Angles inscribed in a circle; squaring curved-sided pyramids
The smaller the arc, the thicker the angle — an inverse proportion; building falcate figures from circle fractions
A largely red-chalk page reasoning about the size of an angle inscribed in a portion of a circle. Leonardo argues that the right angle, standing on the diameter, is of middling stature between infinitely growing obtuse angles and infinitely diminishing acute ones, and that an angle is the thicker the smaller the arc it stands in. He concludes that the proportion between larger and smaller angles equals that of the circular arcs in which they are formed, but taken inversely. A heading then opens the quadrature of falcate pyramids of infinitely varied curvature, built by superimposing circle fractions such as a half plus a quarter, a half plus an eighth, and so on.
On this page
The angle inscribed in a portion of a circle
An angle made in the portion of a circle is caused by the half portion, taking the diameter as base, and is like the right angle of half stature among infinite obtuse and infinite acute angles. An angle is the thicker the smaller the portion of the circle in which it is set, and the narrower the greater the portion from which it derives.
Inverse proportion of angle to arc
The proportion found between the greater and lesser angles is the same as that of the portions of the circles in which they are created, but it is an inverse proportion.
Building falcate figures from superimposed circle fractions
Take a half circle and a quarter, double the first; a half and an eighth, quadruple the first; a half and a sixteenth, octuple the first, superimposing them in turn. Alternatively combine a quarter with an eighth, a sixteenth or a thirty-second against the doubled, quadrupled or sixteenfold figure.