DOCSLIB.ORG

The Beginnings and End of the Slide Rule Kendyl Kennard Imagine You

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Beginnings and End of the Slide Rule Kendyl Kennard Imagine You The Beginnings and End of the Slide Rule Kendyl Kennard Imagine you’re a mathematician working for NASA in the year 1953, armed with the task of calculating the trajectories for space shots which determined the timing for launches. You’re sitting at your desk armed with the latest technology at your disposal. What kind of tools are your imagining? After all you’re trying to put a man on the moon. Are you imagining a computer? Wrong. Computers were introduced in the 1940’s, and didn’t become common place until around the 1980’s. Even then, the computers at this time could only add, subtract, and maybe perform basic multiplication. So what about a calculator? Nope. The pocket calculator was invented even later in the 1970’s. So how, you may ask, are you supposed to compute such complicated arithmetic without the modern conveniences we take advantage of today? Just ask Katherine G. Johnson, the mathematician who completed the daunting task at question. Dr. Johnson calculated the trajectory for the Apollo 11 mission to the moon with the help of a simple slide rule. A tool that is foreign to the engineers and scientists of the modern world, yet was vital to the engineers and scientists of the early 20th century and beyond. You may be thinking, what is a slide rule? How does it work? Or even, why have I never heard of it? The rest of this paper aims to answer these questions. In order to understand the invention of the slide rule, we must venture back to the early 1600’s to the works of John Napier. In 1614, Napier published a book titled Mirifici Logarithmorum Canonis Descriptio. This book contained detailed explanations of logarithms, and pages of tables of numbers related to natural logarithms. Napier’s discovery made it possible to perform multiplication and division by simple addition and subtraction. For instance, a*b = 10log(a+b) . Using logs to multiply a and b, one would look up the logs of a and b, add them together, then find the number whose log was the sum. Shortly after the invention of logarithms, a mathematician named Edmund Gunter of Oxford, made the process of multiplication and division even simpler by creating a calculation device using a logarithmic scale. Simply stated, the logarithmic scale is a number line in which the positions of the numbers are proportional to their logs. Instead of having to go through the entire process Napier invented to find the product of two numbers, the user just simply measured the distance from the beginning of the scale to one factor with a pair of dividers, then moved them to start at the other factor and read the number at the combined distance all on the same scale. Let’s take a look at Edmunds scale below6 and perform the simple multiplication problem 2x3. First we start by using the divider to measure the distance from the beginning of the scale, 1 to the first factor, 2. Then, without altering the span of the divider, you add this length to the second factor of 3 to find that the span of the dividers lands on 6, the product of 2 and 3. Division can be performed using the same method, except by subtracting the lengths instead of adding them. Finally, in 1622, William Oughtred, an Anglican minister and mathematician, placed two of these Gunter scales side by side, eliminating the dividers and creating the very first modern slide rule. Let’s take a look at a simplified sketch of Oughtred’s slide rule above2 and perform the same calculation of 2x3 to understand the simplification that brought us the modern day slide rule. First, we start by sliding the top scale to match one of the factors, which in this case is 2. The answer to 2X3 is given by the number on the bottom scale the corresponds to the other factor,3, on the top scale which we see is 6. The slide rule went through a series of manipulations and additions after it’s initial conception, until 1859 when the modern slide rule that we know today was created by French artillery lieutenant Amédée Mannheim. The rapidly evolving world at this time evoked the beginnings of the engineering profession, which in turn spurred a demand for Mannheim’s slide rule. Finally, in 1881 the slide rule became a staple computational tool in the United States. The slide rule was used by various professions’ ranging from bombardiers and navigators in World War II, to rocket scientists and engineers working for NASA. The slide rule slowly became an iconic symbol of the engineering profession in the 1950’s. In fact, if you were an engineering student in the 50’s, chances are, you would be sporting a 10 inch slide rule in your belt holster. This trend carried on for years until the slide rule started its gradual demise in 1960. In 1960, IBM introduced some of the first affordable computers geared specifically for the engineering profession. The computer made calculations even easier than the slide rule. The user could simply type in a formula and receive an answer without the reformulation needed when using a slide rule. However, the slide rule was still convenient at this point because they were accessible on the go and computers weren’t yet common in the home. The real demise of the slide rule came in 1972 when Hewlett Packard introduced the first pocket scientific calculator which combined the convenience of the slide rule and the ease of the computer. In 1976 the pocket scientific calculator could be purchased for less than $25 and the slide rule became nonexistent. In my opinion, the demise of the slide rule and rise of the calculator impacted more than just the science and engineering professions. The education system in America was forever changed with the invention of the calculator and not necessarily in a good way. This past week I asked my Calculus 1 students if they knew what a logarithm was, and I was met with blank stares and confusion. The power of the calculator allowed my college students to perform advanced mathematical computations for years without even second guessing or questioning exactly what they were doing. Unlike the calculator, the slide rule actually provoked thought and understanding when performing complex computations. Sadly, this deeper level of mathematical understanding in most American students met its demise along with the slide rule in the 1970’s. Works Cited 1. Blakemore, E. (2015, February 20). Before Calculators, This Tool Helped Put a Man on the Moon. Retrieved November 1, 2016, from http://modernnotion.com/slide-rules-helped-put-man- moon/ 2. Slide Rule. (n.d.). Retrieved November 1, 2016, from https://en.wikipedia.org/wiki/Slide_rule#History 3. Slide Rule History. (2013, December 13). Retrieved November 1, 2016, from http://www.oughtred.org/history.shtml 4. Kuhman, H. J. (n.d.). Slide Rule. Retrieved November 1, 2016, from http://www.encyclopedia.com/science-and-technology/mathematics/mathematics/slide-rule 5. John Napier. (n.d.). Retrieved November 1, 2016, from https://en.wikipedia.org/wiki/John_Napier 6. Gunter's Rule. (n.d.). Retrieved November 1, 2016, from http://www.nzeldes.com/HOC/Gunter.htm .
Load more

Recommended publications
  • A Reconstruction of Gunter's Canon Triangulorum (1620)
    A reconstruction of Gunter’s Canon triangulorum (1620) Denis Roegel To cite this version: Denis Roegel. A reconstruction of Gunter’s Canon triangulorum (1620). [Research Report] 2010. inria-00543938 HAL Id: inria-00543938 https://hal.inria.fr/inria-00543938 Submitted on 6 Dec 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A reconstruction of Gunter’s Canon triangulorum (1620) Denis Roegel 6 December 2010 This document is part of the LOCOMAT project: http://www.loria.fr/~roegel/locomat.html Introduction What Briggs did for logarithms of numbers, Gunter did for logarithms of trigonometrical functions. Charles H. Cotter [9] The first table of decimal logarithms was published by Henry Briggs in 1617 [3]. It contained the decimal logarithms of the first thousand integers. Before that, in 1614, Napier had published tables of “Napierian” logarithms of the sines [24]. Eventually, in 1620, combining these two ideas, Edmund Gunter (1581–1626) was the first to publish a table of decimal logarithms of trigonometric functions. This is the table which is reproduced here. 1 Edmund Gunter (1581–1626) Edmund Gunter1 studied at Oxford and became famous for several inven- tions, in particular for the sector invented in 1606 [18, 40, 47].
    [Show full text]
  • Slide Rules: How Quickly We Forget Charles Molían If You Have Not Thrown out Your Old Slide Rule, Treasure It, Especially If It Is One of the More Sophisticated Types
    HISTORY OF PHYSICS91 Instrumental Background Slide Rules: How Quickly We Forget Charles Molían If you have not thrown out your old slide rule, treasure it, especially if it is one of the more sophisticated types. It is a relic of a recent age In 1614 John Napier published details of invention of the electronic calculator. Fig 1 top A straight slide rule from A.W. Faber of Bavaria his logarithms which reduced multiplica­ The bigger the slide rule, the more cl 905. (The illustration is from a 1911 catalogue). In this engraving the number 1 on the bottom scale of the slide tion and division to simple addition and accurate the calculations. But there are is placed above 1.26 on the lower scale on the main body subtraction. To make logarithms easier he limits to practicable lengths so various of the rule. The number 2 on the slide corresponds to devised rectangular rods inscribed with ingenious people devised ways of increas­ 2.52 on the lower scale, the number 4 to 5.04, and any numbers which became known as ing the scale while maintaining the device other number gives a multiple of 1.26. The line on the ‘Napier’s Bones’. By 1620, Edmund Gunter at a workable size. Several of these slide transparent cursor is at 4.1, giving the result of the had devised his ‘Gunter scale’, or Tine of rules became fairly popular, and included multiplication 1.26 x 4.1 as 5.15 (it should be 5.17 but the engraving is a little out).The longer the slide rule and numbers’, a plot of logarithms on a line.
    [Show full text]
  • The Concise Stadia Computer, Or the Text Printed on the Instrument Itself, Or Any Other Documentation Prepared by Concise for This Instrument
    Copyright. © 2005 by Steven C. Beadle (scbead 1e @ earth1ink.net). Permission is hereby granted to the members of the International Slide Rule Group (ISRG; http://groups.yahoo.com/group/sliderule) and the Slide Rule Trading Group (SRTG; http://groups.yahoo.com/group/sliderule-trade) to make unlimited copies of this doc- ument, in either electronic or printed format, for their personal use. Commercial redistribution, without the written permission of the copyright holder, is prohibited. THE CONCISE Disclaimer. This document is an independently written guide for the Concise STADIA Stadia Computer, a product of the Concise Co. Ltd., located at 2-16-23, Hirai, Edogawa-ku, Tokyo, Japan (“Concise”). This guide was prepared without the authori- zation, consent, or cooperation of Concise. The copyright holder, and not Concise or COMPUTER the ISRG, is responsible for any errors or omissions. This document was developed without translation of the Japanese language manual Herman van Herwijnen for the Concise Stadia Computer, or the text printed on the instrument itself, or any other documentation prepared by Concise for this instrument. Thus, there is no Memorial Edition assurance that the guidance presented in this document accurately reflects the uses of this instrument as intended by the manufacturer, or that it offers an accurate inter- pretation of the scales, labels, tables, and other features of this instrument. The interpretations presented in this document are distributed in the hope that they may be useful to English-speaking users of the Concise Stadia Computer. However, the copyright holder and the ISRG provide the document “as is” without warranty of any kind, either expressed or implied, including, but not limited to, the implied warranties of merchantability and fitness for a particular purpose.
    [Show full text]
  • W Chapter.Indd
    Wade, John E. Erwin Tomash Library Wakelin, James H. W 2 Wagner, Balthasar Practica Das ist: Kürze jedoch gründtliche Erkläru[n]g der vornemsten hauss und Kauffmans rechnunge[n], beides nach der Regul Ee Tri: und welsche Practic. Year: 1626 Place: Strasbourg Publisher: Johan Erhardt Wagner Edition: 1st Language: German From Recorde, The whetstone of witte, 1557 Binding: contemporary printed paper wrappers Pagination: ff. [32] Collation: A–D8 W 1 Size: 142x91 mm Wade, John E. This small arithmetic was intended for use by merchants. The mathematical velocipede; or, instantaneous method Its one handicap as a text is that only a few of the of computing numbers. operations are illustrated with examples, and these are Year: 1871 only in the problems at the end of the work. The majority Place: New York of the book is spent discussing the rule of three, with a Publisher: Russell Brothers few pages on topics such as money exchange, etc. The Edition: 1st title page is engraved, and each page of the text has a Language: English decorative border. Binding: original cloth-backed printed boards Illustrations available: Pagination: pp. 144 Title page Size: 142x115 mm Text page This work teaches a number of different tricks that can be used to perform arithmetic. There are so many of Wakelin, James H., editor them that it is difficult to remember which to use in any See Engineering Research Associates, High-speed particular circumstance. The last half of the book deals computing devices. with many different trades, their units of measure and elementary operations (how to preserve wood, etc.).
    [Show full text]
  • The Demise of the Slide Rule (And the Advent of Its Successors)
    OEVP/27-12-2002 The Demise of the Slide Rule (and the advent of its successors) Every text on slide rules describes in a last paragraph, sadly, that the demise of the slide rule was caused by the advent of the electronic pocket calculator. But what happened exactly at this turning point in the history of calculating instruments, and does the slide rule "aficionado" have a real cause for sadness over these events? For an adequate treatment of such questions, it is useful to consider in more detail some aspects like the actual usage of the slide rule, the performance of calculating instruments and the evolution of electronic calculators. Usage of Slide Rules During the first half of the 20th century, both slide rules and mechanical calculators were commercially available, mass-produced and at a reasonable price. The slide rule was very portable ("palmtop" in current speak, but without the batteries), could do transcendental functions like sine and logarithms (besides the basic multiplication and division), but required a certain understanding by the user. Ordinary people did not know how to use it straight away. The owner therefore derived from his knowledge a certain status, to be shown with a quick "what-if" calculation, out of his shirt pocket. The mechanical calculator, on the other hand, was large and heavy ("desktop" format), and had addition and subtraction as basic functions. Also multiplication and division were possible, in most cases as repeated addition or subtraction, although there were models (like the Millionaire) that could execute multiplications directly. For mechanical calculators there was no real equivalent of the profession-specific slide rule (e.g.
    [Show full text]
  • Napier's Ideal Construction of the Logarithms
    Napier’s ideal construction of the logarithms Denis Roegel To cite this version: Denis Roegel. Napier’s ideal construction of the logarithms. [Research Report] 2010. inria-00543934 HAL Id: inria-00543934 https://hal.inria.fr/inria-00543934 Submitted on 6 Dec 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Napier’s ideal construction of the logarithms∗ Denis Roegel 6 December 2010 1 Introduction Today John Napier (1550–1617) is most renowned as the inventor of loga- rithms.1 He had conceived the general principles of logarithms in 1594 or be- fore and he spent the next twenty years in developing their theory [108, p. 63], [33, pp. 103–104]. His description of logarithms, Mirifici Logarithmorum Ca- nonis Descriptio, was published in Latin in Edinburgh in 1614 [131, 161] and was considered “one of the very greatest scientific discoveries that the world has seen” [83]. Several mathematicians had anticipated properties of the correspondence between an arithmetic and a geometric progression, but only Napier and Jost Bürgi (1552–1632) constructed tables for the purpose of simplifying the calculations. Bürgi’s work was however only published in incomplete form in 1620, six years after Napier published the Descriptio [26].2 Napier’s work was quickly translated in English by the mathematician and cartographer Edward Wright3 (1561–1615) [145, 179] and published posthu- mously in 1616 [132, 162].
    [Show full text]
  • The Mathematical Work of John Napier (1550-1617)
    BULL. AUSTRAL. MATH. SOC. 0IA40, 0 I A45 VOL. 26 (1982), 455-468. THE MATHEMATICAL WORK OF JOHN NAPIER (1550-1617) WILLIAM F. HAWKINS John Napier, Baron of Merchiston near Edinburgh, lived during one of the most troubled periods in the history of Scotland. He attended St Andrews University for a short time and matriculated at the age of 13, leaving no subsequent record. But a letter to his father, written by his uncle Adam Bothwell, reformed Bishop of Orkney, in December 1560, reports as follows: "I pray you Sir, to send your son John to the Schools either to France or Flanders; for he can learn no good at home, nor gain any profit in this most perilous world." He took an active part in the Reform Movement and in 1593 he produced a bitter polemic against the Papacy and Rome which was called The Whole Revelation of St John. This was an instant success and was translated into German, French and Dutch by continental reformers. Napier's reputation as a theologian was considerable throughout reformed Europe, and he would have regarded this as his chief claim to scholarship. Throughout the middle ages Latin was the medium of communication amongst scholars, and translations into vernaculars were the exception until the 17th and l8th centuries. Napier has suffered badly through this change, for up till 1889 only one of his four works had been translated from Latin into English. Received 16 August 1982. Thesis submitted to University of Auckland, March 1981. Degree approved April 1982. Supervisors: Mr Garry J. Tee Professor H.A.
    [Show full text]
  • Fourier Analyzer and Two Fast-Transform Periph- Erals Adapt to a Wide Range of Applications
    JUNE 1972 TT-PACKARD JOURNAL The ‘Powerful Pocketful’: an Electronic Calculator Challenges the Slide Rule This nine-ounce, battery-po wered scientific calcu- lator, small enough to fit in a shirt pocket, has log a rithmic, trigon ome tric, and exponen tia I functions and computes answers to IO significant digits. By Thomas M. Whitney, France Rod& and Chung C. Tung HEN AN ENGINEER OR SCIENTIST transcendental functions (that is, trigonometric, log- W NEEDS A QUICK ANSWER to a problem arithmic, exponential) or even square root. Second, that requires multiplication, division, or transcen- the HP-35 has a full two-hundred-decade range, al- dental functions, he usually reaches for his ever- lowing numbers from 10-” to 9.999999999 X lo+’’ present slide rule. Before long, to be represented in scientific however, that faithful ‘slip stick’ notation. Third, the HP-35 has may find itself retired. There’s five registers for storing con- now an electronic pocket cal- stants and results instead of just culator that produces those an- one or two, and four of these swers more easily, more quickly, registers are arranged to form and much more accurately. an operational stack, a feature Despite its small size, the new found in some computers but HP-35 is a powerful scientific rarely in calculators (see box, calculator. The initial goals set page 5). On page 7 are a few for its design were to build a examples of the complex prob- shirt-pocket-sized scientific cal- lems that can be solved with the culator with four-hour operation HP-35, from rechargeable batteries at a cost any laboratory and many Data Entry individuals could easily justify.
    [Show full text]
  • PDF Full Text
    Digital Signal Processing: A Computer Science Perspective Jonathan Y. Stein Copyright 2000 John Wiley & Sons, Inc. Print ISBN 0-471-29546-9 Online ISBN 0-471-20059-X 16 Function Evaluation Algorithms Commercially available DSP processors are designed to efficiently implement FIR, IIR, and FFT computations, but most neglect to provide facilities for other desirable functions, such as square roots and trigonometric functions. The software libraries that come with such chips do include such functions, but one often finds these general-purpose functions to be unsuitable for the application at hand. Thus the DSP programmer is compelled to enter the field of numerical approximation of elementary functions. This field boasts a vast literature, but only relatively little of it is directly applicable to DSP applications. As a simple but important example, consider a complex mixer of the type used to shift a signal in frequency (see Section 8.5). For every sample time t, we must generate both sin@,) and cos(wt,), which is difficult using the rather limited instruction set of a DSP processor. Lack of accuracy in the calculations will cause phase instabilities in the mixed signal, while loss of precision will cause its frequency to drift. Accurate values can be quickly retrieved from lookup tables, but such tables require large amounts of memory and the values can only be stored for specific arguments. General purpose approximations tend to be inefficient to implement on DSPs and may introduce intolerable inaccuracy. In this chapter we will specifically discuss sine and cosine generation, as well as rectangular to polar conversion (needed for demodulation), and the computation of arctangent, square roots, Puthagorean addition and loga- rithms.
    [Show full text]
  • Learning from Slide Rules
    Document ID: 06_05_99_1 Date Received: 1999-06-05 Date Revised: 1999-08-01 Date Accepted: 1999-08-08 Curriculum Topic Benchmarks: M1.4.1, M.1.4.3, M1.4.6, M1.4.7, M2.4.1, M2.4.3, M2.4.5, M4.4.6, M4.4.8 Grade Level: [9-12] High School Subject Keywords: slide rule, analog computer, isomorphism, logarithm Rating: advanced Learning from Slide Rules By: Martin P Cohen, Environmental Standards, Inc, 52 Hollybrook Drive , Langhorne PA 19047 e-mail: [email protected] From: The PUMAS Collection http://pumas.jpl.nasa.gov ©1999, California Institute of Technology. ALL RIGHTS RESERVED. Based on U.S. Gov't sponsored research. Introduction -- In the days before calculators and personal computers an engineer always had a slide rule nearby. These days it is difficult to locate a slide rule outside of a museum. I don't know how many of those who have used a slide rule ever thought of it as an analog computer, but that is really what it is. As such, the slide rule can be used to teach the modern view of the relationship between nature and mathematics and about the formalization of this concept known as isomorphism, which is one of the most pervasive and important concepts in mathematics. In order to introduce the concepts that will be used, we will start with the simplest type of slide rule – one that is made up of two ordinary rulers. Using Rulers for Addition -- Two rulers can be lined up as in the figure below to show that 5 + 3 = 8.
    [Show full text]
  • One to One Property of Exponents
    One To One Property Of Exponents Yule is unbidden: she passages pettishly and manumitted her domination. Say remains gonococcoid after Vladimir wonderingly.disburthens contradictively or redintegrating any cloudlets. Mighty Jermain usually treed some zootoxin or glazed The following exercises, everyone else has vocabulary or those with a of one exponents to To attain the PRODUCT of two powers with salt ADD the exponents. Key Investigating Exponent Properties Quotient of Powers. What charge the 7 properties of exponents? In the teacher and to add the property one to of exponents and would have a positive numbers have to see it is related directly to negative exponents using power of. How can use what is one side in his goal to each property one to exponents of the base on, evaluate the same base number is rooted in a certain advantages while giving them? While having property overseas an exponential function with stump base b 1 is the same property particular. Exponential Equations MathBitsNotebookA2 CCSS Math. This step to obtain an advanced trigonometry lesson. You should also refund the properties of exponents in memory to be successful in solving exponential. Intro to Adding and Subtracting Logs Same Base Expii. Expressions using the distributive property and collecting like terms MCC. Solving Exponential Equations Varsity Tutors. Properties of Logarithmic Functions. One pat of exponential equations that is initially confusing to some students is determining how many solutions an in will have Exponential equations. Solving Exponential Equations with Different Bases examples. Example Rewrite 4243 using a wage base and exponent. I sign write here whole tune about themmaybe one day.
    [Show full text]
  • Slide Rules and Log Scales
    Name: __________________________________ Lab – Slide Rules and Log Scales [EER Note: This is a much-shortened version of my lab on this topic. You won’t finish, but try to do one of each type of calculation if you can. I’m available to help.] The logarithmic slide rule was the calculation instrument that sent people to the moon and back. At a time when computers were the size of rooms and ran on tapes and punch cards, the ability to perform complex calculations with a slide rule became essential for scientists and engineers. Functions that could be done on a slide rule included multiplication, division, squaring, cubing, square roots, cube roots, (common) logarithms, and basic trigonometry functions (sine, cosine, and tangent). The performance of all these functions have all been superseded in modern practice by the scientific calculator, but as you may have noted by now, the calculator has a “black box” effect; that is, it doesn’t give you much of a feel for what an answer means or whether it is right. Slide rules, by the way they were manipulated, gave users more of an intrinsic feel for how operations were performed. The slide rule consists of nine scales of varying size and rate – some are linear, while others are logarithmic. Part 1 of 2: Calculations (done in class) In this lab, you will use the slide rule to perform some calculations. You may not use any other means of calculation! You will be given a brief explanation of how to perform a given type of calculation, and then will be asked to calculate a few problems of each type.
    [Show full text]