I recently bought a caliper rule, a device to accurately measure the size of some object or the size of a hole in some object –see photo. The main reason why it is such an accurate device is that the size is determined using a Vernier scale. A Vernier scale is a wondrously clever invention: it is one of those rare inventions where a very large gain is achieved for only very little added complexity. It really is the engineer’s version of having your cake and eating it.
Vernier scales are used on caliper rules, micrometers, barometers, thermometers, and no doubt several other devices where accurate reading of a scale is of the essence. They can be on a linear scale, like on my caliper rule, or on a cylindrical scale, like on a micrometer.
The first time you see a Vernier scale it seems to work like magic: on the caliper rule you read where the index line (at 0) on the sliding scale points on the fixed scale (about 30mm in the photo). More often than not the index line does not point exactly to a whole millimetre value, but between two whole millimetre values (say, between 30 and 31mm in the photo). Now look at the sliding Vernier scale (which goes between 0 and 10) and see which of these lines exactly aligns with a millimetre line on the fixed scale. Say it was the “6” line on the sliding scale; the distance between the caliper jaws then is 30.6mm. In fact, the sliding scale has five more indicator lines between each of its ten subdivision, allowing reading of the scale with an accuracy of 0.02mm. Looking at the close-up photo, the Vernier scales suggest that the jaws are 30.64mm apart. In fact, you can see that 30.62 and 30.66 match up about equally well, so we can say that the caliper jaws are 30.64±0.02mm apart.
Yes, that is an accuracy of 1/50th of a millimetre! This is crazy: 1/50th of a millimetre is 20μm, or about the width of a human hair. (A quick websearch suggests that the width of human hair varies between 17μm and 181μm —numbers that are way too precise to be believable, but do give a sense of the width of a hair.) Vernier scales change a simple sliding rule into a device that can be read with a precision of a width of a hair.
Actually using Vernier scales is much easier than describing how they are used. Explaining how Vernier scales work is in fact even harder. Let’s try it anyway.
The main fixed scale has millimetre divisions at a distance xm = m Δx from the zero line. Here m is an integer, and Δx is the distance between divisions on the fixed scale, in our case exactly 1mm.
The sliding Vernier scale has 10 divisions a distance yn = n Δy from its zero line. Here n is an integer less than 10, and Δy is the distance between divisions on the sliding scale. We can slide the jaws a distance s apart so that the divisions of the sliding scale can now be found a distance s + n Δy along the fixed scale.
The clever thing now is that the Δy on the sliding scale are not equal to the Δx=1mm of the fixed scale. In fact they are chosen such that 10Δy = (10K-1)Δx, with K some fixed integer usually bigger than 1. In my calipers the value of K equals 5, so the whole sliding Vernier scale is exactly 49mm long (as can be seen in the photo), divided in ten subdivisions, each Δy=4.9mm wide, and each of those Δy subdivisions divided again in five sub-subdivisions, each 0.98mm wide.
If the zero line (y0) on the sliding scale aligns exactly with division line m on the fixed scale we know that the jaws are exactly m Δx apart; this is how a normal ruler works. But suppose the jaws are a bit further apart, say a fraction (p/10)Δx further, with p some integer between 1 and 9. In this case the jaws are exactly s = (m+p/10)Δx apart. How can we accurately determine the value of p?
We do this by finding which of the divisions on the sliding scale exactly align with any division on the fixed scale. In fact I claim that this is the pth division on the sliding scale. The pth division on the sliding scale is found exactly s + p Δy along the fixed scale. But the Δy were chosen cleverly such that we can see that s + p Δy = (m+p/10)Δx + p Δy = m Δx + pK Δx = (m+pK)Δx. Which is exactly on an integer number times the basic division of the fixed scale Δx, so, indeed, the pth division on the sliding scale exactly lines up with the (m+pK)th division on the fixed scale.
So all we need to do is find which of the divisions on the sliding scale lines up with a division on the fixed scale. There is only one division lining up, because the Δy on the sliding scale are different from the Δx on the fixed scale.
The setup described above is suitable for the situation where any small interval on the fixed scale is divided up in 10 main sub-intervals. Of course, this can be chosen whichever way you want. If you want 12 basic subintervals then the Δy and Δx are related as 12Δy = (12K-1)Δx, and it is obvious how to generalize this for any number of subintervals. In fact we could argue that the Vernier scale on my calipers corresponds to 10×5 = 50 subintervals, and the integer K equaling 1 (instead of 10 subintervals and K=5). In this case, the we find 50Δy = (50-1)Δx, or Δy=0.98mm, which is indeed the distance between the smallest subdivisions on the sliding scale of my caliper rule.
The Vernier scale also has a link with the concepts of group velocity and phase speed: the Vernier scale is read off where two index lines on two scales with different spacing exactly line up. This is the same as positive interference between two waves of slightly different wavelength. The phase speed corresponds to the speed at which the jaws of the calipers open, while the group speed corresponds to the speed at which the aligning index line moves along the scales. The Vernier scale is set up such that for a “phase propagation” of 1mm the “group propagation” is 50mm. This allows reading off of the Vernier scale by a factor 50 more precise than the basic fixed scale on the rule, i.e. a precision of 1mm/50=20μm.
There apparently is also a physiological advantage to Vernier scales: it is much easier to judge which two lines exactly line up, as you do with reading off Vernier scales, rather than to estimate where between two index lines a third line falls, as you would do with a normal ruler.
If you google “Vernier calipers” you find several devices which are nothing like Vernier calipers: they are sliding rules which are read off with the help of some digital device, which no doubt is full of clever technology but probably nothing to do with Vernier scales. It seems an over-engineered solution to a problem that never was.
Reading physics 