Archimedes

context: en: Archimedes died c. 212 BC during the Second Punic War, when Roman forces under General Marcus Claudius Marcellus captured the city of Syracuse after a two-year-long siege. According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives a lesser-known account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the soldier thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable scientific asset and had ordered that he must not be harmed. Marcellus called Archimedes "a geometrical Briareus".The last words attributed to Archimedes are "Do not disturb my circles", a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. This quote is often given in Latin as "Noli turbare circulos meos," but there is no reliable evidence that Archimedes uttered these words and they do not appear in the account given by Plutarch. Valerius Maximus, writing in Memorable Doings and Sayings in the 1st century AD, gives the phrase as "...sed protecto manibus puluere 'noli' inquit, 'obsecro, istum disturbare'" – "... but protecting the dust with his hands, said 'I beg of you, do not disturb this.'" The phrase is also given in Katharevousa Greek as "μὴ μου τοὺς κύκλους τάραττε!" (Mē mou tous kuklous taratte!).
qas: question: zh_tw: 羅馬士兵拒絕和誰見面？
answers: en: text: General Marcellus

context: en: Archimedes was able to use infinitesimals in a way that is similar to modern integral calculus. Through proof by contradiction (reductio ad absurdum), he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay. This technique is known as the method of exhaustion, and he employed it to approximate the value of π. In Measurement of a Circle he did this by drawing a larger regular hexagon outside a circle and a smaller regular hexagon inside the circle, and progressively doubling the number of sides of each regular polygon, calculating the length of a side of each polygon at each step. As the number of sides increases, it becomes a more accurate approximation of a circle. After four such steps, when the polygons had 96 sides each, he was able to determine that the value of π lay between 31/7 (approximately 3.1429) and 310/71 (approximately 3.1408), consistent with its actual value of approximately 3.1416. He also proved that the area of a circle was equal to π multiplied by the square of the radius of the circle (πr2). In On the Sphere and Cylinder, Archimedes postulates that any magnitude when added to itself enough times will exceed any given magnitude. This is the Archimedean property of real numbers.
qas: question: zh_tw: 哪種現代數學研究與阿基米德使用無窮小量分析方法相類似？
answers: en: text: integral calculus

question: zh_tw: 縮小範圍的另一種證明最大的方式是什麼？
answers: en: text: reductio ad absurdum

context: en: Archimedes was born c. 287 BC in the seaport city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years. In The Sand Reckoner, Archimedes gives his father's name as Phidias, an astronomer about whom nothing else is known. Plutarch wrote in his Parallel Lives that Archimedes was related to King Hiero II, the ruler of Syracuse. A biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he ever married or had children. During his youth, Archimedes may have studied in Alexandria, Egypt, where Conon of Samos and Eratosthenes of Cyrene were contemporaries. He referred to Conon of Samos as his friend, while two of his works (The Method of Mechanical Theorems and the Cattle Problem) have introductions addressed to Eratosthenes.
qas: question: zh_tw: 阿基米德在哪個市鎮出生？
answers: en: text: Syracuse, Sicily

question: zh_tw: 阿基米德的朋友叫什麼名字？
answers: en: text: Heracleides

context: en: This is a description of a planetarium or orrery. Pappus of Alexandria stated that Archimedes had written a manuscript (now lost) on the construction of these mechanisms entitled On Sphere-Making. Modern research in this area has been focused on the Antikythera mechanism, another device built c. 100 BC that was probably designed for the same purpose. Constructing mechanisms of this kind would have required a sophisticated knowledge of differential gearing. This was once thought to have been beyond the range of the technology available in ancient times, but the discovery of the Antikythera mechanism in 1902 has confirmed that devices of this kind were known to the ancient Greeks.
qas: question: zh_tw: 何時發現了古代的機械？
answers: en: text: 1902

question: zh_tw: 為「球體制造」建立了什麼其他機械？
answers: en: text: Antikythera

context: en: While Archimedes did not invent the lever, he gave an explanation of the principle involved in his work On the Equilibrium of Planes. Earlier descriptions of the lever are found in the Peripatetic school of the followers of Aristotle, and are sometimes attributed to Archytas. According to Pappus of Alexandria, Archimedes' work on levers caused him to remark: "Give me a place to stand on, and I will move the Earth." (Greek: δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω) Plutarch describes how Archimedes designed block-and-tackle pulley systems, allowing sailors to use the principle of leverage to lift objects that would otherwise have been too heavy to move. Archimedes has also been credited with improving the power and accuracy of the catapult, and with inventing the odometer during the First Punic War. The odometer was described as a cart with a gear mechanism that dropped a ball into a container after each mile traveled.Cicero (106–43 BC) mentions Archimedes briefly in his dialogue De re publica, which portrays a fictional conversation taking place in 129 BC. After the capture of Syracuse c. 212 BC, General Marcus Claudius Marcellus is said to have taken back to Rome two mechanisms, constructed by Archimedes and used as aids in astronomy, which showed the motion of the Sun, Moon and five planets. Cicero mentions similar mechanisms designed by Thales of Miletus and Eudoxus of Cnidus. The dialogue says that Marcellus kept one of the devices as his only personal loot from Syracuse, and donated the other to the Temple of Virtue in Rome. Marcellus' mechanism was demonstrated, according to Cicero, by Gaius Sulpicius Gallus to Lucius Furius Philus, who described it thus:
qas: question: zh_tw: 除了設計滑輪和滑輪系統外，阿基米德還負責什麼？
answers: en: text: improving the power and accuracy of the catapult, and with inventing the odometer during the First Punic War

question: zh_tw: 誰演示了馬塞勒斯機制？
answers: en: text: Gaius Sulpicius Gallus