Notes I–VII: Synthetic vs. Analytic Geometry, Space-Perception, Non-Euclidean Metrics & Binary Forms¶
A Comparative Review of Recent Researches in Geometry — Author’s Appendix Notes I–VII (Trans. M. W. Haskell, 1893, pp. 242–249)
I. On the Antithesis Between the Synthetic and the Analytic Method in Modern Geometry¶
The distinction between modern synthesis and modern analytic geometry must no longer be regarded as essential, inasmuch as both subject-matter and methods of reasoning have gradually taken a similar form in both. We choose therefore in the text as common designation of them both the term projective geometry. Although the synthetic method has more to do with space-perception and thereby imparts a rare charm to its first simple developments, the realm of space-perception is nevertheless not closed to the analytic method, and the formulae of analytic geometry can be looked upon as a precise and perspicuous statement of geometrical relations. On the other hand, the advantage to original research of a well formulated analysis should not be underestimated, — an advantage due to its moving, so to speak, in advance of the thought. But it should always be insisted that a mathematical subject is not to be considered exhausted until it has become intuitively evident, and the progress made by the aid of analysis is only a first, though a very important, step.
II. Division of Modern Geometry into Theories¶
When we consider, for instance, how persistently the mathematical physicist disregards the advantages afforded him in many cases by only a moderate cultivation of the projective view, and how, on the other hand, the student of projective geometry leaves untouched the rich mine of mathematical truths brought to light by the theory of the curvature of surfaces, we must regard the present state of mathematical knowledge as exceedingly incomplete and, it is to be hoped, as transitory.
III. On the Value of Space-Perception¶
When in the text we designated space-perception as something incidental, we meant this with regard to the purely mathematical contents of the ideas to be formulated. Space-perception has then only the value of illustration, which is to be estimated very highly from the pedagogical stand-point, it is true. A geometric model, for instance, is from this point of view very instructive and interesting.
But the question of the value of space-perception in itself is quite another matter. I regard it as an independent question. There is a true geometry which is not, like the investigations discussed in the text, intended to be merely an illustrative form of more abstract investigations. Its problem is to grasp the full reality of the figures of space, and to interpret — and this is the mathematical side of the question — the relations holding for them as evident results of the axioms of space-perception. A model, whether constructed and observed or only vividly imagined, is for this geometry not a means to an end, but the subject itself.
This presentation of geometry as an independent subject, apart from and independent of pure mathematics, is nothing new, of course. But it is desirable to lay stress explicitly upon this point of view once more, as modern research passes it over almost entirely. This is connected with the fact that, vice versa, modern research has seldom been employed in investigations on the form-relations of space-configurations, while it appears well adapted to this purpose.
IV. On Manifoldnesses of Any Number of Dimensions¶
That space, regarded as the locus of points, has only three dimensions, does not need to be discussed from the mathematical point of view; but just as little can anybody be prevented from that point of view from claiming that space really has four, or any unlimited number of dimensions, and that we are only able to perceive three. The theory of manifoldnesses, advancing as it does with the course of time more and more into the foreground of modern mathematical research, is by its nature fully independent of any such claim. But a nomenclature has become established in this theory which has indeed been derived from this idea. Instead of the elements of a manifoldness we speak of the points of a higher space, etc. The nomenclature itself has certain advantages, in that it facilitates the interpretation by calling to mind the perceptions of geometry. But it has had the unfortunate result of causing the wide-spread opinion that investigations on manifoldnesses of any number of dimensions are inseparably connected with the above-mentioned idea of the nature of space. Nothing is more unsound than this opinion. The mathematical investigations in question would, it is true, find an immediate application to geometry, if the idea were correct; but their value and purport is absolutely independent of this idea, and depends only on their own mathematical contents.
It is quite another matter when Plücker shows how to regard actual space as a manifoldness of any number of dimensions by introducing as space-element a configuration depending on any number of parameters, a curve, surface, etc. (see § 5 of the text).
The conception in which the element of a manifoldness (of any number of dimensions) is regarded as analogous to the point in space was first developed, I suppose, by Grassmann in his “Ausdehnungslehre” (1844). With him the thought is absolutely free of the above-mentioned idea of the nature of space; this idea goes back to occasional remarks by Gauss, and became more widely known through Riemann's investigations on manifoldnesses (1854/1867), with which it was interwoven.
Both conceptions — Grassmann's as well as Plücker's — have their own peculiar advantages; they can be alternately employed with good results.
V. On the So-Called Non-Euclidean Geometry¶
The projective metrical geometry alluded to in the text (Cayley 1859, Klein 1871) is essentially coincident, as recent investigations have shown, with the metrical geometry which can be developed under non-acceptance of the axiom of parallels (Lobachevsky, Bolyai), and is to-day under the name of non-Euclidean geometry widely treated and discussed. The reason why this name has not been mentioned at all in the text, is closely related to the expositions given in the preceding note. With the name non-Euclidean geometry have been associated a multitude of non-mathematical ideas, which have been as zealously cherished by some as resolutely rejected by others, but with which our purely mathematical considerations have nothing to do whatever. A wish to contribute towards clearer ideas in this matter has occasioned the following explanations.
The investigations referred to on the theory of parallels, with the results growing out of them, have a definite value for mathematics from two points of view.
They show, in the first place, — and this function of theirs may be regarded as concluded once for all, — that the axiom of parallels is not a mathematical consequence of the other axioms usually assumed, but the expression of an essentially new principle of space-perception, which has not been touched upon in the foregoing investigations. Similar investigations could and should be performed with regard to every axiom (and not alone in geometry); an insight would thus be obtained into the mutual relation of the axioms.
But, in the second place, these investigations have given us an important mathematical idea, — the idea of a manifoldness of constant curvature. This idea is very intimately connected, as has already been remarked and in § 10 of the text discussed more in detail, with the projective measurement which has arisen independently of any theory of parallels. Not only is the study of this measurement in itself of great mathematical interest, admitting of numerous applications, but it has the additional feature of including the measurement given in geometry as a special (limiting) case and of teaching us how to regard the latter from a broader point of view.
Quite independent of the views set forth is the question, what reasons support the axiom on parallels, i.e., whether we should regard it as absolutely given, as some claim, or only as approximately proved by experience, as others say. Should there be reasons for assuming the latter position, the mathematical investigations referred to afford us then immediately the means for constructing a more exact geometry. But the inquiry is evidently a philosophical one and concerns the most general foundations of our understanding. The mathematician as such is not concerned with this inquiry, and does not wish his investigations to be regarded as dependent on the answer given to the question from the one or the other point of view [1].
VI. Line Geometry as a Study of a Manifoldness of Constant Curvature¶
In combining line geometry with the projective measurement in a manifoldness of five dimensions, we must remember that the straight lines represent elements of the manifoldness which, metrically speaking, are at infinity. It then becomes necessary to consider what the value of a system of a projective measurement is for the elements at infinity; and this may here be set forth somewhat at length, in order to remove any difficulties which might else seem to stand in the way of conceiving of line geometry as a metrical geometry. We shall illustrate these expositions by the graphic example of the projective measurement based on a quadric surface.
Any two points in space have with respect to the surface an absolute invariant, — the anharmonic ratio of the two points together with the two points of intersection of the line joining them with the surface. But when the two points move up to the surface, this anharmonic ratio becomes zero independently of the position of the points, except in the case where the two points fall upon a generator, when it becomes indeterminate. This is the only special case which can occur in their relative position unless they coincide, and we have therefore the theorem:
The projective measurement in space based upon a quadric surface does not yet furnish a measurement for the geometry on the surface.
This is connected with the fact that by linear transformations of the surface into itself any three points of the surface can be brought into coincidence with three others [2].
If a measurement on the surface itself be desired, we must limit the group of transformations, and this result is obtained by supposing any arbitrary point of space (or its polar plane) to be fixed. Let us first take a point not on the surface. We can then project the surface from the point upon a plane, when a conic will appear as the boundary curve. Upon this conic we can base a projective measurement in the plane, which must then be transferred back to the surface [3]. This is a measurement with constant curvature in the true sense, and we have then the theorem:
Such a measurement on the surface is obtained by keeping fixed a point not on the surface.
Correspondingly, we find [4]:
A measurement with zero curvature on the surface is obtained by choosing as the fixed point a point of the surface itself.
In all these measurements on the surface the generators of the surface are lines of zero length. The expression for the element of arc on the surface differs therefore only by a factor in the different cases. There is no absolute element of arc upon the surface; but we can of course speak of the angle formed by two directions on the surface.
All these theorems and considerations can now be applied immediately to line geometry. Line-space itself admits at the outset no measurement, properly speaking. A measurement is only obtained by regarding a linear complex as fixed; and the measurement is of constant or zero curvature, according as the complex is a general or a special one (a line). The selection of a particular complex carries with it further the acceptation of an absolute element of arc. Independently of this, the directions to adjoining lines cutting the given line are of zero length, and we can besides speak of the angle between any two directions (Math. Ann. 5, 1872, 257–277).
VII. On the Interpretation of Binary Forms¶
We shall now consider the graphic illustration which can be given to the theory of invariants of binary cubics and biquadratics by taking advantage of the representation of $x + iy$ on the sphere.
A binary cubic $f$ has a cubic covariant $Q$, a quadratic covariant $\Delta$, and an invariant $R$ [5]. From $f$ and $Q$ a whole system of covariant sextics
may be compounded, among them being $\Delta^3$. It can be shown [6] that every covariant of the cubic must resolve itself into such groups of six points. Inasmuch as $\lambda$ can assume complex values, the number of these covariants is $\infty^2$ (Beltrami, Memorie dell’ Accademia di Bologna, 1870).
The whole system of forms thus defined can now be represented upon the sphere as follows. By a suitable linear transformation of the sphere into itself let the three points representing $f$ be converted into three equidistant points of a great circle. Let this great circle be denoted as the equator, and let the three points $f$ have the longitudes $0^\circ$, $120^\circ$, $240^\circ$. Then $Q$ will be represented by the points of the equator whose longitudes are $60^\circ$, $180^\circ$, $300^\circ$; $\Delta$ by the two poles. Every form $Q^2 + \lambda R f^2$ is represented by six points, whose latitude and longitude are given in the following table, where $\alpha$ and $\beta$ are arbitrary numbers:
In studying the variation of these systems of points on the sphere, it is interesting to see how they give rise to $f$ and $Q$ (each reckoned twice) and $\Delta$ (reckoned three times).
A biquadratic $f$ has a biquadratic covariant $H$, a sextic covariant $T$, and two invariants $i$ and $j$. Particularly noteworthy is the pencil of biquadratic forms $i H + \lambda j f$, all belonging to the same $T$, among them being the three quadratic factors into which $T$ can be resolved, each reckoned twice.
Let the centre of the sphere now be taken as the origin of a set of rectangular axes $OX, OY, OZ$. Their six points of intersection with the sphere make up the form $T$. The four points of a set $i H + \lambda j f$ are given by the following table, $x, y, z$ being the co-ordinates of any point of the sphere:
The four points are in each case the vertices of a symmetrical tetrahedron, whose opposite edges are bisected by the co-ordinate axes; and this indicates the rôle played by $T$ in the theory of biquadratic equations as the resolvent of $i H + \lambda j f$.
ERLANGEN, October, 1872.
Footnotes (1893 Edition)