De ludo geometrico: squaring curved surfaces

The square is the goal of all geometry

Leonardo writes that the square is the end of all the labour of geometrical surfaces, and that every surface, whether bounded by curved or straight lines, aspires to its own squaring. Because curved-sided surfaces are little known, he has toiled with a new science and various rules to reduce infinite varieties of them to their quadrature, which he calls the goal of geometry.

De ludo geometrico: the process of infinite quadratures

A marginal heading names the work: 'On the geometrical game, in which is given the process for infinite varieties of quadratures of surfaces with curved sides.' It titles the programme the whole sheet illustrates.

Squaring the horns of the sickle-lunes

In a square holding four double circles, the sickle-shapes are squared: drawing a squarable part from a square leaves a wholly squarable remainder. The two horns of the sickles are worth the largest square of the smallest circle enclosed within those lunar horns.

Seven hexagons of lunes joined together

An instruction directs the reader to make seven of these hexagons joined, with this one in the middle. In the large hexagon of seven shaded inner figures, the six lunes are squarable in themselves, as is the hatched rectilinear field, since removing a squarable part from a rectilinear figure leaves a squarable remainder.

Sketch plan of a building

A floor plan of a building is drawn in the upper centre of the sheet. Beneath it, two groups of three circles have their centres joined by a line, prompting the note that a line through the centres of circles set in straight contact will be straight.

Nel margine destro:

Dal margine superiore:

Quadrato contenente quattro doppi cerchi:

Altro quadrato con quattro doppi cerchi e le lettere:

Lungo il margine, esagono iscritto:

Figura:

Figura legata alla precedente con tratto di penna, e le lettere:

Continuando sul margine inferiore; triangolo iscritto:

Tornando a destra, doppio cerchio con quadrato iscritto:

Entro il cerchio con esagono iscritto:

Fuori della figura:

Bisangolo entro un cerchio:

Figura sotto la precedente:

A sinistra, dall'alto.

Sopra la figura successiva:

Grande esagono con sette figure interne tinteggiate:

Sotto la pianta d'edificio, due gruppi di tre cerchi coi centri uniti da una linea:

Margine sinistro: doppio cerchio entro un quadrato:

Triangolo iscritto, colle lettere:

In the right margin:

From the upper margin:

Square containing four double circles:

Another square with four double circles and the letters:

Along the margin, an inscribed hexagon:

Figure:

Figure joined to the previous by a pen-stroke, with letters:

Continuing on the lower margin; an inscribed triangle:

Returning to the right, a double circle with an inscribed square:

Within the circle with an inscribed hexagon:

Outside the figure:

A bi-angle within a circle:

Figure below the previous:

On the left, from the top:

Above the following figure:

Large hexagon with seven shaded inner figures:

Below the plan of a building, two groups of three circles with their centres joined by a line:

Left margin: a double circle within a square:

An inscribed triangle, with letters:

On this page

The square is the goal of all geometry

De ludo geometrico: the process of infinite quadratures

Squaring the horns of the sickle-lunes

Seven hexagons of lunes joined together

Sketch plan of a building