We come now to Pythagoras's achievements in geometry. There is a story that, when he came home from Egypt and tried to found a school at Samos, he found the Samians indifferent, so that he had to take special measures to ensure that his geometry might not perish with him. Going to the gymnasium, he sought out a well-favoured youth who seemed likely to suit his purpose, and was withal poor, and bribed him to learn geometry by promising him sixpence for every proposition that he mastered. Very soon the youth got fascinated by the subject for its own sake, and Pythagoras rightly judged that he would gladly go on without the sixpence. He hinted, therefore, that he himself was poor and must try to earn his living instead of doing mathematics; whereupon the youth, rather than give up the study, volunteered to pay sixpence to Pythagoras for each proposition. 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 = 2^2). 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 = 3^2); and so on. The successive odd numbers 1, 3, 5 ... were called _gnomons_, and the general formula is 1 + 3 + 5 + ... + (2n - 1) = n^2. Add the next odd number, i.e. 2n + 1, and we have n^2 + (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 = m^2; then n = 1/2(m^2 - 1), and we have {1/2(m^2 - 1)}^2 + m^2 = {1/2(m^2 + 1)}^2, where m is any odd number; and this is the general formula attributed to Pythagoras. Proclus also attributes to Pythagoras the theory of proportionals and the construction of the five "cosmic figures," the five regular solids.