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Christiaan Huygens (1629–1695) connected mathematical mechanics with the design of a more accurate pendulum clock, and worked on astronomy, probability and the wave account of light. His optical account treats a wavefront as generating secondary wavelets, the geometric idea behind Huygens' principle.^[ref]

The clock provides a useful scientific motivation: a small-amplitude simple pendulum has period approximately $2\pi \sqrt{\ell /g}$, so a stable length supplies a repeatable time scale. Finite-amplitude corrections show why an approximate classroom formula does not by itself produce a perfect timekeeper. In wave propagation, the wavefront construction addresses a different problem: how a disturbance advances through space and changes direction under refraction.

Huygens' principle treats each point of a wavefront as the source of a secondary wavelet. The later wavefront is the common forward envelope of those wavelets. Consider a plane front reaching a flat interface between media with speeds $v_1$ and $v_2$. In the same interval $\Delta t$, the incident front advances $v_1\Delta t$ while the first transmitted wavelet grows to radius $v_2\Delta t$.

The two right triangles sharing the interface segment of length $d$ give $d\sin i=v_1\Delta t$ and $d\sin r=v_2\Delta t$, where $i$ and $r$ are ray angles to the normal. HenceThis is Snell's law. The frequency is unchanged at a stationary interface, while wavelength changes with the propagation speed.

Huygens' principle treats every point of an advancing wavefront as a source of secondary wavelets; their common envelope forms the next wavefront. Consider two points of an incident plane wave reaching the interface at different times. During the same interval $\Delta t$, the remaining incident portion travels $v_1\Delta t$ while the first transmitted wavelet expands a distance $v_2\Delta t$. The two wavefront right triangles share their interface segment, soThis is Snell's law, with both angles measured from the surface normal. The speeds are phase velocities at the common frequency in stationary isotropic media. If the equation requires $|\sin {\theta }_2|>1$, there is no propagating refracted ray and total internal reflection applies instead.