Theodosius of Bithynia

Theodosius of Bithynia (Greek: Θεοδόσιος; c. 160 BC – c. 100 BC) was a Greek astronomer and mathematician who wrote the Sphaerics, a book on the geometry of the sphere.

Theodosius was the author of Sphaerics, a textbook on the geometry of the sphere, and minor astronomical and astrological works. Strabo, in giving a list of Bithynians worthy of note in various fields, mentions “Hipparchus, Theodosius and his sons, mathematicians.”1 Vitruvius mentions Theodosius as the inventor of a sundial suitable for any region.2 Strabo’s references are usually in chronological order in their respective categories; and since Hipparchus was at the height of his career in 127 b. c., while Strabo and Vitruvius both flourished about the beginning of the Christian era, these statements could refer to the same person and probably do. They harmonize with the fact that Theodosius is quoted by name as the author of the Sphaerics by Menelaus (fl . a.d. 100). To allow sufficient time for his sons to be recognized as mathematicians in their own right before Strabo, Theodosius may best be regarded as a younger contemporary of Hipparchus, born in the second half of the second century B .C. and perhaps surviving into the first century; indeed, it is unlikely that such a work as the Sphaerics would have been written long after the development of spherical trigonometry by Hipparchus, for this development makes it look old-fashioned.

Confusion has been created, however, by the notice or notices in the notoriously unreliable Suda Lexicon. The passage reads:

Theodosius, philosopher, wrote Sphaerics in three books, a commentary on the chapter of Theudas, two books On Days and Nights, a commentary on the Method of Archimedes, Descriptions of Houses in three books, Skeptical Chapters, astrological works, On Habitations. Theodosius wrote verses on the spring and other types of works. He was from Tripolis.3

It seems probable that the first sentence in this passage confuses the author of the Sphaerics with a later skeptical philosopher, for Theudas flourished in the second century of the Christian era;4 and it also is probable that the second and third sentences should be regarded as a separate notice about a third Theodosius. This would be unimportant if the third sentence had not given rise to the belief that the author of the Sphaerics was born at Tripolis in Phoenicia, and in almost all editions until recently he has been described as Theodosius of Tripolis.5

Spherics, the geometry of the sphere, was needed for astronomy and was regarded by the ancient Greeks as a branch of astronomy rather than of geometry. Indeed, the Pythagoreans called astronomy “spherics”; and the stereometrical books XII and XIII of Euclid’s Elements, which lead up to the inscription of the regular solids in a sphere, contain nothing about the geometry of the sphere beyond the proof that the volumes of spheres are in the triplicate ratio of their diameters. Euclid treated this subject in his Phaenomena, and just before him Autolycus had dealt with it in his book On the Moving Sphere. From a comparison of propositions quoted or assumed by Euclid and Autolycus, it may be inferred that much of Theodosius’ Sphaerics is derived from some pre-Eu-clidean textbook, of which some have conjectured that Eudoxus was the author.6 There is nothing in it that can strictly be called trigonometry, although in III .11 Theodosius proves the equivalent of the formula tan a = sin b tan A for a spherical triangle right-angled at C.

Two of the other works mentioned by the Suda have survived. On Habitations treats the phenomena caused by the rotation of the earth, particularly what portions of the heavens are visible to the inhabitants of different zones. On Days and Nights studies the arc of the ecliptic traversed by the sun each day. Its object is to determine what conditions have to be satisfied in order that the solstice may occur in the meridian at a given place and in order that day and night may really be equal at the equinoxes.

One reason why these three works have survived must be that they were included in the collection that Pappus called “The Little Astronomy”7–in contrast with “The Great Astronomy” or Almagest of Ptolemy. Pappus also annotated the Sphaerics and On Days and Nights in some detail.8 All three works were translated into Arabic toward the end of the ninth century.9 The translation of the Sphaerics up to II .5 is by Qustā ibn Lūqā and thereafter by Thābit ibn Qurra. The Sphaerics was translated from Arabic into Latin in the twelfth century by Plato of Tivoli and Gerard of Cremona.

There is no reason to doubt that the Theodosius who wrote the Sphaerics was also the author of the commentary on the Method of Archimedes mentioned by the Suda, for the subject matter would be similar. It may be accepted also that he wrote astrological works. It is tempting to think that the Διαγραφὰι οἰκιω̑ν, Descriptions of Houses, mentioned in the Suda, dealt with the “houses of the planets”; but the latter term is always οἰκοι, not οἰκἰαι. It must be considered an architectural work, which could, however, be by the author of the Sphaerics. The other works mentioned in the Suda must be regarded as by another person of the same name. Theodosius’ discovery of a sundial suitable for all regions– πρòς πα̑ν κλίμα – may have been recorded in a book, but nothing is known about it.

You can access and download, for free, from our Online-Library the original Ancient Greek text ΘΕΟΔΟΣΙΟΥ ΤΡΙΠΟΛΙΤΟΥ – Σφαιρικά. You can download it by following the link below:



1. Strabo, GeographyXII , 4, 9 c 566, A. Meineke, ed. (Leipzig, 1853), II , 795, 13–14.

2. Vitruvius, De architecturaIX , 8, 1, F. Krohn, ed. (Leipzig, 1912), p. 218.7.

3. Suda Lexicon, under θєοδοϳσιοζ, Ada Adler, ed., 11 (Leipzig, 1931), θ 142 and 143, p. 693.

4. Diogenes Laërtius, Vitae philosophorumIX , 116, H. S. Long, ed. (Oxford, 1964), II , 493.14. He was, according to Diogenes, the fifth skeptical philosopher in succession to Aenesidemus, who flourished at the time of Cicero.

5. Even the definitive ed. by J. L. Heiberg (1927) is entitled “Theodosius Tripolites Sphaerica” but the first entry in the corrigenda (p. xvi)is “Tripolites deleatur ubique.”

6. The following propositions in the Sphaerics are certainly pre-Euclidean; bk. l , props. 1,6,7,8,11,12,13,15,20; bk. II , props. 1,2,3,5,8,9,10,13,15,17,18,19,20; bk. III , prop. 2.

7. ‘Ο μικροζ ἀστρουομουμευος, Pappus, Collection VI titulus, F. Hultsch, ed., Pappi Alexandrini Collectionis quae supersunt, II (Berlin, 1877), 475.

8. Sphaerics, ibid., VI .1–33, props. 1–26, F. Hultsch, ed., 475–519; On Days and Nights, Ibid., VI . 48–68, props. 30–36, F. Hultsch, ed., II , 530–555.

9. H. Wenrich, De auctorum Graecorum Versionibus et commentarüs Syriacis Arabicis etc. (Leipzig, 1842), 206; H. Suter, Die Mathematiker und Astronomen der Araber und ihre Werke (Leipzig, 1900), 41.


I. Original Works. The three surviving works of Theodosius are in many MSS , of which the most important is Codex Vaticanus Graecus 204 (10th cent.). The Sphaerics was first printed in a Latin ed. translated from the Arabic (Venice, 1518), which was followed by Voegelin’s Latin ed. (Vienna, 1529), also taken from the Arabic. The editio princeps of the Greek text (with Latin trans.) is J. Pena, Theodosii Tripolitae Sphaericorum libri tres (Paris, 1558). Subsequent eds. are F. Maurolico (Messina, 1558); Latin trans. only); C. Dasypodius (Strasbourg, 1572; enunciations only in Greek and Latin); C. Clavius (Rome, 1586; Latin trans. only, with works of his own); J. Auria (Rome, 1587); M. Mersenne (Paris, 1644); C. Dechales (Lyons, 1674); I. Barrow (London, 1675); J. Hunt (Oxford, 1707); E. Nizze (Berlin, 1852). The definitive ed. is J. L. Heibert, “Theodosius Tripolites Sphaerica,” in Abhandlungen der Gesellschaft der Wissenschaften zu Göttingen, phil.-hist, Kl ., n.s. 19 , no. 3 (1927), which contains notes on the MSS (i–xv), text with Latin trans. (1–165), and scholia (166–199).

The Greek enunciations of On Habitations and On Days and Nights were included by Dasypodius in his ed. (Strasbourg, 1572) and Latin translations of the two texts were published by J. Auria (Rome, 1587 and 1591, respectively), but the Greek texts were not printed until the definitive ed. by R. Fecht, “Theodoii De habitationibus liber De diebus et noctibus libri duo,” in Abhandlungen der Gesellschaft der Wissenschaften zu Göttingen, phil.-hist, Kl n.s. 19 , no. 4 (1927), which contains notes (1–12), text and Latin trans. of On Habitations (13–43), scholia on Habitations (44–52), text and Latin trans. of On Days and Nights (53–155), and scholia on Days and Nights (156–176). The scholia were first edited by F. Hultsch, “Scholien zur Sphärik des Theodosios,” in Abhandlungen des philosohisch-historische Classe der K. Sächsischen Gesellschaft der Wissenschaften, 10 , no. 5 (1887).

There is a German trans. of the Sphaerics by E. Nizze. Die Sphärik des Theodosios (Stralsund, 1826). There are French translations by D. Henrion (Paris, 1615); J. B. du Hamel (Paris, 1660); and Paul ver Eecke, Théodose de Tripoli: Les sphériques (Paris–Bruges, 1927).

II. Secondary Literature. The most useful material on Theodosius is Thomas Heath, A History of Greek Mathematics (Oxford, 1921), II, 245–252; the Latin intro, to R. Fecht’s ed. (see above), 1–12; and K. Ziegler, “Theodosius 5,” in Pauly-Wissowa, Real-Encyclopädie der classischen Altertumswissenschaft, n.s. V, cols. 1930–1935. Other sources, listed chronologically, are A. Nokk, Über die Sphärik des Theodosios (Karlsruhe, 1847): F. Hultsch, “Die Sphärik des Theodosios und einige unedierte Texte,” in Berichte der Sächsischen Gesellschaft der Wissenschaften(1885); R. Carra de Vaux, “Remaniement des Sphériques de Théodose par Jahia ibn Muhammed ibn Abī Schukr al-Maghrabī al Andalusī,” in Journal asiatique, 17 (1891), 287–295; P. Tannery, Recherches sur l’histoire de l’astronomie ancienne (Paris, 1893), 36–37; and A. A. Björnbo, “Studen über Menelaos’ Sphärik: Beiträge zur Geschichte des Sphärik und Trigonometrie der Griechen,” in Abhandlungen zur Geschichte der mathematischen Wissenschaften, 14 (1902), 64–65; and “Über zwei mathematische Handschriften,” in Bibliotheca mathematica, n.s. 3 (1902), 63–75.



[1] "" by Ivor Bulmer-Thomas

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