Book of Equivalence: Squaring Circular Sectors and Portions

Sectors of equal value and their diminishing portions

Three small circles compare portions a b, c d and e f: in the first the portion carries away more than half the sector, in the second much less, in the third the excess is known because it is a square. Making the sectors always of equal value, one can see how the portions diminish, and their diminution is "squarable" because the square remains in the rectilinear triangle left when the portion is removed.

Four equivalent surfaces: Half, Quarter, Eighth, 1/16

Along the upper margin four figures are labelled Half - a, Quarter - b, Eighth, and 1/16. Leonardo notes that all of these are of one and the same value.

Half-octagon inscribed in the circle

Removing from the semicircle a p o its four portions leaves a square, which he draws from the equal quarter-circle below, so that the remainder of that quarter equals the four portions; the same is done in the eighth, and so on to infinity. This is his method for giving the value of one portion in different figures that never depart from a single curved side; the next book will treat the lune (falcate) between two curved lines.

Doubling of successive circles

In the "judgment triangle" at lower right he states the scaling rule: if the second circle is set double the first, all the succeeding circles will be double one to another.

Ratios along the circumference (the braccio of twelve ounces)

From the third to the fourth of a circle's circumference is a third of a quarter, illustrated on a braccio divided into twelve ounces. Better stated, from the fourth to the fifth is a fifth of a quarter, as in 20, whose quarter is 5 and whose fifth is 4, echoed in the column 20 / 5 / 4 near the lower margin.

In alto a destra:

Piccola figura di cerchio con:

Sotto la precedente, piccolo cerchio con:

Tratto di penna che conduce ad altro piccolo cerchio con:

In basso a destra, triangolo giudiziale:

Lungo il margine superiore. Quattro superfici equivalenti, con:

Mezzo ottagono iscritto nel cerchio, colle lettere:

Tre figure di porzioni colle lettere:

Due porzioni con:

A sinistra:

Grande settore di cerchio colle lettere:

Cerchio con quadrato e triangolo iscritti, colla porzione:

Presso il margine inferiore:

Top right:

Small circle figure, with:

Below the previous, a small circle with:

A pen-stroke leading to another small circle, with:

Lower right, the judgment triangle:

Along the upper margin; four equivalent surfaces, with:

Half-octagon inscribed in the circle, with the letters:

Three figures of portions, with the letters:

Two portions, with:

On the left:

Large circle sector, with the letters:

Circle with inscribed square and triangle, with the portion:

Near the lower margin: