Hemisphaerium dissectum opus geometricum: in quo obiter tractatur de maximix inscriptibilibus & minimis circumscribentibus: ratio etiam discutitur quare aliquæ propositiones non admittant solutionem per media plana vel Euclidis elementa: cum methodo novâ geometricâ quâ ad aequationem reducitur propositio de sectione hemisphærij in ratione datâ: accessit appendix de inscriptione in sphaæra coni scaleni & de superficie eius: demum cubatio cuiusdam partis cylindri dissectæ plano

Roe: Manelfo Manelfi, 1648. First edition.

COMMENTARY ON ARCHIMEDES' ON THE SPHERE AND CYLINDER

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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 study Archimedes' writings on the sphere and cylinder, resulting in the present work. Its publication, some years later, was partly at the urging of Michelangelo Ricci. Thomas White, a Jesuit, was a prolific writer. He was acquainted with prominent British intellectuals, including Thomas Hobbes and Sir Kenelm Digby, and belonged to the Mersenne circle in Paris in the 1640s.

Richard White was instrumental in correcting a statement made by Galileo in his theory of the tides, 'Discorso sul flusso e it reflusso del mare,' which he submitted in the form of a private letter to Cardinal Orsini in 1616. Galileo believed that the tides were the result of the differential motions of different regions of the Earth, according to whether in a particular region the diurnal rotation of the Earth around its axis was in the same or the opposite direction as its annual rotation around the Sun; he used the analogy of a vase containing water - when the vase undergoes irregular motion the water itself acquires a motion relative to the vase. In the 1616 letter Galileo stated that the tides at Lisbon were half as frequent as those on the Mediterranean, which he explained from the fact that the Atlantic Ocean is twice as wide as the Mediterranean. These statements were omitted from the printed version in the Fourth Day of the Dialogo (1632), because in 1619 Richard White had informed Galileo that his data on the Lisbon tides were incorrect. In the same year White brought to England copies of Galileo's books and manuscripts, leaving them with Francis Bacon. This probably led to Bacon omitting from his Instauratio magna (1620) his essay on tides, 'On the ebb and flow of the seas,' written about 1611 and sent to Galileo in 1618. Bacon's essay had explained the tides as the effect on the oceans of the daily east-west motion of the Earth relative to the heavens, in opposition to Galileo's conclusion in the Orsini letter.

According to some sources (e.g., Encyclopaedia Britannica, Vol. 20 (1842), p. 554), White published a quadrature of the circle, which he later retracted, in a work entitled Chrysaespsis, seu Quadratura Circuli, but we have been unable to locate any copy of this book.

BL Italian, 17th cent, p. 972; Macclesfield 2122. Drake, "Galileo in English literature of the seventeenth century," in Essays on Galileo and the History and Philosophy of Science, Vol. I, University of Toronto Press, 1999. Netz, The Works of Archimedes. Vol. I. The Two Books On the Sphere and the Cylinder, 2004.

4to (250 x 190 mm), pp. [xii], 294, [2], title page within double-ruled border, text within single rule, full-page woodcut coat-of-arms (probably the author's) on a4 verso and on verso of last leaf, numerous woodcut diagrams and initials in text, printed marginal notes, printed over-slip on P1r and manuscript annotation to P1v as apparently in all copies (closed tear to upper margin of title (no loss), small single wormhole to fore-edge margin, faint damp-stain to upper margin). Eighteenth-century brocade paper wrappers (rubbed and a bit worn at extremities).