De iis quae vehuntur in aqua libri duo. A Federico Commandino Urbinate in pristinum nitorem restituti, et commentariis illustrati. [Bound with:] COMMANDINO. Liber de centro gravitatis solidorum.

Bologna: Alessandro Benacci, 1565.

First edition of both works, a spectacular copy in a mid-seventeenth century red morocco armorial binding from the Library of Felipe Ramirez de Guzmán (ca. 1600-1668), Duke of Medina de las Torres, Viceroy of Naples. The first work is the first complete edition of the foundation work of hydrostatics, Archimedes’ On Floating Bodies, which includes the eponymous ‘Archimedes’ principle’ of buoyancy; the second is the first published work on centres of gravity of solid bodies. “Archimedes − together with Newton and Gauss − is generally regarded as one of the greatest mathematicians the world has ever known, and if his influence had not been overshadowed at first by Aristotle, Euclid and Plato, the progress of modern mathematics might have been much faster … In hydrostatics [Archimedes] described the equilibrium of floating bodies and stated the famous proposition – known by his name – that, if a solid floats in a fluid, the weight of the solid is equal to that of the fluid displaced and, if a solid heavier than a fluid is weighed in it, it will be lighter than its true weight by the weight of the fluid displaced” (PMM, p. 44). For his edition of On Floating Bodies, Commandino (1509-75) used a Latin translation, from a now lost Greek text, by Flemish Dominican William of Moerbeke (1215-86) in 1269 (Moerbeke's holograph remains intact in the Vatican library, Codex Ottobonianus Latinus 1850); for this work he had no access to a Greek text, unlike the five other Archimedean works he had previously translated. But the Greek text used by Moerbeke was corrupt and lacked the proofs of two crucial propositions. In addition, Archimedes used certain results on the centres of gravity of solid bodies, but his work on this subject has not survived. Commandino not only cleaned up the corrupted text, he also supplied the missing proofs and further took it upon himself to prove the necessary results about centres of gravity in the form of a self-contained treatise, De Centro Gravitatis Solidorum. Commandino uses the Archimedean methods of exhaustion and reductio ad absurdum, and De Centro Gravitatis Solidorum may justly be regarded as a reconstruction of Archimedes’ lost work on centres of gravity. Their very close relationship makes it particularly appropriate to find these two works bound together, as here. A second translation of On Floating Bodies, published by Curtius Trioianus from the legacy of Niccoló Tartaglia (1499-1557), appeared in the same year (the brief Book I had been published in 1543), but according to Rose (p. 153) this is a direct transcript of a copy of the Moerbeke translation, retaining all the errors and making no attempt to fill in the lacunae. Commandino’s “masterful version … was far more influential than the version of On Floating Bodies … published under Tartaglia’s direction” (Clagett). “In the sixteenth century, Western mathematics emerged swiftly from a millennial decline. This rapid ascent was assisted by Apollonius, Archimedes, Aristarchus, Euclid, Eutocius, Hero, Pappus, Ptolemy, and Serenus – as published by Commandino” (DSB).

Born in Urbino, Commandino studied Latin and Greek at Fano, then returned to Urbino where he studied mathematics. Later he studied medicine at Padua, and after returning home again he became personal physician to the Duke of Urbino. There he met Cardinal Ranuccio Farnese, the brother of the Duke's wife, who was to become his most important patron. In the early 1550s the Cardinal persuaded Commandino to move to Rome as his personal physician; while there he became friendly with Cardinal Cervini, who was elected Pope Marcello II in 1555. But following Cervini’s death shortly after his election, both Commandino and Farnese returned to Urbino, where Commandino continued in the service of the Duke and Cardinal. But Commandino’s true love was mathematics, and in 1558 he published his edition of Archimedes’ Opera, which he dedicated to Farnese (this did not contain any of Archimedes’ works on mechanics). Also in 1558 Commandino published a work he had begun in Rome, namely Commentarius in Planisphaerium Ptolemaei, in which he gave an account of Ptolemy’s stereographic projection of the celestial sphere. In 1562 he published his edition of Ptolemy’s work on the calibration of sundials, De Analemmate.

“In July 1564 Ranuccio Farnese was appointed to the see of Bologna and by 1565 the Cardinal and Commandino were settled there. At Bologna in that year Commandino published his edition of Archimedes’ On Floating Bodies together with his own De Centro Gravitatis. Since the Greek text of the Archimedean work was then unknown, Commandino availed himself of the same manuscript of the Moerbeke translation that he had used for his 1562 edition of Ptolemy’s De Analemmate … in the dedication to De Centro Gravitatis, Commandino states that Cervini, when still a Cardinal, had given Commandino the early Latin version. This immediately raises the problem of precisely which manuscript Commandino received from Cervini. There are indeed very few codices of the Moerbeke translation … the probability seems to me to be that Cervini loaned, rather than gave, Commandino the autograph Ottob. Lat. 1850 …

“The dedication to Ranuccio Farnese explains that the delay in publishing On Floating Bodies is due to the far greater difficulty of the material and the corruption of the text. Here especially he has felt the lack of a Greek text. But even so, Commandino has seen that the earlier translator’s (Moerbeke’s) Greek text must have been corrupt and defective since two of the proofs are missing, thus disturbing the admirable sequence of mathematical argument. This problem was enhanced by Archimedes’ accepting as evident a great many proofs and facts on conics which had been discovered by earlier mathematicians. The ideas in question, however, are not so evident to moderns and so in order to render the text fully intelligible Commandino has had to resort often to the Conics of Apollonius. But many of the writings of the other pre-Archimedean mathematicians have now been lost, a matter of great regret to Commandino who confesses that he cannot admire too much the skill of the Greek mathematicians … For this reason, Commandino has now with great effort prepared the present edition of Archimedes’ book. The errors of the anonymous translator are now emended, the corrupt passages have been cleaned up, and the lacunae filled in. Commandino has also used Apollonius to illuminate many of the facts taken for granted by Archimedes” (Rose, pp. 200-201). The study of conics necessitated by Commandino’s work on On Floating Bodies led in the following year to the publication of his edition of Apollonius’ Conics, which served as the standard edition until the 18th century.

Book I of On Floating Bodies introduced the concept of fluid pressure and initiated the science of hydrostatics. It contains the result on which the rest of the work rests, Archimedes’ Law of Buoyancy (Propositions 6 & 7). As Heath (pp. 259-261) suggests, this principle led to Archimedes’ fabled Eureka! moment, when he realised how to determine whether a certain crown supposed to have been made of gold did not in reality contain a certain proportion of silver (by weighing the crown both in air and when immersed in water, he could determine the specific gravity of the crown, which could then be compared with the known specific gravity of gold). Book I concludes (Propositions 8 & 9) with a simple, elegant geometric proof that a floating segment of a homogeneous solid sphere, the planar base of which is either completely above the fluid surface or completely below it, is in stable equilibrium when and only when its base is parallel to the surface of the fluid. The mechanical tools used were the Law of Buoyancy, the Law of the Lever (from De Aequeponderantibus), and the equilibrium condition that the centre of gravity of the floating body must lie on the same vertical line as its centre of buoyancy (the centre of gravity of its submerged portion). The proof of Proposition 8, which deals with the case in which the base is above the fluid surface, was absent from Moerbeke’s translation (and hence from Tartaglia’s edition), and was supplied by Commandino.

Book II of On Floating Bodies contains many sophisticated ideas and complex geometric constructions and is considered to be Archimedes’ most mature work, commonly described as a tour de force (Clagett, Biographical Dictionary of Mathematicians, vol. 1 (1991), p. 95). In this book Archimedes extended his stability analysis of floating bodies from segments of a sphere to segments of a paraboloid of revolution (or conoid) of various shapes and relative densities, but restricted to the case in which the base of the conoid lies either entirely above or entirely below the fluid surface (the general case can only be completed by using modern mathematical and computational techniques). Although applications are not indicated, it is surely probable that this study was motivated by the problem of the stability of ships. The crucial result is Proposition 2, which gives the condition for stability of a floating segment of a conoid. Moerbeke’s translation contains only the first three introductory paragraphs of the proof of this difficult proposition; the remainder was supplied by Commandino (Heath, pp. 264-266). The proof of Proposition 2 “was especially important and difficult for Commandino since it required fore-knowledge of the determination of the centre of gravity of a paraboloid segment. Archimedes evidently knew the method for determining this, but in none of his then extant works, nor in any other known Greek text, is this method described. In order to rediscover this particular method and to complete the proof, Commandino was therefore compelled to undertake an investigation of the theory of the centres of gravity of solids.

“Commandino’s researches on this are embodied in De Centro Gravitatis, published with the Archimedean edition at Bologna in 1565, and dedicated to Cardinal Alessandro Farnese, the brother of Ranuccio. Commandino begins the dedication by pointing out the notable lack of any classical text on the centres of gravity of solids although Archimedes has dealt with the centres of gravity of planes in his De Aequeponderantibus. Referring to Cervini’s ‘gift’ to him some years previously of On Floating Bodies, Commandino says that a reading of that text had convinced him, however, that either Archimedes or some other mathematician had written a treatise on the subject of solids. This was especially obvious in the case of a certain proposition (book II, prop. 2). An assiduous investigation of Archimedes and other writers persuaded Commandino that he might undertake some sort of treatise on the subject, if not a complete account. When working on this, a certain book of Francesco Maurolico (1494-1575), in which the author affirmed that he had already written a treatise on the centre of gravity of solids, came into the hands of Commandino. (This was probably the spherics collection of 1558 [Theodosii sphaericorum … Messina: Pietro Spira] which contains an Index Lucubrationum that includes the relevant treatise.) Hearing this, Commandino delayed his book for some time in expectation of the appearance of the work of Maurolico, whom he names honoris causa [Maurolico’s work on centres of gravity was not published until 1685]. But after long delay Commandino has now decided to publish his own work, particularly as it complements his edition of On Floating Bodies which is now in press. Since he is the first mathematician ever to treat of the subject in print, Commandino hopes that any errors will be ascribed to his desire to benefit other students” (Rose, pp. 201-202).

The most important (and most difficult) result proved by Commandino in De Centro Gravitatis is that the centre of gravity of a segment of a conoid is situated on the axis two-thirds of the distance from the vertex to the base – this is the result that was assumed by Archimedes in his proof of Book II, Proposition 2 of On Floating Bodies. To prove it, Commandino uses the method of exhaustion: he divides the axis of the conoid into n equal segments, each of length h, say, and then constructs segments of circular cylinders with axis each of these line segments, which are as large as possible subject to being contained in the conoid. This results in an inscribed solid composed to sections of circular cylinders. Using the fact that the extremities of these cylindrical segments lie on a parabola, Commandino shows that the volume of the kth segment from the vertex of the conoid is proportional to k. Taking moments about the vertex he deduces that the distance of the centre of gravity of the inscribed body from the vertex is


where H = nh is the height of the conoid. Similarly, Commandino constructs a circumscribed body consisting of segments of circular cylinders and finds that its centre of gravity is at a distance

2H/3 + h/6

from the vertex. We would now use a convergence argument, allowing n to tend to infinity and h to zero, to reach the desired conclusion that the centre of gravity of the conoid is at a distance 2H/3 from the vertex, but Commandino followed the method of reductio ad absurdum used by Archimedes: the centre of gravity of the conoid must be between those of the inscribed and circumscribed solids, so assuming that its distance from the vertex is other than 2H/3 leads to a contradiction by taking h sufficiently small (or, equivalently, n sufficiently large). We now know that Archimedes had determined the centre of gravity of a conoid using a similar technique in the Method, which was unknown in Commandino’s time. The only extant copy is contained in the Archimedes Palimpsest, discovered by Heiberg in 1906; this also contains the only extant Greek text of On Floating Bodies.

Although Commandino’s work on centres of gravity was motivated by his efforts to complete On Floating Bodies, it led to later developments in the theory of indivisibles and integral calculus by Cavalieri, Torricelli, Wallis, Leibniz, Newton and others. “The application of the method of moments to centre of gravity determinations was important for the development of the calculus in that it provided a valuable field for the deployment of infinitesimal methods through which the concept could be approached both arithmetically and geometrically. It linked volumetric and area determinations, thereby providing a basis for the geometric transformations which played such a fundamental role in integration before the development of any general concept of function. An important paper of Leibniz, the Analysis tetragonistica ex centrobarycis, shows clearly that these processes were not only important in anticipation of the calculus but also that they played a significant role in its actual invention” (Baron, p. 91).

Rose (p. 185) emphasizes the importance of Commandino to the mathematical renaissance of the sixteenth century: “Perhaps the clearest perception of the mathematical renaissance is to be found in the writings of the Urbino school. Not only did Commandino, Guidobaldo dal Monte (1545-1607) and Bernardino Baldi (1533-1617) pursue the revival of Greek mathematics and the restoration of mathematical certainty, but in their thought there also emerged a strong sense of the historical development of mathematics. The idea of a mathematical renaissance is especially evident in the tributes paid to the founder of the Urbino school by his two important pupils. Guidobaldo writes in 1577: ‘Yet in the midst of that darkness (though there were also some other famous names) Federico Commandino shone like the sun. He by his many learned studies not only restored the lost heritage of mathematics, but actually increased and enhanced it. For that great man was so well endowed with mathematical talent that in him there seem to have lived again Archytas, Eudoxus, Hero, Euclid, Theon, Aristarchus, Diophantus, Theodosius, Ptolemy, Apollonius, Serenus, Pappus and even Archimedes himself, for his commentaries on Archimedes smell of the mathematician’s own lamp. And lo! just as he had been suddenly thrust from the darkness and prison of the body (as we believe) into the light and liberty of mathematics, so at the most opportune time he left mathematics bereft of its fine and noble father and left us so prostrate that we scarcely seem able even by a long discourse to console ourselves for his loss.’ And Baldi: ‘Commandino with the greatest diligence and insight restored to light, to dignity and to splendour the works of nearly all the principal writers of the age in which mathematics had flourished.’”

It is highly unusual to find an important scientific book bound as elaborately as the present volume. J. Basil Oldham in Shrewsbury School Library Bindings (Oxford, 1943, pp. 120-121) notes the following regarding an almost identical binding on another book bound for de Guzmán: On both covers there is a “narrow border formed by a simple conventional foliage roll, with a foliage ornament in each angle; in centre, an heraldic stamp; a shield, surrounded with the following letters in circles CGDDMMAHPPMIGPCLA, and surmounted by a coronet under which is a scroll bearing the letters FEI. On the upper cover: arms: two coats impaled: Dexter (arms of Felipe Ramirez de Guzman, Duke of Medina de las Torres, Marquis of Torrel): Two caldrons checky with snakes issuing therefrom, flanked in saltire by ten ermine-tails (5 and 5), within a bordure gobony of Castile and Leon; Sinister (arms of Anna Caraffa, Duchess of Sabbioneta, Mondragone and Trajetto, Princess of Stigliano): Quarterly of six (two in chief and four in base): 1. Per fesse (a) three bars (Caraffa) and (b) a band counter-embattled between six stars (Aldobrandini); 2. a cross patty between four eagles crowned, and over all an escutcheon quarterly of three bars and a lion rampant (Gonzaga); 3. four pallets (Aragon); 4. per fesse a castle (Castile) and a lion (Leon); 5. four pallets flanked in saltire by two eagles crowned (Sicily); 6. a column ensigned by a crown (Colonna). On the lower cover: arms (unidentified): Upon a terrace in base, a plant growing between reeds or tufts of grass; in chief an arched band inscribed REVOLUTA FOECUNDANT, with, beneath it, and ranged in the same manner, three rows of stars.” Ramiro de Guzmán’s arms impale those of his second wife, “Anna Caraffa, daughter of Antonio Caraffa, Duke of Mondragone, and Elena Aldobrandini. He had previously married Marie de Guzman, daughter of Gaspar de Guzman, Count of Olivares, Philip IV’s minister, to whose titles, through his marriage, he succeeded on Olivares’ death in 1645, for which reason he used the acrologic inscription round the shields which Olivares had used as an adjunct to his armorial insignia. The letters (C and G being transposed towards the end) stand for: ‘Comitatui grandatum ducatum ducatum marchionatum marchionatum arcis hispalensis perpetuam praefecturam magnam Indiarum chancellariatum primam Guzmanorum lineam addidit.’ The letters FEI stand for: ‘Fortuna etiam invidente.’ As the owner of the book would not be likely to use the boastful inscription of his father-in-law until he had, by the latter’s death, succeeded to his titles, the book was probably not bound till after 1645, and in Spain, not Naples, because by that time the owner had ceased to be Viceroy of Naples.”

De Guzmán’s library was acquired en bloc by the English diplomat Sir William Godolphin (1635-96), who spent the years 1667-96 in Spain, serving as ambassador from 1672 to 1678.

Adams A1533; Honeyman 131; Riccardi, I, 42-5. II. Adams C2467; Bibliotheca Mechanica, p. 78; Honeyman 739; not in Riccardi. I. & II. Macclesfield 183. Baron, The Origins of the Infinitesimal Calculus, 1969. Heath, The Works of Archimedes, 1897. Rose, The Italian Renaissance of Mathematics: Studies on Humanists and Mathematicians from Petrarch to Galileo, 1975. For a critical edition, translation, and detailed commentary on Commandino’s version of ‘On Floating Bodies,’ see Clagett, Archimedes in the Middle Ages III, pp. 607-681.

Two works bound in one volume, 4to (189 x 137 mm). I. pp. [viii], 43, [1, blank]. II. pp. [viii], 47, [1, blank], with numerous woodcut diagrams in both works. Mid-seventeenth century red morocco, with central gilt arms of the Duke of Medina de las Torres, Felipe Ramirez de Guzmán, on both covers, roll tool borders and corner pieces enclosing the arms, surrounded by an acrostic inscription, lower cover with the emblematic device of three plants growing between reeds with a starry sky and motto ‘Revolvta Foecundant’ within a shield and the same acrologic inscription, spine gilt lettered and tooled, gilt edges (light damp staining at front cover, some sporadic foxing). A splendid copy.

Item #4790

Price: $17,500.00