Interpretation of Non-Euclidean GeometriesFebruary 12, 2022
vanishes the objection so far as two-dimensional geometries are concerned. It would be easy to extend Beltrami’s reasoning to three-dimensional geometries, and minds which do not recoil before space of 4 dimensions will see no difficulty in it; but such minds are few in number.
I prefer, then, to proceed otherwise. Let us consider a certain plane, which I shall call the fundamental plane, and let us construct a kind of dictionary by making a double series of terms written in two columns, and corresponding each to each, just as in ordinary dictionaries the words in 2 languages which have the same signification correspond to one another:
Space ………….. The portion of space situated above the fundamental plane. Plane ………….. Sphere cutting orthogonally the fundamental plane. Line …………… Circle cutting orthogonally the fundamental plane. Sphere …………. Sphere. Circle ………….. Circle. Angle ………….. Angle.science and hypothesis Distance between two points . . . . . . . . . .
Logarithm of the anharmonic ratio of these two points and of the intersection of the fundamental plane with the circle passing through these two points and cut- ting it orthogonally. Etc. . . . . . . . . . . . . . . . . Etc.
Let us now take Lobatschewsky’s theorems and translate them by the aid of this dictionary, as we would translate a German text with the aid of a German-French dictionary.
We shall then obtain the theorems of ordinary geometry. For instance, Lobatschewsky’s theorem: “The sum of the angles of a triangle is less than two right angles,” may be translated thus: “If a curvilinear triangle has for its sides arcs of circles which if produced would cut orthogonally the fundamental plane, the sum of the angles of this curvilinear triangle will be less than two right angles.”
Thus, however far the consequences of Lobatschewsky’s hypotheses are carried, they will never lead to a contradiction; in fact, if two of Lobatschewsky’s theorems were contradictory, the translations of these two theorems made by the aid of our dictionary would be contradictory also.
But these translations are theoremsnon-euclidean geometries.
of ordinary geometry, and no one doubts that ordinary geometry is exempt from contradiction. Whence is the certainty derived, and how far is it justified? That is a question upon which I cannot enter here, but it is a very interesting question, and I think not insoluble. Nothing, therefore, is left of the objection I formulated above. But this is not all. Lobatschewsky’s geometry being suscep- tible of a concrete interpretation, ceases to be a useless logical exercise, and may be applied. I have no time here to deal with these applications, nor with what Herr Klein and myself have done by using them in the inte- gration of linear equations. Further, this interpretation is not unique, and several dictionaries may be constructed analogous to that above, which will enable us by a simple translation to convert Lobatschewsky’s theorems into the theorems of ordinary geometry.
Are the axioms implicitly enunciated in our text-books the only foundation of geometry?
We may be assured of the contrary when we see that, when they are abandoned one after another, there are still left standing some propositions which are common to the geometries of Euclid, Lobatschewsky, and Riemann. These propositions must be based on premisses that ge- ometers admit without enunciation. It is interesting toscience and hypothesis
try and extract them from the classical proofs.
John Stuart Mill asserted 1 that every definition contains an axiom, because by defining we implicitly affirm the existence of the object defined. That is going rather too far.
It is but rarely in mathematics that a definition is given without following it up by the proof of the existence of the object defined, and when this is not done it is generally because the reader can easily supply it; and it must not be forgotten that the word “existence” has not the same meaning when it refers to a mathematical entity as when it refers to a material object.
A mathematical entity exists provided there is no contradiction implied in its definition, either in itself, or with the propositions previously admitted. But if the observa- tion of John Stuart Mill cannot be applied to all definitions, it is none the less true for some of them. A plane is sometimes defined in the following manner:—The plane is a surface such that the line which joins any two points upon it lies wholly on that surface.
There is obviously a new axiom concealed in this definition. It is true we might change it, and that would be preferable, but then we should have to enunciate the axiom explicitly. Other definitions may give rise to no less important re- 1 Logic, c. viii., cf. Definitions, §5–6.—[Tr.]non-euclidean geometries. 53 flections, such as, for example, that of the equality of two figures. Two figures are equal when they can be super- posed. To superpose them, one of them must be displaced until it coincides with the other. But how must it be dis- placed? If we asked that question, no doubt we should be told that it ought to be done without deforming it, and as an invariable solid is displaced. The vicious circle would then be evident. As a matter of fact, this defini- tion defines nothing. It has no meaning to a being living in a world in which there are only fluids. If it seems clear to us, it is because we are accustomed to the properties of natural solids which do not much differ from those of the ideal solids, all of whose dimensions are invariable. However, imperfect as it may be, this definition implies an axiom.
The possibility of the motion of an invariable figure is not a self-evident truth. At least it is only so in the application to Euclid’s postulate, and not as an an- alytical à priori intuition would be. Moreover, when we study the definitions and the proofs of geometry, we see that we are compelled to admit without proof not only the possibility of this motion, but also some of its prop- erties.
This first arises in the definition of the straight line. Many defective definitions have been given, but the true one is that which is understood in all the proofs inscience and hypothesis
which the straight line intervenes. “It may happen that the motion of an invariable figure may be such that all the points of a line belonging to the figure are motionless, while all the points situate outside that line are in motion.
Such a line would be called a straight line.” We have deliberately in this enunciation separated the definition from the axiom which it implies. Many proofs such as those of the cases of the equality of triangles, of the possibility of drawing a perpendicular from a point to a straight line, assume propositions the enunciations of which are dispensed with, for they necessarily imply that it is possible to move a figure in space in a certain way.