An instrument that leaves the screen
Two scales that disagree
Make a folded-paper caliper whose vernier displays tenths of a millimetre. Then give it a scale with five times as many divisions and ask the paper, not the typography, whether the result improved.
Move the jaw. Read the whole millimetres just left of the vernier zero, then find the numbered vernier line that aligns best with a main-scale line.
Live vernier
The published anchor, before the print button
ISO 13385-1:2011 clause 4.3.2, Table 2 gives these five graduating methods for a 1 mm main-scale interval. The 2011 edition is withdrawn. ISO lists the 2019 edition as current and confirmed in 2025, but its public page does not expose this table, so we have not confirmed that these rows continue unchanged.
| main interval a | vernier length Lᵥ | divisions n | pitch b=Lᵥ/n | nominal interval v |
|---|
Built scale: a = 1.000 mm; Lᵥ = 9 mm; n = 10; b = 9/10 = 0.900 mm; v = a − b = 0.100 mm.
Comparison: Lᵥ = 49 mm; n = 50; b = 49/50 = 0.980 mm; v = 0.020 mm.
Error budget, before you measure
The likely dominant errors are your final jaw cuts and jaws that rack or meet out of parallel, not the scale arithmetic. Printer x scale and closed-jaw zero are observable and correctable. Ruler transfer, line width, parallax, paper movement, clearance, squeeze and jaw depth remain. The provisional reporting floor is ±0.5 mm, or larger when half the repeated range, card discrepancy, reversal shift or jaw-depth shift is larger. This is a project acceptance rule, not an ISO accuracy class.
corrected length Lc = (indication I − zero error e₀) / sx
reported U = max(0.5 mm, repeatability range / 2, |mean card length − 85.60 mm|, reversal shift, jaw-depth shift)
The sheet must earn trust
Print at 100% / Actual Size. Disable Fit, Scale to Fit, and Shrink Oversized Pages. Measure the shared 150 mm ruler, plus the separate 100 mm horizontal and vertical bars on sheet 1. A preview on screen is not calibration.
Not calibrated. This page cannot vouch for a sheet it has not seen measured.
Build the paper instrument
- Choose ordinary paper first. You need scissors or a craft knife, transparent tape, a straightedge, pencil, a metric ruler and an unwarped ID-1-format wallet or payment card. Card stock is optional.
- On sheet 1, accordion-fold the 64 mm blank on its three score lines into four 16 mm plies. Tape the plies, then cut the outer contour. Make the fixed measuring face with one final straight cut on the red line.
- On sheet 2, cut one complete slider-and-jaw contour and its scale window. Score it, fold it around the beam and tape only the tab to the wrap itself. Do not tape it to the beam. Two 9/10 versions provide 0.4 and 0.6 mm clearance; the 0.8 mm version carries the 49/50 comparison scale.
- Make the moving face with one final straight cut. Close the jaws ten times, always approaching gently from open. Record the indicated zero. Do not trim it until the printed zeros happen to agree.
- Close the jaws around an object, switch the computer off, read whole millimetres at the vernier zero, find the coincident vernier line, then add its fraction. The paper is now doing the work.
Let the physical spread overrule the digits
Enter ten indicated card-length readings from the 9/10 scale, then reverse the card and enter ten more. Repeat with the 49/50 comparison strip if it is readable. Use commas or spaces. The 85.60 mm card length is nominal, not an exact property of your card, and the accessible ISO page did not supply tolerances.
Why ten lines can name ten phases
For the direct scale, one vernier step is 0.9 mm while one main step is 1 mm. Each step loses 0.1 mm, so the alignment walks through all ten tenths before returning. At jaw position z = m + r/10, vernier line j = r coincides with a main line.
The broader rule needs care. If b=(p/q)a, coincidence at offset r·a/q requires r+jp ≡ 0 (mod q). Multiplication by p visits every residue exactly once only when gcd(p,q)=1. Coprimality makes q phases distinguishable. It does not by itself make the familiar adjacent-line direct scale. The choice p=q−1 does that.
What is established, and what is not
Established from the cited 2011 table: the five numerical layouts above and the 9/10 scale's 0.1 mm nominal interval. The main graduated length here is 109 mm for a 100 mm range plus the 9 mm vernier length.
Established by arithmetic: the pitches, modular coincidence rule and coprimality condition. The 49/50 scale can display 0.02 mm increments.
Not established in advance: your printer scale, cuts, jaw parallelism, card dimensions, or completed tool accuracy. A 0.1 mm interval is not 0.1 mm accuracy. A household ruler transfers its own uncertainty and does not create traceability. The 49/50 strip is deliberately finer than ordinary printed line width and paper construction errors.
Prototype status: this geometry has not been confirmed across the three printer paths proposed by the scout. No example observations are preloaded: your closure, reversal, jaw-depth and repeat-placement tests decide the result. Failure of the ±0.5 mm provisional criterion is a result, not permission to hide the spread.
Sources and audit trail
- ISO 13385-1:2011 full-text mirror, clause 4.3.2 and Table 2. Used for the five graduating layouts. This edition is withdrawn.
- ISO 13385-1:2019 official record. Used only for current-edition status, title and scope. Its numerical table was not publicly confirmed.
- ISO/IEC 7810:2003 official abstract. Used for nominal ID-1 dimensions 85.60 by 53.98 by 0.76 mm. Its public abstract does not expose tolerances and the edition is withdrawn.
No historical numerical construction is attributed to Pierre Vernier or Pedro Nunes here. The primary 1631 scan was located but the relevant passage was not securely transcribed. The verifier extracts the shipped scale rows and checks their geometry, the physical contours, refusal states, uncertainty terms and visible claims.