The Britannica guide to the history of mathematics by Gregersen E. (ed.)

By Gregersen E. (ed.)

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In sum the Elements gathered together the whole field of elementary geometry and arithmetic that had developed in the two centuries before Euclid. Doubtless, Euclid must be credited with particular aspects of this work, certainly with its editing as a comprehensive whole. But it is not possible to identify for certain even a single one of its results as having been his discovery. Other, more advanced fields, though not touched on in the Elements, were already being vigorously studied in Euclid’s time, in some cases by Euclid himself.

Encyclopædia Britannica, Inc. 62 7 Ancient Western Mathematics 7 spheres, cones, cylinders, and other solids of revolution), assigning to such figures a weight proportional to their area. For example, to measure the volume of a sphere, he imagined a balance beam, one of whose arms is a diameter of the sphere with the fulcrum at one endpoint of this diameter and the other arm an extension of the diameter to the other side of the fulcrum by a length equal to the diameter. Archimedes showed that the three circular cross sections made by a plane cutting the sphere and the associated cone and cylinder will be in balance (the circle in the cylinder with the circles in the sphere and cone) if the circle in the cylinder is kept in its original place while the circles in the sphere and cone are placed with their centres of gravity at the opposite end of the balance.

Such hints about the nature of early Greek practical mathematics are confirmed in later sources—for example, in the arithmetic problems in papyrus texts from Ptolemaic Egypt (from the 3rd century BCE onward) and the geometric manuals by Heron of Alexandria (1st century CE). In its basic manner, this Greek tradition was much like the earlier traditions in Egypt and Mesopotamia. Indeed, it is 41 7 The Britannica Guide to the History of Mathematics 7 likely that the Greeks borrowed from such older sources to some extent.

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