From: djoyce@black.clarku.edu (Dave Joyce)
Subject: Eratosthenes' measurement (was Re: Diameter of earth?)
Organization: Clark University (Worcester, MA)
Date:  6 Oct 92 21:33:01 GMT

A good reference for Eratosthenes' measurement of the diameter of the earth is
Thomas Heath's _A History of Greek Mathematics_, originally published in 1921,
but reprinted by Dover, 1981.

In volume II, pages 106-107, Heath describes this feat.

The Greeks had a real variety of theories on the cosmos.  Some as silly as the
epicurean claim that the sun was really only a foot in diameter, just as it
appears, to heliocentric theories.  Several who believed the earth was a
sphere estimated its size.  Eratosthenes came very close to an accurate
figure, perhaps even closer to the figure than his publicized method
justifies.  Maybe he had more than one reason to believe his figure, but for
some reason only described the one.

From Heath:
  But the most famous scientific achievement of Eratosthenes was his
  measurement of the earth.  Archimedes mentions, as we have seen, that some
  had tried to prove that the circumference of the earth is about 3000,000
  stades.  This was evidently the measurement based on observations made at
  Lysimachia (on the Hellespont) and Syene.  It was observed that, while both
  these places were on one meridian, the head of Draco was in the zenith at
  Lysimachia, and Cancer in the zenith at Syene; the arc of the meridian
  separting the two in the heavens was taken to be 1/15th of the complete
  circle.  The distance bewteen the two towns was estimated at 20,000 stades,
  and accordingly the whole circumference of the earth was reckoned at 300,000
  stades.  Eratosthenes improved on this.  He observed (1) that at Syene, at
  noon, at the summer solstice, the sun cast no shadow from an upright gnomon
  (this was confirmed by the observation that a well dug at the same place was
  entirely lighted up at the same time), while (2) at the same moment the
  gnomon fixed upright at Alexandria (taken to be on the same meridian with
  Syene) cast a shadow corresponding to an angle between the gnomon and the
  sun's rays of 1/50th of a complete circle or four right angles. ...

Heath continues with the mathematics based on similar triangles to compute the
estimate.  I've used this example for years as an application of precalculus
mathematics to the real world, as well as in my history of mathematics course.

Heath goes on to explain the sources for our knowledge of Eratosthenes'
measurement.  According to Heron, Eratosthenes wrote it down in a work, now
lost, entitled _On the Measurement of the Earth_.  Galen also mentions it and
describes its content which includes a fair amount of astronomy and geography.

In other works Eratosthenes estimates the distance to the moon and the sun.
He wrote a lost book on geography.  His name is also remembered in the
algorithm to list primes -- the sieve of Eratosthenes.


-----------------------------------------------------------------------

In <ld39j5INNanc@news.bbn.com> ncramer@bbn.com (Nichael Cramer) writes:

>karr@cs.cornell.edu (David Karr) writes:
>|ncramer@labs-n.bbn.com (Nichael Cramer) writes:
>|>The version I've always heard is that one of his spots happened to be
>|>located smack on the Tropic of Cancer.  So that at exactly one time (i.e.
>|>noon) on exactly one day of the year (i.e. the Summer Solstice) the sun
>|>shown all the way to the bottom of the local well.
>|The Tropic of Cancer runs across the lower corner of Libya, across the
>|southern frontier of Egypt, and then across the middle of the Arabian
>|peninsula.  Seems pretty far south for a Greek philosopher to have
>|ventured.  Greece is all above the 35th parallel; the Tropic is at
>|23.5 degrees.
>
>But (again, if I recall correctly) Erastophenes was based in Alexandria
>(wasn't he the Librarian?)  Alexandria was, of course, a major Hellenistic
>center.  In that event, he would have been starting out from northern
>Egypt.
>
>In any case the Hellenistic Empire (or at least its influence) certainly
>stretched as far south as the Tropic of Cancer.  Erastophenes needn't have
>visited the site himself to be aware of this phenomenon.
>
>|On the other hand, it may be that what you wrote is exactly what
>|happened.  What's the source of your story?
>
>One place it was told was on the PBS series "Cosmos".  (They did something
>similar in "The Ring of Truth" but I don't recall whether they spoke about
>Erastophenes".)
>
>I know I've seen it elsewhere.  Unfortunately I'm away from my books at the
>moment.  I'll check when I get home tonight.
>
>N
-- 
David E. Joyce				Dept. Math. & Comp. Sci.
Internet:  djoyce@black.clarku.edu	Clark University
BITnet:    djoyce@clarku		Worcester, MA 01610-1477
Newsgroups: rec.puzzles