Logarithmic Scale and the Slide Rule

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150 Extra Engineers
150 extra engineers thumb.jpg
Field: Algebra
Image Created By: IBM
Website: Global Nerdy --- Tech Evangelist Joey deVilla on software development, tech news and other nerdy stuff

150 Extra Engineers

This was a picture of an IBM advertisement back in 1953.


Basic Description

This was a picture of an IBM advertisement back in 1953 during an age when "computer" referred to human who did calculations and computation solely. The advertisement boasted that


"An IBM Electronic Calculator speeds through thousands of intricate computations so quickly that on many complex problems it's just like having 150 Extra Engineers. No longer must valuable engineering personnel...now in critical shortage...spend priceless creative time at routine repetitive figuring. Thousands of IBM Electronic Business Machines...vital to our nation's defense...are at work for science, industry and armed forces, in laboratories, factories and offices, helping to meet urgent demands of greater production."


Doesn't one just want to go back to that age when engineers were in "critical shortage" and "If you had a degree, you had a job. If you didn't have a job it's because you didn't want one." Oh well...


Notice that in the picture, in addition to the lack of women engineers back in those days, the male engineers almost all have receding hairlines (possibly due to constant overwork and the disproportional allocations of oxygen and other nutrients between brain cells and scalp skin). It was also typical of the male engineer to be defined by a standard uniform: "white shirt, narrow tie, pocket protector and slide rule." I need to do footnote here. But I don't know how. Yeah! Pocket protector! I did not know what it is until I saw the picture in the Scientific American Article, When Slide Rule Ruled, by Cliff Stoll, who is a brilliant engineer, physicist and educator, and whose TED talk is both hilarious and inspiring. However, it is neither Cliff Stoll, nor the pocket protector that I want to talk about. (It is just a thing that, despite making the engineers look very geeky and very attractive in author's opinion, protected their shirt pockets from being worn out so quickly due to the numerous engineering essentials they had in there.) Rather, it is the ruler looking stuff that is the engineer's hand that I want to talk about. The Slide Rule.


Born into the digital and automatic age, hardly anyone of our generation ever gets to know anything that is analog or manual. Not only that, we hate to have anything to do with stuff that is analog. We have been cultured to base our life and happiness on gadgets that allow us to access the world and all the information with a finger tip. We don't realize that in fact, things that are analog laid the foundation for our modern society and ushered us into the digital age. Slide Rule is one of "those analog things". In the pre-computer age, it was one of those ingenious tools that enabled engineers, mathematicians and physicists to do calculations, and because of it, we have witnessed the erections of skyscrapers, harnessing of hydroelectric power, building of subways, advancement in the aeronautics, beyond that, it helped us won the WWII, sent astronauts into the space (as a matter of fact, a Picket 600-T Dual base Log Log slide rule was carried by the Aopollo crew to the space and moon should the on-board computers fail) and eventually, hastened its own demise.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Algebra

Logarithmic table and Logarithmic Scale

Imagine you work at a petroleum company and your boss wa [...]

Logarithmic table and Logarithmic Scale

Imagine you work at a petroleum company and your boss wants you to calculate the mass, m, of the petroleum that is stored in a cube tank. You have no calculator and no knowledge of the existence of the logarithmic table by the way and you are given the density \rho = 893 kg/m^3 and the sides of the cubic tank, s = 83.52 m. Naturally, you will use m = \rho \times s^3 and before you even start to realize that this calculation is really tedious, your boss asks you to calculate how much profit the company can make given that 67% of the petro can be distilled into gasoline at 11 dollars per kilo and subsequently shipped at 14 dollar per kilo and sold at 54 dollar per kilo (in some really weird country where they sell by the kilo). So now, things are getting really bad. There is no way you are going to calculate that by hand! Then you boss comes back with some information he forgot to tell you. The petroleum cannot be distilled at once because the distillation plant works at only a certain rate and the finished gasoline cannot be shipped at once thus they require additional storage time which costs money. Before he finishes with the whole story, you have quit your job.


Now, you should really appreciate logarithmic table. Say now you have a logarithmic table and recall that profit, p, is the difference between sales and cost. Hence, p = m \times 67% \times (54-11-14). Taking logarithm of both sides yields


\begin{align}
log p & = log (m \times 67% \times (54-11-14)) \\
      & = log (893 \times 83.52^3 \times 67% \times 29) \\
      & = log 8.93 + 2 + 3log 8.352 + 3 + log.67 + log2.9 + 1
\end{align}

Checking the logarithmic table, you find that the right side of the equation is 14.0044 and hence log p = 14.0044. To find p, you go back to the table again and find that p = 1.01018 \times 10^{14} dollars. Logarithms made your life so much easier. Though this situation is hypothetical but engineers and scientists in the pre-computer and pre-digital calculator age faced similar problems everyday. You can imagine the amount of work that is required if there is no logarithms.


If you have fully understood the use of logarithms, you would have realized that for every multiplication, we have to check the table three times. For example, if we want to calculate 3 \times 8, we have to check log3, log8, add them together and then refer back to the table. Is there anyway that we could do without the table?


There is a way and that is the where the logarithmic scale come in. We intend to create a scale with values on it whose lengths equal to its logarithms. That is to say we mark a point on the ruler "1" since log1 = 0, "2" a distance log2 = 0.3010 from "1" and "3" a distance log3 = 0.4771 from "1" and etc. Thus we have the scale below with scale L as a linear scale and D as the new logarithmic scale.

Scale.png

Further, we can subdivide the scale using the same means and thus we have the following scale.


to the could mark on a ruler the position of log3 and name it "3" and the position of log8 and name it "8", then We are familiar with the linear scale, namely the equally divided scale, from the number 0 to whichever number as shown in scale A. We then use the identity x=log_22^x to relabel scale A to produce scale B. Then what we do to transform the linear scale to logarithmic scale is simply to take away the log_2, turning scale B to scale C. Calculating the numbers on scale C, we end up with scale D.

Sliderule001.png


Now, notice that the given numbers are equally divided but the differences between consecutive numbers are not the same. Our goal is to produce a logarithmic scale that has all the integers from 1 to 8. For example, if we want the number 3 on scale D, there will be a number x on scale A that corresponds to the number 3 on scale D. We have <template>AlignEquals




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About the Creator of this Image

International Business Machines (IBM) is a multinational computer, technology and IT consulting corporation headquartered in Armonk, North Castle, New York, United States. IBM is the world's fourth largest technology company and the second most valuable by global brand (after Coca-Cola). IBM is one of the few information technology companies with a continuous history dating back to the 19th century. IBM manufactures and sells computer hardware and software (with a focus on the latter), and offers infrastructure services, hosting services, and consulting services in areas ranging from mainframe computers to nanotechnology.


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