Non-euclidean Geometries

PART 2: SPACE

Every conclusion presumes premises which are either:

• obvious, or
• can be established only if based on other propositions

We cannot go back in this way to infinity, every deductive science, and geometry in particular, must rest upon a certain number of indemonstrable axioms.

All treatises of geometry begin therefore with the enunciation of these axioms. But there is a distinction to be drawn between them. Some of these, for example, “Things which are equal to the same thing are equal to one another,” are not propositions in geometry but propositions in analysis.

I look upon them as analytical à priori intuitions, and they concern me no further. But I must insist on other axioms which are special to geometry.

Of these most treatises explicitly enunciate three:—(1) Only one line can pass through two points; (2) a straight line is the shortest distance between two points; (3) through one point only one parallel can be drawn to a given straight line.

Although we generally dispense with proving the second of these axioms, it would be possible to deduce it from the other two, and from those much more numerous axioms which are implicitly admitted without enun- ciation, as I shall explain further on.

For a long time, a proof of the third axiom known as Euclid’s postulate was sought in vain.

It is impossible to imagine the efforts that have been spent in pursuit of this chimera. Finally, at the beginning of the nineteenth century, and almost simultaneously, two scientists, a Russian and a Hungarian, Lobatschewsky and Bolyai, showed irrefutably that this proof is impossible.

They have nearly rid us of inventors of geometries without a postulate, and ever since the Académic des Sciences receives only about one or two new demonstrations a year.

But the question was not exhausted, and it was not long before a great step was taken by the celebrated memoir of Riemann, enti- tled: Ueber die Hypothesen welche der Geometrie zum Grunde liegen. This little work has inspired most of the recent treatises to which I shall later on refer, and among which I may mention those of Beltrami and Helmholtz.

The Geometry of Lobatschewsky

If it were possible to deduce Euclid’s postulate from the several axioms, it is evident that by rejecting the postulate and retaining the other axioms we should be led to contradictory consequences. It would be, therefore, impossible to found on those premisses a coherent geometry.

Now, this is precisely what Lobatschewsky has done. He assumes at the outset that several parallels may be drawn through a point to a given straight line, and he retains all the other axioms of Euclid.

From these hypotheses he deduces a series of theorems between which it is impossible to find any contradiction, and he constructs a geometry as impeccable in its logic as Euclidean geometry. The theorems are very different, however, from those to which we are accustomed, and at first will be found a little disconcerting.

For instance, the sum of the angles of a triangle is always less than two right angles, and the difference between that sum and two right angles is proportional to the area of the triangle. It is impossible to construct a figure similar to a given figure but of different dimensions.

If the circumference of a circle be divided into n equal parts, and tangents be drawn at the points of intersection, the n tangents will form a polygon if the radius of the circle is small enough, but if the radius is large enough they will never meet. We need not multi- ply these examples. Lobatschewsky’s propositions have no relation to those of Euclid, but they are nonetheless logically interconnected.

Riemann’s Geometry

Let us imagine a world only peopled with beings of no thickness, and suppose these “infinitely flat” animals are all in one and the same plane, from which they cannot emerge.

Let us further admit that this world is sufficiently distant from other worlds to be withdrawn from their influence, and while we are making these hypotheses it will not cost us much to endow these beings with reasoning power, and to believe them capable of making a geometry. In that case they will certainly attribute to space only two dimensions.

But now suppose that these imaginary animals, while remaining without thickness, have the form of a spherical, and not of a plane figure, and are all on the same sphere, from which they cannot escape. What kind of a geometry will they construct?

In the first place, it is clear that they will attribute to space only two di- mensions. The straight line to them will be the shortest distance from one point on the sphere to another—that is to say, an arc of a great circle. In a word, their ge- ometry will be spherical geometry. What they will call space will be the sphere on which they are confined, and on which take place all the phenomena with which they are acquainted.

Their space will therefore be unbounded, since on a sphere one may always walk forward without ever being brought to a stop, and yet it will be finite;

the end will never be found, but the complete tour can be made. Well, Riemann’s geometry is spherical geometry extended to three dimensions. To construct it, the German mathematician had first of all to throw overboard, not only Euclid’s postulate but also the first ax- iom that only one line can pass through two points.

On a sphere, through two given points, we can in general draw only one great circle which, as we have just seen, would be to our imaginary beings a straight line. But there was one exception. If the two given points are at the ends of a diameter, an infinite number of great circles can be drawn through them.

In the same way, in Riemann’s geometry—at least in one of its forms—through 2 points only one straight line can in general be drawn, but there are exceptional cases in which through 2 points an infinite number of straight lines can be drawn. So there is a kind of opposition between the geometries of Riemann and Lobatschewsky.

For instance, the sum of the angles of a triangle is equal to two right angles in Euclid’s geometry, less than two right angles in that of Lobatschewsky, and greater than two right angles in that of Riemann.

The number of parallel lines that can be drawn through a given point to a given line is one in Euclid’s geometry, none in Riemann’s, and an infinite number in the geometry of Lobatschewsky. Let us add that Riemann’s space is finite, although unbounded in the sense which we have above attached to these words.

Surfaces with Constant Curvature

One objection, however, remains possible. There is no contradiction between the theorems of Lobatschewsky and Riemann; but however numerous are the other consequences that these geometers have deduced from their hypotheses, they had to arrest their course before they exhausted them all, for the number would be infinite; and who can say that if they had carried their deductions further they would not have eventually reached some contradiction?

This difficulty does not exist for Riemann’s geometry, provided it is limited to two dimensions. As we have seen, the two-dimensional geometry of Riemann, in fact, does not differ from spherical geometry, which is only a branch of ordinary geometry, and is therefore outside all con- tradiction.

Beltrami, by showing that Lobatschewsky’s 2-dimensional geometry was only a branch of ordinary geometry, has equally refuted the objection as far as it is concerned. This is the course of his argument: Let us consider any figure whatever on a surface.

Imagine this figure to be traced on a flexible and inextensible canvas applied to the surface, in such a way that when the canvas is displaced and deformed the different lines of the figure change their form without changing their length.

As a rule, this flexible and inextensible figure cannot be displaced without leaving the surface. But there are certain surfaces for which such a movement would be possible.

They are surfaces of constant curvature. If we resume the comparison that we made just now, and imagine beings without thickness living on one of these surfaces, they will regard as possible the motion of a figure all the lines of which remain of a constant length. Such a movement would appear absurd, on the other hand, to animals without thickness living on a surface of variable curvature.

These surfaces of constant curvature are of two kinds. The curvature of some is positive, and they may be deformed so as to be applied to a sphere.

The geometry of these surfaces is therefore reduced to spherical geometry—namely, Riemann’s. The curvature of others is negative.

Beltrami has shown that the geometry of these surfaces is identical with that of Lobatschewsky. Thus the two-dimensional geometries of Riemann and Lobatschewsky are connected with Euclidean geometry.