= Solution
An optically active <glucose> solution rotates the direction of <linear polarization>. With monochromatic illumination, rotating the <polarimeter analyzer> again produces a maximum and an extinction, but at angles shifted from the water-filled reference tube. In an ideal transparent sample, <Malus's law> gives
$$
I(\phi)=I_0\cos^2(\phi-\alpha),\qquad \boxed{\alpha=[\alpha]^T_\lambda\ell c.}
$$
The <specific rotation> $[\alpha]^T_\lambda$ is defined for a stated <temperature>, <wavelength>, solvent and concentration convention. With $\ell$ in decimetres and $c$ in grams per millilitre, the conventional numerical value of $[\alpha]$ must use the matching units. A broadband source generally gives coloured appearances because the rotation varies with <wavelength>; one analyser angle need not extinguish every colour.
<Biot's saccharimeter> is an angular <polarimeter>: a polarizing element, a tube of accurately known optical path length holding the sample between plane end windows, and a rotatable analysing element with an angular scale. The original arrangement used a black-glass polarizing mirror and a calcite analyser; later two-prism designs implement the same angular measurement. It should be distinguished from a later quartz-wedge compensating saccharimeter, which cancels the sample's rotation instead of directly reading the analyser rotation.<A Handbook of Sugar Analysis by Charles Albert Browne Jr.>[pp.84–85, original Biot polariscope and simple angular polariscope; chapter VI, quartz-wedge compensation]{ref}
For a historical <urine> estimate, clarify a sample without changing its sugar inventory, fill the tube without bubbles, make a solvent blank at the same settings, and measure the signed rotation relative to that blank. With a known dilution factor $D$ and an appropriate equilibrium <glucose> calibration,
$$
\boxed{c_{\rm original}=D\frac{\alpha_{\rm sample}-\alpha_{\rm blank}}{[\alpha]^T_\lambda\ell}.}
$$
Repeat the reading and check a known <glucose> solution. <Mutarotation> makes the time since preparation relevant. The inference is selective only to the extent that other optically active solutes have been excluded or accounted for; a net rotation is not proof that every contribution is <glucose>. This is a historical analytical application, not a replacement for present clinical assessment of <diabetes>.
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