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 = ? 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 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: log b AC = log b A + log b C Ex: log 4 64 = log log 4 16 = log 4 (416) Quotient rule: log b (A/C) = log b A − log b C Ex log 3 27/9 = log log 3 9 = 3 – 2 = 1 Power rule: log b A C = C(log b A) Ex: log 3 9² = 2log 3 9

A LOGARITHM TABLE

MULTIPLYING 9 X 8 Look up logarithms of the factors ln 9 = ln 8 = Add logarithms together = Find the number who’s anti-log matches ln 72 =

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

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 The length of 0.4 is The sum of those lengths is I can now look where these add together The value is the length of 0.6 (= ) 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

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 The length of 4 is The sum of those lengths is I can now look where these add together The value 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 rule.htmlPicket Model N600-EShttp:// rule.html Not all slide rules are straight

SCALES ON SLIDE RULES (1) Scale Label Value Relative to C/D Scale Label Value Relative to C/D Scale Label Value Relative to C/D Scale Label Value Relative to C/D Ax2x2 DFπxπxHCLL1 Bx2x2 DF/MKLL2 CxDI1/xKZ360xLL3 CFπxπxDIF1/πxLlog xLL00 CF/MEexex Lglog xLL01 CI1/xHLnln xLL02 CIF1/πxH1, H2LL0LL03

SCALES ON SLIDE RULES (2) Scale Label Value Relative to C/D Scale Label Value Relative to C/D Scale Label Value Relative to C/D Mlog xSh1,Sh2sinh xThtanh x PSq1,Sq2VVolts P%SRTsin x, tan x W1,W2 P1,P2STsin x, tan x Zx R1/xTtan x, cot x ZZ1 exex R1,R2T1,T3tan x, cot x ZZ2 exex ST2tan x, cot x ZZ3 exex

INTERESTING SITES Slide Rule Museum A digital repository for all things slide rule and other math artifacts What can you do with a slide rule? Just what the name imples Derek’s Virtual Slide Rule Gallery Software demos for a variety of slide rulers.