How to stay entertained using nothing but a brain

I decided to go out for Japanese food, and for company I decided to bring and re-read Simon Singh’s book, Big Bang. Of course, I’ve read the book before so I knew full well that it would start with a description of how the Greek philosopher, Eratosthenes, measured the circumference of the Earth using a hole in the ground and a stick (possibly to within 1–2% of the right answer!). This is one of my favourite stories from the history of science, as it shows such a purity of acumen and intelligence. However, on the way there it was too dark to read, so I had only my thoughts for entertainment.

As I walked, a snippet of song lyrics from Monty Python’s The Meaning of Life ran through my head: We’re orbiting at ninety miles a second, so it’s reckoned, the Sun that is the source of all our power… I caught myself wondering, is that accurate? More particularly, what is the speed of the Earth as it moves around the Sun? My immediate instinct was to pull out my phone and look up the answer on the internet, but then I decided it was far more in the spirit of Eratosthenes and the book I was reading to at least do the math for myself. Besides, as a member of the mathematically challenged calculator generation, I like to do some mental arithmetic once in a while, half to keep my brain in shape and half to see if I can.

Sometimes people wonder how I do the modest mental arithmetic that I do perform; I do it not by being brilliant but by breaking tricky calculations down into pieces so simple even I can do them.

Happily, I remembered enough facts as trivia to calculate the speed I wanted. I had a brain fart, and one of them was rather wrong, but I’ll get to that. My equipment included the following facts:

1. The Sun’s light takes eight seconds to reach the Earth. (This was the brain fart; it’s actually about eight minutes.)
2. Light moves at very close to 300,000 km/h.
3. The Earth’s orbit is elliptic, but its eccentricity is very low. That is, it’s almost a perfect circle.
4. A circle’s radius r is of course related to its circumference c as c=2πr.
5. The Earth orbits the Sun once a year, which is to say that it takes it about 365.25 days to travel the whole circumference (365.25 rather than 365 because every 4th year is a leap year, the subtleties being too small to worry about).

Okay, so how do we combine all these things to figure out how fast the Earth moves along its path?

Well, if the sun’s light takes 8 seconds to reach the Earth (again, this is wrong, but this was in my mental calculations), and if that light travels at 300,000 km/h, then it covered a distance of 2,400,000 km—2.4 million km. If the Earth’s orbit is almost a perfect circle, this is the radius.

The circumference is 2×π×r = 2×r×π. 2×r = 2 × 2.4 million km = 4.8 million km. π=3.14159…, that is, it’s 3 + 0.14159…, and 1.14≈⅙ (remember, I did this in my head, so I rounded and approximated, often and freely). 3 × 4.8 million km is 14.4 million km, ⅙ of 4.8 million km is 0.6 million km—because as you’ll know from grade school, 6×8=48!—so the circumference of the Earth’s orbit is about 15 million km.

Now we know the Earth’s speed! It’s…15 million km per year. Somehow that doesn’t have much of a ring to it. I’d much rather see it in km/h. Let’s pare it down to km/day for starters. There are about 365 days in a year, so the speed is 15/365 million km/day. That’s an ugly number, but 3.65 is about ¼ of 15 (3.65×4 = 3×4+0.65×4 = 12+2.6 = 14.6). Close enough. Let’s divide our distance by 365 by first dividing it by 3.65, and then by 100: The distance the Earth covers in a day is 4/100 million km or 40,000 km.

Fantastic! Now if we could only take this from days to hours we’d really be set. There are 24 hours in a day. 40,000/24 looks tricky. 40/24 is not an obvious fraction. You can simplify it by dividing both sides by 4 to get 10/6, then by 2 to get 5/3. Unfortunately I didn’t think about that as I was walking—it’s a lot more obvious written down! Instead, I figured that 40/24 is less than 2, because 2×24 = 48. It’s more than 1.5, because 1.5×24 = 1×24+24/2 = 24+12 = 36. 36 is closer to 40 than 48, so the answer must be between 1.5 and 2.0, but closer to 1.5. I’d made a lot of approximations already, so I wasn’t troubled about it, and went with 1.7. (5/3=1.666…, so 1.7 is pretty good.) So, 40,000/24 km/day turns out to be about 1.7×1000 km/h: 1,700 km/h.

Now I had an answer. Now I pulled out my phone and checked online: The Earth moves at…ouch, the internet tells me about 100,000 km/h (more precisely, 107,000 km/h). I expected to be off by a fair bit since I was making rough calculations in my head, but not by that much.

That was when I realised that the sun’s light taking 8 seconds to reach is was a brain fart, and the real figure is about 8 minutes. I should multiply my answer by 60. That’s simple enough: 6×17 = 6×10+6×7 = 60+42=102. So, 1,700×60=102,000, and my arithmetic with a less silly figure gave me an answer of 102,000 km/h, as compared to the internet’s 107,000 km/h. (This is actually within 5% of the correct answer.)

With all my approximations I should give my answer huge error bars, but I still feel confident in asserting that the calculator generation can be taught mental arithmetic.

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How does this compare to the Monty Python lyrics? I didn’t do this in my head earler, so I’ll cheat and use a calculator now. I know that 1 mile is about 1.6 km, so my 102,000 km/h make 63,750 mph. There are 60 minutes to an hour and 60 seconds to a minute, which makes 3,600 seconds per hour, so the speed is…17.7 mps. That’s nowhere near ninety. With the better figure of 107,000 km/h I get 18.6 mps. Perhaps the song should have run: It’s orbiting at nineteen miles a second, so it’s reckoned…

But still, the song is great.