Motion Is Not Simple!

By Jonathan Tennenbaum

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 B’s.

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 in next Tuesday’s briefing. [jbt with ap ]


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 “time-reversal.”

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)]