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5. Archimedes' Magnum Opus On the Sphere and the Cylinder
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39 ARCHIMEDES’ MAGNUM OPUS ON THE SPHERE AND THE CYLINDER V On the Sphere and the Cylinder is Archimedes’ most extensive and well-argued scientific work that has come down to us. This work is concerned with geometric shapes that are not confined simply to flat surfaces, such as the sphere and the cylinder [plate 10a]. The ancients link the image of the Syracusan scientist to these shapes both through this work and, tangibly, through Archimedes’ tomb, on which there were two precisely carved geometric figures, mentioned by Cicero (Tusculan Disputations 5, 64–66, cited more fully in chap. 10). Other works of Archimedes, too, that have survived reveal themselves to have been driven by the question of measuring curvilinear shapes (such as spirals, parabolas, conoids, and spheroids). Admittedly, these objects are theoretical and encompass symbolic meanings: the circle and the sphere had particular mathematical and philosophical significance, as Netz has noted, linked to their roles in the various cosmologies of antiquity.1 The treatise of 40 THE GREAT ARCHIMEDES V Archimedes includes some theorems on the volume of the cone and pyramid already outlined by Democritus at the end of the fifth century BC and, in the fourth century, demonstrated by Eudoxus: “each pyramid is the third part of the prism having an equal base and equal height, and each cone is the third part of the cylinder having equal base and equal height” (preface to On the Sphere and the Cylinder). But Archimedes was able to enrich these theorems with “infinitesimal” arguments which specifically analyze solid figures by breaking them down into lines and circles of extreme thinness or, in the case of disproving a thesis, argumentum ad absurdum, which shows the inaccuracy of a hypothesis by bringing it to an absurd, even “meaningless,” conclusion. By his infinitesimal argument, he was able to push his calculations to such a degree as to specify rather precisely the area and the volume of the sphere, the area of a section of the parabola, and the volume of the paraboloid as it spins. He was also able to identify their respective centers of gravity. Archimedes gave some attention as well to ellipsoids and paraboloids in rotation, in his On Conoids and Spheroids , which was dedicated to the Alexandrian astronomer Dositheus. This treatise was one of the most original and mature works in Archimedes’ corpus. In this work, which begins with complicated terms, Archimedes imagines and measures objects, in effect, far from daily experience. It treats solids obtained by performing a complete rotation of a flat curve around a fixed axis. In particular: • the elliptic paraboloid is achieved by rotating a parabola around its axis [fig. 13]; • the hyperboloid of two sheets is achieved by rotating a hyperbola around its transverse axis [fig. 14]; • the spheroid is achieved by the rotation of an ellipse (Archimedes distinguishes between a flattened [54.81.185.66] Project MUSE (2024-03-19 02:49 GMT) ARCHIMEDES’ MAGNUM OPUS 41 V spheroid and an elongated spheroid, depending on whether the rotation takes place around the greater or lesser axis) [fig. 15]. As we said at the end of chapter 2, conic sections are curves described by the intersection of a cone with different planes: • if the plane is parallel to one of the generators of the cone, it is considered a parabola. • if the plane intersects both halves of a right circular cone at an angle parallel to the axis of the cone, it is considered a hyperbola; • if the plane is not parallel to the axis, base, or generators of the cone, it is considered an ellipse; • if the plane is perpendicular to the axis of the cone, it is considered a circle. In addition to determining their volume, Archimedes also established the notion of segments of conoids and spheroids , i.e., of the parts of these shapes when limited by a plane. Archimedes’ work On the Sphere and the Cylinder should be considered as an ideal continuation of Euclid’s Elements, published a few decades earlier, in which the main results of Greek mathematics were presented in a strictly deductive manner. That work did not treat the sphere, a solid shape that is considered more relevant to astronomy than geometry, in a comprehensive fashion. Indeed, whereas the Elements of Euclid is an amalgam of the results and previous knowledge of different authors, the work of Archimedes is almost completely original. On the Sphere and the Cylinder is divided into two books, which may have originally...