## 1. A Brief History of Descartes' Scientific Work



Despite his fame as a philosopher of purely metaphysical problems,
such as the relation of the soul and body, or God's existence,
it would not be incorrect to conclude that Descartes was a scientist
first and a philosopher second. Not only did Descartes' interest
and work in science extend throughout his entire scholarly career, but
some of his most important metaphysical works (e.g., the
*Meditations*) were prompted by a perceived need to equip his
science with a metaphysical foundation that would be acceptable to the
Aristotelian-influenced Scholastics. Yet, one must be careful not to
impose modern conceptions on the "natural philosophy" of
earlier centuries, for much seventeenth century science was
practically indistinguishable from the more speculative metaphysics of
the era (and thus the label "natural philosophy" is
particularly apt for describing seventeenth century science). In fact,
much of Descartes' science is only part of a much larger system
that embraces all areas of philosophical investigation, including both
his physics and metaphysics.



The awakening of Descartes' interest in physics is often dated
to late 1618, when Descartes first met Isaac Beeckman, an amateur
scientist and mathematician who championed the new
"mechanical" philosophy. The mechanical philosophy's
explanation of natural phenomena, which Descartes quickly adopted,
rejected the use of Scholastic substantial forms (see
 Section 2).
 Rather, the mechanical approach favored a contact or impact model of
the interaction of small, unobservable "corpuscles" of
matter (which possess only a limited number of mainly geometric
properties, such as size, motion, shape, etc.). Over the course of the
next decade, Descartes worked on a large number of problems in both
science and mathematics, with particular emphasis on the theory of
light, mechanics (including hydrostatics), and the free-fall of
terrestrial bodies. Much of Descartes' output at this time is
both highly mathematical and concerns only specific physical problems,
not unlike the work of his contemporary Galileo. One of the
accomplishments of these years includes his discovery of the law of
refraction, often called Snell's law: when light passes from one
medium into another, the sine of the angle of incidence maintains a
constant ratio to the sine of the angle of refraction. By the
beginning of the 1630s, however, Descartes embarked on a more
ambitious plan to construct a systematic theory of knowledge,
including physics. The result was *The World* (1633), an
important text in that it essentially contains the blueprints of the
mechanical/geometric physics, as well as the vortex theory of
planetary motion, that Descartes would continue to refine and develop
over the course of his scientific career. Before publishing the
treatise, however, he learned of the Church's (1633)
condemnation of Galileo for promoting Copernicanism, which prompted
Descartes to withdraw his work from publication (since Descartes also
advanced Copernicanism in *The World*). In the 1630s, the
publication of the *Geometry,* the *Optics,* and the
*Meteorology,* along with a philosophical introduction,
*Discourse on the Method* (1637) further presented Cartesian
hypotheses on such topics as the law of refraction, vision, and the
rainbow. Yet, besides a brief sketch of his metaphysics and physics in
the *Discourse* (Parts IV and V), a comprehensive treatment of
his physics had to await the 1644 publication of the *Principles of
Philosophy.* This work not only represents Descartes' most
fully developed and exhaustive investigation of physics, it also
provides the metaphysical underpinnings of his physical system (in
Part I). As an embodiment of his mature views, the *Principles*
will thus form the basis of our examination of Cartesian physics.


### 1.1 A Note on the Primary Texts



The translations, with minor variations, are from Descartes 1979,
1983, 1984a, 1984b, 1991, but the passages are identified with respect
to the Adam and Tannery edition of the *Oeuvres de Descartes*
(1976) according to the standard convention: "AT",
followed by volume and page number. Passages cited from the
*Principles*, however, will be identified by "Pr",
followed by volume and article, and with a final "F"
indicating the inclusion of new material from the French translation
of 1647.


### 1.2 Suggested Readings



For recent in-depth examinations of Descartes' physics, see
Garber 1992a and Des Chene 1996. Schuster 2014 covers Descartes'
early physics, from 1618 to 1633. A concise survey of Cartesian
physics can be found in Garber 1992b. The scientific career of
Descartes, with special emphasis on his physics, is presented in Shea
1991; see also Gaukroger, Schuster, Sutton 2000 for the many aspects
of his natural philosophy. Gaukroger 2002 examines the *Principles
of Philosophy*, especially the physics, whereas Slowik 2002
focuses primarily on Cartesian space and relational motion. The
historical background to much in Descartes' physics is also
treated in Ariew 2011. On methodology in Descartes' natural
philosophy, see Smith 2009, while Hattab 2009 and Machamer and McGuire
2009 cover the development of various ideas important to his physics.
On the reception of Descartes' work and early forms of
Cartesianism, with special attention devoted to physics, see Schmaltz
2005, 2017, and Dobre 2017.


## 2. The Strategy of Cartesian Physics



Like many of his contemporaries (e.g., Galileo and Gassendi),
Descartes devised his mechanical theory in large part to refute the
widely held Aristotelian-based Scholastic explanation of natural
phenomena that employed an ontology of "substantial forms"
and "primary matter". Briefly, Scholastic natural
philosophy viewed a material body as comprising both an inert
property-less substratum (primary matter) and a quality-bearing
essence (substantial form), with the latter providing the body's
causal capacities. A quantity of matter, for example, possesses
weight, color, texture, and all of the other bodily properties, only
in virtue of being conjoined with a determinate form (of a billiard
ball, chair, etc.). Descartes admits that he had earlier held such a
view of gravity, envisioning the substantial form as a kind of
goal-directed (teleological) mental property of bodies: "what
makes it especially clear that my idea of gravity was taken largely
from the idea I had of the mind is the fact that I thought that
gravity carried bodies towards the centre of the earth as if it had
some knowledge of the centre within itself. For this surely could not
happen without knowledge, and there can be any knowledge except in a
mind" (AT VII 442). In a revealing passage from *The
World*, Descartes declares the Scholastic hypothesis to be both an
unintelligible and inadequate methodological approach to explaining
natural phenomena:



> 
> If you find it strange that I make no use of the qualities one calls
> heat, cold, moistness, and dryness..., as the philosophers [of
> the schools] do, I tell you that these qualities appear to me to be in
> need of explanation, and if I am not mistaken, not only these four
> qualities, but also all the others, and even all of the forms of
> inanimate bodies can be explained without having to assume anything
> else for this in their matter but motion, size, shape, and the
> arrangement of their parts (AT XI 25-26).
> 



Descartes' plan is to reduce the class of metaphysically suspect
properties, such as heat, weight, taste, to the empirically
quantifiable attributes of size, shape, and motion. In other words,
Descartes intends to replace the "mentally" influenced
depiction of physical qualities in Scholastic natural philosophy with
a theory that requires only the properties of extension to describe
the manifest order of the natural world. Consequently, Descartes was
an early exponent of what came to be known as the
"primary/secondary" property distinction, a concept that
was very much "in the air" among the critics of
Scholasticism.



Nevertheless, even if Descartes' mechanistic natural philosophy
shunned the metaphysics of substantial forms, his underlying
methodology or approach to science remained very close to the
Scholastic tradition. By the time of the composition of the
*Principles*, Descartes had formulated a method that, like the
Scholastics, strived to explain natural phenomena based on the
allegedly simple and irrefutable "facts" and/or
observations, drawn from rational reflection on concepts or from
everyday experience, about the most fundamental aspects of reality.
These supposedly basic facts thereby provide the requisite
metaphysical foundation for his physical hypotheses: in other words,
one proceeds from our "clear and distinct" knowledge of
general metaphysical items, such as the nature of material substance
and its modes, to derive particular conclusions on specific types of
physical processes, for instance the laws of nature. This method of
conducting science is quite contrary to the modern approach, needless
to say, since modern scientists do not first engage in a metaphysical
search for first principles on which to base their work. Yet, this is
exactly the criticism that Descartes leveled at Galileo's
physics (in a letter to Mersenne from 1638): "without having
considered the first causes of nature, [Galileo] has merely looked for
the explanations of a few particular effects, and he has thereby built
without foundations" (AT II 380; see, also, the Preface to the
French translation of the *Principles*, AT IXB 5-11). The
structure of the *Principles*, Descartes' most
comprehensive scientific work, reflects these priorities: Part I
recapitulates the arguments (well-known from the *Meditations*)
for the existence of God, mental substance, and other metaphysical
topics; whereas the remaining Parts proceed to explain the nature of
material substance, physics, cosmology, geology, and other branches of
science, supposedly based on these fundamental metaphysical truths.
This preoccupation with metaphysical foundations, and the causal
explanations of natural phenomena derived from them, might also
account for the absence in the *Principles* of Descartes'
more mathematical work in physics, such as his discovery of the law of
light refraction. As he argued in the *Rules for the Direction of
the Mind* (1628), pure mathematicians are only concerned with
finding ratios and proportions, whereas natural philosophers are
intent on understanding nature (AT X 393-395). The development
of modern physics, which is inextricably intertwined with modern
mathematics, thus stands in sharp contrast to the latent Scholasticism
evident in Descartes' metaphysical approach to physics.


## 3. Space, Body, and Motion



Descartes' many hypotheses concerning space and body are best
appreciated when viewed as a continuation of a long debate within
Medieval/Renaissance philosophy centered upon the Aristotelian dictum
that whatever possessed dimensionality was body (see, Grant 1981).
While some philosophers, such as Telesio, Campanella, and Bruno, held
space to be always filled with matter (i.e., a plenum) yet somehow
independent of matter, others, like Patrizi and Gassendi, endorsed a
more absolutist notion that allowed spaces totally devoid of matter
(i.e., vacuum). Rejecting these anti-Aristotelian ideas of empty
space, Descartes equated the defining property, or
"essence", of material substance with three-dimensional
spatial extension: "the extension in length, breadth, and depth
which constitutes the space occupied by a body, is exactly the same as
that which constitutes the body" (Pr II 10). Consequently, there
cannot exist a space separate from body (Pr II 16), since all spatial
extension simply is body (and he rejects the possibility of a vacuum
that is not extended). If, for example, God removed the matter within
a vessel (such that nothing remained), then the sides of the vessel
would immediately become contiguous (but not through motion; Pr II
18). It is important to note that Descartes' claim, that the
vessel's sides must become contiguos, was a fairly common view
among the Scholastics, since they also accepted the idea that spatial
extension required a substance to ground extension (see, Grant 1981,
122). Descartes argues that, with respect to the empty vessel, that
"nothingness cannot possess any extension", since
"all distance is a mode of extension, and therefore cannot exist
without an extended substance" (Pr II 18). While Descartes
accepts a substance-property metaphysics, the rejection of the empty
vessel scenario seems more motivated by his nominalism, which is the
view that only particulars exist, whereas universals are only names
and abstractions from particluar things. As he states in an earlier
section of the *Principles* as regards numbers, "when
number is not being considered in any created things, but only in the
abstract or in general, it is merely a mode of thought" (Pr I
58).



Descartes' actual concept of "space" can be regarded
as a sort of conceptual abstraction from this bodily spatial
extension, which he also dubs "internal place":



> 
> We attribute a generic unity to the extension of the space [of a
> body], so that when the body which fills the space has been changed,
> the extension of the space itself is not considered to have been
> changed or transported but to remain one and the same; as long as it
> remains of the same size and shape and maintains the same situation
> among certain external bodies by means of which we specify that space.
> (Pr II 10F)
> 



Relative to an arbitrarily chosen set of bodies, it is thus possible
to refer to the abstract (generic) spatial extension of a portion of
the plenum that different extended bodies successively
"occupy"; and, presumably, by this process of abstraction
the internal place of the entire plenum can be likewise constructed.
Descartes takes a similar view of time, which is judged to be a
generalized abstraction from the "durations" of particular
bodies (where duration is an attribute of substance; Pr I
55-57; see Gorham 2007 for more on time in Descartes). Also like
the Scholastics, Descartes rejects any form of atomism, which is the
view that there exists a smallest indivisible particle of matter.
Rather, he holds that since any given spatially extended length is
divisible in thought, thus God has the power to actually divide it (Pr
II 20). The material entities that interact in Descartes'
physics come in distinct units or corpuscles (see
 Section 7),
 which explains the "corpuscularian" title often
attributed to his mechanical system, but these corpuscles are not
indivisible.



Descartes' *Principles of Philosophy* also presents his
most extensive discussion of the phenomena of motion, which is defined
as "the transfer of one piece of matter or of one body, from the
neighborhood of those bodies immediately contiguous to it and
considered at rest, into the neighborhood of others" (Pr II 25).
Descartes attempts to distinguish his "proper" conception
of motion, as a change of the "neighborhood" of contiguous
bodies, from the common or "vulgar" conception of motion,
which is change of internal place (Pr II 10-15, 24-28).
The surface of these containing bodies (that border the contained
body) is also called the "external place" of the contained
body. Descartes notes that the vulgar concept of motion allows a body
to simultaneously take part in many (possibly contradictory) motions,
as when a sitting passenger on a ship views himself as at rest
relative to the parts of the ship, but not at rest relative to the
shore (Pr II 24). Yet, when motion is viewed as a translation of the
contiguous neighborhood, a body can only partake in one motion, which
dispels the apparent contradiction (since the body must either be at
rest, or in translation away from, its contiguous neighborhood).



Nevertheless, Descartes' hypothesis of motion may sanction a
species of relative motion, since his phrase, "considered at
rest", implies that the choice of which bodies are at rest or in
motion is purely arbitrary. According to the "relational"
theory (or at least the more strict versions of relationism), space,
time, and motion are just relations among bodies, and not separately
existing entities or properties that are in any way independent of
material bodies. Motion only exists as a "relative
difference" among bodies: that is, the bodies do not possess
individual, determinate properties of speed, velocity, acceleration
(e.g., body *C* has the speed property of "5 miles per
hour"); rather, all that really exists is a difference in their
relative speed, velocity, and acceleration (e.g., there is a speed
difference among bodies *C* and *B* of "5 miles
per hour"). Several passages in Descartes' analysis of
motion seem to support this strong variety of relationism: "we
cannot conceive of the body *AB* being transported from the
vicinity of the body *CD* without also understanding that the
body *CD* is transported from the vicinity of the body
*AB*" (Pr II 29). Hence, "all the real and positive
properties which are in moving bodies, and by virtue of which we say
they move, are also found in those [bodies] contiguous to them, even
though we consider the second group to be at rest" (Pr II 30).
This form of relational motion has been dubbed the "reciprocity
of transfer" in the recent literature. Yet, as will be discussed
in a later section, Descartes also holds that rest and motion are
different bodily states, a view that is incompatible with a strict
relationism as regards motion. Therefore, Cartesian reciprocity of
transfer only satisfies relationism (along with its ban on individual
bodily states of motion) for moving bodies (i.e., when there is a
translation manifest between a body and its contiguous neighborhood).
Many of the difficulties associated with Cartesian physics can be
traced to the enormous ontological burden that Descartes places on his
hypothesis of motion. In a later section we will examine the problem
of integrating his account of motion with the Cartesian laws of
nature, but a brief discussion of the apparent circularity of
Descartes' definitions of motion and body is required at this
point. After describing motion as the transference of a body from the
surrounding neighborhood of bodies, Descartes states that by
"one body, or one part of matter, I here understand everything
which is simultaneously transported" (Pr II 25). The problem, of
course, is that Descartes has defined motion as a change of contiguous
bodies, and then proceeds to define body as that which moves
(translates, transports). Although this circularity threatens the
entire edifice of Cartesian physics, it is possible that Descartes
intended both motion and body to possess an equal ontological
importance in his theory, such that neither is the more fundamental
notion (which serves as the basis for constructing or defining the
other notion). Yet, their intrinsic interrelationship entails that any
attempted definition of one must inevitably incorporate the other. The
problem with this reconstruction of Descartes' reasoning,
however, is that Descartes explicitly deems motion to be a
"mode" of extension; where a mode is a lesser ontological
category that, roughly, can be understood as a way that extension
manifests itself, or as a "property" of extension (Pr I
53; shape is also mentioned as a mode of extension). Finally, another
difficulty implicit in Descartes' theory is the fact that a
resting body, according to the definition of body and place, would
seem to "blend" into the surrounding plenum: that is, if a
body is "everything which is simultaneously transported",
then it is not possible to discern a resting body from the surrounding
plenum matter that forms the external place of that resting body. In
addition, Descartes rejects any explanation of the solidity of a body
that employs a bond among its particles (since the bond itself would
be either a substance or property, and thus the solidity of the bond
would presumably need to be explained; Pr II 55). A macroscopic
material body is, essentially, held together just by the relative rest
of its constituent material parts. This raises the obvious difficulty
that the impact of such bodies should result in their dispersion or
destruction (for there is nothing to hold them together). These sorts
of complications prompted many later natural philosophers, who were
generally sympathetic to Descartes' mechanical philosophy, to
search for an internal property of matter that could serve as a type
of individuating and constitutive principle for bodies; e.g.,
Leibniz' utilization of "force".



Related to the alleged circularity of the definitions of motion and
body, as well as the problem of resting bodies, is the difficulty in
reconciling Descartes' definition of "substance"
with his claim that individual bodies are substances. If, as Descartes
believes, substances are not dependent on other things in order to
exist (Pr I 51), then any part of extension (which is a body, via Pr
II 10, as explained above) would not qualify as a substance since it
depend on its contiguous neighbors to delimit and define its boundary.
Yet, Descartes often declares that individual bodies are substances;
e.g., "the two halves of a portion of matter, no matter how
small they may be, are two complete substances" (AT III 447).
One of the most popular replies to this difficulty, from Spinoza
(*Ethics*, Part I, Prop. 15) to many contemporary commentators
(e.g., Keeling 1968, Lennon 1993, Sowaal 2004, Schmaltz 2020, and
numerous others), is to declare that only the whole plenum is a
substance, and not any of its constituent parts. The problem with this
attempted solution, however, is that it lacks textual support, as is
evident in the Pr I 51 quotation above. Likewise, some of these
reconstructions, such as Lennon's, would seem to violate central
aspects of Cartesian physics and metaphysics, for he interprets motion
as a phenomenal contribution of the mind, such that the plenum and its
parts do not move or change at all. Along these same lines, some
scholars (e.g., Schaffer 2009) have concluded that Descartes was a
supersubstantivalist, i.e., the view that space (spacetime, in the
modern setting) is the only predicable or fundamental substance. While
Descartes' identification of corporeal substance and space (see
above, Pr II 10) might seem to support this reading,
supersubstantivalism takes space as primary, and matter as secondary
or derived from space (see Sklar 1974, 222). Descartes, on the
contrary, takes matter or body as primary and space as a derived,
abstract concept: "the same extension which constitutes the
nature of body also constitutes the nature of space, and . . . these
two things differ only in the way that the nature of the genus or
species differs from that of the individual" (Pr II 11). Whereas
space is a genus or species concept for Descartes (which is a
universal; Pr I 59), space is the individual for the
supersubstantivalist, and thus ascribing supersubstantivalism to
Descartes violates his nominalism (Pr II 8). Indeed, the reason that
Descartes seeks to equate bodily and spatial extension in this part of
the *Principles* is that he strives to reject any view that
treats space as a separate, usually incorporeal, entity that is
independent of matter (e.g., the popular imaginary space tradition,
which was a forerunner to the absolutist or substantivalist
conception): "That corporeal substance, when distinguished from
its quantity or extension, is confusedly conceived as if it were
incorporeal" (Pr II 9).


## 4. The Laws of Motion and the Cartesian Conservation Principle



Foremost among the achievements of Descartes' physics are the
three laws of nature (which, essentially, are laws of bodily motion).
Newton's own laws of motion would be modeled on this Cartesian
breakthrough, as is readily apparent in Descartes' first two
laws of nature: the first states "that each thing, as far as is
in its power, always remains in the same state; and that consequently,
when it is once moved, it always continues to move" (Pr II 37),
while the second holds that "all movement is, of itself, along
straight lines" (Pr II 39; these two would later be incorporated
into Newton's first law of motion). By declaring that motion and
rest are primitive *states* of material bodies without need of
further explanation, and that bodies only change their state when
acted upon by an external cause, it is not an exaggeration to claim
that Descartes helped to lay the foundation for the modern theory of
dynamics (which studies the motion of bodies under the action of
forces). For the Aristotelian-influenced Scholastics who had
endeavored to ascertain the causal principles responsible for the
"violent" motions of terrestrial bodies (as opposed to
their "natural" motions to specific regions of the
plenum), the explanation for these forced, unnatural motions seemed to
lie in some type of internal bodily property, or external agent, that
was temporarily possessed by, or applied to, a body--an
explanation that accounts for the fact that the body's motion
both originates and concludes in a state of rest (since, while on the
earth's surface, the terrestrial element has no natural
motions). According to the medieval "impetus" theory, for
example, these violent motions occur when a quality is directly
transferred to a body from a moving or constrained source, say, from a
stretched bow to the waiting arrow. This property causes the observed
bodily motion until such time that it is completely exhausted, thus
bringing about a cessation of the violent movement (and the
arrow's fall back to earth). Implicit in the Scholastic view is
the basic belief that a terrestrial body continuously resists change
from a state of rest while situated upon the earth, since the
depletion of the impetus property eventually effects a corresponding
return of the body's original motionless, earthbound condition.
Descartes, on the other hand, interpreted the phenomena of motion in
an entirely new light, for he accepts the existence of inertial motion
(uniform or non-accelerating motion) as a natural bodily state
alongside, and on equal footing with, the notion of bodily rest. He
argues, "because experience seems to have proved it to us on
many occasions, we are still inclined to believe that all movements
cease by virtue of their own nature, or that bodies have a tendency
towards rest. Yet this is assuredly in complete contradiction to the
laws of nature; for rest is the opposite of movement, and nothing
moves by virtue of its own nature towards its opposite or own
destruction" (Pr II 37). While one can find several natural
philosophers whose earlier or contemporary work strongly foreshadowed
Descartes' achievement in the first and second
laws--namely, Galileo and Isaac Beeckman (see Arthur
2007)--the precise formulation put forward in the *Principles
of Philosophy* is quite unique (especially as regards the second
law, since both Galileo and Beeckman appear to sanction a form of
circular inertial motion, which possibly betrays the influence of the
Scholastic's circular motion of the celestial element). A
fascinating blending of Scholasticism and the new physics is also
evident in the above quotation, since Descartes invokes the logic of
contrary properties in his statement that "nothing moves by
virtue of its own nature towards its opposite or own
destruction". That is, rest and motion are opposite or contrary
states, and since opposite states cannot (via the Scholastic
principle) transform into one another, it follows that a body at rest
will remains at rest and a body in motion will remains in motion.
Consequently, Descartes has employed a Scholastic/Medieval argument to
ground what is possibly the most important concept in the formation of
modern physics, namely inertia. Yet, it is important to note that
Descartes' first and second laws do not correspond to the modern
concept of inertia, since he incorrectly regards (uniform,
non-accelerating) motion and rest as different bodily states, whereas
modern theory dictates that they are the same state.



While Descartes' first and second laws deal with the rest and
motion of individual bodies, the third law of motion is expressly
designed to reveal the properties exhibited among several bodies
during their collisions and interactions. In short, the third law
addresses the behavior of bodies under the normal conditions in his
matter-filled world; when they collide: "The third law: that a
body, upon coming in contact with a stronger one, loses none of its
motion; but that, upon coming in contact with a weaker one, it loses
as much as it transfers to that weaker body" (Pr II 40). In the
following sections of the *Principles*, Descartes makes
explicit the conserved quantity mentioned in this third law:



> 
> We must however notice carefully at this time in what the force of
> each body to act against another or resist the action of that other
> consists: namely, in the single fact that each thing strives, as far
> as in its power, to remain in the same state, in accordance with the
> first law stated above....This force must be measured not only by
> the size of the body in which it is, and by the [area of the] surface
> which separates this body from those around it; but also by the speed
> and nature of its movement, and by the different ways in which bodies
> come in contact with one another. (Pr II 43)
> 



As a consequence of his first law of motion, Descartes insists that
the quantity conserved in collisions equals the combined sum of the
products of size and speed of each impacting body. Although a
difficult concept, the "size" of a body roughly
corresponds to its volume, with surface area playing an indirect role
as well. This conserved quantity, which Descartes refers to
indiscriminately as "motion" or "quantity of
motion", is historically significant in that it marks one of the
first attempts to locate an invariant or unchanging feature of bodily
interactions. To give an example, if a body *B* of size 3 and
speed 5 collides with a body *C* of size 2 and speed 4, then
the total quantity of motion of the system is 23, a quantity which
remains preserved after the collision even though the bodies may
possess different speeds.



Moreover, Descartes envisions the conservation of quantity of motion
as one of the fundamental governing principles of the entire cosmos.
When God created the universe, he reasons, a certain finite amount of
motion (quantity of motion) was transmitted to its material occupants;
a quantity, moreover, that God continuously preserves at each
succeeding moment. (For more on the difficult issue of God's
continuous recreation or preservation of the material world, see,
e.g., Gorham 2004, Hattab 2007, and Schmaltz 2008).



> 
> It is obvious that when God first created the world, He not only moved
> its parts in various ways, but also simultaneously caused some of the
> parts to push others and to transfer their motion to these others. So
> in now maintaining the world by the same action and with the same laws
> with which He created it, He conserves motion; not always contained in
> the same parts of matter, but transferred from some parts to others
> depending on the ways in which they come in contact. (Pr II 62)
> 



In the *Principles*, Descartes conservation law only recognizes
a body's degree of motion, which correlates to the scalar
quantity "speed", rather than the vectorial notion
"velocity" (which is speed in a given direction). This
distinction, between speed and velocity, surfaces in Descartes'
seven rules of impact, which spell out in precise detail the outcomes
of bodily collisions (although these rules only describe the
collisions between two bodies traveling along the same straight line).
Descartes' utilization of the concept of speed is manifest
throughout the rules. For example:



> 
> Fourth, if the body *C* were entirely at rest,...and if
> *C* were slightly larger than *B*; the latter could
> never have the force to move *C*, no matter how great the speed
> at which *B* might approach *C*. Rather, *B*
> would be driven back by *C* in the opposite direction:
> because...a body which is at rest puts up more resistance to high
> speed than to low speed; and this increases in proportion to the
> differences in the speeds. Consequently, there would always be more
> force in *C* to resist than in *B* to drive, ....
> (Pr II 49F)
> 



Astonishingly, Descartes claims that a smaller body, regardless of its
speed, can never move a larger stationary body. While obviously
contradicting common experience, the fourth collision rule does nicely
demonstrate the scalar nature of speed, as well as the primary
importance of quantity of motion, in Cartesian dynamics. In this rule,
Descartes faces the problem of preserving the total quantity of motion
in situations distinguished by the larger body's complete rest,
and thus zero value of quantity of motion. Descartes conserves the
joint quantity of motion by equipping the stationary object *C*
with a resisting force sufficient to deflect the moving body
*B*, a solution that does uphold the quantity of motion in
cases where *C* is at rest. That is, since *B* merely
changes its direction of inertial motion, and not its size or degree
of speed (and *C* equals zero throughout the interaction), the
total quantity of motion of the system is preserved. For Descartes,
reversing the direction of *B*'s motion does not alter
the total quantity of motion, a conclusion that is in sharp contrast
to the later hypothesis, usually associated with Newton and Leibniz,
that regards a change in direction as a negation of the initial speed
(i.e., velocity). Thus, by failing to foresee the importance of
conjoining direction and speed, Descartes' law falls just short
of the modern law for the conservation of momentum.



In this context, the complex notion of "determination"
should be discussed, since it approximately corresponds to the
composite direction of a body's quantity of motion. In some
passages, Descartes apparently refers to the direction of a
body's motion as its determination: "there is a difference
between motion considered in itself, and its determination in some
direction; this difference makes it possible for the determination to
be changed while the quantity of motion remains intact" (Pr II
41). Yet, a single motion does not have just one determination, as is
clear in his critique of Hobbes' interpretation of
determinations: "What he [Hobbes] goes on to say, namely that a
'motion has only one determination,' is just like my
saying that an extended thing has only a single shape. Yet this does
not prevent the shape being divided into several components, just as
can be done with the determination of motion" (April 21, 1641;
AT III 356). In the same way that a particular shape can be
partitioned into diverse component figures, so a particular
determination can be decomposed into various constituent directions.
In his *Optics*, published in 1637, Descartes' derivation
of his law of refraction seemingly endorses this interpretation of
determinations. If a ball is propelled downwards from left to right at
a 45 degree angle, and then pierces a thin linen sheet, it will
continue to move to the right after piercing the sheet but now at an
angle nearly parallel with the horizon. Descartes reasons that this
modification of direction (from the 45 degree angle to a smaller
angle) is the net result of a reduction in the ball's downward
determination through collision with the sheet, "while the one
[determination] which was making the ball tend to the right must
always remain the same as it was, because the sheet offers no
opposition at all to the determination in this direction" (see
Figure 1).



![Refract](refract.jpg)

Figure 1. Diagram from the
*Optics.*




Descartes' determination hypothesis also incorporates a certain
quantitative element, as revealed in a further controversial
hypothesis that is often described as the "principle of least
modal action". In a letter to Clerselier (February 17th, 1645),
Descartes explains:



> 
> *When two bodies collide, and they contain incompatible modes*,
> [either different states of speed, or different determinations of
> motion] *then there must occur some change in these modes in order
> to make them compatible; but this change is always the least that may
> occur.* In other words, *if these modes can become compatible
> when a certain quantity of them is changed, then no larger quantity
> will change* (AT IV 185).
> 



This principle can be illustrated with respect to our previous example
involving the fourth collision rule. If both *B* and *C*
were to depart at the same speed and in the same direction after
impact, it would be necessary for the smaller body *B* to
transfer at least half of its quantity of motion to the larger
stationary body *C*. Yet, Descartes reasons that it is easier
for *B* in this situation to merely reverse it direction than
to transfer its motion:



> 
> When *C* is the larger [body], *B* cannot push it in
> front of itself unless it transfers to *C* more than half of
> its speed, together with more than half of its determination to travel
> from left to right in so far as this determination is linked with its
> speed. Instead it rebounds without moving body *C*, and changes
> only its whole determination, which is a smaller change than the one
> that would come about from more than half of this determination
> together with more than half of its speed (AT IV 186).
> 



Consequently, reversing *B*'s direction of motion, a
change of one mode (determination), constitutes a lesser modal change
than a transference of motion between two bodies, which alters two
modes (speed and determination). In this passage, it is important to
note that if *B* were to transfer motion to *C*, it
would change both half of *B*'s speed and half of its
determination, even though the direction of *B*'s
quantity of motion is preserved. As a result, a body's
determination is apparently linked to its magnitude of speed.


## 5. The Problem of Relational Motion



As discussed in previous sections, there are various ways in which
Descartes' laws of motion violate a strict relationism. One of
the most problematic instances involves the relational compatibility
of the fourth and fifth collision rules. Whereas the fourth rule
concludes that a large object remains at rest during impact with a
smaller moving body, such that the smaller body is deflected back
along its initial path, the fifth rule concludes that a large body
will move a smaller stationary object, "transferring to [the
smaller body] as much of its motion as would permit the two to travel
subsequently at the same speed" (Pr II 50). From a relational
standpoint, however, rules four and five constitute the same type of
collision, since they both involve the interaction of a small and
large body with the same relative motion (or speed difference) between
them. One might be tempted to appeal to the basic Cartesian tenet that
motion and rest are different intrinsic states of bodies, or the
reciprocity of transfer thesis, to circumvent this difficulty (see
section 3): i.e., there is an ontological difference between a body
that is, or is not, undergoing a translation with respect to its
contiguous neighborhood, and this is sufficient to distinguish the
case of rule four from rule five (since the large body is really at
rest in four, and really in motion in five).



The problem with this line of reasoning, however, is that it only
works if one presupposes that the two bodies are approaching one
another, and this is not a feature of the system that can be captured
by sole reference to the contiguous neighborhood of each individual
body. Even if there is reciprocity of transfer between a body and its
neighborhood, it is still not possible to determine which collision
rule the impact will fall under, or if the bodies will even collide at
all, unless some reference frame is referred to that can compute the
motion of both bodies relative to one another. Suppose, for instance,
that a certain spatial distance separates two bodies, and that one of
the bodies is, and the other is not, undergoing a translation relative
to its neighboring bodies. Given this scenario, it is not possible to
determine if; (i) the translating body is approaching the
non-translating body, or (ii) the spatial interval between them
remains fixed and the translating body simply undergoes a change of
neighborhood (i.e., the neighborhood moves relative to a stationary
body). In short, Descartes' reciprocity of transfer thesis
underdetermines the outcome of his bodily collisions, as well as the
capacity to apply, and make predictions from, the Cartesian collision
rules. The context of the collision rules also supports the view that
the motions of the impacting bodies are determined from an external
reference frame, rather than from the local translation of their
contiguous neighborhoods. In elucidating the fourth rule, for
instance, Descartes states that *B* could never move *C*
"no matter how great the speed at which *B* might
approach C" (Pr II 49)--and only an external perspective,
not linked to the bodily reciprocity of transfer, could determine that
*B* "approaches" *C*. Such admissions make
it very difficult to reconcile Descartes' physics with a strict
relational theory of space and motion, although it may be compatible
with weaker forms of relationism that can countenance various external
reference frames, structures, or other methods for determining the
individual states of bodily motion. These weaker relationist
strategies (or even non-relational, absolutist interpretations) of
Descartes' physics come at a high price, however, since the
reciprocity of transfer thesis must be abandoned. For these reasons,
it is more likely that Descartes' reciprocity of transfer thesis
is intended to counter any interpretation that regards motion as
caused by a bodily property, as some Scholastics had held (such as
Buridan), rather then defend relational motion (see Maier 1982 on
these earlier views of motion). That is, if there is nothing in the
moving body that differs from its neighborhood of contiguous bodies
(see Pr II 30), then a body's motion is not due to it possessing
a special property that its neighborhood lacks.


## 6. "Force" in Cartesian Physics



Despite the mechanistic, non-teleological character of
Descartes' analysis of motion and bodily interactions, there are
many seemingly metaphysical and qualitative traits in Cartesian
physics that do not sit comfortably with his brand of reductionism
(i.e., that material bodies are simply extension and its modes). In
fact, returning to the Cartesian laws of nature (section 4), it is
evident that Descartes has allotted a fundamental role to the action
of bodily "forces" or "tendencies": for
example, the tendency of bodies to follow straight lines, the
resistance to motion of a large resting body (to a smaller moving
body), etc. In *The World*, he states: "the virtue or
power in a body to move itself can well pass wholly or partially to
another body and thus no longer be in the first; but it cannot no
longer exist in the world" (AT XI 15). As an early remark
concerning his conservation principle, this explanation seems to
envision force much like a property or "power" possessed
by individual material objects, similar to the qualitative,
metaphysical properties of the Scholastics (as in the
"impetus" theory). For these reasons, the nature of bodily
forces or tendencies is a philosophical question of much interest in
the study of Descartes' physics.



In order to better grasp the specific role of Cartesian force, it
would be useful to closely examine his theory of centrifugal effects,
which is closely associated with the second law of nature. Besides
straight-line motion, Descartes' second law also mentions the
"center-fleeing" (centrifugal) tendencies of circularly
moving material bodies: "all movement is, of itself, along
straight lines; and consequently, bodies which are moving in a circle
always tends to move away from the center of the circle which they are
describing" (Pr II 39). At first glance, the second law might
seem to correspond to the modern scientific dissection of centrifugal
force: specifically, the centrifugal effects experienced by a body
moving in a circular path, such as a stone in a sling, are a normal
consequence of the body's tendency to depart the circle along a
straight tangential path. Yet, as stated in his second law, Descartes
contends (wrongly) that the body tends to follow a straight line away
from the center of its circular trajectory. That is, the force exerted
by the rotating stone, as manifest in the outward "pull"
on the impeding sling, is a result of a striving towards straight line
inertial motion directed radially outward from the center of the
circle, rather than a striving towards straight line motion aimed
along the circle's tangent. Descartes does acknowledge, however,
the significance of tangential motion in explicating such
"center-fleeing" tendencies, but he relegates this
phenomenon to the subordinate status of a composite effect. By his
reckoning, the tendency to follow a tangential path exhibited by a
circling body, such as the flight of the stone upon release from the
sling, can be constructed from two more basic or primary inclinations:
first, the tendency of the object to continue along its circular path;
and second, the tendency of the object to travel along the radial line
away from the center. Thus, Descartes is willing to admit that
"there can be strivings toward diverse movements in the same
body at the same time" (Pr III 57), a judgment that seems to
presuppose the acceptance of some type of "compositional"
theory of tendencies analogous to his dissection of determinations.
Yet, since he believes that "the sling, ..., does not
impede the striving [of the body along the circular path]" (Pr
III 57), he eventually places sole responsibility for the production
of the centrifugal force effects on the radially directed component of
"striving". He states, "If instead of considering
all the forces of [a body's] motion, we pay attention, to only
one part of it, the effect of which is hindered by the
sling;...;we shall say that the stone, when at point *A*,
strives only [to move] toward *D*, or that it only attempts to
recede from the center *E* along the straight line
*EAD*" (Pr III 57; see Figure 2).



![Stone](stone1.jpg)

Figure 2. Diagram that accompanies Pr
III 57.




Descartes' use of the terms "tendency" and
"striving" in his rotating sling example should not be
equated with his previous concept of a determination of motion. A
determination is confined to a body's actual motion, whereas a
body's tendency towards motion only occurs at a single instant.
He states: "Of course, no movement is accomplished in an
instant; yet it is obvious that every moving body, at any given moment
in the course of its movement, is inclined to continue that movement
in some direction in a straight line,..." (Pr II 39). In
another passage in the *Principles*, Descartes identifies these
strivings as a "first preparation for motion" (Pr III 63).
Hence, while determinations necessitate a span of several instants,
tendencies towards motion are manifest only at single instants. This
is a crucial distinction, for it partitions Cartesian dynamics into
two ontological camps: forces that exist at moments of time, and
motions that can only subsist over the course of several temporal
moments. In many parts of the *Principles*, moreover, Descartes
suggests that quantity of motion is the measure of these bodily
tendencies, and thus quantity of motion has a dual role as the measure
of non-instantaneous bodily motion as well as the instantaneous bodily
tendencies (see Pr III 121).



Given his rejection of the Scholastic qualitative tradition in
physics, Descartes' depiction of centrifugal effects as due to a
"tendency" or "striving" of moving bodies thus
raises a host of intriguing ontological questions (and may even reveal
a vestigial influence of his earlier Scholastic training). That is,
even as his penchant for a geometrical world view increased, as
manifest in the identification of extension as matter's primary
quality, Descartes continued to treat inertial motion and its
accompanying force effects as if they were essential characteristics
of bodies. Descartes' own remarks on the ontological status of
inertial force, furthermore, disclose a certain degree of ambiguity
and indecision. In a 1638 letter, (six years before the
*Principles*), he concludes:



> 
> I do not recognize any inertia or natural sluggishness in
> bodies...; and I think that by simply walking, a man makes the
> entire mass of the earth move ever so slightly, since he is putting
> his weight now on one spot, now on another. All the same, I agree
> ...that when the largest bodies (such as the largest ships) are
> pushed by a given force (such as a wind), they always move more slowly
> than others. (AT II 467)
> 



In this passage, Descartes seems to deny the existence of inertial
force if conceived as a form of Scholastic quality that material
bodies can possess; rather, bodies are "indifferent to
motion", so even the slightest weight should move the entire
earth. On the other hand, he is willing to acknowledge the commonly
observed fact that larger objects are much harder to set in motion
than smaller objects. Consequently, although Descartes finds the
existence of "forces of resistance" (or "natural
sluggishness") problematic, as is the case with such similar
properties as weight, he does not entirely relegate inertia to the
phenomenological status of the so-called secondary properties of
matter (such as color, taste, etc., which only exist in the mind). The
main reason for this inclusion of inertial force effects into
scientific discourse can probably be traced to Descartes'
classification of motion as an intrinsic characteristic or
"mode" of extension (see
 Section 3).
 As the concluding sections of the *Principles* state: "I
have now demonstrated [there] are nothing in the [material] objects
other than...certain dispositions of size, figure and
motion..." (Pr IV 200). Since inertial forces are a
consequence or a by-product of motion, as the product of the size
times speed of bodies, Descartes apparently did not object to
incorporating these phenomena within the discussion of the modes of
material substance.



Yet, even if Descartes described force as an intrinsic fact of
material interactions, the exact nature of the relationship between
force and matter remains rather unclear. In particular, is force a
property actually contained or present within bodies? Or, is it some
sort of derivative phenomenal effect of the action of speed and size,
and thus not present within extension? On the former interpretation
(as favored by Alan Gabbey 1980, and Martial Gueroult 1980), forces
exist in bodies in at least one important sense as "real"
properties or modes whose presence occasions the Cartesian laws of
nature. While many of Descartes' explanations might seem to
favor this interpretation (e.g., "[a body] at rest has force to
remain at rest", Pr II 43), Daniel Garber charges that such
views run counter to Descartes' demand that extension alone
comprise the essence of matter. Garber suggests that we view Cartesian
force as a sort of shorthand description of the dynamical regularities
maintained in the world by God, and not as some form of quality
internal to bodies: "The forces that enter into the discussion
[of the Cartesian collision laws] can be regarded simply as ways of
talking about how God acts, resulting in the law-like behavior of
bodies; force for proceeding and force of resisting are ways of
talking about how, ..., God balances the persistence of the state
of one body with that of another" (Garber 1992a, 298; see also
Hatfield 1979, Des Chene 1996, and Manchak 2009, for more approaches).
In various passages associated with the conservation principle,
Garber's interpretation apparently gains credibility. For
instance: "So in now maintaining the world by the same action
and with the same laws with which He [God] created it, He conserves
motion; not always contained in the same parts of matter, but
transferred from some parts to others depending on the ways in which
they come in contact" (Pr II 42). In retrospect, however, it
must be acknowledged that Descartes' classification of material
substance with extension, as exemplified in his demand that there
exists nothing in bodies except "certain dispositions of size,
figure and motion", is so open-ended and equivocal as to easily
accommodate both of the interpretations surveyed above. All that can
be safely concluded is that Descartes envisioned the forces linked
with bodily inertial states as basic, possibly primitive, facts of the
existence of material bodies--a broad judgment that, by refusing
to take sides, opts out of this difficult ontological dispute.


## 7. Cartesian Cosmology and Astrophysics



Descartes' vortex theory of planetary motion proved initially to
be one of the most influential aspects of Cartesian physics, at least
until roughly the mid-eighteenth century. A vortex, for Descartes, is
a large circling band of material particles. In essence,
Descartes' vortex theory attempts to explain celestial
phenomena, especially the orbits of the planets or the motions of
comets, by situating them (usually at rest) in these large circling
bands. The entire Cartesian plenum, consequently, is comprised of a
network or series of separate, interlocking vortices. In our solar
system, for example, the matter within the vortex has formed itself
into a set of stratified bands, each lodging a planet, that circle the
sun at varying speeds. The minute material particles that form the
vortex bands consist of either the atom-sized, globules (secondary
matter) or the "indefinitely" small debris (primary
matter) left over from the impact and fracture of the larger elements;
tertiary matter, in contrast, comprises the large, macroscopic
material element (Pr III 48-54). This three-part division of
matter, along with the three laws of nature, are responsible for all
cosmological phenomena in Descartes' system, including gravity.
As described in Pr III 140, a planet or comet comes to rest in a
vortex band when its radially-directed, outward tendency to flee the
center of rotation (i.e., centrifugal force; see
 Section 6)
 is balanced by an equal tendency in the minute elements that comprise
the vortex ring. If the planet has either a greater or lesser
centrifugal tendency than the small elements in a particular vortex,
then it will, respectively, either ascend to the next highest vortex
(and possibly reach equilibrium with the particles in that band) or be
pushed down to the next lowest vortex--and this latter scenario
ultimately supplies Descartes' explanation of the phenomenon of
gravity, or "heaviness". More specifically, Descartes
holds that the minute particles that surround the earth account for
terrestrial gravity in this same manner (Pr IV 21-27). As for
the creation of the vortex system, Descartes reasons that the
conserved quantity of motion imparted to the plenum eventually
resulted in the present vortex configuration (Pr III 46). God first
partitioned the plenum into equal-sized portions, and then placed
these bodies into various circular motions that, ultimately, formed
the three elements of matter and the vortex systems (see Figure
3).



![Plate](plate.jpg)

Figure 3. Plenum vortices in the
*Principles*, Pr III 53.




Besides the ontological economy of only requiring inertial motion and
its attendant force effects, Descartes' choice of circularly
moving bands of particles may have also been motivated by worries
over, for lack of a better term, "plenum crowding". In the
*Principles*, he argues: "It has been shown...that
all places are full of bodies.... From this it follows that no
body can move except in a complete circle of matter or ring of bodies
which all move at the same time" (Pr II 33). Circular motion is
therefore necessary for Descartes because there are no empty spaces
for a moving object to occupy. Although the world is described as
"indefinitely" large (Pr I 26-27, with only God
receiving the more positive description, infinite), the non-circular
motion of a single body could violate the Cartesian conservation
principle by resulting in an indeterminate material displacement. As
an aside, it is enormously difficult to reconcile Descartes'
collision rules with his claim that all bodily motion occurs in
circular paths; moreover, since the bodies that comprise the circular
path all move simultaneously, it seems to follow from the definition
of "body" (see
 Section 3)
 that there is only one moving body (and not many).



Returning to the vortex theory, Descartes allots a considerable
portion of the *Principles* to explicating various celestial
phenomena, all the while adopting and adapting numerous sub-hypotheses
that apply his overall mechanical system to specific celestial events.
One of the more famous of these explanations is the Cartesian theory
of vortex collapse, which also provides an hypothesis on the origins
of comets (Pr III 115-120). Briefly, Descartes reckons that a
significant amount of first element matter constantly flows between
adjacent vortices: as the matter travels out of the equator of one
vortex, it passes into the poles of its neighbor. Under normal
conditions, primary matter flows from the poles of a vortex into its
center, i.e., the sun, which is itself comprised of primary matter.
Due to centrifugal force, these particles press out against the
surrounding secondary globules as they begin their advance towards the
equator (Pr III 120-121); the pressure exerted by the primary
and secondary elements (on a person's optic nerve) also serving
as the cause of light (Pr III 55-64, IV 195). Since the adjacent
vortices also possess the same tendency to swell in size, a balance of
expansion forces prevents the encroachment of neighboring vortices. On
occasion, however, a buildup of larger elements on the sun's
surface, identified as sunspots, may conspire to prevent the incoming
flow of first element matter from the poles. If the sunspots
ultimately cover the entire surface of the sun, the vortex's
remaining primary matter will be expelled at the equator, and thus it
no longer has a source of outward pressure to prevent the encroachment
of neighboring vortices. Once the vortex is engulfed by its expanding
neighbors, the encrusted sun may become either a planet in a new
vortex, or end up as a comet passing through many vortices.



On the whole, the vortex theory offered the natural philosopher a
highly intuitive model of celestial phenomena that was compatible with
the mechanical philosophy. The theory was regarded as superior to
Newton's theory of universal gravitation since it did not posit
a mysterious, occult quality (gravity) as the cause of the planetary
orbits or the free-fall of terrestrial objects. The vortex theory
likewise provided a built-in explanation for the common direction of
all planetary orbits. Additionally, the vortex theory allowed
Descartes to endorse a form of Copernicanism (i.e., sun-centered
world) without running afoul of Church censorship. Since the alleged
motion of the earth was one of the Church's principal objections
to Galileo's science, Descartes hoped to avoid this objection by
placing the earth at rest within a vortex band that circled the sun,
such that the earth does not undergo a change of place relative to the
containing surface of the neighboring material particles in its vortex
band (Pr III 24-31; and section 3). Through this ingenious
maneuver, Descartes could then claim that the earth does not
move--via his definition of place and motion--and yet
maintain the Copernican hypothesis that the earth orbits the sun.
"The Earth, properly speaking, is not moved, nor are any of the
Planets; although they are carried along by the heaven" (Pr III
28). In the long run, however, Descartes' vortex theory failed
for two fundamental reasons: first, neither Descartes nor his
followers ever developed a systematic mathematical treatment of the
vortex theory that could match the accuracy and predictive scope of
the (continuously improving) Newtonian theory; and second, many
attempts by Cartesian natural philosophers to test Descartes'
various ideas on the dynamics of circularly moving particles (e.g., by
using large spinning barrels filled with small particles) did not meet
the predictions advanced in the *Principles* (see Aiton
1972).