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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 asorof 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’:
    The Moon receives its light from the Sun.
The Moon’s movement describes a sphere and the Earth is at the central point of that sphere.
At the time of ‘half Moon’, the great circle that divides the dark portion of the Moon from the bright portion is in the direction of our eye. (In other words, we are viewing the shadow edge-on.)
At the time of ‘half Moon’, the angle (at the Earth as shown in Figure 1.3 ) is 87°.
The breadth of the Earth’s shadow (at the distance where the Moon passes through it during an eclipse of the Moon) is the breadth of two Moons.
The portion of the sky that the Moon covers at any one time is equal toof a sign of the zodiac.
    Aristarchus’s fourth and sixth assumptions are both far from accurate. The actual angle at the Earth in Aristarchus’s triangle would be 89° 52’, not 87°, and 89° 52’ is very close to 90°. The angle at the Moon in Aristarchus’s triangle
is
90°. That makes lines B and C so close to parallel that, on a drawing, the triangle would close up and be no triangle at all. The portion of one sign of the zodiac that the Moon covers is not, and it isn’t clear why Aristarchus, who must have known