The second difficulty is more serious. If the truth of geometrical judgements presupposes that space is only a property of objects as perceived by us, it is a paradox that geometricians should be convinced, as they are, of the truth of their judgements. They undoubtedly think that their judgements apply to things as they are in themselves, and not merely as they appear to us. They certainly do not think that the relations which they discover apply to objects only as perceived. Not only, therefore, do they not think that bodies in space are phenomena, but they do not even leave it an open question whether bodies are phenomena or not. Hence, if Kant be right, they are really in a state of illusion, for on his view the true geometrical judgement should include in itself the phenomenal character of spatial relations; it should be illustrated by expressing Euclid I. 5 in the form that the equality of the angles at the base of an isosceles triangle belongs to objects as perceived. Kant himself lays this down. "The proposition 'all objects are beside one another in space' is valid under[59] the limitation that these things are taken as objects of our sensuous perception. If I join the condition to the perception, and say 'all things, as external phenomena, are beside one another in space', the rule is valid universally, and without limitation."[60] Kant, then, is in effect allowing that it is possible for geometricians to make judgements, of the necessity of which they are convinced, and yet to be wrong; and that, therefore, the apprehension of the necessity of a judgement is no ground of its truth. It follows that the truth of geometrical judgements can no longer be accepted as a starting-point of discussion, and, therefore, as a ground for inferring the phenomenal character of space. [59] A. reads 'only under' [60] B. 43, M. 27.