A STUDY OF THE PROPERTIES OF INTEGRAL 
NUMBERS.” 
Francis Ky. NrpuHer. 
In the 1917 edition of A History of Elementary Mathe- 
maties, pp. 32-3, Cajori refers to the work of Nicomachus, 
who livedfipout A. D. 100, as follows: 
‘‘He gives*the following important proposition: Al! 
cubical numbers are equal to the sum of consecutive odd 
numbers. Thus, 8=2°==3+5; 27=3?=74+9+11; 
64 = 4# = 138+ 15+ 17+ 19.”’ 
It seemed probable. that such interesting relations did 
not end so abruptly as this statement might indicate, 
although information along this line seems to be lacking 
in modern text-books and mathematical dictionaries. 
A series of consecutive numbers and likewise of con- 
secutive even numbers yielded no results of this character. 
It was found that a series of consecutive odd numbers 
from unity to any odd number y satifies the equation 
ass: 
154 he to y= san 2 == N? 
where N is the number of terms in the series. The table 
of numbers here given shows the relation of these num- 
bers. In Chrystals’ Algebra, Vol. I, p. 
467, it is stated that this proposition was 
known to the Greek geometers. It is fe ees 
TABLE Tf. 
stated as follows: ‘‘The sum of any num- a ae 
ber of consecutive odd integers beginning 3| 4=-2 
with unity is the square of their number.”’ 5| 9=3* 
A table of squares furnished the neces- 7\16=# 
sary data for a study of the subject. It 9 | 95 = 5? 
was at once apparent that any number 2 11 | 36=—6 
raised to any degree where 7 is an even 
number could be represented as the sum of an arithmeti- 
*Pragented before the Academy of Science of St. Louis, Jan. 19, 1920. 
(373) 
* 
