Essay 3 | Page 1 - UHshanyuji/History/E3-7.pdfEssay 3 | Page 2" theoretically can be done quickly by hand, mentally, or the abacus.3 Depending on the size and length of the slide rule,

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Dr. Shanyu Ji

Math 4388, Section 18577

01 November 2016

AN ANALOG COMPUTING DEVICE: THE SLIDE RULE

Ever wonder what mathematicians, engineers, and scientists used to crunch numbers

before electronic calculators? I know I did not give it much thought since I am used to pulling

out my TI-84 or my pocket-sized Casio calculator to do quick simple or complex calculations.

Computing tool-wise, most individuals have seen or heard about abacuses, which were used to

perform mathematical computations such as addition, subtraction, division, multiplication, and

extract square-roots and cubic roots.1 There are even children’s toys that have an abacus attached

to it.2 But there was a more powerful computing tool after the abacus that was invented in the

1600s that was smaller in and had more computational functionality. This analog computing

device is known as the slide rule and depending on your age, you may or may not been exposed

to it. The slide rule is a mechanical tool used to do quick calculations involving multiplication,

division, exponents, roots, logarithms, and trigonometry. In this paper, the origin of the slide

rule, functionality and explanation of how it is used, and historical achievements attained with

the slide rule such as NASA

missions outer space will be

discussed.

What is a slide rule? A slide

rule is a simple computing device that is capable of calculating complex mathematical equations

with the limitation of not being able to be used for simple addition and subtraction, which

                                                                                                               1 Morris, Vernon. "What Was Used before Calculators?" Meta Calculator Blog. July 27, 2014. Accessed October 31,

2016. Notes: Provided quick description of an Abacus. 2 See Figure A and Figure B for examples of toys with built in Abacus in Illustrations section.

 Figure 1: Slide Rule

Taken from www.vintagecalculators.com/html/slide_rule.html

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theoretically can be done quickly by hand, mentally, or the abacus.3 Depending on the size and

length of the slide rule, the accuracy of the number of significant figures vary. For example, 6-

inch slide rules are the easiest to work with, but it is not as accurate as longer slide rules such as

10-inch slide rules that have up to three significant digits. The original slide rule that was first

proposed by William Oughtred was based on the logarithmic work of John Napier in the 1600s

and has been since modified by mathematicians over many decades to its most well-known form

in the late 1800s. But how exactly did the modern slide rule come about?

The mathematical notion of natural logarithms were invented by John Napier in 1614.

Napier’s purpose was to develop a more efficient way to solve multiplication and division

problems. To do this he essentially designated artificial numbers as real numbers in his

calculations. The next

contributor to the

development of the first

slide rule was Edmund

Gunter in 1620, who visually expressed the logarithmic scale described by Napier. As seen in

figure 2, logarithms begin widely spread apart from 1 to 2, then from 2 to 3, and quickly squishes

towards the right end from numbers 4 to 10. William Oughtred was the one who took Gunter’s

visual representation and placed two of them opposite of each other as linear sliding scales in

1630, which is historically considered as the first slide rule.4

The next set of contributors were Seth Partridge and Edmund Wingate in 1950 and they

improved the tool by adding in upper and lower stator and slider to Oughtred’s design. The next

major advances with the slide rule were the addition of cursor and additional scales, which were

                                                                                                               3 "Slide Rule." Encyclopedia.com. 2002. Accessed October 31, 2016. 4 Calvert, James B. "Slide Rule." Engineering and Technology. January 10, 2004. Accessed October 31, 2016.  

 Figure 2: Visual of Logarithmic Scale

Taken from http://www.sliderule.ca/intro.htm

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made by Amedee Mannheim and is credited for

the modern slide rule. The flaw in this addition

was that the slide rule had to be flipped

depending on the calculation, which is why Max

Rietz in 1902 modified the design by having the

all the scales on one side. To advance the slide

rule further in it computational capabilities,

Alwin Walther added in the Pythagorean scale in

1934. With these significant contributions to the

first slide rule, the modern slide rule was born and overall it allowed continuous mathematical

computations for users.

Modern slide rules have three major components: the body, the slide, and the cursor. The

body is a stationary

component that is

supported by clamps at

each end of the rule.

The slide component is

moveable center section that can be slid throughout the body of the slide rule. Finally, the cursor

also known as the hairline, is a moveable indicator that has two lines with transparent faces on a

mounted frame around the main body of the rule.5 Generally, slide rules are made from various

materials such as wood, bamboo, plastic, and metal. On the slide rule itself there are roughly

between 7 to 33 different scales depending on how advanced and type of rule it is. These scales

                                                                                                               5 Lasater, Judy. "Calculators and Calculating Devices." Wichita State University Department of Mathematics and

Statistics. Accessed October 31, 2016.  

Figure 3: Close Up of Slide Rule Parts

Taken from http://sliderulemuseum.com/SR_Course.html

 Figure 4: Main Components of Slide Rule Labeled Taken from sliderulemuseum.com/SR_Course.htm#Scales

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include: A, B, C, D, CI (C-inverted), DF (D-folded), S for Sine, T for Tangent, ST for Sine-

Tangent, K for cubes and cube roots, and L for logarithms using the mantissa of common logs.

Scales A and B are used to calculate squares and square roots, while scales C and D are used for

multiplication and division.6 The CI scale is used to compute chain multiplication and division.

With all these scales and computational possibilities, how does one use a slide rule?

According to several sources, basic slide rule computations are relatively easy to understand but

more complex computations such as calculating the trajectory of Apollo 11 into outer can only

be done by advanced users. To have a clear example of how to use a slide rule, the equation

2.3 x 3.4 = ?, will be solved. Essentially, a slide rule works by adding or subtracting logarithms

of numbers, which deals directly with exponents and powers. Since this example requires

multiplication, the logarithms will be added rather than subtracted.7 First, the slider is moved

until the leftmost index ‘1’ on the C scale over 2.3 on the D scale. Next, the cursor is moved to

3.4 on the C scale. At this point, the cursor is now on the D scale over the value 7.8.8 This

                                                                                                               6 Davis, Richard, and Ted Hume. "All About Slide Rules." The Oughtred Society. 2012. Accessed October 31, 2016. 7 Ibid. 8 Ross, Derek, and Mike Konshak. "Illustrated Self-Guided Course On How To Use The Slide Rule." International

Slide Rule Museum. 2006. Accessed October 31, 2016.  

 Figure 5: Slide Rule Example For 2.3 x 3.4 = ?

Taken from http://sliderulemuseum.com/SR_Course.htm#Scales

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computation is shown in figure 5, which is correct in consideration of having one significant

digit. The slide rule itself contains no decimal point markers, so it is up to the individual using

this tool to estimate the location of the decimal point.

During the reign of slide rules, individuals all over the world were able to design and

build incredible structures and vehicles, and calculate complex mathematical engineering

problems. A well-known example of the use of slide rules that changed history was its use by

NASA to build rockets for the Apollo 11 mission. Slide rules were used by engineers and

mathematicians to calculate every single trajectory for the mission. With this tool, the United

States was the first country to land a rocket on the moon in 1969. Another instance where the

slide rule was the heroic tool was during the Apollo 13 crisis when an unexpected explosion

occurred onboard causing the team to abort the mission and fall back into the Earth’s

atmosphere. Engineers used slide rules to quickly recalculate data and trajectories to get the crew

safely home.9

Slide rules were the most powerful mathematical tool for about 342 years, from when it

was first invented in 1630 until Hewlett -Packard came out with the first handheld calculator in

1972.10 After these calculators came out, the slide rule reign instantaneous ended. Although,

                                                                                                               9 Blakemore, Erin. "Before Calculators, This Tool Helped Put a Man on the Moon - Modern Notion." Modern

Notion. February 20, 2015. Accessed October 31, 2016. 10 Nadworny, Elissa. "The Slide Rule: A Computing Device That Put A Man On The Moon." NPR. October 22,

2014. Accessed October 31, 2016.

 Figure 6: Slide Rule, 5-Inch, Pickett N600-ES, Apollo 13

Taken from https://airandspace.si.edu/collection-objects/slide-rule-5-inch-pickett-n600-es-apollo-13

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slide rules became obsolete there are manuals available online to help those who want to

understand and calculate more complex problems using a slide rule. Slide rules can still be

purchased from specials retailers or used virtually online.11

Even before electronic calculators, individuals were able to crunch numbers using slide

rules as mathematical computing devices. With the invention and advancement of the slide rule

originally created by William Oughtred, society was able to land on the moon and recalculate

trajectories of a falling rocket without the use of calculators, which is an amazing feat

considering the fact that any miscalculation could have resulted in a major tragedy. Although,

using a slide rule may seem outdated, it is a beneficial way for an individual to increase mental

calculation speed and skill, to develop comfort with decimal estimation, and to save money on

batteries. It is a tool that will be known for its simplicity in design and its wide computational

flexibility.

                                                                                                               11 A list of slide rule emulators can be found at: http://www.antiquark.com/sliderule/sim/ which has a selection of slide rule versions.  

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ILLUSTRATIONS

Figure A : Children’s Abacus Toy

Taken from http://www.webzubra.com/wp-content/uploads/2/2-winning-sensory-toys-for-blind-sensory-toys-for-kids-with-autism-

Figure B: Children’s Toy with Abacus

Taken from http://image.dhgate.com/0x0/f2/albu /g3/M00/46/39/rBVaHVabkmuAXm3IAAPn9wnTm7c370.jpg

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REFERENCES

Blakemore, Erin. "Before Calculators, This Tool Helped Put a Man on the Moon - Modern

Notion." Modern Notion. February 20, 2015. Accessed October 31, 2016.

Calvert, James B. "Slide Rule." Engineering and Technology. January 10, 2004. Accessed

October 31, 2016.

Davis, Richard, and Ted Hume. "All About Slide Rules." The Oughtred Society. 2012. Accessed

October 31, 2016.

Dunbar, Brian. "Calculating By Hand." NASA. October 15, 2003. Accessed October 31, 2016.

Morris, Vernon. "What Was Used before Calculators?" Meta Calculator Blog. July 27, 2014.

Accessed October 31, 2016.

Lasater, Judy. "Calculators and Calculating Devices." Wichita State University Department of

Mathematics and Statistics. Accessed October 31, 2016.

Nadworny, Elissa. "The Slide Rule: A Computing Device That Put A Man On The Moon." NPR.

October 22, 2014. Accessed October 31, 2016.

Ross, Derek, and Mike Konshak. "Illustrated Self-Guided Course On How To Use The Slide

Rule." International Slide Rule Museum. 2006. Accessed October 31, 2016.

"Slide Rule." Encyclopedia.com. 2002. Accessed October 31, 2016.

Williams, Al. "Slide Rules Were the Original Personal Computers." Hackaday. November 05,

2015. Accessed October 31, 2016.