COMBINATIONS AND PERMUTATIONS. 209 



On carefully examining these numbers, we shall find 

 that they are connected with each other by an almost 

 unlimited series of relations, a few of the more simple 

 of which may be noticed. 



1. Each vertical column of numbers exactly corre- 

 sponds with an oblique series descending from left to 

 right, so that the triangle is perfectly symmetrical in its 

 contents. 



2. 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 tri- 

 angular numbers, because they correspond with the 

 numbers of balls which may be arranged in a triangular 

 form, thus 



o 



O 00 



O 00 000 



o oo ooo oooo 

 o oo ooo oooo ooooo 



These numbers evidently differ each from the previous 

 one by the series of natural numbers. Their employment 

 has been explained, and the first 20,000 of the numbers 

 calculated and printed by E. de Joncourt in a small 

 quarto volume, which was published at the Hague, in 

 1762. 



The fourth column contains the pyramidal numbers, 

 so called because they correspond to the number 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 



