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This biography examines the life and work of Archimedes, the ancient mathematician and inventor. It details his early education in Alexandria, his connection to the royal family of Syracuse, and the mechanical and mathematical contributions that defined his career before his death in 212 B.C.

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Table of Contents

Archimedes

finding the circular sections. Prop. 9 gives a similar proof that each elliptic section of a conoid or spheroid is a section of a certain oblique circular cylinder (with axis parallel to the axis of the segment of the conoid or spheroid cut off by the said elliptic section). Props. 11-18 show the nature of the various sections which cut off segments of each conoid and spheroid and which are circles or ellipses according as the section is perpendicular or obliquely inclined to the axis of the solid; they include also certain properties of tangent planes, etc.

The real business of the treatise begins with Props. 19, 20; here it is shown how, by drawing many plane sections equidistant from one another and all parallel to the base of the segment of the solid, and describing cylinders (in general oblique) through each plane section with generators parallel to the axis of the segment and terminated by the contiguous sections on either side, we can make figures circumscribed and inscribed to the segment, made up of segments of cylinders with parallel faces and presenting the appearance of the steps of a staircase. Adding the elements of the inscribed and circumscribed figures respectively and using the method of exhaustion, Archimedes finds the volumes of the respective segments of the solids in the approved manner (Props. 21, 22 for the paraboloid, Props. 25, 26 for the hyperboloid, and Props. 27-30 for the spheroids). The results are stated in this form: (1) Any segment of a paraboloid of revolution is half as large again as the cone or segment of a cone which has the same base and axis; (2) Any

segment of a hyperboloid of revolution or of a spheroid is to the cone or segment of a cone with the same base and axis in the ratio of AD + 3CA to AD + 2CA in the case of the hyperboloid, and of 3CA − AD to 2CA − AD in the case of the spheroid, where C is the centre, A the vertex of the segment, and AD the axis of the segment (supposed in the case of the spheroid to be not greater than half the spheroid).

On Spirals.

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