# Slide rule

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A typical ten-inch student slide rule (Pickett N902-T simplex trig).

The slide rule, also known colloquially in the United States as a slipstick,[1] is a mechanical analog computer.[2][3][4][5][6] The slide rule is used primarily for multiplication and division, and also for functions such as roots, logarithms and trigonometry, but is not normally used for addition or subtraction. Though similar in name and appearance to a standard ruler, the slide rule is not ordinarily used for measuring length or drawing straight lines.

Slide rules come in a diverse range of styles and generally appear in a linear or circular form with a standardized set of markings (scales) essential to performing mathematical computations. Slide rules manufactured for specialized fields such as aviation or finance typically feature additional scales that aid in calculations common to those fields.

The Reverend William Oughtred and others developed the slide rule in the 17th century based on the emerging work on logarithms by John Napier. 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 around 1974 the electronic scientific calculator made it largely obsolete[7][8][9][10] and most suppliers left the business.

This slide rule is positioned to yield several values: From C scale to D scale (multiply by 2), from D scale to C scale (divide by 2), A and B scales (multiply and divide by 4), A and D scales (squares and square roots).

## Basic concepts

Cursor on a slide rule.

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

Scales may be grouped in decades, which are numbers ranging from 1 to 10 (i.e. 10n to 10n+1). Thus single decade scales C and D range from 1 to 10 across the entire width of the slide rule while double decade scales A and B range from 1 to 100 over the width of the slide rule.

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 generally done mentally or on paper, not on the 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 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 (which can usually be pulled out, flipped over and reinserted for convenience), 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

A logarithm transforms the operations of multiplication and division to addition and subtraction according to the rules ${\displaystyle \log(xy)=\log(x)+\log(y)}$ and ${\displaystyle \log(x/y)=\log(x)-\log(y)}$. Moving the top scale to the right by a distance of ${\displaystyle \log(x)}$, by matching the beginning of the top scale with the label ${\displaystyle x}$ on the bottom, aligns each number ${\displaystyle y}$, at position ${\displaystyle \log(y)}$ on the top scale, with the number at position ${\displaystyle \log(x)+\log(y)}$ on the bottom scale. Because ${\displaystyle \log(x)+\log(y)=\log(xy)}$, this position on the bottom scale gives ${\displaystyle xy}$, the product of ${\displaystyle x}$ and ${\displaystyle y}$. For example, to calculate 3×2, the 1 on the top scale is moved to the 2 on the bottom scale. The answer, 6, is read off the bottom scale where 3 is on the top scale. In general, the 1 on the top is moved to a factor on the bottom, and the answer is read off the bottom where the other factor is on the top. This works because the distances from the "1" are proportional to the logarithms of the marked values:

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 2×7. In such cases, the user may slide the upper scale to the left until its right index aligns with the 2, effectively dividing by 10 (by subtracting the full length of the C-scale) and then multiplying by 7, 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 2×7, but instead we calculated (2/10)×7=0.2×7=1.4. So the true answer is not 1.4 but 14. Resetting the slide is not the only way to handle multiplications that would result in off-scale results, such as 2×7; some other methods are:

1. Use the double-decade scales A and B.
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 inverted 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, one of them will always be on-scale.
4. Use both the CI inverted scale and the C scale. Line up the 2 of CI with the 1 of D, and read the result from D, below the 7 on the C scale.
5. Using a circular slide rule.

Method 1 is easy to understand, but entails a loss of precision. Method 3 has the advantage that it only involves two scales.

### 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. There is more than one method for doing division, but 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.

### 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, used for finding square roots and squares of numbers C, D single-decade logarithmic scales K three-decade logarithmic scale, used for finding cube roots and cubes of numbers 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 is not 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, CIF, DIF "inverted" scales, running from right to left, used to simplify 1/x steps S used for finding sines and cosines on the C (or D) scale T, T1, T2 used for finding tangents and cotangents on the C and CI (or D and DI) scales ST, SRT 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 ${\displaystyle e^{x}}$
The scales on the front and back of a Keuffel and Esser (K&E) 4081-3 slide rule.

The Binary Slide Rule manufactured by Gilson in 1931 performed an addition and subtraction function limited to fractions.[11]

#### Roots and powers

There are single-decade (C and D), double-decade (A and B), and triple-decade (K) scales. To compute ${\displaystyle x^{2}}$, for example, 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; to find the square root of nine, use the first one; the second one gives the square root of 90.

For ${\displaystyle x^{y}}$ problems, use the LL scales. When several LL scales are present, use the one with x on it. First, align the leftmost 1 on the C scale with x on the LL scale. Then, find y on the C scale and go down to the LL scale with x on it. That scale will indicate the answer. If y is "off the scale," locate ${\displaystyle x^{y/2}}$ 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 (or D) scale; though on many closed-body rules the S scale relates to the A scale instead, and what follows must be adjusted appropriately. 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 the C (or D) scale for angles less than 45 degrees. For angles greater than 45 degrees the CI scale is used. Common forms such as ${\displaystyle k\sin x}$ 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. For angles below 5.7 degrees, sines, tangents, and radians are approximately equal, and are found on the ST or SRT (sines, radians, and tangents) scale, or simply divided by 57.3 degrees/radian. Inverse trigonometric functions are found by reversing the process.

Many slide rules have S, T, and ST scales marked with degrees and minutes (e.g. some Keuffel and Esser models, late-model Teledyne-Post Mannheim-type rules). So-called decitrig models use 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.

## Compared to electronic digital calculators

Most people find slide rules difficult to learn and use. Even during their heyday, they never caught on with the general public.[28] Addition and subtraction are not well-supported operations on slide rules and doing a calculation on a slide rule tends to be slower than on a calculator.[29] This led engineers to take mathematical shortcuts favoring operations that were easy on a slide rule, creating inaccuracies and mistakes.[30] On the other hand, the spatial, manual operation of slide rules cultivates in the user an intuition for numerical relationships and scale that people who have used only digital calculators often lack.[31] A slide rule will also display all the terms of a calculation along with the result, thus eliminating uncertainty about what calculation was actually performed.

A slide rule requires the user to separately compute the order of magnitude of the answer in order to position the decimal point in the results. For example, 1.5 × 30 (which equals 45) will show the same result as 1,500,000 × 0.03 (which equals 45,000). This separate calculation is less likely to lead to extreme calculation errors, but forces the user to keep track of magnitude in short-term memory (which is error-prone), keep notes (which is cumbersome) or reason about it in every step (which distracts from the other calculation requirements).

The typical precision of a slide rule is about three significant digits, compared to many digits on digital calculators. As order of magnitude gets the greatest prominence when using a slide rule, users are less likely to make errors of false precision.

When performing a sequence of multiplications or divisions by the same number, the answer can often be determined by merely glancing at the slide rule without any manipulation. This can be especially useful when calculating percentages (e.g. for test scores) or when comparing prices (e.g. in dollars per kilogram). Multiple speed-time-distance calculations can be performed hands-free at a glance with a slide rule. Other useful linear conversions such as pounds to kilograms can be easily marked on the rule and used directly in calculations.

Being entirely mechanical, a slide rule does not depend on electricity or batteries. However, mechanical imprecision in slide rules that were poorly constructed or warped by heat or use will lead to errors.

Many sailors keep slide rules as backups for navigation in case of electric failure or battery depletion on long route segments. Slide rules are still commonly used in aviation, particularly for smaller planes. They are only being replaced by integrated, special purpose and expensive flight computers, and not general-purpose calculators. The E6B circular slide rule used by pilots has been in continuous production and remains available in a variety of models. Some wrist watches designed for aviation use still feature slide rule scales to permit quick calculations. The Citizen Skyhawk AT is a notable example.[32]

## Finding and collecting slide rules

Faber-Castell slide rule with pouch

There are still people who 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.[33]

A popular collectible 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 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, made of celluloid, tend to break down chemically over time.

There are still a handful of sources for brand new slide rules. The Concise Company of Tokyo, which began as a manufacturer of circular slide rules in July 1954,[34] continues to make and sell them today. In September 2009, on-line retailer ThinkGeek introduced its own brand of straight slide rules, which they described as "faithful replica[s]" that are "individually hand tooled" due to a stated lack of any existing manufacturers.[35] These are no longer available in 2012.[36] In addition, Faber-Castell has a number of slide rules still in inventory, available for international purchase through their web store.[37] Proportion wheels are still used in graphic design.

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## Notes

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11. instruction manual pages 7 & 8. Retrieved March 14, 2007.
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13. At least one circular rule, a 1931 Gilson model, sacrificed some of the scales usually found in slide rules in order to obtain additional resolution in multiplication and division. It functioned through the use of a spiral C scale, which was claimed to be 50 feet and readable to five significant figures. See http://www.sphere.bc.ca/test/gilson/gilson-manual2.jpg. A photo can be seen at http://www.hpmuseum.org/srcirc.htm. An instruction manual for the unit marketed by Dietzgen can be found at http://www.sliderulemuseum.com/SR_Library_General.htm. All retrieved March 14, 2007.
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15. "Cameron's Nautical Slide Rule", The Practical Mechanic and Engineer's Magazine, April 1845, p187 and Plate XX-B
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17. The Polyphase Duplex Slide Rule, A Self-Teaching Manual, Breckenridge, 1922, p. 20.
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20. Charles Overton Harris, Slide rule simplified, American Technical Society, 1961, p. 5.
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22. The Wang LOCI-2
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24. J. E. Volder, "The Birth of CORDIC", J. VLSI Signal Processing 25, 101 (2000).
25. Stoll, Cliff. "When Slide Rules Ruled," Scientific American, May 2006, pp. 80–87. "The difficulty of learning to use slide rules discouraged their use among the hoi polloi. Yes, the occasional grocery store manager figured discounts on a slipstick, and this author once caught his high school English teacher calculating stats for trifecta horse-race winners on a slide rule during study hall. But slide rules never made it into daily life because you could not do simple addition and subtraction with them, not to mention the difficulty of keeping track of the decimal point. Slide rules remained tools for techies."
26. Watson, George H. "Problem-based learning and the three C's of technology," The Power of Problem-Based Learning, Barbara Duch, Susan Groh, Deborah Allen, eds., Stylus Publishing, LLC, 2001. "Numerical computations in freshman physics and chemistry were excruciating; however, this did not seem to be the case for those students fortunate enough to already own a calculator. I vividly recall that at the end of 1974, the students who were still using slide rules were given an additional 15 minutes on the final examination to compensate for the computational advantage afforded by the calculator, hardly adequate compensation in the opinions of the remaining slide rule practitioners."
27. Stoll, Cliff. "When Slide Rules Ruled," Scientific American, May 2006, pp. 80–87. "With computation moving literally at a hand's pace and the lack of precision a given, mathematicians worked to simplify complex problems. Because linear equations were friendlier to slide rules than more complex functions were, scientists struggled to linearize mathematical relations, often sweeping high-order or less significant terms under the computational carpet. So a car designer might calculate gas consumption by looking mainly at an engine's power, while ignoring how air friction varies with speed. Engineers developed shortcuts and rules of thumb. At their best, these measures led to time savings, insight and understanding. On the downside, these approximations could hide mistakes and lead to gross errors."
28. Stoll, Cliff. "When Slide Rules Ruled", Scientific American, May 2006, pp. 80–87. "One effect was that users felt close to the numbers, aware of rounding-off errors and systematic inaccuracies, unlike users of today's computer-design programs. Chat with an engineer from the 1950s, and you will most likely hear a lament for the days when calculation went hand-in-hand with deeper comprehension. Instead of plugging numbers into a computer program, an engineer would understand the fine points of loads and stresses, voltages and currents, angles and distances. Numeric answers, crafted by hand, meant problem solving through knowledge and analysis rather than sheer number crunching."
29. Citizen Watch Company – Citizen Eco-Drive / US, Canada, UK, IrelandCitizen Watch
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33. Template:Cite web