693:, whose job was to precisely measure the extent of the territory of Egypt for agricultural and taxation-related purposes. Furthermore, the fact that Eratosthenes' measure corresponds precisely to 252,000 stadia (according to Pliny) might be intentional, since it is a number that can be divided by all natural numbers from 1 to 10: some historians believe that Eratosthenes changed from the 250,000 value written by Cleomedes to this new value to simplify calculations; other historians of science, on the other side, believe that Eratosthenes introduced a new length unit based on the length of the meridian, as stated by Pliny, who writes about the stadion "according to Eratosthenes' ratio".
663:
659:. Depending on whether he used the "Olympic stade" (176.4 m) or the Italian stade (184.8 m), this would imply a circumference of 44,100 km (an error of 10%) or 46,100 km, an error of 15%. A value for the stadion of 157.7 metres has even been posited by L.V. Firsov, which would give an even better precision, but is plagued by calculation errors and false assumptions. In 2012, Anthony Abreu Mora repeated Eratosthenes's calculation with more accurate data; the result was 40,074 km, which is 66 km different (0.16%) from the currently accepted polar circumference.
844:
39:
651:, Egypt), the Sun was directly overhead, as the gnomon cast no shadow. Additionally, the shadow of someone looking down a deep well at that time in Syene blocked the reflection of the Sun on the water. Eratosthenes then measured the Sun's angle of elevation at noon in Alexandria by measuring the length of another gnomon's shadow on the ground. Using the length of the rod, and the length of the shadow, as the legs of a triangle, he calculated the angle of the sun's rays. This angle was about 7°, or 1/50th the circumference of a
540:
840:. They calculated the Earth's circumference to be within 15% of the modern value, and possibly much closer. How accurate it actually was is not known because of uncertainty in the conversion between the medieval Arabic units and modern units, but in any case, technical limitations of the methods and tools would not permit an accuracy better than about 5%.
748:
due north of
Alexandria, and the difference in the star's elevation indicated the distance between the two locales was 1/48 of the circle, he multiplied 5,000 by 48 to arrive at a figure of 240,000 stadia for the circumference of the earth. It is generally thought that the stadion used by Posidonius
1052:
of the Earth, making the prototype about 0.02% shorter than the original proposed definition of the metre. Regardless, this length became the French standard and was progressively adopted by other countries in Europe. This is why the polar circumference of the Earth is actually 40,008 kilometres,
883:
formula. This was the earliest known use of dip angle and the earliest practical use of the law of sines. However, the method could not provide more accurate results than previous methods, due to technical limitations, and so al-Biruni accepted the value calculated the previous century by the
543:
Illustration showing a portion of the globe showing a part of the
African continent. The sun beams shown as two rays hitting earth at Syene and Alexandria. Angle of sun beam and the gnomons (vertical sticks) is shown at Alexandria which allowed Eratosthenes' estimate of the circumference of
804:
intended by
Aryabhata is in dispute. One careful reading gives an equivalent of 14,200 kilometres (8,800 mi), too large by 11%. Another gives 15,360 km (9,540 mi), too large by 20%. Yet another gives 13,440 km (8,350 mi), too large by 5%.
681:
Eratosthenes' method was actually more complicated, as stated by the same
Cleomedes, whose purpose was to present a simplified version of the one described in Eratosthenes' book. Pliny, for example, has quoted a value of 252,000 stadia.
1590:
504:). The accuracy of measuring the circumference has improved since then, but the physical length of each unit of measure had remained close to what it was determined to be at the time, so the Earth's circumference is no longer a
1008:
of a circle – such that the polar circumference of the Earth would be exactly 21,600 miles. Gunter used
Snellius's circumference to define a nautical mile as 6,080 feet, the length of one minute of arc at 48 degrees latitude.
749:
was almost exactly 1/10 of a modern statute mile. Thus
Posidonius's measure of 240,000 stadia translates to 24,000 mi (39,000 km), not much short of the actual circumference of 24,901 mi (40,074 km).
761:
had added some 26,000 stadia to
Eratosthenes's estimate. The smaller value offered by Strabo and the different lengths of Greek and Roman stadia have created a persistent confusion around Posidonius's result.
655:; assuming the Earth to be perfectly spherical, he concluded that its circumference was 50 times the known distance from Alexandria to Syene (5,000 stadia, a figure that was checked yearly), i.e. 250,000
904:, he chose to believe that the Earth's circumference was 25% smaller. If, instead, Columbus had accepted Eratosthenes's larger value, he would have known that the place where he made landfall was not
1449:) stated that if the distance is measured by some other number the result will be different, and using 3,750 instead of 5,000 produces this estimation: 3,750 x 48 = 180,000; see Fischer I., (1975),
556:, with an error on the real value between −2.4% and +0.8% (assuming a value for the stadion between 155 and 160 metres; the exact value of the stadion remains a subject of debate to this day; see
753:
noted that the distance between Rhodes and
Alexandria is 3,750 stadia, and reported Posidonius's estimate of the Earth's circumference to be 180,000 stadia or 18,000 mi (29,000 km).
613:
He assumes the simplified (but inaccurate) hypothesis that Syene and
Alexandria are on the same meridian. Syene was actually about 3 degrees of longitude east of Alexandria.
469:, by comparing altitudes of the mid-day sun at two places a known north–south distance apart. He achieved a great degree of precision in his computation. Treating the Earth as a
1394:"The Eratosthenes-Strabo Nile Map. Is It the Earliest Surviving Instance of Spherical Cartography? Did It Supply the 5000 Stades Arc for Eratosthenes' Experiment?"
496:), which is 21,600 partitions of the polar circumference (that is 60 minutes × 360 degrees). The polar circumference is also close to 40,000 kilometres because
1012:
In 1793, France defined the metre so as to make the polar circumference of the Earth 40,000 kilometres. In order to measure this distance accurately, the
1327:
757:
mentions
Posidonius among his sources and—without naming him—reported his method for estimating the Earth's circumference. He noted, however, that
1048:
was based on these measurements, but it was later determined that its length was short by about 0.2 millimetres because of miscalculation of the
847:
Diagram showing how al-Biruni was able to calculate the Earth's circumference by measuring the dip of the horizon from a point at a known height.
398:
488:
in the eighteenth. Earth's polar circumference is very near to 21,600 nautical miles because the nautical mile was intended to express one
875:
top, which made it possible for it to be measured by a single person from a single location. From the top of the mountain, he sighted the
220:
859:(1037). In contrast to his predecessors, who measured the Earth's circumference by sighting the Sun simultaneously from two locations,
388:
356:
1025:
935:
473:, its circumference would be its single most important measurement. Earth deviates from spherical by about 0.3%, as characterized by
366:
952:
to determine latitude at sea. He reasoned that the lines of latitude could be used as the basis for a unit of measurement for
1726:
1705:
1684:
1661:
1628:
1266:
1200:
1140:
944:
assessed the circumference of the Earth at 24,630 Roman miles (24,024 statute miles). Around that time British mathematician
346:
1105:
1502:
1310:
326:
1516:
127:
1544:
622:
426:
1282:
921:
825:
480:
In modern times, Earth's circumference has been used to define fundamental units of measurement of length: the
212:
1743:
Carl Sagan demonstrates how Eratosthenes determined that the Earth was round and the approximate circumference
1605:
1033:
1560:
Behnaz Savizi (2007), "Applicable Problems in History of Mathematics: Practical Examples for the Classroom",
1017:
744:
between the latitude of the two locales is actually 5 degrees 14 minutes). Since he thought Rhodes was 5,000
666:
Measure of Earth's circumference according to Cleomedes' simplified version, based on the approximation that
228:
1328:"Eratosthenes and the Mystery of the Stades – How Long Is a Stade? – Mathematical Association of America"
766:
used Posidonius's lower value of 180,000 stades (about 33% too low) for the earth's circumference in his
1013:
610:
the Sun was directly overhead. Syene was actually north of the tropic by something less than a degree.
191:
1762:
1175:
900:
studied what Eratosthenes had written about the size of the Earth. Nevertheless, based on a map by
745:
500:
to be one ten millionth (i.e., a kilometre is one ten thousandth) of the arc from pole to equator (
489:
158:
927:
1569:
702:
296:
256:
1045:
567:, which has not been preserved; what has been preserved is the simplified version described by
276:
1190:
662:
1716:
1649:
1620:
The Measure of All Things: The Seven-Year Odyssey and Hidden Error That Transformed the World
949:
419:
336:
86:
1167:
879:
which, along with the mountain's height (which he determined beforehand), he applied to the
557:
897:
773:
571:
to popularise the discovery. Cleomedes invites his reader to consider two Egyptian cities,
497:
236:
186:
112:
1434:
8:
1451:
Another Look at Eratosthenes' and Posidonius' Determinations of the Earth's Circumference
1168:
941:
767:
521:
148:
96:
843:
465:
since ancient times. The first known scientific measurement and calculation was done by
1654:
Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Ancient World
1482:
1413:
1393:
821:
517:
68:
38:
598:
He assumes the simplified (but inaccurate) hypothesis that Syene was precisely on the
548:
The measure of Earth's circumference is the most famous among the results obtained by
1722:
1701:
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1657:
1624:
1540:
1417:
1262:
1196:
1136:
1077:
632:
455:
168:
153:
1446:
1021:
1767:
1573:
1405:
1356:
931:
671:
599:
529:
525:
501:
412:
163:
1695:
1673:
1618:
1466:
1109:
1067:
754:
1757:
973:
711:
656:
584:
553:
376:
143:
117:
1742:
876:
710:
calculated the Earth's circumference by reference to the position of the star
454:, it is 40,075.017 km (24,901.461 mi). Measured passing through the
1751:
1535:
Mercier, Raymond (1992). "Geodesy". In Harley, J.B.; Woodward, David (eds.).
1072:
945:
481:
443:
1577:
1062:
1041:
915:
885:
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864:
817:
741:
628:
583:
Cleomedes assumes that the distance between Syene and Alexandria was 5,000
549:
505:
493:
466:
101:
19:
1462:
1163:
1082:
790:
58:
1409:
1049:
901:
758:
723:
707:
675:
603:
572:
474:
462:
1675:
Circumference: Eratosthenes and the Ancient Quest to Measure the Globe
1345:
640:
1037:
909:
860:
852:
785:
715:
686:
618:
568:
137:
63:
1259:
Measuring the Universe: Cosmic Dimensions from Aristarchus to Halley
1503:"Journal of the Royal Asiatic Society of Great Britain and Ireland"
1360:
953:
872:
690:
607:
588:
316:
133:
122:
1487:
1028:
to attempt to accurately measure the distance between a belfry in
776:
in order to underestimate the distance to India as 70,000 stades.
1029:
829:
796:
763:
451:
196:
53:
30:
1283:"Astronomy 101 Specials: Eratosthenes and the Size of the Earth"
718:, Posidonius observed Canopus on but never above the horizon at
458:, the circumference is 40,007.863 km (24,859.734 mi).
814:
750:
719:
652:
636:
539:
470:
306:
266:
204:
1468:
Before Galileo: The Birth of Modern Science in Medieval Europe
1311:"How did Eratosthenes measure the circumference of the earth?"
956:
and proposed the nautical mile as one minute or one-sixtieth (
794:, in which he calculated the diameter of earth to be of 1,050
868:
837:
833:
667:
648:
644:
576:
485:
447:
91:
1697:
Measuring the Earth: Eratosthenes and His Celestial Geometry
1592:
Measuring the Earth: Eratosthenes and His Celestial Geometry
1195:. Springer Science & Business Media. pp. 638–640.
905:
461:
Measurement of Earth's circumference has been important to
393:
286:
552:, who estimated that the meridian has a length of 252,000
916:
Historical use in the definition of units of measurement
563:
Eratosthenes described his technique in a book entitled
1188:
784:
Around AD 525, the Indian mathematician and astronomer
1606:
Why Nautical Mile and Knot Are The Units Used at Sea?
1539:. The University of Chicago Press. pp. 175–188.
639:
and under the previous assumptions, he knew that at
1453:, Ql. J. of the Royal Astron. Soc., Vol. 16, p.152.
1672:
851:A more convenient way to estimate was provided in
587:(a figure that was checked yearly by professional
922:History of the metre § Meridional definition
1749:
1481:Kak, Subhash (2010). "Aryabhata's Mathematics".
1429:
1427:
1182:
624:On the Circular Motions of the Celestial Bodies
1189:Shashi Shekhar; Hui Xiong (12 December 2007).
1559:
1261:. University of Chicago Press. pp. 4–5.
1158:
1156:
1154:
1152:
627:, around 240 BC, Eratosthenes calculated the
420:
18:For modern definitions and measurements, see
1648:Krebs, Robert E.; Krebs, Carolyn A. (2003).
1595:, p. 6 (Berkeley Heights, NJ: Enslow, 2010).
1537:The History of Cartography, Volume 2, Book 1
1424:
1135:. New York, NY: Watson-Guptill. p. 31.
1133:The Illustrated Encyclopedia of the Universe
948:improved navigational tools including a new
1044:through Dunkerque. The length of the first
867:calculations, based on the angle between a
1656:. Greenwood Publishing Group. p. 52.
1647:
1256:
1149:
928:Meridian arc § History of measurement
427:
413:
394:Spatial Reference System Identifier (SRID)
389:International Terrestrial Reference System
1610:
1562:Teaching Mathematics and Its Applications
1530:
1528:
1526:
1517:"The_Aryabhatiya_of_Aryabhata_Clark_1930"
1486:
1103:
1670:
1374:
1372:
1322:
1320:
896:1,700 years after Eratosthenes's death,
842:
661:
538:
1671:Nicastro, Nicholas (25 November 2008).
1650:"Calculating the Earth's Circumference"
1534:
1391:
1130:
936:Arc measurement of Delambre and Méchain
703:Posidonius § Earth's circumference
1750:
1714:
1553:
1523:
1338:
1715:Lowrie, William (20 September 2007).
1616:
1398:Archive for History of Exact Sciences
1369:
1317:
1305:
1303:
1162:
1104:Humerfelt, Sigurd (26 October 2010).
828:to measure the distance from Tadmur (
808:
1693:
1480:
891:
484:in the seventeenth century and the
399:Universal Transverse Mercator (UTM)
361:European Terrestrial Ref. Sys. 1989
13:
1300:
1245:De nuptiis Philologiae et Mercurii
992:of a circle, one minute of arc is
635:. Using a vertical rod known as a
271:Ordnance Survey Great Britain 1936
237:Discrete Global Grid and Geocoding
128:Horizontal position representation
14:
1779:
1736:
1346:The Length of Eratosthenes' Stade
1215:
863:developed a new method of using
689:trips conducted by professional
685:The method was based on several
498:the metre was originally defined
187:Global Nav. Sat. Systems (GNSSs)
37:
1641:
1598:
1583:
1509:
1495:
1474:
1456:
1439:
1385:
740:degrees above the horizon (the
534:
351:N. American Vertical Datum 1988
1721:. Cambridge University Press.
1275:
1250:
1237:
1224:
1209:
1124:
1097:
1040:to estimate the length of the
976:of latitude. As one degree is
772:. This was the number used by
522:Spherical Earth § History
381:Internet link to a point 2010
311:Geodetic Reference System 1980
229:Quasi-Zenith Sat. Sys. (QZSS)
1:
1350:American Journal of Philology
1090:
1018:Jean Baptiste Joseph Delambre
696:
508:in metres or nautical miles.
371:Chinese obfuscated datum 2002
1174:. Berlin: Springer. p.
940:In 1617 the Dutch scientist
779:
674:and on the same meridian as
321:Geographic point coord. 1983
7:
1617:Alder, Ken (October 2003).
1257:Van Helden, Albert (1985).
1056:
932:Earth radius § History
726:he saw it ascend as far as
565:On the measure of the Earth
530:Meridian arc § History
526:Earth radius § History
281:Systema Koordinat 1942 goda
10:
1784:
1718:Fundamentals of Geophysics
1700:. Enslow Publishing, LLC.
1106:"How WGS 84 defines Earth"
1014:French Academy of Sciences
925:
919:
700:
643:on the summer solstice in
629:circumference of the Earth
515:
511:
341:World Geodetic System 1984
17:
1694:Gow, Mary (1 July 2009).
331:North American Datum 1983
301:South American Datum 1969
1392:Rawlins, Dennis (1983).
1218:The Forgotten Revolution
1170:The Forgotten Revolution
820:commissioned a group of
192:Global Pos. System (GPS)
159:Spatial reference system
1570:Oxford University Press
1364:(subscription required)
1679:. St. Martin's Press.
1623:. Simon and Schuster.
1382:, Book 2, Chapter 112.
1344:Donald Engels (1985).
848:
678:
545:
450:. Measured around the
1578:10.1093/teamat/hrl009
1131:Ridpath, Ian (2001).
926:Further information:
846:
665:
542:
440:Earth's circumference
87:Geographical distance
898:Christopher Columbus
800:. The length of the
774:Christopher Columbus
261:Sea Level Datum 1929
113:Geodetic coordinates
1287:www.eg.bucknell.edu
1243:Martianus Capella,
1192:Encyclopedia of GIS
1053:instead of 40,000.
1046:prototype metre bar
942:Willebrord Snellius
291:European Datum 1950
249:Standards (history)
149:Reference ellipsoid
97:Figure of the Earth
1410:10.1007/BF00348500
1380:Naturalis Historia
849:
822:Muslim astronomers
809:Islamic Golden Age
714:. As explained by
679:
575:and Syene (modern
546:
518:History of geodesy
169:Vertical positions
1728:978-1-139-46595-3
1707:978-0-7660-3120-3
1686:978-1-4299-5819-6
1663:978-0-313-31342-4
1630:978-0-7432-1676-0
1268:978-0-226-84882-2
1202:978-0-387-30858-6
1142:978-0-8230-2512-1
1078:Geographical mile
602:, saying that at
492:of latitude (see
437:
436:
385:
384:
164:Spatial relations
154:Satellite geodesy
109:
1775:
1732:
1711:
1690:
1678:
1667:
1635:
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1614:
1608:
1604:Marine Insight,
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1160:
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1128:
1122:
1121:
1119:
1117:
1112:on 24 April 2011
1108:. Archived from
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985:
982:
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968:
965:
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892:Columbus's error
739:
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734:
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672:Tropic of Cancer
600:Tropic of Cancer
502:quarter meridian
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253:
252:
232:
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140:
99:
41:
27:
26:
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1763:Units of length
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1447:in Fragment 202
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1161:
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1143:
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1115:
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1098:
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1068:Spherical Earth
1059:
1034:Montjuïc castle
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993:
986:
983:
980:
979:
977:
966:
963:
960:
959:
957:
938:
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918:
908:, but rather a
894:
857:Codex Masudicus
813:Around AD 830,
811:
782:
755:Pliny the Elder
736:
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633:Ptolemaic Egypt
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1737:External links
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1404:(3): 211–219.
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1361:10.2307/295030
1355:(3): 298–311.
1337:
1316:
1313:. 3 July 2012.
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1216:Russo, Lucio.
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1073:Nautical mile
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1054:
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1035:
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1027:
1026:an expedition
1023:
1019:
1016:commissioned
1015:
1010:
975:
955:
951:
947:
946:Edmund Gunter
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865:trigonometric
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617:According to
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211:
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205:BeiDou (BDS)
203:
201:
195:
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188:
185:
184:
178:
177:
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107:
106:circumference
103:
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32:
29:
28:
25:
21:
16:
1717:
1696:
1674:
1653:
1642:Bibliography
1619:
1612:
1600:
1591:
1585:
1565:
1561:
1555:
1536:
1511:
1497:
1476:
1467:
1458:
1450:
1441:
1435:fragment 202
1433:Posidonius,
1401:
1397:
1387:
1379:
1352:
1349:
1340:
1331:
1290:. Retrieved
1286:
1277:
1258:
1252:
1244:
1239:
1234:, i.7.49–52.
1231:
1226:
1217:
1211:
1191:
1184:
1169:
1164:Russo, Lucio
1132:
1126:
1114:. Retrieved
1110:the original
1099:
1063:Earth radius
1042:meridian arc
1011:
939:
895:
888:expedition.
881:law of sines
856:
850:
836:, in modern
826:Al-Khwarizmi
812:
801:
795:
789:
783:
768:
742:meridian arc
706:
684:
680:
623:
616:
592:
564:
562:
550:Eratosthenes
547:
535:Eratosthenes
506:round number
494:meridian arc
479:
467:Eratosthenes
460:
439:
438:
181:Technologies
136: /
105:
48:Fundamentals
24:
20:Earth radius
15:
1589:Gow, Mary.
1463:John Freely
1445:Cleomedes (
1332:www.maa.org
1292:19 December
1230:Cleomedes,
1083:Arabic mile
791:Aryabhatiya
722:, while at
59:Geodynamics
1752:Categories
1091:References
1050:flattening
902:Toscanelli
759:Hipparchus
724:Alexandria
708:Posidonius
697:Posidonius
676:Alexandria
670:is on the
641:local noon
604:local noon
573:Alexandria
516:See also:
475:flattening
463:navigation
1572:: 45–50,
1488:1002.3409
1418:118004246
1247:, VI.598.
1232:Caelestia
1038:Barcelona
1030:Dunkerque
972:) of one
910:New World
886:al-Ma'mun
877:dip angle
861:al-Biruni
853:Al-Biruni
818:Al-Ma'mun
786:Aryabhata
780:Aryabhata
769:Geography
716:Cleomedes
691:bematists
687:surveying
619:Cleomedes
589:bematists
569:Cleomedes
138:Longitude
64:Geomatics
1166:(2004).
1116:29 April
1057:See also
1024:to lead
954:distance
950:quadrant
873:mountain
647:(modern
608:solstice
444:distance
317:ISO 6709
215:(Europe)
213:Galileo
199:(Russia)
197:GLONASS
134:Latitude
123:Geodesic
81:Concepts
1768:Geodesy
1505:. 1907.
1378:Pliny,
1006:
994:
990:
978:
970:
958:
830:Palmyra
824:led by
797:yojanas
764:Ptolemy
735:⁄
712:Canopus
558:stadion
512:History
452:equator
446:around
442:is the
377:Geo URI
347:NAVD 88
257:NGVD 29
231:(Japan)
223:(India)
207:(China)
69:History
54:Geodesy
31:Geodesy
1725:
1704:
1683:
1660:
1627:
1543:
1471:(2012)
1416:
1265:
1199:
1139:
974:degree
934:, and
815:Caliph
802:yojana
788:wrote
751:Strabo
746:stadia
720:Rhodes
657:stadia
653:circle
637:gnomon
585:stadia
554:stadia
544:Earth.
528:, and
490:minute
471:sphere
367:GCJ-02
357:ETRS89
337:WGS 84
327:NAD 83
307:GRS 80
267:OSGB36
221:NAVIC
102:radius
1758:Earth
1568:(1),
1483:arXiv
1414:S2CID
1178:–277.
1003:21600
869:plain
838:Syria
834:Raqqa
832:) to
668:Syene
649:Aswan
645:Syene
577:Aswan
486:metre
456:poles
448:Earth
297:SAD69
277:SK-42
92:Geoid
1723:ISBN
1702:ISBN
1681:ISBN
1658:ISBN
1625:ISBN
1541:ISBN
1294:2017
1263:ISBN
1197:ISBN
1137:ISBN
1118:2011
1032:and
1020:and
906:Asia
871:and
287:ED50
104:and
1574:doi
1406:doi
1357:doi
1353:106
1176:273
1036:in
987:360
855:'s
631:in
579:):
560:).
1754::
1652:.
1566:26
1564:,
1525:^
1465:,
1426:^
1412:.
1402:26
1400:.
1396:.
1371:^
1348:.
1330:.
1319:^
1302:^
1285:.
1151:^
967:60
930:,
912:.
621:'
595:).
591:,
524:,
520:,
477:.
1731:.
1710:.
1689:.
1666:.
1633:.
1576::
1549:.
1519:.
1491:.
1485::
1420:.
1408::
1366:.
1359::
1334:.
1296:.
1271:.
1205:.
1145:.
1120:.
1000:/
997:1
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981:1
964:/
961:1
737:2
733:1
730:+
728:7
428:e
421:t
414:v
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100:(
22:.
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