The Rise and Fall of Alexandria

by Justin Pollard

  The Hermes and the Erigone have themes we might perhaps associate more with Eratosthenes, as both are bound up with tales of the stars. In his Hermes he tells us of the birth of the messenger god, his miraculous infancy (he invented the lyre on the first day of his life), and his ascent to the planets. The Erigone tells the tragic tale of the daughter of Icarius. The latter had been given the first vine by Dionysus and made wine from it, but when he had given this to his neighbors to drink, the effects of the alcohol made them believe they were being poisoned, and they killed him. His daughter Erigone was guided by their dog, Maera, to the body and was so overcome with grief that she hanged herself on the spot. All the other virgins of Athens then began to follow suit until her ghost was appeased and she, her father, and her dog were taken up into the heavens to form the constellations Boötes (Icarius), Virgo (Erigone), and Canicula (Maera).

  But it was in the realms of mathematics and science that this most versatile of Alexandrian librarians really shone. In his Platonicus Eratosthenes set out the whole mathematical system that, as he saw it, underlay Plato’s philosophy. This included not only basic definitions of geometry, as first propounded by Euclid, but also the nature of arithmetic, and even the meaning and construction of music. He also produced practical tools for understanding some of the more arcane areas of mathematics, some of which are still used today. Among these is “Eratosthenes’ sieve,” a reiterative method of calculating prime numbers that is still used by mathematicians who study the branch of the subject known as number theory.

  Strangely enough, what made Eratosthenes most famous in his own day was a mechanical solution to a problem we would today consider ethereal: the Delian problem, or the “doubling of the cube.” This was one of three mathematical problems that deeply concerned ancient Greek mathematicians, the other two being “trisecting the angle” and “squaring the circle.” Although we have so little of Eratosthenes’ original work surviving, we do have his own explanation of this problem, preserved in a forged letter (nominally from himself to Ptolemy III Euergetes) cobbled together sometime after his death from pieces of his own writing. Here then we can just briefly hear the voice of an Alexandrian librarian:

  When the god proclaimed to the Delians through the oracle that, in order to get rid of a plague, they should construct an altar double that of the existing one, their craftsmen fell into great perplexity in their efforts to discover how a solid could be made the double of a similar solid; they therefore went to ask Plato about it, and he replied that the oracle meant, not that the god wanted an altar of double the size, but that he wished, in setting them the task, to shame the Greeks for their neglect of mathematics and their contempt of geometry.

  Quoted in Theon of Smyrna, On Mathematics

  Useful for the Understanding of Plato

  So pleased was Eratosthenes with his own solution to this problem that he set up a column in Alexandria inscribed with an epigram explaining how he’d done it. It ends, however, with a wry comment on his work teaching the recalcitrant Ptolemy IV:

  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 he 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.”

  Quoted in T. Heath, A History of Greek Mathematics

  If it was mathematics that made Eratosthenes’ name in his own day, it is an entirely different area of study that makes it still stand out today—geography, a subject which on his arrival in the city barely existed. There had of course been travelers throughout the ancient world for many centuries before. Herodotus had extensively explored Egypt, Babylon, and the Aegean and left books detailing what he had seen, but these were more travelers’ tales, more anthropology with a sprinkling of mythology, than geography. Equally, traders and merchants had traveled long distances over established trade routes, but these had been arrived at by trial and error rather than cartography, local guides covering each part of the route but none truly knowing it in its entirety. Aristarchus may have set this earth in motion, but no one yet knew exactly what it was that we were traveling on. With Eratosthenes that changed forever.

  For him the fundamental question was: What was the shape and size of the earth? Was it, as Thales had claimed, just a flat disk floating in an infinite ocean? Did it have an edge? What happened if one continued walking forever in one direction? Where and how did it end? Until this time these questions may have seemed purely academic or perhaps more a matter of faith than observation, but Eratosthenes knew there were clues to the answers, and clues that lay not at the far ends of the earth, but on his doorstep in Alexandria.

  From the top of the forty-story Pharos lighthouse, where, no doubt, the philosophers of the museum sometimes gathered to gaze across the vastness of the sea and debate the nature of the world, a number of strange phenomena were visible, phenomena that would make any Skeptic question his senses. On extremely clear days, as they watched ships sail north from the harbor across the Mediterranean, they would have seen that the hull of the ship would slip beneath the horizon before the mast. Likewise when ships were first spotted heading toward the beacon of the Pharos, their masts were the first things to appear, and only later could the hull be made out. A similar problem faced those energetic enough to run up the Pharos after first spotting a ship traveling along the horizon, for it was clear that a ship that appeared little more than a mast top from the base of the lighthouse appeared as a complete vessel from the top. Somehow, between the top and bottom of the lighthouse, something was getting in the way of the view. Could that obstruction be the curvature of the earth itself ? In the night sky astronomers had also seen a further clue in the form of lunar eclipses. When the earth was between the sun and the moon and cast her shadow across it, the encroaching dark red gloom was clearly curved at its edge, implying that the shadow of the earth was curved. That, and the observations from the Pharos, could mean only that the earth itself was a sphere. The later geographer Strabo, drawing on Eratosthenes’ work, thought the explanation was obvious:

  It is obviously the curvature of the sea that prevents sailors from seeing distant lights at an elevation equal to that of the eye; however, if they are at a higher elevation than that of the eye, they become visible, even though they be at a greater distance from the eyes; and similarly if the eyes themselves are elevated, they see what was before invisible. This fact is noted by Homer, also, for such is the meaning of the words: “With a quick glance ahead, being upborne on a great wave [he saw the land very near].” So, also, when sailors are approaching land, the different parts of the shore become revealed progressively, more and more, and what at first appeared to be low-lying land grows gradually higher and higher.

  Strabo, Geography, book 1, chapter 20

  Of course the idea that the earth was a globe had profound implications. It implied that there was something that made us, and everything else on the planet, stick to its surface—something we now call gravity. It also implied that there were two ways of getting from any one place to another—by going in opposite directions—and this meant that the whole nature of geography had to be changed. And the man to start that ball rolling was Eratosthenes.

  The problem that Eratosthenes set himself was one which even today seems impressive. If the earth was a sphere, then he wanted to know just how big that globe was. And he set out to measure it without the help of artificial satellites, laser range finders, Global Positioning Systems, or even a theodolite. Instead, without ever leaving Egypt, he measured the earth by applying simple logic to careful observation, aided only by a stick and a well.

  Eratosthenes’ solution was as stunning as it was simple. He had heard among travelers’ tales that there was a deep well at Syene (modern Aswan), far to the south on the very edge of Ptolemy’s Egypt, near the first cataract of the Nile. The well, or at least an ancient well, is
still there, just a short felucca trip from Aswan itself across to Elephantine Island. Here a wide stone-lined shaft, lined with spiral stairs, plunges down to the water table. On just one day of the year, at midday, the sun shines directly down this well, illuminating the bottom. As water always lies flat, Eratosthenes knew that for the sun to be reflected back required it to be directly overhead, its rays hitting the surface of the earth there at 90 degrees at just that moment. Now all he had to know was the distance from Syene to Alexandria.

  We do not know whether the librarian personally made the six-hundred-mile journey up the Nile in the blazing midsummer sun to see for himself if the story was true. However, for a man taught to take nothing as read but to observe and question everything, it seems highly likely that he did. Having ascertained that the story was true, Eratosthenes then dispatched a royal official to begin the laborious job of making the next calculation he needed. Among Ptolemy II’s huge civil service staff were the royal pacers, men trained to walk, or run, with exactly even, measured steps, so that large distances could be accurately measured. At Syene one of these men now began the long walk back to Alexandria while Eratosthenes, as royal tutor and librarian, no doubt enjoyed a more leisurely journey home down the Nile on one of Ptolemy’s barges.

  Eventually the royal pacer returned to Alexandria, carrying in his head the number of paces he had taken from the well to the place in a royal park where Eratosthenes had asked him to stop. This place was then marked and the distance recorded. The librarian knew that all he had to do now was wait.

  The following summer on the very same day—the summer solstice—when the sun was directly overhead at midday on the Tropic of Cancer, and at the very time when its rays penetrated the bottom of the well at Syene, Eratosthenes returned to the point he had marked in the Alexandrian park. Removing a small stick from his clothing, he placed it vertically at the point. This stick, or gnomon (itself an invention of Thales’), cast a shadow, just as the vane on a sundial does, and he simply measured the length of the shadow it cast on the ground. Happy with the result, he now returned to the museum knowing he had everything he needed to calculate the circumference of the earth.

  He made two assumptions: first, that Alexandria was due north of Syene, and second, that the sun was so far away from the earth that its rays were effectively parallel. Hence on that summer’s day when a ray fell vertically at Syene, the fact that it cast a shadow in Alexandria meant the earth was angled away from it at that point. Having measured this angle (which he could do using Pythagoras’s theorem, since he knew the height of the gnomon and the length of the shadow), and knowing the distance between the two cities, he could then turn to the works of Euclid in the library and find the method to calculate the circumference of the sphere.

  Of course there was plenty of room for errors in his calculation. Syene was not exactly on the Tropic of Cancer, so at the summer solstice the sun was not exactly overhead at midday. However, the well was quite wide, allowing some of the sun’s rays to pierce to the bottom even though it was not exactly overhead. A further problem lay with the question of how accurately the distance from Alexandria had been measured, although knowing the Ptolemies’ love of bureaucracy, we might imagine that the royal pacers were startlingly accurate. Finally, there was the fact that Alexandria was not quite due north of Syene but on a more westerly longitude. Compared with the size of the earth, however, these errors were trifling, and his result was still extraordinary.

  The angle of the shadow in Alexandria had been 7 degrees, so Eratosthenes calculated that the distance between Syene and his point must be 7/360th of the circumference of the earth. As he knew that distance was 4,900 stadia, he calculated that 1 degree was 700 stadia and hence that the circumference of the world was 700 ✕ 360, or 252,000 stadia. Sadly we do not now know exactly which measurement of a stadion he used (there were several), but the “walking” stadion which we might imagine a royal pacer to use was about 516.8 feet. This would give a circumference for the earth of 24,662 miles. The circumference around the poles is now known to be 40,008 kilometers (24,860 miles). Using just a stick, a well, and a royal pacer, he had proved the earth was a globe and measured its circumference to within 318 kilometers (198 miles) of its true diameter while never leaving Egypt.

  But Eratosthenes had only just started. Having noted that the sun reached the zenith at Syene on the summer solstice but on no other day, he calculated that the earth’s axis must be tilted toward the sun. This would explain how at different times of year the sun rose to a different height above the horizon. To measure this he also created another first in the history of science, the armillary sphere—a model of the celestial globe constructed from a skeleton of graduated metal rings (or armillae), linking the poles and representing the equator, ecliptic, meridians, and parallels, with a ball representing the earth at its center. So pleased was he with this device that he was said to have set one up in “that place which is called the square porch” (quoted in Charles Kingsley, Alexandria and Her Schools, lecture 1), almost certainly the portico of the museum, and it was here that he then used it to calculate the angle of the earth’s tilt at 23 degrees 51 minutes. Its true value is 23 degrees 46 minutes. He was just a twelfth of a degree off. His proof provided the first scientific explanation of a phenomenon that every Greek and Egyptian experienced: the seasons. When the Northern Hemisphere is tilted away from the sun, it is winter there; when it is tilted toward the sun, it is summer.

  Emboldened by the success of these measurements, he now went on to attempt to span truly astronomical distances. Using the same lunar eclipses which had first hinted at the shape of the earth, he first calculated the distance between the earth and the moon and then the distance now known as the astronomical unit, between the earth and the sun. Sadly, due to mistakes in his understanding of how large the sun and moon were relative to the earth—something even Aristarchus had had trouble calculating—these figures were somewhat less accurate. His distance from the earth to the moon came out at 122,300 kilometers (76,437.5 miles), as opposed to the real figure of approximately 384,600 kilometers (240,375 miles), and his calculation of the astronomical unit was, somewhat more accurately, 125.5 million kilometers (78,437,500 miles) as opposed to the true figure of about 150 million kilometers (93,750,000 miles).

  Inspired by his new understanding of the shape of the world on which he lived, Eratosthenes now turned his attention to mapping it, creating a detailed drawing of Egypt as far south as Khartoum and correctly hypothesizing that the annual Nile flood was caused by rains in unknown hills far to the south where the Nile had its source. Previously it had been believed that the flood was caused by the Etesian winds blowing water up the delta and preventing the Nile from draining into the Mediterranean. Eratosthenes’ correct hypothesis would not finally be proved until the eighteenth and nineteenth centuries AD, when the sources of the Blue and White Nile were finally discovered by Speke, Burton, and Bruce.

  But for a man who had measured the world, mapping Egypt alone would never be enough, so he set out, with his usual breathtaking self-confidence, to draw up something even more impressive. He had measured the world; now he intended to map it. This lost masterpiece was a landmark in both mapmaking and geography. Covering the world from Iceland and the British Isles to Sri Lanka and from the Caspian Sea to Ethiopia, it was built around a daring new concept of prime meridians and their offsets, or what we would call longitude and latitude. For his prime line of longitude he chose his own city, drawing the north-south line from the mouth of the Borysthenes (Dnieper River) south through the Euxine (Black) Sea, through Rhodes and Alexandria, down to Syene. Had his map and his prime meridian survived, perhaps we would today use Alexandrian mean time rather than Greenwich mean time as the world’s chronological baseline.

  For his prime meridian of latitude, Strabo tells us, Eratosthenes takes the world and

  divides it into two parts by a line drawn from west to east, parallel to the equatorial line; and as ends of this lin
e he takes, on the west, the Pillars of Heracles [Straits of Gibraltar], on the east, the most remote peaks of the mountain-children that form the northern boundary of India. He draws the line from the Pillar through the Strait of Sicily and also through the southern capes both of the Peloponnesus and of Attica, and as far as Rhodes and the Gulf of Issus. Up to this point, then, he says, the said line turns through the sea and the adjacent continents (and indeed our whole Mediterranean Sea itself extends, lengthwise, along this line as far as Cilicia); then the line is produced in an approximately straight course again for the whole Taurus Range as far as India, for the Taurus stretches in a straight course with the sea that begins at the Pillars, and divides all Asia lengthwise into two parts, thus making one part of it northern, the other southern; so that in like manner both the Taurus and the Sea from the Pillars up to the Taurus lie on the parallel of Athens.

  Strabo, Geography, book 2, chapter 1:1

  And he had a new word for this new work, a new description of what it was to understand the nature of the world and where things are placed upon it. This idea he called “geography”—the first time the term had ever been used.