Inserting two mean proportionals; the cube root
The ancients' method of placing two quantities in continuous proportion between two given ones.
This leaf treats the classic problem of inserting two mean proportionals between two given quantities, the key to extracting a cube root, which Leonardo says requires 'the methods of the ancients.' Taking two quantities c and f, he sets a circle a b g d whose diameter a g equals f and a chord g d equal to c, then erects a half-column orthogonally over the semicircle and turns a second semicircle a e g upon it. The passage is continuous but breaks off at both the top and the bottom of the transcription, and the large lettered circle diagram at the lower right carries the construction.
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Two mean proportionals yield the cube root
When between unity and another number two further numbers fall in continuous proportion, the first of them is the cube root of the last, as geometry demonstrates. To work geometrically one must place, between two given quantities, two others so that all four stand in one proportion, following the methods of the ancients.
Construction with circle a b g d and a half-column over the semicircle
For quantities c and f (f the greater), Leonardo draws circle a b g d with diameter a g equal to f and chord g d equal to c. Over the semicircle a d g he raises a half-column orthogonally, its base-diameter falling on a g between points a and g, then turns a semicircle a e g upon the erected plane, e standing straight above the circle's surface.