SLIDE RULER

WHAT’S THAT???TIM JEHL – MATH DUDE

CONTENTS

• The Fundamental Problem

• Development of Logarithms

• Basic Properties of Logarithms

• History of the Slide Rule

• Building a slide rule with lumber, a ruler and a marker

• More History

• Scales found on Slide Rules

THE FUNDAMENTAL PROBLEM

• Adding is relatively easy.

• 542 + 233 +187 = ?

• Most students (and a couple of us adults) can solve this problem in a timely manner.

• Multiplication is a bit more difficult

• 542 x 233 x 187 = ?

• This could take a bit…

THE FUNDAMENTAL PROBLEM (CONTINUED)

• In ancient times (like when I went to high school), we had to do problems like this without the benefit of a calculator because, well, they didn’t exist.

• The Slide Ruler was developed as a mechanical aid to assist in a variety of calculations

DEVELOPMENT OF LOGARITHMS

• Logarithms were invented by the Scottish mathematician and theologian John Napier and first published in 1614.

• Looking for a way of quickly solving multiplication and division problems using the much faster methods of addition and subtraction.

• Napier's way invented a group of "artificial" numbers as a direct substitute for real ones, called logarithms (which is Greek for "ratio-number", apparently).

• Logarithms are consistent, related values which substitute for real numbers.

• They were originally developed for base e.

• In1617, Henry Briggs adapted Napier's original "natural" logs to the base 10 format.

BASIC PROPERTIES OF LOGARITHMS

• Product rule: logbAC = logbA + logbC

• Ex: log464 = log44 + log416 = log4(4•16)

• Quotient rule: logb(A/C) = logbA − logbC

• Ex log3 27/9 = log327 - log39 = 3 – 2 = 1

• Power rule: logbAC = C(logbA)

• Ex: log39² = 2log39

A LOGARITHM TABLE

Number Log Number Log Number Log Number Log Number Log

1 0 3 1.098612 5 1.609438 7 1.94591 9 2.1972251.1 0.09531 3.1 1.131402 5.1 1.629241 7.1 1.960095 9.1 2.2082741.2 0.182322 3.2 1.163151 5.2 1.648659 7.2 1.974081 9.2 2.2192031.3 0.262364 3.3 1.193922 5.3 1.667707 7.3 1.987874 9.3 2.2300141.4 0.336472 3.4 1.223775 5.4 1.686399 7.4 2.00148 9.4 2.240711.5 0.405465 3.5 1.252763 5.5 1.704748 7.5 2.014903 9.5 2.2512921.6 0.470004 3.6 1.280934 5.6 1.722767 7.6 2.028148 9.6 2.2617631.7 0.530628 3.7 1.308333 5.7 1.740466 7.7 2.04122 9.7 2.2721261.8 0.587787 3.8 1.335001 5.8 1.757858 7.8 2.054124 9.8 2.2823821.9 0.641854 3.9 1.360977 5.9 1.774952 7.9 2.066863 9.9 2.2925352 0.693147 4 1.386294 6 1.791759 8 2.079442 10 2.3025852.1 0.741937 4.1 1.410987 6.1 1.808289 8.1 2.091864 10.1 2.3125352.2 0.788457 4.2 1.435085 6.2 1.824549 8.2 2.104134 10.2 2.3223882.3 0.832909 4.3 1.458615 6.3 1.84055 8.3 2.116256 10.3 2.3321442.4 0.875469 4.4 1.481605 6.4 1.856298 8.4 2.128232 10.4 2.3418062.5 0.916291 4.5 1.504077 6.5 1.871802 8.5 2.140066 10.5 2.3513752.6 0.955511 4.6 1.526056 6.6 1.88707 8.6 2.151762 10.6 2.3608542.7 0.993252 4.7 1.547563 6.7 1.902108 8.7 2.163323 10.7 2.3702442.8 1.029619 4.8 1.568616 6.8 1.916923 8.8 2.174752 10.8 2.3795462.9 1.064711 4.9 1.589235 6.9 1.931521 8.9 2.186051 10.9 2.388763

MULTIPLYING 9 X 8

• Look up logarithms of the factors

• ln 9 = 2.197225

• ln 8 = 2.079442

• Add logarithms together

• 2.197225 + 2.079442 = 4.276667

• Find the number who’s anti-log matches

• ln 72 = 4.276666

HISTORY OF THE SLIDE RULER

• In 1620, English astronomer Edmund Gunter drew a 2 foot long line with the whole numbers spaced at intervals proportionate to their respective log values.

• A short time later, Reverend William Oughtred placed two Gunter's scales directly opposite each other, and demonstrated that you could do calculations by simply sliding them back and forth.

BUILDING A SLIDE RULE WITH LUMBER, A RULER AND A MARKER

• Materials

• A couple of 4-foot lengths of hardwood

• Pine won’t do… it warps and won’t hold it’s shape properly

• A ruler, square and permanent marker

• Used for measuring lengths and marking the wood

• A set of common log tables (or a handy calculator)

• Time

• About 30 minutes if you know what you’re doing.

• All night if it’s your first try at it

• Notes

• The more accurate the measurements, the more accurate the instrument

• Constructing a fixture to hold the slide would be nice, but might require carpentry skills

• Let’s do mechanical addition

• Mark a length of board for some fixed distance

• For some bizarre reason, the marks in this demo were 89.6 cm apart

• Divide the distance evenly into tenths, and attempt to mark accurately.

• Total length times decimal value is the linear length to mark on your board

• See table to the right

LINEAR SCALE Number Length

0 00.1 8.960.2 17.920.3 26.880.4 35.840.5 44.80.6 53.760.7 62.720.8 71.680.9 80.641 89.6

ADDING ON THE LINEAR SCALE

• Using the cleverly pre-fabricated matching board, I can add two numbers together by adding their lengths

• The length of 0.2 is 17.92

• The length of 0.4 is 35.84

• The sum of those lengths is 53.76

• I can now look where these add together

• The value 53.76 is the length of 0.6 (= 0.2 + 0.4)

• Demo

• Let’s do mechanical multiplication

• Mark a length of board for some fixed distance

• For some bizarre reason, the marks in this demo were 89.6 cm apart

• Divide the distance based on the logarithms from 1 to 10

• Total length times decimal value is the linear length to mark on your board

• See table on right

LOGARITHMIC SCALE Number Log Length

1 0 02 0.30103 26.972293 0.477121 42.750064 0.60206 53.944585 0.69897 62.627716 0.778151 69.722357 0.845098 75.720788 0.90309 80.916869 0.954243 85.5001310 1 89.6

ADDING ON THE LOGARITHMIC SCALE

• Using the cleverly pre-fabricated matching board, I can add two numbers together by adding their lengths

• The length of 2 is 26.97

• The length of 4 is 53.94

• The sum of those lengths is 80.91

• I can now look where these add together

• The value 80.92 is the length of 8 (= 2 x 4)

• Demo

MORE HISTORY

• Calculators did not appear until the mid-1970’s.

• This is the sort of device your grand-parents used

• This was what the Apollo mission astronauts used to do their calculations while orbiting the moon. Picket Model N600-ES http://www.antiquark.com/sliderule/sim/virtual-slide-rule.html

• Not all slide rules are straight

SCALES ON SLIDE RULES (1)

ScaleLabel

ValueRelative to C/D

ScaleLabel

ValueRelative to C/D

ScaleLabel

ValueRelative to C/D

ScaleLabel

ValueRelative to C/D

A x2 DF πx HC LL1

B x2 DF/M K LL2

C x DI 1/x KZ 360x LL3

CF πx DIF 1/πx L log x LL00

CF/M E ex Lg log x LL01

CI 1/x H Ln ln x LL02

CIF 1/πx H1, H2

LL0 LL03

SCALES ON SLIDE RULES (2)

ScaleLabel

ValueRelative to C/D

ScaleLabel

ValueRelative to C/D

ScaleLabel

ValueRelative to C/D

M log x Sh1,Sh2 sinh x Th tanh x

P Sq1,Sq2 V Volts

P% SRT sin x, tan x

W1,W2

P1,P2 ST sin x, tan x

Z x

R 1/x T tan x, cot x

ZZ1 ex

R1,R2 T1,T3 tan x, cot x

ZZ2 ex

S T2 tan x, cot x

ZZ3 ex

INTERESTING SITES

• Slide Rule Museum

• http://www.sliderulemuseum.com/

• A digital repository for all things slide rule and other math artifacts

• What can you do with a slide rule?

• http://www.math.utah.edu/~pa/sliderules/

• Just what the name imples

• Derek’s Virtual Slide Rule Gallery

• http://www.antiquark.com/sliderule/sim/index.html

• Software demos for a variety of slide rulers.