In the days before digital readouts, a vernier scale was a common thing for precision tools like micrometers, calipers and other precision measurement tools. The principle is simple. First create a regular scale such as the one shown above. On the top it ithe finest division is 16ths of an inch, on the bottom we have inches, 1/10ths of inches and that is divided even further in to 4 divisions leaving us with 1/40th of an inch (0.025 inches) for the smallest scale. Since we are reading the scale by eye, it is not useful to go much smaller unless we are looking with a magnifying glass - but there is an easier way to get a finer measurement. We use a secondary scale.
The secondary scale is divided such that it has the same number of increments as you would want between divisions, but instead of squeezing them in to a small space, they are spread out almost at the same spacing as the other divisions. I say almost because they are spread so that there is one more of them than there would be normally. Look carefully at the scale below:
The eight divisions on the secondary scale take the same space as seven do on the primary scale. What this means is that only one of the marks is going to line up at a time. We can see here that the mark at 5 on the secondary scale is the one closest to aligning. Since the 0 is between 0 and the first 1/16th mark on the primary scale we know that the object being measured is 5/128ths of an inch. Actually it is a tiny bit more as the marks are not completely aligned.
We can confirm this on the 1000th's scale. There the zero mark is between the first and second marks on the primary scale. Looking further we see that the 15 mark is the closest to aligning. Add up the 0.025" from being past the first primary division and the 0.015" from the vernier scale and you have 0.40, slightly more than 5/128ths (0.039 inches).
