WALLACE HOOPER
4 Inertial problems in Galileo's preinertial framework
Galileo made essential contributions to the development of inertial mechanics. His two most basic contributions were to collect the set of problems that held the keys to inertial mechanics and then address them all with an effective, consistent mechanics.
Classical mechanics is still taught by referring new students to the core set of problems that had to be solved by the original investigators like Descartes, Gassendi, Huygens, Wallis, Wren, Hooke, and Newton, all following Galileo's original line of attack. These problems include the analysis of motion on an inclined plane, the motion of a pendulum, the action of a lever, the force of a spring or pull in a rope, the result of collisions between impacting and moving bodies, and so on.
Inertial mechanics was extended to a far wider range of problems, but no writer before Galileo had put so many of the basic problems together in a single, articulate discussion. For that reason alone we may describe Galileo's work as modern in character and properly within the bounds and spirit of classical mechanics, even though the elements of the latter system were not successfully elaborated for almost fifty years after his passing and in spite of the fact that he sometimes proposed mistaken ideas to solve the basic problems.
Galileo's two major works (1632 and 1638) first defined space, time, and speed, and then moved on to uniform acceleration.1 Galileo analyzed projectile motion into two component motions, the first horizontal and uniform, the other vertical and accelerated. Galileo discussed the motions of bodies upon the moving Earth and of planets around the Sun. He asked questions that led his fellows and successors directly toward inertial mechanics and gave them some of the essential tools to build it.
Yet his own terrestrial and celestial mechanics were not fully inertial. He did, for example, think in terms of impressed forces and the impetus acquired in descent, and he continued to speak of intrinsic motions, both of which were banished from inertial mechanics.
Galileo is best known in mechanics for contributions to kinetics, the analysis of motion in terms of distance, time, speed, and acceleration. About 1602 or, at the latest, 1604, he discovered the times squared law for distance fallen, s α t2. This rule says, for example, that during the first five units of time, that is, 1, 2, 3, 4, and 5, the distances fallen are as the squares of the times, 1, 4, 9, 16, and 25. The differences between those distances are the distances fallen in equal successive times, and they are as the odd numbers, 1, 3, 5, 7, and 9. In the Discourses on the Two New Sciences of Motion and Mechanics (1638), he stated the definition for uniformly accelerated motion from which he derived the times squared law and the odd number rule as deductive results:
…we shall not depart from the correct rule if we assume that intensification of speed is made according to the extension of time; from which the definition of the motion of which we are going to treat may be put thus:
We shall call that motion equably or uniformly accelerated which, abandoning rest, adds on to itself equal moments of swiftness in equal times.2
The amount of speed acquired in the first second is added again in the second second, and in the third, and so on. Thus, for example, after a descent of two seconds, the body has acquired twice the speed as it had at one second (2:1). By the times square law, it has also fallen four times as far (22:12). The speeds acquired in vertical descent are in direct proportion to the times of descent, vacqnired α t.
If the descending body were deflected onto the horizontal plane, it would stop accelerating and continue moving uniformly with the final velocity it acquired. The accelerated motion of the first two seconds is transformed into a uniform horizontal motion that, in the next two seconds, travels twice the distance just fallen.
In other words, half of the final uniform speed would cover the distance fallen in the same time as the fall itself. This relation is called the mean speed theorem, and the expression of it given in the previous sentence is called the double distance rule. The mean speed theorem is Theorem One of On Accelerated Motions, and is the first result derived from Galileo's definition and postulate in the Latin treatise De Motu Locali, in the Third Day of the Two New Sciences. There is evidence that Galileo performed experiments to work with these ideas, especially folio 116 verso in Volume 72 of the Manoscritti Galileiani.3
Galileo's double distance rule and mean speed theorem allow him to compare uniform and accelerated motions by the measures of time and distance. Working without the calculus, he reduces all accelerated motions to their uniform equivalents by this theorem.
Their uniform equivalents can then be compared with other accelerated motions or directly with uniform motions by the simple rules of uniform motion. Galileo uses his rules for uniform motion to resolve important problems of accelerating motion, and in this he is rather Aristotelian in his understanding of velocity.4
Galileo also postulated the equality of the speeds of all motions falling through equal vertical descents:
Salviati: This definition established, the Author requires and takes as true one single assumption, that is: [Postulate]
I assume that the degrees of speed acquired by the same movable over different inclinations of planes are equal whenever the heights of those planes are equal.5
If several inclined planes have the same height, bodies descending them would all acquire the same velocity when they reached the bottom – they simply take different amounts of time to reach bottom and acquire the velocity.
Galileo drew the new science of motion out of these beginnings as Euclid had drawn the Elements out of its opening propositions. Galileo's Theorem One, the mean speed theorem, and Theorem Two, the times squared law, follow immediately from the definition of uniform acceleration. Historically, Galileo knew the times squared law (1602–4) before he learned to define uniform acceleration (about 1608–9), but the postulate of equal velocities for equal descents appeared very early in Galileo's work, in the Paduan De meccaniche 1597.
THE ADVENT OF INERTIAL MECHANICS
Galileo put his new mechanics and physics before the educated world in 1612–1613, in On Bodies That Float Atop Water and Letters on the Sunspots. He undoubtedly discussed them again during the various stages of his public defense of Copernicus at Rome from 1612 to 1616. He wrote about them again, at length, in his two major works published in the 1630s.
His new mechanics did inspire further enquiry. Some of his readers investigated the laws of motion and mechanical action along lines he had suggested, including Marin Mersenne, Pierre Fermat, and Pierre Gassendi in the late 1630s and 1640s and Christian Huygens in the 1640s and 1650s.
These investigators did find some discrepancies between Galileo's descriptions of events and the results of their own experimental trials, but many of his offerings held up under scrutiny. René Descartes also read the Two New Sciences in Mersenne's translation in the late 1630s and remarked that though Galileo philosophized rather better than most, he had failed to begin from first principles and thus could not arrive at a full understanding of matters.6
In 1613, the same year as the Letters on Sunspots, Isaac Beeckman had quietly rejected the ideas of impressed forces in notes in his journal. He proposed that a body continued to move as it had been moving as long as there was no cause acting to slow it down or to stop it.
Beeckman was the rector of the Latin School at Dordrecht, a steady investigator of natural phenomena and a keeper of scientific journals. Prior to 1620, Beeckman had befriended Simon Stevin, court mathematician to Prince Maurice of Orange and notable natural philosopher, who had argued in 1586 that all bodies fall at the same rate.
Beeckman developed the elements of his new approach in discussions in 1619 and after with his younger colleague, Rene Descartes. Descartes was, in those years, a soldier in the army of Prince Maurice and already a well-regarded mathematician. After those initial discussions, Descartes quietly worked with Beeckman's principle of indifferent conservation and improved it in his successive mechanical systems.
Pierre Gassendi had taken up Galileo's research almost as soon as it had been published in the Two New Sciences in 1638. He made experiments with inclined planes and dropped stones from the mast of a moving ship and confirmed Galileo's results and predictions. On paper, he studied Galileo's unaccelerated and unretarded uniform horizontal motion in an imaginary space outside the world and succeeded in abstracting the first statement of the principle of inertia from both the intrinsic gravity and circular motion that had enthralled Galileo. Horizontal motion in such a space would be rectilinear in the absence of intrinsic accelerating tendencies. Gassendi published this work in De Motu Impresso a Motore Translato (1642).7
Descartes published his laws of motion, including the principle of inertia, in Principles of Philosophy (1644) in two laws that Newton subsumes under his own first law.8 Descartes's first rule said that a body will persevere in its state of rest or motion in the absense of resisting or impelling forces. The second rule said that a body is conserved in rectilinear motion. Descartes's Principles set out an extensive system of philosophy and nature and included his theories of impact and planetary vortices. His solution for impact was not correct and led to further enquiry.
Problems of collision engaged European mathematicians in the 165 as and 1660s. Huygens merged the mechanical approach of Galileo with the algebraic methods and inertial directions indicated by Gassendi, Descartes, and Beeckman in his analysis of shared and exchanged motions. Wren, Wallis, and Huygens independently worked out the solution and presented it to the Royal Society as Newton related.9 Newton embraced all their problems and undertook many more original ones in the 1680s in his completed inertial mechanics.
NEWTON'S INERTIA
Newton built his universal mechanics on a small, rigorous logical structure. The eight definitions (and scholium) and the three laws of motion – the principle of inertia, the proportionality of force and mass and acceleration, and the equality of actions and reactions – and their corollaries are all stated in the first forty pages of the Mathematical Principles of Natural Philosophy (1687). The definitions are followed by a scholium on the measurement of time and space, while the third law is augmented by six corollaries outlining the composition of forces. His laws are:
Law 1. Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impress'd thereon.
Law II. The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.
Law III. To every Action there is always opposed an equal Reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.10
The first law, the principle of inertia, connotes several sets of problems. There is a recognition that motion persists in its current uniform state, and that it tends in a right direction as it persists, and that it incurs forces that accelerate or otherwise change it. The principle even includes a notice of the fundamental equivalence of rest and motion.
The first two corollaries to the third law demonstrate the parallelogram of forces and the composition of forces from two others. Galileo's Two New Sciences had demonstrated a parallelogram of motions and had shown how to compose motions when more than one motion was applied to a body at the same time. Important successors also used the same principle.
The three laws are followed by a second scholium resolving the problems of simple machines. This scholium is said to have concluded the enquiries of the science of mechanics as it had been known for the previous two millennia.
Kepler (1609) and Leibniz (1710) had previously published the term inertia as a principle of inactivity, but mentions of it also appeared in Descartes's Correspondence, where Newton probably learned it.11 Newton did, however, change its meaning when he employed the term himself.
For Kepler, the vis inertiae was a force that kept a body at rest or brought it to rest if in motion. The classical vis inertiae stood for an accelerating or decelerating force in the equations of action and reaction, and it was regarded as the principle by which uniform motion continued.
Definition III. The Vis Insita, or Innate Force of Matter, is a power of resisting, by which every body, as much as in it lies, endeavors to persevere in its present state, whether it be of rest, or of moving uniformly forward in a right line.
This force is ever proportional to the body whose force it is; and differs nothing from the inactivity of the Mass, but in our manner of conceiving it … This Vis Insita may, by a most significant name, be called Vis Inertiae, or force of inactivity … a body exerts this force only, when another force impress'd upon it, endeavors to change its condition; … it is resistance in so far as the body, for maintaining its present state withstands the force impressed; … it is impulse in so far as it endeavors to change the state of that other.12
In the Opticks (1717), Newton remarked that the vis inertiae by itself does not add to motion but rather acts to conserve motion or rest:
The Vis inertiae is a passive Principle by which Bodies persist in their Motion or Rest, receive Motion in proportion to the Force impressing it, and resist as much as they are resisted. By this principle alone there never could have been any Motion in the World.13
Although it explains why bodies persist in their motions, the vis inertiae is nothing like a moving force that would push the body along, as in the medieval theories of virtus impressa or impetus.
Newton's view of the impressed force was distinguished explicitly from the medieval theories of virtus impressa or impetus. Those arguments had been advanced against Aristotle's theory of projectile motion, first by John Philoponus, a Greek neo-Platonist of sixth century AD, then in another version by Fransiscus de Marchia in the eleventh, and then restated by John Buridan and Nicole Oresme in fourteenth-century Paris.
The medieval virtus impressa was imparted to a projectile by its projector and continued to be present in the projectile; it served to move it after contact with the projector was broken. Newton's persisted only during the contact, doing all its work then. Their virtus impressa could keep a body moving at a given speed but Newton's changed the speed as long as it was applied.
Definition IV. An impress'd force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line.
This force consists in the action only; and remains no longer in the body, when the action is over. For a body maintains every new state it acquires, by its Vis Inertiae only.14
The fourth definition gave a historic new meaning to an old term, impressed force, by assigning its traditional role in projectile motion – sustaining the flight of the body – to the force of inertia or inactivity. Under the fourth definition, impressed force, which “consists in the action only,” did not remain in the body when the action was over. The definition rescued the traditional term by giving it an exact meaning that includes its intuitive, useful, and traditional sense as “a cause of action” in mechanics.
Recent works of interest on the history of the idea of inertia argue that the three laws have a strong experimental base in Newton's thought and work. It had become useful and common to think of the laws as a set of definitions and axioms in a rational mechanics, yet Newton went to great pains to produce the phenomena of inertia experimentally for observers.15
INERTIA-LIKE IDEAS IN GALILEO'S MECHANICS
Newton gave us his own exegis of the laws and corollaries in the scholium immediately following Corollary VI of the third law. There he gave Galileo credit for some important contributions to mechanics – credit that was largely owed to himself:
Hitherto, I have laid down such principles as have been received by mathematicians, and are confirmed by the abundance of experiments. By the first two laws and the first two corollaries, Galileo discovered that the descent of bodies observed the duplicate ratio of the time, and that the motion of projectiles was in the curve of a parabola; experience agreeing with both, unless so far as these motions are a little retarded by the resistance of the air.16
Galileo did not, however, work with Newton's notion of an accelerating force. Indeed, Galileo's concept of force was closely tied to ideas of static force. And in the broader realm of physics, Galileo did not regard gravity as an external force but always regarded it as an intrinsic property of a body, and that had many consequences for the development of his views.17
Alexandre Koyré often said that there were two strong indications that Newton had not read Galileo, this attribution of the first two laws and corollaries being one of them.18
I. B. Cohen said that only a Newton could have seen his laws in Galileo's work.19
Galileo did say that a body would persist perpetually in its current motion on a horizontal plane if there were no cause for deceleration or acceleration. The idea that gives Galileo's system its inertialike properties is derived from his work with inclined planes. A body set in motion on a horizontal plane would suffer no acceleration and no deceleration and could have the same uniform motion indefinitely. From Letters on Sunspots:
I have observed that physical bodies have an inclination toward some motion, as heavy bodies downward, which motion is exercised by them through an intrinsic property and without need of a special external mover, whenever they are not impeded by some obstacle. And to some other motion, they have a repugnance, as the same heavy bodies to motion upward, wherefore they never move in that manner unless thrown violently upward by an external mover.
Finally, to some movements they are indifferent, as are heavy bodies to horizontal movements they are indifferent as are heavy bodies to horizontal motion, to which they have neither inclination … nor repugnance. And, therefore, all external impediments being removed, a heavy body on a spherical surface concentric with the earth will be indifferent to rest or to movement toward any part of the horizon. And it will remain in that state in which it has once been placed, that is, if placed in a state of rest, it will conserve that, and if placed in a movement toward the west, for example, it will maintain itself in that movement.
Thus a ship, for instance, having once received some impetus through the tranquil sea, would move continually around our globe without ever stopping; and placed at rest it would perpetually remain at rest, if in the first case all extrinsic impediments could be removed, and in the second case no external cause of motion were added.20
For Galileo, uniform horizontal motions imperceptibly become circular motions. The global circular dimension is much larger than the local scope of the laws for free fall, where the horizontal plane merely appears flat and rectilinear, as it does in his f.116v trajectory experiments.
Galileo's analysis of motion on the terrestrial horizontal plane is actually a composition of two component motions. The first component of a terrestrial motion is the downward, center-seeking, and accelerated motion of vertical descent. This component is governed by the times square law for distances fallen. The second component is a horizontal motion, usually uniform, and tending in the direction that its impetus or impressed form impels. That composition of motions is at the heart of Galileo's inertialike thinking.21 Galileo first produced a limited version of this construction about 1590 in his Pisan mechanics, the De Motu Antiquiora, which he abandoned by 1602. It appears there in the absence of any discussion of the Earth's motion. While he was working with balance analogies circa 1590, he had used the arrangement to demonstrate that a body can be moved by the least possible force on a horizontal plane:
A body subject to no external resistance on a plane sloping no matter how little … will move down in natural motion … And the same body on a plane sloping upward, no matter how little, above the horizon, does not move up except by force. And so the conclusion remains that on the horizontal plane itself the motion of the body is neither natural nor forced. But if its motion is not forced motion, then it can be made to move by the smallest of all possible forces.22
Notice that this reflection on inclined planes leads to a challenge of the distinction between natural and forced motions for the horizontal plane. Galileo added an interesting marginal note on mixed motions:
From this it follows that mixed motion [“except circular” is canceled] does not exist. For since the forced motion of heavy bodies is away from the center, and their natural motion toward the center, a motion which is partly upward and partly downward cannot be compounded from these two; unless perhaps we should say that such a mixed motion is that which takes place on the circumference of a circle around the center of the universe. But such a motion will be better described as “neutral” than as “mixed.” For “mixed” partakes of both, “neutral” of neither.23
This was an idea that Galileo would return to throughout his working life. Galileo presented very similar views prominently in the illfated Dialogue on the Two Chief World Systems (1632), and a version of it appears in the Two New Sciences (1638).
In Galileo's view, the impetus imparts a uniform speed to the body, and the speed is proportional to the amount of impressed force or impetus acquired and present in the body. Its presence is properly measured from the body's speed and weight. Accelerations add impetus and decelerations consume it, but in the absence of them, the impetus and the state of motion it entails are ineradicable.
Within a year of the trajectory experiments, Galileo launched his telescopic discoveries and began to think of Copernicus in great seriousness. When Galileo put the Earth into uniform rotation in his mind's eye, he used the circular horizontal construction we just saw in Letters on the Sunspots. The version in Sunspots is a case that does not incorporate the motion of the Earth – the horizontal impetus imparted to the ship has come from something else. Once the Earth is assumed to be rotating, as in the Two Chief World Systems, the horizontal impetus can be seen as due to the Earth's motion.24
The motion of a body at rest on the surface would be circular on the global scale. Among terrestrial bodies that share the Earth's motion, the only motions we can perceive or participate in are those made in addition to the Earth's rotation. A body in motion over the surface or one falling to the center adds any horizontal motions to its intrinsic vertical tendencies and the general rotational motion of the Earth to produce a circular path.
Galileo compares the motions of a body falling uniformly along a circular path from a tower to the center of the Earth with the motion of another ball that remains at rest at the top of the tower while it turns with the Earth.25 From the geometry, the path of the body on the tower is the same length as the path of the body falling to the center of gravity. Finally, after one rotation of the Earth, the two paths are completed uniformly and in the same time. An accelerated vertical motion, freefall to the center, becomes a uniform circular motion when the rotation is taken into account. Galileo is led to declare that nature prefers to use uniform circular motions and that neither rectilinear nor accelerated motions ever occur in nature.
The Two Chief World Systems worked to show that when bodies share the same motion, the shared motion is “as if it did not exist” in relations between them. As Galileo explains, motion is made and perceived relative to other objects that stayed fixed or do not share the motion:
Motion, in so far as it is and acts as motion, to that extent exists relatively to things that lack it; and among things which all share equally in any motion, it does not act, and is as if it did not exist.26
Galileo uses the example of the cargo at rest on a ship bound from Venice to Aleppo. All the boxes and bundles were transported equally and yet the boxes were less affected by the ship's motion than by the small changes of position among themselves. All the bodies in our common experience share the rotational motion of the Earth, but the massive effect of that rotation is insensible to us optically and mechanically.
Galileo believed that his formula had important advantages for the defense of Copernicus. It does admit noninertial results chiefly because it views the circle as the path of conservation of impetus or motion on the surface of the Earth and in the heavens. A circular orbit is not a continually accelerated motion in Galileo's mind; rather it is a uniform motion capable of enduring eternally. Galileo contrasted circular motion with the rectilinear natural motion proposed by Aristotle. Aristotle's rectilinear natural motion was generally vertical and accelerated.
There are other, usually less successful analogies and formal homologies between elements of classical inertia and the structure of Galileo's definitions, theorems, and mechanical ideas. When systems of bodies are Galilean-invariant according to classical mechanics, for example, they preserve the properties of space and time and relative speed defined by Galileo's definitions of uniform motion and acceleration.
Galileo's own version of invariance, as stated in the Dialogue on the Two Chief World Systems, is not Galilean-invariant in the classical sense, as Alan Chalmers (1992) has recently shown, yet Galileo's analysis at least shows in broad strokes what a good invariance conclusion would have to look like.27
WHETHER GALILEO HAD INERTIAL IDEAS
Alexandre Koyré’s Études Galiléennes (1939) was probably the most influential treatment of Galileo's mechanics written in the twentieth century.28 Koyré pointed out that Beeckman and Descartes had written statements that were, word for word, very similar to Newton's first law, while there were no such statements, expressis verbis, in Galileo's works. Koyré said that Descartes's clear expression of the idea of rectilinear inertia marked a real advance over the suggestive and incomplete work of immediate predecessors including Beeckman, Gassendi, and Galileo. The view that Galileo did not grasp the idea of inertia has generally been accepted by the learned world for several reasons.
Nevertheless, Koyré argued that Galileo did succeed in working his way out of the old medieval and Parisian theories of impressed force and impetus and out of the old division of motions into natural and violent. Koyré described how Galileo broke down the distinction between natural and violent.
The discussion of Aristotle's arguments picks up at the point where it had been left by Copernicus: namely, with a qualitative distinction between natural and violent motion as the explanation for the difference between their effects. Now there is subtle modification, and the earth's natural motion (which, logically, is explained by its ‘nature’ or ‘form‘) comes to be attributed to bodies which are on earth, no longer as a result of a commonness of nature but solely because of the fact that they participate in this motion. Another subtle change and now the earth's motion is no longer seen as having any special status over and above the fact that it is circular, and this property, by yet another shift, is attributed by extension to the motion of a ship moving across the sea. The special status of natural motion has now completely vanished. Henceforth, motion is conserved not because it is natural but simply because it is motion. It is motion as such which is conserved and which is ineradicably impressed on the moving body.29
Similarly, Koyré sees reflected in Galileo's description and analysis of accelerated motions implicit classical and inertial views even though Galileo continued to use the language of impetus and impressed forces and of the natural and the violent in Two Chief World Systems.
The same tactics are applied to the transformation of the idea of impetus. Galileo opens his attack on Aristotelian physics with the help of objections and ideas accumulated and developed by ‘Parisian’ physics. The time comes, however, when being convinced of its hybrid and muddled character, Galileo abandons the concept of impetus, seen as the origin and cause of motion. So as the Dialogue progresses impetus can be found identified with moment, with motion, with speed – these successive subtle modifications which imperceptibly guide the reader toward the conception of the paradox of motion which is conserved by itself in the moving body, and of speed, which is ‘ineradicably impressed’ on bodies in motion.
In theory the special status of circular motion is now ready for destruction. It is motion as such which is conserved and not circular motion. But this is in theory. In practice, the Dialogue does not take this step. Regardless of what others have claimed, this move is not in fact taken, and not [neither] is the move to the principle of inertia.30
Koyré maintained that Galileo did not reach the principle of inertia but defended him against Duhem's conclusion that his system was an impetus physics throughout. Koyré believed that a great “mutation of thought” occurred in the works of Copernicus, Kepler, Galileo, Descartes, and Newton that swept away much of what had stood as astronomy, physics, and natural philosophy. Something was at work in Galileo's mechanics besides a repetition of the scholastic's virtus impressa and impetus.
Koyré believed that when Galileo used the term impetus he had in mind not an impressed mover like Buridan's but the product of a body's weight and speed. Everywhere a reader looks in the Two Chief World Systems, one encounters discussion of impetus and impressed forces. Koyré writes:
Thus the proof of Galileo's postulate, the relation between distance and duration, depends on dynamical concepts; the speed of the descending body is explicitly related to the magnitude of the initial impetus.
Have we, then, reverted to impetus physics? Or have we, as Duhem thinks, never left it at all? This is a serious problem, and it requires very close examination. What, in fact, is this Galilean impetus?
‘Let us consider first of all,’ says Galileo, ‘the well known fact that the moments or speeds of a given moving body are different on planes at different inclinations. The speed reaches a maximum along a vertical direction, and for other directions diminishes as the plane diverges from the vertical. Therefore, the impetus, ability, energy [l'impeto, il talento, l'energia] or, one might say, the momentum of descent of the moving body is diminished by the place upon which it is supported and along which it rolls…’
So the impetus of the moving body is nothing other than the dynamic impulse given to it by its gravity. It is no longer in any way the internal cause producing the motion, as it was in Parisian physics. It is the same thing as its ‘moment,’ i.e., the product of its weight and speed. In the moving body at the end of its descent it is the total energy, or total impetus; in the body at the beginning of its motion it is the product of its weight and its initial speed… Finally, for the body at rest the impetus is none other than the virtual speed.31
For example, a reader of either major work will readily find examples of impetum seu gradum velocitatis, or momentum seu gradum velocitatis, which express a practical equivalence or homology between the three terms impetus, momentum, and gradum velocitatis.
Galileo's proof of the postulate in Two New Sciences is his bridge between the science of weight and the rules of accelerated motion. In motion on inclined planes, the momenta gravitatis, which are due to the angle of descent, are shown to be congruent to the momenta velocitatis given by the rules of speed, and are taken as the explanation and cause of the latter. Momentum gravitatis appears to have more in common with the classical acceleration vector than it does with Galileo's momentum velocitatis or with Descartes's momentum, or Newton's. Galileo had first discussed the relations of planes and forces on which the idea of momentum gravitatis was framed in On Motion (ca. 1590). Paolo Galluzzi has written an important study of the semantics of Galileo's use of momentum and shown that there are stages in the development of meaning of the term.32
There is some justice in what Koyré intends here, but there are also some important problems. First, in an important paper in 1951, Ernest Moody pointed out that Koyré had not fully explored or understood the two versions of impetus and impressed force theory of the middle ages and, thus, had misread Galileo's change to his definition of virtus impressa, between 1590 and 1604.33
Galileo discussed the virtus impressa at some length in On Motion. There it appeared as a “praeternatural” lightness, sufficient to overcome the body's intrinsic gravity and carry it away. This view was in tune with his idea that motions could be understood as operations on a balance.
A body's intrinsic motion of descent balanced against the media in which it moved and progressed with an arithmetically reduced speed. All bodies have gravity. A force of projection clearly overcomes the tendency of the gravity and the projectile moves.
Galileo had a stronger view of natural and violent in On Motion than he possessed later. The projecting force was not a natural one and so would cease of its own accord. It tended to decay while the motion continued, and when it was exhausted, its influence stopped, even without deceleration. Galileo's later uses of virtus impressa, as found in his scientific papers, especially after 1604, and in publications in 1612–1613, all reflect the usage in the Letters on the Sunspots. The later version of virtus impressa does not decay unless it is forced into motion away from the center of gravity of the Earth.
Koyré identified Galileo's first version of impressed force with the medieval Parisian impetus theory. He identified Galileo's second version with a new, emerging terminology that was freeing itself from the ambiguities of the other, presumably Parisian impetus.
As Moody showed and Clagett's work sustains, however, Galileo's second version is similar to the Parisian version (ca. 1360) (fourteenth century) of impetus theory, whereas his first, decaying, version is similar to the views of Franciscus de Marchia (ca. 1320) and others who spoke of a vis derelicta decaying over time in projectiles.34 In other words, Galileo was working within the bounds of impressed force theories before and after his important change of position on “impetus” and “impressed force.” This has important consequences for our view of and intuitions for his dynamics.
Earlier in the Études, Koyré argued that Galileo had modified the meaning of impetus, to identify it with motion itself, stripping its sense of ‘cause of motion’. Koyré say this about certain passages in the Second Day of the Two Chief World Systems:
The Aristotelians’ strongest objection against the impetus theory was an ontological objection: An accident does not pass from one body to another. Therefore impetus cannot do this. This is true, replies Galileo, if impetus means a force which causes a motion; but the motion itself can be transmitted.35
This is, perhaps, too strong a reading of Galileo, whose actual reply to the ontological question in the passage just discussed took a much more traditional turn. He began with this indicative exchange:
Salv.: Patience all in good time. Tell me: Seeing that your objection is based entirely upon the nonexistence of impressed force, then if I were to show you that the medium plays no part in the continuation of motion in projectiles after they are separated from their throwers, would you allow impressed force to exist? Or would you merely move on to some other attack directed towards its destruction?
Simp.: If the action of the medium were removed, I do not see how recourse could be had to anything else than the property impressed by the motive force.36
Galileo actively took the same side of the traditional antiperistasis argument that Buridan and others had, and he would retain his commitment to it. The property impressed here was now, as Koyré says, functionally the quantity of motion imparted. Galileo makes this clear immediately in his treatment of a rider dropping a ball. The terms were being defined anew here but little of the traditional language had been discarded yet.
Historians of medieval philosophy have repeatedly argued against Duhem's claims that Galileo plagiarized the Parisians.37 Koyré was right to look instead to Benedetti, Bruno, and Tartaglia as reflections of the contemporary usages of impressed force and impetus.38 This group was recasting the scholastic language in a fundamentally new, geometry-based discourse about terrestrial and celestial motions. The results were, as we know, revolutionary.
For all its medieval connotations, Galileo's later impetus is consistent enough to be precisely the product of the body's weight and its speed of motion, wv, compared with momentum, mv, in the classical sense. Of course, the existence of the agency of motion in a projectile was a distraction from the real issue, but in Galileo's time, there persisted a perfunctory debate over the adequacy of Aristotle's account of projectile motion. The reader can hear Galileo's reluctance to rehearse the antiperistasis argument in the Two Chief World Systems, yet he was consistently anti-Aristotelian enough to pick up the cudgels on each occasion.39
Koyré said Galileo tried to mathematize impetus and found it impossible. Galileo clearly did attempt to mathematize his impetus theories. From his point of view, however, his mathematization continued to succeed within his expectations as far as he tried to push it. His results for impetus and speed are usually closely homologous to classical momentum and speed and are familiar enough to a modern reader to seem identical.
BEECKMAN, GALILEO, AND THE CONSERVATION OF MOTION
Beeckman made the crucial step to a properly inertial perspective by denying the existence of any virtus impressa in a moving body and citing indifference to change as the cause of continued uniform motion. A comparison of Beeckman's and Galileo's views will highlight some of the differences between impetus and inertial views.
Beeckman followed the argument initiated by William of Ockham almost four centuries earlier to deny the existence of the virtus impressa. Beeckman explained the cause of persevering uniform motion by saying there was insufficient reason for any change of motion (note that, like Newton, he does use the verb perseverare):
The stone which has been thrown form a hand persists (perjit) in moving not because of some force (vim) which comes upon it, nor because of abhorrence of the vacuum, but because it cannot not persevere (perseverare) in that motion, arising in that hand by which it was moved.40
In 1613, six years before he met Descartes, Beeckman wrote out the following principle, which differs from the impetus-based principles of perseverance offered by Jean Buridan, Nicole Oresme, and Galileo insofar as it denies a motor cause or force of continued motion, the virtus impressa, with a lack of cause of change of motion. Beeckman wrote:
Once moved things never come to rest, unless impeded. Once any thing is set in motion it never comes to rest, except because of an external impediment. Furthermore, as the impediment is weaker, by that is the moved thing moved of greater duration; truly, if it is projected on high and moved circularly at the same time, it is evident to the senses that it does not come to rest before its return to earth; and if it were to come to rest at length, that would not come about because of an equable impediment, but because of an inequable impediment since one and another parts of the air touch the moved thing in succession.41
Koyré says that both the usual trajectory motion and the circular motion are conserved according to Beeckman.42 In other words, Beeckman's principle admits the persistence not just of uniform, straight line motion, but of motion in general, regardless of the immediate direction and curvature of their path (i.e., their determinations). Many forms of motion could persist under this principle. Beeckman argued for example that if a candelabra once received “the form of swinging motion,” it would persist therein until brought to rest by an overwhelming, inequable resistance set up by the air.
Beeckman's principle does concern itself with the conservation of motion and the formal causes of the indifferent persistence of motion. “Once a thing is set in motion, it never comes to rest, except because of external impediment.” The parsimonious nature of Beeckman's version of perseverance emerges clearly in the passage, which he composed in 1614, where he denies the existence of the virtus impressa.
…Truly what philosophers say about a force which is impressed in the stone, is seen to be without reason; who truly can conceive, that if it should be thus, or how does the stone continue to move, then in what part of the stone does it make its seat? However, by an easier mind one conceives that in a vacuum, a moving object never comes to rest since no cause mutating (changing) the motion occurs; truly nothing is changed without some cause of mutation.43
Beeckman's views offer an instructive contrast for Galileo's answers to the same question, “in what part of the stone does the impressed force sit?” as he gave it explicitly in his Paduan On Mechanics, circa 1600, or as he answered implicitly in the Two Chief World Systems. The seat of the impetus is the body's center of gravity in this definition from the Mechanics:
Center of Gravity is defined to be that point in every heavy body around which parts of equal moments are arranged … And this is that point which would go to unite itself with the general center of all heavy things – that is, with the center of the earth – if it could descend in some free medium.
Whence let us draw this supposition: Any heavy body will move downward in such a way that its center of gravity will never depart from the straight line produced from this center (placed at the first point of the motion) to the general center of heavy things …
And in the second place we may suppose: Every heavy body gravitates principally upon its center of gravity and receives therein as its proper seat, every impetus, every heaviness, and in sum every moment.44
Much later, in the Two Chief World Systems, Galileo discussed the center of gravity of the earth again in very similar terms:
Salv.: “And I shall say that I believe that heavy things exist prior to the common center of gravity; hence it is not a center (which is nothing but an indivisible point and therefore incapable of acting) that attracts heavy materials to itself, but simply that these materials, cooperating naturally toward a juncture, would give rise to a common center, this being that around which parts of equal moments are arranged.”45
Then we come to the question of what the impressed force or impetus actually was in Galileo's view. In 1590, the young Galileo had also tried to describe what precisely passed from hand to stone and was called impressed force, saying,
Do you wonder what it is that passes from the hand of a projector and is impressed upon the projectile? Yet you do not wonder what passes from the hammer and is transferred to the bell of the clock, and how it happens that so loud a sound is carried over from the silent hammer to the silent bell, and is preserved in the bell when the hammer which struck it is no longer in contact … A sonorous quality is imparted to the bell contrary to its natural silence; a motive quality is imparted to the stone contrary to its state of rest. The sound is preserved in the bell, when the striking object is no longer in contact; motion is preserved in the stone when the mover is no longer in contact. The sonorous quality gradually diminishes in the bell; the motive quality gradually diminishes in the stone.46
Beeckman and young Galileo were kindred spirits, but Galileo still supposed that impressed force and impetus were physical entities – similar to the vibrations of the bell – that converge on a body's center of gravity and sum together to cause the resultant motion. Even in 1590, the quality that was passed to a projectile was a simple extension of a familiar mechanical property (i.e., weight). A sufficient projecting force overcomes the intrinsic gravity of a body and literally impresses a praeternatural lightness on it, and so moves it. In 1632, the quality was usually speed, but Galileo was less interested in what precisely the impetus was – it was in fact a mystery, like the true nature of gravity.
Salv.: “Simplicio … what you ought to say that every one knows that it is called gravity. But we do not really understand what principle or what force it is that moves stones downward, any more that we understand what moves them upward after they leave the thrower's hand, or what moves the moon around.”47
What mattered was that we could know how they operated. We may not know what these principles or forces are, but we do understand their consequences and can derive the laws of descent toward the center and the continuation of acquired motions.
There is no evidence that Galileo had abandoned his realist interpretation of impetus and impressed forces by 1609, when he did the trajectory experiments. Whether he had abandoned it in 1638 is unclear, even if his usage of “impetus,” “impressed force,” and “velocity” paralleled classical usage for simple “inertia,” “momentum,” and “velocity.” The traditional words and ideas appear everywhere there. Yet, in spite of this extra baggage of “exchanged accidents,” his formulations and analysis of the facts of motion are wide ranging and deeply perceptive – he often points directly at the large features of the classical solutions without being in a position to properly solve them himself.
RESISTANCE AND THE VIS INERTIAE
One of the themes that is basic to inertia is the idea of resistance. There is a clue that the use of perseverare to mean “persist indifferently,” which Beeckman, Descartes, and Newton all adopted for the first law, may have had its origins in scholastic discussions of resistance, especially in regard to the intension and remission of forms.
Among Galileo's papers there is an interesting collection of notebooks on questions in Aristotelian natural philosophy that he probably composed at Pisa in the late 1580s and early 1590s. Galileo is working his way through discussions of resistance found in lectures of Jesuit professors on physics, and perhaps choosing his positions in his notes for his De Elementis, in a specific collection we now call the Notebooks on the Physical Questions.48 At the end of the discussion, there appears this summary conclusion:
[7] It follows, third that three factors can be found in any resistance. The first is what it formally connotes, and this is permanence in a proper state; the second is what it implies connotatively and this is the impeded action of the contrary; the third is the cause of such permanence, i.e., the cause that makes the thing persevere in its state easily and resist the contrary action. And this cause can be manifold, e.g., the act of resisting, as when an animal by its own powers guards itself through appropriate action; or weight and hardness, as in a stone; or the binding of matter by which the action of a contrary is slowed down, etc.49
Here we actually find, in Galileo's own hand, what are tantamount to the words of the first law, expressive verbis, or at least Descartes's first law. But with what effect? Galileo did not see the full blinding importance of this idea, even as he wrote it out.
Yet Galileo was very interested in resistance then, especially the resistance of media in local motion. His chief aim was to refute Aristotle's rules of motion, and these passages may be considered as a source of reflection when reading his analysis of Aristotle's rules in On Motion.
He sought to show that the main phenomena of motions, their direction upward or downward, and their speeds, were readily resolved by applications of Archimedean mechanics. Here, in the De Elementis, however, Galileo is working with Aristotelian forms in a larger discussion about intension and remission of forms – another order of discourse entirely. This is the problem and replies that led to the summary statement just quoted:
[1] The first problem is what is resistance? Vallesius, in the first Controversies, chapter 5, and others say that resistance is action and that to resist is somehow to act…
[2] I say, first: Resistance is not action formally, because a stone resists a hand pressing on it and yet there is no action; because the least heat resists the greatest coldness – for otherwise alteration would take place in an instant – and nonetheless heat does not react on cold; because the medium resists in local motion and yet it does not react per se; and finally, because bodies here below resist the action of the heavens and nonetheless do not react on them.
[3] I say, a second; resistance is not reception. For, when iron is pressed it does not receive, though it resists …
[4] I say, third, resistance is permanence in a proper state against a contrary action. I say “against a contrary action,” for resistance, while not an action, nonetheless connotes the action of the contrary that it impedes. I say “is permanence in a proper state,” because I do not differentiate resistance from the things’ very existence whereby it endures; indeed resistance formally bespeaks this permanence of a thing in its state and connotes the impeding of a contrary action …50
Some of the most important properties we associate with inertia are linked in these passages with resistance – the permanence in a state and the opposition to contrary actions. And resistance was firmly linked in Galileo's work then with the problem of motion in a medium. There is no overt suggestion here that a body's resistance to change could be its cause of motion, except for that tantalizing hint that a body could “persevere in its state easily.” Either the seed had fallen too soon, or perhaps, his idea of indifferent horizontal motion in On Motion may be an incomplete echo of this passage.
William Wallace has determined the various origins from which Galileo drew this material, including many Jesuit writers. There is a consensus that the Notebooks on the Physical Questions were either reading notes or preparations for lecture notes. Did he ever believe them or defend them? Historians have not decided fully what to make of the Physical Questions. Yet here are the indicative phrases of the later statement of the principle of inertia.
This discussion of resistance came from a literature already in place in the Jesuit system of colleges across Europe. With all of its own foretaste of the context in which inertia was found, this passage indicates that all of the basic elements of the inertial view were coming into hand all over Europe. It only needed the spark of geometry to bring it to life.
In Two Chief World Systems, Galileo provided a more developed discussion of resistance in motion:
Now fix it well in mind as a true and well-known principle that the resistance coming from the speed of motion compensates that which depends on the weight of another moving body, and consequently that a body weighing one pound and moving with a speed of 100 units resists restraint as much as another of 100 pounds whose speed is but a single unit.51
Here are the inklings of the idea of conservation of momentum, but based on and measured in terms of resistance.
CIRCULAR OR RIGHT? THE IMPOSSIBLE MOTION
Koyré regarded Galileo's fascination with uniform circular motion as a basic stumbling block that kept him from grasping the principle of inertia. Specifying the rectilinear direction is an important feature of the principle.
One may recall Gassendi's correction of Galileo's views, or Descartes's amendment, or Newton's insistence on it. Stillman Drake once argued that if one wished to say that Galileo's concepts were inertial, it would be sufficient to show that his motions persisted indifferently, and the direction did not matter, but it appears to be important. Galileo did say in Two Chief World Systems that all motions were circular and denied that rectilinear motions ever occurred in nature. He also said nature really never uses accelerated vertical motion but achieves everything by uniform circular motion.
Yet his circular motions were always composed from two components, and one of those was a rectilinear tangential tendency to persist in the line of the impetus or impressed force – the same tangent that figures in the classical analysis of slings and orbits. Galileo admits the tangent and works with it in all of the important examples in the Second Day of the Two Chief World Systems. Rectilinear motions are never manifest for noticeable distances in nature, either for Galileo or for Newton. Koyré explains:
Contrary to what has often been said the law of inertia does not have its origin in common sense experience, and is neither a generalization nor an idealization of it. What we find in experience is circular motions, or more generally curved motion. We never see rectilinear motion, except in the untypical case of free fall, and this is precisely not a case of inertial motion. Yet it was curved motion that classical physics would struggle to explain on the basis of the latter, rectilinear motion. This is a very strange approach … what it involves, strictly speaking, is the explanation of that which exists by reference to that which does not exist, which never exists, by reference even to that which never could exist.52
In fact, the situation is less paradoxical than it seems. The inertial tangential motions defined in Newton's first law are, of course, found all around us, subsumed as components in all normal motions. Therefore, in one sense, the best example we have of inertial motion is a properly accelerated trajectory, rather than yards of pure inertial motion found only far from the Earth or in our imaginations.
Galileo was, however, well aware of the rectilinear tangential initial tendency of motions, of a body in a sling for example. In Two Chief World Systems, he says:
Salv.: “Up to this point you knew all by yourself that the circular motion of the projector impresses an impetus upon the projectile to move, when they separate, along the straight line tangent to the circle of motion at the point of separation, and that continuing with this motion, it travels farther from the thrower. And you have said that the projectile would continue to move along that line if it were not inclined downward by its own weight, from which fact the line of motion derives its curvature. It seems to me that you also know by yourself that this bending always bends toward the center of the earth, for all heavenly bodies tend that way.”53
Galileo recognizes the existence and meaning of the rectilinear tangential tendency. He does understand this feature of basic classical motion. Yet he argues strenuously for circular motion, consciously refuting Aristotle's rectilinear natural motions, and unwittingly clouding his own appreciation of rectilinear motion.
The most astonishing thing about the world of Copernicus is that all naturally occurring motions are curved motions because they share the Earth's rotational motion. That draws Galileo's focus, especially because of the rhetorical contrast it offered to Aristotle's rectilinear natural motions of the elements.
The rectilinear tendencies in the line of the impressed force at the point of departure of the stone from its sling, which Galileo identified, are deemphasized and neglected. They are not, however, rejected by Galileo either.
It is Galileo's misfortune that his campaign against Aristotle's rectilinear natural motions was carried to the extreme point that he says that straight motions never occur in nature. This leads us to question whether he could have been an inertial thinker at all. But his large-scale circular motions are always composite motions and the inertial tangent is always present, of course.
Galileo didn't see the importance of stating the inertial tangential tendency as a basic principle of motion or analysis. His neglect is borne of the exuberance of a discoverer of new worlds. There was much that was new to see at every turn, and much that was more provocative than this tangent that he admitted, even if it were never produced in nature because of intrinsic and extrinsic forces, much like Newton's right inertial motion. Yet Galileo's analysis always noticed that there was potential straight motion “in the line of the impressed force,” even if he did not write it out.
THE HISTORICAL PRINCIPLE OF INERTIA
The principle of inertia, as stated here in the first law, consists of four historically separable elements. At the base is Galileo's contention that moving bodies will continue in their unchanging motion when there is no cause to decelerate or accelerate them. Over that lies Beeckman's insistence that the body perseveres – not because of an impetus or impressed force, but because of a lack of sufficient cause to change its motion.
Galileo had thought the conservation of unchanging motion or rest was due to the action of the impressed force or impetus persisting in the absence of accelerations and decelerations. That impetus actually connected the body to the vast shared motion of the Earth.
Beeckman saw, however, that the world rested or moved uniformly – persevered was Beeckman's word – precisely in the absence of such impressed forces, as would Descartes and Newton who followed him. But, like Galileo, Beeckman thought a body could persevere in a circular orbit.
With Beeckman's principle of perseverance as its basis, Descartes's principle of right perseverance declares that perseverance only occurs in uniform and unvarying straight line motions – the crucial amendment made by Descartes. He argued that because God sustains Creation from instant to instant, He uses the simplest and most direct means of recreating motions, which is to have bodies continue in right motion to the next point in the next time frame or instant. In Descartes's view, any circular planetary orbit was sustained by outward pressing centrifugal force that balanced pressures from outside the planet's orbit.
Finally, there is Newton's clear recognition that the actions and reactions of the vis inertiae occur as accelerations, that is, as forces. In the works of Descartes and Galileo, the action of an impressed force sustained the speed of a body and did not change it. Newton's forces cause accelerations, and inertial forces change the speeds and directions of motion of colliding bodies. Newton is clear on all the points in Definition III, stated as a preamble to the three laws of motion.
In conclusion, the evidence indicates that Galileo did understand the conservation of motion, but he did not reject the idea of impetus and impressed forces. Impetus had many of the functions of classical inertia, including sustaining motion. He had understood the composition of motions, and his fascination with circular motion was a fascination with composite motions.
In essence then, in the end, he almost had all of the principles of inertia: conservation, rectilinear tangential tendency, and the equivalence of rest and motion. But he still thought in terms of impetus and for that reason failed in understanding the operation of dynamic forces. That, in turn, hindered his analysis of the forces and agencies of nature like gravitation.
NOTES
1 Galileo Galilei, Two Chief World Systems, 1632, and Two New Sciences, 1638.
2 Two New Sciences, pp. 161–2.
3 The document was first published by Stillman Drake, Galileo's Experimental Confirmation of Horizontal Inertia: Unpublished Manuscripts. A considerable literature had developed around this document and others in Volume 72 of the Manoscritti Galileliani, preserved at the Biblioteca Nazionale Centrale, Firenze. Other authors to consult concerning the experiment include Ronald Naylor, Winifred Wisan, James Maclachlan, and Kenneth Hill. Other treatments are cited elsewhere in this article. From the early years of the twentieth century until the 1960s, scholars believed that Galileo had not performed any experiments; Thomas Settle was the first historian to challenge this conclusion, cf. “Galileo's Use of Experiment,” in Galileo. Man of Science. Edited by Ernan McMullin. (New York: Basic Books, 1967), pp. 315–337. Volume 72 contains undated and unordered notes on motion and mechanics that are currently the subject of much scholarly interest.
4 P. Damerow, G. Freudenthal, P. McLaughlin, and J. Renn, Exploring the Limits of Preclassical Mechanics (New York; Springer-Verlag, 1992), pp. 243–7.
5 Galileo, Two New Sciences, pp. 161–2.
6 Rene Descartes, letter to Mersenne, Oct. 11, 1638. Cf., S. Drake, Galileo at Work (1978), p. 387.
7 Alexandre Koyré, “Gassendi and the Science of His Time,” in Metaphysics and Measurement. (London: Chapman and Hall, 1968), pp. 126–7.
8 Descartes, Principes de Philosophie, II, 37, in Oeuvres de Descartes. C. Adam and P. Tannery, eds. (Paris: L. Cerf, 1897–1913). Cf., Koyré, Galileo Studies translated by J. Mepham (Atlantic City, N.J.; Humanities Press, 1978), pp. 262–3.
9 Isaac Newton, Mathematical Principles, pp. 31–2.
10 Newton, Mathematical Principles, pp. 19–21.
11 I. B. Cohen, Newtonian Revolution (Cambridge; University Press, 1980), pp. 182–93.
12 Newton, Mathematical Principles, pp. 2–3.
13 Newton, Opticks, cf. Cohen, Newtonian Revolution, p. 397.
14 Newton, Mathematical Principles, p. 3.
15 Cohen, Newtonian Revolution. Bernd Ludwig, “What is Newton's Law About?” Science in Context 5 (1992), 139–62. Another scholar writing about inertia in Newton's work is Zev Bechler, “Newton's Ontology of the Force of Inertia,” in The Investigation of Difficult Things. P. Harmon and A. Shapiro, eds. (New York; Cambridge University Press, 1992) urging that the natural verus enforced motion distinction not be applied to Newton.
16 Newton, Mathematical Principles, p. 31.
17 Cf. R. S. Westfall, Force and Newton's Physics (New York; American Elsevier, 1971). Chapter 1.
18 Koyré, Newtonian Studies, pp. 207ff.
19 I. B. Cohen, Birth of a New Physics (Garden City, N.Y.; Anchor Books, 1960), pp. 158–61.
20 Letters on the Sunspots, pp. 113–14.
21 Two Chief World Systems, p. 149.
22 On Motion and On Mechanics, pp. 66–8.
23 Ibid., p. 67.
24 Two Chief World Systems, pp. 164–5.
25 Two Chief World Systems, pp. 163–7.
26 Two Chief World Systems, p. 116.
27 Alan Chalmers, “Galilean Relativity and Galileo's Relativity” in S. French and H. Kamminga (eds.), Correspondence, Invariance, and Heuristics (Amsterdam; Kluwer, 1993), pp. 189–205.
28 Koyré, Etudes Galileennes 3 vols., (Paris; Hermann and Cie, 1939).
29 Koyré, Galileo Studies, p. 174.
30 Koyré, Galileo Studies, p. 174.
31 Koyré, Galileo Studies, pp. 184–5.
32 Paolo Galluzzi, Momento: Studi Galileiani. Cf. Galileo, On Motion, and On Mechanics. Translated by I. Drabkin and S. Drake (Madison; University of Wisconsin Press, 1960). Especially Chapter 14 of On Motion, and the discussion of the screw in On Mechanics.
33 Ernest Moody, Galileo and Avempace.
34 Moody, “Galileo and Avempace,” Journal of the History of Ideas 12 (1951): 163–93, 375–422, pp. 392–6. Clagett, Science of Mechanics in the Middle Ages (Madison; University of Wisconsin Press, 1959), pp. 505–32.
35 Koyré, Galileo Studies, p. 169.
36 Galileo, Two Chief World Systems, p. 150.
37 Cf. James A. Weisheipel, “Galileo and his precursors,” in Galileo. Man of Science. Ernan McMullin, ed. (New York: Basic Books, 1967). Especially pp. 87–9. Also, Murdoch, John E. and Edith D. Scylla. “The Science of Motion,” Chapter 7 in Science in the Middle Ages. David C. Lindberg, ed. (Chicago: Chicago University Press, 1978), especially pp. 246–51.
38 Cf. S. Drake, Mechanics in Sixteenth Century Italy (Madison; University of Wisconsin Press, 1969).
39 Galileo, Two Chief World Systems (translated by S. Drake. Berkeley; University of California Press, 1953, 1962, and 1967), pp. 150–1.
40 Isaac Beeckman, Journal, I, wan der Wald edition (LeHayre; Nijhoff), p. 24. The translation from the Latin is my own.
41 Ibid. The translation from the Latin is my own.
42 Koyré, Galileo Studies, pp. 117–18, n. 61.
43 Beeckman, Journal, I, wan der Wald edition (LeHaye: Nijhoff), p. 24. The translation from the Latin is my own.
44 Galileo, On Motion and On Mechanics, pp. 151–2.
45 Galileo, Two Chief World Systems, pp. 245–6.
46 Galileo, On Motion and On Mechanics, pp. 79–80.
47 Galileo, Two Chief World Systems, p. 234.
48 Galileo, De Elementis, SY2 in Wallace, Galileo's Early Notebooks (Notre Dame; University Press, 1977), pp. 243–4.
49 Ibid.
50 Ibid.
51 Galileo, Two Chief World Systems, p. 215.
52 Koyré, Galileo Studies, p. 155.
53 Galileo, Two Chief World Systems, pp. 189–94.