# Eratosthenes

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Eratosthenes (Ερατοσθένης)

Eratosthenes (Ερατοσθένης) (276 BC - 194 BC) was a Greek mathematician, geographer and astronomer. His contemporaries nicknamed him 'Beta' because he was the second best in the world in many subjects.

## Short summary of Eratosthenes' life

Eratosthenes was born in Cyrene (now in Libya), but worked and died in Ptolemaic Alexandria. He is noted for devising a system of latitude and longitude and computing the size of the Earth.

Eratosthenes studied at Alexandria and for some years in Athens. In 236 BC he was appointed by Ptolemy III Euergetes I as librarian of the Alexandrian library. He made several important contributions to mathematics and science, and was a good friend to Archimedes. Around 255 BC he invented the armillary sphere, which was widely used until the invention of the orrery in the 18th century.

He is credited by Cleomedes in On the Circular Motions of the Celestial Bodies with having calculated the Earth's circumference ca. 240 BC, using trigonometry and knowledge of the angle of elevation of the Sun at noon in Alexandria and Syene (now Aswan, Egypt). In recent years this has become controversial for a number of reasons.

There are two theories concerning Plato's machine to solve the doubling of the cube problem. One theory is that Plato invented the mechanical solution to show how easy it is to devise such solutions, but the more widely held theory is that Plato's machine was invented by one of his followers at the Academy.

Eratosthenes is important in the story both because the history of the problem has been communicated through him and also for his own contribution to the problem. He erected a column at Alexandria dedicated to King Ptolemy with an epigram inscribed on it relating to his own mechanical solution to the problem of doubling the cube [2]:-

"If, good friend, thou mindest to obtain from any small cube a cube the double of it, and duly to change any solid figure into another, this is in thy power; thou canst find the measure of a fold, a pit, or the broad basin of a hollow well, by this method, that is, if thou thus catch between two rulers two means with their extreme ends converging. Do not thou seek to do the difficult business of Archytas's cylinders, or to cut the cone in the triads of Menaechmus, or to compass such a curved form of lines as is described by the god-fearing Eudoxus. Nay thou couldst, on these tablets, easily find a myriad of means, beginning from a small base. Happy art thou, Ptolemy, in that, as a father the equal of his son in youthful vigour, thou hast thyself given him all that is dear to muses and Kings, and may be in the future, O Zeus, god of heaven, also receive the sceptre at thy hands. Thus may it be, and let any one who sees this offering say "This is the gift of Eratosthenes of Cyrene".

Eratosthenes attempts to solve the three classical problems of Greek Antiquity may have led him to the geometry for which he is most famous

## Measurement of the Earth

Eratosthenes was told that, at noon on the longest day of the year, the sun shone straight down a well in the city of Syene, in southern Egypt. He deduced that Syene lay exactly on the tropic; and he measured the angle of the sun (on noon of that day) in Alexandria. That angle must represent the difference in latitude between Syene and Alexandria. Since he knew the distance between Syene and Alexandria, dividing by the number of degrees would give him the length of a degree of latitude; multiplying this by 360 would give the circumference of the Earth.

There are several systematic errors in this calculation. Syene is not exactly on the Tropic of Cancer, and is not directly south of Alexandria; nor is the Sun at infinite distance. (Eratosthenes knew the last, but we are not told he corrected for it.) More seriously, angles in antiquity could be measured only to degrees or quarter-degrees, and measurement of overland distances was worse.

We have several different, and incompatible, ancient accounts of the details of Eratosthenes' calculations. The Greek sources mostly agree that his result was 700 stadia; but what was a stadion? The Olympic stadion was 606 ft; almost exactly a 600th part of a degree — and we have no direct evidence that any Hellenistic geographer used any other. An error of one-seventh is not bad for a first attempt; but some modern scholars, however, have conjectured a shorter, Egyptian, stadion to save Eratosthenes from error. (Some Latin accounts record a final figure of 25,000 Roman miles, which would be about a twentieth too short.)

About 200 BC Eratosthenes is thought to have coined or to have adopted the word geography, the descriptive study of the Earth.

## Eratosthenes' contributions

Eratosthenes' other contributions include:

"Eratosthenes made major contributions to geography. He sketched, quite accurately, the route of the Nile to Khartoum, showing the two Ethiopian tributaries. He also suggested that lakes were the source of the river. A study of the Nile had been made by many scholars before Eratosthenes and they had attempted to explain the rather strange behaviour of the river, but most like Thales were quite wrong in their explanations. Eratosthenes was the first to give what is essentially the correct answer when he suggested that heavy rains sometimes fell in regions near the source of the river and that these would explain the flooding lower down the river. Another contribution that Eratosthenes made to geography was his description of the region "Eudaimon Arabia", now the Yemen, as inhabited by four different races. The situation was somewhat more complicated than that proposed by Eratosthenes, but today the names for the races proposed by Eratosthenes, namely Minaeans, Sabaeans, Qatabanians, and Hadramites, are still used."

• Since Eratosthenes never traveled to Yemen or the source of the Nile it is likely that at least this part of his writings came from the shelves of his library.

Eratosthenes was known under the name β, because he supposedly proved himself to be the second in the world in many fields. He was also reputedly known for his haughty character. In 195 BC he became blind and a year later he starved himself to death.

The fragmentary collection of Hellenistic sky-myths called Catasterismi (Katasterismoi) was given an attribution to Eratosthenes, a name to conjure with, to add to its credibility.

## Strabo on the geography of Eratosthenes

From the description of Eratosthenes' geography given by Strabo it would appear that he had access to accounts in his library which had measured the earth itself as far away as India with measuring lines in units he could express in stadia. (Such claims to actual measurement are not uncommon in ancient geography, especially in secondary sources like Strabo. Some modern geographers treat them all with skepticism, as exaggerations or misunderstandings in antiquity.)

• " it was particularly apparent from my former discussion that the summary account set forth in the third book of his geography by Eratosthenes of what was in his time regarded as India, that is, when Alexander invaded the country, is the most trustworthy; and the Indus River was the boundary between India and Ariana, which latter was situated next to India on the west and was in the possession of the Persians at that time; for later the Indians also held much of Ariana, having received it from the Macedonians. And the account given by Eratosthenes is as follows:

"...India...it has been measured with measuring lines, and there is a royal road of ten thousand stadia."

• "...Of the officials, some are market commissioners, others are city commissioners and others are in charge of the soldiers. Among these, the first keep the rivers improved and the land remeasured, as in Egypt, and inspect the closed canals from which the water is distributed into the conduits, in order that all may have an equal use of it. The same men also have charge of the hunters and are authorized to reward or punish those who deserve either. They also collect the taxes and superintend the crafts connected with the land -- those of wood-cutters, carpenters, workers in brass, and miners. And they make roads, and at every ten stadia place pillars showing the by-roads and the distances.
• "The entire length of the country, at its minimum, will be sixteen thousand stadia, as taken from the Register of Days' Journeys that is most commonly accepted, according to Eratosthenes; and, in agreement with him"
• This is also taken to suggest that the Register is measured in days' journeys, or perhaps in parasangs, and Eratosthenes converted it into stadia.