Editing Augustin-Jean Fresnel (section)

==== Second memoir (1822–26) ====

Having presented the pieces of his theory in roughly the order of discovery, Fresnel needed to rearrange the material so as to emphasize the mechanical foundations;<ref>Grattan-Guinness, 1990, pp.&nbsp;896–897.&nbsp; Silliman, 1967, pp.&nbsp;262–263; 2008, p.{{nnbsp}}170</ref> and he still needed a rigorous treatment of Biot's dihedral law.<ref>Buchwald, 1989, pp.&nbsp;286–287,{{tsp}}447.</ref> He attended to these matters in his "second memoir" on double refraction,<ref name=fresnel-1827>Fresnel, 1827.</ref> published in the ''Recueils'' of the Académie des Sciences for 1824; this was not actually printed until late 1827, a few months after his death.<ref>Fresnel, 1866–70, vol.&nbsp;2, p.&nbsp;800n. Although the original publication (Fresnel, 1827) shows the year "1824" in selected page footers, it is known that Fresnel, slowed down by illness, did not finish the memoir until 1826 (Buchwald, 1989, pp.&nbsp;289,{{px2}}447, citing Fresnel, 1866–70, vol.&nbsp;2, p.&nbsp;776n).</ref> In this work, having established the three perpendicular axes on which a displacement produces a parallel reaction,<ref>Fresnel, 1827, tr.&nbsp;Hobson, pp.&nbsp;266–273.</ref> and thence constructed the surface of elasticity,<ref>Fresnel, 1827, tr.&nbsp;Hobson, pp.&nbsp;281–285.</ref> he showed that Biot's dihedral law is exact provided that the binormals are taken as the optic axes, and the wave-normal direction as the direction of propagation.<ref>Fresnel, 1827, tr.&nbsp;Hobson, pp.&nbsp;320–322; Buchwald, 1989, p.&nbsp;447.</ref>

As early as 1822, Fresnel discussed his perpendicular axes with [[Augustin-Louis Cauchy|Cauchy]]. Acknowledging Fresnel's influence, Cauchy went on to develop the first rigorous theory of elasticity of non-isotropic solids (1827), hence the first rigorous theory of transverse waves therein (1830)—which he promptly tried to apply to optics.<ref>Grattan-Guinness, 1990, pp.&nbsp;1003–1009,{{tsp}}1034–1040,{{tsp}}1043; Whittaker, 1910, pp.&nbsp;143–145; Darrigol, 2012, p.&nbsp;228.&nbsp; Grattan-Guinness offers evidence against any earlier dating of Cauchy's theories.</ref> The ensuing difficulties drove a long competitive effort to find an accurate mechanical model of the aether.<ref>Whittaker, 1910, chapter&nbsp;{{serif|V}}; Darrigol, 2012, chapter&nbsp;6; Buchwald, 2013, pp.&nbsp;460–464.</ref> Fresnel's own model was not dynamically rigorous; for example, it deduced the reaction to a shear strain by considering the displacement of one particle while all others were fixed, and it assumed that the stiffness determined the wave velocity as in a stretched string, whatever the direction of the wave-normal. But it was enough to enable the wave theory to do what selectionist theory could not: generate testable formulae covering a comprehensive range of optical phenomena, from ''mechanical'' assumptions.<ref>Fresnel, 1827, tr.&nbsp;Hobson, pp.&nbsp;273–281; Silliman, 1967, p.&nbsp;268n; Buchwald, 1989, p.&nbsp;288.</ref>