Equating a Curvilinear Right Angle, and a Retractable Bridge
A geometric proof that a curvilinear angle can equal a rectilinear one, with a rack-and-pinion drawbridge
The upper half of the folio pursues a geometry problem: a small figure at the top right (a right triangle with an inscribed circular arc) accompanies a proof that a right angle a can be treated as both rectilinear and curvilinear, by cutting the portion a c t from one side and transposing it to a p b. Leonardo argues that a curvilinear angle can be made equal to a rectilinear one, and its third part likewise divided, 'by taking away and giving back.' The lower half turns to a machine: two long horizontal sketches show a bridge concealed in the thickness of a wall, running on rollers and driven in and out by a toothed rack and pinion.
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A right angle proved both rectilinear and curvilinear
Angle a, formed by straight lines c a and b a, is treated as a right angle whether measured by straight lines or by curves. Leonardo removes the portion a c t from side a c and lays it on side a b as a p b, so that base b c and the half f l of the rectilinear triangle match the half p o of the curvilinear one. Returning what was taken leaves the surface 'in the first degree', so the two angles are equal.
Dividing a curvilinear angle into thirds
Every curvilinear angle can be set equal to a rectilinear angle, and a third of one to a third of the other. If you can give a third of the rectilinear angle, you have thereby given a third of the curvilinear angle, 'that is, by taking away and giving back.'
A rack-and-pinion bridge hidden in the wall
The bridge is built concealed within the thickness of the walls; rollers fixed in the masonry let it slide in and out, driven by a pinion (rocchetto) engaging teeth cut along the middle of its whole length, as the drawing shows. In front it rests one end on a box floated like a boat, while at the rear it is thrust forward like a winch.