.] COMBINATIONS AND PERMUTATIONS. 185 



Examining these numbers, we find that they are con- 

 nected by an unlimited series of relations, a few of the 

 more- simple of which may be noticed. Each vertical 

 column of numbers exactly corresponds with an oblique 

 series descending from left to right, so that the triangle is 

 perfectly symmetrical in its contents. The first column 

 contains only units; the second column contains the 

 natural numbers, I, 2, 3, &c. ; the third column contains 

 a remarkable series of numbers, I, 3, 6, 10, 15, &c., which 

 have long been called the triangular numbers, because they 

 correspond with the numbers of balls which may be 

 arranged in a triangular form, thus 



o 



o o o 



o o o o o o 



O 00 000 0000 



o oo ooo oooo ooooo 



The fourth column contains the pyramidal numbers, so 

 called because they correspond to the numbers of equal 

 balls which can be piled in regular triangular pyramids. 

 Their differences are the triangular numbers. The numbers 

 of the fifth column have the pyramidal numbers for their 

 differences, but as there is no regular figure of which they 

 express the contents, they have been arbitrarily called the 

 trianguli-triangular numbers. The succeeding columns 

 have, in a similar manner, been said to contain the 

 trianguli-pyramidal, the pyramidi-pyramidal numbers, 

 and so on. 1 



From the mode of formation of the table, it follows that 

 the differences of the numbers in each column will be 

 found in the preceding column to the left. Hence the 

 second differences, or the differences of differences, will be 

 in the second column to the left of any given column, the 

 third differences in the third column, and so on. Thus 

 we may say that unity which appears in the first column 

 is the first difference of the numbers in the second column ; 

 the second difference of those in the third column ; the third 

 difference of those in the fourth, and so on. The triangle 

 is seen to be a complete classification of all numbers 

 according as they have unity for any of their differences. 

 Since each line is formed by adding the previous line 



1 Wallis's Algebra, Discourse of Combinations, &c., p. 109. 



