The slide rule: how it works, who invented it, and why it vanished

For about 350 years, an engineer who needed to multiply two numbers quickly reached for a ruler with a sliding middle. Slide rules designed bridges, aircraft and power stations, and they flew to the Moon. Then, between about 1972 and 1976, they disappeared.

Where it came from

The idea rests on logarithms, published by the Scottish mathematician John Napier in 1614. A logarithm turns multiplication into addition: log(a × b) = log a + log b. Adding is far easier than multiplying by hand.

Around 1620 Edmund Gunter, professor of astronomy at Gresham College in London, marked a logarithmic scale on a two-foot stick. To multiply, you measured lengths along it with a pair of dividers and added them.

William Oughtred, the rector of a parish in Surrey and a keen mathematician, saw that you could drop the dividers by sliding two of Gunter's scales against each other. He said he had the idea around 1621. He published his circular slide rule in 1632 and the straight version in 1633, but one of his former pupils, Richard Delamain, had already published a circular rule in 1630. The two men spent years accusing each other of stealing it. Oughtred, incidentally, also gave us the × sign.

Two later changes produced the slide rule most people picture. In 1815 Peter Mark Roget, who later compiled Roget's Thesaurus, invented the log-log scale for powers and roots. In the 1850s Amédée Mannheim, a French artillery officer, settled the standard layout and popularised the sliding glass cursor.

How it multiplies by adding

On a slide rule the distance from the 1 mark to any number is proportional to that number's logarithm. So sliding one scale along another adds two logarithms, which multiplies the two numbers. The two main scales are called C (on the sliding part) and D (on the fixed body).

1. To work out 2 × 3, slide the 1 on the C scale until it sits above 2 on the D scale.
2. Find 3 on the C scale.
3. Read the number beneath it on the D scale: 6.

Division is the same move backwards: put 3 on C over 6 on D, and the 1 on C points at the answer, 2.

Where does the decimal point go?

The slide rule does not know. 2 × 3, 20 × 300 and 0.2 × 0.03 all look identical: the rule says “6” and leaves the rest to you.

Take 34 × 0.62. The rule reads 2-1-1. You round in your head: 34 × 0.6 is about 20, so the answer is 21.1. (A calculator says 21.08.) Engineers did this estimate for every sum, which is why people who learned on slide rules tend to notice at once when a calculator answer is ten times too big.

Why three significant figures?

Because a logarithmic scale spaces numbers by ratio, not by size. On a standard 10-inch rule, the gap between 1.00 and 1.01 is about 1.1 mm. The gap between 9.9 and 10.0 is also about 1.1 mm. A 1% difference is always the same distance, wherever you are on the scale.

A sharp eye with a good cursor can split that 1.1 mm into about five, which means you can read a result to roughly 0.2%. That is three significant figures, with the third one partly estimated. For most engineering it was plenty: the loads, measurements and material strengths going into the sum were rarely known more precisely than that.

The standard lived on long after the rule. Texas's school Calculator Applications contest, which replaced its slide rule contest around 1980, still asks for answers to three significant digits, because that was the accuracy a slide rule could give.

Where slide rules went

• To the Moon. The Smithsonian says a Pickett N600-ES slide rule was carried on the first five Apollo flights, and it holds one used by the crew of Apollo 13. Buzz Aldrin's Apollo 11 slide rule sold at auction in 2007 for $77,675.
• Into the cockpit. The E6B flight computer, a circular slide rule invented by US Navy Lieutenant Philip Dalton in the late 1930s, is still in the FAA's handbook for trainee pilots. It works out wind correction, fuel burn and ground speed.
• Onto the wrist. Breitling patented a slide-rule chronograph in 1941. Its Navitimer, made in 1952 for the Aircraft Owners and Pilots Association, is still in production.

How many were made

Nobody counted, but collectors estimate around 40 million slide rules were made worldwide in the 20th century. Japan's Hemmi, the largest maker, was producing about a million a year at its peak in 1963. In the United States, Keuffel & Esser's sales were already falling through the 1960s, years before the pocket calculator arrived.

The end came fast

• 1968Hewlett-Packard releases the HP 9100A, a desktop machine that does trigonometry and logarithms.
• 1972The HP-35 puts the same functions in a pocket for $395, about $3,000 in today's money.
• 1976The TI-30 scientific calculator sells for $24.95. On 11 July, Keuffel & Esser makes its last slide rule and gives it to the Smithsonian. Faber-Castell stops the same year.
• 1980Texas replaces its school slide rule contest with a calculator contest.

A tool that had been standard kit for three and a half centuries went from essential to obsolete in about four years. Some makers kept going for specialist uses: circular rules and medical dose rules were still being made decades later, and pilots still learn the E6B.

Quick answers

Sources

1. Wikipedia, “Slide rule”
2. MacTutor History of Mathematics, Edmund Gunter; William Oughtred
3. Florian Cajori, On the History of Gunter’s Scale and the Slide Rule (1920).
4. Robert A. James, “The Deaths of the Slide Rule”, Journal of the Oughtred Society 23:2 (2014)
5. IEEE Pulse, “The Slide Rule”
6. Smithsonian NMAH, Pickett N600-ES; National Air and Space Museum, Apollo 13 slide rule
7. Heritage Auctions, Buzz Aldrin's Apollo 11 slide rule (2007)
8. FAA, Pilot’s Handbook of Aeronautical Knowledge, chapter 16; Wikipedia, “E6B”
9. University Interscholastic League, Calculator Applications contest manual
10. Breitling, Navitimer history

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