Surfaces of Double and Quadruple Proportion Superimposed
Conceptions on overlapping surfaces and squaring their differences
The verso sets out a series of geometric 'conceptions' on surfaces in proportion, illustrated by lettered rectangles, circular sectors and triangles. Leonardo states that if two surfaces stand in double proportion and the whole smaller is laid over the larger, the part of the larger left outside equals the smaller; and if the smaller only partly overlaps, the uncovered remainder equals the smaller plus the non-overlapping part. Further figures treat circles of quadruple proportion (labelled 'eighth and thirty-second'), showing how the square of a difference can be extracted as a rectangle. A closing note reasons about how much of each quantity is diminished by downward movement and how the missing matter can be calculated.
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Two surfaces of double proportion superimposed
Leonardo's 'conception' states that if two surfaces stand in double proportion and the whole smaller is laid upon the larger, the part of the larger remaining outside the smaller equals the smaller. He illustrates with a double b: if all of b is placed over a, what is left uncovered of a equals b.
Partial overlap of the smaller surface
If the smaller surface only partly overlaps the surface double to it, the part of the smaller that touches the larger equals that same part of the larger. What of the larger is not touched by the smaller equals the smaller, and so much the more as is the part of the smaller that fails to touch the larger.
Squaring the difference of quadruple-proportion figures
For a circle of quadruple proportion (marked 'eighth and thirty-second'), Leonardo takes a b equal to c and treats a as squarable: subtracting the square of b from the square of c leaves a square part of c equal to a. In a companion figure a c equals b d, and subtracting c from d leaves a square part equal to the difference between a and b, drawn out of a as an orthogonal rectangle.