COMMENTARY ON ARCHIMEDES' ON THE SPHERE AND CYLINDER . First edition of this rare Galileianum, an explication and extension of the two books of Archimedes' On the sphere and the cylinder, which gave the first exact determination of the area and volume of a curved figure. They "are, of course, exceptional masterpieces. According to a testimony by Cicero, whom there is no reason to doubt, Archimedes' tomb had inscribed a sphere circumscribed inside a cylinder, recalling the major measurement of volume obtained in [the first book]: if so, either Archimedes or those close to him considered [this book] to be somehow the peak of his achievement. The reason is not difficult to find. Archimedes' works are almost all motivated by the problem of measuring curvilinear figures, all of course indirectly related to the problem of measuring the circle . Measuring the sphere is the closest Archimedes, or mathematics in general, has ever got to measuring the circle. The sphere is measured by being reduced to other curvilinear figures. Still, the main results obtained - that the sphere as a solid is two thirds the cylinder circumscribing it, its surface four times its great circle - are remarkable in simplifying curvilinear, three-dimensional objects, that arise very naturally" (Netz, p. 19). Archimedes' proof is extraordinarily ingenious. "A circle with a polygon inscribed within it is imagined rotated in space, yielding a sphere with a figure inscribed within it. The inscribed figure is made of truncated cones . Furthermore, with the same idea extended to a circumscribed polygon yielding a circumscribed figure made of truncated cones, proportion inequalities come about involving the circumscribed and inscribed figures . such proportion inequalities can be manipulated to combine with the measurements of the inscribed and circumscribed figures, reaching, indirectly, a measurement of the sphere itself" (ibid., pp. 20-21). In this, probably White's only published work, he uses Archimedes' techniques to study other solids that can be inscribed in, and circumscribed about, a hemisphere. He also studies the relation between the areas and volumes of parts of a hemisphere and corresponding parts of a cylinder or a cone. White was personally acquainted with Galileo and he and his younger brother Thomas were instrumental in bringing Galileo's discoveries and opinions to the attention of British scientists, notably Francis Bacon. In the preface White praises Galileo as the outstanding investigator of the heavens above, the waves below (a reference to the tidal theories of the Dialogo - see below), and mechanics on Earth, "though it was published in Rome under license during a period in which many Italian writers found it prudent to forgo any favourable reference to Galileo in their published works. White also conducted some elaborate experiments concerning specific gravities and made accurate observations of Halley's comet" (Drake, p. 245). Richard White was elected a fellow of the Royal Society in 1661. OCLC lists Columbia, Harvard, Huntington and Linda Hall in US. RBH lists four copies since 1931. Richard White (1590-1682) was born to a prominent Catholic family in Essex; his mother was the daughter of the celebrated jurist Edmund Plowden. In his preface, White explains how the urge 'to cross the furious ramparts of the ocean which surround Britain' led him through France to Florence in the company of one James Clayton (probably a member of another Recusant family). He lived for most of his life in Italy: he tells us that he first studied Aristotle in Florence (beginning with the Organon and thence to the Physics), where he came into contact with Galileo and his circle, especially Benedetto Castelli, who taught him Euclid, and Bonaventura Cavalieri, famous for his Geometria indivisibilibus (1635), which introduced the method of indivisibles, an important precursor of calculus. Richard was joined by his brother Thomas in Italy who encouraged him to stud
