Astronomy 161:
An Introduction to Solar System Astronomy |

- Flat Earth and World Ocean

The Spherical Earth

- Appeal to perfect symmetry
- Demonstration by Aristotle

Measuring the Earth's Circumference:

- Eratosthenes of Cyrene
- Claudius Ptolemy

- Homeric: A flat disk surrounded by a world ocean.
- Inca: Called their land Tahuantinsuyu: "The Four Quarters of the Earth"
- Ancient Egyptian: The sky was a tent canopy stretched between mountains at the four corners of the Earth.

- The great myths and metaphors make the world seem intelligible and beautiful to the people who invented them.
- They serve cultural purposes motivated by an aesthetic we can only guess at.

A sphere is the most perfect geometric solid

**500 BC**:- Pythagoras proposed a spherical earth purely on aesthetic grounds
**400 BC**:- Plato espoused a spherical earth in his 4th and final
dialogue
*Phaedo*, giving it wider circulation (the Pythagoreans were somewhat disreputable in Athenian circles)

- Persons living in southern lands see southern constellations higher
above the horizon than those living in northern lands.
- The shadow of the Earth on the Moon during a lunar eclipse is round.
- The fact that objects fall to Earth towards its center means that if it were constructed of small bits of matter originally, these parts would naturally settle into a spherical shape.

Aristotle's demonstration was so compelling that a spherical Earth was the central assumption of all subsequent philosophers of the Classical era (up to ~300 AD).

He also used the curved phases of the moon to argue that the Moon must also be a sphere like the Earth.

**Challenge**: How do you measure something really big?

- Mountains too high to climb...
- The Earth too large to trail a string behind you...

**Solution:** Apply the methods of *Geometry*.

It was known that on the day of the Summer Solstice in Syene Egypt (modern Aswan), the Sun was straight overhead at noon and did not cast shadows. Syene is on the lower Nile in southern Egypt.

On that same day, the noon Sun cast shadows at Alexandria, located north of Syene on the Nile delta.

By measuring the length of the shadow in Alexandria at noon on the Summer Solstice when there was no shadow in Syene, he could measure the circumference of the Earth!

High Noon on the Summer Solstice

[Click on the image to view full size (34k)]
(Graphic by R. Pogge)

**At Syene**:- The Sun is directly overhead, no shadows are cast at that moment.
**At Alexandria**:- The Sun is 7
^{12}/_{60}degrees south of overhead, casting shadows.

Since a full circle is 360 degrees, the arc from Alexandria to Syene is
thus approximately **1/50th** of a full circle (the sun angle above
divided by 360).

Therefore, the circumference of the Earth is 50 times the distance from Alexandria to Syene.

**Question 1**: How far is Alexandria from Syene?**5000 stadia****Question 2**: How big is 1 stadion?**600 Greek Feet**(length of a foot race in a Greek "stadium")

The best modern guess is that 1 stadion = 185 meters, based on the "Attic Stadion" measured from the Stadium at Athens. [Note: The definition of a stadion in modern units has been very contentious. A very compelling case has been made by Donald Engels (1985, American Journal of Philology, 106, 298, full article on jstor.org) for 1 stadion = 184.98 meters (based on 8 Roman miles to 1 stadion). Two numbers reappear in the literature: 148 meters/stadion, which was a mis-calculation by d'Anville in 1759, and 157.4 meters/stadion given by A. Letronne in 1851 (published posthumously). I used to quote the 157m/stadion number in my notes, which gives a circumference of the Earth much closer to that of the modern value. Engels' arguments, however, are very persuasive that Eratosthenes would have used the Attic Stadion based on the Stadium of Athens, which gives the conversion of ~185m/stadion that I quote in these notes.]

Putting Eratosthenes result into modern units, his estimate of the circumference of the Earth is as follows:

Circumference = 50 x 5000 stadia = **250,000 stadia**

250,000 stadia x 185 meters/stadion = **46,250 kilometers**

The modern measurement is **40,070 kilometers**.

Eratosthenes' estimate is only about 15% too large!

For more information, see the wikipedia entry for Eratosthenes

Ptolemy made a similar geometric estimate based on stellar (rather than
solar) measurements made earlier by Marinus of Tyre (by way of
Posidonius). This estimate yields a circumference of 28,800 kilometers,
which is ~28% *smaller* than the correct circumference (40,070 km).

[**Note:** By Ptolemy's time, we are actually on better grounds for
converting Classical Roman units to modern units, largely because many
Roman roads and measuring techniques have survived from antiquity.]

For more information, see the wikipedia entry for Ptolemy

- Early Christian rejection of the "pagan absurdity" of a spherical earth.
- This view was held sporadically until about 1300 AD.

By 1300, the works of Ptolemy and others arrived in Europe by way of Islamic Spain, and fully restored the Spherical Earth to respectability.

Contrary to popular myth, very few educated people after about 300 BC doubted that the Earth was a sphere. While a few early Christian thinkers did try to reject the idea, there is nothing in Christian beliefs that dictates a Flat Earth, in fact it says virtually nothing at all on the matter.

Eratosthenes' work was lost, except for a description of his method in an obscure source. [Note:The only description of Erathosthenes' method that survives from antiquity is from On the Orbits of the Heavenly Bodies written by Cleomedes in the 1st or 4th century AD (also known by the first word of its Greek title as the Meteora). We know little or nothing else about Cleomedes, not even his date or place of birth. It is clear that the numbers 5000 stadia and 1/50th of a circle have rounded off for convenience, contributing to the inaccuracy of the final result. However, while the derived circumference is off by ~15%, the actual difference in latitude between Alexandria (31°13' N) and Syene/Aswan (24°05' N) is 7°08', or about 0.0198 of the arc of a circle, within 1% of the value 0.02 (1/50th) quoted by Cleomedes for Eratosthenes. This is well within any measurement errors expected for the time.]

Ptolemy's estimate survived in his influential writings on geography. His estimate makes the eastern tip of Asia closer to the western tip of Europe than it would be otherwise, which (interestingly!) in part convinced Columbus that he might be able to reach Japan by sailing West from the Canaries.

Unlike many others of his time, however, Columbus not only argued for a smaller Earth, he also convinced the Spanish government to provide him the means to put his claims to the test.

Updated: 2011 September 25, Todd A. Thompson

Copyright © Richard W. Pogge, All Rights Reserved.