B I N 



h)T)"rbola, and the reft of the altern tite curves in the fcrles 

 i+ir)»> r + x')i> f+s'vi' r+Pl^ &c. ; and in a fimilar 

 way might other feries be likewifc interpellated, and that 

 tven if they ftlould be taken at two or more intervalp. 



" This was the way by wiiich I firil opened an entrance 

 into thel'e fptculations, which 1 (hould not have remem- 

 bered, but that, in turning over my papers, a few weeks 

 ago, I, by ciiance, call my eyes upon ihofe relating to this 

 matter. 



" After I had proceeded fo f ar, it imm ediate ly occurr ed 

 to me that the terms i — .v'li> i - x]h 1 — x')i' l — x"\i' 

 tec. that is, I, i—x\ i — zx'+x*, i — ix'+$x* — x'', Sec. 

 might be interpolated in the fame m;nr,ier as 1 liad done ni 

 the cafe of the areas generated by them : and for this, there 

 required nothing more than to leave out the denominators, I, 

 3) 5> 7> &c. in the terms that exprcfslhe areas ; the n the c o- 

 elficients of the terms to be interpolated (i —x'ji> i — x'V' 

 or univerfally i —x'-^") will be had by the continual nuilti- 



,• • r , r 1 r ■ m—l 111—2 



plication or the terms ot the leries ra x X 



' i 3 



Sic. 



" Thus, for example, l — x', = i — 



I 

 16* 



Sec. ; and i — .v 



3 ' 



I — - .V 

 2 



V&c. 

 i6 



-X — —X &C. 



and I— .v' 7=1 



3 9 



" Thus, I difcovered a general method of reducing radi- 

 cal quantities into infinite feries, by the binomla! theorem, 

 which 1 fent in my lall letter, before I obferved that the 

 fame thing might be obtained by the extraftion of roots. 



" But after I had difcovered this method, the other way 

 could not long remain unkrtown ; for, in order to prove the 



truth of thefe operations, I multiplied i .;' ;x^ — 



—ix^ &c. by itfelf, and found the produft to be i— *•', 

 i6 



all the terms after tlicfe ad injimlum vanifhing : in like man- 



2 I C 



ner l x' .v* — — .v^ &c. beinc twice multiplied into 



3 9 8i 



itfelf, produced i— .x". And as this was a certain proof of 

 the truth of thcfe eoncliifions, I was tl'.ereby naturally led 

 to try the converfe of it, viz. whether thcfe feries, that 

 were now knovv-ii to be the roots of the quantity I— .v', 

 mi^ht not be produced by the rule for extradtioii of roots in 

 arithmetic ; and, upon trial, I found it fucceed to my 

 widies. 



" Tiiis being found, I laid afide the method of interpola- 

 tion, and alfumed thefe operations, as a more genuine foun- 

 dation to proceed upon. In the mean time, I was not 

 ignorant of the way of reduftion by divifion, which was fo 

 jnucli eafier." 



From this account, as given by Newton himfelf, it ap- 

 pears that his difcovtry of the law for the areas, witli irra- 

 tional ordinates, preceded that of the law for the cxpanfion 

 of thofe ordinates ; although the latter, as Montucla ob- 

 ferves, might have been cxpcfted to precede the former, if 

 inventive genius always purfued the mod eafy method. But, 

 in tracing the progrefs of the human mind, it may generally 

 be obfcrved, tliat a collection of difcoveries, in any branch 

 of fcience, is feldom found to be a feries of regular dtduc- 

 tioDS ; but, on the contraiy, we often difcein therein many 



B I N 



anticipatlone, and fometimes even a icverfion of the natural 

 and logical order of ideas. 



It is worth while here to remark, that Newton had made 

 thcfe difcoveries, as wx-ll as many others, fev^rcil years be- 

 fore Mercator had publifhed his " Logarithmotechnia," 

 which contains a particular cafe of this theory ; but, frotil 

 an excefs of modelly and indifference for thcfe fruits of his 

 genius, he delayed making them known to the world : and, 

 even after the above-mentioned work hsd appeared, which 

 would have operated as a powerful motive with moll other 

 men, in exciting them, to (hare in the glory of thefe brilliant 

 inventions, he was ftill more confirmed in the refolution he 

 had taken, of net making himfcif known as an author till 

 he was of a more mature age. He conceived, that Mercator 

 having difcovered, as it was faid, the feries for the hyper- 

 bola, would not be long before he extended his method to 

 the circle, and other curves ; or, if this fhould not be don-^ 

 by him, the invention would be readily perceived by others. 

 In (hort, it appears rather fingular, that as Mercator had 



converted the expreffion — — into an infinite feries, by the 



ordinary method of divifion, he (liould not have tried to dif- 

 cover the feries for ^/ i +x' by the known method of ex- 

 trading the fquarc root ; but this, though extremely ob- 

 vious, efcaped his notice : and many circnmllances, of A 

 fimilar kind, are to be found in the hillory of the feiences. 



Newton, as has been already obfcrved, left no demonftra- 

 tion of this theorem ; but appear; to have form;:d it merely 

 from an iuduiSion of particular cafes ; and though no doubt 

 can be entertained of its truth, having been found to fucceed 

 in all the inftances in which it has bicn applied ; yet, agree- 

 ably to the rigour that ought to be obfervcd in the eilablidi- 

 ment of every malhcjnatical theory, and efpecially in a fun- 

 damental theorem of fuch general ufe and application, it is 

 neceffary that as regular and llridl a proof Ihould be given 

 of it as the nature of the fubjcft, and tlie Ihite of analyfis, 

 can aiford. 



One of the firfl demonflrations of this kind that appears 

 to have been given, is that of James Bernoulli, which is to 

 be found, among feveral other curious things, in a fmall trea- 

 tife of his, entitled " Ars Conjeclandi," which has been very 

 improperly omitted in the collection of his works, publifhed by 

 his nephew, Nicholas Bernoulli. But this is only applied to 

 the cafe of integral and affirmative powers, and is nearly 

 the fame with that which was afterwards given by Mr. John 

 Stewart, in his commentary on fir Ifaac Newton's quadra- 

 ture of curves. It is founded on the doC?irlne of combina» 

 tions, and the properties of figurate numbers, which arc 

 there'lhewn to involve in them the generation of thefe co- 

 efficients ; and in the inftance before mentioned, where the 

 index of the binomial is a whole pohtive number, it is clearly 

 and fatisfaftorily explained. 



Since that time, many attempts have been made to demon-, 

 flrale the general cafe, or that where the index of the bino- 

 mial is either a whole number or a fraftion, pofitive or nega- 

 tive ; but moll of thefe dcmonftrations having been con- 

 dueled, either by the method of increments, the mnhino- 

 mial theorem of De Moivre, or by fluxions, are com- 

 monly thought to be unfatisfaftory and imperfeft ; and it 

 (hould feeni not without reafon ; as, independently of other 

 objeftions, it appears contrary to the principles of fcience, 

 as well as to jull reafoning, to employ, in a matter purely aU 

 gebraical, notions and doftrines derived from, other branches, 

 or from an analyfis which is in fome fort tranfcendental. 



For thefe reafons, feveral eminent mathematitians liave en- 

 deavoured to inveftigate this formula on pure analytical prin- 

 ciples. 





