Slide Rule ABCs ... and Ds
Multiplication and Division with A/B or C/D scales
In our last post we found that a multiplication of two numbers could be performed by finding a logarithm for each of the numbers, add the logarithms together, and see what number has that sum as its own logarithm. This final number will be the product of the original two numbers. By having scales on a rule that are marked off according to the logarithms of the numbers on the scales, then sliding the scales next to each other can perform the same operation. The early slide rule was simply such a device. Below is a slide rule, from the early 1700s, that recently sold on eBay. It has two rules held together by brass stators, one attached to each rule, allowing them to slide against each other. Each rule has inch scales on the outer edges, and logarithmic scales on the inner edges, allowing for both length measurements and calculations.
The Logarithmic Scale
So how are the appropriate marks made for a logarithmic scale? The scale might run from 1 to 10, since log 1 = 0 and log 10 = 1. If we want the scale to be 10 inches in length, then we would place a mark for the number z at a distance (10 inches) × log z away from the left end. The result of such a procedure is shown in the next image. The red lines indicate a few intermediate numbers to give a feel for how to read the scales.
The slide rule shown next is a more modern version (1930s) of a beginner’s model, with two sets of logarithmic scales. The top two run from 1 to 10 to 100, and the bottom two run from 1 to 10. This is called the Mannheim scale arrangement, which became popular in the middle to late 1800s.
To multiply two numbers we can use one set of scales, either A and B, or C and D. Let’s take A and B first. Suppose we want to multiply two numbers, each of which are between 1 and 10, for example 2.3 × 6.8. To do so, …
slide the 1 on the B scale to line up with 2.3 on the A scale; you may wish to use the hairline on the cursor to line things up more precisely.
Slide the cursor to line up with 6.8 on the B scale using the hairline.
Follow the cursor hairline to view the number under the hairline on the A scale; this is the final answer: 15.6 (to three digits)
We can see that what is happening is that the distances being added correspond to the addition of logarithms:
Note that it may be hard to tell on the A scale whether the answer is closer to 15.5 or 15.6, due to the granularity of the scale one is reading, especially for numbers on the right-hand end of the scale. So, one may have to “guess” at that third decimal place. And this is one reason for the C/D scales. Since one decade (factor of 10) on these logarithmic scales is twice as long as those on the A/B scales, using the C/D scales to perform a calculation often can give a more accurate reading of that final digit.
But going back to our A/B-scale calculation, and noting that division is the “inverse” of multiplication, we can see from this same setting above that if we take log 15.6 and subtract log 6.8 we will arrive on the rule a distance of log 2.3 from the left end. So, setting two numbers opposite each other on A and B represents a division of the two numbers — “A” / “B” — and the result will be on the A scale opposite the 1 or 10 of the B scale, whichever can be most easily read. Hopefully this will become more clear in the next section; along with a little practice.
Setting Ratios
In the early days of the slide rule, the procedure of aligning two numbers opposite each other on the rule was referred to as “setting a ratio.” Let’s take another simple example for illustration. Let’s set the slide to calculate 3 / 4 using the A and B scales. To do so, move the slide to align 4 on B underneath 3 on A, as illustrated below. As per our earlier directions, the answer should lie on A, located above the 1 or the 10 on B. As the 1 is off scale to the left, we use the 10.
We see that opposite the 10 on A is 7.5 on B, which can be interpreted as 3/4 = 0.75. Another way to write this is as a ratio:
But now, take a look at our last image again. The result 7.5/10, or 0.75/1, could have been reached by a host of many other divisions. We see that combinations like 1.5/2, 4.5/6, 6/8, and 9/12 all give the same result — 7.5/10. That is, they are all equivalent ratios. In fact, for this setting, for any number on A one can find at a glance the number on B that produces a ratio of 75%.
This also allows one to quickly solve a problem like x = 23 × 7 / 12. Such a problem can be re-written as a ratio:
So we make 23 on A be opposite 12 on B. Then, without having to reset the slide, we will find x on A opposite 7 on B. I find x = 13.4. (By computer, 13.41667.) Slide rule users of the 1800s tended to be taught how to perform calculations in terms of the setting of ratios, as is found in many of the early books on slide rule use.
A/B vs. C/D
The C and D scales are each a “1-decade” scale, i.e., they go from 1 to 10, whereas the A and B scales are 2-decades (1 to 10 to 100) over the same length. Either set can be used to perform multiplication and division, so let’s perform our original multiplication calculation using the C/D scales this time. We begin in the same way as before…
slide the 1 on the C scale to line up with 2.3 on the D scale
slide the cursor to line up with 6.8 on the C scale. Oops! What just happened?
The C/D scales only go from 1 to 10, so the above operation takes us off scale!
No problem; since every decade of a logarithmic scale has the same spacing of divisions, we simply need to imagine starting the problem from the left and then keep track of our factors of 10.
So, let’s repeat the operation, but this time …
slide the 10 on the C scale to line up with 2.3 on the D scale instead of using the 1
slide the cursor (to the left!) to line up with the 6.8 on the C scale
now, follow the cursor line to view the number under the cursor on the D scale; this is the answer (within a factor of ten).
You’ll see that the answer that you read off will be 1.56 on the D scale. But because we lined up with the right-hand-edge of the scale rather than the left, there is an extra factor of ten involved. Also, since you were multiplying 2.3 and 6.8 you know that the answer should be something closer to 15, so the correct answer must be 15.6.
Note also that the final digit is much easier to discern than it was when using the A and B scales. Two things to learn here: The C/D scales can be read more accurately in most cases than the A/B scales. So, they’d be the “go-to” scales. Also, it is always important to take a guess at what the answer should be so that one gets the right power of ten in the end. The slide rule will keep track of the digits in the answer; the user will have to keep track of the powers of 10.
General Rule: To multiply, find the value of the first number on the D scale, then line up a “1” on the C scale with this first number, using whichever end of the C scale gives access to the second number (also on the C scale). Slide the cursor to the second number as found on the C scale. Follow the cursor line to read off the answer on the D scale directly below.
As for division?
General Rule: To divide, line up the cursor with the numerator on the D scale; slide the C scale so that the denominator also lines up with the cursor; the answer will be on the D scale directly below the “1” on the C scale.
If you think about the above operations for a while, you should realize that you are either adding logarithms to perform multiplications, or subtracting logarithms to perform divisions. Once you realize that that is what you’re doing, then you’ll be able to perform sequences of multiplications and divisions with lightening speed.
As an illustration, let’s estimate the average speed in miles per hour of the earth in its orbit (average radius = 93.0 million miles) about the sun using a slide rule. Set up the problem by gathering factors of ten as follows:
It is good practice and often useful to order the numbers in the numerator and denominator such that factors of near-equal value are above/below each other. First, this practice gives you an early impression of what the result will be. From our numbers above, we see that multiplying and dividing these numbers should give us an answer somewhat less than 9. Also, by performing a multiplication followed by a division and then repeating using the numbers from left to right, this tends to keep the result fairly centered on the rule. So, here are the next steps using the slide rule:
Move the cursor to 2 on the D scale
+ which, technically, is multiplying 1 by 2.Set the C scale to 2.4 at the cursor
+ thus dividing by 2.4, according to our rules; the answer thus far would be found on the D scale under the “1” on the C scale; but we don’t really need that answer at the moment.Move the cursor to π (3.142) on the C scale
+ thus multiplying the previous result by π.Set the C scale to 3.65 at the cursor
+ thus dividing the previous result by 3.65.Move the cursor to 9.3 on the C scale
+ thus multiplying the previous result by 9.30.
The result is read on the D scale under the cursor; it should be about 6.67. This is consistent with our estimate that it be somewhat less than 9.
The FINAL answer is this number times 10,000, or 66,700 mi/hr.
Compare with result from a computer calculation:
## [1] 2*pi*93.0e6/24/365 = 66705.0494940299By the way, I found that I could compute the answer on the slide rule in about the same amount of time it took me to type the above (and correct one typo) on the computer…
So when is it better to use A/B as opposed to C/D? As noted earlier, the C/D scales being longer per decade can provide more accurate readings of settings and results. So these are typically preferred. However, if a ratio is set on C/D, much of the possible results for that ratio are likely to be “off scale”, requiring a re-set of the slide. So, when looking for a quick answer for a particular ratio setting, using the A/B scale guarantees that the answer can be found somewhere on these scales, though factors of ten — as always — will need to be interpreted correctly.
Next time we will go over another — perhaps the major — purpose for having both the A/B scales and the C/D scales on a slide rule, namely finding squares and square roots. We’ll also discuss two other related scales often found on standard slide rules: the log scale L, and the cubic scale, K.