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Why Does a Slide Rule Work?

Joe Pasquale Department of Computer Science and Engineering

University of California, San Diego

J. Pasquale

Geometric Basis

•  Two in-line line segments: AB and BC

2

A B C

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Geometric Basis

•  Two in-line line segments: AB and BC •  Sum of lengths = length of sum

L(AB) + L(BC) = L(AB+BC) = L(AC)

3

A B C

Length (AB) Length (BC)

Length (AC)


Slide Rule Scales

•  Multiple scales •  Represent various functions •  Parallel to each other

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Slide Rule Scales

•  Slide allows us to add lengths •  But labels are not lengths!

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1.2 2.5

A B C

J. Pasquale, UCSD

What is a Slide Rule Scale?

•  Scale: finite line with graduated marks

6

log x

x

1 2 3 4 5 6 7 8 9 10

f(2)

x

f(x)



•  Scale: finite line with graduated marks •  Each mark corresponds to a value x

7

log x

x

1 2 3 4 5 6 7 8 9 10

f(2)

x

f(x)



•  Scale: finite line with graduated marks •  Each mark corresponds to a value x •  Located at a distance f(x) from origin

8

log x

x

1 2 3 4 5 6 7 8 9 10

f(2)

x

f(x)



•  Scale: finite line with graduated marks •  Each mark corresponds to a value x •  Located at a distance f(x) from origin •  f(x) is the function of the scale

9

log x

x

1 2 3 4 5 6 7 8 9 10

f(2)

x

f(x)


From 2 Dimensions to 1

10

100 1 2 3 4 5 6 7 8 9

1

0

0.2

0.4

0.6

0.8

x

log x

log x

x

1 2 3 4 5 6 7 8 9 10 x

log x


Example: Graph for f(x) = x2

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50 1 2 3 4

0

5

10

15

20

25

x

y5

4

3

2

10


Copy x-axis Marks to y-axis

12

50 1 2 3 4

0

5

10

15

20

25

x

y5

4

3

2

10

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x-ax

is m

arks

y-ax

is m

arks

J. Pasquale, UCSD

Next, Flip About Its Diagonal

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50 1 2 3 4

0

5

10

15

20

25

x

y5

4

3

2

10


x-ax

is m

arks

y-ax

is m

arks

J. Pasquale, UCSD

Resulting Graph

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0 5 10 15 20 25

5

0

1

2

3

4

y

x

543210


x-axis marks y-axis marks

J. Pasquale, UCSD

Focus on New x-axis

15

0 5 10 15 20 25

5

0

1

2

3

4

y

x

543210


Remove Original y-axis Marks

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0 5 10 15 20 25

5

0

1

2

3

4

y

x

543210


Resulting Scale

•  Scale for f(x) = x2 •  Notice

– Each mark is labeled by x – Distance from left end is f(x) = x2

17

543210


Calculating Using Scales

•  2 scales, each f(x) = x2

•  Resulting function:

•  In the above,

18

543210

543210

f (x, y) = x2 + y2

32 + 42 = 5

x

y h


Key Property of Sliding Scales

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g(y´)

g(y˝)

f(x´)

f(x˝)

g(y˝) – g(y´)

f(x˝) – f(x´) x´ x˝ f(x)

y´ y˝ g(y)



20

g(y´)

g(y˝)

f(x´)

f(x˝)

g(y˝) – g(y´)

f(x˝) – f(x´) x´ x˝ f(x)

y´ y˝ g(y)



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f ( !!x )− f ( !x ) = g( !!y )− g( !y )

g(y´)

g(y˝)

f(x´)

f(x˝)

g(y˝) – g(y´)

f(x˝) – f(x´) x´ x˝ f(x)

y´ y˝ g(y)


Mathematical Principle of the Slide Rule

•  A slide rule with two scales that are defined by the functions f(x) and g(y) can calculate any function of the form:

22

h(x, y, z) = f −1( f (x)+ g(y)− g(z))


Some Examples: Addition

•  Given f(x) = x, g(y) = y

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h(x, y, z) = f −1( f (x)+ g(y)− g(z))

h(x, y, 0) = f −1(x + y− 0) = x + y


Hypotenuse of Right Triangle

•  Given f(x) = x2, g(y) = y2

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h(x, y, z) = f −1( f (x)+ g(y)− g(z))

h(x, y, 0) = f −1(x2 + y2 − 0) = x2 + y2x

y h


Resistors in Parallel

•  Given f(x) = 1/x, g(y) = 1/y

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h(x, y, z) = f −1( f (x)+ g(y)− g(z))

h(R1,R2,∞) = f−1( 1R1+1R2− 0) = 1

1R1+1R2


What about Logarithms?

•  Most slide rules have logarithmic scales •  Why?

– Because they lead to a wide range of very useful types of calculations

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Multiplication

•  Given f(x) = log x, g(y) = log y

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h(x, y, z) = f −1( f (x)+ g(y)− g(z))

h(x, y,1) =f −1(log x + log y− 0) =10log x+log y = xy


Raising to a Power

•  Given f(x) = log log x, g(y) = log y

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h(x, y, z) = f −1( f (x)+ g(y)− g(z))

h(x, y,1) =f −1(loglog x + log y− 0) =10y log x = xy


Raising to a Power

•  Given f(x) = log log x, g(y) = log y

29

h(x, y, z) = f −1( f (x)+ g(y)− g(z))

h(x, y, z) =

f −1(loglog x + log y− log z) =10yzlog x

= xyz


UCSD Slide Rule Seminar

•  Started seminar in 2003 •  Students come from all areas

– Engineering, physical sciences, some humanities

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Why Teach about Slide Rules?

•  Engineering heritage

•  Power of slide rules vs. computers

•  Interesting mathematics

J. Pasquale, UCSD


Why Are Students Interested?

J. Pasquale, UCSD



“I have no idea what a slide rule is, but I know engineers used them to build great things”

J. Pasquale, UCSD



“I thought it would be cool to learn how to use a slide rule and show my friends”

J. Pasquale, UCSD



“My dad used a slide rule, and I want to learn about it (and show him)”

J. Pasquale, UCSD


Topics

•  How the slide rule works

•  The math behind the slide rule

•  Precision vs. accuracy

•  How to construct all the scales

•  Advanced Topics J. Pasquale, UCSD


Larger Lessons

•  Economy of calculating – slide rules – calculators – computers

•  Estimation, approximation •  Social value

– parents, grandparents

J. Pasquale, UCSD


Quotes

“My skills of estimation are getting better. I like being engrossed in the calculations, instead of just punching them into my calculator. I make less mistakes, and find I know what I am talking about.”

J. Pasquale, UCSD


Quotes

“I was in physics class, and the professor explained how tan and sin are close for really small angles. The class didn’t show much reaction, but my first thought was “Hey, I learned that from my slide rule seminar.”

J. Pasquale, UCSD


Quotes

“This slide rule seminar is the only thing saving me from a quarter full of literature writing, and other humanitarian monotony. After hours of ‘theory of literature,’ I realized I still had slide rule homework. Hurray!”

J. Pasquale, UCSD


Quotes

“The slide rule is truly an extension of a person, not something completely separate such as the calculator. I actually had to think before, during, and after getting the answer on the slide rule.”

J. Pasquale, UCSD


Quotes

“The more I use the slide rule, the greater the insight I have into how ingeniously the scales were put together. I hope I can re-teach my parents how to use it.”

J. Pasquale, UCSD


Masters of the Slide Rule ‘03

Masters of the Slide Rule, Winter ‘03

J. Pasquale, UCSD


FOR MORE INFO

Joe Pasquale Dept. of Computer Science & Engineering University of California, San Diego [email protected] http://www-cse.ucsd.edu/~pasquale Paper is available at

http://cseweb.ucsd.edu/~pasquale/Papers/IM11.pdf J. Pasquale, UCSD

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