Constructing the Cube Root by Two Mean Proportionals
A geometric proof placing g t and g h in continuous proportion between c and f
A page filled edge to edge with a single dense geometrical proof (the upper-right corner torn away). Leonardo rotates a semicircle a e g about a fixed point g against the surface of a cylinder, dropping perpendiculars and identifying a chain of similar right-angled triangles that share a common angle at g. From their proportional sides he obtains three continuous proportionals between g m and g a, reducing the problem to inserting two mean proportionals between the given quantities c and f. He concludes that, if c is taken as one, the segment g t is the cube root of the number f.
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A rotating semicircle on a cylinder builds a chain of similar triangles
A straight line a z is raised at right angles on g a and made to meet g d at z; the semicircle a e g is then turned about the fixed point g until its arc meets line g z on the surface of the cylinder at h. Perpendiculars h t and l m are dropped, and because the right triangles g h a, g t h, g l t and g m l all share the common angle at g they are shown to be similar. From this it follows that a g is to g h as h g is to g t, and c g to g l as l g to g m.
Two mean proportionals between c and f give the cube root
The similar triangles yield three continuous proportional quantities between g m and g a, namely g l, g t and g h. Since g d is set equal to c and g a equal to f, the segments g t and g h fall as two continuous proportionals between c and f. Therefore, taking the quantity c as one, g t is the cube root of the number f, which was the thing to be demonstrated.