= Solution
A <liquid-in-glass thermometer> has a bulb connected to a narrow graduated capillary. <Thermal expansion> of its liquid relative to the glass raises or lowers the column. A simple linear model is
$$
\Delta\ell\simeq\frac{V_0(\beta_{\rm liquid}-\beta_{\rm glass})}{A}\Delta T,
$$
with capillary area $A$ and volumetric expansion coefficients $\beta$.
For the historical Celsius graduation, establish an ice point in an equilibrium mixture of pure ice and water, and a steam point in saturated steam at standard atmospheric <pressure>, here a barometer height of $760\,\mathrm{mm}$. Mark $0^\circ\mathrm C$ and $100^\circ\mathrm C$, then divide the interval into one hundred equal parts. This is a historical two-fixed-point instrument calibration, not the modern definition of the <kelvin> or of Celsius <temperature>. Nonuniform capillary area and nonlinear expansion limit exactness.
\b[Bulb contraction raises the column and makes the reading too high; retained expansion makes it too low.] Glass volume changes with age and thermal history; heating can cause temporary zero-point depression followed by slow recovery. Rechecking the ice point detects a shifted zero, although it does not diagnose every scale error.<Maintenance and Validation of Liquid-in-Glass Thermometers by Christina D. Cross and others>[NIST SP 1088, printed p. 4 and §§7.4–7.5: permanent and temporary bulb-volume changes]{ref}
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