Superposition of Equal Surfaces, Curved and Straight
What touches stays equal, and reducing a curvilinear area to a square
This page states the principle that when two surfaces equal in quantity are laid one upon the other, the parts that touch are equal and so, in consequence, are the parts that do not touch. Curvilinear surface-pieces are superposed on rectilinear ones, and the leftover fragments are repeatedly re-superposed until the remainder resolves into what is sought. A worked example with the surfaces a c b and c d carries out three successive superpositions, the contact c 'remaining squared'.
On this page
The rule of superposed equal surfaces
If two surfaces equal in quantity are laid one upon the other, whatever of them touches is equal, one to the other, and therefore whatever of them does not touch remains equal between them. Curvilinear figures may be superposed on rectilinear ones, and the pieces re-laid one on the other repeatedly until the intent is reached.
Three superpositions ending in a square
Here two equal surfaces are superposed: the surface a c b and the surface c d. The part c is common to both, and the excess a b of one equals the excess d of the other; from these pieces a b and d a second superposition is made, leaving equal parts n o, whence a third superposition makes c square from a.
