Geometry: Sinuous-Sided Triangles Equal to Straight Ones
Quarter-circles in double ratio, pivoting sectors, and nested lunes and 'pyramids'
Folio 56r, again headed 'Geometry', develops Leonardo's method for equating curvilinear and rectilinear areas. A quarter-circle figure lettered a, b, c, d compares the quarters of two circles in double ratio, while further diagrams show how the two sectors of a circle, pivoted at the centre, can be pressed together so that a 'sinuous-sided' triangle opens up equal in area to a rectilinear triangle a b c. Related sketches extend the idea to a matching pair of figures a m o and p n q and to nested lunes with stacked triangular 'pyramids'. Two of the marginal figures are drawn without any accompanying text.
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Quarter-circles of two circles in double ratio
Of the quarter parts of two circles double one to the other on the same centre, the excess of the greater equals a quarter of the smaller. Removing an equal part from each leaves equal remainders, so that d c are of equal value, and likewise a b.
Sinuous-sided triangles equal to rectilinear ones
Pivoting the two sectors of one circle about its centre produces triangles with sinuous sides equal to straight-sided triangles. As the lips a b c are drawn into continuous contact, a triangle with sinuous sides opens on the opposite side, equal to the rectilinear triangle a b c.
A matching pair of straight and sinuous triangles
The two figures marked a b are of the same nature as the ones above, with a common rectilinear base on each surface. The rectilinear triangle a m o is declared equal in area to the sinuous-sided triangle p n q.
Nested lunes and stacked pyramids
Of all the lunes similar to the first portion, one is to count the number and set below just as many triangular 'pyramids' stripped of their portions. This imagines the portions removed just as they were from the greatest lune above.