Vernier scale

We will make measure­ments of lengths . But, before measuring some­thing, one should always define the units in which the measure­ments are made. The Russian guys remember a song that said: “But in ’parrots’ I am much taller!”

How to measure ’in parrots’ with the calliper? The tips that stick out below the main bar with the rule are used for external measures when one needs to measure, for example, the outer diam­eter of a tube. The tips for the inner measures, used to measure, for instance, the inner diam­eter of a tube, stick out over the bar. Finally, there is a ’tail’ that protrudes at the end of the bar, and serves to measure the depth, for example, of a hole. These three parts are fixed to a sliding bar measuring 15 parrots.

Let us measure a hexag­onal lock nut. Unfor­tu­nately, the refer­ence nick of the sliding bar showing the size of the nut, falls between two divi­sions of the main rule. This measure allows us to say only that the nut measures more than seven parrots, but less than eight. It is diffi­cult to define its size more precisely by eye.

One method, which do not compli­cate at all the construc­tion of the same calliper, but allows to do more accu­rate measures, was invented a few centuries ago. This only adds on the moving bar a new grad­uate scale, the vernier scale.

Let us take on this sliding bar ten nicks after the nick of refer­ence (the zero), that coin­cide with the nicks between 0 and 10 of the rule. Now compress evenly these ten divi­sions to occupy nine divi­sions of the rule. The vernier scale was invented in its modern form in 1631 by the math­e­mati­cian Pierre Vernier (1580–1637). In some languages, this device is called a nonius. Nonius is the Latin name of the Portuguese astronomer and math­e­mati­cian Pedro Nunes (1502–1578) who in 1542 invented a similar system.

It happens that the addi­tional rule built in this way allows to measure with an accu­racy of $0{.}1$ parrots. But how?

Let us measure the nut again. The zero refer­ence nick will be, as before, between the seventh and the eighth divi­sion. This means that our measure contains seven whole parrots. Now observe the divi­sions of the vernier scale from left to right and look for the nick that coin­cides with one of the divi­sions of the main scale. In our case this happens at the fifth nick. Conse­quently, the measure of the nut is equal to $7 + 5 × 0{.}1 = 7{.}5$ (parrots).

Explain math­e­mat­i­cally the argu­ment above. Think how to use the same idea to get measures with more preci­sion.