Teeth and their alignments in meshing wheels
A tooth's centre line must aim at the wheel's centre; matching a wheel's teeth to its pinion
Under the heading "Teeth and their alignments," Leonardo sets out the geometry of gear teeth: the line running through a tooth's length must lie on the straight line born at the middle of its face and ending at the centre of the wheel, and every such line must aim at the wheel's centre (text 1). He adds that the line of motion transmitted from one wheel to another begins at right angles on the centre line joining the two wheel centres (text 5), and warns that anyone wanting motion by toothed wheels must first perfectly match a wheel's toothing to its pinion's, or his labour is wasted (text 6). Two diagrams of paired meshing wheels carry the letters a n and a, r, c, s, f, g, n, m to show how tooth c leaves tooth r as wheel g drives wheel f.
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The tooth's centre line must aim at the wheel's centre
The line passing through the middle of a tooth's length must be part of the straight line born in the middle of its face and ending at the centre of its wheel. All such lines through the middles of the teeth must aim at the centre of their wheel. When contact between the middle of a tooth's face and the middle of the notch that receives it falls on that line running between pinion-centre and wheel-centre, the tooth's flanks touch neither the opposing teeth nor the sides of the notch.
Line of motion begins at right angles on the centre line
The beginning of the line of motion that one wheel imparts to the other must always be born between two right angles upon the central line, that is, the line extending between the centres of the two wheels.
Matching wheel to pinion; tooth c leaving tooth r
You who want motion by toothed wheels must first perfectly learn to match the toothing of the wheel with that of its pinion, or your labour is vain. When wheel g drives wheel f, and their teeth meet on the central line f g, tooth c must just leave tooth r, and below, tooth n leave tooth m when f drives g.
Face and notch meet only at the central line
The face of the tooth must never touch the bottom of its notch until it reaches the site of the central line. The side of the notch will not touch the side of its tooth before it reaches the site of the centric line.