The game of geometry: circle transformations and sphere optics

Right face, along the right margin. 'Cart wheel':

Greatest square of the greatest circle.

Figure with:

b c – g a – f e

Circle with double eight-pointed star:
The crossed-out part is worth the 4 greatest.

Circle with double eight-pointed star:

The crossed-out part is worth the 4 greatest.

Circle with eight empty biangles:

These 8 biangles are greater than the 4 biangles by as much as are the values of the 8 least white biangles enclosed within them.
The greater biangles, for not touching the ends of the circle's semidiameter, are worth half of the 4 biangles that would touch those ends of such semidiameters; and so I make 8 biangles of the value of the 4 greatest of the greatest circle, and as much more as are the 8 least white biangles, which are worth 1/4 of that greatest circle. The ovals drawn from the greater biangles leave the empty biangles of the value of the greatest square of the greatest circle.

Left column. Nine proportional circles within a circle:

Here the lune is worth half of the greatest circle, and the other half of the circle remains included in it. Now draw the 7 lesser circles from the said lune, and draw the least middle circle from the middling circle, and the remainder of the lune stays equal to the remainder of the said middling circle. Therefore, the greatest circle being double the middling circle, the middling one is worth the lunar remainder of the greatest circle; and if from equal things you take equal, etc. It follows that, taking from the lune the 7 circles and taking from the middling one the least circle sub-double to it, the remainder is equal.

Sixteen biangles within a circular crown:

Here the 8 greater biangles are double the 8 lesser biangles. Therefore all are worth 24 lesser biangles, that is the 4 greatest of the greatest circle.
* 24 biangles that are worth the 4 greatest of the greatest circle, and the greater are double the lesser.

Beside the preceding figure:

24 biangles.

Two squares and a cube, with:

i a c b; e d g f; i h L K

Circular crown with sixteen teeth:

This one lacks two greatest portions.

Circle inscribed in the square, with:

b - c a – d

a b c d are the 4 greatest of the greatest circle.

Circle with eight empty biangles:

Squarable by the removal of the 4 greatest of the greatest circle.

Transformation of a half-portion, with:

f - d b d - c a - e

I wish to make a transmutation of the half-portion of circle a b c d into a b d, with the help of a [proposition] of Euclid in his Elements. Therefore I shall extend the line a b, which is the chord of the arc a d b, to b f, and I shall have made the straight line a b d divided equally at b. Then I shall draw the straight line, equidistant to a f, which will be e d, and I shall have made the parallelogram a f e d. Then I shall extend the two lines a c and b d so...

Two small figures, with:

penultimate - last

Sheet upside down:

* The first magnitude that can be drawn from the circle ...
The 3 ...

Left face. Right column, along the central fold. Group of curved rays, biangles and portions, one with: a

Only the portion of a half-circle is equal to a circle and no more.
On the transmutation of the portions of circles, that is a.

Biangle diminished by a quarter:

Here the angle can stand under any part of the curved side, without diminishing or increasing its quantity.

Sector and circular segment with two bases, with:

a; b
a b are 2 greatest of the greatest circle.

Circle with sixteen biangles:

The biangles are 16, of which the 8 greater are worth 16 of the lesser. Therefore the circle lacks 24 biangles similar and equal to the lesser biangles, whose value is equal to the 4 greatest portions of the greatest circle. It follows that each of these greatest portions is worth 6 of the lesser biangles.

Four triangles and four half-biangles applied to an octagon within a circle, with:

f e – b o a - d c

Here are eight triangles similar to the triangle e f o, which has two curved sides similar and equal, and each curvature contains within it the eighth part of the circumferential line of this circle. Therefore a b c d are worth the 4 greatest of the greatest circle, and the remainder is squarable in itself.

Biangle in a square inscribed in the circle, with:

a – m n – e r q – p o - b

The 4 lesser portions n m o p are worth the two greater portions q r. It is proved in r m, where m is made upon the side of a square and r upon the diameter of the same square.

Transversally above the preceding, biangle in a circle, with:

b c a

a b lunes are worth the greatest square that can be drawn from the circle a b c.

Central column.

On the geometric game.
On the transformation of circles into various figures of surface.
On the transformation of various surfaces into various sizes and figures of circle.

Group of small geometric figures, among which: Part of a circular crown, with:

a

The same with an interval between two crossed-out sectors, with:

a

Half of the preceding, with:

b

Sector with:

a

Two circles in a circle, with:

a - b

b is 1/2 circle.

Two semicircles in a semicircle, with:

a

a quarter of a circle.

Along the vertical line, circle with four crossed-out sectors:

Half circle.

Circle with two crossed-out semicircles:

Half circle.

Circle with eight crossed-out sectors:

Cart wheel of various number of spokes.

To the left of the preceding figures:

Spherical bodies of bumpy and polished surface are those whose surface is composed of various bumpinesses (protuberances).

Within a circle with a double curved ray:

circle — circle

Figure repeating one of the parts of the preceding circle, with:

a

Two figures of a sphere and a 'bumpy' sphere, with:

a; b
A body clothed with a surface of uniform sphericity is one of which every part is equally distant from the centre of its whole.

Left column.

Q...

The nearer the moon is to the sun, the less quantity of light it has from it, and conversely.

Figure of luminous rays joining the earth, the sun and the eye that looks at them, with:

sun - world - eye

Analogous figure, with

: sun - world - eye
The uniform sphericity of the polished body renders a single simulacrum to the single eye that sees it. It follows that the polished and bumpy surface renders as many simulacra to the eye that sees it as there are bumps seen of the real thing and by the eye that stands before them.
The numbers of the simulacra of any object seen upon spherical bodies of bumpy and polished surface will be so much the greater or lesser as the eye beholding such simulacra is more remote from or near to the said polished body.
Of that spherical body one sees less quantity that is nearer to the eye that sees it.
* The earth and the moon ...
The simulacrum of splendid bodies upon bodies of bumpy and polished surface grows as the eye is moved to a greater distance.
The earth and the moon lend each other their light.

Facciata destra, lungo il margine destro. 'Ruota di carro': 1)

Figura con:

Cerchio con doppia stella a otto punte:

Cerchio con otto bisangoli vacui:

Colonna sinistra. Nove cerchi proporzionali entro un cerchio:

Sedici bisangoli entro una corona circolare:

Di fianco alla figura precedente:

Due quadrati e un cubo, con:

Corona circolare con sedici denti:

Cerchio iscritto nel quadrato, con:

Cerchio con otto bisangoli vacui:

Trasformazione di una mezza porzione, con:

Due piccole figure, con:

Foglio capovolto:

Facciata sinistra. Colonna destra, lungo la piega mediana. Gruppo di falcate, bisangoli, porzioni, di cui una con:

Bisangolo diminuito d'un quarto:

Settore e segmento circolare a due basi, con:

Cerchio con sedici bisangoli6:

Quattro triangoli e quattro semibisangoli applicati a un ottagono entro un cerchio, con:

Bisangolo in un quadrato iscritto nel cerchio, con:

Traversalmente sopra la precedente, bisangolo in un cerchio, con:

Colonna centrale.

Gruppo di piccole figure geometriche, tra cui: Parte di corona circolare, con:

Idem con intervallo tra due settori depennati, con:

Metà della precedente, con:

Settore con:

Due cerchi in un cerchio, con:

Due semicerchi in un semicerchio, con:

Lungo la linea verticale, cerchio con quattro settori depennati:

Cerchio con due semicerchi depennati:

Cerchio con otto settori depennati:

A sinistra delle precedenti figure:

Entro un cerchio con doppia falcata:

Figura che ripete una delle parti del cerchio precedente, con:

Due figure di sfera e sfera 'globulente', con:

Colonna sinistra.

Figura di raggi luminosi congiungenti la terra, il sole e l'occhio che li guarda, con:

Figura analoga, con

Right face, along the right margin. 'Cart wheel':

Figure with:

Circle with double eight-pointed star:

Circle with eight empty biangles:

Left column. Nine proportional circles within a circle:

Sixteen biangles within a circular crown:

Beside the preceding figure:

Two squares and a cube, with:

Circular crown with sixteen teeth:

Circle inscribed in the square, with:

Circle with eight empty biangles:

Transformation of a half-portion, with:

Two small figures, with:

Sheet upside down:

Left face. Right column, along the central fold. Group of curved rays, biangles and portions, one with: a

Biangle diminished by a quarter:

Sector and circular segment with two bases, with:

Circle with sixteen biangles:

Four triangles and four half-biangles applied to an octagon within a circle, with:

Biangle in a square inscribed in the circle, with:

Transversally above the preceding, biangle in a circle, with:

Central column.

Group of small geometric figures, among which: Part of a circular crown, with:

The same with an interval between two crossed-out sectors, with:

Half of the preceding, with:

Sector with:

Two circles in a circle, with:

Two semicircles in a semicircle, with:

Along the vertical line, circle with four crossed-out sectors:

Circle with two crossed-out semicircles:

Circle with eight crossed-out sectors:

To the left of the preceding figures:

Within a circle with a double curved ray:

Figure repeating one of the parts of the preceding circle, with:

Two figures of a sphere and a 'bumpy' sphere, with:

Left column.

Figure of luminous rays joining the earth, the sun and the eye that looks at them, with:

Analogous figure, with

On this page

On the geometric game: transforming circles and surfaces

Transmuting a half-portion of a circle with Euclid

One image from a polished sphere, many from a bumpy one