Shortly after his work on the foundations of geometry David Hilbert turned his attention to finding a suitable statement of the Dirichlet principle, from which to prove the Riemann mapping theorem and vindicate the topological program for complex analysis. Based on comments made in letters to Frege a major motivation for Hilbert's foray into geometry and independence proofs was to investigate the Archimedean axiom. Specifically, Hilbert mentions (to Frege) Dehn's dissertation on the Archimedean axiom and Legendre's theorem. This leads me to think that conformal mappings were on Hilbert's mind and to guess that Weierstrass's counterexample somehow concerned the Archimedean property. But I can't find anything in the secondary history/philosophy of math literature that quite puts all the pieces of the puzzle together--that Weierstrass had a counter-example is mentioned but details are skirted--and qua philosopher I'm bumping up against my mathematical horizons in piecing it together myself.