SQUAEES INTO WHICH ODD NHMBEES MAY BE DIVIDED. 
55 
if the second set of roots be squared, the sum is which 
is 2pm more than the former ; if, therefore, the first sum of squares equals n — mp, the 
second will equal n-\-mp ; therefore the pair of terms that are at the distance (m) from 
the centre will have their middle roots the same, and their exterior roots one less by m, 
the other greater by m than those of the other. 
In this proof p, the common difierence, may be odd or even, and w, the middle term, 
may be odd or even ; thus 
22 30 410 ^14 ns 
-1,0, 0,1 -1,0, 1,2 -2, 1,1, 2 -2, 0,1, 3 -3, 0,0, 3 
is a series composed of even numbers, all of which obey the theorem ; but frequently 
an even number is not so divisible as to form the required difference. To form every 
difference is a property which belongs universally to odd numbers only, not to even 
numbers; the common difierence may be only 1; the numbers from 25 to 41 are all 
(both odd and even) divisible into four squares, whose roots conform to the theorem, 
33 being the middle term. 
-^25 
-626 
-^27 -"28 -^29 
-^30 -"31 «32 
4,0,0-3 5,0,0,-l 
5,1,1,0 2,2,4,-2 5,0,0,2 
3,2,4,1 3,3,3,2 0,4,4,0 
"33 
0,4, 4,1 
941 
0 
00 
^39 ®38 ®37 
"36 ^35 ^34 
-4, 0,0, 5 -2, 0,0, 6 
-1,1, 1,6 -3,2,4,3 +1,0,0,6 
0, 2,4,4 1,3,3,4 -1,4,4,1 
To apply this theorem as a proof of the matters stated in the beginning of the paper, 
all the examples, whether of the odd squares, or of the even squares +1, or those 
numbers increased or decreased, may be made terms in an arithmetic series ; 4w^ — 4n+l 
and 4w^+4%-|-l (which represent any 2 adjoining odd squares) have a difierence of 8w, 
which is divisible by 4, and therefore they may form terms in an arithmetical series ; thus 
4w"-4w+l, 4?^^-2?^4-l, 4w^+l, 47^^+2?i+l, 4w^+4^^+l, 
the common difierence being 2n, and the odd squares will be two places from the middle 
term ; their exterior roots will therefore be greater and less by 2. So any 2 adjoining odd 
squares, 4^^^ and 4w*-}-8w4-4, difier by 8w-|-4, which is divisible by 4 ; and the adjoining 
even squares, +1, may in like manner be made terms in an arithmetic series. 
I propose now to apply the theorem generally to other instances of odd numbers 
having 2 roots equal and the other 2 roots differing by any number whatever, the 
one root being greater, the other less by that number; for example, the alternate odd 
squares may be divided into 4 squares in such manner that 2 roots of the one may 
equal 2 roots of the other, and the differences of the remaining roots will be 4 : 
19 949 
0,2, 2,1 -4, 2, 2, 5 
+ 1,0, 2, 2 -3, 0,2, 6 
