The Vernier Scale

The vernier scale is a clever way of improving the precision of a reading on an instrument scale. It has been used on quite a number of instruments, many of which have nothing to do with surveying. Machinists’ calipers come to mind, for example. The scale has been especially useful for the so very fine angle scales of a theodolite. Although it has been largely replaced by optical and electronic micrometers, it is still in use.

In theory only, it should be possible to add another decimal place of precision to a scale by making the graduation intervals one-tenth as wide and ten times as dense. The problem soon becomes clear as the graduation marks are anything but clear. Even if it is possible to etch such a scale accurately, who could read it? The scale would become an indiscernible blur.

Below is an image of a simplified linear scale. Never mind what the units are. The bottom section is graduated in units and tenths. It can be read directly to one decimal place of precision. The vernier scale fits above it and has ten graduation intervals. These intervals are actually 0.09 units. The zero mark of the vernier scale is the index pointer for the main scale. Use this to read units and tenths. Count the graduation marks on the upper scale until reaching the one that best aligns with a mark on the lower scale. That upper mark indicates the hundredths. Drag the red point to move the scale.


So how does it work? When the image loads the reading is 1.27. Each interval on the vernier scale has a width of 0.09, so seven intervals cover a distance of 0.63. Add that to the 1.27 of the lower scale. The sum is 1.90, which falls dead on one of the tenth marks of the lower scale. None the other multiples of 0.09 can produce that result.

The same concept can be applied to a circular scale like that used on a theodolite. In the image below, the main outer scale is graduated in intervals of 30 minutes. Each interval on the vernier scale has an arc measure of 29 minutes. Again, count graduations up to the point at which a mark aligns with one on the lower scale. This will show the number of minutes to add to the direct reading.


This scale is a bit more complex because the older-style instruments could read angles either right or left. There are two sets of numbers on the outside, stationary scale. The top numbers give the angle right, which is actually read to the left. This may be counterintuitive, but from this perspective the vernier moves to the left as the instrument turns to the right. The scale is read with a magnifying glass, often attached to the instrument itself. There was a time (never to return) when I could read a one-minute scale without a magnifying glass. Verniers have also been used for a least-count precision of 30, 20, or 15 seconds.

Web Sketchpad Files

The animated sketches on this page were created with Web Sketchpad, a developmental version of The Geometer’s Sketchpad.