Quadrature studies and a note on Leonardo's treatise

Matching a curved area to a rectilinear one across two circles

Because the greater circle is worth four of the lesser, the four portions of the lesser equal one of the greater; removing two portions from the greater and eight lesser portions leaves the surface 'a b' equal to the surface 'c d'. The argument is the recurring exchange of portions that lets a curved figure be squared.

Leonardo's treatise: 113 books and 33 ways of squaring the circle

Leonardo writes that his last work, of one hundred and 13 books, sets out 33 varied ways of giving rectilinear squares equal to circles. The whole rests on a single first proposition of motion: the thing that moves acquires as much space as it leaves, the motions being either rectilinear or circular.

A lathe with a constant-power counterweight

Amid the geometry is a lathe driven by a counterweight that takes up little room and is 'always of equal power'. Its pulley-block wheel is specified with a steel pivot turning in bronze.

A worked multiplication

A small arithmetical operation multiplies 20 by 17, laid out as partial products 140 and 20 summing to 340. It reads as a computation supporting one of the proportional arguments on the sheet.

Classifying flexuous (wavy) squarable surfaces

Leonardo distinguishes the 'flexuous' or wavy square surfaces: some are squarable with parallel equal sides, some with unequal sides, and some with equal sides of similar inequality, where the curvature throws itself now to one side and now to the other. It is a small taxonomy of the curved shapes he is trying to square.

Ruotato il foglio di 90° verso sinistra. Metà destra [b]. Lungo il margine superiore, primo e secondo cerchio da destra - biffato - con:

Quarto cerchio:

Ultimi due cerchi:

Dalla destra, cerchio con:

Cerchio con:

Tra le due precedenti, settore di cerchio con:

Sul margine, tra le figure:

Cerchio, semicorona e porzioni staccate:

Semicorona circolare con:

Sotto i precedenti, due cerchi entro un cerchio, con:

A sinistra del precedente:

Piccola lunula quadrata con:

Figura flessuosa quadrata con:

Figura flessuosa con:

Lungo la piega mediana. Due semicerchi biffati con:

Sotto il precedente:

Porzione divisa in due metà, con:

Figure di porzioni:

Operazione aritmetica:

Metà sinistra [a]. Angolo superiore sinistro. Triangolo iscritto e circoscritto:

Quadrato iscritto e circoscritto:

Figura flessuosa entro un rettangolo, con:

Figura analoga, con:

Lungo la piega mediana. Figura flessuosa con:

Figura di tornio:

Cerchio, due semicerchi entro un quadrato, con:

The sheet rotated 90 degrees to the left. Right half [b]. Along the upper margin, first and second circle from the right, struck through, with:

Fourth circle:

Last two circles:

From the right, a circle with:

Circle with:

Between the two preceding, a sector of a circle with:

On the margin, among the figures:

Circle, half-ring and detached portions:

Circular half-ring with:

Below the preceding, two circles within a circle, with:

To the left of the preceding:

Small square lunule with:

Square flexuous figure with:

Flexuous figure with:

Along the middle fold. Two semicircles struck through, with:

Below the preceding:

Portion divided into two halves, with:

Figures of portions:

Arithmetical operation:

Left half [a]. Upper left corner. Triangle inscribed and circumscribed:

Square inscribed and circumscribed:

Flexuous figure within a rectangle, with:

Analogous figure, with:

Along the middle fold. Flexuous figure with:

Lathe figure:

Circle, two semicircles within a square, with:

On this page

Matching a curved area to a rectilinear one across two circles

Leonardo's treatise: 113 books and 33 ways of squaring the circle

A lathe with a constant-power counterweight

A worked multiplication

Classifying flexuous (wavy) squarable surfaces