Chapter I: A Short History of Metageometry

10. Metageometry began by rejecting the axiom of parallels 7

11. Its history may be divided into three periods: the synthetic,
the metrical and the projective 7

12. The first period was inaugurated by Gauss, 10

13. Whose suggestions were developed independently by
Lobatchewsky 10

14. And Bolyai 11

15. The purpose of all three was to show that the axiom of parallels
could not be deduced from the others, since its denial did
not lead to contradictions 12

16. The second period had a more philosophical aim, and was
inspired chiefly by Gauss and Herbart 13

17. The first work of this period, that of Riemann, invented two
new conceptions: 14

18. The first, that of a manifold, is a class-conception, containing
space as a species, 14

19. And defined as such that its determinations form a collection
of magnitudes 15

20. The second, the measure of curvature of a manifold, grew out
of curvature in curves and surfaces 16

21. By means of Gauss's analytical formula for the curvature of
surfaces, 19

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

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