Hippocrates of Chios
470 BC – 410 BC

Hippocrates was a teacher of geometry in Athens. He is known for working on the classical problem of squaring the circle and also the problem of duplicating the cube. It is often said of Hippocrates that he was an excellent geometer, although he lacked much common sense in real life.

Through his studies of squaring the circle, Hippocrates learned how to find the area of lunes. He is accredited with the discovery of the quadrature of the lune. The first of his theorems states that “similar segments of circles have the same ratio to one another as the squares on their bases.” He showed this by showing that the squares on the diameters have the same ratio as the circles. In doing this, Hippocrates is trying to square a lune (construct a square which has the same area as the lune). He went on to show that it was possible to square a lune, where the outer circumference is that of a semicircle.

To see more of what Hippocrates did with squaring the lune, click here.

In an attempt to make progress on the problem of squaring a circle, Hippocrates studied other cases of squaring the lune, where the outer circumference of the lune is not the arc of a semicircle. He studies cases where the outer arc was less than that of a semicircle and greater than that of a semicircle. He managed to show, in each of these cases, that the lune could, in fact, be squared.

Another achievement of Hippocrates was that he showed that “a cube can be doubled if two mean proportionals can be determined between a number and its double.”

Hippocrates was the first to write an Elements of Geometry, although this work is now lost. It probably contained much of what Euclid included in his books 1 and 2 of the Elements. His book also included geometric approaches to solving quadratic equations and early methods of integration.

References:

Greenberg, Marvin J. Euclidean and Non-Euclidean Geometries:

Development and History. 3rd ed. New York: W.H. Freeman and Company,
1993. 6-19.

Historical Topics for the Mathematics Classroom. Washington D.C.:

National Council of Teachers of Mathematics, 1969.

http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Thales.html (March 2005)

http://geometryalgorithms.com/history.htm (March 2005)

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