Proportions of circles: portions, bisangles and squaring
Rose and star figures inscribed in circles used to compare portions of unequal circles
A page densely covered with circles inscribed with hexagons, squares and multi-petalled rose and star patterns, used to study how the curved "portions" of circles compare when the circles stand in double or quadruple proportion. Leonardo repeatedly states equivalences of portions and bisangles (curved two-cornered lunes): four portions of the greater are worth eight of the lesser, six worth twelve, six of the greatest worth twenty-four of the least. A marginal rule adds that comparisons must be made between similar portions, and that the number of portions grows as the proportion grows. The lower text turns to squaring the circle by removing portions of equal value, whether from within or without, ending with the figures 84 and 42.
On this page
Portions of circles in double and quadruple proportion
Leonardo compares the curved portions of circles standing in fixed ratios: of circles in double proportion the 4 portions of the greater are worth the 8 of the lesser, and the 6 worth the 12; of circles in quadruple proportion the 6 portions of the greatest are worth the 24 of the least. The figures at the top and right show these divisions as inscribed hexagons and squares.
Curvilinear stars, bisangles and portions
A circle drawn with seven rose-like stars is said to contain 42 bisangles (curved two-cornered figures) and 84 portions, while a related figure gives 60 portions. Leonardo notes that the proportion from circle to circle is as that from square to square formed by multiplying their diameters into themselves.
Equal-value circles a and b
For the lettered figures c b a, Leonardo argues that circles a and b are of equal value and that the 12 bisangles of circle b are worth the 24 bisangles of circle a. He proves it by noting that the two inner circles are each worth half their containing circle, so the proportion between the bisangles is the same as between the circles.
Rule for calculating portions and squaring the circle
The marginal note insists that portions be reckoned only against similar portions, and that the number of portions must increase as the enclosed circles shrink. The lower passage explains squaring the circle by removing portions of equal value from inside or outside, it not mattering whether the removed quantity is taken around the center or as a lune, so long as its value is equal.