81. ACADEMIA_INDICE He appears to have used Aristotle s method for the square root of 2. Theodorusof cyrene (390 BC). Theaetetus of Attica (414 BC 369 BC). http://descartes.cnice.mecd.es/ingles/maths_workshop/A_history_of_Mathematics/Gr

PLATO: THE ACADEMY History 1. BACKGROUND TO THE PERIOD Aristotle and Plato in the centre of Raphael's painting "The School in Athens". The Vatican Museum. The Peloponnesian Wars took place in the IVth century B.C. Sparta fought against Athens and behind them other Greek towns followed them into warfare. Sparta called on Persia to help them keep control of the towns they had occupied. Athens and Thebes became allies and together managed to defeat Sparta. King Philip of Macedon took advantage of the situation and became ruler of Greece. His reign lasted from 360 B.C. to 336 B.C. when, upon his death, his son Alexander took the throne. Alexander the Great was responsible for the invasion of the Persian empire, which included Syria, Palestine, Egypt, Mesopotamia and Iran. This century began with the death of Socrates (399 B.C.) The two great philosophers Aristotle and Plato , one of Socrates students and admirers also belonged to this period along with Archytas. Aristotle was Alexander the Great's private tutor and instilled in him the superiority of the Hellenic culture and encouraged him to go East and extend his empire. Plato managed to bring the greatest thinkers of the time together at his Academy in Athens. His contributions to mathematics include his rigorous method of justifying solutions through logical reasoning, his

82. Cyrene And The Cyrenaica cyrene was the hometown of several famous Greek scholars and scientists. The mathematicianTheodorus (c.465399) developed the theory of irrational numbers (eg http://www.livius.org/ct-cz/cyrenaica/cyrenaica.html

home index ancient Greece Cyrene and the Cyrenaica Cyrene: Greek city in Libya, modern Shahhat. Cyrenaica: the country surrounding Cyrene. Cyrene was founded in c.630 BCE as a colony of the Greek island town Thera, which had become too crowded. The first colonists settled at an island called Platea in front of the Libyan coast (modern Bomba). Later, they occupied a coastal strip called Aziris, and finally, after concluding a treaty with the native Libyans, they founded the town Cyrene. The leader of the settlers was Aristoteles, but he was called Battus. (Which means 'stammerer' in Greek, but is probably a Libyan royal title.) In the following centuries, Battus' descendants ruled Cyrene. Aristoteles Battus I c.631-c.599 Arcesilas I c.599-c.583 Battus II the Blessed c.583-c.560 Arcesilas II the Tough c.560-c.550 Battus III the Lame c.550-c.530 Arcesilas III c.530-c.514 Battus IV the Fair c.514-c.470 Arcesilas IV c.470-c.440 Although Cyrene was founded after a treaty with the natives, the relations between the Greeks and Libyans were often strained, and the settlers sometimes felt threatened. As a consequence, during the reign of Battus II, new settlers were invited from the homeland. They received Libyan land, which caused great resentment. The Libyans requested the Egyptian king Apries to assist them in a war against the Greeks, but the pharaoh was defeated (570).

83. Irrational Numbers This contradicts the notion of an atom as an indivisible basic unit. Theodorusof cyrene (465398 BC) was a teacher of Plato and Thaetetus. http://www.mlahanas.de/Greeks/Irrational.htm

Irrational numbers Prior Analytics 1.23.41a26 and 1.44.55.a37), and is clearly older than him. Pythagoras and his students believed the essential unity of things was not in a physical substrate. For them, the one thing that formed the substrate of all things in the universe was number and numerical relations. According to the theorem of Pythagoras the sum of the squares on the sides of a right-angled triangle is equal to the square of the hypotenuse. If we have a right-angled triangle whose sides are equal, the square of the hypotenuse is twice the square of one of the sides. The problem is that the square of one whole number cannot be twice the square of another. The hypotenuse cannot be a whole number whatever the length of the two equal sides. If the adjacent and opposite sides contain the same number of atoms then the hypotenuse must contain one incomplete atom. This contradicts the notion of an atom as an indivisible basic unit. Theodorus of Cyrene (465-398 BC) was a teacher of Plato and Thaetetus. He provided the proof of the irrationality of all integer numbers between 3 and 17 except the square numbers 4, 9 and 16 (the case for n = 2 was well-known before him). His contribution to Mathematics is part of Euclid'd Element, Book X and XIII. Hippasus also showed that the ratio (diagonal / side) of a regular Pentagon cannot be expressed as a ratio of integers.

84. Livia Giacardi-pubblicazioni Translate this page Pubblicazioni di Livia Giacardi. A. Articoli e Libri 1977 On Theodorusof cyrene’s problem, Arch. Int. Hist. Sci., 27, 101, pp. 231-236. http://www2.dm.unito.it/paginepersonali/giacardi/pubbli.htm

Pubblicazioni di Livia Giacardi A. Articoli e Libri On Theodorus of Cyrene’s problem, Arch. Int. Hist. Sci., 27, 101, pp. 231-236. Un’anticipazione del metodo di risoluzione approssimata di un’equazione f(x)=g(x) per via geometrica in un trattato del sec. XVII , Atti Acc. Sci. Torino, 111, pp. 101-107. Il calcolo del volume del tronco di piramide nella matematica egizia (Discussione sulle ipotesi più importanti già proposte), Atti Acc. Sci. Torino, 111, pp. 441-453 (con T. Viola). Saggio su un possibile calcolo dei volumi di alcuni poliedri nella matematica egizia, Atti Ac. Sci. Torino, 111, pp. 523-537. (con T. Viola). Le possibili origini di un teorema di aritmetica noto ai Greci, alla luce dei rapporti con le civiltà che li hanno preceduti, Atti Ac. Sci. Torino, 112, pp. 185-191, (con T. Viola). Contribution à l’interprétation d’un passage du dialogue "Théétète" de Platon, concernant Théodore de Cyrène, C. R. Acad. Sci. Paris, 287, pp. 41-46, (con T. Viola e C.S. Roero). La matematica delle civiltà arcaiche (Egitto, Mesopotamia, Grecia)

85. All Sides Of The Story entirely alone. Finally we can say with confidence that theodorus ofCyrene was an atheist from the contents of his work On the Gods. http://www.teachingreligion.com/atheism/history.html

Agnosticism Atheism Buddhism Christianity ... Links History of Atheism Ancient Times Some men have always disbelived in gods or supreme powers. The only problem is that the phenomenon of atheism could not be easily described early on, as primitive languages had no way to symbolize negation, or existence. Hence saying "gods don't exist" would be a daunting task indeed. The arrival of phonetic language changes that. Atheistic views started to emerge in India, then Greece. India : Probably the first sign of skeptic thought comes from the Rig-Veda, a text which is thought to have been written around 1000 BC. The philosophy promoted in it could be said to be atheistic by omission, as shows us this creation hymn : "Who knows for certain? Who shall here declare it? Whence was it born and whence came this creation? The gods were born after this world's creation. Then, who can know from whence it has arisen? None know whence creation has arisen and whether he has or has not produced it. He who surveys it in the highest heaven, he only knows, or happily, he may know not". Around 500 BC, Buddhism, inspired by the Rig-Veda, became a theistic philosophy. Jainism, an atheistic religion, also began around that time.

86. GIGA Chronological Author List "Before 300 BC" general under Alexander (fl. 335 BC) BUY AMAZON BOOK Theodorusof cyrene, Greek philosopher (fl. 340 BC) - BUY AMAZON BOOK http://www.giga-usa.com/gigaweb1/quotes2/quayb300.htm

Home Biographical List Reading List Links ... Quote Links CHRONOLOGICAL AUTHOR LIST Before 300 BC Hammurabi, Babylonian king (c. 1792 BC - 1750 BC) CHECK READING LIST (1) BUY AMAZON BOOK Ikhenathon (Akhenaton), Egyptian king of Egypt and religious reformer (c. 1375 BC - 1358 BC) BUY AMAZON BOOK Solomon, king of the ancient Hebrews and son of David (died c. 930 BC) READ QUOTES (1) BUY AMAZON BOOK Lycurgus, Spartan semi-mythical law-giver (fl. c. 850 BC) READ QUOTES (2) BUY AMAZON BOOK Hesiod, Greek pastoral poet (c. 800 BC - c. 720 BC) READ QUOTES (13) BUY AMAZON BOOK Homer ("Smyrns of Chios"), Greek poet (fl. 750 BC or earlier) READ QUOTES (200) CHECK READING LIST (2) BUY AMAZON BOOK Aristodemus, Messenian semi-legendary ruler of Messenia (reigned c. 731 BC - 724 BC) READ QUOTES (1) BUY AMAZON BOOK Archilochus, Greek poet and satirist (c. 680 BC or 700 BC) READ QUOTES (2) BUY AMAZON BOOK Periander of Corinth, Greek tyrant, one of Seven Sages (665? BC - 585 BC) READ QUOTES (1) BUY AMAZON BOOK Pittacus of Mitylene, Greek one of Seven Sages, statesman, philosopher and poet (c. 652 BC - 569 BC)

87. Relations And Relationships Of Xena, The Warrior Princess As for grandparents, it may be safe to assume that both cyrene s and Atrius Theodorusdied in an attempt to kill Hercules, and Xena killed Estragon with her http://www.whoosh.org/issue24/mcfar1.html

R ELATIONS AND R ELATIONSHIPS OF X ENA, THE W ARRIOR P RINCESS IAXS project #535 By Grant McFarlane 2129 words Kin Kith: The Lovers Petracles Caesar (and M'Lila) ... Biography Relations And Relationships Of Xena, The Warrior Princess Kin One thing we're certain of... well, pretty much... Cyrene is Xena's mother. Xena was born in the small village of Amphipolis to Cyrene, an innkeeper, and Atrius, a warrior. She was one of three children, with two brothers: Toris, the elder, and Lyceus, the younger. She also had a son, Solan, now deceased, with Borias, another warrior. Xena's family history was tragic, but as she is Greek, this is sort of fitting, literarily-speaking. Lyceus was killed when he battled, at Xena's side, against the cruel warlord, Cortese. Xena thought Atrius abandoned his family when she was young. She later learned that Cyrene killed Atrius because he was going to kill Xena. Finally, Gabrielle's daughter, Hope, murdered Solan. Also, there have been questions raised on the issue of Xena's paternity. This writer believes that Xena is indeed Ares' daughter, as this lineage explains many of her abilities and also goes a long way toward explaining the remarkable number of Xena lookalikes that pepper ancient Greece. Presuming Ares' genes are dominant, like the rest of him, any offspring would tend to look like him. Thus, Xena may have at least three, possibly four (the fourth season will tell), half sisters in Meg, Leah, Diana and ?. As for grandparents, it may be safe to assume that both Cyrene's and Atrius' parents are dead. Now, if Xena is Ares' daughter, then Zeus and Hera would be her grandparents.

88. Bryn Mawr Classical Review 2003.02.16 10 The Life of the Spirit and the Flowering of Art. Here too C. s interest in Cyreneis discernible as he concentrates on theodorus, Callimachus, Aristippus http://ccat.sas.upenn.edu/bmcr/2003/2003-02-16.html

Bryn Mawr Classical Review 2003.02.16 . Malden, MA and Oxford: Blackwell, 2003. Pp. xii + 452. ISBN 0-631-22242-1. $34.95 (pb). Reviewed by Michael D. Dixon, University of Southern Indiana (mdixon@usi.edu) Word count: 1877 words Considering the amount of scholarship on the hellenistic world that has been published over the last two decades, one might question the need to produce an English translation of a survey of the hellenistic world that was published originally in French twenty-two years ago. There is much, however, in Chamoux's Hellenistic Civilization that still makes reading it worthwhile. C. presents an attractive picture of the hellenistic world: one that is diverse and complex and deserves to be considered according to its own merits. His hellenistic world is one of significant continuity from the classical period, while at the same time displaying considerable innovation and vitality. It was definitely "not an age of decadence" (p. 393). The book is designed as a general survey of both the history and culture of the hellenistic world from Alexander the Great's ascension to the Macedonian throne to the death of Mark Antony and Cleopatra after the battle of Actium (336-30 B.C). Chapters 1-5 treat the history of the period chronologically as opposed to geographically, while chapters 6-10 deal with conceptual and cultural aspects of the hellenistic world such as monarchy, the city, literature, philosophy, science, and art. Necessarily there is considerable repetition between these two sections, but C. has provided cross-references. The text lacks notes, with the exception of a few random parenthetical citations to primary sources. While it is clearly designed as a survey for an undergraduate course, several problems with this English edition (discussed below) detract from its overall value, leaving any other recent survey of the hellenistic world a preferable option.

89. THEAETETUS The scene opens with Socrates enquiring of the visiting geometer, Theodorusof cyrene, if there were any young men in Athens who had impressed him. http://caae.phil.cmu.edu/Cavalier/80250/Plato/Theatetus/Theat.html

THE PATH OF KNOWLEDGE: THE THEAETETUS The Theaetetus can be considered a Socratic dialogue, since in it we do not arrive at any definitive answers to the questions which are posed. Its central concern is the problem of knowledge, yet its main conclusions all serve to show us what knowledge is not. Be this as it may, the Theaeteus rightfully belongs to the later set of dialogues since it prepares the way for the truly Platonic analyses of knowledge which are found in the Sophist. The Theaeteus, by clearing away many false opinions, allows Plato to introduce his own full-blown theory, a theory which connects the problem of knowledge with the realm of the Forms. Because of this interconnection between the two dialogues, and because the analyses of the Sophist presuppose the negative critiques of the Theaeteus, we shall begin our path of knowledge with the Socratic problem. The dialogue opens with a brief prologue which serves to date the time of the supposed conversation. An introduction then guides the reader into the setting for the discussions which were to have taken place between an aging Socrates and a youthful Theaetetus. It ishere that the dialogue is given its direction through the posing of its central question: "What is the nature of knowledge?" Theaetetus makes three general attempts to answer this question, and his responses form the major divisions of the work. The first attempt tries to equate knowledge with sense perception; the second speaks of knowledge as true judgement (but how do we know that a judgement is true?); the third response augments the second by saying that knowledge is true Judgement accompanied by an explanation. Yet Socrates is able to show Theaetetus that each attempt to arrive at an absolute answer to the problem of knowledge is fatally flawed. In the end, we are left with an awareness of our ignorance concerning the nature of knowledge (and the way is prepared for the more thoroughgolng analyses of the Sophist).

90. ÇáÍÇÏ ÏÑÏæÑÇä ÊãÏä åáäی The summary for this Arabic page contains characters that cannot be correctly displayed in this language/character set. http://www.kaafar.com/maghale/alhad/helen epikor.htm

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91. ”ŠwŽj The summary for this Japanese page contains characters that cannot be correctly displayed in this language/character set. http://www2m.biglobe.ne.jp/~m-souda/mysouda/math/smith/chapter3/math7.htm

”ŠwŽj [ƒtƒBƒƒ‰ƒIƒX] [ƒGƒŠƒX‚̃qƒbƒsƒAƒX] [ƒƒgƒ“Aƒtƒ@ƒGƒCƒmƒXAƒGƒEƒNƒeƒ‚ƒ“] [Exhaustion‚Ì•û–@] ... [ƒAƒeƒl‚̃eƒAƒCƒeƒgƒX] ƒtƒBƒƒ‰ƒIƒX A”ނ̊֐S‚͐”Šw‚Æ‚¢‚¤‚æ‚è“NŠw‚É‚ ‚Á‚½BƒOƒmƒ‚ƒ“‚ð•`ŽÊ‚µ‚Đ”Šw‚Ì•ª–ì‚ɐG‚ê‚Ä‚Í‚¢‚邪B ƒGƒŠƒX‚̃qƒbƒsƒAƒX ƒƒgƒ“Aƒtƒ@ƒGƒCƒmƒXAƒGƒEƒNƒeƒ‚ƒ“ @ƒsƒ^ƒSƒ‰ƒX‚ƃvƒ‰ƒgƒ“‚ÌŠÔ‚ÌŽž‘ãAƒAƒeƒl‚ł͐”Šw“I“V•¶Šw‚É”ñí‚Ȋ֐S‚ª‚ ‚Á‚½‚±‚Æ‚ªAŽOl‚Ì“V•¶ŠwŽÒAƒƒgƒ“Aƒtƒ@ƒGƒCƒmƒXAƒGƒEƒNƒeƒ‚ƒ“ ‚Ì’˜ì‚Ì’†‚ÉŒ©o‚¹‚éB‚µ‚©‚µA”Þ‚ç‚̍vŒ£‚ð‹æ•Ê‚·‚邱‚Æ‚Í‚Å‚«‚È‚¢B“NŠwŽÒ‚̃eƒIƒtƒ‰ƒXƒgƒX‚Í ƒGƒOƒ][ƒXƒ`ƒ‡ƒ“‚Ì•û–@ ƒAƒ“ƒeƒBƒtƒHƒ“‚ƃGƒOƒ][ƒXƒ`ƒ‡ƒ“‚Ì•û–@ @ƒAƒŠƒXƒgƒeƒŒƒX‚́AƒAƒ“ƒeƒBƒtƒHƒ“ ƒAƒ‹ƒLƒ ƒ^ƒX @ƒvƒ‰ƒgƒ“‚ÌŽž‘ãA’˜–¼‚ȃsƒ^ƒSƒ‰ƒX”h‚Ì“NŠwŽÒAƒAƒ‹ƒLƒ ƒ^ƒX ‚́A”ŠwŽÒA«ŒRA­Ž¡‰ÆA”Žˆ¤Žå‹`ŽÒA‹³ˆçŽÒ‚Æ‚µ‚Ä–¼º‚ðŸ‚¿“¾‚Ä‚¢‚½BƒLƒPƒ(BC106-43)‚́A”Þ‚Ì‚±‚Æ‚ðˆÌ‘å‚ÈŽt‚Ì—F‚Æ‚µ‚ÄŒê‚Á‚Ä‚¢‚éBƒzƒ‰ƒeƒBƒEƒX(BC65-8)‚́AƒAƒhƒŠƒAŠC‚Å“ï”j‚µÀ‚Á‚½”Þ‚ÉŒ¾‹y‚µ‚āAŽŸ‚̂悤‚ÈŒ¾—t‚ðŽc‚µ‚Ä‚¢‚éB @ƒGƒEƒfƒ‚ƒX(Eudemus)(c.335BC)‚́AŽ©‚ç‚ÌŠô‰½Šw‚Ì’˜ì‚ɂ‚¢‚ÄŒê‚èAŽ„‚½‚¿‚Ɂu“ÆŽ©‚ÌŒö—‚ÅŠw–â‚ð–L‚©‚É‚µAŒ’‘S‚È‘ÌŒn‰»‚ð‚¨‚±‚È‚Á‚½vl‚½‚¿‚̈êl‚Å‚ ‚Á‚½‚ÆŒ¾‚Á‚Ä‚¢‚éB•Ê‚̒q‚©‚çAŽ„‚½‚¿‚́A”ނ́AŽŸ‚̂悤‚È–½‘è‚ð’m‚Á‚Ä‚¢‚āA‹^‚¢‚È‚­Ø–¾‚µ‚½‚Ɛ„˜_‚µ‚Ä‚¢‚éB @‚PD’¼ŠpŽOŠpŒ`‚Ì’¼Šp‚Ì’¸“_‚©‚çAŽÎ•Ó‚ɐ‚ü‚ðˆø‚­‚ƁA‚»‚ꂼ‚ê‚̕ӂ́AŽÎ•Ó‚Æ‚»‚ê‚ɗאڂ·‚éü•ª‚Æ‚Ì‘Šæ•½‹Ï‚Å‚ ‚éB @‚QD‚ü‚́AŽÎ•Ó‚É‚Å‚«‚½“ñ‚‚̐ü•ª‚Ì‘Šæ•½‹Ï‚Å‚ ‚éB @‚RDŽOŠpŒ`‚Ì’¸“_‚©‚ç~‚낵‚½‚ü‚ªA‘Εӂ̓ñ‚‚̐ü•ª‚Ì‘Šæ•½‹Ï‚Å‚ ‚é‚È‚çA‚»‚Ì’¸“_‚ÌŠp‚Í’¼Šp‚Å‚ ‚éB @‚UD“ñ‚‚̕½–Ê‚ªA‘æŽO‚Ì•½–ʂƐ‚’¼‚Å‚ ‚é‚È‚çA“ñ‚‚̕½–Ê‚ÌŒðü‚à‚»‚Ì•½–ʂɐ‚’¼‚Å‚ ‚邵A‚Ü‚½A‚»‚Ì•½–Ê‚ª‘æŽO‚Ì•½–Ê‚ÆŒð‚í‚é’¼ü‚Æ‚à‚’¼‚Å‚ ‚éB

92. Human Indexes Of My Books On Mathematics; Te(de) To(do) In Japanese The summary for this Japanese page contains characters that cannot be correctly displayed in this language/character set. http://www.com.mie-u.ac.jp/~kanie/tosm/humanind/jinmeit4.htm

TOSMŽOd‚̃z[ƒ€ w‰ðÍ‹³’öx w”—‰ðÍ‚̃pƒCƒIƒjƒA‚½‚¿x w”Šw–¼ŠˆÄ“àx ... w“V‘‚̏ؖ¾x @l–¼õˆø@‚Ä‚Æ l–¼õˆø‘–ÚŽŸ ƒeƒAƒCƒeƒgƒX ƒfƒBƒIƒNƒŒƒX ƒfƒBƒIƒtƒ@ƒ“ƒgƒX ... ƒeƒCƒ‰[ H.ƒeƒCƒ‰[ ƒfƒBƒ‰ƒbƒN ƒfƒBƒŠƒNƒŒ ƒeƒIƒhƒƒX ƒfƒJƒ‹ƒg ... ƒgƒbƒv ƒeƒAƒCƒeƒgƒX AƒAƒeƒl‚Ì(Theaetetus of Athens, ‹IŒ³‘O415”N -369”N )D @ƒMƒŠƒVƒƒ‚̃Aƒeƒl‚ɐ¶‚Ü‚êAƒAƒeƒl‚ÉŽ€‚·B @ƒLƒ ƒŒƒl‚Ì ƒeƒIƒhƒƒX ƒvƒ‰ƒgƒ“ ‚̃AƒJƒfƒ~ƒA‚ÅŒ¤‹†‚·‚éB ³8–ʑ̂Ɛ³20–Ê‘Ì‚ðÅ‰‚ÉŒ©•t‚¯‚½l‚ŁA ƒ†[ƒNƒŠƒbƒh wŒ´˜_x‚Ì 10Í(–³—”˜_)‚Æ13Í(³‘½–Ê‘Ì)‚͔ނ̋Ɛтðq‚ׂ½‚à‚Ì‚Æ‚³‚ê‚Ä‚¢‚éB ‚Ü‚½ ‚̏‘’†‚Ì”ä—á˜_‚à”ނ̋Ɛтƍl‚¦‚ç‚ê‚Ä‚¢‚éB ƒgƒbƒv ƒfƒBƒIƒNƒŒƒX (Diocles, ‹IŒ³‘O240-180). ƒAƒ‹ƒLƒƒfƒX ‚Ì–â‘è‚ð‰ð‚­B•ú•¨–Ê‚ªÅ“_‚ðŽ‚Â‚±‚Æ‚ðŽ¦‚µ‚½‚±‚Ƃ̓MƒŠƒVƒƒl‚©‚ç‚Í–Y‚ê‚ç‚ꂽ‚ªAƒAƒ‰ƒu‚̐”Šw‚É‚Í‹­‚¢‰e‹¿‚ð—^‚¦‚½B [‰ðII.1] ƒgƒbƒv ƒfƒBƒIƒtƒ@ƒ“ƒgƒX AƒAƒŒƒLƒTƒ“ƒhƒŠƒA‚Ì(Diophantos = Diopantus, 246?-330? (200?-284?). @ƒAƒŒƒLƒTƒ“ƒhƒŠƒA‚ɏZ‚ñ‚Å‚¢‚½‚±‚Æ‚ª‚ ‚邱‚Æ‚µ‚©•ª‚ç‚È‚¢B‘¼‚É‚Í33Ë‚ÅŒ‹¥‚µA‘§Žq‚ª42Ë‚ÅŽ€‚ñ‚¾Žž‚©‚ç4”NŒã84Ë‚ÅŽ€‚ñ‚¾‚Æ‚¢‚¤‚±‚Æ‚ª‰ð“š‚Å‚ ‚éŽZp‚Ì–â‘肪Žc‚Á‚Ä‚¨‚è(5‚È‚¢‚µ6¢‹I‚́wŒ‘ãƒMƒŠƒVƒƒŽ‰ØWx)A¶–v”N‚»‚Ì‚à‚Ì‚Í‚Ü‚Á‚½‚­‚ ‚Ä‚É‚È‚ç‚È‚¢‚Ì‚¾‚ªA84”NŠÔ¶‚«‚Ä‚¢‚½‚¾‚낤‚Æ‚¢‚¤‚±‚Æ‚É‚Í‚È‚Á‚Ä‚¢‚éB ƒoƒVƒFEƒhEƒƒWƒŠƒAƒN –ó‚̃‰ƒeƒ“”Å–ó‚Ì—]”’‚É ‚ª‘‚«ž‚Ý‚ð‚·‚éB‚»‚Ì‘¼‘½Šp”‚ÉŠÖ‚·‚é’f•Ð‚ªŽc‘¶‚µ‚Ä‚¢‚éD [‰ðI.1, •¶], [–¼3, •¶], [Žì2.1, à3.9, •¶]

93. HPM³q°T²Ä¤»¨÷²Ä¤Q¤G´Á The summary for this Chinese (Traditional) page contains characters that cannot be correctly displayed in this language/character set. http://math.ntnu.edu.tw/~horng/letter/vol6no12c.htm

±q´X¦ó­±¦V¬Ý ¦èªQ°ª¤¤ Ĭ´f¥É¦Ñ®v ¤@¡B«e¨¥ ¦b¡m HPM ¡r¤¤¡A§i¶D§Ú­Ì¤@­Ó¥O¤HÅå³YªºÆ[¹îµ²ªG¡C¦b¬Ý¨ì¥Lªº¤å³¹¤§«e¡A§Ú±q¨Ó¨S·Q¹L ©M ¡B ªº¯È±i¤j¤p¦³ö¡C¦pªG±q ¡A ªºäªø¨Ó¬Ý¡A ¯È±iªº¤j¤p¡G ¡A 1.414285714¡K ¯È±iªº¤j¤p¡G ¡A 1.414141414¡K ¡B ªºªø»P¼eªº¤ñ¤j¬ù¬O ¡I ­C¡H ¡H ¬OµL²z¼Æ¡v³o­Ó¼Æ¾Ç·§©À¡A±qµL²z¼Æªº´X¦ó·N¸q¡A¨ì¡u ¬OµL²z¼Æ¡vªº´X¦óÒªk¡A³Ì«á¬O ªº´X¦ó¹Gªñ¤èªk¡C¸ÕµÛ±q´X¦ó­±¦V¨Ó¬Ý ¡A¥H¸É¥R½Ò¥»±q¥N¼Æ­±¦V¨Ó¬Ýªº¤£¨¬¡C ¤G¡B The Meno ¬f©Ô¹Ïªº¼Æ¾Ç­õ¾Ç½×­z¡A¥D­nµoªí¦b¥LªºµÛ§@¡m¦Ì¿Õ¡n (The Meno) Anytus »P¦Ì¿Õ®aªº¤@¦ì¡]¥£Áõ¡^¨k«Ä (slave boy) ¡A´«¥y¸Ü»¡¡A ¡C ³o¤@­Ó¡u¼Æ¦r¡v¡A¨ì©³»Päªø¡B¹ï¨¤½uµ¥¡u´X¦ó¶q¡v¦³¦óö«Y¡H ¤T¡B ¥i¤½«×¶qªº»P¤£¥i¤½«×¶qªº ¦pªG§Ú­Ì±q¦³²z¼Æªº­^¤å rational number ¨Ó¬Ý¡A rational ¤@¯ëªº²z¸Ñ³£¬O¡u¦³¹D²zªº¡v¡A¨º»ò¡AÀ³¸Ó¬O¤°»ò¹D²z¡H±q ration ªº©Ô¤B¤å»y·½¨Ó¬Ý¡A rational ¬O±q ratio ºt¤Æ¦Ó¨¥¡A­ì·N¬O¡u¤ñ¡vªº·N«ä¡F¥ç§Y¥i¥H¼g¦¨¨â­Ó¾ã¼Æ¤§¤ñªº¡A¥s°µ¦³²z¼Æ¡C¨º»ò¡AµL²z¼Æ©O¡H irrational number ¡A¦ÛµM´N¬O¤£¯à¼g¦¨¨â­Ó¾ã¼Æ¤§¤ñªº¼Æ¡C³o­Ó·§©À¤Î§Î¦¡¡A¬O±q²¦¤ó¾Ç¬£ªº¡u¥i¤½«×¶qªº¡v»P¡u¤£¥i¤½«×¶qªº¡vÆ[©Àºt¤Æ¦Ó¨Óªº¡C (Pythagoreans) ¬O«ü²¦¹F­ô©Ô´µ¡]¤j¬ù¦è¤¸«e (number) (commensurable) ¡A¦p¥k¹Ïªº »P ¡F¤Ï¤§¡A¦pªG¤£¯à¦P®É¶qºÉ¡A´NºÙ¡u¤£¥i¤½«×¶qªº¡v (incommensurable) ¬OµL²z¼Æ¡vªºÒªk¡C¦Ó¼Æ¾Ç¥v®a Heath ªº¤£¥i¤½«×¶q©ÊªºÒ©ú¤èªk¡C¨È¨½´µ¦h¼w¤Þ¥Î³o­ÓÒ©ú·í¨Ò¤l¡A¨Ó»¡©ú¥Lªº¡uÂkÂÕÒªk¡v reductio ad absurdum (on the assumption that the diagonal of a square is commensurable with its side, it is proved that odd numbers are equal to even)

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