Leibniz and Dynamics:
Motion Is Not Simple

by Jonathan Tennenbaum

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        Things are not what they appear, nor does the world function
as the naive varieties of "common sense" (horse sense) would have
it do.  Those who subscribe to Rene Descartes' doctrine of "clear
and distinct" truths, and pride themselves on not listening to
anything that smells of "theory," or cannot be explained in five
words or less, are liable to be ripped off by the nearest used
car dealer (or stock broker?).  For a pedagogical exercise,
consider the following sales pitch, invented by wily old
Descartes himself.
        As every simpleton thinks he knows, the universe consists of
"matter and motion."  (In fact, the famed J.C. Maxwell marketed
his famous textbook on physics under that title.)  To measure the
performance of your used car engine, Descartes tells you, just
ask  "how much car (weight in pounds or tons) it moves how fast
(miles per hour)."  You just multiply the pounds together with
the miles per hour, to get the handy performance rating at Rene's
Used Car Lot.  For example, how would you choose between:
        Car A:  a two-ton "super classic," with wall-to-wall marble
ashtrays and other extras.  Flooring the accelerator, it reaches
40 mph in 30 seconds.  Rene urges us to buy this "hell of a car."
        Car B:  a lower-class model, weighs half as much as the
"super-classic," but reaches much less than twice the speed,
namely 60 mph in the same 30 seconds.
        A glance at both cars tells you, that their bodies are
essentially junk.  If anything, the only items of significant
value are the engines.  Now, Rene will let you have Car A for the
same price as Car B, which ("as a friend") he points out, is a
"fantastic deal."  Car A is a "bit slower" but, as you can easily
calculate yourself, with two-thirds the speed but twice the mass,
its engine performance rating is more than 30% larger than Car
        Rene adds another generous offer: If you prefer the smaller,
faster car, he will switch the engines for you, and install Car
A's motor into Car B, free of charge.  Could you turn down such a
deal?  After all, with Car A's engine and half the weight, other
things being equal, Car B should zip up to 80 mph in the same
time it took to bring the heavier car up to 40.
        Rene's enthusiasm makes you a bit suspicious, on several
points.  Simplest of all: does the product of mass and attained
velocity really represent the work performed in accelerating a
car or other massive object up to a given state of motion?
"Clear as day!" Descartes explains, appealing to our "horse
sense" with the following argument:
        Suppose we have a two-ton object.  For it to have a speed of
40 miles per hour means, that in any given hour, those two tons
move a distance of 40 miles.  Dividing the object into two parts,
each of 1 ton mass, we see that each of those has been moved 40
miles by that same motion.  Obviously, it would be the equivalent
amount of motion to move the two halves one at a time, instead of
simultaneously, over the same 40 miles.  In other words, in the
first half hour we move the first half 40 miles, and then during
the second half hour we move the other half 40 miles, the result
being to move the whole mass 40 miles in the course of that hour.
Or, again, since the two halves are identical in terms of mass,
it represent the same effort to take only one of them, and move
it 40 miles in the first half hour, and then just continue to
move it another 40 miles in the second half hour.  Thus, with an
equivalent process we have moved one ton, 40 plus 40 = 80 miles
in the given hour.  We repeat this for every succeeding hour.
Thus, two tons moving at 40 mph is equivalent to one ton moving
at 80 mph.  QED.
        Corollary: Car A's motor is a better buy than Car B's.
        An admirable specimen of deductive-type reasoning.  But, if
you swallow the axiomatics of this argument, you are going to be
cheated!  Can you prove them wrong?  Such a demonstration will be
given next.  [jbt with ap_]


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     In refuting Descartes on the measure of "quantity of motion"
and related points, Leibniz pointed out three interrelated
fallacies.  First is the implicit assumption, that physics can be
subsumed within a deductive form of mathematics.  Second is the
implicit assumption of "linearity in the small," that physical
action has the form of singularity-free continuous motion or
extension in a three-dimensional Euclidean-like space.  Third is
the assumption that matter is characterized by nothing but such
passive qualities as space-filling (extension), inertia, and
resistance to deformation.  In fact, in his 1686 piece on "A
memorable error of Descartes," and in other locations, Leibniz
gave a simple demonstration of physical principle, showing that
the process of change of velocity (acceleration) of material
bodies involves something which is absolutely incompatible with
Descartes' assumptions.  Leibniz demonstrated, for example, that
the work of acceleration is NOT proportional to the mere product
of the mass with the velocity attained, but (to a first degree of
approximation) increases as the SQUARE of the velocity!  To
accelerate a mass to twice a given velocity, we need, not twice,
but FOUR times the work.
     If you have never stopped to consider, how utterly
incomprehensible such a result is from the standpoint of naive
sense-certainty and "horse sense," do yourself that favor now.
     Review the Huygens-Bernouilli-Leibniz discussion on the
cycloid-brachistochrone for a richer development of the same
point.  Also Leibniz's discussion of Descartes' error and of the
required notion of "anti-entropic" substance, in his "Treatise on
Metaphysics" (Article 18 and preceding and following articles).
Consider the relevance of Nicolaus of Cusa's treatment of the
Archimedes problem, and review the whole matter again from the
higher standpoint of Lyn's writings, including on the issue of
     Here is Leibniz's paper of 1686, referred to above: "Seeing
that velocity and mass compensate for each other in the five
common machines, a number of mathematicians have estimated the
force of motion by the quantity of motion, or by the product of
the body and its velocity.  Or, to speak rather in geometrical
terms, the forces of two bodies (of the same kind) set in motion,
and acting by their mass as well as by their motion, are said to
be proportional jointly to their bodies or masses and their
velocities.  Now, since it is reasonable that the same sum of
{motive force} should be conserved in nature, and not be
diminished--since we never see force lost by one body without
being transferred to another--or augmented, a perpetual motion
machine can never be successful, because no machine, not even the
world as a whole, can increase its force without a new impulse
from without.  This led Descartes, who held motive force and
quantity of motion to be equivalent, to assert that God conserves
the same quantity of motion in the world.
     "In order to show what a great difference there is between
these two concepts, I begin by assuming, on the other hand, that
a body falling from a certain altitude, acquires the same force
which is necessary to lift it back to its original altitude, if
its direction were to carry it back and if nothing external
interfered with it.  For example, a pendulum would return to
exactly the height from which it falls, except for the air
resistance and other similar obstacles, which absorb something of
its force, and which we shall now refrain from considering.  I
assume also, in the second place, that the same force is
necessary to raise a body of 1 pound to the height of 4 yards, as
is necessary to raise a body of 4 pounds to the height of 1 yard.
Cartesians, as well as other philosophers and mathematicians of
our times, admit both of these assumptions.  Hence it follows,
that the body of 1 pound, in falling from a height of 4 yards,
should acquire precisely the same amount of force as the body of
4 pounds, falling from a height of 1 yard.  For, in falling 4
yards, the body of 1 pound will have there, in its new position,
the force required to rise again to its starting point, by the
first assumption; that is, it will have the force needed to raise
a body of 1 pound (namely, itself) to the height of 4 yards.
Similarly, the body of 4 pounds, after falling 1 yard, will have
there, in its new position, the force required to rise again to
its own starting point, by the first assumption; that is, it will
have the force sufficient to raise a body of 4 pounds (itself,
namely) to a height of 1 yard.  Therefore, by the second
assumption, the force of the body of 1 pound, when it has fallen
4 yards, and that of the body of 4 pounds, when it has fallen 1
yard, are equal.
     "Now let us see whether the quantities of motion are the
same in both cases.  Contrary to expections, there appears a very
great difference here.  I shall explain it in this way.  Galileo
has proven that the velocity acquired in a fall of four yards, is
twice the velocity acquired in a fall of one yard.  So, if we
multiply the mass of of the 1-pound body, by its velocity at the
end of its 4-yard fall (which is 2), the product, or the quantity
of motion, is 2; on the other hand, if we multiply the mass of
the 4-pound body, by its velocity (which is 1), the product, or
quantity of motion, is 4.  Therefore the quantity of motion of
the 1-pound body after falling four yards, is half the quantity
of motion of the 4-pound body after falling 1 yard, yet their
forces are equal, as we have just seen.  There is thus a big
difference between motive force and quantity of motion, and the
one cannot be calculated by the other, as we undertook to show.
It seems from that that {force} is rather to be estimated from
the quantity of the {effect} which it can produce; for example,
from the height to which it can elevate a heavy body of a given
magnitude and kind, but not from the velocity which it can
impress upon the body.  For not merely a double force, but one
greater than this, is necessary to double the given velocity of
the same body.  We need not wonder that in common machines, the
lever, windlass, pulley, wedge, screw, and the like, there exists
an equilibrium, since the mass of one body is compensated for by
the velocity of the other; the nature of the machine here makes
the magnitudes of the bodies--assuming that they are of the same
kind--reciprocally proportional to their velocities, so that the
same quantity of motion is produced on either side.  For in this
special case, the {quantity of the effect}, or the height risen
or fallen, will be the same on both sides, no matter to which
side of the balance of the motion is applied.  It is therefore
merely accidental here, that the force can be estimated from the
quantity of motion.  There are other cases, such as the one given
earlier, in which they do not coincide.
     "Since nothing is simpler than our proof, it is surprising
that it did not occur to Descartes or to the Cartesians, who are
most learned men.  But the former was led astray by too great a
faith in his own genius; the latter, in the genius of others.
For, by a vice common to great men, Descartes finally became a
little too confident, and I fear that the Cartesians are
gradually beginning to imitate many of the Peripatetics at whom
they have laughed; they are forming the habit, that is, of
consulting the books of their master, instead of right reason and
the nature of things.
     "It must be said, therefore, that forces are proportional,
jointly, to bodies (of the same specific gravity or solidity) and
to the heights which produce their velocity or from which their
velocities can be acquired.  More generally, since no velocities
may actually be produced, the forces are proportional to the
heights which might be produced by these velocities.  They are
not generally proportional to their own velocities, though this
may seem plausible at first view, and has in fact usually been
held.  Many errors have arisen from this latter view, such as can
be found in the mathematico-mechanical works of Honoratius Fabri,
Claude Dechales, John Alfonso Borelli, and other men who have
otherwise distinguished themselves in these fields.  In fact, I
believe this error is also the reason why a number of scholars
have recently questioned Huygens' law for the center of
oscillation of a pendulum, which is completely true."  [Adapted
from {Gottfried Wilhelm von Leibniz: Philosophical Papers and
Letters}, LeRoy E. Loemker, ed. (Chicago: University of Chicago
Press, 1956); vol. I, pp. 455-458)]  [jbt with ap_]

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The preceding article is a rough version of the article that appeared in The American Almanac. It is made available here with the permission of The New Federalist Newspaper. Any use of, or quotations from, this article must attribute them to The New Federalist, and The American Almanac.

For questions or comments: klebes@infi.net.