Squaring the circle: lunes and circle-square proportions

Squaring a figure by removing two lunes

The square nmop is known, and the quadratures of the two lunes are known as well. Removing those two lunes from the square leaves a square equal to the square below it, that is, a square equal to half the square above. It is a single step in the game of reducing lune-and-circle figures to equivalent squares.

Common notion: equals taken from equals

If from equal quantities you take away equal quantities, the remainders stay equal; if from a known quantity you take a known quantity, the remainder stays known. This axiom, written at the head of the sheet, is the engine of every proof below it, where figures are shown equal by subtracting matching parts.

Circle-squares in double and half ratio

Many notes value a figure as double or 'subduple' (half) of the greatest square of a circle. Here ab is squarable into a square half the greatest square of the greatest circle. Leonardo tracks each figure purely through its proportional value rather than by measurement.

The greatest square of the greatest circle

A recurring standard on the sheet is the largest square inscribed in the largest circle: the marked areas a, ab and abc are each declared worth this greatest square. In one figure, taking away four rectangles from it leaves the value of the greatest square of the smallest circle.

Dall'angolo superiore destro scendendo verso il basso.

Figura — collegato colla successiva — di quadrato colle lettere:

Quadrato con due semicerchi:

Cerchio colla lettera:

Doppio cerchio con quadrato e lunole:

Doppio cerchio colle lettere:

Semicerchio colle lettere:

Colonna centrale. Quadrato colla lettera:

Cerchio — connesso con altri della stessa colonna e della successiva — con:

Cerchio connesso a quello adiacente:

Cerchio — connesso con altri — colle lettere:

Penultima colonna. Quadrato con cerchio e quattro lunole:

Cerchio — connesso con altri— colle lettere:

Dall'alto lungo il margine sinistro. Triangolo con lunola:

Cerchio — connesso con altro a destra — e quadrato con:

Cerchio con hisangolo traforato:

Doppio cerchio e doppio quadrato con:

Cerchio — connesso con altro a destra — con:

A destra del precedente e ad esso congiunto, colla lettera:

From the upper right corner, descending.

Figure — linked with the next — of a square, with the letters:

Square with two semicircles:

Circle with the letter:

Double circle with a square and lunes:

Double circle, with the letters:

Semicircle, with the letters:

Central column. Square with the letter:

Circle — connected with others of the same column and the next — with:

Circle connected to the adjacent one:

Circle — connected with others — with the letters:

Penultimate column. Square with a circle and four lunes:

Circle — connected with others — with the letters:

From the top along the left margin. Triangle with a lune:

Circle — connected with another to the right — and square, with:

Circle with a pierced bi-angle:

Double circle and double square, with:

Circle — connected with another to the right — with:

To the right of the preceding and joined to it, with the letter:

On this page

Squaring a figure by removing two lunes

Common notion: equals taken from equals

Circle-squares in double and half ratio

The greatest square of the greatest circle