A Study of the Properties of Integral Numbers. 



379 



The sum of the series, or 625, will then be x times the 

 square of the number of terms, as is shown in the former 

 discussion. 



Any number which is in the second 'column of Table I 

 can be thus treated. 



TABLE V. 



The series having a sum 3 3 appears in Table V as com- 

 puted from equations (4). The first and last terms are 

 a = 3 and 1 = 7. This is one of the results of Nico- 

 machus. 



It is therefore apparent that any integral number 

 raised to any power can be represented as the sum of a 

 consecutive series of odd numbers. 



In the discussion which followed the presentation of 

 this paper, Dr. C. H. Danforth suggested that the numeri- 

 cal values represented in Table I might be represented 

 graphically, as is done in the diagrams at the top of 

 Fig. 1. Starting with a small square at the upper left- 

 hand corner, having unit area, a series of equal squares 

 representing consecutive odd numbers are added in the 

 three diagrams on the right. The added areas are in each 



