LEMMA I.
To find two square numbers such that their sum is also square.
Let two numbers AB, BC be set out, and let them be either both even or both odd.
Then since, whether an even number is subtracted from an even number, or an odd number from an odd number, the remainder is even, [IX. 24, 26] therefore the remainder AC is even.
Let AC be bisected at D.
Let AB, BC also be either similar plane numbers, or square numbers, which are themselves also similar plane numbers.
Now the product of AB, BC together with the square on CD is equal to the square on BD. [II. 6]
And the product of AB, BC is square, inasmuch as it was proved that, if two similar plane numbers by multiplying one another make some number the product is square. [IX. 1]
Therefore two square numbers, the product of AB, BC, and the square on CD, have been found which, when added together, make the square on BD.