Geometry of angles inscribed in circular segments
Similar segments, a line divided at a, c, b and its rectangle, with a column of figures
This geometry page argues about angles inscribed in circular segments: Leonardo holds that similar inscribed angles stand on equal portions of a circle and conversely, while warning that mere similarity does not prove equality. A line is divided at points a, c, b, its part c b is taken, and the rectangle contained by the parts is considered; a segment on the chord o-a-n is imagined folded upward to compare its size. A column of numbers (48, 12, 8 . 4, 144, 16, 160) accompanies the constructions.
On this page
Similar inscribed angles stand on equal segments
Two angles are inscribed in two circular segments to test the claim. Leonardo states that if the angles are similar, the portions of the circle in which they are made are equal, and conversely.
Similarity does not prove equality
A cautionary maxim heads the left column: similarity does not argue equality. A further note on two inscribed angles adds that the inner (intrinsic) angle is greater than any inner one.
Line divided at a, c, b and its rectangle
A straight line is divided into segments marked a, c, b, and the part c b is singled out. The rectangle contained by the preceding lines is then constructed, labelled c b - a.
Folding a segment to compare portions
An analogous figure of circular segments on a single chord is lettered o, a, n. Leonardo asks the reader to imagine the portion n turned upward at a and, in that position, to prove that it is smaller than n, o.
Column of figures at right
At the right stands a worked column of numbers: 48 and 12, 48 and 8 . 4 giving 144, 64 and 16, totalling 160 against 160. The figures accompany the geometrical reasoning on the sheet.