Myths vs Math: Columbus and Eratosthenes

Novelist Washington Irving is probably responsible for misinforming more American schoolchildren than any other single individual.  In 1828, he published The Life and Voyages of Christopher Columbus, a heavily romanticized account of Columbus’s life in which events that occurred only in Irving’s imagination were treated as historical fact.

One such item was the story that the good admiral was bravely voyaging to prove his assertion that the earth was not flat but a sphere. Because Irving was the Stephen King of his era (the book went to 175 editions before the century was out), his version of events took hold as part of the historical record. Textbooks printed as recently as 1987 perpetuated this myth, as did the Encyclopedia Britannica, and well-known pundit Thomas Friedman repeated it only a few years ago in the New York Times.

The hallmark of a good story is not only that it’s plausible, it has elements that we want to believe. There is a whole category of these stories, cataloged by Jan Brunvand, Barbara and David Mikkelson, and other folklorists as “urban legends.” You’ve probably seen several wend their way through the internet via email and Facebook statuses. While these are often calculated to generate goosebumps (organ thieves harvest kidneys from unwary travelers) or provoke political outrage (senators receive outsized lifetime pensions), on some occasions revisionist histories appear as well.

Consider the oft-emailed claim that the original model for the Statue of Liberty was a black woman. This is captivating because it appeals to our sense of justice – how symbolic that liberty be portrayed by a former slave, now free. It’s plausible because the timing fits, the statue was begun less than a decade after the Civil War, and was intended to be a monument to freedom. In truth, however, sculptor Auguste Bartholdi used two women as models for the statue: his mistress, and his mother, both of whom where French and white.

So too, Irving’s tale about Columbus follows this template. It appealingly presents him as a force for progress, echoing the relationship between Galileo and the Catholic church. The explorers becomes a light shining in opposition to a dark age of religious intolerance. It also hews to the beloved American narrative of the lone outsider fighting against all odds; that movie trailer we’ve heard a dozen times before, the baritone voice growling, “but there was one man…”. And given how much the Church dominated pre-Renaissance Europe, the story does seem plausible. After all, it was Queen Isabella who established the Spanish Inquisition, forcing Jews and Muslims out of Spain, and the rest of the country into strict doctrinal orthodoxy.

Contrary to that progressive image however, Columbus was himself something of a religious fanatic. He referred to himself as the “Christ-bearer”, wrote a book of prophesies, and viewed his voyages as one step in God’s master plan culminating in the second coming of Christ, which he predicted would happen in 1672. Columbus had some other odd ideas as well.  He once wrote that the earth was not a globe, but shaped more like a woman’s breast and that a terrestrial paradise could be found at the nipple. Admittedly, he spent a lot of time at sea without the benefits of female companionship.

Columbus, like the King and Queen of Spain, and every other educated individual in Europe, was aware that the earth was round. The evidence had been piling up for centuries. Aristotle’s Meteorology, from the 4th century BCE, listed several observations supporting a spherical earth, and even then, the Greeks had been compiling corroborating data for at least 200 years. We also know that this information was not lost during the dark ages. For one thing, the Church itself looked highly on Aristotle’s books, and there are writings by the Venerable Bede and Thomas Aquinas, as well as Jewish and Islamic scholars, citing evidence for a planetary globe. What Columbus got wrong was not the shape of the earth, but it’s size and geography, and his biblical literalism that played a role in those errors.

The books that make up the present-day Roman Catholic bible were canonized by the church in councils around 393-397 CE. During the Reformation, Martin Luther removed a set of books he found objectionable, and so the Protestant bible differs from the Catholic bible. The set of books he excised are now referred to as the Apocrypha or the Deuterocanonical books.  (Luther thought that the Book of Revelation should have been removed as well. Had he been successful, the history of the Protestant church would have been quite different.)

Columbus, being a Catholic, had no such qualms about the Deuterocanonical books, and it was the Fourth Book of Ezra in which he found inspiration. According to 4 Ezra 6:42-52, which Columbus shared with Ferdinand and Isabella, the oceans only covered 1/7th of the earth. Since he had an approximate idea of the extent of Asia thanks to Marco Polo’s diaries, and a number for the circumference of the earth, he figured there was relatively little water to navigate if he sailed west from Spain towards the orient. There were two small problems: one, the biblical estimate for the amount of water on the planet had no grounding in reality, as 2/3 of the planet is covered in water, and two, he grossly underestimated the earth’s circumference.

The primary source of the latter error was a conversion problem.  (NASA scientists should sympathize.  A metric-to-English conversion error caused the Mars Climate Orbiter to disintegrate in the atmosphere over the red planet in 1999 instead of landing safely.) In Columbus’s case, he converted degrees of latitude to miles using the Arabic miles instead of Greek miles. While this can be viewed as a purely numerical mistake, it’s likely that wishful thinking played a part as well.

This particular error was unfortunate, for Eratosthenes had correctly measured the planet’s size nearly two centuries earlier. His calculations were simple, elegant, and correct. So simple, in fact, you can do them yourself. All you need is a stick, a protractor, and a camel.

At the time, Eratosthenes was living in the Egyptian city of Alexandria. The Ptolemaic dynasty established after Alexander the Great conquered Egypt made the city a center of learning, and it was home to a library dedicated to collecting every book ever written. Around 240BCE, Eratosthenes ran across a curious story. In the city of Swenet (now called Aswan), there is a well, and if you look into the well at noon on the summer solstice, you can see the reflection of sun directly overhead.

Now Eratosthenes had done some traveling. He was born in what is modern-day Libya, crossed the Mediterranean to study in Athens, and was at the time, studying at the magnificent Library of Alexandria. So he was aware, that east-west travel differed from north-south travel.

You’ve probably noticed that as you go further north, the shadows get longer because the is lower in the sky. (This is inverted for anyone in the southern hemisphere.) What Eratosthenes realized was that he now knew a place and time where the sun rays were perfectly perpendicular to the earth, an angle, relative to the sun’s rays, of zero degrees. So the next June 21st, when high noon rolled around, he went into his back yard, drove a stick into the ground, and measured the angle of its shadow. It turned out to be just over seven degrees.

Now if you think of the earth as a pie for a moment, with Swenet and Alexandra two points on the crust, that means that the slice of pie Eratosthenes measured was a chunk spanning just over seven degrees. There are 360 degrees in a circle, so that slice encompasses about 1/50th of the pie, since 360 divided by seven point something is about 50. (We’ll stick with integers to keep things simple.)

At this point, Eratosthenes had the measure of Swenet to Alexandria in degrees, he just needed to convert that to miles. (Actually, distance at the time was measured in stadia, but let’s stick with miles.) This is where the camel comes in. Eratosthenes headed into the suq, searching for the merchants who would load their camels with papyrus and gold and return bearing ivory and spices from Kush. How far is it to Swenet? It’s a trip of twenty-five days. And how far do you travel in a day? Twenty miles, if we do not have to contend with thieves.

Eratosthenes whips out his abacus. 20 times 25 is 500 miles for the slice from Alexandria to Swenet. That times 50 for the whole pie is 25,000 miles. With modern-day technology, we can more precisely pin it down to 24,901, so you can see that Eratosthenes was pretty darn close, and he did it with nothing more than basic arithmetic and middle-school geometry.

Given that Eratosthenes had come up with an accurate measure seventeen centuries before Columbus, why wasn’t it used? It turns out, it was used, but not by Columbus. The experts that Ferdinand and Isabella hired to evaluate his claims advised the king and queen (correctly, it turns out) that Columbus would never make it to the orient, Eratosthenes had shown that the distance was simply too great. Columbus, arguing with his miscalculated figures, might never have carried the day on his own, but his appeal citing of the book of Ezra may have been the deciding factor. The pious queen overruled her own experts and financed the voyage.

Fortunately for Columbus and his crew, he ran into two previously unknown continents. If not for that interfering land mass, he and his crew would have run out of provisions long before reaching Asia. However, even in his dying years, Columbus maintained he’d actually sailed to “the Indies” even in the face of mounting evidence to the contrary. The famed Waldseemüller map, bearing the continental name America appeared the year after Columbus’s death.

Things went somewhat better for Eratosthenes, his reputation for scholarship, not only in mathematics, but geography, poetry, and music, got him appointed administrator of the great library of Alexandria, a position he held until his eyesight began to fail, and he retired at age 78.

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