206 THE PRINCIPLES OF SCIENCE. 



of m things may be chosen out of a total number of n 

 things, in a number of combinations denoted by the formula 



n. (n-i) (n-2) (n-%) . . . . (n-m+ T) 

 1.2.3.4 .... m 



The extreme importance and significance of this formula 

 seems to have been first adequately recognised by Pascal, 

 although its discovery is attributed by him to a friend, 

 M. de Ganieres/ We shall find it perpetually recurring 

 in questions both of combinations and probability, and 

 throughout the formulae of mathematical analysis traces of 

 its influence will be noticed. 



The Arithmetical Triangle. 



The Arithmetical Triangle is a name long since given to 

 a series of remarkable numbers connected with the subject 

 we are treating. According to Montucla g * this triangle is 

 in the theory of combinations and changes of order, almost 

 what the table of Pythagoras is in ordinary arithmetic, 

 that is to say, it places at once under the eyes, the numbers 

 required in a multitude of cases of this theory/ As early 

 as 1544 Stifels had noticed the remarkable properties of 

 these numbers and the mode of their evolution. 11 Briggs, 

 the inventor of the common system of logarithms, was so 

 struck with their importance that he called them the 

 Abacus Panchrestus. Pascal, however, was the first who 

 wrote a distinct treatise on these numbers, and gave them 

 the name by which they are still known. But Pascal did 

 not by any means exhaust the subject, and it remained for 

 James Bernouilli to demonstrate fully the importance of 

 the Jigurate numbers, as they are also called. In his 

 treatise De Arte Conjectandi, he points out their appli- 



f ' CEuvres Completes de Pascal' (1865), vol. iii. p. 302. Montucla states 

 tlie name as De Gruieres, 'Histoire des Hathdmatiques/ vol. iii. p. 389. 



e 'Histoire des Math&natiques/ vol. iii. p. 387. 



h Leslie, ' Dissertation on the Progress of Mathematical and Physical 

 Science,' Encyclopaedia Britannica. 



