## Quinto libro degli Elementi d’Euclide, ovvero Scienza universale delle proporzioni speigata colla dottrina del Galileo, con nuov’ ordine distesa, e per la prima volta pubblicata … Aggiuntevi cose varie, e del Galileo, e del Torricelli; i ragguagli dell’ultime opoere loro, con altro, chedall’ indicie si manifesta.

Florence: Condotta, 1674.

The true first edition, very rare, **signed and inscribed by Viviani to Edme Mariotte**, of this important Galileanum, **an assembly of previously unpublished writings by Galileo**, together with texts by Torricelli and Viviani himself. Most importantly, it contains the first printing of a Galileo manuscript concerned with the Eudoxian theory of proportion which may be regarded as a supplement to the *Discorsi *(1638); the manuscript had been given to Viviani (1622-1703), Galileo's distinguished pupil, amainuensis, and biographer, by Cardinal de' Medici. Galileo had composed this manuscript in response to Viviani’s demand that Galileo provide a more secure geometrical basis for his theory of falling bodies. As Galileo wrote to Benedetto Castelli in 1639, ‘Objections made to me many months ago by this young man [Viviani] who is now my guest and disciple, against that principle postulated by me in my treatise on accelerated motion … made me think about this again in such a way as to persuade him that that principle might be conceded as true. Finally, to his and my great delight, I succeeded in finding a conclusive demonstration” (Boschiero, pp. 43-44). The manuscript had been dictated, in dialogue form, by Galileo to Torricelli in November 1641 (Galileo died on January 9, 1642). Since Torricelli’s book on the theory of proportion (written in 1647) was not published, this edition by Viviani of Galileo’s Fifth Day of the *Discorsi* (pp. 61-78) is the first printing of Galileo’s reform of Book V of the *Elements*. It contains “Galileo’s reflections on two definitions found in Euclid’s *Elements*, that of ‘same ratio’ in Book V, and that of ‘compound ratio’ on Book VI. These were the two most important keys taken from antiquity in creating Galileo’s mathematical physics … Galileo’s critique of these definitions is by no means trivial. His discussion of Book V, Definition V, shows how a rigorous theory of irrational magnitudes can be built on the natural numbers by means of equimultiples … That is a large step in formulating a rigorous analysis of the continuum. His discussion of the spurious Book VI, Definition V illuminates the nature of mathematical definitions in general, essential to foundational analysis” (Drake, pp. 421-422). Galileo’s treatise is prefaced by Viviani’s own attempt to render more rigorous the theory of proportions, which includes ten new principles (pp. 1-60). The work also includes 12 letters from Galileo to an unnamed French scholar, an unpublished treatise by Galileo on the ‘angle of contact’, other unpublished works by Galileo on topics including the construction and use of telescopes and the movement of animals, and writings by Torricelli on Euclid, Book VI, and by Viviani on Euclid, Book I. A second edition was issued in 1676, with a second part containing the solution of 36 problems which had been published as challenges by a Leiden student in 1675, and with the dedication dated 16 May 1676, although the title page date remained 1674. The first edition is very rare: ABPC/RBH list only this copy since 1961. Institutional holdings are difficult to assess as collations are rarely given and the title page carries the same date in both editions.

*Provenance*: Edme Mariotte, French physicist known for formulating Boyle’s law independently – “honored as the man who introduced experimental physics into France, Mariotte played a central role in the work of the Paris Academy of Sciences from shortly after its formation in 1666 until his death in 1684” (DSB) (inscribed by Viviani on title verso ‘A Monsieur Mariotte’ and ‘Vinc. Viviani’ at bottom right); inscription of a Jesuit seminary on half-title recto, shelf-marks on front endpapers.

“Viviani (1622-1703) was the son of Jacopo di Michelangelo Viviani, a member of the noble Franchi family, and Maria Alamanno del Nente. He studied the humanities with the Jesuits and mathematics with Settimi, a friend of Galileo’s. His intelligence and ability led to his presentation in 1638 to Ferdinand II de’ Medici, grand duke of Tuscany. Ferdinand introduced him to Galileo, who was so impressed by his talent that he took him into his house at Arcetri as a collaborator in 1639” (DSB).

“Often described as ‘Galileo’s last student,’ Viviani is a principal actor in the standard account of the reception of the *Two New Sciences*, included for his devotion to his late teacher and for the volume of textual material he preserved and produced … Viviani is often considered in tandem with another of Galileo’s students, Evangelista Torricelli. Torricelli and Viviani served successively in Galileo’s former position as mathematician to the Grand Duke of Tuscany … After Galileo’s death, both Viviani and Torricelli continued to work with Galileo’s unpublished manuscripts. They developed some of these writings into two additional ‘days’ to augment the 1638 publication. The Fifth Day was published in 1674 by Viviani as *Quinto Libro degli Elementi d’Euclide ovvero Scienza Universale delle Proporzioni* (Raphael, pp. 47-48).

“During the last months of Galileo’s life, as Viviani embarked on a career inside the Tuscan Court, they worked together in order to strengthen the role of mathematics in natural philosophy … The topic they worked on together to achieve this was the geometrical demonstration of accelerating falling bodies. Galileo’s and Viviani’s combined efforts to illustrate accelerated motion on inclined planes, resulting in the *scholium* Viviani added to the Third Day of *Two New Sciences*, became a major part of Viviani’s education, shaping the natural philosophical skills, commitments, and agendas he was to use during his entire career …

“To complete his dynamical analysis of the ratios of speeds of bodies falling down differently inclined planes, [Galileo] wrote the following postulate; a critical part of his mathematical demonstration of the physical phenomenon of accelerating falling bodies: the body always reaches the same speed, or final velocity, at the bottom of each plane. For Galileo, this is the final piece in the puzzle for understanding the speeds of falling bodies. In his attempt to verify this postulate, Galileo relied upon an experiment with a pendulum, described in the text by Salviati, the interlocutor representing Galileo …

“However, when Viviani came across this topic in his readings of *Two New Sciences *during the early months of 1639, he doubted that Galileo’s pendulum experiment, and his additional geometrical corollaries, provided a convincing explanation of the postulate … Viviani, therefore, insisted that Galileo provide more convincing demonstrations of the postulate … In *Le Meccaniche *(c.1594) Galileo used the claims by Pappus of Alexandria as a springboard for his own discussion regarding ratios of force and weight on inclined planes … Pappus claimed that … as the inclination of the plane varies, the opposing force needed to resist the body from falling would always be the same …

“Following this recall of Galileo’s previous reflections concerning inclined planes, he and Viviani discussed Eudoxian proportion theory as propounded by Euclid in Book V of the *Elements*. They were seeking a geometrical and mechanical demonstration of the notion that bodies descending along differently inclined planes, but from equal vertical heights, reach the same speed at the bottom of the plane. That is, that two bodies descending a vertical and an inclined plane accelerate uniformly, but at different rates, so that the greater distance needed to cover the inclined plane is proportional to the time needed to reach the same speed as in the vertical. This way, through their search for a geometrical demonstration of the postulate, Galileo and Viviani were establishing a new dynamical theory that relied upon Galileo’s early thoughts as expressed in *Le Meccaniche*, as well as a reconsideration of Euclid’s version of Eudoxus’ proportion theory.

“As Drake argues, the use of ancient geometers, whose works had come to light as crucial to natural philosophical studies only during the sixteenth century, ‘lies at the basis of most of Galileo’s applications of mathematics to physics’. As a result, they also formed the basis of Viviani’s education in natural philosophy. Galileo and his young student were particularly interested in exploiting Euclid’s concept of ‘same ratios’, as given in Book V, Definition 5 of the *Elements*. How this definition is used in the Galilean arguments concerned with natural motion along inclined planes, and in particular, Galileo’s postulate regarding final speeds, is clear from the scholium composed by Galileo with Viviani’s assistance after 1638, and added by Viviani to subsequent editions of *Two New Sciences* … Galileo’s work and Viviani’s contribution, therefore, were based largely on the ancient readings regarding Eudoxian proportion theory. This was a rigorous mathematical and geometrical exercise which had occupied a great deal of Galileo’s career since his beginnings in Pisa. Thanks largely to Viviani’s enthusiasm for further exploring this notion of ratios between weight and force on inclined planes, Galileo arrived at a more convincing demonstration for the postulate” (Boschiero).

In another section of the present work, ‘Ragguaglio delle ultime opere del Galileo’ [‘Summary of Galileo's last works,’ pp. 86-106], Viviani provides valuable information regarding Galileo’s thoughts of the dynamical similarity between projectile motion and the shape of hanging chains, which he had introduced in the *Discorsi *and had promised to develop in the Fifth Day which, however, was never finished.

“In contrast to the modern historian, Viviani was in the unique position to simply ask Galileo what these obscure references to the utility of the chain in the *Discorsi* really meant – which is what he actually did when he was living with Galileo from late 1638 until the latter ’s death in 1642.143 When he later included recollections of this period in a book which he published in 1674 [i.e., the present work], Viviani explained what he had learned from Galileo himself about the utility of the chain and its relation to projectile motion (pp. 105-106):

‘Now all I have left to say is how much I know about the use of chains, promised by Galileo at the end of the Fourth Day, referring to it as he intimated when, he being present, I was studying his science of projectiles. It seemed to me then that he intended to make use of some kind of very thin chains hanging from their extremities over a plane surface, to deduce from their diverse tensions the law and the practice of shooting with artillery to a given objective. But of this Torricelli wrote adequately and ingeniously at the end of his treatise on projectiles, so that this loss is compensated. That the natural sag of such chains always adapts to the curvature of parabolic lines, he deduced, if I remember well, from a reasoning similar [to this]: Heavy bodies must naturally fall according to the proportion of the momentum they have from the places from which they hang, and these momentums of equal weights, attached to points of a balance [which is] supported by its extremities, have the same proportion as the rectangles of the parts of that balance, as Galileo himself demonstrated in the treatise on resistances. And this proportion is the same as the one between straight lines, which from the points of that same balance [taken] as the base of a parabola, can be drawn in parallel to the diameter of this parabola, according to the theory of conic sections. All the links of the chain, that are like so many equal weights hanging from points on that straight line which connects the extremities where this chain is attached and serves as base of the parabola, have in the end to fall as much as permitted by their momentums and there [they have to] stop, and [they] must stop at those points where their descents are proportional to their momentums from the places where those links hang, in that last instant of motion. These then are those points which adapt to a parabolic curve as long as the chain and whose diameter, which raises from the middle of the said base, is perpendicular to the horizon.’

“The first part of Viviani’s text confirms that Galileo intended to introduce the chain as an instrument by which gunners could determine how to shoot in order to hit a given target. The main part of Viviani’s text represents the sketch of a proof. If his report on that proof is reliable, it implies that Galileo had planned to crown his life-long reflections about the relation between projectile trajectory and chain with a proof of the alleged parabolic shape of the catenary, a proof that would have become a key subject of the never-finished Fifth Day of the *Discorsi*” (Renn *et al*., p. 90).

‘Parere del Galileo intorno all'angolo del contatto’ (‘Galileo’s opinion about the angle of contact’, pp. 107-113) is a letter written in 1635 to the mathematician Giovanni Camillo Gloriosi (1572-1643), who had succeeded him in the chair of mathematics at Padua in 1613. It concerned the ‘angle of contact’ or ‘angle of contingence’, a topic discussed by many seventeenth century mathematicians. The question was whether the angle between the tangent to a circle and the circle at the point of contact was a true angle, like the angle between two intersecting straight lines, a view taken by Clavius, or a quantity of a different kind, not comparable to ordinary angles, as advocated by Peletier. “Only two letters of scientific interest written by Galileo in the latter part of 1635 survive. One dealt with the purely mathematical question of the angle of contact addressed Glorioso” (Drake, p. 370). Galileo sided with Peletier, providing a number of different geometrical arguments in support of his position.

“Edme Mariotte, a Roman Catholic priest and prior of Saint-Martin-sous-Beaune, was in 1666 one of the founding members of the Academy of Sciences, in Paris. In his *Discours de la nature de l’air* (1676), in which he coined the word barometer, Mariotte stated Boyle’s law and went farther by noting that the law holds only if there is no change in temperature. From his studies of plants, he concluded that they synthesize materials by chemical processes that vary from plant to plant—a theory verified long after his time. He also observed the pressure of sap in plants and compared it to blood pressure in animals. The first volume of the *Histoire et Mémoires de l’Académie* (1733) contains many papers by him on such subjects as the motion of fluids, the nature of colour, and the notes of the trumpet” (Britannica).

Viviani’s connection to Mariotte may, of course, have been simply a result of Mariotte’s important position in the Academy of Sciences in Paris. But both men were at the time interested in the science of the resistance of materials, which Galileo had sought to found on the principle of the lever.

Carli-Favaro 339; Cinti, 151; Riccardi II.625-7.2. Boschiero, *Experiment and Natural Philosophy in Seventeenth-Century Tuscany*, 2007. Drake, *Galileo at Work*, 1974. Gatto & Palladino, ‘The ‘Dutch’s Problems’ and Leibniz’s point of view on the ‘Analytic Art’,’ Studia Leibnitiana 24 (1992), pp. 73-92. Raphael, *Reading Galileo**: Scribal Technologies and the* Two New Sciences, 2017. Renn *et al*., ‘Hunting the white elephant. When and how did Galileo discover the law of fall?’ *Science in Context* 13 (2000), pp. 299-419. On the angle of contact, see Palmieri, ‘Galileus Deceptus, Non Minime Decepit: A Re-appraisal of a Counter-argument in *Dialogo* to the Extrusion Effect of a Rotating Earth,’ *Journal for the History of Astronomy* 39 (2008), pp. 425-452.

4to (232 x 165mm), pp. [xii], 149, [3], woodcut head- and tailpieces and diagrams to text (browning, spotting, minuscule worm trace in blank upper margin of first few leaves). Contemporary vellum with manuscript lettering on the spine (some soiling to boards).

Item #5827

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Price:
$9,500.00
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