Theoreme Eighth.

EQUAL THINGS WHICH ARE PERPENDICULAR GIVE THEIR OWN SHORTENINGS.

Let the perpendicular line be divided into equal parts, such as B C, C D, D E, E F, and let the eye be marked A. The visual rays will be drawn from each point of the sizes up to the eye. After drawing one or several perpendicular lines between the eye and the engraved line, all the sizes will be equal according to the thirty-seventh proposition of the first of the Elements of Euclid, where he states that all triangles being on the same bases and between the same parallels are equal among themselves. The triangle B C A is therefore equal to C D A, and to all the others also which are made on the same line. After, according to the second proposition of the sixth, he states that if a straight line cuts the two sides of a triangle, the angles will be proportional among themselves, and if the angles are proportional among themselves, it is necessary that all the sides of the line G H be equal. Thus, it can be seen by this reason that all sizes being perpendicular do not give any foreshortening.

A perspective diagram illustrating geometric optics and the theorem described. It shows a vertical line divided into equal segments (B, C, D, E, F) on the left. Lines of sight (rays) converge from these points toward an eye at point A on the right. A vertical "cut line" (ligne taillée) marked G H intersects these rays. A human figure is sketched on the right side near point A, observing the geometry. Labels include B, C, D, E, F, G (cut line), H, and A.

I will give yet another example, inasmuch as this deserves to be understood: let there be a building or a tower raised to one hundred feet in height, and let there be windows marked, those high up being B and those down low being C. Let the eye be A. Let the visual rays be drawn afterwards, which will pass through the cut line. It is certain that the foreshortening which will be on the cut line, both of those high up and those down low, will be equal. But it is very certain that those down low, which are seen under a larger angle, appear larger according to the fifth Theorem. And to show by how much, let the circle F D H I be made, of which the eye will be the center. Now, the size F D is smaller than H I, as much in proportion as the window B appears smaller than C.