Turning curved surfaces into straight; arcs, chords, sagittae

The radius steps six times round the circumference

From a hexagon inscribed in the circle (a-c-b): the compass-opening that describes the circle enters six times into its circumference and twice into the diameter. It follows that the semidiameter, or radius, likewise steps six times round the circumference.

Portions of a circle and biangular surfaces

A portion of a circle is the biangular surface enclosed between an arc and its chord; equal arcs and chords enclose equal biangular surfaces. Such surfaces are of two kinds — bounded by two curved lines, or by a curved and a straight line — joined at their ends.

The sagitta and the equalities of arc, chord and sagitta

The sagitta (versine) is the shortest line reaching from the middle of the arc to the middle of the chord. Leonardo lists the reciprocal equalities: equal sagittae and chords give equal arcs, equal chords and arcs give equal sagittae, and so through every pairing of the three.

Parts shrink as their number grows

If a whole is successively subdivided into equal parts, the parts diminish in size as much as they increase in number: a is the whole, b halves it, c is divided into thirds. Similar parts hold among themselves the same proportion as their similar wholes hold in themselves.

Figura di esagono iscritto nel cerchio con:

Figura di porzione di cerchio:

Due bisangoli colle lettere:

A sinistra:

A destra:

Figure colle lettere:

Figure of a hexagon inscribed in the circle, with:

Figure of a portion of a circle:

Two biangular figures, with the letters:

On the left:

On the right:

Figures with the letters:

Turning curved surfaces into straight; arcs, chords, sagittae

On this page

The radius steps six times round the circumference

Portions of a circle and biangular surfaces

The sagitta and the equalities of arc, chord and sagitta

Parts shrink as their number grows