Of teeth and of toothed wheels and of the pinion.

Always the tooth that drives stands behind the tooth that is driven, from the beginning of contact to the end. But if they meet touching point to bottom before they have come opposite the centers of their wheels, then the motion of such wheels will have no effect. Again, if such teeth at the start of their contact meet point to point, at once the motion will cease and can have no effect. When the teeth of wheels are finer, the bottoms of their hollows are farther from the pivot of their wheels; and for this their contact will last little and will be good. And if such teeth are large, they will be deeper than the small ones, and every effect of theirs will be contrary to the small; it is true that they are more durable. When the teeth are finer, make their wheel thicker, of iron, so that, the contact being wider, they wear the less. The teeth of the wheels and of their pinions should be of equal thickness among themselves, but the spaces should be varied, inasmuch as the one that is driver of the other should have its teeth somewhat more open than the one that is driven, especially when between the wheels and pinions there is great difference; but if the difference is small, make little variety in those spaces; and if the wheels are equal, make the tooth-spaces of the one and the other equal. And if the driving tooth enters the slot of the driven wheel to a greater depth than the space that is found between the points of the teeth of that slot, then that tooth cannot come out of the slot and the motion of the wheels will at once cease. The teeth in contact wear the more on the pinion than on the wheel, by as much as those of the wheel are more in number than those of the pinion; for if the wheel has 60 teeth and the pinion 6, in one whole turn of the wheel each of the pinion's teeth is touched ten times, while those of the wheel are touched but once by the pinion; so that the wearing of the teeth of the wheels will hold the same proportion between them as the numbers of the teeth of those wheels. Wherefore for this a harder temper should be given to the wheel that has fewer teeth. Some have made the wheels of brass and the pinions, or little pinions, of tempered steel. (The pivots of the wheels thrust and wear their hole in lines opposite, parting from the place of contact of their teeth.) Wherefore into such a hole should be set a ring of tempered steel. The little pinion does not turn easily of itself in its wheel, but with respect to the wheel it is easy; yet it is dangerous because with few teeth it touches its wheel.