[106] This is proved by Helmholtz's remark at the end of a detailed attempt to make spherical and pseudo-spherical spaces imaginable (l.c. p. 28): "Anders ist es mit den drei Dimensionen des Raumes. Da alle unsere Mittel sinnlicher Anschauung sich nur auf einen Raum von drei Dimensionen erstrecken, und die vierte Dimension nicht bloss eine Abänderung von Vorhandenem, sondern etwas vollkommen Neues wäre, so befinden wir uns schon wegen unserer körperlichen Organisation in der absoluten Unmöglichkeit, uns eine Anschauungsweise einer vierten Dimension vorzustellen."
[107] Cf. Grassmann, Ausdehnungslehre von 1844, 2nd Edition, p. xxiii.
[108] See especially Stallo, Concepts of Modern Physics, International Science Series, Vol. XLII. Chaps. XIII. and XIV.; Renouvier, "Philosophie de la règle et du compas," Année Philosophique, II.; Delbœuf, "L'ancienne et les nouvelles géométries," Revue Philosophique, Vols. XXXVI.-XXXIX.
[109] M. Delbœuf deserves credit for having based Euclid, already in 1860, in his "Prolégomènes Philosophiques de la Géométrie," on this axiom—certainly a better basis, at first sight, than the axiom of parallels.
[110] This meaning of homogeneity must not be confounded with the sense in which I have used the word. In Delbœuf's sense, it means that figures may be similar though of different sizes; in my sense it means that figures may be similar though in different places. This property of space is called by Delbœuf isogeneity.
[111] For a full proof of this proposition, see [Chap. III.]
[112] See [Chap. III., especially § 133.]
[113] For a criticism of this view, see the above discussions on Riemann and Erdmann.
[114] Cf. Couturat, "De l'Infini Mathématique," Paris, Félix Alcan, 1896, p. 544.
[115] The following is a list of the most important recent French philosophical writings on Geometry, so far as I am acquainted with them.