[161] As is Helmholtz's other axiom, that the possibility of superposition is independent of the course pursued in bringing it about.
[163] This deduction is practically the same as that in Sec. A, but I have stated it here with more special reference to space and to metrical Geometry.
[164] The question: "Relations to what?" is a question involving many difficulties. It will be touched on later in this chapter, and answered, as far as possible, in the fourth chapter. For the present, in spite of the glaring circle involved, I shall take the relations as relations to other positions.
[165] Wiss. Abh. Vol. II. p. 614.
[166] Cp. Grassmann, Ausdehnungslehre von 1844, 2nd ed. p. XXIII.
[167] Delbœuf, it is true, speaks of Geometries with m/n dimensions, but gives no reference (Rev. Phil. T. xxxvi. p. 450).
[168] In criticizing Erdmann, it will be remembered, we saw that Free Mobility is a necessary property of his extents, though he does not regard it as such.
[169] Cf. Riemann, Hypothesen welche der Geometrie zu Grunde liegen, Gesammelte Werke, p. 266; also Erdmann, op. cit. p. 154.
[170] This is subject, in spherical space, to the modification pointed out below, in dealing with the exception to the axiom of the straight line. See [§§ 168–171.]