Let It Slide

My older brother did a fair bit of high school math and science work assisted by a slide rule. I was too young at the time to understand the nuances of this remarkable tool. By the time I was in high school, the pocket calculator was the norm, if any calculation assistance was allowed at all. I recall my chemistry teacher encouraged us to use a calculator in his class because “this is hard enough without messing up the arithmetic”.

Between an undetermined number of calculators of various complexity and more recently, the entire internet full of specific purpose calculators, I never had cause to acquire or learn about the slide rule. I am at a point in my life (an age?) where such mechanical devices are fascinating.

A couple of years ago, I was watching something on YouTube and the subject of sliderules came up. Shown in that video was a sliderule configuration that I found quite compelling, the circular slide rule. Specifically, the Russian KL-1. Turns out they are not stupid expensive and a bit of eBay shopping revealed one from the early 70’s for a price I was willing to pay.

A slide rule is, at it’s most basic, a device inscribed with markings spaced according to a logarithmic scale. Logarithms are their own subject and there is much more to it than this, but the layman can think of logaritms as a numeric series where each next major number is a multiple of the previous, such as 1, 10, 100, 1000, etc where each mark is 10 times the previous mark.

We use computers to calculate logarithms these days, but hundreds of years ago, math and science practicioners generated books full of logarithms so that they could be looked up in a table rather than calculated each time it was needed. John Napier, a Scottish polymath, is credited with introducing the use of logarithms to simplify other calculations. In 1614. So, not yesterday.

One of the most helpful properties of logarithms is that the logs of any two numbers added together will equal the log of those same two numbers multiplied together.

This has the effect of simplifying multiplication problems down to addition, though you need some way to know the logarithms.

For our purposes, b is 10, so we are working with base 10 logarithms.

The log of x plus the log of y equals the log of x times y.

For simplicity, lets use 2 and 3 for x and y.

Log(2) + Log(3) = Log(6)

Now, Log(2) = 0.30102999 and Log(3) = 0.47712125. Those added together is 0.77815125. Since we are using base 10, 10 to the power of 0.77815125 turns out to be….. 6!

The scales on our slide rule gives us an easy way to add those two scary small numbers because someone else marked the scale in logarithmic intervals, forming a lookup table of sorts.

For the moment, let’s focus on the C and D scales, shown by the green arrow. These are identical log base 10 scales. Logarithms for numbers less than 1 are negative numbers, so the scale starts at 1, shown by the red arrow. 2 is shown by the yellow arrow, 3 by blue and 4 by purple.

Next, lets say that our scale is some arbitrary number of units long. It doesn’t matter what units unless you are actually making a slide rule, which I want to do now. 🙂 As luck would have it, the slide rule in this picture has the L scale, which shows the actual numeric value of of the log shown in scales C and D. The L scale is linear, beginning at 0 and proceeding evenly to the right. For our purposes, we can consider the L scale to be a ruler.

Log 2 is 0.30102999. The distance from the red arrow to the yellow arrow on the L scale is 0.3 or so.

Log 3 is 0.47712125. The distance from the red arrow to the blue arrow is 0.47 or so.

Log 4 is 0.60205999. The distance from the red arrow to the purple arrow is 0.6 or so.

By itself, that seems maybe obvious. But what this means is that we can represent the logs of our various numbers by the physical distance they are from 1.

2 x 3 = 6

log(2) + log (3) = ?

0.30102999 + 0.47712125 = 0.77815125

Find 0.77815125 on the L scale (no arrow shown) and it is [drumroll] …. 6! 6ish, anyway, because we are really only able to see 0.775ish on the scale.

This reveals what people accustomed to the instant precision of electronic calculators may find challenging about slide rules. In practiced hands, they are very fast, but not super precise. However, they give an answer that is almost always going to be close enough for most purposes. If someone needs to figure out how many degrees 1/7th of a circle is so they can cut a wooden circle into 7 pieces, 360 / 7 = a hair less thatn 51.5 is close enough for a saw.

Multiplication on the sliderule shown is accomplished by sliding the inner movable scale until the multiplier on C scale lines up with the 1 on the D scale, then move the cursor (the clear slide with the line across it) to the multiplicant on the D scale, then read the result on the C scale.

I don’t have a classic slide rule, but I found a wonderful site with slide rule simulations.

The arrow on the left shows the multiplier 3 lined up with 1 on the D scale. The arrow on the right shows the cursor lined up with the multiplicant 2 on the D scale and the answer 6 revealed on the C scale.

Doing larger multidigit numbers can be done with fractional distances. 30 x 20 = 600 for example. Using 3.0 (times 10 in your head) and 2.0 (times 10 in your head) = 6.0 (times 100 in your head; 10 x 10). If you need large numbers with high precision, just use paper to do long form multiplication and use the slide rule to do the arithmetic in each step.

33 x 21 = 690-something-between-0-and-5. It’s 693 precisely.

Nothing says that the scales need to engraved on something straight. Enter the KL-1.

There are many other circular slide rules and even other models that emulate a pocket watch like this one does. In this case, this side has what is essentially the C/D scale printed on a movable face and two cursors. One cursor is fixed, here aligned with the 1 and the knob which turns the face. The other cursor is the red hand, which is moved with the other knob.

The procedure is slightly different, but still works under the exact same principles. The outer scale is essentially A scale in typical slide rule parlance. Turn the face until the multiplier aligns with the fixed cursor, 3.9 or 39 in this case.

Note that the inner scale is labeled C and the written instructions say it is to be used for basical calculation. If I use the inner scale, the multiplication math still works, but the outer scale has more divisions and is thus is more precise. The inner scale become more important used in division and square root calculations, covered a little later. The A and C scales correlate. The A scale numbers are the squares of the C scale numbers. I have NO understanding how this helps with division. I have much to learn.

The hand is rotated to 1 on the scale. These two steps are equivalent to sliding the scale to line up the mulipler on the C scale with 1 on the D scale.

Then the scale is rotated until the multiplicant, 2.1 or 21 here, is aligned with the red hand. The result is read at the fixed cursor, a hair less than 8.25 or 825. The precise answer is 8.19 or 819

The other side of the KL-1. Interestingly, written instructions call this side 1. It has three scales and another red hand cursor. The scales don’t move, but the hand does. The outer scale is the inverse of the inner C scale on side 2 and is called the DI scale. The other two are the S and T scales, used for sine and tangent trigonometry functions that I don’t yet know how to use. Note that the T scale is a 630 degree spiral. I have much to learn.

Division is similar in that Log (x) minus Log (y) = Log (x/y). The procedure for division is similar to multiplication.

Rotate the outer scale to put the dividend under the fixed cursor, 16 in this example.

Rotate the red cursor to the divisor on the inner scale, 2 in this case.

Rotate the inner scale to put the dividend, 16, under the red cursor. The quotient result, 8, is read on the inner scale under the fixed cursor.

125 / 65 = ?

Put 125 in the outer scale under the fixed cursor and 65 in the inner scale under the red cursor.

Rotate the scale to put the outer scale 125 under the red cursor and read the quotient from the inner scale under the fixed cursor. 125 / 65 = 1.92. The precise answer is 1.92307692, but I think 1.92 is definitely close enough.

The markings on the inner scale show greater precision (0.12 per division) for values between 1 and 2, slightly less from 2 to 6 (0.25 per division) and even less from 6 to 10 (0.5 per division). This might be due to the range of expected results. The closer to 1, the more precision needed. However, I think it is really a horological artifact. The divisions are physically similar in size, but the logarithmic intervals are progressively smaller as the values go up. The lines are all about the same size in degrees of arc, but this is simply room for more divisions between 1 and 2 than there is between 6 and 10.