The Origins of Statics/Chapter 1
From their profound researches touching the laws of equilibrium, the ancients have left us monuments little numerous, it is true, but worth an eternal admiration. Of these monuments the most beautiful, without contradiction, are the book consecrated by Aristotle to the questions of mechanics and the treatises of Archimedes.
The name of "Treatise of Statics" would be unjustly given to the writing where Aristotle examines diverse questions relative to mechanisms (Mηχανικὰ Pροβλήματα); the Stagirite, in effect does not separate the theory of equilibrium from the theory of motion; he does not assign to the first proper, autonomous principles which do not refer at all to the second; he treats in a general manner the motions that can occur in a mechanism; when no motion occurs, the mechanism stays in equilibrium.
The axiom which gives the solution of the diverse mechanical problems is the fundamental law that Aristotle assigns to local motion and which, explicit or hidden, dominates all of what he has written on the subject of this motion. The force of the mover which moves a body is measured by the product of the weight of the body moved (or of its mass, for the two notions of weight and mass are then indistinct) by the speed of the movement impressed on this body. The same force can therefore move successively a heavy body and a light body; but it will move slowly the heavy body and quickly the light body; the speeds of the motions impressed on the two bodies will be inversely proportional to their weights.
This thought is expressed in many passages. We cite only this one (1), of which the clearness is extreme: "Whatever be the force which produces the motion, what is less and more light receives from the same force more motion. . . . In effect, the speed of the least heavy body will be to the speed of the most heavy body as the most heavy body is to the least heavy body.—Ἐπεὶ λὰρ δύναμίς τις ἡ κινοῦσα, τὸ δ’ἔλαττον καὶ τὸ κουφότερον ὑπο τῆς αὐτῆς δυνάμεως πλεῖον κινηθέσεται. . . . Tὸ γὰρ τάχος ἕξει τὸ τοῦ ἐλαττονος πρὸς τὸ τοῦ μείζονος ὡς τὸ μεῖζον σῶμα πρὸς τὸ ἔλαττον."
This fundamental principle of peripatetian Dynamics is, it seems, the faithful and immediate translation of the most obvious givens of our quotidian experience. Modern Dynamics reputes it as grave error. But, to reject this error it has been needful for the science two thousand years of meditations conducted by the greatest spirits who have succeeded each other from Aristotle to Galileo. We will try someday to retrace the principal phases of this gigantic intellectual effort. But today we will exert ourselves to forget what modern Mechanics has taught us and to impress upon ourselves the laws accepted by the peripatetian Mechanicist. By this condition only can we understand the thought of the geometers who, from century to century, will make progress in Statics.
Two forces will therefore be regarded as equivalent if, moving unequal weights with unequal velocities, they give the same value to the product of the weight and the velocity; this product will be the measure of the force.
Let us conceive, now, a rectilinear lever that a fulcrum point divides into two unequal arms, at the extremities on which two unequal masses weigh; when the lever turns around its fulcrum point the two weights move with different velocities, the one which is the furthest from the fulcrum point describes, in the same time, a longer arc than the one which is nearest to the same point; the velocities which animate the two weights are to each other as the lengths of the arms on the end of which they weigh.
When, therefore, we want to compare the forces of these two weights we must, for each of them, take the product of the weights and the length of the arms of the lever; that one will transport it which corresponds to the greater product; and if the two products are equal, the two weights will stay in equilibrium.
- The weight which is moved, says Aristotle (2), is to the weight which moves in inverse ratio to the lengths of the lever arms; always, in effect, a weight will move as much more easily as it is more distant from the fulcrum point. The cause of it is what we have already mentioned: the line which makes way any further from the center describes a greater circle. Therefore in employing the same force, the mover will describe a course as much greater as it is removed from the fulcrum point.—Ὃ οὖν τὸ κινούμενον βάρος πρὸς τὸ κινοῦν, τὸ μῆκος ἀντιπήπονθεν. αἰεὶ δ’ὅσῳ ἂν μεῖζον ἀφεστήκῃ τοῦ ὑπομοχλίου, ῥᾷον κινήσσει. Αἰτία δ’ἐστὶν ἡ προλεχθεὶσα, ὅτι ἡ πλεῖον ἀπέχουσα ἐκ τοῦ κέντρου μείζονα κύκλον γράφει ὥστ ἀπὸ τῆσ αυτῆς ἰσχύος πλέον μεταστήσεται τὸ κινοῦν τὸ πλεῖον τοῦ ὑπομοχλίου ἀπέχον.
These considerations, developed with respect to the lever, are not a particular remark of which the efficacy binds itself to this case; they constitute a general method; they contain a principle which applies itself to nearly all mechanisms; by this principle, the geometers will be able render account of the varied effects produced by these diverse engines in considering simply the velocities with which certain circle arcs are described. "For the properties of the balance (3) are brought back to those of the circle; the properties of the lever to those of the balance; finally the most part of the other particularities offered by the movements of mechanics bring themselves back to the properties of the lever.—Tὰ μὲν οὖν περὶ τὸν ζυγὸν γινόμενα εἰς τὸν κύκλον ἀνάγεται, τὰ δὲ περὶ τὸν μοχλὸν εἰσ τὸν ζυγὸν, τὰ δ’ἄλλα πάντα σχεδὸν τὰ περὶ τὰσ κινήσεις τὰς μηχανικὰς εἰσ τὸν μοχλὸν."
Had he formulated but this sole thought, Aristotle would merit being celebrated as the father of rational Mechanics. This thought, in effect, is the seed from which will be brought out, by a development coming twenty times once in a century, the powerful ramifications of the Principle of virtual velocities (4).
Aristotle was not a geometer; from the principle that he had posed, he did not know to draw with entire rigor all the consequences which provided themselves to be deduced from it; sometimes, also, he believed himself able to apply it to some problems of which the complexity greatly exceeded the means by which he claimed to resolve them. Elsewhere, from the beginning of his researches, he had struck himself with a grave difficulty; the line described, in a movement of the lever, by the application point of the force or of the resistance is a circle's circumference; it does not coincide with the right vertical according to which this force or this resistance acts. Touching this difficulty, Aristotle had given some solidly obscure considerations (5), more appropriate for glossing on the commentators than for satisfying the geometers.
The geometers love to see a long chain of reasonings unroll in a perfect order and form a link without defect which unites some very simple and very certain principles to some remote and complicated conclusions. No work is more capable of satisfying their need of rigor and of clarity than the writings where Archimedes treats of Mechanics.
These writings comprise the Treatise on the Equilibrium of Planes or of their Centers of Gravity (Ἐπιπέδων ἰσορροπικῶν ἢ κέντρα βαρέων ἐπιπέδων) and the Treatise on Floating Bodies (Περὶ τῶν ὀχουμένων). Our intention is not at all to study, in this writing, the origins of Hydrostatics; we will leave aside therefore the Treatise on Floating Bodies to arrest our attention on the other treatise.
Archimedes knew to exclude from the foundations on which he seated his doctrine all propositions of which the solidity could seem doubtful; he will not therefore be going, in the imitation of Aristotle, to be requiring his fundamental hypotheses from the science of motion; for the laws which preside over the motions of weighted bodies seem profoundly hidden under some complex appearances; the analysis of these phenomena, so varied and so difficult to observe exactly, seem little proper to furnish the propositions which win over all the approbations. On the contrary, the daily use of very simple instruments, of the balance for example, reveals to us, on the subject of the equilibrium of heavy objects, some rules of which the truth and the generality would not be able to be made into the object of any doubt. Following the method of which his master Euclid has made usage in the Elements, Archimedes will require of whomever wants to follow his teaching from him to agree with the certitude of these few propositions, from which he will deduce all of his theory.
Here are what are the requirements (6) of Archimedes:
1° Equal weights suspended at equal lengths are in equilibrium.
2° Equal weights suspended at unequal lengths are not at all in equilibrium; and that which is suspended at the greater length is carried down.
3° If weights suspended out from certain lengths are in equilibrium, and if one adds something to one of these weights, they are no longer in equilibrium; and the one to which one adds something is carried down.
4° Likewise, if one subtracts something from one of these weights, they are no longer in equilibrium; and the one from which has not subtracted anything is carried down.
From these postulates and from some others, of which the evidence is too great for it to be useful to recount them here, Archimedes draws, by a method imitated from Euclid, a long series of propositions. Among these propositions, let us cite only the sixth and the seventh (7), which formulate the conditions of equilibrium of the straight lever; these propositions are the following:
Proposition VI. Commensurable magnitudes are in equilibrium between themselves when they are reciprocally proportional to the lengths at which these magnitudes are suspended.
Proposition VII. Incommensurable magnitudes are in equilibrium when these magnitudes are reciprocally proportional to the lengths at which these magnitudes are suspended.
These two propositions contain the properly mechanical consequences of the writing of Archimedes; the theorems which follow them and where the illustrious Syracusian determines the centers of gravity of diverse spaces are worth the meditations of the geometer, who admires the elegance and ingenuity of it, and of the algebrist, who discovers there the first integrations which have been made; but they do not offer to the mechanicist any new clarification on the questions which preoccupy him.
Archimedes has therefore arrived, in studying the equilibrium of weights, at the same point as Aristotle; but he has arrived there by a way entirely different. He has not drawn his principles from general laws of motion; he has made the edifice of his theory repose on some simple and certain laws relative to equilibrium. He has thus made out of the science of equilibrium an autonomous science, which owes nothing to the other branches of Physics; he has established Statics.
By that, he has assured to his doctrine a perfect clarity and an extreme rigor; but, it is quite necessary to recognize, this clarity and this rigor has been bought at the expense of generality and fecundity. These laws which rule the equilibrium of two weights suspended at the arms of a lever have been drawn from hypotheses special to this problem; when the mechanicist will have to treat another problem of equilibrium, distinct from that one, it will be necessary for him to invoke new hypotheses, heterogenous to the first, and the analysis of the first hypotheses will not give him any indication which can guide him in the choice of seconds. Thus, when Archimedes would like to study the equilibrium of floating bodies, he will be obliged to take recourse to principles without analogy with the requirements that he has formulated at the beginning of the Treatise Ἐπιπέδων ἰσορροπικῶν.
An admirable method of demonstration, the way followed by Archimedes in Mechanics is not a method of invention; the certitude and the clarity of his principles hold, in great part, to what they have collected, for thus to say, at the surface of phenomena and not at all uprooted from the very bottom of things; according to a phrase that Descartes (8) applies less justly to Galileo, Archimedes "explains full well quod ita sit, but not at all cur ita sit"; also we will see progress the most marked from Statics come forth well sooner from the doctrine of Aristotle than from the theories of Archimedes.
(1) Aristotle, Περὶ Οὐρανοῦ, Γ, β. Didot edition, Book II, p. 414.
(2) Aristotle, Mηχανικὰ Pροβλήματα, Δ. Didot edition, Book IV, p. 55.
(3) Aristotle, Mηχανικὰ Pροβλήματα, Α. Didot edition, Book IV, p. 55.
(4) In a certain epoch, it was the style to hold as nothing and having not happened the science of Aristotle and of his commentators; this prejudice sufficed to render incomprehensible several of the most important intellectual advancements; thus in the brief historical survey, elsewhere so beautiful, which opens Analytical Mechanics, Lagrange has written what follows on the subject of the Principle of virtual velocities: "As little as one examines the conditions of equilibrium in the lever and in other machines, it is easy to recognize this law, that the weight and the force are always in inverse ratio to the spaces that the one and the other can travel through in the same time. However it does not appear that the ancients had had familiarity with it. Guido Ubaldi is possibly the first who glimpsed it in the lever and in moving pulleys or blocks and tackles."
(5) Aristotle, Mηχανικὰ Pροβλήματα, B. Didot edition, Book IV, p. 55.
(6) Œvres d'Archimède, translated literally with a commentary, by F. Peyrard. Paris, 1807, p. 275.
(7) Loc. cit., pp. 280-282.
(8) Descartes, Lettre à Mersenne du 15 November 1638 (Œvres de Descartes, published by Ch. Adam and P. Tannery, Book II, p. 433).