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 _r_² = _x_² + _y_² + _z_², 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 12 × 12 = 144, 4 × 4 = 16, 3 × 3 = 9, 144 + 16 + 9 = 169 = 13 × 13.