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XXV. On the Mysteries of Numbers alluded to by Fermat.— Second Communication. 
By the Bight Hon. Sir Frederick Pollock, Bart., M.A., F.B.S., F.S.A . , F.G.S., &c. 
Received January 14, — Read February 13, 1868. 
In the last paper I described the mode of constructing a square (which, for brevity, I 
shall call The Square) which would necessarily be made up of three different series ; 
the indices in the margin and in the small squares were explained, and I must refer to 
that paper for the explanations. 
I propose in this paper to show (from that square and a supplemental one) that all 
the odd numbers possess the properties that I have ascribed to them. The Square also 
proves the first theorem of Fermat, viz. that every number is composed of three trian- 
gular numbers or less from the 2nd theorem (relating to the squares), which I believe 
has not hitherto been done. 
If any one will take the trouble to examine any odd number, he will find that it may 
be divided into four squares (generally in several ways or forms) in some set of four roots 
of the squares composing the number; two of the roots will be equal, two of the same, 
or another set of four will differ by 1 ; two will differ by 2, two by 3, and so on, as far 
as the number is large enough to have roots of sufficient magnitude to furnish such dif- 
ferences. This, of course, cannot be done by one set of roots, except by a few of the low 
numbers*. If the four roots do not furnish all the differences, then he will discover 
that there is another or more forms that will furnish the whole of them. 
The algebraic sum of the four roots will also, in some or other of the forms, be equal 
to 1, 3, 5, 7, &c., that is, to every odd number within the compass of the given number. 
These properties, which may be discovered in any odd number, I propose to show belong 
to all odd numbers. 
It may be well, in order to make this statement quite clear, to take an example. 
73 is one of the terms in the series 1, 3, 7, 13, 21, &c., a series increasing by 2, 4, 6, 8, 
&c. ; every term of the series has roots of the form n, n, n, n+ 1. 
73 is composed of four squares in four different forms; the four sets of roots are 
4, 4, 4, 5 
2, 2, 4, 7 
0 , 1 , 6 , 6 
1 , 2 , 2 , 8 
Differences of roots. 
0, 1, 8, 9 
2, 3, 4, 5, 6, 11 
5, 7, 12, &c. 
10, &c. 
* 3, 5, 7, 11, 15, and 23 have each but one form, and the last (23) is the highest odd number that has only 
one form of roots. 
MDCCCLXVIII. 
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