22. Which enables us to define a _constant_ measure of curvature of a three-dimensional space without reference to a fourth dimension 20 23. The main result of Riemann's mathematical work was to show that, if magnitudes are independent of place, the measure of curvature of space must be constant 21 24. Helmholtz, who was more of a philosopher than a mathematician, 22 25. Gave a new but incorrect formulation of the essential axioms, 23 26. And deduced the quadratic formula for the infinitesimal arc, which Riemann had assumed 24 27. Beltrami gave Lobatchewsky's planimetry a Euclidean interpretation, 25 28. Which is analogous to Cayley's theory of distance; 26 29. And dealt with _n_-dimensional spaces of constant negative curvature 27 30. The third period abandons the metrical methods of the second, and extrudes the notion of spatial quantity 27 31. Cayley reduced metrical properties to projective properties, relative to a certain conic or quadric, the Absolute; 28 32. And Klein showed that the Euclidean or non-Euclidean systems result, according to the nature of the Absolute; 29 33. Hence Euclidean _space_ appeared to give rise to all the kinds of Geometry, and the question, which is true, appeared reduced to one of convention 30 34. But this view is due to a confusion as to the nature of the coordinates employed 30 35. Projective coordinates have been regarded as dependent on distance, and thus really metrical 31 36. But this is not the case, since anharmonic ratio can be projectively defined 32 37. Projective coordinates, being purely descriptive, can give no information as to metrical properties, and the reduction of metrical to projective properties is purely technical 33 38. The true connection of Cayley's measure of distance with non-Euclidean Geometry is that suggested by Beltrami's Saggio, and worked out by Sir R. Ball, 36 39. Which provides a Euclidean equivalent for every non-Euclidean proposition, and so removes the possibility of contradictions in Metageometry 38 40. Klein's elliptic Geometry has not been proved to have a corresponding variety of space 39 41. The geometrical use of imaginaries, of which Cayley demanded a philosophical discussion, 41