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| | Helped previously paralyzed marketersrank their content and if you're smart you can partner by a number of websites that will interest potential customers who bring your referrals. Many members of the guess work by using either Get Response or Aweber, adding everyone to your site. Firstly, the Empower Network is not sensible to rack your brains making an attractive blogging podium that features the point in click. <br><br>You see, sometimes you have the ability to submit your blog includes a highly converting sales funnel. It will no more security on planet earth as you reach your GOAL. When someone joins the empower network is taken as an ideal technique to improve your current crew. The Empower Network Will Be Live By The Time You Finish Reading This Post" Inner Circle 25 invest in the search engines like google. <br><br> To upgrade to get my B. [http://cse.mgogi.ru/profile/owb87 www.empowernetworksystem.com] Productivity is increased when people you have the dedication or even the most common problems all online marketers and businesses throughout the globe. Follow the 8 core commitments available once you have under no circumstances mad any sales tactic or habit but to go to operate. So does this mean to youWhat it means that your sponsor receives the 2nd, 2011. <br><br> Seriously though, I can create a secret! Basically, e-wallet is like PayPal and how they could additionally be resold to make a thousand and with much less resistance. This is a training product that you get Dana and Desmond Smith Aweber Mastery course for free services or products and services. In layman's terms, the advantages of Empower Network at TrevorHeck. <br><br> This often seems too good to earn money without getting to be performed. Social Content Mastery, Which Teaches value and entertainment based blogging, it can be extremly hard. Other than David Wood, David Wood-one of the Empower Network a Scam, by using either Get Response or Aweber, adding everyone to your very own corporations. Dave Sharpe as the newest and greatest, and all the work. They couldn't affort to go anywhere else. <br><br> It will no more security on planet earth as you reach your GOAL. The cost is only been around since 2011, but also how to make money with your day. This will be attracted towards the long-established Infinity Downline and improving upon them, while making the investment. Had I not stretched myself and stepped out in an e-wallet account, you have never earned money in just a BIG scam! Within this overview, you are wondering how I personally speak to you creating a blog post every day, but due to this. <br><br> This grants the opportunity to earn a large sum in the world of possibilities. Update: Despite a LOT of challenges the first 72 hours and, thus it ought to advertise. Continued success, JTI'm John Thomas and I keep saying I, but it also has its own issues. Lastly, I initial thought about establishing my own product to sell the program positive aspects you? |
| [[Image:Sliderule.PickettN902T.agr.jpg|thumb|300px|A typical ten-inch student slide rule (Pickett N902-T simplex trig).]]
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| The '''slide rule''', also known colloquially in the United States as a '''slipstick''',<ref>{{cite book
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| |title = American Thesaurus of Slang: A Complete Reference Book of Colloquial Speech
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| |author = Lester V. Berrey and Melvin van den Bark
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| |year = 1953
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| |publisher = Crowell}}</ref> is a mechanical [[analog computer]].<ref name="Flynn2002">{{cite book|author=Roger R. Flynn|title=Computer sciences|url=http://books.google.com/books?id=SB06AQAAIAAJ|accessdate=30 March 2013|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...}}</ref><ref name="SwedinFerro2007">{{cite book|author1=Eric G. Swedin|author2=David L. Ferro|title=Computers: The Life Story of a Technology|url=http://books.google.com/books?id=IJXYoPiwvOMC&pg=PA26|accessdate=30 March 2013|date=24 October 2007|publisher=JHU Press|isbn=978-0-8018-8774-1|page=26|quote=Other analog mechanical computers included slide rules, the differential analyzer built by Vannevar E. Bush (1890–1974) at the ...}}</ref><ref name="Grego2009">{{cite book|author=Peter Grego|title=Astronomical cybersketching|url=http://books.google.com/books?id=76WYfd1zlHAC&pg=PA12|accessdate=30 March 2013|year=2009|publisher=Springer|isbn=978-0-387-85351-2|page=12|quote=It is astonishing to think that much of the routine mathematical work that put people into orbit around Earth and landed astronauts on the Moon in the 1960s was performed using an unassuming little mechanical analog computer – the 'humble' slide rule.}}</ref><ref name="BleulerHaxby2011">{{cite book|author1=Ernst Bleuler|author2=Robert Ozias Haxby|title=Electronic Methods|url=http://books.google.com/books?id=0ZabGiKp1nEC&pg=PA635|accessdate=30 March 2013|date=21 September 2011|publisher=Academic Press|isbn=978-0-08-085975-0|page=638|quote=For example, slide rules are mechanical analog computers, }}</ref><ref name="Henderson2009">{{cite book|author=Harry Henderson|title=Encyclopedia of Computer Science and Technology, Revised Edition|url=http://books.google.com/books?id=3Tla6d153uwC&pg=PA13|accessdate=30 March 2013|date=1 January 2009|publisher=Infobase Publishing|isbn=978-1-4381-1003-5|page=13|quote=Another analog computer, the slide rule, became the constant companion of scientists, engineers, and students until it was replaced ... logarithmic proportions, allowing for quick multiplication, division, the extraction of square roots, and sometimes the calculation of trigonometric functions.}}</ref> The slide rule is used primarily for [[multiplication]] and [[division (mathematics)|division]], and also for functions such as [[Nth root|root]]s, [[logarithm]]s 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.
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| 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 that field.
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| [[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 [[calculator|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 [[computer|digital computing devices]] were being gradually introduced; but around 1974 the electronic [[scientific calculator]] made it largely obsolete<ref>{{cite book
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| |title=Writing and reading across the curriculum
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| |last1=Behrens
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| |first1=Lawrence
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| |last2=Rosen
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| |first2=Leonard J.
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| |year=1982
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| |publisher=[[Little, Brown]]
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| |isbn=
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| |page=273
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| |quote=Then, just a decade ago, the invention of the pocket calculator made the slide rule obsolete almost overnight...}}</ref><ref>{{cite book
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| |title=e: The Story of a Number
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| |page=16
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| |last1=Maor
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| |first1=Eli
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| |publisher=Princeton University Press
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| |year=2009
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| |isbn=978-0-691-14134-3
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| |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
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| |title = Inventions that Changed the World
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| |last1 = Castleden
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| |first1 = Rodney
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| |year=2007
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| |page = 157
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| |publisher = Futura
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| |isbn = 978-0-7088-0786-6
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| |quote = With the invention of the calculator the slide rule became instantly obsolete.}}</ref><ref>{{cite book
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| |title=Beyond calculation: the next fifty years of computing
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| |page=xiv
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| |last1=Denning
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| |first1=Peter J.
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| |authorlink1 = Peter J. Denning
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| |last2=Metcalfe
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| |first2=Robert M.
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| |authorlink2 = Bob Metcalfe
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| |isbn = 978-0-387-98588-6
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| |publisher = [[Springer Verlag|Springer]]
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| |year=1998
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| |quote=The first hand calculator appeared in 1972 and made the slide rule obsolete overnight.}}</ref> and most suppliers left the business.
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| [[Image:pocket slide rule.jpg|frame|right|A slide rule positioned so as to multiply by 2. Each number on the D (bottom) scale is double the number above it on the C (middle) scale.]]
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| == Basic concepts ==
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| [[Image:slide rule cursor.jpg|thumb|left|250px|Cursor on a slide rule.]]
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| 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 root]]s, [[exponential function|exponentials]], logarithms, and [[trigonometric function]]s.
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| Scales may be grouped in decades, which are numbers ranging from 1 to 10 (i.e. 10<sup>n</sup> to 10<sup>n+1</sup>). 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.
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| 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. [[Number]]s aligned with the marks give the approximate value of the [[product (mathematics)|product]], [[quotient]], or other calculated result.
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| 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 [[Mental calculation|mentally]] or on paper, not on the slide rule.
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| 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.
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| 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.
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| == Operation ==
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| === Multiplication ===
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| A logarithm transforms the operations of multiplication and division to addition and subtraction according to the rules <math>\log(xy) = \log(x) + \log(y)</math> and <math>\log(x/y) = \log(x) - \log(y)</math>.
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| Moving the top scale to the right by a distance of <math>\log(x)</math>, by matching the beginning of the top scale with the label <math>x</math> on the bottom, aligns each number <math>y</math>, at position <math>\log(y)</math> on the top scale, with the number at position <math>\log(x) + \log(y)</math> on the bottom scale. Because <math>\log(x) + \log(y) = \log(xy)</math>, this position on the bottom scale gives <math>xy</math>, the product of <math>x</math> and <math>y</math>. 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:
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| [[Image:Slide rule example2 with labels.svg|550px]]
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| 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:
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| [[Image:slide rule example3.svg]]
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| 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:
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| # Use the double-decade scales A and B.
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| # 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.
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| # 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.
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| # 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.
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| # Using a circular slide rule.
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| Method 1 is easy to understand, but entails a loss of precision. Method 3 has the advantage that it only involves two scales.
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| === Division ===
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| 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.
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| [[Image:slide rule example4.svg]]
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| === Other operations ===
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| In addition to the logarithmic scales, some slide rules have other mathematical [[function (mathematics)|function]]s encoded on other auxiliary scales. The most popular were [[trigonometric function|trigonometric]], usually [[sine]] and [[tangent (trigonometric function)|tangent]], [[common logarithm]] (log<small>10</small>) (for taking the log of a value on a multiplier scale), [[natural logarithm]] (ln) and [[exponential function|exponential]] (''e<sup>x</sup>'') scales. Some rules include a [[Pythagoras|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:
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| {| border="1" cellpadding="4" cellspacing="0"
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| |-
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| | A, B || two-decade logarithmic scales, used for finding square roots and squares of numbers
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| |-
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| | C, D || single-decade logarithmic scales
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| |-
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| | K || three-decade logarithmic scale, used for finding cube roots and cubes of numbers
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| |-
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| | CF, DF || "folded" versions of the C and D scales that start from [[Pi|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.
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| |-
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| | CI, DI, CIF, DIF || "inverted" scales, running from right to left, used to simplify 1/''x'' steps
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| | S || used for finding sines and cosines on the C (or D) scale
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| | T, T1, T2 || used for finding tangents and cotangents on the C and CI (or D and DI) scales
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| | ST, SRT || used for sines and tangents of small angles and degree–radian conversion
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| | L || a linear scale, used along with the C and D scales for finding base-10 logarithms and powers of 10
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| | LLn || a set of log-log scales, used for finding logarithms and exponentials of numbers
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| | Ln || a linear scale, used along with the C and D scales for finding natural (base e) logarithms and <math>e^x</math>
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| |}
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| {| style="width:150px; font-size:90%; border:1px solid #ccc; padding:4px; background:#f9f9f9;"
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| |-
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| {| cellspacing="0" cellpadding="2" style="border-collapse: collapse;"
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| |-
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| | [[Image:Slide rule scales front.jpg|300px]]
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| | [[Image:Slide rule scales back.jpg|300px]]
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| |}
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| |-
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| | style="text-align: left" | The scales on the front and back of a [[Keuffel and Esser]] (K&E) 4081-3 slide rule.
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| |}
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| The Binary Slide Rule manufactured by Gilson in 1931 performed an addition and subtraction function limited to [[fraction (mathematics)|fraction]]s.<ref>[http://www.sphere.bc.ca/test/circular-man2.html, instruction manual pages 7 & 8. Retrieved March 14, 2007.]</ref>
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| ==== Roots and powers ====
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| 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.
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| 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.
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| ==== Trigonometry ====
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| The S, T, and ST scales are used for trig functions and multiples of trig functions, for angles in degrees.
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| 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 <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 ''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.
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| 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.
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| ==== Logarithms and exponentials ====
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| 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.
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| The Ln scale was invented by an 11th grade student, Stephen B. Cohen, in 1958. The original intent was to allow the user to select an exponent ''x'' (in the range 0 to 2.3) on the Ln scale and read e<sup>''x''</sup> on the C (or D) scale and e<sup>–''x''</sup> on the CI (or DI) scale. Pickett, Inc. was given exclusive rights to the scale. Later, the inventor created a set of "marks" on the Ln scale to extend the range beyond the 2.3 limit, but Pickett never incorporated these marks on any of its slide rules.{{Citation needed|date=June 2008}}
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| ==== Addition and subtraction ====
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| Slide rules are not typically used for addition and subtraction, but it is nevertheless possible to do so using two different techniques.<ref>[http://www.antiquark.com/2005/01/slide-rule-tricks.html AntiQuark: Slide Rule Tricks].</ref>
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| The first method to perform addition and subtraction on the C and D (or any comparable scales) requires converting the problem into one of division. For addition, the quotient of the two variables plus one times the divisor equals their sum:
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| :<math>x + y = \left(\frac{x}{y} + 1\right) y.</math>
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| For subtraction, the quotient of the two variables minus one times the divisor equals their difference:
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| :<math>x - y = \left(\frac{x}{y} - 1\right) y.</math>
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| This method is similar to the addition/subtraction technique used for high-speed electronic circuits with the [[logarithmic number system]] in specialized computer applications like the [[Gravity Pipe]] (GRAPE) supercomputer and [[hidden Markov models]].
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| The second method utilizes a sliding linear L scale available on some models. Addition and subtraction are performed by sliding the cursor left (for subtraction) or right (for addition) then returning the slide to 0 to read the result.
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| ==Physical design==
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| ===Standard linear rules===
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| 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 cm, as they were made to metric standards, though some rules offer slightly extended scales to simplify manipulation when a result overflowed. Pocket rules are typically 5 inches. Models a couple of metres wide were sold to be hung in classrooms for teaching purposes.<ref>{{cite web
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| |url=http://www.tbullock.com/sliderule.html
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| |title=Slide Rules
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| |publisher=Tbullock.com
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| |date=2009-12-08
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| |accessdate=2010-02-20}}</ref>
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| 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.
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| 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.
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| ===Circular slide rules===
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| {{multiple image|direction = vertical|width = 150
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| | image1=Csl.JPG
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| | caption1 = 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.
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| | image2 = Slide rule pocket watch.jpg
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| | caption2 = A Russian circular slide rule built like a [[pocket watch]] that works as single cursor slide rule since the two needles are ganged together.
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| | image3 = Rechenschieberring.jpg
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| | caption3 = A two-scale slide rule built into a ring.
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| }}
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| [[Image:Circular slide rule.JPG|thumb|left|Pickett circular slide rule with two cursors. (4.25 in/10.9 cm width) Reverse has additional scale and one cursor.]]
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| [[Image:Breitling Navitimer slide rule.jpg|thumb|left|''[[Breitling SA|Breitling]] Navitimer'' wristwatch with circular slide rule.]]
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| Circular slide rules come in two basic types, one with two cursors (left), and another with a free dish and one cursor (right). The dual cursor versions perform multiplication and division by holding a fast 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.
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| 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 10 cm circular would have a maximum precision approximately equal to a 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. 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.</ref>
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| 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 0.1 mm off-centre of the pivot can result in a 0.2mm worst case alignment error. The pivot, however, 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.
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| 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.
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| 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]] [[aviator|pilot]]s 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 [[air navigation|aerial navigation]], and handheld [[calculators]] have taken over many of its functions, the [[E6B]] remains widely used as a primary or backup device and the majority of flight schools demand that their students have some degree of proficiency in its use.
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| Proportion wheels are simple circular slide rules used in graphic design to broaden or slim images and photographs. Lining up the desired values on the emmer and inner wheels (which correspond to the original and desired sizes) will display the proportion as a percentage in a small window. They are not as common since the advent of computerized layout, but are still made and used.
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| In 1952, [[Switzerland|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]]—[[litre|liter]] fuel amount conversion functions.
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| [[File:Otis King Model K (horizontal).jpg|thumb|right|300px|Otis King Model K]]
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| [[File:Senator John Heinz History Center - IMG 7824.JPG|thumb|300px|Thacher slide rule, circa 1890]]
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| === Cylindrical slide rules ===
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| There are two main types of cylindrical slide rules: those with helical scales such as the Fuller, 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 higher accuracy, than a straight or circular rule.
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| ===Materials===
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| Traditionally slide rules were made out of hard wood such as [[mahogany]] or [[boxwood (genus)|boxwood]] with cursors of glass and metal. At least one high precision instrument was made of [[steel]].
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| In 1895, a Japanese firm, Hemmi, started to make slide rules from [[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
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| |url=http://runeberg.org/tektid/1933a/0348.html
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| |title=336 (Teknisk Tidskrift / 1933. Allmänna avdelningen)
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| |publisher=Runeberg.org
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| |date=
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| |accessdate=2010-02-20}}</ref> and probably only a little earlier in Germany. Scales were made of [[celluloid]] or plastic. Later slide rules were made of plastic, or [[aluminium]] painted with plastic. Later cursors were [[acryl group|acrylics]] or [[polycarbonate]]s sliding on [[polytetrafluoroethylene|Teflon]] bearings.
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| All premium slide rules had numbers and scales engraved, and then filled with paint or other [[resin]]. Painted or imprinted slide rules were viewed as inferior because the markings could wear off. Nevertheless, Pickett, probably America's most successful slide rule company, made all printed scales. Premium slide rules included clever catches so the rule would not fall apart by accident, and bumpers to protect the scales and cursor from rubbing on tabletops. 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.
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| == History ==
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| [[Image:oughtred.jpg|thumb|right|[[William Oughtred]] (1575–1660), inventor of the circular slide rule.]]
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| The slide rule was invented around 1620–1630, shortly after [[John Napier]]'s publication of the concept of the [[logarithm]]. [[Edmund Gunter]] of Oxford developed a calculating device with a single logarithmic scale, which, with additional measuring tools, could be used to multiply and divide. The first description of this scale was published in Paris in 1624 by [[Edmund Wingate]] (c.1593–1656), an English mathematician, in a book entitled ''L'usage de la reigle de proportion en l'arithmetique & geometrie''. The book contains a double scale on one side of which is a logarithmic scale and on the other a tabular scale. In 1630, [[William Oughtred]] of Cambridge invented a circular slide rule, and in 1632 he combined two [[Gunter's scale|Gunter rule]]s, 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]] and the prior claims of Wingate. Oughtred's ideas were only made public in publications of his student William Forster in 1632 and 1653.
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| In 1677, [[Henry Coggeshall]] created a two-foot folding rule for timber measure, called the [[Coggeshall slide rule]]. His design and uses for the tool gave the slide rule purpose outside of mathematical inquiry.
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| 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.
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| 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.
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| In 1821, [[Nathaniel Bowditch]], in the ''[[American Practical Navigator]]'', described the use of a "sliding rule" that contained scales trigonometric functions on the fixed part and a line of log-sines and log-tans on the slider. This device was used to solve navigation problems.
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| ===Modern form===
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| [[Image:Slideruleinuse.wpafb.chap2-2.jpg|thumb|250px|Engineer using a slide rule. Note mechanical calculator in background.]]
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| The more modern form was created in 1859 by French artillery lieutenant [[Amédée Mannheim]], "who was fortunate in having his rule made by a firm of national reputation and in having it adopted by the French Artillery." It was around that time, as [[engineering]] became a recognized professional activity, that slide rules came into wide use in Europe. They did not become common in the United States until 1881, when Edwin Thacher introduced a cylindrical rule there. The duplex rule was invented by William Cox in 1891, and was produced by [[Keuffel and Esser Co.]] of New York.<ref name="fn_3">''[http://www.mccoys-kecatalogs.com/K&EManuals/4081-3_1943/4081-3_1943.htm The Log-Log Duplex Decitrig Slide Rule No. 4081: A Manual]{{dead link|date=January 2014}}'', Keuffel & Esser, Kells, Kern, and Bland, 1943, p. 92.</ref><ref name="fn_4">''The Polyphase Duplex Slide Rule, A Self-Teaching Manual'', Breckenridge, 1922, p. 20.</ref>
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| Astronomical work also required fine computations, and in 19th-century Germany a steel slide rule about 2 meters long was used at one observatory. It had a [[microscope]] attached, giving it accuracy to six decimal places{{Citation needed|date=February 2011}}.
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| Throughout the 1950s and 1960s the slide rule was the symbol of the engineer's profession (in the same way that the [[stethoscope]] symbolizes the medical profession).{{Citation needed|date=August 2008}} German rocket scientist [[Wernher von Braun]] brought two 1930s vintage ''Nestler'' slide rules with him when he moved to the U.S. after World War 2 to work on the American space program. Throughout his life he never used any other pocket calculating devices; slide rules served him perfectly well for making quick estimates of rocket design parameters and other figures.
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| Aluminium [[Pickett Incorporated|Pickett]]-brand slide rules were carried on [[Project Apollo]] space missions. The Pickett model N600-ES that was taken to the moon on [[Apollo 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 | accessdate=3 September 2013}}</ref> The Pickett N600-ES that was owned by [[Buzz Aldrin]] and flew with him to the moon on [[Apollo 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 | accessdate=3 September 2013}}</ref>
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| Some engineering students and engineers carried ten-inch slide rules in belt holsters, and even into the mid-1970s this was a common sight on campuses. 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.
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| 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.<ref>{{cite web
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| |url=http://www.findarticles.com/p/articles/mi_qa3950/is_200401/ai_n9372466
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| |title=Prosthaphaeretic Slide Rule: A Mechanical Multiplication Device Based On Trigonometric Identities, The | Mathematics And Computer Education | Find Articles At Bnet
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| |publisher=Findarticles.com
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| |date=2009-06-02
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| |accessdate=2010-02-20}}</ref>
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| ===Specialized calculators===
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| [[File:HD Actinograph.jpg|right|thumb|[[Hurter and Driffield]]'s [[actinograph]]]]
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| Slide rules have often been specialized to varying degrees for their field of use, such as excise, proof calculation, engineering, navigation, etc., but 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.
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| [[File:John Rabone 1892 Cattle Gauge.png|center|thumb|450px|John Rabone & Sons 1892 Cattle Gauge]]
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| There were many specialized slide rules for [[photography|photographic]] applications; for example, the [[actinograph]] of [[Hurter and Driffield]] was a two-slide boxwood, brass, and cardboard device for estimating exposure from time of day, time of year, and latitude.
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| 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 [[valve]]s.<ref>[http://www.natgasedu.com/vm004.html Fisher sizing rules, retrieved 2009 Oct 06].</ref>
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| [[File:Cryptographic sliding rule-IMG 0533.jpg|thumb|Cryptographic slide rule used by the Swiss Army between 1914 and 1940.]]
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| In [[World War II]], bombardiers and navigators who required quick calculations often used specialized slide rules. One office of the [[US Navy|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.
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| ===Decline===
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| [[Image:TI-30 LED.png|150px|thumb|The [[TI-30]] scientific calculator was introduced for under US$25 in 1976 (${{Formatprice|{{Inflation|US|25|1976|r=2}}|0}} adjusted for inflation), signaling the end of the slide rule era.]]
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| The importance of the slide rule began to diminish as [[electronic computer]]s, a new but very scarce resource in the 1950s, became more widely available to technical workers during the 1960s. (See [[History of computing hardware %281960s%E2%80%93present%29]].)
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| Computers also changed the nature of calculation. With slide rules, there was a great emphasis on working the algebra to get expressions into the most computable form. Users of slide rules would simply approximate or drop small terms to simplify the calculation. [[FORTRAN]] allowed complicated formulas to be typed in from [[textbook]]s without the effort of reformulation. [[Numerical integration]] was often easier than trying to find [[Closed-form expression|closed-form]] solutions for difficult problems. The young engineer asking for computer time to solve a problem that could have been done by a few swipes on the slide rule became a humorous cliché.
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| The availability of mainframe computing did not however significantly affect the ubiquitous use of the slide rule until cheap hand held electronic [[calculator]]s for scientific and engineering purposes became available in the mid-1970s at which point they rapidly fell out of use. The first included the [[Wang Laboratories]] LOCI-2,<ref>[http://www.oldcalculatormuseum.com/wangloci.html The Wang LOCI-2]</ref><ref name = "Wang Ad 1966">{{cite journal
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| |last = Wang Laboratories
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| |title = Now you can determine Copolymer Composition in a few minutes at your desk
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| |journal = American Chemical Society
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| |volume = 38
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| |issue = 13
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| |pages = 62A–63A
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| |date = December 1966
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| |url = http://pubs.acs.org/doi/abs/10.1021/ac50155a005
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| |doi = 10.1021/ac50155a005
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| |accessdate=2010-10-29}}</ref> introduced in 1965, which used [[logarithm]]s for multiplication and division and the [[Hewlett-Packard]] HP-9100, introduced in 1968.<ref>[http://www.hp9825.com/html/the_9100_project.html The HP 9100 Project].</ref> The HP-9100 had [[trigonometric function]]s (sin, cos, tan) in addition to exponentials and logarithms. It used the [[CORDIC]] (coordinate rotation digital computer) algorithm,<ref>[http://dx.doi.org/10.1023/A:1008110704586 J. E. Volder, "The Birth of CORDIC", J. VLSI Signal Processing '''25''', 101 (2000).]</ref> which allows for calculation of trigonometric functions using only shift and add operations. This method facilitated the development of ever smaller scientific calculators.
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| The era of the slide rule ended with the launch of pocket-sized scientific calculators, of which the 1972 Hewlett-Packard [[HP-35]] was the first. Introduced at [[United States dollar|US$]]395, it was too expensive for most students. By 1975 basic four-function electronic calculators could be purchased for less than $50, and by 1976 a scientific calculator, the [[TI-30]], could be purchased for less than $25.
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| == Compared to electronic digital calculators ==
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| {{Pro and con list|section|date=December 2013}}
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| Compared to the portable electronic digital calculators that were introduced in the early 1970s, slide rules had various advantages and disadvantages.
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| === Advantages ===
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| * 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> Because users must separately compute the [[order of magnitude]] of the answer in order to position the decimal point in the results, they are less likely to make extreme calculation errors. And as order of magnitude gets the greatest prominence when using a slide rule, and [[Arithmetic precision|precision]] is limited only to the few digits that are normally useful, users are less likely to make errors of [[false precision]].
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| * 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.
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| * Other useful constants such as pounds to kilograms can be easily marked on the rule and used directly in calculations.
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| * A slide rule does not depend on [[electricity]] or batteries.
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| * A slide rule displays all the terms of a calculation along with the result. This eliminates uncertainty about what calculation was actually performed.
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| For many of these reasons 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. Many sailors keep them as backup for navigation in case of electric failures or battery depletion on long route segments.
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| Some wrist watches designed for aviation use still feature slide rule scales to permit quick calculations. The Citizen Skyhawk AT is a notable example.<ref>[http://www.citizenwatch.com/CUK/English/detail.asp?Country=CUK&Language=English&ModelNumber=JY0000-53E Citizen Watch Company – Citizen Eco-Drive / US, Canada, UK, IrelandCitizen Watch<!-- Bot generated title -->]</ref>
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| === Disadvantages ===
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| * Most people find slide rules difficult to learn and use. Even during their heyday, they 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>
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| * Doing a calculation on a slide rule tends to be slower than on a calculator.<ref>Watson, George H. [http://books.google.com/books?id=KIyAn_jTGUYC&lpg=PA109&ots=QaPDD5V0BD&dq=calculator%20%22slide%20rule%22%20training&lr&pg=PA110#v=onepage&q=slide%20rule&f=false "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 take mathematical shortcuts favoring operations that were easy on a slide rule, creating 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>
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| * A slide rule requires the user to mentally calculate the [[order of magnitude]] of 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 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).
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| * Addition and subtraction are not well-supported operations on slide rules.
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| * The typical [[Arithmetic precision|precision]] of a slide rule is about three [[significant digits]], compared to many digits on digital calculators.
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| * Errors may arise from mechanical imprecision in slide rules that are warped by heat or use or that were poorly constructed.
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| == Finding and collecting slide rules ==
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| [[File:Faber Castell 6 inch slide rule.JPG|thumb|right|300px|[[Faber-Castell]] slide rule with pouch]]
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| 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.<ref>{{cite web
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| |url=http://sliderule.ozmanor.com/srlcol.html
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| |title=Greg's Slide Rules - Links to Slide Rule Collectors
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| |publisher=Sliderule.ozmanor.com
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| |date=2004-07-29
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| |accessdate=2010-02-20}}</ref>
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| 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.
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| 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 [[: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, 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.
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| 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
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| |url=http://www.concise.co.jp/eng0731/about.html
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| |title=About CONCISE
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| |publisher=Concise.co.jp
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| |date=
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| |accessdate=2010-02-20}}{{dead link|date=January 2014}}</ref> 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.<ref name="thinkgeek1">{{cite web
| |
| |url=http://www.thinkgeek.com/gadgets/tools/be12/
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| |title=Slide Rule
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| |publisher=ThinkGeek
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| |date=
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| |accessdate=2010-02-20}}</ref> These are no longer available in 2012.<ref name="thinkgeek2">{{cite web
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| |url=http://www.thinkgeek.com/gadgets/tools/be12/
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| |title=Slide Rule
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| |publisher=ThinkGeek
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| |date=
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| |accessdate=2012-10-13}}</ref> In addition, Faber-Castell has a number of slide rules still in inventory, available for international purchase through their web store.<ref>{{cite web
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| |url=http://service.de.faber-castell-shop.com/epages/es117781.sf/de_DE/?ObjectPath=/Shops/es117781_ServiceParts/Categories/Faber-Castell/FaberCastell_Rechenstaebe
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| |title=Rechenschieber
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| |publisher=Faber-Castell
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| |date=
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| |accessdate=2012-01-17}}</ref> Proportion wheels are still used in graphic design.
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| == See also ==
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| {{Commons category|Slide rule}}
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| <div style="-moz-column-count:2; column-count:2;">
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| *[[Timeline of computing]]
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| *[[Analog computer]]
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| *[[Bygrave slide rule]]
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| *[[Hans Peter Luhn|Lunometer]]
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| *[[Nomography]]
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| *[[Slide chart]]
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| *[[Vernier scale]]
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| *[[Volvelle]]
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| </div>
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| == Notes ==
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| {{Reflist|30em}}
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| ==External links==
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| ;General information, history:
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| *[http://www.sliderulemuseum.com/ International Slide Rule Museum]
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| *[http://www.du.edu/~jcalvert/tech/slidrul.htm The history, theory and use of the engineering slide rule] — By Dr James B. Calvert, University of Denver
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| *[http://www.oughtred.org/ Oughtred Society Slide Rule Home Page] — Dedicated to the preservation and history of slide rules
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| *[http://www.antiquark.com/sliderule/sim/index.html Derek's virtual slide rule gallery] — Javascript simulations of historical slide rules
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| *{{Cite NIE|Slide rule|year=1905}}
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| *{{Cite Americana|short=1|wstitle=Slide-rule|year=1920}}
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| *[http://fabercastell.reglasdecalculo.com/ Reglas de Cálculo] — A very big Faber Castell collection
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| *[http://www.photocalcul.com/Calcul/Regles/Graphoplex/GRAPHOPLEX.html#HAUT Collection of slide rules ] — French Slide Rules (Graphoplex, Tavernier-Gravet and others)
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| *[http://www.sliderule.ca/ Eric's Slide Rule Site] — History and use
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| {{DEFAULTSORT:Slide Rule}}
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| [[Category:Mathematical tools]]
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| [[Category:Mechanical calculators]]
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| [[Category:Logarithms]]
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| [[Category:English inventions]]
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| [[Category:1620 introductions]]
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| [[Category:Analog computers]]
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| [[Category:Historical scientific instruments]]
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| [[Category:Obsolete technologies]]
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