Pierre Duhem

“ 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. ”

“ 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 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. ”

“ 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. ”

“ As this velocity, nothing when the plane is horizontal, grows at the same time as the angle of inclination of the plane, Cardan believed himself able to state the following proposition: The weight of a sphere which
descends an inclined plane is to the weight of the same sphere falling in free fall as the angle of the plane inclined from the horizontal plane is to the right angle. ”

“ If therefore one suspends a given weight in some point from a solid moving around a horizontal axis, the motive power of this weight will be so much the greater as the point of suspension lowers itself more rapidly by the effect of a given rotation, imparted to the support; hence, the motive power will be so much the greater as the point of suspension is more distant from the vertical plane containing the axis. ”

“ 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 ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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 ”

“ This is the object of the article entitled, in the De Subtilitate (18) , "Staterae ratio" and that his translator Richard Le Blanc designates in these terms: "The manner of the lever vulgarly called in Paris a sleigh, of what customarily the weavers make use, in Latin Statera (19) ."
Cardan made his analysis repose on two propositions taken for axioms. He admits, in the first place, that one segment AB' (fig. 12) , equal to the small arm AB of the beam and taken on the great arm, makes equilibrium with small arm AB ”