Chapter 8
Appendix²⁹

The investigations by Riemann and Helmholtz of the foundations of geometry led Lie to take up the problem of the axiomatic treatment of geometry as introductory to the study of groups. This profound mathematician introduced a system of axioms which he showed by means of his theory of transformation groups to be sufficient for the complete development of geometry.³⁰

As the basis of his transformation groups, Lie made the assumption that the functions defining the group can be differentiated. Hence in Lie’s development, the question remains uninvestigated as to whether this assumption as to the differentiability of the functions in question is really unavoidable in developing the subject according to the axioms of geometry, or whether, on the other hand, it is not a consequence of the group-conception and of the remaining axioms of geometry. In consequence of his method of development, Lie has also necessitated the express statement of the axiom that the group of displacements is produced by infinitesimal transformations. These requirements, as well as essential parts of Lie’s fundamental axioms concerning the nature of the equation defining points of equal distance, can be expressed geometrically in only a very unnatural and complicated manner. Moreover, they appear only through the analytical method used by Lie and not as a necessity of the problem itself.

In what follows, I have therefore attempted to set up for plane geometry a system of axioms, depending likewise upon the conception of a group,³¹ which contains only those requirements which are simple and easily seen geometrically. In particular they do not require the differentiability of the functions defining displacement. The axioms of the system which I set up are a special division of Lie’s, or, as I believe, are at once deducible from his.

My method of proof is entirely different from Lie’s method. I make use particularly of Cantor’s theory of assemblages of points and of the theorem of C. Jordan, according to which every closed continuous plane curve free from double points divides the plane into an inner and an outer region.

To be sure, in the system set up by me, particular parts are unnecessary. However, I have turned aside from the further investigation of these conditions to the simple statement of the axioms, and above all because I wish to avoid a comparatively too complicated proof, and one which is not at once geometrically evident.

In what follows I shall consider only the axioms relating to the plane, although I suppose that an analogous system of axioms for space can be set up which will make possible the construction of the geometry of space in a similar manner.

We establish the following convention, namely: We will understand by number-plane the ordinary plane having a rectangular system of co-ordinates $x,y$.

A continuous curve lying in this number-plane and being free from double points and including its end points is called a Jordan curve. If the Jordan curve is closed, the interior of the region of the number-plane bounded by it is called a Jordan region.

For the sake of easier representation and comprehension, I shall in the following investigation formulate the definition of the plane in a more restricted sense than my method of proof requires,³² namely: I shall assume that it is possible to map³³ in a reversible, single-valued manner all of the points of our geometry at the same time upon the points lying in the finite region of the number-plane, or upon a definite partial system of the same. Hence, each point of our geometry is characterized by a definite pair of numbers $x,y$. We formulate this statement of the idea of the plane as follows:

Definitions of the Plane. The plane is a system of points which can be mapped in a reversible, single-valued manner upon the points lying in the finite region of the number-plane, or upon a certain partial system of the same. To each point $A$ of our geometry, there exists a Jordan curve in whose interior the map of $A$ lies and all of whose points likewise represent points of our geometry. This Jordan region is called the domain of the point $A$. Each Jordan region contained in a Jordan region which includes the point $A$ is likewise called a domain of $A$. If $B$ is any point in a domain of $A$, then this domain is at the same time called also a domain of $B$.

If $A$ and $B$ are any two points of our geometry, then there always exists a domain which contains at the same time both of the points $A$ and $B$.

We will define a displacement as a reversible, single-valued transformation of a plane into itself. Evidently we may distinguish two kinds of reversible, single-valued, continuous transformations of the number-plane into itself. If we take any closed Jordan curve in the number-plane and think of its being traversed in a definite sense, then by such a transformation this curve goes over into another closed Jordan curve which is also traversed in a certain sense. We shall assume in the present investigation that it is traversed in the same sense as the original Jordan curve, when we apply a transformation of the number-plane into itself, which defines a displacement. This assumption³⁴ necessitates the following statement of the conception of a displacement.

Definition of Displacement. A displacement is a reversible, single-valued, continuous transformation of the maps of the given points upon the number-plane into themselves in such a manner that a closed Jordan curve is traversed in the same sense after the transformation as before. A displacement by which the point $M$ remains unchanged is called a rotation³⁵ about the point $M$.

In accordance with the conventions setting forth the notions “plane” and “displacement,” we set up the three following axioms:

Axiom I. If two displacements are followed out one after the other, then the resulting map of the plane upon itself is again a displacement.

We say briefly:

Axiom I. The Displacements Form a Group.

Axiom II. If $A$ and $M$ are two arbitrary points distinct from each other, then by a rotation about $M$ we can always bring $A$ into an infinite number of different positions.

If in our geometry we define a true circle as the totality of those points which arise by rotating about $M$ a point different from $M$, then we can express the statement made in axiom II as follows:

Axiom II. Every True Circle Consists of an Infinite Number of Points.

As preliminary to axiom III, we make the following explanations:

Let $A$ be a definite point in our geometry and $A_1$, $A_2$, $A_3$, …any infinite system of points. With the same letters we will also denote the maps of these points upon the number-plane. About the point $A$ in the number-plane take an arbitrarily small domain $\alpha$. If then any of the map-points $A_i$ fall within the domain $\alpha$, we say that there are points $A_i$ arbitrarily near the point $A$.

Let $A$, $B$ be a definite pair of points in our geometry, and let $A_1{B}_1$, $A_2{B}_2$, $A_3{B}_3$, $\dots$ be any infinite system of pairs of points. With the same letters we will denote the maps of these pairs of points upon the number-plane. Select about each of the points $A$ and $B$ in the number-plane an arbitrarily small domain $\alpha$ and $\beta$, respectively. If then there are pairs of points $A_i{B}_i$ such that $A_i$ falls within the domain $\alpha$ and at the same time $B_i$ falls within the domain $\beta$, we say that there are segments $A_i{B}_i$ lying arbitrarily near the segment $AB$.

Let $ABC$ a definite triad of points in our geometry, and let $A_1{B}_1{C}_1$, $A_2{B}_2{C}_2$, $A_3{B}_3{C}_3$, $\dots$ be any infinite system of triads of points. With the same letters we will also denote the maps of these triads of points upon the number-plane. About each of the points $A$, $B$, $C$ in the number-plane take an arbitrarily small domain $\alpha$, $\beta$, $\gamma$, respectively. If then there are triads of points $A_i{B}_i{C}_i$ such that $A_i$ falls in the domain $\alpha$, and likewise $B_i$ in the domain $\beta$ and $C_i$ in the domain $\gamma$, then we say that there are triangles $A_i{B}_i{C}_i$ lying arbitrarily near to the triangle $ABC$.

Axiom III. If there are displacements of such a kind that triangles arbitrarily near the triangle $ABC$ can be brought arbitrarily near to the triangle $A\prime B\prime C\prime$, then there always exists a displacement by which the triangle $ABC$ goes over exactly into the triangle $A\prime B\prime C\prime$.³⁶

The content of this axiom can be briefly expressed as follows:

Axiom III. The Displacements Form a Closed System.

We call special attention to the following particular cases of axiom III.

If there are rotations about a point $M$ of the kind that segments lying arbitrarily near the segment $AB$ can be brought arbitrarily near the segment $A\prime B\prime$, then there is always such a rotation about $M$ possible by which the segment $AB$ goes over exactly into the segment $A\prime B\prime$.

If there are displacements of the kind that segments arbitrarily near the segment $AB$ can be brought arbitrarily near to the segment $A\prime B\prime$, then there is always a displacement possible by which the segment $AB$ goes over exactly into the segment $A\prime B\prime$.

If there are rotations about the point $M$ of the kind that points arbitrarily near the point A can be brought arbitrarily near the point $A\prime$, then there is always such a rotation about $M$ possible by which $A$ goes over exactly into the point $A\prime$.

I now prove the following proposition:

If we wish to obtain only the euclidean geometry, it is necessary merely to make in connection with axiom I the additional statement that the groups of displacements shall possess an invariant sub-group. This additional statement takes the place of the axioms of parallels.

In what follows, I will briefly outline the general idea of my method of proof.³⁷ Within the domain of a certain point $M$ construct in a particular manner a certain point-configuration $kk$, and upon this configuration construct a certain point $K$. We then base our investigation upon the true circle $k$ about $M$ and passing through $K$. It may be easily shown that the true circle $k$ is an assemblage of points which is closed and in itself dense. It constitutes, therefore, a perfect assemblage of points.

The next objective point in our demonstration is to show that the true circle $k$ is a closed Jordan curve. We do this in that we first show the possibility of a cyclical arrangement of the points of the true circle $k$, from which it follows that we may map in a reversible, single-valued manner the points of $k$ upon the points of an ordinary circle. Finally, we show that this map must necessarily be a continuous one. Furthermore, it follows also that the originally constructed point-configuration $kk$ is identical with the true circle $k$. Moreover, the law holds that each true circle inside of $k$ is likewise a closed Jordan curve.

We turn now to the investigation of the group of all the displacements which by the rotation of the plane about $M$ transforms a definite true circle $k$ into itself. This group possesses the following properties: (1) Every displacement which leaves one point of $k$ undisturbed, leaves all points of $k$ undisturbed. (2) There always exists a displacement which changes any given point of $k$ into any other given point of $k$. (3) The group of displacements is a continuous one. These three properties determine completely the construction of the group of transformations of all the displacements of the true circle into itself. We set up the following proposition: The group of all the displacements of the true circle into itself, which are rotations about M, is holoedric, isomorphic with the group of ordinary rotations of the ordinary circle into itself.

Moreover, we investigate the group of displacements of all the points of our plane by a rotation about $M$. The law holds that, aside from the identity, there is no rotation of the plane about $M$ which leaves every point of the true circle undisturbed. We now see that every true circle is a Jordan curve and deduce formulæ for the transformation of that group of all the rotations. Finally, the proposition easily follows that: If any two points remain fixed by a displacement of the plane, then all points remain fixed; that is to say, the displacement is the identity. Each point of the plane may be indeed made to go over into any other point of the plane by means of a displacement.

Our further important objective point is to define the idea of the true straight line in our geometry and deduce those properties of it which are necessary in the further development of geometry. First of all, the notions “semi-rotation” and “middle of a segment” are defined. A segment has at most one middle, and, when we know the middle of one segment, then every smaller segment possesses a middle.

In order to pass judgment as to the position of the middle of a segment, we need particular propositions concerning true circles which are mutually tangent, and indeed the question depends upon the construction of two congruent circles tangent to each other externally in one and only one point. We derive also a more general proposition concerning circles which are tangent to each other internally and consequently a theorem covering the special case where the circle which is tangent internally to a second passes through the centre of that circle.

Moreover, a sufficiently small definite segment is taken as a unit segment, and from this by continued bisection and semi-rotation a system of points is constructed of the kind that to each point of this system a definite number $a$ corresponds, which is rational and has as denominator some power of $2$. By setting up a law concerning this correspondence, the points of the above system are so arranged that the above laws concerning mutually tangent circles are valid. It is now shown that the points corresponding to the numbers $\frac{1}{2},\frac{1}{4},\frac{1}{8},\dots$ converge toward the point 0. This result is generalized step by step until it is finally shown that every series of points of our system converges, so soon as the corresponding series of numbers converges.

From what has been said, the definition of the true straight line follows as a system of points which arise from two fundamental points, if we apply repeatedly a semi-rotation, take the middle point, and add to the assemblage the points of condensation of the system of points which arises. We can then prove that the true straight line is a continuous curve, possessing no double points and having with any other true straight line at most one point in common. Furthermore, it can be shown that the true straight line cuts each circle drawn about one of its points, and from this it follows that any two arbitrary points of the plane can always be joined by a true straight line. We see also that in our geometry the laws of congruence hold, by which however two triangles are proven to be congruent if they are traversed in the same sense.

With regard to the position of the systems of all the true straight lines with respect to one another, there are two cases to distinguish, according as the axiom of parallels holds, or through each point there exists two straight lines which separate the straight lines which cut the given straight line from those which do not cut it. In the first case we have the euclidean and in the second the bolyai-lobatschefskian geometry.