 52' 30''+ 27^ 1' 57''=) 880 54' 27", and angle B={61^ 52' 3(/'- 27° r57"=) 34® 50' 33"; next, (Art. 67.) C^ ; ^B : : sin B : sin C, ... ^B-:£:!:i!^= 4.36606 miles, sm B Lastly, through the centre C draw ab parallel to AB, and measure the circumference Net, and it will be found to contain 46® 6', which, by refi^rring to the table (Art. 87.)» will be found to answer to the N W point nearly; that is, B bears from A N W 1® 6' W distance (4.36606 miles=) 4 miles 3 furlongs nearly. VOL. 1|. F f 4ft4 FLAKE TBIGONOHEniT. Part IX. Prob. C. A general wriring wHh his army on the b2nk of a river is deairoiu irf crawiDg it, but there are two of the enetaft fortresses, jI and B, on the opposite shore, and he wishes to know their bearings and distance from each other; for this pur- pose two stations C and D are chosen close to the river side, C being directly east, from D at-i mile disljmce ; at C the angles are as follow, viz. ACB=6eP, BCD=3Vi at D the angles are jlDB=e2', jtDC=Si*. Now suppose be crosses directly froni the point D, required the bearings and distance of ^ and 5 from each, other i the width of the rirer at the point of croesiDB;, Part IX. INACCESSIfiLE HEIGHTS & DISTANCES. 435 and thp distance of the point wbere he proposes to land from AarndB} First In the triangle DACy there are given DC=^ mile=r .75, the angle ^DC=64% DC^= (32'*+ 68°=) 100, and DJC = (180—164=) 16*^5 to find DA. By Art. 67. DC : DA : ; ' ^.r. . T^ry. « ^ DCxsiu DCA .75 X sitt 100° »in DAC : sin DCA, •/ DA^ _-__= r— r^^— = sin DAC sm 16° 3.67963 miles. Secondly, In the triangle BDC, there are given DCzsjB, ^JDC=(62°+64°=) ,126°, i>C£=32°, wad I>J5C= ( 180°— 126°-f 32°=) 22°, to find BD. By Art. 67. DC.BD:: sin T^Di- • r^^D i>n I>C X sin DC5 .75 X sin 32° DBC : sm DCS, •.• 5D= : — ~ -^, = — . ^^^ — = sinDJ^C 8in.22° 1.06095 miles. Thirdly. In the triangle BDA there are given DA=s 2.67963^ j!?D= 1.06095, and the included angle AD£=:eQ'* ; to find the angles DBA, BAD, and the side BA. Now r 180°— 62° = 59°=half the sum of the angles DBA, BAD at the base} also ^D+I^^— 2. 67963 +1.0609$ =s 3. 74058= sum of the sidfes, and ^D— D J? =2. 67963— 1.06093 =1.61 868as diff. of the sides. But (Art. 72.) AD^DB : AD-^BD : : tan DBA+BAD DBA-^BAD , -=^-^^ : tan -, that is, 3.74058 : 1.61868 : : 2 2 ,^ 1.6 i 868 X tan 59° , «„« .^ ^„ ,. ,^ .^ ^.«. tan 59° : — - — =tan 35<» 42^ 5"=half the difference 3.74058 of the angles DBA, BAD at the base. • ^A t 150 ^ /59°+35° 42' 5''= 94° 42' 5'' =the angle DBA. •/ (Art. 09.) 1 5^p_35. 42, 5//-230 17/ 55'/=,the angle BAD. Also (Art. 67) BD : B^ : : sin BAD : sin B£>^, •.• jB^= .BD X sin BD^ 1 .06095 x sin 62° ^ „ •^^ ., :^ — K-7T^ — = . ^00 ,^/ .>// =^'36842 miles. 8in:J9.iJ> sin 23° 17' 55'" Fourt^y. In the triangle DBE there are given the angle E a right angle D-8£=(180°— i)B^=180°-94° 42 5''=) 85« IT 55", the angle BD£=(9Q°— DB£=90°-85° I7' 55"=) 4° 42' 5", and the side BD= 1.06095 -, to find the sides BE and DE. By Art, 63. DB : BJE : : rad : sin BDE, '.' BE = F f 2 43« PLANE TRIGONOMETRY. Part K. DBx&lnBDE 1 .06095 x sin 4M^' 5'' ^«^^,« ., ' ; = ; = .086958 mile = rad. rad. somewhat more than 150 yards. ^ I)B X sin DBE Also DB : DE :: rad : sin DBE, •.' D£= —z rad. 1.06095 X sin 85^ 17' 55 , . ^^^ „ = — 1 .05738 Diile. rad. Lastly. Since the line CD lies directly east sind west, any line CN drawn perpendicular to it wiH represent the meridian^ and the acute angle BNC, which AB makes with CN, will be the bearings of B from A ; this angle may be very readily determined in the present instance ; for since the two opposite angles DCN and DEN of the quadrilateral DENC are two right angles, the two remaining angles EDC-^ENCsz^ right angles (cor. 1* 3«. \.)', but EZ>C=(4« 4«' 5''4-«2<' + 64o=5) 130^ 42' 5'', v EiVC=(l800-J5Z>C=rlSO°-130'* 42' 5''=) 49^ 17' 55'^ which in the table (Art. 87.) answers to S W 4*» 17' 55'' W or S W i W nearly j for the bearings of B from A, Prob. 7. Required the perpendicular height of the spire of a church, the angular altitude of which is 40^ ; the observer being 187 feet distant, and his-eye 54^ feet from the ground ? Answer K0AS7feet, 6. The angular altitude of an observatory is 53**, its perpen- dicular height 129 feet, and the height of the eye 5 feet ; re- quired the distance of the observer? Ans. 93.4407 /eef. 9. A ladder 30 feet long reaches 23 feet up a bdiiding ; re- quired the angle of inclination at the foot, and its' distance from the wall? Ans. inclination 50> 3' SO''; distance 19.261 3 /ciif. 10. A shore 1 1 feet long, in order to support a wall, is placed so that the angle at bottom is double the angle at tc^, how high tip the v^all does it reach, atid how fhr distant from the wall is its foot ? Ans. heigHt 9.52628 feet ; distance %rf^eL 11. Required the altitude of the sun, when the 'length of a iDan*8 shadow is double its height, and likewise when it is^ODly half its height? Ans. 26° 34' 5'' in the first case, and 63^ 25' 55''' in the second. 12. A maypole being broken by a sudden gust of \Vind, the Upper par* (which still adhered by some splinters to the stumps inade with the ground at 15 feet distance from the stunip, an PARxlXi INACCESSIBLE HEIGHTS & DISTANCES. 437 AOgle of 7^ 30'; required the height oi the maypole and the leqgth of each of the pieces ? Ans. stump 29.2072 feet, upper €nd 80,46^6 feet, whole length ^9-6696 feet. 13. A ship having sailed 234 miles between the south and WjBst^ finds herself 96 miles distant from the meridian she sailed from i required her course and difference of latitude ^ ? Ans. course SSW 2* 13' 15'^ west; diff, of latitude 213.401 miles 9outh. . 14. There are three towns A, B, and C; from 5 to C the distance is 7.625 miles 3 at B the towns A and C subtend an angle of 51° ^5', and at C the towns A and B make an angle of 37° 21^5 required the distance from A to each of the other two ? Ans. from A to B 4.6275 miles, from A to C 5.9482$ miles. *' 15. Within sight of my house there is a church and a mil], ^e former is distant 2.875 miles^ the latter 4.24625 miles, and they subtend ^n angle of 47° 23' 3 required the distance from the mill to the church ? Ans, 3.125 miles. 16. A &rmer has a triangular field, the sides of which are as follow, viz. AB:^7S0 yards, -4C=690, and JBC=8505 he is desirous of dividing it into two pails by a l^dge from A, per- pendicular to BC; required its length, and likewise whereabouts it will meet the hedge BC ? Ans. length 585.31 yards; distance from C 365.2942 yards. 17. "^ A man travels from ^ to jB 5^ miles, then bending a little to the right hand of the direct road, he arrives at C distant from B 3 miles -, from C both A and B are visible under an angle of 25»4- -, what is his distance from home by the shortest cut ? Ans. 7.796 miles. 18. A man having ti-avelled from ^ to ^ 5-4- miles, attempts p The angle wbich the directum m sk^ soils nm makes with the meridiaD, n called her course, whence in the present case, constract a right angled triangle, the bypothenuse of wbich is=2d4, this will be her distance, the ba8^»S6 will be her departure, and the perpendicular will be her difference of latitude ; and the same in all cases of plain sailing. 4 Problems similar to this and the following one, are given by Ludlam, to shew how the apparent ambi|^ity of a problem is sometimes corrected by the wording ; particular attention mast be paid to ' bending a little to the right" in prob. n . 2in\ * attempt* to return* in prob. 18. and the solution will be attended with no difficulty. Ff3 k . 438 PLANE TRIGONOMETRY. Pakt IX, to return, but a thick fog coming on, he roistakefii bis way, and takes a road which tends a little to the right hand of bis proposed rout 5 arriving at C, 3 miles from B, he discovers his mistake, and the fug clearing up, he sees both A and B under an angle of 154% ) how far is he distant from home ? Ans. 2.38 miles. 19. In order to measure the breadth of a harbour's mouth, a station was taken at its inner extremity, where the angle made by the two projecting points which form the harbour was ob« served, viz. 33® 40' -, the line bisecting this angle being pro- duced 1900 yards backward and another observation made, the fore-mentioned points were found to subtend an angle of 17* SO'; required the breadth of the said entrance, and how for the harbour extends inltoid? Ans. breadth 751.904 yards, perp. extent inland 124*2.6 j^ards. 'SO. Three trees are planted in such a manner that the angle at A is double the angle at B, and the angle at B double that at C, and a line of 234 yards wiU just reach round them ; required their respective distances ? ' Ans. ABss46,346B yards, ACsz 83.5135 yards, BC= 104.14 yards, 21. in order to determine the distance between two inaccessi- ble batteries A and B, two stations X and Z were chosen, distant from each other 4541.8 yards ; at AT the following angles were taken, viz. AXDszW 34'-, BXZ=i46» 16' 5 at Z the angles were XZA=^96<> 44', XZB:szmo 23'; required the distance of the batteries from each other? Ans. 3373.1 yards. 22. Two ships leave a port together^ A steers S W; 6SE, and sails twice as fast as A: at the end of they arrive at ports 55S miles apart ; now, supposing to have blown equally from one point during the wh^Kflime; at what rate per hour did the ships run ? ' Ans. A 3.l^k miles per hour, B 6.243. • » If *5stbc least angle, viz C; then 2x=» B, and As^A, whence 7*=' 180, and jp:fe-"-5^^= 25« 42'^. Assume either of the sides of any convenient length, and find (by Art. ^7.) the two remaining sides ; then say, as the sam of these three sides : to the given snm 234 : : either of the sides : the corresponding side of the proposed triangle. * From any point draw two indefinite lines in the proposed directions, from the table ( \rt. 87.) Assame any length in the S W line for A*% distance, and take double that length in the other line for ^s ; join these points by a straight line, and fad its length (Art. 72y 69, and 67.) ; then say^ as this line : P&BTIX. INACgfiSSIBLE HEIGHTS & DISTANCES. 430 :,-9t. From one of the aoglea of a rsctangular met .are two straight foot paths, ooe leading to the oppc and the other to a stile 1 10 jaiila distant from it } thi with the two patiis, forms a triangle, of which the as the numbers 9, 3, and 10 j what sum will pay fbrth making;, and carting of tiie said meadow at 37«. Sd. J.n$. 7L Si. ^d. 24. There are three seaport towns J, B, xdA C £ S £, atid Cj £ by N from J : a telegraph is erected, for the purpose of speedy communication with the metrtqxdis, at 5 miles distance from each of the towns, and in the line 4Ci required the distance of B from J and C, and its bearings from the lel^raph ? Ant. from B to A 8 J147 mifc», Jirom B to C 5.55S7 milet} and B heart S £ b S/rom the lekgr^ph. 35. Aflag-staffisplacedon acaetlewalll63 feet long, in sm^ A situation ib^t a line of 100 feet in length will reach fh>m its 4op to one end of the wall, and a line of 89 feet from iu top to the other j required the height of the flag-staff, and its dis- tance from the extremities of the wall } Ant. height 47.7344 jtel; dittance from une extremity 87.8773 feet, front the oilier 75. 1237 /ee*. - 36. la the hedge of ^a drctilar inclosure 500 yards in diame- ter three tixes A, B, and C vere planted 'in such a roauter, that if straight, lines be drawn from each to the other two, the Angle at A will be double the angle at B, and the angle at C douUe of A >ad B together j required the distance between <»ery two of the ti«es '? Am. from A to B 433.013 y(ird*,_/rMn BtoC 321.394 yardt, and from A to C 171 01 yarrff. Jti atamed ilistaoce : ; SS8 : jft real dittantc ; wheoce alu B't diMance wiU be fuaod ; uul the iJislBncc dirided by the nainbei of boun, will give tbc rate o( lailiog per hanr. 9 ' To find tbc aagles, sf tbe Dotc an prob. SO. Ta find the t\6.tt ; Firit,, nUh the ruliuiSSO d«Kribe a cirels, and frum it cut off > setimeat canUiBing SB ai^ equid to the grealett angls of tbe proposod trimf le (34. 3.), draw ■troight lines rrnio the extremities of thU chord to the ceotie, and an Uoyelei triangle will he formed by the)e three lines, of Khich the vertical an^e {M tl^ centre) vKI lie duubk the lupplemeM of (he laid greatut angle (SO and 84. a.), and the three angles of this isosceles triangle will be known (39. I.). Secosily, find tbe b ise (Art. 67.) which will be tbe greatest side of the pro- pmcd triaugle (19. 1 ), whence the two remaining >idei irill likewise be found by Art. 67. Ff4 440 PLANB TBICK>NOMBTRT. Paut IX. 27. An £ogl]^ sloCip of war having orders to survey an enemy's port, placed two boats A and B at 1100 &thoms dis^ tance apart^ A being directly east from B : at the inner ex* tremity of the harbour there is a spire visible from the boats^ likewise a castle on one point of the entrance^ and a light-house on the others at J the castle bore SSW, the spire S W by S, and the light-house W S W. At J9 the castle bore S B, the spire south> and the light-house S by W ^ required the kmgth and breadth of the harbour ? Ans. length from middle of entrance loss futhoms; breadth of Entrance 9iO.S9 fathoms. 2S. On the c^posite sides of an impaasil^k wood, two citisB A and B are situated ^ C is a town visible from A and B, dis- tant from the former 3 miles, and from the latter 2, and they make at C an angle of 'iSP 5 now, it is desirable to cut a passage lh>m A to B, and an engineer undertakes to make one, 19 feet wide, at 7«- 6<i. per square yard; the inhabitants of A agree te furnish 4 of the expense, which th^ can accomplish, by ev^ 7 persons paying 31 shillings 5 those of B can make up the remainder, by every six persons subscribing 33 shillings ; re- quired the number of inhabitants in A and B ? Ans. A 43626, B 8839, to the nearest unit. S9^ An isosceles triangle has each of the angles at the base double that at the vertex ; now, if the vertical ai^le be bisected, and either of the angles at the base trisected, the segment of the trisecting line, intercepted between the opposite side and the bisecting line, will be three inches ; required the sides of the triangle? Ans. each of the equal &ides 13.8314 inches; the base 8.35371 inches. 30. In a circle, whose radius is 5, a triangle is inscribed, and the perpendiculars from the centre of the circle to the sides of the triangle are as 1, 3, an^4 -, required the sides and angles of the triangle ? 31. The altitude of a balloon as seen from A was 47°, and its bearings SE; from B, which is ^4- miles south of A, it bore NE b N'j required the perpendicular height of the balloon, and its distance from B ? / <_ J PART X. THE CONIC SECTIONS. HISTORICAL INTRODUCTION. If a solid be cut into two parts by a plane passing through it, the surface oiade jn the solid by the cutti^og plane, is called A. SECTION. If a fixed point be takep above a plape, and one of fhe extremities of a atraigbt line parsing through it b^ made to describe a circle <>n the plane, then will the seg* ments of this line by their revolution, describe two solids (one on each side of the fixed poipt) which are called OPPOSITB CONES '• A plaDe may be mad(& to cat a cone five ways;^rs/t, by passin g through the vertex and the base ; secondly, by passing through the cone parallel to the base ; thirdly, by passing through it parallel to its sides; fourthly^ by passing through the side of the cone and the base, so as likewise to cut the opposite cone; and^thly, so as to cut its opposite sides in unequal angles *^, or in a posi- tion not parallel to the base. ~ • - — • ■ — —*: - . * If the segment of the geDerating line between the fixed point and the base be o!P>^>givea length, the cone described by ita motion will be A right COKE, ha^iog. Hs m» peipendicuUr to the bate ; but if the Ungth of i\yt segment be variable in any given ratio, so as to become in one revolution a fnaximum and a minimum, the Cone produced will be an oblique coke, and Hs axis will make an oblique angle with the base. ** Of course a right oone is hare understood ; for if the cone be oblique, the base, which is a circle, will <ut the opposite sides in unequal anglrs, and the segment made by cutting them in eqtial angle* will evidently be an ellipse. 44« qONIC SECTIONS. Pabt X. If the plane pass through the vertex and the base, the section is a triangle ; if it be parallel to the base, the section is a circle ; if- parallel to the side of the cone, the section is called a pababola; if the plane pass through the side and cut the opposite cone, the section is called an hyperbola; and if it cut the opposite sides of the cone at unequal angles, the section h called AN ELLIPSE. The triangle and circle pertain to common elementary Geometry, and are treated of in the Elements of Euclid; the parabola, the ellipse, and the hyperbola, are the three figures which are denominated the conic sections. There are three ways in which these curves may be conceived to arise, from each of which their properties may be satisfactorily determined ;^r9f, by the section of a cone by a plane, as above described, which is the genuine method of the ancients ; secondly y by algebraic equations, wherein their chief properties are exhibited, and frooi whence their other properties are easily deduced, accord- ing to the methods of Fermat, Des Cartes, Roberval, Schooten, Sir Isaac Newton, and others of the moderns; thirdly y these curves may be described on a plane by local motion, and their properties determined as in other plane figures from their definition, and the principles of their construction. This method is employed in the following pages. » _ W H E N, or f rom whom the ancient Greek geometricians first acquired a knowledge of the nature aqd properties of the cone and its sections, we are not fully informed, al* though there is every reason to suppose that the discovery owes its origin to that inventive genius, and indefatigable application to science, which distinguished that learned people above all the other nations of antiquity. Some PartX. ^ INTRODUCTION. 443 of the most remarkable properties of these curves were in all probability known to the Greeks as early as the fifth century before Christ, as the study of them appears to have been cultivated (perhaps not as a new subject) in the time of Plato, A. C. SQO. We are indeed told, that until his time the conic sections were not introduced into Geometry, and to him the honour of incorporating them with that science is usually ascribed. We have nothing remaining of his expressly on the subject, the early history of which, in common with that of almost every other branch of science, is involved in impene- trable obscurity. The first writer on this branch of Geometry, of whom we have any certain account, was Aristaeus, the disciple and friend of Plato, A. C. 380. He wrote, a treatise con- sisting of five books, on the Conic Sections ; but unfor- tunately this work, which is said to have been much valued by the ancients, has not descended to us. Me- nechmus, by means of the intersections of these curves (which appears to have been the earliest instance of the kind) shewed the method of finding two mean proper- tionals, and thence the duplication of the cube; others applied the same theory, with equal success, to the tri- section of an angle; these curious and difficult problems were attempted' by almost every geometrician of this period, but the solution (as we have remarked in another place) has never yet been effected by pure elementary Geometry. Archytas, Eudoxus, Philolaus, Denostratus, and many others, chiefly of the Platonic school, pene- trated deeply into this branch, and carried it to an amazing extent; succeeding geometers enriched it by the addition of several oiher Curves as the cycloid, cissoid, couchoicl, quadratrix, spiral, Seethe whole form- ing a branch of science justly considered by the ancients 444 CONIC SECTIONg. Part X. AS possessing a more elevated nature ihan- common Geo- metry, and on this account they distinguished it by the name of TH£ moHBR or sublime geometry. Euclid of Alexandria^ the celebrated author of the BlementSji A. C. 280; wrote four books on the Conic Sections, as we learn from Pappus and Proclus ; but the work has not descended to modern times. Archimedes was profoundly skilled in every part of science, es- pecially Geometry, which he valued above every othet pursuit ; it appears that he wrote a work which is lost^ expressly on the subject we are considering, and his writings which remain respecting spiral lines, conoids, and spheroids, the quadrature of the parabola, &c. are sufficient proofs thai he was deeply skilled in the theory of the Conic Sections. In his tract on the parabola he has proved by two ingenious methods, that the area of the parabola is two-thirds that of its circumscribing rectangle ; which is said to be the earliest instance on record of the absolute and rigorous quadrature of a space included between right lines and a curve. But the most perfect work of the kind among the ancients is a trfsatise originally consis-ting of eight books by ApoUonius Per- gaeus of the Alexandrian School, A. C. (230. The first four only of these, have descended to us in their original Greek, the fifth, sixth, and seventh, in an Arabic version ; the eighth has not been found, but Dr. Halley has sup- plied an eighth book in his edition, printed at Oxford, in 1710. This excellent treatise is the most ancient work in our possession, on the subject; it supplied a model for the earliest writers among the moderns, and still maintains' its classical authority : the improvements on the system of ApoUonius by modern geometricians are comparatively few, except such as depend on the application of Algebra Part X. INTRODUGtION- 445 and the Newtonian Analysis. Hitherto the ancients had admitted the right cone only (of which the axis is per- pendicular to the base) into their Geometry ; they sup- posed all the three sections to be made by a plane cutting the cone at right angles to its side. According to this' method, if the cone be right angled (dcf. 18. 11.), the section will be a parabola; if acute angled, the section will be an ellipse; and if obtuse angled, an hyperbola; hence they named the parabola. The section of a right angled cone; the ellipse, The section of an acute angled cone; and the hyperbola. The section of an obtuse angled cone. But Apollonius first shewed that the three sections depend only on the diiSerent inclinations of the cutting plane, and may all be obtained from the same coiie, whether it be right or oblique, and whether the angle of its vertex be right, acute, or obtuse. Pappus -of Alex* andria, who flourished in the fourth century after Christ, wrote valuable lemmata and observatrons on the writings of Apollonius, particularly on the conies, which ^re to be found in the seventh book of his Mathematical Collet'^ tiotts: and Eutocius, who lived about a ocndtury later^ composed an elaborate commentary on sevitm\ of the propositions. In I J£e John Werner published, at Nofreitiberg, 0ome tracts on the subject; and drboiit thresame time Frtmcia-* cus Maurolycns, Abbotof St. Maria del Porta^ id Sicily/ published a treatise on the Conic Sections^ which has been highly spoken of by somre oSf oor be^t geometers for its perspicuity and eleganoe. The applicacioA of Algebra to Geometry, first generally intrbdnced by Vieta^ and afterwards improved and extended by Dest^Cartesy Fermat, Torricellius, and others, furnished means for the further developement of the nature and properties of Curves. The indivisibles of Roberval and Cavalerius; 446 CONIC SECTIONS. Part X, the AriUmitic of L^iiet, by Dr. Wallis; die Theory of Evoiuies, by Huygens; the Method of Tangents, by Dr. Barrow, &c. were discoveries which supplied additional means for extending the theory or facilitating the several applications of the doctrine ; bat that which rendered the most complete and essential service to this department of science, was the discovery of the method of Flaxions by Sir Isaac Newton, which took place about the year \66S. The principal modern writers on the Conic Sections are, Mydorgtus, Trevigar, Gregory St. Vincent, De Witte, De la Hire, De 1' Hopilal, Dr. Wallis, Milne,. Dr. Simson, Emerson, Muller, Steel, Jack, Dr. Robertson, &c* The ProperticM of the Conic Sections, by Williain Jones, Esq, F. R S. published by Mr. John Robertson, in 1774, is a tract in which is coflnprised a very great number of properties deduced in a most compendioos and general manner, within the narrow compass of 24 pages. Dr. Hamilton's Conic Sections is a very elegant and ample work ; Dr. Hutton's treatise on the subject will be found easy and useful. The introductory tracts of the Rev. Messrs. Vince and Peacock are the shortest and plainest elementary pieces which have been put into the hands of students ; on the plun of these (especiaUj the latter) the following compendium was drawn up, in wbich it is hoped there wiU be found some improve* ments. A coarse of Lectures on the Conic Sections has lately been published by the Rev. Mr. Bridge, of the East India College, I have not seen the work, and therefore cannot speak of it, but the talents of the author are well known. ' PaxtX. THE PARABOLA. 447 I THE PARABOLA. BBFINITIONS. strai^t moving parallel to itself at right angles to xy ; and if another straight line FP revolve about F, so that FP be always equal to MP J the point P will trace out the curve DVPb, which is called A PABABOLA. 2. The straight line xy is called the dirsctbix^ and the poiAt Fthbvocus. 3. If through the focus F, a straight line BZ be drawn per- pendicular to the directrix xy, cutting the parabola in V, VZ is called THE AXIS of the parabola, and V, the vertex. U^ CONIC SECnOlSB; PIky X.: Car. Hence, because jFP is alwayssPJIf (Art. 1,)^ when P by its motion arrives at V, FP becomes FF, and PM beccmies VH, \'Fr=zVH. 4. A straight line drawn through the focus F, perpen^cular to the axis VZ, and meetings the cunre both ways, is called THE LATUS RECTUM, Or PRINCIPAL PARAMETER. ThU8 DB is tfie latus rectum. In some of the following articler^ the latus rectum is denoted by the letter L, 5. Any straight line perpendicular to the axis TZ, meeting the curve, is called an ordinate to the axis 3 dnd the part of the 2LXh intercepted between the vertex Fand any ordinate, is called the abscissa. Thtis NP is an ordinate to the axis, and NV its corresponding abscissa. 6. A straight line meeting the curve in any point, and which being produced does not cut it» is called a tangent to the parabola at that point. Thus FT is a tangent at the point P. 7* A tangent drawn from the eixtremity of the latus rectum^ is called the focal tan«ent» Thus DHis the focal tangent. 8. If an ordinate and a tangent be drawn from the same point in the curve^ that part of the axis produced^ which is intercepted between their extremities^ is called the .sub-tan- gent. Thus P being any point from whence the tangent FT and the ordinate FN are drawn, NT is the sub-tangent to the point P. 9. A straight line drawn perpendicular to the tangent from the point of contact^ and meeting the axis^ is called th& NORMAL. Thus PG is the normal to the point P. 10. If a normal and an ordinate be drawn to the same point in the curve^ that part of the axis intercepted between them, is called THE SUB-NORMAL. Thus NG is the sub-normal to the point P. ^ 11. A straight linei drawn from any point in the curve, parallel to the axis> is called a diameter to that point -, and the point in which iit meets the curve, is called the vertex to THAT DIAMETER. Thus PX is a diameter to the point P, mid P is its vertex. 12. A straight line drawn through the focus F, parallel to the tangent at any point, and terminated both ways by the curve, is called THE PARAMETER TO THE DIAMETER of which that point is the vertex. Tlius db b the parameter to the diameter PX. Paw X; TB£ PAVABOLA, 44^ 13. A atraigfat line ^brawn from any diamet^> parallel to a tangent at its vertex, and meeting the enrre, U called an OBOiNATB tff that diai^ter. Thus vn U an ordinate to ike diamttir PX, PROPERTIBS OF THE PARABOLA \ 14. The straight line FP, drawn from the focus F, to any point P in the curve, is equal to tbe sum of the s^noents FF and FiVof the axis inteit:epted between the vertex and the {bcus» and between the vertex and the ordinate -, that is, JRP=s FJV+ FR For FPz=iPM (Art. 1.) =:HN (34. 1.) sFiyT+FHa (cor. Art. 3.) VN^ VF. Q. E. D. Cor. 1. Hence, when ^ cdneides with 3, N will coincide >«^ith F, Fi\r will become VF, and FP wiU become FB; -.' FB^ • ^^F, and D£^4FF, or the latus rectum is equal to four times ths distance of the focus from the vertex. Cor. 2. Hence FP—Fi^= FSsshalf the latus rectum, for FP - (FF+FiV=) %VF-^FNi V FP-^FNzsL^VF^^FB. 15. The straight line PT, which bisects the angle FPM, is a ttn^eiit to the parabola at P. See the following figure. For if not, let it cut the curve in P and p, join Fp, FM, pM; draw jwit perpendicular to HM, and join FM cutting PT m Y. Then in the triangles FPY, MPY, tPz=iMP (Art. 1.), '^Fis oomaion, and the angle FPFaaHPF (by hypothecs), •/ It will be proper to iafona the student before he begiat to study the Cooi« S«ctioQs, that he ought to be thoroughly nuutor of the first six books of Euclid^ ^d to know tomething of the elerenth and twelfth } the doctrine of propor- tion, as delivered in part 4. paf • 49 U 8f of this vofmt tntMt liktwise be ^ell understood, as its apflicatton cootinnally occurs i« tha foUowimg yaigea. 'OL, H. G g t> 460 CONIC SBCTIOI®. Pam X. FKss JIfK Mid the angle jyp= If FP (4. 1.) */ io the triaDgles FVp, MYp, the two sides FV, YpTs^MY, Yp, and the included angles fTpsMFp, -.* Fjl>=spM(4. 1)5 but fp=pm (Art. 1.), *.' pM^pm: '.* the angle pmM^ pMm (5. l.)> hut pmM b a right angle (Art. 1.), '.* pMm is also a right angle, which is impossible (17* l.)? \' PT does not cut the parabola, consequently it is a tangent (Art. 6.) Q. £. D. 16. The tangent FF at the vertex F, is perpendicular to the as^s FZ. For since the tsuigent PTcuts Fift at right angles in what- ever point of the curve P be taken (Art. 15.), •,• when P coin- cides with F, FP will coincide with FF, Pilf with Fft and FM with FH; •/ the tangent FY is perpendicular to (FJIf, that is, to)JFfl: Q,E.D. Cor, I. Hence, because TPand ilfP are parallel FTP^TPM (29. l.)=PPr (Art. 16.), / FT:izFP. Cor. 2. Hence FM, FY, and PT intersect each other in the point Y. For fy=F^,and (cor. Art. 3.) FF^^FH, / (9.6.) FY is parallel to HM^ and consequently perpendicular to TZ; ': FY is a tangent at F. 17. The focal tangent DH, the dir^trix xy, and the axis TZ, intersect each other in the point H. (See the figure to Art. 3.) For FC=FH (Art. 14.), •.• by the preceding corollary, the tangent meets the axis at the point H, where the axis and directrix intersect. Q. E. D. 18. Jf »r be an ordinate to the diameter PJST cutting FP in r, (see the figure to Art. 3.) Pr^Pv : for Prv^^rPT (29. l.)aa TPM (Art. 15.)= P»r (39. 1.) •.• Pr::>>.Pv (6. 1.). Q. E. J>. 19. The straight line PFis a mean proportional between FP and FF. See the figure to Art 16. For since FYT is a right angle (Art 16.) and YF perpea- dicular to FT, l^F: FY :: FY : FF (cor. 8. 6.), but rP=PP (cor. Art. 16.), .' FP : FY :: FY : FF. Q. E. D. Fjlkt X, PARABOLA. «1 Cor. Hence FP :FF::FI^:f¥^ (coir. 1, da 6.) > consequently FY^^FP.FF{16. 6.), and 4FF»«b4FF.FP; but 4l^=:the latus rectum (Art. 14. cor.) whidi beiog denoted by L, we have 4FF* 20. The line fP varies as FY^. For, let P and p be two points in the curve, from whence the tangents PT, pt, are drawn, and let FY and Fy Be perpendiciilar to the tangents re- spectively. Then, because FY^ssa TP,FV, and Fi/^—Fp.FV (by the pre- ceding cor.) ••• FY^xFy^:-, (FP.FF: fy.Fr : : by 1. 6.) FT : Fp^, v FP ec FT*. Q. E.D. iVWf. Tbe figure tothls Art. is inaccinately n cot; ^^moit be understood as a straig^ht lino at right angles to TZ, 21. If PP be produced through F and njeet the curve again in p, then will 4PP. JFp =»I.PP-hl^. For FP^FB—PM^ FH=:NH'^FH=FN. And JFB— fp=JFH— pw=Pff— H»==jRi, / FP-FB : FJ? -l^E) :: Py:P/* :: (4.6.) FPxFp, / (16.6.) PP.Fp -FJ.i^= FP.FB - FP.Fp; or 2FP.Fp =FP.Fg + FB.Fp^FBFP-itPp \' (since 2FB ^L by A rt.- 4.) 4FPFp=L.FP+f3[>. Cor. Hence, if 4a=: L, A'= FP, and j?2= Fp, the last expression will become 4J&=4a.A'+*, or J&=a-X'+ax, •.• — ^ — +-v^. a X .A 22. If c be the co-sine of the angle FFP to radius 1, then 2FF 1 — c > \ ' For -PP»fW+FF(Art.l4.)=iW^+FiV^+FF==2rFiFy og2 4» CONIC 88CI1QHB. Pa&tX BQt±FN : IvP : : (sia FTN: miam i : cos PFN i niditiB : : oos VFP the 8upp. of PiV: xaditti : : )+€ : 1 b^r Art. 63. |wrt 0. ••• (l«. «.) ±FN:=^c.FF ; .• fF^jVJS^ VF^^FF^FV (Axt. 14.) =) SFF+cfP,- •• (fP— c.FP> or) 1— c.PP=:9FF, or «P= . Q. E. D. 1— c ' - 23. The sub-tangent NT=:2VN. See the figure to ^rf . 20. Let rr be a tangent at F meeting P Tin T, then FF being perpendicular to PT (Art. 15.), an4 FP=Fr (cor. Art. 16.); also FF common, to the two triangles FPK, FTY, theae trian- gles are similar and equal (47 and 4. 1), •.• PY=yT. But F^ is perpendicular to the axis VZ (Art. 16.), •/ it is parallel to the ordinate NP ; \' PY : YT : : ^iNT : VT (2. 6.) ; byt PY=rT/ VN= FT (prop. -4.5.) -, '.• Ae sub-tangent Ntr=zivN, Q. ]E,D: 24. If P»r be parallel to the tangent PT, and vM perpen- dicular to the axis VZ, (tee the figure to Art. 30.), then RM=^ 2FN; for the triangles TNP, RMv being equiangular (29. 1.) TN iNP'.iRM: Mv (4. 6.). But NP=Mv (34. 1.) / JR3f= TN ( 14. 5.) =2 FJY (Art. 23.) Q. E. D. 25. If two parabolas VR and VK be described on the same axis FZ, and the ordinate NQ meet Ffl, FK in P and Q, then will the tangents at P and Q intersect the axis FZ produced in the same point T; for FN is the common abscissa to the or- dinates NP, NQ of both parabo- las, and NT—2FN in both (Art. 23 ) Q. B. D. 26. The square of the ordinate is equal to the rectangle contained by the latus rectum and abscissa^ or PN'^^L.FN For FP=: FN+ FF(Art. 14.) ••• FP^^FN^^FF^+^FEFN (4.2.). But FN''+FF'-z=^PT.FN 4-FA« (7.2.), V FP^^<jtVF.VN + FZV» + 2 FF. FiVr=4 FF riV-f- FN^, But FP^^PN^+Fm (47. 1.), / Pm+FN^:^4FRFN ^EN' 5 •.• PN^^4FF.FN^(cot. 1. Ait 140 L.FN. Q. B. D. Pa»t X THE PARABOLA. 458 Cor. HtiMe^ 41 my cardinate i'N»y> it& abeetea Or^Ar, and the latus rectum=4a, the expi-ession P^=L. FN will beeom^ y'«4a4r; whifih &» the equation of the parabQla» conaideced as a geooietrical curve. 27. llie abscissa varies as the square of the ordinate. Let PN and pn be any two ordinates to the axis VZ; then because PN^=zL.FN, and pn^—LTn (Art. 36.), PiV» : p»« : : L.FN : I..F« : : (15. 5.) FN : Tw, •/ (Art. 97. part 4.) FN cc PA*. Q. E.D. 9S. If two parabolas FR and FK be described on the same a3ds rz, and the ordinate NQ meets FP in P ; then will PN and QA have to one another a given ratia Produce np to q, then (Art. 27.) PiV» :pn*:: FN : Fn:: QN^iqn^i '-: (22. 6,) PN : pn :: QN : qn, and (16. 5.) PN : QiV : : p» « qu. Q. E. D, 29. The area FATP : the area FNQ iiPNi QN. For> let the abscissa FZ be divided into the equal parts Nn nrf'rm, &c. and qomplete the parallelograms Pn, Qt^, pr, qt, <m> tfiiy &c. these ha^ng equal altitudes (Nn, nr, rm, &€.) are to Otte ' another aa their basea (1. 6.)» •.• PmQni: NP'.NQ pr : qr -.: np : nq :: (Art. 28.) NP : NQ itnibmi: rs : rh:: (Art. ^8.) NP : NQ V (12.5,) P»4-|ir+«n^+&c. : Qw+^r-fftm-f &C. :: iVP : NQ (15. 5.). Wherefore^ if the magnitude of the parts An, nr, rm, &C. be diiaifiishjBdt aod their number increased indefinitelyi the 61^ of ^ tb? parallelograms between Faiid mx will approxi-* mat^ iade^ni^ely near to the ar^e^ of the ourvitineal space Fxm ; as ihe sum of the parallelograms between F and ytn willa to the (nirvil^neal $]^e Fjfmi '.: the area FPxm ; the area FQym : ; NP : AQ. Q. E. D. Cpr. Hen^^e, if from my P^int P iu the ws, straight lines FP^ FQ be drawn, the curvilio^ fu^ FFP i the curvilineal area FFQ : : itf^ 5 NQ. yor tbetrijuigle P^^: PCA:: NP : AQ (I. 6.) And FP A : FQA : : AP : NQ (as shewn above.) aW FPAiPPA;: FOA:PQA(U. 5) ... Fpjff^PPN : FQN--FQN : AP : NQ (19. 6.) . TJiat i*, the area FFP : the area FPQ ; : AP : NQ. Gg3 454 CONIC SECTIONS. PartX. SO. The sub-normal is equal to lialf the latus Tectum, that is. For TPG 18 a right angle (Art. 9. 10.), from which NP U drawn perpendicular to the base TG (Art. 5.), '.• TN : NP :•" NP : NG (cor. 8. 6.) ; v T TNNG^Nf'^ (17. 6.)== L.VN (Art. ^6.) y\' Tff: FN r II L : NG (16. 6.). But TN=^VN (Art.23.). •.*£=: QNG (prop. D.5.), and their lialves are ecjuaj, or ffG^ -^ L. Q. E. D. Qyr. X, If from F as a centre with the distance FT =FP a circle be described^ it will pass through G; for M T and P being in the cir- ^ ouniference, and JP G a right angle, the point G will like- wise be in the circumference <31. S.)i •/ FP=FG, and the angle FPG=zFGP(5. 1.). Cor. 2. Hence- also the angle rPP=FGP+FPG (32. I.) =:2FGP. 31. If GA* be drawn perpendicular to FP, then will PK For the triangles PGJT, PGN having PJTG. PNG right angles, GPK=PGN (cor. 1. Art. SO.), and PG common, are wmilar and equal (26. 1.); •/ PK=NG=z^L (Art. 30.) Q. E. 1>. 32. If nv be an ordinate to the diameter PX, then wiU «©«= 4FP.Po. Because the triangles RAn, RMv are similar, jRJtf» : RJ* : : (M»« =) N P' : An^ (4. 6. and 22. 6.) : : FiV : F^ (Art. 27.) j but iiM» = ieXMiW)«=/J^»4.^M»+2fi^.^i«f (4.2,), v (prop. E. 5.) RM' : JM' +^SJ.AM : : FN: AN : : (2FN^) RMiStAN;: RM' : ^RM.AN (15.5); v AM'+^RA.AM^ 9,RM.AN (9.6.), or AM' =±<iRM,AN^^RA.AM. But R,i=^ RM^AM, and AN=s=AM^MN\et these values of RA and AN be substituted in the foregoing expression, and it will be- Pj.«T Xv THE PARABOLA. 4S5 come ^iV'-='iRM.AM-MN—^RM~AM.JM^<iRM.Aia- ^R3LMN-'2RM JM+2.1M')=-^RM MN+^AM' , or AM' =B=^RM.MS. But since TB=:zPv=^MN (34. \.), ■.• RM^^TN ^fW(Art. 23.); ■-■ tbe above expression AM'^IRM.Hti^ 4VN:MN=4rN.Pv. Now m* : Xb£*=) am* : : Re* : BJIf (4. 6. and 22. 6.) : : iIM*+Jlfp' (47. I.) : BM» :: (siDce RM=^VN, and Jtft>»= yP^=4rN.VF. by Art. 26.) 4rJV*+4FW.rF : 4rAf« ;: 4FN+4rF.FN : AVN* : : ^Art. 14.) 4fP : 4rN : : 4FP.Pv : 4VN.Pvi that ia, no* : AM* :: 4FP.i'o : AVN.Pv, but it has been proved ubove that AM'=4FN.Pv, ■.■ (14. 5.) no»=;= 4FP.FV. Q. E. D. . And in like manner, if RI—IM be substituted for RM, and compoiition be used instead of conoeraion, it may be shewn that fcp*=4fy./*i' ; con^quenlly ni'=if; that is, any djameter jpX bisects its ordinates. Cor. I. Because ■kFP.Pii=nv*. and FP is constant, ■.■ Pv «e nt-'. •.■ also On « OP*. Cot. "2. If from any point p in the diameter PA", ordinates itfi b6 drawn cutting P.ir in a given angle, and having a givea ratio to vb I the curve passing through all the points B will be a para- bola. For rb : vB being by hypothesis a given ratio, vb" : vS* ia likewise given i but (cor. 1,) c6« (^nC) « Po, ■.■ pB» « Pv. Cor. 3. Since AM:'=iVN.Po, as shewn above, and Pv=MN (34. 1.), '.■ Am=4yN.NM. Cor. 4. Let Pbe the parameter to the diameter PX then when n&pa3sesthroughthefbcuBF,itbecumes the parameter (Art. 12.), uid the point r coincides with F: ■.■ Pr^Pv=FP (Art. 17.), and because nr*=4FP.Pr, ■.■ n6'=4»r'' (4.4.)=4x4fP.Po= 16 FP» (since Pu=fr), that is P*==16PP'. -.■ P=4F/'. 33. If no he an ordinate to the diameter PX, and nTn tan- gent at », the sub-tangent tt 7" will ^..,-4— _ T n be bisected by the vertex P. ^'"^ Produce nv to meet the carve in b, produce nT to E, and draw Eb parallel to TX. Then (cor. 1. Art. 38.) PTibE :: nT>tnE* ;■. • (4. 6. and 22. 6.) oT* : bE'; :■ , (l6.6.)Pr.6JS"=6E.tir',orPr.&£ I =vT^, V (ir.6.) PPirr:: cT: ^ Gg4 ^ CONIC SBCTION& FAmxi bE : : (4. 6. and 16. B,)w:nb:: (Art. 32.) 1:3; that is^ the sub^tangent vT is bisected in the point P. Q. E. D. Cor. Hence, if 67* be a tangent at b, the two tangents nTt 5T and the diameter TXwiW intersect each other in the same point T; and in like manner^ if other parabolas be described upoi^ the diameter PX, by either increasing or decreasing the ordinate nv, or its inclination to the diameter, the tangents will all pw through the point T, as appears from the precedioip demon* strati on. 34. If several circles be described upon as many diametext of different lengths, these circles will have different d^rees of curvatmre, as is plain ; and if the diameter be increased and decreased indefinitely, and circlea be described from the same centre through every point of the increased or diminished dia-* meter, these circles will possess all possible degrees of curvature, Hence it follows, that if a point be assumed in ^y curvc^^ circle may be found which will coincide with an indefinitely small portion of that curve at the assumed point, so that the curve and the circle will have the same tangent^i and the sane djsfiection from the tangent at that point ; this circle is called Tus ciKCL£ OF cuEVATURa to the proposed point. PabtX; TUB PASABGHLA. 4&t 36* If P be tba loew of a paralx^ and P any point m the curva, the chord of curvature to the point P which pasees through f h equal to 4/P. ^ Let Fr be an indefinitely imall «r«of the paraMn^ coin- ciding with the circle of «u#vsiture FHK (Art. S4.)| then the Hoe nR may be considered as common to both 5 join iiP» nH, produce the latter to M, and draw ae parallel to the tan^nt PY. Then ainoe the angle RPnz^^RHP (99. S.)> and nP k indefinitely near a coincidence with RP, the triangles PHt^, PnR may be considered as equiangular, *.* PH i Pn:: Pni nR (4.6.) and (27.6.) Pn'h^PHjnRi bqt mce the arc is in its nascent state (or indefinitely small) Pn^s^nv, '.* (ae^asby cor. Art. 19.) 4W.P»=Fn«=Plf aHj but nfi=Prs« (Art. 18.) Pv, *.' 4FP.Pv:^PH.Pv,oxPB^^^FP. Q.£.Di. C^r. 1. Hence, because 4fP=:^he panuneter (cor. 4* Artf ^0,>» •.' tbe.i^ord of c«qrtat\ire passing t}iro«!gh the focua ia equal lo the parameter. Cor. 9. If the diameter PK be drawn, HK jQined, and fY drawn perpendicular to PY, the triangles PHK, PKF will he equiangular, since YFP^szHPK (99. 1.) end the angles at H and Fright angles (31. 3. and by construction) •.• FYi FP : : PH, :PK:: (because 4FF=PH) AFP : PK. Hence, if a tangent be drawn to any poipt in the parabola, and a perpendicular to the tangent, he drawn from the fecus, the <Kameter of the circle of Gunratme to that point, will be readily determined. 36. If a cone be cut by a plane parallel to its side^ the sectioA- will be a parabola. Let ABO be a cone, and let the plane VHK pass through it; parallel to the side AB, the section HPVQK will be a parabola. tiet the plane HVK be perpendi- cular to the plane BAG, the common section being VS: PDQE a section of the cone parallel to the base, conse- quently a circle, PQ and DE its com- mon sections with the fbre«mentioned planes, and draw FF parallel to DE, •/ since the planes BVK, PDQE are perpendicular to BAC, their comnoift section PQ will be peffwipdjcmlar !• 458 CONIC sEcrroNs. Part X.' BAO (19. 1 1 .) and coaseqOBatly to the lines DE, V$ (def;3. 11.) ^ and because DE the diameter of tliejarcle FDQE cuts FQ at right angles, FC=CQ (3. S.), v DC.CEsiFO (14. 3.) Now the triangles VCE, AFF being nmilar FC : C£ : : AFi FV (4- 6.) Let AFiFVtiFF'.L (11.6.) v FC: C£ :: FFiL (11. 5.)} V FC.LsiCE.FF (16.6.) ^DC.CE (34. 1.) =PC», •.• (Art. 26.) HFJi: is a iiambohi of wUdi PC is an ordinate to the axis, FC the correspondent abscissa, and. L the latus rectum. Q. £. a THE ELLIPSE. BBFINITIONS. 37' If two straight lines PP, SF intersecting each other in P, revolre about the fixed points Pand S, so that PP+5P be always the same, the point P will trace out the curve PFKU, which is called an ellipse. 38 The points P and 5 about which FP and SP revolve, arc called THE FOCI. 39. The straight line which joins the Ibci being produced both ways to the curve, is called the major axis *". Thus VU. is the major axis. 40. If the major axis VU be bisected in C, C is c^led the CENTRE of the ellipse. 41 . The straight line drawn through the centre perpendicular to the m^or axis, and terminated both ways by the curve, is called THE MINOR AXIS \ Thus EK is the minor axis. 42. Any straight line passing through the centre, and ter- minated both ways by the curve, is called A DIAMETER. ThuS BX is a diameter of the ellipse. j: d c It b also named 7%e irmuverte axU. 4 It is likewise frequently named Z%0 etmJvigmU a«i». Part X. TttE BLLIKS. 460^ 43. The eiftremity of any ittametar is ctUed its viunxJ Thus V and U are the vertices of the major axis, E and K of the minor axiSy and B and X of the diameter BX, 44. A straight line drawn throii^ the focus^ i^rpendicQlar to the nisgor axis, and terminated both ways by the curve, is called TBS latus rectum or principal parameter. Thus 3I> is the latus rectum. 45. A straight line meeting the ellipse- in any point, and which being produced does not ait it, is called a tangent to that point. Tims BT is a tangent at the point B. 46« The tangent to the point B or D, the extreipity of the latus rectum, is called the focal tangent. Thus BTis the focal tangent, 47* The atraiight line drawn perpendieular to the major ajus produced, through the point in which the focal tangent meets it> is called the directrix. Thus xy is the directrix, 48. Any strai ht line drawn from the curve, perpendicular to tiie major axis, is called an ordinate to the axis. Thus FN is an ordinate to the ajiis, 49. The parts of the axis intercepted between its vertices and tiie ordinate, are called abscissas. Thus VN and NU are ah' mssas to the ordinate PN, 50. If from the vertex of any diameter a tangent be drawn/ suay sttaigfot line paraUel to the tangent terminated by the dia- ineter and the curve, is called an ordinate to that diameter } and the intercepted parts of the diameter are called abscissas. Thus dv is an ordinate to the ^meter BX, and Bv, vX abscissas* 51. If the ordinate pass through^ the centre* and meet the curve. botli ways, it is called the conjugate d.iam£ter *; and if it pass through the focus, it is called the parameter to that diameter. Thus DG is the conjugate diameter^ and db thepara* meter, both to the diameter BXr PROPERTIES OF THE ELLIPSE. 52. The sum of the two straight lines drawn from the foci of an ellipse to any point in the curve, is equal to the major axis. • And ID general, if each of two diameters be parallel to the tangent at the vertex of the other, these diameters are called conjugate* to each other, ^^c 8ub.taDf.enty normal^ and sab-Dormal, are the same as in the parabola. 460 COOnC SKTidllS. Fakt Y. TIhv^ VPUmKy poiol in the €«¥•, Ibea FF-i- PS^ VUt=i%VC. For (Art. 37.) ^^4^ ra«p/'l7+ I7«i that ii» 2i?r+JPS» atrS4*iJ^& V ^FVmWS, aad fF^USf and (Art. 37) i^F+ J^«»i^i»+«Fw, F«^r US:nfVU^SlVa Q. E, J>. Csr. 1. Hoiee, beGMiK FV^ VS^^WC} bf adding VT to both> 57+ TF:=2CT; and by taking 27^ frun thia^ ST^TF 9s%C7-^%TFmStCF. Car.^, Hanee, bacmiaB <Art.40«) CWsb^CU, and FF^sUS (as proved abov#) v CV^FV^CU^ US^ or CF^CS. Car. 3. Hence, SF^FU-^FFm^FC^FF^ a^d in like oMn- Mf it wppem tbat JPasd FC«^ &P. Cor. 4. Hence, because jFP+5P=9rC, by taking %SF fh>» bcxth FF^SB^2FC^2SF, or (ttnca ^Pisrji FC-^i^p, by 0OT.a,)»«FP-.9FC 53. Tbe latua rectum is less tban 4Fi^i £air BF-^BS=^VU (Art. 37,)7xt^FF^FSXATU 52.) ; and since BS i& greater than FS, BF must be less than S^VF, and {^BF^) BD 1^ than 4Ff • Q.B.D. M, A straight Una dnMivn frooi tkefQct»»^ili« v^rtw of tli# minor axis is equal to half the msgor a^,.or FB^ F€% Sw ikf folhwiHgJtgwre. For since fCs^CS (ear. 1. Art. ^%) and Cfi ^to^imm te the two lri«»glea FCE^ SCS (and tbe anglea at C rigbt a«gi«i (Art. 41.) ••• FE^ES (4. i.)i U»t (Art. »7.) I'fi^JE* t*at is 2irjS5»Fi;fe»«FC, / FZ^FC. a S. D< Cor. 1. And in like manner it may be shews that FK^tzKSts ti8:izEF^FC,\'^^ tbe triangles FEC, FKC, FK^FE, tbe angles trt C «Fe right anglss, and the side FC is common, whence (««. 1.) EC^CK. Cor. 2. Hence £0 =: Fj E* .-. flC (47. 1) aFCT^-^FC*™ (cor. 5.2.) FC- FC. FC-k- FC;= VF. FU. 55. If on the major axis as a diameter a ciftJe be d^cribed, and the latus rectum be produced to meet the circymferenoe in k, then will Ffc=EC. For (14.2.) P%*=FF.Fr= (cof. 2. Art. 54.) EC*, ••• Fk=EC. 5$. Jim btus rectum k ^ Uurd proportional to the ms^or apd minor eves, er FU : JSJT: : EK: BD. Part X. THE SLLIPSZL 461 ^VC'^BF (cop.i. Art. 59.) •.• BS" 35: 4 FC8 -f. JfF« -* 4rCJiF. Bat BS^ = BF« + FS* (47. 1.) •/ 4F0«+ #F»^4FC.aF=tt JBF*-fF5t/4FC« — 4rc.BF=js«= (4. «.) 4inc« •.* (Vc«—FC« by cor. 3. Art. 54.«) EO«rC.JBP (Art, 56;) | •.' (IT- 6.) FC 1 EC : : EC : BF; whence (15. 6.) VU : EK i : EK : BD. Q. E. D. Cor. 1. If L (=BD) be the Utui recttiiii» theo (lioce Fl/jft 2 FC) L.2KC=E^ (17. 6.) Cor. 2. Hence, of the major and minor aKBA and ktus rectum* ; any two being given, the third may be found. 57. If FP and SP be drawn from the foti, to any point P in the carve, and FP be produced to M, the straight line PT which bisects the exterior angle FPM is a tangent to the dfipse. Make PM:= PF, join MF, let P T if possible, intersect the curve in p, and join Mp, Fp. Then because MP^FP, the angle PJkrF=Pf3»f (5. 1.) MPr:=^FPr by hypothesis, and Pr common, / (4. 1.) ilfrssFr, and the angle MrP^FrP; >r then in the triangles Mpr, Fpr, Mr=^Fr, pr common, and the angles at r are equal, •.• (4. 1.) Mp—Fp. But (20. 1.) 5p+pi)f y SM, that is > SP+PM, that is > SP^PF (because PF^PM) that is ). Sp-hpF (because Sp4"pF= SP-fPP by Art.37.)5;.* since .Sp+pM J> *^+pF, if 5^ be taken from both pM )> pF; but it has been shewn that pM=pF; V Mp and pFare both equal and unequal to each other, which is Absurd? •.' PT does not intersect the curve in any other point p; PT is therefore a tangent at P. Q.E. D. ^ 4Gi CONIC SECTIONS. Past X. Cor. 1. It 18 plain that the nearer the point p be to F, the greater will be the angle FpM; and therefore when p coin- cides with V, the lines Fjp, pM will coincide with fP^, FT, and the angle FpM will become = two right angles > but the tangent at (p which now coincides with) F bisects this angle, */ the tangent at T is at right angles to the axis FU, Car:^. Hence (prop. A. 6.) STiTF:: SP : PP. ' Car. 3. Hence, straight lines drawn from the ibd to any point jn the curve> make equal angles with the tangent at that pdnt, for the angle iPS^^MPT (15. l.)=FPT. Cor. 4. Hence the triangles FPY, SPi will be dimilar, and (4.6.) SP'.Stii FP'.FY. 68. Let P be any point in the ellipse; join FP, SP, then if SG and FG be drawn parallel to these respectively, the point G where they meet will be in the curve. For since FPSG b a parallelogram* FG ^ ^GSz=^SP + FP (34.1.)-.' G is a point in the ellipse by Art. 37. Q.E.D. Cor. Since PG and FS bisect each other in C (part 8. Art. 241. cor.)> C is the centre of the ellipse (cor. 1. Art. 59.), and PG a diameter (Art. 42.)> *.' all the diameters of the ellipse are bi- sected by the centre. 59. if /2r be a tan- gent at G, it will be parallel to Tt. For since SGr+SGF+FGRzsz^ right angles (13. and cor. 1. 15.1.), =5P^+5PP4-PPr, and SGF^SPF (34.1.), by taking the latter equals from the former, the remainders SGr +PGR=5Pe+PPr, that is, (cor. 3. Art. 57.) ^FGR=:^2SPt, or FGR^SPts but PGF^GPS (29. 1.)} add these equali to the preceding, and FGR+PGF^zSPt-^^ GPS^ that is, PGRsi GPt, •.• (27. 1.) Rr is parallel to Tt. Q. B. D. Pa>tX- THE EliLIFSE. Cor. Hence, if HD be a ccmjtigate diameter to PO, taagents at D and H will be parallel^ and the four tan^ntt r/> tr, rR, and RT will form a parailelflgram circumMvibed about the ellipse. 60. If HD be drawn through the centra, parallel to Tl a tangent at P> cuUiag SP in the point E, then will P£^ UC. Draw FN parallel, and Pa perpendicular to HD, Because NF ia parallel. to tT (30. 1.),. and the angles at o right angles, '.' the angles oPT, oPt are right angles (99. l.)» or oPTssoPt, but FPT=:iSPt (cor. 3. Art. 57.), •.* by. taking the latter fmn the former FPq^NPo, ': PNz^PFz (33. 1.), the aisles at z (=sthe angles at o by, 99. 1 ) right angles, and Pz is com- mon to the triangles PzN, PzF, •/ (26. 1.) PN:=^PF. And shice EC is parallel to NF a side of the triangle SNF, and SC^ :=CF (cor. 1. Art. 53.), v 5£=£i\r (2. 60 i •.* SP+PF (=:^ SiV^-|-iVP-f.pjF=2£i\r+2 NP) == 2PE. But «^P + Pf =2 l/C (Art. 52.), •.• 2 PJS= {SP + PP=) 2 l^C, and PE^UC. Q. fi D. 61. If perpendiculars be drawn from the foci to any tangent, axikd a circle be described on the major axis as a diameter, the points in which the perpendiculars intei^ct the tangent shall be in the circumference of the circle. Let P^,<Sr be drawn perpendicular to er a ^ tangent at P, join SP and produce, it to meet Ft produced in F, and join Ct Then in the triangles PtF.PtY, the angle tPFssztPY (Art. 57.)» the angles at t are right angles, and Ft is^ common, V (26.1.) FP =3 PF and P*=^F; also PCszCS (cor. 1. Art. 62.) ••• Ct is parallel to S^ (2. 6), and the triangles FCt, FSY are similar, •.• P C; Ctzi FS i SY (4.6 ,). But PC = ^PS, V a = i5Fx=4.5P+PF=4SP+PP = (Art. 52.) 4- VUszFCj / since Ct^CF, the points ^ and Fare in the circumference of the cirde whose centre is C, and in like manner it may be proveil that T is in the circumference. Q. E. D. COMK flBcTIOKS. FAstJL 01.& Hie i«6tttigte TtAT^EC^. Vrtiifaoil » to JT uid join CA, tben beoauBe tTR U a right ukgle, the segmetit iTM h 4 umUAn^ (SL.3.), v <C and CJ2 tnoetinj^ at thecMitM, will constitute the diameter, and be ia the same strai^t Um, V the aa^ tCF^SCR (15. 1.) and IC CJPccAa CS respec tivdy, V (4. 1.) f]t«&B| •/ Fi.Srfc35JtS2te(a5.3.) rS.8Um (Art. 54. cor. 3.) £C*. Q. fi. D. Car. 1. Hedce Fi .EC .t JBC: ST (17. d.), •.• IV : EC» i: J?# : 5r (eor.«. 20.«.) ! : FP : SP (4. 6. because the triaiigiei KP, STP are similar) : : FP r %VK:^tP (because FP+^P;* «I^C, Alt. 59) Whetefoie pntting FCiiso, ECisA, FP^jf, and Ft^^y the analogy Fl« : EC» :: fp t ^rC^FP becomdi y* : I* : : « : 2a— x, •/ ^^ which «i|uatioa expreasts Um 2a-- J? imtfire of the ellipse considered as a ^piM, described by the retohition of FF about the centre F. Cbr.2. Because Ft* :EO::FF: SP {car. 1.) •.• 4Fh : 4£0 (=Fi:»=I.2rC, cor. 1. Art. 56.) :: l.FP : LSP, v (16.5.) 4Ft^ : LSP : : U^fC i l.SP : : 2FC : SP : : ^VC : gFC—FP. 62. If BT be the focal tangcttt, thu tdU CF.CT^VO. See the following Jigure. Because (cor. 2. Art. 57) ST.tFz.SB: BF» •/ by eoifr» position and division (18. and 17. 5.) STf TF : JST*^TF i : iKH +BF : SB'-BF, or (cor. 1. Art. 52.) 2Cr- 2CF: : SB-¥^St i Sg^BF.v (15.5 .) 2Cr.2CF: 4CF« : j (SA^AP.SB^BF : SB-^BF.SB--BP : : ) SB+bt]^ : 5B«-BP» (^. 5.2.). Bwt BFS is a right ang^k, / (47. 1 .) SB^^ hP^^PS^^ (4.2.) 4CF*, •.• (prop. A. & ) 2Cr.2CF=S^4-jgrFp«= F&» (Art: 62.)* 4ro^4.2.), V cr.CF=:rc*. Cor. Hence, because Cr=CF+Fr,;-.- (CT-CF=) f:F^+ CF.FTz^VL\ V CF.Fr=FC«-CF»=£C^ (w.2. Art. 54.)' t .J > t . -i . - fkUT X; THB £LUMB. 4€6 ^ T M y- \ Ti y^ V /«\\ • J. \ / F N ^ • / "" K C / /r \ r / tt. If m be Amm perpendicvkr t6 the dinetrix yr^ tken wMl PP : i»lf : : i?C : VC. Let PiV^ be perpen* dicuhur to V\J, th en SP-i-FF.SP—PP*, V (16. 6.) SP^PF : SN:- i^P :: SN+NF : SP+ PP. Bui (Art. 52. cor. 4.) 5P— PP = « PC -« PP ; and SN^NF^SC'^CN -iVPss CP-2^P+ Ci^=: ^CN; likewise SN-i^NF s^^CF; and SP+PP= 2FC (Art. 52.) j by sub- stituting these values for tbeir equals in the above analogy^ it becomes 2FC ^^FP : 2CiV : : 2cP : ^ 2FC : : (Art. 62.) 2PC : 2Cr; . (15. 5 ) FC'^FP zCNi: VC: Cr, subtract the former antecedent from the latter, and the former consequent from the latter, then (yC—VC+FP: CT-^CN :: VCiCT-, that is,) FP : (Nr=) PMiiVCiCT:: (Art. 62.) CF : VC. Q. E. D. Cor. Hence, if the centre C be supposed at an infinite distance from Vj CF may be considered as equal to FC, •.* FPssPM, and the curve in this case at every finite distance, becomes a para- bola. See Art. 1. 64. If PF be produ ced to meet the curve in p, then will SFP.Fp=FB.FF-i^Fp. Because FFiNT:: CF: CV (Art. 63.), if P be supposed to coincide with B, the point N will coincide with P, and the straight line FP will become P6 ; *.* the above proportion will become FB :FT::CF: CP; ••• since ^^ \^^j i:CF : CF, ^^ * Por (47. 1 .) SP* -5iV« + A TPg a n4 PF^^lf F^ ■¥ XPu '- SP*— PFi' ^SIV^^JVF* or (cof . S. «.) SP-i- PF,SP^jHF»SN+ /VF.dN-^NF as above. VOL. u. ah M* CONIC SECnOMS. Paut X rPzNT::FB:Ft (11. 5.) ; K«t NTtsPM (S4. l.>, •> F^ : PM :: FB: FT, •/ FBiFT:: FP^FB : (PM-Fr=) FIT. In like maBDCT it may be sbewn tliat'FB : FTi : Fp s fm, v FBzFT: : PB-^Fp : (FT-^pm^J Fn; v (H. 5 ) FP-^FB : FN::FB^Fp<Fn. Bat tlie triangleft FPN, Fpii are simikr, ••• (4.6.) FN : FP :: Fn : Fp, and ex ^uo («. 5.) FP^FB : FP : : FB^Fp : Fp, •.' (16. 6.) FP.Fp-'FB.Fp==JPBJPP'^ FP ^Fp, / b y transposition «FP.Fp=(FB.FP+FS.Fp3:) FB.FP^Pp. Q.KD. Cor. Hence, if Ffi=/, FP=sX^ and Fpas3% the atove con- clusion expressed algebraically will be 2Xx^lJC+x, or —=: 1 1 Jir"*"x' 65. If c be tbe co-sine of the angle UFP to radius 1, then will FP :EC::EC: VC-c.CF. Because ( Art. 63.) FP : PM : : FC : VC, / (16.6.) FPJ€ zsFaPMz=iFC.FT±FN=zFaFT±FC.F^= (bec2Lme FC.FT= EO Art. 62. cor.) EC*±FC.FN. But FY : FP : : -j- c : i; / (16. 6.) TFN=c.FP, and ±FC.FN=c.FaFP, / by sub- stituting this latter quantity for its equal in the above equa- tion, it becomes FP.FC=EC*+c.FC.FP; v(?P.n:— c.JTFF =) FP. FC— c.fr=£C, V (17. 6.) FP:EC::EC: VC-^c.FC. Q. E. D. Cor. If VC be infinite, FC and VC mav be considered as equal, and Aie above analogy becomes FP : EC : : EC: 1— c.FC But (Art. 56.) EC :^L : : FC : EC, '.' ex <gy w (2 vg. 5.) FP : ^L :: (FC: 1-c.FC :: ) 1 : 1-c, or (16.6.) 1— c.]RP=r4.I, and FP=- as in the parabola, see Art. ^S. * 1— c €6. If on the major axis as a diameter, a circle be described, and P^T an ordinate to the'm^jor aoiis be produced* to meet the circumference in Q, and if .c be the co-6ine of the angle FCQ to radius 1 ; then will FP=: VC^cFC. P#»»X. THB £LLTfSB. 4fr a*^ «*«»Mpa» Beoftwe (Art. 63.) S?r?W.5N-2VjFs==SP+i?P.SP-JPP, (16.6.)5V(seeArt. 63.) SCF:2rC^2PP::2FC = SCAT, or CF : FC^FP :: VC : CN :: (because QCzzVQQCiCN. Butu (Art. es. part 9.) QC : Ci^::l:c,vCF: TC- W i : 1 : c, •. (16. «.) c.CF=Fc~FP, and FP ==FC-c.CF. Q.E. D. 67. If PN be an ordinate to the axis, then will UN.NV : PN^ iiFC^iEC^. For (Art. 63.) SN+NF . 5>^-2^F=5P+ FP . SP--FF, '-'SN^NFz SP-^PF :: SP-^PF : SN+NF (16.6.); but 8N^NF=z (5C+CArr-]^^F=:CF#-NF+CAr=) 2Ci^; Ukcwise 5P+PF=^arC (Art 52.) j also SP-PF=:2SP— 2 FC (Art. 52. cor. 4.); and lastly, SN-^NPss^SCf •.• substituting these four values for their equals in the above analogy, it becomes ^CN : ^VC : : 2S|>-.2rC : 25C; / (15. 5.) CNiVC:: SP-^FC ; SC. . . f (18. 5.) UN :VC.: SC-^SP-- VC ; SC. * «- {17. 6.) FNiFC:: SC-SP-i- FC : SC. From the former of these (12. 5.) UN :FC:: IW4-5C+ 5P-FC: FC'^'SC:: SP + SN: UF } And from the latter (19.5.) FN i PC:: FN-SC+SP^ rC: FC^SC::8PS Ni US; V (prop. G. 5.) UN.NF: FC^ :: SPi-SN.SP^SN : UF.US :: (cor. 5.2. SP^-SN^^sz by 47. 1.) PN^ : JSC* (because UF.USz=zFF.FU see cor. 2. Art. 54.) •.• (16. 5.) UN. NF : PiST* : : FC* : ECK Q. E. D. Cor. y. Hence, because UN =VC^CN, fs^d NF =^ (FC-r-CN ^) UC-CN,'.' UN.NV=^VC^CN,UC'-CN-^{coT.^.%.) UC^^CNi '.' also FC*--CN^ : PiV* : : FC* : ECK fjor. 2. Hence, if FQ;ssa, ECssb, CNssx, and PiV^jf, the expression FC^^CN* :PN^::FC* : JEC* becomes aJ»-«* : y* : i« •/ ifisz— . a*—i* which is the equation of the ellipse. Hh 2 : : 02 4m come sECTioi^. part x. Ow. S. Hence VN.'NV^ PN^i that is, the rectangle con- tained by the abscissae varies as the square of the ordinate. 68. If Pn be an ordinate to the minor axis EK, then in like manner En.nK : /V : £C* : TC*. For Pn=CN, and PN^Cn-, VC-Pn^ . Cn^ : FC* : EC* (Art. 67. cor. 1.), •.• (16. 5.) CV* ^Pn"^ : FC« ; : Cn« : ECS . - (17. 5.) P»« : FC« : : EC*-Cfi« : £C« : : (cor. 5.2.) EC+Cn, EC^Cn :EO :: En,nK : EC*-, •.* (16. 5.) P«» : En.nJST : : VC* : EC^ and (prop. B.5.) EumK : P»« : : £C« : VC\ Q. E. D. 69. If on the major axis UK as a diameter, a circle UQVhe described and NQ an ordinate to the axis be drawn cutting the ellipse in P, and the circle in Q ; then will PN : QN : :. EC : VC. For OAr«= UN,NV (14. 2.) •.* (Art 67.) QN^ : PiV^* : : FC* : ECS ••• (22. 6)QN:PN::VCi EC, -,' (prop. B.5.) PiV ; <?iyr : : EC: VC. Q. E. D. Cor, 1. In like manner, if on the minor axis EIIl as a diame- ter the circle EqK be described, it mayi)e shewn that Pn : qn ::VC: EC. Cor. 2. Hence the area VPN : UQN : : (UC=^) EC : VC as in the parabola, (Art. 29.)} in like manner VPN : VQN : : EC : VC, •/ UPV : UQV : : (2EC : 2FC : : ) EC : VC. Also, if any point S be taken in the axis, and SP, SQ be joined, the area UFS : area UQS :: EC : {UC=^) VC as in the parabola, cor. Art. 29. 70. If a mean proportional R be found between VC and EC, and with it as radius, a circle be described > the area of thit, circle will be equal to the area of the ellipse. For the area UPV : area UQV ::EC: VC (cor. 2. Art. 69.) and since VC : R :: R: EC, •.• (2. 12, and cor. 2, 20. 6.) area of circle UQV "whose radius is VC : area of circle who^e radius is -S : : VC : EC; this proportion being compounded with the first, we have UPV.UQV : UQVx2Lrea, of circ. whose rad. is R : : EC.VC :, VC.EC; that is, (15.5.) elliptical area UPV: circu- lar area whose rad. \s R:: {EC.VC : ECVC : : ) 1 : I ; or the area of the circle is equal to the area of the ellipse. * Q. E. D. Cor. 1. Since (cor. 2. Art. 69.) UPV-, UQV : : EC: VC :: (15. 5.) EC. VC : VC^ ••• (16. 5 ) UPV ; EICVC : ; UQV : VC^; Pam X. TH£ ELLIPBB. 4M -.' <15. 5.) atfea of ellipse : ECFC : : area of oirc. whose diam. Is UF : FC^. But the area of the circle varies as FC* («, 12.) j •/ the are^ of the ellipse varies as ECFC. Car. 2. Because FC : EC : : EC : ^L (Art. 56), •/ FCi^Li: FC^ : EC* (cor. 2, 20. 6.) j but VN.NF : PN* : : FC* : £C* <Art. 67.), •/ UN.NF (cm- FC^^CN^, Art. 67. cor. 1.) : PJV^« : t /^C : 4-L; -/ since f'C and ^L are constant quantities VN.NF Cor. 3. Hence, if the major axis UF become infinite, the curve at all finite distances fkom the vertex U vdll be n parsd>o]a3 for NF being infinite will be constant^ and *.* UN oe PN^ which (Art. 27-) is the distinguishing property of the parabola. Cor. 4. The curve UPF which arises by diminishing the ordi- nates NQ oi the circle in a given ratio^ is an ellipse. For, let EC: UC:: PN: QN, then if an eUipee be de^ scribed on UFaa the major axis* having EK for its minor axis, we shall have (Art. 69.) UC : EC : : QN : ordinate of the ellipse ; and from the preceding analogy (prop. B. 5.) UC : EC : : QN i PN ••• PiV^=an ordinate of the ellipse (9. 5.), or the curve passing through P is an ellipse. In like manner it may be shewn, that if the ordinates QN of the circle be increased in any given ratio, the curve described upon UF as a minor aju8> and passing through the extremities of the increased ordinates^ will be an ellipse. 71. If a plane be inclined in any angle to the plane of a cir- cle, and if straight lines be drawn from every poiut in the cir- cumference, perpendicular to the inclined plane, the curve which passes through the extremities of all the perpendicalars will be an ellipse. Let C7£r«rbeacir- u^ >^^ cle, and the perpendi- culars Uu, Ee, Fd, Kk, &c. meeting the inclined plane GuvO in the points tt, e, r, k ; the figure ueok will be an ellipse. Let UF be a diame- ter of the circle parallel U> QO the common sec- tion of the planes^ and 470 OOMIC SICTfQN& MP St right moglet to UV; draw GU, OFevh pcndU to JtfP« join Ov and dnmr Mp, Qu panAel to it, join Nn, Fp, IBecasxtt MFisa, paralkkgnm Jro is innIM to 1^ (94. l.)« tat ITF is perpendicular to the plane iiifn hj eonstradion (4. 11.) '•' MO u abo perpendicnlar to the plana MMn (8. 11.) *•' w is perpen- iiicniar to MNn (19. il.) .- no is panUel to UFifi. 11.) $ aai dnce the planes MpP, VuFv aie hath at right aisles to the plane GF, their common section Nn is at right ang^ to it (19. lU), V Nn 18 parallel to Pp (6. 11.)$ v jm : nJf : : PNi NM («.6.)and fmiPN.i nMz JVJf (1«. 5.) : : radial 2 cMiae FMp (part 9. Art. 63.) the angle of inclination of the planes, or pn : PN in a given ratio, *•* (cor. 4. Art. 70.) a^nlp is an ellipse. Q. £. D. Cor. Hence the oblique section of a cinder is an fSlipse, ef winch the minor axis is the diameter of the cylinder. 79. If a circle be described on the major axis as a diameter, and any ordinate ^TP be drawn meeting the cirde in Q» tangeate at P and Q will meet the axis prodoeed in the same point T. For if possible, let Qr be a tangent to the drde in Q, and PT not a tangent to the ellipse, but cut it in P and p; draw np and produce it to meet TQ produced in in;* then since the triangles TNPy Tup, as also TNQ, Tnm are similar (32. \) PN :pn:i NT inT 11 QN x vm(4. 6.). But PiV : QAT: : pa : qn{hxi. 69.), -.- PN :pn ::QN:qm (16. 5.). But by the first analogy PN : pa : : QN : ma, -.* QNi qn : : QN: ma, v (9.5.) qnszmu, the less to the greater, which is impossible ; *.* TP m which meets the ellipse in P does not cut it, it must therefore be a tangent to the ellipse. In like manner (s^ the figure to Art. ee) ; since Pn : qn (^nC) : : FC : EC (cor. 1. Art. 69.), it may be shewn that tangents at P and q cut the minor as^ in the same point t Q. £. D. Cor. 1. Because CQris a right angle (IS. a. see the iguie to Art. 72 ), CN:CQ::CQ:CT (cor. 8. 6.) 5 but CFarOQ, / CN.CF.iCF: CT. In like manner it is shewn that fsce the figure to Art. 66.) Cn iCEiiCE: Ct. PA&rX. fFHB ELLIfSE. 4n Cor. 2. TN.NCszQN^ (cor.S.e, and iy.6.)«CQ*— CW» 07. 1.)= FC«-CiV«= (cor. 1. Art. 670 VN.NU. Cer, 3, The sub-tang«nt NT i« greater than ^FNg for sinc^ <ly the precedHig) TN.NCszFNMUr-' {16.6.) NTiFN i: NUiNCi bat CU > 2^^C, •/ (JVC+Cl^ss) i^I^ > ai^C, %' 2Vr > Cor. 4. If PG be the normal^ then <cor. 8. 6. and 17- 6.) TNMQ^PNK and TJV.l^C : TN.NG :: FC : ^L (cor.l. Art, 67. and eor. 1. Art. 7«.) •/ NC: NG;:FUiL (15. 5.). 73. mrP be a diamettr and JTO its conjugate, then PM being drawn perpendicular to KO catting the aw FU in G> thfi sect^ngle PM.PG^EC\ For if Cy be drawn parallel to PM, the angle PGNsttyCG (89. L)» bnt yCG+yC<=(CC/=) a right 'iM^gle, and ytC-\-yCt=:A right angle (32. 1.), / yCG+yCt s=yiC-\-yCt; take away the com- mon angle yCtf and the remainder yCG=:yiC, / PGNsx(yCG^) ytC, and PNG:ss Cyt being right angles ; •/ the triangles PGN, Cyt are equi- angular (32. 1.) J and PG : (PN:sl by 34. 1.) C» : : Ct i (Cyzs) PM (4. 6.) 5 ••• PM.PGz=Cn.Ct (16. 6.) iss JSC* by cor. 1. Art. 72. Q. E. D. 74. Join PS, th^n if PO be drawn perpendicular to T^^and Gk perpendicular to PS, Pkss^L. For the angles at k and M being right angles, and the angle kPM common, the triangles PMR, PkG are equiangular (32. 1.) •.• PRiPM :: PG: Pk (4.6.), and PR.Pkz=zPM.PG (16. 6.)=EC^ (Art. 73), •.• (PR-hy Art. 60.) FC : EC : : EC : Pk (16. 6.). But FCzEC : : EC : 4.L (Art. 56.), •.' Pk^iL (9. 5.). O. E. D. 75. If PC; CO be semi'Conjugate diameters, and PN, Om be perpendicular to the axis, then will CN*+ Cm<ss FCK For FC^-^Cm^ : Om* : : FC* : EC* (cor. 1. Art. 67.) : : FC^'-CN^ : Piyr* (Art. 67.) But OC being parallel to tT, and the angles at m and A^ right angles, -.• (29. 1.) the triangles COm, nh 4 4fe CONIC 8BCn(»«. Pamt X. PNT are similar, and (4. 6.) Om : Cm : : PN : NT; / («. «.) Oni< : Cm* : : PN* : NT\ '.' from this and the Brst analqgy (2^. 5.) VC*-Cm* : Cm* :: VC^-^CN* : NT*. But CN.NT i lVr« ::CN:NT (1.6.) / hy iiiTenion Cm* : FC*— Cm* : : NT : CN, and by companUon FC* : FC*^Cn^ ziCTzCNiz (1. 6.) CA^^CT : CiV* But FC*=CN.CT (cor, I. Art 73.)* *•' ''^'- Cm«s= CN« (14. 5.), •.- FC*=: C»r»+ Cm* Q. £. D. Car. 1. Hence FC»— Cy=:0»»», v Cm* : FN* :: VC* I EC* by the first anak^ in the proposition, and Cm : PN:: FC : EC (23. 6.). In like manner, because FC^^Cm^szCN*, V CN* : Om* :: FC« : £CSand CN^t Om:: FC i EC. Cor. 3. Henoe also Cm : FN ::CN: Om, •/ (16. 6.) Cm.Oni zsPN.CN. 76. If PN, Om be perpendicular to the axis FU, and PC, CO semi-conjugate diameters* then will PN' -{^Om* szEC* . For CN' : Om* : : FC* : EC* (cor. 1. Art. 75.), : : FC«- CN' : PN* (cor. 1 . Art. 67.) *•* summing the antecedents and consequents (13. 6.) FC* : Om*'\'PN* n FC*^CN* : PN* :: (Art 67) FC* : EC*, / Om* + PN*=zEC* by 14. 5. Q. E. D. Cor. 1. Because CP and CO are semi-conjugate diameters to. each other, '.* CP will be parallel to a tangent at 0; and Cn*-^ Cr'= (Om* -^PN' 34. 1.=) EC'-, and hence the same relation subsists between the ordinates and abscissas to the minor axis^ that does between those to the msyor axis. 77. CP*'^CO*z=:FC*-}-EC*. For FC*=iCN* + Cm* (Art. 75.), and JBC«=PJV* + Om» (Art. 76.) 3 V FC +EC*=z{CN* + PN* + Cm* + 0m*=:) CP* -^ CO' (47.1.). Q. E. D. 78. If Fe a tangent to the major axis, be made e^pal to the semi-minor axis, and eC be joined cutting PN, any ordinate to the msgor axis in Jf ; then will MN' -k-PN* = Fe*. pART'i. THE ELLIPSE. 473 For the triangles eFC and JUNC being rimilar (2 6.) Ve : MN : : CF : CN, and Fe' : MN' : : CF* : CN^ (92. 6.), / Ve' : Fe'- Jtfiyr* : : CF» : Cr»-CiV' (prop. E. 5.) :: Fe» : P2V* (cor. 1. Art. 67.) } '.• Ve' -^MN* as / 2\r»(l4. 5), and consequently MN* + PN' sFc Q. ED. Cor. Because ArN' + PJV»=:(Fe»=) EC', i£e afeo <Ae fgure to Art, 73. and Oi»' + iW 5=£C' (Art. 76); •• MN=Om, and JIfO being joined^ it will be parallel to the axis vil (33. I.). Hence, if a straight line OC be drawn from the extremity of the parallel JIfO, through the centre C, it will be the conjugate diameter to PC; and henoe by this proposition, having any diameter of an ellipse given^ the position of its conjugate may be readily determined* 78. If PC, CO be semi-conjugate diameters, and PM be drawn perpendicular to CO (see the figure to Art, 73.) then will CCPM^FCEC. Because PNt Om are perpendicular to the axis, and Cy perpendicular to the tangent, *.* (cor. 1. Art. 75.) CN : Om : : FC : EC, and (16. 5.) CN : FC : : Om i EC; and the jtriaogles TCy, OCm, being similar CT :Cy i : CO i (ha (4. 6), the two latter analogies being compounded (prop. F.5.)'CW.Cr: FCCy iiCO: EC; but (because CN.CT^FC\ cor. 1. Art. 72.) TCV : FCCy ::FC:Cy ::C0 : EC; V (16. 6.) FCEC^OCCy^ OC.PM {S4. 1.) Q. £. D. Car. 1. Let FC:=^a. EC^h, PCzzx, Cyszy, then (Art. 77) CO»a:(FC*+ JSC'-PC'ss) a»4.6»-x% v y« = (Cy»=: FC»,EC' aH* CO' + 6*— «•* Cor, 2. Hence, if at the vertices of two diameters which are coiQugates to each other, tangents be drawn, a parallelogram will be circamsei'ib(*d about the ellipse, the area of which is 4C0.PM a constant quantity. See the figure to Art. 58. 79. If CP, CO be senii^oo^jugate diameters, then will FP,SP, szCO». 474 CXnUG 8BCTK>NS. PastX For the trianglM SJH, PRM, FPTvre uoukr, because TF, PM, and iS are parallel^ the apgl^ at r^ ilf, and I right angles, and TPF:sztPS (cor. 3. Art.57.) = PBM (^ l.)| / SP :Si::PR: PM, and FPiFT .: PR: PM (4. 6.)^ these analogies being com- pcmnded (prop. F. 5.) SP.FP : St.FT : : PR* : PM*. But (Art. 78.) rC.J5C= OC.PM, '.' (rC==by Art. eo.) PR : PJf : : OC : JBC (16. 6.) ; and FB« : PM* : : 0C« : EC* (23. 6.) j / from above SP.FP : 8t,FT : : 0C\: JSC* ; but StFT^EC* (Art. 61. B.) V SP.FP=:zOC' (14.5.) Q. £. D. 80. l^et OX be the eoGJugate aad <2o an ordinate to the dia« meter PG, then wUl Pv.vG : Qv* : : PC* : CO*. Draw PA'^ tTn, QH, and Om perpendicular to the axis FIT/ and or parallel to it. Then because PN is par^lel to Qr, or to TN, and <?o to PT, the triangles PTiV; Qvr are equiangular, and (4. 6.) Qr : (rj?=by 34. 1,) Hn:: ' CN PN : J^r, V Qr : j^Bn :: PN. CM (~.i^:5) CN (part 4. Art. 75.) }• bat vhi Cn :: PNiCN («. 6.) j •.• by adding the antecedents together, and tlie consequents together (12. 5.) q in the two last analogies, Qr-f on : CN * ^. Hn+ Cn : : ^PN : 2CN, or QH NT CN NT andQH .iJbi+Cn :: PiV^ : CJV (15. 5.), 'NT .Hn+Cn]* :: PiV : CN* (92. 6.). But (cor. 1. Art. 6?.) FC»-CH« : QH • : : FC» ^CiV' : PJVr* (being each as VC* : EC) / ex aqw («. 5.) CN . VC-^CfI^:—.Hn^i^^::FC'^CN':CN* :: (cor.9. ArtW.) CNNTiCN':-. (15.50 f^TiCNs v (since. C^: JPaktX. THXBLLIPSB. -♦W rc 3 : FC i CT bf eot.l. Axt.nst iribnice, by oor. 9, 90. d, Cy : CT:: CN' : rC' = ^.CiV^*) FC'- CH* or its equal * cr ^_ ^ . _ NT CN ' (16. 6j and {wt 4. cr CT actually squaring and muUiplyuig j) •/ ^k^ . C-AT' — ^^ . Cn' s9 cr C2y CN ~.Hn' (by reduction, and from the figure); •/ CN*'^Cn'=:i CN CT j^.Hn* (by dividing by ^), or NT.CJS'-^Cn^szCNJSn* -, \' (16. 6.) CN'-^Cn' :Hh* :: CN : NT : i (by inversion in th* 7th analogy^ above) CN» : FC'^CN^i :• (16.5.) CN'^Cn* : CiSr* : : H»» : VC'-^CN'-, but (^.6.) CJV : Cn : : CP : C», V CN'^Cn* : CA^« : : CP«-C»» : CP' (part 4. Art. 69.). Also, (by similar triang. and 22. 6.) rv^szHn* : (Cw'ssby Art. 75.) VC'-^CN' :: $»• : CG'j •.• (CP« — Oi7«=cor. 5.2.) Pv.vG : CP* :: Q©« : C0«, and (16.5.) Pto.cG : Q©* : : PC* : CO*, Q. £. D. Cor. Hence it may likewise be shewn by similar reasoning^ that if Q« be produced to meet the curve again in 9» Pv.vG : qv :: PC* : CJT', -.' Qv : qo : : CO : CX. But CO^CX (car. Art. 58.), •-• Qv^qv. 81. The parameter P to any diameter PG is a third propor-* tional to the major axis and conjugate diameter; that is, FU : OX:: OX: P. Let the ordinate Qv passing tfaroi^fa the fbcus F meet the curve 9gain In 9; thea will Qq be the parameter to the dimw* q ter PG, and (cor. Art. 80) Qvx^P. Because (Pv.vGx) PC'-^Cv^: Qv* : : PC* : CO* (Art. 80.) / Qv* : PC'^CV : : CO* : PC«(iHX>p. B.5.) But because Ce is parallel to vP ^^ (Art. 60.) Pe^FC, v PC*^Cv* : (Pe^ wi8e««=) ?€•—€♦" J : P^ : Pr» t ; PC* : Pe* -.* «r ^equto (08. 5.*] 47« COHK aBCTMNS. PAttX. Fe*^er* ;; CO* : (Pe'») FC>. . But JV»— (Se'ae) er»: Pe+er . Pe— er (cor. 5.*2.)=(Ait. 60.) CP. 5P=(Art. 79.) OO'*, •/ Qo- : CO* : : C0» : VC* and (««. 6.) 0© : CO : : CO : VC, v(l5.5.)2CP:«CO::«CO:SrC,]lhati8P:OJir:: OJT: FU orFUiOXiiOXiP. Q.B.D. 82. If two ellipses RPZ, RQZ bave a common diameter RZ, from any point N in which iVP and NQ an ordinate to each of tliem be drawn, then will the tangents at F and Q meet tbe diameter RZ produced in the same point T, Draw TP a tangent to the ellipse RPZ and join TQ; TQ shall be a tangent to the eQipse RQZ. For if not^ let 7X2 meet the curve again in g and draw the ordipates nq, np and produce np, TP to meet in r. Then PN' :pn' :: RKNZ : Rn^Z : : QN' : qn' (cor. S. Art.70.)a •.' P^ i pn :: QN ; qn (92.6.). But the triangles PNT, mT are siBiilar, as are also QNT, qnT; -.' PN i m : : NT : uT (4. 6.) i: QNi qn, •/ PAT :pn:: PN : rn (11. 5.), '.' pn^rn (14.5.)« the less equal to tiie greater^ which is absurd ^ -.* TQ meets the curve no where but in Q, conse- quently touches it in Q. Q. £. D. Cor. Heni», if RZ be bisected in C, the point C will be the centre of both ellipses^ and (cor. 1. Art. 72) CN :CR ::CR: CT. 83. If RPZ be an ellipse, of which RZ is a diameter, atiid if from every point in RZ, straight lines QN be drawn, having aoy given ratio to the ordinates PN, and cutting the diameter RZ in any given angle^ then shall the curve passing throtj^ It, 2, and all the points Q be an ellipse. For since by hypothesis PN iQNiiOC: oC, (22. 6.) PN* ; QN' :: OC* : oC\ But (Art. 80.) RN.NZ : PN' : : CR' : 0C\ / ex aquo (22.5.) RNSZ : QN' :: CR' : Co' which (by Art. 80.) is the property of the ellipse 5 '.- the curve RQoZ is an ellipse. Q. E. D. 84. If PQJIf be the drcfe of cnrvatuie .at the point P in the eUipse PFU, PG the diameter of curvature^ and PH, Pv Part X. THB ELLIPSE. 477 tJie chords of curvfttore paasiog thretigh the centre C> and focus F respectively i then wiU CP:CO::CO: \PH. PK:CO: : CO : ^O. VCiCOxiCO : \Pv. Join PC and produce it to M, and join Gt), HQ and QP; draw the tangent TP, and through I? and C draw Qr, OCK each parallel to TP, then will OC he the semi-conjugate diameter and Qr an ordinate to PH^ and let QP be the arc in its nascent state, which may therefore be considered as common to the circle and ellipse. Then be- cause the angle TPQ=zPHQ (32. 3.)=PQr (29. 1.) and QPr is common4o the two triangles QPr, QPH, these triangles are equiangular (a^.l.)^/ Pr : PQ :: PQ: PH (4,6.) r^' Pr.PH saPO'sr (since the arc QP is indefinitely small, see Art. 35.) Qr'; •• Pr.rM : Pr.rH : : PC* : CO' (Art. 80.), •/ (rM : rH, that is since r and P are indefi- nitely near coinciding) 2PC : PH : : PC : CO'-, •.• (15. 5.) PC : ^PH : : PC : C0\ •.• (cor. 3, 20. 6.) PC : CO : : CO : ^PH. Since CK is parallel to TP, and TP perpendicular to PG (cor, 16. 3.), CKP is a right angle (29. 1.), also PHG is a right angle (31.3.), and the angle ffPG common to the triangkt PKC, PHG •.• these triangles are equiangular, and (4. 6.) PK : PC :: PH : PG :: ^PH : •i.PG. But PC : CO : : CO : ^PH, '.' ex aquo (22. 5.) PK : PC : : CO ; \PH, and PC : CO :: ^PH : ^PG, •.' PK : C(f : : CO: ^PG. Again, the triangles PnK, PvG having the angles at K and r right angles, and the angle at P common^ are similar (32. 1.) j •/ (P«=by Art. 60.) VC : PK :: PG : Pv : : ^PG : ^Pv, and PK : CO : t: CO : ^PG '.' ex aquo FC : CO : : CO : ^Pv. Q. E. D. Cor. Hence VU : ^CO : : ^CO : Pv, that is, the chord of cnrvature Pv which passes through the focus F, is a third proportional to the major axis, and the conjugate diameter, and is consequently equsd to the parameter of the diameter PM.^ (Art. 81.) 478 COme^ SBCntmS. Past X. *> A' ^ £^ ' — >A7 %\j ^/^yv^,. ••/5Cr^ %- -^ ^^^ -A u t ^ 66« If n plmiB out a oone oo u ndtber i^OMot the base nor be parallel to it^ the section wiU be an dUpse. Let ABD be a ooae» and let tbe section VEUK be perpen- dicular to ABC the plane of the geneiBtiiig triangle, VU being their-common section, and the section FiXid be parallel to the base and therefore a circle, and let its common sections with ABD and VEUK be cd and PQ-, let oEKb be a section likewise parallel to the base, bisecting FU in C, having EK and ah for its common sections with the planes F£27J^and ABD. Because ABD and FcQd are both p^^ndicu- lar to VEUK, their common section FQ is perpendicidar to ABD (19. J 1.) b^^^ ^d and therefore perpendicular to VU and cd (conv. 4. 11.), in like manner it may be shewn that EK is perpendicular to VU and ah, '.* EK. and PQ are bisected in C and iV (3. 3.) j and since cd and ah are parallel (16. 11.), '.* the triangles UNt, UCa are ecpiianguhur, and UNiNci. UC: Ca, also AF : IVd : : (CF=) UC : C6, / by compounding the terms of these aoalqgiesT/M^r: Ncfid t: 17C» :CtLC6. But Nc.NdsiFN' and Ca.CbszEO (14.3,), V i/iV.NK : PiV :: l/C* : £C' which (by Art. 67.) is the propwty of tbe ellipse ; therefore VEUK is an elKpse, come- quently if acone be cut by a plane which neither meets the base nor is^paraUel to it^ the section will be an elHpse. Q. £. D. THE HYPERBOLA. DEFINITIONS. 86. If two straight lines JPP, SP revolve about the fixed points F and S, and intersect each other in P, so that SF-^-FF . may alwi^ e«[ual any given straight line Z, the point P witf describe the figuiie PVR which is called am HYPEaaoLii. 87. H two straight lines Fp, Sp revcdve in liloe manner about F and S, so that Fp^Sp may always equal the given stra^ht line Z, the point p will likewise describe an hyperbola pUr^ «this figure and the former, with respect to each otber^ are called OFPOSITB HTrBKBOLAS. Pabt X« THE HYPBBBOLA. 479 86. Tte fisBd poinu Fund 5ftboat whtab tb0 itaiigiit FP and 5P^ jF)» and Sp revolve, are called thb foci. m. \i F, S km joined, the itnogbt line l^K intercepted between the oj^poiite hyperbolae ie allied Tsa major axis, and the pointB {/, FarecaUedTHs ratircirAi. vbbtxoss. 90. If UV be biBected in C, the point C is called thb CENTEB. IT. 91. If through the centre C the straight line £1^^ be drawn perpendicular to the major axis UV, and if from F as a centre, with the distance CF a circle be described, cutting EK in the points £ «9d K» the straight line ££ is called the mi^ob axis. Cor. Hence £C=Ci^ (3.3.). 9^. If JBC=€F, that is, EjP= UV the hyperbola is called EaUII.ATBBAL. 98. H *wtth EK as a major axis, «nd UVt& a minor axis two •pforite hyfierboks GEH, gKh be desoiibed, these are called CO«nEr«AT£ HYPEBBOLAS. 94. Any straigl^ line passing throu^ the centre C, and teiw^ated by liie two opposite hyperboks, Ib called a 9»A1IBTE«. Thtt9 Pp u a diameter to the point P^ or p, 95. A straight iisie soeetlng the ewrve at any p^nt, and which being produced does not cut it, is called a tangent to that poivit. Thus PT \Ba tangent at the point P. 9CL If JP|p be a4iameter, and PTa tangent at thepoint P> and tl^ough the centre C a straight line Hg be drawn paralM 480 CCfMIC SBCnONS. PAMtiL. to the tangoit FT, Uie Due Bg is Gdkd tu cmjugatb PIAMBTER to Fp. 97. If through the focus F a straight line DB be drawn, perpendkmlar to the axis i% meeting the curve in B and D, DB is called thb latds rbctum or principal parameter. 98. A tangent at th^ extremity of the latus rectum produced to meet the axis> is called the focal TANGENT. Thus BT is the focal tangent. 99. A straight line drawn through the point where the focal tangent meets the axis, and parallel to the latus rectum^ is called thb DIBBCTKIX. * TT^iw xy is the directrix. 100. A straight line drawn from any pcHnt in the curve, perpendicu- lar to the axis, is called an ohdi- NATB TO THB AXIS at that pOtUt. Thus FN is an ordinate to the axis at the point P. 100 B. The segments of the axis, ii^rcepted between the ordinate and the vertices of the opposite hyperbolas, are called ABSCISSAS. Thus V and V being the vertices, and FN the mr^^naie, VN and NU are the abscissas. 101. if PG be a diameter and Pr the tangent at the point P» a straight line drawn from any point Q in the curve, puaUel to FT, and meeting FG produced in v, is called an obdinatb to the diameter PG; see the figure to Art. 141. 103. If the ordinate to any diameter pass through the focus, and meet the curve on the opposite side, the ordinate thus produced is called thb parambtbb to that diameter. Thus bd is the parameter to the diameter FG, See the figure to Art. 141. 103. An asymplote is a straight line passing through the centre, which continually approaches the curve, but does not meet it, except at an Infinite distance from the vertex; or, it is a tangent to the curve at im infinite distance. TAid (tte tM Sguni tp. Act,. 134) CX, Cs arc <Ae PBOFERTIES €& TH£ HYPERBOLA. 104. Tlie diflerence of the two strw^t lines drawn from (he Ibd to any point in the curve, is equal to the ms^or axisj that; is, SP^FFss UV^SLFC. (See the figure to Art. 89.) For since SP-^FP is aconstant quamRy in whatever point of the curve P be talcen (Art. 86.)« let the points P, p be sup- posed to arrive at F and U respectively, then SP will become SF, and FP wiU become FF, ••• 5P— FP will become SF^FF; in like manner fy^Sp will (by the arrival of the point p at C7) become FUSU, v SF^FF^FU^SU (Art. 87.) 5 but 5F= FU-^-SUand FU^FU-^FFr* FU-^-SU-FF^zFU-k^FF^SU '.'^SV=i^FFwad SU::^FFi v SP^BP^SF-^FF^SF^SUz^i VFz=i(hn. 90.) 9^FC. Q. B.D, Cor, 1. Hence the foci are equally distant from the centre and likewise ttom the vertices, that b, SC^FC, SV^FF, and SF^FU. Car, 2. Hence SC^UF'\'FPt=9V€-\^FP : and 5P+FP=s 2FC-f-2FP. Car. 3. Because BS-^iTsxVF (see the figure to Art. 97.) =sfS-2FF, and BS^ >* F9 •/ JIP >.«rP and («BFsr> BD ^ 4FF '.* the latus rectum is greater than four times the distance of the focus F from the vertex F. 105. The rectangle FF.FUssEC* (see the figure to Art. 89.) For EC'^FE'-- FC* (47. 1.) =rC'-. FC» (Art. 91.) = FC-^FC. FC-'FC (cor. 5. «.). But FC + ^'C = FU (cor. 1. Art. 104.) and FC'-FCz:^ FF, .• FF. FUz^EC'. Q. E. D. For the same reason C75.SF=EC». 106. The latus rectum is a third proportional to the m^or. and minor asus; or FU: EK :: EKi BD (see the figure to Art. 97.). Because B5»=2FC+FB)« (cor. 2. Art. 104.)=a4rC»-f FB* +4FC.FB(4.«.). And BS«=?FS»+F£» (47. l.)=4FC' + FB»(4.2.), •/ 4FC»+4rC.FB=s4FC»j and FO-^FC.FB^ FC*i \' Fe.FB=sFC« — FC» »(Art. 105.) £ <>, ••• FC E EC : FB (17. «.), ':FU .EK: : £« : BjD (lo. 6.) Q. E. D VOL. It. 1 i « • • 4M €DNICi^CTIOK& BkwmXi Cor I . Henoe J2C» ss^L.TC, vtiA HT* «1»>K - Car. 2. Hence, in the equilateial hypefbdia, because fHtTatt JUT (Art. 99.) ••• BDszEK (prop. A.5.)i that is, the nugor axis, minor axisi and latys lectam, are etfotl to eskth iflkldr. lor. If FP, 8P be drawii from the im to any fioCnt P in the curre, the itraight liaa Pr whkh Useda the angle J^M will be a tafcigent at P. For if not, let Prmeet the hyper- bola again in p, drale FF perpendicular to Pr meeting it in Y, prodooe FV to m, and join pS, pm, afid pF. • In the tHengles FPF, mPT, the angle mPY^FPT by hjrpothdeli, the angles at Frigfat angles by coa6tir0di6n^ and PFcOnnnon, ••• {9SA.) FFasihFj '.' in the triangles Ff^Y, mpY, thfe sides FY» YpctmY, Vp each tOr eMh, and the ineliMkd aogies at Fright aagtasi, V (4.1.) Fp^mp; \' 5p--pF=:5|p— pm. Bat 5p— pFae5P-*PF<Aft Sa) s=5F— Pm=5i», ••• Sp-^pmssSm, and <8fp:a:i$«f+jMn which (30* 1.) i^ afannd^, -/ TP cannot possibly iwet the hyp** b<4a aginn In any point ;p^ \* JP tonchai the curve. Q. E. D. Cor. I. Hence the tangent at the vertex Vis perpendicular to ihe axis SF, See cor, 1. Art 5j'. Cor. 2. Hence (3. 6.) ST. TP .: kP : PP.' lOS. All the diameters of the brperbola ana bi00Bffadi bf dr centre C. (See the figiu*e to AiC W.) Complete the paralldegiam PSpF, then (34. 1.) SpssPP and SPsxpF, •.• Fp-'-^BdbSP^PF, / (Alt. 87) the point p is in the oppo^te hypeibok; join Pp, \* (pai*t 8. AR. 84i. cer.) SC^CFand pCsstCP, and the like may be shewn of any othe^ diameter. Q. E. D. ' Cor^ 1. Hence the tangents PTy pi at the points P ami p are ]pmllel« fcir since (84. 1.) SFF±zSpF and these an^ are bfsecCkl by PT and pi (Art. 167) their halves will bee^lfal; that'is^rPpss l>P, /<^, 1.) Prfelparallel iopi. ?^mmXi THS PYf BRSQ^ m Cor. «. PeipDe, if taffgenits U dirawn at the ejttieiuUiw of tW9 GQi^u^^ dis^meti^ ff , ^* the four Isw^g^Mts will fwii 9.ffkr»lle}pgca^« 109. If CR be paraUel to a tangent PT, cut^iig FB prodimd in H, then will PHas FC. Ihw.^lpaimWtoCiB.aniiwtt;'*. XJw tiWW th» angle PSi^SFV (99. 1.) * yPF (Art. 107 ) « *f5 • • (29. 1.). P/(«;i.)=rJf*«+ it/; the» (€.«.) HL But (XXM*. 1. Ajt. 104.) K?=fc CS, '.* (prop. A.5.) fP4-PK=2P«+ PP. But (cor. 3. Art. 104) P55= «r04-FP,v«P« + FP^^FC-i^FP, V PRtaiVa Q. B, D. 1 10. If the tangent PT he produced, and «traig!it lines SZ, FY be drawn from the foci parpendieukr talt, the points Y and Z will be in the circumference of the circle described ea the major axis UF as a diameter. Join CFand produ^ FY to meet SP In ^m, then since the triangles, mPY, FPY are equal and similar (Art. 40f .), PF* mY and FPrzMPi v Sin = <SP— Pwi=) SP-^BP^sz^VG (Art. 109.). Next, because FC=CS and i>r=r«, •.• (2. 6.) CF is pairallel to Sm, and the triangles PCF, FSm at« equi- angular (29. 1.), s- PC : CF : : FS : «« (4. 6 ) ) but ^^Crcj-IV, •/ CYs:^Sm (16. and prop. J>.^.)x^FC by what has been shewn above, •.• Fand F are in the circumference «f the circle of wbieh C is the centre. Produce I^ to n, then since CY^'^m^ •.' nYssSm, and they are paralteli ••• (33. 1.) Sn, mY are equal and parallel, and if Sn be produced to meet YZ in Z, then SZYJtmYZi;^% right w^les (29. 1.) 5 buj mYZ is a rijjjit angle ii2 ' 484 CONIC SBCTIOKS. PrwitX. '.' SZV is a right angle; that is, the straight line passing through S and n Is pefpendleolar to YZ; and since nY is a diameter of the circle, and nZY a right angle, Z is in the dicun^renoe (31. 3.) Q. K. D. HI. The rectangle FY, SZ= EC'. ' For since Z is alight angle (Art. IIQ.)* and nC, CY meet at the centre C, they are both in the same straight line (31. 3.) •.' FCYzsSCtt (15. 1.). also SC^CF (cor. 1. Art. 104.), and nC=zCY> •• (4. 1.) FY=zSn. But (cor. 3a.3.) S^&iznVS.SU; that is, Pr.5Zr=F5.Sl7=(Art. 105.) EC*. Q. E. D. Cor. 1. Because the triangles fPF, SPZ have the angles at P equal (Art. 107.) and the angles at Y and Z right angles, ihey are equiangular (3^. 1.), and FY : FP : : 8Z : SP (4. 6.), •/ FY : SZ:: FP : SP (16. 5.). But FYJ8Z=EC^i '.-(17. 6) FY: EC:: EC : 5Z, and (cor. 2, 20. 6.) FY^ : KC^ :: FY : SZ^ But since PF : SZ : : FP : SP •/ FF* : EC* :: FP i (SP^) '2VC-^FP (cor. 2. Art. 104.) If VC=:ia, EC:=zb, FP=zx. ajid FY==y the last pn^KH-tion becomes y* : A* : : a? : 2a +x, •/ y*=: 6*3: 2a+j:* Cor. 2. Hence 4Jpy« ; 4EC* : : FP : SP. : : L.FP ; X.5P (15.5). V4fy«: L.FP:: 4EC« : L.5P (16. 5,) ::Lx2KC' : L.SP : : %VC : (SP=) ^FC-^FP (car. Art. 104.) 112. If £D be the focal tan- gent, then wEl the rectangle CECr= VC*. For since (cor. 2. Art. 107.) STiFT :: SB: BF, \' (18. and 17.6.) 5r+JFTor2CF: 5r— Er or2Cr :: SB-^BF: SJSSF, v (15.6.) 4CF« : 4C E Cr :; SB+ BF . SB " BF : (SB — BF. SJB-^BFsz) SB'-^BF]^ :: 5B«- BF^: (I7F«=) 4rC«, see Art. 86. But since SFB h a right angle, 4Ci^ (=SE«)=6'JB*— J5E* (47. 1.). V 4CF..Cr=x4rC* (14. 5.) and CF.Cr^FC*. Q.E. D. -• For «/^C: 2EC : : sEC: /* (Art. 106.) v L.^FC^aEC^ , PAtt X. THE HYPERBOLA. *m Cor. 1. Hence (IT.C.) CF.VC:: VOi CT. Car. 9. Because CT^CF-FT, v €F. CF-^CF.IT^CRCf «FC«5 •/ CF.FTraCF'-FC^^ (Art. 105.) JSC* v CFi EC ECiFT. 113. If from any point P in the carve, PM be drawn per- pendicular to the directrix xy, then wttl FP i PM :: CF: CV. • Join SP and draw PN peri)endicular to the aitis UV, pro- duced, then because (47. 1) SP*=SN' +NP' and FP«=JW« +^^P^ by taking the l atter from the form e r .SP*— JFP«= ^jy« -IW*, that is (cor. 5. 2.) SP-^FP. SP.--FP=:8N^FN.SN>^FN; V (16.6.) SP^FP : SN-^-FN :: SN-^FN : SP^FP. But (cor. 2. Art. 104.) 5P+JFP=3FC+2PP/ also SN+FN=:^Se + CN + FN=:CF+CN + FN=2CN, and SN'-FN=:SF=t 2CF, likewise (Art. 104.) SP^FP=z^VC ; '.• if instead of the terms of the above analogy, their equals be substituted, we shall have ^VC-\'^FP : ^CN. : : ^CF : 2FC, or FC^FP i CN 'i CF: VC :i (cor. 1. Art. 112.) VC : CT, •.• (cor. 19. 5.) FP i {NT=:)PM:: VCiCTiiCFiVC. Q. E. D. Cor. Hence, if P be supposed to coincide with B, FP will become FB and PM wiU=:Fr; *.* the above analogy becomes (FP : PM::)FB:FT::FC: VC. 114. If PF be produced to meet the curve again in p, then will 2PP. Pp=FB.FP-\-Fjf>. Because (cor. Art. 113.) FP : PM : ; FB . FT, •.• (16 and cor. 19. 5.) FP -FB : {PM--FT^) FN : i FB : FT. 3ut (con Art. 113.) FB: FT :: Fjp : pm, ': FB-^Fp : (FT— pni=) Fh : : FB : JFT} / FP'-FB : FN:-. FB--Fp : Fn. But the triangle^ FPN.Fjm are similar, r FN : FP :: Fn. Fp, •/ (22. 5.) FP-^FB : FP : ; FB-^Fp : Fp, '.' (1 6. 6.) FP.Fp^FB.Fp^^FB.FP'-FP.Fp; or 2FP,Fp^FB.FP+Fp. Q. E. D. . Cor. Hence, if PJ5=t FP^X, and Pp=sx, we shaU have — 2 jsr+« 11 2Xj;=?.A'+x, and— =--^=~+jr. , 115. If c be the co-sine of the angle PFU to the radius l^ then wiU FP : EC : : EC : VC-^cCF. For (Art. 113.) FP : PM : : CF : VC, \' (16. eyppyc^^ CF.PM^ (34. l.y CF.TN=CF.TF+FN=^CFTF^^CF.FN. But (cor. 2. Art. 112.) CF.TF=EC\ and (Art. 63. Pai^ 9.) PA^^ I 1 3 4M cONtc st/(yti6m. t^«#£. FF :: ±c : Ij'v itTz^ ^e.Pf(l6.6.), and ¥CF.1^:= ^c.FP,Cf, •/ frdfii the flrst'cqaa«6ii by sabstittitidn FP.VC=i tC^-^cFF/t, or /!P. r(?+(f.K>.CF=lSC*, that is /P FC+d.CF =EC«5 .• (16. 6.) FF: EC:: EC: FC-^c.CF. Q. E. 1>. 116. If PN Ife aa ontinatt to tke nu^r aods FC7, thm& t?iU VN.NUi PiV« : : F C* : £( » . P<* (Aft. 113) iF7?>. 5P- PFtti ^^i5^ IVF. N8^ NF, V (15. 6.) m-^NFi BP^PF: : SF^PF: NS^NFi^hkh by I^Utltution (as in tbe htttf part Of Art. il5.) becomes ^CN : itrC : : ^FC+fi/P : ^CF, •.• CJV : FC : : FC-^FP : CF; whence by coiApoaition (ir. 5.);6iid divbion (17. 5.) ^« obiyn the iblldwio^ analogieli, viz. 15f«# CJT- FC : FC : : VC+FP^CF: CF. SMkdIlf CN^ VC .VCii VC-^FF^ CF: CF By adding^ tbe anteo^cfepts Udd cotts^qu^tltd Uk the flwt, abd siibtnictin^ in tba s^cobd (IS. and 19. 5.) we have CN^VC: VC :: CN^FP^CF :<:F+FC :: FP^-rt^iCF-k^VC CN-^FC :FC :: FP^CF-^CN : CF-FC :: FP-^FN: CF-^VC '.' coropo unding the ratios (23. 6.) gJg-- FC . C?y+FC- : FC^ : yTF^TpN. FP^FN : CF+FC.CF^IfC, or m. m: VC^t: iFP-^FN^^)PIf^: (CF«^F^7*«.byArt. 106,) BO •/ alteirftalely FiV. JVi7 : Pm iiVC^i ECK Q. E. D. Cor. Heaoe» because VN. NU= CN^^VC . CN-k- FCa* CJV«r- ^C* (oor. 6.8.) •/ by substitution C!iy«— FC« : FiV* : : FC* : iC«i wherefore, if VC^a, ECs^h CN^:^, wd Pi^^ry^ ^ shall have a*— a« : y» : : a« : I^, whence y*;±:^ . JrZ^. « • . 117. If two hyperbolas F/\ VQ b« described m the same mioor axis, having eC Mi, EC respectively for the temi-roinor axes ; and if NP be produced to Q, then will QN • FN • • afc:£C For (Art. 116.) ( ^'f • ^^ ^ ^^' : : ^'C» : JEC* V kar li^iio CJV* : FJV» : ; gC» : EC*, &nd r«6. t5. j QN : FN ; : Cor. L Hence it may be shewn, as in lxUf%. Ihil tangents at P and Q will meet the axis produced in the samepoint T; that the area VQN : am^ FPN : : eC : EC, and that if i^ be aqy point in the axis, the area VQF : area FPF : : «C : EC. Car. 2. Hence, if VQ be an equilaterai hyperbola, or VCss^eQ (Art. 92.) J then since VN. Ntl » QN' : : TC r #C* (Art. 11«.) FN.NU=z QN' (prop. A.6.) 118. In the equilaterai hyperbola, the latus rectum is equal to the minor axis, that is ^FbssSteC. For since (Art. 105.) Vt.FU^eC*, if the point N be supposed to coincide with JP, the expression (cor. 9. Art \\T^ VN.NU^QN* will become VF.FU^Fb\ %• F6'=cC',>t= eC, and gPiasaeC. Q. £.D. Cor. 1. Hence it again appears that the miyqr axis, minor axis, and latus rectum of an equilateral hyperbola, are e<^ual to each other. Cor. «. Hence, because (Art. 106.) VC.ECiiEC : BF, %• (cor. 2, 20. 6.) VC xBF'.i VC* : EC'. But (Art. 116.) VC : EC* :: VN.NU or CN'-^CV : P^^ v Fli.NU or CA^«- CF'.PN'::VC:BF.. 119. If Fit be an ordinate to the minor axis £C, then wili Cn* + EC' '< FBP : : £0 : FC' (see the fisBowuig figure.) fkir (34. 1.) Fm=xNCBod Ckm^NP \' (eor. Art 116.) P*f f!f FC : 0»' : : FC^ : £C, *.* by addh^ anteccdenU and CQHr sequentB Pn' : Ca'.-fCC' : : FC^ ; £C^' And by invei]aiQn Cn^ 120. If PN any ordinate to the majm* axis be produced to meet the conjugate hyperbofta In Bf, then wiH ii^^*— PjW*=» 2EC». . . • > • • • • ; V ^ •■• .^.- • . . • . • • ' . • ^ • - ' '. • » 1 . » ' * ' • . ■ ■ \ ii4 4tt CONK SBGTIOBli. Fait 1, B0CM»« (cor* Alt. 116.)a«-JBC»:n^:: JBO : VO •/ (16. 5.) a* -EC* : EO i: (n6«=) CJV» : FC*, ittul (17.5.) C6*-.2EC« : CJB« :: CN*^CF* : CV^ : : (by alternation and inversion in cor. 2. Art., lib.) FN^: EC*, V (9.5.) a«-2i;c«= 2£C«, but (34.1.) a- niyr«— pjvr«=: 2 EC« 5 and in like manner it maybe shewn, that if , hn be produced to meet the hyperbola VP in (he point w^ fr6«— n6»=2FC'. Q.E.D. 121. If PT be a tangent at the point P, tben wfll OJV^cr zsiFO. ^ ' Because (cor. 2. Art. loy.) STi TF :: 5P; PF, v divi- dendo et componendo) Sr— TF : 574- TF : ; SP'^PF : SP4- PF\ that i s (sec Art. liS.) %CT r gjF ; ; 2F C : 5P+ PF But (Art. 113.) SN^NF.SN^NF^SF:Iff.sFTpF, -.• since SN^NF=.SF, SP^PF^^rC (Art. 104.)> a nd SN^NF^ 9CH, by substitution 5E.2CJV^=2rC.SP+PF / (16.6,) 5E ; 2VC:: SP-hPF : ^CN; but it has been shewn that ^CTiSP : : 2FC ; SP^PF '.• fjr dr^uo 2Cr : 2rC : : 2FC ; 2CW that is CT',VC:,VC: CN, •.• (17. 6.) CW.CTs FC». Q. E. D. Cor. 1. Because NT^^rCN-^CT, •.• €N.NTs:CN.CN^ CT« tiV» - CN.CTsz CN' - FC . Cor, 2. Because in the equilateral hyperbola CN^-^FC'ss PN* (because FCs^EC, see the cor. to Art. 116.) • • CN NT= (fN'-^-FC'^z) PN\ Ccr. 3. Hence also^ in the conjugate hyperbola En, if ps be an ordinate to the axis Eg, and pT a tangent at p, then will C«.Cr=EC'. Past X. THE UYFES»9Lh. 48B 1^. If Pit be an ordinate to the minor v&% BC,9Bd the tangent Pt meet EC in t, then will Cn.C^s£C'. Be'cause (Art. 121.) CN.CT=FC', v (17. 6.) €N : FC : : FC : Cr, •/ (cor. 2, 20 6.) Ci\r : Cr : : CN' '• VC\ / (17. 5.) NT: CTi: CN' — FC« :VC':: (because bf cor. Art. 116. CN' "-VC^ : PiV* : : FC : i:c», by alternation) PJV* : EC. But the triangles TPN, TtC are similar, •.• (4. 6.) NT : CT:: PN: Ct; '.' (from above) PN : Ct:: PN» : EC% '/ (16. 6.) PN.EC^ rrO.PiV, or EC'=^Ct.PN; But (34. 1.) P.VssC», v Oi.C/= -EC. Q. E. D. Tor. Hence, because Cn.CfssEC'icor, 3. Art. 121.) •.• 0».C« = C«.Cr and Ct^Cty that is. if the perpendicular Pn cut the conjugate hyperbola in p, and tangents be drawn at P and p, the points i and T where they meet the minor axis^ will be equally distant from the centre. C; and conversely, if Ct=CTy the perpendicular Pn will pass through the point p, 123. The same things remaining nt :nT:: nP' : np». For by the preceding corollary Cn.Ct:=iEC\ '.• (17. 6.) Cn : EC :: EC: Ct, \- (cor. 2, 20. 6. ) Cn:Cf :: Cn' : EC, v (componendo et dividendo) Cfi+CTornf : Cn-^-CfoTnt:: Cn» +EC' : Cn' — ECK But (Art. 119.) Cn'-^-EC' ; Pn* : EC r FC» and (cor. Art. 116.) Cil?-r-C£' : up' :: EC : FC' •.• (11.5.) Cn*+EC' : nP' :: Oi»-«£C» : «pS •.' (alternando) Cn« + £C* : Cn' ---EC' ; : nP' : np» 5 that is, nt : nt : : nP' : up'. Q. £. D. 124. The normals at P^and p will meet the minor axis in the same point g. • For the angles gpT, gPt being right angles nP^=znt.ng and np'2s:nT,ng (14.2.) \- «P« : «p* ; : nt.ng : nt.ng,^y {Art, 123.) nt : nT :: nt.ng : nTng : : ng : ng ; that is, the normals at P and p cut the minor* axis at equal distances from rt or in the satiie point g, Q. E. D. Cor. In like manner it is shewn, that if NP be produced to meet the conjugate hyperbola in n^ the normals from these points will meet the major axis in the* same point G. 125- If CR be parallel to a tangent at P, and MPG perpen- dicuiar to it* then will the rectangle PMPG^EC'. Let PN be the ordinate^ and di*aw Cm perpendicular to the tangent Pt* Because in the triangles PTG^ CTt, the angles at 496 CONIC 8BCT10NS. Pakt X. 7d. * / 1 / 1 X— / > <? K S /: / ^ ^ \ /^ v\ ^ Ni a P and C are right angle»^ and the vertical angles at Teqaal, \*Ctm ssPGi^t and the angles at m and N being right angles^ the* remam- ing angle tCm^sNPG, \' CnU, PNG are equiangtdar^ and (4. 6.) Cm : C* : : P^ : PG, •.• (16. 6.) Cm.PG^Ct,PN, but Cmz=:PM (34.1.), •• PM.PG^Ct.PNx=:EC' (Art. 122.) Q. la. D. 126. If from the point P the normal PG be drawn, PF joined, and OH drawn perpendiccdar to ^ PP, then will PH=^L. Produce GP, FP to M and It, then because the angles at H and M are right angles and those at P vertical, the triangles PHG, PUR ape equiangular, and (4. 6.) i^PG : FH : : PH : PJf , V (16.6.) Pfl.P«:='PG,Pilf=^ (Art. 1^50 JEC'=;?{'»or.l; Art 106,) +L.FC. But (Art 109.) Pfi= TC, v PHJ^R=xi iZ.PR, ov rH=^. Q.£.P. 127. If CR be paraUel totliBtangisnt at P, and PN, RH perpendicular to the im^or axis, then vm. CN^ ^CE* v^VC* . Draw tR an ordinate ta the unnor aais, and produce it ijb Q, and draw the ordiBale Qui, Then (43or. Art. 116.) Cn^^CF^ : Qn* :: CN'-^CV* : PN and Qf-^VC : RM» : : CN't^ CF' I PN'. But (Art. 120.) Qr»— 2^»=2CF% •.• Qr'^VC^ =5 FC + flr«= (34. 1 .)^rC» + Cif*, •/ by substitution yC'JtCW :RH' ::CN' — rC» ; PN'. But the triangles CitH, TPN are similar, •.• (4. 6.) EH : CfT : : PiV : TN, and (22. 6.) JR/P: CW :; PN". TN\ ••• eo? €equo VO + CH^ s CH' ; : CN'-'FC : TiST^ ? : (cor. 1. Art. 121.) CN.NT : TN* itCNi TN, ••• by conversion (prop. E. 5.) FC'^CH* : F€» : : CN : (CN- r2yr=) cr -. : (l. 6.) C2^»^ : ^iV^.CT. But (AitrWl.) rC» = CN.CT. •/ (14.6.) FC' + Cff'±=:CN', r CN\-€H'=^ B. D. «. • H ^ X y Tm _^f^f' Q V s wt^^*^* -^^^ * (^^^ f^Mr" .^.. \ c ^ • • • • . NvX -^T. "K «2r ^ i y^ J % • Pah* It. THfi trft^tOlA. 491 C(ff. Hence CH^ {^tClf* *^r&) t PN' tr FC* : «C» (cor. Art, 1 16.) and fff : PIf ::f^€: EC (24. (?.) 1^8. The same f hixigs remaifting CN -. KH : : VC : EC, For (Art. 127) JP^C' + CJ?' f RW :: CN'-^P^C' : JPiV' :: (cor. Art. 127.) ^C* : £Cv and rC» + C^' = CiSrs v C2^» : RW : : rC : EC* and (22. 6?.) CNiRlt: : VC : EC. Q. E. D. 12a If CR be parfllkl to the tangent PTand PN, RH ordi- nates to the major axis, then will RH'-^PN's^EC'. Because (Art. 128) CN' : RH* :: FC' : EC : : CiV^»— VC' : PiV' by subtracting antecedents and consequents VC* : RH'^PN' :: CiV^» — FC : PN' : : FC« : EG*, V (14.5.) JIH« ^-Pm^zECK Q. E. D. Cor. Because rv*-Ct>«=liH«— P^" (34. l.)s±JEC«> and CiV* ^CH^=zyC* (Art. 127.)» •• i^ ^i* be conjugate to CR. CR k also conjugate to CP, ' 130. If CP and Cil be semi-conjugate diameters^ then will CP«— CB«=FC«— J5:c«. Because (Art. 127.) CN^—CH^^FC^, and (Art. 129.) RB^-^PN'^EC*, ••• by subtracting the latter from the former CN* + PN»-'CH*'-RH'==:FC*'--BCi. But (47. 1) CP'=: CfPj^^PN*, and Cfi«= CH« + RH\ •.• (Ci^" + PiST* - C£P4-JI£/«=)CP«— C«»=KC*— EC*. Q. B. D. 131. The same things remaining, if PL be drawn perpendi- cular to CR, then will CR,PL=FC.EC Draw Cm parallel to PL, then because (Art 128.) CAT : RH ::FC: EC, \' (16 5,) CNiVCiiRH: EC. Bat the triangles CTin, RCH (having the alternate angles RCH, CTm equal t29. 1.), and the angles at H and m right anglers) are similar, and (4. 6.) CT: Cm : : CR : RH, '.' (compounding the two latter proportions,) CiVT.CT (=by Art. 121.) VC : VCXm :: RtfCR : RH.EC : : CR : EC^ \' (15. 5.) VC : Cm . , CR i EC, '.' {\6.6.)=:CR.Cm=^VCEC i but Cm= PL (34. 1.), •.• CRPL ^VCEC. Q. E.l>. Cor. 1. Hence (16.6.) VC : PL :: CR : £C, and (22.6.) rC« :Pi*::Cft* :£C^ Cor. 2. Let VC—a, EC=::b, CP^x, and PL=y; then because ah^CR4f, '•* V*==^^- But <Art. 130.) ;!P»-.C£«ara»-i», .• 4» CONIC SSCFIONS. Part X. ■ Cor. 3. Heooe^ If Umgento be dravm «t tbe ^tremities of any two conjugate diaoieten (cor. % Art 108.) a paraUelogram wOi be formed, and all the panillelogramB that can be formed by the tangents in thb manner are equal to each other, as appean from the foregoing demonstration, being each equal to 2FC2£C= VU.EK: see the figure to Art. 133? 133. If C^ be a semi-conjugate to Cl\ then wiU FP.FS Let FP and CA be produced to meet in R, and draw FY, SZ per- pendicular to the tangent at P. Then the triangles FPY, PRL, and SPZ being equi- angular,- (4. 6.) FP :FY::PR: PL and SP : SZ :: PR : PL, '.' compounding ' these proportions FP.SP : FYSZ : : PR^ I PL' :: (Art. 109.) VC^ : PL^ : : (cor. 1. Art. 131.) CJ* : ECK But (Art. 111.) FYSZ:=EC\ v (14.5.) FP,SP^CJ^. Q.E.D. • 133. If through the vertex V the straight line €k be drawn equal and parallel to the minor axis EK, and from the centre C straight lines GM, Cm be drawn through e and k meeting any ordinate {PN) to the major axis, produced in M and m-, theq willPM.Pw=rc». See the following figure. Because (cor. Art. 116.) CN'-^VC' : PN' :: FC» : EC and (4. aid 22.6.) CiV* ; iVilf» :: TC/ : (FcV=:) EC, v (19.5.) FC : NM'-^PN' :: FC=' : EC', •/ (14.5.) W'- Pjy^ = EC* = Fe*. But (cor. 5.2.) iVM« - PiV* = NM+PN. NM-PN^PMPmi •.• PM.Pm=z Ve^ Q. E. D. Cor. 1. Hence, in like manner pfn.pM may be shewn to be equal to Vk^=::Ve^, ••• PM,Pm=ipm.pM ; and if any other line Paht X . THE HYPERBOLA. 493 J^ be drawn parallel tm^Mm cut- ting the curve in Qq, then by similar reasoning it is shewn tibat FM.Pm^QX,Qx=qx.qX. 134. The straight lines CM, Cm continually approach the curve but do not meet it at any finite distance from the centre C, and therefore (Art. 103.) CM and Cm are asymptotes to the hyperbola. Because PM.Pm^iFe'^ (Art. 133.), PH « 4- (Art. Ill Part 4.) that is PM and Pm are inversely as each other, or as Pm increases, Pilf decreases ; and when Pm be- comes infinitely great, PM be- comes infinitely small 3 that is, at any finite distance it does not entirely vanish. For the same reason as pM increases, pm decreases ; and at an infinite distance ^XHn C becomes infinitely small, . but does not vanish >. '.' CM and Cm continually approach the curve, but do not meet it at any finite distance, they are therefore asymptotes. Cor, 1. Hence it appears that CM. Cm are likewise asympto- tes to the conjugate hyperbolas > for Te, Vk being respectively equal and parallel to EC, CK, %• (33. 1.) Ee, Kk will each be equal and parallel to VC; and by the same reasoning it is plain that CMt Cm continually approach the conjugate hyperbolas, but do not meet them at any finite distance from the centre. Cor.^. If VE be joined, the right angled triangles FfiC, FeC having CE= Fe and VC common, are equal in all respects (4. 1.) •/ VE^eC, and the angle CVE^FCe. In like manner it foUows that VKzs:Ck, and since £C= CIT (Art. 108.) / the right angled parallelograms CEeF, CKkF are equal (36. 1.) and consequently similar, and the four diameters Ce, BF, Ck, KF are equal, •.• (cor. Art. 241. Part 8.) CD, De, ED, DF, CZ, Zk, KZ, ZFnre equal to each other 5 and because FkzsCK iszEC \' (33. 1.) Brand Ck are parallel 5 in like manner it is plain that JlTrand Ce are paralkL 494 CONIC sscmom. P4&t:s. IW. The pasitkm of anjr dMuD^lor ^^ nsftBCt to the «9(i9 li^iiig: given, that of its conjuigaie inajr ^ ^etermiiiedi for (Art. 133.) NM^--FN*^EC*, md (Art. 1^,) RU^^PN*:^;^ EC^ -r NM^RH, \' if CP be a semMliMiieter^ fX^ w m^ nate at P to the major axis produced to the point ilf in the asymptote, and MR be drawn peraMel to I9ie nugor aaaa, tlien if RC be joined, MC win be oot^jtigale toCPhj^ eat. to ^rt. If9, And in the same manner the position of 'tiie oonjugatte to any other diameter is known. Q. fi. I>. 136. If a straight line Xx be drawn in any position cutting the curve in Qq, and the tangent TPt be parallel to it, then win QX.Qxz=iPT.Pt See the figure to Jrt. 141. Through (^ and P draw ¥f», Zt fMrpeodkiihr to the 4ids| then the triangles XQfV, TPZ, wQx, and zPi being similar QW : QXi: PZ : Pr(4.6.) and Qm : Qx n Pz : Pt^ these propor- tions being compounded QW.Qw : QX,Qx t: PZ.Pz ; PT^t. But (cor. Art 133.) QfV.QwzsPZP^- (14.5.) QX.Qx.=PT.Pt Q. £. D. Cor. By simihu* reasoniiig gjr.^jr^P7JP<>// QXQx=f4;i?.9X 137. The same eonstrutftion miMMiing QXs^x, For QX Qx= QXQ9 -h qx^QX.Qq + QX.qx. And ^x.g3r= qx.qQ+QX±zqx.qQ'^qx.QX; •/ (since Qiir.<?j?=:^x.gJr by the preceding corollary) QXQq + QX.qxss qx,qQ + qx,QXj from these equals take away QX.qx, and the remainders are equal, viz. QX.Qq:s:qx.Qqy divide both sides by Qq, and QJTs^x. Q. E. D. Cor. Hence, if ^ move parallel to itself so as to coincide with Tty the points Q and q will each coincide with P, and Q^ will vanish -, also QXand qx will coincide with, and be equal to TP and <P respectively ; •.• (since QX=zqx) T/>=<P, •.• QX.Qx :=zTP*. ■ 138. The same construction remainiiig if through P, the clia- meter Gv he drawni Qvssqv. Becanae the trimi^ei XvC, TPC are dimilar^ and ako xoC, tF<^; / (4.^.aiwi W. 5.,) fJT: Pr : : i?C: PC : : we: PL BMt PTszPi by the pneoeding eor. *.- (14. &.) o2r=s«a;. But (Art. 1370 CJ^=^*i •• ivX-QX:^»x^qx or Q»5»«v. 'O; £. D. Cor. Hence cJT' -t?jQ» =Pr». For (eor. 6. 2.) nX^-^^vQ^zs; vX^vQ . »JK^.f©Q=QXQa?=(cor. Art. 137.) TP*. FUtX. THS 13TPBSBDU. 49» IS9. If PB. VD bo .pmlW to u Mymiitolt Cs, tbea nlB PB.Cltssf'D.CD Sae4iuJigwnto,.M.l»i. TlAough the pomta F and F dnw the suaigbt lines ek, Umtmtix perpendlcalitr M theftxli CN, and fd, Vo psnUel to CX DacauH the triantlea Plffi; PeD, i^dm, and Ttufc u« liorHar, -■-(4.«.)i'a:l'M:: PJ) : FAaad (Prf») CHi Pn u (»'«>=) CD ^ n aad bf oonponnding i>if.C« : PJtf.Pm :: VD.CD : fe.yk. But (Art. 1S3.) P«.P«»:(r«'s«)re*ffc, -., (14.5.) PB.CH=yD.CD. Q. E, D. Cor. X. Hence, became (cor. S. Art. 134.) CD^FD, v Cor. t. Hence aho, if PSbe produced to meet the conjugate hyperbola in R, RH,Ca=^ED.CD==FD.CD=CJy* or riJ«. Cor. 3. Hence, because PH.Ca={CD's=) RB-CH, hj dU vidiog these equab by CB, PB=fRB. I40. If PT be a tsogent at P mestinp Ibe asymptotes is T and a; andCRbejiMDed,tikaaniltCAaitd TX* be paiaJlel and CJt=TP-PX. For P^^bdngperallcl tnCToMwdeof thetmngle CXT. :• (8. 6) PX : PTi-.XB: «C. But (cor. Art. i37.) PXi=PT, V (prop. A.6.) ABr«HCr.Inthe triai^lesPJfff, flCHthare are the two sUea XB, BP= (;B, BR respeotivelyj and the vei^ tical angles at B equnl (15. 1,) ■-■ PX^iPTif) CR; also tl»e angle HRC^BPX (4. 1.) ■. CR and fPX are pandlel (27. 1.) «. E. D. .Ml. If PG «ni DO be con- JBgate dime- teib, and Qt> as orainatetoPG, then will Pv.vO ;'<ie»:: CP' : At the point P draw the tangent Pr, and f»T)dure the ordinate vQ to oteet the asymptote in X. 4M CXmiC SECTIOK& PaktX. tbetf^ mace CD, PT, aad vlTaieptfiifel (Art. 96> ibl.), TP is therefore parallel to Aa a akle of the triangle XO^ */ (3.6.) r» : »X^: : CP : PT, afid <«. 6.) Co* : eJI? : : CP* : FT^ •/ (19.6.) rp*-»CP * : Pjr^— Pr» :; CP* : PST*. But l.Cb^— CP«as: (cor. 6. «.) Cb— C*P . Op + CP= P».t>0. «. (cor. Art. 1S8.) vJf« ^ Oi>«=t: PT* or «;if»- PT»a: ©1^. S. (Art. 140.) P T* CD ; *.' subetHuting theee results^ for their equals in the above aaa* logy, it becomed Pv.vG : <?»« : : CP^ : CD^. Q. E. D. Cor. Hence Pv.vG « Qtr*. 14^. The parjimeter P to any diameter PG is a third propor- tional to the major axis VU, and .the conjogate DO to the dia- meter PG; that isrP : DO ::D0: VU. Let ilfiii be the ordinate to the diameter PG which passes through the focus F, which 19 therefore the parameter P (Art. 10^.) 5 then will Mv^^P (Art. 138.). Then because CD, PJIf are parallel, Cr : CPi: Fe : Pe (9. 6.), and Cr^ : CP^ : : F^ : Pe« (««.6.), ♦.' dividendo C^^^CP^ : CP« : : F^^P^ : Pe\ But (Art. 141.) Pr.rO : Mr^:: CP' : CD» j \- alternando {PrrGzs:) Cr*-CP» : CP« : : Mr^ : CD\'/ M r^ : CD* : : Fc« — Pe* : Pie'. But /c* — Pe» a Fe^Pe . Pe-fPc (cor. 5. 2.) =: PP. PS (Art. 109.) ::=:CD' (Art. 13^2.); / 3fr« : CD' : • CD' ; Pe» and (22. 6.) Mr : CD : : CD : (Pesby Art. 169.) PC; '.• (15. 5.) 2Mr or P : DO :: DO : Ptf. Q. B. D. 143. If two hyperbolas PQq, PW^ be described on the same diameter GP and from any point N in it the ordinates .A^Q^ A7F be drawn, A'Q shall have a given ratio to NW. In GP produced take any other point n, and from it draw the ordi nates nq, nw ; then (cor. Art. 141.). PiV:iV<^ : Pn.nG :i NQ* : nq' : « NH^^ : nw^-, \' NQ : nq : iNfF : nw (22. 6.), and A^Q : NfV :: nq : nw" (16. 5.). O. E. D. Cor. 1. Hence, as in the parabola (Art. 29, and cor.) and the ellipse (Art. 69. cor. 2.) the area NQP: area NWP in a given ratio. Abo, if any point v be taken in the axis and vQ, vW be joined, the area PQt> : thcarea PWv in a given ratio. J ^AKT X. THE HYPERBOLA. 4d7 Cbr. 2. Hcnce^ if FQq be an hyperbola, and ham erery point N, n, &€. in the diameter, ordinatee NQ, nq, &c. be drawn, and if fitiaJgbt lines NW, nw, &c. be drawn irom the points N^ n, &e. making a given angle with NQ, nq^ &c. and having a given ratio to each other, the curve FWio passing through P, and the ex* treoiitiea of those line$, will be an hyperbda, iiaving FG for its diameter. For NQ* : NW* : : nq* : nw« : : PNNO : Pn.nO, that is, nq^^PN.NG (cor. Art. 141.) which is the property of the hyperbola. 144. If two hyperbolas PQq, PWw be described on the same diameter PG, and NQ, NWan ordinate to each be drawn from the same point N, tangents at Q and fV will intersect the dia* meter PG in the same point T. Let QTbe a tangent at Q, and join TW; TW]& a tangent ; for if not, let it meet the hyperbola again in to, draw the ordi- nates nw, nq, and produce nq to meet the tangent TQ produced in t. Then because the triangles QTN, sTNare similar, as also TNfF, Tnw, v (4.6.) NQim (:: TN : Tn) :: ISWinw, But (Art. 143.) NQinq:: NWx nw, \' NQ:n$:: NQ:nq •.• (9. 5.) ns^nq, the greater equal to the less, which is absurd; *.* T9V which noeets the hyp^i)ola, cannot cut it ; T9F is therefore a tangent. Q.E.D. Car. Hence, if GP be the major axis of the hyperbola PQp, since (cor. 1. Art. 117.) tangents at Q and FT will in like man- ner meet the axis 6P in the same point T, -.* (Art. ISl.) CN.CT szCP*, '.' (17. 6.) CN :CP::CP: CT. 145. If PM be the diameter of curvature at the point P, and PL, PR chords of curvature, the former passing through the centre C, and the latter throogfa the focus F, then wiU AfP pro« duced be perpendicular to the semi-conjugate diameter EC, and PCiCE::CE:^PL PH'.CBiiCEi^PM FC:CE:iCE:^PR FirMt Let FQ be a nascent arc common to the hyperbola and circle of curvature, draw Qv parallel to the tangent PT, join VOL. II. K k 496 CONIC SECTIONS. Part X. CF,9nA draw the chords PQ, QL, LM, MR. Then the triangles QPv» QPL having the angle QPv commoo, and (99. 1.) PQv^ rP0=(32.3.) QLP, are equiangular, •/ (4.6.) F» : jPQ : : PQ : PL, '.' (l7.6.) Pv.PL ^siPQ^l but since the arc PQ is indefinitely small, Qv and PQ will be indefinitely near a coincidence, and there- fore may be considered as equal, •.• Pv,PL^ PQ'=zQv*, also for the same reason oC=s PC. But (Art. 141.) Pv.vG:{Qv*=)Pv,PL :: PC' : C£S V (15.5.) (rG=) 2PC: PL :: PC: ^PLi: PC* : CE*, / (cor. 2, 90.6.)FC:C£::C£ : 4^FL. Secondly. The tri- angles PCfl, PML having the vertical an- gles at P equal (15. 1.) and likewise the angles at H and L right angles (31. 3. and construction), are equiangular, and PH : PC :: PL : PM ii^PLi^ PM ; but by the former case PC : CE :: CEz ^PL, / ex aquo PH : CE : : CE i^PM. Thirdly. The triangles PKH, PMR are »milar (15. 1, 31. 3. and construction) / PK : PH :: PM : PR (4. 6.) : : i^PM 1 1 PR (15. 5.). But, as in the preceding case PHiCEzzCEi i PM, \' c» aquo {PKsiby Art. 109.) FC i CE : : CE : ^PR. Q. E. D. Cor. Hence, because 2rC : 2CE : : 'ZCE : PiJ by the above, and ^FC:^CE::^CE : the parameter (Art. 142.) '.* the chord of curvature PR, passing through the focus, b equal to the parameter. Pakt X. THE HYPERBOLA. 499 146. If a cone ABD be out by a plane PFp which meets the opposite cone Md in any point U except the rertex, the section FFp will be an hyperbola. Let dHhKA be the opposite cone, let BD be perpendicular to pP ; bisect UV in C, draw VL, CF, US, and bd parallel to the diameter BD of the base, then will the section passing through FL, CF, US, and bd ]}arallel to the base be circles (13. 1^.) and HK, Pp the inter- sections of the cutting plane with the planes of the circles HbKd, pBPD will be parallel (16. 11.). Draw Cr a tangent to the circle TFs, then (36.3.) BN.ND=PN^ and bn.nd=:Kn^, also 8C.CF=zCT\ Now the triangles FNB, sCF are similar, as are UND, UCF, •.. (4. 6.) VN: NB:: FC'.Qt and UN : ND:: UC: CF, / (com- pounding these analogies) FN. UN : BN.ND : : FC.UC : Cs.CF. that is, FNNU : PA« : : FC» : CT^ '.• (Art. 116.) the figure PFp is an hyperbola, Cthe centre, CFthe semi-miyor axis, and CT the semi-minor axis. Q. E. D. Cor. Hence the section HUK will be the opposite hyperbola to PTp and similar to it -, for Fn : nd :: FC : Cs and Un : nb ii UC : OF, •.• (compounding) Vn.nU : dn^nb : : UCVC : Cs.CF, or (as above) Fn.nU :nK^:: FC^ : CT*. The foregoing are the principal and most useful properties of the Conic Sections ; a branch of knowledge^ which is abso- lutely necessaiy to prepare the Student for the Physico Mathe- matical Sciences; many more properties of these celebrated curves might have been added, if our prescribed limits had per- mitted ', but it would require a large volume, to treat the subject in that comprehensive and circumstantial manner, which its im- portance demands) we must therefore refer the reader^ for a MO COMIC SECTIONS. 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