Vernier scale

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A vernier scale is an additional scale which allows a distance or angle measurement to be read more precisely than directly reading a uniformly-divided straight or circular measurement scale. It is a sliding secondary scale that is used to indicate where the measurement lies when it is 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.

When a measurement is taken by mechanical means using one of the above mentioned instruments, the measure is read off a finely marked data scale (the "fixed" scale, in the diagram). The measure taken will usually be between two of the smallest graduations on this scale. The indicating scale ("vernier" in the diagram) is used to provide an even finer additional level of precision without resorting to estimation.



The vernier scale was invented in its modern form in 1631 by the French mathematician Pierre Vernier (1580–1637). In some languages, this device is called a nonius. It was also commonly called a nonius in English until the end of the 18th century.[1] Nonius is the Latin name of the Portuguese astronomer and mathematician Pedro Nunes (1502–1578) who in 1542 invented a related but different system for taking fine measurements on the astrolabe that was a precursor to the vernier.[1][2]


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

Direct and retrograde verniers are read in the same manner.


When a length is measured the zero point on the indicating scale is the actual point of measurement, however this is likely to be between two data scale points. The indicator scale measurement which corresponds to the best-aligned pair of indicator and data graduations yields the value of the finer additional precision digit.


On instruments using decimal measure, as shown in the diagram below, the indicating scale would have 10 graduations covering the same length as 9 on the data scale. Note that the vernier's 10th graduation is omitted.

On an instrument providing angular measure, the data scale could be in half-degrees with an indicator scale providing 30 1-minute graduations (spanning 29 of the half-degree graduations).

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