Calculating the Moon: The Slide Rule

Rocket science is a serious business. Clearly the popular phrase, “it’s not rocket science” has been proven true on many occasions. For one thing, there is quite a bit of math involved in rocket science. Look at the size of a rocket! It weighs 6.2 million pounds, carries thousands of pounds of fuel, equipment, and a few humans (“Saturn V”)! The rocket needs to launch at just the right angle and reenter the Earth’s atmosphere at yet another specific angle. Lives depended upon it, literally!

In the 1960s, when rockets were headed towards the moon, one tool was used perhaps more than any other.

In the 1960s, when rockets were headed towards the moon, one tool was used perhaps more than any other. Every engineer carried it and most students did as well. The tool was simple yet it allowed man to visit the moon and return safely to the Earth. That tool was the slide rule.

Today slide rules are seen as archaic, an artifact from a different time of pocket protectors and typewriters. However, as outdated as these little tools are, they represent a brilliant understanding of mathematics, its place within modern life, and the need and many uses for math. I’ve spent the past week or so researching and learning more about the slide rule, and how to use one. So, this week’s blog may become a little technical, but I’m not a mathematician, so keep that in mind.

The slide rule can claim three men as its inventor; the first, John Napier, invented the logarithmic function in 1617 (“Slide Rule History.”). In short, a logarithmic function describes how many of one number are multiplied to get another number (“Introduction to Logarithms.”). For example, 2 x 2 x 2 = 8. In this example, we multiply three 2s to get the answer 8. This same problem could be represented using a logarithmic function. For example, log2(8) = 3. The subscript 2, or, the base, is the number we multiply. 8 is the result. And the answer to the function is 3, the logarithm, because we multiply 2 three times. Therefore, the equation, 2 x 2 x 2 = 8, is equal to log2(8) = 3. They both describe the same thing. The logarithmic function can be stated as “the logarithm of 8 with base 2 is 3 (“Introduction to Logarithms.”).

To state it in a simpler way, given the above example, we could have asked the question, how many times should we multiply 2 together to get 8?

To state it in a simpler way, given the above example, we could have asked the question, how many times should we multiply 2 together to get 8? The answer is 3. If you would like to see additional examples I will direct you to this simple article on logarithms: https://www.mathsisfun.com/algebra/logarithms.html.

John Naiper first demonstrated his new logarithmic function in a mechanical device that has come to be known as Napier’s Bones (Garfinkel and Grunspan 30). His device allowed easy multiplication and division of multiple digit numbers by any digit between 2 and 9 (Garfinkel and Grunspan 30). It was on the logarithmic function that the principles of the slide rule were based.

The second inventor is Edmund Gunter, who, shortly after Napier invented the logarithmic function, invented his own device using a logarithmic scale on a ruler-like device (Garfinkel and Grunspan 30). A regular ruler is linear, meaning that the space between 1 and 2 is the same as the space between 2 and 3, or 3 and 4, and so on. However, on a logarithmic scale the space decreases the further up the scale the number is moved. Therefore, the space between 1 and 2 may be the equivalent of 2 inches, but the space between 2 and 3 is the equivalent of 1 inch (example for simplicity). The nature of the logarithmic scale allowed the distance to be measured between any two points while multiplying the numbers in question. If a regular ruler, representing a linear scale, was used the user could only add the numbers involved.

Let’s review the third inventor and then we will consider an example of the logarithmic scale versus a linear scale.

Let’s review the third inventor and then we will consider an example of the logarithmic scale versus a linear scale. The third inventor was William Oughtred, who created the first slide rule in 1622 (“Slide Rule History.”) He placed two of Gunther’s scales next to each other making the distance much easier to measure. Initially his ruler was round, however, by 1650 he had invented the modern slide rule by fixing two stationary rulers on either side of a sliding ruler, hence, the slide rule (Garfinkel and Grunspan 30).

Alright, that’s the invention of the slide rule and it probably still sounds complicated, but how does the slide rule work? Well, consider this example, look at the image below. On the top is a linear scale, using two regular rulers. Then look at the slide rule on the bottom. As you can see, the numbers are not evenly spaced. The space between the numbers decreases as the numbers get bigger. This allows for much more complex math, including multiplication and division.

Using these two scales, let’s consider some mathematical examples. First, the linear scale. One could add or subtract with such a device, however, complex math is rather limited. To add for example 3 to 2, align the first mark on the top ruler with 3 on the bottom ruler, now find where two aligns. 5 is the answer.

Now look at the slide rule using a logarithmic scale. Let’s use, for example, 2 x 2. To complete the problem, align 1 (the index) on the sliding ruler with 2 (the number we are multiplying by) on the stationary ruler. Now find two on the sliding ruler and note where it aligns with the stationary ruler, the 4, 4 is the answer.

It would be important to note that now you can easily see the multiplication scale for the number 2. What’s 2 x 3? Find where 3 on the slide rule aligns with the stationary rule, 6, 6 is the answer.

If the numbers on the slide rule are found past the stationary rule, simply slide the slide ruler the other direction and align 1 (the index) on the right with 2 on the right and again align the results (“How To Use a…”).

Division would simply be carried out in the opposite manner.

For larger numbers, decimals would be used to reduce numbers by 100s, 1000s, etc.

For larger numbers, decimals would be used to reduce numbers by 100s, 1000s, etc. For example, 300 x 200, could be reduced to 3 x 2. Align 1 on the sliding rule with 3 on the stationary ruler and find where 2 aligns, 6, 6 x 100 = 600, the answer is 600. For answers between, say, 200 and 300 the small hash marks between the large ones would be aligned instead of the large marks.

For more information on how to use a slide rule, consider watching the films produced in the 1940s by the US Office of Education. I found the first one particularly helpful: https://youtu.be/rJKmc4PVdh4.

The slide rule changed the way math was done.

The slide rule changed the way math was done. I’ve only shown simple problems here, however, experienced individuals could quickly compute complex mathematical problems more accurately and more quickly than they ever could in their heads or on paper.

Although the device was invented in the 17th century, its use extended well into the 20th century. Engineers at NASA used such devices to put man on the moon. Many movies of a 1960s NASA during the Space Race show use of the device. It has been said that Buzz Aldrin used one to figure the correct landing for the Apollo 11 mission to the moon (Nadworny, “The Slide Rule…”). Students, high school and college, architects, designers, accountants, and other professions that depended upon math used the device regularly until the digital calculator was invented in 1972. However, due to the calculator’s price, many used the slide rule well beyond. Slide rules were once so commonplace many wore them clipped to their belts in cases (Calvert, “Slide Rule.”) or carried smaller versions in their pockets.

The slide rule is a simple device that depends upon complex mathematical principles.

The slide rule is a simple device that depends upon complex mathematical principles, but it makes powerful calculation easy and more accurate. The slide rule is the seventh major milestone in the history of modern computing.

Works Cited

Calvert, J B. “Slide Rule.” Slide Rule, 10 Jan. 2004, mysite.du.edu/~jcalvert/tech/slidrul.htm.

Dunbar, Brian. “Saturn V.” NASA, NASA, www.nasa.gov/centers/johnson/rocketpark/saturn_v.html.

Garfinkel, Simson, and Rachel H. Grunspan. The Computer Book: from the Abacus to Artificial Intelligence, 250 Milestones in the History of Computer Science. Sterling, 2018.

“Introduction to Logarithms.” Math Is Fun, www.mathsisfun.com/algebra/logarithms.html.

Nadworny, Elissa. “The Slide Rule: A Computing Device That Put A Man On The Moon.” NPR, NPR, 22 Oct. 2014, www.npr.org/sections/ed/2014/10/22/356937347/the-slide-rule-a-computing-device-that-put-a-man-on-the-moon.

PeriscopeFilm. “HOW TO USE A SLIDE RULE (C&D SCALES) ANALOG COMPUTER MULTIPLICATION & DIVISION 99134.” YouTube, 2019, www.youtube.com/watch?v=rJKmc4PVdh4.

“Slide Rule History.” Oughtred Society Slide Rule History., www.oughtred.org/history.shtml.

“Slide Rules.” National Museum of American History, Smithsonian Institution, americanhistory.si.edu/collections/object-groups/slide-rules.