18. It is next to impossible for the geometrician to meditate upon the triangle without revolving in his imagination, the image of a triangle as he has seen it drawn a thousand times; and he will, for this reason, be disposed to believe that the idea of the triangle is nothing else than this sensible representation. Were it thus, Condillac's assertion that the idea is only the recollection of the sensation would be verified in the idea of the triangle. In fact, this representation is the sensation repeated: the only difference between the two affections of the mind is that the actual sensation is caused by the actual presence of its object, wherefore it is more fixed and vivid. To prove that the difference is not essential, but consists only in degree, it is sufficient to observe, that if the imaginary representation attain a high degree of vividness we cannot distinguish it from sensation, as it happens to the visionary, and as we have all experienced in our dreams. 19. By noticing the following facts, we shall readily perceive how different the idea of the triangle is from its imaginary representation. I. The idea of the triangle is one, and is common to all triangles of every size and kind; the representation of it is multiple, and varies in size and form. II. When we reason upon the properties of the triangle, we proceed from a fixed and necessary idea; the representation changes at every instant, not so, however, the unity of the idea. III. The idea of a triangle of any kind in particular is clear and evident; we see its properties in the clearest manner; the representation on the contrary is vague and confused, thus it is difficult to distinguish a right-angled from an acute-angled triangle, or even a slightly inclined obtuse-angled triangle. The idea corrects these errors or rather abstracts them; it makes use of the imaginary figure only as an auxiliary, in the same manner as we give our demonstrations when we draw figures upon paper, abstracting their exactness or inexactness, often when we know that they are not exact, which they cannot always be.