slide rule | Teaching Calculus

As I wrote last week, I found an old spiral slide rule last summer. It is about the size of a rolling pin and in fact has a handle like a rolling pin’s at the bottom. The device consists of a short wide cylinder that slides around, and up or down on a longer thin cylinder.

: Figure 1

: Figure 2

The short wide cylinder has a spiral common (base 10) logarithm scale starting at the top at the 100 mark after the words “slide rule” (see Figure 3). The scale runs around the cylinder 50 times ending precisely under the starting mark. By my measurement the scale is about 511 inches or 42.6 feet long. (1.30 meters). The scale is marked for 4 digits reading with a 5th digit that can be reasonably estimated. (By way of comparison, the common 10-inch slide rule scale discussed last week allows for 2 digits reading with the third digit estimated.) These are the mantissas of the common logarithms from 1 at the zero point (since log (1) = 0) to 1.0 (log (10) = 1) at the lower end.

The thin cylinder is marked with several formulas and other information including a table of natural sines from 0 to 45 degrees, from which you can have the value of any trig function if you’re clever enough. This cylinder is not used for calculations; it is there to allow the wider cylinder to move.

There are also two pointers. The shorter one is attached to the bottom and fixed. The cylinder is moved into position for this pointer. The longer pointer is attached to the thin cylinder and can be moved to the position needed – up, down left or right. Both the top end and the bottom end of the long pointer may be used. The pointers are made to slide past each other if necessary. If the long pointer covers the number needed the other side of it may be used instead (just don’t switch back-and-forth in the same computation).

Here is how it works. For the multiplication problem 15.115 x 439.65.

For the moment we ignore the decimal points.

: Figure 3: Setting top marker at 1 and the first factor 1.5115.

: Figure 4: Setting the second factor 4.3965 and reading the product 6.645

The top, “T” shaped, pointer is moved to the start value after “Slide rule.”
The bottom pointer is first set at 15115 (the 151 is marked, the next 1 is the first mark following 151 and the 5 is estimated. See Figure 3 (Click to enlarge). The distance between the two measured almost 9 times around the cylinder is log (1.5115)
Next the cylinder is moved without disturbing the pointers so that the top pointer is at 4.3965 and again estimating the last digit. Figure 4 upper long pointer.
The product is at the fixed pointer: 6.645 Figure 4 lower pointer.
Finally, we put the decimal in the proper place. The product is 6645.
The full value is 6645.30975 by calculator. So the answer is correct to 4 digits, good enough for most practical work.

By moving the top pointer to log (4.3965) and using the pointers to add to it log (1.5115) we have performed the calculation log (1.5115) + log (4.3965) = log (1.5115. x 4.3965) = log (6.645)

To divide the procedure is reversed. $\frac{{6645}}{{439.65}}$

Set the fixed pointer to the dividend and move the top pointer to one of the divisors. (Figure 4)
Without moving the pointers, move the cylinder so the top pointer is at 1.
The quotient is at the fixed pointer (Figure 3)
Adjust the decimal point for the quotient: 15.115.

If the cylinder is moved so that the pointer is off the bottom of the cylinder, the bottom pointer is used instead of the top. (This is the reason it is directly below the top pointer.)

If this seems like a lot of trouble, it is. But remember, a working computer was not available until near the end of World War II and filled a room. Electronic calculators were not available until around 1970. Computations before then were done by hand or with logarithms.

When I was in college in the early 1960s, I worked for an engineer on my summer vacations. My boss had and occasionally used a large table of logarithms. Large, as in a whole book! As I recall, it was good without interpolating for at least 6 digits accuracy. I used a large desktop mechanical calculator that had a hand crank to do calculations. Hence the term “crank out the answer.”

As for teaching: In those old days before about 1970, you spent 3 to 4 weeks in Algebra 2 teaching students how to use logarithm table and compute with logarithms. I gave that up when the students started using calculators to do the adding and subtracting of their logarithms.

The one advantage of the spiral slide rule is that it doesn’t need batteries!

Happy Holidays!

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 19^th 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.