Eratosthenes method
of measuring the Earth
Credit for the first determination of the size of the Earth goes to the
Greek polymath Eratosthenes, who lived from around 276 BC to 195 BC.
Born in the Greek colony of Cyrene, Libya, he was a mathematician, an
astronomer, the inventor of geography, a cartographer and a poet. He is
given credit for developing the system of latitude and Longitude measure
still in use today. As students of Sacred Geometry we might question how
many of the discoveries attributed to Eratosthenes were actually
rediscoveries of pre-existing traditions that have been lost to the
historical record. It does seem increasingly likely that sophisticated
geodetic knowledge was in existence long before Eratosthenes . That he
had access to ancient sources of knowledge is highly plausible, given that
he was one of the chief librarians of the great library of Alexandria,
Egypt, which was ultimately destroyed along with several hundred
thousand scrolls, documents and books, under historically controversial
circumstances. The most likely explanation is that it succumbed to the
pogrom against paganism initiated by Emperor Theodosius in 391 AD,
but may have suffered degrees of destruction prior to that date. In any
case, whether originator or purveyor of knowledge, Eratosthenes played
a vital and important role in the advancement of science.
Around the year 240 BC Eratosthenes used his knowledge of
geometry, astronomy and geography to calculate the diameter of the
Earth. As students of Geometry the means by which he accomplished
this task is of great interest to us. His method demonstrates the power of
even a modicum of geometric knowledge when coupled with the human
imagination.
So how did Eratosthenes measure the size of the Earth using simple
geometry?
Eratosthenes method of measuring the
Earth, continued
The distance between cities in ancient Egypt had been determined
through numerous surveys undertaken by Pharaonic engineers.
Eratosthenes himself probably traveled from Alexandria to the city of
Syene, at the site of modern day Aswan. So through personal experience
or by reference to surveys extant at his time he knew the distance
between Alexandria and Syene to be close to 500 miles.
Of course our modern statute mile of 5280 feet was not in use in
ancient Greece. The unit of measure was the stade, from which we
derive the word stadium. The length of the stade varied widely
depending upon the location. Its length is usually given as approximately
606 feet, sometimes 608 feet, but has varied all the way up to 738 feet.
The exact length of the stade used by Eratosthenes is not known with
certainty, and therefore his determination as to the size of the Earth is
also not known exactly. We will be discussing the use of the stade in
greater detail when we study ancient metrology, or systems of
measurement, and its connection to Sacred Geometry.
Eratosthenes’ knowledge of geography led not only to knowing the
distance from Syene to Alexandria, it also included his awareness of a
spherical Earth. His knowledge of Astronomy led to his knowing that the
Earths rotational axis was tilted about 23 ½ degrees out of perpendicular
to its orbital plane around the Sun (the Plane of the Ecliptic). The term
used to describe this tilt of the Earths’ axis is obliquity.
His great insight, assuming that it did, in fact, originate with him, was
to imagine the Sun so far away, and so large relative to Earth, that its rays
were traveling in parallel lines when they reached the Earth.
To unite these ideas — the spherical Earth with tilted axis, the
distance between the two cities, and sunlight traveling in parallel lines —
it remained only to introduce Proposition 17: A straight line falling upon
parallel lines makes the alternate angles equal.
Eratosthenes method of measuring the Earth, continued
As the story goes, when Eratosthenes visited Syene, he saw a deep well in
the middle of the city. Whether he actually visited Syene or read about the
well in the great library is not known. In any case, this well sat directly upon
the Tropic of Cancer, a latitude 23.5 degrees north of the equator and the
most northerly latitude at which the Sun can occupy the zenith. The zenith
is the point in the sky directly overhead from a given location on the Earth.
If you held up of plumb line with one end of the vertical line pointing
towards the Earth’s center, the other end would point directly to the
zenith. Another way to visualize the zenith is to imagine a perfectly flat
horizon, such as may occur during a calm sea, then measuring up 90
degrees from every point on the horizon to a meeting point straight
overhead. Every point on the surface of the Earth has its own zenith. In
both the DVD Cosmic Patterns and Cycles of Catastrophe and in several
articles posted on the SGI website additional information is given about
these concepts.
So back to the story. On June 21, the summer solstice Sun is directly
perpendicular to the Tropic of Cancer. On that day at exactly local noon
time at Syene, which sits on the Tropic of Cancer, a vertical pole, column,
shaft, or obelisk, as the case may be, would, for a few moments, cast no
shadow. It was during this short interval that the reflection of the Sun could
be seen at the bottom of the aforementioned deep well, and at no other
time. But Eratosthenes knew that at that same moment on the summer
solstice noon, there WERE shadows cast from vertical objects in Alexandria,
500 miles or 800 km to the north. This fact Eratosthenes had observed for
himself. His challenge was to try and explain this phenomenon.
Most accounts have him simply placing a vertical stick into the ground
in order to measure the angle of the shadow thrown by the Sun at noon.
While this humble method would certainly work, it should be born in mind
that Ptolemy II, who reigned from 283 BCE to 246 BCE had massive
obelisks brought to the city of Alexandria, and they would have been
standing during the time when Eratosthenes was serving as director of the
great library there.
It is likely that the presence of these immense obelisks casting large and
impressive shadows that changed noticeably and predictably throughout
the course of the year would have inspired Erastosthenes to his great
insight. The fact that there was no shadow at Syene on this particular day,
and a very definite shadow at Alexandria could only be explained by a
spherical Earth. Were the Earth flat the same shadow would be cast at both
locations. To understand how these accumulated insights allowed
Eratosthenes to calculate the Earth’s diameter reference to the illustrations
below will prove very helpful.
Alexandria, Egypt:
~30º 42’N
Tropic of Cancer
Syene,
Egypt:
~23º 30’N
Equator
This illustration shows the situation described, the well at Syene and the
vertical shaft, obelisk or otherwise, at Alexandria. In the next illustration
we see what Eratosthenes saw in his imagination. The same purpose
could be served by any vertical object at Syene, the lack of shadow
demonstrating the presence of the Sun on the local zenith at high noon.
NOTE: The arc distance from Syene to Alexandria has been exaggerated
in this diagram for clarity.
Eratosthenes, assuming that the Earth was spherical, could conceive of
the well at Syene and the Obelisk at Alexandria as extending into the
Earth as imaginary lines until they met at the center forming an angle.
Knowing the distance from Alexandria to Syene, he knew that if he could
determine the angle formed by those two lines he could easily calculate
the size of the Earth. From the angle he could determine what part of a
circle was equivalent to the distance from Alexandria to Syene.
Alexandria, Egypt:
~30º 42’N
Tropic of Cancer
Syene,
Egypt:
~23º 30’N
Equator
In the next illustration we see the rays of the Sun striking the Earth in
parallel lines on the day of Summer Solstice. Note that the rays are
parallel to the vertical well shaft at Syene but strike the obelisk at
Alexandria at an angle, thereby casting a shadow.
Here we see the Sun’s rays striking the Earth on the Summer Solstice
vertical and parallel to the Tropic of Cancer. This important line of latitude
is so called because on this day the Sun enters into the sign of Cancer. Six
months later, on Winter Solstice, the Sun will occupy the high noon zenith
at 23.5 degrees south latitude, the Tropic of Capricorn, so called because
on that day the Sun enters that sign.
Tropic of Cancer
Equator
Tropic of
Capricorn
This figure represents the situation at Alexandria at noon on the
Summer Solstice. The Sun would have occupied a position offset
towards the south from the local zenith. The shadow from the shaft
would point due north at that moment and also be shorter than at any
other time of day.
This diagram illustrates, how, be means of a protractor, the angle
between a vertical shaft and the Sun’s rays could be measured. A cord
stretched from the apex of the shaft to the apex of the shadow would
make an angle with a horizontal surface, marked here by the letter b,
that could easily be measured. This angle would be complementary to
the angle a formed between the cord and the vertical shaft. When
Eratosthenes measured these angles he found that the angle between
the Sun’s rays and a vertical line was 7.2 degrees. Remember that in the
diagram the angle is drawn larger than this for easier visualization.
a
b
In this diagram the geometric relations have been abstracted for
simplification. The two red lines represent parallel lines of sunlight. The
lower red line is parallel and vertical to the well shaft at Syene. Here you
easily see that the black line can be visualized as a transversal
intersecting two parallel lines, meaning angle a and angle b are equal.
The angle at a is the angle formed by the cord and the vertical shaft,
pole, obelisk, gnomon, stick, etc. as described. The angle at b is the
angle formed between a vertical at Syene and a vertical at Alexandria
extended to the Earth’s center.
a
b
To conclude his reasoning, Eratosthenes would divide the number of
degrees in a full circle — 360 — by the number of degrees in the angle
a, in this case 7.2. The result is 50. In other words, 7.2 degrees is one
fiftieth of the circle. Therefore the distance from Syene to Alexandria,
measured by angle b, is one fiftieth of the distance around the Earth.
If the distance between the two cities was 500 miles the circumference
of the Earth would be 50 times as great or 25,000 miles, very close to
the actual circumference. However, see the discussion that follows.
According to some sources the distance that we have taken as about 500 miles
between the cities was closer to 497.6 miles. If this was the case the distance
becomes significant in terms of Sacred Geometry and the ancient Canon of Sacred
Numbers. If the stade employed by Eratosthenes is taken as 608.17 feet, a measure
essentially matching one of the common values given for the Greek stade, the
distance between the two points, the well and the obelisk, is 4,320 stades. 4,320, as
we shall learn, was an important number in the ancient Canon. The Earth’s
circumference in stades is then 50 times this number or 216,000 stades, another
significant number. Translating these numbers into miles gives us:
Earth’s circumference in stades
x number of feet in the stade
=
= Number of feet in Earth’s circumference
This number divided by 5,280
(the number of feet in a mile)
= Number of miles in Earth’s circumference
216,000
608.17
131,364,720
131,364,720
5,280
24,879.68
24,879.68 divided by π yields the Earth’s diameter.
24,879.68 ÷ π = 7919.448 miles, or in round numbers 7920 miles.
It is highly recommended to take your calculator in hand and perform these simple
calculations for practice. Bear in mind that the Earths diameter and circumference
vary depending upon where they are measured. While the Earth’s diameter varies
between about 7900 and 7926 miles, the number 7920 is a member of the Ancient
Canon and is very close to the diameter of a perfect sphere that has the same surface
area as Earth. It is also close to the diameter taken through the Earth from the Tropic
of Cancer to the Tropic of Capricorn. If 608.17 was the value used for the stade by
Eratosthenes, that too has additional significance in that it is virtually one tenth of the
old American Nautical Mile of 6080.2 U.S. feet, which represents the length of a
minute of arc measured along the meridian at latitude 48 degrees north.