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{{Short description|Mechanical analog computer}}
{{Other uses}}
{{Use dmy dates|date=September 2023|cs1-dates=y}}
[[File:Sliderule.PickettN902T.agr.jpg|thumb|220x220px|Typical {{Convert|10|in|cm|adj=on|spell=in}} student slide rule (Pickett N902-T simplex trig)]]

A '''slide rule''' is a [[hand]]-operated [[mechanical calculator]] consisting of [[Sliding (motion)|slidable]] [[Ruler|rulers]] for evaluating mathematical operations such as [[multiplication]], [[division (mathematics)|division]], [[exponents]], [[Nth root|roots]], [[logarithm]]s, and [[trigonometry]]. It is one of the simplest [[analog computer|analog computers]].<ref name="Flynn2002">{{cite book |author=Roger R. Flynn |url=https://books.google.com/books?id=SB06AQAAIAAJ |title=Computer sciences |date=June 2002 |publisher=Macmillan |isbn=978-0-02-865567-3 |volume=1 |page=175 |quote=The slide rule is an example of a mechanical analog computer... |access-date=30 March 2013}}</ref><ref name="BleulerHaxby2011">{{cite book |author1=Ernst Bleuler |url=https://books.google.com/books?id=0ZabGiKp1nEC&pg=PA635 |title=Electronic Methods |author2=Robert Ozias Haxby |date=21 September 2011 |publisher=Academic Press |isbn=978-0-08-085975-0 |page=638 |quote=For example, slide rules are mechanical analog computers |access-date=30 March 2013}}</ref>

Slide rules exist in a diverse range of styles and generally appear in a linear, circular or cylindrical form. Slide rules manufactured for specialized fields such as aviation or finance typically feature additional scales that aid in specialized calculations particular to those fields. The slide rule is closely related to [[nomogram]]s used for application-specific computations. Though similar in name and appearance to a standard [[ruler]], the slide rule is not meant to be used for measuring length or drawing straight lines. It is not designed for addition or subtraction. Maximum accuracy for standard linear slide rules is about three decimal significant digits, while [[scientific notation]] is used to keep track of the [[order of magnitude]] of results.

English mathematician and clergyman Reverend [[William Oughtred]] and others developed the slide rule in the 17th century based on the emerging work on logarithms by [[John Napier]]. It made calculations faster and less [[Human error|error-prone]] than evaluating on [[paper]]. Before the advent of the [[scientific calculator|scientific pocket calculator]], it was the most commonly used calculation tool in science and [[engineering]].<ref name="MIT-Museum">{{cite web |title=Slide Rules |url=https://mitmuseum.mit.edu/object-collection/slide-rules |website=MIT Museum |publisher=Massachusetts Institute of Technology |access-date=2019-05-01}}</ref> The slide rule's ease of use, ready availability, and low cost caused its use to continue to grow through the 1950s and 1960s, even as desktop [[Digital electronic computer|electronic computers]] were gradually introduced. But after the handheld [[scientific calculator]] was [[HP-35|introduced in 1972]] and became inexpensive in the mid-1970s, slide rules became largely [[obsolete]], so most suppliers departed the business.

In the [[United States]], the slide rule is colloquially called a '''slipstick'''.<ref>{{cite book |last1=Berrey |first1=Lester V. |title=American Thesaurus of Slang: A Complete Reference Book of Colloquial Speech |last2=van den Bark |first2=Melvin |publisher=Crowell |year=1953 |edition=2nd |oclc=319462}}</ref><ref name="petroski">{{cite book |last1=Petroski |first1=Henry |url=https://books.google.com/books?id=oVXzxvS3MLUC&q=slipstick+slide-rule+-wikipedia&pg=PA47 |title=An Engineer's Alphabet: Gleanings from the Softer Side of a Profession |date=2011 |publisher=Cambridge University Press |isbn=9781139505307 |pages=46–47 |access-date=21 March 2017}}</ref>

==Basic concepts==
[[File:Slide rule-homemade teaching base 2.jpg|thumb|Simple slide rule made from [[index cards]] marked with powers of 2, calculating 8x4 by aligning the bottom ruler to start where the top ruler is 8, and then reading at the number above where the bottom ruler is 4.]]Each [[Slide rule scale|ruler's scale]] has [[Graduation (scale)|graduations]] labeled with [[precomputed]] outputs of various [[mathematical functions]], acting as a [[lookup table]] that [[Mapping (mathematics)|maps]] from position on the ruler as each function's input. Calculations that can be reduced to simple addition or subtraction using those precomputed functions can be solved by aligning the two rulers and reading the approximate result.

For example, a number to be multiplied on one [[logarithmic-scale]] ruler can be aligned with the start of another such ruler to sum their logarithms. Then by applying the [[List of logarithmic identities#Logarithm of a product|law of the logarithm of a product]], the [[Product (mathematics)|product]] of the two numbers can be read. More elaborate slide rules can perform other calculations, such as [[square root]]s, [[exponential function|exponentials]], [[logarithm]]s, and [[trigonometric function]]s.

The user may estimate the location of the decimal point in the result by mentally [[Interpolated|interpolating]] between labeled graduations. [[Scientific notation]] is used to track the decimal point for more precise calculations. Addition and subtraction steps in a calculation are generally done mentally or on paper, not on the slide rule.

=== Components ===
[[File:slide rule cursor.jpg|thumb|Cursor on a slide rule]]Most slide rules consist of three parts:
* Frame or base{{snd}} two strips of the same length held parallel with a gap between.
* Slide{{snd}} a center strip interlocked with the frame that can move lengthwise relative to the frame.
* Runner or glass{{snd}} an exterior sliding piece with a hairline, also known as the "cursor".

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 [[Wikt:cursor|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.

=== Decades ===
[[Slide rule scale|Scales]] may be grouped in [[Decade (log scale)|decades]], where each decade corresponds to a range of numbers that spans a ratio of 10 (i.e. a range from 10<sup>''n''</sup> to 10<sup>''n''+1</sup>). For example, the range 1 to 10 is a single decade, and the range from 10 to 100 is another decade. Thus, single-decade scales (named C and D) range from 1 to 10 across the entire length of the slide rule, while double-decade scales (named A and B) range from 1 to 100 over the length of the slide rule.

==Operation==

=== Logarithmic scales ===
The following [[logarithmic identities]] transform the operations of multiplication and division to addition and subtraction, respectively:

<math display="block">\log(x \times y) = \log(x) + \log(y) \, ,</math><math display="block">\log(x/y) = \log(x) - \log(y) \, .</math> 

==== Multiplication ====
With two logarithmic scales, the act of positioning the top scale to start at the bottom scale's label for <math>x</math> corresponds to shifting the top logarithmic scale by a distance of <math>\log(x)</math>. This aligns each top scale's number <math>y</math> at offset <math>\log(y)</math> with the bottom scale's number at position <math>\log(x) + \log(y)</math>. Because <math>\log(x) + \log(y) = \log(x \times y)</math>, the mark on the bottom scale at that position corresponds to <math>x \times y</math>. With {{Math|1=x=2}} and {{Math|1=y=3}} for example, by positioning the top scale to start at the bottom scale's {{Math|2}}, the result of the multiplication {{Math|1=3×2=6}} can then be read on the bottom scale under the top scale's {{Math|3}}:

[[File:Slide rule example2 with labels.svg|550px]] 

While the above example lies within one decade, users must mentally account for additional zeroes when dealing with multiple decades. For example, the answer to {{Math|1=7×2=14}} is found by first positioning the top scale to start above the 2 of the bottom scale, and then reading the marking 1.4 off the bottom two-decade scale where {{Math|7}} is on the top scale:

[[File:Slide Rule Duplex.svg|A duplex slide rule set to multiply any 2 by any number up to 50.]]

But since the {{Math|7}} is above the ''second'' set of numbers that number ''must'' be multiplied by {{Math|10}}. Thus, even though the answer directly reads {{Math|1.4}}, the correct answer is {{Math|1=1.4×10 = 14}}.

For an example with even larger numbers, to multiply {{Math|88×20}}, the top scale is again positioned to start at the {{Math|2}} on the bottom scale. Since {{Math|2}} represents {{Math|20}}, all numbers in that scale are multiplied by {{Math|10}}. Thus, any answer in the ''second'' set of numbers is multiplied by {{Math|100}}. Since {{Math|8.8}} in the top scale represents {{Math|88}}, the answer must additionally be multiplied by {{Math|10}}. The answer directly reads {{Math|1.76}}. Multiply by {{Math|100}} and then by {{Math|10}} to get the actual answer: {{Math|1,760}}.

In general, the {{Math|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 {{Math|1}} mark are proportional to the logarithms of the marked values.

==== Division ====
The illustration below demonstrates the computation of {{Math|{{sfrac|5.5|2}}}}. The {{Math|2}} on the top scale is placed over the {{Math|5.5}} on the bottom scale. The resulting quotient, {{Math|2.75}}, can then be read below the top scale's {{Math|1}}:

[[File:slide rule example4.svg]]

There is more than one method for doing division, and the method presented here has the advantage that the final result cannot be off-scale, because one has a choice of using the {{Math|1}} at either end.

With more complex calculations involving multiple factors in the numerator and denominator of an expression, movement of the scales can be minimized by alternating divisions and multiplications. Thus {{Math|{{sfrac|5.5×3|2}}}} would be computed as {{Math|{{sfrac|5.5|2}}×3}} and the result, {{Math|8.25}}, can be read beneath the {{Math|3}} in the top scale in the figure above, without the need to register the intermediate result for {{Math|{{sfrac|5.5|2}}}}.

==== Solving Proportions ====

Because pairs of numbers that are aligned on the logarithmic scales form constant ratios, no matter how the scales are offset, slide rules can be used to generate equivalent fractions that solve proportion and percent problems.

For example, setting 7.5 on one scale over 10 on the other scale, the user can see that at the same time 1.5 is over 2, 2.25 is over 3, 3 is over 4, 3.75 is over 6, 4.5 is over 6, and 6 is over 8, among other pairs. For a real-life situation where 750 represents a whole 100%, these readings could be interpreted to suggest that 150 is 20%, 225 is 30%, 300 is 40%, 375 is 50%, 450 is 60%, and 600 is 80%.

===Other scales===
{{Main article|Slide rule scale#Scales}}

[[File:Pickett slide rule.jpg|frame|center|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).]]
In addition to the logarithmic scales, some slide rules have other mathematical [[function (mathematics)|functions]] encoded on other auxiliary scales. The most popular are [[trigonometric function|trigonometric]], usually [[sine]] and [[tangent (trigonometric function)|tangent]], [[common logarithm]] (log{{small|10}}) (for taking the log of a value on a multiplier scale), [[natural logarithm]] (ln) and [[exponential function|exponential]] (''e<sup>x</sup>'') scales. 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.<ref>{{Cite web|last=Marcotte, Ph.D.|first=Eric|date=2002|title=Eric's Types of Slide Rules and their Scales|url=https://www.sliderule.ca/scales.htm|access-date=2021-07-13|website=www.sliderule.ca}}</ref>
{| class="wikitable"
|-
| C, D || '''single-decade logarithmic''' scales, single sections of the same length, used together for multiplication and division, and generally one of them is combined with another scale for other calculations
|-
| A, B || '''two-decade logarithmic''' scales, two sections each of which is half the length of the C and D scales, used for finding square roots and squares of numbers
|-
| K || '''three-decade logarithmic''' scale, three sections each of which is one third the length of the C and D scales, used for finding cube roots and cubes of numbers
|-
| CF, DF || '''folded''' versions of the C and D scales that start from '''pi''' ([[Pi|π]]) rather than from unity; these are convenient in two cases. First when the user guesses a product will be close to 10 and 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 [[Multiplicative inverse|reciprocal]] ({{Fraction|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
|-
| R1, R2 || '''square root scales''' – setting the cursor to any value <math>r</math> on R1 or R2, find <math>{\pi}r^2</math> ([[area of a circle]] of radius <math>r</math>) under the cursor on the DF scale
|-
| ST, SRT || used for '''sines''' and '''tangents''' of '''small angles''' and '''degree–radian''' conversion
|-
| Sh, Sh1, Sh2 || used for finding '''hyperbolic sines''' on the C (or D) scale
|-
| Ch || used for finding '''hyperbolic cosines''' on the C (or D) scale
|-
| Th || used for finding '''hyperbolic tangents''' on the C (or D) scale
|-
| L || '''linear scale''' used for addition, subtraction, and (along with the C and D scales) for finding base-10 logarithms and powers of 10
|-
| LL0N (or LL/N) and LLN || '''log-log folded''' <math>e^{-x}</math> and <math>e^x</math> scales, for working with logarithms of any base and arbitrary exponents. 4, 6, or 8 scales of this type are commonly seen.
|-
| Ln || '''linear scale''' used along with the C and D scales for finding natural (base <math>e</math>) logarithms and <math>e^x</math>
|-
|P
|'''Pythagorean''' scale of <math>\sqrt{1-x^2}</math> to (1) solve the [[Pythagorean theorem]] and (2) to accurately determine cosine for small angles (with the S scale)
|}

{| style="width:150px; font-size:90%; border:1px solid #ccc; padding:4px; background:#f9f9f9;"
|-
|
{| style="border-collapse: collapse;border-spacing:0;padding:2px;"
|-
| [[File:Slide rule scales front.jpg|300px]]
| [[File:Slide rule scales back.jpg|300px]]
|}
|-
| style="text-align: left" | The scales on the front and back of a [[Keuffel and Esser]] (K&E) 4181-3 slide rule
|}

The Binary Slide Rule manufactured by Gilson in 1931 performed an addition and subtraction function limited to fractions.<ref>{{cite web|url=http://www.sphere.bc.ca/test/circular-man2.html|title=instruction manual|pages=7–8|access-date=March 14, 2007|work=sphere.bc.ca|archive-date=April 2, 2007|archive-url=https://web.archive.org/web/20070402204923/http://www.sphere.bc.ca/test/circular-man2.html|url-status=dead}}</ref>

====Roots and powers====
There are single-decade (C and D), double-decade (A and B), and triple-decade (K) scales. To compute <math>x^2</math>, 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 <math>x^y</math> 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 <math>x^{y/2}</math> and square it using the A and B scales as described above. Alternatively, use the rightmost 1 on the C scale, and read the answer off the next higher LL scale. For example, aligning the rightmost 1 on the C scale with 2 on the LL2 scale, 3 on the C scale lines up with 8 on the LL3 scale.

To extract a cube root using a slide rule with only C/D and A/B scales, align 1 on the B cursor with the base number on the A scale (taking care as always to distinguish between the lower and upper halves of the A scale). Slide the slide until the number on the D scale which is against 1 on the C cursor is the same as the number on the B cursor which is against the base number on the A scale. (Examples: A 8, B 2, C 1, D 2; A 27, B 3, C 1, D 3.)

====Roots of quadratic equations====
[[Quadratic equations]] of the form <math>ax^2 + bx + c = 0</math> can be solved by first reducing the equation to the form <math>x^2 - px + q = 0</math> (where <math>p = -b/a</math> and <math>q = c/a</math>), and then aligning the index ("1") of the C scale to the value <math>q</math> on the D scale. The cursor is then moved along the rule until a position is found where the numbers on the CI and D scales add up to <math>p</math>. These two values are the roots of the equation.

====Future value of money====
The LLN scales can be used to compute and compare the cost or return on a fixed rate loan or investment.
The simplest case is for continuously compounded interest. Example: Taking D as the interest rate in percent,
slide the index (the "1" at the right or left end of the scale) of C to the percent on D. The corresponding value on LL2 directly below the index will be the multiplier for 10 cycles of interest (typically years). The value on LL2 below 2 on the C scale will be the multiplier after 20 cycles, and so on.

====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. (On many closed-body rules the S scale relates to the A and B scales instead and covers angles from around 0.57 up to 90 degrees; 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 <math>k\sin x</math> 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&nbsp;''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 (Doric duplex 5" models, for example), 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. Logarithms to any other base can be calculated by reversing the procedure for calculating powers of a number. For example, log2 values can be determined by lining up either leftmost or rightmost 1 on the C scale with 2 on the LL2 scale, finding the number whose logarithm is to be calculated on the corresponding LL scale, and reading the log2 value on the C scale.

====Addition and subtraction====
Addition and subtraction aren't typically performed on slide rules, but is possible using either of the following two techniques:<ref>{{cite web|url=http://www.antiquark.com/2005/01/slide-rule-tricks.html|title=AntiQuark: Slide Rule Tricks|work=antiquark.com}}</ref>

# Converting addition and subtraction to division (required for the C and D or comparable scales):
#* Exploits the [[Identity (mathematics)|identity]] that the quotient of two variables plus (or minus) one times the divisor equals their sum (or difference):<math display="block">\begin{align}
\left(\frac{x}{y} + 1\right) y &= x + y \, \text{ (addition)}, \\
\left(\frac{x}{y} - 1\right) y &= x - y \, \text{ (subtraction)}.
\end{align}</math>
#* This is similar to the addition/subtraction technique used for high-speed electronic circuits with a [[logarithmic number system]] in specialized computer applications like the [[Gravity Pipe]] (GRAPE) supercomputer and [[hidden Markov models]].
# Using a linear L scale (available on some models):
#* After sliding the cursor right (for addition) or left (for subtraction) and returning the slide to 0, the result can be read.

===Generalizations===
[[File:Cikk-skalak-r-en.gif|thumb|alt=|Quadratic and reciprocal scales]]
Using (almost) any strictly [[Monotonic function|monotonic scales]], other calculations can also be made with one movement.<ref>{{cite journal |last=Istvan |first=Szalkai |title=General Two-Variable Functions on the Slide Rule |url=http://www.oughtred.org/jos/pages/JOS_2018_Vol_27_1_TOC.jpg |journal=Journal of the Oughtred Society|volume=27|issue=1|pages=14–18 |year=2016 |arxiv=1612.03955}}</ref><ref>{{cite arXiv |last=Istvan |first=Szalkai |title=General Two-variable Functions on the Slide-rule |eprint=1612.03955 |class=math.HO|year=2016}}</ref> For example, reciprocal scales can be used for the equality <math>\frac{1}{x} + \frac{1}{y}=\frac{1}{z}</math> (calculating [[Series and parallel circuits#Parallel circuits|parallel resistances]], [[harmonic mean]], etc.), and quadratic scales can be used to solve <math>x^2 + y^2 = z^2 </math>.

==Physical design==
{{further|Slide rule scales}}
===Standard linear rules===
[[File:Teaching sliderule comparison.jpg|thumb|A {{convert|7|ft|m|adj=on}} teaching slide rule compared to a normal-sized model]]
The width of the slide rule is quoted in terms of the nominal width of the scales. Scales on the most common "10-inch" models are actually 25&nbsp;cm, as they were made to metric standards, though some rules offer slightly extended scales to simplify manipulation when a result overflows. Pocket rules are typically 5&nbsp;inches (12&nbsp;cm). Models a couple of metres (yards) wide were made to be hung in classrooms for teaching purposes.<ref>{{cite web |url=http://www.tbullock.com/sliderule.html |title=Slide Rules |publisher=Tbullock.com |date=2009-12-08 |access-date=2010-02-20 |archive-url=https://archive.today/20130203202414/http://www.tbullock.com/sliderule.html |archive-date=2013-02-03 |url-status=dead }}</ref>

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 magnifier 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 model.

Various other conveniences have been developed. 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. For example, instead of reading from an A scale to a D scale to find a square root, it may be possible to read from a D scale to an R1 scale running from 1 to square root of 10 or to an R2 scale running from square root of 10 to 10, where having more subdivisions marked can result in being able to read an answer with one more significant digit.

===Circular slide rules===
Circular slide rules come in two basic types, one with two cursors, and another with a free dish and one cursor. The dual cursor versions perform multiplication and division by holding a constant angle between the cursors as they are rotated around the dial. The onefold cursor version operates more like the standard slide rule through the appropriate alignment of the scales.

The basic advantage of a circular slide rule is that the widest dimension of the tool was reduced by a factor of about 3 (i.e. by [[pi|π]]). For example, a {{cvt|10|cm}} circular would have a maximum precision approximately equal to a {{cvt|31.4|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 reoriented when results are near 1.0—the rule is always on scale. However, for non-cyclical non-spiral scales such as S, T, and LL's, the scale width is narrowed to make room for end margins.<ref>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 {{Webarchive|url=https://web.archive.org/web/20061230062132/http://www.sphere.bc.ca/test/gilson/gilson-manual2.jpg |date=2006-12-30 }}. 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 {{Webarchive|url=https://web.archive.org/web/20070214152423/http://www.sliderulemuseum.com/SR_Library_General.htm |date=2007-02-14  }}. All retrieved March 14, 2007.</ref>

Circular slide rules are mechanically more rugged and smoother-moving, but their scale alignment precision is sensitive to the centering of a central pivot; a minute {{convert|0.1|mm|abbr=on}} off-centre of the pivot can result in a {{convert|0.2|mm|abbr=on}} worst case alignment error. The pivot does prevent 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 more complex operations like log-of-log scales. One eight-inch premium circular rule had a 50-inch spiral log-log scale. Around 1970, an inexpensive model from B. C. Boykin (Model 510) featured 20 scales, including 50-inch C-D (multiplication) and log scales. The RotaRule featured a friction brake for the cursor.

The main disadvantages of circular slide rules are the difficulty in locating figures along a dish, and limited number of scales. Another drawback of circular slide rules is that less-important scales are closer to the center, and have lower precisions. Most students learned slide rule use on the linear slide rules, and did not find reason to switch.

One slide rule remaining in daily use around the world is the [[E6B|E6-B]]. 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 flight shops, and remains widely used. While [[Global Positioning System|GPS]] has reduced the use of dead reckoning for aerial navigation, and handheld calculators have taken over many of its functions, the E6-B remains widely used as a primary or backup device and the majority of flight schools demand that their students have some degree of proficiency in its use.

Proportion wheels are simple circular slide rules used in graphic design to calculate [[aspect ratio]]s. Lining up the original and desired size values on the inner and outer wheels will display their ratio as a percentage in a small window. Though not as common since the advent of computerized layout, they {{cns|are still made and used.<!-- please add {{as of}} or equivalent when sourcing -->|date=March 2018}}

In 1952, Swiss watch company [[Breitling SA|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 of climb|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.
<gallery>File:Vintage Concise Model 28 Circular Slide Rule, Made In Japan, Circa 1960s (16240893439).jpg|A simple circular slide rule, made by Concise Co., Ltd., Tokyo, Japan, with only inverse, square, and cubic scales. On the reverse is a handy list of 38 [[Metric system|metric]]/[[Imperial units|imperial]] conversion factors.
File:Slide rule pocket watch.jpg|A Russian circular slide rule built like a pocket watch that works as single cursor slide rule since the two needles are ganged together
File:Rechenschieberring.jpg|A two-scale slide rule built into a ring
File:Circular slide rule.JPG|Pickett circular slide rule with two cursors. (4.25&nbsp;in/10.9&nbsp;cm width) Reverse has additional scale and one cursor.
File:Breitling Navitimer slide rule.jpg|''[[Breitling Navitimer]]'' wristwatch with circular slide rule
File:RotaRule-Photo-Front-04.jpg|alt=RotaRule 510 Front|The front side of a Boykin RotaRule Model 510
File:RotaRule-Photo-Back-04.jpg|alt=RotaRule 510 rear side|The rear side of a Boykin RotaRule Model 510
File:Keuffel & Esser - Sperry, Model 4016 Pocket Calculator - MIT Slide Rule Collection - DSC03581.JPG|Sperry 4016 Pocket Calculator
</gallery>

{{anchor|Cylindrical slide rule}}

===Cylindrical slide rules===
There are two main types of cylindrical slide rules: those with helical scales such as the [[Fuller calculator]], the [[Otis King]] and the [[Bygrave slide rule]], 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 greater precision, than afforded by a straight or circular rule.
<gallery>
File:Fuller's calculator.jpg|Fuller calculator, 1928
File:Otis King Model K (horizontal).jpg|Otis King Model K
File:Suwak Bygrave.jpg|Bygrave slide rule
File:Senator John Heinz History Center - IMG 7824.JPG|Thacher slide rule, {{circa|1890}}
</gallery>

===Materials===
Traditionally slide rules were made out of hard wood such as mahogany or boxwood with cursors of glass and metal. At least one high precision instrument was made of steel.

In 1895, a Japanese firm, Hemmi, started to make slide rules from [[celluloid]]-clad bamboo, which had the advantages of being dimensionally stable, strong, and naturally self-lubricating. These bamboo slide rules were introduced in Sweden in September, 1933,<ref>{{cite web|url=https://runeberg.org/tektid/1933a/0348.html|title=336 (Teknisk Tidskrift / 1933. Allmänna avdelningen)|publisher=Runeberg.org|access-date=2010-02-20}}</ref> and probably only a little earlier in Germany.

Scales were also made of celluloid or other polymers, or printed on aluminium. Later cursors were molded from [[acrylyl group|acrylics]] or [[polycarbonate]], sometimes with [[polytetrafluoroethylene|Teflon]] bearing surfaces. 

All premium slide rules had numbers and scales deeply engraved, and then filled with paint or other [[resin]]. Painted or imprinted slide rules were viewed as inferior, because the markings could wear off or be chemically damaged. Nevertheless, Pickett, an American slide rule company, made only printed scale rules. Premium slide rules included clever mechanical catches so the rule would not fall apart by accident, and bumpers to protect the scales and cursor from rubbing on tabletops.

==History==
[[File:oughtred.jpg|thumb|upright|[[William Oughtred]] (1575–1660), inventor of the slide rule]]
[[File:EveradeSlidingRule.pdf|thumb|1763 illustration of a slide rule]]
The slide rule was invented around 1620–1630, shortly after [[John Napier]]'s publication of the concept of the [[logarithm]]. In 1620 [[Edmund Gunter]] of Oxford developed a calculating device with a single logarithmic scale; with additional measuring tools it could be used to multiply and divide.<ref>{{cite book|url=https://books.google.com/books?id=eq5kDQAAQBAJ&pg=PA205|title=History of Mathematics|last=Smith|first=David E.|date=1958|publisher=Courier Corporation|isbn=9780486204307|pages=205}}</ref> In c. 1622, [[William Oughtred]] of Cambridge combined two handheld [[Gunter's scale|Gunter rules]] to make a device that is recognizably the modern slide rule.<ref>{{cite book|chapter-url=https://books.google.com/books?id=k43Q9RHuGXgC&q=William+Oughtred+circular+rule&pg=PT831|title=Encyclopedia of the Scientific Revolution: From Copernicus to Newton|last=Applebaum|first=Wilbur|date=2003-12-16|publisher=Routledge|isbn=9781135582555|chapter=Slide Rule }}</ref> Oughtred became involved in a vitriolic controversy over [[Scientific priority|priority]], with his one-time student [[Richard Delamain]] and the prior claims of Wingate. 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]], expanding the slide rule's use beyond 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 Mark 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.

In 1821, [[Nathaniel Bowditch]], described in the ''[[American Practical Navigator]]'' a "sliding rule" that contained scaled trigonometric functions on the fixed part and a line of log-sines and log-tans on the slider used to solve navigation problems.

In 1845, Paul Cameron of Glasgow introduced a nautical slide rule capable of answering navigation questions, including [[right ascension]] and [[declination]] of the sun and principal stars.<ref>"Cameron's Nautical Slide Rule", ''The Practical Mechanic and Engineer's Magazine'', April 1845, p. 187 and Plate XX-B</ref>

===Modern form===
[[File:Slideruleinuse.wpafb.chap2-2.jpg|thumb|Engineer using a slide rule, with mechanical calculator in background, mid 20th century]]

A more modern form of slide rule was created in 1859 by French artillery lieutenant [[Amédée Mannheim]], who was fortunate both in having his rule made by a firm of national reputation, and its adoption by the French Artillery. Mannheim's rule had two major modifications that made it easier to use than previous general-purpose slide rules. Such rules had four basic scales, A, B, C, and D, and D was the only single-decade logarithmic scale; C had two decades, like A and B. Most operations were done on the A and B scales; D was only used for finding squares and square roots. 

Mannheim changed the C scale to a single-decade scale and performed most operations with C and D instead of A and B. Because the C and D scales were single-decade, they could be read more precisely, so the rule's results could be more accurate. The change also made it easier to include squares and square roots as part of a larger calculation. Mannheim's rule also had a cursor, unlike almost all preceding rules, so any of the scales could be easily and accurately compared across the rule width. The "Mannheim rule" became the standard slide rule arrangement for the later 19th century and remained a common standard throughout the slide-rule era.

The growth of the [[engineering]] profession during the later 19th century drove widespread slide-rule use, beginning in Europe and eventually taking hold in the United States as well. The duplex rule was invented by William Cox in 1891 and was produced by [[Keuffel and Esser Co.]] of New York.<ref name="fn_3">{{cite book |url=http://www.mccoys-kecatalogs.com/K&EManuals/4081-3_1943/4081-3_1943.htm |title=The Log-Log Duplex Decitrig Slide Rule No. 4081: A Manual |publisher=Keuffel & Esser |first1=Lyman M. |last1=Kells |first2=Willis F. |last2=Kern |first3=James R. |last3=Bland |year=1943 |page=92 |archive-url=https://web.archive.org/web/20090214020502/http://www.mccoys-kecatalogs.com/K%26EManuals/4081-3_1943/4081-3_1943.htm |archive-date=14 February 2009 |url-status=dead }}</ref><ref name="fn_4">''The Polyphase Duplex Slide Rule, A Self-Teaching Manual'', Breckenridge, 1922, p. 20.</ref>

In 1881, the American inventor Edwin Thacher introduced his cylindrical rule, which had a much longer scale than standard linear rules and thus could calculate to higher precision, about four to five significant digits. However, the Thacher rule was quite expensive, as well as being non-portable, so it was used in far more limited numbers than conventional slide rules.

Astronomical work also required precise computations, and, in 19th-century Germany, a steel slide rule about two meters long was used at one observatory. It had a microscope attached, giving it accuracy to six decimal places.{{Citation needed|date=February 2011}}

===20th century===
[[File:Buzz and his pipe.jpg|thumb|[[Buzz Aldrin]] with slide rule during [[Gemini 12]] mission.]]

In the 1920s, the novelist and engineer [[Nevil Shute|Nevil Shute Norway]] (he called his autobiography [[Slide Rule: Autobiography of an Engineer|''Slide Rule'']]) was ''Chief Calculator'' on the design of the British [[R100]] airship for [[Vickers#Aviation|Vickers Ltd.]] from 1924. The stress calculations for each transverse frame required computations by a pair of ''calculators'' (people) using [[Fuller's cylindrical slide rule]]s for two or three months. The simultaneous equation contained up to seven unknown quantities, took about a week to solve, and had to be repeated with a different selection of slack wires if the guess on which of the eight radial wires were slack was wrong and one of the wires guessed to be slack was not slack. After months of labour filling perhaps fifty [[Paper size#Traditional inch-based paper sizes|foolscap]] sheets with calculations "the truth stood revealed (and) produced a satisfaction almost amounting to a religious experience".<ref>{{cite book |last= Norway |first= Nevil Shute |title=[[Slide Rule: Autobiography of an Engineer|Slide Rule]] |year= 1954 |publisher= William Heinemann |location= London |pages= 76–78 }}</ref>

In 1937, physicist [[Lucy Julia Hayner|Lucy Hayner]] designed and constructed a circular slide rule in [[Braille]].<ref>{{Cite journal |last=Witcher |first=C. M. |date=1954-12-01 |title=Physics without sight  |journal=Physics Today |volume=7 |issue=12 |pages=8–10 |doi=10.1063/1.3061483 }}</ref>

Throughout the 1950s and 1960s, the slide rule was the symbol of the engineer's profession in the same way the [[stethoscope]] is that of the medical profession.<ref>{{Cite journal |last=Stoll |first=Cliff |date=2006 |title=When Slide Rules Ruled |url=https://www.jstor.org/stable/26061456 |journal=Scientific American |volume=294 |issue=5 |pages=80–87 |doi=10.1038/scientificamerican0506-80 |jstor=26061456 }}</ref>

Aluminium Pickett-brand slide rules were carried on [[Project Apollo]] space missions. The model N600-ES owned by [[Buzz Aldrin]] that flew with him to the Moon on [[Apollo&nbsp;11]] was sold at auction in 2007.<ref>{{cite web |url=http://historical.ha.com/c/item.zx?saleNo=669&lotIdNo=41035 |title=Lot 25368 Buzz Aldrin's Apollo 11 Slide Rule – Flown to the Moon. ... 2007 September Grand Format Air & Space Auction #669 |publisher=Heritage Auctions |access-date=3 September 2013 |archive-date=3 September 2013 |archive-url=https://archive.today/20130903142931/http://historical.ha.com/c/item.zx?saleNo=669&lotIdNo=41035 |url-status=dead }}</ref> The model N600-ES taken along on [[Apollo&nbsp;13]] in 1970 is owned by the [[National Air and Space Museum]].<ref>{{cite web |url=http://airandspace.si.edu/collections/artifact.cfm?id=A19840160000 |title=Slide Rule, 5-inch, Pickett N600-ES, Apollo 13 |publisher=Smithsonian National Air and Space Museum |access-date=3 September 2013 |archive-date=9 October 2013 |archive-url=https://web.archive.org/web/20131009225758/http://airandspace.si.edu/collections/artifact.cfm?id=A19840160000 |url-status=dead }}</ref>

Some engineering students and engineers carried ten-inch slide rules in belt holsters, a common sight on campuses even into the mid-1970s. Until the advent of the pocket digital calculator, students also might keep a ten- or twenty-inch rule for precision work at home or the office<ref>Charles Overton Harris, ''Slide rule simplified'', American Technical Society, 1961, p. 5.</ref> 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. However, there has been little practical interest in constructing one beyond the initial prototype.<ref>{{cite web |url=http://www.findarticles.com/p/articles/mi_qa3950/is_200401/ai_n9372466 |archive-url=https://web.archive.org/web/20050510072825/http://www.findarticles.com/p/articles/mi_qa3950/is_200401/ai_n9372466 |url-status=dead |archive-date=2005-05-10 |last1=Sher |first1=David B. |last2=Nataro |first2=Dean C. |title=The Prosthaphaeretic Slide Rule: A Mechanical Multiplication Device Based On Trigonometric Identities |work=Mathematics And Computer Education, Vol. 38, Iss. 1 (Winter 2004): 37–43 |via=Findarticles.com |date=2009-06-02 |access-date=2010-02-20}}</ref>

===Specialized calculators===
Slide rules have often been specialized to varying degrees for their field of use, such as excise, proof calculation, engineering, navigation, etc., and some slide rules are extremely specialized for very narrow applications. For example, the John Rabone & Sons 1892 catalog lists a "Measuring Tape and Cattle Gauge", a device to estimate the weight of a cow from its measurements.

There were many specialized slide rules for photographic applications. For example, the [[actinograph]] of [[Hurter and Driffield]] was a two-slide boxwood, brass, and cardboard device for estimating [[Exposure (photography)|exposure]] from time of day, time of year, and latitude.

Specialized 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.<ref>{{cite web|url=http://www.natgasedu.com/vm004.html |archive-url=https://web.archive.org/web/20100106031649/http://www.natgasedu.com/vm004.html |archive-date=6 January 2010 |title=Fisher sizing rules |access-date=2009-10-06 |work=natgasedu.com |url-status=dead }}</ref>

Pilot balloon slide rules were used by meteorologists in weather services to determine the upper wind velocities from an ascending hydrogen or helium-filled pilot balloon.<ref>{{cite web|url=http://www.pilotballoon.com/slide.htm|title=Pilot Balloon Slide Rules|work=www.pilotballoon.com|access-date=28 September 2016|archive-url=https://web.archive.org/web/20160928041013/http://www.pilotballoon.com/slide.htm|archive-date=2016-09-28|url-status=dead}}</ref>

The [[E6B|E6-B]] is a circular slide rule used by pilots and navigators.

Circular slide rules to estimate ovulation dates and fertility are known as ''wheel calculators''.<ref>{{cite journal |doi=10.1080/14767050412331312200 |pmid=15061314 |title=Circle of time: errors in the use of the pregnancy wheel |journal=Journal of Maternal-Fetal and Neonatal Medicine |date=2003 |volume=14 |issue=6 |pages=370–372 |first=M. G. |last=Ross }}</ref>

A Department of Defense publication from 1962<ref>{{cite web|url=https://www.fourmilab.ch/etexts/www/effects/|title=The Effects of Nuclear Weapons|access-date=2021-05-02}}</ref> infamously included a special-purpose circular slide rule for calculating blast effects, overpressure, and radiation exposure from a given yield of an atomic bomb.<ref>{{cite web|url=https://www.fourmilab.ch/bombcalc/|title=Strangelove Slide Rule|access-date=2021-05-02}}</ref>

<gallery>
File:E6b-front.jpg|An E6-B aviation computer
File:John Rabone 1892 Cattle Gauge.png|John Rabone & Sons 1892 cattle gauge
File:HD Actinograph.jpg|[[Hurter and Driffield]]'s [[actinograph]]
File:Cryptographic sliding rule-IMG 0533.jpg|Cryptographic slide rule used by the Swiss Army between 1914 and 1940
Keuffel & Esser Model 4180, Fraction Adder - MIT Slide Rule Collection - DSC03616.JPG|Rare fractional adder
</gallery>

===Decline===
{{See also|History of computing hardware (1960s–present)}}
[[File:TI-30 LED.png|thumb|upright=.5|The [[TI-30]] scientific calculator, introduced for under US$25 in 1976]]
The importance of the slide rule began to diminish as electronic computers, a new but rare resource in the 1950s, became more widely available to technical workers during the 1960s.

The first step away from slide rules was the introduction of relatively inexpensive electronic desktop scientific calculators. These included the [[Wang Laboratories]] LOCI-2,<ref>{{cite web|url=http://www.oldcalculatormuseum.com/wangloci.html|title=The Wang LOCI-2|work=oldcalculatormuseum.com}}</ref><ref name="Wang Ad 1966">{{cite journal |last=Wang Laboratories |title=Now you can determine Copolymer Composition in a few minutes at your desk |journal=Analytical Chemistry |volume=38 |issue=13 |pages=62A–63A |date=December 1966 |doi=10.1021/ac50155a005}}</ref> introduced in 1965, which used logarithms for multiplication and division; and the [[Hewlett-Packard]] [[HP 9100A]], introduced in 1968.<ref name="Leibson_2010">{{cite web |title=The HP 9100 Project: An Exothermic Reaction |date=2010 |first=Steven<!-- Steve --> |last=Leibson |url=http://www.hp9825.com/html/the_9100_part_2.html |access-date=2024-06-16}}</ref> Both of these were programmable and provided exponential and logarithmic functions; the HP had [[trigonometric function]]s (sine, cosine, and tangent) and hyperbolic trigonometric functions as well. The HP used the [[CORDIC]] (coordinate rotation digital computer) algorithm,<ref name="Volder_2000">{{cite journal |first=Jack E. |last=Volder |title=The Birth of CORDIC |journal=Journal of VLSI Signal Processing |issn=0922-5773 |volume=25 |issue=2 |pages=101–105 |date=June 2000 |url=http://late-dpedago.urv.cat/site_media/papers/fulltext_2.pdf |archive-url=https://web.archive.org/web/20160304064804/http://late-dpedago.urv.cat/site_media/papers/fulltext_2.pdf |url-status=dead |archive-date=2016-03-04 |access-date=2016-01-02 |doi=10.1023/a:1008110704586 }}</ref> which allows for calculation of trigonometric functions using only shift and add operations. This method facilitated the development of ever smaller scientific calculators.

As with mainframe computing, the availability of these desktop machines did not significantly affect the ubiquitous use of the slide rule, until cheap hand-held scientific electronic calculators became available in the mid-1970s, at which point it rapidly declined. The pocket-sized Hewlett-Packard [[HP-35]] scientific calculator was the first handheld device of its type, but it cost US$395 in 1972. This was justifiable for some engineering professionals, but too expensive for most students.

Around 1974, lower-cost handheld electronic scientific calculators started to make slide rules largely obsolete.<ref>{{cite book |title=Writing and reading across the curriculum |last1=Behrens |first1=Lawrence|last2=Rosen|first2=Leonard J.|year=1982|publisher=[[Little, Brown]]|page=273|quote=Then, just a decade ago, the invention of the pocket calculator made the slide rule obsolete almost overnight...}}</ref><ref>{{cite book |title=e: The Story of a Number |page=16 |last1=Maor |first1=Eli |publisher=Princeton University Press |year=2009 |isbn=978-0-691-14134-3 |quote=Then in the early 1970s the first electronic hand-held calculators appeared on the market, and within ten years the slide rule was obsolete.}}</ref><ref>{{cite book |title=Inventions that Changed the World |last1=Castleden |first1=Rodney |year=2007 |page=157 |publisher=Futura |isbn=978-0-7088-0786-6 |quote=With the invention of the calculator the slide rule became instantly obsolete.}}</ref><ref>{{cite book |title=Beyond calculation: the next fifty years of computing |page=xiv |last1=Denning |first1=Peter J. |author-link1=Peter J. Denning |last2=Metcalfe |first2=Robert M. |author-link2=Bob Metcalfe |isbn=978-0-387-98588-6 |publisher=[[Springer Verlag|Springer]] |year=1998 |quote=The first hand calculator appeared in 1972 and made the slide rule obsolete overnight. |url=https://archive.org/details/beyondcalculatio00pete}}</ref> By 1975, basic four-function electronic calculators could be purchased for less than $50, and by 1976 the [[TI-30]] scientific calculator was sold for less than $25 (${{Formatprice|{{Inflation|US|25|1976|r=2}}|0}} adjusted for inflation).

1980 was the final year of the [[University Interscholastic League]] (UIL) competition in [[Texas]] to use slide rules.{{cn|date=April 2023}}  The UIL had been originally been organized in 1910 to administer literary events,{{cn|date=April 2023}} but had become the governing body of school sports events as well.

==Comparison to electronic digital calculators==
[[File:IBM 150 Extra Engineers 1951.jpg|thumb|upright|December 1951 advertisement for the IBM 604 [[IBM CPC|Electronic Calculating Punch]] explicitly comparing electronic computers to engineers calculating with slide rules]]
Even during their heyday, slide rules never caught on with the general public.<ref>Stoll, Cliff. "When Slide Rules Ruled," ''Scientific American,'' May 2006, [http://sliderulemuseum.com/Ephemera/ScientificAmericanMay2006_SR_byCliffStoll.pdf 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."</ref> 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.<ref>Watson, George H. [https://books.google.com/books?id=KIyAn_jTGUYC&q=slide+rule&pg=PA110 "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."</ref> This led engineers to use mathematical equations that favored operations that were easy on a slide rule over more accurate but complex functions; these approximations could lead to inaccuracies and mistakes.<ref>Stoll, Cliff. "When Slide Rules Ruled," ''Scientific American,'' May 2006, [http://sliderulemuseum.com/Ephemera/ScientificAmericanMay2006_SR_byCliffStoll.pdf 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."</ref> 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.<ref>Stoll, Cliff. "When Slide Rules Ruled", ''Scientific American'', May 2006, [http://sliderulemuseum.com/Ephemera/ScientificAmericanMay2006_SR_byCliffStoll.pdf 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."</ref> 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. It has thus been compared with [[reverse Polish notation]] (RPN) implemented in electronic calculators.<ref name="Williams_2023"/>

A slide rule requires the user to separately compute the [[order of magnitude]] of the answer to position the decimal point in the results. For example, 1.5&nbsp;× 30 (which equals 45) will show the same result as {{val|1500000}}&nbsp;× 0.03 (which equals {{val|45000}}). This separate calculation 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 [[arithmetic 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 [[Tachymeter (watch)|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 [[grid (electricity)|grid electricity]] or batteries. 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 being replaced only by integrated, special purpose and expensive flight computers, and not general-purpose calculators. The [[E6B|E6-B]] 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 and the Seiko Flightmaster SNA411 are two notable examples.<ref>{{cite web|url=http://www.citizenwatch.com/en-us/watches/watch-detail/?model=JY0000-53E#filters=0000-53E|title=Citizen Watch Company – Citizen Eco-Drive / US, Canada, UK, IrelandCitizen Watch<!-- Bot generated title -->|work=citizenwatch.com|access-date=2014-04-21|archive-url=https://web.archive.org/web/20140422232006/http://www.citizenwatch.com/en-us/watches/watch-detail/?model=JY0000-53E#filters=0000-53E|archive-date=2014-04-22|url-status=dead}}</ref>

==Contemporary use==
{{Original research|section|date=February 2015}}
[[File:Faber Castell 6 inch slide rule.JPG|thumb|right|[[Faber-Castell]] slide rule with pouch]]

Even in the 21st century, some people prefer a slide rule over an electronic calculator as a practical computing device. Others keep their old slide rules out of a sense of nostalgia, or collect them as a hobby.<ref>{{cite web|url=http://sliderule.ozmanor.com/srlcol.html|title=Greg's Slide Rules – Links to Slide Rule Collectors|publisher=Sliderule.ozmanor.com|date=2004-07-29|access-date=2010-02-20}}</ref>

A popular collectible model is the [[Keuffel & Esser]] ''Deci-Lon'', a premium scientific and engineering slide rule available both in a ten-inch (25&nbsp;cm) "regular" (''Deci-Lon 10'') and a five-inch "pocket" (''Deci-Lon 5'') variant. Another prized American model is the eight-inch (20&nbsp;cm) Scientific Instruments circular rule. Of European rules, [[Faber-Castell]]'s high-end models are the most popular among collectors.

Although a great many slide rules are circulating on the market, specimens in good condition tend to be expensive. Many rules found for sale on [[:Category:online auction websites|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, expensive, and generally available only 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 chemically break down over time. [[Conservation and restoration of plastic objects|Methods of preserving plastic]] may be used to slow the deterioration of some older slide rules, and [[3D printing]] may be used to recreate missing or irretrievably broken cursor parts.<ref name="ISRM-Restore">{{cite web |title=Restoring and Repairing Slide Rules and Notes |url=https://www.sliderulemuseum.com/ |website=International Slide Rule Museum}}</ref>

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,<ref>{{cite web |url=http://www.concise.co.jp/eng0731/about.html |archive-url=https://web.archive.org/web/20120312213959/http://www.concise.co.jp/eng0731/about.html |archive-date=2012-03-12 |title=About CONCISE |publisher=Concise.co.jp |access-date=2010-02-20}}</ref> continues to make and sell them today. In September 2009, on-line retailer [[ThinkGeek]] introduced its own brand of straight slide rules, described as "faithful replica[s]" that were "individually hand tooled".<ref name="thinkgeek1">{{cite web |url=http://www.thinkgeek.com/gadgets/tools/be12/ |title=Slide Rule |publisher=ThinkGeek |access-date=2015-04-08 |archive-url=https://web.archive.org/web/20100327174231/http://www.thinkgeek.com/gadgets/tools/be12/ |archive-date=2010-03-27}}</ref> These were no longer available in 2012.<ref name="thinkgeek2">{{cite web |url=http://www.thinkgeek.com/gadgets/tools/be12/ |title=Slide Rule |publisher=ThinkGeek |access-date=2015-04-08 |archive-url=https://web.archive.org/web/20120415123554/http://www.thinkgeek.com/gadgets/tools/be12/ |archive-date=April 15, 2012 |url-status=dead}}</ref> In addition, Faber-Castell had a number of slide rules in inventory, available for international purchase through their web store, until mid 2018.<ref>{{cite web |url=http://service.de.faber-castell-shop.com/epages/es117781.sf/de_DE/?ObjectPath=/Shops/es117781_ServiceParts/Categories/Faber-Castell/FaberCastell_Rechenstaebe |title=Rechenschieber |publisher=Faber-Castell |access-date=2012-01-17 |archive-url=https://web.archive.org/web/20131121041445/http://service.de.faber-castell-shop.com/epages/es117781.sf/de_DE/?ObjectPath=%2FShops%2Fes117781_ServiceParts%2FCategories%2FFaber-Castell%2FFaberCastell_Rechenstaebe |archive-date=2013-11-21 |url-status=dead}} It is believed they may still have some slide rules, but their new website and online store does not carry them.</ref> Proportion wheels are still used in graphic design.

Various slide rule simulator apps are available for Android and iOS-based smart phones and tablets.

Specialized slide rules such as the E6-B used in aviation, and gunnery slide rules used in [[Gun laying|laying artillery]] are still used though no longer on a routine basis. These rules are used as part of the teaching and instruction process as in learning to use them the student also learns about the principles behind the calculations, it also allows the student to be able to use these instruments as a backup in the event that the modern electronics in general use fail.

==Collections==
[[File:Comically_large_slide_rule,_MIT_Museum.jpg|thumb|upright|A classroom-sized slide rule displayed at the [[MIT Museum]]]]
The [[MIT Museum]] in [[Cambridge, Massachusetts]], has a collection of hundreds of slide rules, [[nomogram]]s, and [[mechanical calculator]]s.<ref>{{cite web |url=https://commons.wikimedia.org/wiki/Category:MIT_Slide_Rule_Collection |title=Commons category MIT Slide Rule Collection}}</ref> The [[Keuffel and Esser]] Company collection, from the slide rule manufacturer formerly located in [[Hoboken, New Jersey]], was donated to MIT around 2005, substantially expanding existing holdings.<ref name="MITnews">{{cite news |title=MIT Museum measures up |url=https://news.mit.edu/2005/sliderules |access-date=2019-05-01 |work=MIT News |publisher=Massachusetts Institute of Technology |date=January 11, 2005}}</ref> Selected items from the collection are usually on display at the museum.<ref name="MITM">{{cite web |title=Slide Rules |url=https://mitmuseum.mit.edu/installation-or-collaboration/slide-rules |website=MIT Museum |publisher=Massachusetts Institute of Technology |access-date=2019-05-01 |archive-date=2019-05-01 |archive-url=https://web.archive.org/web/20190501192812/https://mitmuseum.mit.edu/installation-or-collaboration/slide-rules |url-status=dead }}</ref><ref name="Proun">{{cite web |title=MIT Museum — Slide Rules |url=http://proundesign.com/MIT-Museum-Slide-Rules |website=proundesign.com |publisher=Proun Design |access-date=2019-05-01 |archive-date=2019-05-01 |archive-url=https://web.archive.org/web/20190501192803/http://proundesign.com/MIT-Museum-Slide-Rules |url-status=dead }}</ref>

The [[International Slide Rule Museum]] is claimed to be "[the world's] most extensive resource for all things concerning slide rules and logarithmic calculators".<ref>{{Cite news |last=Turner |first=Kimberli |date=2009-10-07 |title=Slide Rule: 101 |work=Colorado Hometown Newspapers}}</ref> The museum's Web page includes extensive literature relative to slide rules in its "Slide Rule Library" section.<ref>{{Cite web |title=Slide Rule Library Main Desk |url=http://www.sliderulemuseum.com/SR_Library.htm |access-date=2023-01-07 |website=International Slide Rule Museum |archive-date=2023-01-07  |archive-url=https://web.archive.org/web/20230107151122/https://www.sliderulemuseum.com/SR_Library.htm |url-status=dead }}</ref> 

== See also ==

* {{annotated link|Abacus}}
* {{annotated link|Computer (occupation)}}
* {{annotated link|Curta}}
* {{annotated link|Flight computer}}
* {{annotated link|Floating point}}
* {{annotated link|Hans Peter Luhn}}
* {{annotated link|Nomogram}}
* {{annotated link|Sector (instrument)}}
* {{annotated link|Slide calculator}}
* {{annotated link|Slide chart}}
* {{annotated link|Timeline of computing}}
* {{annotated link|Vernier scale}}
* {{annotated link|Volvelle}}

==References==

{{refs|refs=
<ref name="Williams_2023">{{cite web |title=In Praise Of RPN (with Python Or C) |author-first=Al |author-last=Williams |date=2023-06-21 |work=Hackaday |url=https://hackaday.com/2023/06/21/in-praise-of-rpn-with-python-or-c/ |access-date=2023-09-23 |url-status=live |archive-url=https://web.archive.org/web/20230923090403/https://hackaday.com/2023/06/21/in-praise-of-rpn-with-python-or-c/ |archive-date=2023-09-23}}</ref>
}}

==External links==
{{Commons category|Slide rules}}

* [http://www.sliderulemuseum.com/ International Slide Rule Museum]
* [http://mysite.du.edu/~jcalvert/tech/slidrul.htm The history, theory and use of the engineering slide rule] — By Dr James B. Calvert, University of Denver
* [http://uksrc.org.uk/ United Kingdom Slide Rule Circle Home Page] {{Webarchive|url=https://web.archive.org/web/20150928114254/http://uksrc.org.uk/ |date=2015-09-28  }}
* [http://www.oughtred.org/ Oughtred Society Slide Rule Home Page] — Dedicated to the preservation and history of slide rules
* [http://sliderules.lovett.com/ Rod Lovett's Slide Rules] – Comprehensive Aristo site with many search facilities
* [http://www.antiquark.com/sliderule/sim/index.html Derek's virtual slide rule gallery] — Javascript simulations of historical slide rules
* {{cite NIE|wstitle=Slide rule|year=1905 |short=x}}
* {{cite Americana|wstitle=Slide-rule|year=1920 |short=x}}
* [http://fabercastell.reglasdecalculo.com/ Reglas de Cálculo] — A very big Faber Castell collection
* [http://www.photocalcul.com/Calcul/Regles/Graphoplex/GRAPHOPLEX.html#HAUT Collection of slide rules ] — French Slide Rules (Graphoplex, Tavernier-Gravet and others)
* [https://archive.today/20121208211557/http://www.sliderule.ca/ Eric's Slide Rule Site] — History and use
* [http://www.hpmuseum.org/sliderul.htm Slide Rules] — Information from The Museum of HP Calculators
* {{usurped|1=[https://web.archive.org/web/20181220044459/http://www.decadecounter.com/vta/tubepage.php?item=5 Descriptions, alphabetical by brandname, with images]}} (Vintage Tech. Assoc.)

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