Squaring the Circle: Lunes and Circle-Portions

Lunes and portions worth the greatest inscribable square

Removing the four greatest portions of the double circle equals removing the eight greatest of the sub-double circle. Once the four bi-angles are taken away, worth the sixteen portions into which the four greatest portions resolve, the remainder equals the largest square that can be drawn from the greater circle.

The cross a b c d and the leftover square

The cross a b c d together with the circular ring is worth the greatest square drawable from the larger circle, because 32 portions taken from the middle circle equal the four greatest portions of the larger one. Whatever of the larger circle is not occupied by those 32 portions is the square that remains once its four greatest portions are removed.

Sector equal to a semicircle: lune a equals portion b

With the sector a c set equal to the semicircle c b and their common contact at c, the lune a is proven equal to the portion b. Removing equal parts (c) from equals leaves equals, so two equal and similar surfaces are superimposed exactly one on the other.

Portions of circles in quadruple and sub-double proportion

Because the two circles stand double one to the other, the sixteen portions of the lesser (sub-quadruple) circle are worth the four greatest portions of the greater. The relations are governed throughout by simple ratios — double, quadruple and their halves — between the areas of the nested circles.

The helix as a squarable curved surface

Let there be a surface of uniformly non-uniform curvature equal to a rectilinear square. Leonardo states that such a surface is found in the helix, one of the curved figures he treats as reducible to a square.

Squarable remainder by the 'known minus known' rule

The ring with the diamond n is squarable, and the field remains squarable by the rule that if from a known quantity you remove a known part the remainder is known. The whole square being known and the ring-with-triangle n being known, the remainder is known and therefore squarable.

Prima colonna. Cerchio con quadrato:

Terza figura successiva:

Cerchio colle lettere:

Doppio cerchio colle lettere:

Seconda colonna. Doppio cerchio con quattro bisangoli:

Settore soprapposto a semicerchio, con:

Al centro del margine superiore:

Figura di elica:

Terza colonna. Doppio cerchio con otto bisangoli:

Altro cerchio con otto bisangoli:

Cerchio iscritto nel quadrato:

Figura successiva:

Figura successiva:

Cerchio con bisangoli assottigliati:

Doppio cerchio con 8 bisangoli:

Quarta colonna. Cerchio iscritto colle lettere:

Figura sottostante:

Cerchio colla lettera:

Altro cerchio colla lettera:

Cerchio iscritto nel quadrato, colla lettera:

Doppio cerchio con 8 bisangoli:

Corona circolare con 8 bisangoli:

Doppio cerchio iscritto, depennato:

Quinta colonna. Doppio cerchio con quadrato e le lettere:

Figura a sinistra:

Quadrato quadripartito con cerchi:

Quadrato contenente doppio cerchio con appendice triangolare:

First column. Circle with a square:

Third figure in sequence:

Circle with the letters:

Double circle with the letters:

Second column. Double circle with four bi-angles:

Sector laid over a semicircle, with:

At the centre of the upper margin:

Figure of a helix:

Third column. Double circle with eight bi-angles:

Another circle with eight bi-angles:

Circle inscribed in the square:

Next figure:

Next figure:

Circle with thinned bi-angles:

Double circle with 8 bi-angles:

Fourth column. Inscribed circle with the letters:

Figure below:

Circle with the letter:

Another circle with the letter:

Circle inscribed in the square, with the letter:

Double circle with 8 bi-angles:

Circular ring with 8 bi-angles:

Double inscribed circle, struck through:

Fifth column. Double circle with a square and the letters:

Figure on the left:

Square divided in four with circles:

Square containing a double circle with a triangular appendage: