Dividing wheels into teeth and sizing wheels to pinions
Tooth counts, compass methods, and the half-tooth correction
This recto sets out how to lay out gear teeth by dividing a circle into equal parts and how to match a wheel to its pinion (carello), with two small figures at the right showing hatched tooth-arcs and their diameters. Leonardo poses problems in proportion, for instance sizing to 36 teeth a wheel whose six-tooth pinion is a sixth of an inch across, and describes the compass (sesto) method masters use, measuring across three teeth to set the pinion's diameter. He explains the half-tooth correction that must be added or subtracted according to whether the wheel drives the pinion or the pinion drives the wheel, and notes that straight-toothed racks like jack-scales follow no such rule because they move both ways. The figures are lettered a, n and n, m.
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Dividing a circle to lay out gear teeth
A general rule is sought whereby a given circle can be divided, with a given line, into as many parts as one wishes. Leonardo poses the case of a six-tooth pinion a sixth of an inch across and asks what diameter to give a wheel of 36 teeth, and inversely how to enlarge or shrink a 20-tooth wheel to 21 or 19 teeth of the same size.
Scaling a wheel's diameter to its number of teeth
The problems ask, for a general rule, how much the diameter must grow or shrink so that a wheel keeps teeth of a fixed size while changing their count. He works from a known six-tooth pinion toward a 36-tooth wheel, and from a 20-tooth wheel toward one of 21 or 19 equal teeth.
The masters' compass method and the half-tooth correction
If the wheel is driven by its pinion, take with the compass the span of three teeth at their root and let that be the pinion's diameter; if the pinion is driven by the wheel, close the compass by half a tooth. This half-tooth is added or subtracted because the driving tooth leans to one side of the slot it drives and away from the other; straight racks like jack-scales keep no such rule since they run both ways.
Pinion diameter a–n for six teeth
In the figure, a n is the diameter of a pinion of 6 teeth when the wheel is driven by the pinion. But if the pinion is driven by the wheel, n m is its diameter, that is, diminished by half a tooth; the same is done for a pinion of 8 teeth.