This second and fourth figure proves the first and fourth to be correct, which re-proof is drawn from the elements of geometry.

I will extend the length of a given rectangle according to the length of a given line: it is asked what will be its narrowness (new width). Let a b c d be the given rectangle, let d f be the given length to which the said rectangle is to be extended; therefore I will continue the side of the rectangle c a as far as g or higher, at will, then I will produce the straight line from f to e, so that it is tangent to the angle b of the aforesaid rectangle, and doing thus we will have given the width to the extension of the rectangle that is sought, that is a e, and the width to which the extended rectangle d f is reduced. Therefore the rectangle a b c d, extending itself in the length of the line d f, is narrowed according to the length of the line e a. It is proved in the 2nd. The same rectangle is divided equally by its diagonal, and one part is the upper triangle b f d, the other part is the lower triangle a e c; I will remove from the upper part the triangle b, and from the lower I will remove the triangle a, and these two triangles are similar and equal; and if from similar and equal things similar equal parts are removed, the remainders will be similar and equal. Then I will remove from the upper part the triangle d, and from the lower part the triangle c, which is equal and similar to triangle d; there remains for us the rectangle f in the upper part, equal to the rectangle e below, which is the proposition. Let the rectangle be shortened according to the brevity of a given line: it is asked what will be its width; a b is the given rectangle, b c is its shortening, n m is the acquired width, as will be shown below. Here is the proof of what is proposed in the penultimate, that is, that the rectangle n has been shortened and widened as the square m shows, without diminishing in its quantity, as was defined in the second above, etc.

Of many rectangles of various widths and lengths I wish to make a single one of any given width or length. And here is shown the way to make these four rectangles n m o p into a single rectangle of a given length, or if you wish, of a given width. I wish to unite them. For now I will give the width marked a b at the fronts above each rectangle, and having done this I will give the length marked c d below the tail of each rectangle. And thus, you giving me the width of such a rectangle, I will give you the length, and if you give me the length, I will give you the width of the union of all the rectangles. a b shows the widths of each rectangle; c d shows the length of the same rectangles, etc.

This strip, marked with its divisions and letters, has gathered and joined in itself the value of these four various rectangles n m o p. Give me the width of the rectangles at the fronts a b and I will give you the length of the same rectangles at the corners c d of the same rectangle. And so conversely, if you give me the length of these rectangles I will give you their width, etc. — because the soup is getting cold.