The user determines the location of the decimal point in the result, based on mental estimation. Scientific notation is used to track the decimal point in more formal calculations. Addition and subtraction steps in a calculation are done mentally or on paper, not on the slide rule.
Even the most basic student slide rules have more than two scales. Most consist of three linear strips of the same length, aligned in parallel and interlocked so that the central strip can be moved lengthwise relative to the other two. The outer two strips are fixed so that their relative positions do not change.
Some slide rules ("duplex" models) have scales on both sides of the rule and slide strip, others on one side of the outer strips and both sides of the slide strip, still others on one side only ("simplex" rules). A sliding cursor with a vertical alignment line is used to find corresponding points on scales that are not adjacent to each other or, in duplex models, are on the other side of the rule. The cursor can also record an intermediate result on any of the scales.
The figure below shows a simplified slide rule. It consists of two scales that can move with respect to each other. A numeral x is printed on each scale at a distance from the "index" (the left side number 1) equal to its base-10 logarithm (logx) times the length of the scale. Tick marks between each numeral are similarly placed according to logarithmic distance.
A logarithm transforms the operations of multiplication and division to addition and subtraction according to the rules log(xy) = log(x) + log(y) and log(x / y) = log(x) - log(y). Sliding the top scale rightward by a distance of log(x) aligns each number y, at position log(y) on the top scale, with the number at position log(x) + log(y) on the bottom scale. Since log(x) + log(y) = log(xy), reading this position on the bottom scale gives xy, the product of x and y.
The illustration below shows the slides arranged for multiplication of 2 with any other number. The index on the upper scale is aligned with the 2 on the lower scale. This shifts the entire upper scale rightward by log(2). The numbers on the upper scale line up with the multiplication-by-2 result on the lower scale. For example, the 3.5 on the upper scale is aligned with the product 7 on the lower scale, the 4 with the 8, and so on:
Operations may go "off the scale". For example the diagram above shows that the slide rule has not positioned the 7 on the upper scale above any number on the lower scale, so it does not give any answer for 2 x 7. In such cases, the user may slide the upper scale to the left, effectively multiplying by 0.2 instead of by 2, as in the illustration below:
Here the user of the slide rule must remember to adjust the decimal point appropriately to correct the final answer. We wanted to find 2 x 7, but instead we calculated 0.2 x 7 = 1.4. So the true answer is not 1.4 but 14. (see note 1)
The illustration below demonstrates the computation of 5.5/2. The 2 on the top scale is placed over the 5.5 on the bottom scale. The 1 on the top scale lies above the quotient, 2.75. (see note 2)
In addition to the logarithmic scales, some slide rules have other mathematical functions encoded on other auxiliary scales. The most popular were trigonometric, usually sine and tangent, common logarithm (log10) (for taking the log of a value on a multiplier scale), natural logarithm (ln) and exponential (ex) scales. Some rules include a Pythagorean scale, to figure sides of triangles, and a scale to figure circles. Others feature scales for calculating hyperbolic functions. On linear rules, the scales and their labeling are highly standardized, with variation usually occurring only in terms of which scales are included and in what order:
Roots and powers
There are single-decade (C and D), double-decade (A and B), and three-decade (K) scales. To compute x2, for example, we can locate x on the D scale, and read its square on the A scale. Inverting this process allows square roots to be found, and similarly for the powers 3, 1/3, 2/3, and 3/2. Care must be taken when the base, x, is found in more than one place on its scale. For instance, there are two nines on the A scale, and to find the square root of nine, we must use the first one; using the second one gives the square root of 90.
For angles between 5.7 and 90 degrees, sines are found by comparing the S scale with C or D. The S scale has a second set of angles (sometimes in a different color), which run in the opposite direction, and are used for cosines. Tangents are found by comparing the T scale with C, D, or, for angles greater than 45 degrees, CI. Sines and tangents of angles smaller than 5.7 degrees are found using the ST scale. Inverse trigonometric functions are found by reversing the process. The ST scale could also be used to convert between degrees and radians and was sometimes labeled SRT for this reason.
Logarithms and exponentials
Base-10 logarithms and exponentials are found using the L scale, which is linear. For base e, the LL scales are used.
Standard linear rules
The length of the slide rule is quoted in terms of the nominal length of the scales. Scales on the most common "10-inch" models are actually 25 cm in length, as they were made to metric standards, though some rules offer slightly extended scales to simplify manipulation when a result overflowed. Pocket rules are typically 5 inches. Models a couple of meters long were sold to be hung in classrooms for teaching purposes.
Typically the divisions mark a scale to a precision of two significant figures, and the user estimates the third figure. Some high-end slide rules have magnifying cursors that make the markings easier to see. Such cursors can effectively double the accuracy of readings, permitting a 10-inch slide rule to serve as well as a 20-inch.
A number of tricks can be used to get more convenience. Trigonometric scales are sometimes dual-labelled, in black and red, with complementary angles, the so-called "Darmstadt" style. Duplex slide rules often duplicate some of the scales on the back. Scales are often "split" to get higher accuracy.
Specialised slide rules were invented for various forms of engineering, business and banking. These often had common calculations directly expressed as special scales, for example loan calculations, optimal purchase quantities, or particular engineering equations. For example, the Fisher Controls company distributed a customized slide rule adapted to solving the equations used for selecting the proper size of industrial flow control valves.
Circular slide rules
Circular slide rules come in two basic types, one with two cursors, and another with a movable disk and a cursor. The basic advantage of a circular slide rule is that the longest dimension was reduced by a factor of about 3 (i.e. by p). For example, a 10 cm circular would have a maximum precision equal to a 30 cm ordinary slide rule. Circular slide rules also eliminate "off-scale" calculations, because the scales were designed to "wrap around"; they never have to be re-oriented when results are near 1.0 - the rule is always on scale.
Circular slide rules are mechanically more rugged, smoother-moving and more precise than linear slide rules, because they depend on a single central bearing. The central pivot does not usually fall apart. The pivot also prevents scratching of the face and cursors. Only the most expensive linear slide rules have these features.
The highest accuracy scales are placed on the outer rings. Rather than "split" scales, high-end circular rules use spiral scales for difficult things like log-of-log scales. One eight-inch premium circular rule had a 50 inch spiral log-log scale!
Technically, a real disadvantage of circular slide rules is that less-important scales are closer to the center, and have lower precisions. Historically, the main disadvantage of circular slide rules was just that they were not standard. Most students learned slide rule use on the linear slide rules, and never found reasons 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 all 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. As noted below, at least one high precision instrument was made of steel.
In 1895, a Japanese firm, Hemmi, started to make them from bamboo, which had the advantages of being dimensionally stable, strong and naturally self-lubricating. These bamboo slide rules were introduced in Sweden in the fall of 1933, and probably only a little earlier in Germany. Scales were made of celluloid or plastic. Later slide rules were made of plastic, or aluminium 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 are inferior because the markings wear off.
Premium slide rules included clever catches so the rule would not fall apart by accident, and bumpers so that tossing the rule on the table would not scratch the scales or cursor.
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. 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. Oughtred's ideas were only made public in publications of his student William Forster in 1632 and 1653.
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. (see notes 3 and 4)
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 aluminium 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 symbolized the medical profession). As an anecdote it can be mentioned that 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 obviously served him perfectly well for making quick estimates of rocket design parameters and other figures. Pickett brand slide rules were the standard in the Apollo program; Pickett's slide rules of the era often included a NASA or Apollo logo to promote the fact. A Pickett N600-MES (6 inch, magnifying cursor, "Eye-Saver" yellow) was standard equipment on all Apollo flights.
Some engineering students and engineers carried ten-inch slide rules in belt holsters, and even into the late 1960s this was a common sight on some 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. Another mechanical pocket calculator, the Curta a.k.a. "pepper grinder", was also popular among scientists.
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.
The importance of the slide rule began to diminish as electronic computers, a new, but very scarce resource in the 1950s, became widely available to technical workers during the 1960s. The introduction of Fortran in 1957 made computers practical for solving modest size mathematical problems. IBM introduced a series of more affordable computers, the IBM 650 (1954), IBM 1620 (1959), IBM 1130 (1965) addressed to the science and engineering market. John Kemeny's BASIC programming language (1964) made it easy for students to use computers. Wang Laboratories introduced desk-sized calculators starting in 1965. The DEC PDP-8 minicomputer was introduced in 1965.
Computers also changed the nature of calculation. With slide rules, there was a great emphasis on working the algebra to get expressions into the most computable form. Small terms were approximated or dropped. Fortran allowed complicated formulas simply to be typed in from textbooks. Numerical integration was often easier than trying to find closed form solutions. More difficult problems could be attacked. The young engineer asking for computer time to solve a problem that could have been done by a few swipes on the slide rule became a humorous cliché. Many computer centers had a framed slide rule hung on a wall with the note "In case of emergency, break glass".
The last nail in the coffin for the slide rule was the launch of scientific pocket calculators; i.e., models featuring trigonometric (sin, cos, ...) and logarithmic functions, of which the Hewlett-Packard HP-35 was the first, in 1972. By that time slide rules were mostly used in schools, where textbook and exam problems were designed to be solved on them. That market dried up quickly once scientific calculators became affordable.
One advantage of using a slide rule in addition to an electronic calculator is that an important calculation can be checked by doing it on both; because the two instruments are so different, there is little chance of making the same mistake twice.
Finding and collecting slide rules
For the reasons given above, some people still prefer a slide rule over an electronic calculator as a practical computing device. Many others keep their old slide rules out of a sense of nostalgia, or collect slide rules as a hobby.
A popular model is the Keuffel & Esser Deci-Lon, a premium scientific and engineering slide rule available both in a ten-inch "regular" (Deci-Lon 10) and a five-inch "pocket" (Deci-Lon 5) variant. Another prized American model is the eight-inch Scientific Instruments circular rule. Of European rules, Faber-Castell's high-end models are the most popular among collectors.
Although there is a large supply of slide rules circulating on the market, specimens in good condition tend to be surprisingly expensive. Many rules found for sale on online auction sites are damaged or have missing parts, and the seller may not know enough to supply the relevant information. Replacement parts are scarce, and therefore expensive, and are generally only available for separate purchase on individual collectors' web sites. The Keuffel and Esser rules from the period up to about 1950 are particularly problematic, because the end-pieces on the cursors tend to break down chemically over time. In many cases, the most economical method for obtaining a working slide rule is to buy more than one of the same model, and combine their parts.
Note 1: Resetting the slide is not the only way to handle multiplications that would result in off-scale results, such as 2 x 7; some other methods are: (1) Use the double-decade scales. (2) Use the folded scales. In this example, set the left 1 of C opposite the 2 of D. Move the cursor to 7 on CF, and read the result from DF. (3) Use the CI scale. Position the 7 on the CI scale above the 2 on the D scale, and then read the result off of the D scale, below the 1 on the CI scale. Since 1 occurs in two places on the CI scale, and one of them will always be on-scale. Method 1 is easy to understand, but entails a loss of precision. Method 3 has the advantage that the it only involves two scales.
Note 2: There is more than one method for doing division. The method presented here has the advantage that the final result cannot be off-scale, because one has a choice of using the 1 at either end.
Note 3: The Log-Log Duplex Decitrig Slide Rule No. 4081: A Manual, Keuffel & Esser, Kells, Kern, and Bland, 1943, p. 92.
Note 4: The Polyphase Duplex Slide Rule, A Self-Teaching Manual, Breckenridge, 1922, p.20.