Vernier scale

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Vernier caliper scales; main at top, vernier at bottom. It reads 3.58 mm ± 0.02 mm by adding 3.00 mm (left red mark) on the fixed main scale to vernier 0.58 mm (right red mark). The main scale reading is that to the left of the zero on the vernier scale. The vernier reading is found by locating the best aligned lines between the two scales. The 0.02 mm engraving indicates the caliper's readability and is the "Vernier constant" for this scale.

A vernier scale is a visual aid to take an accurate measurement reading between two graduation markings on a linear scale by using mechanical interpolation; thereby increasing resolution and reducing measurement uncertainty by using Vernier acuity to reduce human estimation error.

The vernier is a subsidiary scale replacing a single measured-value pointer, and has for instance ten divisions equal in distance to nine divisions on the main scale. The interpolated reading is obtained by observing which of the vernier scale graduations is co-incident with a graduation on the main scale, which is easier to perceive than visual estimation between two points. Such an arrangement can go to higher resolution by using higher scale ratios, known as the vernier constant. A vernier may be used on circular or straight scales where a simple linear mechanism is adequate. Examples are calipers and micrometers to measure to fine tolerances, on sextants for navigation on theodolites in surveying, and generally on scientific instruments. The Vernier principle of interpolation is also used for electronic displacement sensors such as absolute encoders to measure linear or rotational movement, as part of an electronic measuring system.


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 (1580–1637). 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, a precursor to the Vernier scale developed by Portuguese mathematician, cosmographer Pedro Nunes (1502–1578), latinised as Petrus Nonius. It was known by this name in English until the end of the 18th century.[4]

The name "vernier" was popularised by the French astronomer Jérôme Lalande (1732–1807) through his Traité d'astronomie (2 vols) (1764).[5]

Mechanism of a vernier scale[edit]

Vernier caliper with a vernier constant of 0.1 for clarity of operation. The standard for a caliper is usually a constant of 0.02
Vernier caliper scale with the normal 0.02 vernier constant, showing measurement of object at 19.44 mm to two decimal places

The use of the vernier scale is shown on a vernier caliper which measures the internal and the external diameters of an object.

The vernier scale is constructed so that it is spaced at a constant fraction of the fixed main scale. So for a vernier with a constant of 0.1, 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.

Least count or vernier constant[edit]

The difference between the value of one main scale division and the value of one Vernier scale division is known as least count of the Vernier. It is also known as Vernier constant. 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 scale, and are also called the vernier constant.

Vernier acuity[edit]

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'.[6] Historically, none of the alternative technologies exploited this or any other hyperacuity, giving the Vernier scale an advantage over its competitors.[7]

Zero error[edit]

Zero error is defined as the condition where 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. 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 to be thrown off at the 0.00 mm when the jaws are perfectly closed or just touching each other. Perfection is not necessarily equal (or scalar) to zero error. "Knocks" seem an excellent example of mathematical imperfection. Alignment of linear and rotational mathematics is a difficult but interesting task as previously described by Pierre Vernier and much elaborated in later ages.

Vernier micrometer reading 5.783 mm ± 0.001 mm. made up from 5.5mm on main screw lead scale, 0.28mm on screw rotation scale, and 0.003mm added from vernier.
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 case when the jaws of the vernier caliper are just closed and 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.

Negative zero error refers to the case when the jaws of the vernier caliper are just closed and the reading is a negative reading away from the actual reading of 0.00mm. If the reading is 0.08mm, the zero error is referred to as +0.08mm. If positive, the error is subtracted from the mean reading the instrument reads. Thus if the instrument reads 4.39 cm and the error is +0.05, the actual length will be 4.39-0.05=4.34 cm. If negative, the error is added to the mean reading the instrument reads. Thus if the instrument reads 4.39 cm and as above the error is -0.05 cm, the actual length will be 4.39+0.05=4.44 cm. (Considering that, the quantity is called zero correction which should always be added algebraically to the observed reading to the correct value.)

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

Direct and retrograde verniers[edit]

Direct verniers are the most common. The indicating scale is constructed so that when its zero point coincides 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−1 graduations of the data scale.

Retrograde verniers are found on some devices, including surveying instruments.[8] A retrograde vernier is similar to the direct vernier, except its graduations are at a slightly larger spacing than on the main scale. N graduations of the indicating scale cover N+1 graduations of the data scale. The retrograde vernier also extends backwards along the data scale.

Direct and retrograde verniers are read in the same manner.

See also[edit]


  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". Archived from the original on 31 August 2014. 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. ^ Vernier acuity definition at the Online Medical Dictionary
  7. ^ Kwan, A. (2011). "Vernier scales and other early devices for precise measurement". American Journal of Physics. 79 (4): 368. doi:10.1119/1.3533717.
  8. ^ Davis, Raymond, Foote, Francis, Kelly, Joe, Surveying, Theory and Practice, McGraw-Hill Book Company, 1966 LC 64-66263

External links[edit]

Classroom activity on the Vernier scale at Wikiversity