Geometric Figures Composed from Portions of Circles
Studies toward squaring the circle: lunes, sectors and the six-fold division of a circle
The sheet sets out a program for constructing geometric figures around given points out of portions of circles, the crescent-shaped 'bisangoli' (lunes) Leonardo counts and rearranges. He divides a circle into six sectors and argues that the lune-portions enclosed in a sector can be transformed into figures of various shapes and sizes yet remain of one and the same value, working toward a squaring of the circle. The drawn figures include hexagonal flower-of-life patterns, rosettes, and star-and-triangle lattices built of tiny circle-portions, annotated with counts of lunes (24, 30, 84) and small doubling calculations. A closing note observes that no more than six lunes, or twelve circle-portions, can meet at a single point without their twelve sides intersecting.
On this page
A five-part program for figures made of circle-portions
Leonardo outlines successive demonstrations: composing a figure from the six portions of circles; measuring their true magnitude by a square divided into equal and unequal parallel strips; showing that the portions in a sixth of the circle can be arranged, cut and transformed at will yet keep one value; a rule for composing them pleasingly; and finally perforated circles reduced to the squaring of the circle.
Lunes ('bisangoli') counted in the central figures
The central column of figures is captioned with the number of lunes each contains, running 24, 30, 30 and 84. The count expresses how many crescent circle-portions are packed into each hexagonal or star-shaped composition.
The greatest portion equals all portions in the triangle
In the lettered figure m - a n - o, the greatest portion a, belonging to the figure of the smallest portions, is always worth all the portions enclosed within the whole triangle. It is a statement of equal value between one large lune and the many small ones.
No more than six lunes may meet at a point
The composition cannot bring more than 6 lunes, that is 12 portions of a circle, together at one and the same point without the intersection of their 12 sides. This limits how the crescent pieces can be tiled around a center.
Doubling calculations beside the figures
Small arithmetic accompanies the diagrams: 75 doubled is 150, 77 doubled is 154, 26 doubled is 52, 38 doubled is 76. Further figures are simply tagged with the numbers 30, 52, 762 and 154.