Chapter 10 Chapter 8 Kepler, Giber, and Galileo

A revolution in physics? Scholars who write about the Copernican revolution often conclude that it did not happen before the innovations of Kepler and Galileo.In fact, the bold and original ideas of these two scientists went far beyond the scope of the naive Copernican theory.Galileo was an ardent advocate of Copernican's theory, which he tried to confirm based on his discoveries with his telescope.However, his contribution to kinematics was done by means of mathematical analysis and experimentation, which was more revolutionary than the work of his predecessor Copernicus.Kepler was also said to be a disciple of Copernicus, although he eventually abandoned all but two of the most general Copernican axioms: the sun is stationary; Not only rotation, but also revolution.In place of the complicated approach of the Devolution of the Celestial Spheres, Kepler proposed a new and entirely different astronomical system of the universe that is still largely accepted today.He also proposed a new mechanical basis for the whole of astronomy.

Kepler's dual-purpose reformulation of astronomy is by far the most revolutionary.But we must ask, was this a silent or covert revolution, or an overt one?If the latter, did it happen in its day?Is it, in essence, a scientific revolution unaffected by outside influences?Does it stay at the stage of discussing revolution?The same question must be asked of Galileo.We should also briefly consider the work of William Gilber; a contemporary of Kepler and Galileo, and the oldest of the three.He was a revolutionary, not only in his advocacy of experimental techniques but also in his idea that the Earth was a giant spherical magnet.This point of view led Kepler to get such a hint that the magnetic force of the planets might be the dynamic factor causing the motion of the planets.

Kepler: The Incredible Revolutionary Johannes Kepler devoted himself to the study of planetary dynamics (that is, the analysis of the effects that cause planetary motion) and a kind of astronomy based on the factors of physics rather than on the dogma of kinematics, from some In a sense, he was indeed a modernist.However, he still carries a deep brand of tradition.A great believer in astrology (in fact, he was the last important combined astronomer and committed astrologer), his scientific thinking was imbued with so-called number mysticism, and he The argument begins with the basic principles of cosmological inevitability.He was particularly proud of his early "discovery" that there was a direct relationship between the number, size, and order of planetary orbits and the existence of five (and only five) regular geometric bodies.Among his greatest discoveries was one in which he had the good fortune to eradicate the effects of an important mathematical error, but by counteracting the first with another error.Kepler was one of the greatest astronomers who ever lived; yet we can easily put together some of his writings, which show how unscientific his thinking and his theories were.

The title of Kepler's treatise, published in 1609, boldly indicates the revolutionary nature of his astronomy; he says he has created a new astronomy.There are several reasons why this science of astronomy is new.But Kepler only emphasized in the title of this work that this new astronomy is "based on various causes," and that it is a "new astronomy" (Kepler printed this word into Greek).Or, keep the title, but say, the book is a Phpica Coelestis or a work of astrophysics.By using this term Kepler seems to have indicated that he was taking a step beyond Aristotle.Aristotle's metaphysics was developed after his physics, and Kepler wanted to replace Aristotle's metaphysics with his own new astrophysics.As Kepler noted in his letter to Johann George Brengel on October 4, 1607 (1937, 16:54), in his forthcoming book he proposes his new "Philosophy or astrophysics, so as to replace astrotheology or Aristotelian metaphysics." In the Introduction to the New Astronomy, Kepler makes a similar statement, and he further explains that he has already made a point of "the nature of motion Cause" is explored and studied (3:20).This book is quite a radical program, so radical that it uses the actuation of celestial bodies to explain the motion of planets. If you want to understand how radical it is, you only need to note: in this respect, Kepler can be said to be unprecedented, and at that time. No bosom friend.Even the great Galileo never conceived of celestial mechanics such a dynamical system leading to motion.No wonder Alexander Koyre (1961, 166) passionately wrote, "What the title of Kepler's work suggests is not a symptom but a revolution."

Kepler's astronomy is a comprehensive reconstruction of the discipline according to its purpose, method and basic principles.Before Kepler, the aim of astronomers was purely photographic, that is, they aimed at creating a celestial geometry (based on a circle within a circle) that gave The positions of the planets are consistent with observations.Kepler wanted to find out the real physical cause of motion, that is, the reason for motion, not just to invent or perfect a geometric system.Because he believes that the sun is the center of the power mentioned here, and the sun must be located at the center of the universe.Thus, the real sun - not Copernicus' "flat sun" - lies at the intersection of the common orbital planes of all the planets.

As for the method, Kepler was concerned with finding, by the use of mathematics, the actual force due to the action of the sun, without any arbitrary or restrictive restrictions on orbits, on uniform motion, etc., at all. The curve (size, shape, direction) of a planet's orbit.After some painstaking efforts, he discovered that each planet moves on an elliptical, simple convex curve orbit.For most planets (with the exception of Mercury), the shape of the elliptical orbit is not much different from a perfect circle, but the sun is not at the center, or even close to it; A circular orbit (or, rather, a quasi-circular elliptical orbit) in which the Sun is distinctly off-centre (or off-centre).Kepler also discovered that the motion of planets along elliptical orbits is not uniform, but directly coincides with the law of area.This law also explains why each planet moves very quickly at apohelion (or in an orbit close to the sun), but very slowly at aphelion (farther from the sun).

Kepler's astronomy was an astrophysics of forces directly related to its concept of bodies, based on a new set of principles of motion.In his view, a planet or a satellite of a planet ("satellite" is the word he introduced into astronomy), or some physical object, is like a large inanimate rock; it has no intrinsic or dynamic forces of its own. Because of this disposition (Kepler called it "inertia"), the body can neither set itself in motion nor keep itself in motion. To move, the body needs a push. Obviously, because of this passivity or inertia, objects must stop moving whenever and wherever power is lost or ceases to function. This may not seem like a radical conclusion to a reader of the double century, but it is in line with 2000 The views of science and philosophy, which have been conditioned by Aristotle's thought for many years, are tit-for-tat; according to Aristotle's thought, an object will stop moving only when it has reached its "natural position".This natural position theory posits a hierarchical space in which heavy objects "naturally" move toward a center below and light objects move upward.The space in which objects move in the kingdom of heaven is different from the space in which objects move or remain still in the "world" because such objects have different levels in nature and their ultimate composition is also different.Obviously, a person like Kepler who believes in the Copernican theory has adopted the viewpoint of the earthquake theory, so he must abandon the natural position creed and the related hierarchical space theory.Kepler proposed the following new basic principles: space is isotropic, space is not hierarchical, there are no natural positions, and matter is inert.In proposing new principles, he revealed what Copernican thought suggested, that the earth itself, as well as the moon and other planets, belonged to the same physics.Kepler's physical principles of inertia, force, and motion heralded the end of Aristotelian cosmology and the dawn of Newtonian science.

If the motions of all the planets were directly governed by the motion of the sun (since all the planets moved in ellipses of which the sun was at one focus, moreover, the motions of all the planets in their orbits were governed by Constrained by the area law of the planet), then there must be a force acting on the planet pointing towards the sun, which is deduced from Kepler's idea: the planet is inert in nature, so there must be a force to keep them on track.Kepler concluded that this force must be magnetic.He knew, William Gibb proved, that the earth was a gigantic spherical magnet.Since the earth is a planet, why can't other planets be magnets.Wouldn't the sun be a magnet?The orientation of the magnetic poles of the sun and planets determines that the orbits are elliptical rather than purely circular.

Kepler's concept of inertia differed from that developed by Galileo (later refined by Descartes) and Newton.However, his astronomy was more like that of Newton than that of Galileo or Descartes, in that he related orbits and orbital motions to causative forces.Kepler may have been wrong about the function of force (thinking that force varies inversely with distance rather than the square of distance), but that is not important, except that perhaps he was the first to conceive of a celestial body , and realized that the effect of this force must be some kind of function that is inversely proportional to distance.

In the preface to Rudolph's Tables, Kepler pointed out that one of the main features of his (we would say innovative and revolutionary) work was that the whole of astronomy "moved from unreal circular reasoning to Look for transformations of natural causes." Kepler said that Copernicus created his system a posteriori on the basis of observations, but he asserted that the true arrangement of the universe can be a priori obtained from cosmology, from the physical confirmed in its nature and properties.Indeed, Kepler thought such a proof would have satisfied even Aristotle had he been alive.Therefore, Kepler believed that he had far surpassed Copernicus in pursuing the ultimate cause.In his letter to Fabricius of July 4, 1603 (1937, 14:412), he wrote that his astronomy had been tested and confirmed by astronomical observations.In this sense, as Eric Ayton pointed out in his letter to me on March 17, 1979, Kepler's "transcendental reasoning contains not necessary consequences but only possible result."

There is no doubt that Kepler provided a revolutionary program for the development of astronomy.Because he is a person who likes to reflect, he has recorded in detail the development process of his ideas and methods.We have already described in detail the occasion for his discovery of, for example, the third law of planetary motion.In his "New Astronomy" he sets out with great care and detail the various periods of his revolution in thought and belief; And the evolution of computing, these developments led him to eventually abandon traditional circular motion astronomy and start exploring the curves of other possible classes of orbits.While the reader will be weary of the imperiled calculations on the following folios, Kepler reminds himself not to forget how much he suffered by doing them by hand.After he got the answers, he published them as samples.With his major works—or Mysteries of the Universe (1596), New Astronomy (1609), Rudolph's Tables (1627), or Cosmic Harmony (1619) and Copernicus The publication of "Summary of Astronomy" (1618-162) and so on, a revolution in thought completely turned into a revolution in treatises, the book has been published, and anyone can read and use it. So, has there been a scientific revolution?Did the revolution in Kepler's treatises change the practice of the great astronomers, and had it become so fundamental to astronomical thought that a Kepler revolution in science would follow?In my opinion, the answer is no.First, the generations of astronomers from Kepler to Newton did not fully embrace the new Kepler astronomy.The dominant astronomical thought, for example, soon changed, and the system of Cartesian vortices became its center rather than the study of the dynamics of the action of celestial bodies advocated by Kepler.In part, this was a result of Kepler not being able to invent a new mechanics sufficient for the needs of astronomy as Newton eventually did.Kepler tried, but failed (and could not) to create a celestial mechanics based on a modified version of Aristotle. Secondly, there are also people who hold diametrically opposed views on the idea that there may be some solar force stretching hundreds of millions of miles in the sky.For example, Galileo neither recognized nor applied Kepler's three laws of planetary motion when explaining Copernican astronomy.In his Dialogue Concerning Two World Systems, Galileo specifically criticized Kepler for suggesting that controlling forces could move outward through space in the same way that the moon might cause the tidal motion of our oceans.Although the law of elliptical orbits (Kepler's first law) is generally accepted by practicing astronomers, the role of the second or "idle" focus of the ellipse remains puzzling, and, due to the However, there are still considerable and quite "natural" objections to the fact that planetary orbits are not circular in shape.To many astronomers, the area law (Kepler's second law) seems to confuse them conceptually rather than help them.In any case, as Kepler himself noted, this law cannot be used as a basis for exact calculations of planetary positions unless some approximation is used. In order to replace Kepler's law of areas, the Astronomers intend to borrow a direct approximation based on the uniform rotation of the moment of magnitude centered on an idle focus (which can be used as a kind of bisection point), even for those who are willing to accept and use The laws themselves are also oddities to those who know them, since they do not seem to have any causal or deductive connection to the accepted rationale. Many astronomers did recognize Kepler's third or harmonious law (it was published in the 1619 Cosmic Harmony rather than in the 1609 New Astronomy), where Kepler The law shows that the ratio of the square of a planet's sidereal period to the cube of its mean distance from the sun is constant.However interesting this third law is, it is of no practical use, since it makes no predictions, has no obvious physical reason, justification, or proof, and, moreover, it appears to be nothing more than Kepler's idea of ​​numbers. An expression of many kinds of curiosity.This law helps neither in calculating the positions of the planets nor in determining their orbits.In principle, it could be used to predict the cycles of planetary appearances at any known distance from the sun, but this is a theoretical rather than a practical problem.This law, like the law of elliptical orbit and the law of area, does not seem to be able to play any obvious role in physical principles. Furthermore, when considering Kepler's astronomy, we must remember that in the final summary (in the "Synopsis of Copernican Astronomy"), Kepler stated more than just the three laws of planetary motion, which we have today. The well-known Kepler's laws.There are many such laws in the book, including the relationship between the size and order of planets and the size and order of orbits, and the non-circumferentiality of planetary orbits. considerations.Also included in this volume was Kepler's first discovery: the law relating the number and size of planetary orbits to the geometrical bodies of Plato's five rules.To accept Kepler's astronomy, there is also the problem of mixing the principles of mechanistic and animistic physics.The mixture of the two results in a dynamic that is not a pure study of physical action and the resulting physical motion.For example, the orbital motion or revolution of the planets is accounted for physically by the sun-planetary (magnetic) force, while the regular and continuous rotation of the earth and sun is said to be animistic" The result of the soul principle".In Kepler, "animist and mechanistic principles explain motion compete with each other" (Max Casper 1959, 296). The fact is that, before Newton's Principia (1687), very few works of theoretical or practical astronomy mentioned Kepler's three laws of planetary motion, let alone Kepler's relation to the actions of celestial bodies that cause orbital motion. thought out.It thus appears clear that prior to 1687 there had been no Kepler revolution in science.In retrospect we can conclude that Kepler's program constituted only a revolution in treatises - not because, intellectually, Kepler had not quite succeeded in developing a The motions of the planets he discovered were not in the dynamical system of the laws, but because he failed to convert most of his contemporaries and subsequent successors to either his astronomy of planets in elliptical orbits or his astrophysics. William Gibb: The Experimentalist and His Spokesperson Like Kepler, William Gibb must be included among the revolutionary scientists of the early seventeenth century.He showed the originality of his science in his work De Magnete (1600); he said in the subtitle of the book that his book was a "Physiologia nova, plurimis & argumentis & experimentis demonstrat. "That is to say, he created a "new physiology" or natural philosophy.A new natural science, a doctrine "proven by many arguments and experiments." This new natural philosophy was magnetism, and the title of the book tells the reader that Geber was concerned with lodestones or natural magnets, "Magnets" (such as magnets) and "magnets of the earth".Throughout the book, Gilbert emphasizes the idea of ​​experimentalism, the notion that knowledge is based on experience, field experience, or empirical proof.In later classical Latin, the words "experimentum" and "experientia" meant both "experience" (even "well-known" experience) and "experiment", just like the French "expeence" It is the same meaning that "espertenza" in Italian still contains.It can be seen that Gilber was emphasizing practical experience in the field (such as that of blacksmiths and navigators), direct study of nature through experiments, and knowledge based on experience rather than intuition or speculation. In addition to calling attention to the book's features, as indicated by its subtitle, Gilber collected a great deal of new experimental information, and he added numerous notes in the margins of the book.in order to show what he describes "our discoveries and experiments" more or less "according to the importance and subtlety of things" (1900, ii).Gilber's account of the phenomenon of gravitation in post-friction amber is an example of the originality of his experimental approach to the problem (Ch. 2, bk. 2. He severely criticizes " The "philosophers of our time who have discovered nothing of their own, are not supported by any practical experience, ... have made no progress" (p. 48): not only amber and charcoal (as they supposed that) attracts small objects, but also diamonds, sapphires, garnets, iridescent gems, opals, amethysts, and bristolite (an English gemstone or spar), emeralds, and crystals .Having a similar gravitational force are glass (especially light-transmitting and transparent glass), artificial gemstones made of glass or crystal, antimony glass, and various spars extracted from various ores, and arrows Stone, etc. In addition, sulfur, scented butter, and hard sealing waxes synthesized from shellac dyed with various colors also have gravitational effects. Even hard resins, such as orpiment, also have this effect, although of course, its effect is not very strong ; In relatively dry weather, rock salt, muscovite and alunite are difficult to produce gravitational effects, and even if they do, their effects are very weak. The preface to "The Magnetite," addressed to an "impartial reader," is one of the most vocal statements of the principles of the scientific revolution.In it, the author proudly states that those "reliable experiments" and "proved arguments" are superior to "the possible conjectures and opinions of ordinary philosophers." Here, Gilber speaks of "our philosophical ...to...the tireless observation of things," he also spoke of "true proofs and ... experiments of obvious significance," and "a great deal of experimentation and discovery (that apparently makes every philosophy flourish)." He Also describes the correct method of philosophical inquiry by which knowledge moves from "easy questions" to "other questions of more noteworthy" and finally to "hidden questions about the earth." the most mysterious problems of the world "thus continually developing, and thus" understand the causes of those problems which, either through the ignorance of the ancients or through the negligence of the moderns, were unrecognized and missed "(fol. ii). Gilbert made empirical notes; he also eventually invented some theories and formulated some hypotheses.Gilber's own most important scientific insight was that the earth itself was a great magnet, with two magnetic poles, north and south.He concluded that he had shown experimentally that a perfectly spherical magnetite with two poles would rotate about its axis, and he therefore concluded that the earth must do so, as Copernicus had told the people.However, Gilber was not very interested in the revolution of the earth, because for him it was a problem that had nothing to do with magnetism. In this sense, he was not a Copernican. It will be noted that in Gilber's program the theme of the "On the Magnetite" is not always carried through in great detail, although this does not give him the weight of the definite assertion that a new science is about to emerge. And reduce.Like Kepler, Gilbert lived in a time of transition, so it is not desirable to see "Gibber's big talk and grandiosity, but he was a moderate peripatetic scholar, and never did what he criticized. Plagiarism" (Heilbrunn 1976, 169), we should not be surprised.Although Heilbrunn quite rightly refuses to admit that "Gieber was a revolutionary hero" and refuses to believe that his "Renaissance rhetoric is true," he praises Gieber for publishing "one of the earliest A monograph of a particular branch of physics, "a" first published report of a large number of interconnected and reconfirmed experiments." Yet despite his revolutionary zeal, Gilbert did not create a new science.Neither the signs at that time nor the works on magnetism published in the next half century or more showed that the subject had undergone a drastic transformation.Nor did his writings on the burgeoning research topic of electric gravity lead scientists to establish a new subdiscipline of physics; it was only in the next century that it emerged.In this light, Gilber's work fails the first two tests for identifying scientific revolutions, and neither scientists nor historians envisage a Gilbert revolution in science.So, although Gilber was indeed a revolutionary, at best he caused a revolution in writing.There is no doubt that his "On the Magnet" contained the seeds of a revolution, but it did not spark a revolution after all. Even if Gilber did not cause or initiate a revolution, his work was a symptom or foreshadowing of one to follow.In that later revolution, science gradually changed from a primarily philosophical and abstract discipline to one based on experience, on the special kind of experience gained by directly questioning nature through experiments. Galileo's Revolutionary Science The scientist who advocated new experimental science and technology before anyone else was Galileo.Galileo's scientific program was as revolutionary indeed as Kepler's, and it is even more important in that it contains methods and results that have the potential to affect all science. significance.Unlike Kepler, Galileo's writings were widely circulated (and translated into other languages), and his writings had a huge impact on the scientists and scientific thought of his day.This influence was even amplified by his famous trial and conviction. Like Lillo, he made a large number of discoveries, but his revolutionary activity is mainly famous in four unique fields, namely, telescopic astronomy, principles and laws of motion, models of the relationship between mathematics and experience, and experimental science or science. Experimental science. (It may well be argued that Galileo was also well-known in another field, this fifth field being the philosophy of science, as one might very well cite, however, Galileo's revolutionary ideas in this respect are contained in both experimental science and The relationship between mathematics and experience.] There is much evidence that Galileo did revolutionary work in the field of kinematics.Moreover, the editors and writers of the works of physics in the mid-seventeenth century—Christian Huygens, John Wallis, Robert Hooke, Isaac Newton—all recognized and used Galileo's laws and principle.For at least two centuries, many historians and philosophers of science have hailed the Galilean revolution.Moreover, physicists and other scientists have long regarded Galileo as a revolutionary hero, and have even exaggerated his role to the point of describing him as the founder of modern science and the scientific or experimental method, a pre-Newtonian Discoverer of two laws of motion.In short, Galileo seems to have easily passed all the tests for identifying a scientific revolution. Galileo's first public demonstration of his revolutionary science was in 1610, when he published some of the first results of exploring the heavens with a telescope.In Chapter 1 of this book, I talked about the transformation process of Galileo's view of the sky, that is, the transformation process from personal observation experience to rational conclusion.He used the principle of analogy and physical optics to explain that the surface of the moon is also like the earth, with many cliffs and ups and downs.He discovered that the earth makes the moon shine.He saw that the Jupiter system had four moons and that Venus had phase changes.His telescope not only revealed some new information about previously known objects such as the sun, the earth and the planets, but also revealed in the visible range a large number of stars (and moons) never seen by the naked eye. ). Galileo's discoveries, along with those of others, showed everyone what the sky was like for the first time.The phases of Venus, if related to the apparent size of the planets, would prove that Venus orbits the Sun and not the Earth, and thus prove Ptolemy wrong.All these discoveries are consistent with Copernicus' proposition that the earth is just another planet; that is, all discoveries show that the earth is more like a planet than something different from a planet.Galileo thus at once proved that he had shown the correctness of the Copernican system (despite the fact that his findings were also quite compatible with Tycho Brahe's, and that in Tycho Brahe's system , the Earth is still seen as being at the center, the other planets circle the sun, and the sun circles the Earth). These discoveries revolutionized observational astronomy and fundamentally changed the level of discussion of Copernican astronomy.Before 1610, the Copernican system might have seemed like a thought experiment, a hypothetical system of calculations, to those who denied that the Earth looked like a planet (i.e., a planet that we think shone extremely brilliantly) To some, it is something philosophically absurd.After the revolution of 1610 happened and borne its fruits, scientists could (and did) prove that the Earth was indeed similar to the other planets and supposedly moved in the same way.Copernicus quite rightly pointed out that the earth is just "another planet".The only way to deny this new empirically corrected Copernican theory is to refuse to observe with a telescope, or to assert that what is seen through a telescope must be an optical illusion or a product of the telescope's lens. an aberration rather than the true face of the planet.The fact that some very sensible philosophers have adopted this attitude shows how radical and innovative it was at the time to base knowledge of nature on the basis of empirical evidence. The second field in which Galileo revolutionized was kinematics.This subject has always been considered central to natural philosophy; so, in the opening remarks of his dialogue on the third day of The Two New Sciences (1638), Galileo boasted that he was introducing "a brand new subject on a very ancient subject." the disciplines" (Galileo 1674, 147).Perhaps many new laws and principles of motion are due to Galileo.He discovered the isochronism of pendulums - when a free-swinging pendulum travels along an arc of shorter and shorter arc lengths, its speed also slows down, but it completes the full length of each swing. The time required is (always) constant.He proved by exciting experiments that in the air, objects of different weights fall at almost the same speed, and does not (as Aristotle before and most people uneducated in physics still think) It is proportional to the weight of the object. He found that free fall is a case of uniformly accelerated motion, in which case the speed of motion increases as time continues, and the distance traveled is proportional to the square of time. He Proposed the principle of independence of vector velocities and adopted the method of combination (synthesis) of vector velocities, he used this principle to solve the problem of the trajectory of the projectile: he found that the course of this motion is a parabola. Therefore, he pointed out that When the cannon of the cannon is at an angle of 45° to the horizon, the range of the cannon is the farthest. In his analysis of the parabolic course of a projectile, Galileo sketched out the initial development of the principle of inertial motion.A series of successively modified concepts led to Newton's law of inertia in 1687, the first of which was apparently proposed by Galileo.It must be remembered, however, that Galileo analyzed motion primarily in terms of kinematics.That is to say, although Galileo's discussion has or contains some problems of the action of forces, he neither tries to find out the force that causes (or causes) the motion, nor does he try to discover the strict mathematical relationship between the force and the motion. relation. Galileo's third contribution was in the field of mathematics.Modern science, especially physics, is characterized by the expression of its highest principles and laws in mathematics.In the 17th century, this feature of science began to show importance, and the importance of this feature reached its first peak when Newton's "Mathematical Principles of Natural Philosophy" (ie "Principles") was published.From Galileo's discussion of "natural acceleration" on the third day of "Two New Sciences", we can see the revolutionary side of Galileo's methodology.Bringing up the topic, Galileo explained that it was perfectly reasonable (as was often done before) to postulate any kind of motion and to explain its nature mathematically.不过，他愿遵循另一种方针，亦即"找出并阐明与大自然所进行的那种运动［加速运动烬可能完全一致的定义。"在考虑"在某一高度静止不同的"石头是怎样下落之后，他得出结论说，"新增值的速度"的连续获得，是由"最简单和最明显的规律导致的"（伽利略1974，153-154），这就是说，这种增值总是以同样的比率持续进行的。因此，（a）在下落的每一连续相等的特定距离内，或（b）在所消逝的每一连续相等的时间间隔内，速度的增加肯定总是相等的。伽利略出于逻辑上的理由对等距规则不予考虑，转而着手阐述等时规则的各种数学推论，其中有这样一个结论：在匀加速运动中，"物体在任何时间内所通过的距离都与各自所用的时间成倍比"（也就是说，它们各自都与那些时间的平方成正比）。伽利略随后对"这是否就是大自然在她的下落的物体上施加的加速作用"提出了疑问。 答案是通过一项实验找到的，这一实验程序"在把数学证明应用于物理学推论的那些学科中是非常有用和非常必要的"（伽利略1974，169）。实验也许看起来是相当容易的，但实验设计和对实验结果的解释，需要对现代科学的基本原理有高水平的理解（参见下文）。要正确地评价伽利略程序具有何等的革命性和创新性，我们应当把它与中世纪的数学家一哲学家们的活动加以比较和对照。在12、13和14世纪，数学家一哲学家们一直在积极探讨运动问题（参见第5章｝，他们的数学发展处于一种抽象的水平。在这里，运动问题属于一般的范畴，这一范畴包含了从"潜在性"到"实在件"（亚里土多德的定义）的任何一种可以量化的变化，这里的"潜在性"和"实在性"包罗万象，从爱、仁慈到（从一处向另一处的）地点的变化。所以，伽利略要根据（并举例说明）自然界中实际出现的运动来阐述有关运动的数学定律，这的确是一个大胆的举动。以前同样也没有人发展到用实验检验来证明物理学定律——而这里正是伽利略为科学做出重要贡献的第四个领域。 伽利略在数学上阐述了诸多运动定律，其中包括匀速运动定律，匀加速运动定律，以及抛物运动定律等等。这例证了17世纪科学的一个（可以毫不过分地说）普遍特征，亦即这一思想：基本的自然规律必须是用数学阐明的。在17世纪中，对数学的这种强调有着多种多样的形式。例如，从最初级的水平上讲，数学也许仅仅意味着数量的确定，计数作用。也许存在着这样的柏拉图教条：宇宙中的真理将借助数学而不是借助观察和实验来发现，首先应该考虑的是数学方面的特性，而不是与经验世界的一致。我们已经看到，在相当一段人类的历史中，人们感到圆是一种完美的体现，天体运动最应表现出这种完美的特点。咖利略驳斥了所有此类抽象的几何属性观，他认为，也许有些不同的几何特征最能说明某些特殊情况。当然，从数学上阐述科学是对科学的最高级的表述这种观点，在17世纪并不是十分新鲜的东西；托勒密曾把他的伟大的天文学杰作取名为《数学的综合》或《综合》。对伽利略而言，这些传统的数学观与新科学的数学观之间的差异意味着，在经验世界与知识的数学形式之间将会有一种和谐，这种和谐可以通过实验和批评性观察来获得。 不过，在伽利略撰写的数学著作中，他所阐述的并不是通常我们所想到的那种数学，亦即代数方程的应用，混合比例（例如"距离与时间的平方成比例"），流数，或微积分等。他所论述的是数列。以下规则即为其中一例：若取自由落体在第一段时间间隔末的速度值作为速度单位，则它在相继且相等的时间间隔末的速度为从一开始的自然数（或整数），或者说它在相继且相等的时间间隔内所走过的路程彼此的比为奇数，或曰，在这一系列时间间隔末所走过的总距离按平方律变化。在《试金者》中（伽利略1957，237－238），伽利略对自然界的数学问题作了精彩的陈述，他指出，应该把几何学看作像有关数的法则一样重要。"哲学［自然科学，或科学］写在宇宙——这部一直向我们敞开的伟大著作中"；但是，"我们如果不先学会书里所用的语言、掌握书里的符号，就不能了解它。这部书是用数学语言写出的，它的字母是三角形、圆和别的几何图形。不借助它们，那就一个字也读不懂。"所以，谈到伽利略与数学的关系时，重要的并不在于数学本身的水平有什么创新之处，而在于他清晰而引人注目地表述了用数学来阐述自然现象的必要性，以及以实验和观察为基础确立自然界的数学规律的必要性。 谈到伽利略与科学实验方法论的关系，有必要谨慎一些。近年来有一项值得注意的事业（主要集中在小约翰·赫尔曼·兰德尔的著作中），这就是对伽利略科学方法论的先驱者进行探讨。我发现，很多的历史学家都犯了一个根本性的错误，即没有分清有关方法的那些抽象的陈述或格言与实际的科学工作之间的区别。在许多16世纪的作者的著作中，确实有听起来像是讨论实验或从事科学研究的方式的论述，然而，了解到这些作者中没有一个人曾完成过任何一项科学研究工作这一事实，我们就不太相信它们真是有关实验问题的阐述了。附带说一句，在拉丁语和罗曼语中，用来表达实验、经验的词都是相同的，而且大体上每个人都知道。 据说，伽利略曾在一个高塔上抛下重量不等的物体这一著名的实验，解决了一个独特的问题。所谓伽利略在众目睽睽的情况下在比萨斜塔上进行演示，公开与亚里士多德学说对抗之说，不过是些过于耸人听闻的虚构之词，无疑，它们都是杜撰出来的。不过，伽利略确实在自己的笔记本中记录过他"从一个高塔上"把重物抛下来的情况。伽利略这样做是为了了解：传统的"常识"观是否正确，重物在空气中自由下落时的速度是否与它们各自的重量成比例。伽利略用另一种实验来检验他的假说——自由下落的物体的运动是匀加速的。我们要问的是：自由落体的速度的增值是否与消逝的时间成正比呢？我们会看见，在进行一项人们会在其中提出这类有关大自然的问题的实验中，将会产生出许多问题。要直接检验这种比率是不可能的。所以，伽利略检验了另一个定律，一个他希望检验的逻辑推论，这就是：距离与时间的平方成正比。即使这一检验也超出了伽利略的能力所为，因为自由下落的物体运动得太快，以致于他难以进行测量。因此，正像他所说的那样，他"冲淡重力"，在一个斜面上进行了实验。他在实验中发现，时间平方律确实经受住了实验的检验。当然，伽利略是位伟大的实验家，他充分认识到，进行大量不同角度的斜面实验是很重要的；在所有这些斜面实验中，定律都经受住了检验。我不想详细地讨论伽利略是怎样根据斜面角度的增大用数学来表述重力沿斜面的分量的。只要说明以下这一点就足已了：伽利略在所选择的例子中表明，随着思想的发展和"科学"的日益复杂，必须要设计出一个实验用来检验那些哪怕看起来最简单的定律如：距离与时间的平方成正比。 伽利略不仅认识到对运动所作的抽象的数学推理一般均可适用于自然界中所观察到的真实的运动，并且通晓用实验来检验数学规则的技术，他也熟知怎样说明思想状态与实验状态的差距。例如，他通过实验发现，从一个高塔上下落的重的物体比轻的物体略微早一点接触地面；他把这个微小的差额归因于空气阻力以及重的物体和轻的物体克服这种阻碍作用的相对能力。他提出结论说，在理想状态下，在真空中或自由空间内，它们下落的情况是完全相同的。 在进行实验设计以便对假说加以检验的同时，伽利略还对自然现象作了实验探讨。斯蒂尔曼·德雷克对伽利略的手稿进行了仔细研究后，重现了这类探讨实验。这类实验很有可能就是伽利略解决惯性问题的关键，而且，它们几乎已经使伽利略以一种与他在《两种新科学》中所描述的方法略有不同的方式得出了匀加速运动定律。 伽利略的确不是第一位进行实验的科学家，但他是头等重要的科学家之一，他在进行数学分析的同时，使实验成了他的科学的一个组成部分。事实上，他把实验技术与数学分析相结合（例如在斜面实验中所做的那样），使他名符其实地成了科学的探究方法的奠基人。 伽利略大量的实验和天文学观察包含了他的科学的哲学中两个革命的特征（与斯蒂尔曼·德雷克的通信为我澄清了这个问题）。一个是，伽利略所表明的信念："感性经验和必要的证明""不仅优于哲学信条而且优于神学信条。"很有可能，直到19世纪，"大多数科学家才采取了与他相同的立场。"第二个特征与伽利略的探讨工作有关（德雷克称，伽利略的探讨是"他的科学中主要的富有创新性的部分，而且，伽利略在许多地方都提到过这种探讨"），这就是"在裁决任何科学问题时权威不足为据。"在《水中的物体》中，伽利略更进一步评论说："阿基米德的权威并不比亚里士多德的权威更加重要；阿基米德之所以正确，是因为他的结论与实验相符。"德雷克怀疑"除了他那些自身就可以说明问题的发现外，伽利略对其科学中任何新颖的问题都要考虑。"我们可以同意德雷克的看法，即伽利略仅仅"把他自己看作是把托勒密很成功地运用于天文学上的方法用在了物理学上；也就是，在不考虑古老的［亚里士多德的］意义上的因果条件或［借助于〕形而上学原则的情况下，用几何学方法和算术方法把辛勤测量的结果运用在可检验的预见之上。" 伽利略的成果广为人知，人们也都因此承认，他使运动学得到了改革和更新。沃尔特·查尔顿1654年出版了《自然科学》一书，该书主要涉及的是新老原子论的自然哲学，而且，它以介绍伽利略、伽桑狄以及笛卡尔等人在运动学方面的成就而闻名。查尔顿在这部书中毫不怀疑地认为，伽利略的研究是全新的研究。他认为"伟大的伽利略""奠定了运动本质的…基础"，正是这一成就导致了"亚里士多德的有关学说"的"覆灭"（p．435）。他认识到了，"没有一位古人的探讨"深入到"物体向下运动"时速度增加的"比率或速率"，而伽利略却发现了这个问题，此外，正是这位"伟大的伽利略"完成了"对大自然最鲜为人知的奥秘的探讨，这种探讨是无与伦比的"（35，455）。 在17世纪的科学文献中，伽利略似乎不仅是运动定律的发现者和亚里土多德的驳斥者，而且还是最早用望远镜观察天空的探索者。约瑟夫·格兰维尔在其论文《现代实用知识的改进》中（1676，18－19），用了整整一页的篇幅来论述伽利略用望远镜所做出的发现： 也许可以把这段会令读者窒息的说明与格兰维尔对开普勒一带而过的叙述加以对照： 牛顿在《原理》中指出，伽利略之所以名扬天下，不仅是因为三项运动定律中的头两项定律，而且还是因为这头两项定律的推论，它们涉及到了向量速度的组合问题及其解决办法。所以，牛顿为伽利略欢呼，说伽利略是他自己的理论力学最初的奠基者，同时却贬低了开普勒的作用：说他只是行星运动的第三定律或和谐定律的发现者，彗星的观察者。他甚至怀疑开普勒是否发现了椭圆轨道定律和面积定律。（有关牛顿和开普勒的讨论，参见科恩1975）17世纪的天文学无疑就是伽利略天文学。伽利略倡导使用望远镜，从而使天文学的观察基础发生了革命，并使他以现代科学奠基者之一的身份赢得了主导地位。他对自由下落问题的研究。他对抛射体运动和沿斜面向下的运动的分析，业已成为与实验相结合的数学分析的典范。他所发现的有关匀速运动和匀加速运动的定律依然是这门科学的基础。实验方法，尤其是那些每次可能只改变一个参量的实验方法，仍旧以他的名字命名。伽利略比开普勒（他没有伽利略那种用实验获取知识的惊人才能）和吉伯（他缺少伽利略的那种数学知识）更胜一筹，他的研究体现了科学的新的特点，这些特点则是科学革命的表征。伽利略是现代科学最伟大的奠基者之一，他是科学革命中的一位英雄人物。 然而，伽利略革命并没有完成。在其运动问题的研究中，伽利略把他的注意力主要集中在我们今天会称之为运动学的那部分。他已经开始思考地球运动中力的作用，但他所取得的最重要的进展并不是在这方面。与开普勒不同，伽利略本人完全没有注意到，宇宙中的作用力、地球的作用力或太阳的作用力，有可能是行星运动现象的原因。他无视开普勒行星运动定律的发现，而且嘲弄开普勒的这一见解：月球远距离的作用力有可能是导致海洋中潮汐运动的原因。在科学中，伽利略革命的完成还需要有另一个阶段的革命，那就是对惯性、对加速度产生的地球的和天体的作用力的认识，伽利略本人在这些问题方面的思考尚处于萌芽阶段。牛顿革命使伽利略已经完成的工作中的潜力得以实现，而且取得了远远不仅如此的成就。当然，在此之前还需要有半个世纪的发展时期。说伽利略科学革命的完成还需要有一场更为深入的革命，而伽利略在运动原理和运动定律方面所做出的那些伟大发现——就其所达到的程度而言——只是有可能成为科学革命顶峰的宇宙动力学的发现的初级阶段，这一结论对这位曾在科学史上享有如此高的声望的人来讲，并不是什么不光彩的事情。