Hippocrates of Chios
c. 470-c. 410 B.C.
Greek Mathematician
Designated as Hippocrates of Chios to distinguish him from the better-known physician of the same name, Hippocrates has been cited as the greatest mathematician of the fifth century B.C. He wrote the first textbook on geometry, in which he addressed problems such as squaring the circle and doubling the cube.
Hippocrates's achievements were particularly notable in light of the fact that he started his career as a mathematician late in life. Certainly he does not seem to have set out in his youth to pursue such a career; on the contrary, his involvement in mathematics was the indirect outcome of misfortunes. Apparently he had a successful business as a merchant until he was attacked near Byzantium by Athenian pirates, though another version of the story depicts his assailants as corrupt customs officials who seized his goods and threatened imprisonment if he complained.
It seems that Hippocrates then went to Athens itself to seek legal redress, and while waiting for his case to come to court, he attended lectures on mathematics and philosophy. During this time, he came under the influence of the mathematical school based on the principles of Pythagoras (c. 580-c. 500 B.C.), and in time he reached such a degree of proficiency as a mathematician that he opened a Pythagorean school of his own. Pythagoras had forbidden his students to earn money through their mathematical knowledge, but in view of Hippocrates's recent financial hardships, the Pythagoreans of Athens made an exception.
Among the legacies of Hippocrates was a mathematical textbook, long since lost, called the Elements of Geometry. The first work of its kind, it would have an enormous impact on another book of a similar title, the highly influential Elements of Euclid (c. 325-c. 250 B.C.). In his work, known through the writings of Aristotle (384-322 B.C.), Proclus (410?-485), and others, Hippocrates became the first mathematician to adopt scientifically precise and logical methodology for developing geometrical theorems from axioms and postulates. It is also likely that Elements of Geometry contained the first written explanation of Pythagorean principles, since the Pythagoreans who preceded him did not believe in committing their ideas to writing.
Other issues addressed in the book included the Delian problem of doubling the cube. In attempting a solution, Hippocrates became the first mathematician to use a method of reduction: that is, altering a difficult problem to a simpler form, which, once it was solved, made it possible to apply the solution to the original problem. Dinostratus (c. 390-c. 320 B.C.) would later apply this idea to the construction of the quadratix for squaring the circle, and Tartaglia (Niccolò Fontana; 1499-1557) would use much the same principle to the solution of cubic equations two millennia after Hippocrates. In Hippocrates's case, by reduction he discovered that finding mean proportionals was the key to doubling the cube—a principle that revolutionized Greek mathematicians' approach to the problem.
Hippocrates's quadrature of the lune, a crescent shape, represented an attempt to solve another problem of long standing among Greek mathematicians, the squaring of the circle. Starting from the principle that the ratio of area between two circles is the same as that of the squares of their diameter, Hippocrates set out to square the lune—that is, to find a square equal in area to a given lune. The process he followed was a complicated one, but in essence it put to work his idea of reduction by finding a correlation between a given straight segment and a curved one.
Also significant was Hippocrates's use of reductio ad absurdum, a method he may have introduced. The reductio, which is reflected today in the scientific method, sets out to prove the opposite of what is desired; and by so doing, it shows that the original proposition is true. In similar fashion, a scientist makes every effort to disprove his or her propositions, and only if an idea stands this test is it formalized as a theory. (By contrast, a pseudo- or anti-scientist starts with a conclusion and then looks for evidence that corroborates it, simultaneously ignoring all facts that do not.)
In addition to his mathematical work, Hippocrates also conducted astronomical studies. In this he was hampered, however, by his allegiance to Pythagorean ideas such as the claim that there was only one comet, which simply reappeared from time to time.
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