Descartes' Constants

                            ____________________

 

                             Timothy Paul Smith

                         University of New Hampshire

                             tim.smith@unh.edu

 

                          (received June 20, 1996)

 

 

 

			*What is it that angelic music,

		papal decrees, and the resistance of a ball

		in motion to change, all have in common?*

 

	To the medieval thinker the world was divided into two major regions;

heaven and earth. Up there, starting at the orbit of the moon,  was the world

of constants.  The stars stayed where they should be.  The planets continued

on tirelessly in their orbits.  Even if the reason for the details of their

orbits was not clear,  they still continued to trek the night sky.

 

	Here below the orbit of the moon, the world seemed to not be bound by

those divine laws.  We not only can choose mistake,  but we can also slow

down, wear out and stop.  With this viewpoint it was nearly hopeless to try

and construct repeatable laws of motion,  or a science with accurate

predictions. Secular (or terrestrial) laws of motion were viewed as being

inherently imperfect and inconsistent.  Thus any dynamical system on the earth

would always resist a complete description.

 

	This is where the scientific revolution of the seventeenth century

changed things.  Sir Isaac Newton is credited with showing that the laws of

motion for `up there' are the same as the laws of motion for `down here'; the

trajectory of a baseball pitch is governed by the same laws as the orbit of

the moon,  except that whereas the baseball hits the earth, the moon misses

*Terra Firma*. The term "Newtonian Mechanics" and "Classical Mechanics" are

almost synonymous,   and have dominated the field ever since the publication

of *Principia* in 1686.  In that book Newton laid out his three laws of

motion:

 

	I) Every body continues in its state of rest, or of

	uniform motion in a right line,  unless it is compelled

	to change that state by forces impressed upon it.

 

	II) The change of motion is proportional to the motive

	force impressed; and is made in the direction of the

	right line in which that force is impressed.

 

	III) To every action there is always opposed an equal

	reaction: or, the mutual actions of two bodies upon

	each other are always equal,  and directed to contrary

	parts.

			[Newton - Principia]

 

One might describe the laws as: 1) a description of motion, 2) a prescription

for calculation (F = m a) and 3) the conservation of force.

 

	What the enlightened thinkers where fighting was the Greek idea of

motion.  Aristotle had left all of Europe thinking that 1) there was an

absolute rest frame,  with the earth at the center,  and 2) motion was caused

by a body seeking to obtain its natural and preordained position.  A rock fell

to the earth,  because it was earth like.  Fire rose to the sky,  to join the

sphere of the sun.  Once an object had obtained its ideal position, it would,

of course cease its motion - it had realized its goal.

 

	Galileo challenged both of these ideas with what we now refer to as a

`Galileian Transformation":

 

	Shut yourself up with some friend in the main cabin

	below decks on some large ship,  and have with you

	some flies,  butterflies, and other small flying animals.

	Have a large bowl of water with some fish in it;  hang

	up a bottle that empties drop by drop into a wide

	vessel beneath it.  With the ship standing still,

	observe carefully how the little animals fly with

	equal speed to all sides of the cabin.  The fish

	swim indifferently in all directions; the drops fall

	into the vessel beneath; and, throwing something to

	your friend, you need throw it no more strongly in

	one direction than another, the distance being equal....

	When you have observed all these things carefully,...

	have the ship proceed with any speed you like, so long

	as the motion is uniform and not fluctuating this way

	and that.  You will discover not the least change in

	all the effects named, nor could you tell from any of

	them whether the ship was moving or standing still.

			[Galileo - Two Great World Systems]

 

	Rene Descartes [1596-1650] was hesitant to follow in the footsteps of

Galileo,  perhaps because of where those foot steps led - if not the label

heretic,  at least ostracization from the Church.  In any case,  Descartes did

not reject an absolute rest frame.  However he went to great lengths to define

a `proper-motion'.  We would refer to this as `relative-motion',  where the

motion of an object is measured not in terms of some universal reference

frame, but rather relative to another local object.  In making this

distinction I think he hoped sidestep the political-religious problems of

Galileo and Copernicus.  There is a normal/Aristotelian-motion which involves

the earth and God and the Church,  but that is not what Descartes is going to

write about.  He will instead write about his special case - or 

`proper-motion'. This is not in conflict with Aristotle - it is something 

different.

 

	Descartes wrote on a wide range of topics,  in a series of books which

spanned the last two decades of his life.  The two primary sources for

Descartes' laws of motion are *The World* and *Principles of Philosophy*.

*The World* was near completion in 1633,  but Descartes suppressed it in 1633,

because in that year all copies of Galileo's *System of the World* were burned

in Rome.  Descartes feared that unless his whole philosophical system was

presented at one time, its parts would be misunderstood,  and the whole opus

condemned.  Therefore he waited to publish his complete system, *Principles of

Philosophy*,  in 1644. Later *The World* was published posthumously in 1664.

 

 

	In Descartes' `Principles' he laid out his three laws of motion:

 

	1) Each and every thing, insofar as it is simple and

	undivided, always remains, insofar as it can, in the

	same state,  nor is it ever changed except by external

	causes .... And therefore we must conclude that whatever

	moves, always moves insofar as it can.

 

	2) Each and every part of matter,  regarded by itself,

	never tends to continue moving in any curved lines,

	but only along straight lines.

 

	3) When a body comes upon another, if it has less

	force for proceeding in a straight line than the other

	has to resist it, then it is deflected in another

	direction, and retaining its motion, changes only

	its determination.  But if it has more,  then it moves

	the other body with it,  and gives the other as much of

	its motion as it itself loses.

			[Descartes - Principles of Philosophy -

				part II 37, 39, 40]

 

	Descartes' first and second law are referred to as his "law of

persistence".  Their purpose is captured in Newton's first law,  the "law of

inertia". Descartes recognized that a rock in a sling would travel straight,

except for the sling.  At the moment it is released it travels without

`external causes' - in a straight line.  I think the reason that Descartes

breaks these into two laws is that the first tells us that motion as magnitude

is conserved.  The second tells us that the direction of normal motion is

straight.  He sees motion as having two properties;  it is constant and it

goes straight.  In a real sense we could derive these from Galileo's story of

a ship.  There is always a reference frame where the object is at rest,  let

"the ship (reference frame) proceed with any speed you like, so long as the

motion is uniform and not fluctuating this way and that".  So motion must be

"uniform and not fluctuating".  Since Descartes' notion of motion is more

complicated then most people this distinction is not surprising - as we will

later see.

 

	The third law of motion is referred to as the "law of impact".  In

Descartes earlier book "The World",  it appeared as:

 

	When a body pushes another, it cannot give it any

	motion without at the same time losing as much of

	its own,  nor can it take any of the other's away

	except if its motion is increased by just as much.

			 [Descartes - The World - ch 7, p. 437]

 

	The earlier version of the law of impact, from *The World* is

essentially correct and in consort with our modern physics.  By the time

Descartes penned *Principles* he understood that this was the hardest of the

three for this contemporaries to understand.  Therefore he wrote seven rules

to accompanied this law.  These rules were observationally oriented,  but not

always correct,  as we will later see.  Thus the law of impact in *Principles*

is reduced to the role of introducing the seven rules to follow.  In this

introductory role the law has lost much of its earlier simplicity and

symmetry which we now consider essential to a law of motion.  In losing its

symmetry it lost its general correctness.  Therefore I will confine myself to

the law of impact as it appeared in *The World*.

 

	We might at this stage sum up Descartes' law as; 1&2) a description of

motion, 3) a suggestion of how to calculate motion, using a motion

conservation.

 

	In all fairness,  although it was not the first "Laws of Motion",

Newton's are unique in their ability to be used to calculate with potentially

infinitesimal detail the course and trajectory of any particle in the

universe.  Newton identified "force" as the unique quantity which drives

motion.  His contribution,  as a tool for calculation and prediction, is

unprecedented.

 

	But Descartes' are not an inferior version.  Rather Descartes is

trying to answer a number of questions which "Principia" never touches.

Descartes is writing to challenge the philosophical foundations set down by

Aristotle,  and taught throughout Europe at that time.  But in challenging

Aristotle, Descartes was aware of the recent event surrounding Galileo,  and

he tip-toed around the whole problem of Copernicanism.

 

	Let me now return to the third law of motion.

 

	When a body pushes another, it cannot give it any

	motion without at the same time losing as much of

	its own,  nor can it take any of the other's away

	except if its motion is increased by just as much.

 

	We are left asking ourselves what is the "motion" in this law.

Descartes spends a great deal of time distinguishing motion from the Greek

understanding of motion.  Motion is a `mode' of a body. A `mode' seems to be

something like size,  or mass, or color or any other property.  He tells us

that motion is not the opposite of rest. Finally after all these hints about

motion,  he tells us;

 

	Although motion is nothing in moving matter but its

	mode, yet it has a certain and determinate quantity,

	which we can easily understand to be able to remain

	always the same in the whole universe of things,

	though it changes in its individual parts.  And so,

	indeed, we might, for example, think that when one

	part of matter moves twice as fast as another,  and

	the other is twice as large as the first, there is

	the same amount of motion in the smaller as in the

	larger.

			[Descartes - Principles of Philosophy -

				part II 36]

 

	Apparently Descartes really means volume when he says "twice as large"

since the physical extension of a body was so important to his thinking.

However it is but a small intellectual extension to assume a consistent

density among the bodies in these systems,  in which case "twice as large"

also means "twice as massive".  Then we realized that what Descartes has

quantified for us is "momentum". The word "momentum" doesn't appear in the

English language until half a century later,  according to the Oxford English

Dictionary, but that is clearly what Descartes is referring to by his

"motion".  We can then reread the third law as;

 

 

	When a body pushes another, it cannot give it any

	[momentum] without at the same time losing as much of

	its own,  nor can it take any of the other's away

	except if its [momentum] is increased by just as much.

 

 

Which is a simple statement of the conservation of momentum.

 

	Descartes followed up his laws of motion with seven "Impact Rules".

The rules consisted of the conditions and results of two bodies colliding:

 

	The first rule.

 

		First,  if these two bodies, for example B and C,

	were completely equal in size and were moving at equal

	speeds, B from right to left, and C toward B in a straight

	line from left to right;  when they collided, they would

	spring back and subsequently continue to move,  B toward

	the right and C toward the left,  without having lost any

	of their speed.

			[Descartes - Principles of Philosophy -

				part II 46]

 

or in the language of freshman physics:

 

	If:

		m(B) = m(C) and v(B) = -v(C)

 

	Then after the collision:

		v'(B) = -v(B) = v(C) = -v'(C).

 

In fact some of his rules are wrong or at least unclear. For example, the

second rule states;

 

 

	If:

		m(B) > m(C) and v(B) = - v(C)

 

	Then after the collision:

		only the lighter one (C) is reflected,

		and they move off together,  so (to conserve momentum)

		v(B) = v'(B) = v'(C) = -v(C).

 

which is not right (if the mass of C is not zero). In light of what we all

have been taught in our introductory physics, it is interesting to note

what Descartes did.  First,  he conserved the magnitude of the momentum for

each object.  We would write instead:

 

	momentum of the total system = 	

		m(B) * v(B) + m(C) * v(C) =

		[ m(B) - m(C)] v(B) =

		m(B) * v'(B) + m(C) * v'(C)

 

So in his rules Descartes conserves the sum of the magnitude of momentas of

each part, instead of conserving the magnitude of the sum of the  momentas of

each part.  Or:

 

		|P(B) + P(C)| not = |P(B)| + |P(C)|

 

Again - this is something which Newton got right.

 

	Still,  he has a "conservation of momentum" law,  and that is

significant.  It is not that it is the first conservation law.  In fact the

atomist had been conserving matter for a long time.  What is new and unique to

Descartes, (as far as this author knows) is the idea that what is conserved is

a constructed quantity.  Momentum is not directly measurable.  One can

directly measure velocity and time and distance. But not momentum. 	

 

	So what is Descartes' Conservation principle?

 

	And as far as the general cause is concerned, it

	seems obvious to me that it is nothing but God himself,

	who created motion and rest in the beginning, and now,

	through his ordinary concourse alone preserves as much

	motion and rest in the whole as he placed there then.

			[Descartes - Principles of Philosophy -

				part II 36]

 

	In the Greeks,  the conservation principle takes a very different

role:

 

	By rights, if things can perish, infinite time 	

	And ages past should have consumed them all 	

	... Nothing can 	

	Disintegrate entirely into nothing, 	

	.... 	

	Matter is indestructible 			

				[Lucretius  235-240]

 

	The conservation law for the atomist is a necessary property of atoms,

 but it only goes as far as explaining why matter doesn't vanish.  You really

can not start with the conservation of matter and derive the atomic model.

 

	Descartes realized the importance of his conservation,

 

	And from this same immutability of God, certain rules

	or laws of nature can be known, which are secondary and

	particular causes of the different motions we notice in

	individual bodies

			[Descartes - Principles of Philosophy -

				part II 37]

 

	I can not really get inside of Descartes' head. We only have his books

and letters - which really are his public face. I would really like to know if

it was absolute necessary to Descartes for there to be this "immutability of

God",  in order for this conservation to work.  Or was it that he could not

imagine the world without God playing the most essential and critical role?

Or perhaps he was just trying to avoid the fate of Galileo?

 

	In the twentieth century we are satisfied that conservation is a

grounded in 1) observation and 2) symmetry. Descartes' reason is started

simply as "Thus we know from experience" - it is observationally grounded.

We can only speculate what his observations were.  His "Rules of Impact"

are all presented as descriptions of collisions.  Yet the details of

these observations were incorrect.  Somehow he has gone form incorrect

observations,  to the correct general law. This draws me to suspect that

the general character of the collision,  and not the details, were enough

to draw Descartes attention to momentum.  Momentum by itself must be a

powerful enough explanation of collisions that it would not be lost in

the details of the observation.  In retrospect it is the primary explanation

of elastic collisions.

 

	What has happened to `momentum' in today's physics?  In the language

of classical, relativistic and quantum mechanics,  the term 'momentum' is

generalized beyond just velocity times mass.  It is the constant of the

system:

 

	Define the quantity

                                       d L

                             p(i) = ___________

                                    d(dq(i)/dt)

 

	know as the *generalized momentum* or *canonical

	momentum*.

			[Fetter and Welecka - Theoretical Mechanics]

 

in other words,  a canonical momentum is a quantity which is constant with

respect to some generalized velocity or evolution of the system.  In some

sense conservation laws are just the definition of another *canonical

momentum*.  All physical systems have canonical momentas,  and one of the

chief goals of an investigation is to identify these.  Familiar ones include;

linear momentum,  angular momentum,  energy,  even quantum numbers.

 

	If one wanted to describe the angular velocity of a planet in orbit

one could start out with the reams of data Tycho Brahe tabulated, but there

is little knowledge in raw numbers.  Perhaps one could turn to one of Kepler's

laws of planetary motion: "equal area in equal time".  This law applied to

satellite and binary stars as well as the planets in Brahe's data. But in

the end it is the conservation of angular momentum which is most general.

Once we have identified what is constant,  then we can apply it to planets and

galaxies and figure skaters spinning on ice,  or even quarks bound in a

neutron.

 

	It is the constants or canonical momentas which are the most general

characteristics of a system. Particular events are merely the results of

certain initial or boundary conditions,  and do not in themselves tell us

about the dynamics of a generalized system.  The "conservation of momentum"

does transcend the particulars of a system.  This is what Descartes first did,

for when we know what is conserved, only then do we have truly fundamental

and general laws of nature.

 

	So *What is it that angelic music, papal decrees, and the resistance

of a ball in motion to change, all have in common?*  They are canonical.

 

	By "canonical" we mean generally accepted, standard,

	conventional.  It derives from the meaning "according

	to the canons."  The word is used, as we shall see,

	quite technically in mechanics.

 

			[Saletan & Cromer - Theoretical Mechanics]

 

Laws dictate the rises and falls,  the cycles and progressions of Pachelbel's

and Bach's Canons (Kanons),  the papal canons,  the canons of motion.  All of

them are below the sphere of the moon.  All of them continue in great

immutable constants.  Because that which describes the ten thousand musical

notes of a canon,  or the scattering of the objects in any system (even a

chaotic system), are the constants.

 

 

REFERENCES

__________

 

Garber, Daniel,  "Descartes' Metaphysical Physics",

	University of Chicago Press, 1992.

 

Newton, Isaac, "Principia" (original: 1686),

	trans. Andrew Motte (1729), revised Florian Cajori,

	University of California Press (1962).

 

Galilei, Galileo, "Two Great World Systems" (original: 1633)

 

Descartes, Rene, "The World", (original: 1664),

	trans. Mahoney, M. S., Abaris Books, N. Y. (1979).

 

Descartes, Rene, "Principles of Philisopy", (original: 1644)

	trans. Miller, V. R. & Miller, R. P., D. Reidel Pub. (1983).

 

Lucretius, "On The Nature Of Things" or "The Way Things Are",

	trans. Rolfe Humphries, Indiana University Press (1969).

 

Fetter, A. L. & Welecka, J. D.,

	"Theoretical Mechanics of Particles and Continua",

	McGraw-Hill (1980).

 

Saletan, E. J. & Cromer, A. H., "Theoretical Mechanics"

	John Wiley & Son (1971).