This essay combined the ideas of several non-Euclidean geometries into one work expressing the ideas of the field. Most importantly, this essay showed that curved space could seem like a Euclidean geometry, that curved lines could seem straight and that spherical topology could seem flat. The next century would soon realize the practical applications of non-Euclidean geometries, especially within physics and astronomy. Einstein realized that Newtonian physics, based upon Euclidean geometry, failed to consider the curvature of space, and that this simple matter constituted for major errors in the equations of planetary motion and gravity. With Einstein's declaration, a new era for physics and mathematics was soon born.
Today there are three non-Euclidean geometries of particular interest to students of mathematics.
They are Riemannian, Poincare, and Taxi-Cab geometries. Other geometries exist, but many of them are complex versions of these geometrical forms. A Riemannian geometry is also known as a spherical geometry. The plane of a Riemannian geometry occurs on the surface of a spherical object. The applications of this geometry, as well as its necessity, are best seen when thinking of a globe or the surface of some planet.