[101] See Lectures and Essays, Vol. I. p. 261. [102] On the meaning of geometrical possibility, cf. Veronese, Grundzüge der Geometrie (German translation), pp. xi.-xiii. [103] Compare Calinon, "Sur l'Indétermination géométrique de l'Univers," Revue Philosophique, 1893, Vol. XXXVI. pp. 595-607. [104] Vorträge und Reden, Vol. II. p. 9: "Parallele Linien würden die Bewohner der Kugel gar nicht kennen. Sie würden behaupten, dass jede beliebige zwei _geradeste_ Linien, gehörig verlangert, sich schliesslich nicht nur in einem, sondern in zwei Punkten schneiden müssten." (The italics are mine.) The omission of _straight_ in such phrases is a frequent laxity of mathematicians. [105] It has been suggested to me that Lotze regards the meridians as projected on to a plane, as in a map. If this be so, there is an obviously illegitimate introduction of the third dimension. [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. Andrade: "Les bases expérimentales de la géométrie euclidienne"; Rev. Phil. 1890, II., and 1891, I. Bonnel: "Les hypothèses dans la géométrie"; Gauthier-Villars, 1897.