Primarily consisting of theorems which were known at least informally a couple centuries earlier, theSpherics was a foundational treatise for geometers and astronomers from its origin until the 19th century. It was continuously studied and copied in Greek manuscript for more than a millennium. It was translated intoArabic in the 9th century during theIslamic Golden Age, and thence translated intoLatinin 12th century Iberia, though the text and diagrams were somewhat corrupted. In the 16th century printed editions in Greek were published along with better translations into Latin.
Several of the definitions and theorems in theSpherics were used without mention inEuclid'sPhenomena and two extant works byAutolycus concerning motions of the celestial sphere, all written about two centuries before Theodosius. It has been speculated that this tradition of Greek "spherics" – founded in the axiomatic system and using the methods of proof of solid geometry exemplified byEuclid'sElements but extended with additional definitions relevant to the sphere – may have originated in a now-unknown work byEudoxus, who probably established a two-sphere model of the cosmos (spherical Earth and celestial sphere) sometime between 370–340 BC.[1]
TheSpherics is a supplement to theElements, and takes its content for granted as a prerequisite. TheSpherics follows the general presentation style of theElements, with definitions followed by a list of theorems (propositions), each of which is first stated abstractly as prose, then restated withpoints lettered for the proof. It analysesspherical circles as flat circles lying in planes intersecting the sphere and provides geometric constructions for various configurations of spherical circles. Spherical distances and radii are treated as Euclidean distances in the surrounding 3-dimensional space. The relationship between planes is described in terms ofdihedral angle. As in theElements, there is no concept ofangle measure ortrigonometry per se.
This approach differs from other quantitative methods of Greek astronomy such as the analemma (orthographic projection),[2]stereographic projection, or trigonometry (a fledgling subject introduced by Theodosius' contemporaryHipparchus). It also differs from the approach taken inMenelaus'Spherics, a treatise of the same title written 3 centuries later, which treats the geometry of the sphereintrinsically, analyzing the inherent structure of the spherical surface and circles drawn on it rather than primarily treating it as a surface embedded in three-dimensional space.
Inlate antiquity, theSpherics was part of a collection of treatises now called theLittle Astronomy, an assortment of shorter works on geometry and astronomy building on Euclid'sElements. Other works in the collection includedAristarchus'On the Sizes and Distances, Autolycus'On Rising and Settings andOn the Moving Sphere, Euclid'sCatoptrics,Data,Optics, andPhenomena,Hypsicles'On Ascensions, Theodosius'On Geographic Places andOn Days and Nights, and Menelaus'Spherics. Often several of these were bound together in a single volume. During theIslamic Golden Age, the books in the collection were translated intoArabic, and with the addition of a few new works, were known as theMiddle Books, intended to fit between theElements andPtolemy'sAlmagest.[3]
Czinczenheim, Claire, ed. (2000).Édition, traduction et commentaire des Sphériques de Théodose (PhD thesis) (in Greek and French). Université de Paris IV, Paris-Sorbonne.
Spandagos, Vangelēs, ed. (2000).Ta Sphairika tu Theodosiu tu TripolituΤα Σφαιρικα του Θεοδοσιου του Τριπολιτου (in Greek). Athens: Aithra.ISBN9789607007889.
Kunitzsch, Paul; Lorch, Richard, eds. (2010).Theodosius, "Sphaerica": Arabic and Medieval Latin Translations (in Arabic, Latin, and English). Stuttgart: Franz Steiner.ISBN9783515092883.
^Berggren, John L. (1991)."The relation of Greek Spherics to early Greek astronomy". In Bowen, Alan C. (ed.).Science and Philosophy in Classical Greece. Garland. pp. 227–248. For more about the two-sphere model, see:Goldstein, Bernard R.; Bowen, Alan C. (1983). "A New View of Early Greek Astronomy".Isis.74 (3):330–340.JSTOR232593.
Lorch, Richard (1996). "The transmission of Theodosius'Sphaerica". InFolkerts, Menso (ed.).Mathematische Probleme im Mittelalter: Der lateinische und arabische Sprachbereich. Wiesbaden: Harrassowitz. pp. 159–184.
Malpangotto, Michela (2010). "Graphical Choices and Geometrical Thought in the Transmission of Theodosius'Spherics from Antiquity to the Renaissance".Archive for History of Exact Sciences.64 (1):75–112.doi:10.1007/s00407-009-0054-1.JSTOR41342412.
Thomas, Robert S.D. (2013). "Acts of Geometrical Construction in theSpherics of Theodosios".From Alexandria, Through Baghdad. Springer. pp. 227–237.doi:10.1007/978-3-642-36736-6_11.
Thomas, Robert S.D. (2018). "The definitions and theorems ofThe Spherics of Theodosios". In Sidoli, Nathan; Brummelen, Glen Van (eds.).Research in History and Philosophy of Mathematics. CSHPM Annual Meeting, Toronto, Ontario, May 28–30 2017. Springer. pp. 1–21.doi:10.1007/978-3-642-36736-6_11.
