In the concluding paragraph Kant says that this is the only explanation which can be given of the possibility of geometry. He does not distinguish between pure and applied geometry, though the proof which he has given of each differs in a fundamental respect. Pure geometry presupposes only that space is an _a priori_ intuition; applied geometry demands that space be conceived as the _a priori_ form of external sense. Only in reference to applied geometry does the Critical problem arise:--viz. how we can form synthetic judgments _a priori_ which yet are valid of objects; or, in other words, how judgments based upon a subjective form can be objectively valid. But any attempt, at this point, to define the nature and possibility of applied geometry must anticipate a result which is first established in _Conclusion b_.[458] Though, therefore, the substitution of this transcendental exposition for the third space argument is a decided improvement, Kant, in extending it so as to cover applied as well as pure mathematics, overlooks the real sequence of his argument in the first edition. The employment of the analytic method, breaking in, as it does, upon the synthetic development of Kant’s original argument, is a further irregularity.[459] It may be noted that in the third paragraph Kant takes the fact that geometry can be applied to objects as proof of the subjectivity of space.[460] He refuses to recognise the possibility that space may be subjective as a form of receptivity, and yet also be a mode in which things in themselves exist. This, as regards its conclusion, though not as regards its argument, is therefore an anticipation of _Conclusion a_. In the last paragraph Kant is probably referring to the views both of Leibniz and of Berkeley. CONCLUSIONS FROM THE ABOVE CONCEPTS[461] =Conclusion a.=--_Thesis_: Space is not a property of things in themselves,[462] nor a relation of them to one another. Proof: The properties of things in themselves can never be intuited prior to their existence, _i.e. a priori_. Space, as already proved, is intuited in this manner. In other words, the apriority of space is by itself sufficient proof of its subjectivity.