The height of the pyramid grows and the base diminishes. The base grows and the height diminishes. The height diminishes and the base grows. The base diminishes and the height grows.

e e a d – n – c c m b; f r q – t – m S n – p o; g n f – r o – c e a – h d b; i c h a – r o p – n g – f d b. Let a pyramid be diminished in its height by a given irrational lowering, and let the increase of its base be found. Let the length of a given pyramid be increased, and let the diminution of its base be given. Let the pyramid be a b c, which is truncated in pyramidal form at the height m n, and I wish to know how much it widens in its base. In this case I will clothe the said pyramid with a cylinder made from its base and height, which cylinder shall be d e b c. Then I will set that cylinder beneath this one at n m o p, and by the fifth of the first I will make its squaring q r n S, which will have the form of a slab, because the side S t is not as great as the side S n, since S t is similar to the thickness of the cylinder n o, which is of equal thickness and width. Therefore S t and S n are similar to n o and n q, so that in this proof you have the certainty of the inequality of the sides of that square q r n S. Now the square of the side r f S t will be set beneath it at a b c d, which has the side a b c d similar to the side r f S t of the figure above, that is similar to n o, that is the first cylinder; and the side n g e o is similar above to the side q r n S, whence by the fifth of the first I will make its squaring, and it shall be f n a e, which remains in the form of a cylinder, because the side that the squaring f n a e shows is above the side r f S t, and the side n g e o is as long as r q S n is above. Having thus far made the cylinder f n a e, I will set it below and put into its proper form the true quantity of the side n g e o, which will be a b c d, and the side f n a e shall be h i a c, the upper part of the cylinder a b c d, which cylinder contains within itself the whole quantity of the cylinder proposed above. Now I will mark on that cylinder the given height above of the pyramid m n, and it shall be placed at g b. Then with the help of the seventh of the first I will extend the cylinder a b c d into the cylinder g b n f, which cylinder I will extend not with a square face, but it all remains in the form of a long slab, because the side of the face n r is of the width b d, that is of the thickness of the cylinder a b c d, and the other side of the face n f is such as it shows itself. Now I will turn the wider side of the slab, that is g p n r, toward me, and thus beneath it I will put it into its proper form, and it shall be n m o p.

Of a cylinder ... The required pyramid, truncated above at the height in t v below, and the width of the base is likewise below between n m. Therefore the given height of the pyramid at the parallel n has the quantity of the cylinder a reduced into the equally-lying parallel b. But because that quantity of b is understood to be only the surface of the cylinder a and not the body of that cylinder, it is necessary to take its thickness or depth and add it beneath the squaring of the cylinder. But because that thickness exceeds the height of the given pyramid, it is necessary to lower the excess thickness and reduce it to the given height of the pyramid, and it will be done according to the rule marked here.

From a cube let a pyramid be made.