FIGURATE NUMBERS.* 



Edmund A. Englek. 



A consideration of the following properties of natural 

 numbers leads to the sets of series of numbers which have 

 come to be known as figurate numbers^ and provides a 

 simple method of disclosing some of the properties of 

 these numbers. 



If we write a series, each term of which is the number 

 one (1), thus 



11111 , 



it is evident that the sum of any number of terms of this 

 series is expressed by the formula 



' ^ = n (1) 



where n represents the number of terms considered ; that 

 is for 

 ^ = 1,2,3,4, , 8 = 1,2,3,4, , 



or the n^^ term of the series 1, 2, 3, 4, is equal 



to the sum of the first n terms of the series 1, 1, 1, 1, . . . . 

 Write a number of rows of the number one (1), as fol- 

 lows: 



Tahle 1. 



* Presented before The Academy of Science of St. Louis, April 3, 1911. 



^ Figurate numbers were so called by Nicomachus {circa 100 A. D.) because 

 of the possibility of arranging points in regular figures (plane or solid), 

 according to certain rules, to represent them. (37) 



