But from the time of Kant’s adoption, in 1770, of the Critical view of space as being the universal form of our outer sense, he seems to have definitely rejected all such possibilities. Space, to be space at all, must be Euclidean; the uniformity of space is a presupposition of the _a priori_ certainty of geometrical science.[481] One of the criticisms which in the _Dissertation_[482] he passes upon the empirical view of mathematical science is that it would leave open the possibility that “a space may some time be discovered endowed with other fundamental properties, or even perhaps that we may happen upon a two-sided rectilinear figure.” This is the argument which reappears in the third argument on space in the first edition of the _Critique_.[483] The same examples are employed with a somewhat different wording. “It would not even be necessary that there should be only one straight line between two points, though experience invariably shows this to be so. What is derived from experience has only comparative universality, namely, that which is obtained through induction. We should therefore only be able to say that, so far as hitherto observed, no space has been found which has more than three dimensions.” But that Kant should have failed to recognise the possibility of other spaces does not by itself point to any serious defect in his position. There is no essential difficulty in reconciling the recognition of such spaces with his fundamental teaching. He admits that other races of finite beings may perhaps intuit through _non-spatial_ forms of sensibility; he might quite well have recognised that those other forms of intuition, though not Euclidean, are still spatial. It is in another and more vital respect that Kant’s teaching lies open to criticism. Kant is convinced that space is given to us in intuition as being definitely and irrevocably Euclidean in character. Both our intuition and our thinking, when we reflect upon space, are, he implies, bound down to, and limited by, the conditions of Euclidean space. And it is in this positive assumption, and not merely in his ignoring of the possibility of other spaces, that he comes into conflict with the teaching of modern geometry. For in making the above assumption Kant is asserting that we definitely know physical space to be three-dimensional, and that by no elaboration of concepts can we so remodel it in thought that the axiom of parallels will cease to hold. Euclidean space, Kant implies, is _given_ to us as an unyielding form that rigidly resists all attempts at conceptual reconstruction. Being quite independent of thought and being given as complete, it has no inchoate plasticity of which thought might take advantage. The modern geometer is not, however, prepared to admit that _intuitional_ space has any definiteness or preciseness of nature apart from the concepts through which it is apprehended; and he