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 All will find a practical way to grasp a body of knowledge that, if long neglected, is never out of date.

Contents

List of Figures

Preface

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