96. The objections to non-Euclidean Geometry which have just been discussed fall under four heads:
I. Non-Euclidean spaces are not homogeneous; Metageometry therefore unduly reifies space.
II. They involve a reference to a fourth dimension.
III. They cannot be set up without an implicit reference to Euclidean space, or to the Euclidean straight line, on which they are therefore dependent.
IV. They are self-contradictory in one or more ways.
The reader who has followed me in regarding these four objections as fallacious, will have no difficulty in disposing of any other critic of Metageometry, as these are the only mathematical arguments, so far as I know, ever urged against non-Euclideans[108]. The logical validity of Metageometry, and the mathematical possibility of three-dimensional non-Euclidean spaces, will therefore be regarded, throughout the remainder of the work, as sufficiently established.
97. Two other objections may, indeed, be urged against Metageometry, but these are rather of a philosophical than of a strictly mathematical import. The first of these, which has been made the base of operations by Delbœuf, applies equally to all non-Euclidean spaces. The second, which has not, so far as I know, been much employed, but yet seems to me deserving of notice, bears directly against spaces of positive curvature alone; but if it could discredit these, it might throw doubt on the method by which all alike are obtained. The two objections are:
I. Space must be such as to allow of similarity, i.e. of the increase or diminution, in a constant ratio, of all the lines in a figure, without change of angles; whereas in non-Euclid, lines, like angles, have absolute magnitude.
II. Space must be infinite, whereas spherical and elliptic spaces are finite.