Global Positioning System Reference
In-Depth Information
Eratosthenes assumed that the sun was so far away that a ray of sunlight
striking the bottom of the well at Syene was e√ectively parallel to a ray of
sunlight grazing the top of a gnomon at Alexandria. This is true. However,
he did make a number of errors, which account for the 16% error in his
measurement. Syene is not due south of Alexandria (it is 2\ 58% east); it is
not quite on the Tropic of Cancer (it is north by 22%). The sun has a finite
angular width—it is not a point source of light—and so the length of the
gnomon shadow is di≈cult to estimate accurately. In fact, Eratosthenes
measured the angle
a
of figure 2.4a as being
1
⁄
50
of a full circle, or 7\ 12%,
whereas it is actually 7\ 8%. Distances between cities were more di≈cult
than angles to estimate in those days, and his distance of 5,000 stadia
between Syene and Alexandria is 10% too large.
5
A hundred and fifty years later, Posidonius of Apamea was aware of
Eratosthenes' work and produced a di√erent estimate of the radius of the
earth. On the Mediterranean island of Rhodes, where he lived, he noted
that the star Canopus (the second brightest in the sky) just grazed the
horizon, whereas farther south in Alexandria, it rose
1
⁄
48
of a circle, or 7\
30%. Knowing the distance between Rhodes and Alexandria, Posidonius
could then estimate the earth's radius using the geometry shown in fig-
ure 2.4b. His estimate was 11% too large, though possibly so close to the
true value only because of a fortuitous cancellation of errors. The method
he used was reasonable, except that it did not take into account the atmo-
spheric refraction of starlight.
Fast-forwarding a millennium and moving east, we find the caliph Al-
Mamun, seventh sovereign of the Abbasids, in Baghdad, sending surveyors
north and south of his city, across the plain of Sinjar. Al-Mamun was greatly
interested in astronomy, geodesy, and medicine; and Baghdad reached its
scientific zenith under his reign (786-832 CE). His surveyors traveled
until they reached locations where the angle to the Pole Star was one de-
gree di√erent from the angle observed at Baghdad. The geometry is shown
in figure 2.5a. The distances from Baghdad to the two sites were measured
with either wooden rods or knotted ropes. From the geometry, the radius of
the earth was determined to be 6,364 km (or possibly 6,409 km; the
5. That is, 5,000 stadia corresponds to 925 km if we are right in assuming 1 sta-
dion = 185 m. The distance between Alexandria and Aswan is 842 km, as I found after a
two-minute interrogation of Google Earth. Here is an eloquent illustration of the vast
increase in geodetic information; how much e√ort would it have taken to ascertain the
distance between these two cities 20 years ago—or 200 years ago?
