2911:
658:
1349:
2898:
507:
38:
262:
was addressed to a lady of noble birth, at whose request
Nicomachus wrote the book, which suggests that he was a respected scholar of some status. He mentions his intent to write a more advanced work, and how the journeys he frequently undertakes leave him short of time.The approximate dates in which
468:), on the Pythagorean mystical properties of numbers in two books is mentioned by Photius. There is an extant work sometimes attributed to Iamblichus under this title written two centuries later which contains a great deal of material thought to have been copied or paraphrased from Nicomachus' work.
765:, Nicomachus points out that if one writes a list of the odd numbers, the first is the cube of 1, the sum of the next two is the cube of 2, the sum of the next three is the cube of 3, and so on. He does not go further than this, but from this it follows that the sum of the first
358:. Nicomachus distinguishes between the wholly conceptual immaterial number, which he regards as the 'divine number', and the numbers which measure material things, the 'scientific' number. Nicomachus provided one of the earliest Greco-Roman
581:
centuries earlier. Propositions are typically stated and illustrated with one example, but not proven through inference. In some instances this results in patently false assertions. For example, he states that from
633:'s epiphany outside of a smithy that pitch is determined by numeric ratios. Nicomachus also gives the first in-depth account of the relationship between music and the ordering of the universe via the "
1004:
1667:
937:
894:
851:
810:
2771:
1557:
242:
writing commentaries on it. A Latin paraphrase by
Boethius of Nicomachus's works on arithmetic and music became standard textbooks in medieval education.
1608:
The treasury of mathematics: a collection of source material in mathematics edited and presented with introductory biographical and historical sketches
2849:
1200:
554:) is the only extant work on mathematics by Nicomachus. The work contains both philosophical prose and basic mathematical ideas. Nicomachus refers to
2152:
1688:
270:) can only be estimated based on which other authors he refers to in his work, as well as which later mathematicians who refer to him. He mentions
2950:
2945:
1660:
1634:
2970:
2696:
1350:"Boethian Number Theory - Michael Masi: Boethian Number Theory: A Translation of the De Institutione Arithmetica (with Introduction and Notes)"
2955:
2426:
1587:
1227:
2391:
495:
has not survived. Among his known lost work is another larger work on music, promised by
Nicomachus himself, and apparently referred to by
314:
properties rather than their mathematical properties, citing an extensive amount of
Pythagorean literature in his work, including works by
2985:
2980:
2759:
2371:
2162:
664:
states that a square whose side length is a triangular number can be partitioned into squares and half-squares whose areas add to cubes
2826:
1653:
1521:
1314:
1271:
1119:
641:
and
Pythagorean concerns, normally regarded as antitheses. In the midst of theoretical discussions, Nicomachus also describes the
1476:
Musici scriptores graeci. Aristoteles, Euclides, Nicomachus, Bacchius, Gaudentius, Alypius et melodiarum veterum quidquid exstat
2795:
2728:
2361:
2241:
114:
2975:
2864:
2621:
2509:
1684:
1511:
2965:
2504:
362:; the oldest extant Greek multiplication table is found on a wax tablet dated to the 1st century AD (now found in the
20:
2573:
942:
2990:
2200:
2016:
1991:
1547:
429:
2960:
2735:
2706:
2066:
1921:
1177:
219:
2869:
2146:
1639:
1361:
323:
107:
2603:
2831:
2807:
2681:
2616:
2557:
2494:
2484:
2220:
2139:
2001:
1911:
1791:
1211:
756:
661:
2854:
2802:
2701:
2529:
2479:
2464:
2459:
2230:
2031:
1966:
1956:
1906:
1015:
719:
476:
461:
409:
195:
298:'s work on music, implying that they were either later contemporaries or lived in the time after he did.
2902:
2754:
2539:
2416:
2339:
2287:
2089:
1996:
1846:
1534:
359:
147:
2910:
2598:
2879:
2819:
2783:
2626:
2449:
2396:
2366:
2356:
2265:
2128:
2021:
1936:
1891:
1871:
1716:
1701:
1020:
683:
of
Nicomachus was a standard textbook in Neoplatonic schools, and commentaries on it were written by
271:
1629:
2940:
2859:
2778:
2766:
2747:
2711:
2631:
2549:
2534:
2524:
2474:
2469:
2411:
2280:
2172:
2101:
2036:
2026:
1926:
1896:
1836:
1811:
1736:
1726:
1711:
1625:
1266:. Source books in the history of the sciences. Vol. 13. Harvard University Press. p. 17.
517:
223:
649:, ten extracts survive from what appear to have originally been a more substantial work on music.
2935:
2915:
2874:
2814:
2742:
2608:
2401:
2376:
2344:
1941:
1886:
1851:
1746:
1373:
1345:
1206:
642:
637:." Nicomachus's discussion of the governance of the ear and voice in understanding music unites
634:
425:
401:
388:
2583:
2182:
2588:
2499:
2349:
2275:
2248:
2011:
1831:
1821:
1756:
1676:
1583:
1517:
1320:
1310:
1304:
1285:
1277:
1267:
1223:
1219:
1115:
899:
856:
815:
772:
396:
and the sensible world, however, he also makes more
Pythagorean distinctions, such as between
307:
187:
119:
1261:
2790:
2661:
2578:
2334:
2322:
2270:
2006:
1561:
1391:
Pengelley, David (2002), "The bridge between continuous and discrete via original sources",
1365:
1215:
574:
realm. The work consists of two books, twenty-three and twenty-nine chapters, respectively.
379:
2431:
2421:
2315:
2041:
1243:
688:
571:
567:
521:
413:
393:
335:
291:
239:
42:
Plato (left) and
Nicomachus (right), as inventors of music, from a 12th-century manuscript
1427:
1112:
History of
Mathematics, Volume I: General Survey of the History of Elementary Mathematics
2691:
2686:
2514:
2406:
2386:
2214:
1771:
1741:
1606:
1553:
539:
375:
363:
251:
231:
163:
657:
210:, Nicomachus wrote about the mystical properties of numbers, best known for his works
2929:
2651:
2519:
2489:
2310:
2118:
2061:
1377:
469:
331:
230:. Nicomachus' work on arithmetic became a standard text for Neoplatonic education in
184:
1006:
Many early mathematicians have studied and provided proofs of
Nicomachus's theorem.
2593:
2381:
2056:
1816:
1806:
1474:
1257:
674:
618:
397:
227:
207:
1577:
1446:
1445:
D'Ooge, Martin Luther; Robbins, Frank Egleston; Karpinski, Louis Charles (1926).
2207:
2095:
1801:
1786:
1195:
744:
638:
622:
563:
63:
506:
37:
2193:
2112:
2051:
2046:
1986:
1971:
1901:
1856:
1796:
1781:
1761:
1731:
1696:
1392:
1369:
1324:
739:
732:
706:
684:
630:
559:
500:
480:
339:
235:
131:
1281:
1946:
1931:
1881:
1776:
1766:
1751:
1721:
1645:
351:
315:
16:
1st century AD Neopythagorean philosopher, mathematician, and music theorist
1289:
378:
says that Nicomachus's philosophy "fits comfortably within the spectrum of
2083:
1861:
1706:
1543:
593:
496:
405:
347:
319:
311:
287:
1479:(in Ancient Greek). Lipsiae, in aedibus B. G. Teubneri. pp. 209–282
1400:, National Center for Mathematics Education, Univ. of Gothenburg, Sweden
2656:
1981:
1976:
1876:
1866:
1841:
1530:
420:
as the supreme first principle is both the demiurge and the Intellect (
295:
1951:
1826:
1303:
Arnold, Jonathan; Bjornlie, Shane; Sessa, Kristina (April 18, 2016).
645:
of his time, also providing a valuable resource. In addition to the
626:
578:
327:
255:
203:
199:
191:
59:
1114:, New York: Dover Publications (a reprint of the 1951 publication),
737:
The work of Boethius on arithmetic and music was a core part of the
456:
survive in fragments, epitomes, and summaries by later authors. The
570:
and basic mathematical ideas are eternal and unchanging, and in an
2327:
2305:
1961:
1429:
Nicomachi Geraseni Pythagorei Introductionis arithmeticae libri II
703:
656:
555:
505:
383:
355:
343:
283:
250:
Little is known about the life of Nicomachus except that he was a
135:
421:
1649:
1499:(Cambridge: Cambridge University Press, 1989), pp. 245–69.
1309:. Brill's Companions to European History. Brill. p. 332.
629:. It provides the earliest surviving record of the legend of
417:
717:
also became the basis of the Boethius' Latin treatise titled
543:
167:
1432:(in Ancient Greek). Lipsiae : in aedibvs B.G. Teubneri
1134:
1132:
1130:
1128:
1068:
1066:
1064:
1062:
1049:
1047:
1045:
1043:
1041:
1039:
1037:
1035:
342:
is ontologically prior to the other mathematical sciences (
448:
are extant in a complete form, and two others, a work on
404:, Nicomachus makes no attempt to distinguish between the
1426:
Nicomachus, of Gerasa; Hoche, Richard Gottfried (1866).
392:
to make a distinction between the intelligible world of
945:
902:
859:
818:
775:
577:
Nicomachus's presentation is much less rigorous than
382:." In his work on arithmetic, Nicomachus quotes from
400:
numbers. Unlike many other Neopythagoreans, such as
2842:
2720:
2670:
2644:
2566:
2548:
2447:
2440:
2296:
2258:
2075:
1683:
600:is in large part a Latin translation of this work.
549:
141:
125:
113:
103:
83:
70:
47:
28:
1605:
1199:
998:
931:
888:
845:
804:
743:liberal arts and had a great diffusion during the
590:, only because this is true for a=6, b=5 and c=3.
374:Although Nicomachus is considered a Pythagorean,
1576:Dillon, John M. (1996). "Nicomachus of Gerasa".
1510:Iamblichus (January 1989). Gillian Clark (ed.).
812:odd numbers, that is, the odd numbers from 1 to
310:based on his tendency to view numbers as having
999:{\displaystyle {\bigl (}n(n+1)/2{\bigr )}^{2}.}
562:can only be possible if one knows enough about
1582:. Cornell University Press. pp. 352–361.
1560:from the original on May 16, 2021 – via
424:), which Nicomachus also equates to being the
1661:
1394:Study the Masters: The Abel-Fauvel Conference
982:
948:
499:in his comment on the sphere and cylinder of
8:
853:. The average of these numbers is obviously
290:,while he makes no mention at all of either
1201:"Nicomachus [Nikomachos] of Gerasa"
491:, referred to by Nicomachus himself in the
2674:
2444:
1668:
1654:
1646:
1579:The Middle Platonists, 80 B.C. to A.D. 220
36:
25:
987:
981:
980:
971:
947:
946:
944:
921:
901:
878:
858:
817:
794:
774:
432:from which all actualities are created.
1612:. Philosophical Library. pp. 15–16.
1072:
1635:MacTutor History of Mathematics Archive
1220:10.1093/gmo/9781561592630.article.19911
1031:
19:For other people named Nicomachus, see
2697:Latin translations of the 12th century
1448:Nicomachus' Introduction to Arithmetic
1138:
1053:
408:, who acts on the material world, and
2427:Straightedge and compass construction
218:, which are an important resource on
7:
2392:Incircle and excircles of a triangle
524:: Oriental Manuscripts, Add MS 7473.
475:was one of the main sources used by
330:, especially on the significance of
1099:The European Mathematical Awakening
1473:Jan, Karl von; Nicomachus (1895).
769:cubes equals the sum of the first
14:
1539:Anonymous, Theology of Arithmetic
1263:A Source Book in Medieval Science
709:and a partial translation of the
306:Historians consider Nicomachus a
2909:
2896:
1360:(1). The Classical Association,
1306:A Companion to Ostrogothic Italy
761:At the end of Chapter 20 of his
566:. Nicomachus also describes how
1604:Midonick, Henrietta O. (1965).
282:was apparently translated into
140:
2951:2nd-century Greek philosophers
2946:1st-century Greek philosophers
2729:A History of Greek Mathematics
2242:The Quadrature of the Parabola
1516:. Liverpool University Press.
1184:. Vol. 1. pp. 97–98.
1182:A History of Greek Mathematics
1086:The development of mathematics
1084:Eric Temple Bell (1940),
968:
956:
918:
906:
875:
863:
834:
822:
791:
779:
440:Two of Nicomachus' works, the
1:
2971:Ancient Greek music theorists
558:quite often, and writes that
264:
173:
74:
51:
2956:Ancient Greek mathematicians
2510:Intersecting secants theorem
1497:Harmonic and Acoustic Theory
1244:UK public library membership
727:Medieval European philosophy
412:which serves as the supreme
234:, with philosophers such as
2505:Intersecting chords theorem
2372:Doctrine of proportionality
1549:De institutione arithmetica
700:De Institutione Arithmetica
621:treatise since the time of
598:De institutione arithmetica
550:
493:Introduction to Arithmetic,
326:. He writes extensively on
21:Nicomachus (disambiguation)
3007:
2986:2nd-century mathematicians
2981:1st-century mathematicians
2201:On the Sphere and Cylinder
2154:On the Sizes and Distances
1419:Introduction to Arithmetic
763:Introduction to Arithmetic
754:
730:
711:Introduction to Arithmetic
681:Introduction to Arithmetic
672:
544:
536:Introduction to Arithmetic
530:Introduction to Arithmetic
513:Introduction to Arithmetic
465:
442:Introduction to Arithmetic
286:in the mid 2nd century by
280:Introduction to Arithmetic
212:Introduction to Arithmetic
168:
90:Introduction to Arithmetic
18:
2903:Ancient Greece portal
2892:
2707:Philosophy of mathematics
2677:
2622:Ptolemy's table of chords
1677:Ancient Greek mathematics
1413:Editions and translations
1370:10.1017/S0009840X00107462
939:of them, so their sum is
617:) is the first important
586:it can be concluded that
294:'s work on arithmetic or
220:Ancient Greek mathematics
153:
99:
35:
2574:Aristarchus's inequality
2147:On Conoids and Spheroids
1640:University of St Andrews
1362:Harvard University Press
932:{\displaystyle n(n+1)/2}
889:{\displaystyle n(n+1)/2}
846:{\displaystyle n(n+1)-1}
805:{\displaystyle n(n+1)/2}
613:(Ἐγχειρίδιον ἁρμονικῆς,
489:Introduction to Geometry
466:Θεολογούμενα ἀριθμητικῆς
108:Ancient Roman philosophy
2682:Ancient Greek astronomy
2495:Inscribed angle theorem
2485:Greek geometric algebra
2140:Measurement of a Circle
1513:On the Pythagorean Life
1491:Andrew Barker, editor,
1212:Oxford University Press
1110:David E. Smith (1958),
1097:Frank J. Swetz (2013),
750:
615:Encheiridion Harmonikes
196:Roman province of Syria
2916:Mathematics portal
2702:Non-Euclidean geometry
2657:Mouseion of Alexandria
2530:Tangent-secant theorem
2480:Geometric mean theorem
2465:Exterior angle theorem
2460:Angle bisector theorem
2164:On Sizes and Distances
1493:Greek Musical Writings
1016:Superparticular number
1000:
933:
890:
847:
806:
720:De institutione musica
665:
525:
458:Theology of Arithmetic
450:Theology of Arithmetic
380:contemporary Platonism
2604:Pappus's area theorem
2540:Theorem of the gnomon
2417:Quadratrix of Hippias
2340:Circles of Apollonius
2288:Problem of Apollonius
2266:Constructible numbers
2090:Archimedes Palimpsest
1001:
934:
891:
848:
807:
660:
584:(a-b) ∶ (b-c) ∷ c ∶ a
510:Arabic manuscript of
509:
483:, for their (extant)
360:multiplication tables
148:Multiplication tables
2820:prehistoric counting
2617:Ptolemy's inequality
2558:Apollonius's theorem
2397:Method of exhaustion
2367:Diophantine equation
2357:Circumscribed circle
2174:On the Moving Sphere
1626:Robertson, Edmund F.
1354:The Classical Review
1021:Superpartient number
943:
900:
857:
816:
773:
757:Nicomachus's theorem
751:Nicomachus's theorem
702:) of Boethius was a
662:Nicomachus's theorem
635:music of the spheres
551:Arithmetike eisagoge
160:Nicomachus of Gerasa
30:Nicomachus of Gerasa
2976:2nd-century writers
2906: •
2712:Neusis construction
2632:Spiral of Theodorus
2525:Pythagorean theorem
2470:Euclidean algorithm
2412:Lune of Hippocrates
2281:Squaring the circle
2037:Theon of Alexandria
1712:Aristaeus the Elder
1624:O'Connor, John J.;
1466:Manual of Harmonics
1141:, pp. 353–358.
1056:, pp. 352–353.
715:Manual of Harmonics
605:Manual of Harmonics
545:Ἀριθμητικὴ εἰσαγωγή
446:Manual of Harmonics
276:Manual of Harmonics
260:Manual of Harmonics
224:Ancient Greek music
216:Manual of Harmonics
94:Manual of Harmonics
2966:People from Jerash
2599:Menelaus's theorem
2589:Irrational numbers
2402:Parallel postulate
2377:Euclidean geometry
2345:Apollonian circles
1887:Isidore of Miletus
1346:Ivor Bulmer-Thomas
1207:Grove Music Online
1101:, page 17, Courier
996:
929:
886:
843:
802:
687:(3rd century) and
666:
611:Manuale Harmonicum
526:
487:of Pythagoras. An
472:Life of Pythagoras
454:Life of Pythagoras
416:. For Nicomachus,
402:Moderatus of Gades
2991:Middle Platonists
2923:
2922:
2888:
2887:
2640:
2639:
2627:Ptolemy's theorem
2500:Intercept theorem
2350:Apollonian gasket
2276:Doubling the cube
2249:The Sand Reckoner
1589:978-0-8014-8316-5
1348:(April 1, 1985).
1242:(subscription or
1229:978-1-56159-263-0
1075:, pp. 15–16.
190:philosopher from
157:
156:
120:Neopythagoreanism
2998:
2961:Neo-Pythagoreans
2914:
2913:
2901:
2900:
2899:
2675:
2662:Platonic Academy
2609:Problem II.8 of
2579:Crossbar theorem
2535:Thales's theorem
2475:Euclid's theorem
2445:
2362:Commensurability
2323:Axiomatic system
2271:Angle trisection
2236:
2226:
2188:
2178:
2168:
2158:
2134:
2124:
2107:
1670:
1663:
1656:
1647:
1642:
1613:
1611:
1600:
1598:
1596:
1565:
1562:Internet Archive
1527:
1488:
1486:
1484:
1460:
1458:
1456:
1441:
1439:
1437:
1402:
1401:
1399:
1388:
1382:
1381:
1342:
1336:
1335:
1333:
1331:
1300:
1294:
1293:
1254:
1248:
1247:
1240:
1238:
1236:
1203:
1192:
1186:
1185:
1174:
1168:
1161:
1155:
1148:
1142:
1136:
1123:
1108:
1102:
1095:
1089:
1082:
1076:
1070:
1057:
1051:
1005:
1003:
1002:
997:
992:
991:
986:
985:
975:
952:
951:
938:
936:
935:
930:
925:
896:, and there are
895:
893:
892:
887:
882:
852:
850:
849:
844:
811:
809:
808:
803:
798:
768:
691:(6th century).
589:
585:
553:
547:
546:
518:Thābit ibn Qurra
516:, translated by
467:
354:), and is their
338:and argues that
269:
266:
182:
178:
175:
171:
170:
86:
79:
76:
56:
53:
40:
26:
3006:
3005:
3001:
3000:
2999:
2997:
2996:
2995:
2926:
2925:
2924:
2919:
2908:
2897:
2895:
2884:
2850:Arabian/Islamic
2838:
2827:numeral systems
2716:
2666:
2636:
2584:Heron's formula
2562:
2544:
2436:
2432:Triangle center
2422:Regular polygon
2299:and definitions
2298:
2292:
2254:
2234:
2224:
2186:
2176:
2166:
2156:
2132:
2122:
2105:
2071:
2042:Theon of Smyrna
1687:
1679:
1674:
1623:
1620:
1603:
1594:
1592:
1590:
1575:
1572:
1556:. p. 110.
1542:
1524:
1509:
1506:
1504:Primary sources
1482:
1480:
1472:
1469:
1454:
1452:
1444:
1435:
1433:
1425:
1422:
1415:
1410:
1405:
1397:
1390:
1389:
1385:
1344:
1343:
1339:
1329:
1327:
1317:
1302:
1301:
1297:
1274:
1256:
1255:
1251:
1241:
1234:
1232:
1230:
1196:Levin, Flora R.
1194:
1193:
1189:
1176:
1175:
1171:
1162:
1158:
1149:
1145:
1137:
1126:
1109:
1105:
1096:
1092:
1083:
1079:
1071:
1060:
1052:
1033:
1029:
1012:
979:
941:
940:
898:
897:
855:
854:
814:
813:
771:
770:
766:
759:
753:
735:
729:
689:John Philoponus
677:
671:
655:
608:
587:
583:
568:natural numbers
533:
522:British Library
438:
414:first principle
372:
336:perfect numbers
304:
292:Theon of Smyrna
267:
248:
240:John Philoponus
180:
176:
144:
128:
92:
84:
77:
66:
57:
54:
43:
31:
24:
17:
12:
11:
5:
3004:
3002:
2994:
2993:
2988:
2983:
2978:
2973:
2968:
2963:
2958:
2953:
2948:
2943:
2938:
2928:
2927:
2921:
2920:
2893:
2890:
2889:
2886:
2885:
2883:
2882:
2877:
2872:
2867:
2862:
2857:
2852:
2846:
2844:
2843:Other cultures
2840:
2839:
2837:
2836:
2835:
2834:
2824:
2823:
2822:
2812:
2811:
2810:
2800:
2799:
2798:
2788:
2787:
2786:
2776:
2775:
2774:
2764:
2763:
2762:
2752:
2751:
2750:
2740:
2739:
2738:
2724:
2722:
2718:
2717:
2715:
2714:
2709:
2704:
2699:
2694:
2692:Greek numerals
2689:
2687:Attic numerals
2684:
2678:
2672:
2668:
2667:
2665:
2664:
2659:
2654:
2648:
2646:
2642:
2641:
2638:
2637:
2635:
2634:
2629:
2624:
2619:
2614:
2606:
2601:
2596:
2591:
2586:
2581:
2576:
2570:
2568:
2564:
2563:
2561:
2560:
2554:
2552:
2546:
2545:
2543:
2542:
2537:
2532:
2527:
2522:
2517:
2515:Law of cosines
2512:
2507:
2502:
2497:
2492:
2487:
2482:
2477:
2472:
2467:
2462:
2456:
2454:
2442:
2438:
2437:
2435:
2434:
2429:
2424:
2419:
2414:
2409:
2407:Platonic solid
2404:
2399:
2394:
2389:
2387:Greek numerals
2384:
2379:
2374:
2369:
2364:
2359:
2354:
2353:
2352:
2347:
2337:
2332:
2331:
2330:
2320:
2319:
2318:
2313:
2302:
2300:
2294:
2293:
2291:
2290:
2285:
2284:
2283:
2278:
2273:
2262:
2260:
2256:
2255:
2253:
2252:
2245:
2238:
2228:
2218:
2215:Planisphaerium
2211:
2204:
2197:
2190:
2180:
2170:
2160:
2150:
2143:
2136:
2126:
2116:
2109:
2099:
2092:
2087:
2079:
2077:
2073:
2072:
2070:
2069:
2064:
2059:
2054:
2049:
2044:
2039:
2034:
2029:
2024:
2019:
2014:
2009:
2004:
1999:
1994:
1989:
1984:
1979:
1974:
1969:
1964:
1959:
1954:
1949:
1944:
1939:
1934:
1929:
1924:
1919:
1914:
1909:
1904:
1899:
1894:
1889:
1884:
1879:
1874:
1869:
1864:
1859:
1854:
1849:
1844:
1839:
1834:
1829:
1824:
1819:
1814:
1809:
1804:
1799:
1794:
1789:
1784:
1779:
1774:
1769:
1764:
1759:
1754:
1749:
1744:
1739:
1734:
1729:
1724:
1719:
1714:
1709:
1704:
1699:
1693:
1691:
1685:Mathematicians
1681:
1680:
1675:
1673:
1672:
1665:
1658:
1650:
1644:
1643:
1619:
1618:External links
1616:
1615:
1614:
1601:
1588:
1571:
1568:
1567:
1566:
1554:Erhard Ratdolt
1540:
1537:
1528:
1522:
1505:
1502:
1501:
1500:
1489:
1468:
1463:
1462:
1461:
1442:
1421:
1416:
1414:
1411:
1409:
1406:
1404:
1403:
1383:
1337:
1315:
1295:
1272:
1249:
1228:
1187:
1169:
1156:
1143:
1124:
1103:
1090:
1077:
1058:
1030:
1028:
1025:
1024:
1023:
1018:
1011:
1008:
995:
990:
984:
978:
974:
970:
967:
964:
961:
958:
955:
950:
928:
924:
920:
917:
914:
911:
908:
905:
885:
881:
877:
874:
871:
868:
865:
862:
842:
839:
836:
833:
830:
827:
824:
821:
801:
797:
793:
790:
787:
784:
781:
778:
755:Main article:
752:
749:
728:
725:
670:
669:Late antiquity
667:
654:
651:
607:
602:
532:
527:
437:
434:
376:John M. Dillon
371:
368:
364:British Museum
308:Neopythagorean
303:
300:
254:who came from
247:
244:
232:Late antiquity
188:Neopythagorean
155:
154:
151:
150:
145:
142:
139:
138:
129:
127:Main interests
126:
123:
122:
117:
111:
110:
105:
101:
100:
97:
96:
87:
81:
80:
72:
68:
67:
58:
49:
45:
44:
41:
33:
32:
29:
15:
13:
10:
9:
6:
4:
3:
2:
3003:
2992:
2989:
2987:
2984:
2982:
2979:
2977:
2974:
2972:
2969:
2967:
2964:
2962:
2959:
2957:
2954:
2952:
2949:
2947:
2944:
2942:
2939:
2937:
2934:
2933:
2931:
2918:
2917:
2912:
2905:
2904:
2891:
2881:
2878:
2876:
2873:
2871:
2868:
2866:
2863:
2861:
2858:
2856:
2853:
2851:
2848:
2847:
2845:
2841:
2833:
2830:
2829:
2828:
2825:
2821:
2818:
2817:
2816:
2813:
2809:
2806:
2805:
2804:
2801:
2797:
2794:
2793:
2792:
2789:
2785:
2782:
2781:
2780:
2777:
2773:
2770:
2769:
2768:
2765:
2761:
2758:
2757:
2756:
2753:
2749:
2746:
2745:
2744:
2741:
2737:
2733:
2732:
2731:
2730:
2726:
2725:
2723:
2719:
2713:
2710:
2708:
2705:
2703:
2700:
2698:
2695:
2693:
2690:
2688:
2685:
2683:
2680:
2679:
2676:
2673:
2669:
2663:
2660:
2658:
2655:
2653:
2650:
2649:
2647:
2643:
2633:
2630:
2628:
2625:
2623:
2620:
2618:
2615:
2613:
2612:
2607:
2605:
2602:
2600:
2597:
2595:
2592:
2590:
2587:
2585:
2582:
2580:
2577:
2575:
2572:
2571:
2569:
2565:
2559:
2556:
2555:
2553:
2551:
2547:
2541:
2538:
2536:
2533:
2531:
2528:
2526:
2523:
2521:
2520:Pons asinorum
2518:
2516:
2513:
2511:
2508:
2506:
2503:
2501:
2498:
2496:
2493:
2491:
2490:Hinge theorem
2488:
2486:
2483:
2481:
2478:
2476:
2473:
2471:
2468:
2466:
2463:
2461:
2458:
2457:
2455:
2453:
2452:
2446:
2443:
2439:
2433:
2430:
2428:
2425:
2423:
2420:
2418:
2415:
2413:
2410:
2408:
2405:
2403:
2400:
2398:
2395:
2393:
2390:
2388:
2385:
2383:
2380:
2378:
2375:
2373:
2370:
2368:
2365:
2363:
2360:
2358:
2355:
2351:
2348:
2346:
2343:
2342:
2341:
2338:
2336:
2333:
2329:
2326:
2325:
2324:
2321:
2317:
2314:
2312:
2309:
2308:
2307:
2304:
2303:
2301:
2295:
2289:
2286:
2282:
2279:
2277:
2274:
2272:
2269:
2268:
2267:
2264:
2263:
2261:
2257:
2251:
2250:
2246:
2244:
2243:
2239:
2237:
2233:
2229:
2227:
2223:
2219:
2217:
2216:
2212:
2210:
2209:
2205:
2203:
2202:
2198:
2196:
2195:
2191:
2189:
2185:
2181:
2179:
2175:
2171:
2169:
2165:
2161:
2159:
2157:(Aristarchus)
2155:
2151:
2149:
2148:
2144:
2142:
2141:
2137:
2135:
2131:
2127:
2125:
2121:
2117:
2115:
2114:
2110:
2108:
2104:
2100:
2098:
2097:
2093:
2091:
2088:
2086:
2085:
2081:
2080:
2078:
2074:
2068:
2065:
2063:
2062:Zeno of Sidon
2060:
2058:
2055:
2053:
2050:
2048:
2045:
2043:
2040:
2038:
2035:
2033:
2030:
2028:
2025:
2023:
2020:
2018:
2015:
2013:
2010:
2008:
2005:
2003:
2000:
1998:
1995:
1993:
1990:
1988:
1985:
1983:
1980:
1978:
1975:
1973:
1970:
1968:
1965:
1963:
1960:
1958:
1955:
1953:
1950:
1948:
1945:
1943:
1940:
1938:
1935:
1933:
1930:
1928:
1925:
1923:
1920:
1918:
1915:
1913:
1910:
1908:
1905:
1903:
1900:
1898:
1895:
1893:
1890:
1888:
1885:
1883:
1880:
1878:
1875:
1873:
1870:
1868:
1865:
1863:
1860:
1858:
1855:
1853:
1850:
1848:
1845:
1843:
1840:
1838:
1835:
1833:
1830:
1828:
1825:
1823:
1820:
1818:
1815:
1813:
1810:
1808:
1805:
1803:
1800:
1798:
1795:
1793:
1790:
1788:
1785:
1783:
1780:
1778:
1775:
1773:
1770:
1768:
1765:
1763:
1760:
1758:
1755:
1753:
1750:
1748:
1745:
1743:
1740:
1738:
1735:
1733:
1730:
1728:
1725:
1723:
1720:
1718:
1715:
1713:
1710:
1708:
1705:
1703:
1700:
1698:
1695:
1694:
1692:
1690:
1686:
1682:
1678:
1671:
1666:
1664:
1659:
1657:
1652:
1651:
1648:
1641:
1637:
1636:
1631:
1627:
1622:
1621:
1617:
1610:
1609:
1602:
1591:
1585:
1581:
1580:
1574:
1573:
1569:
1563:
1559:
1555:
1551:
1550:
1545:
1541:
1538:
1536:
1532:
1529:
1525:
1523:9780853233268
1519:
1515:
1514:
1508:
1507:
1503:
1498:
1494:
1490:
1478:
1477:
1471:
1470:
1467:
1464:
1450:
1449:
1443:
1431:
1430:
1424:
1423:
1420:
1417:
1412:
1407:
1396:
1395:
1387:
1384:
1379:
1375:
1371:
1367:
1363:
1359:
1355:
1351:
1347:
1341:
1338:
1326:
1322:
1318:
1316:9789004315938
1312:
1308:
1307:
1299:
1296:
1291:
1287:
1283:
1279:
1275:
1273:9780674823600
1269:
1265:
1264:
1259:
1253:
1250:
1245:
1231:
1225:
1221:
1217:
1213:
1209:
1208:
1202:
1197:
1191:
1188:
1183:
1179:
1178:Heath, Thomas
1173:
1170:
1166:
1160:
1157:
1153:
1147:
1144:
1140:
1135:
1133:
1131:
1129:
1125:
1122:, pp 58, 129.
1121:
1120:0-486-20429-4
1117:
1113:
1107:
1104:
1100:
1094:
1091:
1087:
1081:
1078:
1074:
1073:Midonick 1965
1069:
1067:
1065:
1063:
1059:
1055:
1050:
1048:
1046:
1044:
1042:
1040:
1038:
1036:
1032:
1026:
1022:
1019:
1017:
1014:
1013:
1009:
1007:
993:
988:
976:
972:
965:
962:
959:
953:
926:
922:
915:
912:
909:
903:
883:
879:
872:
869:
866:
860:
840:
837:
831:
828:
825:
819:
799:
795:
788:
785:
782:
776:
764:
758:
748:
746:
742:
741:
734:
726:
724:
722:
721:
716:
712:
708:
705:
701:
697:
692:
690:
686:
682:
676:
668:
663:
659:
652:
650:
648:
644:
640:
636:
632:
628:
624:
620:
616:
612:
606:
603:
601:
599:
595:
591:
580:
575:
573:
569:
565:
561:
557:
552:
541:
537:
531:
528:
523:
519:
515:
514:
508:
504:
502:
498:
494:
490:
486:
482:
478:
474:
473:
470:Nicomachus's
463:
462:Ancient Greek
459:
455:
451:
447:
443:
435:
433:
431:
427:
423:
419:
415:
411:
407:
403:
399:
395:
391:
390:
385:
381:
377:
369:
367:
365:
361:
357:
353:
349:
345:
341:
337:
333:
332:prime numbers
329:
325:
321:
317:
313:
309:
301:
299:
297:
293:
289:
285:
281:
277:
273:
268: 100 AD
261:
257:
253:
245:
243:
241:
237:
233:
229:
225:
221:
217:
213:
209:
206:). Like many
205:
201:
197:
193:
189:
186:
185:Ancient Greek
181: 120 AD
165:
161:
152:
149:
146:
143:Notable ideas
137:
133:
130:
124:
121:
118:
116:
112:
109:
106:
102:
98:
95:
91:
88:
82:
73:
69:
65:
61:
50:
46:
39:
34:
27:
22:
2907:
2894:
2736:Thomas Heath
2727:
2610:
2594:Law of sines
2450:
2382:Golden ratio
2247:
2240:
2231:
2225:(Theodosius)
2221:
2213:
2206:
2199:
2192:
2183:
2173:
2167:(Hipparchus)
2163:
2153:
2145:
2138:
2129:
2119:
2111:
2106:(Apollonius)
2102:
2094:
2082:
2057:Zeno of Elea
1916:
1817:Eratosthenes
1807:Dionysodorus
1633:
1630:"Nicomachus"
1607:
1593:. Retrieved
1578:
1552:(in Latin).
1548:
1512:
1496:
1492:
1481:. Retrieved
1475:
1465:
1453:. Retrieved
1447:
1434:. Retrieved
1428:
1418:
1408:Bibliography
1393:
1386:
1357:
1353:
1340:
1328:. Retrieved
1305:
1298:
1262:
1258:Edward Grant
1252:
1235:25 September
1233:. Retrieved
1205:
1190:
1181:
1172:
1164:
1163:Nicomachus,
1159:
1151:
1146:
1111:
1106:
1098:
1093:
1085:
1080:
762:
760:
738:
736:
718:
714:
710:
699:
695:
693:
680:
678:
675:Neoplatonism
646:
639:Aristoxenian
619:music theory
614:
610:
609:
604:
597:
592:
576:
535:
534:
529:
512:
511:
492:
488:
484:
471:
457:
453:
449:
445:
441:
439:
430:potentiality
398:Odd and even
387:
373:
305:
279:
275:
259:
249:
228:Roman period
215:
211:
208:Pythagoreans
159:
158:
93:
89:
85:Notable work
2803:mathematics
2611:Arithmetica
2208:Ostomachion
2177:(Autolycus)
2096:Arithmetica
1872:Hippocrates
1802:Dinostratus
1787:Dicaearchus
1717:Aristarchus
1535:Bibliotheca
1451:. Macmillan
1167:, ii. 6. 1.
1165:Arithmetica
1139:Dillon 1996
1054:Dillon 1996
745:Middle Ages
698:(in Latin:
643:instruments
623:Aristoxenus
564:mathematics
370:Metaphysics
252:Pythagorean
64:Roman Syria
2941:120 deaths
2930:Categories
2855:Babylonian
2755:arithmetic
2721:History of
2550:Apollonius
2235:(Menelaus)
2194:On Spirals
2113:Catoptrics
2052:Xenocrates
2047:Thymaridas
2032:Theodosius
2017:Theaetetus
1997:Simplicius
1987:Pythagoras
1972:Posidonius
1957:Philonides
1917:Nicomachus
1912:Metrodorus
1902:Menaechmus
1857:Hipparchus
1847:Heliodorus
1797:Diophantus
1782:Democritus
1762:Chrysippus
1732:Archimedes
1727:Apollonius
1697:Anaxagoras
1689:(timeline)
1570:References
1325:1016025625
1210:. Oxford:
1088:, page 83.
740:Quadrivium
733:Quadrivium
731:See also:
707:paraphrase
696:Arithmetic
685:Iamblichus
673:See also:
631:Pythagoras
560:philosophy
520:(d. 901).
501:Archimedes
481:Iamblichus
340:arithmetic
324:Androcydes
302:Philosophy
278:, and his
272:Thrasyllus
263:he lived (
236:Iamblichus
183:) was an
179: – c.
132:Arithmetic
78: 120
2936:60 births
2316:Inscribed
2076:Treatises
2067:Zenodorus
2027:Theodorus
2002:Sosigenes
1947:Philolaus
1932:Oenopides
1927:Nicoteles
1922:Nicomedes
1882:Hypsicles
1777:Ctesibius
1767:Cleomedes
1752:Callippus
1737:Autolycus
1722:Aristotle
1702:Anthemius
1378:125741349
1364:: 86–87.
1282:1556-9063
1246:required)
838:−
352:astronomy
316:Philolaus
194:, in the
177: 60
169:Νικόμαχος
55: 60
2880:Japanese
2865:Egyptian
2808:timeline
2796:timeline
2784:timeline
2779:geometry
2772:timeline
2767:calculus
2760:timeline
2748:timeline
2451:Elements
2297:Concepts
2259:Problems
2232:Spherics
2222:Spherics
2187:(Euclid)
2133:(Euclid)
2130:Elements
2123:(Euclid)
2084:Almagest
1992:Serenus
1967:Porphyry
1907:Menelaus
1862:Hippasus
1837:Eutocius
1812:Domninus
1707:Archytas
1595:16 April
1558:Archived
1546:(1488).
1544:Boethius
1483:16 April
1455:16 April
1436:16 April
1260:(1974).
1198:(2001).
1180:(1921).
1010:See also
594:Boethius
572:abstract
497:Eutocius
477:Porphyry
444:and the
406:Demiurge
348:geometry
320:Archytas
312:mystical
288:Apuleius
2860:Chinese
2815:numbers
2743:algebra
2671:Related
2645:Centers
2441:Results
2311:Central
1982:Ptolemy
1977:Proclus
1942:Perseus
1897:Marinus
1877:Hypatia
1867:Hippias
1842:Geminus
1832:Eudoxus
1822:Eudemus
1792:Diocles
1531:Photius
1495:vol 2:
1330:May 16,
1290:1066603
1152:Timaeus
1150:Plato,
410:The One
389:Timaeus
328:numbers
296:Ptolemy
274:in his
226:in the
2875:Indian
2652:Cyrene
2184:Optics
2103:Conics
2022:Theano
2012:Thales
2007:Sporus
1952:Philon
1937:Pappus
1827:Euclid
1757:Carpus
1747:Bryson
1586:
1520:
1376:
1323:
1313:
1288:
1280:
1270:
1226:
1118:
713:. The
653:Legacy
647:Manual
627:Euclid
588:ab=2bc
579:Euclid
452:and a
428:, the
350:, and
322:, and
258:. His
256:Gerasa
204:Jordan
200:Jerash
192:Gerasa
115:School
60:Gerasa
2870:Incan
2791:logic
2567:Other
2335:Chord
2328:Axiom
2306:Angle
1962:Plato
1852:Heron
1772:Conon
1398:(PDF)
1374:S2CID
1027:Notes
704:Latin
556:Plato
540:Greek
485:Lives
436:Works
426:monad
394:Forms
384:Plato
356:cause
344:music
284:Latin
198:(now
164:Greek
136:Music
2832:list
2120:Data
1892:Leon
1742:Bion
1597:2023
1584:ISBN
1518:ISBN
1485:2023
1457:2023
1438:2023
1332:2021
1321:OCLC
1311:ISBN
1286:OCLC
1278:ISSN
1268:ISBN
1237:2021
1224:ISBN
1116:ISBN
694:The
679:The
625:and
479:and
422:nous
334:and
246:Life
238:and
222:and
214:and
71:Died
48:Born
2734:by
2448:In
1366:doi
1216:doi
1154:27D
723:.
418:God
386:'s
366:).
134:,
104:Era
2932::
1638:,
1632:,
1628:,
1533:,
1372:.
1358:35
1356:.
1352:.
1319:.
1284:.
1276:.
1222:.
1214:.
1204:.
1127:^
1061:^
1034:^
747:.
596:'
548:,
542::
503:.
464::
346:,
318:,
265:c.
202:,
174:c.
172:;
166::
75:c.
62:,
52:c.
1669:e
1662:t
1655:v
1599:.
1564:.
1526:.
1487:.
1459:.
1440:.
1380:.
1368::
1334:.
1292:.
1239:.
1218::
994:.
989:2
983:)
977:2
973:/
969:)
966:1
963:+
960:n
957:(
954:n
949:(
927:2
923:/
919:)
916:1
913:+
910:n
907:(
904:n
884:2
880:/
876:)
873:1
870:+
867:n
864:(
861:n
841:1
835:)
832:1
829:+
826:n
823:(
820:n
800:2
796:/
792:)
789:1
786:+
783:n
780:(
777:n
767:n
538:(
460:(
162:(
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
