§§ 3–5: Projective Geometry, Transfer by Representation, and Choice of Space-Element

A Comparative Review of Recent Researches in Geometry — §§ 3–5 (Trans. M. W. Haskell, 1893, pp. 221–226)

§ 3. Projective Geometry

Every space-transformation not belonging to the principal group can be used to transfer the properties of known configurations to new ones. Thus we apply the results of plane geometry to the geometry of surfaces that can be represented (abgebildet) upon a plane; in this way long before the origin of a true projective geometry the properties of figures derived by projection from a given figure were inferred from those of the given figure. But projective geometry only arose as it became customary to regard the original figure as essentially identical with all those deducible from it by projection, and to enunciate the properties transferred in the process of projection in such a way as to put in evidence their independence of the change due to the projection. By this process the group of all the projective transformations was made the basis of the theory in the sense of § 1, and that is just what created the antithesis between projective and ordinary geometry.

A course of development similar to the one here described can be regarded as possible in the case of every kind of space-transformation; we shall often refer to it again. It has gone on still further in two directions within the domain of projective geometry itself. On the one hand, the conception was broadened by admitting the dualistic transformations into the group of the fundamental transformations. From the modern point of view two reciprocal figures are not to be regarded as two distinct figures, but as essentially one and the same. A further advance consisted in extending the fundamental group of collinear and dualistic transformations by the admission in each case of the imaginary transformations. This step requires that the field of true space-elements has previously been extended so as to include imaginary elements, — just exactly as the admission of dualistic transformations into the fundamental group requires the simultaneous introduction of point and plane as space-elements. This is not the place to point out the utility of introducing imaginary elements, by means of which alone we can attain an exact correspondence of the theory of space with the established system of algebraic operations. But, on the other hand, it must be remembered that the reason for introducing the imaginary elements is to be found in the consideration of algebraic operations and not in the group of projective and dualistic transformations. For, just as we can in the latter case limit ourselves to real transformations, since the real collineations and dualistic transformations form a group by themselves, so we can equally well introduce imaginary space-elements even when we are not employing the projective point of view, and indeed must do so in strictly algebraic investigations.

How metric properties are to be regarded from the projective point of view is determined by the general theorem of the preceding section. Metrical properties are to be considered as projective relations to a fundamental configuration, the circle at infinity [1], a configuration having the property that it is transformed into itself only by those transformations of the projective group which belong at the same time to the principal group. The proposition thus broadly stated needs a material modification owing to the limitation of the ordinary view taken of geometry as treating only of real space-elements (and allowing only real transformations). In order to conform to this point of view, it is necessary expressly to adjoin to the circle at infinity the system of real space-elements (points); properties in the sense of elementary geometry are projectively either properties of the configurations by themselves, or relations to this system of the real elements, or to the circle at infinity, or finally to both.

We might here make mention further of the way in which von Staudt (Geometrie der Lage, Nürnberg, 1847) develops projective geometry, — i.e., that projective geometry which is based on the group containing all the real projective and dualistic transformations [2].

We know how, in his system, he selects from the ordinary matter of geometry only such features as are preserved in projective transformations. Should we desire to proceed to the consideration of metrical properties also, what we should have to do would be precisely to introduce these latter as relations to the circle at infinity. The course of thought thus brought to completion is in so far of great importance for the present considerations, as a corresponding development of geometry is possible for every one of the methods we shall take up.

§ 4. Transfer of Properties by Representation (Abbildung)

Before going further in the discussion of the geometrical methods which present themselves beside the elementary and the projective geometry, let us develop in a general form certain considerations which will continually recur in the course of the work, and for which a sufficient number of examples are already furnished by the subjects touched upon up to this point. The present section and the following one will be devoted to these discussions.

Suppose a manifoldness $A$ has been investigated with reference to a group $B$. If, by any transformation whatever, $A$ be then converted into a second manifoldness $A'$, the group $B$ of transformations, which transformed $A$ into itself, will become a group $B'$, whose transformations are performed upon $A'$. It is then a self-evident principle that the method of treating $A$ with reference to $B$ at once furnishes the method of treating $A'$ with reference to $B'$, i.e., every property of a configuration contained in $A$ obtained by means of the group $B$ furnishes a property of the corresponding configuration in $A'$ to be obtained by the group $B'$.

For example, let $A$ be a straight line and $B$ the $\infty^3$ linear transformations which transform $A$ into itself. The method of treating $A$ is then just what modern algebra designates as the theory of binary forms. Now, we can establish a correspondence between the straight line and a conic section $A'$ in the same plane by projection from a point of the latter. The linear transformations $B$ of the straight line into itself will then become, as can easily be shown, linear transformations $B'$ of the conic into itself, i.e., the changes of the conic resulting from those linear transformations of the plane which transform the conic into itself.

Now, by the principle stated in § 2 [3], the study of the geometry of the conic section is the same, whether the conic be regarded as fixed and only those linear transformations of the plane which transform the conic into itself be taken into account, or whether all the linear transformations of the plane be considered and the conic be allowed to vary too. The properties which we recognized in systems of points on the conic are accordingly projective properties in the ordinary sense. Combining this consideration with the result just deduced, we have, then:

The theory of binary forms and the projective geometry of systems of points on a conic are one and the same, i.e., to every proposition concerning binary forms corresponds a proposition concerning such systems of points, and vice versa. [4]

Another suitable example to illustrate these considerations is the following. If a quadric surface be brought into correspondence with a plane by stereographic projection, the surface will have one fundamental point, — the centre of projection. In the plane there are two, — the projections of the generators passing through the centre of projection. It then follows directly: the linear transformations of the plane which leave the two fundamental points unaltered are converted by the representation (Abbildung) into linear transformations of the quadric itself, but only into those which leave the centre of projection unaltered. By linear transformations of the surface into itself are here meant the changes undergone by the surface when linear space-transformations are performed which transform the surface into itself. According to this, the projective investigation of a plane with reference to two of its points is identical with the projective investigation of a quadric surface with reference to one of its points. Now, if imaginary elements are also taken into account, the former is nothing else but the investigation of the plane from the point of view of elementary geometry. For the principal group of plane transformations comprises precisely those linear transformations which leave two points (the circular points at infinity) unchanged. We obtain then finally:

Elementary plane geometry and the projective investigation of a quadric surface with reference to one of its points are one and the same.

These examples may be multiplied at pleasure [5]; the two here developed were chosen because we shall have occasion to refer to them again.

§ 5. On the Arbitrariness in the Choice of the Space-Element. Hesse’s Principle of Transference. Line Geometry

As element of the straight line, of the plane, of space, or of any manifoldness to be investigated, we may use instead of the point any configuration contained in the manifoldness, — a group of points, a curve or surface [6], etc. As there is nothing at all determined at the outset about the number of arbitrary parameters upon which these configurations shall depend, the number of dimensions of our line, plane, space, etc., may be anything we like, according to our choice of the element. But as long as we base our geometrical investigation upon the same group of transformations, the substance of the geometry remains unchanged. That is to say, every proposition resulting from one choice of the space-element will be a true proposition under any other assumption; but the arrangement and correlation of the propositions will be changed.

The essential thing is, then, the transformation group; the number of dimensions to be assigned to a manifoldness appears of secondary importance.

The combination of this remark with the principle of the last section furnishes many interesting applications, some of which we will now develop, as these examples seem better fitted to explain the meaning of the general theory than any lengthy exposition.

Projective geometry on the straight line (the theory of binary forms) is, by the last section, equivalent to projective geometry on the conic. Let us now regard as element on the conic the point-pair instead of the point. Now, the totality of the point-pairs of the conic may be brought into correspondence with the totality of the straight lines in the plane, by letting every line correspond to that point-pair in which it intersects the conic. By this representation (Abbildung) the linear transformations of the conic into itself are converted into those linear transformations of the plane (regarded as made up of straight lines) which leave the conic unaltered. But whether we consider the group of the latter, or whether we base our investigation on the totality of the linear transformations of the plane, always adjoining the conic to the plane configurations under investigation, is by § 2 one and the same thing. Uniting all these considerations, we have:

The theory of binary forms and projective geometry of the plane with reference to a conic are identical.

Finally, as projective geometry of the plane with reference to a conic, by reason of the equality of its group, coincides with that projective metrical geometry which in the plane can be based upon a conic [7], we can also say:

The theory of binary forms and general projective metrical geometry in the plane are one and the same.

In the preceding consideration the conic in the plane might be replaced by the twisted cubic, etc., but we will not carry this out further. The correlation here explained between the geometry of the plane, of space, or of a manifoldness of any number of dimensions is essentially identical with the principle of transference proposed by Hesse (Borchardt’s Journal, vol. 66).

An example of much the same kind is furnished by the projective geometry of space; or, in other words, the theory of quaternary forms. If the straight line be taken as space-element and be determined, as in line geometry (Plücker), by six homogeneous co-ordinates connected by a quadratic equation of condition, the linear and dualistic transformations of space are seen to be those linear transformations of the six variables (regarded as independent) which transform the equation of condition into itself. By a combination of considerations similar to those just developed, we obtain the following theorem:

The theory of quaternary forms is equivalent to projective measurement in a manifoldness generated by six homogeneous variables.

For a detailed exposition of this view I will refer to Über die sogenannte Nicht-Euklidische Geometrie (Math. Ann. 6, 1873), and to a note at the close of this paper [8].

To the foregoing expositions I will append two remarks, the first of which is to be sure implicitly contained in what has already been said, but needs to be brought out at length, because the subject to which it applies is only too likely to be misunderstood.

Through the introduction of arbitrary configurations as space-elements, space becomes of any number of dimensions we like. But if we then keep to the (elementary or projective) space-perception with which we are familiar, the fundamental group for the manifoldness of $n$ dimensions is given at the outset; in the one case it is the principal group, in the other the group of projective transformations. If we wished to take a different group as a basis, we should have to depart from the ordinary (or from the projective) space-perception. Thus, while it is correct to say that, with a proper choice of space-elements, space represents manifoldnesses of any number of dimensions, it is equally important to add that in this representation either a definite group must form the basis of the investigation of the manifoldness, or else, if we wish to choose the group, we must broaden our geometrical perception accordingly. If this were overlooked, an interpretation of line geometry, for instance, might be sought in the following way. In line geometry the straight line has six co-ordinates; the conic in the plane has the same number of coefficients. The interpretation of line geometry would then be the geometry in a system of conics separated from the aggregation of all conics by a quadratic equation between the coefficients. This is correct, provided we take as fundamental group for the plane geometry the totality of the transformations represented by the linear transformations of the coefficients of the conic which transform the quadratic equation into itself. But if we retain the elementary or the projective view of plane geometry, we have no interpretation at all.

The second remark has reference to the following line of reasoning: Suppose in space some group or other, the principal group for instance, be given. Let us then select a single configuration, say a point, or a straight line, or even an ellipsoid, etc., and apply to it all the transformations of the principal group. We thus obtain an infinite manifoldness with a number of dimensions in general equal to the number of arbitrary parameters contained in the group, but reducing in special cases, namely, when the configuration originally selected has the property of being transformed into itself by an infinite number of the transformations of the group. Every manifoldness generated in this way may be called, with reference to the generating group, a body [9].

If now we desire to base our investigations upon the group, selecting at the same time certain definite configurations as space-elements, and if we wish to represent uniformly things which are of like characteristics, we must evidently choose our space-elements in such a way that their manifoldness either is itself a body or can be decomposed into bodies. This remark, whose correctness is evident, will find application later (§ 9). This idea of a body will come under discussion once more in the closing section, in connection with certain related ideas [10].

Footnotes (1893 Edition)