Doubling the cube: diagonal sections and mean proportionals

The cube's three dimensions

Because the cube has three dimensions, width, length and thickness, a half-cube parallelepiped that is to be made into a cube must be corrected by those three sides.

Squaring the diagonal section of the cube

Here m is the diagonal section of the whole cube and L is its squaring; n is the squaring of the diagonal section of the half-cube. o, born of n, is likewise the diagonal section of a cube made from half the greater cube, and placed between two rectangles gives one whole square side of the cube.

Rule of three by lines for the proportionals

A rule of three set out with lines: if L gives m, what will n give? It gives o, proportional to m. The right-angled intersection is made by the two lines c h and e K.

Scales of square and cube roots on two circles

With the circle b r a scale of square roots up to twenty is to be drawn, and with another circle a scale of cube roots up to twenty, so as to show the difference between the one scale and the other.

A cylinder equal to the cube, and its half

If the cylinder a b c d e f g h descends from the whole cube beside it, half the cylinder equals half the cube. A cylinder two cubes high leaves, in its half, a cube sub-double to the first given cube.

Successive halvings of the cube

The cube is divided in succession, 1/2, 1/4, 1/8, down to 1/16; then the face of the sixteenth is squared and compared, at the 'judicial angle', with the diagonal cut of the sub-eightfold cube n to m.

Facciata destra. Margine superiore, presso la linea mediana:

Figura a destra colle lettere:

A sinistra della precedente, figura colle lettere:

Quarta figura, con:

Margine destro; angolo della proporzionalità:

Figura delle due medie proporzionali:

Figura colle lettere:

Due figure, con:

Figure di un cubo e la sua metà, con:

Triangolo rettangolo con sei ipotenuse, le lettere e f numeri:

In basso, da destra a sinistra cubo e sue frazioni, con:

Facciata sinistra. In alto, angolo giudiziale, con:

Figura per trasformare un cubo in parallelepipedo con:

Altro angolo giudiziale, con:

A sinistra del precedente, solido a base triangolare:

Altro angolo giudiziale:

A sinistra:

Quadratura di un quadrato dimezzato, con :

Sotto il precedente, cubo segato, con:

A destra, cubo e parallelepipedo, con:

Angolo inferiore sinistro, figure con:

Right side. Upper margin, near the middle line:

Figure on the right with the letters:

To the left of the preceding, a figure with the letters:

Fourth figure, with:

Right margin; angle of proportionality:

Figure of the two mean proportionals:

Figure with the letters:

Two figures, with:

Figures of a cube and its half, with:

Right triangle with six hypotenuses, the letters and numbers:

Below, from right to left, a cube and its fractions, with:

Left side. At the top, a judicial angle, with:

Figure for transforming a cube into a parallelepiped with:

Another judicial angle, with:

To the left of the preceding, a solid with a triangular base:

Another judicial angle:

On the left:

Squaring of a halved square, with:

Below the preceding, a cut cube, with:

On the right, a cube and a parallelepiped, with:

Lower left corner, figures with: