In the history of mathematics, it can be difficult to match the sheer audacity of the great minds of Ancient Greece. At a time before the invention of notebook paper—even before the invention of zero—several visionary Greeks managed to calculate aspects of the natural world that remain staggering to this day.
Today, We will focus on two such audacious calculations from two of the greatest minds ever to grace this Earth with their consideration. Both lived in the 3rd Century BCE, and though neither one lived in modern-day Greece (one was born in Libya and worked in Egypt, the other was from Sicily), they both fall squarely under the umbrella of Greek thought and study. Both had the audacity to reach for the impossible, and both used unparalleled brilliance to reach their conclusions.
Eratosthenes of Cyrene
Eratosthenes may not have been the most attractive guy on the block, but boy was he ever smart. In fact, smart doesn’t really cover what Eratosthenes was. Born in Cyrene, on the northern coast of Africa, Eratosthenes’ education brought him eventually to Athens, where he studied philosophy first under the Stoic Zeno of Citium and then under the Stoic/Cynic Aristo of Chios. During this time, he began to write mathematical analysis of Plato’s postulates and even began writing poetry on the side.
His poetry was so good that it began to get him some recognition, and in time even crossed the sea to reach the halls of the Pharaohs in Egypt. Ptolemy III Euergetes invited him to be the head Librarian of the Great Library at Alexandria. As head Librarian, he did a great deal to expand the library’s already-unparalleled collection of knowledge. He also invented an early cataloguing system, so that scrolls could more easily be found within the library’s extensive archives.
It is clear that, throughout his tenure as librarian, he did more than simply copy and tag texts. Finding himself with access to more knowledge than perhaps any other man on Earth at the time, Eratosthenes seems to have recognized his privilege, and at the same time allowed his curiosity to run wild as he sought to amass as much knowledge and understanding as could be fit within a human mind.
His studies led to some answers but, like any great scholar, he always found himself with a surplus of questions that surpassed even his his knowledge to address. One question in particular would lead him to a calculative effort that is truly exquisite in its audacity. It is a question that would likely tax even you—dear reader—living in the modern world of calculators, textbooks, and mathematical concepts that would not be devised until centuries after his death:
How large is the Earth?
Take a moment to consider how you might try to answer that question if you were deprived of the access to any authority that could simply hand the answer to you on a platter. If you had to calculate the size of the Earth without looking it up, how confident do you feel that you could do so, even with modern technology? How would you prove it?
Now imagine that you are living in a world without the internet, without calculators, without pens and paper, without zero, and without anyone who has ever even tried to make this calculation before, and you can perhaps begin to understand the conditions which were met by Eratosthenes. Would you even begin to know where or how to start?
It is often taught in America that back in Columbus’s day, everyone thought that the world was flat. That is, for lack of a better word, a lie. People had understood that the Earth was a sphere since well before Eratosthenes, and by his day every educated Greek was well aware of the globular Earth. That is remarkably important if we are to sensibly examine his calculation of its circumference.
It is hard to say how exactly the process began, but Eratosthenes set out to make this unfathomably audacious calculation for himself. There was a single piece of information that passed through his grasp as he studied in his library, and he had the presence of mind to recognize it as the pivotal fact that it was in his pursuit.
At some point, Eratosthanes read the account of someone who had travelled through the city of Syene, in southern Egypt (a short way south of the modern city of Aswan, the location of Syene is now under the waters of Lake Nasser, which was created by the Aswan Dam in order to bring hydroelectric power to Egypt). The account said that there was a deep well in Syene in which, at noon on the Summer Solstice, the sun shone all the way to the bottom without casting any shadow.
That might not sound like much to you, dear reader, but Eratosthenes recognized it for the exquisite datum that it was. That meant that on the Summer Solstice, the sun stood directly above the city of Syene at noon, meaning that Syene must lie exactly on the Tropic of Cancer. From there, all it would take to calculate the circumference of the Earth was patience and mathematics.
Those unfamiliar with Greek often spend their lives blissfully unaware of the fact that the word “geometry” literally means “earth-measuring,” and it was Eratosthenes who first exemplified its capability to accomplish that end. He commissioned a long wooden pole to be crafted, perfectly straight and with a length that was measured exactly. This pole he had erected in Alexandria as close to perfectly vertical as the tools of his day would allow. Now he only had to wait.
On the Summer Solstice, he had the greatest masters of measurement in the land wait for the sun to reach its apex. At exactly noon, measurements were taken of the shadow cast by the pole. More mathematically-minded readers will likely have worked out the rest of the calculation for themselves, but I am more than happy to explain it for the rest of us.
The length of the pole (which he knew) and the length of the shadow (which had just been measured) provided two sides of a right triangle. From there, the trigonometric principles established by Pythagoras and embellished upon by those who came after him allowed Eratosthenes to calculate the angle at which the same sun that sat vertically above Syene cast its rays upon Alexandria, and thereby to determine the angular degrees that divided the two cities in the greater sphere of the Earth.
Once he had calculated the angular distance between Alexandria and Syene from the perspective of the Earth’s core, all Eratosthenes needed was the distance between the two to extrapolate the total circumference of the Earth. In this regard, he had a great deal of luck on his side. For all its twists, turns, and cataracts, the Nile River, on which both cities were situated, runs a remarkably longitudinal course throughout all of Egypt. Taken as a whole, its adherence to a perfectly south-to-north course is almost uncanny. Furthermore, as the most trafficked river of trade in the world (at the time) there were already extensive and continually-updated documents measuring the distances between all of its major ports of trade.
And Eratosthenes was the overseer of the greatest compilation of knowledge on Earth, and in Egypt most particularly.
So it was that Eratosthenes, using only the facts he had read, his own grasp of the principles of mathematics, and a bit of on-the-fly thinking, calculated the circumference of the entire planet more than seventeen centuries before Columbus would drastically underestimate that same calculation and believe that he had found India when he was actually in the Dominican Republic.
By modern calculations, Eratosthenes’ calculation was only off by about 41 miles, roughly the distance from San Jose to San Francisco.
Archimedes of Syracuse
When it comes to ultra-badass scientists/mathematicians, there really may be no parallel to Archimedes. His brilliance—matched only by his ultra-badassery—is so intense that there is still some scepticism regarding the assertion that anyone could ever have truly been as much of a baller as Archimedes appears to have been every single day.
Among his myriad* accomplishments not to be examined in this answer, you will find:
- An efficient water-pump that still bears his name (and which, for the record, was apparently invented to pump water from the hull of a warship he had invented, which just so happened to be the largest in the history of the Ancient World).
- The use of mirrors to literally set enemy ships on fire, which still—twenty-three centuries later—continues to bear the unbelievably sci-fi name of the “Archimedes Heat Ray.”
A Renaissance Painting of the Heat Ray from Wikimedia Commons, still inspiring fear and awe 1800 years later.
- Devising a way to calculate the density of an object by combining the measurement of its displacement in a measured quantity of water with its mass, now known as Archimedes’ principle (the discovery of which was apparently made while he was in the bath, leading him to famously shout “Eureka!” (or, more properly, “εὕρηκα,” meaning, “I’ve found it,”) while running naked around the building which contained the bath that no longer contained the ecstatic Archimedes).
But, worthy as all those discoveries were, they are subjects for another day. Now we are looking at something that somehow manages to be more audacious than sprinting naked through the halls of academia, pumping water from the bellies of gargantuan warships, or even—somehow, bafflingly—setting enemy fleets on fire with nothing but a mirror and the Sun.
As I have already mentioned, the mathematicians of the Hellenistic period, for all their brilliance and far-sightedness, were working with a vastly inferior set of tools than those with which I empower my Second-Graders every day in math class. Indeed, every day I teach place value to a group of seven-and-eight-year-olds who already seem to think that it is kids stuff. The concept of zero, and the magical way that it allows a mere ten digits (itself included) to express an infinite spectrum of the numerical value is so ingrained on us these days that it seems quite basic, even to children.
See, in Archimedes’ day, the numbers didn’t go on forever like they do now. Of course, people recognized that there was no “highest number,” but even within the culture of mathematics that had the likes of Euclid and Pythagoras for inspiration, it was widely accepted that some numbers were just too high to be discussed in any kind of a constructive manner. Greek Numerals were drastically different than the numbers we know and love today. The concept of zero and the place-value it empowered would not even be invented in India until about six centuries after Archimedes’ death, and the Arabs wouldn’t bring it to prominence in Europe until centuries thereafter.
The Ancient Greeks did math with letters. While they used the Greek alphabet, their system can be transferred to our own Latin alphabet without much effort for the ease of explanation. To put it in our own letters, the Greek system would use the first nine letters (a, b, c, d, e, f, g, h, and i) to represent the numbers 1–9. Then the following letters could be used to represent the next order of magnitude, so that j, k, l, m, n, o, p, q, and r would represent 10–90 respectively. As numbers got higher, it was decided that not every one of the nine non-zero numerals requires representation. But still, the problem of simply running out of letters remained.
As such, Greek Numerals had an upper bound, above which the numerals had yet to be invented to express such a quantity. For the Greeks, as for a remarkable number of other contemporary cultures, the greatest expressible number was 10,000, which they called “μυριάς” (myrias). Anything above 10,000 was considered an uncountable multitude, which is why we still use the term “myriad” to mean “innumerable” in modern English.
One of the common analogies in Archimedes’ day was that the number of grains of sand on a beach was something that could never be mathematically expressed with any degree of exactitude. It was simply, “a lot,” and everyone should be content to leave it at that.
As it happens, Archimedes didn’t like other people telling him what was beyond the bounds of his ability to calculate or express. Not only did he scoff at the idea of the grains of sand on a beach defying his calculative abilities, but he decided to take the proposed impossibility about ten-trillion steps further.
“The grains of sand on a beach are too numerous to be calculated, you say?” Archimedes quipped. “Hold my fermented grape mash. I’m gonna calculate how many grains of sand it would take to fill the universe!”
Of course, those weren’t his exact words, nor even any kind of close translation, but I nonetheless feel that the sheer audacity of Archimedes’ mathematical cojones are accurately conveyed in the lines above. Not only did he reject the innumerable quantity of sand on a beach, he jumped straight to the whole damn universe (as he understood it, of course).
But to do that, he first had to invent his own number system.
Without spending too much time going into the details of his invention, Archimedes saw that the numerical systems of his day could be extrapolated out exponentially to the point that they could encompass numbers on a scale that we still haven’t come up with specific names for.
(Note: as much of a Greek purist as I tend to be, I will transition all of the terms in the following section into English for you, dear reader. You’re welcome.)
Archimedes figured that if myriad was the highest number, then he could start working from there. Myriad—which is to say 10,000—became his “number of the first order.” You could have one myriad, two myriad, three… I’m just playing. You could go all the way up to a myriad myriad, also known as 108108, which could then be used as a new substitute for myriad. He named this new quantity as his “number of the second order.” And then the pattern continued.
This system allowed Archimedes to succinctly write numbers ranging up to ((108)(108))108((108)(108))108 . As someone with very little experience with LaTeX math coding, I hope that comes out coherently, but in case it doesn’t, that is a one with eighty-quadrillion (80,000,000,000,000,000) zeroes.
That’s a pretty effing huge number, and the fact that Archimedes took the Greek Numeral system all the way from 10,000 to 1079,999,999,999,999,9961079,999,999,999,999,996 times as much is pretty hardcore if but that is not all.
Now that he had invented super-huge numbers, Archimedes had to calculate the size of the universe, and even in that small step, he was so far ahead of his time that it is difficult to even fathom just how insanely right he was.
But first, let’s take a moment to examine what a word like “universe” would mean to an ancient Greek. To Archimedes, the “universe” was generally confined to what we would now call the solar system. The universe had a single sphere at its center, and a whole host of planets circulating around it. Beyond that was a spherical edge painted with stars. The very concept of foreign galaxies was yet to be devised.
Indeed, for the greater part of the next two millennia, it would be held as an indisputable fact that the Earth was the centre of the universe, and even 1800 years after the death of Archimedes, the assertion that it might be otherwise would remain so revolutionary as to merit oppression against those who spoke it. When Copernicus’ heliocentric theories were first printed—more than eighteen centuries after the death of Archimedes—they were enormously controversial.
Indeed, the works of the first great heliocentric astronomer, Aristarchus of Samos, were lost to history entirely, and it is entirely through the work of Archimedes that we know of his brilliance at all. Was it not for his incredible work, even educated people would still blithely assert that Copernicus was the first to posit that the Earth orbited the Sun, and the depth of Greek insight into the matter would lie in the darkened vaults of unrecorded history?
While we may not be able to solidly attribute Archimedes’ reason for believing that the Earth revolved around the Sun, it is undoubtedly true that he accepted the heliocentric model of Aristarchus as the one by which he ought to make his calculations.
Armed with his new form of advanced large-number mathematics and his unbelievably-prescient Aristarchan Heliocentric model, Archimedes determined that the number of grains of sand that it would take to fill the universe (the Solar System) was approximately 10641064.
