60 Royal Society. 



1 = 1 2 



1 + 3 =2» 



1 + 3 + 5 =3« 



1 + 3 + 5 + 7 = 4 2 



1 + 3 + 5 + 7 + 9 =5 2 



1+3 + 5 + 7 + 9+11 =6 2 



1 + 3 + 5+7 + 9 + 11 + 13 = 7 2 



Thus, every square number is formed by the addition of a series 

 of odd numbers commencing with unity ; a result universally 

 known. 



The difference of any two squares is either an odd number, or the 

 sum of consecutive odd numbers. 



Each series may be resolved into two others consisting of alter- 

 nate odd numbers, the respective sums of which are two adjacent 

 triangular numbers, the addition of which it is well known forms a 

 square. Ex. : . 



1+5 + 9 + 13=28 

 3 + 7 + 11 = 21 



49=7* 



B. 



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



b4- 1 

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



1 =1* 



1|+2| =2* 



2 + 3 + 4 =3* 



2i + 3| + 4i + 5i =42 



3+4 + 5 + 6 + 7 =5 2 



3£ + 4i+5i + 6i+7i+8i = 6 2 



4 + 5 + 6 + 7 + S + 9+10 = 7 2 



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 (3«+ 1) 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 ; 



