B I N 



r.esrly the fame thing, yet the advancement in ar.alyfis de- 

 pended on the circumftance of the law which tliey obferve, 

 btiiig exprefied by means of a general fymbol (m) ; with- 

 out which, its exrenfion would never have bt-en made to 

 thofe cafts in which the index is negative or fradlional : fo 

 that Briggs, even in the cafe of integral powers, does not 

 appear to be fully entitled to the invention of the binomial 

 theorem, properly fo called. 



Cut howev-r this may be, it is uiiiverfa'Iy agreed that no 

 0"c before I'ewton liad ever thought of extradting roots by 

 means of infinite fentj. He was the firll who happily dil- 

 covered, thit, by confidering roots as powers having frac- 

 tional exponents, the fame binomial feries would equally 

 fer.-c for ihem all, whether the inicx {hould be fraclional or 

 integral, or the feries finite or infinite ; and from this ex- 

 tenfion of the theorem, forr.e of the moft; important improve- 

 ments, in the higher departments of mathematics, have 

 ariftn ; particularly in the conllruflion of logarithms, and 

 the doftrine of fcri^-s in gencr^.l, which have fiiice been car- 

 ried to a great degree of perfection, and now form fome of 

 the moft curious and interefting branches of analytics. 



It may alfo be farther obfervrd, with refpeft to the claim 

 of Newton as an origipn! inventor of this highly ufeful 

 theorem, that he had probably never feen the Arilhinetica 

 Loger'uhmica of Briggs ; for it is well known that he was 

 not an esteniive reader of mathematical works, depending 

 more on the powers of his own geiiins than upon any helps 

 of this kind : fo that there can be but little doubt of his 

 having made the difcovery himfelf, without receiving any 

 light from wh^t had been done by Briggs ; and that he 

 conceived the theorem to be new for ail powers in general, 

 as it was for roots and quantities with fraftional indices. 



But though this appears to be the cafe with refpeft to 

 Newton, it is yet fnrprifing tJ^at Dr. Wallis, who was a 

 general reader of moft mathematical works, and who had 

 actually feen Briggs's Anthmetka Logarithrmca, as he men- 

 tions it in page 60. chap. xii. of his Algebra, ftiould not 

 have attended enough to this curious treatife, to know that 

 it contained fuch a new and excellent theorem, as it fully 

 appears he did not ; fince, in the S5th chapter of the above- 

 mentioned work, he afcribes the invention entirely to New- 

 ton ; and adds, that he himfelf had fought after fuch a rule, 

 but without luccefs. It is alfo no lefs fmgular, that John 

 Bernoulli, not half a centur)- fince, (hould firll difpute the 

 invention of this theorem with Ne«ton, and afterwards give 

 the difc«very of it to Pafcal, who was not born till long 

 after it had been taught by Briggs. (See Bernoulli's works, 

 vol. iv. p. 173). 



Dr. Wallis's Algebra was published in the year 1685 ; 

 and it was here, for the firil time after Newton's difcovciy 

 of it, that the binomial theorem, according to his general 

 manner of exprcfling it, appeared in print, and was made 

 known to the learned world ; though Leibnitz, and pro- 

 bably Dr. Barrow (who was Newton's great friend and 

 patron in his youth), as well as fome other mathematicians 

 of that time, had feen it, in a letter addreflcd to Mr. Olden- 

 burgh, of Oclober 24th 1676, {which was given in the 

 Comnunium Epifolicum), foon after the laid letter was 

 written. But he no where tells us his manner of invcftigat- 

 ing it ; nor is any demouftration of it to be found, even in 

 the cafe where the index i^ a whole number, in any part of 

 his works. He fayi, indeed, in his next letter to Olden- 

 burgh, to be fou".d in the fame work, that the occafion of 

 its difcovery was as follows : 



" Not long (he obferves) after I had ventured upon the 

 ftudy of the mathematics, wliilft I wa?; perilling the works 

 of the Celebrated Dr. Wallis, and coiifideiing the feries of 



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univcrfal roots, by the interpolation of which we exhibit the 

 area of the circle and hyperbola: for inftance, in this fericJ 

 of curves, who f'- co mmon bafe or ax-s is x, and the r-.fpec- 

 tivc ordinates i~x'-.-> 1— .x'-j> \^*h r-^' r> i-x~!-» 

 1— *=)». &c, 1 obferved that if the areas of the alternate 



curves, which are x, :t x', x - — x' -f -' x", x-^ x^ + 



3 3 5 3 



3 I 

 — x^ k', Sec, could be interpolated, we (hould by thij 



means, obtai n the areas of the intermediate one?, the firft of 

 which I — x4 2 is the area of the ciicle. In order to this it 

 vras evident, that in each of thtfe feries the firft term was x. 



and that the fecond terms • 



12 3 



— a', —x', — x', ice. were in 

 3 3 3 3 

 anthmetical progreluon ; and coufequer.tly the firft three 



terms of the feries to be interpolated r.iuft be x ( -x' ) , 



'-T(^)-^-?(f:')'- 



" Now, for the interpolation of the reft, I confidercd that 

 the denominators I, 3, 5, 7, &c, were, in all of them, in 

 arithmetical progreflioa ; and confequently the whole diffi- 

 culty confifted in difcoverir.g the numeral co-fficicuts : but 

 thefe, in the alteniatc areas which are given, I obferved 

 were the fame with the figures of which the feveral afcend- 

 ing powers of the number ir corfift, viz. 11°, 11', 11% 

 n', llS &c. that is, the firft, 1 ; the fecond, I, 1 ; the 

 third, I, 3, I ; the fourth, i, 3, 3, I ; the fifth, I, 4, 6, 4, 

 I, &c. 



" I applied myfelf, therefore, to difcover a fliethod by 

 which the firft two figures of this feries might be derived 

 from the reft ; and I found, that if for the fecond figure, 

 or numeral term, I put m, the reft of the terms would be 

 produced by tlie continual multiphcation of the terms of ihii 



fenes, X X X ? X 2, &c. 



12245 



" For inftance, if the fecond term be put for 4, there 



will arife 4 x 



that is 6, which is the third term ; 



the fourth term will be 6 X 

 3 



-, that is 4 ; the fifth term 



will be 4 X 



, that is I ; a;id the fixth term will be 



4 X , that is o, which (hews the feries is hire termi- 



5 

 nated, in this cafe. 



" This being found, I applied it, as 3 rule, to interpolate 

 the above-mentioned feries. And fince, in the feries which 

 expreffcs the circle, the fecond term was found to be 



— ( — .V M , I therefore put n: = — , and there was produced 

 3 V 2 / 2 



-j-— ; — X— |-— 31 or — — ^» and fo on aJ injir.iluin 



or 



the terms — X— ; I 1 or 



2 2 V 2 / 



I I /I 



-7 ; -7X- I - 



16 i6 4 V2 



Hence I found that the area of the fegment of the circle 



i_ 

 Hi 



.„„,u..-L(H-i(r')-^(H 



" In the fame manrcr, the areis to be interpoLtcd of th- 



other curves might be produced ; as alia t'<e area of the 



3 C 2 hyperbola. 



