ALLUDED TO BY EEKMAT. 
635 
term in the series be correct, that is, have the proper sum of two roots, or the proper 
sum of all the roots, then every other term in that series will also be correct (see last 
paper) and may be derived from that term, and it will therefore have all the qualities 
that belong to the different portions of The Square, and in every one it will be divided 
into four squares, and the roots will appear. It may be said the first does not appear to 
have two roots equal, or two differing by 1, but (see Diagram No. 1) where the top row 
is called A, the second B, the third C, the fourth D, and the terms in each row are 
distinguished by 1, 2, 3, 4, and because 25 in A 1 has two roots differing by 1, 29 in 
A 2 will have two roots differing by 3; and because 17 in A1 has two roots differing 
by 1, 29 in A 3 will have two roots differing by 5 ; and because 5 has two roots differing 
by 1, 29 in A 4 will have two roots differing by 7. But in A 2, 29 (which is derived 
from 27 the sum of whose roots is 1) will have the sum of its roots 3 ; and for a similar 
reason in A 3 it will have the sum of its roots 5 (derived from 23 the sum of whose roots 
is 1 in B 2), and in A 4 it will have the sum of its roots 7. 
But if the sum of the four roots =2w+l and the difference between two of them 
=2w+l, the other two must be equal to one another, or they would prevent the sum 
of the roots being 2n-\-l. 
A similar mode of reasoning applies to show that 29 must have two of its roots differing 
by 1. For, because 27 has two roots equal, therefore 29 in B1 (derived from 27) has 
two roots differing by 2, and in Cl (derived from 21) has two roots differing by 4, and 
in D 1 (derived from 11) has two roots differing by 6. But if any number has the sum 
of its roots equal to 2w+l and to 2w+3, and two of the roots differ by 2^+2, the other 
two must differ by 1, or the sum of all the roots could not be both 2w+l and 2w-j-3. 
I propose now to point out some of the results of what has been already stated. 
There can be no doubt that every number, odd or even, is composed of four squares 
or less. And whether this be proved by the gradation series , or by the combination of 
The Square and the Supplemental Square (if these furnish a proof), or by Lagrange’s 
method from the prime numbers, or by any other, the result is the same, the proposition 
is true ; and it follows that a number of the form 4n -f- 2 must he composed of two odd 
squares, or of two odd squares and one even one, or two odd squares and two even ones. 
In the first case 4w + 2 = 4a 2 + 4a+l -\-4b 2 -\-4b-{-l ; deducting 2 from each side and 
dividing by 4, n=a 2 -\-a-\-b 2 -\-b \ that is, four trigonal numbers, of which two are equal 
and the other two are equal. In the second case 4w-j-2 = 4a 2 +4a + l + 45 2 +45 + l+4c? 2 , 
n=a 2 a-\-b 2 -{-b c 2 ; but d?-\-a-\-c 2 equals 2 trigonal numbers, therefore n equals four and 
trigonal numbers, of which two are equal. In the third case n=a 2 -4-a-\-b 2 -\-b-\-c 2 -\-d 2 ; 
and as a 2 -{- and b 2 -\-b-\-d 2 are each equal to 2 trigonal numbers, n in every case is 
equal to four trigonal numbers, that is, to a 2 -\-a-\-l) 2 -\-c 2 -4-c-\-d 2 . 
In Euler’s 4 Opuscula Analytical vol. ii. p. 4, he says, “ Lagrange’s demonstration as 
to the square numbers is deduced from principles of such a kind that clearly no assist- 
ance can be expected from that to prove the rest of Fermat’s theorems*. 
* Quae autem ex ejusmodi principiis estdeducta, ut inde nullum plane subsidium ad reliqua demonstranda 
expectari possit. 
MDCCCLXVIII. 4 S 
