Project Gutenberg has David Hilbert’s Foundations of Geometry available. It is a translation of Hilbert’s *Grundlagen der Geometrie*, which is famous as the first modern axiomization of Euclidean geometry. The difference between Hilbert’s approach and that of Euclid is that Hilbert fills in all of the fiddly little details required to meet modern standards of rigor.

The book is elementary, and (as translated by Townsend) is a pleasant read. Much of the book centers around constructing the field of real numbers in terms of the axiomized geometrical constructions. This in turn allows Hilbert to show that the set of axioms is complete. The topic leads naturally to one of the main themes of research in plane geometry in the early part of the last century, which is to consider different algebraic objects and how they can serve as coordinates for different notions of affine or projective planes. The reals can be replaced with an arbitrary division ring, for example. For a projective plane, the most general object is a planar ternary ring, with has a ternary operation that serves as a hybrid of addition and multiplication. Determining the projective planes with a finite number of points is still an open question.