147 



B. 



Every square n- is the sum of an arithmetical progression of n 



I 1 



terms, the first term of which is "T and the common difference 1 . 



m 



This arrangement renders evident that every square of an odd 

 number is the sum of as many consecutive natural numbers as the 

 root has units. 



Every square of an odd number is the difference between two tri- 

 angular numbers the bases of which are respectively (3w+l) and n. 

 For, the sum of any series of natural numbers is the difference of 

 two series of natural numbers commencing with unity ; and since, 

 as it is shown above, every square of an odd number is the sum of a 

 series of natural numbers, it is also the difference between two tri- 

 angular numbers. 



It is also evident that series, the sums of which are squares of odd 

 numbers, may be so taken that, when placed in succession, they will 

 form an uninterrupted progression of natural numbers commencing 

 with unity, the sum of which is a triangular number ; 



...&c.= 



a triangular number the base of which is the series 

 (1+3 + 9 + 27 ..... + 3"). 



2. CUBE NUMBERS. 



* 

 If S= 3 , the first term =n+-(\ n). 



