A Method Before It Is a Subject

The Elements is remembered for its geometry and was influential for its shape. It opens with definitions, then five postulates, then the common notions, and after that every proposition is derived from what has already been granted, in order, with nothing smuggled in from outside. That structure was the innovation. It states what is being assumed and confines the argument to what follows from it, and it made available the idea that a body of knowledge can be certain because of the way it is built rather than because of who is asserting it.

The content carries its own weight. The fifth postulate, about lines that never meet, is longer and less obvious than the other four, and two thousand years of failed attempts to derive it from them ended in the nineteenth century with the discovery that consistent geometries exist in which it is false. Book ten treats magnitudes that have no common measure, the discovery that had unsettled the Pythagoreans. And the work ends, in book thirteen, by constructing the five regular solids and proving that there are no others, which puts the whole treatise in the same territory as the Timaeus, where those five shapes are the forms of the elements and of the cosmos.