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HUTTON’S HISTORICAL INTRODUCTION TO HIS NEW EDITION OF SHERWIN S MATHEMATICAL TABLES: TOGETHER WITH SOME TRACTS ON THE BINOMIAL THEOREM AND OTHER SUBJECTS CON- NECTED WITH THE DOCTRINE OF LOGARITHMS. VaOoks Ulivh Ese tt. ESO UN) DON: PRINTED BY J. DAVIS, AND SOLD BY B. WHITE AND SON, FLEET-STREET. MDCCXCI, ri ¥ 1 ~~ ¥ eo v hee A. | wp t: ’ a j a Lad hi ny ae id at i This ty eb, Ff ! Faeries : ar. ree - n ier ¥ a i cost ‘ei i % ‘ 1 . A een J ’ . Whe 4 : PER . a *s re Tek UES iy Lite rer dy ‘ Se ak MERLE PLS One ett | i 7 ; ‘ “4 a er a. tr Bae PN. & ee) “ 7 ¢ wo bas j “ Ais on r] 7 rt’ i) ik | ' i . 4 Yh fi ! : ‘ 4 Ja ie 4 if ig: A. ‘ / ve, 4 A aa ; 1 y ’ Fahd a) Ay en » + 4 j ‘ ‘ “t ae us - ; § ' Babe. r 7y 3 ‘ : , ‘ ‘ ) e . t } ‘ . ' } a , ‘ ® ‘ ; ; ‘ § x \ oo ‘a Pe : " Po ¢ f “e f ; ¢ ‘j ’ ’ as , a +¥1 Ae | a > < 4 ~ ; a ‘ ’ ‘ ¢ 4] Lee “ a4 | - 4 a -— 2 he . ~~ we Raye if ; * + ’ j b ‘ao J 4 ; , - . ' si . > ’ ‘ «~ - ’ { ‘ ha iA ; : ? 7 P A. * > P ” as * st we ney. & ‘ ® - ag? | ' ; . : 3 : ¢ ‘ 4a . i] bey thi » ~* : i ) ' , P4 é & * M : ta Ff . tl oy Pe 4) Vy * et . +a : ; ; : aa ea hh i F) ° oe or ' a ' e. Ai) Re eke ee ’ ; . ; ‘ ' ] ‘2 . * ; 4 rs f ‘ ar ‘peed 1 i i ie oe Cte3e S/o. 7'2 3 ei > OF TFHE ORS eE HE mete GO NONE? OF be’ UM: E. J} TRACT of Mr. James Gregory, of Aberdeen in Scotland, written in Latin, and intitled, Nicolai Mercatoris Quadratura Hyperboles Geometricé demonftrata; which was firft publifhed at London in the year 1668, with fome other {mall tracts, under the title of Exercitationes Geometrice; con- taining a geometrical demonftration of the method of {quaring the Hyper- bola, by means of an infinite feries of decreafing quantities, then lately publifhed by Mr. Nicholas Mercator. In pages 2—5. II. Another Tract of the fame author, intitled, 4valogia inter Lineam Meridianam Planifpherii Nautici et Tangentes Artificiales, Geometricé demonftrata: feu, Quod Secantium Naturalium Additio efficiat Tangentes Artificiales: Item, Quid Tangen- tium Naturalium Additio efficiat Secantes Artificiales: Item, Quadratura Concho- éidis, et Quadratura Ciffo-éidis : taken from the fame collection of traéts, in- titled, Exercitationes Geometrica. In pages 6—15. Ill. A third Tra&t of the fame author, intitled, Methodus componendi Tabulas Tan- gentium et Secantium Artificialium ex Tabulis Tangentium et Secantium Natura- lium, exaétiffimé et minimo cum Labore: taken from the fame collection of tracts, intitled, Exercitationes Geometrica. In pages 16—18. IV. An vi COUN TAS IN it toe IV. An Extra& from a Letter of the fame author to Mr. John Collins, formerly Secretary to the Royal Society of London: dated on the 15th day of Fe- bruary in the year 1670-1, and firft publithed in the Commercium Epiftolicum Domini Fohannis Collins et aliorum de Analyfi promotd, in the year 17123 con- taining fome infinite feriefes relating to the tangents and fecants of circular arcs, and to the Logarithms of the ratios of fuch tangents and fecants to the Radius. In pages 18 and 19. v. An Extract from Mr. Ifaac Newton’s firft Epiftle to Mr. Henry Oldenburgh, Secretary to. the Royal Society of. London; with a direction to communi- cate the contents of it to Mr. Godfrey William Leibnitz : dated on the 13th of June, in the year 1676, from Cambridge, where Mr. Newton y ho was afterwards Sir Ifaac Newton, knight) was at that time Profeffor of the Mathematicks upon Dr. Lucas’s foundation : containing a difcovery relating to Logarithms. In page 20. VI. An Extract from an Epiftle of Mr. Godfrey William Leibnitz, of Hanover, to Mr. Henry Oldenburgh, Secretary of the Royal Society of London ; with a dire€tion to communicate the contents of ic to Mr. Ifaac Newton: dated the 27th day of Auguft, in the year 1676: containing a paflage re- lating to Logarithms. In pages 21, 22. VIl. An Extraét from Mr. Ifaac Newton’s Second Epiftle to Mr. Henry Olden- burgh, Secretary of the Royal Society of London; with a direétion to communicate the contents of it to Mr. Godfrey William Leibnitz: dated on the 24th day of October, in the year 1676: containing fome difcove- ries relating to Logarithms. In pages 22—26. VIII. The twelfth Chapter of Dr. John Wallis’s Treatife of Algebra, intitled, Of Logarithms, their Invention and Uje. Publifhed in the year 1685. In pages 27—34. IX, A Ga Of TN) Tr END Tr. S: Vil IX. A Letter from the Reverend Dr. Wallis, Profeffor of Geometry in the Uni- verfity of Oxford, and Fellow of the Royal Society of London, to Mr. Richard Norris; concerning the Collection of Secants, and the true Di- vifion of the Meridians in the Sea Chart. Publifhed in the Philofophical Tranfagtions of the year 1686, Number 176. In pages 35—4I. X. Logarithmotechnia: or the making of the numbers called Logarithms to twenty-five places of figures, from a geometrical figure, with {peed, eafe, and certainty. By Euclid Speidell, Philomath. Publifhed at Jondon in the year 1688. In pages 44—75. XI. An eafy Demonftration of the Analogy of the Logarithmic Tangents to the Meridian Line, or Sum of the Secants; with various methods for com- puting the fame to the utmoft exactnefs. By Dr. Edmund Halley. Pub- lifhed in the Philofophical Tranfactions for the year 1692; Number 219. In pages 76—84. XII. A moft compendious and facile Method of conftructing the Logarithms, exemplified and demonftrated from the nature of Numbers, without any regard to the Hyperbola: with a fpeedy method for finding the number from the Logarithm given. In pages 84—og1. By Dr. Edmund Halley. Publifhed in the Philofophical Tranfactions for the year 1695, Number 215. XIII. Notes on fome of the more difficult Paffages of the foregoing Difcourfe of Dr. Edmund Halley. In pages 92—122. By Francis Maferes, Efg. F.R.S. Curfitor Baron of the Court of Ex- chequer. . XIV. An viii GO EM Ay Bi ; XIV. An Appendix to the foregoing Traé&t of Dr. Edmund Halley upon Loga- rithms, being a direct method of computing the logarithms of ratios, either in Briggs’s fyftem, or any other that may be propofed, by the help of Sir Ifaac Newton’s Binomial Theorem, without the intervention of the Hyperbola, or the Logarithmick Curve, or any other geometrical figure, and likewife without having recourfe to the method of Indivifibles or the Arithmetick of Infinites. In pages 123—152. By the fame. ; XV. A Demonftration of Sir Ifaac Newton’s Binomial Theorem in the Cafe of Integral Powers, or Powers of which the Indexes are whole Numbers; to- gether with an extenfion of the faid demonftration to Sir Ifaac Newton’s Refidual Theorem, relating to the powers of a refidual] quantity as a — 4, in the cafe of the integral powers of fuch quantity, or when the indexes of the faid powers are whole numbers. In pages 153—169. By the fame, XVI. A Demonttration of Sir Ifaac Newton’s Binomial. Theorem, in the cafes of roots and the powers of roots, as well as in the cafe of integral powers ; publifhed by Mr. John Landen in the year 1758. In pages 170—175. XVII. An Explanation of the foregoing Demonftration of the Binomial Theorem, in the cafe of the fraCtional index =, invented by Mr. John Landen. In pages 176—193. By Francis Maferes, Efq. F.R.S. Curfitor Baron of the Court of Ex- chequer. | | XVII. A Difcourfe concerning the Binomial Theorem, in the cafe of fractional powers, or powers of which the indexes are fractions ; containing a demon- {tration of the faid theorem in that cafe of it. In pages 194—344. By the fame. , 6 ALK Ses re, Wy ey Tes et? ix XIX. A Difcourfe concerning Sir Ifaac Newton’s Refidual Theorem, or Theorem for raifing the powers of the Refidual Quantity 1 — w, in the cafe of frac- tional powers, or powers of which the indexes are fractions; containing a demonttration of the faid theorem in that cafe of it. In pages 345-378. By the fame, XX. A Method of extending Cardan’s Rule for refolving the Cubick Equation. y+ ger, orgy +7? =7, tothe refolution of the Cubick Equation . . 3 3 qy — jy? = 7, when = is of any magnitude lefs than Ae or than - x ae ‘or when ¢ is lefs than “2 x Iv, by the help of Sir Ifaac Newton’s binomial and refidual theorems, which have been demonftrated in the two preceeding difcourfes. In pages 381—440. By the fame. XXI. A Method of extending Cardan’s Rule for refolving the Cubick Equation y? — qy =7, inthe firft cafe of it, or when r is equal to, or greater than, £ 3 ht re is equal to, or greater than, to the other cafe of the fame 29/9 faid rule is not naturally fitted to refolve; provided that the abfolute term r (though lefs than wee be greater than 4/2 xX £1, or that ot (though 3 3 3 lefs than =) be greater than ~ x oe or than eo In pages 443—575. . . . . TF -« q3 . equation, in which r is lefs than , or — is lefs than a and which the By the fame. N. B. This Tra& was firft publifhed in the Philofophical Tranfactions, for the year 17783; but is here very much enlarged. XXII. A Conjeture concerning the Method by which Cardan’s Rules for refolv- ing the Cubick Equation x3 + qv = 7 in all cafes (or in all magnitudes of the known quantities g and r) and the Cubick Equation «3 — gx =r in the firft cafe of it Cor whens is greater than “it Vou. I. b were or “is greater than £) LaevCah hoe rie oa: @ NM 2, EM’ DS were probably difcovered by Scipio Ferreus of Bononia, and Nicholas Tar- talea, or whoever elfe were the firft inventors ofthem. In pages 579—586. By the fame. N.B. This Tract was firft publifhed i in the Philofophical Tranfactions for the year 1780. XXIII. An Appendix to the Tra& contained in the foregoing part of this fecond volume of Mathematical Tracts, in pages 153, 154, 155, &c, to page 169, intitled. ** A Demontftration of Sir een Newton’s Binomial Theorem in ‘« the cafe of Integral Powers, or powers of which the Indexes are whole <¢ Numbers :” containing an Inveftigation of the Law by which the co- efficients of the third and fourth and other following terms of the feries which is equal to any integral power of a binomial quantity, are derived from the co-efficient of the fecond term of the faid feries ; grounded on a probable induction from particular examples. In pages 587—~591. By the fame. BE ReR Ayer aA, In the figure in page 3 the line XZ ought to pafs through the point Y. In page 4, line 6, inftead of *¢ NCDH,”’ read “ NODH.” In the fame page 4, line 9, inftead of ‘¢ infinitas,” read “ infinitis.” In the fame page 4, line 15, inftead of ** egualis,” read ** equales.” In page 55, lines 9, 13, 14, 15, 16, inftead of “ trapezia,” read “ trapezium.” In page 67, line 17, inftead of “ to bis doctrine,” read “ according to his doétrine.”” In page 84, line 17 from the bottom, inftead of * conntryman,” read “ countryman.” In page 85, the laft line, inftead of “ zufinite infinite,” read ** zufinité infinite.” In page 88, line 10 from the bottom, inftead of ‘* demonftratiou,” read demonftration,’” In page 100, line 11, inflead of *¢ the ratio of 22 x 4,’’ read “ the ratio of 22 x 24.” In page 105, line 8 from the bottom, inftead of “ Suppo/e b to be,” read * Suppofe k to be.? . e I In page 110, line 6, inftead of ** ——-,”” read * ; ”? 2 2 2 And in the fame page 110, in the laft line but one and the laft line, inflead of * the fecond 1 -- &e,”’ read * the feries bl — oe &Se,”? In page 111, note ix, line 3, inflead of «3 — read 5 — - ee : “i | sai: mix In page 112, line 4, inftead of ¢ yee ele read ae I I I I aE Beal be all pa um m 2 - : ‘ bd bd And in the fame page 112, line 15, inftead of ¢ Utne read ¢¢ See —+-—d —+-—d Mm i 2 . ° . H Z And in the fame page 112, line 10 from the bottom, inftead of * : >” read §¢ pants And in the fame page 112, line 8 from the bottom, dele the mark ) of a parenthefis after ad d (3 2 , and fubftitute a femicolon ; in its ftead. e e 1 —T!I mn—-2 == T FF, In page 154, line 2, inftead of ‘or to— X Xi me? read OF ta Hie K me m 2 3 I 2 3 4 In 9? ° Er VAR Pak ONE. ea In the fame page 154, line 3, inftead of “ fewaral,”’ read ** feveral.’? In the fame page, line 14, inftead of 1 x am,’ read “1 x a”. In page 156, line 11 from the bottom, inftead of “ or am” read “or a”? In page 157, line 5, inftead of * a + bm,” read “a + 3)”.” In the fame page 157, line 7, inftead of ** 1 -+ 1}m,’? read 14% In page 158, line 5, after the words ‘¢ to do which” place a comma. In page 164, line 7 from the bottom, inftead of * 73) read « S39 In page 190, line 7th from the bottom, after ‘* G, H Lk,d, geo infert the word * egual.’? And in the fame page 190, line 6th from the bottom, after the word * refpectively” infert the mark of a parenthefis, ). In page 197, line 8th from the bottom, after the word “ refpeétively,”’ infert the mark of a paren- thefis, ). In page 202, article 11, line 1ft, inflead of “ 1 x” read “1 +4 4.” Tt In page 226, line 8, inftead of « x y” read « =.” In page 322, line 4th from the bottom, in the numerator of the laft fraction in the faid line, inftead of €¢ 48,000” read ** 48,000 «””. teiy 2207 ” et 207 by In page 341, line 6, inftead of peed aed read ¢¢ saat § I In page 363, line 10th from the bottom, inftead of «1 — xz” read 1 + x|\2.” In page 388, article 9, line 6, inftead of Ue one terms,” read “other following even terms.” In page 396, line 3d from the bottom, after —, &c, dele the word which. 5? In page 418, the laft line, inftead of *¢ the fign” read “ the fign +.” In page 439, the laft line, inftead of * ,/3(a — 4/06” read “ ,/3a — ,/ — 0b.” In page 449, in the running title, inftead of “ THE RESOLUTION OF CUBICK EQUATIONS,” read 4¢ RESOLVING THE CUBICK EQUATION.” 4 In page 584, line 11, inftead of * 29 5s ren $6 W455 3 And in the fame page 584, line 20, inftead of, “if #3 — ga = 1, or 1,” read if x3 — gx, orr.”? FROM MR. JAMES at (olay Sica le PRERCITATIONES GEOMETRICG, RELATING TO Pet. ee gee Fe pee eye yrs PRINTED AT LONDON IN 166% Vor. Il. B Neh Mes Ro eiG Ar a Oty eae QUADRATURA HYPERBOLES GEOMETRICE DEMONSTRATA. PROP. I. S! fuerint quantitates continue proportionales a, B, c, D, E, F, &c, numero infinite, quarum prima & maxima a; erit a — B ad « ut A ad fummam omnium ; hoc enim paffim demonftratur apud geometras. BP: REOTP seus Eifdem pofitis que antecedente; dico a + B efle ad a ut a eft ad exceffum omnium a, Cc, E, G, &c, in locis imparibus, fupra omnes B, D, F, &c, in locis paribus: eft enim dictus exceflus fumma feriéi infinite continue proportionalium in ratione A ad c, nempe A — B, c— D, E — F, &c, & ideo (ex precedente) uta —c ad a vel a? — B? ad a*, itaa — B ad fummam dicte feriei, quam vocamus z; & priores analogie terminos applicando ad a — B, a + B eft ad 2 aa ut A— Bad z, & ideo az + Bz = a’, & proinde a + B eft ad a ut aad z exceflum omnium 4, ¢c, E, G, &c, fupra omnes B, D, F, &c. Quod demon- {trare oportuit. PROP. GRECORIL ,DEMONSTRATIO, &c. 2 PURUOGR sw Tite Sit hyperbola sz3, cujus vertex zB, afymptote AR, a4, afymptote RA du- catur parallela Bx, & altera ad libitum inter rectas BK, RA, utrique parallela yp; dico yp efle fummam infinite feriéi continue proportionalium, cujus primus terminus BK = Ka & fecundus KD; eft enim BK — KD = aD ad BK ut BK ad py ; & ideo ex hujus prima patet propofitum. Eifdem rectis RA, BK, ultra punc- tum « fiat parallela 34; dico rectam ,A~-HD K A A 34 equalem efle exceffui omnium ter- minorum imparium fupra omnes terminos pares infinite feri€i cujus primus ter- “ minus kB, & fecundus x4: eft enim kB + k4 = a4 ad BK ut BK ad 34; & ideo ex hujus 2 patet propefitum. | Pik Oise IV. Sit s8KH fpatium hyperbolicum, contentum a curva hyperbolica sz, afymptote portione HK, & rectis. sH, BK, alteri afymptote parallelis, pofito 8 hyperbole ver- tice : fit parallelogrammum BKHs, et producatur B5 in R, jungattrque KR que 5H fecet in 6: deinde continuetur feries infinita continué proportionalium nempe 5H, OH, LH, NH, & fic deinceps; fitque 58KH parallelogrammum, K6H trian- gulum KH trilineum quadraticum, KNH trilineum cubicum, & ita deinceps in infinitum. Dico f{patium hyperbolicum, sxus xquale effe dicto parallelogrammo, dicto triangulo, una cum infinitis illis trilineis, quorum omnium fummam vo- camus w. Si figura BKHs & w non funt equales, fit inter illas differentiaw; & dividatur recta HK in tot partes equales a rectis afymptote ra parallelis, ut rec- tangula (ab illis & portionibus rete KH contenta) figura BKHs circumf{cripta, nempé vH, zD, differant 4 rectangulis figure BKHs infcriptis, nempe YH, BD, minore intervallo quam « ; hoc enim fieri poteft ab indefinita divifione recte KH. Quoniam B eft hyperbole vertex, parallelogrammum Bk ar eft equilaterum ; & proinde recta 6H ad. libitum eft equalis recte HK, clmque 5H, 6H, LH, NH, &c, fint continué proportionales in infinitum, ex hujus 3 erit recta sH #qualis fumme omnium, & parallelogrammum sp equale fumme omnium parallelo erammorum 5D, 6D, LD, ND, &c, in infinitum ; atque fumma omnium parallelo- grammorum 5D, 6D, LD, ND, &c, in infinitum, major eft parallelogrammo 5p una cum portione trianguli 6FpH una cum portione trilinei quadratici tcp una cum portione trilinei cubici NcpH, &c, in infinitum, quoniam predicte portiones dictis parallelogrammis infcribuntur, & ideo parallelogrammum sp majus eft parallelo- grammo §D una cum di¢tis portionibus ; eodem modo demonttratur parallelo- grammum yk majus effe rectangulo 7K und cum infinitis numero portionibus FKD, CKD, OKD, &c, & proinde rectilineum svyzKH majus eft quam a. De- B 2 indé 4 GREGORTI DEMONSTRATIO indé recta rp eft zqualis recte DK}; atque 7D, FD, cD, oD, &c funt rece continué proportionales in infinitum, & igitur recta yp eft equalis ipfarum fumme, & parallelogrammum xp equale parallelogrammis 7H, FDH, CDH, ODH, &c, at fumma,parallelogrammorum 7H, FDH, e€DH, opH, &c minor eft quam reCtangulum 7#*una cum portione trianguli 6FDH una cum portione trilinei qua- dratict L¢DH una cum portione trilinei cubici ncpu, &c, quoniam dicta paral- lelogramma dictis portionibus infcribuntur, & ideo paraHelogrammum yx minus eft parallelogrammo 7H una cum dictis portionibus ; eodem modo demonftratur parallclogrammum gp minus efle parallelogrammo sp una cum infinitas nu- mero portionibus FKD, CKD, okD, & ideo rectilineum xy7BKH minus eft quam w: quatuor igitur funt quantitates, quarum maxima & minima funt rec- tilinea svVYZKH, XY7BKH, intermediz autem w & {patium hyperbolicum ssxH; & ideo differentia intermediarum, nempé & minor eft quam differentia extrema- rum, quod eft abfurdum, ponitur enim major; nulla igitur eft differentia inter figuram sBKH & w, & ideo equalis funt, quod demonftrandum erat. Conf. 1. Et proinde fi fuerit feries infinita quantitatum continué proportiona- lium in ratione KB ad KH = 6H, cujus primus terminus eft parallelogrammum BH; erit prinius terminus + 4 fecundi + 4 tertii + + quartt + 4 quinti + &c in infinitum = fpatio hyperbolico ssKH, hoc enim evidenter fequitur ex qua- dratura trilineorum. Conf. 2. Si autem ultra x fumatur {patium B34K, pofita 34 parallela recte zx, & fit feries infinita quantitatum in continua ratione BK ad K4, cujus primus terminus eft parallelogrammum kQ; erit primus terminus — 4 fecundi + 4 tertii — + quarti -- &c in infinitum = {patio hyperbolico 834k : poterit hoc confec- tarium eodem fer¢ modo demonftrari geometricé ex fecunda conclufione hujus rertia, quo antecedens ex ejufdem conclufione priore ; utrumque autem ex me- thodo indivifibilium Cavalleriana nullo negotio demonftratur ; fed quoniam magni funt momenti, placuit methodum rigorofam adhibere. Conf. 3. Quod fi k4 = ku, & fuerit feries infinita quantitatum in continua ratione BK ad K4 = KH, cujus primus terminus eft BH = B4; erit exceflus {patil sBKH fupra f{patium 834k = toti fecundo termino + 4 quarti -+ + fexti + + octavi + &c in infinitum: nam ex primo confectario, fpatium sBKH = primo termino + + fecundi + + tertii -+ + quarti + &c in infinitum: & ex fecundo confectario {patium B34K = primo termino — i fecundi + 4 tertli — 4 quarti + &c; at manifeftum eft horum differentiam = toti fecundo + 4 quarti + + fexti + + octavi + &c; & ideo patet propofitum. Conf. 4. Eifdem pofitis que in antecedente confectario, manifeftum eft {fpatium hyperbolicum sa43 = duplo primi termini + } tert 4+ % quinti + } feptimi +- 2 noni + &c in infinitum, PIR O Bie Ve Sit hyperbola cz, cujus vertex E, & afymptote aB, ay; in qua fumantur duo fpatia hyperbolica ad libitum H1rR, KLYv, contenta a curva hyperbolica,. una afymptota & rectis alteri afymptote parallelis: dividantur rete RT, vy, bifariam in s & x punétis. Dico fpatium nuirr effe ad fpatium KLyy, ut ns x AO rs? X AO Rs5 X AO Rs? X AO ay ee . vx X ao ae wae Ae + Foie + &c in infinitum ad - + QUADRATURE HYPERBOLES, 5 Né x Vist S BQ PLO IN ler Px vx? xX AO vx5 X AO vx? X AO aoe ‘ ar ae Sea ae BtyCae + &c in infinitum. Afymptote ap fiat recta parallela 0; fitque ut asad ar ita ao= z0ad am; & ut asad Ar ita ao ad AQ; fimiliter, fit ut ax ad Av ita ao adan, & aoad ap ut ax ad AY; mani- feftum eft mo = oc & No= op. Dnuttis rectis Mc, ND, PF, ag, evidens eft (ex hyperbole proprietatibus) {patium ceam effe equale {patio uitR & {patium DFPN {patio KLYV; at patet (ex confectario 4 antecedentis) {patium ccam effe ad {pa- ; Mo3 MoS Mo? No} Nos um DFPN, ut MO —, + &c ad no cae? at 3A07 ie 5aot as 740° + i 3.4027 if 5.A0* + xo? : Tot + &c, que analogia eadem eft cum propofita, quod demonftrandum erat. Conf. Hinc manifeftum eft (ob analogiam inter f{patia hyperbolica N A & logarithmos) differentiam inter logarithmos numerorum a, B, effec B ad differentiam inter logarithmos numerorum D, E, (pofito c medio arithmetico inter a, 8B, & F medio arithmetico inter D, E, item N diffe- Oo D er : oor N n3 né rentia inter c & a, & o differentia inter F & p) ut— + — +— F 7 c 3c3 5c> 07 N7 re) o3 o5 . eo ee te + &c, & ideo fi ponatura =p =1, FS Fv hinc patet methodus inyenventh iobarithmotn quemcunque ex uno dato, abfque ulla hyperbole confideratione, fed calculo plerumque nimis laboriofo. Quod fi ponatur A = 999, B = Joo1, cum datis logarithmis, item p, £, numeri dati majores, unitate vel parvo aliquo intervallo differentes, nullo negotio invenietur differentia logarithmorum numeris p, £, debitorum ; hac methodo non difficulter computatur integra logarithmorum tabula ad quotvis notas. Facilé quoque deducitur (ex 3 confect. antecedentis prop.) differen- G tiam fecundam logarithmorum numerorum in ratione arithmeticaG, H,1, P 4H effe ad differentiam fecundam logarithmorum numerorum in ratione arith- I metica K, L, M, (pofito p differentia inter c, H, & q differentia inter kK, L) Beene Ft ead & 4 SX 4. x, non diffimili feré “ ut H? 7 ope mp 3 ot ca L2 + 2.4 ae Cc; MMs rere Gis BD methodo determinantur differentie logarithmorum tertia, quart, quinte, M & fic deinceps ; fed non eft opere pretium illas profequi : prime enim differentiz compofitioni logarithmorum abundeé fufficiunt, ANALOGIA AOUN SA OTE Oo. Ga INTER ~LINEAM MERIDIANAM PLANISPHERII NAUTICI, TANGENT. EVWS VAR Tie CoA ia GEOMETRICE DEMONSTRATA, &c. PROP. I. THEOREMA. IT circuli quadrans rez, cujus pars fit arcus Qt: fuper arcu Qi imaginetur portio fuperficiei cylindrici recti talis nature, ut ({umpto in arcu ar quolibet puncto ©) perpendicularis ad planum Tz@ ex punéto o ad fum- mitatem portionis fuperficiei cylindrice excitata femper fiat equalis fecanti arcus 0g, Deinde fit mixtilineum ALYTQ talis nature, ut (ducta in eo recta xp radio ar parallela & arcum quadrantis fecante in puncto ad li- bitum 0) reéta px, fecans arcus oq, & radius Tq fint continué proportionales. Dico mixtilineum ayTq effe zequale dicte portioni fuperficiei cylindrice: fi pre- dicte figure non funt zequales, fit inter illas differentia «, & dividatur recta yr in tot partes equales a rectis Bx, EV, radio ar parallelis, ut (completis reCtangulis ax, DV, GT, CX, Fv, At,) differentia reCtilineorum ABDEG HTY, CDFGAQTY, fit minor quam «; manifeftum eft enim hoc fieri poffle ab indefinita divifione recte yr. Ducatur in puncto o recta tangens op, fitque in rectam GREGORII ANALOGIA, &c. 7 EV perpendicularis os: triangula ops, TMo, rectangula ads & o funt fimilia ob zqualitatem angulorum pos, orm; nam ab angulis equalibus nempe rectis por, sox, auferendo eundem angulum soT, relinquuntur anguli equales oe XOVs atque ob parallelas ox, ar, angulus xor equalis eft angulo orm, & ideo equalis funt anguli pos, orm ; & proinde ut os ad op, ita or ad rTM, ut autem or ad Tm ita TM ad xp, & igitur rectangulum os in xp, nempe rectangulum DV, equale eft rectangulo Tm in op, fed rectangulum TM in op majus eft portione fuperficiei cylindrice recte infiftente curve ON, quoniam (TM exiftente maxima ejufdem portionis altitudine) recta po major eft quam curva No, ut patet ex prop. 1. Geomet. part. univerfalis; & proinde rectangulum pv majus eft quam portio fuperficiei cylindrice*recte infiftens curve on: eodem modo demonttratur er majus portione ejufdem fuperficiei infiftente curve nq & ax majus portione fuper curva 10, & ideo re¢ctilineum aBpEGHTY majus eft integra propofita portione fuperficiei cylindrice ; idem quoque rectilineum majus eft mixtilineo aary. Tangens in puncto wn ducatur tnx, 4 parallelis proximis utrinque terminata in L, K; ex prop. 1. Geomet. part. univer. patet rectam tw feu NK minorem eile arcu on, fed (ut hactenus) demonftratur rectangulum cr vel Fv equale effe rect- angulo nx vel NL in KT; atque re@angulum NL in KT minus eft portione fu- perficiei cylindrica recta infiftente curve oN, quoniam (TK exiftente minima ejufdem portionis altitudine) recta tn minor eft curva on: & proinde rect- angulum Fy minus etiam eft quam predicta portio: eodem modo demon- ‘ftratur reCtangulum Ar minus effe portione fuper curva nq & rectangulum cx minus portione fuper curva 10, & ideo rectilineum cpFcAery minus eft in- tegra propofita fuperficiei cylindrica recte2 portione, idem quoque re¢tilineum minus eft mixtilineo aery : ex predictis ergo manifeftum eft quatuor efle quantitates, quarum maxima & minima funt reCtilinea AaBDEGHTY, CDFGAQTY, intermediz autem fuperficiei cylindrice portio propofita & mixtilineum AQTY 5 8 ideo differentia intermediarum minor eft quam differentia maxime & minime, differentia autem maxime & minimz ex conftructione minor eft quam &, & pro- inde differentia intermediarum nempe fuperficiei cylindrice & mixtilinei aary multo minor eft quam «, quod eft abfurdum, ponitur enim &, non igitur differunt mixtilineum aary & propofita fuperficiei cylindrice portio ; "8 ideo. equalia funt, quod demonttrare oportuit. PROP. ff, THEOREMA, Sit circuli quadrans QML, fitque mixtilineum AHLQ@M talis nature, ut (ducta recta ad libituin HN radio rqparallela & quadrantis arcum fecante in kK) recta HN equalis fit fecanti arcus LK, fitque mixtilineum ABLQMA talis nature, ut (producta arbitraria nH in B) recte LQ, NH, NB, fint continue proportionales : : ‘deinde fit femihyperbola 1pz cujus axis MA, vertex 1 & afymptoton MnzE: ducatur ad libitum radio rq parallela recta nB, curvas LKM, LHA, LBA, fecans in punctis K, H, B; & per punctum u ducatur radio Merecta parallela HF hy- perbolz occurrens in punéto p. Dico fectorem hyperbolicum imp equalem effe femifi figure BLQN, qua figura (ut in antecedente demonftratum eft) equalis eft 8 GREGORIL ANALOGIA AA A N a eft fuperficiei cylindrice conflate ex omnibus fecantibus arcuum infinitorum o% plano remain debitis fuis pundtis o normaliter infiftentibus. Ex Gregorii a S. Vincentio dogtrina duétuum, truncus cylindrici recti fuper figura LHNQ@_refectus & plano bafin feminormaliter fecante in recta qm, equalis eft femiffi cylindrici recti cujus bafis eft figura tpnq_& altitudo recta tq, quoniam tre femper eft ad NH ut NH ad NB: manifeftum autem eft eundem truncum oriri ex ductu trianguli rectanguli ifofcelis rem in figuram LHNQ, hoc eft ex ductu trianguli rectanguli ifofcelis wim in rectangulum vLen & ex ductutrapezii Te Iw in figuram HLV; atque folidum factum ex ductu trianguli rectanguli wim in rectangulum vLen equale eft prifmati cujus altitudo tq & bafis triangulum 1AM, quod fic probo; Mc’: MI*:: MI”: KN’, & per converfionem rationis MG* : MG? — MI* = GD? :: M1*; MI” — KN? = NQ?, & ideO MG: GD:3 MI: NQ, ut autem MG ad DG, ita que- libet mp = pr ad ductam Ps ipfi cp parallelam, & ideo ut pr ad Ps ita m1 ad NQ, cumque hoc femper fiat, manifeftum eft prifma cujus bafis Arm & altitudo im equale effe folido ex ductu trianguli wim in rectangulum vLan: item foli- dum factum ex ductu trapezii ré1 in figuram uv equale eft cylindrico cujus bafis eft figura 1DA & altitudo im, quod fic probo; GM: GD :: MI: Qn, & per convertionem rationis GM : DF ?; MI: GH; ducatur recta ad libitum y£& radio Mq_parallela & curvas fecans ut in figura, eritque GM: DF i: XM: y&, & ideo XM = XY: yt: MI: GH = xz, & ideo x¥ X XZ = MI X y&; ut autem ante demonflratum eft cm effe ad pF ut M1 ad GH, eodem modo nunc demon- ftrari poteft xm = xy effe ad e& ut M1 ad x9, & ideo xy x x9 = MI X oof, camque XY X Xz = MI X ¥&, fi ab zqualibus equalia auferantur, xy x z9= MI X ay, camquehze equalitas femper contingat, manifeftum eft trapezium Tetw ductum in figuram Hiv efficere folidum zquale cylindrico cujus bafis 1DA & alti- tudo 1m; & proinde totus truncus fuper figura HLQn ortus ex ductu trianguli rem 2 In nes INTER LINEAM MERIDIANAM, &c. 9 in eandem figuram HL QNn eft zqualis cylindrico cujus bafis eft feCtor hyperbolicus mip & altitudo m1, atque idem truncus zequalis eft dimidio cylindrici cujus bafis eft figura stan & altitudo eadem 1m, & ideo femiffis cylindrici fuper figura Br QN cum altitudine 1m equalis eft cylindrico cujus bafis mrp & altitudo eadem 1m, eft igitur figura BLan dupla fectoris hyperbolici mip, quod demonftrare oportuit. CONSECTARIUM. Hinc fequitur, quod figura shen femper fit dupla logarithmi differentiz inter tangentem & fecantem arcus KL, pofito radio 11 loco unitatis, quod fic probo. Ex punctis 1, D, in afymptoton mz demittantur perpendiculares 1c, pg ; ex de- monftratis in Circ. & Hyperb. Quad. manifeftum eft feCtorem mip effe a2qualem figure 1Dre, item figuram tpqe effe logarithmum recte pa pofita 1c unitate ; ut autem Ic ad.Dz ita 11 radius ad pF differentiam inter tangentem & fecantem, & ideo pofita 11 unitate erit idem fector mip logarithmus recte2 pF, nempe exceffus quo fecans arcus KL fuperat ejufdem tangentem. PROP. III, THEOREMA. Linea Meridiana Planifpherii Nautici ef Scala Logarithmorum Exceffuum, quibus Secantes Latitudinum /uperant earundem Tangentes, pofito Radio Loco Unitatis. Suppono ex fcriptoribus nauticis in eorum planifpherio arcum LK in zquatore effe ad eundem Lx in latitudine, ut reCtangulum ex Lq@_in kz ad portionem fu- perficiei cylindrice conflate ex omnibus fecantibus arcuum infinitorum o1 Pek. Vou. Il. C plano - Se 10 GREGORII ANALOGTIA plano LMqin debitis fuis punctis o normaliter infiftentibus, feu figuram Br Qn ; demonftratum autem eft in antecedente fectorem hyperbolicum. m1p feu loga- rithmum rect2 pF (pofita az unitate) differentie inter tangentem & fecantem arcus Kx effe femiflem figure BLN, patet quoque fectorem circularem QKL effe femiffem reCtanguli rq in ki; & ideo nk in equatore eft ad Lx in latitu- dine ut fector ext ad logarithmum exceffus quo fecans arcus KL fuperat eyufdem tangentem; & ideo folidum ex Lx in logarithmum dicti exceflus equale eft folido ex Lx in latitudine in feCtorem akL, & utrumque folidum applicando: ad Lx, logarithmus dicti exceffus equalis eft reCtangulo ex Lx in latitudine *; eodem modo demonftratur logarithmum exceffus, quo fecans arcus cujuf- 2 cunque ox excedit ejufdem tangentem, aqualem effe rectangulo ex ox in latitue dine in =; & proinde logarithmus exceffus quo fecans arcus KL fuperat ejufdem tangentem, eft ad reCtangulum ex Lx in latitudine in =, ut logarith- mus exceffus quo fecansarcus or fuperat ejuf{dem tangentem ad reCtangulum ex or in latitudine in =; & permutando, & utrumque rectangulum applicando ad ae logarithmus exceffus quo fecans arcus KL fuperat ejufdem tangentem, eft ad logarithmum exceflus quo fecans arcus ot fuperat fuam tangentem, ut arcus KL in latitudine ad arcum ot in latitudine ab equatore planifpherii nautici in linea recta computatis ; et ideo linea meridiana planifpherii nautici analoga eft logarithmis excefluum, quibus fecantes latitudinum fuperant fuas tangentes, quod demonftrare oportuit. S$CHOLIUM. Ex hoc theoremate evidens eft methodus defcribendi integram meridianam, etiam ignota arcus dati in equatore menfura; quam menfuram ex hujus 2 talt praxe invenimus. Sit Qu radius 10000000000, DF 1000000000, & proinde ex noftra Circl. & Hyper. Quadratura prop. 32 invenietur fector hyperbolicus MID 1151292546497022812008, qui eandem habet rationem ad fectorem akE quam arcus Kx in latitudine ad eundem in equatore, & dividendo utrumque fectorem per ~, erit ut 2302585092994 045624 ad arcum KL ita KE in latitu- dine ad eundem in equatore; qualis autem fit arcus KL ita invenimus, ut pF exceffus fecantis arcus KL fupra ejufdem tangentem ad tq radium, ita tq ad 100000000000 fummam tangentis & fecantis ejufdem arcus KL, erit ergo Ke tangens arcus KL 49500000000, cq ejufdem fecans 50500000000, & ne ejufdem finus 9801980198+2,7, € quibus non difficile erit invenire ipfum arcum ope noftre Quadr. Circ. & Hyp. prop. 30, vel (fi quis rudiore calculo contentus fuerit) é tabula finuum. Ad datam autem arciis dati in equatore menfuram, meridianam nauticam conftruere, praxis effet hujus inverfa, que nullo negotio ex hac colligitur. Ex predictis manifefté patet lineam meridianam planifpherii nautici effe {calam tangentium artificialium arcuum, qui funt femiffes complementorum Jatitudinum, INTER LINEAM MERIDIANAM, &e, ay Jatitucinum, pofito radio loco unitatis, quoniam (ut patet ex trigonometria) predicte differentie funt eedem cum didtis tangentibus. Si autem comple- mentorum fecantes radio infiftant, erit figura ex illis conflata nempe aHLQMA ad quadratum radii ut quadrans circumferehtia ad radium, ut patet ex noftra ‘Geom. part. univerfali prop. 2da. PROP. IV. THEOREM A, Sit circuli quadrans voay cujus fit pars 6w: fuper arcu 6w imaginetur fuper- ficies cylindrici recti talis natura, ut (fumpto in arcu 6w quolibet puncto 0) perpendicularis ad planum vax ex puncto o ad fummitatem fuperficiei cylin- drice excitata femper fiat equalis tangenti arcus oa. Sit hyperbola cra, cujus vertex c (nempe fuppofita ac parallela & wquali radio xv) & afymptote xa, xv; ducantur rete pwy, 267, radio vx parallele : dico fpatium hyperbolicum 2py7 zquale effe dicte fuperficiei cylindrice ; fi dicte figure non fint equales, Sit inter illas differentia @, & dividatur recta 7v in tot partes zquales a planis recte@ 7¥ perpendicularibus & fpatium hyperbolicum in rectis Ey, 6z, fecantibus, item fuperficiem cylindricam in diverfas portiones dividentibus, ut omnia rect- angula cylindrica fimul hifce portionibus infcripta differant ab omnibus rect- angulis cylindricis fimul eifdem circumfcriptis minore quantitate quam @, hoc enim abfque dubio fieri poteft ab indefinita divifione recte 7y ; intelligo autem rectangulum cylindricum portioni infcriptum effe fuperficiem cylindricam rectam 2 cuyus Ag GREGORDICANALOGIA cujus eadem eft bafis cum portione cui inferibitur, & altitudo ubique eadem cuny minima altitudine portionis, item circum{criptum cui eadem etiam eft bafis cum portione, & altitudo eadem cum portionis altitudine maxima. Compleantur rect- angula 27, By, EZ, FZ, GY, Ny ; & ducatur in puncto x tangens LKé&, fitque in rectam Lz perpendicularis km. Triangula LKM, kx, rectangula ad k & funt fimilia ob angulos equales KEx, LKM ; & ideo ut KM ad KL, ita Kad xé, & ideo KL X KE = KM X X§&, atque yE eft equalis recte x€, quod fic probo ; xy eft ad xk vel ac, ut xx ad x€, fed ob hyperbolam xy eft ad ac ut ac ad ve, funt ergo equales ye, x&, & ideo rectangulum ex Kt in K& equale eft rectangulo ex KM in Ey nempe £2, atque reCtangulum) ex KL in Kg mayus eft rett- angulo cylindrico infcripto portioni fuper KO, quoniam eandem cum ‘te habens al- titudinem nempe x bafem habet majorem (eft enim recta KL major quam curva ko) & ideo reCtangulum £z majus eft re€tangulo cylindrico infcripto portioni fuper ko; eodem modo probatur rectangulum cy majus eile rectangulo cylin- drico infcripto portioni fuper ow, & reCtangulum 2 majus effe rectangulo cylin- drico inf{cripto portioni fuper 6k ; & proinde rectilineum 72DEHGRY majus eft omnibus rectangulis cylindricis infcriptis fimul fumptis, & ideo fpatium hy- perbolicum 2Py¥7 eifdem rectangulis cylindricis mult6 majus eft. In puncto o ducatur tangens 30s; demonftratur ut anté rectangulum os vel or in 03 zquale effe rectangulo cy feu Fz, atque reCtangulum 03 in or minus eft rectangulo cy- lindrico circumfcripto portioni fuper ko, quoniam eandem cum illo habens alti- tudinem, bafem habet minorem (eft enim recta or minor quam curva ko) & proinde rectangulum Fz minus eft rectangulo cylindrico circumfcripto portioni fuper Ko; eodem modo probatur rectangulum ny minus effe rectangulo cylin- drico portioni fuper ow circumfcripto, & ream By minus effe rectangulo cylindrico portioni fuper 6k circumfcripto ; & ideo re¢tilineum 7BEFGNPY minus eft omnibus rectangulis cylindricis circumf{criptis fimul fumptis ; & ideo {patium hyperbolicum 2py7 eifdem reCtangulis cylindricis multo minus eft. Quatuor igitur funt quantitates, quarum maxima & minima funt rectangula cylindrica circumfcripta fimul fumpta, & reétangula cylindrica infcripta fimul fumpta, intermedie autem fpatium hyperbolicum 2py7 & fuperficies cylin- drica fuper curva 6w, & ideo differentia intermediarum minor eft quam diffe- rentia maxime & minime ; at differentia maxime & minime ex conftructione minor eft quam a, & ideo differentia intermediarum nempe {pati hyperbolici & fuperficiei cylindrica: eft multo minor quam @, quod eft abfurdum, ponitur enim @; non igitur differunt quantitates intermedia, & ideo equales funt, quod demonftrandum erat. Quod fi fuperficies cylindrica producatur ufque ad terminum quadrantis a : dico adhuc illam effe equalem {patio hyperbolico correfpondenti cpya; fi non funt zquales, fit fuperficies cylindrica fuper aw major {patio hyperbolico cpya, & ab{cindatur plano 367 recte ax normali portio fuperficiei cylindrice, ita ut relicta nempe fuperficies cylindrica fuper curva 6w fit zequalis {patio hyperbolico cpya, atque fuperficies fuper 6w equalis eft {patio hyperbolico 2py7, ex hac- tenus “femouitidue {patia igitur hyperbolica cpya, 2P¥7, funt equalia, quod eft abfurdum ; fuperficies ergo cylindrica fuper aw non eft major {patio hyper- bolico cpyA; fit (fi fieri poteft) minor & a {patio hyperbolico cpya auferatur recta 27 ipfi ca parallela pac c27a, ita ut relictum 2py7 fiat equale sae ciel INTER LINEAM MERIDIANAM, &c. Lz ficiei cylindrice fuper aw, atque fpatium hyperbolicum 2py7 equale eft fuper- ficiei cylindrice fuper 6w ; & ideo fuperficies cylindrica fuper 6 & fuper Aw inter fe funt equales, quod eft abfurdum ; & ideo fuperficies cylindrica fuper ace non eft minor {patio hyperbolico cpya, fed (ut ante demonftratum eft) nec eft major; reftat igitur, ut illi fit equalis, quod demonftrandum erat. GONSECTARIUM. Hinc manifeftum eft fuperficiem cylindricam ex tangentium fumma conflatam (v. g. fuper curva Aw) effe fecantem artificialem ejufdem arcus aw, pofito radio loco unitatis ; nam (ut demonftratur in noftra Circuli & Hyp. Quadr.) pofita ca feu radio loco unitatis, erit fpatium hyperbolicum acpy logarithmus recte ye fecantis arcus Aw. . PR.O?P.. Ve. THEOREMA. Sit circuli quadrans ecw bifariam divifus in puncto Dd. Sit arcus DM minor quam pw, fuper quo imaginetur fuperficies cylindrici recti talis nature, ut (fumpto in arcu pm quolibet puncto Gc) perpendicularis ad planum ecw ex puncto c ad fummitatem fuperficiei cylindrice excitata femper fiat equalis fecanti arcus cc. In radium ac demittantur recte perpendiculares pv, mrp, A que producantur in 2 & @; fiatque mixtilineum v2@r talis nature, ut (ducta recta, quacunque crs recte pv parallela) tangentes arcuum Gc, Gw, fimul, nempe recta ao, equales fint recte rs. Dico ‘mixtilineum v26r zxquale effe fuperficiei cylindrice fuper curva pM: fi prediéte figure non fint equales, fit inter 14 GREGORTI ANALOGIA inter illas differentia a, & dividatur recta vR in tot partes equales planis recta vr perpendicularibus, & mixtilineum v2@r in rectis T5, s8, fecantibus & fu- perficiem cylindricam in diverfas portiones dividentibus, ut omnia rectangula cylindrica fimul, hifce portionibus infcripta, differant ab omnibus rectangulis cy- lindricis fimul, eifdem circumfcriptis minore quantitate quam-« ; hoc enim abfque dubio fieri poteft ab indefinita divifione rete vr. Compleantur reét- angula v4, V5, T7, T8, sg, s@, & ducatur in puncto c tangens aco, fitque in rectam Ls perpendicularis ck. Triangula ick, oaq, funt fimilia, & ideo ut ex vel vt ad G1 vel GE, ita qa ad ao feu TS; & proinde re€tangulum VTINTS nempe v5 zquale eft recta ingulo EG in aq, atque rectangulum &e In AQ minus eft rectangulo cylindrico circumfcripto portioni fuper GD, quoniam eandem cum illo habens altitudinem nempe aq, bafem habet minorem ; & ideo rectangulum v5 minus eft reétangulo cylindrico circumfcripto portioni fuper GD; eodem modo demonftratur rectangulum +8 minus efle rectangulo cylindrica circum- {cripto portioni fuper c1, & re€tangulum s@ minus efie rectangulo cylindrico circum{cripto portioni fuper bm; & “ideo retilineum v 3 568788 minus eft om- nibus rectangulis cylindricis circumfcriptis fimul fumptis, & igitur mixtilineum v2r eifdem rectangulis cylindricis multo minus eft. Ob fimilia triangula pF, BPQ.; DH vel vr eftad DF ut BQ_ad BP vel v2, & ideo rectangulum v4. xquale eft rectangulo DF in Bq; at rectangulum DF in B@_majus eft rectangulo cylin- drico portioni fuper De infcripto, quoniam eandem cum illo habens altitudinem Bq, bafem habet majorem; & ideo reGangulum v4 majus eft rectangulo cylin- drico infcripto portion fuper Dc ; eodem modo probatur rectangulum 17 majus effe rectangulo cylindrico infcripto portioni fuper cL, & reétangulum sg majus effe rectangulo cylindrico infcripto portioni fuper tm, & ideo re¢ctilineum v245789R majus eft omnibus reétangulis cylindricis infcriptis fimul fumptis ; eft ergo mixtilineum v26r eifdem reétangulis cylindricis multo majus. Quatuor igitur funt quantitates, quarum maxima & minima funt reGtangula cylindrica circumfcripta fimul, & re€tangula cylindrica infcripta fimul; intermediez autem mixtilineum vt6x & fuperficies cylindrica fuper curva pm, & ideo differentia maxime & minime eft. major differentia intermediarum, differentia autem maxime é& minime ex conftructione eft minor quam #, & ideo differentia in- termediarum eft multo miner quam #, quod eft abfurdum, ponitur enim &; non igitur differunt quantitates intermedia & ideo equales funt, quod, &c. Quod fi arcus. pm fumeretur verfus c, adhuc ftaret prop. demonftratio autem effet paulo diverfa, que tamen nullo negotio ex priore colligeretur. Si punctum in ipfo c caderet, verum etiam effet heorema, fed negativé per duas pofitiones demon- freinds ficut in fine antecedentis. Sit mixtilineum vxzR talis nature, ut (duéta recta quacunque cry recte Dx parallela) recta ry femper equalis fiat tangenti co ; manifeftum eft (ex Geomet. part. univer. prop, 4.) mixtilineum vxzR zequale effe fuperficiei trunci cylindrici recti refeéte 4 plano bafem feminormaliter fecante in recta pq, atque (ex Geom. part. univerf. prop. 3.) evidens eft eandem trunci fuperficiem zqualem effe reét- angulo N& in aa & ideo mixtilineum vxzr eidem reQangulo eft equale. Ex predictis evidens eft’ mixtilineum x2@z talis effe nature ut y5 femper fit equalis tangenti GA; & proinde mixtilineum X2/8Z una cum portione qua- drantis DVRM zequale eft {patio conchoidali refeto a rectis Dv, MR, cujus con= choidis INTER LINEAM MERIDIANAM, &C. 18 choidis vertex eft c, norma Pq, polus 7, ca = ag: ex hac prop. & hujus 2 evidens eft fequens confeétarium {patio conchoidali quadrando fatis expeditum. CONSECTARIUM. Si in prediéta conchoide accipiatur fpatium contentum a curva conchoidali & reétis cr, T6; erit predi€tum fpatium equale duplo {pati hyperbolici (cujus afymptote Qa, QP, femiaxis a¢) contenti 4 curva hyperbolica una afymptota & reétis ac, QA — GA, alteri afymptote parallelis, una cum femifegmento circu- lari cor dempto reGtangulo ec in et. Aliarum conchoideon {patia (nempe quando vertex & polus non equaliter diftant 4 normali) poffunt menfurari per Analogiam A Wallifio in Epift. Com. pag. 171 demonftratam. PROP. VI. THEOREMA.? Sit ciflois KLM cujus afymptoton cM, femicirculus cox. Diametro cK fit nor- malis pit, & jungantur recte DK, GL; dico fpatium ciffoidale cxu triplum effe fegmenti circularis pNx. Ducatur curva Kes talis nature, ut (ducta ad libitum a puncto c recta Gpa que tangenti occurrat in A) cD1 perpendicularis recte cx, fiat equalis rect Ka. Recta pr femicircu- lum tangat in puncto p, & ideo xquales funt recte FD, FK; cumque angulus KDA fit rectus, patet FA, FD, effe aequales ; camque ci femper fit equalis duple ipfius pF, manifeftum eft (ex Geom. parts univer. prop. 4) mixtilineum cx? efle duplum fuperficiei trunci cylindrici recti fuper curva pNx refetz a plano bafem feminormaliter fecante in recta KA; atque eadem fuperficies trunci equalis eft rectangulo ex curva DNK in radium KH ablato reCtangulo ex p1 in Hx (ut fatis patet verfatis in fuperficierum ftudio) hoc eft duplo fegmenti circularis pnx ; eft igitur mixtilineum cx1 quadruplum fegmenti circularis pNK. Ut p1 ad 1x ita eft 1k vel pE ad Ea vel cp, & proinde aquales funt rectze cp, 11, cfimque hoc femper fiat, patet {patium ciffoidale 1ku equale effe mixtilineo pyxe, & idem utrinque addendo nempe femifegmentum circulare p1kN, mixtilineum cx1, feu quadruplum fegmenti circularis DNK, equale eft mixtilineo pNKL, & utrinque auferendo fegmentum circulare pNK; triplum fegmenti circularis DNK zquale eft mixtilineo DoKL, feu (ob equalitatem triangulorum pki, GIL) ipfi fpatio ciffoidali propofito GkL, quod demonftrandum erat. Hic fupponitur arcus pNk quadrante minor, quod fi quadrante effet major, nullo negotio variari poteft demonftratio ut illi inferviat: at KNp exiftente femi- circumferentia, multd facilior effet demonftratio, nempe quéd fpatium ciffoidale infinite extenfum ezquetur femicirculi triplo. METHODUS Meo ot ve OnE) a lee COMPONENDI Tabulas Tangentium & Secantium Axtificialium EX Tabulis Tangentium & Secantium Naturalium EXACTISSIME ET MINIMO CUM LABORE. IT Az arcus quadrantis in lineam rectam extenfus, fitque figura AHI conflata ex tangentibus naturalibus fingulorum arcuum a puncto a, in debitis fuis punétis, rectis a1 normaliter infiftentibus: fit ap pars minima, in cui equales dividitur quadrans, nempe so vel +3, gradus, fintque illi zequales po, ON, NM, &c, & ducantur rect a1 perpen- diculares PB, oc, ND, ME, &c, manifeftum © eft ex confeétario 4 hujus, figuram ABP effe fecantem artificialem arcus aP item figuram aco effe fecantem artificialem arcus ao (pofita cyphra loco radii artificialis), &c, manifeftum eft rectangula Bo, cn, pM, &c, inveniri ex multiplicatione minimee partis quadrantis ap in fingulas tangentes naturales; at in menfurandis figuris ABP, BCx, cpv, pauld major eft difficultas; primo igitur fi tangentes conveniant in differentiis primis, non differunt linez as, Bc, cp, &c, a rectis, & ideo figura aBP, Bcx, cov, &c, erunt He Ngseh reétangula, & proinde, fies GHQ GREGORII METHODUS COMPONENDI TABULAS TANGENTIUM, &c. 17 HQ_X QG . ° ° . GHA aS : quod fi differentie fecunde fuerint equales, erunt dicte fi- ‘gure portiones trilineorum quadraticorum, e. g. erit GH@_ portio trilinei quadra- tici, cujus axi H@_eft parallela, differentia: illa inter fe equales fint z; & pro- H X G 2s S ; ° : 4 cebeseasabeeired = *. fi autem differentie tertie fuerint equales, inde erit GHQ, = : erunt dicte figure portiones trilineorum cubicorum, eritque e. g. GHq = et Vy Heine. se so gnc quando differentie_ quarte funt zequales, erunt ditte figure portiones trilineorum q-quadraticorum, & diffe- rentiz quarte erunt equales 24Plo q-quadrati ipfius Gq divifo per cubum lateris recti, item quando differentie quinte funt equales, erunt ditte figure portiones trilineorum furfolidorum, &-~differentie quinte erunt equales 1202! furfolidi ipfius Gq_divifo per e-quadratum lateris reéti, & fic in infinitum. Que hic diximus de compofitione fecantium artificialium ex tangentibus naturalibus eodem modo intelligi velim de compofitione tangentium artificialium ex fe- cantibus naturalibus fecundum hujus tertiam. Animadvertendum tangentes & fecantes artificiales fupra computari, pofito o Jogarithmo unitatis, 1pe9000000000 radio, & 230258 SN eT eae lite logarithmo denariu: facilius autem (ni- mirum fol4 additione) pofito ~, grad. = G@= AP = 1, computabimus tan- gentes & fecantes artificiales ad 7915704467897819 denarii logarithmum ; nam H G HQ: Wd4%, 1G Zz X.GQ, HQ. aa GHQ.= <= item Se ati? PnP Oe GHy va BO hac ratione 12 2 sae GQ, HQ_X ZX GQ? ef Ba EOS v9 72 1728 HQX z z . H => item = HGQ= —— / 9 ( a) : fe tandem unica fola divifione invenimus tangentes & fecantes artifi- ciales ad logarithmum denarii 1000000000000000, pofito femper radio unitatis loco, que funt differentiz tangenttum & fecantium artificialium in tabula ab ipfo radio artificiali; & proinde diviforis multiplices, ad facilitandam opera- tionem, in tabella fubfidiaria hic reponimus. Quod fi +3, grad. = Ga, tan- gentes & fecantes artificiales debentur logarithmo denarii 13192840779829703, cujus etiam multiplices in fubjecta tabella notantur. Quod, fi quis velit hos numeros potius reprefentare radium artificialem quam denarii logarithmum, addatur cyphra & -habebit-intentum. ~Notandum hos numerds convenire radio naturali 1009090000000, hinc enim patebit numerus notarum in ad{criptis artificialibus. Vou. Il. p> 18 GREGORII SERIES 1} 7915704467897819 | 13192840779829703 | ee 15831408935795538 | 26385681559659406 | 3 | 23747113403693457 | 39578522339489109 | 4 |31662817871591276 | 62771363119318812 | 5 |39578522339480095 6 5904203899148 515 |. 6 147494226807 386914 791§7044678978218 | Sil eae cokes SE erate oe ae 9 55409931275284733 | 92349885458807921 |. 63325635743182552 | 105542726238637624 | 71241340211080371 | 118735567018467327-}, Ex Epiftcla D. Jacobi Gregorii ad D. Collins, 15 Feb, Ano nee data, 10h habetur Autograpbon. X quo epiftolam ad te dedi, tres a te accept, unam Dec. 15, alteram Dee. 24, tertiam 21 Fanuarii nuper elapfi datam. Quod attinet Newtont methodum univerfalem, aliqua ex parte, ut opinor, nuhi innotefcit, tam quoad geometricas quam mechanicas curvas. Nihilo tamen minus ob feries ad me miffas gratias habeo, quas ut remunerem- mitto que fe- quuntur. Sit radius = 7, arcus = a4, tangens = ¢, fecans = 5, ° #8 4s 27 1 Et erit@=t — a oF ee + oe &e. . 5 7 9 Eritque ¢ = a4 isting Dp erp Es 62a &e. t 1or* gigre.. 283578 gat 610% 27748 Ee Sie) rat = + 24r3 in 720r5 + 806477 Sit nune tangens artificialis = ¢, & fecans artificialis = s, & integer : seebiaindd Z {cilicet, erit 2 = — + f : . . I 1 + # fit major unitate, erit I+ 2 ‘ ; r+ 2° ac inventa m, habebitur & 1 + # numerus quefitus. 3. An Extra from Mr. Ifaac Newton's Second Epiftle to Mr. Henry Oldenburgh, Secretary of the Royal Society of London ; with a Direction to communicate the Con- tents of it to Mr. Godfrey William Leibnitz; dated the 24th Day of OGober, in the Year 1676: containing fome Difcoveries relative to Logarithms. O tempore peftis ingruens (que contigit annis 1665, 1666), coegit me hine fugere, & alia cogitare. Addidi tamen fubinde condituram quandam lo- garithmorum ex area hyperbole, quam hic fubjungo. . Sit drp hyperbola, cujus centrum c, vertex F, & quadratum interyyeCtum cAFE = 1. Inac cape aB, Ad hinc inde = +5 feu'o.1: Et, erettis perpendiculis sp, dd ad hyperbolam terminatis, E erit femi-fumma f{patiorum ap & ad =0.1 + O,OO!L 0.00001 0.000000 1I ° ° itatig jifle + ——— &c, & femi-diffe- 3 5 7 Cc bAB “ . 0;000 OQOOO00r 0.000080 rentia = = + WR Rae &c. - Que reducte fic fe habent, ©.1000000000000 0.00 50000000000 3333333333 2a OND 20000000 1666666 142857 12500 1I1I 100 9 I 0.1003 353477310 0.0050251679267 Horum fumma 0.1053605156577 eft ad; & differentia 0.0953101798043 eft ap. Et eadem ratione pofitis as, ad hinc inde = 0.2, obtinebitur ad = 0.2231435513142, & AD = 0.1823215567939. Hiabitis fic logarithmis hy- perbolicis numerorum quatuor decimalium 0.8, 0.9, 4.1, & 1.23 cum fit ty Mme; & 08 & 0.9, fint minores unitate: adde logarithmos eorum 0.8 0.9 E ; t : ad duplum logarithmi 1.2, °& habebis 0.6931471805597 logarithmum hyper- I bolicum COMMERCIUM EPISTOLICUM. 23 bolicum numeri 2. Cuyus triplo adde log. 0.8 (fiquidem fit a wml} Woy VAe habebis 2.3025850929933 logarithmum numeri 10: Indeque per additionem fimul prodeunt logarithmi numerorum g & 11: adeoque omnium primorum shorum 2, 3, 5, 11 logarithmi in promptu funt. Infuper, ex fola depreflione nu- merorum {uperioris computi per loca decimalia & additione, obtinentur logarithm decimalium 0.98, 0.99, 1.01, 1.02; ut & horum 0.998, 0.999, 1.001, 1.002. Et inde per additionem & fubdutionem prodeunt logarithmi primorum 7, 13, ¥7, 37, &c. Qui una cum fuperioribus, per logarithmum numeri to divifi, eva- dunt veri logarithmi in tabulam inferendi. Sed hos poftea propius obtinul. Pudet dicere ad quot figurarum loca has computationes, otiofus eo tempore, produxi. Nam tunc fane nimis deleCtabar inventis hifce. Sed ubi prodiit in-. geniofa illa * Nicolai Mercatoris Logarithmotechnia (quem fuppono fua primum inveniffe), ccepi ea minus curare; fufpicatus, vel eum noffe extractionem radicum zeque ac divifionem fractionum ; vel alios faltem, divifione patefacta, inventuros reliqua, prius quam ego etatis effem mature ad fcrigendum. Eo ipfo tamem tempore quo liber ifte prodiit, communicatum eft per amicum D. Barrow (tunc Mathefeos Profefforem Canta.) cum D. Collinio, 4 compendium quoddam methodi harum ferierum ; in quo fignificaveram areas & longitudines curvarum omnium, & folidorum fuperficies & contenta, ex datis rectis; & vice verfa, ex his datis rectas determinari poffe : & methodum ibi indicatum illuftra- yeram diverfis feriebus. Suborta deinde inter nos epiftolari confuetudine; D. Collinius, vir in rem ma- thematicam promovendam natus, non deftitit fuggerere ut hec publici juris facerem, Et ante annos quinque (1671) cum fuadentibus amicis confilium ceperam edendi tractatum de Refractione Lucis, & Colorrbus, quem tunc in promptu habebam ; coepi de his ferebus iterum cogitare; & { tra¢tatum de lis etiam con{cripfi, ut utrumque fimul ederem. Sed, ex occafione’ telefcopii catadioptrici, epiftola ad te miffa qua breviter explicui conceptus meos de natura lucis, inopinatum quiddam effecit ut’ mei intereffe fentireny' ad te feftinanter fcribere de impreffione iftius epiftole. Et fuborte ftatim per diverforum epiftolas (objetionibus aliifque refertas) crebre interpellationes me prorfus @ coniilio deterruerunt;. & effecerunt ut me arguerem imprudentie, quod umbram captando, eatenus perdideram quietem meam, rem prorfus fubftantialem. * Mathematici priores invenerunt hoc theorema, quod /umma terminorum progrefionis geomeiricé in infinitum pergentis eff ad terminorum primum’S maximum, ut bic terminus ad differentiam duorum ter- minorum primorum. Idem demonftratur arithmetice multiplicando extrema & media. Demonttravit Wallifius dividendo rectangulum fub mediis per extremumultimum. Vide Wallifii opus arithme- ticum anno 1657 editum, cap. 33 § 68. Per Wailifii divifionem Mercator demonftravit & auxit quadraturam hyperbole 4 D. Brounker prius inventam.. Et Gregorius idem demonftravit geomc- trice. Sed horum nemo methodum generalem quadrandi curvas per divifionem invenit. Mercator hoc nunquam profeffus eft. Gregorius ejufmodi methodum, licet vir acutiffimus & literis Collinii admonitus, vix tandem invenit. . New*onws invenit per interpolationem ferierum, & poftea divi- fionibus & extractionibus radicum, ut notioribus, ufus efts- + Analyfin intelligit per equationes infinitas. } Hujus tractatus meminit D. Collins in epiftolis duabus impreffis, pag. 101, 102, Commere. Epiftol, Et Newz¢onus in epift. impreffa, p. 205 ibid, 4 Su 24. JOHANNIS COLLINS Sub eo tempore Yacobus Gregorius, ex unica quadam ferie é meis, quam D. Collinius ad eum tranfmiferat, poft multam confiderationem (ut ad Collinium refcripfit) pervenit ad eandem methodum, & tractatum de ea reliquit quem {peramus ab amicis ejus editum iri. Siquidem, pro ingenio quo pollebat, non potuit non adjicere de fuo nova multa, que rei mathematice intereft ut non pereant. Ipfe autem traétatum meum non penitus abfolveram, ubi deftiti a propofito ; neque in hunc diem mens rediit ad reliqua adjicienda. ‘Deerat quippe pars-ea qua decreveram explicare modum folvendi problemata, que ad quadraturas re- duci nequeunt ; licet aliquid de fundamentis ejus pofuiffem. Czterum in trac- tatu ifto, feries infinite non magnam partem obtinebant. Neque majori labore eruitur area totius circuli ex fegémento cujus fagitta eft quadrans diametri. Ejus computi fpecimen, fiquidem ad manus eft, vifum fuit apponere; & una adjungere aream hyperbole quz eodem calculo prodit. Pofito axe traniverfo = 1, & finu verfo feu fegmenti fagitta = «; erit femi- hyperbole E. wan i lash itn xt ! fegmentum 7) P& L eH in $x —-— +— &c,-. Hee atitem Janes ie circull 5 28" 972 2 9 3 3 3. d. fic in infinitum produeitur, fit 2¥° = a, < es rs == a, se dite i = . hyperbole a b c d e =f, &c. Et erit femi-fegmentum Moa te CO oe reuli 3 5 7 9 shat c e » hn's nail aa &c. Eorumque femi-fumma = TRE &c, & femi-differentia te +< + Z + &c. His ita preparatis, fuppono * = 4, quadrantem nempe 0.26 1x8 oS = 0,001953525: d(= > =e) = = 0.000244140625. Et fic procedo ufque dum venero ad tte deprefifimum, qui poteft ingredi opus. Deinde hos terminos per 3, 5, 7, 9, 11, &c, refpective divifos difpone in duas tabulas: ambiguos cum primo in unam; & negativos in aliam; & addo ut hic vides, eianyh 25 5, A ke 368 axis; & prodit a (= ) = 0.03125: ¢ (= == 0.083 3333333333333 0.0002790178571429 62 500000000000 34679066051 271267361111 834465027 5135169396 26285354 144628917 961296 4954581 38676 _ 190948 1663 7963 75 352 4 ie 0.000282 571938 0.089610988 5646618 Tune COMMERCIUM-EPISTOLICUM. 25 Tunc a priori f{umma aufero pofteriorem, & reftat 0.089328416625%7043 area femi-fegmenti hyperbolici. Addo etiam eas fummas, & agoregatum aufero a primo termino duplicato 0.1666666666666666, & reftat 0.0767731061630473 area femi-fegmenti circularis. Huic addo triangulum iftud quo completur in fectorem, hoc eft 3.1/3, feu 0.0541265877365274, & habeo fectorem 60 graduum, 0.1308996938995747, cujus fextuplum 0.7853081633974482 eft area totius circuli: que divifa per 4 five quadrantem diametri, dat totam pe- ripheriam 3.1415926535897928. Si alias artes adhibuiflem, potui per eundem numerum terminorum feriei pervenifle ad multo plura loca figurarum, puta vi- ginti quinque aut amplius: fed animus fuit hic oftendere, quid per fimplex feriei computum preftari poffet. Quod fane haud difficile eft, cum in omni opere multiplicatores ac divifores magna ex parte non majores quam 11, & nunquam majores quam 41 adhibere opus fit. Neque obfervaffe videtur [clariflimus Leidnitius] morem meum generaliter ufurpandi literas pro quan- titatibus cum fignis fuis + & — affectis, dum dividit : z Zw 23 z4 hanc feriem Palette ergs: sa + &c, Nam cum area hyperbolica sr, hic fignificata per z, fit affrmativa vel negativa, prout jaceat ex una vel al- tera parte ordinatim applicate Bc}; fi area illa in nu- meris data fit /, & / fubftituatur in ferie pro z, orie- A oD Z I] 13 rs i tur vel = Users + Ae + yam &C, vel ey + il 73 i4 poe) —)_ eS e i ] r¢ 1 7 ° e€ ance mre &c; prout / fit affrmativa vel negativa. Hoc eft pofito a=1= 4, & /\ogarithmo hyperbolico; numerus ei correfpondens erit 1 +- i ll 13 4 : 1 1] 3 [4 = fae 5, &o 1 fit afirmanvus; & 1 ——- + SS eto &c, fi / fit negativus. Hoc modo fugio multiplicationem theorematum, que alias in nimiam molem crefcerent. Nam v. g. illud unicum theorema, quod fupra pofui pro quadratura curvarum, refolvendum effet in 32 theoremata, fi pro fignorum varietate multiplicaretur. Preterea, que habet vir clariffimus de inventione numeri unitate majoris per yey, il i3 Is datum logarithmum hyperbolicum, ope feriei — — —— + : Halt 3 Eee ON ee ea ka . ay 1 il 13 it 2° tius quam ope feriei — —— &c, non- Beeccs potius, quam ope terie Tit Fapera tab esdinae Mitac Doeg oC eT ? dum percipio. Nam fi unus terminus adjiciatur amplius ad feriem pofieriorem quam ad priorem, pofterior magis appropinquabit. Et certe minor eft labor com- putare unam vel duas primas figuras adjecti hujus termini, quam dividere unitatem per numerum prodeuntem ex logarithmo hyperbolico ad multa figurarum loca extenfum, ut inde habeatur numerus quefitus unitate major. Utraque igitur s © be 1 [3 feries (fi duas dicere fas fit) officio fuo fungatur. Poteft tamen — + ——~—+ Ud. < WAP 3 is ae’ ; Beas ———— &c, feries, ex dimidia parte terminorum conftans, optime adhiberi ; IX2X3X4X5 Voz, II. 1D fiquidem 26 JOHANNIS COLLINS, &c. fiquidem hec dabit femi-differentiam duorum numerorum, ex qua et rectangulo ; . 7 is ; i . oe —_— ——— a - dato uterque datur. Sic & ex ferie 1 + Te ha GRP. may &c, datur femi fumma numerorum, indeque etiam numeri. Unde prodit relatio ferierum inter fe, qua ex una data dabitur altera. Conftruétionem logarithmorum non aliunde peti debere credetis forte, ex hoc fimplici proceffu qui ab iftis pendet. Per methodum fupra traditam querantur logarithmi hyperbolici numerorum 10, 0.98, 0.99, 1.01, 1.02: id quod fit {patio unius & alterius hore. Dein divifis logarithmis quatuor pofteriorum per logarithmum numeri 10, & addito indice 2, prodibunt veri logarithmi nume- rorum 89, 99, 100, 101, 102, in tabulam referendi. Hi per dena intervalla interpolandi funt, & exibunt logarithmi omnium numerorum inter 980 & 1020: & omnibus inter 980 & rooo iterum per dena intervalla interpolatis, habebitur tabula eatenus conftructa. Tunc ex his colligendi erunt logarithmi omnium primorum numerorum & eorum multiplictum, minorum quam 100: ad quod nihil requiritur preter additionem & fubtraCtionem. Siquidem fit (/——— eo ors = 2, / 7 = 3, = 5; S=7, Bau, nb = I75 Sas = 195 “936. 32 Sor = 29, B31, Ma 37, a4, 3 = 43> 2 = 47, acne Exo / oe = 61; $10. = 6 re pee. a 9928 9954 9968 __ 9804 8X17 — 733 7X18 193 7 ish, = 83; Bie ee 98, 6X Fe = 97- Et habitis fic logarithmis omnium numerorum minorum quam 100, ‘reftat tantum hos etiam femel atque iterum per dena intervalla interpolare. END OF EXTRACTS FROM COLLINS. - THE THE Wi EsbuboE AH. «GHA P TERR DR. WALLIS’S TREATISE OF ALGEBRA. Of Logarithms : Their Invention and Ufe. HE other improvement which I mentioned (as added to the algorifme of the Arabs fince we borrowed it from them), is that of Logarithms; an improvement of our own age and nation. This was firft of all invented (without any example of any before him, that I know of) by John Neper, Baron of Merchifton in Scotland ; and by him firft publifhed at Edinburgh, in the year 1614: and foon after by himfelf (with the affiftance of Henry Brigges, Profeffor of Geometry, firft at London, in Grefham College, and afterwards at Oxford) reduced to a better form, and perfected. The invention was greedily embraced (and defervedly) by learned men. Mr. Brigges, upon the firft publication of it, was fo pleafed with it, that he prefently repaired into Scotland, to confult the author, advife with him, and be affiftant to him in the perfecting of it, and in calculating tables for it; which was a work of great labour, as well as fubtile invention. And it was embraced and promoted abroad by Benjamin Urfinus, John Kepler, Adrian Ulack, Petrus Crugerus, and others. And at home by Henry Gellebrand, who perfected the Trigonometria Bri- tannica, which Mr. Brigges began, but died before it was perfected. So that, in a fhort time, it became generally known, and greedily embraced in all parts, as of unfpeakable advantage ; efpecially for eafe and expedition in trigonometrical calculations. 2 The 28 OR. WE DI ss The foundation of it is this : : If to a rank of continual proportionals in a geometrical progreffion from 1 : fuppofe LorBet ator 16. 32. 64. &c. We accommodate a rank of exponents in an arithmetical progreffion, from o : fuppofe DMN Ie Bie CREP IS 8 stone apie’ sao It is manifeft, that for every multiplication or divifion of thofe terms one by another, there is an anfwerable addition or fubduction of the exponents. For as (inthe terms) 4 multiplied by 8 makes 32, fo (in the exponents) if to 2 we add 3, it makes 5 ; and as 32 divided by 8, gives 4: fo if from 5 we fub- dué& 3, there remains 2 : and fo every where. Terms. Tit 2at As Pat LOpeG 2A TeX PONCNES On teat eel ems AK O soem. shah iearsby yes HY: Twat OB | 5-3 = 2. (Not much unlike to what we before fhewed out of Archimedes’s Arenarius, concerning his #, 8, y, 6, &c, in continual progreffion geometrical from 1, at- tended by a feries of exponents in arithmetical progreffion ; the foundation of that and this being all one.) And the fame holds, if between any two of thofe terms, interpofe one or more means proportional ; and between their exponents, as many arithmetical means. As if between 4 and 8 (or between 2 and 16) we interpofe a mean propor- tional 4/ 32, that is 44/2; and between 2 and 3 (or 1 and 4) an arithmetical mean, 23; thenas 44/2 by 8 makes. 32\/2 (a mean proportional between 32 and 64): fo adding their exponents 23 and 3, makes 5+, an arithmetical mean between 5 and 6; and fo every where. And univerfally, (whatever be the values of r. e.) fuppofing The terms, vtote tie Fi hei dtar ue oes j Exponents, 0. ¢. 2¢.\ 3¢. 46. 56 6¢. &c. Then,as rr x= 75, andar/r xX rrr = rV/7;3 So 2e€ + 3¢ = 5¢, and 23¢ + 3¢ cea teeey And fo every where. | And confequently whatever term we interpofe between any of thofe continual proportionals ; if we alfo interpofe between their exponents, a like arithmetical mean, as that is a proportional mean (as if that be the firft or fecond of two means proportional, this accordingly is the firft or fecond of two means arith- metical; if that the fecond of five means proportional, this the fecond of as many TREATISE OF ALGEBRA. 29 many arithmetical means, &c) then to every addition or fubduétion of thefe -one with another, will anfwer a like multiplication or divifion of thofe. And if for 0, e, 2¢, 3¢, &c, (taking e = 1) we put o, 1, 2, 3, &c, then doth this exponent always give us the number of rations or dimenfions in the term to which it belongs. | . rents iWin ter? N8ce, Grae Bee an eG (as 3 inr*, 6 in r®, and fo every where) or fhews how many fold (quam multi- plicata) the proportion (for inftance) of 7° to 1 is of rto 1. That is, how many rations or proportions of 7 to 1 are compounded in 7* to 1, to wit 6; to which the name Logarithmus fitly anfwers, that is Ac'yay cpi$zos, the number or propor- _ tions fo compounded. | Now this foundation being laid, their defign in the logarithms is this: Having felected (as moft convenient) a rank of continual proportionals, in a decuple progreffion ; to wit, I. 10. 100. 1000. 10000. 100000. 1000000, &c.. they fit hereunto (as their exponents) in arithmetical progreffion, ©. 0she 2: a. A. - 5. 6. &c. (And confequently the logarithm of any fractions lefs than 1 is to be a negative number.) And then for each of the numbers interpofed between 1 and 10, between 10 and 100, and fo of the reft (as 2, 3, 4, &c, 11, 12, 13, &c); they feek out (between o and 1, between 1 and 2, &c) an exponent (to be exprefled in decimal parts), which is fuch a mean arithmetical as the other is a mean proportional. And thefe exponents they call logarithms, which are artificial numbers, fo anfwering-to the natural numbers, as that the addition and fubduction of thefe anfwers to the multiplication and divifion ‘of the natural numbers. By this means (the tables being once made) the work of multiplication and divifion is performed by addition and fubduction; and confequently that of {quaring and cubing, by duplation and triplation, and that of extracting the {quare and cubick root, by bifection and trifeCtion; and the like in higher owers. Of thefe logarithms we have printed tables, for all numbers as far as one hundred thoufand , fo that if any two numbers (not exceeding 100,000) be propofed to be multiplied or divided one by the other, the logarithms of thofe numbers (to be found in thofe printed tables) being accordingly added or fubducted, will give the logarithm of that natural number (to be found by thofe tables), which is the product or quotient of fuch multiplication or divifion, And the double or treble of fuch logarithm, is the logarithm of its fquare or cube. And the half or third part of it, is the logarithm of its quadratick or cubick root ; and the like of higher powers, which, in large numbers, is matter of great expedition. And go DR. WALLIS’S And (becaufe a main end of this defign was to facilitate aftronomical and other trigonometrical calculations) befide thofe logarithms for numbers in their natural order, we have alfo tables of artificial or logarithmical fines, tangents, and fecants ; the addition and fubduction of which anfwers to the multiplication and divifion of the natural fines, tangents, and fecants: which is a very com- pendious advantage for expediting fuch calculations, and is not lefs accurate than the operation by tables of natural fines, tangents, and fecants. Thus in a plain triangle; fuppofing the angles ! < given, A 60 degrees, B 50 degrees (and confe- quently c 70 degrees), and the fide aB 31323 paces: for finding the fides ac, or Bc, we have this proportion : As the fine of c, 70 degrees, 9396926 To the fine of 8B, 50 degrees, 7660444 x So is the fide aB, 31323 paces. To the fide ac. 25535—paces. For finding which, we are to multiply 7660444 by 31323, and then divide by 9396926 ; which gives for the fide ac (almoft) 25535 paces. And, as the fine of c, 70 degrees, 9396926 To the fine of a, 60 degrees, 8660254 So is the fide aB, 31223 paces, To the fide zc. 288674 .paces. For finding which, we are to multiply 8660254 by 31323, and divide by 9396926, which gives for the fide sc, 288674 paces, proximé. Now (to prevent thefe tedious multiplications and divifions) by logarithms, we proceed thus : Log. fine c, 70 degrees, — 9.9729858 Log. fine B, 50 degrees, + 9.8842540 Log. aB, num. 31323, + 4.4958633 Log. ac, num. 25535, + 4.4071315 where fubdudting the firft logarithm from the fum of the fecond and third, gives the fourth, which (the tables tell us) anfwers to the number 25535, feré. So many paces therefore is the fide ac. | Again, Log. fine c, 70 degrees, — 9.9729858 Log. fine a, 60 degrees, + 9.9375306 Log. aB, num. 31323 + 4.4958633 Log. sc, num, 288674 + 4.4004081 Where TREATISE OF ALGEBRA. 31 Where fubduéting the firft logarithm from the fum of the fecond and third, gives the fourth; which (the table tells us) anfwers to the number 288673, proximé. So many paces therefore is the fide 8c; which operations are much more expeditious than multiplying and dividing fuch large numbers. And in like manner, in fpherical triangles, fave that there all the logarithms are to be taken out of the tables of fines, tangents, and fecants ; which in this example are taken partly from thence, partly from the table of numbers; but the expedition is alike in both. ; This was firft publifhed by the Lord Neper (the firft inventor of it), in the year 1644, under the title of Mirificus Logarithmorum Canon, with its defcription and ufe; but referving the manner of conftruction, and its demonttration, to be after publifhed; this being but an eflay fet forth to fee the judgment of learned men concerning this defign, and how it was like to be received. In this we have a canon or table of natural and logarithmical fines for each degree and minute of the quadrant. And whereas it was at his choice to give to what number he pleafed the logarithm o, and whether to proceed by way of increafe or decreafe, he chofe to make o the logarithm of the whole fine 10000000, that fo the multiplication or divifion by the whole fine (frequent in trigonometrical calculation) might be difpatched without trouble, requiring here but the addition or fubduction of o. And becaufe the ufe of leffer fines and numbers, lefs than the radius or whole fine, were likely to be of more frequent ufe than of tangents, fecants, and other numbers greater than the radius, he chofe to give to thofe leffer numbers affirmative logarithms (increafing the logarithms from o, as the fines decreafe), which he calls abundantes: and confequently negative logarithms (which he calls defeffives ) to greater numbers. Defigning thofe by +, thefe by —. And, by this means, he directs how this table of fines (with the differences there inferted) may ferve alfo for a table of tangents and of fecants: fo that this canon is a complete canon of natural fines, and of logatithmical fines, tangents, and fecants. | He fhews alfo how this table may be applied to the logarithms of abfolute numbers : but becaufe with fome trouble, he referves the fuller account hereof to a farther treatife. In the year 1619, the Lord Neper being then dead, the fame was again publithed by his fon, Robert Neper, with fome poflhumous treatifes of his father, concerning the conftruction of this Logarithmical Canon, and concerning his defign (after communication had with Mr. Brigges) of changing the form of logarithms, making o to be the logarithm of 1 (of which he had before given notice in the preface to his Raddologia, publifhed in the year 1617), and con- cerning fome things pertaining to trigonometry ; with fome lucubrations of Mr. Brigges on the fame fubject. But, the Lord Neper being dead, the whole work was devolved on Mr. Brigges, who (according to their joint advice) making the logarithm of 1 to be o, and of 10, 100, 1000, &c, to be 1, 2, 3, &c, which he calls indices, or charaéterifiics, and which we may repute as integer numbers, with fourteen Ciphers annexed, which we may repute as fo many places of decimal fra¢tions 7 below 32 DRe. 1! VAL eee below the place of units, or of the charatteriftick : and between thefe he fits the intermediate logarithms for the intermediate numbers. And confequently the logarithm of 1 being o, the logarithm of fractions lefs than 1, or of numbers intermediate between 1 and o, muft be negative numbers, or numbers lefs than o (which he calls defective logarithms), denoted by — (the note of negation) prefixed. Now thefe defective logarithms may be two tways expreffed; either fo as that the note of negation fhall affect the whole logarithm, or fo as to affect only the characteriftick (leaving the reft of the logarithm to be underftood as affirmative). As, for example, the fraftion 3, or (which is equivalent) 0.375. This fraction fuppofeth the numerator 3 to be divided by the denominator 8, which in logarithms is to Log. 3. 0.4771212 be performed by fubtracting the logarithm of 8 oe .b. 0.9030899 from that of 3, and the remainder will be the loga- Log, 3. — 0.4259687 rithm of 2, which will then be the negative num- ber, —0.4259687. Or thus; for as much as the logarithm of 375 (fuppofing it to be an integer number) is 2.5740312. And the deprefling this to the firft, fecond, or third, or further place of decimal fraction, doth (without altering the figures) divide — the value by 10, 100, 1000, &c; which in logarithms is done by fubgraéting 1,2, 3, &c, from the characteriftick or place of . integers (1, 2, 3, &c, in that place, being the lo- ) set of 10, 100,: 1000, &c). Such alter- es ce eee ation of the value (the figures remaining) is Log. 37/5 145740312 done by altering the characteriftick of the loga- Log. 3l75 0.9740312 rithm, without varying the other figures in this AT — 3 manner. 8 0/375 1+5740312 Which two forms, though they feem different, Log. olo375 2.5740312 and fome may rather choofe the one, fome the other; or in fome cafes the one, in fome cafes the other; yet they are in fubftance or value the fame. For — 1.0000000 + 0.5740312 + is = — 0.4259687 And every one is left to his liberty, whether of the two ways (or what other equivalent thereunto) he fhall pleafe to ufe. In this method Mr. Brigges hath calculated a table of logarithms (publifhed in the year 1624) for twenty chiliads of abfolute numbers (from 1 to 20,000); and again for 10 more (from 90,000 to 100,000), and one chiliad fupernumerary (to wit, the hundred and firft chiliad), that is 31 chiliads in all. Before which is prefixed, a large account of the nature and conftruction of this Logarithmical Canon, and the ufes thereof; and direction how to fupply the TREATISE OF ALGEBRA, 33 the intermediate chiliads which are here wanting. The whole intituled 47:4- metica Logarithimica. The fame is again publifhed in the year 1628, by Adrian Vlacq (or Flack), with a fupplement (as Mr. Brigges directed) of the chiliads before omitted ; that is, in all, of 100 chiliads, with one fupernumerary. But in fhorter num- bers, extending but ‘to 10 places below that of the integers, or the characteriftick, And he fubjoins alfo a logarithmical canon of fines, tangents, and {ecants (for degrees and minutes of the quadrant) of as many places. Mr. Brigges proceeded to calculate a trigonometrical canon logarithmical, fuited to that for abfolute numbers, to the logarithms extending (as in that other) to 14 places, befide the characteriftick. And having before calculated a table of natural fines, tangents, and fecants (for degrees “and centefines of degrees) in numbers extending to 15 places, he fitted thereunto a canon of loga- rithmical fines and tangents (becaufe thofe of fecants might be f{pared) ; and a treatife prefixed concerning the conftruction thereof, with other things pertinent thereunto ; intending a farther treatife concerning the ufe of it. But dying before this laft was finifhed, or the reft publifhed, Mr. Henry Gellibrand fupplied this latter, and publifhed the whole, with the title of Trigonometria Britannica, in the yeare1633. To which is fubjoined another canon of logarithmical fines and tangents, by Adrian Vlacq, for degrees, minutes, and tenth feconds, extending (as his former did) to 10 places, befide the characteriftick ; and Mr. Brigges’s 20 chiliads for logarithms of abfolute numbers. So that the whole doétrine of logarithms was by this time fufficiently per- fected, with convenient canons or tables fitted thereunto, in large numbers : of which alfo Petrus Crugerus gives an account in the preface to his Trigono- metria Logarithmica, printed in ‘the year 1634, with his logarithmical tables, but in fhorter numbers. And the tables of logarithms above mentioned (for 100 chiliads of abfolute numbers, and for fines and tangents to degrees and centefmes) were the fame year, 1633, contratted into a leffer form, and more manageable (but in fhorter numbers, the former not extending to above feven places, befide the charac- teriftick, but the latter to 10) ; by Nathaniel Roe; with directions for the ufe of them (in trigonometry, geometry, aftronomy, geography, and navigation), by Edmund Wingate. In the mean time, Benjamin Urfinus did alfo publifh tables of logarithms in the year 1618, and again in the year 1625, in his Trigonometria ; and Johannes Keplerus alfo in the year 1624, in his Chilias Logarithmorum (which he applies alfo to his Rudolpbhine Tables, publifhed in 1627), and Claudius Batfchius about the fame time, or foon after ; and Georgius Ludovicus Frobenius, in the year 1634, and perhaps fome others. But all, or moft of them, in fhort numbers, and conformable to the Lord Neper’s firft defign; not to that form, which, upon fecond thoughts, he and Mr. Brigges acreed upon as moft eligible, and which hath fince been received in common practice. Since which time, much hath not been added to the doétrine of logarithms ; nor was it neceflary, that work having obtained fuflicient perfection, Vo.. II. } But 34 DR. WALLIS’S TREATISE, &c. But in cafe logarithms on any emergent occafion be defirable with greater exacinefs, and in larger numbers than thofe printed tables do afford, Mr. Nicholas Mercator, in a fimall treatife, called Logarithmotechnia, printed in the year 1668, fhews (with great fubtilty) how it may be effected, in numbers of - whatever length defirable, with much more eafe than heretofore. ats Nor fhall I need to add more concerning logarithms: thofe who defire farther, may find it in the authors above mentioned; efpecially Mr. Brigges’s Arithmetica Logarithmica and Trigonometria Britannica, with Adrian Vlac’s ad-~ ditions to both. Without farther infifting, therefore, on the algorifm by numeral figures (with the improvements thereof fince we had them from the Arabs), I fhall return to what doth more immediately concern Algebra. | END OF THE TWELFTH CHAPTER OF WALLIS’S ALGEBRA, EXTRACT FROM THE PHILOSOPHICAL TRANSACTIONS. A Letter from the Reverend Dr. Wallis, Profeffor of Geometry in the Univerfity of Oxford, and Fellow of the Rayal Society, London, to Mr. Richard Norris ; con- cerning the Colleétion of Secants, and the true Divifion of the Meridians in the Sea Chart. N old inquiry (about the fum or aggregate of fecants) having been of late moved anew, I have thought fit to trace it from its original, with fuch folution as feems proper to it; beginning firft with the general preparation, and then applying it to the particular cafe. GENERAL PREPARATION. 1. Becaufe curve lines are not fo eafily managed as ftraight lines: the ancients, when they were to confider of figures terminated (at leaft on one fide) by a curve line (convex or concave), as AFKE, did oft make ufe of fome fuch expedient as this fol- lowing (but diverfely varied as occafion required), namely, F a2 2. By 36 WALLIS’S COLLECTION 2. By parallel ftraizht lines, as AF, BG, cn, &c (at equal or unequal diitances, as there was occafion), they parted it into fo many fegments as they thought fit; or fuppofed it to be fo parted. 3. Thefle fegments were /o many wanting one, as was the number of thofe parallels. To each of thefe parallels wanting one; they fitted parallelogranis, of fuch- breadths as were the intervals (equal or unequal) between each of them refpectively, and the next following; which formed an adfcribed figure made up of thofe parallelograms. 5. And if they began with the greatelt’ (and therefore neglected the leaft) fuch figure was circumfcribed (as fig. 1), and therefore bigger than the curvi- linear propofed. 6. If with the leaft (negledting the greateft), the figure was infcribed (as fig. 2), therefore lefs than that propofed.., - | Ha 7. But, as the number of fegments was increafed (and thereby their breadths diminifhed), the difference of the circumfcribed from the infcribed (and there- fore of either from that propofed) did continually decreafe, fo as at laft to be lefs than any afiigned. 8. On which they grounded their method of exhauftions. g. In cafes wherein the breadth of the parallelograms, or intervals of the pa- rallels, is not to be confidered, but their length only; or, which is much the fame, where the intervals are all the fame, and each reputed — 1; Archimedes (inftead of infcribed and circumfcribed figures) ufed to fay, ** All except the ereateft, and all except the leaft :” as prop. 11 Lin. Spiral. he PARTICULAR CASE, 10. Though it be well known, that, in the terreftrial globe, all the meridians meet at the pole, as Ep, EP, whereby the parallels to the equator, as they be nearer to the pole, do con- tinually decreafe. 11. And hereby a degree of longitude in fuch _ parallels, is lefs than a degree of longitude in the equator, or a degree of latitude. 12. And that, in fuch proportion, as is the co-fine of latitude (which is the femidiameter of fuch parallel), to the radius of the globe, or of the equator. 13. Yet hath it been thought fit (for fome reafons) to reprefent. thefe me- ridians, in the fea chart, by parallel ftraight lines, as EP, Ep. 14. Whereby each parallel to the equator (as LA) was reprefented in the fea chart (as /a) as equal to the equator EE, and a degree of longitude therein as large as in the equator. TS. By : | OF SECANTS, &c¢. 37 } 5. By this means each degree of longitude in fuch parallels was increafed beyond its juft proportion, at fuch rate as the equator, or its radius, is greater than fuch parallel, or the radius thereof. 16. But in the old fea charts, the degrees of latitude were yet reprefented (as they are in themfelves) equal to each other, and to thofe of the equator. 17. Hereby, amongft many other inconveniencies: (as Mr. Edward Wright obferves, in his Correétion of Errors in Navigation, firft publifhed in the year 1599), the reprefentation of places remote from the equator was fo diftorted in thofe charts, as that, for inftance, an ifland in the latitude of 60 degrees (where the radius of the parallel is but half fo great as that of the equator) would have its length, from Eaft to Welt, in compari(on of its breadth, from North to South, reprefented in a double proportion of what indeed it is. 18, For rectifying this in fome meafure (and of fome other inconveniencies), Mr. Wright advifeth, that (the meridians remaining parallel, as before) the degrees of latitude remote from the equator, fhould, at each parallel, be pro- tracted in like proportion with thofe of longitude. 19. That is, as the co-fine of latitude (which is the femi-diameter of the parallel) to the radius of the globe (which is that of the equator), fo fhould be a degree of latitude (which is every where equal to a degree of longitude in the equator) to fuch a degree of latitude fo protracted (at fuch diftance froin the equator), and fo to be reprefented in the chart. 20. That is, every where in fuch proportion as is the refpective fecant (for fuch latitude) to the radius. For as the co-fine to the radius, fo is the radius to the fecant ‘(of the fame arch or angle), as 2.Ri:R./. 21. So that, by this means, the pofition of each parallel LANG Bp in the chart fhould be at fuch diftance from the equator, compared with fo many equinottial degrees or minutes (as are thofe of latitude), as are all the fecants (taken at equal diftances in the arch) to fo many times the radius. 22. Which is equivalent (as Mr. Wright there notes) to the projection of the {pherical furface (fuppofing the eye at the center) on the concave furface of a cylinder, erected at right angeles to the plain of the equator. 23. And the divifion of meridians, reprefented by the furface of a cylinder ereéted (on the arch of latitude) at right angles, to the plain of the meridian, or a portion thereof. The altitude of fuch projeétion, or portion of fuch cylindrick furface, being, at each point of fuch cir- cular bafe, equal to the fecant of latitude anfwering to fuch point. 24. This. 38 WALLIS’S COLLECTION 24. This projection (or portion of the cylindrick fur- face) if expanded into a plain, will be the fame with a plain figure, whofe bafe is equal to a quadrantal arch ex- tended, or a portion thereof, on which, as ordinates, are R erected perpendiculars equal to the fecants, anfwering to Td the refpective points of the arch fo extended ; the leaft of which (anfwering to the equinodtial) is equal to the radius, and the reft continually increafing till, at the pole, it be infinite. i re 25. So that as ERsw (a figure of fecants erected at right angles on EL, the arch of latitude extended) to ERRx (a rectangle on the fame bafe, whofe altitude ER is equal to the radius), fo is EL (an arch of the equator equal to that of latitude) to the diftance of fuch parallel (in the chart) from the equator. 26. For finding this diftance anfwering to each degree and minute of lati- tude, Mr. Wright (as the moft obvious way) adds all the fecants (as they are found calculated in the ‘Trigonometrical Canon) from the beginning, to the degree or minute of latitude propofed. 27. The fum of all which, except the greateft (anfwering to the figure in- {cribed), is too little; the fum of all, except the leaft (anfwering to the circum- {cribed), is too great (which is that he follows; and it would be nearer to the truth than either, if (omitting all thefe) we take the intermediates ; for min. 2, 14, 23, 34, &c, or (the doubles of thefe) min. 1, 3, 5, 7, &c: which yet (be- caufe on the convex fide of the curve) would be fomewhat too little, 28. But any of thefe ways are exact enough for the-ufe intended, as creating no fenfible difference in the chart. He ae 29. If we would be more exatt, Mr. Oughtred direéts (and fo had Mr. Wright done before him) to divide the arch into parts yet {maller than minutes, and calculate fecants fuiting thereunto. 30. Since the arithmetick of infinites introduced, and, in purfuance thereof, the doctrine gf infinite feries (for fuch cafes as would not, without them, come to a dettrminate proportion), methods have been found for fquaring fome fuch figures, and particularly the exterior hyperbola (in a way of continual approach), by the help of an infinite feries. As in the Philofophical Tranfactions, Numb. 38, for the month of Auguft, 1668; and my book, De Motu, cap. 5, prop. 31. | 31. In imitation whereof, it hath been defired (I find) by fome, that a like quadrature for this figure of fecants (by an infinite feries fitted thereunto) might be found. ' 32. In order to which, put we for the radius of a circle, R; the right fine of an arch or angle, s; the verfed fine, v ; the co-fine (or fign of the complement) X= R-v = :Rq — sq: the fecant, /; the tangentr. Fig. 4. 2 an. Chen is 2 PR rR sy anal deg) eC — the fecant. LOA - Nd 1 Ss aa PR . aA, Ati 2. 352° hy thabis, >) eRigr ee eat the tangent. 4 35. Now OF SECANTS, &c. 39 33. Now if we fuppofe the radius cp divided into equal parts (and each of them = +3, R), and on thefe to be erected the co-fines of latitude La : 36. Then are the fines of latitude in arithmetick pro- greffion. 37. And the fecants anfwering thereunto, Lf = ~. 38. But thefe fecants (anfwering to right fines in arith- -metical progreffion) are not thofe that ftand at equal dif- tance on the quadrantal arch extended. Fig. 6. 39. But ftanding at unequal diftances on the fame extended arch; namely, on thofe points thereof whofe right fines (whilft it was a curve) are in arithmetical S progreffion. 40. Yo find therefore the magnitude of REL/, fig.. 6, which is the fame with this (f{uppofing Ex of the R fame length in both, however the number of fecants therein may be unequal), we are +» confider the fe- cants, though at unequal diftances here, to be the fame with thofe at equal diftances in the preceding figure, anfwering to fines in arithmetical progreffion. — 41. Now thefe intervals, or portions of the bafe, are the faine with the intercepted arches, or portions of the arch, in the preceding figure ; for this bafe is but that arch extended. 42. And thefe arches, in parts infinitely fmall, are to be reputed equivalent to the portions of their refpec- tive tangents intercepted between the fame ordinates. As in fig. 7 and 9. 43. That is, equivalent to the portions of the tangents of latitude. 44. And thefe portions of tangents are to the equal intervals in the bafe, as the tangent of latitude to its fine. 45. To find therefore the true magnitude of the paral- lelograms, or fegments of the figure, we mutt either pro- tract the equal fegments of the bafe (fig. 7), in fuch proportion as is the refpec- tive tangent to the fine, to make them equal to thofe of fig. 8. 46. Or elfe, which is equivalent, retaining the equal intervals of fig. 7, pro- tract the fecants in the fame proportion ; for either way the intercepted rect~ angles or parallelograms will be equally increafed ; as Lo, fig. 9. 47. Namely; as the fine of latitude to its tangent, fo is the fecant to a fourth ; which is to ftand on the radius equally divided, inftead of that fecant. Lye Patty R3 wee f Deir Cat x. R) bv tees?) Seatac SLES) Gr LM, Mg. Q.. 48. Which therefore are as the ordinates. in (what I call Arith. Infin. prop.. 104) Reciproca Secundinorum, fuppofing 2* to be fquares in the order of fecundanes, Fi ans +S 49.. This: 49 WALLIS’S COLLECTION east) ei (Rn +po poy 49. This becaufe of &* = r* — s?, and R?—s*R the fines s, in arithmetical progreffion, sina iccla dah is reduced by divifion into this infinite : sf ferics : Unite 2 4 6 pumrmmrrnrmes oi rt tae Ste +e 530. That is (putting rR = 1); mae aS % + s* -+- s* +..9° &c. R R3 5t. Then (according to the arithme- M EY tick of infinites) we are to interpret s, a3 fucceffively, by 15s, 28, 3s, &c, till we come to s, the greateft; which therefore reprefents the number of all. 52. And becaufe the firft member doth reprefent a feries of equals, the fe- cond of fecundans, the third of quartans, &c; therefore the firft member is to be multiplied by s, the fecond by 4s, the third by {s, the fourth by 4s, &c. 53- Which makes the aggregate s + is} + 15° + Js’ + Js° &c = ECLM, fig. g. $4. This (becaufe s is always lefs than R = 1) may be fo far continued, till fome power of s become fo fmall as that it, and all which follow it, may be fafely neglected. 55- Now (to fit this to the fea chart, according to Mr. Wright’s defign) having the propofed parallel of latitude given, we are to find, by the Trigono- metrical Canon, the fine of fuch latitude, and take, equal to it, cL = s, And, by this, find the magnitude of EciM, fig. 9; that 1s, of REL/ fig. 8; that is, of REL/, fig. 6. And then, as RRLE, or fo many times the radius, to REL/ (the aggregate of all the fecants), fo muft be a like arch of the equator (equal to the latitude propofed) to the diftance of fuch parallel (reprefenting the latitude in the chart) from the equator ; which 1s the thing required. 56. The fame may be obtained, in like manner, by taking the verfed fines in arithmetical progreffion. For if the right fines, as here, beginning at the equator, be in arithmetical progreffion, as 1, 2; 3, &c, then will the verfed fines beginning at the pole (as being their complements to the radius) be fo alfo, THE COLLECTION OF TANGENTS. 37. The fame may be applied in like manner (though that be not the prefent bufinefs) to the aggregate of tangents, anfwering to the arch divided into equal parts. 58. For thofe anfwering to the radius fo divided, are =; taking s in arithmetical progreffion. 59. And or SECANTS, &c, AL 59. And then enlarging the bafe (as in fig. 8), or the tangent (as in fig. 9), in the proportion of the tangent to the fine. etl ge SR SR? 2 Teed Gey? Fal) wien fide = See 2 0 Teel a R* aes 2 33 6S Rims) srt (s+ Spey 60. We have by divifion this feries, SR? — s® 'g3 s 7 ——————— ? stotoat+atake. +3 61. That is (putting rR = 1), + sia _ s+ s3 + s§ + 57 + 99 &, . 62. Which (multiplying the refpec- + tive members DY) aia asp 260059, 2g. as 37 &c) becomes RT ST, ES” + Ast ish 4 Es? 4 ts Sc, $7 R*# Which is the aggregate of tangents to the arch, whofe right fine is s. 63. And this method may be a pattern for the like procefs in other cafes of like nature, Vot. Il. G ‘ sietbah * We : a ed ai oy Sa ye deat NEY OF prt et Pheest bly’ ne ae wd x re re ree Ah aor HG gee ae ep 5 +. | wa meyer ‘ - ; 0 - ie %y ache ty. eS | at “3 nag daz scien be ich t5 Arm ; d ; ‘ bay . hee ees “v . . eee to se. “ se ~ . i ‘ : a. ; ies By or ot es ata , ee “- Aad, “de coy E ak . | : 95 shh o: ait a Wo rigs x ated os i ca a Ms Mal i it ary Ayia te ot ae gir a deprt ye” Ti ohne. hit aia yolg yal di aoe Bt vd casi aan be a yor . ‘ ' ‘ £ bh. | « ing ‘ , ra? g oe é 7 ’ E , ¥ \ * { ~ P d ‘. a - = : ‘ a ' = i“, F ‘ : ‘ ‘ » . ‘ . . "4 i : “ad t .. hi eee ? ~ ¥ : 4 ii ‘ , I tt t ; . | “ae : - ra * ’ 4 2?! : ° - . ‘ . e as 4 ‘ ~< 4 : Ay , : Poke S: LOGARITITHMOTECHANITIAS OR, THE MAKING OF NUMBERS CALLED ate ECsy olie Aes Rie geht ines Rie) Beira Mhag io BtyO. Toa. 1 Yin Ri V EBL ACES, FROM A Pye. TR ol CAL, of. 1.G UR. WITH SPEED, EASE, AND CERTAINTY. By EUCLID SPEIDELL, Puirtomarna. PRINTED AT LONDON, IN THE YEAR 1688, TO THE READER. E AVING for fome years paft fhewn to feveral perfons the praxis of the following Treatife, and alfo communicated to them fomewhat of the doétrine leading thereunto, I was often defired not to let them fleep in oblivion, but to publifh the fame; which was firft promoted by my honoured friend Mr. Peter Hoot, merchant, and feconded by my loving friend Mr. Reeve Williams; or elfe they had not feen the publick: What I have done therein I defire thee to take in good part, being alfo at proportional charges myfelf, befides my compofing thereof, to make it communicable to thee, rather than fuch an eafy and certain way to make logarithm numbers (to fo many places) fhould not be known in our native tongue. I have called them Geometrical Logarithms, for that the firft inventors of thofe numbers had not adapted geometrical figures to them. But the fcheme hereunto annexed having fuch properties and affections as logarithm numbers have, hath made me fo ftyle them. What 1 have done herein is to gratify fuch who have a curiofity to examine lo- garithm tables, and to make logarithm numbers to fo {mall radiufes as are fo often printed for common ufes with brevity and exactnefs. Two fheets of the praxis hereof were, printed fome time before the reft; which having found kind acceptance with divers, induced me alfo to let the remainder be publifhed. And before the printing thereof, one was writing upon thofe two fheets, and was fo fair to defire my confent to publith it; which I readily gave; for that I knew him able enough to do. it, and when to be at leifure myfelf to attend the publifhing of the re- fidue, I knew not. But that not being performed by him, I defire thee to accept of what is done herein as time and leifure hath permitted. I fhall not need to write how needful logarithm numbers are in thofe great and ufeful arts of Navigation, Aftronomy, Dialling, Fortification and Gunnery, Surveying, Gauging, Intereft, and Annuities, &c, when, as there are fo many books written and publifhed thereof, not only in our own language but in many others. And truly the firft inventors thereof are not a little to be had in reverence for making and perfeCting thofe numbers with fo much labour as thofe methods by which they derived them did require. Here thou mayft make a logarithm to 7 or 8 places readily and ealily ; but to 25 places would have been very difficult, if not impoflible, for the firft inventors to have produced after their ways. If any thing herein fhall offer whereby thou mayft make farther improve- ment, let the publick fhare of the benefit thereof. Thus wifhing thee good fuccefs in all thy ftudies, is moft earneftly defired by London, March 26, 1688, F E. SPEDE e GEOMETRICAL LOGARITHMS, Cskie Ae By by BOU T the year 1672, being in company with Michael Dairy, a citizen of London (who had for moft part of his life-time addiéted himfelf to mathematical {tudies, and hath publifhed divers. practical pieces of feveral parts of the mathematicks, of good ufe and delight) ; and difcourfing about making hyperbolical logarithms, I defired him to give me a rule to make the hyper- bolical logarithm of 10, from the confideration of an hyperbola infcribed within a right-angled cone, who. gave me this rule following. ‘To the number propofed, viz. 10, add an unit, and fubtra& from it an unit, and there will be a refult of =; then divide 1, or 100000000 &c, by 2, which is 818181818 &c, which cube én infinitum, and divide every one of them (which will be a rank of proportional numbers) by the proper indices of their refpective powers, that is to fay, by 3, 5, 7, 9, 11, &c, then the addition of thofe quotes will make the logarithm of ro. Finding, then, that 10 divided by 4% maketh 81818181818181 &c, and to cube it iz infinitum, was very difficult, I ‘rejected the rule, and thought it then not much more eafy than Briggs’s way: neither did he “tell any reafon or de- monftration for the faid rule; and becaufe in this example I found it fo intricate, I did not much care to profecute it, but neglected it. Not long after he de- parted this life, and fince his death refuming the faid thing, and trying if it were ferviceable in any other part of the hyperbola, [ foon found it a jewel, and could make the hyperbolical logarithm of 10 at twice, that is to fay, from two parts numbered in any afymptote, whofe fact is Io, with eafe, certainty, and delight, and have made the hyperbolical logarithm of 2 to 25 places, in order to fee if the learned and laborious Henry Briggs’s logarithms were true to 15 places, which were made after a moft laborious and difficult way of extracting {quare roots, and, as I have heard, was the work of eight perfons a whole year, and that without any proof but ee, if any two or more agreed in their extractions, line bh Vv 46 SPESDELL’$ by line, ftep by ftep, it was taken de bene effe; which was a work of very great pains and uncertainty. However, they did effect it ; and I do find they made the logarithm of 2 to 15 places very true, as by my operation, hereafter fol- lowing, will appear, being done to 25 places, and afterwards from thefe hyper- bolical logarithms deduced Briggs’s logarithms ; both which figurative operations were performed and examined by me in 8 hours’ time. I took this pains to make the hyperbolical logarithm to 25 places, in order, alfo, to fee if the moft ingenious and laborious James Gregory’s hyperbolical logarithm would agree with this of mine, which he hath, in his Quadratura Circuli © Hyperbole, printed at Padua; but I find that his logarithm of 2 correfponds with mune but to 17 places. I mutt confefs, I did not take the pains to raife the logarithm of 2 to 25 places, according to the doctrine he hath delivered in that moft learned piece, but am contented that this eafy and certain way I deliver here, and by the operation thereof the hyperbolical logarithm of 2 to 25 places, is as true in the laft as in any where, and may be examined in a few hours; fo that any body, if he pleafe, may be his own examiner and judge, if this way be not eafy, certain, and f{peedy. Having made feveral logarithms for digit numbers, and mixt numbers, as for 12, 14, which are hereafter inferted, I find the rule delivered by Michael Dairy is of admirable ufe and benefit in fquaring the hyperbola, and making loga- rithms from it. | Some time fince the death of the faid Michael Dairy, I fhewed unto Mr. John Collins, who I knew had been a great familiar and friend to the faid Michael Dairy, the figurative work of my making the hyperbolical logarithms, ~ according to the faid Dairy’s rule; who feemed very well pleafed with it, ac- knowledging it to be the {peedieft way could poffibly be of fquaring the’ hyper- bola, and making the logarithms from it; and, after a little paufing on it, re- plied, ** That Dairy muft have had this rule out of the faid James Gregory’s works.” I made anfwer, * Not from his faid Quadratura Circuli Hyperbole.” He anfwered “* No, from his Exercitationes Geometrice, printed at London, 1668.”——A book I had not feen or heard of till then. And as he the faid Mr. Collins had been always very frank and free to communicate any mathematical thing to me, fo I held myfelf obliged to acquaint him firft with this work. He feemed to admire that Michael Dairy fhould keep fuch a thing from him, who had been fo great a familiar with him in thefe fludies. Not long after my difcovery hereof to the faid Mr. Collins, he alfo departed this life ; whofe death, all that were mathematical, and knew him, lamented not a little: for he was not only excellent in mathematical arts and f{ciences, but of a very good, affable, and frank nature to communicate any thing he knew to any lover and inquirer of thofe things; and hath left behind him thofe mathematical works which will continue his fame amonett the lovers and ftudents therein. He alfo, in his life- time, promoted the publifhing of other men’s mathematical works, as the elaborate Algebra of the learned John Kerfy, who was my father’s difciple about 1643 and alfo of the learned Baker’s Algebra, and feveral others. He was a man of great correfpondence with mathematical perfons in foreign parts, and thereby could give information of any new’or old mathematical book ; and, ull my acquakatance with him, I was ignorant of foreign authors; being but LOGARITHMOTECHNIA. 47 but young when my father died, and not then having taken any pains in thefe ftudies : fo that, by the faid Collins’s information and means, I have heard of, and feen, fome mathematical authors of note and efteem. After the faid Mr. Collins had told me of James Gregory’s faid Exercitationes Geometrica, fold by Mofes Pitts, in St. Paul’s Church-yard, I bought there one of them; and do find, that Michael Dairy had deduced this rule from the faid book: wherein the faid James Gregory hath made the fquaring of the hyper- bola, an exercife geometrically demonttrating the quadrature of the hyperbola, fome time before publifhed by the induftrious and lucky Nicholas Mercator; who, by the happy difcovery of fome properties in the hyperbola, hath made all the ways of {quaring the hyperbola flowing from the fame, very eafy, cer- tain and delightful: and becaufe neither of them have exemplified their doctrine and rules with figurative work, fo large as to 25 places. I have here, to illuf- trate their admirable works, inferted divers figurative operations, whereby the reader and ftudent may fee, and have that fatisfaction in fact and operation, which is fo pleafing and defirable by every one. I fhall not here trouble the reader with any fections of the cone, whereby he may fee the rife and geniture of an hyperbola from that body, but content my- felf to fhew him from a fquare and an infinite company of oblongs on a fu- perficies, each equal to that fquare, how a curve is begotten which fhall have the fame properties and affections of an hyperbola infcribed within a right- angled cone: and feeing a-curve made after this manner following doth be- come fuch an hyperbola, the dodtrines and analogies delivered and difcovered by thofe two ingenious artifts, Mercator and Gregory, may be applied to this curve as often as need and occafion doth require. And, not to detain the reader any longer from knowing, how to make this curve, we proceed to defcribe the fame accordingly. There is a fquare ascp, whofe fide or root is 10; let ps be prolonged im infinitum, and continually divided equally by the root, or pg, and thofe equal divifions numbered by 10, 20, 30, 40, 50, 60, 70, &c, i infinitum : upon thefe numbers let perpendiculars be erected, which call ordinates, and each of thofe perpendiculars of that length, that perpendiculars let fall from the aforefaid: perpendiculars to the fide or bafe cp (which call complement ordinates), the oblongs made of the ordinate perpendiculars, and complement ordinate perpen- diculars, may be ever equal to the fquare ap, which may eafily be done thus; for it is 4,22, 49:2, +22, 1422, &c, produces the length of the ordinate perpendicu- lars ; for 100 divided by 20, maketh 5 for the length of the ordinate perpendi- cular 202. And 100 divided by 30, giveth 3333333 &c for the ordinate perpendicular 30F ; and 100 divided by 40, produceth 25 for the ordinate 40e,. and fo of the reft. And, geometrically, it is as 20D is to BD, fo is AB to an equal to 20, as before; for that the angle acu is equal to c2zop, and fo of the reft. And, for the length of the next ordinate, you fay, as gop to gb, fo aB to AK, which is equal to gor. And, for the ordinate 40s, fay, as 40c to BD, fo aB to AM, which will be equal to 406, and fo of all the reft ;. whereby you have all the perpendiculars upon the prolonged fide rp, both geometrically and arithmetically. The fame proportion is to be obferved for any interme~ diate parts. Now,. 48 SPEIDELLE’S Now, for all the perpendiculars which are let fall from the aforefaid perpen diculars or ordinates to the bafe cp, which call complement ordinates, the geo metrical proportion for NE, equal p20 is as HA to Ac, fo cD to cL20 equal to we; and for the complement ordinate or equal p30, it is as KA to Ac, fo cp to p30 equal or, and fo of the reft. Now, for nx arithmetically, fay, as 5 to to, fo 10 to 20 equal to NE, equal to p20; and for oF, fay, as 33333333 to 10, {0 10 to 30 equal or, which is equal to cL3o, and for pc equal to D4o, fay, as 26 is to 10, fo 10 to 40 equal to pc, equal to p4o; and {fo for all the reft of the com- plement ordinates ftanding upon the bafe cp, whereby it doth appear, that all the oblongs made of the ordinates, and complement ordinates, are each of them equal to the fquare ap, which is here roo ; for the oblong Ep being made of £20 and p20, is by the 13 of the 6 Euclide, equal to the fquare ap; for @20 is a mean proportional between p20, and 2or, and Q20 Is found to be equal as, fo is the oblong or parallelogram ED equal to the fquare ap, and the like de- montftration ferves for all the oblongs or parallelograms ftanding upon the bafe cp, by the tips or angular points of thofe parallelograms, or from the ends of all the ordinates ftanding upon 20, 30, 40, 50, 60, 70, in infinitum, draw the curve line from a towards £, fo fhall you defcribe the curve arFes, which curve you fee is begotten without any confideration or refpect to the fection of a cone, and yet becomes the fame in all refpects, to have the fame affections and properties of an hyperbola derived from the interfedting of a right-angled cone, as fhall be fhewed in the next chapter. You may obferve the complement ordinate nz, being equal to p20, is equal to twice radius. And if cp be made the radius of a circle, then is NE equal to p20, equal to the tangent of twice radius; for p20 becometh the tangent of twice radius. Alfo, it is manifeft that the complement of the tangent equal to twice radius is alfo equal to half the radius; that is, the tangent complement of D20 is 20£ equal to 5. And feeing the radius is ever a mean proportion between the tangent and the tangent complement, therefore each oblong is equal to the f{quare AD. Sewers Cob AP. II. N the former chapter, we have fhewed the begetting of a curve, without any regard to the fection of any folid body; and now it remaineth to prove that this curve hath the fame properties and affections that an hyperbola, deduced trom the fection of a right-angled cone. | I remember fome time before the death of John Collins, he told me, it was a great work of the learned Vincent, or Magnan, to prove, that diftances reckoned in the afymptote of an hyperbola, in a geometrical progreffion, and the fpaces that the perpendiculars thereon erected, made in the hyperbola, were equal the one to the other. This property is now very well known ; the hyperbola hath, and this curve hath the fame property; which is difcernible almoft ituitz. In the hyperbola, they call the prolonged line pz i infinitum, from the point B, an afymptote. And here in this prolonged line from B, on 20, 40, 80, 160, 3 320, LOGARITHMOTECHNIA. 49 320, 640, &c, let the ordinates touch the curve in EFes &c; I fay, that thofe trapezias with the curve line (or hyperbolical f{paces) are all equal the one to.the other. In the right-lined trapezias thereon, it is manifeft, they are all equal the one to the other, by feveral propofitions of the 6th book of Euclid: for in the right-lined trapezia zEAB, the fide aB is twice Ez; and, by the former chapter, it was found, that cy is half Ez, by faying, as azB to Ez, fo Ez tocy. And the right-lined trapezia zEaB fhall be therefore equal to 75. Now, forafmuch as in the right-lined trapezia ycrz, the bafe of that yz isdouble to zz, but the per- pendiculars are in the ratio of a8 to EZ; for, as before, it is as AB : EZ 72 EZ? Gy; therefore the right-lined trapezia yczz equal to the right-lined trapezia zeEAB, and fo will all right-lined trapezias, fo bafed and perpendiculared, be equal the one to the other. The trapezia LyEZ is equal to the fquare ap, becaufe zy xX ZE is equal to AB X AB, as in the foregoing chapter, the oblong cz is half the parallelogram Lz, and the triangle cx half the parallelogram 1. Now the parallelogram cz + the triangle cx1 (half the parallelogram Lr) 1s equal to the right-lined trapezia zzaB, for in numbers 20 x 25 = 50 + half 50 equal to 75. Thus you fee the right-lined trapezias, numbered upon the prolonged fide in geometrical proportion, are equal the oneto the other. It remaineth now to prove the mixed trapezias; that is, the trapezias ftandiné upon the fame bafis, but joined aloft with this curve, are alfo equal the one to the other. Firft, let it be obferved, that thefe curvilined trapezias (or hyperbolical fpaces) are ever lefs than the right-lined trapezias, becaufe all the points in the curvilined trapezias fall within the right ‘line that joins the right-lined trapezias ; and is thus proved in the right-lined trapezia Bzza: let there be in the bafe zB upon the point 5 erected a perpendicular to touch za in T, then is T5 equal to aB lefs wr, which is half qx, that is, 10 lefs 2, 5 equal 7, 5 = 1s. But, by the foregoing chapter, if a perpendicular be erected upon the faid point 5 to v (to touch the curve in v) fo that the parallelogram vp fhall be equal to ap, as in the former chapter ; then will it be ap divided by pv = 5v, which is 100 divided by 15, produceth 6,666666 for the true length of 5v; whereas before st is 7,5. By the fame means may all the intermediate points in this curve line Eva be found to fall within the right line az, that is, between the line Ea and zp; and therefore the right-lined trapezia zeETaB greater than the curvi- lined trapezia (or hyperbolical fpace) zEvapz. Now, forafmuch as we have proved that the aforefaid right-lined trapezias are ever equal the one to the other, it will now follow, that feeing the curve paffing by all thofe points which are extremities of the right-lined trapezia (as well as the curvilined f{paces, being upon the fame bafis always), and this curve being generated continually by one and the fame ratio, as in the former chapter. That therefore the curvilined trapezias flanding upon geome- trical proportional bafes, fhall be alfo equal the one to the other; which is the affection and property of the hyperbola. And fo the dottrines and pre- cepts delivered by thofe two famous Geometers, Mercator and Gregory, for the fquaring of the hyperbola, be applied to this geometrical curvilined figure, and from it derived logarithms, which may be called hyperbolicat logarithms. The way and means to find the hyperbolical fpaces in numbers fhall be fhewed in the following chapters. Wor, IT. Hi . | ot oes we ot $0 + eS PB EB ey, s Oe a Wy av 8 Ill. N this chapter we will confider that moft admirable difcovery (I fuppofe Mercator made) upon drawing the diagonal cs, which, by conftruction, cutteth all the perpendiculars ftanding upon the bafe cp at equal angles, and in fuch diftances from the bafe cp as doth unravel the myftery of his infinite feries, and make the quadrature of the hyperbola more eafy and certain than any I ever faw or heard of. , The diagonal cz being drawn, doth give the firft term of a geometrical pro- ereffion, or infinite feries, between 10 and 20, or 30, 40, 50, 60, 70, 80, go, &c. That is to fay, would you know the firft term of an infinite feries (or num- bers geometrical proportional continued) between 10 and 20, the fum of all which fhall be juft 20. Having from z drawn the line zc, to cut Ba into H, which taken off and applied to cp, from c to N equal ns, becaufe the angle BCN is equal to the angle can; I fay, that np is the firft term of an infinite feries between cp equal ac, and the perpendicular nz equal pz; which may be done by fquaring ac, and dividing it by the fide or number given, the comple- ment whereof to 10 is the firft mean or term of that infinite feries, fo fhall the firft term of the infinite feries between to and 20 be found 5. Thus in numbers, 10 X 10 = 100, 4,%:* = 5, the complement whereof to ro is 5, equal cn, equal np for the firft term of an infinite feries between 10 and 20, whofe fum is 20, as by the arithmetical work in the margin, where a, b, ¢, d, e, f, &c are a rank of geometrical pro- a.10 greffional numbers, whofe infinite fum would make but b. 0 8 20, and is demonftrated by the 7th and 8th of Euclid. C2255 And in numbers, thus: as 10 lefs 5 isto 10, what 10 ? dia bse the quotient will be found 20 for the whole fum of that cj hOB infinite feries between 10 and 20 whofe firft term is 5. Pac one tae In like manner, would you know the firfttermofanin- 4,....15625 finite feries between 10 and 30, divide the fquare of ac ~ 7..... 78125 = 100 by 30, the quotient will be found 3 333333, whofe Ke. isis 390625 complement to 10 is 6666666; I fay, that 6666666 is iakish - 1953125 the firft term of an infinite feries between 10 and 30, as m........ 9765625 by the arithmetical operation in the margin; and briefly ao thus, as 10 :: 3333333 = 6666666 : 10 :: 10: 30, fo is 19. + 99. 0234375 30 the whole fum of all thofe infinite progreffional num- bers between 10 and 30. In the figure you draw the line a,10 @c, which cutteth Ba in x; I fay, that Bx transferred b..6. 666666666 from c to o equal or is the firft term of an infinite feries 6..4.444444444 between ac and oF equal p®. And ps be the firft term Gd, .2. 962962962 equal 7,5, equal pc between ac and pc equal Dy = 4o €..1.975308641 between ro and 40; for, as before, 10 — 7,5 = 2,5: f.. 1. 316872427 10 :: 10 = 40, fo is 4o the whole fum of an infinite £....877914958 feries between 10 and 40, whofe firft term is 7,5, and b.... 585276634 fo of all the reft. On this great myftery depends much i wees 390184422 the following {quaring of the hyperbola, and. hath made it : fo intelligible and eafy. 4 Hence LOGARITHMOTECHNIA, st Hence you may note, that if you would know the firft mean of an infinite feries between any other number, and that number doubled, tripled, qua- drupled, &c, you may from this root 10 deduce it; as, for example, let cp be 12, and I would know the firft term of an infinite feries between 12 and 24, the double of 12, I fay, as 10 is to 5 what 12 facit 6; for, fuppofing cp = 10 it was before found 5. Therefore between 12 and 24 you fay 10: 5 :: 12: = 6 for the firft mean of an infinite geometrical progreffion between 12 and 24; and is thus, by the former analogy, proved by faying, as 12 — 6 = 6 212°: 12% = 24, fo is 24 found to be the total fum of an infinite feries between 12 and 24, as by the operation in the margin will appear. Alfo, if it were required to find the firft mean between 12 and three times that number, viz. 36, fay, as 10 : a.12 6666666 :: 12:8. And fo by tabulating or working b..6 this example as you do the former, the total fum of that Gy 383 infinite feries between 12 and 36 (the firft term being Avil. § found 8), will amount to 36; and, for proof, you fay, COR AS Peete As 12 32 12): — 36, fo is 36 the wholefum. /f....375 of that infinite feries, or geometrical progreffion, between = g.... 1875 12 and 36, the firft term being 8, as was defired. CBE 9375 If it fhall be required to know an infinite feries*between hea) 46875 ro and any other number ; as, to know the firft term be- OE tween 10 and 15, that is, between pc and ps, draw 5c, Fe hs ik Be ts rd pe and it cutteth Ba in £; I fay, that sz is the firft term ee 5859375 of an infinite feries between 10 and 15, as by the opera- eile GATT aT tion in the margin, and as before taught, +°° = 6666666 ; 235994140625. which fubtraé from ro, leaveth 3333333 for the firft term. And by the former rule is proved thus: as 10 — 3333333 wee = 6666666 : 10 !: 10: = 15. Thus may you find the b. « 3333333333 firft of any term of an infinite feries between 10 and any ce ce tera d. «370370370 other number. And if it fhould be defired to know the firft term of an infinite feries between any two other numbers; as, for ex- ample, I would know the firft term of an infinite feries €..+123456703 Svoe 0 41152264 Zee. 13717421 between 12 and 16: to do it geometrically, you muft : moe hy 4572473 fuppofe Ba = 12; and then counting 16 from p in the Pot pa line pz, prolonged by that point, and c draw a line which mont 5 ; O52 will cut the line Ba (now reprefenting 12) in a point, A SE ; rei ei tidihe 5 which taken from B, will be the length of that line geo- metrically; and arithmetically, it will be found by fay- ing, as 16D to Dc, fo 16 = 4 to BK in the line Ba, 145999971774 numbered from B to a when Ba is 12. In numbers the proportion ftands H 2 thus: 52 SPEIDELL’S$ thus :).165: 12, f2-4 329g; vwhich? (9 asthe Grit 4 woe ; term of an infinite feries between 12 and 16, bi 3 as was required, and is manifeft by the opera- Gi TG : tion in the margin; which to prove ie the fore- a. JJ1875 going rules, you : fay, as (PN BueenO Tae Se 2.40875 16 for the whole fum of that infinite feries between oso. 29290875 12 and 16, the firft term being found, as before, a aloes § 7 3240 bOeae toghe /3: be wseee 18310546875 Thus have you that great myftery unfolded, of ——-—_—__-- finding geometrically and arithmetically the “fictt 155¥99993896484375 term of a geometrical progreffion with the whole fum of that infinite progreffion, or feries, between any two numbers; which is the main thing I conceive that famous Mercator was fo lucky in difcovery thereof, and doth unravel the myftery of fquaring the hyperbola, as will be manifeft in the next chapter following. é = aa ate antes? is to fay, as 9 to 12, Shee 12 facit fess Tir Bee r hb i RB eS eR RE TS TOR RR A RTE EIA BE ESET LA ACAD TE SNES SIDE L LES TILER IO AC AE ETO i Ores > Beer. ha IV. Obferve from the faid learned Gregory’s Exercitationes. Geometrice, he giveth three quantities, or {paces, contiguous to the vertex a, which fhall be all equal the one to the other, which is very true and perfpicuous ; and then fhews how to find the areas of them feverally, as in page 9, 10, 11, and 42, of. {aid book. And here we muft confider them all three before we come to uwnderftand Dairy’s rule, which is but a deduCtion from thefe, as will appear hereafter. And now I begin to confider the faid three feveral quantities, or fpaces, all contiguous to the vertex a, and of a different form, and yet equal. the one to: the other. Let it therefore now be fhewn.thofe three feveral fpaces differing in form, and’ yet equal the one to the other, contiguous to the vertex a, which fhall reprefent the curvilined trapezia, or hy perbolical {pace for 2. The firft curvilined tra- pezia, or hyperbolical {pace for 2, let be zpave, which is intelligible intuitu: the fecond let be avenc, which is equal to the former zBave, by the 43 of the 1 of Euclid, becaufe the parallelogram EzBH (BH being equal to zz) 1s equal to the parallelogram HACN (Ha being equal to HB). Now, forafmuch. as the curvilined triangle avEH is common to both the faid curvilined trapezias, or hyperbolical fpaces, it remaineth therefore, that thefe two. curvilined’ trapezias,, or hyperbolical fpaces, zEvap and avENc, are equal the one to the other. And now to find out the third curvilined ‘fpace contiguous, to the vertex a, and yet equal to either of the other two, but differing in form, doth require a little further confideration, which from him is directed thus: And is manifeft by the figure, divide nz in two equal parts in 5; then, as before taught, wilk it be as 5D : Dc 3! 5B : BK = Dx make cx equal to cll, or find pDII, it is as: D5:DzZ%; pc: DIT upon I erect a perpendicular to touch the return or con- 3 tinuation LOGARITHMOTECHNIA, 63 tinuation of the curve on the other fide of a, from a towards © in =; {0 is this curvilined figure, or hyperbolical fpace, 2 avx (differing in form) equal to either of the other two zevaB or cavn. And for finding the area of this curvilined figure, or hyperbolical fpace, 2 avx, is derived Dairy’s rule, which is but a dedu@tion from the finding of the areas of the other two, as will here- after appear. Arithmetically, px is found by faying, as 15 to 10; what 10? facit 66666666, and 10 lefs 66666666 reft 33333333 for cI, ‘fo is xII equal to 66666666; or if may be found thus: as D5 : pz :: pc: DIT, which in num- Bere i9,04S'1 53 20-22 10 = 1.39333393 for\pH: James Breit § in the 4th propofition, page to and 11, of his Exercitationes Geometrice, doth contemplate firft, the fecond of thefe three curvilined trapezias, or hyperbolical {paces ; that is to fay, the curvilined trapezia, or hyperbolical {pace, CAVEN}; and in that fourth propofition, after a long and learned demon- ftration,: doth prove the {pace caven to be equal to his fuppofed quantity w; and then refers you to Cavalerius’s method of indivifibles; a book I have not yet feen.. Which briefly, I conceive, may be thus eafily demonttrated.. It is manifeft, : that all the per pendiculars let fall from the ordinates (ftanding upon the prolonged fide ps) to the bafe cp, doth not only defcribe the curve, but would alfo fill the whole hyperbolical fpace, were number, and the curve definitive. And thofe perpendiculars let fall from the ordinates (ftanding upon Bp prolonged, numbered 20, 30, 40, 50, 60, &c), doth divide the bafe pc ini t2224&c. And whereas the diagonal cs, by croffing all thofe perpen- diculars, doth give the firft term of an infinite feries between the root, or fide, ac (= nw), and the length of each of thofe perpendiculars ftanding upon the bafe cp. Therefore, to know the hyperbolical {pace caven, divide the paral- lelosram cu, making it the firft term of an infinite feries by the ratio of nz = Nc to NH in infinitum, and each of thofe quotes, or proportional numbers, byt, 2, 3, 4, 5, 6, 7, 8, &c, alfo in infinitum. The quotes of all the laft di- vifions added, ‘will give you the area of the hyperbolical {pace caven, and fo for any other curvilined or hyperbolical {pace ftanding upon the bafe cp, as by the calculation following. And before I handle any of the other two curvilined fpaces, differing in form, and yet equal to this hyperbolical fpace caven, we will exemplify this de- monttration in operation; and the figurative work thereof fhall be the work of the next chapter, Ga bbl sAos Rx V. E T it be required to calculate the area of the curvilined trapezia, or hyperbolical {pace, caven to 15 places, and hereafter you will fee it done to 25 places, according to Dairy’s rule, in chapter VII. but with greater difpatch. Thus by the calculation do you find the area of the curvilined trapezia, or hyperbolical fpace, cAvEN to the 15 place, to be 693147180559945 equal eS the 54 SPIDER L Ls the curvilined trapezia zEvaB, and alfo equal to the curvilined trapezia, or hyperbolical fpace, WXavx; the calculation of the areas of any part of thefe two latter fhall be fhewn hereafter, which will differ in operation, yet bring out the fame number; and in calculating the laft, we fhall ufe Dairy’s di- rections. It having been before fhewn, that the hyperbolical fpace zavaB, equal to the curvilined trapezia, or hyperbolical fpace, cavEN, is equal to the curvi- lined trapezia, or hyperbolical {pace ye¢rrez; that therefore the faid zEvaB Is a {pace, or quantity, to reprefent the logarithm of 2. So then the aforefaid number 693147180559945 is an hyperbolical logarithm of 2; and having the logarithm of 2, you have alfo the logarithm of all the powers of 2. And, by this calculation, you have not only gotten the logarithm of 2, but gained alfo the logarithm of 3: for if you add all the quotes marked with this afterifm (*), the addition of them fhall be half the fum of the hyper-: bolical logarithm of 3, agreeable to the 4th confectary of the 4th propofition, and firft inference on the 5th propofition of faid James Gregory’s Exercitationes Geometrice ; from whence, ’tis plain, that Michael Dairy had his rule, as will appear more manifeft after we have contemplated the two other curvilined tra- pezias, or hyperbolical fpaces, zDvazB and TDavx. The addition of the quotes marked with (*), make 549306144334055, which doubled, is 1098612288668110 for the logarithm of 3; and now, having gotten the logarithm of 3, you have alfo the logarithm of all the powers of 3, and of all the compofites of 2 and 3. . Again, if you fhall from so fubtract 125, and to that add 4166666666666, and from that fubtraé 15625, and fo on throughout, you fhall have the loga- rithm of the difference between 2 and 3, or the logarithm of 1 and. 4, or 1 and 3, correfpondent to the inference on the 5th prop. of James Gregory ; all which fhall be fully exemplified hereafter. The calculation of the logarithm of 2, according to the method before going, is the ground-work of all the calculations following; and I fhall only give the calculation of one more fpace to reprefent the logarithm of 3, after that method, though we have, you fee, gotten already the logarithm thereof ; but fuppofing we had not, and were to find that firft according to the faid method. From F you let fall a perpendicular, as ro; fo is the curvilined trapezia, or hyperbolical fpace, aEFoc equal to the curvilined trapezia, or hyperbolical {pace, AEF®z, for the oblong azoc, as before fhewn, will be equal to the parallelogram EF®s, and the curvilined triangle azFE common to both pa- rallelograms. ‘Therefore the curvilined trapezia aEFoc equal to the curvilined trapezia, or hyperbolical fpace, azr@s, to calculate whofe area, which will be the logarithm of 3, you proceed as followeth, which work is but partly done, to fhew the way thercof, the logarithm of 3 being hereafter done to 25 places, but with far greater difpatch than this method will permit, ‘5 LOGARITHMOTECHNIA. 666666666666666 1 666666666666666 444444444444444 I] 222222922222223 296296296296296 III = .98765432098765 197530864197530 IV. 49382716049382 1316872427983 52 V. 26337448559671 . 87791495198902 VI. 1463191 5866484 55 - - 8361094780848 -» 4877305288829 » 2890254985973 1734152991583 »+ 1051001813081 2+. 642278885771 - §8527663465934 : VI - 39018443310630 : VIII = 26012294873752 1,30 «1 -17341529915834 : X -11561019943888 : XI -- 7707346629258 : XII 1 have but gone twelve fteps in the calculation of the logarithm of 3 after faid method, which will, if it were added, but give the logarithm of 3 to 5 places. I have left it unfinifhed for the exercife of thofe who fhall take delight herein, and finifh it throughout, to the intent of making the logarithm of 3 to 15 places, according to this method. By adding the quotes of fo much as is done, the firit five figures will be 10986, correfpondent to the logarithm of 3: this method being fomewhat flow, I fhall not calculate the logarithms of any other numbers according to it. And, by thefe two examples, the reader may fee enough to calculate any other curvilined trapezia, or hyperbolical fpace, ftanding only in or upon the bafe ep, equal to an hyperbolical {pace ; reckoned in the prolonged fide ps. And fo we will contemplate, in the next chapter, the affections and properties of the hyperbolical fpace, or curvilined trapezia, aZ@yc, equal to the curvi- lined trapezia, or hyperbolical fpace, aBZzEv. Cues AS ep, VII. EFORE we fhew how to calculate any part of the curvilined trapezia, or hyperbolical fpace, aXOYc, equal to the curvilined trapezia, or hyperbo- licai fpace, Av EzB, we will infert tables to illuftrate the 1, 2, and 3 propofitions of James Gregory’s Exercitationes Geometrica. THE a re lea) SPEYTDPLL*s 6666666666 — z SZ1g9zgQ0QP The rule to find z, 2,5 = 2,5:5355: = 10=2Z. eee ee tee eee aah; e i a fe) Q > The rule to find z is 5185 375 5S 66666666 impares = z. — en ee ~ Be GN ORE ay ~) ww The proportion to find z, 25—625=1875:25::26: = ‘Be 33333333 pares =z. Zz os — The rule is 25 — 625 = 187%: 25+ = 33333333 Equalis paribus. ae Flt+lti+i+ PRM Pea ta The proportion is 52575585 233333933 Equales exceffui imparium fupra omnes pares. a plas ml ibe rl fia Pls wal 2E° AY Tanah on The aC HARTBA is 5m 125=375:5325:— 66666666 Equalis paribus. 7 | + 4 wn P | 3 — a + Ww od +t+++44+ Q Qa y phakic hie Q —a+ FIRST TABLE OF ILLUSTRATION, I] Ill AI TA ITA ‘TIA THE _ all LOGARITHMOTECHNIA, $7 THE EXPLANATION OF THE FOREGOING TABLE. This table, confifteth of eight columns. The firft is a fuppofed literal rank ef quantities continually proportional. The fecond is of numbers correfpondent to the firft in a ratio, as 2 isto 1; or 5 to2,5: What2,5? And fo fuppofed continued im infinitum ; with the rule how to find out the whole fum of thofe numbers fo continually proportional. The third column fheweth how to find the whole fum of the odd quantities or numbers, The fourth teacheth how to know the whole fum of the even quantities or numbers. The fifth telleth how to find the whole fum of the difference of the quantities of the firft column. The fixth fuppofeth a — B + c — Dp + £, and teacheth how to find the folution thereof. The feventh fuppofeth a +B —c +4 Dp —E, and giveth a rule to refolve the fame. The eighth and laft fuppofeth the whole rank, firft affirmative, and the fecond evenly or alternately lefs; and giveth a folution thereof. A further explanation of this table will be when we come to calculate the area of any part of the curvilined trapezia aZ@yxc, THE SECOND TABLE, CaS eas il anes Ee Sav od o L SG =e 4, 7 Ge Ribs 20 |[3desoasse a Ge | COOOCNGE. | FE oe SES 40] 255 a 735 Sold pee $5 Bie ee ore Se 48, e4f Ba BEDe 60] 1,66666666 gs . | 8,33333333 ¢] i a8 Eig 2 es Tete ads, Be tae eae a! un 32 Oia Ge 9 ras 3 Sa 8 oof 1111111 6 § 8 | 8,88ssssss S| 3 E82 E58 100) 1. gE 0 ER This table confifteth of four columns. The firft is equal fpaces, numbered in the fide pz prolonged, or the tangents greater than radius. The fecond fheweth the length of the perpendiculars ftanding upon the fide DB prolonged; which are tangents lefs than radius; and by the tops pafs the curve or hyperbolical line. The third column is the arithmetical complements. , The fourth column fheweth what proportion the fecond column hath to radius. The reétangle, or parallelogram, of the firft and fecond column, is equal always to the {quare ap. Vot. II. I This 58 pe hw ARES Oe oes : This table is of ufe to find points to defcribe the curve or hyperbolical Tine, or to examine if the curve pafs through fuch points as the table mentions. The making of this table hath been formerly fhewn, when it was taught how to defcribe the curve. We now come to fhew, how to make a table to find the length of the bales, of the compound curvilined trapezias, or hyperbolical fpaces. We call that a compound curvilined trapezia, or hyperbolical fpace, isk Ae is in the middle of that bafe, ~ So ac flanding upon the middle of I[x hath perpendiculars or fides TIX and xv, fo is the curvilined trapezia I1X-avx, to be hereafter under{ftood a compound curvilined trapezia, or hyperbolical fpace, and will be fhewn as followeth to be equal to the aforefaid {paces cavEN, and to avezs for the logarithm fpace- of 2. And the compound curvilined trapezia A@ AvEN will be equal to the cur- vilined trapezia avFoc, and to avrés for the logarithm {pace of 3. The compound curvilined trapezia, or hyperbolical {pace TZavx, we may prove to be equal to avEzB, thus: by the 43 of the firft of Euclid, ‘the paral-. lelogram ck is equal to K§, and the curvilined triangle avK common to both ; fo then is avKxe equal to av58. Andthe parallelogram H# equal to the fquare vz, and the curvilined triangle 24. equal to the curvilined triangle VeazE; and fo the compound curvilined trapezia WX avx equal to the curvilined trapezia AVEZB for the logarithm {pace of 2. For, by the 4th table following, look what proportion the perpendiculars. or fides of the compound curvilined trapezias have one to the other, the like proportion have the fides or perpendi-- culars of the other two curvilined trapezias. So in this compound. curvilined trapezia 1X and xv, the fides or perpendi-. culars are in a proportion as 2 is to 1 defcending, or as 1 is to 2 afcending; fo hikewife in the curvilined trapezia caven (equal to the aforefaid compound curvilined trapezia I1Xavx), the fide or perpendicular we is double to ca, And. alfo in the curvilined trapezia avEzB (equal, as before, to the compound cur-- vilined trapezia HXavx), the fide, or perpendicular, na, twice ze, as before. taught. Thus, by the ratio of the 2 tables Sher s you may make a compound curvilined trapezia equal to either of the other two curvilined trapezias, or hyperbolical {paces ; and the calculating the area of the compound curvilined: trapezias, will be found to be of far greater difpatch than the former method s. by which we fhall make ufe of Dairy’s rule, or rather the learned James Gregory’s, from his firft inference on his 5th propofition. We come now to infert the third table, which is a table of ratios, to find the length of the bafes of the compound curvilined trapezias. _ You may note, that in all the three different forts of curvilined trapezias, or hyperbolical fpaces, equal the one to the other,. if on the middle of their bafes you fhall erect perpendiculars to touch the curve, the greater part or fegments in ahs is equal to either greater fegment of the “other, and fo is the leffer part ‘or fegment of the one equal to the Teffer fegment of either of the other. THE LOGARITHMOTECHNIA, 69 THE THIRD TABLE. \ a a Table of Ratios to find the Length of the Bafes of the compound curvilined Trapezias, or hyperbolical Spaces. WP IP 2 vy 5D:DC%. DC: DX 15:10 3210: 666666666) For (Length of the 5D. ZD tf} Dc: ou 2 bafe 153 20°22 To! 1, 333333333 666666666 Go fee CI. DO aU NS, 20:30 ;:;T0 : 15 ue g TO as Ons ZO seDG ss Css DN ge 5. TO: |'5 ‘ Oe dO ees EOn S410 ae JO % 10082 EO% : 333333333 30 3503310: 1660666006 5 { 1333333333 35:10 7 Tos | 286714285 35 2 60. S Toos 1, 714285715 6 | 1428571430 This table confifteth of fix columns. The firft four fhew the proportion or ratio to find the lengths of the bafes; and the number in the fixth column is the length of the bafe for fo many {paces as the fifth column fignifies. And, by the fame reafon, you may find the lengths of the bafes for any other curvilined tr apezia, or hyperbolical {pace. Thus is 666666666, of the fixth column (the difference of the two firft numbers in the Furth column), the length of the bafe, for the curvilined compound trapezia, or hyperbolical {pace, to reprefent the logarithm of 2. And 10 the length of the bafe for 3,-fo is 12 for 4; and 1,333333333 for s; and fo 1s 1428671430 for the length of the bafe for the hyperbolical {pace for 6. And thus may you do for any other {pace or number. The numbers in.the fourth column for 2, 35 4, 5 6, &c, are in proportion as 2,2,4,4,4, &c; and added are equal to twice radius, or 20 = DY. We proceed next to fhew how to make a table of ratios, to find the lengths of both the perpendiculars, or fides, of the compound curvilined trapezias. 69 SPEIDELL’S THE FOURTH TABLE. a Being a Table of Ratios to find the Length of both the Perpendiculars, or Sides, of the compound curvilined Trapezias, or hyperbolical Spaces. 10 — 666666666 cD + cm (= ch Io + 666666666 : DBi: DB: Ms 1666666666: 10 :: 10 : 6 | iy THiiy Thy) TV V cD — cx renfl = a9 $ MBI DB Lp Sis) ae 10 — 333333333 = 666666666 : 10 3:10:15 F cD + cll (= cx) =. pII > DB i? DB: DY 10°: 333333333 = 1333333333 +\19°+: 10 of 725 cD — CN = ND : DB.«? DB: Den IO — 5 iis * (LO msc sO ae ae cD +c4 (= cn) = vdII > DB i: DB : S40 3 10 + 5 mans! FF 2 10 33 10 : 666666666 cD — cS — aD : DB3; DB: DR= OB 10 — 6 =e 2; 00 Sasol Oy tsiraie cD +c8 (=DS)= DB 3 DBs cB ie es 4 10 + 6 — ee 3) FOC, gOws 625 cp — c& = (slp > DB: DB: D@P= FL 333333333 : 10 t: 10 : 30 i Dung *s1) 5 This table confifteth of five columns. The firft contains 35 + Qiteg the quantities and numbers of the firft term in the propor- 583333333 tion. The fecond column, the quantities and numbers of 40 the fecond term in the proportion. The third column, 571428571 7 the quantities and numbers of the third term in the pro- 45 | } 8 portion. The fourth column, the quantities and numbers 5625 of the fourth proportional number or term; wherein are’ 50 numbers for the length of both, the perpendiculars for 5555555 a 9 2, 3,4, 5, &c. The fifth column, is the numerical order ) of the compound curvilined trapezias, or hyperbolical 55. * {paces, s0f2, 3,04, 5, here. And, by the fame ratio, you may find the lengths of both the perpendiculars for any other compound curvilined trapezias to reprefent the logarithm of any | other number. By the fourth column, you may perceive the perpendiculars, or fides, of the compound curvilined trapezias, or hyperbolical fpaces, are in fuch proportion the one to the other, as the number they reprefent are to unity. That is to fay, in the compound curvilined trapezia TEX avx to reprefent the logarithm {pace of 2, the perpendicular xv is to n& as 2 is to 1. And in the compound curvilined trapezia A@aveEn to reprefent the loga- rithm {pace of 3, the perpendicular NE js in proportion to 4©@ as 3 is to I. And fo the perpendiculars of the fourth logarithm fpace as 4 to 1; and of the fifth {pace as 5 to 1, &c; as by the fourth column of this fourth table appeareth. And LOGARITHMOTECHNIAs 61 And the perpendiculars of both the other forts of the curvilined trapezia, or hyperbolical {paces, are likewife in the very fame proportion the one to the other ; as you may note from what hath been faid before of them. By thefe tables, and by what hath been faid formerly, thefe three curvilined trapezias have the fame properties and affections as thofe have in an hyperbola derived from the fection of-a right-angled cone. We fhall now, therefore, come to calculate fome part of this latter hyperbo- lical fpace, before we fhew how to do it all at once; that is, of the hyperbolical {pace 4@aveEN to calculate the area of the fpace A@ac, which is equal, as before fhewn, to the fpace of svas. And when we have fhewn to calculate this part, we fhall, from this, and what hath been taught how to do the other part, come to derive Dairy’s rule, or rather James Gregory’s, which is com- prifed in the firft inference on his 5th propofition, Ouyabi Ahi Ps VIE. FE, have, in the fifth chapter, calculated the area of the curvilined trapezia, or hyperbolical fpace, caven equal.to avezs for the loga- rithm of 2. In this. curvilined trapezia cAvEN all the perpendiculars ftanding upon the bafe cn, are each more than radius ca (or greater than the tangent of 45° 00’), being {till afcending and affirmative; and therefore, by the firft table, to be con- tinually adding ; as by the calculation thereof is alfo manifeft, We are now come to calculate the curvilined trapezia cAOA, part of the curvilined trapezia cY©Oa equal to AvEzB. In this curvilined trapezia cA@a (equal to Bsva) all the perpendiculars {tanding upon the bafe ca, are each lefs than the radius ca, being ftill de- fcending and negative; and, therefore, to be handled by the firft table ac- cordingly. The bafe ca is equal to: cw of that {pace, calculated, as before, in the sth chapter. | ‘ If by the vertex a you draw a parallel to the diagonal cp as zaY, it is a tangent to the curve touching it in the point a, and az doth cut all the perpen- diculars contrarywife to cz. Forcx = x is not equalto VN = Ud, but nd is equal to Km = xB, becaufe cmis equal to cx, and the anglenY¢ equal to the angle ksm; fo, therefore, by the 1 and 4 tables, all the perpendiculars ftanding upon cy are lefs than radius. And feeing, by the fixth column of the firft table, and alfo by the fourth of the fourth table, we may find the length of A@.. Therefore, to know the area of cAga, making ce the firft term of an infinite feries in a continual proportion, as ca is to c4, that is, as 50 to 25: What 25? Facit 125, asin the infinite feries of numbers continually propor- tional for the calculation of the logarithms of 2, in chap. 5. You do therefore, as there faid, from 50 (of the fecond part, under the title, The Quotes to be 62 SPEIDELL?S added) fubtra&t 125, and to that add 416666666666666, and from that fub- trac 15625, and fo on throughout, you fhall have 405465108108165 for the area of the {pace cAOa equal to Bavs. And thus may you find any other part ofcYma. We hall thew how to do it for cata and cxva, becaufe from them we fhall derive Dairy’s rule, or rather James Gregory’s; for from them we have derived and calculated the logarithm for two to twenty-five places, as by the calculation, following next after this will appear. Now to calculate the area of the curvilined trapezia, or hyperbolical {paces, cura and cxva, you make cx, or cH, the firft term of any infinite feries, and the fecond term in fuch a ratio as mc is toca for your proportionals of your in- finite feries, and fo proceed on, as in chap. V. and as here appeareth. The Infinite Series of Numbers The Quotes to be added for cxva. proportional. @ 333333333333333- 1 A+ 333333333333333 +4 a@urirwziri1zi1z111t. «Il B—.55555555555555 + B Qaa «37037037037037- III c +.12345679012345 +c a* .12345678012345. IV vD—.. 3086419753086 + Dp a* ..4115226337448. V £ +...823045267489 + E a® ..1371742112483. VI ¥F —...228623685414 4+ F a7 ...457247370827. VIl Ge +....65321052927 +c a* ...152315790275. VIII H—....19051973784 +H a°® ....50805263425. IX 12 +..... 5645029269 + 1 a> ....16935087808. X K-—.....1693508781 + K TI AE 5645029269. XI Lt +......513184479 + L at Wh ots bs bes 1881676423. XIT m—......156806369 + M We have but gone through twelve fteps of this calculation, to fhew the manner thereof; but fhould you proceed to go through it till it works off, as in chap. V. you may have both the fegment cxva, and cmra : for, if you finifh the calculation, and add up all the quotes, that fum will be the area of cxva, and be found 405465108108165, as in chap. V. and is equal to the greater fegment WNHE ) in the curvilined trapezia, or hyperbolical {pace, cNHED A, ~ and alfo WNHE ) is equal to cA @a equal to ABSv. And if from 333333333333333 you fhall fubtract the fecond number 555555555555555, and to that add the third number 2345679012345, and then from that fubtra¢t the fourth, and fo add and fubtraét according to the fiens +- and — throughout, you will have . . 287682072451780 for the area of the leffer ‘feginent of the compound curvilined trapezia wmra ), that is, the area of cza, equal to cwD4A, equal to v5zE. And you have not only gotten, by this calculation, the area of each fegment feparately, and fo, confequently, the area of the whole fpace, by addition of thefe two, but you have alfo obtained the half of the whole area at once; fot if you fhall (corre{pondent to the column of the firft table) add the numbers with -- affirmative, they will give you half the area of the compound curvilined’ trapezia LOGARITHMOTECHNIA, | 63 trapezia xurav for the logarithm of'2, which you will fee prefently exemplified, and done to 25 places: and this is the fum of James Gregory’s inference on his fifth propofition of his Exercitationes Geometrice ; and fo agreeable to the rule delivered to me, as before declared, by Michael Dairy. Having acquainted * feveral perfons with Dairy’s rule, in page 45, and fhewn to them fome figurative work thereupon, in order to make a logarithm, I was, notwithftanding, fome time, through inadvertency, almoft difcouraged of ever knowing how to cube in infinitum fuch a number as there {poken of; neither did any of thofe to whom I had communicated the fame take any fuch notice thereof (that I know of), fo as to do it. And now I come to fhew, how I overcame that difficulty of eubing a range. of figures for 25 places, which he told me I muft do i infinitum, before 1 could make the logarithm of fo many places; and to re- move the ftumbling block, Ido confefs, took up fome time; for Dairy had not then told mea word of fuch an author as James Gregory, and I had not known his works but for John Collins, fome years after Dairy’s death. But, before I ever met with Gregory’s book, I had obtained my defire to cube iz infinitum twenty-five figures ; that is, twenty-five 3, by dividing by g continually, as in the calculation following, to find the logarithm of 2 all at once; which manner difpatcheth the calculation much more {peedy than the method of calcu- lation in the fifth chapter. ; And now the reafon of cubing. twenty-five 3, by dividing only by 9, doth ollow. Forafmuch as Dairy’s rule, before declared, to make the logarithm of 2, doth bid you to 2, add 1, and from 2 fubtract 1, fo fhall there be a refult or fraction of +; and then divide 1, or 100,000,000,000,0000, by 4, whofe quotient is 333333333333333» Which cube zm infuitum, it had been as much as if he had faid cube + fractionally, which is .1,, and divide 10000000000000000 by 33,, the quotient will be 37037037037037 for the cube of a, or 4, as in the ope- ration beforegoing. Now, forafmuch as you would cube,the number for 4, VIZ. 333333333333333 (which is 1, or 100,000,000,000,0000, divided by 3), it is as if you. fhould fay, as. 27 to 1000000000000000: What 1? the quotient will be 37037037037037 for the cube of 4, or 333333333333333, as before. Now, if you fhall, as in the operation beforegoing, fet down 3333 33333333333 (which is equal to +), you have no more to do but to divide by 9, for that 2 of + is equal to .3,; and, therefore, dividing 333333333333333 by 9, the quotient will be 370370370370370, as before, for the cube of 4, or a; and feeing + x+ 1s equal to $, you have no more to do but to divide continually by 9, and they fhall all be proportional numbers, by 7th of the 8th of Euclid, and confequently corre{pondent to the odd powers ; for if the root be multiplied by the fquare that begets the cube, and the cube again by the {quare that begets the fifth power, andfoon. So here, forafmuch as dividing by 9 doth beget the third power; if you fhall therefore continually divide. by 9, you fhall have the refpective odd powers accordingly,. as is alfo manifeft by the laft figurative calculation ; and all for that a 1. doth neither multiply nor divide, and that 2 of + is equal to 43,3; and if you fhall divide .4 of 1 by 9g, the quotient will. be 333333333333333 Which is equal to a4 for the firft number or root, as. efore.. . 4 Now, 64 SPEIDELL’S Now, forafmuch as to make a compound curvilined trapezia equal to an un- compounded; as, for inftance, to make the compound curvilined trapezia wnra ) to be equal to the uncompounded caveEn, equal to anzE = aQy for the logarithm of 2, and to find the length of the bafe, and both the perpen- diculars, hath been difcourfed, and may be feen, as in the third and fourth tables beforegoing. We come to handle and calculate the area of this com- pounded curvilined trapezia wur) a, for to make the half logarithm of 2 at once. Seeing, by the fixth columnofthethird table, the bafe wt is6666666666666666, whofe half is .. 333333333333333 for cw, or ci, equal to the firft term in the former operation (and alfo the fame as Dairy’s refult, or fraction, of 4), and that . I mut divide in the ratio of ac to cn, or cw, iz infinitum, asin the fifth chapter, and alfo as in this is fhewn and taught, for to make the infinite feries of numbers proportional. It will appear, that if I do divide 33333333333333 by 9, it will give me the cube of the firft term, and fo dividing continually by 9g, will pro- duce the numbers appertaining to the odd powers, as by the large calculation to 25 places, next following. And feeing I am, by Dairy’s rule, or rather James Gregory’s, to divide each of the numbers of the infinite feries by the indices of the odd powers, it is manifeft, that this rule of Dairy’s is derivable from the 8th column of the firft table ; For at+B+c+dD+E+F4+64+H 41 Anda—-B+c—dD+E—F+G—H+I1 Doth make *a + *c + 7E + *o + *1 + 7L + *N; And therefore every other line of the quotes to be added in the former opera- tion, doth make half the logarithm of 2. In making the infinite feries, as in page 66, in order to make the half logarithm of 2 to 25 places, be very careful to fet the figures in their due places, and to make that feries you are to divide continually by 9, which being done throughout, you may then prove your work by multiplication, in multi- plying each line by 9; and if thofe multiplications produce the foregoing numbers, you may conclude that part of the work to be well prepared. And feeing, by the direction over the figurative work in the fecond column of page 66, you are to divide each of the numbers in the firft by 1, 9, 5, 7, 9, &c, you muft fo order the quotes of the fecond, that they may lie in the fame line, or range with the refpective dividends or numbers in the firft. For the better preventing miftakes, the letter figures do reprefent the divifors proper to each line. And, would you make the logarithm of 2, according to the fol- lowing method, for 7 or 8 places only, you may very well produce it in half an hour’s time, as by that calculation 1s very perceptible. And fome that have had thofe two fheets I formerly printed as a {pecimen hereof, have told me, they have done the fame; and were very folicitous I would, as foon as I could, publith the remainder; which, at length, as time and leifure hath permitted, is done. And though I have not here inferted many examples, yet, by what are herein done, you may perceive how to proceed for any other number pro- pofed. And, with the direction and reference at the end of this chapter, thofe that are willing, and curious herein, may make a logarithm for any natural number defired. I have not added hereto any table of logarithms at this time ; | and LOGARITHMOTECHNIA. . 65 and what I may do-hereafter, in order thereunto, I do not prefume to promife. I doubt not but fome may both examine fome table or other, or make by this me- thod one de novo, and fatisfy themfelves about the fame; and fome have told me, fince my communicating this method unto them, that if the firft makers of tables of thefe.numbers had made them by fuch eafy ways, they did not doubt but their tables might have been. fomewhat more exact. Howfoever, it pleafed God, who is the giver of every good and perfect gift, to raife and endue fuch men with great ability and patience, to. perform thofe tables, with fo much difficulty and labour as their methods did require, and for common ufes fufficient. And with fuch eagernefs did that age embrace and purfue the invention of thefe numbers, that Ullach, a Dutchman, had exhibited a table of logarithms to 10 places for 100000, before the learned Henry Briggs’s table, which he had in part done to 15 places, could be accomplifhed by him.—So exceeding glad were they of the invention. And the learned Henry Briggs, in’ his Epiftle Dedicatory to our moft gracious King’s father, when Prince of Wales, faith, that amongft the ancients there is not found any footfteps of thefe numbers. Of whofe conftructon and ufes the faid Henry Briges hath written, in his Arith- metica Logarithmica, moft learnedly and copioufly. And now follows. the figu- ratiye part of making the logarithm of 2 to 25 places. . Ee | aed K The 66 The Infinite Series, or Numbers conti- nually proportional, Thefe numbers are continually divided by 9, in order to make the half logarithm of z. ( @ ~—-3333333533333333333333333 3793793793 79370370370370 — 4115226337448 5596707819 a!" 4572473708276177411980 a® 508052634252908601 331 a® 56450292694767622370 a" 627225474386 30691 52 a” 696917193762563239 as 77435243751395915 a 8603915972377 324 @. 955990663597480 106221184844164 118023 53871574 1311372652307 145708072489 16189785832 1.79886 5092 199873899 22208211 2467579 Differentia 1 27 jae i Unitas 1 30464 Numerus] , ¢— A isetche arin 30 5 Propofit. —FAT—s | 376 Summa - 3 You may perceive, that if 1 be added to 2, and fubtracted from it, it leaveth a refult of 3; which multi- plied into itfelf, maketh 2; and therefore thefe numbers are conti- nually divided by g. SPEIDELL’S Thefe numbers are quotes from thofe on the oppofite fide, by divi- ding them byt, 3, 5, 7, 9, &c, and 7A #¢ 2B *G % *L Shara a ak oot metre 9 Parser sp si = + &c, correfpondent to the laft preceding calculation ; which added, make half the area of the compound curvilined trapezia xnrav for the half logarithm of 2 to 25 places. bs 33333333333333333383300aue III 123456790123456790123457 C v 8230452674897119341564 vin 653210529753739030283 IX 56450292694767622370 I XI 5131844790433420216 L XIII 482481134143313012 N Pay 46461146250837549. P XVII 4555014338317407 RB XIX 452837682756702 T XXI 45523364933213 W XXIII | 4618312384529 ¥ XXV 472094154863 XXVII 48569357496 XXIX 5024416293 XXXI 522251156 XXXIII 54511063 XXXV 5710683 XXXVII 600222 XXXIX 63271 xUr 6687 XLIII 708% XLV rh XLVII 8 XLIX 346573 5902799726547086160°% half the logarithm of 2, 69314.71805599453094172321 the logarithm of 2. Thus LOGARITHMOTECHNIA, 67 Thus have we calculated the logarithm for 2 to 25 places, after Dairy’s rule, or rather James Gregory’s, which method maketh far greater difpatch than that in chap V ; for this calculation, though to 25 places, is fooner performed than that of 15 places, in chap. V; as, by comparing them, is very perfpicuous and manifeft. ) And now we have exemplified the rule Dairy declared; and I am apt to believe he had ftudied well Gregory’s faid Exercitationes, though he was not pleafed to tell any more thereof, but that others fhould take pains therein as well as he; and, truly, if John Collins had not acquainted me with Gregory’s works, I had done the work, but not with that fatisfaction I met with from James Gregory’s books : and here you have it in a more familiar difcourfe and diale& than that of James Gregory’s, being altogether analitical; and if any letter or fymbol be miftaken in his, it is very great ftudy and labour to find, and to fet it to rights. I find James Gregory had calculated the hyperbolical logarithm for 2, in his Vera Circuli § Hyperbole Quadratura, to 25 places, which agreeth with this cal- culation but to 17 places. I have not raifed the logarithm for 2 to his do¢trine in that book, but am fatisfied this calculation for the logarithm of 2, in this chapter, is true, to an unit, in the 25th place, and may be, in two hours, very well examined by any one that will take the pains to do it, and they fhall find it to be as herein calculated. And to this I have the concurrence of the moft ingenious and laborious Mr. Abraham Sharp, who (from the occafion of my pub- lifhing formerly two fheets of the praxis hereof as a f{pecimen) hath fhewn me his calculation of the logarithm of 2, and fome others, to 40 places; the like, I fuppofe, not hitherto heard of or feen. Without all doubt, Gregory found, that Mercator’s lucky invention of fquaring the hyperbola, was of far more difpatch than that of his Vera Circuli & Hyperbole Quadratura, or elfe he would not have wrote on Mercator: but fo excellently hath Gregory illuftrated Mer- cator, that a better way of fquaring the hyperbola, I fuppofe, hath not, nor may be found. | : We fhall, for the future, keep to this laft method of calculating thofe ex- amples hereafter following. And now we proceed to calculate a logarithm for 1i;-for having got the logarithm of 2, we have the logarithm of 4, and all the powers thereof ; fo then, if we make a logarithm of 14, and add to it the lo- garithm of 4, we fhall have the logarithm of 5; becaufe that 4 x 14, makes 5; that is, 4 multiplied by 12, makes 5: and this method I took to make the logarithm of 5; and having got the logarithm of 5, to the logarithm ‘of 5 add the logarithm of 2, and you have the logarithm of ro. And, when I have fhewn this, I fhall produce Briggs’s logarithm of 2, by one fingle divi- fion ; for that all forts of logarithms are proportional the one to the other. We therefore now haften to make the loganthm for 14. K 2 The om. SPELDELL’S The Calculation of the Logarithm for 11 to 25 Places, The Infinite Series, or Numbers con- tinually proportional, Thefe numbers are continually divided by 81. We Lil REL eel Lr ict’ Ltt Thefe numbers are quotes from thofe on the oppofite fide, they being divided by 1, 3, 5, 7, 9 &c; which added, make half the logarithms for 12 to 25 places. ITIIIQIIIIIIIIIIIIIIIIIII “A a = 137174211248285 32235940 IIL 4572473708276177411980 C a’ 169350878084302867111 Vv 33870175616860573422 E qa’ 2.090751581287689718 VII 298678797326812817 G 2§811747917131972 IX 2867971990792441 I 318663554532493 XI 28969414048408 L 2034117957191). \-X11t 302624458245 N 48569357496 xv 3237957166 P 599621697 XVII 35271865 R WANA RT a Rk 389618 T 91392 |. XXI 4352 Ww 1128 XXIII 49 X 14 : 11157177$6571048778831475 _ 2231435513142097557062951 This number is 4 the logarithm for 12, This number 1s the logarithm of 14, Adding 1 to 14, it maketh 23 and fubtracted from it, leaveth +, which!refult maketh a fraction of 3; for 2+ being reduced into fourths, make 2, fo the:re- fult of $ and 2 9 (rejecting the denominators) is 3, as above; which {quared, maketh galt 6 are thefe numbers, therefore, continually divided by 81, to make the infinite feries. By decimals, it prefently fheweth itfelf to be a fraétion of 4. Thus the dif- ference, or numerator, is 255 the fum, or denominator, Differentia - 25 ee) 2,25; which fraction 2, is equal to 3. Unitas 1 To divide the numbers on the other fide (to make Numerus the infinite feries) by 81, is eafy enough; for it is but Propoltenat 7225 dividing twice by 9, or taking the ninth part twice, and rejecting or cancelling the firft; fo is it very readily Summa - 22/5 done ; and the whole operation “hereof may very well be performed i in two hours’ time: and thus have we got the logarithm for '12'to 25 places; and now fhall proceed to make the logarithm of 10, which is, by adding together the logarithm of 2, 3 times, and that makes the logarithm of 8; and that added to the logarithm of 13, makes the logarithm of 10 ; and the logarithm of 2 fubtraéted from the logarithm of 10, leaveth the logarithm of 5, and is to the fame effect as is before. Logarithm LOGARITHMOTECHNIA, 69 Logarithm of 2. 6931471805599453094172321 | Logarithm of 8. 20794415416798359282516963 Logarithm of 13. 2231435513142097557662951 Logarithm of 10. 23025850929940456840179914 Logarithm of 5. 16094379124341003746007593 We have now made and exhibited the logarithms of 2, 5, and 10, and from thefe you may make all their compofites. _ And now we proceed to make the logarithm of 3 to 25 places, which we fhall thew two ways; firft, all at once, from a compound curvilined trapezia, or hyperbolical fpace; fecondly, by a compofition of two logarithms, viz. of 2 and 14, for that 2 x 14, maketh 3; and this latter we chiefly recommend. ‘The compound curvilined trapezia, or hyperbolical {pace, nA © AVE, we have, in the’foregoing chapter, fhewn to be equal to the uncompounded cavro, and alfo equal to avr®s; we fhall calculate the logarithm for 3 according to the compounded fpace, and by the third and fourth tables you may know the lengths of the bafe and perpendiculars, The bafe nA is 10, therefore ca = § = cx. Now, forafmuch as Dairy’s rule of adding to, and fubtracting 1 from 3, produces the half length of the bafe, agreeable to the third table, we fhall fhew how to calculate the half of the logarithm for 3, as we did for the half logarithm of 2. “Adding 1 to 3, it makes 4, and fubtrating 1 from 3, leaveth 2; which ‘maketh a refult or fra€tion of 2 = 2. Now, dividing 1 and 25 cyphers by 2, the quotient is 5, for the firft term of your infinite feries. Now, forafmuch as + x + maketh 4, the fecond term mutt be, therefore, + of the firfl, and fo on; as was difcourfed before, in making the half logarithm of 2. Having made the infinite feries as followeth, you divide each of thofe numbers (which, as before taught, are proportional) by 1, 3, 5, 7, 9, &c, and thefe added, make half the area of nA®aveE for the half logarithm of 3. The 70 SPEIDELL’S The Infinite Series, or Numbers conti- nually proportional. Thefe numbers are continually divided by 4, in order to make the 4 log. of 3 to 25 places. a SO000000D00009N0090000000009 aaa 125 aaaaa 3125 4 78125 Tech | epee a 48828125 1220703125 30517578125 762939453125 ¥y 1907 3486328125 476837178203125 11920928955078125 298023223876953125 74505805969238281 18626451492309570 _ 465661287 3077392 1164153218269 343 2.91038 304567336 72759570141834 1818989403 5458 4547473508864 1136868377216 284217094304 71054273576 17763568394 4440892099 1110223025 277555759 69333936 17333485 .& S88 nw WwW VLA | iS) Differentia 2 1083343 _ 270836 Unitas 1 67709 Numerus Propofitus | _— 1058 Summa - 4 264 Thefe numbers are quotes from thofe on the oppofite fide, thofe being divided by 1, 3, 5, 7, 9, &c; which added, make 4 the log. of 3 to 25 places. i 5000000000000000000000000 III 416666666666666666666666 Vv 625 VII 11160714295714285714285 IX 2170138888888888888888 at 443892045454545454545 XIII 93900240384615384615 XV 2034505208 3333333333 XVII 4487879136029411765 XIX 1003867701480263158 alas 227065 313430059523 XXIII 51830125891644022 XXV 11920928855078125 XXVII 2759474295156975 re apd 642291430769295 XXXI 150213318486367 XXXIII 32247067220283 XXXV 8315389130495 XXXVII 1966475030861 SXXIX 466407539294 ° XLI 110913988025 XLIIE 26438799470 XLV 6315935429 XLVII 1511793055 XLIx 362521804 LI 87076316 LIIL 20947604. LV 5046469 LVIE 1216385 Lik 293788 LXI 71039 LXIII 17196 LXV 4167. LXVII Iolo LXIX 245 LXXI 59 LXXII 15 LXXV 3 Half the log. of 3. 5493061443340548456976226 The log. of 3. 10986122886681096913952452 We ve ‘ ~ Ze ‘ 4 ’ hy x ; repeat = ee eo fl) s@adit) aa ad ‘ Seriniéa oe i ar PR RH 8 ee ; Jeo 4 aie is aap Se ee 3_ +2 2 Pal pape eit ee] pale eeetebeme test lel | oT Sen Sl "See ese. oes SSS Bees eReSeNeee, as PPT TT ha ie ' . SS eRERaLERSSRRESes BEEEEEECCHCEE TSE ERS _ SESS SRBRSERZanSN ao tL et Bae z SECC RGSS Ee 8 caSenem CTS BEeBemede Bs M ete Alia) | |. | 2° 42S 22n Ree errr terrier ree elt pee ieee Ov 'o) TeV 21 To be placed tn VotSI facing Page 70 a FO glo LOGARITHMOTECHN?A-» 71 We hall now proceed to make the logarithm for 3 the fecond way, which is from the logarithms of 2 and 4. The Infinite Series, or Numbers conti- nually proportional. Thefe numbers are continually multiplied by 4, in order to make the half of the logarithm of 14 to 25 places. @ 2000000000000000000000000 aaa 8 aaaaa 32 a’ 128” a° ex ONT ft) a™ 2048 8192 32768 131072 524288 Differentia 3 2097152 RL? — 8388608 Unitas - 1 554432 Numerus I Aiea tod! Sabie Books | Bene er 53687 ne | 2147 Summa - 2J 85 To make this infinite feries, I fhould divide by 25 continually; but if you multiply by 4, and transfer it anfwer- ably, it will be the fame thing :_becaufe + of 1 is +3,, and that multiplied in it- felf, is -¢,. Therefore, multiplying by 4 and dividing by 100, is the fame thing as multiplying by 25. And thus this in- finite feries is made very fpeedily, in order to make the half of the logarithm for 1 and 2. Thefe numbers are quotes from thofe on the oppofite fide, they being divided by 1, 3, s, 7,9, 11, &c; and added, make half the logarithm for 14 to 25 places. I 2000000000000000000000000° III 26666666666666666666666 Vv: 64 VII 18285714285714285714 IX 568888888888888888 XI 18618181818181818 XII 630153846153846 XV 2184.5333333334 XVII 771011764706 XIX 27594105204 XXI 998643809 XXIII 36472209 XXV 1342177 XXVII 49711 D4 4s 1851 XXXI 69 - XXXIII 3 2027325540540821909890065 half the logarithm of 14, 4054651081081643819780131 the logarithm of 12, 6931471805 59945 3004172321 the logarithm for 2, 1098612288668 1096913952452 the logarithm for.3. Having now made the logarithm for 13, you add to it the logarithm of 2, and that makes the logarithm for 3, which will be found, as before, to be the fame number. And now we proceed to make the logarithm for 7, and then we fhall have all to 11: in order thereunto, we make the logarithm for 12, or 1,4, and add that to the logarithm of 5, and it will produce the logarithm of 7, for that 12 multiplied by 5, maketh 7, or 14, X § = 7. The m2 SPEIDELL’S The Infinite Series, or Numbers conti- nually proportional. Thefe numbers are continually divided by 36, in order to make half the logarithin of TH Onis. a 166666666666 6666666666666 277777777771777777771777 46296296296296296296296 7716049382716049382716 1286008230452674897119 21433470507 5445816186 a7 35722450845907636031 ° 5953741807651272672 a° 99229030127§212112 165381716879202019 ae" 2.7563619479867003 4593936579977864 765656096662972 127609349443829 21268224907 304 3544704151217 590784025203 98464004200 16410667 366 2735111228 455851871 eR 12662552 2110425 SORE aaa 58623 — I IT meme = sex $6. aaa aaaaa 4) Unitas 1 Numerus a Propofit. 4 Differentia Summa - 2 4 This feries is made by dividing twice by 6, which is all one as if you divided at once by 36; and fo every other number is the proper number of the feries to be divided by i, 35 5) 7, 9, 11, &c, as in the op- pofite. column, to make the half lo- garithm for 144. Thefe numbers are quotes from thofe on the oppofite fide, thofe being divided by 15 3, 5, 7,9, 11, &c; and added, make half the logarithm of: I}, oF I=4- I 1666666666666666666666666 III 15432098765432098765432 Vv 2572016460905 34979424 VII 5103207263701090863 Ix 110254477819468014. XI 2505783589078819g XIIL 58896622820229 XV 1417881660487 XVII 34752001483 XIX 863719335 XXI 21707237. XXIII 550546 XXV 14069 XXVII 360 XXIX 9 168236118 3106064652522972 half the logarithm, of I+, 3364722366212129305045944 the logarithm of 12, or 144, 16094379124341003 168895254 the logarithm of 5, 19459101490553132473941198 the logarithm of. 7. N.B. The logarithm of 5, in page 69, being put in the room of this in the fecond column,, will produce 1945910149055 313305105353 for the true lo- garithm of 7 ; thofe two laft numbers being part miftaken. Having LOGARITHMOTECHNIA., 74 bbe by this calculation made the half logarithm of 1,4,, if we double it, and to that add the logarithm of 5, that addition will produce the logarithm of 7, as was required. And now we have all the logarithms to 11, and to make the logarithms from 10 to roo, it will not be much difficulty to proceed after the foregoing methods; as to make the logarithm of 11, you have for the firft term a, the refult or fraction 2, and for aa, it will be 43, which is very eafy to work ; and for the logarithm of 13, you make it of 12 multiplied by oe and fo it is for the firft term a, the refult or fraction zros cand fou the fet cond aa, it is 5, which 626 is = 4; and fo may you make many eafy compendiums for the prime numbers between to and roo, and alfo, not with great difficulty, from tooo to 10000; and when you have made fome loga- rithms, you will perceive how the differences arife; and having for compofites, logarithms in a readinefs, greater and leifer than the prime or incompofite very near, it will be, by the difference, no great difficulty to make a logarithm for fuch a prime very readily and eafily. And they that’are curious herein, may have compendiums hereof in James Gregory’s aforefaid Vera Circuli F Hyperbole Quadratura, to make logarithms for prime or incompofite numbers, to which I fhall refer him. And here I fhall content myfelf to have exemplified James Gregory’s method in his Exercitationes Geometrice to fo many examples of lo- garithms as I have herein calculated to 25 places, and fhall, in the next chapter, fhew how to produce from thefe geometrical logarithms Briggs’s logarithms, C cEbaohd cP: VIII. AVING, inthe preceding chapter, made the logarithms for 2, 3, 4, 5, 6, 7, 8,9, and 10, according to the geometrical figure, or hyperbola, I require the logarithm of 2, according to Briggs’s table. Forafmuch as all lo- garithms are proportional, it is as the hyperbolical logarithin of 10, Is to its logarithm of 2 :: fo is Briggs’s logarithm of 10 to this logarithm of 2. The operation followeth. By this divifion it doth appear, that this quotient doth agree with Briggs’s table of logarithms for his logarithm number of 2, whereby it is apparent he did produce the logarithm for 2 to 15 places very true; though I have been told it was eight perfons’ work for a year’s time after his method, which was by large and many extractions of the fquare root ; and if it was fo to 1 5 places, it would have been very tedious, if not impoffible, for them to have produced the logarithm of 2 to 25 places, as before herein is fhewn, and done by us ; and both the hyperbolical and Briggs’s logarithm to 25 places, may very well be calculated and done, according to the foregoing method, in half a day’s time; by which method herein beforegoing one may make a table of loga- rithms, in a fhort fpace, to what Pardie (a French author), in his Elements of Geometry, hath declared ; for he faith, he knew more than 20 perfons engaged for 20 years, with indefatigable afiduity, to calculate the logarithms. He doth not fay to how many places; but the greateft radius that I have feen of any Vou, IT. L, French 94 SPELIDELL’S French author is but 11 places, which, I fuppofe, muft be the fame as Vlacq’s. And the logarithm for 2, 3, 4, or 5, &c to 11 places, according to the method in this book, may be very well done and performed in lefs than two hours’ time. " This divifor is half the logarithm of 10, according to the hyperbola. This quotient is the logarithm of 2, according to Briggs’s table. This dividend is compounded of half the hyperbolical logarithm of 2, and Briggs’s logarithm of to. Divifor. Quotient. 115129254649702281) (301029995663981190 | Dividend. 34657359027997264 , C0CCDDD00000000000 11858203 30865797. 3453378436877419. 11 507933438833 7338. 1146 30052036052851 110137228513207981 65208993284759281 76443659599081405 73561068092600364 45835153027789954 112963766328792697 93474371440605431 13709677208436262 21967517434660339 104445919096901109 8295905121690568 The reader may now fee, that logarithms derived from this figure, or the hyperbola, are not only more perceptible and intelligible, but with far more cer- tainty and expedition produced, than what was known in former times, The divifor in the foregoing work differs 2 units in the 18th place from the half logarithm of 10 before herein calculated; and the reafon is, that I took Gregory’s logarithm of 10, in his Vera Circuli S Hyperbole Quadratura de bene effe; and having calculated the half logarithm of 2, as before, I was very de- firous to fee if we could produce Briggs’s logarithm of 2 to 15 places,.as by the divifion is manifeft; and this I did long before I met with Gregory’s other book of his Exercitationes Geometrice ; for, fince I got that book, I did calculate, de novo, the logarithm of 10 to 25 places, according to his dottrine in that book, and as before herein is done. And the calculation of the loga- rithm of 10, as before, doth agree with Gregory’s former book but to 17 places. Howfoever, the divifion beforegoing is fufficient to produce Briggs’s logarithm for 2 to 15 places; and if any fhall be fo curious as to produce Briggs’s LOGARITHMOTECHNIA, 78 Brigegs’s logarithm for 25 places, he may rely on the foregoing examples » herein, and may, in 4 hours’ time, examine the foregoing calculations there- of, and in as little time produce Briggs’s logarithm for them to the like number of places. Having this divifion ready done, long before the publifhing hereof, I have contented myfelf to infert it here, whereby the ftudious may foon perceive what to do further to gratify himfelf herein. I do not add hereto any table of logarithms, that being not my defién at this time, but only to fhew how Briggs’s, or any other logarithms, may be de- rived from the doctrine beforegoing ; and alfo for the curious, at his will and pleafure, to examine, whether any logarithms formerly publithed be truly made or not. _As for the various ufes of logarithms I add none here, but refer the reader to fuch authors (whereof there are plenty) who have, long before, written largely and learnedly, as the firft inventor, the famous Lord Napier, Henry Briggs, Edmund Gunter, Richard Norwood, Wingate, and divers others; as alfo my father, John Speidell; in which the reader may meet with many excellent ufes of the logarithms in all parts of the mathematicks. And I do find, my father printed feveral forts of logarithms, but at laft concluded, that the decimal, or Brigegs’s logarithms, were the beft fort for a ftandard logarithm; and did alfo print the fame feveral ways, fo ordered, whereby they might be applied to arithmetical queftions, and other operations for the folution thereof, with eafe and readine(s. END OF SPEIDELL’s LOGARITHMOTECHNIA, maa ee ce FROM THE PHILOSOPHICAL TRANSACTIONS, V,O La» XIX. 1. dn eafy Demonftration of the Analogy of the Logarithmick Tangents to the Me- ridian Line, or Sum of the Secants; with various Methods for computing the Jame to the utmoft Exatinefs. By Dr. E. Halley. T is now near one hundred years fince our worthy countryman, Mr. Edward Wright, publifhed his Correction of Errors in Navigation ; a book well de- erving the perufal of all fuch as defign to ufe the fea, Therein he confiders the courfe of a fhip on the globe, fteering obliquely to the meridian; and having fhewn, that the departure from the meridian is, in all cafes, lefs than the difference of longitude, in the ratio of radius to the fecant of the latitude, he concludes, that the fum of the fecants of each point of the quadrant being added fucceffively, would exhibit a line divided into f{paces, fuch as the intervals of the parallels of latitude ought to be in a true fea chart, whereon the meri- dians are made parallel lines, and the rhombs, or oblique courfes, reprefented by right lines. This is commonly known by the name of the meridian line, which, though it generally be called Mercator’s, was yet undoubtedly Mr. Wright’s invention, as he has made it appear in his preface. And the table thereof is to be met with in moft books treating of navigation, computed with fufficient exactnefs for the purpofe. It was firft difcovered by chance, and, as far as I can learn, firft publifhed by Mr. Henry Bond, as an addition to Norwood’s Epitome of Navigation, about HALLEY’S DISCOURSE ON THE MERIDIAN. a about fifty years fince, that the meridian line was analogous to a fcale of loga- rithmick tangents of half the complements of the latitudes. The difficulty to prove the truth of this propofition, feemed fuch to Mr. Mercator, the author of Logarithmotechnia, that he propofed to wager a good fum of money, againft whofo would fairly undertake it, that he fhould not demonftrate either that it was true or falfe.* And, about that time, Mr. John Collins, holding a cor- refpondence with all the eminent mathematicians of the age, did excite them to this inquiry. The firft that demonftrated the faid analogy, was the excellent Mr. James Gregory, in his Exercitationes Geometrice, publifhed anno 1668 ; which he did not without a long trath of confequences, and complication of proportions, whereby the evidence of the demonftration is in a great meafure loft, and the reader wearied before he attain it. Nor with lefs work and apparatus hath the celebrated Dr. Barrow, in his Geometrical Lectures (Lect. XI. app. 1.), proved, that the fum of all the fecants of any arch is analogous to the logarithm of the ratio of radius 4- fine to rad. — fine; or, which is all one, that the me- ridional parts an{wering to any degree of latitude, are as the logarithms of the rationes of the verfed fines of the diftances from both the poles. Since which, the incomparable Dr. Wallis (on occafion of a paralogifm committed by one Mr. Norris in this matter) has more fully and clearly handled this argument, as may be feen in number 176 of the Tranfactions. But neither Dr. Wallis nor Dr. Barrow, in their faid Treatifes, have any where touched upon the aforefaid relation of the meridian line to the logarithmick tangent; nor hath any one, that I know of, yet difcovered the rule for computing independ- ently the interval of the meridional parts anfwering to any two given la- titudes. Wherefore having attained, as I conceive, a very facile and natural demon- {tration of the faid analogy, and having found out the rule for exhibiting the difference of meridional parts between any two parallels of latitude, without finding both the numbers whereof they are the difference, I hope I may be intitled to a fhare in the improvements of this ufeful part of geometry. And firft let us demonftrate the following PROPOSITION. ‘The meridian line is a fcale of logarithmick tangents of the half complements of the latitudes. For this demonftration it is requifite to premife thefe four lemmata. Lemmat. In the ftereographick projection of the fphere upon the plane of the equinoétial, the diftances from the center, which, in this cafe, is the pole, are laid down by the tangents of half thofe diftances; that is, of half the comple- ments of the latitudes.—This is evident from Eucl. 3, 20. Lemma 2. In the ftereographick projection, the angles under which the circles interfeét each other are, in all cafes, equal to the fpherical angles they repre- Yent ; which is, perhaps, as valuable a property of this projection as that of all the 79 HALLEY’S DISCOURSE: the circles of the {phere thereon appearing circles. But this not being vulgarly known, mutt not be aflumed without a DEMONSTRATION, Let EBPL be any great circle of the {phere, z the eye placed in its cir- cumference, c its centre, P any point thereof, and let Fco be fuppofed a plane erected at right angles to the circle EBPL, on which Fco we - defign the fphere to be projected. Draw EP croffing the plain Fco in 2g, and p fhall be the point Pp projected. To the point p draw the tangent ape, and on any point thereof, as a, ereé a perpendicular ap at right angles to the plane EgpL, and draw the lines PD, AC, Dc, and the angle app fhaill be equal to the f{pherical angle con- tained between the planes apc, prc. Draw alfo aE, pe, interfecting the E plain Fco in the points a and d, and join ad, pd; I fay, the triangle adp is fimilar to the triangle apr, and the angle apd equal to the angle app. Draw pL, AX, parallel to Fo, and, by reafon of the parallels, ap will be to ad as aK to ap, But (by Eucl. 3, 32) in the triangle axp, the angle akPp = LPs, is alfo equal to aPK = EPG; wherefore’ the fides AK, ap, are equal, and it will be as ap to ad, fo ap to ap; whence the angles Dap, dap, being right, the angle app will be equal to the angle apd; that is, the {pherical angle is equal to that on the projection, and that in all cafes ; which was to be proved. This lemma I lately received from Mr. Demoivre, though I fince under- ftand, from Dr. Hook, that he long ago produced the fame thing before the fociety. However, the demonftration, and the reft of the difcourfe, is my own. Lemma 3. On the globe the rhomb lines make equal angles with every me- ridian, and, by the foregoing lemma, they muft likewife make equal angles with the meridians in the ftereographick projection on the plane of the equator ; they are therefore, in that projection, proportional {pirals about the pole point. {Lemma 4. In the proportional fpiral it is a known property, that the angles ppc, or the arches BD, are exponents of the rationes of Bp to pc; for if the arch Bp be divided into innumerable equal parts, right lines drawn from them to the centre Pp, fhall divide the curve Bece into an infinity of propor- tionals; and all the lines pe fhall be an infinity of proportionals between psp and pc, whofe number is equal to all the points dd in the arch Bp; whence, and by what I have delivered in the next enfuing difcourfe, it follows, that as Bp to Bd, or as the angle ON THE LOG. TANGENTS AND MERID. 79 BP¢ to the angle spc, fo is the logarithm of the ratio of PB to Pc, to the loga- rithms of the ratio of PB to Pc. From thefe lemmata our propofition is very clearly demonftrated ; for by the firft, pp, pc, pc, are the tangents of half the complements of the latitudes in the {tereographick projection ; and by the laft of them the differences of longitude, or angles at the pole between them, are logarithms of the rationes of thofe tangents one to the other. But the nautical meridian line is no other than a table of the longitudes, anfwering to each minute of latitude on the rhomb line, making an angle of 45 degrees with the meridian; wherefore the meridian line is no other than a fcale of logarithmick tangents of the half complements of the latitudes ; quod erat demonftrandum. * Coroll. 1. Becaufe that in every point of any rhomb line the difference of la- titude is to the departure, as the radius to the tangent of the angle that rhomb makes with the meridian ; and thofe equal departures are every where to the differences of longitude, as the radius to the fecant of the latitude ; it follows, that the differences of longitude are, on any rhomb, logarithms of the fame tangents, but of a differing {pecies, being proportioned to one another as are the’ tangents of the angles made with the meridian. Coroll. 2. Hence any fcale of the logarithm tangents (as thofe of the vulgar tables made after Briggs’s form, or thofe made to Napier’s, or any other form whatfoever) is a table of the differences of longitude to the feveral latitudes, upon fome determinate rhomb or other; and therefore as the tangent of the angle of fuch rhomb to the tangent of any other rhomb, fo the difference of the logarithms of any two tangents to the difference of longitude on the propofed rhomb, intercepted between the two Jatitudes, of whofe half complements you took the logarithm tangents. And fince we have a very complete table of logarithm tangents of Briggs’s form, publifhed by Vlacq, anno 1633, in his Canon Magnus Triangulorum Loga- rithmicus, computed to ten decimal places of the logarithm, and to every ten feconds of the quadrant (which feems to be more than fufficient for the niceft calculator) I thought fit to inquire the oblique angle with which that rhomb line croffes the meridian, whereon the faid Canon of Vlacq precifely anfwers to the differences of longitudes, putting unity for one minute thereof, as in the common meridian line. Now the momentary argument or fluxion of the tan- gent line at 45 degrees is exa¢tly double to the fluxion of the arch of the circle, as may eafily be proved; and the tangent of 45 being equal to radwus, the fluxion alfo of the logarithm tangent will be double to that of the arch, if the logarithm be of Napier’s form; but for Briggs’s form, it will be as the fame doubled arch multiplied into 0,43429 &c, or divided by 2,30258 &c. Yet this muft be underftood only of the addition of an indivifible arch, for it ceafes to be true if the arch have any determinate magnituce. Hence it appears, that if any one minute be fuppofed unity, the length of the arch of one minute being ,000290883208665721596154 &c, in parts of the radius, the proportion will be as unity to 2,908882 &c, fo radius to the tangent of 71° 1’ 42”, whofe logarithm is 10.46372611720718 325204 &c; and under that angle is the meridian interfected by that rhomb line, on which the diferences of Napier’s logarithm tangents of the half complements of the latitudes are the + true Ye) HALLEY’S DISCOURSE true differences of longitude eftimated in minutes and parts, taking the firft four figures for integers. But for Vlacq’s tables we muft fay, | As 2302585 &c to 2908882 &c, fo radius to 142633114387424.466Q812 &c, which is the tangent of 51° 38’ 9”, and its logarithm 10,1015 104285077 20941162 &c; wherefore in the rhomb line which makes an angle of 51° 38’ 9” with the meridian, Vlacq’s logarithm tangents are the true differences’ of lon- gitude. And this compared with our fecond corollary may fuffice for the ufe of the tables already computed. But if a table of logarithm tangents be made by extraction of the root of “a infiniteth power whofe index is the length of the arch you put for unity (as for minutes the ,ooo2z9g0888ath &c power) , which we will call a; fuch a fcale of tangents fhall be the true meridian line, or fum of all the fecants, taken in- finitely many. Here the reader is defired to have recourfe to my little Treatife of Logarithms, in the enfuing difcourfe, that I may not need to repeat it. By what is there delivered, it will follow, that putting ¢ for the excefs or defect of any tangent above or under the radius or tangent of 45, the logarithm of the ratio of radius to fuch tangent will be I 5 ° in eit) Sib epi ith add 42 - gr into ¢ —tt + itt zitit Jp =! &c when the arch is greater than 458", or — into t + —tt Bese “tit + Pi &c when it is lefs than 458. And, by the fame doé¢trine, putting Tr for the tangent of any arch, and ¢ for the differ- ence thereof from the tangent of ROR TGs arch, the logarithm of their ratio will be TS t tt #8 t# — — + — 4+ —— 4+ — c wh o m sage diy chia Creme ae + par pected se & en T is the greater term, or tits t tt e t4 peta Wiese Pec omer p camehUayet> sy ios ghey &c when T is the leffer term. 54 T 2TT 3T 4T oT And if m be fuppofed ,0002908882 &c = 4a, its reciprocal — will be 34375 7467707849392526 &c, which multiplied into the aforefaid (erie: fhall give precifely the difference of meridional parts between the two latitudes to whofe half complements the affumed tangents belong. Nor is it material from whether pole you eftimate the complements, whether the elevated or de- preffed, the tangents being to one another in the fame ratio as their comple- ments, but inverted. In the fame difcourfe I alfo fhewed, that the feries might be made to converge twice as {wift, all the even powers being Sha and pone t for 3 the fum of the two tangents, the fame logarithm would be — — or si into * 7 5 z7 7? 6 aE ants a &c ; but the ratio of 7 to ¢, or of the ies of two tangents 3 Tr ors to their difference, is ie farae as that of the fine of the fum of the arches to the fine of their difference: wherefore, if s be put for the fine complement of the middle latitude, and s for the fine of half the ate of latitudes, the 2 5 : = oo = Bo ar &c; wherein, as the differences of latitudes are fmaller, fewer fteps will fuffice. And if the equator be put for the middle latitude, and confequently s = Rr, and s tothe fine of 4 the ‘ : ar 5 $3 fame feries will be — into = + — ON THE LOG. TANGENTS AND MERID. Si 53 the latitude, the meridional parts reckoned from the equator will be -- ane: rra _ i, &c, which is coincident with Dr. Wallis’s folution, in number 176 of the Philofophical Tranfaétions.. And this fame feries being half the logarithm of the ratio of R + 5toR— 5, that is, of the verfed fines of the dif- tances from both poles, does agree with what Dr. Barrow had thewn in his XIth lecture. The fame ratio of + to’¢ may be expreffed alfo by that of the fum of the co-fines of the two latitudes to the fine of their difference, as likewife by that of the fines of the fum of the two latitudes to the difference of their co-fines, or by that of the verfed fine of the fum of the co-iatitudes to the difference of the fines of the latitudes, or as the fame difference of the fines of the latitudes to the verfed fine of the difference of the latitudes; all which are in the fame ratio of the co-fine of the middle latitude, to the fine of half the difference of the latitudes. As it were eafy to demonftrate, if the reader were not fuppofed capable to do it himfelf, upon a bare infpection of a {cheme duly reprefenting - thefe lines. This variety of expreffion of the fame ratio I thought not fit to be omitted, becaufe by help of the rationality of the fine of gogr. invall cafes where the fum or difference of the latitudes 1s gogr. 6ogr. gogr. 120gr. or 150 degrees, fome one of them will exhibit a fimple feries, wherein great part of the labour will be faved. And befides, I am willing to give the reader his choice, which of thefe. equippolent methods to make ufe of ; but for his exercife fhall leave the pro fecution of them, and the compendia artiing therefrom, to. his own induftry ; contenting myfelf to confider only the former, which, for all ufes, feems the moft convenient, whether we defign to make the whole meridian line or any REET Ae 5 33 ss s7 5? - kr i part thereof, vizs)— into hb fy + —, &c; wherein @ is the length of any arch which you defign fhall be the integer or unity in your me- ridional parts (whether it be a minute, league, or degree, or any other), s the co-fine of the middle latitude, and s the fine of half the difference of latitudes : but the fecants being the reciprocals of the co-fines, —- will be equal to fe 7 putting / for the fecant of the middle latitude; and — into — will be = a : ar This multiplied by SP that is, by will give the fecond ftep ; and that again by ali the third ftep, and fo forward till you have completed as many srrrr’ ; » , ; places as you defire. But the fquares of the fines being in the fame ratio with the verfed fines of the double arches, we may, inftead of ef , affume for our multiplicator = or the verfed fine of the difference of the latitudes, divided by thrice the verfed fine of the fm of the co-latitudes, &c, which is the utmoft compendium I can think of for this purpofe, and the fame feries will become 45r. ~v yt v3 v4 Gale sek ce hire + — &e. as Vou. IT. M Hereby $2 HALLEY’S DISCOURSE Hereby we are enabled to eftimate the default of the method of making the meridian line by the continued addition of the fecants of equidifferent arches, which, as the difference of thofe arches are {maller, does {till nearer and nearer approach the truth. If we affume, as Mr. Wright did, the arch of one minute to be unity, and one minute to be the common difference of a rank of arches, it will be, in all cafes, as the arch of one minute to its chord :: fo the fecant of the middle latitude to the firft ep of our feries. This, by reafon of the near equality between a and 2s, which are to one another in the ratio of unity to I—OoO GRO PROIOSSASROES 7913 é&c, will not differ from the fecant f but in the ninth figure, being lefs than it in that proportion. The next ftep being +25 ei - will be equal to the cube of the fecant of the middle latitude multi- olied into Fat = 0,000000007051329087153 which therefore, unlefs the fecant exceed ten times radius, can never amount to 1 in the fifth place. Thefe two {teps fufice to make the meridian line, or logarithm tangent to far more places than any tables of natural fecants yet extant are computed to; but if the third ftep be required, it will be found to be +./§ into = == 0,0000000000000000 89498: by all which it appears, that Mr. Wright's S ‘table does no where exceed the true meridian parts by fully half a minute 5 which finall difference arifes by his having added continually the fecants of Ty 2 5°35 OC, INitead OF Op, taeammaaee 3x, &c; but, as it is, it is abundantly fufficient for nautical ufes. That in Sir Jonas Moor’s ‘New Syftem of the Mathematics is much nearer the truth, but the difference from Wright is fcarce fenfible till you exceed thofe latitudes where navigation ceafes to be practicable; the one exceeding the truth by about half a minute, the other being a very fmall matter deficient therefrom. For an example eafy to be imitated by whofoever pleafes, I have added the true meridional parts to the firft and laft minutes of the quadrant ; not fo mueh that there is any occafion for fuch accuracy, as to fhew that I have obtained, and laid down herein, the full doétrine of thefe fpiral rhombs which are of fo creat concern in the art of navigation. The firft minute is — 1.00000001410265862178. The fecond = - 2 ,00000005641063806707. The laft, or 89° 59’, is 30374,963431 1414228643. And not 32348,5279, as Mr. Wright has it, by adding the fecants of every whole minute ; nor 30249,8, as Mr. Oughtred’s rule makes it, by adding the fecants of every other half minute ; nor 30364,3, as Sir Jonas. Moor had con- cluded it, by I know not what method, though in the reft of his table he follows Oughtred. And this may fuffice to fhew we to derive the true meridian line from the fines, tangents, or fecants, fuppofed ready made; but we are not deftitute of a method for deducing the fame independently from the arch itfelf. If the lati- tude from the equator be eftimated by the length of its arch a, radius being unity, and the arch put for an ner be a, as before, the meridional parts anfwering to that latitude will be — — into A + A tay A+ sa A’ OF soqc Al 1 ON THE LOG, TANGENTS AND MERID. $3 _-- $ va . . + Tire AU OO s335h0 4° &c; which converges much fwifter than any of the former feries, and befides has the advantage of A increafing in arithmetical progreflion, which would be of great eafe, if any fhould undertake, de xovo, to make the logarithm tangents, or the meridian line, to many more places than now we have them. The logarithm tangent to the arch of 45 + 4A being no other than the aforefaid feries a + 143 + 1, a5 &c, in Napier’s form, or the fame multiplied into 0,43429 &c, for Briggs’s. But becaufe all thefe feries toward the latter end of the quadrant do converge exceeding flowly, fo as to render this method almoft ufelefs, or, at leatt, very tedious, it will be convenient to apply fome other arts, by affluming the fecants of fome intermediate latitudes ; and you may for s, or the fine of a, the arch of half the difference of latitudes, fubftitute # — ta} + 71.05 — rere ok ssreso” &c, according to Mr. Newton’s rule for giving the fine from the arch ; and if « be no more than a degree, a very few {teps will fuffice for all the accuracy that can be defired. And if « be commenfurable to a; that is, if it be a certain number of thofe arches with which you make your integer, then will — be that number; which ° ' ¢ . . . . ° ¢ if we call 1, the parts of the meridional line will be found to be art et meet “ae | 4 6 nl into ee tal is Pres cs &c. . 4 2,6 a ——— PB sho &c. § ye ids Tn this the firft two fteps are generally fufficient for nautical ufes, efpecially when neither of the latitudes exceed 60 degrees, and the difference of latitudes doth not pafs 30 degrees. : But I am fenfible I have already faid too much for the learned, though too fittle for the learner: to fuch I can recommend no better treatife than Dr. Wallis’s precedent Difcourfe, wherein he has, with his ufual brevity and that perfpicuity peculiar to himfelf, handled this fubje& from the firft principles, - which here, for the moft part, we fuppofe known. I.need not fhew how, by regreffive work, to find the latitudes from the me- widional parts, the method being: fufficiently obvious: I fhall only conclude with the propofal of a problem which remains to make this doctrine complete, -and that is this : A fhip fails from a given latitude, and, having run a certain number of deagues, has altered her longitude by a given angle; it is required to find the courfe fteered. The folution hereof would be very acceptable, if not to the public, at leaft to the author of this traét; being likely to open fome fur- ther light into the inyfteries of geometry. Mz To 84 HALLEY S$ DISCOURSE To conclude, I thall only add, that unity being radius ; the co-fine of the arch a, according to the fame rules of Mr. Newton, will be 1 — 4a* + 4 at we a GA? be aot A® — seergrocA” &c; from which, and the former feries exhibiting the fine by the arch, by divifion, Hf is cally to conclude, that the na- tural tangent of the arch a is A + a3 igo ms + 3 eae 9 + repsk ; iG and the natural fecant to the fame arch 1 He 2A -f2A* + oSh aie &c ; and from the arithmetick of infinites, the number ob thefe fecants being t fhe arch a, it follows, that the fum total of all the infinite fecants on that arch is A+ 2a? + A’ + Seal 247,a° &c; the which, by what foregoes, is the logarithm tangent of Napier’s form, for the arch “of 45gr. + ZA, as before. : | And collecting the he fum of all the natutal tangents on the faid arch a, there will arife aa + ~biat - Z,a® + aha’ + 534.4” &e, which:will' be found tu be the "Dakine oF the see of the fame arch a. 2. moft compendious and facile Method of conftrucéting the Logarithms, exemplified and demonftrated from the Nature of Numbers, without any Regard to the Hyper bola ; with a fpeedy Method for finding the Number from the raped te given. By Dr, E. Halley. HE invention of the logarithms is juftly efteemed one of the moft ufeful: difcoveries in the art of 1 numbers, and accordingly has had an univerfal re- ception and applaufe ; and the. great Geometricians of this age have not been wanting to cultivate this fubjeét with all the accuracy and fubtilty a matter of that confequence doth require; and they have demonftrated feveral very admi-. rable properties of thefe artificial numbers which have rendered their conftruc- tion much more facile than by thofe operofe methods at firft ufed by their truly noble inventor, the Lord Napier, and our worthy conntryman, Mr. Briggs, But, notwithftanding all their endeavours, I find very few of thofe who make . conftant ufe of logarithms, to have attained an adequate notion of them, to know how to make or examine them, or to underftand the extent of the ufe of them; contenting themfelves with the tables of them, as they find them,;- without daring to queftion them, or caring to know how to reétify them, fhould they be found amifs; being, I fuppofe, under the apprehenfion of fome great dificulty therein. For the fake of fuch, the following tract is. principally intended ; but not without hopes, however, to. produce fomething that may. be acceptable to the moft knowing in thefe matters. But firft it may be requifite to premife a definition of logarithms, in order to. render the enfuing Difcourfe more clear, the rather, becaufe the old one, Nume- rorum Proportionalium equi Differentes comites, feems too fcanty to define them fully. They may more properly be faid to be Numeri Rationum Exponentes 5 wherein we confider ratio as a quantitas fui generis, beginning from the ratia of equality, or 1 to 1 = 0; being affirmative when the ratio is increafing, as of unity to a. greater number, but negative. when decreafing ; and thefe rationes 2, WE: ON LOGARITHMS.. 8 § we fuppofe to be meafured by the number of ratiunculz contained in each.. Now thefe ratiuncule are fo to be underftood as in a continued fcale of pro- portionals infinite in number, between the: two terms of the ratio, which infinite number of mean proportionals is to that infinite number of the like and equal ratiuncule between any two terms, as the logarithm of one ratio is to the loga- rithm of the other. Thus if there be fuppoied between 1 and 10 an. infinite fcale of mean proportionals, whofe number is 100000 &c, im infinitum, between iz and 2 there fhall be 30102 &c, of fuch proportionals, and between 1 and 3 there will be 47712 &c, of them; which numbers therefore are the logarithms of the rationes of 1 to 10, 1 to 2, and 1 to 3; and not fo properly to be called the logarithms of 10, 2, and 3. But if, inftead of fuppofing the logarithms compofed of a. number of equal ratiuncule proportional to each ratio, we fhall take the ratio of unity to: any number, to confift always of the fame infinite number of ratiuncule, their magnitude in this cafe will be as their number in the former; wherefore if between unity and any number propofed there be taken. any infinity of mean proportionals, the infinitely little augment or decrement of the firft. of thofe means from unity, will be a ratiuncula ; that is, the momentum or fluxion of the ratio of unity to the faid number. “And feeing that in thefe continual pro- portionals all the ratiunculz are equal, their fum, or the whole ratio, will be as the faid momentum is directly ; that is, the logarithm of each. ratio twill be. as the fluxion thereof. Wherefore if the root cf any infinite power be extracted out of any number, the differentiola of the faid root from unity fhall be as the logarithm of that number. So that logarithms thus produced may be of as. many forms as you pleafe to aflume infinite indices of the power. whofe. root you. feek ; as if the index be fuppofed 1to0000 &c, infinitely, the roots fhall.be. the. logarithms invented by the Lord Napier; but: if the faid index were 2302585 &c, Mr. Briggs’s logarithms would immediately be produced. And if you pleafe to ftop at any number of figures, and not to continue them on, it will fuffice to affume an index of a figure or two more than your intended logarithm is to haye, as Mr. Briggs did; who,.to have his logarithms true to 14. “places, by continual extraction of the fquare root, at laft came to have the room of the 1407374883 55328th power; but how operofe that extraction was will be eafily: judged by whofo fhall undertake to examine his Calculus. Now, though the notion of an infinite power may feem very ftrange, and, to. thofe that know the difficulty of the extraction of the roots.of high powers, perhaps > impracticable; yet, by the help of that admirable invention of Mr. Newton, whereby he determines the unciz or numbers prefixed to the members com- pofing powers (on which chiefly depends the dottrine of feries), the infinity of the index contributes to render the expreffion much more eafy; for if the the infinite power to be tefolved be put (after Mr. Dee method)- —-30 2mm : p+ 29, P + pq” or1 + g\"> inftead of 1 + 49 + "79 + - arena ome A ‘om 6 * . « - - - . aN perenne &c Cri is the root when m is finite), becomes 1 -+ m i eer 28 nd wild + eh eB rat sty mat &c, mm "being infinite infinite ; and‘confe quently> 86 HALLEY’s DISCOURSE quently whatever is divided thereby vanifhing. Hence it follows, that - multiplied into g — 399 + 3999 — 49* + 495 &c, is the augment of the firft of our mean proportionals between unity and 1 + q, and is therefore the loga- rithm of the ratio of 1 to 1 + 4g and whereas the infinite index m may be taken at pleafure, the feveral fcales of logarithms to fuch indices will be as —, or reciprocally as the indices. And if the index be taken 10000 &c, as in the cafe of Napier’s logarithms, they will be fimply ¢ — 99 + i9qq —iq* tigi — 37° &e. Again, if the logarithm of a decreafing ratio be fought, the infinite root of 4 hy he I I I I I ) eee | it PSO Mina A RH Rens oat here | ao cmt Gattee &c ; whence the decrement of the firft of our infinite number of propor- tionals will be ~ into g + 399 + ig? + igt + i¢8 + 29°, &c; which there- fore will be as the logarithm of the ratio of unity toz1—g. But if m be put 10000 &c, then the faid logarithm will be g + igg + 493 + 394 + 49° + 2g° BCs | Hence the terms of any ratio being @ and 2; ¢ becomes nae, or the differ- . . . . . . b — a ’ . e ence divided by the leffer term when it is an increafing ratio; or a i when it is decreafing, or as 6 to a. Whence the logarithm of the fame ratio may be doubly expreffed ; for putting » for the difference of the terms @ and 4, it will be either itis Shs $2 x? xs Pad #s x t*, ee Fi haeeh ge Tipe Tigh, cee or into Sh yeh Ae gine Sate wt a 2aa' 3a3 gat sas sa But BS the ratio of ato 6 be fuppofed divided irito two parts, viz. into the ratio of a to the arithmetical mean between the terms, and the ratio of the faid arithmetical mean to the ether term 4%, then will the fum of the logarithms of thofe two rationes be the logarithm of the ratio of 4 to; and fubftituting iz” inftead of +@ + 70, the faid arithmetical mean, the logarithms of thofe rationes will be, by the foregoing rule, I PLT i HEE: GL apts amt GRAN Tc TERE I x HH xs xt *s x and — in — ote Pam ae cones &c, fum fie ee * wal Ci. 2% © deca the whereof — in— + ++ ai + f+ ree ay 43 wai Cy will be the logarithm of the ratio of @ to 4, whofe difference is w and fum 2. ~ And this feries converges twice as {wift as the former, and therefore is more proper for the practice of making logarithms; which it performs with that ex- pedition, that where « the difference is but the hundredth part'of the fum, par ste ON LOGARITHMS.. 87 firft ftep = fuffices to feven places of the logarithm, and the fecond ftep to twelve. But if Briggs’s firft twenty chiliads of logarithms be fuppofed made, as he has very carefully computed them to fourteen places, the firft {tep alone is capable to give the logarithm of any intermediate number true to all the places of thofe tables. After the fame manner may the difference of the faid two logarithms be very fitly applied to find the logarithms of prime numbers, having the loga- rithms of the two next numbers above and below them: for the difference of the ratio of a to iz and of iz to J, is the ratio of a6 to tzz, and the half of that ratio is that of 4/ad to iz, or of the geometrical mean to the arithmetical. And confequently the logarithm thereof will be the half difference of the lo- garithms of thofe rationes, viz. = into — s + a +- = &c; which is a theorem of good difpatch to find the logarithm of 4z. But the fame is yet much more advantageoufly performed by a rule derived from the foregoing, and beyond which, in my opinion, nothing better can be hoped. For the ratio of ab to izz, or 34a + 4ab + 4b, has the difference of its terms jaa — 4a) 4- 4b, or the fquare of 44 — 44 = 4x, which, in the prefent cafe of finding the logarithms of prime numbers, is always unity, and calling the fum of the terms Izz + ab = yy, the logarithm of the ratio of «fab to ia + 3, or iz will be found — in — +—+-5+-,+—-», &c, which converges very much fafter than any theorem hitherto publifhed for this purpofe. Here note — is all along applied to adapt thefe rules to all forts of logarithms.. If m be roo00 &c, it may be negleéted, and you will have Napier’s logarithms, as we hinted before; but if you defire Briggs’s logarithms, which are now ge- nerally received, you mutt divide your feries by 2,30258509299404568401 799 1454684364207601101 4886287729 76033328, or multiply it by the reciprocal thereof, viz. 043429448 190325182765112891891660508.2294397005803666 566114454. But to fave fo operofe a multiplication (which is more than all the reft of the work), it is expedient to divide this multiplicator by the powers of z or y con- tinually, according to the direction of the theorem, efpecially where x is fmall and integer, referving the proper quotes to be added together, when you have produced your logarithm to as many figures as you defire; of which method L will give a {fpecimen. | If the curiofity of any gentleman that has leifure would prompt him to under- take to do the logarithms of all prime numbers under tooooo to 25 or 30 figures, I dare affure him, that the facility of this method will invite hifi thereto; nor can any thing more eafy be defired. And, to encourage him, I here give the logarithnis of the firt prime numbers under 20 to 60 places, computed by the accurate pen of Mr. Abraham Sharp (from whofe induftry and capacity the world may in time expect great performances), as they were com- municated to me by our common friend, Mr. Euclid Speidell. | N°. 88 HALLEY’s p1scovurst N°, Logarithm. 2 0430102999566398119521373889472449302676818988 1462108 541 310424 0,4.77121254.719662437295027903255115309200128864.19069 5864829866 7 0,8450980400142568 307122162585 9263619340357239632 3965406503835 1,041 392685 15822504075019997 1243024241 706 70219046645 3094596539 13 1,11394335230083776920654189 502624625456118900505 3673288598083 oy De 230448921 37827 302854016989432833 7030007 56737842 5046397380368 19, 1,27875 3600952828961536333475756929317951129337394497 598906819 The next prime number is 23, which] will take for an example of the fore- going doctrine ; and, by the firlt rules, the logarithm of the ratio Baz 22 to 23 will be found to be either I I ng I I Piss 968 + 31944 4 937024 + 25708160 &c, z I OF spalnao aga +> I I ots E c: 30501 1119364 32181715 : as likewife that of the ratio of 23 to 24 by a like procefs, T I 23 4 1058 +- &cy I I I Oe a I I 1 36c01 «1119364 + 32181716 Ce u I I I152 3 41472 1327104 3 39813120 And this is the refult of the doétrine of MeRater, as um played by the learned 2x3 Dr. Wallis. But by the fecond theorem, viz.— + ea + = ——>, &c, the fame logarithms are obtained by fewer fteps ; to AE 2 2 2 2 45 a 293375 z 922640625 + 2615686171875 ac, re peti pee Ree E bk d Mibbslebin d theine, anc a7 31149 T Frqo7a5035 + 3546301843243 . which was invented and demonftrated in the hyperbolick fpaces analogous to the logarithms, by the excellent Mr. James Gregory, in his Exercitationes Geometrice, and fince further profecuted by the aforefaid Mr. Speidell, in a late Treatife, in Englith, by him publifhed on this fubject. But the demonftratiou, as I con- ceive, was never till now perfected without the confideration of the hyperbola, which, in a matter Hektor arithmetical, as this is, cannot be fo properly ap- lied. But what follows, I think, I may more juftly claim as my own, viz. that the logarithm of the ratio of the geometrical mean to the arithmetical be- tween 22 andl 24, or of % be to 23, will be found to be either I He + Tee tT Bake teaaw ahs | ea &c, I I I PLEA ce ee oe emt * Tos7 + 3542796579 | 659676558485285 All thefe feries being to be multiplied into 0,4342944819 &c, if you defign to make the logarithm of Briggs, But, with great advantage in refpect of the work, Cc; av | ON LOGARITHMS 89 work, the faid 4342944819 &c, is divided by 1057, and the quotient thereof again divided by three times the fquare of 1057, and that quotient again by 4 of that fquare, and that quotient by + thereof, and fo forth, till you have as many figures of your logarithm as you defire; as, for example, the logarithm of the geometrical mean between 22 and 24 is found by the logarithms of 2, 3, and 11, to be 1.361,316,961,266,906,129,450,091,726,698,05 1057 ) 43429 &c (5...3410,874,628,101,468,143,473,158,863,68 Belen t7249>) 41087 &C (5.2.5... 50 + 51225506,215,441,918,294,000,74 Mumma 22.9))) T2258 OCC (5. 0. yess ye os gs + «peels 90 0S50325951504357 01975 I 2 OC O82 OCC cee ts nie nas (4 gers 35s 8 cigs ce bee vas 42,000,297 ,08 MI PLA ZOO NOLO (rag ccc ee ont yaw tigen c gras 358 2 gees 8 3°25 90 Summa 1.361,727,836,017,592,878,867,777,112,251,17 Which is the logarithm of 23 to 32 places, and obtained by five divifions with very {mall divifors ; all which is much lefs work than fimply multiplying the feries into the faid multiplicator 43429, &c. Before I pafs on to the converfe of this problem, or to fhew how to find the number appertaining to a logarithm affigned, it will be requifite to adver- tife the reader, that there is a {mall miftake in the aforefaid Mr. James Gre- gory’s Vera Quadratura Circuli &F Hyperbole, publifhed at Padua, aano 1667; wherein he applies his quadrature of the hyperbola to the making the loga- rithms. In page 48, he gives the computation of the Lord Napier’s logarithm of Io to 25 places, and finds it 2,302,585,092,994,045,624,017,870, inftead of 2,302,585,092,994,045,684,017,991, erring in the eighteenth figure; as 1 was affured upon my own examination of the number I here give you, and by com- parifon thereof with the fame wrought by another hand, agreeing therewith to 57 of the 60 places. Being defirous to be {fatisfied how this difference arofe, I took the no {mall trouble of examining Mr. Gregory’s work, and at length found, that in the infcribed polygon of 512 fides, in the eighteenth figure, was ao, inftead of 9; which being rectified, and the fubfequent work corrected therefrom, the refult did agree to an unit with our number. And this I pro- pofe, not to cavil at an eafy miftake in managing of fo vaft numbers, efpe- cially by a hand that has fo well deferved of the mathematical fciences, but to fhew the exact co-incidence of two fo very differing methods to make loga- rithms, which might otherwife have been queftioned. From the logarithm given to find what ratio it expreffes, is a problem that has not been fo much confidered as the former, but which is folved with the dike eafe,-and demonttrated :by a ‘like procefs, from the fame general theorem of Mr. Newton. For, as thelogarithm of the ratio of 1 to1 + g was proved to be we Re 1 + g\” — 1, and that of the ratio of 1 to 1 — g tobe 1 — 1 — q\”; fo the lo- garithm (which we will from henceforth call 1) being given, 1 + B will be equal.to 1 + nN" in the one cafe, and 1 — 1 will be equal to 1 — 9” in theother; confequently 1 + 1)” will be equal to r 4+ 7, and 1 — 1)” tor — q; that is, Vor. Hi, N - according 90 HALLEY 6. DESCOUR SE according to Mr. Newton’s faid rule, 1 -- mu + 3m?*L* -+-- gm'L? + = m4 + aiomsis &c, will be = 1 + 9, andi — mi + 4m*L? — Em't} 4- am'*L* — +iom’ &c, will be equal to 1 — g, m being any infinite index whatfoever, which is a full and general propofition from the logarithm given to find the number, be the fpecies of logarithm what it will, But if Napier’s logarithm be given, the multiplication by m is faved (which multiplication is indeed no other than the reducing the other {pecies to his), and the feries will be more fimple, viz. 1 +L + LL + iL? + 414 +43.1n° &c, or 1 — Lb + ZLL — EL? + ALt — Fiok &c. This feries, efpecially in great numbers, converges fo flowly, that it were to be wifhed it could be contracted. If one term of the ratio, whereof x is the logarithm, be given, the other term will be-.eafily had by the fame rule; for if L were Napier’s logarithm of the ratio of a the leffer to 4 the greater term, 4 would be the product of a4 into1 +4 +- itn + itty &c, Sa+at + iatt + tar? &c. But if J were given, 2 would be = 6— 6, + 26LL — 36L? &c. Whence, by the help of the chiliads, the number appertaining to any logarithm will be exaétly had to the utmoft ex- tent of the tables. If you feek the -neareft next logarithm, whether greater or leffer, and cal] its number a, if leffer, or 4, if greater, than the given L, and the difference thereof from the faid neareft logarithm you call Z; it will follow, that the number anfwering to the logarithm t will be eitherw into 1 + 27 + i] + 211 4+ 14 + +315 &c, or elfe 4 into1 —/] + 34] — 21] 4+- 4 — hI &ce; wherein, as / is lefs, the feries will converge the fwifter. And if the firft 20000 logarithms be given to fourteen places, there is rarely occafion for the three firft fteps of this feries to find the number to as many places. But for Vlacq’s great canon of 100000 logarithms, which is made but to ten places, there is fcarce ever need for more than the firft f{tep @ + a/ or a + mal, in one cafe, or elle 4 — 4/ or 6 — ml, in the other, to have the number true to as many figures as thofe logarithms confift of. If future induftry fhall ever produce logarithmick tables to many more, places than now we have them, the aforefaid theorems will be of more ufe to reduce the correfpondent natural numbers to all the places thereof. In order to make the firft chiliad ferve all ufes, I was defirous to contract this feries, wherein all the powers of / are prefent, into one, wherein each alternate power might be wanting; but found it neither fo fimple nor uniform as the other. Yet the firft ftep thereof is, I conceive, moft commodious for practice, and withal exa& enough for numbers not exceeding fourteen places, fuch as are Mr. Briggs’s large table of logarithms ; and therefore I recommend it to common ufe.—It is al bl . he vive thus: @ + ——,or 6 — ey will be the number anfwering to the logarithm given, differing from the truth by but one half of the third ftep of the former feries. But that which renders it yet more eligible, is, that, with equal facility, it ferves for Briggs’s, or any other fort of logarithm, with the only variation of I I at ; ; ] —a + —la writing — inftead of 1; that is, @ + = eee by ieee ay ale OF ee m7 04 Ay I tt: on I bg 2 2 J 2 ON LOGARITHMS, gI cy gel and aR ; which are eafily refolved into analogies, viz. As 43429 &c 1 Biased = 21 to 43429 + 1/:: fo is ato the number fought ; or, as 43429 &c -+- 3/ to 43429 —i1::foisd to the faid number fought. If more fteps of this feries be defired, it will be found as follows, a pie Say, © 3 t_als te usa , &c, as may eafily be demonftrated by working out the di- oe 1 r—/ I— vifions in each ft and collecting the quotes, whofe fum will be found to agree with our former feries. Thus, I hope, I have cleared up the doétrine of logarithms, and fhewn their conftruction and ufe, independent from the hyperbola, whofe affections have hi- therto been made ufe of for this purpofe, though this be a matter purely arith- metical, nor properly demonftrable from the principles of geometry. Nor have I been obliged to have recourfe to the method of indivifibles, or the arithmetick of infinites, the whole being no other than an eafy corollary to Mr. Newton’s General Theorem for forming Roots and Powers. END OF DR. HALLEY’S DISCOURSES. N 2 N On a E S bs SOME OF THE MORE DIFFICULT PASSAGES sy ae FOREGOING DISCOURSE O F DR EDMUND.) HB A Lileeee - By FRANCIS MASERES, Esa CURSITOR BARON OF HIS MAJESTY’S COURT OF EXCHEQUER. oe ats Se eS te EE TTR AR a pS et HE foregoing difcourfe of Dr. Halley on the conftruction of logarithms has always been confidered, even by perfons of great fkill in the mathe- maticks, as a very obfcure and difficult tract: I have therefore thought it might be ufeful to draw up the following explanatory notes upon many of the more difficult paffages of it, by which, I hope, I have rendered them fufficiently in- telligible. The reader will, however, obferve, that I have made no notes on fome of the moft important parts of this difcourfe, to wit, thofe parts in which the author gives us the inveftigation of the two logarithmick feriefes g — ee a ie + Cainer ak + &c, ad infinitum, andg + oa + Dchay aR ile atid. & 5 6 2 3 4 + 7m + a7 + &c, ad infinitum, and the two anti-logarithmick feriefes 1 -+ L 12 L3 4 LS 25 ; ; tlk 7s sn &c, ad infinitum, and 1 — L ry 2 a 2.3 + 2.3.4 7 zidtaie 5 223.4.5.0 sr ? if ? af 74. 13 i+ LS L° os + — &c, ad infinitum. The reafon of my 2 243 26304 2.30405 243040526 ‘ omitting to make notes on thefe important paflages, is, that I never have been able perfe@tly to underftand them. I have, however, given full inveftigations of the two former of thefe infinite feriefes, in the ** Remarks on the faid Seriefes” (which were firft publifhed by Mr. Mercator and Dr. Wallis), con- tained in the former volume of this Collection of Tracts, from page 235 to page NOTES, &c 93 page 344; and I have given the like inveftigations of the two latter of thefe L3 1° ry rn 263 2.34 15 Le peas L* L? L4 LS 203-4.5 i 3.3-4.5.6 + Sc, ad‘infinitum, and 1 ur 2 2.3 t 2.304 2BeAnS + ah — &c, ad infinitum, in.the fubfequent difcourfe, intitled, «* An Ap- pendix to the foregoing Remarks.” And it feems-probable, that thefe invefti- gations of thefe feveral feriefes are either the fame with thofe given, or intended to be given, by Dr. Halley, in the preceeding difcourfe, or very analogous to them, and founded on the fame principles of arithmetick and abftract reafoning on the nature of ratios, without having recourfe to the hyperbola, or any other geometrical figure. I therefore refer the reader to thofe two difcourfes, in the former volume of thofe Tratts, for the inveftigations of thefe four feriefes; and I flatter myfelf, that, by the attentive perulal of thofe two difcourfes, toge- ther with.the following notes on thofe paffages of Dr. Halley’s preceeding traé& which I have been able to underftand, and which feemed to ftand in need of explanation, he will be able to underftand, and perceive the truth of, all the conclufions contained in the foregoing tract of Dr. Halley, notwithftanding its obfcurity.. : 4 4 * L2 feriefes, or the two anti-logarithmick feriefes 1 + L + sprigs N yOu Tai Ls N page 86, line 16, &c.—Hence the terms of any ratio being a and b, q —“, or the difference divided by the. leffer term, when it is an in- becomes : 6-4 AAS sreafing ratio, or , when it is decreafing, or-as b to a. b In this paflage, Dr. Halley fuppofes 4 to be greater than a, and he calls a ratio of minority, as that of @ to d, an increafing ratio, becaufe it proceeds from a leffer term to a greater; and he calls a ratio of majority, as that of J to a, a@ decreafing ratio, becaufe it proceeds from a greater term to a leffer. This feems to be an odd kind of language: but definitions and the meaning of words are arbitrary, or what we pleafe to make them; and this is evidently the fenfe in which Dr. Halley here ufes the expreffions of am increafing and a. de- creafing ratio. The propofitions therefore that are affirmed by Dr. Halley in this paflage of | his difcourfe, are thefe two, to wit: firfl, that if a2 beto das 1isto 1 +97, ¢ acc will he = “; and, fecondly, that if d be toaas1 isto1 —g,g will be = a. Now thefe propofitions may be proved in the manner following. Tn ga NOTES ON THE FOREGOING In the firft place, if ais to b ast is to r+ q, we fhall have, ievértendo, brat: i+ gia, and, dividendsyb—-a:anhitg—i:1,o0 ba: Mire 73. Los ale confequently g xamb—axi,andg= ad Fe = “=, Q. E. é And Peedi if Bistogastistor = g; we fhall have, dividends, b= a: ait -—(1—qgit—g,orbsa:acigit=g. But, by the {uppofition, Pepe PL Gee therefore, ex eqno, we fhall have d= a@:6::q: 1; and buaXt b—a . confequently gx d= 6—a|x 1; and g = jo SH FOG ED. NSO: .DouTh N page 86, line 18, &c.—Whence the logarithm of the fame ratio may be doubly exprefjed: for, putting x ie the di rcs of the terms a and b, it will be either x3 — xX the see ST + — an + ao = +5, ab jac + &e, or — x the xz a xt “xs ra re) mPa Series = PL rie PIs & FO SEMIS: a re For if 1 be to 1 + g as ais to S, the 1garichai of the ee of a to 4 will be equal to the logarithm of the ratio of to 1 +- g, that is, to = x the infinite 2 3 4 5 6 Ste le : feries g — 4 a joer Sao rary ti +- &c. But g is in this cafe = —— e . . z 3 5 bas =. Therefore = x the infinite feries g — 4 + 9 aes A fa ths a 9° poten fa fnghite feries 2 > et oe ee et oe se 4 + &cis=— xX the infinite feries — — — Teal gate Praca tas 1 . : PEE x x3 xt xs a — x the infinite f we eet eels abe, eR em, eel &c. Cond ana ab nite cries = => i hve Ga + &c, is the logarithm of the ratio of 4 to 4. QE. D. Secondly, if 1 is to1 — g as dis to a, the logarithm of the ratio of b to a will be equal to the were of the ratio of 1 to 1 — g, that is, to — x the infinite feries g a peters ie + + £ 2 Bg) se — ae 3 But g is in this cafe = oi = =. " Therefore 2 x the tes g+& LE 4. £ ce us isiro 2.4. = fue aL) aes : + iLv&eo istzs x the feries cs 2B 7 +e i xs +5 fs &c. ets Boe — xX i feries = +5 Ee oy od 7 hima Previa roy + x +- &c is the logarithm of the ratio of 4 to a. But the logarithm of the ratio of & to a is equal to the logarithm of the ratio of a to 4; becaufe thofe two ratios are equal, DISCOURSE OF DR. HALLEY. 95 equal, though contrary, to each other. Therefore -. x the infinite feries = 4. _ ve 5 vis = BY, 5 + ae +. &c is equal to — x the infinite feries = t. ~ = — aS fa ox a — +- &c, which is the logarithm of the ratio of @ to b. Therefore the logarithm of the ratio of a to 3 is equal either to = x the infinite feries = — os a 5 —_ a -. a — a + &c, or to — x the infinite feries = ‘4 - + af + a + = + oe + &c, agreeably to Dr. Halley’s affertion in the text. Q. E. D, Ni303' fe E Ill. N page 86, line 23, &c.—But if the ratio of a to b be Juppofed to be divided into two parts, viz. into the ratio of a to the arithmetical mean between the terms, and the ratio of the faid arithmetical mean to the other term b, then will the Jum of the logarithms of thofe two rationes be the logarithm of the ratio of a to b; and, fubftituting +z inflead of 4a + 4b, the faid arithmetical mean, the logarithms of thofe rationes will be, by the foregoing rule, I a ° Fe a? 3 at xs x — into the feries — + —, + 7 sled ae ore &Fe, i. Saat’ x at a xs x? and — into the Series — rier ca e, cme ee + &e, and the fum of thefe feriefes, or tae ot 2a 243 2x5 227 — into the feries — +- — + ai 4+ a + €9e; willbe the logarithm of the ratio of a to b, whofe difference is x, and fum 2. This paflage requires fome explanation. Dr. Halley affirms in it, that if a and 4 are two given quantities, of which 4 is the greater, and » be put for their difference 6 — a, and z for their fum 4+ a, the logarithm of the ratio of @ to 4 b +2 . ERLE S . “ie A x x? x3 x4 i} a sor to-—_—, will be = = ON the infinite feries = teh coe ich = wi 3+s 6 } : . : 7 = + &c ad infinitum, and that the logarithm of the ratio of = re ve By I b will 96 NOTES ON THE FOREGOING ‘ t hats vt x x3 xt xs Me ae bwill be =— x the infinite feries — — = + re Saat xe car tee + &c ad infinitum. Now this ais be fhewn in the manner following. b+ : B ies Let the ratio of a to = , or to = (which is evidently a ratio of minority, bya. becaufe J is greater than @, and confequently. amt ereater than * + ¢, or than a) be equal to the ratio of 1 — g to 1; then will 24 be to z in the fame propor- tion of 1— gto1. Therefore, dividendo, z — 2a will be to z as 1 —[1 —g, or pageaal = a q, isto 1; and confequently.g will be = a = - see . <= b-—a i « = 2 ihe q* Aja gh — =: Therefore — x the feries ¢ + ; ner ee a i ole AY Px at xs 3S &c, ad infinitum, will be = — x the feries = — + = on a + re + pari 3 5 iG 4 &c ad infinitum, But — x the feries g +2424 2 4 fe m1 2 2 4 5 6 ‘&c,. ad i is the mtg ia the Bee of 1 —.g to 1; therefore ~ x the : aw Penies a at ne = haere - pape ea a + — ~ + &c, ad infinitum, will be is Ic za- rithm of the fame ratio of 1g to-r, and confequently of the ratio of @ to =? or ate which is equal to it. Q. E. D. ; x b fo a : 1 ‘ Secondly, let the ratio of ee » to b (which is evidently a ratio of minority, becaufe 4 is ae Dep a, and confequently 24 is greater than 2b ba, and-—, or b, than —*) be equal to the ratio of 1 to 1 + a Then will z be to 24 in sie fame proportion of 1 to 1 4+ @, and confe- quently, dividendo, 2b— zwillbetozas1+Qq-—1,0rq,istor. Therefore 25 —2z\X1 2b—2 2b—(b +a 23-6 —a«a b—a x il be = 0 Ee ee ne feos ——, =: aoe foe qwill z C yaat Fey zB % z x tee eee ee 2 bgt 4 s 6 Therefore — x the infinite feries 9 — = -- 3 — 7 -- = — — + &c, ad x2 “3 4 5 6 ie Al aes 12) Peas a mpd infinitum, willbe = — x the irffinite feries = ia Tear ed + aig Zz a3 ob 5 6 &c ad infinitum. But — esl a gens Pia agp + ufinitu at the feries @ = ; = a5 r a + &c, ad infinitum, is the logarithm of the ratio of 1 to 1 + @; therefore — x the: . x we - x3 x x? ¢? ri s “ feries estes + ie nied Lae SM +°&c, ad infinitum, is alfo the loga- b+a — rithm of the ratio of 1 to 1 4+ q, and confequently of the ratio of =, or to 2, which is egual to it. Q. Es-D. And hence it’ follows that the logarithm of fie ae ‘of ato ’ (which is equal to the fum of the two ratios of a to aia and ok = to 6) will be sequal to the fum DISCOURSE OF DR. HALLEY. 97 4 5 6 } F fum of ~ x the feries = — + = a Lame nm rt — + get + as + &c ad infinitum, x x3 as xs and — — x he feries = — — =e i Beats a ea jai &c ad infinitum, or 3 5 2x7 nti 1 feries Sys ae a ee oe to — x the -4= t ag —— + Pec ad infinitum ; ee is, the logarithm of eM ratio of a to b, whereof 2 is the greater, and of which the So pH Fs b — ais denoted EY x, and the fum 2+ 4 Ff Menoted by 3 5 7 Ik z, is equal to — x the infinite feries — = 4~ 2 4 — + pot - + += &c, agreeabl tO Dr. Halley’s laft afiencigd in the alae e te explained. g y 3 paiag: P Q..Fai De Dr. Halley obferves that this feries oe oe Fadl = te ba i Atte &c converges twice as : : . x oe ood a* x“s x {wift as either of the two former feriefes — 4- — sarithiqee® as bee + a x x3 x4 xs « &c and — re at ace + nrg geet Sardar &c; and fays, that, on that ac- count, it is more proper than thofe other feriefes for the prattice of making lo- garithms. But this circumftance is not of any great importance, becaufe there is almoft as much apple in computing four eal or any other given number 3 2x5 a7 of terms, of the feries = Ba ae ae ae i &c, as in computing twice x3 xt xs xo the fame number of terms of the feries — — + = a Lofts o +f aah era begat a ” x” x3 xt xs x x7 Be a feo or te fein a Se Gee 8 i. . *~ 4 &c. For in both cafes we muft compute the fame number of powers of Bn Pp p the fraction = ; in which the principal labour of the bufinefs confifts; for when anh es Fad ° xv . x” once we have computed the firft eight powers of —, to wit, =, =, Vater) xs xe x? x8 . . . ost heer ike and —z» or any other number of thofe powers, the dividing them by their indexes 2, 3, 4, 5, 6, 7, 8, &c, which are very {mall and eafy numbers, is a matter of very little difficulty; and the paradise half of thefe 2x3 2x5 2x7 operations of divifion, by making ufe of the third feries — = a as a3 ag ce -+- &c, is but a fmall faving of trouble. ap ull, O | NOTE 98 NOTES ON THE FOREGOING Nar et) Eee C page 87, line 8, &c.—-For the difference of the ratios of a to 4%, and of 12 to b, is the ratio of ab to 12x; and the half of that ratio is that of fab to 4%, or of the geometrical mean to the arithmetical ; and confequently the logarithm thereof will be the half difference of the logarithms of thofe rationes, viz. = into the wx xt x8 Series — + 7 eS tT on + &c ad infinitum; which is a theorem of good dif- patch to find the logarithm of 42. The propofitions contained in this paflage may be proved in the manner following. The ratio of @ to 4z is equal to the ratio of a x 4 to Iz x 4, or of ab ta 326z; and the ratio of iz to 4 is equal to the ratio of tz x iz tod x iz; that is, to the ratio of = to 24z. Therefore the excefs of the ratio of 2 to iz above the ratio of $z to 4 will be equal to the excefs of the ratio of ab to 4bz above the ratio of i to tdz. But the excefs of the ratioof as to +bz above the ratio of vi to tbz is evidently the ratio of ad to mal Therefore the excefs of the ratio of @ to +z above the ratio of 4z to d is equal to the ratio of aé to = which is the firft propofition affirmed in the foregoing paflage. ospln, Secondly, the ratio of sad to +z is equal to half the ratio of ab to =; that is, the ratio of the geometrical mean proportional between the two quantities a and & to the arithmetical mean between the fame quantities is equal to half the ratio of ab to rap But fince the ratio of ad to = has been fhewn to be equal to the difference of the ratios of 4 to +z and of iz to J, it follows that the ratio of ad to = will be equal to half the difference of the faid ratios ; that is, the ratio of the geometrical mean proportional between the two quan- tities ¢ and 4 to the arithmetical mean between the fame quantities will be equal to half the difference between the ratios of a to 3z and of tzto d. Therefore the logarithm of the ratio of ./ ad to 42, or of the faid geometrical mean to the faid arithmetical mean, will be equal to half the difference of the logarithms of the ratios of a to 4z and of tz to 4. But the logarithm of the ratio of @ to ° ° . e od 2z has been fhewn, in Note III. to be = — x the infinite feries = — obs 3 4 Ss 6 t 7 4 z aa os -F ze + = + &c ad infinitum ; and the logarithm of the ratio of . I . . . 4z to 6 has been fhewn, in the fame note, to be = res the infinite feries = od we x x x oe - £8 é perth Me saaer irre theese hse er &c ad infinitum. ‘Therefore the excefs of the logarithm DISCOURSE OF ‘DR. HALLEY. 99 logarithm of the ratio of 2 to 2% above the logarithm of the ratio of $z to é will x4 xs be = 4 x theferies > +5 ++ Se Se * + &c — + x the fe- x x* x x4 xs * res Rann ast, cant a a, + &c = — x theferies= —-= + x x xe x at at a xs x x ee or _ a 4z* oes ain eae the feries = + _ ed 8 be ey &c 3 and half the faid excefs will be = — — x the feries 5 + -— Pe oa ° 4. &c. Therefore the logarithm of the ratio of V ab ia to 42, or to —, or - of the geometrical mean proportional between the quan- Fe a and 3d to ithe i Shaped Bae mean between the fame Diab pa will be = x xe x? xro — x the infinite feries + — ae tice tig hoe t pee &c ad inf- ipl Q. E. D. Thirdly, by this feries we may derive the logarithm of the ratio of 4, or 2 ci: “, to 1, from the logarithms of the ratios of ¢ to 1, and of d to 1, For when thefe two logarithms are known, we need only add them together, and we fhall thereby obtain the logarithm of the ratio of ah to 1, the half of which will be the logarithm of the ratio of Wad ab, or the geometrical mean between aand é,to1. And if we add to this logarithm the logarithm of the ratio of ~ to +/aé, or of the arithmetical mean between @ and 3B to the {aid geometrical sae are “ei be Heed ti means of the expreffion ~ x the feries ae Bi = + i acr 5 T ay ere iow -- vie ad infinitum, a fum will be the logarithm of the ratio of = = tol, or of 2 =! to 1, or of the faid arithmetical 12% repay mean between a and 4 to 1. Q.E. D. If the number 4 exceeds the number a by 2, we fhall have » (= 4 — 4) —= 2, andz(=b++a=>a+2 fe a) == 2a+ 25 and ee == ni viz 1 Toate + &c, ad ~—, Therefore the feries = — Bitte at = — + = = hc aha ae I x z a Toxt0 I . . - As se le rs SNe I infinitum, will, in this ae bes ; ara a Seat Sy t+ i ~ tiara pore Seetaat c ad ; 8$xatnr® P houdean ne Hiisgisie + ht + & infinitum ; and confe- . 2a — quently the logarithm of the ratio of = >» oF ts or @ + 1, to ./Z), or Va X 4+ 2, or “aa + 24, or of the arithmetical mean between a and 4, or between 2 and a + 2, to the geometrical mean between them, will be = I . s ° I I I I — > the infinite feries de 22 Fines Oe SPT ah elias CON Lek beans Oy tb 2xXa+ui) Pg anit 6s ote vi eee oes — = = is — =F + &c ad infinitum, O2 Thus, 100 NOTES ON THE FOREGOING Thus, for example, if ¢ was = 22, and 6 = 24, and we had already com- puted the logarithms of the ratios of all the prime numbers under 23 to 1, we inight make ufe of the laft-mentioned feries to find the logarithm of the ratio of the prime number 23 to I. For fince we had already computed the logarithms of the ratios of the prime numbers 2 and 3 and 11 to 1, we might from thence derive the logarithms of the ratios of 22 and 24 to 1, by mere addition; becaufe the logarithm of the : . 22 It . ratio of 22 tor is = L.—- +L.-= L. = + L. =, and the logarithm of a alg OF aoe Eien ib, = + L. = = L.2 + L2= L. 2 + L. 3 a ese It +3 xL. =. And having thus found the logarithms of the ratios of 22 and 24 to 1, we fhould have the logarithm of the ratio of 22 x 4; me: LA + <= 3; or the fum of the logarithms of the ratios of 22 to1 and 24 to 3, would be that of the ratio of 528 to Te Therefore half the fum of thofe logarithms would be the logarithm of the ratio of the fquare-root of 528, or of the geometrical mean proportional between 22: and 24,to 1. And, laftly, if to this logarithm of the ratio of 47528 to 1 we fhould add the logarithm of the ratio of 23, or the arithmetical mean between 22 and 24, to 4% 528, which may be computed by means of the expreffion Zi te 8 By al I i I 2X at.) 4xa+i1\ 6x a@+1) 8xa+n\* I I ° . I . I ee + Xe, ad infinitum, or — X the feries higues p° 1zxXa@+ i)” , ws 2X23) I 1 I I I bale Ge eee ee ee Ht OC, 2d infinitum,, ea Sxa” ERE oa ee 3 fi 3 I ; I I I hI I or — x the ferles ——— + —=S 3 + ———= Ft SSS Oe murs ok Fie) Picco te cee 8x 529) 10%! 529% + &c, the fum will be the logarithm of the ratio of 23 to 1. or 528, to 1, = x the infinite feries 4. I 6 I2 X $29 Q: E. I. This method of computing the logarithm of the ratio of 23 to 1 is the fame in fubftance with that by which we computed the fame logarithm in the 9th: Example of the foregoing Remarks on Mr. Mercator’s and Dr. Wallis’s Se- riefes, Art. 32, 80, and 95, pages 264, 304, and 328. N.O) Tuk DISCOURSE OF DR. HALLEY. ToT is fins 2 kote! Hal o V. N page 87, line 13.—But the fame is yet much more advantageoufly performed by arule derived from the foregoing, and beyond which, in my opinion, nothing better can be hoped ; for the ratio of ab to 422%, or taa + tab + 4b), has the difference of its terms 44a — 3ab + ibb, or the fquare of 34 — 7b = ixx; which, in the prefent cafe of finding the logarithms of prime numbers, is always unity; and, calling the fum of the terms 12% +- ab = yy, the logarithm of the ratio of ./ ab to 44a + ib, or iz, will be found = — X the feries be + — ++ =, + —, ++ ae Sc, ad infinitum ; which converges much fafter than any theorem hitherto publifoed for this purpofe. Since z is = d+ a, zz will be = 4b + 2ba + aa. Now 4, ba, and aa, are three quantities in continued geometrical proportion, 40 being to ba as } is to a, and da being to aa alfo as 4 istoa. Therefore, by El. 5, 25, the fum of the two extreme terms will be greater than twice the middle term ; that is, 25 + aawill be greater than 2ab. Therefore 4d + aa + 200, or bb + 24b + aa, will be greater than 24h + 2¢b, or 4ab, Therefore zz (which is = 33 + 2ab + aa) will be greater than 444, and confequently 7 will be greater than ab. Now let 2 be put = 4, and — =; then will 3 be greater than a, Let ° z B— A, or the difference of B and a, or = and ab, be called p, and gB + a, or the fum of z and a, or + and ad, be called s. It is fhewn above, in Note III. that, if ¢ and 4 be any two unequal quantities, of which #4 is the greater, and x be put equal to their difference 6 — a, and z to their fum 4 + a, the logarithm of the ratio of 2 to 4 will be equal to i : : weet 28 a7 245 27 249 oa mix the infinite feries — + —[ + —+—>,+ us +; + &c. It fol- lows therefore, that, in the prefent cafe, in which there are two unequal quan- tities a and 8, of which 8 is the greater, and the difference of the faid quan- tities is denoted by p, and their fum by s, the logarithm of the ratio of a to B 2p3 2D5 2D? 2D? va pet aaa or oe a ae ~ But the ratio of a to 8 is equal to the ratio of ab to a becaufe a is = ab, and q 7 2 will be = — x the infinite feries = +4 74 : : ‘ 2 2% r pis ——. Therefore the logarithm of the ratio of 2d to > will be = — x 4 3 : - 2D 2p3 2D 2p? 2D9 2p™ & - —~+ sate t+5o54+5o5+ c..% Therefor the infinite feries — poe es “\ et a ortig ore the logarithm of half the ratio of ad to = or of the ratio of «/aé to =, will be = t . : : D p3 p> Dp? De p** : peat, el ate gees ape. | os hitigas Ss . t 1S ; x the infinite feries 7 + 383 + ss + 761 {- 93° -| eT “+ &c3 tha F the logarithm 102 WOTES ON THE FOREGOING logarithm of the ratio of 4/ad, the geometrical mean between the numbers &% a and 4, to = fee 4 ps — xX the feries — — + 3 + — rr + — 3 + = + —— Now let the aveamte of the numbers 4 and @ be 2, or let d be =a 4-2. a+ 6\* 4 + &c, ad infinitum. Then will ab be(=axa+2)> 44+ 24; and = will be (= _a@+a+ 2)’ 2a+ 2\" _ g4a%+ 8at+4, _ = Stet ee ee et) = 4 4 a will be (= @ +24 +1 —[a? 4+ 24) = 1. But = — abis = D. Therefore, in this cafe, or when 4 is = @ + 2, Dv will be = 1 ; and confe- quently D3, pS, p?, p°, p™, and all the following powers of p, will alfo. be equalto1. Therefore the logarithm of the ratio of “ad, or the geometrical b " 2 ea “, the arithmetical mean bes a* + 24 + 13 and confequently 5 % mean between the numbers a and 3, to =, or tice the fame numbers, wt, in this cafe, or when 4 if = 4-+ 2, become I : : 4+- — + &e ad infini- + the infinite feries — iis a ++ ss, tum. ee. if, inftead of s, we fubftitute yy in the terms of this feries, Toe Tt per T os or fuppofe yy to be equal to as + ab, or the fum of x and ab, we fhall have — the feries — 4- — a a oe &c, ad infini- nen Sis aye EPs “pe —, + 3 inf as , — b tum, for the logarithm of the ratio of «/aé to —, or “+ or of the geometrical 2 2 5F + 1% mean ren the two numbers a and 4, or 4 and a +2, to the arithmetical a a 2 a 2a 2 mean (+ waled dy ates, or aa > or) @ + 1, between the fame numbers. OF aT An Example of the Computation of a Logarithm by Means of the foregoing Series. Let it be required to find, by means of this laft feries, the logarithm of the ratio of the geometrical mean between the numbers 22 and 24 to the arithme- tical mean between the fame numbers ; that is, to the number 23. Flere: we have -@s= 22,6 == 24, ab (=.22.X 24) > 528; and 47 gue Vf 528, andz(= 5+ 4 = 24 + 22) = 46, and confequently ~ —=.23, and oy (= 23]*) = 529, and yy (=> ++ ab = 529 + 528) = 1057, and y* (= . 4 I I I I I I 1057) = 1,117,249. Therefore the feries — + — + — + — + —4-—— 57 3 7» Ss ay by 3y° ape ayi4 + gy?® + I 1y** "th 1057 5 x tos? Sinisps7 © Se tile gy gx 105719 tT I . I I T a &c, and — x the feries — . — eet rs ae cs — xX the eries = h a6 0 aoe art gt Tt ryat &cec i | DISCOURSE OF DR, HALLEY. 103 é&e will be = = x the feries —— + —_— -+- wn Pay TS 1057 3 X 1057" 5 X 1057 7 X 1059)? aa > Loma ee =F + &c; that is, the logarithm of the ratio of Wa to » or of 44528 to 23, will be = — x the feries - —— + ——— = °57 3X 1057)" +- &c ad spe z b “3? or I I I 5 x sxiosF 7X 1057V" yi 9 x 1057)” Titx 1057\" Rt agtt eect ei ae eos XY 3 x 1057’ 5 x 1057) = 7 X 106+)” LE 9 X 1057)” I ; + &c is = the feries — hy gba abi Res cot mipuiont Il rs ar + B 1057 3m x 1057)° 7 $m X 1057) a —_—. Le eee ie + &c; which, if we put c for the qm Xx 1057)" + gm xX 1057) on tim X 1057) firtt term ———_ the fecond term - : and & fort i ft term reper and p for pie kaey? he third =~, and F, c, H, &c, for the fourth, fifth, fixth, and other fol- term 5m X 1057 1 c the feries, will be = the feries —~— pee oe lowing terms of ; AE eer AY aixnaent ~}. 3D se SE Re gaan iy" § X 1057)* ry 7 X 1067)" z g x 1057)” 11 x 1057)" * m X 1067 199 c Boe Sire eo tet lh ue AE) id es OC 3 X 1,117,249 a 5 X 1,117,249 £ Xb 1575249 ii 9 X 1,117,249 11 X 1,117,249 . ie b + &c. Therefore the logarithm of the ratio of ad to =, or a, or of : a EP itil ey eg Adel a raanareei i y Aiae < t aba" 4/528 to 23, is = the feries — eaets + PSE I S29 + earnest + arse ead Oe tae Ae &c ad infinitum aM 1,117,249 + 9 X 1,117;249 10 BY MP UET7 249 sh if 5 : ’ r I p aes But in Briggs’s fyftem of logarithms — ‘is = I 2+302,585,092,994,045,684,017,991,454,084,364,207, &c = 0-434,294,481,903,251, 827, 651,128,918,916,605,082, &c. Therefore Sere is (= = x a ©+4345294,481,903,251,827,651,128,918,916, 607,082, CCL sore Seaton ee = 0.000,410,874,628, 101, 468,143,473,158,863,68; Seta borg en earemarert 7 winds 4 — “oie a Sarin 3 ae ney + -+ &c is = the feries 0.000, 410,874)628,101,468;143:4739158,86368 + "+ B Soy LS ae + &c= 7 * 1,117,249 Q X 131175249 iI X 1,117,249 4 03000, 104 NOTES ON THE FOREGOING 03000,410,874,628,101,468,143,473,158,863,68 e's “EL OFS 0g. aig. = wig l R295 0852 1 5,449 0 16,294,000,7m as.e £.92eae® 46 te 500 55052. aet apa Ot TS “FroMPas s+ 92 cin 90 8s 30-9, 9190 8 e300 9s 0 9 350A SoG ULES S2 siege m tia Syergge Sagem ried «ls dpe tan sre deams2 O90 +- &c = 03000,410,874,750,686,749,417,085,385,5535,12 &c. Therefore the logarithm of the ratio of Wab to =, or 2 - “, or of 4/528 to 23 IS = 0.000,410,874,750,686,749,417,685,385,553,12, &c. Q..E. i, The logarithm of the ratio of 1 to 23 is equal to the fum of the logarithms of the ratios of 1 to W528 and of 4/528 to 23. The logarithm of the ratio of I to 47528 is equal to half the logarithm of the ratio of 1 to 528, or to half the fum of the logarithms of the ratios of 1 to 22 and of 22 to 528, or of 1 to 22 and of 1 to 24, or to half the fum of the logarithms of the ratios of i to 2, and of 2 to 22, and of 1 to 3, and of 3 to 24, or to half the fum of the loga- rithms of the ratios of 1 to 2, and of 1 to 11, and of 1 to 3, and of 1 to 8, or to $L¢+4L4+3L.1+4+41L-.4, and therefore may be derived from the logarithms of thofe leffer ratios by addition. And, if it be fo derived, it will be found to be = 1.361,316,961,266,906,129,450,091,726,698,05, &c. If there- fore we add to this logarithm the logarithm of the ratio of 4/528 to 23, which has juft now been found to be = 0.000,410,874,750,686,749,417,085,385, 553,12, &c, the fum, to wit, 1.361,727,836,017,592,878,867,777,112,251517y &c, will be the logarithm of the ratio of 1 to 23, or of 23 to 1, in Briggs’s fyftem, or, according to the common way of expreffing ourfelves on this fubject, the lo- earithm of the number 23. NO) ae VI. N page 90, line 11.——lf one term of the ratio, whereof 1 is the logarithm, be given, the other term will eafily be bad by the fame rule: for, if L was Napier’s bogarithm of the ratio of a, the leffer, to b, the greater, term, b would be the produ of 2 3 + $ 6 @ into the feries 1 Bes ph pha F * ot J Burd 2 a 203 "¥ 26304 ii 2030405 i 203042500 + &, a0 aut au3 a4 aus av® infinitum, = the feries a + au + PHL, or + Say - EWE, i brent: + &e, ad infinitum. But if b was given, a would be = the feries b — bu + on _ te 3 bu bus bi ‘ : —— = ————- + ——— — &e, ad infinitum. 2e3e4 2324S iF 2+3.4.5.0 ? , The DISCOURSE OF DR. HALLE Y¥. ; 104 The firft propofition afferted in this paffage is, ‘* that, if any number, or quantity, called a, be given, of the ratio of which to another quantity called 4, which is greater than it, L is the logarithm in Napier’s Syftengy the faid greater quantity 4, may be derived from a and L, by computing the value of the feries EN ied ag vide lai ob &c, ad infinitum, t . a 4 + gt tk 2.3 243-4 2034.5 7345.6. ¢ nin gi 2. RO which it will be equal.” Now this may be proved in the manner following. Suppofe & to be of fuch a magnitude, that 1 + & fhall be to 1 as & is to a. Then will L (which is thelogarithm of the ratio of to a) be alfo the logaritim of the ratio of 1+ to1. Therefore 1 + & will be =the feries 1 +L | L* L3 74 L$ 1° : : + ——— + &c. ad infinitum, and confequently that 2 + 2.3 $5352.00) 3.9.40 22304500 ay Ui : q y feries will be to 1 as distoa. Therefore dX 1, or 4, will be = a@ x the faid P 2 3 i LS L® c ‘ fenes mds + + + &c, ad infinitum, = + “A 2 a3 + aae8 + “4.5 + 2-3:4-5.0 hi z a au 2 G& a au e ° at+taL+— ote F —— + &c, ad infinitum. miso ch 2.3 + 2.3.4 a 2.3.4.5 + 2.3.4,5.0 Bs, Ss uf 23 LS Q. E. De The Second Propofition afferted in the foregoing paflage is, ‘* That, if an¥ number, or quantity, called 4, be given, of the ratio of which to a leffer quan- tity, called a, the logarithm, in Napier’s fyftem, is known and denoted by L, the faid leffer quantity @ may be derived from the faid greater quantity 4, and the logarithm L, by computing the value of the feries 4—dbL+4 Oe 613 4 s S ee cer btcbee tc a PL EL _. , aad infinitum, to which it will be 2 a4 2.3.4 2.3.4.5 2+3+42526 equal.” This may be proved in the manner following. Suppofe 4 to be of fuch a magnitude that 1 fhall be to 1 ~& inthe fame propor- tion as toa. Then will L be the logarithm of the ratio of 1 to 1—, as well as of the ratio of toa. Therefore 1— & will be =the feries 1 — L + 2 3 6 v a al ns! -- maT: — &c. ad infinitum. Therefore 1 will be se to this feries in the fame proportion as J is to a; and confequently @ x 1, or L® 203 2.304 2030405 L” L3 Ls LS a, will be = 4x the feries 1 —-L+-— Ch oer Ra oE +; a, rd Z . . ‘ é 2 b 3 b 4 3 5 s &¢, ad infinitum = the feries b— 2. + — pls L mere mae : — &c, ad infinitum. 2.3 2.34 2+36465 Q. E. D. Vou. Il. P NOTE 106 NOTES ON THE FOREGOING pn N 70.7 eB Vil. N page 90, line 15th—Whence, by the help of the Chiliads, the num- ber appertaining to any logarithm will be exatily bad to the utmoft extent of the Tables. If you feek the neareft next logarithm, whether greater or leffer ; and call its number a, if leffer, or b, if greater, than the given L, and the diffe- rence thereof from the faid neareft logarithm you call |; it will follow that the number anfwering to the logarithm L will be either a x the feries 1 +1+ aT ti tad et Gace 4 an + €fc, ad infinitum, or e//e b x the feries 1 —1 + PUR sie ene ge), Maphstiy 4 1s I ieoy gals aees RAE WIS, ht kad tt seats: ¢, ad infinitum. The firft propofition contained in this paflage is as follows; to wit, ¢¢ That, if any logarithm in Napier’s fy{tem of Logarithms be given, and the fame be denoted by L, and we feek in a Table of Napier’s Logarithms’ the logarithm which is neareft to L of all thofe which are lefs than L, and denote the faid logarithm by the Greek letter A and the number correfponding to it (or of the ratio of which to unity it is the logarithm) by a, and the excefs of L above A by the fmall letter /, the number correfponding to the given logarithm L, (or of the ratio of which to unity the faid given logarithm L is the logarithm,) will be = 4x the feries 1 + / +5 + ~ -- ~ + ee + — + &c. ad infinitum.” This may be fhewn in the manner following. Let 6 be put for the number hitherto unknown, which correfponds to’ the given logarithm L, or of the ratio of which to unity L is the logarithm, We . Mi [2 73 [4 7s fhall then be to prove that J is = @ X the feries 1 + / + - SS ee is ; Bap £ at. af - ah gh. ape az + &c ad infinitum, or= the feriesa + al + HS HE ry lane Se wm &c ad infinitum. Now, fince L is the logarithm of the ratio of 4 to 1, and Ais the logarthm of the ratio of ato 1, it follows that /, or L — A, will be the logarithm of the ratio which is equal to the excefs of the ratio of 6 to 1 above the ratio of 4 to 1; that is, / will be the logarithm of the ratio of to a. But it has been fhewn above in Note 6th, that if 7 is the logarithm of the ratio of 4 to a, (the quan- tity 4 being greater than a,) 4 will be equal to the feries 2 + a7 + st + ~ alt als a i6 pei pses . al? — + — 4+ —— + &c, ad infinitum, or the feries a + a] + — 22304 - 21345, 2-3 405-0 ns > if ) a ot Pi oH hes ai 1 <— 4- on + &c, ad infinitum. Therefore the ratio of 6to 1, or the ratio correfponding to the given logarithm L,, is that of the feries@ + a/ + al? als a l|4 als aig ; : ; 73 + 112 NOTES ON THE FOREGOING «¢ bles correfponding to a, or of the ratio of which to unity A is the logarithm ; <¢ and A be Napier’s logarithm of the fame ratio of ato 1; anddbe = B—a, «and / be= L — A; the ratio correfponding to the given logarithm B will be <¢ very nearly equal to that of @ + ~ ora + eacets to 1; and, fecondly, ‘¢ That, if B be any given logarithm in Briggs’s fyftem, or in any other fyftem ** of logarithms different from Napier’s, and L be the logarithm of the fame “© ratio in Napier’s fyftem whereof B is the logarithm in the other fyftem ; and «* the proportion of the logarithms of any given ratios in Napier’s fyftern, “<< to the logarithms of the fame ratios in the faid other fyftem be that of m to 1; «¢ and A be the next greater logarithm than B that is found in the tables, and } «* be the number found in the tables correfponding to a, or of the ratio of «¢ which to unity A is the logarithm ; and A be Napier’s logarithm of the fame ‘¢ ratio of to 1; and d be =A—B, and / be= A —L; the ratio corre- « {ponding to the given logarithm B, will be very nearly equal to that of bd Thefe two propofitions may be proved in the manner following. By what is {hewn in Note 7th it appears that, if L be any given logarithm of Napier’s fyftem, and A the next lefs logarithm fet down in a table of Napier’s logarithms, and a is the number correfponding to the faid tabular logarithm A, (or the number of the ratio of which to unity Ais the logarithm,) and 7 be = L— A, the ratio correfponding to the given logarithm L, will be that of . 2 $ 4 5 6 f ’ the feries a+ a] 4 a ++ =a + = am es +. — + &c, ad infinitum to 1, or, als al* als (becaufe the feries a ++ a] + fi + “Ey el tg se + &c, ad infinitum, 24 120 720 al is nearly equal to a +- ——,,) nearly, that of @ + ; ah = ‘ iz, correfponding to the logarithm L in Napier’s fyftem, is the fame as the ratio corre{ponding to B in the other fyftem, whofe logarithms are to thofe of the fame ratios in Napier’s fyftem as 1 tom. And /, orL.— A, is(=mB— ma > b bad or fuze to I. to 1. Now the ratio ie =m xIB —A)= md; and confequently a/ is = amd; and 8 =» and y Cale m d 2—-md Pe heb amd 2 — — = l[l— —) = See a techy > m a = are isi( = thet ee) rane Mesaeerore pvc erat UEP, =z zamd pont amd ad ) and therefore a a at 1$,2c @ + <<. Therefore the 2—mdad 1—md — 1—/ b Aplid td d ar 2 Te z ™ Es ratio correfponding to the given logarithm B in the fyftem of logarithms in which the logarithms of any given ratios are to Napier’s logarithms of the fame ratios in the proportion of 1 to m, is, nearly, thatof + ““< tor. 4 eed Q. E. D Secondly, it appears likewife by what is fhewn in Note 7th, that, if L be any given logarithm in Napier’s fyftem, and A be the next greater logarithm fet down in a table of Napier’s logarithms, and 4 is the number correfponding to the faid tabular logarithm A, (or the number, of the ratio of which to unity A 1$ 4 nearly, that of d — DISCOURSE OF DR. HALLEY, Ng is the logarithm,) and /be = A—L, the ratio correfponding to the given 2 4 logarithm L will be that of the feries 6 — > /+- fag las LR EAE AB pel? 2 6 PA 45 n° — &c, ad infinitum to 1, or (becaufe the faid feries — 57 + — BE AN eS ta 6 2
  • —, and 73 (= lie bmd __ 2) 5 abd _\bmd s+ bd RIG See Ok Dime Eee Pee es z and 4 — 77 : pale Stes te = 4 — —,. Therefore the ratio correfponding to the given logarithm B m2 Hee ets et . in the fyftem of logarithms in which the logarithms of any given ratios are to thofe of the fame ratios in Napier’s fyftem in the proportion of 1 to m, is, b nearly, that of d — —— tol. Q: E. D. —+— m 2 Befides the two propofitions which we have juft demonftrated, the foregoing paflage contains the two following propofitions, to wit, 1ft, “That @ + iB ++ la - is equal to — ct ,»” and adly, « That b — — — is equal ma mn 2 aehetie ss: to ———— ;” or, according to the notation ufed in the foregoing part of — + — 1 wm 2 this note, (in which d anfwers to the logarithm denoted by / in Dr. Halley’s ad + sae & da expreffion,) “* That @ + — — is equal to — -————,” and “ That 4 | san = a” ba hao db — ———— is equal to ~__——..”_ Now this may be eafily fhewn in the ninth “aa ee manner following. Vou, II. Ou: The 414 NOTES ON THE FOREGOING The former oe @ + ——— is (= : : meee ae was iui ake tr ere ois Mie atekden ier’: _ wetted . Ph te ey Aa av ay A ye a ue 2 bia 2 7/3 2 773 I I bd bx|> +5 bd the latter quantity 2 — ere = ——— - —— = wie wat tm, 1 ee I I I I I 1 I I I 1 Mite! ag I . se See aint tei ray a tig Q. E. D. —a+-—ad Laftly, the expreffion ———, which is equal to 4, affords us the fol- me 2 lowing analogy for arias the value of 4; to wit, as - _ ~ d is to = — a a —ad » or) 4, or the number fought; or, if d de- + ~ 4, fo is a to (— note a logarithm in Briggs’ S fyftem, and confequently m be = 2.302,585,092, 994,045,684, &c, and — — be SCE TERIOR or, 0.434,294,481, 903,251,827, &c, we fhall have 0.434,294, &c, — — ~ 4 tO 0.434,294, ii + ~ d as ato, or the number fought. —h——bd And the expreffion = : ; , which is equal to @, will afford us the . a PS - following analogy for aia the value of 4; to wit, as — + 7 d is to —ib—'—ba —— = 4, fo is & to (—-——., or) 4, or the number fought; or, if 2 Wt 2 be a peti ee in Briggs’s fyftem, and confequently ~ peje 0-4345294,4815 903,251,827, &c, we fhall have 0.434,294, &c, + — d tO 0.434)294, &e, — _ @ as & is to a, or the number fought. NOTE ; ’ DISCOURSE OF DR. HALLEY. ars N OTT) E Xx. N page o1, line 4th—————Jf more feps of this Series be def red, it will al js 1s be found to be as follows, to wit, ‘a 5 heen amare seas + sree Ses as may eafily be demonftrated by working out the divifi ons in each frep, a colletting the quotes, whofe fum will be found to agree with our former feries. 1 5. al3 is Dr. Halley here affirms that the feries @ +- ——- rw — ut ; + ms &c, is 120 720 This Afertion | is true as far as the fourth power ie ey dnd pee but not accu~ rately, fo with refpect to the tern ap which involves the fifth power of J, al veal | shals m bP 7 el? Tat into feriefes by dividing their numerators by their denominators, and then adding together the firft and third quotients, or feriefes fo obtained, and fub- tracting the fecond quotient from their fum. For, if this be done, we fhall This will appear by converting the three fractions - al al a I” alt. alt als have Ti? Ci torn tatty Ae ime armarcate Sane aN + &c. and J = == 2 ; 1 @]3 tals Bee = 2268 + ~al4 +820615 4 4.415 + &e, and ue = l szals eh bee. val goal? + &e. Therefore ——,— 245 4+ 22" will be = a 13 a 14 lidues a 16 al + ot. 5 ++ 5 Beh sxgrts-Ae taa% + &c. — et GB am Dal — ak me 5 al® — &e, : is hy ee he gral + &c, a a a a a =al+— +3 shar Sey PET AG - + &c. a ls zal ais a I 8 Tes Ma Wh Ae en ot als 2al 35 24/13 a it als a l° maby it ia eereda reve cay 3 + &c. als a I Te a ae als 221° Bee eee he. 35 35 re al a lt 3a/8 3a iP =ail+* Kp ssoer is ; Be ita 4als 8al & — — — — — &e 8 6 als zal BUT ABT be ca 116 NOTES ON THE FOREGOING ait ag Pi als cng aera ICMR MRE TS TITS is gg tH ig spe age bi OT fea an aot ay “88 ob + &c; and confequently the al dials als ait : ze pe Dill be = «healer CO +- TET 13als 17al® 1680 3360 a i TE als a l§ ; : $ ; -+ mri air co + &c ad infinitum in the five firft terms, but ee re : 1s re ; ‘ it in the fixth and feventh terms oo and aes , of which the former is a : is - 16 Si. little lefs than —, and the latter is vids Loeiit than 2—. For £5 is 120 720 120 14 als _. 4als “34 X 120. , 1680. feries a2 ad » which agrees with the original feries ¢ + a/ + ae +’ a ses the difference 4"; and ie 1680 GIN ie TA alot dal? gi 3K 17s 17a 1® rt berliernn . spenitas pare ot which is lefs than ae or “3 x 3360 2+” “ga6a “Hidis or, nearl ltt which is more than double of 25. Dees oats y 10,080? ~ ? Ya 272? Qs Fai De , which is greater than ~ l bi : I In order to make the feries 4 Bie - Srl -2 _ 7 “ ae i ~~ &e, which is afferted by Dr. pat to be equal to the ee feriesa@ ++ al + — me - ce 5 oa fea ha a2) ie &c ad infinitum, agree with the faid original “ees in 120 20 ti gx.al5 gr als its Acie term = we muft increafe the fourth term ih from pe zx als 2 als 7 nals bys were, to = 22~-- For then we fhall have +224 240 4 = 1-—a/ r—2t GEE, c, mag, gay a i? a i3 a I+ als bg Nt rbreiearretan wicca ocr (a I2 I2 12 oie al’ +14. al® &c. ay a |? ai3 a lt 3als 3al% yea tine Ne at a ch ania + &c, 4als 8alé 48 “06 — &c, 7als yal + ar —— + &c. a |* a 13 a l4 als 5 als "Ua Ge eee a a a See 7 als zal a + 240 120 + &e, _ aP / a8. ahs saks | soul® am G Z + 6 24 a Gt Saad 960. ee: &c. ] 4. 2° a + &c. DISCOURSE OF DR. HALLEY. -117 a |? a l3 a l4 2al1s 6a16 a It a [3 a |4 hia ai ree 4 240 u 960 + * Bree ieee oe 5 trie ior &c, and confequently ¢ + ST Por Sika Pe = 4 + ai 4. <= + a -- ae 2. a + &c, which agrees with the original feries a + al] + mite eat $444 &c ad infinitum in the fix firft terms, or as far as the fixth term << inclufively. Q. Es D. Dr. Halley feems to think that there is much lefs trouble in computing the ris als 1—2/ l x four terms a + — aint ree far than in computing the firft ix terms = Zz _ of the original feries a -- a@/ + << aE ae + a + ae + &c, to which fix - terms the foregoing four terms are equal. But I much doubt whether this opi- nion is well founded; becaufe the divifors 1 —i/, 1 —/, and1—2/, of the fecond, third, and fourth terms of the faid new feries will be much longer numbers, and confequently more troublefome to ufe as divifors, than the num- bers 2, 6, 24, and 120, which are the divifors of the third, fourth, fifth, and fixth terms of the faid original feries: and in both feriefes it will be neceflary sia als ais ' : t1—2/ 120 finding which power of / a confiderable part of the difficulty of the computa- tion will confift, I fhould therefore, in a computation of this kind, incline to make ufe of the original feries ¢ + @] + << + = + e -+ eae -. = -. &c ad infinitum, rather than of Dr. Halley’s faid new feries. to raife / to the fifth power, in order to obtain the terms + Taina aye Big a XY, N page 91, line 8th.— Thus, J hope, I bave cleared up the doétrine of loga- rithms, and fhewn their contruction and ufe independent of the hyperbola, whofe affettions have hitherto been made ufe of for this purpose, though this be a matter purely arithmetical, nor properly demonftrable from the principles of geometry. Nor have I been obliged to have recourfe to the method of Indivifibles, or the arithmetick of Infinites; the whole being no other than an eafy corollary to Mr, Newton's general theorem for forming roots and powers. ArTicLE I.—Dr. Halley here feems to affert, that he was the firft perfon: - who had fhewn the conftruction of logarithms without. the affiftance, or inter- vention, 118 NOTES ON THE FOREGOING vention, of the hyperbola. But this affertion is not true in its full extent. For both Lord Napier, the firft inventor of logarithms, and Mr. Henry Briggs, who invented, or firfl computed, thofe of the fyftem now in ufe, which is called by his name, had conftruéted logarithms, and explained their manner of conftructing them to the world, without any mention of the hyperbola. And fo did the learned and fagacious John Kepler afterwards in his two tracts, firft publifhed in the years 1624 and 1625, and now republifhed at the beginning of this colleCtion. And fo likewife did Mr. Nicholas Mercator in the former and larger part of his tract, intitled, Logarithmotechnia, which was firft pub- lifhed in the year 1668, and has now been republifhed in this collection. Dr. Halley muft therefore be fuppofed to mean only that the two logarithmick feriefes 9 — o + eo 4 Lf + 8c, ad infinitum, and q. + . +04 q* + qe + a + &c, ad infinitum, (which had been publifhed by Mr. Mercator and Dr. Wallis for the purpofe of conftru€ting, or computing, logarithms more conveniently and expeditioufly than by the methods ufed by Napier and Briggs, and the other old computers of logarithms,) had been cerived by Mercator and Wallis, and other writers, (as, for inftance, James Gregory,) in the tracts they had publifhed on the fubject, folely from the hyperbola. And this, I fuppofe, was true. | _ Ii. Dr. Halley fays further, ‘* That the doétrine of logarithms, being a matier merely arithmetical, is not properly demonftrable. from the principles of geometry.” But this is by no means a juft conclufion ; fince numbers may as well be re- prefented by geometrical lines, or areas, (fuch as the afymptotick {paces, or areas, of an hyperbola,) as by any other quantities whatfoever, and the in- ferences obtained by means of fuch a reprefentation of numbers to the eyes, or © fenfes, are at leaft as clear and fatisfactory as thofe obtained by confidering them merely in the abftrat. Indeed, when we fpeak of abj/tract numbers, we can — mean only commenfurable quantities in general, or commenfurable quantities of any kind, without confining our ideas to any one particular fort of quantities. Thus, for example, when we fay, ‘ that 3 times the number 7 is equal to the num- ber a1,” the only meaning of the propofition is, ‘that, if we take 3 parcels of men, or of horfes, or of houfes, or of cannon-balls, or of lines a mile long, or of any thing elfe, and each parcel confifts of 7 individuals, or units, of the things under confideration, that is, 7 men, or 7 horfes, or 7 houfes, or 7 cannon-balls, or 7 miles, and we add the faid 3 parcels together, the fum thereby arifing will be 21 of the faid individuals, or units, to wit, men, horfes, houfes, cannon-balls, or miles; and that this will be equally true, whether we fpeak of one of thefe forts of things, or of any other of them.” Now certainly our ideas upon this fubject will not be the more clear, and our conclufions the more certain, for our not confining ourfelves to the confideration of only one of thefe fubjects, but extending our thoughts to all of them at once, as we do when we confider them in the abftraét: but they will, on the contrary, be rather lefs fo, it being a plainer and eafier propofition to under- ftand, <« That 3 times 7 miles make 21 miles,” than “ That 3 times 7 things of any kind make 21 things of the fame kind.” When therefore it is fhewn, 4 that Fd DISCOURSE OF DR, HALLEY. it9g that the afymptotick {paces in an hyperbola that are intercepted between ordi- nates that are in the fame ratio to each other are equal, and confequently that the afymptotick areas that are intercepted between ordinates whofe ratios to each other are unequal, are proportional to, or meafures of, the ratios of fuch - ordinates, we have a clearer idea of thefe meafures of the faid ratios, or, in \ other words, of thefe logarithms of the faid ratios, than when we endeavour to form a conception of the meafures, or logarithms, of the ratios of mere abftra& numbers to each other. And the conclufions that are drawn concerning thefe afymptotick fpaces, or vifible logarithms, are likely to be more clear and fatif- factory than thofe that might be drawn concerning abftraét logarithms, and at the fame time muft evidently be full as certain, It is not therefore true, as Dr. Halley has afferted, ‘‘ That the doétrine of logarithms, being a matter purely arithmetical, is not properly demonftrable from the principles of geometry ;” but it is true, on the contrary, ‘* that the faid do¢trine may be very clearly and fatisfactorily demonftrated in that manner.” III. In addition to what has been here advanced concerning Dr. Halley’s affertion, we may obferve, that Euclid himfelf, the moft clear and accurate writer on the mathematicks that ever lived, has, in the fifth book of his Ele- ments, (which is of the moft abftract nature poffible, and treats, not of lines and angles, and the areas of right-lined figures, as the former books do, but of quantities in general, and their proportions,) thought fit to reprefent abftrac& quantities by right lines, in order to affift his readers to conceive and underftand what he fays of them ;- thereby calling in the imaginations of his readers in aid, and to the eafe and relief, of their underftandings. And with refpect to the fubject now under confideration, to wit, the nature and conftruction of loga- rithms, I have always found right lines to convey to my mind the cleareft con- ception both of logarithms, or the meafures of the ratios of quantities to each other, and of the numbers, or quantities, themfelves, of whofe ratios to each other they are meafures ; and therefore I am of opinion that this doétrine can- not be better explained than by the affiftance of the logarithmick curve, in which the ordinates increafe on one fide, and decreafe on the other, ad infinitum, and confequently bear all poffible ratios to each other, and therefore are capable of reprefenting all forts of numbers, however great or fmall, and the abfciffes of the axis, or afymptote, belonging to any two different pairs of ordinates, are proportional to, or meafures of, the ratios correfponding to the faid ordinates. And further, I confider this curve as being fomewhat preferable even to the hyperbola for the purpofe of illuftrating the doétrine of logarithms, becaufe right lines, (fuch as the abfciffes of the axis, or afymptote, of this curve,) are fill more eafily conceived in the imagination than the mixtilinear afymptotick areas of an hyperbola. IV. However, though the illuftrations and demonftrations of the properties of logarithms that have been given by Mr. Mercator and Dr. Wallis, and Mr. James Gregory, by means of the hyperbola, appear to me (for the reafons that have been mentioned,) to be very legitimate and {atisfactory, yet I agree with Dr. Halley in thinking, that fo ufeful and important a doctrine as this is, ought likewife to be explained, if poffible, without the help of the hyperbola, and upor i120 NOTES ON THE FOREGOING upon the abftra&t and pure principles of arithmetick, to the end that it may Be feen and underftood in every poffible light and mode of confidering it: and . therefore I have, in a preceding tract of Vol. I, intitled, ‘* Remarks on the two foregoing infinite Seriefes of Mr. Mercator and Dr. lWallis,” endeavoured to de- duce thofe two feriefes from. principles of pure arithmetick and the abftraét na- ture of proportion, and in a manner that I conjecture to be fimilar to that in which Dr. Halley has deduced them in the foregoing difcourfe which is the tubject of thefe notes, though I have not been able, after the moft earneft endeavours, perfectly to comprehend the Doétor’s own inveftigation of them : and I have likewife, in the next following difcourfe, in Vol. I, intitled, 4 Appendix to the foregoing Remarks,” endeavoured to give a fimilar inveftigation of the two anti-logarithmick feriefes contained in the foregoing difcourfe of Dr. Halley, that is grounded likewife on the principles of pure arithmetick, and which I conjecture to be, if not the very fame with, at leaft very fimilar to, the inveftigation of thofe two feriefes given by Dr. Halley in the foregoing difcourfe, which, however, from the extreme brevity with which it is exprefled, I have not been able to underftand. The other parts of the foregoing difcourfe of Dr. Halley, (that is, all but the inveftigations of Mr. Mercator’s and Dr. Wallis’s two logarithmick feriefes and the two anti-logarithmick feriefes derived from them,) I think I have been able to underftand, (though not without great labour and difficulty,) and I hope I have explained them fufficiently to the reader in the courfe of the ten foregoing notes; for the extraordinary length of which, the uncommon degree of obf{curity which runs through the text, muft be my apo- logy. V. Dr. Halley tells usin the latter part of the paflage cited in the beginning of this note, ‘* That in the foregoing difcourfe he has not been obliged to have recour fe to the method of indivifibles, or to the arithmetick of infinites ;? for which he feems much to applaud himfelf. But furely this applaufe is not deferved. For what can be a recourfe to the arithmetick of infinites, if the following paflages are not fo? Firft, in page 85, the Dodtor’s words are as follows: << And thefe rationes we Juppofe to be meafured by the number of ratiuncule con- << tained in each, Now thefe ratiuncule are fo to be underftood as in a con- “ tinued feale of proportionals, infinite in number, between the two terms of “ the ratio; which infinite number of mean proportionals is to tbat infinite num- “ber of the like and equal ratiuncule between any other two terms as the «¢ logarithin of the one ratio is to the logarithm of the other. Thus, if there be “© fuppofed between 1 and to an infinite feale of mean proportionals, whofe number “is 100,000, Se. in infinitum; between 1 and 2 there fhall be 30,102, &e. of “¢ fuch proportionals; and between 1 and 3 there will be 47,712, €Fe. of them; “¢ which numbers therefore are the logarithms of the rationes of 1 to 10, 1 to 2; “and 1 to 3; and not fo properly to be called the logarithms of 10, 2, and 3. ‘« But, if, inftead of Juppofing the logarithms compofed of a number of equal “* vatiuncule proportional to each ratio, we fhatl take the ratio of unity to any num- «< ber to confift always of the Jame infinite number of ratiuncule, their magnitudes ‘* in this cafe will be as their numbers in the former. Wherefore, if between unity << and any number propofed, there be taken any infinity of mean proportionals, the “¢ infinitely little augment, or decrement, of the firft of thofe means from unity “ will DISCOV RSE OF DRsa HALLE: Y, 121 “* will be.a ratiuncula, that is, the momentum, or fluxion, of the ratio of unity to “< the faid number... And, feeing that in thefe continual proportionals all the ratiun- “ cule are equal, their fum, or the whole ratio, will be as the feid momentum is *¢ direéily; that is, the logarithm of each ratio will be as the’ fluxion thereof. “ Wherefore, if the root of any infinite power be extrafted out of any number, tire “¢ differentiola of the faid root from unity fhall be as the logarithm of that number. “< So that logarithms thus produced may be of as many forms as you pleafe to affume “* infinite indices of the power whofe root you feek; as, if the index be fuppofed “‘ rooo00 €S¢, infinitely, the roots shall. be the logarithms invented by the Lord “‘ Napier; but, if the faid index were 2302585 Sc, Mr.. Briggs’s logarithms “© qwould immediately be produced. And, if you please to flop at any number of <‘ figures, and not to continue them on, it will fuffice to affume an index of a figure «< or two more than your intended logarithm is to have, as Mr. Briggs did; who, to ‘© have his logarithms true to 14 places, by continual extraction of the fquare root, “ at laft came to have the root of the 140,737,488,355,328tb power; but bow “« operofe that extraétion was will be, eafily. judged by whofo faall undertake to & examine bis Calculus. . _ & Now, though the notion of an infinite power may feem very ffrange, and, to “¢ thofe that know the difficulty of the extraction of the roots of high powers, “perhaps, imprafticable: yet, by the help of that admirable invention of Mr. << Newton, whereby be determines the uncie, or numbers prefixed to the members ““compofing powers (on which the doétrine of Jeries chiefly depends), the infinity. of “ the index contributes to render the expreffion much more eafy; for if the. infinite cc power _ be refolved be bat “eirad Mr. Newton's method) p + pq, p+ pq = 3" -- 2mm “ qi elidel SG ee —q+- are fea roel 1—6m+i1imm— mii aes ce qt. Be, (which is the root when, m. is Sinite), becomes 1 + ~q— —qq + $a eg? — ne + q? Fc, mm being infinite infinite; and confequently what/oever “45 divided thereby vanifbing. Hence it follows, that — multiplied into q — 44q fe aes eo iq’ Se, is the augment of the np of our mean proportionals & segye be unity and 1 -+- q, and is therefore the logaritkm of the ratio of 1 to “1+ q. And, whereas the infinite index m may be taken at pleafure, the several << Yeales of logarithms to fuch indices will be as —; or reciprocally as the indices: “¢ And, if the index be taken 10000 Se, as in the coe of Napier’s logarithms, they “ will be fimply q — 399 + 3q? — 29° +.3q° — 3 Ge, A ses Again, of the he a of a decreafing ratio the fraghts the infinite root of ©r— qyort— aig, ist — SID TET pet — Ruse A alarming sin es whence the. decrement of the firft “of our vets bh ber of proportionals will be © into q = Taq ae ag .cktgtct aq’ + 30%.Ge;..which therefore will be as the logarithm of the ratio of unity to 1 — q: But, if m be put 10000 ce, “© then the faid REGED, will be q + 49q + 3q° + 49% + $q°> + sq 7h, Vi Vou. IT. R Tn 522 NOTES ON THE FOREGOING DISCOURSE’ OF DR. HALLEY. In this paflage we have frequent mention made of infinite numbers, infinite indices of powers, different infinite numbers bearing proportions to each other, fluxions, momenta, and differentiole, and a quantity, mm, that is:imfinite infinite. Surely thefe expreffions partake both of the method of indivifibles and of the arithmetick of infinites, notwithftanding the author’s boaft. of the contrary. And, fecondly, the following paflage makes mention of infinite indexes of powers, and fuppofes that there may be as many different indexes of this kind as we pleafe; which certainly is a branch of the arithmetick of infinites.. “* From the logarithm given to find what ratio it expreffes, is a problem that bas <¢ not been fo much confidered as the former, but which 1s Jolved with the like eafe, “< and demonftrated by a like procefs, from the fame general theorem of Mr. New- “© ton. For, as the logarithm of the ratio of 1 to 1 + q was proved to be Bc 1 cy + g\™ — 1, and that of the ratio of 1 to1 —q tobe 1 —1—q|”; fo the lo- “ garithm (which we will from henceforth call L) being given, 1 + u will be ae = ss « egual to 1 + Q\™ in the one cafe, and 1 — u will be equal to 1 — q\™ in the other ; “¢ confequently 1 + u\" will be equal tor + q, and 1 — 1\"to1 — q; thatis, “* according to Mr. Newton’s faid rule,1 +mve + ~m°L* +2m'?L* + 4 m* sors + 12. m5 &c, will be = 1 + q, and 1 — me + im’? L? — im?13 + «om*L* — ~1,.m>1' &e, will be equal to 1 — q, m being any infinite index “¢ qwhatfoever: which is a full and general propofition, from the logarithm given to “© find the number, be the fpecies of logarithm what it will. But, tf Napier’s lo-. ‘* garithm be given, the multiplication by m is faved (which multiplication is “* indeed no other than reducing the other /pecies to bis), and the feries will be * more fimple, Vi. 1 --L + ZLL + iL? + =4Lt + 535L° &c, or — L + OORT me FL Lei ke. After reading thefe two paflages of this difcourfe of Dr. Halley, (which are thofe which, with the greateft attention I could beftow upon them, I have not been able perfectly to underftand) I prefume, it will be readily allowed that the Doctor is not intitled to the merit he feems to claim, of having explained the doétrine of logarithms without having had recourfe to the method of indivifi- bles, or to the arithmetick of infinites. VI. It is, however, poffible by another method of applying Sir Ifaac New- ton’s excellent binomial theorem, to compute a table of logarithms either of Briggs’s fyftem, or any other that may be thought fit, in the manner which Dr. Halley recommends, or without the intervention of the hyperbola or Jogarith- mick curve, or any other geometrical figure, and likewife without having -recourfe to the method of indivifibles or to the arithmetick of infinites, and by the help of the principles of pure and finite arithmetick only, and the contem- plation of the nature and properties of ratios, which are the quantities of which logarithms are the meafures. The method of doing this I fhall now proceed to explain in the following difcourfe; which, as it has taken its rife from the reflections and obfervations of Dr, Halley contained in the foregoing traét on logarithms, I thal] intitle an Appendix to.the faid tract. AN Peet EN. Do Lox TO THE FOREGOING TRACT aL OF | | Ma. DM UND HAL LEY uPOR a O GAR TF. AH, Ms: BEING A dire& Method of computing the Logarithms of Ratios either in Briggs’s Syftem, or any other that may be propofed, by the help of Sir Iaac New- . ton’s Binomial Theorem, without the intervention of the Hyperbola, or the Logarithmick Curve, or any other Geometrical Figure, and likewife without Ahaving recourfe to the method of Indivifibles or the Arithmetick of Infinites. By FRANCIS MASERES, Esq, CURSITOR BARON OF HIS MAJESTY'S COURT OF EXCHEQUER, [Soe oa aT eee ARTICLE 12. 'H E logarithms of ratios in Briggs’s Syftem are the numbers that exprefs the magnitudes of the faid ratios, or their proportions to each other, upon a iuppofition that an unit is taken for the reprefentative, or logarithm, of the ratio of 10 to 1. And the logarithms of ratios in any other fyftem are the numbers that reprefent the magnitudes of tthe faid ratios, or their proportions to each other, upon a fuppofition that an unit is taken for the reprefentative, or Jogarithm, of fome other ratio than that of 10 to a. If therefore we can find the proportion of any other ratio to the ratio of 10 to 1, we fhall thereby bs A 2 enable 124. AN APPENDIX "TO ERE FOREGOING Th ACT enabled to difcover Briggs’s logarithm of fuch other ratio. For, if the mare found for the reprefentative of the ratio of 10 to 1 is 1, (which is Briggs’s » logarithm of that ratio,) the number found for the reprefentative of the other ratio will be Briggs’s logarithm ef the faid other ratio: and, if the number found for the reprefentative of thé ratio of 19 to 1 is not 1, but fome other number, which we may call x, either greater or lefs than 1, we need only diminifh or increafe the number found for the reprefentative of the faid other ratio in the proportion of » to 1, and the number thereby obtained will be Briggs’s loga- rithm of the faid other ratio, Thus, for example, if we fhould feek the propor- tion between the ratio of 11 to 10, and’the;ratio of 10 to 15, and fhould find it to be that of 1 to 24.158,857,928,096,6, (as we fhall find it to be in the courfe of the following pages,) of which two ‘numbers the lefier, to wit, 1, is the re- EU api of the fmaller ratio of 11 to 10, and the greater number LAL SOs 857,928,096,6, is the reprefentative of the greater ratio of 10 to 1, we need only reduce the {maller number 1, (which; is the reprefentative of thesratio of 11 to 10,) in the proportion of 24.158,857,928,096,6, (the firft reprefentative of the ratio of 10 to 1,) to 1, (which is its fecond reprefentative, or its repre-~ {entative in Briggs’s fyftem of logarithms,) and the number thereby obtained, I ——.— —.———,, OF 0.041 24+158,857,928,096,6 reprefentative of the ratio of 11 to 10, or the logarithm of the faid ratio in Briggs’s fyftem. And the like obfervation may be extended to the logarithms of any other fy ftem, in which 1 is not the logarithm of the ratio of io to 1, but of fome other ratio. Our bufinefs therefore on the prefent occafion is to find a method of inveftigating the proportion between any. given ratio (as, for example, the ratio of 11 to '10,) and the ratio of 10 to 1, which. 4is the fundamental, or ftandard, ratio in Briggs’s fyftem of logarithms, or that. to ‘which all other ratios are, in that fyftem, compared and referred. Or, if any other fyftem of logarithms were to be computed, it would be neceffary to find a method of inveftigating the proportion between any given ratio and the ratio of which i was the logarithm in that fyftem. But, as no fyftem of logarithms befides that of Briggs is now in ufe, it is needlefs to think of any other on this occafion. 2. Now in order to find the proportion between the ftandard ratio of 10 to 1 and any other ratio by the method which we are now going to explain, it will be moft convenient (though not abfolutely neceflary) to fuppofe the faid other ratio to be lefs than the ratio of 2 to 1, and indeed, for the moft part, _to be a great deal lefs than that ratio, and not to exceed the ratio of 11 to10 or that of 10 tog. But this will be fufficient for the purpofe of computing the logarithms of all forts of ratios, how great foever, becaufe the greateft ratios are compounded of, and may be divided into, a number of other leffer ratios, of which fome fhall be exa&t multiples of the ratio of 10 to 1, and others fhall be fmall ratios of the magnitudes that have been juft defcribed. Thus, for example, the ratio of 1788 to 1 is compounded of the ratio of 1788 to 1780, that of 1780 to 178, that of 178 to 89, that of 89 to 80, that of 80 to 8, and that of 8 to 1; of which the firft is a very {mall ratio, the fecond is equal to the ratio of 10 to 1, the third is equal to the ratio of 2 to s, the sd a mall to wit, »392,685,158,228,29 &c, will be the new OF “DR. “EDMUND HALLEY UPON LOGARITHMS. 125 finall ratio, very little greater than that of 88 to 80, or rr to 10, the fifth is equal to that of 10 to-1, and the fixth is equal to three times the ratio of 2 to 1. Therefore the logarithm of the ratio of 1788 to 1 is equal to the fum of the logarithms of the ratios of 1788 to 1780, or of 447 to 445, and of 10 to 1, and of 2to1, and of 8&9 to 80, and of 10 to 1, and of three times the ratio of 2 788 : 8 to 1; or Log. = will be = Log. — + Log. — + Log. -- + Log. = 8 + Log. =" Pos Log. = = Log. + Log. =? + 2 Log. = + 4 Log. 447 445 ++ 2 +4 Log. —. And confequently, when the logarithm of the ratio of 8 3 = Log. B+ Log. a +2 x 1 + 4 Log. — = Log, + Log. -% 2to1, and thofe of the {mall ratios of 447 to 445, and of 89 to 80, have ‘been computed, that of the ratio of 1788 to 1 may be derived from them by _mere addition. . And, in like manner, the logarithm of any other great. ratio may be obtained by adding together the logarithms of the fmaller ratios of which it is compounded. It is not therefore neceffary, in the prefent invefti- gation, to take into our confideration any other ratios but fuch {mall ones as have been mentioned, to wit, thofe of 10 to 9, and 11 to 10, or others of a- ftill {maller magnitude.. 3. Now,.as the reafonings'by which we may find the proportion of any one {mall ratio, (fuch as thofe of 10 to 9 and of 11 to 10,) to the ftandard ratio of 10 to 1, are exactly the fame as thofe by which we may find the proportion of any’ other fuch {mall ratio to the fame ratio of 10 to 1 ;—and, as particular examples. are fimpler and eafier to underftand than general problems, I fhall confine myfelf, in the courfe of the following pages, to the inveftigation of the pro-- portion which a fingle particular fmall.ratio, namely, the ratio of 11 to Io, bears to the faid ftandard ratio.. This therefore will be the fubject of the fol-- lowing Problem.. PROBLE M.. 4: To find in numbers the proportion between the ratio of 11 to 10 and the. greater ratio of 10 to 1. 5.0.L,.U.T 1 O.N.. . ° . b F% 10° 10 I 10 The ratio of 11 to 10:is equal to the ratio of — to —, or of 19? to 22, or 10 10 19 10- "a I P ° . of 1 + — tor. Weare therefore to find in numbers the proportion between the ratio of 1 + = to 1 and the ratio of 1oto1.. Let this proportion be that of 1 to the unknown: quantity x. > ° . I ° Then, fince the ratio of 10 to.1 is to the ratio of 1 + ~5 tO 1 as # is to I,, and, from the nature of the powers of quantities, (which are only continual - proportionals to unity and the quantities themfelves, of which they are-faid e 126 AN APPENDIX TO THE FOREGOING TRACT: be powers,) the ratio of 1 4 an to 1 is tothe ratio of 1 -- « to x in the fame proportion of «to 1, it follows that the ratio of 1 + =) to 1 will be equal to the ratio of 10 to 1; and confequently 1 + a will be = 10. We mutt therefore endeavour to find the value of the index » in this equation 1 + = " — 10. . « ° wv. ° | s. Now, by Sir Ifaac Newton’s binomial theorem, 1 ++ +| is = the feries 1 te edeextzix daex Sst x SS? ec ee eee Sire igi aigeragl ashesrsresta Bh aBeast te Remmi sire se Bint PUG Y ae Mabeas i e ES eI TRA IE ape Ce + & m1 + S++ ORE + CEES + cE + ‘96 S48 ey4— ats oe P : ST TRS apa been Ok 4 &c. Therefore this laft feries 1 4. — 720,000,000 a PAP sseraia « is 190 xm w3—3xxex , x4—6x3+ Liar — pS me 35+ — Soxr-+ 24% a 200 i 6000 a 249,090 ; + 12,000,000 = Ser cy 4 — ° AUm gat + Sst a 2asatboqgee— 120" 1 ge will be = 103; and confequently (fub- 720,000,000 . tracting 1 from both fides) the feries =. + = + ee Xo —10A4 3 G43 — Sone 24e , xo — 15x84 85x4—22543427444— 120K . i 12,Q00;000 + : 7 20,000,000 + &c will be == 9. This equation we muft now endeavour to refolve. 6. Now the true value of « in this equation, fo far as it can be expreffed by eighteen decimal figures, is 24.158,857,928,096,805,5; as may be collected from Mr. Abraham Sharp’s computation of Briggs’s logarithm of 11 to 61 places of figures. For the firft twenty-two figures of that logarithm are 1.041,392,68 5, 158,225,040,750; that is, the firft twenty-two figures of the logarithm of the ratio of 11 to1 in Briggs’s fyftem are 1.041,392,685,158,225,040,750; and confequently the logarithm of the ratio of 11 to 10 will be equal to the excefs af 1.041,392,685,158,225,040,750 above the logarithm of the ratio of 10 to 1, that as, above 1, or willbe = 0.041,392,685,158,225,040,750. There- fore the prapertion of the ratio of 11 to 10 to the ratio of 10 to 1, is that of 0.043, 392,085,158,225,040,750 to 1, and confequently is equal to that of 1 to ‘I .000,000,000,000,000,000,000,000 dent 4 a Se ey ory pero eee 58,857,928,096,805,5 &c. This therefore is the true value of x in this problem; which I have here fet down before-hand, to the end that we may fee, in the courfe of the following inveftigation of it, to what degree of exactnefs every new ftep in our gradual approaches towards it,’ will exhibit it. .. : ’ I x aX ~—%¥ ak — 300+ 27 7- To find the value of x» in this equation sare Th WE re A ge ¥ M4 6x9 $1 Ta — O« a5 — 1014+ 4543 —coxx +244 ip xo — 5a%at 8544 — 22543427477 — 1208 240,090 42,000,009 720,000,000 OF DRs EDMUND HALLEY UPON LOGARITHMS, 127 + &c = 9, we mutt take a few of the firft terms of the feries which forms the left-hand fide of the equation, and {uppofe them to be equal to the whole feries, and confequently to the abfolute term 9, and then refolve the finite equations refulting from fuch fuppofitions. And it is evident that, in this way of pro- ceeding, the more terms of the feries we retain, the nearer will the value of x thereby obtained approach to its true value; but that it muft always be fome- what greater than the faid true value, on account of the fubfequent terms of the feries which have not been retained. The values of » arifing from the retention of the firft term only, the two firft terms, the three firft terms, and the four firft terms, will be as follows. : 8. If we fuppofe the firft term — alone to be equal to the. whole feries, and confequently to 9, we fhall have x = 10 X 9 = go. This value of » is more than triple of its true value, which is 24.158,857, &c. g. If we fuppofe the two firft terms = + == 200 feries, and confequently to 9, we fhall have = + —— to be equal to the whole ‘-— =~ a: ——} = 200 200- 9, or 925 = 9, and confequently 19 ¥ + «* (= 200 X 9) = 1800. Therefore 5 + 19% + xx willbe (= = + 1800 == + ea — 758! ry. 86.95 2 and confequently 2 + * will be (= Sa, and x willbe (= - 2 = erat =. 33.97. This fecond value of x is much nearer to its true value than the former was, but is ftill a good deal too large. 10. L.et us then, in the third place, fuppofe the three terms = + i pote to be equal to the whole feries, and confequently to the abfolute fofote) term g, and inveftigate the value of « refulting from this fuppofition. | Fe el 200 x et * 23 — 34x-+ 24 _ S00% 30x% — 30% PK Lie en Now To 1h 200 + 6000 ae 6000 + 6000 + 6000 6724+ 2744-443 ae ; a . Therefore sree tite will, upon this fuppofition, be = 9, and confequently 572%: 27x« + x3 will be (=.6000 X 9) = 54,000. We muft therefore endeavour to refolve this cubick -equation. Now, if we fuppofe * tobe equal to 30, and fubftitute 30 inftead of x in the compound quantity 572” +-27%% +- x°, (which: forms: the left-hand fide of the equation 572” 4+-27xx +«3'= 54,000,) we fhall have xv = goo, and «7 = 27,000, and confequently 27% (= 27 X 900) = 24,300, and 572” (= 572 X 30) = 17160, and 572K" + 27K" + x? (= 17160 + 24,300 + 27,000) = 68,460; which is greater than :the abfolute term, 54,000, of the cubick equation 572% -- 27x% 4- x?'= 54,000. Therefore 30 muft be greater than the true value of x in that equation. Let us therefore fuppofe the faid true value to be 30 — 2; and let 30 —z be fubftituted inftead of » in the terms of the faid equation, but with an omiffion of all the quantities that would involve eit7her zz or z*. And we fhall then have i ty oo AN, APPENDIX. 7T0 THE FOREGOING TRAC Tra xx (= 30 — zl? = 30!? —~ 2 X 30 X z + &c) = 900 — 60 X z + &C, “, randisa(— 30 — zi} = 3018 — 3 x 30)" xX 2+ &c is ee SE = 30° — 3 X 900 X z + &c) i) = 27000 — 2700 X z + &c, and confequently 27%" (= 27 x 900 — 60 X z+ &c = 27 X 900 — 27 X 60 Xz + &c) = 24,300 — 1620 X z+ &c, and 572% (= 572 X 30 — z= 572 X 30 — 5972 X 2) = 17160 — 572 X 2, and §72¥ + 27%% + x? = 17,160 — 572 X2 + 24,300 — 1620 xX z + &c + 27,000 — 2700 Xz4+ &c= 68,460 — 4892 x z +-.&c. Therefore this laft quantity 68,460 — 4892 x z + &c, will be = 54,0003; and confequently, (adding 4892 x 2 to both fides) we fhall have 68,460 = 54,000 + 4892 x z, and (fubtracting 54,000 from ‘both fides,) 4892 x z = 14,460, and, laftly, (dividing both fides by 4892) z = 2.95. Therefore x, or 36 — z, will be (= 30 — 2:95) = 27.053 thats, the root of the cubick equation 572% 4-274" +- x3 = 54,000 will be = 27.05, or (neglecting the decimal fraction .o5) = 27. Therefore 27 is the third aX — x : 200 + ae be - ans = 4 &e = 9; and it is confiderably nearer to the truth than the foregoing, -or fecond, approximation, 33.95, which was derived from only the two firft terms of the feries. 11. And if, in the fourth place, we fuppofe the four firft terms of the feries, tH — x x? — 3xe + 2% xt — 603 + rive — 6% 200 a: 6000 240,000 whole feries, and confequently to the abfolute term 9, we fhall have another approximation to the value of *, which will come within lefs than a 25th part’ approximation to the value of « in the original equation ~ + . xv to wit, — + , to be equal to the : ; * XH x3 — 34K 4 aR of its true value. For we fhall then have ate Epy th Sten + a4 — 603+ 1144 — Ge 9 g72 4 + 27 Kee + x3 it. at+— 623 + 114% — 64 240,000 SES BE: 6000 240,000 40 X $724 + 40 X 274% + 4023 4 x4 — 623 + 11ee— 6% 22880% + 1080%x 4+ 404% 240,000 240,000 i’ 240,000 ; 4 6 3 ip Be 3 4 : 4 x e+ tiawe — Ox ey 22,874 * + 109! ¥.4 + 34.2% + x »and confequently < 240,000 : 240,000 - | 4 a” UU Be dd Lr ha will be = 9; and 225874 x IOgI xx - 34x83. 240,000 ; + x«* will Se (= 240,000 X 9) = 2,160,000. - We. mutt therefore now en- deavour to refolve the biquadratick equation 22,8744 + 1091 xv + 34«3 + x* 2,160,000. . Peaess's 12. Now the root of this equation muft be lefs than 27. For, if we fuppofé x to be equal to-27, we fhall have x* = 729, and ¥? = 19,683, and «* = | | 531,442, OF DR. EDMUND HALLEY UPON LOGARITHMS. 129 $31,441, and confequently 22874. (= 22874 x 27) = 617,598, and 1ogixx (= 109% X 729) = 7951539, and 34”* (= 34 X 19683) = 669,222, and 22,8746" + 1091 xv 4+ 3443 + x* (= 617,598 + 795,539 + 669,222 + $31,441) = 2,613,800; which is greater than 2,160,000, or the abfolute term of the equation 22874” + 1og1xx + 34x? ++ x* = 2,160,000; and con- fequently 27 is greater than the true value of x in that equation. 13. Let us therefore fuppofe the true value of » in this equation to be 27 —2z. And let 27 — z be fubftituted in its terms inftead of x, but with an omiffion of all the quantities that fhall involve either zz, 2°, or 2*. And we fhall then have XN ee RE ea) oe X 27 x 2"&c), = . 729 — 54 xX z+ &c, and.*? (= 27 — 23 = 29\3,— 3 x 27) x z+ &e 273 — 3X 729 X z+ & = 27 — 2187 x z+ &c) 19683 — 2187 x z+ &c, and x4 (= 27 — 2 = 27 — 4 x 2713? x x + &e ave— 4x 19683 X z+ &C = 27 — 78,732 XZ + &c) 531,441 — 78,732 X 2+ &e, 22874. X 27 — 2% = 22,874 X 27 — 22,874 X 2) 617,598 — 22,874 2, Iogl X 729 — 54 X = &c 1091 X 729 — 1091 X 54 XZ + &c) 7959539 — 58,914 X z+ &C, and 22874% ( and 1ogixx ( We TP = 669,222 — 74,358 xX = + &c, and confequently 22874¥ + 1091x™ -- 34x° +4 x* = 617,598 — 22,874 x z + 9539 — 58,914 X 2+ &e £: ks 4 ae Pa jae ne la = 2,613,800 — 234,878 Xz + &e. + 531,441 — 78,732 X % + &c . Therefore this laft quantity 2,613,800 —°234,878 x z + &c, will be equal to the abfolute term 2,160,000; and confequently (adding 234,878 x z to both fides,) 2,613,800 will be = 2,160,000 + 234,878 x z, and (fubtracting 2,160,000 from both fides,) 234,878 x z will be = 453,800, and confequently z will be = ae = 1.93. Therefore x, or 27 — 2, will be (= 27 — 1.93) = 25.07, that is, the root of the biquadratick equation 22,874x +- 10gIK™ ++ 34”3 + x* = 2,160,000 is = 25.07; which is therefore a fourth approxima- 4 L oe : x xn —~ *% x3 — 34K + 24 tion to the true value of » in the original equation hice -+- car xt — 6x3 + 11x% — 6x oN * 240,000 + &c = 9. Vo.. Il. S 14. As 330 AN APPENDIX (TO WHE (FOR BiG OTNIG: PRACT 14. As this fourth approximation, 25.07,°to the value of » in the ‘orginal equation agrees with the third approximation to it, 27.05, in the firlt, or higheft, figure 2, we might conclude (if we did not already know the true value of «,) that the firft figure of the true value of » in the faid original equation muft be 2, though, perhaps, the fecond figure of it may be lefs than 5. And, further; as the difference between the third’ and fourth approximations, 27.05 and 25.07, is lefs than 2, whereas the difference between the fecond and third approxima- tions, 33.97 and 27.05, 1s more than 6, we have reafon to fuppofe, (without taking the pains to try it,) that, if we were to take in another term of the feries Fd ne — x x — 3xK + 2H) f, ; eich Sete C fuppofe th firft spate st aie ++ &c, and fupp e five firft terms of it, to : x xi xi — 34x + 20 «4 —623+ ive — 6% x5 — 10x4-+ 3643 — cove + 244 wit, — + + ———— ft ) 10 200 6000 240,000 12,000,000 to be equal to the whole feries, and confequently to 9, and were to refolve the equation of the fifth order thence refulting, the difference between the value of x that would be thereby obtained, and the laft, or fourth value of it, 25.07, would be lefs than 1, and confequently that the faid fifth approximation to the value of # would be greater than 24. We might therefore conclude, without | refolving any more equations for the purpofe, that the true value of # in the XY — Kw faid original equation — + + &c = 9g, or of the index » in the 200 C x > . equation 1 + pal == 10, 1s greater than 24, but lefs than 25; and this conclu- fion we fhal]l now proceed to verify by raifing the faid binomial quantity 1 + = to the 24th and 25th powers. 15. Now 1 + = is = 1.1; of which the fquare is 1.21, and the fourth power is (= 1.21") = 1.4641, and the eighth power is (= 1.4641\*) = 2.143, 588,81, and the fixteenth power is (= 2.143,588,81\") = 4.594,972,986,357, 216,1. Therefore 1.124 is (= 1.1)° x 1.1% = 4-594,972,986,357,216,1 X 2.143,588,81) = 9.849,732,675,807,611,094,711,841, and 1.11 is Cx T.1\* X I. = 9.849,732,675,807,611,094,711,841 X 1-1) = _10.834,705,943, 388,372,204,183,025,1. It appears therefore that wae ori + =" (which is {uppofed to be = 10,) is greater than 1.1)", or 1 + 2B but lefs than 7,1)*5, 1 \?5 ori -+ =| , and nearer to the former than to the latter. Therefore the index « is greater than 24, but lefs than 25, as we had concluded it to be in confe- quence of the foregoing inveftigations. We therefore now know for certain that the two-firft, or higheft, figures of the true value of the index x in the 5 x equation I + ~ =) 1o\are:.24, 16, We might now proceed to inveftigate the value of » to a greater degree of exactnefs by taking in five, or fix, or feven, or more, terms of the feries = 4. *#=% 10 200 OF DR. EDMUND HALLEY UPON LOGARITHMS, 131) x3 — 30x + 24 a 6000 confequently to 9, and refolving the equations refulting from fuch fuppofitions. But the computations neceflary to thefe refolutions oe be found exceffively la- KX — ¥ H3 — Bau + 2x Ar ~~ 6000 + &c = g, and thall proceed to obtain, in its fread, another equation, of which the root, or unknown quantity, fhall be a much fmaller quantity than x, and fhall even be lefs than 1, and in which the powers of the unknown quantity fhall confequently form a decreafing progreffion, and therefore a few of the firft terms of the feries will approach much nearer to an equality with the whole feries than in the former equation, and confequently our approaches to the true value of the faid unknown quantity by retaining firft one term, then two terms, then three terms, and, laftly, four terms, of the feries, and fuppofing them to be equal to the whole feries, and refolving the equations refulting from thefe fuppofitions, will be much {wifter than our approaches to the value of x in the foregoing operations. This may be done in the manner following. 17. Since # is greater than 24, let us fuppofe it to be = 24 +z. Then will 1 + =\"4 lige be-== 17+ =l", and confequently = tro. But 1 + - taal is = 1 + = 214 xr-al.. Hiicboiis I + +\" x I ey will be = ro. But we have feen that 1 + avi is = 9.849,732,675,807,611,094,711, 841. + &c, and fuppofing them to be equal to the whole feries, and borious. We fhall therefore lay afide the equation — Sa Therefore 9.849,732,675,807,611,094,711,841 X I +=) iS) a Opsnianel confequently 1+ a ¥ mite (= 10.000,000,000,000,000,000,000,000, aie 10 9-84.9,732675,807,611,094,711,841 1.015,255,979,947,706,347,941, &c. We mutt therefore endeavour to find . . . . z the value of the index z in this new equation 1 + =| mee 4-01 55255,070.047, 706,347,941, &c. 18. ‘alt ie the binomial oe r+ ss is = the feries 1 +z x —~ Io 2— 2 I Z—2 100 3 1000 rs rid PH & ZB— &B ia scoala a ee ae pad OTS f c =r ad deci eis etoile Mer meme SS 3 x 10000 + Wee = 1 6000 += 240,000 + &c, or (to exprefs the tay mae honealy, becaufe z is, in this CALC, lefs than 1, and confequently zz is lefs than z, and z+ than z',) 1 + — — (2 — = 2% — 32% + 23 __ (6% — 112% + 623 — 24 Rint: Jane ene ee + &c. Therefore this aur I +. % f—x 2% — 32% + 23 6z — 11zz + 623 — x4 ; ae Sue ech Sone Se + &c will be = 1.015,255, 97929472796,347,941, &c; and pontcauen uy (fubtracting 1 from both fides,) S 2 the 132 AN APPENDIX T.O.. THE FOREGOING TRACT { Zi z — % + 33 6% — 11zz + 623 — xt ‘ the feries —— — wadN oer de —_ {Gas cringe Get + &c, will be = 6000.” : 240,000 0.015,25 ea aii be ae 9415 &c. This equation we muft now endeavour to refolve. xX — x xo me 3x0 + 24 19. Now here, as in the former equation = dsttopaae ds oo + v3 Any eS lo 2981 dosha + &c = 9, we may make continual approaches to the 240,000 true value of the root by taking in more and more terms of the feries, and fuppofing them to be equal to the whole feries, and confequently to the abfo- lute term 0.015,255,979,947,7069,347,941, &c, and then refolving the equa- tions that will refult from thefe fuppofitions. This may be done in the manner following. 20. If we fuppofe the firft term, =, of the feries to be alone equal to the whole feries, and confequently to 0.015,255,979,9, &c, we fhall have z = 10 X 0.015,255,979,9, &C = 0.152,559,799, &c; of which the two firft figures .15 are exact. For, if z be = 0.152,559,799, occ, we fhall have x (= 24° 4 Zz) = met 152,559,799, &c, which agrees with the more exact value of x; to wit, 24.158,857,928,096,805,5, in the four higheft figures. This is no incon- fiderable approximation to the true value of 2, “and is obtained with hardly any trouble. 21. This firft approximation 0.152, &c, to the value of zin the equation ~ | i op 28 ee ee = oon 6000. 240,000 &c, is lefs than its true value, becaufe the firft term of the feries, — BF. is not lefs than the whole feries, (as was the cafe with — the firft term of the Pen feries x HY — # #3 — 3xu + 2% «+ — 643 + rine — 6% & X —=—_ eo Cc bu 1S OT 10 peste 6000 1 240,000 + &C,) but is greater. than the whole feries, on areooae of the diminution of its value by the fubtra¢ction of —— the following term — And for a like reafon the value of z derived from the two firft terms Bf the feries will be greater than the truth, and the value of it derived from the three firft terms will be lefs than the truth, and the value of it derived from the four firft terms will be greater than the truth, and fo on alternately, becaufe any odd number of the terms will be greater than the whole feries, and any even number of them will be lefs: all which is the con- fequence of 2’s being lefs than unity. 22. In the fecond place, let us fuppofe the two firft terms df the feries, to wit, — — ——, to be equal to the whole feries, and confequently to 0.015, 255,979,9 &c. We fhall then have = — = ou — cia (abe —=) = medio >! 2553979,9 &C, or tS are 9 bd delta. &c, and confequently 192 + 74 rae Tale ith Of oF +01 51255197999 &c) = 3.051,195,800, &c, and v + 192 a Zz OF DR. EDMUND HALLEY UPON LOGARITHMS., 133 7 6 ri. ‘ + iz20(= oh -- 3.051,195,800 = a ae LASSE Byi382 1) 4 peedo-s7 831200, _. 373+204,783,200 973,200, &Sc 4 4 ,and confequently = +2(= EE ee 15.318 gy: Ss 10.9 IGE _ 0.318 as ee, and'z (= oo _ “2 fm aS, = 0.1592. Therefore 0.1592 is the fecond approximation to the value of z in the equation Bed at — 2% 190 200 2% — 32% + 23 6z.— 112% + 623 — 24 oad Pete tt WSTiideoo + &c = 0-01552555979,9473700,347, 941, &c, or in the original equation 1 + =I. = 1.015,255,979,947,706,347, 941, &c. ! OL Essie 23. In the third place let us fuppofe the three firft terms of the foregoing . . z Bm ZZ 2% — 32% +- x3 feries, to wit, the three terms Foyt eric — = to be equal to the whole feries, and confequently to the abfolute term 0.015,25 529793947,706, 347;941, &c. , We fhall then have eee Sere Te, bach Mat pay lid cota a ate it 6000 : 6000 6000s” * 570% + 302% 2% — 32% + 23 572% + 27ze ze sé if 6000 Pk 6000 Lue 6000 » = 0-0155255,979,947,706, 347,941, &c, and confequently 572% + 27zz -- z3 (= 6000 x 0.01,5,2 555 979:947706,347:941, &C) = 91.535,879,686,238,087,646, &c. We mutt therefore now refolve this cubick equation, 572% + 272% -+ 2* = 91.535,879, 686,238,087,646, &c. Now, if we fubftitute 0.1592, or the value of z derived from the two firft terms of the feries, (and which we know to be not very different from the true value of z in this cubick equation,) inftead of z in the compound quantity 572% + 2722 + 2° (which forms the left-hand fide of this equation) we fhail have Be (2 '0.t 592 |") 1=2 0.02.53344,64, and 2? (= 0.1592)*°) = 0.004,034,866,688, and 272% (== 27 X 0.025,344,64) = 0.684,305,28, and 672% (= 572 X 0.1§92) = 91.0624, and confequently 5722 + 2722 + 2? = g1.062,4 +- 0.684,305,28 + 0.004,034,866,688 = 91.750,740,146,688 ; which is greater than 91.535,879,686,238,087,646, &c, or the abfolute term of the cubick equation 572% 4 2722 -+ 2? = 91.§35,879,686,238,087,646, &c. Therefore 0.1592 muft be greater than the true value of z in that equation. Let us therefore fuppofe the value of z in that equation to be 0.1592 — v. | Then we fhall have zz (== 0.1592 — v)’ = 0.1592)" — 2 X 0.1592 X V+ & = 0.1592)" — 0.3184 x v + &c) = 0.025,344,64 — 0.3184 x v + &c, anc 134. AN APPENDIX TO THE FOREGOING TRACT) and z? (='0.1592 — v}’ = 0.159213 — 3 X 0.1592" x v + &e = 0.1592|? — 3 X 0.025,344,64 X UV + &c = 0.1592)? — 0.076,033,92 X v + &c) = 0.004,034,866,688 — 0.076,033,92 x v + &c, and confequently 5722 (= 572 X 0.1592 — Uv = 572 X 0.1592 — §72 x V) =I Q1.0024 159.23: Vs and 272z'(== 27 X 0.025,344,64 — 0.3184 X v + &c = 27 X 0.025,344,64 — 27 x 0.3184 x v + &c) = 0.684,305,28 — 8.5968 X v + &c, and 5722 2722 +2? = 91.062,4 — 572 XU + 0.684,305,28 — 8.5968 x v + &c ++ 0,004,034,866,688 — 0.076,033,92 X v + &c = 91.-750,740,146,6838 — 580.672,833,92 X v + &c. Therefore this laft quantity 91-7505740,146,688 — 580.672,833,92 x v + &c will be equal to the abfolute term 91.535,879,686,238,087,046, &c; and confequently (adding 580.672,833,92 X wv to both fides,) 91.750,740,146,688 will be = 91.535,879, 686,238,087,646, &c + §80.672,833,92 xX v, and (fubtra&ing 91.535,879, 686,238,087,646, &c from both fides,) 580.672,833,92 x v will be = 0.214, 860,460,449,912,354, and confequently z will be (= 0:21 pBO0 Aah eee ; 580.672,833,92 = 0.000,370. Therefore z, or 0.1592 —v, will be (= 0.1592 — 0.000, 370) = 0.15883. Therefore 0.15883 is a third approximation to the true . . z (3 — 2x 2% — 32% + 23 6% — 112% + 623 — x4 value of z in the equation rea perg rr Jecle eeaigiai aceaae Ic sy oc + 200 6000 240,000 &c — 0.01 5525 5,979,9475706,347,941, &c, or in the original equation 1 ++ =f = 1.015,255,979,947,706,347,941, &c. And it is a pretty near approxima- tion to it, the four firft figures of it, 0.1588, being exact. For, if we fuppofe z to be = 0.15883, we fhall have x, or 24 + %, = 24.15883, of which the firft fix figures 24.1588 are true, the more accurate value of x being (as has been already obferved,) 24.158,857,928,096,805,5. 24. In oe to obtain a ftill nearer approximation to the value of z in the z— Vw 2B — 32% + 23 6z — 112% + 623 — 24 equation —— —\—— A ea Glaiea 3% —— + &c = 0.015, 2.555979, mae Wt 941, &c, let us fuppofe the four firft terms of the faid feries to be equal to the whole feries, and confequently to the abfolute term 0.015,2553,97939475705,347,:941, &c, and refolve the biquadratick equation that will refult from that fuppofition. { — _ x3 anit —1 6z3 — zt The four terms —~ pete sto meted once Boer eS ee ne equal 200 ~ 6000 240,000 Pag7a8 cr! + 272% + 23 _(o% — 112% + 623 — 24 40 X 572% + 40 X 27zz + 4023 6000 240,000 -i 240,000 a __ 22880z + 1080zz ++ 40z3 (6% — 1122 + 623 — 24 oy 240,000 240,000 240,000 OF DR. EDMUND HALLEY UPON LOGARITHMS, 135 ga7ae Oka ch 34h) Therefore this lait quantity Sremeaebeeg tae 348! Ju ; 240,000 240,000 Is = 0.015,255,979,9475706,347,941, &c; and confequently 228742 4 1091 2% -+- 342° + 2* Is (= 240,000 X 0.015,255,979,947,706,347,941, &c) = 3661.435,187,449,523,505,840,000, &c. We mutt therefore now endeavour to refolve this biquadratick equation 228742 + 10912% + 342? + 2* = 366r. 43 59187,449,523,505,840,000, &c. 25. Now we know that the root of this equation cannot differ much from the root of the laft cubick equation 5722 + 27zz + 2) = 91.535,879,686, 238,087,646, &c, which was found to be 0.15883. We will therefore fubfti- tute 0.15883 inftead of z in the compound quantity 22874z + 1091zz + 3423 +-2* (which forms the left-hand fide of the faid biquadratick equation, ) in order to fee whether the value of the faid compound quantity refulting from fuch fubftitution will be greater, or lefs, than the abfolute term, 3661.435,187,4.49, 523,505,840,000, &c, of the faid biquadratick equation, and confequently whether 0.15883 is greater, or lefs, than the root of the faid equation. Now, if z be fuppofed to be = 0.15883, we fhall have zz = 0.025,226, 968,9, and z* = 0.004,006,799,470,387, and z* = 0.000,636,399,959,881, 567,21, and 22,874z (= 22,874 X 0.15883) = 3633.077,42, and 10912% (== 1091 X 0.025,226,968,9) = 27.522,623,069,9, and 3427 (= 34 X 0.004,006,799,470,387) = 0.136,231,181,993,1585 and confequently 22,874z +- 109122 + 342% -+- z4 = 3633-07742 + 27-522,623,069,9 + 0.136,231,181,993,158: + 0.000,636,39.9,959,881,567,21 = 3660.736,910,651,853,039,567,213 which is lefs than 3661.435,187,449, 523,505,840,000, &c, or the abfolute term of the equation 22,874z + 10g1zz + 342° + z* = 3661.435,187,449,523,505,840,000, &c. Therefore 0.15883 is lefs than the true value of z in that equation. 26. Let us therefore fuppofe z to be = 0.15883 + w, and let this quantity be fubftituted inftead of z in the terms of the faid biquadratick equation, but with an omiffion of all the quantities that fhall be found to involve either w’, or w*, or w*, And we fhall then have Z% (= 0.15883 + wl) = 0.15883)" +2 X 0.15883 x w+ &c = 0.15883)?--+ 0.31766 x w+ &c) » = 0.02 5,226,968,9 + 0.31766 X w + &c, and 23 (= 0.15883 + w\? = 0.15883)? + 3 x 0.15883)" x w+ &e = 0.15883)? + 3 X 0.025,226,968,9 X w + &c = 0.15883]? + 0.075,680,906,7 X w -- &c) = 0.004,006,799,4.70,387 + 0.075,680,906,7 X w -- &c, and 136 AN APPENDIX TO THE FOREGOING TRACT and z* (= 0.15883 + wilt = 0.15883\* + 4 X 0.15883)) x w+ &e = 0.15883]|* + 4 X 0.004,006,799,470,387 X w + &c = 0.15883\" + 0.016,027,197,881,548 X w 4+ &c) = 0.000,636,399,959,881,567,21 + 0.016,027,197,881,548 x w+ &e, and 228742 (== 22,874 X.0.15883 + w . 22,874 X 0.15883 + 22,874 xX w) = 3633.077;42 -- 22,874 X w, and 109122 (= 1091 X 0.02 5,226,968,9 -+4- 1091 X 0.317,66 xX w) = 27.522,623,069,9 -- 346.567,06 X w+ &c and 3423 (= 34 X 0.004,006,799,470,387 + 34 X 0.075,080,906,7 X w + &c) = 0.136,231,181,993,158 + 2.573,150,827,38 xX w+ &c, and confequently 22,8742 -+ 109g12z + 34 2°+4+ 2* = 3633.077:42 + 22,874 xX w + 27.522,023,069,9 + 346.567,06 X w + &c fc 0.136,231,181,993,158 - 2.573,150,827,8 X w+ &c + 0.000,636,399,959,881,567,21 + 0.016,027,197,881,548: X w + &c = 3660.736,910,651,853,039,567,21 + 2.3522 3-1 56,238,025,081,548 XW + &c. Therefore this laft quantity 3660.7 36,910,651,853,039,567,21 + 23,223. 156,238,025,681,548 X w -+ &c will be = the abfolute term 3661.435,187,449, §23,505,840,000, &c; and confequently (fubtracting 3660.736,910,051,853, 039556721 from both fides, ) 23,223.156,238,025,681,548 X e tates 0.698, : -698,276,797,070,460,272, 2.76,797,670466,272,790, and therefore w will be = eee = 0.000,030,0. Therefore z, or 0.15883 + w, will be (= 0.15883 + 0.000, ©30,0) = 0.158,860,0, or 0.15886, and x, or 24 +4 2, will be = 24 + 0.15836, or 24.158,86; that is, the index » in the original equation 1 + a tog 24.15886. 27. This value of x is exact in the firft fix figures 24.1588, and differs only by an unit in the next figure from the true value of x, which is (as we have before obferved,) 24.158,857,928,096,805,5. | 28. The labour of refolving the laft equation was fo confiderable, and the progrefs made by it in our approach to the value of z was fo fmall, (the value of z obtained by it, to wit, 0.15886, being not more exact than the next pre- ceeding value of it, to wit, 0.15883, by fo much as one figure, but only ap- proaching nearer to the truth in the fifth, or laft, figure,) that it would be by no means expedient to feek for a nearer approximation tothe value of z by : . ° z — 22— - 23, ze 6z3 — x4 taking in more terms of the feries — Bl eto as orcas bd (ox eee 200 6000 240,000 + &c, (as, for example, five terms, or fix terms, or feven terms of it,) and fuppofing them to be equal to the whole feries, and confequently to the abfo- lute term 0.015,255,979,947,700,347:941, &c, and refolving the equations refulting from fuch fuppofitions, But this purpofe will be much better anfwered yi by | OF DR. EDMUND HALLEY UPON LOGARITHMS., 139 2B mm BUX +. 23 by dropping all further confideration of the equation — me re lM —— — 112% + 623 — x4 6z3 — z* a + &c = 0.015,255,979,947, fa is &c, and 240,000. deriving from the value of z already found, to wit, 24.15886, another equation in which the unknown quantity fhall be a quantity much lefs than 1B or 0.15886 ; x, — as we before dropped the confideration of the firft equation — + —— xP es Som 6 4 — 643 4+ Tine — 6 pereeeeect O° + placate eke Pra + &c = 9, in order to enter upon that 6000 240,000 4 Pa 2% — 2: x3 6z — r1zz% + 6z3 — xt of the fecond equation = ae nee dere nee ene. Pear ia oe Oe rea 10 200 6000 240,000 &c = 0.015,255,979;947,706,347,941, &c. This may be done by com- puting the value of the 24.15886th power of the binomial quantity 1 + “ by means of Sir Ifaac Newton’s binomial theorem, in order to difcover whether ote 3 ig the faid power is greater, or lefs, than 10, or 1 + ~ , and confequently whe- ther ‘the index 24.15886 is greater, or lefs, than the index #; after which we may fuppofe x to be equal either to 24.15886 — y, or 24.15886 + 9, (that 1 1S, to 24.15886 — y, if it is lefs than 24.15886, and to 24.15886 + y, if it is greater than 24.15886) and thereby obtain another equation of which y -fhall be the root. This may be done in the manner following. Ba 1\ 886 29. By the binomial theorem 1 + Mis Aer is = the feries 1 + 24.15886 2315886 EL a - + 241 $886 x a x as Y, -+- &c = (if the firft, eal third, fourth, and other following terms of the x hay. t 015886 , Paar 1000 feries be denoted by the capital letters A, B,C, D, &c, refpectively) 1 + 24.15886 23.15886 22.15886 2115886 20.15826 si Bs * aaa seas 885 On Paes Satits 19-15886 18.15 19.15 16.15886 . Ng “6X 10, bane 7X10 x ye 8.x 10 ESE 9 X 10 * ih 15.1588 14.158 13.1588 12.15886 18 & Zs x0 X 10 Bete yi outace “Serene 14X10 10.15886 9.15 15886 7.15886 me Re 1§ X10 led ox10 A px xX 10 ee NP a sere. we aS 5.15886 | 4.15 3.15886 2.15 19 X 10 Tetris VE ie * Wo Ss Kp Rte x Y 4 abi Nerty xX Z + misses xX A’ + &c ad infinitum 24 X10 25X10 <= 1.000,000,000,000,000,000, + 2.415,886,;000,000,000,000, + 2.797,458,282,498,000,000, + 2.066,282,881,257,121,076, “+ 1.093,004,755,122,9001,221, + 0.4.40,6745596,7571305979s Vot. II. $b sec @38 AN APPENDIX TO THE FOREGOING TRACT + 0.140,713,715,080,440,686, + 0.036,502,866,460, 365,873, + 0. 007,829,34430892901 2409s + 0.001,405,703,163,731,7825 a 0.000,21 3,088, 574,605,671, + 0.000,027,428,102,685,8295 ++ 0.000,003,007,688,027, 570s + 0,000,000,281,308,135,776, ++ 0,000,000,022,421,986,457,5 ae 0,000,000,001,518,545,475 -++ 0.000,000,000,086,925,908, + 0.000,000,000,004,175,519» -+ 0.000,000,000,000,166,066, -+ 0.000,000,000,000,005,383; ++ 0.000,000,000,000,000,1 38 -+ 0.000,000,000,000,000,0025 + &c mt 10.000,001,974,734,858,821, &c; which is fomewhat greater than 106 Therefore 24.15886 is a little greater than the index x in the equation 1 + cg —— it toe 30. In order to obtaim the value of » to a ftill greater degree of exattnefs, let us fuppofe y to be the difference by which 24.15886 exceeds it. Then will 1 te ; Dee be = 1+ te = it Os ue \2 15886— cy \ 243 886 Tee te Ea | 2401 886 Butr +2)" “t ism + a)" x1+— ee ; t bat a ei) 24015886 I ’ xX ===. Therefore 1 + xX === is = 10, and confe- f 10 15886 . \ quently 1 + ayes wis as 10..% 13 ae : "y | 15886 , But we have juft feen that 1 + = ae is = 10.000,001,97 457343858; 821, &c. Therefore 10.000,001,974,734,958,821, &c, is = 10 X I + z > and confequently 1 + a 16 = HOON Tb TSA ES eS = Ps 1.000,000;! 97> 473,485,882,1, &c. We muft therefore aad endeavour to find the value of the index y in the equation ee == 1.000,000,197,473:485,882,1, &c. 31. Now, ps the binomial theorem, 1 a 2) is = the feries 1 + y X = +y oe I = _ mins moo 7 Sl Tits seas) 26. goon cho he 2000 ne “9-9 HypLsawhs yo - 6) + ny ns 10,000 + F ier ot a6 200 6000. + 240,000 + ie = OF DRe EDMUND HALLEY VPON LOGARITHMS. 139 J oPaD 4 Br wt _ [yy — ry + ty" ; - h Toh BEE bag oe Tr ren ee. Fherefore, the x beatis 240,000 , 2 SU (eR | hg ty? (6 — tty -F by? — 94 y ferie +r 10 200 Te 6000 inode ws &e will be = 4°, T.000,000,197,473,485,882,1, &c ; and confequently (fubtracting 1 from both files ‘eg FEE: Dis BEY” », (by = tip + Or —y" ide ») the feries 10 200 ie 6000 240,000 + .&e will be = 0.000,000,197,473,485,882,1, &c, This equation we muft now endeavour to refolve. 32. Let us therefore, in the firft place, fuppofe the firft term 2 alone of the laft-mentioned feries to be equal to the whole feries, and confequently to the abfolute term 0.000,000,197)473,485,882,1, &c. Then will y be (orm cee ©.000,000,197, &cC) = 0.000,001,97, &c, and confequently «, or 24.15886 — y, will be (= 24.15886 — 0.000,001,97) == 24.158,858,03 ; of which the firft feven figures, 24.158,85, are exact: fo that this firft approximation to the value of y (which has been obtained by the refolution of an eafy fimple equa- tion,) enables us to determine the value of 24.1 5886 — y, or x, to one figure more than we had found before. 33. In the fecond place, let us fuppofe the two firft terms of the laft-men- tioned feries, to wit, the terms + —P—2 confequently to 0.000,000,197,4.73,485,882,1, &c, and then feek the value of y refulting from this fuppofition. , to be equal to the whole feries, and. We thall then have 2 —(2=2% — 2% [poy — M+ ang confequently JO 200 200 200 200 toy ty Soo = 0+000;000;1975473,485,882,1, &c, and 19y + yy (= 200 X 0.000, 000,197,473,485,882,1, &C) = 0.000,039,494,697,176,420,0, &c. There- fore S + roy + yy will be (= n + 0.000,039,494,697,176,420,0, &c = 361 4 4 X_0.000,039,494,697,176,420,0, & __ 361 4 ape ests 78: ZS SIERO 4 4 4 "50 361.000,157,978,788,705,680,0 | ear, Seekers Ol 7s ; and confequently “5 + y will be (= 4 af 391.000, 157:978,798,705,680,0, _. 19+000,004,157 19.000,004,15 7 2 here 2 2 i ae Saaschcetulh Yi see , and y will be (= =_ ons 0.000,002,078; and confequently *, or 24.15886 2 2 — y, will be (= 24.158,860,000 — 0.000,002,078) = 24.158,857,922; of which number the firft ten figures, 24.158,857,92, are exact, the more accu- 5 : : 1)* : rate value of the index x in the equation 1 + —! = 10 being (as we have already obferved,) 24.158,857,928,096,805,5. 34. Of thefe ten figures, 24.158,857,92, which are exact, the 7th figure 5 was obtained by the refolution of the fimple equation + = 0.000,000,197,4473s 10 485,882,1, &c, and the three next Ate 7192 have been obtained by the refolu- & {10 140 AN APPENDIX TO THE FOREGOING TRACT tion of the quadratick equation a _ 2 = 0.000,000,197,4.73,485;882,1, &c. And more figures of the value of « might in like manner be obtained by ee . — 2y— 3 retaining three, or four, or more terms of the feries 2 p= Blige Aa has 10. 200 6000 oi (Seed ek Oren + &c, and fuppofing them to be equal to the whole 240,000 { ? feries, and confequently to 0.000,000,1975473,485,882,1, &c, and refolving the equations refulting from fuch fuppofitions. But the labour of performing thefe refolutions would be very confiderable, and the number of new figures of the value of » which we fhould thereby obtain would be but {mall. For I have tried the two next equations, to wit, the cubick equation that refults from a fuppofition that the three firft terms of the faid feries are equal to the whole, and confequently to 0.000,000,1975473,485,882,1, &c, and the biquadratick equation that refults from a fuppofition that the four firft terms of the faid feries are equal to the fame quantity, and I have found that the root of fuch cubick equation has been = 0.000,002,071,400, and that the root of fuch biqua- dratick equation has been = 0.000,002,071,941,6, and confequently that the value of *, or 24.15886 —y, obtained by means of the faid cubick equation, has been (= 24.158,860,000,000 — 0.000,002,071,400) = 24.158,857,928, 600, of which the firft eleven figures, 24.158,857,928, are exact, and that the value of w, or 24.15886 — y, obtained by means of the faid biquadratick equa- tion, has been (= 24.158,860,000,000,0 — 0.000,002,071,941,6) = 24.158, 857,928,058,4, of which the firft twelve figures, 24.158,857,928,0, are exact. So that the refolution of thefe two equations gives us only two figures of the true value of x more than we had before obtained by means of the foregoing quadratick equation. In order therefore to obtain the value of x exact to a few more figures than the ten figures 24.158,857,92, which were obtained by the refolution of the foregoing quadratick equation, we will have recourfe to another method of proceeding, which will be lefs laborious than the refolution of either the cubick or the biquadratick equation that have been juft men- tioned, and, @ fortiori, lefs laborious than the refolution of any of the higher equations that would refult from the fuppofition that more than four terms of the foregoing feries were equal to 0.000,000,197,473,485,882,1, &c, and which. will give us four additional figures of the true value of « above the ten figures, 24.158,857,92, already obtained by means of the foregoing quadratick equation. ‘This method of proceeding is as follows : 35. Let all the quantities that form the left-hand fide of the equation = — f=», y-wts Gp thio a 2 DT ear Sopot yt -+- &¢ = 0,000,000,197,473,485, 882,1, &c, be ranged in feparate lines according to the feveral powers of y, thofe involving the fimple power of y being placed in the firft, or higheft line, and thofe involving yy, or the fquare of y, being placed in the fecond line, and thofe involving y%, or the cube of y, being placed in the third line, and thofe involving y*, or the fourth power of y, being placed in the fourth line, and fo on of the following powers of y. And we fhall then have OF DR. EDMUND HALLEY UPON LOGARITHMS. 14% 2 2 120 ge ee aie ely rth aie eos gator +, io 200 6000 240,000 12,000,000 720,000,000 ARs Aen Sn) 2 SOM ori t *"* 200 6000 240,000 12,000,000 720,000,000 iby een csc ees 0 a Ae Oa -" 6000 240,000 I 2,000,000 720,000,000 8. AE ey sik teat Es RR! 240,000 12,000,000 J 20,000,000 ni ucts aga OLS ll 12,000,000 420,000,009 6 a y 720,000,000 = 0.000,000,197,473,485,882,1, &c. But, becaufe y is an exceeding {mall quantity in comparifon of unity, (being I ore . ———., or ———.,) it is evident that all the powers TO000,000 §00,009 of y will be extremely {mall in comparifon of y itfelf, and confequently that all the quantities contained in the fecond, third, fourth, fifth, fixth, and other fol- lowing lines of the compound quantity that forms the left-hand fide of the foregoing equation, will be extremely fmall in comparifon of the quantities contained in the firft, or upper, line of it, which involve only the fimple power of ys We may therefore, without erring much from ‘the truth, confider the quantities contained in the upper line alone as being equal to all the quantities contained in alt the lines together, and confequently to the abfolute term 0.000, 000,197,473,485,882,1, &c; and then the foregoing equation will be con- = to about 0.000,002, or : a . 6 2 verted into the following one, to wit, = —_ — im - me ae —_— ’ 3 2 120y s J F tes di + &c = 0,000,000,197,473,485,882,1, &c, or = — == vat 2y by 4 24y 120 2X3xX1000 2X3X4X10,000 2xX3X4X5X 100,000 2.3.4.5.6 X 1000,000 + &c = 0.090,00051975473,485,882,1, Reno yyetopias pads 0 GANS AL Io 2 X 100 3 X 1000 [eee Pee J J Boe -= — lai = 4 X 10,000 + 5 X 109,000 6 X 1000,000 + &C = 0,000,000,1973473,485,882, f, &c, or the Intnne ferieee ees eee 3 é ee ‘ fo0 ax ie 3X 1000 4 X 10,000 + 1 I , Senin —— +- — ri 5 X 100,000 6 X 1000,000 + &c 0:000,000,1974.735485,882,1, &c, 0 ( RB; I I we denote the feveral terms —, ———, ———, ont — ass aa —_—— 190° 2 X 100 3 x 1000° 4 X 10,000 5 xX 100,000 I ° . . . ° Bodo ag? KE Of the foregoing infinite feries by the capital letters A, B,C, D, 3 E, F, &c, refpectively,) y x the infinite feries ~ _ —; CAL : — x B +e x C+ Sate iM red ~ = x E + &c = 0.000,000,197,473; 485,882,1, &c.. We mutt therefore, in the next place, compute the value of this infinite feries to as great a degree of exattnefs.as we fhall think neceflary ; which will not be very difficult, becaufe the faid feries evidently converges with a confiderable degree of fwiftnefs, every new.term of it being lefs than atenth part ee 142 AN APPENDIX TO THE FOREGOING TRACT. part of the foregoing term. And when this value, which we will call S, is obtained, we fhall have y x S = 0.000;000,197,473,485,882,1, &c, and con- 0.000,000,197, 85,882,1, &c fequently y = ee ATS I Nee de ee Bie . tine ¢ ° I The Computation of the infinite Series —— — 3 X 10 3 Xx 10 4X 10 4 s “fut x C+ ACMA ESET x E + &e ad infinitum. ° . . . | stn I so 3 36. NOW the infinite feries cesn brpedeare sr fis A + ae x B ee x Pig Lai ye a x E + &c may be computed in the following Bata Xx 10 manner. ° I Ais= <5 = 9.100,000,;000,000,000,000, ° I Bis = cares Oe A = 0.005,000,000,000,000,000, — 2 — C = sym X B = 0.000,333,33323332333333 as 3 aes D = Fie X © = 9-00050243999,99929993999s pat als res = Fx 10 % D = 0.000,001,999,9993999;999s - Fo; 3 — x E = 0.000,000,166,666,666,666, 6 é Gas Teens x F = 0.000,000,014,285,714,285, el ‘pd H= ss X G = 9.000,000,0012495999,999s 8 1 pe aie xX H = 0.000,000,000,111,111,11T, Gunes tee STIG —_019003000,000;009,99929993 10K TTI To =_01000;000,000,000,909;090, . iL M = 75 = 0:000,000,000,000,08 3,333, Ne oe: eee 6 — 13 X 10 ca 0.000,000,000,000,007, 25 TRON ce O TERT On 0.000,000,000,000,000,714, Les Oe re tee 0.000,000,000,000,000,066, PS (4 SB fal z Oi= Fis T_01000,000,000,000,000,006, bret) oy — ir erie 0.000,000,000,000,000,000. ThereforeA+ C+tE+G+I4+L+4+N+P4Rarez ©-.100,000,000,000,000,000, Hr 9 + 93332333233333323332 OF DR. EDMUND HALLEY UPON LOGARITEMS, 143 Po oseeegee dak NS GSO ete Hoses eee 96 14,285,714,285, “Trifesc a beet suse whidst lio Lit + . 26 90 © e920 © 99093090, oa SOE Ay TOR NN Pe 70 Prats. Wetgits ebm leiad of elgetes yd OOS erithictenate A aes a ett = 0.100,33523473731,075,576; andB+D+F++H+K+M+04Q aici = 0-00 5,000,000,000,000,000, ve ee 9° 245999399959993999> ee er #0100 666,666,666, ti3s pat 99 7929005 Peal oF oe Tae sakes 2 ee ee Sa ° 3° fee ae AT FEE Js ae i it Pa DENeP PR Oe De F—~H—K—M—O glee or, its equal, A—B+C—D+E—F+4+G—H+I—-K+L—~M+N—0O +P—-Q+R— &¢, will be = Q+10053 35534737 3130753576 — 0.005,025,167,926,750,716 afer 09533 1031 7938045324,860; ii Therefore the infinite feries. — rromy ber lbapcrctacd bette ts uit nix 5 ee 2 i xC r 2 3 tpt AP Toie XEt te wena 3X10 4X10 rie ghee aks &c, or the feries S, is = 0. PASSRE RES HE THRNC &Co. Q. E. I 37. Therefore y x the feries Sis = y X 0.095,310,179,8043324,860, &c3 and confequently y x 0.095,310,179,804,324,860, &c is = 0.000,000,19 754735 0.000,000;197,473,485,882,1, &C __ = 0.000,002,071,9035. 485,882,1, &c, and y is = 0.095,310,179,804,324,860, &c 403. Therefore x, or 24. 4.15886 — y, is = 24.158,860,000,000,000 — 0.000,002,071,903,403 = 24. 158; 857,928,096,597, or the ‘value of the index in the original equation: I +=! = 10 is 24.158,857,928,096,5973; of which number the firft 14 figures 24.158,857,928,096, are exact, the more accurate value of x being 24.158,857,928,096,805,5. 38. Therefore the proportion of the ratio of ro to r to the ratio of r1 to ip is that of the number 24.158,857,928,096,597 to. 1, or (becaufe 597 is near! ny = 600) that of 24.158,857,928,096,6 to I. Q: E. 39. Coroll, 144 AN APPENDIX TO THE FOREGOING TRACT. 39. Coroll. The proportion of 24.158,857,;928,096,6 to 1, is equal to that { of 1 to OF 0.041,392,085,158,228,29. Therefore, if 1 be 24-1 58,85 7,928,090,0" + ape taken for the reprefentative of the magnitude of the ratio of 10 to 1, (as it is in Briggs’s fyftem of logarithms,) 0.04.1,392,685,158,228,29 will be the reprefenta- tive of the leffer ratio of 11.to 10; or, in other words, the logarithm of the fmall ratio of 11 to 10 in Briggs’s fyftem of logarithms will be 0.041,392,685, 158,228,293 of which number the firft thirteen figures 0.041,392,68 5,158, 22, are exact, the more accurate value of this logarithm (as computed by Mr, Abraham Sharp,) being 0.041,392,685,158,225;040,7 50. 40. And in the fame manner we may compute Buiggs’s logarithms of the . I I ratios of to tog, or of r ow 1, and of 81 to 80, or of t + ao Or Es and I I of 121 to 120, or of 1 =s tot, and of 2401 to 2400, or of 1 + tof, 2400 and, in general, of m+. 1 to m, (m being any whole number whatfoever,) or of 1 + — to 1, by firft finding the values of the index » in the feveral equa- . \~ 1\* x I \"~ tions 1 + + ——= + xv were 3 pent . aa ee ee <* were equal to the whole feries, and confequently to 9, there refulted the cubick equation 572%” +4 27x" +- x* = 54,000, by the refolution of which we had x = 27.05. This therefore was the third approximation to the Vou. IJ, U value 146 AN APPENDIX TO THE FOREGOING TRACT . 42 : 1 4 2° ° value of x in the original equation 1 + —! = 10; and it is true in the firft, or higheft, figure 2, the more accurate value of w being 24.158,857,928,096, 805,65. 3 a E x xX — From the fourth fuppofition, to wit, that the four terms — 13; and confequently v = - — 1, ands (= S 24.158,857,928,096 + V) = 24.158,857,928,096 + = — 1; which will be S true to 27 or 28 places of figures. 55. Or, if it fhould be required to find the value of x to only a few figures more than thofe which were found in Art. 37, to wit, 24.158,857,928,096,597, (of which the firft 14 figures, 24.158,857,928,096, were exact,) it may be done without raifing the binomial quantity 1 -+ = to the 24.158,857,928,096th power, (which is a very laborious operation,) by retaining fome of the terms that involve the fquare of y, as well as thofe that involve the fimple power of : ARIE AAGULD Caine! 60 Whe! Paaal Ve dina aM calm Din aban) 2 y, in the laft equation =~ Ps OE eR &c = 0,000,000,197,473,485,882,1, &c, and fuppofing the terms fo retained to be OF DR. EDMUND HALLEY UPON LOGARITHMS. Ist be equal to the whole feries that forms the left-hand fide of the faid equation, and confequently to be equal to the abfolute term 0.000,000,197,473,48 5,882, 1, &c, and then refolving the quadratick equation which will refult from fuch {uppofition. This maybe done in the manner following. 56. If we retain only the terms that involve y and yy in the foregoing equa- tion, it will be converted into the following equation, to wit, Jy J ah pha) 24y 1209 10-200 6000 240,000 12,000,000 720,000,000 + &c Fay 5 agg AK yy Lil gory 2RUOP TOA Sol 200 6000 240,000 12,000,000 720,000,000 = 0,000,000,197,473,485,882,1, &c, es os J J J PA ! OF yo 209 a 3000 40,000 500,000 6000,000 | #0 SO bn AA dle Mala 2a ote oe LR Sy a?) SO RR oe 200 2000 240,000 240,000 7 29,000,000 = 0.000,000,197,473,485,882,1, &c, oF Yo nOo Ia, I a asent et? a 10 200 3000 40,000 500,000 6000,000 -- 0.005,000,000,000,000,000 X yy — 0.000, 500,000,000,000,000 X By + 0.00030453833,3 3353332333 WY — 0.000,004,166,666,666,666 x xy TH 9-000,0005380555 555553555 X I — 0.000,000,;035,000,000,000 X yy + &c == 0.000,000,197,473,485,882,1, &c; J Ps es J #4 M4 rie 10. 200 1 3000 yi 40,000 500,000 a2 6000,000 aF &e + 0.005,046,213,888,888,888 x yy — 0.000,504,201,666,666,666 x »y = 0.000,000,19734.73;485,882,1, &c BA Pd a "4 Ps J 4+ &c + &c or — — + —— — 10 200 3000 40,000 500,000 Ais 6,000,000 + 0.004,542,012,222,222,222 X Vy = 0.000,000,197,4.73,485,882,1, &c, or 0.09 5,310,179,804,324,860, &c. xX ¥ “+ 0.004,542,012,222,222,222 X yy = 0.000,000,197,473,485,882,1, &c, or (neglecting all but the four higheit figures of the co-efficient of yy, ) 0.095,310,179,804,324,860, & xX y + 0.004, 542 X WY = 0.000,000,197,473,485,882,1, &c. 57. This quadratick equation may be moft conveniently refolved by approxi- mation, by fubftituting, inftead of y, in the quantity 0.004,542 x jy the value of y derived from the fimple equation 0.095,310,179,804,324,860, &c X y = 0.000, 197,4.73,485,982,1, &c 0.09 5,3 10,179,804, 324,860, &c” ) 0,000,002,071,903,403. We fhall then have yy = 0.000,002,071,903,403]” = 0.000,000,000,004,292,783,711,362,980,409, and confequently 0.004,542 X WY == 0,004,542 X 0,000,000,000,004)292,783, &C = 0.000,000,000,000, o1g, ©.000,000,197,473,485,882,1, &c, which is = ( Ci 152 AN APPENDIX TO THE FOREGOING TRACT, &e, 019,497,820, &c. Therefore 0.095,310,179,804,324,860 X ¥ + 0.000,000,. 000,000,019,497,820, &c, will be = 0.000,000,197,473,485,882,1, &c, and (dividing all the terms by 0,.095,310,179,804,324,860,) y -+ 0.000,000,000, 000,204, will be = 0.000,002,071,903,403, and (fubtracting 0.000,000,000, 000,204, from both fides) y will = 0.000,002,071,903,199; and confequently x, Or 24.158,86 — y, will be = 2.4.1 58,860,000,000,000 — 0.000,002,071,903,199 = 24.158,857,928,096,801 ; of which all the figures, except the laft, are exact, the more accurate value of « being (as we have before obferved in Art. 6,) 24.158,85.7,928,096,805,5. We have therefore now found the value of . . ° I |\+ the index # in the equation 1 + <5! = 10 exact to. fixteen places of figures. Qik. o3s 58. The value of juft now obtained, to wit, 24.158,857,928,096,801, is : 1.000,000,000,000,000,000, &c __ a to I, as 1 isto Far TB Ber qetcyeet Gee 0.0415392,085,158,225,484, &Co : fs . I Therefore the proportion of the ratio of 10 to 1 to the ratio of 1 + — to 1, or to the ratio of 11 to 10, 1s that of 1 to 0.041,992,685,158,225,484, &c, or,. in other words, Briggs’s logarithm of the ratio of 11 to 10 1s = 0.041,392,685,.. 158,225,484, &c; of which number the firft fifteen figures, (reckoning from. the place of units,) to wit, 0.041,392,685,158,225, are exact, the more accu-. rate value of this logarithm (as computed by Mr. Abraham. Sharp,) being. 0.041,392,68 5,158,225 ,040,7 50s. A DEMON- A DAN: Ou NoS-ebi Re AT 1 ON . Pet Oat WA Cab IN, He Wee To O..)N.7 Bel: INO ®eM. Dae io aT ice E OO Re EUM In the Cafe of Integral Powers, or Powers of which the Indexes are whole Numbers. BY .fRiANCLIS MASE R ES,.. Es: CURSITOR BARON OF HIS MAJESTY’S COURT OF EXCHEQUER. en ate TELS DV EGS ee LS ROT ETS Ach Segal 0 tigi esd Ah Spies gs H E binomial theorem of the great Sir Ifaac Newton has been fo much reforted to in the preceeding Difcourfe of Dr. Halley, and in the laft fore- going Tract, intitled, “ An Appendix to the faid Trad,” for the purpofe of computing logarithms, and is fo clofely connected with that fubject, that it will probably be agreeable to the readers of this collection to fee a demonftration given of it in the fame volume with the other tracts, of which it is the chief foundation. I fhall therefore now proceed to give a demonftration of it in the firft and fim- pleft cafe, or that in which the indexes of the powers, to which the binomial quantity is raifed, are pofitive whole numbers; which may be done with fuffi- cient perfpicuity and exactnefs within the compafs of a few pages. 2. This theorem is as follows. TRHSESOSRE EF. -M If m be any whole number whatfoever, a + 2\", or the mth power of the binomial quantity 2-4, will be equal to the following feries of terms, to wit, a + xa” *b+ 2x7 x a”? Bat x tx tx" : m m— I m— 2 m— 3 m—4 74 m m—t1 m— 2 ot rae Seer Seren etd iT rite dete aU ace aaiy er M3 xy ZSt y g@”~ 598 + &c continued to the term" x “= x 4 5 VoL. It. x 154 A DAE LM OCN Sp aeak ee eee oo ee Oo F fad aa abated Vy pg geceikt bee SNH aK eyelet SE Or toe eee % 4 5 2 I Z m1 — 2 m— 3 mt — 4. ’ ee ie ay m—t1 m—2 arate anes MOR ae! DIO x ve fin bm. ae? ex eX ——; in which feries the law by which the fevaral terms, S : : after the firft term a”, are generated from the faid firft term, is as follows. The literal parts of the fecond, and third, and fourth, and other following terms, to m.— 1 b . Wi— 2 wi— wit, @ a Gere 3 23, &c, are generated from the firft term 4” F BS ee, Mae they : by the continual multiplication of the fraction —, the index of the power of 4 (the firft term of the binomial quantity 4 + 4) in every new term decreafing con- tinually by an unit, and the index of the power of 4 (the fecond term of the bino- mial quantity ¢ + 4,) increafing by an unit at the fame time. And the feveral co-efficients of the fecond, third, fourth, and other following terms of the feries, t a 72 Ma iy, cepa we AS 6 SISO ae 12 M1 m—_— tT m1 — 2 m—% O Wit, 5 = eae 5 eu? oy See 3 x ele -— I W— 2 YW — Ht— By SS SEE Ke AES LE, Be," are generated from ti ye teens I 2 3 ; A> Oe ie 4 cient of the firft term a”, or 1 X am,) by the continual multiplication of the fraCtions - m—-l m—2 m— mm —~ . : wrod | ete he eee 2 ik “+4, &c, of which the denominators are the numbers Te. 3 I) 25 3) 4, 5, &c, in their natural order, and the numerator of the firft fraction 2 ie . . . w— 1 ME wen — is the index m, and the numerators of the following fractions Be aa I 3 ‘ 3 cdive 111 yeh 4+ &c, are derived from that of the firft, to wit, m, by the continual 4 3 : fabtraction of an unit. Examples of the faid Theorem, 2. Thus, for example, if m is = 2, or it is required to find, by means of this theorem, the fquare of the binomial quantity a + 2, we fhall have @ + 4? = e@pexa bp txt xa pee +2 x asp yt 2 x ox Pa + 20b+— x1 x ma +2ab + 3 And, in like manner, if m is =: 3, we fhall have 2 + 3 = a? + oe Var $f be hx St x et et 2 x ot y eee Asi ee ee XPD+RKXEXPPR EEX DK EXO RS a+ gab + 3 ab + &. And, if mis = 4, we-fhall have a + 3} = at 4 x atte pF x sat ee Bey fe? eat 4 AY A 2 I o 3. I 2 3 4— 3 AEA 7S lek eng oe 3 mf Mgten 2 2? tae > oareruaa lt i Gee Ope A GREE Ak G0 ee Co yea SIR TSAAC NEWTON'S BINOMIAL THEOREM. 155 he x a B+ e xixixtx IX¢= a +agi@s+ 6a t4ab3 And, if m is = 5, we fhall have a + Js = a° + o hy a +e x Se ah Sy Sot yx Sat y GS-35 4 Sy SEL y Sot 2 3 bir 3 5-4 5 per if! Thee “2 5 39 §—4 i 5 5 Xa b+ 2. Sates Eph: geese pe Ea oe Sg >—a a “+ : x é : x i x F min, + 2 =< x eaateer bey LEI BE Sei D pe Ol MA Biogera tgs pe Sy lalyu ad I I 2 I 3 I 2 3 2 2 Seas BR Ra wie Me sep iat o 7s — 43 : 3 BP 7 xe it OS einer dm ee «ob? aa? obi 52 6+10 &b + 104 8 + 5 ab+ + DB, 4. Thefe examples are fufficient both to illuftrate the meaning of the fore- going theorem, and fo prove the truth of it in thefe few eafy cafes. For “that ** the theorem is true in thefe cafes,” will appear by raifing the fecond, third, fourth, and fifth, powers of the binomial quantity @ + 4 in the common way by multiplication; which will produce the very fame quantities for the faid powers as have been juft now obtained by means of the foregoing theorem. This may be done in the manner following. a+b a+b aa + ab + ab+ bb 4a+24ab+ bb =a+ db" a+b a + 2aab +. abb + aab + 2abb + BF a+ 3406+ 3466+ =a-t d a+b +3@6 +4 3476 +.ab5 + 05+ 32h + 3a)? + b* 4+4R@b+ 6070 +400 + 6*=a+ b\ +b a@+4a+*64+ 600? + 4a Hh + ab* +44*b64+407 +670 + 4ab+4+ 35 eit @+ 5a@b6+ 1000+ 10064 506+ 45 = a+ os Thefe values of the feveral powers a + 6%, a2 + 413, a + d+, anda+ Ais, are the very fame with thofe that have been obtained juft above in Art. 3 by means of the foregoing theorem. And confequently the faid theorem is true in thefe four inftances. Of the literal Parts of the Terms of the foregoing Produéts, and the Law of their Generation one from another. 5. The law by which the literal parts of the fecond and other following terms of the feries that is equal to @ + 4\» are derived from the firft term a”, ae X 2 rom 7 156 AOD BUM. O NWSE RB Aol Nol from each other, to wit, ‘* that they are generated by the continual multiplica~ tion of the fraction —,” will be fufficiently evident from the foregoing multi- plications of a + 3b and its powers by a+ 4 For it is evident that the in- dexes of the powers of @ in thofe feveral produéts decreafe continually by an unit, and that the indexes of the powers of: 4 in the fame produéts increafe by n unit at the fame time; fo that the literal part of the fecond and every fol- tiaials term in each of the faid products is generated from the term next before b | : n into — at it by multiplying the faid preceeding term into —. And it is eafy to fee that the fame thing muft take place in any higher powers of a + 4 whatfoever, if the faid multiplications by a + b were to be continued till fuch higher powers were produced. This part, therefore, of the aforefaid binomial theorem ftands in need of no further demontftration. Of the numeral Co-efficients of the two firft Terms of the faid Produés. 6. And with refpect to the numeral co-efficients of the feveral terms of thefe products obtained by the foregoing multiplications by a + 4, (which co-efficients are inthe {quare 1,2, and 1, and in the cube 1, 3, 3, 1, and in the.fourth power 1,4, 6, 4, 1, and in thé fifth power 1, 5, 10, 10, 4, I,) it 1s eafy to fee that the co-efficient of the firt term of every new power of 4 +4 mutt always be 1; becaute it arifes by sd the next preceeding power of a by a, or 1 Xa, which cannot alter the co-efficient of the faid next preceeding power, which at firttwas1. And itis alfo manifeft that the co-efficient of the fecond term of every new power of 4 + 4 muft always be the index of the faid new power, or, in our prefent notation, muft be m; becaufe it is produced by adding the produat of the multiplication of the firft term of the next preceeding power of a + 4, (of which firft term 1 is always the co-efficient,) by 4 to the product of the multi- plication of the fecond term of the faid next preceeding power of a + 4 bya; the effect of which addition is, to increafe the co-efficient of the fecond term of every new power of a4+d byanunit. And this, it is eafy to fee, muft be the cafe in any higher powers s whatfoever of a -+- 4, if we were to continue the faid multi- plications by a +- 4 tll fuch higher powers were produced. We may, therefore, conclude that the co-efficients of the two firft terms of the feries that is equal to a+ mn mutt always be 1 and w, and confequently that the two firft terms of the faid feries muft always be 1 X a” and mx a” * b, or ¢mand =~ x2 ee 7. It remains that we fhew that the co-efficients of the third, fourth, fifth, fixth, and other following terms of the feries that is equal to a + d\™ are gene- rated from the co-efficient m, or =, of the fecond term by the continual multi- — I M— 2 Ww — % mh — plication of the fracttonsi-——, 9, a0 &c; which is indeed by no F 3 a means obvious. Of the numeral Co-efficients of the third and fourth, and other following Terms of the Jatd Produé#s, and the Law of their Generation from the Jecond Term, and from each oiber, 8. Now, in order to demonftrate the law of the generation of thefe co-effi- cients, it will be convenient to get rid of the powers of a and 4 in the terms of the SIR ISAAC NEWTON’s BINOMIAL THEOREM. 157 the feries that 1s equal to a + é\*, and to fix our attention only on the gene- ration of the numeral co-efficients of the third, fourth, fifth, fixth, and other following terms of the faid feries. This may be done by fuppofing @ and 4 to be, each of them, equal to 1, and confequently a + 4 to be equal to 1 + 1; and 4 + dm to*be equal to 1 + 11%. For, as all the powers of both @ and 2 will, on this fuppofition, be equal to 1, our theorem will then be reduced to this, to wit, that 1 + 11,, will be equal to the feries 1 +—+— Ne — a wa Pap Le opto 712 42 mM m— I UE mm ay 3 m Mh aa m— 2 i es re tte Mee yer 9 &c, continued to the tem ~-~ 24 x» 27 x atte 4 5 ; 2 3 4 2 —(m— 1 4 = —_ — Mee Kio XS “~——, or to the term — x = Wwe = bia 3 >¢ 5 Wm i 2 3 4 m— 4 I e . ts le —, or to the term 1. For the laft term of this feries muft al- ways be 1; becaufe the numerators of the feveral factors in it form a decreafing progreffion of numbers from m to 1, and the denominators of the fame factors form an increafing progreflion of numbers from 1 to m, and confequently, the product of all the denominators is equal to the product of all the numerators, and therefore the produét of all the faid factors, or the faid laft term, muft always be equal to 1; which we have feen to be the cafe in the laft terms of the values of a+ b’,a+\, 2+ o\, a + 4, as derived from the feries in the theorem in Art. 3, which laft terms were — x —, and ~. x — x —, and +. ye x 3 2 aiae I : x it x =e and = SZ oa x ma x os a4 = which are, each of them, equal to1. We are therefore now to demonttrate that 1 + 1 is equal to the feries m mm m— 2 ma m— 1 m— 2 me ma — m— 2 m— 3 MNGatCo fs eS Pee Se Laan Pe Nee aceon icra itch att Of Mr. Fames Bernouilli’s Demonftration of the Law of the Generation of the faid numeral Co-efficients. 9. Now the cleareft and beft demonftration that I have ever feen of this ufe- ful propofition is, that which is given us by the learned and fagacious Mr. Fames Bernouilli in the third chapter of the fecond book of his excellent Treatife on the Doctrine of Chances, intitled, ‘* De Arte Conjecéfandi,” which 1s writ- ten in Latin, and was publifhed in a fmall quarto volume at Bafi/l, or Bafle, in Switzerland, in the year 1713, eight years after the author’s death. This demonftration is founded on the doétrine of combinations and the proper- ties of the’ figurate numbers, which are there fhewn to involve in them the generation of thefe co-efficients. And the moft important properties of the faid numbers are in the fame chapter fet forth and demonitrated by that great author in a very perfpicuous and mafterly manner, though with rather too much concifenefs to be eafily underftood by beginners in thefe ftudies. Thofe readers, therefore, who are defirous of feeing this theorem demonttrated from its natural and fundamental principles, and in the cleareft and moft fatisfactory 1 manner 158 A DEM ON ST RAE Ow Oe manner that, as I believe, the nature of the thing will admit of, muft be referred to.the {aid third cha ipter of the fecond book of that learned treatife ; which, to- gether with the two preceeding chapters of the fame book, I would ‘advife them to perufe with the clofeft attention, ‘and to make themfelves thorough matters of their contents; to do which they will find, will require a confidorable exertion of their diligence. Another Demonftratien will be here given of the faid Law. 1o. But, though that is the beft and moft fatisfactory demonftration that has ‘been given, and, probably that can be given, of this theorem, yet it may alfe be demonttrated in a fhorter and eafier manner, and with the fame degree of ‘certainty as by that method of Mr. James Bernouilli, though not with the fame degree of originality and elegance. And fuch a demonitration I now propofe to give of itin the courfe of the following pages. ir. The demonttration which is here intended to be given of this important _ theorem is founded on the obfervation ‘ that the faid theorem is found by ‘‘ trial to be true in fome of the loweft powers of the binomial quantity ‘* 7 + 1,3” as has been feen above, in Art. 4, in the cafes of the fecond, third, fourth, and fifth, powers of 2 + 4. For, if this theorem is true when the index m is of any particular value, as, for example, when it is equal to 5, it may be fhewn by abftract and general reafonings, derived from the nature of multiplication, that it muft likewife be true when the index m is increafed by an unit, or that, af be taken = m+ 1, the quantity 1 + 1)", or the ath power of the binomial quantity 1 + 1, will be equal to the feries 1 +. — + = pom) n nu— tI a— 2 ay id x aptvery tao ere ed Bilt arf ae we Sek y fS8 y 223 y Bt & &c contifined t the term — Ss eee 2 3 sy 5 ; ; x Aisrse te ean Oe ae bas ay —, or 1. And this is what I fhall now en- 4 deavour to demonftrate. 12. To facilitate the demonftration of this propofition it will be convenient to premife the following Lemma, ° A ly ExMaMi A: If the terms of the feries 1 +- — + = x x Biers te | &c + 1 (in which m reprefents any whole number whatfoever,) be fet down twice together i in two parallel lines, or rows, one under the other, but with the terms in the lower row advanced one ftep further to the right-hand than the terms in the upper row, fo that the firft term in the lower row fhall ftand under the fecond term of the upper row, and the fecond term in the lower row fhall ftand under the third term in the upper row, and the third, fourth, ‘fifth, fixth, and other following terms in the lower row fhall ftand tinder the Fourth! fifth, fixth, feventh, and other following terms in the upper row, refpectively ; t 7 and SIR ISAAC NEWTONS BINOMIAL THEOREM. 159 and both rows are continued to the fame number of terms, namely, to the whole number of terms in the faid feries, or to m + 1 terms; and then the terms in the lower row, (which, it is evident, will confift of one factor lefs than the cor- re{ponding terms, or terms ftanding immediately above them in the upper row, ) be reduced to the fame denomination as the terms that ftand immediately above them in the upper row, and, after being fo reduced, are added to the faid terms that fland immediately above them in the {aid upper row ;—upon thefe fuppofi- tions the new feries of terms arifing from this addition of the faid two rows of m+ I m m m+ I terms to each other, will be as follows, to wit, 1 + + — on ae SE NR de m— 1 er ae ae Va ae I 2 3 3 4 i 2 3 my — 3 m+ ° : - : x —$5 ->K& —— + Me 4- 23 44n Ae feries the laft term is 1, as well as in the two feriefes from the addition of whichthis feries arifes ; and the numerators of the lait factors in all the terms, except the laft,- are always equal to m + 1, inftead of being equal to m—1, m— 2, m— 3, m— 4, &c, as in the two foregoing feriefes ; and the number of terms in the faid new feries ism + 2, inftead of m + 1, which is the number of terms in each of the faid foregoing fericfes. ”~ DUPE MOON Sah Re AT. TL Om N. Mmmm I 13. This will appear by fetting down the faid feries 1 + = eaters m m— 1 TT P95 72) <4 5 i= 2 m— 3% Buinas x += x x x + &c + 1 twice 3 2 over, in the manner that bes been juft eae te which may be done as follows : ™m ri ae Sea eae Sua Ba Res A 2 3 I r+ pees yten ones Rae + &c. In thefe two rows of terms it is evident, in nk firft place, that the terms in the upper row, after the two firft terms 1 and =, confift of two, three, and four, and more, factors, every new term having one more factor than the term next: before it; and, zdly, that the terms in the lower row that ftand immediately under the third, fourth, fifth, and other following terms in the upper row, con- fift of one factor lefs than the correfponding terms, or terms immediately over them in the upper row; and, 3dly, that the terms in the lower row confift of the very fame factors as the correfponding terms in the upper row, excepting that they want the laft faétors of the faid terms in the upper row.. And hence: it follows, that, in order to reduce the terms in the lower row to the fame de- nomination as the terms in the upper row, we mu{t multiply them by factors that fhall have the fame denominators as the la{t, or additional factors in the upper row, and which mutt have their numerators equal to their denominators, fo as to make each of them equal to 1, to the end that the magnitudes of © the faid lower terms may not be altered by the multiplication of them by the . faid - r6p: ho Obi MPOENUS AR AST TOON slo faid new factors. Thus, for example, the fecond term of the lower row, to wit, =, muft be multiplied into the factor =» in order to bring it to the fame deno- aie mination as the third term in the upper row, to wit, — X , without alter- m— tI » muft .* . e % . . . ing its magnitude ; and the third term in the lower row, to wit, be multiplied into the factor =~ in order to bring it to the fame denomination m— I 27) hemes x . 2 ° as the fourth term of the upper row, to wit, = x » without al- i aL 2. . e . e . 7,74 tering its magnitude; and the fourth term in the lower row, to wit, x Aa— x =, muft be multiplied into the faétor oe in order to bring it to the fame Carag | — 7 m x ™m™ ¥ 3 —3, without altering its magnitude; and, for the like reafon, the fifth, and f . . . i . i denomination as the fifth term in the upper ‘row, to wit, ratte x fixth, and feventh, and other QUO vURE terms : the lower row muft be multi- plied into the feveral factors rE and = =, and ~2) &e, re{pectively; after which multiplications the two rows of terms san are to be added to each other, will be as foes: to ‘ap m Mm—T m—2 m m—t1 m— Ll I aT A 3 I 2 ob st &c x nid Mm— 1 Be m m—T rt “+ Le - = 6 NP ct OS ase 108 3 ‘ m + &c. 14. And, if thefe two rows of terms, (being now brought to the fame de- nominations, ) are added together in the manner above defcribed ; that is, every term in the lower row to the term that is immediately above it, the fum thence refulung will i the feries m+ Sear I m m— 1 m+-1 1-7 += Fae pee herefee p shes + &c, in ich the pe etae of the laft factor in every term is always m + 1, inftead of m—1, m—2, m—3,m—4, &c. And « That this muft be the cafe in all the following terms of the faid new s¢ {eries as well as in the few terms of it that have been here fet down,” will be evident from this confideration, to wit, That the denominator of the laft faétor of every term in the upper of the two rows of terms that are added together is always greater by an unit than the number which is fubtracted from m in the numerator of the fame factor. For from thence it follows that the denominator of the new multiplying fraction in the correfponding term of the lower row (which is always equal to the denominator of the faid laft factor in the upper row,) muft always be greater by an unit. than the number which is fubtraéted from m in the numerator of the lat faétor. of the faid upper term, And, eps therefore, the numerator of the faid new multiplying fraction in the lower row (which is always equal to its denominator,) muft alfo always be greater by ‘an unit than the number which is fubtracted from m in the numerator of the Jatt, factor of the faid upper term ; the confequence of which, in adding the lower SIR ISAAC NEWTON’S BINOMIAL THEOREM, 161 lower term to the upper term, is to convert the numerator of the laft fa€tor in the upper term from m — 1, orm — 2, or m — 3, or the excefs of m above fome other number, into m + 1. ele iar. 15. And the number of terms in the new feries, arifing from the addition of the two former in the manner that has been defcribed, will be greater by one than the number of the terms in either of the two added feriefes: becaufe the lower row of terms, confilting of the fame number of terms as the upper row, and being placed one term further to the right-hand, muft extend one term beyond it; and confequently, as the number of terms in each of the two rows of terms is m +1, the number of terms in the new feries, arifing from the addition of the two rows together, muft be m + 2. Q. E. D, 16. And, laftly, the laft term of the faid new feries muft be the fame as the laft term of the old feries, or of the lower row of terms; becaufe, as the lower row of terms extends one term beyond the upper row, the laft term in the lower row will not have any term over it in the upper row to which it is to be added, and confequently will continue the fame inthe new feries 1 + ao + — ole * < + x SS BE Sx Fw x BE + Be as in the old . Mm ZA a F 1 m2 MNi—T 12. 72 Precise g Uh as 2 Mea aes tim oe ge : x ite +- &c. But we have feen above, in Art. 8, that the laft term of the . Mm mW 73 — I Mm —= I MW — 2 m MW == TI umi— 2 Oo hes Sees a eae SM boas x rE sr ae 3 x ae + &cis1. Therefore the laft term in the new feries 1 + nat +. 2x 244% 7 ee tees eerie 7234 &ewill alfo be 1. Q. .E.>D, 17. Coroll. 1. Now let the order of the numerators m, m — 1, m— 2, m — 3,m — 4, &c, and m + 1, of the factors of the feveral terms of the laft feries 1 + “27> 4 2 mas le later se tht periael coet kaa m+ 2 I 2 3 2 3, x -- &c, after the two firft terms, be changed, by making m + 1 th numerator of the firft factor of every term inftead of being the numerator of the laft fa&or. The faid feries will then be as follows, to wit, 1 + — - : +. 4 mM - mi Mm 1 m— tI m—2 =a. : ES m X ets Ba Nes Epc x ; + &e. Now this change in the order at the numerators of the feveral factors of the terms will create no change in the values, or magnitudes, of the feveral terms themfelves; becaufe the produéts arifing from the multiplication of the fame nuinbers are always the fame, in whatever order the numbers are multiplied. T eo. the foregoing feries, after this change in the order of the uci oe ox. Il. . Y re) 162 A DEMON S'TRACT LON 40.F of the feveral factors of its terms, will {till be of the fame magnitude as before, and confequently will be equal to the fum that arifes from the addition of the aforefaid two rows of terms in the manner above defcribed; that is, the feries m+. m+I mt m+ 1 m wt — T m-- I m m— 1 1% LIS REREAIIC ENERGIE ma 2 4. &c + 1 will be equal to the fum that arifes from the addition of the aforefaid two rows of terms in the manner above defcribed. 18. Coroll. 2. Now let 2 be = m+ 1. Then will » —1 be = m, and n—2 willbe = m—1, andz— 3 will be = m—2, and » — 4 will be = m — 3, and, in like manner, »— 5, 7—6,u—7, &c, will be equal to it — 4, mM — 5, m— 6, &c. refpectively. And confequently the feries ri 4. 2 ES pkgs x 2x Betyg Bey Sy Met Me saith confifting of m+ 2 I 2 I 3 ; uZz— I n / ten | 4o.— 2 terms, will be equal to the feries 1 + ~ js) ae HIE | 7M obtained in the foregoing Corollary, to wit, 1 +- a nt— 1 im—m 2 3 n x = + &c + 1, confifting of z 4+ 1 terms. There- — I fais x tm I Nm 2 te ah x A— 1 I I roma: HI vi + I, aeaatine of 2 + 1 terms, will be equal to the fum 3 : ee. : : that arifes by adding the two aforefaid rows of terms together in the manner above defcribed. fore Ne heise I-+— x Zz a——Z The Demonftration of the principal Propofition. 19. Thefe things being premifed, the main propofition ftated at the end of Art. 8, to wit, that, if m denote any whole number whatfoever, the quantity 1 + i\”, or the mth power of the ace rae ree 1 + 1, ie gh equal to 1). 5 i the feries1 + = 4+ x == Bini Io m—2 3 ite Pee ae key hee x ees terms, or to the term 1, may be demontftrated in the manner following. > c ° : — J 20. The product that arifes by multiplying the feries 1 + a 4- = 3 : —— J — _— — — . . + =x 2 xD ex Se x BS x 8 + &c into 1 + 1 is the fum “that arifes by fetting down the faid feries twice following in two parallel rows, one under the other, with the terms in the lower row ad- vanced one term further to the right-hand than the terms in the upper row, in the manner above defcribed, and then adding the terms in the lower row to the correfponding terms in the upper row. And the m + 1\” power of 1 + 1 is the produdt of the multiplication of the mth power of 1-+-1 into 1 +1. Therefore, if in any particular value of m the mth power of 1 + 1 is equal to the SIR ISAAC NEWTON'S BINOMIAL THEOREM. 163 m 4 M— I m2 Woman J Th ome 2. mM Mi— the feries 1 + — prec eX Wi — 2 x m x ‘i 34. &c 4 se of m 4+- 1 terms, the m+ 1)? power of 1 + 1 will be equal to the fum that arifes by fetting down the faid feries twice follow- ing in two parallel rows in the manner above defcribed, and adding the faid two rows of terms together. But, by the fecond Corollary of the “fores going Lemma, if # be = m + 1, the fen arifing from the addition of the faid two aZa— tI 2 u— TT Z—— 2 4. 3 rae x x fe + &c +1 confifting of w+ 1 terms. There- fore, if in any particular Gene of mthe mth Ep HEY of 1 + 1 is equal to the feriest += 4% arm -+— x BE Tons, 19° Glee -4+% x7 sr bhagh gq hse! Lh m— 3 es . 7 7 rows of terms is the feries 1 -+- = + ax 74 at "2 2 3 x + &c + 1, ik arith of m + 1 terms, it will follow that the m +- 1\, or mth, or next higher power, of 1 + 1 will be equal to the feries ik a a hg gtd men ee Mo eet I I 2 I 2 3 “I 2 aM 4 + &c + 1, confifting of 2+ 1 terms. But it has been fhewn in Art. 3 and 4 that, when m is equal either to 2, or to 3» or to 4> or to 5, the 7 the mth power of 1-1 is equal to the feries 1 + ~ + = I Mi —— JT Wt— 2 772 m— tT i == 2 2 3 I 2 - terms. Therefore, if x be equal to 5-+-1, or 6, the 5 + 1\4 power, or 6th power, or wth power, of 1 + 1 will be equal to the feries 1 4 = are 99 uae tate slinged es — + &c + 1, confifting of m-+-1,o0r 6 + 1, o0r7, terms, And in the fame manner it may be proved that, fince, when m is = 6, the mth power of 1 +1 1s equal to the feries 1 + — + 2x2 i = Sx 7ST x Pat x 284 &e + 1, confifting of m+ 1, or 6 +1, or 7, terms, the m + 1\*, or 6 + 1\%, or 7th, or (putting # = eae 1.2 6 +41 = 7) the ath, power inge I ee I ea be equal to the feries 1 += ae — x ee — x eee 2 2 “Beet — irs i Ba vce lepine acai terms. And fo we may proceed from number to number ad tujinitum. And confe- quently, whatever be the whole number denoted by m, it will always be true that 1 -+ 1” is equal to the feries 1 + — _—- =~ SD a mm— 2 m pL Mn — 2 (1d hee la um iL eowrat wh m— 2 m— 3 alae” * x —7 1X x Sie 3 i 2 3 AU E a 3 4 m— 3 spk &e ot 1, confifting of m -- 1 terms. Q. E. D. ag 2, The 164 4 DEMONSTRATION OF The foregoing Demonftration expreffed in a more concife Manner. 21. The foregoing reafonings may be expreffed in a more concife manner as follows. If x be = m + 1, and it be true in any paren a value of m that 1+ iis = the feries 1+=4+—x7 lis re eae x ——- x aad x pak + &c, it will alfo be true se 1 +i will be = n 2 Aw I n \/ oe | n2—2 xh. — TI Uwe 52 tera Barbican dies RON ator TT 20 oe 7h x * 3 iE add fy + &c. eh TA Te ee 4 Ww — I mn UW —» 1 (Meee mm m— it! Th a2 1 PISMO RT HTFERM EGET Men 1 me ihe multiplied into 1 + 1 = r+—4 2x2 ty 2xUE xy a (72S Siar xe 1 at es ft =x a axa DY bed Eh: 2 + &e, bs4 M—T1 —I!I Mm—2 in Ni— I W—2 — oil t Siti Kure ty patie gh Sa ee Rae 2 = —tI hr ES EE ESSE i x srpTy ty My axe OEE Dy TOE BO? x TH Be e3 m+1 m+ we, m+i m= m—tI m+1 m m—-I. m—2 aati) rabneania nm raminb casita oases kai ARE eS x nica J n u—I 2 n—I u—2 nu a—tI 1—2 =ItTt— 3) didah oaAC Rp tae Pest But it has been fhewn in Art. 3 and 4 that, when m is ae cer to ies or to Li Somes 3, or to 4, or to 5, 1 +-1\" is equal to the feries 1 += m—tI m—2 m m—I m—2 ees San 3 aah Pa ie cerita! ; x + &e. Thevbre ibe sa or 6,1 + 11", or 1 +. 1)°, will be = x ooh A at yt sat ted | ee ue its maybe eae 4 in like manner that, if z be put for 7, 8, 9, 10, &c ad infinitum fucceflively, 1 + 1\* will in all thefe tate be always equal to the feries 1 +4 ~ oh % a ——" 1 Mi 2 2 na— TI a— 2 mH els —-+ = eer etae sar Weal adolae Vii (1 0: 3 ean therefore the Pe pee is univerfally true, weatenee be the whole number denoted by the letter 2. Q. E. D. 22. This demonftration of the binomial theorem in the cafe of integral pow- ers, is nearly the fame with that given by Mr. John Stewart, of Aberdeen, in the 6th Section of his Commentary on Sir Ifaac Newton’s curious little Tra&, intitled, Analyfis by Equations of an infinite number of Terms. See his edition of SIR ISAAC NEWTON'S BINOMIAL THEOREM, 165 of Newton’s Treatife on the Quadrature of Curves, and of the faid Tract intitled Analyfis, Fc, with his learned Commenis oa both, in one volume, quarto, publifhed at London, in the year 1745, page 471, Art. 155. Of the Invention of the foregoing Theorem. 23. This famous theorem ts ufually afcribed to Sir Ifaac Newton: and in the cafe of roots, or fractional powers, of a binomial quantity, it is generally agreed that he was the firft inventor of it. But in the cafe of integral powers, (which has been the fubject of this Difcourfe,) it feems to have been difcovered many years before by Mr. Henry Briggs, the ingenious computer of the loga- rithms that are called by his name, For it has been lately obferved by Dr. Hutton, (the learned Profeffor of Mathematicks at Woolwich Academy,) in his very curious hiftorical Introduction to his new edition of Sherwin’s Ma- thematical Tables, that there is a paflage in the faid Mr. Briggs’s valuable Treatife, intitled, Arithmetica Logarithmica, (which was publifhed in the year 1624,) in which Mr. Briggs fhews how to derive the third, and fourth, and fifth, and other following terms of any integral power of a binomial quantity from the fecond term of it without raifing all the intermediate, or lower powers of the binomial by continual multiplication ; which is the operation of the bino- mial theorem. But Mr. Briggs has only defcribed the method of doing this in words, and has not expreffed it in Algebraick fymbols, as Sir Ifaac Newton has mu—l m—2 “a eA CR > > 2 done, by affigning the fractions = as the factors, by the continual multiplication of which ane co- dmidtcnts of the faid terms may be produced. The merit, therefore, of being the firft inventor, or publifher, of this ufeful difcovery in the cafe of integral | powers, mutt be allowed to Mr. Briggs, and only that of giving the invention a more convenient form, by exprefs- ing it by a fhort and fuitable algebraick notation, mutt be afcribed to Sir ‘Tfaac Newton, together with that of extending it by his fagacious conjectures from the cafe of powers of which the indexes are pofitive whole numbers to the vari- ous other cafes of it in which the indexes of the Peaks are either fractions, as bets 5 4, iP and Bf or, in general, — ~, OF negative whole numbers, — 3, —4, — 5, &c, or, in sens — m, or negative fractions, tia — a ar ; —m 3 rie Bh; —3, =F i or, In general, PE which extenfion " cy able? : , of it is of aH arf utility in various branches of the higher parts of Algebra and mathematicks, as we have feen in the remarks that have been publifhed in the former volume of thefe Traéts on the Logarithmick Seriefes invented by Mr. Mercator and Dr. Wallis, and in the foregoing Tract concerning the refo- 2 fe j |\* it ‘ 3 mi. jution of the equation 1 + =| == 10, or the inveftigation of the proportion . : : TiAas “ of the ratio of 19 to 1 to the leffer ratio of 1 + =; tof, or of 11 ta fo. 24. Yet it may reafonably be conjeCtured that Sir aac Newton was likewife an inventor, as well as Mr. Briggs, of this ufeful theorem even in the cafe of integral powers, though wot the firft inventor of it. For it is well known that he ae | was 166 A DEMONSTRATION OF was not an extenfive reader of mathematical works; and he appears to have applied himfeif principally in his younger years to the ftudy of Des Cartes’s Geometry, with Schooten’s Commentary on it, and the other Tracts publifhed by Schooten with it, and of Dr. Wallis’s Arithmetica Infinitorum, and his other works on mathematical fubjects then publifhed. And therefore he may well be fuppofed not to have feen Mr. Briggs’s Arithmetica Logarithmica, in which this o . . method of deriving the co-efficients of the terms of the powers of a binomial quantity one from another is contained, at the time of his difcovering this famous theorem himfelf, which was about the year 1665, or when he was only 23 years old. And, if he had feen that book, and obferved this difcovery to be contained in it, it is hardly to be conceived that, when he was {peaking of this theorem, he would have omitted to make mention of this difcovery, and to acknowledge that it contained the fubftance of the faid binomial theorem in the cafe of in- tegral powers, though not expreffed in algebraick fymbols. For thefe reafons I-am inclined to think that Sir Haac Newton had not feen Mr. Briggs’s Arithmetica Logarithmica, when he invented the binomial theorem, and. there- fore, that he was truly am inventor of it even in the cafe of integral powers, though wot the firft inventor. 25. But it feems more furprifing that Dr. Wallis, who was a much more copious reader of mathematical works than Sir Ifaac Newton, and who actually had feen Mr. Briges’s Arithmetica Logarithmica, and makes mention of it in his Algebra, chap .er xl, page 60, fhould not have attended to the contents of that ingenious treatife enough to have obferved that it contained this moft ufeful theorem. Yet this appears to have been the fa&, from what the Dodtor tells us in the 85th chapter of his Algebra, page 319, where, in {peaking of Mr. Newton’s method of generating, or deriving, the ‘co-effi- cients of the third and fourth, and other following terms of the mth power of a binomial quantity from m, the co-efficient of the fecond term of it, by the ree — oe 7= 4 &c, he con- ~ feffes that he had fought after this method of generating, or deriving, thefe co- efficients himfelf, but without fuccefs. His words are thefe, after {peaking of fome other excellent inventions in the mathematicks contained in a letter of Mr. Haac Newton (at that time Profeflor of Mathematicks in the Univerfity of Cambridge, and who was afterwards better known by the ttle of Sir Maac Newton,) to Mr. Oldenburgh, (the Secretary of the Royal Society at London,) dated October 24, 1676. ‘* He[ Mr. Newton] then obferves (what I bad for- “« merly fought after, but ynfucce/sfully,) that the following numbers are, from the “two firft, to be found by continual multiplication of this feries 1 X oe x m— 4 BoM Mae BBL - x &c.” Thefe are the words of Dr. Wal- continual multiplication of. the fraCtions 66 x 4 3 lis in his Algebra, page 319; from which, I think, we may conclude that, . though he had feen Mr. Briggs’s 4rithmetica Logarithmica, he had not read it with fufficient attention to difcover that this method of generating the co-effi- cients of the terms of the mth power of a binomial quantity, when m was any whole number whatfoever, was contained in it: though it feems indeed un- accountably ftrange that he fhould not have taken notice of it. Tas 26. Dr. SIR ISAAC NEWTON'S BINOMIAL THEOREM. 167 26. Dr. Wallis’s Algebra was publifhed in the year 1685, And that was the firft time, after Sir Ifaac Newton’s difcovery of it, that the binomial theorem (in Sir Ifaac Newton’s manner of expreffing it) was publifhed in print, and made known to the learned world in general, though Mr. Oldenburgh and Mr. Leibnitz, and probably Dr. Barrow, (who was Sir Ifaac Newton’s great friend and patron in his youth,) and fome other learned mathematicians of that time, had feen it in that letter to Mr. Oldenburgh, of Otober 24, 1676, foon after the faid letter was written. But, how Sir Ifaac difcovered this theo- rem, is not known; nor is any demonftration of it, even in this eafieft cafe of it, (in which the index m of the power to which the binomial quantity is to be raifed, is a whole number,) any where to be found in all his works. Of the Powers of a Refidual Quantity a —b, when their Indexes are whole — Nambers. 27. We have hitherto been confidering the integral powers of a binomial quantity @ + 4, or of the /um of two fingle quantities a and 4; and we have feen that, if the faid binomial quantity ¢ + 4 be raifed to any power of which a whole number denoted by m is the index, the quantity a + 4", or the faid mth power of a + 4, will be equal to the feries 2” + — goa. Gee — x M2 == 1 _ m—2 m m—1 m m—1 m— 2 mn — 3 —.' 2 Bri me eis 2 I 2 ig t 2 3 4 ee <2 Xi xX x haraariec he bs + 8c + om, or (if weputA = 1, B=7A,C=7—B,D=7—C,E=7—D, Ee Meena G) Ho 1 K, 1. xo, — 2 FZ =" G, “2H 22 1, 452 5 6 7 8 9 19 K, &c, refpettively,) to the feriesa” + ~Aa”™~'b +7 Ba”? o 4 2 Beige Dy set a"? oS + &e +4 bs in which 3 all the terms after the firft term a” are marked with the fign +-, or are added to the faid firft term. We will now proceed to confider the value of a4 — 2\", or the mth power of the refidual quantity a — 4, or the difference of the two quantities 2 and 4, upon a {uppofition that a is the greater of the two. 28. Now, if @ be fuppofed to be greater than 4, and m be any whole num- ber whatfoever, the quantity 2 — 4”, or the mth power of the refidual quantity, . . . 4 772 2 =~ I mn or difference, a — J, will be equal to the feries@° — —a Jes wits seaita’ yam N12 Mm i= T Mmw—-2 m= wn wmi—~- 1 = 2 tem % —— a b* — — X .b i=> &e, or at =A Rin Be + 7 Ba”. bo Ca™ Fp p23 Da” 4 MF Ba S 4 Be, QE. De You. It. Z A DEMONS. A DE'M O-N S-T R A T 1200 Srat oT! S Ar AS CoN OE Weel © Oe aes Be LigN® © UME Aor) ST eve In the Cafes of Roots and the Powers of Roots, as well as in the Cafe of Integral Powers; publifhed by Mr. John Landen in the Year 1758. N the year 1758 the very learned Mr. John Landen, of Walton, near Peterborough, in Northamptonfhire, publifhed a mathematical Traét in quarto, containing 43 pages, intitled, Difcourfe concerning the Refidual Ana- Lyfis: anew Branch of the Algebraick Art, of very extenfive Ufe both in Pure Mathematicks and Natural Philofophy. Th this Difcourfe he has given us a de- monftration of the famous binomial theorem of Sir Ifaac Newton, that extends to the cafes of roots and the powers of roots of a binomial quantity, as well as to the cafe of its integral powers; and this, without having recourfe to the doctrine of fluxions, which had ufually been employed for this purpofe by the writers that had gone before him. The propofition he demonftrates is as follows, to. wit, “* That, if m and # be any two whole numbers whatfoever, the quantity m 1 + x)”, or that nk of the binomial quantity t ne x oe is a by 2 fractional index = —, will be Bi: to the feries 1 + —* + — x ——owe + — cas mm 1 M—2n 3 WA— in Wi—— 22 m—3n 4 Mi—— x 22 x 3n ape x . 3% va 42 yeh aoe 2” x Bibaaiaee Se) Ze caesar ne zt we oe Ate or (putting A = 1, and ns — =, and C 37 4 mi ae B, and D= Sora, C, and E =~ D, and F = free F, and G, m— tn m— 6n m— 71 m— 3n m— H, 1, KL, &c, for ——* F, “== G, =a H.-J, “SEK, bee, tee fpectively,) SIR ISAAC NEWTON'S BINOMIAL THEOREM. r7t {pectively,) to the feries 1 + ~ Ng aes ie sia Tee ae oe D x‘ 2n 3n 4 rs as oa Ext + == a F x + -—= =e Ga’? + &c. And it is the firft propofi-. ° : . oe 4 / ; tion contained in the faid Difcourfe. 1 fhall therefore here give the reader both the propofition itfelf and the demonftration of it, in the learned author’s own words, together with the reflections with which he introduces it; and, as the demonftration is in itfelf rather fubtle and difficult, and is alfo expreffed in a very concife manner, | fhall afterwards fubjoin fuch an explanation of it as will, I hope, enable the reader to comprehend it eafily. The author’s words are as follows : “¢ Before Sir Ifaac Newton invented the Method of Fluxions, mathematicians *¢ had made confiderable improvements in the algebraic art, and had devifed *¢ feveral curious rules for refolving certain problems relating. to the greateft *¢ and leaft ordinates, points of inflexion, tangents, curvature, and quadrature ** of curve lines, and the cubature of folids, &c. Thofe rules, however, were «© efteemed of little value, upon the appearance of the fluxionary method; “‘ which, being of far more extenfive ufe, was, by the mathematical world, *¢ received with great admiration, and ftudied with great eagernefs. Highly ‘S indeed has that method been extolled by many writers; yea, a certain «< gentleman has gone fo far as to fay, The method of fluxions is capable of *¢ refolving fuch difficulties as raife the wonder and furprife of all mankind, and «* which would in vain be attempted by any other method whatfoever. So that “* itis juftly efteemed the greateft work of genius, and the nobleft thought that ** ever entered the human mind. Pref. to EMgrrs. Flux. <¢ Yet, notwithftanding the method of fluxions is fo greatly applauded, I *¢am induced to think it is not the moft natural method of refolving many «¢ problems to which it is ufually applied.—The operations therein being chiefly *¢ performed with algebraic quantities, it is, in faét, a branch of the algebraic <¢ art, or an improvement thereof, made by the help of fome peculiar principles *‘ borrowed from the doctrine of motion: which principles, I muft confefs, to << me feem not fo properly applicable to algebra as thofe on which that art was, “¢ before, very naturally founded. We may indeed very naturally conceive a «¢ line to be generated by motion; but there are quantities of various kinds, «¢ which we cannot conceive to be fo generated. It 1s only in a figurative fenfe *‘ that an algebraic quantity can be faid to increafe or decreafe with fome ‘¢ velocity or degree of {wiftnefs; and, by the fluxion of a quantity of that ‘¢ kind, we muft, 1 prefume, to have a clear idea of its meaning, underftand ‘© the velocity of a point fuppofed to defcribe a line denoting fuch quantity. ‘© Fluxions therefore are not immediately applicable to algebraic quantities ; “¢ but in fluxionary computations made by means of fuch quantities, we, to ‘¢ proceed with perfpicuity, muft have recourfe to the fuppofition of lines being «¢ put to denote thofe quantities, and the generation of thofe lines by motion. *¢ It therefore, to me, feems more proper, in the inveftigation of propofitions by «‘- algebra, to proceed upon the anciently received principles of that art, than to <« introduce therein, without any neceffity, the new fluxionary principles, de- ‘* rived from a confideration of motion; and the rather, as the introduction a Z 2 ** thole 172 A DEMONSTRATION Oo F 6e cc € ~ ¢ nn ee €¢ 6 € nn €< ec ce é n ce ¢ n ¢ n ec ¢ nn €¢ &¢ thofe new principles is not attended with any peculiar advantage.—That the borrowing principles from the doctrine of motion, with a view to improve the analytic art, was done, not only without any neceffity, but even without any peculiar advantage, will appear by fhewing that, whatever can be done by the method of computation, which is founded on thofe borrowed princi- ples, may be done as well by another method founded entirely on the anciently-received principles of algebra: And that I fhall endeavour to thew, as foon as I have leifure, in the treatife I lately propofed to publifh by fub- {cription.—In the mean time, this eflay is intended to give the inquifitive reader fome notion of the new method of computation, which is the fubject of that treatife.x—Which method I call the Refidual Analyfis; becaufe, in all the enquiries wherein it is made ufe of, the conclufions are obtained by means of refidual quantities. ‘© In the application of the Refidual Analyfis, a geometrical or phyfical pro- blem is naturally reduced to another purely algebraical; and the folution is then readily obtained, without any fuppofition of motion, and without con- fidering quantities as compofed of infinitely {mall particles. <¢ It is by means of the following theorem, viz. bial liad v “w\? v\* ; a ee | bee (m) ceed? Tale hae "y ve 4 d PT on : 3” aA leit Blom ks rear sh sy © (where m and 7 are any integers, ) that we are enabled to perform all the principal operations in our faid Ana- lyfis; and Iam nota little furprifed, that a theorem fo obvious, and of fuch vaft ufe, fhould fo long efcape the notice of algebraitts ! «<] have no objection againft the truth of the method of fluxions, being fully fatisfied, that even a problem purely algebraical may be very clearly refolved by that method, by bringing into confideration lines, and their generation by motion. But I muft own, I am inclined to think fuch a problem would be more naturally refolved by pure algebra, without any fuch confideration of lines and motion.—Suppofe it required to inveftigate the binomial theorem; i. ¢. to expand 1 + *” into a feries of terms of x, and Known co-efficients. ‘¢ To do this by the method of fluxions, we firft affume 1 + ~*” = 1 + ax 4 bx? +- cx? + dx* &c. We, to proceed with perfpicuity, are next to conceive *, and each term of that affumed equation, to be denoted by fome line, and that line to be defcribed by the motion of a point: then, fuppofing « to be the velocity of the point defcribing the line *, and taking, by the rules taught by thofe who have treated of the faid method, the feveral contemporary velocities of the other defcribing points, or the fluxions of the feveral terms —_—— I : . : m n . : af og in the faid equation*, we get a Mile xX x = ax + 2bex + 30x*x 4 « adx’x &c. becaufe, when the {pace defcribed by a motion is always equal to *¢ the * See Mac Laurin’s Fluxions, vol. 2, page 585, Art. 714. SIR ISAAC NEWTONS BINOMIAL THEOREM. 173 ** the fum of the {paces defcribed in the fame time by any other motions, the ** velocity of the firft motion is always equal to the fum of the velocities of the ** other motions. <¢ From which laft equation, by dividing by x, or fuppofing x equal to unity, ane i “ we have — Kate” xmata2bx + 3cx* + 4dx* &c. Confequently, m — é< iplyi 2S TE Gah abd fis multiplying by 1 + *, we have — X 1 + # , or its equal — + — ax +4 — d 6 Dx? ~ cx? 80 = 2. + “ash x + a x™ + ay x? &c, From whence, by <«* comparing the homologous terms, the co-efficients a, 4, ¢, &c. will be found. «« The fame theorem is inveftigated by the Refidual Analyfis, in the following ** manner : “¢ Affuming, as above, 1 4+ «” =1 + ax- dx* 4 cx? &c, wehave 1 +y" = bid bd 1 + ay + by* + cy* & c; and, by fubtraction, 1 + i +" =auK—y C4 bx — y* + Coxe ye d.x* — y* &c. ‘¢ If, now, we divide by the refidual x — y, we fhall get m nd etd ae) si m ie a Mar aT ae are ib I ae od Pn se PY Bee: SRI aa ERR ess ear ree SE | - am a=} y\* 1 +222 | r ea Si, lia (2) I+ 1+-* S~—atb.ntytpe. te tu ty + dx + x*y + xy? + y? &3 which *€ equation muft hold true, let y be what it will: from whence, by taking y mt — J s© equal to x, we find, as before, = Ke we ee a + abe 36x? + 4adxs «© &c. The reft of the operation will therefore be as above fpecified. ** Now, as to either of thefe methods of inveftigation, 1 fhall not take upon ** me to fay any thing in particular; it is fubmitted to the reader to compare <¢ one with the other, and judge which of the two is moft natural.” Upon this pafflage, at the place marked with an afterifk *, the author has fubjoined, at the end of his difcourfe, in page 41, the following note. m ™ v\ wv \3 7” vo” “7 being = «2 xa— V 8 3m ; me tl) t+ +y ** (as 1s obferved in page 5) we have, by writing r + wand 1 + y refpectively, &¢ Sh a — fi + ‘— Her x : +i", I+we—l1+y n—y ; | 4 174 A DEM (OW RR. RAG Hores Serr 1+ I . at. Ht yw, 8ST abet x I + y\# 1 + y\*% n ‘at tg piheale e? This is Mr. Landen’s demonttration of the foregoing celebrated theorem, ex- prefled in his own words in the above-mentioned Tract, intitled, 4 Di/courfe concerning the Refidual Analyfis, Sc, which was publifhed in the year 1758. Six years afterwards, to wit, in the year 1764, Mr. Landen publifhed a fecond Tract on the fame fubjeé&t, containing 218 pages in quarto, intitled, The Ref- dual Analyfis; Book 1. To which the former Traét, intitled, 4 Difcourfe con- cerning the Refidual Analyfis, Sc, had. been only an introduétion. In this fecond Tract, pages 5 and 6, Mr. Landen informs us that one of the theorems which chiefly enabled him to perform a certain divifion of one refidual quantity by another, which is of great ufe in his Refidual Analyfis, was the following, to MW 72 wit and Te Wena ie chai Bact egy tee ee (™) 2 a= Tp) Wi— i W——2m m Mm 31 2m Mwi— 47h 2m 9 Tf fy rwr4ty rwrty rye (fr) w w aw \3 eras, Iephyem i ot (#) os r Vv Vv UV Sore x 5) a agar aie wir wr aw \* r+—| $e 42) © Vv Vv Vv mand r being pofitive integers. And in the bottom of the faid page 6 he fubjoins the following note, containing an inveftigation of the faid theorem. Nore. This theorem may be inveftigated as follows: It is well known, that 773 mm whe : m—t1 m—2 m— = v7 isu + v wevuv 3 wy? (™) 5 U—W and that Zu Be r—I r—2 r—3 ist aa +4 b+a b? (7); Pe mand r being pofitive integers. Mm m In the fecond equation write v? and w ’ inftead of a and 3 re{pectively ; and you will have m wn mM—=mm W— 2001 m M—2m 2m v —WwW oe aaoe ise ra 4) r+tv rwriery POT m mn AL Vv" —w Then, from the firft and third equations, it will appear by divifion, that m mm tay cep © [ ott y@—-2 x (m) isi U—W : m—m m—2m : yo f4y reer (7) SIR ISAAC NEWTON’Ss BINOMIAL THEOREM, 175 which being fo obvious, it is matter of furprife to me, that algebraifts have not before obferved it, and fhewn its fingular ufe in analytics. e Thefe are all the paffages that I have found in thefe two Tracts of Mr. Landen concerning the foregoing demonftration, of which I will now proceed to give as full and clear an explanation as ‘I can. 4s EXPLA: AN EX POL Aegis =e pe FOREGOING DEMONSTRATION OF THE Bul NvO M A°AvLiy ? Vek OR eee In the Cafe of the fractional Index =, invented by Mr. John Landen. By FRANCIS MASERES, Esa CURSITOR BARON OF HIS MAJESTY’S COURT OF EXCHEQUER, a a OR ce SP Fe ene amr A, Boat EC LE os HE propofition demonftrated by Mr. John Landen is as follows: If m and # be any whole numbers whatfoever, and » be any quantity not m greater than 1, the quantity 1 + «)”, or that power of the binomial quantity 1 -- x which has for its index the fraction —, will be equal to the feries 1 + = a % = x “~—- x ae Xx sor x es eat &c, or (putting A= 1,and B= = A, and C = =—B, and D =" ¢, and E ="=¥p, and F = 2? B, and G, H, I, K, L, & = "=* F, and 7= °"G, and ==7" H, and m — Sn n= I, and ——* K, &c, refpectively,) to the feries 1 + = Aw + On Bat + == C at + 2% m —3#AD ye 2-44 ps, A- 58 1g | BORK ye Dex ahi LON ia toy a I Mi c= G x? DEMONSTRATION OF THE BINOMIAL THEOREM. 177 iB a H #* + spat I «9 + ae K x? + &c. His demontftration of . this propofition is deduced from the two following Lemmas. L Gey My Mi A I. 2. If a and 2 be any two quantities whatfoever, of which a is the greater, and m be any whole number whatfoever, the difference of the mth powers of a and 4, to wit, a*— 4”, will be capable of being exactly divided by 4 — 4, or the difference of the faid quantities themfelves, fo as to leave no remainder ; and the quotient that arifes by fuch divifion, or the value of the fraction a” ee bm = s . . ° . Be)? will be a feries of terms in continued geometrical proportion, confift- ing of m terms, of which 4 *—~* will be the firft term, and 2” * will be the latt term, and the common ratio of the terms will be that of a to 2. EXAM PLE S. Thus, for example, if we divide a*—4* by a—4, the quotient will be a+. The divilion is performed as follows : a—b)a *—## (ats a’ — ab +-2b — $* And, if we divide a? — 3? by a — 4, the quotient will be a* + ab + 2%. The divifion is performed as follows: _ @—b)a * * — 3 (a’+ad+ 0 Bien fp * 4 27 b ok a ab ap Seem ys +*ab* — $3 * * And, if we divide at — d* by a — 4, the quotient will be a* + a4 4+ ab + 43, The divifion is performed as follows : a—b) as # % * —~4 (@+ab+ar°+h3 a+&—aeb *4+eb * +ab—a’P + a’ b?— ab} cn Fea aah i D> + ab> — b+ * * Vou. I. teh dared And, 178 AN EXPLANATION OF THE FOREGOING And, if we divide a5 — 2’ by a — 4, the quotient will be a* + 4b + a’ B + ab? + &*. The divifion is performed as follows : a—b) a’ a * & — 8 (a#+4b+P7R% + ad3 + Pi ee ¥* tatd # ele a me a : * + gif? * eg Dein Be Ae am ae Yh AR fen ee +-ab+ — JF 3. In afl thefe examples we fee that this Lemma is true; fince each of thefe quotients 4 + b, aa+ab+ bb, a+ ab 4+ ab’ + 53, and a* + ab+ ah + ab? + 6*, isa geometrical progreffion of terms, of which the common ratio is that of @ to 6, and the number of terms in each feries is equal to the number of units in m, or the index of the powers of a@ and 2 in the dividend; and the . . — I . . 2~=~I firft term of each feries isa” °, to wit, in the firft example, @~, or a, and 4" sen 7 . —fJ - in the fecond example, 237°, or a*, and in the third example, a*~* ora, and in the fourth example, a 5~* or a*; and the laft term of each feries is : By eae —l[ , to.wit, in the firft example, 2*~°, or 4, and in the fecond example, 63", or 2°, and in the third example, 247", or 23, and in the fourth exam- ple, 257", or B+, DEMONS FR AF I_@Ns« 4. And that the fame thing muft take place when m is equal to any greater whole number whatfoever, will appear by performing the divifion of a* — b* by a — din general terms; which may be done in the manner following. — . a—b) ge as ee ee ee Cae 4 Pa tel Oia 4474 gt eee an — a” "b bY 4. &o 4 Ma Ok 4 >| ae PN ie’ Pa adh, Soe iy be 13 eas dig thee eae a a Li. 3B: ag San ata ay, Be % tga. tbs saa A 54 nv. ags is oe 555 DEMONSTRATION OF THE BINOMIAL THEOREM. 179 Aa = 5 é 5 om f 7 Now it is evident that the terms of this quotient, a sel 2 Oe a a m—6 hs wi ‘a i— ' . . p= SP yl ah 5 ed tr dps 1@ + &c, decreafe in the continual propor- tion of ato 6. For the index of the power of 4 in every new term of itis greater by an unit than the index of the power of 4 in the term next before it; and the index of the power of a in every new term of it is lefs by an unit than the index of the power of a in the term next before it; fo that every new term is equal Py MP oo hea tee; to the term next before it multiplied into the fraction —. And, further, the firft ° . . . —TI . . ca term of this feries, or quotient, isa” ° ; and its laft term will be 5” * for the following reafon. The index of the power of 4 in every term is lefs by an unit than the number fubtracted from m in the index of the power of a in the fame term. Thus, for example, in the fourth term, @ “~4-33, the index of 4 is 3, which is lefs by an unit than the number 4, which is fubtrated from m in the index m— 4 of the power of a. It follows therefore that, by continuing the divifion, and obtaining more and more terms of the quotient, we fhall come at laft (whatever be the magnitude of the number m,) tothe terma” ” x ei gore x be On TX b™~*, orb” *; and this term, being multi- plied into the divifor ¢ — 4, will produce the terms ab” * — 8”, which being fubtracted from the laft dividend, (which will alfo be 2b” ~* — b”,) will leave no remainder. Therefore 2” * will be the laft term of the quotient, as is afferted in the propofition, And, laftly, fince 2” * is the laft term of the faid quotient, and the terms of the faid quotient, beginning with the fecond term a” * b, involve all the powers of J, from 4, or J, tod” *, in their regular order, it follows, that there will be as many terms in the faid quotient that will involve fome power of 3, as there are units in the index m — 1, that is, m — 1 fuch. terms; and confequently, the whole quotient, including the firft term wT : . ere & a » (which does not involve 4 in it,) will confit of m-— 1 terms and one term, that is, of m# terms. It appears therefore, in the firft place, that the terms of this quotient will conftitute a decreafing geometrical progreffion, in which the common ratio of the terms will be that of ato 4; and fecondly, that the firft erm of the faid progreffion will be a” ", and that the laft ferm of it will be b”~* ; and, thirdly, that the number of the terms of the {aid progreffion will > be m; which are the feveral points which were to be demonftrated. * LEMMA” Il 5. If mand ~ be any whole numbers whatfoever, and @ and 4 be any two quantities of which a is the greater, and p and g be any two quantities of which | ¢ fuz PJs 180 AN EXPLANATION OF THE FOREGOING p is the greater, the fraction ——, or the quotient of the divifion of pee g? by 2 — g, will be equal to the fraction tH P ate a pein NS a Panes q+ rade g? + &c, continued to m terms, i— mM Wi— 201 1 uh 3 Ld 27 tt — AM 3 — —— —S ? a4 p mgr? +p *xXq"™ +p “2 xq # +&c, continued to# terms. DEMONSTRATION. % % ; a—b : whi ae By the foregoing Lemma we fhall -have Tp the feriesia” |b aaa gG— b+a” 3 4+ 4"*33 4 a” 5 b+ + &c, continued to” terms. Therefore, “i 772 if we fubftitute pe inftead of a, and gq? inftead of 4 in this equation, we fhall a ie i“ m\t—1 m\n—2 oe ays m\2 have’? oem foe fertes | 4 pel a = ry oe 1ay a ? 2 Ae a a a Kog © ie ii Wi pe — 4" mam a3 La m4 mt 2° xX 9” | +p? x g%! + &c, continued to # terms; that is, - ea TH siete 3— 271 re Mi — 270 = Mm — 402 “3m will be = the feriesp *+p * xXq* +p *@XG@* +p * KG" m — G7 Am +p xq” + &c, continued to w terms. ‘Therefore, the reciprocal of 2 m y hse — q” M4 this fraction, or the fra€tion , will be equal to the quotient that arifes Wii — mM— 2% 2 1 — 272 2 Mm — Ait by dividing 1 by the feries p *+ p aX gq ep gq? +? Pp 4m w— Sm 4m q? +p * Xq* + &c, continued to # terms; that is, the fraction will be equal to the fraétion Wi — i2 Vi— 2 2 i= 371 272 ? a ? aX q? +p Xx q* + &c, continued to # terms. iat gp orig pr ueaes Se Bike g* + &c, continued to m terms, Further, by the foregoing Lemma, the fraction is equal to the 74 feries hae = a DEMONSTRATION OF THE BINOMIAL THEOREM. 181 m4 m pe g* mterms. ‘Therefore, if we multiply the fraction =>, by) tha «fraction 1A See m m , a and the fraction ‘ dmg I Wi — iT i — 271 ” i — 372 27m Petes xX gz +p 7” xX gt + &c, continued to w terms, by the mee +p gp Seas 7 ie g3 +p” > q* + &c, continued to m terms, the products thence arifing will be equal. But the produét of the wz mW eee by the fraction Py ae 2p” ans ae multiplication of the fraction is the fra€tion m 714 — — ae il ; and the produ& of the multiplication of the fraction ! alk! T Dy 7 — 71 Wi— 22 Mm Wi — 27% 2h —- Pp *+) 2 x gq” +? = x q? + &c, continued to # terms, by the > feriesp" +p gtp %gtt+p” *q3 +p” °g* + &c, continued to m terms, is the fra¢tion W— 1 Wi—2 ui— M— WM—SF ° ?P eee peg EP aoe cw 43 +? ; qg* + &c, continued to m terms, 1 — tL m—2m m m—3m 2m i p +p *q"* +p *¢* + &c, continued to z terms, m Mm P rag Gl Therefore the fraction —— will be equal to the fraction id pate og tp OG tp *g2 +p” > ¢* + &c, continued to mterms, W—m Wi— 272 2 1 — 3748 2m Wit — 472 3m eet tp FF gt tp | 8 gt +R 2 g@* + &c, continued to # terms, Q. Ee D. Mt M2 ait pram m 2 Pattee ot Ae hevtaids GaGionk=. nu —_——T isalfoequalto p* x the fra¢tion 9g) 7 \ a\e, (q\ 1+ i + 4/ + val + ay + at + &c, continued to m terms, Tae. pie cae I+ a + ok + ay -+- 4" + &c, continued to # terms. p p p | p! ge pg tp tgs + pgs + &e, con- "x the feries1 +p q+ prt Beep, 9 Age 4. ” q* -+ &c, continued to m terms, For the feries p” ° + p A ‘ m— tinued to m terms, is = p 182 AN EXPLANATION OF THE FOREGOING — 7" he feri ae ele 4. bat pay & ontinued to m# terms pa x the ieries 1 yo Py e rs f c, con ’ =p” " x the feries 1 er a ksi Li ‘ahs & a + my + &c, continued to m terms. m1 — 7m M—20t in —3m 2m wz—Am 3m _-- —_—_ Andthe feriesp *+p * g® + Me ag*%™ +p %*g* + &c, conti- nued to # terms, is = Lhe sh he wane —2 27 ~— Bin 3% ? mse tHE REIS ER agh +p %G% +p % g% 4 &c, continued to # terms, <— Phos m 2m 3% Te SeaG he teri hee MS a q” ? x the feries r+ —— + med Bees re + &c, continued to z terms, = p % y) a p® m— Im ig 2m hdd 2 “2 x the feries 1 4+. ay “ 40 + ey + &c, continued to z terms._ Therefore the aes =f 509 en ' : He +2” “9 tf p- 34 * + Be 493 = Ie - Dh g* + &c, continued to m terms, Pp c Pip ” gi” hip 8 g® +p = gt + &c, continued to # terms, W=-T will be = the fraction = i the fraction . Por s alk + re a + &c, continued to m terms, ae I+ oucr a + ay + &c, continued to # terms. peat BYNES : wie —m+m thm Ym teh 8 But the fraction —— is = p xX——=) Xp t= Dp a mH Py es mana . be —1 Desc, , Therefore, if we fubftitute p* — inftead of ——— in the laft equation, we ? 2" fhall have the fraction Pr gp 7g tp” 3 gt 4p” *g3 +p” > 94 + &c, continued to mterms, m—m m—2m° mm m—3I 2m m— 4m 37 p "+p *q™> +p *%9g% +p %*4* + &c, continued to # terms, mn ——I =—p"%” % the,fraction. DEMONSTRATION OF THE BINOMIAL THEOREM. 183 Bey 7). 73 7} ie “ 1 + ; + dl ++ ai + ‘7 + &c, continued to m terms, I g” q \* f[ a —— a | “+ a, re . . oh . a" a + &c, continued to # terms. 22 4 a aw Therefore, the fraction seer Ca (which is fhewn in the foregoing Lemma to be equal to the fraction M— 1 Mn -~-2 —2% _ _ : p + Pp qt py 3g +p” 493 +p” 5 94 + &c, continued to m terms,» wa—m W—2tt mM pes i 2m Wi A702 3m : ea q” 8 Sod len q FP aq +. &c, continued to m terms, - i will be equal to pe x the fraétion 4 q | r+ f+ 2} 4 L m 2m 3m 4m 4 4 : + &c, continued to m terms, eatet tates Thefe things being premifed, Mr, Landen’s Demonftration of the Binomial Theorem will be as follows : 7. In the firft place, Mr. Landen fuppofes that the quantity 1 + «\*, or the —th power of the binomial quantity 1 + x, will be equal to a certain feries of quantities, of which. the firft term will be 1, and the following terms will in- volve the feveral powers of x in their natural order, to wit, #, x”, «3, 74,45, &c, multiplied by certain fixt numbers, as co-efficients, which may be denoted by isa the letters B, C, D, E, F, G,-H, &c, or that 1 + x|” is = the feries 1 -- Bw + Cx? + Dx? + Ext+ + Fri + &c, in which the co-efficients B, C, D, E, F, &c, are hitherto. unknown, and are now to be inveftigated. And this he fuppofes to be true, of whatever magnitude, not greater than 1, the quantity « be taken. i 8. Now, if 1 + x|* is always equal to the feries 1 + By -+ Cx? + Dx? + Ex* + Fx 8c, whatever be the magnitude of », (fo long as it is not greater than 1,) it follows that, if we fuppofe » to decreafe from its firft magnitude: (which we will fuppofe to be denoted by x,) to a magnitude fomewhat lefs than. wa x, which we will call y, we fhall have 1 + y\* equal to a feries containing the fame powers of y, combined with the fame co-efficients B, C, D, E, F, &c, refpectively, as there were powers of « in the former feries which was equal to in . I xl”; ort + )\* will be equal to the feries 1 + By +Cy?+ Dy? 4+ Ey? +Fy> + &c. Therefore, if we fubtract this latter equation from the former, we 184 AN EXPLANATION OF THE FOREGOING tid we { thall haver +x” —(7 + ne = the feries 1 + Be + Cx? + Dx? + Ex* + Fat + &c — the feries 1 + By Cy? + Dy? + Ey* + Fy' + & = Dosey —ytCxw—y + tD x #P—y EEK w+ — yt + FE KH — ys + &c; and ow enuly if we divide both fides by « — y, we fhall have LUT 2 2 3 3 Te al" ge eg x — y ol hces * 4 ar $$ oS er Bis x SS + F x 5% + &e = (by the fit Lemma) B+ C x #494 D x P47 +B x8 yay ey XE oy of Bre But x —yis=1+"—(t +. Therefore oi will be = I+ x —(1 + 7 xt + xy wry? + xy? + y+ + &c. Now it has been fhewn in the Corollary sich ee pe —qt : gem | I Tie, a. of 4; + i+ 2} + &c, continued to m terms, m 2m sis oe mt is = p” XX the fraction to Lemma 2, that I i ou ay ax 4! + ni + &c, continued to z terms. Piibcelste! if we fubficae 1+. for p, and 1 + y for q, it will follow that idk mM seal) oy will be = 1 + Pe x the fra&tion Ii+*— by + ¥ 2, 3 “ 4+ Lt) +4 &c, continued to m terms, — 1+ mt 2m 3m ae = Se 1 1+» ‘7 I = nt I +y ‘74 & . c, continued to # terms. fer ares Targners Remi, 7 2 m : rep isla But it has been fhewn that —————-——— 1+«—(1+y7 e+ytyt+Ex® ery tay ty tF xa pay ty py Fy m 4 is==B+Cxxr+y+Dx + &c. Therefore 1 + ne x the fraction 1+ 2 bit aha + — +- &c, continued to m terms, v 1 ve + # 2m 3m ~ arnt) eee ae ie ay" PTE cad abt ine : tinue A bara: Sl inee Le es + &c, continued to # terms, will DEMONSTRATION OF THE BINOMIAL THEOREM. 185 willbe =B+Cxx+ty+tDxe¥+y+y+tExv pep ty + Fox xf + xty + xy? + xy? py + y* + &c. 9. Now let y (which was fuppofed to be lefs than *,) be fuppofed to increafe gradually till ic becomes equal to x. Then, fince the laft equation is always true, however little y may differ from #,. it will alfo be true when y becomes abfolutely equal tox; as might eafily be proved by a demonftration ex ab/urdo, if the matter were not too evident to make {uch a demonftration neceffary. But, when y is equal to x, we fhall have x + y = 2x, and ** + xy + y? = 3x7, and x? + xy + xy? + 9? = 4x3, and xe xty + x?y? + xy? + y* = 5x43 and, in like manner, in the following terms of the feries B+ Cx«+y +DXx° 4 xy +¥ +EX«# + xy Wes +93 + FX xt + xy + 47y? + xy? + y4 + &c we fhall have 6x5, 7x°, 8x7, &c. And 2 I +2 will, in this cafe, be? = = a= I; and awe a ed and : 3} : rea and all the following integral powers of - 2, and lkewife i ae =F and * a) and + 2" , and all the following fractional powers of - = , will become equal to 1. Therefore the laft equation, to wit, 1 + w > the fraction I+y ty st Babes + I Ey =a as z ntinued to ” terms | a aa teas x tee pias ‘ ae Cex by Dak Foch psy PBX eo et ot ty ce a “+ xy? + y* + &c, will, in this cafe, be transformed 3 + ii continued to m terms, x {Q into the equation 1+ mi "x the pueied r+1+1+1 + &c, continued to m terms, se nimi 10 tea I+1+1+1 + &c, continued to # terms, m ——r1 +E x 4x3 + FX pxt 4 &c, or 1 + +\" X>=B+4+C%X2x+D x = Bik C % a” +-D>x 30° rae ‘ gx? + EX 4x* 4+ Fx gx* + &c, or— xX 1 + a\" = BoeyC ox ae D x 3x*° +E x 40? + F x 5x* + &c. Therefore, (multiplying both fides of this laft equation by 1 + x,) we fhall have = XT+alt* = B+C x2ae+D x 3x7 + EX 4x7 + Fx oxt + & +Bx«+C x 2x? + DX 3x7 + E x get + &e = B + 2Cw + 3Ds + 4x3 + 5Fx+ 4+ &c + Bw + 2Cx? + 3Dx? + 4Fx* + &e. Vou. I, 2B But 186 “AN EXPLANATION OF THE FOREGOING But 1 + x is = the feries 1 + Bx + Cr? + Dx? + Ext + Fr' + &e, mm and confequently = Xi + x is = - x the feries 1 +--+ By + Cx? + Dx? + Ext + Fei 4- &c-= the feries — aE — Bx - = Cx? — Dx: + — Ex* + “s Fx' + &c. Therefore the feries — + — Bx + = Cx? + — Dx? + = Eoet — ~ Fs + &c will be = B + 2Cr + 3Dx* + 4Ex? - 5Fxt + &c + Bw + 2Cw? + 30e + 4x4 as Ree 10. By the help of this equation — ate — Be +. — — Cx? BBE 2s — Dx: piters Ewt+ + — Fx: FO Lt eg Moma '. B+ 2Cx + 3Dx? + 4Ex? + 5Fxt + &c + Bw + 2Cr? + 3Dx? + 4Ex* + &c we may determine the values of B, C, D, E, F, &c, or the co-efficients of the powers of x in the affumed feries 1 + By. + Cr? + Dx? + Ext + Fro + &c m (which is = 1.+ at ») by proceeding in the following manner : | 11. This equation is always true, how fmall foever we fuppofe the magnitude ; of x to be: and therefore, it will alfo be true nee ® 1s = 0. But, when x is = o, all the terms on both fides of the equation = os = Be + = Cx? + = Dx? + — Ex* a — Fes + &= B + 2Cw + 3Dx? + 4Ex3 4+ 5Fx* + &c + Bre + 2Cx* + 3Dx? + 4Ex* + &c . - that involve any power of x, will become equal to o likewife, that is, all the terms, except - on the left fide of the equation, and B on the right fide of it, will become equal to o; and confequently, = and B will be the only remain- ing terms of the equation. Therefore B will be = ar that is, the co-efficient of x in Bx, the fecond term of the aflumed feries 1 + Bx + Cx? +. Dv3 4° Ex¢ 4 Fxs -- &c, which is = 1 + md » willbe = =. SOS OBE tata 12. To find the value of C, the co-efficient of *? in the third term Cx? of the faid affumed feries, we muft proceed as follows : Since = -f- — Be + — Cx? + =—De += Ex + &c, is = B+ 2Cx + 3Dx* + 4Ex 4+ Fx + & + Bw + 2Cx* + 3De374+ 4Ex* + &C, and B has been fhewn to be equal to =; it follows that, if we fubtract — and B on DEMONSTRATION OF THE BINOMIAL THE@REM. 187 on the oppofite fides of the equation, the remaining quantities will be equal to each other; that is, the feries = Be + — Cet < Dw? + — E«* + &c will be equal to the feries 2Cx + 3D" + 4Ex + 5Fet + &c +Be + 2Cw? 4+ 3D} 4+ 4 Ext 4+ &c. Therefore (dividing all the terms by x,) we fhall have — B+ —Cx steal Dw? + Ex? + &ce= 2C + 3Dx+4En? + 5 Fx + &c +B+2CK% + 3Dx* + 4Ex3 + &c, and (fuppofing x to become = o) = B=2C+4 8. Therefore 2C will be — Be B= Hx B =o x B; and C will be = ~~“ B = =—* x Zn _ mM ni . . ° é : eer Xs —— ~; that is, the co-efficient of «? in the third term Cx? of the afflumed feries 1 + Bw + Cx? + Dx} + Ext + Fx’ + &c, (which is = mm & + xl” ,) will be =— x, B, or — x —. Onghants 13. To find the value of D, the co-efficient of x? in the fourth term of the faid affumed feries 1 + Bx + Cw? + Dx3 + Ext + Fes + &c, we mutt proceed as follows : It has been feen in the laft article that — ee B+ — Cx + = Dx? + — ~ Ex? + a Als 2C + 3Dxe + 4Ex* 4+ 5 Fx? + &c Satie m * &e is = da Bee ape 1 wae digo}? 204 likewife that 2 'B is = 2C-+4B. Therefore, if we ek ee this laft equation fom f the t ier. we {hall have — Cx Dax mm Ex? + &c = 4 Dee spd Bax -- s Fx + &c +2Cx + 3Dx* + 4Ex? + &c by «,) —C +=De +2 Ex + &c= 3D+4+4Ex + 5Fx* + &c +2C +3D* + 4Ex? + &c =a -+.2C. Therefore 3 D will be = —C —2C= Pax cats al Op and D will be = “—* x C, ie ah ae cg ed a my mas m— 22 ; that is, the co-efficient of x? in the fourth term of the feries 1 + Bx , and confequently (dividing all the terms i, and (fuppofing * to become = o,) = G 20 2m x — 2% + Cut + Da? + Ext 4 Fx? 4 &c (which is = T+ -\",) will be = = m—k m= 2n x ERT Q. E. f. pt teh tae: 14.°To pY aap. OF or — x 186 4¢2 AN EXPLANATION OF THE FOREGOING 14. To find the value of E, the co-efficient of «* in the fifth term of the m feries 1 + By + Ca’? + De’? + Ext + Fut + &c (which is = 1 + x: ») we muft proceed as follows :. It has been feen in the laft article that — C a ~ Dw + ~ Ex? + &cis= e & ay tan ine and likewife that — C is = 3D-+2C. 3 n Yherefore, if we fubtract this laft equation from the former, we fhall have m m : Bak 4x + 5 Fx’ + &c = — De +— Ex? + &c= fe Daibia Base Ge , and confequently (di viding all the terms by x,) =D a — Ex + &c= ZE + 5 Fx + &c + 3D+4Ex + &c 3 D. Therefore 4 E will be = ~D—3D= = 4! x D= xD; and i and (fuppofing « to become = o,) =~ D=4E + confequently E willbe = ae x D; that is, the co-efficient of «* in the fifth term of the affumed feries 1 + Bx + Cx? + Dv? + Ex* + Fx + &c, mm -@ Pe Ses TUR Toe eee : m — 34 aa m—n Mm — 2n Mm — 38 (which is = 1 + «{*,) will be te % Dj Or =X ay XK ctaehs x =a? Q1)'Es Hed 1s. Laftly, to find the value of F, the co-efficient of «* in the fixth term of the affumed feries 14+ Bw + Cw? + Dx? + Ext + Fu’ + &c, we mutt proceed as follows : It has been feen in the laft article that = D+ = Ex + &cis= 4h + 5Fx + &c +3D+4E"+4+ &c fore, if we fubtract this laft equation from the former, we fhall have -- Ex + CO 5 EOL 6Fx + 6Gxe? + 7H «3 + &c A — Es ch gents Seer ate 44 Ee +5 Fx" 4 6G 44 &e » and con fequently (dividing all the terms by w,) — E + — Fe + = Ge? + &c = F+6Gx+7H«? + &c , na val a AE + oFx + pers Me ete and (fuppofing x to become = 0,) —E = 5F + 4E. Therefore 5 F will be = E—4E = a 41x Py eee 8. \, and likewife that ~ D is = 4E + 3D. There- m— 4n x E, and confequently F will be = x E; that is, the co-efficient of x5 Peis in the fixth term of the affumed feries 1 + Bw 4+ Cx? + Dx? + Ent + Fe mm + &c (which is = 1 + x]",) will be -—® x E, or — x 2 x Fo Sn an 3 mm — 3 i — Gn S) ‘42 x TS gnee Q. E. T. . 16. And, DEMONS. TRATION OF THE BINOMIAL THEOREM. 189 16. And, in like manner, we fhall have for the determination of the values of the following co-efficients G, H, I, K, L, &c, the following equations, to wit, —F=6G + 5F, —G=7H+ 6G, ~H=81+ 7H, —I=9K + 8], and —K = 10L + 9K, SECs And confequently 6G==F— sFa2 5x P= 7o* y F, and 7H == G —6G= 27 x eee iG! n 2 foe et Sy el # a nt a 2 aprdhonK Sst al gi8 Tne Be Po and1oL = 2K —-9K =" %y K = 2=* x K, &c; and confequently Gx rate ay x F, On and H = =" x G, andl = 7" ae ss _ m— 8x and K = i eee and Mice eK 10% 17. And in the fame manner we may determine as many more of thefe co- efficients as we think proper, the law of their generation, or continuation, from each other being manifeft from the manner in which the final equation = + —Bx + = Cy? + —D x? + —E x’ + “ es co 3 Dx? + g4Ew? + SF xt + &c pC x? + 3D? + 4Ew+ + &c 4+ Be i in Art. 9, (by means of 18. We may therefore conclude that, if m and # be any whole numbers 74 =. "1 4 was um me tm — nh ae whatfoever, 1 +-«|* will be equakto the feries 1 + Pe. Bee. Marea es ceo iil m — 2n m m—n m— 2n Mm — 3M m Nm 2 x opens 3 Ly ee © x A id ee od ger = hyn tha hia eae man a Gay Sua ae rea RT dite 199 “ AN EXPLANATION OF THE FOREGOING $2 — 298 Mt — 3n “Mi — 4h. ys atid ey aL Ati oh ua i Ar ia oP te <5 &c, or (putting A = »B= Sy eis BAD eon Pp te and G, H,1, K, L, &c, for 1 the co-efficients of the following powers of w, to-wit, x°, x7, x°, 4°, «7°, 8c, derived from the former co-efficients by the fame law of generation or continua- Nim 12 Sees 2” tion,) to the feries 1 + —Avw +- ——" Bx? + ret Cs + eee “P mM—-4n wp. m— on 6 m— 62 For had 8 m—82z 7 6 m — On . Exe + “a Fx + mae Gwe? + Bi Hx" +- - Ix? + ee Kw? + &c. Q; WEv dhs 19. Coroll. 1. If mis lefs than », (as, for example, if m is 6 and z is 17,) #7 — n will be a negative quantity, and therefore the third term of the feries, to wit, -— x — “x, will bea negative quantity likewife, and will be equal to. 7: _ x = «*, and confequently muft be fubtracted from the two firft terms 1 2% Hi— 22 and and — x, inftead of being added to them. And, in like manner, gal ef . a : ; : he powers of x in the following terms of the feries, will alfo be negative quantities, on account of the excefs of 2”, 3”, 4, &c, above m: and confequently, thofe terms in which thefe negative factors occur an odd number of times, will be m— An , and all the following factors in the feveral co-efficients of the negative, and muft be fubtra¢ted from 1 and — x, the two firft terms of the feries ; and thofe terms in which the faid negative factors occur an even number of times, will be pofitive, and muft be added to the faid two firft terms; that is, the third, fifth, feventh, and other following odd terms of the feries will be negative, and mutt be fubtracted from 1 and - «; and the fourth, fixth, eighth, and other following even terms of the feries will be pofitive, and muft be added to the faid two firft terms. Therefore, when m is lefs than x, the foregoing feries, when its terms are fet down correctly, with the proper figns prefixed to ' F e Mm mM n 2 (%”% them, will be as follows, to wit, 1 + me Re 2n — Mm m Eek ys iii gy its al Lag Sl «ae ne coor ht eee 3u4— m 3n n 2n 3n 4n 2 2m 3n 4n arnt — &c, or (if Abe put =1,B= =, C= ee BD ae = GE 4 — Mm n1— mM n—m 62—m hw = 3 ee u = E, and G, H,1,K,L, &c,to o—" F, — G, “— H, 8 l— Mm Qu — m . m2 a— m 2n —— 18 sae I, —— K, &e, refpectively, 1 ay pre a, ee rete oe tae = forked pee re 4 — Bs Me Beater Feo 4 On mitt ics od aos = Be i. 82 — m 4n . 5 6n qn 82 Qn I x? aaa K x? + &c. _ 20. Coroll, 2. If m is an exact aliquot part of m, and is contained in it p times, or, if 7 is = pm, p being put for fome whole number, the foregoing feries will exhibit the pth root of the binomial quantity 1 + w. : I | . . For — Mm u— DEMONSTRATION OF THE BINOMIAL THEOREM. Igi For we fhall then have == ry, and “—~ ct so ae re and there _ pum—-m _ 2p—t 3S —1 6—1 — 8s — a ; &c, equal to ame si : a a eae and res -, &c, refpectively ; and 174 i confequently 1 + x\’ will be = 1 + «\?, and the feries 1 + —x— ~ x= 2n mw Wm IE SD mes: T78 Besa aks. Diab gk yy kn Wr ais alt 20 fl obata ik ny x 22 x 3n " n x 22 x 32 x An apie 2 x a— Mm 24 — m 32 — m At os : "uae I ov iey Pies eer tiaan x eae x ote < &c will be = 1 + z x x 53 A I pot 2p vx I) vs I Poe 2p — 1 3p — 1 I jieee ll x + — x D— x S— we — — x S— x A x Dit tp = gene vs ap 3p Bee aap) Slee Oa ep ap I x rg * a x —— — &c. Therefore 1 + «\?, or the pth root of the binomial quantity 1 + x, will be = the feries 1 +4 5 na : p-!l io I p-i 2p-—!t , I p-! 2p—1 Vat ere ——— — xX — X S— — — x — xX Oe xx a eee P ite NEE aan re ws 9 Sa ep ret MT s — &c, or I = tAS mig eae ge + 2p 3p wie: SP P 2p ee Cit 22132 2 — 4 ee SF ETc bes oy OC, : 3p 4p ny 5p ies An Example to the foregoing Corollary. 21. Let it be required to find by means of the laft-mentioned feries the cube - I root of the binomial quantity 1 + x, or the value of 1 + “3. Here p is = 3. We fhall therefore have B= —xA=—A, gel itens 4 and fs 2p Biss 2x3 pees ~ B, AND Yn = saul 9 C= '2.C, 3p a9 4 and E = 32-- D = 2=*~ D=-—D, 4p 12 12 ind Pa ES Ee EB, ay nas ‘ TY aes _¥ ~ and, G, =r B= AG F= QF; ad ty = ogra Brig wag; ip a1 21 and |] = 2—*yH-2%"H= 4H, [92 AN EXPGANATION OF THE FOREGOING and -K = Re = ie ; ne ae ee ahd phen —s 0) | ek a = 75K; and Mo PE Le BS a, 11p oe 33 se hp I oar 5 L_-3% andi - = a 36 M===M, and @ a RENE Shoe ine 13p 39 | And 2 ee a, ee a 4p 42 42 Tap. st Ae Pi 4! and LS Pa ee Q Sp 45 45. 4 : — oT oe ae oe SB andere 53 y= 8 Q= me and S ponte NE ee SE eel eae 5 SO ara ee are 18) 54 & ba 34 Therefore the feries 1 + : Ax —— SB ok 4 SE Seine vo Dxwt + Bee eR Bo ae ~ Fxé hac bes vs beat +— Cr? — ede Ene ~BRes dor Be +e See 1z : . 16 pe 30 Ext 12> Mia? Stl 2 Ox" P x" +2 Rx? — 2 “ " 2 a: 3 ras a 5 # Qs st 7 8 18 Leu aay HM ee ee: aac 37427 935% St RO. le wise 243 ' 729-6561 19683 59049 21505x%9 553913479 | 147,407.47? 1170725 64"4 41,27 3,960x73 15945323 4,782,909 ' _ 14,348,907 129,140,163 5,036,466, 357 112,020,3202"4 918,640,424475 2,526,261, 166x7° 118,734,274,802x17 15,109,399,071 135,984,5911039 427,9539774:917 20,805,642, 520,767 I > 68, 6, 870,060: 18 ’ ‘ — aa sf a neat a ae + &c, which therefore is = 1 + x)’, or the cube root of the 9/5°9 o/ binomial quantity 1 + #. Q. E. Ie 22. Coroll. 3. The foregoing propofition demonftrated by Mr. Landen, to m wit, that, if m and be any whole numbers whatfoever, the quantity 1 + «\*, or the —th power of the binomial iri I + «x, is equal to the feries 1 + — nm m un : m | nm— Mt — 2n m2 w—n 1 — 20 et — X et —— x SS + SX x x nu 7 3n ¥/ 4 22 3n — a? as e ops A SO AMT Me Wag eg ie* De. main Se. co Be eee ee &c, includes within it arts 2n 32 42 a the cafe of the integral powers of r + x which was the fubje& of the preceeding difcourfe. For we need only fuppofe m to be greater than #, and to contain it ptimes, DEMONSTRATION OF THE BINOMIAL THEOREM. 193 Py : . ” ? times, or to be equal to pz, and we fhall have 1 + «/’ = ¥’. Therefore in this cafe the foregoing feries will be = 1 + x. But, ee — is = p, and m is = pn, we fhall have (iia) epi se, Bhim ft inh PAs FS er he eT ee Ui — 218) LPP — 2 Np — 2 and oer aaeiae mae and: ee esi PTI 4n eran w ay and a NY ce date can pre 2 52 ah ee and, in like manner, ~—, 7— ‘3 (Dai lsd ahs gaa SEN ea Set A mit at vAD 82 gn 10” : 62 tgs gt dalla a jam aa a8) por Ricahat son? and » &c, refpectively, or to—= ; —— , = ——* f =, and 4 —, &e, silpedtively and nee, the tics I--— x bank « 2+ 2x 2 aa ep = Ga Ce ss Mh oc Bt Se EE x RG e+ Heyl be he fs 4 pe + p x ee hase SEARLS TRS es SRN yy x Pat eee SNe 7a x at ys + &c. Therefore Taw will be = the Re “x? + p xP x Ft ep x ES x FH x a3 xs +p xf WE a x bt y +. &c; which is the bino- mial theorem in the cafe of integral powers, which was demonftrated.in the foregoing difcourfe. VoL. as mG A DIS- D TDS «Cw On Us Ro 23 CONCERNING THE BL. NSO Mp A cbr To CE SOL LR ae In the Cafe of fractional Powers, or Powers of which the Indexes are Fractions. By FRANCIS MASERES, Esq FR.S. CURSITOR BARON OF HIS MAJESTY'S COURT OF EXCHEQUER. crane EER grees ED A BRO Cortes ion poke HE Propofition known by the name of the Binomial Theorem, to wit, ‘¢that 1 + x)", or the mth Paice of the cama quantity 1 + 4, is equal to the feries 1 + oepox Sluis tivcleeee Se x — 2s — { aa mae Sse Mirah ORE Sete tx Be em mee aX eee 5 2 near? 3 is found to be true, not only in the cafe of the integral. powers of 1 + *, or when the index m of the power to which the faid quantity is to be raifed, is a whole number, but alfo in the cafe of its roots, or when the index m is a fraction, of which 1 is the numerator, and any whole number whatfoever is the & . I 5 I I I I . . e denominator, fuch as the fractions gh oO eee RSPR LE aes &c; and likewife in 4 the cafe of the powers of any roots of the faid binomial quantity 1 + »¥, or the roots of any powers of it, (which comes to the fame thing,) or when m is equal to a fraction of which any whole number whatfoever is the numerator, and any other whole number whatfoever is the begat {uch as the fractions 6 let 3, a 4, & | &c, or 3,4, 4, =, 4, &c. The truth of this theo- rem in the former cafe, or when the index m is a whole number, has been already fhewn in the laft tract but one in this collection, from page 153 to page 169; DIVHIEe WIN OM I ASL? SPH SO CR IECM. 195 169; and therefore will here be taken for granted, and made ufe of as a flep in the reafonings by which we fhall endeavour in the prefent Difcourfe to eftablith the truth of it in the other cafes. But, before we attempt to demontftrate the faid theorem in thofe other cafes, it will be proper to fet it forth in the ex- preffions and forms in which it will appear in thofe cafes, upon a fuppofition that the faid theorem extends to the {aid cafes, or to fet down thofe expreffions of the firft theorem above-mentioned, which will arife from a fuppofition that it continues to be true when the index of the power to which the binomial quantity 1 + ~ is to be raifed, is a fraction, as well as when it is a whole num- ber. And, in doing this, it will be convenient to divide the fubjeé into three cafes, according as the index of the power, to which 1 + »# is to be raifed, fhall be a fraétion of which 1 is the numerator, and any whole number what- foever is the denominator, or a fraction of which any whole number whatfoever is the numerator, and any greater whole number is the denominator, or a frac- tion of which any whole number whatfoever is the numerator, and any /efer whole number is the denominator. The expreffions that will be derived from . . — iI = — the original feries 1 + ~x4+— x 7 Pe PSE 8 ee x re oe i eS x Ot a fe > &c, according to thefe different fuppofitions, will be as follows: Of the Binomial Theorer in the Cafe of Roots, as derived from the fame Theorem in the Cafe of Integral Powers. 2. When the index m of the power to which the binomial quantity 1 + * is to be raifed, is equal to a fraction of which 1 is the numerator and a whole number is the denominator, let # be put for the faid denominator, or let m be fuppofed to be = ~. Then, by fubftituting oa inftead of m in the terms of the foregoing theorem, 1 + 4|\”"=1 + —x+4 — at elie SE i Snack RN een 2+ 3 ie m— 1 m— 2 m—3 4 m m—tI m—2 m— 3 ey Mat yes < LETT are sik EET Ey : ny F ea ER = = 4%5 +. &c, we fhall (if the faid theorem extends to the cafe of roots,) have I 1+ x” = to the feries a ee ea a ee n Sit n . n # n 3 — —— == eer 7 sak saglgi' ghgiy a tan es aR pokes =r Raeeaer aa I I I I eet eee a ein — Sa eS 4 tise a iaaetal aS Xx are I I ? ¢ I Pop) CIty In eee & Bhs -s Toews h ras 3 x ri x 13, _ +. 196 A MDA Sic ‘O0U'R SHEA CMON CER NOIUNTS I1—2 I t-—2 I — 2” te Sg Say sae ee +—x x “a n 2 % n n 3 Te 4 2 2 I—2 I — 2” I— 32 An “ n 7m n tS x «4 4 3 4 I— - I — 37 I~ +ix = baat 3" aA x x x —— x + &e = 3 4 Ma i ey RS T2205 I I-— 2% L 28 is ES One Eanes se Beer hae 3% a Ag in esrila eve basics Do yee n 2n 3n 4n aE dg dno A LG, vg alate“ ipeasanl ing are npr pats £ nt 2n 3n 4n 5a PR elit ply Sci alta tt a et hs co EAE eA , n 2 22 3% gaa b ny tT eb Sat Ser ine a 2n 3n 42 PEO BYT Ae A 32 — 1 4n Sys Soca 7 20 3n 4n that is, 1 + «|", or the wth root of the binomial quantity 1 + #, will (if the . . . I binomial theorem extends to the cafe of roots,) be equal to the feries 1 + — I 2 me I a—I 2n— I A— I 24u— 1 an— 1 _— ae xX x — — 3 — = oe cee 2 iar - 2n ce 32 3 n x 2n * 32 os 42 Pr, ip ny >. ype SIS ails guoneribiana wf mail iV Rac oiAbay Oyo! bye (if, for the fake of 73 2n 3n 4u 52 brevity, we put A= 1, aad B= —A and C = 21 B, and D = ~— C, 2 ” 22 3% and’, = 2" DP), and F = 2" b.. and. Ge L, Rae 42 gz gz —I Gee ¥ Gx. Oe “nm — 1 82 — 1 gz — I : SAG Reg j clon aS G, ——H, e i fe K, &c, Sete ely,) to the feries 1 + : Ax [= By ham a 4D x45 Vo Exs —V— Fé mci Bote y bee trys hey faire (et eo one in which the nears term -- at or x, 1S atc with Hit fign +, or is to be added to the firft term 1, but all the following terms are marked alternately with the figns — and +, or are to be ab Par from, and added to, the faid firft term 1, or the two firft terms 1 ese — x, alternately. This is, as ] apprehend, the cleareft and moft convenient way of expreffing the binomial theorem in the cafe of roots, or of fering down the feries which foal is equal to 1 + RY, or to ig 1 + #, or the #th root of the binomial quantity I+. 6 Of THE BINOMIAL ‘'TH-F*O'R' EM. 4 197 \ Of the Binomial Theorem in the Cafe of the Powers of Rootey when the Index of the Po: wer of the Root is lefs than the Index of the Root. 3. When the index m ftands for a fraction, of which a whole number is the numerator, and another whole number, greater than the former, is the denom1- nator, we mutt fubftitute ~ inftead of m in the theorem ftated in Art. 1, to wit, the theorem, 1 + x)” = 1 Ea Nato sili a ad ere wef Bx At yx Baty May eS ev er Wa" x 7S? x 7S3 x 74 x* + &c; and then (if the binomial Hieafeta eater to the alee) we fhall 2/4 have 1 +- x\’ = the feries 1 eee i si at ae a as 7 n n Fs n 7m na Sek tf cigs 1 oh aes x . I 2 3 a ~ —-—2 ft | saben ——2 peek ay, vA i iL — % x x aes a x pepe I 2 3 I iS 4 5 met & m= p+ e+ xe a A ee ia - i 5 3 mi — Wt — 2n Wt — 3n 4 "idl Wi— 2 TE == OF Wi — 3n Wt — An 5 2 x nu x x 1 n x 2 a A x x un m. 8 2 3 4 2 3 4 5 Leg a) he’ MW A 2 Gy 2 A 2 e=1 272 3 sees Wi — &c = 1 + x + Se Sy + x - x : xe + ra) nec 71 — 2n i Bn 4 aa 7 — Wi — 2n mM — 30 1 4n 5 pes x a x . 2 fale site tn Whewnae thes = x o ie ; “> + &c= oy x ri/4 pa paseaw e+ SB a allie We Rae 9 GARR 1/2 22 2n 3u 2 22 32 Re ere SC i oe ge Feo, or GE for the fake of brevity, we put A = 1, and B = =, or = Agana Gr — ——B ~B, and D aa, and E = = D, and F = re PbpandG) Hy K peo — Sn —m F, Sa vities Gt May coe K, &c, refpectively, to. the feries? 6n 70 Sz 1 patio a pie C a PO Das + Ee — |e 2 a; mye. F x° +25" “Gxt oes Hx? + == In? — FHF K yo 4&4 iff which 2 the Ce term — Ax, or — x, is always to be marked with the ‘fen ++, or to be added to the firtt Sahin bes but the third, and fourth, and fifth, and other following terms, are marked alternately with the fign — and the ion. ee , or are to be fpbrected from, and added to, the faid firft term 1, or the two firft terms i+ ~ «, alternately. This. 198 AS Sp: TBXCrO WU Ro sig, . CHOW CEPR Ne TNS This is, as I apprehend, the cleareft and moft convenient way of expreffing the binomial theorem in this cafe of the power of a root of the binomial quantity 1 + x, when m, or the index of the power to which the root of 1 + « is to be raifed is lefs than #, or the index of the root, Of the Binomial Theorem in the Cafe of the Powers of Roots, when the Index of the Power of the Root is greater than the Index of the Root. 4. When the index m ftands for a fraction, of which a whole number is the numerator, and another whole number, lefs than the former, is the denomi- nator, we muft, as in the laft cafe, fubftitute = inftead of m in the theorem ftated fa — ¥ ° % Art. 1, to wit, the theorem, 1 + «| = 1 +4 — x + = x um — 1 x m— 2 x? 4. m x m—t Y Mt —2 nt 2 3 I 2 3 ge: x Be Ror 1 pA xm = Ba ob Cw + Dx* +— Ex: a: aw i x3 Gti le 7253 26 2 - : : — & = 1+ se +. See oh as ar 556 &c. - Therefore, if the binomial I theorem is true in the cafe of roots, 1 + x’, or the {quare-root of the binomial *. ue ‘3 : quantity 1 + x, will be equal to the feries 1 + -— + A Brie — &e. g. Now that this feries is really equal to the fquare-root of 1 + x, will appear by multiplying it into itfelf. For we fhall find that the product of the faid multiplication will be 1 + ». This may be done in the manner following : x 2 3 nu xs * x bee 4 Paes &e OPE Res Capen TAL APT, Be eee ape cage EC Pier et eo $24 5-54 5-8 + & a2 hs ; ~ $84 Rake ee Sie 14 5 - <= st &e +2 + &e I+w * . *® 4 &c. Vou. Il. 2D I Vv 202 A DISCOURSE CONGERNING . x Yd we cet VE bd ¥ It appears therefore that the feries 1 + Tet tor tier ire + 6 &c is really and truly, as far as relates to the faid fix firft terms of it, the {quare-root of the binomial quantity 1 +- ¥, and confequently that the binomial theorem is true in the cafe of fquare-roots. I 10. In the next place we will inveftigate the value of 1 + «#3, or the cube- root of the binomial quantity 1 + *, by means of the fame feries 1 + -- Ax a ae Bx? + ae Ayah ee woe ~Dw4 eae > Exs — &c. Now in this cafe z is = 3, and confequently we fhall have 22 = 6, 32 = 9, 44 = 12, and 5u=15,andu—1=2, 27#—1= 5, 3u—1 = 8, andgu—1 a— iI 22 — I Bx” + — Cx? — 2n pga = 11. And therefore the feries 1 + — Ax — ae Dwt + ear Ew«s — &c will, in this cafe, be — 1 + = Av — = B x? 3 Rea A 2. VAPOR one palit he 2 2 er 4 eat ie PASE Aa pols ne 243 “929 — &c. 11. Now that this feries is really equal to the cube-root + of 1x, will appear by multiplying the faid feries twice into itfelf. For we fhall find that the pro- duct of the faid multiplications will be equal to 1 +. Thefe multiplications will be as follows : 4 x x? 5x3 Tox 22%5 gS NC Br 243, tas x x oe e 10% 226 I — — = - — —— — aaa 4 9 81 243 729 &c x x 5x3 10%4 22%5 DOs 9 81 243% 729 sas * x w 5x 10% Se PRUNE RS por + &e 3 9 27 243 729 + & x“? “3 ie x & 1 ge 27 Suniel + Cc i fe 5x3 Sa as & tn Sh wy amas 7260 gence 10%* 10%5 — a _ ke 243 Tai 22 + a — &c 2 3 4 5 rfpS—- 24-4 Ke, THE BINOM™MIAE THEORE Mg 208 veal fat de A algal yr dpc 3 2. a} aay. 729 Co Pree ere SE FOR 22%". . ‘Bne 9 Sr 243729 3 4 5 pee ied EE a ee Wa te SINS Be 3 % ie 243 729 x 24 x 4x 7% ois ee wy ger adam — &c 3 9. 2) ys “as 729 aS pais tye fags yA ‘ 9 27 81 729 + &c 5«3 10¥4 ae 5x5 MET 243 729 AS ih 10%4 20% § agi aye pT 224” &c 129 a I % : ie * * &c. : 2 3 4 - It appears therefore that the feries 1 + ~ —~2 4 S —i* 4 2288 Br 243729 9 &c is really and truly, as far as relates to the firft fix terms of it, the cube-root of the binomial quantity 1 + «, and confequently that the binomial theorem is true in the cafe of cube-roots as well as in that of {quare-roots. Examples of the Extraétion of the Roots of fome particular Powers of the Binomial ——- Quantity 1 + x by means of the Series given above in Art. 3; in which the Numerator of the Fraéional Index = is fuppofed to be lefs than its Denominator. 2 12. In the next place we will inveftigate the value of 1 + a3, or of the cube-root of the hae of Ms pee quantity 1 + *, by means of the feries rie geal # ax (tS = Bx* +-—— Cx! —B—*p a wes Ex’ — &c given wm in Art. 3 as the value of 1 + «,”. Now in this cafe # is = 2, and” is = 3. And therefore we fhall have 2x ae 38 9, 4% 12, and 67, = if: and z—m(=m= 3—2)=1, an—m(= 6—~—2)= 4, gn —m(= 9-2) =7, and 44 —m (= 12 — 2) = I0, Lain qf bined tai pyr vari 3n 4n n— Mm and confequently 1 + = A t— = Boot yal = : Atapepaliy «tyre SN ie Zo ae co tt Ae ais ee SD es 2 3 4 5 . : oo ee = S = ae — &c. Therefore, if the bino- mial theorem is true in the cafe of the powers of adh or the roots of powers, 2 D2 204 AWD fs COC Oo UR FF Eo ORO ry eres oie eee 2 . t + x\3, or the cube-root of the fquare of the binomial quantity 1 + *, will be equal to the feri pe eit i foe eee e equal to the feries 1 oir maori 7 Dt Vines” Fy — &c. 2 13. Now that this feries is really equal to 1 + x\3, or to the cube-root of the {quare of the binomial quantity 1 + #, or to the cube-root of the trinomial quantity 1 + 2” + «x, will appear by multiplying the faid feries twice into itfelf. For we fhall find that the product of thefe multiplications will be the faid trinomial quanuty. Thefe multiplications will be as follows : 2% x? Ans vee 1445 rie eas eet Pe Sandor) You E ae ye), 3 yee Q: AAR EE 24 e 4x 7% 14x 1-2-2 4 ee ee 3 9 81 243 at 729 2x i Be An? ead 145 r+ 4+ tie ALPE aA SE 6, 9 81 243 729 26 4x? 2e8 8x4 1445 on8 xt 4x5 y lcsapeilan hols at lee 4x x 4x +3- 4+ &c 81 os hyace Ex 72 14% 243 729. Ree 144 se ab Oe 729 Ax zee Axe cee 8a 5 3 9 81 243 729 4 2x? 4g? nat 8x eS Sr 24g 72g) Hes 25 Lee ae eee ust ages el 3 9 81 243 729 ee 4x 2x 4x3 cat 8a eT ; ot 2, er 243 729 + &c 24 x 4x eC 104 er fe eae — &c 3 a ie 243 729 See ae ee ee used 9 27, Sr 729 &c 4x 164 x i — &c, 81 + a) 129, ms Se ee Re 243 bd 14%” &c 729 + I + 2” + xv sf ** ® &c. . 4 It appears therefore that the feries 1 + 2% — © + 4c I eee mat 8: 243 | 729 is really and truly the cube-root of the trinomial quantity 1 ++ 2x + xx, .or of the fquare of the binomia quantity 1 + +, and confequently that the binomial EEN: theorem T- HE SIN OM? AL TRHmiE sOnRti Be Ms 205 theorem is true in the cafe of the cube-root of the fquare of a binomial quan- . . M1 . - * . tity, or when m, the numerator of the fraction orp which is the index of the power of 1 + x, is = 2, and x, the denominator of the faid fraction, is = 3. m 14. As another example of the inveftigation of the value of 1 + al” by : : | — 24 — x iz means of the foregoing feries 1 + soak We ys SRR ad Mme Qe ge a 2m 32 4n Dx* 4- “ Ew’ — &c, we will fuppofe m to be = 3, andz to be = 5, or 1 + x)” to be equal to.1-4 «|S, or to the fifth root of the cube of the bino- mial quantity 1 + x, or to the fifth root of the quadrinomial quantity 1 + 3+ + 3x7 + 2°. . In this cafe we fhall have 22 = 10, 34 = 15, 4m = 20, and 5m = 25, and confequently _ : n—m(= 5—3)=2, and 24 —m(= 10-3) = 7; and 374 —m (= 15 — 3) = 12, and 4n —m(=20—-3)= 17, 4 — 71 ee fe oS a Ce rE De + Ew? — &c 2 a iS OD Sides Vor ES a 4. 14 oe gel ies _— Ser ist Ae = Bx zak ta: =, Dx jae Sere Me 3x? rte ALAS te LY cee : . : ; le a5 eT Or eC CERMETTAT &c. Therefore, if the binomial theorem is true . 3 in the cafe of the powers of roots, or the roots of powers, the quantity 1 + x\5, or the fifth root of the cube of 1 +, or the fifth root of the quadrinomial quantity 1 + 3x + 3x* + x*, will be equal to the feries 1 + a Sea Peel 25 125 as2% B57 eh ec 625 15625 3 15. Now that this feries is really equal to 1 + w\5, or to the fifth root of the cube of the binomial quantity 1 + «, or to the fifth root of the quadrinomial quantity 1 + 3x + 3«° + x°, will appear by raifing the faid feries to the fifth power by multiplying it firft into itfelf, whereby we fhall obtain its {quare, and af- terwards multiplying the faid {quare into itfelf, whereby we fhall obtain its fourth power, and, laftly, multiplying the faid fourth power of it into the feries itfelf, whereby we fhall obtain its fifth power. For we fhall find that the produ& of thefe three multiplications will be the faid quadrinomial quantity 1 + 3x + 3x? + «3. Thefe multiplications will be as follows : ge get, pet atat | gs7es ss 5 aC WH u26 =a 625 15625 aaeeG - oo ET aia baste LT de - ick 5 25 oh 126 625 > 16625 Wee 206 ALDILSCGMmR SE (CLOGN CHORD Owe 3% 3x? 7 21x 357%*% ~ cite 25. ie 125 625 a 15625 &c gx 3 aie 63x 2 a Pate Free Speen + gt Gag ~ ag t 4 5 iam aie y ee rite 6x i og 4x3 gx 12045 * 25 125 625 15625 af ao BE cc 8 wt ba cine a 25 125 re620 7 ee eee Sy Se ee ee ee ey ee 2 3 4 + Foes oe + 2p tae © Gy 5 og tS com eee ee 625 NETS 1265 15626 12x Aga® 28x3 21x64 168%5 rps t28 a Th &c. 5 25 125 625 a Ese? + } an Sate : bid Pell DS wilh Abe 3oT* This is the fourth power of the feries 1 + pairs + iit aoa + of yy 12H 4207 28%3 2104 168«5 oy aie 25 Br (ag ie Oke 15625 “3 3x ae ae ar? 3574x ir tt ee 125 625 + 15625 124 ee 28%3 zix* 16845 3e 36.2 rsbe4 B4x4 = =. 6345 Be 5 26 + 125 626" Bing “= &e 3x3 36x3 12644 84x5 + &c 25 125 £28. 326 _ 74 4x 294 a 4 125 ar 625 3125 3 < 4 5 MEE inal Seb ls a eS 625 3125 35745 — &c 15625 I ++ 3x + 3x7 + x * &c. THe, SPI NO Mr AL ern ao RW FM. 207 ax3 2tx4 35745 25° 125 ee 625 15625 &c, is really and truly the fifth_root of the quadrinomial quantity 1 ++ 3x + 3x* + *%, or of the cube of the binomial quantity 1 -+- *, and confequently that the binomial theorem is true in the cafe of the fifth root of the cube of a bino- It appears therefore that the feries 1 + = = « . z 74 4 . . mial quantity, or when m, the numerator of the fraction —, which is the index of the power of 1 + %, is = 3, and #, the denominator of the faid fraétion, is Seog. Examples of the Extraétion of the Roots of fome particular Powers of the Binomial Quantity 1 + x» by means of the Series given above in Art. 4 and 5, in which the Numerator of the Fra&ional Index = 5 Seppyfed to be Stas than its De nominator. a 16. In the next place we will inveftigate the value of 1 + x\?, or of the {quare-root of the cube of the binomial quantity 1 ad * by means of the feries fet down in Art. 4 and 5. Now inthis cafe mis = 3, and” is = 2, and con- fequently m is greater than z, but lefs than 27. Therefore, by Art. 6, the feries fhat is equal to 1 + x\” will, in this cafe, be 1 + — Ag+ fin AD yt 2 7s cD x (M7 bx 4 Be, Naw, Gntew if oc aid wig = > we fhall have m — 2(= 3 — 2) =1, fue 4, 3b — 0) 47 — 6, and <# — 10, and nm—-m(—4—3)=—1, 3 — m (= 6 — 3) = 3; Be $B §, a ei +e Dx* wee Brees 2 be “Re ——C# 4+ + 20x — SBe teen Be Po geS iB 3 128 256 + &c. Therefore, if the binomial theorem fs is true in the cafe of the roots of powers, the quantity 1 + x|\?, or the fquare- root of the cube of the binomial quantity 1-++%, or the {quare-root of the quadrino- | mial quantity 1-+-+-3% + 3x*+.4%, will be equal to the feries 1 + — Wage et x —2 oh 3x* 3x* | 128 256 gt, Oe 17. Now ts this feries is really equal to the fquare-root of the cube of 1 + *, or to the fquare-root of the quadrinomial quantity 1 + 3% + 3%* + x’, will appear by multiplying the faid feries into itfelf. For we fhall find that the preduct of the faid multiplication will be the faid quadrinomial quantity. This multiplication will be as follows : 6 208 Aus To S.0.0,UaR BoE, sCeOgh CMR Se Ta au 2° gyi" iS ee ae fae & 8 a aA ze Sc ty Bu SEL A Re bE oy (ee aay: 2 ae: 8 16 128 meh, &c aS ot oe Se Ree i aiditr'g 8 16 a ae ee 30 poge tip tosfosnigatum prt ia 2 “e 4 16 32 266 ee get 1 get) owt gas tg Theor ene x3 3x4 3x5 TE ae ee art “929 + 128 266 hatte enSEHPL $a) 256 &c I + 3K + 3x7 + x * *® + A ‘ 2 3 xt It appears therefore that the feries 1 + = 4- a ie Bee ele 128 et Beis really and truly the fquare root of the anda toni quantity 1 -+ 3% + 3x* + «3, or of the cube of the binomial quantity 1 + x, and confequently that the bino- mial theorem is true in the cafe of the {quare-root of the cube of a binomial quantity, or when m is = 3, anda is = 2. 18. I fhall add one more example of the feries given in Art. 4 and 5, in which it is fuppofed that the numerator m 1s ‘greater than the denominator z. Let it be required to find, by means of the faid feries, the value of 1 a or of the cube-root of the fifth power of the binomial quantity 1 + am or of the cube-root of the fextinomial quantity 1 + 5” + 10x% + 10x? 4+ 5x4 + x5. In this cafe mis = 5, andwis = 3. ‘Therefore m (though greater than n,) py, . is lefs than 2”, and confequently, (by Art. 6,) 1 + ao will be equal to the feries 1 + = Ax + —— ~ Byte oe Cx + ae " D x4 t= Exs + ce. Now, fince mis = 5, and#is = 3, we fhall have have m—” (= 5 — 3) <= 2,.and. 24 = 6,37 0, 4" —2 i125 he PRS ane 22 — m (= Gea" 5) Sry gn — m (= 9 = 5) = 4; and 42 —m(=12—5) = 7, 22 — im # vo (22 28 Cgniop 32 3n — m rae DT aie SLE femelle hare EBs — CH +g Det = oixtet- yf Bs Saye ye CMTS _ 8 di ——! i * 4 &c. Therefore, if the 15 binomial theorem is true in the cafe of the roots of powers, the quantity ‘Eh ; 1 + x3, or the cube-root of the fifth power of the sib iiial quantity 1 + ¥, or the . Mm and confequently the feries 1 + — Aw THE BINOMIAL Tayi 0. Re E- Me 209 the cube-root of the fextinomial quantity 1 + 5% + 10x° -b ro? 4 gxt 4 45, will be equal to the feries 1 + 3 + SE i + = an te Bee, UR + item 743 a? 19. Now that this feries is really equal to the cube-root of the fifth power of 1 +, or to the cube-root of the fextinomial quantity 1 + sv ++ 10x” + rox? + 5x* + x5, will appear by multiplying the faid feries twice into itfelf. For we fhall find that the product of the faid multiplications will be the faid fexti nomial quantity. Thefe multiplications will be as follows : | 5x cae sus 5+ ee : sae sgn: Fe 243 799 1 ov it 54 Sw 7x 1+ a —— —- & 3 9 Si 243 729 ‘Cada <3 oe (t% Sat to Miley he et P ae . 7 243 729 + &c it: 25% 20x 254 2x & _ — &c a] 3 as 9 + 2} Sie thy Le 204 254 25% a ae 27. i Br 729 a ME SRR TESA _ 52 by 81 243 729 vikeg te 250° = &c 243 ‘he zy ae EC 729 OS OEE ROLL OSI II II I OLENA EE: SO cS TES 10% 35%" 140%8 35+* 14% 1+-—+e.4 2°42" ke, Boo gay. ter 243729 I0¥W id T40n 3 at r405 See See aio Mh std ot 742 ” x ox 5x 7% OE a Ray ect or ares oon ape ea &e 3 9 a 243 729 a 10+ a Se 140%3 350% 1445 ee ee ee ec ne 3 a" 9 as ar ian 243 | 729. Sw 50x” 1754 700w 175% & es eae ee a =e Cc + 3 * 9 ce =a 243. oak I "al mE Lt ae SOx 75” me JOO" — &c 9 27 81 729 5x3 sox 4 17545 $1 243 729 reas 5«* 5Ox & a Cc ig tet ee ee BCC art ive : 729 TI 5x - 10x" + 10%? - 5x* + x &e. ies DR aR PI Se : {t appears therefore that the feries 1 + ~ + alerts ee + &e is really and truly the cube-root of the fifth power of the binomial quantity 1 + x, or the cube-root of the fextinomial quantity 1 + 5 -- 10%* ++ rox® + 5x* +. x5, and confequently that the binomial theorem is true in the cafe of the cube-root of the fifth power of a binomial quantity, or when mis = 5, and # is 5113 Vout. II, 2E 20, Thefe 210 ASD" T°SverorUFtR Shek” TORO Mee wae cer se 20. Thefe fix examples of the application of the binomial theorem to the inveftigation of the fquare and cube-roots of the binomial quantity 1 + x, and of the cube-root of the {quare of 1 -+- *, and of the fifth root of the cube of 1 +, and of the fquare-root of its cube, and the cube-root of its fifth power, may fuffice for the purpofe of ifluftrating the meaning of that celebrated theo- rem, and the manner of applying it to particular cafes, when the index of the power of 1 + # is either the fraction J or the fraction =. And the proofs that have been given of the truth of the feriefes obtained by means of the binomial theorem in thefe examples, by raifing the feriefes fo obtained to the proper powers, and fhewing that the powers thereby produced are exactly what they- ought to be, afford a very {trong \prefumption, ** that, fince the binomial theo- <¢ rem is true in thefe particular values of m and 7, it mutt alfo be true in all «‘ other values of them whatfoever.” But in the mathematical fciences we. ought to aim at fomething more than this high degree of probability, and to endeavour, if poffible, to demontftrate the propofitions we advance.. And this is what I fhall now proceed to do with refpect to this theorem in the cafes men- tioned in Art. 2, 3, 4, and 5.. And, that we may not fall into confufion by making our inquiries too extenfive, and embracing too many objects at the I. fame time, I fhall firft confider the cafe of 1 + x\”, or the ath root of the binomial quantity 1 + x, and endeavour to demonftrate, that, whatever be < the Pe number denoted by z, ca value of 1 + xt will be equal to the feries I Bak Thim—ni I 2 =e 3 1 hak atti 2 — _ — x — KF —— — J : + * 2 x 2n + x 2n 3% a * 2% an n—I n— TI 2n— 1 32 — I 4a — 1 ¢ xetet ox? x a iA <7 a” An 3n 42 bit 2 a a al > Cx a r “Det in Art. 2, and fechas fhall proceed to aoe a like demontftration of the truth of the other feriefes fet forth in Art. 3, 4,-and 5, for the values of mt — &c, fet forth above i+ x" . Before we enter on the inveftigation of this feries, it will be neceffary to make the following obfervarions. s Olfervations preparatory to the Inveftigation of the Series that is equal to 1 +- x\", or the nth root of the binomial quantity 1 + x. a1. In the firkt place, we muft obferve that the firlt term of the feries that i is I equal to 1 + xe or o/*i + x, muft always be 1. This follows from the common manner of extracting the roots of a binomial, or other more compound quantity, of which 1 is the firft term. For, if we were required to extract the fquare-root, or the cube-reot, or any other root, of THE BINOMIAL THEOREM, 211 of fuch compound quantity, according to the’ common rules laid down in books of arithmetick, or algebra, for fuch extractions, the firft ftep of fuch an-ex- traction would be to extract the {quare-root, or the cube-root, or other higher root, of its firft member 1; which {quare-root, or cube-root, or other higher root, of 1, would be 1; fo that 1 would be the firft term of the faid {quare- root, or cube-root, or other higher root, of the faid binomial, or other more compound, quantity 1 +», or 1 + «+ &c. GE De The truth of this obfervation may alfo be deduced from the following confi- I deration, to wit, ‘ that the feries that is equal to 1 + x\”, or (/"1 + x, mutt “¢ be equal to it in all magnitudes of «, or while w is of any magnitude lefs than ‘¢ 1, which is fuppofed to be the firft and greater member of the binomial quan- “tity 1 +x.” For, it follows from hence that the faid feries muft be equal to I I I +x", 0rto/"1 +, when wis = 0. But, when wis = 0, 1 + wx” is = I I 3 - 1+ 0” =1f =1. Therefore, the faid feries muft, in that cafe, be = 1. Therefore, the firft term of the faid feries cannot be involved with any of the powers of x, (by which it would be made to become equal to o when w was equal to o,) but muft be = 1. hbk Feige 22. Inthe next place we muft obferve, that the fecond and third, and other f following terms, of the feries that is equal to 1 + x)", or "1 ++ x, will con- tain the feveral powers of ~, to wit, x, 7, #9, #*, #5, &c, in their natural or- der, without interruption, and confequently that the faid quantity will be equal to a feries of the following form, to wit, 1, Bx, Cx’, D3, Ext, Fxs, &c; in which the fecond and third, and other following terms, Bx, Cx’, Dx’, Ex«+, Fx‘, are to be either added to the firft term 1, and in that cafe to have the fign + prefixed to them, or to be fubtracted trom the faid firft term, and in that cafe to have the fign — prefixed to them. But, as it is not yet certain to which of the faid terms the fign +- is to be prefixed, and which of them are to be marked with the fign —, or which of them are to be added to the faid firft term 1, and which of them are to be fubtracted from it, I have not, on this occafion, (where it was only neceflary to mention the form of the feries which I is equal to 1 + x\”,) prefixed either of thefe figns to any of them, but have only feparated the feveral terms of the faid feries one from another by placing a comma after each term. . ‘ = 23. Now, ‘that this obfervation is true, or that 1+ x, or 4/714 #, €¢ will be equal to a feries of the faid form, 1, Bw, Cx’, Da’, Ext, Fx’, &c,” has already been fhewn in two inftances, to wit, when z is = 2, and when z is = 3, or in the cafes of the fquare-root and the cube-root of 1 —-+ #. ~ For it has been f ; x x8 he the 7%? fhewn by Art. 9, that the feries 1 + >— 3+ [=~ + Pe DS) ua Soa to — &c is equal 12 BV BFC OS UR BE C ovN'*C ER NING to the fquare root of 1 ++ x; and it has been fhewn in Art. rr, that the feries : je aged by f diab 2245 9 81 2 iy And both thefe feriefes are of the fame form as the general feries 1, Bw, Cx*, Dw3, Ext, F x5, &c above-mentioned. And, ‘ that all other roots of the bino- «¢ mial quantity 1 + x muft likewife be equal to feriefes of the fame form,” will appear from confidering the manner of extra¢ting any root of fuch a binomial quantity. For the firft ftep of fuch an extraction gives us the firft term of the root fought; which firft term (as we have already obferved) is always 1: and the fecond ftep of fuch an extraction is to fquare the faid firft term, 1, of the root fought, if we are performing the extraction of the fquare-root of 1 + #3; or to cube the faid firft term, 1, of the root fought, if we are performing the extrac- tion of the cube-root of t + 3; or, in general, to raife the faid firft term, 1, of the root fought to the mth power, if we are performing the extraction of the ath root of 1 + #3; and then to fubtract fuch fquare, cube, or wth power of the faid firft term, 1, of the root fought (which will always be 1, becaufe all the powers of 1 are equal to 1), from 1 + w, or the original quantity of which the root is fought: in confequence of which fubtraction there will remain the quantity « (which is the fecond and leffer member of the binomial quantity 1 + w,) for the ground- work of the gradual evolution of the fecond and third and other following terms. of the feries, or root, fought. And the third ftep, or procefs, of fuch extraction (whereby we fhall obtain the fecond term of the feries that 1s equal to the root fought) 1s to double the firft term of it already found, to wit, the term 1, in the cafe of the extraction of the {quare root ; and to treble the faid firft term 1, in the cafe of the extraction of the cube-root; and, in general, to multiply the faid firft term, 1, by # in the cafe of the extraction of the ath root; and then to divide the faid remainder x by the faid produ&; to wit, by 2 x 1, or 2, 1n the cafe of the fquare-root; and by 3 x 1, or 3, in the cafe of the cube-root; and by # x 1,. — &c is equal to the cube-root of 1 + #. or #, in the cafe of the #th root : which will give us = in the cafe of the fquare- root, and in the cafe of the cube-root,, and — in the cafe of the xth root, for 1 I >t+-x*\3,or14 4)”: fo I that the two firft terms of the feries that is equal to 1 + x\2 will ber + =, i 2 the fecond term of the feries that 1s equal to 1 + *«}) : T and the two firft terms of the feries that is equal to 1 + «)3. will be 1 + me and I in general, the two firft terms of the feries that is equal to 1 + x|* will be 1 4 =. And the next operations of thefe extractions will be to raife 1 + = to the fe- 2 cond power in the extraction of the {quare-root, and to raife 1 + a to the third power in the extra¢tion of the cube-root, and, in general, to raife 1 + = to the a nth power in the extraction of the wth root; and then to fubtraéct thefe powers from the original binomial quantity 1 +- x, of which we are feeking the root; or, q to THE BINOMYAL THEOREM. arg to fpeak more correctly, (becaufe thefe powers of 1 + , t Sand f+ = will be greater than 1 + «,) to fubtract 1 + x from the faid powers : after which I I I the third term of the feries that is equal to r+ a2, or t+)3, ort} x\* ; will be obtained by dividing the remainder of the laft fubtraction by 2 x frpick. or 2 + «, in the extraction of the fquare-root; or by 3 x{r + 4, or 3 +x, in the extraction of the cube-root; or, in general, by z x {1 +—,ora2+-, inthe 1 extraction of the mth root. Now, it it is evident, that, by raifing the aforefaid powers of the binomial quantities 1 + =, I++, andi + =, (in the fecond Bi 2 member of each of which the fimple power of » occurs,) we fhall obtain quanti- ties in which the powers of * will occur in their natural order, to wit, w, x’, x, x*, x°, &c, without any interruption. Thus 1 + =\' i~mi+t+se + os and w)3 . # i a3 we x? Pee OKI XH 8B XI KK tH) SIP est Kt “| ( 3 ae ek renter) ae ope Sr n e e . e . ° and 1 + = is (by the binomial theorem in the cafe of integral powers, which has been already demonttrated above, in the laft tract but one, from page 153 to page 169,) =r + 2x2 42x 2x2 pg Pat Baby ZY ; P : : Ax x x ex ate tx tx tx tx tx y+ mest tixSeixttxteixtaixtixgsis oa oS SK eS XS x x x St ke = ae ee x Saat + x SS x oe —3 x 224 45 4 &c; in all which powers of the faid binomial quantities 4n 52 P q wv ve ° . Resi Lct —, and 1 + = the powers of « occur in their natural order, to wit, w, «”, *, x*, «5, &c, without any interruption. Therefore, when 1 + x has been fubtracted from each of thefe powers, the remainders, (which will be ae <2 in the cafe of the {quare-root, and es + es in the cafe of the cube-root, and a—t1 ms a—1 n—2 3 a—tI a—2 n— 3 4 a— I 22 x 4 + 22 x 32 x “ a5 22 x 30 x 472 x rs an 2% = x wi x — x ws + &c in the cafe of the wth root,) will confift of the regular powers of x, beginning with xx, combined with certain numeral co-effici- ents; and confequently the quotients ef the divifions of thefe remainders by 2 + x, and 3 + x, andz + x, refpectively, will be quantities that involve in them the fquare of ¥; to wit, in the cafe of the {quare-root, the faid quotient will pO AM 1.5 COUR S'E. ClO N CER NINec 4 2+ 2 will be = — a — &ce= > — &c; and in the cafe of the cube-root the ar A . 7 3 » ‘ {aid quotient will be == + = = == &c = = &c; and, in the cafe of the mth - 5 ee A 9 : 34a 3 anes root the faid quotient will be SMH plo omen Ca Sat See eee 2n 2” 30 2n a+ew F774 ue— Tt x % S &c = — ax &c; fo that the third term of the feries that is equal to 1 + x)? I will be = and the third term of the feries that is equal to 1 + m3 will be ve I and, in general, the third term of the feries that is equal to 1 -+ «)* will be —— : 2un «x; all which third terms involve in them the fquare of x. And thus we fee I that the three firft terms of the feries that is equal to 1 + mi will be 1 + = _ ; I “5 , and the three firft terms ofthe feries that 1s equal to 1 + w«)3 will ber + E I xx , : : wee — ——, and the three firft terms of the feries that is equal to 1 x\* will be ae q a 2 1+ - — — x 3 in all which feriefes the powers of » afcend regularly. And as the following terms of thefe feriefes are derived from the three firft terms of them by the repetition of the fame procefles of multiplication, fubtrac- tion, and divifion, by which the third term is derived from the firft and fecond terms, and the fecond term is derived from the farft, it follows that the fourth and fifth and fixth and other following terms muft involve in them the cube and fourth power and fifth power, and other following powers, of x, in their natural order, without interruption, and confequently that the form of the feries that is I equal to1 + x)\" will ber, Bx, Cx*, Dx, Ex4, Fx5, &c, or, rather, (fince we now know that the fecond term of the feries 1s to be added to the firft term, 1, and that the third term of it is to be fubtra¢ted from them,) 1 + Baw — Cw, Dix; Vs iat’ &c. Q. EL 24. The foregoing reafonings may, perhaps, feem rather abftracét and diffi- cult, as they are expreffed in words in the laft article. But the force of them will become more apparent by actually exhibiting to the reader’s view an ex- ample of the extraction of a root of the binomial quantity 1 + x in the manner alluded to in them. Now the extraction of the fquare-root of 1 + «# is per- formed in the following manner. The y Tey OT NO OOM ACL 2 Toe EF OO RVE™M, 21 tur The Extraétion of the /quare-root of the binomial quantity 1 + w. x He “3 at 7 #2 rH od I x Th plat ie oat ROU ua ade Ea cae y rte. of 2 8 aR 160 128 a 256 1024 + &c x om : es 4 og A 2) 8 ih. a x3 xf ; TS eee BE HH x3 x3 se 2 oi jel she 4 sty 76) Hh 8 64 ‘gf on us x? Ff 8 15 16 64 256 ed gx xs xo 2+% 2 ep or hee he) # ey a Ogi) vase sey a? |, sat 64 128 + 512 + &e ux xs he 7465 eo f 2+ * 4 2 zy — &e ) AEH gat, 512 : 7% 7% a 128 Pe 266 &c ' xe 2 4 21x° ree ae ). merre &c sa 214° ec 512 25. It appears from the foregoing extraction of the fquare-root of 1 + x, that the faid {quare-root is equal to a feries of quantities of which the firft fe- xe “3 kav ry 4 are LR ES 128 x 256 5% 1024 of x increafe regularly from x to ¥°,. And it is plain from.the manner in which x” ‘ 5 ven terms are I + Aim in which the powers he fecond of thofe terms, to wit, =; and all the terms that follow it, are gradu- ally derived from x, or the remainder of the firft fubtraction, that the following terms of the feries, to whatever number they fhould be continued, would in- volve in them the next following powers of «x, to wit, *7, x*, ¥®, w™°, x", &c in their natural order. And the fame thing would appear in the extraction of the cube root of 1 ++ x, or of its fourth root, or its fifth root, or any higher root of it whatfoever ; though the operations necefiary to thefe extra€tions would be much more complicated and laborious than thofe that occur in the foregoing extraction of the fquare-root. And therefore we may conclude in general from the nature of thefe extractions, that the #th root of the binomial quantity 1 + w will in all cafes be equal to a feries of the before-mentioned form, to wit, 1, Bw, Cx*, Dx?; Ext, Fx’; &c, in which the powers of # occur in their natural order, or increafe gradually by the continual multiplication of x. Q: EnLD. 26. In 2i60% A D-I1S8SGC60U 8S 2 .GO0-MC/BR NINE 26. In the third place we muft obferve, that the fecond term Bw, of the feries I 1, Bx, Cx*, Dw?, Ex*, Fx’, &c, which is equal to 1 + x|", or /*{1 + x, muft be added to the firft term 1, and confequently have the fign + prefixed to it. Ox his obfervation has been already fhewn to be true in the reafonings ufed in Art: 23 to demonttrate the fecond obfervation. But, as it will be referred to in the following inveftigation as a neceflary preliminary propofition, it will not be amifs to give the following additional demonftration of it before we begin that inveftigation. 7 Now, fince 1 -+ x is greater than 1, it follows that 1 + «\” , or the th root of 1 + «, or the firft of ~ — 1 geometrical mean proportionals between 1 and 1 +4, muttalfo be greater than 1, Confequently the feries 1, Bx, Cx”, Dx3, I Ex‘, Fx’, &c, which is equal tor + .«\*, muft alfo be greater thanr. And this muft always be true, of how fmall a magnitude foever we fuppofe x to be taken, fo long as it has any magnitude at all. But, in order that the feries 1, Bx, Cx’, Dx’, Exs, Fx’, &c, may be always greater than 1, however {mall the magnitude of « be taken, it is neceflary that Bx fhould be added to the firft term 1, and not fubtracted from it. For, if Bw were fubtra¢ted from 1, it would be poffible, by diminifhing the magnitude of x, to make all the terms Cx”, Dx’, E x4, Fx’, &c that follow Bx, put together become lefs than B x, however great we may fuppofe the magnitudes of the co-efficients C, D, E, F, &c, to be; in which cafe it is evident that the whole feries 1 — Bx, Cw*, Dx3, Ext, Fx, &c would (even though all the terms after Bw were to be added to the firft term 1, and the feries were to ber — Bx +Cwx? + Dvx?+ Ex* + Fu 4+ &c,) be lefs than the firft term 1: which is contrary to the fuppofition. Therefore the fecond term, Bw, cannot be fubtracted from the firft term 1, but muft be added to it, and confequently muft be marked with the fign + ; and therefore I the feries that is equal to 1 + x)" 5 will be of this form, 1 + Bw, Cx”, Dx, Eut, Fes, &c. . Q. E. D. 27. By the help of thefe three preliminary obfervations we may now proceed I to the inveftigation of the feries that is equal to 1 + )#, or the ath root of the binomial quantity 1 + «, when x is fuppofed to ftand for any whole number whatfoever. This inveftigation may be made as follows. ; 2i7e AoW wotoD. Dal Eu .OoN T)/O* T HE Difcourfe on the Binomial Theorem. [The Binder is defired to place this and the other fheets with ftarred Jiguatures and folios after page 216 in the fecond volume. | Artic.ie i. The foregoing conclufion contained in articles 22, 23, 24, and I 25, to wit, “that 1 -+ «|, or Vali + x, or the zth root of the binomial quantity 1 + w, is equal to a feries of the before-mentioned form, to wit, 1, Bw, Cx*?, Dx?, Ex*+, Fx’, &c (in which the powers of x follow each other in their natural order without any interruption, or increafe gradually by the continual multiplication of «) or that a feries of this form may always exift (to whatever whole number we fuppofe the letter ~ to be equal) which fhall be [ equal to the faid quantity 1 + «|*, or /*/1 + «,” may be alfo fhewn in the following manner. 2. Whatever whole number may be fuppofed to be denoted by x, it is evi- dent that a feries of the before-mentioned form, to wit, 1, Bx, Cx?, Dx, ’ Ext, Fx5, &c, will be equal to it, if the feveral numeral co-efficients B, C, D, E, F, &c, of «, «*, «3, xt, «5, &c, in the fecond and other following terms of fuch feries, are of fuch magnitudes, and the faid fecond and other fol- lowing terms, to wit, Bx, Cx*, Dw3, Ew*, Fx’, &c, are fo connected with the firft term 1, by the figns ++ and -—-, or by addition and {ubtraction, that, when the faid feries 1, Bx, Cx*, Dw*?, Ex+, Fu’, &c is raifed to the mh power, or multiplied n — 1 times into itfelf, the compound feries which, it is evident, will be produced by fuch multiplication, fhall be equal to the binomial quantity 1 + x, or that the two firft terms of the faid compound feries fhall be 1 and x, and that the fign prefixed to the faid fecond term « fhall be 4-, and that ghe third, and fourth, and fifth, and fixth, and other following terms of the faid compound feries fhall, each of them, be equal to nothing, or that the mem- bers of each of the faid terms of the faid compound feries, after the fecond term, fhall be marked with both the figns + and —, that is, fome of them with the fign +, and others of them with the fign —, and that the fum of thofe of the faid members of each term which are marked with the fign — fhall be equal to the fum of the other members of the fame term which are marked with the fign +, fo as thereby to counterbalance, or deftroy, them, and make the whole of the faid compound term be equal to nothing. , Vou. Il. aS ad 3. Thus #218 AN ADDITION TO THE DISCOURSE ON THE BINOMIAL THEOREM 3. Thus, for example, if # is <= 2, the-feries 1, Ba, Cx¥*, Da?, Ext, ' I F F xs, &c, will be equal to 1 + «]2., or 7p + x, or the fquare-root of the- binomial quantity 1 -+ x, if the numeral co-efficients B, C,.D, E, F, &c, of x, «7, v3, x*, x5, &c, in the fecond and other following terms.of the faid fe- ries, to wit, Bx, Cw?, Dx3, Ext, Fw’, &c, are of fuch magnitudes, and the faid fecond and other following terms of the faid feries are fo connected with the firft term 1 by the figns -- and —, or by addition and fubtraétion, that, when. the faid feries 1, Bw, Cw?, Dx?, Ex*+, Fx, &c, is raifed to the {quare, or fe- cond power, or multiplied 2 — 1 times, or once, into itfelf, the compound fe- ries which, it is evident, will be produced by fuch multiplication, fhall be equal: to the binomial quantity 1 + x, or that the two firft terms of the faid compound feries fhall be 1 and x, and that the fign prefixed to the faid fecond term « thall: be -++, and that the third, and fourth, and fifth, and fixth, and other following: terms of the faid compound feries fhall, each of them, be equal to nothing, or that the members of each of the faid terms of the faid-compound feries, after the: faid fecond term, fhall be marked with both the figns + and —, that is, fome of them with the fign +, and the others with the fign —, and that the fum of thofe members of each term which are marked with the fign — fhall be equal. to the {um of the other members of the fame term which are marked with the fin -++-, fo as thereby to counterbalance, or deftroy, them, and to make the. whole of the faid term be equal to nothing. . 4. And; ifn’ "3.the feries 1). Bx, C x75 Dies, ee &c, will be. I equal to 1 + #13, or 7? [1 + x, or the cube-root of the binomial quantity 1 + x, if the numeral co-efficients B, C, D, E, F, &c of x, #7, #3, #4, #°, &c,. in the fecond and other following terms of the faid feries, are of fuch magni- tudes, and the faid fecond and other following terms of the faid feries, to wit,, Bx, Cx?, Dx3, Ext, Fx, &c, are fo connected with the firft termr by the figns + and —, or by additioy and fubtraction, that when the faid feries 1, Bw,. Cx?, Dx3, Ext, Px, &c, 1s raifed to the cube, or third power, or multiplied’ — 1 times, or twice, into itfelf, the compound feries which, it is evident, will be. produced by the faid two multiplications, fhall be equal to the binomial quantity 1 +, or that the two firft terms of the faid compound feries fhall be 1 and #, and that the fign prefixed to the faid fecond term w# fhall be +, and that the third, and fourth, and fifth, and fixth, and other following terms of*the faid’ compound feries fhall, each of them, be equal to nothing, or that the members of each of the faid terms of the faid compound feries, after the faid fecond term, fhall be marked with both the fiens ++ and —, that is, fome of them with the fion +, and the others with the fign —, and that the fum of thofe members of’ each term which are marked with the fign — fhall be equal to the fum of the other members of the fame term which are marked with the fign +, fo as there- by to counterbalance, or deftroy, them, and to make the whole of the faid term. . be equal to nothing. , 5. We “. IN THE CASE OF FRACTIONAL POWERS, 219* 5. We mutt therefore endeavour to fhew that, if 7 be equal to 2, or to 3, or to any other whole number whatfoever, it will be poffible for a feries of the be- fore-mentioned form, to wit, 1, Bx, Cx?, Dv?, Ex*, Fxs, &c, to exift, in which the magnitudes of the numeral co-efficients B, C, D, E, F, &c, of x, x*, w*, «+, «5, and the following powers of x, in the fecond and other following terms of the faid feries, to wit, Bx, Cw?, Dx, Ext, Fx, &c, are fuch, and the faid fecond and other following terms of the faid feries are fo connected with the firft term 1 by the figns ++ and —, or by addition and fubtraétion, that, if the faid feries be raifed to the fquare or the cube, or the wth power, or be multi- plied into itfelf2 — 1, or 3 — 1 times (that is, once or twice) or 7 — I times, the compound feries that will arife from fuch multiplications fhall be equal to the binomial] quantity 1 + w, or that the two firft terms of the faid compound feries fhall be 1 and w, and that the faid fecond term w fhall have the fign +- prefixed to it, or fhall be added to the firft term 1, and that the third, and fourth, and fifth, and fixth, and all the following terms of the faid compound feries, fhall, each of them, be equal to nothing, or that fome of the members of each of the faid terms of the faid compound feries, after the fecond term, fhall be marked with the fign +, and the others with the fign —, and that the fum of the latter members which are marked with the fign’—, fhall be equal to the fum of the former members, which are marked with the fign +, fo as thereby to counterbalance, or deftroy, them, and to make the whole of the {aid term be equal to nothing. ee RETR NE RennemeTt; ‘Of the Square-root of 1 + x. | i EEE 6. Now that 1t is poffible for fuch a feries of the before-mentioned form, 1, Bx, Cx*?, Dx, Ex+, Fx', &c, to exift in the cafe of the {quare-root of I + x, or Puhen n1S equal to 2, may be fhewn in the following manner. Let the feries 1, Bu, Cx*, Dx3, Ext, Fx’, a be multiplied into itfelf; which may be done as follows. Ey ee ee Ce ED Ew E'x3, .&c eR ee SC RASt © DFS Batt Pye oc pred Se FPA Ee ee (OEE Fes, &c Bik san 0 Aes aa esy sgt Dl Ms DEM, aC MDa ei Eko) cw na a” Chis: 8c Dw}, Bt, Cbxr, *&c Bx‘, BEws, &c Fxs, &c ere nt ee SR NN Lae Bay 2 Cw? paint) hes 24, (S2.10x 550 Se Bex 2 BE 474t2 BD xl, a BEX > &e C44 2 CD ¥s,<&e 2° 2. * This #220 AN ADDITION TO THE DISCOURSE ON THE BINOMIAL THEOREM This compound feries is the fquare-of the fimple feries 1, Bx, Cx*, Dx’, Ext, Fx’, &c. Therefore, if we can prove that it is poffible that the co-effi- cients B, C, D, E, F, &c, may be taken of fuch magnitudes, and that the fe- cond and other following terms of the fimple feries 1, Bx, Cx*, Dx3, Ex’, Fx, &c, may be fo connected with its firftt term 1 by the figns -+- and —, or by addition and fubtraction, as to make the faid compound feries become equal to 1 + x, it will follow that the faid Gmple feries 1, Bx, Cx, Dx3, Ext, F «°, &c, will be equal to the {quare-root of 1 +- ». We muft therefore endea- vour to fhew that it is poffible for the numeral co-efficients B, C, D, E, F, -&c, to be of fuch magnitudes, and for the terms Bx, Cx*, Dw, Ext, Fx, &c, to be fo connected with the firft term 1 of the fimple feries 1, Bx, Cx*, De’, Ex*+, Fws, &c, by the figns +- and —, or by addition and: fubtraction, that the whole compound feries above-mentioned fhall be equal to1 -+- x. ‘This may be done in the manner following. “7, In the firft place it is evident that the firft term of the faid compound fe- ries is 1. This is too plain to admit-of any proof. ; In the next place it is plain that, if we prefix the fign - to the fecond ternr Bx of the fimple feries 1, Bx, Cx*, Dx3, Ex*, Fx’, &c, fo as to convert the faid feries into the feries 1 -- Bx, Cw?, Dx?, Ext, F x5, &c, the fign + will likewife be prefixed to 2 Bw, or the fecond term of the compound {feries. Toews 50 242 ig eG B*x?, &c) above-mentioned, fo as to convert it into the feries. t+ 2Bxw, 2Cx’, &c a) io And, in the 3d place, it 1s evident that, if we fuppofe 2 B to be equal to 1, or B to be equal to —, the fecond term, 2 Bw, of the faid compound feries will be: equal to2 X < x *, OF = x x, or #, and confequently the two firft terms of the faid compound feries|1, 2 Bw, 2 C*?, &c . B*x?, &c), orlt + 2 Bx, 2 C7, &c ; \B* x7, we wilk be r 4 x. E If, therefore, we can prove that it is poffible to take the following numeral co- eficients C, D, E, F, &c, of fuch magnitudes, and to connect the third and fourth, and other following terms of the fimple feries 1, B x, C «*, Dx3, Exs, Fxs, &c, to wit, the terms Cx?, Dx3?, Ex*, Fx, &c, with the two firft terms 1, Bw, or 1°-+ Bw, or t + =, by the figns + and —,. or by addition: ae fubtraction, in fuch a manner, that every following term of the compound: erles t +.25B 4. 2CGer*,, 2D«3,.. 2B wt, . gE xt, &c Bax. 2iIBOx 3). VB x*y 2 BE x, &c (3204 a See Daas 5 See or IN THE CASE OF FRACTIONAL POWERSs 221% ort + x, 2Cx*, 2Dx3, 2Ext, 2F x5, &c ) Bix a BC x; 2 BD se,” 2 BE v?, &c Chats CD e- &c] fhall be equal to nothing, or that fome of their members (for each of thefe terms is evidently a compound quantity, .or a quantity confifting of more than one member) fhall be marked with: the fign +, and others of them fhall be marked with the fign —, and that the {um of the latter members in each term, which are marked with the fign —, fhall be equal to the fum of the former members in the fame term, which are marked with the fign +, fo as to exa@ly counterbalance, or deftroy, the faid former members, and make the whole term become equal to o, it will then be evident that the whole compound feries: above mentioned will be equal to its two firft terms 1 + #, and confequently that the fimple feries 1, Bw, Cx, Da3, Ext, Fx', &c, ori + Ba, Cx?, Dw3, Ext, Fx’, &c, ori + =, Cx*, Dx, Ext, Fes, &c, (of which the faid compound feries is the fquare) will be equal to the fquare-root of 1 + «. This therefore is what we muft new endeavour to prove.. 8. In order to this, we muft obferve, in the firft place, that in the firft, or upper, horizontal row of terms in the faid compound feries, to wit, 1, 2 Bw, @Gsrtyi2 D rtj:2cBy ty 2? yp 8oe,cor 1-pwy 2 C #7519 Divi, 2 Bet) oF 25), &c, the fign + or —, that is to be prefixed to every term of it, muft be the fame which is prefixed to the correfponding term of the original fimple feries 1, j we ms ete x, bat, Fess occ, or re > Ce Lae Doerr ar Oe term) which involves the fame capital letter, C, or D, or E, or F, &c. Thus, if the fign — is'to be prefixed to the term C.? of the fimple feries 1, Bw, Cw*, Dx3, Ext, Fx, &c, the fame fign — muff likewife be prefixed to the term 2Cw#*, in the upper horizontal row of terms in the compound feries, Seo Ge, 2 Cx*, aD.x*y Ko B*x75 2 BE x3). &c); offi! + 5.2 C #7, 2 Dx, &c a pew BC x3, &c)\; becaufe the faid term, 2 C «?, in the faid compound feries, arifes from the addition of the two like terms C x* and Cx* to each other, of which the firft Cv? is the pro- duct of the multiplication of C x? by 1, and the other Cx is the product of the multiplication of 1 by Cx, and each of thefe produéts muft evidently have the fame fign -- or — prefixed to it as-is prefixed:to its factor Cx”, or the third term of the fimple feries1, Bw, Cw*, Dx?, Ext, Fx’, &c, its other factor be- ing 1. And for the fame reafon each of the following terms 2 D#?, 2 Ex*, 2 Fw, &c, in the faid firft, or upper, horizontal row of terms muft have the fame fign ++ or — prefixed to it as is prefixed to the correfponding term, or term involving the fame capital letter D, or E, or F, &c, in the fimple feries 1,. ~ Bw, Cx?, Dx3?, Ext, Fw’, &c, ori + —, Cw?, Dx?, Ext, Fx', &e. 3 3 3 3 3 p | 2 3 3 b) 3 In the fecond place, it is evident from the laft obfervation that, if we can de L termine #222 AN ADDITION TO THE DISCOURSE ON THE BINOMIAL THEOREM termine the-fign +.or—, which.ought .to-be prefixed:to any term .in the faid upper horikantal row of terms 1,..2,Bixy.2-4aw*, 2 iw? coll ata 22 as peer Tem 2 ON ot aoe Rye te 2E x, &c,-we-fhall at the fame time deseudaas the fign + ae —, which ought to be prefixed to the corre{ponding term (or term involving the fame capital letter, D, or E, or F, &c) in the fimple feries AOR ye Ey ES EES Re eee Or I +=, Cx?; Dix?, Ejx4. Fees occ, tne fions to be prefixed.to the faid two, corr Weiatae terms-being always the fame. In the third place it is evident that in the firft, or upper, “horizontal. row lof terms in,the above-mentioned compound feries Vig 25 Bits 1.2. Ge ste) Hae ccd High egltr ankuaass &c.) Bw? 42 BC #34; 2.BD«9A4; 2: BEd «ccd \Cixt, 2 CDx*, x) | _(which.is equal to the fquare of the fitiple feries 1, Bx, C De?; Ex, Fx, &c) the co.efficients Of w, -% 2s¢ 0? X, OF 5 80G, Are -29B,2 irs 2D, 2 E, 2s "kee, or exaétly the doubles of the co-efficients of the fame powers of * In the fimple feries'1, Bx, Cx?, Dx3, Ext, Fx’, &c, fo that'a new capital letter C, or D, or E, or F, &c, enters into every new term of the faid firft, or upper eaten row of terms. For each of thefe terms in the faid upper horizontal row’is the fum of two equal terms, of which one is placed in the higheft herizontal row of terms of the original fet of feparate products arifing from the multiplication of the feries 1, Bx, Cx?, Dx3, Ext, Fix’, &c, into itfelf,, before the-fimilar terms in .the faid feparate products are added together at the bottom, and the other is placed at the bottom of the fame vertical ‘column of terms in which the former is the upper term. Thus, in the firft vertical column of terms in the faid original fet of products arifing from the multiplication of the feries r, Bx, Cx", Dx3, Ext; F x5, &c, into itfelf, there are two equal terms, to wit, Bx and Bx, whale ae forms the term 2 B» in the upper horizontal row of terms of the compound fe- ries at the bottom, to wit, the feries luo, Dar. Per Oe a ae ie Oe Se ncs &c Bs’, %2, BGaig 2 BD et) oe BRwet &c CAystererCiywer. toes which is equal to the fquare of the feries 1, Bx, Cot »Dw«3, Ext, Fins, dos and in the fecond vertical column of the fume original fer of products, there are the terms Cw*, B*x*, and Cx’, of which the higheft and loweft are equal to each other, and their fum is confequently equal to 2 C«?; and in the third vertical column there are the terms Dx3, BC «3, BC x3, and Dx?, of which the higheft and the loweft are equal to each other, and their fum-is confequently equal to 2 D«?; and in the fourth vertical column there are the terms Ex*+, BD x+, C*x*, BD w+, and Ew*, of which the higheft and the loweft are equal to each other, and their fum is confequently equal to 2 Ew*; and in the fifth vertical column there are the terms Fx’, BEx’, CDx5, CDw«5, BE, and Fx‘, of which the higheft and the loweft are equal to each other, and their fm is con- Lanny equal to 2 F ws. And it is evident that the fame thing would take place in all the following vertical columns of terms, to whatever number of rerms IN THE CASE OF FRACTIONAL POWERS: 2.23% terms the faid feries 1, Bx, Cx*, D3, Ext, Fx5, &c, fhould be continued ; So that the following terms of the upper horizontal row of terms after the term 2Ex> mutt be 2 Goes (2 TA ID xt co K #512 bows} 2a wt, & 0, ad: inf» nitum. And in the fourth place it muft be obferved,. that the capital letter C, or D; or E, or F, &c, which enters into the higheft term of any of the faid vertical columns of terms (which higheft terms are 2Cw?, 2Dw3, 2 Ewt, 2F es, 2Gw°, &c) will not enter into any of the other terms of the; fame vertical ok luma, but the faid other, or lower, terms of the faid vertical column will in- volve in them only fuch capital letters as had appeared in the upper terms of the Soak sane vertical columns. -Thus for example, in the vertical column Blgirs J, (which is the firft vertical column of the compound feries in which af Bir terms have been added to each other) the letter C enters only: inthe upper term 2 C #*, and. the other term B*x* irivolves only the preceed- ing letter B; and in the fecond vertical column of terms in the fame compound feries, to wit, the vertical column 2 Dw, 2 BC w3, the letter D enters only in the upper term 2 D«3, and the other term 2 BC «3 involves only the preceed- ing letters B and C; and in the third vertical column 2 Ew+,.2 BD wx*, C743, ‘the letter E enters only in the upper term 2 Ew‘, and the two other terms 2 BDx*, and C’w*, involve only the preceeding d etters B,.C, and D; and in the fourth vertical column 2.Fw5, 2B x, 2.CD «5, the letter F enters only in the upper term.2 Fx’, and the other two:terms 2-BE «5 and 2-CDx° involve only the preceeding letters B, C, D, and E. And it is ealy to. perceive, fron the nature of the ealinheatnn by which the faid’ compound feries is obtained, that the fame thing wculd take place in all the following vertical columns of terms in the faid compotind feries, to whatever: number of terms the faid feries fhould. be continued. _We may therefore con- clude that there will be one, and-but one, new capital letter B, or C, or D, or E, or F, &c, contained in every new vertical column of terms in the above- mentioned compound feries, which is equal to the {quare of the fimple feries 1, Bx, Cx*, Dx3, Ex*, Fx’, &c, and that this.new letter will appear only in the upper term of fuch vertical column. This is a moft-important remark, and ought to be well underftood and ree membered. Tn the sth place it is evident, that when the figns + and —, which are to be’ prefixed to Bx and Cx” and Dx?, or any greater number of the terms of the Menpic teries 1, Bix, Cr* "Dy, Ext, Fx’, &c, after the firtt tefm 1,-have been determined, the figns that are to be prefixed to all the feveral terms in the next following vertical columns, except the higheft term, will be thereby deter- mined likewife. Thus, if the figns +- and —, which are to be prefixed to the’ fecond, third, and fourth terms, Bx, “Cx, and Dx*, of the fimple feries' r, Bw, Cwr?, D3, Ext, x5, &c; are determined, the figns of all. the terms, ex-: cept the higheft, in the next following vertical column of terms, which involves in it the next co-efficient E, to wit, the column containing the terms 2 Ew?,. 2. BDx*, C*«+, will be determined likewife ; becaufe they ‘will involve in them only the old letters B and C.and D, which. are the co-efficients of the powers of: x In. #224 AN ADDITION TO THE DISCOURSE ON THE BINOMIAL THEOREM xin the terms Bx, Cx*, and Dx}, of which the figns are already fuppofed to be determined. For example, if the firft four terms of the fimple feries 1, Bx, Cw?, Dx3, Ex*, Fu’, &c, are 1 + Be — Cx* + Dw? (as in truth they are found to be) the faid following vertical column of terms, which involves in it the letter E, to wit, the column containing the terms 2 Ex*, 2 BDw*, and C*x*+, will be 2 Ex* + 2BDx*t + C*x*. For 2 BDx* and C*«* being pro- duced by the multiplications of Bw into Dx and of Cx? into Cx’, it is evi- dent that, when it is afcertained that the fign ++ is to be prefixed to Bx and to D x3, and that the fign — is to be prefixed to Cx?, it will neceffaril, follow that BD x* will be (= + Be x + Dw?) = + BD*x*, and confequently that 2BD«* will be = + 2BDx«*, and that Cu? x Cx” will be (= — Cx? x — Cx?) = + C’x*, or that the fign 4- muft be prefixed both to the term 2 BD x* and to the term C*x*, This is a neceflary confequence of the com- mon rules of multiplication in Algebra, according to which when the figns of the factors of any product are known or determined, the fign to be prefixed to the product may be thence determined likewife. Thefe five obfervations being well underftood and affented to as fufficiently. demonfirated, we may now deduce from them a proof of the propofition that was {tated in the latter part of art. 7, to wit, that it is poffible for a feries of the form 1, Bx, Cx*, Dx?, Ext, Fx’, &c, to exift, in which the co-efficients C, D, E, F, &c, of the third, fourth, fifth, fixth, and other following terms of it, to wit, Cx*, Dx’, Ext, Fx’, &c, fhall be of fuch magnitudes, and the faid terms themfelves fhall be conneéted with the two firft terms 1 + Bw, or I + oa by the figns -+ and —, or by addition and fubtra@tion, in fuch a man- ner, that every following term of the compound feries t+ a Bat,’ 2 Cw2p 12 De3,)\ 2 Bett Bind fh occ B*x7,02 BCws, 2 BD e+ 0a: BE 457 Bee C*x4, 2CDx*, &c), or Pe eee a De ee Pex we ne eee } B75 2°BC oh) aS Bene ie ee . C*x*, 2CDx, &c), fhall be equal to nothing, or that fome of its members (for each of thefe terms is evidently a compound quantity, or quantity confifting of more than one mem- ber) fhall be marked with the fign +, and others of them fhall be marked with the fign —, and that the fum of the latter members in each term, which are marked with the fign —, fhall be equal to the fum of the former members in the fame term, which are marked with the fign +, fo as to exactly counterba- lance, or deftroy, the faid former members, and make the whole term be equal to nothing ; in confequence of which the whole of the faid compound feries will be equal to its two firft terms 1 + 2Bwx, or1 + 2X = ¥- Or t +40 This propofition may be proved in the manner following. g- Since the fecond term Bx of the fimple feries 1, Bx, Cw?, Dx?, Ext, Fx, &c, is to have the fign + prefixed to it, fo that the faid feries will be 1 tr Be, —S SS IN THE CASE OF FRACTIONAL POWERS, 225* $+ Be, Cx’*, Dx, Ext, Fx, &c, it follows (by art. 8, obf. 5) that the quantity B*~*, which forms the fecond member of the third term, 2 Cv*, B*x?, of the foregoing compound feries ty) 2D; 2 Cwe?;2-2°De3;! 2 Ext, io2F ety &c B*x7, 2 BCx?, 2 BDx*, 2 BEx', Cie srase Ac pat mutt alfo have the fign + prefixed to it. For it is the product of the multipli- cation of +- Bx into + Bx, which is + B*x?. And, fince the faid fecond member B?x? of the third term 2 C «*, B*«? of the faid compound {eries is to have the fign + prefixed to it, it is evident that, in order to make the whole of the faid term 2C x*, B*x?, or2 Cx? + B*x’, be equal to o, we muft prefix the contrary fign — to its firft member 2 C x7, and at the fame time muft take C of fuch a magnitude that the faid firf’ member 2 Cx* fhall be equal to the fecond member B*«x*, that is, we mutt take C equal to -, or (becaufe B has been already found to be equal to =) we muft take C equal to ~ x of or = And then we fhall have the whole third term 2 Cx’, 2ut — 2 D aq?! mem 7's iin Rete & Pe. i ase, MRSS PT as | Bee ie 2 C 21 - B2 x? == 2X 9a bo XK pes qt Ta = “7 == 0. And, becaufe the fign — is to be prefixed to 2 C* in the aforefaid com- pound feries, which is equal to the fquare of the fimple feries 1, Bw, Cw, Dw3, Ext, Fxs, &c, or 1 + Bx, Cx?, Dv?, Ext, iF x5, &c, it mutt like- wife (by art. 8, obf. 1) be prefixed to the correfponding, or third, term C x of the faid fimple feries ; and confequently the threc firft terms of the faid fimple I ” re of g *¥, OF I Ti So b's And therefore, if the three firft terms of the fimple feries 1, Bx, Cx?, Dw, Ew«*, Fx*, &c, are 1 + = — = the three firft terms of the foregoing com- feries will be 1 + Be — Cx’, ori +—« = pound feries 1,2 D4, 2C #*):23Dy?, 2Ext, 2-Fx',. &c Bes? 2 BC x. 2 BL as pe Dirk. 5» Sc Ce ye, “BCD we (which is equal to the fquare of the faid fimple feries 1, Bx, Cx*; Dix?, Ex*, F x5, &c) will be I+ 2 Bx — Cx? + Bex? Jy ori+2x —#%#— 0; or 3 + * — 03; which are equal to1 + ». Vo. II, 25 * Tose *226 AN ADDITION TO THE DISCOURSE ON THE BINOMIAL THEOREM 10. The fourth term of the foregoing compound feries is 2 D x3, 2 BC v3, or (becaufe the three firft terms of the fimp}e feries 1, Bw, Cw*, Dx, Ext, Fx, &c, have been fhewn to be 1 + Bx — Cx?) 2 Dx? — 2 BCx!, Now, fince the fecond member, 2 BCw3, of this fourth term, 2 Dw} — 2BCw?, has the fign — prefixed to it, it is evident that, in order to make the whole term be equal to 0, we muft prefix the fign ++ to its firft member 2 D3, and we muft at the fame time take D of fuch a magnitude as fhall make 2 Dx? ‘be exactly equal to 2 BC x, that is, we muft take D = BC, or (becaufe - ° I . it has been fhewn that Bis = —, and that Cis = =) we muft take D = é It > 4 = or eo Ab becaufe the fign + is to be prefixed to 2 Dx? in the aforefaid com- pound feries! which is equal to the fquare of the fimple feries 1, Bx, Cx, Dx?, Ext, Fxs, &c, or 1 + Be —Cx?, D3, Ext, Fx’, &c, it mult hkewife {by oe 5. ObL 1) be prefixed to the correfponding , or fourth, term, Dx}, of the faid fimple feries, and confequently the four firft thin fe the pes fi imple feries will be 1 + Bx —Cx* 4+ Dew, or r + <= = xt + | x, Or ceteue And therefore, if the me at tres of the fimple feries 1, Bw, Cx?, Dw, Eix*, Fx, &c, are 1 += Pith g's = + — z, the four firft terms of the foregoing compound feries (which 1 z ae to tie {quare of the faid fimple feries) will be 1+o2Bys—Cr? + 2D»: } + Bex? — 2BCw? f, ori +2 Xe -0-+o0,0r1 + %*% —o + 03 which are equal to 1+ # 11. The fifth term of the foregoing compound feries 150 a Bx, 2iCxtia Dwg) si2ihatio 2 as, woes Bex?,) 2 BC wt) 2 BD x42 BE 3s ace C*x4, 2CD«5, &c or (prefixing the proper figns to the feveral terms that involve only the letters B, C, and D, which have been already inveftigated) of the compound feries 1+ 2Be —9CHw* 2Dx3, 9 2 Eat, 2k, &c | J + Bx? —2BCx3 +2BDx«*, 2 BEws, &c: &c + C*x*+ — 2CD*#, is2Eut + 2BDx*t + C44. Now, fince the fign + is prefixed to the fecond and third members, 2 BD x«* and C’x*, of this fourth term, it is evident that, in order to make the whole tN THE CASE OF FRACTIONAL POWERS. 224% whole term be equal to o, we muft prefix the contrary fign — to its firft mem- ber 2 Ex*, and muft likewife take E of fuch a magnitude that 2 Ew* fhall be equal to the fum of the other two members 2 BD x* and C7? ; hae is, we muft = prea Bs Eine 2 T= h a! Saf eg PO Se Ng toa eo SS pire ST Ast. a8 ay — 723° -And, becaufe the fign — is to be prefixed to the quantity 2 E x* in the fore- going compound feries which is equal to the fquare of the fimple feries 1, Bx, Cr, Dx?, Ext, Fxs, &c, ort + Be—Cw? + D3, Et, Fx’, Sei it mutt likewife (by art. 8, obf. 1) be prefixed to the correfponding, or fifth, term, Ei x*, of the faid fimple feries; and confequently the five firft terms of the faid fimple feries will be 1 + Be — Cx* + Dx? — Ex‘, or 1 + = x — 5 tt ete R pee om yt rr-s—S 45 pape ante 1) 128% ? z 128° ‘ And therefore, if the five firtt terms of ne fimple feries 1, Bay Cx*, Dx?, Ex*+, Fx, &c, arer + - ake = ier’ _ = — the five firft terms of the fore- going compound feries (whictri is Cal to the fquare of the faid fimple feries) will be 4 + 2 Bem ox? +2D«e3 —2Ex* as + Bex? — 2 BCw? + 2 BDx* +. Cex, ort +2X= 4-0 Hegre 3 OF Ui chee en oH Ours 03 which are equal to 1 4 Ne 12, The fixth term of the foregoing compound feries é 2, 2: Bey 2:Cr??: 2D ee, Me ERS] 22 Fivss) tee Bix?, 2BCx!, 2 BDx*, 2 BEx', &c Crt), wee CDR 59 bec or (prefixing the proper figns to the fevera] terms that involve only the letters B,C, D, and E, which have been already inveftigated) of the compound feries 4- Bix* —2BCKx3 + 2BD«+ —2BEx &c 1+2Bxe —2Cxr*? +2D%? ~—~2Ext 2Ex &c &c 4+ ° C*x4 —2CDx:, is2Fxs —2BExs — Bore yt) Now, fince the fign — is prefixed to the fecond and third members, 2 BE w? and 2 CD x, of this fixth term, it is evident,- that, in order to make the whole term be equal to nothing, we mutt prefix the contrary fign + to its firft mem- ber 2 Fx*, and muft likewife take F of fach a magnitude that the faid firft Wo Se i member #228 AN ADDITION TO THE DISCOURSE ON THE BINOMIAL THEOREM member 2 F x5 fhall be equal to the fum of the other two members, 2 BEx** and 2 CDw ; that is, we muft take F caval to BE + CD, or De canRS it has been fhewn that B is = =. and that ‘@oist& es and that D is = =, and that E isi ive muft take F equal to— x a += <75 OF tO == i en eral , or to mr aby 7 256 ser Ot ee ab 266" And, becaufe the fign is to be prefixed to the quantity 2 Fw’ in the fore- going compound feries which is equal to the {quare of the fimple feries 1, Bx, Cx?, Dx3, Ext, Fut; &c; or 1,+ Be — Cx* + Dx? + Ext; Fx, &e, ir mutt likewife (by art. 8, obf. 1) be prefixed to the correfponding, or ‘fxth, term, F x’, of the faid fimple feries; and confequently the fix firft terms of the - faid fimple feries will be 1 + Bw —.Cw* + Dx3 — Ex* + Fx, or1+ ~ x wD gto a ia Set HEAL Opie ae a Bee Gee ee gi 25.) 200) a8.) slqcentGar lo crush flail aed ocd tis Come na And therefore, if the fix firft terms of the faid fimple feries. 1, Buy Cuz Div 35 3 4 ‘ Ete) Hat ra OCG Ane + -—-> ae lmgaaitoes -, the fix firft terms of the foregoing compound feries (which is equal to the ase of the faid fimple feries) will be + B*x? —2BCxw? + 2BDx* — 2 BEX: + Crt — 2CDx ori +2X~x—~o+0-0+40,0r1-+x*—0+0—0 4+ 0; which are equal to 1 -+- *. 1+2Be—2Cx? +2Dx3> —2Exwt +2F x! | 13. And in like manner it will evidently be poffible to take the following nu- meral co-efficients G, H, 1, K, L, M, &c of x*, x7, x*, «°, w7°,.*, and of the other following powers of x, in the feventh, eighth, ninth, tenth, eleventh, and twelfth, and other following terms of the feries 1, Bx, Cx*, Dx3, Ext, Fix’, Gx«S, Hx’, Ix®, Kx’, Lx, Mx", &c, of fuch magnitudes, and fo to con- nect aie fail feventh, eighth, ninth, tenth, eleventh, twelfth, and other follow- ing terms of the faid feries with its firft term 1 by the fi figns +- and —, or by ad. dition and fubtra¢ction, as to make the whole of the feventh, and of the eighth, and of the ninth, and of the tenth, and of the eleventh, and of the twelfth, and of every following term of the compound feries which is the fquare of the faid fim- ple feries*a5B 4) /Ch%, cD-v*, Eex*) F xf, (Gx", Hae’, Tx", Kx, Lee, Wie eee be equal to 0, and confequently to make the whole of the faid compound feries be equal to its two firft terms 1 -+ 4. And therefore we may conclude that there is a certain infinite feries of the aforefaid form, to wit, 1, Bx, Cx*, Dx3, I. E x*, Fx’, Gx, Hx, Ix*, Kx®, Lax’®, Ma”, &c, which is equal to 4 + 4 2, or tof” {rt + «, or the fquare-root of the binomial quantity 1 + ¥. QE. De 14. Having IN THE CASE OF FRACTIONAL POWERS. 229* 14. Having thus proved the truth of this propofition in the cafe of the {quare- root of the binomial quantity 1 + x, or when the number z in the general ex- I preffion 1 + x2, or /" fr + x, is equal to 2, we will now proceed to fhew that it is alfo true in the cafe of the cube-root of 1 +- x, or when zis equal to 3; after which we will endeavour to fhew that it will alfo be true in the cafe of any other root of 1 + x, or when z is equal to any other whole number whatfoever. Of the cube root of the binomial quantity. + x. : ; I 15. Now it is evident (as has been already obferved in art. 4) that 1 + +43, or ¥?{1 + «x, or the cube-root of the binomial quantity 1 + x, will be equal to an infinite feries of this form, to wit, 1, Bx, Cx*, Dx’, Ext, Fx’, &c, if it is poffible to affign fuch magnitudes to the numeral co-efficients B, C, D, E, F, cc, of the powers of « in the fecond and other following terms of this feries, and fo to connect the faid fecond and other following terms of the faid feries with the firft term 1, by the figns + and —, or by addition and fubtra¢tion, that the fe- cond term of the compound feries which is the cube of the faid feries, or is the produé that arifes by multiplying it twice into itfelf, thall be equal to x and thall have the fign + prefixed to it, or fhall be added to 1 (which muft evidently be the firft term of the faid cube or compound feries), and that the third term of the faid compound feries fhall be equal to nothing, and that the fourth term of it fhall alfo be equal to nothing, and that every following term of it fhall in like manner be equal to nothing, or that each of the faid terms, after the faid fecond term, fhall confift of two, or more, members, and that fome of the faid mem- bers of each term fhall be marked with the fign +, and the other members of: the fame term-fhall be marked with the fign —, and that the fum ef the latter members of each term, which are marked with the fign —, thall be equal to the fum of the former members of the fame term, which are marked with the fign +, fo as to counterbalance, or deftroy, them, and to make the whole of the faid term be equal to nothing. We mutt therefore fhew that it is poffible to affign fuch magnitudes to the numeral co-efficients B, C, D, E, F, &c, and fo to con- nect the terms Bx, C x*, D x3, Ext, Fx, &c, of the fimple feries-1, Bx, C x’, Dx’, Ext, Fx’, &c, with its firft term 1 by the figns +-- and —, as to produce the effects juft now. defcribed.. | : 16. It has been fhewn above in art. 6, that, if the fimple feries 1, Bx, Cw’, Dx’, Ex*, Fx’, &c, be multiplied once into itfelf, the product thence arifing, or the fquare. of the faid fimple feries, will be the following compound feries,, to wit, te Bt er oe eo Los | 23 Ma Abe wortengs. ee. B? x. 2, BC ¥3;-2-BD ++, 2 BE xs, &c CLE ee OA FBT a ee Therefore, *230 AN ADDITION TO THE DISCOURSE ON THE BINOMIAL THEOREM Therefore, in order to obtain the cube of the fimple feries 1, Bx, Cx’, Dx’, Ext, Fx, &c, we muft multiply this compound feries (which 1s equal to its fquare) into the fimple feries 1, Bx, Cx*, Dx’, Ext, Fx', &c, itlelf. This may be done in the manner following. Fy h2 hye hob Ne ay eke es 2F x5, &c B*x?; 2-BOwss 2: BDiatss &BEa yoke Ci ete 2 Ce eee 75.1) Et, fins ies thw Les Ex, Eig, ec Bee bes 2: Rig GOIN is 2 Paden un) eat eee ‘B?x?, 2 BC x, 2 BDx*; 2 BE x» & CAgt 2 CDs ore Bx, 2 B*x*, 2 BC x, 2 BDx*, 32 BEX, oe Bite, 2 DiC vs, 2) a ee ee BC2x%5, &c Cts 2 BC w. 2 Co 2 Cee ees BC 8850. 5C es ae Dx) 2. BD#*,” 2 CDR", oc BoD xs ore Ex*, “SBE XR, tac Fuxs, &c I, 3Bxe,3Cx*, 3Dx3, 3Ext, 3 Fxs,. &c ! 3. Be?) OBC x3... 6. BD wt ) 6 Bx? 1 hee < / Bie ais Beta OLN. bec 2B 7 C ie ots ae ae Cet he ee Therefore this laft compound feries tT, 3 Bw, 3Cx*, 3Dx3; gEwt, ~ 3 Fx, &c 3 B’x*, 6 BCx?, 6BDx*, 6 BEx:, &c 2 ale pl Oe ante le oy BL arc de BCR ae DD eee 4 BC? x5 5 d¢ will IN THE CASE OF FRACTIONAL POWERS, 231% will be the cube of the fimple feries 1, Bx, Cw?, Dx3, Ext, Pixs, &c, ad in- Junitum, Vet this compound feries for the fake of brevity) be denoted by the Greek capital letter T. We are therefore now to prove that it is poffible to aflign {uch magnitudes to the numeral co-efficients B, C, D, E, F, &c, of the powers of « in the fecond and other following terms of the fimple feries 1, Bx, Cw*, Dx3, Ex*, Fx, &cc, and fo to connett the faid fecond and other following terms of the faid feries with its firft term 1 by the figns 4+ and —, that the fecond term 3 Bx of the faid compound feries I fhall be equal to x and fhall have the fign + prefixed to it, or fhall be added to the firft term 1.of the faid feries, and that the whole compound third term of the faid feries, to wit, 3 Cx”, 3 B?x?, thall be equal to o, and that the whole compound fourth term of the faid feries, to wit, 3 Dx3, 6 BC «3, B:x?, fhall alfo be equal to 0, and that the whole compound fifth term, to'wit, 3 Ev+,6BDxt, 3C*x*, 3 B*C x4, fhall alfo be equal to o, and that the whole compound fixth term, to wit, 3 Fx’, 6 BEx’, 6CDx«‘, 3 B*Dx', 3 BC*x', fhall alfo be equal to o, and, in like manner, that every following compound term fhall alfo be equal to o, and confequently that the whole of the faid compound feries F fhall be equal to its two firft terms, or to the binomial quantity 1 -- #. ; 147. Now, to the end that the fecond term 3 Bx of the compound feries & may have the fign + prefixed to it, we need only prefix the fame fign + to the fecond term Bw of the fimple feries 1, Bw, Cw*, Dx3, Ex+, Fu’, &c; it being evident from the rules of multiplication that, if the fign + be prefixed to the term Bw in the feries 1, Bx, Cx*, Dx?, Ext, Fx',. &c, it muft like- wife be prefixed to the term 3B « in the feries IY, which arifes by multiplying the faid feries twice into itfelf. And to the end that 3 Bx, or the fecond term of the compound feries F, may be equal to w, we need only fuppofe that .3 Bis equal to 1, or that B is equal to =. For then we fhall have 3B4¥ = 3 x —~Xw¥=2xe =m. ‘Therefore, if the two firft terms of the finple fenes 1, By, Cx*, Div?, Ex, Fs, &c, ber -- — x, or r + > the two firlt terms of the compound feries F will be 1 + x. 18, In the next place we muft obferve that in the compound feries (which is equal to the cube of the fimple feries 1, Bx, Cx*, Dw3, Ext, Fx’, &c) as well as in the former compound feries (which was equal to the fquare of the faid feries) one new capital letter C, or D, or E, or F, &c, and but one, will enter into every new compound term, or vertical column of terms ; and this new letter will appear in the higheft term of every fuch vertical column, and only in the faid higheft term, and the other terms of fuch vertical column, or thofe which are placed immediately under fuch higheft term, will involve in them only fuch of the capital letters B,C, D, F, F, &c, as had occurred in the preceeding com- 8 pound “232 AN ADDITION TO THE DISCOURSE ON THE BINOMIAL THEOREM pound terms, or vertical columns of terms. And, further, when the magnitudes of any number of the faid capital letters, BgC, D, E, F, &c (or co-efficients of the powers of x in the terms of the fimple feries 1, Bx, Cxv?, Dx3, Ext, Fes, &c) have been determined, and the figns ++ and —, which are to be prefixed to thofe of the terms Bx, C x*, Dx3, Ex+, Fx’, &c of the faid fimple feries, in which the faid letters are involved, have been likewife determined, the magni- tudes of all the quantities in the compound feries T which involve the faid letters of which the magnitudes have been determined, and the figns -+ and —, that are to be prefixed to fuch quantities, will all be determined likewife. Thefe obfervations are obvious confequences from the rules of algebraick multiplication and the manner in which the compound feries I. has been derived from the fimple feries 1, Bx, Cw?, Dw, Ext, Fx’, &c, to wit, by multiply- ing it twice into itfelf. And they are true likewife, with refpec& to all higher powers of the feries 1, Bx, Cx*, Dx’, Ex*+, Fx’, &c, whatfoever, as well as with refpect to its.cube. 19. Since, by art. 17, it appears that Bx, or the fecond term of the fimple feries 1, Bx, Cx*, Dx}, Ent, Fx’, &c, 1s to have the fign + prefixed to it, it follows that the fame fign + mutt alfo be prefixed to the quantity 3 B*«?, which is the fecond member of the third term, 3 Cx”, 3 B*«*, of the compound feries TY, fo that the faid third term will be 3 Cx? + 3 B*x’. Therefore, to the end that the whole of the faid third term 3 Cx* + 3 B*x* may be equal to nothing, we mutt prefix the contrary fign — to the firft member of it, to wit, 3 C«?, and muft likewife take C of fuch a magnitude that 3 C x? fhall be equal to 3 B*x*, or that 3 C fhall be equal to 3 B*, that is, we muft take C equal to B’, orto = >< = oF a And then we fhall have the faid third term 3 Cw? 4+ 3 B*x? = — 3Cx* +3 B’*x*? = — 3 aes 3 7M =o. And, becaufe 3 C x? in the compound feries I has the fign — prefixed to ir, the third term C.w«* inthe fimple’feries 1, Bx, Cw*, Dx?, Biv*i hoe which correfponds to it, mufl alfo have the fign — prefixed to it, and confe- quently the three firft terms of the faid fimple feries will be 1 + Bx — Cx?, I I x ar orl + —x* ——wv,or!I + —-——. Bites S35 3 9 | And, therefore, if the three firft terms of the faid fimple feries 1, Bw, Cx?, Dw?) Esch y's occ, are 4 * —=) the three firft terms of the compound feries F' (which is equal to the cube of the faid fimple feries) will be 1+ 3Be — 3 Cx" + 3 B*x*, Or 1 + 3 X= — 0, 0r1 -+- « — 0; which are equal tor + w. 20. The IN THE CASE OF FRACTIONAL POWERS. 233% 20. The fourth term of the compound feries Tis 3 Dx?, 6 BC x3, B33, or (becaufe the three firft terms of the fimple feries 1, Bx, Cx*, Dx?, Ext; Fxs, &c, have been found to be 1 + Bw — Cx?) 3 Dx? — 6BCws + Biwx?, ot (becaufe B has been found to be equal to ns and C has-been found to be equal : ~ %— -oa ye Koy Rint wees trisgs OL: Pees DOK Xe RX ee ss re ar %?, or 3 Dx? — 2 x3. Now, fince the fign — is prefixed to the fecond mem- 3 27 ber of this term, to wit, 2 «3, itis evident that, in order to make the whole of the faid term be equal to o, we mutt prefix the fign ++ tothe firft member of it, 3 Dx, and we muft alfo take D of fuch a magnitude as will make the faid firft member of it 3 Dw? be equal to the fecond member 53, that is, we mufttake D equal to i x = or = And, becaufe the fign + 1s to be prefixed to the quan- tity 3 Dx? in the compound feries T, we muft prefix the fame fign + to the correfpondent term D «? of the fimple feries 1, Bx, Cx*, Dx3, Ew*, Fx, &c, and confequently the four firft terms of the faid fimple feries will be 1 + Bw | — Cr? + Dx, ort pS ee Ew, of pS ue a4 8. And therefore, if the pate ie ee of the fimple feries 1, fs ae! ib ads Ex*, Fw, &c,arer ~2—= mp AS *, the four-firft terms of the compound fe- ries I (which i is equal to the cube of lie faid fimple feries) will bet + 3 Bx cx, -4 O, OF I + e x z*—o +0, ort + x—o + 0; which are equal to I et a _ a1. The fifth term of the faid compound feries I is 3 Ew*, 6 BDx*, 3 C*x*, 3 B*C x*, or (prefixing the proper figns +- and — to the three laft members of this term, to wit, 6BDx*, 3C*x*, and 3 B*C«*, which involve only the 1co.efiicients ‘B, -C, and D, which have already been inveftigated) 3 Ex* + 6BD«* +3C*«* — 3 B*Cx*, or (becaufe B has been found to be = i> and C has been found to be = = and D has been found to be = =) 3 Ex* + 6 ganar ie ra Poe eae as eh aa 3 = = “a = ae ars or 3 Ext s+ a ee, Now, in order to make this whole term, 3 Ex* + = «+ (of which the fecond member = - = a4 is marked with the fign +) be equal to o, we mutt prefix the contrary fi zat — to its firft mem- her gk x*, an we muft likewife take E of fuch a magnitude that 3 Ew* thall be equal to = x*, that is, we muft take E = = * = tr; And, becaufe Vou. Il. 2 i *% the *234 AN ADDITION TO THE DISCOURSE ON THE BINOMIAL THEOREM the fign — is to be prefixed to 3 Ex* in the compound feries TP, it muft alfo be prefixed to the correfponding term, Ew*, in the fimple feries 1, Bx, Cx?, Dx, Ext, Fw’, &c; and confequently the five firft terms of the faid fimple feries will be 1 + Bx — Cx? 4+ Dx? — Ext, or 1 + > _ > +} a 3 4 xo — 2 x, OG cae aoa Tv And therefore, if the is firft terms of ‘id fimple feries 1, Bx, Cx*?, Dw3, 3 Fix 42 Rigi ie sare Pete oes oh a _ es the firft five terms of the com- 3 pound feries T (which is Zaual to the cube of the faid feries) will be 1 + 3 Bx —O+o—oQorr1+3X—#—0 4 0.0, of Itv~—-orto cog: which are equal to 1 + x. 22. And the fixth term of the faid compound feries T is 3 Fx*, 6 BEw’, 6CD«', 3 B°Dx*, 3BC’x’, or (prefixing the proper figns + and’— to the fecond, dnd third, and fourth, and fifth members of this term, to wit, 6 BE #5, 6CD x 73 B* Des, and 3 BC*x*, which involve only the co-efficients B, C, D, and E, which have already been inveftigated) 3 F xs —6BExs — 6 CD x + 3B*Dx> + 3 BC’w#', or (becaufe B has been found to be = > and C to nae sien her bao $13 Jey Gate avrapteaat iasan bane eda PUT shed 6.28 See =O XK Ges Eg ashame bee ta xd xg xe a 8 Gti a Loree wt or 3 Fx! — 2 4s eee 3 243243 steer ss 3 243." oh Bag? or 3 Fx — an x. Now, in order to sires this whole term 3 Fat — ag x (of which the fecond member roe «* is marked with the fign —) be equal to o, we mutt prefix the contrary fign -f to its firft member 3 F x*, and we mutt like- wife take F of fuch a magnitude that 3 F «5 fhall be equal to a «*, that is, we | mutt take F = = x Weng or ap And, becaufe the fign + is to be prefixed to 3 F «° in the compound feries T, it muft alfo be prefixed to the correfponding term F'x5 of the fimple feries 1, Bx, Cx*, Dx, Ew*, Fs, &c; and confe- quently the firft fix terms of the {aid feries will ‘be 1 "a Be — Cx? + Dx? — Ext + Fx',ori hig mck ox} — xs 3 v5. Oh eee P si 243° Tt Fag" ti x s= 10.44 22 o —— $B, ot Serr 243 And eure if shee oe firft terms of the fimple feries 1, Bx, Cw?, Dx?, Fint,Faxs, Guo; Hix? ,&c,:are1 i nds BFE Dig hOS gene the firft 81 2 2 fix terms of the compound feries T (which is equal to the cube of the faid feries) 8 will » IN THE CASE OF FRACTIONAL POWERS. 235™ wil be t+ 3Bx—o + o—-0 +0, or api db ia Ori ++—o+0—0 +403 which are equal to 1 + x. 23. And in the fame manner it will evidently be poffible to take the following numeral co-efficients, G; H, I, K, L, M, &c, of 7°, #7, «*, 79, «7°, x", &c, in the 7th, 8th, 9th, roth, 11th, 12th, and other following terms of the fimple fe- Tees, Cr. ae, ee ye, biel, la, RXOy liek, Mx &c, or fuch magnitudes, and fo to conneét the faid 7th, 8th, gth, roth, rith, rath, and other following termis of the faid feries with its firft term 1 by the figns + and —, or by addition and fubtraction, as to make the whole of the feventh term, and the whole of the eighth term, and the whole of the ninth term, and of the tenth term, and of the eleventh term, and of the twelfth term, and of every following term, of the compound feries T (which is equal to the cube of the faid fimple feries) be equal to nothing, and confequently to make the whole of the faid compound fe- ries be equal to its two firft terms 1 + +. And therefore we may conclude that there is a certain infinite feries of the aforefaid form, to wit, 1, Bx, Cx?, Dx, Pee Pe GR Od eS Re Te? Mat, &e, < which is” equal’*to Vi . 1+ x3, orto /*f1 + x, or the cube-root of the binomial quantity 1 4- x. mie Q. Re Ds 24. We have now fhewn that an infinite feries of the foregoing form, 1, Bx, I Cx*, Dx?, Ext, Fx’, &c, may exift, that fhall be equal to 1 + 4)2, or /*{r + «, or the fquare-root of the binomial quantity 1 + *; and that ano- ther infinite feries of the fame form, 1, Bw, Cw?, Dx3, Ext, Fx’, &c, but with different values of the co-efficients B, C, D, E, F, &c, may exift, that I fhall be equal to 1 + «13, or ?.(1 + *, or the cube-root of the binomial quantity 1 +: It remains that we prove that, if » be any other whole num- ber whatfoever greater than 3, it will always be poffible for an infinite feries of the fame form 1, Bx, C«*, Dx, Ex+, Fx°, &Xc, to exift, that fhall be equal I tor + x\#, or o/*(1 + 4%, or the wth root of the binomial quantity 1 + wx. Of the nth root of the binomial quantity 1 + x, when n is any whole number whatfoever. 25. Now the poffibility of the exiftence of fuch a feries in all values of ‘the - index a may be deduced from an attentive confideration of the two foregoing compound feriefes (which are equal to the fquare and the cube of the faid fimple feries 1, Bx, Cw’, Dx?, he Poe &c) and of the feveral properties bate see) oe *236° AN ADDITION TO THE DISCOURSE ON THE BINOMIAL THEOREM of their terms defcribed. above 1n art. 8, and refulting from the nature of multi- plication. For, it will appear, upon fuch a confideration of thofe feriefes, that, if we were to raife the faid fimple feries 1, Bx, Cw?, Dx?, Ext, Fx, &c, to the fourth power, and to the fifth power, and to the fixth power, and to the fe- venth power, and to any greater number of its following powers whatfoever,. by continual multiplications of the next preceeding powers of it into the faid feries 1, Bx, Cx*, Dx3, Ext, Fx, &c, itfelf, the obfervations contained in art. 8 would always be true of all thefe powers, or products, as well as. of the two for- mer and lefs complicated compound feriefes which are equal to its fquare and cube. 26. And hence it follows that, if the faid fimple feries 1, Bx, Cx*, Dx, E.x*, Fx, &c, were to be raifed to the uth power (» being any whole number whatfoever) the firft, or upper, horizontal row. of terms in the compound feries that would. be equal to fuch th power of it, would be 1, 2Bx, »Cwx?, 2 Dx3, nE xt, nF x°, &c. ad infinitum; juft as, when” was = 2, we have feen. that.it wasil,.2.Bx, 2C«?, 2D«3; 2Ex«*, 2 Fm, &c 3 and, whem owas == 3,.we have feen that it was 1, 3 Bx, 3 Cx”, 3 Dx?, 3Ex*, 3 Fx’, &c. And. it will likewife be evident that the fecond term, or term involving x, in the faid compound feries will be only a fingle quantity, to wit, » Bw, but that every following term of the faid compound feries involving any of the following powers of x, to wit, xv, «3, «*, x5, %°, x7, &c, will be a compound term, or will confaft of two, or more, fingle quantities. And, 3dly, it will be evident, that in the third term, or term involving xx (which will be the firft compound term, or vertical column of terms) the capi- tal letter C, which enters in the higheft term of the faid vertical column, to wit, in the term # C x?, will not be contained in the other, or lower, terms of the fame vertical column; but the faid lower terms will only involve the preceeding ca- pital letter B. And, in like manner, the capital letter D, which enters in the higheft term of the next vertical column of terms, involving x3, to wit, in the term 2 D x3, will not be contained in the other, or lower, terms of the fame ver- tical column ; but the faid lower terms will only involve the preceeding capital letters Band C. And, in like manner, the capital letter E, or F, or G, or H, &c, that will enter in. the higheft term. of any fubfequent vertical column of terms, will not be contained in the other, or lower, terms of the fame vertical column; but the faid lower terms will involve only the capital letters preceed- ing the faid capital letter that enters in the higheft term of the faid vertical co- Jumn. And therefore, 4thly, if the magnitudes of the capital letters contained in any given number of terms in the firft, or upper, horizontal row of terms of the faid compound feries, to wit, 1, 7Bx,nCx«’?,nDx3,nEx*,nF x*, &c, have been determined, and the figns ++ and —, which are to be prefixed to the faid terms. In IN THE CASE OF FRACTIONAL POWERS. 23am in the faid horizontal row, have alfo been determined, and confequently the figns which are to be prefixed to the corre{ponding terms of the fimple feries 1, B x, Cx?, Dx3, Ex*+, Fx’, &c (which are always the fame with the former figns, by art. 8, obf. 1.) have alfo been determined, the magnitudes of all the terms of the next following vertical column of terms, except the higheft term, will be thereby determined; and likewife the figns + and — which are to be prefixed to all the faid terms of the faid next following vertical column of terms, except the higheft term, will alfo be thereby determined. And therefore, sthly, fince both the magnitudes of all the terms in the faid next vertical column of terms, except the higheft term, and the figns + and —, that are to be prefixed to the faid terms, are determined, in the cafe here fup- pofed, or when the preceeding terms of the firft, or upper horizontal row of terms 1,”2Bx,nCx’,n Dx, n Ext, nF x5, &c, and the figns + and —, that are to be prefixed to them, have been determined, it follows that in the fame cafe the refulting value of all the terms of the faid next vertical column of terms, except the higheft term, or the value refulting from the computation of their feparate values, and the addition of thofe feparate values to each other, or the fubtraction of fome of them from the others, or from the firft term 1, of the faid compound feries, according as the figns + or — are prefixed to ther, will likewife be deter- mined. 27. If thefe conclufions are allowed to be juft, they will enable us to prove that the faid compound feries (which is equal to the mth power of the fimple fe- ries 1, Bx, Cx?, Dx?, Ex*, Fx5, &c) may be made to be equal to the bino- mial quantity 1 + x, and confequently that the faid fimple feries may be made to be equal to the th root of the faid binomial quantity, by taking the co-effi- cients B, C, D, E, F, &c, of certain proper magnitudes, and by connecting the fecond, and third, and other following, terms of the faid fimple feries, to wit, the terms Bx, Cw?, Dx3, Ex*, Fx’, &c, with its firft term 1 by the figns + and —, or by addition and fubtraction, in a certain proper manner ;—I fay, the foregoing conclufions, if admitted to be juft, will enable us to prove this propo- fition in the manner following. In the firft place, let the fign + be prefixed to the fecond term B~« of the fimple feries 1, Bx, Cw*, Dx, Ex’, Fx, &c. And it will follow that the fecond term 2 B x of the compound feries which is equal to the zth power of the fimple feries 1, Bx, Cx*, Dx3, Ext, Fx’, &c, or 1 + Bx, Cx, Ds:, Eix+, Fx, &c, muft likewife have the fame fign -+- prefixed to it; and con- fequently the two firft terms of the faid compound feries will be 1 + “Bx. Secondly, let us fuppofe B to be equal to =. Then will x Bx be = 2” x -. * = x, and confequently the two firft terms 1 + »Bx of the compound feries which. *238 AN ADDITi0N TO THE DISCOURSE ON THE BINOMIAL THEOREM which is equal to the wth power of the fimple feries'1 + Bx, Cx*, Dx?, Ext, Fx, &c, willbe 1 + 2 x= %, orl +x. j Thirdly, fince the fign + is prefixed to the fecond term Bw of the fimple fe- ries.1 Bx, Cx?, Dx? dict, bes. &c, or 1 + Bas C x4) Das). eee &c, and B has been taken = to =, let the refult of the value of the lower term of the firft vertical column of terms in the faid compound feries (in which lower term only the capital letter B will enter) be computed, and be marked with its proper fign + or —; and then let the contrary fign be prefixed to mC x, or the upper term of the faid vertical column ; and let C be taken of fuch a magnitude as to make mC w* equal to the faid refult. And, laftly, Jet the fame fign + or —, which is prefixed to Cx’, be alfo prefixed to the correfponding term, Cw, of the fimple feries 1, Bx, Cxv?, Dwt, Ext, Fes, &c. And the confe=, quence will be that the whole of the faid vertical column, or third term of the faid compound feries, will be equal to o, and confequently that the three firft . terms of the faid compound feries will be t + x, 0; which are equal tor + x. Fourthly, fince the values of B and Care now determined, and likewife the figns +- and —, which are to be prefixed to the terms Bw and C x? in the fimple feries 1, Bx, Cx?, Dx3?, Ex*+, Fx’, &c, let the values of the feveral lower terms of the vertical column of terms which involves x3, and of which the higheft term is x D x3, be computed ; which will evidently be poffible, becaufe all the faid lower terms will involve only the two capital letters B and C, which have been already determined : and let the values of the faid feveral lower terms be added to each other, or fome of them be fubtra¢ted from the others, or from the firft term 1 of the faid compound feries, according as the fign + or the fign — is prefixed to them ; and let the refult of fuch additions and fubtractions be marked with its proper fign + or —. And then let the contrary fign be pre- fixed to the higheft term, z D x’, of the {aid vertical column, which involves the new. capital letter D ; and let the faid capital letter D be taken of fuch 2 magni- tude that the faid higheft term z Dx: fhall be equal to the faid refult of the values of all the lower terms placed under it in the fame vertical column. And, laftly, Tet the fame fign which is prefixed to 2 D3, be alfo prefixed to the correfpond- ing term ‘D«: of the fimple feries 1, Bx, Cx?, Dx3, Ex*, Fx*, &c. And itis evident that the whole of the faid vertical column of terms of which z D x? is the higheft term, or the whole of the fourth term of the faid compound feries will be equal to o, and confequently that the four firft terms of the faid compound feries will be 1 ++ #, 0, 0; which are equal to 1 + wx. . And, in like manner, by computing the feveral lower terms of the next verti- cal column of terms, of which z E x* is the higheft term, and finding the refult of them, and prefixing the proper fign ++ or — to fuch refult, and then prefix- ing the contrary fign to the faid higheft term »Ex*, and fuppofing E to be of fuch a magnitude as to make m Ex* be equal to the faid refult, and, laftly, pre- fixing to the correfponding term, E.x*, of the faid fimple feries 1, Bx, Cx’, Dx, Ext, Fx’, &c, the fame fign as was prefixed to the faid higheft term, wee, IN THE CASE OF FRACTIONAL POWERS. 239% nE.x*, of the faid vertical column involving «+, we fhall make the whole of the faid vertical column, or the whole fifth term of the faid compound feries, to be equal to o, and confequently the five firft terms of it to be 1 + *, 0,0, 03 which are equal to 1 + x. And in the fame manner it will be poffible to affign certain proper magni- tudes, or values, to the following capital letters F, G, H, I, K, L, M, &c, which are involved in the following terms Fx', Gx°, Hw’, Ix*, Kiv9, Lav, mameecc, of the faid fimple feries 1, Bx, Cx*, Dxv?, Ext, Fut, Ge*, Hx7, Ix*, Ko, Lx*®, Mx", &c (to whatever number of terms the faid feries may be continued) and to conneét the faid terms with the firft term 1 of the faid fe- ries by the figns + and —, or by addition and fubtra¢tion, in fuch a manner, that the whole of the fixth term, and of every following term, of the compound feries that is equal to the wth power of the faid fimple feries, or that arifes by the multiplication of the faid feries 7 — 1 times into itfelf, fhall alfo be equal to o, and confequently that the whole of the faid compound feries fhall be 1 + x, 0, O, Oy 0, O, O, O, 0, O, O, &C ad infinitum ; which are equal to 1 + x. And therefore an infinite feries of the foregoing form, 1, Bx, Cw?, Dwx*, Ex+, Fx', Gx, I H-«7,Ix*, Kx°, Lx, Mx", &c, may exift, which fhall be equal to 1 + x)”, or to 4/"{t + x, or to the wth root of the binomial quantity 1 + x. Q. E.5D, = ae) * , “ane: 7 A te is edito:giod'« ult hats Hasan ah os) ay. Sager? TS: ORY. einrsay. aie G} ae og of fiw a FOS a SS q lignes fn eu - =—/ od a) pained) paticpenttin Bask odo uray ge close ae versie — be oa scr to"bari9} Sui ond edly Abie t Sidox br eM wh RAl The @ Des teh Trane goieok Ses UD : e tak ote aD a anion gruvrallgt: via Ee : REL RE aa Rh Ge ne Gi BOR Se garrat sts petti-antis! Ties sm} ams t Tacherton T¥BWsAH OF): 2, awe: = at er bey E ir) b> di sbi W, EOE bind opi) Baan Tatar § ah) RY ottag rid bap SOURS AQ. yer has eo: Seis i ay pia. ten wee Me basi 5 (SEBO, dix ont yd Bes n j aS toto) signin b eb oil? rte reyog, di 96 7 eo ¢ fktrps ; Sd'e is lis dts as) N33 1, emi iti a aatiat. ish be ot or Socet Oo va rod Ba f é9i war baliogae: + fish oda } to Slow oil ‘Ted? 3 “Locka clap siciotod bak, dock Y o3.lsvon cade e » aoa oath eas oe 19.9 Sale Be Te vel ey RG: ot, cE! oy neh git kc ‘ cotinine MoD « Ci AS p* +101 Hige pe chaishy Aine yaa ge. oe Lop leinigatd ‘salt do 2e01.dee oly OFS. * « { be i ~~ « : ; - teh 7 i et ack .*) \, ‘ 4 + x a ie } (si / ’ - . % [ 4 » de 7 — , , ». s . . : how ’ : . * . em . : * ce 4 4 2 v - ge Fy b » ~ $ , a) . ae 7 } a x , ay } x by i i , »? “ j * 4 . i . ‘. i ay . ‘~ > ¥ ’ . 2 ‘ > . > ae J MP ee: ' > Fr \ cae mn THE BiIPN*OUM ITAL YTV REC KR EM. 217 An Inveftigation of the Series > si ees Bx? 9s ame Se ie Otek Ewe eke, aon is fet oe above in Art. 2, as being equal to I 1+ x\nor Vt + x Bemiect i, Bx, Cx*,, Dwi, Evt,-d x, &c, be the teres that is equal to Li 1 + x)", or4/"1 + #; in which feries the capital letters B, C, D, E, F, &c, ftand for certain numbers, or numeral co-efficients of the feveral powers of x, which are hitherto unknown, and which it is the object of this inveftigation to difcover. And further, fince we have feen that the fecond term, Bw, of this feries, is al- ways to be added to the firft term 1, let us prefix the fign + to the faid fecond I term; and then we fhall have 1 + ~)* = the feries r +B x;,C x’, Dx, Ex, F x5, de: 29. Further, let all the ‘terms of the faid feries 1 + Bx, Cxv*, Dx’, Ext, I Fx°, &c, except the firft, be reprefented by the lettery; fo that 1 + «| thall be equal 1 + y. : ngs Then will the mth power of 1 + x\# be equal to the mth power of 1 +; that is, 1 + x willbe = 1 +)" = (by the binomial theorem in the cafe of integral powers, which has been already demonftrated ks in Hiss sie tract Buy one, in fa ——05 pages 153,154, &c,---- 169,) to the feries u—2 u—TI Z—2 mT a—2 2— 3 eet oe Ke “yey 2 peel MGS co 4 Blais sp Bre ead conlequently (faberating I from both fides ») x will be equal a im tI t— 2 Fee ae mo aN Syst 2 yx PO yx Dey 3 yx ys + &e. For the fake of brevity, let the fmall letters ¢, d, e, and_f, &c, Ey fubftituted in- se Eps Waa agai: 4 i 2 tia x rs feries — = ew = x ZA ui—tT 2 == t Nt—2 (3 a—T A—2 Z— itcad of — x ——, — x — Vale eee CUPS u4—2 2—3 x “pa refpectively, in the laft equation. And we fhall “aah have « = my + cy” + dy? + ey* + fy® + &e. 30. In the next place let the feries Bx, Cx’, Dx’, Ex*, Fx’, &c, which is equal toy, be raifed to the fecond, third, fourth, fifth, and other following Vou. II. Zi Of powers, 218 4u'D1S:G OUR S E YWOsOuNuCrEeR N Ie h CG powers, in order to have the values of y*, y?, y*, y°, &c, expreffed in the powers of x, This may be done in the manner following. yar Bx 4 § Caxt, | Dont ert elu. Sere 5 Pets Ce Ge Dat, wala ee eioce. Bx?,. BCx, BDx*, BEx', &c BC eC et Cee BD x4, CDx5, &c | BEx, &c age of DBgAt 2 BOiwe a BD ate on La) Cit, 2CDaxs, &c PB ep Cows Dies) so Bis rar baw s:) SB bec BiG, 1BCixijika 2BC*x5, &c B? Dx', &c el) Boyt sa BA eg te ee ae 3 BC* #5, Se: 1D one Bi xt, 9B Gwe, Scct B?Cws, &ec y+ = Bret, 4BICxs, &o 7 Be ork y > —— Biraaeee Now let thefe values of y, y?, 9, y*, y°, &c, be fubfticuted inftead of y, »*, y%, y*,y%, &c, refpectively, in the equation w = ay + cy? + dy? + ey* + fys + &c. And we fhall have: (1 Bx,~ “Ce, —aDa% n Ev x*,. Bie &e: Ht B'x*s2 BC x2 26 Beto. Be we D640 Ho BCI MS ace -+ 2 Bix?) 3 dB Ce 2dB De, ae CW os amt Ee +P Bit aeB Oxy &e + f/Bixt, &c {| ee By means of this equation we may determine the values of as many. of the co-efficients B, C, D, E, F, &c, as we think proper, by proceeding in the manner following. 31. To THHIE BIN Ow ol ae eT eee Ov RD Mo. 2t9 31. To find the value of B, we mutt proceed as follows. Divide all the terms of this equation by ; and we fhall have ay Rene, n\) x, nF x’, mE x*, &c | EOD. x, “2tee p88 Ghlinis > 2 6: xt, Sc | AYA Cin epee Sina OF DIE ae rd Css PUA ake, 809 Gets LD es OC ; SABC Mey SC | ei Gseke, A ea 4 Bcc + fBix*t, &c of equation is always true, of how {mall a magnitude foever we fuppofe « to betaken. And therefore it will be true when «is = o. But, when x is = 0, all the terms that involve « in them will be equal to 0 likewife, and confequently the equation willbe 1 = 2B. Therefore B will be = =; and confequently the twoO firft terms of the feries 1 + Bw, Cx*, Dax?, Ex*, Fx’, &c, which ts equal e I tor + x\ willbe 1 +—«, ort + =, Q, (Bs T. This value of the co-efficient B agrees with that which we before found for it in Art. 23, in which it appeared that the three firft terms of the feries that is I a x 73 a2=— | equal to 1 + x\2 would be 1 peg - = — * x: v, OY 'Y += eee 32. To find the value of C, the co-efficient of the third term, Cw’, of the © : I faid feries 1 + Ba, Cx’, Dx’, Ext, Fx’, &c, which is equal to 1 + x|*, and likewife to determine, which of the two figns + and — is to be prefixed to the faid term, or whether the faid term is to be added to the two foregoing terms ee etor 1 - — —, or to be fubtracted from them, we mutt proceed as follows. Since 1, which fers the left-hand fide of the equation obtained in Art. 31, is equal to 2 B, which is the firft term of the upper line of terms on the right- hand fide of that equation, it follows that all the other terms on the right-hand fide of that equation, taken together, muft be equal to 0; that is, the compound {eries A Os nD x’, ~a Ex, n¥ «4, &ce 61D MH, .) £2 GDC Xi) 2 DDFs); 32,6 BE XG (&e 2 0G Be G Doty csc + dB x, 3d B*Cs3, gd B*Dx+, &c ant aie Grws, bcc +- ¢Btx3, 4¢B3 Cx, &c . + fBix#*, &c will be = 0; that is, fome of the terms of this feries muft be fubtracted from the others, and the fum of thofe that are fo fubtracted muft be equal to the fum of the other terms from which they are fubtracted, 2 2 Now . 220 AU Dm s'c!olv'R Sim. Icio NC E R Ni tite Now let all the terms of this compound feries be divided by w. And it is evident that the terms that were fubtracted from the others, and which were equal to them before fuch divifion, will fill be equal to them after it. There~ fore the new feries that will refult from fuch divifion, will ftull be = 0; thatis, the compound feries nC, nx, Oe HA RL raids © + ¢ B’, ROBE, (2% BAS | yom Bw fe 26 ee omni One 4d BY ey) 3 BO Ue gn 24D eee 34 BCA Ke, Acce ie BY 2" 5 A Cie Oo. ore + fBix3, &c will be =o. And this equation will be true in all the poffiible magnitudes of «; and therefore it will alfo be true when x is = 0. But, when # is = o, all the terms of the faid feries that involve « in them will be equal to o likewife, and the whole feries will be reduced to the two terms »C 4+ ¢B*. Therefore thefe two terms 2C + ¢ B’ will be = 0; and confequently zC muft be fubtracted from cB’, and marked with the fign —. For, if it were to be added to ¢B’, it would not be poffible that 2 C ++ ¢ B’, (which on that fuppofition would mean. the fum of the two quantities 2 C and ¢ B*) could be equal too. Therefore 2 C muft have the fignm — prefixed to it; and confequently the third term, Cx’, of I the affumed feries 1 + Bx, Cx*, Dw3, Ext, Fx, &c, which is = 1 + x\2> (from which third term the faid quantity 7 C has been derived by multiplication and divifion in the courfe of this inveftigation,) mutt likewife have the fign — prefixed to it, and muft be fubtracted from. the two former terms 1 + Bw, or 1+ =; fo that the three firft terms of the faid affumed feries, which is equal to I 1 + x|\2, will ber + Be — Cx?, ort + = — Cw. And, for determining the magnitude of the co-efficient C, we. fhall have the equation — # C+ ¢B* = 03 whence ¢ B* willbe = #C, and C will be = — x n—I a—T I ; a—tI rE : ab? ins x ees 2 cB?=—-x —x na I 2 2 72 nn 22. 2 Therefore the three firft terms of the affumed feries 1 + Bw,C v?, Dx?, Exe, I Fx’, &c, which is equal to 1 + x|#, will ber + ~ —_ ~ x = “x, Or L + 1 al fet Ph ies — PS Bs, ’ ._ E. I, : A me GQ. ek 33. To find the value of D, the. co-efficient of D x3, the fourth term of the — aT i1-= I aflumed feries 1 + Bx, Cw?, Dx3, Ex*, Fx’, &c, which is equal 1 + #|7, and THE BEN @M EyA rt TBE oO RAELM. 221 and to determine whether the fign + or the fign — is to be prefixed to the faid fourth term, we muft proceed as follows. In the courte of the laft article we obtained the following equation; to wit, 7G, nD*x, nE x, nEx3, &c +c By, 12 ¢ Bis 2o¢ BD x, oc BE x2,..&c Or ta ee.) DEC 48 kc dee eed Ow, od Re D xt! ber ) = 0s Sd We er 8ce + @¢Bt x, 2eB?Cx3, &c + fiBd-w3,i;&e- | And it was alfo found in the laft article, that the term ”C in this equation was to have the fign — prefixed to it. Therefore the terms 2¢BC wx and 3 dB? Cw*, and 4eB? Cw, in the fame equation, (which contain the fimple power of C, or are produced by multiplications into C,) mutt likewife have the fign — prefixed to them; but the terms 2¢C*«? and 3d BC*w«3, (which contain the {quare of C,) muft have the fign + prefixed to them, becaufe — C x — Cis = +C*. Therefore, if thefe feveral terms have their proper figns prefixed to. them, the faid equation will be as follows ; to wit, — nC, aDx, nF. x?, nE x3, &c +¢B* — 2¢ BCyw, ar Bi) a4 2¢BEw3, &c Ze Ces at CDive , 8c + ad Bi¢—3dB*Cx*, 3¢2B* Dx?;, &c +3 ‘d'BC? x3,- &e +eBtx? —4¢ B*Cx, &c + fBx«3,, é&c But it has been alfo fhown in the laft article,, that — ~C ++ cB’ is = o, Therefore, if we drop thefe two terms, the remaining terms on the left-hand fide of the equation will {till be equal to.o; that is, nDx, n Ex, WE #3, , &c — 2¢BCyw, PMA Dik 2¢eBEx3, &c Pore Cents 2¢CD'%*’, - 8 + dBix—3dB*Cx, 3dB*Dx?, &c /willbe=e. + 3dBC’* x3, &c +e Btw? — 4eBiCw3, &c | + fBix3, &c And, if we divide all the terms on the left-hand fide of this equation by x, the equation will {till be true, or the terms on the left-hand fide of it will {till be equal to o, or thofe of the faid terms which are marked, or to be marked, with the fign —, and are to be fubtracted from the other terms, will ftill be equal to the terms from which they are fubtracted ; and therefore the equation will be as follows, to wit, I nD, I 2 222 AMD 1 SHO OU R SE: Cap NOC RE SR eS eG nT); nEx, Eg >? 8c6 — 2c¢BC, 2¢BDx, Bt BE We Re a er oa iat 8 BeCD Fac —-25Bi— 3¢B* Cx, 32B* Dx*) kc Soeers + 3d4BC* x*, &c + eBtx — 4eB?Cx’, &c + fB>x*, &c And this equation will be true in all the poffible magnitudes of x: and there- fore it will alfo be true when x is = 0. But, when x is = o, all the terms that involve w will be equal to o likewife, and confequently the equation will be z D —2¢BC +dB% =o. Therefore, by adding 2¢ BC to both fides, we fhall have 2D + dB? = 2¢ BC, that is, (becaufe dis = = x — oy —¢.xX ait aD + = x «Bs =.2¢ BC, or. (becaufe Bis = —)2D oa rin x) 623 —= 26 x ~~ x C, or (becaufe C is = — x sap? Disa x cx— = dK ea eee es OF DE xeo= x xe, or naD+ Eee Kies — xX ¢ ah erg x ¢, and (adding rey x ¢ to both fides, ) 20D + a AY Siam a x c.. Now, becaufe z is a whole number, and confe- quently greater than 1, 2 + m mutt be greater than z + 1, that is, 2 » muft be greater than # + 1, and, @ fortiori, 3 m (which is greater than 2 z,) will be greater than ” + I. Therefore ma x ¢ will be greater than ae x ¢; and n3 a+1 32 gar € may be equal to al X ¢, It 1s ne- ceflary that z D fhould be added to “th x c, and not fubtracted from it. We muft therefore prefix the fign + to 2D, and confequently to the fourth term, D3, of the affumed feries 1 + Bx, Cx*, Dx?, Ext, Fx’, &c, (which is equal I to 1 + x|#,) from which fourth term the term # D was derived by the opera- tions of multiplication and divifion in the courfe of this inveftigation. There- I fore the four firft terms of the faid feries, which is equal to 1 +x)”, will be 1 +. Bex — Cx? + Dx?,orr + —x —— + ints the points we were to determine. And to determine the magnitude of the co-efficient D, we fhall have the faid equation + #D + a 328 a+1 _ 2"—1 : PRINT, a—TI : _ 2n—1I [cs XC = x ¢, or (becaufe ¢ is = — X =) #D will be = “Sar x @ confequently, to the end that #D + y/4 ° se + Dw«*; which was one of x om 2 x ¢; whence 2D will be = 22 xc = 32 3ns Pee SEIONIOUM-? A.D bit. Bee oO 8 zon, 223 n #—t 227—TI m—-t 24—1 tm. : 6 7 = ia = +e ees and confequently D will be 2z— I na—I I 2%uz— I eg Or lg C. Therefore Dx?, or the fourth term of I the feries 1 + Bx, Cx*, Dx3, Ext, Fx’, &c. which is equalto1 + x lx , will be I i— TI 22—I 2 8, I. —— ——Eees 3 3s fz — xX ——. X tee or io C x3; and confequently the four firft terms of tlie faidferies will ber SL = wi ch ey pe 2 yg ZZ! si BE ds Ha, nt 2 2% 2 2n 3% vx OF I a—tT 2n—I - — —_—\—— B 5 eT et ETS wig mecprigh tetas = eS os Ger Evors 34. By reafoning in the fame manner as in the foregoing articles on the equa- tion obtained in the laft article, to wit, the equation 2D, nE x, nE x?, &c — 2¢BC, 2¢BDx, 2¢BEx*, &c + 2¢C?* x, 2¢CDx?, &c +d2B:—3dB*Cw, 3dB* Dx*, &c yt DOL Ks. SC +e¢Btx —4eB2 Cw, &c + fB>x?, &c or (if we prefix the fign ++ to the terms 7 D, 2¢ BD x, and 3 dB? Dx’, and the fien — to the term 2 ¢CD x?,) the equation + 2D,. nw E x, 2 ee OG —2¢BC + 2¢ BDx; a2cBEx?, ‘&c » +2cC*x —2¢CDx?, &c + €B? — 3dB* Cw + 3dB’* Dx’, &c ae + 3dBC* x, &c + e¢Btx — 4eB3 Cx, &c + fB'x?, &c we may obtain the two following equations for the determination of the values of E and F, and of the figns that are to be prefixed to them; to wit, the equation nE+ 2¢BD+4+2¢C* — 3dB*C + e¢Bt =o, or7z#E + 2¢BD+ 2¢C* + ¢B* = 3dB’C, for the determination of E; and the equation n¥ —2¢BE — 2¢CD + 3dB7 D + 3dBC? i416 DEC. -- fB* = 0, or nF + 3dB*D + 3dBC* + fBs = 2cBE + 2¢CD + 4¢B:C, for the determination of F. But the labour of refolving thefe equations by fubftituting in them, inftead of the fmall letters / Dame | n Ni tT C2. 2 aoe eg x pee 2 3 t— I]. 0» . . ° 2 ¢, d, e, f, their refpective values, to wit, as 2 224. ASD -is ee OU RSE * CRONE C Pee es on He si a— 2 3 n n—T1 n—2 n—3 ae —, and — x x od “ 3 * £.% I x 2 2 4 wes . ; 2s the capital letters B, C, D and E, their refpective ay to wit, ~, rhb — + ~ x “= x 4, and = x -x ix. ~ (for this laft quantity air a n 2n a #2 will be found to be the value of E ,») will be found to be very great; more efpe- cially in the latter equation, by which the value of F isto be determined. And in the inveftigation of the following co-efficients of the powers of «, to wit, the co- efficients Ge H, I, K, L, &c, the intricacy of the calculations becomes fo ex- ceffive as to make the difcovery of thefe co-efficients in this method. become ab- folutely impracticable. 3s. And further, if this method of inveftigating the values of the co-efficients of the terms of the feries 1 + Bw, Cw?, Dx3, Ew*t, Fx', &c, and the figns that are to be prefixed to them, was not, after the firft four or five terms, fo ex- ceedinely troublefome as foon to become impracticable, it would ftill be liable to another objeGtion. For, to whatever number of terms we had carried the in- veftigation,—as, for example, if we had difcovered twenty terms of the faid feries,—it would flill be impoffible to fee, from this method of obtaining thefe terms, that the next, or twenty-firft, term (which we had not actually invefti- gated by refolving the fimple equation that belongs to it,) would obferve the fame law of generation, or derivation from the preceeding terms, which had been found to take place amongft the twenty terms which had been inveftigated : fo that we fhould not be able to difcover with certainty any more terms of the faid I feries 1 + Bx,Cw?, Dx3, Ext, Fw’, &c, (which is equal to 1 + «|#,) than we fhould actually have inveftigated. And therefore I fhall dwell no longer on the foregoing method of inveftigating the values of thefe co-efficients, but fhall proceed to explain another method of inveftigating them, which will be both much eafier to practife in the few co-efficients we may think it neceflary to in- veftigate, and will, from the fimplicity and regularity of the feveral fimple equa- tions involving the co-efficients C, D, E, F, &c, by the refolution of which the values of thofe co-efficients are to be determined, enable us to perceive that the law of the generation, or derivation, of the load co-efficients one from another, which takes place in the firft four or five terms which we fhall have actually in- veftigated, muft likewife take place in all the remaining terms of the feries, to whatever number they may be fuppofed to be continued, Another . THE BINOMIALSTHEOREM, | nie Ts Another Inveftigation of the Series + SD um 1 Wi ee rt—Ke—— K — r+ — x x 22 2% 3% I n—tI a oT 3: tao I YF on oe 2 xt = 22 2n 3% x 42 " Te x n—t 2n— I a—t a—T ee Stole vs == &c, which is equal tor + x\2,or f1 + x3 by which the law of the continuation of the co-efficients A, B, C, D, E, F, G, H, I, K, L, &&c, will be apparent. 36. In the courfe of the foregoing inveftigation of the feries that is equal te I 1+ xl", or / ny + x, of which we have found the four firft terms to be 1 + 4— I 2mm] T I a—tI I 3 I n= Mad a deal xn a “3, or I —Axr— B x? 0 Soe. aa tf ane ‘ By n 22 4. 2"=* C3, the principal difficulties that we had to encounter related to the 3 n 2h ; » — I 22n— p difcovery of the third and fourth terms —- B ww, and —— Cw? of which the former is to be marked with the fign —, or to be fubtracted from the two firft . I : . _— . terms 1 + - Ax, ort + — #3 and the latter, to wit, ———— Cw’, is to be marked with the fign +, or to be added tothe faid two firft terms. But it was clear beyond a doubt, both from the {aid foregoing inveftigation and from the three preliminary obfervations in art. 21, 22, 23, —- — 26, that the two firft terms of the faid feries would be 1 and - ¥;. OF =, and that the fecond of thefe terms is to be added to the firft; and alfo that the form of the faid feries (which I is equal to 1 + w\t, or 44% 1+ x) would be 1, Br, Cx?, Dw3, Ext, Fx, &c, or 1 + — x, Cw?, Dx3?, Ex+, Fx, &c, in which the feveral powers of x 4 follow each other in their natural order, without any interruption. We may therefore affume it a8 a truth fufficiently afcertained, anda legitimate ground- Vor. il. 2G work hee 226 A. DIGEOUR ESE LClOOW CIFIR WAH G : - work of the inveftigation we are now going to explain, * that 1 + «l#, or /* apache TORR . ; . I : b «7 + «, will be equal to the feries 1 + — H, Cx*, Dv, Ext, Fiavt, Seerea eae being premifed, the faid inveftigation will be as follows. 37. It having been proved in art. 21, 22, 23, — — 26, that, if wis of any I magnitude lefs than 1, the quantity 1 + win, or4/" 1 + x, will be equal to: . * I 7? . . the feries 1 4- —w, Cx*, Dus, Ext, Fus, &c; it follows that, if y be any T n quantity greater than », but lefs than 1, the quantity 1 + yl", or "1 + Jy. z will, in like manner, be equal to the feries 1 + * ¥, Ca, Dy, Ey*, Fy’, &cs Now let d be equal to the difference by which y exceeds x, fo that » -+- d thall I be = y; and let w + d be fubftituted inftead of y in the laft equation ae ee = 1+ =, Cy?, Dy?, Ey*, Fy’, &c. And we fhall then have ay ene = the feries 1 —+- — x osha: Cax « + a), Dx x + @\3, E xx + d)*,, F x 7+ d\s, &c = the feriest ++ x ¥ + dy Cx xe + 20d + dd, D Kee + 3K? d+ gxd* +43, EX wt + 443d + On d* + 4nd > + d+, Bx wi + Seid - 10n3d* + 10n*di + §Sxd* 4+ di, &c = the coms pound feries I, + =x, C5 ee E ms; Fw, &e + —d, 2€xd, 3De' ds 4 Ex ds” oR ed, he Cd4*, 3 Dxd*, 6Ex* d*; 10 Fx d*,, Se: D.d3,, 48h xd3,:, rob «? d*, Se h.a:*, 5B xd+, &c: Fd’, &c. 38. Let f be = 1 +x; and we fhall have f + d= 1 + 4% +d) and! : I I fea =1+«+4+ 4)", But, becaufef+dis =f x [a ++ =, it follows. I I: I I. that f + d\n will be = fin xr + ae But, becaufe 1 + «|# is equalito. i ay: the feries 1 -- - “, Cw?, Des, Ext, Fut &c; it follows that 1 + will, in like. manner, be. equal to the-feries 1 +. ~- x 5 Cx ay ae the ail ode d+ d\4 5 d 2 a3 a> 6. ; ; &c. 2 THE BINOMIAL THEOREM 22% I I * &e: Therefore f * x I+ ag will be fe a x the feries 1 + —*x f OR Sra x Fo FX fe = the fries f Pisoniige 40x se x po DS x EEX se K FOE XSF X Fo be }: al fod e sat ieee. f- —xXi+ a will alfo be equal to the wien fF x hah 53 BD xs x Rx ff x - "? F ? 5 $3? S F x ee x = &c. T herefore, if we fubftitute 1 + x in this laft equa- I tion a Sat of f, to which it is equal, we fhall ses 1 +x + d\*= the Mage Fay 4 +—XI eee a3 Ex tpi xaos XT Ea ee I 3g. But it has been fhewn, in art. 37, that 1 + x + ad” is equal to the. compound feries 1+ — x, Cea Vlas E x4, Fix, &e +—d, 2Cwxd, 3Dw7d, 4Exwtd, 5F xed, &c Cay, wiaetd, © xyes 16 Bix? d*, F xt, &c. Therefore, if we multiply all: the terms of this equation by”, we fhall have 1 -- x\2 x : ~ = the feries pe anCx, 34Dx?, 40 Ex3, gun F x*, &c; and, if we multiply both fides of: i this laft equation by 1 -+ +, we fhall have i he “|” = the compound feries tr, 2”2CHw, 32 Dx, 4nEx?, ga Fx*, &c + # 2nCw?, 34Dw3, 4nE xt, &c 4I ) Bit: THE BINOMIAL THEOREM. 229 I AL But 1 + x |e is.equal tothe feries 1 + a Henn, D3 wt, Fx, &c. Therefore the feries 1 + — x, Cx, Dx?, Ext, Fx’, &c, will be equal to the compound feries bya 0 Ow) i9% Dx abe. a8 e+, bcc ein Ma tieeiee, aii La? st am Koes GrC: and confequently (fubtracting 1 from. both fides of the equation) we fhall have the feries — x,Cx*, Dx?, Ext, Fx’, &c = the compound feries 2%Cx, gnDe*,4n EK x3, nF x*, &c +a 2eGn.) on Din, ain hx4;ece3 and, laftky, dividing all the terms by x, we fhall have the fimple feries —, in, Le Ewe, Fx+, &c = the compound feries an, ta Dia, An Bows 60 Fixi,y &c tee 9) es Com oud ep amet e yc ec S by the help of which equation we may determine both the values of the feveral, coeficients C, D, E, F, &c, of the terms Cx”*, Dw?, Ex*+, Fxvs, &c, of the affumed feries 1 + Bx, Cw*, Dx3, Ext, Fw’, &c, or 1 + = ree OF: Shed De ‘ , Ex*, Fes, &c, which is equal'to.1 + x\n, and the figns + and — that are to be prefixed to the faid terms refpectively, by reafonings fimilar to thofe ufed in the former inveftigation, in deriving the values of the co-efficients B, C, and D from the much. more complicated equation obtained above in art. 30. This may be done in the manner following :. 42. In the firft place we may determine both the fign to be prefixed to the term C %? in the feries 1 +- =~ eC kU et, Lath x, coc (Which is equal , . to I + x|), and the magnitude of the co-efficient C, by proceeding as follows.. The final equation obtained in the laft article, to wit, the equation between the fimple feries =, Cx, Dx, Ex3, Fx+, &c, and the compound feries nC, anD x, 49 WDinr, 6 wi #3, &e uty) 28.C x, ton Dnt, 4 aks 72, &c, 1s always true, how fmall foever we may fuppofe x to. be. And therefore it will be true when w is = o. But when w is = 0, all the terms in the foregoing equation that involve w in them will be equal to o likewife; and the faid equa- . . a , . i tion will confequently then be — = 20 Geter or 2 2 Rue aC Ser i — 2 72 will be equal to 1, together with 2 C either added to it, or fubtracted from it, as may be neceflary to produce the faid equality. But, becaufe z is a whole : I : number, — muft be lefs than 1, and therefore cannot be equal to 1 togetlier with: 240 aM res?ctotu'’r ste fet W ] BR wirtartc with 2C added to it; therefore it muft be equal to 1 with 2 ”C fubtraéted ° . . I from it. Therefore the term 2 #C, in the equation —= 1,20 C, muft have the fign — prefixed.to it: and confequently the terms 2” Cw and Cx, in the final equation obtained in art. 41, and Cx’, the third term of the feries 1 + —x, Cx?, Dx3, Ext, Fx’, &c (from which third term the term 2 #C has £3 been derived, by various multiplications and divifions, in the courfe of the pre- ceeding invefligation), muft likewife have the fign — prefixed tothem. So ° . ‘ . I that the {aid third term C «? of the feries 1 + —, Cw, Dx}, Ext, Fas, I &c, which is equal to 1 + x|2 , muft be fubtraéted from the two firft terms of the faid feries ; and the three firft terms of the faid feries will therefore be 1 + ts x pastas Sem Gar Q. E. I. w And, to determine the magnitude of C, we fhall have the equation — = 1 » n — 2%C3; whence (adding 2 nC to both fides) we fhall have - +2"2C=1; and (fubtraGiing — from both fides) 27C = 1 — = = 7; and (dividing a 2 a both fides by 2 2) C =x — = = x =, or — x —. Therefore the three firft terms of the feries 1 + ~ tC) Wat hae I ‘ Fx, &c (which is equalto1 + xz) will be x + - x “ x = = 4, OF a T a— | — —— Bwx. ° E, iF reps AK | a Q: 43., Secondly, to find the value of the co-efficient D, and the fign that is to be prefixed to Dw’, the fourth term of the fenes 1 + ~ x, Cw?) Deg ies I F x5, &c, which is equal to 1 + x|, we muft proceed as follows. Since it has been fhewn in the laft article that the fign — 1s to be prefixed to the terms 2”C and 2”Cwx and Cx, in the final equation obtained in art. 41, to wit, the equation =, Cx, Dx?, Ext, Fx*, &c = the compound feries BON 8 eA ea ao ee ee +1, 24a Cx, 327 De", aw ae, ~ ore, let it oe accordingly prefixed to the faid terms; and then the faid equation will be as follows, to wit, = — Cx, Dex, Ex?, Fx+, &c = the compound feries — 220, 32D anther, cn Bws,, &c +1—2nCyx, 32Dx’?, 4nEu?, 8c. Now let 2 C w be added to both fides of this equation; and we fhall'then have Pie = T.2E (BION OMIEAL THEOREM, 231 =. + Cx, Dw*?, E x3, x4, &c = the compound feries —2nC, 9"Da, 4nEu*, gnE x3, &c +1—2nCw, 3nDx?, 4nEx3, &c. + 2Cx% ° . . I ° ’ p . But it was fhewn in the laft article that St er 2 C. Therefore, if I . ° we fubtract — and 1 — 2#C from the oppofite fides. of this, equation, the re- mainders will be equal; that is, the feries ++ Cx, Dx?, Ex3, Fut, &c, will be = the compound feries ga.Din, An Ewer, 5 a F ess) &e me 2B 0, 2 BD) "An EE w?,. &e + 2Ce Therefore (dividing all the terms by ~) we fhall have C, Dx, Ex’, Rx?, ac - = the compound {eries 265: Atle 54 BHD WAS, Cec ma 2NC, 3nDx, 4nEx*, &c +2C This equation will be true, how finall foever we may fuppofe w to be. And therefore it will be true alfo when w is == o. But, when x is = 0, all the terms which involve « will be equal to o likewife; and confequently the equation will be C = 32D — 22C 4+ 2C. Therefore, if we add 2%C to both fides, we fhall have C + 22C = 32D +2C; and (fubtraGting C from both fides) anC = 32D +C, or2“uC =C, 32D; that is, 2%C will be equal. to C, together with 3 7D ‘either added to it, or fubtracted from it, as may be necef- fary to produce the faid equality. But, fince z is a whole number, and con- fequently greater than 1, 2”C muft be greater than C, and therefore can- not be equal to C with 3 2D fubtracted from it, but’muft be equal to C with 32D addedtoit. Therefore the fign, + muft be prefixed to 3 D, and con- fequently to the terms 3% D« and 32D x? in the laft equation, and to the term D x? in the final equation obtained in art. 41; and likewife to the fourth 2 I . A : term, Dw:, of the feries 1. + = Coxe, Lows, Eid, Powd , oe, which is equal I tot -- «\*; from which fourth term D 3 all the other terms that involve D were derived, by the operations of multiplication and divifion,. in the courfe of the foregoing inveftigation., Therefore the four firft terms of the faid feries 3 I ~ x, Cx, Dwi, Ext, Fu, &c, whichis equalto1 + x|*, will be 1 + I I tm I I I — it — — xX oF we + Da3,,0r 3 + —# — Cxx + Dex, or I + — An. P| Q. E. % And, to determine the magnitude of D, we fhall have the equation 22C = C+ 32D; whence 34D will be = 2nC —C = 2%4—1|)xC, andD will: 2 32 AM DES GOUVRS EF CO WC BR NENG 2n—I _ 24—I1 u— I I I na—tI sn—tI will be = —— Gis hese Tg 6h = ke : 3% 2%n a a 2n 32 4 . I Therefore the four firft terms of the feries 1 + —*; C#*5°Dx3, Eee rae I ; r oo : I I na— I t &c, which is equal to 1 4+ «|, will be 1 + = t= XK a n—tI CF ee HC , I a—1 a 2n—I ori +—Aw —/2— Bye —— Cx, et oer 2n x 32 a? ag n 2n Tj 32 Q: E.1 44. To determine the fign that 1s to be prefixed to Ex‘, or the fifth term of I . I . . — the feries 1 + — # Cx, Dk?, Ext, Fxs» &c, which is equal to 7:4 way. and to find the magnitude of the co-efficient E, we muft proceed as follows. In the laft article we obtained the equation C, Dx, Ex?, Fw3, &c, = the compound feries og Th An Het, or ean ea — "3270, 32 Dx, an x5” ae +2C; which, if we prefix the fign + to the terms that involve D (agreeably to what was found in the laft article to be neceffary), will become C + Dx, Ex?, Fx3, &c = the compound feries +3”D, 4nEx, 5uFx*, &c —2nC + 3nDx, 4nEnx’*, &e + 2C. Now it has been feen, in the laft article, that Cis = + 32D —2nC + 2C. Therefore, if we fubtrat C and 32D ~— 2uC + 2C from the oppofite fides of this laft equation, we fhall have + Dw, Ew?, Fx?, &c = the coms pound feries | 40 ty 5 ke 5 OCG + 32Dsx, 4nE x7, &c3 and confequently (dividing all the terms by x) + D, Ew, Fw?, &c = the ¢ompound feries 4n¥E, guku, &e + 32D, 4”Ex, &ce. And this equation will be true, how fmall foever we fuppofe x to be: and therefore it will alfo be true when « is = 0. But, when x is = o, all the terms that involve w will be equal to o likewife; and the equation will be as follows, towit, + D=4xzE + 32D, or + D= 3zD, 4zE; that is, D will be equal to 3 7 D, together with 4” E either added to it, or fubtraéted from it, as may be neceffary to produce the faid equality. But becaufe z is a whole num- ber, 2 D muft be greater than D; and therefore 3 x D muft, @ fortiori, be greater than D. Therefore, in order to make 3 D be equal to D, the quantity 4” E muft not be added to 3 ”D, but muft be fubtracted from it; and confequently the fign — muft be prefixed to 4” E; and the laft equation + D = 32D, 42nE willbe + D=3”2D—4nE, Therefore the fame fign — mutt like- wile TSE BIN OMI AL “RR EO RFR EM. 233 wife be prefixed to Ew*, the fifth term of the affumed feries 1 ++ =, Cx* F I D3, Ext, Fx’, &c (which is equal to 1 + mln); from which fifth term the faid term 4 # E is derived, by the operations of multiplication and divifion, in the courfe of the foregoing inveftigations. Therefore the faid fifth term, Ex*, of the faid feries, muft be fubtracted from the firft term 1; and confequently the five firft terms of the faid feries will be 1 + ~ xv—Cxx + Dei — Exti,ori + — na— I A 22 — TI Ax —| aps Be pry rete GRE NS Q. E. tT) _And, to determine the magnitude of the co-efficient E, we fhall have the equation + D= 3”2D — 4uE; whence D + 4z7E will be = 32D, and 4nE willbe = 3xD—D= 3u—1 X D, and E will be ==> DD) = SA 1 7 Ne 8 nz—I 42 x 30 x 2n . mer ° ‘ four firft terms of the feries 1 ++ = ey Ce, Dx, Ex*, Fes, &c, which is equal I a pa ba x -, or— X a x eda ale, 3*=", Therefore the “a % 42 2n 3u I a2=— TI Ve. wi ‘sete ven ate ie 3 tor+als, willber + 7x NTN SW i MS ieX Hr A a I , a— I x 2nu— 1 x 32 — stim bc ee hoe I Bx? 4 2n ee 45. To determine the fign that 1s to be prefixed to Fw’, the fixth term of I the feries 1 + — xe, Cw, Dx?, Ext, Pixs, &c, which is equal tor + x|\@, and to find the magnitude of the co-efficient F, we muft proceed as follows. In the laft article we obtained the equation + D, Ex, Fx*, Gw3, Hx*, &c = the compound feries | 4. Es, 6-2 ty 2G 75-70 Tw, ae BO DL) Ag bot, oe ws, 6 are.) &c or (if we prefix the fign — to the terms 4”E, 4” Ew, and Ex, agreeably to what was found in the laft article to be neceffary) + D — Ew, Fa*y7 Ge, H+, &c = theicompound feries nA ant we OSS Woe gd EL OCC +3nD—4nEw, Sn x*, 6n2Ge3, &e. Therefore, if we add 2 E » to both fides of this equation, we fhall have 4+-D + Ex, Fx*, Gv; Hx, &c = the-compound feries ABs nici dat, | Ot Goes a EL ts 8 Sco + 32D — 47Ex, SOE wy 6 ns x3 SORE 24 i, And it has alfo been fhewn in the laft article, that Dis = 32D — 4nE. Therefore, if we fubtrac&t D and 3% D — 47”E from the oppofite fides of the Vo, II. | | 2H laft 234 A.i2D 198,4C 70:0 UR Ssk.. 3010 NE EIR Werlonac laft equation, the remainders will be equal; that is, the fimple feries Ex, Fx’, G «3, Hx*, &c will be equal to the compound feries ie ee Pea RES gee MCE LRM Cd woe Lise aE, Omir e st tek +2 Ex ; and confequently (dividing all the terms by «) the fimple feries E, F x, Gx?, Hx, &c will be equal to the compound feries ~ 600 B50 On Ces A 7il & ys oe — ARE, suk xy j6inG 025) dc +2E. And this equation is true, of how fmall a magnitude foever we fuppofe ¥ to be taken: and therefore it will alfo be true when x is== 0. But, when x is = o, all the terms that involve x will be equal to o likewife, and confequently, the equation will then be E = 52F — 4nE +24. Therefore (adding 42 ¥ to both fides) we fhall have E+ 4” E = 5”F + 2 E, and (fubtraG. ing E from both fides) we fhall have 4xE = 5”F + E, or 4nE = hy 5nF; that is 4” E will be equal to E, together with 5 7 ¥ either added ‘to it or fubtracted from it, as may be neceflary to produce fuch equality. - But, be- caufe 4 is greater thant, 4” E muft be greater than EK, ‘Therefore, in ones to make E be equal to 4” E, it will be neceflary to add 5” F to it; and confe- quently the fgn + muft be prefixed to 5 #F, and the laft equation. 402-35 -=3, Bey gsn¥F will be 4nE=E + guF. Therefore the fign + mutt alfo be pre- fixed to the fixth term, F x‘, of the aflumed feries 1 + -. x, C#*) Dx *spiraes I Fx, Gx, Hx’, &c, which is equal to 1 + x\#; from which fixth term the term 5 7 F has been derived by the operations of multiplication and divifion in the courfe of this inveftigation. Therefore the faid fixth term, Fx‘, of the {aid feries is to be added to the firft term 1, and confequently the fix firft terms. of the faid feries will be 1 + — x — Cx? + Dx? — Ext + Fr, or 1 + I I a— 1 x warmed" B ISA Ms 5 n—tT 22— 1 a 22 2n 3% 73 2% an n— TI T A—T° 2n— I u— TI uaz—~ I Hf n—? Si y* 4+ —xX cmaty wan 4 w5,ort +—Ax— 44 a 2n 3% 42 $2 a 24 I u— I By? + CMa oo ee ee bee tigen 4% And to determine the magnitude of F, we fhall have the aforefaid equation 4nE = E + put; pee 52F will be = 4nE — E=4n—1)x E, SA eS Po 32 — 1 22 — 1 a—t I and F will be = Tair x ae x ar x = Mh ine or —- yy Sty 2 x 2". Therefore the firft fix terms of the 2n 3n 47 52 I . ° feries 1 + — H, Cax, Dx?, Ext, Fx', Gxt, Hw’, &c, which is equal to t — Jae I pend eae t+ a2, willber + —* — > x ~ xa — x A ee —x* THE BINOMIAL THEOREM. 235 a—t 21 — TI an — 1 I a—t 2n— 1 ant 4n—1 ~— & 2 —— | 0 Tg anes 2.6 aie 2n 3n 4n n 24 3% 4n Sn 46. The reader will obferve, that in this way of inveftigating the values of C, D, E, F, &c, there is no more difficulty in determining the values of E and F, than in determining the values of C and D; becaufe the fimple equations, by the refolution of which the feveral new co-efficients are to be determined, confift always of only three terms, to wit, of one term derived from a term of the fimple {eries —, Cw, Dx?, Ex, Fut, &c, (which forms the left-hand fide of the final equation obtained in art. 41,) by dividing it by the power of « with which it involved, and of two terms derived in like manner from: two terms of the compound feries anC, 3nDx, 4nEx*, snF x3, . &c +1, 22Cx, 3nDx*, 4n Ex}, &c (which forms the right-hand fide of the fame final equation), by dividing them by the fame power of x: whereas, in the former way of inveftigating the values of thofe.co-efficients, fet forth in art. 31, 32, and 33, the feveral fimple equations» by the refolution of which the faid co-efficients are to be determined, to wits the equations 1 = »B,andcB* —”zC = 0, or cB? = uC, anduzD — 2¢BC + dB?-= 0, or“D + dB? = 2¢BC, and zE + 2¢BD + 2¢C? — 34B°C + ¢B* =0, or nE+ 2¢BD + 2¢C* 4+ ¢Bt = 3dB°C, and nF —2¢cBE — 2¢CD + 3dB°D + 3dBC* —4eB3C +/B* =0, or nF + 3dB*D + 3¢dBC* +/B* = 2¢ BE + 2¢CD + 4eB3C, in- creafe continually in the number of their terms by the addition of two terms in every new equation; and likewife confift of terms that are more and more com- plicated continually, fo as foon to make the labour of refolving them become intolerable. Therefore the prefent method of inveftigating thefe co-efficients is, in a practical view, very much to be preferred to the former method; though _ the reafonings ufed in that former method: feem to be rather more direét and fimple than thofe which we were obliged to have recourfe to in art. 37, 38, 39, 40, and 41, in order to obtain the final equation fet forth in the laft of thofe ar-_ ticles. Both the methods therefore have their feparate merits, and are worthy of our attention; and the former method ferves as a proper introduction to the latter. 47. From the manner in which the grand final equation, —, CAL De} Ex}, Fx+, &c = the compound feries, onCii 38 Dixe od wEns | ion Bx3, .&c + 1,2” Gey 93.2. DK, 4B 7, &c,) in art. 41, was obtained (which was by the multiplication of the feries 1, 2 2 C x, g32uDx*,4n Ex, 52 F x*, &c into 1 + # inart. 40), it is ealy to fee that the feveral equations for determining the values of the following co-efficients G, 2H2z tra GD 236 A DISCOURSE CONCERNING H, I, K, L, &c (which we have not yet inveftigated), and the figns + and —, that are to be prefixed to them, would be as follows. The firft equation would be F = 6”G, 5xF, orF=5uF, 6”G, or (fince we have already feen in art. 45, that 5 #F and F muft have the fign ++ Beste to them) + F = + sa, 6G, and confequently (in order to make the faid equation poffible) + F = 5uF — 6”G; Wee it will follow that Gw*, or the feventh term of the affumed feries 1 + _ x, C#m I - Ex‘, Fx, &c (which is equal tor + x\”), muft be marked with the fign —, or be fubtra€ted from the firft term 1, and that G will be = — t (Serle, And the fecond equation would be G = 7” H, 6G, or (becaufe it has jutt a been fhewn that the fign — is to be prefixed to6”GandG) —G = 7nH — 62G, and confequently (in order to make the faid equation pof- fible) — G>+7nH — 6%G; whence it will follow that Hw 7, or the r ‘ r ey . eighth term of the affumed feries 1 + — ty Cx, De, Ex, Fixt, &e Grhich I is equal to 1 -++ xl”), will be marked with the fign +, or added to the firft term 1, and that H will be = “= aie G, And the third equation would be H = 82], 7”H, or H = 7a#H, 3 ni, or (becaufe it has been juft now fhewn that the terms 7 7 H and H muft have the fign + prefixed tothem), + H = + 7”H, 8 ee and confequently (in order to make the faid equation poffible,) + H = 72H — 8 #1; whence it will follow that Iw®, or the ninth term of the feries 1 + — i, Cw, Dat, Et, m) us Fx*, &c (which is equal to 1 + «\7), will be marked with the fign —, or fub- tracted from the firft term 1, and that I will be = ie eter And the fourth equation would be I = gzK, 871, or (becaufe it has been juft now fhewn that the terms § #1 and I muft have the fign — prefixed ta them) — I= 9”K — 8x1, and confequently (in order,to make the faid equation pofble,) — I= ss gnK — 8x13; whence it will follow that K «2%, “= I . ° or the tenth term of the feries 1 + — — x, COR, 1) fy ae, Fxs, &c (which is equal to 1 + AP n), will be marked with the fign +, or.muft be added to the firft term 1, and that K will be = =" y I. And the sth equation would be K = 10”L, 97K, or Seat it has Bea juft now fhewn that the terms gzK and K rivult have ‘the fign + prefixed to them) + K = 10“L + 9”K, or + K= + 9”K, 10”L, and confe- quently (in order to make the faid equation pofible) + K=-+ go#K — 191.3; whence it will follow that Lw*°, or the eleventh term of the Cosicn I +f * ¥ a THE BENOMItAL THEOREM, 237 : L = ”, Cx’, Dx*, Ex+, Fx’, &c (which is equal to 1 + x\n), will be marked with the fign —, or fubtracted from the firft term 1, and that L will be = aa —1 x K, “And the like fhort and eafy equations, —L==—i10“zL+i1124M, +M= + 114aM— 122N, —~N =— 12“2N 4+ 1320, +O2+ 1320 — 14”7P, — P = — 14HP + 157Q, &e, would be found for the determination of the values and the figns of the following co-efficients M, N, O, P,Q, &c, ad infinitum, or as far as we pleafed to continue the invettig ote We may therefore now conclude with certainty, that the quantity 1 + als my OF rs ath root of the binomial quantity 1 + ¥, 1s nz—il I 2 —- 5 2H =-\F ee. es a 2 } ast x3 metal a to the feries 1 ++ x ra act 24h Carpe re Fe —I 2a — I ol: biti Bs a—tI 2n —1 32 — 1 4n —1 n x - 2n * 3% x 42 = 4 mh en x 2n 3n x 42 % 52 ~ =, 1 Se - 2" —~1 Dh totam ae 4n— TF a &c, or 1 + — Ax = By? + eae Cry ite Det Eee Ed ig a 62 —1 Gis hts ope ey 82—1 Tye 2 et 62 70 Sz gn + &c ad infinitum, in which the figns of the terms that come after the fecond I : ° term — ~ are alternately — and +, and the law of the generation, or continu- ation os the terms one from another, is very manifeft, every new generating fraction of the co-efficients C, D, E, F, G, H, 1, K, L, &c, being derived from the generating fraction which immediately preceeds it, by the addition of x te both its numerator and its denominator. (her diea te 48. This feries agrees exactly with that which was fet forth in art. 2, as be- I ing equalto 1 + x|#, or \/* 1 + ~, and which had been derived from the binomial theorem in the cafe of integral powers. A Review of the foregoing Inveftigation of the Series which is equal to I I + x|", or f® 1+ x. 49. I have now completed the fecond inveftigation of the feries 1 + Bw, Cx, Dx?, Ext, Fi, &c, ort — = i; Cx, Dx3, Ext, F #5, 8c, which is equal 238 A 'DIS}C OUR Se” ClO N'C*EOR) NIT G I equal to 1 ++ wn, or 4% 1+ x. The eafe with which it enables us to find the values and the figns of the feveral co-efficients C, D, E, F, G, H, I, K, L, &c, gives it a great {uperiority over the former inveftigation explained above in art. 28, 29, 30, &c— 34, and raifes it to an equality with any method of invefti- gating this feries that ] have yet feen. Yet, from the number of the fteps which it contains before we obtain the final equation in art. 41, it may, per- haps, be thought to be rather too abftrufe and difficult; though none of the fteps in it, taken feparately, feem to deferve that character. ‘To explain it, therefore, as fully as I can to the reader, and make the connection of the rea- fonings ufed in it as apparent as poflible, I will here ftate them over again with fomewhat more brevity- than at firft, in order to bring all the parts of the invefti- gation into view at once. Now the principal artifice of this inveftigation confifts in finding two diffe- ‘ I rent expreffions for the value of 1 + *« + d|*. The firft of thefe expreffions is obtained by confidering the trinomial quantity 1 + « + das being a bino- mial quantity, of which the firft member is 1, and the fecond isw +d. By fuppofing y to be equal to x + d, and confequently 1 + y to be equal to 1 ++ x -++ d, and affuming the feries 1 + —¥, Cy?, Dy3, Ey*, Fy, Gee toe I the value of 1 + y]# (as we before affumed the feries 1 + # C wet Deere i I Ex*, Fx', &c for the value of 1 + «\), and then fubftituting in the faid feries 1 +- ~ 9, Cy*, Dy*, Ey*, Fy*, &c, anitead of ¥, 973 a yews Bec, the like powers of x + d, to wit, + d, x +d), + d\', w+ d\‘, x + d\*, &c, or & +d, ow? + 2nd + d*, x? + 3nd + 3d? + d3, «* Laxid + 6x7d? + 4xd3 4+ d*,45 4+ 5xtd + 103d + 10nd? + Sxd* + 45, I &c, we obtained the equation, 1 + « + d\*, = the compound feries i ee ORL DA | Ci aes F243, &c n +—4, aCxd, 3Dx«*d; 4B wid, is Bxtd,vk&e C?Z*, 3Dxd*, 6Ex*d?, 10F x3d*, &c mL IGr Sy Ae es LO eae ore Batt ku da eee Fds, &c. I The fecond expreffion of the value of 1 + x + d\~ was obtained by confi- dering the trinomial quantity 1 ++ « --+ d as being a binomial quantity of which 1 + « is the firtt member, and d the fecond. By fuppofing f to be equal to 1 + «, and confequently f + d to be equal to 1 + * + d, and therefore f x eins y; (which is = f + d) to be alfo equal to 1 + » + d, and affuming the feries 3 7 HOR BINOMIAL THEOREM, 239 feries 1 + = « 4 Cx a, D x 7, E x 4" F x a, &c, for the value of 1+ (as we before affumed the feries 1 + ~ Mees, Divs ext, Fes, ~ &c, for the value Baar ty and the feries 1 + — ¥, Cy?, Dy3, Ey*, Fys, &c, for the value of Ea ae we found oa EESTI to be equal to f = x tas 3 d ; the feries 1 + “. x 4, se SI's D x 4| mae mols F x 4)’, &c, and I sa : dt a3 confequently to f » x the feries 1 + — x e. CR rat Dax FF? Eo a I rT I F x = Sc, and therefore to the feriesf » + ~- x fx “, CORK A Se a SOS ao ics es ee a PE fo BMX se x fi" X Fes ay he: x 4 Cy. thaters Cf we {ubftitute 1 + x in the terms of this feries inftead of f; which is equal to it), I I I . Ei oe d — d* to the feries 1 +- x} += xX 1H ieee Ole dae a i be x —-——=.D 1 wn)? ey d3 ae d* 2 das amet reg S020 F Tae STRAT Koo e. - Y We then equated this latter expreffion of the value of 1 + ¥ + d\n to the former, to wit, the compound feries ba — x, Cr, Dx’, E x*, Fxs, &c +—d, 2Cxd, 3Dx*d, 4Ewd, 5F rtd, &c G73 D4, 6 Ewe aio EH xed? les&e Dixi,% gE w d3; eto Fix2d3) &c gs med tc F ds, &c; and ‘ a then fubtraéted 1 + x)* from the left-hand fide of this laft equation, and the ; sf Bi feries I + —x, Cx, Dx3, Ext, Fx’, &c (which is equal to 1 + x\~), from the right-hand fide of the fame equation; whereby we obtained the equation = 2 d3 I I I ee d ee _—_—_ se me eae x1 +x\2 Simeoarees ber CA Shes a Xe Ee i d4 ws ds “1+ x2 X a Fx1-+«\" xX neat &c = the compound feries i —d. wa >) 240 & DISCOURSE CONCERNING td, 2Cw#d, 3Dxid, AEKd, ~sFxtd, &c 4 é ge IT Be ls 64 x*d*, tok «di, . &e Dd’, 4 Wd: St6F x7.d35\, Boe Ed, Cd ben sl ZS BEG and (by dividing all the terms by @) the equation I I I ned Sale e a d? ~x 1+ x2 Xap © em agate x > DX 1+" Xe 1 + x? aS d3 = d+ ; Siro ae ex Tae Fx1+wizx paar &c = the compound feries =, 2Cxu, eDx*, 4.E #; & FE xt, &c Cd, 3D xa, GE x*dy 10F x34, &e Dd*, Ane ds, Po Piet ae ye E 43, Si xd), de | \ AE d*, ~é&o3 ands fuppofing d to become = o, and confequently all the terms that involve d to : vA become equal to o likewife) the equation a xX I+ x\ x E = the feries —, 2Cx, 3Dx*,4En, 5F xt, &c; and (by multiplying both fides into 7) # _—~ ; the equation 1 + n| = Xx — = 1, 2nC x, 32 De, 4aEu swe T and, laftly (by multiplying both fides into 1 + ), the equation 1 + | =a, 2a, 96 Dt ae tae) oct ee “+ ¥, 2@nGux?, 3nDxri, San Eats Sc It Having thus obtained a fecond expreffion of the value of 1 + «|, involv~ ing the feveral powers of x, to wit, x, x*, «°, «*, 4°, &c, in their natural order, without interruption, and involving likewife the co-efficients C, D, E, F, &c, . : I < of the powers of x in the feries 1 + — x, Cx*, Dx’, Ext, FxS, &c, that was n I at firft affumed for the value of 1 ++ w|#, we equated thefe two expreffions of 1 the value of t + x| 2 to each other, and thereby obtained the equation 1 + ~ x, Cx’, Dx’, Ext, Fx’, &c = the compound feries lj tahoe, SR DRT 4 TKs Cn BAe wre -+ pe) arent,” aaa Lae nate ee een Ores and (by fubtracting 1 from both fides) the equation ~ ¥; Cw", Dirk, Ewes Mees &c = the compound feries anCw, THE BINOMIAL THEOREM. 24K an Cw: 32 D.x*5 gn Eni, wn a4, 8c + v0: 22x50 Su Dnt, man Lv, ec; and, laftly (by dividing all the terms by w), the equation =, Cx, Dx’, Ex, > F «+, &c = the compound feries 2G eRe. ee ee es eA MPS” Occ ml. f5 -ha-Cw,-.997 Duly smi x? yi &cs which is the grand final equation from which we are to deduce the feveral particu- lar equations (confifting of only three terms each), by means of which the figns of the terms Cv*, Dx?, Ext, Fx, &c, of the feries 1 + _ x, Gietgalin?, I Ext, Fx’, &c (which is equal to 1 + *|*), and the magnitudes of the co- efficients C, D, E, F, &c, are to be determined. 50.. Thefe were the feveral fteps of the foregoing inveftigation, which, the reader will probably obferve, bear fome refemblance to Mr Landen’s inveftiga- P/4 tion of the more general feries that is equal to 1 + x\ 7, which has been printed above in this collection, in pages 170, 171, 175, and of which an expla- nation has been given in pages 176, 177, 178, 193. And I readily ac- knowledge that it was after a very clofe and attentive perufal and confidera- tion of that inveftigation of Mr Landen, and in confequence of the ideas which that perufal fuggefted to me, that I difcovered the inveftigation that has been here explained. It feems, however, to be in many refpects different from that of Mr Landen, though it had been originally fuggefted by it; and, in particu- lar, I hope it will be found much lefs abftrufe and difficult. An Exainple to the foregoing Series. 51. Let it be required to find, by means of the foregoing feries, the cube-root » I of the binomial quantity 1 + »*, or the value of 1 + x 3. Here z is = 3. We fhall therefore have Bo-= - A= AN a 3 baw ae and C = —— = 2B =58 2n—I _ 6—1 Bas shee and D=-—>-C = 9 'G Tad 7 — 321 _ 9-1 at welt and, oes fies 1B Pies =r j 1 Pa = Dy adF —~“@ oI FAK BVOIE — SE, 5% 15 15 andG=*—-F = So F = -tF, . iA I I ee st le Ts atte wet, fF D wi Es eT a SO and t= o = H=3 H, Vou. Il. 2] and 242 A DISCOURSE CONCERNING and K = @=2 y — 45% p = 237, g” 27 27 and = “=*K = 2K = 2K, andM= = L = SLA, and N = —- M= #3" M= = M, ang (r= ae N= on N-— 3 N, and P = “=O = 2-0 = 20, and Q = “=p —.=—2p == Pp ay bons Sas D Hy? Eye Wt | Ujees = RS na swe ee nd $= R=f— RAR, and Ta i@o sae Uris Bes, 182 54 54 Therefore the feries 1 + — Aw — ~ Bat + = Cus oe — Det hing pe tee + SS Gar — ates 4 MoE ge 70 82 gz | I | I ° 102 IIa 12” 32 42 4 MIR Dyas Aoi apa ota ae + &c will, T7 Yb 18 in this cafe, be = 1 soe = But Bible £ Cx — 2 Det + Ex? mera ae: pe ge lok ig soditad By! Bae =F Exe ‘e OR 3 Hw* + a res ia Kx Sa - mee Mx i 5 Nis < 3ee Oat ite puss anae AD Ree 7 39 x nee 5x5 cf 10«4 #6 ae. 374%7 te So 4 kes 1g Tog Be Teas Ti gay Tp ser eae 935° 21,505.49 $5,953,470 147.4074" 15179256"? 35174,920%"? 59049 155945323 4,782,969 14,348,907 129,140,163 387,420,489 8,617,640%74 70 _70,664,648415_ 5 1945327,7824°° 5379259,102477 rad 1,102,267,467, 5° "10,460,353, 203 ~~ 31,381,059,609 94,143,178,827 I 18 — Te eanOSR EES he *; which therefore, is = 1 + »#|3, or the cube-root of the 255 ’ ? binomial quantity 1 + x. 7 Q, E. Ie 52. Note. This feries for exprefling the value of 1 + ae or the cube-root of the ety quantity 1 -+ x, and a fimilar feries for expreffing the value of L— ov or the cube-root of the refidual quantity 1 — x, will enable: us to * 'The co- efficients of #13, art, “wt5, wt, v17, wt® here given are equal to the co-efficients of the fame powers of x given above in page 192, but are reduced to fomewhat {maller numbers by di- viding both the numerators and denominators of the faid former co-efficients by 13 and 17. q exten THE BINOMIAL T’ Ho Eso: Rk BM, 243 extend the well-known rules for refolving cubick equations of thefe two forms, to wit, vw? + gx = rand x3} — gx =r, called Cardan’s Rules, to that cafe of the fecond of thefe equations which they naturally are not fitted to folve, and which therefore has obtained the name of the irreducible cafe; which is the cafe in which = is lefs than re This cafe muft be fubdivided into two branches, 3 2X27 . . 3 3 . . according as = is greater than , or 7 (though lefs than ra and as it is 3 j 3 3 z lefs than “. When — is greater than one though lefs than ee the equation «* — qx = r may be refolved by extending to it, by a certain peculiar train of reafoning, the fecond of Cardan’s rules, or that by which the fame equation is to ae 3 : be refolved when — is greater than =; which extenfion may be performed by the help of the two infinite feriefes that exprefs the cube-roots of the binomial quantity 1 4- « and the refidual quantity 1 — w, in a manner that is explained at confiderable length in a Paper of mine publifhed in the Philofophical Tranf- actions for the year 1778, without any mention of impoffible roots, or impoffi- ble quantities of any kind, or even of negative quantities. But when a is lefs than $ the method explained in that Paper will not enable us to find the value of x in that equation, becaufe the feries obtained for the faid value will not be a converging feries. In this branch therefore of the irreducible cafe of the equa- tion “3 — gx =r, we are obliged to have recourfe to another method of pro- ceeding, and to determine the value of » by deriving it from that of the leffer of the two roots of the oppofite equation gx — w? = r, which leffer root may be obtained by a fimilar extenfion of Cardan’s firft rule, or that by which he finds the root of the equation v3 + gv = 7, or gx + «3 =r, to the equation qx — x3 = r, by means of the faid two feriefes for expreffing the cube-roots of 1 + «and 1 — x; which extenfion may be made in a clear and intelligible manner, without any mention of impoffible roots, or other impoffible quantities, or even of negative quantities, any more than in the extenfion of Cardan’s other rule. But this extenfion of Cardan’s firft ruie to the refolution of the equation gx — x3 = rhas not yet been publifhed. i Of the Binomial Theorem in the cafe of + + x), or of the nth root of the mth power of the bi- nomial quantity 1 + x, or the mth power of its nth root, when m is.any whole number whatfoever, and n any other whole number greater than m. 53. Having now fhewn in a pretty full, and, I hope, fatistactory manner, that I the quantity 1 + *\*, or "1 + «, or the ath root of the binomial quantity 212 t+ x, 244 A DISCOURSE CONCERNING ” Ke eX Z n 2% n , 22 3n 7 I = EK a . . I I ++ x, is equal to the feries 1 + —# — I 2 — F 21k 4— TF x ner 2 2% 30 4 a—1 42 xs — &e ad infinitum, ort + — — Ax — ve a — 1 xo TK, K x*° + &c, ad Acedia ate to what was afferted in im art. 2; I thall now proceed to confider the quantity 1 + x)”, or the ath root of the with power of the binomial quantity I -- x, or (which comes to the fame thing) the mth power of its mth root, in the firft cafe of it, or when x, the index of the root, is greater than m the index of the power, and fhall endeavour to fhew that 2 in this cafe, the faid quantity 1 + «| will be equal to the feries 1 + — vw — ‘'7n nu — Mm " Mm 2 — M1 2z—=—m m t/a //4 2i— im Toate cae Se — x 19 oe a ant tee Plot < phar Maley ar glee - aa Sr as m n—m 2n — m 3n — m 42 — m : 2 4 rad mma Seneca > - tre PO se ae ae Xx regan Fe &c ad infinitum, or (put a — mM ting A = 1, and B= =, or the co-efficient of x, and C = = x , or the co-efficient of #7, and D, E, F, G, H, &c, for the co- efioients As Pep Tere x®, «7, and the following powers of «, in the fourth, fifth, fixth, Pe eighth, “Bx? and other following terms of the feries,) to the feries 1 + —~ Ae 2 — m 3n — m 4n —™ C3 — (> — Dx + —— Exs — uy 3n we o 7iz—m Hx? mh LN inp: pial a Kw? + &c, ad infinitum, rae |e to what was afferted 1 in rater 3. Now this may be done by methods of reafon- ing exactly fimilar to thofe which we have already made ufe of in nating the I foregoing feries, which is equal to 1 + x \*, or /"1 +4. Thefe mestiods may be explained as follows. Obfervations preparatory to the Inveftigation of the Series fet forth in the foregoing agile as being equal to the quantity 1 Fae a, when n is greater than m. 34. In the firft place, we muft obferve that the quantity 1 + mp or the mth power of the #th root of the binomial quantity 1 + ~, mutt be equal toa feries Sef at TN OM Ta Ly. eo EO RUE M, 245 . I feries of quantities of the fame form with the feries that is equal to 1 + x), or the ath root itfelf of the fame binomial quantity,‘that is, to a feries of the form 1, . Bx, Cx?, Dx3, Ext, Favs, &c, of which the firft term is 1, and the fecond and third, and other following terms, involve in them the feveral powers of «, to wit, v, x7, #3, ~*, 5, &c, in their natural order without any interruption. For, if a feries of the faid form 1, Bx, Cw*, Dwx3, Ext, Fxs, &c, be mul- tiplied into itfelf any number of times, or raifed to any power denoted by the whole number m, it is evident that the product, or mth power of the faid feries, muft always be a feries of the fame form, or a feries of which 1 will be the firft term, and of which the fecond and third, and other following terms, will involve the feveral powers of w in their natural order without any interruption. And I therefore, fince 1 ++ x|* has been fhewn to be equal to a feries of the faid form m 1, Bx, Cx*, Dx?, Ex*, Fx’, &c, it follows that 1 + x)”, or the mth power I of 1 + x\%, mutt alfo be equal to a feries of the fame form, 1, Bx, Cx?, Dw, Ewt, F xi, &ce. ‘ Os EvDy ss. In the fecond place, we muft obferve that the fecond term, Bw, of the 3 feries 1, Bx, Cx?, Dx?, Ext, Fx, &c, which is equal tor + x\@, mutt be added to the firft term 1, and confequently marked with the fign +. For, fince 1 + # is greater than 1, the wth root of 1 + x, or the firlt of x — 1 geometrical mean proportionals between 1 and 1°+- x, mutt alfo be greater than 1; and therefore, @ fortiori, the mth power of the faid wth root (which is greater than the faid th root itfelf) muft alfo be greater than 1. Confequently the 74 feries 1, Bx, Cx, Dx?, Ext, Fx5, &c, which is equal to1 + xl2, mutt alfo be greater than 1. And this mutt be true, of however {mall a magnitude we fuppofe w to be taken, fo long as it has any magnitude atall. But, in order that the feries 1, Bx, Cx, Dw3, Ex*, Fx’, &c, may be always greater than 1, how- ever fall the magnitude of » be taken, it is neceflary that the fecond term Bw fhould be added to the firftterm1. For, if B « were fubtracted from 1, it would be poffible, by diminifhing x, to make all the terms Cw*, Dx}, Ext, Fx’, &c, that come after the fecond term Bw, become fo much lefs than Bx, that the fum of them all put together thould be lefs than B x, whatever might be the magnitudes of the numeral co-efficients C, D, E, F, &c 3 in which cafe it is evident that the whole feries 1 — Bw, Cw*, Dx3, Ex*t, Fx, &c, would (even though all the terms after B x fhould be added to the firft term 1, and the feries fhould confequently be 1 — Bx + Cw* + Dwi + Ext + Faw? + &c) be lefs than the firftterm 1; which, we have fhewn, it never can be. Therefore B x cannot be fubtracted from 1, but muft be added to it; and the feries that is i equal to1 + x\7, muft confequently be of this form, 1 + Bx, Cx*, Dw’, Ext, Fx, &c. | GQ. ftuD. 56. The 246 A DISCOURSE CONCERNING 56. The truth of the foregoing obfervation, to wit, that the fecond term, B «, m2 of the feries 1, Bx, Cx*, Dv3, Ex*, Fx, &c, which is equal to 1 + xin, is to be added to the firft term 1, and not fubtra@ted from it, will likewife appear ye : I I — | by raifing a few of the powers of the feries 1 + — « — 5 we - 8c 2% I (which is equal to 1 -- x)” ), by actual multiplication, For we fhall find, in all the powers that we fo raife, that the fecond term is always added to the firt ; and we fhall at the fame time be eafily able to perceive, from the manner of the operation, that the fame thing muft take place in all higher powers whatfoever of the fame feries, and confequently in the mth power of it, or in the feries that. 4 is equal to 1 + x| 2 Now the fecond, third, fourth, and fifth powers of the feries 1 + — c— “ x a «* -+- &c, carried only to three terms, may be raifed in the manner following. 2 27 ap he a ES ae Se _ n ae i 2n : C I I I — a Ke i ee BL arm sts Po 7 a I n= il pila Skt nn — &c 2 22 2 re 2 I 2n—4 xe oooer —im_m — — wn & I+ ae ates 2x & Cc C+ eer poem — xX ax + &e 2n 2 I 21— Lt SW Asai oe Gin I I 2 a pia rie eh ab tonlc aereeat Ae ery [ RS Xx Rd | R re) 2% +oa—w t+ — x 2x? &c #2 2 I —— xX x — &e° a 2 I+-e Te 2 BINOMIAL THEORE MN. 247 —16 r+ ear +e -— x Dx & n Tee I I w= Tt 1+ a2 = 1 +—xv——xX eo wes 1 eC —16 Teel etn sete SA Ee, Bree 77 n 2n Bees Sk Shy ee NA at ae a n a I a—I . ——« + oa — OC 2 x 2n 3 Eee re eta Meteee shoe ne Ea | nm n 2n In all thefe powers of the feries 1 + _ x— ~ x —- w? - &e (which is I equal to 1 + «|#), the fecond term is added to the firft term 1; and from the manner of raifing thefe powers (which is dene by multiplying the laft feries, or : 3 I the feries equal to the laft power of 1 + x\~, by the feries 1 + - Wo re gran ttt 4 &c, and confequently by adding the produét of the laft feries mul- 2n tiplied by -~ x to the product of the fame feries multiplied by 1, or, properly fpeaking, not multiplied at all), it is evident that the fame thing will always take place, or the fecond term will be always added to the firft term, in all higher powers of the fame feries whatfoever, and confequently in the mth power of it, or in the feries which is equal to 1 + w|#. Os Bend: 57- And, in the 3d place, it is evident likewife from the foregoing operations (whereby we have obtained the three firft terms of the feriefes that are equal to 2 4 1+ x)”, I + Ae I+ x) ® , and 1 + ney refpectively), that the co-effi- cient of the fecond term of each of thefe powers of the feries that is equal to I I + x2, is always a fraction, of which the index of the power to which the va- T lue of + x\~ is raifed is the numerator, and # is the denominator. Thus, the . 2 co-efficient of the fecond term of the feries that is equal to 1 + x) ® , to wit, I 22 — . 2 . . the feries 1 + “Sc feline + x? &c, is the fration —; and the co-effi- nu 3 cient of the fecond term of the feries that is equal to 1 + x\#, to wit, the feries BS — . . . 1+ =. Hmm — — *? &c, is the fraction =; and the co-efficient of the fecond ’ 248 aA“ &S PS o'u rn ge “lon “cee W Ihc 4) fecond term of the feries that is equal to r + «|, to wit, the feries 1 4 = e pase 4n — 16 Se Xe *? &c, is the fraction 4; and the co-efficient of the fecond I A 5 term of the feries that is equal to 1 + x|*, to wit, the feries 1 + 3 x— x —*5 v* &c, is the fraction 2, And it is evident that the fame thing will ‘ I take place in all higher powers of 1 + x), or of the feries t + x— — + {ie tt = «* + &c, and confequently that the co-efficient of the fecond term of the 73 feries that is equal to 1 ++ we will be =, and that the faid fecond term itfelf U2 will be ~ x. Therefore the two firft terms of the feries that is equal to 1 + «\2 will be 1 + - «. And thefe three obfervations will always be true, whether z be greater than m™, or m be greater than z. Thefe things being premifed, the values of the following co-efficients, C, D, E, F, &c, of the feries 1 + — #, Cu, Dx?, Eat, bes, Sec, whichis m2 equal to 1 -+ ~\* (m being any whole number whatfoever, and # any other whole number greater than m), may be inveftigated in the manner following. An Inveftigation of the third, fourth, fifth, fixth, and otber following terms of the feriest + — ie, Gi Des, Eeus, boe?, Cie. wbicy. 16. egaar Ba , tor + x|% or the mth power of the nth root ’ of the binomial quantity 1 + x, when m is any whole number whatfoever, and n is any other whole number greater than m. 58. Lety be put = the feries = w, Cx, Dat, Ext, Fix’; \occyhor eee be equal the feries 1 +. = x, Cx?, Dx3, Ext, Fxs, &c, which is equal to mn mM 1-+ «2, Then will 1 + y\" be equal to the ath power of 1 + x)", or to T + xy. But, by the binomial theorem in the cafe of integral powers (which has been demonftrated above in the preceeding tract in pages 153, 164, &c — ° . Mm wm W— I — — 169,) 1 + #|™is = the feries 1 + — 4% + — x : x? 2” See 4 m—2 THE BINOMIAL THEOREM, 249 wum—2 m Vit am Li! 1 Ph dn 2 m= % m m— 1 %— 2 : Von RESERV Loe SO Rae ta YX — 3 4 i Ps iticg 3k Daas fic 3 a is + &c, and 1 + yl" is = the feries 1 + y+ pyle 2 1S ae 9 f “a—Z2 nu 2—T Nwm Z ai 1 u4— I % ae = 3 ora +. &c, will be equal to the feries ~y + — m+ox 2 et | 9 “aZ—tI Dull ae A ——4-2 3 la Wi T 2 m2 \ileaee 4 Saal ao ; aa X gee te ebay: eek i -W Srareee Ae e = Gea eer J, 59. To fhorten the expreffions of thefe co-efficients, let us put Q = = x Tf — TI t1I— TI MmM~— 2 Mm m— 1 Ww—2 m— 3 »,R=—x 3 I 2 3 4 I 2 3 pe Xx ~yet And we fhall then have m « a Qu? + Rw? + Sat + Tas ee ae + ry3 + sy* + ty’ 4+ &e. 60. We mutt now raife the feveral powers of the feries = x, Ce Det cx; Fxs, &c, or Bx, Cx?, Dx?, Ex*, Fx, &c (which is equal toy), in order to have the values of y?, y*, y*, y°, 8c, exprefled in powers of x», This may be done in the manner following : we = Bom 20ix*y Dxt, Ext, Fes, 8c gx Bw rt; D 933250 oF av iy ke Bin?) BO wis, oBD2*,. BENS, 4 8c BG Cay, CD 45. bcc BO sty Gx, oc BEx, &c Wore Dees, 2 BC x, 2 BD x4, 2 BE x, &c C2x4, AC Das &c Vor. Il. 2K ic 250 A- DIS COU-:R Sut Coo NC DESK N IN'G YB, Cx, Dw3, &c Bie?) 2 BO. 43. BD xt, secc BC? 235. &e B7C 44, 2) BCtx? ae B*Dx, &c 93. VB? x5) eB tC v2. Re? ceo TUS e oeeerc © bhbor deel SWF OF deg 4° ; BtK aA Cee, eee BeCw? &oecm ty 4 es ee de Coch, eee b tigress ) Ca Meee ie B x56 S&C. 61. By fubftituting thefe powers of the feries Bx, Cx*, Dx3, Ext, Fs, &c, inftead of y, y*, 3, y*, y°, &c, in the equation mx + Qw? + Rw} + Sx? + Tw + & = ny + ay? + ry? + sy* + ty’ + &c, we fhall have. the fimple feries mx + Qu? + Rat? -+S xt + Tx’ + &c = the compound feries Rib inlaws 4.) #3,% 0: #4; ee PG hg Bin? ss 2 abOwss 2) BONE wed beeen g.C*a%, ~& 2g OLens, ee + 7 Bix, 37 B*Cxt, 3) Bale ee 27 BO*x? ,)-éce +sBtxy*. 45B3C x5, &e | $+ ¢ BSx5> bce _ Therefore, if we divide all the terms of this equation by w, we fhall have the fimple feries m + Qv + Rx? + Sx? + Tx* + &c = the compound feries n B, nie, DP gon Bes n¥Ext, &c +97 Bx, 2¢BC2?, 2¢ BD x3 o7 Bw ae PR hs eg SS ee + 7 Bx, gr B*Cx3, grB*Dxt, &e 27 RO eee Sa kg a Re eee + ¢Bix*+, &e. And this equation will be true, of however fmall a magnitude we fuppofe « to be taken: and confequently it will be true alfo when x is equal too. But, when x is = 0, all the terms in this equation that involve w in them will be equal to o likewife, and confequently the equation will be as follows, to wit, m —nB. Therefore B will be = =, agreeably to what has been already fhewn in art. 57, and confequently the two firft terms of the feries1 + Ba, Cx?, Dx3, Ext, Fx', &c, which is equal to1 + «| will be r + = He Oak. 7 62. Since THE BINOMIAL THEOREM. 251 62. Since m is equal to 7 B, it follows that, if we fubtract m and z B from the oppolite fides of the laft equation, the remainders will be equal; that is, the fimple feries Qv + Rw* + Sw? + Txt, + &c, will be equal to the com- pound feries n.C'x, ae CES 25.«3, nEwt, &c eG Is tien POL BC fe ee ey) So DN, ? OCC gq Cree SCD whan Sec + rB3x?, Bie es let, ee Sneaker OCC a kl as Babs Cxr i &e + ¢ Bix, &c. Therefore, if we divide all the terms by x, the quotients on both fides will be equal; that is, the fimple feries Q + Rw + Sx? + Tx? + &c, will be equal to the compound feries nC, nD x, nE x, nEw3, &c +9 B*,» 2q¢BCe, - 2¢BDx*, 2¢BEx3, &c g Geer 2941) #*,.. &C Baia s 37 BR C#t,o 39% BED ,- &c arc OCC + 5 B*x’, AS BaC-wt, 8 + #B> x3, &c. And this equation will always be true, to how fmall a quantity foever we fup- pofe x to be diminifhed ; and confeqiently it will be true alfo when « is = o. But, when x is = 0, all ‘the terms on both fides of the equation, that involve w,, will alfo be equal to 0, and-confequently the equation will be as follows, to wit, Q= aC + ¢B’, that is, arches Qe = x “—, and g is = — x —) = x “~—= will be = C ee Kae X84, OF (ovale Bis= 5) 4 x sells bile = aC 4 2x FLtx Bx 5, oS x 5 ~ will be =2C yt x m*, or mn x ~— pba | 2n be=r2C +7" ; nee sade —- — to both fides) we fhall have ~ nC + igiides aan tn ———, and (adding = SCO oe, fides) ——— a atm hee ‘ ote 2n m*n and Mabiidting - from both fides), = eae 5 A = ; that is, “ will be equal to —; together with #C, either added to it or fubtracted from it, as may be necefflary to produce fuch equality. But, becaufe x is fuppofed to be greater than m, — will be greater than —; and confequently z C muft be fubtraéted mn - . mm m* m* from — in order to make it equal to —-, or —. We fhall therefore have — 2% =—2C+ =, and ean (adding #C to both pe ~ +2C= Fy? and (fubtracting = — — from both fides) »C = —— — — _ ire ma 2K2 m xX 252 A DISCOVWVRSE CONCERNING wxXna— n-— mm Mm “a-— — “=m*x ee and confequently C = ae Therefere the third term, C x”, of the feries 1 ++ — te, Cx*, Dx3, Ext, Fxs, &c, which is equal to 1 + «|, muft have the fign — prefixed to it, and will be — ~ x 2 — 2n na— Mn 2 ——— x7. Q. EL tr: x *?, and the three firft terms of the faid feries willbe r + ~y — ” a 72: 63. It has been fhewn, in the laft article, that Qis = »C + g B*, or (becaufe it has alfo been fhewn that # C is to be fubtracted from g B’, or to have the fign — prefixed to it) —#2#C + 7B*. Therefore, if we fubtract Q and — 2C + q B* from the oppofite fides of the equation which was obtained in that article: by the divifion of », the remainders will be equal to each other ; that is, the fim- ple feries Rw + Sw* + Tw? + &c will be equal to the compound {eries nD x, n TS 5 2.4 ¥3 Ble aq¢BC #, 2¢gBD 4*,)) 2 ¢ BEx,t&e Cixt “eg Diey oes +e Bie, > 37 B*Cxt, 37 B*Day &c 9.7 BC An seerecc +s Btx?, 45B3C x3, &c + #Biwv3, &c: or, if we prefix the fign — to the terms 2¢ BC x, 37 B°Cx*, and4sBiCx? (in which the fimple power of C occurs with the powers of B), and the fign + to. the terms g C**, 3 7 BC*x? (in which the fquare of C occurs by itfelf and with B), the fimple feries Rx + Sx? + T x3 + &c will be equal tothe com~ pound feries | aD x, nE x *, mE ¢*, &e = 2q9BCx, 2¢BD«*, 2¢BEx, &e + ¢ C*x’,. 2'¢ CD ety kc + 2 Be — 37 BC, 3798 x ke +37 BC*ws, . &e + 5sBtx? — 45B32Cwx3, &c + ¢B°x3, &c. Therefore, if we divide all the terms of this equation by «, we fhall' have: the fimple feries R + Sx -+-- T «* + &c = the compound feries nD, nE x, aE w?; &e —2q¢BC, 2¢gBDx, 2qgBEx’, &c Hg CAM 20 CL nace +7B? —3rB°*Cx, 3rB*Dx*, &c +97 BC2x2 Sc + 5Bte — 45B%Cwx*?, &e + 7Bix7, &c And this equation will be always true, to how fmall a quantity foever we fup- pofe x to be diminifhed : and therefore it will be true alfo when x is = 0, But, when THE BINOMIAL THEOREM. 253 when « is = o, all the terms in the equation that involve x will be equal to o: ‘likewife, and confequently the equation will be as follows, to wit, R = 2D — 2qBC + 7 B*; by examining and refolving which equation we may both de- termine whether the fign. +. or the fign — is to be prefixed to the quantity D, and confequently to D «3, or the fourth term of the feries 1 + Bx, Cw’, Dx’, Pe E «+, Fx’, &c, which is equal to 1 + x\* (from which fourth term the faid quantity 7 D has been derived by the operations of multiplication and divifion in the courfe of the foregoing proceffes), and likewife what is the magnitude of D, and confequently that of Dx. Now this refolution will be found to be a matter of fome intricacy. It may, however, be performed in the manner fol- lowing : w%— TT Wr — 2 Xx t= tT 64. By fubftituting = rth cape in this equation inftead of R, and # oe rE 2 B, and — x m— 2 inftead of g, and = x x ae inftead of r, and — inftead of 2 —™m mw— “ a inftead of C, it will become as follows, to wit, = x 2 4 = T L—Z x n— Mm x =2D—»xn—-1x—x—x + — x ms m3 —3m*+-2m __ a— 23 — 32? -|- 22 Fei ch Teh ish mt ncaa ty tact ate ie nia Vit RTE Re 3 13n® — 3m? n® + 2mn™ m™ Sar t— Mt un—3n+2 ey Bye Sa Sg ge PORT An EE BOS n3 622 37 2n 6nn 1n—nm—n +m min? — 3732+ 2m3 __ gD? 3m? n? +- 33.2 + 3m? 2 — 3m3 6nz 6x2 6nn 2 n® —3m3n+ ta Gy — 3m? n + 3m? n —m3 + mn 22 2 = 2D — 3m” x ; and confequently (adding 6nn Onn 34,2 2 249 ee 9993. 372 3" to both fides) we fhall have =* OR og Diyos E Re and; (ad- Onn 6un 6nn 3 34% 2 3 2 By * ding = to both fides) =* = tra nD + ee and (fubtracting ehtpe ttl ign 2 mn from both fides) = = wie + 3 6nn- 6nn° Now est is greater than ig as may be thus demontftrated. Since x is greater than m, mn” will be greater than mz, and confequently 2m*s + mn* will be greater than 2m’n + m*n, or than 3m*n. Further, fince mn’, mn, and m? are in continued proportion, the common ratio being that of z to m, it follows from Euclid’s Elements, Book 5, Prop. 25, that the fum of the two extreme terms will be greater than twice the middle term; that is, #n* + m? will be greater than 2m’2. Therefore mn* + mn” + m? will be greater than mn* + 2m'n, or 2mn” + m' will be greater than mn + 2m’n. But it has been fhewn that mn® +-2mn is greater than mn + 2m7n, or 3m*n. Therefore 2mn* + m?> (which is greater than mn” + 2m°n) will, 4 fortiori, be greater than 3 m’n. 2mn® + m> amin Therefore —~—-— will be greater than —— Q; E. De oun: ; Since 254 Ae Ds THRO ON RSE CoO. Ne BOR NEIONG . » 2 3 Since therefore 272 +” F 2 on is greater than —, but equal toxD + 22%, it fol- ’ 6nn? 3m7n . ; : ~, in order to produce the faid equality. Therefore the fign + muft be prefixed to xD, and confequently to Dx?, or the fourth term of the feries 1 + Bx, Cw?, Dwx3, Ext, Fx5, &c, which is mM equal to 1 + x\ or the faid fourth term muft be added to the firft term 1. Thereforé the four firft terms of the faid feries will be 1 + Bx — Cx? + Dx», 2 Hz na=—- Mm ori +—-x* —— X un a lows that 2D muft be added to oe «x? +. Dwx3; which was the firft point that was to be determined. And, fecondly, the magnitude of D may be determined by means of the . : - 2mn--m> 3m™n 298 n* ye equation laft-obtained, to wit, —~—— = 2D + ane ee D 3m n : 3m*n + =. For, by fubtracting ¢— from both fides, we fhall have »D = 2mn®—3m*n-+m> _ n—mX 2am—m __ n—m 22 — m , -——— = re = Xa rede and confequently (di- eye . m a—— 2 22 — mM 22 — Mm viding both fides by 7) D = — x BD I secsOU-R SE, CH AL ER Ne ieNrs whatever may be the magnitude of x. It follows, therefore, that if we fuppote x to be increafed from any one particular magnitude denoted by x, to “ other magnitude (lefs ee 1) that is denoted by y, the quantity 1 + y|#, or the mth antes of 1 + 9\” or of the mth root of 1 +.y, will be equal to the feries np Pent > macy an » Dy?, Ey*, Fy', &c. Now let d be equal to the difference by which y exceeds w; fo that w + d fhall be equal toy: and let « + d be fubftituted inftead of y in the laft equation 1 + y)* = 1 4 = 9, Cy*; Dy; m E y+, Fys, &c. And we fhall then have T+txtdn = the feries 1 + = 4 x+d,C xx + 4p, Dx«+d43,E x x + dj', F xx + dy, &c = the feries 142x544 CxP pod ta, DK Tadd aR oe, 1ow’d? + 5xd* +d’, + &c = the compound feries i + —x, Ox 19x25 Eat, Fx, &c + —d, Mh OH MRE Me BEF AEP EY: §Htd,: n8c0 Cd*, | 3 Ded?) 6 Extd, 10 Pixsd*, cbse Dd?, sExd, -10oF x7d3, &e Ef‘, oh od *ss ee Fd‘, &c. 68. Letfbe = 1 + *. And we fhall have f +d = 1 4+ # + d, and m mM S £4 (ee ee + 4). But f + dis =f x FE + <. Therefore f + d|x will be = fz X1+ ie But, becaufe 1 + x) is equal to the feries m 1 -+- = x, Cx?, Dx3, Ewt, Fx’, &c, it follows that I el ab will, in like manner, be equal to the feries 1 + — x =, C x a. “PAB, x am tae Pa ex sai poy 8 x EAE oh + *)5? 69. But it has been fhewn-in art. 67, that 1 + « + ap is equal to the com- pound feries m 1+ —*, Gung) 1D Bvxs) Fex>,—8cc + —d, ih OCH pk UE re Gah Se ed, VEC Oden Dae, Olea”) 10D x3 d*oc&c Dd, Bex d* tole? dt k&u Ed‘, ga Kae eC Fd5, &c. Therefore the feries 1 + am LE aahe 3 I+ ar x —, = C x ee x d* d+ 1/4 ae x I ar 4 ates a oa ated oh I xn x aap? &c, will be equal to the faid compound feries ifs I + = x, Clee eel; Ex*; Fes, &c + —d, 2Cxd, 3Dx7d, 4Exd, 5F xd, &c Cdames Dads nk wid ieto HK x3d*, dc Dd?; “4E#d37 “1oF «*d3,’ &€ Ed‘, od OE Rae hd tg Fds, &c. m 70. Now let 1 + x) be fubtracted from the left-hand fide of this laft equa- tion; and the feries 1 + = x, Cx?, Dx3, Ext, Fx5, &c (which is equal to E/4 1+ x\*), be fubtracted from the right-hand fide of it. Then, it is evident, mm the remainders will be equal to each other; that is, the feries = x 1 + a\= Vout. Il. aL d xem I+, 258 AMD HS CGvR FE LCOINUCIECR B PMS 1/4 ‘ wm MD - 8) ies d* nt ds pani nig ae RS 0 ee xa BX 1+ xe ds x a peal ae rE Fs tae , &c, will be equal to the compound feries Are oC #5203 Dix*d, . 4 Exid, gkixtdy &e¢ C a°,.% Dxd*, (Oe a4 io bee D d? 64) aged a OP ds oe Ed‘; 6 Fowdt)) eae Ka, ec: And aa if we divide all the terms by 4, we fhall ae the feries. ox ipa ae Cea: Sep? ae . x rar cx, = Ud et yale n~ X a So —— tite Cee feries =, aC x, (2a. “Ere; cc Fax, fee Cd, 3D«d, 6Ex*d, 10F #3d, & Dd?, 4B xa?) “joe da eee Edt) sien kee Eod* dec. 41. This equation is always true, how fimall foever we may fappofe d to bez and therefore it will be true alfo when dis = 0. But, when dis = 0, all the terms on both fides the equation that involve d will be equal to o likewife ; and mM confequently the equation will then be as. follows, to wit, i Xi twa x I . 17/4 ° ra the feries one Cr, 3D, 4E x, 5 Pixt, &c. ) Theretoreyaaawe wm multiply all the. terms by the fraction oe we fhall have 1 + x\ x E x Fi - the feries 1, —— Cx, 3 = D «7, +4 =~ Bx, a Fix*, &c; and, if we multiply MH both fides of this laft equation by 1 -+ x, we fhall have 1 + | = the com- pound feries ia Cx » Dx, A" Ex 2= F x4, &e. 1/3 2n ne ———— 498 x3 a + e ieee nee ee 2 72. Buti + %| 2 is alfo equal to the feries 1 ++ = «, Cx, Dx, Ext, F «5, &c. Therefore the faid feries 1 + — x, C eos D CL OK is Fes, &c,, will be equal to the compound feries 6 2 I, nm Y ‘ THE (BIN -OtI 2 ESD wR Bo RE M; 259 ; 2n 3n j iy a C#, 2] Dentyite Pate 2 Ets &c. WW 2 m m2 2 + x, — OF 22D, Ex*, &c; and confequently (fubtracting 1 from both fides of the equation) we fhall have the feries = bk, Cx? > Dw? Ext, Fx, écc — the compound feries Kees Quip be ppm be Ex, 2 Fx, &c, Mn 7/4 7/4 7m +n, —— Cx?; 22 Dery 4® Ext) &c: Wz Wt id and, laftly (dividing all the terms by «), we fhall have the fimple feries —, CB, Dx’, Ex3, Fx*, &c, equal to the compound feries 2x a 3 n Wt im mM ca 3 2n Bn 4% 7 5 “Tait Shed Wy pnd Ae ta Bx, &c; by the help of which equation we may difcover both ** which of the figns + and — are to be prefixed to the feveral co-efficients C, D, E, F, &c (and con- fequently to the correfponding terms Cx*, Dw?, Ex*, Fxs, &c, in the feries 17/4 1+ — x, Cx*, Dw3, Ext, Fx’, &c, which is equal to 1 + x)*),” and alfo, <¢ what are the magnitudes of the faid co-efficients C, D, E, F, &c, refpectively.” This may be done by proceeding in the manner following : 73. In the firft place, fince the fimple feries =, Ce, Dx*, Es, Fi sfc; is equal to the compound feries ~C, = Dx, Ex, 2" Fx, &c Ht 2m Qn 4n ; se Soot 49 tn x*,—— Ex, &c, and this equation is always true, of how fmall a magnitude foever we fuppofe « to be taken, it follows that it will alfo be true when «iso. But, when «is = o, all the terms in the equation that involve x will be equal to o likewife; that is, all the terms on the left-hand fide of the equation, except the firft term =, Z and all the terms on the right-hand fide of the equation, except the two firft . : 2 . terms of the two lines of terms, to wit, — C + 1, will be equaltoo. There- fore — will be = to — C + 1, orto, —C, or to 1 together with oe Cc, 1 either added to it or fubtra¢ted from it, as may be neceflary to produce fuch equality. Now, becaufe is fuppofed to be greater than m, = will be lefs than 1, and confequently “= C mutt be fubtracted from 1, in order to make it equal 2L2 to 4 260 A’ p T's clo'urrn £2. TOON ER FPN . * MM to =. Therefore the fign — mutt be prefixed to =~ C in the equation — = 2 —— C + 1, 0r a= tr =— ; and confequently the fign — mutt alfo be pre- V7: 7 fixed to the third term C «? of the feries 1 + — wx, Cw?, Dx3, Ext, Fx', &c PR Ret: 2 b) 4 > (which is equal to 1 + x\#), from which third term the faid quantity == C has been derived by the operations of multiplication and divifion in the courfe of the foregoing proceffes. Therefore the three firft terms of the feries 1 + m1 m1 . . te . 2 —«x,Cx’, De’, Ext, Fxs, &c, which is equalto1 + «\7, will be p + ad — Cy. Oi te BA And the magnitude of the co-efficient C may likewife be determined by ‘means of the equation = am it — Cosor — = Iie = C. For, by adding == C to both fides, we fhall have = + = = 1; and, by fubtraéting = from both fides, we fhall have = C = 1 — = = “—; and (by dividing both fides by =, or multiplying them into ~~) Sip os = x = = == = x —. There- ; ° m1 fore the three firft terms of the feries 1 + — «, Cx*, De, Batt Rymeeees mt ° ° et . m id n— mt which is = 1 + x|7, will be 1 a ieee ie tend 0°, OF TS 2H a— 2 a Bw. Qu) Baek 74. To determine the fign that is to be prefixed to the fourth term, Dx?, of . mM . . ; 14 the feries 1 + — &; Cwx?, Dx?, Ext, Fx5, &c, which is equal to 1 + X\—=» and to find the magnitude of its co-efficient D, we muft proceed as follows : Since the fign — 1s to be prefixed to the co-efficient C, it mult likewife be prefixed to all the terms which involve it in the grand, fundamental equation obtained in art. 72, to wit, the equation between the fimple feries —, Cady Dix E «3, F x4, &c, and the compound feries 2 n Cee oie Pie teed Bonds Ex, &c m m m m 2 uw - 1,—~ Cx, = De*, S Ex’, &c; which equation will therefore be as follows, to wit, the fimple feries Pi Cyn D xv?, Ex?, Fx*+, &c = the compound feries n n 2 LEC, Et ena, &c m7 r4 mn 24 pik de TE hear macley Op + 1 ae ee Dw, —- Ex, &e. Let Tiptg- BIN o:M Pr AeLs HH BO REM, 261 Let 2 Cx be added to the two fides of this equation; and we fhall then have the fimple feries = + Cx, Dx*, Ex}, Fx*t, &c = the compound feries : oe 2 Ct Be ye AS bee, SF 3; Bc WH 7 °// 4 Wn 2n 3” a 4h 4+-I,— — Cx, ae » —— Ex}, &c +2Cx, But it has been fhewn in the laft article, that - Ca ees ee = C. Therefore if we fubtra& — and 1 — —C from the oppofite fides of the iat equation, the remainders = be equal me each other; that is, the fimple feries C ey Dace E «3, Fx*, &c, will be. equal to the compound feries 2 ie Be, Hx 5 8c esis 7% e/4 nie 2" De’, Ex}, &c 2€ m3 and confequently (dividing all the terms by x), we fhall have the imple feries C, Dx, Ex?, F x3, &c = the compound feries 2“ D, 4" Ex, 22 F x, &c —-—C, 2 Dx, # Est, &e ete: _ And this equation will be true, how fmall foever we may fuppofe « to be: and therefore it will alfo be true when v is = 0. But, when xv is = 0, all the terms that involve w will be equal to o likewife, and confequently the equation willbe C = 23> D— C4 2C. Therefore C + +=C will be == D + 2C, and = C will be = 22D +, or C, 3" D; that is, = C will be equal to C together with 3a D either added to it or fubtra¢ted from it, as may be neceffary to produce the faid equality, But, becaufe is greater than m, aoa C will be greater than = C, or than 2 C, and, @ fortior?, greater than C. Therefore, in order to make C and 4% D be equal to - G3 sm D muft be ‘added to C, and confequently muft haves the fign + prefixed to a Therefore the fign + muft alfo be prefixed to the fourth term, D3, of the feries 1 + = x, Cwx?, Dwx3, Ex*, Fx*, &c (which is equal to 1 + 4|*), from which fourth term the quantity a D has been derived by the operations of multipli- cation and divifion in the courfe of the foregoing procefles. Therefore the four firft terms of the faid feries will be 1 + — x— Cx? + Dy, Kos AP And 262 AOD IS €C OURS E. COMCER NoIWie And the magnitude of the co-efficient D may likewife be determined by means of the equation = C=C + a D. For, by fubtraéting C from both ied we tall are T= #26 EOE HE == Ae 2n2—m (by dividing both fides by 3 2, or multiplying them into —) D meee x C, or (becaufeCis= = x“) D2 = x aoe = Qs, or Therefore the four firft terms of the feries 1 4 — #, ©x?, Dw, Bit, ee, mM te a al aret + =" —— x 2" — x? += x* — 2n a= 77 2% —- Mm 3 . iad SES) iene 2 3 x ~ x), O11 4+ — Ax ee Bx* + - A OSs Q, Hike 4s. To determine the fign that is to be prefixed to the fifth term, Ex‘, of 8 the feries 1 + = x, Cx?, Dx3, Ext, F «3, &c, which is equalto1 + a\2, and to find the magnitude of the co-efficient E, we muft proceed as follows : It has been fhewn in the laft article, that the ‘fimple feries C, Dix, Erwin &c is = the compound feries =D, Ex, 3” Fy, &c — =, 2 Ds, A” Ex’, &c + 2 iG or : (peahsine the fign + to the terms which in- volve the co-efficient D), that the fimple feries C + Dx, Ex*, Fx}, &c, is = the compound feries +d, 2 Ex, 4 Fe, &e ——c+2 Me Dex, = Ex’; &c + 2C. And it has been fhewn alfo in the fame article, that C is = + a D— a C-+2C, Therefore, if we fubtraét C from the left-hand fide of the laft equation, and + oS D —- = C + 2C from the right- hand fide of it, the remainders will be equal; that 1 4 the fimple feries 4+- Dx, Die ab it, ba &c, will be = the compound feries Ex, 2 Fx, &e + — Dx, = ~ Ex, &c; and sontea een (dividing all the terms by «) the fimple feries ay D, Ex, F x7, &c, will be = the coms pound feries 4" E, “- Fx, &c 22D, Ex, &e. And THE SBSINDOCMOI £2 ES ATE EDO BE M; 263 And this equation is always true, of however fmall a magnitude we fuppofe x to be taken: and therefore it will alfo be true when xis = o. But, when « is = o, all the terms in the equation,that involve » will be equal to o likewife, and confequently the equation will be + D= a E+ 3" D, or + D= + 3" p, 42 E; that is, D will be equal to a D together oy < E, either add- 1/4 72 ed to it or fubtraéted from it, as may be neceflary to produce the faid equality. But, becaufe w is greater than m, = D will be greater than D, and confequently _ E mutt be fubtracted from it, in order to make it equalto D. We mutt therefore prefix the fign — to the quantity i E inthe equation + D= =". E -. -- D; which equation will therefore be + D= — Ae E +- 2“D, or-+D —+ 2 D— <2 E. Therefore the ign — mutt alfo be prefixed to the fifth . m . ° term, E+, ofthe feries 1 + — a, Cx; Dwi; Ext, Fx’, &c, whichis equal _ ma | Na 7 + *\#; and confequently the five firft terms of the faid feries will be 1 + ie Cx aD x3 ie Bieta is ail tr He ie ides taecs recfureel the magnitude of the co- efficient » may ‘likewife be determined by means of the equation + D= — + E42 = D, seit a0 ay hatees E. For, by adding a E to both fides, we hall have D + + is — an D, and, by fubtra¢ting D from both fides, = | Diem Jn D—-D=+_, x D — <= & D, and confgieny B= xo" yx DaMEE De 4% 4n 3%—m 2n—m n—m a n— mM 22 —m 3n—m ———— gee = Dormer eS oe —_— tC—_— x \ 4n x 32 iaa x ” Soap opeytapiesty* Bn 4n pfctl Therefore the five firft terms of the feries t + =i, Cx?, Dx3, Ext, Fx’, m &c, which is equal to 1 + x|7, will ber 4+- —% re = x i e? ob “5 - a—m oni Oy 2) a—m 2n — m 32m ” 2n an #2 n a 2n x 3 4 e*, orl + a Ax 76. To determine the fign that is to be prefixed to Fw, the fixth term of | 1.) the feries 1 + = wy C v7. Dive? ats Eve, occ, which is equal toa + x, and to find the Paeaitade of the co-efficient F, we mutt proceed as flows: In the laft article it was fhewn that the fimple feries + D, Ex, Be Mes as é&c, was equal to the compound feries A 264. A'D 1S COfUoR BLE ECO WH RAR LN rENRG 6 a2 BE, 2= Fz, —Gw?, toe, Bec ™m //4 M1 m Qn An $2 sults ; eee eee i en Fes; — Gx, &c, or (prefixing the fign — to the terms that involve the co-efficient E) that the fimple feries + D — Ex, Fx?, Gx}, &c, was equal to the compound feries 6 7. 72 Ht ™m BOT) SAR ee PS BE Tr OES es “+b D — Ex, 2— Fx*, — Gwe, &e. m ¢ Add 2 E x to both fides of this equation. And we fhall then have the fimpte feries D + Ex, Fx?, Gx*, &c = the compound feries 6 Bran ie (era ave Ne So lye Mn Mm 72 6 + 2 D—* Ex, * Fx, G x3, 8c MH WW 3 +2En. But it has been fhewn in the laft article that Dis = — i E + as D, or — a D — += E. Therefore, if we fubtraét D from the left-hand fide of the laft equation, and = D— + E from its right-hand fide, the remainders will be equal; that is, the fimple feries + Ex, Fx, Gx', &c, will be = the com- pound feries SP ye os Gee a geese In oi m2 anes Hae 3” By, sa Gx3, &e’ m1 m ™m oy (7. + 2Ew; and confequently (dividing all the terms by. «) the fimple feries E, F x, G x7, &c will be = the compound feries 2* £2 Gx, Hex, &c mM m Mm be Bad: OMe On Get &e MW 7 we oe I Oe ; And this equation is always true, of however {mall a magnitude we fuppofe x to be taken; and therefore it will alfo be true when wis = 0. But, when w is = 0, all the terms in the equation that involve x will be equal to o likewife, and confequently the equation will be E = i F— x E+ 2£E. There- fore (adding = E to both fides) we fhall have E + *. bees u F + 2&E, and (fubtraéting E from both fides) we fhall have i” | iim “ F + E, or =” ‘Hoong UA aa F ; that is, = E will be equal to E together with — F either ad- ded to it or fubtraéted from it, as may be neceflary to produce the faid equality. But, becaufe w is gréater than m, = E will be much greater than E ; and there- fore THE “BION OMIALS PABORE M 265 fore - F muft be added to E in order to make it equal to 4° E. We mut - wn : therefore prefix the fign + to the quantity 2. F in the equation ~- E=_E; 7 _ F; and confequently the-faid equation will be a EE + °2°F. There- fore the fign + muft alfo be prefixed to the fixth term, F x', of the feries that m2 - [= : : 72 -is equal to 1 + x|*, to wit, the feries 1 + EOyhs, DS, Ett) bse oes t é 2 2 from which fixth term the quantity 2. F has been derived above by the opera- tions of multiplication and divifion in the courfe of the foregoing proceffes. v2 Therefore the firft fix terms of the faid feries will be 1 + — ~ — Cw? + Dw; a ee? Ex. Git. '1. And the magnitude of the co-efficient F may likewife be determined by means of the equation = E = E + F. For, by fubtracting E from both 4 ox, ) fides, we fhall have ** F = + E—E= i \x Eat x E;: and | oar See 4 — m _ 44—m _ 4"%—m 3n — m confequently F will be = Rear at x 5 = x E= x . 2n — 2— Mm mn in 2 — 1 un — Qn — ih 4% — mm ie str —- __ —, he ee es 37 ao? bw tide 32 ” 4n a 5a eee s mM Therefore the firft fix terms of the feries r + — x, Cw*, Dx3, Ext, Fx’, m * . aa In (e4 a — 2 i n2— Mm &c, which is equal to 1 + w|%, are I + ee ee 72 21 —m mM n— m2 2% —m 3n — m2 mn 2 — mM 2n — m a wast a x — xX = 1* + — X A X 3% * 2% 3% 4% 2 22 3% —m n— mM m nt — 72 22 — Mt — m8 ee x’, ort + —Aw — Tine cha siete” (ce 73 3 2% 32 77. Having thus gone through the inveftigations of the values of the four co- efficients C, D, E, F, at confiderable length, I fhall treat more concifely of the inveftigation of the following co-efficients G, H, I, K, L, M, N, O, P,Q, R, S, T, &c, and fhall only obferve concerning them, that the figns + and —, which are to be prefixed to them, may be determined, and their magnitudes dif- covered, by means of the following fhort and eafy fimple equations, which may be eafily derived from the grand, fundamental, equation obtained above in art. 72; to wit, the equations F=—G, “Ff, G “4, & Vor. Il. | 2M = 266 A DISCOURSE CONCERNING II S eh Saas & | = at S YN “ee |e BS to 7 a ee 8x = m K, m I, Kyi: Salas “Kk, ae M, Be o M = — N, —- M, Noe Biro. ein ae Pay jain ei Ones) 221 Sp. Sng ne OE 5 ’ Pp Q R S = T, a S, &c, or, (becaufe F has the fign + prefixed to it, and = F is greater than F, and confequently = G mutt be fubtracted from a F in order to make it equal to F), Fro— ee G + > F, and (fgr the like reafons), oe Abe = ee ze Sf ad+H=—-— 1+ 44H, -~-l=y+2#K—— 1, +K=—--L+ “Kk, ~-Li+~“M--“L, +M=—-—N+—M, —-N=4—“0-—N, +O=>—-—“~P+“0, Lio pale: -eoeesealgt) tae ph + Quine SS oF nil tres (att iS — es R, 181 17K &c¢ TCH EY “BS I'N OM 1 ACCS ow SO 2 TEM. 264 &c ; in all which equations the figns to be prefixed to the feveral co-efficients G, H, I, K, L, M, N, O, P, Q,R, S, T, &c, are alternately — and +. And the fame thing, it 1s evident, mutt take place i in all the following co-eflicients of the powers of « in the feries 1 + — eee. es “*, Fixs, &c, which is - equal to 1 + X\ # to whatever number of terms the faid feries fhall be conti- nued. We may therefore conclude that the faid feries will, when z is greater than m, be 1 +x — Cx? + Dx? — Ext + Fe? — Gre + He? — Ix? +.K xo — Lx? + Me™ — Na? + Ox? — Px™® + Qa — Rx? + S#? — Tx" + &c, in which all the terms after the two firft are marked with the figns — and + alternately, or are to be alternately fubtracted from, and added to, the {aid two firft terms. ve eae And, fecondly, the magnitudes of the co-efficients G, H, I, K, L, M, N, O, P,Q,R, S, T, &c, may be difcovered by refolving the foregoing (hort fimple equations, to wit, 6n eG Sp, —-G=+H-—G, +H= — = I + me H, SS ee eee +K=>-—L+ kK, —L=+—-"M-—-—L, +M=—-=~N+-=M, — Nom. +) 8 ON, +-Orz—-—“P+“0, . ~-P=i“Qq-—p, — +Q=—-—-R+ Q, HRs 128s ol R, Spee x Bee EB be Sp 8c For, fince + Fis = — “*G + R is = TF 5, 3 ERetwe fhall:have 22 § — 2° .R a , a v7 fA 74 6 6n — “4 R = ——1\x R = —— x R, and confequently S (= —- x 4S= f4 1/4 ht 17 2 L3 m 162 —m _. 164 —m 2 x >a R) —_— ea x R. Q. es And, fince + Sis= — 2" 17 + 12", we thall have S + S27 = 224 V2 ay and =? T= w*s—-S= ei} x 1S = 2=2 XS, and confe- arth 18% atic 172 — m __ 19n—m Re ra, ae a XS) = ee Q. E. I. mn ' Therefore the quantity 1 + x|*, or the mth power of the wth root of the bi- nomial quantity 1 +..«, will, when mis greater than m, be equal to the feries m pp pe 2s = 24. — m fe igt te : 4u— m Th 1+ - Ax = Bx ore Cx = Dx* + ee Ex as = Sx — ne Bam pet 4p BBG — Ot Het + =" xo — Pea Kye 4 102 — 71 11z—-m I2nz— mM I2n — mM I44 =— 72. ee Ma ee Nat Ory “here 11%” 12” ” 16% — m 16 1621 —m 17 17n—™m 78 r : : a5 —— R x"? — |——_ Sx ad infinitum Px 16% Q« zr 172 + 182 S a &ec if ide which all the terms, after the two firft, are marked with the figns — and + alternately, or are alternately‘to be fubtracted from, and added to, the two firlt terms, and the co-efficients of the third, fourth, fifth, fixth, and other following terms are generated from =, the co-efficient of the fecond term, and from each a—tm@ 2n—t 3n— mM un 3n ? 4n 3 , &c, in which every new frac- other, by the continual multiplication of the fractions es tion a ‘ 270 A.D. Woh GO,U SR SE CON CERNING tion is derived from that which immediately preceeds it, by adding # to both the numerator and its denominator. This feries is exactly the fame with that which is derived above in art. 3 from the binomial theorem, in the cafe of integral powers. +8. When-the denominator 2 of the fraction — (which is the index of the binomial quantity 1 + #), is an exact multiple of the numerator m, the forego- e . rss u—- mM 22 — mM se Ty 4p ; = ae B x? —— Cv — | — De* ing feries 1 + : ee a 7 x : + —— Ex’ — &c, will co-incide with, or may be metered to, the feries found in the former part of this difcourfe for the value of 1 + EL or /* 1 + x, or the ath root of the quantity 1 + x. For, if # is an exact multiple of m, let it contain m exactly p times, or be I 2 71 » equal to p times m, orpm. Then will 1 + x|2 be == 1 + e\pm =r! + ale ; and the feries 1 + < Ax — Fe Bw? + Sane Cx — on Dx* + oe x > Wi I vif ae Ax of x - ——— 5 tae x gag is rea Exs SC 1 w AK ~ Be Siete 7p — I or. Es — &c. Therefore t + x)/, or /? 1 + xy or the pth root of the Boe quantity 1 + #, will be equal to the feries 1 + < Ax — PBs + 4S" Cas — Et D»x ee Ex«s — &c; which is the fame feries that was found above, in art. 47, for the value of any root of 1 + «, excepting that the letter p is here ufed, inftead of the letter #, to denote the index of the root that is to be extracted. Qe Bil dhe 79. And, if we fuppofe # to be exaétly equal to m, and confequently = to be mn mn m ee ee . 5 equal to >, or 1, andi + «| to be equaltor + hi or 1 + x\", or1 +4, the foregoing ferles 1 ++ A eh te ee xk ee ee Oe fe D «+ 7 nea Mr Re: : —e 7 n> [ar F: 20 — mM —r,, + ee Ex &c will be = 1 + 52 =~ Ax ce — Bx + 7 Cx 3m — mm Det + SA Ew —& 1+ I x Aw — [St Bat $258 Ca — PEt Det 4 S24 Ext — &e =1+1x Ax —= Bx? ais Cx — 7 Det she Ext = be 11 x 1s — 0 x eee XP — x OK xt bE x OK KE oe BEET tHe 014.0 OED — &c = 1 + #3; as it ought to be. 80. We TeH*ED STI WO MM AGE “TURE JO° REE M, 27t 80. We come now to the third cafe of fractional indexes, in which the nume- . m . . . ° rator m of the index — of the power of the binomial quantity 1 + +, is greater than the denominator. mM Of the Binomial Theorem in the cafe of 1 + «|, or of the nth root of the mth power of the bi- nomial quantity 1 + x, or the mth power of its nth root, when m is any whole number whatfoever, and n any other whole number le/s than m. Me 81. The methods of inveftigating the feries that is equal to 1 + x)» when the numerator m is greater than the denominator z, are the fame with thofe which have been employed in the inveftigation of the feries that is equal to the fame 43 : quantity 1 -- «| when the numerator m is lefs than the denominator ». But mn the feries that will be obtained by thefe methods for the value of 1 + x)” will not be always exattly the fame, of whatever magnitude greater than » we fup- pofe v to be taken; as was the cafe with the feries that was found for the value mM ofr + | when m is lefs than #: but it will be different for every new multi- ple of that is contained in m. For, if # is greater than x, but lefs than 22, the third term of the feries will be marked with the fign +, or added to the firft term, and all the following terms will be marked with the figns — and +- alter- nately: and, if m is greater than 2”, but lefs than 3%, the third and fourth terms of the feries will be marked with the fign +, or added to the firft term, and all the following terms will be marked with the figns — and + alter- nately : and, if m is greater than 37, but lefs than 4”, the third and fourth and fifth terms of the feries will be marked with the fign +, or added to the firft term, and all the following terms will be marked with the figns — and + alter- nately: and, if m is greater than 4%, but lefs than 5x, the third, fourth, fifth and fixth terms of the feries will be marked with the fign +, or added to the firft. term, and all the following terms will be marked with the figns — and + alter- nately; and in general, if m is greater than p times #, or pz, but lefs than p + 1 times ”, or pw + x», the third, fourth, fifth, fixth, and other following terms of the feries, to the number of p terms, will be marked with the fign +, or added to the firft term, and all the following: terms will be marked with the figns — and + alternately; as was fet forth in the beginning of this difcourfe in art. 5 and 6. Now, if we were to apply the two foregoing inveftigations, contained in art. 54, 55, 56, 57, 58, &c — 65, and in art. 67, 68, 69, 70, 71, 472, &c — 77, to the difcovery of the feries which is equalto 1 + «|”, when 6 M13 272 A w D, TgS,.C.0,U.R SS, E C.0 NCE Ro N LNG m is greater than ”; and, in the tofsberuae of each of the fimple equations by which the co-efficients of the feveral powers of v would be to be determined, were to attend to all the different relative magnitudes of # and, and to fap. pofe m, firft, to be greater than 7, but lefs than 2n, and, fecondly, to be greater than 22, but lefs than 3uz, and, thirdly, to be greater than 37, but lefs than an, and, fourthly, to be greater than 42, but lefs than 52, and fo on with re- f pect to all the co- efficients we were to inveftigate 5; s we thoutd find the exami- nation of fuch a variety of cafes intolerably tedious and laborious. And there- fore, in applying the foregoing methods of inveftigation to the difcovery of the mn feries which is equal to 1 +- «|# in this fecond cafe, in which m is fappofed to be greater than z, I fhall make only the firft of thefe fuppofitions, to wit, that m, though it is greater than z, is lefs than 2%; which will reduce the inveltiga- tion of this feries to the fame degree of difficulty, and no more, as was found in 1 the inveftigation of the feries that is equal to 1 + a in the former cafe, in which m was fuppofed to be lefs z» Now upon this fuppofition, the feries that mM is equal to 1 + x)” may be inveftigated in the following manner. 82. The reafonings ufed above in art. §4, 55, 56, 57, in the three obferva- tions preparatory to the foregoing inveftigations, are evidently true when m, or Mm . wn ° . : — the numerator of the index — of the power of 1 + « in the quantity 1 + a|”, is of any magnitude greater than the denominator z, as well as when 2 is greater than m. And therefore it follows that in both cafes the quantity 1 + «” will : a 7 . 4 be equal to the feries 1 + — x; Cx, Dx3, Ext, Fx, &c, of which 1 is the firft term and ~ x is the fecond term, and is to be added to the firft terra, and confequently ida 28 with the fign +, and the following terms are Cv*, D x3, Eawt, Fas, &c, or the fecond and other soa powers of w in their natural oniers without 4 interruption, to wit, x*, #3, “+, “', &c, multiplied into certain fixed numeral co-efficients aenbted by the capital letters C, ‘DE, (Fy Oceana are to be connected with the two firft terms 1 ++ = « either by addition or fub- traction, and confequently are to have either the fign + or the fign — prefixed to them. It therefore only remains that we determine which of the faid en Cx, D3, Ext, Faws, &c, are to be added to the two firft terms 1 + — «x, 2 and confequently are to have the fign + prefixed to them, and which of them are to be fubtracted from the faid two firft terms, and confequently are to have the fign — prefixed to them, and what are the feveral values, or magnitudes, of the co-efficients C, D, E, F, &c, refpectively. . mm . 3. Now let y be put = the feries — «, Cv’, Dx3, Ext, Fxs, &c (as in . rLi4 . art. §8), or 1 + y be = the feries 1 + — Wy C x2, Dit E vt, Bivss Gocco mage 1s oe WS Vn" Ov Ms IS Pe eS. O RS Me 273 mt 2 is equal to1 + x\*. Then will 1 + y} be = the wth power of 1 + x)”, that is, tor + «”. And confequently the feries 1 + —J os - x “= y? + -- Z— tT Bim 2 a =o 4 = 2 Feces ie un u—t z— 2 Beye ya: dy ey lag x3 x 3 J tale heey: kes 3 * 4 oak a ee 3 x 4 x = xy! t+ &, will be = the feries x + e+ — x ——— 7 m m—t1 Mea a 4 a mm — 1 nl — 2 mS gue 4 mn x” = mis,» ; ll a ; 3 ar aie a nt : — x icc x i x a «> + &c, and confequently (fubtracting 1 from both fides of the equation), the feries = x + — Sah Rai a es — x =i m— 2 bs (Mage i | Mm-— 2 vi — TI i_— “—" x x3 + — x “x x Sat pox Ay TE ge I 2 3 4 I 2 3 x am x =F + &c, will be = the feries — y at = x —— y? + = a il 4—2 3 Hs a2—T a— 2 CS ae | 4 we nam I en Dee Pi x SG TS att he XK a —42 Hh —) A : bs ne 72 wt — 1 x sae x wae + &c, or (if, for the fake of brevity, we make — Xi sao m1 m— I Ls fet ee Mi el Wm — T Ww— 2 Di Si Gp 2. iid =Q,7*x : Somes = Rar | Soe res tem = S, and — m— 1 m— 2 m— 3 Mt— 4 a % 1 2. a a—t ae Si pee eg eg eds Cg es a Se... n 5 poe tn Miri bat nasa § antl ei dP pi allan ce ale. 30% 8 2 3 tits i 2 3 4 av’ ~ 254 = t), the feriesmx + Qu* + Rw? 4+ Sxt+ + Tx? + &c, will be = the feries wy + gy* + ry? + 5y* + ty’ + &c. 84. Now let the feveral powers of the feries ~ mp Gx LD x3, 18 ee Binds &c, or (putting B = =) of the feries Bx, Cx?, Dw, Ext, Fx, 8c (which is equal toy), be raifed by multiplication, to the end that we may have the va- lues of y*, v3, y*, 9°, &c, exprefled in powers of x; and let the faid values of 975 9359*, 9%, Sc, obtained by the faid multiplications, be fubftituted inftead of y*, y3, y*, 7°, &c, refpectively in the laft equation. And we fhall then have the fimple feries mv + Qu? + Ra? 4+ Sx* + Tx? + &c = the com- pound feries eens tC? « nD x3, nkix+, nE xs, &c gir, mtr geeks .2.9 DL 8) 2g bik; ae GA Re 27°C) x6, eee Pee, $1 Ce*, 3 Bi Dixie iic 3°97 BO" 44;—<6c + sBtxt, 45B3:Cw#5, &c + ¢Bxs; Vout. Il. 2N and 274 AxwiD PS Gor’. R Sie. pQyOsNyciye RD Pea le and confequently (dividing all the terms by w) the fimple feriesm + Quy + Rw? + Sx? + Txt + &c = the compound feries “B, ex. REX nEx 3, aE x+, occ +7 Bx, 129 BC x, 29 BDxes 29 BE, Goeem Os 54 2¢CD #4; *&c + 7 Bie, 8 a7 Bie xt, 37D Dae . a'r BCtKA, Mace + 5s Bx, 45 BC cS cee +t Bi xt; &e; as is fhewn more at large above, in art. 58, 59, 60, and 61. 85. From this general and fundamental equation we may, by repeating the reafonings ufed above in art. 61, 62, 63, 64, derive the following particular equations for the determination of the magnitudes of the feveral co-efficients B, C,D, E, F, &c, and of the figns 4+ and —, that are to be prefixed to them ; to wit, Ti.) ot -—— 2s adiy, = Q —*4.C-+ 9B*s aaly, R= #'D,')2 7 BC. Bs): Athy, S°¢= # E927 8D)" 7O4) 3 ri O + wae and sthly, T = #F, \2¢:BE, 29 CD,, 37 BaD, erBes 45B:C+7B5. 86. From the firft of thefe equations, to wit, m = ”B, it follows that the co-efficient B is = =, as it has already been fhewn to be in art. 56 and 57. 87. From the fecond of thefe equations, to wit, Q= ”C + ¢B’*, oQ= q B*, aC, we may deduce both the fign that is to be prefixed to #C and the magnitude of C, by proceeding in the manner following : I Q : 1 —~ 1 : 1— mm ; Since,O isi mx , and gis # x , and B is = —, we fhall have 2 3 Mim I a— I ea r MH4 — MH mn — Mm W178 —— 2 mx — = 2 x —— x —B, aC, or — B, #C,; ‘or —— B nin — Mt mit — Mt ” mn — nt m*n — mn x— = B, #2 C,-or x — x B = —— B, #C, or ——— B B 2 2 m 2 2m 73°F 8 May ; 2 2 cco — 2A" % B, aC, and (adding ~* B to both fides) 77 B = 7b" 2m 2% 2m 2 B, # C, and (adding ”" B to both fides) sl iy il ales B, “C, and 2m 2 2m (fubtra@ting “* B from both fides) — B = “B,C, or — B= — B, nC; 2m 2m 2m 2 2 and therefore (becaufe m is now fuppofed to be greater than #, and confequently ~ B is greater than = B) = B will be = = B + uC; and confequently 2C 2 M— 2 . Mm — nt Wi — 2 mm B — —B = — B, and C will be = —— B = —— x —-, g 2 2 22 £4 will be = 2 wi—n aroma ie Therefore C ~*, or the third term of the feries 1 + Bx, Cx’, mM Dw?, Ewv*, F x5, &c, which is equal to1 + xn , muft be added to the two firft mM OF 5% 2 THE’ BEFNOMPA‘L? PH EO R'E M 275 firt terms 1 + Bx, ori + = x, and the three firft terms of the faid feries will Mm — nn mH" Ww—n x7 Or T eee BY? lee OPES ber +—x + — x 88. From the third of the eee a equations, to wit, R= 2D,2¢qBC + r B}, or (becaufe we now know that the fign + is to be prefixed to the term 29gBC),R =2D + 2q BC +7 B3, we may deduce both the fign that is to be prefixed to 2D, and the magnitude of D, by proceeding in the manner fol- lowing : By fubRituting m x os + x “== in this equation inftead of R, and x ——— inftead of g, and 2 x 2a" x 222 inftead of r, and = inftead of B, and a — inftead of C, it will become as follows, to wit, m x 2——- x 27? ets x pee Se x of x >= Ey ant a n 22 2 ns ee =—=2D+mxn—1 x = x ea + pee: x =, or nae aaa 2m DL ae x (ete mth G + 3m x — ae Rasa A6 yao pecensee tl hal a aaa = 2D — ees and confequently (adding = ll to both fides) we fhall have in = Date a ae ———, and (adding ae to both fides) a ike kes ea tia Bp and (fubtracting g from both fides) te =nD+ ae “i oem ae nD; that is, a will be equal to _ together with D either added to it, or fubtracted from it, as may be neceffary to produce fuch equality. 89. Now, becaufe m (though greater than #) is fuppofed to be lefs than 2, it follows that m — 2 will be ‘lefs than 22 — u, or than ”; and confequently that mx m—n X m—n will be lefs than m xX m—n Xn, or thatm x m*—2mn+n* will be lefs than mu x m—x, or that m3—2m*n+mn* will be lefs than m*n—mn*. Therefore (adding mn? to both fides) m3 —2i0°n + 2mn* will be lefs than m*n, and (adding 2m to both fides). m*-+-2ma* will be lefs 2mn? +-m3 than 3m*n, and cconfequently Sa Will be lefs than oe - Therefore, in or- der to make 2 —_, 2 D be equal to sted aaa , we mutt fubtract # D from — a and confequently muft prefix the fign — to nD; fo that the equation aa aD i he fo 3 1). Therefore the fign — mutt alfo be 622 6 2% Ona ZN 2 prefixed - _2a4—m 276 A DISCOURSE CONCERNING prefixed to D3, or.the fourth term of the feries 1 + Bw, Cx*, Dx?, Ext, Mm Fx, &c (which is equal toy + «\2); from which fourth term the quantity n D was derived, by means of the operations of multiplication and divifion, in the courfe of the foregoing inveftigation. Therefore the four firft terms of mt the faid feries. 1 + Bx, Cx?, Dx}, Ex*, F x5, &c, which is equal to 1 + x) ‘ will be 1+ Bx + Cx? — Dxw3},or1 + ox 4 2 x —— x? — Dx, or 2 2 ; 1 Wit —- 2 — at —— ra B % Saag a5 e 1+ Att = Be Ds Q@ EI. go. And to determine the magnitude of D, we fhall have the aforefaid equa- 2mn*+-m3 _ 3m7n Sa = Ga 72 Ds whence (by adding »D to both fides) we thall 2mn? +-m3 47 py tion am*n bun 2mn? + m3 have , and (by fubtracting from both fides) 37n—20? —m> mx m—2 X 2n— Mm nl — 2 6 un 62 3m n— 2mn* — m3 21) = 24 — mM , and confequently (by dividing both fides by x) D = — x a Oe 7_—_—_-_— x On 2n wai~—n x . Therefore the four firft terms of the feries 1 + Bx, Cx?, Dx3, Ext, Onn 6 un 2n 32 7 F w>, &c (which is equal to 1 + x|7), will be 1 4 = aq 4 = % a on 2 2 mm — 2 Sih ssa Ww—n 22.7 B x? ~~ Cr, 2n 32 Qf Ew) To m2 ee OFT + Kage Ax + 3 2h 3% Br g1. Thefe four terms of the feries which is equal to 1 + w|” in this cafe of the relative magnitudes of m and, in which m is greater than #, but lefs than 22, are the fame with the four firft terms of the feries given above in the be- c mM ginning of this difcourfe, in art. 6, for the value of 1 + «|# in-the fame cafe. Therefore the faid feries, in art. 6 is true, at leaft in its four firft terms. 92. I fhall not attempt to find the values of the co-efficients E and F by re- folving the fourth and fifth equations fet down in art. 85, to wit, the equation Smn2E, 2¢BD, ¢C’*, 37 B*C + 5 B*, or (becaufe it has been fhewn that the co-efficient C is to have the fign + prefixed to it, and the co-efficient D is to have the fign — prefixed to it) S = #E — 2¢gBD + gC* + 37 B*C + sB*, and the equation T = “F, 2q¢ BE, 2¢CD, 37rB*D, 37 BC*, 45B:C +72Bs,orT oak, 2q¢BE — 2gCD— 3rB°D-+ 3rBC* + 45BsC + #B*: J fay, I fhall not attempt to find the values of E and F by refolving thefe two equations, for the reafons given above in art. 65 and 66; but fhalk proceed to apply the other method of invefligating the values of the third and . tiie ™m fourth, and other following terms of the feries 1 + —«, Cx*, Dx, Ext, 6 i 7m FREE cRUNOMI AL BA 20 Be My 277 Mt Fx’, &c, which is equal to 1 + «|*, to the prefent cafe, in which m is fup- pofed to be greater than #, but lefs than 2%. This may be done in the man- ner following : 93+ It has been fhewn in art. 82 that the two firft terms of the feries that is W2 _— . 72 ° 5 aw —* V 7 e j equalto 1 + «| ill be rt and c when m 1s greater than 2, as well as when . iA . mis greater than m; and that the fecond term — x Is to be added to the: firft term 1, and confequently to have the fign ++ prefixed to it, in this cafe as well as in the former; and likewife that the following terms of the faid feries will be Cx?, Dx?, Ext, Px>, &c (or the {quare and cube and other following powers of x in their hattieal order, without interruption, multiplied into certain fixt numeral co-eflicients, which may be denoted by the capital Testers CT): E, F, &c), and connected with the two firft terms 1 + — w by either addition 4 m or fubtraétion : fo that the feries that is equal to 1 + «|# is inthis cafe} as well . Mt \ . . as in the former, 1 ++ — He, Cx?,:Dxe3, Ext, Fx', &c3 in which the feveral e terms C x*, Dx*, Eu*+, Fx, &c, have a comma prefixed to them, inftead of either of the figns os and — , becaufe we do not as yet know to which of them we are to prefix the fign +, and to which we are to prefix the fign —. m 94. er fince i + *\# is in this cafe, as well as in the former, = the fe- ries I + ~ a Gx, _ #3, Ext, Fx>, &c, it follows from art. 67, 68, 69, 70, 71, 72, that the fimple feries =, Cx, Dx*, Ex3, Fx, &c, will be equal to the compound feries *C, 22 De, iar Fx3, &e Bie uhh Sate 4" Ex, &c3 by the help of which equation we may di huver: both “¢ which of the figns + and — are to be prefixed to the feveral co-efficients C, D, E, F, &c (and confe- quently to the correfponding terms C x7, Dw’, Ex, Fx, &c, 1n the feries { 1+ = #, Cw?, Dx?, Ext, Fs, &c, whichis equal to 1 + x}7),” and alfo «¢ what are the magnitudes of the faid co-efficients C, D, E, F, &c, refpect- ively.” ‘This may be done by proceeding in the manner following : 95., In the firft place, fince the fimple feries = ev) ns peediidy Bytes s RECs IS equal to the compound feries En Sev ee Ex?,22F x3, &c m2 mt Fatt dol, 2 i Bie 3 ° ta3 Cx . Px, re E x3, &c; and this 278 AG ip Visi chotui Kn see. Echo oN Ge Sele Gy Bon Gc this equation is always true, of how {mall a magnitude foever we fuppofe x to be taken, it follows that it will alfo be true when wis = o. But, when xis = 0, all the terms in the equation that involve x will be equal too fikenafes ; that is, all the terms on the left-hand fide of the equation, except the firft term =, and all the terms on the right-hand es a the equation, except the two firtt terms of the two lines of terms, to wit, oa C + 1, will be = o, Therefore = % e 2 . 2 e will be = to = C + 1,o0rtoT, — C, orto 1, together with — C, either added to it, or F faberaewea from it, as may be neceflary to produce fuch equa- snk Now, becaufe m is fuppofed in the eel cafe to be greater than z, — will be greater than 1, and confequently = C muft be sadee to 1, in order to make it equal to =. Therefore the fign + mult be prefixed to -—- C in . bed 22 7 news — — — —_- = Cc the equation — pS BE ENE Sor 1, —- C; and confequently the fign +- mutt alfo be prefixed to the third term C x? of the feries 1 + ~ x, Cx?, Dx, m2 Ext, F a &c (which is equal to 1 + x\7); from which third term the faid quantity — —" C has been derived, by the operations of multiplication and divifion, in the Hot of the inveftigation contained in art. 67, 68, 72. There- fore the three firft terms of the faid feries 1 + = x, Cx?, Dx3, Ex+, Fxi, &c, Mm which is equal to r + *)#, will be 1 + = “+ Cx’. Q, E. J And the Husa, os the eure es C may likewife be determined by means of the equation — -- = 1,— -C, or— = 1 + = C. For (by fubtraé- ing 1 from both aed we thall hae — =C ah ea 5 “—; and (by di- viding both fides by — — vor Win rie ie into —~) we fhall have C = G x Mm—n __ DEG “ Therefore the three firft terms of the feries 1 + — x, 2% mn Cx?, Dei, Ext, Fxs, &c, which is ae to 1 +- al”, will be 1 + ua yt m i = tt —e?,ort + — Aw Os aaie 96. To determine the fign that is to be prefixed to the fourth term, Dx?, of 7/4 the feries 1 + — — i, Cx’, Dx, Ext, Fx', &c, which is equal tor 4 x\z, and to find the magnitude of its co-efficient D, we muft proceed as follows : . Since tes. 8 BINOMIAL t. HakO Rae M,. 279 Since the fign + is to be prefixed to the co-efficient C, it muft likewife be ~ prefixed to all the terms which involve it in the grand fundamental. equation fet down in art. 94, to wit, the equation between the fimple feries —, Cx, Dx?, £7) E «3, F x*, &c, and the compound feries 2h AN nM + —C, = he Dx, i bp ae “ ev, occ 2 7. un 4 4 ok. —— Cx, = Dr, * Ex«?, &c3 which yi equation will therefore be as follows, to wit, the fimple feries = + Cx, Dx’, Ex, Fx+, &c = the compound feries 2% 0 38H, 4% BD yr St Ey +¢, Dx,“ Ext, Fx, &c 4 2H +r1+—Cx, LE De, Exes, &e. mm 23 7% - But it has been fhewn in the laft article that ~ is — 1 + = C. Therefore, 2 “if we fubtrac ~. and 1 -E —< C from the oppofite fides of the laft equation, the remainders will be equal to each other ; that is, the fimple feries Cx, Dx’*, Ew, F «+, &c, will be equal to the compound feries 3° Dx, VM Ex, 2 Fx, &c WH ™ Mm +=Cr, 2 Dw, Ew, &c: Mm Mm mM , and confequently (dividing all the terms by v) we fhall have the fimple feries Cee x*, I x3?, &c = the compound {eries peri yie Heat 3” EF x, &c 1 r//3 m + =C, 2" Dx, S Ex’, 8c. And this equation will be true, of how {mall a magnitude foever we fuppofe « to be taken: and therefore it will alfo be true when x is==o. But, when » is equal to o, all the terms that involve x will be equal to o likewife, and confe- quently the equation will be C = ~ D+ — C,orC = + <= C, a2 D; that is, C will be equal to — C, together with D either added to it or fubtracted from it, as may be neceffary to produce the faid equality. But, becaufe m is fuppofed to be lefs than 2”, C will be lefs than = C; and confequently -- D muft be fubtracted from — C. in order to make it equaltoC. Therefore the fign — muft be prefixed to the quantity s- D; and confequently the fame fign mutt alfo be prefixed to the fourth term Dw?, of the feries 1 + =, Cx, Dx}, 280 A UVpirScc DOW BR sige OD HE ER NING m Dwi, Ext, Fxs, &c (which is equal to 1 + n\”)5 from which fourth term the faid quantity - D has been derived, by the operations of multiplication and divifion, in the courfe of the inveftigation by which the fundamental equation, fet down in art. 94, was obtained. ‘Therefore the four firft terms of the feries mM r+ ~ x, Cx, Dx3, Ext, Fx5, &c, which is equal to 1 + x\*, will be t—n m . mn 7 Z/ Tc rm he Cpt = 2, or 1 oe eel 4 x? — Dw, ort hie ve 2 ba! D Me ‘ Q; B. 2 Me And, to determine the magnitude of D, we fhall have the equation c= = — = D; whence (adding = “ D to both fides) we fhall have C + > = -“D= = C, and (fubtracting C from both fides) 2" ee ~~ C—~CH= -- —11xC an Pre is GE Ya 1 (multiplying both fides by > Do eG eee x m 9 BA oredeo ek ode ee the four firft terms of the a 2a. 2 2% 3n : ” feries 1 + = x, Cx, Dx?, Ext, Fx, &c,) which is equal tor + x) , will mM m—-n . 1/4 Ww— Kn 21 —mM mm ber +—* + — x —*xv? = — w3,orr + —Aw + 2 2% r/4 Wm 2 — m —— Bw? — = Cw, Q: Bef PAE 3% 97- To determine the fign that is to be prefixed to the fifth term, Ew*, of rif ° Mm s . ° — the feries 1 + — wt, Cx", Dx, Ext, Fees ec, which 15 cqualyw see x) 2, and to find the magnitude of the co-efficient E, we mutft proceed as follows : It has been fhewn in the laft article that the fimple feries C, Dx, Ex?, Fwx3, &c, is equal to the compound feries 2"'D, are, 5 By?) &c mW mW +—C, Dx, a Ex’, &c; and that the fign — is to be ole fed to the terms that involve the co-efficient D. It therefore follows that the fimple feries C ~ Dx, Ex*, Fx, &c, will be equal to the compound feries 7 — =D, bine @ &c + —C— —- 1" Ex’, &c. Therefore, if we add 2 Dx to both pres we “thal have the fimple feries C + Dx, | Ew, Fx?, &c = the compound feries “THE BINOMIAL THEORE M, —~2D, 2 Ex, Fa, &e ASA Pit, hes al ee +—C — Dx, — Ex*, &c +2Dx. Further, it has alfo been fhewn, in the laft article, that C is = ~ erent). m2 Therefore, if we fubtract C and = Cc — an D from the oppofite fides of the laft equation, the remainders will be equal; that is, the fimple feries Dv, Ex’, F x3, &c, will be equal to the compound feries | 7 Ex, 2 Fx, &c r//4 Mm — 22 Dx, @ Ext, &c m mn . + 2Dx. Therefore (dividing all the terms by «) we fhall have the fimple feries + D, Ew, F x’, &c = the compound feries 47 E, Fx, &e 4 i? —~p, Ex, &c mM" i +: 2D. And this equation will be true, of how {mall a magnitude foever we fuppofe x to be.taken: and therefore it will alfo be true whenxis == 0, But when x is = 0, all the terms that involve x will. be equal to o likewife, and confequently the equation will then be + D= #E— 24D4 2D. Therefore (adding 2° D ‘i i//4 Mm . m to both fides) we fhall have D + a De “2 E + 2D, and (fubtracting D from both fides) “ Dis a E:+°D,- or. 2° D ~ D, = E, or 2" D = D, together with x E, either added-to it, or fubtracted from it, as may be neceflary to pro- duce fuch equality. But, becaufe m is fuppofed to be lefs than 27, 2 will be greater than 1, and 2 D will be greater than D; and confequently +2 E mutt be added to D, in order to make it equal to = D.. Therefore the fign -- muft be prefixed to the quantity a E in the equation 2 pr Dy tL E, and therefore it mutt alfo be prefixed to the fifth term, Ex*, of the feries 1 + = x, Cx, Dx}, 1 Ext, Fx, &c (which is equal to 1 + x\” ), from. which fifth term the faid — quantity = E has been derived above in art. 67, 68, ..... 72, by the opera- tions of multiplication and divifion. Therefore the five firft terms of the feries r + m x, Cx*, Dx?, Ex*, Fas, &c, which is equal to 1 + x\*, willbe + — x id n Vout. Il. 20 47 282 A BESCOUVUR SB “GHONVC BER BW RNG ti— m mm n 72 h— 28 2i =m m i am 8 x oer Beord + 2 n a n 2 Bx A= Cat + Ew. OQ: share And to determine the magnitude of the co-efficient E, we fhall have the equa- tion = D=D+ *“E: whence (fubtracting D from both fides) — E will m be = 22 D—D= 34 —1|x D= *— x D, and confequently (multi- 7 mt m7 ? plying both fides into re E will be = a xD=*— x - x ~—— x 4TT or ty ty, x FS. Therefore the five firft terms of nm 2n 3n 4n mt . 1 e . ‘ee the feries b ++ — x, Cw, Dwi, Ext, Fx', &c, which is equal to1 + x], : ” m m—n m m—n 22 —m m n— hn r San ae ge A ik fa ete Bets See, Pee tess 3 a will be 1 tr n tb n x 2% e n x 2n x 32 Ait 2 x 22 2u—m tee salen Mm m—n _ (24—-# ; 3n—m Say a “Wien 5 Ot T Tb eee tt ah Bx i aw Bae er Dx. Q. Beaks 98. To determine the fign that is to be prefixed to the fixth term, Fx’, of 3 the feries r + — x, Cx, Dxv3?, Ex*, Fx>, &c, which is equal tot + x|\*, and: to find the magnitude of the co-efficient F, we muft proceed as follows: It has been fhewn, in the laft article, that the fimple feries D, Ex, Fx?, &c, is equal to the compound feries 4* FE, 22 Fx, &e 7/4 mM i a Le ae Ex, + 2D, &c, and alfo that the fign + is to be prefixed to the terms = E, and a Ex,,and Ex, Therefore the faid equa- tion will, when thefe three terms have their proper figns prefixed to them, be as follows; to wit, the fimple feries D + Ex, F x*, &c = the compound ‘feries 4 cate : 3" Fx, St Gx, &c > =~ D + Ex, 22 FB x’, &e +2D. | Further, it has been fhewn, in the laft article, that D is equal to + — E— a2 D+.2D. » Therefore, if we fubtract D and + aS E— a D + 2D from the oppofite fides of the foregoing equation, the remainders will be equal, that is, the fimple feries + Ex, Fx’, &c, will be equal to the compound feries a 57 Fy ws ? THE BINOMIAL THEOREM. 283 rr 6” ‘ — Fw, — Gu, &c hm Mm + = Ex, a Fix*, &c; and confequently (dividing all the terms on both fides by x) the fimple feries + E, Fx, &c, will be equal to the compound feries ‘ F, = Gwe, &c 4n 5” + a FE, oe 1x; Orc. And this equation will be true, of how {mall a magnitude foever we may fup- pofe « to be taken; and therefore it will alfo be true when wis = o. Bur, when w is = 0, all the terms that involve x will be equal to o likewife, and confequently the equation will then be + E = ~ ines in By, cor 4. Bia be 47 £, =F; that is, E will be equal to a E, together with 42 F, either added ™m m m to it or fubtracted from it, as may be neceffary to produce fuch equality. But, becaufe 2 # is fuppofed to be greater than m, 4 will, @ fortiori, be greater than m, and confequently a E will be greaterthan E. Therefore, in order to make 7s ON i F be equal to E, we mutt fubtract - F from ** E, or prefix the on 3 fign — to the term a. F. Therefore the fame fign — mutt alfo be prefixed to the fixth term, F w°, of the feries 1 + = x, Cx*, De3, Ext, Fx5, &c (which ™ . is equal to 1 + x\* ), from which fixth term the faid quantity F has been de- rived by the operations of multiplication and divifion in the courfe of the fore- going inveftigation. Therefore the firft fix terms of the feries 1 + = eee, Dex?, Ext, Fx, &c (which is equal to 1 + «\*), will inthis cafe be 1 " = x“ A SRE i lca Digs OO RRR, ilar Vlad 2n 3% % 2n 3” 4” td mwn—n +—x—- % 2H be wt — Fy5, or 1 + ~ Ax + =— Be? _ ae. Oe bets And to determine the magnitude of the co-efficient F, we fhall have the equa- tion E = 42 EB — 2 F; whence (adding 5" F to both fides) we fhall have E m1 mM ™ + St = 4” E, and (fubtra¢ting E from both fides) = ig, (ae — EWE = ™m™ m2 pias: |x E) = &—" x E, and (multiplying both fides into Sek Eng, 2 m m gn ga 4n—m 32 — mM 24 =; w— nt lsd ye os al x Sor x aaa he peat dart Therefore the firft fix Loam @ lar) terms 284 at’ Tis’ ChORUR: SSE CVO NS IR NOP . m . 2 terms of the feries 1 + — ty Cx’, Dw3, Ext, Fx, &c (which is equal to ™m T + x\*), will (upon the fuppofition here made, that m is greater than 2, but lefs than 2”) ber + =x + = x Sat? x AU" x Gree es m—n 24 —— ™ 3u4— im face We 22 — mM 3n —m 42 m Agee Gh ony mai. mae ie te ae Pee Ae or 1 + Ax + = Bx Ee 43 ek S™ DT xs oo Ew. Q. E. I. 99. In like manner we may derive from the general equation in art. 94, to wit, the equation between the fimple feries = —, Cx, Dx*, Ex?, Fat, Gus, 8c, and the compound feries 2n ~ Bn _ An 7 _ 3 7 C2 Dx, Ex, 2° Fw, "Gat, Hw, &e Pert “Cx, a Dy, Ee, EE oe, 2 Gx, &c, 62 the following particular equations for determining both the figns that are to be prefixed to the following terms, G «°, Hw7, Ix*, Kw9, Lx, Mx™, &c, of the feries 1 + =H, Cx, Dx, Ext, Fat, Geo, He7, Le*, K x2, Dix) Meats ™m &c (which is equal tor + x|n ), and the values of the co-efficients G, H, I, K L, M, &c; to wit, the equations Pa.? Bitlet Gina G >) ie ote G, HS re Tie I = 26-K; S274, Rees — oF a K, pe = — M, = L, Cy or == ~ F, ee G, THE BINOMIAL THEOR EMe 285 ales Qu 10” Ke Kgs = 10” 117 ee Teme &c, as far as we pleafe to continue them. 100. And, in all thefe equations, the fingle term which forms the left-hand fide of the equation, will always have the fame fign + or — prefixed to it as is pre- fixed to the term on the right-hand fide of the equation that involves in it the fame letter. Thus, the two terms F and “= F, in the firft equation, will have the fame fign prefixed to them; and the two terms G and “2 G, in the fecond equation, will have the fame fign prefixed to them; and the two terms H and i H, in the third equation, will have the fame fign prefixed to them ; and the following letters, I, K, and L, in the fourth, fifth, and fixth equations, will, in like manner, have the fame figns prefixed to them as are prefixed to the terms L iF = K, — L, refpectively. This follows evidently from the obfervation that has been often repeated in the foregoing articles, to wit, that the terms that involve the fame letters in thefe latter, or particular equations, are all.derived from the ots . . . m fame terms of the fame original feries, to wit, the feries 1 + — x; Cix7,- Dix? , eee, tue, rx’, Fiat le, Kao, Lx, Mx", &c (which, 1s*eqnaliito m 1+ x\|”), by the operations of multiplication and divifion in the courfe of. the foregoing inveftigation ; by which operations the figns + and —, that are to be prefixed to them, cannot be affected. toi. Further, the terms denoted by the fingle letters F, G, H, 1, K, L, &c; which form the ‘left-hand fides of thefe equations, are uniformly lefs than the terms a F, = G, a H,. ae les 2" K, oat L, &c, on the right-hand fides of the fame ™m™ equations which involve the fame letters refpectively ; becaufe m is fuppofed to be lefs than 2”, and, 2 fortiori, lefs than 57, 6”, 7n, 8u, gn, 10n, &c. It follows, therefore, that, in order to make the right-hand fides of thefe feveral equations be equal to the left-hand fides of them refpectively, it is neceflary that the fign + or —, that is to be prefixed to the fecond term on the right-hand fide of each of thefe equations, fhould be contrary to the fign which is prefixed to the firft term. And hence we may determine the feveral figns that are to be prefixed to the feveral terms of the faid particular equations, in the manner following. 102. It having been fhewn, in art. 98, that the fign — is to be prefixed to the * 3 . . . ° co-efficient F, and te the term 42 F, which involves it, it follows, from the laft iE: . as 62 : : article, that the fign + mutt be prefixed to the remaining term — G, in the firft Z; en QO 286 Ae’ p Ps%clotutr sip tc ROVER WN PNG of the foregoing particular equations, to wit, the equation F = ‘- F, a fem and confequently the faid equation, when its terms have their proper figns pre- ‘ 6 fixed to them, willbe — F = — a B+ — G. And, fecondly, fince Ee G has the fign + prefixed to it, it follows, from art. 100, that the term G in the fecond equation, G = nd G, a H, mutt alfo have the fign + prefixed to it; and confequently, by art. 101, that the remaining term, i H, muft have the fign — prefixed to it; and confequently the faid fecond equation, when its terms have their'proper figns prefixed to them, will b+G=+“G—-2H. And, thirdly, fince a H has the fign — prefixed to it, it follows, from art. 100, that the term H in the third equation, H = <= H, 3 I, muft alfo have the fign — prefixed to it, and confequently (by art. 101) that the remaining term, 2” I, muft have the fign + prefixed to it; and confequently the faid third equation, when its terms have their proper figns prefixed to them, will be er eNO IS Ey doe Soh ™m 2 And in like manner it may be fhewn that the fourth equation, I = =" I; 2” K, when the proper figns are prefixed to its terms, will be + I= + BA snd m a <. K; and that the fifth equation, K = ea K, ae L, when the proper figns are prefixed to its terms, will be —K = — 2" kK + -27 L; and that mm 1/4 the fixth equation, L = a L, — M, when the proper figns are prefixed to mm its terms, will be + L=+—"L— <* M. Therefore the fix foregoing equations, when their proper figns + and — are prefixed to their terms, will be as follows; to wit, —-Fro—-“F+ y G, +Ga + ns G — en H, ~-H=-“~H+—1], 82 n + I my eptilbe GoeRe t— 2 K, oh ae Ae 10” 11” f= AL arco ia ee anv Ty In > TH B BION OcM I & Bo Ti MEO RE M; 287 In thefe equations the feveral co-efficients F, G, H, I, K, and L, have the figns — and + prefixed to them alternately ; and it is eafy to fee that the fame thing muft take place in all the following co-efficients, M, N, O, P,Q, R, S, T, &c, to whatever number of terms the feries be continued. We may there- fore conclude that in the feries 1 + = x, Cx? Dina? Wee x85 Gx *, Pry, Seem, ie, Mx Nx, Ox'3, Pe, Ox, Rx®, Si, Tx &c m (which is equal to r + «|*), when m is greater than z, but lefs than 22, the third term, Cw, is to be added to the two firft 1 + = «, and confequently marked with the fign +-, and all the following terms Dx3, Ewv*+, Fx5, Gx, eee aw, NOT Ne Ont, Pat Qty Ro Se) Tae &c, to whatever number of terms the feries be continued, are to be marked with the figns — and + alternately; and confequently that, with refpect to the figns of its. terms, the faid ferres will be as follows, to wit, 1 + = e+ Cx? — Dx + Ext — Fwi + Gre — He? Iv? — Keo + Lv? — Mat® + No? — Ox + Px — Qu + Rx? — Sx? + Ta — &c. Q, BvLT. 103. It remains that we determine the magnitudes of the feveral co-efficients F,G, H, 1, K,L, &c. Now thefe may be eafily found by refolving the fhort fimple equations obtained in the foregoing article ; to wit, the equations 6 Mm Mm +G=4—c¢_“H, -H=-—-CH+— 1, +1= + = I — z K, —-K=—~“K+L, +La+"*L-—-—™, &c; which may be done in the manner following : To refolve the firft equation, — F = — a F + = G, let af F be added to both fides ; and we fhall have 2* G = F-F=+*— I xF= x F, and confequently (multiplying, both fides into =) eu: is moe ) Q. Eerie To refolve the fecond equation +- G = ne (gees. fo H, let aa Hl he added 62 to both fides; and we fhall have + G + i H = —-G, and confequently (fubtracting G from both fides) Z?’H= pial Ga Qa nee | “xG= Gan — ms ° im mM int 1/3 x G, and (multiplying both fides by 8 {pee —— Sa Cy Qs Ee Ts To i” 288 AT DISCOURSE CONCERNING ‘i ; , ae VEL 82 an Lal To refolve the third equation — H = — —- H + — I, let —- Hl be added to both fides; and we fhall have =" I = a H =H = in — | x a == x H, and confequently (multiplying both fides by oe — 4 82 Q Ent, To refolve the fourth equation + I= 52 1 — 2K, let 2* K be added to both fides; and we fhall have I + a Kins = 1, and confequently (fubtraé- 82 —m ing I from both fides) 2=° K = Be Ueenil ogee gt oe a Mm ™ £14 x7I5.and Q. E. I, 82 — m Emig 7 (multiplying both fides by an) K= To refolve the fifth equation — K = — 2° K +—*L, let 2* K beadded to both fides; and we fhall have —L = 2° K—K == _ i]x Ke “— x K, and confequently (multiplying both fides by ae ie fox x K. | pei 102 I1#” 112 And, to refolve the fixth equation, me Toscte+ ‘ee L— a M, let a M be added to both fides; and we fhall have L + ~— M = a L, and confe- quently (fubtraéting L from both fides) —* M = ?L—L=?%—1i\xL Ld m m = =" x L, and (multiplying both fides by) M = ““—" x L, Q. E. I. It appears therefore that G will be — “4=” x | 6 And that H will be = a % G: andI = ee x H, and: aad ae i and LL, = ~—* x K, and M = 24>” x L. And it is eafy to fee that, if we were to inveftigate the values of the following co-eficients, N, O, P, Q, R, S, T, &c, by the refolution of the following fhort, fimple, equations which relate to them refpectively, we fhould find _ N to THE BINOMIAL THEOREM. 289 N to be = ~——" x M, 12% andO = —— xN, 4 13% 13% —m 100 Ba = x O, 142 . 142 — m and 2 Bese Te ae is cut Cl Setre> m8 and Ri TRE AAS Oi); ford poh = irae 8 2 ; 172 Ande at ee BS 182 and fo on ad infinitum, every new generating fraction being derived from that which immediately preceeds it by adding z to both its numerator and its deno- minator. 104. We may therefore now conclude that when m is greater than #, but lefs mM than 22, the quantity 1 + we| # , or the mth power of the th root of the binomial quantity 1 + wx, will be equal to the feries 1 +- — Ax + = Bw — ies ° . 3n— m 4. \44#—-” ; aa 6 fats (OR > 72 — me Cue + re Dx are Exs + “Fx a Gui + —2 Bx? is added to the two firft terms 1 ++ — x, and the fourth, fifth, fixth, feventh, and other following terms are alternately fubtraéted from the three firft terms 2 LD ee i+ = Ax + the figns — and + alternately; and the feveral generating fractions ak = Bx, and added to them, and confequently marked with — 3 {n—m s4n—m si—m br—m Pere teca cs?! GR? 7x nual addition of z to both their numerators and their denominators. , &c, are formed one from the other by the conti- ey oe a 105. We have now fhewn the binomial theorem to be true in the cafe of mm 1+ x|", or the mth power of the wth.root of the binomial quantity 1 + «, when m, the numerator of the index —, is greater than #, its denominator, but lefs n than 2”. It remains that we fhew it to be true alfo when m is greater than 22, Vot. II. 2 .P but 290 Av) DT StcloluiR sim (CO -N ic tRoR NES but lefs than 3 2, and when it is greater than 3” but lefs than 4, and when itis greater than 4 but lefs than §, and, in general, when it is of any greater mag- _ nitude whatfoever. 106. Now for this purpofe we need only obferve that in all thefe different magnitudes of m with refpect to 7, as well as when m is either lefs than z, or 7 greater than #, but lefs than 2”, the quantity 1 + «| will be equal to the fame feries 1 + =~ x, Cw?, De3, Ext, Fx', @c; of which the firt/termmes and the fecond term is = x and is added to the firft term, and the third and fourth and fifth and fixth, and other following terms, Cw?, Dx?, Ex+, Fx, &c, confift of the fquare, and cube, and fourth power, and fifth power and other following powers of w in their natural order without interruption, multi- plied into certain fixt numbers, or numeral co-efficients, which may be denoted by the capital letters C, D, E, F’, &c, and are to be connected with the two firft terms 1 + — « either by addition or fubtraction, and confequently marked with either the fign + or the fign —. For from hence it follows that, if we apply the reafonings ufed above in art. 67, 68, 69, 70, 71, 72, to the invefti- gation of the terms C%*, D3, Ex*, Fx, &c, in the feries 1 + - ¥; ORes Dx?, Ex*, Fxs, &c, and of the figns + and —, which are to be prefixed to. them, in any of thefe mew relative magnitudes of m and u, we fhall always come to the fame final equation which was obtained for this purpofe in art. 72, to wit, . . nm the equation between the fimple feries ~, Cx, Dx*, Ex?, Fx*, &c, and the compound feries == C,'2— Diy ti akteeenasensec 2m wat We > im “ ; 72 + 1, a= ‘ “ ioc o E x3, &c. And from this general and fundamental equation we may derive particular equa- tions for the determination of the figns that are to be prefixed to the feveral terms Cx’, Dw?, Ex*, Fx’, &c, and of the magnitudes of the feveral co-effi- cients C, D, E, F, &c, which will confift of the very fame terms as the particu- lar equations derived from the faid general equation in the former relative mag- nitudes of m and 7; to wit, the particular equations following : m1 22% 7 adly, C = 2D, = Ch 3dly, D = = E, 3" D, 4thly, E= 42°F, +E, Or (as they may be exprefled with more convenience). ift, Tees: 98 IN 10 MX SATEY 9 cr "0 RYE" M. 29% m2 2n ¥ ift, ig boeas I, aI; odly, C =—C, #*D, gulysy Ds = oS D, = E, athly, E = +E, 2 F, iG &c. And the only difference between thefe particular equations in the cafes of thefe new relative magnitudes of m and # (in which m is fuppofed to be greater than 27”) and the former particular equations, confifting of the fame terms, in the cafes of the former relative magnitudes of m and ” (in which m was fuppofed to be either lefs than , or greater than 2, but lefs than 27) will be in the figns + and — that are to be prefixed to their terms. This dif- ference we muft now proceed to inveftigate. 107. In the aforefaid particular equations, by means of which the figns to be prefixed to the terms Cx”, Dx3, Ex*, Fx’, &c, and the values or magnitudes of the co-efficients C, D, E, F, &c, are to be determined, to wit, the equations mt 2% a Mm F = 2F, 3 ? ° a 7A n nh &c, the common denominator m of the feveral fra¢tions —, 22, 42 5” && > m 3 m 2 m 3 2 > 3 will, on the prefent fuppofition, be greater than the numerator, 2, of the firft of the faid fractions; and it may be greater than 3”, 4”, 5”, 6”, 7m, and many more of the numerators of the following fractions, 3", am S, a, Z, 8c. But, however great we may fuppofe m to be in comparifon of 7, we may always, by continuing the aforefaid particular equations to a fufficient number, come at Jaft to an equation in which the numerator of the fra¢tion involved in the firft term on the right-hand fide of the equation will be greater than the denomina- tor m. For let pz be the greateft multiple of # that is lefs than m: then will p + 1\x 2, or px + 2, be greater than m; and confequently, when the fractions ? 6 . 1|x 2 n+n 2m gn 4n gm On 7% wc, are continued to the termftU**, or MF" (45 m? mm? m? m? m? m m ms which, it is evident, they may be continued), the numerator of the faid fraétion put will.be greater than its denominator. Now, fo long as the numerators of m : 6 ; Y thefe fractions, 24, 32%, 42, £2, -*, 2, &c, are lefs than their denominator m, ASE ee RE RL DO the fractions themfelves will be lefs than 1: but when we come to the fra¢tion Bebe 2 puta . > 4 292 A Di 17SCC 86 OR SIE CONCERNING mi ~, of which the numerator is greater than the denominator, the faid fraétion 1 will be greater than 1; and fo will, @ fortiori, all the following fraCtions ; to wit, Pern ip ex a pF 4k 2 OP PosK a & put2n pat 3n pnt 4n i Ee ST Pe ACL PES CE. RT SRN Ye ws Cc, or yer. ae he) Ht : m 72 Mm ; mM fat 5" &c. Therefore, fo long as the numerators of the faid fraCtions =, = MH 4u $2.62 ‘fn —— —_—_— -— ee m? m? m? ua 3 , &c, continue to be lefs than their common denominator m, the ee ee 6 - ; - quantities — ce a D, “~ i = F, “< Gc; 2 H, &c (which are the firft terms on the right-hand fides of the 2d, 3d, 4th, 5th, 6th, 7th, &c, particular equa- tions above-mentioned) will be lefs than the fingle quantities C, D, E, F, G, H, &c, which form the left-hand fides of the fame equations: but when we are come to the fractions os, 20-12%, eee PETAR a &c (which are mm greater than 1), the quantities of which thefe fractions will be the co-efficients, and which will be the firft terms on the right-hand fides of the feveral particular equations in which they will appear, will be greater than the correfponding fingle quantities, which will form the left-hand fides of the fame particular equa- tions; or, if we fuppofe the faid particular equations to be as follows, to wit, Cc’ = purr Ce put 2n D, m mm Do — ft 7; PAF 3" pe! Sa mm a m 4 a oe E, oe F, and FY ee EEG ; wey 73 Mm ‘4 &c (in which the capital letters C, D, E, F, G, &c, have an accent “ placed over them, in order to diftinguifh them from the former co-efficients C, D, E, F, G, &c), the quantities “+ @') SER’ py, Pecan) Shee ee F’, &c, m um ms m which are the firft terms on the right-hand fides of the faid particular equations, will be greater than the correfponding fingle terms C’, D’, E’, F’, &c, which form the left-hand fides of the fame equations. 108. Further, it follows, from what has been repeatedly obferved in the courfe of the foregoing inveftigations, that in all thefe particular equations the terms that involve the fame co-efficients, or capital letters, C, D, E, F, G, &c, or C’; D’, BE’, F’, G’, &c, muft have the fame fign +- or — prefixed tothem; becaufe they are derived from the fame original terms Cx?, Dx?, Ewx*, Fus, &c, in the affumed feries 1 + ~ x, Cw, Dx, Ext, Fx’, 8c, by the operations of multiplication and divifion, by which the figns + and —, that are to be prefixed to them, cannot be affected. Therefore the firft terms on the right-hand fides of all thefe particular equations muft always have the fame figns + and — pre- fixed to them as are to be prefixed to the fingle quantities which form the left- hand fides of the fame equations. 2 109. And THE BINOMIAL THEOREM. 293 109. And hence it follows that, fo long as the firft terms on the right-hand fides of thefe particular equations, to wit, the terms ——C, 2° D, “ E, 2 F i m 3 m > m > = G, &c, are lefs than the fingle terms that form the left-hand fides of the fame equations, to wit, the terms C, D, E, F, G, &c, refpectively, the fecond terms on the ae fides of the faid particular equations, to wit, the terms a D, 2" e, “ Bee — — G, = H, &c, mutt be marked with the fame figns as 7 firft terms on the Ore ene kab fides of the fame equations, to wit, =" er By (1 = E, F, = G, &c, or muft be added to the faid firft terms, fo as thereby to increafe their magnitudes. For otherwife the two quantities on the «right-hand fides of the faid equations cannot be equal to the fingle quantities on the left- hand fides of them. But, when the firft terms on the right-hand fides of thefe - particular equations become greater than the correfponding fingle terms that form the left-hand fides of the fame equations, which happens in the equations ries n + pat+2n , Cc Foor} ie er, D = et2# + 2n LY, ts E, Eo — pz + per Be E’, pt 4% a 42 Fy 4 and F = pe des E> ells SL Mm &c, the fecond terms on the right-hand fides of thefe equations, to wit, the terms ==" = = 1 DE fer 38 E, a se Fe prt. 5 G’, &c, mutt be marked with the contrary figns to ote which ae to be piohred to the firft terms on the right- hand fides of thofe equations, to wit, the terms oe = (A peste ae D, pa + 32 = 32 FE, Pm TAR EY Bc, refpettively, fo as to leffen the too great Laitinen of th faid firft terms, and reduce them to an equality with the fingle terms C’, D’, EF’, F’, &c, which form the left-hand fides of the faid equations. t1o. From thefe obfervations it follows that, if m is greater than 2”, but lefs pet is a ——, or x the parti- than 3%, or, if pis = 2, and confequently cular equations fet down above in the beginning of art. ee , to wit, the equa- tions Wee 2n THUG G pat 2n 3” ‘| 2 2 Deei25 Dy tip 7% 774 204, Aig. D HeSy Cy OgUeR Suk. (Cao NEC SEaR. wi ho weg, Bo SR, SRS 2 n 62 Ho 2 8, seas &c, will, when the proper figns + and — are prefixed to their terms, be as fol- lows, to wit, mm 2 ~=i1+C, Mw +C=>+—C+#D, +D=4+#¥D-#E, -~E=-—-fE+*+ D+, as Oy Se eed Link pt hl kag 2 +F=> 4 UF 4 +D= +e p- n+ 32 E’, i 296 AMD IeS:C OP BS. CO MSC ER NIN SS , 2+ 32 v7 +- +F => Ee Fb’ — pr” CG, &c; and thefe equations may be refolved in the manner following : 112. To refolve the firft equation, = 1+ — C, fubtraé&t 1 from both fides ; and we fhall have =< (Oh a - —i= —, and confequently (multi-. . m a — n plying both fides by —) C= — x= == x, QB T¥: To refolve the fecond equation, + C = RBA IN UBT T EBs 3=D D, fubtract =~ from C; and we hail have 3° D= C — cat 2)xca2=# x C, and confequently (multiplying both fides by oe 1) ee Be 3G. Q. Bats To refolve the third equation, + D—=-+ a. D + a E, fubtraét a D from D; and we fhall have 42 E = dp- = =1— #|xD= "58 x D, and confequently (multiplying both fides by cal Eon Q."Ea To refolve the fourth equation, + E = + a E+ — Jn F; fubtraét 4° = from both fides; and we fhall have 5% FrE-~-#E=-1- x x ee — “= x E, and confequently (multiplying both fides by <2) Ble moat ce Cai Beate To refolve the fifth equation, + F = + 47 F sige, ~~ G, fubtract 4° F from both fides; and we fhall have <2 GiF— se id Peet ya 54) x pe x F, and confequently (multiplying both fides into =) (= - pea x re | Q. E. I. Thus it appears that, fo long as the numerators of the feveral fraétions —, 37, mM m4 2 a . . . <, 3, &c, continue to be lefs than their common denominator m, the feveral terms Cx*,; Dw3, Ext, "Fx Gxt, &c, of the feries r -+ ~ x, OF ered BE f Ex, T HE BOIoN OrMo! A L sToOH EO Rk EM. 297 Ex*, Fx’, Gx*, &c, which is equal to 1 + x|#, will be all marked with the fign +, or added to the two firft terms 1 + = , and that the co-efficient C will be = — x =; or ~=" x B; and D will be = ate x D, shar WM be Se By -and G will be = KEELER EN' 6 42 hy 3 On J which co-efficients are derived from — 4p OF B (the co-efficient of the fecond term Wi—~— h mn . a) ONE ' x, or Bw), bythe continual multiplication of the generating fractions : 2% —— M—- 22 mM—%n m—~An m— Sn ak im bK ea \ = x Tepe ~, &c, of which the numerators decreafe by the con- tinual fabtr aétion of m, and the denominators increafe by the continual addition of the fame quantity. 413. And, when the numerators of the fractions —*, 34 48 SR) Df 6 a , &ey ww m? wm m? are greater than the common denominator m, and rf 1€ equations to be bee are confequently ra, —|F**c Tg tectaesigg f V2 7 4. [Y= oe Ty ae, m ite n+ 32 n+ 4h rv cate 3 =—-F'er yer, ee pe en a ea &c, thefe equations may be refolved i in the manner followings To refolve the firft equation, — C= — |e aan 6 ET cal We t 8 ender Sek t " C’ be added to both fides; and we fhall have @ =" a oD = pen =~ By his ps me ae i. ma x C*, and content (multi wi both fides Mm ¢ — pr +n2u—m , wae) D haPaePaus ovis ni eo To refolve the fecond equation, + D’ = iS = BR fy a CHS? pe fg Pee 2 3% BY at Pieters) ™m 2 E’ be added to both fides; and we fea tate 1b Aree D’; and (fubtraéting D’ from both fides) i SY ae Bi M28 Dee Doe a -1xD= me x D’, and ee both fides into the fiaétion —*—) paneiate xD. Q. E. I. Wo.. IT... ae. To 298 A DISCOURSE CONCERNING To refolve the third equation, — EF’ = — la x E 4 oa EMP ter pes 3” EY be added to both fides; and we fhall have ewe "i 4% yy —. $3 5 3% + 3% ta r= — i —1| xb Mt oN x E’, and sircnetty Cassi paying ; _ prat+zn-—m , i Se Sidi QE. I. both fides into the fraction er And, to refolve the fourth equation, ++ FY’ = +4 ete F’ — pnw st G’, let pu + 5% y G’ be added to both fides; and we fhall have F’ + — GC '= 72 fat oe. and (fubtraéting EF” from both fides) Cis GP ae E, a1 : v2 mw = mee #8 1] eat che longo x F’, and confequently (multiplying both fides by the fraction \G= ae Pr sda ates Qs BI. It therefore appears that, when we are come to the equation — C = — eer er orm D’, in which the firft term on the right-hand fide of the m 2 pa a equation, to wit, ~ C’, is greater than the fingle term on the left-hand fide of the fame spienon to wit, C’, the feveral terms, D’, E’, F’, G’, &c, will be generated, or derived, from C’ by the ee multiplication of the fractions petn—m pat 2n—m pn+3n—m pa+4n— put 22 : pn + 32 : put 4a : pi sn and denominators increafe by the continual addition of ~; whereas, before we putau~m , put2na x C “—— a et, A &c, decreafed by the continual fabpaghon S wpa). 4a sph ar be be i, ie their denominators ets by the continual addition of i It. ~, &c, of which both the numerators came to the term , the numerators of the’ generating fractions ( ° . — os 2 114. The numerators of the firt fet of generating fractions, “—, oC » &c, till we come toa multiple of w that is greater than m, mM— 32 Mm 4n Mt — on Ae Vaiig a 66 # are the exceffes of m above the feveral fucceffive multiples of n, taken in their natural order, to wit, ”, 2”, 3%, 4n, 5”, &c3. and the numerators of the fecond pn aon —m pr+2n—m pn+3n—m puban—m &e } putan ? .pat3n 2? pn+gns? pu+sn ’ ? after we are come to p + 1|X #, or pa +4, or the firft multiple of z that is ereater than m, are the excefles of the feveral following multiples of 2, to wit, p+i\x a, p+ 2\x 2, p+3\X 4, p+4lx 2, &e, or pm + m, pn + an, « + 3%, pn + 4n, &c, taken in their natural order, above m: fo that in both {ets of generating fraCtions the numerators of the faid fractions are the differences of m from the feveral fucceffive multiples of #, taken in their natural order, to 6 | Wit, fet of, generating fractions, STeHIED BIN O Md Ale .;T HE (O)R; EFM.» 299 wit, 2”, 3”, 4", §u,6n, &c, andpu,p+1)x np +2|x 2, p+ 3) x n, p+4)x "p+ 5\ x nm, &c, or2n, 3n, 4m, 5m, 6n, &c, and pn, pu+n, pit + 2M, pn + 3n, pn + 4n, pn + 5n, cc, adinfinitum | f 115. It therefore has now been demonftrated in art. 105, 106, &c.... 114, that, when m is greater than 2”, and pz is the greateft multiple of x that is lefs m than m, the quantity 1 + wx), or the mth power of the th root of the binomial 3 Mm ~~ 27 3% m— 3n it — 4n M— Str 6 . (ether o pt+* Cre + nin., Dx* + Tm Exs + ae Fxro+ &c arpere Cx prt2n—m D’ pt+3 dé pn+3n—m REL eih4 put+4n—m Fp’ b+5 ane pee seae om pa + 32 i ieee . a put 5x 4 pn + bx G’x?t® +- &c, ad infinitum; in which feries all the terms after the firft term 1 prtu—m Clee , 3 quantity 1+ ~, will be equal to the feries 1 af os Ax + ——— Boxee are to be added to the faid firft term till we come to the term oo which isto be fubtracted from the faid firft term; and all the terms after the faid Neem C’ x’? are to. be added to and fubtraéted from the faid firft 2% term 1 alternately. _ All which is agreeable to what was afferted above in the be- ginning of this difcourfe, art. 5 and 6, concerning the feries that was equal to mn 1 + x) upon thefe fuppofitions of the rélative magnitudes of m and x. 116. We have therefore now compleated the demonftration of Sir Ifaac New- ton’s famous binomial theorem, in all the cafes of fractional: powers whatfoever; to wit, 1ft, in the cafe of the roots of a binomial quantity 1 + x, in art. 21, 22, 23, 24, 25, &C..+.- 51, where we invefligated the terms of an infinite feries I that would exhibit the value of 4/7 1 -+- x), or 1 + w\n, when #, the index of the root, was equal to any whole number whatfoever; and, edly, in the firft cafe of the powers of the roots of a binomial quantity, in art. 54, 55, 56, 57; 58, 59). +¢+.++79, where we inveftigated the terms of an infinite feries that m would exhibit the value of 1 + x]*, or of the mth power of the wth root of the binomial quantity 1 ++ x, when z, the index of the root, was equal to any whole number whatfoever, and m, the index of the power, was equal to any other whole number Jefs than 7; and 3dly, in the fecond cafe of the powers of the roots of a binomial quantity, in art. 80, 81, 82, 83,.....,.. 404, where we invefti- wn gated the terms of an infinite feries that would exhibit the value of 1 + “\7, or of the mth power of the ath root of the binomial quantity 1 + #, when x, the index of the root, was any whole number whatfoever, and m, the index of the power, was any other whole number greater than, but lefs than 2”; and, 4thly, in the laft cafe of the powers'of the roots of a binomial quantity in art. 105, 1065 2 Q2 107; 300 A =D T°8°C4O OU sR. SY E-- COW €C ER NOTANTG 107, 108, &c.s....115, where we inveftigated the terms of an infinite feries ™m that would exhibit the value of 1 + |, or of the mth power of the #th root of the binomial quantity 1 + «, when z, the index of the root, was any whole number whatfoever, and m, the index of the power, was any other whole num- ber whatfoever greater than 2”. Here therefore we might with propriety put an end to this difcourfe ; but as the laft cafe above-mentioned of this theorem, in which m is greater than 2, is attended with rather more difficulty.than the former cafes of it, on account of the different numbers of terms of the feries ar a, Cx?, Dx?, Ex*, Fw’, &c, after the two firft terms 1°+ = x, which are to be marked with the fign +, or added to the faid two firft terms, according to the different magnitudes of m with ref{pect to z, I fhall now proceed to lay before the reader another demonftration of this laft cafe of the faid theorem (in which m is greater than 2”), and likewife of the next preceeding cafe of it (in which m is greater than ”, but lefs than 2”); which is grounded on the fuppofition that : ™m the faid theorem is true in the firft cafe of the quantity 1 + «|, or when m, the . ™m . . . . numerator of the index —, is lefs than its denominator z.. Another demonftration of the Binomial Theorem 72. in the cafe of the quantity 1 + x|7 , or the mith power of the nth raot of the binomial quantity 1 + «x, when m, the index of the power, is greater than n, the index of the root; deduced from the feries that is equal ta m” 1 + x\7, when m is lefs than n. 117. It has been fhewn above, in a pretty full, and, TF hope, fatisfatory manner (in art. $4, 55, 59, 575 58, 59, &C.-+~- 79), that when m, the numerator of mt the index = , is lefs than ”, its denominator, the quantity 1 + x|7, or the mth power of the mth root of the binomial quantity 1 + #, will be equal to the feries i+ Ae 2a" Be? 22 Cx — (Mo Des + ne 2n 4n &c, ad infinitum in which all the terms after the two firft terms 1 + — Aw are na marked alternately with the figns — and +, or are to be alternately fubtracted from, and added to, the. faid two. firft terms. Now from this propofition we may deduce a proof that, when m is greater than z, but lefs than 2%, the quan- ; tity THE*+BINOMIAL THEOREM. 301 ~ 7 tity 1 + «)* will be equal to the feries above found for it in art. 80, 81, 82, 83, &c, ..... 104; and that, when m is of any magnitude greater than 22, the m quantity 1 + «\* will be equal to the feries found for it above in art. 10 23 106, 107, 108, &c..... 115. ‘This may be done in the manner following: 118. In the firft place let us fuppofe the index -- of the power to which the uw +* inftead of —; a a m m+ 2 at aa and let p be put =m +. Then willis + «} > OL Tt Ss] Fo POeras £ 1+ x)” ; in which quantity p, the numerator of the index £, will be greater binomial quantity 1 + is to be raifed, to be ~ + 1, or than its denominator #, but lefs than 2”. For m + m (to which p is equal.) is greater than 2, but (becaufe'm alone is fuppofed to be lefs than 7) lefs than 22. | ci We mutt therefore fhew that 1 + x|# will, upon thefe fuppofitions, be equal to the feries 1 + 4 Ax + oa “Bx? = etn a ete a er a Ext + &c, ad LF inte in high the third term#—* By? is marked with the fign +, or is added to the two firft terms 1 “he A as and all the following terms are marked with the figns — and + alternately, or are to be alternately fubtracted from, and added to, the faid two firft terms, agreeably to art. 104. omen _ oa x ) a Me z 119. Nwt+xeis=rexl* = i¢ae** = r+ale x m2 T+ afort+* x1 4e. mt But, by the fuppofition 1 + | i is = the feries 1 + ae 2 ES m a—m 2nu —m ™m 2 —m™m 2u—m 3u—m mm a—m 2” 3 7. oe zn i an 4n n 22 a yk Dt &c, as is hewn inart. 54, s5, 56, 57> 58; 595 3% ae 52 " &c.....-793 Or, if we denote the feveral co-efficients of x, «*, #3, w*, #5, &c, m in this feries, by the capital letters B, C, D, E, F, &c, refpectively, 1 -+ x) will be = theferies 1 + i — Cw? + Dx? — hele: + Fei — &c. Tecesiore I + aa (which is = 1 + ia Xit+ x) wi will be = the feries 1+ Be —Cx? 4+ Des — Ext + Fxs — & x 1 + # = the compound feries 1+ 302 £.,.DTS.COQUR SS E .C,O—Nt Ci BR Nd, 6 1+ Bs — Cx? + De? — Ext + Fes — &c + x +Bwe?7 —Cx3 + Det — Ex + &e. But, becaufe B is greater than C, and C is greater than D, and D than E, and E than F, and every following co-efficient of a power of x in the fecond line is greater than the co-efficient of the fame power of x in the firft line, it is evident that this compound feries will be equal to the feries 1 + 1 + B) xX w+ B—C|x « —(C—Dxx#+D—E x «+ —(E—F x x’ + &c3 in which feries the third term B—C)| x «* is added to the two firft terms r + 1 + B| x «*, and the feveral following terms C — D| x x’, D-— £\ x x4, E— F\ x «’, &c, are marked with the figns — and + alternately, or are al- ternately fubtracted from, and added to, the faid two firft terms. Fed: | Therefore 1 + x|# will be equal to the faid feries 1 + 1 + BJx «+ B—C} x x? aaa (OED) x xi + D—FE| x x* —(E—F x «> + &c, in which the third term B — C) x x? is added to the two firft terms 1 + 1 4+ B|x a, and the following terms are alternately fubtracted from, and added to, the faid two firft terms; which is one of the properties that ought to belong to the feries - p that is equal to 1 + x|*, according to art. 102. 120. It remains that we fhew that the co-efficients 1 + B, B—C, C—D, -D—E,andE—F, &c, of the powers of # in the faid feries 1 -R 1 + B) x # + B—C\|x «7 —(C—D x «3 + D—E)X x* —(E—F X x &c (which p . is equal to 1 + «\#) are equal to f, é Rider ins [hal eines Ie bes Px on ee 22 32 7 prt mp. Bt ph LV pH, mp ep | amp at, Me ere and — en X = Sn eas a » &c, re- fpectively ; agreeably to what is fhewn in art. 104. This may be done in the manner following: In the firft place it is evident that 1 + B (being = 1 + =) wil be = atm ip rs 5 % Q. E. D. A=— Mm Secondly, fince Cis = B x , it follows that B — C will be = B—~Bx 24 adie a 2n B a1—m ae 2n * Faas 22—n-+-m —_—_— — as — a aa eee mm ony 2. | 2n- 2m 22 — mM Thirdly, fince Dis = C x ——, we fhall have C+D=C—C x 2n4— Mm dens? jal 22 — im St Ol NOR ek gn ontm 32 rier e3 3% gt, Bee 32 ater 3 a C és ~=C x=. 3% 3% Fourthly, THE BINOMIAL THEO R EM 303 Fourthly, fince E is = D x ere we fhall hve D—E=D—D v4 3a—m 4n Qn — m2 ee S At —3n+m 4a wey Dx 42 = Dx 42 4% =Dx 4n ray Dx "=D 4. 4% 18 And, fifthly, fince F is = E x2 ae we fhall hve E—- F =E-~—EX 4n— nm ag Ee ee ge 52—4e+m 52 ee a EX Sn met a ia oe 52 =Ex 52 no 5 And it is eafy to fee that, if we were to proceed in the fame manner to examine the following differences, F —G,'G — H, H — 1], 1—K, K — L, &c, con- ag to any number of terms, we fhould find them to be refpectively equal to F a oe -G 4 om ix é ,Ilx= -: ,Kx ~ Cite in which quantities the de- nominators of i tdi eho fs 29 ?, PP.) &c, of which p is the n 82 gn’ ron? numerator, increafe continually by the addition of z. _ It follows therefore that the feries 1 +1 + B) x «+ B—C\|x«* —[(C—D ra x03 + D—E)X x —(E—F xX *' + &c, whichis = 1 + ie will be =i1t+fe+BxFw—-Cx bei dDxint we ph aS ta &c; and confequently this laft feries will be = 1 + w\z. aly muft ek fhew that the feveral co-efficients B x £ iC ax “ns ,»D KG -, and E x a &c, of the powers of x in this laft feries are Bea ref ‘pda to af x — cou = _ ae poe Pld at ap te ap pe ep aa ws gn 7 2 pr a 3” . 4 oz 7 and 4 as ocr 3% x | = x rr &c. This may be proved in the manner following : 121. Since pis = # + m, we fhall have 22 —p (= 24 —u—m) =nN—M, and confequently (adding # continually to both fides) an —p = 2n— mm, and 4n—p = 3n—m, and 51 —p = 4n—m, and 64—p = 5u—m, and 7z7—p = 6n—m™, and 8u—p = 7n—m, and fo on ad infinitum. We fhall therefore have pit-date: 304 AWD WSOCHONUIR SOE CIO NC ER NW ey CS wz ZH nu 2n . NL aE ‘nm — wm 2ft = x Poe 2n — p 3n—p and D (= = x 2n x 3” , = a x 3n * an ? 7 n— Mm 2n — m : ine ad en Doel Spe. Jhaae 3 32 4i — and E(=— marere ons 32 x 4n diet TS Sees 3n 4¢ 0 re ee ae Si na — mM 21 — im 3n — Mm 4a — m ~P-s 22 —_p andl (= 5 xX ove 32 x 4n x 52 i rt ahah’ 1 — _— oN liga silk ager caren a n An 5% and fo on ad infinitum, every new capital letter being equal to the bees preceed- ing capital letter multiplied into a new generating fraction (as sa r none p ale 68 ply i de &c), which is derived from the generating fraction next "TB 2 gv 102 before it, by adding x to both its numerator and its denominator. It follows therefore that the co-efficients B x -, Gx a Dx a E xX A &c, of x7, 3, x*, x5, and the following powers ef x, in the fue Pa fifth, fixth, and other following terms of the feries 1 +4 ~* +B x —#* —CX - sg? < ++ &c, ad infinitum, agreeably to what was fhewn above in art. 104. QE. D. 122. Having a4 found that, when p is greater than #, but lefs than 2», a el. ee the quantity 1 + its is equal to the feries 1 Beata Sarai my = 2n £Su* Beep gy. BP: Pr 2n — p eyed i weary p-* 2n =p ah alla Gin acams fae) ge 3°81. Taal AGEs LENE pee x oP y eat y «§ + &c, ad infinitum, in which the third term 'e Px te 7d 4n is marked with the fign +, or added tothe two firft terms, and all ag following terms are marked with the figns — and + alternately, or are alternately fubtract- ed from, and aacies to, the faid two met terms, we muft proceed to confider the remaining cafes of the quantity 1 + AF a, in which m, the index of the power to which the mth root of the binomial quantity 1 + * is to be raifed, is greater than 2 n, or twice the index of the faid root. And, in order to do this with the more diftinétnefs, I fhall ufe the letter g, inftead of the letter m, to denote the index of the faid power, and fhall continue to fuppofe m to be lefs than # (as it has been fuppofed to be in the courfe of the five laft articles, 117, 1 J » I1g, 120, and 121), and p to be = m + 2, as before, and confequently 4 — to be = ae = + 1, and fhall fuppofe the new index g to be a top-+-m,orm+n-+ua, ne or m + 2n, and confequently 4 =. to be = —+—, of 47; ; and, upon thefe fup- "9. pofitions, fhall proceed to fhew that 1 + : a will be = the My 1 + = x + py link a lgae Eee Sem > Samet & GP re, Q-28 YY BO A in ! ” x poe ai a eS 2” eS 3n ‘ aie 2n ~ 32 < An hag BEI Tend TAP or: x sr co ts in which. the fourth term, as wel 3 as the third, 1s Be toart with che viet +, or is added to the firft term, agreeably to what is fhewn above in art. 115, and afferted in the beginning of this dif- _ courfe in art. 5 and 6. a 123. Now, upon thefe fuppofitions, the quantity 1 + x|*, or the gth power pts of the wth root of the binomial quantity 1 + x, will be = 1 + x 2. 2 Sm Vou. Il. 2R p rae 306 AW) D ES! ChLONUER S EB. LCHOGNECTEGR N LNG ae a p Het } | 1 fp x\2 =i1+eaje X1 +x)’ > 1--ale X 1 + « = the feries 1 —n p —n 2n—p —1 2n— n—?p Bhy pei Ca ee Sia et ON ry ag ae Be gt BE Sev ABE g ST Bit ee Bs +&ex1+te = Cif the co- n 22 Qn 42 5” efficients of x, 7, x3, «*, «5, &c, in the terms of this feries, be denoted by B,C’, D’, E, F’, &c, or the capital letters B, C, D, E, F, &c, with an accent placed near them at the top) the feries 1 + Bx + Cx? — De + ie — F’xs — &c X 1 + ~ = the compound feries t+ Be + Cx? — D’e? + Ent — Fes + &e +xe+ Ber + Ce? — Det +: Exes & &e, But, becaufe C’ is greater than D’, and D’ is greater than E’, and E’ is greater than F’, and every following co-efficient of a power of « in the lower line of this. compound {feries is greater than the co-efficient of the fame power of x in the up- per line of the fame, it is evident that the {aid compound feries will be equal to the feries 1 + 1 + BX «+ Bo + C)X x? + C — D|x «3 —([D—E x fs K+ +E — Fx x’ — &c. Therefore the quantity 1 + x) will be equal to the faid feries 1 + 1+ B\X x + B+ C|X«? + C—D)x x3 —(D-— E XK xt E’— F\x x? — &c; in which feries the fourth term C’— Dx a. as well as the third term B’ + C) xX x*, is marked with the fign +, or is to be added to the two firft terms 1 + 1 + B’) x w; and all the following terms are marked with the figns — and + alternately, or are to be alternately fubtracted from, and added to, the faid firft terms; which is one of the properties of the a of a C phase a 2 a5 ' gn Y—2n pe z q—n ST Oi ed dea ome Sheen Pui vripe Dates weve. 2 yea. at Ciera ea aes — &c, to which q—2n 22 30 42 , Crate Rg aR AE Nie ne a ERY, 32 4n n 7 we are now to fhew the faid quantity 1 + «|# to be equal.. 124. It remains that we fhew that the co-eflicients 1 + B’, B’ + C’, C’— D, D’ — EF, EF — F, &c, of the feveral powers of + in the feries 1 + 1 + Bi x x +B C\x x + C—D)|x «3 —(D— Ex «* + BE — F\X @ — &e 2! (which has been fhewn to be equal tor + x\2 ), are refpectively equal to the PGi A ds ITB PSB, Pg Pe Ne oe co-efficients os di Mire ayatbes thee x st ols 4) Terns a x ae and £ x 22% x 2B y 21x BEL &c, of the fame powers of « in the n 2n 3% 4% (2 coat Pie £2 a aL ye LER aes ee Mat ae ant My ie BASS tals POG Terbalad Shobak Gc eda: Swi. ; 2 rap + a Ra ies as Shue x cn he &c, to which we are THE BINOMIAL WHE O KEM. _ 307 q are now to prove the quantity 1 + “1 to be equal. For then it will follow that this laft feries being thus fhewn to be equal to the foregoing feries 1 + 1 + B xv + B+C]X x? 4+C — D|\ x «3 —(D’— E)x «+ + E — Fx xs q &c, which has been fhewn to be equal to 1 + x\#, will alfo be equal to q 1 + *\*. We muft therefore endeavour to fhew that 1 + B’ will be = 4, r/ and B’ 4 C’ will be = 4 x 2, and C’— D’ will be =+ a, SeGtle R fos 27 3% and D’ = E‘will’be = 2x To2iy fa , oF n 2” 32 4n and FE’ — F’ willbe = 2 x 2”? » 4™* y 1 hie fy ro 2n 3% 42 52 and that the following co-efficients F’—G’, G’—H’, H’—I’, ’ -K’, K’ —L’, &c, of the powers of x in the former feries will, in like manner, be equal, refpectively, to the following co-efficients of the fame powers of x in the latter feries. ‘This may be done in the manner following. 125. Since p + wis = q, we fhall have P=— 7-4, andp—u(=q—n—12) =| — 2%, and 24 —p (= 2u—(g—nu—=2an—g+n)l=yn—4y and (adding x continually to both fides) ty 2 Sy eds 4u—p = 5u— | su—p = 6u—¥, 6n—p = JX — Ys n—p = 81— 4, 8u—p = on— 43 and fo on ad infinitum. Therefore B’ will be (= 4) = 1%, and C willbe (= 2°x 2 = 0%, f*y)=B x t* " 2n n 2n 2n and D/ will be (= £ x 4" x M=P=C' x Sf) SC x BaF, é rp ate bs BB and (= 4 x2 x ah yy BP Lp x Mf) =D x #4, 3n 4n 4n 4n and F (=4 x So", SUP, BAY Moho py, ff =n x MS San 2n 3% An Sa. ie, Pe, Sad doo be Yas And, in like manner, 7 os _— Ww N—b. ry 62 —gq G' will be (= F x Sf) =F x 4, and H’ will be (= G’ x “=4) = G’ x #24, : 7% 7% Pes Me and 308 “AS WsicWWwR sies. FO we BR yer Ss and I’ will be (= H’ x Esky —H x aaa 82 Sn? me le a4 and K’ willbe(= VY x “=4)= 1 x 4-4; 9% g% and fo on ad infinitum. Therefore the co-efficient of w, to wit, ae —n yeh Q—-8t » atq-—ay — ca 1 +B, will be'(= 1 + 2— oie — =— = 3 ABC wile (a4 B x GEE MA ETDs == B x ia aad ee BEE Vs 2% 2n and C’ — Dy will bet CS €’ x oT G 6 Fs ie x ee 3% 3° t - BRE ev oee hee ir eee eae) =C x L; 3? an? and DY — E iN be (= Dp D x Moti Mop y Hota -*" 4n 4n 4n — 4 —D' x nolo —D x port) = 1 x an vibeg "4a An 4% and E’ — F’ will be (=P -P x SA POL OL SE A 52 5% 5A SAOANG adie: GARE Pos 5n —(sn —4 sls ohe Hime ble SB es 57 | Sa 5% 5” < and, in like manner, F —G' willbe = F x & 6” and G’ — H’ will be = G’ x me and H’ — IV will be, = H’ x J, and I’ — K’ will be = Fx & and fo on ad'infinitum. Therefore the feries 1 +1 +B)x «e+ B4C)\x «7 + C=D}xX'x? —(Do— Eh x «* + BE —F)x «5 — &c (which has been fhewn to be equal g to I Ci (i eve Ae ae cn +C x 2 Oe 4” we tE xt xe —-&er14+4 eg Aon = Dye pity day gn 22 ‘ 2 22 Lae SN De See 5% = pi % Sak 2 ee 32 a 2” an 42 a 2n 32 4n xtew —&oa 1+ tet x $3! 4 Sy Sy toe See 52 22 n ae)". 3% nn q—-% q — 2” bLediend Laer 7 q—-% q—2n 34 —@ fp = gee * 20 a 3% s 4n Plan n * 2% ‘. 3n - 42 * 52 ‘ 4 &c. Therefore the quantity 1 + «)#, orthe-gth, or p+ x)th, or m + 2z)th, power THE BINOMIAL THEOREM. 309 power of the mth root of the binomial quantity 1 + «, will be equal to the feries ay IS EL Ng Mah AE EE a eli ye AP 2 Sli RPE a | Bera q — 2" Re We ce ee 8 x x SAE Y DOGO E 7 i Wai 2% 3.2 a 2n 32 neal oi — 2” a — = Mss Mir pry cae flank Oye ys 1 «5 — &c, or (if we de- 4u 2% 3” 7 A note the co- Diente OL AN 5 oye, wo OCC. ID thie feries, by B”, C’, D”, E”, F”, &c, or the capital letters B, C, D, E, F, aire with two accents placed near them at the top) to the feries 1 + 2 + 25" Bg? 4 —— 22°C gt sch Sa D’ 4 +p oo LE 5 — &c, ad pee 3 in pee the niles term ay Ctx, as well as the third term i—* B’ x*, is marked ath the fign +, or is to be added to the firft term 1, and all the following terms 34 —2 re: =! Ty’ x4, niet By, x5, &c, are marked with the figns — and + alternately ; or are to be alternately fubtracted from, and added to, the faid firft term, agreeably to what is fhewn above in art. 115, and afferted in the beginning of this difcourfe in art. 5 and 6. 9 pet PN BY 126. And, in ee manner, ris be = +4, orp+u+n, orm + 32, we Z Le (SO ee pelierene rae ores pale T' =i +ee x Tt ay= ELEN I+ x\x x 1 + =the feries 1 4+ BY x + C%x* + DY x3 — EY’ x* + F% x: —&,x1i+%*= the compound feries t+ Bx + C% x? + DY’ 9? — E’ et + FY’ = &e ye BY x? 42 COC’ xs + DD’ et — ES et ee &e =r re Bx 2 + BY 4 Cx x? + C74 Du +, D? — B7) 5 77 7 — Ae “7 Wh Gee 2 aN x4 —[E PSA Ag pigeons aa xx + BY+B x 122) x « + C%4-C” x oon x «3 -+ D” — D” x eg 374) 3 pt — |B” — BE” x eo 5” FY n -, 2n+q— pete oH iy + BY x REIN" ye 4 C” x wera s 2% yr ee 8 ae ye SEI ee Wide Bale ae ath ate Ea ear el a x Efe $&eS toe Beer tC” x. — #2 Sh ee x Geib &c sit ox a : 2% 3% 42 Bee 7k Pr he a) oe te re lp Tig 3,” ms 2n Fue a on 32 OF ao ae Pome to? Le ee veccos Gee ee + 2n 32 4n 52 2n r—-2n r r—n r—2n r— 3% re Var shoe) 943 alae ae oe * 27 x an + a x 2M x 37 “r= 30 3% 318 A cD. InS (CuO oR Suk CON CC ER Nyvflexec ¥ — 32 4u—?r . ex sn Ca ae te ae se fum@Z 30 42 x at ei &e = 1 cg wt x 22 a t 2 x 2n 3 ral Bet Bg Dey toot eda A Zn 3% 4n n 2n 32 42 x S— x x5 + &c, or (if we put BY”, C’”, D”, E”, F”, &c, or the capital letters B, C, D, E, F, &c, with three accents placed near them at the top, for the co-efficients of x, *7, x3, x4, v5, and the following powers of « in the terms of this tee 1+ BY «6 + C% xt + DY x3 4 EY’ xt — Fx + &e, or r—n UT 42 — 2n M1 Roe or i WM vg + Oe ee I + — x + — B SS Et Sa CM 93 + — D brie se E ns + &c. Q. E. D. 127. pee if s be a r +n, org + 2n, or p 4- 3, or m + 4n, we fhall have ie n r r r = -L _— — Tray stra = 1+ «\2 “= 1+aextT+ay srt x 1+ 4 =the feries1 + BY’ x + C/ xn? + Des EM 5+ — FY» - &c x 1 + « = the compound feries I “Lt Be ” + + C” x aaa D” x 3 PE | Ad Con, tote FY’ x5 i &e uF x “1 BY” x2 if C/" 43 a's D”” «4 =. | iki ae &c 2 ET PEA ex PBT RC" & ne OED et D7+P x xt + E77 icles 06 BY’ + BY” x a Kx Ts Cee Cc“ a x x3 at D” + D’” x x4 a E” _ E” x —- ¥\x yee ee tpi a4 CM x mee x34 D” x 4n +7 — 3n yt Ngee adit sade ass ee 4% : n Bop EE et BY x tr gy OM EEE or 4 DY yp SEE get xt — ke tp ow + BY x at + CY x = es + DY eye ; 5 r s r Bx at Romi + Se fo wt + Ok XS aT agit ch GX coor Xi ei Dhow bt xX Ce +S kX Set ee ak ot tae oe a sat eal A mip ox A x ay Sy Et x Ee — 32 s eri ee S— 3n 5 ~~ 4” ee as ees 8 a 1 Iv x ——— Xi ie em Sec, or (putting a 4 3 4 b ’ Cc", D’’, E’’, F'", &c, for the feveral co-efficients of w, «7, #3, «*, «5, 8c, in the Dene ST NOM. FF ACLS PR SE OCRAEYM. 3ir the terms of this eee Y -+-iB"x a Cx? 4 DD x3 a Ege Per pies &c, or 1 + -- x + — ” Bry 5 neers Cr x3 gs = DY xt + aaa Z E!" v5 aed, &c. ih Sa 128. We have now fhewn in the courfe of the foregoing articles, from art. 117 to art. 127, how from the feries 1 + ~ Ax — — Bw? + Set Cri — mM 3 —m 4n—m : . — : ~ et ua WED Ex’ — &c (which is equal to 1 + x) when m is lefs m m+n than 7), we may derive the feries which is equalto 1 + m e * oi-+x|", P or I + x}z, to wit, the feries 1 atti LZ le 4 + a ae pe oat C/43 + = D’x* — — |" LE’ «s + &c; ind from this oan feries, which is equal 4 to1 + x|*, we may derive the feries gies is equal to 1 + x\ ce ae yates m+ 2n pr T+ x) 2 > OF I + + x) 2 en scala aint, the feries 1 + 4 x +i B’ x? + Spiny Coys = DD” «* + a EE” «5 — &e; aif from ba % third ipa which is equal to 1 + x\*, we ae ee the feries ioD is equal m +- 3n wTbae to TP A TER GIA iG feries I += — op, ala a Be 2 4 = CMAs... =e D” 4+ — ee PE” 95 A + &c; and from this fourth feries, which is equal to. 1 + x], we may derive om 4 m+ 4n ren the feries which is equal to 1 + x\# *e Ores hen aE wai, or ‘ S$ I— I + x7, to wit, the feries 1 + — x + — BY x? MORE su 2 C'¥ y3 Ge a ee DD? x4. TE Bits nes &o: . Andsin ae of thefe fap after the ye 3 (which is equal to 1 fe x\ ), we may obferve, 1ft, that there is one more term marked with the fign +, or added to the firft term 1, than in the feries next before it; and, adly, that, after the feveral terms in the beginning of the feries which are thus to be added to each other, the following terms are marked with the figns — and + alternately, or are to be alternately fubtra¢ted from, and added to, the firft term; and jdly, that the co-efficients B, C, D, E, F, &c, and B’, C’, Di; re Ee; Bre and BY} Sag: Dp’ ‘Ey Re: &c, ana BY”, Cc”, bg BE”, FY, &c, and ne SEG: D", EX, Er veces in all thefe feriefes, are de- rived 312 A DISCOURSE CONCERNING rived from the firft'term 1 by the fame law, or by the continual multiplication a= mM 22 — Mm Py x 3” -ssaares ese &c, and f, fs, et ee == &c, and 4; — - . . . . . . mM of fimilar generating fractions, to wit, the generating fraCtions =) ; 5) ? 2 an? 42 2n 3 42 52 _ _ _ — fi— 2h Ff — 22 fim s 1 Naan he SG Ra LOE SEN Ys Pee a an PEL a 3n 42 § 2 n 22 32 42 gz £7) 22 phabeMP EAE ay a 4”, &c; the denominators of all which generating fractions 37 4” n ] : ; : are 2, 2%, 3”, 4u, 5”, &c, or mand its feveral fucceffive multiples in their natural order ; and the numerators of the firft fractions in each fet are m, p, 9, r, 5, or the indexes of the powers to which the wth root of i + w is to be raifed; and the numerators of the following fractions. are the excefles of 2, 2, 3%, 4n, &c, above the faid indexes m, p, ¢, r, 5, or of the faid indexes above a, 2n, 3n, 4n, &c, according as m, 2%, 3m, 4, &c, happen to be greater or lefs than the faid indexes. And 1 apprehend that, from the manner in which &, ws & thofe feveral feriefes, which are equal to 1 + *\27, 1 + x\7, 1+ 47%, and m+n m-+2n m+ 3n m+ An 1 + x)”, or 1 +x)", r+ x] ,i +x n ,andi+x| # , have s . been derived from the feries which is equal to 1 + x\ 4; and from each other, it will be fufficiently evident that the following feriefes, that will be equal to m+ ¢n m+ 6x m+ 72 m+8n m-+- on r+e) * ,r+s] @ >the) * Pr tx) * , 1 4 xm eed infinitum, will have the fame three properties which have been found to be- : m+n long to the four above-mentioned feriefes which are equal to 1 + «| 2 , m+2n m+ 3n m+ 42 1+x| * > 1+x| 2 , andi +x| # 3; and confequently that, if Q. be any whole number, how great foever, and M be = m+ qu, the quantity m-+Qn M E I +s} 2, ori + «|2, will be equal to the feries 1 + = x 4. — “eee na 22 . M M—%” M— 2” fan! n M ETE ITE Le A EC Re Pe X —T er — ie: rit — x ai x’ +. &c, or (if we put B, C, D, E, F, &c, for the feveral co-efficients of «, ~*, 13, w+, v5, &c, in the terms of this feries) to the feries 1 + — x ++ “—* Bx? ae = Cx: ate —2" Det + as Ex + &c, till we come to the term in which the multiple of z that enters the numerator of the generating fraction is greater than M, which multiple we will fuppofe to be P + 1|X 2”, or Pu--x. The term in which this happens will be marked with the fign —, or will be fubtracted from the firft term 1, and all the following terms will be marked with the fign + and the fign — alternately, or will be alternately added to, and fubtraéted from, the faid firft terms; and the numerators of the following generating fractions will be P + 2)x% — M, : Pry M—2 M— 2%” Mag “THE BINOMYL AL TWHEOR EM, , gee P+3\x2—M, P+4|)x“z—M, P+ 5|xz—M, P + 6|xx—M, &c, or Pz + 2n — M, Px + 3n— M, Pu+ qn —M, Pu + sn — M, Pa + 6x — M, &c ad infinitum. All which is agreeable to what has been fhewn above in art. 105, 106, 107, &c.... 115, with only a fmall variation in the nota-~ tion, which the different methods of inveftigation ufed in the two places had made neceflary ; the capital letters P and M being ufed in thefe latter articles inftead of the fmall letters p and'm, refpectively, which were employed in art. 105, 106, 107, &c ...-115. And it alfo agrees with what was ftated and afferted above in the beginning of this difcourfe, in art. 5 and 6. Another Inveftigation of fo many of the firft terms of the feries 1 + Bx, Cx*, Dax3, Exe, ™ F x5, €c (which is equal to 1 —- «\”) as are to be added together before any terms are fub- trated from the firft term 1, when m, the nu- mevator of the index =, 1s greater than n, its denominator ; deduced from the binomial theorem in the cafe of integral powers. 129. But there is {till another method of difcovering many of the firft terms ™m of the feries which is equal to 1 + *\*, in the cafe that has been laft under . . m1 . confideration, or when m, the numerator of the index —, is greater than z, its denominator, which the lovers of thefe fubjedls will, I imagine, be glad to WH fee. It is founded on a fuppofition that the quantity 1 + x)” will be equal to a feries of the following form, to wit, 1 + Bw, Cx*, Dw3, Ext, Fx, &c, of which 1 is the firft term, and all the following terms confilt of the feveral fucceffive powers of x in their natural order, to wit, x, «?, x3, wt, x5, &c, multiplied into certain numeral co-efficients, which may be denoted by the ca- pital letters B, C, D, E, F, &c, and that the fecond term, B., is to be marked with the fign +, or added to the firft term 1, and all the following terms C x, Dw}, Ex*, Fxs, &c, are to be marked either with the fign + or the fign —, or to be connected with the faid firft term 1, either by addition or fubtra¢tion, as fhall be hereafter determined in the courfe of a proper inveftigation of the mm fubjec&t. That this fuppofition is true, or “that the quantity 1 + x\* will be equal to fuch a feries,” has been fhewn in a pretty ample manner in the fore- going part of this difcourfe, in art. 54, 55, 56, 57, and art. 82; to which I fhall therefore now refer the reader. Vou. II. 200 _ 130, Upon ft, » a gif." |) ap Wei ckolura se BE clo Wie EER Wi DN . mt 130. Upon this fuppofition, * that 1 + ~|# is equal to the feries 1 + Bx, Cx?, Dx?, Ex+, Fx, &c,” it was fhewn above, in art. 58, 59, 60, and 61, and in art. 84, that, if we putQ = m x —— 2 ad R =mx 7 x= en OL Pd 2 3 3 and S =m 2S xorg KS, and T = m xX x MG te NC . x 2. I t—2 im 2 ATIC aoe ee —— x am Mein 3 4 a—t a—2 2—3 2. U3 aC. San x 7% x . x ; x 4 4 > “= xt x te xtta=t xs, we fhall come to the following general and fundamental equation, to wit, the fimple feries w+ Quv + Ruw + Sx} + Tx* + &c, is = the compound feries ane ¢) eran nB, uC x, BL) 6. nie, RE Rg eee WG, Be xsi Renata Geleaadade gamma Renkin ine , g. Gi x? > 2 CD anes + Bae? si), 3 BEC #3 0 307 Bo Dwaygetee 3.7 BCix+,) te +5 Bx3, .) 45 BIC vty) Se + FBT es pee by the help of which we may determine both the figns which are to be prefixed to the feveral terms Cw?, Dx?, Ex*, Fx5, &c of the feries 1 + Bx, Cx, Dwx3, Ew*+, Fx’, &c, and the values, or magnitudes, of the co-efficients B, CG; DES ES tc: 131. And the manner in which thefe points are to be determined, is by de- ducing from the foregoing general equation feveral particular fimple equations, involving the feveral co-efficients, B, C, D, E, F, &c, fingly, or feparately from the others that have not yet been difcovered, and refolving the faid fimple equations; which fimple equations will, by art. 85, be as follows; to wit, rit, " m ='2 Bs 2dly, Q = »C + qB’; gdly, “RR nD, 2q¢BC + 7B:; 4thly, S eco BD gC. 37 Bot ee and sthly, T — 4 Ff, 2'¢ BE, 9¢CD, 37 B*D, 37 BC",” 44 Bee And, by refolving the firft of thefe fimple equations, to wit, m= 7B, it was found, in art. 86, that B was = =; and. by refolving the fecond of thefe equations, THEE UNOMIAL IT © Eo R E-M: 315 equations, to wit, Q=”C + g¢B°, it was found, in art. 87, that, if m was greater than m (as we have here fuppofed it to be), the term #C mutt have the fign + prefixed to it, or muft be added to g B*, and confequently that the tétm C x’, in the feries 1 + Bx, Cw*?, Dvi, Ext, Fxi, &c, (which is = m 1 +x\#), mutt alfo have the fign + prefixed to it, or be added to the firft term 1; and it was likewife found that the co-efficient C would be equal. to m— 12 1/4 w= 2 xB or =— e 2% ? nt x 2n And, in like manner, it would be poffible, by refolving the third and fourth and fifth of thefe equations (if we would go through the labour of doing fo), to determine which of the figns +- and — was to be prefixed to each of the follow- ing terms, Dx?, Ex+, Fx*, 8&c, inthe feries 1 + By, Cx?,.Dx3, Ex*, Fx, &c, (which is equal to 1 + «\~), and what would be the values, or magnitudes, _ of the co-efficients D, E, and F, &c. And, further, it is evident that all the values of the faid co-efficients B, C, D, E, F, &c, which would be obtained by the refolutions of thefe fimple equations, would be derived from m and #, the numerator and denominator of the index o, by various additions, fubtractions, multiplications, and divifions; becaufe all the known quantities that enter thofe equations, to wit, the quantities Q , R, S, T, &c, and g, 7, 5, ¢, &c, are only different multiples, or parts, or fums, or differences, or, in general, different combinations, of the faid original quanti- ties # and m. And therefore all the values of the co-efficients B, C, D, E, F, &c, mutt themfelves alfo confift of certain combinations of the fame quantities ## and #2; as has been found to be the cafe with the co-efficients B and C, which are equal to — and > x “— 7 a 2 re{pectively. 7% 132. Thefe things being premifed, it will, I think, be evident, ‘¢ that, if “¢ be greater than p times #, or pz, but lefs than p + 1 times”, or pu +” (p *¢ being any whole number whatfoever), the feveral values of the co-efficients D, “«E, F, G, H, &c, will confift of the very fame combinations of the original ‘¢ quantities m and z, when m is of any one magnitude greater than pz, but lefs ** than pz +, as when m is of any other magnitude lefs than its former magni- *< tude, but yet greater than pw, and confequently as when the excefs of m above *¢ pz becomes equal to a, or m is exactly equal to pz.” And confequently, if we can difcover to what combinations of m and x the faid co-efficients, D, E, F, G, H, &c, will be equal when m is exactly equal to pz, we may conclude that the {aid co-efficients will be equal to the fame combinations of # and when m 1s of any other magnitude greater than py, but lefs than px + 2. This is the prin- ciple of the prefent inveftigation, which may be eafily deduced from it, by the help of the binomial theorem in the cafe of integral powers, in the manner following. Boe 133~ Let to ae - _*. 316 ACAD I s°Ccoru RSE EC OUNTC! BPR a wc 133. Let us then fuppofe m to be exactly equal to p times #, or pm, and try to difcover what will be the values of the co-efficients B, C, D, E, F, G, H, &c, upon this fuppofition. mm Now upon this fuppofition = will be = = = p; and confequently 1 + x\z, or the mth power of the wth root of the binomial quantity 1 + x, will be equal toi + «?, or that integral power of the fame binomial quantity 1 + «, of which the whole number 7 is the index. But, by the binomial theorem in the cafe of integral powers (which has been demonftrated above in the tract con- tained in pages 153, 154, 155, 156, &c....169) 1 + xf is equal the feries A Mis Sid Ea Sa Aina las AEE Ae INES So r+4u4F OG Sa ay ity ad LP 60h bas = 3 x ins Po Ne eas xi A et w' + &c, continued to p + 1 terms. Therefore the quantity 1 + \* will, when m is = pz, be equal to the Sane: xf +2 x MR ance Ng am ON damit 07% | Mors x oe ———» elie! de 4 2 +4 55 3 x i Kio 5 terms, and idl ehadiaty Cif we fubftirute = in thefe terms inftead of p, to which x’ + &c, continued top +1 13 ™m Mm mM 7A — — mf it is equal), to the feries 1 +- “ x x + ~ Sha : Pee ~ x & Mm Mi m mH SE eS = ks a ° na v/4 cs 2 2 1 I wt + 2 3 4 sc Ea Suge 3 n x 2 4 ? . Scar ial x -- &c, continued to p + 1 terms, or the feries 1 4+ = x + M1 —n mM—n Mm— 2K w~ Nn Hw— 2n Mw — 3% 2 veto x x r+ xX x >. n Pa n 2 3 a 2 PUY Re. 4 Him 7 Mm — 2n w— 32 m=— An x Mi 2 1/4 ae x< +- ; ; 72 m ni— 2 WH terms, or the Hd Ry ars iy ys cc x? = Pe: 72 “> + &c, continued to p +1 N-—-n m— 22 m. m— Nn Mm — 20 m — 3 ‘ m—n m— 2n nae n 2” * n 4 4” " al 2n “es x x x° -- &c, continued to p + 1 terms, sianary by *. principle laid fe cait in the laft article, when m is of any magnitude greater than pz, but lefs than p+.11x 2, or pa +n, it will alfo be true that the firit p +1 terms of the a m feries that is equal to 1 + xls will ber + —-x + = mn Sao me += — x W— 2 ii— 22 4d} M2 2 Ii -— 22 bin tet 74 Timm m—2n rs ———ee SL SS a ta eee Tre 20 Qn E Raints 2” 30 x“ 4” T n x 2% x 32 mi — 3n x ennieens 4n T HE BINOMIAL TH EO Ret Ms 317 x x aoe continued to p + 1 terms, or 1 +- = Ax + ——* ax 1 m — 2n : m— 3H 4 mi — An ae “vie + - Cs + ass Det + rare Ex’ + &c, continued to p +1 terms. Pe ate Thus, for example, if z be 7, and m be 373, which is greater than 371, or 53 times 7, but lefs than 378, or 54 times 7, the firft 54 terms of the feries % 373 that is equal tor + x)*, or 1 + ]7 or to the 373d power of the 7th root of the binomial quantity 1 + », will be 1 + ov Ax + — Bur + eed Cues + B= Det + BIBI OB re ys ee: Pere ue Ceunsaee ¥ Atk s tee He + aes Ino + a3 "3 Kx" + &c, continued to 54 terms. 134. This inveftigation of the values of the co-efficients B, C, D, E, F, G, H, &c, of «, «*, «3, «+, «5, *°, «7, and the following powers of « in the feries ™m t+ Bx, Cx?, Dx*; Ex*; Fx?, Guo, Hx’, &c, which is equal to 1 + x\-, is fhorter and eafier than the foregoing ones. But it relates only to the p + 1 firft terms of the faid feries, or thofe terms of it which are all to be added to- gether. For to fo many terms only will the feries that is equal to 1 + «|p 1 (from which feries the feries that is equal to 1 + x)” has been here derived) extend. We cannot therefore difcover by it what will be the co-efficients of the terms of the feries in queftion that come after the p + 1]th term (and of which the number will be infinite), nor whether the faid terms are to be added to, or fubtracted from, the firft term 1 of the feries. But, when p isa very great number, or # (which is greater than p times ”, or pz) is very much greater than z (as in the foregoing example, in which m is fuppofed to be 373 and z to be 7), the firft p + 1 terms of the feries 1 + ~ Aw + “— Bivge eee e 2 3 n 2 Cx3 + ea Dx«* + ae Ew’ + &c, (which is equal to 1 + « 7), will be very nearly equal to the whole feries, and confequently very nearly equal to 4 1 + x\” ; unlefs when x is very nearly equal to 1, in which cafe it would be neceflary to take in more than the firft p + 1 terms of the feries, and confe- quently to have recourfe to fome of the former inveftigations of it. An 418 A’ DISCQUR SE CONCER NENG § An example of the binomial theorem in raifing the m@¢h power of the nth root of a binomial quan- tity, when m and n are very great whole numbers. ca 135. Let m be = 970,877, and be = 10,000. And let it be required to =<, or i - --, to thé,—th, of 970-877 4h, or 1000 12 n 10,000 97-0877th power, or to find the 970,877th power of the 10,00oth root of the {aid binomial quantity. 7 raife the binomial quantity 1 + 136. Now it has been fhewn in feveral different ways in the courfe of the fore- going articles, that, if x be any quantity not greater than 1, and » be any whole number whatfoever, and m any other whole number greater than ”, and m be greater than pv, or p times #, but lefs than p + 1| x m, or p +1 times”, the ™m ‘ * quantity 1 + x\ * , or the mth power of the mth root of the binomial quantity 1 + «, will be equal to the feries 1 + —~Ax tr —— Bat + = Shot ba ee py" xT FIP = 10. Butr + 2197 is 1 + —— x 1+ — = 9.979)201,547,673, 5993050) X I + ie Therefore 9.979;201,547,073,599,050 X I -+ aelr 1S 328 A, DISCOURSE, CONCERNING ; iy % 10.000,009,000,000,000,000 __ __ is == 10; and confequently 1 qe — |S is ——— = 1.002, 1000 S:99A2O SATIOT SOV OS 084,180,004,486,389. But, by the binomial theorem, ae “is = the . 24 Z—I oy ae i Z—1 Z—2 24 |3 feries I zx — x — xX — 2x — a. ee RE 1000 Bd ay SS Sarid ons a 3 1000 a fe 2—I Z—-2 3 24 |4 aes 2 24 VB— 24 | 2 3 8 4 * 1990 RS Bh a i oe a 1000 mre ti 1000 ts 73 — 32% -} 2% 24 13 x4 — 623 + rizz—6z 24, \+ me : eo TmTp ed cance he vanes xX | + &c = (becaufe z is lefs than s, and confequently zz is lefs than z, and 2? than zz, and z* than 2) Fe 24 BB 24? seeeeser ee _24 8 eee x r+ 2X —j——— 1K aa a 1c00 2 1000 1000 feries tb gic Rees NE Te ~ + &c. Therefore the faid fe 1-2 x a - =u = as 2%—32%z-+ 23 24 724.18 6z%— 112% 4 6z3 —24 24 |4 . = altel i a RET ET a + &c, will be = 1.002,084, 180 Sioa hater and gor ay (fubtraGing 1 from both fides) the feries %— Bw _—s 56 24 |3 6z —11zz+ 623 — 24 x ey ay 1000 1000 1000 24 rane} + as will Be = 0.002 raat seeyaBe ato! &c. This equation we muft’ now endeavour to refolve. , : ree ; 4 150. To find the value of z in this equation we will, firfl, fuppofe the firft term 2 X —* of the laft-mentioned feries to be, alone, equal to the whole feries, and confequently to the abfolute term 0.002,084,180,004,486,389, &c ; and we fhall then have z X 24 = 1000 X 0.002,084,180,004,486,389, &e 8 = 2.084,180,004,486,389, &c, and confequently z = 2 hy bacon tH 89, 6 &e, = 0.086,840, &c. i; 0 QotE aa Therefore x, or 97 + 2, will be = 97 + 0.086,840, &c, or 97.086,840, é&c; of which number the firft four figures 97.08 are exact, the more accurate value of « being 97.087,787,353,856,001,437, as I have found by dividing 1 by 0.010,299,956,639,811,952,13, which is the logarithm of the ratio of 1024. to 1000. And of thefe four figures, 97-08, which are exact, the two laft, to wit, .o8, have been obtained: by the Pe(olinan of this very ealy fimple equation. . This firft value of z (which is obtained by refolving the fimple equa tion z X ot = 0.002,084,180,004,486,389) muft be lefs than the true value of z in the infinite equation fet down in the latter part of art. 149; becaufe the fecond and other following terms of the infinite feries contained in that equation are marked alternately with the fign — and the fign +, or are alternately fub- ' 4 tracted THE BINOMIAL .T.H.E OR EM. 329 eracted from, and added to, the firft term of it, the confequence of which is (as the terms form a decreafing progreffion) that the firft term alone mutt be greater than the whole feries, and the firft and fecond together, that is, the excefs of the firft above the fecond, muft be lefs than the whole feries, and, in like manner, that the three firft terms muft be greater, and the four firft terms muft be lefs, and every following odd number of terms muft be greater, and every following even number of terms muft be lefs, than the whole feries. Therefore, when the firft term is fuppofed to be equal to the whole feries, or to the abfolute term 0.002,084,180,004,486,389, it is fuppofed to be lefs than it really is, and the value of it derived from fuch fuppofition will be lefs than its true value. There- fore 0.086,840 1s lefs than the true value of z. bide Sain: PID 12. In the next place we will fuppofe the two firft terms, z x ee Bim BBW 24° \7 Zz 1000 fequently to the abfolute term 0.002,084,180,004,486,389, and will refolve the quadratick equation refulting from ee fuppofition, which (from what has been fhewn in the foregoing article) will give us a value of z fomewhat greater than the truth, but which will come nearer to it than the laft value 0.086,840. 24 B— BB 2 B—- ZS Now thefe two terms z xX eee a 4 ss are ze X SS eS 1000 r1000 eh A 2% 5 — 762% me paar. as ee as — See we i Gnk saa LST ooo)? 1000 = ee ery 000 1000 2 X 1000,000 te 24 2X 1000 5762—5 762% ak 48,000% 576% — 576zx 1000 2X 1000 2X1000,000 — 2X 1000,000 2 X 1000,000 48,000z—5762-+.5762% _— -47,4242%-4+5762% _ 23,712z-+ 28822 Phere 23,7122 288zz 2 X 1000,000 TSK IOQ0,000 = '*~"1600,000 1000,000 will be = 0.002,084,180, 0045486,389, and confequently 23.4128 + 28822 will be (== 1000,000 X 0.002,084,180,004,486, ROOF eave 2084. 189 dei cing 389, and (dividing all the terms by 288) zz + 8. 3°93,929,cC, axe will be , of the foregoing feries to be equal to the whole feries, and con- §702.36;,7390,1 26, ae aig (adding 41.166,606, é&c}* to both fides) we fhall have zz + 82. 31333, Bc; X Bf 41-166,666, Sec|? (= 7.236,736,126, 688 + 417106, 566, ae p47 <2 WO> 736,126, 688 + 1694. 89413 8955555550) &. 1701.93.15 12.5,68242 44, and confequently z + 41.1 66,666, fo ory Gas VY 1701.931,125,682,244) = 41.254,467, and z (= 41.254,467 — 41.166, 666) = 0,087,801. Therefore o. 087,801 is a fecond approximation to the true value of z in.the infinite equation fet down at the end of art. 149, or in the equation 1 + 1" = 1.002,084,180,004,486, 389.. TEA 153-/Lhe arithmetical mean between 0.086,840, and 0.087,801 is 0. 087 320. But the true value of z is much nearer to 0.087,801 than to 0,086,840. Therefore it muft be greater than the {aid arithmetical mean between them, to wit, 0,087,320, and probably not much) lefs than 0.087,801,, the latter and greater of the two foregoing approximations to it. It feems reafonable there_ Vou. Il. y aoe : fore 330 wae. Dil S$. ¢. 0 UCR.S.E CO NiO: Ee RN ae fore to conjecture that it is equal to 0.08%7,7. We will therefore fuppofe it to be equal to 0.087,7, and confequently fuppofe x, or 97 + z, to be equal‘ to 97-0877, and fhall proceed to make a trial whether this number is greater or lefs Py . . than the true value of # in the equation 1 ++ — 03 = 10, by raifing the bino-_ ee mial quantity 1 + = eke 5 to the 97.0877th, or th power by means afl the binomial theorem. 154. Now this has been done above in art. 136, and it has appeared that the faid power of the binomial quantity 1 -+ <= is equal to the mixt number g. 999, 9791282219557 346, &c, which is very little lefs than 10.. There- fore the number 97.0877 muft be very httle. lefs than the true value of the in- co : ion I 4 = ELOe dex x in the equat ay as Let y be the excefs whereby the true value of x exceeds 97.0877. Then will 97.0877 + y be = *, and 1 + 4 |* will be = 1 + se anit confequently 1 + a will be = 10, Buti + - ash a OO 7 Tae 2 8 * ieee” 7 x1 + SEP = sodhic wssiyodere eee fe x 1 + 24}, Therefore 9.999,97952825720,950,507:346, &e X 1 + 2 10.600,000,000,000,000,000,000, will be — ro, and confequently 1 see Zt will bet==— —_— ; q y ‘S 9-999979:282 120,950,507:346, > = 1.000,002,071,732,197,014, &c. But, by the binomial theorem, 1 +- ak ? aa 1000 ett: ta a ae sad: is == the’ tnftniteferies <2 919 y 2% ts yes Xi eee Ore x OC Bis 240 mal — B Sart a. 5g oa 3 Se ey 1 OPAL oe. 1000 a ok "x 3 1000 ‘+ Jers i $ 4 lye ny —_ obi gps Aas ee eA Lis WON 282 Sone 2 24 $4 x : Kier + y X ; ri x : x a + &e = chy JY a mae yt hy = 3yy-b2y 24.|3 J — 693 + Ly — by 7 2 J x 1000 a ts 1000 TF 6 x 1000 -F 24 x 24 1 J DPI Miah as x W274 yuk 1+ Boy = 25597 + 27499 — 1207 vi 1000 120 1000 720 any + &c = (becaufe y is lefs than 1, and confequently yy 1s lefs thany, © 1000 and y? than yy, and y* than »%, and every following power of y lefs than the next preceeding power of of it), To 0h ee = ee power 1000 1000 x 24)" — Jal Bees Lae siptinictl Y Heol 250 Soy EIST ey NOP? ok OCalt Toco 1000 120 1000 THE BYN OM 1SACLA T HE ORE M 338 ol shame) . me 720 ¥ob0 ey &c. Therefore this feries 1 +y x aod? anne 24 24) eter 24 )* 1000 2 1000 + 1000 1000 rT cc will be = 1.000,002 ace Prataboned and Edesanentti (fubtracting 1 from both fides) the feries y x — - (2 POF eka j i EEE ¥ fa 4 satin ie ac 10900 2 19000 6y — 1 1yy+ by3 —y* x 24 24 1000 equation we muft now endeavour to refolve. Ler &c will be = 0.000,002,071,732,197,014. This 155. Now, if we fuppofe the firft term, y x —, of this feries to be equal to the whole feries, and confequently to the abfolute term 0.000;002,071,732, 197,014,-we fhall have y xX 24 (= 1000 X 0.000,002,071,732,197,014) = 0.002,071,732,197,014, and confequently y (= oes) = = 0.000, 4 086,322,1. Therefore x, “or 97.0877 + ¥, will be = 97.087,786,322,1, of which the firft feven figures 97.087,78 are exact, the more accurate value of # _ being (as we before obferved in art. 150) 97.087,787,353,856,001,437. 156. In the next place we will fuppofe the two firft terms, y x —— -~* x — LY’ of the foregoing feries, to be equal to the whole feries, and confequently to the abfolute term 0.000,002,071,732,197,014, and will refolve the quadra- tick equation thence refulting. 24 yyy 24 2 X 1000 aa —_- ——S ——— j cates os Now y X tddo | ase 24} on aS hicco %. kaeos x 576 = 48,0007 [szor— sr — — "48,0009—'576y + 576yy panes 1900,000 ~—-2 X 1000,000 2 X 1000,000 2 X 1000,000 2 X 1000,000 2357! 2y + 288yy_ Therefite 222” 23, 712y + 288yy 1000,0G0 1000,000 - -and confequently 23,712y + 2.88yy will be (= 1000,000 X 0.000,002,071, 432,197,014) = 2.071,732,197,014, and (dividing all the terms by 288) 82. 3332333, & XY + yy will be = 0.007,193,514,572. ‘Therefore, if we add 41.166,660, &c\*, or 41 + Bi » to both fides, we fhall have 41.166,666, &c}* + 82.333,333, &C X ¥ + IY (= 0.007,193,514)572 + 41 + =| = 0.007, 193,514,572 - iaglh Pater ts, oe oe it - x ae 0.007 5193,514,572 -f- 1681 + 41 X — + 0.166,666,666,666, &c x F = 0.007319315145572 + 1681 + 13. 666, 666,666,666, + 2185556 866,565 = 0.007,193,514,572 1694.666,666,666,666, + 0.027,777, ae 77F* = 0.0075193,514, 572 0 1694.694,444,444,444) = 1694-701,637,959,015.. Therefore 41,166,666, Zale 2 666,, will be = 0.000,002,071,732,197,014, 332 aA DASE DT RSE CON CERNING 666, &e + y will be (= 1 1694-701,637,959,015) = 41.166,754,037, and y will be (= 41.166,754,037 — 41.166,666,666) = 0.000,087,371, and con- fequently *, or 97.0877 - y, will be = 97.087,787,371; of which number the firft nine figures 97.087,787,3 are exact, the more accurate value of x being 97:0875787,353,850,0015437- This is a very confiderable degree of exactnefs. 157. In order to obtain the value of y to a {till greater degree of exactnefs, it will be convenient to fet down the infinite equation that is obtained in art. 154, ina different manner, as follows; to wit, 24 y cP a be A aaeeitam meee t tt 1000 2 1000 Wy 24 + -—- 2 1000 <= 0.000,002,0715732,197,014. Now, fince y is a very {mall are e I fortiori, \efs than 0.0001, or —— 10,000 + 2x Bey & x Al + &e — Bx BY 4 My 28h we +x Ht Sy 2h tec + x Sh) — &c quantity (being lefs than 0.000,088, and, @ ), it follows that yy and y? and »* and all the following powers of y will be very {mall in comparifon of y; and confequently, if we omit all the terms in the foregoing equation which involve yy, or any higher power of y, the feries which forms the left-hand fide of the faid equation will not be much affected thereby, but the remaining terms of it (which are thofe that involve only the fimple power of y) will be very nearly equal to the abfo- lute term 0.000,002,071,732,197,014; that is, the upper line of terms alone, 24 1600 malig 2 24 1000 to wit, y¥ X I 20y nv AE? aa by 2+ ase: Se athes Tages apantene 24 \° x ra + &c, will be very nearly equal to 0.000,002,071,732,197, 24y 1290 5 x | 1000 720 be Sa es a4 J 24 \? y 24 | : e of terms is = Neth lag TY, ide pam Tet 3 "Flee 014. But this lin ms 1s = y X ——= — = mare Poa® : reser a 24 \4 y 24 \5 ye 24 |§ s : 24 I 1000 a 5 1000 6 x 1000 + &c= Paes the feries 1000. 43 x anals I 34. cee Ee 24 29 24 |S 1 24 \ ay Path 1000 3 robe a 1000 i 5 Xx a 6 x por + &c = (ifwe put A for the firft term =. of the faid feries, and B for its fecond term = x 2 . . 24] , and C for its third term pe 1000 3 = i andD,E, F,G, H, &c, for its fourth, 1000 fifth, fixth, feventh, eighth, and’ other following terms refpectively) y x the Ter Ayer 3 6 be eae 7 24 1000 A 2 feries —~ — 1000 a a rae Ve! 1000 +—K x 28 ASCE ANT ee 1000 4 1000 5 1000 PY CRY hak tig v bid cinta 4.9/5 ae 8 1000 9 1000 10 1000 > ee: sae 2 SUN OM t 2b Fr ASE oO ROM, 333 0.024,000,000,000,000,000,000, A = 0.000,288,000,000,000,000,000, = 0.000,004,608,000,000,000,000, — ©.000,000,082,944,000,000,000, 0.000,000,001,592,524,800,000, 0.000,000,000,031,850,496,000, 0.000,000,000,000,6 55,210,203, 0.000,000,000,000,01 3,759,414, 0.000,000,000,000,000,293,534, 0.000,000,000,000,000,006,222, 0.000,000,000,000,000,000,1 35, 0.000,000,000,000,000,000,002, rat een 0.024,004,609,593,180,303,872 x ¥ — 0.000,288,082,031,864,261,638 x y = 0.023,716,527,561,316,042,234 X Je Therefore 0.023,716,527,561,316,042,234 X ¥y will be very nearly equal to the abfolute term 0.000,002,071,732,197,014; and confequently y will be ery nearly equal to —2.2002007:07%:7329197,014 i i nets ty 0.023,716,5275561,316,042,2347 OF 0.000,087,3535943,011 Therefore x, or 97.0877 + y, will be = 97.087,787,353,943,0113 of which number the firft eleven figures, 97.087,787,353, are exact, the more accurate OB “ w PAP TOADS +l ert ti+ S value of « in the equation 1 + —4)|* = 10 being 97.087,787,353,856,00I5 Too 437- 158. We have now obtained the value of the index » in the equation 2 x ‘ . . — = 10 exact to eleven places of figures. And, if we wifh to ob- tain it to a {till greater degree of exactnefs, we need only retain a few of thofe terms of the compound infinite feries fet down in art. 157 which involve the {quare of y, and which conftitute the fecond horizontal row of terms in the faid compound feries. Now the firft five terms of the faid fecond horizontal ; } I 4 ° 2 } row are 4- pd x ee coats 3 x Ta 4- mie x Et — boy x was. + 2 1000 6 1000 24 Iooc 120 1000 2749y 5, 24 24° aeisth§ - 24 43 6 ° : I » which are equal to yy x the feries — x —~ 720 1000 2 1000 6 1000 II 24 |4 50 24 \$ 274 34°45 POO Ir ; ate 24 se nee | palin pin et we POS | Eh 24 3 1000 120 s 1000 aC 720 s 1000 = 4 x the feries 2 Ree as 3 at 4 5 s 137 at eo _— sQZ —_—_ Fy ——_—— . Steers, x -O2 — x = x 0.024)) + 7 x 0.024| me 0.024]§ + 6 0.024| yy x the Baer A I II feries es 0.600,576 — pee 0.000,013,3824 + = re D-COC DR Sat a7) a. ot X 0.000,000,907,962,624 + as X 0.000,000,000,031,850,496 = yy x the feries 0.000,288, 334 A D-T 8S COU RS ECO NOC 2 8 FN 1 Che — 0.000,006,912,000,000,000,000, + 0.000,000,1 52,064,000,000,000, — 0.000,000,003,317,760,000,000, © + 0.000,000,000,012,120,883,200, ] 0.000,288,152,076,120,883,200 — yy X 0.000,006,915,317,760,000,000 = XY X 0.000,281,236,758,360,883,200. Therefore the two upper horizontal lines of the compound infinite feries fet down in art. 157 are = 0.023,716,527, 561,316,042,234, xX ¥ + 0.000,281,236,758,360,883,200, x yy; and con- fequently thefe two terms 0.023,716,527,501,316,042,234 x y + 0.000,281, 236,758,360,883,200 x yy will be very nearly equal to the abfolute term © 000,002,071,732,197,014. This quadratick equation we muft now endea- vour to refolve. This equation may be moft conveniently refolved by approximation, by fub- ftituting, inftead of y, in the quantity 0.000,281,236,758,360,883,200 x yy the value_of y derived from the fimple equation 0.023,716,527,561,316,042,234 0.000,002,071,732,197;014 0.023,716,527,561,3 16,042,234? Or) 0.000,087,353,943,011. We fhall then have yy = 0.000,087,353,94\* = 0.000,000,007,630,710,833,523,6, or (dropping the laft feven figures), ©.000,000,007,630,710, and confequently 0.000,281,236,758,360,883,200 x XY (= 0.000,281,236,758,360,883,200 xX ©.000,000,007,630,710) = 0.000, 000,000,002,146,036,144,391,975,043,072. Therefore 0.023,716,527,561, 316,042,234 X y + 0.000,000,000,002,146,036,144,391,975,043,072 will be (= 0.023,716,5275561,316,042,234 x y + 0.000,281,236,758,360,883, 200 X Jy) = 0.000,002,071,732,197,014, and confequently (dividing all the terms by 0.023,716,527,561,316,042,234) y ++ 0,000,000,000,090,487 will be = 0.000,087,353,943,011, and y will be = 0.000,087,3 535943,011 0.000,000,000,090,487 = 0.000,087,353,852,524. Therefore x, or 97.0877 + y will be = 97.087, 787,353,852,5243 of which number the firft thirteen figures 97.087,787,353, 85 are exact, the more accurate value of x being (as we have before obferved) 97-08 7,787,353:856,001,437. We have therefore now obtained the value of 0.000)288,000,000,000,000,000, —I xX X Y = 0.000,002,071,732,197,014, which is = ( #, or the index of the power of the binomial quantity 1 -+ —t in the original equation 1 + 24" = 10, exact to thirteen places of figures. ; Q. Bet Es : 24 \# . . 24 159. The ratio of 1 + = > OF 10, to 1 is tothe ratio of 1 + ssop tO 1 as x isto 1, and confequently as 1 isto the fraction =; fo that, if 1 be taken for the reprefentative of the ratio of 10 to 1 (as it is in Briggs’s Syftem of Loga~ rithms), the fraction — will be the reprefentative of the ratio of 1 + —. tO I, that THE BINOMIAL THEOREM. 335 ghat is, in other words, the fraction -- will be Briggs’s logarithm of the ratio of 24 f a ; o aa 24 — : : the ratio o ark i ee tO T Therefore Briggs’s logarithm of ratio of r + =~ to ty 1.000,000,000,000,000,000 or of the ratio of 1024 to 1000, will be (—= ———> ——-——- J) “9.010 Z : ( 97-087, 7875353985 255 24 : 299,956,639,812. Therefore the logarithm of the ratio of 1¢24 to 1 will be (= log. + log. = = 0.010,299,956,639,812 + log. os 1000 I 1 956,639,812 + 3) = 3.010,299,956,639,812; and confequently the loga- rithm of the ratio of 2 (which is the 10th root of 1024) to 1 will be (= .010,299,956,6 . 32910»290:950-039,812 ) = 0.301,029,995,663,981,2 ; or, according to the common == 0.010,290% 10 mode of expreffion, the logarithm of the number 2 will be = 0.301,029,995, 663,981,2, Oss Hoole This value of the logarithm of 2 is exaé to fifteen places of figures, and errs only in the 16th, or lait, figure, which fhould be an unit inftead of a 2, the more accurate value of this logarithm (according to Mr. Abraham Sharp’s com- putation) being 0.301,029,995,663,981,195,21357 385894,7245493- _ 160. As the folution of the foregoing problem confifts of a great number of fteps, which, for the eafe of the reader, have been fet forth diftinély and at confiderable length, it will not be amifs to take a fhort review of all the fore- going proceffes, and of the feveral gradual approximations, to the value of x, (or of the index of the power of the binomial quantity 1 + — in the equation r+ + * = 10) which have been obtained by means of them. 4A review of the feveral feps of the foregoing refo- lution of the equation 1 + a Se 1. GHA computation of the logarithm of 2. 161. The firft ftep towards finding the value of the index w in the equation 24 \% : 24 — \ and the quantit + Sool = 10 was to exp quantity I + a by means of Sir Ifaac Newton’s binomial theorem, by which we obtained the : 24 AY 24 ) x3 —34e+ 24x 24 {3 ho led or he 1000 ti 2 x 1000 AE 6 x 1000 WV - : ° ° into an infinite feries x4—6x3+1147—6% 241|* #5 — 10444-3543 — 50a7 + 244 24 |s x 240 x 1000 R 120 1000 PL &e = ao and confequently (by fubtracting 1 from both fides) the equation x x —L + ; 1000 t, 336 A’ DISCOURSE CONCERNING xX a4, \* rs #3 — 3 Hx + 20 24 \3 4 a4 — 643 + 1147 —6x 24 |* + . 1000 6 1000 24 1000}. : x5 — 10x4-+ 3643 — SOxn + 24% 24 |5 Ly a x — -{. &c ~~ Qo 120 1000 162. We then fuppofed, firft, one term, then two terms, then three terms,. and laftly, four terms, of the feries which forms the left-hand fide of this laft equation, to be equal to the whole feries, and confequently to the abfolute term 9, and we refolved the feveral equations refulting from thofe fuppofitions. By refolving the fimple equation refulting from the firft of thefe fuppofitions, : - } ony eo ’ to wit, the fimple equation x X —“= = 9g, we found « to be equal to 375; which was therefore our firft approximation to the value of x in the original . Zz : . . . . equation 1 + i = 10. This approximation is very wide of the true value of x, being more than three times as great. From the fecond fuppofition, to wit, that the two terms x x —L 4 4% »& ; y 4000 2 24 2 . —— | Were equal to the whole feries, and confequently to 9, there refulted the 100 quadratick equation xx + 82.333,333, &c X « = 31,250; by the refolution: of which « appeared to be = 140.34; which was therefore our fecond approxi-. ‘ : ei" ° 4 24 \4 ; mation to the value of x in the original equation 1 -+ —=- ==)fo:/4ee ap-. proximation is much nearer to the truth than the former, but yet is confiderably too large. From the third fuppofition, to wit, that the three terms » X = + a ° ad : ; . 4 x aie ie ee x 24] were equal to the whole feries, and confe- quently to 9, there refulted the cubick equation «7 + 122%” + 10,293.666,. 666, &c X « = 3,906,250; by the refolution of which in a grofs manner, by conjecturing x to be equal, firft, to 100, and then (upon finding it to be greater than 100), to 110, we found it to be fomewhat lefs than 110; which was there-- fore our third approximation to the value of » in the original equation: 24 \’ Cees el pee 10. at 1000 And from the fourth fuppofition, to wit, that the four terms x x a a 1000 2 4 a3 — 300 -b 2% 24)\3 | «4t—6*3+1144—6% 24 \4 Pad psig 6 x 4 gle iS x —-| are equal to the whole feries, and confequently to g, there feinhed the biquadratick equation «* + 160,666,666, &o°x x* + 20,344.333,333, &c KX we + 1571§;005.005. ir, &c X # = 6§1,041,666.666,666, &c; by the refolution of which in a grofs manner, by a conjecture and trial, we found « to be fomewhat greater than 100; Which was therefore our fourth approximation to the value of » in the . ° . 24 * Jue, * original equation 1 + oh = ive 163. Having ' : Tt Ho BIgoNcOoMT ALATDEHEQRE Ms 337 163, Having thus obtained the numbers 375, 140, 110, and too for our . / . . 2 4 four firft approximations to the value of « in the equation 1 + = == 10, we obferved that the difference between the fecond and third approximations, to wit, 140 and 110, was 30, and that the difference between the third and-fourth ap- proximations, to wit, 110 and 100, was only ro, or one third part of the pre- ceeding difference; and we were thereby led to conjecture that the difference between the fourth approximation 100 and the true value of x in the equation otaeea = 10 would probably be lefs than a third part of the laft difference to, or would be nearly equal to 3, and confequently that the true value of » in “ 244 { the equation 1 + ——-) = 10 would be nearly equal to 100 — 3, or 97, or that 97 would be a fifth approximation to the faid true value of x. And we then tried whether the faid true value of » was greater or lefs than 97, by raifing the . . bi 24 . . —\tot - binomial quantity 1 + —- to the g7th power by means of the binomial theo rem; and we found the faid 97th power of 1 + = to be equal to 9.979,201, 547,6735599,050, which 1s fomewhat lefs than 10; whence it followed that 97 mutt be fomewhat lefs than the true value of » in the equation 1+ 2} r, to. We then multiplied the number 9.979,201,547,673,599,050 (which is equal to the 97th power of 1 + = into I +} oa, or 1.024, and found the product to be == 10.218,702,384,817,765,427,;200, which is greater than 10, and we therefore concluded that the 98th power of 1 + = was greater than 10, and confequently that the true value of w in the equation 1 + a eas was greater than 97, but lefs than 98. 164, We then put z for the unknown difference by which the true value of x 24 in the equation 1 een = 10 exceeds 97, fo that «” was = 97 +2. And 24 197+2 -__ set WS ge wethereby had 1 + a aah 24 pisecer Bay 24.\97+% - 24 \97 24 \%, 24 \97 But 1 + io 1S Sepa OF 4 xt 4 =| 3 andi + | had been found to be = 9.979,201,547,673,599,050, &c. Therefore 9.979, 244g" “eo 24 |97 24 | 201,54.750735599,050 x I + = 1S es I+ pee xX 1 + aay — . = 1 + se ata = 10,'/and confequently 1. + +" 1Soeat I000 10,000,000,000,000,000,000 ; eee | SS T:002,094,1 $0;0048080;98 0.1. Ahdi thus we eb- 9-979»201, 547467 35 599,05 we. »004, 900454.0033909 tained a new equation, in which the unknown index 2 of the power of the bi- Vou, Il, 2X nomial 338 A’ DISCOURSE CoN'CERNING ° . 2 ° . . . . ory nomial quantity 1 + + is lefs than an unit, inftead of the original equation i+ value of « in the original equation 1 + TTH EY BIN OM TO ACL” THE Of RTE. Me 339 3333333 &e X 2 = 7.236,736,126,688; by the refolution of which we had % = 0.087,801, and confequently ~, or 97 + 2%, —= 97.087,801, of which number the five firft figures, 97.087, are exact. Therefore ore was a wv — 10. feventh approximation to the value of v in the equation 1 + an 167. We then found the arithmetical mean between the two laft values of z, to wit, 0,086,840, and 0.087,801, which was 0.087,320; and we obferved that this mean muft be lefs than the truth, becaufe the fecond value of z, to wit, 0.087,801, muft be much nearer to its true value than the firft value of it, to Wit, 0.086,840. And therefore we conjectured that the true value of z (being greater than 0.087,320, and, probably not much lefs than 0.087,801) might be ey nearly equal to 0.087,7, and confequently that the true value of #5 OD 97 + 2, might be very nearly equal to 97.087,7. And thus we obtained 97.0877 for an eighth approximation to the true value of « in the equation 2 1 + ay = 7, 1co0o 168. We then dropped all further confideration of the equation z x Z— VS 2%— aaa 5 “24 | fa See x 24 i Rian os x tl + 1000 — a O. 2S ae ae &c, and ede a me a the exactnefs of the laft value of « obtained by the foregoing procefles, to wit, 97.0877, by raifing the binomial quantity 1 ++ i. to the power of which 97.0877 is the index; which was done by the help of Sir Ifaac Newton’s binomial theorem. And we found that the faid power of 1 + a WAS = 9.999,979,282,720,950,507,346, which is very little lefs than 10; and we thence concluded that 97.0877 muft be a ° . > Zz e very little lefs than the true value of w in the equation 1 + zal ZT 169. We then fuppofed x to be = 97.0877 + y, and confequently po Zep 087747 to be = 10. 1000 Mon Mad (ayo Tory we 24 seh oe 77 a, Then, fince 1 +- —— ore is eecat 1 4 — x1 += we had 24 |97-0877 4\y 24 |97.0877 tick 24 XI+ we = 10, and confequently (becaufe 1 ++ zt has been found to be = 9.999,9793282,720,950,5075340) 9-999,979282,7203 10.009,000,000,000,000,000,000, 6X1 24 )7 = 10, and 1 aay = acai! Le ine 95935073340 XI + = eden 9-9992979s2825720;950,5075340 == 1.000,002,071,73 oft 97,014, &c. 170. Having thus obtained a third equation 1 + = 732,197,014, &c, in which y is much fmaller than z in the former equation, FAs we = 1,000,002,071, 340 A MDIIS £O VW RR SIE CONCERN ING “ we proceeded to expand 1 + - ie into an infinite feries by means of Sir Ifaac Newton’s binomial theorem, ms pres obtained the equation 1 y x —& ~ 1000 Uma 24 ] ay—3yy +8 2 “ 1000 x 6 “ 7000 — Pewee iol 24 1000 4 24y — Sopy + 35 y3— 1Oy4-f yS y, 24 \5 OT 5 24 120 1000 1000 + &c = 1.000,002,071,732,197,014, &c, and (by fubtracting t from both y— 3 fides) the equation y x 24 - ie a Sighs Je = ae x aa 10090 ‘ 1000 Be g —11yy + 6y3 — y4 24 24y — Soy + 3y3— toys $99 a ot 24 1000 120 1000 ’ 120y—274yy +235 y3 —S85y*t+ 16 y5 —y? 24 aks Pa ipod 2 : as: 720 1000} =: - — 0.000,002,07 73 5 97> 014, &c. 171.° We then proceeded to approximate to the value of y in this equation, by firft fuppofing the firft term, y x —, alone, of the feries y x 1000 Pp x a4 +- &c (which forms the left-hand fide of this equation), to be equal to the whole feries, and confequently to the abfolute term 0.000, 002,071,732,197,014, &c; and, fecondly, by fuppofing the two firft terms, yx = — us x —+ i of the faid feries to be equal to the fame quantity, and refolving the equations refulting from thefe fuppofitions. From the firft of thefe fuppofitions we had the fimple equationy x 24 = 0.002,071,732,197,014, &c; by the refolution of which we had y = 0.000, 086,322,1, and confequently « (= 97.0877 +- y) = 97.087,786,322,15 of which number the firft feven figures 97.087,78 are exact, the more accurate value of « being (as we have before obferved) 97. 087578753 53,850,001 ,437. Therefore the number 97.087, clea is a ninth approximation to the true value of « in the equation 1 + al" = =<60, 1000 And from the fecond of thefe fuppofitions there refulted the quadratick equa- tion yy + 82.333,333, &c x y = 0.007,193,514,5723; by the refolution of which we had y = 0.000,087,371, and confequently ¥ (= 97.0877 +7) = 97:087,787,371; of which number the firft nine figures, 97.087,787,3, are exact. This number 97.087,787,371, is therefore a tenth approximation to the true value of x in the equation 1 4- 24)" wall (a 1000 172. We then, in order to obtain the value of y toa ftill greater degree of ; ° . : 2 exactnefs, had recourfe to a different method of refolving the equation y x = | 4 : Sis ’ : 2 in FU-Ree BIN OoMIMA. LL > EEO R EM. RAL 2 < 3 3 6y— idigl a. o' —p2 Batt or 27 ay) poate: = TUyytby3 —y* 24\4 4 eS 1000 1000 24 1000 ©.000,002,071,732,197; ue &c; which was grounded on the omiffion of all the members in each term of the feries that involved either yy, or y*, or y*, or any other power of y, except the fimple power, or y itfelf: by which means the ; ; . : : ; 24 {aid equation was ee into the peeled ‘sete ee tO. Wiksty ix Toa y ey 24 |* 24 \3 524\t Ph lis «sph E20 24 |6 2 1000 1000 1060 I 120 1000 7 20 1000 24 y 4 |? y + &c = 0.000,002,071,732,197,014, &c, ory X ——~ — — xX at : ay Spe heyg EE yt JH} myatishe eee me | * 1000 4 x 1000 a 5 x 1000 6 X 1000 + &c — 0.000,002,07T, 24 I 24 |? I 732,197,014, &c, a uke: ame 2 4 A x 24 l I 24 ork 24 \5 I 24,\° = 1oool 4 os rool | 5 "s a EtrGwr «Too mn &c = 0.000,002,071, ; 24 . ; 732,197,014, &c, or (if we put A for the firft term 1 X — of this feries, and B for its fecond term -- x A and C, D, E, F, &c for its third, fourth, fifth, fixth, and other following terms, refpectively), y x the feries 1 x +. e090 erase libaticp ye tt alg Ye al ooh} 24 Sim sg 2 1000 3 1000 4 : 1000 5 1000 6 ey = Lily ane wal i 2 SARE OPEC SR Ore b dolore) ot 7 Fx 1060 Fe G 1000 T 9 H x 1000 antes s 1000 + 1k 24 11 Beer K X ooo a X Fog F KE = 9.000,002 ey ite &c, We then computed the value of the faid infinite feries 1 os Sh spre oe 2 AX +. ex — 2 C x ++Dx 3b Te By 2: + &c,. and 1000 4 1000 5 1900 s| 1000 Sand it to be = 0.023,716,527,561,3.16,042,2345 “wehich gave us the fimple equation y X 0.023,716,527,561,316,042,234 = 0.000,002,071,732,197;014, &c, by the ‘refolution of which we had y (==_ ——ORORO7 137 TOD OT 0.023,716,527,561, 3 10,04 042,234 0.000,087,353,943,011, and confequently x (= 97.0877 + y) = 97. eye 987,353,943,011 ; of which number the firft eleven figures, 97.087,787,353, # are exact, the more accurate value of « in the equation. 1 ++ 84) = 10 being (as we have before obferved) 97.087,787,353:856,001,437- This number 97.087,787,353,943,011, is therefore the eleventh cee ae tion to the true value of x in the equation 1 + ay waht qf 173. And, laftly, to obtain the value of y to a ftill greater degree of exact- uae we retained in art. 1 58 the-five firftiterms of the infinite feries fet down in ) arte 342 AV FDAT Sib BS 10+ WS ClLOON CME CREN “EAS art. 157 and 170, that involved the fquare of y, to wit, the five terms x 24 |? 3y 24.\3 , 1 24 |* Ow ., 24 )% | 274y7 24) 24 Nia Pie +2 x 24 Barr gat + 2B x 28) whicls ee 424°? 24.4% II 24 \* fe) are equal to yy X the feries ey x <4 _ ~ Pe 4 Gs i x 24) _ — x 24 |$ 274 ty — : 6.728. 260.88 re + se x foo]. — X 0.000,281,236,758,360,883,200. And hence we obtained the quadratick equation 0.023,716,527,561,316,042,234.X y + 0.000, 26 1,236,758,360,883,200 X yy == 0.000,002,071,732,197,014, &c; which (being refolved by approximation by fubftituting, inftead of y, in the quantity 0.000,281,236,758,360,883,200 X yy the value of y before obtained by the refolution of the fimpie equation, to wit, 0.000,087,353,943,011) gave us ¥Y = 0.000,087,353,852,524, and confequently « (= 97.0877 + ¥) he 087,787, 353,852,524; of which number the firft thirteen figures, 97.087, 7875353955 are exact. This number 97.087,787,353:8525524 is therefore the twelfth approximation 24 \% 1000 to the true value of « in the equation 1 + =< STs 174: We then divided 1 by this laft, or twelfth, near value of *, to wit, 97. 087,787,353,852,524, in order to obtain the value of =, or the logarithm of the ratio of 1 + + to 1 in Briggs’s fyftem; and we found the quotient to be == 0.010,299,956,639,812. And hence it followed that the logarithm of 2, or 1024 loo, 1024 1000 . . I I of the ratio of 2 to 1 (being = 7; log. —- = 5 log. = + x log. — I~ I I 1024 I RVATS 1024 fee 1.024 hd 10 ©" 1000 a Toh pe teil log. roc0, 1 03 = to OS: a, ee Sages 7 log. 1 + ~4 + 0.3 == X 0.010,299,956,639,812 + 0.3) would be = . ) 0.3 + 0.001,029,995,663,981,2, OF 0.301,029,995,663,981,2 ; which is exact to 15 places of figures, the more accurate value of that logarithm (according to Mr. Abraham Sharp’s computations) being 0.301,029,995,663,981,195, 213,738,894,724,493- | End of the review of the feveral fieps of the foregoing refolution of the equation 1 + = = 10, and computation of the logarithm of 2. ST a RT RE a ee Le Ie Wee LIE NOL INCI LIENS A SCHOLIUM, 175. The foregoing method of computing Briggs’s logarithm of 2 is cer- tainly fomewhat laborious, but much lefs fo than the methods ufed for the fame purpofe by Mr. Briggs himfelf, which required many very long stray of the quare~ Xx wal 4B BEN GO M"h. AL T H'E OR EM. 343 fquare-root. For the difficulty of performing the operations that were neceflary in thofe methods was fo great, that (according to what Mr. Euclid Speidall in- forms us (fee above, page 73) he had been told) it was the work of eight perfons for a whole year to compute the logarithm of 2 by thofe methods exaét to 15 places of Jigures, or to the degree of exactnefs to which is been obtained in the foregoing articles. This affertion of Mr. Speidall feems, I confefs, a little ftrange. Yet, as he publifhed his traét on logarithms (which has been printed above in the for- mer part of this volume) fo long ago as in the year 1688, it feems probable that in his youth (perhaps, about the year 1660) he might have converfed with fome old men who had been acquainted with Mr. Briggs himfelf, who publifhed his Arithmetica Logarithmica in the year 1624, which was lefs than 40 years be- fore that time; and this feems the more likely to have been the cafe, as his father, Mr. John Speidall, was an eminent mathematician, and had very much cultivated the, at that time, new invention of logarithms; which muft have given both him and his fon an opportunity of hearing many remarkable parti- culars relating to them. We may further obferve that the foregoing method of computing logarithms by the help of the binomial theorem, and Mr. Briggs’s methods of computing them by repeated extractions of the {quare-root, are equally founded on the pure and genuine principles of arithmetick, without any reference to the hyperbola, or the Jogarithmick curve, or any other geometrical figure, and alfo without any recourfe to the doétrine of infinitefimals, or of fluxions, or of the limits of ratios, or in general, of the arithmetick of infinites in any of its modifications; which 1s, in Dr. Halley’s opinion, the proper way of treating this fubjet, and the way in which he boafts (though without being fufficiently authorized in his pretenfions), that he himfelf has treated it in the foregoing difcourfe reprinted above in this volume in pages 84, 85, 86, 87, 88, 90, and o1. ; a ee ee CLO UNC CeLaUlS IO UN 176. I have now completed the inveftigation of the famous binomial the- orem in all the cafes of fractional powers; which was the propofed fubject of this difcourfe. This, however, is but a part of that important and moft com- prehenfive propofition. For it is found to be true likewife in the cafes of nega- . ° : ° . ° — 7% tive powers, both integral and fractional, that is,.in the cafes of 1 + | -, mm I See ge I es Hi ite 7 « the knowledge of thefe and 1 + sor of ye and But, as no g cafes is not neceffary to the underftanding any of the foregoing methods of computing logarithms, I fhall not on this occafion enter into any inquiries concerning them. And therefore I here conclude what I meant to offer to the reader’s confideration concerning the binomial theorem properly fo called. But, as the theorem concerning the fractional powers of @ refidual quantity, fuch as I=, 344. A DISTCOUVRSE Conecrrwimneey &c. 1 — x, is very nearly related to the foregoing theorem concerning the fraétional powers of the binomial quantity 1 -+- x, infomuch that it is ufually confidered ~ as a branch of it;—and, as the faid refidual theorem, in the firft cafe of it, or the cafe of the wth root of the refidual quantity 1 —.«, is made ufe of (as welk as the binomial theorem) in the fAveftigations of fome of the foregoing methods - of computing logarithms ;—I fhall now proceed to fhew how we may derive from the theorems above demonftrated concerning the roots, and the powers of the roots, of the binomial quantity 1 + %, the like theorems concerning the roots, and the powers of the roots, of the refidual quantity 1—wx. But this fhall be the fubjec&t of a feparate tract. And therefore I here conclude this difcourfe concerning the roots, and the powers of the roots, of the binomial ‘quantity 1 + *. End of the Difcourfe concerning the Binomial Theorem in the cafe of Frattional Powers. A DISCOURSE Fi. Lae a A Dee's Cel OVW RISE Cor N.C Eek Wear NG Sm ISAAC NEWTON’s RESIDUAL THEOREM, OR THEOREM FOR RAISING THE POWERS OF THE RESIDUAL QUANTITY 1~—%, IN THE CASE OF FRACTIONAL POWERS, OR POWERS OF WHICH THE INDEXES ARE FRACTIONS, By FRANCIS MASERES, Esa. F.R.S. CURSITOR BARON OF HIS MAJESTY’S COURT OF EXCHEQUER. Art. 1. JN the foregoing difcourfe we inveftigated the feriefes that were equal to the wth root of the binomial quantity 1 + and to the mth power of its wth root. In the prefent difcourfe we are to inveftigate the feriefes that are equal to the wth root of the refidual quantity 1 — x, and to the mth power of its wth root. Now thefe feriefes may be inveitigated by the fame me- thods which were employed in the foregoing difcourfe to inveftigate the feriefes I m2 which are equal tor + x\# andi + xc) : but they may likewife be derived from thofe former feriefes (obtained in the foregoing difcourfe), by a juft and legitimate train of reafoning, with much lefs trouble than would be neceffary to the difcovery of them by a new application of all the methods of inveftigation ufed in the foregoing tract. And therefore I fhall, for brevity’s fake, have re- courfe to this derivative method of obtaining them, rather than to the methods employed in the foregoing difcourfe. And, firft, ‘I thall confider the 7th root of the refidual quantity 1 — *, and endeavour to fhew that it is equal to an in- finite feries confifting of the very fame terms as the feries which is equal to the ath root of the binomial quantity 1 + #, but with the fign — prefixed to all the terms after the firft term 1, inftead of only the third, and fifth, and feventh, and other following odd terms of the feries, as in the feries which is equal to ns /(i +x, ort + xiz. 2. Now it has been fhewn in the E TEOe tract, that, if z be any whole number whatfoever, the quantity 1 + a, or (/"[t + *, will be equal to the Ver. Il. infinite 9 346 A DISCOURSE CONCERNING - : 2 I = ; a ar infinite feries 1 + Ax — P=! Bet - 2 Ce: ae ” 2n 3% 4% a > Ex? — oe -~Fx° + &c ad infinitum. We will therefore now proceed : I to fhew that the quantity »/*{1 — x, or 1 — x\*, will be equal to the infinite feries 1 — 2 Ax — P= Be Bay peri (ines 1 Dt — eS ys n 2% 3% 4” 5a ; _ “— Fx° — &c ad infinitum, which confifts of the very fame terms as the I ~ former feries (which is equal to 1 + «\, or "(1 + «,) but with the fign — prefixed to every term after the firft term, inftead of every otherterm. This may be fhewn in the manner following. 3. Since the feries 1 + SUN ee [4 Be 4 2D Ce Bee f a7 3% 4” I RE a Ext — &c is equal to1 + x\", or "ft +x, it follows that, if we raife the faid feries to the wth power, or multiply it 7 — 1 times into itfelf, the produét of thefe multiplications will be equal to 1 + «; that is, the product of the feries 1 + Bx — Cux + Dx* — Et + Fx — &c ad infinitum multi- plied » — 1 times into itfelf (upon the fuppofition that the co-efficient B is = ~Ax -- sChr, OF -, and that the co-eficient C is = — B, and that the co-efficients D, E, F, &c, are equal, refpectively, to * = Gy tn — ots ets E, &c) will be equal to 1 +; and confequently the co-efficient of ¥ will be equal to 1, and the compound quantities which will be the co-efficients of x”, «3, x*, x, and of all the following powers of x in the terms of the faid product, will be, each of them, equal to 0, or will confift of feveral members, of which fome will be marked with the fign +, and others with the fign —, and which will be of fuch magnitudes that the fum of the terms, or members, which are marked with the fign — will be equal to the fum of the terms, or members, which are marked with the fign +. Some of thefe multiplications will be as follow. I 1+ Bree Cx? + De? —& =1 + xe. I 1+ Be— Cx? + Dx? — &c = 1+ x2, I+ Be — Cer? + Dx? — &c + Bxe+ Bx? — BC? + &c — Cr? — BCwr? + &e + Dr? + &c 1+ 2Be—2Cx* + 2D»; i 2 + B*x? —2BCw3 + &c 1+ By THE RESIDUAL THEOREM. 347 r+ Be— Cx + Dei — &c =1+ 4)". 1+2Byse— 2Cx* + 2D: &ec + B*x* — 2BCx: &c + Bxe+ 2B*x*? — 2BCx? + &c + Bix? — &c — Cr? — 2BCxe? + &c + Dx? + &c 1+3Be— 3Cx* + 3D: &c ra 2 + 3B*x* — 6BC%3 cc =i + «}2. + B33 &c I 1+ Be— Cx? + De? —& =1+ ux)". 1+ 3Be — 3Cx* + Dini &c + yb ams 30° DC.x3 &e + B?x3 &c + Be+ 3B*x* — 3BCx3 + &c + 3Bixt — &c — Cx? — 3BCxi &c + Dx? + &c 1+4Bsy — 4Cx*? + 4D»: &c 4 + 6B*x? — 12BCw; sc =1+x\. + 4B?x3 &c r+ Be~ Cx* + De? —& =I+ xs. r+4B.e — 4Cx*? + 4D; &c . + 6B*x? — 12 BCx:; &c t+ 5Be— 5Cx* + 5 Dx? &c $y: +10B7*x? — 20 BC x? &$ =I + x42. + 10 B3x3 &c 4. By thefe multiplications it appears that the fquare of the feries 1 + Bx — Cx? + Dx? — Ext + Fx — &c (carried only to the third power of «) is equal to the compound feries I1+2Be—2Cxr? +2Dx? &ec + B*x? —2BCx: &c; and that its cube is equal to the compound feries 1+ 3B,e—3Cx* +3Dx3 — &c + 3B?*x*—6BCx? &c + Bx: &c; ga ee” and 348 As iD AD BC OW R) Seen CoN CoB R Nowe and that its fourth power is equal to the compound feries 1+4Bxe —4Cxr? + 4Dx &c + 6 B’x* — 12 BC x3 &c + 4Bix>? &c; and that its fifth power is equal to the compound feries 1+ 5Bxe = 5 Ce? + 5 Dx? adc + 10 B*x* — 20BC x? &c + 10B3x3 &ec. s. Now, when is = 2, the firft of thefe compound feriefes (which is equal to the {quare of the feries 1 + Bx — Cw? + Dx? — &c), to wit, the feries 1+2Be-—-2Cx* +2Dx> &c , B*x? —2BCx? &c, mutt be equal to 1 + x, and confequently 2 B (the co-efficient of x) muft be = 1, and B* —2C (the compound co-efficient of **) muft be = 0, and 2D — 2 BC (the compound co-efficient of x?) muft alfo be = 0, and every following co-efficient of one of the powers of « in the faid feries muft, in like manner, be = o, if the co-efficients of the terms of the feries 1 + Bx — Cx? + Dx? — &c, or 1 + = Ax — = Baz + =" Cy3 — &c, have been. rightly affigned. For otherwife the faid compound feries cannot be equal to 1 + #, as it-ought to be upon the prefent fuppofition that is = 2, becaufe upon this fuppofition the feries r + ~ Ax — = Bx? + eee Cae eae Bec H | I (which is univerfally equal tor + x\#, or /{t + «), will be = 31 + “2, or, 47 1 + «, or the fquare-root of 1 +- *, and:confequently the {quaiqiof the faid feries muft be equal to 1 4 x. And accordingly we fhall-find that, if 2 be fuppofed to be = 2, the co-effi- cient 2 B, of the fecond term B.x, -of this compound feries, will-be = 1, and B? — 2Cand 2D — 2 BC (the compound-co-efficients of the two following powers of x, to wit, ** andw? in the faid feries) will each of them be equal to o.*n Mor, Bhs aces we fhall have B(= —X A=+x AS xi} tT —I I _ go and C (ar Be KX EE eS) gq and D(=> SSC enh Reiter ET Sh, eee 2— 3 pale ca’ J): te Ey uently 2B (= 2 X ~) =1,and B? = 2C(=2— a et ee ee ( x =) ; ae z acta rea 3 = — coet a4 Et Rois fi. ZT 7) = % and 2D 2BC(=2x = Jah lead, Os — <4) jaro 6. In like*manner, when # is = 3, the fecond of thefe compound {eriefes (which is equal to the cube of the feries 1 - Bw = Cx* + Dw? — &c), to wit, the compound feries 1+ 3Bx PE WRAESSSLED IU Mod o oR ot fobRn eM, 349 1+ 3Bry~—3Cx* +3D*? &e + 3 B**x* — 6 BCw &e + Bx? &c muft be = to 1 + x, and confequently 3B muft be = 1, and 3B? — 3C muft be = o, and 3 D — 6 BC + B* muft be = 0, or 3 C muft be = 3 B’, and 6 BC mutt be = B? + 3D. | —lia-—t —-+ — + a oe Ben erate Sete x I) = 7, and (= =i Brome Saf 2 pe: ‘F; a | —ti oe oe hall | eatin se cae 3 8 sae 1D ¢ Sian re sic ea pee Di, =P. I ; 5 = 3% a = g73 and confequently 3B (= 3 x *, = 1, and 3B’ —3C(=3 x ©— 3 x >) =o, andgD—-6BC 4+ BI(=3 x S — 6 ale ge, oa a RIN RA IRE COR PAs Bie Soe chs Ae CUR i ane ihe a ie 7. And, when w is = 4, the third of the foregoing compound feriefes. (which is equal to the fourth power of the feries 1 + Bx — Cx* + Dx — Ex«+ + Fx — &c), towit, the compound feries 1+4Bse—4Cx* + 4Dx? &c + 6B*x? — 12 BCx? &c + 4B3x3 &c, muft be equal to 1 4+ «; and confequently 4B muft be = 1, and 6B? — 4C muft be = o, and4D — 12 BC + 4 B? mutt likewife be = o. And.fo we fhall find thefe quantities to be. For, if z is = 4, we fhall have B(=— — 2 and C (= 2" B= gee: hikes f— 2% —1 _ 2X 4—1 3° B41 "3 , F ; es CG — 3. Eee See b 2 — —_-- >. Oe — aL x = ae x ( 3 3X 4 Gey Mal ae a = sg) and confe quently 4B (= 4 x Z) = rand 6B* — 4C (=6 x = —4 x 2 =F —3)=0, and 4D — 12 BC +.4B3 (=4 x + - 125% - x o+4 x oe ee See 8. And, when is = 5, the laft of the foregoing compound feriefes (which is equal to the fifth power of the feries 1 + Bx — Cx* + Dw? — Ext + Fx — &c), to wit, the compound feries 1+ 5Be— 5Cx* + 5Dx3? &e +. 10 Bw? — 20 BC x? &c | : + 10 Bx? &c, muft be equal to 1 -+ x, and confequently 5 B muft be = 1, and 10 B* — 5 C muft be = o, and 5 D — 20 BC + 10 B? muft likewife be = o, And 350 AD TSC oO URS BF CON CE R NTN G And fo we fhall find thefe quantities to be. | For, if is = 5, we fhall have Bigs A ot eee ap and C (= =“ B= i= x-= m 5 5 oe 2X5 3 ; Box Sk oe and D (= = CS SX x= h ease aby = and confequently 5 B(= 5 x =) = 1, and 10 BX — 5C(= 10 X= X-—— 5 XH THs and Sa 20BC + 10B' (= 5 x G2 —20X = X 3g ok 19K a ee 10 — 4 #)= 6 €26- FH A25 125 9. Thus we fee in all thefe inftances, that, when the feries 1 + Bw — Cx? I + Dx: —~ Ext + Fx = &c (which is equal to 1 + x|7, or "1 + x) is raifed to the wth power, or multiplied into itfelf 2 — 1 times, the co-efficient of « in the compound feries obtained by fuch multiplication is always equal to 1, and the co-efficients of «* and x? (which are compound quantities, or quanti- ties confifting of more than 1 term) are, each of them, equal to o, the term, or terms, which are marked with the fign —, or fubtra¢ted from the other terms, being equal to the term, or terms, which are marked with the fign +, and from which they are to be fubtraéted. And the fame thing muft take place in the co- efficients of all the following terms of the compound feries obtained by fuch mul- tiplication. For otherwife the faid feries could not be equal to 1 ++. We may therefore lay it down univerfally, as an undoubted truth, refulting from the I nature of powers and roots, ‘ that, if the feries which is equal to 1 + «|", or —ae F aon if tk «4/1 + 4, to wit, the feries 1 + - Aw — I Bx? + ary Cxi — _ — I ° «(eo Det + oe Ex«xs — &c, ortheferies1 + Bx — Cx? + Dx? — n 4 5 ; : , < Ex4¢ + Fxs — &c (in which latter feries the co-efficients B, C, D, E, F, “¢ &c, are re{pectively equal to the tormer co-efficients — A,-or =, and ral B, peeiin Wc Oe ry 48-1, &c), be raifed to the mth power, or multiplied into n a n cc itfelf n—1 iss the Greene of « in the compound feries produced by fuch «© multiplication will be = 1, and the co-efficients of «*, 3, «*, «5, and all the << following powers of x in the faid feries, will be, each of them, equal too; the « faid co-efficients being compound quantities confifting of fimple terms, or mem- «© bers, connected with each other, by the figns + and —, and the fum of the << members of each of the faid co-efficients that are marked with the fign —, or ‘¢ fubtraéted from the other members of them, which are marked with the fign +, «« being always equal to the fum of the faid other members from which they are « fubtracted.” This fundamental propofition being well underftood, we may de- rive from it a proof of the propofition afferted above in art. 2, concerning the quantity THE RESIDUAL THEOREM. 351 rey; 1 quantity 1 + x|#, or 4/7" 1 — x, or the wth root of the refidual quantity 1 —«, to wit, that it will be equal to the feries 1 —— Ax — [ Bx? 2n—1 nN—-— I nm I “2 banat QF SE “— Det — o— Ex? — &c, or 1 — Be — Cx? — a” : «Dx? — Ext — Fx’ — &c3 which confifts of the very fame terms as the ‘¢ former feries which is equal to the wth root of the binomial quantity 1 + x, <¢ but with the fign — prefixed to every term after the firft term 1, inftead of “¢ every other term.” 10. To render the proof of this propofition as eafy as poffible, it will be con- venient to multiply each of the two feriefes 1-+ Bx — Cx* + Dx} — Ext + Fx’ —&c,andi—Bwr — Cx* — Dx} — Ext — Fx — &c, into itfelf, and then to compare together the f{quares, or produ¢ts thence arifing, Thefe multiplications will be as follow. Poesy ae Cx? + Date Extn eRe d) ~ &e E+ Bw — Cw? +: De?) —: Ext + Fei. — &c ra Be — Cx? + De? — Ext 4+ Fes —.&e + Bre+t+ B*x?— BCx? + BDe+ — BEx? + &c — BCw3 + Cert — CDi + &e + Dx? + BD«+ — CDxr + &c — Ext — BEx«: + &c + Fx + &c 1+: 2Be—2Cx* +2De3 ~—~o2kut + 2F x: &c + Bex? —2BCxKw? + 2BDx*+ — 2BEx &c 4+ Crt — 2CDx &c Poe DB he Cyt DY pele eee LR Eo = bre 1 —- Be—° Ca? — -Dxe — OER* (=~ OF es = &e t— Be— Cr? — Dei *—" Ext > Fes -— &c — Bxe-+ Bx? +: BCx? + BDx* + BEw + &c — Cx> + BCw? + C2x* + CDs + &c — De? + BDxwr+ +. CDxe: + &c — Exvt + BEx + &c — Frys + &c 1—2Be—-2CHe? —2Dx? —2Ewt —2F ew: &c + Btw? + 2BCw3? + 2BDx* + 2BEx: &c + C*x+ + 2CDx> &e. 11. Now, if we compare this laft compound feries (which is equal to the fquare of the fimple feries 1 — Bx — Cx* — Dx? — Ext — Fx’ — &c) with the former compound feries (which is equal to the fquare of the fimple feries1 + Bx — Cw? + Dx? — Ex* + Fx’ — &c), we hhall find the fol- lowing obfervations to be true concerning them.. Tir 352 A“ WstctiolutTR siM © f.0R mR NE ete In the firft place, all the terms of the fecond of thefe compound feriefes are the very {ame with the correfponding terms of the former of them, but are dif- ferently connected with each other by the figns + and —. And this, it is evi- dent, muft be the cafe with all the following terms of both thefe compound’ feriefes (to whatever number of terms thefe feriefes may be continued) as well as with the few terms here computed. For, fince the terms of the fimple feries 1— Bx — Cx? — Dx? — Ext — Fv &c, are the fame with the terms of the fimple feries 1 + Bw — Cx* + Dx? — Ex* + Fx — &c, though differently connected with each other by the figns'+ and —, it follows that the products of the multiplication of the terms of the feries 1 — Bx —, Cw — © Dx3 — Ext — Fx’ — &c, into each other muft be equal to the produéts of the multiplication of the correfpondent terms of the feries 1 + Bx — Cu? +4 Dei — Ext + Fx* — &c, into each other, that is, the feveral members: of the {quare of the feries 1 — Bx — Cx? — Dv — Ext — Fai = &e; muft be equal to the correfponding members of the {quare of the feries 1 44 Buy: —Cxr + Dxi — Ext + Fes — &c, though differently connected with each other by the figns + and —. . And fecondly, we may obferve that the firft term in both thefe compound feriefes is 1. , And, thirdly, we may obferve that the third and fifth terms in the fecond of thefe products, or compound feriefes, which is equal to the {quare of the fecond © fimple feries 1 — Baw — Cx? — Dx? — Ex+ — Fx’ — &c (in which all the terms after the firft term 1.are marked with the fign —) to wit, the terms — 2C x” + Bex? and —2Ex*+ + 2BD«* + C*x*, have the fame figns + and — | prefixed to their feveral members 2 Cx’, B*x*, 2 Ext, -2 BDx*, and C*x*, re{pectively, as are prefixed to the fame members of the third and fifth terms of the former of thefe products, or compound feriefes, which is equal to the {quare of the fimple feries r + Bx — Cw 4+ Dwi — Ext + Fwi — &c (in which the fecond and other following terms are marked with the figns + and — alternately), to wit, the terms — 2Cx%x? + B*x? and — 2Ex* + 2BDx«*t + C*x*. And the fame obfervation is true likewife of the feventh and ninth, and eleventh, and all the following odd terms of thefe two com- pound feriefes (to whatever number of terms the faid feriefes may be con- tinued), to wit, that the figns + and — that are to be prefixed to the feveral members of any of the faid odd terms in the latter of thefe compound feriefes will be the fame that are to be prefixed to the fame members of the fame odd terms in the former of thefe compound feriefes; as will appear from an attentive - confideration of the two foregoing operations of multiplication fet down in the foregoing article. 3 | And, 4thly, we may obferve that the fecond, fourth, and fixth terms of the fecond of thefe products, or compound feriefes (which is equal to the fquare of the fecond fimple feries 1 — Bx — Cx? — Dx? — Ex* — Fax’ — &c), «o wit, the terms — 2 Bx, — 2 Dx? + 2BCw3, and — 2F ses 4+ 2 BEw: -+- 2CD«x‘, have their feveral members 2Bx, 2Dw3, 2 BCw?, 2 Fx, 2 BE x, and 2CD«:, marked with contrary figns to thofe which are pre- fixed to the fame members of the fecond, fourth, and fixth, terms of the for- 2 . mer Tole RESID VU AcL oTeH E-o-R BM. 353 mer of thofe compound feriefes (which is equal to the fquare of the firft fimple feries1 + Be — Cx? + Dx? — Ext + Fxs — &c), towit, the terms -+ 2Bxe, + 2D«? — 2BCxw3, and + 2Fxs — 2BExw' —2CDx5. Andthe fame obfervation is true likewife of the eighth, and tenth, and twelfth, and other following even terms of thefe two compound feriefes (to whatever number of terms they may be continued), to wit, that the figns ++ and — that are to be prefixed to the feveral members of any of the faid even terms in the latter of ‘thefe compound feriefes will be, refpectively, contrary to thofe which are to be prefixed to the fame members of the fame even terms in the former of thefe compound feriefes; as will appear from an attentive confideration of the two foregoing operations of multiplication fet down in the foregoing article. 12« And if we were to repeat the foregoing multiplications of the two feriefes 1+ Be — Cx? + Dei? — Ext + F xi — &c and 1 — Bx — Cx? — De: — Ext — Fx’ — &c into themfelves any number of times whatfoever, fo as to obtain the cubes, andthe fourth powers, and the fifth powers, or any higher powers of the faid feriefes, the foregoing obfervations would be true of all the compound feriefes, or of all the powers of the faid two fimple feriefes, which would be thereby obtained ; to wit, 1ft, That the terms of every power of the one feries would be the very fame with the correfponding terms of the fame power of the other feries; and, 2dly, That the firft term of every power of both feriefes would be 1; and, adly, ‘Chat the figns -+- and —, which would be prefixed to the feveral members of the third, fifth, feventh, ninth, eleventh, and other following odd terms of any’ power-of one of thefe feriefes, would be the very fame that weie to be prefixed to the fame meinbers of the fame odd terms, refpectively, of the fame power of the other feries; and, gthly, That the figns + and —, which would be prefixed to the fecond term, and to the feveral members of the fourth, fixth, eighth, tenth, twelfth, and other follow- ing even terms of any power of one of thefe two feriefes would be, refpectively, contrary to the figns that would be prefixed to the fecond term, and to the fame members of the fame following even terms of the fame power of the other feries. 13. Of the truth of thefe obfervations in the cafe of higher powers of the two foregoing fimple feriefes 1 + Bx —Cx* + Dx? —Ew* + Fut — &c and 1— Be —Cxr? — Dx? — Ext — Fx — &c than the fquare, I fhall here give one example, by raifing the faid two feriefes to the cube, or third power. We have already feen in‘ art. 10 that the fquare of the feries 1 4+ Bx — Cx’ + Dx? — Ex* + Fx — &c is the compound feries 1+ 2Be—2Cx*? +2Dx? —2Ex* +2Fx5 — &c + Bx? —2BCx? + 2BDx* —2BEx* + &c + Crxt —2CDs* + &e. ‘Therefore to find its cube, we muft multiply this compound feries into the original feries 1 + Bx — Cw? + Dx? —Ewt + Fx' — &c; which may be done as follows. Vou. II. 27, 1+ 2B 454 A DISCOURSE CONCERNING r+ 2Be—2Cr*? +2DKe* —2Exwt + oFxes — &e + Bx? —2BCwr? + 2BDx* —oaBExs + &c é' + C*x* —2CDs 4+ &c 1+ Bee Cx? + De? — Ext + Fre — &c 1+ 2Be—2Cxr? +2Dxe3 —2Ewt +2Fe5 —&e + Btx? —2BCxe? + 2BDx* —2BEx® + &c + C*x* —2CD«%5 + &e + Bw+2B*x? —2BCx? + 2BDx* —2BEx® + &c + B?x3 —2B*Cxt* + 2B*Dxs — &c + BC*xs —&c- — Cx? —2BCx? + 2C*x* —2CDxi + &c — B*Cx* + 2BC*xs —&c + De? -2BDse+ —2CDx> + &e + B*D«s — &c me Ext = —2BEx® + &e + Fx: — &c 1+ 3Bxe—3Cx* + 3Dx? —3Ex* +3Fx° —k&c + 3 Bx? —6BCx? 4+ 6BDx* —6BEx> + &c + Bix? + 3C*x* —6CDx> + &c — 3B*Cx* + 3B°Dxs — &c + 3 BC*xs — &c. Therefore the cube of the feries 1 + Be — Cx? + Det — Ext + Fx® > &c is the compound feries 1+ 3Be—3Cx? + 3Dxe? —3Ex* +3Fx5 —&e + 3 B*x* —6 BCx? + 6BDx* —6BEx® + &c + Bix3 +—+ 3C*x* —6CDx> + &c — 3B°*Cx* 4+ 3 B*Dx — &c + 3BC*xs — &e. And we have already feen in art. 10 that the fquare of the fecond feries 1 — Ba — Cx? — Dx? — Ex* — Fx — &c is the compound feries’ 1—2B.e—2Cx*? —2De? —2Exw+ —2o2Fews — &e + B*x? + 2BCw? + 2BDxw* + 2BExs + &c + C*x* +2CDs> + &c. Therefore, to find the cube of the faid fecond feries 1 — Bx — Cw? — Dx? — Ext — Fx’ — &c, we muft multiply this laft compound feries into the faid fecond feries itfelf; which may be done as follows. I—2Byxr THE RESIDUAL THEOREM 355° tr—2Buy—oCx*? ~2Dx? —2Ext —2Fxs —&c + Bx? + 2BCwi + 2BDx* + 2BExs) + &c + C*x+ +2CDxe> + &c 1— Be— Cx? — De? — Ext — Fre — &c ¥—2Bse—2Cx? —2Dxe3 ~2Ex* —22F xs — XC + Bex? + 2BCx? + 2BDxe* + 2BEx5 + &c + C€C*xt+ + 2CDxs + &c — Br +2B*x* + 2BCw? + 2BDxe+ + 2BEx5 + &c — Biys —2B*Cx* —2B* Dx’ — &c — BC*xs —k&c — Cr? +2BCx? -2C*x* +2CDx5 4+ &e — B?Cx+ —2BC*xs — &c — Dx? + 2BDx+ + 2CDx* + &c — B*Dxs + &c — Ext +2BExs + -&c — xs +. &c 1—3Bx —3Cx*? —3Dxr? —3Ex+ —3Fx &c + 3B*x* + 6BCx? + 6BDxt 4+ 6BEx' + &c — Bix? + 39C*x* + 6CDxes + & — 3B*Cx*—3B*Dxs &c . — 3BC*x’ — &c Therefore the cube of the faid fecond feries 1 — Bx — Cw? — Dx3 ~ Ext — Fx’ — &c is the compound feries 1—3Bxe—3Cr* —3Dx? —3Ex* —3Fx5 —&c » | + 3B*x* + 6BCw? 4+ 6BDx* + 6BEx® + &c — Bes + 3C*xrt + 6CD«5 + &c — 3B°Cx* —3B*Dx' = &c — 3BC*xs — &c. 14. The four obfervations made in art. 11 and 12 are evidently true of the two compound feriefes obtained in the foregoing article 13, and which are equal to the cubes of the two fimple feriefes 1 + Bx — Cx? + Dx? — Ext + Fx’ — &c and 1 — Be — Cx? — De? — Ext —Fxs — &c. For, in the 1ft place, the feveral terms of the latter compound feries are exaétly the fame with the correfponding terms of the former compound feries; and 2dly, the firft term in both thefe compound feriefes is 1; and, gdly, the figns -- and — that are prefixed to the feveral members of the third and fifth terms, — 3 C x* + 3 Bex? and — 3 Ex* + 6 BDx* + 3 C*x* — 3 B*C x* of the latter com- pound feries (which is equal to the cube of the feries 1 —- Bx — Cv? — Dx? — Ext — Fx — &c) are the fame with thofe which-are prefixed to the fame members of the third and fifth terms, — 3 Cw? + 3 B’x? and —3 Ex«* + 6 BDxt + 3 C2x* — 3 B*C x‘, of the former compound feries, which is equal to the cube of the feries 1 + Bu — Cx? 4+ Dwi —Eut + Fes — &c; and, 4thly, the figns +- and — which are prefixed to the fecond term, — 3 Bx, and to the feveral members of the fourth and fixth terms, — 3D? +- 6.BC x3 — Bixs and — 3F xe) + 6BEw! 4+ 6CDx> — 3 B*Dw — 2Z 2 3 BC*x5y 356 Aw D TT SVCTONUTR SIR. “CS CO INF CT ESR WOH C6 3 BC*w’, of the latter compound feries (which is equal to the cube of the fim- ple feries 1 — Bu — Cx* — Da} — Ext — Fx — &c) are, refpectively, contrary to thofe which are prefixed to the fecond term, + 3 Bx, and to the feveral members of the fourth and fixth terms, + 3 Ds} — 6 BC w#3 4+ B?x: and + 3Fx«* —6BEx*’ —6CDx> + 3 B*D«’ + 3 BC’*x5, of the former compound feries, which is equal to the cube of the fimple feries 1 + Ba — Cx? + Dx? —Ext + Fas — &e. And the fame obfervations will be found to be true.of the feveral members of all the following terms.of the faid.two compound. feriefes after the fixth terms (to whatever number of terms the faid feriefes may be continued) and of the fions 4+-.and — that are to be prefixed tothe faid. members. . And they will be true alfo of the terms of. the feveral.compound feriefes that are equal to the fourth powers, and to the fifth powers, and to all higher powers, of the faid two fimple feriefes 1 + Bx —Cx? 4+ Dx} — Ext + Fx’ — &c and 1 — Bx — Cx?.— Dx —- Ext — Fes —&c. 15. This relation between the figns that are to be prefixed to the fame*mem- bers of the. feveral terms of the faid compound feriefes which are equal to the {quares and the cubes and other higher powers of the two fimple feriefes 1 + Bx — Cxrte+ Deri — Ext ot) Fai — &c and 1 — Be — Cx? — Dx? — Ext —Fx’ — &c 1s the confequence of the known rules of Algebraick mul- tiplication; according to which the product of the multiplication of two.quan- tities which are both marked with the fign — is to be marked with the fign ++; as well as the product of the multiplication of two quantities. which are both marked with the fign +; and the product of the multiplication of two«uan- tities, of which one is marked with the fign -+, and the other with the fign —, is to be marked with the fign —. .For it will follow from hence that in the compound feries which is equal to the fquare, or cube, or fourth power, or fifth power, or other higher power, of the feries 1 + Bx— Cx? + Dx3 — Ext + Fxt — &c, all the terms that will involve the odd powers of *, to wit, #, #3, «5, w7, «9, x'*, &c, will be marked with contrary figns, re+ fpectively, to thofe which are to be prefixed to the fame terms in the com: pound feries which is equal to the fquare, or cube, or fourth power, or fifth power, or other correfponding higher. power, of the feries 1 — Ba — Cx? — Dx? — Ext —Fx' — &c; and all the terms that will involve the even powers of x, to wit, ~*, x*, *°, x*, «'°, xt7, &c, in the former compound feries, which is equal to the fquare, or cube, or other higher power, of the feries 1 + Bx —Cx* + Dy?--Ext+ Fut — &c, will be marked with the fame figns + and —, refpectively, as are prefixed to the fame terms in the latter compound feries, which is equal to the fquare, or cube, or other correfponding higher power, of the feries 1 — Bw — Cx* — Dx? — Ext — Fx — &e. 16. To make this more evident, let the capital letter P be put = Bx + Dei + Fes + He? + Kxe + Max" + &c, or the fum of all the terms in the firft feries 1 + Bx —Cx* + Dx? — Ext + Fx' — Gao 4+ He? — Tx? + Keo —Lx? + Myx" — &c, which involve the odd; powers of x, and which in this feries have, all of them, the fign + prefixed to them; and let e Ti He E REST DUAL Ty. H EO R EF M. 357: let the capital letter Q be put = Cx? + Bet + Gut + Le? + Le? + &, or the fum of all the terms in the fame feries which involve the even powers of «, and which in this feries, as well as in the fecond feries 1 — Bu — Cx? — Dw? — Ext — F xe — Gut — He? —I xe? —Keo — Lx? — Mat — &c, are all marked with the fign —. Then will ihe feries 1 + Bu — Cx? + Dx? —Ewt + Fxs ~— & be — 1+ P—Q,, and the feries 1 — Be — Cv? — Dx? — Ext — Fu — & will be = 1 — P —Q;; and confequently the fquare of the former feries will eat a Pi OP = Lefp2s=Q\} = ST eg. cir x{[P—Q + P—Q)’) = 1+2P—2Q+ P*— 2 PQ + Q’, and the {quare of the erences ail) sbe- (=. PO) ee P+Q Sicepy EY 6 ay ops. P+ Qi+ P + Q’) = 1—2P—2Q4 P?+2PQ+ Q. 17. The terms of this latter quantity (which is equal to the fquare of the lat- ter feries) are exactly the fame with the terms of the former quantity 1 + 2 P — 2Q + P*—2PQ+ Q‘, (which is equal to the fquare of the former fe- ries) but are not connected with each other by the figns + and — in the fame manner as the terms of the faid former quantity. And the difference between the terms of thefe two quantities in this refpeé is as follows. The third term 2 O., and the fourth term P*, and the fixth, or laft, term Q?, in both the quantities r+2P—2Q 4 P?’—~2PQ+4+ Qi and 1—2P —2Q+ P* + 2PQ + Q’ are marked with the fame figns + and —, being —2Q + P* + Q?in both quantities; but the fecond term 2 P, and the fifth term 2 PQ , have dif- ferent figns in the two quantities, being + 2P and — 2 PQ in the former quan- tity, and — 2P and + 2 PQ in the latter quantity. 18. Now the three terms 2Q, P*, and Q’, which are marked with the fame figns in both thefe quantities, will be equal to feriefes that will involve only the even powers of x. ' For, in the firft place, the fecond term 2 Q is = 2 x the feries Cx* + Ext +1Greoqe le? + Lx? + &c = the feries 2Cw2.4+2Ex* + 2Geo 4+ 21xv* + 2 Lx'® + &c, which contains only the even powers of x. And, in the fecond place, the fixth term Q?’ (being equal to the fquare of the feries Cx? + Ex*+ + Gwe + Ix? + Lx + &c) will evidently be equal toa feries that will contain only x*, *°, x°, x"°, and the following even powers of x. And, laftly, the fourth term P* (being equal to the {quare of the feries B x +Dxe 4+ Fo + He? + Kx? + Me™ + &c, which contains only the odd powers of x) will alfo be equal to a feries that will contain only the even powers _ of x; becaufe its terms will be the products of the multiplication of the terms of the feries Bx + Dx? + Fes + Hx? + Kao + Ma™ + &c (which involve_only the odd powers of x) into each other: it being evident that the multiplication of two odd powers of x into each other muft always produce an even power of it. daig wT Therefore the three terms 2 Q., P*, and Q’, which are marked with the fame figns + and — in both the quantities 1 + 2P —2Q-+ P*—2PQ 4+ Qi and 458 Ae F PY slctoBu’ Ri se “CO WhOSEIR NPN GS andi —2P—~2Q + P? + 2PQ + Q’, will be equal to feriefes which will involve only the even powers of «. Q. E. D. 19. And the fecond term 2 P, and the fifth term 2 PQ, of the faid quanti- ties1 + 2P—2Q+ P?~2PQ+ Qandi —2P—2Q+ P* + 2PQ_ + Q*, which are marked with contrary figns in the latter of thefe two quanti- ties to thofe with which they are marked in the former quantity, will be equal to feriefes which will contain only the odd powers of x. ; For, in the 1ft place, 2 P is = 2X the feries Bx + Dw? + Fuss + Hx? + Kwo + Ma™ + &c = the feries 2Bx + 2D? + 2Fxe3 + 2H? + 2K x9 +- 2 Mx" + &c, which contains only the odd powers of ». And, 2dly, 2PQ is = Q. x 2P=Q -™ the feries 2Bx + 2Dx? + 2Fxe5 +2Hx7 + 2Kxe° + 2Mx" + &c = the feries Cx? + Ext + Gee + Ix? + Lx’? + &c (which involves the even powers of x) x the feries 2Be+2Dx3 + 2Fxe' 4+ 2He? + 2Ke? + 2 Mx" + &c (which in- volves the odd powers of «) = a feries confifting of terms which will involve only x3, x5, «7, ¥°, x, and the following odd powers of x; becaufe all the terms of it will be the products of the multiplication of fome of the terms of the feries Cv? 4+ Ext + Guo + Ix* + Lx? 4+ &c (which involve only the even powers of x) into fome of the terms of the feries 2 Bx + 2 Dx? + 2Fe5 +2H«? + 2Ke° + 2Mxe" + &c, which involve only the odd powers of w; it being evident that the product of the multiplication of an even power of x into an odd power of it muft always be an odd_power of it. Therefore the two terms 2 P and 2 PQ of the two quantities 1 + 2P — 2Q+ P?—2PQ+ Q* and 1 —2P—2Q+ P* + 2PQ + Q?, which are marked with contrary figns + and — in thofe two quantities, will be equal to feriefes which will contain only the odd powers of x. Q. E. D. 20. It appears, therefore, from the two preceeding articles, that, to whatever number of terms the two feriefes 1 + Bx — Cx? + Dx? — Ext 4 Fxs — &c and 1— Bx — Cx? — Dx? — Ex* — Fx’ — &c, and the fquares of thefe two feriefes, may be continued, the terms involving the even powers of x in the fquares of both feriefes, will be marked with the fame figns + and —, and the terms involving the odd powers of « in the {quare of the latter feries will be marked with the contrary figns to thofe with which the fame terms are marked in the fquare of the former feries. 21. The reafonings ufed in the fix preceeding articles to prove, ‘ that, «* in the two compound feriefes which are equal to the {quares of the two fimple “‘ feriefes 1 + Bx — Cx* + Dx? — Ext + Fus — &c andi — Bye — “Cx? — Dx? — Ext — Fxw' — &c, the feveral terms, and members of ‘© terms, which involve the even powers of x will be marked with the fame “ fions + and — refpectively, and that the feveral terms, and members of “* terms, which involve the odd powers of « in the latter of the faid compound fe- «© riefes will be marked with the contrary figns to thofe which are to be prefixed “to the fame terms, and members of terms, in the former of the faid com- «* pound SME aReFaSulipev A LB ¢T\i> E.0 8 EB M. 359 “* pound feriefes,”” may be extended to the more complicated compound {eriefes which are equal to the cubes of the faid fimple feriefes 1 + Bu — Cx? + Dx; — Ext + Fes — &c andi — Be —~ Cx? — De? — Ext — Fw — &e, and to the fourth powers of the faid feriefes, and to the fifth powers of the faid feriefes, and to all higher powers of the faid feriefes, fo as to prove, ‘that in ** all thefe compound feriefes the feveral quantities which involve the even ** powers of » will be marked with the fame figns in every two correfponding “* feriefes, and that in the compound feries which is equal to the cube, or the “* fourth power, or the fifth power, or any higher power, of the feries 1 — Bu — “Cx? — Dei — Ext — Fx’ — &c, the quantities which involve the odd ** powers of x will be marked with the contrary figns to thofe which are prefixed ‘* to the fame quantities in the compound feries which is equal to the cube, or “« the fourth power, or the fifth power, or other correfponding higher power of “the feries 1 + Be — Cw? + De? — Ext + Fw’ — &c.”? And there- fore I think, we may now confider the four obfervations fet forth above in art. 11 and 12 as fufficiently eftablithed. 22. Now, if thefe obfervations are admitted to be true, it will follow that, whenever the co-efficients of the third, fourth, fifth, fixth, and other following terms of the compound feries which is equal to the fquare, or the cube, or the fourth power, or the fifth power, or any higher power, of the feries 1 + Bu — Cx? + Dx? — Ext + Fx’ — &c are all equal to 0, or thofe of the mem- bers of the faid co-efficients which are marked with the fign —, taken together, are equal to the other members of the faid co-efficients which are marked with the fign +, and from which the former members marked with the fign —, are to be fubtracted, the fame thing will alfo take place in the compound. feries which is equal to the {quare, or the cube, or the fourth power, or the fifth power, or the other correfponding higher power, of the feries 1 — Bx — Cx*— Dw: — Ex*+ — Fx — &c; towit, that the co-efficients of the third, fourth, fifth, fixth, and other following terms of the faid compound feries will likewife all be equal to o, or that thofe members of the faid co-efficients which are marked with the fign —, or are fubtracted from the other members of them which are marked with the fign +-, will be equal to the faid other members. For in the faid’correfpondent compound feriefes the members of the feveral correfponding terms, or terms involving the fame powers of x, will be the very fame quantities, by the 1{t obfervation in art. 11 and 12. And-in the third, fifth, feventh, and other following odd terms of the faid compound feriefes (which will involve the even powers of x) the figns + and —, to be prefixed to the feveral members of the faid terms, will be the fame in both feriefes; and in the fecond terms and the feveral members of the fourth, fixth, eighth, and other following even terms of the faid compound feriefes (which will involve the odd powers of «) the figns + and —, to be prefixed to the faid fecond terms, and to the feveral members of the fourth, fixth, eighth, and other following even terms of the faid compound feriefes, will be, refpectively, contrary in one of thofe compound feriefes to what they are in the other ; as is evident from the third and fourth obfervations in art. 11 and 12. Therefore, if in ge 2 ourth, 260 A DISCOURSE, CONCERNING fourth, fifth, fixth, and other following terms of the compound feries which is equal to any power of the firft fimple feries 1 + Bx — Cx? + Dx? — Ext -+ Fws — &c, the fum of the members of each of the faid terms that are marked with the fign — is equal to the fum of the members of the fame term which are marked with the fign +, the fame thing will alfo take place in the compound feries which is equal to the fame power of the fecond fimple feries 1 — Be — Cx? — Dx? — Ext — Fxs — &c, or the fum of the members which are marked with the fign — in the third, and the fourth, and the fifth, and the fixth, and every other following term of the faid fecond compound feries, will be equal to the {um of the members of the fame term which are marked with the fign +; that is, in other words, when the third, fourth, fifth, fixth, and other following terms of the compound feries which is equal to any power of the fimple feries 1 + Bx —Cx? + Dx? — Ext + Fx? — &c, become, all, equal to o, and the whole feries confifts of only the two firft terms, the third, fourth, fifth, fixth, and other following terms of the compound feries which is © equal to the fame power of the feries 1 — Bw — Cx? — Dx} — Ex* — Fx — &c will alfo be, all, equal to o, and the whole feries will confift of only the two firft terms. 0 iE. aD. 23. And hence we may derive a proof of the propofition afferted above, in 1 art. 2, concerning the quantity 1 — |”, or ¥/*{1 —.«, or the ath root of the refidual quantity 1 — x; to wit, that, if A be = 1, and B be = - A, and C be = = B, and D be = ore ~C, and E, F, G, H, &c be equal to - -D, E, ie a “= > G, &c, refpettively, the faid quantity 1 — « =, or n a : 3 5 2 ne 2 Roe CL: 210 /"{t —«, will be equal to the infinite feries 1 —Ax —= Bx er Cxi — [a= LD ¥* pS Ext — &c, ort — Bx — Cx? — Dx? — Ex+ — Fxs — &c, in which all the terms after the firft term 1 are marked with the ign —, or are fubtracted from the faid firft term. For, fince the feries 1 + — Ax fms By 4 20 Cys ee % 2% 3% 42 Dist + 2— Ex: — &c, or 1 + Be — Cx? + De? — Ext +Fes — &c, has been fhewn in art. 47 of the preceeding difcourfe concerning the Bino- I mial Theorem, page 237, to be equal to 1 + x|7, or "(1 + x, it follows that, if the faid feries were to be raifed to the mth power (” being any whole number whatfoever) or to be multiplied # — 1 times into itfelf, the compound feries thence arifing would be equal to t + x, which would be the two firft terms of it, and confequently the co-efficients of «*, «3, #*, *', &c ad infi- nitum in the following terms of the faid compound feries would be, each of them, equal too, or thofe members of the faid co-efficients which would be marked with THO RYE SHI Dov AG «HH Eo: R EM. 301 with the fign —, or would be to be fubtracted from the others which would be mark- ed with the fign +, would be equal to the faid other members, from which they would be to be fubtracted; of which we have given fome examples above in art. 5, 6, 7; and 8. But, by the foregoing article 22, whenever the co-efficients of the third, fourth, fifth, fixth, and other following terms of the compound feries which is equal to any power of the feries 1 + Bx —Cx* + Dw? — Ex* + Fx’ — &c, are all equal to o, or the fubtratted members of each of the faid co- efficients are equal, taken together, to the members from which they are fubtrafted, the co-efhi- cients of the third, fourth, fifth, fixth, and other following terms of the compound feries, which is equal to the fame power of the feries 1 — Bu — Cx? — Dx — Ex*+ — Fx* — &c, willalfo be, all of them, equal to 0, or the fubtratted nem- bers of each of the faid co-efficients will be equal, taken together, to the mem- bers from which they aré fubtracted. Therefore, if the feries 1 —— Ax — ete Bet on Cory oh 2 De ELS dee See or ie Be 2% 32 4% 5% ° c . I I — Cx? — Dx? — Ext —F xs — &c (in which Bis =— A, or— x 1, or —, and C is = ~—" B, and D is = ~—~ C, and E is = 2~— D, and F, n 22 32 44 oe x 6re “ss G, H, I, &c, are equal to ae a oF vs F, a ~G, and = - H, &c, re- fpectively), be raifed to the wth power, or multiplied 2 — 1 times into itfelf, the co-efficients of the third, fourth, fifth, fixth, and other following terms of the compound feries which will be produced by fuch multiplication, and which will be equal to the faid mth power of the feries t - Bx — Cwx* — Dei — Ext —Fxs — &c, will, each of them, be equal to 0, or the fub- tracted members of each of the faid co-efficients will be, all taken together, equal to the other members of them, from which they are to be fubtrac- ted; and confequently the whole of the faid compound feries will be equal to its two firft terms; which will be 1 — x, becaufe the two firft terms of the compound feries which is equal to the wth power of the feries 1 + Bx —- Tell 4 Pi) Seto | Cz + Dr? — E.x* 4+ Fwi — &c, or 1 + LAx — -Et Be -- Cxi — meg Dwt + ne Exs — &c (to which the compound feries that is equal to the wth power of the feries 1 — Bw — Cx? — Dx? — Ew* — Fx’ — &c, has been fhewn to be analogous in the manner above-defcribed), are Lr. And, fince the wth power of the feries 1 — Bx — Cw* — Dv? — Ext — I n— tI 22 — 1 32 —— I Fxs — &c, or 1 — — Aw — |— Bx? — Cri — ~— Dx* — 2 22 32 4u Ew’ — &c, will be equal to 1 — w, it follows that the faid feries itfelf 4n—1 gu I will be equal to 1 — x|*, or "(1 — «, or the ath root of the refidual quan- tity I — x. 0. E. D; Vous II. MeN 24. We 362 AOD LT stigh oh ulm ise FC ROANe CHER (N Fee G Pc nm 24. We muft now proceed to confider the quantity 1 — x|#, or the mth power of the wth root of the refidual quantity 1 — x. mn Of the feries which ts equal to the quantity 1 — x\7, or the mth power of the nth root of the refidual quantity 1 — x, when m and n are any whole num- bers whatfoever. 25. It has been fhewn in the foregoing difcourfe concerning the binomial theorem, art. 77, p. 269, that, ifm be any whole number whatfoever, and z WW any other whole number greater than m, the quantity I + x”, orthe mth power of the zth root of the binonal quantity 1 + «w, will be equal to the feries 1 + = Ax — Fee Bx? + =—2Cws — |2=* Des + P=" Ewxs — &e ad Lymipe ; in which feries the fecond, and third, and fourth, Endl other following terms, are alternately en with the fign + and the fign —. And it has been fhewn in art, 104 of the faid difcourfe, page 289, that, when m is greater than Wm n, but lefs than 2”, the quantity 1 + *|*, or the mth power of aa nth root of the binomial quantity 1 + x, will be equal to the feries 1 + — = Ax = —* ee — Cus + a Teste fe Ews ++ &c; in which the three firft terms of the feries are added together, and the fourth and fifth, and other following terms of it are marked with the fign — and the fign + alternately, or are alternately fubtracted from, and added to, the faid three firft terms.. And it has been ‘fhewn in art. 115 of the faid diteourte: page 299, that, if m be of any magnitude greater than 2%, -and pz is the greateft multiple of m that is lefs than m,{o that m is greater than pz, but lefs ave? + 1] X #, or pz + n, the quan- 7 / tity 1 + x\a , or the mth power of the zth root of the binomial quantity 1 + x, will be equal to the feries 1 + = Ae eeepc eee Tae Cua hee 4a th m— 4n 2- = ie Gitte 2 PP *.2 pu+2nu—m Dx Fee a Eas 2 Fixe + &c, nate Cx eae | 4 prt3 “EE py ra pict: 4 1 Paria pbs _ [pet su—m 1 p+6 See n+ 4n sa pat bn pn + 6x sh +- &c ad infinitum; in which feries all the terms after the firft term 1 are to be added to the faid firft term, till we come to the oe Cig hie > which 2 is - x THkHER ARWEVSATKDW AL «THE OUR er Ms 363 pr+tn—m pz + 22 are to be added to, and fubtracted from, the faid firft term alternately. is to be fubtracted from it; and all the terms after the faid term Ciyh +? m1 Now the quantity 1 — x\*, or the mth power of the th root of the refidual quantity 1 — x, will always be equal to a feries confifting of the very fame « ™m terms as the feries that is equal to the quantity 1 + x\ 7 , or the fame power of _ the fame root of the binomial quantity 1 + *, but with the figns + and — changed into their contraries in all the terms that involve «, w3, #5, “7, ¥°, x**, and the other following odd powers of w, or in the fecond, and fourth, and fixth, and eighth, and tenth, and other following even terms of the feries. This I fhall now endeavour to demonttrate. I 26. It has been fhewn above in art. 23, that 1 — «|, or the th root of the . ° . : . I ae refidual quantity 1 — x, is equal to the infinite feries 1 — — Aw — |—— Bx? 3 74 2% — — iI — I . > = yt “— Dx — |= = Ex’ — &c, which confifts of the 2 very fame terms, only differently connected together by the figns + and —, as the feries 1 + ~ Ax — oe BNO BEC == Det + ts Bf 3n An 5 Ex’ — &c, which is equal to r + x\”, or the wth root of the binomial quan- tity 1 + *. And the difference between thefe two feriefes with refpec& to the figns + and —, that are to be prefixed to their fecond and other following I : terms, is that in the feries which is equal to 1 — x|# the fign — is to be pre- fixed to all the terms after the firft term, whereas in the feries which is equal to I ‘ 1 — x\# the fign — is to be prefixed only to its third, and fifth, and feventh, and other following odd terms, and the fign + is to bé prefixed to its fecond, and fourth, and fixth, and other following even terms. ae eT os : : : I 3 i At A Now, fince 1 — x|* is equal to the infinite feries 1 — — Aw —|—— Bx — 2% — — J - . es — | yt — | Bt — &e, ‘it follows that the quan- 3n 42 G2 \ m I tity 1 — x\”, or the mth power of 1 — «| will be equal to the mth power of - ae : : I nt 2 — 1 a— I the faid infinite feries 1 — — Aw — |—— Bx? — [= Ce? — 2 —— Des 7 22 3n 4a — |" Exs — &c, or (as it will be convenient to denote it for thie fake of Seba h : brevity), of the feries 1 —Bw — Cx* — Dx} — Evt — Fx? — &c ad in- finitum, We mutt therefore inquire what will be the terms of the feries that is BAe equal 364 A Det Sve ONUTR & B “CHOMNECTESR W. Tew G equal to the mth power of the feries 1 — Bw — Cx? — Dx? — Ext —Fxi =) &e, and what will be the figns + and — that will be to be prefixed to them, or what will be the terms of the feries that will be produced by multiplying the faid feries 1 — Be — Cx* — Dx} — Ew* — Fx’ — &c m — 1 times_into itfelf, and which of the figns + and — willbe to be prefixed to each of them: and we mult compare the feries fo produced with the feries which is produced F 3 sy : I —il 2nu— tI by the multiplication of the feries 1 + = Ax — = By? + toe Cart. a. read go gore 2h ee &c, OG + Bx Thy Cx? + D x3 Exe I F «xs — &c (which is equal to 1 + x)” ) m —1 times into itfelf, and which is mM therefore equal to the quantity 1 4+ «|*, or the mth power of the mth root of the binomial quantity t + «. 27. Now, from what has been fhewn above in art. 11, 12, —-—-—-— 27, it is evident, in the firft place, that if we raife both the feriefes 1 + B® — Cx? + Dx? — Ewt + xi — &c,.and 1 —Bxr — Cxr?'=— Dx? — Bvt — &c, to the mth power (m being any whole number whatfoever), or muluply each of them m — 1 times into itfelf, the produéts of the faid multiplications will be two compound feriefes confifting of exaétly the fame terms, of which the firft term will be 1; and, 2dly, that the figns + and —, that will be pre- fixed to the members of the feveral terms of the faid compound feriefes, will be the fame in all the terms which will involve the even powers of » in both feri- efes, but will be different in thofe terms of the faid two feriefes which will in- volve the odd powers of «. Therefore the compound {feries which is equal to the mth power of the fimple feries 1 — Bx — Cx? — Dx? — Ext —F ys — é&c, or 1 — a Av — — Be? — a Crs — — Det — oe Ex’ — &c, may be derived from the compound feries which is equal to the mth power of the fimple feries 1 + Bx — Cw* + Dx? — Ext 4 Fas — &c, or 1 — Ax — lament ats caspeeenes Sid ee i Det j+ Ae 22 4 : n Ex! — &c, by changing the figns of all thofe terms of the faid latter compound feries which involve the odd powers of x, into their contraries. But the com- pound feries which is equal to the mth power of the faid fimple feries 1 + Bx 5 / id — Cx? + Dx? — Ext + Fx) — &c, or 1 +— Ax — ~Bu? 4 2a ae NE | ua— I bd ae Rice Aoi = Det + ag — &c, is equal to the mth power z mm of 1 + |”, orto the quantity 1 + +)", and confequently (by what is fhewn in art. 115 of the foregoing difcourfe, page 299) is alfo equal to the fimple feries 1 “Bx? + po Diet 5 Be eee E +a—m Cite tn -+- 22 " m Fi es Ml — 24 —C % TlH EB.) ARAEOSIA Www A ke sTYHOESO:R BM. 365 jae ey Cp? b? os, Pes A Dy? t > car 8c. _ Therefore, if the figns + + 32 and — of thofe terms in this latk fimple feries which involve the odd powers of », that is, of the fecond, fourth, fixth, and other following even terms of it, be changed into their contraries, the fimple feries thereby produced will be equal to the compound feries produced by changing the figns of thofe terms of the compound feries that is equal to the mth power of the fimple feries 1 + Bx — Cx? + Dx? — Ext + Fx’ — &c, or 1 + ~ Ax _ o> Bx* + ae _ wee De+-+ “atm E «> — &c, which involve the odd-powers of «, or the figns of its fecond, fourth, fixth, and other following even terms, into their: contraries. But it has been juft now fhewn, that the compound feries pro-= duced by changing the figns of thofe terms of the compound feries that is equal. to the mth power of the fimple feries 1 + Bx — Cx* + Dx? — Ext 4+ Fas — &c, orb + —Ax — a Bw? + Sop Ce — a Dxt + ee E «> — &c, which involve the odd’ powers of «, into their contraries, is equal to the compound feries, or rather is the compound feries, which is equal to the mth power of the fimple feries 1 — Bx — Cx? — Dx? — Ex* — Fux’ — &c, meee eA | Bix? — |—— Cx? — eee of 8 RE ah PE as po 7 2n n 2 n &c. Therefore the fimple feries.that is produced by changing the figns of thofe terms of the fimple feries 1 + - Ax + “—* Bx? eee oe Ce n putn—m cy pt2 a pr+2n—m pu + 2n pr +32 Mm — 42 Dx +- Exs + &c, — 4n 7 Dae? t 3 _ ae E’x? 4 4. &c, which involve the odd powers of, or the figns of the fecond, fourth, fixth, and other following even terms of it, into their contraries,, will be equal to the compound feries which is equal to the mth power of the fimple feries 1 — Bw — Cx* — Dx? — Ext — Fx’ — &e, I Pars 24 — 1 A Sn wae: m ze 3n 4n 5m I &c, and confequently will be equal to the mth power of 1 — x) (which is equal to the faid laft-mentioned fimple feries), or to the mth power of the ath root of the refidual quantity 1 — x. On He ws 28. The reafonings in the foregoing article appear fomewhat perplexed in confequence of the multitude of words which have been made ufe of in defcrib- ing the feveral infinite feriefes mentioned in them. I will therefore, now re- peat them in a concifer manner, which will, 1 hope, remove all obfcurity from them; and for this purpofe 1 fhall denote all the feriefes that we fhall have oc- cafion to confider, by fingle letters. a0. bce 366 AUD MSOciOUUNR Sl El OO WICSEAR N PNG 29. Let the Greek capital letter P be put for the feries 1 -- An — = — Be? + cane Cet at Det + — Exs — &c, sd infeoten ori + By — trees + Dei — Evt + Fe! — Ke ad infinitum, which is equal to © 1+ Be , or the wth root of the binomial quantity 1 + »; and let the Greek re 24%— I Co x3 3% Ew’ — &c ad infinitum, or 1 — Be — Cx? — Dx I — Ext — Fx’ — &c ad infinitum, which is equal to 1— x} , or the ath root of the refidual quantity 1 — #. And let I™ f{tand for the compound feries which is equal to the mth power (m being any whole number whatfoever) of the fimple feries 1 +. By — Cx? + Dx? — Ex* 4+ Fx’ — &c, orI, or the produé which - arifes by multiplying the faid fimple feries # — 1 times into itfelf; and let A” {tand for the compound feries which is equal to the fame, or the mth, power of the fimple feries 1 — Bx — Cx* — Dx? — Ex+ —Fx* — &e, or A, or the product which arifes by multiplying it # — 1 times into itfelf. And let ‘the “Bee + Pe Na iy PE manne i Eile ‘ey * 4a Greek capita letter A denote the fimple feries1 + = Aw 4° m— 28 OL, “gel m= 3" D ys eo Eade ae Biyel)poigye ale (=e pt2 2 a put 2n pet 2n—™m yy, Fes pega py, P+4 Ig ita Sy Fo PAE yn pa &c ad put 32 pn + 42 pat su ™m infinitum, which is equal to the quantity 1 + x), or the mth power of the ath root of the binomia’ quantity 1-+*. And, laftly, let the Greek capital letter II denote the fimple feries which is derived from the fimple feries A by chang- ing thé figns of thofe terms in it which involve the odd powers of «, that i is, of the fecond, fourth, fixth, eighth, and other following even terms of it, into their contraries. With this notation the reafonings contained in the foregoing article 27 will be as follows. 30. From what has been fhewn above, in art. 11, 12, &c, —-— — 21, it is evident, in the 1ft place, that the compound feriefes I and A” will confit of exactly the fame terms, of which the firft term will be 1; and, 2dly, that the fizns + and — that will be prefixed to the members of the feveral terms of the faid two compound feriefes will be the fame in all the quantities, or members of the terms of the faid feriefes, which will involve the even powers of «, that is, in the third, and fifth, and feventh, and other following odd terms of the faid fe- riefes; but will be different in thofe terms of the {aid two feriefes which will in- volve the odd powers of x, that is, in the fecond, and fourth, and fixth, and other following even terms of the faid feriefes. Therefore the compound feries A” may be derived from the compound feries I™ by changing the figns of all the members of thofe terms of the feries F” which involve im them the odd powers of ToH B ROE VSI DV AL (Tuo Por HM, 307 of ¥, that is, the figns of all the members of the fecond, fourth, fixth, and other following even terms of it, into their contraries. But, becaufe the fimple feries : ji I is equal to 1+ x\2 , the compound feries I will be equal to the mth power 13 LZ of 1 + x\”, orto the quantity 1 + x)|*, and confequently (by what is thewn in art. 115 of the foregoing difcourfe, page 299) will be alfo equal to the fimple feries A. Therefore, if the figns + and — in thofe terms of this fimple feries A. which involve the odd powers of «, that is, in the fecond, fourth, fixth, and other following even terms of it, be changed -into their contraries, the fimple feries thereby produced will be equal to the compound feries that is produced by changing the figns + and — of all the members of thofe terms of the com- pound feries I, which alfo involve the odd powers of x, or the figns of all the members of the fecond, fourth, ‘fixth, and other following even terms of the faid compound feries I, into their contraries. But it has been juft now fhewn, that the compound feries which is produced by changing the figns + and — of all the members of thofe terms of the compound feries T™ which involve the odd powers of w, or of all the members of the fecond, fourth, fixth, and other following even terms of it, is the compound feries A”. There- fore, if the figns + and — in thofe terms of the fimple feries A which in- volve the odd powers of «, or in the fecond, fourth, fixth, and other fol- lowing.even terms of it, are changed into their contraries, the fimple feries thereby produced will be equal to the compound feries A, and confequently — will be equal to the mth power of 1 — x\*, or of the mth root of the refidual . quantity 1 —.*, which is equal to the fimple feries A ; that is, the fimple feries ; I TT will be equal to the mth power of 1 — |”, or of the wth root of the refidual ™m quantity 1 — x, or to the quantity 1 — ~|”. bee Eg 31. And, if fill greater brevity be defir’d, this demonftration of the equality between the fimple feries II and the quantity 1 — «|» may be expreffed in the manner following. I Since the fimple feries T is = 1 + x|*, it follows that the compound feries a... i * T will be = the mth power of 1 + w|#, or = 1 + x\ 74 But (by the foregoing difcourfe, art 115, page 299) 1 + *\# is equal the fimple feries A. Therefore the compound feries F” is equal the fimple feries A. Therefore, if we change the figns + and — of all the even terms, or terms involving the odd powers of «, in both thefe feriefes 1” and A, into their con- traries, the ferieles thereby produced will be equal to each other. ~ But 368 Aw D InSocwOuUTR 408 1CO NICs RM Ne Nes But the feries produced by this change in the figns of the terms of the com- pound feries I will be the compound feries A”, by what has been fhewn above in art. 11, 12, 13, &c —— — 21. And the feries produced by this change in the figns of the terms of the fimple feries A is the fimple feries TI. Therefore the fimple feries II will be = the compound feries A”. I But becaufe the fimple feries A is = ng Inte it follows that the compound feries A” muft be = the mth power sie aaa or = the qos catdaaae Therefore the fimple feries IT will be equal the quantity esas y deren merne 32. We have now demonftrated in a manner that, I aie will be thought fatisfactory, 1ft, that the quantity 1 — x\*, or the zth root of the refidual quan- . . . . . — 22) —IT tity 1 — x, is equal to the infinite feries 1 — — Aw — |- - By? — | m 2% 32 ae ih Cyr 422 Dx* — (A= yy — &c ad infinitum, in which all the terms that come tae the firft term 1, are marked with the fign —, or fubtracted from 712 the faid firft term; and fecondly that the quantity 1 — x\#, or the mth power of the nth root of the faid refidual quantity, is equal to a fimple feries (which we have called II) confifting of the very fame terms as the feries (which we have mM called A) that 1s equal to the quantity 1 + «|z, or to the fame mth power of the fame nth root of the binomial quantity 1 + », and derived from the faid feries A by changing the figns of thofe terms of it which involve the odd powers of x, or the figns of its fecond, fourth, fixth, and other following even terms, into their contraries. Nothing therefore feems now to remain to be done with re- fpect to this fubject, but to illuftrate thefe two feriefes by applying them to a few particular cafes, or examples, in the fame manner as we illuftrated the fe- if m riefes which are equal to the quantities 1 + x|# and 1 + a” in the beginning of . the foregoing difcourfe concerning the binomial theorem in art. 8, 9, 10, 11, 12, &c. — — — ig of the faid difcourfe, pages 201, 202, 203, &c. — — — 209. “Thefe examples may be as follows. pila cee ala eat Examples of the extraction of fome particular roots of the refidual quantity 1 — x” by means of the feries given above in art. 2. ER seems 33. In the firft place we will extract the {quare root of the refidual quantity . . . . . I a — x by means of the feries given in art. 2, to wit, the feries 1 — —> Ax — 4—YI 22 HH RO sRaboselDW AL WHE OR BM. 369 fn 2u—~— 1 3n — 1 jn — 1 - = Bee — ¥ Cre = lcqul? Det — a Ews — &c ad ine Jinitum. I I Now in this cafe 1 — x\s is = I—wl2, ormis = 2. Therefore 27 is = 29X29) ="4,-and 3” 1s (= 3 X 2) = 6, and 4” is (= 4 x 2) = 8, and £2 6 £= 5.x; 2) aozjand confequently 2—1 is (= 2—1) = 41, and 24-1 is(= 4—1) = 3, and 3” —1 is ae Oper) Lome SANG AW mm 15 (= 8—1) = 7. We fhall therefore have 1 — ~Ax yaw a panes PSV OF 1 34=* p ys SPR ee ay A te i a pes 42 52 2 4 3% a pls ras GA, ce ee ae gut 7x5 = Cx 3 Dx ot Re ea : Ph ATE EE Se ae ie &c. Therefore, if the feries given above in art. 2 is really equal to 1 — x)”, the I quantity 1 — x«|2, or the fquare-root of the refidual quantity 1 — x, will be ; w “ x + ae 9x5 equal to the feries 1 -- —->—3Z— TB se6 &e. Q; E. I. 34. Now « that this feries is really equal to the fquare root of 1 — «,” will appear by multiplying it into itfelf. For we fhall find that the produé of the faid multiplication will be 1 — x. This multiplication may be performed in the manner following. 2 3 4 5 5 aay TRI ra 56 Bec (EHF E-§- Be z 3 4 s (En F- 5-8 Bk i 4 St 4+ + &e ae taste hase + &e 3 4 5 “Hie Figs + aa + Be = a + & — Ee + &e It appears therefore that the feries 1 — a 7-2-3 &c, is really and truly, as far as relates to the fix firft terms of it, the fquare-root of the refidual quantity 1 —*; and confequently that the feries given above in VoL, Il. 3 B art. 37° AS DF SAO O.UIR SF CAO INGCORSER WM Bee iG I art. 2 for the value of 1 —w, or /"[1 — w#, is true in the cafe of fquare- roots. I 35. In the next place we will inveftigate the value of 1 — x\3, or the cube- root of the refidual quantity 1 — 4, by means of the fame feries 1 — + Aw — a — I 2u2— 1 n= T 1 — . . — Bur — C3 — a7 Dat — (eo Ex’ — &c ad infinitum. 3n Now in this cafe “is = 3; and confequently we fhall have 22 = 6, and 3n = 9, and 4m = 12, and 5u= 15, andu—1i = 2, 2n—1 = 5, 3n—1 = 8, and 4mn—1= 11. And therefore the feries 1 —~ ~Ax — |= By? a 22 2u4— 1 3 3n— I ni vas 5 : : : Sie Cwe ee Sey Dx Ps Ex’ — &c, will in this cafe be = 3 a §x* 10x4 2245 & ees ees ee ee ee a Ce z ‘ cy D gueht. 243) gag728 pee 36. Now “that this feries is really equal to the cube root of 1 —¥w,” will appear by multiplying the faid feries twice into itfelf. For we fhall find that the product of the faid multiplications will be equal to1——w. Thefe multiplica- tions will be as follow. & & wi & we -_ [@) > & > is} N S wm T ——— et ee ee Eee eee ss & 3 Laide weer hive ues, : 3 9 81 243 729 x “? gz? 10x4 2245 THE RESTDUAL THEOREM, 27% 2 2 3 a 4 aw S ra nd lade. ee Sa 3 9 Be 7ath Zao we Linge 3 9 81 243 72 CO (ae NR ea OTE OPN oe Sa 3 om 81 243 729 2x X 4x 7x —— + — + &c 2. + a ot 243 729 a a we a 4x hy sng Byers Sige &c 9 ai Pe ee a 5* 10% Sx 4 81 243 729 oer ox* ox § Oye 10% 20% we &c 243 iat) 22% ~ + &c 729 if ; I—vx * Fi * * &c. : 3 uv 2 3 4 a8 fap herchore tha he hia 1 Se SO eee 9 Br 243 729 &c, is really-and truly, as far as relates to the firft fix terms of it, the cube-root of the refidual quantity 1 — +, and confequently that the feries given above 1 in art. 2 for the value of 1 — «|”, or /* 1 —w, is true in the cafe of cube- roots as well as in that of {quare-roots. Examples of the extraétion of the roots of Jome particular powers of the refidual quantity 1 — x by means of the Series given above in art. 25, and which in art. 29, 30, and 31 is denoted by the Greek capital letter Tl. RS ee AE 2 _37. In the next place we will inveftigate the value of the quantity 1 — *|3, or the cube-root of the fquare of the refidual quantity 1 —., or (which comes to the fame thing) the fquare of its cube-root, by means of the feries given above in art. 25, and denoted by the Greek capital letter IT in art. 29, 30, and he 3 : 2 Now it is fhewn in the foregoing difcourfe, art. 12, page 203, that 1 + *\35 or the cube-root of the fquare of the binomial quantity 1 ++ *, is equal to the 3 4 5 . ° O44 ea ME _ Kc, ad infinitum. Therefore the f{eries 1 Thea Sg eiaall ee 9 81 243 729 2 ’ quantity 1 — x\3 muft, according to art. 25, be equal to a feries derived fom aes this ~ 37% ARS TD EMSFER OU TR ae CQ 'NLCLECR ® PNG this feries by changing the figns of its fecond, fourth, fixth, and other follow- . ; : . . (ori 26 x 4x3 ing even terms into their contraries, that 1s, to the feries 1 — ree ward »$ 4 . ‘ x 14 __ &c, in which all the terms after the firft term 1 are marked with 243 i7y : ; the fign —, or are fubtracted from the faid firft term. Therefore, if the propo- fition contained in art. 25, is true, and the feries juft now fet down, and which, ” in the foregoing articles, is denoted by 1, is really equal to 1 — x\”, the quan- 2 tity 1 — x\3, or the cube-root of the fquare of the refidual quantity 1 —w, will ; 2 3 sith der pres be equal to the feries 1 —=* —~ —— F-— 7 Be, 38. Now “ that this feries is really equal to 1 — x\3, or to the cube root of ‘¢ the {quare of the refidual quantity 1 — «, or to the cube-root of the trinomial “ quantity 1 — 2x” + xx,” (for both the quantities 1 + 2” + «x and 1— 24 + xx are equally called ¢rinomial quantities, though in the latter quantity the middle term 2 is marked with the fign —, or fubtracted from the fum of the two other terms) will appear by multiplying the faid feries twice into- itfelf. Thefe multiplications will be as follow. : ee ee ee eet eee ee ce eee ee eee ee me . bes 2 ne Bea ht "Haw how am Jas page a9, 8 ee ings + & me ea aoe HP Bec Benya exci aahigg Ha tg ofa &c. THF RLEVS'T DM AYL TRE OR i Me 373 nt! 20% 4x3 gue 8x5 por Donets et len es BT a ee a EO gilp toy a: Sind 243 ui 729 BN ig ok BAS 7* 14a , Toes a gts eal 3 9 81 243 “fT 4x pias x3 xt Sy 5 pe OER BSE as + 2 &c Td . a3 h29, 2x z ke a wey woe LORE el pth 3 9, ¥is 243 729 ad pani g li talah ala gaa Bp 9 27 81 92 nee = 16x4 Sx 5 “at oe : 243 729 par: 7° 28%5 iy ae ; ‘ 243 ‘fe 14% — c 729 + I—2K” + XK * # % &ce. . 2 3 4 5 It appears therefore that the feries 1 — aie Abe A 784 aE ae 81 243° 729 : des» o . is really and truly the cube-root of the trinomial quantity 1 — 2% + xw, or of the fquare of the refidual quantity 1 — x, and confequently that the propofition contained in art. 25 is true, or that the feries II is really equal to the quantity m 1 — x\”, in the cafe of the cube-root of the fquare of a refidual quantity, or when m, the numerator of the fraCtion ~ (which is the index of the power of mh 1 — x in the quantity 1 — «|*) 1s = 2, and #, the denominator of the faid faction, 1s = 3. 1 39. As another example of the inveftigation of the value of 1 —x)* by means of the feries II, we will fuppofe m to be = 3, and z to be = 5, or = 3. 1 — x|# to be equal to 1 — «5, or to the fifth root of the cube of the refi- dual quantity 1 —«, or to the fifth root of the quadrinomial quantity 1 — 3% + 3x7 — x7. Now it appears by art. 14 of the foregoing difcourfe concerning the binomial 3 2 theorem, page 205, that the quantity 1 + x]5 is = the feries 1 + $ — 5 4 $s . . + 7H" 23") 4 3574" __ & , ~Therefore the feries II will be 1 — = — ote 125 625 15625 F 25 qa 2txt 35785 MEE Ose ic6a5 &c, and confequently, by art. 25, the quantity 3 A 2 3 4 G 7—x\5 will be equal to the feries 1 — = —.— Pb ype! wa a 5 25 125 625 15625 “? DLP S¥CTONUTR S* Ev CO WTC IER Ni EPG 3 40. Now * that this feries is really equal to the quantity 1 — x|5, or to the «© fifth root of the cube of the refidual quantity 1 — x, or to the fifth root of the «* guadrinomial quantity 1 — 3x + 3x* — #’,” will appear by raifing the faid feries to the fifth power by multiplying it, firft, into itfelf, whereby we fhall obtain its fquare, and afterwards multiplying the faid fquare into itfelf, whereby we fhall obtain its fourth power, and, laftly, multiplying the faid fourth power of it into the faid feries itfelf, whereby we fhall obtain its fifth power. For we fhall find that the produét of thefe three multiplications will be the faid qua- drinomial quantity 1 — 3x + 3** — «3. Thefe multiplications will be as fol- low. ahi La fi hs “e¥ spp Accolia Seah 5 25 125 625 15625 Paid at CY Slip ep tinea fae PY GIN RR! fs 4 Se ae at ; ee. — 3 + BE spi atte 7 Sere Ae ease Gina ahs pA Oe ae 6x 3x 4x3 ox4 12645 ag aa 25 r res + 625 15625 8c 6x 3x 4x3 ox* 126«5 Ngee 5 + 25 126 625 15625 &c ‘ Ox i 30 ay) 4u3 4 gx+ RY 126% Soe 5 25 125 625 15625 _. O*. 36x 1873 2444 54x58 &e 5 25 125 625 3125 hy 1843 gx4 124% 25 125 625 mis 3125 ar Sc 4x3 24x 12x75 4+ &e 625 aia. 126% &c 15625 12 42x pel 28x%3 __. 21«4 nt 168% 5 arc, 5 25 125 625 15625 This y eS RS SL DW AL ,.8. 2.0.28 BM. 375 Pe) This is thé fourth: power of the feries 1 — = — Xai Sia ee ASN. -OCFs ee . 5 25 125 625 15625 : 12¥ 424% 28x 3 2t¥* 168%5 5 25 eT 625 15625 . py cia cae ae LAA or 1. ae SRM ee STS ils &c 5 25 125 625 16625 ; 124 420% 2843 21x04 16845 5 25 126 625 Ae 16625 wy ax 36x? 126%3 8444 6345 5 25 125 625 rt 3125 bike le Ge 36x 126x4 84x5 25 125 625 3125 plas 943 8404 2945 125 x cee ahaa + &e I 3 le rie ts &c 625 arse 357x5 15625 7 ade I— 3K + 3x7 — 8 % # , 2 3 4 $ It appears therefore that the feries 1 — Be cls Sy ana a eal ala PR 26 126 625 15625 &c, is really and truly the fifth root of the quadrinomial quantity 1 — 3x +. 3x° — «3, or of the cube of the refidual quantity 1 — x, and confequently that the refidual theorem laid down in art. 25, to wit, that the quantity 1 — x\~ is equal to the feries Tl, is true in the cafe of the fifth root of the cube of a refidual quantity, or when m, the numerator of the fraction = (which is the index of G m the power of 1 — # in the quantity 1 — x]*) is = 3, and #, the denominator of the faid fraction, is = 5. 41. Inthe foregoing examples the denominator z of the fraction = (which ts 74 the index of the power of the refidual quantity 1 — in the quantity 1 — x\”) has been greater than m, the numerator of the faid fraction. We will now give m an example, or two, of the expreffion of the value of the quantity 1 — x)» by means of the feries II, when the denominator az of the fraction, or index, —, is lefs than its numerator m. 3 And, firft, we will inveftigate the value of 1 — x|2, or of the fquare-root of the cube of the refidual quantity 1 — « by means of the faid feries IT. Now it appears by art. 16 of the foregoing difcourfe concerning the binomia] . 7 theorem, 376 A DISCOURSE CONCERNING x Ih . ’ a! 7 theorem, page 207, that the quantity 1 + x)? is equal tothe feries r + as 3 “! - . . ee 4 3% __ 3*" 4 &c. =Therefore the feries II will be r — 3% = + sae 8 16 720 0Ne He 206 x3 3x4 3x5 f 4 ¢ 6 . a3 a6 128 as Sy 256 + &c, and confequently, if the propofition Be ey: In 3 art. 25 is true, 1 —.«\2, or the fquare-root of the cube of the refidual quantity I —w«, or the fquare-root of she obit Mane Tage I — 3x + 3x7 — x3, 2 4 = cae + &c. will be equal to the feries 1 — = = + a2 hoe = 42. Now “ that this feries is really equal to the {quare-root of the cube of ‘¢ 7 —w, or to the {quare-root of the quadrinomial quantity 1 — 3”. + 3«7 — 66 x3,” will appear by multiplying the faid feries into itfelf.. For we fhall find that the produc of the faid multiplication will be the faid quadrinomial quantity. This multiplication will be as follows. Boe ae eo ee ater aes hy ig 256 + &C 8 ‘ cla GAY siege a Es Sica am 8 TaeiG abst sacha OF 3h «9 SN ae tghe deinen nipie’ air iat iss Sud 2: 128 SAL Nagar LN aS DIN at MS «IE 2 4 16 32 266 &e ll aligh Sc Fy La + 3 76 toseqe ares ee e eS poe jst oss 128 per te ils 8 s + =e 256 Rare Re pee oh pers *% bd hd It appears therefore that the feries 1 — ae een ee + + oe + && is really and truly the {quare-root of the ie ett’ quantity 1 —= jx + 3x — x°, or of the cube of the refidual quantity 1 —w, and confequently that the m refidual theorem laid down in art. 25, to wit, that the quantity 1 — «| is equal to the feries II, is true in the cafe of the fquare-root of the cube of the refidual quantity 1 — x, or when m, the numerator of the fraction — — (which is mm the index of the power-of 1 — w in the quantity 1 — x\7) is = 3, and #, the denominator of the faid fra€tion, is = 2. 43. I TOMKE IRIE Sel Dw AL. THE OR EM. 379 * . 43. I thall add one more example of the feries 1, in which the numerator 3 2 . : of the index — fhall be greater than its denominator a, Let it be required to find, by means of the faid feries II, the value of | — x)3, or of the cube-root of the fifth power of the refidual quantity 1 —.~, or of the ‘ cube-root of the fextinomial quantity 1 — 5% + 1047 — 1048 + 5xut — x, It appears by art, 18 of the foregoing difcourfe concerning the binomial 5 theorem, page 208, that the quantity 1 + x\3 is equal to the feries 1 +4. x a 3 io a a ——— — ——— — = Ge 9 Sr 243°. 729 = . : : Therefore the feries I will be 1 — = + S ++ _ ae Ptah ay MT we &es 2] +5 and confequently 1 — w|3, or the cube-root-of the fifth power of the refidual quantity 1 — #, or the cube-root of the fextinomial quantity 1 — 5¥ + 10x* — 10%% + 5x* — x*, will be equal to the feries 1 — 2 a he coer — ED Sade bs 9 ey. 7*° -+- + &c, © 729 44. Now “ that this feries is really equal to the cube-root of the fifth power « of 1 — w, or to the cube-root of the fextinomial quantity 1 — 5” + 10% ~ « 1ow%% + 5x* — x°,” will appear by multiplying the faid feries twice into itfelf. For we fhall find that the product of the faid multiplications will be the faid fextinomial quantity. ‘Thefe multiplications will be as follow. 5 xt 7x5 ne LF pitica ot lant ie = + &e Et Ee Bae 5a 5H 52 Tie: r it 2 be b 5 fe ee 0 33 aecrbest (27, 243 729 abet? ) ee Serr ag os ¥ 26x mg 7p + Be tei oC OE EI 2g ae Gra 3 9 81 243 729 ae: - | 3 374 A DISCOURSE CONCERNING THE RESIDUAL THEOREM, . ae er ee Ut ) 3 243 129 kd a Sees 54 14 wo CE Wee TUE aan 10% 35.4% 140*3 ght L4«° Sosa Fak 9 81 2.43 729 &e esa if 50x” ay 175%3 700.«4 175% ee 3 iB af pa | 729 ba% 508% 175% 700% T a a} 81 729 wx: 5x3 50x44 Tha & — XC a $1 243 720, 5 Sox pr — Cc w 243 a9 si hae T Syag eae eT i — 5x + 10x" — 10K? + 5x* — x? &c. : 2 3 4 5 It appears therefore that the feries 1 — Sey 3 a + ne + = Occ. is really and truly the cube-root of the fifth power of the refidual quantity 1 — «, or the cube-root of thesfextinomial quantity I — 5” + 10%* — 104% - 5.x* — x, and confequently that the refidual theorem laid down in art. 25, to wit, that 712. the quantity 1 — «| is equal to the feries II, is true in the cafe of the cube- root of the fifth power of the refidual quantity 1 — », or when m, the numera- tor of the fraction — (which is the index of the power of 1 — # in the quantity mM 1 —«\*) is = §, and#, the denominator of the faid fraction, is = 3. 45. Thefe examples will, I apprehend, be fufficient to illuftrate the two theorems delivered above in art. 2 and 25 of this difcourfe concerning the I 2 quantities 1 — x|* and 1 — |, or the mth root of the refidual quantity 1 — ~ and the mth power of the faid mth root. And therefore, having already given demonftrations of thefe theorems in art. 11, 12, 13, &c....and 26, 27, 30; 31, 32, &c, I thall here. put an end to this difcourfe. End of the difcourfe concerning the refidual theorem in the cafe of Sraétional powers. ae ‘eel A Oe Silas Rell G ae Be OF EXTENDING fee LD) A Nis ROOD ULE FOR RESOLVING THE CUBICK EQUATION Keyan wr yt Par TO THE RESOLUTION OF THE CUBICK EQUATION 3 rs P g* VW—yr=r, when —- is of any magnitude le/s than % 4 I q3 : a/9 or — X +, or when ¢r ts lefs than f2 ——; pies a5" = aes 3/3 By the Help of Sir Ifaac Newton’s binomial and refidual Theorems in the Cafe of Roots, which have been demonftrated in the two preceeding Difcourfes. By FRANCIS MASERES, Esa F.R.S. CURSITOR BARON OF HIS MAJESTY’S COURT OF EXCHEQUER. 3C2 3 Duras TAR AO x 3 | ~ ae rhe 3 ; <4 ‘ " Pte 4 ie ot") iat fT ty ; | sey) a Fal - iN Cf & MALS SMM Diet ACMA Rh j ( 4 i ; : Y wt uy Hes i, a . | . ¥ i -_ CE PT a eee a Fy Lys oer Us M% RW P| sep aa ie sia ~ = ey ps j ‘ ad ; 4 4 Sis in ‘ “es it Oe aA or aa ey ET eG io Yew Oe ea oe : . . | ak cae aoe Se a Lins Ait wade ce ils Bhat aah ee Wy) eh atria al ae whe Ch ry | Sate, Seu, | spk” Deans is weres E ’ r N : Ef ey i> pet ; | 3 4 ; oe Rr Me AH GAA Re URGE Nom gt SHS aceon essa apa ) a % _ Db S Mabal as sav ., oe Asesia) gaameaany Men Ne wi romain vet > a . yf ett 14 - : Pita we ring or ; oi ; a Nee ag : pO Lil ahr : roan LP _" ] . SF x ry *¥ ’ 7 oe ie , 2 ‘ ' s _ \ A 5 as . > is ; my: . ‘ vi -! hs wt si he aa wn pat TATA diowhnag a he ' i et a os '¢ ty ry 2 & ¥ te EAM. at 40 MOKA a iy te ) | ' 4 ; af pairs Wi f A he “4 rein Si ay { uf ane Sis *" 2M, et S0F ae Mi 2 . ae Ee FREE DORIS 8 “ed - M E T H O D OF EXTENDING eR BA.” Nise fe Bol hb. eke. ART. I. HE binomial and refidual theorems, which have been demon- {trated in the preceeding tracts both with refpect to integral and to fractional powers of the quantities 1 + » and 1 — w#, are of very extenfive ufe in many other branches of the mathematicks as well as in the conftruction of the logarithms, or meafures, of ratios. And, amongft other fubjedts, they may be applied to the refolution of cubick equations in which the {quare of the unknown queue is wanting, to wit, of the equations y* + gy =r, and y? — gy =r, and i) — 7; fo as to enable us to find the values of y, in all the poffible ‘cafes of thefe equations, or in all the different relative magnitudes of the co-efficient g, and the abfolute term 7, that can be fuppofed, not excepting that cafe of the fecond equation y? — gy = r in which ri is lefs than ans or rr is lefs than an, or ris lefs than 242, and which (from its not being capable of a dire& and immediate refolution by the help of Cardan’s rule for refolving the faid equation 3 y? — qy =7, in the firft cafe of it, or when ot is greater than at or 77 is 3 ° greater than ee or 7 is greater than 24v1) has obtained amongft algebraifts the name of the irreducible cafe. Even in this cafe we may always, by the help of Sir Ifaac Newton’s binomial and refidual theorems in the cafe of roots (which have been inveftigated in the two preceeding difcourfes) derive from one or other, of Cardan’s rules an expreffion of the true value of y in the faid equa- tion y3 — gy =r. The method of doing this, when — is lefs than E, but 3 3 ; ; greater than = . 27? OF Ve has been explained at confiderable length in a paper of mine 982 A METHOD OF EXTENDING CARDAN’S FIRST RULE »mine publifhed in the Philofophical TranfaCtions for the year 1778, which pro- ceeds upon clear and intelligible principles, without any mention of impoffible roots or impoffible Sa of any kind, or even of negative quantities. But when 2 is lefs than , or © orrris lefs than ( 4g" 4 2X27) 54 2 ye V4 ee the method explained in that paper will not enable us to find the value of ag: ot or) a? tT is lefs than /2 y in that equation; becaufe the feries obtained in that paper for the faid value will not in that cafe be a converging feries. Nor had I, at the time of publifhing that paper, difcovered any method of deriving from Cardan’s rules an expreffion of the value of y in the faid equation y? — gy =~ 7 in this fecond branch of the . ° . 3 3 irreducible cafe of it, or when "i was lefs than - - 3? OF ei , or 7 was lefs than 7 aK vs, But I have fince found out a way of doing it by the help of Car- dan’s firlt rule, which gives us the value of y in the equation y* + gy == 7, or qy +y* =r. For from the expreflion of this value of y in the faid equation . 1 ? qy +9? =r we may, in the cafe fuppofed, or when * is lefs than ae or ae : A _, or r is lefs than 4/2 X it, derive an expreffion of the value of the leffer of the two roots of the equation gy — y? = 7, by the help of the two fe- riefes for exprefling the cube-roots of the Gitiotad quantity 1 + » and the refi- dual quantity 1 — » obtained by means of the binomial and refidual theorems : and from the value of the faid leffer root of the equation gy —y? = 1, we may afterwards derive the value of y in the equation y? — gy = r by the e refolution of a quadratick equation. And this extenfion of Cardan’s firft rule Oy hick gives uis the value of y in the cubick equation y? + gy = 1, or q) + 3 = r), to the difcovery of the leffer root of the cubick equation gy — y* =r, when_r is lefs G/4 than 4/2 x =“+, may be made in aclear and intelligible manner, without any mention of impoifible roots, or other impoffible quantities, or even of negative quantities, as well as the former extenfion of Cardan’s fecond rule (which gives us ce value of y in the equation y? — gy =r in the firft cafe of it, or when it) to the difcovery of the value of g? — is greater than 2 re or r is greater than 4 4; in the firft branch of the fecond cafe of that equation, or when - is lefs than, 3 3 . . ore or-“. The manner of making this extenfion of atte s firft rule to ae difcovery of the leffer Toot of the SAAR qgy—y=r, A fhall now endeavour to explain. , but greater than The FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. - 98¢ a I I The values of the quantities 1 + x\ 3 and 1 — x\3, or 73 (i + w and-4/%% (x - — x, expreffed in infinite -Jeriefes by means of the binomial and refidual theorems. 2. It has been fhewn above in the difcourfe concerning the binomial theorem I é in the cafe of fra€tional powers, art. 51, page 242, that 1 + «|3, or ft + x or the cube-root of the binomial quantity 1 + «x, is equal to the infinite feries I 2 5 8 TI 14 1 — v— — Bx — C x3 —_— — x4 faa oe ea wo uy 1} gg hee Ax z 0 hie peed Te aeuer pnts a = 6 f 2 Gua — £2 Fx + 23° T 9 “3a Ki 2? at satuas — = Mx" + 35 Nx 24 27 3° 33 39 39 sO A ; = Ox" he a Px — ag Qe" + f Rw? — 3 Sx* +. &c ad infinitum, or ee eS ga MC rox.“ 22kse rege 37447 93548 21,5059 pe eg, ex 729 7 6561 + 19,683 =a f 243 729 SMA BQ5093 592949 155945323 55,913x7° T47,4074'* —_ ¥4179,256.477 32174920019 8,61 7,640.r'4 45782,909 14,348,907 129,140,163 387,420,489 1,162,261,467 70,664,648 x75 1945327578247% , §379259,102477 — 13,431,479,050478 ana a — ee See | Ott Cc ad n- 10,460,353,203 — 31,381,059,609 = -944143,178,827 2541, 865,828,329 Jnitum. Therefore (by what is fhewn in the foregoing difcourfe concerning the refidual theorem in the cafe of fractional powers, art. 23, page 360) the quantity z I — “3, or4/*f1 — x, or the cube-root of the refidual quantity 1 — x, will be equal to a feries confifting of the very fame terms as the foregoing feries, but with the fign — prefixed to every term after the firft term 1, inftead of every 3 ; I z 8 other term, that is, to the feries 1 — = A om = 1 Bn A aa il 2 al = Dx+ ae hc FOB! ES SE SIAR eee 22 Boke +23 iD +4 = Ex = Fx is Gr oa Hx P Ix ie eS Mars By Boge sper Hee Re 42 me 32 Sx* — &c ad infinitum, or 1 PER eRe we Rigs See ea EK Gh | 2A Lally GE 54 ; a ea 243 729 6561 3747 935%" 21,505%2 555913499 147,407.44 1,179,250"? 19,683 59,049 155942323 4,782,969 14,348,907 129,140,163 3174,920473 8,617,640.%14 70,664,648 «15 194,327,78247° 5379259,1024%7 387,420,439 1,162,261,467 10, 46073 53,203 31,381,059,009 94,143,178,827 1394315479050%1" 25541,865,828,329 The ~e. 384 A METHOD OF EXTENDING CARDAN’S FIRST RULE The value of /3{i +x — V2[1 — x expreffed in an infinite feries. ~ 3. Therefore the difference of the two quantities »/*/1 + wand o/*[{r — x, or the excefs of »/*{1 + x, (or the cube-root of the binomial quantity 1 + x,) above /{1 — «, (or the cube root of the refidual quantity 1 — wx, will be) equal to the excefs of the former of thefe two feriefes above the latter, that is (it, for the fake of brevity, we denote the co-efficients of the fecond, and third, and fourth, and other following terms of thefe two feriefes by the fingle letters B; C, D, E; F, G,'H, 1;K; L, M, N,'O,;P,; QR; S,'T, &c)Ptorthe exeefs of the feries 1 + Bey — Cx? + Dx? — Ex* + Fei — Ge 4) Hy — Iw? + Kw? — Lx? + Mat — Nx? 4+ Ox3 — Px + Qas — Rav? + Sw? — Tx" + &c above the feries 1 — Be — Cx? — Dxs — Ext — Fy — Gr — He — le? Ke = List? ~ Me" — Nx? Oa Qx — Rat — Sx’? — Tx" — &c, and confequently to the feries 2 Br ++ 2Dx3 + 2Fx + 2H»? + 2Kee + 2M" + 2 On + 2Qaus 4 2 Sx? + &c. Of the root of the cubick equationy? + qy =r, orgy +yi? =P, according to Cardan’s firft rule. er RE LEE TR re NS 4. The firft of the two rules for the refolution of certain cubick equations ufu- ally known by the name of Cardan’s rules was not invented by Cardan himfelf, but, about 30 years before the publication of his book, by one Scipio Ferreus, of Bononia, or Bologna, in aly ; as Cardan himfelf informs us. It relates to the re- folution of the cubick equation y? ++ gy = 1, and is true :in all the poffible cafes of that equation, or in all the different relative magnitudes of g (the co-efficient of the unknown quantity y), and the abfolute term, and confequently when 7 is b f ; lefsthan 42 x 2%%, which is the cafe that we are about to confider. And it gives us two different expreffions of the value of y in the faid equation, to wit, eres. i ate v[e+2 Us aac £ +4“, and, 2dly, 3 r q. orr 3 r a (ESS . Pee g v {e a a ay: + /f SC ++ re (if, for the fake of bre- vity, we:put ee = re ants £ + 7 it, fele+s 3% *[e + 5, and, F 7 adly, Lo) FOR THE RESOLUTION OF CUBICK EQUATIONS, &c, 385 odty, lets — {(—e +s, or, rather, (becaufe ss is- greater than re or e¢, and confequently s is greater than —, or ¢, and therefore ought to- be af Ne 8 placed before it) 1ft, 1/2? +e 3/3 +e, and, adly; W3ls te — of? (s —e. See my differtation on the ufe of the negative fign in Algebra, art. 208, 209, 210, 211, 212, pages 178, 179, 180. It is the fecond of thefe two expreffions, to wit, the expreffion (/3{5 + e— \/ 3s — ¢, that we fhall have occafion to confider in the courfe of the following articles, The value of the root of the faid cubick equation expreffed by an infinite feries by means of the binomial and refidual the- orems. . e : A Ge Nows eis =s x1 + =) and $ —\e¢ is = 5 x1 —-—. Therefore ‘ RY VWs +e willbe = Vis x 31 +, and fis —e will be = /* 5 x Vad —-; and confequently (ste — VW? [s—e will be = > 5 x Va 35 x Vit—t= Vis ae +in-yi{r-4 Therefore the root of the cubick equation)? + gy =7, orgy +y3 =7, will ber 4/3's x? ppin-ys|p—s. But, fince /*{1 + « — o/?{1 —« is equal to the feries 2Bx +2Dx* + 2Fxe> +2H«x? + 2K ae + 2Ma™ 4+:2 Ox8 + 2Qxa 4+ 25477 + & ad infinitum (as is fhewn in art. 3) it follows (by fubftituting = inftead of w in the terms of the faid equation) that veh +- = _ y>| I — < will be equal to aco. 2 Be 2D 2Fes 2He7 2 K e? 2mMe™ 2063 2Q0¢5 the feries tse zt se + a 57 5? git 533 st of il + &c ad infinitum. "Therefore Y?s xl¥?{1 + —-—v3{1 — ~ will be iT A 5 5 2D ee 2F¢é 2He!7 2K¢ 2mMe™ 20e3 s3 $5 s7 59 ie i3 equal to 7? 5 x the feries — + p 208% 4 28 | 8c ad infinitum; and confequently the root of the cubick + mage + &kca infinitum ; and confequently : equation y? + gy =r, or gy +.y3 =F, will be equal to Y? s x the faid feries 2Be PEIY GS ol) 3 Bes 2He? 2Ke? 2mMei! 20¢'3 2 gets 2set7 & ae + me ea? -- —— = ae ae = - -c 53 55 7 5? ches gt3 5 : JS Vor, Il. 3 D ad 386 A METHOD OF EXTENDING CARDAN’S FIRST RULE D F 3 H Hf K e? Met ad infinitum, or to 24/3 s X the feries ~ — = + = a tept oot oa Fae 0 33 aes $é p38 pis cents B, C, D, E, Fy G, Hy, bec) ay#s-x the ies 4. MOP la 2 or fer ie ie a the co-effi- 81 53 2958 374.67 SIT, COk 2S Se 1479407 et! 3,174,920e83 70,664,648 e%5 19,683 57 1,594,3 235? 14,348,907 s** 387,4.20,489573 10, 460,35 3,203 st5 ,»162 e?7 Ty RY eae a + &c ad infinitum. ; ? 2 We will now proceed to give an example of the refolution of a cubick equa- tion of the faid form y? + 9y = 7, or gy + y? = 7, by means of this expreffion. An example of the refolution of a cubick equation of the foregoing form y* + ay =r, r qg+y3 = 7, by means of the expreffion 2/35 xX the Series - oo ie ee 8153 we “aie 374¢e7 21,505 e9 eprehye: 7298° 1968357 1594532389 -14,348,9075"* 39174,929¢73 70,664,648 e'S §375259,162 677 3875420,4895'% ° — 10,460,353,2035"° 9.4, 1435178,827 577 + &c ad infinitum, 6. Let the equation that is to be refolved by means of this expreffion, be ye 15y = 4, or 15y + ys = 4 i Here g is = 15, andris = 43 and confequently & IN Cee Sy = 5, and . as 3 ; ; r\2 4 is C= +) te fe and 7 1s (= 4 i fe = 195, aga = is (= ae 2] ) + 4. We hall therefore have ss (= % _ ot oa = 125 -- 4) =) 129;nenelieeee Y 129) = 11.357,816,691, and 3 s et 7? 11.357,010;091) = meee 835, and confequently 2 of? s (= 2 X 2.247,835) = 4.495,670. ~ And we fhall alfo have e (= =) = 2, and confequently = c= 2.000,000,000 ) 11.357,816,691 = 0.176,090,181, and — {== _ 0.176,090,181|*) = c.opi congue. and <> (ey = = 0.176,090,181 X 0.031,006,752) = 0.005,459,9845 5 3 2 and = om 7 x ~ = 0.005,459,984 X 0.031,006,752) = 0.000,169,296, e7 e5 A Rona Ure and > (= = X = = 9.000,169,296 X 0.031,006,752) = 0,000,005,249, 9 7 2 and > (= = X = = 0,000,005,249 X 0.031,006,752) = 0.000,00051625 7 and FOR THE RESOLUTION OF CUBICKR RQVATIONS, &e. 38% It e? et and => (= -5 X = = 0.000,000,162 % 0.031,006;752) = 0.000;000;00%, er3 eit e” ‘ and saz (= a X = 0.000,000,005 X 0.031,006,752) = 0.000,000,000, é a I é Lk I be and 4 (Hs ace i Rr OTE, Ogo; Tor), 7 0.058,696,727, caer. /) 5 TEP 8 — 5X 0.005,459,984 and ¢> (= aq % FW = HX 0-005,459.984 = ees = 0.0275299,920, __ 2205 el ge ae Ar, See a dys ee 036, and TOF (= Ly MO Fg % 01000: 22X0.000,169,296 __ 0.003,724,512 ‘ 74e7 169,296 == ———_—_—_——— = ete) = 0.000,002,1fo0, and BAe 9529 729 aes gO 519091 an ae Lea 37 5 iste —_ 374%X0,000,005,249 EE rR sd reer ols 2.00 A 2 5 as = 19000,600,090s and ee (ae eee LO A Ln EOS 19,053 15943238 32592373 fe 1, 5945323 _. 21,505 X0,000,000,162 __ 0.003,483,810 X 0.000,000,162 = —-- ee SE " ) = 0.000,000,002 : ‘ a? Sh ha WA 755949323. Sie wa 47,4076 — 1475407 eH gs L447 _ RR 14,348,907 Uae aus ea 147,407 X 0,000,000,005 0.000, 737,035 ‘ eS a) = 0.000,000,;000. Therefore 142348,907 1493481997 ) soda pect e@ Se 22¢ 374e 21,505¢ 147,407e"* : 35 ae 8153 72955 19,683 57 155943235? 14,439,907 5" + &c will be (= 0.058,696,727 + 0.000,337,036 ++ 0.000,005,109 + 0.000,000,099 -F 0.000,000,002, + 0.000,000,000, + &c) = 0.059,038,973; and confe- quently 2 /? s multiplied into the faid feries will be = 2 »/? s multiplied into 0.059,038;973 = 44953670 X 0:059,038,973 = 0,265,419,739- There- fore y, or the root of the cubick equation y3 + 157 = 4, or 157 + y3 = 4, will be = 0.265,419;739- Qs Es dy 47. This value of y approaches very nearly to the truth. For, if we fuppofe y to be equal to 0.265,419,739, we fhall have yy = 0.070,447,637,850, and 3 = 0.018,698,193, and 15y (= 15 X 0.265,419,739) == 3-981,296,085, and confequently y? + 15y (= 0.018,698,193 -- 3.981,296,085) = 3.999, 994,278 ;. which differs from the abfolute term, 4, of the propofed equation y? + 15y = 4 by only 0.000,005,722, or lefs than 0.000,006, ORs rer 791 OF > 6 millionth parts of an unit, or 6 four-millionth parts, or 3 two-millionth parts, of the faid abfolute term 4 itfelf. 8. If we were to make the value of y already found, to wit, 0.265,419,7395 the bafis of a further approximation to its true value according to Mr. Raphfon’s method of refolving equations, by fuppofing y to be equal to 0.265,419,739 + z, and fubftituting this compound quantity inftead of y in the propofed equa- tion y? + gy = 4, and then refolving the equation refulting from fuch fubfti- tution in the fame manner as a fimple equation by omitting all the terms that involve either the fquare or cube of z, we fhould find that z was equal to 3 Lys 0,000,005, 72 2,000,000,000 I§0211,342,913 388 _ A METHOD OF EXTENDING CARDAN’S FIRST RULE 0.000,005,722,000,000,000 150211,342,913 419,739 + 2, would be (= 0.265,419,739 + 0.000,000,376) = 0.265,420, 115. We may therefore confider this laft number, 0.265,420,115, as being the true value of y as far as the faid true value can be expretied in nine places of figures. = 0.000,000,376, and confequently that y, or 0.265, Pn en ete mare eee 2S End of the refolution of the equation y? +- 159 = 4, orisy +y*? = 4. re EET a email g. It has been fhewn in the foregoing articles, that, if e be put = -, and ss 3 3 ° . ° beim = aa = or ss + e¢, the root yof the cubick equation y? + gy = r will ale ge 22¢§ a7de? 21,506.69 3 _ ae - ee be equal.to, 2 yi £25 ine AMG Se: iat tegen gaay 155943238 De Fe He heer ee 53 147,407 €™* : : ‘ : Be MA ABD c ad infinitum, or 2 S the feries — ¥4,348,9075** + & off My VA xX 5 + K 6? Met o e%3 Qe § et7 : ; 4 ode Soma Hh cae phos Sia eh iG ad infinitum, the terms of which are the fecond, fourth, fixth, eighth, tenth, and other following terms of the feries I . : aa e te. : which is equal to 1 + — 35 or ¥3{t + —, or the cube-root of the binomial quantity I + <. Now from this expreffion, which is equal to the value a J in the equation y? + gy = 7, orgy + 9° = 7, we may, by a peculiar train of reafoning, derive another expreffion, very much refembling the former, which fhall be equal to the leffer of the two roots of the equation gv — x3 =r in which the letters g and 7 denote the {ame known quantities as in the foregoing equation y? + gy = 7, orgy +y3? = 7. The method of doing this I fhall now endeavour to explain. ©3 Of the cubick equation qv —x? =r. 10. The equation gv — x? = r is not always poffible, whatever be the mag- nitudes of g and 7, but only when 7 is equal to, or lefs than, the quantity svt, or that value of the compound quantity gx — x? which refults from the 3 ' fubftitution of nae or ¥ fz, in its terms inftead of «. If r is equal to this 4 quantity, the equation gv — «3 =, will have only one root, to wit, vt, or / [z ; and, when the abfolute term 7 is lefs than the faid quantity 299 the equation gv — #3 = r will have two roots, of which the leffer will be lefs than vq /3° FOR THE RESOLUTION OF CUBICK EQUATIONS, &c, 389 v4. and the greater will be greater than “1, but lefs than 7. See my differtation on the ufe of the negative fign in Algebra, art. 114, page 92. It is the leffer of thefe two roots of the equation gv — x3 =r that I now propofe to find in thofe cafes of the faid equation in which the abfolute term is lefs not only than tet (which 3’ is its greate{t pofible magnitude), but than 2 x vt, by means of an expreffion hag ° . . 3 to be derived from the foregoing expreffion 2 Y? s x the feries = -} oa + RK) Fes He? K e? Me! o e%3 qets sei7 ; ‘ ‘ X Mi BP. +. TA. ne -+- pyize + Ym + WW -1 &e ad infinitum, which is equal to the root y of the equation y3 + gy = 7, or gy +93 =7, the letters ¢ and 7 being fuppofed to ftand for the fame quantities in both equations. The value of the leffer root of the faid equation expreffed in an infinite feries. J . 2q3 “, Fi 11. Now, when ris lefsthan ¥ 2 X i, or vr is lefs than Soe or i is lefs 2 3 3 3 3 . than (ye os IS a al or) J. let oo be taken — 2 o> 2 and) let’ z. be 4 29 2% 27” 54 ; Rie? 4 fubftituted every where, inftead of 5, in the foregoing expreffion 2 ¥* 5 X the oe BR Des Fe He? K @7 Me" 0 e's ig : : ‘ feries a + — “ie hearer ir ee gore Seggcae ath &c ad infinitum, whereby the faid expreffion will be changed into the expreffion 2/3 z x the feries Be De? Fe H e7 K e? M eT Gle*4 : : nen “- PSs Ft 4 + + rr + cagnts a erry + &c ad infinitum ; and, laftly, let the fign — be prefixed to the fecond, fourth, fixth, eighth, tenth, and every following even term of this laft feries, inftead of the fign +. The new . . . Be expreffion thereby obtained, to wit, the expreffion 2 7? z X the ferles tone pe Fe He? Kk ¢9 Me" oe%3 + => — sr t+ ar — &c will be equal to the lefler root of the equation gx — x? =r. This propofition we will now endeavour to demon- ftrate. ge RE OI TT ES TE ET eS SE A proof that the infinite feries fet forth in the foregoing article is a converging fertes. err nS I 12. In the firft place it will be proper to fhew that this feries will be a con- verging feries. Now this may be {hewn in the manner following. - Since —, or ¢@, is lefs than oe it follows that >, or 2¢¢, will be lefs than. cn or che Therefore (fubtracting ¢e, or ane from both fides) 2¢e — ee will 54 2 | | be lefs than = — “, that is ce will be lefsthan zz. Confequently ¢ will be lefs 2) a : than nae 4 39° . A METHOD OF EXTENDING CARDAN’S FIRST RULE 3 . ¢€ . 6 than z, and the a — will be lefs than 1. Therefore the fractions <, 5 es ¢7 e? |e ? 2 52 oF? 9? ott “as, &c will form a acbnehlnis progreflion, and confequently 9 sags Be De Fe He? Ke? M eT a fortiori, the feveral terms of the feries — — —— bth om ie hcl z 23 2S 27 29 zit 4+ 25 — &c (of which the numeral co-efficients B, D, F, H, K, M, O, &c, alfo Hee a decreafing progreffion) will alfo form 3 a decreafing progreffion, or a converging feries. ; Git: De : ae & Des Fe H e7 K ¢? Therefore the expreffion 2 /? z X the feries — mon le ecg, OP oa ee a 2% 3 25 z7 2? m e7* or a a " = — &c will converge toa certain aie magnitude, and ‘confe- - 5 quently may be equal to the leffer root of the cubick equation qx—xi or, It remains that we fhew that it is fo. Preparations for demonftrating that the foregoing infinite feries is equal to the leffer root of the equation gx —x* =r. pes Be 13. In order to demonftrate that the expreffion 2 /* z x the feries > — >> Fé He? K e? M ex 0 e783 . ’ er eer din Miners emdesery muna re ree Ao mci ce the leffer root of ny equa tion gx — «3 =r, it will be neceffary to ref{ume the confideration Ni the ex- pe} F H e7 K e? mM ™ preffion 24/3 5 x the feries = + <> + > (Meera) awheret Le eto. of + &e 40 infinitum, from which it is derived. oo Bie Des Fes He7 Ke Now, fince the expreffion 2 35 x the feries — tingle + ae or Ir be vale niinkse as +- &c ad nage is equal to the root of the cubick equation J ykqg= a or gy +93 =¥F, it follows that, if we were, firft, to multiply the De3 Fe 7 09 AE oes {aid expreffion 2 /* s x the feries + RU + — + <> += += ay + &c ed infinitum into the oO ecient 2 ariel then to raife the faid expreffion Met 20/35 the feries 22 + 25 4 22 Sahel oe aa Ne SS + Bec ad in-. git Jinitum to its cube, or third. sere and lafly, to add the faid cube to the faid product, the fum hence abang would. be eaual to r, to whatever number of ni ifs Seta terms the faid feries = + = sf ee += continued. For, if this Pa were dick equal &0 a it HES not cg true that the H : K €9 0 e73 expreflion 24/3 5 X the feries 425 re dhicarick op ote epee —— tar + &c ad infinitum, was equal to ne root of the” ahh qgitye=en Thefe operations may be performed in the following manner. 14. Since FOR THE RESOLUTION OF CUBICK EQUATIONS, &c, 3gr d : 3 rr: 14. Since = + = is = 85, or (becaufe = fa is == ¢e), 2 ~ ee is = 55, we | : 3 aw as Bp ee fhall lhc hapa 6, and 7?'S"47 X [ss — ee = 27 X $5 x1 color and ‘s.. I 2 cat confequently g = 3 x 5s)3 xr 3 a3x59 x1 ——|3 =3s a aa oe I — 2 ee) 3 Bee ce D E¢€ F et° Gel? _ 433 the feries:1 == —— ee = : 3S pare 3 x 55 54 5° 58 10 siz 2 &c ad infinitum. ‘Therefore the product gy will be = 35 3 x the feries 1 — 2 Bee’ cet. De Fe F el? G elt af So Spee tegere Some &c ad infinitum x y = 353 x the (eri Bee ce pe Ee F et? Ge eens s+) 1). aw O: stexiaghsPncogar — &c ad infinitum x the ex- I — $5 Be pe F eS He? K e? Me re) Prcinoar 0-5 —9<-the-fenes —— —- pnee—b pe e — + 4+ &c ; ; } Bee c é p Ee? F et? G et ad infinitum = 6s x the feries 1 — Paes So Sao Fes He Ke Met 0 e%3 Beclad sss x the feries — “try otter tees be ba &c ad infinitum = 6 5 into the followitte compound feries, to wit, Be n*e3 Bc eS BDe? BE €? BF e™? BG et %& as hah 5 Mk eines tap Game ial, sl? De BDe> cDe? D7e9 DE e*t DF et3 8 oF fe? MOTTE RE! OE ae Te Me aes S ear ae he st Fe BF e? CF ec? DE ex* EF e'3 & i wae he arp He7 BH e9 cH ett DH et3 & aA S68) SUT por noo: y Re BK et CK e*3 * 59 Fy ATE ee oc M et! BM ets git gi3 - &c 0 ¢7s 4. Os ecg | or, if, for the fake of brevity, we denote this compound feries by the Greek ca~ pital letter I, we fhall have the product gy = 65 x the compound feries I. 15. In the foregoing compound feries T, which is the product of the multi- Bee ca ee Ee® F et? vets e (of Peeroncof the feries 1 me ma me ee ome which all the terms after the firft term 1 are coco with he fign —, or fub- ae ue? K e? tracted from the faid firft term) into the feries = — = pS is ake 7 ay 13 pia = sar b &c (of which all the terms ire Hy fut term — are marked e with a see 492 A METHOD OF EXTENDING CARDAN’S FIRST RULE with the fign +, or added to the faid firft term), it is evident that the firft term of every horizontal row of terms wall be marked with the fign +-,-and that every following term of the fame row will be marked with the fign —, to whatever number of terms the faid horizontal rows of terms may be continued, and like- wife that the firft term of every new herizontal row of terms is placed imme- diately under the fecond term of the next preceeding horizontal row, becaufe they both involve the fame power of the fraction <; whence it follows that all the terms in every vertical column of terms, except the loweft term, will be marked with the fign —, and that the faid loweft term will be marked with the fign +. This will appear moft evidently upon performing the multiplica- tion of the former of the faid feriefes into the latter. 16. If we multiply the foregoing compound feries called I into 65, the product will be the following compound feries, to wit, ‘ 6 B2¢3 6Bces 6 BDe?7 6 BE e&? 6 BF elt 6 BG e%3 & BE a ee ie re Cc 6 pe} 6 BpeS 6-cp e7 6 D269 6 DE et! 6 pF e?3 & aa ah Lae Be IRS 510 —o 73 eed G 6 Fe 6 BF e7 6 cF e9 6 pFelt 6 EF e3 & Day See ET EE So 2 6 He7 6 BH e? 6cH e¥ 6 pH e3 & ot 6 Se aie oe ee c 6 Ke? 6 BK el! 6 cK e383 & F g8 rs) Iz Cc 6 mM e®t 6 BmMet3 far Io T2 — &£¢ 5 5 + 6 o e!3 Se? ell e gi Let this compound feries, for the fake of brevity, be denoted by the Greek capital letter A. Then, fince the produ& qy is equal to 6s x the compound feries T', it will alfo be equal to the compound feries A. 17. In the next place we mutt find the cube of the expreffion 2 /?5 x the 3 5 7 9 1 13 mn ite feries — + == ale = st — at = uF — + a -+- &c ad infinitum, which 1s equal to y. j : . 3 Now the cube of this expreffion is = 8s x the cube of the feries “ + — + orale wien ght =" Sees - ++ &c; which cube may be found by multi- plying the faid feries twice into itfelf in the manner following. The y tio FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. 393 The multiplication of the Series * _

    4. nae a a S6 ae aaa 4: ee of ma sl odoc po 4 TE + &e ae — zie a a a “ ae ve aa 4. 8c x ae aa Z = + &c a ae + &c ee ai oe + plas ABB ELE we 2 Vou. Il. 3 E ged 394 A METHOD OF EXTENDING CARDAN’S FIRST RUBE B7H e9 2 BDH et? 2 BFHe'3 fay tt ee ee oe +—,— + &c B7xK ett Nees Sioa anand + = Ges + &c B2M ¢t3 G ommarumce arte” B3¢3 3 B*De5 3 BF e? 3 BH e? 3 BK e™ 3B at an Eos Eaai aN aak Gea ee ae ee BD7e7 6 BDF e? 6 BpHe!t 6roKe + ao + Sree es at eee + —— + &c D3¢9 3 BF" 6 bFH re a ee tee ea ee 3 D* Fe" ee ers + Sa, ob 3 4+- &c © wis err jt sort Le —_-— This laft ge feries is the cube of the feries ~ — = + = bee rey Ae Es Spe Ke? Met SAE eae — + &c ad infinitum. Therefore 8 s into the cube of the feries ~ + ie FA She —— mh x =< ae a Lena oe yet &c will be = 8s x the foregoing eke aaa 5 cor if for the fake ‘of brevity, we denote the eae aii Seis Py the ie capital letter A, 85 X the cube of the feries = — “+= — + if apepaae = ee ay a git be —— = &c wall be =) 8s < the ee fie Aa ac i, < cube of the OFe Be ood Met! sane 27% s xX the feries — + esa ~—e =< ae xs &c, or the cube of y, will be = 8 5 se the a suet feries A. 0 e3 gil 513 18, If the foregoing compound feries A be multiplied into 8s, the produ& will be the following compound feries, to wit, — v 24 oe ue a = 24 —— He 24 pe a tee Pa 9 IL : 3 as 24 2: e 4s 48 ne ve Nic of ee Re 3,9 2,1t y 13 — in aa 48 == se z It 2 13 $e + ES + &e i ae - + &c, which St FOR THE RESOLUTION OF CUBICK EQUATIONS, é&c. 395 which we will denote by the Greek capital letter I. Then will y* be equal the compound feries II. 19. Since the compound feries A, obtained in art. 16, is equal to the pro- duct gy, and the compound feries II, obtained in art. 18, is equal toy, it follows that the fum of the two compound feriefes A and II will be equal to gy + y3, and confequently to its equal, the abfolute term r; that is, the fe- riefes A + It will be = 7. Therefore the feries TI will be — 7 — the feries A. Butz is equal to the firft term, 6 Be, of the feries A. For B is > and re =, and confequently 6 Be is (= 6 x — x ~ = 2 = yi Pheterore the feries II is = 6 Be — the feries A; that is, the compound feries IT is equal to the excefs of the firft term, 6 Be, of the compound feries A above the whole of the faid feries. But 6 Be — the feries A, or the excefs of the firft term, 6 Be, of the feries A above the whole of the faid feries, will be a compound feries confifting of all the terms of the compound feries A, except the firft term 6 Be, with their figns + and — every where changed. Therefore the compound feries II will be equal to a compound feries confifting of all the terms of the compound feries A, except the firft term 6 Be, with their figns +. and — every where changed ; that j is, the compound feries 8 B33 24 87D e5 24. B7F 7 be 24 B7H e9 24 B?K e7% 24 BM et3 “3 a aie + 6 Sate ig Sear t Pern we &c 24 BD7e7 48 BDF e? 48 BDHe™ 48 Bpk e™3 re Taseropl ated Vay orca rir ele i" 8 p3¢9 2pIl 8 w = HDPE fe ts 24 BFe big teen BFHets + 8c 58 mar ore 24 D? F et! 24 ote" gt? Wa DFe"3 a Tee + &c will be equal to the compound feries 6 Be3 6 Bc eS 6 Bp e? 6 BE ee? 6 Brett 6 BG e!3 aa 54 56 rh re of 510 oF 22 0 &e 6 pe3 6 pepe 6cpde7 6 p29 6 pE e™™ 6 pFe%3 & Ts g% g*# 56 58 gO an gid 7c Cc 5 qi 9 It 13 1 Oke 6 Bre ‘, Gere _ 6vFe 6EFe 4. Be s* 5° : 99 10 yt? 6 He? 6 BHe? 6 cu et 6 pels 56 FPHEY: gre T a + &c 6 Ke 6 BK e™ 6 cK e%3 i] er 38 1 #0 > 4m &c 6 Me™ 6 BM e%3 ay gf? = gi rE &c 6 as a + &c; sz which, for the fake of brevity, we will denote by the Greek capital letter =. And then we fhall have the compound feries I1 = the compound feries &. EA es Of 3.96 A METHOD OF EXTENDING CARDAN’S FIRST RULE Of the figns + and — that are to be prefixed to the feveral terms of the foregoing compound feries &. 20. In this compound feries = all the terms of the firft horizontal row of terms are to be added together, and confequently all the terms after the firft B*¢3 : , are to be marked with the fign +, to whatever number of terms s” 6 term, the faid horizontal row of terms may be continued; and the firft terms of the fecond, third, fourth, fifth, and other following horizontal rows of terms (to whatever number of horizontal rows the faid compound feries may be conti- nued) will be marked with the fign —, and all the following terms of the faid horizontal rows after the firft terms will be marked with the fign + ; and con- fequently all the terms of every vertical column of terms in this compound {e- ties, except the loweft term, will be marked with the fign +, and the faid loweft term of each vertical column will be marked with the fign —. This follows neceffarily from art. 15 and 16; becaufe the fizns + and — that are prefixed to the terms of the compound feries A, fet down in art. 16, are the fame with the figns of the correfponding terms of the preceeding com- pound feries T°, fet down in art. 14, from which the compound feries A is de+ rived by only multiplying its terms into 6 5; and the compound feries © is de- rived from the compound feries A by omitting its firft term 6 Be, and chane- ing the figns of all its following terms. Confequently the figns to be prefixed to the terms of the compound feries 2 mutt be contrary to thofe which are to be prefixed to the correfponding terms of the compound feries T, and there- fore muft be contrary to thofe which are defcribed in art. 15, or mutt be fuch as they are defcribed to be in the prefent article. Of the equality between the co-efficients of the terms of the compound feries 1 and the co-efficients of the correfponding terms of the compound feries x. ° . . 3 . 21. Since ee is = ne and ss is = a + = we may, by leffening the value of = or of r, without altering that of g, leffen the value of the fraction La fe — rR “, and confequently that of =, as far as we pleafe. Yet in all a7 4 thefe values of the fraction < it will always be true that the compound feries II, which involves in its terms the fractions 5, ss e o = =, &c, which will be equal to the compound feries ©, which involves in its terms the fame fractions. it therefore follows from this conftant equality between thefe two feriefes in all poffible FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. 397 x m 2 e3 e e7 e? eit et3 poflible magnitudes (how fmall foever) of the fractions “a PEP res) eS &c, that the term, or terms, that involve any given powers of ¢ ands in one of the two feriefes muft be equal to the terms that involve the fame powers of them in the other feries. And confequently the feveral co-efficients of the terms which involve any given powers of ¢ and sin one of the two feriefes muft be equal to the feveral co-efficients of the terms that involve the fame powers of e and 5 in the other feries. Thus, for example, the co-efficient of the fraction = in the feries II, to wit, 8 B*, muft be equal to the two co-efficients of the fame fraction £ in the feries X, to wit, 6 BX — 6D; and in like manner the co-efficient of the fraction s in the feries TI, to wit, 24 B*D, muft be equal to the three co-efficients of the fame fraction = in the feries 5, to wit, 6BC + 6 BD —6F; and the two co-efficients of the fraGtion = in the feries II, to wit, 24 B°F + 24 BD*, muft be equal to the four co-efficients of the fame fraction < in the feries ©, to wit, 6BD + 6CD + 6 BF — 6H; and the fame thing mutt take place with refpe& to the co-efficients of the following fractions ae > i &c ad infinitum, or to whatever number of terms the two feriefes may be continued. — Examples of the faid equality of the co-efficients of the terms of the Jaid two compound Jeriefes. 22. Of this equality between the co-efficients of the fame fractions, confifting of the powers of ¢ and s, in thefe two compound feriefes II and &, it may not be amifs to give a few inftances by actually computing the values of the faid co-eflicients ; which may be done in the manner following. The capital letters B, C, D, E, F, and H, are equal to = Be TS chat pce Qn, Br 243 ee and 324, refpectively. Therefore B? will be = —, and 8 B? will be = —; 19683 27 ¥ and 6 B* will be (= 6 x —jo and 6 D will be (= 6 x 2=2x2)= 9 3 81 27 ~“, and confequently 6 B*— 6 D willbe (= = —~— = neuer y= £ si.that 27 : 3 ll 27 27 27 is, 8 B3, the co-efficient of the fraétion = in the compound feries II, and 6 B? } Ab he — 6D, the co-efficient of the fame fraction = in the compound feries 2, will, each of them, be equal to the fame quantity = 4 In 398 A METHOD OF EXTENDING CARDAN’S FIRST RULE Tn like manner 24 B’D, or the co-efficient of the fraction & in the compound . ; 8 Teresa ty 1S (> 2a aay x es eae x =) = Fa: and 6 BC + 6BD — . . Sey e 6 F, or the co-efficient of the fame fraction = in the compound feries 2, will be found to be equal to the fame quantity. For 6 BC + 6 BD —6Fis=6 x I I I 5 228 7 t2 10 44 84 30 44 —X—-+6xX=—-xXe—-6xX— (= -+ ep — ee St SK 3 as REPS 729 § 9 8 243 243 243 243 ma) Be Baath 942 (was 243 243 Q. E. D. And 24 B’F + 24 BD’, or the co-efficient of the fraction 5 in the com- ae + yy 55S eee 26 Oye Os pi ube ne be 200. Cees 72 0s ee SET TIRBUTT TU TM TMT e oo and6BD + 6CD + 6BF . . 7 . . -— 6H, or the co-efficient of the fame fra¢tion = in the other compound feries pound feries IT, is (= 24 X 5 x =, is equal to the fame quantity. For itis = 6 x _ “4 ma + 6x *5 x = hp 204 cs di 8 374 (= TA ge adh Sa 222X374 =D Sard eaters ie Serer rR Se eH ey 6561 PATO pL AO oy glad 748 __ 810 270 396 748 _. 1476 748 » He ae 243. | 729 +6561 6561 ‘ 6561 ' 6561 6561 — 6561 6661 6561 Q. E. D. And the fame equality will be found to take place between the co-efficients of s ; oF OR sia dette ; : . : the following fractions = 3° ee &c, ad infinitum, in the compound feries II and the co-efficients of the fame fractions in the compound feries &, refpec- tively, to whatever number of terms the faid feriefes may be continued. The reduction of the compound feries Il to a fimple Series “f “4 aes Re7 8 e9 Tet v e333 , : rps dice ihe tear ue ear caer sine ESc ad infinitum. 23. In the compound feries TJ, obtained in art. 18, all the terms after the x are marked with the fign + and added to the faid firft term. 8 B3 firt term, —— s Therefore, if we reduce the faid compound feries to a fimple feries, with fingle letters for the co-efficients of its terms, by denoting the co-efficient, 8 B3, of 8 B3¢3 , by the fingle letter P, and the co-efficient, 24 B*D, . 3 . the fraction = in the firft term —; POR THE RESOLUTION OF CUBICK EQUATIONS, &c, 399 . ec. 24 B7pD eS - ¢4 B*D, of the fraction role the fecond term, oaks by the fingle letter Q , and the compound co-efficient, 24 B*F + 24 BD’, of the fraction & in the third term, by the fingle letter R, and the compound co-efficients of the fol- ee eg lowing fractions rr &c, in the fourth, fifth, and fixth, and other fol- po yi? lowing terms of the fame feries, by the fingle letters S, T, V, &c, the fimple . . . ( 4 3 S 7 feries that will be equal to the faid compound feries II will be —— + -5- J c s 6? Tet v e%3 : : Pe Ag = eat ate &c, in which all the terms after the firft term, => are marked with the fign ++, or are added to the faid firft term. dame? aes Re? s 09 pe vers The fame fimple Series 2 + 7, +- a + n> +- ao, iia + Se ad infinitum will alfo be equal to the compound feries =. 24. But by art. 19, the compound feries IT is equal to the compound feries aes Re? se? Tene v et3 bo, Therefore the fimple feries = + aren -b pare ol a + oN a be via 4. &c s will be equal to the compound feries 2, which is fet down. in thefatter part of arte 1Q- And the feveral co-efficients P, Q, R, S, T, V, Se, of the terms of the faid fimple feries will be refpectively equal to the Several compound co-efficients of the corre/ponding terms of the Said compound feries x. 25. And, further, it is fhewn in art. 21, that the co-efficients of the feveral 5 It I fractions S, => 5, a = S &c, in the compound ferjes II are refpec- tively equal to the co-efficients of the fame fractions in the other compound fe- ties &. Therefore the co-efficients P, Q, R, S, T, V, &c, of the feveral 5 Ir I 3 s fractions 5, = 5 5 aor a &c, in the fimple feries — SB < + ae +. $9 Tet s8 tO fame fractions in the compound feries II) will be alfo refpectively equal to the co-efficients of the fame fractions in the compound feries =. And hence it fol- lows that, if we were to reduce the faid compound feries = to a fimple feries by the addition and fubtraction of the feveral members of each of its vertical com lumns Ee + &c (which are refpectively equal to the co efficients of the: 400 A METHOD OF EXTENDING CARDAN’S FIRST RULE lumns according to the figns -+ and — which are prefixed to them, we fhould find the fum of the co-efficients of the terms that are marked with the fign + in each of the vertical columns of the faid compound feries (and which are all the terms of the column, except the laft, or loweft) to be greater than the co- efficient of the laft, or loweft, term in the column, which is marked with the fign —, and the refulting differences, or excefles, of the fums of the co-effi- cients of the terms marked with the fign + in each column above the co-effi- cients of the loweft terms in them, which are marked with the fign —, to be equal to the co-efficients P, Q, R, S, T, V, &c, of the fractions = salle 9 11 13 : i 3 s 7 9 Ir 13 oe <5) ae &c, in the fimple feries = +++ + —— + &c refpectively ; that is, we fhould find the compound co-efficient of the frac- tion a in the compound feries =, to wit, 6 B* — 6D, to be = P; and the aed ; compound co-efficient of the fraction — in the fame feries 2, to wit, 6 BC + 6 BD — 6 F, to be = Q; and the compound co-efficient of the fraction 5 in the fame feries, to wit, 6 BD + 6CD + 6 BF — 6H, to be = R; and the compound co-efficient of the fraction - to wit, 6BE -+- 6D* - 6CF + et 6 BH — 6K, to be = S$; and the compound co-efficient of the fraction —, towit, 6BF + 6 DE + 6DF + 6CH + 6BK — 6M, to be = T; and the compound co-efficient of the fraction ee) to wit, 6BG + 6 DF + 6EF + 6DH + 6CK + 6BM — 60, tobe = V; and, in like manner, 15S et? el9 ert 623 the compound co-efiicients. of the following fractions sth? G16 GHA “Gao? as? &c, ad infinitum in the fame compound feries % to be refpectively equal to the co-efficients of the fame fractions in the following terms of the fimple feries re oe Re? se? ia eg v x3 : ; +> sta tar tae t &c ad infinitum. 5 § Conclufions contained in the three foregoing articles. 26. It appears from the three foregoing articles, that the fimple feries pe Re? se? Tune v e33 — = ta tar tar + &c will be equal to each of the two compound feriefes, II and 2, and likewife that each of its terms will be equal to the correfpondent term, or term that involves the fame powers of ¢ and s, in each of the faid two compound feriefes, and confequently that the co-effi- cient of each of its terms will be equal to the co-efficient of the correfpondent term, or term involving the fame powers of ¢ and s, in each of the faid two compound feriefes. ae “et Examination FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. 401 B Examination of the expreffion 2»f%? z X the Series —— — e x3 z z Fe H é7 K ¢9 m étt oe 13 pepe Nr cocks plinerr stom &Pe¢ ad infinitum, 27. Thefe things being thoroughly underfteod, we muft now turn our at- , , 3 ee De Fes H e7 K ¢? tention to the expreffion 2 4/72 x the feries fan ts ee mM ett 0 e%3 Var . 39x%23 X1+—13 = (by the binomial theorem in the cafe way mee rt Se | 2 of roots) 3 X 213 x the feries 1 ++ — p Aa EBX Ce ee ok ze 1I jee 14 17 ae Le els Spx S+EEx GBP x S =e Gx -RHx 3+ Bee c et pe Ee F e!0 G et &G,.==43 estas x the feries 1 + = — a Sb fie an + &c. fairest ~ product of the multiplication of g into As : 3 5 1 9. Pe on > ad ° Be De Fe He’ Ke Me 9 the expreffion 2 % 3 X the feries — — —| meet ene ated er oes Ue ig eae bar ce de® Ee cwillbe = 3 ar x the feries 1 -+- — =; ——S rec Cr, H et 1e%6 u De Fes piety oe at eh rig 8 BH te He? K 22 Meter eye Bee: ce — Sit Gaines ee OFX PReASTES Hh aoa n e® Ee F et? 6 e™ H 14 So De F eS H ¢7 K e9 M ett 0 e33 fe ; EERE PRET RPA ee Be X the compound feries Be B%e3 Bo e> BD e7— BE ¢?" BF et® BG e*3 - & ada zt nr Zilia an tans ze gts + Xe De3 BDe> cD e? D%e9 DEe™ DF et3 eee DE ee leew eierr a Oe" ata — &c ad infinitum equal to 6 z X the compound feries y. 4 com- FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. 493 A comparifon between the foregoing compound feries y and the compound Jeries 1 obtained above in art. 14. 30. This compound feries y will confift of terms that will involve the feveral * Be. g3:. | Give etens eta ett tere fractions yh any! eB a al oe as aes &c, or the feveral odd powers of the frac- tion —, in the fame manner as the terms of the compound feries F, obtained . . . ° é e3 5 7 9 Ir above in art. 14, involve the correfponding fractions —, Se Sl eis fet3 gt3? gg) FB? Gp &c, or the feveral odd powers of the fraction =: and the eo efficients of the feveral powers of = in the prefent feries y will be the fame with the co-effi- cients of the fame powers of - in the former feries I, though they will not be every where marked with the fame figns 4+ and —. This equality, or, rather, identity, of the co-efficients of the correfponding terms in both thefe compound feriefes arifes from the equality, or identity, of the co-efficients of the terms of the two fimple feriefes, by the multiplication of which into each other the faid compound feries y is produced, with the co-efficients of the correfponding terms of the two fimple feriefes, by the multiplication of which into each other the for- mer compound feries [is produced, For, fince the co-efficients of the terms : Bee ce De® E Be Gel® Be of the two feriefes 1 + — — 17; REY aS ae Ce cand D e3 Fé H e7 K € Me" 0 e%3 ei cas : (1 Set Silrceeen ser eheorreiey &c (by the multiplication of which into each other the compound feries y is produced) are the fame with the co- : : . Bee ce D ef efficients of the correfponding terms of the two feriefes 1 — ve te eieey oo A) Ee F ef? Gel” Be Des Fe> He? Ke? M ett 0 et3 “5 age) ae akan a a Ca le + &c (by the multiplication of which into each other the compound feries T° was produced) it is evident that the co-efficients of the feveral correfponding members of the two compound feriefes T and y produced by thefe multiplica- tions muft be the fame combinations of the co-efficients B, C, D, E, F, G, &c and B, D, F, H, K, M, O, &c of the terms of the two original feriefes, or in other words, muft be the fame quantities in both feriefes. a EE ee Of the figns + and — that are to be prefixed to the terms of the compound feries y. 31. But the figns +- and — that are to be prefixed to the feveral terms of the compound feries y will not be every where the fame as thofe that are to be pre- fixed to the correfponding terms of the other compound feries I’, but only in the oun, terms 404 A METHOD OF EXTENDING CARDAN’S FIRST RULE terms of the third, fifth, feventh, and other following odd donot: columns of - “ : es 0? er3 e%3 ; terms, which involve the fractions —, Sue ce &t and ——, —-, =) eae J 2? 233 the fecond, fourth, fixth, and other rolTewhay even bere columns of terms . . . A 3 7 tr in the faid compound feries y, which involve the fractions =, =; = , &c the figns + and —to be prefixed to the feveral terms in the nn vertical columns will be refpectively contrary to thofe which are to be prefixed to the correfpond- ing terms of the fecond, fourth, fixth, and other following even vertical columns Pe of the compound feries I’, which involve the correfponding fractions = [pigs er d git tical columns of Wie {aid two compound feriefes, or the terms which involve the &c. This will appear to be the cafe as far as the terms of the feventh ver- er3 13 fraCtions > and —, upon the infpeétion of the faid two compound feriefes, as herein before fet dows in art.14 and 29. For in the former of thefe feriefes the . . . . BC e3 terms contained in the third, fifth, and feventh, vertical columns are — BD e°? F e> BE e? pe? CF e? BH e? K e? d BG ef3 DF ef3 Thea ss EE Ok NTF BEE ee Ere WP MenbedtMELER Tun EF e73 DH e?3 CK e33 BMe!3 Ole eens : SF. RT GTC Ae BAP Baa © oho Suse? Fa hee each of which columns, or fets, of terms, the gn — is prefixed to every term, except the laft, and the fign + is prefixed to the laft term; and in the latter of thefe feriefes thie terms contained BCce> BD eS Fe5 in the third, fifth, and feventh vertical columns are — rs = —) z BE €9 D9 CF e? BH e9 Ke? My BG e%3 DF et3 EF e'3 DH 73! BOB go go he egg A eg cre? BM e13 oes “f <3 in each of which columns, or fets, of terms the fign — is, in like manner, prefixed to all the terms, except the laft, and the fign + is prefixed to the laft term. And in the former of thefe foricles: the terms con- B7e3 De? tained in the fecond, fourth, and fixth, vertical columns are — =, — J BD e? cp e? BF ¢7 He? A BF e** DE e*? DF et one* BK e™* Seo fonivaT alt ae cian? Ton Sar TT ares par Se ee It 2 nf 4 oe See in each of which columns, or fets of terms, the fign — is prefixed to all the terms, except the laft, and the fign ++ is prefixed to the laft term: and in the latter of thefe feriefes the terms contained in the fecond, fourth, and fixth, : Be3 pe} BD e7 cpe?7 — Bre? H e7 BF a vertical columns are -+- —— wre ye, me a = ——) and + — DE e* DF et! CH yi BK ett Melty. A s iserreatin irae bs (oo sete ert sci Fe 2) ae of which eolumns, or fet of terms the fign + is prefixed to all the terms, eacens the laft term, and the fign —— is prefixed to the laft term. And that the fame thing will take place in the terms of all the following ver- tical columns of thefe two compound feriefes, to whatever number of terms they may be continued, may be fhewn i in the manner following. 7 $7. 4k “ / FOR THE RESOLUTION OF CUBICK EQUATIONS, &e. 405 32. It is fhewn above in art. 15, that, in the compound feries I, all the terms in every vertical column of the feries, except the laft, or loweft, term, are marked with the fign —, and the loweft term 1s marked with the fign +. And we have feen that the fame thing takes place in the terms of the third vertical column and of the fifth vertical column, and of the feventh vertical column, of the com- I pound feries y, which involve in them the fractions < <, and — to wit, that all the terms in each of thefe vertical columns, except the loweft, have the fign — prefixed to them, and that the loweft term is marked with the fign + ; but that the contrary rule takes place in the terms of the fecond vertical column, and of the fourth vertical column, and of the fixth vertical column, of the faid It : mth Ff ; ‘ ne € ‘ compound feries y, which involve in them the fractions —-, —-, and —-; to wit b] 23 > 27 ») gil > that all the terms in each of thefe vertical columns, except the loweft, are marked with the fign +, and the loweft term is marked with the fign —. We are now to prove that the fame rules will hold as to the figns of the terms con- tained in the oth, 11th, 13th, 15th, and other following odd vertical columns of terms, and the figns of the terms contained in the 8th, 1oth, 12th, rqth, and other following even vertical columns of terms of the faid compound feries y, to whatever number of terms, or columns of terms, the faid compound feries y may be continued. 33. Now in the faid compound feries y (which is fet down in art. 29), it is evident, that in the 1ft, and 3d, and 5th, and 7th, and other following odd ho- rizontal rows of terms, the two firft terms will be marked with the fign +, and all the following terms will be marked with the fign — and the fign + alter- nately ; and in the 2d, and 4th, and 6th, and other following even horizontal rows of terms, the two firft terms-will- be marked with the fign —, and all the following terms will be marked with the fign + and the fign — alternately. This is a.neceffary confequence of the order in which the figns + and — follow each other, in the two fimple feriefes, by the multiplication of which the com- pound feries y is produced. For the fimple feries which is the multiplicand of Bee art Eé Res G el? ‘ eg : : De® his multiplication, is the feries 1 + ie. Sori. gerr Sapper werner, cn mere heer ce &c; in which the fecond term — is marked with the fign +, and all the follow- ing terms are marked with the figns — and + alternately: and confequently, whenever this feries is multiplied by a quantity to which the fign + is prefixed, the product will be a feries of terms, in which the figns + and — will follow each other in the fame order as in the multiplicand, that is, the fign + will be prefixed to the two firft terms of the faid product, and the fign — and the fign + will be prefixed to the third, fourth, fifth, fixth, and other following terms of the faid product alternately ; and, whenever this fimple feries is multiplied by a quantity to which the fign — is prefixed, the product will be a feries of terms in which the figns will be, refpectively, contrary to the figns of the correfponding terms of the multiplicand, and therefore the fign — will be prefixed to the two firft 406 A METHOD OF EXTENDING CARDAN’S FIRST RULE firft terms of the faid produ&t, and the figns + and — will be prefixed to the third, fourth, fifth, fixth, and other following terms of the faid product alter- eS ; ; : : Bee 1c e4 nately. But the-fimple feries into which-the faid fimple feries 1 + — ate 7 BS pe E F 20 G et : ae J Be a ie ageae tie teats &c is multiplied, in order to produce the com- * oe ‘ eS eae De F e§ He? K e Mert 0 33 pound feries 7, ds the feries — —— Be ee &c, in which the terms are marked with the fgn + and the fign — alternately. Therefore in the 1ft, and 3d, and 5th,-and 7th, and other following odd hort- zontal rows of terms in the compound feries-y (which is produced by the faid multiplication) the fign + will be prefixed to the two firft terms of the faid ho- rizontal rows, and the fign — and the fign + will be prefixed to the third, and fourth, and fifth, and fixth, and other following terms of the faid rows, alter- nately ; and in the 2d, and 4th, and 6th, and 8th, and other following even ho- rizontal row’ of terms in the faid compound feries y, the fign — will be pre- fixed to the two firft terms of the faid horizontal rows, and the fign + and the fign — will be prefixed to the third, and fourth, and fifth, and fixth, and other following terms of the faid horizontal rows, alternately. 34. Now from this order of fucceffion of the figns + and — to each other in the horizontal rows of terms of the compound feries y, we may deduce the order of their fucceffion in the terms contained in the feveral vertical columns of the faid compound feries. This may be done in the manner following. Since the two firft terms of the feveral odd horizontal rows of terms in this compound feries y are marked with the fign +-, and the two firft terms of the feveral even horizontal rows are marked with the fign —, and the firft terms of athe fecond, and third, and fourth, and other following horizontal rows of terms are the laft, or loweft, terms of the fecond, and third, and fourth, and other fol- owing vertical columns in the faid feries, it follows that the laft, or loweft terms of the fecond, and third, and fourth, and other vertical columns will be marked with the fign — and the fign + alternately ; and it follows likewife that the fign prefixed to the‘fecond term of every horizontal row of terms (being the fame as the fign prefixed to the firft term of the fame horizontal row) will be contrary to the fign of the term placed immediately under it, or of the firft term of the next horizontal row, and confequently that the fign prefixed to the laft term but one of every vertical column (which is always the fecond term of one of the horizon- tal rows of terms) will be contrary to the laft, or loweft, term of the fame verti- cal column, which is the firft term of the next lower horizontal row of terms. Therefore, fince the firft terms of the fecond, and fourth, and fixth, and other following even horizontal rows of terms, or the laft or loweft terms of the fe- cond, fourth, fixth, and other following even vertical columns, are marked with the fign —, the laft terms but one of the fame columns, to wit, the fecond, fourth, fixth, and other following even.columns, will be marked with the fign + ; and, fince the firft terms of the third, and fifth, and feventh, and other fol- Jowing odd horizontal rows of terms, or the laft, or loweft terms of the third, fifth, FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. 407 fifth, feventh, and other following odd vertical columns, are marked with the fign +, the laft terms but one of the fame columns, to wit, the third, and fifth, and feventh, and other following odd columns, will be marked with the fign —., 35- In the foregoing article 34, we have fhewn that the loweft terms of the fecond, fourth, fixth, and other following even columns of terms in the com- pound feries y will be marked with the fign —, and the next higher terms, or the loweft terms but one, of the fame columns will be marked with the fign + ; and that the loweft terms of the third, fifth, feventh, and other following odd columns of terms in the fame feries will be marked with the fign +, and the next higher terms, or the loweft terms but one, of the fame columns will be marked with the fign —. We muft now examine the figns that are to be pre- fixed to the other terms of the feveral columns that are higher than the two loweft terms. Now the figns to be prefixed to all the terms of every column above the two loweft terms are the fame with each other and with the fign of the lowelt term but one of the fame column; as may be fhewn in the manner following. The fecond term of every horizontal row of terms in the compound feries y is placed direétly under the third term of the next horizontal row above it, and under the fourth term of the fecond horizontal row above it, and under the fifth term of the third horizontal row above it, and fo on ad infinitum, the number of terms in every new horizontal row above the faid fecond term that preceed the term that is placed immediately above fuch fecond term, increafing continually by an unit. But, becaufe the figns + and — are prefixed to the fecond, and third, and fourth, and fifth, and other following terms of the firft, the third, the fifth, the feventh, and other following odd horizontal rows of terms, alter- nately, and the figns — and + are prefixed to the fecond, and third, and fourth, and fifth, and other following terms of the fecond, the fourth, the fixth, the eighth, and other following even horizontal rows of terms, alternately, it is evident that the fign prefixed to the fecond term of every horizontal row of terms mutt be the fame with the fign prefixed to the third term of the next ho- rizontal row above it, and with the fign prefixed to the fourth term of the fe- cond horizontal row above it, and with the fign prefixed to the fifth term of the third horizontal row above it, and with the figns prefixed to the fixth, feventh, eighth, &c, terms of the fourth, fifth, fixth, &c, horizontal rows above it, ad infinitum. And therefore the fign + or —, that is to be prefixed to the fecond term of every horizontal row of terms in the faid compound feries y will be the fame with the figns of all the terms placed immediately above the: faid fecond term, or of all the terms of the fame vertical column that preceed it : or, the figns to be prefixed to all the terms of every vertical column of terms in the faid compound feries y above the two loweft terms of the faid column will be the fame with each other, and with the fign of the loweft term but one of the faid column, Qe ows A réca- 408° A METHOD OF EXTENDING CARDAN’S FIRST RULE A recapitulation of what bas been foewn in the fix foregoing articles concerning the analogy between the.compound feries YT’ and the compound feries-y. 36. It follows from the fix foregoing articles, 30, 31, 325 33, 34, and 35, that, if we compare the compound feries y, obtained in art. 29, with.the com- pound feries T, obtained above in art. 14, the analogy and the differences be- tween them will be as follows, to wit: In the 1ft place, in the compound feries Sr? Saye Se Ga eee &c, or the e3 e e7 e? elt ‘ ; j > exes < e odd powers of the fraction —, inftead of the fractions ~, 3, = > > aT Te : Be Nee es ee? ee eee y the terms will involve the fractions —, &c, or the odd powers of the fraction — —, which are ‘avenge in the terms of “t3? the compound feries T. In the next place, the co-efficients of the feveral terms in the feries y that in- volve in them the feveral odd powers of the fraction < in the compound feries y will be the very fame with the co-efficients of the feveral correfponding terms, or terms involving the fame odd powers of the-fraction ~, in the feries I. refpec- tively. And, laftly, the figns — and -+- to be prefixed to the feveral terms of the 3d, sth, 7th, and other following aur vertical columns of termsan the feries y — (which involve the fractions xa &c) will be the fame as are to be pre- 39? => fixed to the correfponding terms of the 3d, sth, 7th, and other following odd ; : F - 3 : e eo 3 vertical columns of terms in the feries T, which involve the fractions 73g : 5 &c; but the figns + and —, which are to be prefixed to the feveral terms of the 2d, 4th, 6th, 8th, and other Paints even vertical columns of terms in the el ea bay Peat E? feries y (which involve the fractions - &c) will be refpectively con- trary to thofe which are to be ppeeaced to che correfponding terms of the 2d, 4th, 6th, 8th, and other following even vertical columns of terms in the feries oF eee ° i 3 I’, which involve the fra€tions S &c. HEE TE 37. If we multiply the foregoing compound feries cailed y, (which was ob- tained in art. 29) by the quantity 6z, the product will be the following com- pound feries ; to wit, 6 Be.+ - FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. 409 6 6 Bces 6 Bp e7 6 BE e? 6 Brett 6 pc el3 Be 6 B é +- 2 rary zt 76 Fhosk 23 zto Ps ae “gg l2 — &e 6 pe} 6 BpeS 6 cp e7 6 p69 6 DE elt 6 pF et? pk zt ze 28 P<) wiz 6 Fes 6 BF e7 6 cr e? 6 pFe™! 6 EF el3 & + z4 zo Lees + St) Tite + c 6 He? 6 BH e? 6 cH e™! 6 DH et3 ae, 2° ay: <8 > i Pra ce) atin t giz ole &c 6 Ke 6 BK e™ 6 cK e¥3 + 28 + gto > git + &c M eX! BM et3 aang: zt? ae wiz p &e 0 33 to erat ie, OCs Let this compound feries, for the fake of brevity, be denoted by the fmall Greek letter 5. Then, fince the produc of the multiplication of g into the ex- I — é 3 5 q i Ir 13 pefion a='5 x the ries = BP 4 PEF BP BE 2 — ad infinitum is equal to 62 X the compound feries y, it will alfo be equal to the compound feries 6. Of the analogy between the foregoing compound feries 3 and the compound feries & obtained above in art, 16. @ 38. Now fince the compound feries A, obtained above in art. 16, is equal to 65 x the compound feries T, and the compound feries 3 is equal to 62 x the compound feries y, it follows that there will be the fame analogy and the fame differences between the two compound feriefes A and 3 as between the two compound feriefes P and y. And confequently the analogy and the differences which are fhewn in art. 36 to take place between the two compound feriefes T and y, will take place alfo between the two compound feriefes A and 0, and therefore, In the firtt place, the fecond and other following terms of the compound feries ie . Berea. (et gem ete . 8 will involve the fractions >, 4, =» se» cis» sa» &c, inftead of the cor- : 2 e3 ae EIR 8 pd. ptt ers A 3 . refponding fractions ae Mt gil este 1 qge hae &c, which are involved in the fecond and other following terms of the compound feries A. Coe Fhe Coa Cee i ; 3 And 2dly, the co-efficients of the feveral fractions “nO So eee z 12? & &c, inthe terms of the fecond and other following vertical columns of the com- pound feries 9 will be the fame with the co-efficients of the correfponding frac- . vot) a nT 2 td . . , : Seige ilies errs &c, refpectively, in the terms of the fecond and other following vertical columns of the compound feries A, Vor. II. 3G And, 410 A METHOD OF EXTENDING CARDAN’S FIRST RULE~ And, 3dly, the figns — and +, that are to be prefixed to the feveral terms of the 3d, sth, 7th, and other following odd vertical columns of terms in the compound feries 3 (which involve in them the fractions <, <, = &c) will be the fame as are to be prefixed to the correfponding terms of the 3d, 5th, 7th, and other following odd vertical columns of terms in the compound feries & which . ; . 5 e e? er3 involve in them the correfponding fractions =, 4, sz, &c; and the figns + giz? and — which are to be prefixed to the feveral terms of the 2d, 4th, 6th, 8th, and other following even vertical columns of terms in the compound feries é . . . . 3 e7 = et . . (which involve in them the fraétions = —e = =az> &c) will be, refpectively, contrary to thofe which are to be prefixed to the correfponding terms of the 2d, 4th, 6th, 8th, and other following even vertical columns of terms in the com- Ve : . : F e3 e7 eit ets pound feries A, which involve in them the fractions ie? aed) pao &c. ee ee nem ements oncemmmaeeanee The proauct of the multiplication of the co-efficient q into the ; 3 s 7 exprefion 2/3 (z x the feries - ea z3 zs 2? K e? Me™ o e%3 : Sonia esc is equal to the fimple ; P 3 aes Re? $e? a JOS OBC sae hoattita de ARR vers raat + &c. 39. The compound feries mentioned above in art. 19, and denoted by the Greek capital letter 2, is equal to 6 Be — the compound feries A. And it is fhewn in art. 25 and 26 that the compound feries X, or 6 Be — A, is equal to P ¢3 ae Re? se? pe ls v e%3 the fimple feries —- + =-+ [+ 4#+—ae+ ae + &c. Therefore (adding A to both fides) we fhall have 6Be = A425 4 26 4 S24 28 + ia sanee + &c, and A = the fimple feries 6pe— if GS 5 5 ma hag 54 ° e? bay ers : : “ = _ _— = — &c, in which all the terms after the firft term 6 Be are marked with the fign —, or fubtracted from the faid firft term. But it has been fhewn in the foregoing art. 38, that the co-efficients of the fractions ORs BE He eee Vert. er ets . ; meat Mhecet Wierd) Seas ottng ber &c, in the compound feries 3 are equal to, or the : a : ‘ eae @& 4 se eae fame with, the co-efficients of the correfponding fractions =) =p) a eee ae &c in the compound feries A, refpectively; and that the figns — and +, that are to be prefixed to the feveral terms of the 3d, 5th, 7th, and other following odd vertical columns of the compound feries 3, (which involve in theni the fractions e> ry FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. 411 ¢5 9 e313 $ : —e = =z» &c) are the fame with thofe which are to be prefixed to the corre- {ponding terms of the 3d, 5th, 7th, and other following odd vertical columns of the compound feries A, which involve in them the correfponding fractions _ 4 13 x oe &c; and that the figns which are to be prefixed to the feveral terms of the 2d, 4th, 6th, 8th, and other following even vertical columns of the compound feries 3 . . . - e3 e7 etl ers £ (which involve in them the fractions ee a eS SP &c) are, refpectively, contrary to thofe which are to be prefixed to the correfponding terms of the 2d, 4th, 6th, 8th, and other following even vertical columns of the compound fe- ries A, which involve in them the fractions 5, , = = &c. And in both’ the faid compound feriefes A and 6 the firft term is 6 Be. It follows therefore, that, fince the compound feries A is equal to the fimple feries 6 Be — retin aes Re? $e meth vets Se eee er eC ad infinitum, the compound feries 3 muft s s 5 > P e3 qe Re? se? Ter* Vv et3° Deequal to the fimple ferics 6 Be Pp &c ad infinitum. But, by art. 37, the product of the multiplication of the I . ‘ — = ernie De} Fe H e7 co-efficient g into the expreffion 2 z 3 x the feries ee Ue rene eT 9 M Ir oO 13 = = — i a — — &c is equal to the compound feries 3. Therefore the I produét of the multiplication of the co-efficient g into the expreffion 2 x Bax Pee pe Fé H e7 K e? Me™ o e%3 ; eer enes i eth ee tse + a a &c will alfobe equal : P e3 aes Re7 se? i dee v et3 Saeumple feries 6 Be = + ae t+ &e ad infinitum. " Of the relation of the product of the multiplication of the co-efficient q into the exprefion 2 /*(z x the Series Be D e3 Fes He7 Ke M ett oe &? % x3 2 a! zg zis f to the cube of the faid expreffion. 40. We are now to fhew that the product of the multiplication of the co-effi- I : : “ts AaB & De? Fes He7 K e9 cient g into the expreffion 2% 3 x the feriles— ——> +> —-=+5>- oy ea 13 i . ve 4 -— — &c is greater than the cube of the faid expreffion, and that the agit 213 . ° : . ° 6 difference is equaltor, orto 6 Be, whichis (= 6 x a XLSexn=ar. 2G 2 Now , A12 A METHOD OF EXTENDING CARDAN’S FIRST RULE Now this will be evident, if we can thew that the fimple feries 6 Be, = ps o¢5 R e7 s 9 Tet wars Sg oF BEE MAP SUE — <0 zz z K e? Met! 0 e?3 + 5-45 +a &c is = 82 x thecube of ae faid feries. We mutt De Fe H e7 K e9 Melt 0 e3 therefore raife the faid feries = Se Eiger art oe é&c to its cube, or third Bowen by multiplying it twice into itfelf This may be done as follows. rpg TA 5. Be Dé Fes He? The multiplication of the feries = — —- + —- — = K e9 M.e%t o e%3 aS SS te ei twice into itfelf, in order x9 gil I Be De - Fe> He7 K e? M ert 0 e%3 —— — aoe eee SP ee = == z zs ge 2 2! nL 2? zit zi3 Bec Be Des Fes H e7 wi Ke? Met 2 &e B7¢2 RD e4 BF e° BHe® BK et BM et? & a rps ot 2° ae 23 210 a7 7h 1Z aL c BD e+ p2e° DF e® DH e?° DK el &e roma x x =, = zo om par Pr + BF e® DF é® Fre FH e'2 + &c + 6 mart 8 210 me giz 8 10 Iz BHe DHe FHe z gio el BK e?° DKe + — SE + &e BM e7* = ze ob &ec Be? . rat FOR THE RESOLUTION OF CUBICK EQUATIONS, &c, Be” 2 BD et 2BFe®° 2BHe?® + 2 BK el 2BM rma + &¢ _—_—_ ——s wae — 3 xt 26 28 zt De® 2 DF e® 2 DH ele 2 DK eet 6 ag 8 ar ro ae 12 + &c z z z x Rees 2 FH at, + + &c Be De F eS H e7 4 K e? Mel! 0 33 z 23 25 ie 27 zo gil xi3 B3¢3 2 B*pe> 2 B7F ¢7 2 B7H e? 2 Ber 2 B2M e}3 & 0 aa a Mee CM eT BIR BD7e7 2 BDF e? As 2 BDHe™ 2 BDK ets 13 ae are Pag i z? Za BF7e™ 2 BFH Ee! 13 stores pr osehre SEROCE B7D es + 2 BD7e7 2 BDF e? 2 BDH el! 2 BDKel3— 8 D309 2 p*Fe™ 2 D*H e'3 if & as eee -+ =F ae az c DF2e%3 elas Ser &c B7F e7 2 BDF 9 2 BF2e! 2 BFH et3 Bec D7F e™ 2DF se ply ees cee agg muons ah we mA B7H e?9 2BDH et 2 BFHe?3 ee gt yg pee _D?H ets i — + &c B?K et 2BDK ce le a eer ee + &c ce eens ry eer + &c eS B3¢3 3 B*pDeS i" 3 BF e? 3 BH e? Hf ZB’Ker 3 B*Mets a 4 &e x3 z9 z? 2? pat! 3 BDe7 6 BDF e? 6 BDH e™ prate 3 + & x? x x? As zi3 ¢ D3¢9 3 BF7e™ 6 brHe™3 x) zit p?F elt ra 13 + say Vani ta pe DF2e%3 = sores + &e. . " ‘ 3 B Des 5 This laft compound feries is the cube of the feries — -_ — = K e? M er 0 ef3 = ace aT > mone &c ad infinitum. zs 413 — &c He? gl Therefore 7 AI4 A METHOD OF EXTENDING CARDAN’S FIRST RULE ‘ iB Des 5 7 9 Therefore 8 z x the cube of the faid feries— ——=— + 22 — 2 4 22 _ Zz 23 zs Pau 2? ue _ os — &c will be = 8z x the foregoing compound feries; or, if, for the fake of brevity, we denote the faid compound feries by the fmall Greek - Be De Fes He? letter A, we fhall have 8 z x the cube of the feries — — 2 K e9 mM ett 0 et ae = ar tor — &c equal to 8z x the compound feries~; and confe- I : cae : Be De Fe> He? quently the cube of the expreflion 22 3 xX the feries — — — =F Ke M eT Oe? + => — = + ar — &ec ad infinitum will be = 8% xX the compound feries A. A comparifon between the compound feries A, obtained in the foregoing article 41, and the compound feries A, ob- tained above in art. 17. 42. We muft now compare the compound feries A, which is equal to the “ape De Fé He? K e? M ett o et3 cube of the fimple feries TA 7s on ob ern a: + ee — ao +. pers &c, with the compound feries A, obtained above in art. 17, which is equal to the ren Be De F és He? K e9 M ett 0 e%3 P cube of the fimple feries SP Saha gs eg to &c, in order to difcover the analogies and the differences that fubfift between them. Now, upon comparing together thefe two compound feriefes A and a, we fhall find that, In the firft place, wherever there is any power.of the fraction = in the com; pound feries A, there will be the fame power of the fra¢tion < in the correfpond- ing term of the compound feries A. And, in the fecond place, the co-efficients of the terms of the compound fe- ries A are equal to, or the fame with, the co-efficients of the correfponding terms of the compound feries A. And thefe two analogies between thefe two compound. feriefes muft take place not only in the few firft vertical columns of terms of the faid feriefes which are fet down above in art. 17 and art. 41, but throughout all the terms of the faid feriefes, to whatever number of terms the faid feriefes may be continued : ae Te Bie Des F eS H e7 Ke? M ett oe becaufe the original feries yr sas oe erie Poa a 2 hip eaee zis aes &c (by the multiplication of which twice into itfelf the compound feries A is pro- duced) involves in its terms the fame powers of the fraction =; and the fame co-efficients B, D, F, H, K, M, O, &c, combined with thofe powers refpec- tively, 4 i a FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. A418 . “ite ve uh Bre De} F eS He? Ke M ett tively, as the other original feries —+— + >7+—47+ 4+ + aa + &c (by the multiplication of which twice into itfelf the compound feries A was produced) involves.of the fraction <. And, in the 3d place, all the terms of the compound feries A, after the firft term a, are marked with the fign +, or added to the faid firft term: but in the compound feries A only the terms of the 3d, and sth, and other following odd vertical columns of terms (which involve the fractions =, ss =, ae &c) are marked with the fign +, or added to the firft term <3 and the fecond B7D e> ‘ J ‘ term 3 , and the terms in the 4th, 6th, 8th, and other following even verti- 25 . . . ° . = 5 e? 13 cal columns of terms in the faid feries (which involve the fractions —, —, —, 2 z z er? a7? &c) are marked with the fign —, or fubtracted from the faid firft term. 43. That in the compound feries a the terms in the fecond, third, and other following vertical columns of terms are marked with the figns — and + alter- nately, is evident upon infpection as far as the multiplication is carried above in art. 41, that is, as far as the fix firft vertical columns of terms. And that the - fame alternate fucceffion of thofe figns will take place in all the following co- lumns of terms in the faid feries, to whatever number of terms the {aid feries may be continued, will be evident from the following confiderations, : ; eth AR ee ohe De Fes H e7 K €9 Mert Since in the multiplication of the feries — ——— + — — — +—— — — % 2 z z? z? git 13 mal 3 ae ~— — &c into itfelf in art. 41, in order to obtain its {quare, the figns — a and + follow each other alternately both in the multiplicand and the multipli- — cator (which are both the fame feries = — teh sf ae a nde &c), it is evi- % x3 zs x7 : dent that in the feveral products of the multiplicand by the firft term “ of the multiplicator, and by its third term <, and its fifth term xe, and its feventh term oo. and all its following odd terms (to which odd terms the fign + is prefixed) the order of the figns + and — in theterms of the faid products muft be the fame as in the multiplicand itfelf; that is, the terms of the faid feveral products which conftitute the 1ft, 3d, sth, 7th, and other following odd hori- zontal lines of the faid general product (before the fimilar terms are collected together by addition at the bottom) will be marked with the figns + and — alternately. And it is likewife evident, that in the feveral products of the faid multiplicand by the fecond term = of the multiplicator (which is marked with the 416 A METHOD OF EXTENDING CARDAN’S FIRST RULE the fign —), and by the fourth term Be, and the fixth term ap and the fol- lowing even terms of the multiplicator (which are all marked with the fign —), the order of the figns + and — inthe terms of the faid products muft be con- trary to the order of them in the terms of the multuplicand itfelf; that is, the terms of the faid feveral products which conftitute the 2d, 4th, 6th, 8th, and other following even horizontal rows of terms in the faid general produc (be- fore the fimilar terms are collected together by addition at the bottom) will be marked with the figns — and + alternately. Therefore the firft term of every new horizontal row of terms will be marked with a contrary fign to that of the firft term of the horizontal row next above it, and confequently with the fame fign as the fecond term of. the faid horizontal row next above it, or as the term under which it is placed; the firft term of every new horizontal row being placed immediately under the fecond tern of the horizontal row next above it. And thus it appears that the laft term of every vertical row and the laft term but one will be marked with the fame fign. And hence it follows (from the alternate fucceffion of the figns ++ and — to each other in the terms of all the horizontal rows of terms in the faid generaliproduct, before the terms are col- lected together at the bottom) that all the terms in every vertical column of terms will be marked with the fame fign as the loweft term, and confequently that the terms in the fecond, third, fourth, fifth, fixth, and other following ver- tical columns of terms will be marked with the fign — and the fign + alter- nately. Q. E. D, 44. And, fince all the terms in each of the vertical columns of the general produgt in art. 41 (before the fimilar terms are colle&ted together at the bottom by addition) are marked with the fame fign + or —, and the terms in the 2d, 3d, 4th, 5th, 6th, and other following vertical columns of terms in the faid ge- neral produét are marked with the fign — and the fign ~- alternately, it follows that in the compound feries fet down under the faid. general product (and which is equal to it and derived from it by adding together the fimilar terms in each vertical column fo as to convert them into fingle terms) the feveral terms of the fecond, and third, and fourth, and fifth, and fixth, and other following vertical columns of terms will alfo be marked with the fign — and the fign + alter- hately. ae as 45. And, fitice in the compound feries laft mentioned, which is equal to the D e3 Fe5 H ¢7 K e? mM e*? o e33 : Be fquare ofthe fertes — — =e th se 5 5 ar thepps tg, “theres cond and other following vertical columns of terms are marked with the fign — - and the fign -+ alternately, and the compound feries A (which is the cube of the e De Fé He? Ke? wert 0 73 j —_—_ Pe PS Se og eS See” Ee ; , j a SELIGS ee mn Saapeh Sees ee &c) is produced by mul é , ee AE cae os” BE De Fes He? tiplying the faid compound feries into the original feries - — ~- -+ = — 5 K €? m e*t oes : . ‘ += — > + = — &e, in which the fecond and other following terms 2 are Y FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. Al] are alfo marked with the fign — and the fign + alternately, it will follow, by repeating the reafonings ufed in the two laft articles 43 and 44, that in the pro- duct of this laft multiplication, that is, in the compound feries a, the terms of the fecond, and third, and fourth, and fifth, and fixth, and other following ver- tical columns will alfo be marked with the fign — and the fign + alternately, agreeably to what we have feen by infpection of the faid compound feries.a (as fet down in art. 41) to take place in the firft fix vertical columns of it, or as far ° ‘ ex3 ‘ : as the terms that involve the fraction —;, and, agreeably likewife, to what was afferted, in art. 42, concerning all the following vertical columns of the faid compound feries. Q: E. D. Of the compound feries 3, which is equal to 8z xX the compound feries r, and confequently to the cube of the ne “pes Fes He? exprelfio 4 fz peiiaeiae: at preffion 2/3 |z x the feries = Se awe x K 69 M ett 0 e™3] ie Ss ser tar — Se ad infinitum. 46. If the foregoing compound feries A, obtained in art. 41, be multiplied into 8 z, the product will be the following compound feries, to wit, 8 Bse3 24 B*peS 24 B7F e7 24 B7H e9 24 BK e™! 24 BM et$ 4. & 2 zt ze 28 zie at2 Cc 24 BD%e7 48 BDF e? 48 BpHe™ 48 BDKet3 & i 6 8 a5 10 Iz + Cc % z & Zz 8 p39 24 BFZet 48. BFHe™3 tg win Ba gt fae At Spa tla EE ad z8 Paid zi2 24.D7F eT 24 D7H e783 & ot seror amin a cara Cc 24 DFe™3 & -B fT + XC; which, for the fake of brevity, we will denote by the fmall Greek letter 7. I De3 Fe5 Then will the cube of the expreflion 2 x 3 x the feries ~ ere a He? Ke? M ett 0 el3 » eel : ! ES eatin sgt p oh rps &c (which is equal to 8 z x the compound feries a) be = the compound feries z. 47. Since the compound feries 7 is = 8% x the compound feries A ; and the terms of the fecond, and third, and fourth, and other following vertical columns of the compound feries A are marked with the fign — and the fign + alter- nately ; it is evident that the terms of the fecond, and third, and fourth, and other followiag vertical columns of the compound feries x will alfo be marked with the fign — and the fign + alternately: becaufe the multiplication into 8 z can make.no change in the figns of the quantities that are multiplied. Vo. Il. pM | A com- 4 , 418 A METHOD OF EXTENDING CARDAN’S FIRST RULE A comparifon between the compound feries mr juft now ob- tained in art. 46, and the compound feries 11, obtained above in art. 18. 48. It has been fhewn above in art. 42, that the co-efficients of the fevera} terms of the compound feries A (fet down in art. 41) are refpectively equal to, or the fame with, the co-efficients of the correfponding terms of the compound feries A, fet down in art. 17. And therefore the co-efficients of the feveral terms of the compound feries 7, or 8 2 x the compound feries A, which are ~ equal to 8 times the co-efficients of the correfponding terms of the compound feries A, muft be equal to the co-efficients of the correfponding terms in the compound feries II, or 8s x the compound feries A (fet down above in art. 18), which are equal to 8 times the co-efficients of the correfponding terms of the compound feries A. ———————————————————————————————————e A proof that the compound feries x is equal to the fimple rd aes R e7 se? eu v e33 Series perietres sooth ee A arden 7 EF¢ ad infinitum. 49. It appears from the laft article 48 that the co-efficients of the feveral terms of the compound feries 7, which is fet down in art. 46, are refpectively equal to, or the fame with, the co-efficients of the correfponding terms of the compound feries H, which is fet down in art. 18; and it appears from art. 47 that all the terms in the fecond and other following vertical columns of the compound feries 7 will be marked with the fign — and the fign + alternately, to wit, all the terms in the 2d, 4th, 6th, 8th, and other following even vertical columns of it with the fign —, and all the terms of the 3d, sth, 7th, oth, and other following odd vertical columns of it with the fign -++ ; whereas in the com- pound feries II all the terms in the 2d, 3d, 4th, 5th, and all the following ver- tical columns, both odd and even, are marked with the fign +. It follows therefore that, if the compound feries II be converted into a fimple feries by adding together into one fum all the terms of each feparate vertical column, and the compound feries 7 be likewife converted into a fimple feries by adding together into one fum all the terms of each feparate column, the co-efficients of the terms of the fecond fimple feries, which is equal to the compound feries 7, will be refpectively equal to the co-efficients of the correfponding terms of the former fimple feries, which is equal to the compound feries II: but in the fim- ple feries that is equal to the compound feries 7, the fecond, and fourth, and fixth, and other following even terms will be marked with the fign —, and the third, and fifth, and feventh, and other following odd terms wilt be marked with the fign + ; whereas in the fimple feries that is equal to the compound feries II all the terms after the firft term will be marked with the fign.» There- fore, FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. 419 fore, if we fuppofe (as was done above in art. 23) the co-efficients of the terms of the fimple feries that is equal to ats Py feries ue taba PO... R, S, 5 7 3 ee, oc, oo eat ish — ett vers Mashia a. to ad HU the fimple feries ait is haig to iv com- aes R e7 $ 69 Tes v eX3 pound feries x will be ~ a SIR Cle coc ak Sh i ye eee tum. A proof derived from the foregoing articles, that the ex- ; 3 5 7 preffion 2/3 (z x the feries = —s A aa aang pint & pA vA Ke? M ett oet3 Bi ieag ahaa? te ia ua er is equal to one of the roots of the cubick equation qa—x} =r. go. It is fhewn in art. 46 that the compound feries 7 is equal to the cube of I = ox B D e3 F eS He7 K e? m et® 0 e%3 the expreffion 2 z 3 A NS ol nei oar og sien pile wee or RPC aerry . » pee Qe Re? se? Tet v e%3 —@c. Therefore the fimple feries — ~ <> tise — gry tige 7 GE &c ad infinitum (which is equal to the compound feries 77) will alfo be equal to : 3 S 7 9 == ° Be De Fe He Ke the cube of the expreflion 22 3 xX the feries — —— + —- —- ato % 23 2 27 Ad met 0 e'3 eT &c ad infinitum. g1. It was fhewn above in art. 39 that the product of the multiplication of gh De Fe’ He? the co-efficient gq into the expreffion 2 3 & the feries = screen eae Ke? Me™ 0 et3 . : ; pe aoe aero &e ad Nee. 1s fa to * feries 6 Be 4 —- — 5 7 9 Piet Ww es fh ae And now we have feen, in A x ze zie I : _ a, eRe Des the laft art. 50, that the cube of the faid expreffion 2 z 3 x the feries et Fe He? Ke? M er 0 et3 ‘ ope aes Regetprot Iota ge ce =z — &c is equal to the feries — 7 7 9 II 13 4 : net Se 4 Te 4 &c ad infinitum, which confifts of the very fame c Qe Re? s 9 aTett a e%3 terms as the feries 6 B é- = EVs ait pg apc pete PT + &c ad in- finitum, except the firft term 6 Be, and therefore is lefs than fie faid laft-men- gq Hz tioned 420 A METHOD OF EXTENDING CARDAN’S FIRST RULE tioned feries by the difference 6 Be. It follows therefore, that the cube of the ia De Fes H e7 Ke? Met o et3 expreffion 2 2 3 xX the feries = ——> + + Soo tic = &c ad infinitum, is \efs than the product of the multiplication of the co-efficient g into the faid expreffion, and that their difference is 6 Be, or r. And confe- : quently the faid expreffion 2z 3 x the feries = — zs a — es a + xe at es = oe — &c ad infinitum mutt be equal to one of the roots of the cubick equation gx — «° = Tr. Osama? I G2. Having now fhewn that. the expreflion 2 z 3. x the feries a nt ae ae ~- ae “+ ae a a a nae — &c ad infinitum is equal to one of the roots of the cubick equation gx — «= 7, it remains that we fhew that it cannot be equal to the greater of the two roots of that equation, being always lefs than the leaft poffible magnitude of the faid’ greater root; whence it will follow that it muft be equal to the leffer root of the faid equation. ‘This may be fhewn in the manner following. Of # the leaft poffible magnitude,. or the lower limit of the magnitude, of the greater root of the cubick equation qx —=x3 <= r upon the fuppofition that r is lefs than V2 xX AT 3V3 53- The greateft poffible magnitude of the abfolute term 7 of the cubick 29/9 equation gx — «7 = ris —=¥* reyzene that value of the compound quantity gx — x° which refults from a fuppofition that wis = “A tity 219 the equation gx — «* =r 1s impoffible; and if ris exa€tly equal to 3/3 sti , the equation gx — «* = r will have but one root, which will be = wt; and, if 7 is lefs than “v4, the equation gx — «* =r will have two roots, of V9 If r is greater than this quan- which the leffer will be lefs than “=, and the greater will be greater than “4 bes 3 but lefs than /g. But in the foregoing articles it is fuppofed that the abil ae 294 v3 term 7 is lefs, not only than 5; (which is its greateft poffible magnitude), but likewife than4/2 ivf, which is lefs than wrk in the proportion of (2 to. 2, or of) 1toW2. Therefore the greater root of the equation gw — x? =-r muft, on the Wn FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. 421 the prefent fuppofition, be greater, not only than wt, but than the greater root , ee qV4q , iat of it when7 is = 4/2 X is Now, when rv is = 2 X , the greater root of the at qu — x =r, will be 4/2 X vo For, if we anes * to be = W2xX “2, we thall have «? = 2-4/2 X ff aac ig nie alae Fa Y2xX ty = 3VW%2*x Me and confequently gx — #7 (= 3 V2 X ivt 24/2 x ivt) a Ie poe ——— iv, or (42 X ive. Therefore, on the pre- eh fuppofition, to wit, that 7 is 8 nthe f2%X it, the greater root of the equa- tion gx — «* =r muft be greater than 4/2 X x4, or than ve X /7, or than eV [= x 77, or than /0.666,666, &c xk g, or than 0.8165 x o/¢3 or, in other words, / (2 x Yq, or 0.8165 xX +g, will be the leaft poffible magni- tude, or the lower limit of the magnitude, of the greater root of the equation qx — x? = r upon the prefent fuppofition. A proof that the faid lower limit of the magnitude of the greater root of the cubick equation qxn—w«? =r is ee than the boi pofft ble magnitude of the ex- De3 Fe H e7 K e Met? 0 et3 + $Me 4 SS ee Now this quantity x V7, or 0.8165 X 49, is greater than the greateft I pDe3 Fes He? : = ° Be poffible magnitude of the expreffion 2 z 3. x the ferles = — —- + => — => Ke: Met” “Gel neaty ; —— 4 — & aad infinitum as may be fhewn in the manner fol- ze aa git zi3 lowing. z 3 S 7 9 Ir ° en ° Be De Fe He Ke Me 544 Chis expreffion 24.3 ox; the fonies ep Sth See 0 et3 : : 2Be 2Dde3 2Fe5 2He? 2K e . Ai az Bz Bz 25 23 2 me 20e % 3,2 =e 3 a &c. M4 Ky vr Now, fince 2% is =f = re. or £ — ee, it is evident that, if ce, or —, be 2 fuppofed 22 A METHOD OF EXTENDING CARDAN’S FIRST RULE 4 - 3 y ; . fuppofed to increafe, zz, or the excefs of = above ¢¢ or 3 will at the fame time decreafe. ‘Therefore, if ce, or 5 is fuppofed to increafe from o to its greateft . : 3 3 poffible magnitude on the prefent fuppofition, to wit, to ri or ——, the nume- 2X27 v2 Be 2Dé3 2¥Fes 2He 2Ke° Cat gs rators of the terms of the feries —- — —; — oa tar rae 2 : ; 20e%3 ; : : i : J m : A : _ =z; — &c (which involve in them the powers of ¢) will continually increafe, 5 and their denominators (which are powers of the decreafing quantity 2) will at the fame time continually decreafe: and confequently the terms of the faid fe- ries will, on account both of the increafe of their numerators and the decreafe of their denominators, continually increafe at the fame time. Therefore the faid s1) 2B 2De 2Fes 2He7 2Ke?9 2Me™ 2 0e%3 : feries EE ETRE Sry hel Oe Sipe a a Le ee &c will have 3 3 3 3 3 3 Ki ; 3 : rr ~ 2 : attained its greateft magnitude when ee, or re has attained its greateft magni- . 3 : , $s 3 rr , 3 tude, or is = og at which time zz, or f — ¢, or f ——, will be = — MRED LT and confequently will be equal to ee. Therefore the : 3 S 7 ] Ir 13 ae . - Be De Fe He Ke Me oe expreffion 2 2 3 x the feries' — — — Tanti —>; tn ‘4: - ‘2Be 2Ddeé 2¥Fes 2He 2K 9 2 Me™ &c (which is equal to the feries —- — —— + —{ — —— + —— — — > % F zs ait % 8 ur? ay 20 e%3 : : . = tr. + —- — &c) will alfo have attained its greateft magnitude when ee or 73 x 3,8 3 < ° equal to me and zz is confequently equal to ee. But, when zz is equal to ee, De3 Fes He7 K €? M eT 0 e%3 Pe er wee ee ee — &c hecomes = B—D-+F—H+K—M+ 0 &¢, and z is (=e = T I eh q° Series 9 en se =) = tale and confequently z 3 = Fs and:! 2.23, <2 ee an aS‘? _ 2/79 _ (64x 2g __ 9/864 ~ 7+ (64 — 7s(32 USA Gea eda We Cl Pats he | X og = Vv [1.185,185,185 KX Yq = 1.028,7 X ¥g3 and confequently the f e 3 ° and ¢é is equal to a the feries — — z — BS De Fes He? Ke? Melt 0 ¢%3 expreffion 22°30 x’ the leries 0 ee a <> &¢e becomes in this cafe = 1.0287 & Wq x the feries B— D+ F—H-+ I K—M + O— &c; or the greateft poffible magnitude of the expreffion 2 z 3. ; cB e De F es He? K 9 Melt 0 33 . the {ergs A

    =r be lefs, not only than 294 (which is its greateft pofiible magni- 34/3 1g ey ete Wa) Tees q? tude) but than 2 X ae? or if r be lefs, not only than = but than reser or 424 A METHOD OF EXTENDING CARDAN’S FIRST RULE 3 3 or a? and ¢ be taken = -, and zz be = r os or ne — ee; the leffer root I piss / of the equation gx — x? = r will be equal to the expreffion 2 z 3 xX the feries = = =f — - 9 ae ahs sae aa — &c ad infinitum. It is in- z z & z z ze deed a very long and complicated eee But I know not how to make it fhorter without taking from the perfpicuity of the reafonings ufed in it, which are both various and abftrufe. And ‘ that they fhould be fo” will appear the lefs furprifing, if we confider that they fupply the place of thofe very obfcure and intricate operations by which many writers of Algebra endeavour to find the roots of impoffible quantities, fuch as 81 + — 2700 and 81 — ¥ — 2700. See upon this fubject Monfieur Clairaut’s Elémens d Algébre, Part V, art. ix, pages 286, 287, 288, 289, and a paper of Monfieur Nicole in the Memoirs of the Academy of Sciences at Paris for the year 1738, pages 99 and 100, from which Monfieur Clairaut has extraéted what he has delivered upon this fubjeét in the pages of his Algebra juft now cited. And fee alfo Dr. Wallis’s Algebra, chapter 48, pages 179, 180, of the folio edition at London in 1685, and Pro- feffor Saunderfon’s Algebra, pages 744, 745, 746, 747, and Mac Laurin’s Al- gebra, Part I. the fupplement to the 14th chapter, pages 127, 128, 129, 130, and the Philofophical Tranfactions, No 451. — — 57- We will now proceed to give an example of the refolution of a cubick ae of the foregoing or qx—w«3 =r, when s is lefs than ./2 X ii, or — is lefs than a 7 or , by means of the expreffion 2 32 x the feries / Be Dé Fe H e7 K e? Met 0 e338 . ey Re TE ie eck wit ee which we have fhewn in the foregoing articles to be equal to its leffer root. An Example of the refolution of a cubick equation of the foregoing form, qx—x =r, by means of the expref- > Be De F ¢ He7 fin 2? =X the feries — — — = ate zs K 69 Mert © ¢t3 ———- + <3 — &e. 5? git Let the Cae that is to be refolved by means of this expreffion be 15% PACte.g sae "ae and ris = 4; and confequently = =is(= = ae and — — is ee pte oe eS Per C= +) = 2, and 4 ; is (= 4 oh) 125, and ere orf, isyaee >= > 62.5, and — x is ed = 2\*) = 4, which is lefs than 62.5, or e. Therefore 5 this 1 " Bika FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. 425 this equation may be refolved by means of the foregoing expreffion 2 ¥?2 x ve BEe De Fé H e7 K €? M elt 0 e%3 the feries 2 ae Gp alters keer bamiees “oe ea psig —r — &c. 8 Now, fince r is = 125, and — is = 4, we hall have zz (= . as = 125 — 4) = 121, and confequently z (= VY 121) = 11, andy? z (= y? 11) = 2.223,980,090,569,361, &c, and 2? z(= 2 X 2.223,980,090, 569,361, &c) = 4.447,960,181,138,722, &c. And we fhall have ee (= = = 4, and confequently e = 2, and = meypes, Se aie Ba Sigh 0.181,818,181,818, and = = oT io Therefore —~ will be (= = x — = o 181,818,181,818 x eri oA 0.727,272,727,272 wei) ay 01000,010,5 15,407": 5 s 3 i And = will. be (= = Xx = = 0.006,010,518,407 X “hi 0,02 4,042,073,6 Ey —O.OOCm Ob, 0045023" e7 r es e* 4 — And = will be (= = x = 0.000,198,694,823 xX Tr = SEY = 0.000,006,568,423 ; Py ' 7 2 4 a And | will be (= S xX = = 0.000,006,568,423 X 7 = Oo. by 6 ,6 as = 0.000,000,21 751 37; elt . e? em 4 —s And a will be GS ai ee 0.000,000,217,137 xX eT ee: : 868,48 ocencconen sa) = 0.000,000,007,178 ; 13 . an ert 2a by 5 4 oa And <= will be (= = X = = 0,000,000,007,178 xX T= POP 028712 = 121 >) Therefore = Will be (= B® x o.1o1,o18,181,318 = S x 0.181,818,181, 818,181,818 818 = —e) = 0,060,606,060,606 ; 0.000,000,000,237- And 22 will be (= D X 0.006,010,518,407 = = X 0.006,010,518,407 is Sai 25273972035) = 0.000, 371,019,654 3 SI And ££ will be (= F x 0.000,198,694,823 = => X 0.000,198,694; C2 esaaeg 289110) = 0.000,005,996,277 5 729 Vou. II. 3 1 And 426 A METHOD OF EXTENDING CARDAN’S FIRST RULE He? . And ale be: (="H-X 0:000,006;565,423 = tte X 0.000,006, 568, =. 0,002,456,590,202, __ ‘ i ge ee cases ) = 0.000,000,1 24,807 5 Kg? is 150 And Pe will be (0K 3<, 0.0GGsn00,2% 77137 a ee X 0.000,000, ’ ? i.) 01004,669, 5315185, g. . 8 By Se ears ie Gael = 0.000,000,002,928 ; : M : 1. 7 And — will be (= M xX 0.000,000,007,178 = ae X' 0.000,000, ie 0.001,058,087,446. __ ; ee COs ly oie a GoT ee = 0.000,000,000,073 5 oet3 é : 5I 3920 And Te will be (= O X 0.000,000,000,237 = pois oqid X 0-000,000, ? > 000,237 = wait Bake a | = 0.000,000,000,001 9°37 = 387,420,489 FET OS RRA GR has p> ‘ Therefore the feries — ——— + —— ——— 4 = = 4 = Siena z 23 25 a Ad ou 213 be = 0.060,606,060,606 — 0.000,371,019,654 -+ 0.000,005,996,277, — 0.000,000,124,807 + 0.000,000,002,928 —= 0.000,000,000,073 = 0.000, 000,000,001 — &c = 0.060,612,059,812 — ©.000,371,144,534 = 0.060, 240,915,278; and confequently the expreffion 2 ./3 z x the feries oo Fes He? rn Ee as ae “< — &c will be = 4.447,960,181,138, &c X 0.060,240,915,278 = 0.267,949,192,431, &c. Therefore this number 0.267,949,192,431, &c is the lefler root of the propofed equation 15 «— x? ah 44 Qs. Ee is 25 ‘ zi The foregoing number 0.267,949,192,431, agrees with the true value of the lefler root of, the equation 15% — x? = 4 in all its twelve figures, the faid root being equal to 0.267,949,192,431,122, &c, or 2 — 1.732,050,807,568,877, &c, or 2 — 3. For, if we fuppofe «to be = 2 — ¥3, we fhall have 15% (= 15 X{2—v3) = 30 — 15 V3, and #*® (= 8 — 3 X4XA3+3X2*X 3—3 V3 = 8— 12 93 + 18 — 373) = 26 — 15 V3, and 154-7 (= 30 — 1573 —[26 — 1573 = 30 — 1573 — 26 + 15 73 = 30— 26) calf Another Example of the refolution of a cubick equation by means of the fame expreffion. 58. Asa fecond example of the foregoing method of refolving cubick equa- tions, let it be propofed to refolve the equation gov — x? = 98. Here g, the co-efficient of «, is = go, and the abfolute termris = 098. Therefore a will be (= = 30, and " will be (= 4 = 301?) = 27,000, ea ie and FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. 427 3 3 ; ; : 8 and ve or mere will be (= a) = 13,5003 and — will be (= =) = 40, and * will be (= 40\°) = 2401, which is lefs than 13,500, or ma Therefore the equation 90“ — «* = 98 ar be refolved by means of the foregoing ex- De F é§ H e7 K e9 M eT 0 3 3 ERY” yb Te, SRG Se ee DEAR athens $e ah preffion 2 ,/* z x the feries * paia aren as me = ane &c. / q Now, fince fis = F127 5Q005 and — is = 2401, we fhall have ez (= £ — vit a | rr — = 27,000 — 2401) = 24599, ie Z(= VW 24599) = 156.840,683, and VE % (= 4/3 156.840,683) = 5.392,865,326,078, and 2/2 z(= 2 X 5. 392,865,326,078) = 10.785,730,652,156. | _ And we fhall have ee (= a = 2401, and confequently e= 49, and = ae sete | Sey iA 4OU 156.840,683 ~~ Peg Oe Bee and 5 "" 24599° e ; ee é 2A0T es Therefore —= will be (= z Xo FS 0-312,418,940,435 X 24599 750.117,875,984,4 ee. 1$2.017.575,984835) — 0.030,493,836,1713 - zs : _* 33 e2 (ae : 2401 And — will be (=. 5. X..m =. 0.030,493,836,171 °X hay 0 73-115,600,646,571 : O06." =~ 1) = 0.002,972,299,713 : e7 ° es e 2 And om will be (= a XX Fw = 0.002,9725299,713 X Fn a 6,491,6 ee) = = 0.000,290,113,078 ; 0, 7 » And <'will be (= 5 x, S = 0,000,290,113,078 x 2° = 0.696,561,500,278, __ ere = 0.000,028,316,659 Ir q ; 9 g@ ; And = willbe: {= =. x ~ = 0.000,028,316,659 x EES =. ee —o. 000,002, 763,864 ; | oe he ‘. : And 5; will be (= S x SF = 0.000,002,763,864 x 2 = Sieh = 0.000,000,269,768 ; ae ips tid - 2401 0.000,647,712,968 And = (= az X gq = 0.000;000,269,768 x 24599 24599 = 0.000,000 30265330 Therefore — will pee B X 0.312,418,940,435 = — X 0.312,418,940, 435 = astethqntssy = 0.104,139,646,811 ; 3 1.2 And — 428 A METHOD OF EXTENDING CARDAN’S FIRST RULE 03 ° And ~- will be (="DD "oid 36,4039;036,171 = = X 0.030,493,836,17% : »469, 180,8 MN pore Sot Seeds) ae 0.001,882,335,566 ; gs 2 And — will be (= F X 0,002,972,299,713 = = X 0.002,972,299,713 _ 0.065, 399,593,586 a ) = 0.000,089,699,031 ; And 4£ will be (= H x 0.000,290,113,078 = ah X 0.000,290,113, o78 = ab Op 29' tI") == 0.000,005,512,487 And a ae oe X 0.000,028,316,659 = Seer X 0.000,028, 316, Ce Pom, oh deae a = 0.000,000,381,948 ; And aes Wil be 4 =. Vise 0.000,002,763,864 as raya. X 0,000,002, 763,864 = senses ) = 0.000,000,028,393 5 And a will be (= O xX 0.000,000,269,768 = eRe: X 0-000,000, 0.856,491,318,560 387,420,489 And & = will be (= Q_X 0.000,000,026,330 = 269,768 = = 0.000,000,002,2103 70,664,648 10, 460,3 § 3,203 ) == 0.000,000,000,177. X 0.000, 1.860,600,18 1,840 10,460, 3531203 De3 Fe H e7 K e9 Mert 0 e753 qets Therefore the feries ~ —— Sb Se eee ee é&c will be = a. eae 646,811 — 0.001,882,335,566 + 0.000,089,699, O31 — 0.000,005,512,487 + 0.000,000,381,948 — 0.000,000,028,393 + ©.000,000,002,210 — 0.000,000,000,177 + &cC = 0.104,229,730,000 — 0.001,887,876,623 == 0.102,341,853, 3773 and ope ees the expreffion e e e? ell el3 ae 2 V/?x x the feries 2 — 2S 4 Ff Ae —=r, when the abfolute term 7 is lefs than 42 x iv, Oe is lefs 3 3 than Me or > - 5» by means of the ama 27> z x the feries < — — + Fes He? KO M e™™ 0 e%3 OG ee rea Ae ters ir aT "4 &e ad infinitum: and eee | fhall not add to the length of this coils (which is already longer than I could have wifhed) by applying the faid expreffion to the refolution of any more ex- amples. Another expreffion of the value of the leffer root of the cubick equation qx —«* =r, derived from the foregoing ex~ preffion of tt. 61. But there is another expreffion for the value of the leffer root of the cu- bick equation gx — x? = r in the cafe here fuppofed, which, as it may be eafily F, . & ° . ° Be De} derived from the foregoing expreffion for it, to wit, 2 /3z X the feries — Stile LU — Fes He? K €9 M ett 0 e78 eaTsREe aS so See ee ebaabrere ag as &c, ought not, I fete to be omitted. This expreffion does not confift entirely of an infinite feries (as the foregoing expreffion does), but partly of a finite algebraick expreffion, and partly of an infinite feries; and fewer terms of the infinite feries are neceflary to be computed and added together, in order to obtain the value of ae expretf- De fion to any propofed degree of exactnefs, than of the infinite feries = —[£ + z x3 Fes He? K e? Me™ 0 e%3 qets - : . er te ae Pigs “aa P&C contained an’ the’ foregoing exe preffion. It is as follows, to wit, /*(z Aa em V{z—e—4vV7'%z x the infi- Me™ qets v er9 a ~ . De He nite feries oar ait aaa ent waite: a &c ; the terms of which feries are : ° i aaeee . taken from the feries that is equal to 1 + 2 or the cube-root of the binomi- ; e : . ce pe Ee Fes Ge al quantity 1 uae —, to wit the feries 1 - ere cconsgaepeh ss EMleoragh va -lherere bx pence oracle zt 25 ze H e7 168 K oe Ler? M el N on o 33 P et4 qets R er z7 28 ea eet zt ik, Wane ott mish ca wit 1S) cyte Keene 430 A METHOD OF EXTENDING CARDAN’S FIRST RULE 8 e> neni oer ts — &c, by beginning with the fourth term ~~, and taking Big eT every fourth term from it. iis expreflion may be derived al the foregoing D e3- F eS H e7 K e? Me?! oet3 3 —$ ee —__ ee SS ee oe expreffion 2/722 xX the feries = 3 are, mI a) iE Py ane = “+ &c in the manner a i zt The derivation of the expreffion of the value of the leffer root of the equation qx —x* == r given in the preceeding ar- ticle 61, from the former expreffion of it. 62. By the binomial theorem in the cafe of roots we have 4/3 F 4 = Ce : Be c e De Ee Fes G 6 H é7 re® K 9 Let fenenapchig a Qockha rah ge tie ca M ett N el” 0 e3 Pp v4 qe R er s e%7 ig e8 v et9 fant gay glk ce gia hee gh ie gt a up Be ce pe3 E e+ F eS Ge He7 re Vee a ENCLEIICS < + &c is (= 0.000,371, 019,654 + 0.000,000,124,807 -- 0.000,000,000,073 + &c) = 0.000,371, 144. A432 A METHOD OF EXTENDING CARDAN’s FIRST RULE 144,534, &c, and 4 /? 2 xX the faid feries ze + a: ne <> + &¢is (=m X 2.223,980,090,569,361, &c X 0.000,371,144,534 = §.895,920,362,2775 A444, X 0.000,371,144,534) == 0.003,301,672,217,358, &c.° And z + els (= 11 +2) = 13, and g—eis (= 11 — 2) = g; and confequently /3(z + eis(= WV * 13) = 2.3515334,087,721, and V3 {z—eis(= V3 9) = 2,.080,08 3,823,052. Therefore /? (z + e — /3 z—e will be (= 2.3-7, 334,687,721 — 2.080,083,823,052) = 0.271,250,864,669, and 3 [z+ ¢ e3 e ‘ — /3\z--e— 4 Viz X the feries — +— + +oiexc_ will Degtees 0.271,250,864,669 —- 0.003,301,672,217) = 0.267,949,192,452. And confequently this laft number 0.267,949,192,452 will be equal to the lefler of the two roots of the equation 15“—w#«3 = 4, Qe Es as This number 0.267,949,192,452 is exact in the firft ten figures 0.267,949, 192,4, the more exact value of the leffer root of the faid equation being 0.267, 949,192,431,122, &c, or 2 — 1.732,050,807,568,877, &c, or 2 — 473, as was fhewn above in art. 57. In the other equation gox— x? = 98 we have feen above that eis = 49, and 2 is (= / 24599) = 156.840,683, &c, and < 1s? ct 3. be 418,940,435, and —- 1S =-0.001,882,335,566, and — IS = 0.000,005,512, ) = 0.312, 1S /* % is (= 7% 156.840,683) = §.392,865,326,078. ‘Therefore the memes Met. ets , 487, and ar IS = 0.000,000,028,393, and ~ 1S == 0.000,000,000,177, and e 1S : ae a +> + = will be (= 0.001,882,335,565 + 0.000,005,512, 487, + 0.000,000,028,393 + 0.000,000,000,177) = 0.001,887,876,623, He7 M ett qe and 4/3 2 X the faid feries a + + te will be 4 a ee 865,326,078 X 0.001,887,876,623 = 21.571,461,304,312 X 0.001,887,876, 623) = 0.040,724,25 7,520. And z +e will be (= 156.840,683 + 49) = 205.840,683; and z—e will be (= 156.840,683 — 49) = 107.840,683; and confequently Y? (z + e will be (= 3 205.840,683) = 5.904,417,671,968, and 3 (z—e will be (= WV? 107.840,683) == 4.759,860,337,980. Therefore “3 (z+e — / 3{z—e will be (= 5.904,417,671,968 — 4.759,860,337,980) = 1-144,5 575 333,988; and 73 (ze — J/3(z—e—4 V3 x the feries + ae + Me™? gets << + Sy will be (= 1.144,557,333,988 — 0.040,724,257,520) = 1.103, 833,076,468. Therefore this laft number 1.103,833,076,468 will be equal to the leffer of the two roots of the equation gox — #3 = 98. Qs Ect This value of the leffer root of this equation is exact in the fix firft figures 3.103,83, and exceeds the true value of the faid root (which is 1.103,832, QII,1;) ¥ , FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. 433 911,15) by only the fmall quantity 0.000,000,165,3, which is-fomewhat lefs than the difference whereby the former value found for this lefler root in art. . 2 3 5 4 58 by means of the expreffion 2 /? 2 X the feries ee Ee i nd mae rs K e? Melt 0 et3 qets , eer ar — ar &c, to wit, 1.103,831,664,9, falls hort of the faid true value, that difference being 0.000,001,246,2. A third expreffion for the value of the leffer root of the equation qx—x3 =r, derived from the expreffion obtained for it above in art. 55. 64. We may alfo derive another expreffion for the value of the leffer root of the equation gx — «* =r in the cafe here fuppofed, from the foregoing ex- Des Fe He? K e? M ert 0 ¢3 . Be aes) —_—_— reffio Ps.5 sh Race A (AO, oe 8 Me p merce’ 2 x the feries sree ee =A os mE zis Qe <== + &c, in the manner following. b. . . é The quantity /3 (2 +e— /3(z —eis = /3 2X velt “ha Siri BS e ‘ 2 3 E e+ Fes Ge J? |1—L = fz x the feries 1 + 2 — SE bg al! AU wig et at = 23 xt 2s = He7 Wu K 6? Le M ett N el2 o e%3 Pp el4 qets 8 Pim ste ee et ee Pa ae Pe . Be ce pe Ee Fes Ge He7 1 K 6 3 / feries 1 : = =; 3 * “e 3 = = L eto Melt Netz 0 et3 Pp elt qets : 32 2B€ ets ge 3) en Pe ok ae = V3 ‘ies —— et a: Si “73 a me &c = /3 z X the feries = 2pe 2Fe5 2He!7 2K e? 2 Met 2 0¢%3 2aQe5 — 3 Dy ag tee ty ts tee yin e De Fe He K e9 M ett oe ae feries — + “3 Gather aan < Sphor santos fe Paalirers = + &c. Therefore, : ite BE De Fes He? Ke Me™ 0 e3 _ — te —e_ _—_—_ = _— — —7_ neal if we add 2 ? z & the feries = = = = = hs hte — + &c to both fides, we fhall have WY? |z +e — yi (z—e + 2732 xX eee oe Bre Des Fe H e7 K e9 M ett 0 e%3 Qu eres — 3 the feries — — — A eR eee ee —=7 +&em2y%z 23 zs he et a git zi3 : ° 2 9r° 13 is ce x the feries — ‘de ~~ + — +. — Eee LA eee LE feries — + Fe K €? 13 f yh = a + &c. Therefore, if we fubtratt 3 (x +e — Vile De Fes He? * . B from both fides, we fhall have 2 47? z x the feries — TX Saga hee Ke M ett mens aes -.* Be Fes Ke? 0 e73 —_— — —— a pe ee aS 3 as i= ee a =3 eee et tors ae &c = 47? zx the feries 7 ob aaa e) ee De +&e + f3(z—e — V3 [z +e. But 2/32 x the feries — — — + Fes H e7 K @? M ett 0 3 ers esi or —— — oe — + &c has been fhewn above to be & 2 & 2 Vou. Il. ak. equal 434 «A METHOD OF EXTENDING CARDAN’S FIRST RULE equal to the leffer root of the equation gv — x? = r in the cafe here upp or when oy is or r., Therefore the expreffion 4./32 x the feries = — +o + << += —— ay &c ad infinitum + ae (z—e — +f? (z “ae or the 5 excefs of the oe 4/32 xX the feries — els a a = 4 and gor — #3 = 98 in the manner following. In the egret 15¢%— x3 =r we have feen above that eis = 2, and 2 fs = 11, and - is = 0.060,606,060,606, and ae i is = 0.000,005,996,277, and Kee? Fen 3 —~ is = 0,000,000,002,928, and —- * is = 0.000,000,000,001, and 3 2 is a ae Bt a/R R123 28 3500, eran &c. Therefore the feries - hee ae K e? a Pi aaa = cap &c will be (= 0.060,606,060,606 + o.cisctod ASME + ©.000,000,002,928 + 0. ciopehaaeer 000,001 + &c) = 0.060,612,059,812, &e, and 4/3 X the faid feries = = + — = + = a + &c will be (54 2..223,980,090,569,361, &c X o. Beate, one a &e = 8. 895,920,362,277%5 444 X 0,060,612 i & faid equation, or when — is equal°to, or greater than Z, another expreffion fomewhat different from the former, but bearing a great refemblanee to it, that fhall exhibit the true value of y in the fecond cafe of that equation, or when 2 3 - reecness a PSR MV ae Sale tiat apace ney stein mim an 3 3 ~ is lefs than oat provided it be not alfo lefs than “ x pr or than 2. And 2 54 this may be done by a train of juft and clear reafonings, and without any men- tion of. impofible,. or even of negative quantities, To fhow how, this may be effected is the.defign of the following pages. NORE 3. That the’ whole of this matter may’ be {een at one view, it sill be conveni= ent to fet forth the foundation and inveftigation of Cardan’s rule for refolving the equation y? — gy = 7 in the firft'cafe of it, ‘or when the ‘abfolute term’ rts équal to, or greater than, ava, or = is equal to, or’ greater than, fa which may be done in the manner following. Obfervations preparatory to the inveftigation of Cardan’s\ rule for refolving the cubick equation y* —.qy = 7, when r is equal to, or greater than, vt, or oe is 3 equal to, or greater than, - 4. Previoufly to the inveftigation of this celebrated rule, it will be proper'to make the following obfervations: OxssERVATION 1. In the cubick equation y3 — gy = 7 (which is a propofi- tion affirming that y?, or the cube of the unknown quantity y, is greater than gy, or the product of the multiplication of y by the known co-efficient g, and that the excefs, or difference, is.equal to the known quantity 7) it is evident that, fince y? is greater than qy, Pad or yy, mutt be greaterthan 2 or gy and confe- quently that y muft be greater than 4/7. af Oxs. 2. While y increafes from «/q ad infinitum, y? will increafe continually from 9/q.ad infinitum, and gy will alfo increafe continually: from the fame quan- tity 91/7 ad infinitum, Obs. 3, a ————EEE——E FOR RESOLVING THE CUBICK EQUATION &c. 445 Oss. 3. And, while y increafes from 4/g ad infinitum, the excefs of y? above gy will increafe continually from nothing ad infinitum, without ever de- cveafing. For, if we put y (or the letter y with a point placed over it) to denote the in- crement which y receives in any given portion of time, either {mall or great, during its increafe, gy will be the increment which gy will receive in the fame time, and 3y*y + 3yy” +? will be the increment which y? will receive in the fame time; bécaufe, when y is increafed to y + ys qy will be increafed to g x 'y + y, or togy + y, and y* will be increafed from y? toy + #3, or yi + 39°97 + 39 + y?. Now, fince y* is always greater than ¢ during the whole increafe of y from being equal to gq ad infinitum, it follows that y* x 9, or yy, will be greater than g X y, or gy, during that whole increafe. But ay is greater than y*y, and 3y*y + 3yy* + y? isitill greater than 3y*y. Therefore, & fortiori, 39° 9 + 399" + y? mutt be greater than gy during the whole increafe of y; that is, the increment of y? will be greater than the contemporary incre- ment of gy during the whole increafe of y.. Therefore the refidual quantity y3 — gy, or the excefs of y? above gy, will continually increafe from o, without ever decreafing, while y increafes from 4/q to any greater magnitude. Further, fince 3y7 y+ 3yy + y? is the increment of y*, and gy is the incre- ment of gy, and yy is greater than ¢¥, it follows that the excefs of the increment of y? above the increment of gy will be greater than the excefs of the increment of y* above y’y, or than the excefs of 3y*y + 39y° +? above y*y, or than the quantity 2y*y + 3777 + y%. - But the excefs of the increment of y* above thencrement of ¢,y is the.increment of the refidual quantity y3 —qy. | There- fore.the increment of the refidual quantity y3 — gy 1s greater than the quantity 2y°9.+- 399? + y3. But it is evident that the quantity 2y*y + 3 yy” + 3 will increafe continually ad infinitum, while ‘'y increafes ad infinitum; {fo that no quantity can be affigned, how great foever, which the faid quantity’ 2 y*y + 3.7°*y + 9%, or either of its two firft members 2y*y and 3 yy”, may not, by in- creafing the quantity y continually, be made to exceed. Therefore the incre- ment of the refidual quantity y*? — gy (which increment is greater than the quantity 2y*y + 3997 + fe will inereafe continually ad infinitum, or fo as to become greater than any finite quantity, how great foever., And confequently the refidual quantity itfelf (which receives the faid continually-increafing incre- ments) will increafe continually from,o.adinjuitum, or fo-as to become greater than any finite quantity, how great foever. Q; E. D. -. Oxps. 4. Since the compound quantity y? + gy increafes continually at the fame time as y increafes; and, when y is equal.to : SS, | 89/9 2gr/q Sag __ 64/9, 29K/9, « To equal to(—*+ — =, or —*- — =~, or) —~; it follows that, ify 1s greater AMG: GV V3 OBB? 33? 373 , . ng 5 is lef than 29 the {aid compound quantity will be lefs than 744. and 2 con- the faid compound quantity is and, if ver/o, if the compound quantity y?— gy, or; itsequal; the abfolute term 'r, is wor 6 greaten, 446 A METHOD OF EXTENDING CARDAN’S RULE greater than “iy, ‘the value of y will be greater than 21, ify? — q yy or ‘ is lefs than “v4, the value of y will be lefs than ne geOr ih — is greater than 274 f, y will be greater than rt and, if = = is lefs than = Pre, will be lefs than 29 3 29 3/3 guently (by the laft ance, y is greater than wp and confequently ~ will be. greater than <. - But i 1s ths fuare of 2. 29/7 Therefore, when 7 is totes than years iP is greater than 4 c, the {quare of “Oss. 5. When =z is greater than gs , or — is greater than £, and confe- 4! 27 2V4 /3 » 2 will be greater than = <, or of half the unknown quantity y, will be greater than 2. But (by Euclid’s Elen Book II, Prop. 5) it is always, pofible to divide a line, as y, into two unequal parts of fach magnitudes that the re€tangle under the faid parts fhall be equal to any quantity that is lefs than the UES of it’s half. Therefore, when =I¥4,. or = — is greater than 4 -=» it is poffible to divide the line, or root, y into two unequal parts of fuch magnitudes that their rectangle, or pro- y is greater than dud, fhall be equal to 4. This lait obfervation is the foundation of Cardan’s rule for the refolution of the cubick equation y? —gy = 7, in the firft cafe of it, or when r is greater 7 3 ‘ 4 r . ' than =v, or — is greater than f; the inveftigation of which rule I fhall now proceed to explain by the folution of the following problem. PF ROS a ME I. 5. To refolve the cubick equation y? — gy = 7, when the abfolute term 7 is 24/4 3/3” TF ae g ereater than or — 18 greater than at sO LU “ON. Since ¢ is fuppofed to be greater than “tvs, q ey it is poflible for y to be divided into two whequal and confequently (by Obferva- tion pike is greater than parts of fuich magnitudes that their rectangle, or produét, fhall be equal to 4, Now, let it be conceived to be fo divided; and let the greater of its two parts be called v, and the lefler be called zx. Then will vz be = ‘, and confequently 3vu2 willbe = g,and37z x v+2willbe= gy xX v+2z. Now, FOR RESOLVING THE CUBICK EQUATION &c. 447 Now, fince v + zis = y, we hall have y3 = v+ 2\3 = v3 +3072 + BUz* + SF = UP +z b 3BU°S Hb 3BUz* = Vi + Bf 3ZVEX V+ Z; and we fhall have gy = 9 X Vv XU+2. Therefore y3 — gy will be = v3 + 23 + 30% xX) V+%—gX vu +2; that is (becaufe 3vz x w+ 2 is equal to gx v+z), 95 — gy will be = v? + 23. Therefore the abfolute term ¢ (which is equal to y3 — gy) will be = v3 +4 23. 4g ahs Suerge ie and confequently z3 = ee But, fince 3 vz is = 9, we fhall ey; pm Therefore v? + z? will be = mn THF oe iene, and confequently r (which has been g3 2703" fhewn to be equal v? + 23) will mye =v! + Therefore (multiplying both fides of the equation by v3) rv3 will be = v®& + a and fubtracting v° from both fides). rv? — v° will be = Z. But rv? — v° is the product of the multiplication of 7 — v3 into v?, which are together equal tor. Therefore (by El. H, 5,) rv’ — v° muft be lefs than the fquare of half of r, that is, than =; and soaeaneny may be fubtratted from 7 Let it be fo fubtracted ; and, let its equal, = =, be alfo fubtracted from the fame quantity ra And the remainders will be en to each other ; that is, 7 — fru? — v*, or z — rv3 + v°%, will be equal to * —£. Therefore the f{quare-root of the trinomial quantity - — rv + v® will be equal tothe fquare- root of — — r. But the fquare-root of the trinomial quantity n —rvi + v%is the difference of the fimple quantities = and v3, that is, either = — vor v3 —-, according as = — or v3 is the greater quantity. But it has been already mew that v? and ty together are equal to r; and v is fuppofed to be greater than z, and confequently v3 is greater than z3. Therefore v? muft be greater than the half of v3 + z3, and confequently than the half of 7, or than — > There- fore the difference of “ and v3 muft be v3 — -, and not = — vv; and confe- quently vs — = muft be the fquare-root of the trinomial quantity 7 — 7v3 + v*. We fhall therefore have v? — = =VE rants and confequently (ad- ding < to both fides) v3 will be = > + / nn — £. and therefore (extracting. the cube-roots of both fides) v will be = f sft + of ce ce t. But 448 A METHOD OF EXTENDING CARDAN’S RULE But z has been fhewn above to be = Therefore v + x will be & vw + : aentieeee , I, — ml ila a: > WE ‘) =— / f ay val : -+- Weer ru and confequenely Ms 27. or the root of the cubick equation y* — g := = r (which is equal to v + 2) will ane CET b= Vint ¥ (2 i oper cae | +y¥[2—-£ 27 Q, E. I, 6. This expreffion of the value of y in the equation y3 — gy = r may be ren- & it dered more fimple by fubftituting in it the fingle letter e inftead of > and mr fingle letter s inftead of v|= wa Lr. For then it will bef? [e + 5 + — == ee pri Therefore, if e be put for half the abfolute term 7 of the cubick equation y? — gy =1, and ss be put for — aie £ or § be put for the {quare-root of = —— Es the te de aE Qe Ep is root y of the faid equation will be equai to" fe +5 + Ai fynthetick oi eae of the truth of the foregoing Solution. : | . Since this expreffion e + s}3 + re —S is equal to the root’ y of the sbatlintys — 7y = 7, it is evident that, if it were fubftituted inftead of y in the compound quantity y3 — @.y, which forms the left-hand fide’ of. that equation, it would make the faid compound quantity be equal to the abfolute term r, And fo we fhall find it will, if we make the faid fubftitution ; which may be done in the manner following. Ewe fuppole tobe = e+ iF + Sepa Me fal have y3 = ete ane eae x vd

    and confequently the compound quantity y? — qy (which has been thewn to be equal to the faid fraction) will alfo be equal to the fraction oe ri rk add é 275 I 2 2e x ee therefore = 77" — (XAT 2m x =r. Therefore e + 3)3 muft be equal to the root y of the equation y? — gy = r. menace q b q Mone Gy, Q, E. De Another expreffion for the root of the foregoing equation ye— qr. 8. By refuming the folution of the foregoing problem we may find another expreffion for the root y of the aes y? —47y =7, to wit, the expreffion rr eat ahr dag Zamnit ers res ame 1 Sunes 4 This expreffion may be found in the following le — sJF or ¢— manner. The inveftigation of the faid fecond expreffion for the value of the root y of the cubick equation yi —qy =r. g. In art. 5 we fuppofed the line y to be divided into two unequal parts, wv and x, of which vw was fuppofed to be the greater; and we firft found the value of the greater part v, and then determined that of the leffer part z by its relation to v, which is expreffed by the equation 3 vz == q. But we may with the fame Vou. II. 32M eale ‘ 450 A METHOD OF EXTENDING CARDAN’S FIRST RULE ‘ 4 eafe firft determine the value of the leffer part z, and then derive from the faid value of z the value of the greater part v by means.of the fame equation 3 vz = g, Which exprefles their relation to each other; whereby we fhould obtain the fecond expreffion of the value of v + z, or z + v, ory, which was fet forth in q ; : : : 3 =e the foregoing article, to wit, /3|—-— yj/— —2 + 2 S| aie =a I 7 q oT q ; : bd 3 ae ent gine =e — ———~\ Te or 1/? {e Seen Ge ore—4l 3 t=5 t> Rae This may be done in the manner following. : : ; 3 . Since 3 vz is = g, and confequently v 1s = ro and v3 1s = 2 and it has 3 g PP) + 27, 2723 3 3 Therefore rz? will be = = + 2°, and rz? — z° will be = 7 Therefore, if been fhewn in art. 5 that ris = v? + 2%, it follows that 7 will be = we fubtraét both fides of this equation from a (which is fuppofed to be greater than c, and confequently muft alfo be greater than, its equal, 7z* — 2°) the remainders will be equal to each other; that is, sa — frzi — z°, or a — rz? + 2°, will be equal to oe — So Therefore the fquare-root of the trinomial quantity ie — 7z> -+- z° will be equal to the {quare-root of = ~ r But the {quare-root of the trinomial quantity i — rz> + 2° is the difference of the fim- °° r . . - ia : r ple quantities > and z*, that is, either — — 23 or z*— —, according as — or x? is the greater quantity. But, becaufe 7 is equal to v* + 2°, and ¥? is sreatee than 2, it follows that 2? muft be lefs than one half of v3 + Bs or than one half of r, or than — =; and confequently the difference. between — and 2? will be — > — 2°, and not 2° — =. Therefore = — 2° will be the fquare-root of the tri- nomial quantity a — rz? + 2°, and ge cca will be equal to the fquare- root of — eer or — — 2? will be = v\7 —_— i Therefore (adding 2? to both fides) = > will be = 2? 4 ¥ ceo ZL, a (fubtracting 4/ 33 —f from both fides) 2? will be = —— ¥ (- —f, and confequently ene the cube-roots of both fides) z will be = vit—yv{z ait Therefore z + v 8 to as HA phere e. OS Tale eS: PL wihbe = Wt —vE-f+o0e vE-vy--f£445) af 3 — = OO _— : FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. 451 r rr g3 tion y?—¢y =r will be equal to the expreffion W3/— — yj— — © + 7 =, or to the expreffion 4/3 e—s panier 1e The oy | 3 Vd een eae i ¥ gfiens 2 2 I > 7 ae i). ~ Q. E. bi A fynthetick demonftration of the truth of the foregoing exprefion. , r 10. oo again we may demonftrate fynthetically, that this expreffion e — 5] 3. "ya = > s\3 by fubftituting it for y in the compound quantity y3 — gy, which forms the left- hand fide of the faid equation. For, if we make this fubftitution, we fhall find that the value of y* — gy thence arifing will be equal to the abfolute term r. This may be fhewn in the manner SENS is equal to the true value of y in the propofed equation y3 — gy = 7, If y is {uppofed to be equal to pa daa aan 6 <=) we fhall have y3 2 a himcalsacilel < iensoam a g (=e¢—s +3 x ¢—s53 x : Beaty deena ute fT) ae 35 9Xe—sF 93 aia qxer VF Magen taixene Me Os Se ee sl 3: + Sp =s oe pear er + = can: ee and confequently y: 3 I Sel es, —W(=e¢—-s+ qx a3 + in: + 27 Xe—s ei IN Ga Sk a ea Se = Tie BN atoet seeds 27@ — 278 276% —i54es + 275" + 93 peer mew 2ps But it has been fhewn in art. 7 that g? is = 27¢*—-275. Therefore the «| 27e% — 54es + 2757 + (Bors 2 age? — 540s 2757 2762 2759 a She — Sages fraction er ae nog oo, Th. IMCS and confequently the compound quantity y3 — gy (which has been fhewn to be equal to the faid fraction) will alfo be equal to the fraction Sea Sa = 47 3 M 2 $4 Xx a sees 452 A METHOD OF EXTENDING CARDAN’S FIRST RULE 25 x = ae aS ae lS 2X SS a2 Xe 52 xX = Sr; thats, 27xKe-s ae ie value of the compound ane y3 — gy arifing from the fubftitution of the expreffion e— s| 3 han Tye in its terms inftead of y, is equal to the abfo- lute sae r of the cane equation y? —gy =r, Therefore the faid expreffion e—A|Z aes TI muft be equal to the value of y in the faid equation. Qi E. D. A third expreffion for the root of the foregoing equation yeor-gy=r. 11. We may alfo, by refuming the folution of the foregoing problem con- tained in art. 5, find a third expreffion for the root of this equation y? mm is to wit, the expreflion 3 = + ie iat a Naa bier k aes £ A I Ne bck Sirk? GaSe ODS stad Zick tear dl Sap bis expreffion may be found in the following manner. The inveftigation of the faid third expreffion for the value of the root y of the cubick equation y3 —qy =Tr. 12. Since v3 - z3 is = 7, it follows that z3 will be > r—v?. But v3 is iy fhewn in art. 5 to be = = + 7 = Piet Therefore r —v? will be = r— 3 3 . . -— WV 4 —La=-— v/|—— ‘ Confequently z3 (which is equal tor — 2 Aenitete 7 ez 4 v3) willbe = —— V 7. Therefore v will be = 3) “+ vie, and z will be = vie — v[=—£, and confequently v ++ z, or y, will be Wipe te aeeae ~—foyhts+ hoe AT + e@—s Be Q. E. I, A [y= FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. | 453 A fynthetick demonftration of the truth of the foregoing exprefion. 13. Here again we may demonttrate fynthetically that this expreffion I T @+5\3 + ¢—s\3 is equal to the true value of y in the propofed equation y? —qy =r, by fubttituting it for y in the compound quantity y* —gy, which forms the firft, or left-hand, fide of the faid equation. For, if we make this fubftitution, we fhall find that the value of y3 —- gy thence arifing will be equal to the abfolute term 7 of the faid equation. This may be fhewn in the following manner. ae} ij is peg to oa = ene +. A+—2B> + joao Ss Ha — ‘S 11.G s? 20 17 38 23 5 26 K hag +2 I eMS F e 21 e7 24 ee 27 J e 30 rs a ~ I 4 1s 3 16 ne 18> Bn5-Hop+Erpl—#an+8RE- oss ah ts 39 es 2 eX ets 48 ee infinitum, or (if, for the fake of brevity, we a ‘the feveral at crane fractions a Ne, Cee Rae @ Dey BES co D 53 E s# F 55 ~~ awe — —_— — — — I —_— — os —— ee — == rel et tte otek Comte ee &c) to the feries oe € e at 3 et 57 es G 5° H 57 15° K 59 Abe M st¥ N 5? 0 513 Pp st4 asis R tS Ce Myelin ee “eo ere eu git ak ers gl es gt & 577 ake - <= — ++ &c ad infinitum, And, by the refidual theorem in the cafe of id 53 Bey fe =; i eatal to the Gia 3 8 s4 II ss 14 Cd 17 20 $8 28 wit 26 12 x0 15 o 18 é 21 7 24 e 27 ae, 39 st? 29 pil 32 gt 35 g33 38 git 4! gis 44 sl& ero 33 eit 5ay 36 ez 39. Pry ——— 42 el4 45 ik ae 48 Qa gif ie) st 4 * « i . Bis. oO 5% a ecvtire S ier &c ad infinitum, or to the feries 1 — — — — — é e~ ps3 E 54 F 55 G 56 H 57 ts K 59 Ls? M sit N si2 ost3 psi# e. - en e an Pee, RE gk ee Q sts wane s 517 To epee. . —S oe a ae &c ad infinitum, which. confifts of. the very, fame terms as the feries which. is equal. to »/ f ote -; but with the fign — pre= fixed to every term after the firft term 1 inftead of every other term. Now, if thefe two feriefes (which are equal to /? 1 + Land fr — 2) be c ; : 2cs Est 6 g- added together their fum will be the feries 2 — —— — ==> — 742 2° é€ e* e& . ee ae 2 Ls! 2N 5% 2 pst4 2Rnstt 28 Wey SF sa : . . Sn ee ae a — =~ ad infinitum, in: which all the terms, after the firft term, 2, are marked with the fign —,, or fubtra¢ted from the faid firft term. Therefore /° ot — + 3 f — ~ will be equal to the faid FOR RESOLVING THE CUBICK EQUATION &c. 459 fa ferles 2 — 2cs 2E 54 2656 215° a SA 2 Ss ope 2 Pott 2R 56 a oa er Ta ae et art es par nr graw" wiz lz ce elt | ero 27st Spe ; $4 ; a &c ad infinitum. Therefore \/3{e x Iv: Bs ib Ia vt sels . : ° 2 cs 2554 265 215° x0 will be = 43 fe 9¢ the faid feries 2 — “= — why Be a Rees e et Po ° er? 2Nns'% aps 24ais 27 st ; . ; . ee es — ee ad infinitum = 2 / (ex the feries Cs? E s+ Gs® 13? hee N 5? P 5%4 R st6 vst! Seer Ree A ek pt gt Oa finitum. Therefore the expreffion /3(e +5 + W#fe—-s (which is = ¥2f x1? [: a = ie vei —-) will be equal to oVile x the feries I a E 54 G 5° 15° L 5? N 5? p st4 R 5t6 T 58 : J eg ag eg ee a Ca Q. E. F. 25. It follows from the foregoing article, that the root y of the cubick equa- tion y? — gy = 7 (which has been fhewn above to be equal to the expreffion z Vife+s + Y3[e—s) will be equal to 27? (e x the infinite feries 1 — = E s+ Gs® 158 ie N st? p st4 R58 7 st8 pds Sais Sa PE recency ai &c ad infinitum. . . . 3 ° 26. This feries will always be a converging one; becaufe # — Sy Or ss, is ‘Tr 5 d { 1 f ge SS 5* 56 58 g10 always lefs than —, or ee, and confequently the fractions >, 4, 3 > aes gi? 4 gi8 g8 git? “eis? 16? G8? ries), will form a decreafing progreffion, and therefore the terms themfelves, which are produced by the multiplication of the faid fractions into the numeral co-efhcients C, E, G, 1, L, N, P, R, T, &c (which likewife are a decreafing progreffion), will alfo form a decreafing progreffion. &c (which are the literal parts of the terms of the faid fe- OS. GA ERT PETE ALE a nae a — ne = — a + > — a + &c ad-infinitum, which is equal to y° f + -; we hall find them to be as follows, to wit: i 35 and B (= = => x => and C (== = = XT =ETXPVHp and D (= 2 C= > x -) = ae ies tied and 460 A METHOD OF EXTENDING CARDAN’S RULE\ ; 8 8 2 wt O18 Decrees ry x §) =a on, EE a ER SQ EDs SS a ay eee eee oo a and P(e Ee - X 343 3x5 243 Fx ogee ag A Wag Stes be 7 lod a and G es gS g * ag > Mya? = Bree 2 7 17 $y en TX 22) S17 R52 See 374 and H (= 21 Ga 21 x 6561 3x x 6c6r 7 seb sth a 19,683? + 225 +20 20 3 Tn nS Bide: te oat 187 X2 _. 5X 187 and 1 (= 24 24 “* 19,683 — 6 as 19,683 — 3x2 19,683 — 519,683) 2835 592049” 523 7 ne agg POS RN oe ase and 27 27 59,089 145945323" 26 26 21,505 13 21,505 13 § X 4301 =2K ae ~ — H Se xO SK HS and L ( 30 30 * 1,594,323 15 15594323 3X5 s 13594323 13X 4301) __ _ $5,913 3 X 1,594,323 4,782,969" 42 26 = 29. de $5,913 — 147,407 CG cera By 15782,969) 145348,907” __ 32 — 32 147,407. _. 8 147,407 151 79,256_ and N (= 36 M= 36 ~~ 14,348,907 149348,907) 129,140,163” Pasay gat as _15179)256 35 13% 90,712 35 007s and O (= 39 N= 39 129,140,163 ~ 3X13 129,140,163 — Shea iia 397749920 | ™ 387,420,489” : 85 — 38. ., 35174920 __ 19 39174920 19 39174,920 and P (= 4 42 3875420,489 21 387,420,489 3X7 3875420,489 JAG 7% 4539560 __ _19X 453,560) _ __ 8,617,640 _ Tag 387,420,489 3X 387,420,489" ~ 1,162,261,467? set PIAL 8,617,540 _ a 8,617,640. _ | ar and Q (= near ee 45° 3;162,261,407 —~ $x9 1,162,261,467 — 5x9 * 5X1,723,528 _ 41 X 1,723,528 )= 40, 664,648 _ 1,162,261,467 ~ 9X 1,162,261,4677 10, 460,35 3)203° 7 L pup eee ee! 70,644,648 11 70,644,648 ort and R(=3Q= 10,460,353,203 2 10,460,353,203 3% 4 x 4% 17,666,162 __ 11 X 17,666,162 Me 19453 27;782 10,460,353,203 3X 10,460,362,203’ ~ 31,381,059,609° 2 Rarer 104,327,782, 47 _ 17% 11,431,046 and § (= 51 31,381,059,609 3X17 31,381,059,609 47 X 11,431,046 ae 5372259162 3 X 31,381,059,609 945143,178,827" ° 50 5372259162, __ 35 §375259,162 and T (= &S = ei SE stata ¢ 54 945143,178,827 27 949143178,82)> 13943 19479:050 2) 541,805 ,928,329 - r Note, FOR RESOLVING THE CUBICK EQUATION &c. 461 Note, thefe values of the co-efficients B, C, D, E, F,G, H, I, K, L, M, N, O, P,Q, R,S, and T, are-expreffed in the fmalleit poffible numbers. 28. It follows from art, 25 and'27 that the root y of the equation y? — gy =r ° rn . . . 2 4 is equal to the expreflion 2 4/?(¢ x the infinite feries 1 — — — —- ge 243.6% 1545° 935 5° §50913.57° 1,179,256 s*? 8,617,640 5%4 6561e° 59,049 4,782,959 e?° 129,140, 163 e 1, 162,261,467 e%4 194,327,782 576 13,431,479,00 538 te as ELE ou Di nk dat i Mia Se a — &c ad infinitum. 31,381,059,009 e 25541;865,825,3 29 e% : ‘ css E s4 G 56 29. It has been obferved in art. 26 that the feries 1 — $e — — = é é tg us N 5%? p st4 R 516 rst t , eee , as a eae &c ad. infinitum will always con- verge; from whence it follows that the expreffion 2 /? fe x the faid feries 1 — gal 5S E 54 6 5° 15° 1s s?9 Nn s!? p st4 Fs feat Ch Ree = wr tyes arta amr er pre Ela gece glean OEE: ——s sas | =a — &c a infinitum will always truly exhibit the value of the root y of the cubick equation y*? — qy ; 3 =r in the aforefaid firft cafe of it, or when - is greater than a . And, when ss 4 on ’ is confiderably lefs than ee, or m — - is confiderably lefs than re or mi is 3 . . . very little greater than = , the convergency of the terms of this feries will be fuf- ficient to make it ufeful. But in other cafes, or when = is much greater than 3 7r . ° 3 “- (as, for example, when — is triple, or quadruple, or quintuple, of i, or of fome ftill greater magnitude), the terms of this feries will decreafe fo flowly as to render it very unfit for practice. And, indeed, in the moft favourable cafes, this expreffion of the value of the root y of the equation y? —¢y =r will be lefs convenient in practice than the expreflion f'(e+s5 + Y?(e—s, from which it is derived. But, though its merit in a practical view be but fmall, yet, as it is the foundation of the method, which is here intended to be explained, of extending Cardan’s rule to the fecond cafe of the equation y3 — gy = 7, | thall now proceed to illuftrate the truth of it by applying it to the refolution ofa fingle numeral equation of the foregoing form y? — gy = r, in the firft cafe of the faid : rk. : equation, or when - is greater than 5? in which I have taken care to choofe fuch numbers for g and 7 as fhall make = be but little greater than f, and confequently fhall give us only a {mall quantity for the value of the fraction =, by the continual multiplication of which the terms of the above feries are generated. Pa 73 462 A METHOD OF EXTENDING CARDAN’S RULE An example of the refolution of a cubick equation of the : foregoing form y? —qy = 1, in the firft cafe of it, (or . . 3 when r is greater than ud AOE "is greater than 7 ) by means of the exprefion 2 /3(2 x the infinite A Css E 54 c 56 158 157° N st SEES A pp Sg age Fog gta 14 16 18 ae &c ad infinitum, or 2 f 3(e ear eA ; I SS 10 tag Xx the infinite Series Oh praia Sry okt ey 154. & 935 s _553913 eer 6561 ee 59049 eé 4,782,969 ero 1,179,256 git 8,617,640 58 129,140,163 * ee 1,162,261,467 ee Oe 1943275782 | iad 13,431,479,050 se 31,381,059,609 eX -34541,865,828,329 en — Sc ad infinitum, San 30. Let it be required to refolve the equation y? — 3009 = 2108 by means of this expreffion. Here g is = 300, and 7 is = 2108. Therefore - is. == to0,8ane VA is he s ENE Bef pl = (== v{t = yY 100) = 10, and confequently iyaes (= 100 X10) sium and uN is (= 2 X 1000) = 2000, which is lefs than 2108, or. Therefore a Ja A 4 this equation comes under the firft cafe of the general equation y* — gy =7, and confequently may be refolved either by one of the three expreffions ob- tained by Cardan’s fecond rule above explained, or by the foregoing tran{cen- ff > fe x the feri Cf ERS Oe eee dental expreffion 2 4/? |e X the feries 1 — —) >= => >= Ht Ge eee 12 r ; : : , a+ — &c ad infinitum, which was derived from the third, or laft, of them. ei ; f rr 31. Now, fince r is = 2108, we fhall have —, or ¢, = 1054, and reps: ee, (= 1054]*) = 1,110,916, And, fince gis = 300, we fhall have < om 3 : 100, and ©, or the cube of > (= 100]*) = 1000,000, and confequently ss, 8) 3 or — —£ (= 1,110,916 — 1000,000) = 110,916. Therefore the frac- 4 27 ? : ss ; eye TO,OLO wu ! tion — will be Gz a aye,GIE = 0.099,341,932,23 and as will be (= 0.099,8413932,2)) = 0.009,968,411,4 5 5* $ and + will be (= = et 0.009,968,411,4 X 0.099,841,932,2) a e* ee 0,000,99 5320554 3 and a - —- FOR RESOLVING THE CUBICK EQUATION &c. 463 GEO 56 2 and =; will be (= = x oa = 0.000;995,265,4 X 0.099,841,932,2) = * ; 0.000,099,369,2 and = will be (= | x = = 0.000,099,369,2 X 0.099,841,932,2) = 0.000,009,92F,2 3 gi? a! 10 2 and = will be (= a x = = 0,000,009,921,2 X 0.099,841,932,2) = 0.000,000,990,5 5. s* 7 and = — — will be (= ar X J = 0.000,000,990,5 X 0.099,841,932,2) = : 0.000,000,098,8 ; L : 14 2 and — will Deve a x > = 0.000,000,098,8 X 0.099,841,932,2) = 0.000,000,009,8 ; 16 z : and = will be; (<= a x => = 0.000,000,009,8 X 0.099,841,932,2) = ©.000,000,000;0 ; And confequently -~ will be (= C X 0.099,841,932,2 = = X 0.099,841, 932,2 = o-299 019822) = 0.081,0933,548,0 5 and = will be (= E XX. 0.009,968,411,4 = << X 0.009,968,411,4 = LO X 0,009,968, 411,4. __ pom Panete, hal De ime zeph rir 4 243 ay 223 itp aan ? > and = will be (= G X 0.000,995,265,4 = a X 0.000,995,265,4 = 154 X 0.000,99 5126554 — 2:1532270,871,6 (= 0.000,023,360,9 ; RS Sy ar a ae Caos Aa a . 9 BS Ye a7 2 6561 6561 8 . and <; will be (= I X 0.000,099,369,2 = ee X 0.000,099,369,2 = > 935 X 0.000,099,369,2 __ 0.092,910:202,0, __ 222 a ——)) = 0.000, 001 ‘. 593049 593049 ; Recipe aah? Se 55292 HSS 7 teil an (= L X'0.000,009,921,2 = 47825989 X 0.000,009,921,2 = 55913 X 0.000,009,921,2 0-554,724;055,0 22 ae) 0.000,000,11 5,93: 4,782,909 4,782,969 ) 12 toe » am 19179925 and ~ 5 ~ will be (= N X 0.000,000,990,5 = Mae Oe X 0.000,0003;990, 5 __ 14179,256 X 0.000,000,990,5 __ 1.158,053,068,0, __ . : . eRe aT coe PEETRNET HS ) = 0:000;,000,009,0 5. en a . oie $619,640 and Ara will be (= P X 0.000,000,098,8 = WRC cn X 0.000,009,098,8 __ 8,617,640 X 0.000,000,098,8 __ 0.851,422,832,0. ___ oy = 1,162,261,467 7 2,162,261,467 ) == 0.000;000,00057 3 — _1949327,782 | and == e will be (= R X 0,000,000,009,8 = ST x Cet a 194,327,782 X 0.000,000,009,8 __ 1.904,412,263,6 ) = 0.000,000,000,0. | 31)381,059;609- “~ 315381,059,609 Therefore 4.64 A METHOD OF EXTENDING CARDAN’S RULE : cz? Es G 5° as? L st? N st? pist4 R515 eh, Therefore praia itereiuD ory vinelp: ta us sda Were och, ci lta x &c will be = 0.011,093,548,0 + 0.000,410,222,6 + 0.000,023,360,9 + 0.000,001, 57354 -- 0.000,000;115,9 -- 0.000,000,009,;0 + 0.000,000,000,7 = 0.000; = 3 : : c s* 000,000,0 = 0.011,528,830,5 + &c; and confequently the feries 1 — = # G56 158 x s*S N 57 Pp st4 R.st6 & itt be = eh GAD RTT gia Son Re Ne 000,0 — 0.011,528,830,5 &c = 0.988,471,169,5 — &c. Further, fince ¢ is == 1054, we fhall have 3 fe (= (1054) = 10.176, 853,833,7, and confequently 2 3 bhi 2 X 10.176,853,833,7) = 20.353, ix.s4 Css Est G 56 707,667,4. Therefore'2 3 (e X the feries 1 oe —, aie laa NW 512 Pp si4 R. s#6 ai ; > al gerree iat sy mo — &c ad infinitum will be = 20.353,707,667,4 0.988,471,169,5 = 20.119,053,221,6. Therefore the root of the propofed equation y3 — 300y = 2108 is = 20.119,053,221,6. Q. Bets 32. This value of y is true to nine places of figures. For its true value is fomewhat greater than 20.119,053,2, as will appear by fubftituting 20.119, ©53,2 inftead of y in the compound quantity y? — 300y. For, if we fuppofe y to be = 20.119,053,2, we fhall have y* = 404.776,301,664,430,24, and y3 = 8143.715,9475285,920,546,248,768, and gooy(= 300 X 20.119,053,2) = 6035.715,960,0, and confequently y* — 300y (= 8143.715,947,285,920, 546,248,768 — 6035.715,960,0) = 2107.999,987,285,920,546,248,768, which is fomewhat lefs than the abfolute term, 2108, of the propofed equation y? — 3007 = 21083; and confequently 20.119,053,2 muft be fomewhat lefs than the true value of y in the faid equation. And, if we were to profecute the value of y fomewhat further by means of Mr. Raphfon’s method of approxima- tion, by fuppofing y to be = 20.119,053,2 + 2, and fubftituting this quantity inftead of y in the equation y3 — 300y = 2108, and refolving the new equa- tion that would refult from fuch fubftitution, as if it was a-fimple equation, (to wit, by omitting all the terms that involve either the fquare, or cube, of x) we fhould find z to be equal to the fraétion 2°22 2070-4 53°75 023" or to 0.000, _ 914+328,904,9934290,72 000,013. And confequently the firft eleven figures of 20.119,053,2 + 2, or of the true value of y in the propofed equation y? — 300 y = 2108, would be 20.119;053,2 + 0.000,000,013, OF 20.119,053,213- 33. It appears therefore from the foregoing example, that this expreffion o/?(e »« the infinite feries 1 — Etro B Shon ig out < ss = —*S —— ae — “eS — &c does truly exhibit the root y of the cubick equation “—¢y = rin that cafe of it which falls under Cardan’s fecond rule above ex- plained, or in which 7 is greater than ive, or a is greater than c. 6 Of FOR RESOLVING THE CUBICK EQUATION &c. 465 Of the fecond cafe of the cubick equation y>—qy =r, in which the abfolute term r is lefs than “iv, or “ is % 9g lefs than mr 34. We muft now proceed to confider the other cafe of the cubick equation ° : “s Bae 3 ; y? — gy =7, In which z is lefs than 209 or — is lefs than < . This cafe (as . 7 we have already feen) cannot be refolved by the aforefaid rule of Cardan, be- caufe it is impoffible in this cafe to divide the line, or root, y into two fuch parts vand z, thatthe product, or rectangle, under the faid parts, to wit, the product v2, thall be equal to%, and confequently that 3 vz fhall be = g, and 3vz.X v + 2 fhall be = ¢ X v + 2, which is a fundamental ftep in the folution of Problem 1 given above in-art-5. Andon this account this cafe of the equation ¥? —47y =r has obtained amongtt Algebraifts the name of the irreducible ca/e; and particularly it is often fo denominated by the French writers of Algebra. Monfieur Montucla in his Hiffoire des Mathématiques, Tom. 1. page 482, {peaks of it in thefe words: On doit @ Cardan la remarque de la limitation d'un cas des équations cubiques, ou il arrive que Vextraftion de la racine quarrée qui entre dans la formule, weft pas poffible. Cet ce que nous appellons \e cas irréductible ; dout la difficulté a donné et donne encore la torture aux Analyftes. lt may, however, be re- folved by means of a certain tranfcendental expreffion (or expreffion containing an infinite feries of terms) which bears a great refemblance to the foregoing tranfcendental expreffion which we have fhewn to be equal to the value of y in the firft cafe of the equation y3 — gy =r, to wit, the expreffion 2 »/: {e x the =. glans . os E s# G 5° ¥s' Loe ie igs p st4 yi a infinite feries -1,— — — '—- — —— — — = = SS SS eS é — &c, and which was derived from the finite expreffion 4/3 ets + V73fe—s _ by the help of Sir Ifaac Newton’s binomial and refidual theorems. To affign fuch a tranfcendental expreffion, and to demontftrate that it will be equal to the root v of the cubick equation y3 — gy =r in the fecond cafe of it, or when r is lefs 29/9 than at 3 : Wak eat ; , or — is lefs than £, with a certain limitation which we fhall men- tion prefently, is the chief object of the remaining part of this difcourfe. 35. In order to preferve the two cafes of the cubick equation y* —qy = r 29/9 : 34/3” and in the fecond of which it is fuppofed to be lefs than the faid quantity), diftinét from each other, it will be convenient to denote the root of it in thefe_ two cafes, and likewife the abfolute term of the equation, by different letters. I fhall therefore henceforward denote the root of this equation in the firft cafe / . ord. ur by the letter y, and in- 3/3 (in the firft of which the abfolute term 7 is fuppofed to be greater than of it (or when the abfolute term is greater than Vot. Il. 3 O the 466 A METHOD OF EXTENDING CARDAN’S RULE the fecond cafe by the letter x‘, and fhall denote the abfolute term of the hi tion in the firft cafe by the letter 7 (as in the. foregoing articles) and in the fecond cafe by the letter ¢; fo that the two cafes of the equation y? — eA pean i will now be exprefied by the two feparate equations y3 — gy = rand «3 — gx = f, in the former of which 7 1s greater than = he and in the latter of which tis lefs than nt, the letter g being fuppofed to denote the fame quantity. in qs ay (as before) by the detters ss, but fhall put the letter g for =, and the letters zz both equations. And I fhall denote < (as before) by the letter e, and as ase 3 tt for % Sg? eh = — gg. With this notation the propofition which I fhall now endeavour to ddmonttrates will be as follows. PROP. 3+) [A TH E.OR E M,. 36. If in the foregoing tranfcendental expreffion, to wit, 2 4/3 (e x the infi- f C55 E54 6 56 15° arse Ns? pst R 576 ce TE eS Pesci aBh of ncthcatn Geen eke ME) &c ad infinitum (which has been fhewn to be equal to the root » of the cubick equation y? — gy = 7), we make the following changes, to wit, firft, infertthe letter g every where inftead of the letter e, and, fecondly, infert the letter x every where inftead of the letter 5, and, gdly, change the fign —. into the fign -- in the fecond, and fourth, and Fath: and eighth, and every following even term of the infinite feries contained in the {aid expreffion, the new tranfcenden- tal expreffion that will be thereby obtained, to wit, the expreffion’2 /3{9 « C2z% E24 G2? 128 Lato N 2i2 Pp xt4 eee the infinite feries 1 +} — — — +4 — Sj + -3 oOo ore toa roe SS & & & BS s & § ¥8 f 4 Ae . 1 + — = &c ad infinitum will be equal to the root « of the cubick equation «3 — gx = +t; provided that the abfolute term ¢ of this equation (though it be lefs: than a4) be greater than 472 X 174, or that oa or gg, (though it be lefs 34/3 g than £) be greater than — — xX 52 or than 2 aE Cz Ext 37. This limitation is neceflary to the end that the feries 1 + <7 aca G x6 12° Est N xt P zit R2t t aire ree eae ett et — &c (which forms a part of the expreffion of the value of the root ) may be a converging feries. For if ¢ is . 3 . 3 « lefs than 2 X 901, or # is lefs than =e Ene RA quantity = we 33 4 on 4 q 3 hei oteatersthanst-cak Sek OF an pein tet ecaree anal and, 2 ane reater e 27-54 7 54 a Sorters % ; 54 than FOR RESOLVING THE CUBICK EQUATION &c. 467 : #3 335 “1 than ri that is, zz will be greater than gg; and confequently the terms of the 4 ; 4 G x6 I pau L xto N giz Pp alt R iS T 18 Na ey a hy ——- — — + — 4 6 8 gl I2 14+ 16 18 . . +. . 5 - &c will diverge, inftead of converging, and the faid expreffion 2 1/3 8 e 8 6 8 oP Iz & : : C2 E G2 1% Ds N& Pp zit x the faid feries 1 -+ — — = + — — — + ert ES “ gg ie 2° 2° £9 Ys pee R& T . } aes —- — &c, will confequently become ufelefs. But, when ¢ is lefs oS 2 ff. 3 than =i, and yet greater than 4/2 et, Oior—-Is lefs than a and yet 3 . . . greater than ©, the terms of the aforefaid feries will converge, or decreafe, and the aforefaid tranfcendental expreffion will be equal to the root x of the equation ah anne qx — 7. , 38. Now, in order-to-demonftrate this propofition, it will be neceffary to make fome further obfervations on the former tranfcendental expreffion 2 4/7? (2 x ’ ; : C55 E54 Gs® 1s? Ls NW st ne Rs the infinite feries,1.— —— a ee Hi Si By He NR eI ca! Gla Sula (4 y 518 i &c ad infinitum, which has been fhewn to be equal to the root of the equationy? —4¢.y =r. Odfervations on. the-exprefiion 2 »/3 (e x the infinite feries Css Es* Gs® rs* rise N st re Se Par te ee eee ad infinitum, «which is equal to the root of the equa- tiny? —qy=r. e? e ert 6 8 “uy 2 Sia Ty ec. % ee Be ae ie a a cc 6 5 ont e x 4 + as Bi oe 4) &e ton LA _ 24 a a + Later 2,10 if Iz Pus a is a sche bey Vt er 48 = aay we —— — &c; which, for the ke of brevity, we will denote by the capital letter A. Then will y? be = tothe compound feries A. 42. We mult now multiply the expreffion 2+/?{e x the infinite feries 1 — C35 E 5# G 5° re Es? N 512 he one ae — & ad infinitum (which is equal to the ce e* e root y of the cubick equation y? — gy = 7) into the co-efficient 4. , 3 3 3 Now, fince ss is = = & a? we fhall have ss + = ——. and = = = ¢¢—55, an ly g™he= —s = — Saves fee , and confequently ¢ 27 X €@— 55 27: 4X Utena 7 2 and 9.337% Vilee x {1 Sars 6.3, Kit 2)-5 = (by the refdual . rom . : : B Ss crm theorem in the cafe of roots) 3 x e 3 & the infinite feries 1 — <> see ps9 ES? F37° Gs? H 514 Fi cre 2 nhs en ae — SAM as So giee ny aaih eee O a at — &c ad infinitum. Therefore : ‘ ; : Css E s4 G 5° 15° q x the expreffion 2 /*e x the infinite feries 1 — — — => —~= — 4 — ee e*. é é 10 2 2 Ls ns? . mie : . . BSS ——. — —— — &c will be equal to 3 x e 3 X the infinite feries 1 — —: — e é ee c'st ps° E s§ ere G st? H 514 ae K 518 er er ast Se ay a er a as ee ad gina x the Css E s4 G 56 158 Ly? expreffion 2 /3(e x the infinite feries 1 — — — —- — — ee e# e és ee ws? FOR RESOLVING THE CUBICK EQUATION &c. 471 N st? ‘ : . : . 4 6 8 —— — &c ad infinitum = 6e X the infinite feries 1 —2* — =~ Pe ES e ee e4 e e® Fst? Gst2 F 3 y 4 6 8 = sr — &c ad infnitum x the feries1 ~~ SL EL e e ee e* ra e E gto N gi2 ‘ * ° * se a &c ad infinitum. We mutt therefore multiply thefe two feriefes together; which may be done as follows. The multiplication of the infinite feries 1 — — — — — . e J Dns E 5° F 570 G st A : ie ; Bete pe Sa EFc into the infinite feries Css E s+ G 56 z,3° L st? N 512 ce € e€ e € eu BSS c 54 D 5® ESs® F 510 G 512 & — ——. =< — — — ere lhe I ce et e® e er? 12 Cc Css E st G 5° 15° L st? N 5% & = aa ra et e® rw heptane 7 ero oe et2 a c BSS c s* p s° Es® F st G st ee e* e e er° el2 Css BC 54 ¢75° cp s8 CE s?° CF si? ee e* e® ee 7% er? ei ~f- &c E 54 ae BE s® cE s® DEs?° E252 & oe. 4 2° 28 ero el2 + Cc 6 8 10 12 Gs BGS cGs DGs @ e ere = ei2 a7 &c 8 10 Iz Is BIS CIs ae é ero = elz “Fe &c L.ste ie 1o 12 oe Gre e€ é N gi2 + &c. Therefore the product of the multiplication of g into the expreffion 2 4/?(e x . Css E 54 cue oe 2187 es 21847 € 2187 e 2187 3 2187 25 162 ie 180 c 308 say TILE = 342 59. Nie Bat se ies 2187 x 7 hn 2187 e 2187 ee 2187 es 2137 Poe teeth 6ps® 6 056 6 BE s® 6c65° q Phersfore —— ——_- == sake —, or the third term of the compound é 8 cE 5§ 8 356 3 feries 6 ¢ — I, is equal to Le cate ile = : + “= » or the third term of the compound feries 8e — A. Q. ED: : 8cc 38 49. The fourth term of the compound feries 8e¢ — A is —— a se —i : 24 £75* PAGES. 4 we 935 Po t 154 iv ee aa which is = 24 X Paaaa a a ar x — x Gaba ier 10 10 s8 I 10 8 ~ Bogs . 16X154 Be arash ay cet 28S Rye 343 Xa ™ 19683 e7 3x 6561 3 P 2 i. 476 A METH 7 ETHOD OF EXTENDING CARDAN’S RULE 7 8x 8 = BEDS we: = soma Ls! s __ 7480 38 2464 38 800 hind SI X 243 e7 81x 243 @? — 1968 a ao pe ey x ie 4 240 shee ayy eR 58 3264 3 58 eye is 19683 —_—- i ee ee aa 58 ii 19683 e7 19683 e7 19683 ef — Fare 7? and the fourth term of the compound feries 6 e — II is had 8 aL yeb.o0 TNE S 6 BG st 6158 é . e e7 aT a, \ en t which is = sin ee 5 He : ee 6X ig ST Oke XX 7 — —6 xX a Ope eee ja My sabes 5's ES i 6 aide i é 3, 265 Gh saa OTe AL KGS a ee ag Bae —_ — 6X27X5 58 6xQxXx10 538 5% 49 3 é SIX 243 oleae tad ale Leh ee 6X 154 3 2X 935 8 86 chek e7 f SIX 243 le 3 x 6561 “a = 19683, err — a. x “8 e 1 TRO Fie Je ee Se oro yA peas 2 ted, OR 0 en 79683 x tae 19683 x a Therefore — cma 6cpst 6 cE s® 62658 8 7 é 7 hd Bee is — 241s — 48cos® 2g B*s8 | 24 C7 ESE : g é€ e? e7 e7 mir e7 ° Q: .EeaDe 50. The fifth term of the compound feries § e — A is kana iy t 4S cag] 48 EGsTOo 24.0%G 5%° a4 CR? ; e? i Me ae aa which is = 24 x 2223. yx oe 3 935 v4 as 2 x JOU Dal G4 10 497829 19°95 e 9 5 Rae Sa ea 4? X ae spony waway orate Bt oi X roeasik Bio nae) “ I Bese dE Seu FO) Sg EES ee ee BAGO ES Fcc A peere 935) ee 9 243 243 eS 15945323 =) 48 X¥ > xX x — 48X1OX 154 gr 3 1¢4 ;t0 27 597949 e? —_—_—_——_—O —_—_— 9 apne pee eS. <<, 1, 5945325 e? rte 4 243 ‘a 6561 a e + 24 4 F x “ 36) ee x os oe 447,304 v gz0 134,640 570 44,920 si 43 243 e? = Hex a eK SS ue ere. 9594323 7594323 ° 1 15594323 T5493 Or 7200 Ag! 465,592 es 208, 56 10 MOE: ; Arye CMO Bah TEV O __ 208,560 5, 1° __ 257,032 st ; 155949323 ee 155949323 er a A, 1,594,323 x ~>? and the fifth term of the compound feries 6 ¢ — II is ~~ ix _ =, Optay 6 ve st° 6 cGs%° 6Brs x65? 1 S ee - eS ee ae ees 5 . 7 ram Li Nich DEER AK ee LBRO OS ae ey ots ea ik, hed eee 1 mee |e : 81 243 e? 6 x“ 9 “ 6561 x ci Oe — 6 x Ez x Re xX ee = $5913 ca _ 6x 22% 2187 gre 6X 10X 729 st0 ae : 4,782,969 e° 729 X 2187 re Va 9X 243X729 ‘oye ae sr0 6x2 10 Licey : 81 x 81 x 243 aT Bates S nay 6X9 X 935 jo 2x 10 e 9x 27x6561 FY ees 3 X 9 X $9,049 og -- lina SI x Zt am 288,684 10 4440 rs im bi e of 155941323 a ——— >? AEE Ay ore 5) J 355940323 e? 1,594,323 eG ene re 24,948" v bis : 155949323 e? 1,594,323 “~ Jp? ee 50,499, ae ib Nee: 1,826 s 400,10 AS 95942323 e 1594323 °° ® ' 359323. — H59n323 °° AE += 257,032 ,10 ray * é bs 25945323 pe 1,594)323 “yeas Therefore Soa aT: — — 6 Es" eae 6 cG st° a 621s d ¢? reg swiee BE 0 61st e ? FOR RESOLVING THE CUBICK EQUATION &c. 477 6 1 5% 10 aoe 5 OF the fifth term of the compound feries 6¢ — II, is equal to a — 48 cI 57° 8 EG st 24 c2G 51° 24 CE? fe Oe aD “+ rth Sit or the fifth term of the sae feries 8e — A. UG Get EP 51. Laftly, the fixth term of the compound feries 8e — A is 4 Ab us 33? 48 E151 24G29%) ait2njo74's34 48 cEGs™* | 85351? ; a7 Bp SER E ae W e aa aeree > which is = 24 1,179,256 ri. I I st 10 ? 79> 5 a7 fe 48 x es s< 5599 3 x _ a A8 x x 93 bee é a 22991400163, et 9 “4,782,969 a 59049 7 6c61 “* 6561 et 593049 ‘ et : 1 Iz 12 iz Meme hy yg tO Bae So nee pies Wace sete in POPES OE SO ae ae 243 6561 en 243 243 243 ent 43,046,721 as 48 X 55,913 act 48 X TOM 3 X935 lye 24K M4 K BS 4 PAR NOS 4.39040,721 ert 243. X 3 X $9,049 Ag 43,046,721 er" Br x9 x 59,049 sy, 48X3X10X 154 5 cgi 8 X 3 X 1000 ‘ ga” °9343.4;048 i et 9x3 X 243 X 0508 pu Se ade Meds x 248 et 43,046,725 “elk 2,683,824 ag 1,346,400, 57 569,184 5 on Phe 201,000 2 ja alee Te aaa rhe: eae ORE bog. Reece eee bal ip we ope, a Se 43040,721 ett 43,040,721 ee 43,049,721 en 43,046,721 ert 221,760 pe 24000 Pe ee 9 By fate” s 4,599,408 s?2 43,046,721 es 43,046,721 eh EAs CAO; 72% cee 43,040,721 ett 282,36 12 : ; i as ~—; and the fixth term of the compound feries 6¢ — I is 35 72 é 6 G 5% 6 crs? 6 y20% 6 DG s™ Grek 3?* 6.3L 5s) Gms eer er ee ee in ce a ais Gh sgt) pec IS eG 154 PE I 22 CAs 10 10 ce 5 Xk a —xX — x — — x — x ——6 = 4 6561 43 au dee 9 " 729 git oe 243 i 243 en Ce SL 154 st 6 I 935 es 6 I 555913 le Tt. — -. ——-—060 x — x ——o x — x 6561 x eit 5 X 9 59,049 eit 3 43782,969 eit 6X 1,179,256 si. 6% 154x6561 He 6X 22x 6561 rr a 120,140,163. et 6561 X 6561 ert 9X 729 x 6561 ATE AT eceaoeio 4729 sat? 6x5xX154x81 4 art 6x81 xX 935 - giz 243% 243X729 © et Six 81x 6561 el 9 X 81 X 59,049 ell 6X 3°X'55,913 RPO Ae EST, 2 ODIO Pies 07 Osof 4, ley ehoile somes2 3X 3% 41782,969 ~ eT 43,046,721 er 43,040,721 ext 43,046,721 ES EI te ee Laat hogy gy ols siege a oI aE 8 ert 43,046,721 ag 43,040,721 er 43,040,721 er 43,046,721 had i 2,358,512 sad Tete IO SS Bits __ 39138,516 x 552824360 ert 43,040,721 ge 43,046,721 é 43,046,721 ett 43,046,721 st? c 6G 5% Gerst 6 E75" 6 pest? 6 crs? 6 Bi s™ OL mar ats.) Sa Iz 12 ee , or the fixth term of the compound feries 6 ¢ — TI, is equal to ay — 3 cps? 48 EI 5? 24 G75™ 24 OnE st 48 cEG s@ 8 £3512 Bou — omen T waa — aT ta ota eae. gto + ata or the A, term of the compound feries 8¢ — A. CPE OD, A ge- 478 A METHOD OF EXTENDING CARDAN’S RULE “4 general demonftration of the equality between the cor- refponding terms of the two compound fericfes 8e — A and 6e — Il. SS Ee ae a I RN 52. Wehave feen in the fix preceeding articles that each of the firft fix com- pound terms (or vertical columns of fimple terms involving the fame powers of s and ¢) in the compound feries 6 ¢ — II is equal to the correfponding com- pound term, or vertical column of fimple terms, in the compound feries 8¢ — A. We now proceed to fhew that the fame equality muft of neceflity take place between all the correfpondent terms of the faid two feriefes, as well as between the firft fix terms of them, to whatever number of terms the faid feriefes may be continued. { 53. Now this equality of the correfpondent terms of thefe two feriefes will ap- pear from this confideration, to wit, that the compound feries 6¢ — TI is con- {tantly equal to the compound feries 8¢— A in all the different values of ss and ee that are poffible, that is, when ee, or re is of any magnitude greater than r (which is its leaft poffible magnitude) and confequently when ss, or ve Ree : 3 3 : a es is of any magnitude greater than ° — oe, or o, or, in other words, of any magnitude, how {mall foever. For from hence it may be fhewn of each of the compound terms of the compound feries 6 e — II fucceffively, beginning with the firft term, that it is equal to the correfponding term of the compound feries 8e — A. This may be done in the manner following. 54. In the firft place, fince the whole compound feries 6 ¢ — IT is equal to the whole compound feries 8¢ — A (as has been fhewn in art. 44) it follows that, if we divide all the terms of both feriefes by the fraction =; the quotients of thefe divifions will be equal to each other, that is, the compound feries 6css 6 pst 655° 6 8 Ee Ee eee : 6Bcss 6 c?54 6cps® 48 sei hee Be OE Ss 6 BE 54 6 cE 5° ipa alata ob nk Br €e e* e 6654 6 BGs® + eyed Greig &c 6156 e will be equal to the compound feries 24.C =) FOR RESOLVING THE CUBICK EQUATION &c. 479 aed 246 54 ak 24.¢ + Aer er + &¢ 24 0? 8 cr s4 8 , gh abe 274 cEst — 48cGs Risto ee e e 8 c354 24 E256 tet a ee olin sia - &c. And this will be true, of how {mall a magnitude foever we may fuppofe ss to be taken. Siig eonfequently it will be Bhs likewife, when ss is equal to 0, or 3 when — — oi is equal to o, or when — is equal to oa But, when ss is = 0, all the terms of thefe two compound feriefes that involve any power of ss, that is, all the terms, except the firlt terms, of the faid feriefes, will likewife be equal too, and the faid two feriefes will become equal to their firft terms 6 B + 6C and 24 C refpectively. Therefore the faid firft terms muft be sae to each other, that is, 6 B + 6C, the firft term of the compound feries Bs » will be é By? 4 vv . Therefore, if we equal to 24 C, the firft term of the compound feries a, 6 Bss 6css 4 C SS 5 ahce or the firft 5-or. the multiply both thefe firft terms into —, it will follow vat firft term of the compound feries 6¢— WN, will be equal to term of the compound feries 8¢ — A. Qo) Bs Ds 6OxBss 6css + 7a OF 55. Secondly, fince it has been fhewn in the laft article that the firft term of the compound feries 6 e — IT, is equal to *, or we firft term of the compound feries 8 e — A, it follows that, if we fubtract thefe firft terms of thefe two feriefes from the Wile feriefes, the remainders will be equal to each other, that is, the compound feries 6c 54 4 6ps® Eee eb ae e3 e 6 Bc s4 6.c®s® ee Sagar ar rear ary 6 s# 6 BE Ss® 6 cES WV EPS a ATR RT nae 6cs° 6zc6s° a ee &c 615° +i —a7 77 &e will be equal to the compound feries 2 24E s*# 480 A METHOD CF EXTENDING CARDAN’S RULE 4 6 $ yet | ot sat gy, i 4 Be 48 S $9 _ 48 - is? gas 4 8 si 28 an ek ey, au 24 oe 4. &c, Now let all the terms of thefe two laft compound feriefes be divided by the . 4 . . . 7 . . Y fraction — ; and it is evident that the quotients thence arifing muft be equal to e? ‘ each other, that is, the compound feries Opss b. OF ss 6c By oo fh 7 et ‘Te &c 6 C255 6 cps4 — 6 3C — ap afer dep sore &c 6 BESS 6 cE s* +65— Se ee a &c 6G 68G54 ae : gins —_ pars | (4 ‘, 61s &c ; 24 will be equal to the compound feries , 24G55 24154 csi te ee et + &c 3 ra) ie RESETS co ABR ge as €e e+ 8 c3ss C254 + Sot HOt te 24.c7Es* 2a -\- &c. And this will be true, of how fmall a magnitude foever we may fuppofe ss to be taken. And confequently it will likewife be true, when ss is = 0, or when oe Tiss $ > ° : - =" cant Ss =o, or when rr > But in this cafe all the terms of thefe \ 4 . . a . two compound feriefes that involve any power of ss, that is, ail the. terms ex- cept the firft, muft likewife be equal too. Confequently thofe firft terms of the {aid two feriefes (being equal to the faid whole feriefes re{peCiively) muft be equal to each other, that is, 6C —6BC + 6E, or the firft term of the com- pound feries derived from the compound feries 6 ¢ — U1, will be equal to 24 E — 24(C’, or the firft term of the compound feries derived from the compound feries 8¢ — A. Therefore, if we multiply both thefe firft terms into the frac- 6c s4 6 Bc s4 6E s4 245 s4 24 0754 equal to — — = a ; that 4s, the fecond term of the compound feries 6¢ — IT will be equal to the fecond term of the compound feries 8 e — A. Q: E. De ; s4# tion =, we fhall have 6 Bss 6 css + 56. Thirdly, fince it has been fhewn in art. 54, that » or the é . . 2 firft term of the compound feries 6e¢ — TI, is equal to aon or the firft term of the FOR RESOLVING THE CUBICK EQUATION &c. asi : : : 6cs4 . the compound feries §¢ —- A; and it has been fhewn in art. gs, that oo oo eae 4 65 : t Et ca or the fecond term of the compound feries 6 ¢ — II, is equal to ., E ct 24 Cs een that, if we fubtract thefe two firft terms of thefe two feriefes from the whole fe- riefes, the remainders will be equal to each other, that is, the compound feries , or the fecond term of the compound feries 8e — A; it follows oes gone ha wir f os 2,6 8 ~_ — _ 6204 5, 6 g a 25! paces Aa Re 6 8 wat at fi ag ys will be equal to the compound feries 24158 246 5° pw Bay —— + &c é é 8 cEs® 8ccs® —_ re — S— — &e é é 8 c356 24 E758 ao &e e e ieee + &c. Now let all the terms of thefe two laft compound feriefes be divided by the frac- 6 . tion = ; and it is evident that the quotients thence arifing muft be equal to each other; that is, the compound feries 6ESs 6D + + &c — 6 c’ — $204 ye 6 cEss — 6 BE — — &c ee 6 8G ss need ewes SOC ee +6G¢ — en eee ce ‘will be equal to the compound feries ok iss + &c hid BG He ti os. ee PN! iy ies te eae — 48 CE— at gc — AEE _ &e + st SN 4 &e. 45 eee OL. II. 3 Q. ‘ And 482 A METHOD OF EXTENDING CARDAN’S RULE And this will be true, of how fmall a magnitude foever we may fuppofe ss to be . taken; and confequently it will likewife be true, when ss is = 0, or when oper is = 0, or when — is = r. But in this cafe all the terms of thefe ive compound feriefes that involve any power of ss, that is, all the terms except the firft, muft likewife be equal to o. Confequently thofe firft terms of the faid two feriefes (being equal to the faid whole feriefes refpectively) muft be equal to each other; that is, 6 D— 6C* —6BE + 6G, the firft term of the compound feries derived from the compound feries 6e¢ — II, muft be equal to 24 G — 48 CE + 8 C%, the firft term of the compound feries derived from the compound feries 8¢e — A. Therefore, if we mul- 6p 5° 6c** (a e tiply both thefe firft terms into the fraction s we fhall have 6 6 6 G 6 . . aL ty oe equal to ; that is, the third term of the compound feries 6 e — II will be equal to the third term of the compound feries 8e — A. 0. Ewe 245G) 5° 48 cE s® 8 0356 e ay és —_- 57. And in the fame manner we may fhew that the fourth term of the com- pound feries 6¢ — II is equal to the fourth term of the compound feries 8 ¢ — A, and the fifth to the fifth, and the fixth to the fixth, and the feventh to the feventh, and every following term of the one feries to the correfponding term of the latter feries, to whatever number of terms the faid feriefes may be continued. The reduétion of the compound feries 8e — A to a fimple wep Se Q s# ie S's* T st v st Jries -h ces- Thatan i oot bisa naa ose ea ad infinitum. ST a eae Ears : é 4 6 8 58. Now let the feveral numeral co-efficients of the fra@tions =, =, =, -, e é é € gyro gi% >? sz» &c in the terms of the compound feries 8 e — A (which is fet down above in art. 44) be reduced (by performing the neceffary arithmetical opera- tions of multiplication, addition, and fubtraétion) to fingle numbers, fo as to convert the faid compound feries into a fimple feries, or feries of fimple, or fingle, terms. This reduction has been already made for the firft fix terms of this feries in art. 46, 47, 48, 49, §0, and 513; in which it has been fhewn that the firft term of the faid compound feries, to wit, #°“, is = oe 3 and that its é é 2465¢ 24E 54 240754 . 6 s4 A J fecond term, ad a +=, ices ‘terete and that its third term, € e 81 e3 48 cE s® 8 6356 pees F570) 2 LE PTR e° 6° ———R . ‘ i and that its fourth term, a eee —- — « é , 24 E758 e] FOR RESOLVING THE CYBICK EQUATION &c. 483 24 £75° Ber ay 4456 3° 24 L st0 48 cr st? a + wos 1S. == 19,683 <7? and that its fifth term, 2 a a ace 48 EG s*° 24 Cc? G@s?° A CEesO 5 257,0325 ‘ 24.N 51? ——— Ct i, VE teen. a5 70g SEN. and that its fixth term, oh é é e 1,594,3230? ae ay! 48 ci s%2 48 EI s?? 24.G7st? | 24. 07157 48 cEG s!% 8 E3512 su gil Fy ell —y eit i ett =e elt np eu ? ap poe 5,282,360 5% fo that the firft fix terms of the faid compound feries 8e — A are j 776 56 43,046,721 ef! ; equal to the fix following fimple quantities, to wit, = + 2— 4456 5° 257604 25"° 5,282, 360 5% 19683 e? 1,594,323 €9 1 432046972 Ter gi? Pace. s° eae see 572i? => &c, in the faid fimple feries be (for the fake of bre- vity) denoted by the capital letters P, Q, R, S, T, be sen meee ey ts compound feries 8 e — A be equal to the fimple feries ~ path es - +5 ~— += ins = or + —— + ~- + &c ad infinitum. e? And let thefe co-efficients of the frachoi The reduction of the compound feries 6 e — VI in like man- ner to a feries of fimple terms. 59. And, in like manner, let the feveral numeral co-efficients of the fame fracki Bee eased mage p25 ght or ght ractions —, Tat ord Nocatee me &c in the terms of the other compound feries 6 e — II (which is alfo fet down above in art. 44) be reduced (by performing the neceflary operations of multiplication, addition, and fubtraétion) to fingle num- bers, fo as to convert the faid compound {feries into a fimple feries, or feries of fimple, or fingle, terms. This reduction has been already made for the firft fix terms of the feries in art. 46, 47, 48, 49, 50, and 513; in which it has been 6 6 css fhewn that the firft term of the faid compound feries, to wit, — + 8 ss wig E Ns well as the firft term of the former compound feries 8e¢ — A; e >) and that the fecond term of this compound feries, to wit, aoe. 5 s a | ea as 3» as well as the fecond term of the faid former compound feries; and that the third term of this compound feries, to wit, Sas oe Es ae “ a — ae -, as well as the third term of the faid former compound feries; and that the fourth term of this compound feries, to wit, of ae ss a 6G 58 CT sh 44565° as well as the fourth term of the faid former Bre wae 19083 e7? ao By compound f ¢7 e? 484 A METHOD OF EXTENDING CARDAN’S RULE 6 Fst compound feries; and that the fifth term of this compound feries, to wit, —— e 6 cE 5%° 6 DEs?° 6 ces? Gp13te Orso). Seemer.07 2 17° ee etiiiiened e 9 e? 69 go? 1,594,323 9” fifth term of the faid former compound feries; and that the fixth term of this 6G5% 6 cr 57 6 E25! 6pDGs"" 6.cis* 6 Bi s™ compound feries; “tO Wit, — > at ee as well as the hs 6 f + sand is — 5:25%300" as well as the fixth term of the faid former com- é 43,046,721 e* pound feries; fo that the firft fix terms of the Gpakeud feries 6e¢ — II are 6 65° S++ 81 218765 as well as the firft me terms of the formér equal tothe fix following fimple quantities, to wit, — =p 4456 3° 257,032 5'° §3282,360 5% 19683¢7 "1,594,323 | 43,046,721 6” compound feries 8e — A. And it is evident, from art. 57, that if the feventh, and eighth, and ninth, and other following terms of the compound feries 6 ¢ — II were in the fame manner to be reduced to fimple terms, the faid fimple terms would be equal to the fimple terms to which the feventh, eighth, ninth, and other following terms of the faid former coe ogee fees 8¢— A would be ast R 56 s3* reduced. Bre confequently the fame fimple feries = 4- 37 ba ee a YS" + &c ad infinitum, will be equal to both the compound feriefes 8 ¢ — Aand 6e — II. e7 “The values of y° wigs 1 eapreyed by means of the fimple P ss pst T st v st? Series = + = ae Seek a mes yates len palbine e ad pa Seiten 60. We have feen, in art. 41, that the compound feries A is = 53; and we have feen, in art. 43, that the compound feries Il is = gy. Therefore 8e — Ais = 8e¢ — y?, and 6e—Tlis = 6¢ — gy. We hall therefore have R x s - 4 oF had se — ey (= 8e — A) = the fimple feries + = ae + —+ — . BE: + &c ad infinitum, and confequently (adding A to both fides) Se J; -. Pss qst R 56 ss T 510 wit ‘A ate -+- the feries — ape pune ieiocplbs ney oe margin mosepeb mt) infinitum, and ¥% pa P ss Q 54 Rs s 58 ste vs? ee : Riel Lah GEE ee a a, ee pe we PSS fhall alfo have 6¢ — gy (= 6¢— TI) = the fame feries — + = af + = ss° ~~ 5t9 SO = + &c ad infinitum, and confequently (adding qy to “path fides) bi ‘ FOR RESOLVING THE CUBICK EQUATION &c, 485 A BSS. ast Rs gs% 0) gga vy st ee ee Op et op Seige ae + &c ad infinitum, LE: PSS Qs4 Re? s 58 Tet? v yt raed and i Oe Sh pelea Sow tae oppst eral te ar tert imag @ ad infinitum. Examination of the expreffion 2 »/? ¢ X the infinite feries C 2% E 24 G28 128 Lio N zt? et aid i Tigh oat vik etwas ee ad infinitum. 61. We muft now turn our attention to the other tranfcendental expreffion . : . a 6 8 ° Bede setherinfinite feries 1-40 —— — ——- <2 —- ciel ta &Z gt g g gi rae &c ad infinitum, in which g is put for -, or half the abfolute term of the equa-- tion «7 — gx = ¢, as e was before put for = , or half the abfolute term of the equation y3-— gy =r, and in which zz is put for E — > or the difference between f and the fquare of half the abfolute term ¢, as ss was before put for rr 3 : 3 i ae “ae or the difference between é and the fquare of half the abfolute term: 4 r. ‘This expreffion we have afferted above, in art. 36, to be equal to the root of the equation x3 — gx = ¢, in which the abfolute term # is lefs than yt, but . . F se 3 greater than 4/2 X iu, or in which * is lefs than re but greater than = x 3 3 . re . <, or than me This affertion we muft now endeavour to prove. : 3 : . : C2z% E2** 62. Now ‘¢ that this expreffion 2 4/? (g X the infinite feries 1 ++ gran mcpee g ; 6 8 Io N 12 2 z pS A 3 ee ~ + &c is equal to the root x of the equation «* — qx = t, when fis of the- magnitude here fuppofed,” will be evident, if we can. fhew that this expreffion, being fubftituted inftead of « in the compound quan- - tity «3 — gx, will make that quantity be equal to the abfolute term ¢ of the faid equation; or that, if the faid expreffion be cubed, or raifed to the third power ~ by multiplying it twice into itfelf, and alfo be multiplied into the co-efficient g, . the faid cube of the faid expreffion will be greater than the faid product ‘of ‘its multiplication into g, and that the excefs, or difference, will be equal to the abfolute term ¢. This therefore is what I fhall now endeavour to demon- - {trate. 2 62. The ; 486 A METHOD OF EXTENDING CARDAN’S RULE 63. The cube of the expreffion 2 «/?(g x the feries 1 + = - = - = 12° L zr° N 21? a a + &c ad infinitum is = 8g X the cube of the faid feries, & ao We mutt therefore raife the cube of this feries by multiplying it twice into itfelf; which may be done as follows, 4 The multiplication of the infinite feries 1 + as ~ a 4 G25 128 Ete N 212 4 : 7 Soe at Soo, RR + &e ad infinitum into itfelf, in order to obtain its fquare. C xz E x4 G2 128 L zie WN zi? I + Lene 4 + 6 yas 8 10 12 + &c && i & = &§ & C Zz E x4 Gx I 28 L zi? Nz !2 T + Paar 4 6 2 8 10 12 &c &S y 3 & & Zz CUZ EZ G ze i238 Lite N zl I = a5) a a Sg + ge oe g ; g : Seok are Cz% C2z CEZ CGZ C1Z CL2Z + ae 4 ataad 6 ts} rere 10 12 saa &c &S i a § = =) E x4 CE 2® E228 EG z!° EI 2!” & om = - = —— Gc gt g° 3 pie gi? GZ CG EG 27° G?z* + oo 8 Pr 10 12 —. &c & & & 128 CI zt EI z!2 ery 2 Rees gie Poe ies &c fi L200 Crist & : — Cc Io I2 & ee — = — &e 2022 2E2* 262 2128 21 z10 2Nx12 ‘ a ae 4 Ce BY es g 7 10. .: Genes 12 ch &c 8 & - fy S C224 2 CEZ® 2CGz 2 CI 2% 2 CL2!2 gt ae: a ie fo * ors &c E2z8 2.EG2Z!I° 2 EI zi gf ge 12 —&c Grzl2 says : E 2+ ‘G28 8 10 This is the fquare of the feries 1 + = menial ea bok — yh ne + &c. FOR RESOLVING THE CUBICK EQUATION &c. 487 A comparifon between the foregoing compound feries, which “ . c ar is equal to the fquare of the feries 1 + —— — a + 28 1 28 Ligt? N x12 : a ; : — ey te + OCS ad infinitum, and } 3 the compound feries which is equal to the /quare of the : Cc 5s E s# G 59 rs Ea"? fe SJ €é et e° 2 er? N gid sz — Ge ad infinitum, and which is fet down above in art. 40. . ‘ ‘ 4 6 64. Now, if we compare this fquare of the feries 1 | “= —= 4 @ — 3S et ks, rz? é 5 4 4 + &c with the compound feries which is equal to the fquare of the former ; css E 54 G 56 is : : F feries 1 — ae pet oreter atamertin: a" &c, and which is fet down above in art. 40, we fhall find that there are the following refemblances and differences be- tween them. In the firft place, the firft terms of both thefe compound feriefes are equal to the fame quantity, to-wit, 1. Secondly, the fecond, and third, and fourth, and other following terms of the latter compound feries obtained in the foregoing article 63, involve in them the : Bes yao Phe? wreFO. ., att ‘ fractions —, —-, =, > —s>» >, &c, or the feveral powers of the fraction g gt 2 2 ig ‘ee 3 BB re juft as the fecond, and third, and fourth, and other following terms of the former compound feries, obtained in art. 40, involve in them the fractions Fe ae SO end? Sh, 4», 5* : : SS ans ra Sime ar La Te &c, or the feveral powers of the fraction —. And it is evident that this obfervation will be true of all the following terms of thefe ‘ : Czz E x4 Gx 1° two compound feriefes, becaufe the fimple feries 1 + eon ge argc ere 2) ° . . . BB 4+. &c contains in its terms the very fame powers of the fraction aa the for- 5 : css E54 Gs R13? cg NW 5? ; tier imple feries t—— ———— = a - — ec contains of the fraction = in its feveral corre{ponding terms. Thirdly, the figns of the feveral members of the third, fifth, and feventh terms of the latter compound feries obtained in the preceeding art. 63, are the fame with the figns of the correfponding members of the third, fifth, and fe- venth terms of the former compound feries obtained in art. 40. For the third pies 2ES* crt term of the former compound feries is — —- + —-, the two members of which are marked with the figns —- and + ; and the third term of the latter compound 458 A METHOD OF EXTENDING CARDAN’S RULE 4 paki bas ate the two members of which are likewife marked with the figns: — at +. And the fifth term of the former compound 2ca st, gst ’ : feries is — at —— -- =s the three members of which are marked with compound feries is — the figns —, +, arn + ; and the fifth term of the latter compound feries is — aP 2cc2° : ‘ P mia, ae + 7: the three members of which are likewife marked with the figns a +, and +. And the feventh term of the former compound feries . 2N 5% 2 CLSs** 2 EL s** eo . is — —— + —- + a + a> the four members of which are marked with the figns —, +, -+, and +; and the feventh term of the latter compound 2N 212 2 CL2%* Zila G72" Iz Sid £2 Iz likewife marked with the fiens —, +, +, and +. And the fame analogy will take place between the figns of the feveral members of the ninth, eleventh, thirteenth, and other following odd terms of the former compound feries ob- tained in art. 40, and the figns of the correfponding members of the ninth, eleventh, thirteenth, and other following odd terms of the latter compound fe- ries obtained in art. 63; as we fhall prefently endeavour to make appear. feries is — ,» the four members of which are BB ° Fourthly, the fign of the fecond term, , of the latter compound feries, ; 2G2° 2 CE 2° and thofe of the feveral members of its fourth term, ++ so np pay nd . 2L2%° 2 cI z!° EG xl thofe of the feveral members of its fixth term, -- ——- — ——~ — >= are & Side he refpectively contrary to the fign of the fecond term, — =<, of the former compound feries, and to thofe of the feveral members of its fourth term, — 2Gs" 2 CE Ls? oe % a » and to thofe of the feveral members of its fixth term, — — +- — +5 — . And the fame contrariety will take place between the figns of the feveral members of the eighth, tenth, twelfth, and other following even terms of the latter compound feries obtained in art. 63, and thofe of the feveral correfponding members of the eighth, tenth, twelfth, and other following even terms of the former compound feries obtained in art. 40; as we fhall now en- deavour to make appear. 4 65. It is evident, from the rules of multiplication|in algebra, that, whenever a feries of algebraick quantities is multiplied by either the fame, or another, feries of algebraick quantities, a]ll thofé-horizontal lines of terms in the product, which arife from the multiplication of the firft feries, or multiplicand, into thofe terms of the multiplicator which are marked with the fign +, will have the fame figns + and — prefixed to.their feveral terms as are prefixed to the cor- refponding terms in the multiplicand; and that all thofe horizontal lines of terms in the produ which arife from the multiplication of the. firft feries, or multiplicand, into thofe terms of the multiplicator which are marked with the fign FOR RESOLVING THE CUBICK EQUATION &c, 489 fign —, will have contrary figns prefixed to their feveral terms to thofe which are prefixed to the corref{ponding terms of the multiplicand. It follows there- ; CSS fore that in the general product of the multiplication of the feries 1 — “oa een Est | asf 158 Lgt° N 5% rs ? see ae es &: (Of:~ which all the terms after the firft é é € é é 40 (I mean by the general product the firft product, before the fimilar terms in each vertical column of terms are added up together at the bottom, fo as to make but one term), the firft horizontal line of terms (which arifes from the PTY : Hee : css E s# G s€ Ny multiplication of the faid multiplicand, or feries 1 — ofa ane ee ee @ — a er? el? feries itfelf) muft have the fign + prefixed to its firft term (or rather, no fign at all; becaufe, being the firft term of the whole produdt, it is that to which all the other terms are to be referred, and to be added to it when they are marked with the fign +, and to be fubtracted from it when they are marked with the — &c, into its firft term 1, or, in other words, which is the faid _ fign —) and muft have the fign — prefixed to all the following terms; and the fecond, and third, and fourth, and fifth, and fixth, and feventh, and every fol- lowing horizontal line of terms in the faid produ& (which arife from the multi- eh re . re i ess E s4 Gis6 rs8 ste plication of the faid multiplicand, or feries 1 — — — —— — — — — — = ee et 2 e ean N 5% : Grp, Eat, G3 ps® x, 88. ars : —= &c, into the feveral terms A le ke era aCe, &c, which are all marked with the fign —) muft have the fign — prefixed to their feveral firft terms, and the fign + prefixed to all their following terms, to whatever number of terms the faid horizontal lines may be continued. And it follows likewife, that in the general product of the multiplication of the feries 1 + = 8 10 1z _ dy + chet aed eas ust ail — ~= + &c (of which the fecond, and fourth, and fixth, and all the following even terms, are marked with the fign +, and the third, and fifth, and feventh, and all the following odd terms, are marked with the fign —) into itfelf, which is fet down in art. 63, the firft horizontal line of terms (which arifes by the multiplication of the faid multiplicand, or 4 C zz E24 Gx 128 EB zt0 N x72 : : feries 1 + — — et en tee oes t+ C&;, into its firft term.1, or,in other words, which is the faid feries itfelf) muft have the fign + pre- fixed to its fecond, fourth, fixth, and other following even terms, and the fign — prefixed to its third, fifth, feventh, and other following odd terms (to what- ever number of terms the faid horizontal line may be continued), being the fame figns as thofe of the feveral correfponding terms of the faid multiplicand itfelf; and, in like manner, the fecond horizontal] row of terms in the faid product, and the fourth, and fixth, and eighth, and tenth, and every following even horizontal row of terms in it (which arife from the multiplication of the faid multiplicand, : C xz B x4 G2 128 ca N xl : or feries 1 + — — — 4+ — + ae ~t «C&e, into the fecond, g & g g*° é Vot. II. gR fourth a 490 A METHOD OF EXTENDING CARDAN’S RULE ; ozz G2® r2% ‘ fourth, and fixth terms, —-, —, —,, and the other following even terms, of the faid feries, to all which the fign + is prefixed) will have the fame figns | -+ and — prefixed to their feveral terms as are prefixed to the correfponding terms of the faid multiplicand, or feries itfelf, to wit, the fign + prefixed to their’ firft, and fecond, and fourth, and fixth, and eighth, and tenth, and other fol- lowing even terms, and the fign — prefixed to their third, and fifth, and fe- venth, and ninth, and eleventh, and other following odd terms; and the third horizontal row of terms in the faid produét, and the fifth, and feventh, and ninth, and eleventh, and every following odd horizontal row of terms in it (which arife from the multiplication of the faid multiplicand, or feries 1 + = &S E x4 G 2® 12° Lt N zit eens -+- &c, into the third, and fifth, and fe- 6 38 aT) re «8 ars venth, terms ak an “=~, and the other following odd terms of the faid fe- ries, to all which the fign — 1s prefixed), will have contrary figns prefixed to their feveral terms to thofe which are prefixed to the correfponding terms of the {aid multiplicand, or feries itfelf, and therefore will have the fign — prefixed to their firft, and fecond, and fourth, and fixth, and eighth, and tenth, and other following even terms, and the fign + prefixed to their third, and fifth, and feventh, and ninth, and eleventh, and other following odd terms. 66. It has been fhewn, in the laft article, that in the general product of the c 5s E 54 G 58 eke Last N si multiplication: of the feries: 1, —\—="—= = =a ae ee &c into itfelf, fet down above in art. 40, all the terms in the firft, or higheft, hori- zontal row of terms, except the firft term 1, will have the fign — prefixed to them, and that in the fecond, and third, and fourth, and other following hori- zontal rows of terms in the faid produét, the firft terms of the faid rows will have the fign — prefixed to them, but all the following terms in them will be marked with the fign +. Now the firft terms of the feveral horizontal rows of terms in the faid general produc are the loweft terms of the feveral ver- tical columns of terms in the faid product, which involve the fame powers of the fraction =; and the fecond, and third, and fourth, and other following terms of the firft, or higheft, horizontal row of terms in the faid produ, are the higheft terms of the fecond, and third, and fourth, and other following vertical columns of terms in the faid product. ‘Therefore the higheft term and the loweft term of the fecond, and the third, and the fourth, and every fol- lowing vertical column of terms in the faid produét, will have the fign — pre- fixed to them, and all the other terms of the faid vertical columns will be marked with the fign -+-. Now this will likewife be the cafe with the third, and the fifth, and the feventh, and the other following odd vertical columns, of the gene- whee . . 4 6 ral product of the multiplication of the other feries 1 + — -: = m ea 1z° Lio N i ~ + &c into itfelf, fet down in art. 6 3» For the firft, or higheft, terms ——= ——— 8 ; 10 é & FOR RESOLVING THE CUBICK EQUATION &c. 491 terms of the faid odd vertical columns are the third, fifth, feventh, and other following odd terms of the firft, or higheft, horizontal row of terms, to wit, the a iz terms =. —, ——, &c, which are all marked with the fign — ; and the loweft terms of the fame odd vertical columns are equal to their higheft terms, and are marked alfo with the fign —, becaufe they are the firft terms of the feveral odd horizontal rows of terms, which have been fhewn to be marked with the faid fign —. And the intermediate terms of the faid odd vertical columns between the higheft term and the loweft mutt all be marked with the fign +, becaufe they are, all of them, the products of the multiplication of factors, which are both marked either with the fame fign + or with the fame fign —. Thus, for example, in the feventh vertical column of the faid product, of which the higheft term is — 12 Cay aa hi he : N*, the next term is the product of the multiplication of the factors 12 ? 12 p Pp & & 10 b 2 Z 2? and —, which are both marked with the fien and therefore it muft ay 43 e ; giz be likewife marked with the fign + ; and the third term is the product EI ce “erodes 128 E x4 : % ; of the multiplication of the factors — and —-, which are both marked with the fign —, and therefore it muft be marked with the fign + ; and the fourth G2zzi2 . A G 2x G 2° : term —— is the produét of the multiplication of — by —=, which are both marked with the fign +, and therefore it muft be marked alfo with the fign EI %7* Poy is the product of the multiplication of the factors +; and the fifth term 4 8 - . . — and kite which are both marked with the fign —, and therefore it muft be & & 1z A c marked with the fign + ; and the fixth term, een Sh or the loweft term but one, or the laft of the intermediate terms between the higheft and loweft terms of the C 2% cme Ae {aid vertical column, is the product of the factors 7m and ——, which are both marked with the fign +, and therefore it muft likewife be marked with the fizn + ; and confequently all the five intermediate terms of the faid feventh vertical column between its higheft and loweft terms muft be marked with the fien 4+. And it is eafy to perceive that, for the fame reafons, all tle intermedi- ate terms between the higheft and loweft terms of any other vertical column of which the higheft term was marked with the fign —, that is, of any other odd vertical column, muft be marked with the fign +. We may therefore con- clude that in all the odd vertical columns of terms in the general product fet down in art. 63, the figns -+- and —, which are to be prefixed to the feveral terms of the faid columns, will be the fame as thofe which are to be prefixed to the correfponding terms of the fame odd vertical columns of terms in the gene- ral product fet down above in art. 40, 402 HL Ds 67. Since the figns of the terms in the feveral odd vertical columns of the general product fet down in art. 63, are the fame as thofe of the correfponding Bae terms 492 A METHOD OF EXTENDING CARDAN’S RULE terms of the like odd vertical columns of the general product fet down in arts 40, it is evident that, when the fimilar terms contained in the faid odd vertical columns are added up together at the bottoms of the faid general products, fo as to make but fingle terms, the fame analogy between the figns of the terms of thefe vertical columns will continue; that is, in the feveral odd vertical columns of the reduced compound feries, which is equal to the fquare of the feries 1 + C2, E24 Gz? 128 L 2? Nz? — Hee te ett «Ce, the figns of the feveral terms, 4 g® 8 10 S$ or members of the faid columns, will be the fame with the figns of the feveral corref{ponding terms, or members of the like odd vertical columns in the re- ‘ and oS ‘of which the former is marked with the fign +, and the latter is marked with the fign —, and confequently their product muft be marked with the fign — ; EGZ rte) ; ro , a ae et es 4 and the fourth term is the product of the multiplication of the faétors = and oe of which the former is marked with the fign —, and the latter is marked with the fign +-, and confequently their produ&t muft be marked with the fign — ; and the fifth term esa or the loweft term but one, or the laft of the faid intermediate terms between the higheft and the loweft terms of the faid vertical column, is the produc of the multiplication of the factors = and on of which the former is marked with the fign 4+, and the latter is Stier with the fign —, and confequently their product muft be marked with the fign —. And it is eafy to perceive that, for the fame reafons, all the intermediate terms between the higheft and the loweft terms of any other vertical column, of which the higheft term was marked with the fign +, that is, of any other even vertical column, muft be marked with the ign —. But inthe general product fet down in art. 40, all the intermediate terms between the higheft and loweft terms of all the vertical columns, both odd and even, are marked with the fign +. There- fore all the intermediate terms between the higheft and the loweft terms of all the even vertical columns, in the general product fet down in art. 63, are marked with contrary figns to thofe of the correfponding intermediate terms of the even vertical columns of the general product fet down in art. 40. And the fame thing has been fhewn concerning the higheft and loweft terms of the faid even vertical columns of the faid two general products. We may therefore conclude that in all the even vertical columns of terms in the general product fet down in art. 63,. the figns + and —, which are to be prefixed to the feveral terms of the faid co- lumns, will be every where contrary to thofe which are to be prefixed to the correfponding terms of the like even vertical columns of terms in the general product fet down above in art. 40. Oi Es De 69. Since the figns of the terms in the feveral even vertical columns of the general product fet down in art. 63, are refpectively contrary to thofe of the correfponding terms of the like even vertical columns of the general produé fet down in art. 40, it is evident that, when the fimilar terms contained in the faid even vertical columns are added up together at the bottoms of both the faid ge- neral products, fo as to make but fingle terms, the fame analogy, or rather con- trariety, between the figns of the terms of thefe vertical columns will continue ; that is, in the feveral even vertical columns of the reduced compound {eries, Aas : C XB EZ G2 12° bat? which is equal to the fquare of the feries 1 + re ot a SOM Ae RN &: 8 iz i “~~ + &c, the figns of the feveral terms, or members of the faid columns, — Iz will be every where contrary to the figns of the feveral correfponding terms, or 7 members 494 A METHOD OF EXTENDING CARDAN’S RULE members of the like even vertical columns, of the reduced compound feries, CSS E s# G s° 158 L 51 which is equal to the fquare of the feries 1 — — — — — =~ — — — —_ — el? + (iB . . . “~~. — &c, to whatever number of terms the faid feriefes may be continued. el? Therefore the fourth obfervation made above in art. 64, concerning the figns of the members of the even vertical columns of thefe two compound feriefes, is uni- verfally true. Q. Es D. 70. In the fifth place we may obferve concerning thefe two compound feri- : . . 7, c E s# efes, which are equal to the fquares of the two infinite feriefes 1 — = — — ee 4 e G 56 158 L st N 572 8: \ C zz E x4 G 2° 123 L zie “seeistiss damian ets eR Cy ANG To en re a &8 8 "5 ‘4 & NZ — — + &c, that the co-efficients of the feveral terms, or members of the fe- veral vertical columns, of the compound feries which is equal to the fquare of the latter fimple feries, and which is fet down in art. 63, will be equal to the co-effi- cients of the correfponding terms, or members of the feveral vertical columns of the former compound feries, which is equal to the {quare of the former fimple feries, and which is fet down above in art. 40; though the figns + and — that are to be prefixed to them will not every where be alike. For, fince the two fimple fe- 3 Css E 54 G56 158 L st? N 512 Cz E x4 G 26 12° L zie? N2i2 & age ‘ ‘ coving et eres = irre riety + &c, contain the very fame co-efficients C, E, G, I, L, N, &c, combined with the fame powers of the two fractions — and =, css Gs®& pst? though with different figns -- and — prefixed to the even terms —, wie c2z2 G2 pete . ‘ &c, and —, —, ——, &c, of the two feriefes; and fince it has been fhewn that, in the third, and fifth, and feventh, and other following odd vertical columns of terms, in the two compound feriefes which are equal to the fquares of the faid fimple feriefes, the figns + and —, that are to be prefixed to the correfpond- ing terms of the faid odd vertical columns, are the fame in both feriefes; and that in the fecond, and fourth, and fixth, and other following even vertical co- lumns of terms in the faid compound feriefes, the figns 4+ and —, that are to be prefixed to the correfponding terms of the faid even vertical columns, are uni- formly contrary to each other in the two feriefes; it follows that the co-effi- cients of the feveral terms, or members of the vertical columns, of one feries, mutt arife from the fame combinations of the original co-efficients C, E, G, I, L, N, &c, by multiplication and addition, by which the co-efficients of the correfponding terms, or members of the feveral vertical columns, of the other feries are produced: and confequently the faid co-efficients muft be the fame in both feriefes, though the figns + and — that are to be prefixed to them will, in all the terms of the feveral even vertical columns, be different. @. E. D. ee) A ,* FOR RESOLVING THE CUBICK EQUATION &c. 495° 71. We now proceed to multiply the compound feries obtained above, in art. 4 6 8 10 63, for the fquare of the fimple feries 1 + — == a el pe ay a ee ah & & N21 —— + &c, into the faid fimple feries itfelf, in goa. to obtain the cube of the fdid fimple feries, This multiplication is as follows. The multiplication of the compound feries (obtained above in art. 63) which is equal to the Square of the fimple C 2 E x G25 123 L2te N zi@ Series 1+—_—— ree + Se ep tck & + a on, comet be & tf 4 + €3c, into the Said fi Jimple Series, in order to obtain the cube of the faid fimple feries. 2C2zz 2E24 2625 2128 2L2z!° 2N2! & &S Bak '£ » g & g& am C224 2CEZ 2cG6 28 2 C1 27° Fe 2 CL 2%” Rc yj ry Par! 6 8 ier To 12 . ie? 2 EG z7° 2 E12 2 2° ee 1A Iz G2z12 tT git act &c C zz E% G 2° 12 L zie N ZI? 8 + met 4 6 at, ai 8 10 ca 12 + c §& 2 4 z "4 2 C22 2E24 262° 21° 24a." 2N 212 & ne “a 4 7 eae 8 as 10 12 + XC &S ie ty 8 & & c2z4 2CE2® 2CG% 2 c1z?° 2cL2!2 a + 24 aT 2 8 gO fas Cc E228 2 EG 2% 2 EI 2! & hi g° Pre ‘pte Yd ar G7z12 +—— — &e C2zz 2074 2 CE 25 2 cG x8 2 c12z%° 2 CL 31? & ot aN oh gk a CPU “te 77 a + = — & 4. C36 2c7Ex? 2 c2Gz1° 2 c21 zl? is ae 6. xe 8 10 12 S & CE2z1° 2 CEG 2!” & 190 ca, oi c Poy E x4 2CEZ 2-E72z8 2 BG z!° 2 EI 22 eae at g° 5 8 ie. Io “te 12 Pe &c c?Ex® 2 CESRrC 2 CEG 2" ore =e a ‘ei? 12 c E3zt2 — —— + &e G2 2cG28 2 EG Z!° 2 G*z12 “f pe + 2 mae or tie gt ens &c 2 10 ” oI2 c*G62% 2CEGZ + — SS + ke 496 A METHOD OF EXTENDING CARDAN’S RULE 128 2cr2te 2Erz'® &e a i 2 ae 10 Seger ce Batons é é c*I eit 8 L 2x0 2 eon us ene, 12 eee &c N zi — — &c 2 C 2% 3 E2t 3G2% 312 21k 3.N zt iP ; og ot F aaa + To —— + &c k 3 C728 6cEz 6cEx 6 c1z%° 6ci 2% ee IP: 6 8 7 Io “Iz — s* § 8g 4 c3z6 Sea" 6 EG x? 6 EI zl Bc 8 aS Io Iz eer’ ‘ gt Fs ial 3 c2zz? 3.07Gzr? 3 Gtzl2 ie ro s to 12 & 3 osareg 3 C71 git & +r pc = + &c 6 cEGz™ ae 12 EIzt2 ae Ss This laft compound feries (which, for the fake of brevity, we will denote by 4 6 & 2 the {mall Greek letter ) is the cube of the feries 1 + —~ —== + =a sil a Io 1z && & gf. & ——— a + &c ad infinitum. Therefore 8g x the cube of the faid feries a Cc 2z E24 G2® 12° rut N 2 ot ecg ee Pe + &c ad infinitum will be = 8g x the compound feries y ; see confequently the cube of the expreffion} 2¥3(¢ E 24 G 2° 12° x2 N xi? x the infinite feries 1 + — — ray + Set aage oh es iets &c will he = Se5% the compound *(eries Ye 72. If the foregoing compound feries y be actually multiplied into 8g, the - product will be the following compound feries, to wit, 1 24524 24.6 2° 2412° 24 L zt eee eee 8g + —— a er ke a aa a Mae eyes eo mm + &c 24.c7%4 48 cE x® 48 ce x® 48 cr 27 48 cLz™ WT VL Awa RS ee a — &c fo 8 c3z6 24 5728 8 EG x? Ser zt2 of pee ey eae eas Oe Z g! a 24c7E 2° 24.C07G27° 24 G?z% an 24 CE7Z1° OU ek vlad a5 2° aE gt + &c __ 48cEc2™ nw ee 8 E3z12 sn “> — &c: 5 which, FOR RESOLVING THE CUBIGK EQUATION &c. 497. which, for the fake of brevity, we fhall denote by the fmall Greek letter 3. And 4 1 ‘ : : zt then the cube of the expreffion 2 ¥?(g x the infinite feries 1 + — — ° eesti Le? . Nal AeA : : Me. ga os a &c ad infinitum will be equal to the faid compound feries é. A comparifon between the compound feries 3, obtained in the foregoing art. 72, and the compound feries A ob- tained above in art. 41. 73. If we compare the_laft-mentioned compound feries 3 with the compound feries A obtained above in art. 41, we fhall find that the co-efficients of the cor- refponding terms are the fame in both feriefes, and likewife that the figns pre- fixed to the correfponding terms, or members, of the third, and fifth, and fe- venth, vertical columns of terms in both feriefes are the fame, but that the figns prefixed to the’ fecond terms of the faid two feriefes, to wit, the terms anes “4°, and to the correfponding terms, or members, of their fourth and and fixth vertical columns of terms, are contrary to each other. And the fame agreement of the figns of the terms of thefe two feriefes will take place in all the following odd vertical columns of terms after the feventh; and the fame con- trariety between the figns of the terms of the faid feriefes will take place in all the following even vertical columns of terms in them after the fixth terms, to whatever number of terms the faid:feriefes may be continued; as might be fhewn by reafonings fimilar to thofe contained in art. 65, 66, 67,68, and 69. 44. Since the co-efficients of the feveral terms of the compound feries 3 are equal to the co-efficients of the corre{pondent terms of the compound feries A; and the figns prefixed to the correfponding terms of the {aid two fericfes are exactly the fame in thé third, and fifth, and feventh, and other following odd vertical columns of terms in the faid feriefes, but are contrary to each other in the fecond terms of the faid feriefes, and in the correfponding terms of the fourth, and fixth, and other following even vertical columns of the faid feriefes ; it follows that, if we reduce the feveral columns of terms in the compound feries A into fingle terms by making the neceffary multiplications, additions, and fub- tractions (as is done above in art. 46, 47, 48, 49, 50, and 51), and denote the co-efficients of the fingle terms thereby obtained by the capital letters P, Q, R, S, T, V; &c, and, if we likewife reduce the feveral columns of terms in the compound feries 6 into fingle terms by making the neceflary multiplications, additions, and fubtractions, fo as to convert the faid compound feries 3 into a fimple feries; the co-efficients of the terms of this latter fimple feries, which will be equal to the compound feries 6, will be equal to the co-efficients of the correfponding terms of the former fimple feries which is equal to the compound Wo, II. wo feries 498 A METHOD OF EXTENDING CARDAN’S RULE ap feries A; and the figns + and — that are to be prefixed to the third, and fifth, and feventh, and other following odd terms of the latter fimple feries, will be the fame as thofe which are to be prefixed to the third, and fifth, and feventh, and other following odd terms of the former fimple feries, which is equal to the . compound feries A; but the figns to be prefixed to the fecond, and fourth, and fixth, and other following even terms of this fecond fimple feries will be- contrary to thofe which are to be prefixed to the fecond, and fourth, and fixth, and other following even terms of the former fimple feries, which is equal to the compound feries A. 75. Now it has been fhewn in art. 60 that y* is equal to the fimple feries 8 ¢ 4 6 8 b fe) ry (12 ~ ; a = —= — — > —_ = — &c ad infinitum. Therefore the é é compound feries A (which is equal to y*?)muft likewife be equal to the fimple 3 4 6 6 8 10 IZ ; b feries 8e — — —= = “— _ = — — — = — &c ad infinitum. It fol- lows therefore, from the foregoing article 74, that the compound feries 3 will be ‘ + 5 14 6 P 10 12 . P Zz QZ RZ $% TZ Vz equal to the fimple feries 8g + uae Wa pian: Gl i eee &c ad infinitum. ‘Therefore the cube of the expreffion 2 /?(g x the infinite : Cc E vt 6 I 2% 10 N zie ; pi feries 1 + as ate + a a a aca eee + &c (which is equal to ; : : P 2% Q 2* = ] — ome - the See a eae will om be equal to the fimple feries 8g + res y + ee — on + oa — far + &c ad infinitum. : So .. The value of the co-efficient q of the fimple power of x in the cubick equation x? — qx = t expreffed by an inft- nite feries involving the powers of z and g. 76. In the next place we will find a tranfcendental expreflion, involving the powers of z and g, for g, the co-efficient of » in the propofed cubick equation x3 — gx xt; after which we fhall multiply the faid value of g into the expref- C22 E 2 G2 i Lae Nae fion 2 +/?(g x the infinite feries 1 + —-— —- + —-} oe -- &c, which we have afferted to be equal to the value of « in the equation x3 — gx == t. And, if it can be fhewn that the faid product of the multiplica- tion of the expreffion which is equal to g into the faid expreffion 2 ¥/3(g x the Cx E 24 G 2° pe ta N xi? mnfinite feries. 1 bhi se a oe oe ee ° 65 . ~ 4 . the cube of the faid laft-mentioned expreflion, or than the fimple feries 8¢ + % 4 “ s 2° toh, tie Vv zi ; 5 fo SA eae EG alah lo haa a anh, 4- &c (which has been fhewn in the Pe 5 7 9 pit late article 75 to be equal to the faid cube), and that the difference by which it 3 , falls FOR RESOLVING THE CUBICK EQUATION &c. 4.99 falls fhort of the faid cube, or of the faid feries 8g + eS — “= a — a T z%° ¥2tF . . : : , ++ 8c, Is ons to # (which is the abfolute term of the equation i a oe x — qx = ¢) orto2 X —, or to 2g, or that the faid product is equal to the 4 6 28 10 2 : fimple feries 6 ¢ + $s — “: 2 ss - re + &c, which is lefs than the feries - + Pe sath ae a adn tM gt dn 2 5 7 9 the difference 2g, or#, we thal] be able July to conclude that nee fee expret- 3g | fi f, Ex G2 12° L zt? N zi? fion 2 4/?(g x the infinite feries 1 + —= — — 4 “> — = 4 + &c is equal to the root x of the propofed equation «#— gx =f. This heres fore is what we muft now endeavour to prove: and, for this purpofe, we mutt, in the firft place, find a tranfcendental expreffion for the co-efficient g, which fhall involve in it only the powers of g and z. Now this may be done in the manner following. is = 2, and ge is = 4, we thall | = £ — gg, and Since zz is = 5 - and gg is = Fi we fhall have zz = Ff fg, an confequently gg + zz = £, and.9* 27k ge Se lees 27 xX gg X E fs a andg= 3X W3(gg x Ane fa 2 rae x 1+ 25. But, by the I ° . e Zz eraet <3 . . . I binomial theorem in the cafe of roots, er: rises ee infinite feries 1 + + Bed 2 zt paa 8 gto I ene Carts Cox fe = Fe Dex S+EExi- BZ c 24 oa Ez? F zt G zi4 F x +- &e ad injinitum, or 1 + —— — + = = ” fs & & & & 2 +- &c ad infinitum. Therefore q will be = 3 x Ql 3 the infinite feries 1 + Br c2t D 2° E x8 F zt0 it Gis — -- &c ad infinitum. yr irae oe Bye jos tige ge Q: Eve 77. Therefore the’ product of the multiplication of g into the expreffion ZB nxt Gx 128 Ez? N zi? 273 ee ae ap = oor st oS Y (z £e gt e° 2 pr eee P oe c x4 D 2 . Seaawill bea <3 ara 3 x the infinite feries 1 + —— — aa + a “it SS 5} 10 1z z a4 6 wit 3 Ey + &e x 246/% x the infinite feriesr - = = = + 2% ge & : & 128 L zt? N zt - ce? I . . mS we cage cee x gl3_X 2X g> X the infinite feries 1 + 2 : 4 6 8 FE zo 12 i ‘e 2 " ad ae ee — ee — ——= + &c x the infinite feries 1 + —= & & & of & 7% F 24 @3° 1x8 se Aa * N 27 & 6 x th Gj fe Se pes “ci 0.8 the infinite feries 1 + Z00 2 B2z ‘ 500 A METHOD OF EXTENDING CARDAN’S RULE BU% cz p2° Ex? | FRt0 G2" 2 APR soem a fe en + Se serge + &c x the infinite feries 1 + = 9 een at ‘ Gx 4 128 ne N z?2 ss i —- + Te ope eee + &c. We muft therefore now proceed. 2 ‘ “ BX c zt p 2° E x® F z!0 G zi ‘ : to multiply the feries 1 + a ae Hie ae ad + &c into 3} ro) & , C Zz E x4 G x9 Le Lmt° N 272 : the feries 1 -- — —-—> + Pra + ears + &c; which may be . . ro} done in the manner following. a. ¥e . . e 4 3 4 The multiplication of the infmite feries 1 + —— —= D z° E x8 Fa" Gazi oF . ‘ ages Wie + ur a oe + &e ad infinitum 5 5 : ty Pas ; C2zz E 24 G26 Iz into the infinite feries 1 + ae + ye 10 12 " Z oy So ee eee taal imei 10 iz & & B 2% cat D 2x E x? F x0 cu 1+-_—-— et 26 = gs gi? 12 &c s C 2% E x4 GB 12° ae N& I -+- —— a &c 4 a g & a oo B 2% c x4 DZ E 2° Bat? Gaz! i pease s bea ae “aece } ge gt - 2 : iat ane ( CZz BCZ C2 CD2 CEZ CF2Z + 04 + <> 6 8 + — &C o pt g g 10 est i E x4 BE 2° cE 2° DE 21° ore & | — cor — &c gt 9° 28 p “apts Sei) hk G2 BGZ CG 2% DG2 : 55 a a 0 1 Pete rae &c fs 128 BI gt? C1 x2 — — + — &c : 8 10 Iz & & 4 at L ute a BL2 & , : — &c Io Iz s ou, : ies Ta he cae &c. a ( Therefore the product of the multiplication of the expreffion 2 /?{g x the : : : C zz E 2* G Lh = : I infinite feries 1 + <= ——> + -Z—~> 4+ — — AS + &c into the co- ) . . 6 . . . @ efficient q will be equal to 6g x the compound feries juft now obtained, to wit, 4 the feries : P t+ ~~ FOR RESOLVING THE CUBICK EQUATION &c, 501 BUS c zt D 2x8 E28 F 21° G 21% I oh ele 4 = Gla a 8 eT 12 2 &c &3 g iz & & C2z BC 24 c2z cpz® CE z?° CF zi2 & — rr et —! eeionenll el o 4 6 8 gio = Hh) Cc és & & = & 8 E x4 BE x® CE 2? DE 2!? E2212 es A 6” aE 8 10 ti giz &e Dy ss My el o G2 BG 28 CG zt DG 2" & eae ast era CR aa ce ee > & 8 & 10 S 12 Iz BIZ cIl2 8 olO + Wie &c s is Lo é rz LZ BL2 silencer ry emer: sree & 8 N zi” pb wanker DEC: giz or, if, for the fake of brevity, we denote this compound feries by the {mall Greek letter A, the faid product will be equal to 6g x the faid compound feries A. Qubni-t. A comparifon between the foregoing compound Jeries % and the compound Jeries obtained above in art. 42. “8. If we compare this compound feries A with the compound feries A ob- tained above in art. 42, we fhall find that there are the following refemblances and differences between them. In the firft place, the firft terms of both thefe compound feriefes are equal to the fame quantity, to wit, 1. ; Secondly, the fecond, and third, and fourth, and fifth, and all the following vertical columns of terms of the latter compound feries A, obtained in the fore- F ‘ : " ‘ an a es Oe A going art. 77, involve in them the feveral fractions ini GP haa . ZS : &c, or the feveral fucceflive powers of the fraction —, juft as the fecond, and 3 third, and fourth, and fifth, and other following vertical columns of terms of the former compound feries A, obtained in art. 42, involve in them the feveral , 0 Raa RA II AMOI : fractions —, => <> Gr Gor Gar OF the feveral fucceflive powers of the frac- tion —. And it is evident that this obfervation will be true of all the following ee terms of thefe two compound feriefes, to whatever number of terms the faid feriefes may be continued, as well as of the terms fet down in art. 42 and 77, a ' B 2% c 24 p-2° E x8 F zr G zi2- becaufe the two fimple feriefes 1 + = anima dita ss ie ee 6 CBZ Ex‘ Gz? 1z ie wi? ge &c and 1 + Be oh ol rT et Eteach i + &c (by the multipli- cation of which the compound feries A is produced) contain in their terms the B S§ very fame powers of the fraction = as the two former fimple feriefes 1 — €eé 502, A METHOD OF EXTENDING CARDAN’S RULE c 54 p 6 E 38 F570 G 5 s C35 E s* G 36 1st Bias N 3?? Pe ee ; 3 “ : Ss — a — &c (by the multiplication of which the former compound feries . . . . 5S A was produced) contain in their terms of the fraction =a Thirdly, the co-efficients of the feveral terms in the compound feries A will be equal to the co-efficients of the correfponding terms, or terms placed in the {ame fituations in the feveral horizontal lines, of the compound feries A, though they will not every where have the fame figns + and — prefixed to them. For thefe co-efficients are the products of the co-efficients of the terms of the fim- ple feriefes, by the multiplication of which into each other the faid compound feriefes A and A are produced: and therefore, as the co-efficients of the powers of B 2% c.zt D2 . ZB os : the fraction pore the terms of the two fimple feriefes 1 + ct ae E 2? F zte G zi? C zz E xt G 26 12° Lae N aie ee od 8 FS %) 12 + &c (by the multiplication of which the compound feries A is produced) are 8c and ee -+- Wc an Sater ot 6 8 10 I2- . . . SS 6 the very fame with the co-efficients of the powers of the fraction — in the terms BSS 6 54 D 5® ES of the two fimple feriefes 1 — —— me ae css. Est G 56 rs? LE st? nist eee” De ea at ST ee which the compound feries A is produced) it follows that the feveral co-efficients of the terms of the compound feries A muft be the fame combinations of the fe- veral original co-efhcients B, C, D, E, F, G, I, L, N, &c, as the co-efficients of the correfponding terms of the compound feries A are of the fame original co-efiicients, and confequently that they muft be equal to the faid co-efficients of the correfponding terms of the compound feries A. Q, E. Ds Fourthly, the figns + and — that are prefixed to the terms contained in the third, and. fifth, and feventh, vertical columns of the compound feries A, are the fame as thofe which are prefixed to the correfponding terms contained in the third, and fifth, and feventh vertical columns of the compound feries A, as will be evident upon the infpection of the faid two compound feriefes. And the fame fimilitude will take place between the figns prefixed to the terms contained in the following odd vertical columns, after the feventh, of the compound feries a, and thofe prefixed to the correfponding terms of the following odd vertical columns, after the feventh, of the compound feries A; as we fhall prefently en- deavour to make appear. And, fifthly, the figns + and — that are prefixed to the terms contained in the fecond, and fourth, and fixth vertical columns of the compound feries A are every where contrary to thofe which are prefixed to the correfponding terms of the fecond, and fourth, and fixth vertical columns of the compound feries A; as will be evident upon the infpection of the faid two compound feriefes. And the fame contrariety will take place between the figns to be prefixed to the terms contained in the following even vertical columns, after the fixth, of the compound ; FOR RESOLVING THE CUBICK EQUATION &¢, 503 compound feries A and the figns to be prefixed to the correfponding terms of - the following even vertical columns, after the fixth, of the compound feries A; as we fhall now endeavour to make appear. 79. Inthe compound feries A, or the produc of the multiplication of the -s B ss c st pD 5° E 58 Ad G 5% : Ts eS a — Sec (of which all the terms, after the firft term 1, are marked with the fign —) into the feries 1 —““ — ee Es gs rs? 550° N 57? 6 pants a) el2 term 1, are likewife marked with the fign —) fet down above in art. 42, it is evident that all the terms, after the firft term 1, of the firft, or higheft, hori- zontal row of terms muft be marked with the fign —, to whatever number of terms the faid horizonta! row of terms may be continued ; and it is evident like- wife, that in all the following horizontal rows of terms in the faid produét the firft terms of the faid horizontal rows (being the produéts of the multiplication of css Es G 56 hy iy ba Me : ee eS ore) muft be marked with the fign —, but that all the following terms of the faid horizontal rows, to what- ever number of terms the faid rows may be continued, will be marked with the fien +, becaufe they are the products of the multiplication of the feveral terms — &c (of which all the terms, after the firft BSS c s# p 5° E 58 F sto G 5* 8 fe Fi dark: — es rs ae _ ace ae? ——w a? Cc, as mu tip icands, into the Css E 54 ex? 158 Es N 5? yl feveral eee bir gah Poe or ere asa Taal? &c, as multiplica- tors, and, wherever the fign — is prefixed to both the fators of any multipli- cation, the fign + is to be prefixed to the product. But the fecond, and third, and other following terms of the firft, or higheft, horizontal row of terms in the ~ faid product, are the higheft terms of the fecond, and third, and other follow- ing vertical columns of terms in it; and the firft terms of the fecond, and third, and other following horizontal rows of terms of the faid product are the loweft terms of the fecond, and third, and other following vertical columns. of terms inthe fame. ‘Therefore, the higheft and the lowe(t terms in the fecond, and the third, and every following vertical column of the faid product will be marked with the fign —, and all the intermediate terms in the faid vertical columns between the higheft and the loweft terms will be marked with the fion +. rhele will be the figns to be prefixed to the feveral terms of the compound feries A, or the product, fet down above in art. 42, of the multiplication of the BSS c 54 seth E 58 F ste sea feries I — PREPAID Pe aes ca ear — &c ad infinitum into the fe- ‘ C55 E 54 G 5° rs es yar age #2 ; es oe ee ae &c ad infinitum. We mutt now enquire what figns are to be prefixed to the feveral correfponding terms of the compound feries A, or the product, fet down in art. 77, of the multiplication . : B Uz c x4 p 2° E2° zr en itor of the feries 1 + age + Fr tier: am oly sierra bry + &c ad infinitum into 504 A METHOD OF EXTENDING CARDAN’S RULE : ; C 23% E 24 G28 128 L2xteo alt into the feries 1 + —_—— 3 - aa — my + aa —— + &c ad in- rey jinitum. So. Now the firft, or higheft, horizontal row of terms in this produc, to wit, " B 2% c zt px E28 F zt id I + — —~ 4 e+ ce, ‘confifts of the very fame terms as the multiplicand, being the srodeee of the multiplication of the faid miultiplicand into 1, which will make no change either in the terms of the mul- tiplicand themfelves, or in the figns to be prefixed to them. Therefore in this firft, or higheft, horizontal row of terms, the fecond, and fourth, and fixth, and eighth, and tenth, terms, and all the following even terms (to whatever number of terms the faid horizontal row of terms may be continued) will be marked with the fign +, and the third, and fifth, and feventh, and ninth, and eleventh, terms, and all the following odd terms (whatever be their number) will be marked with the fign —. But the fecond, and fourth, and fixth, and other fol- lowing even terms of the firft, or higheft, horizontal row of terms in the faid product, are the firft, or higheft, terms of the fecond, and fourth, and fixth, and other following even vertical columns of terms in the faid product; and the third, and fifth, and feventh, and other following odd terms of the faid firft, or highett, horizontal row of terms in the faid produ& are the firft, or higheft, terms of the third, and fifth, and feventh, and other follawing odd vertical co- j lumns of terms in the faid product. Therefore the firft, or ‘highett, terms of the fecond, and fourth, and fixth, and other following even vertical columns of the faid produé will be marked with the fign + ; and the firft, or-higheft, terms of the third, and fifth, and feventh, and other following odd vertical co Jumns of terms in the faid produc will be marked with the fign —., n= ee) a 81. We will next enquire what figns muft be prefixed to the laft, or lowett; terms of the feveral vertical columns of the faid product. Now, becaufe the firft term of the feries which is the multiplicand in the multiplication fet down above in art. 77, is 1, and the fecond, and fourth, and — fixth, and other following even terms of the feries which is the multiplicator in that multiplication, are marked with the fign +, and the third, and fifth, and feventh, and other following odd terms of the faid multiplicator are marked with the fign —, it follows that in the fecond, and fourth, and fixth, and other follow- ing even horizontal rows of the faid produa, the fins -+- and — that are to be prefixed to the feveral terms in the faid rows will be the fame as thofe of the correfponding terms. of the multiplicand, and confequently the two firft terms of each of the faid even horizontal rows will be marked with the fign +, and ~ the third, and fourth, and fifth, and fixth, and other following terms of the faid even horizontal rows will be marked with the fign —~ and the fign + alter- nately ; and it follows alfo, that in the third, and fifth, and feventh, and other following odd horizontal rows of terms, the figns of the terms will be contrary to thofe of the correfponding terms of the multiplicand, and confequently the two firit terms of each of the faid odd horizontal rows of terms will be marked with the FOR RESOLVING THE CUBICK EQUATION &c. 505 the fign —, and the third, and fourth, and fifth, and fixth, and other following terms of the faid odd horizontal rows of terms will be marked with the fign + and the fign — alternately. But the firft terms of the fecond, and third, and fourth, and other following horizontal rows of terms are the loweft terms of the fecond, and third, and fourth, and other following vertical columns of terms. Therefore the loweft terms of the fecond, and fourth, and fixth, and other fol- lowing even vertical columns of terms in the faid prody& will be marked with the fign + ; and the loweft terms of the third, and fifth, and feventh, and other pes odd vertical columns of terms in the faid product will be marked with the fign —. 82. It appears therefore that in the third, and fifth, and feventh, and other following odd vertical columns of the faid compound feries A, or product, fet down in art. 77, both the higheft and the loweft terms will have the fign — prefixed to them, as is the cafe in the compound feries A, or the product fet down above in art. 42 ; and that in the fecond, and fourth, and fixth, and other following even vertical columns of the faid compound feries A, or produc fet down in art. 77, both the higheft and loweft terms will be marked with the fign +, which is contrary to the fign which is prefixed to the higheft and loweft terms of the fecond, and fourth, and fixth, and other following even ver- tical columns of terms in the compound feries A, or product fet down above in art. 42, It remains that we enquire into the figns that are to be prefixed to the feveral intermediate terms between the higheft and the loweft terms of the feve- ral vertical columns of terms in the faid compound feries a, or product fet down in art. 77. 83. Now, wherever the higheft term of one of the vertical columns of the product fet down in art. 77 is marked with the fign — (which has been fhewn to be the cafe in the third, and fifth, and feventh, and all the following odd vertical columns), all the intermediate terms of the faid column between the higheft term and the loweft will be marked with the fign +, becaufe they will be the product of two factors which are both marked with the fame fign, to wit, firft with the fign +, then with the fign —, then with the fign +, and then with the fign — again, and fo on alternately. Thus, for example, in the fe- : » ; < Ge : - venth vertical column of the faid produét, the higheft term is — Fea » which is zil2 ~ the produét of the multiplication of — ==, the feventh term of the multipli- Iz cand, by 1, the firft term of the multiplicator; and the fecond term of the faid cr", which is marked with the fign +, becaufe it is the g* vertical column is + F wre = > a the fecond term of the multiplicator, which terms are both marked with the the fixth term of the multiplicand, by product of the multiplication of 12 fign + ; and the third term of the faid vertical column is + a » which is Vou. Il. gu] marked 506 A METHOD OF EXTENDING CARDAN’S RULE marked with the fign +, becaufe it is the product of the multiplication of — the fifth term of the multiplicand, by iad the third term of the multiplicator, which terms are both marked with the fign —; and the fourth term of the faid DG zi? 12 & vertical column is ++ , which is marked with the fign +, becaufe it is the product of the multiplication of =, the fourth term of the multiplicand, 6 . . : sf by ——, the fourth term of the multiplicator, which terms are both marked with & Crzt » which is in like manner marked with the fign +-, becaufe it is the ‘produ& of the the fign + 5 and the fifth term of the faid vertical column is + 12 . . . 4 . . . 8 multiplication of =, the third term of the multiplicand, by ~>, the fifth term of the multiplicator, which terms are both marked with the fign —; and the fixth term of the faid vertical column, or the loweft term but one, of the faid column, or the laft of the intermediate terms of the faid column between ‘its higheft and its loweft terms, is + = which is alfo marked with the fign +, IZ y-3 becaufe it is the product of the multiplication of a the fecond ‘term of the 10 . . . multiplicand, by —{, the fixth term of the multiplicator, which terms are both Io 9 marked with the fame fign +. And thus it appears, that all the intermediate terms between the higheft and loweft terms of the faid feventh vertical:colamn muft be marked with the fign ++, becaute the two factors, by the multiplication of which they are produced, are always marked with the fame fign, whether -- or —. Andi it is eafy to fee, from the alternate fucceffion of the figns +. and — to each other, in both the multiplicand and multiplicator, that the fame thing muft take place with refpe& to the intermediate terms of any other vertical co- lumn of terms of which the higheft term is marked with the fign —, that is, of any other odd vertical column. We may therefore conclude that in all the odd | vertical columns of the compound feries A, or the product fet down in art. 77 (to whatever number of terms the faid product may be continued), the higheft and loweft terms of the faid vertical columns will be marked with the fign —, and.all the intermediate terms will be marked with the fign +, and confequently (by what is fhewn in art. 79) that the figns of the feveral terms of all the faid odd vertical columns of this product, or of the compound feries A, will be the fame with thofe of the correfponding terms of the like odd vertical columns of terms _ in the product fet down above in art 42, or in the compound feries A; or that the fourth obfervation made above in art. 78 is univerfally true of the terms:con- tained in all the odd vertical columns of the faid two products, or compound fe- riefes A and A, fet down in art. 42 and 77 after the feventh vertical columns (to whatever number of columns the faid feriefes may be continued), as well as of the terms contained in their third,’and fifth, and feventh vertical columns. Q. E. D. 6 84. In FOR RESOLVING THE CUBICK EQUATION &c. $07 84. In the feveral even vertical columns of the compound feries A, fet down in art. 77, the figns + and —, that are to be prefixed to the terms of the faid columns, will be « every where contrary to thofe which are to be prefixed to the corref{ponding terms of the like even vertical columns of the compound feries A, fet down above in art. 42. For it has been already fhewn in art. 80 and 81, that the higheft and loweft terms of the fecond, and fourth, and fixth, and all the following even vertical columns of terms in the faid compound feries A (to whatever number they may be continued) will be marked with the fien +, which is contrary to that with which the higheft and loweft terms of the fecond, and fourth, and fixth, and other following even vertical columns of terms, in the compound feries A, are marked. And all the intermediate terms of the fecond, and fourth, and fixth, and other following even vertical columns of the compound feries A, between the higheft and the loweft terms of the faid co- lumns, muft be marked with the fign —, becaufe they will be the produéts of two factors which are marked with different figns -- and-—. Thus, for ex- ample, in the fixth vertical column of the faid PHO the higheft term is + the fixth term of the Io 9 =~, which is the product of the multiplication of + - 10 multiplicand, by 1, the firft term of the multiplicator; and the fecond term of ; ; : GE ate oa . the faid vertical-column is — ; bee is marked with the fign —, becaufe Ze it is the product of the multiplication of = roe the fifth term of the multiplicand, which is marked with the fign —, by ox the fecond term of the multiplica- tor, which is marked with the fign + ; and the third term of the faid vertical 10 ° e . . ’ column is — = , which is alfo marked with the fign —, becaufe it is the pro- duct of the multiplication of a the fourth term of the multiplicand, which is marked with the fign +, by Ex! the third term of the multiplicator, which is marked with the fign — ; and the fourth term of the faid vertical column is — CGE2*° 10 ES , which is alfo marked with the fign —, becaufe it is the product of the x¢ : he ay multiplication af —. ae the third term of the multiplicand, which is marked with the fign —, by = *, the fourth term of the multiplicator, which is marked with the fign + ; and sie fifth term of the faid vertical column, or the loweft term but one of the faid column, or the laft of the intermediate terms of the {aid co- which is alfo marked lumn between the higheft term and the loweft, is — eas gi? 3 with the fign —, becaufe it 1s the product of the multiplication of ee the fe- cond term of the multiplicand, which is marked with the fign +, by Sa the fifth term of the multiplicator, which is marked with the fign —. And thus it appears that all the intermediate terms between the higheft and the loweft terms Cai ter of 508 “A METHOD OF EXTENDING CARDAN’S RULE of the fixth vertical column muft be marked with the fign —, becaufe the two factors, by the multiplication of which they are produced, are always marked with contrary figns. And it is eafy to fee, from the alternate fucceffion of the fions -- and — to each other in both the multiplicand and the multiplicator, that the fame thing muft take place with refpeét to the intermediate terms of every other vertical column of which the higheft term is marked with the fign --, that is, of every other even vertical column. We may therefore conclude, that in all the even vertical columns of the compound feries A, or the produc fet down in art. 77 (to whatever number of terms the faid product may be con-’ tinued) the higheft and loweft terms will be marked with the fign +, and all the intermediate terms wiil be marked with the fign —, and confequently (by what is fhewn in art. 79) that the figns of the feveral terms of all the faid even vertical columns of terms in this produ, or compound feries a, will be every where contrary to the figns of the corre{ponding terms of the like even vertical columns of terms in the compound feries A, or the product fet down above in art. 42; or that the fifth obfervation made above in art. 78 is univerfally true of the terms contained in all the even vertical columns of terms of the faid two products, or compound feriefes A and A, fet down above in art. 42 and 77 (to whatever number of columns of terms the faid feriefes may be continued), as well as of the terms contained in their fecond, and fourth, and fixth vertical columns. Q-TED: 85. If the foregoing compound feries a, fet down in art. 77, be actually mul- tiplied into 6g, the product thence arifing will be the following compound fe- ries, tO Wit, 6 B2z 6 c2* 6 px 6528 6 Fx! 662% RP Signa imall ate ues nay os Pie ee ee 6 c zz 6 Bc x4 6 c2z6 6cp28 6 CE 27° 6cr2% & au @ am Pe CERNE TS ghee 9 Tm a 6E x4 6 BEZ® 6 cE x8 6 DE x 6 E2z12 & pt Tips hiigaC ime = tee 662° 6 BG 2 6 cG x? 6 DG 27” 33 g7 pre a ge sik&c 6128 6 BI zt? 6 cI 2! i igre ayes "rma #:.{° 6 L zt 6 BL 2 6 + + — —&c - a mi? BETH &c ; which, for the fake of brevity, we will denote by the {mall Greek letter 7. Then will the product of the multiplication of the expreffion 2 vv , (¢ x the feries E24 G28 12° L zt N zi? poe ; ek czz pe et rene -+- &c ad infinitum into the co- 5 4 - efficient g be equal to the compound feries =. | We muft now compare this compound feries x with the compound feries II obtained above in art, 43. yy: Om= FOR RESOLVING THE CUBICK EQUATION &c. 509 A Comparifon between the foregoing compound feries mw and thé compound fertes Il obtained above in art. 43. 86. The compound feries TI is equal to 6e¢ X the compound feries A; and the compound feries z is equal to 6g x the compound feries A. - Therefore the figns + and — which are to be prefixed to the terms of the compound feries II will be the fame with the figns which are prefixed to the correfponding terms of the compound feries A; and the figns + and —, which are to be pre- fixed to the terms of the compound feries 7 will be the fame with the figns which are to be prefixed to the correfpondent terms of the compound feries a; becaufe the multiplications of the terms of the two compound feriefes A and A into 6¢ and 6g cannot make any change in the figns of the terms which are multiplied. It follows therefore that there will be the fame fimilitude and the fame differences between the figns + and — that are to be prefixed to the feveral terms of the two compound feriefes II and w (which are equal to 6e x A and 6g x A) as there are between the figns which are to be prefixed to the terms of the two for- mer compound feriefes A and a. But it has been fhewn in art. 78, 79, 80, 81, 82, 83, 84, that the figns + and — that are to be prefixed to the feveral terms of the third, fifth, feventh, and other iollowing odd vertical columns of terms in the compound feries A are the fame with thole which are to be prefixed to the correfponding terms of the third, and fifth, and feventh, and other following odd vertical columns of terms in the compound feries A ; and that the figns -+ and — which are to be prefixed to the feveral terms of the fecond, and fourth, and fixth, and other following even vertical columns of terms in the compound feries A are uniformly contrary to thofe which are to be prefixed to the corre- {ponding terms of the fecond, and fourth, and fixth, and other following even vertical columns of terms in the compound feries A. It therefore follows that the figns ++ and — which are to be prefixed to the feveral terms of the third, and fifth, and feventh, and other following odd vertical columns of terms in the compound feries 7 will be the fame with thofe which are to be prefixed to the correfponding terms of the third, and fifth, and feventh, and other following odd vertical columns of terms in the compound feries IT; and that the figns + and —, which are to be prefixed to the feveral terms of the fecond, and fourth, and fixth, and other following even vertical columns of the compound feries + will be uniformly contrary to thofe which are to be prefixed to the correfpond- ing terms of the fecond, and fourth, and fixth, and other following even vertical columns of the compound feries IT. 87. It has been fhewn in the third obfervation of art. 78, that the co-effi- cients of the feveral terms of the compound feries A are equal to the co-efficients of the correfponding terms of the compound feries A. Therefore, if the.co- efficients of all the terms of both thefe feriefes A and A be multiplied by the fame number, the new numbers, or co-efficients, thereby produced in the one feries will be equal to the > corre{ponding new numbers, or co-efficients, there- by produced in the other feries. But the co-efficients of the terms of the com- pound feries Il are produced by the multiplication of the co-efficients of the terms 510 A METHOD OF EXTENDING CARDAN’S RULE terms of the compound feries A by 6, becaufe the feries II is = 6¢ X the fe- ries A; and the co-efficients of the terms of the compound feries 7 are produced by the multiplication of the co-efficients of the terms of the compound feries a by the fame number 6, becaufe the feries x is = 6g x the feries A. Therefore: the co-efficients of the terms of the compound feries 7 will be equal to the co- efhcients of the correfponding terms of the compound feries IT. 88. It appears therefore from the two laft articles, that the co-efficients of the feveral terms of the compound feries 7 are equal to the co-efficients of the cor- refponding terms of the compound feries Il; and that in the third and fifth, and feventh, and other following odd vertical columns of terms in the come pound feries 7, the figns + and — that are to be prefixed to the feveral terms of the faid columns will be the fame with thofe which are to be prefixed to the correfponding terms of the third, and fifth, and feventh, and other following odd vertical columns of terms in the compound feries 1; and, laftly, that in the fecond, and fourth, and fixth, and other following even vertical columns of terms in the faid compound feries 7, the figns + and —, that areto be prefixed to the feveral terms of the faid columns, will be refpeétively contrary to thofe which are to be prefixed to the correfponding terms of the fecond and fourth, and fixth, and other following even vertical columns of terms in the compound feries II. Now from hence it follows that, if all the vertical columns of terms in the compound feries II be reduced to fingle terms, by the addition, or fubtrac- tion, of their feveral members according to the marks +- or — which are pre- fixed to them, and the compound feries II be thereby converted into a fimple feries, or feries confifting of fingle terms; and, if all the vertical columns of terms in the compound feries 7 be in like manner reduced to fingle terms, by the addition and fubtraction of their feveral members according to the figns + and — that are prefixed to them, and the faid compound feries 7 be thereby con- verted into a fimple feries, or feries confifting of fingle terms;—I fay, it follows that, if thefe two compound feriefes II and x be fo converted into fimple feriefes, the co-efficients of the fecond, and third, and fourth, and all the following terms of the fimple feries which is equal to the compound feries 7, will be refpectively equal to the co-efficients of the fecond, and third, and fourth, and other follow- ing correfponding terms of the fimple feries which is equal to the compound feries II; and the figns + and — that.are to be prefixed to the third, and fifth, and feventh, and other following odd terms of the fimple feries which is equal to the compound feries 7 will be the fame with thofe which are to be prefixed to the third, and fifth, and feventh, and other following odd terms of the fimple feries which is equal to the compound feries IT; but the figns + and — that are to be prefixed to the fecond, and fourth, and fixth, and other following even terms of the fimple feries which is equal to the compound feries 7 will be refpectively con- ‘trary to thofe which are to be prefixed to the correfponding terms of the fimple feries which is equa] to the compound feries I. 89. It has been fhewn above in art. 60, that the compound feries II (which is equal to y? in the equation 7>— gy = 7) is equal to the fimple feries 6¢ — PSs é FOR RESOLVING THE CUBICK EQUATION &c, 51 Pss ast R 5° $54 mg tO ats Pye ae — ec ad infinitum. At follows therefore from the foregoing art. 88, that the compound feries x will be equal to the fim- P 2% qz4 R26 s 28 mT wtO V xi nes + &c ad infinitum, ple feries 6g + ob haba st cae ai, ew a go. But it has been fhewn in art. 85, that the product of the multiplication C Bz E x4 of the tranfcendental expreffion 2 4/3 (g x the infinite feries 1 + “% 22 &§ ge G28 12° conte N ai? AED : ; : Set se ee t+ «Cc ad infinitum into g (the co-efficient of x in the equation x* — gx = ¢) is equal to the compound feries 7. Therefore the product arifing.from the faid multiplication of the expreffion 2? {g x the in- ‘ 4 CR E x4 G xe 12° 1 N 272 By , finite feries 1 + Pear i + sehr 4+ Epo IH -- &c ad infinitum in- P 2% Qz4 to the co-efficient ¢ will be equal to the faid ‘ample feries 6 g + —=— = & & R 6 5 8 Tv 10 Iz x . = os a vg nolo + &c ad infinitum. g1. We have juft now feen that, if the tranfcendental expreffion 2? {¢ xX the 0 . : C 2% Ex* G2° 12° Lae N zt cabot infinite feries 1} —— — —> + — —-— 4+ — — et Oe ad infinitum be multiplied into ¢ (the co-efficient of w in the equation x? — gy = #) the pro- duct arifing from the faid multiplication will be equal to the fimple feries 6g + 4 6 S 8 10 Iz i r a “! ae — _—— + — — 4+ &c ad infinitum. And it was thewn above in art. 75, that the cube of the fame tranfcendental exprefflion 2 «/?(¢ x E xt G2 128 Lio N zl? 4 Bec ad inf —- —- oo eee ULo Fad g pai Ree iia So hut a 4 . P BZ az RZ $2 T% Vv zi tum is equal to the fimple feries 8g -+ —- ——— + — — — as) ll q P a te g? g F g! "2 ore . ‘ . Piss, z + &c.ad infinitum. But the fimple feries 8g + oa — =. +- — ao a rate Nie Gaus é : ° a a oe + &c ad infinitum is.greater than the fimple feries 6g 4-4 — — —- 2 R 2S S <3 T zlo Vv giz the infinite feries 1 -+- = _— + .&c, and the difference is 8 g — 6g, or 2g, or ET ge 2x ~, or ¢. ‘Therefore the cube of the faid tranfcendental expreffion will be ‘greater than the.product.of its multiplication into the co-efficient g, and the ex- .cefs-ordifference will be equal to ¢, or the abfolute term of the equation «* — qx = t. Therefore the faid tran{cendental expreffion 2 /*(g x the infinite fe- . C zz E24 G 2° 12° Lint? Nt Epatgh Mest -- ——— 4+ ee ee teres s+ CC ed infinitum mult be equal.to the root w of the faid equation, agreeably to the aflertion made above in art, 36. Q. .E. D. | End of the Demonftration of the Propofition, or Theorem, laid down in art. 36. 99 Fae gu 4 Re- 512 A METHOD OF EXTENDING CARDAN’S RULE A Recapitulation of the Contents of the foregoing Demon- ftration of the Propofition laid down above in art. 36. 92. The foregoing demonftration of the propofition laid down above in art. 36 is in itfelf very fubtle and intricate, and therefore has been fet forth in the foregoing articles at very great length. But now, after it has, I hope, been ren- dered intelligible and fatisfactory to the attentive reader, by the full expofi- tion that has been given of it, he will probably be glad to fee the matter of it comprefied into a fmaller compafs, by which the connection of the feveral fteps of which it is compofed will be more clearly and eafily perceived. This there- fore I fhall now proceed to do in the following articles. 93. It has been fhewn in art. 25 that_y, the root of the cubick equation y? — gy = r (in which the abfolute term 7 is greater than iat, or at is greater than £) is equal to the tranfcendental expreffion 2 4/? (e x the infin feries 1 — page ral Ta a7 Fe er Re es &c ad infinitum, in which all the terms, after the firft term 1, are marked with the fign —. Let this infinite feries be de- noted by v; and then we fhall have vy = 2 +/(e x the feries v. 94. In this expreffion 2 ? [e x the feries v the letter ¢ is put for =, or half . p 3 the abfolute term 7 of the equation y3 — gy = 1, and ss is = os bre or €@ — Te age ita 1 le Pain ae ss aye and confequently ss + 7 is = ee, and = 1S. =. €2.=— Ss. = 66 x TE ve 21 I and gis 27 x ce x [r—, and gis = 3 x eo x |r — £3 = (by 2 the refidual theorem in the cafe of roots) 3 x e 3 X the infinite feries 1 — BSS c 54 p s° E 58 B, 51°, G st? é 95. Since y is = 2 /?(e x the feries v, it follows that 3 will be = 8e x the feries v?, which will be a compound feries, or feries confifting of feveral dif- ferent horizontal lines, or rows, of terms placed one under another. And the firft term of this feries v? muft evidently be 1, becaufe the firft term of the feries v is 1. Call this compound feries [. And we fhall have y?, or 8e x v3, = 8e x the compound feries I. 96. Let all the terms of the compound feries T be actually multiplied into 8 e; and we fhall thereby obtain another compound feries, of which the firft term will be 8 ¢ (becaufe the firft term of the compound feries I is 1), and which. will be equal to y3. Let this fecond compound feries be called 4, And we fhall then have y* = the compound feries A. 97. Now * Bd FOR RESOLVING THE CUBICK EQUATION &c. 513 x ’ . ’ 4 97. Now let the expreflion 2 4? (e x the infinite feries 1 — m a kh et Gs, scit1s® Est? W st aaah SEY CR ae ae &c (which 'is'equal to y) be multiplied into the ex- 2 prefon 3.x ¢ 3,.X,the infinite feries\t — —-—= = = erp &c.ad infinitum, which is equal to the co-efficient g._ And we fhall have el . 2 . Qa=2/'fe x3 « e 3 X the produa of the faid two infinite feriefes, which product will be a certain compound feries, of which the firft term will be 1. Let 2 this compound feries be called A. Then will gy be (= 2/3 ex3xe3 x I ey 2 the compound feries A = 2 x ¢ 3 X 3.x ¢ 3 X the compound feries A) = 6¢ X the compound feries A. 98. Let all the terms of the compound feries A be actually multiplied into 6¢; and we fhall thereby obtain another compound feries, of which the firft term will be 6¢ (becaufe the firft term of the compound feries A is 1), and which will be equal togy. Let this new compound feries be called II. And we fhall then have gy = the compound feries I. 99. Since y3 — gy is = 7, and y3 is = the compound feries A, and gy is = the compound feries IT, it follows that the excefs of the compound feries A above the compound feries II will alfo be equal to 7, that is, A — TI will be = r= 2x = =2¢e. But the excefs of the firft term of the compound feries A (which is 8e) above the firft term of the compound feries II (which is 6¢) is alfo = 2é¢, Therefore the excefs of the whole compound feries A above the whole com- pound feries II is equal to the excefs of the firft term of the former compound feries above the firft term of the latter compound feries ; that is, A — II is = 8% — 6e. Confequently (adding 6 eto both fides) we fhall have A + 6¢ — 1 = 8¢. But the compound feries A is lefs thanvits firft term 8 e, becaufe the fecond term of it is marked with the fign —. Therefore in the laft equation A + 6¢ — Tl = 8e it is poffible to fubtract A from both fides. Luet it be fo fubtracted ; and we fhall then have 6e.— II = 8e — A. too. Now let the compound feries 8¢ — A be reduced to a fimple feries, or feries confifting of fingle terms, by making the feveral multiplications, additions, and fubtractions that are neceflary for that purpofe; and let the fimple feries, to ast R 56 hich it is thereby reduced, be ** ae ap ee EE ee send which ut is thereby reduced, be — + | +-- +> +-—-+ar + &c a4 : 2 ee ; 2 . we SE aq s+ R 56 Ss T09 vst infinitum. ‘Then will the faid fimple feries Wee by ea Ey IPL, ay 4 &c ad infinitum be equal to the compound feries 6e¢ — II. And, becaufe me OL. II, ah 8 this 514 A METHOD OF EXTENDING CARDAN’S RULE : ; an wale Qqs* this equality between the fimple feries Het hy OE ROS ie + &c ad infnitum, and the compound feries 6 ¢ — II takes place in all the dif- . . rT 3 , ferent relative magnitudes of ss and ee, or of “ ie and me and confequently rr é d q3 rr gq when ri approaches as near as we pleafe to an equality with = and pitt 3 3 approaches as near as we pleafe to & — £, or to o, and = approaches as near 27 83 ee ~ re) : . . PSS as we pleafe to — or too, it follows that every term in the fimple feries = - 3 qa s4 R 5° Vv 53% eit pet steer vn - -- > + —— + &c ad infinitum will be equal to the corref- ponding vertical column of terms, or column of terms involving the fame powers of ss and ¢ in their numerators and denominators, in the compound fe- ries 6¢— II. And therefore, if the compound terms, or vertical columns of terms of the faid compound feries 6 e — II be reduced to fingle terms by making the feveral multiplications, additions, and fubtractions, that are neceflary for that : . . . : P SS purpofe; the fimple feries thence arifing will be the foregoing fimple feries — qst ws? s 58 Phi v st atop é us Pie iL ot Ane aenieirees ottSae ad infinitum, which arofe from a fimi- lar reduction of the terms of the compound feries 8 e — A. ; : es eT Qs Rs $58 zs? 101. Since 8¢ — Ais equal to the fimple feries — + = += +7+—> + oe -- &c, we fhall have 8¢ =A + the faid feries, and A = 8e — the faid ; ; PSS s# nue a Fs , feries — the feries 8 ¢ ——— — = ——--—— Caen — &c. There- e e é e7 e? i fore y? (which is equal to the compound feries A) will alfo be equal to the feries P ss o25 R 5® s 5° Tst° v si ; A Pareto SW apt ary Sear: ibe rw pmcay yy a P ss Q 54 ns ss®crri or bre, Fa oe ren er memes = — + &c ad infinitum, we fhall have 6¢ = P ss ast R 56 353 T st° vis P SS fore the product gy (which is = HM) will alfo be equal to the feries 6¢ — = — ast R 56 s 58 © iss v si? sr iheretetaae asain ae — &c ad infinitum. 103. It appears therefore that y? is equal to the feries 8e — — — = —— s 5° oe Visor WPT : : pe a — &c ad infinitum, and that gy 1s equal to the feries 6e a3: wr ss é FOR RESOLVING THE GCUBICK EQVATION &c. . 515 P ss 5 R 56 s 58 T 510 vst Lee PEE ety — = Se oa — &e aad infinitum, which is lefs than ‘ P ss 54 R 55 $58 p st v si : ‘i sie former feries 8-6—— at ee ee ee eed ein e 03 e e7 e? Be tum, by the quantity, or difference, 8¢— 6¢, or 2¢, or 2 X =, or 7, as it ought to be, agreeably to the original equation y3— gy =r. 104. We then endeavoured to prove that the tranfcendental expreffion 2 /? [¢ : : . C 2% E 24 G2 128 Ex N 2%? : X the infinite feries 1 ceranh he para tet hice trmerspae ee? ad in- : . . . z . Jinitum (in which g is equal to =, or half the abfolute term of the equation x* Z ch et oP ¢ 1 to th t x of hae are aa; gg) was equal to the roo the fame equation. And the method we took to prove the faid equality, was to fhew that, if the faid tranfcendental expreffion was fubftituted inftead of « in the compound quantity «3 — gx (which forms the left-hand fide of the faid equa- - tion) it would make the faid compound quantity equal to the abfolute term ¢, or that, if the faid expreffion were to be cubed by multiplying it twice into it- felf, and were likewife to be multiplied into the co-efficient 7, the faid cube would be greater than the faid product, and the difference would be equal to the abfolute term ¢, Thefe fubftitutions, and the reafoning grounded on them, are as follows. — qx = t, and zz is equal to \ e ° ° ° 4 105. Let us, for the fake of brevity, denote the infinite feries 1 + nit — G 2° 12° L xt° N 2!2 MPSA + — = t+ tse aed infinitum, by the letter w, and the pro- Bick bf the multiplication of 2 3 (¢ into the faid feries by the letter m. Then it will be neceffary for us to prove that the tranfcendental quantity m will be equal to the root » of the equation x* — gx = ¢, or that, if we raife the tranf- cendental quantity m to its cube, fo as to obtain the quantity m, and alfo mul- tiply m into the co-efficient g, or its value expreffed in powers of g and z, fo as to obtain the value of the produ&t q x m, the faid cube of m will be greater than the faid product gm by the quantity, or difference, 2g, or 2 X ~ st Ob fy the abfolute term of the equation «3 — gx = ¢; from which it will follow that the faid tranfcendental quantity ™ is equal to x. 106. Now, fince m is = 2 «/?(¢ x the feries w, it follows that m? will be = 8¢ x the feries «3, which will be a compound feries, or feries confifting of feveral different horizontal lines, or rows, of terms placed one under another. Call this compound feries y. And we fhall have m* = 8g x the compound feries y. : : C 2% E x* G 2° 128 L2zt? 107. Since the terms of the feries w, or 1 ++ —- —— + ——-> 4+ —— i ; 3 eriipups* Insbisigiph go) 128 — one +- &c ad infinitum, have the fame co-eflicients C; E, G, I, L, N, &c, 70 2 as 516 A METHOD OF EXTENDING CARDAN’S RULE ' . : : J css E 5# G56 rs* as the corre{ponding terms of the feries.v, or 1 ~~—~ —= > —= a Bin? N 53% la Nr infinitum (though differently connected together by the eh? ei figns + and —), it follows that the co-efficients of the feveral terms of the compound feries w*, or y, will be equal to the.co-efficients of the correfpond- ing terms of the compound feries v3, or I’, both fets of terms being fimilar combinations, or products, of the fame original factors, or co-efiicients, C, E, G, 1, L, N, &c.. And from the continued repetition of the fign — in the fe- . eo C 5s E 54 Gist Siti we cond and other following terms of the feries v, or r — — — — — — — =| . ee e* e wie 1” gl? N giz . — <> — sr — &c, and the alternate fucceflion of the fign + and the fign 4 ; . - CRE Ex4 — in the fecond and other following terms of the feries w, or 1 + — aa Ser 1 G pad I 28 L glo ? N 2% + ——— + —— + — + &c, it follows that, in the third, and fifth, and 6 8 to. g* feventh, and other following odd vertical columns of terms of the compound feries w*, or y, the figns ++ and — that are to be prefixed to the terms of the faid columns will be exaétly the fame with thofe which are to be prefixed to the corre{ponding terms of the third, and fifth, and feventh, and other following odd vertical columns of terms.in the compound feries v*, or I'; and that in the fe- cond, and fourth, and fixth, and other following even vertical columns of terms in the compound feries w*, or y, the figns + and —, that are to be prefixed to the terms of the faid columns, will be every where contrary to thofe which are to be prefixed to the correfponding terms of the fecond, and fourth, and fixth, and: other following even vertical colunins of terms in the compound feries v3, or iF. And, as to the firft terms of thefe two compound feriefes v? and w’, or TP and y, they will be equal to each other, becaufe each of them will be equal to 4, 108, Let all the terms of the compound feries w?, or y, be multiplied by 8 g. And we.-fhall thereby obtain another compound feries of which the firft term will be 8 g (becaufe the firft term of the compound feries w3, or y, is 1) and which will be equal to m*. Let this new compound feries be called 3. And we fhall then have m? = the compound feries 0. 109. Now there will evidently be the fame analogies and the fame differences _ between the compound feries. A and 6, or 8e x IT. and 8g x y, with refpeét to the co-eficients of their feveral correfponding terms and the figns + and —, which are ta be prefixed to them, as between the compound feriefes T and y themfelves ;- becaufe the faid multiplications by 8eand 8 g can make no diffe- rence at all in the figns of the terms multiplied, and will increafe the co-effi- cients of all the faid terms in the fame proportion of 8 to 1, which will not af- fect their antecedent ‘equality. For as to the quantities e and g (which‘are in- volved inthe multiplicators 8 ¢ and 8 g) the multiplications of the terms by them will not affect the numeral co-efficients, of the terms arifing from the combina- tions FOR RESOLVING THE CUBICK EQUATION &c. 517 tions of the original numeral co-efficients C, E, G, I, L, N, &c, but will only affect the literal parts of the faid terms by removing one power of e¢ and g from their feveral denominators, It therefore follows from art. 107, and 108, in the firft place, that the firft term of the compound feries 3, or 8g x y, will be (8g xX 1, or) 8g; and, fecondly, that the co-efficients of the fecond term, and all the following terms of the compound feries 38, or 8g x y, will be ref{pectively equal to the co-efficients of the fecond term, and all the following correfpond- ing terms of the compound feries A, or 8e x I’; and, thirdly, that the figns + and —, which are to be prefixed to the feveral terms of the third, and fifth, and feventh, and other following odd vertical columns of terms in the compound fe- ries 0, or 8g X ¥, will be the fame with thofe which are to be prefixed to the correfponding terms of the third, and fifth, and feventh, and other following odd vertical columns of terms in the compound feries A, or 8¢ x T'; and, fourthly, that the figns + and —, which are to be prefixed to the feveral terms of the fecond, and fourth, and fixth, and other following even vertical columns of terms in the compound feries 3, or 8g x y, will be every where contrary to thofe which are to be prefixed to the correfponding terms of the fecond, and fourth, and fixth, and other following even vertical columns of terms in the compound feries A, or 8e X T. t10. It follows from the laft article, that, if the compound feries 3, or 8g x y, be reduced into a fimple feries, or feries of fingle terms, by performing all the multiplications, additions, and fubtractions that are neceflary for that purpofe, the co-efficients of the fecond and other following terms of the fimple feries thereby obtained will be refpectively equal to the co-efficients of the fecond and other following terms of the fimple feries which is equal to the compound feries A, or 8e XT, that is (by art. 101) to the co-efficients P,Q, R, S, T, V, &c; and the figns to be prefixed to the third, and fifth, and feventh, and other fol- lowing odd terms of the faid fimple feries, which is equal to the compound fe- ries 3, or 8g x y, will be the fame with thofe which are prefixed to the third, and fifth, and feventh, and other following odd terms of the fimple feries which is equal to the compound feries A, or 8e XT, or to the odd terms of the fimple feries Geer tse AS nS BS Le ye Bee; and. the figns to be prefixed to the fecond, and fourth, and fixth, and other following even terms of the fimple feries which is equal to the faid compound feries 6, or 8g x y, willbe refpectively contrary to thofe which are to be prefixed to the fecond, and fourth, and fixth, and other following even terms of the fimple feries which 1s equal to the.compound feries A, or 8e X I, or to the even terms of the fimple feries 8 ¢ 6 8 10 12 eo BE ht ee vet And ‘confequently the Tit ple 22 e oy ef e? aoe) feries that will be equal to the compound feries 3, or Sg X y, will be 8g + P UZ Q xt Rx s x8 Tmt? v zi? ——— —— ee g 3 5 - 9 the cube of the tranfcendental expreffion m, or 2 Vv>gs the feries w (which, by art. 108, is equal to the compound feries 3) will be equal to the fimple feries 4 6 8 10 giz j : Bg + ee ee aL eld ast pan yy Pinel 4. &c ad infinitum. 2 g & & BF # at — —— + &c ad infinitum. Therefore m’, or g Bie kL 518 A METHOD OF EXTENDING CARDAN’S RULE : C B% E 24 aetyr! 111. In the expreffion 2 f(g x the feries w, or 1 + pei hs ee igh Jif roi Se LE. + &c ad infinitum, the letter g is put for -, or half the ab- g eT. ona ‘ } : t 3 equation x? —=-gw = #, and zz is = H — =, or L — gy. folute term ¢ of the eq q ; , PF ge = && rie eae oe VX Wiese BB Therefore > 1s = og + zz = ge X F + oe and g* is = 27 X ge X |t + i 2 I and 7 is =='"3) & gS KX I+ =| 3 = (by the binomial theorem in the cafe of 2 —er . . . Bw cz Dx® Ex® F xt? ts x the infinite feries 1 + —- — — + ———| + => — roots) 3 X £3 oe ro o z = G zt ie + &c ad infinitum. ° . : Czz E x4 112. Now let the expreffion 2 /?{g x the infinite feries 1 + Pas ce + 2 G2 12° Let? N xj? ce —i% 4 2% — lS + &c be multiplied into the expreflion 3 x g 3 X ; : a : i c x4 D z® Ex? F 210 G x2 xotrte the infinite feries 1 + ye iinet 1 Haligelgh =F Sioa Gare + &c, which is 2 equal to g; and the product will be (= 2 (g x 3 X g 3 X the product of I 2 ‘ the faid two infinite feriefes = 2 x g 3 X 3 X g 3 X the product of the faid two infinite feriefes) = 6g x the product of the faid two infinite feriefes. Therefore g x the expreffion 2 3 (g x the infinite feries 1 + = _ Sy 8 10 12 ae a — A= + &c, org X 2¥7[e x w, org x m, will be = 6g x the product of the faid two infinite feriefes, which product will be a cer- tain compound feries, or feries confifting of feveral different horizontal rows of terms, placed one under another. Let this compound feries be denoted by the fmall Greek letter A. And then¢g x 2¥'(g x the fimple feries w, org x m, will be = 6g x the compound feries a. 113. Now let the feveral terms of the compound feries A be multiplied into 6g, and let the new compound feries thence arifing be denoted by the {mall Greek letter x. Then willg x 24/?(g x the fimple feries w, or g x m, be = the new compound feries x. And in this compound feries 7 it is evident that the firft term muft be 6g, becaufe the firft term of the compound feries A (which is the product of the oe bul. : EB 2z% c x4 D ze E x8 ; 5 multiplication of the feries 1 + —- —— + -—> al -+- &c into the feries 4 6 &s Be 4 C2zz E z+ G2 yee cada ach +> Bae + &c) muft be 1. And the following terms of the faid compound feries 7 will involve the fucceffive even powers of z, to wit, BB FOR RESOLVING THE CUBICK EQUATION &¢, 519 8 . . Sz, z*, ZY, Z, z'°, 2, &c, in their numerators, and the fucceffive odd powers of g, to wit, g, 2°, 2°, £’, 2°, g , &c, in their denominators. 114. Since the co-efficients of the terms of the two infinite feriefes 1 + 2 ez p28 Ex? F x10 G x1? czz Ext , Gx U era 5 et eumeallitene v — — — + &candi + —-- — — ase he g g et ee &E at £ s bed N 272 BO has <0 7 3 i é . et «Oc (by the multiplication of which the compound feries A is Io produced) are the fame with thofe of the correfponding terms of the two infinite . B S$ cst Dp s® E 58 F st° Gs css E 54 BI cg et PE a ran Tas ae TI NE: oma aa aan a est OF N 5%? tig lis / SO Oe es — &cc (by the multiplication of which the compound feries A is produced in art. 97), though the terms of the two former feriefes are not connected with each other by the figns + and — in the fame manner as the terms of the two latter feriefes; it follows that the co-efficients of the feveral terms contained in the compound feries A will be equal to the co-efficients of the correfponding terms contained in the compound feries A; becaufe the co-efii- cients of the terms of both the faid compound feriefes are fimilar combinations, or products, of the fame original factors or co-efficients, B, C, D, E, F, G, H, I, K, L, M,N, &c, of the terms of the fimple feriefes by the multiplication of which into each other the faid compound feriefes are produced. And from the continued repetition of the fign — in the fecond and other following terms of the two fe- By ers* D s® E 58 Fst G 51 Cc 5s Es © OAS tov hipaa cad mth ili thet gt Ht! Pedra orator 6 8 rT 51° N st2 ao ¢ . = —*S wat = _ > — &c (by the multiplication of which the compound feries A is produced), and the alternate fucceffion of the fign ++ and the fign — Z : : BZ C u* in the fecond and other following terms of the two feriefes 1 + eran D 2x5 E2® F zr G 2% & Czz EB xf G 2° iz Ba"? sae oe to c, and 1 + — — — + — —— +5 os Ba pel ype es ek DIGcips 8 lig? Dom EI td geP et ac (by the multiplication of which into each other the compound feries 12 A is produced), it follows that in the third, and fifth, and feventh, and other fol- lowing odd vertical columns of terms in the compound feries 4 the figns + and — that are to be prefixed to the feveral terms of the faid columns, will be the fame with thofe which are to be prefixed to the correfponding terms of the third, and fifth, and feventh, and other following odd vertical columns of terims in the compound feries A; and that, in the fecond, and fourth, and fixth, and other following even vertical columns of terms in the faid compound feries A the figns + and —, that are.to be prefixed to the feveral terms of the faid columns, will be refpectively contrary to thofe which are to be prefixed to the correfpond- ing terms of the fecond and fourth, and fixth, and other following even vertical columns of terms in the faid compound feries A. 115. Since the compound feries If is = 6¢ x the compound feries A, and the compound feries 7 is = 6g X the compound feries A, it is evident that there will 520 A METHOD OF EXTENDING CARDAN’S RULE will be the fame analogies and the fame differences between the two compound feriefes II and sr, with refpeét to the co-efficients of their feveral correfponding terms, and the figns -- and — which are to be prefixed to them, as between the two compound feriefes A and A themfelves; becaufe the faid multiplications by 6 ¢ and 6g can make no difference at all in the figns of the terms multiplied, and will increafe the co-efficients of all the faid terms in the fame proportion of 6 to 1, which will not change their antecedent equality. For, as to the quantities e and g (which are involved in the multiplicators 6 ¢ and 6 g) the multiplications of the terms by them will not affect the numeral co-efficients of the terms arifing from the combinations of the original numeral co-efficients B, C, D, E, F, G, H, I, K, L, M, N, &c, but will only affect the literal parts of the faid terms by re- moving one power of e and g out of their feveral denominators. It therefore fol- lows from the two laft articles 113 and 114, in the firft place, that the firft term of the compound feries 7, or 6g X A, willbe (6g x 1, or) 6g; and, fecondly, that the co-efficients of all the terms of the compound feries 7, or 6g x A, will be, refpectively, equal to the co-efficients of the correfponding terms of the com-. pound feries TI, or 6e X A; and, thirdly, that the figns + and —, which are to be prefixed to the feveral terms of the third, and fifth, and feventh, and other fol- lowing odd vertical columns of terms in the compound feries #7, or 6g X A, will be the fame with thofe which are to be prefixed to the correfponding terms of the third, and fifth, and feventh, and other following odd vertical columns of terms in the compound feries Tl, or 6¢ & A; and, 4thly, that the figns + and —, which are to be prefixed to the feveral terms of the fecond, and fourth, and fixth, and other following even vertical columns of terms in the compound fe- ries 7, or 6g X A, will be, refpectively, contrary to thofe which are to be pre- fixed to the correfponding terms of the fecond, and fourth, and fixth, and other following even vertical columns of terms in the compound feries II, or 6¢ X A. 116. It follows, from the laft article, that, if the compound feries 7, or 6. “% A, be reduced into a fimple feries, or feries confifting of fingle terms, by making all the multiplications, additions, and fubtractions, that are neceflary for that purpofe, the co-efficients of the fecond and other following terms of the fim- ple feries thereby obtained will be refpectively equal to the co-efficients of the fecond and other following terms of the fimple feries which is equal to the com- pound feries II, or 6e X A; and it follows in the fecond place, that the figns -++ or —, that are to be prefixed to the third, and fifth, and feventh, and other following odd terms of the fimple feries which is equal to the compound feries z, or 6g X A, will be the fame with thofe which are to be prefixed to the third, and fifth, and feventh, and other following odd terms of the fimple feries which is equal to the compound feries TT, or 6¢ X A; and it follows, in the third place, that the figns ++ or —, that are to be prefixed to the fecond, and fourth, and fixth, and other following even terms of the fimple feries which is equal to the compound feries 7, or 6g X A, will be contrary to thofe which are to be pre- fixed to the fecond, and fourth, and fixth, and other following even terms of the fimple feries that is equal to the compound feries II, or 6¢ x A. But it has been fhewn in art. 100, 101, 102, that the fimple feries that is equal to the com- 2. pound FOR RESOLVING THE CUBICK EQUATION &c. 52 . ° 4 6 8 10 y (i% pound feries II, or 6¢ x A, is G0 mad ab RAS ee ‘ . : evar rer a e7 e en &c ad infinitum. ‘Therefore the fimple feries that is equal to the compound {e- F : P XZ ax R 26 s x8 na Vv st2 &c ad infinitum. But by art. 112, ¢ X m, org x the tranfcendental expreflion asf 3 (¢ x the feries w, org X the tranfcendental expreffion 2 W/?(g > the in- E 24 G2 12? L zr N 2%? . ° C 2S ° finite feries § —— — Oe te est &e, is equal to the . cod . . compound feries 7, or 6g x A. Therefore g x m is equal to the fimple feries P 2% Q x4 Rx® s x8 ~1Zt° Vv xi yd eee ts er bar Fe, 20 iniitam. ~ 117. But it was fhewn in art. 110, that m*, or the cube of the tranfcendental Cz E xt expreffion m, or 2»? (¢ x the feries w, or 2 4/3 (¢ x the feries 1 4+- —— — — 8&5 & G 28 128 27? N zi : : + paar + areas ++ &c is equal to the fimple feries 8z 4- = — & xt R xo s x8 T x%° v 2 it oerss fo a : ea a platen Ce + &c ad infinitum, which is greater than the PLS azt R x6 s 28 T zt ees Vv 2 , : : : ——— + &c ad infinitum, by the difference (8 ¢ — 6g, or) 2g. Therefore m’ is greater than ¢ X m by the fame difference 2g, or the refidual quantity m’§—¢ Xmis=2g, or2 x ~, or ¢, the abfolute term of the equation x* — gx = ¢; and confequently m, or 2 i Cz E 2 the tranfcendental expreffion 2 4/7 {g x the infinite feries 1 + gesagt G2® 12° L zte N x? oo = ”_—_-_— 6 3 to g* faid equation. Q: E. D. + &c ad infinitum, mutt be equal to the root « of the End of the Recapitulation of the foregoing Demonftration of the Propofition laid down above in art. 36. ! Another Recapitulation of the Subjtance of the foregoing Demonftration, fomewhat foorter than the former. 118. The fubftance of the foregoing demonftration may be exprefled in a more concife manner as follows. | . ° ‘ 3 Having fhewn in art. 25, that, if “3 be greater than = or y be greater than v4 the root y of the cubick equation y?— gy =r will be equal to the tranf- 373 6 : 3 : . : _ css E s*# PE is cendental expreffion 2 /? * x the infinite feries 1 —--->— = ——3— 3B 522 A METHOD OF EXTENDING CARDAN’S RULE — —— — —- — &¢, ad infinitum, in which the letter e denotes =, or half the é : : : rr ere é abfolute term r of the equation y3 — gy = 1, and ss is = — — i, we raifed the faid value of y to its cube, or third power, by multiplying it twice into itfelf, whereby we obtained a certain compound feries, of which the firft term was 8 ¢, : ; y $5. 54: SB gS, ee and the following terms involved the feveral fractions 7) a? 9s) a? ee &c, or the feveral fucceflive powers of ss divided by the fucceffive odd powers of e. This compound feries we called A, and fo we had y* = the compound feries A. 119. We then found an expreffion of the value of g (the co-efficient of y in the equation y3 -— gy =r) in terms involving the quantities e and s, to the end that we might multiply the faid value of ¢g into the foregoing tranfcendental ex- preffion which is equal to y, and thereby obtain a quantity expreffed in powers of e and s, that fhould be equal to the product gy, which is fubtraéted from 9% 2 in the equation y3 —gy==r. This value of g we found tobe 3 x g 3 & the ; ‘ : BSS c 54 p 5° E 5° OP he G st ; : infinite feries yrs: oe os ee ee ees &c, which, being multiplied into the foregoing tranfcendental expreffion of the value of y, pro- duced a certain compound feries, of which the firft term was 6 e, and the fol- : , . sy 54 SR eee lowing terms involved the feveral fractions wri &c, or the feveral fucceflive even powers of s divided by the fucceffive odd powers of ¢. This compound feries we called FH, and fo had the produét ¢y equal to the compound feries IT. 120. We then obferved that, fince the compound feries A was equal to 9%, and the compound feries II was equal to gy, the excefs of A above II would be equal to 73 — gy, and confequently to the abfolute term 7, or2 x =, or 2 ¢, But this, we obferved, was the excefs of 8 ¢, the firft term of the compound fe- ries A above 6e, the firft term of the compound feries Hl. And hence it fol- lowed that 8e — A, or the excefs of the firft term of the feries A above the whole of the faid feries, would be equal to 6e — II, or the excefs of the firft term of the feries II above the whole of the faid feries. And thus we obtained two compound feriefes, 8¢ — A and 6¢ — II, which were equal to each other. 121. We then obferved that, fince this equality between the compound feri- efes 8¢ — A and 6e¢ — II took place in all the poffible magnitudes of ss and ee, it followed that each feparate compound term, or vertical column of fingle terms, involving any powers of ss and e, in the compound feries II, muft be equal to the correfponding compound term, or vertical column of fingle terins, involving the fame powers of ss and e, in the compound feries A; and confequently that, if thefe two compound feriefes A and II were to be reduced to fimple feriefes, > or a ae, a! FOR RESOLVING THE CUBICK EQUATION &c. 523 or feriefes confifting of fingle terms, by making all the multiplications, additions, and fubtractions, that were neceffary for that purpofe, the terms of the two fim- ple feriefes that would be thereby obtained, would be exactly equal to eac other, and marked with the fame figns + or —; and confequently that, ip the fimple feries that was equal to the compound feries 8¢ — A was = += 6 8 10 — ~ —— + vas + &c ad ns the be. es i 6¢— II lc 8 10 saint be equal to “ fame fimple fenigs ee A ae ee — Carag pad Ev ~.- -|- ae + &c ad infinitum. 123. ee so eG copay ot te sy ran feries 8e-— A to the fimple P SS feries 24 be Poet oe = — +- meat ig hs —5- + —— + &c ad infinitum, we inferred that : P ss s# the WL ue A Ba be ae to the fimple feries 8¢ — — — or _—: R 56 s 58 mist i es ety ig pe a Cee a oad infinitum; and, in like manner, from the t eae i the compound feries 6 e — II to the fame fimple feries — pee R56 rh Fa — + Be — = + &c ad infinitum, and that the product of the multiplication of ¢ P 2 into the faid tranfcendental expreffion is equal to the fimple feries 6 g + ra ye Qzt R 2° s x8 7 are V2 Poe Ee mer fimple feries by 8g — 6g, or by 2g, or 2 X -, or?¢. Therefore the cube + &c ad infinitum, which is lefs than the for- CZ E x* of the tranfcendental expreffion 2 ./3(g x the infinite feries 1 -- — —— gg en G 2° I 8 L xo N 22 ; c p ee + &c aed infinitum is greater than the product of the multiplication of the faid tranfcendental expreffion into g, and their diffe- rence is equal to ¢, the abfolute term of the equation x3 — gx = +. Therefore the faid tranfcendental expreffion muft be equal to the root » of the faid equa- tion. QesE. De End of the fecond and fhorter Recapitulation of the Subftauce of the foregoing Demonftration of the Propofition laid down above in art. 36. 134. I have now compleated the demontftration of the propofition, or theorem, laid down above in art. 36, and which is the principal fubject of this difcourfe, to wit, ** That, if the abfolute term, ¢, of the cubick equation «* — gx = ¢ be SI lefs than vt, but greater than W2 X mre a (or ¢# be lefs than aa but greater 3 3 3 3 than —/), or = be lefs than £, but greater than — x ©, or than £, and £ 27 4 27 2“ 27 54 $ be put = —, or half the abfolute term ¢ of the faid equation, and zz be = - 3 F < . . == = or i — gg, the root « of the faid equation will be equal to the following : : : : fo: z 4 tranfcendental quantity, to wit, 2 /* (g x the infinite feries 1 + or — = + ro 6 8 ro r2 . . . . . 4 pa — a 4 A — A + &c ad infinitum, in which feries the numeral co- efficients C, E, G, I, L, N, &c of the fecond and other following terms. are the co-efficients 528 A METHOD OF EXTENDING CARDAN’S RULE co-eflicients of the third, and fifth, and feventh, and ninth, and eleventh, and thirteenth, and other following odd terms of the infinite feries which is equal to the cube-root of a binomial quantity, as 1 + «, and the fecond and third, and fourth, and fifth, and fixth, and feventh, and other following terms are marked with the fign + and with the fign - alternately.” This‘demonftration has been, I confefs, exceedingly long and intricate; but I did not know how to make it fhorter or eafier, without leflening the evidence of the reafonings contained in it. And I hope that the reader will have found it convincing and fatisfactory, fo as to have no doubt remaining on his mind of the truth of the propofition it is in- tended to demonftrate. But it may poffibly be afked, «‘ How could fuch an ex- preffion be difcovered to be equal to the root of the cubick equation x* — gx =r, if we did not already know that it is fo? feeing that, when we have found out that it is equal to the faid root, it requires fuch a long train of fubtle and abftrufe reafoning to prove that it is fo.” In anfwer to this queftion I will now proceed to {tate a method of inveftigating the value of ~ in this equation x* — qx = t by which we might have been led to this difcovery. A method of inveftigating the value of the root « of the cubick equation «3 — qu = t, in which the abfolute 3 term t is lefs than ays, or — is le/s than £, by means of the tranfcendental expreffion 2 6/{e x the ‘ ; , Css E 54 G 5° 13° z 579 infinite feries 1 — a i ee N pie ei ; in art. 25 to be equal to the root y of the cubick equa- tion y>—qy =r, in which the abfolute term r is . 3 greater than 2400 op - is greater than = 303 — &¢ ad infinitum, which has been fhewn above 135. Since the tranfcendental expreflion 2 /%{e x the infinite feries 1 — 4 6 8 - 10 12 s 4 Cai RRC ESN OES ~~ — &c is equal to the root y of the cubick ce e+ og eo” go gt equation 7? — gy ='r, it follows that, if the faid expreffion were to be raifed to its cube, or third power, by multiplying it twice into itfelf, and if it were alfo to be multiplied into’the co-efficient g, or into the value of the faid co-efficient expreffed in powers of ¢ and s, the faid cube would be greater than the faid pro- 5 : a T duct, and their difference would be equal tor. Now, becaufe ¢ is = > and ss : 3 3 3 3 3 is = 7 —Z, or ee — Z, we fhall have ss + = ce, and £ = em ssse 4 27 27 27 27 2 x fatty and g? = 27 x ee X 4 - —, and confequently g= 3 xX eg x » eo FOR RESOLVING THE CUBICK EQUATION &c. 529 nh 2 x f — =| 3 = (by the refidual theorem in the cafe of roots) 3 x ¢ 3 X the : . : BSS c 54 D 56 ES® F st0 G 5 infinite feries 1 — — — — — — — ——- — —~— — —~ — &e. ee e+ e 3 (os ei It follows therefore that the product of the multiplication of this laft expreffion into the : : . oy E st Gs° ne op eet 2 4/? (ores the infinite. feries 5 Ah me ee N git ere — &c will be lefs than the cube of the {aid expreffion 2 /?(e x the infinite Css E 54 Gi rs* L st? N 5%” feries 1 — a ee &c, and that their diffe- : te rence will be equal to r, or to 2 X > 0r 2X ¢, or 2¢, 136, Now from hence it feems natural to conjecture that the root x of the fecond equation x? — gw = ¢, in which the abfolute term ¢ is lefs than “fat far ° 3 . = is lefs than 7? will be equal to a tranfcendental expreffion that fhall bear a 4 : : : ; C55 fg great refemblance to the expreffion 2 \/?(e x the infinite feries 1 — ee 6 8 L st N st2 5 i ; pa ao — = —=> -- Ts — &c (which is equal to the root y of the former equation ys — gy =r), and fhall differ from it only in fuch circumftances as fhall be the confequences of the abfolute term’s being lefs than a inftead of being greater than that quantity, as it was in the foregoing equation y? — gy =r. . . . z Therefore, if in the equation x* — gx = ¢ we were to put g for =» or half the abfolute term ¢ (as we before put e for =, or half the abfolute term 7) and were 3 A to put zz for the excefs of rr above bd or gg (as we before put ss for the ex- 20. cefs of —, or ce, above <) it feems reafonable to conjecture that the root x of 4 the equation x3 — gx == ¢ will be equal to a tranfcendental expreffion of this # ; 10 12 kind, to wit, 2 y? A x the infinite feries 1, a ra ae pra eese My bs &c, 5 in which I have not prefixed any figns + or — to the fecond and other follow- ing terms, becaufe it does not hitherto appear whether they ought all to have the fign — prefixed to them (as is the cafe with the fecond and other following ; Css E 4 G 56 15° Iga N st? terms of the former feries 1 — es ee ge as re &c), or whether fome of them are to be marked with the fign —, and others of them to be marked with the fign +. And we fhall have reafon to fuppofe that thefe terms are not, all of them, to be marked with the fign — (as the correfponding terms of the former feries were) becaufe the figns prefixed to the terms of the in- finite feries which enters into the expreflion of the value of gq in this cafe, will not be the fame as thofe which are prefixed to the terms of the infinite feries Vou. II. aoy: wo PONE _o8 ? to ? 127 530 A METHOD OF EXTENDING CARDAN’S RULE which enters into the expreffion of the value of g in the former cafe. For we 2 have feen in art. 135, that the value of ¢ in the former cafe is 3 x ¢ 3 x the ; : : BSS c s+ ps? E 53 F st G st : ; infinite feries r — —'— ——- &- Se ee a ee e* 6° e® ¢i0 ei the terms after the firft term 1 are marked with the fign —. But in the prefent 3 3 ? 3 P tt cafe we have zz = = ave an — gg, and confequently os = oe + Reve 2 -¥ BB Ze — 22) eg x {1 + —, and g? = 27 x ge X It +—,andg=3Xg3X|t + —/3 &S5 && &8 - 2 ; == (by the binomial theorem in the cafe of roots) 3 x g 3 x the infinite feries B ZZ c zt DZ Ez? awe G Shale hichatha rie St er ee I ee £ & other following terms are marked with the fign + and the fign — alternately. Therefore, if this value of g were to be multiplied into the expreffion 2 /3{g¢ : , : C 2% E x4 Cx) eae Las N zi x the infinite ferles 1 — — —- — — —- ~ > — — — =~ — &, the && & figns + and —, that would be prefixed to the feveral terms of the produét, or compound feries, thence arifing, would in many inftances be different from thofe which would be prefixed to the correfponding terms of the compound feries or 2 product, arifing from the multiplication of the former value of ¢, to wit, 3 Xe 3 . ‘ : B Ss c 54 D s® E 58 gare G st : x the infinite feries 1 — — ——- —-— —-; — —~- —— > — &c, into the €é é é é é e Css E 54 G 56 158 L st , ate ' expreffion 2 4/3(e x the infinite feries 1 — primer uhiemiior wiles ib 5 WN Iz ° . ° = — &c. Therefore, fince the expreffion 2 /?{e x the infinite feries 1 — Css E 54 G 3° oS oe ae Ww st : : : : Pe MY SL GPUF Lo eo her Le Li takes fubftituted inftead of y in compound quantity y? — gy makes the faid quantity equal to 7, or 2 e, it follows . . ° ~ as 6 s that the expreffion 2 »/*(g x the infinite feries r — = —~ = —~-4__ 4 see = L zlo N giz re Me &c, being fubftituted inftead of x in the compound quantity x* —— gw, cannot (on account of the different figns prefixed to the terms of the value of the product g* from thofe which are prefixed to the correfponding terms of the value of the product gy) make the faid quantity equal to 2g, or #, and confequently the faid expreffion 2 3.(g x the infinite feries 1 — — _ E 24 G 2% rz en “4 en ee ae a — ~“&C cannot be equal to the root # of the ges Og g g g equation x? — qx = 4. 137. Having thus found that the expreffion 2 /3(g x the infinite feries 1 — 4 *< ~8 10 Iz < C 2% Es fe 2 2 3B sha fe is not equal to the root w of the rae egy § . equation x? — gx = ¢;—and having obferved that the figns of the fecond and other FOR RESOLVING THE CUBICK EQUATION &c. 531 other following terms of the infinite feries that enters into the expreffion of the value of ¢ in this cafe are alternately -+ and —, whereas in the former cafe the fecond and all the following terms of the infinite feries which entered into the expreffion of the value of g, were marked with the fign — ;— it feems reafon- able to conjecture, in the next place, that the root « of the equation #7 — gx = ¢ C Zz E 24 may be equal to the expreflion 2 ?(g x the infinite feries 1 + at ee G x6 12° Ext? Naz : : 5 . ee ae te est «Cc, in which the fecond and all the following } . . terms are marked with the fign + and the fign — alternately. And to this ex- preffion we fhould find that the faid root was really equal, if, when we had made this conjecture, we had tried it in a few numerical examples, by computing the . y 4 values of 2 4/? [g and of a few of the firft terms of the feries 1 + aa — a -- 6 8 10 12 t > s — ee “> -- ao + &c, and multiplying 2 1/3 {g into the refult of the faid terms, and then fubftituting the produét thence arifing inftead of x in the compound quantity «?—— gx. For it would always appear, upon fuch fubftitu- tions, that the values of x? — gw thence arifing would be equal to the abfolute terms, ¢, of the faid equations. And thus we might have difcovered that the . ° : . j E24 2 28 72 faid expreffion 2 »/?{g x the infinite feries 1 + ee ree 4 a a - 4: a N x12 . . . . — == + &c is equal to the root x of the equation x? — gv = ¢; after which it would have become neceflary to our further fatisfa€tion on the fubject, to endea- _vour to demonftrate the truth of this propofition fynthetically ; which we have done at great length, and with as much exactnefs as poffible, in the foregoing articles of this difcourfe. 138. But there is alfo another method of difcovering that the faid tranfcenden- : : f C 2z E 24 G x6 x8 tal expreffion 2 /° (¢ x the infinite feries 1 + penal =" au a a ail ¥ pet Nz? . F Pi : > eo + &c is equal to the root x of the equation x7 — gx =f. And this method is more direct than the former, and carries with it a proof that the faid expreffion mutt be equal to the faid root, inftead of affording us only a probable conjecture that it is fo, which is afterwards to be confirmed by trials, in particular numerical examples, and by a fynthetical demonftration, This method I fhall now endeavour to explain. Ee Gey) Another es 532 A METHOD OF EXTENDING CARDAN’S RULE Another method of difcovering that the tranfcendental ex- pure ; 4 preffion 273 (gz x the infinite feries 1 + ae me &S§ Gz feu fae N ZI f ee eb Se ee + (Se ts equalsto, the é / root x of the cubick equation x* — gx = t, in which ¢ is lefs than =i, but greater than f2 X it. V3 139. We have feen, that in the equation y3 — gy = 7 (in which rf is greater 2 : wey than 4 Lee or - is greater than =a) the tranfcendental expreffion 2 3 (e x the : we ; Car E 54 G 5° a past? Nn 5?? . infinite feries 1 — — ———- —-> — + — => —-—; — &c is equal to the ee é cs E 54 G 36 15° di N 51? : : : ‘ -— — — -- = — — —-— — ~~ — &c contained in this expreffion, which (though not equally fhort and convenient in practice) are, neverthelefs, equally jalt,and true: and therefore they muft both produce the fame refult for the va- lue of the faid feries. ‘Thefe different methods of computing the faid feries are as follows. 140. The firft way of computing it is the common one, which confifts of the following proceffes ; to wit, firft, to compute the quantities 7 and f; as was done in the foregoing example in art. 30, 31, where a was found to be = 1,110,916, and z was found to be = 1000,000 ; and, fecondly, to fubtra& t from —, in order to obtain the quantity ss, which is equal to their difference = mone £, and which in the foregoing example was 110,916; and, thirdly, to di- vide ss by ee, fo as to obtain the value of the fraction =; as in the foregoing ex- : .g16 ample we found the fraction ane to be = 0.099,841,932,2; and, fourthly, to compute the powers of the value found for the fraction =; as in the foregoing xample we computed the powers of the decimal fraction 0.099,841,932,2, and found its fquare to be = 0.009,968,411,4, and its cube to be = 0.000,995, 265,4, and its fourth power to be = 0.000,099,369,2, and its fifth, fixth, fe- venth, and eighth powers to be equal to 0.000,009,921,2, and 0.000,000,990,53 and 0.000,000,098,8, and 0.000,000,009,8, refpectively ; and, fifthly, to mul- " TG 4 st 56 58 gto gi gl4 6 518 > tiply.—, and its powers =a? Vigdd (ede eo Daperad Gia? Gre? 7? &c, into the co-ef- ficients C, E, G, I, L, N, P, R, T, &c, refpectively; as in the foregoing ex- ample we multiplied 0.099,841,932 2 into the fraction = (which is = C), and 0.009, ¢ FOR RESOLVING THE CUBICK EQUATION &c, 533 0.009,968,411,4 into the fraction ir (which is = E), and 0,000,995,265,4 - into the fraction an «which is = G), and 0.000,099,369,2 into the fraction 935 592949 = L), and 0.000,000,990,5 into the fraction (which is = I), and 0.000,009,921,2 into the fraction Sees 3 (which is 1,179 .256 a by ares, 129,140,163 (which is = N), and 9917,040 : ao FRE NATL; (which is = P), and 0.000,000, 009,8 into the fraction tee al (which is = R), and found the produds 31,391,059,609 to be 0.011,093,548,0, 0.000,410,222,6, 0.000,023,360,9; 0.000,001,57354, 0.000,000,115,9, 0.000,000,009,0, 0.000,000,000,7, and 0.000,000,000,0 ; and, fixthly, to fubtract the fum of all the products fo obtained from 1, the firft term of the feries. This is the common, and the proper, way of computing the : Css E s4 G 56 15° L 5? N 5?” p 54 R 5t6 T st8 eg I Si ah Te a Ne asa ae LEE fy op ee 0.000,000,0098,8 into the fraction when we want to make ufe of it in practice. But it may alfo be computed in another manner, which may be defcribed as follows. 141. Inftead of ss let us infert the compound quantity ee is itfelf, to which 55 is equal, in all the terms of the faid feries. And it will thereby be c E converted into the following feries, to wit#1 —— x ye hE ts srt ee 27 Fee $x (22) “i seo et HG tes af q3\s N Gers? 27 ool y 27 a se Gi. omen a x a a — &c, or (if, for the fake of ten te we fubftitute ee for —, and Ff for £) into the feries 1 — — x lee f= — X [ee—f\ —- = x Ge — f\% rary fe—f\ —= x fe—f i —— X (ee— fF" — &c bien or (fubftituting inftead of ee — f\’; ef , e@—F)*, cece — #5, ee —ff)* , &c their expanded values) into the feries 1 — — x fe — Ff — = K (Aan Dek? +f —= x pl Tad i) 36 f* — f fa xfer + oe ae oP L ee x“ oP es ee? Et 10 6° f* — 10 6 a. 5ef* — fr + 15¢f* — 20ef° + 15ef* — O8f + f* : : Sek : G T> ()te3 oe — &c ad infinitum, or (multiplying the fra¢tions = OLR, oo esr RES BU Sie ee? 67. 8) 410). 4127 N Iz 10 £2 na | ¢ — oe f into the feparate terms of the compound pence or powers of ee — ff, with which they are connected, refpectively) into the feries 1—C 534 A METHOD OF EXTENDING CARDAN’S RULE cH 1—C+—— ; 2E E f* — Err ee ae G -} 3 cH 3 cf* pas ee e# ef of dub ABE gia gale ane: ee et ie é poe MR Coe re, 10 Lf® A Re ae ie ce et "i a e uk ge Lad, 15 N/* 20n f° 15 nf 6n ft? ay fi® Fone Thay a) PORES AEE) & 9'> earmes — &c ad infinitum ; vehi feries confifts of a much greater number of terms ; : C ss E 54 G 56 {iS aee than the correfponding part of the feries 1 — — = = ep Thal N Rake . ° ° . . ; a Pad = — &c, from which it is derived, and many of its terms are more compli- cated, and more difficult to compute, than the terms of the faid former feries. . rr 3 . Neverthelefs, fince the compound quantity iam r, or e@ — ff, 1s equal to ss, the infertion of it inftead of ss in the terms of the faid former feries cannot alter its value, or magnitude, though it will make it much more difficult to compute. It muft therefore be true of the new and complicated feries juft now obtained, as Css E 54 G 5° i Lge? Nest: well as of the former feries 1 — 7 mires So &e (from which it was derived), that, “if it be multiplied into 2 3 [2, or 2 y? aa the quantity thereby produced will be equal to the value-of y inthe equation y? — qy == r, or that, if the faid laft-mentioned quantity be raifed to its cube, or third power, by multiplying i it twice into itfelf, and alfo be multiplied into the co-efii- cient g, the product of the faid multiplication will be lefs than the faid cube, and their difference will be equal to the abfolute term 7. ~ 142. Having fhewn that the product of the multiplication of 2 VJ? fe or 27% (foi into the new and complicated feries laft obtained in the foregoing ar- ticle (which feries we will, for the fake of brevity, denote by the Greek capital letter I) is equal to the root y of the cubick equation y* — gy = 7 in the firft : TT cs 3 . cafe of that equation, or when 7 18 greater than = or eé is greater than ff, as 5S E 34 well as the product of the multiplication of 2 4/° {e into the feries Ls pe e* cs 15° i539 N si ; — S —* _ *_ — 5 — &c, from which the feries P was derived ;—we é é é é may, by means of this equality between thefe two produ¢ts and the root of the equation y3 — gy = r in the faid firft cafe of that equation, deduce from the fe- Cc 5s E 5* G 5° 14° aah N 5™? Hes Boy ee oa oe a = ke anothers femed aiteleme bling it in the powers of the literal quantity involved in its fecond and other fol- lowing a gwas g ereater than ff, or cr. But the fiens prefixed to the faid terms will be contrary to thofe which were prefixed to thém in the former cafe. Thus, for example, the cube of the difference ee —ffin the former cafe was eo — 3e4*f? 4 26e7f* — f°; and the cube of the SCLC ASS ee in the pre- fent cafe is f° — 3 fte? + 3 f%e* —e*®, which confifts of the fame terms (or the fame powers, and products of ee and #) as are contained in the cube of ee — ff, to wit, e°—getf* + 3e°*f* —f*; but they are placed ina contrary order to that in which they fends in the former cafe, and the figns, which are prefixed to them, are contrary in every term to what they were before. And the fame thing is true of the fifth power of the difference ee — ff, to wit, e° — [gw AS ep tke tare ae ar ibe *f* —f*°, and the fifth power of the oppofite difference ff — ee, to wit, f° — 5f%e? + 10 f%et —10f%e® + 5f%e? — e%, and of all the following odd powers of the faid oppofite diffetentess ee — ff and if —¢. 3 145. It follows, therefore, that, if zz be put for fr — e or ff — ee, in the / ; ; 3 ; fecond cafe of the equation y* — gy = 7, in which — is lefs than f, or éé is lefs than ff, the even powers of zz, to wit, zzl’, e@z\4, z2ls, zzl®, &c, or zt, z*, 2, 27°, &c, will reprefent, or be equal to, the fame terms (or the fame powers, products, and multiples of the two original quantities ee and r, or ¢é and fF) in the prefent cafe, as were reprefented by the fame even powers of ss, to wit, salt, ssl, ol, sie &c, or st, 5°, 37, 3”, &C,) 1m the tomer cares when = or ee, was greater than - or ff, and ss was put for the difference p ; ode Z, or ee — ff; and the feveral terms (or powers, products, and multiples, of 7 the original quantities ee and #/) reprefented by the i even ee of zz in the fecond cafe of the equation y3 —gy =r, when — is lefs than 2 r, or e¢ is lefs than ff, will have the fame figns ++ aig oo nrchced to them as iis had in the former cafe of the faid equation, when — was greater than fi, or ¢é was greater than f#; and the faid terms were reprefented by the even powers of ss; and the only difference between the terms reprefented by the even powers of zz in the latter cafe of the faid equation and the terms reprefented by the even powers of ss in the former cafe of the faid equation will be in the order in which the {aid terms will be placed in the two cafes, and in which the letters e and fin the feveral middle terms of the values of the faid even powers of zz and ss (which will involve both the quantities ce and #) will follow each other. 146, And. . FOR RESOLVING THE CUBICK EQUATION &c, 537 146. And it likewife follows, in the fecond place, that the odd powers of zz, to wit, zz itfelf and z2\?, zz)', Zz\7, zz\®, &c, or 2°, 2, 2 ‘ot &c, will alfo reprefent, or be equal to, the fame terms (or the fame powers, products, and ° . . re e 3 = multiples of the two original quantities — and ce or ee and ff) in the prefent cafe as were reprefented by the fame odd powers of ss, to wit, ss itfelf, and s.)3, = Yr me ss\7, ss°, &c, or $%, 57°, 5%, 5°, &c, in the former cafe, or when —, or > 3 3 4 > ee, was greater than Lor ff; and. ss was put for the difference — — for 2 4 e7 ee— ff. But the figns + and —, that will be prefixed to the terms that are re- prefented by the faid odd powers of zz, to wit, zz, 2°, 2°, x, 2", 8c, will be refpectively contrary to thofe which were prefixed to the fame terms in the former cafe, when they were reprefented by the fame odd powers of ss, to wit, 55) 5°, $*°, 5%, 5, 8c: and the terms reprefented by the faid odd powers of zz will likewife differ from the terms reprefented by the fame odd powers of ss in the order in which they will be placed, and in which the letters ¢ and / in the feveral middle terms of each of the faid powers of ce — ff and ff — ee (which middle terms will involve both the quantities ee and #) will follow each other. 147. If therefore, in the fecond cafe of the equation y? — gy =r, in which ‘ 3 é 3 . + is lefs than 35 or e¢ is lefs than f, we put zz for - — vs or ff — ce, the fe- é C 2B E x4 Gx 12° 1 cseae Nn 2 aes ; en &c, ad infinitum, will re- prefent, or be equal to, a fyftem of terms derived from the two original quan- tities — and ©, or ce and ff, that will be the very fame in point of compofition, 4 2 that is, will be the very fame powers, products, and multiples of ee and ff, as the terms of the complicated feries T', which was derived above, in art. 141, from ei E s4 G 56 158 es Wis? RS oe tae wore owner poamee is oe utc vs as &c, ad infinitum, by fub- ftituting ee — ff in its terms inftead of ss in the former cafe of the equation y3 — 3 ° gy = 1, or when —, or ee, was greater than f, or f. But the terms of the faid two complicated feriefes, or fyftems of terms, fo reprefented by the two fericfes Cc 55 E s4 G 5° z38 Lee n si? ni C 2% Pee to tas ee 8c, ad infinitum, and 1 — — = pst Se ee _ XS _ kc, ad infinitum, will not all have the f ae Oe ee CS infinitum, will not a// have the fame figns + and — prefixed to them; but thofe terms only of the latter complicated : ; : : C 2% Be feries, or fy{tem of terms, which is reprefented by the feries 1 — ee ee I 8 10 N xl2 3 . kee — &c (which latter fyftem of terms we will, for the e e ero ez fake of brevity, denote by the Greek capital letter A), which correfpond to, or are reprefented by, the terms which involve the even powers of zz, to wit, the Vou. Il. 3 2 terms 538 A METHOD OF EXTENDING CARDAN’S RULE Ext, 2°) Nz ‘ . Aaah Baraat athe &c, will have the fame figns + and — prefixed to them as were prefixed to them in the former fyftem.of terms, or complicated feries, — ~ : . F ‘ 4 called I (which was derived. above in art. 141 from the feries 1 — £2 — = — : 2 é G 56 re he B st? N si ; = — =f as — ae — &c), when they correfponded to, or were repre- boi es 4: fented by, the terms which involved the even powers of ss, to wit, the terms > : Lie ee —=—, -—_ & ek2 ) or are reprefented by, the terms which involve the odd powers of zz, to wit, the Caz, G2 E 2° . terms, 31, ate Ks will have contrary figns. prefixed to them to thofe ee which were prefixed to them in the former complicated feries T, when they correfponded to, or were reprefented by, thofe terms of the ferieés 1 — [© t ee &c; and thofe terms of the complicated feries A which correfpond to,, E 5* G 5° 158 I a N 51 : : cial —z — &c, which involved the odd powers: oe — 8 ero er —— — — e et € ‘ Cle Gs hth ae , of ss, to wit, the terms Heyes 2 eds &c. And, laftly, the order in which the feveral terms, or powers, products and multiples of the two original quantities ee and fare placed in the complicated feries A, will be contrary to that in which the fame terms are placed in the complicated feries F:. But this laft circumftance will make no change in the magnitude of the terms, nor in that of the whole feries that is compofed of them, and therefore need not be attended to. any further. 148. It follows from the laft article that, if the terms of the complicated feries. A be placed in the fame order. as the terms of the complicated feries T, the faid complicated feries A will be as follows ; to wit, | eee 2Ef E f* Fcky bey Fo Maio I + pn terig i, eo é A gen tg cn og nad ae 8 Or is afr ce et e? Ps elo N oni 15 nf* 20 Nn f° r6nf? 6nf? y= Sadat il red peace MOL as oo ee + &c, ad infinitum eae : 4 6 ts Io T2: 449. Therefore, if in the feries 1 — — _ = —_ — _ ~~ — Me _ — = &c, we change the figns of thofe terms which involve the odd powers of, zz,. . czz Gz? rez z i to wit, the terms vad Shy 3 ar &c, and in the terms of the new feries thereby produced,, » FOR RESOLVING THE CUBICK EQUATION &c. 539 Cxzz E 24 1 2° a produced, to wit, the feries 1 + —— — = EE. —s+ Soot &c, we fubftitute ff — ee inftead of : 2%, and sce produce a third complicated feries (which, for the fake of brevity, we will denote by the Greek capital letter A), this third complicated feries A will confift of the fame terms in point of compofi- tion (or the fame powers, products, and multiples, of the two original quantities . ee and ff) as each of the two former complicated feriefes denoted by the Greek capital letters T and A, and its terms will have the fame figns + and — prefixed to them as are prefixed to the fame terms in the complicated feries I’, which is fet down above, in art. 1413; and the only difference between the faid two com- plicated feriefes T and A will be in the order in which their terms are ranged, thofe of the feries proceeding according to the powers of the letter ¢, and thofe of the feries A proceeding according to the powers of the letter f. 150. Now, fince this laft complicated feries A confifts of the fame terms in point of compofition (or of the fame powers, products, and multiples of the ori- ginal quantities ee and f#/) as the firft complicated feries T, which is fet down in art. 141, and has every where the fame figns -- and — prefixed to its terms ; and it has been fhewn above, in the fame arts 141, that if the faid feries P be multiplied into 2 /?(e, or 24/3 (-, and the feries thereby produced, to wit, the complicated feries 2 4/? (? x T, be cubed, or raifed to its third power, by mul- tiplying it twice into itfelf, and alfo be multiplied into the co-efficient ¢, the faid product, to wit, g x 2 +/3(e x TF, will beefs than the faid cube, to wit, 8e x T°, and their difference, to wit, 8e x T? —g x 2y? fe » I, will be equal to the abfolute term 7 ;—it will follow that, if the third complicated feries A (which agrees fo entirely in all its terms, and the figns which are prefixed to them, with the complicated feries T) be multiplied into 2 »/*(e, or 2 73 i. and the complicated feries thereby produced, to wit, the complicated feries 2/'le x A, be cubed, or raifed to its third power by multiplying it twice in- 2 itfelf, and alfo be multiplied into the co-efficient g, the faid produé, to wit, x 2 See x A, will be lefs than the faid cube, or 8¢ « A, and their dif- foe to wit, 8¢ x Aig X 272 x A will be equal. to the abfolute termirt, < E 2* 11. But the complicated feries A is equal to the feries 1 + < poelgetrett 6 g 10 12 GAR, 4 RE AA a ge) ad infinitum, from which it was nme by fub- e° e cn 3 ftituting ff — ¢e in its tert inftead of zz. Therefore 2 ?(e x the faid feries 6 8 Io 12 Tar +t Batata > i +i x= + &c will be equal to 2 3 {e x €é the complicated feries A, or to the Spmetickita feries 2 Yate (ex A. And confe- 4 6 quently, if the quantity .2 vfe x the infinite feries 1 + —— — > +r oe see 3 Lb 2 : ra* ™~ rule 540 A METHOD OF EXTENDING CARDAN’S RULE 8 10 12 ays Jas a ~=— + &c (which is equal to the complicated feries 2 V/ife x é é A) be raifed to its ies 2 or third power, by multiplying it twice into itfelf, and likewife be multiplied into the co-efficient g, the faid produét will be lefs than the faid cube, and their difference will be equal to the abfolute term 7. There- C 2% E x4 6 x fore the faid aera 2/*(e x the infinite feries 1 + —~ — — i = 8 ' te) N tel “> —X*" 4 &c, ad infinitum, is equal to the root y of the Nae equa- é tion y3 — gy = r in the fecond cafe of it, or when - ~ is lefs than 4 me or ¢¢ is lefs : ‘ 3 rr i : than ff, and zz is put for the difference a4 sale f— ee; or, if in this fe~ cond cafe of the equation y3— gy = 7, we fubftitute ¢ inftead of r for the ab- folute term of the equation ; and put g, inftead of e, for half the abfolute term ¢ ; and denote the root of the equation by the letter x inftead of the letter y, the: . ° ° . + 6 tranfcendental quantity 2 «/?/[g x the infinite feries 1 + a — <= et & & + &c, ad infinitum, will be equal to the root x of the cubick. 323 L zt0 N 2? “ge gt equation x3 — gv = 7. Q@ Es Ik. End of the Inveftigation of the Tranfcendental Exprefio on 2 Va fe Xx the infinite C Zz E24 G 2x8 1 x8 Le erie a CE ee Ne + &e which is: equal te f te pt ss s g ; rhs Ne the root x of the cubick equation x? —-qx = t, in sobich t is lefs than — we, Ly 3 le Ae reater than f2 x 1¥4 g of ia A Remark on the foregoing Inveftigation. 152. The foregoing inveftigation i is very abftrufe and difficult, and therefore: has been fet forth at great length, in order to make the reafonings ufed in it as: clear as poffible: and I hope the attentive reader will have found them, in ge- neral, intelligible and fatisfactory. There is, however, one part of the deduc=. tion which is more fubtle than the reft, and may therefore require fome further elucidation ; I mean that part of it which is contained in art. 150.. In that ar- ticle the reafoning is as follows: ‘* Since the complicated feries A confifts of the very fame terms in point of compofition (or the very fame powers, produtts,. and multiples of the two original quantities ee and #) as the complicated feries. I, and with the fame figns + and — prefixed to them; and fince it has been fhewn in art, 141, that, if the expreffion 2 /? (e x the complicated feries Tbe: cubed, or raifed to its third power by multiplying it twice into itfelf, and alfo. be multiplied into the co-efficient g, the faid produ@t g x 2? (e x T will be lefs. than the faid cube, or 8¢ x I°*, and their difference will be equal to the abfo- lute: FOR RESOLVING THE CUBICK EQUATION &c. 541 lute term r, or 2 ¢ ;—it will follow that, if the expreffion 2/3 (¢ x the compli- cated feries A be cubed, or raifed to its third power by multiplying it twice into itfelf, and alfo be multiplied into the co-efficient g, the faid product g X 2 / 3(e x A muft in like manner be lefs than the faid cube, or 8e x A3, and their difference muft be equal to the abfolute term 7, or 2 ¢.” An Oljeétion to the Conclufion ftated in the laft Article. a ta near 153. Now to this conclufion it may be objected, “¢ That the letters 7 and e do not denote the fame quantities in the two expreffions 2 ,/3 (e x T and 2 /*fe x A, and in the two cafes of the equation y? —gy = r which corre- fpond to thofe expreffions ; but that in the Pas of the equation y3 —gy=r 294/49 ’ BA 3s | preflion 2 ¥* (e x F the letter e fignifies a quantity greater than f, and that in the fecond cafe of the equation y3 —qy =r the letter r fignifies a quantity lefs than aang and in the correfponding expreffion 2 4/3 (e x A the letter ¢ figni- fies a quantity lefs than f; and that confequently the fuppofed refemblance be- tween the expreffion 2 f?fe x I and 2/3 fe x Ais only apparent, and not real, and therefore will not warrant the conclufion drawn from it.” the letter 7 fignifies a quantity greater than , and in the correfponding ex- An Anfwer to the faid Objettion. 154. In anfwer to this objection we muft obferve that it never has been af- ferted, that the expreffion 2 /* (e x the feries A was equal to the expreffion avi(e x the feries T. For that would not be true; becaufe 2 ¥* fe x Ais always lefs than 2? (e x I. But it was only faid that thefe two expreffions confifted of terms compofed in the fame manner of the two original quantities ee and ff, and that therefore, fince the cube of 2 ¥*{e x T’ was greater than the rodu& of the multiplication of 2 3 (e x T into the co-efficient g, and their difference was equal to 7, or 2¢, that is, to the greater value of r or 2e, which belongs to the firft cafe of the equation y3 —gy = 7, it followed that the cube of 2+/2(e X A muft be greater than the product of the multiplication of 2./?(e x A into the co-efficient g, and that their difference mult be equal to the correfponding, or leffer, value of 7, or 2¢, which belongs to the fecond cafe of the faid equation y* —qgy =r. And this conclufion is moft certainly juft and true, notwithftanding the inequality of the two values of y in the two different cafes of the equation y? — gy = 7, and of the two values of ¢ in the two expreffions 2 of? {¢ x T and 2 /3lex A. A fur- 542 A METHOD OF EXTENDING CARDAN’S RULE A further Proof of the Conclufion ftated in Art. 152. 15s. But that the truth of this. conclufion may be made as plain as poflible, let us fubftitute the letter g inftead of e in the expreffion 2 /* fe x the feries A. Then will this expreffion become 2 ./*(g x the feries A, or 2 o/° (¢ x the following complicated feries, to wit, ; — C I + o A ee ci 88 Ee sg 8 g - jy 4 61/4 41f iss Bp GER ea! ER ae L IO L 10 L L Lo ee Rear eo 7m 6 “4 8 °. [2 “ah | -N4 SMP ey ey Te — &c, ad infinitum. We mutt therefore endeavour to prove that, fince the cube of the expreffion 2 4? {e x the complicated feries I (which is fet down above in art. 141) is greater than the product of the multiplication of the faid expreffion 2 3 (xr ‘nto the co-efficient g, and their difference is equal to 7, or 2¢, it muft follow that the cube of the other expreflion 2 /: {¢ x the complicated feries A (which has been juft now fet down) will be greater than the produét of the multiplica- tion of the faid expreffion 2 4/3 (g x 3 (which has been explained in art. 139, 140, 141, &c..... 157), was that by which I difcovered this expreffion to be equal to the root x of the equation — gx = ¢, after having feen it afferted to be fo by Monfieur Clairaut in his Elé- mens d’ Algébre. But, as the propofition appeared to me a very curious one, and worthy to be eftablifhed by more than one method of proof, I afterwards fought for, and difcovered, the long fynthetical demontftration of it which has taken up fo great a part of this difcourfe, and which, I apprehend, will have confirmed the truth of it beyond any poffibility of doubt. 161. I will now proceed to give a few examples of the refolution of cubick equations of the aforefaid form x* — qx = #, in which ¢ is fuppofed to be lefs . 3 than sit, but greater than /2X iv, or ¢é is f{uppofed to be lefs than ea aggre taat lefs than £, b 4 coe is e lefs than ~, but greater than but greater than eben ds fuppofed to b See ae rere) or —, or £, by means of the foregoing tran{cendental expreffion 2 /?(g x 4 r 2X27 F d : ‘ C 2% E zt G2 128 mime WN xt? 1 phe the infinite feries 1 + ——— tees Ot er es +s«CKS, cd infini-- tum, in order to confirm the truth of the reafonings by which the faid feries has been obtained. Wot. Il. 4A Examples, 546 A METHOD OF EXTENDING CARDAN’S RULE ' Examples of the refolution of cubick equations of the forego= ing form, x3 —-qx==t, when the abfolute term t is les than ws but greater than he 2x v4 or — is Vid mn +, but greater th £ lefs than 4 = but g an rr or Fi by means of the tranfcendental exprefion 2 »/3(g Xx the infinite fe- Ds C Uz E 24 G 2° 12° Bate N zi Pes Nate ar ae Serge OW paloOT gia +. &c, ad infinitum. EX -A.-M..P-E -E I, 162. Let it be required to refolve the equation #3 — 50” = 120 by means of the faid tranfcendental expreffion. . . if . tt . .- Tere g is = 503 418 = 120; —, or g, is = 60; 70 0! SEs TS ae 3600; g3 Is ne 125,000 = 124,000; and iis 3 i) = 4629.629,629,629,629, &c, which is 4] tt . 4 ; greater than 3600, or ve Therefore this equation cannot be refolved by Car- dan’s rule, but may by the setae 24/3 fg x the infinite feries 1 + <= — && : of 6 x3 Le N ye ee ae = + &c, provided that the faid feries con-- verges. Now, fince 5 is = 4629.629,629,629,629, &c, and a is 3600, we fhall 27 have 2z (= 4 a -—-i = = 4629.629,629,629,629, &c — it 1029.6295 629,629,629, &c, which is much lefs than 3600, or gg; and confequently the feries will converge. .629,629,629,6 163. We fhall therefore have — a Si Bee ay Bes = 0.286,008,2 30,4, and confequently 5 (= 0:286,008,230,4 |") — 0.001,000;707,00 and = ome = x ae = sania eiubabeiosbeiie x Sikes fo akg = 0,023,395 6 x8 25 Sa (os 756 and Fb Gee PLT Hanae: X 0.286,008 ina = 0.006,691, i : 355973 7 and pet i a x a = 0.006,691,355.7 X 0.286,008,230,4) = 0,001,913, 7828 3 i Joyand FOR RESOLVING THE CUBICK EQUATION &c. 547 Fa Cad ay and ge (=p x aa 0-001,913,782,8 X 0.286,008,230,4) = 0.000,547, 357205 =e 4 at aod and pes X 0.000,547)357,6 X 0.286,008,230,4) = 0,000,156, 543573 zd zit % and (= aa x pate fone X 0.286,008,230,4) = O.000,044, 77452 x8 zi6 zz and a Cas ge X a = 2900 pO NY X 0.286,008,230,4) = 0.000,012, 80557. = CX 0.286,008,230,4 = — x 0.286, 86,008, pene ey ie = ae tay ore. ane Berk 10 ' and — will be (= E x 0.081,800,707,8 = Fe X 0.081,800,707,8 = 10 X 0.081,800,707,8 _ 0.818,007,078,0 aw ; ; ie. Paes RL, = 0.003,366,284,2; G2e 154 De A © TX 0,0235395,075,0 = a. XK 0.025,995)67 5:6 154 X 0.02 3,395,675,6 3:602,934,042,4, Zea GELS te oe = abel 5) aaa ') = 0.000,549,144,05 and will be (= I x 0.006,691,355,7 = 85. x 0.006,691,355,7 = 935 X0.006,691,355,7 __ — 6-256s41757995 ee ee IID) — 6.000310 2 59,049 504g fia Mids — 55,913 Hae =" LX 6200739137782 50 orn X 0.001,913,782,8 = 6 696 $5913 X 0,001,913,782,8 __ 107,005,337,69 2°94) = 9,000,022,372,1 ; 4,782,969 4,782,969 ; meee e197 240: and + ro — will be (=n NS ‘0,000, 54453597,6 = 129,146,164 7+ OO25 47 35736 — 15179,256 X 0,000, 547535755 __ 645-4745733294596) 0.000,00 8,2; 129,140,163 rey 129,140,163 ) ioe 4599 3© 9 ie 1 8.67'7,620 6 8 and - = P X.9.000)1 §6,54857 :=— tedjnsis6u, oF X 0-000,150,545,7 :080, 68, zk _ 8,617,640 x 0.000.19654847 = 1:349:080,33906810) — 4, 5400,001,16059 ; 1,162,261,467 1,162,261,467 E will be (= R a 10413270782 yg 000;0 and = or will be (= X 0.000,044,774,2 = pies 059,609 9944577452 — 194,327,782 xo. 000044577492 _. SRT TIOrERAIE) == _0.000,000,277,2 31,381, 059,009.” 31,381,059,609 a a — _131431:4791050 and ro will be (= T X 0.000,012,805,7 = ; EA OOS, Edn2G SOSRE2, — 1354315479,050 X 0.000, 012,805,7 __ —_ 371,999+4915270.585,0 = 0.000.000 805.7 = 25541,905,928,329 2,541,865,828,329 ) ; : 067,6 ; ye And 548 A METHOD OF EXTENDING CARDAN’S RULE ‘ C zz E 2x4 G2 12° L2to N xt And confequently the feries 1 + — — ee 4+ 4 ee Bott R2to Tene 5 e > 5 ms ‘i ——- — —_ — — &c will be ol4 gto 28 C7 fay 4 == 1.000,000,000,0 +- 0,031,778,692,2 — 0.003,366,284,2 + 0.000,549,144,;0 — 0.000,105,952,9 + 0.000,022,372,1 — 0.000,004,998,2 -- 0.000,001,160,7 — 0.000,000,277,2 + 0.000,000,067,6 — &c = 1.032,351,436,6 — 0.003,477,512,5 1.028,873,924,1. Further, fince g is = 60, we fhall have 3 [g = ? (60 = 3-914,867,641,1 and confequently 2 s/?(g (= 2 X 3.914,867,641,1) = 7.829,735,282,26 Therefore the tranfcendental expreffion 2 /?(g x the infinite feries 1 + on oes I] E v4 G x8 12° rete N xi Pp zit R 215 ro 282,2 X 1.028,873,924,1 = 8.055,810,464,4. Therefore the root # of the propofed equation x*—— 50%” = 120 is = 8.055,810,454,4. ee zr8 ‘ “L ae — &c is = 7,829,735, 6. pee 8 TO' Linke, Iz ere 16 & & « 164. This value of the root x in the equation x3 — 50” = 120 is exact in the firft feven figures 8.055,810, its more accurate value being 8.055,810,345, 702, as may eafily be found by Mr. Raphfon’s method of approximation. N. B. This equation «7 — 50% = 120 expreffes the relation between the diameter of a circle and three chords in it that lie contiguous to each other, and together take up the arch of a femicircle, and form a trapezium of which the diameter of the circle is the fourth fide. For, if the three chords are called 4, k, and ¢, and the diameter of the circle is called x, the relation between them will be expreffed by the cubick equation «7 — 24 | —kkp X * = 2bkt3 which, if the — tt numbers 3, 4, and 5 are fubftituted inftead of the letters 4, k, and ¢, will be- come 7— 50% =120. See Sir Ifaac Newton’s drithmetica Univerfalis, edit. 2d, A. D. 1722, page 101. Exe Avi ee Leer IT. 165. Let it be required to find by means of the fame tranfcendental expref- Vo ; : . C 2% E xt Gx 12° L xie N23 fion 2,/3j{¢ xX the infinite feries 1 -- —- — -—— pha I tel es ee Vig ag gh te ge hee ee + &c, the root of the equation «7 — «= ~~ In- FOR RESOLVING THE CUBICK EQUATION &c. 549 . . . . I Zr ° In this equation g is = 1, and fis = ri and confequently —, or g, is = -" it : : rar Te x x = = and ie or gg, 1s = 563 and A is = = and - Se =, which is greater than 6 or i Therefore this equation cannot be refolved by Cardan’s CUZ rule, but may by the tranfcendental expreffion 2 4/?(g x the feries 1 + eam E x4 eer ko Vee 2 nett s ; = +> = + &c, in cafe that feries is a converging one. roy Te bhi 3 ti Now, fince {is = —, and — is = —, we fhall have zz, or 2. — = = — ; 27 aa 4 36 27 4 27 ee 7) = = = , which is lefs than , or — Ba 29 X30 F327 X36 37 RG BHO 1 K10k 36” 4? C ZZ E xt or gg, in the proportion of 1 to 3. Confequently the feries 1 + rice ps, + x : ; 3 : a: a = +- &c will be a converging feries, and the expreffion 2 /?([g x the faid feries will be equal to the root of the equation x3 — x = —. 3 . . t ° I ZB I 166. Since zz is = ——, and —, or gg, is = —, we thall have = =— = 3 x 36 rots 307 re pins, } 9-3335333233353 3 and confequently = (= = x = = 0.37253333,3 ee 2:33 99333933303 Bere Nee spy gg 3335333333393 % = ; Sleds deb] pL oT 2 x4 a I O.EVIgl ii, titst Coe 0.5 1d Ei sius — 22. 0102702 eH F7%*e ital TAs tas 5 j 3720373 , O350 5 z $ Ig es" 1.050374049,503 7,0 and = (= = x = = 0.037,037,037,0 X ~ = 3234 2 wees =+0,0125 & & && 3 3 : 345207930 3 : x z I 0.012,345,679,0 and — (= 7 x Fe = 0.012,345,679,0 X -— = Sao ee ==4.0,0045 2 . b Pigs22 0545 x* ae ae 1) 79'6.004, 1155220 bs and = = re pes 0.004,11§,226,3 X es ac ope = 0,001, 3715742515 Mh a Ls eae TY __ 0.001,371,742,1 and — (= —- X — = 0.001,371,742,1 X — = ——**— 0.000 gt git x ge 937 274 2 3 3 ) b : 457924723 3 at a a IT __ 0.000,457,24753, __ Mase saa x Bika 0:000,4.57:247,3 X at SD : ~—:) = 0.000, 1525415573 wr8 16 Sei ee, Bn 0.000, 18 AT 7. 8 Oe and — (= = X Fe = 0:000;152,41557 X = BALE GY ey ry 050;80552. Therefore will be (= C X 0.333,333,33323 => X .0-3335333:93393 5 a aa, = 0.037,0373037303 and 350 A METHOD OF EXTENDING CARDAN’S RULE ~ 4 Pe and => will be (=" Hx corre yet, era 6s re X O.1LI,III,III,1 = TO XO.1LLI PIE TL, 2 UF TEV,ELIGIII,T : i EE Te OUST 2 47 Si GRP) se ike sek teas he and ~~ will be’ (= G K°6169730375037,0 “<= 6561 % 0:03720372037,0 = 15 4 X 0.037,03 7203759 __ 5-793,703,698,0) __ SA \6.500 deere cela PRIOR A ae a 0.000,869,344,5 5 8 °. and > will be (= EASE 0.01 2,345,679,0 = SRG x 0.012,345,679,0 = is > 935 X0.012,345,679,0 __ I1.543,209,865,0 pe Meshal Sic bs A BLN GR ae Ure EA LE oe 85,23 593049 snaps 1F 95:40 599 552913 ona an hls oe sere De F52,966 X 0.004,1153226,.3 = 55913 X0.004,115,226,3 _ 230.094,648,111,9, __ at hn LNA ls ui INA ak tnd ah Pd | See is aie ote) 6 ; 4,782,969 4,782,969 J Roe Wa. ate _._- 1,179,256 and sar Will Deis ses 0.001,3715742,1 = TERME: X 0.001,371,742,8 pat 1,179,256 X 0.001,371,74251 pee 1617. 635,101,377,6 37756 oe gates oa =" aye 00O, OI2 526, i yo 129,540,163 Pk F i ye Ma 8,617,640 and — wil be. (= BP. x"0.660,457-247 53-6 ican ie X 0.000,457,24.753 — 8,617,640 X 0.000,457247)3 __ 3940-392,622,37240 1,162,261,467 ~~ -15162,261,467 Retin. i —— 19453275782 _ and 76 will be (= R X 0.000,152,415,7 = UAT RRNA x (2927782 X0.0R0. 1S 41507 ge 298 TES s92 29977 eo a900, 08m, eee one . 3 > b Dea) ) = 0:000,003,390,2 ; X 0.000,152,415,7 a 31,381,059,609 ~~ - 31,381,059,609 is — _13,431479,050_ | and = ~ will bé (==-T. X 0:000,050; 805520 = Finite; 86-435 X 0.000,050, __ 139431,479,050 X 0.000,050,805,2 _ 682,388.979,431,060,0. __ aoe ra 2.541,805,828,329 ~ —-2,.641,865,828,329 7% O:00G; ORs 268,45 : CUR E x4 G x6 128 L zt N x72 and confequently the feries 1 + elon BLL ty ee "e 2, = Pp z4 R wd sez . = porn ree aor — &c will be == 1,000,000,000,0 ++ 0-037;037;037,0 —= 0.004,672,473,7 + 0.000,869,334,5 —- 0.000,195,485,2 ++ 0.000,048,107,0 — 0.000,012,526,1 + 0.000,003,390,2 — 0.000,000,943,8 + 0.000,000,268,4 — &c = 1.037,958,137,1 == 0.004,781,428,8 SS 1.03 331 705;708,3. Further, fince g is = = we fhall have Vile =a [é = ie WAG °% = ane an ! FOR RESOLVING THE CUBICK EQUATION &c. 551 ; , . Therefore the tranfcendental expreffion and confequently 2fif{e= 1,817,121 ; : : C 2% E x4 G 2° i L x0 N x2 2/’\g x the infinite feries 1 onl a ee as he ee ae (g + £2 a ae 2 pe iad p zit rz T zt8 . a ee 2 = peo -- ee — WXc wil cd 1,817,121 x 1:033,176,708,3 — 2.066,353,416,6 _ en 1.137,157,853,8, &c. Therefore the root x of the propofed equation «7 —x* = = will be equal to 1.137,157,853,8, &c, @: Es. De * 167. This value of the root x of the equation x? — x = — is exact in the firft fix figures of it, to wit, 1.137,15, the more accurate value of the faid root being 1.137,158,104, as may eafily be found by Mr. Raphfon’s method of approxi- mation. Eee acm Nie Esau III. 168. Let it be required to find the root of the equation *7 — 5x = 4 by means of the fame tranfcendental expreffion:2 /*(g x the infinite feries 1 + C 2% E x4 Gz 128 ue N x12 -+- &c. 5 a i eee meeherg is*—)6 53°F 15'=> 4s =, Oe; terezie sy or gg, is = 43 9° is = 125, and £ Ey cw os = 4.629,629,629,629, &c, which is greater than 4, or = Therefore this equation cannot be refolved by Cardan’s rule, but may by means of the tranfcendental expreffion 2 3 (g x the infinite feries 1 + a —= G2 rz List? N.wi? ; : oy ee y + &c, in cafe the faid feries is a converging ne Cy eee 2 & & eee twit one. | 2 tt Now, fince = is = 4.629,629,629,629, &c, and mi or gg, 18 = 4, we thal 3 , . . . have = — a Or ZZ, = 0.629,629,629,629, &c, which is lefs than 4, or gg, in the proportion of about 6 to 40, which is a pretty large proportion of minority, and much larger than the proportion of zz to gg in either of the former ex- E24 G 2 rx amples; and confequently the faid feries 1 + es) =~ + est + &c will converge with a greater degree of {wiftnefs than in either of thofe examples. Therefore the aid tranfcendental expreffion 2 4: (g x the infinite feries 1 + E zt G 6 I 8 * ¢ oe — — - + &C will be equal to the root « of the equation «3? — & Pe ce £ 5*=4. This expreflion may be computed as follows. 169. Since aga A METHOD OF EXTENDING CARDAN’s RULE. 169. Since gg is = 4, and - — a Or Z%, 1s = 0.629,629,629,629, &c, we % 0.629,629,629,629 __ fall have = = EOE = 0.157,407,407,45 BZ x4 LB and confequently m3 eee X Ty = 9-157,407,40754 XK 0.157,407,407,4) $5 && =01024,7997,091595 26 net zt BB _ and <5 (= Ge X ge = 1024277720919 X 0-1575407:407,4) = 0.003,900, 09773 ee Zs and (cer e x ap > PR ASR BO ye 4 X 0.157;407,407,4) = 0.000,613, , 90452 5 zr? 4 ZL P and = (3 ra x ae 0.000,613,904,2 X 0.157,407,407,4) = 0.000,096, 653,03 i wre LN and a — ~~ x ee = 0.000,096,633,0 X 0.157,407,407,4) == 0.000, “: O15,240,75 and <; (= a x i = 0.000,015,210,7 X 0.157,407,407,4) == 0.000, 002394525 zi6 zit Dad and =z (= aa X je, =. 9:000,002539432 X 0.157,407,407,4) == 0.000, roy 000, 376,8 ; zi8 zi6 8 F¥ and a (= ae XZ = 0.000,000,376,8 X. 0.157,407,407,4) = 0.090, 0000593. Therefore will be (= C X 0.157,407,407,4 = - X 0.157,407,407,4, a 9915 79497249704, She tat) 5 OT ARG ti ag 9 ack, and —— wil be. (=e AF: x 0,024,7.77,001,0 “== et X 0,024,777,091,9 = f=) 10 X0.024,797;091,9 ___ 0-24.7,770,919,0, __ , and <= wil be (= G.& -0.003,900,097,7 = rat X 0.003,900,09737 = 154 X 0.003,900,097;7 _ 0.600,615,045,8, __ : To ngnGsGing Deak widane his CEOS a 0.000,091,54352 $ s P j and al be (=-1,:% 50,000,613,9004;2)— 935 x 0.000,613,904,2 = & 592049 | 935 X 0.000,613,904,2 __ .0.6'74;000,427,0, __ A tla ai ciate Seti (ee Tinea aval day PE Mg alone lee el 59,049 39,049 PMA ae Laie ia — _55:913 Nate and re will be (= L. X 0.000,096,633,0 = RA PRR 0.000,096,633,0 = «403,040.92 30 we —— ©,000,001.1 29,6 : and FOR RESOLVING THE CUBICK EQUATION &c. 553 I, 151 795256_ POP CED X 0,000,015,210,7 IMs 4; and a will be (= N X 0.000,015,210,7 = _ 1,179,256 X 0.000,016,210,7 &917-93 75309523952 Pan 129,140,163 Wp. £20;840;003 ) = 0,000,000,1 38,8 5 8 1617,640 _ 1,162,261,467 _ 8,617,640 X 0.000,002,394,2 __ 20.632,353,688,0, __ wT 1,162,261,467 = tpit, 162,267,467 ) = 0.000,000,017,7 5 = R X 0.000,000,376,8 = praee yt 31381 059,609 — 194,327) 782 X 0.000,000,376,8 _— 73.222,708,257,6, __ , 31,381,059,009 “Hs, 9353827059;609 727 Akg Weneas Bekah CX te and will be (== \Ts X0.000,000,059,3 = sts oot Raith _. and aes will be (= P X 0.000,002,394,2 = X 0.000,002,394,2 X 0.000,000,376,8 2,541,865,828,3 29 X 0.009,000, 13,43 154.79:050 X 0-900,000,05953 __ — 796.486, 707,66550 ©5993 = 2,541, 865,828,329 _ ENTE ERE ib ig soe cient 4 8 Io Iz and confequently the feries 1 + = — = 4 i sal i ip rl tke && & & g ae ee P uk Bat Ta Mas 5.9 ole Na ls &c will be f = 1.000,000,000,0 + 0.017,489,711,9 — 0.001,019,63 3,4 + 0,.000,091,543,2 — 0.000,009,720,7 + 0.000,001,129,6 — 0.000,000,138,8 -- 0.000,000,017,7 —- 0.000,000,002,3 + 0.000,000,000,3 == 0.000,000,000,0 = 1.017,582,402,7 —= 0.001,029,495,2 =e HOLG, 5.52,90755. Further, fince g is = 2, we fhall have 4/3 (¢ = f3 (2 == TZ 605021,049;05 and confequently 2 /3 (g = 2 X 1.259,921,049,8 = 2.519,842,099,6. There- fore the tranfcendental expreffion 2 3 (g X the agate feries 1 + caleaeen +}. 6 8 10 12 xl4 16 ee — &c will be equal to 2.519, § g & & & 842,099,6 X 1.016,552,907,5 = Senta tied Therefore the root of the propofed equation #3 — 57 = 4. is = 2.561,552,812,7. Q, Ev Ie 170. This number 2.561,552,812,7 is true to ten places of figures, and errs only in the laft figure, which pent to bean 8 inftead of a 7. For the accurate ett ot et 1+4.123,106,625,6 , OF gees Yt We value of w in this equation is ; 2.561,552,812,8. For, if we fubftitute at V7 inftead of w in the compound quantity #3 ts 5x, we fhall find that the faid s@uasiticy will be equal to 4, which is the abfolute term of the equation ¥* —5x—= 4, For, if wis = wt, we Vou. II. 48 fhall 554 A METHOD OF EXTENDING CARDAN’S RULE TH ZX UX SITAR ZX EX ITAA UAB IT EA XITEIDAIY : Ce . vie! a Eh Ee cca CS ON dk A RM fhall have «3 (= ———— 7 oa ‘ — IAB SITES IT IZ 52+ 20/17 au ee baba = _ 139+5,/ 17 was a 8 m4 amo opal” "3 ee y and 5x (= 5% att) ae SAE ated confequently #3 — 54 (= tS a el 9 ot 5 ae fa sosGm 3p or oe 171. Thefe examples fufficiently prove that the expreffion 2 4/* (g x the in- C2 E 24 G-2x° 1.28 Lat0 N zi? P ait Rzt6 T zt finite feries I - eg . gt i ¢ "eS o® gt? Ses: ar f pers Y a o gi8 z roy ~ — &c (which we derived from the former expreffion 2 (e x the infinite fe- Cc ss E s4# G 56 15° L 5% nisi? _. pst4 R 516 T zr8 & The Ke GOT gag gh ee ee ge ae by the peculiar train of reafoning ufed in the inyeftigation fet forth in art. 139, T40,° 447, Sey. 4158) B hie the true Foot of the equation #3 — gx =f, 29Vv'9 3/3 Sep be refolved by Cardan’s rule ; stided that ¢ (though lefs than Pry: is greater when ¢ is lefs than = or — is lefs than 4 f or when the faid equation cannot than 4/72 X if, or that = (though lefs than f) is greater than 2, I will, however, fubjoin one more example to the fame purpofe; which fhall be that of the equation x? — 63" = 162, which both Dr. Wallis and Mr. De Moivre have refolved by extracting what they call the impoffible cube-roots of the impoffible binomial quantities 81 + / — 2700 and 81 — — 2700. Now this equation may be refolved by the foregoing expreffion 2/3 (¢ x the C 2% E 24 G 28 12° Lie N zi” infinite feries 1 + er er Ci oS oe gt ty Sabie EF + &c, in the man- ner following. ED Xr Ae MP bi EB IV. 172. Let it be required to find the root of the equation v3 —634 = 162 by means of the tranfcendental expreffion 24/3 (g¢ x the infinite feries 1 -+ C 2% E.2* Gz 128 Toh N 2i% Pp zi4 RxI5 T x8 ee os a ey —- — —-— Ht ee — XC. aa ge ‘* ise gt ve fn pe Pas Pine : °o infinitum. , Tete 71s. 02 sf 18 ce LO a =, or g, is = 813, or gg; is = 65615 — . Ee : : tt : iS 21 3 and = is = 9261, which is greater than 6561, or oi, Therefore this equation cannot be refolved by Cardan’s rule, but may by the expreffion C 2x E24 G28 128 so Nizt 273(¢ X the infinite feries t + eeipiaiersty 7 por isifueiges ar &c, in cafe that feries is a converging one. FOR RESOLVING THE CUBICK EQUATION &C. 555 Now, fince 2 =— is = 9261, and - or gf, is = 6561, we fhall have 2 “ ees (= 9261 — 661) = = 2700, that is, zz will be = 2700, which is lefs than 6561, or gg, in the proportion of 100 to cS, or) 243. Therefore the feries C 22 E z* G 26 12° L zt Nz Efe = tH eae to a —— +. &c will be a converging feries, and the expreflion 2 f(g x the faid feries will be equal to the root of the. equation w* — 637= peal %% ” | 2700 (ee -— 173. Since 2% is = 2700, and = is = 6561, we fhall have — = sor a ee 5ag) = Or42 14522563 NN oe 4 % and confequently a C= a x a mar Org. t 1552250535 June ak = 00 XO. 633, Te 26 : a age) = 041690350,878,05 x6 za BB 100 100 X 0.169,350,878,0 and — (= — a ny BO 0,878,0 So C88 eee oe wapemry rama = 0.069,691,719,35 28 26 B® : 100 100 X 0.069,691,719,3 and — (= — — = 0.069;691,71 es Ce terreno ee EES s ( 2° x g : 9> 9 27 953 x 243 243 171, $9691719392) = 9,028,6795719585 are x8 RR 100 100 X 0.028,679,719,8 and — (= -— — = 0.028,6 19,9 Ed 2 fn tte et a Re EARL, gio ( 2 x 8 , 79971930 X 243 243 2 ? pn) F 0.0115802,353,8 5 eR ae i . 100 __ 100X0.011,802,353,8 __ and os (= ——°xX Bx Sent ah Rese! 9 sees ere rr rom ee) = 0.004544554130 5 zis axe Bw 100 100 X 0,004,445,413,0 __ and — (= — — = 0.00 13,0 — 3. eS ope 45445,413,0 X so re eee = 0.001 pM Uy Bind as BR ___ 100 ¥ 0.001,829,388,0 and =e (= Fa X a, ti 829,388,0 x = ——— = toc: ee way ee) | 2a = 0.000,7 32,894,655 py sea ae BX 100 __ 100 X 0.000,752,934,5 __ and ro i ~ x in — ine 000,752, 834,55 om aT ped oe ey L: = ee = 0.000,309,808,4. Therefore as Will De"(== CO Oty, 221039, 7 ne X 0.411, 522,63357 411562250335 me ESET) = 0.045,724573720 5 4B2 and 556 A METHOD OF EXTENDING CARDAN’S RULE I | and == will be (= E x 0.169,350,878,0 = ro X 0.169, 350,878,0° == gt 10 X0.169,350,878,0 ___ 1.693,508,780,0 Sey 243 dy 243 dS will be & G x 0.069,6 = a4 6 and —- will be (= G x 0.069,691,719,3 = gs; X. 0.069,691,719,3 = ea ESP hile = 0.006,969,1 71393 = 0.001,635,806,2 ; 6561 Les a a ap and ~F will be Se I x Mee eae a5,08h 0.028,679,719,8 =. 935 X 0.025,079,710, 26.015,530,013,0 safe le LX Bar AK SION eae Lae at, tae WR ah 5 129,45 ‘ eye | CARE 24545123,43 LZ ; Se.Q1 and —— will be (= L X 0.011,802,353,8 = aaRe X 0.011,802,353,8 = & > 555913 X 0.011,802,353,8 _ 659.905,008,019,4 22 eS I *) = 0.000, 1 6 4,782,969 4,782,469 cay : 27379999973 z eee 117025 . and - “ay ~ will be (= NX> 0.0045445 545 350 = Pid ae 0.004,445,413,0 Lia ey ecern, fore a — 52242-2791952,728,0, __ : 129,140,163 129,140,163 ) = 0,000,040) 59357 3 8,617,6 ) (=P x 0.001,829,388,0 = Mita NEY, 0.001,829,388,0 14 aR pniae __ 8,617,640 X 0.001,829,388,0 _ 15765.007,204,320,0 Hd 1,162,261,467 ve 1,162,261, 467 has 9.000;01. 955 Ose 2 _ 194,327,782 and ~ a = will be (= R X 0.000,7 52,834,5 = casa cee 3 a, adage 27,782 X 0.000,752,834,5 _ 146,296.658,598,079,0, __ Y 3.1,381,059,009 3.1,381,059,009 ) = 0.000,004,661,93 ana _1314315479,050 and — ae = will be (= T X o. 000,309,808,4 = afar, 2,541,305,828,329 X 0.000,7 52,834, 5. X 0.000;3095 13)431,479)050 X 0.000 000,309,808,4 __ 4,161,185.034,114,020,0 oa saa Reaa bec bekees a atbeeatg me —) 0.000,001,, 637,0; C xz E 24 G2 128 B Zio N 27 and confequently the feries 1 + —— — — + ett eee Pp 24 R z16 ptt - = s > 5 * Cambie 7 CET &c will be é I.000,000,000,0 + 0.045,724,737,0 —= 0.006,969,171,9 + 0.001,635,806,2 —- 0.000,454,123,4. + 0.000,137,969,7 — 0.000,040,59357 ++ 0.000,013,564,0 — 0.000,004,661,9 + 0.000,001,637,0 —- &c = 1.047,5135743,9 — 0.007,468, 55059 ext 3.040,04 6410750. Further, fince gis = 81, we fhall have /?{g = /?[8 (81 = 4.326,749, and confequently 2 /?{g (= 2 X 4.326,749) = 8.653,498. Therefore the tranf- cendental = > FOR RESOLVING THE CUBICK EQUATION &c. oy , ate ke vers ve ‘ E 4 G <6 s cendéntal expreffion 2 4/?(g x the infinite feries 1 + <= — oc pose AEs [ha && & & Ls Pee ib, “ad <2 ; Bee Crt eae ae Tree pe ae &e will be = 8.653.498 x 1.040, 045,103,0 = 9.000,028,737,9, and confequently the root of the propofed €quation x? — 63 x = 162 is equal to 9.000,028,73739. Q: Ew Te 174. This number 9.000,028,737,9 1s true to five places of figures, the true value of « in the equation «7 — 63* = 162 being the whole number 9, as will appear by fubfiituting 9 inftead of x in the compound quantity x3 — 63 «. For, if we fuppofe « to be equal to 9, we fhall have «3 = 729, and 63 x(= 63 X 9) = 567, and conilequently «* — 63 (= 729 — 567) = 162; and confe- quently g is equal to the value of x in the equation «* — 63 x = 162. Q. E. Ds seeentenenemmnanemennammmenininiat ESS SCTE TIT ET ne ny A S SC ere oz ols Uns 175. This refolution of the equation x* — 63 x = 162 anfwers to Dr. Wallis’s refolution of it by extracting the cube-roots of the impoffible binomial quanti- ties 81 + y — 2700 and 81 — 4“ — 2700, in as much as both refolutions are originally derived from Cardan’s rule. But the difference between them is, that the method here delivered is intelligible in every ftep of it, whereas Dr. Wallis’s method treats of impoffible quantities, or quantities of which no clear idea can be formed, in the whole courfe of the procefs by which the value of w is invefti- gated, though it concludes with a refult that is intelligible, to wit, that w is equal to the fum of the two impoffible quantities — as - X / — 3 and 2 pe — x ¥ — 3; of which quantities the impoffible members + = xX ¥ — 3 and —— x VY —3 are equal to each other, and are marked with the contrary figns 2 + and —, and therefore (when added together in order to obtain the fum of the faid two impoffible quantities ~ + V — 3 and 2 — a — 3) will deftroy each other, and leave us only the two poffible members of the faid two quanti- ties, to wit, 2 and 2, of which the fum is the whole number 9. Doctor Wallis’s method of finding “ -- = V¥ — 3 and aa _ -. / — 3 to be the cube-roots of the impoffible binomial quantities $1 +4 /— 2700 and 81 — / — 2700, is confidered by both Profeffor Saunderfon and Mr. De Moivre as only sentative, and not likely to fucceed in equations of which the roots are incommenfurable to unity, which is the cafe with ninety-nine equations out of a hundred, when the equations are taken at random, and not framed on purpofe with rational numbers for their roots). But Mr. De Moivre has fupplied this defect, and given a certain method of finding the cube-roots of fuch impoffible binomial quantities : but not without the trifection of an angle, or finding (by the help of a table 558 A METHOD OF EXTENDING CARDAN’S RULE a table of fines, or otherwife) the cofine of the third part of a circular are of which the cofine is given; by means of which trifection it is well known (inde- pendently both of Cardan’s rule, and of Mr. De Moivre’s procefs) that the fe- cond cafe of the cubick equation y? — gy = r (in which = is lefs than £) may be refolved. So that Mr. De Moivre’s method of doing this bufinefs, though more perfect than Dr. Wallis’s, does not feem to be of much ufe in the refolu- tion of thefe equations. And both methods are equally liable to the objection above-mentioned, of exhibiting to our eyes during the whole courfe of the pro- ceffes, a parcel of algebraick quantities, of which our underftandings cannot form any idea ; though, by means of the ultimate exclufion of thofe quantities, the refults become intelligible and are true. It is by the introduction of fuch needlefs difficulties and myfteries into algebra (which, for the moft part, take their rife from the fuppofition of the exiftence of negative quantities, or quanti- ties lefs than nothing, or of the poffibility of fubtracting a greater quantity from a leffer) that the otherwife clear and elegant fcience of algebra has been clouded and obfcured, and rendered difgufting to numbers of men of learning, who are poflefled of a juft tafte for reafoning, and could therefore, if they pleafed, make great advances in the mathematical {ciences, but who are apt to complain of this branch of them, and defpife it on that account. And, doubt- lefs, they have too much reafon to do fo; and to fay, in the words of the famous French mathematician and philofopher, Monfieur Des Cartes, in his diflertation De Methodo, page 11—Algebram vero, ut folet doceri, animadverti certis regulis et numerandi formulis ita effe contentam, ut videatur potiis ars quedam confufa, cujus ufa ingenium quodam modo turbatur et obfcuratur, quam fcientia, qué excolatur et per- fpicacius reddatur. If this complaint was juft in Des Cartes’s time, there is cer- tainly much more reafon for it now. _ 1496. The paflage above alluded to in Dr, Wallis’s Algebra, is in the 48th chapter, pages 179, 180, of the folio edition printed at London in 1685. And Mr. De Moivre’s method of extracting the cube-root of an impoffible binomial quantity, as 81 + / — 2700, or a ++ / —4, is publifhed in the appendix to the fecond volume of Profeifor Saunderfon’s Algebra, pages 744, 745, 746, 74.7. It is very ingenious, and fhews that author’s great fkill in the ufe and manage- ment of algebraick quantities. See alfo on this fubject Monfieur Clairaut’s Elémens d’ Algébre, and a paper of Mr. Nicole in the Memoirs of the French Academy of Sciences for the year 1738, pages 99 and too. See alfo Mac Laurin’s Algebra, part 1, the fupplement-to the 14th chapter, pages 127, 128, 129, 1303; and the Philofophical Tranfactions, No. 451. Another q A ee ct FOR RESOLVING THE CUBICK EQUATION &c, 559 Another expreffion of the value of the root of the equation RIMS: but greater 3 3 x? — gx —=t, when t is lefs than than /2 xX asd or — ; is lefs than + a, om greater than 373 ey os i ree tA than pr derived from the foregoing expref- Sion of it. 177- But there is another expreffion for the value of the 1 root # of the equa- tion «* — gx = ¢ in the cafe here fuppofed, which, as it may be derived from C2 the foregoing expreffion of it, to wit, 2 /” [¢ x the infinite feries 1 + Lae: 12° L zr N zl? p zis R to a . - e 5 think, to be omitted. This expreffion does not confift seri or an infinite fe. ries (as the foregoing expreffion does), but partly of a finite algebraick expref- fion, and partly of an infinite feries; and fewer terms of the infinite feries are ne- ceflary to be computed and added together in order to obtain the value of the : ; . 4 feries to any propofed degree of exactnefs, than of the feries 1 + ri -—- #3 “|. = ie follows, to wit, /; erties 4-4/3 [g—z+4av? (gx the infinite feries © = = < — “o a =f a + &c, ad infinitum; of which expreffion the firft part, to wit, ‘ist th V3 Ae — 2, is algebraick ; and the latter part, to wit, AS Pp zi4 et 47% (¢ x the feries — a AR a a - pe 4. =r + &c is tranfcen- dental. Bees! Pp zt4 A . CUZ The terms of the feries — + —> + —> ge a eo gio Lr N 2 ++ &c, contained in the foregoing expreffion. It is as + &c, are taken ¢ I from the feries that is equal tolr + =)3 , or the cube root of the binomial quan- : z : : BZ Cz DZ E 2* FS Gz tity I + +; to wit, the feries 1 + = — =~ on ee ae arf H27 128 4 K 29 Tear? M 2?! N xi? 0 233 P zi4 Q2ts nia a $ x7 g! 2 2? gr pr? yt 23 git gs gre gi 18 Ca — —_ + &c, ad infinitum, by beginning with the third term, a —,*and ae aay “fourth term reckoned from it. sie Si eee v3 x6 L ze ate as oie Toe Ps Tay 7s + ee may be deriv os eee the foregoing tranfcendental expreffion 2 4/3 (fz. x the 4 E 2 Gz 1S, sh v2 p xi4 x16 Pmenteicreas | —s — tt ee ee — — <5 oY 88 g g z g g git — “gis —- — &c, in the manner following. ie . The 560 A METHOD OF EXTENDING CARDAN’S RULE The derivation of the mixt expreffion of the value of the root of the cubick equation x3 — gx = t, given in the pre- ceeding article 177, from the former tranfcendental ex- preffion of it. 178. By the binomial theorem in the cafe of roots we have /3 [1 4+-— = : ; . B& C zz D 23 E x4 F 25 Gxt H 7 rae the infinite (ents 7 Tye tea te tes yer le ee K 29 i ztO Mit N 31? o 213 Pp t4 Qzts R xt6 $zt7 T 218 9? fae Pei i ee g3 a git gis 7% “gio gt — ge &c; and, by the refidual theorem in the cafe of roots, we have /3 ae = | are t BS Cz D x3 Ext F 25 G26 H 27 ref the in nite EW Miceekivertie wien Ty 2G EAP TT io: vee ae ae K 2? Li zr° M zit N xt 0 213 Pzt4 Q2i5 AE Sule s zt? Ts vy fle giidiheag ool a ea ee toe ae &c. Therefore, if we add /? F446 BY ae = and add the latter of the two foregoing feriefes (which is det toa F =) to the former feries (which is equal to 4/3 jt + =) the fums Se a obtained wall be equal to each other ; that is, /3 (1 tee + /3{1 — 2 will be equal to the infinite feries 2 — == &S 2E x4 26% 212° Bae 2 Nght 2 P xi4 ee 2723 ue sgt nt 2 a I & I z , . and confequently = x /* F -- , ta X Ve f ae will be equal to the in- ae C zz E x4 G 2 128 1. Re N x72 P zi4 R 216 finite alo Peete th een pep ee ee 18 2 C2% S 7 as z: 14 E oe Z . . : CZ 2 & ZB & 272 “= — &c, Let the infinite feries toe +e tH ——— 2 ge g g g g + &c be added to both fides of the laft equation; and we fhall have = x % 1 z . ‘ Taz CE 262° sp eonl Vir bot xv [: — — + the infinite feries eee Thats 2 ey aay one 14 18 ° . . iz 26 8 To SO EAeiel) Oem ae OC = te ee feries 1 + —— — — 4 — 4 4 4 6 8 ro gh ne mts ne && & & & & NZ zt —3 we tak ae &c. Therefore (multiplying both fides by Rem Span Zp awe xt x 2625 ane

    + + a t+ C&c will be See F : : C 2% E x4 G 2° 1z ceo N 2i® —_ 3 Su? 44 G x the infote frien 1 4 Ey pe ae 88 4 & p zit R x6 T 218 : — es + er — &c, and confequently will be equal to the root of the gt g Ren x3 — gx = ¢, when ¢ is lefs than rae but greater than /2 X i 2 g when = is lefs than 2 —, but greater than — ~ x 4 37? than 4 a Q2tE.AD, » OF An application of the laft expreffion of the value of the root of the equation x*—qx = t to the refolution of the above-mentioned numeral equations x* — 50 = 120, x py me 5 = 4s and HF — 63%= 162. 179. This new expreffion ae (g ams 2 +V73{g—z+4y:3 (ze x the infinite C 2% G 2 tS Pp zi4 feries — + — +> + pa aie 7m + &c ad infinitum may be applied to the refolution of the four foregoing cubick equations «7 — 50” = 120, x3—« I . . = —, 0° — 5” = 4, and «* — 63 x = 162, in the manner following. 3 < . In the firft equation «3 — 50 = 120 we have feen above in art. 162, 163, that g is = 60, and zz is = ie he Nigh pay &c, and * Is = 0.031, ie he I, and 6 778,692,2, and gr {S70 Peek et 73 ee .. Therefore the ies “F — + a + =n &e will be (= 0.031,778,692,2 + 0.000,549,144,0 +4- 0.000,022,372,1 + 0. 000,001,160,7 -- 0.000,000,067,6 + ae 4 = 0,032,351,436,6 &c, ang Biyori. 4C 4? (ge = 0.000 ROwEe & and Y3(g is = 62 A METHOD OF EXTENDING CARDAN’S RULE 5 43 (z will “9 (= ons 3. 914,867,641, 1) = 15.659,470,564,4, and 40/3 (g x the feries = Z -++ “+ ie + ar + — + &c will be (= 15. 6593470, 564,4 X 0.032,351,436,6) =0.506,606,369,1. And z wi rm be (= V tae 629,629,629,629, &c) = 32.087,842,395,93 and confequently g + z will be (= 60 + 32.087,842,395,9) = 92.087,842,395,9, and g — 2 will be (= 60.090,000;000,0 — 32:0875842,395,9) = 27-91231575 604,1, and and /% (ge + z will be (= V3 92.087,842,395,9) = 4-515,793,760,9 and: 3 {g — z will be (= Vv? 2'7.9125157,604,1) = 3 303324105 ee There- fore /3 (z +z2+Vv¥3(g> —xz+4/3(g x the infinite feries — Sar ane Pa We 1c a 2 Roe + + *5 - - &c will be = 4.515,793576059 + 3.0335 feats + 0.506, 606,369,1 = 8.055,810,284,2 ; and confequently the root of the equation «3 — 50% = 120 will be = $.055,810,284,2. Q. Es I This value of ~ is exaé in the firft feven figures 8.055,810, its more accurate value being 8.055,810,345,702. 180. In the‘next equation «3 — « pe we have feen above, in art. 165, 166, that 2 is — —, and zz is Sopp gz. T, tO rae ae and =z 18 = 0.000 POSH 5 and a = 0:000,048,107,0, To and ~ — picts = 0+000,003,39052, and ar waar A, — 0.000,000,268,4, and that /: (z i se i ; C zz G 26 is = aT Tea and 2 W3(g is = ee Therefore the feries = + jae 4. Eze p zit T Zt a + + &c is (= 0.03730373037;0 + 0.000,869,344,5 + 0.000,048,107,0 +--+ 0.000;003,390,2 + 0.000 Ogee Sate oh kos 0.037, CB P zit 958,147,1 and confequently 4 /? (g x the feries — +5 ays ah aC i. a “= + &e will be (= AVE x cosnastinan Mra Io g 2 af 1471 = V3(g x 0.151,832,58854 = O.1ST, 83295884) 1,817,121 X 0.151,832,588,4 = saree 0.08 3355667457. And z will be (= yae = ¥V 0.009;259,259,2) = 0.096,225,044,8, and confequently g + z will be (= + + 0.096,225,044,8 = 0.166,666,666,6 + 0.096,225,044,8) = 0.262,891,711,4, and g —z will be (= 0.166,666,666,6 — 0.096,225,044,8) = 0.070,;441,621,8, and /3 (¢g + z will be (= V3 0.262, 891,711,4) = 0.640,607,911,4, and /?/g — z will be (= /? 0.070,441, 621,8) = 0.412,993,403,9. Therefore /*(g +2 - /3 (p—z + 4 v* le x the FOR RESOLVING THE CUBICK EQUATION &c. 563 ¢ ° Czy Gz L xt p zt4 >» the infinite feries —— + —— +-- =— 4 — er er er 607,911,4 + 9.412,993,403,9 + 0.083,556,674,7 = 1-137,157,989,0; and confequently the root of the equation x? — v = =“ will be = 1.137,157,989,0. Q. E. I. OM be . + — + &c will be = 0.640, This value of is exact in the firft fix figures, 1.137,15, its more accurate value being 1-137,158,164. 181. In the third equation «3 — 5% = 4 we have feen above, in art. 168, 169, that gis = 2, and /3 fg is = 1.259,921,049,8, and zz is = 0.629,629, 629,629, &c, and = is = 0.157,407,407,4, and oa Is = 0.017,499,711,95 rates pzié , fi 1S == d Gu. 1 ‘ é 4 ee 0:000,091554352, anc IS = 0.000,001,129,6, an oon T mee Fs : 1S = 0.000,000,000,3 ; and confequently the feries ©.000,000,017,7, and 7 CxS Gx L zre Pp zit T zt8 ef os oo 4. a a 3 + Ir + &c will be (= 0.017,489,711,9 +- 0.000, _ 091,543,2 + 0.000,001,129,0 -- 0.000,000,017,7 -+ 0.000,000,000,3 + &c) = 0.017,582,402,7 &c, and 4 /3 [g x the feries — + = ax = + P ait here . q ; a a + &c will be (= 4 X 1.259,921,049,8 X 0.017,582,402,7 = 5-039,684,199,2 X 0.017,582,402,7) = 0.088,609,757,0. Further z will be (= / 0.629,629,629,629,629,629,629) = 0.793,492, 047,60, and confequently g + 2 will be (= 2 + 0.793,492,047,6) = 2.793, 492,047,0, and g — z will be (= 2.000,000,000,0 — 0.793,492,047,6) =, 1.206,507,95254, and /?/g + 2 will be (= ? 2.793,492,047,6) = 1.408, 366,911,5, and W3(g—xz will be (= 7% 1.206,507,952,4) = 1.064,576, 143,3- Therefore /3(g+%+ v3 e—z + 43g x the infinite feries eee G xo Ee Pp 14 Ta" += +-—-+——+H+ ro + &c will be (= 1.408,366,911,5 + 1.064, 576514353 + 0.088,609,757,0) = 2.561,552,811,8; and confequently the root of the equation v3 — 5 * = 4 will be = 2.561,552,811,8. ee a This value of the root*of the equation v3 — 5.“ — 4, is exact in the firft nine figures 2.561,552,81, its more accurate value being 2.561,552,812,8. See above, art. 170. 182. In the fourth equation v3 — 63 « = 162 we have feen above in art. 172, . . i : 173, that gis = 81, and #?(g is = 4.326,749, and 22 1s = 2700, and = is L ge 710 3 ¢ Gz. = =, and = is = 0.045,724,737,0, and os Is = 0.001,635,806,2, and ec Pa Oo: is 564. A METHOD OF EXTENDING CARDAN’S RULE “, pzi4. Pak IS = 0.000,137,96957, and v3 1S = 0.000,013,564,0, and = > is = 0.000, 001,637,0. Therefore the feries = + <5 + = a Se -L = + &c will be (= 0.04.5,724,737,0 + 0.001,635, ane 2 + 0.000,137,969,7 + 0.000,013,564,0 + 0,000,001,637,0 + &c) = 0.047,513,713,9, and 4 73 (g x the feries a +++ +4 + &c will be (= 4 V7? (¢ X 0.047,513,713,9 = +x 4:526,749 X 0.04755135713,9 = 17-306,996 X 0.0475513,71359) = 0.022,319,050,4. Further z will be (= eas / (900 3 = V900 X 73 = 30 X ¥3 == ERO.) Kp iT 2792505000755). = 5 Ts Weyer ; and confequently g + z will be (= 81 + '51-961,524,225,0) = 132. 19015524225 ,0, and g—2z will be (= 81.000,000,000,0 — 51.961,524,225,0) = 29.038,475,775,0, and ¥i(g+2 will be (= W? 132.961,524,225,0) = 5-103,976,447,9, and nia iz — 2 willbe (= /3,29.038,475,775,0),=— 3-075 674,958, 2. Therefore hos “aie aad ata (fx the feries 22 iop 2 See Ps € = $-103,975,44759 + 3:0732674,958,2 + O. Sie 33 70905 6,4. = 8.999,971,062,5 ; and confequently the root of the equa- tion «7 — 63% = 162 will be = 8.999,971,062,5. Go Tbe ee This number 8.999,971,062,5 is exact in the firft five figures, 8.9999, the true value of the root # in this equation being the whole number 9. See above, ATEN AS End of the refelution of the four equations x? — sox = 120, 43 —x = —, «3 — 5x = 4, and x? —63x = 162, by means Behl aad: expreffion /3(g + % 4 26 wad I I ¥3{[g—z+4Vv73 (¢ x the infinite feries — oat yeep oe we i + &e. om A third expreffion of the value of the root of the equation x3 — qu = t, derived from the ee ton obtained for it aboue ttt. 47t..139,\1A40, LAI fFC, «005 61570 183, We may alfo derive another expreffion for the value of the root of the 247 3/3” 4g? or when = is lefs than 4 f, but greater than 4 Pr. from the former tranf{cendental but greater than /2 x 2¥%%, equation #3 — gx =f, sa t is lefs than E x4 G 2° expreffion of it, to wit, 2 Ayes (z x the infinite feries 1 ot i oa a = re Fe FOR RESOLVING THE CUBICK EQUATION &C. 565 12° Lint? Wx p 24 R zt6 Ta zt — wa be ae te «Ce, which was obtained by the Oe a g g g g inveftigation fet forth in art. 139, 140, 141, &c....157. This expreffion E 24 iz Wai? R xt6 w xte is 4/3 (g x the infinite feries 1 — — — —~ ——> — —~ — — — &cad gt 2 DEP Taos 2° infinitum — /3(g + %— V3 |g—x; which (like the fecond expreflion of the value of w given above in art. 177) is partly algebraick, and partly tranfcen- dental; but in this expreffion the tran{cendental part, to wit, 4 /3 [g x the in- : ; E 24 128 N x32 R x76 wx? oh. ae § : finite feries 1 — —> — —- — —- — er — eC «& ec aad infinitum, is greater g ism fy than the algebraick part, to wit, /? (g +. z+ /3(g — 2, whereas in the fecond expreflion before-mentioned, to Pe the expreffion /3(g+z2+ VW3(g—z + G 6 Io 14 18 473 (z x the infinite feries = ete tet oes a 1 eaeay arate + &c, the alge- braick part, /? fg + z + a Ge ae was greater than te tranfcendental part ro 14 4./3(g x the infinite feries — a — + ee +. aare oo ast &c ad infinitum. And, further, in the laft, or na Seite to wit, 4/3 é x the infinite feries E x4 128 nN zt R ZI W 22° es er ig ee Te hg — %, it is the difference of the tranfcendental part and the algebraick part, or the excefs of the former above the latter, that is equal to the root of the equation 7? — gx = ¢; whereas in the former, or fecond, expreffion, to wit, /?(g +2 + 6 10 I V3 (g—2z +473 [g x the infinite feries > + -b Bae te = + = + &c ad infinitum, it is the fum of the algebraick and the tranfcendental parts of the expreffion that 1s equal to the root of the faid equation. In both expref- fions the indexes of the powers of z and. g in the numerators and denominators of the terms of the infinite feriefes contained in the tranfcendental parts of them, increafe continually by 4. This third expreffion 4 /*(g x the infinite feries 1 — Renae ssamaee 16 20 pert OR vate — &c ad infinitum — /? \g + = — V3 (g — 2, may be derived from the foregoing tranfcendental expreffion 2 /3 (¢ x the infinite feries 1 ++ C 2% E24 G2 128 LE zr N 212 p zi4 R i T ats at — &c ad inf- ra AT a a GE a cd nitum in the manner following. The 566 A METHOD OF EXTENDING CARDAN’S RULE The Derivation of the third expreffion of the value of the voct of the equation x? — qx = t, given in the pre- ceeding article 183, from the former tranfcendental ex- preffion of it. 184. Since V2 fg + zis(= V3 (ex V3 (rt + my = (by the binomial the- Z : A ; B& CZ D 23 orem in the cafe of roots) /3 (¢ x the infinite feries 1 -+ —— os +> 3 a E 24 F295 ez Hz? 128 K 2? Lz M 22! N 272 0233 p zi4 pteT gt eo te oe te 7 ee QzKs R xte s zt7 Tate Vv x9 WwW 27° & bins (G—-z . en + gis ae on ah g 7 Soe fe: + gt? EG, g?° + Cy an Se 1S ¢= J? (g x /3{t — =) = (by the refidual theorem in the cafe of roots) /3 {g¢ i , i BZ Cx D 23 E x4 F 25 G 29 H 2? ix® x the infinite feries 1 — — — — — —> — SO OS SS & K 29 L 27° M zt! N 214 o 213 P zr4 azis R x16 s x17 Tx Zhytpo 4 WADA? AGEL TN, lglSdoyp pp lekSur Dy raekbe | bier BOIS ep 4) kG) Pe ete eee rn ibis in *-5 Si : & & s is § § bay é pr pe 77 &e it follows that Vilg+tz+ V3(g—z will be = V3 (g x bh assens ivitch (93 2 C2% 2E x4 2626 2128 2 L229 2N2% 2P2z14 t € in nite eries 2 — 28 = gt —= 2 (Soweto canoe a a 2Rxr6 2T 2%? 2 Ww 220 : 202% 2Ex* See eC > Ce. =~Now the feries 2 — -— & is 262° 2128 2% 21° 2N2"2 2 P24 2 R Zr 27 xis 2 W229 &biae g° 2 gt eed Fer E. : 3 st . . ‘ . . . 2-E.% 212 2N2 2RZ infinitum is evidently lefs than the feries 2 —- —— — —~ ———- — —— — § & & & 2W 22° 5 4 a Stes 4 Z —— — &c ad infinitum, and therefore, 4 fortiori, is lefs than twice that feries, Z X ; E x4 128 N 22 R zt6 Ww 22° 7 or than the feries 4 — 4 sR Soir EEL SERA nee — &c ad infi- 4 s gt? 16 20 ; \ ; E x4 12° N zi? R zr6 nitum. Therefore /?(g x the feries 4 —4-= — #2 '_ 4°% eee oe g Be é Ware < ; : E 2C2Z ant — &c ad infinitum will be greater than 4/3 (g x the feries 2 — = 2E 24 2G2° 2128 2 Lt 2N x12 2Pxi4 2 R216 2728 2 W 229 rT Se a ay ee ea a TST 5 — &c ad infinitum, and confequently will be greater alfo than /? {g + z+ 5 : 4 . ze : BCs 2Ex* and, its equal, the quantity /% (¢ x the infinite feries 2 — ps ES } 262° 212° Be Fe 2N x12 2Pp2'4 2R ez? 2 T28 2 Ww x20 Se Ne cr, ey ie ae Tih MRR IE ira Pees 16 18 ogee 00) =a &c ad & 5 & & & & & Alb ah 4 , : ° . : 4Ex* infinitum, be fubtracted from the quantity 4/3 (¢ x the infinite feries. 4 — = & Ate! Nn 2! R z16 Wer? : : ; —- : : aioe aol Rr ereu 0 infinitum; and the remainders Baie ne g will “go FOR RESOLVING THE CUBICK EQUATION &c, 567 4E 24 ot c 4128 4N 2% 4 R x15 4 w 20 ‘ : se, ae &c ad infinitum — 3 (g+2z2— /3(g—2% will be equal to cach other; that is, /% (¢ x the infinite feries 4 —- 2 C2z 2E24 26° 128 2-b2* will be equal to /?(¢ x the feries 2 + ——; — oe lt S58 & 5 & & 2 wat? ap att eg eae By See 2 w 220 : a hg BOT Ee aN ead + &c ad infinitum, and confe- & & . C 2% E 24 G 2° ris? Lt? N 212 uently to 2 /3 (> x the feries 1 hf jaca MOA Sa gg 20 wp Ba ea sth sae ea SS q ai 14 7 oO 2 is ge gt ee te gt? 7 tie P% RZ TZ wee i ae et eee + &c ad infinitum, or to the tranfcendental ex- preffion for the root of < equation #3? — gx = ¢ difcovered by the inveftiga- tion contained in art. 139, 140, 141, &c....157. Therefore the expreffion E 24 8 zi2 16 20 /*[g x the infinite feries 4 — ~> hii com tar TRL, Baer Re &c ad infinitum — J} (g ee — V3 (g— 2, orthe expreffion 4 /3(g x the in- E zt N zi R x16 Ww 20 finite feries 1 — —- — — — ee ee ee &c ad infinitum — Vile+z— V3 i rari, * will be equal to 2 /3{g »x the infinite feries 1 + C 2% Ez* Lae N xi? Pp i4 1a igh matt w 370 eR Serie EE tpn Ea NEE GPS Re ag oe &c RE ge La Re Ia: oe pete os ea ad infinitum, or to the root of the equation 3 — gv = ¢. a E. D. An application of the laft, or third, expreffion of the value of the root of tbe equation x* — qx = t to the refolution of the above-mentioned numeral equations x3 — 50x = I Cee vary (hae K3 — 5x = 4, and x3 — 63x ==162. 4 8 185. This laft expreffion, 4 /3 (e x the infinite feries 1 — = ee er — 12 16 20 SS Speier ad infinitum — gtze /?(g—2z, may be applied to the refolution of the foregoing equations v3 — 50” —= 120, #3 — % = —, #3 — 5x4 =4, and v3 — 63% = 162 in the manner following. In the firft equation x? — 50 = 120 we have feen above, in art. 162, 163, that g is = 60, and zz is = 1029.629,629,629,629, &c, and = IS 0,003,366, Le) y ; 6 arallt ole alle . 284,2, and — is = 0.000,105,952,9, and — is == 0.000,004,998,2, and al E is = 0.000,000,277,2, and that /3 fie = 3.914,867,641,1. Therefore E z* Eee N 2? R zt : — the feries 1 — [Sargeras tee &c will be (= 1.000,000,000,0 & 2 : — 0.003, 368 A METHOD OF EXTENDING CARDAN’S RULE — 0.003,366,284,2 — 0.000,105,952,9 — 0.000,004,998,2 — 0.000,000, 277,2 — &C = 1.000,000,000,0 — 0.003,477, Hh he se =0.996;522,40 7e90 and 4 /3(g will be (= 4 xX 3. DF BOOT 4 she = ch 6593470356454, and 4/2 (gz x the infinite feries 1 — coe i —*> _ “a — &c ad infinitum will be (= 15.659,470,56454 X 0.996,522,487,5) = 15.605,014,55957- And we have feen above, in art. 179, that g + z Is = 92.087,842,39539, and g— 2 is = 27.912,157,604,1, and W73(g +2 is = 4.515,793,760,9, and /3(g—z is = 3.033,410,154,2. Therefore /slg+z + V3(¢—2 will be (= 14:51 §27939760:9 b+, 30332410315452) =" 7-549:208,08 55 teams 4”7(g¢ x the infinite feries 1 — ee Sa aa ihe at — &c ad infinitum é* & & & mete Rend ar a rele be (= 15. 605,014,55927 — 7+5495203, 915,1) = 8.055,810,644,6. Therefore the root of the equation x*— so¥ = 120 will be = 8.055,810,644,6. 0, JE ae This value of w is exact in the firft feven figures 8.055,810, its more accurate value being 8.055,810,345,702. . I . 186. In the next equation 43 —x = el have feen above, in art. 165, : I . I I BBs I E 2* 5° ay : ge and’ 1S —is = 166, that ¢ is =? and 22 Is eae or —; nd = e and re is Rae Nisin. 0.004,572,473,7, and = is = 0.000,195,485,2, and ae = 0.000,012, gid i haa IT £2657, and = Is = 0.000,000,943,8 and 3 (¢ is =a TPA Ty Therefore ? ? : 4 8 12 16 s the feries 1 — —— — — —-— —-~— — &c will be (= 1.000,000,000,0 & & & 5 —o. 004357247397 — 0.000,195,485,2 —- 0.000,012,526,1 — 0.000,000, 943,8 —- &c = 1.000,000,000,0 — 0.004,781,428, 8) = '0.995,218s69aeas 4 ; (e Wega? E 24 Coe NZ 2 238 & aT and 4 Suite ries 1 — SE PE TE c will be C= aa? [ge XNO.00 532 Less flee aon A ee X -0.9955;2100 571 s20 toe 4X 0.995,218,671,2 __ 3.980,874,284,8. _. 16817,121 ae 1,817,121 cleian oS eso ea es And we have feen in art. 181 that z is = 0.096,225,044,8, and g + zis = 0.262,891,711,4, and g—2 is = 0.070,441,621,8, and /3 [Z+ 2 iS Us 0.640,607,911,4, and /3 [g — z is = 0.412,993,403,9. Therefore /* |g + 2 + v3 {¢—z will be (= 0.640,607,911,4 + 0. 4123993 403; 9) 1,059,601, z iz 16 315,3, and 4 /? (g x the feries 1 — =F — WS EP _ BE 8c ad inf nitum — Rh [e+ z2— V/*(g—z will be (= 2.190,759,055,0 — I. 053,601, 31553) == 1613751577391 3; and confequently the root of the equation x3 — x = = will bei 20s y50n eos Q, E. D. This FOR RESOLVING THE CUBICK FRY ATION &c. 569 This value of » is exact in the firft fix figures, 1.137,15, its more accurate value being 1.137,158,164. 187. In the third equation «3 — 54 —= 4 we have feen above, in art. 168, 169, that g is = 25 ene V3 Rel 18: S/T .259407B, mee and 2% is = 0.629,629, pve tig &c, and = —is = 0. wg ae and = —= * is = 0.001 cia a. and - = is = 0.000 AS Must and — Nz" is = 0.000 eas 8, and ~ ©-000,000,002,3. Therefore the feries Cher eee: Bal pea AO PO will be (= 1.000,000,000,0 —= 0.001,019,633,4 — 0.000,009,720,7 — 9.000,000,1 38,8. — 0.000,000,002,3 — &c = 1.000,000,000,0 — 0.001, 029,495,2) = 0.998,970,504,8, and 4 /3(g x the faid feries will be (= 4v°(g X 0.998,970,504,8 = 4 X 1.259,921,049,8 X 0.998,970,504,8 = 5+0393084,1992 X 0.998,970;504,8) = 5.0342495,868,5. And we have feen in art. 181 that /?(g + 2 is = 1.408,366,911,5, and that /? (e—2z is = 1.064,576,143,3. Therefore 4 4/3(g x the feries 1 — a= te SE AE — &e — (E+ —Y*[g—xz will be = 5.034 gt 8 gr 495,868,5 — 1.408,366,911,5 — ara 14353 = 5.034,495,868,5 —. 2.4.72,0435064,8 == 2.561,552,813,7-;-and confequently the root of the equa- tion ** — 5% = 4 will be = 2.561,552,813,7- act ts This value of « is exact in the firft nine figures 2.561,552,81, the more ac- curate value of it being 2.561,552,812,8. Sec above, art. 170. 188. In the 4th and laft equation x? — 63 * = 162 we have feen above, in art. 172, 173, that gis = 81, and 4/?(g is = 4.326,749,. and zz is = 2700, ; 8 ZB- 100 Em . E2" and — is = —, and — is = 0.006,969,171 and —- is = 0.000 sn 553’ oe 29995171395 a 24545 ine. RRM hs %23,4, and a is = 0.000,040,593,7, and yw 8 = 0.000,004,661,9. 4 = . a4 128 N z1? a0 ts Therefore the infinite feries r — —— lr oe pe — ae — &c will be (= 1.000,000,000,0 — 0.006,969,171,9 — 0.000,454,123,4 — 0.000,040,593,7 — 0.000,004,061,9 — &c = 1.000,000,000;0 — o. 0075468,550,9) = “5 E x4 128 WN zi weie ©.992,531,449,1, and 4/3 f(z x the feries 1 — a eRe — &c ad infinitum will be (= 44643 (¢ X 0.9925531,449,1 = 4 X.4.326,749 X 0.9925531,449,1 = 17-306,996 X 0.992,5315449,1) = 171775737581 954. And we have feen in art. 182, that 7? [g + 2 is = 5.103,976,447,9, and that /? (g — z is = 3.073,674,958,2. Therefore /? [ge +z2+ Yi (g—z will be (= 5.103,9763447,9 + 3:073> 1674958; 2) == 8.2797;051,400,5. 2 eres mat 12 R zr fore 4 7? (ge x the feries 1 — —- — = —- — Se aad infinitum — Vou, IL. 4D ; va Epic, A METHOD OF EXTENDING CARDAN’S RULE SJiets—V¥ fe —z will be = 17.177,737,819,4 — 8.177,651,406,1 = 9.000,086,413,3 ; and confequently the root of the equation x5 — 63« = 162 will be = 9.000,086,413,3. q. "a. T. This value of w is exact in the firft five places of figures, 9.0000, the true value of « being the whole number g. A foort view of the three exprefions that have been here obtained for the value of the root of the cudick equation w—gx mt. 189. It appears therefore that the root of the cubick equation x? — gx = #, 29/9 b qv q tu. ; ut greater than 4/2 x £%4, or when — is lefs than Rahs” ss a 3/3" 4 L, but greater than a will be equal to either of the three following expref- when f£ is lefs than fions ; to wit, ~ . “ : 3 Cx E24 G2e 12° L2te Firft, to 2/3 {g x the infinite feries 1 + ———— + —- —— + — & & Ss & & Wat? Patt ne ed perk . : pout cred" Bi enert elon Ole re —- —= &c ad infinitum; which is an expreffion wholly tranfcendental ; And, 2dly, to the expreffion /3 |g +2z2+ 73 fr—z+4y? (zx the in- 6 zo I 18 5 - . . finite feries + ol + se + Bai +. pa +- &c ad infinitum ; which is an expreffion partly algebraick and partly tranfcendental, and in which the tran- {cendental part is added to the algebraick part ; $ And, 3dly, to the expreffion 4 /3 {¢ x the infinite feries 1 — = aa —— = — <> — &c ad infinitum — f3{g + 2 — V3 {g— 23 which is alfo an expreffion partly tranfcendental and partly algebraick, but of which the greater part is tranfcendental, and the algebraick part is fubtracted from the tranfcen- dental part. And we may obferve that the algebraick part of the two latter expreffions, to wit, the quantity /? fg + z + 3 [g—g, is fimilar to the algebraick ex- prefion Wife + 5 + #3 e —s, given by Cardan’s fecond rule for the root 29/4 of the equation y3 — gy = r, when r is greater than SA £. See on this fubje& Monfieur Clairaut’s Elémens d’ Algébre, pages 282, 283, &c, to page 297. 2 a or when 7 is greater than Of FOR RESOLVING THE CUBICK EQUATION &c, 57t Of the refolution of the equation gy — y3 =: 7 by means of either of the three foregoing expreffions of the value of the root of the equation x*'— qx = t, when t is les than 22¢2, but greater than /2 x L244. at * 3/3 190. The greateft poflible magnitude of the abfolute term r in the equation qgq—y=ris ve, which is the value of the compound quantity gy — y? when Jix= i And, when r is equal to this quantity 204 the equation gy —y? 373 . =r will have but one root, to wit, 4. and, when r is lefs than 294 Hn iy equation gy — y? =r will have two roots, of which the leffer will be lefs than 71 and the greater will be greater than ¥9° but lefs than /q. See my Differ- tation on the Ufe of the Negative Sign in Algebra, art. 114, page 92. There- fore, whenever we have an equation of this form, gy —y? = 7, propofed to us to be refolved, the abfolute term 7 of fuch equation will be always equal to the abfolute term ¢ of fome, equation of the oppofite form v3 — gx = ¢, in which the 29/9 abfolute term ¢ is lefs than , and which therefore cannot be refolved by means of the expreffions given by Cardan’s fecond rule; and confequently we may fuppofe the compound quantity gy — 9°, which forms the firft fide of the . equation gy -—y* =r, to be equal to the compound quantity x? — gx, which forms the firft fide of the equation x* — gx = 7. Now, upon this fuppofition that gy — y? is equal to x7 — gx, we may reduce the equation gy — 3? =r toa quadratick equation, and derive the values of its two roots from the value of (the only root of the-equation #3 —.gx =+) by proceeding in the manner fol- lowing. Since qy —y? is(—=r=t) = x3 — gx, add y3 to both fides; and we fhall have gy = «3 —qx + y3. Now let gx be added to both fides; and we fhall have qx + gy = «3 + y3. Therefore pr will be = ame , and confequently J a q will be = «xv -—wxy + yy. Add «xy to both fides; and we fhall have g + xy = ww + yy, and (fubtracting yy froin both fides) 7 + xy —yy = xx. Further, fince q is lefs than xx, and confequently than the other fide of the equation, let it be fubtracted from both fides; and we fhall have «xy — yy = wx — g. And, becaule *, or the fquare of =, is greater than « —.») X y, or than xx — xy, and confe- quently than the other fide of the equation, both fides may be fubtracted. from xx rx Let them be fo fubtra€ted ; and we fhall have a —vyty(= os — Xe xx Axe 4g 49 — 34% 197 4 4 4 ) 4 49 = 3% Therefore the fquare-root of = — xy 4 + yy will be = But the quantity < — «y+ yy has two fquare- 4 D 2 roots, 572 “A METHOD OF EXTENDING CARDAN’S RULE . x ° ° roots, a greater and a leffer, to wit, ery and y — =, according as arts 2 greater, or lefs, than y. Therefore for the determination of the greater value of y we fhall have the equation y — > = sh and confequently y = we -|- V2 ia pins , ° - cae ; and for the determination of the leffer value of y we fhall have — v 4g = 3% 2 the equation =. —y , and confequently < —=y + vt 3ee and Piet ay rip aaah eae lit As MAE DN confequently the greater root of the cubick equation I= z gy —y3 =r will be pi 3s, and the leffer root of the fame equation will be Sw oe Therefore, if ¢, or 7, 1s lefs than at, but greater than /2 q/4 x 373 values of the two roots of the cubick equation gy —y3 = r by firft computing the value of the root « of the oppofite equation v3 — gx = ¢ by means of either of the three expreflions fet down in the preceeding article 189, and then com- a+ sate DER ee a, eat -¢ tt : 3 : , or if —, or —, is lefs than f, but greater than £, we may obtain the puting the two quadratick expreffions Q.\ Es. 1, Of the Trifection of a circular arc by means of either of the three foregoing exprefions for the value of the root of the cubick equation x} — qx = t, when t is lefs than =A. 3 ip greater than / “s x A, or when - is le/s than ©, but greater than L. 44 54 191. If @ be the radius of a circle, and confequently 24 be its diameter, and . k be the chord of any arc in it, and y be the chord of the third part of the lefler of the two arcs whereof & is the common chord, the relation between the chords k and y will be expreffed by this equation 3 aay — y* = aak, of which the faid chord y will be the leffer root; as is demonftrated in many books of mathema- ticks, and, amongft the reft in my Differtation on the Ufe of the Negative Sign in Algebra, publifhed in the year 1758, pages 183, 184, art. 220, 221. Now let g be fubftituted in this equation inftead of 3 aa, and let r be fubftituted in it inftead of aak; and the equation 3 aay —y* = aak will thereby be converted into the equation gy —y*? = 1, in which “ will reprefent a, or the radius of the circle, and =v will reprefent 2a, or the diameter of the circle, and (=, or r di- vided FOR RESOLVING THE CUBICK EQUATION &c, $73 vided by 4, orr x 3, or) 2 will reprefent &, or the given chord of the greater arch, which is to be trifected, or of the third part of which we are to find the chord y ; and the chord y will be the leffer root of the faid equation gy —y? = 7. We muft therefore endeavour to find the leffer root of this equation gy — y? = 7. Now it appears from the foregoing art. rgo, that, if the abfolute term ¢ of the equation x* — gx = ¢ be equal to the abfolute term 7 of the equation gy —y? =r, the leffer root of the equation gy —y? =r will be equal to the quadratick expreffion Fai 4d 35 Therefore the value of the chord of the arc which is €qual to one third part of the leffer of the two arcs of which the given chord k, LS OG, ° ° ° . ° or re is the common chord, in the circle of which a, or a, is the radius, 2 : . . . r and 24, or a? is the diameter, will be equal to the quadratick ex#¥effion — _ “s : £ Sa Ag 3 Therefore, if r, or ¢, is lefs than a, but greater than /2 X ie or if aak is lefs than 24’, but greater than 4/2 x a, or if & is lefs than 3 3 ° . . (—, or) 24, but greater than (W2 xX <, or) 2 xX a, that is, if the given chord & is lefs than the diameter 2, but greater than »/2 X a, or the chord of a quadrantal arc, or of an arc of go degrees, or, if the given arc that is to be trifected, is lefs than the femicircumference, but greater than the arch of a qua- drant, we may find the value of the chord y by firft finding the root w of the op- pofite equation «3 — gx = ¢, by means of one of the three expreffions fet down in art. 189, and then computing the quadratick expreffion tov, Q. E. I. Of the analogy, or harmony, between circular arcs and lo- garithms, or between the meafures of angles and the meafures of ratios. TE Es IMS reer cae 192. This application of the expreffions obtained in the foregoing articles to the trifection of a circular arc is an inftance of the analogy, or harmony (as Mr. Cotes calls it) that fubfifts between logarithms and circular arcs, or between the meafiires of ratios and the meafures of angles. For from the expreffion V3 fe +5 + 3 le — 5s given by Cardan’s fecond rule for the root of the equa- tion y3 —gy =rin the firft cafe of that equation, or when 7 is greater than 3 ' ' : 20v9 or — is greater than 5 (the value of which expreffion is to be obtained by sea: megane . St pram extracting the cube-roots of the given quantities ¢ -+ s and e — 5, that is, by tri- fecting the ratios of e+ s to 1 and of e —s to 1) we have derived the three ex- preffions fet down in art, 189, by either of which we are enabled to find the value of 574 ' A METHOD OF EXTENDING CARDAN’S RULE —— of the root x of the equation «3 — gx = ¢, when’¥ is lefsthan 10/4, but greater than 42 X iv, from which value we may afterwards, by computing the qua- J dratick expreffion Sn 3 A", obtain the value of the leffer root of the equa- tion gy —y? =7, or fie value of the chord of an arc that is equal to the third. part of the leffer of the two arcs of which the given quantity oo ork, is the com- , mon chord, in a circle of which the given quantity vt, or 2, is the radius; .or, in other words, may trifeét the faid leffer of the two arcs of which 2, or &, is the common chord ; provided that the faid arc 1s greater ‘than the arch of a quadrant. And cgnfequently problems that require the trifection of.a circular arc, or of an angle, may, by means of the method here explained, ‘be folved by the trifection of a ratio, or by the help of a table of logarithms. . Of the refolution of the cubick equation x — qx = t by means of expreffions derived frags Cardan’s neh when t gp EMoody Eb i is lefs than sf 2 R a or os is lefs than 193. When the abfolute term # of the equation #3 — gw = #, or the abfolute term 7 of the pcr equation gy —y3 =r, is lefs than /2 x if or oa 7”, is lefs than 4, the leffer root of the equation gy — 3 =r may be found by or means of the expreffions inveftigated in the preceeding difcourfe contained in pages 379, 380, 381, &c..... 440 of this volume of tracts; which expreffions are derived from the expreffion given by Cardan’s firft rule for the refolution of the cubick equation y? + gy = 7, or gy + y* =r: and, when the value of the faid leffer root of the equation gy —y* =r has been fo "obtained, the value of the root x of the equation «3 — gw = +t may be derived from it by computing yo 4¢- 3y Por ° Jy! 2 the quadratick expreffion fince x3 — gas «<= gy — y*, we fhall (by adding y+ to both fides) have «3 —gx + y3 ==qy, and (by adding gx to both fides) w? + 9% = ge +.gy = X [« +5. “Therefore + 7 Gok we AY oo aty will be = = q; that is, xv — xy + xy will be = g; and con- J eanenty, fubtracting yy (which is lefs than g, and therefore than the other fide of the equation) from both fides, we (hall have wv — xy = ¢— yy, or xe — yr =q—yy5 and (adding = to both fides) xxv — yx + a = =g—yy + 2 — 49 — 4yy boy 4 FOR RESOLVING THE CUBICK EQUATION &c. 575 = 4 ne — ——— . Therefore x — = will be = a and « will be I a aR a a Q. E. D. lr en Conclufion concerning both branches of the fecond or irre- ducible cafe of the cubick equation y? —qy =r. 194. It appears therefore that in both the branches of the equation x3 — gx = #, or of the fecond cafe of the cubick equation y3 — gy = 7, that is, both when f, or 7, is lefs than ate, but greater than 4/2 xX v4, and when it is lefs ‘ qv Be ee es | ee mie, Be EE than /2 X WE or both when q? 8 To Is lefs than , but greater than ry . . 3 . and when it is lefs than at the value of the root x, ory, of the equation 3 — gx = t, ory? — gy =1r, may always be found by means either of the expreffions derived in the prefent difcourfe from Cardan’s fecond rule, given for the refolu- tion of the equation y? — gy = 7 in its firft cafe, or of the expreffions derived in the laft preceeding difcourfe in this volume of traéts from Cardan’s firft rule, given for the refolution of the equation gy + y3 =r, ory? +qy=r. And thus that which is called the irreducible cafe of the equation y3 — gy = r is fhewn to be always capable of being refolved by means of expreffions derived from one or other of Cardan’s rules by the help of Sir Ifaac Newton’s binomial and refidual theorems, which therefore feem to form a fort of bridge of communica- tion between the problems belonging to the trifection of a ratio, and the pro- blems belonging to the trifeCtion of an angle, or between the problems depend- ing on, or refolvable by means of, the afymptotick areas of an hyperbola, or a table of logarithms, and the problems depending on, or refolvable by means of, circular arches, or a table of fines and tangents. AS ee Se Cri Orr, 15-0 5M. 195. The firft perfon who feems to have difcovered the poffibility of thus ex- tending Cardan’s rules to the fecond, or irreducible, cafe of the cubick equation y3? —qy =r, was the famous German mathematician and philofopher, of the latter part of the laft century, Mr. Leibnitz of Hanover, as is fhewn in the fcho- . lium at the end of the laft preceeding difcourfe in this volume of tracts, pages 438, 439, 440. But he has done it by the means of what are called impofidle quantities, or quantities involving the {quare-roots of negative quantities, which, in the expreffions to which his calculations lead him, are of fuch magnitudes, and fo conne¢ted by the figns + and —, as ultimately to deftroy, or expunge, each other ; whereas in this and the preceeding tract I have carefully avoided all mention of thefe unintelligible myfteries. ; 6 “i | 4 x i MRE ey ee ee aA £ Fs Alar gif? ot #51i9 5 sta * bite ayot 2a 2 mer gape st svissh: enomiargcs sit Is 40, re fit eh AES 0p sles SHA washing mot shin orauloy 2idi ai Shull bt ALI ie 1A ac te v5 "toh sity eens Sb corer 40% = 1p Se nolReps ‘oth Yo Seo SG vuleyest osly Bais aid: ent Devit = 2.0L lov e9 to artnet te “bavictorgniad t0ot Ret mperer © ndenbers ya Sitié tetow 10 tgs, gcibnayod amoldorg. 989: pits cers eat i yiifidiieg. sit botsvesib. snd oF ecrisot osl ai 8 3 if} to eed: otolidg bas oaiusmcslitem osgreo euottt od an i df oijeit mwah A os verse 1 Bis : *) nS Su 2612 Wo seviloy ris ni Shuboltb fete ge: Salvoile ‘ae THe pad boligo 3; rnctadwto ances od} yd ti_saob ead: ost i ete tle 2 ai) pt Op. oviangag. 1 sens 3 Achinionsed du ao wa tid! beet Rev tac $. dd os 2y iat es to vari by of ytetentialis ont a ca Ha hy bioys eieie> ree cea. ° sie Rel itstip? “gab? to sodganid A od diets ue hora 1 Gh 2eglt gnc Up eaSe, fyeLpg axidua ont 4o},9) ie | <4 ctyhwintt iosbe be ae ‘4 Moy rings sitet: . » am af ae <, a a i BOG? sahasg fuck ‘Se easels. al eh ms WO iste =, dale paest Foe ay bias S sila to 18 “ ioe ott to rath sey asd dod ah ; 5 eh 4 , ; 8 thi ine ese . BT toes o io ~ *, re ‘ “,% sy ices wie Lk aes fifi S alilag ie (EDR Sd BME Ye te Net let COLIN IND 16 5 sani ih re ee & Sic m) andat otic toi viet = cose entig sels: baa usta ko cvbiietia ot &o-eaignobseh -aresidoy 9 «fb z oC} 3h emial: 16 evney sdt FS Ss baal! a oq 4 “yf SOS az: to noimelin. 33 o7 iyo ‘alddasd ta as to nAas Aes Serie Soe te"biesny Xo Steevie cig ni 4 ny at Ae “oteggal ba eam taidad sm Es bs a ald sr Hy Carag” a bat Fu. ta, te: ai cet Wid Wr: an the OF ioe egal Bt aS ae ft sbaeeet Gie ips doideo adi io plest Sidhels yo ebnos3t ads 9 + aglua. # od *- di oaivlovl: asik @ CRONE | ME GIT Ue RE CONCERNING THE METHOD sy wuicnu BeAr RAD aAgeNisoy Re Ue L Bx FOR RESOLVING The Cubick Equation x? +- gx =r in all Cafes (or in all Magnitudes of the known Quantities q and r) And the Cubick Equation x? — qx =r in the firft Cafe of it 29/9 3/3 were probably difcovered by Scipio Ferreus of Bononia, and Nicholas Tartalea, or whoever elfe were the firjt Inventors of them. ; : ’ 3 (or when r is greater than suk oi is greater than =) By FRANCIS MASERES, Esa. F.R.5. CURSITOR BARON OF HIS MAJESTY’S COURT OF EXCHEQUER. N. B. This Trac was firft publithed in the Philofophical Tranfactions for the year 1780. Vou. I. | 4 © ‘poe va ‘@OnTaM : ir oy ; + ~ ., ’ . | P i on} ane pre: ic , | ipniearsees i, ASAT wh ARE Sy NE ®) ed: Vy WO wa. Oe tee . \ + : ayn ‘ a ? wvitiatanyg2 We cram aht ~ Whe 3s Had % Sede a a ek ~e winay hi e Te Mad os (ome awh “esting te yo PAMES Lind ss Geary ya SA ENR sao f ict sinowodl param einind rh Pre ire x _ Mrpkaaadh Ral adit sony Shs “samnodiany; me iwi w : 4 ean i my] r : 5 \ oh a ‘ 4 ; 2 . pe : : ‘ n > cm oh Dee} Groat cy 2 ean ab vg 210MAaT, a i ; BELO sida 10: rEvos e ‘erexiaar ent 10 nouaa 49518 (OY tasy afd wat enotiarerD Ispidgaholidt ody ni ; bs 3 ea % y wert oy - " 4 § : i’ \G * ee a 7 $i ry z } Le 4 ah OMIM Ta cae OTS id ; "et lk a a4. ‘ee aes Wes ; \- Lote A € = Caney’ bee eer MyEsR oe CONCERNING em PP eB) ERASING as?) Roe Epes Bb? Gy Art. 13 HERE is nothing more amufing, or more grateful to an inquifi- tive mind, in the ftudy of the {ciences of Geometry or Algebra (for, if we banith from it the ridiculous myfteries arifing from the fuppofition of negative quantities, or quantities /e/s than nothing, the latter may deferve the name of a /cience as well as the former), than to contemplate the methods by which the feveral ingenious and furprifing truths that are delivered in the books that treat of them were firft difcovered. This we are fometimes enabled to do by the au- thors themfelves to whom we are indebted for thefe difcoveries, who have can- didly informed their readers of the feveral fteps, and fometimes of the accidents, by which they have been led to them: but it alfo often happens, that the au- thors of thefe difcoveries have neglected to give their readers this fatisfaction, and have contented themfelves with either barely delivering the propofitions they have found out, without any demonftrations, or with giving formal and po- fitive demonftrations of them, which command indeed’the affent of the under- ftanding to their truth, but afford no clue whereby to difcover the train of rea- foning by which they were firft found out; and confequently contribute but little to enable»the reader to make fimilar difcoveries himfelf on the like fub- jects. This feems to be the cafe with thofe ingenious rules for the refolution of certain cubick equations, which are ufually known by the name of Carpan’s Rules. We are told to make certain fubftitutions of fome quantities for others in thefe equations x? + gv = 7 and «3 —— qx ==r (which are the objects of thofe rules), and certain fuppofitions concerning the quantities fo fubftituted; by do- ing which, we find that thofe equations will be transformed into other equations which will involve the fixth power of the unknown quantity contained in them, . but which (though of double the dimenfions of the original equations «* + gx = rand x? — gx = 7, from which they were derived) will be more eafy to re- folve than thofe equations, becaufe they will contain only the fixth power and the cube of the unknown quantity which is their root, and confequently will be of the fame form as quadratick equations ; fo that by refolving them as quadratick equations we may obtain the value of the cube of the unknown quantity which is their root; and afterwards, by extracting the cube-root of the faid value, we 4h 2 may 580 A CONJECTURE CONCERNING ~ may obtain the value of the faid root, or unknown quantity, itfelf; and then, at laft, by the relation of this laft root to x, or the root of the original equation (which relation 1s derived from the fuppofitions that have been made in the courfe of the preceeding transformations), we may determine the value of x. And, if we pleafe to examine the feveral fteps of this procefs with fufficient at- tention, we may perceive, as we go along, that all thefe fubftitutions are legiti- mate and practicable, or are founded upon poffible fuppofitions ; though I can- not but obferve, that the writers on algebra, for the moft part, have not been fo kind as to fhew us that they are fo. But ftill the queftion recurs, “‘ How came “¢ Scrpro Ferreus, of Bononia (who, as Canpan'tells us, was the firft in- <¢ yentor of the former of thefe rules, and NicHoLas TarTALEa, who was the << firft inventor of the latter of them), or the other perfons, whoever they were, <¢ that invented them, to think of making thefe lucky fubftitutions which thus ‘* transform the original cubick equations into equations of the fixth power, ‘¢ which contain only the fixth and third powers of the unknown quantities «© which are their roots, and confequently are of the form of quadratick equa- “¢ tions?” To anfwer this queftica as well as I can by conjeéture (for 1 know of no hiftorical account of this matter in any book of algebra), and in a man- ner that appears to me to be probable, is the defign of the following pages. Art. 2. The moft probable conjecture concerning the invention of thefe rules, called Carpan’s Rules, by Scipio Ferrevs, of Bononia, and: by Nicuo- LAs TARTALEA, or whoever elfe were the inventors of them, feems to be this : that the faid inventors tried a great variety of methods of reducing the three cu- bick equations of the third clafs, to wit, «3 + gx =r and x?—gx = 7, and qx —«? = r (to fome one of which all other cubick equations may, by proper {ubftitutions, be reduced), toa lower degree, or to a more fimple form, by fub- {tituting various quantities in the ftead of «, in hopes that fome of the terms arif- ing by fuch fubftitutions might be equal to others of them, and, having contrary figns prefixed to them, might deftroy them, and thereby render the new equa- tion more fimple and manageable than the old one. And, amongft other trials, it feems natural to imagine, that they would fubftitute the fum or afte of two other quantities inftead of », as being the moft fimple and obvious fubftitutions that could be made. And, by making thefe fubftitutions, the above-mentioned rules would of courfe come to be difcovered, as well as the aforefaid limitation of them in the refolution of the equation «* — qx =r (which reftrains the rule to ‘ : . 2 : - thofe cafes only in which ¢ is greater than 20Ng or — is greater than ae and their utter inutility in all the cafes of the equation gx —«? =r. This will ap- pear by examining each of thefe equations feparately in the following manner. CS a SO OLA SA EL SE SS IG ie ARR AND Of the equation x? + gx =r. Art. 3. In the equation x? + gx =~, the inveftigator of thefe rules would naturally be inclined to fubftitute the difference of two quantities (which we will here THE INVENTION OF CARDAN’S RULES, &c. 584 here call y and z, and of which we will fuppofe y to be the greater) inftead of «, rather than their fum, or would fuppofe « to be equal to y — z, rather than to y + 2; becaufe, if he was to fuppofe « to be equal to the fum of the two quan- tities y and z, and was to fubftitute that fum, or the binomial quantity y + z, inftead of « in the equation «? + gx = 7, it is evident that (as the figns of x3 and gx are, both of them, affirmative) the terms of the new equation, arifing- from fuch fubftitution, would all of them be likewife affirmative; and confe- guently none of them, though they fhould happen to be exaétly equal to each other, could exterminate each other, and thereby render the new equation more fimple than the old one, which was the only view with which the fubftitution would have been made. He would therefore fuppofe « to be equal toy — z; and by fubftituting this quantity inftead of » in the original equation x? + gx = r, he would transform that equation into the following one, to wit, FFAS aT By ene Gy Rg eas ory>— 372 XVJ—-Z—Bt+y Xly—sor. Now in this equation it is evident that the terms 3 yz X y— zand¢ xy—~z have contrary figns; and therefore, if their co-efficients 3yz and ¢ can be fup- pofed to be equal to each other, thofe terms will mutually deftroy each other, and the equation will be reduced to the following fhort one, y3— 2? =r. And, if in this equation we fubftitute, inftead of z, its value me derived from the fame fuppofition of the equality of g and 3,yz, the equation will be y? — Fai = By and, by multiplying both fides by y’, it will be y*° — £ = ry*; and, by adding £ to both fides, it witl be y® = ry? + ae and, by fubtracting ry? from both fides, it will be y® — vy? = ; which equation, though it rifes to the fixth power of the unknown quantity y, is evidently of the form of a quadratick equa- tion, and may therefore be refolved, fo far as to find the value of the cube of y, in the fame manner as a quadratick equation ; after which it will be poffible to find the value of y itfelf by the mere extraction of the cube root; and then at laft, from the relation of to « (derived from the foregoing fuppofitions that y — % was equal to v, and that 32 was equal tog, and confequently z equal to =) we fhall be able to determine the value of «. Art. 4. It would therefore remain for the inveftigator of this method to en- quire, whether or no the fuppofition ‘* that 3 yz was equal to ¢,” was a poffible fuppofition ; that 1s, whether it was poflible (whatever might be the magnitudes of g and r) for two quantities, y and z, to exift, whofe nature fhould be fuch that their difference y — z fhould be equal to the unknown quantity win the equa- tion x? + gx = r, and that three times their product fhould at the fame time be equal to g, or their fimple produc to the third part of g. And this fuppofition he would foon find to be always poffible, whatever may be the magnitudes of qgand r; becaufe, if the leffer quantity z is fuppofed to increafe from 0 ad infini- tum, and the greater quantity y is likewife fuppofed to increafe with equal {wift- ne{s,, 382 A CONJECTURE CONCERNING nefs, or to receive equal increments in the fame times, and thereby to preferve their difference y — z always of the fame magnitude, or equal to w, it is evident that the product or rectangle yz will increafe continually at the fame time from o ad infinitum, and confequently will pafs fucceffively through all degrees of magnitude, and therefore muft at one point of time during its increafe become equal to x | And having thus found this fuppofition of the equality of yz and rm or of 3yz and q, to be always poffible, whatever might be the magnitudes of g and 7, our inveftigator would juftly confider his folution of the equation «3 + gx =r (which was founded on that fuppofition) as legitimate and complete. And thus we fee in what manner it feems probable, that Carpawn’s rule for refolving the cubick equation «* -+ gx = r may have been difcovered. | Of the equation x* — qx =r. Art. 5. In this fecond equation x3 — gx =r, in which the fecond term gy is fubtracted from the firft, or marked with the fign —, it feems to have been na- tural for the perfon who invented thefe rules to fubftitute the /um as well as the difference of two other quantities, y and z, inftead of x, in the terms x? and gx, in hopes of fuch an extermination of equal terms, and confequential reduc- tion of the equation to one of a fimpler and more manageable form, as was found to be fo ufeful in the cafe of the former equation «3 + gx =r. We will therefore try both thefe fubftitutions ; and, as that of the difference y — z has in the former cafe proved fo fuccefsful, we will begin by that. — Art. 6. Now, by fubftituting the difference y— x inftead of x in the equa- tion «3 -- gx = r, we fhall transform it into the following equation, to wit, PF = 3.99% +b 3y2z— 2 — |g Xly — aA = 7, ory? — 3yz x [y—z— zi —g x y — % =r; in which the terms 3yz X y— 2% andg X y—2z have both of them the fame fign — prefixed to them, and confequently can never extermi- nate each other, whether 3yz be equal or unequal to g. This fubftitution therefore is in this cafe of no ufe. . Art. 7. We will now therefore try the fubftitution of the /um of y and g, in- ftead of their difference, in the equation «3 —gx =r. Now, if x be fuppofed to be equal toy + z, and y + 2 be fubftituted inftead of it in the equation «3 — gx = 7, that equation will be thereby transformed into the following one, to wit, Ji SEIS WS BAe Bier Dee ory3 + 372 XVH+RZtz—gxXy+er=ns Now in this equation, the terms 3 yz x{y + zandg x y + z have contrary figns. Confequently, if they can be fuppofed to be equal to each other, they will deftroy each other, and the equation will be thereby reduced to the follow- Ing ~ ~ THE INVENTION OF CARDAN’s RULES, &c. 583 ing fhort one, y? +- z3 = 7; that is, if 3yz and g can be fuppofed to be equal to each other, or if yz can be fuppofed to be equal to £, the equation will be re- ~ duced to the fhort equation y? +23 =r. And, if in this fhort equation we fubftitute inftead of z its value (derived from the fame fuppofition of the equality of 3yz and ¢), the equation thence refulting will be 3 +- mr =e anc by multiplying both fides by y?, it will be y® + > = ry?; and, by fubtract- ing y° from both fides, it will be ry? — y® = £ ; which, though it rifes to the fixth power of y, is evidently of the form of a quadratick equation, and confe- quently may be refolved in the fame manner asa quadratick equation, fo far as to find the value of y3, or the cube of the root y.; after which it will be poffible to find the value of y itfelf by the mere extraction of the cube root ; and, laftly, from the relation of y to « (contained in the two fuppofitions, that_y + z is equal to x, and that 3 yz is equal to g, and confequently that x is equal i we may determine the value of x. Art. 8. The only thing, therefore, that would remain for the inveftigator of thefe rules to do, in order to know whether the foregoing method of refolving the equation «3 — gx = 7 was practicable or not, would be to enquire, whether it was poffible in all cafes, that is, in all magnitudes of the known quantities ¢ and r, for 3.yz to be equal to g, or for yz (or the product, or rectangle, of the two quantities y and z, whofe fum is equal to *) to be equal to £; and, if it was not poffible in all cafes, but only in fome, to determine in what cafes it was pofs fible, or what mutt be the relation between g and 7 to make it poffible. Art. 9. Now, in order to determine this queftion, it would be proper and na- tural to obferve, that the quantity yz, or the product of the multiplication of the two quantities y and z, whofe fum is fuppofed to be equal'to x, can never'be greater than the fquare of half that fum, that is, than the fquare of =, er than 7 by El. 2, 5, but may be of any magnitude that does not exceed that fquare. Therefore, if r is greater than at it will be impoflible for yz to be equal to it; but, if - is either equal to, or lefs than, =, it will be poffible for yx to be equal to it ; and, if 4 is exactly equal to re z will be exactly equal to y, and each of them equal to one half of x. We mutt therefore enquire what is the . . Df a . magnitude of « when 7 is equal to 7%. Now, when aa ffm 5 we will. be = ° 2\/ Pu « = 5 | <2. and.% = wrt therefore, when » is-lefs than ae it will be inpofibie for 3 { 3: VS. 584 A CONJECTURE CONCERNING : 2/9 yz to be equal to me ; but when w is greater than ae it will be poffible for yz to be equal to — But when x 1s = =v4, mt willicb elem st, and gx will be = “1¥4 on vt, and confequently x3 — gx will be = 89d _ O99 — 29/9 Ce aude ha eae fe Therefore, if it be true (as we fhall prefently fee that it is) that, while # in- creafes from being equal to 4/g (which is evidently its leaft poffible magnitude) to any other magnitude, the compound quantity v? — gx, or the excels of #3 above gx, will alfo continually increafe from 0 (to which it is equal when « is= 4/q, or «x is = g) to fome correfpondent magnitude without ever decreafing ; A it will follow that, when w is lefs than , the compound quantity «3 — gw will 24/9 V4 be lefs than Therefore, it «3 — gx, or 7, is lefs than sea or — is lefs than 4 _ the foregoing method of refolving the cubick equation #3 — Tee =r will fe PA Cas but, if #3 — gx =r, orr, 1s greater than SWE or re is greater than 4 o it will be practicable. Art. 10. It now only remains to be proved, that while » increafes, from be- ing equal to 4, ad infinitum, the compound quantity «3 — gx will likewife in- creafe from 0 ad infinitum, without ever decreafing. Now this may be demon- {trated as follows. Art. 11. It is evident that while w increafes from being equal to ~/q ad infi- nitum, both the quantities x3 and gx will increafe ad infinitum likewife. But it does not therefore follow, that the excefs of * above gx will continually increafe at the fame time. This will depend upon the relation of the contemporary in- crements of #3 and gw: if the increment of x3 in any given time is equal to the contemporary increment of gx, the compound quantity «? — gw will neither in- creafe nor deécreafe, but continue always of the fame magnitude during the faid time, notwithftanding the increafe of w; if the former increment is lefs than the 2 latter, THE INVENTION OF CARDAN’S RULES, &é. 585 latter, the faid compound quantity will decreafe; and if it is greater, it will in- creafe. We mutt therefore enquire whether the increment of x? in any given time is greater or lefs than the contemporary increment of gx. Art. 12. Now, if % be put for the increment which « receives ‘in any given time, the increment of #3 in the fame time will be the excefs of « + x]? above x3, that is, the excefs of ** + 3v7W + 3xX* + 43 above x3; and the incre- ment of gx in the fame time will be the excefs of g x * + X, or gx + gx, above gx ; that is, the increment of x? will be 3¥*4 + 3xv? + 43, and that of gw will be gx. Now in the equation x3 — gx = r it is evident that wv mutt be greater than g; for otherwife «* would not be greater than gx, as it is fuppofed to be. Confequently, «« x % muft be greater than gx’; and, @ fortiori, 3x°X + 3x” ++ %3 (which is more than triple of ~*~) muft be greater than gv; that is, the in- crement of «3 will be greater than the contemporary increment of gx. Therefore the excefs of «3 above gx, or the compound quantity «3 — gx, will increafe con- tinually, without decreafing, while « increafes from /q ad infinitum. QE. D. Art. 13. It follows, therefore, upon the whole of thefe enquiries, that, if the compound quantity x? — qx, or, its equal, the abfolute term 7, is lefs than 2409 , or — is lefs than f, it will be impoffible for yz to be equal to, and 3/3 4 : confequently the foregoing method of refolving the equation «3 — qx = 7 will be impracticable ; but, if «3 — qv, or r, is greater than 2909" or = is greater 3/3 than r it will be poffible for yz to be equal to 2, and confequently, the fore- going method of refolving the equation v3 — gx =r will be praéticable. And thus we fee in what manner it is probable that Carp awn’s rule for refolving the cubick equation w? — gx =r in the firft cafe of it, or when r is greater than . 3 . . . . =i, or oa is greater than = together with the reftrition of it to that firft cafe, may have been difcovered. Of the Equation qx — x? =r. ee ee RR IE IR a ae aes Art. 14. In the third equation gx — x? = r the terms x? and gw have diffe- rent figns, as well as in the fecond equation «? — gx =r; and therefore it feems: to have been natural for the inventor of CarpDAwn’s rules to try both the fubfti- tutions of yy — z and y + 2 inftead of « in this equation, as well as in that fe- cond equation, in hopes of an extermination of equal terms that are marked with contrary figns, and a confequent reduction of the equation to another which, though of double the dimenfions of the equation gx—w«3= 7, fhould have been of a fimpler form, and more eafy to.be refolved. But it will be found, upon trial, that neither of thefe fubftitutions will anfwer the end propofed. Vou. Il. 4 F Art. 15.. 586 A conjecturr, &e, Art. 15. For, in the firft place, let us fuppofe «tobe =y—z. Then we. fhall have w= y? — 3972 + gyz—2oy? — 39e X ¥—Z— 23, and gx =9¢ xX y—z and confequently gy —x?7 = 9X y—%—)' + 392 X fy —Zz +23, Therefore, gX y—z—y? + 39% X¥ (Y¥—2+ 2? willbe =r. Now in this equation it is evident, the terms g x [y —z and 39z X [y —z have the fame figns, and therefore can never deftroy each other. Therefore, no fuch me- thod of refolving this equation gx—wx> =r as was found above for refolv- ing the two former equations x? + qx = r and «3? —qv = 7, can be obtained by fubftituting the difference y — z in it inftead of w. Art. 16. We will now try the fubftitution of y + z inftead of w in the terms of this equation. Now, if « be fuppofed to be = y + z, we fhall have «$ = y3 + 3yyz + 3yzz +2 =y + 372 X yt e2t2, andgoqyx yt 2, and confequently gx —ogxy+2—y—3yeX yy +z—2*: Therefore,g x y+ z—y* —372 X Yt Z—2 will be = 7. | In this equation it is true indeed that the terms g X y + zand 39% X y— 2 have different figns. But they cannot be equal to each other: for, fince the three terms y? and 3yz x y—2z and 2? areall marked with the fign —, or are to be fubtracted from the firft term g x y + 2, and the remainder is = 7, it is evident that ¢g X y + z mutt be greater than the fum of all the three terms y3, 3yz X y + 2, and z3, taken together, and therefore, @ fortiori, greater than 392 X y + zalone. Therefore, no fuch extermination of equal terms marked with contrary figns as took place in the transformed equations derived from the two former equations v3 -+-- gv—=r and «3— gx = 7, can take place in this transformed equation derived from the equation gx — x3 = 7 by fubftituting y + 2 in its terms inftead of «; and confequently no fuch method of refolving the equation gx — «3 = r as has been found for the refolution of the equations #3 4+ gx = rand «* —gx =7, can be obtained by means of that fubftitution. Art. 17. Thefe are the methods of inveftigation by which I conceive it to be probable that Carpan’s rules for the refolution of the cubick equations x3 ++ gx =rand «3 —gx =r, together with the limitation of the rule relating to the latter of thofe equations, and their inapplicability to the third equation ga —— *> =r, may have been difcovered by the firft inventors of them. AN AN re Pes GB NG: Dee X TO THE Brig ned rs g's West SAT CONTAINED IN THE FOREGOING PART OF THIS SECOND VOLUME OF MATHEMATICAL TRACTS, IN PAGES 153, 154, 155, &C, To 169; INTITLED, €¢ A DEMONSTRATION OF SIR ISAAC NEWTON’s BINOMIAL THEOREM IN THE CASE OF INTEGRAL POWERS, OR POWERS OF WHICH THE INDEXES ARE WHOLE NUMBERS:” Containing an Inveftigation of the Law by which the co- efficients of the third and fourth and other following terms of the feries which is equal to any integral power of a binomial quantity, are derived from the co-efficient of the Second term of the faid feries, grounded on a probable in- duction from particular examples. By FRANCIS MASERES, Esa. F.R.S. €URSITOR BARON OF HIS MAJESTY’S COURT OF EXCHEQUER, Art. 1. WT is fhewn in art. 5 of the foregoing tract, pages 155, 156, that, in: all integral powers whatfoever of the binomial quantity ¢ + 4, the literal parts of the terms of the feries which is equal to @ + 4)” (in which the letter m denotes any whole number whatfoever), will always be P toda Shean | a” *b, a” 33, &c, of which every term is generated from the next be- fore it by the multiplication of the fraction a And it is-alfo fhewn in art. 6 of the faid tract, page 156, that the numeral co-efficient of the firft term of the feries that is equal to a + 2)” muft always be 1, or that the firft term of the faid feries will always be a”, and that the numeral co-efficient of the fecond term of the faid feries will always be m, or the index of the power to which a + d is raifed, or that the fecond term of the faid feries will always be m x a” ‘6 to whatever whole number the lettgr m may be fuppofed to be equal. 3 ) Art. 2. And it is further fhewn in the faid tra& that the numeral co-efficients of the third, and fourth, and fifth, and other following terms of the feries which is: Ar equal: 308 ON THE BINOMIAL THEOREM. equal to @ + 2 ke may always be derived from m, the co-efficient of the fecond term of the faid feries, by the continual multiplication of the following fractions, M—l m—2 M—-% mM—4 M—SF . . ° Mm to wit, ——; ae aa ase Coes &c, till we come to the fraction ree which is = 0, or till the faid feries of fraftions is terminated or exhaufted ;. which fra¢tions, ma ee ara —. &c, are therefore called she ge= nerating fractions of the co-efficients of the third and other following terms of the feries which is equal to a + 4)”. Art. 3. And the method by which it is fhewn in the faid tra& that the faid frace Fotis eel ae mn 4 M3 &c, are in all cafes, or when m is fuppofed 2 to reprefent any whole number whatfoever, the generating fractions of the co- efficients of the terms of the feries that is equal to a + b)”; or the fractions by which the co-efficients of the third and fourth, and other following terms of the {aid feries, are derived from m, the co-efficient of the fecond term mx a” ‘3, confifts of the three following parts; to wit, firft, of a demonftration that, if it be true that thefe are the generating fractions of the co-efficients of the third and other following terms of the feries that is equal to a + 4|” when m is equal to any one whole number whatfoever, it will alfo be true that they will be the gene- rating fractions of the co-efficients of the third and other following terms of the feries that is equal to a + b\” when m is equal to any other whole number greater than the former; and, fecondly, of a proof, by actual trials of the co-efficients of the terms of the feveral feriefes that are equal to a + d|*, 2+ 43, a+ 4, and a@ + 4}5, that in thefe four feriefes, or when m is equal either to the number 2, or the number 3, or the number 4, or the number 5, the faid fractions — ma? 273, and =>, are the generating fractions by which the co-efficients of 3° 4 6 s the third and other following terms of the faid feriefes (that are equal to 2 + dl*, a+ 3, a + d+, and a+ d)5) are derived from the co-efficients of the fecond terms of the faid feriefes refpectively ; and, thirdly, of a conclufion evidently following from the former two propofitions, to wit, that, to whatever whole number the index m be fuppofed to be equal, it will always be true that the 3 . : 42 — I _ _ we — ° . . faid fractions “—, ==", 223, 7=4 2—5 wc, will be the generating fractions 2 4 by the continual multiplication of which the co-efficients of the third and other following terms of the feries that is equa! to @ + 4}” will be derived from m, the co-efficient of the fecond term. ‘Thefe reafonings I take to be juft and clear, and fuch as muft give every reader full fatisfaction as to the truth of the propofition, or conclufion, obtained by means of them. Alt ay ON THE BINOMIAL THEOREM. 58g Art. 4. But it may be afked, ‘* How came it to be fufpected that the fractions oe =3, &c, were the generating fractions by which the S 4 5 “¢ co-efficients of the third and other following terms of the feries that is equal 7 . ° ° £©to a + b| are derived from m, the co-efficient of the fecond term, in any “¢ one value of the index m, fince it is by no means apparent from the mere *¢ infpection of the terms of the feriefes that are equal to a + 2\” when m is ** equal to the {mall numbers 2, 3, 4, and 5?” This is a very natural and reafonable queftion, and well deferves to be con- fidered ; more efpecially if we recollect that Dr. Wallis informs us that he had fought for thefe generating fractions without being able to difcover them. And till a perfon had firft fufpected, and then found upon trial, that thefe are the ge- nerating fractions of the co-efficients of the terms of the feries that is equal to a + 2)” in fome of the lower values of m, he could never think of fhewing, in the method above defcribed, that the fame generating fractions would enable us to find the co-efficients of the terms of the like feriefes in all other integral values of m. Art. 5. Now, inanfwer to this queftion it may be obferved, that thefe fractions will occur to our notice as the generating fractions of the co-efficients of the third and other following terms of the feriefes that are equal to a + 2)” in fome of the lower values of m, if we divide the feveral fucceffive terms of thefe fe- riefes by the terms next before them, in order to difcover the generating frac- tions by which they are derived one from another, and then reduce the gene- rating fractions fo obtained to their loweft deneminations. Thus, for example, the fixth power of the binomial quantity ¢ + dis = 4° + 64a'b + 15a*d* + 204353 + 154a°b* + 6ab5 + O°; 29 eas une) 4 we divide a fecond term 645d by the firft term 2°, prOLeOr x a and, if we divide in like manner the third term by the conc: and the fourth term by the third, and the fifth term by the fourth, and the fixth term by the fifth, and ash the feventh, or laft, term by the fixth, the a will be ———— a rere r 2 x = 20 a353 20 15 a7b4 ue ay 6 abs _ ‘ d se 5 x ~ and ——3 0 x and TET Bh read and > or— X- Ee And confequently, if we muldaly the firft term a° by the a quotient i x eo we fhall thereby produce the fecond term 6454; and, if we multiply the fecond term-6 4°d by “3 x i) we fhall thereby produce the third term 152d; and, if we multiply the third term 15 4*d* by = x a we fhall thereby prague i fourth term 2047453 ; and, if we UNICENTS the ame term 20473 by 4 potas 5, we fhall thereby produce the fifth term 15 470+ 3 and, 2 if 590 ON THE BINOMIAL THEOREM, if we multiply the fifth term 15 2° by = Ss =, we fhall thereby produce the fixth term 6 a)‘ ; and, if we multiply the fixth term 6 bs by = — X 4 we hall thereby produce the feventh, or laft, term 4°. Therefore es generating frac- tions, by the peaual multiplication of which the fecond, and other following terms of the feries 2° 4+ 645d + 15a*b” + 204363 + 15307b* + 6455 + 6° (which is equal to a + 4°) are pan from the fins term @°, are © x se 3 sf = x 4 - une hi Klee and ess and confequently the batcipati fractions, by the eit frlatuiplicatial bf which the numeral co-efficients of the fecond and other following terms of the faid feries 4° + 6a°b + 154a*b* + 2043b3 + 1507b* 4+. 6 abs + 6° (independently of the literal parts of the faid yep are derived from 1, oe numeral co-efficient of the firft term a°, will be ut | Hil 28) Py and +. Now let thefe co-effi- Bi 140.° 6 67) 207/915 6 : : : 6 cients be reduced to their loweft terms; and they will then be —, 4, 2) , 6". ; pat = of which the numerator of the firft term — Is the index 6 of the power of the binomial quantity to which the faid feries of terms 1s equal, and: the nu- . . 2 I . ‘ acy merators of the following fractions 3, Eero ree are derived from the faid index 6 by the continual fubtraction of 1, and the denominators of the faid frac- tions are the natural numbers 1, 2, 3, 4, 5, 6, which begin with an unit, and increafe by the continual addition of 1. This obfervation on the increafe and. decreafe of the denominators and numerators of the fractions + 2 al - and 6 ; ; > and. their derivation from the index 6, or oa in the cafe of the feries which. is equal to a + 4%, is fufficient to have induced the perfon who. fhould have made it, to conjecture, that pofibly, when the index m was equal to any other number (fuch as 5, or 4, or 7, or 8), the generating fractions whereby the nu- meral co-efficients of the third and fourth and other following terms of the feries that was equal to @ + 4\”, were derived from the numeral co-efficient m of the fecond term, and from each other, might likewife, when properly reduced, be found to confit of numerators and denominators that did in like manner de- creafe from the index m by the continual fubtraétion. of an unit, and increafe from 1 by the continual addition of an unit; or, in other words, might be equal. wi las ee as “=, &c. And this conjeCture might have pro- 5 duced a trial whether this ae et place in fome particular examples, and more efpecially in the feriefes that were equal to @ + Alaa ah 135 a+%l+,a+43, a+ d\°, a+", anda -+ d\%, and perhaps a few more of the lower inte- gral powers of a + 4; after which trials, and’ the fuccefs that would have at- tended them, it would have become fo highly probable that the fame rule would take- ON THE BINOMIAL THEOREM. 59or take place in the feriefes that were equal to any other integral powers of a + 3, that it would have been almoft impoffible to doubt of it. And then it would have been natural to endeavour to find fome general demonftration of the truth of the rule in all integral powers of the binomial quantity @ + 2 whatfoever, ’ which might have led to fuch a demontftration as that which is given in the tract above-mentioned, which is contained in pages 153,154, 155, &c.... 169, of this volume. ; N. B. This method of difcovering (by a conjecture grounded on atrial or two, in fome particular examples), that the generating fractions by which the co-effi- cients of the third and fourth and other following terms of the feries that is equal to a + 4)” (or any integral power of the binomial quantity 2 + 4) are derived from m (the index of the power to which the faid binomial quantity is raifed) or from the co-efficient of the fecond term of the faid feries (which is equal to the faid index) are =, se on HRS a &c, is fuggefted by profeffor Saunderfon, in the fecond volume of his Algebra, in the chapter on the bino- mial theorem, where the reader will find a good explanation and illuftration of the faid celebrated theorem, by a variety of examples, both in the cafe of inte- gral powers and in the cafe of roots, and other fractional powers, and even in the cafe of negative powers, and of powers that are both fractional and negative ; but no demonftration of it in any cafe, not even in that of integral and affirmas tive powers. 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