# Eratosthenes

Eumenis Megalopoulos | Jan 1, 2023

## Table of Content

## Summary

Eratosthenes of Cyrene († c. 194 B.C. in Alexandria) was an exceptionally versatile Greek scholar in the heyday of Hellenistic science.

He worked as a mathematician, geographer, astronomer, historian, philologist, philosopher and poet. On behalf of the Egyptian kings from the Ptolemaic dynasty, he directed the Library of Alexandria, the most important library of antiquity, for about half a century. With its excellent facilities, the library offered him excellent working conditions. He is most famous as the founder of scientific geography. His determination of the circumference of the Earth, based on careful measurements, is one of the most famous scientific achievements of antiquity. In addition to his research activities, collecting and organizing already existing knowledge was one of his main concerns. Of his numerous lost works, only a tiny fraction is known from quotations and reports of later authors, which makes an appreciation of his life's work very difficult.

Eratosthenes was the first ancient scholar to call himself a "philologist". By philology he understood not only occupation with linguistics and literature, but in a more general sense a versatile scholarship. Characteristic for his unbiased attitude towards ingrained convictions is his criticism of the poets, which did not spare even a highest-ranking authority like Homer. He did not believe that the poets' descriptions had any truth in them, since their aim was only to entertain and not to instruct.

Eratosthenes came from the city of Cyrene in present-day Libya. His birth can be narrowed down to the period between 276 and 273 BC. He went to Athens to study. His teachers were the grammarian Lysanias of Cyrene, the Stoic philosopher Ariston of Chios and the Platonist Arkesilaos. Ariston, who was only interested in ethics and considered scientific studies unimportant, does not seem to have had a lasting influence on Eratosthenes. Far stronger were apparently the impressions Eratosthenes received from the thinkers of the Platonic Academy, for his later statements on philosophical subjects prove him to be a Platonist. However, he does not seem to have been a regular member of the Academy. In addition, the famous scholar Kallimachos of Cyrene is also mentioned in ancient tradition as a teacher of Eratosthenes, but this information is hardly credible. Other philosophers who impressed Eratosthenes were Arkesilaos' student Apelles of Chios and the Cynic Bion of Borysthenes. An unclear and controversial, chronologically problematic remark of Strabon about a relationship of Eratosthenes to the Stoic Zenon of Kition does not have to be interpreted in the sense of a teacher-pupil relationship.

Soon after his accession to power, probably around 245, the Egyptian king Ptolemy III Euergetes brought Eratosthenes, who was only about thirty years old, from Athens to his residence in Alexandria. Apparently, the young scholar already enjoyed an excellent reputation at that time, whereby his poetic and mathematical-philosophical achievements were in the foreground; his geographical, philological and historical works were only written later. The king appointed him director of the Library of Alexandria after his predecessor in that post, Apollonios of Rhodes, resigned over disagreements with Ptolemy III. From about the mid-thirties, Eratosthenes taught the king's son and future successor, Ptolemy IV Philopator, who ascended the throne in 222.

There is a lack of reliable news about Eratosthenes' later life. He remained in charge of the library until the end of his life. There are different reports about his death. The Suda, a Byzantine encyclopedia, reports that he ended his life by refusing to eat because of blindness. Such a death was considered worthy of a philosopher at the time. The poet Dionysius of Kyzikos, on the other hand, who shortly after the death of Eratosthenes wrote a poem on the deceased - probably as an epitaph - wrote: "Quite mild old age extinguished you, not debilitating disease". Dionysius thus assumed that the cause of death of the approximately eighty-year-old was old age; perhaps he wanted to counter the rumor that it was a suicide. Eratosthenes was buried in Alexandria.

Despite his fame and extraordinary erudition, Eratosthenes did not become the founder of his own school. Of the four persons named in the Suda as his students, three cannot be identified with certainty, and were thus hardly important scholars. The fourth is the prominent grammarian Aristophanes of Byzantium, who succeeded Eratosthenes as director of the Library of Alexandria.

Eratosthenes wrote numerous works, but only fragments of them have survived. His views and achievements are therefore known only from these fragments and other information in ancient literature. In his intellectual development three phases can be roughly distinguished. In the first phase he dealt intensively with philosophy (especially Platonism), in the second natural science came to the fore, and in the third his interests shifted to philology. Constant features of his work were the preoccupation with scientific problems and the special attention paid to cultural-historical aspects of his fields of research.

## Astronomy

Three astronomical writings of Eratosthenes are known, but only fragmentary preserved:

The Greeks assumed a spherical shape of the earth long before Eratosthenes. Already Aristotle dealt with the question of its circumference. He referred to "mathematicians" not mentioned by name, who had determined a circumference of 400,000 stadia, a probably rather estimated than calculated number. The exact length of the stadium used by the "mathematicians" is unclear, so different numbers are cited when converting to kilometers. A few decades later (after 309 BC), an explorer - possibly it was Dikaiarchos - determined a circumference of 300,000 stadia. Eratosthenes is the only scholar of antiquity for whom a scientifically substantiated measurement is attested. The conditions for it were excellent: He had excellent knowledge in both mathematical and geographical fields, had access to the relevant literature already available in the library, and could rely on the king's support in carrying out the elaborate measurements. The result was 250,000 stadia; later he changed it to 252,000.

The procedure of Eratosthenes has been handed down in a summarizing and simplifying description by the imperial astronomer Kleomedes. It consisted, if one follows this account, of the following steps: He assumed that the Egyptian cities Alexandria (on the Mediterranean coast) and Syene (today's Aswan, the southernmost city of the country) lie on the same meridian (longitude). The distance between two measuring points in the two cities, determined by Eratosthenes, was 5000 stadia, according to his knowledge. Since Alexandria had only been founded in the 4th century, he could not rely on data in ancient Egyptian literature for the distance, but probably had the distance of his two measuring points accurately measured by royal pedometers. At both places he set up a gnomon, a metal hemisphere equipped on the inside with a graduation and a vertical pointer to read the shadow that was created. The measurement of the sun's altitude above the horizon was made with these devices at noon on the day of the summer solstice. It showed that the shadow pointer cast no shadow in Syene, so the sun was exactly in the zenith there. In Alexandria at that time the sun was the "fiftieth part" of a full circle from the zenith, that is, according to the present division of circles into 360 angular degrees 7° 12′. Thus, one had to travel 5000 stadia south to cover one fiftieth of the circumference of the earth. This resulted in a value of 50 × 5000 = 250,000 stadia for the circumference of the earth.

A considerable inaccuracy results from the fact that Alexandria and Syene in reality do not lie on the same meridian; Syene is located about 3° east of Alexandria. Since a value of 5000 stadia was measured for the distance between the two cities, a point lying exactly on the meridian arc of Alexandria would have resulted in a distance of 4615 stadia and thus for the circumference of the earth an amount of 50 × 4615 = 230,750 stadia. The error thereby is 7.7%.

It is unclear how long the "stadium" length measure used for the measurement was. It can hardly have been the "Olympic" stadium, which was about 185 meters long, because then the pedometers would have made a mistake of several days' travel when determining the distance between the two cities, which was actually 835 km as the crow flies. Therefore, numerous researchers assume that the measure of length used was much shorter. Conjectures vary between 148.8 and 180 meters. A particularly often mentioned number, which is derived from a statement in Pliny the Elder's Naturalis historia, is 157.5 meters. If one assumes the actual distance of 835 km, one comes to 835,000 m : 5000 = 167 m for the stadium.

For the accuracy of the determination of the circumference of the Earth, however, the unit of length used does not matter: According to the experimental concept and the measurement, it is fifty times the distance from Alexandria to Aswan, i.e., according to today's units, 835 km times 50 equals 41,750 km, which is very close to the actual value (40,075 km at the equator, 40,008 km over the poles). The error is about 4.2 %.

Two inaccuracies in the assumptions underlying the calculation are not significant:

Eratosthenes determined the obliquity of the ecliptic. The ecliptic is the apparent circular path of the sun projected on the imaginary celestial sphere in the course of a year; its obliquity is the inclination of its plane with respect to the plane of the equator. The value of this angle (in the time of Eratosthenes it was 23° 43′ 40″. Already in the 5th century BC Oinopides of Chios had arrived at 24°; Eratosthenes improved the accuracy of measurement. He determined as the angular distance between the two tropics 11 83 {\displaystyle {\tfrac {11}{83}} of the full circle (360°), i.e. 47° 42′ 40″, from which a value of 23° 51′ 20″ is obtained for ε by halving. How he arrived at this result is unknown, the hypotheses considered in research on this matter are speculative.

## Geography

Eratosthenes wrote only one geographical text, the Geography (Geōgraphiká) in three books. This writing, which was considered a standard work throughout antiquity, has also been preserved only in fragments. It was the most famous and influential of his works, since scientific geography began with it. It was probably he who coined this previously unattested term. For him, geography literally meant "the drawing (gráphein) of the earth", by which he meant not only the mere description of the earth's surface, but also cartographic recording, measuring, dividing and localizing. In doing so, he built on the knowledge that he had already presented in the treatise On the Measurement of the Earth, which dealt with geography from an astronomical point of view.

First he described the basics of geography including its history. In his argument with the views of earlier naturalists, he allowed only the mathematical-physical approaches and rejected the claims of the poets. He insinuated to the poets that they aimed only at entertainment and not at instruction. Therefore, he considered their geographical indications worthless. This criticism was directed especially against the authority of Homer, who had not been familiar with the geographical conditions outside Greece.

Then Eratosthenes presented his own views. Apparently, he explained the geographical consequences of the findings presented in his treatise on earth surveying. He presented probably all known lines of evidence for the spherical shape of the earth and discussed the distribution of water and land on the earth's surface. That the ratio of water and land is not constant was clear to him thanks to geological observations; from findings of fossilized shells he concluded that the Libyan Desert was once a sea. He shared the idea, already widespread in the time of the Pre-Socratics, that the Oikumene (the known, populated part of the Earth's surface) was a huge island surrounded by the ocean. From this he concluded that, theoretically, one could reach India by sea from the Iberian Peninsula across the Atlantic Ocean, if the size of the ocean permitted such a journey. He tried to determine the longitude and latitude of the island. For the maximum length, he arrived at 77,800 stadia by adding known or estimated distances, and for the maximum width, he arrived at 38,000 stadia. He designed a coordinate system with meridians and parallel circles, which provided the basis for his map of the inhabited world, which he presented and explained in the third book.

He drew his knowledge of distant countries from the voyage reports that were available to him. He critically reviewed their often inaccurate or erroneous information, and then evaluated them for his cartographic project, as far as he found them credible and coherent. His position as head of the extraordinarily well-equipped library of Alexandria - the best in the ancient world - gave him the unique opportunity to make use of the entire wealth of information available at that time in the descriptions of seafaring and countries.

He divided the Oikumene by the diaphragm, a parallel to the equator, which ran through the columns of Heracles, into a northern and a southern part. Thus he abandoned the conventional division into three continents. In the further division he distinguished at least four large country complexes, which he called "seals" (sphragídes, plinthía). Africa he considered as a right-angled triangle. He was less informed about southwestern Europe than about the Orient, about which relatively detailed information was available since the campaigns of Alexander the Great and the Diadochi. For the northwest, he relied on the travelogue of Pytheas, which was resented by ancient critics, because Pytheas was not considered very credible. As a cause for the lack of reliable reports about the west he called the xenophobia of the Carthaginians. His knowledge of the north and northeast was inadequate; he considered the Caspian Sea to be a gulf of the northern world ocean. He did not limit his description of the earth to topographical facts, but included cultural and economic geography as well as historical and political circumstances.

## Mathematics, Music Theory and Metaphysics

The philosopher and mathematician Theon of Smyrna quotes two passages from a work by Eratosthenes entitled Platōnikós, which has not survived. To which literary genre the Platonikos belonged is disputed. Some scholars have thought of it as a commentary on Plato's dialogue Timaeus, but Eratosthenes does not seem to have limited himself to a discussion of only a single work of Plato. It has often been assumed that it was a dialogue in which Plato was the main sub-speaker, but then, according to ancient usage, the writing would have to be called Platon and not Platonikos. Probably Platonikos is to be understood in the sense of Platonikos logos (writing about Plato). It was probably a manual intended to facilitate access to Plato's works for a wider audience by clarifying terms and explaining difficult passages.

It dealt primarily with mathematical questions; among the concepts discussed were distance, ratio, continuous and discontinuous proportion, mathematical mean, prime number, and point. The focus was on the theory of proportions, in which Eratosthenes saw the key to Platonic philosophy. Mathematical knowledge meant for him at the same time philosophical. The tool of the ratio equation ("a relates to b like c to d"), which he called "analogy", should also help to gain extra-mathematical knowledge. He generally aimed at solving problems by finding analogies in the sense of ratio equations. He thought that he had found the connecting link between the "mathematical" sciences (arithmetic, geometry, astronomy, music theory) in proportion, since all statements of these sciences could ultimately be traced back to statements about proportions.

As the one is the starting point (archḗ) and the primal element (stoicheíon) of the numbers and thus of the quantity and as the point is the not resolvable, not retraceable element of the length, for Eratosthenes the equality (as primal ratio 1 : 1) is the element and the origin of all ratios and proportions. The numbers originate by addition and the different ratios by enlargement of the members of the initial ratio; the line, on the other hand, cannot be brought forth as a joining of single points, since the single point has no extension, but it originates by a continuous movement of a point. This view was later criticized by the skeptic Sextus Empiricus.

Eratosthenes proposed a mathematical approximation solution for the problem of cube doubling, the "Delic problem", which could not be solved with compass and ruler. For the prime number research he used an algorithm, which permits to separate out all prime numbers from the set of all odd natural numbers, which are smaller than or equal to a given number. This method is known as the Sieve of Eratosthenes. However, he did not invent it - as one believed in former times -; it was rather already known, only the designation "sieve" comes from him.

A secondary topic of Platonikos was music theory, in which Eratosthenes transferred the theory of proportions to music. He succeeded so convincingly that he was counted among the most important authorities in the field of music in antiquity. The scholar Ptolemy handed down Eratosthenes' calculations for the tetrachord, which show that he used the "Pythagorean" tuning, which he refined. Eratosthenes also knew and took into account the system of the music theorist Aristoxenos. How he proceeded with his calculations, however, Ptolemy does not communicate.

Furthermore, Eratosthenes also dealt with metaphysical matters such as the doctrine of the soul in the Platonikos. Like the Platonist Krantor, by whom he was probably influenced, he held the view that the soul cannot be purely immaterial, but must also have something corporeal about it, because it is in the world of sensually perceptible things; moreover, it is always in a body. This is based on the consideration that the soul can grasp sensually perceptible objects only if it has a corresponding disposition in its own structure. According to this it is a mixture of two components, an incorporeal and a corporeal one.

The late antique mathematician Pappos mentions a mathematical writing of Eratosthenes with the title On Middle Members (Peri mesotḗtōn). Since this work is not mentioned anywhere else in ancient sources, it can be assumed that it is identical with Platonikos. In 1981, a medieval Arabic translation of a text by "Aristanes" (Eratosthenes) on mean proportionals was published. However, this is not the lost work On Mean Proportions mentioned by Pappos, but an alleged letter of Eratosthenes to King Ptolemy III on the doubling of cubes, also preserved in the original Greek text. The authenticity of the letter is disputed.

## Smaller philosophical writings

In addition to the Platonic, Eratosthenes wrote a number of smaller philosophical works, some of them in dialogue form, of which only the titles and some isolated quotations have survived:

## Historical works

According to the Suda, Eratosthenes wrote historical works (historíai). Only one work known by name can possibly be attributed to him: the History of the Galatians (Galatiká). Only sparse fragments have survived. This work is usually denied to Eratosthenes, but his authorship cannot be excluded. Since it could not have been written before 205, it would have to be an older work, if he is the author.

Eratosthenes is considered to be the first chronographer and the founder of scientific chronography (the creation of a temporal framework into which historical events are placed). However, his interest was apparently directed more towards the collection of news of cultural-historical interest than towards the determination of an absolute chronology. Therefore, his role in this field is not as prominent as often assumed in older research. Three relevant writings of his are mentioned in the sources:

## Seal

While Eratosthenes is famous today only as a scientist, in antiquity he was also appreciated as a poet and even compared with the lyricist Archilochos. He was recognized in this field for the elegance and formal flawlessness of his verses, but he was chalked up to a certain lack of inspiration. Six poems are mentioned in the sources:

## Philology

Apart from his geographical work, Eratosthenes' philological works attracted the most attention in antiquity. However, they have survived only in (relatively numerous) fragments. He was considered an authority in this field and was the first ancient scholar to call himself a "philologist," by which was meant not only philology in the modern sense, but scholarship in general. His extensive philological magnum opus was entitled On Ancient Comedy. In it, he discussed questions of textual criticism, the authorship of individual plays, performance time and practice, and explained historical backgrounds. He was primarily concerned with linguistic phenomena, with the study of individual words and expressions and dialectal peculiarities, which provided him with criteria for clarifying questions of authenticity and attribution. He dealt critically and sometimes sharply with the views of earlier authors. On the Old Comedy became a standard work.

Another writing was entitled Grammatiká (Grammatical). He also wrote a treatise on terms from the world of crafts, the Architektonikós (Craftsmanship), and one on the names of household utensils, the Skeuographikós (Outfitting), as well as a commentary on Homer's Iliad from a particular point of view that has not survived. According to the Suda, his grammatical works were numerous.

Eratosthenes also wrote letters in which he addressed philological and cultural-historical questions. Two fragments have been preserved.

## Ancient

The famous mathematician Archimedes was in correspondence with Eratosthenes. He honored him by dedicating to him his writing Methodology, his only work on methodology. There he called him an outstanding scholar, strongly emphasizing his philosophical merits and at the same time implying that he considered the mathematical achievements less important. Furthermore, Archimedes apparently sent Eratosthenes the poem The Cattle Problem, consisting of 22 distichs, about a difficult mathematical problem, which he wanted to present to the mathematicians in Alexandria; the authenticity of the verses, however, is not beyond doubt.

Eratosthenes' versatility was noticed by contemporaries and posterity, but it was not viewed only positively. Critics were of the opinion that he distinguished himself more by the breadth of his interests and his erudition than by depth of understanding or groundbreaking achievements in the individual fields. This assessment was also expressed in his nicknames or epithets, which were probably already common in his environment during his lifetime; the inhabitants of Alexandria were famous for their mockery. Among his opponents, he was considered a "know-it-all" (as opposed to a true philosopher). In this sense, they called him a "pentathlete" (péntathlos) - someone who is remarkable in several fields but is not the best in any of the individual disciplines. The nickname Beta - "the second" in the sense of "second-rate" - was also common. In view of this background, it is possible that the designation as "Second Plato" or "New Plato" was not only meant positively, but at the same time was meant to imply a lack of originality.

Apparently, he was only grudgingly acknowledged in wide circles. Scholars looked for and found weak points, which they used to criticize him, sometimes excessively. Strabon and Pliny the Elder generally praised his competence in various fields of knowledge, but when it came to concrete individual questions, Strabon found much to fault in his expertise and judgment. Sharp criticism of Eratosthenes was made by Polemon of Ilion, who wrote a multi-volume pamphlet On the Presence of Eratosthenes in Athens for this purpose. The surviving fragments indicate that Polemon accused his opponent of lacking knowledge of the cultural history of Athens. Other Eratosthenes critics were the famous astronomer and geographer Hipparchus of Nikaia and the mathematician Nicomedes. Hipparchus blamed the unreliability of the world map, and Nicomedes wrote a book On Conchoids against Eratosthenes, in which he polemicized against Eratosthenes and portrayed his inventions (such as the Mesolabos) as impractical. Polybios strongly reproached him for having trusted the report of Pytheas, criticized his localizations and distance indications in the Mediterranean area and defended the division of the Oikumene into three continents, which had been abandoned by Eratosthenes. Also the accusation of plagiarism, very popular in antiquity, was raised against Eratosthenes.

To what extent unfavorable assessments by ancient critics who applied a strict standard were justified, despite his undoubtedly significant achievements, is difficult to judge, since little of his works has survived. He liked to polemicize, expressed himself sarcastically, and in turn became the target of attacks.

In the 2nd century, the geographer Dionysius of Alexandria (Dionysios Periegetes) wrote a doctrinal poem that offers a description of the world, for which the poet relies on information from Eratosthenes, among others. The poem received much attention in antiquity, in the medieval Byzantine Empire, and in the early modern period. Whether Dionysius had access to the original text of the Geographika of Eratosthenes or drew his knowledge from a middle source is unknown.

The poems Hermes and Erigone were famous in antiquity. The aftermath of Hermes was considerable, even among Roman authors. Cicero's Somnium Scipionis was probably inspired by Hermes, Vergil in his Georgica exploited Eratosthenes' depiction of the five zones of the heavens that Hermes perceived during his ascent. A contemporary of Eratosthenes named Timarchos wrote a commentary on Hermes in at least four volumes.

It is unclear whether a pictorial representation of Eratosthenes from antiquity has survived. A fresco was found in Villa Boscoreale depicting an ancient philosopher who is believed to be Eratosthenes. According to a controversial hypothesis, the Boscoreale frescoes, painted around the middle of the 1st century BC, are copies of a cycle of paintings commissioned by Ptolemy III, and are thus based on contemporary portraits of the persons depicted. Speculative is a conjecture of Konrad Gaiser, who thinks he can recognize Eratosthenes on a famous mosaic from the 1st century AD, which was found in Torre Annunziata in 1897 and is now in the National Archaeological Museum of Naples. Gaiser believes it is probably a copy of a painting made in Alexandria soon after Eratosthenes' death, where it decorated either his tomb or a room in the Museion.

## Modern times

When in the 17th century the Dutch astronomer and mathematician Willebrord Snel van Royen published a new method of determining the circumference of the Earth, he chose the title Eratosthenes Batavus (The Dutch Eratosthenes) for his work published in 1617. His contemporary Claude de Saumaise (Claudius Salmasius), an eminent classical scholar, was praised as the Eratosthenes of his time.

In 1822, the year of his doctorate, the philologist Gottfried Bernhardy published the first and to this day only collection of Eratosthenes fragments aiming at completeness. In the following period and even in the 20th century, research concentrated on individual questions. Bernhardy's youthful work, at that time a brilliant achievement, is today completely obsolete, but has not been replaced.

From today's perspective, Eratosthenes' consistently scientific way of thinking and working is particularly striking and has earned him special esteem in modern times. In the research literature, his pioneering achievements and his impartiality, conscientiousness and comprehensive education are appreciated. However, it is also pointed out that Eratosthenes did not excel in all the fields to which he turned; in a part of his works he shows himself mainly as a material compiling book scholar.

The asteroid (3251) Eratosthenes and a lunar crater are named after Eratosthenes. In addition, Eratosthenes Point in Antarctica has borne his name since April 2021.

A surveying expedition commissioned by Eratosthenes is the subject of Arno Schmidt's story Enthymesis or W.I.E.H..

The Förderkreis Vermessungstechnisches Museum awards the Eratosthenes Prize for outstanding work in the field of historical research in surveying, especially for theses and dissertations, as well as the Eratosthenes Honorary Prize for outstanding book publications.

Astronomical, Geographical and Mythographical

Seal

Historical

Mathematical and philosophical

Philological

## Sources

- Eratosthenes
- Eratosthenes
- Zur Datierung Klaus Geus: Eratosthenes von Kyrene, München 2002, S. 10–15; Pedro Pablo Fuentes González: Ératosthène de Cyrène. In: Richard Goulet (Hrsg.): Dictionnaire des philosophes antiques, Bd. 3, Paris 2000, S. 188–236, hier: 190f.; Giorgio Dragoni: Introduzione allo studio della vita e delle opere di Eratostene. In: Physis Bd. 17, 1975, S. 41–70, hier: 46–48.
- a b c et d Catastérismes C.U.F., p. VII-VIII.
- Catastérismes C.U.F., p. VII-IX.
- Ferguson 2001, s. 16
- Ratkaisuehdotuksen hän antoi kirjeessään Egyptin kuningas Ptolemaios III:lle. Ongelmaan liittyvän tarinan hän kertoo Platonikoksessa.
- Otros autores estiman su nacimiento en 273 a. C.: Manuel Lozano Leyva, De Arquímedes a Einstein, Edit: de Bolsillo, pág. 37.
- a b Roller, Duane W. Eratosthenes' Geography. New Jersey: Princeton University Press, 2010.