The Seven Liberal Arts.

The art of words and their right form.

Grammar is the foundation. Before logic, before rhetoric, before any mathematics: how language is built, how it carries meaning, how a sentence holds together. The classical tradition took grammar seriously as a discipline — not the schoolroom drill it later became, but the careful analysis of language as a system.

• Marcus Terentius Varro — the Latin grammarian, De Lingua Latina.
• Latin Syntax — the Cicero-Varro composite specialist, the topological algorithm of Latin word order.
• Attic Word Order — the four-orator composite specialist, the topology of Greek syntax as a working system.
• Pāṇini — the systematic grammar of Sanskrit, four millennia ahead of its time.
• Quintilian — the educational treatment, where grammar meets rhetoric.
• Mark Aronoff — morphology as a working modern science.

The art of right reasoning. Also called Dialectic.

Logic is the discipline of the well-formed argument. From the syllogism through the medieval disputation to the symbolic logic of the nineteenth century, it is the same project — to distinguish what truly follows from what only seems to. The Universitas teaches logic in its classical and medieval forms, where it was a tool of the philosophical and theological tutorial.

• Aristotle — the Organon, the syllogism, the foundations.
• Chrysippus — Stoic logic and the bridge to the propositional tradition.
• Peter Abelard — medieval dialectic and the Sic et Non method.
• William of Ockham — the late-medieval terminist tradition.
• Leibniz — the dream of a universal logic.
• Gödel — the limits of formal systems.

The art of right speech — of persuasion under truth.

Rhetoric completes the trivium. Where grammar gives you form and logic gives you proof, rhetoric gives you the means to move an audience — not by deception but by the clear and well-ordered presentation of what is genuinely true. The Greek and Roman tradition treated this as the highest verbal art.

• Cicero — the orator's manual, De Oratore; the practising statesman.
• Latin Rhetoric — Cicero's specialist composite for Latin rhetoric, the figures and operations of his own oratorical practice made explicit.
• Quintilian — the Institutio Oratoria, the great educational treatise.
• Demosthenes — the Greek master, oratory in practice.
• Isocrates — the Greek theorist, rhetoric as paideia.
• Attic Rhetoric — the four-orator composite specialist, the topological figures of Attic oratory taught as a working system.

Number as such — number at rest.

Arithmetic in the classical sense is not calculation. It is the study of the integers and their properties — primes, factors, ratios, the structure of number itself. The Greek tradition (Euclid, Diophantus, Eratosthenes) treated this as a contemplative discipline; the modern descendant is number theory, still pursued in the same spirit.

• Euclid — Elements, books VII through IX: the foundations of number theory.
• Diophantus — the Arithmetica, the prefiguration of algebra.
• Eratosthenes — the prime sieve, computational arithmetic at its origin.
• Nicomachus of Gerasa — the Introduction to Arithmetic; perfect numbers, figurate numbers, and the proof that arithmetic comes first.
• Boethius — who named the quadrivium and carried Nicomachus into Latin.
• James B. Thomson — the schoolmaster's art: how arithmetic is actually taught, one difficulty at a time.
• Donald Knuth — the art of computation.

Number in space — magnitude at rest.

Geometry is the application of number to extension. The Greek geometers built the discipline that every later mathematics presupposes — lines, surfaces, solids, the relationships between them. The Elements alone is a complete education in deductive reasoning, taught in tutorial as it was written: proposition by proposition, from common notions to the regular solids.

• Euclid — the Elements proper, the foundation of all geometry.
• Archimedes — mensuration, mechanics, the method of exhaustion.
• Apollonius — the Conics, the highest achievement of Greek geometry.
• Felix Klein — the Erlangen Programme.
• Henri Poincaré — topology, the modern grammar of space.

Number in time — proportion, consonance, the hexachord.

Music in the quadrivium is not composition or performance. It is the study of proportion expressed in sound — the Pythagorean ratios, the consonances, the hexachordal system that ordered Western music for seven centuries. The lineage we teach runs from Guido of Arezzo's invention of solfège and the staff, through the late-medieval consolidation of mensural notation, to the Renaissance summa of Zarlino, and on to Bach, in whom the proportional tradition is still alive as a working method.

• Guido of Arezzo — the hexachord, solfège, the staff — the system itself.
• Franco of Cologne — mensural notation, proportion in time.
• Tinctoris — consonance, the close of the medieval system.
• Glareanus — the Dodecachordon, the modal system at its fullest.
• Zarlino — Renaissance proportional harmony, the summa.
• J. S. Bach — counterpoint as living hexachordal proportion.

Number in motion — magnitude moving through time.

Astronomy closes the quadrivium. It is the application of number, geometry, and proportion to the heavens — the discipline that gave the West the calendar, the clock, the map, and eventually a working theory of the cosmos. The classical tradition (Hipparchus through Ptolemy) and its Islamic continuation (al-Battānī, al-Ṭūsī) is taught here before the Copernican turn, because the geocentric astronomy is the discipline as the classical tradition understood it.

• Hipparchus — the first systematic astronomer, the founding star catalogue.
• Aristarchus — the early heliocentric proposal, eighteen centuries before Copernicus.
• Ptolemy — the Almagest, the long-canonical synthesis.
• al-Battānī — the great Islamic refinement of Ptolemaic astronomy.
• al-Ṭūsī — the Maragha school, the late medieval transmission.
• Copernicus — the heliocentric turn.
• Kepler — the laws of planetary motion.