Postulates. / To cut from a larger surface a part equal and similar to a given smaller surface. / To add to a surface another surface. To superpose upon a surface another surface. To take a part of a surface from one of its sides, and restore it on another side. / If two equal and similar surfaces are wholly superposed one on the other, they will be, throughout their whole, all in contact, whole with the whole of the other. / And if 2 equal and similar surfaces are partly superposed one on the other, that of them which touches will be equal in quantity and similar in figure, and the like will happen to the parts of them that do not touch.

a, b

Such will be the proportions that the same members of the surfaces have among themselves as that of their wholes among themselves. Say that the portion b, being subdouble to the portion a, [means] that the pyramids from which they are born are likewise double one to the other.

Common notions. / The thing that moves acquires as much space as it leaves. / If to 2 equal surfaces I add two equal surfaces, the outcomes will be equal. / If from 2 equal surfaces I remove two equal parts, the remainder will be equal. / If I restore to a surface what I took from it, then it returns to its first being. / If from 2 double surfaces double parts are removed, the remainder will be double. / If from 2 unequal surfaces unequal parts are removed, the remainder can be equal. As 8 and 4, which is unequal; remove 6 from the 8, and remove 2 from the 4, so that 6 and 2 are unequal parts, and there will remain 2 and 2. / If from the parts that make a proportion you remove a quantity of the same nature, the remainder will be in the first proportion.