## On the Hypotheses Which Lie at the Bases of Geometry in Nature 8, 1873, pp. 14-17 and 36-37

London: Harrison & Sons, 1873. 1st Edition. FIRST EDITION IN ENGLISH OF RIEMANN’S CLASSIC LECTURE ON CURVED SPACE, the foundational work of Riemannian geometry. Riemann’s lecture, presented on the 10th of June, 1854 and his first as an unsalaried lecturer at Göttingen is a seminal piece and one of the most important in mathematics. The first publication of the lecture, “Über die Hypothesen welche der Geometrie zu Grunde liegen” did not appear in print “until 1868, two years after the author’s death, in part since Riemann made no particular effort to publish it” (Knoebel, Mathematical Masterpieces, 217).

The ideas Riemann presents here “entirely reformulated geometry as being about spaces (sets of points which he called manifolds) and argued that the geometric properties of a space were its intrinsic ones. He noted that there are three constant-curvature spaces in two dimensions and showed how the idea of constant curvature can be extended to higher dimensions” (Princeton Companion to Mathematics, 775). “Riemann's idea was to introduce a collection of numbers at every point in space (i.e., a tensor) which would describe how much it was bent or curved. Riemann found that in four dimensions, one needs a collection of twenty numbers at each point to describe the properties of a manifold, no matter how distorted it is. This is the famous construction central to his geometry, known now as a Riemannian metric” (Wikipedia).

Riemann began by “considering geometry as an empirical science, boldly regarding the parallel postulate not as an axiom, but as a hypothesis, whose truth depends on the nature of space. “These data [the datea of Euclidean geometry] are – like all data – not necessary, but only of empirical certainty; they are hypotheses…” In the Euclidean plane the shortest path between two points is, of course, a straight line, while on a curved surface, Gauss (Riemann’s teacher) and others had shown that a geodesic (a curve of shortest distance between two given points) depends on the curvature of the surface. With the introduction of a paradigm-breaking concept, Riemann then generalizes the ideas of space and distance from two-dimensional surfaces to objects of arbitrary dimension” (Knoebel, 217).

His lecture posed “deep questions about the relationship of geometry to the world we live in. He asked what the dimension of real space was and what geometry described real space. The lecture was to far ahead of its time… [and] only Gauss was able to appreciate the depth of Riemann’s thoughts” (University of St. Andrews Biographies). In fact, Riemann’s work, though respected was “not understood until sixty years later.  [About Riemann’s lecture, the mathematician Hans Freudenthal wroe]: The general theory of relativity splendidly justified [Riemann’s] work. In the mathematical apparatus developed from Riemann's address, Einstein found the frame to fit his physical ideas, his cosmology, and cosmogony: and the spirit of Riemann's address was just what physics needed: the metric structure determined by data” (ibid). Item #556

CONDITION & DETAILS: London: Harrison & Sons. Complete volume. 4to. 10.5 by 7.5 inches (263 x 188mm). [xii], 562, [4]. Ex-libris with two stamps on the title page and no exterior markings whatsoever. In-text illustrations throughout. Handsomely rebound in half calf , gilt-lettered at the spine. Five gilt-ruled raised bands at the spine; each compartment gilt tooled. Small, professionally rendered and all but invisible Japanese paper repair on p. 14. Marbled text block. Bright and clean throughout. Very good condition in every way.

Price: \$575.00