Doubling the cube and squaring lunes and circles

Doubling the cube through a double parallelepiped

Leonardo sets cd as a cube and ab as a solid half its width, equal to the cube in height and double in length; adding as much width again yields a parallelepiped of double the cube's volume. Drawing out its square-faced core equal to half the face gives a parallelepiped half the first and equal to the original cube. He then asks to 'cube' both the larger and smaller parallelepiped so as to obtain one cube double another cube.

Squaring the face of solid a into parallelepiped b

The solid a is treated as worth cube c, keeping the cube's height with half its width and double its length. Leonardo proposes to square that face the way quadrilaterals are usually squared, without shortening a, producing parallelepiped b, then to clothe it with a surface equal to it, as a square surface double another is given, to make surface n.

A square equal to a curvilinear lune

Where the greater semicircle is double the lesser, the excess of the greater equals the whole lesser; removing equal parts leaves the square a equal to the curvilinear region bc. Taking from a semicircle the value of its two greater portions leaves a remainder n which, halved, gives parts each worth the square a, so that n is worth twice the square a.

Axioms of equal remainders and doubled circles

Leonardo records the common notions used throughout: if equal parts are taken from equal things the remainders stay equal, and superposed equal parts are equal and alike. He adds that with two circles one double the other, a quarter of the one equals a half of the other, and that the excess of a doubled circle is worth the smaller circle.

Sketch of jagged mountains

Near the centre of the crowded sheet Leonardo has drawn a small range of steep, jagged mountains, incidental to the surrounding geometry. The transcription notes that a geometric figure lies to the right of and above this drawing of mountains.

A destra e sopra il disegno di montagne, figura con:

Figure di parallelepipedi e di un cubo colle lettere:

Lungo il margine superiore, due cerchi e un settore di cerchio concentrici con:

Figure di cerchi, settori e porzioni, fra cui tre con:

Girato il foglio sul lato sinistro: In alto a sinistra, figure colle lettere:

In basso, figura con:

Figura con:

Due semicerchi colle lettere:

Semicerchio colla lettera:

Due rettangoli colle lettere:

Lungo il margine destro dieci semicerchi distinti da crocette e numeri:

Due semicerchi biffati coi numeri:

Tre semicerchi coi numeri:

Semicerchio biffato con:

Due semicerchi biffati coi numeri:

At right and above the drawing of mountains, a figure with:

Figures of parallelepipeds and of a cube, with the letters:

Along the upper margin, two circles and a circular sector, concentric, with:

Figures of circles, sectors and portions, among them three with:

Turning the sheet onto its left side; at upper left, figures with the letters:

Below, a figure with:

A figure with:

Two semicircles with the letters:

A semicircle with the letter:

Two rectangles with the letters:

Along the right margin, ten semicircles distinguished by small crosses and numbers:

Two semicircles struck through, with the numbers:

Three semicircles with the numbers:

A semicircle struck through, with: