Squaring a rectangle: the gnomon and Euclid's Elements
With a marginal note comparing the body of the earth to a whale that breathes water
A dense geometrical sheet on the quadrature of rectangles, worked out through the figure of the gnomon and the equal parallelograms that stand about the diagonal of a square. Leonardo lays out a full construction for making a square equal to a given right-angled parallelogram a b c d, and justifies each step by citing propositions of the first two books of Euclid's Elements. In the lower right, inside an irregular oval outline, sits an unrelated natural-philosophy aphorism likening the earth to a fish or whale that breathes water instead of air. Faint additional geometric sketches crowd the margins beyond the transcribed blocks.
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The gnomon: parallelograms about the diagonal are equal
The construction builds a gnomon in which the parallelograms lying about the diameter (diagonal) are shown to be equal. The square a equals the parallelogram b together with a small square, the standard relation behind Euclidean quadrature.
Constructing a square equal to the rectangle a b c d
Leonardo cuts the front c d to the rectangle's width, halves the remainder at p, and swings the arc n r c with the compass set to p c. From the intersection n he draws the line n f through the corner c, cutting b f at g, then completes the square m o c e proved equal to the given rectangle by removing matching triangles.
Cited authorities: Elements I.34 and I.43
Marginal notes ground the proof in Euclid: by proposition 34 of Book I the diagonal cuts the square into two equal parts with opposite angles and sides equal, and by proposition 43 the complements about the diagonal are equal.
The earth breathes water like a whale
A brief aphorism set within an oval outline states that the body of the earth is of the nature of a fish, orca or sperm whale, because it breathes water in place of air. It is one of Leonardo's microcosm analogies treating the earth as a living body.