JAMES BERNOULLI. 65 



plafcionibua vacarunt (quos inter Faulliaberus et Remmelini TJlmenEes, 

 Wallisius, Mercator in Logarithmotechnia, Prestetus, aliique)... 



111. We may notice that James Bernoulli gives incidentally 

 on his page 89 a demonstration of the Binomial Theorem for the 

 case of a positive integral exponent. Maseres considers this to 

 be the first demonstration that appeared ; see page 283 of the 

 work cited in Ai't. 47. 



112. From the summation of a series of figurate numbers 

 James Bernoulli proceeds to derive the summation of the powers 

 of the natural numbers. He exhibits definitely 2?i, Sn^ 2n^... 

 up to Xw^" ; he uses the sj^mbol / where we in modern books use S. 

 He then extends his results by induction without demonstration, 

 and introduces for the first time into Analysis the coefficients since 

 so famous as the numbers of Bernoulli. His general formula is that 



^ , n'"-' n' c . ^_, c(c-l)(c-2) J, ,_^ 



c(c-l)(c-2)(c-3)(o-4) _, 

 ^ 2.3.4.5.6 



c(c-.l)(o-2)(c-3)(c-4)(c-5)(c~6) 

 "^ 2.3.4.5.6.7.8 '^'" 



where ^ = 6 ' ^ = " SO ' ^ = A' ^ = - i' - 



He gives the numerical value of the sum of the tenth powers 

 of the first thousand natural numbers ; the result is a number 

 with thirty-two figures. He adds, on his page 98, 



E quibus apparet, quam inutilis censenda sit opera Jsmaelis Bul- 

 lialdi, quam conscribendo tarn spisso volumini Arithmeticae sufe Infijii- 

 torum impendit, ubi niliil prgestitit aliud, quam ut primarum tantum 

 sex potestatum summas (partem ejus quod unica nos consecuti sumus 

 pagina) immense labore demonstratas exhiberet. 



For some account of Bulliald's sjnssum volumen, see Wallis's 

 Algebra, Chap. LXXX. 



113. James Bernoulli gives in his fourth Chapter the rule 

 now well known for the number of the combinations of ti thiners 



