Illustrated Self-Guided Course On
How To Use The Slide Rule
"Dad says that anyone who can't use a slide rule is a cultural illiterate and
should not be allowed to vote.
Mine is a beauty - a K&E 20-inch Log-log Duplex Decitrig" - Have Space Suit - Will
Travel, 1958.
by Robert A. Heinlein (1907-1988)
This self-guided course gives numeric examples of the basic calculations that a slide
rule can do. Just follow the step-by-step instructions and you will be
amazed by the power and versatility of the venerable Slipstick. Click on any of
the images below to get a large, unmarked, blowup of each slide rule as shown in
the problem.
Images created from Emulators designed by Derek Ross.
Initial examples based on Derek's Self Guided demo and expanded by Mike Konshak.
No Slide rule handy?
You can start up this Virtual Slide Rule (opens in new window) to try
these and other problems on your own. (Hint: Teachers can use a computer projector to manipulate the slide rule
in front of the class).
Want to try a real Slide Rule?
Participate in our School Loaner Program.
Sets of up to 25 slide rules are available FREE of charge for teachers to use in class.
1614 - Invention of logarithms by John Napier, Baron of Merchiston,
Scotland.
1617 - Developments of logarithms 'to base 10' by Henry Briggs,
Professor of Mathematics, Oxford University.
1620 - Interpretation of
logarithmic scale form by Edmund Gunter, Professor of Astronomy, London.
1630
- Invention of the slide rule by the Reverend William Oughtred, London.
1657
- Development of the moving slide/fixed stock principle by Seth Partridge,
Surveyor and Mathematician, England.
1775 - Development of the slide rule
cursor by John Robertson of the Royal Academy.
1815 - Invention of the log
log scale principle by P.M. Roget of France.
1850 - Amédée Mannheim, France,
produced the modern arrangement of scales.
1886 - Dennert & Pape,
Germany, introduce white celluloid as a material for inscription of
scales.
1890 - William Cox of the United States patented the duplex slide rule.
c1900 - Engine divided scales on celluloid increases
precision of slide rules.
1976 - The final slide rule made by K&E donated to the Smithsonian Institute, Washington, DC,
USA.
Today, slide rules can be found on eBay, antique stores and estate auctions.
Lost inventories of brand new slide rules turn up every year.
Was there Life Before Computers? Calculations before We Went Digital
The scales on a slide rule are logarithmic, in that the spacing between divisions (the lines on the scale)
become closer together as the value increases. This is why the slide rule is able to do multiplication and division
rather than addition and subtraction. Compare the two sets of offset scales
below in Figure B. In both cases the left index X:1 or C:1 is placed over the first whole
number, either Y:2 or D:2. On a linear scale the value of any number on the X
scale as read on Y is increased by 1. On a logarithmic scale, the value on any
number on the C scale as read on the D scale becomes a multiple of the number
under the index.
William Oughtred discovered the above characteristic in 1630, when he placed two
logarithmic scales that were invented by his contemporary, Edmund Gunter,
alongside each other. Thus the slide rule was born.
At this point it is best to just describe how to read the scales.
On almost all slide rules, the black scales (A, B, C, D, K, etc.) increase from left
to right. The red scales, or inverse scales
(CI, DI), increase from right to left.
The pocket sized Pickett 600-ES will be used in most illustrations. The full
sized Pickett N3-T for others. By the way, the Pickett 600-T (white) was taken by the
Apollo 11 NASA astronauts to the moon.
Except for 'folded', 'trig' or Log' scales, each scale begins
with 1. C and D scales are single logarithmic (1-10) scales. The A and B scales are double logarithmic
(1-10-100) having two cycles of 1-10,
the K scale being triple logarithmic (1-10-100-1000) having three cycles of
1-10. The Primary divisions are whole numbers.
The secondary divisions divide the Primary by 10, the Tertiary divisions divide the secondary by 5.
Of course as you get to the end of each scale the divisions get so close together that the tertiary divisions
disappear. The scales on each side of a slide rule are aligned so that
calculations can be carried from one side to the other.
Its important to become familiar with not only the physical divisions as marked on the scales, but in becoming able to
extrapolate* values when the hairline falls in the spacing between
divisions. Positions of the slide and cursor shown in the examples will mention the label of the scale
and the value on the scale, such as scale C at 1.5 will be referenced as C:1.5.
*In mathematics, extrapolation is the process of constructing new data points outside a discrete
set of known data points. It is similar to the process of interpolation,
which constructs new points between known points, but its results are often less
meaningful, and are subject to greater uncertainty (Ref: en.wickipedia.org).
8. Cos(x) for angles between 5.7° and 90° (uses S and C scales)
Example 8: calculate cos(33°). (Figure 8)
The cos scale shares the sin S scale. Instead of increasing
from left to right like the sin scale, cos increases from right to
left. This is indicated on the slide rule by '<' characters which
remind you that the number in increasing 'backwards'.
Move the cursor to <33 on the S scale.
The cursor is now at 8.4 on the C scale.
We know that the correct answer for a cos in this range is
between 0.1 and 1, so we adjust the decimal place to get 0.84.
13. Sin(x) and tan(x) for other small angles (using C and D scales)
For small angles, the sin or tan function can be approximated closely
by the equation:
sin(x) = tan(x) = x / (180/π) = x / 57.3.
Knowing this, the calculation becomes a simple division. This technique
can also be used on rules without an ST scale.
Example: calculate sin(0.3°) (Figure 13)
Move the cursor to 3 on the D scale.
Slide 5.73 on the C scale to the cursor. Most rules have a tick
labeled 'R' at this point.
Move the cursor to either the leftmost or rightmost '1' on the C
scale, whichever is in range.
The cursor is now at 5.24 on the D scale.
We know that the correct answer is near 0.3 / 60 = 0.005, so we
adjust the decimal place to get 0.00524.
You will notice that the B scale has two similar halves. The
first step is to decide which half to use to find a square root.
The left half is used to find the square root of numbers with odd
numbers of digits or leading zeros after the decimal point. The right
half is used for numbers with even numbers of digits or leading zeros.
Since 4500 has an even number of digits, then we'll use the right half
of the scale.
Move the cursor to 4.5 on the right half of the B scale.
The cursor is now at 6.7 on the C scale.
We know that 70^{2} = 4900, which is in the ballpark of
4500. Therefore we adjust the decimal point to get a result of 67.
Example 15b: calculate √450 (Figure 15b)
Try it a again with a three digit number.
Move the cursor to 4.5 on the left half of the B scale.
The cursor is now at 2.12 on the C scale.
We know that 20^{2} = 400, which is in the ballpark of
450. Therefore we adjust the decimal point to get a result of 21.2
We know that the correct answer is near 5 x 5 x 5, which, to
further approximate, is near 5 x 5 x 4 = 5 x 20 = 100. Therefore we
adjust the decimal point to get a result of 104.
Example 17a: calculate ^{3}√4500 (Figure 17 - left)
You will notice that the K scale has three similar thirds. The
first step is to determine which third to use to find the cube root.
The first third is used to find the cube root is numbers with one
digit. You can cycle through the thirds, increasing the number of
digits by one
for each third, to find which part to use. Just like having a line of people and you count off into 3's.
For the value of 4500, which has 4 digits, we cycle through the
thirds and find that we would use the first third.
Move the cursor to 4.5 on the first third of the K scale.
The cursor is now at 1.65 on the D scale.
We can take a guess that the correct answer is around 10. The
cube of 10 is 1000 and the cube of 20 is 8000. Thus we know that the
correct answer is between 10 and 20, therefore we can move the decimal
place and get the correct result of 16.5.
Example 17b: calculate ^{3}√450000 (Figure 17 - right)
For the value of 450000, which has 6 digits, we cycle through the
thirds and find that we would use the third third.
Move the cursor to 4.5 on the third third of the K scale.
The cursor is now at 7.68 on the D scale.
We can take a guess that the correct answer is around 10. The
cube of 10 is 1000 and the cube of 100 is 1000000. Thus we know that the
correct answer is between 10 and 100, therefore we can move the decimal
place and get the correct result of 76.8.
Log-log scales are used to raise numbers to powers. Unlike many of the
other scales, log-log scales can't be learned simply be memorizing a
few
rules. It is necessary to actually understand how they work. These
examples are intended to gradually introduce you to the concepts of
log-log scales, so you gain that understanding. Hopefully, the power of
10 examples don't bore you, as they lay the foundation for later
examples.
Since there are many slight variations of log-log scales on different
slide rules, I'll refer only to the scales found on the Pickett N3,
Pickett N600 and Pickett N803 slide rules (among others). If you want
to view a virtual N3, click here,
if you want a virtual N600, click here
(opens in a new window.)
Another interesting aspect of LL scales is that the decimal point
is "placed." That is, you don't have to figure out afterwards where the
decimal point belongs in your result. The disadvantage to this is that
LL scales are limited in the numbers they can calculate. Typically, the
highest result you can get is about 20,000, and the lowest is 1/20,000
or 0.00005. One exception to this is the Picket N4 (virtual here),
which goes up to 10^{10}.
To raise a number to the power of 10, simply move the cursor to the
number and look at the next highest LL scale. (These examples are for
numbers greater than 1. )
Example 18a: calculate 1.35 ^{10} (uses LL2
and
LL3 scales) (Figure 18a)
Move the cursor to 1.35 on the LL2 scale.
The cursor is at 20.1 on the LL3 scale. This is the correct
answer.
The reciprocals of the LL scales are the -LL scales. They work the same
way, but you have to make sure that you look for the answer on a -LL
scale. They are RED scales so they increase in
value from right to left.
Example 19: calculate 0.75 ^{10} (uses
-LL2
and -LL3 scales)
Move the cursor to <0.75 on the -LL2 scale.
The cursor is at <0.056 on the -LL3 scale. This is the correct
answer.
As you've seen in the previous examples, to raise a number to the 10th
power, you simply look at the adjacent number on the next highest LL
scale. To find a tenth root, you look at the adjacent number on the
next lowest LL scale. Remember also that finding the tenth root is the
same as raising a number to the power of 0.1.
Example 21: calculate ^{10}√5, or 5 ^{0.1}
(uses LL2 and LL3 scales) (Figure 21)
Set the cursor over 5 on the LL3 scale.
The cursor is now at 1.175 on the LL2 scale. This is the correct
answer.
Slide the leftmost Index '1' on C to the hairline.
Move the hairline over 2.3 on the C scale.
The cursor is now at about 161 on the LL3. This is very close to
the correct answer of 160.6. One of the problems with LL scales is that
their accuracy diminishes as the numbers increase in value.
One of the rules of exponents is that (A ^{B} ) ^{C}
is equal to A ^{B x C}. We can use this fact, along with our
knowledge of powers of ten, to calculate arbitrary powers.
Example: calculate 1.9 ^{2.5 }
(uses LL2, C and LL3 scales) (Figure 24a)
If we try to calculate this the easy way, the power 2.5 is out of
range for the scale.
We can reinterpret the problem as:
Calculate (1.9 ^{0.25} ) ^{10}
Because 0.25 x 10 is 2.5.
Set the cursor hairline to 1.9 on the LL2 scale.
Slide the rightmost Index '1' on the C scale to the hairline.
Shift the cursor hairline to 2.5 on the C scale.
The hairline is now at 1.9 ^{0.25} on the LL2 scale. Since
we want to also raise this to the power of 10, we look "one scale
higher" at the LL3 scale.
The cursor is at 4.97 on the LL3 scale. This is the correct
answer.
Example: calculate 12 ^{0.34} (uses
LL3, C and LL2 scales) (Figure 24b)
Like the previous example, if we try to calculate this the easy
way, the power 0.34 is out of range for the scale.
We can reinterpret the problem as:
Calculate (12 ^{3.4} ) ^{0.1}
Because 3.4 x 0.1 is 0.34.
Move the cursor to 12 on the LL3 scale.
Slide the leftmost '1' on the C scale to the cursor.
Move the cursor to 3.4 on the C scale.
The cursor is now at 12 ^{3.4} on the LL3 scale, which
is about 5000 (which is not the number we're looking for). Since we
also want to raise this to the power of 0.1, we look at the LL2 scale.
The cursor is now at 2.33 on the LL2 scale. This is the correct
answer.
In general, LL scales don't handle numbers extremely close to 1, such
as 1.001 or 0.999. This is not a problem because there is an accurate
approximation for numbers in this range. In general, if you have a very
small number 'd', then:
(1 + d) ^{p} = 1 + d p
Example: calculate 1.00012 ^{34} (uses C and D scales) (Figure
25a)
In this case, if we use the approximation (1 + d) ^{p} =
1 + d p, then:
d = 0.00012, and
p = 34
We must calculate 0.00012 * 34.
Set the leftmost Index '1' on the C scale to 1.2 on the D scale..
Set the cursor hairline on 3.4 on the C scale.
The cursor is now at 4.08 on the D scale.
We know that the correct answer would be near 0.0001 * 30, or
0.003. Therefore we adjust the decimal point to get a value of 0.00408.
Add 1 to 0.00408. The result is 1.00408, which is very close to
the correct answer of 1.004088.
If you are an educator or home schooler wishing to give your students a hands-on experience and instruction with
actual slide rules, the museum is able to supply quantities of up to 25 matching slide rules for temporary use, free of charge, to many countries, courtesy of several collectors and members of the Oughtred Society, Dutch Kring, and German RST. Find out more about it here.
In the beginning, at the time of the great flood, Noah went thru his ark after it landed, and found two small snakes huddled in a corner. Noah looked at these poor specimens - and said "I told you to go forth and multiply - why haven't you?"
The poor snakes looked up at Noah and replied "We can't because we are adders....."
Noah looked a bit perplexed, and then proceeded to tear bits of planking from his ark. He went on to build a beautiful wooden platform. He gathered up the snakes and placed them on the platform, and joyfully told the snakes - "Now go forth and multiply, because even adders can multiply on a log table"
D. Scott MacKenzie, PhD
Metallurgist specializing in Heat Treatment and Quenching