If α be positive, space is finite, though still unbounded, and every straight line is closed—a possibility first recognized by Riemann. It is pointed out that, since the possible values of a form a continuous series, observations cannot prove that our space is strictly Euclidean. It is also regarded as possible that, in the infinitesimal, the measure of curvature of our space should be variable.

There are four points in which this profound and epoch-making work is open to criticism or development—(1) the idea of a manifold requires more precise determination; (2) the introduction of coordinates is entirely unexplained and the requisite presuppositions are unanalysed; (3) the assumption that ds is the square root of a quadratic function of dx1, dx2, ... is arbitrary; (4) the idea of superposition, or congruence, is not adequately analysed. The modern solution of these difficulties is properly considered in connexion with the general subject of the axioms of geometry.

The publication of Riemann’s dissertation was closely followed by two works of Hermann von Helmholtz,[12] again undertaken in ignorance of the work of predecessors. In these a Helmholtz. proof is attempted that ds must be a rational integral quadratic function of the increments of the coordinates. This proof has since been shown by Lie to stand in need of correction (see VII. Axioms of Geometry). Helmholtz’s remaining works on the subject[13] are of almost exclusively philosophical interest. We shall return to them later.

The only other writer of importance in the second period is Eugenio Beltrami, by whom Riemann’s work was brought into connexion with that of Lobatchewsky and Bolyai. As he gave, by an elegant method, a convenient Beltrami. Euclidean interpretation of hyperbolic plane geometry, his results will be stated at some length[14]. The Saggio shows that Lobatchewsky’s plane geometry holds in Euclidean geometry on surfaces of constant negative curvature, straight lines being replaced by geodesics. Such surfaces are capable of a conformal representation on a plane, by which geodesics are represented by straight lines. Hence if we take, as coordinates on the surface, the Cartesian coordinates of corresponding points on the plane, the geodesics must have linear equations.

Hence it follows that

ds² = R²w−4 {(α² − v²) du² + 2uvdudv + (α² − u²)dv²}

where w² = α² − u² − v², and −1/R² is the measure of curvature of our surface (note that k = γ as used above). The angle between two geodesics u = const., v = const. is θ, where

cos θ = uv / √ {(α² − u²) (α² − v²)}, sin θ = aw / √ {(a² − u²) (a² − v²)}.

Thus u = 0 is orthogonal to all geodesies v = const., and vice versa. In order that sin θ may be real, w² must be positive; thus geodesics have no real intersection when the corresponding straight lines intersect outside the circle u² + v² = α². When they intersect on this circle, θ = 0. Thus Lobatchewsky’s parallels are represented by straight lines intersecting on the circle. Again, transforming to polar coordinates u = r cos μ, v = r sin μ, and calling ρ the geodesic distance of u, v from the origin, we have, for a geodesic through the origin,

dρ = Radr / (a² − r²), ρ = ½R log a + r, r = a tan h (ρ / R).
a − r