 ## Thursday, December 29, 2011

### 137: A What What?

In the last episode, I briefly mentioned the tool known as a slide rule.  But after that podcast, I realized that there are a significant number of potential listeners who have never seen a slide rule, and are not sure how one works.  I even found fellow engineers who were not quite sure on the concept.  What was this tool that was so essential for science and engineering half a century ago, but now is virtually forgotten?  Today we're going to find out.

The slide rule was invented in the early 1600s, soon after Scottish mathematician and theologian John Napier popularized the definition of a logarithm.  Napier an odd character, devoting time to both mathematics and theology, and was thought of as a sorceror and magician by many of his contemporaries.  One story about him suggests that he liked to convince people that he had a magic rooster, who would tell him whether a criminal suspect was innocent or guilty after they pet him.  Actually, Napier coated the rooster with soot, and identified the guilty suspect by looking to see who had no soot on his hands, and thus had just pretended to pet it.

But in many ways, the power of the logarithm seemed just as magical.  You may recall from high school that a logarithm, or log for short, is just a fancy way of describing an exponent.  For example, since 2 to the 3rd power is 8,  the log (base 2) of 8 is 3.  Since 2 to the 5th power is 32, the log (base 2) of 32 is 5.  What makes this useful is that, if we hold the base constant, the log of (a times b) is log a + log b.  So as long as we have tables that can tell us the logs and inverse-logs of all the numbers involved, we can transform multiplication problems into addition problems!  So let's suppose we want to multiply 8 * 32.  We can take the log of 8, which is 3, and the log of 32, which is 5, and add them together.  The result is 8, so we just need to find the number whose log (base 2) is 8.  This number is 256, which as we expected, is the product of 8 and 32.

The slide rule takes advantage of this fact by providing two rulers, labelled according to a logarithmic scale, that can be slid relative to each other.  So, for example, assume we're using base 2 logs again.  The leftmost edge of the ruler is marked with a 1, since 2 to the 0 power is 1.  Then at one inch, we see the number 2, since 2 to the 1st power is 2.  At two inches, we see 4, since 2 to the 2nd power is 4, etc.  So to multiply 8 times 32, we would first find the 8 marking on the bottom ruler, which would be 3 inches over, though we don't need to know that fact as we are just looking for the 8.  We would align the top ruler's 1 mark to that.  Then on the top ruler we would look for the 32 marker, which would be 5 inches over.  Now what number on the bottom ruler would be under this 32 marker?  Well, we started at 3 inches on the bottom ruler, and moved another 5 inches-- so we are at 8 inches on the bottom ruler, the exact spot where we see the number whose log is 8.  So the number displayed under the 32 marker is 256, and we have successfully done our multiplication.  Division can be done similarly, through the reverse process.

Of course, there are many additional subtleties to the slide rule as it evolved over the centuries.  To use one effectively, you usually have to normalize your values to the scale on the ruler, usually multiplying or dividing all values by some power of 10 before you start, and remembering to do the proper transformations on your answer.  Over the years slide rules were enhanced with additional scales to ease calculations of roots and 1/x values; sines and cosines; and logs and exponents of numbers.  You can see more details on these topics on the web pages linked in the show notes.

As late as the mid-1970s, a slide rule was an indispensable tool for scientists and engineers:  it allowed them to perform common mathematical calculations much more quickly than they could on paper.  But then came the development of electronic calculators, which could replicate anything a slide rule could do and required less thinking from the user.  For a while there was a period when slide rules and calculators coexisted, and the older engineers would make fun of the young whippersnappers who would bring out a newfangled calculator for problems they could easily solve in seconds with a slide rule.  But as calculators got cheaper and less bulky, younger engineers got the last laugh, and the slide rule became a little more than a historical artifact.

Oddly, they have not gone away completely though:  when researching this podcast, I found several hobbyist sites on the web posted by collectors of various forms of slide rules.  And there is even an organziation known as the Oughtred Society, named after another of the early slide rule pioneers, which to this day still publishes a journal on slide-rule-related topics, has conferences several times a year for slide rule enthusiasts to get together and calculate, and hosts an international slide rule championship for high school students.  Amusingly, slide rules for team practice are apparently so hard to come by that the International Slide Rule Museum sponsors a Slide Rule Loaner Program to encourage potential participants.  You can find these sites in the show notes as well.

And this has been your math mutation for today.
• Slide Rules at Wikipedia
• John Napier at Wikipedia
• Intro to Slide Rules
• Oughtred Society
• International Slide Rule Museum