# Eratosthenes of Cyrene (276 B.C.-194 B.C.)

# Eratosthenes of Cyrene (276 b.c.-194 b.c.)

*Greek astronomer*

Using elegant mathematical reasoning and limited empirical measurement, in approximately 240 b.c., Eratosthenes of Cyrene (in current-day Libya) made an accurate measurement of the circumference of Earth. In addition to providing evidence of scientific empiricism in the ancient world, this and other contributions to geodesy (the study of the shape and size of the earth) spurred subsequent exploration and expansion. Ironically, centuries later the Greek mathematician and astronomer Claudius Ptolemy's erroneous rejection of Eratosthenes' mathematical calculations, along with other mathematical errors, resulted in the mathematical estimation of a smaller Earth that, however erroneous, made extended seagoing journeys and exploration seem more tactically achievable.

Eratosthenes served as the third librarian at the Great Library in Alexandria. Serving under **Ptolemy** III and tutor to Ptolemy IV, the head librarian post was of considerable importance because the library was the central seat of learning and study in the ancient world. Ships coming into the port of Alexandria, for example, had their written documents copied for inclusion in the library and, over the years, the library's collection grew to encompass hundreds of thousands of papyri and vellum scrolls containing much of the intellectual wealth of the ancient world.

In addition to managing the collection, reading, and transcription of documents, Eratosthenes' own scholarly work concentrated on the study of the mathematics related to Platonic philosophy. Although Eratosthenes actual writings and calculations have not survived, it is known from the writings of other Greek scholars that Eratosthenes' writings and work treated the fundamental concepts and definitions of arithmetic and geometry. Eratosthenes' work with prime numbers, for example, resulted in a prime number sieve still used in modern number theory. Eratosthenes' also contributed to the advancement of scientific knowledge and reasoning, including the compilation of a large catalogue of stars and the preparation of influential chronologies, calendars and maps. So diverse were Eratosthenes' scholarly abilities that he was apparently referred to by his contemporaries as "Beta," a reference to the fact that although Eratosthenes was well-grounded in many scholarly disciplines, he was seemingly second-best in all.

Eratosthenes is, however, best known for his accurate and ingenious calculation of the circumference of the earth. Although Eratosthenes' own notes regarding the methodology of calculation are lost to modern historians, there are tantalizing references to the calculations in the works of Strabo and other scholars. There are extensive references to Eratosthenes' work in Pappus's Collection, an a.d. third century compilation and summary of work in mathematics, **physics** , **astronomy** , and geography. Beyond making an accurate estimate of the earth's circumference, based on observations of shifts in the zenith position of the **Sun** , Eratosthenes also made very accurate measurements of the tilt of the earth's axis.

Apparently inspired by observations contained in the scrolls he reviewed as librarian, Eratosthenes noticed subtle differences in accounts of shadows cast by the midday summer sun. In particular, Eratosthenes read of an observation made near Syene (near modern Aswan, Egypt) that at noon on the summer solstice, the Sun shone directly on the bottom of a deep well and that upright pillars were observed to cast no shadow. In contrast, Eratosthenes noticed that in Alexandria on the same day the noon Sun cast a shadow on a stick thrust into the ground and upon pillars.

Based upon his studies of astronomy and geometry, Eratosthenes assumed that the Sun was at such a great distance that it could be assumed that its rays were essentially parallel by the time they reached the Earth. Although the calculated distances of the Sun and the **Moon** , supported by measurements and estimates made during lunar eclipses, were far too low, the assumptions made by Eratosthenes proved essentially correct. Utilizing such an assumption regarding the parallel incidence of light rays, it remained to determine the angular variance between the shadows cast at Syene and Alexandria at the same time of day. In addition, Eratosthenes needed to calculate the distance between Syene and Alexandria.

Given modern methodologies, it seems intuitive that Eratosthenes would set out to establish the values of angle and distance needed to complete his calculations. In the ancient world, however, Eratosthenes' empiricism, reflected in his actual collection of data, reflected a significant break from a scholarly tradition that relied upon a more philosophical or mathematical approach to problems. Moreover, Eratosthenes' solution relied upon a world-view, especially reflecting the spheroid shape of the world, that itself was subject to philosophical debate.

Eratosthenes ultimately determined the angular difference between the shadows at Syene and Alexandria to be about seven degrees. Regardless of how he obtained the distance to Syene (some legends hold that he paid a messenger runner to pace it off), Eratosthenes reasoned that the ratio of the angular difference in the shadows to the number of degrees in a circle (360 degrees) must equal the ratio of the distance (about 500 mi, or 805 km) to the circumference of the earth. The resulting estimate, about 25,000 mi (40,200 km), is astonishingly accurate.

In making his calculations, Eratosthenes measured distance in units termed stadia. Although the exact value of the stadia is not known, modern estimates place it in the range of 525 ft (160 m). Depending on the exact value of the stadia, Eratosthenes' estimate varied only a few percent from the modern value of 24,902 mi (40,075 km) at the equator. It is necessary to specify that this is the circumference at the equator because the earth is actually an oblate sphere (a slightly compressed sphere with a bulge in the middle) where the circumference at the equator is greater than the distance of a great circle passing through the poles.

Although Eratosthenes' calculations were disputed in his own time, they allowed the development of maps and globes that remained among the most accurate produced for

more than a thousand years. The interest in geography and geodesy emboldened regional seafaring exploration using only the most primitive navigational instruments. Moreover, Eratosthenes' calculations fostered the persistence of belief in the sphericity of the earth that ultimately allowed for the development of the concept of antipodes and an early theory of climatic zones dependent on distance from the earth's equator. Eratosthenes' work, encapsulating many of his theories, was reported to have been the first to use the term geography to describe the study of Earth.

# Eratosthenes of Cyrene

# Eratosthenes of Cyrene

**c. 285-c. 205 b.c.**

**Greek North African Mathematician, Astronomer, and Geographer**

Most famous of the librarians at Alexandria, Eratosthenes provided a measurement for Earth's circumference within 1% of the actual number. He also developed a method for finding prime numbers, and made contributions as an astronomer, geographer, philosopher, and poet.

Born in Cyrene, now part of Libya, Eratosthenes's father was named Aglaus, but these are the only known facts of his origins. He studied with the grammarian Lysanias, the philosopher Ariston of Chios, and the poet Callimachus (c. 305-c. 240 b.c.), second librarian of Alexandria. In his teens he traveled to Athens, where he may have studied both at the Academy established by Plato (427-347 b.c.) and the Lyceum of Aristotle (384-322 b.c.).

Eratosthenes, who at a later time might have been called a Renaissance man, displayed his talents as a poet in *Hermes* and *Ergion.* These in turn attracted the attention of Ptolemy III Euergetes (r. 246-221 b.c.), Greek ruler of Egypt, who invited Eratosthenes to Alexandria as tutor for the crown prince. Soon afterward, Ptolemy appointed Eratosthenes director of the city's famed library.

During his fruitful career, Eratosthenes contributed to mathematical knowledge with a technique for finding prime numbers that came to be known as the "sieve of Eratosthenes." Using this formula, a type of algorithm, he could pull out primes from an ordered list of the natural numbers. Not only did this method spark later investigation in number theory, but it could perhaps be cited as an early manifestation of computing techniques.

In working on the Delian problem, which involved finding mean proportionals as a way of doubling the cube, Eratosthenes developed what was called the mesolabe or "mean-finder," which made it possible to determine an infinite number of mean proportionals between two lines. He also wrote a number of mathematical texts, but the only piece of his writing that survives is a letter to Ptolemy III explaining the mesolabe.

Most famous among Eratosthenes's many achievements was his measurement of Earth's circumference. This he did by measuring the distance from Alexandria to the city of Syene in the south, which he determined to be 5,000 *stadia* (489 miles; 783 kilometers). By using calculations of the Sun's position over both cities during the summer solstice, he was then able to estimate that the difference between them represented 2% of Earth's circumference, the total of which he calculated at 252,000 stadia (24,662 miles; 39,459 kilometers).

In fact the true figure at the equator is 24,901.55 miles (39,842.48 kilometers), meaning that Eratosthenes was astonishingly close. At his time, of course, no expedition had ventured below the equivalent of 10° north; nor did anyone in the Old World know about the existence of the New. Certainly his figure implied the existence of a very large unexplored area on the other side of the world, and had Christopher Columbus (1451-1506) been using the numbers derived by Eratosthenes when he made his historic landing in the Caribbean some 1,700 years later, he would have realized that he had not reached Asia. Unfortunately, however, Hipparchus (190-126 b.c.) had rejected Eratosthenes's measurement in favor of a much smaller number, and Hipparchus's follower Ptolemy (c. a.d. 100-170) had propagated this misconception, which remained the received wisdom throughout the Middle Ages.

As for Eratosthenes, his now-lost manuscript *On the Measurement of the Earth* marked the foundation of geodesy, the branch of mathematics that deals with determination of Earth's size and shape, and the location of points on its surface. Among the topics of study within geodesy is the system of latitude and longitude, which Eratosthenes seems to have pioneered in his maps, the most accurate in the world at the time.

Believed to have developed the calendar that remained unchanged in the Greco-Roman world until the time of Julius Caesar (102-44 b.c.), Eratosthenes also established the first reliable method of dating events, by reference to the years of the Olympic festivals. In his later years, he went blind, and perhaps as a result of despair over his inability to any longer read the manuscripts of his beloved library, he starved himself to death in the ninth decade of his life.

**JUDSON KNIGHT**

# Eratosthenes of Cyrene

# Eratosthenes of Cyrene

The Greek scholar of natural history Eratosthenes of Cyrene (ca. 284-ca. 205 B.C.) was proficient in manyfields, but his most outstanding work was probably in mathematics and geography.

The Lexicon of Suidas (ca. 10th century A.D.) records the birth of Eratosthenes as the 126th Olympiad (276-272 B.C.), but since he was a pupil of Zeno (died 262-261) and since a number of authorities describe him as an old man of 80, it is more probable that he was born about 284 B.C. He studied grammar in Alexandria but was educated in philosophy in Athens, where he was influenced by the philosophers Arcesilaus and Ariston. About the age of 40 Eratosthenes was recalled to Alexandria by Ptolemy III to take charge eventually of the famous library of the Alexandrian Museum, succeeding Apollonius of Rhodes.

Eratosthenes wrote works of literary criticism (On Ancient Comedy), philosophy, history (establishing chronology as a scientific discipline), mathematics, astronomy, and geography. He also wrote a short epic dealing with the death of Hesiod, and Erigone, an elegy praised by Longinus. His Geographica comprises a history of geographical ideas, including a section on mathematical geography in which the division of the globe into zones was established and the inhabited portions were delimited. There were also some crude map-making attempts in his memoirs, and it is believed that Eratosthenes compiled a catalog of 675 stars.

Eratosthenes investigated arithmetical and geometrical problems. In his "sieve" method of distinguishing prime numbers, by which "the prime and incomposite numbers are separated by themselves as though by some instrument or sieve," there is the foundation for a logical theory of the infinite. The prime numbers were found by listing all odd numbers beginning with 3, then striking out every third number, every fifth number, and so on, with the remaining numbers being the primes. The much-attempted problem of the duplication of the cube, which dealt with the problem of finding the mean proportional between two lines, occupied Eratosthenes at an early date. To solve it, he constructed a bronze instrument called a mesolabe. He also applied geometrical methods, by ascertaining both the difference of latitude and the distance apart of two places that were supposedly located on the same meridian, to deduce the circumference of the earth. The size of the units of measure (stadia) that he employed is doubtful, but it is assumed that 10 stadia approximates 1 mile; he computed the circumference at 250,000 stadia, or 25,000 miles, very close to today's estimates.

The Alexandrian Age cultivated specialization, and Eratosthenes did not therefore win the approval of his contemporaries. He had achievements, however, which could not be denied, so he was called Beta (the second letter of the Greek alphabet), indicating that he was never "first-rate" at anything. During his last years he developed ophthalmia and became blind. The end came in Alexandria as the result of suicide by voluntary starvation.

## Further Reading

A brief discussion of Eratosthenes's life appears in Carl Boyer, *A History of Mathematics* (1968). Thomas L. Heath, *A Manual of Greek Mathematics* (1931), is a full treatment of Eratosthenes's mathematical research. See also Ivor Thomas, *Greek Mathematical Works,* vol. 1 (1939). □

# Eratosthenes of Cyrene

# Eratosthenes of Cyrene

**276-194 b.c.**

Greek astronomer and mathematician who wrote on geography, mathematics, philosophy, geology, chronology, and literary criticism, though only fragments of his texts remain. Eratosthenes studied at Athens and became head of the library of the Museum of Alexandria. He founded a system of chronology based on the fall of Troy, was an authority on Old Comedy, and introduced mathematics into the field of geography. He estimated the size of the Earth from shadows cast in bowls, and invented a simple method for finding prime numbers (Eratosthene's sieve).

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