=54.= Now it must be admitted, I think, that Metageometry has destroyed the legitimacy of the argument from Geometry to space; we can no longer affirm, on purely geometrical grounds, the apodeictic certainty of Euclid. But unless Metageometry has done more than this--unless it has proved, what I believe it alone cannot prove, that Euclid has _not_ apodeictic certainty--then Kant's other line of argument retains what force it may ever have had. The actual space we know, it may say, is admittedly Euclidean, and is proved, without any reference to Geometry, to be _à priori_; _hence_ Euclid has apodeictic certainty, and non-Euclid stands condemned. To this it is no answer to urge, with the Metageometers, that non-Euclidean systems are _logically_ self-consistent; for Kant is careful to argue that geometrical reasoning, by virtue of our intuition of space, is synthetic, and cannot, though _à priori_, be upheld by the principle of contradiction alone[70]. Unless non-Euclideans can prove, what they have certainly failed to prove up to the present, that we can frame an _intuition_ of non-Euclidean spaces, Kant's position cannot be upset by Metageometry alone, but must also be attacked, if it is to be successfully attacked, on its purely philosophical side. =55.= For such an attack, two roads lie open: either we may disprove the first two arguments of the Aesthetic, or we may criticize, from the standpoint of general logic, the Kantian doctrine of synthetic _à priori_ judgments and their connection with subjectivity. Both these attacks, I believe, could be conducted with some success; but if we are to disprove the apodeictic certainty of Geometry, one or other is essential, and both, I believe, will be found only partially successful. It will be my aim to prove, in discussing these two lines of attack, (1) that the distinction of synthetic and analytic judgments is untenable, and further, that the principle of contradiction can only give fruitful results on the assumption that experience in general, or, in a particular science, some special branch of experience, is to be formally possible; (2) that the first two arguments of the Transcendental Aesthetic suffice to prove, not Euclidean space, but _some_ form of externality--which may be sensational or intuitional, but not merely conceptual--a necessary prerequisite of experience of an external world. In the third and fourth chapters, I shall contend, as a result of these conclusions, that those axioms, which Euclid and Metageometry have in common, coincide with those properties of any form of externality which are deducible, by the principle of contradiction, from the possibility of experience of an external world. These properties, then, may be said, though not quite in the Kantian sense, to be _à priori_ properties of space, and as to these, I think, a modified Kantian position may be maintained. But the question of the subjective or objective nature