THE MATHEMATICAL SCIENCES 123 



but rectangles (Fig. 11). We notice also that the sum 

 1 + 2 + 3 -f • ••• + f» of n consec- 

 utive numbers beginning by one 

 is a triangle (Fig. 12). 



It is not only plane figures 

 which thus correspond to sums of 

 numbers arranged in series, it is 

 also spatial figures. For example, 

 by superposing the triangular 

 numbers we obtain the pyra- 

 midal numbers 1, then 1 +3 =4, 

 then again 1 -f 3 + 6 ■» 10, etc., this being represented 

 as in Figure 13. 



Fig. 10. 



• • 



• • 



• • 



Fig. 11. 



It was probably from these arithmetical-spatial con- 

 ceptions there originated the classification of numbers 

 into squared numbers (obtained by multiplying a num- 

 ber by itself), plane numbers (formed by two factors), 

 and solid numbers such as the cube. Of this classifi- 

 cation only the terms square and cube still remain. 



Further, as numbers were not abstractions, but 

 beings endowed with qualities and almost feelings, 

 there were some which were perfect, that is, equal to 

 the sum of their divisors (for example, 6 = 1 +2 +3), 

 and there were others which were " friendly," that is, 

 such that each was equal to the sum of the divisors 

 of the other. 1 



1 3 Boutroux, Analyse, I, p. 5. 



