On the equality of unequal surfaces.

d, a, b, c, e

d b are equal to each other, and e c also are equal to each other, but they are not all together equal to the other 2 joined together, because e c are greater than b d by about a fourth part.

b, c, a

a is equal to c

d, a b c, n m o

a and c are equal to each other.

This rule of squaring by way of motion always leaves the falcate of equal curvature, because it superimposes figures equal in quantity and in figure.

The whole lunula is equal to the whole square. It follows that what the part of the lunula occupies of the square is equal to that part of the lunula (by the conception). Whence arises that the remainder of the square not occupied by the lunula is equal to the remainder of the lunula not in contact with the square. And because it was first proved that the half-portion e is equal to the whole portion c, because the circle from which that half-portion derived is double the circle from which the lesser portion derived, whence it is concluded that, removing from triangle b e the half-portion e, and from triangle c d the portion e, triangle b remains equal to triangle d, because if from equal things equal parts are removed, the remainder remains equal.

From the superimposition spoken of above there arises a falcata of 2 curved lines and one straight, equal to a rectilinear triangle. This falcata has one line born of a circle double to that of the curvature of the opposite line. And so let the triangles b d of the figure above be taken and superimposed, as you see below, in such a way that triangle d becomes triangle a b and triangle b becomes triangle b c. And thus being, as is proved, equal among themselves, if from the one and the other an equal part is removed, the remainder remains equal. Let therefore b be removed from triangle a b and a will remain. And let b likewise be removed from triangle b c, and c will remain. Which c will be exactly equal in quantity to a, and thus the falcate pyramid a remains squared into the rectilinear figure c.

The same can be done here below by motion, inasmuch as if I move the figure b c into b a, keeping its point above fixed at d, without doubt I shall make the falcate pyramid a equal to the rectilinear figure c, which is proved by the conception spoken of above, that is, that if from equal things equal quantities are removed, the remainder will be equal. Now I will remove from the figure b c the part b, and from the figure a b I will likewise remove b, and the remainder of the one, which is a, remains equal to the remainder of the other, which is c. Here is fulfilled the promise I have made, that I have made a equal to c.