By Peter Pesic

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

These remarkably available papers will entice scholars of contemporary physics and arithmetic, in addition to somebody drawn to the origins and resources of Einstein's such a lot profound paintings. Peter Pesic of St. John's collage in Santa Fe, New Mexico, offers an creation, in addition to notes that supply insights into each one paper.

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**Example text**

The investigations that can be carried out in this case form a general division of the science of quantity, independent of measurement, where quantities are re garded, not as existing independent of position and not as expressible in terms of a unit, but as regions in a manifold. , for the treatment of many-valued analytic functions, and the dearth of such studies is one of the principal rea sons why the celebrated theorem of Abel and the contributions of Lagrange, Pfaff, and Jacobi to the general theory of differential equations have remained unfruitful for so long.

The problem then is to set up a mathematical expression for the length of a line, for which purpose the quantities x must be thought of as expressible in units. I will treat this problem only under certain restrictions, and I first limit myself to lines in which the ratios of the quantities dx—the increments in the quantities x—vary continuously; one can then regard the lines as broken up into elements within which the ratios of the quantities dx may be considered to be constant, and the problem then reduces to setting up a general expression for the line element ds at every point, an expression which will involve the quantities x and the quantities dx.

The quantities dx at the origin of this shortest line, and by the length s of this line. In place of the dx° one now introduces linear expressions da formed from them in such a way that the initial value of the square of the ine element will be equal to the sum of the squares of these expressions, so that the independent variables are: the quantity s and the ratio of the quantities da. , xn proportional to them, but such that the sum of their squares equals s2. ). This quantity remains the same as long as the quantities x and dx are contained in the same binary linear forms, or as long as the two shortest lines from the initial point to x and from the initial point to dx remain in the same surface element, and therefore depends only on the position and direction of that element.