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)
Downloadable Version of this Web Page.
Use it on your classroom computers as
a Seminar on your computer (4MB PDF)

Introduction

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.


Table of Contents (TOC)



Pickett Teaching Guide 17.7MB pdf


Good Illustrated use of Slide Rule. A Pickett 902 is used as example 23.5MB pdf


Presentation on the use of Slide Rule by Prof. Joseph Pasquaule, UCSD 12.5MB pdf


Brief History of the Slide Rule

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.
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Parts of a Slide Rule



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Reading the Scales

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).
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Multiplication

1. Simple Multiplication (uses C and D scales)

Example: calculate 2.3 × 3.4 (Figure 1)
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2. 'Wrap-Around' Multiplication (uses C and D scales)

Example 2a: calculate 2.3 × 4.5 (Figure 2a)


Example 2b: calculate 2.3 × 4.5 (Figure 2b)


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3. Folded-Scale Multiplication (uses C, D, CF and DF scales)

Example 3: calculate 2.3 × 4.5 (Figure 3)
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4. Multiplication by π (uses D and DF scales)

Example 4: calculate 123 × π (Figure 4)


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Division

5. Simple Division (uses C and D scales)

Example 5: calculate 4.5 / 7.8 (Figure 5)


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6. Reciprocal (uses C and CI scales)

Example 6: calculate the reciprocal of 7.8, or 1/7.8 (Figure 6)
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Trigonometry

7. Sin(x) for angles between 5.7° and 90° (uses S and C scales)

Example 7: calculate sin(33°) (Figure 7)

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8. Cos(x) for angles between 5.7° and 90° (uses S and C scales)

Example 8: calculate cos(33°). (Figure 8)

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9. Tan(x) for angles between between 5.7° and 45° (uses T and C scales)

Example 9: calculate tan(33°) (Figure 9).

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10. Tan(x) for angles between between 45° and 84° (uses backward T and CI scale)

Example 10a: calculate tan(63°) (Figures 10a and 10b)


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11. Tan(x) for angles between between 45° and 84° (uses forward T and C scale)

Example 11: calculate tan(63°). (Figure 11)
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12. Sin(x) and tan(x) for angles between 0.6° and 5.7°  (using the ST and C scales)

In this range, the sin and tan functions are very close in value, so a single scale can be used to calculate both.

Example: calculate sin(1.5°) (Figure 12)

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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)

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Squares and Square Roots

14. Square (uses C and B scales)

Example 14: calculate 4.7 2

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15. Square Root (uses C and B scales)

Example 15a: calculate √4500 (Figure 15)

Example 15b: calculate √450 (Figure 15b)

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Cubes and Cube Roots

16. Cube  (uses D and K scales)

Example 16: calculate 4.73 (Figure 16)
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17. Cube Root (uses D and K scales)

Example 17a: calculate 3√4500 (Figure 17 - left)
Example 17b: calculate 3√450000 (Figure 17 - right)
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Log-Log Scales

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 1010.
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18. Raising a Number to Powers of 10 (N>1)

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)
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Example 18b: calculate 1.04 100     (uses LL1 and LL3 scales) (Figure 18b)

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Example 18c: calculate 1.002 1000  (uses LL0 and LL3 scales) (Figure 18c)

Example 18d: calculate sequential powers of ten of 1.002 (uses LL0, LL1, LL2 and LL3 scales) (use the above Figure 18c)

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19. Raising a Number to Powers of 10 (N<1)

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)
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20. Raising a Number to Powers of -10 (N<1), N-10 or 1/N10

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.

Example 20: calculate 1.175 -10 (uses LL2 and -LL3 scales)
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21. Finding the 10th Root (N>1)

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)
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22. Finding the 10th Root (N<1)

Example: calculate 100√0.15, or 0.15 0.01   (uses -LL3 and -LL1 scale) (Figure 22)
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23. Arbitrary Powers (Staying on Same LL Scale)

Occasionally, depending on the numbers, it is possible to calculate a power without switching scales.

Example: calculate 9.1 2.3         (uses LL3 scale) (Figure 23a)
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Example: calculate 230 0.45      (uses LL3 and C scale) (Figure 23b)

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Example: calculate 0.78 3.4      (uses -LL2 and C scale) (Figure 23c)
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Try another:

Example: calculate 0.78 0.45      (uses -LL2 and C scale) (Figure 23d)

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24. Arbitrary Powers (Switching LL Scales)

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)
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Example: calculate 12 0.34      (uses LL3, C and LL2 scales) (Figure 24b)
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Example: calculate 0.99 560    (uses -LL1, C and -LL3 scales) (Figure 24c)
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25. Log-Log Approximations

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)
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Example: calculate 0.99943 21    (uses C and D scales) (Figure 25b)

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ISRM Slide Rule Loan Program

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.
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Additional Resources
Instructions, Manuals, Scales and Construction

You can download additional instructions from the ISRM museum library or visit one of the following sites.

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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

Example set Copyright © 2005 Derek Ross.
Modifications, layout and graphics Copyright © 2006 by Mike Konshak
Copyright © 2003-2012 International Slide Rule Museum