Reading a Vernier Scale on a Linear Main Scale

Vernier scales are typically used to add precision to standard, evenly divided scales (such as the scales on Thorlabs' rotation, goniometric, or translation mounts). A vernier scale has found common use in many precision measurement instruments, the most common being calipers and micrometers. The vernier scale uses two scales side-by-side: the main scale and the vernier scale. The direct vernier scale has a slightly smaller spacing between its tick marks owing to the vernier scale having N ticks for every N - 1 ticks on the main scale. Hence, the lines on the main scale will not line up with all the lines on the vernier scale. Only one line from the vernier scale will match well with one line of the main scale, and that is the trick to reading a vernier scale.

Figures 1 through 3 show a linear vernier scale system for three different situations. In each case, the scale on the left is the main scale, while the small scale on the right is the vernier scale. When reading a vernier scale, the main scale is used for the gross number, and the vernier scale gives the precision value. In this manner, a standard ruler or micrometer can become a precision instrument.

The 0 on the vernier scale is the "pointer" (marked by a red arrow in Figures 1 - 5) and will indicate the main scale reading. In Figure 1 we see the pointer is lined up directly with the 75.6 line. Notice that the only other vernier scale tick mark that lines up well with the main scale is 10. Since the pointer lines up with the main scale’s 75.6, the reading from Figure 1 is 75.60 (in whatever units the instrument measures).

That is essentially all there is to reading a vernier scale. It's a very straightforward way of increasing the precision of a measurement instrument. To expound, let’s look at Figure 2. Here we see that the pointer is no longer aligned with a line on the main scale, but instead it is slightly above 75.6 and below 75.7; thus, the gross measurement is 75.6. The first vernier line that coincides with a main scale line is the 5, shown with a blue arrow. The vernier scale gives the final digit of precision; since the 5 is aligned to the main scale, the precision measurement for Figure 2 is 75.65.

Since this vernier scale is 10% smaller than the main scale, moving the vernier scale by 1/10 of the main scale will align the next vernier marking. This asks the obvious question: what if the measurement is within the 1/10 precision of the vernier scale? Figure 3 shows just this. Again, the pointer line is in between 75.6 and 75.7, yielding the gross measurement of 75.6. If we look closely, we see that the vernier scale 7 (marked with a blue arrow) is very closely aligned to the main scale, giving a precision measurement of 75.67. However, the vernier scale 7 is very slightly above the main scale mark, and we can see that the vernier scale 8 (directly above 7) is slightly below its corresponding main scale mark. Hence, the scale on Figure 3 could be read as 75.673 ± 0.002. A reading error of about 0.002 would be appropriate for
this instrument.

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Figure 1: An example of how to read a vernier scale. The red arrow indicates what is known as the pointer. Since the tick mark labeled 10 on the vernier scale aligns with one of the tick marks on the main scale, this vernier scale is reading 75.60 (in whatever units the instrument measures).

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Figure 2: The red arrow indicates the pointer and the blue arrow indicates the vernier line that matches the main scale. This scale reads 75.65.

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Figure 3: The red arrow indicates the pointer, and the blue arrow indicates the vernier line that matches the main scale. This scale reads 75.67 but can be accurately read as
75.673 ± 0.002.

Reading a Vernier Scale on a Rotating Main Scale

The vernier scale may also be used on rotating scales where the main scale and vernier scale do not share units. Figures 4 and 5 show a vernier scale system for two different situations where the main scale is given in degrees and the vernier scale has ticks every 5 arcmin (60 arcmin = 1°). In each case, the scale on the top is the main scale, while the small scale on the bottom is the vernier scale.

In Figure 4 we see the pointer is lined up directly with the 341° line. Notice that the only other vernier scale tick marks that line up well with the main scale are ±60 arcmin. Since the pointer lines up with the main scale at 341°, the reading from Figure 4 is 341.00°.

There are two ways to determine the reading if the zero on the vernier scale line is between two lines of the main scale. For the first method, take the line on the left side of the pointer on the vernier scale and subtract that value (in arcmin) from the value on the main scale that is to the right on the main scale. As an example, in Figure 5 the vernier pointer is between 342° and 343°; using the left blue arrow of the vernier scale results in 343° - 15 arcmin = 342.75°. The second method is to take the reading from the blue arrow on the right side of the vernier pointer and add that value to the lower value from the main scale; reading the right blue arrow in Figure 5 results in 342° + 45 arcmin = 342.75°.

As we've seen here, vernier scales add precision to a standard scale measurement. While it takes a bit of getting used to, with a little practice, reading these scales is fairly straightforward. Vernier scales, whether they are direct or retrograde*, are read in the same fashion.

*A retrograde vernier scale has a larger spacing between its tick marks with N ticks for every N + 1 ticks on the main scale.

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Figure 4: An example of a vernier scale where the main scale and the vernier scale are in different units (degrees and arcmins, respectively). The red arrow indicates the pointer. This scale reads 341.00°.

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Figure 5: The red arrow indicates the pointer and the blue arrows give the precision value from the vernier scale.
This scale reads 342.75°.