And he proceeded to prove the theorem of Pythagoras--not in Euclid’s way, but by the simpler and more satisfying method which was, in all probability, employed by Pythagoras himself. He had drawn a square and dissected it, by a pair of crossed perpendiculars, into two squares and two equal rectangles. The equal rectangles he divided up by their diagonals into four equal right-angled triangles. The two squares are then seen to be the squares on the two sides of any one of these triangles other than the hypotenuse. So much for the first diagram. In the next he took the four right-angled triangles into which the rectangles had been divided and re-arranged them round the original square so that their right angles filled the corners of the square, the hypotenuses looked inwards, and the greater and less sides of the triangles were in continuation along the sides of the square (which are each equal to the sum of these sides). In this way the original square is redissected into four right-angled triangles and the square on the hypotenuse. The four triangles are equal to the two rectangles of the original dissection. Therefore the square on the hypotenuse is equal to the sum of the two squares--the squares on the other two sides--into which, with the rectangles, the original square was first dissected. In very untechnical language, but clearly and with a relentless logic, Guido expounded his proof. Robin listened, with an expression on his bright, freckled face of perfect incomprehension. “Treno,” he repeated from time to time. “Treno. Make a train.” “In a moment,” Guido implored. “Wait a moment. But do just look at this. _Do._” He coaxed and cajoled. “It’s so beautiful. It’s so easy.” So easy.... The theorem of Pythagoras seemed to explain for me Guido’s musical predilections. It was not an infant Mozart we had been cherishing; it was a little Archimedes with, like most of his kind, an incidental musical twist. “Treno, treno!” shouted Robin, growing more and more restless as the exposition went on. And when Guido insisted on going on with his proof, he lost his temper. “Cattivo Guido,” he shouted, and began to hit out at him with his fists. “All right,” said Guido resignedly. “I’ll make a train.” And with his stick of charcoal he began to scribble on the stones. I looked on for a moment in silence. It was not a very good train. Guido might be able to invent for himself and prove the theorem of Pythagoras; but he was not much of a draughtsman. “Guido!” I called. The two children turned and looked up. “Who taught you to draw those squares?” It was conceivable, of course, that somebody might have taught him.