d'alembeht. 2bo 



S 

 But this is not the same thing as 7,-7 , where S denotes the 



number aUve at the beginning of the year ; for s is a variable 

 gradually diminishing during the year from the value S with 



which it began. But ^7 ^^ ^^^ result which Daniel Bernoulli 



professed to take from observation ; therefore Daniel Bernoulli is 

 inconsistent with himself. D'Alembert's objection is sound ; Daniel 

 Bernoulli would no doubt have admitted it, and have given the 

 just reply, namely that his calculations only professed to be 

 approximately correct, and that they were approximately correct. 



Moreover the error arising in taking sdx and S to be equal in 



value becomes very small if we suppose S to be, not the value of 



s when x = 7i ov n + 1 but, the intermediate value when x = n -\- -^ ; 



and nothing in Daniel Bernoulli's investigation forbids this sup- 

 position. 



517. We have put the objection in the preceding Article as 

 D'Alembert ought to have put it in fairness. He himself however 

 really assumes n = 0, so that his attack does not strictly fall on the 

 whole of Daniel Bernoulli's table but on its first line ; see Art. 403. 

 This does not affect the principle on which DAlembert's objection 

 rests, but taken in conjunction with the remarks in the preceding 

 Article, it will be found to diminish the practical value of the ob- 

 jection considerably. See D'Alembert's pages 312 — 314. 



618. Another objection which D'Alembert takes is also sound ; 

 see his page 315. It amounts to saying that instead of using the 

 Differential Calculus Daniel Bernoulli ought to have used the 

 Calculus of Finite Differences. We have seen in Art. 417 that 

 Daniel Bernoulli proposed to solve various problems in the Theory 

 of Probability by the use of the Differential Calculus. The reply 

 to be made to D'Alembert's objection is that Daniel Bernoulli's 

 investigation accomplishes what was proposed, namely an approxi- 

 mate solution of the problem ; we shall however see hereafter in 

 examining a memoir by Trembley that, assuming the In^otheses of 

 Daniel Bernoulli, a solution by common algebra might be effected. 



