By Peter Pesic

Contents contain "On the Hypotheses which Lie on the Foundations of Geometry" by means of Georg Friedrich Riemann; "On the evidence which Lie on the Foundations of Geometry" and "On the foundation and value of Geometrical Axioms" through Hermann von Helmholtz; "A Comparative overview of contemporary Researches in Geometry" by way of Felix Klein; "On the distance conception of subject" via William Kingdon Clifford; "On the rules of Geometry" by way of Henri Poincaré; "Euclidean Geometry and Riemannian Geometry" by way of Elie Cartan; and "The challenge of house, Ether, and the sphere in Physics" by Albert Einstein.

These remarkably available papers will entice scholars of recent physics and arithmetic, in addition to a person drawn to the origins and assets of Einstein's so much profound paintings. Peter Pesic of St. John's university in Santa Fe, New Mexico, offers an advent, in addition to notes that supply insights into each one paper.

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**Example text**

W. F. Hegel had argued that space as “self-externality ... is therefore absolutely continuous”; see Rosenfeld 1988, 198-199. Riemann’s last thoughts are his most provocative, even today. By directing our attention to the possibility that solid bodies and rays of light “lose their validity in the infinitely small,” he indicates the way in which the presumed continuity of space might finally be grounded on something not continuous. It is purely an assumption, he notes, that bodies exist independently of position and hence can be of any size (as is not true in Lobachevsky’s geometry, for instance).

Riemann’s last thoughts are his most provocative, even today. By directing our attention to the possibility that solid bodies and rays of light “lose their validity in the infinitely small,” he indicates the way in which the presumed continuity of space might finally be grounded on something not continuous. It is purely an assumption, he notes, that bodies exist independently of position and hence can be of any size (as is not true in Lobachevsky’s geometry, for instance). Since this is not necessary, only empirical observation can really tell us whether or not there is some fundamental lower limit to the size of such bodies or light rays.

Consider a meter stick being moved around; suppose it were somehow noted that the length of that stick is not constant. There is a problem: either the distance function is presumed to be uniform and the surface is curved, or the surface is really flat and the distance-function is not uniform. ” It may help here to anticipate the controversy that followed Riemann’s assertion. ] 36 Beyond Geometry 8. [In order to find the general form that the distance function must take, Riemann assumes (1) continuity and (2 ) that distance must be unchanged (to first order) if all the quantities dx are equally displaced.