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How did Euclid compare the areas of circles using their diameters?
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Euclidean geometry
This diagram depicts three basic theorems that triangles are congruentâthat is, they are of equal shape and sizeâif: (1) two sides and the included angle are equal, designated SAS; (2) two angles and the included side are equal, designated ASA; or (3) all three sides are equal, designated SSS.
In his Elements, Euclid utilized the method of exhaustion to demonstrate that the areas of circles are to one another as the squares of their diameters.1
This approach, attributed to Eudoxus of Cnidus, was a rigorous logical procedure rather than a calculation involving limits or infinitesimals. By inscribing polygons within a circle and increasing the number of sides indefinitely, the area of the polygons could be made to "exhaust" the area of the circle, allowing for the comparison of circular areas through the properties of these simpler, rectilinear figures. This method provided a foundational way for Greek mathematicians to handle continuous magnitudes and establish proportional relationships between geometric shapes.234