149 



BINOMIAL THEOREM. 



BINOMIAL THEOREM. 



150 



Before the time of Vieta, no materials for its expression were in the 

 hands of algebraists. That writer first used general symbols of 'deter- 

 minate number : and in his works we find the first rude cases of the 

 binomial theorem, though only in the results of simple multiplications, 

 and without the discovery of any law of connezion among the co- 

 efficients. For instance, in his ' Ad Logisticen speciosarn Notse priores,' 

 we find the following : 



" Sit latus unum A, alterum B. Dico A quad.-quadratum + A cubo 

 in B quater, + A quadrato in B quadratum sexies, + A in B cubum 

 quater, + B quad. -quadrato, sequari A + B quad.-quadrato." This we 

 lil now express thus : 



The coefficients of the binomial theorem, in the case of a whole 

 exponent, had long been derived from the method employed in what 

 Pascal called the Arithmetical Triangle, and Briggs the abacus 

 irii-fXfiriffTos. To trace the history of this method would here lead us 

 too far [FiorRATE NUMBERS] ; it must suffice to say that Lucas Pacioli, 

 Stifel, Stevinus, Vieta, and others, all had in their possession some- 

 thing from which, if we did not know that such simple relations were 

 difficult to discover, we should say a little attention would have 

 enabled them to find the first glimpse of the binomial theorem, which, 

 as we shall proceed to state, occurred to Briggs. 



The abacus of the last-mentioned writer above alluded to is as 

 follows (we have only reversed right and left) : 



1 1 1 1 1 &c. 



2 

 3 

 4 



B 

 &c. 



6 



10 

 15 

 Ac. 



4 



10 

 20 

 35 

 &c. 



5 

 15 

 35 

 70 



6 &c. 



21 &c. 



56 &c. 

 126 &c. 

 &c. 



In which each number is formed by adding that on the left to that 

 immediately above. On which (' Trigonometria Britannica,' 1633, pre- 

 face, p. 22) Briggs remarks, that by ascending obliquely, the coefficients 

 of the several powers are obtained ; for instance, that 4, 6, 4 are the 

 coefficients of the fourth power, 5, 10, 10, 5, of the fifth power, and so 

 on. Briggs therefore knew the dependence of these coefficients on the 

 preceding columns of figurate numbers, but not the algebraical 

 expression for the th of each class. 



The next step was made by Wallis, in hifl ' Arithmetica Infinitorum,' 

 published in 1655. One of the great objects of this work was the 

 determination of the areas of a class of curves, involving a problem 

 amounting to the determination of 



_/(l x?) m dx froma: = tox=l 



where m is a whole number. In this he deduces the algebraical 

 expressions for any figurate number, but not in the form in which 

 Newton afterwards gave. For example, he prefers 



P + 3F + 2 I to jl + l 1 + 2 

 ~6~ ~F~3~' 



though it appears he knew the latter form. But he confined him- 

 self almost entirely to the definite integral, and did not exhibit his 

 results in the form of algebraic series. His work is broken into pro- 

 positions, after the manner of the ancients, and the simple form in 

 which Newton afterwards enunciated his results does 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, and 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 ordinates at its extremities, and the intercepted curve. 



(1 a?) x 



(1^)' x\a? 



(1 i x>)- x - 

 &c. 



Wallis had suggested that the method of determining the area of the 

 circle depended upon finding a mean term between 1 and | in the 

 series, I, |, J^, made by taking the lower sign in the preceding set, and 

 making z= 1 (he was considering the total areas). For the ordinate 

 of the circle being / lx 1 , 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, give the results 1, 5, J,, &c., 

 what will 4 give when operated upon according to the same law ? 

 This interpolation he attempted, and obtained his well known and 

 remarkable expression for the ratio of the circular area to the square 

 i'i] its diameter. But he could not succeed in the interpolation, and 

 as he informs 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 interpolated 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, 1676, speaking of 

 some developments then newly discovered by Leibnitz, gives the 

 i.-il theorem. We shall give bis own words (that is, translated 



from the Latin) : " In the 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 goes on to describe 

 what we have already alluded to . . . . "for interpolating between these 

 I remarked that in all the first term was x, and the second terms were 

 in arithmetical progression .... that the two first terms of the series 

 to be intercalated should be 



For the remaining intercalations I reflected that the denominators 

 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 11, II 1 , II 2 , II 3 , II 4 ; 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, 6, 4, 1. I inquired, therefore, in what manner all the 

 remaining figures could be found from the first two ; and I found that 

 if the first figure be called m, all the rest could be found by the con- 

 tinual multiplication of the terms of the formula 



TO TO 1 TO 2 TO 3 . 



x x x x , &c. 



1234 



" This rule, therefore, I applied to the interpolation of the series. 



And since in the circle the second term is J x 1 ,r j , I made m= J 



whence I found the required area of the circular segment to be 



- &c. 



" 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 weeks since. Jiut when I had learned these things, I pre- 

 sently considered that the terms themselves (1 2 ), (1 a:*) 1 ,^ x-f ,&c., 

 might be interpolated in the same manner as the areas generated from 

 them, and that nothing more was necessary except the omission of the 

 denominators 1, 3, 5, 7, &c. in the terms expressing the areas : that 



is, that the coefficients of the quantity to be intercalated (1 .r 2 ) 2 , 



3 - 



or (1 yfp, or generally (1 a?)" would arise from continual multipli- 

 cation of the terms of the series, 



m 1 TO 2 m 3 . 



TO X - X - , X , - &C. 



234 



Newton then proceeds to relate that he proved these operations by 

 actual multiplication, and afterwards by applying the common rule for 

 the extraction of roots, which gave the same results. He then states 

 that he knew the common logarithnu'c series by the same method, and 

 that being then much pleased with such investigations, he continued 

 them until the appearance of Mercator's ' Logarithmotechnia ' ; 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 June 13, 1676, with more 

 copious examples ; the statement of it is as follows ; " The extraction 

 of roots is much shortened by this theorem. 



(P4PQ)" = p"-r- AQ + m ~ 

 n 2n 



BQ+&C. 



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



It must therefore be noticed, and similar things are common in the 

 history of discovery, that several of those theorems which are now 

 among the simple consequences of the binomial theorem, were in fact 

 discovered before it. Thus Mercator and James Gregory had already 

 used the logarithmic series, and Newton's discovery itself was not a 

 consequence of any attempt at the general development of (1 + ;r) ", 

 but of the series fory (1 3?)* dx, which was (between certain limits) 

 implied in the discoveries of Wallis. 



Newton gave no other demonstration of his theorem except 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 Arte Conjectandi," 

 published after his death in 1713. Maclaurin, in his fluxions, pub- 

 lished in 1742, gave, as we consider, the first general demonstration : 

 for though he employs fluxions, yet he had not, as he himself notices 

 (page 607), " made use of this theorem in demonstrating the rules in 

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

 which the results of modern analysis were sought began so far to 

 subside as to allow mathematicians to look at and discuss the grounds 

 on which the several principles were established, a host of demon- 

 strations appeared, each of which met with objectors : for it is a 



