Vernier effect
Two scales with slightly different spacings create a magnification effect: the point where tick marks align reveals sub-division measurements invisible to the naked eye. Drag the Vernier scale to explore, then see the same principle as an abstract moiré pattern below.
Least count = main division − vernier division = d − (n−1)/n · d = d/n
Practical view — Vernier scale
About this lab
The Vernier principle, published by Pierre Vernier in 1631, uses two scales with slightly different spacings to achieve measurement precision far beyond what either scale provides alone. The main scale has divisions of spacing d, while the Vernier scale has n divisions that together span n − 1 main divisions. Each Vernier division is therefore d(n−1)/n, and the difference (the "least count") is d/n.
When you slide the Vernier scale, exactly one of its tick marks aligns with a main scale tick. The number of that Vernier division tells you how many least-count increments to add to the main scale reading. This is the same as asking: where is the phase match between two slightly different periodic structures?
The moiré pattern below shows the same principle in its purest form. Two gratings with slightly different periods, when overlaid, produce broad bright and dark bands called "beats." The spacing of these beats equals d1 · d2 / |d1 − d2| — much larger than either grating period. Shifting one grating by a tiny amount causes the beat pattern to shift by a magnified amount. This magnification is what makes Vernier measurement possible: a sub-division displacement becomes a visually obvious shift in the moiré fringe.
The Vernier principle appears throughout precision engineering: calipers, micrometers, protractors, theodolites, and optical encoders. The moiré version is used in strain gauges, displacement sensors, and even security printing on banknotes.