Two Mean Proportionals and Cube Roots
Similar triangles from a cut pyramid, means between 1 and 8, and the cube roots of 100 and 101
Leonardo attacks the doubling-of-the-cube problem, seeking two mean proportionals between 1 and 8 by way of similar triangles cut from a pyramid parallel to its base, where the greater side about the right angle is always double the lesser. He then tests the idea numerically with cube roots, noting that 100 is the cube root of a million and 101 the cube root of 1,030,301. The dense mirror-script filling the upper half of the sheet is not included in this transcription; the holding institution's index also associates this leaf with studies of compound shadow (light and dark).
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Two mean proportionals between 1 and 8
He seeks the 2 mean proportionals between f g, which is 1, and f e, which is 8; they are found in g c, which is 2, and a e, which is 4. The proportion of 2 to 1 is the same as of a c, 4, to c f, 8.
Similar triangles from a pyramid cut parallel to its base
There remain 2 pyramids a c f and f c g which, being cut equidistant from the third side, keep the same proportion of sides as the greater triangle, whose greater side about the right angle is double the lesser. Triangle a c f has base c f double the side a c, and triangle f g c has side g f double f g.
Cube roots of 100 and 101 tested
One hundred is the cube root of a million, and a hundred and one is the cube root of a million and 303 thousand and one. Placing one above 100 made a cube root greater by three hundred thirty-one thousand and one.
Untranscribed shadow notes on the sheet
The upper half of the leaf is filled with mirror-script prose not included in this transcription. The holding institution's scholarly index links this sheet with studies of compound shadow, made of light and dark, alongside the geometry.