Squaring the Circle and Circling the Square

The twin problem: square the circle, circle the square

At the top (with the sheet inverted) Leonardo states the paired tasks: given a circle, make a square equal to that circle; and given a square, make a circle equal to that square.

Two octagons in concentric circles made equal

He has two rectilinear triangles, unequal and irrational, and makes them equal by taking from one and giving to the other, found in a b c and c d e. The figure has quadrature because its 8 half-portions, removed outside, are rendered inside as 8 whole portions of the same value, leaving only the interposed fillings to be equalized among the 8 parts.

The full made equal to the void

He asserts that the "full" can be made equal to the "void" — that from which portions are neither taken nor rendered. Making the base of triangle c d e just wide enough to enter exactly the space a b c yields a six-sided figure, within and without, equal to the circle within and without.

"Good": reducing the pieces into a parallelogram

With the sheet turned to normal, he notes "Good": take d r c as it is and put it into the space b c a, untroubled by the space since that piece carries a portion with it. Then reduce the whole quantity into the parallelogram o b r a, aim its face at the centre of the circle, do the same on the opposite side, and note what fraction of its circle the square is.

Squaring the lune squares the portion

A portion divided into lunes (falcate) closes the reasoning: if he gives the quadrature of the lune o, he gives the quadrature of the portion n, which is worth as much as o.

Marginal sketches of an animal and a rider

At the right edge, beside the geometry, are two small, faint pen sketches not accounted for in the transcription: a rearing horned animal above, and a mounted horse-and-rider below. They read as rapid marginal doodles rather than finished studies.

Foglio capovolto. In alto a sinistra:

Entro un tratto di penna circolare, doppio settore:

Parte di corona circolare colle porzioni:

Due ottagoni iscritti in cerchi concentrici, colle lettere:

Sotto la figura predetta e con piccola figura di fianco, con:

Lungo il margine destro:

Con riferimento alla suddetta piccola figura:

Rigirato il foglio in posizione normale. Lungo il margine sinistro:

Figura colla lettera:

A destra settore e cerchio equivalenti con:

Porzione divisa nelle falcate:

Sheet turned upside-down. Top left:

Within a circular pen-stroke, a double sector:

Part of a circular ring, with the portions:

Two octagons inscribed in concentric circles, with the letters:

Below the aforesaid figure, with a small figure alongside, with:

Along the right margin:

With reference to the aforesaid small figure:

The sheet turned back to normal. Along the left margin:

Figure with the letter:

On the right, an equivalent sector and circle, with:

Portion divided into lunes: