(b Cyrene [now Shahhāt, Libya], c276 bce; d Alexandria, c196 bce). Greek scholar. He was educated at Alexandria by the poet Callimachus and the grammarian Lysanias, and at Athens encountered the philosophers Arcesilaus and Ariston of Chios. In about 246 bce Ptolemy III Euergetes summoned him to Alexandria, where he served as tutor to the royal family and director of the library.
Only fragments of his writings survive, including a work entitled Katasterismoi about constellations (although Eratosthenes' authorship of this has been doubted), a didactic poem Hermes (which includes discussion of the harmony of the spheres), expositions concerning the measurement of the earth, and, above all, a three-volume Geography. He covered a wide range of subjects, including philology, grammar, literary history, chronology, geography, astronomy and mathematics, as well as the mathematical theory of music; the latter was the subject of his dialogue Platōnikos, in which he reputedly dealt with the fundamental concepts of mathematics following Plato's Timaeus. Theon of Smyrna (ed. Hiller, 81.17ff) and also Porphyry in his commentary on Ptolemy's treatise on harmonics (i.5: ed. Düring, 1932/R, 91.14ff) cited a distinction drawn by Eratosthenes in the Platōnikos between one calculated arithmetically in the Aristoxenian manner (diastēma) and one calculated by ratio; Theon also quoted a definition of proportion from Eratosthenes (Hiller, 82.16ff).
Eratosthenes' calculation of the tuning of the degrees of the tetrachords was reproduced by Ptolemy (ii.14), Theon and Porphyry and is of particular importance. The intervals are as follows: diatonic tetrachord – 9:8, 9:8, 256:243; chromatic tetrachord – 6:5, 19:18, 20:19; enharmonic tetrachord – 19:15, 39:38, 40:39. According to Ptolemy, Eratosthenes adopted Pythagorean ratios for the diatonic tetrachord: a whole tone is thus equal to a 5th minus a 4th (i.e. 9:8), and a semitone or limma to a 4th minus two whole tones (i.e. 256:243). He then calculated the intervals for the other tetrachords using arithmetical and harmonic means: in the chromatic tetrachord the small whole tone is divided into the two semitones 20:19, 19:18; and in the enharmonic, the semitone 20:19 (90 cents, almost exactly the same as the Pythagorean limma, 256:243) is divided into the quarter-tones 40:39 and 39:38. All these ratios are superparticular (the numerator or antecedent exceeds the denominator or consequent by 1), apart from the limma and the upper interval in the enharmonic tetrachord, a major 3rd (19:15, or 409 cents), whose size approaches the Pythagorean ditonus (81:64, or 408 cents). In the chromatic tetrachord, Eratosthenes included a pure harmonic minor 3rd (6:5), which was subsequently retained by Didymus and Ptolemy, rather than the pure major 3rd (5:4) employed by Archytas in the enharmonic tetrachord.
E. Hiller, ed.: Eratosthenis carminum reliquiae (Leipzig, 1872)
J.V. Powell, ed.: Collectanea alexandrina (Oxford, 1925/R) [extant frags.]
A. Barker, ed.: Greek Musical Writings, ii: Harmonic and Acoustic Theory (Cambridge, 1989), 364–9
E. Hiller: ‘Der Platōnikos der Eratosthenes’, Philologus, xxx (1870), 60–72
E. Hiller, ed.: Theonis Smyrnaei philosophi Platonici: Expositio rerum mathematicarum ad legendum Platonem utilium (Leipzig, 1878/R)
G. Knaack: ‘Eratosthenes (4)’, Paulys Real-Encyclopädie der classischen Alterthumswissenschaft, vi (Stuttgart, 1909), 358–88
P. Tannéry: Mémoires scientifiques, iii (Toulouse and Paris, 1915), 68ff, 299ff
F. Jacoby: Die Fragmente der griechischen Historiker, ii/B (Berlin, 1929–30/R), §241, pp.1010–21
I. Düring, ed.: Porphyrios Kommentar zur Harmonielehre des Ptolemaios (Göteborg, 1932/R)
R.P. Winnington-Ingram: ‘Aristoxenus and the Intervals of Greek Music’, Classical Quarterly, xxvi (1932), 195–208
I. Düring, ed. and trans.: Ptolemaios und Porphyrios über die Musik (Göteborg, 1934), 177–8, 248ff
F. Solmsen: ‘Eratosthenes as Platonist and Poet’, Transactions of the American Philological Association, lxxiii (1942), 192–213
B.L. van der Waerden: ‘Die Harmonielehre der Pythagoreer’, Hermes, lxxviii (1943), 163–99
G.A. Keller: Eratosthenes und die alexandrinische Sterndichtung (Zürich, 1946)
R.M. Bentham: The Fragments of Eratosthenes (diss., U. of London, 1948)
J.M. Barbour: Tuning and Temperament: a Historical Survey (East Lansing, MI, 1951/R, 2/1953), v, 15ff
E.P. Wolfer: Eratosthenes von Kyrene als Mathematiker und Philosoph (Groningen, 1954)
W. Burkert: ‘Hellenistische Pseudopythagorica, II: Ein System der Sphärenharmonie’, Philologus, cv (1961), 228–43
H. Husmann: Grundlagen der antiken und orientalischen Musikkultur (Berlin, 1961), esp. 36ff
M. Vogel: Die Enharmonik der Griechen (Düsseldorf, 1963), i, esp. 33–4, 44–5
R. Pfeiffer: Geschichte der klassischen Philologie (Hamburg, 1970), 191ff [incl. extensive bibliography]
A. Barbera: ‘Arithmetic and Geometric Divisions of the Tetrachord’, JMT, xxii (1977), 294–323
LUKAS RICHTER