A Slide Rule for Seeing Mathematics

Last week, I wrote about what Gabriel’s horn can teach us about mathematics education. That post began with a strange mathematical object whose volume is finite even though its surface area is infinite.

I was reminded of another mathematical idea from when I was younger. I was already reasonably comfortable with mathematics, but a friend once remarked that he found logarithms amazing. The logarithm keeps increasing as its input increases, yet its rate of increase keeps falling. It becomes flatter and flatter, but never reaches a finite upper limit.

I remember thinking that this was an odd thing to find amazing. I had just learnt logarithms at school, so I treated the statement as another fact to remember. I had not looked at it from that angle. The graph described a relationship that kept stretching upwards while becoming increasingly reluctant to do so.

Mathematics is full of these small patterns. The pleasure comes when you stop seeing them as isolated rules and notice the relationships underneath them. A formula can suddenly behave in a way that is completely determined and still surprising.

I suspect that something missing from mathematics education is a way for students to feel scaling relationships directly. We usually encounter them as symbols on a page. We learn that multiplication changes a number by a factor, or that logarithms reverse exponentiation, but those relationships can remain abstract.

This is one reason I became interested in slide rules. When I was younger, I watched Apollo 13 and noticed a strange-looking device being used during the mission. It was a slide rule. I eventually bought one and found that I really enjoyed using it.

The calculation is slower than pressing buttons on a calculator, but the relationship is visible in your hands. The scales encode logarithms, so multiplication becomes the addition of distances. Division becomes subtraction. Powers and roots become movements along the same scales.

I think slide rules could work well as a supplement to high-school mathematics. They would not replace calculators or formal methods. They would give students another way to encounter logarithms and scaling, with enough friction to make the structure visible.

I am going to write more about slide rules as a small side project. There are many mathematical ideas hiding inside this old instrument, and I would like to explore them one at a time.