Transforming rectangles and a theorem on equal triangles
Area-preserving constructions with arcs and tangents, opened by the maxim that a moving body leaves what it gains
Folio 74 recto develops Leonardo's series on transforming rectangles to a chosen height while conserving area. A theorem states that all triangles of equal sides on one base and of equal height are equal to one another, and a demonstration reduces the rectangle a b c d of unknown size to a rectangle of a given height e f using an arc a g, the tangent c o and a perpendicular i K. The long central passage frames one proof with the physical maxim that a thing which moves leaves as much space as it acquires, applied to strips b d e f and n m o p. A closing note gives the converse construction. Small diagrams at upper right, including boxes with serpentine lines, carry further sketching not covered by the transcription.
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Reducing rectangle a b c d to a rectangle of given height e f
The rectangle a b c d of unknown height and width is reduced to a rectangle at an arbitrary given height e f. Leonardo extends e f to e n, draws the arc a g equidistant to angle c and the tangent c o to base c d, then the line c h to the arc's intersection with e n. A perpendicular i K equal to the base is placed between c h and c o, and the perpendicular K L cuts off the part e c L K equal to a b c d.
Theorem: equal triangles on the same base and height
All triangles of equal and similar sides, set upon one and the same base and with equal height, are equal among themselves. A short note adds that it suffices to make a figure similar to o n p by making o p equal to r m.
A moving body leaves as much space as it gains
The demonstration opens with the principle that the thing which moves leaves as much space as it acquires: cutting the surface a b c d at n o and drawing the front b d back to e f occupies space b d e f while leaving space n m o p, exactly equal to it. The equality is justified by removing equal parts from two equal quantities.