Scientific Enlightenment, Div. Two 4: The Problem of Representation Chapter 3 Leibniz' living force and Descartes' quantity of motion ACADEMY | previous section | Table of Content | next section | GALLERY

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L'inuention de tous ces engins n'est fondée que sur vn seul principe, qui est que la mesme force qui peut leuer vn poids, par exemple, de cent liures a la hauteur de deux pieds, en peut aussy leuer vn de 200 liures, a la hauteur d'vn pied, ou vn de 400 a la hauteur d'vn demi pied, & ainsy des autres, si tant est qu'elle luy soit appliquée. Et ce principe ne peut manquer d'estre receu, si on considere que l'effect doit estre tousiours proportionné a l'action qui est necessaire pour le produire: de façon que s'il est necessaire d'employer l'action par laquelle on peut leuer vn poids de 100 liures a la hauteur de deux pieds, pour en leuer vn a la hauteur d'vn pied seulement, celuy cy doit peser 200 liures. Car c'est le mesme de leuer 100 liures a la hauteur d'vn pied, & dereches encore cent a la hauteur d'vn pied, que d'en leuer deux cent a la hauteur d'vn pied, & le mesme aussy que d'en leuer cent a la hauteur de deux pieds. (Oeuvres de Descartes, ed. Adam and Tannery, vol. 1, p. 435 - 6)

The invention of all these engines are founded on only one single principle, which is that the same force which can lift a weight, for example, of a hundred livres to the height of two feet, can also lift one of 200 livres to the height of one foot, or one of 400 to the height of half a foot, and so on with others, if such is how it is applied. And this principle cannot but be received, if one considers that the effect must always be proportional to the action which is necessary to produce it: such that, if it is necessary to utilize the action by which one can lift one weight of 100 livres to a height of two feet, to lift one to the height of one foot only, that must weigh 200 livres. For it is the same to lift 100 livres to a height of one foot, and then this 100 livres still to another height of one foot, as to lift 200 livres to the height of one foot, and the same also as to lift 100 livres to the height of two feet.

In this way the new structural perspective tries to quantitatively represent motion and the work necessary to produce it. Descartes' formula is already modern, e.g., that force (work) is pl (weight [p] x distance [l]); recall that weight = mg, i.e. force, and work W = force x distance. Contrast this with the formula of Descartes' contemporary, Galileo, for momento, pv (weight x velocity), i.e. mgv, almost that for momentum, mv. "Descartes formally claims to have excluded consideration of the velocity [against the tendency of his contemporaries, who always have recourse to velocity when formulating quantitative representation of the structure of motion, i.e. equations], 'which would make it necessary to attribute 3 dimensions to the force'." (Dugas, ibid., p. 157) Distance is two dimensional, as a line; time is thus three dimensional. Another fact of note is that the motivating experience underlying this quantitization of "force" (energy) is the principle of Conservation. F (i.e. W) = 100l-2ft = 200l-1ft = 400l-1/2ft, in all three scenarios the same "force" is conserved; i.e. in the transformation from cause ("action") to effect not a ounce of what is started with will be lost in the end. In Newtonian language this can be expressed as F = m₁a₁ = m₂a₂. There is order in the world, and laws of nature exist, because of Conservation, that the same force has always the same effect, and this because the same is always the same. From now on, in the story of the emergence of mechanics, we will focus on two themes, that of representation and that of the evolution of the principle of Conservation. What we will see in regard to the second is the transformation of the Greek upokeimenon into the Conservation of energy, the first law of thermodynamics. That is, the transformation of the meaning of the Conservation of "everything". Originally, "everything" that is conserved is upokeimenon as the eternally same (amount of) material; then the "everything" that is conserved means -- in accordance with the world view of classical mechanics -- the total amount of "(extended) things changing places", which, when the total clarity of classical mechanics is reached with Lagrange, means "mass" (the extended thing, substantia) plus the "total mechanical energy" (kinetic plus potential energy) generable by the motion (place-changing) of this mass, which is constant, i.e. forever the same in the Universe. This until Einstein's E = mc² which reduces both mass and mechanical energy to aspects of the same thing (energy), and with which the contemporary notion of the Conservation of energy is obtained. To reiterate, Conservation -- whether in the Ionian upokeimenon or in the advanced Parmenidean Is or in its most logically extreme expression of Chinese Buddhist "Emptiness is emptiness" or in the modern sense of energy -- is the expression of the logical tautology of A = A which is the fundamental structure of the Universe. Scientific enlightenment (or rather one task of it) consists in the formulation of this fundamental truth of existence in quantitative and structural terms.

The following two excerpts, on the lifting on an inclined slope and on lever, from the same letter illustrate Descartes' conception of the "action" (energy) transformed into "force" (work) that is needed to lift something.

We see how Descartes' manner of "calculating" or "representing" in a quantitative fashion "work" is still purely geometrical, reminiscent of al Khwarizmi's way seen earlier. This again appears in the manner in which Descartes calculated the increasing velocity (acceleration) of a falling body as seen in a letter to Mersenne dated November 16th, 1629. "Descartes starts by recalling that a body which falls from A to B and then from B to C travels much more quickly in BC than in AB, 'for it keeps all the impetus by means of which it moves along AB and besides, a new impetus which accumulates in it because of the effect of the gravity, which hurries it along anew at each instant.' This is the scholastic doctrine on the accumulation of impetus. 'The triangle ABCDE shows the proportion in which the velocity increases. The line 1 denotes the strength of the impressed velocity at the first moment, line 2, the strength of the velocity impressed at the second moment, etc... Thus the triangle ABE is formed and represents the increase of the velocity in the first half of the distance which the body travels. As the trapezium BCDE is 3 times greater than the triangle ABE, it follows that the weight falls 3 times more quickly from B to C than from A to B. That is, that if it falls from A to B in 3 moments, it will fall from B to C in a single moment. Thus in 4 moments its path will be twice as long as in 3; in 12, twice as long as in 9; and so on." (Cited by Dugas, ibid., p. 158; emphasis added.) It is especially to be noted here that "[b]efore he developed his own analytic geometry, he used the geometrical representation of uniformly varying quantities that was due to Oresme." (Ibid.) This recalls the geometrical proof of algebra that al-Khwarizmi used which we saw earlier. Analytic geometry is more advanced and abstract, as the moments are removed to the x coordinate axis there and the 2 dimensional area of the triangle is reduced there to the tangential properties of the 1 dimension of the curve representing the relationship between time and distance on the Cartesian coordinate. This contraction of the 2 dimensional representation of the variation of velocity to a 1 dimensional curve whose representational properties is expressed by a 2 dimensional space (Cartesian coordinate) on which it lies facilitates the re-representation of this triangular representation by the 1 dimensional algebraic function (tangential properties are algebraic derivatives, (2x³)' = 6x²; v = (x)' and a = (v)'). As seen, Spengler has noted the emergence of a different meaning of number in this reduction. The advantage of analytic geometry as compared with the "pure geometry of shape" as a two dimensional representation is thus found in the ease of translation between the two dimensional geometric representation and the one dimensional algebraic representation. As for the increase of the triangular area downward: "Descartes called the measure of such a quantity the augmentatio velocitatis" (ibid., p. 159), i.e. something like acceleration.

Beeckman elaborates on this "prototypic" notion of "acceleration": "If, during the first moment of time, the body has travelled a moment of distance AIRS, during the first two moments of time it will have travelled 3 moments of distance, represented by the figure AJTURS. The distance travelled in any time whatever is therefore represented by the corresponding triangle supplemented by the small triangle ASR, RUT, etc... which are equal to each other. But these equal triangles added in this way are smaller as the moments of distance are smaller. Therefore these added areas will be of zero magnitude when it is supposed that the moment is of magnitude zero. It follows that the distance which the thing falls in one hour is to the distance through which it falls in two hours as the triangle ADE is to the triangle ACB." (Cited in Dugas, ibid.) Hence if distance travelled in 1 moment is A, in 2 would be 3A; in 4 would be 2 x 3A + 3A = 9A. Acceleration, the rate at which velocity is increasing at a particular instant, is still buried in these burdensome geometric and equational representations.

The next important notion to be noted of Descartes' mechanics is his "conservation of the quantity of motion." In a way, this repeats the law of inertia of his contemporary Galileo. "I suppose that the motion that is once impressed on a body remains there forever if it is not destroyed by some other means. In other words, that something which has started to move in the vacuum will move indefinitely and with the same velocity." (Descartes' letter to Mersenne, 1629; cited by Dugas, ibid., p. 160) Elsewhere he explains: "When a part of matter moves twice as quickly as another that is twice as large, we ought to think that there is as much motion in the smaller part as in the larger. And that each time the motion of one part decreases, that of some other part is increased proportionally." "God in his omnipotence has created matter together with the motion and the rest of its parts, and with his day to day interference, he keeps as much motion and rest in the Universe now as he put there when he created..." (From Principles, 1644; cited by Dugas, p. 161.) In a another place Descartes emphasizes that motion is conserved always in straight line, like the principle of least distance, which, together with the law of inertia, is simply the consequence of the logical tautology of A = A¹. It is interesting that he -- and Leibniz too -- attributes this principle of conservation to God or his creation as its essential property, since, as seen, Conservation has been the chief content of human enlightenment or the mystical experience of divinity of human beings. Descartes' "quantity of motion" as the product of mass and velocity (since 2mv = m2v) thus rivals Galileo's "momento", and moreover the law of inertia, the conservation of the quantity of motion, and the principle of least distance are all aspects of the same thing, the ultimate structure of the Universe as A = A, although the principle of least action (the later form of least distance) may turn out to be problematic in quantum mechanics. Now "Descartes' successors, and in particular the great Dutch physicist Huygens, had shown that Descartes' principle of the conservation of motion was valid only if the products of mass and velocity were understood to be directed quantities: In other words, motions in opposite directions were to be subtracted one from the other. With this correction, Descartes' 'quantity of motion' is equivalent to our modern principle of the conservation of momentum." (Thomas Hankins, Science and the Enlightenment, p. 31) This error of Descartes is to become the fuel of the controversy over the living force of Leibniz. The problem is therefore that while Conservation has been recognized as the Truth of the Universe or existence, what exactly it is that is conserved is the question to which the continually mutating answer is the governing principle of the history of human spiritual consciousness. The Presocratic upokeimenon as the conserved substrate of existence is about to be transformed into the principle of the conservation of energy. We want to see how this works.

First, "Leibniz protested against the Cartesian mechanics in a memoir which appeared in 1686 in the Acta eruditorum at Leipzig, under the title A short demonstration of a famous error of Descartes... Leibniz set out to show that the vis motrix (or, in the words of the XVIIIth century, the force of bodies in motion), was distinct from the quantity of motion in Descartes' sense. Like Huyghens, Leibniz assumes that a body falling freely from a given height will acquire the 'force' necessary to rise again to the same height... On the other hand, like Descartes, he assumes that the same 'force' (...work) is needed to lift a body A, whose weight is 1 pound, to a height DC of 4 ells as to lift a body B, whose weight is 4 pounds, to a height of 1 ell. In falling freely from the height CD the body A acquires the same 'force' as the body B acquires in falling from the height EF. For when it has arrived at D, the body A has acquired the force that it needs to climb again to C, and the body B, when it has come to F, has acquired the force needed to climb to E. By hypothesis, these 2 forces are equal. Now the quantities of motion A and B are far from equal. Indeed, Galileo's laws shows that the velocity acquired in the free fall CD is twice the velocity acquired in the free fall EF. The quantity of motion of A is then proportional to 1 x 2, while that of B is proportional to 4 x 1, and is therefore twice that of A. [A = 1m2v = 4d = W; B = 4m1v = 1d = W; but A = 2mv & B = 4mv.] This contradicts the Cartesian thesis in which the quantity of motion is used to evaluate the 'force' [work]." (Dugas, ibid., p. 220; emphasis added.) Leibniz thinks that it is "force" (work, i.e. energy) that is conserved and "force" is mv², for then A = 1*2² = 4; 4 *1² = 4; distance (d) is then purely a correlative of velocity and so of work ("force").

Descartes arrived at the conservation of the "quantity of motion" through the notion that the same W produced the same mv. "Leibniz recognized that in simple machines (lever...) the same quantity of motion tended to be produced, in one part and the other, when equilibrium [is] obtained. 'Thus it happens by accident that the force can be reckoned as the quantity of motion. [I.e. sometimes W = mv.] But there are other instances in which the coincidence no longer exists." (Ibid., p. 220) The root of the error is that F = ma, and p = mv. But if acceleration is 0, F = ma = mv = p, i.e. when velocity is constant. Since W = fd, here fd = pd and so "force" (W) is mistaken as mv.

The memory of Conservation was active in Descartes. For the ancients the "total amount of everything" must be the same, and so they arrived at upokeimenon as the total amount of matter (material), thus the Ionians. Descartes wanted to arrive at a Conservation of "everything" -- God's creation has to remain the same, as a matter of logical necessity A = A -- but what is this "everything"? In the Cartesian view of the world as "extended objects changing places" the "everything" was the product of mass and motion, mv, "the quantity of motion." Both upokeimenon and mv were decontextualized view of Weltlichkeit, but the combination of the structural perspective with the greater precision in representation had shifted the meaning of "everything" from upokeimenon to substantia in motu.

Now Leibniz expressed the necessary conservational principle in terms: "that there is always a perfect equality between the complete cause and the whole effect." (Ibid., p. 220) That is, in transformation not a ounce of anything transforming is to be lost in that into which it is transformed, always the same starting point as Descartes (or anyone else).

As a consequence of the necessary Conservation, Descartes formulated 7 rules of motion in accordance with it. The third rule would be the occasion of attack by Leibniz: "If 2 equal bodies impinge on one another with unequal velocities, the slower will be carried along in such a way that their common velocity will be equal to half the sum of the velocities they possessed before impact." (cited by Dugas, p. 162) As Dugas remarks: "Nearly all Descartes' rules on impact are experimentally incorrect" although this one passes more or less, as to be seen. (Ibid., p. 163) Descartes eschewed observation and favored "rationality" (logic) because he knew, rightly, that Conservation was the eternal truth of reality. A always equals A and never not-A. The problem is whether he had correctly identified what exactly it was to be conserved, mv.

"Suppose that 2 bodies, B and C, each weighing 1 pound and travelling in the same direction, collide with each other. The velocity of B is 100 units and that of C, 1 unit. Their total quantity of motion will be 101. But if C, with its velocity, can rise to a height of 1 pouce, the velocity B will enable it to rise to a height of 10,000 pouces. Thus the force of the two united bodies will be that of lifting 1 pound to 10,001 pouces. Now according to Descartes third rule, after the impact the bodies will go together in company with a common velocity of 50 and 1/2... But then these 2 pounds are only able to lift themselves to a height of 2550 pouces and a 1/4, which is equivalent to lifting 1 pound to 5100 pouces and a 1/2. Thus almost half the force will be lost according to this rule, without there being any reason and without its having been used for anything." (Cited by Dugas, p. 220) B = 1m100v, C = 1m1v; B + C = 101v or mv, the total quantity of motion in B and C that is conserved after collision. But for Leibniz, 1m100v = 1 * 100² = energy converted to 10,000 pouce, and 1 * 1² = 1 pouce. "In this discussion lies the germ of the controversy about living forces (vis viva) that was to divide the geometers at the beginning of the XVIIth century... We know now that Descartes third rule is correct and is applicable to perfectly soft bodies... The total quantity of motion [momentum] is conserved (no difficulty of sign occurs here) and a part of the living force is transformed into heat." (Ibid., p. 221)

Leibniz then lay down the foundation of the modern notions of mechanical energy (in Specimen dynamicum): "Force is twin. The elementary force, which I call dead because motion does not yet exist in it, but only a solicitation to motion, is like that of a sphere in a rotating tube or a stone in a sling [,which was still held still by the band.]" ("Hinc Vis quoque duplex: alia elementaris, quam et mortuam appello, quia in ea nondum existit motus, sed tantum solicitatio ad motum, qualis est globi in tubo, aut lapidis in funda, etiam dum adhuc vinculo tenetur".) This is like potential energy. "The other is the ordinary force associated with actual motion, and I call it living." (Cited by Dugas, ibid., p. 221; "alia vero vis ordinaria est, cum motu actuali conjuncta, quam voco vivam.") This would become kinetic energy. Vis viva, as mv², would become today's 1/2mv². As said, everyone, since time immemorial, knew that "the total amount of everything" would forever be the same, and with the advent of the mechanics' view of the world as things changing places the "everything" whose total had to be conserved must have had something to do with mass and motion (velocity). Now one side said fundamentally this "everything" was vis viva and the other the quantity of motion; who's right? "Leibniz's suggestion that the fundamental quantity of motion was different from the one Descartes had proposed was rejected out of hand by all good Cartesians. A great controversy ensued between the German school of physical thought, which naturally supported Leibniz, and the French and English schools, whose Cartesians and Newtonians opposed him." (Eric Weisstein) However: "Leibniz vigorously clung to his concept of universal conservation of living force, which had nothing but his metaphysical beliefs ['that it was equivalent to the eternity of God's creation'] to support it, even though it appeared to be violated for inelastic collision and was bitterly opposed by a large segment of the scientific community. Thus, Leibniz serves as the first example of a scientist who vehemently argued the existence of a fundamental conservation quantity based not on experimental evidence, but rather from a belief in the order and continuity of the universe." (Ibid.) Such belief of course was not unjustified, and it was right to maintain it in spite of experimental contraction. If experiments seemed to suggest that A was not totally equal to A, then experiment must have been wrong. The problem to worry about was what A is.²

To quickly conclude the controversy: "Thus Herman considers a perfectly elastic body M, of mass 1, and velocity 2, colliding with a motionless sphere N of mass 3. The body N will take, after the impact, the velocity 1 while the body M will be thrown back with velocity 1. If M then meets a motionless body O of mass 1, it can communicate its velocity to the latter and remain at rest. Therefore the force of M, which has mass 1 and velocity 2, is equivalent to 4 times the force of a body of mass 1 and velocity 1, which verifies the law of living forces and contradicts that of quantities of motion." (Dugas, ibid., p. 238) M = 1m2v; N = 3m0v; now M + N = 1m2v + 3m0v = 1m(-)1v + 3m1v. But 1 * 2 + 3 * 0 = 2 = 1 * -1 + 3 * 1 = -1 + 3 = 2. Again, Descartes was right as long as the direction of velocities be taken into account, i.e. mv-> was conserved. Now then, M = 1m(-)1v, O = 1m0v; 1m(-)1v + 1m0v = 1m0v + 1m1v. Descartes was still right since (-)1v was actually 1v as O was also moving in the direction of -. But with Leibniz, in the first case, 1 * 2² + 0 = 4 = 1 *-1² + 3 * 1² = 4; and the second, 1 + 0 = 0 + 1. Leibniz appeared right (also), especially since the error of Descartes (not m|v|, but mv->) had not been discovered.

"De Mairan observed that this coincidence was accidental and stemmed from the equality 2 + 2 = 2 x 2. For his part, he considered a body M of mass 1 and velocity 4 which he arranged to collide with a body N of mass 3 which was initially at rest. If M communicates a velocity 2 to N, the force of N is as 6. The body M, which keeps the velocity 2, can transfer this to a body O of mass 1, initially at rest. The total force of M is therefore as 6 + 2 = 8, and not as 16 as the law of living forces would require." (Ibid.) M = 1m4v & N = 3m0v; 1m4v + 3m0v = 1m(-)2v + 3m2v = 4 + 0 = (-2) + 6. And then O = 1m0v. 1m2v + 1m0v = 1m0v + 1m2v. Descartes was right. "The reader will easily verify, in all the examples which have been cited -- which are examples of elastic impact -- that if the direction is introduced, that is, if quantities of motion mv-> are considered, then the quantities Smv-> and Smv² are both conserved. Therefore the controversy of living forces was based on a mis-statement of the doctrine." (Ibid.) Descartes' conservation of Smv-> becomes today's conservation of (linear) momentum: the momentum of a system after the collision is equal to the momentum of the system before the collision, Sp_after - Sp_before = Dp = 0; before becoming today's principle of the conservation of energy, Leibniz's conservation of Smv² would first be formulated as the conservation of mechanical energy, that the maximal of kinetic energy of a system is equal to the maximal of its potential energy, or the change of kinetic energy is equal to the inverse change of potential energy, DT = -DV, crystallizing at last into Lagrange's equations. The first law of thermodynamics of today is Conservation of matter-energy and momentum. Note that the (Cartesian) principle of Conservation is the basis of Newton's three laws of motion: 1st law, Dp = 0, i.e. when no (external) force is present, there is forever no change of momentum possible (law of inertia); 2nd law, Dp/Dt = F, i.e. the change of momentum signifies the presence of force whose measure is the rate of change of momentum; 3rd law, Dp₂/Dt = -(Dp₁/Dt), i.e. the force of body 2 on body 1 is equal to (but in opposite direction of) the reacting force of body 1 on body 2. This is the beginning of the completion of the spiritual meaning of history (or of human consciousness in its spiritual dimension) which is the progressive articulation of the memory of Conservation from the Generic Ancestral Ghost as the eternally conserved substrate-soul (including the monotheistic God) through upokeimenon and the substantia transmuted from it to the substantia in motu articulated as various types of the principle of Conservation. At this stage, the quantitative substantia in motu is still deformative of the metaphysical (i.e. spiritual) and awaits its application to the whole of Universe; scientific enlightenment then consists in the recovery of spirituality (i.e. mysticism) within the deformative quantitative concept.

Footnotes:

1. That Conservation (of "force" or energy) necessarily implies the law of inertia and the principle of least distance is commented by the various editors and translators of Leibniz's Specimen Dynamicum (Felix Meiner Verlag, Hamburg, 1982) with regard to his "living force": "Die Kraft wird als die Begruendung fuer das Traegheitsgesetz (c.f. Desc Princ II, 37, 39, New Princ Lex 1a) gesehen; und zwar erfolgt aus der Erhaltung der Kraft nicht nur die Konstanz des Betrages der Geschwindigkeit, sondern insbesondere auch die Erhaltung der Richtung (Geradlinigkeit) (c.f. auch Spec dyn I, 8). Die Kraft naemlich ist Ursache fuer den Drang (nisus), dem als differentielle Groesse nur eine Richtung, aber keine Kruemmung zukommen kann. Eine Abweichung von einer geradlinigen Bewegung kann also nur durch das Zusammenspiel verschiedener Kraefte zustandekommen." (p. 151)

2. Leibniz defines the living force: "in the percussion that is produced by a body which has been falling for sometime, or by an arc which has been unbending for some time, or by any other means, the force is living and born of an infinity of continued compression of the dead force." This definition is to serve as the pivot in classical mechanics, and may be represented by the modern (Dugas, p. 221):

that is, force moves mass m by an increment of distance ds and the work done is therefore Fds which is the change of kinetic energy during that increment of distance (the differential of kinetic energy, its incremental increase; d = differential).