PASCAL AND FERMAT. 17 



The Arithmetical Triangle in its simplest form consists of the 



In the successive horizontal rows we have what are now called 

 the figurate numbers. Pascal distinguishes them into orders. He 

 calls the simple units 1, 1, 1, 1,... which form the first row, num- 

 bers of the first order; he calls the numbers 1, 2, 3, 4,... which 

 form the second row, numbers of the second order; and so on. 

 The numbers of the third order 1, 3, 0, 10,... had already received 

 the name oi triangular numbers; and the numbers of the fourth 

 order 1, 4, 10, 20,... the name oi pyr^amidal numbers. Pascal says 

 that the numbers of the fifth order 1, 5, 15, 35,... had not yet 

 received an express name, and he proposes to call them triangulo- 

 triangulaires. 



In modern notation the if^ term of the r*^ order is 



n(ii + l) ... {n + r - 2) 



r-1 



Pascal constructs the Arithmetical Triangle by the foUowdng 

 definition ; each number is the sum of that immediately above it 

 and that immediately to the left of it. Thus 



10 = 4 + 0, 35 = 20 + 15, 126 = 70 + 50,... 



The properties of the numbers are developed by Pascal with 

 great skill and distinctness. For example, suppose we require the 

 sum of the first n terms of the r^^ order : the sum is equal to the 

 number of the combinations of n + r — 1 things taken r at a 

 time, and Pascal establishes this by an inductive proof 



2 



