210 THE PRINCIPLES OF SCIENCE. 



contain the trianguli-pyramidal, the pyramidi-pyramidal 

 numbers, and so on. k 



3. 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 thus seen to be a complete 

 classification of all numbers according as they have unity 

 for any of their differences. 



4. Every number in the table is equal to the sum of 

 the numbers which stand higher in the next column to 

 the left, beginning with the next line above ; thus 84 is 

 equal to the sum of 28, 21, 15, 10, 6, 3, i. 



5. Since each line is formed by adding the previous 

 line to itself, it is evident that the sum of the numbers 

 in each horizontal line must be double that of the line 

 next above. Hence we know, without making any ad- 

 ditions, that the successive sums must be i, 2, 4, 8, 16, 

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

 the Logical Abecedarium. Speaking generally, the sum 

 of the numbers in the nth line will be 2 n ~ 1 . 



6. 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 1 i ; the first two lines give 3 or 2 2 i ; the first three 

 lines 7 or 2 3 i ; the first six lines give 63 or 2 6 i ; 

 or, speaking in general language, the sum of the first 

 n lines is 2 n i. 



k Wallis's ' Algebra/ Discourse of Combinations, &c. p. 109.^ - 



