538 Royal Society : — 



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"), 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 



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



51 83 



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



and also when they are (witliin 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 I, 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'- + 5- and —4 and 5 differ by 9; but 39 = P+ 

 2"+3-+5^ 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 e<:[ual roots : 39 is however divisible 

 into 1--I-1-+ F-4-6*, and then the equal roots are discovered. It is 

 proved from the known properties of numbers that this property of 

 having 2 roots whose difference will be 0, 1, 2, 3, &c., as far as is 

 possible, belongs to all odd numbers. A new symbol is then sug- 

 gested to represent the division of a immber into 4 squares, such 

 that 2 of the roots will have a given difference, and these are made 

 the exterior roots ; the number or figure denoting the difference is 

 placed above on the left hand : thus ^25 denotes 0, 0, 3, 4 or 

 — 2, 1, 4, 2 ; '25 denotes 1, 2, 4, 2. 



The general theorem is this : — If any odd number of odd numbers 

 be in arithmetical progression (4 being the common difference), as 

 9, 13, 17, 21, 25, then if the common difference be assumed as the 

 index of the difference of roots to the middle term, and the higher 

 terms in the series have as indices (4-|- 1), (4-t-2), &c. in succession. 



