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.
“Nobody.” He shook his head. Then, rather anxiously, as though he were afraid there might be something wrong about drawing squares, he went on to apologise and explain. “You see,” he said, “it seemed to me so beautiful. Because those squares”—he pointed at the two small squares in the first figure—“are just as big as this one.” And, indicating the square on the hypotenuse in the second diagram, he looked up at me with a deprecating smile.
I nodded. “Yes, it’s very beautiful,” I said—“it’s very beautiful indeed.”
An expression of delighted relief appeared on his face; he laughed with pleasure. “You see, it’s like this,” he went on, eager to initiate me into the glorious secret he had discovered. “You cut these two long squares”—he meant the rectangles—“into two slices. And then there are four slices, all just the same, because, because—oh, I ought to have said that before—because these long squares are the same, because those lines, you see….”
“But I want a train,” protested Robin.
Leaning on the rail of the balcony, I watched the children below. I thought of the extraordinary thing I had just seen and of what it meant.
I thought of the vast differences between human beings. We classify men by the colour of their eyes and hair, the shape of their skulls. Would it not be more sensible to divide them up into intellectual species? There would be even wider gulfs between the extreme mental types than between a Bushman and a Scandinavian. This child, I thought, when he grows up, will be to me, intellectually, what a man is to a dog. And there are other men and women who are, perhaps, almost as dogs to me.
Perhaps the men of genius are the only true men. In all the history of the race there have been only a few thousand real men. And the rest of us—what are we? Teachable animals. Without the help of the real men, we should have found out almost nothing at all. Almost all the ideas with which we are familiar could never have occurred to minds like ours. Plant the seeds there and they will grow; but our minds could never spontaneously have generated them.
There have been whole nations of dogs, I thought; whole epochs in which no Man was born. From the dull Egyptians the Greeks took crude experience and rules of thumb and made sciences. More than a thousand years passed before Archimedes had a comparable successor. There has been only one Buddha, one Jesus, only one Bach that we know of, one Michelangelo.
Is it by a mere chance, I wondered, that a Man is born from time to time? What causes a whole constellation of them to come contemporaneously into being and from out of a single people? Taine thought that Leonardo, Michelangelo, and Raphael were born when they were because the time was ripe for great painters and the Italian scene congenial. In the mouth of a rationalising nineteenth-century Frenchman the doctrine is strangely mystical; it may be none the less true for that. But what of those born out of time? Blake, for example. What of those?
This child, I thought, has had the fortune to be born at a time when he will be able to make good use of his capacities. He will find the most elaborate analytical methods lying ready to his hand; he will have a prodigious experience behind him. Suppose him born while Stone Henge was building; he might have spent a lifetime discovering the rudiments, guessing darkly where now he might have had a chance of proving. Born at the time of the Norman Conquest, he would have had to wrestle with all the preliminary difficulties created by an inadequate symbolism; it would have taken him long years, for example, to learn the art of dividing MMMCCCCLXXXVIII by MCMXIX. In five years, nowadays, he will learn what it took generations of Men to discover.
And I thought of the fate of all the Men born so hopelessly out of time that they could achieve little or nothing of value. Beethoven born in Greece, I thought, would have had to be content to play thin melodies on the flute or lyre; in those intellectual surroundings it would hardly have been possible for him to imagine the nature of harmony.
From drawing trains, the children in the garden below had gone on to playing trains. They were trotting round and round; with blown round cheeks and pouting mouth, like the cherubic symbol of a wind, Robin puff-puffed, and Guido, holding the skirt of his smock, shuffled behind him, tooting. They ran forward, backed, stopped at imaginary stations, shunted, roared over bridges, crashed through tunnels, met with occasional collisions and derailments. The young Archimedes seemed to be just as happy as the little tow-headed barbarian. A few minutes ago he had been busy with the theorem of Pythagoras.
Now, tooting indefatigably along imaginary rails, he was perfectly content to shuffle backwards and forwards among the flower-beds, between the pillars of the loggia, in and out of the dark tunnels of the laurel tree. The fact that one is going to be Archimedes does not prevent one from being an ordinary cheerful child meanwhile. I thought of this strange talent distinct and separate from the rest of the mind, independent, almost, of experience. The typical child-prodigies are musical and mathematical; the other talents ripen slowly under the influence of emotional experience and growth. Till he was thirty Balzac gave proof of nothing but ineptitude; but at four the young Mozart was already a musician, and some of Pascal’s most brilliant work was done before he was out of his teens.
In the weeks that followed, I alternated the daily piano lessons with lessons in mathematics. Hints rather than lessons they were; for I only made suggestions, indicated methods, and left the child, himself to work out the ideas in detail. Thus I introduced him to algebra by showing him another proof of the theorem of Pythagoras. In this proof one drops a perpendicular from the right angle on to the hypotenuse, and arguing from the fact that the two triangles thus created are similar to one another and to the original triangle, and that the proportions which their corresponding sides bear to one another are therefore equal, one can show in algebraical form that c2 + d2 (the squares on the other two sides) are equal to a2 + b2 (the squares on the two segments of the hypotenuse) + 2ab; which last, it is easy to show geometrically, is equal to (a + b)2, or the square on the hypotenuse.
Guido was as much enchanted by the rudiments of algebra as he would have been if I had given him an engine worked by steam, with a methylated spirit lamp to heat the boiler; more enchanted, perhaps—for the engine would have got broken, and, remaining always itself, would in any case have lost its charm, while the rudiments of algebra continued to grow and blossom in his mind with an unfailing luxuriance. Every day he made the discovery of something which seemed to him exquisitely beautiful; the new toy was inexhaustible in its potentialities.
In the intervals of applying algebra to the second book of Euclid, we experimented with circles; we stuck bamboos
