What Is a Slide Rule? Multiplying by Adding Lengths

Logarithms convert multiplication into addition, since the logarithm of a product equals the sum of the logarithms. John Napier published the idea in 1614 and it was immediately recognised as transformative for astronomy and navigation, where long multiplications consumed enormous amounts of time. Printed tables of logarithms let a calculator look up two values, add them and look up the result. The slide rule mechanises exactly that: numbers are marked along a scale at positions proportional to their logarithms, so the distance from the start to a number represents its logarithm. Sliding one such scale along another physically adds two distances, and reading where they align gives the product. Because the scale is logarithmic, the markings crowd together towards the higher end, which is why a slide rule looks the way it does. William Oughtred arranged two logarithmic scales for this purpose around 1620, and the sliding design with a fixed body followed.

The capabilities follow directly from the mathematics of logarithms and shaped the practice around it:

• •Multiplication, division, squares, square roots, cubes, reciprocals, logarithms and trigonometric functions, all through appropriately marked scales
• •No addition or subtraction at all, which had to be done on paper, and which surprises people encountering one for the first time
• •Roughly three significant figures of precision, limited by how finely a person can read a printed scale, with a good operator estimating a fourth
• •No indication of the decimal point, since the scales are periodic, so the operator had to know the approximate magnitude of the answer independently
• •That last limitation is the interesting one, because it forced every user to carry a rough mental estimate of every result, which meant a wildly wrong answer was noticed immediately
• •A cursor with a hairline allowed intermediate results to be held while scales were moved, enabling chained calculations without writing anything down

Slide rules were the standard calculating tool of engineering and science for roughly three centuries, and the scale of what was designed with them is worth stating. Structural calculations for major bridges and buildings, the design of steam and internal combustion engines, chemical plant, aircraft from the Wright brothers through to the Boeing 747, and the entire Apollo programme were carried out with slide rules and tables, with astronauts carrying them aboard as a check on the computers. Specialised versions proliferated, with circular and cylindrical forms offering longer scales in a smaller object, and pocket rules for field use. Industry-specific rules existed for artillery, aviation navigation, photography exposure, concrete mixes and medical dosing, each with scales customised to a particular set of formulas, which is a form of embedding domain knowledge in a physical object. The rule was a professional badge as much as a tool, carried in a case on the belt, and the phrase about someone's calculations being on the back of an envelope describes the alternative.

The end was abrupt. Electronic pocket calculators arrived at the start of the 1970s, with the Hewlett-Packard HP-35 in 1972 being the machine usually credited with finishing the slide rule, since it performed trigonometric and logarithmic functions and gave ten digits with an explicit decimal point. Manufacturers who had dominated the market for decades stopped production within a few years, and one American maker reportedly donated its final rule to a museum in 1976. What the transition removed, according to engineers who worked through it, was the constant mental estimation the decimal point problem enforced, along with an intuitive feel for orders of magnitude and for which digits in a result actually mean anything. A calculator returning ten digits from three-digit inputs presents false precision that a slide rule made impossible, and several engineering educators have argued that significant figures became a taught rule rather than an obvious property once the tool stopped enforcing it.

A slide rule marks numbers at distances proportional to their logarithms, so sliding one scale along another adds those distances and multiplies the numbers. It cannot add or subtract, gives about three significant figures and shows no decimal point, which forced every user to keep a mental estimate of the answer. It calculated bridges, aircraft and the Apollo programme for three centuries, and pocket calculators from 1972 ended production within a few years.