London: Macmillan & Sons. FIRST ENGLISH TRANSLATION OF RIEMANN'S EPOCH-MAKING PAPER AND ADDRESS ON THE FOUNDATIONS OF GEOMETRY, Ueber die Hypothesen, commonly known as Riemann’s ‘Habilitation.’ “The importance of this treatise is not confined to pure mathematics. Without it, Einstein would not have been able to develop his general theory of relativity" (PMM 293b). “In the mathematical apparatus developed from Riemann’s address, Einstein found the frame to fit his physical ideas… metric structure determined by physical data” (PMM XI 455).
Riemann was one of the most important and influential mathematicians of the modern era. The paper translated here began as an essay Riemann wrote and then delivered as a lecture at the University of Gottingen; it is a work that “greatly extended the whole idea of what is meant by geometry, isolating the concept of measurement-relations (‘metric’) as fundamental. The concept, as generalized by Riemann, was a development of Gauss’s theory of surfaces [but] was not restricted [like Gauss’s to two dimensions]… [Riemann’s lecture] threw a new light on geometry, isolating the concept of measurement-relations ('metric') as fundamental” (ibid). In this work, “Riemann's approach to geometry… "did more to change our ideas about geometry and physical space than any work on the subject since Euclid's Elements." (Landmark Writings in Western Mathematics, 507).
Riemann’s essay begins with a gauntlet of sorts – “a challenge not just to the scope and reach of mathematics in 1854,” but also to the mathematical philosophy and orthodoxy then prevailing in Gottingen (ibid). His “remarkably bold opening, [promises] nothing less than the overthrow of Euclidean geometry as the source of all geometric ideas” (Gray (ed), The Symbolic Universe: Geometry and Physics 1890–1930).
“[Riemann] starts “with a remark about a darkness that lies at the foundations of geometry, and which, in Riemann’s opinion, is not illuminated by the usual axiomatic presentation. This darkness obscures the connections between what is assumed, which is the notion of space and of constructions in space. It has persisted from Euclid to A.M. Legendre, to name only the most famous of recent authorities, perhaps because the idea of multiply extended magnitudes has not been discussed. Once this is done, said Riemann, it will be seen that even among three-dimensional extended magnitudes there is no unique choice, and so the nature of space itself becomes empirical (ibid).
Riemann’s “underlying idea is very simple. One has a geometry whenever one has a space of points (a manifold) and a way of measuring distance between points, which would be the case if one always knew the distance between infinitesimally close points. So what one wants is a manifold and an infinitesimal ruler” (ibid).
Riemann’s insight “is not an unintuitive concept. A one-fold extended magnitude, or one-dimensional manifold, is a curve. Points on a curve require one measurement or coordinate to determine their position. If a curve moves along a line it sweeps out a surface or two-dimensional manifold, in which points require two coordinates to specify their position… There is no need to stop with three-dimensional manifolds, and indeed Riemann contemplated extended magnitudes of arbitrary multiplicity. So, roughly speaking and without looking for complications, a multiply-extended magnitude is something that is captured or measured by using a certain number of coordinates” – the “frame” to fit Einstein’s “physical ideas, his cosmology and cosmogony… metric structure determined by physical data” (ibid; PMM 455).
“Riemann’s vision of geometry …lead slowly and not always directly to the dominant view of the 20th century which interprets gravity in geometric terms” (Gray). Item #1604
CONDITION: 4to. Complete. Ex-libris with markings confined to title page. Handsomely rebound in half calf. Gilt-ruled bands at the spine; red and black morocco labels, each gilt-lettered. Clean and bright inside and out. Near fine.