52. It is necessary to suppose this existence, otherwise nothing could be explained. Common sense teaches us what has escaped some metaphysicians. To prove it, let us see how a mathematician, who never dipped into metaphysics, would talk. We will suppose the interlocutor to set out to demonstrate to us that in a rectangular triangle the square of the hypothenuse is equal to the sum of the squares of the base and perpendicular; and that we, in order to exercise his intelligence, or rather to make him show us, without himself being aware of it, what is passing in his own mind with respect to the perception of its object, put various questions to him, in reality searching, although apparently asked out of ignorance. We will adopt the form of a dialogue for the sake of greater clearness, and will suppose the demonstration to be given from memory, without the aid of figures. _Demonstration._ Drop a perpendicular from the right angle to the hypothenuse. Where? Why, in the triangle of which we speak, of course. But, sir, if there be no such triangle---- Why then, what are we talking of? We are talking of a rectangular triangle, and the case supposed is that there is none. Is not, but can be. Take paper, a pencil, and ruler, and we will have one right away. That is to say, you speak of the triangle we may make? Yes, sir. Ah, I understand; but then we should have it; now, we have not got it. All in good time. But if we had drawn it, could we not drop the perpendicular? Certainly. That is all I meant to say. But you were saying drop---- No doubt we cannot drop a perpendicular in a triangle unless the triangle exists, since then there is neither vertex of a right angle, hypothenuse, nor any thing else; but when I say, drop a perpendicular, I always suppose a triangle; and as it is evident that the triangle may exist, I do not express the supposition, but understand it. I comprehend this; but then we should drop the perpendicular only in this triangle, but you spoke as if we might drop it in all triangles. I only took this triangle for an example; we can clearly do with all others what we can do with this one. With all? Certainly. Can you not see how, in every rectangular triangle, a perpendicular may be drawn from the right angle to the hypothenuse? Yes, in your figure; but since what is in my head is not a triangle, for I imagine some with sides a thousand miles long, and there is not in my head room enough-- There is no question of what is in your head, but of triangles themselves-- But these triangles do not exist; therefore, we can say nothing of them. Yes; but may they not exist? Who doubts it? Well then, if they do exist, be they large or small, in one position or another, here or there, is it not true that a perpendicular may be drawn from the vertex of the right angle to the hypothenuse? Evidently. I have then only to say that, in every rectangular triangle, this perpendicular may be drawn.