A History of Greek Mathematics and Astronomy

Page 21

Speaking generally, the geometrical works are directed to the measurement of curvilinear areas and volumes; and Archimedes employs a method which is a development of Eudoxus's method of exhaustion. Eudoxus apparently approached the figure to be measured from below only, i. e. by means of figures successively inscribed to it. Archimedes approaches it from both sides by successively inscribing figures and circumscribing others also, thereby compressing them, as it were, until they coincide as nearly as we please with the figure to be measured. In many cases his procedure is, when the analytical equivalents are set down, seen to amount to real integration; this is so with his investigation of the areas of a parabolic segment and a spiral, the surface and volume of a sphere, and the volume of any segments of the conoids and spheroids.

The newly-discovered Method is especially interesting as showing how Archimedes originally obtained his results; this was by a clever mechanical method of (theoretically) weighing infinitesimal elements of the figure to be measured against elements of another figure the area or content of which (as the case may be) is known; it amounts to an avoidance of integration. Archimedes, however, would only admit that the mechanical method is useful for finding results; he did not consider them proved until they were established geometrically.

In the Measurement of a Circle, after proving by exhaustion that the area of a circle is equal to a right-angled triangle with the perpendicular sides equal respectively to the radius and the circumference of the circle, Archimedes finds, by sheer calculation, upper and lower limits to the ratio of the circumference of a circle to its diameter (what we call π {p}). This he does by inscribing and circumscribing regular polygons of 96 sides and calculating approximately their respective perimeters. He begins by assuming as known certain approximate values for √3, namely 1351/780 > √3 > 265/153, and his calculations involve approximating to the square roots of several large numbers (up to seven digits). The text only gives the results, but it is evident that the extraction of square roots presented no difficulty, notwithstanding the comparative inconvenience of the alphabetic system of numerals. The result obtained is well known, namely 3-1/7 > π {p} > 3-10/71.

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T. L. Heath
A History of Greek Mathematics and Astronomy

Learned Freeware

T. L. Heath A History of Greek Mathematics and Astronomy

Learned Freeware

T. L. Heath
A History of Greek Mathematics and Astronomy