Slide Rules

Last summer I bought myself a new calculator. Well, it’s actually an old calculator manufactured in 1914 (if I’m reading the correct information engraved on it). It is called a Fuller Spiral Slide Rule.

Before looking at that, I’ll try to explain how the more standard (flat) slide rule works. Next week, I show you the spiral slide rule. Hopefully, you and your students will find this historical note interesting and it will show you how logarithms used to be used. Slide rules were the standard for mathematics, science, and engineering students from the 19th century up to about 1970 when electronic calculators took over. Everyone in STEM fields used them, there was no other choice.

If you were in high school after 1970 you probably never had to learn how to use a slide rule, but you’re probably heard of them.

Let’s look at the standard slide rule. You can find a working virtual model here. (The model doesn’t work on an iPad; you’ll have to use it on a computer.) This It is called a 10-inch slide rule because the scales are 10 inches long.

You can move the slide (center section) with your mouse. You can also move the piece withthe screws top and bottom, called the cursor. The cursor is used to read non-adjacent scales and scales on the other side. (Click in the upper right to see the other side). In a real slide rule the slide can be turned over and used with the other side if necessary.

The slide rule only gives the digits of the answer. The decimal point must be determined separately.

The main scales are the C and D scales. These scales are identical and are marked so that the distance from the left end is the mantissa of the common (base 10) logarithm of the number on the scale. The mantissa is the decimal part of the logarithm. The numbers on the C and D scales are all between 0  (= log (1)) and 1 (= log(10)). The scales allow for 3-digit accuracy on the left up to 4 where the spacing allows for only 2-digits. In each case an extra digit may be estimated.

To multiply: slide the 1 on the C scale until it is above the first factor on the D scale. Then find the second factor on the C scale and the number below it on the D scale is the product. Figure 1 shows the computation of 4 x 2 = 8. Remember the distance from the ends are really logarithms, so what you are really doing is log (4) + log (2) = log (4 X 2) = log (8).

Figure 1: Showing 4 x 2 = 8

Other products may also be seen such as 4 x 1.5 = 6, or 40 x 17.5 = 700, etc.

If the second factor is off the right end of the scale; put the 1 on the right side of the C scale over the first factor and the product will be under the second factor. The second figure shows 4 x 5 = 20 (and other products with 4 as a factor). Remember you need to properly place the decimal point.

Figure 2: Showing 4 x 5 = 20

Division is just the reverse: 8 divided by 2 is done by putting the 2 over the 8 and reading the quotient, 4, under the 1 on the C scale. (See figure 1 again). The scales are interchangeable so you could also put the 8 over the 2 (looks better) and find the quotient on the C scale over the 1 on the D scale. Can you find 60 divided by 15 = 4? On figure 2 you can see 2 divided by 5 = 0.4 or 2800 divided by 0.07 = 40,000.

Chain computations can be done by using the cursor to mark (without reading) one answer and then move on to the next, either multiplying or dividing.

The other scales give other functions. The lower scale marked with a radical sign gives square roots. Move the cursor to 2 and read the square root of two (1.414) on the top part of the scale and the square root of 20 (4.472) on the lower part.

Figure 3: Showing \displaystyle \sqrt{2}\approx 1.414 or \displaystyle \sqrt{{20}}\approx 4.47

The S scale gives the sines and cosines of numbers in degrees. Reading from the left the black numbers are for sines and reading from the right the red numbers are for cosines. See figure 3. The cursor is on 60/30 for the sin(30) = cos(60) the value is on the C scale 0.5 (remember you need to supply the decimal). Reading in the other direction the sin-1(0.5) = 30 or cos-1(0.5) = 60.

Figure 4: Showing sin(30) = 0.5 = cos(60) or arcsin(0.5) = 30 or arccos(0.5) = 60.

The CF and DF scales are “folded” at \displaystyle \pi to make multiplying by \displaystyle \pi easier. Computations are done the same way. The CI, DI, CIF, and DIF give the reciprocal (I for inverse) and are read right to left.  T is for tangents a double scale from 0 to 45 degrees on the top and 45 degrees on up at the bottom. I’ll leave the others for you to research.

So, that’s today’s history lesson. Next week, the Spiral Slide Rule – a little more complicated, but a lot more accurate.

Spiral Slide Rule









2 thoughts on “Slide Rules

  1. Pingback: Then and Now: Looking Back Back on 30 Years of High School Math Teaching | Math, Teaching, and Teaching Math

  2. Hi Lin,
    I have my fathers slide rule( case, magnifier and all) he used when he worked with NASA in Huntsville, AL I also took a 3 week slide rule class in high school (1973) . Barely remember how to use it though.


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