The A B C of Relativity

by Bertrand Russell

But now suppose that, instead of taking a small piece of the earth’s surface which can be regarded as flat, you consider making a map of the world. An accurate map of the world on flat paper is impossible. A globe can be accurate, in the sense that everything is produced to scale, but a flat map cannot be. I am not talking of practical difficulties, I am talking of a theoretical impossibility. For example: the northern halves of the meridian of Greenwich and the ninetieth meridian of west longitude, together with the piece of the equator between them, make a triangle whose sides are all equal and whose angles are all right angles. On a flat surface, a triangle of that sort would be impossible. On the other hand, it is possible to make a square on a flat surface, but on a sphere it is impossible. Suppose you try on the earth: walk 100 miles west, then 100 miles north, then 100 miles east, then 100 miles south. You might think this would make a square, but it wouldn’t, because you would not at the end have come back to your starting point. If you have time, you may convince yourself of this by experiment. If not, you can easily see that it must be so. When you are nearer the pole, 100 miles takes you through more longitude than when you are nearer the equator, so that in doing your 100 miles east (if you are in the northern hemisphere) you get to a point further east than that from which you started. As you walk due south after this, you remain further east than your starting point, and end up at a different place from that in which you began. Suppose, to take another illustration, that you start on the equator 4,000 miles east of the Greenwich meridian; you travel till you reach the meridian, then you travel northwards along it for 4,000 miles, through Greenwich and up to the neighborhood of the Shetland Islands; then you travel eastward for 4,000 miles, and then 4,000 miles south. This will take you to the equator at a point 4,000 miles further east than the point from which you started.

In a sense, what we have just been saying is not quite fair, because, except on the equator, traveling due east is not the shortest route from a place to another place due east of it. A ship traveling (say) from New York to Lisbon, which is nearly due east, will start by going a certain distance northward. It will sail on a “great circle,” that is to say, a circle whose centre is the centre of the earth. This is the nearest approach to a straight line that can be drawn on the surface of the earth. Meridians of longitude are great circles, and so is the equator, but the other parallels of latitude are not. We ought, therefore, to have supposed that, when you reach the Shetland Islands, you travel 4,000 miles, not due east, but along a great circle which lands you at a point due east of the Shetland Islands. This, however, only reinforces our conclusion: you will end at a point even further east of your starting point than before.

What are the differences between the geometry on a sphere and the geometry on a plane? If you make a triangle on the earth, whose sides are great circles, you will not find that the angles of the triangle add up to two right angles: they will add up to rather more. The amount by which they exceed two right angles is proportional to the size of the triangle. On a small triangle such as you could make with strings on your lawn, or even on a triangle formed by three ships which can just see each other, the angles will add up to so little more than two right angles that you will not be able to detect the difference. But if you take the triangle made by the equator, the Greenwich meridian, and the ninetieth meridian, the angles add up to three right angles. And you can get triangles in which the angles add up to anything up to six right angles. All this you could discover by measurements on the surface of the earth, without having to take account of anything in the rest of space.

The theorem of Pythagoras also will fail for distances on a sphere. From the point of view of a traveler bound to the earth, the distance between two places is their great circle distance, that is to say, the shortest journey that a man can make without leaving the surface of the earth. Now suppose you take three bits of great circles which make a triangle, and suppose one of them is at right angles to another—to be definite, let one be the equator and one a bit of the meridian of Greenwich going northward from the equator. Suppose you go 3,000 miles along the equator, and then 4,000 miles due north; how far will you be from your starting point, estimating the distance along a great circle? If you were on a plane, your distance would be 5,000 miles, as we saw before. In fact, however, your great circle distance will be considerably less than this. In a right-angled triangle on a sphere, the square on the side opposite the right angle is less than the sum of the squares on the other two sides.

These differences between the geometry on a sphere and the geometry on a plane are intrinsic differences; that is to say, they enable you to find out whether the surface on which you live is like a plane or like a sphere, without requiring that you should take account of anything outside the surface. Such considerations led to the next step of importance in our subject, which was taken by Gauss, who flourished a hundred years ago. He studied the theory of surfaces, and showed how to develop it by means of measurements on the surfaces themselves, without going outside them. In order to fix the position of a point in space, we need three measurements; but in order to fix the position of a point on a surface we need only two—for example, a point on the earth’s surface is fixed when we know its latitude and longitude.

Now Gauss found that, whatever system of measurement you adopt, and whatever the nature of the surface, there is always a way of calculating the distance between two not very distant points of the surface, when you know the quantities which fix their positions. The formula for the distance is a generalization of the formula of Pythagoras: it tells you the square of the distance in terms of the squares of the differences between the measure quantities which fix the points, and also the product of these two quantities. When you know this formula, you can discover all the intrinsic properties of the surface, that is to say, all those which do not depend upon its relations to points outside the surface. You can discover, for example, whether the angles of a triangle add up to two right angles, or more, or less, or more in some cases and less in others.

But when we speak of a “triangle,” we must explain what we mean, because on most surfaces there are no straight lines. On a sphere, we shall replace straight lines by great circles, which are the nearest possible approach to straight lines. In general, we shall take, instead of straight lines, the lines that give the shortest route on the surface from place to place. Such lines are called “geodesics.” On the earth, the geodesics are great circles. In general, they are the shortest way of traveling from point to point if you are unable to leave the surface. They take the place of straight lines in the intrinsic geometry of a surface. When we inquire whether the angles of a triangle add up to two right angles or not, we mean to speak of a triangle whose sides are geodesics. And when we speak of the distance between two points, we mean the distance along a geodesic.

The next step in our generalizing process is rather difficult: it is the transition to non-Euclidean geometry. We live in a world in which space has three dimensions, and our empirical knowledge of space is based upon measurement of small distances and of angles. (When I speak of small distances, I mean distances that are small compared to those in astronomy; all distances on the earth are small in this sense.) It was formerly thought that we could be sure à priori that space is Euclidean—for instance, that the angles of a triangle add up to two right angles. But it came to be recognized that we could not prove this by reasoning; if it was to be known, it must be known as the result of measurements. Before Einstein, it was thought that measurements confirm Euclidean geometry within the limits of exactitude attainable; now this is no longer thought. It is still true that we can, by what may be called a natural artifice, cause Euclidean geometry to seem true throughout a small region, such as the earth; but in explaining gravitation Einstein is led to the view that over large regions where there is matter we cannot regard space as Euclidean. The reasons for this will concern us later. What concerns us now is the way in which non-Euclidean geometry results from a generalization of the work of Gauss.

There is no reason why we should not have the same circumstances in three-dimensional space as we have, for example, on the surface of a sphere. It might happen that the angles of a triangle would always add up to more than two right angles, and that the excess would be proportional to the size of the triangle. It might happen that the distance between two points would be given by a formula analogous to what we have on the surface of a sphere, but involving three quantities instead of two. Whether this does happen or not, can only be discovered by actual measurements. There are an infinite number of such possibilities.

This line of argument was developed by Riemann, in his dissertation “On the hypotheses which underlie geometry” (1854), which applied Gauss’s work on surfaces to different kinds of three-dimensional spaces. He showed that all the essential characteristics of a kind of space could be deduced from the formula for small distances. He assumed that, from the small distances in three given directions which would together carry you from one point to another not far from it, the distances between the two points could be calculated. For instance, if you know that you can get from one point to another by first moving a certain distance east, then a certain distance north, and finally a certain distance straight up in the air, you are to be able to calculate the distance from the one point to the other. And the rule for the calculation is to be an extension of the theorem of Pythagoras, in the sense that you arrive at the square of the required distance by adding together multiples of the squares of the component distances, together possibly with multiples of their products. From certain characteristics in the formula, you can tell what sort of space you have to deal with. These characteristics do not depend upon the particular method you have adopted for determining the positions of points.