Quadrature of circles and lunes
Squaring curved figures by transmutation, with proportion arithmetic in the margin
This geometry sheet works on the quadrature of curved figures, turning circles, semicircles and their lune-like portions into equal squares. Circles inscribing squares, triangles and hexagons, with hatched crescent portions, fill the page, accompanied by the axiom that removing equal parts from equal wholes leaves equal remainders. In the left margin are two short proportion problems, finding the sixth of 36 and the quarter of 18.
On this page
A surface with an equal square is itself quadrable
Leonardo states that a surface to which an equal square can be assigned is itself again quadrable, as demonstrated in the figures below the note. The note sits in the upper corner of the inverted sheet.
Removing similar portions from two semicircles
Working with two semicircles, one double the other, and labelled points f, c, d, a, e, m, b, he removes portions each an eighth of the circumference and redistributes them, so that when c is put at f the parallel f e is continued and d f is made equal to the triangle m. Removing similar parts, he argues, leaves the remainder in the original proportion.
Axiom of equal remainders
If from two equal things an equal part is removed, the remainder stays equal. The rule is written inside the larger figure.
Finding the sixth of 36 and the quarter of 18
Two short problems in the margin: give the sixth of 36 (6 times 6 makes 36), and give the quarter of 18 (16 and 2/4; 4 and 1/2).