What are the sources of the quadrivium?
Pythagoras
You have heard of the Pythagorean Theorem. Pythagoras was born around 570 BC on Samos, an island in the Aegean Sea. He moved to southern Italy, where he founded a group. To describe the group in contemporary terms: it was a combination of school, research institute, and religious community.
We do not have any writings of Pythagoras, but there is a body of teaching that is attributable to the Pythagorean school. One thing we get from this tradition is the fourfold division of mathematics in the quadrivium. We also receive from them a deep conviction of the intelligibility of the universe's order.
Euclid
Euclid is our most important source. He lived around 300 BC, probably studied in Athens at the school founded by Plato, and then founded his own school in Alexandria.
Euclid is the author of The Elements. This work covers plane geometry, geometry of solid figures, and arithmetic (i.e. elementary number theory). Sir Thomas L. Heath, a prolific translator and historian of ancient mathematics, says that Euclid's Elements "remains the greatest elementary textbook in mathematics that the world is privileged to possess."
The Elements is not solely Euclid's creation. He contributed new theorems, including a generalization of the Pythagorean Theorem, but also compiled the work of others, organizing and handing on knowledge that had been pondered and transmitted for many years before him. The great success of the Elements is due in large part to the marvelous clarity of its structure, the orderly way in which the contents are presented for a learner.
Ptolemy
Ptolemy is a third significant contributor to the quadrivium, and also one of the latest. He lived in Alexandria from about 100 to 175 AD. He studied and wrote about many things but is especially known for his astronomical work now called The Almagest.
Ptolemy's Almagest is a clear textbook that incorporates prior astronomical observations and theory and also includes new contributions. It begins with an account of geometrical principles, and then goes on to apply those principles to describe the behavior of the sun, the moon, and the planets.
One of Ptolemy's basic principles is the circle. Through circles, certain kinds of repetitive motion, like the daily passage of the sun through the sky and the monthly journey of the moon through its phases, can be understood and modeled. More complex motion can be accounted for using multiple circles. The term 'epicycle' is used when speaking of such compounding of circular motions.
Many others
The account above is hardly comprehensive. Rather than list any other names, let's just say that many other people would deserve mention in a longer story.
Philosophers and the Quadrivium
Other thinkers reflected on and promoted mathematical learning and the quadrivium. Here are two.
Plato
Plato was born around 428 BC. He founded a philosophical school, the Academy, near Athens. Plato required that those who wished to study with him be mathematically prepared. It is said that he put over the entryway that no one should enter who did not know geometry.
Plato made mathematical contributions himself but he is mentioned here especially because he emphasized the significance of mathematical study in liberal education. Mathematics helps us to pass from lower, changing things to higher, unchanging things. Plato adopted the same division of mathematical disciplines as the Pythagoreans, the four parts of the quadrivium.
Plato's works, written as dialogues, are rich, beautiful, and complex. They can hardly be summarized briefly. Some details about how you can begin to explore Plato's thoughts on mathematics and education are included in the notes below.
Aristotle
Aristotle, born in 384 BC, was a student of Plato and the tutor of Alexander the Great. His writings span a striking breadth. In some places what we have today are lectures notes rather than polished writings.
Aristotle wrote about both terrestrial and celestial physics, and these works helped shape the thought of later thinkers like Ptolemy. For now, though, we will consider not Aristotle's specific physical studies but something more general.
Aristotle gave a careful account of the structure of human knowledge and explanation. This broader work, his study of logic, greatly influenced subsequent thinkers, guiding the way that they thought about mathematics and natural science. Aristotle regularly uses mathematical examples in order to reason by analogy, and the larger logical frame makes such comparisons apt.
Conclusion
The quadrivium has a great ancestry. These disciplines, accessible to all of us, present the distillation of the clear thinking of some of our wisest and most profound thinkers. Surely they deserve our attention today.
Notes:
The quotation of Heath is from his book A Manual of Greek Mathematics. This work can be found freely online. Inexpensive physical editions are also readily available.
If you are interested in mathematics in Plato, here are a couple places to look.
There are many more places to turn; this is just a start. Translations of Plato's works are freely available online if you want to get a taste. You should buy real books, though.
Plato's dialogue Timaeus treats mathematical themes. It is discussed in the article Chronology on the transmission of the quadrivium.
One place to get the flavor of Aristotle's logic and its relation to mathematical study is the first section of Book I of his work Posterior Analytics. This section, dense but just a few paragraphs long, includes multiple geometrical examples and a reference to Plato's Meno. You can find free translations online.