Let a lunule be made equal to a triangle of two straight lines and one curved. And again, in the same figure, let a triangle of three straight lines be made, equal to a triangle of two curved lines and one straight.

b, d, a o f g, e, c

As to the first requirement, the lunule made of the parts d o e is equal in quantity to the triangle made of the parts o f g, which is proved thus. First, let the triangle o stand fixed in the middle of four equal portions of the circle, namely d e f g. I say, then, that if the two portions d e are lent to the triangle, I have made the lunule d o e. And if I take those portions d e from the triangle o and give it back two others similar to them, I shall have remade a surface equal to the aforesaid lunule. Therefore let the two portions f g be given back to it, and I shall have made the triangle o f g equal to the lunule o d e. And for the second requirement, I shall remove the portion f from the rectilinear triangle a b c and return it at d, and I shall have made, with d o, the triangle of two curved lines and one straight for its base.

b c, a d, e

If the pyramid of curved sides and straight base be cut by a straight line through the middle of its sides, its upper part will be one third of its quantity. And if it were of straight sides, the upper part would be one quarter of its quantity.

a, b, c