Christiaan Huygens, in his Treatise on Light (1690), paid much attention to the threshold at which the incident ray is "unable to penetrate into the other transparent substance". Although he gave neither a name nor an algebraic expression for the critical angle, he gave numerical examples for glass-to-air and water-to-air incidence, noted the large change in the angle of refraction for a small change in the angle of incidence near the critical angle, and cited this as the cause of the rapid increase in brightness of the reflected ray as the refracted ray approaches the tangent to the interface. Huygens' insight is confirmed by modern theory: in Eqs. (13) and (15) above, there is nothing to say that the reflection coefficients increase exceptionally steeply as θ t approaches 90°, except that, according to Snell's law, θ t itself is an increasingly steep function of θ i.