Conceptions on the equality of superimposed surfaces
Axioms underlying the quadrature method, including a triangle laid over a square
A geometry page setting out foundational propositions for Leonardo's quadrature work. He argues that when a surface is laid over another that is double it, the uncovered part of the greater equals the smaller, and he treats the case where the smaller is only partly superimposed on its double. Two numbered 'Conceptions' follow: that equal and similar surfaces mutually occupy equal areas, and that surfaces equal in quantity but unlike in figure (as a triangle over a square) leave leftover parts unequal in number and shape but equal in total quantity.
On this page
A surface superimposed on its double
When a surface is entirely laid over another that is double it, the whole part of the greater left outside the smaller equals that smaller. With the smaller surface b and the greater a b, the part a left outside b comes to equal b.
The subduple surface partly superimposed
If the smaller surface is only partly laid over its double, the uncovered part of the greater equals the smaller plus as much more as the smaller sticks out beyond the greater. With b subduple (half) to a, a without c equals b c, and moreover as much as b alone.
First Conception: mutual occupation of equal surfaces
The first stated Conception: when two surfaces equal and similar are wholly superimposed, that which of the second is occupied by the first equals that which of the first is occupied by the second.
Equal but dissimilar figures: a triangle over a square
Two surfaces equal in quantity but unlike in figure leave untouched parts that are equal in total quantity though not in number or shape. A triangle laid over an equal square leaves 3 parts outside the square while the square leaves 2 parts outside the triangle, yet the 2 leftover parts of the square equal the 3 of the triangle.