Logarithmic Scale and the Slide Rule

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

Contents

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." [1] 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 at the 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

What Good are Logarithmic table and Logarithmic Scale ?

If you are new to logarithms, check out [...]

What Good are Logarithmic table and Logarithmic Scale ?

If you are new to logarithms, check out Logarithms before you proceed.


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 + 3\log 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 2 \times 4.5, we have to check \log2, \log4.5, 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.

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


Below is a more finely divided scale.

Thus with two such scale, we could thus find 2 \times 4.5 by adding 2 and 4.5 on that scale which give us 9. The operating principles are explained later.

History of Slide Rule and Its Construction

A Bit of History on Slide Rule

Before slide rule (and of course, logarithms for that matter), there was the sector whose inventor was debatable. It was able to solve problems in proportions, trigonometry, multiplication and division, and even taking square root and squaring as well. It was in popular use until mid 19th century.

John Napier
John Napier
Mirifici Logarithmorum Canonis Constructio
Mirifici Logarithmorum Canonis Constructio

John Napier's invention of logarithms was really out of necessity. With the increasing amount of data and advancement in the natural science, mathematician and scientists were constantly occupied by huge stack of calculations which not only took up immense amount of time but also impeded further progress. This is especially true for the study of astronomy. Astronomers usually had to observe and collect data for many hours at night and then perform laboriously repetitive calculations (which was very prone to mistakes and errors) during the day time. This presented an imperative need for a new way of calculations that was fast and reliable. John Napier quietly stepped up for the challenge and excited the community of scientists and mathematician in 1617 with his publication of Mirifici Logarithmorum Canonis Descriptio in which he introduced the Logarithms and came up with the logarithms table. The history and development is a fascinating topic and hence I have dedicated an individual page The Logarithms, Its Discovery and Development to it. In it there are also more detailed and comprehensive information on logarithmic scale. Now, the important things about logarithms are some of its very useful properties some of which are known by even 9th graders, namely \log(xy) = \log(x) + \log(y) and \log \frac{x}{y} = \log(x) - \log(y). So the that means if we have an operation in the linear scale such as x \times y, we can transform it into x+y on the logarithmic scale. Henry Briggs, England's most eminent mathematician of the time, traveled to meet Napier and introduced himself: "My Lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help unto astronomy...I wonder why nobody else found it out before, when, now being known, it appears so easy." [2]


In 1620,payday loans onlinecame payday loan up with the first application of the logarithms. He put the logarithms scale on a rule together with line of squares, cubes, tangent, sine and cosine. It was some two foot long and 1 and a half inch wide and to use it, one had to use a compass (or a pair of dividers or calipers) to transfer distances. It was cumbersome but it worked.


All these were really happening at an exciting time before the enlightenment when Galileo Galilei was making discoveries about the universe using his homemade telescope and Johannes Kepler was figuring out patterns of planetary motions from the huge amount of data he collected. A few decades later after the invention of logarithms, Sir Isaac Newton would be born and change our view of the universe completely.

William Oughtred
William Oughtred
Roget P M
Roget P M

In 1622, William Oughtred made one improvement on Gunter's Scale and placed two such scales side by side then reads the distance relationship from there. He also developed a circular slide rule.


In 1675, Sir Isaac Newton solved cubic equations using three logarithmic scales and suggested the use of a cursor. 2 years later, Henry Coggeshall changed and popularized the timber and carpenter's rule which saw the first specialized application of slide rule.


In the following years, there were many improvements to the slide rule. One major improvement was the increase in scale graduation accuracy. The next major improvement came in 1815 when Peter Roget invented the log log scale which enabled the calculation of roots and powers to any number or fraction which came into great usefulness some fifty years later when advances in engineering and physics required such operations. In 1851, a French artillery officer Amedee Mannheim standardized a set of four scales for the most common calculation problems and his design became the foundation for all the slide rules that were produced in the subsequent years till portable pocket calculators took over what was formerly slide rule's territories, rendering them obsolete.[3]


Operating Principles

An Analogy

Say we have two linear scales S and B. A number x on S will be referred to as S-x and a number y on B will be referred to as B-y. The statement "Move the top ruler to the right until its left end is over number y on the bottom ruler" will be simply be stated as "Place S-0 over B-y"


Now, say we want to do the addition 4+3, how would you use the two ruler to do such an addition? Well we will place S-0 over B-4 and the number opposite S-3 on B is the answer which is 7.

What about 5+7?

The number opposite S-7 went beyond scale B. What should we do then? Well we can extend the B scale like shown below.

Well then the answer is 12. But suppose our original scales are 10 inches, making it longer will force the rule to be inconveniently cumbersome. So what is the next solution? The more observant of you will find that S-0 is over B-5 and S-(1)0 is over B-(1)5. If S-0 were over B-7, then S-(1)0 would be over B-(1)7. This should not come at a surprise since whatever single digit number x you add to the number 10 would end up (1)x. Hence the right hand digit does not change. All we have to pay attention the left hand digit. Hence for the previous case, we can actually place S-(1)0 over B-5 and the number opposite S-7 is thus the right hand digit of the answer. Keeping track of the left hand side of the digit, we get the answer 12 as shown below.

What about 24+58? Then we need a more finely divided scale with ten more division between each consecutive numbers. What we are going to do is to compute 2.4+5.8 and them move the decimal place afterward. Refer to the diagram below. Place S-0 over B-2.4 and the number opposite S-5.8 is 8.2. Move decimal point to the right, we have 82 which is the real answer. Now, you should realize that the same computation is conducted even if the desired calculation is 240+580 or 2400+5800. All we have to do is to pay attention to the decimal place. Even though there is no limits to the subdivision, there is a limit to how finely we con read the subdivisions. Not only that, markings on the ruler have perceptible thickness themselves so there is actually limits to how many subdivisions we could put on the ruler. Therefore, we are condemned to inexactness if you desire to calculate 18578+8473954 in which case we have to approximate.[4]

The reverse process is subtraction.

Now what is the point of this? We all know how to do addition and subtraction without the use of the addition rule. But these operations have close connection to the operations of slide rule though on a different scale. Knowing that 99% of those who read this page won't even have a slide rule, I have found a Virtual Slide Rule from the internet so that you could play around with this and try out the operations described below.

Multiplication

Using the principle \log(xy) = \log(x) + \log(y) , we can transform multiplication into addition on a logarithmic scale.

Example: 578 \times 9849

Use 5.78 \times 9.849 \times 10^5. Place C-10 over D-9.85, read the number opposite C-5.78 which is approximately 5.69. Now, take note that slide rule does not keep track of the decimal place so we have to keep track of it. Since 5.78 \approx 6 and 9.849 \approx 10, the answer is a little less than 60. Therefore the answer 56.9. In addition, we have to multiply 56.9 with 10^5 to get the answer for 578 \times 9849 is approximately 5690000. Use a calculator to check the answer, we have 5692722 which is not that far away from our approximated answer.


Division

Using \log \frac{x}{y} = \log(x) - \log(y), we can transform division into subtraction on a logarithmic scale.

Example: 578 \div 9849.

As before, we use 5.78 \div 9.849 \times 10^{-1} and employ the subtraction principle from before. But then you will realize that we cannot do 5.78 - 9.849 on the slide rule. Therefore, we will do 5.78 \times \frac {1}{9.849} and to do that we will use the CI scale to find \frac {1}{9.849} and then add that number to 5.78 on the slide rule.

We first look for \frac {1}{9.849}.

Read the number opposite to CI-9.849 on the C scale which is 1.011. Keeping track of the decimal, the answer is 1.011 \times 10^{-1}

Now we do the operation 5.78 \times 1.011 \times 10^{-1}

Place C-1 over D-5.78 and read the number opposite C-1.011 which is 5.86. Keeping track of the decimal, the answer is approximately 0.586. The answer from a calculator which gives the answer 0.58686..., we have the answer pretty close.

Squaring

To do squaring, we need a new scale that corresponds to the old C and D scale.

After we have fixed those points, we can then recalibrate the B scale on a logarithmic scale. Then we can do squaring by choosing number on the C scale and read the number on the B scale.

Example: 1965^2

We use 1.965 \times 10^3. Find the number opposite to C-1.965 on the B scale, which is approximately 3.86. Since 1.965 \approx 2 and 2 \times 2 = 4, we don't have to move the decimal point. Therefore, the answer is approximately 3.86 \times 10^6. A calculator gives us 3.861225 \times 10^6 which is really close to our approximation.


Taking Square Root

Taking square root is the reverse process of squaring. Hence we are choosing a number on the B scale and read a the answer on the C scale. However, there is a trick to this process. B scale is divided into two halves. The left half is to find the square root of numbers with odd numbers of digits, i.e. 1, 345, 48529, etc. The right half is used to find the square root of numbers with even numbers of digits. But why is that you may ask. It is easy to see. Note that B scale is divided into two identical parts, with left side running from 1 to 10 and the right side running from 10 to 100. Since the full length of the B scale is denoted as 100, then the left side can denote x with 1 < x < 10, or 1 \times 10^2 < x < 10 \times 10^2, or 1 \times 10^4 < x < 10 \times 10^4, etc and the right side can denote y with 10 < y < 100, or 10 \times 10^2 < y , 100 \times 10^2, or 10 \times 10^4 < y < 100 \times 10^4. That is to say that we need the left side to find numbers with odd numbers of digits and the right to find numbers with odd number of digits.

Example: \sqrt {4850}.

Use 4.85 \times 10^3 and the right half of the B scale. Find the number opposite B-4.85 on the C scale which is approximately 6.96. Since 70^2=4900, the answer is thus approximately 69.6. A calculator gives us answer 69.6419.......

Cubing

Similar to squaring, we need a new cube scale that corresponds the old C and D scales. Hence, we have the new K scale.

Then we recalibrate the scale according to the logarithmic scale and hence we have the new K scale.

Example: 783^3

Read the number on the K scale opposite D-7.83 which is approximately 480. Keeping track of the decimal point, we have the answer 480000000. A calculator gives the answer 480048687 which is really close.

Taking Cube Root

The K scale is separated into three identical section. Left section runs from 1 to 10; the middle section runs from 10 to 100; the right section runs from 100 to 1000. The same argument follows from the Taking Square Root section. Since the full length of the K scale is denoted as 1000, then the left section can denote x with 1 < x < 10, or 1 \times 10^3 < x < 10 \times 10^3, or 1 \times 10^6 < x < 10 \times 10^6, etc, the middle section can denote y with 10 < y < 100, or 10 \times 10^3 < y < 100 \times 10^3, or 10 \times 10^6 < y < 100 \times 10^6, etc and the right side can denote z with 100 < z < 1000, or 100 \times 10^3 < z < 1000 \times 10^3, or 100 \times 10^6 < z < 1000 \times 10^6. Therefore, we use the left section to find numbers with 1, 4, 7...digits, the middle section to find numbers with 2, 5, 8...digits and the right section to find numbers with 3, 6, 9...digits.


Example: \sqrt [3] {783}

Read the number opposite K-783 on D scale which is approximately 9.22. A calculator gives the answer 9.21695.......


Calculating a^x and \sqrt[x]{a}, where x is arbitrary.

Zenith and Downfall

I kind of talked about it in the introduction section. But I will discuss in a little detail here.

What is the point of all this?

Why is is so important to learn about this?




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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.


Related Links

Additional Resources

Notes

  1. Stoll, 2006
  2. Sylvester, 1886
  3. The Oughtred Society
  4. Asimov, 1965, p. 19-38

References

  1. Asimov, I. (1965). An Easy Introduction to the Slide Rule. New York: Fawcett Publications, inc.
  2. Stoll, C. (2006 , May). When Slide Rules Ruled. Scientific American Magazine , pp. 80-87.
  3. Sylvester, J. J. (1886). On The Method of Reciprocants As Containing An Exhaustive Theory of The Singularities of Curves. Nature , 222-223.
  4. The Oughtred Society. (n.d.). Slide Rule History. Retrieved from The Oughtred Society: http://www.oughtred.org/history.shtml

Future Directions for this Page

At least finish the Calculating a^x and \sqrt[x]{a}, where x is arbitrary. I lost patience to read on the operating principles behind this and got too preoccupied with the The Logarithms, Its Discovery and Development. Other than that, you can add anything that you deem necessary.




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