ALLUDED TO BY EERMAT. 
637 
In like manner any 2 roots that differ by 1 may be increased by 4, 12, 24, 40, &c. 
(2a 2 -\-2a) by changing 
Increase. 
n, n- f 1 into n— 1, n-\- 2 4 
n— 2, n-\- 3 12 
n— 3, n+4 24 
&c. &c. {2a 2 -\-2a) 
Another conclusion is, that if The Square have as the first number in it all the odd 
numbers in succession which are to be found in The Square when 1 is the first number, 
then every other odd number will be obtained as some one of the numbers thus formed. 
For every such Square will begin with a number of the form of l-\-2a 2 -\-2a-{-2b 2 ; and 
in forming The Square from that, 2m 2 -f 2m + 2n 2 will be added, therefore all the possible 
combinations of 2a 2 -\-2a-\-2b 2 and 2m 2 -\-2m-\-2n 2 will be found, with every value of a 
and h, m and n ; that is, every odd number will be found. 
But if any odd number whatever be made the first number in The Square , and a Sup- 
plemental Square be formed, and the numbers in the Supplemental Square be successively 
put in the first place in The Square , the assumed number will be found in some of the 
terms at the top, and also in some of those at the side. And the necessary consequence 
is, if we are allowed to notice the numbers of The Square when 1 is the first number, 
that any odd number is not only equal to 1 -\-2a 2 -\-2a-\-2b 2 -\-2c 2 -\-2c-\-2d 2 , but also 
to \-\-2a 2 -\-2a-\-2b 2 -\-2c 2 -\-2c, or l-\-2a 2 -\-2a-\-2b 2 -\-2d 2 •, that is, the four squares may 
have two equal roots, or two roots differing by 1 . 
I now propose to show in what manner The Square can obtain a division of its first 
term into four squares, the algebraic sum of whose roots =1; the result of which is 
that the first term less 1, divided by 2, will be composed of 3 trigonal numbers (see 
Theorem C in Philosophical Transactions, vol. cxliv. p. 315). 
Every odd number is of the form 4%+l, or 4%+3. If it be of the form 4w+3, then 
in the (2w+l)th term of the series which increases by 2, 4, 6, 8, 10, &c., the roots will 
be n, n-\~ 1, n-{- 1, n-\- 1. 
For 4w+3+2n terms of 2, 4, 6, 8, &c. 
2n 
=4ra+3+(2 + 4«) x 7 ) 
n 2 
= 4w 2 4-6%+3= 
(n+iy 
(»+ 1) 2 
I>+1) 2 
the roots of which are n, w-fl, n-\- 1, w+1. The index of this term is 4w+l (since it 
is the (2w-(-l)th term of 1, 3, 5, 7, &c.), that is, 2 less than the sum of the roots. 
In like manner, if the number be of the form 4?i+l, then the 2nth term in the series 
already mentioned will have as roots n, n, n, n-\- 1. 
4 s 2 
