182 THE PRINCIPLES OF SCIENCE. [CHAP. 



the relative order or place in the conjunction, we require 

 the number of combinations. Now a selection of 2 out of 8 



is possible in ~? or 28 ways; of 3 out of 8 in ~f~ 



or 56 ways ; of 4 out of 8 in -'^' ' 5 or 70 ways ; and it 



may be similarly shown that for 5, 6, 7, and 8 planets, 

 meeting at one time, the numbers of ways are 56, 28, 8, 

 and I. Thus we have solved the whole question of the 

 variety of conjunctions of eight planets ; and adding all the 

 numbers together, we find that 247 is the utmost possible 

 number of modes of meeting. 



In general algebraic language, we may say that a group 

 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 3) (nm -f i) 



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 GanieTes. 1 We shall find it perpetually recurring in 

 questions both of combinations and probability, and 

 throughout the formulae of mathematical analysis traces 

 of its influence may 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 z <( 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 re- 

 quired in a multitude of cases of this theory." As early 

 *s 1544 Stifels had noticed the remarkable properties of 

 these numbers and the mode of their evolution. Briggs, 

 the inventor of the common system of logarithms, was so 

 struck with their importance that he called them the 



1 (Euvres Completes de Pascal (1865), vol. iii. p. 302. Montucla 

 states the name as De Gruieres, Ilistoire des Mathematiques, vol. iii 



2 Histoire des Mathematiques, vol. iii. p. 378. 



