Sunday, July 20, 2014

幾何三大問題,Diophantus

------ ----- --------------------------- 所謂幾何三大問題無非是，給定一個單位長度； （方圓問題） 求作一個長度為 $\sqrt{\pi}$ 的線段。 （倍立方問題） 求作一個長度為 $\sqrt[3]{2}$ 的線段。 （三等分角問題） 先給長度為 α 的線段，求作一個長度為 x 的線段，而 x 滿足 $4x^3-3x-\alpha=0$。 ------------------- Diophantus's riddle is a poem that encodes a mathematical problem. In verse, it read as follows: 'Here lies Diophantus,' the wonder behold. Through art algebraic, the stone tells how old: 'God gave him his boyhood 1/6 of his life, 1/12 more as youth while whiskers grew rife; And then yet 1/7 ere marriage begun; In 5 years there came a bouncing new son. Alas, the dear child of master and sage After attaining 1/2 the measure of his father's life chill fate took him. After consoling his fate by the science of numbers for 4 years, he ended his life.' Stated in prose, the poem says that Diophantus's youth lasts 1/6 of his life. He grew a beard after 1/12 more of his life. After 1/7 more of his life, Diophantus married. 5 years later, he had a son. The son lived exactly half as long as his father, and Diophantus died just 4 years after his son's death. All of this totals the years Diophantus lived. Let D=84 be the number of years Diophantus lived, and let S=42 be the number of years his son lived. Then the above word problem gives the 2 equations
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