“A right-angled equilateral triangle” contains no logical contradiction; for the predicates do not by any means cancel the subject, nor are they inconsistent with each other. It is only when their object is constructed in pure perception that the impossibility of their union in it appears. Now if on this account we were to regard this as a contradiction, then so would every physical impossibility, only discovered to be such after the lapse of centuries, be a contradiction; for example, the composition of a metal from its elements, or a mammal with more or fewer than seven cervical vertebra,(14) or horns and upper incisors in the same animal. But only _logical_ impossibility is a contradiction, not physical, and just as little mathematical. Equilateral and rectangled do not contradict each other (they coexist in the square), nor does either of them contradict a triangle. Therefore the incompatibility of the above conceptions can never be known by mere _thinking_, but is only discovered by perception—merely mental perception, however, which requires no experience, no real object. We should also refer here to the proposition of Giordano Bruno, which is also found in Aristotle: “An infinitely large body is necessarily immovable”—a proposition which cannot rest either upon experience or upon the principle of contradiction, since it speaks of things which cannot occur in any experience, and the conceptions “infinitely large” and “movable” do not contradict each other; but it is only pure perception that informs us that motion demands a space outside the body, while its infinite size leaves no space over. Suppose, now, it should be objected to the first mathematical example that it is only a question of how complete a conception of a triangle the person judging has: if the conception is quite complete it will also contain the impossibility of a triangle being rectangular and also equilateral. The answer to this is: assume that his conception is not so complete, yet without recourse to experience he can, by the mere construction of the triangle in his imagination, extend his conception of it and convince himself for ever of the impossibility of this combination of these conceptions. This process, however, is a synthetic judgment _a priori_, that is, a judgment through which, independently of all experience, and yet with validity for all experience, we form and perfect our conceptions. For, in general, whether a given judgment is analytical or synthetical can only be determined in the particular case according as the conception of the subject in the mind of the person judging is more or less complete. The conception “cat” contains in the mind of a Cuvier a hundred times more than in that of his servant; therefore the same judgments about it will be synthetical for the latter, and only analytical for the former. But if we take the conceptions objectively, and now wish to decide whether a given judgment is analytical