By Peter Pesic
Contents contain "On the Hypotheses which Lie on the Foundations of Geometry" by means of Georg Friedrich Riemann; "On the proof which Lie on the Foundations of Geometry" and "On the starting place and value of Geometrical Axioms" via Hermann von Helmholtz; "A Comparative evaluate of modern Researches in Geometry" by way of Felix Klein; "On the gap concept of topic" via William Kingdon Clifford; "On the rules of Geometry" through Henri Poincaré; "Euclidean Geometry and Riemannian Geometry" by means of Elie Cartan; and "The challenge of house, Ether, and the sector in Physics" by Albert Einstein.
These remarkably available papers will entice scholars of contemporary physics and arithmetic, in addition to a person drawn to the origins and assets of Einstein's such a lot 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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Extra info for Beyond Geometry: Classic Papers from Riemann to Einstein
About manifolds in which the square of the line element can be represented as the sum of squares of complete differen tials. 30 Beyond Geometry In a flat n-fold extended manifold the curvature in every direction, at every point, is zero; but according to the preceding investigation, in order to deter mine the metric relations it suffices to know that at each point the curvature is zero in independent surface-directions. The manifolds whose curvature is everywhere = 0 can be considered as a special case of those manifolds whose curvature is everywhere constant.
But secondly, if one assumes with Euclid not only the existence of lines inde pendently of configuration, but also of bodies, then it follows that the curvature is everywhere constant, and the angle sum in all triangles is determined if it is known in one. In the third place, finally, instead of assuming the length of lines to be independent of place and direction, one might assume that their length and direction is independent of place. According to this conception, changes or differences in position are complex quantities expressible in three independent units.
So the remaining 38 Beyond Geometry components are n 2 —Pfot1) = n 21—. ] 11. [In the Euclidean case, the distance function will take on the neat form ds2 = dx2 -f dy2 4- dz2 only if we choose the right variables; in polar coordi nates the expression has a different look (ds2 = dr2 + r2d02 + r2 sin2 0 d4>2) and in other curvilinear coordinates might be even more unrecognizably scrambled. That is, the same distance function might look very different in different coordi nates: how can we distinguish real, physical effects from deceptive appearances that result from choosing different coordinates?
Beyond Geometry: Classic Papers from Riemann to Einstein by Peter Pesic