[6] Huygens eventually was able to solve this problem by means of elliptical waves in 1677 and confirmed his theory by experiments mostly after critical reactions in 1679.

By studying the symmetry of the crystal, Huygens was able to determine the direction of the axis of the ellipsoids, and from the refraction properties of the abnormal ray he established the proportion between the axes.

[3][8] Huygens intended to publish his results as part of the Dioptrica but decided to separate his theory of light from the rest of the work at the last minute, marking the transition from geometrical to physical optics.

[10] In the first chapter, Huygens describes light as a disturbance which moves in a material medium of an unknown nature which he calls ethereal, and which is different from that which propagates sound.

Huygens considers that the structure of matter is atomic, made up of an assembly of particles "which touch each other without composing a continuous solid.

Huygens carefully explains the differences between transparent and opaque media in terms of their particulate composition, specifically exploring atmospheric refraction.

Huygens cuts a piece of the crystal and studies the geometry of light propagation inside it before guiding the reader through a series of step-by-step empirical investigations.

[14] Thus, when light travels through the crystal, it breaks into two wave surfaces that follow distinct paths within it, resulting in two refracted rays being observed.

By reducing the ray to a geometrical construct devoid of physical character, Huygens was able to treat the theory of light kinematically (and thereby mathematically), allow him to succeed where his predecessors have failed.

[10][16] Although the completeness of Huygens's analysis is impressive, he was unable to comprehend the effect that we now recognize as polarization, which occurs if the refracted ray is directed through a second crystal of which the orientation is varied.