Quadrature of Circles: Doubling Circles and Squares

Compass construction for circles and squares infinitely double

To make squares and circles infinitely double one another, Leonardo first builds the right-angled triangle a b c by halving line a b at n, then draws the semicircle a c b with the compass. Opening the compass to successive spans (a b, then d c, then q c) he draws the curves p d q and o S t, and continues the construction to infinity.

Equating a curved annular strip to a truncated sector

The circular parallel (annular strip) p is treated as whole, its removed half-portion o compensated by the portion n. Leonardo takes a sector equal in value to the strip, removes a portion equal to half-portion m, and matches the truncated rectilinear sector to the remainder of the curvilinear strip p S t.

Birds rising in spirals

Beside a small drawing of birds climbing in a spiral, Leonardo notes simply that the bird 'makes itself light.' The remark links the geometric quadrature exercises on the sheet to his continuing study of how birds gain height in the air.

Arithmetical calculations among the figures

Short columns of figures are worked among the diagrams, including a multiplication set out as 12 by 12 with partial products, and a separate operation on 360, 45 and 88. These support the proportional reasoning about eightfold and sixteenfold circles elsewhere on the sheet.

Metà destra. Due cerchi in un cerchio quadruplo; due quadrati in due cerchi concentrici:

Settore e corona circolare:

Di fianco a due cerchi con:

Quadrante di cerchio diviso da archi quadrati, con:

A destra. Uccelli che s'alzano a spirali:

Operazioni aritmetiche e numeri:

Metà sinistra. Corona circolare con:

Corona circolare con due ottagoni iscritti:

Corona circolare con porzioni e bisangoli, con:

Doppio ottagono in corona circolare, con

Figura come la precedente, con:

In basso:

A sinistra. Corona circolare con grande settore diviso da archi e porzioni, con:

Operazione aritmetica:

Girato il foglio sul lato sinistro. Corona circolare modificata dallo spostamento di porzioni, con:

Right half. Two circles within a quadruple circle; two squares within two concentric circles:

Sector and circular annulus:

Beside two circles, with:

Quadrant of a circle divided by squared arcs, with:

On the right. Birds rising in spirals:

Arithmetical operations and numbers:

Left half. Circular annulus with:

Circular annulus with two inscribed octagons:

Circular annulus with portions and lens-shapes, with:

Double octagon in a circular annulus, with:

Figure like the preceding, with:

Below:

On the left. Circular annulus with a large sector divided by arcs and portions, with:

Arithmetical operation:

Sheet turned on its left side. Circular annulus modified by the displacement of portions, with:

Quadrature of Circles: Doubling Circles and Squares

On this page

Compass construction for circles and squares infinitely double

Equating a curved annular strip to a truncated sector

Birds rising in spirals

Arithmetical calculations among the figures