Squaring circular portions into triangles
Lunes and quarter-circle sectors reduced to rectilinear figures; double proportion of quadrilaterals
On folio 61r Leonardo pursues the quadrature of curved figures, setting circular portions (lunes) equal to sectors and then to triangles. A boxed diagram at upper right marks two rectangles a and b in 'double proportion,' with a note that among quadrilaterals of equal width the ratio equals that of their lengths. Below, semicircular figures lettered a to g show the portion a b c equated to a quarter-circle sector and reduced, by removing equal portions, to triangles e, f and g. Additional geometric reasoning accompanies each figure.
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Double proportion of equal-width quadrilaterals
A boxed figure at upper right marks two rectangles a and b and labels them of 'double proportion.' The accompanying note states that among quadrilateral surfaces of equal width the same proportion is found as that of their lengths.
Portion a b c set equal to a quarter-circle sector
The portion a b c is set equal to the sector of the quarter-circle d e. Hence e equals a c, and so is double a and double c, from which it follows that b is equal to d.
Reducing circular portions to the surface e f g
Using the earlier result that the two portions a c of the semicircle a b c equal e and that b equals d, Leonardo removes two equal portions to leave the triangles e and f, then adds them to the triangle g to obtain the rectilinear surface e f g.