Vernier scale
= Vernier scale
{c}
{wiki}
A \b[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.<Quick Guide to Precision Measuring Instruments: Calipers by Mitutoyo>[printed pp. 16–17, scale reading and Types of Vernier Scale]{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 is
$$
v=\frac{N-1}{N}d,\qquad \Delta=d-v=\frac dN.
$$
If 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 $12+7(0.05)=12.35\,\mathrm{mm}$.
\Image[/vernier-scale-coincidence.png]
{title=An ideal Vernier scale reading 12.35 mm: twenty auxiliary intervals span 19 mm, and graduation 7 coincides with a main-scale line}
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.<Quick Guide to Precision Measuring Instruments: Calipers by Mitutoyo>[printed p. 17, standard and long Vernier scales]{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.<Quick Guide to Precision Measuring Instruments: Calipers by Mitutoyo>[printed p. 17, Sources of Error and Moving Jaw Tilt Error]{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.