Quadrature of the Circle: Lunes, Squares and Diameters

Lunes and quarters: a quarter of the great circle equals the whole small circle

With the greater circle double the middle one and quadruple the smaller, the quarter a b c of the greater equals the quarter d f of the middle plus the half b d of the smaller. It follows that the portion d equals the two lunes a c and the two portions e n, and that a quarter of the greater circle equals the whole smaller circle. The surfaces b, m and S are marked as three equal areas.

Circles stand to one another as the squares of their diameters

Leonardo takes as proved that the ratio of circle to circle equals the ratio of square to square built on their diameters. Setting one diameter at 100 (square 10000) against another of 1 (square 1), he subdivides a ten-thousandth of the larger circumference through repeated halvings and tenths to compare a single part, once squared, with the diameter of the unit circle. The construction aims to read the circle's area off a chain of square divisions.

Squaring the diameter: 3000 x 3000 = 9,000,000

To apply the diameter-squared rule, Leonardo sets the smaller circle's diameter at 3000 braccia and multiplies it by itself to obtain nine million square braccia. The columns of figures below work this multiplication, alongside a square of side one thousand set against a second of one million. The arithmetic supplies the numeric backbone for the geometric comparison.

Lamp or candle device

At the upper center a small apparatus is drawn: a horizontal shaft carrying a rounded, disc-like element beside a small flame, with dotted lines suggesting emitted rays or a reflector behind it. The scholarly subject-tag for the sheet names it a candle device. No transcribed text is attached to this drawing, which sits among the geometrical figures.

Memorandum: one ducat on Thursday morning

In the upper margin Leonardo notes a payment of one ducat received or due on Thursday morning. A further line at the lower right records a moral maxim, that the impediments of truth turn into penance. These jottings sit apart from the geometrical work that fills the sheet.

A destra, doppio cerchio con lunola:

Figura analoga colle lettere:

Quadrante di cerchio con lunola:

Sopra il brano precedente:

In basso:

Figura di tre semicerchi con settori:

Angolo inferiore sinistro. Piccola figura di cerchio diviso in settori e iscritto in un quadrato, con quadratino di cui si indicano le proporzioni:

A destra due cerchi colle lettere:

Angolo inferiore destro:

Nel margine superiore:

On the right, a double circle with a lune:

Analogous figure with letters:

Quarter of a circle with a lune:

Above the preceding passage:

Below:

Figure of three semicircles with sectors:

Lower left corner. A small figure of a circle divided into sectors and inscribed in a square, with a little square whose proportions are given:

On the right, two circles with letters:

Lower right corner:

In the upper margin: