Pierre Duhem

“ One would imagine with difficulty that Cardan, so avid to be familiar with the windfall of Tartaglia and so prompt, despite his swearing, to adorn his book of Algebra from it, would not have suffered from the curiosity to be familiar with the thoughts of Leonardo da Vinci on Mechanics and Physics and, having been familiar with them, that he would have resisted the temptation of gleaning from them some or others in order to nourish his own meditations. ”

“ The study of the origins of Statics has conducted us thus to a conclusion; in proportion as we have pushed our historical researches earlier and in directions more varied, this conclusion has imposed itself to our spirit with an increasing force; also we dared to formulate it in its full generality: the mechanical and physical science of which we are well within our rights proud in modern times flows, by an uninterrupted series of scarcely sensible improvements, from the doctrines professed in the heart of the schools of the Middle Ages ”

“ If it is a mechanical problem which often presented itself to the meditations of the great painter, this is assuredly the study of the weight of a heavy object which slides on an inclined plane; one cannot leaf through his manuscripts without encountering at each instant, with variant particulars, a same design: on one pulley, a cord is stretched by two weights which slide on two planes unequally inclined. ”

“ The idea of which the first germ was found in the treatise of Jordanus had grown, following a continuous development, through the writings of the disciples of Jordanus, of Leonardo da Vinci, of Cardan, of Roberval, of Descartes, of Wallis, to attain its form achieved in the letter of Jean Bernoulli to Varignon, in the Analytical Mechanics of Lagrange, in the work of Willard Gibbs. The Science of which we are today so legitimately proud derived, by an evolution which has been given to us to mark the gradual phases, from the Science which was born about the year 1200. ”

“ 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 ”

“ If a force moves a body some time, in some space, it is not necessary that one such power moves a double weight, in a double time, two times this space; because it would be possible that such a force would not be able to move this moving body. Fifth: If a force moves a body so much time, in so much space, it is not necessary that half of this force moves this same moving body in the same time half of one such space, for possibly the force would not be able to move the body. These restrictions express the impossibility of certain motions not to clash with the axiom of Aristotle ”

“ We will speak first of the heat of the sun, which impresses itself in the concave mirror and is reflected from it in pyramidal figure, a pyramid which acquires proportionally as much more power as it compresses itself more. ”

“ 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. ”

“ He supposes, at length, that there is a proportionality between the speed of the descent and the distance from the center of gravity to the vertical of the point of support; from there, he draws without trouble this conclusion: the speed with which a sphere falls on an inclined plane is to its speed in free fall in the same proportion as the height of fall to the the length of the line of greater slope described by the body in motion. Elsewhere, for Leonardo da Vinci as for Aristotle, the intensity of a mechanical action is proportional to the speed that it communicates to a given moving body ”

“ Cardan disdained not at all to exercise his talent for geometry in some demonstrations constructed in the manner of Archimedes and to fill certain lacunae that the illustrious Syracusian had left gaping. Thus Archimedes had always neglected the weight of the lever or of the balance beam to which he was suspending the loads from which he would study the equilibrium; Cardan had in view determining the mechanical properties of a horizontal, homogenous balance beam suspended by any whatsoever of its points. ”

“ It is not at all solely by the doctrines of the School of Jordanus that the Mechanics of the Middle Ages has contributed to the formation of modern Mechanics. In the middle of the XIVth century, one of the doctors who was bringing the most honor to the brilliant nominalist School of the Sorbonne, Albert of Saxony, inaugurated a theory of the center of gravity which was obliging to have the greatest credit and the most durable influence. ”

“ 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. ”

“ As one will see it in Chapter IV, Leonardo da Vinci and Cardan do not agree less than exactly in what touches the impossibility of perpetual motion.
The harmony between them is perfect on the subject of the principles of Dynamics; and it is so much the more significant that their opinions on diverse questions of Dynamics have a form very peculiar that one scarcely finds in the works of their predecessors or their contemporaries. ”

“ 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. ”

“ That is to say that if a wheel moves a machine, it is impossible that without employing it one time more times, the wheel moves two of it; therefore the wheel does as much work in an hour as two machines with a second hour; thus the same wheel can make an infinite number of machines turn; but, with a very long time, the machines will not do more work than the first in one hour." A given weight, falling from a given height, produces therefore a mechanical effect of which the magnitude is independent of the circumstances which accompany this fall ”

“ His respect for the intellectual richesse of others does not go as far as scruple; he does not blush to weight the baggage of his own discoveries by slipping into it some borrowings made from the science of his contemporaries. ”

“ I say that the weight of the heavy body a b shares its gravity in two aspects, that is to say according to the line b c and according to the line n m; why and how much the weight is greater for one than for the other aspect and which obliquity is the one which shares the two weights in equal parts, will be said in the book On weights. ”

“ But for what the proportion of times is to forces and powers, the child will draw by two rolls four times more slowly, by three rolls six times more slowly than he does not draw but would lift from a cord by the same force, thus a little greater, being above, and more slowly more than six times or four times, so much as the length of the cord adds more to the load; therefore it arrives that the child scarcely in one hour will draw and lift the same load by such tackle-block, what a man six times more robust, being above, can lift immediately from a single cord. ”

“ But, from the time of Leonardo and of Cardan down to Descartes and Torricelli, we had been able to follow the development of Statics without the passage of this development having seemed to us essentially different from those that one would attribute commonly to it. ”

“ At the extremity of a lever, one can cause to act a force of which the direction is different from the vertical; it will suffice
to employ a cord extended in this direction, passing over a pulley and pulled then by a weight. A rule similar to the preceding will help to evaluate the motive power of a similar engine ”

“ I say that the loads set at C will be more powerful than if the balance were being set in some other place, for instance than if it were being set at F. Now, in order that we know that C is heavier in such a situation than in F, it is necessary that it be moved in equal time through greater space towards the center [of the world] . For we see that the heaviest things, all things being equal, are carried more lightly [rapidly] to the center. Now that this comes to more from the load and from the balance rather placed at C than at F, I show by two reasons. ”

“ Let there be a sphere a of weight g (fig. 14) , placed at point b, that one can pull on the inclined plane bc, bf being the vertical plane. On the horizontal plane be, a can be moved by a force as small as one wants, according to what has been said above; in consequence, according to the common opinion, the force which will move a following eb will be nil; on the other hand, according to what has been said, a will be moved towards f by a force always constant and equal to g, in the direction bc by constant force equal to k, in the direction bd lastly by a constant force equal to h ”

“ If someone descends (17) step by step in making a jump from the one step to the other, and that you add all the powers of the percussions and weights of such jumps, you will find that the powers are equal to the totality of the percussion and of the weight that one such man would give falling, by perpendicular line, from the head to the feet of said steps. ”

“ Leonardo da Vinci had not applied in detail the axiom of Aristotle but to the lever and to tackle-blocks; in what concerns the screw, he was contented with this brief indication (27) : "The more a force extends itself out of a wheel across the wheel, out of a lever along the lever or out of a screw across the screw, the more the force is powerful and slow."
This indication Cardan develops (28) under this title: The manner of pulling and of pushing all things with little force. ”

“ Let us suppose that one possesses a cause of motion well-determined: for example, a quantity of water, immobile in a reservoir, expecting that one allows it to fall, from a given height, in a smaller canal. ”

“ If, for example, a weight, hung by a cord at the extremity of a lever arm, oscillates in the manner of a pendulum, it will not weigh at each instant on the lever but by the vertical component of its weight ”

“ So much the more therefore will there be curvings in the screw, and so much more will be low, that is to say nearer to the circumference or beyond, so much the more the weight will be light and the movement easy; and so much the more the movement will be easy, so much the more will be slow. The screw therefore is able to be two cubits from these curvings to such a degree large and low, that the weight will be easily lifted by a child of ten years. ”

“ And thus the intention of the load is to be carried in right lines to the center; but for that which it is prevented by ligature, it is moved as it is able."
Thus when a weight descends following the vertical, the power of motion of this weight is, as Aristotle would want it, measured by the velocity with which it falls; but, by the agency of the mechanism which carries it, by the nature of the links [liaisons] , according to the word employed by Cardan and reprised by modern Mechanics, it can happen that the weight does not move itself according to the vertical ”

“ And a force will be more feeble than another by as much as the time between the one from the other." These principles render account very easily of the properties of tackle-blocks; Leonardo da Vinci exposits with the greatest exactitude the properties of these mechanisms. ”

“ He recognized that the weight which pulls the moving body towards the center of the earth could be decomposed into two forces, the one normal to the inclined plane upon which the weight slides, the other tangent to this plane ”

“ This solution was possibly not so arduous to obtain that it were meriting this song of triumph; nevertheless, it had, on the reasonings of the successors of Cardan, an undoubtable influence. Abandoning the requirements that Archimedes had placed at the base of his reasonings on the equilibrium of the lever, Simon Stevin on one part, Galileo on another part, will bring back the study of the lever to the consideration of a homogenous weighted bar, suspended in its middle, and that by means of the same axioms that Cardan has proposed. ”

“ In 1551, Cardan published his twenty-one books On Subtlety (7) ; a second Latin edition (8) of this work, more complete than the the first, appeared as early as 1554 and, in 1556, was translated into French by Richard le Blanc (9) ; the French or Latin editions of this work succeeded themselves, numerously, during the second half of the XVIth century (10) . To this writing, Cardan joined later his Opus novum de proportionibus (11) . All the Mechanics contained in these two works carry, still recognizable, the mark of Leonardo. ”