How many equal segments fit into a sixth of the circle
Dividing the circle into 'bisangles', counted and multiplied by sixes
Leonardo studies the division of a circle into equal two-angled segments (bisangoli) and counts how many fall within one sixth of the circle, then multiplies to reach the total for the whole circle (arriving at figures such as 132 and 792 portions). The sheet is crowded with small circles carrying inscribed six-point stars, a large six-petalled rosette, a many-faced solid built of triangles, and lens-shaped segment figures, all interspersed with columns of multiplication and division. A recurring rule states that valid portion-numbers must be divisible by six without fractions. The reasoning continues on the verso and the following folios.
On this page
Equal bisangles within a sixth of the circle
The central note announces that the sheet describes the number of equal bisangles (two-angled segments) that fit into the sixth part of a circle. The diameter of the reference circle is labelled a-b. This sets up the counting worked through the arithmetic columns.
Portions of the whole circle counted by sixes
The left column treats 66 as one sixth of the whole circle and 132 as the count per sixth, then multiplies by six to obtain 792 portions in the whole circle. The proof rests on the sixth being the natural unit of division of the circumference.
A portion-number is valid only if divisible by six
Leonardo lays down that if a number cannot be divided by six without fractions it is not a valid count of divided portions: 6 divides only by one, 12 by 2, 18 by 3, 24 by 4, and so on by sixes to infinity. Small circles labelled 6, 12, 18, 24, 30 and 36 illustrate the successive multiples.
Inscribed star-circles, rosette and a triangle-faced solid
The margins carry rows of small circles inscribed with six-point stars and hexagons, a large six-petalled rosette at lower left, a starburst rosette at lower right, and a many-faced solid built of triangles near the centre, together with a lens-shaped bisangle figure. These illustrate the segment-and-sixfold division argued in the text.