Squaring Curved Triangles from Circles in Double Ratio
A two-arc triangle equated to a square; superposition proof that a equals b
Leonardo squares a triangle a bounded by two curved lines of unequal curvature, showing it exactly equal to the figure b. His proof rests on the arcs being derived from circles in double proportion — one of 4, the other of 8 — so that an eighth of the smaller circle equals a sixteenth of the larger; removing the shared portion o and then relocating parts c, d and e onto triangle b f leaves a equal to b. A second construction calls for a two-arc triangle from circles in quadruple proportion, and a closing note equates a curvilinear "square" of three curved sides and one straight side to the four-straight-sided square b by removing the equal portions m and n. Labelled diagrams (noting that a and b are equal) and a faint pencil sketch marked "done" fill the page.
On this page
A two-arc triangle squared by superposition
The triangle a of two curved lines is squared with the part b, exactly equal to it. Removing the shared portion o from each leaves c d a equal to b f; then, placing part c at site e and laying the triangle d e upon b f at f, both reduce to a and b alone, proving a equal to b.
Arcs from circles in double and quadruple proportion
The curvatures are drawn from circles in double proportion, one of 4 and the other of 8, so that 1/8 of the circle of 4 equals 1/16 of the circle of 8. A further construction asks for a two-arc triangle from circles quadruple one another.
A three-arc curvilinear square equated to a straight-sided square
The closing note treats a as a square of three curved sides and one straight side, equal to b, a square of four straight sides. Removing the equal portions m from the one and n from the other leaves equal remainders.