Compound motion applied to squaring a circular sector
A false rectification is corrected by the double motion of a rolling wheel
Leonardo argues from the motion of cart wheels that circular circumferences can be straightened: a wheel whose thickness equals its semidiameter leaves a track equal to the squaring of its circle, since a moving thing gains as much space as it loses. He first sketches a sector-over-triangle and labels it "This is false," showing why simply pulling the sector's sides down does not rectify the curve. He then gives the correct construction, marked "This is true," in which a compound (double) motion — straightening the curve d c while it descends into c f — squares the sector a b c. A marginal lemma states that two equal plane figures of different shape, superimposed, exceed each other by equal parts.
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The rolling wheel that rectifies its own circumference
From the motion of carts it has always been shown how to straighten the circumferences of circles: a full revolution of a wheel whose thickness equals its semidiameter leaves a track equal to the squaring of its circle, for the thing that moves gains as much space as it loses.
"This is false": pulling the sector sides down (a b, a c to e f, e g)
Drawing the two sides of sector a b and a c down to e f and e g would supposedly straighten curve b d e out to f g and make surface e f g equal to surface a b c d, so the space lost a b c and o n equals the space gained o f d and e n g d. Leonardo marks this attempt false.
"This is true": squaring the sector by double motion (d c into c f)
The correct quadrature of the circle's sector uses compound motion generated as the curve d c moves into d f: besides the motion straightening the curve, a motion from top to bottom is added at the same time, so d c straightens and descends into c f. This squares sector a b c, matching the rectilinear triangle a e f, since the lateral surfaces equal the lower surfaces.