The Vernier

A tenth of a division, read by coincidence: the caliper, the arc, and the screw.

A scale cut in millimetres tells you the millimetre and leaves you to guess the tenth. Lay a second, shorter scale beside it, with ten divisions in the space of nine, and the guess becomes a question the eye answers exactly: which line meets a line? Line 7 meeting means seven tenths. This is the vernier, and the same trick reads arcs to the minute and screws to the thousandth of a millimetre.

Plate I. The vernier caliper. Below it, two reading glasses: one at the vernier's zero, one where a vernier line meets a main line.

Vernier

Lay in

Zero error Hide the reading, to practise

Drag the sliding jaw, or turn the thumb wheel for fine steps. With the keyboard, focus the jaw and use the arrow keys: one least count a press, a whole millimetre with Shift.

Lay the brass cylinder in the jaws and read it with the 0.1 mm vernier: 21.3. Then change to the 50-division vernier and read it again: 21.34.
Take the cylinder out, choose the 10-division vernier, and set the jaws to exactly 12.4 mm: the zero just past 12, and line 4 meeting a main line.
Find the least count of the 20-division vernier. Close the jaws, then open them until its first line meets a main line. The jaws are then 2 − 1.95 = 0.05 mm apart.
Set a zero error of +0.3 mm and close the jaws: they read 0.3, not 0. Lay in the steel ball and read 13.0; subtract the error: 13.0 − 0.3 = 12.7 mm.
With the 10-division vernier, set a whole number of millimetres, such as 27.0. Lines 0 and 10 both meet main lines, nine millimetres apart.

a.The beam carries the main scale in millimetres, figured every centimetre. The fixed jaws are part of it.

b.The slider carries the vernier on a bevelled edge just under the main scale, so the two sets of lines almost touch. Its zero stands as far from the moving jaw as the main zero stands from the fixed jaw, so where the zero falls on the main scale is the opening of the jaws.

c.The outside jaws grip an object. The inside jaws, above, open in a bore and read its width. The depth rod runs out of the end of the beam as the jaws open and reads the depth of a hole. All three read the same scale.

d.The thumb wheel moves the slider in small steps. The locking screw on top holds it, so a setting can be carried to the light and read there.

e.The reading glasses. The figures in faded italic are the glass's own aids, the count of millimetres above and of vernier lines below; they are not engraved on the instrument.

Plate III. The micrometer, with a vernier on its sleeve; below, a straight scale with a vernier running each way.

Close on Hide the reading

Drag the thimble up or down to turn it, or sideways to spin it fast. Keys on the thimble: ↑ ↓ one thousandth, ← → one hundredth, Shift for fifty times as much. Drag the slider of the straight scale, or use its arrow keys: 0.1 a press.

Close the micrometer on the wire with the ratchet and read it: 1.5 on the sleeve, 12 on the thimble, line 8 of the vernier. 1.5 + 0.12 + 0.008 = 1.628 mm.
Open it and set 7.385 mm: 7 on the sleeve with no half-millimetre line showing, 38 on the thimble, and line 5 of the vernier.
Now set 7.885 mm. The thimble and vernier read as before; only the half-millimetre line, now showing, tells the two apart.
On the straight scale, set the index to 12.7. Line 7 of the forward vernier meets on the right; line 7 of the backward vernier meets on the left, counted the other way.
Find the least count of the backward vernier: put the index just past a whole number, so that its line 1 meets a main line. Its divisions are 1.1, so line 1 stands 0.1 past a main line when the zero does.

a.The sleeve is fixed to the frame. Along its datum line it carries whole millimetres above and half millimetres below.

b.The thimble turns with the screw, whose pitch is 0.5 mm. Its edge is cut in 50 parts, so each is 0.01 mm, read against the datum line.

c.The vernier is ten long lines on the sleeve spanning nine divisions of the thimble. The line that runs straight on into a thimble line gives the thousandths.

d.The ratchet slips once the spindle presses with a set force, so every reading is taken at the same pressure.

e.The reading glass shows the thimble's edge opened out flat: going up the glass is going round the screw.

f.The straight scale has two verniers on one slider. The forward one puts 10 divisions in 9 and is read to the right; the backward, or retrograde, one puts 10 in 11 and is read to the left. Both read to 0.1 division.

¶Commentary

The principle. Let n vernier divisions fill n − 1 main divisions. Then each vernier division is short of a main division by 1/n of one. If the vernier's zero stands k/n of a division past a main line, its first line is (k − 1)/n past the next main line, its second (k − 2)/n, and so on, until line k lands exactly on a main line. So the number of the line that meets is the number of nths. The least count, the smallest step the vernier reads, is one main division divided by n.

Spread and backward verniers. Nothing requires the vernier divisions to be nearly equal to the main ones. Twenty divisions in 39 mm makes each 1.95 mm, short of two main divisions by 0.05 mm; the lines are farther apart and easier to see, and the least count is the same as twenty in 19. A vernier can also be longer than it should be: n divisions in n + 1 makes each vernier division 1/n too long, and then the lines must be counted backward, from the zero against the direction of the main scale. That is the retrograde vernier of Plate III.

Why coincidence. The eye is poor at judging what fraction of a gap a line has crossed, but it is very good at seeing whether two lines are in line. People can see a break in a straight line far smaller than the finest detail they can resolve, a few seconds of arc against about a minute; this is called vernier acuity, after the instrument. The vernier turns a judgement of fraction into a judgement of alignment. The pattern helps too: the lines on either side of the match are out of step by equal amounts in opposite directions, so the eye can find the middle of the pattern even when no single pair looks perfect.

Nunes and Vernier. Pedro Nunes, the Portuguese cosmographer, proposed in De crepusculis (Lisbon, 1542) to draw many concentric arcs inside a quadrant, each divided into one fewer equal parts than the last, so that the plumb line would fall nearly on a division of one of them. The idea was sound but the instrument was hard to divide, and it gave the vernier its other name, the nonius. Pierre Vernier, of Ornans in the Franche-Comté, published the sliding scale in La construction, l'usage, et les propriétés du quadrant nouveau de mathématiques (Brussels, 1631): one short movable arc against the main one. It is the device on every plate here.

The caliper. Sliding calipers are old, but a vernier on one made it a precision tool. In 1851 Joseph R. Brown, of Providence, Rhode Island, made a vernier caliper that read to a thousandth of an inch, and his small shop became Brown & Sharpe. The metric caliper of Plate I, with its 0.1, 0.05, and 0.02 mm verniers, is that instrument's descendant.

The screw. A screw turns a long rotation into a short advance: one turn, one pitch. William Gascoigne, a Yorkshire astronomer, used a fine screw in the eyepiece of a telescope around 1640 to measure the sizes of the planets' discs. Jean Laurent Palmer patented a handheld micrometer caliper in Paris in 1848, and the French still call a micrometer a palmer. The thimble divided into fifty is itself a scale wrapped round a cylinder, and a vernier on the sleeve divides it again, as on Plate III.

Zero error and parallax. Worn or dirty jaws may not read zero when closed. If the closed reading is +0.3 mm, every reading is 0.3 too large, and you subtract it; if the vernier's zero stands behind the main zero, the error is negative, and subtracting it adds. Check the zero before you measure. Parallax is the other enemy: if the eye is not square to the scales, a line on the vernier seems to meet the wrong line on the beam. That is why the vernier is cut on a bevel, so its lines lie almost in the plane of the main scale.

On the planimeter. The measuring wheel of a planimeter has its drum divided into a hundred parts, and a fixed vernier of ten beside the drum reads the thousandth of a turn. You can watch it count on the Planimeter Bench.

Further. Maurice Daumas, Scientific Instruments of the Seventeenth and Eighteenth Centuries and Their Makers (1972), for Nunes, Vernier, and the dividing of arcs; Joseph Wickham Roe, English and American Tool Builders (1916), for Brown & Sharpe and the shop instruments of the nineteenth century.

Drawn for the Wending House, 2026. Drag anything that looks like brass.A vernier at work: the Planimeter Bench.