Another of the profound impulses Gauss gave geometry concerned the general description of surfaces. Typically—with the notable exception of the geometry of the sphere—mathematicians had treated surfaces as structures in three-dimensional Euclidean space. However, as these surfaces occupy only two dimensions, only two variables are needed to describe them. This prompted the thought that two-dimensional surfaces could be considered as “spaces” with their own geometries, not just as Euclidean structures in ordinary space. For example, the shortest distance, or path, between two points on the surface of a sphere is the lesser arc of the great circle joining them, whereas, considered as points in three-dimensional space, the shortest distance between them is an ordinary straight line.

The shortest path between two points on a surface lying wholly within that surface is called a geodesic, which reflects the origin of the concept in geodesy, in which Gauss took an active interest. His initiative in the study of surfaces as spaces and geodesics as their “lines” was pursued by his student and, briefly, his successor at Göttingen, Bernhard Riemann (1826–66). Riemann began with an abstract space of n dimensions. That was in the 1850s, when mathematicians and mathematical physicists were beginning to use n-dimensional Euclidean space to describe the motions of systems of particles in the then-new kinetic theory of gases. Riemann worked in a quasi-Euclidean space—“quasi” because he used the calculus to generalize the Pythagorean theorem to supply sufficient flexibility to provide for geodesics on any surface.

When this very general differential geometry came down to two-dimensional surfaces of constant curvature, it revealed excellent models for non-Euclidean geometries. Riemann himself pointed out that, merely by calling the geodesics of a sphere “straight lines,” the maligned hypothesis of the obtuse angle produces the geometry appropriate to the sphere’s surface. Similarly, as shown by Eugenio Beltrami (1835–1900), who ended his teaching career in Saccheri’s old post at Pavia, the geometry defined in the plane by the hypothesis of the acute angle fits perfectly a surface of revolution of constant negative curvature now called a pseudosphere (see figure)—again, provided that its geodesics are accepted as the straight lines of the geometry.

Since the hypothesis of the obtuse angle correctly characterizes Euclidean geometry applied to the surface of a sphere, the non-Euclidean geometry based on it must be exactly as consistent as Euclidean geometry. The case of the acute angle treated by Lobachevsky and Bolyai required a sharper tool. Beltrami found it in a projection into a disc in the Euclidean plane of the points of a non-Euclidean space, in which each geodesic from the non-Euclidean space corresponds to a chord of the disc. Geometry built on the hypothesis of the acute angle has the same consistency as Euclidean geometry.

The key role of Euclidean geometry in proofs of the consistency of non-Euclidean geometries exposed the Elements to ever-deeper scrutiny. The old blemishes—particularly appeals to intuition and diagrams for the meaning of concepts like “inside” and “between” and the use of questionable procedures like superposition to prove congruency—became intolerable to mathematicians who laboured to clarify the foundations of arithmetic and the calculus as well as the interrelations of the new geometries. The German mathematician Moritz Pasch (1843–1930), in his Vorlesungen über neuere Geometrie (1882; “Lectures on the New Geometry”), identified what was wanting: undefined concepts, axioms about those concepts, and more rigorous logic based on those axioms. The choice of undefined concepts and axioms is free, apart from the constraint of consistency. Mathematicians following Pasch’s path introduced various elements and axioms and developed their geometries with greater or lesser elegance and trouble. The most successful of these systematizers was the Göttingen professor David Hilbert (1862–1943), whose The Foundations of Geometry (1899) greatly influenced efforts to axiomatize all of mathematics. (See Sidebar: Teaching the Elements.)