Circles and Squares of Double Capacity
Overlaid circles, semicircles, a hexagon and inscribed squares compared by area
The sheet is crowded with overlaid geometric constructions — circles, semicircles, quadrants, squares inscribed and circumscribed, and a hexagon — used to compare the areas ('capacity') of figures. Leonardo repeatedly finds that one circle is of 'double capacity' to another: the circle through a square's angles is double the circle touching its sides, and he extends the doubling through a series of quarters, eighths, sixteenths and thirty-seconds. Scattered numbers and a fragmentary note 'to the cloth of the model' also appear. Faint additional constructions fill the margins.
On this page
Inscribed and circumscribed circles of double capacity
For a square, one circle touches the four sides and another passes through the four angles. Leonardo states that these two circles are of double capacity (double area) one to the other.
Circles about a hexagon and a square
If one and the same line serves as a side of the square and a side of the hexagon, the circles through their angles are again of double capacity one to the other. The relation is carried on through quarter and eighth, eighth and sixteenth, sixteenth and thirty-second.
Diameter equal to a square's side
Leonardo asks the reader to 'prove this': if the diameter of a circle equals one side of a square, then the circle that touches the angles of that square is double the first circle. The corresponding figure follows on the lower margin.
Biangular figures n m e and m o
Beside overlaid semicircles labelled o, c, d, b, a, m, n Leonardo notes that a b e is equal to d, and that the two biangular (lune-like) figures n m e and m o are double one to the other.