SQIJAEES INTO WHICH ODD NHMBBES MAY BE DIVIDED. 
61 
16"f*l — 17 
+ 1, 0, 0, 4 
364-1=37 
-1, 2, 4,4 
&c. 
364-1=37 
- 1 , 0 , 0,6 
64:4-1=65 
-3, 2, 4,6 
&c. 
And each of these may in like manner be increased, or (subject to a similar limit) be 
diminished; these, however, are derived from the alternate terms of another series, 
1, 5, 13, 25, &c. (271^4-2^4-1), and 7^^4■(^^-l-l)^ will represent any 
2 alternate terms of the series; and if to 2n^ — 6774 - 6 , and also to 2n^-\-2n-{-l there be 
added 277^—277, they become 477^ — 8774-6, and 477^4"l5 or (277-- 2)^ 4-1? and (277)^4-1? 
or, adjoining even squares 4-1- 
The various examples of these or similar relations among the roots of the 4 squares 
into which numbers may be divided are endless; the increase and decrease of the variable 
roots is not always by 2 ; it may be by any other number. But instead of multiplying 
examples, it will be better to enter on the proof of what has been aBeady stated, which 
will furnish the means of investigating every instance that can be produced. The proof 
depends upon, — 
1st, a general property of all odd numbers which (as far as I am aware) has not hitherto 
been noticed ; and 
2ndly, a general theorem relating to odd numbers in arithmetical progression. 
The property of odd numbers is this : — Every odd number may be divided into 4 
squares, in such manner that 2 of the roots will be equal, 2 of them will differ by 1, 
2 of them will differ by 2, 2 of them by 3, and so on, as far as the number is capable 
(from its magnitude) of having roots large enough to form the difference required. The 
difference may be algebraical, and result from one of the roots being considered as a 
negative quantity. For example, there is only one mode of dividing the number 23 
(a number of the form (8774-7)) into 4 squares; these must be 1, 4, 9, 9, and their roots 
±1, ±2, ±3, ±3:— 
3 and 3 are equal, difference 0, — 3 and 3 differ by 6, 
3 and 2 differ by 1, —2 and 3 differ by 6, 
3 and 1 differ by 2, —1 and 3 differ by 4, 
2 and —1 differ by 3. 
The differences of the roots may therefore be 0, 1, 2, 3, 4, 5 or 6 ; a greater difference 
than 6 is (in the number 23) impossible; the least numbers that would make a difference 
of 7 would be 3 and 4, and the sum of their squares would be 26. 
The same form or mode of dividing a number into 4 squares does not always furnish 
every possible difference (as in the case of 23); thus 13 in one form has the roots 
0, 0, 2, 3 ; these furnish as differences 0, 1, 2, 3, 6. To obtain the difference 4, the other 
form of dividing 13 into 4 squares (viz. 1®, 2^, 2®, 2^) must be resorted to, and then —2, 
4-2 differ by 4: so 39 has two modes of dividing into 4 squares; in one the roots are 
H 2 
