Biot also claimed that the substitutions for and and therefore Fresnel's expressions for and were empirically wrong because when Fresnel's intensities of spectral colors were mixed according to Newton's rules, the squared cosine and sine functions varied too smoothly to account for the observed sequence of colors. That claim drew a written reply from Fresnel, who disputed whether the colors changed as abruptly as Biot claimed, and whether the human eye could judge color with sufficient objectivity for the purpose. On the latter question, Fresnel pointed out that different observers may give different names to the same color. Furthermore, he said, a single observer can only compare colors side by side; and even if they are judged to be the same, the identity is of sensation, not necessarily of composition. Fresnel's oldest and strongest point – that thin crystals were subject to the same laws as thick ones and did not need or allow a separate theory – Biot left unanswered. Arago and Fresnel were seen to have won the debate. Moreover, by this time Fresnel had a new, simpler explanation of his equations on chromatic polarization. Breakthrough: Pure transverse waves (1821) In the draft memoir of 30 August 1816, Fresnel mentioned two hypotheses – one of which he attributed to Ampère – by which the non-interference of orthogonally-polarized beams could be explained if polarized light waves were partly transverse. But Fresnel could not develop either of these ideas into a comprehensive theory. As early as September 1816, according to his later account, he realized that the non-interference of orthogonally-polarized beams, together with the phase-inversion rule in chromatic polarization, would be most easily explained if the waves were purely transverse, and Ampère "had the same thought" on the phase-inversion rule. But that would raise a new difficulty: as natural light seemed to be unpolarized and its waves were therefore presumed to be longitudinal, one would need to explain how the longitudinal component of vibration disappeared on polarization, and why it did not reappear when polarized light was reflected or refracted obliquely by a glass plate. Independently, on 12 January 1817, Young wrote to Arago (in English) noting that a transverse vibration would constitute a polarization, and that if two longitudinal waves crossed at a significant angle, they could not cancel without leaving a residual transverse vibration. Young repeated this idea in an article published in a supplement to the Encyclopædia