from the second paper. For the avoidance of ambiguity, zero will always be a valid possible constituent of "sums of two squares", so for example every square of an integer is trivially expressible as the sum of two squares by setting one of them to be zero. 1. The product of two numbers, each of which is a sum of two squares, is itself a sum of two squares. This is a well-known property, based on the identity due to Diophantus. 2. If a number which is a sum of two squares is divisible by a prime which is a sum of two squares, then the quotient is a sum of two squares. (This is Euler's first Proposition). Indeed, suppose for example that is divisible by and that this latter is a prime. Then divides Since is a prime, it divides one of the two factors. Suppose that it divides . Since (Diophantus's identity) it follows that must divide . So the equation can be divided by the square of . Dividing the expression by yields: and thus expresses the quotient as a sum of two squares, as claimed. On the other hand if divides , a similar argument holds by using the following variant of Diophantus's identity: 3. If a number which can be written as a sum of two squares is divisible by a number which is not a sum of two squares, then the quotient has a factor which is not a sum of two squares. (This is Euler's second Proposition). Suppose is a number not expressible as a sum of two squares, which divides . Write the quotient, factored into its (possibly repeated) prime factors, as so that . If all factors can be written as sums of two squares, then we can divide successively by , , etc., and applying step (2.) above we deduce that each successive, smaller, quotient is a sum of two squares. If we get all the way down to then itself would have to be equal to the sum of two squares, which is a contradiction. So at least one of the primes is not the sum of two squares. 4. If and are relatively prime positive integers then every factor of is a sum of two squares. (This is the step that uses step (3.) to produce an 'infinite descent' and was Euler's Proposition 4. The proof sketched below