Two mean proportionals and the doubling of the cube
Proportional construction between two lines, with matching geometric figures
The text works through finding the two mean proportionals c g, f a between two given straight lines a b, b c, then reasons about the ratios of similar solids, holding that they stand to one another in the tripled ratio of their homologous sides. This chain leads toward the classic problem of multiplying one solid by another, that is, doubling the cube. The facing leaf (178r) carries the matching diagrams: a quarter-circle construction with two tangent circles, triangles crossed by internal lines, and columns of figures.
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Finding two mean proportionals between two lines
For the two given straight lines a b, b c the two mean proportionals c g, f a are found. Taking lines a and b with a double b, one seeks c and d such that a is to c as c is to d and d is to b.
Similar solids stand in the tripled ratio of their sides
The argument uses that similar solids are to one another in the tripled ratio of their homologous sides. Since a has to b double the ratio it has to c, what is made from a is triple what is made from d.
Diagram: quarter-circle with two tangent circles
At the upper right of 178r a right angle is closed by a quarter-circle arc, with two circles set tangent along a diagonal line. Below it are triangles crossed by cevians and a square with its diagonals, all illustrating the proportional construction.