On teeth and toothed wheels: the square-tooth construction
Making teeth as tall as they are wide, with round faces and round pinion staves
Under "On teeth and toothed wheels," Leonardo gives the practitioners' rule for making a tooth appear square: draw a line between the depths of two adjacent notches, then raise a perpendicular of equal length at its middle to set the tooth's height, so that line n m equals line b d (text 5). The tooth's face should be rounded (as shown in a, b, c) with straight flanks, while the pinion's staves should be as round as possible (f c), which is why the iron-rod spindles of mill pinions work well. He declines to give one general rule for the tooth-spaces because their intervals vary with every ratio of wheel-teeth to pinion-teeth, and insists the pinion staves seat exactly in the notch-bottoms, meeting over a half-circle so the wheels cannot wobble. Small right-hand diagrams show the square-tooth construction and teeth ranged on a curved rim, labelled f c and b, a c, n d m, with a note on the "disparity of tooth intervals" (text 4).
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The square-tooth construction: n m equals b d
The tooth is made to look square by drawing a line between the bottoms of two adjacent notches and raising, at its middle, a perpendicular of equal length; this perpendicular is the tooth's height, so that line n m is as long as line b d.
Rounded tooth faces and round pinion staves
The faces of the teeth must be round, as shown in a b c, with their flanks exactly straight; the pinion's staves must be as round as possible, as in f c. For this reason the iron-rod spindles of mill pinions are praiseworthy, since the wheel-tooth finds their space wide at the bottom and turns better.
No single rule for tooth-spaces; exact seating in the notches
Leonardo gives no general rule for the tooth-spaces, since their intervals vary with every ratio of wheel-teeth to pinion-teeth. The pinion staves must imprint themselves exactly in the notch-bottoms, as if cast one into the other, meeting over a half-circle so the wheels cannot wobble; the notch mouth is made wide enough to be measured by the length of the emerging tooth.