(fl Alexandria, c300 bce). Mathematician and theorist. His Elements of Geometry has from the earliest times been the basis for the study of geometry in the West. The definitions in book 5 of ratios and proportions, perhaps attributable to Eudoxus, are of great mathematical importance because they accommodate incommensurable magnitudes. Numerous other mathematical works, some no longer extant, have been ascribed to Euclid, and writings on music have also been attributed to him in several ancient and medieval Arabic, Greek and Latin sources (Proclus even claimed that he wrote an Elements of Music). Of the two treatises on music that have come down to us bearing Euclid’s name, the Harmonic Introduction (Eisagōgē harmonikē), containing Aristoxenian music theory, is now ascribed to Cleonides; the other, Division of the Canon (Katatomē kanonos; Sectio canonis), survives in more than 200 manuscripts in which three distinct traditions are discernible – a Greek version, a shorter Greek version, and a shorter Latin version in Boethius’s De institutione musica.
The Division of the Canon contains a discussion, based on Pythagorean principles, of the relationship between mathematical and acoustical truths. In the philosophical introduction, which possibly quotes Pythagoras, the author draws a parallel between sound and motion, specifically in terms of vibration; he treats musical acoustics as a branch of arithmetic and proposes a definition of consonance limited to intervals built on multiple or superparticular ratios. Other sections of the treatise include arithmetical and acoustical propositions, a passage devoted to the enharmonic genus and another in which the two-octave canon is divided according to the diatonic genus. Like the Elements of Geometry, the Division is largely compiled from a number of different sources, and its varied and sectional nature would suggest that it is not the work of a single author. The Division’s underlying assumptions appear to derive from the Elements, but the acoustical propositions, though similar in style to the latter work, especially to books 7–9, are less rigorous in their logic, even to the extent of displaying false reasoning. By virtue of its brevity and its focus on fundamental issues, the Division has been a source of interest to music theorists since antiquity, and in one form or another the work has been edited, translated and commented upon many times since the 15th century.
A. Barker, ed.: Greek Musical Writings, ii: Harmonic and Acoustic Theory (Cambridge,1989), 190–208
L. Zanoncelli, ed.: La manualistica musicale greca (Milan, 1990), 29–70
A. Barbera: The Euclidean Division of the Canon: Greek and Latin Sources (Lincoln, NE, 1991) [incl. translations, analyses and commentaries]
A.C. Bowen: ‘Euclid’s Sectio canonis and the History of Pythagoreanism’,Science and Philosophy in Classical Greece (New York, 1991), 164–87
ANDRÉ BARBERA