## 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, a fine copy in untouched contemporary vellum, of Galileo’s last and most important work, “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). Copies in fine condition in contemporary bindings are rare on the market – copies are very often found in 18^{th} century bindings (about 50% of those listed on ABPC/RBH)).

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.

4to (199 x 150 mm), pp. [viii], 306 [recte 314], [6], with numerous woodcut diagrams and illustrations in text (former owner’s name written neatly on title and two more leaves, some light browning). Contemporary vellum (a few spots). An excellent copy, completely untouched.

Item #4985

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Price:
$150,000.00
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