Quadrature of lunes with a note on the sea's tides

Curvilinear triangles worth the inscribed square

The four curvilinear triangles a b c d are worth the greatest square that can be inscribed within the circumference. The eight portions are said to be worth the two greatest portions, and a crossed-out figure is worth the two greatest portions. These are steps in reducing lunes and portions to equivalent squares.

When the solid is square the field is squarable

If the solid (filled) part is a square, then the field around it is squarable. In a related crossed-out figure, the greatest square that fits inside a lune is worth that lune, proved from the preceding proposition where the circles are quadruple, since a greater portion is worth four smaller ones.

Quadruple and quintuple ratios; unequal subtraction

A small table sets quadruple (4 and 16, 1 and 1) against quintuple (3 and 15) ratios. The accompanying rule states that if from unequal things I remove equal things, the remainder does not stay in the same proportion. It underpins the reasoning about how portions compare after equal parts are taken away.

Transforming a curved surface by a given proportion

From a squared curved surface let another curved surface be drawn according to a given proportion, transformed into various rectilinear and curvilinear surfaces. Of surfaces in sixteenfold proportion, the sixteenth part of the greater is worth the whole lesser. Leonardo then proposes giving a square equal to any part of the helix, in both thickness and length.

The earth's tides over a 24-hour cycle

On a circle representing the earth, when the highest rising water is at a the lowest falling is at d; when the rising is below b, after 6 hours the space a d is equalized. After 12 hours, with the rising below c, the space a d is motionless; after 24 hours, with the rising at d, below a is the highest falling. Leonardo asks whether the water rising and falling in the flux and reflux of the sea at the shore belongs to the near or the remote part.

Colonna destra, lungo il margine. Terza figura: semicerchio con segmento circolare a due basi:

Ottava figura, rettangolo con due lunule:

Terz'ultima figura, quattro bisangoli, un cerchio entro semicerchio:

Andando verso sinistra. Cerchio con quattro bisangoli e:

Più sotto, figura analoga entro quadrato :

Più sopra, figura biffata con:

Altra figura biffata con lunula:

Più sopra, quadrato in corona circolare e figura aritmetica:

Figura biffata con:

Analoga figura con due figurine sussidiarie e:

In alto, seconda colonna 2).

Cerchio raffigurante la terra colle maree:

Right column, along the margin. Third figure: semicircle with a circular segment on two bases

Eighth figure, rectangle with two lunes

Third-from-last figure, four bi-angles, a circle within a semicircle

Moving toward the left. Circle with four bi-angles and

Below, a similar figure within a square

Above, a crossed-out figure with

Another crossed-out figure with a lune

Above, a square in a circular ring and an arithmetical figure

Crossed-out figure with

A similar figure with two subsidiary small figures and

At the top, second column

Circle representing the earth with the tides