Equal Areas by Cutting and Shifting Parts
A moved part leaves a space equal to the one it gains; surfaces proved equal to squares
Leonardo states a principle of area conservation — the thing that moves gains as much space as it leaves behind — and applies it to a square a b c whose part b c is cut and shifted to the site c f, so the space h l m f left behind equals the space n o m f. Cutting the two "superfluous" pieces a b c d and e f c d by a diametral division yields four mutually equal triangles, of which c d b equals c d f. By the first proposition he proves the surface a b c d e f equal to the square c d g h, since removing the portion c d f from each leaves two equal squares. Several rectangle-and-diagonal diagrams numbered 1a to 5a illustrate the steps.
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A moving part gains as much space as it leaves
Leonardo's opening maxim: the thing that moves gains as much space as it leaves behind. Cutting the part b c from the square a b c and moving it to c f, he claims the space h l m f left behind the moved part equals the space n o m f.
Diametral division into four equal triangles
By the same reasoning, cutting the two superfluous parts a b c d and e f c d along a diameter divides them into four triangles, all equal to one another, of which the two c d b and c d f are equal.
A stepped surface proved equal to a square
By the first proposition the surface a b c d e f is proved equal to the square c d g h: removing the portion c d f from the square and likewise from the surface a b c d leaves two equal squares, and the same holds if the portion c is taken and restored in a.