483 
DIOPHANTUS OF ALEXANDRIA 
hypotenuse. The method is this. Form two right-angled 
triangles from (a, h) and (c, d) respectively, by which Dio- 
phantus means, form the right-angled triangles 
{a 2 + h 2 , a 2 — h 2 , 2 ah) and (c 2 + d 2 , c 2 — d 2 , 2 cd). 
Multiply all the sides in each triangle by the hypotenuse of 
the other; we have then two rational right-angled triangles 
with the same hypotenuse (a 2 4- h 2 ) [c 2 + d 2 ). 
Two others are furnished by the formula above; for we 
have only to ‘ form two right-angled triangles ’ from (ac + hd, 
ad — he) and from (ac — hd, ad + he) respectively. The method 
fails if certain relations hold between a, h, c, d. They must 
not be such that one number of either pair vanishes, i.e. such 
that ad = he or ac = hd, or such that the numbers in either 
pair are equal to one another, for then the triangles are 
illusory. 
In the case taken by Diophantus a 2 + h 2 = 2 2 +1 2 = 5, 
c 2 + d 2 = 3 2 + 2 2 = 13, and the four right-angled triangles are 
(65, 52, 39), (65, 60, 25), (65, 63, 16) and (65, 56, 33). 
On this proposition Fermat has a long and interesting note 
as to the number of ways in which a prime number of the 
form 4 n +1 and its powers can be (a) the hypotenuse of 
a rational right-angled triangle, (b) the sum of two squares. 
He also extends theorem (3) above: ‘ If a prime number which 
is the sum of two squares be multiplied by another prime 
number which is also the sura of two squares, the product 
will be the sum of two squares in two ways; if the first prime 
be multiplied by the square of the second, the product will be 
the sum of two squares in three ways; the product of the first 
and the cube of the second will be the sum of two squares 
in four ways, and so on ad infinitum.’ 
Although the hypotenuses selected by Diophantus, 5 and 13, 
are prime numbers of the form 4 n + 1, it is unlikely that he 
was aware that prime numbers of the form 4 n +1 and 
numbers arising from the multiplication of such numbers are 
the only classes of numbers which are always the sum of two 
squares; this was first proved by Euler. 
(4) More remarkable is a condition of possibility of solution 
prefixed to V. 9, ‘ To divide 1 into two parts such that, if