186 THE PRINCIPLES OF SCIENCE. [chap. 



to itself, it is evident that the sum of the numbers in each 

 horizontal line must be double the sum of the numbers in 

 the line next above. Hence we know, without making 

 the additions, that the successive sums must be i, 2, 4, 

 8, 16, 32, 64, &c., the same as the numbers of combinations 

 in the Logical Alphabet. Speaking generally, the sum of 

 the numbers in the nth. line will be 2"~\ 



Again, if the whole of the numbers down to any line be 

 added together, we shall obtain a number less by unity 

 than some power of 2 ; thus, the first line gives i or 

 2^ — I ; the first two lines give 3 or 2'^ — i ; the first three 

 lines 7 or 2^ — i ; the first six lines give 63 or 2^ — i ; or, 

 speaking in general language, the sum of the first n lines 

 is 2" — I. It follows that the sum of the numbers in any 

 one line is equal to tlie sum of those in all the preceding 

 lines increased by a unit. For the sum of the nth. line is, 

 as already shown, 2"~\ and the sum of the first n — i lines 

 is 2""' — I, or less by a unit. 



This account of the properties of the figurate numbers 

 does not approach completeness ; a considerable, probably 

 an unlimited, number of less simple and obvious relations 

 might be traced out. Pascal, after giving many of the 

 properties, exclaims^: "Mais j'en laisse bien plus que je 

 n'en donne ; c'est une chose etrange combien il est fertile 

 en proprietes ! Chacun pent s'y exercer." The arith- 

 metical triangle may be considered a natural classification 

 of numbers, exhibiting, in the most complete manner, 

 their evolution and relations in a certain point of view. 

 It is obvious that in an unlimited extension of the 

 triangle, each number, with the single exception of the 

 number two, has at least two places. 



Though the properties above explained are highly 

 curious, the greatest value of the triangle arises from the 

 fact that it contains a complete statement of the values of 

 the formula (p. 182), for the numbers of combinations of m 

 things out of n, for all possible values of m and 71. Out 

 of seven things one may be chosen in seven ways, and 

 seven occurs in the eighth line of the second column. The 

 combinations of two things chosen out of seven are 



or 2 1 , which is the third number in the eightli 



1x2' 



' LEuvres Comjdetes, vol. iii. p. 251. 



