The mystery of space

by Robert T. Browne

Schweikart's treatise, dated Marburg, December, 1818, is here quoted in full:

"There is a two-fold geometry—a geometry in the narrower sense, the Euclidean, and an astral science of magnitude.

"The triangles of the latter have the peculiarity that the sum of the three angles is not equal to two right angles.

"This presumed, it can be most rigorously proven: (a) That the sum of the three angles in the triangle is less than two right angles.

"(b) That this sum becomes ever smaller, the more content the angle incloses. (c) That the altitude of an isosceles right-angled triangle indeed ever increases, the more one lengthens the side; that it, however, cannot surpass a certain line which I call the constant."

Squares have consequently the following form:

Fig. 2.

"If this constant were for us the radius of the earth (so that every line drawn in the universe, from one fixed star to another, distant 90° from the first, would be a tangent to the surface of the earth) it would be infinitely great in comparison with the spaces which occur in daily life."

The above, being the first published, not printed, treatise on the new geometry occupies a unique place in the history of higher mathematics. It gave additional strength to the formative tendencies which characterized this period and marked Schweikart as a constructive and original thinker.

The nascent aspects of this stage received a fruitful contribution when Nicolai Lobachevski (1793-1847) created his Imaginary Geometry and Janos Bolyai (1802-1860) published as an appendix to his father's Tentamen, his Science Absolute of Space. Lobachevski and Bolyai have been called the "Creators of the Non-Euclidean Geometry." And this appellation seems richly to be deserved by these pioneers. Their work gave just the impetus most needed to fix the status of the new line of researches which led to such remarkable discoveries in the more recent years. The Imaginary Geometry and the Science Absolute of Space were translated by the French mathematician, J. Hoüel in 1868 and by him elevated out of their forty-five years of obscurity and non-effectiveness to a position where they became available for the mathematical public. To Bolyai and Lobachevski, consequently, belong the honor of starting the movement which resulted in the development of metageometry and hence that which has proved to be the gateway of a new mathematical freedom.

Gauss, Schweikart, Lobachevski, Wolfgang and Janos Bolyai were the principal figures of the formative period and the value of their work with respect to the formulation of principles upon which was constructed the Temple of Metageometry cannot be overestimated.

The Determinative Period

This period is characterized chiefly by its close relationship to the theory of surfaces. Riemann's Habilitation Lecture on The Hypotheses Which Constitute the Bases of Geometry marks the beginning of this epoch. In this dissertation, Riemann not only promulgated the system upon which Gauss had spent more than thirty years of his life in elaborating, for he was a disciple of Gauss; but he disclosed his own views with respect to space which he regarded as a particular case of manifold. His work contains two fundamental concepts, namely, the manifold and the measure of curvature of a continuous manifold, possessed of what he called flatness in the smallest parts. The conception of the measure of curvature is extended by Riemann from surfaces to spaces and a new kind of space, finite, but unbounded, is shown to be possible. He showed that the dimensions of any space are determined by the number of measurements necessary to establish the position of a point in that space. Conceiving, therefore, that space is a manifold of finite, but unbounded, extension, he established the fact that the passage from one element of a manifold to another may be either discrete or continuous and that the manifold is discrete or continuous according to the manner of passage. Where the manifold is regarded as discrete two portions of it can be compared, as to magnitude, by counting; where continuous, by measurement. If the whole manifold be caused to pass over into another manifold each of its elements passing through a one-dimensional manifold, a two-dimensional manifold is thus generated. In this way, a manifold of n-dimensions can be generated. On the other hand, a manifold of n-dimensions can be analyzed into one of one dimension and one of (n-1) dimensions.

To Riemann, then, is due the credit for first promulgating the idea that space being a special case of manifold is generable, and therefore, finite. He laid the foundation for the establishment of a special kind of geometry known as the "elliptic." Space, as viewed by him, possessed the following properties, viz.: generability, divisibility, measurability, ponderability, finity and flexity.

These are the six pillars upon which rests the structure of hyperspace analyses.[5]