PORISMS ANU PROPOSITIONS ASSUMED 481 
In fact YZ±a = {(m+ 1) (2m+ l)+2a} 2 , 
ZX+a = {m(2 m+ 1) + 2 a} 2 , 
17+« = {m(m+l) + a} 2 . 
(3) If 
A r = m 2 + 2, F = (m + l) 2 + 2, ^ = 2 {m 2 + (m + l) 2 + 1} + 2, 
then the six expressions 
YZ—(Y + Z), ZX-{Z+X), XY— (X + Y), 
YZ-X, ZX-Y, XY-Z 
are all squares (V. 6). 
In fact 
7^-(F+^) = (2m 2 + 3m+3) 2 , FF-J=(2m 2 + 3m + 4) 2 , &c. 
2. The second class is much more important, consisting of 
propositions in the Theory of Numbers which we find first 
stated or assumed in the Arithmetica. It was in explana 
tion or extension of these that Fermat’s most famous notes 
were written. How far Diophantus possessed scientific proofs 
of the theorems which he assumes must remain largely a 
matter of speculation. 
(a) Theorems on the composition of numbers as the sum 
of two squares. 
(1) Any square number can be resolved into two squares in 
any number of ways (II. 8). 
(2) Any number which is the sum of two squares can be 
resolved into two other squares in any number of ways (II. 9). 
(It is implied throughout that the squares may be fractional 
as well as integral.) 
(3) If there are two whole numbers each of which is the 
sum of two squares, the product of the numbers can be 
resolved into the sum of two squares in two ways. 
In fact (a 2 + h 2 ) (c 2 -+ d 2 ) = (ac + hd) 2 + {ad + he) 2 . 
This proposition is used in III. 19, where the problem is 
to find four rational right-angled triangles with the same 
1623.2 I j