Euclid

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.

The afterlife of Euclid outside mathematics is large. The medieval Old Charges that English masons copied and read aloud trace the craft back to geometry and name Euclid among those who taught it, so that a working mason's trade was given a descent from a liberal art rather than from manual labour. When speculative freemasonry took shape that inheritance came with it, and the forty seventh proposition of the first book, the theorem usually named after Pythagoras, became one of its emblems, worn as a jewel by a past master.

The influence on argument is wider still. Spinoza set out the Ethics in definitions, axioms, propositions and demonstrations, closing each with the abbreviation for which was to be demonstrated, precisely because that form claimed a kind of certainty theology could not. Medieval magical literature made the same claim more baldly: the Sworn Book presents its author as Honorius, son of Euclid, borrowing the name in order to borrow the authority. In each case what is being invoked is not any particular theorem. It is the promise that a chain of reasoning can compel assent, and that promise is the thing the Elements taught Europe to want.

There is a cost to the borrowing that is worth naming. Geometry earns its certainty by starting from stated assumptions and claiming nothing about the world beyond what those assumptions yield, so a proof of God modelled on a proof about triangles inherits the form without inheriting the guarantee. Critics of Spinoza said as much at the time, and the objection applies equally to the many later systems that took up the diagram and the vocabulary of geometry as a badge of rigour.