Geometry: turning a square into a rectangle of given length
Proportional division of a quadrilateral by its diagonal, proved by Euclid's fortieth proposition
This folio treats the line that orthogonally divides a quadrilateral into two equal halves, showing how it cuts the lateral and longitudinal parallels into proportional parts that recombine into whole parts. Leonardo then poses the problem of transforming a given square into a rectangle whose length equals a given line, asking by how much it must narrow. He solves it by the fortieth proposition of the first book of Euclid's Elements, extending line b f and building right-angled figures to yield the quadrilateral K i h a equal to the given square a b c d. Three diagrams accompany the argument: a quartered rectangle with its diagonal, an elongated square, and a parallelogram divided by diagonal and parallels.
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The orthogonal divider cuts parallels into proportional parts
The line that orthogonally divides the quadrilateral into two equal parts cuts the lateral and longitudinal parallels into proportional pieces. The first joined to the last recomposes the whole last part, and the second below on one side joined to the second above on the other always remakes the whole part.
Problem: extend a square to a given length
From a given square, make a rectangle whose length equals a given line. Leonardo asks how much the figure will narrow when so extended.
Construction by Euclid's fortieth proposition
Taking square a b c d to be extended along the given line d e, Leonardo prolongs b f equal to d b and adds f e, draws e g tangent to the square's corner to form the right triangle g d e, adds an equal right triangle e i g, and by drawing a K and a h obtains the quadrilateral K i h a equal to the given square, proved by the fortieth of the first of Euclid.