Fraso / nomograms

Method

How a nomogram is built

Almost everything written about nomograms online is either a photograph of an old one or an assertion that they are elegant. This is the derivation. It is short, and it has one genuinely surprising result in it.

The shape of the problem

A parallel-scale nomogram has three vertical scales. Two carry the values you know, one carries the value you want, and a straight line laid across any two cuts the third at the answer. That works for any equation you can bend into the form

f₁(u) + f₂(v) = f₃(w)

which is a much larger family than it looks. Multiplication becomes addition under logarithms, so w = k·u^p·v^q qualifies. So does 1/w = 1/u + 1/v, which is resistors in parallel and thin lenses, the same equation twice. So does plain subtraction. What does not qualify is anything where the variables refuse to separate — wind chill, for instance, has a term multiplying temperature by wind speed, and no substitution pulls them apart. Those need a different construction, and this site does not attempt them.

Placing the scales

Put the first scale at x = 0 and the second at x = d. Each carries its variable at a height given by an affine map of its function — a modulus m setting how many millimetres of paper one unit of f is worth, and an offset c sliding the whole scale up or down:

y₁ = m₁·f₁(u) + c₁    y₂ = m₂·f₂(v) + c₂

Now put the third scale somewhere between them, at x = k·d. A straight line from (0, y₁) to (d, y₂) crosses it at the height you would expect from simple interpolation:

y = (1 − k)·y₁ + k·y₂

For the instrument to be correct, that crossing must land on the right w for every pair (u, v) — not approximately, and not just in the middle of the range. So it has to equal m₃·(f₁ + f₂) + c₃ identically. Expand, collect the f₁ and f₂ terms separately, and two conditions fall out:

(1 − k)·m₁ = m₃     k·m₂ = m₃

Solve the pair:

k = m₁ ÷ (m₁ + m₂)     m₃ = m₁·m₂ ÷ (m₁ + m₂)

The surprising bit

The moduli are not free. Each one is fixed by how much paper its scale is allowed: m = height ÷ span of f. Which means k — the horizontal position of the middle scale — is not a design decision at all. It is a consequence. Draw the result scale down the centre of the page because that looks tidier, and the instrument is wrong.

When both outer scales span the same amount, m₁ = m₂, so k = ½ and the middle scale does sit centred. That is the case in every textbook illustration, which is probably why the centring reads as a convention rather than a coincidence.

A worked example nomogram: three vertical scales, u (1 to 10) on the left, w (1 to 100) in the middle and v (1 to 10) on the right. A straight line laid across any two scales cuts the third at the value that satisfies w = u · v.1098765.554.543.63.332.72.42.221.81.61.41.21.11u1098765.554.543.63.332.72.42.221.81.61.41.21.11v1008060504030201613108654321.61.31wWORKED EXAMPLEw = u · vu 1–10 · v 1–10 · w 1–100100 mm — measure thisFRASO · fraso.co
A worked example with a stretched left scale nomogram: three vertical scales, u (1 to 100) on the left, w (1 to 1000) in the middle and v (1 to 10) on the right. A straight line laid across any two scales cuts the third at the value that satisfies w = u · v, u over two decades.1008060504030201613108654321.61.31u1098765.554.543.63.332.72.42.221.81.61.41.21.11v100070050030020014010070503020141075321.41wWORKED EXAMPLE WITH A STRETCHED LEFT SCALEw = u · v, u over two decadesu 1–100 · v 1–10 · w 1–1000100 mm — measure thisFRASO · fraso.co
Left: both outer scales cover one decade, so the moduli match and the result scale lands at the centre. Right: the same equation with the left scale stretched over two decades. Its modulus halves, and the result scale moves to one third of the way across. Nothing was designed differently — the geometry moved because the ranges did.

Why scales bunch

The spacing of a scale is a picture of its function. A logarithmic scale crowds its upper end, which is why the gap from 1 to 2 is wider than 5 to 10. A reciprocal scale crowds far harder — on the resistors instrument the marks above about 400 Ω pile into a few millimetres, and that crush is the physics: a large resistor in parallel with a small one barely changes anything. A linear scale, like the tap drill instrument, is evenly spaced from end to end because subtraction has no leverage anywhere.

This is the argument for nomograms that survives the pocket calculator. The calculator answers the question you asked. The instrument shows you which direction is cheap, which is expensive, and where the returns stop — before you have thought to ask.

Ticks

Placing marks on a scale that bunches is the part that separates an instrument from a picture of one. Dividing the range into ten works only for linear scales. Everything here is generated candidate-and-reject instead: round numbers are proposed in ladders from coarsest to finest, and a candidate is accepted only if it clears every mark already accepted by a minimum physical distance — 0.9 mm for a tick, 3.4 mm for a label. Because those thresholds are in millimetres of paper rather than pixels or percentages, one rule serves screen and print, and no scale can produce marks too close together to read at the size it is actually drawn.

Ends are always marked and always labelled, even when they are not round numbers. Knowing where an instrument stops being valid matters more than a tidy run of tens.

Printing

Every drawing here is emitted in real millimetres, so a browser printing at 100% puts the scales on the paper at the size the geometry intended. That is a claim, though, and claims about print scaling are wrong surprisingly often — a printer handed a page taller than its paper does not refuse, it shrinks to fit, and a chart at 94% gives readings that are all wrong by the same invisible factor with nothing on the sheet looking amiss.

Which is why the sheet is 200 × 270 mm and not A4. That size sits inside both A4 and US Letter with several millimetres to spare in every direction, so no common printer has a reason to scale it at all. Designing for one paper size guarantees the other one gets quietly rescaled.

So every printed instrument carries a 100 mm bar. Measure it with a rule. If it is not 100 mm, the print was scaled and nothing on the sheet can be trusted. It is the only way to make a paper instrument check itself, and it is the reason this project can claim true scale rather than hope for it.

Sources

The geometry above is standard nomography, worked out between roughly 1880 and 1910 and set down definitively by Maurice d'Ocagne, who named the field. The derivation here is written from the mathematics rather than transcribed from a source; the tick engine, the balance rule and the check strip are this project's own. Nothing on this site is copied from an existing chart.