How Do You Calculate With a Ruler and a Chart? Lines That Do the Arithmetic

The diagram carries three or more scales, typically two inputs and one result, positioned and marked so that a straight line crossing all of them always connects values satisfying the relationship being computed. To use it, the operator finds the first known value on its scale, the second on its scale, lays a straight edge or a stretched thread between them, and reads the answer where that line crosses the third scale. No arithmetic is performed and no formula is consulted. The accuracy is limited by how finely the scales are printed and how carefully the line is placed, which is typically around two or three significant figures, and that was sufficient for a great deal of engineering work.

Before electronic calculation the advantages were substantial:

• •No arithmetic skill required, so an assistant could use one reliably
• •No possibility of the arithmetic errors that hand calculation produces constantly
• •One diagram replaced a formula that most users could not manipulate
• •Fast enough to use repeatedly during design work
• •Printable on a card and carried in a pocket
• •Able to handle relationships with several variables at once

Maurice d'Ocagne, a French engineer, established the theory in the 1880s and published a systematic treatment, giving the field its name and the rules for constructing a diagram to represent a given relationship. The mathematics involved determines where each scale must be placed and how it must be graduated, and is considerably more interesting than using the result. The technique spread rapidly through engineering, and by the middle of the twentieth century these diagrams were everywhere, in handbooks for structural design, chemical processing, ballistics, aviation, electrical work and medicine. Whole books of them were published for particular industries, and designing them was a recognised specialisation.

Constructing a diagram to represent a given relationship is a genuine piece of mathematics and the simplest case shows how it works. For a relationship in which one quantity is the sum of two others, three parallel vertical scales suffice, with the two inputs on the outer scales and the result on a scale placed midway between them and graduated at half the spacing. A straight line across any two inputs then crosses the middle scale at their sum. Relationships involving multiplication are handled by graduating the scales logarithmically, which converts multiplication into addition and lets the same arrangement work. More complicated relationships require curved scales and non-parallel arrangements, and determining those is where the theory becomes substantial.

Electronic calculation removed the original purpose and the format persists where its particular advantages still apply. Medicine uses them for calculating drug doses from body surface area, for assessing risk from several measurements at once, and for estimating gestational age, because a printed chart requires no device, no power and no training, and because seeing the whole relationship at once shows how sensitive the answer is to each input. Aviation retains several for weight and balance. Diving tables are a related form. Weather services use them for humidity relationships. And the format is being reconsidered as a way of presenting statistical models to clinicians, since a diagram makes the contribution of each factor visible in a way a computed number does not.

Scales are positioned and graduated so that any straight line crossing them connects values satisfying a relationship, so a calculation is performed by laying a ruler and reading where it crosses. That required no arithmetic and produced no arithmetic errors, which is why engineering handbooks were full of them. Maurice d'Ocagne established the theory in the 1880s. Medicine still uses them where no device and no power are wanted.