A Brief Quadrivium

by Peter Ulrickson
Catholic University of America Press, 2023
Cloth: 978-1-949822-32-8, eISBN: 978-1-949822-33-5

Mathematics holds a central place in the traditional liberal arts. The four mathematical disciplines of the quadrivium-arithmetic, geometry, music, and astronomy-reveal their enduring significance in this work, which offers the first unified, textbook treatment of these four subjects. Drawing on fundamental sources including Euclid, Boethius, and Ptolemy, this presentation respects the proper character of each discipline while revealing the relations among these liberal arts, as well as their connections to later mathematical and scientific developments.

This book makes the quadrivium newly accessible in a number of ways. First, the careful choice of material from ancient sources means that students receive a faithful, integral impression of the classical quadrivium without being burdened or confused by an unwieldy mass of scattered results. Second, the terminology and symbols that are used convey the real insights of older mathematical approaches without introducing needless archaism. Finally, and perhaps most importantly, the book is filled with hundreds of exercises. Mathematics must be learned actively, and the exercises structured to complement the text, and proportioned to the powers of a learner to offer a clear path by which students make quadrivial knowledge their own.

Many readers can profit from this introduction to the quadrivium. Students in high school will acquire a sense of the nature of mathematical proof and become confident in using mathematical language. College students can discover that mathematics is more than procedure, while also gaining insight into an intellectual current that influenced authors they are already reading: authors such as Plato, Aristotle, Augustine, Thomas Aquinas, and Dante. All will find a practical way to grasp a body of knowledge that, if long neglected, is never out of date.
Peter Ulrickson is assistant professor of mathematics at The Catholic University of America.
List of Figures
Part I. Geometry
1. Instruments
2. Procedures
3. The Foundation of a Science
4. Proofs with Triangles
5. Parallels
6. The Composition of Quadrilaterals
7. Ratio
8. The Golden Ratio
Part II. Arithmetic
9. Counting
10. Numbers in Themselves
11. Demonstration with Natural Numbers
12. Primes and Relative Primality
13. Linear Diophantine Equations
14. Numbers in Themselves, Revisited
15. Relations Between Numbers
Part III. Music
16. Sound
17. The Monochord
18. The Tone
19. Approximation
20. The Diatonic Genus
21. Gregorian Modes
Part IV. Astronomy
22. Observation
23. Plane and Spherical Trigonometry
24. Principal Solar Events
25. A Refined Solar Model
26. Terms of Time
27. Elements of Lunar Astronomy
28. Stars, Fixed and Moving
Part V. Beyond the Quadrivium
29. Physics
30. Mathematics
31. Philosophy
General Index
Index of Euclidean Propositions

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