Proportioning Gear Teeth Between a Large and Small Wheel
The number and contact of teeth on meshing wheels, with a pinned drum mechanism
Leonardo lays down a rule for meshing gears: the smaller wheel must never carry the same number of teeth as the larger, but a number proportioned to its size, so that a wheel of double circumference has double the teeth. He shows that the teeth engage correctly only when the wheels' centers lie on the line of equality a b, which passes between one tooth and the next through both centers, and explains how the circles of the outermost teeth (at n) must touch the root of the opposing teeth (at r), while the mid-length tooth circles (at q) meet on that same line. Two labeled wheels of 16 and 8 illustrate the ratio, and the sheet is filled with heavier machine studies including a pinned drum and a lever mechanism.
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Teeth numbered in proportion to wheel size
The smaller wheel must never, and cannot, have a number of teeth equal to the larger, but a number proportioned to the size of the wheel: if the larger wheel has double the circumference of the smaller, it must also have double the teeth. The teeth must be in contact at the site of equality, which is possible only when the wheels' centers lie on line a b, drawn between one tooth and the next through both centers. Two illustrative wheels are marked 16 and 8.
Where the tooth circles touch on the line of equality
The circles of the outermost teeth touch the root of the teeth on the opposing wheel: the outer-tooth circle of the larger wheel at n touches the smaller wheel at the root of its teeth at r, and likewise the larger circle of the small wheel touches the smaller circle of the large wheel. The circles of the mid-length of the teeth, marked q, meet on the line of equality a b.
A pinned drum and lever mechanism
The upper part of the sheet is filled with heavier machine studies, drawn without accompanying transcription: a large drum set with projecting pins and a screw shaft, and below it a lever or trip mechanism engaging a second drum. These belong to the geared apparatus explored on the gearing rule.