The Origins of Statics/Chapter 4
One would arrange more willingly the question of perpetual motion in Dynamics than in Statics; but, for Leonardo da Vinci and for Cardan, no more than for Aristotle, there does not exist between the two sciences any insurmountable barrier. On the other hand, the impossibility of perpetual motion has been admitted, by Galileo and Stevin, as an axiom suitable to base certain demonstrations of Statics; and Galileo and Stevin would have read the writings of Cardan, where they would have drawn their confidence in this axiom; and Cardan, writing against perpetual motion, had not done but summarize the sparse notes of Leonardo da Vinci. We would not have the familiarity therefore to produce an idea neat and complete of the origins of Statics if we did not pass in review the objections that Leonardo da Vinci and Cardan have opposed to the perpetuum mobile.
The search for perpetual motion is the generic name by which one designates two distinct utopias, the search for the perpetual mover and the search for the perpetually moved.
The broadest of these utopias, the search for the perpetual mover, is the error of the miller who, in his reservoir, detains a determined mass of water, allows it to fall from a determined height and who would want without adding a pint to this water, without adding an inch to the height of his reservoir, to combine some marvelous gears which would permit him to grind as much grain as it would please him to.
We have seen with what precision the great hydaulicist that is Leonardo brings back to their just measure the ambitions of our miller. As he sets on his wheel one hundred millstones in the place of one; each one of them to him would grind one hundred times less grain. A given weight, falling from a given height, represents a determined motive power; this power, one can morsel it out, to variegate the work from it to infinity; one will not increase it.
This truth cuts short the hopes of the one who searches for a perpetual mover; it leaves the field free to the dreams of the one who pursues the realization of a perpetually moved.
Without asking from an engine any exterior mechanical effect, but also without exerting on it any action, could one not see this engine, once set in oscillation, moving indefinitely? Could one not, for example, construct a wheel so perfect that once launched, it would turn around its axis without ever stopping? Could one not arrange a clock from weights exactly equal, where the weight which reached the height of its course would descend in its turn while raising again the weight of which the fall had caused its ascension, in such a way that this perpetual clock would wind itself?
It is folly to ask a perpetual movement from an initial impulsion, for the motive power of this impulsion, what Leonardo da Vinci names his forza or his impeto, what Leibniz will name his living force, will be exhausted without cease; it is folly equally to wait for from an arrangement of weights a perpetually moved, for the gravity tends always to equilibrium; all movement produced by gravity has as its end repose:
- "Nothing without life," says Leonardo da Vinci (1), "can push or pull without accompanying the thing moved; these movers cannot be but forza or weight; if the weight pushes or pulls, it does not effect this motion in the thing but because weight desires repose, and anything moved by its falling motion not being able to return to its first height, its motion ends.
- "And if the thing which moves another thing is a forza, this force, it also, accompanies the thing moved by it, and it moves the thing in such a way that it wears itself out; being consumed, none of the things which has been moved by the force is capable of reproducing it. Therefore anything moved cannot have a long operation, because, the causes being missing, the effects are missing."
The contemporaries of Leonardo conceded willingly to him that the motive power of an impulsion communicated to a set of bodies will dissipate; all the peripateticians, in effect, held as an axiom that the violent motion will always wear itself out: "Nullum violentum potest esse perpetuum", they repeated. To portray this continual loss of living force at the heart of a system in motion, Leonardo finds expressions of an inflamed poetry: "I say (2) that the forza is a spirited virtue, an invisible power which, in the midst of an accidental exterior violence, is caused by the motion, introduced and infused in the bodies, which find themselves pulled and diverted from their natural habit; it gives them an active life of a marvelous power, it constrains all created things to change in form and in place, runs with fury to its desired death and goes diversifying itself according the causes. Slowness makes it great and speed makes it feeble; It is born from violence and dies from liberty. And the greater it is, the more quickly it consumes itself. It chases with fury what opposes itself to its destruction, desires to vanquish and to kill the cause of what makes obstacle to it and, vanquishing, kills itself....Any movement made by it is not durable. It grows in hardships and disappears from repose."
With the same richesse of images, Leonardo compares this loss of the living force to the continual tendency of gravity towards repose: "If the weight desires stability (3) and if the forza is always in desire of flight, the weight is by itself without fatigue, while the forza is never exempt from it. The more the weight falls, the more it augments (4), and the more the forza falls, the more it diminishes. If the one is eternal, the other is mortal. The weight is natural and the forza accidental. The weight desires stability and then immobility; the forza desires flight and death of itself."
How does this continual tendency of gravity to a state of final equilibrium (5) manifest itself in a mechanism? It manifests itself through this law that in a mechanism in movement, "always the mover is more powerful than the moved (6)"; it is in virtue of this law, for example, that "the cord which descends from pullies feels more weight and, by consequence, fatigues itself more than the opposing cord which rises." This inequality, of invariable orientation, between the power of the mover and the resistance of the moved, finds itself again in all mechanisms: "For example (7), if you wish that the weight b lift the weight a, the arms of the balance being equal, it is necessary that b be heavier than a. If you were wishing that the weight d would be lifting the weight c, which is heavier than it, it would be necessary for it to effect to make a greater course in its descent than c makes in its ascent; and if it descends more, it is necessary that the arm of the balance which descends with it be longer than the other. And if you were wishing that the small weight f would be lifting the great e, it would be necessary that the weight f would be moved along a greater length and more rapidly than the weight e." It is the excess alone of the power of the mover on the resistance of the moved which determines the movement; the more this excess is great, the more the movement is lively. "No power (8) prevails over its resistance, except with the part of which it exceeds this resistance. Or else: no mover prevails over its moved, except by that of which it exceeds the moved....And so much the more that movement of the moved is joined to the impeto, so much the more the impeto is greater than the moved, which can grow to the infinite." If a pulley carries two equal weights, the weights remain immobile; if they are unequal, the heavier will descend with a velocity proportional to its excess over the lightest: "If a pound of weight falls against a pound of resistance (9), it will not change from its place; it will rest the same way. And if beside is attached another pound, it will descend to earth in a certain quantity of time; if you add to it still another pound, all the weight will descend with a doubled velocity."
Therefore the clock which would wind up itself is a chimera; always the weight which possesses the greatest motive power will begin to descend and, when it will have arrived at the bottom of its course, the clock will stop itself; from there, this conclusion (10) from Leonardo:
- "Against perpetual motion. Nothing insensible could move itself by itself; in consequence, if it moves itself, it is moved by an unequal power, that is to say in time and motion unequal, or in weight unequal. And as soon as the desire of the first mover will have ceased, so soon will cease the second."
These are the thoughts of Leonardo that Cardan resumes when in the books De la Subtilité, "he demonstrates that motion is not perpetual in all things (11)." When one attempts to realize a perpetually moved, "what one asks properly to speak, it is this: does a motion exist which in itself, and outside of all new generation, reconstruct a cause capable of perpetuating it? The problem would be resolved if one possessed some clocks which, in place of setting in oscillation this movement which announces the hours in hitting some blows, would wind up the weights to the top of their course. Now, the motions which can oscillate the heavy objects are of three sorts only: either they tend essentially to the center of the world; or they they are not simply directed towards the center, as the flow of waters; or they flow from a particular nature, as the movement of iron towards the magnet. It is constant that perpetual motion must be searched for among the motions of the two first kinds (12). Now, when one weight is drawn more strongly or retained more energetically than its nature comports it, its motion is natural, it is true, but it is not exempt from violence; from these two circumstances, one finds an example in the weights of clocks....With regard to motion around a circle, it is not naturally fitting but to the heaven and to the air; still the latter is not animated in the form of it in a constant manner; for the other heavy objects, it has always its principle in a motion according to the vertical. The waters themselves are animated from a certain motion according to the vertical; thus, in rivers in proportion as the waters are engendered by the source, they descend without cease following the declivity of the bed. Now, in order for the motion to be perpetual, it would be necessary that the heavy objects which have been deplaced, attained at the end of their course, were being carried back to their initial situation. But they cannot be carried back to it but by a certain excess [of motive power]. Thus therefore, either the continuity of the motion will proceed from what this motion is consistent to its nature (13), or this continuity will not maintain itself equal to itself. Now, what diminishes without cease, unless being accrued by an exterior action, would not be able to be perpetual."
In the considerations of Leonardo da Vinci and of Cardan, there is not only the negation of the perpetually moved, there is more; there is this affirmation as one uniform tendency in all the movements that we observe, the tendency of heavy objects to descend as much as possible, to look for the place of their eternal repose. This thought is constantly present to the spirit of Leonardo da Vinci. "All weights (14) desire to descend to the center by the shortest way; and where there is more weight, there is a greater desire, and the thing which weighs the most, let free, falls the quickest...."—"The weight (15) pushes always towards the place of its departure....And the place of the weight is unique; this is the earth." This proposition can serve as a principle to explain the equilibrium and the motion of waters: "This thing is higher which is more elongated from the center of the world (16), and that there is lower which neighbors more closely this center. Water does not move from itself if it does not descend; therefore the sphere of water not having any portion of surface to be able to descend, it is necessary by the first conception that it does not descend.
Without doubt, water seems sometimes to go up spontaneously and certain hydraulic apparati exploit this property; but, in reality, one does not obtain in these apparati the ascension of a small quantity of water but by the fall of a very great mass; this is what Cardan observes (17), treating of "the Archimedes screw. Therefore it seems that this argument concludes: Water descends perpetually, therefore, in the end, it will be in a place lower than at the beginning. However, it does not descend always, but the part which descends the greatest pushes up the smallest and constrains it from rising."
Such is therefore the general law of the motions produced by gravity; no body rises but it descends by it more heavy. "All heavy objects tend below (18), and the high things will not stay at their height, but with time, they will descend all, and thus with time the world will stay spherical and, in consequence, will be all covered with water."
All this argumentation of Leonardo da Vinci and of Cardan is drawn from the principles of the peripatetician Dynamics: proportionality of the speed to the force which moves the body in motion, of the velocity of fall to the weight of the heavy object. These foundations, the progress of Mechanics are going to remove. And meanwhile, a Mechanics more advanced still will come to fortify the conclusions. Almost constantly, we have left behind the utterance to the authors of the XVIth century; now, what they have said to us has almost a savor very modern; their thoughts are closely neighboring those of physicists who have read Clausius, William Thomson and Rayleigh. This is that Thermodynamics, in completing the too simplified Dynamics issued from the Discorsi of Galileo, has filled up in part the abyss which was separating the first from the Dynamics of Aristotle.
Here is not the place to insist on this reconciliation, which would carry us away well long from the origins of Statics. We have seen how the most essential thoughts of Leonardo da Vinci had been published in the works of Cardan; the great vogue of the second will permit these thoughts to have an influence on the development of the Science.
At the end of the XVIth century, this influence is divided into two currents; the one makes itself felt in Italy, where it inspires the works of Jean-Baptiste Benedetti, of Guido Ubaldo, of Galileo, of Torricelli; the other, canalized by Simon Stevin, makes Flemish science fruitful; these two currents will come to coalesce in Roberval and in Descartes.
(1) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. A of the Library of the Institute, fol. 21, verso. Paris, 1881.
(2) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. A of the Library of the Institute, fol. 34, verso. Paris, 1881.
(3) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. A of the Library of the Institute, fol. 35, recto. Paris, 1881.
(4) Leonardo was familiar with the accelerated fall of heavy objects of which he has treated at length in many passages, notably in Ms. M of the Library of the Institute.
(5) Here again, Leonardo does not but develop the teachings of the School: "Motus simplex terminatur ad quietem", about it said one.
(6) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. E of the Library of the Institute, fol. 20, recto. Paris, 1888. — Cf. Ms. E, fol. 58, verso; Ms. G, fol. 81, recto and fol. 82, recto. Paris, 1890.
(7) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. A of the Library of the Institute, fol. 22, verso. Paris, 1881.
(8) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. E of the Library of the Institute, fol. 21, verso. Paris, 1888.
(9) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. A of the Library of the Institute, fol. 22, verso. Paris, 1881.
(10) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. A of the Library of the Institute, fol. 22, verso. Paris, 1881.
(11) Cardan, Les Livres de la Subtilité, translated from Latin into French by Richard Le Blanc. Paris, L'Angelier, 1556, p. 339. The quotations which follow are translated directly from Latin text and not drawn from the translation of Richard le Blanc, very obscure in this passage.
(12) One will remark that Cardan avoids pronouncing himself on the possibility of engendering perpetual motion by the aid of magnets. The properties so strange of magnets were preoccupying singularly, in this epoch, those who were hoping to realize a perpetuum mobile. In 1558, Achille Grasser was imprinting for the first time at Augsbourg, according one of the numerous manuscript copies which were circulating among the physicists, the celebrated writing composed by Pierre de Maricourt (Petrus Peregrinus), in the army of Charles d'Anjou, before Lucera, August 8, 1269. In this writing (a), Pierre de Maricourt, after having established the laws of magnetic actions, its logician broken in to the experimental method, tries to produce a perpetuum mobile by the aid of magnets.
- (a) Petri Peregrini Maricurtensis, De magnete, seu rota perpetui mobilis libellus. Divi Ferdinandi Rhomanorum imperatoris auspicio per Achillem P. Grasserum L. num primum promulgatus Augsburgi in Suevis, Anno Salutis 1558. —This work is reprinted in: Neudrucke von Schriften und Karten über Meteorologie und Erdmagnetismus, herausgegeben von G. Hellmann. N° 10, Rara magnetica. Berlin, 1896.
(13) Cardan intends to reserve by this phrase the motion of the heaven, which is perpetual by nature.
(14) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. A of the Library of the Institute, fol. 35, recto. Paris, 1881.
(15) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. C of the Library of the Institute, fol. 6, verso. Paris, 1888.
(16) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. F of the Library of the Institute, fol. 27, recto; fol. 26, verso et fol. 30, verso. Paris, 1889.
(17) Cardan, Les Livres de la Subtilité, translated from Latin into French by Richard Le Blanc. Paris, L'Angelier, 1556, pp. 12 and 13.—This passage is not found in the first edition of the De Subtilitate; it has been added in the second edition.
(18) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. F of the Library of the Institute, fol. 84, recto. Paris, 1889.