- Eratosthenes
made the assumption of a spherical Earth.
The ancient Greeks had been aware of
the spherical Earth theory.
since the theory
was recorded (and perhaps discovered) by perhaps
Parmenides (fl. c.500 BCE)
(see, e.g.,
David Furley (1922--2010), The Greek Cosmologists, 1987, p. 41, 56).
Strong empirical evidence for a
spherical Earth was known
by Eratosthenes' time and
maybe since Parmenides.
Perhaps NOT overwhelmingly convincing, but pretty convincing.
For more on the
spherical Earth theory in
Greco-Roman antiquity, see
parmenides_earth.html
and
Wikipedia: Spherical Earth: Antiquity.
- Eratosthenes
also assumed that the Sun was
so remote that light rays
from the Sun were
parallel
when they arrived at the
Earth.
No evidence contradicted this theory.
Attempts to measure the Sun's distance
were probably known to
Eratosthenes
and it was probably known that they were probably failures because the
Sun was too remote for the techniques
available then.
Of the
ancient Greek astronomers, it can be said
their geometry was strong,
but their instruments were weak.
In fact, if the ancient Greek astronomers
had the right idea of the
heliocentric solar system model
and put their minds to it they could have measured the
Earth-Sun distance to fair accuracy just as was done in the
17th century by
Jean Richer (1630--1696) and
Giovanni Domenico Cassini (1625--1712)
in 1672
(see Wikipedia: Astronomical unit: History).
In fact moreover, if
Eratosthenes (c.276--c.195 BCE),
Archimedes (c.287--c.212 BCE)
(arguably both greatest
ancient Greek mathematician
and experimentalist),
Aristarchos of Samos (c.310--c.230 BCE)
(the first heliocentric solar system),
and Ktesibios (c.285--c.222 BCE)
(another great
experimentalist)
had put their great brain
together, they could have done it.
- It was known to Eratosthenes
that in Syene (now Aswan)
(on the Nile in
Egypt)
that the Sun was
at zenith at
solar noon
on the day of the
summer solstice.
This means that Syene is on the
Tropic of Cancer.
See Image 1.
- At the same time in
Alexandria---the
city
founded by
Alexander the Great (356--323)
which he named for himself---the Sun
was 7° south of
zenith
along the meridian.
Of Alexandros:
... coming into his own city Alexandria,
he found no memory of his time,
not even a stone set on stone,
and he rested on a stone,
for a moment.
- The diagram in Image 2
makes the geometical situation clear.
The angle
φ
between the zenith direction
in Alexandria and
a light ray
from the Sun is equal
to the angle between the
radius to
Alexandria and
the radius
to Syene
by the
converse of Euclid's parallel postulate
(see also Euclidean
parallelism axiom).
The radius
to Alexandria is a
transversal that
intersects 2
parallel
straight lines.
- Behold:
φ/360=S/C , and so C = (360/φ)S = 50*5020 = 252000 stadia
C = (252000 stadia) * [0.1577 km/(1 stadion)] ≅ 39700 km
and
r = C/(2π) ≅ 6320 km .
where
φ = 7.2° = 1/50 of a
circle
as measured by Eratosthenes,
S is the arc length between
Alexandria and
Syene,
C is the circumference
of the Earth,
and
the
Eratosthenian stadion = 0.1577 km (modern estimate).
- Alas, we are NOT certain
which station
Eratosthenes used,
but the
Eratosthenian stadion, as we can call it
for the nonce, is our best assumption.
With that assumption,
Eratosthenes'
values are ∼ 0.7 % smaller than modern values for the
Earth meridional circumference = 40007.86 km
and the
Earth mean radius R_me_⊕ = 6371.0088 km.
This is really very good agreement for an ancient measurement.
- By the by,
recall that the Earth is
NOT perfectly spherical.