Colin McLarty writes:
A lot of people long before Tarski, or Hilbert, knew how to extend this kind of axiomatization to any finite dimension. But is it "satisfactory"?
Specifically, what about (n-dimensional) volumes. Euclid 11--13 does not well axiomatize the method of exhaustion he uses for the volume of solids. And Dehn's theorem that already for polyhedra the theory of volume will not reduce to equidecomposition.
I should probably already know this, but what is known about the theory of n-dimensional volume in axiomatic n-dimensional Euclidean geometry?
. . .
Alfred Tarski and Givant, Steven, 1999, "Tarski's system of geometry," Bulletin of Symbolic Logic 5: 175-214.
Of course Tarski's axiomatization had precursors going all the way back to Euclid, and as is discussed in the Tarski and Givant paper none of his axioms are new with Tarski. There is some discussion of the actual development of Euclidean geometry from the axioms in that article, but I believe (though I do not actually have the book) the detailed development was presented in Schwabhauser, W. and Szmielew, W. and Tarski, A., Metamathematische Methoden in der Geometrie, Springer-Verlag, 1983 But the development is in showing how to go from the axioms to analytic geometry not in developing a direct extension of Euclid's methods. -- Bob -- Robert L. Knighten RLK@knighten.org