2 HISTORY OF THE THEORY OF NUMBERS. [CHAP. I 



1 + 3 + 5 = 9, etc. In Ch. 19, p. 47, the triangular numbers are denned 

 to be 1, 1 + 2, 1 + 2 + 3, . In Ch. 20, p. 52, the squares are obtained 

 as before and the pentagonal numbers are obtained by addition of 1, 4, 7, 

 10, . In Chapters 26 and 27, pp. 62-64, pentagonal and hexagonal 

 numbers are shown by dots forming regular pentagons (as in the figure 

 on the preceding page) or hexagons. Ch. 28, p. 65, gives the theorem that 

 the sum of two consecutive triangular numbers is a square. In Ch. 30, 

 p. 66, is denned the pyramidal number P r m = p] n + p 2 m + + p r m . 



Nicomachus 5 (about 100 A.D.) gave the same definitions and results 

 as did Theon of Smyrna and perhaps gave them slightly earlier. Ch. 12 

 gives the theorem on consecutive triangular numbers: 



- r(r + 1) 



<-i > 



also the corresponding theorem that the sum of the rth square and (r l)th 

 triangular number is the rth pentagonal number, just as a pentagon is 

 obtained by annexing a triangle to a square. He gave the generalization 

 (apart from the notation) : 



<n r 4- n r ~ l <n'" 

 Pm T Ps - Pm+1- 



These theorems are illustrated by means of the following table: 



Each polygon equals the sum of the polygon immediately above it in 

 the table and the triangle with 1 less in its side [triangle in the preceding 

 column]]; for example, heptagon 148 is the sum of hexagon 120 and 

 triangle 28. 



Each vertical column is an arithmetical progression whose common 

 difference is the triangle in the preceding column. 



In Ch. 13 he remarked that just as polygonal numbers arise by summing 

 the simple arithmetical progressions, so by summing the polygonal numbers 

 one obtains the like named pyramidal numbers, triangular pyramid from 

 the triangular numbers, pyramid with square base from the squares, etc., 

 the base being the largest polygon. 



* Introductio arithmetica (ed., Hoche), 2, 1866, Book 2, Chs. 8-20. Cf . G. H. F. Nesselmann, 

 Algebra der Griechen, 1842, 202. 



