Part 1 — CHAPTER V AN ALTERNATIVE VIEW OF THE NON-EUCLIDEAN GEOMETRIES

THE type of geometry we obtain is dependent, as we know, on our definition of congruence. If we define as congruent displacements the displacements of those bodies which in ordinary life we consider rigid and undeformable, we obtain Euclidean geometry. If, on the other hand, we define as congruent the displacements of those bodies which in ordinary life we should regard as of changeable shape, we generally obtain some non-Euclidean type of geometry.

In order to simplify what is to follow, we shall first consider the particular case of two-dimensional geometry. Consider, then, a plane surface. If on this plane surface we effect measurements with Euclidean rods, we obtain Euclidean results; if we employ rods which according to the Euclidean point of view squirm in an appropriate way when displaced, we obtain the geometry of Riemann or of Lobatchewski, as the case may be. It is to be noted that the plane is the same in all three cases, and yet the geometry we obtain on its surface may be Euclidean or non-Euclidean. It is not the plane itself but our methods of measurement conducted thereon which have changed.

Now it may be mentioned that there exist a number of alternative ways of representing both Euclidean and non-Euclidean geometries. Which way is to be preferred depends largely on the scope of our investigations. One of these alternative procedures has often been appealed to in popular writings (by Helmholtz in particular) because it enables us to visualise the sequence of theorems involved, without abandoning thereby our habitual Euclidean representations.

Thus three-dimensional Euclidean space is taken as a starting point. It is then proposed to show how, in this three-dimensional Euclidean space, a two-dimensional non-Euclidean geometry can arise. The fundamental space which we are here postulating being Euclidean, distances in this space must be computed with rigid Euclidean rods. Then it is shown that if we apply tiny Euclidean rods to the surface of a sphere and conduct measurements on this surface, we shall obtain a series of numerical results which are representative of Riemann’s geometry. The chief advantage of this method of presentation is that it allows us to foresee at a rapid glance that Riemann’s geometry must be consistent,[17] since in the present case it reduces to Euclidean geometry on a sphere, which is a particular case of Euclidean geometry in three-dimensional space; and Euclidean geometry is known to be consistent. But aside from this advantage the procedure is to be avoided, for it tends to obscure the philosophical importance of non-Euclidean geometry.

In the first place it is not always possible to follow this method. Consider, for example, the case of Lobatchewski’s geometry. Just as Riemann’s geometry was that of the spherical surface, so Lobatchewski’s geometry turns out to be that of a peculiar saddle-shaped surface called the pseudosphere, as was proved by Beltrami. Hilbert, however, has shown that there cannot exist a surface free from singularities which would represent the total spread of Lobatchewski’s plane geometry; hence here is a first reason for being on our guard against a number of unsuspected difficulties.

But this is not all. If we adopt the method of representing two-dimensional non-Euclidean geometry as the geometry obtained by taking Euclidean measurements on a curved surface, how are we to conceive of three-dimensional Riemannian geometry? Obviously we must start with a four-dimensional Euclidean space in which a three-dimensional spherical surface is embedded. But as a four-dimensional Euclidean space transcends our immediate experience and is in the nature of a mathematical fiction, we should be led to the erroneous conclusion that three-dimensional non-Euclidean geometry must likewise be a purely conceptual construction having nothing in common with measurements which might be conducted with material rods.

We should be led into still greater difficulties if we wished to represent a three-dimensional non-Euclidean space possessing various degrees of non-Euclideanism from place to place. Reasoning by analogy, we should be tempted to say that just as an irregularly curved surface embedded in three-dimensional Euclidean space yielded a two-dimensional geometry of varying degrees of non-Euclideanism, so now all we should have to do would be to conceive of an irregularly curved three-dimensional surface embedded in a four-dimensional Euclidean space. But the analogy would be deceptive; for calculation shows that it would be impossible to represent an arbitrarily curved three-dimensional surface in a four-dimensional Euclidean space. In the general case we should have to situate our variously bumped three-dimensional surface in a six-dimensional Euclidean space. In the same way, a four-dimensional non-Euclidean space of variable curvature could be represented only in a ten-dimensional Euclidean space; and so on.[18]

Now in view of these difficulties, in view of the fact that to represent a non-Euclidean space of three dimensions in terms of Euclidean space, we may be compelled to appeal to a Euclidean space of six dimensions, it is certainly exalting Euclidean space unduly to regard it as The Space. This is especially obvious when we remember that we could have conceived of our three-dimensional non-Euclidean space directly, as a result of measurements with squirming rods, without any reference to Euclidean geometry, and without ever having had to introduce any greater number of dimensions.

At this juncture the beginner is often inclined to argue as follows: “You say that non-Euclidean geometry of two dimensions is the geometry obtained by executing Euclidean measurements on a suitably curved surface. But your concept of curvature is meaningless unless we conceive it as contrasted with some pre-existing standard of straightness. Does not your presentation prove, therefore, that Euclidean space as representative of flatness or straightness is logically antecedent to non-Euclidean space, which connotes curvature?”

This argument is radically incorrect, and arises from too loose an understanding of non-Euclidean geometry. In the first place, to assume that the concept of curvature presupposes the concept of straightness is no more inevitable than to assume that the concept of straightness presupposes that of curvature. For if it is true to say that that which is curved is that which is not straight, it is equally true to say that that which is straight is that which is not curved; so that in the absence of curvature, straightness would in turn be meaningless. The question of deciding, for example, whether a circle or a straight line is the more fundamental is strictly a matter of opinion. The Greeks held circular motion to be the noblest of all motions; and between the designations “noblest” and “most fundamental,” the distinction is exceedingly slight. And even this is not all, for thus far we have been arguing as though Euclidean space were truly flat, and non-Euclidean space truly curved. But we must remember that this curvature of which we speak is not to be construed as representing anything absolute; it arises solely from the particular type of representation we have agreed to select.

Thus, in the presentation of non-Euclidean geometry which we have just been discussing, we started from three-dimensional Euclidean space as a basis and considered curved surfaces embedded in this space. This form of presentation was equivalent, as we know, to stating that Euclidean measurements must be adhered to on the curved surface. But we might have proceeded otherwise. We might have started with three-dimensional non-Euclidean space and considered the geometry of curved surfaces (of a sphere, for example) embedded in this space. This would have been equivalent to applying non-Euclidean rods on our curved surface.

Under suitable conditions the geometry of our curved surface (or sphere) would then become Euclidean. Euclideanism itself would thus be linked with curvature, while non-Euclideanism, which would here be the type of geometry obtained on the plane embedded in our three-dimensional non-Euclidean space, would accordingly be linked with flatness. In short, we must not allow the word “curvature” to mislead us into forming a false impression about non-Euclidean geometry. In many respects Riemann’s choice of the word “curvature” has proved unfortunate and the more scientific appellation non-Euclideanism should be adhered to.[19]

There is still another point which we must mention. It might be argued that in Riemann’s geometry of two dimensions, that is to say, in the geometry of the spherical surface when Euclidean measurements are adhered to, a straight line on the sphere, being a great circle, constitutes a closed curve, whereas in Euclidean geometry a straight line can never constitute a closed curve. Here, however, the difficulty which arises does not pertain so much to Euclideanism and non-Euclideanism as to another branch of geometry, Analysis Situs, on which we shall make a few brief remarks in a note at the end of the chapter. The fact is that the geometry of the cylinder embedded in Euclidean space is Euclidean to the same extent as the geometry of the plane, and yet on a cylinder certain straight lines, namely, the rings which surround the cylinder, constitute closed curves.

It might be objected that these rings are not Euclidean straight lines. But here the critic would be confusing geometry and dimensionality. From the standpoint of two-dimensional Euclidean geometry, the rings on the cylinder are perfectly straight lines, and it would never enter the mind of the flat being moving over the cylindrical surface to view them in any other light. Only when we represent the cylinder’s surface in a three-dimensional Euclidean space must we regard the rings as curves. In short, lines that would be straight in two dimensions might be curved when viewed from the standpoint of a higher dimensionality. By analogy, lines which would be regarded as straight in three-dimensional space might be recognised as curved were we to appeal to a fourth dimension. Thus, whichever way we choose to investigate the concept of straightness, we find that it eludes us more and more.

It appears unnecessary to dwell further on these rather special problems, for they would lead us too far afield.

Now, finally, there is still another reason, on which we have lightly touched, which must make us very wary of being led astray by faulty analogies. When, for instance, we consider the two-dimensional geometry of a spherical surface embedded in a three-dimensional Euclidean space, we realise that the surface divides space into an outside and an inside. We should then be inclined to argue that a three-dimensional Riemannian or spherical space must also divide four-dimensional space into an inside and an outside.

This line of reasoning would be correct in the present case, but it is important to note that this conception of an inside and an outside is in no wise essential to non-Euclidean geometry. Had we adopted the more rational method of presenting three-dimensional non-Euclidean space as due to the peculiar behaviour of our measuring rods, the conception of an inside and an outside would have been meaningless. We must understand that this notion arose merely as a result of our particular choice of a mode of presentation, and in no wise constitutes an intrinsically necessary condition.

If these rather delicate points have been understood, no harm can be done by discussing non-Euclidean geometries as the geometries obtained by applying Euclidean rods on curved surfaces. In many respects, indeed, this method of presentation is helpful; for accustomed as we are in everyday life to effect measurements with Euclidean rods, we are able to visualise more easily a series of abstract investigations when this familiar procedure is followed.

Let us now proceed to a more thorough study of the preceding method. We have said that non-Euclidean results are obtained when we confine ourselves to Euclidean rods while conducting our measurements on curved surfaces. In the case of Riemannian geometry of two dimensions, the required surface is that of a sphere embedded in three-dimensional Euclidean space. In the case of Lobatchewski’s geometry, it is a saddle-shaped surface known as a pseudosphere (subject to the limitations mentioned previously).

We will confine ourselves to the study of Riemann’s geometry for the time being. Such a study in two dimensions permits easy visualisation, owing to the fact that since the earth is spherical in shape it is precisely this type of geometry which we obtain when we conduct measurements on its surface with rigid Euclidean measuring rods (assuming the surface to be perfectly smooth, like that of the ocean on a calm day). In fact, we shall see that the postulates of Riemann’s geometry are immediately verified on the sphere.

To begin with, the shortest distance, or, better still, the most direct distance, between two points on the earth’s surface, when Euclidean rigid rods or inextensible tape measures are employed as a means of measurement, is an arc of a great circle. Great circles, such as meridians on the surface of our planet, constitute therefore the straight lines or geodesics of Riemann’s geometry; and when discussing Riemann’s geometry we may call them straight lines.

Consider, then, one such straight line or geodesic, and let it be that particular great circle which we call the equator. According to the axioms of Euclid’s geometry two perpendiculars drawn at two different points of the same straight line can never meet; they lie parallel to one another. On the sphere, on the other hand, two perpendiculars to the equator are two meridians and these meridians always meet at the pole. In fact it is impossible to draw any two great circles on the sphere which do not intersect. Hence we see that no parallel geodesics can exist in Riemann’s geometry. Likewise, whereas in Euclid’s geometry the sum of the three angles of a triangle is equal to two right angles, in Riemann’s geometry this sum is always greater than two right angles. This we could easily verify by stretching ropes between three distant points on the earth’s surface so as to form a triangle, and then summing the values of the angles at the three corners of our triangle.

A similar verification would ensue if we were to effect measurements along a circumference and its diameter. Thus, if at a central point on the earth’s surface we fix one end of a rope and cause the other extremity to rotate around the central point, it will describe a circle on the earth’s surface. We shall then find, upon measuring the length of the circumference marked out by the extremity of the rope, that the ratio of the length of this circumference to the length of the diameter is a variable number, always smaller than and depending on the area covered by our circle; whereas in Euclidean geometry this ratio would be the constant number .

There is still another result of importance which must be mentioned. On a plane, the farther we wander along a straight line, the farther we move from our starting point, whereas on our sphere we see that if we follow a straight line or geodesic we shall finally return to our starting point after having circled the earth. We express this fact by saying that in Riemann’s geometry space is finite, yet unbounded. It is finite since we cannot wander away indefinitely from our starting point; it is unbounded because however far we go, we never come to a stone wall or a gap beyond which we can proceed no farther. On the other hand, Euclidean space, such as that of the plane surface, is infinite and unbounded.

And now suppose the earth on which we were conducting our measurements were to swell indefinitely. All the characteristics of Riemann’s geometry which we have discussed would gradually fade away, provided we limited our measurements to the same restricted area of the surface. Our measurements would still yield Riemannian results, of course, since however large our earth had grown, it would still remain spherical. But the results of our measurements over a definite area would approximate more and more to those of Euclidean geometry. The reason is obvious, since the greater the volume of the sphere, the more nearly would a given area of its surface approximate to an ordinary plane.

In fact a plane surface may be assimilated to the surface of a sphere of infinite radius; so that if our sphere were to grow indefinitely we should find that our measurements lost little by little their Riemannian characteristics, until it would be impossible to distinguish them from Euclidean measurements.

Suppose now that our sphere continued to change in shape after having attained an infinite radius; in particular, suppose that our surface gradually began to assume the shape of a saddle, curved downwards if we intersected it from east to west and curved upwards if we intersected it from north to south. As this double curvature became gradually more pronounced we should find that our measurements lost their Euclidean character and became more and more conspicuously Lobatchewskian.

Thus we see that Euclidean geometry stands at the dividing line of Riemannian and Lobatchewskian geometry. When our surface changes in shape the geometry remains Riemannian so long as there subsists the least trace of sphericity, that is, of positive curvature. Likewise it remains Lobatchewskian so long as there subsists the least trace of saddle-shapedness, or negative curvature. Finally, it is strictly Euclidean only when the surface has become a plane, that is, exhibits zero curvature.

For this reason Euclidean geometry has a uniqueness about it which is denied to the non-Euclidean varieties. These latter constitute a class of geometries; and the precise type of the geometry we may happen to be discussing is determined by the intensity of curvature of the sphere or pseudosphere to which it pertains. We also see why it is so difficult to determine by empirical methods whether the space of the universe is truly Euclidean or not. It is because both Riemannian and Lobatchewskian geometry merge by insensible gradations into Euclidean geometry.

The analysis we have given refers to non-Euclidean geometries as derived from the geometry of surfaces; that is, to the numerical results obtained when we use rigid Euclidean rods to effect measurements on curved surfaces. It must be noted, however, that the surfaces we have discussed are of a very special kind. Both spheres and pseudospheres are known as surfaces of constant curvature. There is no need to go into the mathematical definition of what is meant by this; we will limit ourselves to stating the principal characteristic of such surfaces.

Consider, for example, a net made of inextensible threads (inextensible or undeformable being used in the Euclidean sense). If this net can be applied with perfect contact to any portion of a plane, it can be slid over the entire plane without ever losing its perfect contact. The same is true of the sphere and pseudosphere. Thus, a net which could be fitted with perfect contact to any part of a sphere could be slid over any portion of the surface without ever losing its perfect contact.

The only surfaces which possess this property are precisely the sphere, the plane and the pseudosphere (and those derived therefrom without stretching); and these surfaces are the surfaces of constant positive curvature, of constant zero curvature, and of constant negative curvature, respectively. So far, then, as the shape of nets applicable to the surface is concerned, there is no means of distinguishing one part of the surface from any other part, since the net can be slid over the surface, preserving its dimensions and never losing its perfect contact with the surface. In other words, the surface appears to be isotropic and homogeneous when measurements with rigid Euclidean rods are conducted over it, or when the nets we slide over it are Euclideanly inextensible, This is the property called free mobility.

Such would no longer be the case were we to consider surfaces of variable curvature, e.g., the surface of an ellipsoid or of a trumpet. A net which could be applied with perfect contact to one part of the surface would lose its perfect contact when slid over the surface, unless we deformed it by stretching it or causing it to shrink.

It is because all three types of geometry we have mentioned (Euclidean, Riemannian and Lobatchewskian) hold in exactly the same way for all parts of the space in which we are operating, that the surfaces which portray them in two dimensions are necessarily of the constant-curvature type.

We have yet to show the connection between non-Euclidean geometry of two dimensions defined as the geometry obtained with Euclidean rods on a surface of constant curvature, and non-Euclidean geometry obtained with squirming Euclidean rods, i.e., rigid non-Euclidean ones. For this purpose consider a sphere resting on a plane. We may call the south pole that point of the sphere which stands in contact with the plane; and we shall assume that a point-source of light is located at the north pole. If the sphere is transparent to rays of light all figures traced on the surface of the sphere will cast shadows on the plane. But the same will be true of our little Euclidean rods which we place alongside these figures on the sphere for purposes of measurement.

It is easy to see that a circle, for instance, traced on the spherical surface will, generally speaking, have for its shadow an ellipse on the plane. As this circle is displaced on the sphere towards the north pole, its shadow will grow larger and larger, tending finally, when any point of the circle coincides with the north pole, to become a parabola extending to infinity. But inasmuch as the shadows of the little rods with which we measure lengths on the sphere vary in exactly the same way as do the shadows of the figures measured, we shall obtain on the plane exactly the same Riemannian results that endure on the sphere.

In short, if we assume that on the plane our standard measuring rods are given by the shadows of the standard measuring rods used on the sphere, we shall obtain Riemannian geometry on the plane. It is scarcely necessary to state that with our Euclidean ideas of congruence it would be impossible for us to accept the statement that these successive squirming shadows of a Euclidean length displaced on the surface of the sphere were all congruent to one another; but this is not the point at issue. If we were perfectly flat, two-dimensional beings, living on the plane, and if not only all the flat bodies sliding over the plane behaved as did the shadows in our previous example, but also our measuring rods and our own living bodies behaved likewise, we should have no other alternative than to assume that these shadows represented the displacements of bodies that moved while remaining rigid or without change. We should accordingly regard the geometry of our two-dimensional space as Riemannian.

It is impossible to show more clearly that the geometry we attribute to space is nothing but an expression of the properties of our bodies and measuring rods when displaced, since here we have a plane which is Euclidean or Riemannian in its geometry according to the behaviour of the bodies that glide over its surface. And so we see that two alternative methods of interpreting non-Euclidean geometry are open to us. Either we may regard our rods as Euclidean and applied to a curved surface, or else we may attribute non-Euclideanism to the behaviour of our rods on a flat surface.[20]

Now in this chapter we have discussed only non-Euclidean geometries of two dimensions. Riemann’s geometry and Lobatchewski’s geometry are more particularly these same geometries, extended to the case of three dimensions. We can easily see how Riemann’s geometry of three dimensions would arise.

We have only to suppose that, as contrasted with Euclidean solids, all material bodies expand as did the shadows when displaced from a fixed centre. But the Riemannian observer of course would have no realisation of this expansion, owing to the modified laws of light propagation and owing to his own body’s expanding in company with all other bodies. So he would be of the opinion that all these bodies maintained their same shape and size, that is, remained congruent when displaced. In fact, if he applied the test of practical congruence he would find that these bodies which the Euclidean observer would claim were expanding were, so far as he himself was concerned, always the same both in visual appearance and as regards exploration by the sense of touch. If our Riemannian being were to view bodies which the Euclidean man considered rigid and undeformable, he would assert without hesitation that those bodies were decreasing in size as they approached a certain point. Once again maintenance of shape and size is essentially relative.

All the conclusions which we reached when discussing a Riemannian world of two dimensions can be extended to the case of three dimensions. Thus, a Riemannian universe or spherical space of three dimensions is finite but unbounded. Travelling for a sufficient length of time along a geodesic or straight line, we should return finally to our starting point, and rays of light which follow geodesics would circle round our space.

Just as in the case of two-dimensional geometry, we may also conceive of a three-dimensional non-Euclidean space as of constant or variable curvature. Prior to Einstein’s discoveries mathematicians concerned themselves more especially with the homogeneous types of geometry, because it was assumed (notwithstanding Riemann’s premonitions) that whatever the practical geometry of real space might turn out to be, it would always remain the same throughout, there being no palpable reason for it to vary here from there. But with the advent of the material or metrical field, conditioned by a more or less capricious distribution of matter, the possibility of a variation in the geometry of space from place to place had to be taken into consideration, and it then became impossible to limit our analysis to spaces of constant curvature. It has been proved that the variable non-Euclideanism of the space surrounding matter in Einstein’s theory—say, over the equatorial plane—would be represented by the geometry of a surface of variable curvature, which turns out to be that of a paraboloid of revolution.

There is one additional aspect of Riemann’s geometry which it may be of interest to mention on account of its possible bearing on the shape of the universe in Einstein’s theory. We refer to elliptical space. The type of Riemann’s geometry which we have discussed so far is known as Riemann’s spherical geometry, because in the case of two dimensions it turns out to be the Euclidean geometry of a spherical surface. But there exists another type of Riemannian geometry discovered by Klein. It is called elliptical space, though it has nothing to do with the surface of an ellipsoid. It corresponds to exactly the same geometry as that of Riemann proper, the only difference consisting in the connectivity of the space. That is to say, the paths of continuous passage from a point to a point are different in the two spaces.

In a similar way the geometry of the cylinder is Euclidean, as is that of the plane (since a plane sheet of paper can be rolled round a cylinder); but its connectivity is different since we can go round the cylinder by following a geodesic and yet return to our starting point.

In order to illustrate elliptical space in two dimensions, as we have illustrated spherical space, we must assume that we limit the surface of our sphere to one of its hemispheres. It might appear that in this case the surface would no longer be unbounded since it would stop abruptly at the equator, but the connectivity of elliptical space is such that every point on the equator is identical with its antipodal point on the other side of the equator. It would be as though, in our concrete representation, there were cross-connections of zero length uniting into one sole point, each pair of points situated diametrically opposite one another on the equator.

It is exceedingly difficult to visualise how it can be that a diameter joining two antipodal points of the equator should be vanishing in length, but the difficulty arises from our endeavouring to represent the distance between two points by a line joining these two points. Such a procedure may often be successful, but in the more complex cases of connectivity it breaks down completely. As Eddington remarks, for obscure psychological reasons we find it natural to represent a distance between two points by a line joining them, believing that this line represents a something fundamental in nature. But it is probable that the line we trace does not have the fundamental importance with which we credit it, and that it cannot be regarded as having any intrinsic significance outside of the ease with which it often introduces itself into our graphical description of natural phenomena. Distance would be more akin to a difference of temperature between two points, and it would never enter our minds to represent this difference of temperature by a line joining the two points. Why in the case of distance such a graphical representation appears obligatory is more or less of a mystery. At any rate the graphical representation is not always possible, and our description of elliptical space gives us an illustration to this effect.

Incidentally we see that the problem of space, even when restricted to the particular cases studied by Riemann, is not exhausted by the sole concept of congruence. Connectivity or routes of continuous passage between points must also be taken into consideration. The study of connectivity opens up a new branch of geometry also created by Riemann and known as Analysis Situs.[21]