3² + 4² = 5².

It is supposed that Pythagoras noticed this fact, after he had learned from the Egyptians that a triangle whose sides are 3, 4 and 5 has a right angle. He found that this could be generalized, and so arrived at his famous theorem: In a right-angled triangle, the square on the side opposite the right angle is equal to the sum of the squares on the other two sides.

Similarly in three dimensions: if you take a right-angled [solid block], the square on the diagonal (the dotted line in the figure) is equal to the sum of the squares on the three sides.

This is as far as the ancients got in this matter.

The next step of importance is due to Descartes, who made the theorem of Pythagoras the basis of his method of analytical geometry. Suppose you wish to map out systematically all the places on a plain—we will suppose it small enough to make it possible to ignore the fact that the earth is round. We will suppose that you live in the middle of the plain. One of the simplest ways of describing the position of a place is to say: starting from my house, go first such and such a distance east, then such and such a distance north (or it may be west in the first case, and south in the second). This tells you exactly where the place is. In the rectangular cities of America, it is the natural method to adopt: in New York you will be told to go so many blocks east (or west) and then so many blocks north (or south). The distance you have to go east is called x, and the distance you have to go north is called y. (If you have to go west, x is negative; if you have to go south, y is negative.) Let O be your starting point (the “origin”); let OM be the distance you go east, and MP the distance you go north. How far are you from home in a direct line when you reach P? The theorem of Pythagoras gives the answer. The square on OP is the sum of the squares on OM and MP. If OM is four miles, and MP is three miles, OP is 5 miles. If OM is 12 miles and MP is 5 miles, OP is 13 miles, because 12² + 5² = 13². So that if you adopt Descartes’ method of mapping, the theorem of Pythagoras is essential in giving you the distance from place to place. In three dimensions the thing is exactly analogous. Suppose that, instead of wanting merely to fix positions on the plain, you want to fix stations for captive balloons above it, you will then have to add a third quantity, the height at which the balloon is to be. If you call the height z, and if r is the direct distance from O to the balloon, you will have

r² = x² + y² + z²,

and from this you can calculate r when you know x, y, and z. For example, if you can get to the balloon by going 12 miles east, 4 miles north, and then 3 miles up, your distance from the balloon in a straight line is 13 miles, because