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 [ix] 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 42. Has a merely technical validity, 42 43. And is capable of giving geometrical results only when it begins and ends with real points and figures 45 44. We have now seen that projective Geometry is logically prior to metrical Geometry, but cannot supersede it 46 45. Sophus Lie has applied projective methods to Helmholtz's formulation of the axioms, and has shown the axiom of Monodromy to be superfluous 46 46. Metageometry has gradually grown independent of philosophy, but has grown continually more interesting to philosophy 50 47. Metrical Geometry has three indispensable axioms, 50 48. Which we shall find to be not results, but conditions, of measurement, 51 49. And which are nearly equivalent to the three axioms of projective Geometry 52 50. Both sets of axioms are necessitated, not by facts, but by logic 52