Figure, with: c b – a
On making the pivot that does not wobble. b c, the sheath of the pivot a, turns about with a very slow motion, in such a way that for every hundred turns given by the pivot the sheath gives one; and this is the way that the pivot does not wear its sheath more on one side than on the other. But because such a pivot is not very easy in its revolution, it is necessary that such an instrument be driven by water or by horse.

b a – d c – n e – p o

That part of the pivot wears more which is of greater thickness in an equal number of revolutions. The pivot a b c d must be similar in thickness at the top as at the foot to the pivot e n o p, and for this reason in equal revolution they will make equal wearings of themselves and of their mother, being of equal length.

On the pivots. On the material that most resists in frictions. Frictions are made in three ways, namely: hard with hard, soft with soft, and soft with hard. And of these three ways the contacts vary in two ways, of which the first is simple contact of the rubbed bodies, the second is composite contact. In simple contact no material is interposed, but their surfaces alone touch between themselves; but in composite contact various materials are interposed between the surfaces of the rubbed bodies, which materials wear the said bodies the more, the rougher they are. But if the rubbed bodies are of various hardnesses, the softer wears the harder, and this arises from the roughness of the interposed material, which sticks into the material of less hardness and, having fixed itself in it, acts in the manner of a file, with which it then wears the harder material. But if the materials are of equal hardness, then the interposed material grinds itself, if it is less hard than the rubbed things; but if it is harder, it wears such a friction no otherwise than two files of equal power would act upon one another. Which pivot wears more, the thick or the small. That pivot wears more in an equal number of revolutions which is of greater thickness. It is proved by the fourth, where it says: 'That thing wears more of which the friction makes a longer path'. Therefore our proposition is concluded, because a point marked on the pivot of greater circumference makes a longer path than the point marked on the smaller circumference, and consequently wears more in its friction, and so on. Whether the pyramidal pivot wears equally. The pyramidal pivot set upright wears, but with a pyramidal wearing, which pyramid is of the same proportion as is in the pivot itself. It is proved by the third, which says: 'That part of the pyramidal pivot set upright wears more in its revolution which is of greater thickness'. It is proved: let the thickness of the pivot be four at the top and two at the foot. By the fifth of the Elements: 'such proportion is from circumference to circumference as is from diameter to diameter', therefore the greater circumference is double the lesser. I say that the entire revolution of the pivot will not wear equally, but so much more at the top than at the foot as the point marked on the circumference at the top will, in equal time, make a longer and swifter motion than the point marked on the circumference at the foot. Therefore such wearing is not parallel, as the adversary would have it, but pyramidal.