266 THE PRINCIPLES OF SCIENCE. 



covered by the observation of particular cases, and may 

 again be so discovered. We have Newton's own state- 

 ment, to the effect that he was thus led to the all-impor- 

 tant Binomial Theorem, the basis of the whole structure 

 of mathematical analysis. Speaking of a certain series of 

 terms, expressing the area of a circle or hyperbola, he says, 

 ' I reflected that the denominators were in arithmetical 

 progression ; so that only the numerical co-efficients of 

 the numerators remained to be investigated. But these, 

 in the alternate areas, were the figures of the powers of 

 the number eleven, namely 1 1, 1 1 1 , 1 1 3 , 1 1 3 , 1 1 4 ; that is, 

 in the first i ; in the second i, i ; in the third i, 2, i ; in 

 the fourth i, 3, 3, i ; in the fifth i, 4, 6, 4, i. 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 

 continual multiplication of the terms of the formula 



m o ra i m 2 mi 



X X X - X &c.'P 



1234 



It is pretty evident, from this most interesting statement, 

 that Newton having simply observed the succession of the 

 numbers, tried various formulae until he found one which 

 agreed with them all. He was so little satisfied with this 

 process, however, that he verified particular results of his 

 new theorem by comparison with the results of common 

 multiplication, and the rule for the extraction of the 

 square root. Newton, in fact, gave no demonstration of 

 his theorem ; and a number of the first mathematicians 

 of the last century, James Bernouilli, Maclaurin, Landen, 

 Euler, Lagrange, &c., occupied themselves with discovering 

 a conclusive method of deductive proof. 



These are the figurate numbers considered in pages 206-216. 



P * Commercium Epistolicum. Epistola ad Oldenburgum,' Oct. 24, 

 1676. Horsley's 'Works of Newton', vol. iv. p. 541. See De Morgan 

 in 'Penny Cyclopaedia', art. Binomial Theorem, p. 412. 



