Similar triangles and the proportion 1, 2, 4, 8
A triangle cut parallel to one side yields a continued proportion whose second term is the cube root of the fourth
Mirror-script text beside a lettered right-triangle diagram argues that a triangle cut by a line parallel to one of its sides separates off a triangle whose sides keep the proportions of the whole. From side b c 10 double a c 5, the cut triangle d f 8 is double a f 4, and b e 2 double d e 1, giving 1 2 4 8 in continued proportion. Leonardo concludes that of four proportional numbers beginning from 1, the second is always the cube root of the fourth, linking the figure to the doubling of the cube.
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Triangle cut parallel to a side divides it proportionally
A triangle a b e is cut by a line parallel to one of its sides, separating a triangle whose sides stay in the same proportion as the whole. Side b c 10 is double a c 5; the cut triangle d f 8 is double its side a f 4, and the cut triangle b e 2 is double d e 1.
Continued proportion 1, 2, 4, 8 and the cube root
The lettered lengths yield 1 2 4 8 in continued proportion. From this it follows that of four proportional numbers beginning from 1, the second is always the cube root of the fourth number.