Circles, tangency and the quadrature of lunes

Circles tangent around a central circle

The straight line drawn from the centre of one of two circles to the centre of the other always passes through their point of contact. Of circles surrounding another circle, the more curved the line of their contacts, the farther the centres of the surrounding circles lie from the centre of the circle they enclose.

The sphere's surface and the squaring of curved figures

The sphere's surface is quadruple its greatest circle, and a circle is quadruple the greatest circle of its semicircle. Curvilinear figures are squared by lending a triangle to straighten a curve and then removing the borrowed area: take away a b c d and the remainder b d c o stays a square.

Catalogued among curved-mirror and light-ray studies

Clusters of concentric circles fill the upper sheet. The Ambrosiana classes these among Leonardo's studies for constructing curved (burning) mirrors and the concentration of light rays, the note 'make me the flame, or sun.' No mirror text is transcribed here, so the theme rests on the diagrams and the catalogue.

A profile head in red chalk

A large profile of a face, drawn in red chalk, is laid over the geometry at lower centre, its nose and mouth clearly modelled. It is unrelated to the surrounding mathematical figures and appears to be an independent sketch on the same sheet.

In alto a destra. Dodici cerchi attorno a un cerchio, fra loro tangenti:

Figura analoga:

Colonna centrale.

Entro la figura di combattente, triangolo rettangolo con:

In basso, dalla destra. Quadrato quadrettato, con:

Quadrati costruiti sui lati d'un triangolo, con:

Tangente a settore di cerchio, con:

Figura analoga, con:

Girato il foglio sul lato destro. Presso il margine, figure quadrettate coi numeri:

Foglio capovolto, quadratura di figura curvilinea, con:

Top right. Twelve circles around a circle, tangent to one another:

Analogous figure:

Central column.

Within the figure of a combatant, a right triangle with:

Below, from the right. A gridded square, with:

Squares built on the sides of a triangle, with:

Tangent to a sector of a circle, with:

Analogous figure, with:

Sheet turned onto its right side. Near the margin, gridded figures with the numbers:

Sheet upside down; quadrature of a curvilinear figure, with:

Circles, tangency and the quadrature of lunes

On this page

Circles tangent around a central circle

The sphere's surface and the squaring of curved figures

Catalogued among curved-mirror and light-ray studies

A profile head in red chalk