How did the quadrivium get handed down over time? What did we end up getting?
Let's break this down into two strands. One strand is the structure of the quadrivium, the sense that mathematical knowledge is divided into the four parts of arithmetic, geometry, music, and astronomy. The other strand is the fundamental texts themselves, such as Euclid's Elements.
To be very brief: the structure was preserved in Europe, but the texts had a more complicated trajectory, and often passed through the Islamic world.
The full story is of course more elaborate. See the notes for ways to learn more.
This essay will touch three points on the transmission of the quadrivium.
Boethius
Boethius was a Roman, born around 480 AD, of a noble family. He rose to significant political positions at a young age and was known as a learned man.
A complicated political situation resulted in his death at the order of Theodoric, the Ostrogothic king. Boethius is venerated as a martyr in the Italian city of Pavia.
Boethius wrote The Consolation of Philosophy. This book, written during his imprisonment before his execution, presents a story of how philosophical consideration can lead one from discouragement at the trials of fortune to confidence in the order of divine providence.
In addition to The Consolation of Philosophy, Boethius translated classical works and wrote commentaries on them. His commentaries helped transmit Aristotle to medieval Europe.
Boethius wrote a number of mathematical works on disciplines of the quadrivium.
He wrote a book called de arithmetica which translated the classical arithmetic of a scholar named Nichomachus. He also wrote de institutione musica. This book of music theory handed on the thought of Nicomachus and Ptolemy.
It seems that Boethius translated Euclid's Elements in some form, but this has not survived.
The writings of Boethius, both mathematical and philosophical, were widely read and copied for many years, and he coined the Latin term quadrivium.
Plato's Timaeus
The dialogue Timaeus was the only work of Plato available in Europe in the early medieval period, through a translation into Latin by Chalcidius.
The dialogue is complex. It features extended speeches by the participants, one of whom is named Timaeus. There are accounts of how the universe came to be and what its structure is.
One feature of Timaeus' account is an idea of how material elements are constituted of simple geometric shapes. More generally, the dialogue considers mathematical order and its relation to physical things.
The Timaeus is a philosophical rather than a mathematical text. It is nonetheless important to consider here because it played a large role in forming medieval perceptions: of the world's mathematical order, and of the character of Greek thought, thought to which the medieval thinker had only limited access.
Chartres is one place that Timaeus was read and discussed. It was the location of a significant school in the 12th century. The cathedral at Chartres, constructed around 1200 AD, is an outstanding example of French Gothic architecture. The cathedral is shaped by the view of divine and cosmic order that arose from the richly mathematical Platonism of the school of Chartres.
Astronomy and The Almagest
Many astronomical terms in use today have Arabic roots. Examples include azimuth, zenith, and Betelgeuse.
If we were to use Ptolemy's original Greek title to name his work, we would say something like Mathematical System or Mathematical Treatise. Instead, we say Almagest, a name revealing the contribution made by the Islamic world to the preservation of this book.
Ptolemy's Almagest was used in Alexandria for a couple centuries after his death. At that time it was used in Greek and was not translated into Latin. It was later translated into Syriac and then into Arabic, where it acquired its name Almagest from an Arabic transliteration of the Greek 'megiste' (greatest).
The work started to become known in Europe in the later 12th century through the work of Gerard of Cremona - a man born in Italy, working in Spain, and translating from Arabic.
Notes:
The translation of Timaeus by Peter Kalkavage contains an excellent introduction and appendices to guide the reader through the challenging dialogue. It would be a good work to read with advanced, older students after studying the quadrivium.
The book The Gothic Cathedral by Otto von Simson discusses the relation between geometry, cosmology, and Gothic architecture.
Two names that should be mentioned but could not be included above are Isidore of Seville and Martianus Capella. Isidore was an encyclopedist who transmitted the structure and content of the quadrivium, to some degree. Martianus Capella wrote The Marriage of Philology and Mercury, a complex story that shaped the medieval sense of the liberal arts.