Slide Rules, Logarithms, and Compute Servers

By: Ted Harris
– June 10, 2014
Categories: ,

If any of you have been to PADT’s headquarters in Tempe, Arizona, you probably noticed the giant slide rule in the middle of our building.  You can see a portion of it in the picture below, at the top of our Training, Mentoring, and Support group picture.

PADT-TechSupport-Team-Prop

This thing is huge, over 6 feet (2 m) from side to side, in its un-extended position hanging on the wall.

In theory a gigantic slide rule could provide more accuracy, but our trophy, a Kueffel & Esser model 68 1929 copyrighted 1947 and 1961, was intended for teaching purposes in classrooms.  Most engineers had essentially pocket size or belt holder sized slide rules, also known as slip sticks. 

For the real thing, here is a picture of a slide rule used by Eric Miller’s father Col. BT Miller while at West Point from 1955 to 1958 as well as during his Master’s program in 1964.

Burt-Miller-SlideRule-D2

Why do we care about the slide rule today?  Have you ever seen World War II aircraft, submarines, or aircraft carriers?  These were designed using slide rules and/or logarithms.  The early space program?  Slide rules were used then too.  Some phenomenal engineering was accomplished by our predecessors using these devices.  Back then the numerical operations were just a tool to utilize their engineering knowledge.  Now I think we have a tendency to focus on the numerical due to its ease of use and impressive presentation, while perhaps forgetting or at least de-emphasizing the underlying engineering.  That’s not to say that we don’t have great engineers out there; rather it’s a call to energize you all to remember, consider, and utilize your engineering knowledge as you use your simulation tools.

By contrast, here is a picture of PADT’s brand new server room, with cluster machines being put together in the big cabinets.  Hundreds of cores.

servers

What about the giant slide rule?

My father found a thick book at an estate sale a few months ago.  There are a lot of retirees living in Arizona, so estate sales are quite common and popular.  They occur at a life stage when due to death or the need for assisted living, folks are no longer able to live in their home so the contents are sold, clearing out the home and generating some cash for the family.  This particular estate sale was for a retired engineer.  The book caught my father’s eye, first because it was quite thick and second because the title was, Mechanical Engineers’ Handbook.  Figuring it was a bargain for the amazing price of $1.00, he bought it for me.  This book is better known as Marks’ Handbook.  It’s apparently still in publication, at least as late as the 11th Edition in 2006, but the particular edition my father bought for me is the Fifth Edition from 1951.

marks-handbook

Although the slide rule is mostly a curiosity to us today, in 1951 it was state of the art for numerical computation.  While Marks’ has a couple of paragraphs on “Computing Machines”, described as “electrically driven mechanical desk calculators such as the Marchant, Monroe, or Friden”, the slide rule was what I will call the calculator of choice by mechanical engineers at the beginning of the 2nd half of the 20th century. 

As an aside, these mechanical calculators performed multiplication and division, using what I will describe as incredibly complex mechanisms.  Here is a link to a Wikipedia article on the Marchant Calculator:  http://en.wikipedia.org/wiki/Marchant_Calculator

Marks’ Handbook devotes about 3 pages to the operation of the slide rule, starting with simple multiplication and division and then discussing various methods of utilization and various types of slide rules.  It starts off by stating, “The slide rule is an indispensable aid in all problems in multiplication, division, proportion, squares, square roots, etc., in which a limited degree of accuracy is sufficient.” 

The slide rule operates using logarithms.  If you’re not familiar with using logarithms then you are probably younger than me, since I recall learning them in math class in probably junior high in the late 1970’s.  The slide rule uses common logarithms, meaning the log of a number is the exponent needed to raise a base of 10 to get that number.  For example, the common log of 100 is 2.  The common log table in the 1951 edition of Marks shows us that the common log of 4.44 is 0.6474.  For the sake of completeness, the ‘other’ logarithm is the natural log, meaning the base is the irrational number e, approximated as 2.718.

log-table

Getting back to common (base of 10) logs, the math magic is that logarithms allow for shortcuts in fairly complex computations.  For example, log (ab) = log a + log b.  That means if we want to multiple two fairly complicated numbers, we can simply look up the common log of each and add them together.  Similarly, log (a/b) = log a – log b. 

Here is an example, which I will keep simple.  Let’s say we want to multiple 0.0512 by 0.624.  On a calculator this is simple, but what if you are stranded on a remote island and all you have is a log table?  Knowing the equations above, you can look up the log of 0.0512 which is 0.7093-2 and the log of 0.624 which is 0.7952-1.  We now add:
adding_numbers

Writing that sum as a positive decimal minus an integer is important to being able to look up the antilogarithm or number whose log is 0.5045 – 2.

Looking up the number whose log is 0.5045 we get 3.195, using a little bit of linear interpolation.  The “-2” tells us to shift the decimal point to the left twice, meaning our answer is 0.003195.  Thus, using a little addition, some table lookup, a bit of in the head interpolation, and some knowledge on how to shift decimal points, we fairly easily arrive at the product of two three digit fractional numbers.  Now you are free to look for more coconuts on the island.  Or maybe get back to a hatch in the ground where you need to type in the numbers 4, 8, 14, 16, 23, and 42 every 108 minutes.  Oops, I’m really becoming Lost here…

sliderule-book

Getting back to the slide rule, one way to think of it is a graphical representation of the log tables.  In its most basic form, the slide rule consists of two logarithmic scales.  By lining up the scales, the log values can be added or subtracted.  For example, if we want to multiply something simple, like 4 x 6, we simply look from left to right on the scale on the ‘fixed’ portion of the slide rule to get to 4, then slide the moving portion of the slide so that its 1 lines up with the 4 found above on the fixed portion.  We then move left to right on the movable scale to find the 6.  Where the 6 on the movable slide lines up with on the fixed portion is our solution, 24.  What we’ve really done is add the log of 4 to the log of 6 and then find the antilog of that result, which is 24.  Now that we’ve found 24, we’re not Lost

We don’t intend to give detailed instructions on all phases of performing calculations using slide rules here, but hopefully you get the basics of how it is done.  There are plenty of online resources as well as slide rule apps that provide all sorts of details.  Besides multiplication and division, slide rules can be used for squares and square roots.  There are (were) specialty slide rules for other purposes.  Note that with additional knowledge and skill in visually interpolating on a log scale, up to 3 or even 4 significant digits can be determined depending on the size of the slide rule.

ted-slide-ruleThe author, attempting to prove that 4 x 6 is indeed 24

After having studied the Marks’ section on slide rules, experimenting with a slide rule app on an iPad as well as the PADT behemoth on the wall, I conclude that it was a very elegant method for calculating numbers much more quickly than could be done by traditional pencil and paper.  It’s must faster to add and subtract vs. complicated multiplication and long division.  My high school physics teacher actually spent a day or two teaching us how to use slide rules back in the early 1980’s.  By then they had been made functionally obsolete by scientific calculators, so looking back it was perhaps more about nostalgia than the math needed.  It does help me to appreciate the accomplishments made in science and engineering before the advent of numerical computing.

The preparation of this article has made me wonder what the guys and gals who used these tools proficiently back in the 1930’s, 40’s, and 50’s would think if they had access to the kind of compute power we have available today.  It also makes me wonder what people will think of our current tools 50 or 60 years from now.  When I first started in simulation over 25 years ago, it would have seemed quite a stretch to be able to solve simultaneously on hundreds if not thousands of compute cores as can be done today.  Back then we were happy to get time on the one number cruncher we had that was dedicated to ANSYS simulation.

Incidentally, this article was inspired by my colleague David Mastel’s recent blog entry on numerical simulation and how PADT is helping our customers take compute servers and work stations to the next level:

https://www.padtinc.com/blog/the-focus/launch-leave-forget-hpc-and-it-ansys

If you are ever in our PADT headquarters building in Tempe, don’t forget to look for the giant slide rule.  Now you will know its original purpose.

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