410 SIE F. POLLOCK ON FEEMAT’S THEOEEM OF THE POLYGONAL NOIBEES. 
perties connected with the division of numbers into 4 squares, which probably (in some 
form) were part of the system to which Feemat alluded. 
I have ah’eady stated two properties of the odd numbers (not, I believe, noticed 
before), upon one of which the whole of Feemat’s theorem depends (as will hereafter 
appear). The first is to be found in the Transactions of the Eoyal Society for the year 
1854, p. 313 ; it is there called Theorem C. 
“ Every odd number may be divided into square numbers (not exceeding 4), the alge- 
braic sum of whose roots (positive or negative) will (m some form of the roots) be equal 
to every odd number from 1 to the greatest possible sum of the roots. 
“ Or in a purely algebraic form. If 
and 
CL ~\~h ~\~G -\-(l — 2/'-|-li 
a, h, c, d being integral or nil, n and r being positive, and r a maximum, then if r' be 
any positive integer (not greater than r), it will always be possible to satisfy the pah of 
equations 
-\-x ■\-y +2 =:2/+l5 
by integral values (positive, negative, or nil) of w, sc, y, z” 
The other is to be found in the Koyal Society’s Transactions for 1859, p. 49, and 
relates not to the sum of the roots, but to the difference between two of them. The 
first of these connects together the first and second branch of Feemat’s theorem. 
For if every odd number can be divided into 4 square numbers, so that the sum of 
the roots of two of them being deducted from the sum of the roots of the other two, 
there shall be a remainder of 1, — 
Then evo'y number is divisible into 3 triangular numbers ; for the 2 sums of the roots 
must be of the form 2a-\-l, and 2a, and the four roots will be of the form 
a+i^+l, a—i^, a-\-g, a—g; 
and if 2n-\-l equals the sum of these roots squared, 
2n-\-\ = ^a^-{-2gf-{-2cf-\-2a-\-2])-\-l, and n'=2a^-\-a-\-'p^’\-X)-{-^', 
but 2a^-\-a is a triangular number, and^^+i>+2^ is the general form for the sum of any 
2 triangular numbers*; therefore 7i any number is equal to 3 triangular numbers (nil 
being considered as a triangular number, as some of the terms may become equal to 
nothing). 
There are some theorems worthy of remark arising out of a comparison of the differ- 
ences of the roots of the four square numbers into which every odd number may be 
divided. 
It will appear from the Table that accompanies this paper, that when a number of 
the form 4/i-l-l is dhisible into 2 square numbers (of which one must be even and the 
other odd, ^n-\-\ being an odd number), the roots of these 2 squares furnish the exte- 
* The proof of this is given presently. 
