Circular slide rules are mechanically more rugged and smoother-moving, but their scale alignment precision is sensitive to the centering of a central pivot; a minute 0.1 mm off-centre of the pivot can result in a 0.2 mm worst case alignment error. The pivot, however, does prevent scratching of the face and cursors. The highest accuracy scales are placed on the outer rings. Rather than "split" scales, high-end circular rules use spiral scales for more complex operations like log-of-log scales. One eight-inch premium circular rule had a 50-inch spiral log-log scale.
The main disadvantages of circular slide rules are the difficulty in locating figures along a rotating disc, and limited number of scales. Another drawback of circular slide rules is that less-important scales are closer to the center, and have lower precisions. Most students learned slide rule use on the linear slide rules, and did not find reason to switch.
One slide rule remaining in daily use around the world is the E6B. This is a circular slide rule first created in the 1930s for aircraft pilots to help with dead reckoning. With the aid of scales printed on the frame it also helps with such miscellaneous tasks as converting time, distance, speed, and temperature values, compass errors, and calculating fuel use. The so-called "prayer wheel" is still available in flight shops, and remains widely used. While GPS has reduced the use of dead reckoning for aerial navigation, and handheld calculators have taken over many of its functions, the E6B remains widely used as a primary or backup device and the majority of flight schools demand that their students have some degree of its mastery.
In 1952, Swiss watch company Breitling introduced a pilot's wristwatch with an integrated circular slide rule specialized for flight calculations: the Breitling Navitimer. The Navitimer circular rule, referred to by Breitling as a "navigation computer", featured airspeed, rate/time of climb/descent, flight time, distance, and fuel consumption functions, as well as kilometer–nautical mile and gallon–liter fuel amount conversion functions.
Traditionally slide rules were made out of hard wood such as mahogany or boxwood with cursors of glass and metal. At least one high precision instrument was made of steel.
In 1895, a Japanese firm, Hemmi, started to make slide rules from bamboo, which had the advantages of being dimensionally stable, strong and naturally self-lubricating. These bamboo slide rules were introduced in Sweden in September, 1933 , and probably only a little earlier in Germany. Scales were made of celluloid or plastic. Later slide rules were made of plastic, or aluminum painted with plastic. Later cursors were acrylics or polycarbonates sliding on Teflon bearings.
All premium slide rules had numbers and scales engraved, and then filled with paint or other resin. Painted or imprinted slide rules were viewed as inferior because the markings could wear off. Nevertheless, Pickett, probably America's most successful slide rule company, made all printed scales. Premium slide rules included clever catches so the rule would not fall apart by accident, and bumpers to protect the scales and cursor from rubbing on tabletops. The recommended cleaning method for engraved markings is to scrub lightly with steel-wool. For painted slide rules, and the faint of heart, use diluted commercial window-cleaning fluid and a soft cloth.
The slide rule was invented around 1620–1630, shortly after John Napier's publication of the concept of the logarithm. Edmund Gunter of Oxford developed a calculating device with a single logarithmic scale, which, with additional measuring tools, could be used to multiply and divide. The first description of this scale was published in Paris in 1624 by Edmund Wingate (c. 1593 – 1656), an English mathematician, in a book entitled "“L'usage de la reigle de proportion en l'arithmetique & geometrie”." The book contains a double scale on one side of which is a logarithmic scale and on the other a tabular scale. In 1630, William Oughtred of Cambridge invented a circular slide rule, and in 1632 he combined two Gunter rules, held together with the hands, to make a device that is recognizably the modern slide rule. Like his contemporary at Cambridge, Isaac Newton, Oughtred taught his ideas privately to his students, but delayed in publishing them, and like Newton, he became involved in a vitriolic controversy over priority, with his one-time student Richard Delamain and the prior claims of Wingate. Oughtred's ideas were only made public in publications of his student William Forster in 1632 and 1653.
In 1677, Henry Coggeshall created a two-foot folding rule for timber measure, called the Coggeshall slide rule. His design and uses for the tool gave the slide rule purpose outside of mathematical inquiry.
In 1722, Warner introduced the two- and three-decade scales, and in 1755 Everard included an inverted scale; a slide rule containing all of these scales is usually known as a "polyphase" rule.
In 1815, Peter Roget invented the log log slide rule, which included a scale displaying the logarithm of the logarithm. This allowed the user to directly perform calculations involving roots and exponents. This was especially useful for fractional powers.
The more modern form was created in 1859 by French artillery lieutenant Amédée Mannheim, "who was fortunate in having his rule made by a firm of national reputation and in having it adopted by the French Artillery." It was around that time, as engineering became a recognized professional activity, that slide rules came into wide use in Europe. They did not become common in the United States until 1881, when Edwin Thacher introduced a cylindrical rule there. The duplex rule was invented by William Cox in 1891, and was produced by Keuffel and Esser Co. of New York.
Astronomical work also required fine computations, and in 19th century Germany a steel slide rule about 2 meters long was used at one observatory. It had a microscope attached, giving it accuracy to six decimal places.
In World War II, bombardiers and navigators who required quick calculations often used specialized slide rules. One office of the U.S. Navy actually designed a generic slide rule "chassis" with an aluminum body and plastic cursor into which celluloid cards (printed on both sides) could be placed for special calculations. The process was invented to calculate range, fuel use and altitude for aircraft, and then adapted to many other purposes.
Throughout the 1950s and 1960s the slide rule was the symbol of the engineer's profession (in the same way that the stethoscope symbolizes the medical profession). German rocket scientist Wernher von Braun brought two 1930s vintage "Nestler" slide rules with him when he moved to the U.S. after World War II to work on the American space program. Throughout his life he never used any other pocket calculating devices; slide rules served him perfectly well for making quick estimates of rocket design parameters and other figures. Aluminum Pickett-brand slide rules were carried on five Apollo space missions, including to the moon, according to advertising on Pickett's N600 slide rule boxes .
Some engineering students and engineers carried ten-inch slide rules in belt holsters, and even into the mid 1970s this was a common sight on campuses. Students also might keep a ten-or twenty-inch rule for precision work at home or the office while carrying a five-inch pocket slide rule around with them.
In 2004, education researchers David B. Sher and Dean C. Nataro conceived a new type of slide rule based on "prosthaphaeresis", an algorithm for rapidly computing products that predates logarithms. There has been little practical interest in constructing one beyond the initial prototype, however.