1090:. This also allows the representation of a division of two numbers in geometrical terms, an important feature to reformulate geometrical problems in algebraic terms. More precisely, if two numbers are given as lengths of line segments one can construct a third line segment, the length of which matches the quotient of those two numbers (see diagram). Another application is to transfer the ratio of partition of one line segment to another line segment (of different length), thus dividing that other line segment in the same ratio as a given line segment and its partition (see diagram).
3242:
976:
1356:
3229:
968:
20:
1094:
1761:(around 300 BC), and there it plays an important role in the derivation of other theorems. It is given as proposition 43 in Book I of the Elements, where it is phrased as a statement about parallelograms without using the term "gnomon". The latter is introduced by
957:
1765:
as the second definition of the second book of
Elements. Further theorems for which the gnomon and its properties play an important role are proposition 6 in Book II, proposition 29 in Book VI and propositions 1 to 4 in Book XIII.
1270:
1081:
1343:. Now each of those two parallelepipeds around the diagonal has three of the remaining six parallelepipeds attached to it, and those three play the role of the complements and are of equal
173:
1904:
Vighi, Paolo; Aschieri, Igino (2010), "From Art to
Mathematics in the Paintings of Theo van Doesburg", in Vittorio Capecchi; Massimo Buscema; Pierluigi Contucci; Bruno D'Amore (eds.),
1323:, each parallel to the faces of the parallelepiped. The three planes partition the parallelepiped into eight smaller parallelepipeds; two of those surround the diagonal and meet at
1614:
1998:
1202:
1180:
1158:
1116:
56:
1727:
1698:
1646:
1502:
1473:
1421:
1392:
772:
743:
710:
681:
652:
623:
588:
559:
229:
3102:
1669:
1444:
510:
467:
424:
381:
338:
295:
272:
1747:
1522:
1341:
1321:
1297:
1136:
530:
487:
444:
401:
358:
315:
249:
796:
1086:
The theorem of the gnomon can be used to construct a new parallelogram or rectangle of equal area to a given parallelogram or rectangle by the means of
782:
The proof of the theorem is straightforward if one considers the areas of the main parallelogram and the two inner parallelograms around its diagonal:
3180:
2483:
2019:
786:
first, the difference between the main parallelogram and the two inner parallelograms is exactly equal to the combined area of the two complements;
1991:
1524:
with the inner parallelogram. Now the difference of the areas of those two parallelograms is equal to area of the inner parallelogram, that is:
1365:
The theorem of gnomon is special case of a more general statement about nested parallelograms with a common diagonal. For a given parallelogram
3271:
3027:
2757:
1889:
1860:
1087:
2722:
3090:
2702:
2493:
3157:
1984:
1913:
1795:
1207:
982:
3126:
3059:
2692:
2572:
1749:
is on the diagonal and that the difference of their areas is zero, which is exactly what the theorem of the gnomon states.
3195:
2952:
2840:
2015:
3266:
2835:
2904:
2531:
2347:
2322:
63:
3066:
3037:
2397:
2252:
3200:
2477:
2934:
1815:
Hazard, William J. (1929), "Generalizations of the
Theorem of Pythagoras and Euclid's Theorem of the Gnomon",
3162:
3138:
3012:
2947:
2888:
2825:
2815:
2551:
2470:
2332:
2242:
2122:
3185:
3133:
3032:
2860:
2810:
2795:
2790:
2561:
2362:
2297:
2287:
2237:
1530:
3233:
3085:
2747:
2670:
2618:
2420:
2327:
2177:
3241:
2929:
3210:
3150:
3114:
2957:
2780:
2727:
2697:
2687:
2596:
2459:
2352:
2267:
2222:
2202:
2047:
2032:
1971:
1959:
1758:
1965:
3190:
3109:
3097:
3078:
3042:
2962:
2880:
2865:
2855:
2805:
2800:
2742:
2611:
2503:
2432:
2367:
2357:
2257:
2227:
2167:
2142:
2067:
2057:
2042:
1504:
the sides of which are parallel to the sides of the outer parallelogram and which share the vertex
1185:
1163:
1141:
1099:
3246:
3205:
3145:
3073:
2939:
2732:
2707:
2675:
2272:
2217:
2182:
2077:
1937:
1832:
1619:
This statement yields the theorem of the gnomon if one looks at a degenerate inner parallelogram
2914:
2513:
2919:
2830:
2680:
2606:
2579:
2342:
2162:
2152:
2087:
2007:
1909:
1885:
1856:
1850:
1791:
1879:
3121:
2992:
2909:
2665:
2653:
2601:
2337:
1824:
26:
2762:
2752:
2646:
2372:
1703:
1674:
1622:
1478:
1449:
1397:
1368:
748:
719:
686:
657:
628:
599:
564:
535:
205:
187:
952:{\displaystyle |IPGD|={\frac {|ABCD|}{2}}-{\frac {|AHPI|}{2}}-{\frac {|PFCG|}{2}}=|HBFP|}
1651:
1426:
492:
449:
406:
363:
320:
277:
254:
3022:
3017:
2845:
2737:
2717:
2545:
2102:
2072:
1732:
1507:
1326:
1306:
1300:
1282:
1276:
1121:
975:
515:
472:
429:
386:
343:
300:
234:
1355:
3260:
2982:
2850:
2820:
2641:
2449:
2392:
1303:
of a parallelepiped, and instead of two parallel lines you have three planes through
596:
is the name for the L-shaped figure consisting of the two overlapping parallelograms
183:
1446:
as a diagonal as well. Furthermore there are two uniquely determined parallelograms
2924:
2712:
2387:
2147:
2137:
1828:
1788:
Mit harmonischen Verhältnissen zu
Kegelschnitten: Perlen der klassischen Geometrie
2538:
2426:
2132:
2117:
2524:
2443:
2382:
2377:
2317:
2302:
2247:
2232:
2187:
2127:
2112:
2092:
2062:
2027:
967:
979:
Transferring the ratio of a partition of line segment AB to line segment HG:
2277:
2262:
2212:
2107:
2097:
2082:
2052:
1976:
19:
1906:
Applications of
Mathematics in Models, Artificial Neural Networks and Arts
2414:
2192:
2037:
1942:
2987:
2312:
2307:
2207:
2197:
2172:
1836:
1093:
2282:
2157:
1762:
1344:
16:
Certain parallelograms occurring in a gnomon have areas of equal size
789:
second, all three of them are bisected by the diagonal. This yields:
2658:
2636:
2292:
1354:
1092:
974:
966:
18:
1786:
Halbeisen, Lorenz; Hungerbühler, Norbert; Läuchli, Juan (2016),
532:. Then the theorem of the gnomon states that the parallelograms
191:
1980:
1265:{\displaystyle |\mathbb {B} |=|\mathbb {C} |=|\mathbb {D} |}
1076:{\displaystyle {\tfrac {|AH|}{|HB|}}={\tfrac {|HP|}{|PG|}}}
1275:
A similar statement can be made in three dimensions for
1757:
The theorem of the gnomon was described as early as in
1118:
is the (lower) parallelepiped around the diagonal with
1032:
987:
1928:
Evans, George W. (1927), "Some of Euclid's
Algebra",
1735:
1706:
1677:
1654:
1625:
1533:
1510:
1481:
1452:
1429:
1400:
1371:
1329:
1309:
1285:
1210:
1188:
1166:
1144:
1124:
1102:
985:
799:
751:
722:
689:
660:
631:
602:
567:
538:
518:
495:
475:
452:
432:
409:
389:
366:
346:
323:
303:
280:
257:
237:
208:
66:
29:
3173:
3051:
3001:
2975:
2897:
2879:
2778:
2771:
2627:
2589:
2406:
2014:
1855:(in German), Walter de Gruyter, pp. 134–135,
1741:
1721:
1692:
1671:. This means in particular for the parallelograms
1663:
1640:
1608:
1516:
1496:
1467:
1438:
1415:
1386:
1335:
1315:
1291:
1264:
1196:
1174:
1152:
1130:
1110:
1075:
951:
766:
737:
704:
675:
646:
617:
582:
553:
524:
504:
481:
461:
438:
418:
395:
375:
352:
332:
309:
289:
266:
243:
223:
167:
50:
1908:, Springer, pp. 601–610, esp. pp. 603–606,
59:Theorem of the Gnomon: green area = red area,
1992:
168:{\displaystyle |AHGD|=|ABFI|,\,|HBFP|=|IPGD|}
8:
1881:A Mathematical History of the Golden Number
1351:General theorem about nested parallelograms
3005:
2775:
1999:
1985:
1977:
1873:
1871:
1394:consider an arbitrary inner parallelogram
1941:
1734:
1705:
1676:
1653:
1624:
1601:
1584:
1576:
1559:
1551:
1534:
1532:
1509:
1480:
1451:
1428:
1399:
1370:
1328:
1308:
1284:
1257:
1253:
1252:
1247:
1239:
1235:
1234:
1229:
1221:
1217:
1216:
1211:
1209:
1190:
1189:
1187:
1168:
1167:
1165:
1146:
1145:
1143:
1123:
1104:
1103:
1101:
1064:
1053:
1046:
1035:
1031:
1019:
1008:
1001:
990:
986:
984:
944:
927:
913:
896:
893:
879:
862:
859:
845:
828:
825:
817:
800:
798:
750:
721:
688:
659:
630:
601:
566:
537:
517:
494:
474:
451:
431:
408:
388:
365:
345:
322:
302:
279:
256:
236:
207:
160:
143:
135:
118:
117:
109:
92:
84:
67:
65:
28:
1810:
1808:
1806:
971:geometrical representation of a division
1884:, Courier Corporation, pp. 35–36,
1775:
1648:whose vertices are all on the diagonal
3028:Latin translations of the 12th century
1088:straightedge and compass constructions
2758:Straightedge and compass construction
1781:
1779:
403:. Similarly the parallel to the side
7:
2723:Incircle and excircles of a triangle
1609:{\displaystyle |AFCE|=|GFHD|-|IBJF|}
1878:Herz-Fischler, Roger (2013-12-31),
716:(of the parallelograms on diagonal
654:. The parallelograms of equal area
14:
1817:The American Mathematical Monthly
1361:green area = blue area - red area
3240:
3227:
1849:Tropfke, Johannes (2011-10-10),
1279:. In this case you have a point
3060:A History of Greek Mathematics
2573:The Quadrature of the Parabola
1829:10.1080/00029890.1929.11986904
1790:, Springer, pp. 190–191,
1602:
1585:
1577:
1560:
1552:
1535:
1258:
1248:
1240:
1230:
1222:
1212:
1065:
1054:
1047:
1036:
1020:
1009:
1002:
991:
945:
928:
914:
897:
880:
863:
846:
829:
818:
801:
161:
144:
136:
119:
110:
93:
85:
68:
1:
3272:Theorems about quadrilaterals
2841:Intersecting secants theorem
1197:{\displaystyle \mathbb {D} }
1175:{\displaystyle \mathbb {C} }
1153:{\displaystyle \mathbb {B} }
1111:{\displaystyle \mathbb {A} }
2836:Intersecting chords theorem
2703:Doctrine of proportionality
963:Applications and extensions
3288:
2532:On the Sphere and Cylinder
2485:On the Sizes and Distances
1729:, that their common point
3234:Ancient Greece portal
3223:
3038:Philosophy of mathematics
3008:
2953:Ptolemy's table of chords
2008:Ancient Greek mathematics
2905:Aristarchus's inequality
2478:On Conoids and Spheroids
1967:Definition of the gnomon
3013:Ancient Greek astronomy
2826:Inscribed angle theorem
2816:Greek geometric algebra
2471:Measurement of a Circle
1930:The Mathematics Teacher
3247:Mathematics portal
3033:Non-Euclidean geometry
2988:Mouseion of Alexandria
2861:Tangent-secant theorem
2811:Geometric mean theorem
2796:Exterior angle theorem
2791:Angle bisector theorem
2495:On Sizes and Distances
1743:
1723:
1694:
1665:
1642:
1610:
1518:
1498:
1469:
1440:
1417:
1388:
1362:
1337:
1317:
1293:
1272:
1266:
1204:have the same volume:
1198:
1176:
1154:
1132:
1112:
1083:
1077:
972:
953:
768:
739:
706:
677:
648:
619:
584:
555:
526:
506:
483:
463:
440:
420:
397:
377:
354:
334:
311:
291:
268:
245:
225:
175:
169:
52:
51:{\displaystyle ABFPGD}
2935:Pappus's area theorem
2871:Theorem of the gnomon
2748:Quadratrix of Hippias
2671:Circles of Apollonius
2619:Problem of Apollonius
2597:Constructible numbers
2421:Archimedes Palimpsest
1961:Theorem of the gnomon
1744:
1724:
1695:
1666:
1643:
1611:
1519:
1499:
1470:
1441:
1418:
1389:
1358:
1338:
1318:
1294:
1267:
1199:
1177:
1155:
1133:
1113:
1096:
1078:
978:
970:
954:
769:
740:
707:
678:
649:
620:
585:
556:
527:
507:
484:
464:
441:
421:
398:
378:
355:
335:
312:
292:
269:
246:
226:
180:theorem of the gnomon
170:
53:
22:
3151:prehistoric counting
2948:Ptolemy's inequality
2889:Apollonius's theorem
2728:Method of exhaustion
2698:Diophantine equation
2688:Circumscribed circle
2505:On the Moving Sphere
1733:
1722:{\displaystyle IBJF}
1704:
1693:{\displaystyle GFHD}
1675:
1652:
1641:{\displaystyle AFCE}
1623:
1531:
1508:
1497:{\displaystyle IBJF}
1479:
1468:{\displaystyle GFHD}
1450:
1427:
1416:{\displaystyle AFCE}
1398:
1387:{\displaystyle ABCD}
1369:
1327:
1307:
1283:
1208:
1186:
1164:
1142:
1138:and its complements
1122:
1100:
983:
797:
767:{\displaystyle AHPI}
749:
738:{\displaystyle PFCG}
720:
705:{\displaystyle IPGD}
687:
676:{\displaystyle HBFP}
658:
647:{\displaystyle AHGD}
629:
618:{\displaystyle ABFI}
600:
583:{\displaystyle IPGD}
565:
554:{\displaystyle HBFP}
536:
516:
493:
473:
450:
446:intersects the side
430:
407:
387:
364:
344:
321:
317:intersects the side
301:
278:
255:
235:
224:{\displaystyle ABCD}
206:
182:states that certain
64:
27:
3237: •
3043:Neusis construction
2963:Spiral of Theodorus
2856:Pythagorean theorem
2801:Euclidean algorithm
2743:Lune of Hippocrates
2612:Squaring the circle
2368:Theon of Alexandria
2043:Aristaeus the Elder
202:In a parallelogram
3267:Euclidean geometry
2930:Menelaus's theorem
2920:Irrational numbers
2733:Parallel postulate
2708:Euclidean geometry
2676:Apollonian circles
2218:Isidore of Miletus
1753:Historical aspects
1739:
1719:
1690:
1664:{\displaystyle AC}
1661:
1638:
1606:
1514:
1494:
1465:
1439:{\displaystyle AC}
1436:
1413:
1384:
1363:
1333:
1313:
1289:
1273:
1262:
1194:
1172:
1150:
1128:
1108:
1084:
1073:
1071:
1026:
973:
949:
764:
735:
702:
673:
644:
615:
590:have equal areas.
580:
551:
522:
505:{\displaystyle BC}
502:
479:
462:{\displaystyle AD}
459:
436:
419:{\displaystyle AB}
416:
393:
376:{\displaystyle AB}
373:
350:
333:{\displaystyle CD}
330:
307:
290:{\displaystyle AD}
287:
274:, the parallel to
267:{\displaystyle AC}
264:
241:
221:
176:
165:
48:
3254:
3253:
3219:
3218:
2971:
2970:
2958:Ptolemy's theorem
2831:Intercept theorem
2681:Apollonian gasket
2607:Doubling the cube
2580:The Sand Reckoner
1972:Euclid's Elements
1891:978-0-486-15232-5
1862:978-3-11-162693-2
1759:Euclid's Elements
1742:{\displaystyle F}
1517:{\displaystyle F}
1336:{\displaystyle P}
1316:{\displaystyle P}
1292:{\displaystyle P}
1131:{\displaystyle P}
1070:
1025:
922:
888:
854:
525:{\displaystyle F}
482:{\displaystyle I}
439:{\displaystyle P}
396:{\displaystyle H}
353:{\displaystyle G}
310:{\displaystyle P}
244:{\displaystyle P}
3279:
3245:
3244:
3232:
3231:
3230:
3006:
2993:Platonic Academy
2940:Problem II.8 of
2910:Crossbar theorem
2866:Thales's theorem
2806:Euclid's theorem
2776:
2693:Commensurability
2654:Axiomatic system
2602:Angle trisection
2567:
2557:
2519:
2509:
2499:
2489:
2465:
2455:
2438:
2001:
1994:
1987:
1978:
1947:
1946:
1945:
1925:
1919:
1918:
1901:
1895:
1894:
1875:
1866:
1865:
1846:
1840:
1839:
1812:
1801:
1800:
1783:
1748:
1746:
1745:
1740:
1728:
1726:
1725:
1720:
1699:
1697:
1696:
1691:
1670:
1668:
1667:
1662:
1647:
1645:
1644:
1639:
1615:
1613:
1612:
1607:
1605:
1588:
1580:
1563:
1555:
1538:
1523:
1521:
1520:
1515:
1503:
1501:
1500:
1495:
1474:
1472:
1471:
1466:
1445:
1443:
1442:
1437:
1422:
1420:
1419:
1414:
1393:
1391:
1390:
1385:
1359:general theorem:
1342:
1340:
1339:
1334:
1322:
1320:
1319:
1314:
1298:
1296:
1295:
1290:
1271:
1269:
1268:
1263:
1261:
1256:
1251:
1243:
1238:
1233:
1225:
1220:
1215:
1203:
1201:
1200:
1195:
1193:
1181:
1179:
1178:
1173:
1171:
1159:
1157:
1156:
1151:
1149:
1137:
1135:
1134:
1129:
1117:
1115:
1114:
1109:
1107:
1082:
1080:
1079:
1074:
1072:
1069:
1068:
1057:
1051:
1050:
1039:
1033:
1027:
1024:
1023:
1012:
1006:
1005:
994:
988:
958:
956:
955:
950:
948:
931:
923:
918:
917:
900:
894:
889:
884:
883:
866:
860:
855:
850:
849:
832:
826:
821:
804:
773:
771:
770:
765:
744:
742:
741:
736:
711:
709:
708:
703:
682:
680:
679:
674:
653:
651:
650:
645:
624:
622:
621:
616:
589:
587:
586:
581:
560:
558:
557:
552:
531:
529:
528:
523:
511:
509:
508:
503:
488:
486:
485:
480:
468:
466:
465:
460:
445:
443:
442:
437:
425:
423:
422:
417:
402:
400:
399:
394:
382:
380:
379:
374:
359:
357:
356:
351:
339:
337:
336:
331:
316:
314:
313:
308:
296:
294:
293:
288:
273:
271:
270:
265:
251:on the diagonal
250:
248:
247:
242:
230:
228:
227:
222:
174:
172:
171:
166:
164:
147:
139:
122:
113:
96:
88:
71:
57:
55:
54:
49:
3287:
3286:
3282:
3281:
3280:
3278:
3277:
3276:
3257:
3256:
3255:
3250:
3239:
3228:
3226:
3215:
3181:Arabian/Islamic
3169:
3158:numeral systems
3047:
2997:
2967:
2915:Heron's formula
2893:
2875:
2767:
2763:Triangle center
2753:Regular polygon
2630:and definitions
2629:
2623:
2585:
2565:
2555:
2517:
2507:
2497:
2487:
2463:
2453:
2436:
2402:
2373:Theon of Smyrna
2018:
2010:
2005:
1956:
1951:
1950:
1927:
1926:
1922:
1916:
1903:
1902:
1898:
1892:
1877:
1876:
1869:
1863:
1852:Ebene Geometrie
1848:
1847:
1843:
1814:
1813:
1804:
1798:
1785:
1784:
1777:
1772:
1755:
1731:
1730:
1702:
1701:
1673:
1672:
1650:
1649:
1621:
1620:
1529:
1528:
1506:
1505:
1477:
1476:
1448:
1447:
1425:
1424:
1396:
1395:
1367:
1366:
1360:
1353:
1347:(see diagram).
1325:
1324:
1305:
1304:
1281:
1280:
1277:parallelepipeds
1206:
1205:
1184:
1183:
1162:
1161:
1140:
1139:
1120:
1119:
1098:
1097:
1052:
1034:
1007:
989:
981:
980:
965:
895:
861:
827:
795:
794:
780:
747:
746:
718:
717:
685:
684:
656:
655:
627:
626:
598:
597:
563:
562:
534:
533:
514:
513:
491:
490:
471:
470:
448:
447:
428:
427:
405:
404:
385:
384:
362:
361:
342:
341:
319:
318:
299:
298:
276:
275:
253:
252:
233:
232:
204:
203:
200:
194:of equal size.
186:occurring in a
62:
61:
60:
58:
25:
24:
17:
12:
11:
5:
3285:
3283:
3275:
3274:
3269:
3259:
3258:
3252:
3251:
3224:
3221:
3220:
3217:
3216:
3214:
3213:
3208:
3203:
3198:
3193:
3188:
3183:
3177:
3175:
3174:Other cultures
3171:
3170:
3168:
3167:
3166:
3165:
3155:
3154:
3153:
3143:
3142:
3141:
3131:
3130:
3129:
3119:
3118:
3117:
3107:
3106:
3105:
3095:
3094:
3093:
3083:
3082:
3081:
3071:
3070:
3069:
3055:
3053:
3049:
3048:
3046:
3045:
3040:
3035:
3030:
3025:
3023:Greek numerals
3020:
3018:Attic numerals
3015:
3009:
3003:
2999:
2998:
2996:
2995:
2990:
2985:
2979:
2977:
2973:
2972:
2969:
2968:
2966:
2965:
2960:
2955:
2950:
2945:
2937:
2932:
2927:
2922:
2917:
2912:
2907:
2901:
2899:
2895:
2894:
2892:
2891:
2885:
2883:
2877:
2876:
2874:
2873:
2868:
2863:
2858:
2853:
2848:
2846:Law of cosines
2843:
2838:
2833:
2828:
2823:
2818:
2813:
2808:
2803:
2798:
2793:
2787:
2785:
2773:
2769:
2768:
2766:
2765:
2760:
2755:
2750:
2745:
2740:
2738:Platonic solid
2735:
2730:
2725:
2720:
2718:Greek numerals
2715:
2710:
2705:
2700:
2695:
2690:
2685:
2684:
2683:
2678:
2668:
2663:
2662:
2661:
2651:
2650:
2649:
2644:
2633:
2631:
2625:
2624:
2622:
2621:
2616:
2615:
2614:
2609:
2604:
2593:
2591:
2587:
2586:
2584:
2583:
2576:
2569:
2559:
2549:
2546:Planisphaerium
2542:
2535:
2528:
2521:
2511:
2501:
2491:
2481:
2474:
2467:
2457:
2447:
2440:
2430:
2423:
2418:
2410:
2408:
2404:
2403:
2401:
2400:
2395:
2390:
2385:
2380:
2375:
2370:
2365:
2360:
2355:
2350:
2345:
2340:
2335:
2330:
2325:
2320:
2315:
2310:
2305:
2300:
2295:
2290:
2285:
2280:
2275:
2270:
2265:
2260:
2255:
2250:
2245:
2240:
2235:
2230:
2225:
2220:
2215:
2210:
2205:
2200:
2195:
2190:
2185:
2180:
2175:
2170:
2165:
2160:
2155:
2150:
2145:
2140:
2135:
2130:
2125:
2120:
2115:
2110:
2105:
2100:
2095:
2090:
2085:
2080:
2075:
2070:
2065:
2060:
2055:
2050:
2045:
2040:
2035:
2030:
2024:
2022:
2016:Mathematicians
2012:
2011:
2006:
2004:
2003:
1996:
1989:
1981:
1975:
1974:
1955:
1954:External links
1952:
1949:
1948:
1936:(3): 127–141,
1920:
1914:
1896:
1890:
1867:
1861:
1841:
1802:
1796:
1774:
1773:
1771:
1768:
1754:
1751:
1738:
1718:
1715:
1712:
1709:
1689:
1686:
1683:
1680:
1660:
1657:
1637:
1634:
1631:
1628:
1617:
1616:
1604:
1600:
1597:
1594:
1591:
1587:
1583:
1579:
1575:
1572:
1569:
1566:
1562:
1558:
1554:
1550:
1547:
1544:
1541:
1537:
1513:
1493:
1490:
1487:
1484:
1464:
1461:
1458:
1455:
1435:
1432:
1412:
1409:
1406:
1403:
1383:
1380:
1377:
1374:
1352:
1349:
1332:
1312:
1301:space diagonal
1288:
1260:
1255:
1250:
1246:
1242:
1237:
1232:
1228:
1224:
1219:
1214:
1192:
1170:
1148:
1127:
1106:
1067:
1063:
1060:
1056:
1049:
1045:
1042:
1038:
1030:
1022:
1018:
1015:
1011:
1004:
1000:
997:
993:
964:
961:
960:
959:
947:
943:
940:
937:
934:
930:
926:
921:
916:
912:
909:
906:
903:
899:
892:
887:
882:
878:
875:
872:
869:
865:
858:
853:
848:
844:
841:
838:
835:
831:
824:
820:
816:
813:
810:
807:
803:
791:
790:
787:
779:
776:
763:
760:
757:
754:
734:
731:
728:
725:
701:
698:
695:
692:
672:
669:
666:
663:
643:
640:
637:
634:
614:
611:
608:
605:
579:
576:
573:
570:
550:
547:
544:
541:
521:
501:
498:
478:
458:
455:
435:
415:
412:
392:
372:
369:
349:
329:
326:
306:
286:
283:
263:
260:
240:
220:
217:
214:
211:
199:
196:
184:parallelograms
163:
159:
156:
153:
150:
146:
142:
138:
134:
131:
128:
125:
121:
116:
112:
108:
105:
102:
99:
95:
91:
87:
83:
80:
77:
74:
70:
47:
44:
41:
38:
35:
32:
15:
13:
10:
9:
6:
4:
3:
2:
3284:
3273:
3270:
3268:
3265:
3264:
3262:
3249:
3248:
3243:
3236:
3235:
3222:
3212:
3209:
3207:
3204:
3202:
3199:
3197:
3194:
3192:
3189:
3187:
3184:
3182:
3179:
3178:
3176:
3172:
3164:
3161:
3160:
3159:
3156:
3152:
3149:
3148:
3147:
3144:
3140:
3137:
3136:
3135:
3132:
3128:
3125:
3124:
3123:
3120:
3116:
3113:
3112:
3111:
3108:
3104:
3101:
3100:
3099:
3096:
3092:
3089:
3088:
3087:
3084:
3080:
3077:
3076:
3075:
3072:
3068:
3064:
3063:
3062:
3061:
3057:
3056:
3054:
3050:
3044:
3041:
3039:
3036:
3034:
3031:
3029:
3026:
3024:
3021:
3019:
3016:
3014:
3011:
3010:
3007:
3004:
3000:
2994:
2991:
2989:
2986:
2984:
2981:
2980:
2978:
2974:
2964:
2961:
2959:
2956:
2954:
2951:
2949:
2946:
2944:
2943:
2938:
2936:
2933:
2931:
2928:
2926:
2923:
2921:
2918:
2916:
2913:
2911:
2908:
2906:
2903:
2902:
2900:
2896:
2890:
2887:
2886:
2884:
2882:
2878:
2872:
2869:
2867:
2864:
2862:
2859:
2857:
2854:
2852:
2851:Pons asinorum
2849:
2847:
2844:
2842:
2839:
2837:
2834:
2832:
2829:
2827:
2824:
2822:
2821:Hinge theorem
2819:
2817:
2814:
2812:
2809:
2807:
2804:
2802:
2799:
2797:
2794:
2792:
2789:
2788:
2786:
2784:
2783:
2777:
2774:
2770:
2764:
2761:
2759:
2756:
2754:
2751:
2749:
2746:
2744:
2741:
2739:
2736:
2734:
2731:
2729:
2726:
2724:
2721:
2719:
2716:
2714:
2711:
2709:
2706:
2704:
2701:
2699:
2696:
2694:
2691:
2689:
2686:
2682:
2679:
2677:
2674:
2673:
2672:
2669:
2667:
2664:
2660:
2657:
2656:
2655:
2652:
2648:
2645:
2643:
2640:
2639:
2638:
2635:
2634:
2632:
2626:
2620:
2617:
2613:
2610:
2608:
2605:
2603:
2600:
2599:
2598:
2595:
2594:
2592:
2588:
2582:
2581:
2577:
2575:
2574:
2570:
2568:
2564:
2560:
2558:
2554:
2550:
2548:
2547:
2543:
2541:
2540:
2536:
2534:
2533:
2529:
2527:
2526:
2522:
2520:
2516:
2512:
2510:
2506:
2502:
2500:
2496:
2492:
2490:
2488:(Aristarchus)
2486:
2482:
2480:
2479:
2475:
2473:
2472:
2468:
2466:
2462:
2458:
2456:
2452:
2448:
2446:
2445:
2441:
2439:
2435:
2431:
2429:
2428:
2424:
2422:
2419:
2417:
2416:
2412:
2411:
2409:
2405:
2399:
2396:
2394:
2393:Zeno of Sidon
2391:
2389:
2386:
2384:
2381:
2379:
2376:
2374:
2371:
2369:
2366:
2364:
2361:
2359:
2356:
2354:
2351:
2349:
2346:
2344:
2341:
2339:
2336:
2334:
2331:
2329:
2326:
2324:
2321:
2319:
2316:
2314:
2311:
2309:
2306:
2304:
2301:
2299:
2296:
2294:
2291:
2289:
2286:
2284:
2281:
2279:
2276:
2274:
2271:
2269:
2266:
2264:
2261:
2259:
2256:
2254:
2251:
2249:
2246:
2244:
2241:
2239:
2236:
2234:
2231:
2229:
2226:
2224:
2221:
2219:
2216:
2214:
2211:
2209:
2206:
2204:
2201:
2199:
2196:
2194:
2191:
2189:
2186:
2184:
2181:
2179:
2176:
2174:
2171:
2169:
2166:
2164:
2161:
2159:
2156:
2154:
2151:
2149:
2146:
2144:
2141:
2139:
2136:
2134:
2131:
2129:
2126:
2124:
2121:
2119:
2116:
2114:
2111:
2109:
2106:
2104:
2101:
2099:
2096:
2094:
2091:
2089:
2086:
2084:
2081:
2079:
2076:
2074:
2071:
2069:
2066:
2064:
2061:
2059:
2056:
2054:
2051:
2049:
2046:
2044:
2041:
2039:
2036:
2034:
2031:
2029:
2026:
2025:
2023:
2021:
2017:
2013:
2009:
2002:
1997:
1995:
1990:
1988:
1983:
1982:
1979:
1973:
1969:
1968:
1963:
1962:
1958:
1957:
1953:
1944:
1939:
1935:
1931:
1924:
1921:
1917:
1915:9789048185818
1911:
1907:
1900:
1897:
1893:
1887:
1883:
1882:
1874:
1872:
1868:
1864:
1858:
1854:
1853:
1845:
1842:
1838:
1834:
1830:
1826:
1822:
1818:
1811:
1809:
1807:
1803:
1799:
1797:9783662530344
1793:
1789:
1782:
1780:
1776:
1769:
1767:
1764:
1760:
1752:
1750:
1736:
1716:
1713:
1710:
1707:
1687:
1684:
1681:
1678:
1658:
1655:
1635:
1632:
1629:
1626:
1598:
1595:
1592:
1589:
1581:
1573:
1570:
1567:
1564:
1556:
1548:
1545:
1542:
1539:
1527:
1526:
1525:
1511:
1491:
1488:
1485:
1482:
1462:
1459:
1456:
1453:
1433:
1430:
1410:
1407:
1404:
1401:
1381:
1378:
1375:
1372:
1357:
1350:
1348:
1346:
1330:
1310:
1302:
1286:
1278:
1244:
1226:
1125:
1095:
1091:
1089:
1061:
1058:
1043:
1040:
1028:
1016:
1013:
998:
995:
977:
969:
962:
941:
938:
935:
932:
924:
919:
910:
907:
904:
901:
890:
885:
876:
873:
870:
867:
856:
851:
842:
839:
836:
833:
822:
814:
811:
808:
805:
793:
792:
788:
785:
784:
783:
777:
775:
761:
758:
755:
752:
732:
729:
726:
723:
715:
699:
696:
693:
690:
670:
667:
664:
661:
641:
638:
635:
632:
612:
609:
606:
603:
595:
591:
577:
574:
571:
568:
548:
545:
542:
539:
519:
499:
496:
489:and the side
476:
456:
453:
433:
413:
410:
390:
370:
367:
360:and the side
347:
327:
324:
304:
284:
281:
261:
258:
238:
231:with a point
218:
215:
212:
209:
197:
195:
193:
189:
185:
181:
157:
154:
151:
148:
140:
132:
129:
126:
123:
114:
106:
103:
100:
97:
89:
81:
78:
75:
72:
45:
42:
39:
36:
33:
30:
21:
3238:
3225:
3067:Thomas Heath
3058:
2941:
2925:Law of sines
2870:
2781:
2713:Golden ratio
2578:
2571:
2562:
2556:(Theodosius)
2552:
2544:
2537:
2530:
2523:
2514:
2504:
2498:(Hipparchus)
2494:
2484:
2476:
2469:
2460:
2450:
2442:
2437:(Apollonius)
2433:
2425:
2413:
2388:Zeno of Elea
2148:Eratosthenes
2138:Dionysodorus
1966:
1960:
1933:
1929:
1923:
1905:
1899:
1880:
1851:
1844:
1823:(1): 32–34,
1820:
1816:
1787:
1756:
1618:
1364:
1274:
1085:
781:
713:
593:
592:
201:
179:
177:
3134:mathematics
2942:Arithmetica
2539:Ostomachion
2508:(Autolycus)
2427:Arithmetica
2203:Hippocrates
2133:Dinostratus
2118:Dicaearchus
2048:Aristarchus
714:complements
712:are called
3261:Categories
3186:Babylonian
3086:arithmetic
3052:History of
2881:Apollonius
2566:(Menelaus)
2525:On Spirals
2444:Catoptrics
2383:Xenocrates
2378:Thymaridas
2363:Theodosius
2348:Theaetetus
2328:Simplicius
2318:Pythagoras
2303:Posidonius
2288:Philonides
2248:Nicomachus
2243:Metrodorus
2233:Menaechmus
2188:Hipparchus
2178:Heliodorus
2128:Diophantus
2113:Democritus
2093:Chrysippus
2063:Archimedes
2058:Apollonius
2028:Anaxagoras
2020:(timeline)
1770:References
2647:Inscribed
2407:Treatises
2398:Zenodorus
2358:Theodorus
2333:Sosigenes
2278:Philolaus
2263:Oenopides
2258:Nicoteles
2253:Nicomedes
2213:Hypsicles
2108:Ctesibius
2098:Cleomedes
2083:Callippus
2068:Autolycus
2053:Aristotle
2033:Anthemius
1582:−
891:−
857:−
3211:Japanese
3196:Egyptian
3139:timeline
3127:timeline
3115:timeline
3110:geometry
3103:timeline
3098:calculus
3091:timeline
3079:timeline
2782:Elements
2628:Concepts
2590:Problems
2563:Spherics
2553:Spherics
2518:(Euclid)
2464:(Euclid)
2461:Elements
2454:(Euclid)
2415:Almagest
2323:Serenus
2298:Porphyry
2238:Menelaus
2193:Hippasus
2168:Eutocius
2143:Domninus
2038:Archytas
1943:27950916
426:through
297:through
23:Gnomon:
3191:Chinese
3146:numbers
3074:algebra
3002:Related
2976:Centers
2772:Results
2642:Central
2313:Ptolemy
2308:Proclus
2273:Perseus
2228:Marinus
2208:Hypatia
2198:Hippias
2173:Geminus
2163:Eudoxus
2153:Eudemus
2123:Diocles
1837:2300175
1423:having
1299:on the
198:Theorem
3206:Indian
2983:Cyrene
2515:Optics
2434:Conics
2353:Theano
2343:Thales
2338:Sporus
2283:Philon
2268:Pappus
2158:Euclid
2088:Carpus
2078:Bryson
1940:
1912:
1888:
1859:
1835:
1794:
1763:Euclid
1345:volume
594:Gnomon
188:gnomon
3201:Incan
3122:logic
2898:Other
2666:Chord
2659:Axiom
2637:Angle
2293:Plato
2183:Heron
2103:Conon
1964:and
1938:JSTOR
1833:JSTOR
778:Proof
192:areas
190:have
3163:list
2451:Data
2223:Leon
2073:Bion
1910:ISBN
1886:ISBN
1857:ISBN
1792:ISBN
1700:and
1475:and
1182:and
745:and
683:and
625:and
561:and
178:The
3065:by
2779:In
1970:in
1825:doi
774:).
512:in
469:in
383:in
340:in
3263::
1934:20
1932:,
1870:^
1831:,
1821:36
1819:,
1805:^
1778:^
1160:,
2000:e
1993:t
1986:v
1827::
1737:F
1717:F
1714:J
1711:B
1708:I
1688:D
1685:H
1682:F
1679:G
1659:C
1656:A
1636:E
1633:C
1630:F
1627:A
1603:|
1599:F
1596:J
1593:B
1590:I
1586:|
1578:|
1574:D
1571:H
1568:F
1565:G
1561:|
1557:=
1553:|
1549:E
1546:C
1543:F
1540:A
1536:|
1512:F
1492:F
1489:J
1486:B
1483:I
1463:D
1460:H
1457:F
1454:G
1434:C
1431:A
1411:E
1408:C
1405:F
1402:A
1382:D
1379:C
1376:B
1373:A
1331:P
1311:P
1287:P
1259:|
1254:D
1249:|
1245:=
1241:|
1236:C
1231:|
1227:=
1223:|
1218:B
1213:|
1191:D
1169:C
1147:B
1126:P
1105:A
1066:|
1062:G
1059:P
1055:|
1048:|
1044:P
1041:H
1037:|
1029:=
1021:|
1017:B
1014:H
1010:|
1003:|
999:H
996:A
992:|
946:|
942:P
939:F
936:B
933:H
929:|
925:=
920:2
915:|
911:G
908:C
905:F
902:P
898:|
886:2
881:|
877:I
874:P
871:H
868:A
864:|
852:2
847:|
843:D
840:C
837:B
834:A
830:|
823:=
819:|
815:D
812:G
809:P
806:I
802:|
762:I
759:P
756:H
753:A
733:G
730:C
727:F
724:P
700:D
697:G
694:P
691:I
671:P
668:F
665:B
662:H
642:D
639:G
636:H
633:A
613:I
610:F
607:B
604:A
578:D
575:G
572:P
569:I
549:P
546:F
543:B
540:H
520:F
500:C
497:B
477:I
457:D
454:A
434:P
414:B
411:A
391:H
371:B
368:A
348:G
328:D
325:C
305:P
285:D
282:A
262:C
259:A
239:P
219:D
216:C
213:B
210:A
162:|
158:D
155:G
152:P
149:I
145:|
141:=
137:|
133:P
130:F
127:B
124:H
120:|
115:,
111:|
107:I
104:F
101:B
98:A
94:|
90:=
86:|
82:D
79:G
76:H
73:A
69:|
46:D
43:G
40:P
37:F
34:B
31:A
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
