Estimating some even bigger things

In the last issue we looked at how we might estimate the height of a tall tree (or building), and the distance of a ship from land (or a hill from a road). Next, let's see how we might estimate the diameter of the earth.

This amazing feat was carried out by Eratosthenes, the head of the great Library at Alexandria on the North coast of Africa, and friend of Archimedes, about two centuries BC. (Yes, they did guess the earth was round that long ago!)

What he did was this: he chose a place, Syene, which was known to be 790 kilometres (but he measured distance as 5 040 `stadia') due South of Alexandria (i.e. on the same meridian or line of longitude), and he waited for the day the sun passed directly overhead of Syene - going by when it shone all the way down a vertical well in the ground. On that day he measured the angle the sun's rays made with the vertical at Alexandria at midday (about 7[1/2] ⁰). And knowing that angle and the distance to Syene, he estimated the circumference C and hence diameter of the earth. How? [The answer is is on page .]

Let's think about places nearer to home. A map of Zimbabwe will tell you that Beitbridge is almost directly due South and L kilometres from the village of Kanyemba (on the Zambezi River). If you measure the angle of the sun at midday at Beitbridge on the day the police at Kanyemba confirm (by radio to the Police Station at Beitbridge) that the sun is directly overhead there (I think it will happen early in December) you can estimate the circumference of the earth, and its diameter, and see if your estimate is as good as Eratosthenes!

We would love to publish the story and results of any school - or pair of schools on the same line of longitude - that actually carries out such an estimate. Why not attempt it? It would make a very instructive class activity.

Your first problem will be to estimate the distance `as the eagle flies' between the schools. Unfortunately, no places in Zimbabwe will see the sun directly overhead until around December. You can carry out the estimation at any time by measuring the angles at both places, but the calculations are a little harder. Since we have atlases that tell us much more geography than Eratosthenes knew, here is a short-cut, avoiding communication problems between schools: measure the angle of the sun at your school (or home) at midday on the shortest day (in June) or the longest day (in December) of our year when the sun is directly over the tropic of Cancer, 23⁰ North of the equator, or the tropic of Capricorn, 23⁰ South of the equator. All you need then, is the latitude of your school or home; you don't need to know your distance from the tropic of Cancer, because you know the latter's latitude.

Now, for something really huge - how could we estimate the distance of the sun or the moon from the earth? Or at least the ratio of their distances? The ancient Greek astronomers tried this also. Theoretically, their methods were sound; the trouble was that their instruments were not nearly precise enough. Try to think of ways of doing this yourself - you will probably come up with the same difficulty: for the error limits of all simple technology - about ±2⁰ of angle - the answers will be way out.

Here are some of the best estimates of the ancients for the sun's distance from the earth, in terms of earth-diameters as unit:

Aristarchus (c. 200 BCE) - 180. He was among the first to propound a sun-centred universe.
Hipparchus (c. 140 BCE) - 1245.
Posidionius (c. 130 BCE, teacher of Cicero) - 6545.
Ptolemy (c. 150 ACE) - something much worse than Posidionius.
Modern value is 11 726 earth diameters!

QUOTE from EUDOXUS (408-ca.355 BCE):

Willingly would I burn to death like Phaeton, were this the price for reaching the sun and learning its shape, size and substance. When I trace at my pleasure the winding to and fro of the heavenly bodies, I no longer touch the earth with my feet: I stand in the presence of Zeus himself, and take my fill of ambrosia, food of the gods.