From: msb@sq.sq.com (Mark Brader)
Subject: Re: Diameter of earth?
Organization: SoftQuad Inc., Toronto, Canada
Date: Wed, 7 Oct 92 07:51:47 GMT

> > > 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 shone all the way to the bottom of the local well.
> > 
> > ... 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 ... Erastophenes was based in Alexandria (wasn't he the Librarian?)
> Alexandria was, of course, a major Hellenistic center.  [Then] he would
> have been starting out from northern Egypt.  ... [Anyway, he] needn't have
> visited the site himself to be aware of this phenomenon.

Right on both counts, according to John Noble Wilford in his fine
history of cartography and related technologies, "The Mapmakers"
(1981, Random House, Vintage Books edition ISBN 0-394-75303-8).
Wilford writes (excerpts from pages 19-23):

#  Among the travelers' tales circulating in Alexandria at the time
#  was one about a well in Syene, up the Nile at the first cataract,
#  where the Sun shone directly into its deep waters on the longest
#  day of the year, June 21 [sic].  Nothing so strange had ever been
#  seen in Alexandria.  To Eratosthenes the story meant that Syene
#  (the Greek name for Aswan) must lie at the northern boundary of
#  the tropics ...
#
#  With this in mind, Eratosthenes sensed that he had all he needed
#  to measure the circumference of Earth.  ...  If he could measure
#  the angle of a shadow in Alexandria at noon on that day, Eratosthenes
#  thought, he just might be able to measure the size of the Earth without
#  ever leaving the grounds of the library. ...
#
#  From what Eratosthenes could learn from travelers, Syene was supposed
#  to be due south of Alexandria, which was particularly convenient ...
#  if he determined the distance between the two he would know the exact
#  length of an arc of the meridian ... and therefore a part of the
#  circumference of the Earth.  Eratosthenes was told that a camel caravan
#  needed 50 days to make the journey and that camels usually traveled
#  100 stadia a day.  The distance would thus be about 5,000 stadia.
#  Royal surveyors, according to some versions of the story, may have
#  paced off part of the route as an attempt to confirm the distance.

Eratosthenes did the shadow measurement at Alexandria; Wilford says he
found the sun to be "7 degrees 12 minutes, very nearly one-fiftieth of
a complete circle" off vertical.  Actually that value is exactly 1/50
of a circle, but I don't think his technology would have been up to
measurements to a minute of arc, so this must be slightly garbled.
Anyway, assuming 1/50 gives the Earth's circumference as 250,000 stadia.
Wilford gives this as 46,250 km, or about 16% too high, but I've seen
other sources say that the length of the ancient unit is not known all
that accurately anyway.

#  [Eratosthenes's] one theoretical error, assuming the perfect spher-
#  icity of Earth, made little difference.  More important, however,
#  was the fact that Syene is not exactly on the Tropic of Cancer, but
#  is about 60 kilometers to the north.  Syene and Alexandria were not
#  even on the same meridian, with Syene 3 degrees 3 minutes to the east.
#  And, as could be expected, the camel caravans proved to be less than
#  precise in their measurement of distances; the metropolis and the
#  oasis are less than 5,000 stadia apart -- that is, 725 kilometers
#  apart, rather than 800.  But the various errors must have [sic]
#  roughly canceled each other out.

This paragraph appears to have some errors in it itself; for one thing,
all three principal errors cited are in the same direction, tending to
lead to an overestimate of the diameter.  Also, taking the conversion
factor used above, 5000 stadia is 925 km (exactly!).

My atlas shows the modern cities located as follows:

	              deg min N   deg min E
	Alexandria     31  12      29  54
	Aswan          24   5      32  53

That makes them about 840 km apart, with the north-south component of
this about 790 km.  The actual Tropic of Cancer is at 23 deg 27 min
today, though this varies slightly with time and I don't know what it
was when Eratosthenes was active some 2200 years ago.  Thus the Tropic
is actually 7 deg 45 min from Alexandria, or 1/46.45 of a circle;
this implies that Eratosthenes's angle measurement was also slightly
off, and this time tending to underestimate.

Misplacing Syene on the Tropic introduces an error factor of 1.09;
the error in its east-west position introduces an error factor of
about 1.06; and if the distance error factor is 925/840, then that
is a factor of 1.10; and the angle measurement error factor is
46.45/50 or 0.93.  Multiplying these out, we get 1.18, which is pretty
close to the final result above.

It may seem surprising that Aswan could be as far as 38 minutes north
of the Tropic, given that the observation of the Sun's reflection must
have been genuine.  However, consider first that the Sun is not a point
source; it has a visual radius of about 16 minutes of arc.  Then, even
a deep well has a finite depth and nonzero diameter.  If the well was
1 cubit in diameter, then it would have to be over 50 cubits deep
for the image of the sky as reflected in the water surface to occupy
a visual angle less than 30 minutes of arc.  Eratosthenes assumed that
the entire sun's image appeared centrally in the well, but the travelers'
tale would probably be the same if only a part of the image was visible
anywhere on the water surface.

The experiment was replicated by Philip Morrison in "The Ring of Truth",
to emphasize how simple it really is.  He used a vehicle's odometer to
measure distance and simple equipment for the angle measurements.
I don't remember his final result but he was also at least a few
percent off.
-- 
Mark Brader, SoftQuad Inc., Toronto, utzoo!sq!msb, msb@sq.com
	Until 3,000 million years ago we can say not a lot happened
	although further study would not come amiss.  Then signs of life
	appeared, including some large reptiles and, very recently, bipeds.
	It is too soon to say whether these bipeds will play an important
	part in the world's story.    -- Colin Morris in "History Today"

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