Slide rule

The slide rule (often nicknamed a "slipstick") is a mechanical analog computer, consisting of at least two finely divided scales (rules), most often a fixed outer pair and a movable inner one, with a sliding window called the cursor. Before the advent of the pocket calculator, it was the most commonly used calculation tool in science and engineering. The use of slide rules continued to grow through the 1950s and 1960s even as digital computing devices were being gradually introduced; but in the early 1970s the electronic scientific calculator made it largely obsolete and most suppliers exited the business.

Despite their similar appearance, a slide rule serves a purpose different from that of a standard ruler: a ruler measures physical distances and aids in drawing straight lines, while a slide rule performs mathematical operations by using distances on nonlinearly-divided scales.

Basic concepts

In its most basic form, the slide rule uses two logarithmic scales to allow rapid multiplication and division of numbers, common operations that can be time-consuming and error-prone when done on paper. More complex slide rules allow other calculations, such as square roots, exponentials, logarithms, and trigonometric functions.

In general, mathematical calculations are performed by aligning a mark on the sliding central strip with a mark on one of the fixed strips, and then observing the relative positions of other marks on the strips. Numbers aligned with the marks give the approximate value of the product, quotient, or other calculated result.

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.

Operation

Multiplication

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 (

\log x

) 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 $2 \times 7$ . 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 $2 \times 7$ , but instead we calculated $0.2 \times 7 = 1.4$ . So the true answer is not 1.4 but 14.

Division

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.

Other operations

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 (e^x) 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:

A, B	two-decade logarithmic scales
C, D	single-decade logarithmic scales
K	three-decade logarithmic scale
CF, DF	"folded" versions of the C and D scales that start from π rather than from unity; these are convenient in two cases. First when the user guesses a product will be close to 10 but isn't sure whether it will be slightly less or slightly more than 10, the folded scales avoid the possibility of going off the scale. Second, by making the start π rather than the square root of 10, multiplying or dividing by π (as is common in science and engineering formulas) is simplified.
CI, DI, DIF	"inverted" scales, running from right to left, used to simplify 1/x steps
S	used for finding sines and cosines on the D scale
T	used for finding tangents on the D and DI scales
ST	used for sines and tangents of small angles and degree–radian conversion
L	a linear scale, used along with the C and D scales for finding base-10 logarithms and powers of 10
LLn	a set of log-log scales, used for finding logarithms and exponentials of numbers
Ln	a linear scale, used along with the C and D scales for finding natural (base e) logarithms and $e^x$

The scales on the front and back of a K&E 4081-3 slide rule.

Roots and powers

There are single-decade (C and D), double-decade (A and B), and three-decade (K) scales. To compute

x^2

, 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

x^y

problems, you can use the LL series of scales. There are often several, but we only need worry about the one with x on it. First, you align the leftmost 1 on the C scale with x on the LL scale. Then, you find y on the C scale and go down to the LL scale with x on it. That scale will have your answer. If y is "off the scale," then you can find

x^{y/2}

and square it using the A and B scales as described above.

Trigonometry

The S, T, and ST scales are used for trig functions and multiples of trig functions, for angles in degrees.

For angles from around 5.7 up to 90 degrees, sines are found by comparing the S scale with C. 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 or, for angles greater than 45 degrees, CI. Common forms such as k*sin(x) can be read directly from x on the S scale to the result on the D scale, when the C-scale index is set at k. 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 can also be used to convert between degrees and radians and was sometimes labeled SRT (sines, radians, and tangents) for this reason.

Many slide rules had S, T, and ST scales marked with degrees and minutes. So-called decitrig models used decimal fractions of degrees instead.

Logarithms and exponentials

Base-10 logarithms and exponentials are found using the L scale, which is linear. Some slide rules have a Ln scale, which is for base e.

Physical design

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-labeled, 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 π). 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.

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.

Cylindrical slide rules

There are two main types of cylindrical slide rules: those with helical scales such as the Fuller and the Otis King, and those with bars, such as the Thacher and some Loga models. In either case, the advantage is a much longer scale, and hence potentially higher accuracy, than a straight or circular rule.

Materials

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 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 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.

History

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 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.

Modern form

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 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. 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 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.

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. ^*

Demise

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 solved. 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.

Advantages

A slide rule tends to moderate the fallacy of "false precision" and significance. The typical precision available to a user of a slide rule is about three places of accuracy. This is in good correspondence with most data available for input to engineering formulas (such as the strength of materials, accurate to two or three places of precision, with a great amount—typically 1.5 or greater—of safety factor as an additional multiplier for error, variations in construction skill, and variability of materials). There's an old saying in engineering, "if you care about the third significant digit of tensile strength, you are already in trouble." When a modern pocket calculator is used, the precision may be displayed to seven to ten places of accuracy while in reality, the results can never be of greater precision than the input data available.

A slide rule requires a continual estimation of the order of magnitude of the results. On a slide rule 1.5 × 30 (which equals 45) will show the same result as 1,500,000 × 0.03 (which equals 45,000). It is up to the engineer to continually determine the reasonability of the results: something easily lost when a computer program or a calculator is used and numbers might be keyed in by a clerk not qualified to judge how reasonable those numbers might be.

When performing a sequence of multiplications or divisions by the same number, the answer can be often determined by merely glancing at the slide rule without any manipulation. For example, using the ruler pictured above, the user can compute virtually any multiple of two just by looking, leaving the user's hands free. This can be especially useful when calculating percentages, e.g., for test scores.

A slide rule does not depend on batteries.

Slide rules, unlike electronic calculators, are highly standardized, so there is no need to relearn anything when switching to a different rule.

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 websites 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.

Notes

Resetting the slide is not the only way to handle multiplications that would result in off-scale results, such as

2\times7

; 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.

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.

The Log-Log Duplex Decitrig Slide Rule No. 4081: A Manual, Keuffel & Esser, Kells, Kern, and Bland, 1943, p. 92.

The Polyphase Duplex Slide Rule, A Self-Teaching Manual, Breckenridge, 1922, p.20.

External links

General information, history:

History and collection of slide rules
The history, theory and use of the engineering slide rule – By Dr James B. Calvert, University of Denver
Oughtred Society Slide Rule Home Page – Dedicated to the preservation and history of slide rules
Early Calculators: Slide Rules – At the Museum of HP Calculators
The Slide Rule Universe – A comprehensive slide rule reference and buying/selling site
"How a slide rule works" – From the Slide Rule Universe
International Slide Rule Group – The International Slide Rule Group "ISRG" is devoted to collectors of slide rules and associated mechanical calculating instruments.

Online simulators:

Derek's Virtual Slide Rule Gallery – Gallery of seven virtual slide rule models with examples
Sag Milling's Online Sliderule – Fully functional online version of a slide ruler
A circular slide rule (Flash) – From the Center for Technology and Teacher Education, University of Virginia
Java simulation of a slide rule – By Andrew Davie

Specific rules/manufacturers:

ASA Micro E-6B Flight Computer – A specialized circular slide rule
ASA E-6B Metal Flight Computer – Metal model of the E-6B
Pickett Slide Rules – From the Slide Rule Universe
Breitling Navitimer info webpage – Wristwatch with circular rule
Clark McCoy – Extensive information about K&E slide rules: catalogs, scanned manuals, and historical information
Concise Online Shop – One of the few current manufacturers of circular slide rules; based in Japan

How-To's:

"Make your own slide rule" (PDF) – By Luis Fernandes, Dept. of Electrical and Computer Engineering, Ryerson University
"Make your own circular slide rule" – By Dr. Charles Kankelborg, Department of Physics, Montana State University
How to make your own slide rule, from Sphere Research

Collecting, preserving, trading:

The Slide Rule Universe – A comprehensive slide rule reference and buying/selling site
Oughtred Society Slide Rule Home Page – Dedicated to the preservation and history of slide rules
Eric's Slide Rule Site – Collector's site of Eric R. Marcotte, McMaster University
Ron Manley's Slide Rule site – Collector's site including a statistical analysis of slide rule prices on eBay
Cold War Calculators Slide rules produced for military and civil defence use.

Mathematical tools | Mechanical calculators | Theory of computation | Logarithms | English inventions

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