# Slide rule

The slide rule is an analog computer, usually consisting of three interlocking calibrated strips and a sliding window, called the cursor. It was commonly used until the 1970s, when electronic calculators made it obsolete.

Missing image
Pocket_slide_rule.jpg
A slide rule being used to multiply by 2. Each number on the D scale is double the number above it on the C scale.

 Contents

## Basic concepts

In its most basic form, the slide rule uses two logarithmic scales to allow multiplication and division, common operations that are time-consuming and error-prone when done on paper. The user determines the location of the decimal point in the result, based on mental estimation. In a calculation with steps involving addition, subtraction, multiplication and addition, the addition and subtraction steps are done on paper, not on the slide rule.

Missing image
Slide_rule_cursor.jpg
Cursor on a slide rule.

In reality, even the most basic student slide rules have far 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 lengthways 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 one or more vertical alignment lines can record an intermediate result on any of the scales, and is also used to find corresponding points on scales that are not adjacent to each other.

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.

## Operation

### Multiplication

The figure below shows a simplified slide rule with two logarithmic scales. That is, a number [itex]x[itex] is printed on each rule at a distance proportional to [itex]\log x[itex] from the "index", which is marked with the number 1.

Missing image
Slide_rule_example1.jpg
Image:slide_rule_example1.jpg

A logarithm transforms the operations of multiplication and division to addition and subtraction thanks to the rules [itex]\log(xy) = \log(x) + \log(y)[itex] and [itex]\log(x/y) = \log(x) - \log(y)[itex]. Sliding the top scale rightward by a distance of [itex]\log(x)[itex] aligns each numeral [itex]y[itex], at position [itex]\log(y)[itex] on the top scale, with the numeral at position [itex]\log(x) + \log(y)[itex] on the bottom scale. Since [itex]\log(x) + \log(y) = \log(xy)[itex], this position on the bottom scale is marked with the numeral [itex]xy[itex], the product of [itex]x[itex] and [itex]y[itex].

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 [itex]\log(2)[itex] 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:

Missing image
Slide_rule_example2.jpg
Image:slide_rule_example2.jpg

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 [itex]2 \times 7[itex]. 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:

Missing image
Slide_rule_example3.jpg
Image:slide_rule_example3.jpg

Here the user of the slide rule must remember to adjust the decimal point appropriately to correct the final answer. We wanted to find [itex]2 \times 7[itex], but instead we calculated [itex]0.2 \times 7 = 1.4[itex]. So the true answer is not 1.4 but 14.Template:Ref

### 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.Template:Ref

Missing image
Slide_rule_example4.jpg
Image:slide_rule_example4.jpg

### 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 (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:

 A, B two-decade logarithmic scales C, D single-decade logarithmic scales K three-decade logarithmic scale CF,DF versions of the C and D scales that start from pi rather than from unity CI,DI,DIF inverted scales, running from right to left 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 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 natural logarithms and exponentials
Missing image
Slide_rule_scales.jpg
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 [itex]x^2[itex], 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.

#### Trigonometry

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.

#### 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.

## Physical design

### Standard linear rules

The length of the slide rule is quoted in terms of the length of the scales, not the length of the whole instrument. The most common high-end slide rules are 10-inch duplex rules, while student rules are often 10-inch simplex. Pocket rules are typically 5 inches.

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 effectively double the accuracy, 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.

### Circular slide rules

Circular slide rules come 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 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!

Missing image
Breitling_Montbrillant.jpg
Breitling Navitimer Montbrillant: chronograph certified wristwatch with circular slide rule.

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.

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.

### 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  (http://runeberg.org/tektid/1933a/0348.html), 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

Missing image
Oughtred.jpg
William Oughtred

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, used 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. 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 natinal 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.Template:Ref,Template:Ref

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, or "slipstick," 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.

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.  (http://www.findarticles.com/p/articles/mi_qa3950/is_200401/ai_n9372466)

• 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, e.g., for test scores.
• An important calculation can be checked by doing it once on a slide rule, and once on an electronic calculator; because the two instruments are so different, there is little chance of making the same mistake twice.
• 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.

## 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.

## Notes

1. Template:Note Resetting the slide is not the only way to handle multiplications that would result in off-scale results, such as [itex]2\times7[itex]; 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.
2. Template:Note 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.
3. Template:Note The Log-Log Duplex Decitrig Slide Rule No. 4081: A Manual (http://www.mccoys-kecatalogs.com/K&EManuals/4081-3_1943/4081-3_1943.htm), Keuffel & Esser, Kells, Kern, and Bland, 1943, p. 92.
4. Template:Note The Polyphase Duplex Slide Rule, A Self-Teaching Manual, Breckenridge, 1922, p.20.

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy