 Vernier scale

A Vernier scale is an auxiliary graduated scale whose spacing differs slightly from the main scale. A fine displacement is read by finding which pair of graduation lines coincides. The device converts a difficult visual estimate of a small gap into a more conspicuous coincidence index. A caliper can attach the auxiliary scale to its sliding jaw.^[ref]

For an original ideal direct vernier, let one main-scale division be $d$ and let $N$ auxiliary divisions span $(N-1)d$. Each auxiliary division isIf the auxiliary zero is displaced by $x$ beyond main mark $m$, its graduation $j$ lies at $md+x+jv$. Coincidence with main mark $m+j$ gives $x=j(d-v)=j\Delta$. Thus the reading is $md+j\Delta$, for the stated geometry.

With $d=1\mathrm{mm}$ and $N=20$, the auxiliary divisions are $0.95\mathrm{mm}$ apart and the reading increment is $0.05\mathrm{mm}$. If the auxiliary zero is just beyond 12 mm and graduation 7 coincides, the reading is $12+7(0.05)=12.35\mathrm{mm}$.

This is a constructed example. The manufacturer guide gives the same 0.05 mm spacing-difference principle and distinguishes a longer vernier, whose auxiliary division is 0.05 mm shorter than two main divisions. That longer scale can improve readability while retaining the same increment; the reading rule must match the actual graduation geometry.^[ref]

A Vernier zero correction is needed when the closed or reference setting does not indicate the intended zero. For example, if the instrument reads $+0.10\mathrm{mm}$ at a true zero setting, subtracting that offset changes a later raw reading of $12.35$ to $12.25\mathrm{mm}$. The sign of the correction follows the observed error, not the direction of the object's length.

Fine graduations do not establish measurement accuracy. Jaw tilt, measuring force, thermal expansion, wear and viewing geometry can introduce errors larger than one increment. The primary guide identifies those physical sources explicitly.^[ref] Check reference closure and suitable calibrated artefacts, control contact force and record the relevant measurement uncertainty rather than treating the last readable digit as the whole error budget.