it helps if its the equinox

January 4, 2019 7:53 AM

The talk/workshop I give about computing the distance to the Moon[PDF] uses, it claims, nothing more than a pendulum and a stopwatch. And while it's sort of true that it uses nothing else, it's not really true, because it also uses the period of the Moon, and the size of the Earth. Now it might be possible to persuade you that it's OK to use the period of the Moon, since you can simply look out the window and measure that for yourself, but to use the size of the Earth seems a bit of a stretch. Surely there's no way to compute that from your back garden. Of course, we could claim that since the original definition of the metre was:

"One 10 millionth of the distance from the North Pole to the Equator on the longitude that runs through Paris"

then perhaps we can quite reasonably claim that the Earth's circumference is, by definition, 40 million metres, but even so, perhaps that's not really fair, not really playing the game. But when I showed Distance to the Moon calculation to a friend of mine, it turned out he (a) was really interested, and (b) suggested a method to compute the radius of the Earth from your back garden.

Colin Wright calculates The Radius Of The Earth, from his back garden.

This is not the Method of Eratosthenes

Radius Of The Earth, Part 2

Some nine and a bit weeks ago I posted about a method of calculating (or estimating) The Radius Of The Earth using a stopwatch and watching the Sun at sunrise (or sunset). When "Mike the Sundial" told me the idea I was just stunned at the simplicity and elegance. Colin Beveridge took up the challenge, and you can read his account of his method here:
http://www.flyingcoloursmaths.co.uk/stab-colins-puzzle/
When I read that I was interested to see just how different his take was from mine, but then I realised - it comes down to the phenomenon of "chunking". ColinB was working - as most people would - from first principles.
But I have a piece of knowledge that ColinB doesn't have, and I hadn't realised just how much it affected my approach.

The Size Of The Earth, A Correction

Recently I've been discussing the details of measuring the size of the Earth by watching a shadow descending a wall at sunrise, or ascending at sunset. The sums are easy enough if you do it at either Equinox, and at the Equator, and you can read about it here:

The Radius Of The Earth
Radius Of The Earth Part Two

The challenge then is to do it somewhere other than at the Equator. After all, you can choose your time of year, so you can choose to do it at Equinox, but you can't necessarily travel to the Equator. So how can we do it at a non-Equator location, a non-zero latitude?<>

posted by the man of twists and turns (5 comments total)

This is not the Method of Eratosthenes

Good job anticipating the inevitable "Eratosthenes did it first" comment. (Which I may or may not have been rushing here to post.)
posted by tobascodagama at 8:02 AM on January 4, 2019 [10 favorites]

I realize that he then turns around and does it more generally, but assuming the knowledge that the circumference of the Earth is 40 million meters while trying to calculate the diameter of the Earth is an egregious cheat. Regardless of how the meter was defined - if I know that, then I also know the diameter of the Earth.

So we do have to do the more general calculation while watching the sun rise and the line of sunlight march a known distance down a wall.

And then, g from a pendulum and stopwatch, sure, but you'll need good luck with measuring the sidereal period of the moon (27.32 days) to that precision.

Not bad, though - it's easy to get a ballpark answer (hundreds of thousands of km) even with extreme approximations.
posted by RedOrGreen at 8:28 AM on January 4, 2019 [2 favorites]

On a somewhat related note, Scott Manley measures the distance to the ISS.
posted by rhamphorhynchus at 12:20 PM on January 4, 2019 [2 favorites]

assuming the knowledge that the circumference of the Earth is 40 million meters while trying to calculate the diameter of the Earth is an egregious cheat

So is assuming that the Earth is a globe while trying to demonstrate that the Sun is not, in fact, a relatively small spotlight running on a complicated track above it.
posted by flabdablet at 4:33 PM on January 4, 2019 [1 favorite]

assuming the knowledge that the circumference of the Earth is 40 million meters while trying to calculate the diameter of the Earth is an egregious cheat

I did a double take at that spot too but I think it's a weird rhetorical turn presented unclearly, not a cheat. In a way it's such an odd misreading of his likely audience that I think I find it weirder than a cheat...

My read of his argument there is:

1) I know the circumference, which given this relationship I've just found lets me predict the time.

2) But wait, I'm measuring the time, so a difference between the actual and predicted time allows me to correct my original guess at the circumference.

That's fine, but I think the reason it made all of us here stumble is that it's basically reframing the idea of algebra itself, as if "find relationship, solve for unknown" is such a novel concept that he needs to demonstrate it from first principles. Given the likely background of anybody who's made it that far already I think that's a shaky rhetorical path at best. Though given that he denies himself even a protractor (itself weird since you can make one with a compass and straightedge, which seems in bounds for this kind of exercise) maybe he added denial of algebra as another internal constraint.
posted by range at 10:02 PM on January 4, 2019 [1 favorite]