The new terminology was useful in the memoir of January 1823, containing the detailed derivations of the sine and tangent laws: in that same memoir, Fresnel found that for angles of incidence greater than the critical angle, the resulting reflection coefficients were complex with unit magnitude. Noting that the magnitude represented the amplitude ratio as usual, he guessed that the argument represented the phase shift, and verified the hypothesis by experiment.Lloyd, 1834, pp.369–70; Buchwald, 1989, pp.393–4,453; Fresnel, 1866, pp.781–96. The verification involved
