written by Walter Banzhaf, P.E.
It was fall 1961 and I’d waited until my senior year of high school to take chemistry, having already taken biology and physics. Math didn’t present too many challenges so far, but I knew that logarithms would be part of chemistry, and logarithms were a strange concept to my 15-year-old brain. Sure enough, the topic of “pH” came along, and what a strange and inconvenient introduction to logarithms it was. First, why call this new term “pH”? Turns out because the pH stands for the “potential of Hydrogen”, whatever that might mean. I’d heard of exponents like 103 and x2 but the abstract way logarithms were introduced to me then in chemistry was painful to my adolescent brain. Here’s how “pH” was explained, as best I can recall: “pH is the negative logarithm of the hydrogen ion concentration in an aqueous solution”
It got worse in a hurry; instead of exponents like 2 or 5 or -3, now the exponents were evil numbers that weren’t always integers. So now we were supposed to understand 103.4 and 10-6.5. Mr. Slutzsky, who the class felt was the nicest chemistry teacher ever, explained that a logarithm was a synonym for “power” or “exponent”. That someone decided with two excellent words already well established in English for the small number “x” next to the zero in 10x, a third word with a strange name was needed, made no sense to an honors chemistry class.
Mr. Slutzky wasn’t bothered at all that the class didn’t like the word “logarithm”, and he proceeded to have each student master using positive and negative exponents (OK, yeah, logarithms) without using the huge yellow teaching tool at the front of the classroom. He called it a “slide rule”; that made sense in that it had a part in the middle that slid left or right as needed between two fixed parts. To make it useful there were numbers all over the sliding part and the two fixed parts. So that an entire class of 30-some-odd students (a few were really odd) could easily see the numbers the teaching slide rule was 7 feet long with a yellow-colored background and it hung at the top of the blackboard. It looked like this:
But for the moment, we didn’t need to use a slide rule; that would come in a week or so. We all knew about scientific notation and logarithms that were integers, so for now we learned about pH values that were integers. We learned that pure water has a pH of 7, meaning that one in every 107 water molecules had split into an H+ ion and an OH- ion, and that pure water was neither acidic nor basic (bases are said to be basic). Shortly thereafter we learned, for example, that black coffee had a pH of 5, and that meant the hydrogen ion concentration was 10-5 which meant that coffee has 100 times more H+ ions than pure water. Black coffee is a mild acid. A solution of baking soda has a pH of 9, so its hydrogen ion concentration was 10-9. This meant the concentration of H+ ions is (1/100) that of pure water and that a baking soda solution was basic. So far, we had no need to use a slide rule or learn from the teaching slide rule above the blackboard.
It didn’t take too long before everyone in the class was competent and comfortable with integer values of pH. Smaller slide rules about 12 inches long were given to us to use at our desks to be used when the pH value was not an integer. At this point in the chemistry course, we needed to learn about pH values that were not integers. Mr. Slutzky reminded us that 100.5 was the square root of 10, equal to 3.162, 101 was equal to 10, and that 101.5 was equal to 101 x 100.5 = 10 x 3.162 = 31.62. My reaction to this revelation was “wow”, I think I get it. This was the moment to teach the class how to use a slide rule for non-integer exponents.
At the blackboard, using the 7-foot-long teaching slide rule, we were shown how to use the C scale and L scale, when given “x” to determine the value of 10x. Similarly when given a number (like 316.2) we learned to determine the value of x (2.5). Knowing that 316.2 = 3.162 x 100, we realized that log(3.162) = 0.5 and log(100) = 2. Adding 0.5 and 2 resulted in 2.5, the log of 316.2. The joy and excitement of mastering these concepts was dampened some by realizing that it was up to the slide rule “operator” to keep track of powers of 10. Turns out this is true for not just logs, but for basic multiplying and dividing. 8/2 = 4, but 800/2 = 400, 80/200 = 8/20 = 0.4.
The chore of keeping track of the powers of 10 was offset by the benefit of visually understanding how trigonometric functions varied as the angle changes. For example, how the sine of numbers from 0 to 90 degrees varies. By looking at the “S” (sine) scale and the C (or D) scale (lowest value 0.1, highest value 1.0), one can see why the sine function increases rapidly for small angles like 10 or 20 degrees, and its rate of increase really slows down for angles from 70 to 90 degrees.
Looking at the “S” (for sine) scale below, notice under 90 (degrees) is 1 (on the C scale). Under 70 (degrees) is 0.94 (on the C scale), and under 30 (degrees) is 0.5 (on the C scale). Wait a minute! Under 30 degrees is “5” , not 0.5. You are right; it’s up to YOU to keep track of whether “5” on the C scale means 0.5, or 5, or 0.05, etc.
Since the sine function varies between 0 and 1.0 as the angle varies between 0 and 90 degrees, under 30 degrees on the “S” scale is indeed 0.5. By the way, did you notice that on the “S” scale next to “70” degrees is “<20 degrees in red”. The slide rule lacks a “cosine” scale, but the “S” scale numbers in red tell us that sin(70°) = cos(20°).
A slide rule user, seeking the cosine of 17.4°, would first have to subtract 17.4° from 90°. The difference being 72. 6° (done without using the slide rule because slide rules add and subtract logarithms, not numbers). Then, using the S scale (for sine), put the cursor on (72.6°) which gives the cosine of (17.4°), or “about” 0.954 on the C scale. This example shows a limitation of all slide rules: at most three significant figures can be determined for most operations. Sure, a scientific calculator could tell us the cosine of 17.4° is 0.9542403285, but for many engineering problems three significant figures is very satisfactory.
Now we’re going to look at the tangent scale(s). The upper “T” scale ends in 45 degrees, and the smallest angle shown above is 30 degrees. The lower “T” scale starts at 80 degrees below and ends at 84.3 degrees, because the tangent of 90 degrees is not “wow” as shown in the table below. The tangent of 84.3 degrees is not 1, but in fact is 10. The tangent of 90 degrees is infinity. A math purist would say the limit as the angle approaches 90 degrees the tangent approaches infinity.
Looking at the “T” (for tangent) scale above, notice under 84.3° is 10 (looks like “1” on the C scale). Under 80° is 0.57 (really 5.7 on the C scale), and under 30° is 0.5 (on the C scale). Now, let’s look at the “ST” scale. It ends in 5.72°. Turns out the sine and tangent of “small” angles are so close to being equal that on a slide rule the values or essentially the same. For example, sin(5°) = 0.0871557, and tan(5°) = 0.0874887. The tan(5°) is larger than sin(5°) by only 0.0003329, a tiny difference; that is why a single ST scale works well for both sine and tangent of small angles.
What is up with the “C” scale above (5, 6, 7, 8, 9, 1??)?
Great question; the C scale as it appears varies from 1 to 1, but the second 1 is really 10. Or, the first 1 is really 0.1, so the C scale can be thought of as (0.1, 0.2, …. 1.0) or as (1, 2, …. 10). As with most of the scales on a “loaded for bear, or scientific” slide rule good for engineering students, it is up to the operator to keep track of the decimal point and what the correct value of an answer is.
Below is a picture of the Pickett “log-log-duplex-deci-trig” slide rule I received as a high school graduation present in 1962. It took a long time to learn how to use it for nearly any task it was capable of, including hyperbolic trigonometric functions (useful for radio frequency transmission lines). It was carried to lectures and laboratories in its leather case that hung from my belt, and clearly differentiated me as a student of engineering among the many majors at R.P.I. Before exams I lubricated my aluminum slide rule’s slider with a light coating of petroleum jelly, while students with slide rules made of bamboo used talcum powder. I don’t think adding lubrication helped much if at all to speed the calculations, but it was a time-honored tradition among engineering students at the time.
Above is the side of the slide rule with hyperbolic trigonometric functions (TH and SH scales on the slider) and base 10 and base “e” scales (L and Ln). The sine and tangent scales are on the opposite side of the slide rule.
I began my college teaching career in 1977 at the University of Hartford, 11 years after earning my B.Eng.E.E. degree. I taught electronic engineering technology and related math courses. In every classroom used by the faculty of this program there was a rather large (7 feet long) slide rule hung by two hooks at the top of the blackboard. I thought it strange that these teaching slide rules 16 years later were the same as the one in my high school chemistry classroom in 1961.
Even in 1977 the slide rule was obsolete, and all faculty and students had scientific calculators which became available in the early 1970s. Mine was a Texas Instruments SR-50; SR was for “slide rule”. The venerable faculty were comfortable having antiques such as teaching slide rules in the classroom, although never was anything taught about slide rules, or were they used in any way except to accumulate chalk dust. An administrative decision was made to remove all the teaching slide rules from classrooms, and they were placed temporarily in a storeroom. Soon they were offered to faculty to take for any purpose. Of course, I took one home to treasure and to remind me of the tool that had served me well in my undergraduate years over a decade earlier. A colleague, 20 years my senior, took one as well.
About a week later he proudly brought to his office a desktop bookshelf he had just built to hold all his books, class notes, handouts for lectures and lab reports to be graded. I was horrified to see that instead of building it using lumber, he cannibalized the 7-foot-long slide rule and sawed its wood into pieces for the bookshelf. It stood proudly (for him, not for me) on his desk for many years, the yellow background color and the numerous scales (C, D, L, Ln, S, ST, and √ ). For years afterward when he and I met in his office to discuss curriculum, books for courses or new lab experiments we were developing, I had to avert my eyes to avoid the pain of seeing the yellow bookshelf he made by cannibalizing a slide rule.
