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A caliper measures a dimension by relating the separation of contacting measuring faces to a scale or readout. A sliding-jaw caliper has a fixed jaw and a movable jaw; appropriate faces can measure outside dimensions, inside dimensions or steps, while a depth bar can supply a separate depth measurement. These modes use different contact geometries even when they share one slider.^[ref]

A Vernier scale, dial or electronic readout converts slider position into a measurement indication. It does not remove the need to align the contacts with the desired length. For an outside measurement, the opposed faces must meet the intended surfaces without dirt, excessive force or tilt. An inside measurement uses its designated faces and can require geometry-specific corrections; it is not merely the outside-jaw reading with a different label.

An original small-angle calculation shows why jaw geometry matters. Suppose the displacement reference and contact point are separated by $h=40\mathrm{mm}$, and a moving-jaw tilt produces angle $\theta =0.002\mathrm{rad}$. The resulting offset contribution has magnitude approximatelyIt exceeds a $0.05\mathrm{mm}$ reading increment. These are stipulated dimensions and tilt, not a measured error for a particular caliper. The primary guide gives the same lever-arm mechanism for caliper jaw tilt error and identifies contact force, wear and thermal differences as physical error sources.^[ref]

A useful check closes clean outside faces under consistent light force, then compares readings at several separations with suitable calibrated artefacts such as gauge blocks. Repeating the comparison near different contact positions can reveal a geometry-dependent error that a single zero check misses. Apply a Vernier zero correction when using that readout, while retaining uncertainty from the reference, alignment and repeatability.

The measurement result concerns the specified object dimension under stated conditions, not just slider displacement. Thermal expansion of either the object or instrument, deformation under contact force and the selected faces belong in the measurement model. More displayed digits or a finer scale cannot by themselves establish better measurement accuracy.

Caliper jaw tilt error is a change in the indicated dimension caused by rotation of a caliper's moving jaw relative to its reference geometry. If a contact lies distance $h$ from the displacement reference, a small rotation $\theta$ gives a signed offset contributionThe sign depends on the direction of rotation and contact geometry. The approximation requires small $|\theta |$; it describes this angular contribution rather than every possible measurement error.

Mitutoyo gives a concrete geometry: a jaw slope of $0.01\mathrm{mm}$ over $50\mathrm{mm}$ and a contact depth of $40\mathrm{mm}$ produce a tip error of $0.008\mathrm{mm}$. Excessive contact force and wear of the guide can cause the moving-jaw rotation.^[ref]

An original diagnostic repeats a gauge block comparison at contact depths $h_1=10\mathrm{mm}$ and $h_2=40\mathrm{mm}$. With stipulated constant $\theta =2\times 10^{-4}\mathrm{rad}$ and other effects unchanged, the angular contributions are approximately $2\mu \mathrm{m}$ and $8\mu \mathrm{m}$. Their $6\mu \mathrm{m}$ difference isolates a depth-dependent effect that a single closed-jaw zero reading could miss. Real contacts may also change force or deformation, so this comparison does not uniquely identify rotation without controls.

A Vernier zero correction removes an estimated constant offset; it cannot generally remove an error proportional to contact depth or varying force. Use consistent light contact, a clean reference and a stable temperature, and compare several separations and contact positions. A fine Vernier scale improves reading increments without guaranteeing that these mechanical contributions are small.

This lever-arm mechanism is distinct from merely projecting a fixed length onto a tilted measurement axis: the latter involves $\cos \theta$ and has a quadratic small-angle change, whereas the offset term above is linear in $\theta$. Keeping the geometry in the measurement model prevents applying the wrong angular correction.

A Vernier zero correction removes an indicated offset observed at a known zero setting of a Vernier scale. If the raw indication is $r$ and the indication at true zero is $z$, the additive model givesThe zero error is $z$; the correction applied to a later reading is its negative. This model addresses an offset, not every possible scale or contact error.

For an original signed example, a raw reading $r=12.35\mathrm{mm}$ with closed-jaw indication $z=+0.10\mathrm{mm}$ gives $L=12.25\mathrm{mm}$. If the zero indication is instead $-0.10\mathrm{mm}$, the corrected reading is $12.45\mathrm{mm}$. A negative zero indication therefore requires adding its magnitude. It does not imply a negative object length.

The reference setting must represent the intended zero: clean, properly contacting outside faces on a caliper are one example. Record the actual signed indication and the scale convention, rather than relying on a memorized rule about which engraved zero lies to the left. The manufacturer guide explains the coincidence reading and identifies jaw tilt, force and viewing geometry as separate error sources.^[ref]

For standard uncertainties, the additive model givesWith stipulated independent $u(r)=0.03\mathrm{mm}$ and $u(z)=0.02\mathrm{mm}$, the combined value is approximately $0.0361\mathrm{mm}$. Subtracting a measured offset does not make that offset known exactly. Correlation can matter when both observations share an uncertain reference or drift contribution.^[ref]

A second reference checks the limitation. Suppose true zero reads $0.10\mathrm{mm}$ but a known $50.00\mathrm{mm}$ separation reads $50.60\mathrm{mm}$. Removing the zero offset leaves $50.50\mathrm{mm}$, still $0.50\mathrm{mm}$ high. A model $r=aL+z$ would have $a=1.01$ here. Measurement calibration across the working interval is needed to distinguish scale error from a zero correction, with the relevant measurement uncertainty retained.

William Hyde Wollaston developed optical methods for measuring the refractive index and optical dispersion of materials. His 1802 paper described determining a specimen's refractive behavior from the boundary at which total internal reflection begins in a contacting optical prism.^[ref]

The apparatus made the boundary into a measurement. A specimen contacted a glass prism, and hinged sighting arms located the transition between transmitted and totally reflected rays. Wollaston described a square-prism arrangement with a Vernier scale for reading the required geometric quantity. This contact method avoided having to shape the specimen itself into a prism.^[ref]

In modern notation, Snell's law gives the critical-angle relation $n_s=n_p\sin {\theta }_c$ for a specimen index $n_s$ below a prism index $n_p$. As an original example, if $n_p=1.60$ and ${\theta }_c=60^{\circ }$, then $n_s=1.60\sin 60^{\circ }\approx 1.386$. The angle is measured inside the prism from the interface normal. These stipulated numbers illustrate the method rather than reproducing Wollaston's measurements.

Wollaston also compared oils sold under the same name and reported differing refractive values. This gave the optical measurement a practical motivation: testing material consistency with only a small specimen.^[ref]