A Vernier scale is a small,
moveable scale placed next to the main scale of a measuring
instrument. It is named after its inventor, Pierre Vernier (1580  1637). It allows us to make measurements to a
precision of a small fraction of the smallest division on the
main scale of the instrument. (In the first example below the
"small fraction" is one tenth.) Vernier scales have been used with
many instruments, for example, spectroscopes, supports for
astronomical telescopes etc. One specific example, the Vernier
caliper, is considered below. 



Using a Vernier Scale 

The diagram below shows a Vernier scale reading zero. Notice
that 10 divisions of the Vernier scale have the same length
as 9 divisions of the main scale. 



In the following examples we will assume that the smallest
division on the main scale is 1mm
so the divisions on the Vernier scale are 0.9mm
each.
The position of the zero of the Vernier scale tells us
the number of cm and mm in our measurement. For example, in the
diagram below, the reading is a little over 1.2cm. 



To find a more precise reading, consider
the next diagram (which is a magnified view of part of the previous
diagram). 



We are trying to find the distance marked
x. 

To find x, find the mark on the Vernier scale which
most nearly coincides with a mark on the main scale. In
this case, it is obviously the third mark. 

It is clear that x = d  d' 

Remembering that each division on the main scale is
1mm and that each division on the
Vernier scale is 0.9mm, we have: 

x =
3mm  3(0.9)mm
= 3(0.1mm = 0.3mm 

Therefore, the reading in this example is
1.23mm 

Similarly, if it had been, for example, the seventh mark on the
Vernier scale which had been exactly opposite a mark on the main
scale, the reading would be 1.27cm 



Hence, the level of precision of an instrument which has a
Vernier scale depends on the difference between the size of
the smallest division on the main scale and the size of the smallest
division on the Vernier scale. 



In the example above, this difference is 0.1mm
so measurements made using this instrument should be stated as:
reading ±0.1mm 



Another instrument might have a scale like the one shown in the
next diagram. 



Using the same logic as above we can see that, in this case, the
precision is much better: 

1mm  (49/50)mm
= (1/50)mm = 0.02mm 

and so, results of measurements made using this instrument
should therefore be stated as: reading ±0.02mm. 



This principle is used in the Vernier caliper shown below. 





The diagrams below illustrate how to use a Vernier caliper to
measure: 

A. the internal diameter of a hollow cylinder B. the external
dimensions of an object C. the depth of a hole in a piece of
metal. 



A 



B 



C 



Vernier scales are also use to measure
small angles (or more specifically small angular displacements) on
the supports for astronomical telescopes and many other instruments.
See here for some pictures. 

The same basic idea applies, that is, for
example, 10 divisions on a circular Vernier scale correspond to 9°
on the main angular scale. 
