Squaring the Circle: Lunes, Sectors and Perforated Circles

A circle squared through eight bi-angles

The figure m n – a – p o carries eight bi-angles that together equal the largest square of the largest circle. Half of them fill the four lunes n m o p so the perforated circle is reduced to a square, while an equal square is left outside in the four triangles. Leonardo concludes that half of the outer square equals the whole perforated circle.

Curved figures equal to their triangle or square

Repeated captions assert that a concentric semicircle equals the largest triangle of the largest semicircle, and that a perforated circle equals the largest square of that circle. The same reasoning is applied to right-angled sectors, which are said to equal either a triangle or a square.

Straight rivers and their branchings

A marginal note headed 'Water – a b' states that straight rivers are those composed of their own branchings, and that such straight rivers are counted by their ramifications. These are called fontanili, or 'rozze', as shown in the accompanying figure ab.

Semicircular arch and vault studies

At the lower right of the sheet are several small drawings of semicircular arch segments filled with curved tracery, of the centering-and-truss type. They are drawn but not transcribed in the bundle.

Sketch of a resting animal

In the lower right quarter a small pen study depicts an animal with a rounded body and prominent ear. The drawing is present on the sheet but is not included among the transcribed texts.

Metà inferiore, prima colonna, dalla destra. Figura di semicerchio concentrico:

Figura sotto la precedente:

Seconda colonna. Cerchio iscritto in un quadrato:

Semicerchi concentrici:

Terza colonna. Figura di cerchio:

Figura sotto la precedente:

Quarta colonna. Quattro cerchi e quattro quadrati entro un cerchio:

Settore di cerchio:

Altro settore di cerchio:

Metà superiore, girato il foglio di 90°. Presso la piega mediana, due figure biffate:

Due figure biffate sotto le precedenti:

In alto a sinistra, due figure biffate:

Sotto le precedenti, figura unita con tratto di penna * con altra a sinistra:

Cerchio iscritto in un quadrato:

Sotto la precedente, figura con:

A sinistra, cerchi concentrici e quadrati iscritti, con:

A destra due figure con:

Lower half, first column, from the right. Figure of a concentric semicircle:

Figure below the preceding one:

Second column. A circle inscribed in a square:

Concentric semicircles:

Third column. Figure of a circle:

Figure below the preceding one:

Fourth column. Four circles and four squares within a circle:

Sector of a circle:

Another sector of a circle:

Upper half, the sheet turned by 90°. Near the central fold, two crossed-out figures:

Two crossed-out figures below the preceding ones:

Top left, two crossed-out figures:

Below the preceding, a figure joined by a pen-stroke to another on the left:

A circle inscribed in a square:

Below the preceding, a figure with:

On the left, concentric circles and inscribed squares, with:

On the right, two figures with:

On this page

A circle squared through eight bi-angles

Curved figures equal to their triangle or square

Straight rivers and their branchings

Semicircular arch and vault studies

Sketch of a resting animal