BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 285 



By similar reasoning he shows that if the first player has gained 

 two points and the second none, the division should be 56 to 8 ; 

 while if the first has gained one point, the second none, it should 

 be 44 and 20. 



The calculus of probabilities, when confined within just limits, 

 ought to interest, in an equal degree, the mathematician, the experi- 

 mentalist, and the statesman. From the time when Pascal and 

 Fermat established its first principles, it has rendered, and continues 

 daily to render, services of the most eminent kind. It is the calculus 

 of probabilities, which, after having suggested the best arrangements 

 of the tables of population and mortality, teaches us to deduce from 

 those numbers, in general so erroneously interpreted, conclusions of 

 a precise and useful character ; it is the calculus of probabilities which 

 alone can regulate justly the premiums to be paid for assurances; 

 the reserve funds for the disbursements of pensions, annuities, dis- 

 counts, etc. It is under its influence that lotteries and other shameful 

 snares cunningly laid for avarice and ignorance have definitely disap- 

 peared. Arago. 



With this work connected itself his arithmetical triangle in which 

 successive diagonals contain the coefficients which occur in ex- 

 pansions by the binomial theorem, which Newton was soon to 

 generalize. 



111111 



12345 



1 3 6 10 



1 4 10 



1 5 



1 



He applies the method of indivisibles successfully to the cycloid 

 (the curve generated by a point on the rim of a rolling wheel). 



Pascal invented in 1645 an arithmetical machine, writing the 

 Chancellor in regard to it : 



Sir : If the public receives any advantage from the invention which 

 I have made to perform all sorts of rules of arithmetic in a manner as 

 novel as it is convenient, it will be under greater obligation to your 



