Of mechanics.

Mechanics is the paradise of the mathematical sciences, because through it one comes to the fruit of mathematics.

a; b; c; d

Therefore let the given square b be a square hole, whence the square wax is drawn out, and let such square grow 3 times and you will have the height of the required pyramid.

From a given quantity of dilatable and extensible material, such as wax or other similar things, let a pyramid be made of base equal to a given square, and the quantity of the height of that pyramid is asked.

a b - g h o - e f - c d

The uniform body that uniformly narrows acquires as much length as it loses of its width, in proportion to the size of such sides. For if the sides of the tablet are various, various will be the increases, because each side grows according to its proportion. And in confirmation of this let us say that the length of the tablet to be narrowed be 4 and the width 2. I say that if the tablet is narrowed by one braccio, the length grows by 4, because by narrowing half the width, it grows half the length, etc.

Therefore the tablet a c b d, narrowing to a quarter of its thickness, loses the three quarters and acquires three times its first length.

And because the pyramid is 1/3 of its cylinder, make the length of the pyramid by 3 cylinders, and thus you will consume the material of that cylinder in the required pyramid, that is you will make a pyramid equal to the cylinder, etc.

And do the converse, to make the pyramid according to a given height. To know the base, make a parallelepiped of height equal to the third of that given line, and the base of such parallelepiped will be the base of the proposed pyramid.

Note. Make a parallelepiped of all the material a according to the face b, and multiply it by 3 and you will have the pyramid.