Roller machine with pyramidal beams for shaping plates
Curved guide-channels, converging pyramids, and Euclid's rule of proportion for the winding beams.
Two large drawings show an apparatus of rollers and belts in which the base of the template slides right and left along two curved channels that receive a pair of teeth. The four pyramidal beams (subbi) must all converge to a single point, and the curvilinear guide-channels must share that same centre. Leonardo warns that each belt must give no more than a full turn on its pyramidal beam, since a second turn doubling upon the first would thicken the beam and spoil the convergence of the pyramid's sides. He proves it from Euclid's Elements: adding equal amounts to unequal quantities changes their ratio, so a sesquialtera proportion (2:3) becomes sesquiquarta (4:5). A pair of small wheels a b turns the same way in both directions, unlike those drawn above.
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Template base guided by two curved channels
The base of the template moves right and left, guided by the concavity of two curved channels that receive two teeth within them. These curvilinear channels must be shaped so their curvature converges to one point, at which the sides of each pyramidal beam also meet.
Four pyramids converging to one point
The angles of the four pyramids must all converge in one and the same point. The curved guide-channels along which the template runs are likewise drawn with a curvature that meets at this single common centre.
Euclid's proportion: why a belt must not double
A belt should give no more than one full turn on its pyramidal beam, for a second turn doubling on the first would thicken it and spoil the ratio of the pyramid's converging sides. By the Elements, adding equal amounts to unequal things changes their ratio: add a binary to 2 and to 3, and the sesquialtera 2:3 becomes the sesquiquarta 4:5.
Two rollers turning one way in both directions
The wheels a b turn in the same direction whether driven to the right or to the left, whereas the wheels drawn above them do the contrary. The template base n m must travel back and forth through the curvilinear channels.