Difference between revisions of "Logarithmic Scale and the Slide Rule"

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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  
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  
|e4r=\frac {log_{10}3}{log_{10}2}
In the last step, we used change of base. Hence, the number 3 on scale D is located at <math>\frac {log3}{log2} \approx 1.585</math>.
Using this method, we have constructed the logarithmic scale as shown below. "The scale divisions are not uniform because when the logarithms are in arithmetical progression the numbers are in geometrical progression" and "when the numbers are in arithmetical progression the logarithms are in geometrical progression."
Below is a more finely divided scale.
The logarithmic scales are widely used in science and other applications. For example, the well known pH scale is logarithmic since <math>pH=log_{10}[H^+]</math> where <math>[H^+]</math> denotes the concentration of positive hydrogen ions in the solution.
==History of Slide Rule and Its Construction==
==='''A Bit of History'''===
Before slide rule, there was the [http://en.wikipedia.org/wiki/Sector_(instrument) 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. However, the scale was very poorly divided and using the sector required the use of a compass to transfer distance.
The invention of logarithms was really out of necessity. Since people knew how to do <math>x^y</math> where <math>x,y \in \mathbb{R}</math>, it was natural for people to come up with an operation that tells us the power, knowing the base and the result, i.e. obtain <math>x</math> in <math>a^x=b</math>. The solution, of course, is <math>x=log_ab</math> as we know today. In 1617, [http://en.wikipedia.org/wiki/John_Napier John Napier] came into the picture with his publication of ''Mirifici Logarithmorum Canonis Descriptio''in which he introduced the Logarithms and came up with the logarithms table. <font color=red> '''OK the history is all good but how exactly did he, or Henry Briggs, know what is, for example <math>log_57</math>? None of the Wikipedia article mentioned this. '''</font>. Now, the important things about logarithms are some of its very useful properties some of which are known by even 9th graders, namely <math>log(xy) = log(x) + log(y) </math> and <math>log \frac{x}{y} = log(x) - log(y)</math>. So the that means if we have an operation in the linear scale such as <math>x \times y</math>, we can transform it into <math>x+y</math> on the logarithmic scale. [http://en.wikipedia.org/wiki/Henry_Briggs_(mathematician) 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."
In 1620, [http://en.wikipedia.org/wiki/Edmund_Gunter Edmund Gunter] came up with the first application of the logarithms. He put the logarithms scale on a rule together with scales of tangent, sine and cosine. It was some two feet long and 1 and a half inch wide and to use it, one had to use a compass to transfer distances. It was cumbersome but it worked.
This was really happening at an exciting time before the enlightenment when [http://en.wikipedia.org/wiki/Galileo_Galilei Galileo Galilei] was making discoveries about the universe using his homemade telescope and [http://en.wikipedia.org/wiki/Johannes_Kepler 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, [http://en.wikipedia.org/wiki/Isaac_Newton Sir Isaac Newton] would be born and change our view on the universe completely.
In 1622, [http://en.wikipedia.org/wiki/William_Oughtred 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, [http://en.wikipedia.org/wiki/Coggeshall_slide_rule 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. But the next major improvement came in 1815 when [http://en.wikipedia.org/wiki/Peter_Mark_Roget 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.
==='''An Extra Section on How John Napier Came up and Calculated logarithms'''===
This section, though embedded in this page as a sub section, stands equal to an independent article. I found a very thin volume, [http://www.archive.org/details/johnnapierinvent00hobsiala ''John Napier and the Invention of Logarithms, 1614 --- A Lecture''], by [http://en.wikipedia.org/wiki/E._W._Hobson Ernest William Hobson]. It is a very concise and succinct volume that presented how John Napier presented his original ideas. It is an absolute a pleasant to read because it translated Napier's argument and thought into relatively modern mathematical symbols and notations and at the same time preserved and revealed Napier's ingenuity. In the next section, most of the ideas are from the above mentioned book. In addition, I have supplied some additional proofs and necessary information for understanding.
Though a thorough understanding of the book requires some careful thoughts and ruminations, in the end, you will find that you will appreciate logarithms a lot more than you did before. However, you should not be intimidated and discouraged by the section, in which case you should continue to the next section.
Today, we regard taking logarithms as nothing but the inverse of calculating exponential. i.e. <math>x=log_ab</math> knowing <math>a^x=b</math>. It seems that taking logarithms is as natural as operations involving indices. Then it should come at a huge surprise that at the time of Napier, the notion of index, in its generality, was no part of the stock of ideas of a mathematician, and that the exponential notation was not yet in use. In addition to that, Napier predated Isaac Newton and Leibniz so calculus and subsequently calculation by means of infinite series was not available him as well. It was with these difficulties that Napier invented and calculated the logarithms.
Napier published the ''Mirifici Logarithmorum Canonis Descriptio (The Description of the Wonderful Canon of Logarithms)'' in 1614 which did not contain an account of the methods by which the "wonderful canon" was constructed. It was not until after his death, ''Mirifici Logarithmorum Canonis Constructio (The Construction of the Wonderful Canon of Logarithms)'' was published in 1619. It was later found out that the "Constructio" was written before the "Descriptio".
It should be noted that Napier did not try to obtain the logarithms of natural numbers. Instead, he was trying to obtain the logarithms of sines of angles. Note that Sine was not defined as the ratio as we know it today. It was defined as the length of that semi-chord of a circle of given radius which subtends the angle at the center. Hence we have the relation <math>crd \theta = 2sin \frac {\theta}{2}</math>. Napier took the radius to be <math>10^7</math> units. Therefore, Napier was actually looking for the logarithms of the numbers between <math>0</math> and <math>10^7</math>, not for equidistant numbers, but for the numbers corresponding to equidistant angles. It also should be observed that the logarithms  in Napier's table are not what we know under the name of natural logarithms. Therefore, to           
==='''Other Scales on the Slide Rule and How They Were Constructed'''===
==Operating Principles==
==Zenith and Downfall==
==What is the point of all this?==
[http://www.antiquark.com/sliderule/sim/n909es/virtual-n909-es.html Virtual Slide Rule]
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Revision as of 16:01, 16 June 2010

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."

Don'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, there was a lack of women engineers back in those days. 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. However, it is not 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. As a matter of fact, things that are analog laid the foundation for our modern society and usher us into the digital age. Slide Rule is one of "those analog things". In the pre-computer age, it was the sole tool 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 failed) and eventually, created its own demise by ushering us into the digital age.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Algebra

Logarithm and logarithmic Scale

This section is not to preach you on logarithms since the subjec [...]

Logarithm and logarithmic Scale

This section is not to preach you on logarithms since the subject had been explained exhaustively by Logarithms in Math Images Project and Logarithm in Wikipedia. OR HAS IT? Rather, I will simply introduce logarithmic scale which is less mentioned and understood but widely used and essential to the construction of a slide rule.

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.


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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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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