Slide Rules | Computer Museum | University of Waterloo

by Ayoung Eun and Charlotte Wipp

Slide Rules!

What is this exhibit?

From the late 19th century to the 1970s, Analog slide rules were among the most common computing devices. Cheap to buy, quick to use. However, by the mid 1970s, digital pocket-sized calculators had replaced slide rules as the tool of choice for most scientists, engineers, and mathematicians.

Unique Design You may think that the slide rules only have traditional "ruler" shape. That's wrong! Just like computers, they have more than one geometric shape- They could also be circular or cylindrical.

Parts of a Slide Rule

The Rise and Fall of Slide Rules

Scottish mathematician John Napier developed logarithms in the early 17th century. He also used logarithms to create physical numbering rods known as “Napier’s bones” that simplified multiplication and division of large numbers. However, it was English mathematician William Oughtred who first combined different logarithm scales in the 1620s to create the small, portable, analog computing devices known as slide rules to speed up and simplify various arithmetic operations.

Over time, the popularity of slide rules increased, and they were standard equipment for many occupations. The slide rules that you can see in many of our artifacts used scales meant for general-purpose arithmetic and trigonometry calculations, but many slide rules with different scales were created for specific users such as doctors, engineers, or pilots.

In the mid-1970s, the arrival of portable electronic calculators decreased demand for slide rules. Although the first digital calculators were very expensive and could cost hundreds of dollars, prices fell quickly and by the end of the decade the electronic technology had replaced slide rules as the mathematical tool of choice for almost everyone. The new pocket-sized devices contained microprocessors, and some could be programmed by users to include different mathematical functions. Some calculators, like the Texas Instruments SR-52, could even save short programs to small magnetic cards for later use.

An example of the Napier's bones calculating tool which performs multiplications and divisions by addition and subtraction.

Parts of a Slide Rule

The slide rule is made of several parts: the top and bottom body and the movable slide in between. Both the body and slide contain scales.

The hairline is the movable, clear part. It is used to align values and improve usability.

Many different scales can be used on a slide rule. Some are common, and some are tailored to a specific use or profession. Here is a list of some common scales and their uses:

A and B: Scales of squares.
- These scales have the squares of the D and C scales and can also be used for square roots. The first half of the scale represents 1-10, and the second from 10 to 100, although the significant digits are not on the scale and must be handled mentally.
- In the first photo to the right, A and B scales are highlighted in blue, while the D and C scales are in green.
C and D: Fundamental scales.
- Many of the primary slide rule calculations are done with the C and D scales, which run from 1-10 in a standard log scale format.
- The D and C scales are highlighted in blue.
CI and DI: Reciprocal scales.
- These scales are helpful to simplify division and if the slide extends far past the body.
- The DI and CI scales are highlighted in blue in the photos.
E: Log-log scale.
- The scale for exponential powers. On some rules it is known as the LL3 scale.
- The exponential scales are highlighted in blue
K: Scale of Cubes.
- The scale can be used for cubes and cube roots. It is divided into three sections: 1 to 10, 10 to 100, and 100 to 1000. Like the A scale, significant digits are not on the scale and must be handled mentally by the user.
- The K Scale is highlighted in blue while the D scale is highlighted in green
L: Log scale.
- A linear, non-logarithm scale, often used with C and D scales to determine the logarithmic relation to log base 10.
- The L Scale is highlighted in blue, while the C and D scales are highlighted in green.
S: Sine and cosine scale.
- The scale often presents cosine as decreasing and sine as increasing and usually presents angles from 5.7 to 90 degrees.
- The S Scale is highlighted in blue
ST: Sines and tangents scale.
- The scale exists to work with smaller angles in the range of 0.57 to 5.7 degrees. This scale is used with the C and D scales for values between 0.01 and 0.1.
- The ST Scale is highlighted in blue
T: Tangent and cotangent scale.
- The scale typically ranges from 5.7 to 45 degrees, although sometimes the scale includes values up to 84.5 degrees which run along the bottom in the opposite direction.
- The T Scale is highlighted in blue
  .

Scales

University of Waterloo Computer Museum. The A and B Scales are highlighted in blue, and the C and D scales are in green. 2024.

University of Waterloo Computer Museum. C and D Scales. 2024.

University of Waterloo Computer Museum. CI Scale. 2024.

University of Waterloo Computer Museum. The DI Scale is highlighted in blue. 2024.

University of Waterloo Computer Museum. K Scale. 2024.

More Scales

University of Waterloo Computer Museum. The exponential Scales are highlighted in blue. 2024.

University of Waterloo Computer Museum. The L Scale is highlighted in blue, while the C and D scales are highlighted in green. 2024.

University of Waterloo Computer Museum. The S Scale is highlighted in blue. 2024.

University of Waterloo Computer Museum. The ST Scale is highlighted in blue. 2024.

University of Waterloo Computer Museum. The T Scale is highlighted in blue. 2024.

Using a slide rule

Multiplication and Division

Multiplication X*Y

For multiplication, use the identical C and D scales. The only difference between the scales is that C is on the slide and D is the body.

Choose one value of X or Y to be represented by the D scale, the example will choose X for this.
Move the hairline to X on the D scale, then move the slide so that the C left end of the scale BEGINS at the hairline. Slide the hairline to Y on the C scale. If Y on the C scale exceeds all numbers on the D scale, align the right END of the slide with X on the D scale instead.
Read the resulting number that lines up with Y on the D scale.
1. If the C scale’s right end was used, increase the answer by a magnitude of 10.

Division X/Y

For division use the same identical C and D scales. This will be the same process as multiplication, but backwards.

The divisor will be represented by the C scale, in this example it will be Y, and the dividend will be on the D scale.

Move the hairline to X on the D scale, then move C so Y lines up with X.

Read the solution off the D scale where it lines up with the START of the slide (C scale).

Logarithms

Log(X)

To solve the logarithm of X, the D and L scales are used:

Move the hairline to X on the D scale.

The corresponding value on the L scale is the logarithm of X.

However, you must also keep track of the significant digits in X via the exponent in scientific notation to complete the answer!

To understand more about logarithms, exponents, and scientific notation, visit this page.

Trigonometric Functions

Sin(x)

The D scale will represent the resulting answer, X will be the values increasing from left to right on the S scale.

Line up the beginning and ends of the scales, the slide will not move for trigonometric functions.

Line up the hairline with X along the values that increase from left to right on the S scale.

Find the corresponding value on the D scale to X.
Remember that all values of sin(x) must exist between 0 and 1, change the answer to the proper magnitude.

Cos(x)

The D scale will represent the resulting answer, X will be the values increasing from right to left on the S scale.

Line up the beginning and ends of the scales, the slide will not move for trigonometric functions.

Line up the hairline with X along the values that increase from right to left on the S scale.

Find the corresponding value on the D scale to X.
Remember that all values of cos(x) must exist between 0 and 1, change the answer to the proper magnitude.

Tan(X)

The D scale will represent the resulting answer, X will be found on the T scale.

Line up the beginning and ends of the scales, the slide does not move for trigonometric functions.

Line up the hairline with X on the T scale.

Find the corresponding value on the D scale to X.

Squares/Square Roots

The A scale is the square of the D scale values, and the B scale is the square of the C scale values.

The A and B scales go through two iterations of 1-10 where the right half represents the values 10-100 and the left is 1-10. Significant digits that are adjusted to fit on the scale are handled mentally.

Square X²

Position the hairline at X on the C or D scale, if the value must be changed to scientific notation, keep in mind if it gained or lost an order of magnitude.

Identify the corresponding value on the A or B scale. Remember to adjust the significant digits.

Square roots √X

Position the hairline at X on the A or B scale, if the value must be changed to scientific notation, keep in mind if it gained or lost a decimal place for the answer.

Identify the corresponding value on the C or D scale, and correctly change the significant digits if needed.

Cubes/ Cube Roots

For values of X, the K scale represents D values cubed.

The K scale goes through three iterations of 1-10, split into magnitudes of 1-10, 10-100, and 100-1000 from left to right, corresponding to 101, 102 and 103. Significant digits that are adjusted to fit on the scale are handled mentally.

Cubes

Find X on the D scale and move the hairline to its position.

The cubed solution for X will be the corresponding value on the K scale.

Cube roots

Find X on the K scale, based on its number of digits, and move the hairline to that position.

The cube root of X will be the corresponding value on the D scale.

Roots of Quadratic Equations

Finding roots (x-p)(x-q)=0

From the initial equation ax²+bx+c=0 we will be factoring to (x-p)(x-q)=0.

Use the CI and D scales to find the roots of a quadratic equation, presuming the roots are real.

Move the hairline to the c value on the D scale.

Align the left beginning of the slide with the hairline. The b value may exceed most numbers on the D scale, in this case align the right end of the slide with the hairline.

The product of any pair of numbers on the D and CI scale is equal to c.
1. Notice that CI goes from <10 left to <1 right and D goes from 1 left to 10 right.
For a pair of values (q and p) from D and CI to be roots of the quadratic, they must sum to b. Additionally, the signs of p and q can be either negative or positive.

W.H. Breithaupt's Slide Rule

William Henry Breithaupt was an engineer, entrepreneur, and community leader. First educated in Berlin, Ontario (now Kitchener),his engineering career began in the 1880s, as he worked across the American Midwest, designing railway bridges and tunnels. A cylindrical slide rule such as this one would have been enormously useful for scientific and financial calculations. In 1901 he returned to Waterloo to help manage his family's local businesses and build the first electric railway in the region. He also helped found the Waterloo Historical Society and Grand River Conservation Authority. He is an inductee of the Waterloo Region Hall of Fame.

Keuffel & Esser slide rule

Edwin Thacher's 1881 patentdescribes a slide rule small enough to fit on a desk, but much more precise than a normal one the same width. Keuffel & Esser of New Yorkbegan making Thacher-style slide rules around 1887.

What Came After the Slide Rule?

In the mid-1970s, the arrival of portable electronic calculators decreased demand for slide rules. The new pocket-sized devices contained microprocessors, and some could be programmed by users to include different mathematical functions. Some calculators, like the Texas Instruments SR-52, could even save short programs to small magnetic cards for later use.

Can you measure up to these slide rule questions?

Multiply 3.5*4

Choose one of the two numbers to place on the D scale, the other will later be placed on the other on the C scale
1. It does not matter which number is chosen since the C and D scales are identical, except that C is the slide and D is the body
For this solution, move the hairline to 4 on the D scale
Then move the right end of the slide, with the C scale, to the hairline
1. To find the multiplied value, the value that matches up with 3.5, found on the C scale, will appear on the D scale
2. Since the end of the slide is placed on the hairline, the answer located on the D scale will gain a decimal place
  1. 1.4 becomes 14
3.5*4=14

3.5*4

University of Waterloo Computer Museum. Slide rule. 2024.

Solve Log(8)

Use the D scale on the bottom of the slide rule for the value inside the log (8) and the L scale to find the corresponding log value.
Place the hairline at 8 on the D scale
The log value will be where the hairline lines up with the L scale
Log(8)=0.9

Log(8)

University of Waterloo Computer Museum. Slide rule. 2024.

Factor x²+11x+28

Factor x²+11x+28=0 to find two numbers, u and v, that add to 11 and multiply to 28 using the D and CI scales

Move the hairline to 28 on the D scale
1. In this case there is no 28 on the D scale, so move the decimal one to the left and find 2.8 on the D scale
Place the left side of the slide, which includes CI, at the hairline

The product of any pair of number on scales D and CI will be equal to 28
1. Some products include: 8*3.5, 7*4, 28*14
2. Notice that CI goes from <10 left to <1 right and D goes from 1 left to 10 right
The pair of numbers, u and v, that sums to 11 will be the correct factors for x²+11x+28. Keep in mind when summing u and v that the values can be either negative or positive.
1. From some trial and error: 7*4=28 and 7+4=11
2. The solution will be (x+7)(x+4)=0 or x=-7,-4

D and CI scales

Solve Sin²(62)

Calculating sin(62)
1. Use the S scale for 62 degrees and the D scale to find the answer, for these scales the slide does not need to be adjusted, instead line up the beginnings/ends
2. Place the hairline on the S scale at 62, for sine the values increases from left to right, the other values on the scale that increases from right to left represent cosine
3. Find the corresponding D value at the position of the hairline, note that sin of a value cannot be greater than one
4. Sin(62)=0.88
Calculating 0.88²
1. Squaring a number involves the A and D scales
2. Before placing the value on the D scale, 0.88 must gain a decimal place and become 8.8 as there is no 0.88 on the D scale. Remember for later that an order of magnitude was gained
  1. 0.88 becomes 8.8 x 10
3. Place the hairline on 8.8 on the D scale
4. The corresponding value to 8.8 is located on the second half of the A scale, or the second iteration of 1-10.
  1. 8.8²=77
5. 8.8 gained a order of magnitude then the value was squared so that (8.8x10)²=77x10², to return to the original order of magnitude two decimal places must be lost
  1. 77 becomes 0.77, so 0.88²=0.77

Sin(62)

University of Waterloo Computer Museum. Slide rule. 2024.

0.88²

Slide rule

University of Waterloo Computer Museum. Slide rule. 2024.

Solve log([5²+4³]/5.2³)

First to solve the inside of the expression
1. 5²
  1. Using the A scale and D scale
  2. Place the hairline on 5 on the D scale
  3. The value lines up with the second order of magnitude on the A scale indicating the decimal place will be moved one to the right, so 5²=25
2. 4³
  1. Place the hairline on 4 on the D scale
  2. The cubed answer will be the corresponding value on the K scale, keeping in mind that the answer rests on the 2/3 iteration of 1-10 indicating that the answer will increase by an order of magnitude
    1. 6.4 on the K scale becomes 64 so that 4³=64
3. Solving the numerator 64*25
  1. Move the hairline over D
  2. Choosing either 60 or 25, we will choose 25 but it will appear as 2.5 on the scale
  3. Place the right side of the slide at the hairline since 64 (or 6.4) exceeds the end of the body
  4. Move the hairline to 6.4 on the C scale
  5. Since 64 and 25 both lost a decimal place increase the answer two decimal places, additionally the slide was placed at its end so another order of magnitude is added
    1. 6.4x10 x 2.5x10= 1.6x10³
    2. 1.6 becomes 1600, so that 64*25=1600
4. Solving 5.2³
  1. Place the hairline on 5.2 on the D scale
  2. The cubed answer will be the corresponding value on the K scale, keeping in mind that the answer rests on the 2/3 iteration of 1-10 indicating that the answer will increase by an order of magnitude
    1. 1.406 becomes 140.6, so 5.2³=140.6
Solving the inside of the log, 1600/140.6
1. The divisor will be the C scale and the dividend will be the D scale
2. Move the hairline to 1600 or 1.6 on the D scale, keeping in mind that 1600 becomes 1.6x10^-3
  1. Position 140.6 or 1.406 located on the C scale at 1.6 on the D scale, keeping in mind that 140.6 becomes 1.406x10^-2
3. The beginning of the C scale will line up with the resulting answer found on the D scale
  1. 1.600/1.406=1.14
4. Keep in mind the previously changed orders of magnitude to 10^-3 and 10^-2
  1. Find the difference between these values and change the decimal place of the previously found 1.14 value to restore the places lost
    1. 10^-2-10^-3=10, so 1.14x10 becomes 11.4
  2. 1600/140.6=11.4
Solving the log of 11.4
1. 11.4, the argument inside the log, will be represented by the D scale, while L represents the log scale
2. Place the hairline at 1.14, keeping in mind that 11.4 becomes 1.14x10^-1
3. The log value will be the corresponding position on the L scale, since the value on the D scale moved by a magnitude of one, the answer will have a 1 in front of it
  1. L scale reads 0.06, but with the loss of 1 decimal place it becomes 1.06
4. Log(11.4)=1.06