How to read a Vernier scale
The vernier scale is a decidedly low-tech way to make ultra-fine measurements of length or angle. Today, there are plenty of digital calipers and micrometers, but plenty of machinists still use a trusty Vernier. The interesting thing is that a pair of Vernier calipers is really nothing more than a sliding jaw with one ruler sliding against another - how the HECK can it measure in 0.01mm increments??! After all, if you had a ruler that was marked out in hundredths of millimeters, you'd scarcely even be able to see the marks...
The vernier scale, cleverly devised by Pierre Vernier (1580–1637) makes use of the little known fact that humans are actually very VERY good at seeing how well things are lined up. So good, in fact, that if you are comparing two tiny marks placed end-to-end against one another, it's actually easy to tell if they are misaligned by as little as 1/100th of a millimeter! Vernier put this this perceptual aptitude to use by designing a sliding scale that moves against a ruler. He scaled the sliding scale so that, over its whole length, only one of its marks would ever perfectly align to a mark on the aforementioned ruler. To read a fractional mm, you just look along the sliding scale, and choose the number that is most perfectly aligned to a mark on the ruler above it.
It's actually easier to show you than to explain it!
The picture below is a closeup of the scale on a machinist's calipers. The top part (reading 0...1...2...3...) is the fixed ruler part of the scale. The smaller scale below it (reading 0-6...) is the sliding vernier scale. To read the calipers, first look to see where the zero mark of the vernier scale falls on the ruler above. The zero is the whole-millimeter marker. In this case, it reads 3 (see the top red mark on the left). It is between 3 and 4, so it reads 3... it has not yet gotten to 4.
Now for the fractional millimeter reading and some vernier scale magic! Start by looking at the zero mark of the vernier scale. Notice it doesn't line up with either of the marks on the mm ruler above it? Look to the right, and stop when you see one of the vernier marks perfectly aligned with the mm scale above it... (hint: there is a red mark down there for cheats!) Once you've found a pair of aligned marks, just read the vernier scale's measurement, denoted as reading in increments of 0.02mm above.
It reads 58, which is 0.58mm. Add it to the whole millimeter read we found first, and it reads 3.58 mm. This is even easier to do with a real scale, and not with a pixelated image of a scale!
The Vernier scale was originally invented to measure angles on an advanced protractor that was used to measure positions of stars. Once this proved to work well, the method was adapted to a wide assortment of other measuring devices... basically anything that involved finding the precise distance between a pair of moving parts. The precision can be enhanced by making the vernier scale portion longer, by using narrower lines for marks, and by adding a small lens, something akin to a microscope. Various arrangements of miniature gears can also be employed to scale-up the motion of the vernier scale against the ruler, but the main bit of technology is that human line-alignment-checking aptitude. With all of these innovations at work, mechanical tools were made that can easily measure in mere thousandths of millimeters. In fact, it wasn't until 1880s that a new method for measuring distances began to supersede the vernier scale: Enter the Michelson Interferometer. Interferometry being a full topic in its own right, I'll end with a number of vernier calipers set to various distances. See if you can read them without reading the answers I placed in the captions. How fun! ;->
You probably want to right click and open each image in a separate tab so you can see a full-resolution version.