Ludwig Prandtl is properly credited with the development of the bounda
ry-layer idea in viscous flow, which was generalized to the method of
matched asymptotic expansions. However, the idea of matching a global
and a local approximation was previously applied to specific problems
in the nineteenth century by a number of well-known natural philosophe
rs, starting with Laplace in 1805. The author summarizes their applica
tions in hydrostatics, hydrodynamics, elasticity, electrostatics, and
acoustics, with particular attention to the process of matching.