2278:
336:
2265:
164:
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
125:
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 by
85:
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 into
1034:
207:, 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.
393:
2138:
101:, though the text and diagrams were somewhat corrupted. In the 16th century printed editions in Greek were published along with better translations into Latin.
2302:
2216:
1519:
1055:
234:
370:
Sphera mundi noviter recognita cum commentariis et authoribus in hoc volumine contentis, videlicet [...] Theodosii de
Spheris [...]
1027:
2063:
98:
1793:
566:
Autolykos: Rotierende kugel und
Aufgang und untergang der gestirne. Theodosios von Tripolis: Sphaerik. Übersetzt und mit anmerkungen versehen
1758:
2126:
1738:
1529:
156:, with definitions followed by a list of theorems (propositions), each of which is first stated abstractly as prose, then restated with
2193:
1020:
788:
669:
642:
623:
773:
535:
137:, who probably established a two-sphere model of the cosmos (spherical Earth and celestial sphere) sometime between 370–340 BC.
580:
Kitāb al-ukar li-Thāʾudhūsiyūs: Taḥrīr al-alāma al-faylasūf al-H̱awāǧah Naṣīr al-Dīn Muḥammad ibn Muḥammad ibn al-Ḥasan al-Ṭūsī
2162:
2095:
1728:
1608:
2231:
1988:
1876:
1051:
539:
2307:
1871:
1940:
799:
The Little
Astronomy and Middle Books between the 2nd and 13th Centuries CE: Transmissions of Astronomical Curricula
1567:
1383:
1358:
2102:
2073:
1433:
1288:
2236:
1513:
189:
133:
but extended with additional definitions relevant to the sphere – may have originated in a now-unknown work by
1970:
2198:
2174:
2048:
1983:
1924:
1861:
1851:
1506:
1368:
1278:
1158:
185:
834:
Malpangotto, Michela (2010). "Graphical
Choices and Geometrical Thought in the Transmission of Theodosius'
2221:
2169:
2068:
1896:
1846:
1831:
1826:
1597:
1398:
1333:
1323:
1273:
197:
60:
53:
379:
2269:
2121:
1906:
1783:
1706:
1654:
1456:
1363:
1213:
2277:
1965:
340:
220:
81:
Primarily consisting of theorems which were known at least informally a couple centuries earlier, the
2246:
2186:
2150:
1993:
1816:
1763:
1733:
1723:
1632:
1495:
1388:
1303:
1258:
1238:
1083:
1068:
574:
229:
127:
553:
488:
407:
2226:
2145:
2133:
2114:
2078:
1998:
1916:
1901:
1891:
1841:
1836:
1778:
1647:
1539:
1468:
1403:
1393:
1293:
1263:
1203:
1178:
1103:
1093:
1078:
431:
389:
315:
Authoritative critical editions of the Greek text, compiled from several manuscripts, were made by
122:
908:
592:
578:
203:, a treatise of the same title written 3 centuries later, which treats the geometry of the sphere
2282:
2241:
2181:
2109:
1975:
1768:
1743:
1711:
1308:
1253:
1218:
1113:
998:
895:
855:
714:
512:
287:
91:
64:
46:
1950:
1549:
459:
258:
694:
403:
184:
This approach differs from other quantitative methods of Greek astronomy such as the analemma (
1955:
1866:
1716:
1642:
1615:
1378:
1198:
1188:
1123:
1043:
784:
665:
638:
619:
417:
345:
134:
50:
463:
395:
Theodosii sphaericorum elementorum libri III, ex traditione
Maurolyci Messanensis mathematici
358:
2157:
2028:
1945:
1701:
1689:
1637:
1373:
990:
965:
944:
923:
887:
847:
772:
753:
657:
525:
252:
161:
157:
75:
734:
583:كتاب الاكر لثاوذوسيوس: تحرير العلامة الفيلسوف الخوا نصير الدين محمد بن محمد بن الحسن الطوصي
449:
421:
1798:
1788:
1682:
1408:
739:
35 and analemma methods: An astronomical determination of the distance between two cities"
2058:
2053:
1881:
1773:
1753:
1581:
1138:
1108:
822:
564:
549:
211:
166:
67:
868:
757:
504:
Clavius's
Commentary on the Sphericks of Theodosius Tripolitae: or, Spherical Elements
502:
477:
335:
2296:
2018:
1886:
1856:
1677:
1485:
1428:
705:
Goldstein, Bernard R.; Bowen, Alan C. (1983). "A New View of Early Greek
Astronomy".
445:
368:
174:
26:
1960:
1748:
1423:
1183:
1173:
964:. CSHPM Annual Meeting, Toronto, Ontario, May 28–30 2017. Springer. pp. 1–21.
498:
473:
435:
178:
994:
827:
Mathematische
Probleme im Mittelalter: Der lateinische und arabische Sprachbereich
969:
948:
70:, while the second half of Book II and Book III contain propositions relevant to
1574:
1462:
1168:
1153:
224:, an assortment of shorter works on geometry and astronomy building on Euclid's
1560:
1479:
1418:
1413:
1353:
1338:
1283:
1268:
1223:
1163:
1148:
1128:
1098:
1063:
891:
851:
468:(in Latin and French). Vol. 5. Parisiis: Henry le Gras. pp. 218–329.
193:
192:, or trigonometry (a fledgling subject introduced by Theodosius' contemporary
927:
1313:
1298:
1248:
1143:
1133:
1118:
1088:
1012:
797:
286:. Often several of these were bound together in a single volume. During the
267:
95:
71:
661:
1450:
1228:
1073:
479:
Theodosii Sphaerica: Methodo Nova Illustrata, & Succinctè Demonstrata
308:
42:
1002:
899:
859:
2023:
1348:
1343:
1243:
1233:
1208:
303:
718:
1318:
1193:
935:
Thomas, Robert S.D. (2013). "Acts of Geometrical Construction in the
291:
114:
87:
36:
357:
Valla, Giorgio, ed. (1501). "De sphaericis (book XII, chapter V)".
1694:
1672:
1328:
977:
Thomas, Robert S.D. (2018). "An Appreciation of the First Book of
334:
606:(PhD thesis) (in French). Université de Paris IV, Paris-Sorbonne.
490:
Theodosiou Sphairikōn biblia 3. Theodosii Sphaericorum libri tres
597:(PhD thesis) (in Arabic and English). University of St. Andrews.
426:(in Latin and Greek). Argentorati: Excudebat Christianus Mylius.
1016:
960:
of Theodosios". In Sidoli, Nathan; Brummelen, Glen Van (eds.).
635:
Theodosius, "Sphaerica": Arabic and Medieval Latin Translations
569:(in German). Leipzig: Akademische verlagsgesellschaft m. b. h.
462:, ed. (1637). "Theodosii Sphaerica = Sphériques de Théodose".
451:
Les trois livres des Élémens spériques de Théodose Tripolitain
294:, and with the addition of a few new works, were known as the
956:
Thomas, Robert S.D. (2018). "The definitions and theorems of
604:Édition, traduction et commentaire des Sphériques de Théodose
586:(in Arabic). Ḥaydarʾābād: Dāʾirat al-maʿārif al-ʿuthmānīyyah.
30:
652:
Sidoli, Nathan; Thomas, Robert Spencer David, eds. (2023).
637:(in Arabic, Latin, and English). Stuttgart: Franz Steiner.
148:, and takes its content for granted as a prerequisite. The
695:"The relation of Greek Spherics to early Greek astronomy"
817:
Lorch, Richard (1996). "The transmission of Theodosius'
544:(in Greek and Latin). Berlin: Weidmannsche Buchhandlung.
731:
A description of the analemma method can be found in:
869:"The role of geometrical construction in Theodosius's
218:
was part of a collection of treatises now called the
59:
Book I and the first half of Book II establish basic
2209:
2087:
2037:
2011:
1933:
1915:
1814:
1807:
1663:
1625:
1442:
1050:
290:, the books in the collection were translated into
920:New Perspectives on the History of Islamic Science
363:(in Latin). Vol. 1. Venetiis: Aldus Manutius.
962:Research in History and Philosophy of Mathematics
909:"Naṣīr al-Dīn al-Ṭūsī's revision of Theodosius's
63:needed for spherical geometry using the tools of
594:The Arabic Translation of Theodosius's Sphaerica
440:(in Latin). Rome: Ex Typographia Dominici Basae.
398:(in Latin). Messina: Petrus Spira mense Augusto.
775:The History & Practice of Ancient Astronomy
109:Several of the definitions and theorems in the
633:Kunitzsch, Paul; Lorch, Richard, eds. (2010).
412:(in Greek and Latin). Paris: Andreas Wechelus.
327:is an English translation by modern scholars.
196:). It also differs from the approach taken in
152:follows the general presentation style of the
1028:
530:(in Greek and Latin). Berlin: Georgii Reimer.
527:Theodosii Tripolitae Sphaericorum Libros Tres
409:Theodosij Tripolitae Sphaericorum, libri tres
8:
922:. Vol. 3. Routledge. pp. 355–392.
829:. Wiesbaden: Harrassowitz. pp. 159–184.
324:
320:
703:For more about the two-sphere model, see:
437:Theodosii Tripolitae Sphaericorum Libri III
2041:
1811:
1035:
1021:
1013:
699:Science and Philosophy in Classical Greece
493:(in Greek and Latin). Oxford: H. Clements.
907:Sidoli, Nathan; Kusuba, Takanori (2017).
228:. Other works in the collection included
16:Ancient Greek spherical geometry treatise
616:Τα Σφαιρικα του Θεοδοσιου του Τριπολιτου
465:Cursus mathematicus = Cours mathématique
454:(in French). Paris: Chez Abraham Pacard.
685:
384:(in Latin). Vienna: Joannes Singrenius.
316:
2064:Latin translations of the 12th century
613:Ta Sphairika tu Theodosiu tu Tripolitu
1794:Straightedge and compass construction
880:Archive for History of Exact Sciences
840:Archive for History of Exact Sciences
555:Les sphériques de Théodose de Tripoli
7:
1759:Incircle and excircles of a triangle
838:from Antiquity to the Renaissance".
558:(in French). Bruges: Brouwer et Cie.
160:lettered for the proof. It analyses
867:Sidoli, Nathan; Saito, Ken (2009).
802:(PhD thesis). New York University.
602:Czinczenheim, Claire, ed. (2000).
423:Sphaericae doctrinae propositiones
381:Theodosii de Sphaericis libri tres
14:
758:10.1111/j.1600-0498.2005.470304.x
611:Spandagos, Vangelēs, ed. (2000).
482:(in Latin). London: Guil. Godbid.
2303:Ancient Greek mathematical works
2276:
2263:
941:From Alexandria, Through Baghdad
779:. Oxford University Press. "The
360:De fugiendis et expetendis rebus
591:Martin, Thomas J., ed. (1975).
378:Vögelin, Johannes, ed. (1529).
2096:A History of Greek Mathematics
1609:The Quadrature of the Parabola
943:. Springer. pp. 227–237.
563:Czwalina, Arthur, ed. (1931).
298:, intended to fit between the
90:in the 9th century during the
56:in the 2nd or 1st century BC.
1:
995:10.1080/0025570X.2017.1404798
350:presents his work to the king
113:were used without mention in
94:, and thence translated into
1877:Intersecting secants theorem
970:10.1007/978-3-642-36736-6_11
949:10.1007/978-3-642-36736-6_11
918:. In Iqbal, Muzaffar (ed.).
701:. Garland. pp. 227–248.
618:(in Greek). Athens: Aithra.
1872:Intersecting chords theorem
1739:Doctrine of proportionality
796:Roughan, Christine (2023).
697:. In Bowen, Alan C. (ed.).
373:(in Latin). Venetiis. 1518.
2324:
1568:On the Sphere and Cylinder
1521:On the Sizes and Distances
693:Berggren, John L. (1991).
654:The Spherics of Theodosios
524:Nizze, Ernst, ed. (1852).
517:Die Sphärik des Theodosios
487:Hunt, Joseph, ed. (1707).
325:Sidoli & Thomas (2023)
235:On the Sizes and Distances
31:
2270:Ancient Greece portal
2259:
2074:Philosophy of mathematics
2044:
1989:Ptolemy's table of chords
1044:Ancient Greek mathematics
892:10.1007/s00407-009-0045-2
852:10.1007/s00407-009-0054-1
331:Editions and translations
173:, there is no concept of
51:Hellenistic mathematician
1941:Aristarchus's inequality
1514:On Conoids and Spheroids
928:10.4324/9781315248011-18
615:
582:
339:1529 title page: Editor
190:stereographic projection
121:and two extant works by
2049:Ancient Greek astronomy
1862:Inscribed angle theorem
1852:Greek geometric algebra
1507:Measurement of a Circle
733:Sidoli, Nathan (2005).
519:(in German). Stralsund.
186:orthographic projection
144:is a supplement to the
61:geometric constructions
2283:Mathematics portal
2069:Non-Euclidean geometry
2024:Mouseion of Alexandria
1897:Tangent-secant theorem
1847:Geometric mean theorem
1832:Exterior angle theorem
1827:Angle bisector theorem
1531:On Sizes and Distances
351:
240:On Rising and Settings
99:in 12th century Iberia
54:Theodosius of Bithynia
1971:Pappus's area theorem
1907:Theorem of the gnomon
1784:Quadratrix of Hippias
1707:Circles of Apollonius
1655:Problem of Apollonius
1633:Constructible numbers
1457:Archimedes Palimpsest
771:Evans, James (1998).
662:10.4324/9781003142164
656:. London: Routledge.
541:Theodosius. Sphaerica
536:Heiberg, Johan Ludvig
338:
2187:prehistoric counting
1984:Ptolemy's inequality
1925:Apollonius's theorem
1764:Method of exhaustion
1734:Diophantine equation
1724:Circumscribed circle
1541:On the Moving Sphere
983:Mathematics Magazine
575:Naṣīr al-Dīn al-Ṭūsī
432:Clavius, Christopher
390:Maurolico, Francesco
355:partial edition in:
276:On Geographic Places
244:On the Moving Sphere
41:) is a three-volume
2273: •
2079:Neusis construction
1999:Spiral of Theodorus
1892:Pythagorean theorem
1837:Euclidean algorithm
1779:Lune of Hippocrates
1648:Squaring the circle
1404:Theon of Alexandria
1079:Aristaeus the Elder
783:", pp. 89–91.
507:. London: J. Senex.
321:Czinczenheim (2000)
2308:Spherical geometry
1966:Menelaus's theorem
1956:Irrational numbers
1769:Parallel postulate
1744:Euclidean geometry
1712:Apollonian circles
1254:Isidore of Miletus
418:Dasypodius, Conrad
352:
288:Islamic Golden Age
280:On Days and Nights
92:Islamic Golden Age
74:as modeled by the
47:spherical geometry
2290:
2289:
2255:
2254:
2007:
2006:
1994:Ptolemy's theorem
1867:Intercept theorem
1717:Apollonian gasket
1643:Doubling the cube
1616:The Sand Reckoner
162:spherical circles
2315:
2281:
2280:
2268:
2267:
2266:
2042:
2029:Platonic Academy
1976:Problem II.8 of
1946:Crossbar theorem
1902:Thales's theorem
1842:Euclid's theorem
1812:
1729:Commensurability
1690:Axiomatic system
1638:Angle trisection
1603:
1593:
1555:
1545:
1535:
1525:
1501:
1491:
1474:
1037:
1030:
1023:
1014:
1006:
973:
952:
939:of Theodosios".
931:
917:
903:
877:
863:
830:
804:
803:
794:
781:Little Astronomy
778:
768:
762:
761:
743:
729:
723:
722:
702:
690:
675:
648:
629:
607:
598:
587:
570:
559:
545:
531:
520:
508:
494:
483:
469:
460:Hérigone, Pierre
455:
441:
427:
413:
399:
385:
374:
364:
349:
341:Johannes Vögelin
282:, and Menelaus'
221:Little Astronomy
76:celestial sphere
40:
34:
33:
2323:
2322:
2318:
2317:
2316:
2314:
2313:
2312:
2293:
2292:
2291:
2286:
2275:
2264:
2262:
2251:
2217:Arabian/Islamic
2205:
2194:numeral systems
2083:
2033:
2003:
1951:Heron's formula
1929:
1911:
1803:
1799:Triangle center
1789:Regular polygon
1666:and definitions
1665:
1659:
1621:
1601:
1591:
1553:
1543:
1533:
1523:
1499:
1489:
1472:
1438:
1409:Theon of Smyrna
1054:
1046:
1041:
1010:
976:
955:
934:
915:
906:
875:
866:
833:
823:Folkerts, Menso
816:
813:
808:
807:
795:
791:
770:
769:
765:
741:
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704:
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687:
682:
672:
651:
645:
632:
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617:
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590:
584:
573:
562:
550:Ver Eecke, Paul
548:
534:
523:
511:
497:
486:
472:
458:
444:
430:
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402:
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377:
367:
356:
343:
333:
107:
49:written by the
17:
12:
11:
5:
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2210:Other cultures
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2059:Greek numerals
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2054:Attic numerals
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2015:
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1882:Law of cosines
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1774:Platonic solid
1771:
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1754:Greek numerals
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1582:Planisphaerium
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1052:Mathematicians
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1032:
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1007:
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932:
904:
886:(6): 581–609.
864:
831:
812:
809:
806:
805:
789:
763:
752:(3): 236–258.
724:
713:(3): 330–340.
684:
683:
681:
678:
677:
676:
670:
649:
643:
630:
624:
608:
599:
588:
577:, ed. (1939).
571:
560:
552:, ed. (1927).
546:
538:, ed. (1927).
532:
521:
515:, ed. (1826).
509:
501:, ed. (1721).
495:
484:
476:, ed. (1675).
470:
456:
448:, ed. (1615).
446:Henrion, Denis
442:
434:, ed. (1586).
428:
420:, ed. (1573).
414:
406:, ed. (1558).
400:
392:, ed. (1558).
386:
375:
365:
332:
329:
317:Heiberg (1927)
274:, Theodosius'
212:late antiquity
167:dihedral angle
106:
103:
68:solid geometry
15:
13:
10:
9:
6:
4:
3:
2:
2320:
2309:
2306:
2304:
2301:
2300:
2298:
2285:
2284:
2279:
2272:
2271:
2258:
2248:
2245:
2243:
2240:
2238:
2235:
2233:
2230:
2228:
2225:
2223:
2220:
2218:
2215:
2214:
2212:
2208:
2200:
2197:
2196:
2195:
2192:
2188:
2185:
2184:
2183:
2180:
2176:
2173:
2172:
2171:
2168:
2164:
2161:
2160:
2159:
2156:
2152:
2149:
2148:
2147:
2144:
2140:
2137:
2136:
2135:
2132:
2128:
2125:
2124:
2123:
2120:
2116:
2113:
2112:
2111:
2108:
2104:
2100:
2099:
2098:
2097:
2093:
2092:
2090:
2086:
2080:
2077:
2075:
2072:
2070:
2067:
2065:
2062:
2060:
2057:
2055:
2052:
2050:
2047:
2046:
2043:
2040:
2036:
2030:
2027:
2025:
2022:
2020:
2017:
2016:
2014:
2010:
2000:
1997:
1995:
1992:
1990:
1987:
1985:
1982:
1980:
1979:
1974:
1972:
1969:
1967:
1964:
1962:
1959:
1957:
1954:
1952:
1949:
1947:
1944:
1942:
1939:
1938:
1936:
1932:
1926:
1923:
1922:
1920:
1918:
1914:
1908:
1905:
1903:
1900:
1898:
1895:
1893:
1890:
1888:
1887:Pons asinorum
1885:
1883:
1880:
1878:
1875:
1873:
1870:
1868:
1865:
1863:
1860:
1858:
1857:Hinge theorem
1855:
1853:
1850:
1848:
1845:
1843:
1840:
1838:
1835:
1833:
1830:
1828:
1825:
1824:
1822:
1820:
1819:
1813:
1810:
1806:
1800:
1797:
1795:
1792:
1790:
1787:
1785:
1782:
1780:
1777:
1775:
1772:
1770:
1767:
1765:
1762:
1760:
1757:
1755:
1752:
1750:
1747:
1745:
1742:
1740:
1737:
1735:
1732:
1730:
1727:
1725:
1722:
1718:
1715:
1713:
1710:
1709:
1708:
1705:
1703:
1700:
1696:
1693:
1692:
1691:
1688:
1684:
1681:
1679:
1676:
1675:
1674:
1671:
1670:
1668:
1662:
1656:
1653:
1649:
1646:
1644:
1641:
1639:
1636:
1635:
1634:
1631:
1630:
1628:
1624:
1618:
1617:
1613:
1611:
1610:
1606:
1604:
1600:
1596:
1594:
1590:
1586:
1584:
1583:
1579:
1577:
1576:
1572:
1570:
1569:
1565:
1563:
1562:
1558:
1556:
1552:
1548:
1546:
1542:
1538:
1536:
1532:
1528:
1526:
1524:(Aristarchus)
1522:
1518:
1516:
1515:
1511:
1509:
1508:
1504:
1502:
1498:
1494:
1492:
1488:
1484:
1482:
1481:
1477:
1475:
1471:
1467:
1465:
1464:
1460:
1458:
1455:
1453:
1452:
1448:
1447:
1445:
1441:
1435:
1432:
1430:
1429:Zeno of Sidon
1427:
1425:
1422:
1420:
1417:
1415:
1412:
1410:
1407:
1405:
1402:
1400:
1397:
1395:
1392:
1390:
1387:
1385:
1382:
1380:
1377:
1375:
1372:
1370:
1367:
1365:
1362:
1360:
1357:
1355:
1352:
1350:
1347:
1345:
1342:
1340:
1337:
1335:
1332:
1330:
1327:
1325:
1322:
1320:
1317:
1315:
1312:
1310:
1307:
1305:
1302:
1300:
1297:
1295:
1292:
1290:
1287:
1285:
1282:
1280:
1277:
1275:
1272:
1270:
1267:
1265:
1262:
1260:
1257:
1255:
1252:
1250:
1247:
1245:
1242:
1240:
1237:
1235:
1232:
1230:
1227:
1225:
1222:
1220:
1217:
1215:
1212:
1210:
1207:
1205:
1202:
1200:
1197:
1195:
1192:
1190:
1187:
1185:
1182:
1180:
1177:
1175:
1172:
1170:
1167:
1165:
1162:
1160:
1157:
1155:
1152:
1150:
1147:
1145:
1142:
1140:
1137:
1135:
1132:
1130:
1127:
1125:
1122:
1120:
1117:
1115:
1112:
1110:
1107:
1105:
1102:
1100:
1097:
1095:
1092:
1090:
1087:
1085:
1082:
1080:
1077:
1075:
1072:
1070:
1067:
1065:
1062:
1061:
1059:
1057:
1053:
1049:
1045:
1038:
1033:
1031:
1026:
1024:
1019:
1018:
1015:
1011:
1004:
1000:
996:
992:
988:
984:
980:
975:
971:
967:
963:
959:
954:
950:
946:
942:
938:
933:
929:
925:
921:
914:
912:
905:
901:
897:
893:
889:
885:
881:
874:
872:
865:
861:
857:
853:
849:
846:(1): 75–112.
845:
841:
837:
832:
828:
824:
820:
815:
814:
810:
801:
800:
792:
790:0-19-509539-1
786:
782:
777:
776:
767:
764:
759:
755:
751:
747:
740:
738:
728:
725:
720:
716:
712:
708:
700:
696:
689:
686:
679:
673:
671:9780367557300
667:
663:
659:
655:
650:
646:
644:9783515092883
640:
636:
631:
627:
625:9789607007889
621:
614:
609:
605:
600:
596:
595:
589:
585:
581:
576:
572:
568:
567:
561:
557:
556:
551:
547:
543:
542:
537:
533:
529:
528:
522:
518:
514:
510:
506:
505:
500:
499:Stone, Edmund
496:
492:
491:
485:
481:
480:
475:
474:Barrow, Isaac
471:
467:
466:
461:
457:
453:
452:
447:
443:
439:
438:
433:
429:
425:
424:
419:
415:
411:
410:
405:
401:
397:
396:
391:
387:
383:
382:
376:
372:
371:
366:
362:
361:
354:
353:
347:
342:
337:
330:
328:
326:
322:
318:
313:
311:
310:
305:
301:
297:
293:
289:
285:
281:
277:
273:
272:On Ascensions
269:
265:
261:
260:
255:
254:
249:
245:
241:
238:, Autolycus'
237:
236:
231:
227:
223:
222:
217:
213:
208:
206:
205:intrinsically
202:
199:
195:
191:
187:
182:
180:
176:
175:angle measure
172:
168:
163:
159:
155:
151:
147:
143:
138:
136:
132:
131:
124:
120:
116:
112:
104:
102:
100:
97:
93:
89:
84:
79:
77:
73:
69:
66:
62:
57:
55:
52:
48:
44:
39:
28:
24:
23:
2274:
2261:
2103:Thomas Heath
2094:
1977:
1961:Law of sines
1817:
1749:Golden ratio
1614:
1607:
1598:
1592:(Theodosius)
1588:
1587:
1580:
1573:
1566:
1559:
1550:
1540:
1534:(Hipparchus)
1530:
1520:
1512:
1505:
1496:
1486:
1478:
1473:(Apollonius)
1469:
1461:
1449:
1424:Zeno of Elea
1184:Eratosthenes
1174:Dionysodorus
1009:
986:
982:
978:
961:
958:The Spherics
957:
940:
936:
919:
910:
883:
879:
870:
843:
839:
835:
826:
818:
798:
780:
774:
766:
749:
745:
736:
727:
710:
706:
698:
688:
653:
634:
612:
603:
593:
579:
565:
554:
540:
526:
516:
513:Nizze, Ernst
503:
489:
478:
464:
450:
436:
422:
408:
394:
380:
369:
359:
314:
307:
299:
296:Middle Books
295:
283:
279:
275:
271:
263:
257:
251:
247:
243:
239:
233:
225:
219:
215:
209:
204:
200:
183:
179:trigonometry
170:
169:. As in the
153:
149:
145:
141:
139:
129:
118:
110:
108:
82:
80:
58:
38:tà sphairiká
37:
21:
20:
18:
2170:mathematics
1978:Arithmetica
1575:Ostomachion
1544:(Autolycus)
1463:Arithmetica
1239:Hippocrates
1169:Dinostratus
1154:Dicaearchus
1084:Aristarchus
989:(1): 3–15.
344: [
246:, Euclid's
230:Aristarchus
32:τὰ σφαιρικά
2297:Categories
2222:Babylonian
2122:arithmetic
2088:History of
1917:Apollonius
1602:(Menelaus)
1561:On Spirals
1480:Catoptrics
1419:Xenocrates
1414:Thymaridas
1399:Theodosius
1384:Theaetetus
1364:Simplicius
1354:Pythagoras
1339:Posidonius
1324:Philonides
1284:Nicomachus
1279:Metrodorus
1269:Menaechmus
1224:Hipparchus
1214:Heliodorus
1164:Diophantus
1149:Democritus
1129:Chrysippus
1099:Archimedes
1094:Apollonius
1064:Anaxagoras
1056:(timeline)
811:References
404:Péna, Jean
248:Catoptrics
194:Hipparchus
1683:Inscribed
1443:Treatises
1434:Zenodorus
1394:Theodorus
1369:Sosigenes
1314:Philolaus
1299:Oenopides
1294:Nicoteles
1289:Nicomedes
1249:Hypsicles
1144:Ctesibius
1134:Cleomedes
1119:Callippus
1104:Autolycus
1089:Aristotle
1069:Anthemius
819:Sphaerica
746:Centaurus
735:"Heron's
268:Hypsicles
264:Phenomena
198:Menelaus'
128:Euclid's
123:Autolycus
119:Phenomena
72:astronomy
65:Euclidean
2247:Japanese
2232:Egyptian
2175:timeline
2163:timeline
2151:timeline
2146:geometry
2139:timeline
2134:calculus
2127:timeline
2115:timeline
1818:Elements
1664:Concepts
1626:Problems
1599:Spherics
1589:Spherics
1554:(Euclid)
1500:(Euclid)
1497:Elements
1490:(Euclid)
1451:Almagest
1359:Serenus
1334:Porphyry
1274:Menelaus
1229:Hippasus
1204:Eutocius
1179:Domninus
1074:Archytas
1003:48664899
979:Spherics
937:Spherics
911:Spherics
900:41134325
871:Spherics
860:41342412
836:Spherics
309:Almagest
300:Elements
284:Spherics
226:Elements
216:Spherics
201:Spherics
181:per se.
171:Elements
154:Elements
150:Spherics
146:Elements
142:Spherics
130:Elements
111:Spherics
83:Spherics
43:treatise
22:Spherics
2227:Chinese
2182:numbers
2110:algebra
2038:Related
2012:Centers
1808:Results
1678:Central
1349:Ptolemy
1344:Proclus
1309:Perseus
1264:Marinus
1244:Hypatia
1234:Hippias
1209:Geminus
1199:Eudoxus
1189:Eudemus
1159:Diocles
825:(ed.).
737:Dioptra
304:Ptolemy
135:Eudoxus
105:History
2242:Indian
2019:Cyrene
1551:Optics
1470:Conics
1389:Theano
1379:Thales
1374:Sporus
1319:Philon
1304:Pappus
1194:Euclid
1124:Carpus
1114:Bryson
1001:
898:
858:
821:". In
787:
719:232593
717:
668:
641:
622:
292:Arabic
262:, and
259:Optics
214:, the
158:points
115:Euclid
88:Arabic
2237:Incan
2158:logic
1934:Other
1702:Chord
1695:Axiom
1673:Angle
1329:Plato
1219:Heron
1139:Conon
999:JSTOR
916:(PDF)
896:JSTOR
876:(PDF)
856:JSTOR
742:(PDF)
715:JSTOR
680:Notes
348:]
96:Latin
27:Greek
2199:list
1487:Data
1259:Leon
1109:Bion
785:ISBN
707:Isis
666:ISBN
639:ISBN
620:ISBN
319:and
302:and
278:and
253:Data
242:and
140:The
19:The
2101:by
1815:In
991:doi
981:".
966:doi
945:doi
924:doi
888:doi
848:doi
754:doi
658:doi
306:'s
210:In
188:),
177:or
165:of
117:'s
45:on
2299::
997:.
987:91
985:.
894:.
884:63
882:.
878:.
854:.
844:64
842:.
750:47
748:.
744:.
711:74
709:.
664:.
346:de
323:.
312:.
270:'
266:,
256:,
250:,
232:'
78:.
35:,
29::
1036:e
1029:t
1022:v
1005:.
993::
972:.
968::
951:.
947::
930:.
926::
913:"
902:.
890::
873:"
862:.
850::
793:.
760:.
756::
721:.
674:.
660::
647:.
628:.
25:(
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