"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)

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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.

A Brief Timeline 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.

Was there Life

Before Computers?

Calculations before

We Went Digital

Teaching Aid on the history of mathematics by Nicola Marras, Italy 12.5MB pdf



Parts of a Slide Rule







Reading the Scales



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).

Multiplication

1. Simple Multiplication (uses C and D scales)

Slide the leftmost Index '1' on C over 2.3 on the D scale.

Move the cursor to 3.4 on the C scale.

The cursor is on the D scale just a bit over 7.8 or 7.82. This is the answer.



2. 'Wrap-Around' Multiplication (uses C and D scales)

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

Move the left Index on C to 2.3 on the D scale.

Try to move the cursor to 4.5 on the C scale. The cursor is blocked by the brace (also called strap or bridge).

The target C:4.5 is off the D scale. The right Index must now be used.

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

Move the right Index on C (C:1) to over 2.3 on the D scale (D:2.3)

Slide the cursor to 4.5 on the C scale (C:4.5)

On the D scale, you'll see that the hairline is between divisions. Extrapolate the answer to 1.035.

Do a rough approximation by rounding 2.3 to 2 and 4.5 to 5.

We mentally calculate 2 × 5 = 10, so we adjust the decimal place to get 10.35 or 10.4.

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

Slide the leftmost Index, '1', on C over the 2.3 on the D scale (D:2.3).

We can't move the cursor to 4.5 on the C scale; it's out of range. We can use the folded scales to get this answer.

Move the cursor to 4.5 on the CF scale (CF:4.5).

The cursor is now at 1.04 on the DF scale (DF:1.04).

We know the correct answer is near 2 × 5 = 10, so we adjust the decimal place to get 10.4.

4. Multiplication by π (uses D and DF scales)

Example 4: calculate 123 × π (Figure 4)



Move the cursor to 1.23 on the D scale.

The cursor is now at 3.86 on the DF scale.

We know that the correct answer is near 100 × 3 = 300, so we adjust the decimal place to get 386.

Division

5. Simple Division (uses C and D scales)

Move the cursor to 4.5 on the D scale.

Slide 7.8 on the C scale to the cursor.

Move the cursor to either the leftmost or rightmost '1' on the C scale, whichever is in range. In this case, you would move it to the rightmost '1'.

The cursor is now at 5.76 on the D scale.

We know that the correct answer is near 4/8 = 0.5, so we adjust the decimal place to get 0.576.

6. Reciprocal (uses C and CI scales)

Move the cursor to 7.8 on the CI scale. Note that the CI scale increases from right to left, as indicated by the '<' symbols before the numbers.

The cursor is now at 1.28 on the C scale.

We know that the correct answer is near 1/10 = 0.1, so we adjust the decimal place to get 0.128.



Trigonometry

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

Move the cursor to 33 on the S scale.

The cursor is at 5.45 on the C scale.

We know that the correct answer for a sin in this range is between 0.1 and 1, so we adjust the decimal place to get 0.545.

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

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.

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

Move the cursor to 33 on the T scale.

The cursor is now at 6.5 on the C scale.

We know that the correct answer for a tan in this range is between 0.1 and 1, so we adjust the decimal place to get 0.65.

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

Move the cursor to <63 on the T scale. Note that this range increases from right to left, as indicated by the ' <' before the numbers.



on the T scale. Note that this range increases from right to left, as indicated by the ' before the numbers. The cursor is now at <1.96 on the DI scale. Figure 10a

on the scale. Figure 10a If you flip to the other side the cursor is also at <1.96 on the CI scale. Figure 10b

on the scale. Figure 10b We know that the correct answer for a tan in this range is between 1 and 10, so we don't need to adjust the decimal place.





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

Move the cursor to 63 on the forward T scale. This scale increases from left to right.



The cursor is now at 1.96 on the C scale.

We know that the correct answer for a tan in this range is between 1 and 10, so we don't need to adjust the decimal place.

12. Sin(x) and tan(x) for angles between 0.6° and 5.7° (using the ST and C scales)

Move the cursor to 1.5 on the ST scale.

The cursor is now at 2.62 on the C scale.

We know that the correct answer for a sin in this range is between 0.01 and 0.1, so we adjust the decimal place to get 0.0262.

13. Sin(x) and tan(x) for other small angles (using C and D scales)

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.

Squares and Square Roots

14. Square (uses C and B scales)

Move the cursor to 4.7 on the C scale.

The cursor is now at 2.2 on the B scale.

We know that the correct answer is near 52 = 25, so we adjust the decimal place to get 22.

15. Square Root (uses C and B scales)

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 702 = 4900, which is in the ballpark of 4500. Therefore we adjust the decimal point to get a result of 67.



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 202 = 400, which is in the ballpark of 450. Therefore we adjust the decimal point to get a result of 21.2



Cubes and Cube Roots

16. Cube (uses D and K scales)

Move the cursor to 4.7 on the D scale.

The cursor is now at 1.04 on the K scale.

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.



17. Cube Root (uses D and K scales)

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.

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

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

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.



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



Set the cursor over 1.04 on the LL1 scale.

Flip the SR over

(on some advanced SR's the LL0, LL1, LL2 and LL3 will be on the same side)

(on some advanced SR's the LL0, LL1, LL2 and LL3 will be on the same side) The cursor is at 50.5 on the LL3 scale.

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



Set the cursor over 1.002 on the LL0 scale.

The cursor is on 7.4 on the LL3 scale. This is the correct answer.

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

Set the cursor to 1.002 on the LL0 scale. This the same as 1.002 1

On LL1, the cursor is at 1.002 10 , or 1.020

, or 1.020 On LL2, the cursor is at 1.002 100 , or 1.22.

, or 1.22. On LL3, the cursor is at 1.002 1000 , or 7.37.

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

RED

-LL2

-LL3

Move the cursor to <0.75 on the -LL2 scale.

on the scale. The cursor is at <0.056 on the -LL3 scale. This is the correct answer.

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

Set the cursor hairline to 1.175 on the LL2 scale.

The hairline is over 5 on the LL3 scale. This is 1.175 10

The cursor is also over .2 on the -LL3 scale. This is the reciprocal of 1.175 -10 or 1/1.175 10 and is the correct answer.

21. Finding the 10th Root (N>1)

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.

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

-LL3

-LL1

Set the cursor to 0.15 on the -LL3 scale.

scale. Flip the SR to the other side to read the LL1 scale.

The cursor reads 0.9812 on the -LL1 scale. This is the correct answer.

23. Arbitrary Powers (Staying on Same LL Scale)

Set the cursor hairline to 9.1 on the LL3 scale.

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.



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



Set the cursor to 230 on the LL3 scale.

Since we're raising to a power that's less than 1, we have to go left on the LL scale.

Slide the rightmost Index '1' on the C scale to the cursor hairline.

Move the cursor to 4.5 on the C scale.

The cursor is now at 11.6 on the LL3 scale. This is close to the correct answer of 11.56.

Set the cursor to 0.78 on the -LL2 scale.

Slide the leftmost Index '1' on the C scale to the cursor hairline.

Move the cursor to 3.4 on the C scale.

The cursor is now at 0.43 on the -LL2 scale. This is the correct answer.

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

Set the cursor to 0.78 on the -LL2 scale.

Since we're raising to a power that's less than 1, we have to go left on the LL scale.

Slide the rightmost Index '1' on the C scale to the cursor hairline.

Move the cursor to 4.5 on the C scale.

The cursor is now at 0.894 on the -LL2 scale. This is the correct answer.



24. Arbitrary Powers (Switching LL Scales)

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.

Calculate (1.9 ) 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.

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.



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.

Calculate (12 ) 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.

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.



-LL1

-LL3

We can reinterpret this problem as:

Calculate (0.99 5.6 ) 100

Because 5.6 x 100 = 560

Calculate (0.99 ) Because 5.6 x 100 = 560 Set the cursor Hairline to <0.99 on the -LL1 scale.

on the scale. Shift the leftmost Index '1' on the C scale to the cursor hairline.

Move the cursor to 5.6 on the C scale.

The cursor is now at <0.99 5.6 = <.9452 on the -LL1 scale.

= on the scale. Since we also want to raise this to the power of 100, we look "two scales higher", or the -LL3 scale.

scale. The cursor is now at <0.0036 on the -LL3 scale. This is the correct answer.

25. Log-Log Approximations

In this case, if we use the approximation (1 + d) p = 1 + d p, then:

d = 0.00012, and

p = 34

= 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.