Eratosthenes

• 1. Introduction

  ›

  Eratosthenes of Cyrene looked down into a well and understood the shape of the entire planet. Standing in Alexandria, he noticed that at noon on the summer solstice, a vertical rod still cast a shadow. But in the city of Syene, far to the south, rods cast no shadows at all, and sunlight fell straight to the bottom of the city-center well. From that single observation, he calculated the circumference of the Earth with a result that modern instruments confirm was remarkably close to correct. He was born in 276 BC in Cyrene, a city that is now part of Libya, and he died around 196 BC in Alexandria, at the age of eighty. In between, he coined the word geography, created the first world map drawn with parallels and meridians, founded scientific chronology, introduced the algorithm known as the Sieve of Eratosthenes, calculated the tilt of Earth's axis, and served as chief librarian of the greatest archive in the ancient world. His contemporaries could not quite decide what to make of him. Some called him Pentathlos, after the Olympians celebrated for excelling across many events. Others called him Beta, meaning second, because he was never seen as the leading expert in any single field. Both nicknames tell the same story about a mind that refused to stay in one place.

• 2. A City Built for Ambition

  ›

  Cyrene was founded by Greeks during the second half of the 7th century BCE, on terrain that gave it every advantage: proximity to the sea, a defensible position, abundant water, and rich soil. By the time Eratosthenes was born, the city had already passed through the hands of Alexander the Great, who took control in 332 BC, and then through a local civil war before Ptolemy I Soter, founder of the Ptolemaic Kingdom, seized it. Under Ptolemaic rule, Cyrene prospered on the export of horses and silphium, and it had long been known as a hub of Greek culture. The debate over Eratosthenes' own social origins is unresolved. Roller points to the rarity of both his name and his father Aglaos' name as signs of humble origins, while Matthew argues that his name meaning "lovely strength" points to noble upbringing, as does an education that began early and covered physical training, reading, writing, arithmetic, poetry, and music at the local gymnasium. What is not disputed is that by the late 260s BCE, he had left Cyrene for Athens to continue his studies. Strabo records that one of his teachers there was Zeno of Citium, the founder of Stoicism, though their time together was brief because Zeno died shortly after Eratosthenes arrived. He also studied under the cynic Aristo of Chios, the eclectic Bion of Borysthenes, Arcesilaus of Pitane, who had just been appointed head of the Platonic Academy, and Lysanias of Cyrene, a philologist and grammarian with a focus on Homer. The poet and librarian Callimachus likely crossed paths with him in scholarly debates, though probably never formally taught him.

• 3. The Library and Its Librarian

  ›

  In 246 BCE, Ptolemy III succeeded his father Ptolemy II, and Alexandria entered what would become its peak as an intellectual center. The post of chief librarian carried with it the role of royal tutor to the king's heir, making it the most prestigious academic appointment in the Greek world. The incumbent librarian, Apollonius of Rhodes, was forced into retirement by the new king, and Eratosthenes was summoned from Athens to fill the position. Roller suggests that his roots in Cyrene, the native city of Callimachus and of Queen Berenike, worked in his favor. He had spent nearly twenty years in Athens by that point, during which he had written Platonikos, an inquiry into the mathematics and music in Plato's philosophy, as well as the poetic works Hermes and Erigone. His Chronographies had investigated the key dates of the Trojan War, and his Olympic Victors had compiled a list of winners of the games. In Alexandria, his early focus was mathematics. Archimedes, who was closely affiliated with him, sent him material for comment, praised his contributions, and addressed his Method of Mechanical Theorems to Eratosthenes directly. Archimedes also sent him the famous Cattle Problem to present to the mathematicians of Alexandria. From that mathematical foundation, Eratosthenes expanded into geography, philosophy, rhetoric, literary criticism, grammar, poetry, and star lore, producing a body of work that spanned essentially every discipline the Library touched.

• 4. Measuring the World

  ›

  The measurement of Earth's circumference survives not in Eratosthenes' own words, since his original account has been lost, but in a simplified description preserved by Cleomedes. The method rested on two cities on approximately the same meridian: Alexandria and Syene, which is now Aswan. At noon on the summer solstice, a gnomon in Syene cast no shadow, while the same measurement in Alexandria showed a shadow angle of about 7.2 degrees, which is one-fiftieth of a full circle. The distance between the two cities had been measured by professional bematists and reported at about 5,000 stadia. Multiplying by 50 gave a total of roughly 250,000 stadia for the full circumference of the Earth. Matthew's analysis of the result puts it at approximately 40,338 km; the actual equatorial circumference is 40,075 km. Eratosthenes made two false assumptions: that Syene sat exactly on the Tropic of Cancer, and that it lay directly south of Alexandria on the same meridian. Syene is in fact one degree north of the Tropic and three degrees east of Alexandria. These errors partially cancelled each other out, which is why the final figure landed as close as it did. His method for determining Earth's axial tilt was similarly precise. He measured the angular distance between the two tropics as 47 degrees 42 minutes 40 seconds of arc; halved, that gives an obliquity of 23 degrees 51 minutes 20 seconds. The actual value at his time was 23 degrees 43 minutes 40 seconds, a figure that Greek astronomer Hipparchus, born around 190 BC, later credited to Eratosthenes. Hipparchus also credited him with inventing the armillary sphere, a physical model of the celestial sphere built from rings representing lines of latitude, longitude, and other astronomically significant features.

• 5. Mapping and Measuring the Earth

  ›

  Three volumes of Geography, known as the Geographika, laid out Eratosthenes' vision of the entire known world. In the Library he had access to travel accounts from Alexander's companions, including Nearchos, Onesikratos, and Ptolemy I himself, whose campaigns had opened large portions of the world to Greek knowledge. He divided the Earth into five climate zones: two freezing regions around the poles, two temperate bands, and a zone spanning the equator and the tropics. He placed a grid of overlapping lines across the surface, using parallels and meridians to connect locations and allow distance estimates between remote places. The Geography recorded the names of over 400 cities and plotted their positions, a scale of effort without precedent. The first book reviewed his predecessors. Eratosthenes broke with many contemporaries by rejecting Homer as a geographic authority, which angered scholars who regarded the world of the Odyssey as literally real. He also proposed that the Mediterranean had once been a vast inland lake, only becoming connected to the western ocean when a passage eventually opened. The second book contained his Earth-circumference calculation; Pliny the Elder described this as the moment when "The world was grasped." Julius Caesar later cited the Geographika in De Bello Gallico, referencing what Eratosthenes had written about the Hercynian forest. Strabo, the work's primary surviving commentator, complained that Eratosthenes was too mathematical in his approach; Hipparchus argued the opposite, that he had not made enough use of astronomical data. Of the original three books, only 155 fragments survive: 105 in Strabo, 16 in Pliny the Elder, and the remainder scattered in Byzantine sources.

• 6. Numbers, Primes, and Proportions

  ›

  Platonikos, almost certainly Eratosthenes' main mathematical treatise, survives only in extracts preserved by Theon of Smyrna in the Expositio rerum mathematicarum ad legendum Platonem utilium. Friedrich Solmsen summarizes its central argument: that the theory of proportions, which Eratosthenes called "analogy," was the unifying bond holding together arithmetic, geometry, astronomy, and music theory, because all statements in those fields could ultimately be traced back to statements about proportions. He saw the number one as the starting point and primary element of all numbers and quantity. His view of lines was more unusual: he held that a line cannot be produced by combining individual points, since a point has no extension, and that it must instead arise from the continuous movement of a point. The skeptic Sextus Empiricus later challenged this position. Eratosthenes also tackled the ancient problem of doubling the cube, which is unsolvable with compass and ruler alone. His solution was a mechanical line-drawing device he called the Mesolabio, and he presented a bronze model of it to King Ptolemy along with a letter and an epigram. The Sieve of Eratosthenes, an algorithm for identifying all prime numbers up to a given limit by iteratively marking multiples as composite, carries his name. Hans-Joachim Waschkies has argued that Eratosthenes did not invent the underlying method, which was already known, but rather coined the term "sieve" for it. His application of proportional theory extended into music as well: the scholar Ptolemy preserved his calculations for the tetrachord, showing that he refined the "Pythagorean" tuning system, and he also engaged with the work of the music theorist Aristoxenus.

• 7. The End in Alexandria

  ›

  Toward the end of his life, Eratosthenes served as advisor and companion to Arsinoe, the sister and wife of Ptolemy IV. He wrote a memoir about her, titled Arsinoe, which is cited by Athenaeus in the Deipnosophistae but is otherwise lost. Arsinoe was murdered at the age of thirty after her husband's death, in 204 BCE, at which point Eratosthenes was around eighty years old. The Arsinoe memoir is likely the last work he completed. According to the Suda, as Eratosthenes aged his eyesight failed. Losing the ability to read and to observe nature depressed him deeply, and he chose to starve himself to death. He died in Alexandria around 196 BCE. His epitaph, recorded by Dionysios of Kyzikos, mourns that he was buried far from his birthplace: "Mother Kyrene did not receive you into the paternal tombs, son of Aglaos, but you are buried as a friend in a foreign land, here on the edge of the shore of Proteus." The phrase "shore of Proteus" is a Homeric allusion to Egypt. Among the four students the Suda records as his: Aristophanes of Byzantium, who succeeded him as Librarian of Alexandria; the geographer Mnaseus of Patara in Lycia; the historian Menander, probably of Ephesos; and Aristis, who is otherwise unknown. His dating system, built on Olympiads and preserved in Chronographies and Olympic Victors, supplied dates for the fall of Troy at 1184/1183 BCE, the first Olympiad at 777/776 BCE, and Xerxes' invasion at 480/479 BCE, dates that scholars still regard as authoritative.