Lesson 11: Mathematics in Ancient Greece
- Objectives
- Greek Mathematics and Mathematicians
- Alexandrian Mathematicians
- Vocabulary List
- Vocabulary Practice
- Reflection Questions
Objectives
- Learn an overview of key figures and ideas in Ancient Greek mathmatics
- Recognize various elements of the Greek tradition in the modern day
- Decentralize Greek influence from the narrative of mathematical history
Greek Mathematics and Mathematicians
What we would consider science and mathematics the Greeks would have called natural philosophy. Philosophy itself questions the nature of pretty much anything, from human emotion to the order of the universe. Natural philosophy asks questions and posits answers about the physical world, giving conjectures and reasoning for natural phenomena. For a long time, answers to those questions were given in mythologies and religion. As the Greeks moved away from mythology and into philosophy, they began to develop a method of scientific inquiry (though distinct from our modern scientific method), conjectures, and ultimately a long-lasting tradition that carries into the modern day.
Now, it’s easy to point at the ancient Greeks and credit them as the forerunners of modern science. In some ways, they are. But we have to take into account the fact that the Greek sphere of influence was largely limited to Europe and the Mediterranean. While there was some contact through the centuries with South and East Asia, there was no heavy influence there. There was a strong mathematical tradition developed by scholars in India, China, and the Middle East outside of the theories developed by the ancient Greeks. In fact, what we now call the “Pythagorean” theorem actually appeared in Babylonian and Indian texts far before Pythagoras was even alive!
The Baudhayan theorem, also called Baudhāyana Śulvasūtra, is a statement found in the Baudhayana sutras, a group of Vedic Sanskrit texts which contain mathematical writings. The Baudhayan theorem contains the same statement as the Pythagoren theorem: that the sum of the squares of the legs of a right triangle are equal to the sum of the square of the hypotenuse. The kicker? The Baudhayan theorem dates back to around 1000 BCE, while Pythagoras’s earliest dates are around 570 BCE.
How did the Greek mathematical tradition become so prevalent in the modern day? Part of this is because of the Roman empire, of which the Greek city-states became provinces, and which appropriated and disseminated so much of Greek culture and philosophy. Because Rome eventually grew to swallow nearly all of Europe at one point, the Greeks by proxy would have had greater influence across Europe. However, it’s important to remember that Europe is not the entire world, as much as the ancient Greeks and Romans liked to think that it was. Certainly, significant developments in science, math, and technology were made in Asia, Africa, and the Americas as well. There may well be entire classes devoted to scientific and mathematical achievements outside of Europe, but for now, let’s look at some contributions from some notable Greeks.
Thales
The first of these is a fellow called Thales of Miletus, who lived from about the 620’s to the 540’s BCE. The origins of who really invented or started Greek philosophy are a little messy, but many can agree that Thales was one of the earliest, and the school of thought that he pursued laid the foundation for many later philosophers and mathematicians. Thales is reported to have studied the fundamentals of engineering, some astronomy, and math. He engaged in simple physics problems and calculated the timings of solar eclipses and the solstices. He also created some of the first star charts for maritime navigation. He made a few achievements in mathematics and geometry.
His most famous concept was the theory that all of nature is based on the existence of water. The idea of the first principle, or the idea that everything in the universe comes from one foundational building block, is not unusual– in fact, we might consider this to be hydrogen, the first element in the periodic table. An interesting coincidence that hydrogen is one of the elements that comprises water. It’s unclear where Thales got this idea of water being the first principle from. Aristotle later suggests that Thales might have arrived at this idea because water was naturally observable as a solid, liquid, and a gas; that it could be found in both heat and cold; and that many natural body functions seemed to run on some kind of liquid or water-like substance.
Thales was also credited with a few mathematical achievements, mostly in geometry. Diogenes Laertius writes that Thales learned geometry from the Egyptians, upon which he discovered the aptly-named Thales’ theorem. That’s a special case of the inscribed-angle theorem, which lets him draw a right triangle inside a circle, with proof that its greatest angle is 90 degrees.
The phrase “know thyself” has also been attributed to Thales. This is an expression that basically means to understand your own mind and motivations in order to find peace with what you want and what you need. It’s actually more or less unknown who first came up with this phrase, and it’s variously been attributed to other well-known philosophers like Socrates or Plato. It’s an unfortunately common pattern that much early philosophical work is difficult to trace back to one person. As such, the same idea may often be attributed to multiple people, or in fact to the wrong person.
Pythagoras
One such example is Pythagoras of Samos, who lived from about 570 to 495 BCE. He was a contemporary of Thales, meaning that he lived and worked about the same time. Importantly, we don’t actually have writings that survive from either of these philosophers, which is why attribution occurs with both of them. For example, the aforementioned Pythagorean theorem is perhaps the most famous of the theorems and concepts associated with him. Interestingly, though, there’s no proof that Pythagoras himself actually proved or even discovered this theorem!
In fact, there are quite a few other mathematical discoveries that are attributed to Pythagoras, such as calculations on the five regular solids and the Theory of Proportions (which says that two sequences are proportional if their corresponding elements have a constant ratio). Part of this was due to the fact that in his lifetime, Pythagoras had a reputation more as a mystic than a mathematician. He gained a cult following, and it’s possible that the discoveries and proofs that his followers and colleagues made were later attributed to him because they were all in the same cult or the same school of thought.
We do know that Pythagoras taught and more or less believed in the immortality of the soul and reincarnation. Again, this is not a theory that was original to Pythagoras, as many earlier cultures in Asia believed in reincarnation. However, this was a relatively new idea in Greece at the time that Pythagoras was alive. This was one of the most important things that he and his followers believed in, which greatly influenced later philosophers like Plato. Because Pythagoras believed that souls were immortal, a human soul may very well enter the soul of an animal or insect; the Pythagoreans kept to vegetarianism. (The idea was that you didn’t want to accidentally consume the soul of, say, your grandparent.) Pythagoras also did some work on celestial harmony, which is the idea that the spheres (or orbits of the sun, moon, stars, and planets) all followed some sort of discernible tune that could be discovered and reckoned by mathematics. This was an idea that was also quite influential on later philosophers.
Plato
Perhaps the most famous of Pythagoras’ successors was Plato, who lived in Athens around the 420s to 348 BCE. Plato is best known for his written dialogues and philosophical frameworks, which often feature his teacher, Socrates. We don’t have anything by Socrates, so it’s always an interesting debate to figure out what is actually from Socrates and what’s been amended or filtered by Plato. Plato’s most famous work is probably the Republic, in which he describes the nature of justice by positing a theoretical ideal city-state.
Plato’s allegory of the cave is a rather well-known vignette in the Republic. In this allegory, prisoners are shackled inside a cave, facing a wall, where they can only see shadows cast by objects behind them, illuminated by a fire. These shadows represent their limited understanding of reality. When one prisoner escapes and discovers the outside world, he encounters the sun and the physical objects which have cast the shadows, symbolizing truth and enlightenment. Upon returning to the cave to share his newfound knowledge, he faces resistance and hostility from the remaining prisoners, who remain quite attached to the shadows that they’re used to. Plato uses the prisoner’s journey and frustrations as a metaphor for the philosopher who reaches enlightenment and cannot persuade the unenlightened to change their ways.
Outside of questions of justice and the nature of a city-state, though, the Republic does get into some math and cosmology. Plato was greatly influenced by Pythagoras when it came to the mathematical nature of the universe, to the extent that he recommended that those who would rule city-states be well-educated in mathematics.
If you recall the regular solids that Pythagoras was credited for doing work with, those same solids appear in Plato as well, this time called “Platonic”. They’re so named because in one of Plato’s other works, the Timaeus, Plato goes into much greater detail about math, natural philosophy, and the makeup of the universe. In that treatise, Plato theorizes that the natural elements of the world take the shape of the regular solids. And so, much after the fact, those regular solids were named after him.
Euclid
The first Greek philosopher that we consider a true mathematician is Euclid, who lived probably sometime in the 300s BCE. He is known as the father of geometry because of his extensive and thorough writings and study of the discipline. It’s important to note that in the Ancient Greek world, geometry was seen as synonymous with math, rather than a branch of it. Practically speaking, one needed a good sense of geometry to take measures for land, buildings, and infrastructure. Geometry was also the path to understanding and measuring the natural world and the cosmos.
Euclid’s major contribution was a treatise called the Elements, which has a huge impact on the European mathematical tradition, which has become the prevailing tradition of mathematics today. Euclid’s Elements established the foundations of proof-based mathematical reasoning, laying out not only the things we know but how we know them. If you’ve studied mathematical reasoning or real analysis, you’ve been following in Euclid’s footsteps! The first few chapters of the Elements start with several definitions, or axioms, that lay down solid rules for what points, lines, planes, and angles are, and how they can interact with each other. From there, Euclid gives instructions for how to construct triangles and polygons and circles, and then he gives further corollaries and theorems for their properties.
It’s hard to really pin down Euclid’s dates, since even though most of the Elements survives, not much else does. He does quite a bit of work with triangles and two-dimensional figures, so some scholars believe that he might have been a contemporary of Plato, certainly placed much later than Pythagoras. Unfortunately, as far as exact dates go, we don’t have much to go on.
Archimedes
Archimedes of Syracuse was a successor of Euclid who lived from about 287 to 212 BCE. These dates are a lot more exact since we know that he died in the siege of Syracuse during the Second Punic War, in 212 BCE. A Byzantine scholar by the name of John Tzetzes says that Archimedes was 75 years old when he died, so we can calculate his years almost exactly. Archimedes followed in Euclid’s footsteps but took geometry a step and a dimension further to deal with solid objects, their properties, and their interactions.
Because of this, Archimedes is regarded as perhaps the greatest mathematician of the ancient world. He was basically a Renaissance man before the Renaissance was even invented! Much of his work can be considered the precursor to modern calculus and physics, which is a really big deal given that modern calculus is built on the notion of the infinitesimal—the super small (and this, depending on which flavor of mathematical history you prefer, was alternately developed by Isaac Newton and/or Gottfried Wilhelm Leibniz in the early eighteenth century, nearly two thousand years later). The work of Archimedes more or less gives a precedent of advanced mathematical reasoning and intuition all the way back in the ancient world.
Archimedes is best known for his inventions. He was a geometrist, a theoretical mathematician, and an engineer who worked with spheres, buoyancy, revolutions, and cylinders. It would be difficult to list all the accomplishments and discoveries he made in this lesson, or else we’d be here all day. We don’t have a lot of original work surviving from him in the same way we more or less have most of Euclid’s Elements. We do have some writings that were transcribed and preserved at the Library of Alexandria, but a lot of Archimedes’ work only survives in fragments and in quotations by other writers.
One famous story about Archimedes, though, is his association with the phrase “Eureka!”. The story goes that the king of Syracuse had commissioned a crown to be made out of gold but wanted to check to make sure that the crown was made of solid gold, and that it wasn’t just painted or gilded. So the king asked Archimedes to figure out a way to determine if the crown was solid gold or not, without melting down the crown or breaking off a piece or otherwise damaging it. Archimedes sat on this problem for a while, and one day while he was taking a bath in the bathhouse, he noticed that when he sat down, the water level rose. Archimedes realized that the water displacement could be used to calculate the volume and density of the crown without damaging it, and he was so excited that he ran out of the bathhouse completely naked, yelling “Eureka!”, which is Greek for “I have found it!”
Whether that story’s actually true or not is up for debate, seeing as how it was written nearly three hundred years after Archimedes was alive, but it’s fun to think about nonetheless.
Apollonius
It might be easy to think that Greek mathematics and engineering more or less peaked with Archimedes, but there were still a good number of mathematicians and philosophers following in his and Euclid’s footsteps. One of these was Apollonius of Perga, who was a mathematician and astronomer living around 240 to 190 BCE. He combined the theories and ideas from both Euclid and Archimedes— two-dimensional figures, lines and planes from Euclid; curved surfaces and conic sections from Archimedes— to do more work with ellipses and conic sections, beyond the circle. In fact, his definitions of parabolas, hyperbolas, and ellipses are still the ones that we use today.
Apollonius was the first to propose eccentric orbits for the planets. In other words, he came up with the idea that the planets don’t move in perfect circles. Whether he believed that the planets orbited the Earth or the Sun is difficult to know. Not much of Apollonius’ work survives, either, also mostly in fragments or quotations. Apollonius’ more foundational ideas of astronomy— namely, that the planets had eccentric orbits— were still widely believed up to the Medieval and Renaissance periods. More work on the movement of the planets and towards heliocentrism would be done by folks like Nicolaus Copernicus, Galileo Galilei, and Tycho Brahe. Just like how John Dalton’s education in Greek allowed him to rediscover atomic theory from Empedocles and Theocritus, the astronomers of the 16th and 17th-century Scientific Revolution looked back to their Greek predecessors.
Eratosthenes
Another of these was Eratosthenes of Cyrene, who was regarded as the father of geography. He also did some work with conic sections and angles, but he applied these towards geography (which you may recall from the roots is the process of writing about the earth).
His most notable achievement was measuring the circumference of the Earth, which he did by measuring the length of obelisk shadows in two different cities, calculating the distance between them and the time of day, and then using trigonometry to figure out how many degrees of the Earth’s surface were between the cities. He did this several times, with the assistance of survey data available at the Library of Alexandria. Then he was able to extrapolate the entire circumference of the Earth as roughly 24,660 miles. Seeing as how the actual circumference of the Earth is 24,901 miles, Eratosthenes got pretty close for the tools he was working with!
Beyond the measure of the Earth, Eratosthenes also calculated the Earth’s axial tilt within a few degrees, and he created a map with parallels and meridians of the known world. He was a cartographer as well as a geographer. He also developed a system of scientific chronology, or the practice of dating historical events using records and astronomical calculations. Much of Eratosthenes’ achievements probably owed to his position as a head librarian in the Library of Alexandria.
Alexandrian Mathematicians
The Library of Alexandria (often called the Great Library) was probably the largest and most important library of the ancient Mediterranean, at least as far as everyone hitherto mentioned is concerned. It was established and patroned in Egypt by the Ptolemies, a family descended from Ptolemy Soter, who was a general in the army of Alexander the Great. This is also a reason why the library was established in a city called Alexandria, as it was named after the eponymous Great.
Alexandria was at a strategic location along the Egyptian coast. Its lighthouse, which is unfortunately no longer around, was considered one of the Seven Wonders of the Ancient World. Because it was a port city, scholars and educators from all over the Mediterranean and Middle East–from Greece, Rome, North Africa, Byzantium, Asia Minor–were able to access this city and its incredible library. The Ptolemies created the library as part of a greater research institution with the idea of a universal library in mind, one that housed all the known written works of the ancient world and could be accessible to anyone who wanted to study them.
As such, Alexandria itself became synonymous with its library, a center for learning and the dissemination of knowledge from all over the Mediterranean. Many notable mathematicians, philosophers, and scientists flocked to Alexandria because of its library and rich scholarly tradition, including the aforementioned Eratosthenes. The library’s archives weren’t just limited to mathematical treatises; you could find works on Homer and poetry, mythology, language, grammar, engineering, and history. In fact, the first library catalog of the Western world is believed to have been written for the Library of Alexandria.
Unfortunately, the library doesn’t exist anymore. There is a lovely reconstruction and functional library in Alexandria that you can visit today, but the original library suffered the fate as many libraries in wartime societies; it was variously overrun, underfunded, and burned— the last of which occurred most famously by Julius Caesar in 48 BCE. As a result, many of the writers and scholars who we know were active at the Library of Alexandria have their works now lost to us. Almost all the mathematicians mentioned so far since Euclid have either written their treatises in or at the library, or have had them copied over into its collection. The loss of the Library of Alexandria truly is one of the greatest tragedies when it comes to intellectual history in the ancient Mediterranean.
Ptolemy
The Great Library of Alexandria was in decline under Roman rule for a variety of reasons, but there were other prominent libraries that rose up in Alexandria. It’s likely that Claudius Ptolemy worked in one or more of these, as Alexandria had not completely lost its reputation as a knowledge center. Claudius Ptolemy, of ambiguous relation to the Ptolemies who established the library, was an astronomer and mathematician when Egypt (and by extension, Alexandria) was a Roman province. In fact, Ptolemy’s presence there made his work especially more relevant to later Byzantine, Islamic, and Western European scientific traditions. He’s best known for his work Almagest, which is a combined Arabic-Greek work meaning “the greatest”. It has 13 sections outlining pretty much everything Ptolemy knew about astronomy, the movement of the planets, measurements of time, eclipses, and the structure of the known universe.
Ptolemy also made conjectures about music and cartography in Almagest, since these disciplines were not considered too far from astronomy (a notion dating all the way back to Pythagoras). The Almagest was immensely popular in Medieval and Renaissance Europe and in the Middle East. Translations and editions of his work in Arabic, Greek, and Latin were composed from the 9th century all the way up to the 16th century. Many of the technical terms were abridged or edited, and because of this, some scholars believe that few people actually had the mathematical training to understand everything that was in this book.
Nevertheless, Ptolemy’s work on spheres and planetary motion had lasting influence into the European astronomical traditions, with nearly direct influence on the works of Copernicus, Galileo, Brahe, and Johannes Kepler.
Hypatia
In the ancient Greek-speaking world, women’s roles were extremely limited when it came to philosophy, scientific discovery, and intellectual history. However, Hypatia of Alexandria was a prominent mathematician who lived and worked in Alexandria, and she’s the most famous female mathematician in antiquity. Though she’s not the first female mathematician in Alexandria (that honor goes to a woman named Pandrosion), Hypatia is the first for whom we have a surviving record. Hypatia was a mathematician and astronomer, but a lot of the chronicle of her life gives detail of her as an educator and a teacher.
Hypatia also wrote several commentaries. It’s believed that she edited the surviving copy of Ptolemy’s Almagest that we have today, and she also wrote commentaries on other mathematical works, including one on conic sections by Apollonius of Perga, who I talked about just a few slides earlier. Though it may not seem too impressive, commentaries are vital resources for anyone trying to understand or study a text that was previously written. It speaks a lot to Hypatia’s sense as a teacher that she was able to not only go into technically and linguistically challenging texts, but that she was able to make them accessible to others. As I mentioned earlier, the common consensus was that a lot of people who were making edits and abridgments to the Almagest probably weren’t able to understand it too well. Hypatia, by her reputation, almost certainly did. She was also quite renowned as a philosopher, studying and teaching the works of Plato and Aristotle.
Hypatia’s life was tragically cut short by a Christian mob in 415 CE, quite violently, and likely the result of political or religious motivation. Hypatia herself was non-Christian, but she had gotten along well with Christians and non-Christians alike, so her murder came as a shock to many. In those days, it was unheard of to murder a philosopher, much less a teacher, much less a woman. After her death, Hypatia was claimed as a martyr for philosophy and women’s rights by varying groups.
Vocabulary List
| Root | Language of origin | Meaning | Example |
|---|---|---|---|
| icos(a) | Greek | 20 | icosahedron |
| (a)gon | Greek | angle | pentagon |
| arithm(o) | Greek | number | arithmetic |
| scel(e/o) | Greek | leg | isosceles |
| stoich(io) | Greek | element, part | stoichiometry |
| iso | Greek | equal | isometric |
| gram | Greek | written | telegram |
| hedron | Greek | geometric solid with X faces | hexahedron |
| orth(o) | Greek | straight, regular | orthodontics |
| ster(eo) | Greek | solid, stiff | stereotype |
| pto/pit(e) | Greek | to fall | peripiteia |
| ball/bol(e/a) | Greek | to throw | hyperbole |
| ten(u/e) | Greek | to stretch | tenuous |
| math(e) | Greek | to learn | mathematics |
| lips(e)/laps(e) | Greek | to fall | relapse |
| (h)eur(o) | Greek | to find, to discover | eureka |
Vocabulary Practice
Practice Set A: The following are Greek words that we use almost as loanwords, i.e., their original meanings in Greek are retained in English. (The terms are transliterated for you.) Tell what each of the terms describes, using your familiarity with mathematical terms or by looking them up.
- sphaira
- kylindros
- pyramidos
- kyklos
- konos
- prisma
- kybos
- elleipsis
Practice Set B: Identify the roots in each of the following words, give their language of origin, and their definitions. Also give their part of speech. Then, following the guidelines in Lesson 4, arrange the definitions of the individual roots to create a literal definition.
- isomorphic
- asymptote
- orthogonal
- parallelogram
- heptagon
- icosahedron
- logarithm
- parabola
- hypothesis
- hypotenuse
- polynomial
- heuristics
- polymath
- ellipse
- analogous
Reflection Questions
- Some Greek letters take on specialized meanings in mathematics. Look up and tell the definitions or conventional usages of the following letters: delta (Δ/δ), epsilon (ε), mu (μ), pi (π), sigma (Σ/σ)
- Choose one of the mathematicians in this reading and research more of their contributions to the Western mathematical tradition. Are there concepts in their work that you are familiar with? Have you used/interacted with any?
- The Ancient Greeks believed that mathematics was essential for not just understanding the natural world and the universe, but for learning discipline and mastery over oneself. Do you agree or disagree? Why?
- Intellectual discovery and rationality often clash with religion and politics in history, like Julius Caesar’s burning of the Library of Alexandria or Hypatia’s murder. What are some ways that you see science and politics clashing in the modern day?