In conclusion, I do not, of course, maintain that mankind first arrived at space-intuitions, in agreement with the axioms of Euclid, by any carefully executed systems of exact measurement. It was rather a succession of everyday experiences, especially the perception of the geometrical similarity of great and small bodies, only possible in flat space, that led to the rejection, as impossible, of every geometrical representation at variance with this fact. For this no knowledge of the necessary logical connection between the observed fact of geometrical similarity and the axioms was needed; but only an intuitive apprehension of the typical relations between lines, planes, angles, &c., obtained by numerous and attentive observations--an intuition of the kind the artist possesses of the objects he is to represent, and by means of which he decides with certainty and accuracy whether a new combination, which he tries, will correspond or not with their nature. It is true that we have no word but _intuition_ to mark this; but it is knowledge empirically gained by the aggregation and reinforcement of similar recurrent impressions in memory, and not a transcendental form given before experience. That other such empirical intuitions of fixed typical relations, when not clearly comprehended, have frequently enough been taken by metaphysicians for _à priori_ principles, is a point on which I need not insist. * * * * * To sum up, the final outcome of the whole inquiry may be thus expressed:-- (1.) The axioms of geometry, taken by themselves out of all connection with mechanical propositions, represent no relations of real things. When thus isolated, if we regard them with Kant as forms of intuition transcendentally given, they constitute a form into which any empirical content whatever will fit, and which therefore does not in any way limit or determine beforehand the nature of the content. This is true, however, not only of Euclid’s axioms, but also of the axioms of spherical and pseudospherical geometry. (2.) As soon as certain principles of mechanics are conjoined with the axioms of geometry, we obtain a system of propositions which has real import, and which can be verified or overturned by empirical observations, just as it can be inferred from experience. If such a system were to be taken as a transcendental form of intuition and thought, there must be assumed a pre-established harmony between form and reality. APPENDIX. The elements of the geometry of spherical space are most easily obtained by putting for space of four dimensions the equation for the sphere _x_² + _y_² +_z_² + _t_² = _R_² (1.) and for the distance _ds_ between the points (_x_, _y_, _z_, _t_) and [(_x_ + _dx_) (_y_ + _dy_) (_z_ + _dz_) (_t_ + _dt_)] the value _ds_² = _dx_² + _dy_² + _dz_² + _dt_² (2.)