A Short Slide Rule Primer
                            by Dave Van Domelen
                            copyright 2000, 2004
                               Updated 2/4/04
                        (notes added at the bottom)

     As slide rules slowly move from the desk drawers of scientists and
engineers and into the cabinets of collectors, there's a lot of people out
there who own a slide rule and have no idea how to use it.  True, no one
really needs to use a slide rule for mathematics anymore thanks to the advent
of electronic calculators, but it's nice to be able to see your slide rule as
more than just a collectible.
     This piece is intended as a short lesson on how to use the more common
functions of slide rules...at least to the point that something without
pictures can manage.  It will not teach you how to use the mathematics
involved, but there's plenty of readily-available resources for that.  For a
more in-depth treatment of slide rules, you might check your local university
library or public library.  Materials on slide rules can be found in the QA73
(Library of Congress) classification when they're available at all.

GLOSSARY:

     Cursor - This is the plastic or glass window with a hairline in it, used
to line things up.  Especially useful when trying to compare indices that are
far apart on the rule.
     Decade - One set of numbers from 1 back to 1.  This actually goes from 1
to 10, but slide rules drop powers of ten (see below).
     Index - This is one of the lines of hash marks and numbers, with a
letter for a label at one end.  The plural is indices.  Most slide rules have
indicies labeled A, B, C, D, S and T, and there's dozens of possible indices.
     Scale - The outer parts of the slide rule, the part that doesn't slide.
Usually has the A and D indices.
     Slider - The inner bar of the slide rule, the part that does slide.
usually has the B and C indices.

     Note that these terms have been defined based on a normal straight slide
rule.  Circular rules are built somewhat differently, but the principles of
use are about the same.


LOGARITHMIC PROGRESSION:

     With only a few exceptions (such as the L and Ln indices), indices on a
slide rule are marked off in a logarithmic progression, meaning that there is
no zero.  The logarithms are base ten, which means you really only deal with
numbers between 1 and 10.  In other words, if you write your numbers in
scientific notation (like 3 * 10^8), you only use the number before the power
of ten.  This makes it important to keep track of how many powers of ten are
flying around, since those will have to be put through the operation as
well.  If you're squaring a number, you have to square the whole number
(i.e. if you square 34, you put in 3.4 and square it to get about 18.4, then
multiply that by 10 squared, to get 1840).  
     Some slide rules have special indices to help get around the powers of
ten problem, like the LL indices or specific power calculation indices.
Other indices contain more than one decade, letting you keep things clear as
long as nothing gets more than about ten times bigger.  However, the more
decades you have in an index, the more crowded numbers get, and the harder it
is to get precision.


THE INDICES:

     A and B - As the letters imply, these are pretty basic indices.  Used
mainly for multiplication and sometimes as a "home row" for other
calculations.  A is on the scale, B is on the slider, and they match up
perfectly when the slide is lined up in the scale.  These scales usually have
two decades.
     C and D - Counterparts to A and B, these are more spread out, having
only one decade for every two decades on the A and B scales.  They can also
be used for multiplication, but not as easily as the A and B scales, because
of the single decade.  However, they are better for use as a home row.  C is
on the slider, D is on the scale.
     S and T - Sine and Tangent, these are trigonometic indices.  They run
from about 6 degrees to 45 and 90 degrees respectively.  There is no Cosine
index because you can just use trigonometry to find the Cosine of one angle
from the Sine of another.  There are often red numbers over these scales,
representing the Cosine from 6 to 90 degrees and the Tangent from 45 to 90,
to help people who don't want to do the trig.
     ST - Sine and Tangent are almost the same at very small angles, and this
index covers them for angles of about half a degree to about 6 degrees.  
     K - This has three decades for every one decade in D, and is used for
finding cubes and 3/2 powers (such as found in Kepler's equations of
planetary motion).
     CI and DI - The inverse of C and D.  Usually printed in red, to remind
the user that these indices go backwards.
     L and Ln - Logarithm and Natural Logarithm.  The tick-marks are evenly
spaced on these scales because everything else is already a logarithm.
     LL - Called Log-Log indices, these are used for advanced operations.
This file will touch on one possible use of LL1 through LL3.

     There exist other scales, especially on larger rules or specialty rules
intended for engineers, but the above group should get you pretty far.


MATHEMATICAL OPERATIONS:

     Multiplication: Move the slider so that the leftmost "1" on the B index
is lined up with the number on the A index that you want to multiply by.
We'll call it N.  Now, every number on the A index is N times the number
below it on the B index.  If you just want to multiply two numbers, it
doesn't really matter which you choose as N, but one advantage here is that
if you need to multiply several numbers by the same thing (say, for instance,
you need to scale something up from a blueprint to a real object), leaving
the slide rule set lets to just read off your answers.
     If you need a little more precision, you can use the C and D indices
instead.  Move the slider so that the leftmost "1" on the C index is over the
N you want to multiply by on the D index.

     Division: Same as multiplication, only now N is what you're dividing
by.  The number on your A index is divided by N to get the answer.

     Addition and Subtraction: Contrary to what you may have seen in the
movie Apollo 13, you can't do these things on a slide rule.  

     Squares/Square Roots: To find the square of a number, line the cursor up
on that number on the D index.  The square of that number will be shown on
the A index by the cursor.  Be careful to keep in mind orders of magnitude...
the square of 9 isn't 8.1, after all.
     For square roots, reverse the process, moving the cursor to the number
on the A index you want to take the square root of.

     Cubes/Cube Roots: To cube a number, line the cursor up on that number on
the D index.  The cube (third power) will be found on the K index where the
cursor lines up.
     Cube roots can be found by moving the cursor to the number on the K
index, then reading the cube root off the D index.

     3/2 Power: In the event you're doing celestial mechanics with a slide
rule, you can get the 3/2 power by finding your number on the A index with
the cursor.  Its 3/2 power will be on the K index below.

     Sine/Tangent: Move the cursor to your angle of interest on the S or T
index.  The value of Sine or Tangent will be found on the C and D indices
(use whichever index is on the same piece of slide rule as the S and T
indices, some rules have the S and T on the slider, others put it on the back
of the scale or even on the back of a reversible slider.  Divide the number
on the C/D index by ten to get the final answer, since the values for Sine
and Tangent (for angles less than 45) fall between zero and one.
     When using the ST index, divide your result from the C/D index by 100
instead.  

     Inversion: To get the inverse of a number, find the number on the C or D
index with the cursor.  Its inverse will be on the CI or DI index.  Be
careful to not read the inverse indices backwards, and be sure to invert any
powers of ten you have.

     Logarithm: The base-10 logarithm of a number on the C index can be found
above or below it on the L index.  Thus, the log10 of 2 is roughly .3.
     
     Power of Ten: To raise 10 to a fractional power, find that number on the
L index.  Ten to that number will be found on the C index.  Thus, 10^.3 is
roughly 2.

     Natural Logarithm: The natural logarithm can be found on some slide
rules, using the Ln index instead of the L index.  Thus, the natural log
(base e) of 2 is between .69 and .7.

     Power of e: To raise e (2.71828182845904523536...) to a power between
zero and 2.3, line up the cursor on the desired power on the Ln index (which
goes higher than 1 since the marks are closer together).  The answer will be
on the C index.  Thus, e^2 is about 7.4.

     More Fun With e: If you have the LL indices on your slide rule, one
thing you can do with them is find powers of e (and natural logarithms) over
a much wider range.
     Put the cursor on a number N on the D index.  The number it lines up
with on the LL1 index will be e raised to the power of N/100.  The number it
lines up with on the LL2 index will be e raised to the power of N/10 (this is
slightly better than using the Ln index, because it's spread out more).  The
number it lines up with on the LL3 index will be e raised to the power N.
     So, putting the cursor on 2 on the D index, we get the following
results: 
     LL1 - e^.02 = 1.0202 (really spread out, good precision)
     LL2 - e^.2 = 1.222
     LL3 - e^2 = 7.4

     One bit of warning about the LL3 index...it often uses "M" to mean
"thousand."  e^10 is not over 20 million, it's over 20 thousand.

ADDENDUM:

     Recently, I got email from someone curious as to the nature of a few
scales on his rule: Td, Sd, ITd and ISd.  They were not listed in any of the
books I had available, so I set out to deduce the nature of these scales.
Along the way, I had to decipher some odd markings on the C scale.  Here's
the results of that exercise.

     R: On the C scale, you may have a point labeled with a small R.  This is
the number of degrees in a radian, within an order of magnitude.  It may be
labeled something other than R, it was V on the slide rule I was
investigating.  I felt it was important because it was the zero of the Sd and
Td scales.  Located at about 5.7.
     degree: Also on the C scale, there may be a small degree symbol at about
1.7.  This gives the number of radians in a degree (about 0.017).  This was U
on the rule I was investigating, and does not appear on all rules that have
an R.  This was the zero for the ISd and ITd scales.
     ISd and ITd scales: Without going into a lot of detail, these scales are
used for situations where you have sine of a quantity divided by that
quantity, or tangent of a quantity divided by that quantity.  These are
decaying trigonometric functions, and I suppose the d is for "decay."
     Sd and Td scales: Gives the inverse of the ISd and ITd scales,
naturally.  I'm pretty sure these are x/sinx and x/tanx, but I may be
recalling incorrectly, and ISd is x/sinx.

     6/11/00 Note: I am informed that the d stands for "differential" in
these scales.

     2/4/04 Notes: In addition to the degree and R marks on the C scale, some
rules also have ' and " for minutes and seconds of arc.
     The R tick is also useful for finding sin x and tan x for small angles,
since at small angles those quantities equal the angle in radians.
     Some rules have CF and DF scales, which start and end at pi instead of
1.  The "F" stands for "Folded", as the scales are folded around at pi.
     Despite the letter order, the A and B scales are not "fundamental", the
C and D scales have that role.  On extremely simple rules, you may get C and
D but not A or B.

     Hopefully you will find this file useful in understanding your slide
rules.