How Do You Read Between the Marks? Line Up a Second Scale
By the BrainSnail editorial team. How these articles are written and checked, and how to tell us when one is wrong.
A short sliding scale with marks slightly closer together than the main scale lets a reader resolve a tenth of a division by eye. The trick needs no magnification and no electronics.
The problem it solves
A ruler can only be divided so far before the marks become too close to distinguish, and estimating a fraction of a division by eye is unreliable and differs between people. The device solves that by adding a second short scale sliding along the first, marked so that its divisions are slightly smaller than the main scale's. Because of that difference, exactly one pair of marks on the two scales lines up at any position, and which pair it is tells the reader the fraction directly. Judging whether two lines coincide is something the eye does extremely well, far better than estimating a proportion.
How to read one
The procedure is the same on any instrument carrying one:
- •Read the main scale value just before the sliding scale's zero mark
- •That gives the whole number of divisions
- •Look along the sliding scale for the mark aligning with any main mark
- •The number of that mark gives the fraction
- •Add the two, and no estimation has been made anywhere
- •The resolution is set by how many divisions the sliding scale has
Why the mismatch produces the resolution
The arithmetic behind it is simple and worth following. If ten divisions on the sliding scale are made to span exactly nine divisions of the main scale, each sliding division is nine tenths of a main one, so each successive pair of marks is out of step by one tenth of a main division. Moving the slide by a tenth therefore brings the next pair into alignment, and reading which pair aligns reads off tenths directly. Making fifty sliding divisions span forty nine main ones gives fiftieths instead. The resolution is entirely a matter of that ratio and costs nothing but careful marking.
How it goes wrong
The reading is objective in principle and several errors recur in practice. Reading the main scale from the wrong side of the sliding zero shifts the whole result by one division, which is the commonest beginner's mistake. Looking at the scales from an angle makes a different pair appear to align, so the instrument must be viewed square on. Wear at the jaws of a caliper puts a zero error into every reading, which is why the jaws are closed and the zero checked before use. Dirt between the jaws does the same. And forcing the jaws onto the work springs them slightly, which is why better instruments carry a limited-force thumbwheel.
Where it came from and where it went
The principle was published by Pierre Vernier in 1631, building on an earlier and clumsier arrangement described by Pedro Nunes, and it spread rapidly through surveying, astronomy and navigation because it made instruments dramatically more precise without making them larger. Sextants, theodolites, barometers and microscopes all carried one. Machine shops adopted it on calipers and micrometers. Digital displays have replaced it in most everyday use, and the older instruments remain in service because they need no battery, cannot lose calibration electronically and can be read in conditions where a display fails, which is why they are still made and still taught.
The takeaway
A sliding scale whose divisions are slightly smaller than the main scale's brings exactly one pair of marks into alignment at any position, and reading which pair gives the fraction directly rather than by estimation. Ten sliding divisions spanning nine main ones gives tenths, and fifty spanning forty nine gives fiftieths. Vernier published it in 1631 and it transformed surveying and navigation.