Screw and windlass: obliquity of the thread and lever proportions
Winding a thread at constant slope reaches the same height as the incline
The upper diagrams treat a thread wound around a cylinder at constant obliquity, arguing that its final end reaches the same height as if the thread were stretched straight along that same slope, and that moving a weight by the screw from m to n costs as much as from m to p along the line p m. A central passage compares the screw with the windlass h: if the lever of the screw d has the same ratio to its counter-lever as the windlass's lever to its own, equal loads are moved by equal powers, and the tooth's friction is measured at the middle of its length, as with a millstone's semidiameter. A lower square diagram (m m, n n, f f, d d, b, c) confirms the winding rule for a thread laid at fixed obliquity.
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Screw compared with the windlass
If the lever of the screw d has the same proportion to its counter-lever as the lever of the windlass h has to its own, then in both cases equal loads are moved by equal powers, with the line c b oblique to the obliquity of the screw a b. The counter-lever of the screw runs from the centre of the screw to the middle of its tooth, and the tooth's friction is reckoned at the middle of its length, as with the semidiameter of a mill's stone.
Equal work to raise a weight by the screw
It costs as much to move the same weight by the screw from m to n as it does from m to p along the line p m. The advance of the load along the thread is governed by the constant obliquity of that thread.
A thread wound at fixed obliquity reaches the incline's height
Winding a thread around the square at observed obliquity is confirmed to be equivalent to drawing it straight at the same slope: wherever the thread ends, it stands at the same height as the corresponding point of the extended incline. Thus S m is as high as m opposite, n as n wound, f as f wound, and likewise d to d and b to b.