Squaring circles and lunes; the motions of a twisted cord

At the top, a block of lines beside the figure of a body hung from a cord, with the letters:

e - c - b a - d. Four can be the motions in one and the same body hung from a cord along a straight line, namely: swaying from right to left, incident and rebounding, revolving upon itself, and also circular motion. It is proved: let the cord b e be strongly twisted upon itself and set at a great obliquity. Then, being let free, it at once generates three various motions, of which the first makes a circular and declining motion. The circular one is made by the moving body which turns upon itself about its own centre as the cord that was forcibly twisted unwinds. The second motion is of double obliquity, of which the first is in the incident motion, the other in the cord that lengthens as it unwinds, and this is a motion compounded of two obliquities; and up to here we have three motions. The fourth is the rebounding motion; the fifth is the circular motion the mover can give it at the start of the motion. The sixth will be made in bounds, when the mover throws the moving body upward, and it returns downward and then rebounds, and so it goes bounding in that circular motion — the cord being, as said, thin and well wound and forcibly twisted upon itself.

On the left from top to bottom, a circle inscribed in and a circle circumscribed about a triangle:

The circle drawn outside the triangle is fourfold the circle drawn inside it.

A circle inscribed in and circumscribed about a square:

The circle drawn outside the square is double the circle drawn inside the same square.

A triangle and a hexagon inscribed in a circle:

The triangle whose angles touch the three angles of the hexagon is worth half of that hexagon.

Two squares and an octagon inscribed in a circle:

Let the lune be divided into as many parts as one wishes.

Five circles one within another, with the letter:

b. All the excesses are worth the circle b.

Three circles one within another, with the letters:

a n o. a is worth n o. It arises from the preceding.

Two figures with the letters:

a c. In the quadrangular surface let a circle be made that touches the angles, and outside that circle let a square be made that touches the circle, and such a square will be equal to these three curvilinear surfaces. It is proved by b, the third figure above, and by c, the third figure below, and so on.

Two small figures — with reference also to the next — with the letters:

a b. a b are of equal value and are squarable. It is proved, because the octagon is rectilinear and has the 8 greatest areas less, and has 8 less inside, which are 32. But there is a fallacy, because from the sub-double circle 16 greatest areas are taken away, of which 8 are for itself and 8 are taken away for the circular parallel.

A star-shaped figure with the letters:

b M. This one, marked M, is the same as the one above marked M, and is squarable, because the circles are double and consequently, the parts removed from them being equal, the remainder is equal, and so on. 8 are the greatest areas and 16 are the least.

Second column, from the centre. A double-circle figure with four bi-angles:

Squarable, and it is like to like in the same lune at the side. The four bi-angles are worth the four greatest areas of the circle whose diameter is made the diameter of the enclosed square.

A figure with the letters:

a b. a b is worth a square that contains within itself the circle b, that is, where the cross is made its diameter.

A figure with the letters:

a b - c d - a. The three circles are double one another. The empty square is worth the square on the right-hand side.

Third column, from the centre. Two circular figures, of which the first with the letters:

a b. Here the three greatest areas are missing, and it is worth the greatest triangle of the greatest circle.

A figure with the letter:

M. Less the 8 greatest areas, it remains equal in value to the greatest octagon that can be contained in the greatest circle.

A figure with the letters:

a b. a b is a square, because a lacks the 4 greatest areas of a circle worth the lune and itself.

A circle with four bi-angles:

It lacks the 8 greatest areas of the circle sub-quadruple to the greatest.

Beside the figure:

Less the 8 greatest areas of the circle sub-double to the greatest, and the greatest areas of this greatest fill the 4 empty ones.

Fourth column, from the centre. A double-circle figure with four bi-angles:

Less the 8 greatest areas; and the circles are, in three degrees, double one another. The square of the whole is the greatest square that can be contained in the greatest circle.

A circle containing two circles:

Less the 4 greatest areas, it is worth the greatest square of the greatest circle.

A figure with the letters:

a b c d. a b is a square, and the remainder is a square, because it lacks the 4 greatest areas.

A figure with the letter:

a. a is in itself a square.

A figure with the letters:

a b c. Here the 8 greatest areas are removed.

Last column, from the top. Four little circles within a double circle:

Take away 4 circles from one worth 6 circles. The remainder is worth two of those circles.

A figure with the letters:

a b c d. a b is worth the 4 greatest areas.

The next figure:

Less the 4 greatest areas.

The next figure:

Less the 4 greatest areas.

The next figure:

Less the 4 greatest areas.

A figure with the letters:

a b c d. a b is worth c d.

A double figure with the letters:

a n m b - c d. a b is sub-double to the greater. If you add c d to n m, you have the value of a square equal to the greatest square of the greatest circle.

Figures with the letters:

a b c d - n. a b c d is worth the field, and the circles are double one another. Add to it the quadrangle n, and you have the value of the greatest square of the greatest circle.

In alto blocco di righe accanto alla figura di un corpo sospeso a corda, colle lettere:

A sinistra dall'alto in basso, cerchio iscritto e cerchio circoscritto a un triangolo:

Cerchio iscritto e circoscritto a un quadrato:

Triangolo ed esagono iscritti in un cerchio:

Due quadrati e un ottagono iscritti in un cerchio:

Cinque cerchi interni l'uno all'altro, colla lettera:

Tre cerchi interni l'uno all'altro, colle lettere:

Due figure colle lettere:

Due piccole figure — ma con riferimento anche alla successiva — colle lettere:

Figura stellare colle lettere:

Seconda colonna, dal centro. Figura di doppio cerchio con quattro bisangoli:

Figura colle lettere:

Figura colle lettere:

Terza colonna, dal centro. Due figure circolari, di cui la prima colle lettere:

Figura colla lettera:

Figura colle lettere:

Cerchio con quattro bisangoli:

Di fianco alla figura:

Quarta colonna, dal centro. Figura di doppio cerchio con quattro bisangoli:

Cerchio contenente due cerchi:

Figura colle lettere:

Figura colla lettera:

Figura colle lettere:

Ultima colonna, dall'alto. Quattro cerchietti interni a un doppio circolo:

Figura colle lettere:

Figura successiva:

Figura successiva:

Figura successiva:

Figura colle lettere:

Doppia figura colle lettere:

Figure colle lettere:

At the top, a block of lines beside the figure of a body hung from a cord, with the letters:

On the left from top to bottom, a circle inscribed in and a circle circumscribed about a triangle:

A circle inscribed in and circumscribed about a square:

A triangle and a hexagon inscribed in a circle:

Two squares and an octagon inscribed in a circle:

Five circles one within another, with the letter:

Three circles one within another, with the letters:

Two figures with the letters:

Two small figures — with reference also to the next — with the letters:

A star-shaped figure with the letters:

Second column, from the centre. A double-circle figure with four bi-angles:

A figure with the letters:

A figure with the letters:

Third column, from the centre. Two circular figures, of which the first with the letters:

A figure with the letter:

A figure with the letters:

A circle with four bi-angles:

Beside the figure:

Fourth column, from the centre. A double-circle figure with four bi-angles:

A circle containing two circles:

A figure with the letters:

A figure with the letter:

A figure with the letters:

Last column, from the top. Four little circles within a double circle:

A figure with the letters:

The next figure:

The next figure:

The next figure:

A figure with the letters:

A double figure with the letters:

Figures with the letters:

On this page

The motions of a body on a twisted cord

Circle outside a triangle is fourfold the inside circle

Circle outside a square is double the inside circle

Triangle inscribed within a hexagon is half the hexagon

A squarable star figure of eight and sixteen areas

Toward the greatest square of the greatest circle