5

LAWS OF NATURE AND THE MATHEMATICS OF MOTION

DANIEL GARBER

NATURE CAME TO BE UNDERSTOOD through mathematics in the seventeenth century, when Galileo (1890) famously wrote: “Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth” (6:232; Galilei 1957, 237–38). This, in a way, can be understood as a motto for the century as a whole, or, at least, for those figures in the century who are now recognized as the ancestors of modern science. But it is also the century in which the laws of motion as we know them are first articulated. People before the seventeenth century had certainly talked about nature as being governed by overarching laws. But even so, it is only in this century that specific laws were proposed, and their consequences explored.1

It is quite natural to see these two trends as being closely linked. In contemporary physics, after all, mathematically expressed laws of nature such as quantum theory or general relativity are at the heart of mathematical physics, which would be unimaginable without such structures. One might imagine that this close link between mathematics and laws goes back to the origin of both in early modern natural philosophy. But, I shall argue, however plausible such an assumption might be, the story is more complicated than that. Though both the mathematization of nature and the idea of a law of nature are important to the early modern vision of the physical world, they are to a large extent distinct. In making my case, I would like to explore three key figures from the period: Descartes, Galileo, and Hobbes. In Descartes we certainly have laws of nature: these are central to the project of his physics, both in the early Le monde and in the later Principia philosophiae. But, I shall argue, despite what Descartes sometimes says, mathematics is only marginal to his program. In Galileo, on the other hand, mathematics is central: his application of mathematics to motion was one of the great accomplishments of early modern science. But even so, I would argue, Galileo made no substantive use of anything that could properly be called a law of nature. And finally, with Hobbes we have something of an ambiguous situation. For Hobbes, the subject matter of geometry is body, including its motion, and so there is a sense in which the kind of general statements about motion that Descartes identifies as laws are a part of mathematics, but a mathematics that is very different from what anyone else in the period recognized as such. And while Hobbes offered a number of important general statements about bodies in motion, he was very careful not to call them laws.

DESCARTES AND LAWS OF NATURE

Descartes may not have invented the idea of a law of nature, the idea that nature is structured in accordance with some general laws that order things in the world. But he may well be the first who actually tried to articulate the laws of nature in such a way that their consequences for how nature works can be set out and evaluated.2

For Descartes, of course, body consists only of extension. As a consequence, everything in nature must be explicable in terms of the size, shape, and motion of bodies and the smaller parts that make them up. What he calls the rules or laws of nature govern the motion of bodies, one of its modes. These laws of nature are first given in chapter 7 of his early Le monde, which Descartes suppressed after finding out about the condemnation of Galileo in 1633. But they appear later in his Principia philosophiae, suitably rethought and reorganized. It is in this form that they were best known by his contemporaries.

The laws as given in the Principia are as follows:

[Law 1] Each and every thing, in so far as it can, always continues in the same state; and thus what is once in motion always continues to move. (PP 2.37)3

[Law 2] All motion is in itself rectilinear; and hence any body moving in a circle always tends to move away from the center of the circle which it describes (PP 2.39).

[Law 3] If a body collides with another body that is stronger than itself, it loses none of its motion; but if it collides with a weaker body, it loses a quantity of motion equal to that which it imparts to the other body (PP 2.40).

Preceding the statement of these laws in the Principia, Descartes proposes a principle in accordance with which the total quantity of motion in the world must remain constant:

In the beginning [God] created matter, along with its motion and rest; and now, merely by his regular concurrence, he preserves the same amount of motion and rest in the material universe as he put there in the beginning … For we understand that God’s perfection involves not only his being immutable in himself, but also his operating in a manner that is always utterly constant and immutable. Now there are some changes whose occurrence is guaranteed either by our own plain experience or by divine revelation, and either our perception or our faith shows us that these take place without any change in the creator; but apart from these we should not suppose that any other changes occur in God’s works, in case this suggests some inconstancy in God. Thus, God imparted various motions to the parts of matter when he first created them, and he now preserves all this matter in the same way, and by the same process by which he originally created it; and it follows from what we have said that this fact alone makes it most reasonable to think that God likewise always preserves the same quantity of motion in matter (PP 2.36).

Though this principle is not called a law, it is an important constraint on the behavior of bodies in motion. Following the statement of the third law, Descartes works out a series of examples in which he shows the outcome of direct collisions for two bodies with different sizes and speeds (PP 2.45ff).4 The conservation principle is a key tool that Descartes uses in deriving those supposed consequences of the third law: Descartes treats collisions in such a way that the total quantity of motion in the system of colliding bodies remains the same before and after the collision. From those examples, it is clear that the quantity of motion is measured jointly by the size and the speed of the bodies. Size rather than mass, since Descartes does not have a conception of quantity of matter distinct from size, and speed rather than velocity, since directionality is treated distinctly from the magnitude of the speed, and the conservation principle does not govern it.5

The argument for the conservation of quantity of motion is grounded in a theological doctrine that dates from long before Descartes’ time, the view that God must sustain the world from moment to moment for it to continue to exist. Descartes’ claim is that because God is immutable, and acts in a constant way, the total quantity of motion must remain constant in the world. This same divine immutability is also what he claims grounds the three explicitly named laws of nature. After the second law he notes that “the reason for this second rule is the same as the reason for the first rule, namely the immutability and simplicity of the operation by which God preserves motion in matter. For he always preserves the motion in the precise form in which it is occurring at the very moment when he preserves it, without taking any account of the motion which was occurring a little while earlier” (PP 2.39). And similarly for the third law, which follows from the fact that “since God preserves the world by the selfsame action and in accordance with the selfsame laws as when he created it, the motion which he preserves is not something permanently fixed in given pieces of matter, but something which is mutually transferred when collisions occur” (PP 2.42). The laws of nature are thus grounded in divine immutability and the fact that the created world depends from moment to moment on the power by which he keeps the world in existence.

In what sense are Descartes’ laws of nature intended to be laws of nature? First of all, one can point to their generality: they are true not merely of this or that group of bodies, but of all bodies, of bodies as such. In this way their scope is over nature as a whole. In both Le monde and in the later Principia philosophiae, the general laws are taken to apply to the cosmos as a whole, and are appealed to in a broad and hand-waving way to explain the general features of the cosmos as a whole, such as the fact that there are infinite suns, each of which is a source of light.6 Now, in Le monde Descartes refers to them as “the laws God imposed on it [i.e., nature]” (AT 11 36). This suggests that the laws are chosen by God and then “imposed” on bodies. But this is clearly not exactly what is going on, even in Le monde, which, as in the Principia philosophiae, grounds the laws in the constant and continuous activity of God on bodies in sustaining them in existence. God does not impose them on bodies in the way in which a monarch might formulate laws and then impose them on the citizenry. But there is another sense in which the laws might be said to be “imposed” on bodies. For Descartes bodies are essentially extended. As I understand that, it means bodies contain only their geometrical properties: they are the objects of geometry, made real, made concrete. But as such, considered as the objects of geometry, there is nothing in them that determines that they must obey the rules that Descartes calls the laws of nature. Taken by themselves, they are completely indifferent to motion of any kind. However, when they are realized in nature, as real, existent things, they must behave in a certain way because of the way in which an immutable God sustains them. What Descartes calls the laws of nature are really just a way of formulating how it is that God keeps them in existence. But insofar as these laws do not follow from the nature of body as such, and insofar as they require God’s intervention, we might say that they are, indeed “imposed” on body by God. This can be so even though Descartes claims that “if God had created many worlds, they [i.e., the laws] would be as true in those worlds as they are in this one” (AT 11 47). To that extent, one might want to say that the laws of nature are, in a sense, necessary for Descartes. But if they are necessary it is not because of any necessity in body, or any necessity in the laws themselves, but only because God exists necessarily, and is necessarily immutable, so necessarily will act immutably in any possible world he creates. In this way the laws of nature pertain not to the essence of body but to its continued existence, insofar as the continued existence derives directly from God.

It is interesting to note here that as Descartes understands them, the laws of nature by themselves do not entail that bodies are heavy, that is, that they tend to fall toward the center of the earth. Cartesian bodies, as I earlier noted, are completely indifferent to motion, including gravitational motion. For Descartes, bodies tend to fall to the center of the earth only because of the particular configuration of the vortex of subtle matter that surrounds the earth and the make-up of the gross bodies of our experience.7

Descartes sometimes talks as if his entire physics were just a kind of mathematics. For example, writing to Mersenne on March 11, 1640, Descartes remarked: “I would think I knew nothing in physics if I could say only how things could be, without demonstrating that they could not be otherwise. This is perfectly possible once one has reduced physics to the laws of mathematics. I think I can do it for the small area to which my knowledge extends” (AT 3 39). And at the end of Part II of the Principia philosophiae, he wrote: “The only principles which I accept, or require, in physics are those of geometry and pure mathematics; these principles explain all natural phenomena, and enable us to provide quite certain demonstrations regarding them” (PP 2.64).8 And, indeed, much of Descartes’ writings about nature do involve serious attempts at applying mathematics to the physical world. Famously, the young Descartes and the young Isaac Beeckman attempted to make physics mathematical, in some sense or another. In a famous passage in his journals, which bears the marginal note “Physico-mathematici paucissimi,” Beeckman wrote, with pride, that the young Descartes had told him that “he had never found a man, beside me [i.e., Beeckman] who … had accurately joined physics with mathematics in this way” (AT 10 52). So inspired, Descartes’ early work bristles with various attempts to combine physics and mathematics, including attempts to treat the problem of free fall that Galileo would solve in mathematical terms.9 Later, of course, Descartes will use mathematics essentially in his derivation of the law of refraction in optics, in a famous argument he will give in discourse 2 of the Dioptrique. Sophisticated mathematical arguments will also play a central role in Descartes’ account of the rainbow in discourse 8 of the Météores. In his correspondence in the late 1630s, there are attempts to apply very serious mathematics to a number of problems in the motion of bodies (on this, see Garber 2000).

But for all of that, and despite the statements in which he claims that all his natural philosophy is mathematics, there is no serious mathematics at all either in Le monde, or in the later Principia philosophiae, the canonical presentation of his natural philosophy. As he notes in discussing his thought with Frans Burman in 1648, “You do not … need mathematics in order to understand the author’s philosophical writings [e.g., his physics], with the possible exception of a few mathematical points in the Dioptrique” (AT 5 177). This is largely true of his laws of nature. The laws themselves are given in purely qualitative terms in the text. It is true that a bit of arithmetic enters into the example he works out of the application of the third law, the law of collision to the case of direct collision, as mentioned earlier. There he works out some solutions to the problem of direct collision by applying the principle of the conservation of quantity of motion to the various combinations of size and speed of two colliding bodies. But the mathematics is trivial in comparison with other attempts to join physics and mathematics in Descartes’ corpus and hardly counts as serious “physico-mathematics.”

Figure 1. From Descartes, Principia philosophiae (Amsterdam: Elzevir, 1644), 101.

And their application later in the Principia philosophiae is qualitative as well. For example, in PP 3.59, Descartes addresses the question as to force (vis) associated with the striving (conatus) that a body, rotating, has to escape along the tangent of a circle in which it is rotating, a striving derived from his second law. To treat this striving, he imagines a hollow tube, EY, fixed at E and rotating, with a ball A in the tube (see Figure 1). He writes: “When we first begin to rotate this tube around the center E, the globe will advance only slowly toward Y. But in the next instant it will advance a bit faster, because in addition to retaining its original force, it will acquire new force from its new striving to recede from E: because this striving continues as long as the circular motion lasts and is, as it were, renewed constantly” (PP 3.59).10 It is interesting here—and in radical contrast with a similar analysis in Galileo, as we shall see—that Descartes nowhere ever attempts to represent the motion of A or its acceleration in mathematical terms.

GALILEO AND THE SINTONI OF MOTION

While mathematics enters into Descartes’ natural philosophy from time to time, his treatment of the laws of motion seems to be quite independent of any attempts that he may have made to understand nature mathematically. With Galileo, on the other hand, the application of mathematics to nature is quite central to his project. However, it is not clear that the laws of nature play any substantive role in his account of the physical world. Now, there is no doubt that in some sense Galileo did recognize the idea of the laws of nature. In his important “Letter to the Grand Duchess Christina,” in which he discusses Copernicanism and the Bible, he wrote: “Nature … is inexorable and immutable; she never transgresses the laws [leggi] imposed upon her, or cares a whit whether her abstruse reasons and methods of operation or understandable to men” (EN 5 316).11 But even though in a very general sense he may have recognized the importance of laws of nature, I will argue that his own mathematical science of motion would seem to involve nothing that one can call a general law of nature.

Galileo worried about the behavior of bodies in motion for virtually his entire career, from the time that he was a young professor until his last years under house arrest. It would not be appropriate in this modest essay to try to survey Galileo’s thought throughout his long career. Instead, I would like to begin by looking at the treatment of bodies in motion in his last work, the Discorsi e dimostrazioni matematiche, intorno à due nuove scienze (Two New Sciences).12

The second of the two new sciences treated in the book is the science of motion. This is introduced at the beginning of the Third Day of the dialogue, in which the interlocutors gather to read and discuss a Latin treatise by “the Academician,” Galileo, of course. The treatise is most likely the result of work on motion done during the first decade of the seventeenth century, before Galileo got sidetracked by the astronomical project of the Starry Messenger. It begins as follows: “We bring forward a brand new science concerning a very old subject. There is perhaps nothing in nature older than MOTION, about which volumes neither few nor small have been written by philosophers; yet I find many essentials of it that are worth knowing which have not even been remarked, let alone demonstrated” (EN 8 190). This, then, is the subject matter of the new science: motion, treated in the Third and Fourth days of the dialogue.

In the Third Day, Galileo begins with a treatment of uniform motion. But the centerpiece is the treatment of naturally accelerated motion. For most readers the featured result is often referred to as the “law of free fall”:

Proposition II. Theorem II

If a moveable descends from rest in uniformly accelerated motion, the spaces run through in any times whatever are to each other as the duplicate ratio of their times; that is, are as the squares of those times. (EN 8 209)

According to this theorem, the distance fallen by a body in free fall is proportional to the square of the time. Also important is the so-called odd-number rule, a corollary to this theorem, in accordance with which the distances fallen in equal successive times are proportional to the sequence of odd numbers (210).

Interestingly, though, Galileo’s own presentation is somewhat different and not focused on the times-square rule. He begins with a question as to what the proper definition of accelerated motion is. The definition that he proposes is the following: “I say that motion is equably or uniformly accelerated which, abandoning rest, adds on to itself equal momenta of swiftness in equal times” (198, 205). But, Galileo wonders, is this definition the correct definition for falling bodies as they actually accelerate in nature? He writes: “And first, it is appropriate to seek out and clarify the definition that best agrees with that which nature employs. Not that there is anything wrong with inventing at pleasure some kind of motion and theorizing about its consequent properties … But since nature does employ a certain kind of acceleration for descending heavy things, we decided to look into their properties so that we might be sure that the definition of accelerated motion which we are about to adduce agrees with the essence of naturally accelerated motion” (197). This, then, is the question that he investigates.

The theorem generally called the law of free fall is presented as a direct mathematical consequence of that definition.13 Galileo then goes to nature, and sees if falling bodies actually satisfy that consequence, that is, he goes to nature to see if the distance fallen is proportional to the square of the time. (Actually, it is somewhat more complicated than that. Galileo does not have the means to measure that directly, so he has to slow free fall down by rolling balls down inclined planes, so that he can actually measure the time and compare it with the distance fallen. But he needs to establish that the relation between time and distance fallen in free fall will be the same for a ball rolling down an inclined plane.) Galileo describes in some detail the experiments that he performed: “In a wooden beam or rafter about twelve braccia long, half a braccio wide, and three inches thick, a channel was rabbeted in along the narrowest dimension” (212). Balls were rolled down in the channel, and the relation between distance and time noted. Galileo even felt the need to inform the reader about how exactly the time was measured by way of a water clock. His conclusion is that the balls do, indeed, observe that ratio between distance and time that Galileo’s definition of acceleration requires (212–13). His conclusion is that the definition of acceleration that begins the discussion—“this first and chief foundation upon which rests an immense framework of infinitely many conclusions”—is, indeed, the kind of acceleration that is at issue in naturally falling bodies.

The way in which Galileo derives the mathematical account of free fall from the definition of acceleration shows an interesting contrast with Descartes. Descartes’ account of the acceleration of the ball in the rotating tube, discussed earlier, is entirely qualitative: there is no mathematical reasoning and no mathematical treatment of accelerated motion. Galileo, on the other hand, starts with the idea that in accelerated motion, equal momenta of speed are added in equal times, similar to Descartes’ starting place. But he then represents the relation between time and distance fallen in geometrical terms, and then uses geometrical reasoning to derive an exact geometrical expression of the relation between time and distance.

In the remaining pages of the Third Day of the dialogue, Galileo goes on to draw further consequences from the definition of natural acceleration that he establishes early in the dialogue.14 Among the consequences that Galileo noted was one that we may consider of special interest. In the middle of a scholium to Proposition XXIII Problem IX, Galileo makes the following observation: “It may also be noted that whatever degree of speed is found in the moveable, this is by its nature indelibly impressed on it when external causes of acceleration or retardation are removed, which occurs only on the horizontal plane: for on declining planes there is cause of more acceleration, and on rising planes, of retardation. From this it likewise follows that motion in the horizontal is also eternal, since if it is indeed equable it is not weakened or remitted, much less removed” (243). This looks very much like Descartes’ first and second laws of nature from the Principia philosophiae, the so-called (but incorrectly named) principle of inertia.15 But there are differences. It is important to note here that what Galileo is talking about is not rectilinear motion but horizontal motion: motion on a plane all of whose points remain equidistant from some point toward which the heavy body is attracted. That is to say, what persists is circular motion around the point to which a heavy body tends to fall. And it is important to note that we are dealing with motion on a plane: if the plane were eliminated, the body would simply continue to fall toward the center of attraction. And finally, it should be noted that as salient as this observation is to us, in the context of the Third Day of the Discorsi, Galileo presents this simply as an observation in passing in the course of a scholium. In the context of the Third Day, it does not even get separate designation as a theorem or proposition. Though we may consider it of special interest, in the context of the Third Day, Galileo felt otherwise.

But this observation has a special role to play in the central proposition of the Fourth Day. The very first proposition of the Fourth Day is Galileo’s account of projectile motion: “When a projectile is carried in motion compounded from equable horizontal and from naturally accelerated downward [motion], it describes a semiparabolic line in its movement” (EN 8 269). This proposition is proved quite simply by putting together the uniform motion of a body on a horizontal plane with the accelerated motion of a body in free fall: when you combine the two, it follows straightforwardly that the projectile describes a semiparabola.

In this way, Galileo’s new theory of motion is able to give very sophisticated mathematical descriptions of the motion of bodies in various important circumstances, in free fall, on a horizontal plane, and in projectile motion. But are these laws of nature? I think not. But why not? Fundamentally it is a question of scope.

The facts about motion that we have been examining depend strongly on the assumption that we are dealing with heavy bodies, bodies that have a tendency to fall toward a particular point. In the Discorsi, that point is, of course, the center of the earth. In his earlier book, the Dialogo sopra i due massimi sistemi del mondo (Dialogue Concerning the Two Chief World Systems), Galileo attempts to generalize this. In the First Day of that dialogue he attempts to articulate an alternative to the Aristotelian cosmology that takes the center of the earth as the center of the universe. In that context he writes: “Every body constituted in a state of rest but naturally capable of motion will move when set at liberty only if it has a natural tendency toward some particular place; for if it were indifferent to all places it would remain at rest, having no more cause to move one way than another. Having such a tendency, it naturally follows that in its motion it will be continually accelerating” (EN 7 44).16 It is evident that the center of the earth is such a particular place for bodies on the earth: “the parts of the earth do not move so as to go toward the center of the universe, but so as to unite with the whole earth (and that consequently they have a natural tendency toward the center of the terrestrial globe, by which tendency they cooperate to form and preserve it)” (57–58; TCWS 33). But it is not at all clear how to generalize this to other bodies outside of the earth. In the TCWS, Galileo does offer his famous Platonic hypothesis about the formation of the planetary system:

Let us suppose that among the decrees of the divine Architect was the thought of creating in the universe those globes which we behold continually revolving, and of establishing a center of their rotations in which the sun was located immovably. Next, suppose all the said globes to have been created in the same place, and there assigned tendencies of motion, descending toward the center until they had acquired those degrees of velocity which originally seemed good to the Divine mind. These velocities being acquired, we lastly suppose that the globes were set in rotation, each retaining in its orbit its predetermined velocity. (EN 7 53; TCWS 29)

Galileo claims that if we assume such a hypothesis, we can find a single place from which, should each of the planets be imagined to fall toward the sun and their final speed converted into rotational motion, we would get a system that agrees very closely with the observed speeds of the actual planets. In this way Galileo hypothesizes the sun as a center of tendency for planets, and in this way, in principle, extends his account of motion to the planets with respect to the sun. But this is far from a systematic generalization of the accounts of free fall, the persistence of horizontal motion, and the behavior of projectiles, which remain particular claims about heavy bodies near the surface of the earth. What would happen if a piece of Earth were released ten feet over the surface of Mars, or a Martian rock ten feet over the surface of Earth? What happens to bodies in interplanetary space, or beyond the orbit of the last planet?

Lacking an obvious general applicability outside of a fairly narrow context, it is difficult to see how Galileo’s accounts of motion could play anything like the role in organizing the Galilean world that Descartes’ laws of nature play in his, where they are used to explain the structure of the whole universe. In writing to Mersenne about his impressions of Galileo’s Discorsi, Descartes (1638) noted the following:

Generally speaking, I find that he philosophizes much more ably than is usual, in that, so far as he can, he abandons the errors of the Schools and tries to use mathematical methods in the investigation of physical questions. On that score, I am completely at one with him, for I hold that there is no other way to discover the truth. But he continually digresses, and he does not take time to explain matters fully. This, in my view, is a mistake: it shows that he has not investigated matters in an orderly way, and has merely sought explanations for some particular effects, without going into the primary causes in nature; hence his building lacks a foundation. (Descartes to Mersenne, 11 Oct. 1638, AT 2 380)

It seems true to say that “his building lacks a foundation”: the generalizations that Galileo presents explain things on Earth, but they fail to treat nature as a whole. For that he would need to articulate general laws that unite the terrestrial and cosmological domains.

Now, this may not be entirely fair to Galileo: it is not clear to me that he wanted to do the kind of thing that Descartes was doing, and may have been quite happy to work piecemeal, one problem at a time. In a letter Galileo wrote to Belisario Vinta in 1610, describing the project that would become Days Three and Four of the Discorsi, he projected “three books on local motion—an entirely new science in which no one else, ancient or modern, has discovered any of the most remarkable characteristics [sintomi] which I demonstrate to exist in both natural and violent movement (EN 10 351–52).17 Similarly, in a letter from January 1639, shortly after the publication of the Discorsi in 1638, Galileo describes the project in similar terms: “I’m interested in examining what might be the characteristics [sintomi] which accompany the motion of a moving body, which, starting from a state of rest, it goes on moving with a speed that constantly increases in the same way” (EN 18 12).18As we saw earlier in the “Letter to the Grand Duchess Christina,” Galileo has the concept of an overarching law of nature, something that governs reality as a whole. But his mathematical account of the motion of bodies is not conceived in those terms: his aim is just to give some of the most interesting “sintomi” of accelerated motion. What we have in Galileo, in essence, is a thoroughly mathematical account of at least some aspects of the motion of bodies, but without laws of nature.

HOBBES AND THE GEOMETRY OF MOTION

Motion plays a central role in Hobbes’s natural philosophy: it is the sole determinant of change in his materialistic world of body. And mathematics is central as well. Hobbes, like Galileo before him, was interested in a mathematical account of motion. In fact, for Hobbes, motion is an integral element of his geometry. However, while Hobbes put forward a number of general statements about bodies in motion, his relation to the tradition of laws of motion is not entirely clear.

Hobbes was a great admirer of Galileo, particularly in regard to his treatment of motion. In the 1660 dialogue, Examinatio et Emendatio Mathematicae Hodiernae, Hobbes wrote: “the doctrine of motion is known to very few, notwithstanding the fact that the whole of nature, not merely that which is studied in physics, but also in mathematics, proceeds by motion. Galileo was the first who wrote anything on motion that was worth reading” (Hobbes 1660 quoted in Jesseph 2004). In De corpore (Hobbes 1655) his praise was even stronger, advancing Galileo as the founder of natural philosophy: “After him [i.e., Copernicus] the Doctrine of the Motion of the Earth being now received, and a difficult Question thereupon arising concerning the Descent of Heavy Bodies, Galileus in our time striving with that difficulty, was the first that opened to us the gates of Natural Philosophy Universal, which is the knowledge of the Nature of Motion. So that neither can the Age of Natural Philosophy be reckoned higher than to him” (DC, Epistle Dedicatory, n.p.).19 But despite his high praise for Galileo, his own treatment of motion was radically different from that of his hero, and much closer to that of Descartes, whose philosophy he generally rejected.

While there are a number of treatments of motion in Hobbes’s writings, I will concentrate on what is arguably the canonical treatment in his De corpore of 1655, the treatise on body that begins his Elementa philosophiae, the three-part philosophical project that begins with a physics, is followed by an account of the human being (De homine 1658), and is completed by a politics (De cive 1642). De corpore is divided into four parts. Part I, “Computation or Logique,” is a preface to the Elementa project as a whole, and contains a treatise on logic. Parts II, III, and IV constitute a natural philosophy. Part II is called “The First Grounds of Philosophy.” This First Philosophy consists in “universal definitions … the most common notions [distinguished] by accurate definition, for the avoiding of confusion and obscurity” (DC 6.17, “The Author’s Epistle to the Reader,” n.p.). In one place, Hobbes characterizes Part III as concerning “the expansion of space, that is, geometry” (DC, “The Author’s Epistle to the Reader,” n.p.). But elsewhere he is more expansive. He writes: “Next [i.e., after the First Philosophy], those things which may be demonstrated by simple motion (in which Geometry consists). After Geometry, such things as may be taught or shewed by manifest action, that is, but thrusting from, or pulling towards” (DC 6.17). The final part contains the investigation of “the motion or mutation of the invisible parts of things, and the doctrine of sense and imagination” (ibid.). Unlike Parts II and III, which involve, in principle, only definitions and that which follows directly from definitions, in Part IV, “Physiques,” Hobbes argues from physical effects observed by the senses to conjectured underlying causes, “the finding out by the appearances or effects of nature which we know by senses, some ways and means by which they may be (I do not say, they are) generated” (DC 25.1).

Part II does, indeed, contain a number of important definitions, including definitions of space (7.2), time (7.3), body (8.1), accident (8.2), place (8.5), motion (8.10), and rest (8.11), among other things. And it is in this context that Hobbes introduces certain general truths about bodies in motion. After offering his basic definitions, Hobbes advances a statement very much like Descartes’ first law of nature: “Whatsoever is at rest, will always be at rest, unless there be some other body besides it, which, by endeavouring to get into its place by motion, suffers it no longer to remain at rest” (DC 8.19). This statement is defended as follows:

For suppose that some finite body exist and be at rest, and that all space besides be empty; if now this body begin to be moved, it will certainly be moved some way; seeing therefore there was nothing in that body which did not dispose it to rest, the reason why it is moved this way is in something out of it; and in like manner, if it had been moved any other way, the reason of motion that way had also been in something out of it; but seeing it was supposed that nothing is out of it, the reason of its motion one way would be the same with the reason of its motion every other way, wherefore it would be moved alike all ways at once; which is impossible. (DC 8.19)

That is, if a body at rest were to begin to move, it would have to move in some direction or another, and there is no reason why it should move one way rather than another. And for a similar reason, Hobbes holds that a body in motion will remain in motion: “In like manner, whatsoever is moved, will always be moved, except there be some other body besides it, which causeth it to rest. For if we suppose nothing to be without it, there will be no reason why it should rest now, rather than at another time; wherefore its motion would cease in every particle of time alike; which is not intelligible” (DC 8.19). Here the argument is very similar: if a body in motion were to come to rest there is no reason why it should come to rest in any one moment in preference to any other moment. From this Hobbes infers a more general principle, that the only thing that can cause motion is another motion: “There can be no cause of motion, except in a body contiguous and moved” (DC 9.7).20 Unlike Descartes’ arguments, which appeal to a God who sustains the world from moment to moment, Hobbes appeals to something like a principle of sufficient reason.

These accounts of motion occur in Part II of De corpore, ostensibly about definitions. But in Part III, his “Geometry,” Hobbes offers a general statement that looks a great deal like Descartes’ second law. To understand that law we need to understand Hobbes’s notion of endeavor (conatus in the Latin version): “I define ENDEAVOUR to be motion made in less space and time than can be given; that is, less than can be determined or assigned by exposition or number; that is, motion made through the length of a point, and in an instant or point of time” (DC 15.2). Despite appearances, this is not an infinitesimal motion:

For the explaining of which definition it must be remembered, that by a point is not to be understood that which has no quantity, or which cannot by any means be divided; for there is no such thing in nature; but that, whose quantity is not at all considered, that is, whereof neither quantity nor any part is computed in demonstration; so that a point is not to be taken for an indivisible, but for an undivided thing; as also an instant is to be taken for an undivided, and not for an indivisible time. (DC 15.2)

Endeavor, then, is a genuine motion, the motion of a body through a finite (though “inconsiderable”) distance in a finite time.

It is in terms of this notion of endeavor that Hobbes characterizes (without supporting argument) the motion of a body that is moved simultaneously by two different causes (motions):

And whatsoever the line be, in which a body has its motion from the concourse of two movents, as soon as in any point thereof the force of one of the movents ceases, there immediately the former endeavour of that body will be changed into an endeavour in the line of the other movent. Wherefore, when any body is carried on by the concourse of two winds, one of those winds ceasing, the endeavour and motion of that body will be in that line, in which it would have been carried by that wind alone which blows still. (DC 15.5–6)

And from this he draws the following consequence:

And in the describing of a circle, where that which is moved has its motion determined by a movent in a tangent, and by the radius which keeps it in a certain distance from the centre, if the retention of the radius cease, that endeavour, which was in the circumference of the circle, will now be in the tangent, that is, in a straight line. For, seeing endeavour is computed in a less part of the circumference than can be given, that is, in a point, the way by which a body is moved in the circumference is compounded of innumerable strait lines, of which every one is less than can be given; which are therefore called points. Wherefore when any body, which is moved in the circumference of a circle, is freed from the retention of the radius, it will proceed in one of those strait lines, that is, in a tangent. (DC 15.6; cf. DC 21.9)

This closely resembles what Descartes puts forward in his second law. As in the case of the earlier general statements about bodies in motion, this one does not involve God and his continual conservation. In this case it is taken to follow from an apparently self-evident principle about the combination of motions.

Hobbes only touches on the problem of collision, and unlike Descartes, does not really offer a developed account of impact, or any arguments for his account.21 But even more significantly, Hobbes does not seem to present any kind of conservation principle at all that corresponds to Descartes’ principle of the conservation of quantity of motion. This, presumably, cannot be done without God. Or, at least, Hobbes, I suspect, was unable to figure out how to do it without God. Or simply chose not to.22

There are a number of important ways in which this account of motion is like that of Descartes—and unlike Galileo—despite Hobbes’s extravagant praise of the latter. Like Descartes, Hobbes is treating body as such, and not just heavy bodies: his statements are intended to follow in some way or another from the notions of body and motion, and to hold for all bodies. For Hobbes, as for Descartes, heaviness is not essential to body, but is only introduced later, after the general truths about body and motion are given. In the De corpore, heaviness appears as a physical phenomenon, to be explained in terms of a conjectured underlying physical mechanism, given in terms of bodies in motion that satisfy the constraints Hobbes had set out earlier in that work.23

But, in the context of our questions, are these general constraints on motion laws? And are they mathematical? In both cases it is not entirely clear what to think.

The general statements about motion are not called “laws” by Hobbes, unlike Descartes did in the Principia philosophiae. Descartes’ work was published in 1644, and there is no doubt that Hobbes knew that publication, and knew it well. There are references to Hobbes’s reaction to it in the correspondence in his circle, and direct references to it in the De corpore, though not explicitly by name. One can suppose that the avoidance of the term “law” in this connection was an explicit decision on Hobbes’s part, one that was intended to express a difference between his view and Descartes’. And, as I noted earlier, unlike Descartes, Hobbes very self-consciously does not appeal to God in his account of natural philosophy in general, and these general statements about motion in particular. This, for him, was a matter of principle. In the De corpore, Hobbes argues explicitly that God can play no role in natural philosophy. He wrote:

The subject of [natural] Philosophy, or the matter it treats of, is every body of which we can conceive any generation, and which we may, by any consideration thereof, compare with other bodies, or which is capable of composition and resolution; that is to say, every body of whose generation or properties we can have any knowledge … Therefore, where there is no generation or property, there is no philosophy. Therefore it excludes Theology, I mean the doctrine of God, eternal, ingenerable, incomprehensible, and in whom there is nothing neither to divide nor compound, nor any generation to be conceived. (DC 1.8)24

The question of God comes up in chapter 26 of De corpore, where Hobbes takes up the question of creation and the infinity of the world. Such questions, he argues, are beyond reason to resolve. He wrote:

The questions therefore about the magnitude and beginning of the world, are not to be determined by philosophers, but by those that are lawfully authorized to order the worship of God. For as Almighty God, when he had brought his people into Judæa, allowed the priests the first fruits reserved to himself; so when he had delivered up the world to the disputations of men, it was his pleasure that all opinions concerning the nature of infinite and eternal, known only to himself, should, as the first fruits of wisdom, be judged by those whose ministry he meant to use in the ordering of religion. (DC 26.1)

The questions, in short, are theological and not philosophical. And so, he concludes: “Wherefore I purposely pass over the questions of infinite and eternal; contenting myself with that doctrine concerning the beginning and magnitude of the world, which I have been persuaded to by the holy Scriptures and fame of the miracles which confirm them; and by the custom of my country, and reverence due to the laws” (DC 26.1). Though he does not say so in the De corpore, I suspect that Hobbes’s attitude to Descartes’ grounding of the laws of nature would be the same, insofar as it requires us to know features of God, such as his immutability, that go beyond what we can know through reason.25

Another reason to wonder whether they are laws comes from their role in Hobbes’s natural philosophy. For Descartes the laws are central constraints on the behavior of bodies as such: they are isolated as special principles, and designated as propositions of special importance. But while Hobbes presents these general statements about bodies in motion, he does not have the same ambitions for them. The general statements are presented almost in passing, in chapters entitled “Of Body and Accident,” “Of Cause and Effect,” “Of the Nature, Properties, and divers considerations, of Motion and Endeavour.” There is no sense of these as principles that are intended to structure nature, in any real sense.

Furthermore, their status is very close, if not identical, to that of geometrical truths. Here there is another contrast with Descartes. For Descartes, geometrical truths hold for extension as such, whether it is the extension of purely geometrical bodies that do not exist in the real world of created things, or for the objects of pure geometry, independent of real existence. But the laws of nature hold only for bodies that are created—and sustained—by God: they are truths that depend on God in a way in which geometrical truths do not.26 But their status in Hobbes is rather different. As noted earlier, for Hobbes, natural philosophy begins in first philosophy, and first philosophy begins in definitions. After the definitions, though, “we should first demonstrate those things which are proximate to the most universal definitions (in which consists that part of philosophy which is called “First Philosophy”), and then those things which can be demonstrated through motion simpliciter, in which consists geometry” (DC 6.17).27 Which is to say, these facts about motion are taken to be general truths about motion on a par with geometrical theorems, eternal truths of a sort, either things that follow directly from definitions, or what he calls geometry. It should be noted here that Hobbes’s conception of geometry is somewhat idiosyncratic. Motion, for Hobbes, is part of the subject matter of geometry. Furthermore, for Hobbes geometry is just the science of extended body: unlike Descartes, he recognizes no radical distinction between geometrical bodies and physical bodies. Writing in the Six Lessons to the Professors of the Mathematiques, published in 1656 with the English translation of the De corpore, Hobbes writes: “there is no Subject of Quantity, or of Equality, or of any other accident but Body” (15). That is to say, all mathematical notions pertain to body. And this, for Hobbes, is especially true of motion. This, for Hobbes, determines the proper way of interpreting Euclidean geometry: “And by all these a man may easily perceive that Euclide in the definitions of a Point, a Line, and a Superficies, did not intend that a Point should be Nothing, or a Line be without Latitude, or a Superficies without Thickness … For Lines are not drawn but by Motion; and Motion is of Body only” (Hobbes 1656, 9). For Hobbes, in short, the objects of geometry are bodies, strictly speaking, and there is no real distinction between natural philosophy and geometry, at least at the level of the general and foundational part of natural philosophy.28

That said, there remains some uncertainty about how to understand what Hobbes is doing. Hobbes certainly did not advance laws in the sense that Descartes did. But can we say that these general assertions about the behavior of bodies in motion are not laws of nature, strictly speaking? One might make such a judgment on the basis of a philosophical conception of what constitutes a law of nature, perhaps. But I now feel somewhat reluctant to do so. Any such philosophical conception would seem to be a priori and perhaps a bit arbitrary, and certainly historically suspect. To look at these Hobbesian texts at the moment when the notion of a law of nature in this sense is just being articulated suggests to me that there may not be a clear answer to this question. And is Hobbes’s account of motion mathematical in any sense? Well, it is certainly mathematical in the Hobbesian sense: these general statements about bodies in motion are part of mathematics as Hobbes understood it. But it is hard to ignore the fact that Hobbes’s conception of mathematics is highly idiosyncratic, hardly a conception that we would recognize as mathematical. In this respect, the situation with respect to Hobbes is quite different from the situation with respect to Galileo, whose geometrical treatment of bodies in motion is mathematical in a very classical sense. Which is to say, it is unclear whether Hobbes’s account of motion involves laws, and whether we should say that it is mathematical.

CONCLUSIONS

And so, in the end, the relation between the mathematization of nature and the discovery of the laws that govern the natural world would seem to be more complicated than expected. While there are, no doubt, ways in which they are connected, they are also in many ways independent of one another, as the cases of Descartes, Galileo, and Hobbes show. More generally, I think that there is a temptation to suppress the complexity of the so-called scientific revolution of the early modern period in favor of a simpler narrative. In broad brush, there are a number of important developments in the period, including the mathematization of nature and the development of laws, which we have examined here, but also the development of mechanical models, the growth of experimentalism, the invention of new instruments, including the microscope and telescope, the foundation of new institutions, such as the Royal Society and the Académie royale des sciences, the development of the learned journal as a means of communication, among other innovations. (There are also many “innovations” that arose in the period that did not survive.) There is a strong temptation to think that these different elements march together to produce something that we can call The New Science, which replaced the Aristotelian natural philosophy that dominated the intellectual world in the period before. When we look at these more carefully, I think that we will realize that the transition to modern science was much more complicated than that.29 But that is an argument for another occasion.

ABBREVIATIONS

NOTES

I would like to thank the participants in the discussion of my paper at the conference, “The Language of Nature,” for a lively discussion. I would especially like to thank the organizers, Geoff Gorham, Ed Slowik, Ben Hill, and Ken Waters, both for organizing the conference and for the very detailed comments on my essay. I owe a special debt to Ursula Goldenbaum, my commentator during the conference session, both for her comments and for the very helpful exchanges we had on the paper after the conference. Because of all of these interventions, this paper is much changed, and, I hope, much improved from the version that I had originally submitted.

1. The pioneering study of the history of the laws of nature is Zilsel (1942). Another important earlier study is Milton (1981). Milton develops his views further in Milton (1998). For more recent studies see, for example, Steinle (1995), Roux (2001), Henry (2004), and the essays collected in Daston and Stolleis (2008). Dana Jalobeanu has emphasized to me for many years that the view of nature as governed by overarching law is a central feature of Stoic thought. See her essay (Jalobeanu n.p.). Henry (2004, 79), in defending the priority of Descartes on the laws of nature, makes an important contrast between laws of nature as “merely references to the regularity of nature,” as opposed to “the concept of a law of nature as a specific and precise statement which codifies observed regularities in nature but which is also assumed to denote an underlying causal connection, and therefore can be said to carry explanatory force.”

2. See references cited above in note 1.

3. All translations from Descartes are taken from Descartes 1985–91, except where otherwise noted.

4. For a discussion of Descartes’ account of collision, see Garber (1992, chapter 8).

5. On directionality (what Descartes calls determination) see Garber (1992, 188ff). On the conservation principle, see Garber (1992, 204ff).

6. The argument here is that light is pressure in the ether, which derives from the rotation of the vortices around each sun by way of the second law.

7. On this see Descartes, Le monde, chap. 11 (AT 11:72ff) and PP 4.23ff.

8. Cf. Descartes to Plempius, October 3, 1637 (AT I 410–11 and 420–21).

9. For a discussion of Descartes’ attempts to deal with the problem of free fall, see Koyré (1978) and Jullien and Charrak (2002).

10. Translated in Descartes (1983).

11. Translated in Galilei (1957, 182). Thanks to Ursula Goldenbaum for pointing out this passage to me.

12. References to the Discorsi will be given in EN 8. All translations are from Galilei (1974). Since this translation is keyed to the pagination in EN 8, I won’t cite it separately.

13. The argument goes roughly as follows: In uniformly accelerated motion as Galileo defines it, the speed is proportional to the time. Consequently the terminal speed is proportional to the time. But by the so-called mean-speed theorem, proven in prop. I theorem I (EN 8:208ff), a body uniformly accelerated (on Galileo’s definition) will move a distance in a given time equal to the distance that it would go in the same time were it moving at half the terminal speed. So the distance fallen is proportional to one half of the terminal speed times the time. But by the definition of uniform acceleration, the terminal speed is proportional to the time. And so the distance fallen is proportional to the square of the time.

14. At least one of Galileo’s readers—René Descartes—was not impressed and could not find the patience to read them: “I shall say nothing of the geometrical demonstrations of which the book is full, for I could not summon the patience to read them, and I am prepared to believe they are all correct. When looking at his propositions, it simply struck me that you do not need to be a great geometrician to discover them; and he does not always take the shortest possible route, which leaves something to be desired (Descartes to Mersenne, October 11, 1638, AT 2 388).

15. On the notion of “inertia” in the early seventeenth century, see Garber (1992, 253ff).

16. Translated in Galilei (TCWS 20). Cf. EN 7 56, TCWS 31–32.

17. Translated in Galileo (1957, 63), slightly altered, as discussed in the following note.

18. A key question, of course, is the proper translation of the term “sintomi” in Galileo. In the previously cited passage, Drake translates it as “laws.” This seems clearly wrong. Galileo does not use the term often, but in two passages he pairs it with “accidenti,” suggesting that they are virtual synonyms. See Intorno alle cose che stanno in su l’acqua … (EN 4 115) and Discorso del flusso e reflusso del mare … (EN 5 377). It also appears in the Letters on Sunspots, where Reeves and Van Helden translate it as “characteristics.” (EN 5 117, Galilei and Scheiner [2010]. See also EN 4 698 and EN 7 189 for similar uses of the term.) I am grateful for help on this tricky issue from Eileen Reeves.

19. Throughout, the English is quoted from the anonymous English translation (Hobbes 1656).

20. Though Hobbes goes on at some length in 9.7 to establish this, so far as I can see it follows pretty directly from the considerations in 8.19.

21. See DC 15.8, for example. Hobbes’s account of collision is made particularly complicated by the fact that only motion resists motion, so that a body at rest does not resist the acquisition of new motion. For a very helpful discussion of what amounts to the elements of Hobbes’s account of impact in De corpore, see Morris (2007).

22. In his essay, Gorham (2013) presents a somewhat different view of Hobbes’s physics and its relation to God. Gorham takes seriously Hobbes’s various statements that God is body, arguing that for Hobbes, God is a fluid body that infuses the universe, “most pure, most simple corporeal spirit” (Gorham 2013, 254). He then argues that God, understood in this way, “is the perpetual source of motion, and hence diversity, in a material world governed by mechanical principles” (252). This, in part, would seem to address the problem in Hobbes’s natural philosophy that “any motion in the world must dissipate in no time, like shock waves” (251). And so, Gorham suggests, Hobbes’s physics, like Descartes’, would seem to involve divine sustenance and a kind of conservation principle. This is a fascinating suggestion. But it is worth pointing out a few things. First of all, even if the material God supports motion in this way, Gorham does not suggest that the general statements about motion that correspond to Descartes’ laws are in any way derived from God as cause of motion. Second, even though we might see the appeal to a material God in the passages that Gorham cites as addressing the continual diminution of motion in the world, there is no place where Hobbes articulates a conservation principle. And finally, the support for Gorham’s position is almost exclusively in texts significantly later than the 1655 publication of the De corpore, particularly the “Answer to Bramhall,” written probably in 1668 but not published until 1682, after Hobbes’s death. My focus in this text is on the doctrine in the much more widely read and influential De corpore.

23. See the account of heaviness in DC 30.

24. Gorham (2013) would disagree with this, of course. See note 22.

25. Here, again, we may be dealing with a view that Hobbes gave up in his later writings, if Gorham (2013) is right in its interpretation.

26. This is actually a bit subtle since even geometrical truths for Descartes depend on God by way of his (in)famous doctrine of the creation of the eternal truths. But geometrical truths depend on God for their creation, as do all eternal truths, while the laws of nature depend on God in his moment-by-moment sustenance of bodies in the material world.

27. My translation of the 1655 Latin.

28. On this, see Jesseph (1999), chapter 3. There is, however, a distinction between the project of Parts II and III of the De corpore, his first philosophy and his geometry, which are grounded in definitions and what can be drawn from definitions, and the project of Part IV, which he calls physics proper, which involves conjectured mechanisms.

29. I have tried to sketch out an alternative model of intellectual change in the period in Garber (2016).

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