239 



the other 2 less than the corresponding roots of the other. So 

 comparing 15 2 with 17 2 , 



225 289 



4,3,10,10 6,3,10,12 



6,5,10,8 8,5,10,10 



the result is the same ; and it is true of all adjoining odd squares. 

 The paper contains a Table of odd squares (up to 27 2 ), compared in 

 this manner with the odd square immediately before it and after it. 

 It is then shown that the same property continues when the 2 odd 

 squares are increased by any the same even number 



49 81 



0,2,3,6 -2,2,3,8 



51 83 



-1,3,4,5 -3,3,4,7 



and also when they are (within certain limits) diminished by the 

 same even number. It is then shown that a similar property be- 

 longs to the even squares + 1 , as seen below, 



16 + 1 = 17 36 + 1 = 37 



+ 1,0,0,4 -1,0,0.6 



0,2,2,3 -2,2,2,5 



37 65 



-1,2,4,4 -3,2,4,6 



and also to these numbers increased or decreased by the same even 

 number. 



If, instead of comparing the adjoining squares, the alternate squares 



be compared, a similar result is obtained ; the middle roots are the 

 same, the exterior roots differ by 4 instead of 2. 



The proof of this property depends upon a general property of 

 all odd numbers and upon a general theorem. 



The property of odd numbers is this, that every odd number can 

 be divided into 4 squares in such manner that 2 of the roots will be 

 equal, 2 will differ by 1, 2 will differ by 2, &c. as far as the number 

 is capable (from its magnitude) of having roots large enough to form 

 the difference required : thus in the No. 39 there cannot be roots 

 having a difference of 9 ; for the least number that can have that 

 difference is 41=4 2 +5 2 and 4 and 5 differ by 9; but 39=l 2 -f- 

 2 2 +3 2 + 5 2 , and the difference between 3 and 5 is 8 ; and the num- 

 bers 1, 2, 3, 5, either as positive or negative, give all the differences 

 up to 8, but they do not give 2 equal roots : 39 is however divisible 



VOL. ix. s 



