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<meta name="author" content="Dave Morriss">
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<title>How to use a Slide Rule (HPR Show 2166)</title>
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<header>
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<h1 class="title">How to use a Slide Rule (HPR Show 2166)</h1>
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<h2 class="author">Dave Morriss</h2>
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<h1>Table of Contents</h1>
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<ul>
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<li><a href="#introduction">Introduction</a></li>
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<li><a href="#what-is-a-slide-rule">What is a Slide Rule?</a></li>
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<li><a href="#how-does-a-slide-rule-work">How does a Slide Rule work?</a><ul>
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<li><a href="#what-is-a-logarithm">What is a logarithm?</a></li>
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<li><a href="#john-napier">John Napier</a></li>
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<li><a href="#the-slide-rule-as-a-short-cut-to-using-logarithms">The slide rule as a short-cut to using logarithms</a><ul>
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<li><a href="#multiplication">Multiplication</a></li>
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<li><a href="#division">Division</a></li>
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</ul></li>
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</ul></li>
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<li><a href="#further-study">Further Study</a></li>
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<li><a href="#links">Links</a></li>
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</header>
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<h2 id="introduction">Introduction</h2>
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<p>In my show <a href="http://hackerpublicradio.org/eps/hpr1664" title="Life and Times of a Geek part 1">1664</a>, “Life and Times of a Geek part 1”, I spoke about using a slide rule as a schoolboy. As a consequence, I was asked if I would do a show on slide rules, and this is it (after a rather long delay).</p>
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<h2 id="what-is-a-slide-rule">What is a Slide Rule?</h2>
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<p>A slide rule is an analogue computer which can be used to do multiplication and division (amongst other mathematical operations). Most slide rules consist of a fixed portion with a central slot into which a sliding part fits. The top and bottom areas of the fixed part hold various different scales, and the slider is marked with its own scales. A transparent cursor slides over the top of the other parts and can be used to read from one scale to another.</p>
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<p><a title="By ArnoldReinhold [GFDL (http://www.gnu.org/copyleft/fdl.html), CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/) or CC BY 2.5 (http://creativecommons.org/licenses/by/2.5)], via Wikimedia Commons" href="https://commons.wikimedia.org/wiki/File%3ASliderule.PickettN902T.agr.jpg"><img width="512" alt="Sliderule.PickettN902T.agr" src="https://upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Sliderule.PickettN902T.agr.jpg/512px-Sliderule.PickettN902T.agr.jpg"/></a><br />
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<em>Slide Rule (Wikimedia)</em></p>
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<p>I still have my slide rule from my schooldays, a plastic Faber-Castell version from the 1960’s.</p>
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<p><img src="hpr2166_slide_rule.png" alt="My old slide rule" /><br />
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<em>My old school slide rule looking very much worse for wear</em></p>
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<p>Recently, while contemplating this HPR episode, I checked eBay to see whether slide rules were still available. Within the hour I had found an interesting-looking example, had placed a bid on it for £9.99, and won. It is also a Faber-Castell but mainly made of wood (possibly boxwood or mahogany) with ivory-like (celluloid) facings. It seems quite a bit older than my other one. It is a model 1/60/360, made in Bavaria, apparently from some time after 1935 when this style of model numbering began to be used.</p>
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<p><img src="hpr2166_FC_1_60_360_0.png" alt="Faber-Castell 1/60/360 picture 1" /> <img src="hpr2166_FC_1_60_360_1.png" alt="Faber-Castell 1/60/360 picture 2" /> <img src="hpr2166_FC_1_60_360_2.png" alt="Faber-Castell 1/60/360 picture 3" /><br />
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<em>My newly acquired Faber-Castell 1/60/360</em></p>
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<p>In researching it I found that the slide rule is actually split in two, with a spring steel spine which keeps the two halves together, and tensions the slot in which the slider runs. You can see some of this in the pictures.</p>
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<h2 id="how-does-a-slide-rule-work">How does a Slide Rule work?</h2>
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<p>Slide rules use logarithmic scales to perform multiplication and division.</p>
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<h3 id="what-is-a-logarithm">What is a logarithm?</h3>
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<p>A <a href="https://en.wikipedia.org/wiki/Logarithm" title="Logarithm">logarithm</a> of a number is the exponent to which a base must be raised to produce the number.</p>
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<p>So, if the base is 10 (known as a <em>common logarithm</em>, written as ‘log<sub>10</sub>’) then 100 is 10<sup>2</sup>, so the log<sub>10</sub> of it is 2, and the log<sub>10</sub> of 1000 (10<sup>3</sup>) is 3. The Wikipedia page on the <a href="https://en.wikipedia.org/wiki/Logarithm" title="Logarithm">logarithm</a> does a better job of explaining this than I can do.</p>
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<p>At the time I was using a slide rule, back in the 1960’s, we were expected to know how to use logarithms and were each allocated a book of <a href="https://en.wikipedia.org/wiki/Mathematical_table" title="Tables of common logarithms">log tables</a>. This allowed you to look up the common logarithm of a number, or to convert a logarithm back to a number.</p>
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<p>The great advantage of logarithms is that multiplication can be achieved by addition, and division by subtraction. In other words, the following rules apply for any base <em>b</em>:</p>
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<p><span class="math inline">log<sub><em>b</em></sub>(<em>x</em><em>y</em>)=log<sub><em>b</em></sub>(<em>x</em>)+log<sub><em>b</em></sub>(<em>y</em>)</span></p>
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<p><span class="math inline">log<sub><em>b</em></sub>(<em>x</em>/<em>y</em>)=log<sub><em>b</em></sub>(<em>x</em>)−log<sub><em>b</em></sub>(<em>y</em>)</span></p>
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<p>Provided <em>b</em>, <em>x</em> and <em>y</em> are positive and <em>b</em> is not 1.</p>
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<p>So, at school when multiplying two numbers, the process was to take the first multiplicand, look up its log<sub>10</sub>, write it down, then do the same for the second multiplicand and add the two logarithms together. The result could then be looked up in an “<a href="https://en.wikipedia.org/wiki/Logarithm#Antilogarithm" title="Antilogarithm">anti-log</a>” table to get the product of the two original numbers.</p>
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<p>If you want to go further with this look at the <a href="http://www.wikihow.com/Use-Logarithmic-Tables" title="How to use logarithmic tables">wikiHow article</a> below for details of how to use logarithmic tables.</p>
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<h3 id="john-napier">John Napier</h3>
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<p>As an aside, the inventor of logarithms, <a href="https://en.wikipedia.org/wiki/John_Napier" title="John Napier">John Napier</a>, lived in Edinburgh and was born in 1550 in <a href="https://en.wikipedia.org/wiki/Merchiston_Tower" title="Merchiston Tower">Merchiston Tower</a>, otherwise known as Merchiston Castle. The original grounds of the tower is now the site of <a href="https://en.wikipedia.org/wiki/Edinburgh_Napier_University" title="Edinburgh Napier University">Edinburgh Napier University</a>, and the tower is part of their Merchiston Campus. I live in Edinburgh, and have visited this site on many occasions.</p>
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<p><a title="By Stefan Schäfer, Lich (Own work) [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons" href="https://commons.wikimedia.org/wiki/File%3AMerchiston_Castle.jpg"><img width="256" alt="Merchiston Castle" src="https://upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Merchiston_Castle.jpg/256px-Merchiston_Castle.jpg"/></a><br />
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<em>Merchiston Castle (Wikimedia)</em></p>
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<h3 id="the-slide-rule-as-a-short-cut-to-using-logarithms">The slide rule as a short-cut to using logarithms</h3>
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<p>With a slide rule the process uses logarithmic scales but short-circuits the table look-ups.</p>
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<p>The operation of a slide rule is covered quite well on the <a href="https://en.wikipedia.org/wiki/Slide_rule" title="Slide Rule">Wikipedia page</a> referenced in the <em>Links</em> section below.</p>
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<h4 id="multiplication">Multiplication</h4>
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<p>We have already seen that the process of multiplication using logarithms is transformed into a process of addition. So the example shows the multiplication of 3 by 2. The sliding scale is positioned so that the 1 is positioned over the 2 on the scale below it. Looking at the 3 on the sliding scale the answer of 6 can be seen below it.</p>
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<p>On my <em>Faber Castell 1/60/360</em> I used the upper scale to achieve the same result (since it’s a little bit easier to see):</p>
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<!-- Image showing the 1/60/360 with the 1 on the slider against 2 on the upper scale -->
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<p><img src="hpr2166_FC_1_60_360_3.png" alt="Faber-Castell 1/60/360 picture 4" /><br />
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<em>Calculating 3 times 2</em></p>
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<p>The same can of course be achieved by placing the 1 on the sliding scale against the 3 on the upper scale and reading from the 2 on the sliding scale:</p>
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<!-- Image showing the 1/60/360 with the 1 on the slider against 3 on the upper scale -->
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<p><img src="hpr2166_FC_1_60_360_4.png" alt="Faber-Castell 1/60/360 picture 5" /><br />
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Calculating 2 times 3</p>
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<h4 id="division">Division</h4>
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<p>Taking the Wikipedia example of 5.5 divided by 2, on my <em>Faber Castell 1/60/360</em> again, the 5.5 mark on the slider is aligned with the 2 mark on the upper scale and the result, 2.75 read off the slider under the 1 on the upper scale.</p>
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<!-- Image showing 5.5/2 on the upper scale -->
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<p><img src="hpr2166_FC_1_60_360_5.png" alt="Faber-Castell 1/60/360 picture 6" /><br />
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<em>Calculating 5.5 divided by 2</em></p>
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<h2 id="further-study">Further Study</h2>
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<p>The <a href="http://www.sliderulemuseum.com/" title="International Slide Rule Museum"><em>International Slide Rule Museum</em></a> offers many resources for the slide rule enthusiast. If you are interested in learning more about how to use a slide rule then they have a <a href="http://www.sliderulemuseum.com/SR_Course.htm" title="Slide Rule Course">self-guided course</a> with a virtual slide rule.</p>
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<p>In addition, you could consider obtaining a real slide rule. There are many to be had for not very much money on eBay. Apart from the Faber-Castell I bought myself for £10, and have been demonstrating here, I bought two more Faber-Castell models, costing less than £20 for both.</p>
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<h2 id="links">Links</h2>
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<ul>
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<li>HPR show “<em>Life and Times of a Geek part 1</em>”: <a href="http://hackerpublicradio.org/eps/hpr1664" class="uri">http://hackerpublicradio.org/eps/hpr1664</a></li>
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<li>Wikipedia article on the slide rule: <a href="https://en.wikipedia.org/wiki/Slide_rule" class="uri">https://en.wikipedia.org/wiki/Slide_rule</a></li>
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<li>International Slide Rule Museum: <a href="http://www.sliderulemuseum.com/" class="uri">http://www.sliderulemuseum.com/</a>
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<ul>
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<li>Self-guided course: <a href="http://www.sliderulemuseum.com/SR_Course.htm" class="uri">http://www.sliderulemuseum.com/SR_Course.htm</a></li>
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</ul></li>
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<li>Wikipedia article on the <em>Logarithm</em>: <a href="https://en.wikipedia.org/wiki/Logarithm" class="uri">https://en.wikipedia.org/wiki/Logarithm</a></li>
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<li>Wikipedia article on the <em>Mathematical table</em>: <a href="https://en.wikipedia.org/wiki/Mathematical_table" class="uri">https://en.wikipedia.org/wiki/Mathematical_table</a></li>
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<li>wikiHow article on how to use a Logarithm Table: <a href="http://www.wikihow.com/Use-Logarithmic-Tables" class="uri">http://www.wikihow.com/Use-Logarithmic-Tables</a></li>
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<li>Wikipedia article on <em>John Napier</em>: <a href="https://en.wikipedia.org/wiki/John_Napier" class="uri">https://en.wikipedia.org/wiki/John_Napier</a></li>
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<li>Wikipedia article on <em>Merchiston Tower</em>: <a href="https://en.wikipedia.org/wiki/Merchiston_Tower" class="uri">https://en.wikipedia.org/wiki/Merchiston_Tower</a></li>
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<li>Wikipedia article on <em>Edinburgh Napier University</em>: <a href="https://en.wikipedia.org/wiki/Edinburgh_Napier_University" class="uri">https://en.wikipedia.org/wiki/Edinburgh_Napier_University</a></li>
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<li>More about the <em>Faber Castell 1/60/360</em>:
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<ul>
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<li><a href="http://sliderulemuseum.com/isrm/hmd/fc%20slide%20rule%20pages/fc%2025%201-60-360/fc%2025%201-60-360.htm">Example 1</a></li>
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<li><a href="http://sliderules.lovett.com/fabercastell1-60/fabercastell1-60.htm">Example 2</a></li>
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<li><a href="http://fabercastell.reglasdecalculo.com/1_60_360/1_60_360.html">Example 3</a></li>
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<li><a href="http://www.countbelmiro.com/slides/faber/360/360.html#160360_1">Example 4</a></li>
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</ul></li>
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