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Eratosthenes
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This article is about the Greek scholar of the third century BC. For the crater named after him, see Eratosthenes (crater). For the ancient Athenian statesman of the fifth century BC, see Eratosthenes (statesman).
Eratosthenes
Born
276 BC
Cyrene
Died
194 BC
Alexandria
Occupation
Scholar
Librarian
Poet
Inventor
Known for
Sieve of Eratosthenes
Foundation of Geography
Eratosthenes of Cyrene (/ɛrəˈtɒsθəniːz/; Greek: Ἐρατοσθένης ὁ Κυρηναῖος, IPA: [eratostʰénɛːs]; c. 276 BC[1] – c. 195/194 BC[2]) was a Greek mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria. He invented the discipline of geography, including the terminology used today.[3]
He is best known for being the first person to calculate the circumference of the Earth, which he did by applying a measuring system using stadia, a standard unit of measure during that time period. His calculation was remarkably accurate. He was also the first to calculate the tilt of the Earth's axis (again with remarkable accuracy). Additionally, he may have accurately calculated the distance from the Earth to the Sun and invented the leap day.[4] He created the first map of the world, incorporating parallels and meridians based on the available geographic knowledge of his era.
Eratosthenes was the founder of scientific chronology; he endeavored to revise the dates of the chief literary and political events from the conquest of Troy. Eratosthenes dated The Sack of Troy to 1183 BC. In number theory, he introduced the sieve of Eratosthenes, an efficient method of identifying prime numbers.
He was a figure of influence in many fields. According to an entry[5] in the Suda (a 10th-century reference), his critics scorned him, calling him Beta (the second letter of the Greek alphabet) because he always came in second in all his endeavors.[6] Nonetheless, his devotees nicknamed him Pentathlos after the Olympians who were well rounded competitors, for he had proven himself to be knowledgeable in every area of learning. Eratosthenes yearned to understand the complexities of the entire world.[7]
Contents
[hide]
1 Life
2 Measurement of the Earth's circumference
3 "Father of geography"
4 Achievements
5 Prime numbers
6 Works
6.1 Titles
7 See also
7.1 Eponyms
8 References
9 Further reading
10 External links
Life[edit]
The son of Aglaos, Eratosthenes was born in 276 BC in Cyrene. Now part of modern-day Libya, Cyrene had been founded by the Greeks centuries earlier and became the capital of Pentapolis (North Africa), a country of five cities: Cyrene, Arsinoe, Berenice, Ptolemias, and Apollonia, Cyrenaica. Alexander the Great conquered Cyrene in 332 BC, and following his death in 323 BC, its rule was given to one of his generals, Ptolemy I Soter, the founder of the Ptolemaic Kingdom. Under Ptolemaic rule the economy prospered, based largely on the export of horses and silphium, a plant used for rich seasoning and medicine.[3] Cyrene became a place of cultivation, where knowledge blossomed. Like any young Greek, Eratosthenes would have studied in the local gymnasium, where he would have learned physical skills and social discourse as well as reading, writing, arithmetic, poetry, and music.[8]
Eratosthenes went to Athens to further his studies. There he was taught Stoicism by its founder, Zeno of Citium, in philosophical lectures on living a virtuous life.[9] He then studied under Aristo of Chios, who led a more cynical school of philosophy. He also studied under the head of the Platonic Academy, who was Arcesilaus of Pitane. His interest in Plato led him to write his very first work at a scholarly level, Platonikos, inquiring into the mathematical foundation of Plato's philosophies.[10] Eratosthenes was a man of many perspectives and investigated the art of poetry under Callimachus.[11] He was a talented and imaginative poet. He wrote poems: one in hexameters called Hermes, illustrating the god's life history; and another, in elegiacs called Erigone, describing the suicide of the Athenian maiden Erigone (daughter of Icarius).[12] He wrote Chronographies, a text that scientifically depicted dates of importance, beginning with the Trojan War. This work was highly esteemed for its accuracy. George Syncellus was later able to preserve from Chronographies a list of 38 kings of the Egyptian Thebes. Eratosthenes also wrote Olympic Victors, a chronology of the winners of the Olympic Games. It is not known when he wrote his works, but they highlighted his abilities.
These works and his great poetic abilities led the pharaoh Ptolemy III Euergetes to seek to place him as a librarian at the Library of Alexandria in the year 245 BC. Eratosthenes, then thirty years old, accepted Ptolemy's invitation and traveled to Alexandria, where he lived for the rest of his life. Within about five years he became Chief Librarian, a position that the poet Apollonius Rhodius had previously held. As head of the library Eratosthenes tutored the children of Ptolemy, including Ptolemy IV Philopator who became the fourth Ptolemaic pharaoh. He expanded the library's holdings: in Alexandria all books had to be surrendered for duplication. It was said that these were copied so accurately that it was impossible to tell if the library had returned the original or the copy. He sought to maintain the reputation of the Library of Alexandria against competition from the Library of Pergamum. Eratosthenes created a whole section devoted to the examination of Homer, and acquired original works of great tragic dramas of Aeschylus, Sophocles and Euripides.[13]
Eratosthenes made several important contributions to mathematics and science, and was a friend of Archimedes. Around 255 BC, he invented the armillary sphere. In On the Circular Motions of the Celestial Bodies, Cleomedes credited him with having calculated the Earth's circumference around 240 BC, using knowledge of the angle of elevation of the Sun at noon on the summer solstice in Alexandria and on Elephantine Island near Syene (modern Aswan, Egypt).
Eratosthenes believed there was good and bad in every nation and criticized Aristotle for arguing that humanity was divided into Greeks and barbarians, and that the Greeks should keep themselves racially pure.[14] As he aged he contracted ophthalmia, becoming blind around 195 BC. Losing the ability to read and to observe nature plagued and depressed him, leading him to voluntarily starve himself to death. He died in 194 BC at 82 in Alexandria.[15]
Measurement of the Earth's circumference[edit]
Illustration showing a portion of the globe showing a part of the African continent. The sunbeams shown as two rays hitting the ground at Syene and Alexandria. Angle of sunbeam and the gnomons (vertical pole) is shown at Alexandria, which allowed Eratosthenes' estimates of radius and circumference of Earth.
Eratosthenes calculated the circumference of the Earth without leaving Egypt. He knew that at local noon on the summer solstice in Syene (modern Aswan, Egypt), the Sun was directly overhead. (Syene is at latitude 24°05′ North, near to the Tropic of Cancer, which was 23°42′ North in 100 BC[16]) He knew this because the shadow of someone looking down a deep well at that time in Syene blocked the reflection of the Sun on the water. He measured the Sun's angle of elevation at noon on in Alexandria by using a vertical rod, known as a gnomon, and measuring the length of its shadow on the ground.[17] Using the length of the rod and the length of the shadow as the legs of a triangle he then calculated the angle of the suns rays. This turned out to be about 7°, or 1/50th the circumference of a circle. Taking the Earth as spherical, and knowing both the distance and direction of Syene, he concluded that the Earth's circumference was fifty times that distance.
His knowledge of the size of Egypt was founded on the work of many generations of surveying trips. Pharaonic bookkeepers gave a distance between Syene and Alexandria of 5,000 stadia (a figure that was checked yearly).[18] Some[who?] say that the distance was corroborated by inquiring about the time that it took to travel from Syene to Alexandria by camel. Carl Sagan said that Eratosthenes paid a man to walk and measure the distance. Some claim Eratosthenes used the Olympic stade of 176.4 m, which would imply a circumference of 44,100 km, an error of 10%,[18] but the 184.8 m Italian stade became (300 years later) the most commonly accepted value for the length of the stade,[18] which implies a circumference of 46,100 km, an error of 15%.[18] It was unlikely, even accounting for his extremely primitive measuring tools, that Eratosthenes could have calculated an accurate measurement for the circumference of the Earth. He made five important assumptions (none of which is perfectly accurate):[18][19]
That the distance between Alexandria and Syene was 5000 stadia,
That Alexandria is due north of Syene
That Syene lies on the Tropic of Cancer
That the Earth is a perfect sphere.
That light rays emanating from the Sun are parallel.
Eratosthenes later rounded the result to a final value of 700 stadia per degree, which implies a circumference of 252,000 stadia, likely for reasons of calculation simplicity as the larger number is evenly divisible by 60.[18] 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 of the Earth.[19]
Seventeen hundred years after Eratosthenes's death, while Christopher Columbus studied what Eratosthenes had written about the size of the Earth, he chose to believe, based on a map by Toscanelli, that the Earth's circumference was one-third smaller. Had Columbus set sail knowing that Eratosthenes's larger circumference value was more accurate, he would have known that the place that he made landfall was not Asia, but rather the New World.[20]
"Father of geography"[edit]
19th-century reconstruction of Eratosthenes' map of the (for the Greeks) known world, c. 194 BC
See also: History of geodesy and history of longitude
Eratosthenes now continued from his knowledge about the Earth. Using his discoveries and knowledge of its size and shape, he began to sketch it. In the Library of Alexandria he had access to various travel books, which contained various items of information and representations of the world that needed to be pieced together in some organized format.[21] In his three-volume work Geography (Greek: Geographika), he described and mapped his entire known world, even dividing the Earth into five climate zones:[22] two freezing zones around the poles, two temperate zones, and a zone encompassing the equator and the tropics.[23] He had invented geography. He created terminology that is still used today. He placed grids of overlapping lines over the surface of the Earth. He used parallels and meridians to link together every place in the world. It was now possible to estimate one's distance from remote locations with this network over the surface of the Earth. In the Geography the names of over 400 cities and their locations were shown: this had never been achieved before.[3] Unfortunately, his Geography has been lost to history,[how?] but fragments of the work can be pieced together from other great historians like Pliny, Polybius, Strabo, and Marcianus.
The first book was something of an introduction and gave a review of his predecessors, recognizing their contributions that he compiled in the library. In this book Eratosthenes denounced Homer as not providing any insight into what he now described as geography. His disapproval of Homer's topography angered many who believed the world depicted in the Odyssey to be legitimate.[7][24] He also commented on the ideas of the nature and origin of the Earth: he had thought of Earth as an immovable globe; while on its surface was a place that was changing. He had hypothesized that at one time the Mediterranean was a vast lake that covered the countries that surrounded it; and had only become connected to the ocean to the west when a passage had opened up sometime in its history.
In the second book is his discovery about the circumference of the Earth. This is where, according to Pliny, "The world was grasped." Eratosthenes described his famous story of the well in Syene, described above. This book would now be considered a text on mathematical geography.
His third book of the Geography contained political geography. He cited countries and used parallel lines to divide the map into sections, to give accurate descriptions of the realms. This was a breakthrough, and can be considered the beginning of geography.[25]
Achievements[edit]
Eratosthenes was described by the Suda Lexicon as a Πένταθλος (Pentathlos) which can be translated as "All-Rounder", for he was skilled in a variety of things: He was a true polymath. He was nicknamed Beta because he was great at many things and tried to get his hands on every bit of information but never achieved the highest rank in anything; Strabo accounts Eratosthenes as a mathematician among geographers and a geographer among mathematicians.[26]
Eusebius of Caesarea in his Preparatio Evangelica includes a brief chapter of three sentences on celestial distances (Book XV, Chapter 53). He states simply that Eratosthenes found the distance to the Sun to be "σταδίων μυριάδας τετρακοσίας καὶ ὀκτωκισμυρίας" (literally "of stadia myriads 400 and 80,000") and the distance to the Moon to be 780,000 stadia. The expression for the distance to the Sun has been translated either as 4,080,000 stadia (1903 translation by E. H. Gifford), or as 804,000,000 stadia (edition of Edouard des Places, dated 1974–1991). The meaning depends on whether Eusebius meant 400 myriad plus 80,000 or "400 and 80,000" myriad. With a stade of 185 m, 804,000,000 stadia is 149,000,000 km, approximately the distance from the Earth to the Sun.
Eratosthenes also calculated the Sun's diameter. According to Macrobious, Eratosthenes made the diameter of the Sun to be about 27 times that of the Earth.[25] The actual figure is approximately 109 times.[27]
During his time at the Library of Alexandria, Eratosthenes devised a calendar using his predictions about the ecliptic of the Earth. He calculated that there are 365 days in a year and that every fourth year there would be 366 days.[28]
He was also very proud of his solution for Doubling the Cube. His motivation was that he wanted to produce catapults. Eratosthenes constructed a mechanical line drawing device to calculate the cube, called the mesolabio. He dedicated his solution to King Ptolemy, presenting a model in bronze with it a letter and an epigram.[29] Archimedes was Eratosthenes' friend and he, too, worked on the war instrument with mathematics. Archimedes dedicated his book The Method to Eratosthenes, knowing his love for learning and mathematics.[30]
Prime numbers[edit]
Sieve of Eratosthenes: algorithm steps for primes below 121 (including optimization of starting from prime's square).
Main articles: Sieve of Eratosthenes and primality test
Eratosthenes proposed a simple algorithm for finding prime numbers. This algorithm is known in mathematics as the Sieve of Eratosthenes.
In mathematics, the sieve of Eratosthenes (Greek: κόσκινον Ἐρατοσθένους), one of a number of prime number sieves, is a simple, ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite, i.e., not prime, the multiples of each prime, starting with the multiples of 2. The multiples of a given prime are generated starting from that prime, as a sequence of numbers with the same difference, equal to that prime, between consecutive numbers. This is the sieve's key distinction from using trial division to sequentially test each candidate number for divisibility by each prime.
Works[edit]
Eratosthenes was one of the most pre-eminent scholarly figures of his time, and produced works covering a vast area of knowledge before and during his time at the Library. He wrote on many topics — geography, mathematics, philosophy, chronology, literary criticism, grammar, poetry, and even old comedies. Unfortunately, there are only fragments left of his works after the Destruction of the Library of Alexandria.[31]
Titles[edit]
Platonikos
Hermes
Erigone
Chronographies
Olympic Victors
Περὶ τῆς ἀναμετρήσεως τῆς γῆς (On the Measurement of the Earth)[32] (lost, summarized by Cleomedes)
Гεωγραϕικά (Geographika)[33] (lost, criticized by Strabo)
Arsinoe (a memoir of queen Arsinoe; lost; quoted by Athenaeus in the Deipnosophistae)
Ariston (concerning Aristo of Chios' addiction to luxury); lost; quoted by Athenaeus in the Deipnosophistae)[34]
A fragmentary collection of Hellenistic myths about the constellations, called Catasterismi (Katasterismoi), was attributed to Eratosthenes, perhaps to add to its credibility.
See also[edit]
Atlas portal
Eponyms[edit]
Eratosthenes (crater) on the Moon
Eratosthenian period in the lunar geologic timescale
Eratosthenes Seamount in the eastern Mediterranean Sea
References[edit]
Jump up ^ The Suda states that he was born in the 126th Olympiad, (276–272 BC). Strabo (Geography, i.2.2), though, states that he was a "pupil" (γνωριμος) of Zeno of Citium (who died 262 BC), which would imply an earlier year-of-birth (c. 285 BC) since he is unlikely to have studied under him at the young age of 14. However, γνωριμος can also mean "acquaintance", and the year of Zeno's death is by no means definite. Cf. Eratosthenes entry in the Dictionary of Scientific Biography (1971)
Jump up ^ The Suda states he died at the age of 80, Censorinus (De die natali, 15) at the age of 81, and Pseudo-Lucian (Makrobioi, 27) at the age of 82.
^ Jump up to: a b c Roller, Duane W. Eratosthenes' Geography. New Jersey: Princeton University Press, 2010.
Jump up ^ Alfred, Randy (June 19, 2008). "June 19, 240 B.C.: The Earth Is Round, and It's This Big". Wired. Retrieved 2013-06-22.
Jump up ^ Entry ε 2898
Jump up ^ See also Asimov, Isaac. Asimov's Biographical Encyclopedia of Science and Technology, new revised edition. 1975. Entry #42, "Eratosthenes", Page 29. Pan Books Ltd, London. ISBN 0-330-24323-3. This was also asserted by Carl Sagan 31 minutes into his Cosmos episode The Shores of the Cosmic Ocean
^ Jump up to: a b Chambers, James T. "Eratosthenes of Cyrene." Dictionary Of World Biography: The Ancient World (January 1998): 1–3.
Jump up ^ Bailey, Ellen. 2006. "Eratosthenes of Cyrene." Eratosthenes Of Cyrene 1–3. Book Collection Nonfiction: High School Edition.
Jump up ^ Rist, J.M. "Zeno and Stoic Consistency," in Phronesis. Vol. 22, No. 2, 1977.
Jump up ^ Chambers, James T. "Eratosthenes of Cyrene." in Dictionary Of World Biography: The Ancient World January 1998: 1–3.
Jump up ^ Bailey, Ellen. "Eratosthenes of Cyrene," in Eratosthenes of Cyrene 1–3. Book Collection Nonfiction: High School Edition, 2006.
Jump up ^ Chambers, James T. "Eratosthenes of Cyrene" in Dictionary Of World Biography: The Ancient World (January 1998): 1–3.
Jump up ^ Chambers, James T. "Eratosthenes of Cyrene" Dictionary Of World Biography: The Ancient World, January 1998.
Jump up ^ * p439 Vol. 1 William Woodthorpe Tarn Alexander the Great. Vol. I, Narrative; Vol. II, Sources and Studies0. Cambridge: Cambridge University Press, 1948. (New ed., 2002 (paperback, ISBN 0-521-53137-3)).
Jump up ^ Bailey, Ellen. "Eratosthenes of Cyrene." Eratosthenes Of Cyrene (January 2006): 1–3.
Jump up ^ Balasubramaniam, R. (10 August 2017). "Story of the Delhi Iron Pillar". Foundation Books – via Google Books.
Jump up ^ "Astronomy 101 Specials: Eratosthenes and the Size of the Earth". www.eg.bucknell.edu. Retrieved 2017-12-19.
^ Jump up to: a b c d e f "Eratosthenes and the Mystery of the Stades – How Long Is a Stade? – Mathematical Association of America". www.maa.org.
^ Jump up to: a b "How did Eratosthenes measure the circumference of the earth?". 3 July 2012.
Jump up ^ Gow, Mary. "Measuring the Earth: Eratosthenes and His Celestial Geometry, p. 6 (Berkeley Heights, NJ: Enslow, 2010).
Jump up ^ Smith, Sir William. "Eratosthenes", in A Dictionary of Greek and Roman Biography and Mythology. Ann Arbor, Michigan: University of Michigan Library, 2005.
Jump up ^ Morris, Terry R. "Eratosthenes of Cyrene." in Encyclopedia Of The Ancient World. November 2001.
Jump up ^ 2011. "Eratosthenes." Hutchinson's Biography Database 1.
Jump up ^ Eckerman, Chris. Review of (D.W.) Roller 'Eratosthenes' Geography. Fragments Collected and Translated, with Commentary and Additional Material. The Classical Review. 2011.
^ Jump up to: a b Smith, Sir William. "Eratosthenes," in A Dictionary of Greek and Roman Biography and Mythology. Ann Arbor, Michigan: University of Michigan Library, 2005.
Jump up ^ Dicks, D.R. "Eratosthenes," in Complete Dictionary of Scientific Biography. New York: Charles Scribner's Sons, 1971.
Jump up ^ "Ask an Astronomer". Cool Cosmos. Archived from the original on 2014-07-30.
Jump up ^ Greek Scholar's Work Shows Usefulness of Measurement." Manawatu Standard, June 19, 2012., 07, Newspaper Source Plus
Jump up ^ Zhumud, Leonid. Plato as "Architect of Science". in Phonesis. Vol. 43 (3) 1998. 211–244.
Jump up ^ Chondros, Thomas G. Archimedes Life Works and Machines. in Mechanism and Machine Theory. Vol.45(11) 2010. 1766–1775.
Jump up ^ Dicks, D.R. "Eratosthenes", in Complete Dictionary of Scientific Biography. New York: Charles Scribner's Sons, 1971.
Jump up ^ Mentioned by Hero of Alexandria in his Dioptra. See p. 272, vol. 2, Selections Illustrating the History of Greek Mathematics, tr. Ivor Thomas, London: William Heinemann Ltd.; Cambridge, MA: Harvard University Press, 1957.
Jump up ^ Dicks, D.R. "Eratosthenes," in Complete Dictionary of Scientific Biography. New York: Charles Scribner's Sons, 1971.
Jump up ^ Smith, Andrew. "Athenaeus: Deipnosophists – Book 7". www.attalus.org.
Further reading[edit]
Aujac, G. (2001). Eratosthène de Cyrène, le pionnier de la géographie. Paris: Édition du CTHS. 224p.
Bulmer-Thomas, Ivor (1939–1940). Selections Illustlating the History of Greek Mathematics. Cambridge, MA: Harvard University Press.
Diller, A (1934). "Geographical Latitudes in Eratosthenes, Hipparchus and Posidonius". Klio. 27 (3): 258–269.
Dorofeeva, A. V. (1988). "Eratosthenes (ca. 276–194 B.C.)". Mat. V Shkole (in Russian) (4): i.
Dutka, J. (1993). "Eratosthenes' measurement of the Earth reconsidered". Arch. Hist. Exact Sci. 46 (1): 55–66. doi:10.1007/BF00387726.
El'natanov, B. A. (1983). "A brief outline of the history of the development of the sieve of Eratosthenes". Istor.-Mat. Issled. (in Russian). 27: 238–259.
Fischer, I (1975). "Another look at Eratosthenes' and Posidonius' determinations of the Earth's circumference". Quarterly Journal of the Royal Astronomical Society. 16: 152–167. Bibcode:1975QJRAS..16..152F.
Fowler, D. H.; Rawlins, Dennis (1983). "Eratosthenes' ratio for the obliquity of the ecliptic". Isis. 74 (274): 556–562. doi:10.1086/353361.
Fraser, P. M. (1970). "Eratosthenes of Cyrene". Proceedings of the British Academy. 56: 175–207.
Fraser, P. M. (1972). Ptolemaic Alexandria. Oxford: Clarendon Press.
Fuentes González, P. P., "Ératosthène de Cyrène", in R. Goulet (ed.), Dictionnaire des Philosophes Antiques, vol. III, Paris, Centre National de la Recherche Scientifique, 2000, pp. 188–236.
Geus K. (2002). Eratosthenes von Kyrene. Studien zur hellenistischen Kultur- und Wissenschaftgeschichte. München: Verlag C.H. Beck. (Münchener Beiträge zur Papyrusforschung und antiken Rechtsgeschichte. Bd. 92) X, 412 S.
Goldstein, B. R. (1984). "Eratosthenes on the "measurement" of the Earth". Historia Math. 11 (4): 411–416. doi:10.1016/0315-0860(84)90025-9.
Gulbekian, E. (1987). "The origin and value of the stadion unit used by Eratosthenes in the third century B.C". Archive for History of Exact Sciences. 37 (4): 359–363. doi:10.1007/BF00417008 (inactive 2017-11-01).
Honigmann, E. (1929). Die sieben Klimata und die πολεις επισημοι. Eine Untersuchung zur Geschichte der Geographie und Astrologie in Altertum und Mittelalter. Heidelberg: Carl Winter's Universitätsbuchhandlung. 247 S.
Knaack, G. (1907). "Eratosthenes". Pauly–Wissowa VI: 358–388.
Manna, F. (1986). "The Pentathlos of ancient science, Eratosthenes, first and only one of the "primes"". Atti Accad. Pontaniana (N.S.) (in Italian). 35: 37–44.
Muwaf, A.; Philippou, A. N. (1981). "An Arabic version of Eratosthenes writing on mean proportionals". J. Hist. Arabic Sci. 5 (1–2): 147–174.
Nicastro, Nicholas (2008). Circumference: Eratosthenes and the ancient quest to measure the globe. New York: St. Martin's Press. ISBN 0-312-37247-7.
O'Connor, John J.; Robertson, Edmund F., "Eratosthenes", MacTutor History of Mathematics archive, University of St Andrews .
Marcotte, D. (1998). "La climatologie d'Ératosthène à Poséidonios: genèse d'une science humaine". G. Argoud, J.Y. Guillaumin (eds.). Sciences exactes et sciences appliquées à Alexandrie (IIIe siècle av J.C. – Ier ap J.C.). Saint Etienne: Publications de l'Université de Saint Etienne: 263–277.
McPhail, Cameron (2011). Reconstructing Eratosthenes' Map of the World: a Study in Source Analysis. A Thesis Submitted for the Degree of Master of Arts at the University of Otago. Dunedin, New Zealand.
Pfeiffer, Rudolf (1968). History of Classical Scholarship From the Beginnings to the End of the Hellenistic Age. Oxford: Clarendon Press.
Rawlins, D. (1982). "Eratosthenes' geodesy unraveled : was there a high-accuracy Hellenistic astronomy". Isis. 73 (2): 259–265. doi:10.1086/352973.
Rawlins, D. (1982). "The Eratosthenes – Strabo Nile map. Is it the earliest surviving instance of spherical cartography? Did it supply the 5000 stades arc for Eratosthenes' experiment?". Arch. Hist. Exact Sci. 26 (3): 211–219.
Rawlins, D. (2008). "Eratosthenes's large Earth and tiny universe" (PDF). DIO. 14: 3–12.
Roller, Duane W. (2010). Eratosthenes' Geography: Fragments collected and translated, with commentary and additional material. Princeton: Princeton University Press. ISBN 978-0-691-14267-8.
Shcheglov, D.A. (2004/2006). "Ptolemy's System of Seven Climata and Eratosthenes' Geography". Geographia Antiqua 13: 21–37.
Shcheglov, D.A. (2006). "Eratosthenes' Parallel of Rhodes and the History of the System of Climata". Klio. 88 (2): 351–359. doi:10.1524/klio.2006.88.2.351.
Strabo (1917). The Geography of Strabo. Horace Leonard Jones, trans. New York: Putnam.
Taisbak, C. M. (1984). "Eleven eighty-thirds. Ptolemy's reference to Eratosthenes in Almagest I.12". Centaurus. 27 (2): 165–167. Bibcode:1984Cent...27..165T. doi:10.1111/j.1600-0498.1984.tb00766.x.
Thalamas, A. (1921). La géographe d'Ératosthène. Versailles.
Wolfer, E. P. (1954). Eratosthenes von Kyrene als Mathematiker und Philosoph. Groningen-Djakarta.
External links[edit]
Wikimedia Commons has media related to Eratosthenes.
Wikisource has original works written by or about:
Eratosthenes
English translation of the primary source for Erastothenes and the size of the earth at Roger Pearse.
Bernhardy, Gottfried: Eratosthenica Berlin, 1822 (PDF) (Latin/Greek), Reprinted Osnabruck 1968 (German)
Eratosthenes' sieve in Javascript
About Eratosthenes' methods, including a Java applet
How the Greeks estimated the distances to the Moon and Sun
Measuring the Earth with Eratosthenes' method
List of ancient Greek mathematicians and contemporaries of Eratosthenes
New Advent Encyclopedia article on the Library of Alexandria
Eratosthenes' sieve in classic BASIC all-web based interactive programming environment
Following in the footsteps of Eratosthenes : project fr:La main à la pâte.
Open source Physics Computer Model about Eratosthenes estimation of radius and circumference of Earth
Eratosthenes, video
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Legal status of tattooing in the United States
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In the United States, there is no federal law regulating the practice of tattooing. However, all 50 states and the District of Columbia have statutory laws requiring a person receiving a tattoo be 18 years or older. This is partially based on the legal principle that a minor cannot enter into a legal contract or otherwise render informed consent for a procedure. Most states permit a person under the age of 18 to receive a tattoo with permission of a parent or guardian, but some states outright prohibit tattooing under a certain age regardless of permission, with the exception of medical necessity (such as markings placed for radiation therapy).
In all jurisdictions, individual tattooists may also choose to place additional restrictions based on their own moral feelings, such as refusing any clients under a specific age even with parental consent despite it being legal, or limiting the type and/or location of where they are willing to tattoo (such as refusing any work around intimate parts of the body). They may additionally refuse to perform specific artwork, including if they merely find it inappropriate or offensive, or refuse to work on a client that they suspect may be intoxicated. Artists sometimes claim their personal business restrictions are a matter of law even when it is not true, so as to avoid arguments with clients.
Legal Restrictions on Tattooing & Body Modification by State
State
Minimum Age (with parent/guardian consent)
Form of Parent/Guardian Consent
Notes & Exceptions
Intoxicated/Impaired Individuals
Area of Law/Type of Penalty
Licensing & Regulation
Relevant Statute(s) and Rules
Alabama
none specified[1] informed written consent, signed in presence of the artist/service provider or their agent[1][2] Ear piercing exempt from Alabama body art laws & rules.[2][3] cannot be tattooed, branded, or pierced[1][2] Violations of Alabama Tattoo & Body Piercing laws are a class c misdemeanor, punishable by a fine of up to $100 and/or up to 90 days imprisonment for each violation.[4] Alabama's Department of Public Health licenses Tattoo, body piercing, & body art facilities, and sets standards for their operation.[5]
Individual tattoo artists, piercers, etc. require a permit.[2]
Ala. Code § 22-1-17A[3][6]
State Board of Health rules on Body Art Practices and Facilities.[2]
Alaska
18 (piercing excepted)[7] parent/guardian must both give written consent and be present during the procedure[7] Minors may be pierced with parental consent [7] no restriction[8] Alaska Stat. § 08.13.217[7]
Arizona
none specified[9] parent/guardian must be physically present[9] Minors can have their ears pierced with the written or verbal consent of a parent/guardian, can have procedures prescribed by a health care provider.[9] no restriction[9] Violations are a class 6 felony.[9] There are no health regulations for tattoo shops in Arizona. [10] Tattoos. Ariz. Rev. Stat. § 13-3721[9]
Arkansas
16[11] parent/guardian must both give written consent and be physically present[12] none specified[12] cannot be tattooed, branded, or pierced.[12] Ark. Stat. Ann. §§ 20-27-1501 et seq.[12]
California
18 (piercing excepted)[13][14] parent/guardian must be present during procedure Minors may have their ears pierced with parental consent.[15] They may be pierced elsewhere with parental consent, but not on their genitals or nipples. Permanent cosmetics are permitted on nipples, when done by a cosmetic technician and with parental consent.[14] no restriction Body Art. Cal. Health & Safety Code § 119300 to 119328[6][15]
Cal. Penal Code 653[13]
Colorado
18 Express written consent of parent or guardian[16] parent/guardian must be present during procedure Persons who are "noticeably impaired" cannot have body art procedures[16] Body Artists. Colo. Rev. Stat. § 25-4-2101 to 2103[6]
Connecticut
none specified permission of parent or guardian[17][18] Minors may have their ears pierced without parental consent[17]
Tattooing may only be performed by physicians, or by registered nurses & technicians working under the supervision of a physician.[18]
no restrictions Conn. Gen. Stat. §19a-92a,[6][18]
Conn. Gen. Stat. §19a-92g[6][17]
District of Columbia
18 (ear piercing excepted)[19] written consent (for ear piercing)[19] minors may have their ears pierced with written parental consent[19] no restrictions[19] Body Artists. DC Code § 47-2853.76d[6][19]
Delaware
18[20] notarized written consent to the specific procedure[20] none specified[20] cannot be tattooed, branded, or body pierced[20] Del. Code Ann. Title 11, Ch 5 §1114(a)[6][20]
Del. Code Ann. Title 16, Ch 2 §122(3)(w)[6]
Florida
16[21] Must be accompanied by parent or guardian who can show proof of guardianship, both parent and minor must present photo ID, and a consent form must be notarized.[21] for medical or dental purposes by a licensed physician or dentist[21] no restriction Fla. Stat. § 381.00787[6]
Fla. Stat. §381.0775 et seq.[6] Fla. Stat. §381.075[6]
Georgia
18 (body piercing excepted)[22] For body piercing: prior written consent of parent/guardian[23] Minors may have their ears pierced without parental consent[23]
Minors may be body pierced with written parental consent[23]
Physicians and Osteopaths or technicians working under their direct supervision can give minors tattoos or produce scars for "medical or cosmetic purposes."[22]
No one can be tattooed within an inch of their eye socket.[24]
no restrictions Ga. Code § 31-40-1 to 31-40-10[6]
Ga. Code § 16-12-5[6]
Ga. Code § 16-5-71.1[6]
Hawaii
none specified[25] written consent of parent/guardian[25] Hawaii Rev. Stat. § 321-13[6]
Hawaii Rev. Stat. § 321-379[6]
Idaho
14[26] Written informed consent of parent/guardian, executed in the presence of the person performing the procedure, or their agent/employee.[26] Minors can have their ears pierced without parental consent.[6]
Piercing for medical procedures exempt.[6]
no restrictions Idaho Code § 18-1523 (2004)[6]
Illinois
18 (body piercings excepted)[27] Written consent of parent/guardian (for body piercings)[27] Minors cannot be present in places where tattooing or body piercing is done except in the presence of a parent/guardian.[27]
Minors can have their ears pierced without parent/guardian consent.[27]
no restrictions Ill. Stat. 720 §5/12-10.1[6]
Ill. Stat. 410 §54/1 t0 54/999[6]
Indiana
none specified[28] Parent/guardian must both be present during the procedure and provide written permission.[28] Minors may have their ears pierced without parental consent.[28] no restrictions Ind. Code Ann. §35-42-2-7[6][28]
Ind. Code Ann. §16-19-3-4.1[6][29]
Iowa
18[30] n/a - minors cannot be tattooed[30] Body piercing is not regulated by state law in Iowa, however county or city laws may regulate it.[31] no restrictions Iowa Code §135.37[6]
Iowa Code §135.37[6]
Kansas
16[32] Parent/guardian must give written consent, written consent must be notarized, and the parent/guardian must be present during the procedure.[32] Minors may have their ears pierced[33] no restrictions Kan. Stat. Ann. §65-1953[6]
Kan. Stat. Ann. §65-1940 to 65-1954[6]
Kentucky
16 Parent/guardian must give written, notarized consent.[34] none (ear piercing also requires written, notarized consent).[35] Ky. Rev. Stat. §211.760[6]
Ky. Rev. Stat. §194A.050; 211.760[6]
Louisiana
16 Parent or guardian must consent, and their presence during the procedure is required for tattoos, but not for body piercings.[36] no restrictions La. Rev. Stat. Ann. §14:93.2[6]
La. Rev. Stat. Ann. § 40:2831 et seq.[6]
Maine
18 (piercings excepted)[37] For body piercings: prior legal consent of parent/guardian.[38] Maine law does not require parent/guardian consent for ear piercing[39] no restrictions Me. Rev. Stat. Ann. Title 32 § 4201-4329[6]
Maryland
none specified[40] written consent, which must be retained for 3 years. Piercing of the ear lobe is exempt[41] no restriction Health Department Regulations A license is not required, but any person performing tattooing or piercing must comply with all the relevant health code regulations[40] COMAR 10.06.01.02 and 10.06.01.06
Massachusetts
18 (except body piercing other than genitalia)[42] Minors cannot receive tattoo, branding, or scarification at all. For piercing of areas other than the genitalia, a parent may provide identification and sign the consent document.[42] Physicians performing for treatment reasons and ear piercing are exempt from regulation.[42] no restriction County Health Department regulates body art, but all follow a model regulation set by the state. Artists must have a permit, and establishments are subject to following specific health department regulations[42] Model Regulations for Body Art Establishments[42]
Michigan
none specified[43] Written, informed consent of parent/guardian. Must be executed (signed) in the presences of the person performing the procedure, or their employee.[43] cannot be tattooed, branded, or body pierced.[43] Mich. Comp. Laws § 333.13101 to 333.13112[44]
Minnesota
18 (piercings excepted)[45] For piercings: presence of parent/guardian and a consent form are both required.[45] Minors cannot be tattooed. They can be body pierced, but not in the nipples or genitals. Branding, scarification, suspension, subdermal implantation, microdermal, and tongue bifurcation are also prohibited for minors.[45] cannot have any body art procedures.[45] Minnesota Statutes 146B.07[45]
Mississippi
18 n/a - minors cannot be tattooed piercing also prohibited except for the outer perimeter or lobe of the ear. no restriction Miss.Code Ann. §73-61
Missouri
none specified[46] For both piercings and tattoos, parent/guardian must sign written consent in the presence of the person performing the procedure, or their employee.[46] cannot be tattooed, branded or body pierced.[46] violations are a misdemeanor, punishable with a fine of up to $500 (more for repeat offenses)[46] The state director of the division of professional registration licenses all body artists, and sets rules & regulations for the operation of body art establishments.[47] Mo. Rev. Stat. §324.520[46]
Montana
none specified[48] Parent or guardian must give "explicit, in-person" consent.[48] Montana administrative rules also require that the parent/guardian be present throughout the procedure.[49] cannot be tattooed or body pierced.[49] A person who tattoos or body pierces a minor without parent/guardian consent is guilty of Unlawful transactions with children, punishable by a fine of up to $500 or six months in county jail, or both.[48]
Other violations of Montana body art laws are a misdemeanor, with punishments ranging from a fine of $50–100 for a first offense to $300 and/or 90 days in county jail for third and subsequent offense.[50]
Montana's Department of Health and Safety licenses and regulates body art establishments.[51] Local health boards may establish their own licensing and regulatory schemes in lieu of the state department's.[52] Mont. Code Ann. §45-5-623[48]
Mont. Code Ann. §50-48-101 to 110[53]
Mont. Code Ann. §50-48-201 to 209[54]
Nebraska
none specified[55] written consent, must be present during procedure Piercing does not include the external parts of the ear[56] No restrictions Nebraska Revised Statute 38-1008, 38-1009, 38-1010, 38-1011, 38-1060, 38-10,165
Nevada
n/a[6] n/a[6] Nevada state law does not regulate tattoos, piercings, or other body art. Regional health districts may regulate the practice. The southern Nevada Health District, for example, does not allow minors below the age of 14 to be tattooed, and requires a parent or guardian to provide written consent and be present during the procedure for 14- to 18-year-olds. The southern Nevada district also bans tattoos on those who are under the influence of drugs or alcohol.[57] n/a[6] n/a[6]
New Hampshire
18 (piercings excepted)[58] For piercings: parent/guardian must both sign written consent and be present during the procedure. Minors cannot be tattooed or branded. They may be pierced with parent/guardian consent.[58] Minors may have their ears pierced without parent/guardian consent.[59] no restrictions N.H. Rev. Stat. Ann. § 314[6]
New Jersey
none specified[60] written permission of parent/guardian is required for tattooing and body piercing.[60] Parent/guardian must be present and sign written consent for ear piercings as well as for body piercing. Minors cannot have genital piercings even with parent/guardian consent.[61] no restrictions no restrictions no restrictions N.J. Stat. Ann. §2C:40-21
N.J.A.C. 8:27-1 et seq.
N.J. Stat. Ann. §26-1A-7[6]
New Mexico
none specified[62] Written consent and presence during the procedure. Medical procedures by physicians and ear piercing are exempt[63] no restrictions Administrative Regulations set by the Board of Barbers and Cosmetologists All body artists are required to be licensed through the state, and are subject to regulations from the Board Body Art Safe Practices Act. N.M. Stat. Ann. § 61-17B et seq.,[63] N.M. Administrative Code 16.36 et seq.[62]
New York
18 (piercings excepted) [64] Minors cannot be tattooed with parent/guardian consent. For piercings: parent/guardian must give written consent, signed in presence of the body artist or body art establishment owner.[65] Written parental consent is not required for ear piercings.[66] no restrictions New York Public Health Law, Article 4-A[6][65]
N.Y. Public Health Law §460-466[6]
N.Y. Penal Law § 260.21[6]
North Carolina
18 (piercings excepted)[67] Minors cannot be tattooed, even with parent/guardian consent. "Prior consent" of parent/guardian is required for body piercings.[67] Minors may have their ears pierced without parental consent.[67] no restrictions N.C. Gen. Stat. § 130A-283[68]
N.C. Gen. Stat. § 14-400[69]
North Dakota
none specified[70] Parent/guardian must be present during the procedure and give written consent.[70] cannot get tattoos or other body art[71] N.D. Code §23-01-35[72]
N.D. Code §12.1-31-13[6][70]
Ohio
none specified[73] Parent/guardian must both sign a consent form and "appear in person at the business at the time the procedure is performed."[73] none - parent/guardian consent and presence is also required for ear piercings.[73] no restrictions Ohio Rev. Code Ann. §3730.01 to 3730.99[6][73]
Oklahoma
18 (piercings excepted)[74] Minors cannot be tattooed, even with parent/guardian consent. For piercings: parent/guardian must both give written consent and be present during the procedure.[74] Ear piercing is not governed/restricted by Oklahoma law.[74]
Scleral tattooing is illegal.[74]
cannot be tattooed.[74] Okla. Stat. Title 21 §842.1, 842.2, 842.3[6]
Oregon
18 (piercings excepted)[75] For piercings: parent/guardian must give written consent and present proof of identity.[76][77] Minors can only be tattooed with the authorization of a physician.[75]
Scarification and dermal implants are prohibited[78]
Persons with sunburns or skin diseases or disorders cannot be tattooed or pierced.[75][77]
Minors cannot be pierced on the genitals or nipples, even with parent/guardian consent.[76]
Piercing on testes, deep shaft (corpus cavernosa), uvula, eyelids and sub-clavicle are all prohibited.[77]
cannot be tattooed or pierced.[75][77] Violations of Oregon body art laws are a misdemeanor.[79]
Violations of Health Licensing Office rules are fined (various amounts).[80]
Both body artists and body art establishments are licensed by the Oregon Health Licensing Office.[81] With the assistance of a Board of Body Art practitioners, the Health Licensing Office also sets detailed rules and regulations for body art.[82] Or. Rev. Stat. § 690.350 et seq
Or. Rev. Stat. § 690.401 to 410
Or. Rev. Stat. § 679.500[6]
Health Licensing Office Rules, especially Divisions 900, 905, 915 & 920[82]
Pennsylvania
none specified[83] parent/guardian must both give consent and be present during the procedure.[83] no restrictions Pa. Cons. Stat. Title 18 § 6311[6][83]
Rhode Island
18 (piercings excepted)[84][85] Minors cannot be tattooed.[84] For piercings: parent/guardian must both give consent and be present for the procedure.[85] no restrictions Tattooing a minor is a criminal offense (misdemeanor), maximum fine $300[84] Tattooing and body art are regulated by the department of health, artists must be registered.[85] RI Gen L § 23-1-39[85]
RI Gen L § 11-9-15[84]
South Carolina
18 (piercings excepted)[86][87] Minors cannot be tattooed.[86] For piercings: parent/guardian must either give written, notarized consent or be present during the procedure.[87] Head, face, and neck tattoos are illegal.[86] cannot be tattooed or body pierced[86][88] Under the Health Code, persons violating restrictions on tattooing and body piercing are guilty of a misdemeanor. Upon conviction, they may be fined up to $2500 or imprisoned for a year, or both.[86][87] Tattoo and body piercing facilities are regulated by the Department of Health and Environmental Control. Tattoo facilities are licensed, body piercing facilities require a permit, and artists must be registered[89][90] S.C. Code § 44-32-10 to 44-32-120[88]
S.C. Code § 44-34-10 to 44-34-110[91]
South Dakota
none specified[92] Parent/guardian must sign a consent form.[92] cannot be tattooed or pierced[93][94] Tattooing a minor is a class 2 misdemeanor.[95] Tattooing and Body Piercing are governed by Department of Health rules.[96] Municipalities may set stricter regulations, and have the authority to license body artists.[95] S.D. Codified Laws Ann. §26-10-19[92]
S.D. Laws § 9-34-17[95]
Tennessee
18 (piercings excepted)[97][98] For tattoos that cover an existing tattoo (see Notes & Exceptions): Parent/Guardian must be present during the procedure.[97]
For piercings: Parent/Guardian must give written consent, be present during the procedure.[98]
Minors over the age of sixteen may be tattooed to cover up an existing tattoo, with parent/guardian consent.[97] Minors who lie about their age to be tattooed are guilty of a "delinquent act," are required to pay a fine of $50–$250 and serve at least 20 hours of community service.[97]
Tattooing a minor is a class a misdemeanor,[97] breach of body piercing law is a class b misdemeanor.[99]
Tattoo artists and body piercers are licensed by the state department of health, tattoo shops require a certificate from the local health department.[100][101] Tenn. Code §§ 62-38-201-310[102]
Texas
18 (piercings excepted)[103][104] For body piercings and for covering existing tattoos (see Notes & Exceptions): Parent/guardian must be present during procedure, sign an affidavit[103][104] Ear piercing is exempt from body piercing statutes[105]
Minors may, with parent/guardian consent, be tattooed to cover up an existing tattoo that has offensive, gang-related, or drug-related content, or "other words, symbols, or markings that the person's parent or guardian considers would be in the best interest of the person to cover"[103]
Tongue splitting prohibited.[106]
Cannot be tattooed or pierced.[103][107] Misrepresenting one's age to get a tattoo, or to get a piercing without parental consent, is a class B misdemeanor[103][104]
Violations of Texas tattooing & body piercing statute are a Class A misdemeanor, with each day of violation counting as a separate offense.[108]
Health Commissioner can impose a fine of up to $5000 on artists & shops for each violation[109]
Tattoo artists and body piercers must be registered with the Texas Department of Health[110] Tattoo and body piercing shops must be licensed[111] Texas Health and Safety Code Ann. §146[112]
Utah
none specified[113] Parent/guardian must be present during the procedure, give written permission, and provide proof of identity.[113] Parental consent not required by law for ear piercing[113] Tattooing or body piercing a minor is a class b misdemeanor, shops can be fined up to $1000 for each violation.[113] Utah Code Ann. §76-10-2201[6]
Vermont
none specified[114] Tattooists and body piercers must be registered.[114] Vt. Stat. Ann. Title 26 §4101 to 4109[6][115]
Virginia
none specified[116] Parent/Guardian must be present during the procedure.[116] Virginia law does not limit minors' ability to have their ears pierced.[116] Tattooing or body piercing a minor is a class 1 misdemeanor[116] Tattoo artists and body piercers, as well as the operators of tattoo and body piercing establishments, are licensed by the state.[117][118]
Civic/local governments are empowered to regulate tattoo and body piercing establishments.[119]
Va. Code §54.1-700 et seq.
Va. Code §15.2-912
Va. Code § 18.2-371.3.[6]
Washington
18 (piercings excepted)[120] cannot have tattoos, body piercings, or other body art.[121] Wash. Code §26.28.085[120]
Wash. Code § 18.300[122]
Wash. Code § 246-145-001 to 060[123]
Wash. Code §70.54[124]
West Virginia
none specified[125] parent/guardian must give written consent[125] All persons receiving a tattoo must attest they are not under the influence of drugs or alcohol.[125] Tattoo shop owners and body piercing studios must register with local board of health, and must be registered as a business with the state[126][127][128] W. Va. Code §16-37
W. Va. Code §16-38[129]
Wisconsin
18[130] n/a (tattooing not permitted under 18) Physicians in the course of professional practice are exempt no restriction Tattooing a minor or even offering to do so is subject to a fine of up to $200 Artists must be satisfy training requirements and obtain a license from the health department. Establishments are subject to inspection and licensing[131] Wisconsin Statutes & Annotations 948.70 and 252.235
Wyoming
none specified[132] Verbal consent, presence during procedure.[132] Procedures performed under the supervision of a licensed physician, and piercing of the ear, do not count as body art for this law. no restriction[132] Misdemeanor criminal offense, punishable by fine and prison time[132] Wyoming Statute §14-3-107[132]
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This page was last edited on 23 August 2018, at 21:24 (UTC).
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