# Vernier scale

A vernier scale is a device that lets the user measure more precisely than could be done unaided when reading a uniformly divided straight or circular measurement scale. It is a scale that indicates where the measurement lies in between two of the marks on the main scale. Verniers are common on sextants used in navigation, scientific instruments used to conduct experiments, machinists' measuring tools (all sorts, but especially calipers and micrometers) used to work materials to fine tolerances, and on theodolites used in surveying.

A close-up of a caliper's measurement scales. Assuming the caliper has no "zero error" (that is, it registers 0.00 mm when fully closed) the image shows a reading of 3.58 mm ± 0.02 mm. This is found by adding 3.00 mm (left red mark) read off from the fixed main (upper) scale to 0.58 mm (right red mark) obtained from the sliding vernier (lower) scale. The main scale reading is determined by the rightmost tick on the main scale that is to the left of the zero tick on the vernier scale. The vernier reading is found by locating the closest aligned lines between the two scales. The 0.02 mm inscription indicates the caliper's precision and is just the width that corresponds to the smallest interval on the vernier scale.

## History

Mechanical displacement gauges with vernier scales on wall cracks (Moika Palace, Saint Petersburg).

Calipers without a vernier scale originated in ancient China as early as the Qin dynasty (AD 9).[1][2] The secondary scale, which contributed extra precision, was invented in 1631 by French mathematician Pierre Vernier (15801637). Its use was described in detail in English in Navigatio Britannica (1750) by mathematician and historian John Barrow.[3] While calipers are the most typical use of Vernier scales today, they were originally developed for angle-measuring instruments such as astronomical quadrants.

In some languages, the Vernier scale is called a nonius. It was also commonly called a nonius in English until the end of the 18th century.[4] Nonius is the Latinised name of the Portuguese mathematician Pedro Nunes. The name "vernier" was popularised by the French astronomer Jérôme Lalande (1732–1807) through his Traité d'astronomie (2 vols) (1764).[5]

## Construction

In the following, N is the number of divisions the maker wishes to show at a finer level of measure.

Vernier scale direct vernier

Direct verniers are the most common. The indicating scale is constructed so that when its zero point is coincident with the start of the data scale, its graduations are at a slightly smaller spacing than those on the data scale and so none but the last graduation coincide with any graduations on the data scale. N graduations of the indicating scale cover N−2 graduations of the data scale.

Retrograde verniers are found on some devices, including surveying instruments.[6] A retrograde vernier is similar to the direct vernier, except its graduations are at a slightly larger spacing. N graduations of the indicating scale cover N+1 graduations of the data scale. The retrograde vernier also extends backwards along the data scale.

## Use

Animation of a caliper measurement using a vernier scale. Click to enlarge
Vernier scale use 0.02 scale measurement is 19.44 mm

The main use of the vernier caliper is to measure the internal and the external diameters of an object. To measure using a vernier scale, the user first reads the finely marked "fixed" scale (in the diagram). This measure is typically between two of the scale's smallest graduations. The user then reads the finer vernier scale (see diagram), which measures between the smallest graduations on the fixed scale—providing much greater accuracy.

It is also used in measuring an object to its lowest decimal point.

## Least Count of Vernier scale

Let the measure of the smallest main scale reading, that is the distance between two consecutive graduations (also called its pitch) be S and the distance between two consecutive Vernier scale graduations be V such that the length of (n-1) main scale divisions is equal to n Vernier scale divisions.Then,

the length of (n-1) main scale divisions = the length of n vernier scale division

or, (n-1)S = nV

or, nS-S = nV

or, S = nS - nV

or, S/n = (S-V)

or (Pitch)/(Number of Vernier scale divisions) = (Length of one main scale division - Length of one Vernier scale division)

S/n and (S-V) are both equal to the least count of Vernier Caliper, and are also termed as "Vernier constant".

## Examples

On decimal measuring instruments, as in the diagram below, the indicating scale has 10 graduations that cover the same length as 9 on the data scale. Note that the vernier's 10th graduation is omitted.

On an angular measurement instrument, the data scale might be in half-degrees, with an indicator scale that provides 30 1-minute graduations (that span 29 of the half-degree graduations).

## Mechanism of a vernier scale

Vernier scale How a vernier scale works

The vernier scale is constructed so that it is spaced at a constant fraction of the fixed main scale. So for a decimal measuring device each mark on the vernier is spaced nine tenths of those on the main scale. If you put the two scales together with zero points aligned, the first mark on the vernier scale is one tenth short of the first main scale mark, the second two tenths short, and so on up to the ninth mark—which is misaligned by nine tenths. Only when a full ten marks are counted is there alignment, because the tenth mark is ten tenths—a whole main scale unit short, and therefore aligns with the ninth mark on the main scale.

Now if you move the vernier by a small amount, say, one tenth of its fixed main scale, the only pair of marks that come into alignment are the first pair, since these were the only ones originally misaligned by one tenth. If we move it two tenths, the second pair aligns, since these are the only ones originally misaligned by that amount. If we move it five tenths, the fifth pair aligns—and so on. For any movement, only one pair of marks aligns and that pair shows the value between the marks on the fixed scale.

Vernier acuracy

Main article: Vernier acuity

Vernier scales work so well because most people are especially good at detecting which of the lines is aligned and misaligned, and that ability gets better with practice, in fact far exceeding the optical capability of the eye. This ability to detect alignment is called 'Vernier acuity'.[7] Historically, none of the alternative technologies exploited this or any other hyperacuity, giving the Vernier scale an advantage over its competitors.[8]

## Zero error

Zero error is defined as such a condition when a measuring instrument registers a reading when there should not be any reading. In case of vernier calipers it occurs when a zero on main scale does not coincide with a zero on vernier scale. Rather the zero error may be of two types i.e. when the scale is towards numbers greater than zero it is positive else negative. The method to use a vernier scale or caliper with zero error is to use the formula: actual reading = main scale + vernier scale − (zero error). Zero error may arise due to knocks that cause the calibration at the 0.00 mm when the jaws are perfectly closed or just touching each other.

When the jaws are closed and if the reading is 0.10mm, the zero error is referred to as +0.10mm. The method to use a vernier scale or caliper with zero error is to use the formula 'actual reading = main scale + vernier scale − (zero error)' thus the actual reading is 19.00 + 0.54 − (0.10) = 19.44 mm

Positive zero error refers to the fact that when the jaws of the vernier caliper are just closed, the reading is a positive reading away from the actual reading of 0.00mm. If the reading is 0.10mm, the zero error is referred to as +0.10 mm.

When the jaws are closed and if the reading is −0.08mm, the zero error is referred to as +0.08 mm. The method to use a vernier scale or caliper with zero error is to use the formula 'actual reading = main scale + vernier scale + (zero error)' thus the actual reading is 19.00 + 0.36 + (+0.08) = 19.44 mm
`                                FORMULA for LEAST COUNT of VERNIER CALIPER = leastcount of main scale/number of division on vernier scale`

Zero Error (Z.E) = + or- n* Least Count(L.C).

## References

1. Ronan, Colin A.; Needham, Joseph (24 June 1994). The Shorter Science and Civilisation in China: 4. Cambridge University Press. p. 36. ISBN 978-0-521-32995-8. adjustable outside caliper gauge... self-dated at AD 9. An abridged version.
2. "Bronze Caliper of the Wang Mang Regime". Retrieved 26 November 2013.
3. Barrow called the device a Vernier scale. See: John Barrow, Navigatio britannica: or a complete system of navigation … (London, England: W. and J. Mount and T. Page, 1750), pp. 140–142, especially page 142.
4. Daumas, Maurice, Scientific Instruments of the Seventeenth and Eighteenth Centuries and Their Makers, Portman Books, London 1989 ISBN 978-0-7134-0727-3
5. Lalande, Jérôme (1746), Astronomie, vol. 2 (Paris, France: Desaint & Saillant), pages 859-860.
6. Davis, Raymond, Foote, Francis, Kelly, Joe, Surveying, Theory and Practice, McGraw-Hill Book Company, 1966 LC 64-66263
7. Vernier acuity definition at the Online Medical Dictionary
8. Kwan, A. (2011). "Vernier scales and other early devices for precise measurement". American Journal of Physics. 79 (4): 368. doi:10.1119/1.3533717.