## Discorsi e Dimostrazioni Matematiche, intorno adue nuove Scienze. Attenenti all Mechanica & i Movimenti Locali … Con une Appendice del centro di gravita ad’alcuni Solidi.

Leiden: Elzevier, 1638.

First edition, the extremely rare and virtually unrecorded first issue, of Galileo’s last and most important work; this issue has the final gathering as a bifolium Rr2, with the final leaf blank, before the addition of the index and errata. The *Discorsi* is “the first modern textbook of physics, a foundation stone in the science of mechanics” (Grolier/Horblit); the ‘two new sciences’ were the engineering science of strength of materials and the mathematical science of kinematics. Galileo presented the work in dialogue form, with the same interlocutors Salviati, Sagredo and Simplicio, as those of the condemned *Dialogo*. The results of his trial before the Inquisition for his support of heliocentrism had left Galileo “so crushed that his life had been feared for” (DSB), and it was only at the urgings of his friend and supporter the Archbishop of Siena, Ascanio Piccolomini, that Galileo set about pulling together his life’s work in physics. “Unable to publish this treatise on mechanics in his own country because of the ban placed on his books by the Inquisition, he published it in Leyden. Considered the first modern textbook in physics, in it Galileo pressed forward the experimental and mathematical methods in the analysis of problems in mechanics and dynamics. The Aristotelian concept of motion was replaced by a new one of *inertia *and general principles were sought and found in the motion of falling bodies, projectiles and in the pendulum. He rolled balls down an inclined plane and thereby verified their uniformly accelerated motion, acquiring equal increments of velocity in equal increments of time. The concept of mass was implied by Galileo’s conviction that in a vacuum all bodies would fall with the same acceleration. Newton said he obtained the first two laws of motion from this book” (Dibner). Subject matter includes, among other things, uniform and accelerated motion, parabolic trajectories, the constitution of matter, the nature of mathematics, the role of experiment and reason in science, the weight of air, the nature of sound and the speed of light. The *Discorsi* “underlies modern physics not only because it contains the elements of the mathematical treatment of motion, but also because most of the problems that came rather quickly to be seen as problems amenable to physical experiment and mathematical analysis were gathered together in this book with suggestive discussions of their possible solution” (DSB). The *Discorsi* was only fully appreciated after the publication of Newton’s *Principia *in 1687. “Mathematicians and physicists of the later seventeenth century, Isaac Newton among them, rightly supposed that Galileo had begun a new era in the science of mechanics. It was upon his foundations that Huygens, Newton and others were able to erect the frame of the science of dynamics, and to extend its range (with the concept of universal gravitation) to the heavenly bodies” (PMM).

In 1589, on the recommendation of Guidobaldo del Monte, Galileo (1564-1642) was appointed to the chair of mathematics at the University of Pisa. While in Pisa, in addition to carrying out his alleged demonstration at the Leaning Tower, he composed an untitled treatise on motion, now usually referred to as *De motu*, in which he attempted to destroy the Aristotelian dichotomy of natural versus forced motions. Its opening sections developed a theory of falling bodies derived from the buoyancy principle of Archimedes, an idea previously published by Giovanni Battista Benedetti in his *Diversarum speculationum* (1585). In the same treatise, Galileo derived the law governing equilibrium of weights on inclined planes and attempted to relate this law to speeds of descent. However, the results did not accord with experience—as Galileo noted—owing to his neglect of acceleration, and he withheld the treatise from publication.

Galileo’s position at Pisa was poorly paid, and he was out of favour with the faculty of philosophy owing to his opposition to Aristotelianism. At the end of his three-year contract he moved, once again with Guidobaldo’s assistance, to the chair of mathematics at Padua, where there were several kindred spirits, notably including Paolo Sarpi. To supplement his university income Galileo gave private lessons on fortification, military engineering, mechanics, and the use of the quadrant for artillerists. “The knowledge of artillerists, which he presumably partook of to accomplish his lessons, became the basis for his emerging new science of motion, published in the *Discorsi *in 1638. It was this fundamental knowledge that allowed Galileo and Guidobaldo del Monte to set up the experiment to demonstrate that the trajectory of a projectile follows a parabolic path, Galileo’s first step toward formulating the law of fall” (Valleriani, p. 200). This experiment, which is described in the *Discorsi*, involved rolling an inked ball obliquely down an inclined plane in order to make visible the path of its trajectory.

“Toward the end of 1602, Galileo wrote to Guidobaldo concerning the motions of pendulums and the descent of bodies along the arcs and chords of circles. His deep interest in phenomena of acceleration appears to date from this time. The correct law of falling bodies, but with a false assumption behind it, is embodied in a letter to Sarpi in 1604. Associated with the letter is a fragment, separately preserved, containing an attempted proof of the correct law from the false assumption. No clue is given as to the source of Galileo’s knowledge of the law that the ratios of spaces traversed from rest in free fall are as those of the squares of the elapsed times … It is probable either that he observed a rough 1, 3, 5, . . . progression of spaces traversed along inclined planes in equal times and assumed this to be exact, or that he reasoned (as Christian Huygens later did) that only the odd number rule of spaces would preserve the ratios unchanged for arbitrary changes of the unit time. From this fact, the times-squared law follows immediately. Galileo’s derivation of it from the correct definition of uniform acceleration followed only at a considerably later date …

“Early in 1609, Galileo began the composition of a systematic treatise on motion in which his studies of inclined planes and of pendulums were to be integrated under the law of acceleration, known to him at least since 1604. In the composition of this treatise, he became aware that there was something wrong with his attempted derivation of 1604, which had assumed proportionality of speed to space traversed … [This treatise] *De motu accelerato*, which correctly defines uniform acceleration and much resembles the definitive text reproduced in his final book, seems to date from this intermediate period” (DSB).

For the next several years, Galileo’s attention turned from mechanics to astronomy, following the great discoveries he made with the newly invented telescope, published early in 1610 in *Sidereus nuncius*. These discoveries made Galileo famous, and in June 1610 he returned from Padua to the University of Pisa as Chief Mathematician and ‘Mathematician and Philosopher’ to the Grand Duke of Tuscany. There he observed and developed theories about comets and sunspots, and began the composition of the *Dialogo*, published in 1632. Following his trial and conviction in 1633, he was sent to Siena, under the charge of its archbishop, Ascanio Piccolomini. Within a few weeks Piccolomini had revived Galileo’s spirits and induced him to take up once more his old work in mechanics and bring it to a conclusion. Early in 1634 Galileo was transferred to his villa in Arcetri, in the hills above Florence. Following the death of his elder daughter in April 1634, Galileo briefly lost interest in his studies, but the unfinished work on motion soon absorbed his attention once more, and within a year it was virtually finished.

The final work, *Discourses and Mathematical Demonstrations Concerning Two New Sciences*, is divided into four ‘days’. “The first two days treat the problems of matter. It is often said that these deal with the strength of materials, but claiming this is the topic makes it difficult to see why Galileo would have considered this to be an important new science. More clearly, they are Galileo's attempt to show the mathematics necessary for and the problems inherent in treating the nature of matter. Days Three and Four are a sustained treatment of the problem of local motion, and they contain the results of his research during his earlier time in Padua” (Machamer, p. 24).

“The book opens with the observation that practical mechanics affords a vast field for investigation. Shipbuilders know that large frameworks must be strongly supported lest they break of their own weight, while small frameworks are in no such danger. But if mathematics underlies physics, why should geometrically similar figures behave differently by reason of size alone? In this way the subject of strength of materials is introduced. The virtual lever is made the basis of a theory of fracture, without consideration of compression or stress; we can see at once the inadequacy of the theory and its value as a starting point for correct analysis. Galileo’s attention turns next to the problem of cohesion. It seems to him that matter consists of finite indivisible parts, *parti quante*, while at the same time the analysis of matter must, by its mathematical nature, involve infinitesimals, *parti non quante*. He does not conceal—but rather stresses—the resulting paradoxes. An inability to solve them (as he saw it) must not cause us to despair of understanding what we can. Galileo regards the concepts of ‘greater than,’ ‘less than,’ and ‘equal to’ as simply not applicable to infinite multitudes; he illustrates this by putting the natural numbers and their squares in one-to-one correspondence.

“Galileo had composed a treatise on continuous quantity (now lost) as early as 1609 and had devoted much further study to the subject. Bonaventura Cavalieri, who took his start from Galileo’s analysis, importuned him to publish that work in order that Cavalieri might proceed with the publication of his own *Geometry by Indivisibles*. But Galileo’s interest in pure mathematics was always overshadowed by his concern with physics, and all that is known of his analysis of the continuum is to be found among his digressions when discussing physical problems.

“Galileo’s *parti non quante* seem to account for his curious physical treatment of vacua. His attention had been directed to failure of suction pumps and siphons for columns of water beyond a fixed height. He accounted for this by treating water as a material having its own limited tensile strength, on the analogy of rope or copper wire, which will break of its own weight if sufficiently long. The cohesion of matter seemed to him best explained by the existence of minute vacua. Not only did he fail to suggest the weight of air as an explanation of the siphon phenomena, but he rejected that explanation when it was clearly offered to him in a letter by G. B. Baliani. Yet Galileo was not only familiar with the weight of air; he had himself devised practicable methods for its determination, set forth in this same book, giving even the correction for the buoyancy of the air in which the weighing was conducted.

“Phenomena of the pendulum occupy a considerable place in the *Two New Sciences*. The relation of period to length of pendulum was first given here, although it probably represents one of Galileo’s earliest precise physical observations. Precise isochronism of the pendulum appears to have been the one result he most wished to derive deductively. In discussing resistance of the air to projectile motion, he invoked observations (grossly exaggerated) of the identity of period between two pendulums of equal length weighted by bobs of widely different specific gravity. He deduced the existence of terminal constant velocity for any body falling through air, or any other medium, but mistakenly believed increase of resistance to be proportional to velocity.

“Like the pendulum, the inclined plane plays a large role in Galileo’s ultimate discussion of motion. The logical structure of his kinematics, as presented in the *Two New Sciences*, is this: He first defines uniform motion as that in which proportional spaces are covered in proportional times, and he then develops its laws. Next he defines uniform acceleration as that in which equal increments of velocity are acquired in equal times and shows that the resulting relations conform to those found in free fall. Postulating that the path of descent from a given height does not affect the velocity acquired at the end of a given vertical drop, he describes an experimental apparatus capable of disclosing time and distance ratios along planes of differing tilts and lengths; finally, he asserts the agreement of experiment with his theory. The experiments have been repeated in modern times, precisely as described in the *Two New Sciences*, and they give the results asserted. Following these definitions, assumptions, and confirmation by experiment, Galileo proceeds to derive a great many theorems related to accelerated motion.

“In the last section Galileo deduces the parabolic trajectory of projectiles from a composition of uniform horizontal motion and accelerated vertical motion. Here the concept of rectilinear inertia, previously illustrated in the *Dialogo* (‘Second Day’), is mathematically applied but not expressly formulated. This is followed by additional theorems relating to trajectories and by tables of altitude and distance calculated for oblique initial paths. Because of air resistance at high velocities, the tables assumed low speeds and hence were of no practical importance in gunnery. But like Galileo’s theory of fracture, they opened the way for rapid successive refinements at the hands of others” (DSB).

The *Discorsi* concludes with an appendix on centres of gravity. In 1587, just two years after Galileo started learning Euclid and Archimedes, he was recommended for a lectureship in mathematics at the University of Bologna. The university refused to employ him, however, because Galileo did not then have any written works that demonstrated his ability. Galileo therefore began a new area of research. “The title of the output of this research is *Theoremata circa centrum gravitatis solidorum*, published first in 1638 as an appendix to the *Discorsi. *Most of the *Theoremata *were written between 1587 and July 1588. It is one of Galileo’s ‘unpublished treatises,’ representing a further extension of Archimedes’ mechanics into the context of practical applications. They present theorems that demonstrate where the centers of gravity of certain bodies are located. In particular, Galileo found the center of gravity of ideal balances, whose weights are hung in a few different and well-determined distributions, and also the center of gravity of bodies like those determined by a cross-section of a parabolic conoid. Galileo probably did not approach this study casually. From a letter by Guidobaldo del Monte, it is possible to infer that several mathematicians were trying to improve, or better said, to generalize the final proposition of Federico Commandino’s *Liber de centro gravitatis solidorum*, as one of Galileo’s theorems from the *Theoremata *tries to do as well” (Valleriani, 15-16).

The Roman mathematician Luca Valerio, who had first met Galileo in Pisa in 1590, influenced him to renew his studies on centres of gravity. “Early in 1609 Galileo sent his demonstration that a parabolic line through the corners of a rectangle divided its area in the ratio of one-third to two-thirds to a friend in Rome for delivery to Luca Valerio, whose book on centers of gravity and quadrature of the parabola, *De Centro Gravitatis Solidorum libri tres* (1604), he greatly admired. Galileo had forgotten his meeting with Valerio at Pisa nearly twenty years before, of which the Roman mathematician reminded him in reply, praising Galileo’s demonstration. The correspondence thus opened resulted in Galileo’s sending to Valerio for criticism two principles upon which he intended to establish his treatise on motion, now greatly expanded, in June 1609” (Drake, p. 136). “Galileo repeatedly stated that he had given up the idea of publishing his early work, *Theoremata*, because ‘some time later, he ran across the book of Luca Valerio, a prince of geometers, and saw that this resolved the entire subject without omitting anything; hence he went no further, though his own advances were made along quite a different road from that taken by Valerio’” (Napolitani & Saito, p. 108, n. 1).

With all his writings banned by the Inquisition, Galileo hoped to have the *Discorsi *published in Leiden by the Elzeviers, who had already published the Latin translations of the *Dialogo* and *Letter to the Grand Duchess Christina* in 1635 and 1636. This was agreed when Louis Elzevier visited Galileo at Arcetri in May 1636. By December, with Fulgenzio Micanzio acting as intermediary, Elzevier was back in Leiden and in possession of the first three days. On 16 March of the following year, Elzevier wrote to Micanzio asking for the rest of the book; he was able to forward the fourth day to Elzevier in June. Galileo had considered including a fifth day on percussion, and his indecision over this resulted in some delay in the completion and printing of the *Discorsi*.

Elzevir wrote to Galileo on 4 January 1638 (Opere XVII p. 251, letter 3640), stating that he had sent off gatherings Gg to Pp, and saying that he was awaiting Galileo's index and errata. Elzevir wrote again on 25 January (op. cit. p 311, letter 3702), enclosing proofs of the final pages and again requesting the index and list of errata, along with text for the dedication to Comte François de Noailles, who had been a pupil of Galileo’s in Padua, and was now French ambassador at Rome. Noailles had attempted, but without success, to alleviate Galileo’s detention at Arcetri. In the autumn of 1636, Galileo met Noailles in Poggibonsi and may have given him a copy of the *Discorsi *manuscript there. The dedication was dated Arcetri, 6 March 1638; in it, Galileo praised the publishers for their taste and skill.

“The printed *Two**New **Sciences* appears not to have been offered for sale before June 1638, despite its March dedication. On 7 May Micanzio reported his surprise that the book was not even mentioned in a list of titles sent by Elzevier to his Venetian agent. Noailles did not acknowledge receipt of the dedication copy until 20 July. No copies were sent in advance to the author … From a letter to Diodati supposed to have been written by Galileo on 7 August it appears that only a single copy of his book had been received, and only after he was totally blind and could not see it” (Drake, p. 386).

This is the first issue, exceptionally rare and virtually unknown. The normal issue ends with a four-leaf gathering Rr4, comprising the final leaf of text (pp. 305-6, *recte *313-14) on Rn, followed by two leaves listing the principal topics of the text in alphabetical order, 'Tavola delle case piu notabili in ordine alfubetico' on Rr2-3, followed by errata on Rr4. The verso of Rr1 has the catchword 'Tavo-.' indicating the first word on the following page 'Tavola'. The first issue, as here, ends with the gathering Rr2, with no catchword on Rr1 verso, and with the conjugate leaf, Rr2, blank.

Although a few other copies of this issue are known, it has not been properly recorded in the Galileo bibliographies. This copy came from the science collection of the head of the Royal Swedish Academy of Sciences, and was sold almost 40 years ago by Ove Hagelin (Cat. 9, n. 52, ca. 1977); it was noted by Hagelin that a similar copy was described in Charles Traylen's catalogue 66 in 1966. It was then offered by W.P. Watson (Cat. 19, n. 38, 2013) – part of our description of this first issue is based upon his. The dedication copy to Noailles, sold by Pierre Berge in 2017, was also of this first issue. We have located one other copy (Wellcome), but it is difficult to know exactly how many copies of this issue exist as online institutional collations are often inaccurate.

Carli and Favaro 162; Cinti 102; Dibner 141; Evans, *Epochal achievements in the history of science *27; Horblit 36; Norman 859; Parkinson, *Breakthroughs *pp. 80-81; *Printing and the Mind of Man *130; Riccardi I 516; Roberts & Trent, *Bibliotheca Mechanica*, pp. 129-130; Sparrow, *Milestones *75; Wellcome 2648; Willems 468. Drake, *Galileo at Work*, 1978. Machamer (ed.), *The Cambridge Companion to Galileo*, 1998. Napolitani & Saito, ‘Royal Road or Labyrinth? Luca Valerio’s *De Centro Gravitatis Solidorum* and the beginnings of modern mathematics,’ *Bollettino di Storia delle Scienze Matematiche* 24 (2004), pp. 67-124. Valleriani, *Galileo Engineer*, 2010.

4to (190 x 153 mm), pp. [viii], 306 [recte 314], [2, blank], with numerous woodcut diagrams and illustrations in text (corner of Hh4 with repaired tear without loss of text, some very light waterstaining on margins of a few gatherings). Near-contemporary Dutch quarter sheep spine and marbled boards (joints cracked but sound, very rubbed), housed in a black morocco box. Overall a very fresh, crisp copy, completely untouched.

Item #4980

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Price:
$325,000.00
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