Lunes, Circles and Squares: Geometry of Equal Areas

Along the right margin. Circle superimposed on a semicircle, with:

First - d c b a - f e

a d equals e f.

If from equal things equal parts are taken, the remainder will be equal. Therefore, removing the common contact b c from the circle and also from the semicircle which ..

Of circles in double proportion, half the greater equals the whole lesser. Therefore they are equal to one another.

Of equal things that are partly superimposed one upon the other, what of them touches will be equal, and equal will be what of them does not touch. Therefore the semicircle a b c d, which is superimposed on the circle b c e f equal to it, projects as much outside that circle as that circle projects outside the semicircle — that is, the a d excesses of the semicircle are equal to the e f excesses of the aforesaid circle …

Circle within a semicircle, with:

Second - c b a

ad

a b equals d c.

Of circles quadruple one to the other, half the greater equals twice the lesser. It follows that, the lesser being wholly superimposed on half the greater, the excesses of the greater will be equal to the whole lesser circle. Therefore the lunes (falcate) generated in those parts by which half the greater circle exceeds the lesser circle are equal to the whole lesser circle itself.

Two circles superimposed on a semicircle, with:

Third - b d h c a - e

The lunes a c d b are worth the lune e c d; and if there be two equal surfaces partly superimposed one upon the other, the parts that cover one another will be equal to those they occupy, and equal will be those that remain uncovered. It follows that a b, the uncovered part of the lunes, equals e, the uncovered part of the lune; and the 2 lunes d b are worth half the included circle and the …

To the left of the preceding, two figures with:

man;

b - p o

It is the same for circle a to be surrounded by the semicircle n m as for circle b to be surrounded by the lune, because the lune is worth in itself as much as the two lunes n m; and the circles a b are equal.

Superimpose the lune on the two lunes and the two circles on one same centre, and say: if two equal things are partly superimposed one on the other, what of them touches will be equal to one another, and equal will remain those that do not touch.

Lower right corner, two semicircles on a quadrant of a circle, with:

c a n - b

n a b equals n a c. Remove c, and n a b remains equal to n a c. Remove b, and n a c remains equal to n a b.

Along the lower margin:

In the first above is the value of four equal quantities equal to the lune a, and this we wish to prove, that is a d e f; in the second the same holds in d c, and the greatest half-portion is one and the same in the first and in the second; and because the first has two outer half-portions equal to the two lunes, as will be proved in its place, the void b c is greater than the void a b of the second by as much as the value of the two lunes, that is the two said portions e f, which are worth exactly those two lunes. Now I shall take away the portion e f and put it inside the void b c of the first, and there will remain double lunes in the manner of the second, and the void in this first will contract by as much as is the void of the second. Therefore we have proved, in the third demonstration, the lunes c d to be equal to the lunes a b, which was our proposition. By which proof it remains concluded that the portion e of the third figure is worth the two lunes c d — or, if you will, the two lunes d b to be equal to the portion e, and consequently equal to half the lesser circle h, because it is proved that the lunes a c d b are worth the whole lesser circle h.

Upper margin, at left, referring to the figure of a circle inscribed and circumscribed between two squares.

The circle that touches the corners of the square is double that which touches the sides of the same square.

And the square that with its corners touches the circle is double that which with its sides touches the same circle.

Circle within a semicircle, with:

Second - d c b a

The lunes a d are worth the circle b c.

Two circles on a semicircle, with:

b c - a

a equals b. a with c is worth half the lesser circle, and a with b is worth (half the lesser circle), and c with b is worth half the lesser circle.

Left margin.

If two various surfaces … If two equal surfaces, of various or similar figures, are partly superimposed one on the other, the parts of them that touch will be equal and similar, and those that do not touch will be equal in quantity and rarely equal in figure, unless they be circular.

And if there be two surfaces of double proportion, and the whole lesser is superimposed on the greater, always the excess of the greater equals the whole lesser.

To the right of the preceding:

And if the proportion be triple, the excess of the greater will be double the lesser; and so it continues to infinity.

At the bottom, three figures of circles superimposed on semicircles, with:

a; c a b; a - c e d b - f n

a equals b, because the semicircle is worth the circle and the excesses are equal.

a equals b c, because the circle is worth half the semicircle. Therefore the circle is worth the excesses of the semicircle.

e c is worth half of a.

e f is worth half of a.

f c is worth half of a.

f n is worth half of a.

Following the pen-stroke:

f e equals e c, and f e 4 is worth f c. It follows that e equals c; and this is because, if f by itself alone equals e, it equals c as well.

Following the pen-stroke further:

If f is worth 2 and e one and c 2, it will follow that f with e makes 3, and e with c makes 3 again. But if you join together f c, that is 2 and 2, it makes 4. Therefore f e and e c are less than f c, and we find …

It is proved that the portion n f is worth half the circle a, because I have shown before, beneath the figure, how f c is worth half the circle a, and f e is worth half of a. Therefore e is worth c, because if c f is worth half a, and likewise f e the same, and again e c does the like, it is necessary that what f does with the 2 lunes e c, so too n, being equal to f, does the same as f. Therefore n f is equal to the half-circle a.

After the two lines already transcribed:

But we shall say: if the lune is worth the circle it contains, half the lune f e is worth half the circle; but finding half the circle in the base of the lune n f, the remainder of the lune is worth the remainder of the circle — that is, the other half of the lune d c is worth half the circle.

Lungo il margine destro. Cerchio sovrapposto a semicerchio, con:

Cerchio entro semicerchio, con:

Due cerchi sovrapposti a semicerchio, con:

A sinistra del precedente, due figure con:

Angolo inferiore destro, due semicerchi su un quadrante di cerchio, con:

Lungo il margine inferiore:

Margine superiore, a sinistra con riferimento alla figura di cerchio iscritto e circoscritto tra due quadrati.

Cerchio entro semicerchio, con:

Due cerchi su semicerchio, con:

Margine sinistro.

A destra del precedente:

In basso, tre figure di cerchi sovrapposti a semicerchi, con:

Seguendo il tratto di penna:

Seguendo ulteriormente il tratto di penna:

Dopo le due righe già trascritte:

Along the right margin. Circle superimposed on a semicircle, with:

Circle within a semicircle, with:

Two circles superimposed on a semicircle, with:

To the left of the preceding, two figures with:

Lower right corner, two semicircles on a quadrant of a circle, with:

Along the lower margin:

Upper margin, at left, referring to the figure of a circle inscribed and circumscribed between two squares.

Circle within a semicircle, with:

Two circles on a semicircle, with:

Left margin.

To the right of the preceding:

At the bottom, three figures of circles superimposed on semicircles, with:

Following the pen-stroke:

Following the pen-stroke further:

After the two lines already transcribed:

Lunes, Circles and Squares: Geometry of Equal Areas

On this page

Circles of double proportion: half the greater equals the whole lesser

Two lunes equal to the whole included circle

Circle and square: corners double the sides

Excess of the greater by proportion, to infinity

Marginal quadrature: transferring a portion to equalise the lunes