B00B7H7M2E EBOK (4 page)

Figure 1.1

Because the Sun’s rays are running parallel as they strike the Earth, if a line is drawn from Alexandria (a) where the stick casts a shadow, to the centre of the Earth, and a second line from Syene (s) where there is no shadow, to the centre of the Earth, the angle where those two lines meet will be the same as the angle of the shadow at Alexandria.

Figure 1.2
illustrates Eratosthenes’s measurement.

Figure 1.2

Because the sunlight shone all the way to the bottom of the well at Syene (s), Eratosthenes knew that the Sun was shining straight down on the Earth there. He set up a stick at Alexandria (a), where the Sun wasn’t shining straight down, and he measured the angle (x) of the shadow cast by the stick. He knew that because the Sun’s rays all run parallel as they strike the Earth, the angle (y) where a line drawn straight down from Alexandria and a line drawn straight down from Syene would meet at the centre of the Earth would be the same angle as the angle of the shadow cast by the stick (x). If Syene is due south of Alexandria, then the distance between Syene and Alexandria must be the same fraction of the Earth’s total circumference as the angle at x or y is of 360°

Eratosthenes found that the shadow angle at Alexandria was 7
°, and so he knew that the angle between the ‘Syene–Alexandria lines’ (meeting at the centre of the Earth) was also 7
°. A circle has 360°, and it is a simple process to find out how many of the Syene–Alexandria angles (7
°) it will take to make 360°. Think of the cross-section of the Earth as a pie and the two lines coming from Syene and Alexandria as cutting out a wedge of pie. How many wedges of that size can you cut from the whole pie? Divide 360 by 7
, and it comes out to 50 wedges. If we say (as Eratosthenes did) that the distance between Syene and Alexandria at the surface of the Earth (at the pie-crust edge of the pie) is ‘5,000 stades’, then we can multiply 5,000 by 50 and conclude that the distance all the way around the Earth – the circumference of the Earth – is 250,000 stades. Eratosthenes later fine-tuned this to 252,000 stades.

What is this odd unit of measurement, the stade? That question brings up a problem in evaluating Eratosthenes’s result. Whether or not that result matches modern measurements for the circumference of the Earth depends on the length of ‘stade’ he was using, and it isn’t known exactly what that length was. If there are 157.5 metres in a stade, Eratosthenes’s result comes to 39,690 kilometres or 24,608 miles for the circumference of the Earth. That is very near the modern calculation – 24,857 miles (40,009 kilometres) around the poles and 24,900 miles (40,079 kilometres) around the equator. After he had found the circumference, Eratosthenes calculated the diameter of the Earth as 7,850 miles (12,631 kilometres), close to today’s mean value of 7,918 miles (12,740 kilometres).

Another way of figuring a stade was as
or
of a Roman mile, and that would make Eratosthenes’s result too large by modern standards. There was one additional small difficulty. Eratosthenes assumed that Syene lay on the same line of longitude as Alexandria. Actually, it does not.

But this is nit-picking! No apology need be made for Eratosthenes. First of all, he arguably came astonishingly near to matching the modern measurement. Second, he was probably, for all his curiosity about the world, enough a man of his time to find the puzzle of how to solve this problem by the imaginative use of geometry at least as interesting as the actual measurement. The
method
is ingenious and it is correct. If the numerical result is a little fuzzy because of a lack of agreement about the length of a stade and the impossibility of determining longitude precisely, that does not prevent our recognizing what a brilliant achievement this was or appreciating the intellectual leap involved in recognizing that it
could
be done and
how
it could be done.

Eratosthenes didn’t focus his thoughts only on the Earth. He also raised them above the horizon to consider astronomical questions of his day. When it came to measuring the distances to the Sun and the Moon, he must have realized that he had no tool at his fingertips to equal the news about the well in Syene. Nevertheless, he gave it a try, with far less success than he had in measuring the Earth’s circumference.

Another Hellenistic scholar, Aristarchus of Samos, also tried to measure the distances to the Moon and Sun. Little information exists about him as a person. He lived from about 310 to 230
BC
and would already have been a grown man when Eratosthenes was born. The island of Samos was under the rule of the Ptolemys during Aristarchus’s lifetime and it is possible that he worked in Alexandria. Archimedes was certainly aware of his contributions.

The only written work of Aristarchus that has survived is a little book called
On the Dimensions and Distances of the Sun and Moon
. In it he describes the way he went about trying to determine these dimensions and distances and the results he got.

The book begins with six ‘hypotheses’: