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# Slide rule

The slide rule is a portable, mechanical, analog computer usually consisting of three interlocking calibrated strips and a sliding cursor used to record intermediate results. It was once widely used for rapid, approximate scientific and engineering calculations. It was invented in 1622 by William Oughtred and was very commonly used until the 1970s when it was made obsolete for most purposes by electronic scientific/engineering calculators.

A typical slide rule, with 7 visible scales (4 on the body, 3 on the slide) and an ordinary, linear ruler scale seen on the topmost part.
 Contents

## Basic concepts

Cursor on a slide rule.

Most slide rules consist of three linear strips of the same length, aligned in parallel and interlocked so that the central strip can be moved lengthways relative to the other two. The outer two strips are fixed so that their relative positions do not change. Some slide rules 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. A sliding cursor with one or more vertical alignment lines can record an intermediate result on any of the scales.

In general, mathematical calculations are performed by aligning marks on the sliding central strip with marks on either of the fixed strips and then observing the relative positions of other marks on the strips. The marks engraved or printed on the strips are carefully placed to allow the handler to perform a number of important mathematical operations. The geometry of the markings determines which operations may be performed.

## Theory of operation

The rule has logarithmic scales. That is, a number x is printed on each rule at a distance proportional to logx from the "index", which is marked with the number 1.

A logarithm transforms the operations of multiplication and division to addition and subtraction thanks 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 numeral y, at position log(y) on the top scale, with the numeral at position log(x) + log(y) on the bottom scale. Since log(x) + log(y) = log(xy), this position on the bottom scale is marked with the numeral xy, the product of x and y.

The illustration below shows the multiplication of 2 with any other number. The index (1) 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 (multipliers) correspond with the multiplication 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 as in this diagram:

Operations may go "off the scale". For example the diagram above shows that the slide rule has not positioned the 70 on the upper scale above any number on the lower scale, so it does not give any answer for $2 \times 70$. 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 $2 \times 70$, but instead we calculated $0.2 \times 70 = 14$. So the true answer is not 14 but 140.

Division reverses this process. The illustration below demonstrates division by 2.75. For example, to divide 22 by 2.75, one aligns the index (1) on the upper scale with the 2.75 on the lower scale. The 22 on the lower scale (the mark just to the left of the 22.5 mark) then lies below the quotient, 8, on the upper scale.

## Physical design

### Standard linear rules

Slide rules calibrated on one side are called "simplex". Slide rules calibrated on both sides are called "duplex".

Typically two significant figures of precision are possible, three being obtainable by expert users who can estimate the fraction between gradations. Some high-end slide rules have magnifying cursors that effectively double the accuracy, permitting a 10-inch slide rule to serve as well as a 20-inch.

Slide rules often have other mathematical functions encoded on other auxiliary scales. When they were in widespread use, 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 included a Pythagorean scale, to figure sides of triangles, and a scale to figure circles. Others featured scales for calculating hyperbolic functions.

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.

A number of tricks were used to get more convenience. Trigonometric scales were sometimes dual-labelled, in black and red, with complementary angles, the so-called "Darmstadt" style. Duplex slide rules often duplicated basic scales on the back. Scales were often "split" to get higher accuracy.

### Circular slide rules

Circular slide rules came in two basic types, one with two cursors, and another with a moveable 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".

Circular slide rules were mechanically more rugged, smoother-moving and more precise than linear slide rules, because they depended on a single central bearing. The central pivot did not usually fall apart. The pivot also prevented scratching of the face and cursors. Only the most expensive linear slide rules had these features.

The highest accuracy scales were placed on the outer rings. Rather than "split" scales, high-end circular rules used spiral scales for difficult things like log-of-log scales. One eight-inch premium circular rule had a 50 inch spiral log-log scale!

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 kilometernautical mile and gallonliter fuel amount conversion functions.

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. It is still available in all flight shops, and remains widely used. While GPS has reduced the use of dead reckoning for aerial navigation, the E6B remains widely used as a primary or backup device and the majority of flight schools demand its mastery to some degree.

One significant advantage of a circular slide rule is that it never has to be re-oriented when results are near 1.0 - the rule is always on 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.

### Materials

Traditionally slide rules were made out of hard wood such as mahogany or boxwood with slides of glass and metal. In 1895, a Japanese firm started to make them from bamboo, which had the advantage of being less sensitive to temperature and humidity. These bamboo slide rules were introduced in Sweden in the fall of 1933 [1], and probably only a little earlier in Germany.

The best older slide rules were made of bamboo, which is dimensionally stable, strong and naturally self-lubricating. They used scales of celluloid or plastic. Some were made of mahogany. Later slide rules were made of plastic, or aluminium painted with plastic.

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.

Early cursors were metal frames holding glass. Later cursors were acrylics or polycarbonates sliding on teflon bearings.

Magnifying cursors can help engineers with poor eyesight, and can also double the accuracy of a slide rule.

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

Slide rules came into wide use in the 1850s, as engineering became a recognized professional activity. 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 WWII 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.

Some engineering students and engineers actually carried ten-inch slide rules in belt holsters, or kept 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. All this came to an end in the 1970s, when the advent of miniaturised calculators made slide rules obsolete. The last nail in the coffin was the launch of scientific pocket calculators; i.e. models featuring trigonometric and logarithmic functions, of which the Hewlett-Packard HP-35 was the first, in 1972.

Most slide rules are now collectors' items. A very 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. As recently as 2002, brand new slide rules were being located in the back-shelves of university book-stores, even though production ended almost 30 years earlier, in 1973.

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). 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 "reasonableness" 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, you can compute virtually any multiple of two just by looking, leaving your hands free. This can be especially useful when calculating percentages for test scores or similar.