Mathematicians versus practical men: the founding of the Advisory Committee for Aeronautics -- The air as an ideal fluid: classical hydrodynamics and the foundations of aerodynamics -- Early British work on lift and drag: Rayleigh flow versus the aerodynamics of intuition -- Lanchester's cyclic theory of lift and its early reception -- Two traditions: mathematical physics and technical mechanics -- Technische Mechanik in action: Kutta's arc and the Joukowsky wing -- The finite wing: Ludwig Prandtl and the Göttingen school -- '"We have nothing to learn from the Hun": realization dawns -- The laws of Prandtl and the laws of nature -- Pessimism, positivism, and relativism: aerodynamic knowledge in context