Dividing and Equalising Circles
Comparing circles by removing sectors and portions; restoring a circle's area
The sheet is filled with geometrical constructions comparing two circles by removing matching sectors and portions until equal remainders are found. One marginal problem asks for a quarter-circle, or half a semicircle, to be fitted between two parallel lines. A lower figure shows a circular ring with an inscribed circle and square and argues that an emptied circle can be restored to its original area with the help of two smaller portions equal to one larger, forming two triangles. Circles, squares, lunes and a spiral inscribed in a square are drawn across the page.
On this page
Equalising two circles by removing sectors and portions
Two circles said to be equal are reduced step by step: a circle with its portions is taken on the right and a sector on the left, then the portions p c and the part b are removed, leaving f a on each side. A further variant removes the circle and sector plus three portions from each side, leaving a b.
A quarter-circle between two parallel lines
A challenge problem asks to give one quarter of a circle set between two parallel lines. It is restated as giving half of a semicircle between those same parallel lines.
Restoring a circle's area with two lesser portions
An emptied circle is restored to its first quantity by returning its parts with the help of two smaller portions that together equal one of the greater. Leonardo notes that these make two triangles.
Spiral inscribed within a square
A spiral is drawn coiling inside a square at the upper right of the sheet, among the other circle-and-square studies. It belongs to the same family of constructions relating curved and straight figures.