Archimedes

In geometry Pythagoras set himself to lay the foundations of the subject, beginning with certain important definitions and investigating the fundamental principles. Of propositions attributed to him the most famous is, of course, the theorem that in a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the sides about the right angle (Eucl. I., 47); and, seeing that Greek tradition universally credits him with the proof of this theorem, we prefer to believe that tradition is right. This is to some extent confirmed by another tradition that Pythagoras discovered a general formula for finding two numbers such that the sum of their squares is a square number. This depends on the theory of the gnomon, which at first had an arithmetical signification corresponding to the geometrical use of it in Euclid, Book II. A figure in the shape of a gnomon put round two sides of a square makes it into a larger square. Now consider the number 1 represented by a dot. Round this place three other dots so that the four dots form a square (1 + 3 = 2²). Round the four dots (on two adjacent sides of the square) place five dots at regular and equal distances, and we have another square (1 + 3 + 5 = 3²); and so on. The successive odd numbers 1, 3, 5 ... were called gnomons, and the general formula is

1 + 3 + 5 + ... + (2n − 1) = n².

Add the next odd number, i.e. 2n + 1, and we have n² + (2n + 1) = (n + 1)². In order, then, to get two square numbers such that their sum is a square we have only to see that 2n + 1 is a square. Suppose that 2n + 1 = m²; then n = ½(m² − 1), and we have {½ (m² − 1) }² + m² = {½ (m² + 1) }², where m is any odd number; and this is the general formula attributed to Pythagoras.