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# Eratosthenes Calculates the Circumference of the Earth

## Overview

The measurement of distance became increasingly important to ancient and classical civilizations as their territorial and cultural horizons expanded. Using elegant mathematical reasoning and limited empirical measurement, in approximately 240 b.c., Eratosthenes of Cyrene (276-194 b.c.) accurately measured the Earth's circumference. This feat was more than simply a scientific achievement. Eratosthenes's calculation, and others like it, contributed to the field of geodesy (the study of the shape and size of the Earth) and helped spur subsequent exploration and expansion. Ironically, centuries later the Greek mathematician and astronomer Claudius Ptolemy would reject Eratosthenes's mathematical calculations, which, when combined with other mathematical errors that he made, produced a mathematical estimation of a smaller Earth that, however erroneous, made extended seagoing journeys and exploration seem more feasible.

## Background

Eratosthenes, who served under Ptolemy III and tutored Ptolemy IV, was the third librarian at the Great Library in Alexandria. This post was of considerable importance because the library was the 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. Over the years, the library's collection grew to encompass hundreds of thousands of papyri and scrolls that contained much of the intellectual wealth of the ancient world.

In addition to managing the collection, reading, and transcribing documents, Eratosthenes studied and wrote on many topics. Although all of his writings and calculations have been lost, we know from the work of other Greek scholars that Eratosthenes studied the fundamental concepts and definitions of arithmetic and geometry. One of his discoveries was the "sieve of Eratosthenes" a method for determining prime numbers that's still used. Eratosthenes also compiled a star catalogue that included hundreds of stars, devised a surprisingly modern calendar, and attempted to establish the date of historical events, beginning with the siege of Troy. So diverse were his abilities that his contemporaries apparently referred to him as "Beta"—the second letter of the Greek alphabet—implying that he was well versed in too many scholarly disciplines to be the best at any one of them.

Eratosthenes is best known for his astonishingly accurate and ingenious calculation of the Earth's circumference. Although his own notes on the method of calculation have been lost, there are tantalizing references to them in the works of Strabo and other scholars, including in Pappus's Synagoge or "Collection," a compilation and summary of work in mathematics, physics, astronomy, and geography published in the third century a.d. Beyond accurately estimating the Earth's circumference, based on observed differences in the Sun's zenith position, Eratosthenes also made a amazingly precise measurement of the tilt of the Earth's axis.

## lmpact

Apparently inspired by observations in the scrolls he reviewed as librarian, Eratosthenes noticed subtle differences in the accounts of shadows cast by the midday summer Sun. In particular, he read of an observation made near Syene (near modern Aswan, Egypt) that at noon on the summer solstice the Sun shone directly into 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 upon both pillars and a stick thrust into the ground.

Based on his studies of astronomy and geometry, Eratosthenes assumed that the Sun was at such a great distance its rays were essentially parallel to the Earth by the time they reached it. Although the calculated distances of the Sun and Moon, supported by measurements and estimates made during lunar eclipses, were far too low, the Eratosthenes's assumptions proved essentially correct. Assuming the parallel incidence of light rays, he needed to determine the difference between the angles of the shadows cast at Syene and Alexandria at the same time on the same day. In addition, he had to calculate the distance between the two cities.

Viewed from the perspective of modern science, it seems intuitive that Eratosthenes would try to determine precise values for the angles and distances needed to complete his calculations. In the ancient world, however, this type of objective science was far different from the prevailing scholarly tradition, which took a more philosophical or mathematical approach to problems. Moreover, Eratosthenes's belief that the Earth was round was itself subject to debate.

To perform his calculation, Eratosthenes determined the angular difference between the shadows at Syene and Alexandria to be about 7°. He determined the distance to Syene at about 500 miles (805 km), possibly, as some legends hold by paying a 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°) must equal the ratio of the distance to the circumference of the Earth. The resulting estimate, about 25,000 miles (40,234 km), is astonishingly accurate.

In making his calculations Eratosthenes measured distance in stadia, a unit of measure based on the Greek footrace, or stade. These units varied from place to place in the ancient world. Eratosthenes almost certainly used the Attic stade, which was based on one circuit of the track in the Athens stadium, 606 feet, 10 inches (185 m), or a little over a tenth of a mile. Using this measure, Eratosthenes was able to calculate a circumference that varies only a few percent from the modern value of 24,902 miles (40,076 km) at the equator. It is necessary to specify that this is the circumference at the equator because the Earth is actually an oblate (slightly compressed) sphere with a bulge in the middle, making the circumference at the equator greater than it would be if measured around the poles.

Eratosthenes's theories and calculations were published in his Geography, a title that reflects the first-known use of the term, which means "writing about the Earth." Although his calculations were disputed in his own time, they allowed the development of maps and globes that remained among the most accurate produced for over a thousand years. This, in turn, sparked interest in geography and geodesy, and emboldened regional seafaring exploration using only the most primitive navigational instruments. Eratosthenes's work, moreover, helped solidify belief in a round Earth, and promoted an early theory that the relative warmth or coolness of a locations climate was determined by its distance from the equator. Geography also supported the concept of antipodes—undiscovered lands and peoples on the "other side" of the world.

Eratosthenes's work may have inspired the Greek astronomer and geographer Claudius Ptolemy to make his own determination of the circumference of the Earth in the second century a.d. Unfortunately, he rejected Eratosthenes's calculations and substituted errant values asserted by the geographer Posidonius (130-50 b.c.). In this system, a degree covered what would now equal approximately 50 miles (80 km), instead of Eratosthenes's more accurate estimation of about 70 miles (113 km) per degree at the equator. Although Ptolemy went further than Eratosthenes in his calculations, measuring the movement of shadows over varying time intervals, his inaccurate assumptions and measurements skewed his final values to a less accurate and much smaller circumference of approximately 16,000 miles (25,750 km) .

Ptolemy published these inaccurate numbers in his Almagest, which was written about a.d. 150 and remained the world's most influential work on astronomy and geography throughout the Middle Ages. Ptolemy's well publicized error of a smaller Earth eventually the possibility of surviving a westward passage to India seem more possible. Although the point is disputed by many scholars, Ptolemy's mistake may have played a part in Columbus's decision to seek a westward route to India.

K. LEE LERNER

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Heath, T. L. A History of Greek Mathematics. Oxford: The Clarendon Press, 1921.