On adjusting the pivots. On providing that the pivots wear equally, when they are set in a straight line. How, for the socket that receives within itself the pivot of the rotating bodies, one must provide against its dilation; and of this it will be treated in the book on friction. Which motion is that which most destroys the equality of the pivots. How the socket receiving the pivot wears more than the pivot itself. Which toothed wheels wear the sockets receiving the pivots more. Whether wheels moved by cords wear the sockets of the pivots more or less than wheels moved by the teeth of their pinions. The thinner pivot wears the socket that receives it more, being of equal weights. The wearing of the socket receiving the pivot is made along an oblique line, on the side toward which its rotary motion goes declining. What materials are those that most resist friction: whether dense with dense, or rare with rare, or rare with dense. In instruments where many wheels move, there can be no permanence of perfection except for a brief span of time; and this arises because the wheels are not equal in weight, nor in pivots, nor in motion, nor in force.

On the pivots. On adjusting the pivots of the wheels. In things which must have rotary motion, certainly the pivots of those rotating bodies, by means of their precise equality, are the cause of the perfection of the said motions; therefore below it will be treated of such equality. Which motion most destroys the mother [socket] of the pivot. There are three positions of pivots, namely horizontal, upright and oblique. But the upright wears the mother more than the horizontal; and this arises by the fourth [proposition] of this, which says: 'of pivots of equal weight, that one wears its female most which touches it least'. Therefore the pivot set upright occupies less space with its face than with its sides, except if the mother that sustains it were very narrow, because this in a brief time would cut the pivot itself. On which side the mother is worn by the pivot. The mother is worn on the side toward which its pivot declines its motion, because such a pivot continually moves toward that side, always removing itself from the opposite side. By what aspect the mother is worn by the friction of its pivot. Oblique is the concavity made in part in the mother by the friction made in it by its pivot. It is proved by the fourth of this which says: 'Every friction made by the revolution of horizontal pivots upon one same site will wear that site with an oblique concavity'. And this is caused by the disposition of its two motions, of which one is by the heavy body that weighs toward the center of the world, the second is by the friction that the pivot makes on that wall, where it most approaches with the part that declines in its aforesaid revolution.

c a b – g h - d

Which revolution of the horizontal pivot is that which makes a perpendicular concavity. The horizontal revolution of the pivot made by means of the cord wound about that pivot, as is seen done by jewel-cutters, will always make the path of its wearing along a line perpendicular to the center of the wheel where the driving cord of that pivot is wound. It is proved by the fifth of this, where it says: 'That thing which is bound in the middle of the opposite ends of the cords that bind it cannot draw near to the ends of those cords'. Therefore the pivot a, which turns by means of the two equal cords b h and c g, cannot wear its mother except along the perpendicular line of the pivot where its wheel is fixed, that is, along the line a d and not along the line a h, because the cord b g pulls it toward g and does not let it make friction toward b. And this you will test with the pivot a cut around in the manner of a file or rasp on a wheel of soft stone.

How the pivot of the pinion, moved by the toothed wheel, wears its mother with an oblique concavity. The horizontal pivot that resides in the pinion moved by the toothed wheel generates a wearing of oblique concavity, as is proved above.