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Hippocrates

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Hippocrates of Chios was a Greek merchant turned mathematician who wrote the first textbook on geometry. Hippocrates, who was born on the island of Chios, has been called one of the greatest mathematicians of his time. Legend has it that Hippocrates was a thriving merchant until lost all of his fortune. He is said to have traveled to Athens to try to win his fortunes back in a court of law. Since he was required to stay in Athens for the trials, Hippocrates began to attend lectures on mathematics to pass the time. He eventually became proficient enough in mathematics to open his own school.
Hippocrates is believed to have been influenced by the Pythagorean School of Mathematics. His book contained some of the first written accounts of Pythagorean mathematics since the Pythagoreans themselves did not believe in written texts. Hippocrates was one of the first “paid” mathematics teachers, although the Pythagoreans believed it was taboo to earn money from their knowledge. He may have been able to establish a school because of his financial troubles. Hippocrates wrote the first mathematical textbook, called the Elements of Geometry and even though the book is lost, it had a profound influence on the mathematicians who followed him. Euclid based some of his work on that of Hippocrates in his book, Elements that was written more than a century later. One of the most famous problems faced by ancient Greek mathematicians was doubling the cube, also called the Delian Problem. One of the legends of the Delian Problem has to do with the Greek god Apollo. Apollo, who was displeased with his altar, ordered a second altar built in his honor. The altar was supposed to be double in size but have the same cubical form as the first altar. The people of Athens assumed that all that they had to do was double each of the old altar’s edges. Since this was not the correct way, a plague continued in Athens and this math problem became a very important problem to find a solution.
In his attempts to solve the Delian Problem, Hippocrates used the method of reduction. Plato originally developed a method of reduction for philosophical problems; however, Hippocrates was possibly the first to use this approach in geometry. The method of reduction is altering a difficult problem into a simpler form, solving this simpler form of the problem, and then attempting to apply the solution to the more difficult problem.
In the case of the Delian Problem, Hippocrates proposed that a cube could be doubled by finding the two mean proportionals (geometric means) between two given lines or between a number and its double. The mean proportionals in question are a : x = x : y = y : 2a and the solution to this problem found through the method of reduction is x³ = 2a³. While Hippocrates never completely solved the problem of doubling the cube, others followed up with his methods to develop several solutions to this ancient geometric puzzle.

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