B I N 



412 



B I N 



of tbit work was the determination of the areas of a clans of 

 curves, involving a problem amounting to the determina- 

 tion of 



from x = to x = 1 



where m U a whole number. In this he deduces the alge- 

 braical expressions for any figurato number, but not in the 

 form in which Newton afterwards gave. For example, he 

 prefers 



/ 4- 3 / + 2 / ,1 + 1 1 + 2 



6 to ' ' 



though it appears he knew the latter form. But be con- 

 fined himself almost entirely to the definite integral, and 

 did not exhibit his results in the form of an algebraic, series. 

 His work is broken into propositions, after the manner of 

 the .indents and the simple form in which Newton after- 

 wards enuntiated his results docs not appear (that we can 

 find) in his work. It was as follows, using the notation 

 already adopted, or rather Newton expressed it as follows, 

 ami in the method of expression is the happy simplification 

 which led him to the binomial theorem. In the first column 

 is the expression of the ordinate of the curve in question ; 

 in the second the area included between the abscissa, the 

 ordiuutcs at its extremities, and the intercepted curve. 



(lJF*)" 



&c. Sic. 



\V;ilhs had suggested that the method of determining the 

 area of the circle depended upon finding a mean term be- 

 tween 1 and | in the scries 1, f, f,, &c., made by taking 

 the lower sign in the preceding set, and making x = 1 (he 

 was considering the total areas). For the ordinate of the 

 circle being / \ x', the exponent of which is J, the mean 

 between and 1, the question reduced itself to this: If 0, 

 1, 2, &c., operated upon according to a certain law, (jive the 

 results 1, }, -fg, &c., what will ^ give when operated upon 

 according to the same law? This interpolation he attempt:' I. 

 and obtained his well known and remarkable expression for 

 the ratio of the circular area to the square on its diameter. 

 But he could not succeed in the interpolation, and as he in- 

 forms us himself in his Algebra, afterwards published in 

 1685, 'he gave it over as a thing not feasible,' one difficulty 

 being that he could not imagine a series with more than 

 one term and less than two, which it seemed to him the in- 

 terpolated series must have. And here the question rested 

 till it was taken up by Newton. The latter, in a celebrated 

 letter to Oldenburg, dated October 24, 1C7G, speaking of 

 some developments then newly discovered by Leibnitz, gives 

 the binomial theorem. We shall give his own words (that 

 is translated from the Latin). 'In tin- beginning of my 

 mathematical studies, when I happened to meet with the 

 works of our celebrated Wallis, in considering the series, by 

 the intercalation of which he exhibits the area of the circle 

 and hyperbola,' .... He then itoes on to describe what we 

 have already alluded to. . . .' for interpolating between these 

 I remarked that in all the first term .vas x, and the second 

 (mis were in arithmetical progr i > i . . . .that the two first 

 terms of the scrits to be intercalated sheuld be 



T - 



i- * 

 ,&c. 







For tha remaining intercalations I reflected that the deno- 

 minators were in arithmetical progression ; so that only the 

 numerical coefficients of the numerators remained' to be 

 investigated. But these, in the alternate areas, were the 

 figures of the powers of the number eleven, namely II", 1 ! ', 

 1 1', 1 P. 1 1: that is, in the first 1 ; in the second 1,1; in 

 the third 1, 2. 1 ; in the fourth 1, 3,3, 1; in the fifth 1, 4, 

 ft. 4, I. 1 inquired, therefore, in what manner all the re- 

 maining figures could be found from the first two; and I 

 found thit if the first figure be called m, all the rest roii'.d 

 be found by the continual multiplication of the terms of the 

 formula 



-0 m-J 

 1 X 2 



m 2 m - 3 



-3 X j-X.&e. 



This rule, therefore, I applied to the interpolation of the 

 series. And lince in the circle : t. nn U J X i-r*. 



I made m - $ v. : . , ,,{ the 



sircular segment to be 



X - 



-.Sic. 



3579 

 'This was my first introduction to such meditations, and 

 it would have gone out of my memory, had I not cast my 

 eyes on some of my notes a few . ... j{ u t when I 



had learned these 'things, I presently considered that the 

 term* themsehet (1 -)*, (I -**)', (1 -*")*. Sec. might be 

 interpolated in the same manner at the area* grneratrd 

 from them, and that nothing more was necessary except the 

 omission of the denominators 1, 3, 5, 1, &c. in the terms 

 expressing the areas: that is, that the coefficients of the 



quantity to be intercalated (1 -x'y, or (1 -x 1 )*, or gene- 







rally (1 -x*)* would arise from continual multiplication of 

 the terms of the series, 



m x 



m 1 



w-4 



m 2 



"I" 4 



Newton then proceeds to relate that he proved t: 

 operations by actual multiplication, and afterwards by ap- 

 plying the common rule for the extraction of roots, which 

 he same results. He then states that he knew the 

 common logarithmic series by the same method, and that 

 U'ing then much pleased with such investigations, he con- 

 tinued them until the appearance of Mercator's Lnarilft- 

 moter/inia; when, suspecting that Mercator had made the 

 same discoveries (which however was not the case) before he 

 (Newton) was of an age to, write, he began to care little 

 about prosecuting his researches. 



It must be noticed that Newton had previously given the 

 theorem itself in a former letter to Oldenburg, dated 

 13, 1676, with more copious examples: the statement of it 

 is as follows:' The extraction of roots is much shortened 

 by this theorem, 



(P + P Q) = P + A Q + 



B Q + &,..' 



where A means the first term itself, B the second term, &c. 



It must therefore be noticed, and similar things are com- 

 mon in the history of discovery, that several of those theorems 

 which are now among the simple consequences of the bi- 

 nomial theorem, were in fact discovered before it. Thus 

 Mercator and James Gregory had already used the logarith- 

 mic scries, and Newton's discovery itself was not a con- 

 sequence of any attempt at the general development of 

 (1 + .r), but of the scries for J' (1 - .TI)" il.r, wluch was 

 (between certain limits) implied in the discoveries of Wallis. 



Newton gave no other demonstration of his theorem ex- 

 cept the verification by multiplication or actual extraction. 

 The theorem of Stirling (commonly called after Maclaurin) 

 and that of Taylor, being the general theorems of which 

 the binomial is a particular case, soon diverted the attention 

 of mathematicians. James Bernoulli first demonstrated 

 the case of whole and positive powers by the application of 

 the theory of combinations, in his treatise De .-trie Cnnjn-- 

 tunili, published after his death in 1713. Maelai.vin. in his 

 fluxions, published in 1712, gave, as we consider, the first 

 general demonstration: for though he employs fluxions, 

 yet he had not, as he himself notices (page 607), ' ma>i< 

 of this theorem in demonstrating the rules in the direct 

 method of fluxions.' In later times, when the avidity with 

 which the results of the modern analysis were sought be^an 

 so far to subside as to allow mathematicians to look ai and 

 discuss the grounds on which the several principles were 

 established, a host of demonstrations appeared, each of which 

 met with objectors : for it is a property of all the funda- 

 mental theorems of every branch of mathematics to be in- 

 capablo of establishment in a manner in which all shall 

 agree, though the theorems themselves are held indis- 

 putable. Among these demonstrations arc those of James 

 Bernoulli, Maclaurin, I.anden, Epinus, Stewart, Enler, 

 Lagrange, I.'Uuilicr, Manning, Woodhouse, Hutton, Bon- 

 nvcasile, Knight, Robertson, Creswell, Swinburne, and 

 Tylecotc. We shall not discuss the various objections, be- 

 cause they apply as much to the general doctrine of infinite 

 is to tlie binomial theorem in particular ; and wo 

 must refer the reader to TAYLOR'S THEOREM. We shall 

 however allude to the principal objections after we have given 

 what appears to us a sufficient proof of the theorem, or 

 rather after we have indicated the steps of such a proof. 



Definition. T&y (1 + x) we mean (m and n being whole 



