Area

Abstract

Looking at Euclid’s theory of area in Books I-IV, Hilbert saw how to give it a solid logical foundation. We define the notion of equal content by saying that two figures have equal content if we can transform one figure into the other by adding and subtracting congruent triangles (Section 22). We can prove all the properties of area that Euclid uses, except that “the whole is greater than the part.” This is established only when we relate the geometrical notion of equal content to the notion of a measure of area function (Section 23).