# Aristarchus and Eratosthenes

## Key Ideas:

Angular Units: Degrees (º), Minutes ('), & Seconds of arc ('')

Circles and Spheres: Circumference, Surface area, Volume

• Flat Earth and World Ocean

The Spherical Earth

• Appeal to perfect symmetry
• Demonstration by Aristotle
Aristarchus:
• Derivation of Earth-Moon distance.
• Derivation of Earth-Sun distance.

Eratosthenes :

• Derivation of the Earth's circumference.
• Claudius Ptolemy

## Measuring Angles

The Babylonians started the tradition of dividing the circle into 360 degrees because 360 is close to 365, the days in a year. Also, 360 is divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120 and 180 without having to use fractions.

1 degrees is divided into 60 Minutes of Arc (') and

1 arcminute is divded into 60 Seconds of Arc ("). So, 1 arcsecond is 1/3600 of a degree.

The Babylonians actually subdivided the degree as fractions of 60, for example: 7 14/60 degrees

Claudius Ptolemy introduced the modern notation of expressing angles in terms of minutes and seconds of arc:7º 14' 00"

## Circles and Spheres

Given a circle of radius R, the diameter is 2R, the circumference is 2π R. Given a sphere of radius R, the surface area is 4π R2 and the volume is (4/3)π R3. The simple thing to remember is that an area has dimensions of length squared (e.g., square meters) and a volume has dimensions of length cubed (e.g., cubic meters), so the scaling of surface area and volume for a sphere must scale with R2 and R3, respectively, since R is the characteristic length.

## The Four Corners of the Earth

The most common ancient theme is that of a Flat Earth surmounted by a hemispherical sky. For example,
• 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.
In some cultures, the canopy of the sky was held up by immense trees or gigantic mountains.

## Classical Greece

The Ancient Greeks were intoxicated by geometry, form, and symmetry.

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)

## Aristotle

Aristotle (384-322 BC) also proposed a spherical earth on geometric grounds, but backed up his assertion with physical evidence (described in his On the Heavens of 350BC. [Note: Aristotle's arguments for the spherical shape of the Earth are given in On the Heavens, Book 2, Part 14 (350 BCE). A modern translation by J.L. Stokes may be found online at MIT on the Internet Classics Archive. Aristotle argued that the Earth must necessarily be spherical because the weight of all its parts setting towards the center would naturally form a spherical shape. Today we would recognize the tendency to settle towards the center to be a consequence of gravity (Aristotle does not, of course, use that word). He noted the curved outline of the shadow of the Earth on the Moon, and the different heights of stars between orthern and southern regions, as mentioned in the lecture, but he also put forth a rather odd argument that one finds elephants in Africa ("Pillars of Hercules" is the Strait of Gibraltar) and India, arguing for "continuity of parts", meaning they are close together on the surface of a sphere (they are far apart across the mediterranian, but closer going west from Africa). The idea being the world conceived by Aristotle is small, but not so small that small changes in place result in dramatic changes in the heights of constellations.]
• 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.

## Aristarchus

Aristarchus used simple geometric arguments to show that the Moon was about 70 times the radius of the Earth away, and that the Sun was about 18-20 times the Earth-Moon distance away. Here is an excerpt from the lecture on this topic in .pdf format, provided by Professor Paul Martini. You can find more information here.

## Eratosthenes of Cyrene

Born in Cyrene (now Shahhat Libya) in 276 BC. He was the 2nd Librarian of Alexandria until his death around 195 BC.

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.

Eratosthenes knew that no shadows on the Summer Solstice meant that Syene was on the boundary of the northern tropic zone (the Tropic of Cancer).

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:

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?

Question 2: How big is 1 stadion?
600 Greek Feet (length of a foot race in a Greek "stadium")

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

The modern measurement is 40,070 kilometers.

Eratosthenes' estimate is only about 15% too large!

## Claudius Ptolemy (c. 140 AD)

Claudius Ptolemais (Ptolemy for short) was a Geometer and Astronomer of the late Classical Age in Alexandria. His work was immensely influential in later centuries, as we'll see in later lectures.

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.]

## Return of the Flat Earth

• Early Christian rejection of the "pagan absurdity" of a spherical earth.

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: 2014, Todd A. Thompson