Opus Novum de Proportionibus, Numerorum, Motuum, Ponderum, Sonorum … Praeterea, Artis Magnae, sive De Regulis Algebraicis … Item, De Aliza Regula …

Basel: Henric Petri, 1570.

First edition of Cardano’s mathematical treatment of mechanics, the Opus novum, together with the revised, second edition of his Ars Magna (first 1545) – the greatest work of 16th century algebra – and with the first edition of a supplement to that work, the De Aliza Regula, in which Cardano was the first to make use of imaginary numbers. The Opus novum was the first significant work to examine mechanics from a largely mathematical basis. Here he sets out to determine what effect different densities might have on missile trajectories; explores the connection between medical efficacy of drugs and their dosage (a geometric or arithmetic relationship?); and formulates the earliest estimate of the relative densities of air and water. He is thus described by the DSB as being one of the earliest to “apply quantitative methods to the study of physics”. “His use of the concept of moment of a force in his study of the conditions of equilibrium in balance and his attempt to determine experimentally the relation between the densities of air and water are noteworthy. The value that he obtained, 1:50, is rough; but it is the first deduction to be based on the experimental method and on the hypothesis that the ratio of the distances traveled by bullets shot from the same ballistic instrument, through air and through water, is the inverse of the ratio between the densities of air and water.” Cardano’s major work “was the Ars magna, in which many new ideas in algebra were systematically presented. Among them are the rule, today called ‘Cardano’s rule,’ for solving reduced third-degree equations (i.e., they lack the second-degree term); the linear transformations that eliminate the second-degree term in a complete cubic equation (which Tartaglia did not know how to solve); the observation that an equation of a degree higher than the first admits more than a single root; the lowering of the degree of an equation when one of its roots is known; and the solution, applied to many problems, of the quartic equation, attributed by Cardano to his disciple and son-in-law, Ludovico Ferrari. Notable also was Cardano’s research into approximate solutions of a numerical equation by the method of proportional parts and the observation that, with repeated operations, one could obtain roots always closer to the true ones. Before Cardano, only the solution of an equation was sought. Cardano, however, also observed the relations between the roots and the coefficients of the equation and between the succession of the signs of the terms and the signs of the roots; thus he is justly considered the originator of the theory of algebraic equations. Although in some cases he used imaginary numbers, overcoming the reluctance of contemporary mathematicians to use them, it was only in 1570, in a new edition of the Ars magna, that he added a section entitled De aliza regula (the meaning of aliza is unknown; some say it means ‘difficult’), devoted to the “irreducible case” of the cubic equation, in which Cardano’s rule is extended to imaginary numbers. This was a recondite work that did not give solutions to the irreducible case, but it was still important for the algebraic transformations which it employed and for the presentation of the solutions of at least three important problems” (DSB). The first edition of the Ars magna is very rare in commerce.

Provenance: Königliche Handbibliothek, Tübingen (ink stamp on title).

“In the Opus novum de proportionibus … of 1570 there is much of interest and ingenuity. Some questions of statics are taken up with great insight but the novelty of the work lies in its discussions of problems of motion: the possibility of unifying statics with dynamics or at least of mathematically connecting the two disciplines seems to have captured Cardano’s imagination …

“Cardano’s account of acceleration is wholly Aristotelian; he remarks that in natural motion the body has an appetite to approach some end, whence the end must be good, and therefore the body hastens as it approaches the end. He holds that since the medium is divided and driven aside beneath a falling body, it must force upward with it the neighbouring parts of the medium. Those parts then press in above the body to prevent the formation of a vacuum and in so doing they press down on the body and speed its motion. To this concept of antiperistasis he then adds that in both violent and natural motions there is an increase in speed at least up to some point, by which he explains the need in war machines for space through which to act in order to increase the violence of their projectiles. In one proposition, acceleration is linked to time, but Cardano’s reasoning for this depends again on antiperistasis.

“In discussing the motion of projectiles, Cardano asserts that motion in some part of the horizontal (initial) path is uniform, and he says that, as the path turns downward at the end of that part, the projectile is slowed; hence he believes that it will reach the ground later than it would have reached the corresponding point on the initial horizontal line. This idea is consistent with his argument elsewhere that there is always a conflict between motions of different kinds, rather than a simple composition. But despite its overall Aristotelian orthodoxy, Cardano’s discussion of projectile motion is of interest because of his clearly expressed view of speed as a ratio of space to time. This concept, inspired by Cardano’s algebraic approach to mathematics, was never grasped by Galileo or his contemporaries.

“Still more striking is Cardano’s classification of motion into three kinds rather than two: natural, violent, and ‘voluntary.’ Voluntary motion is exemplified by circulation of the celestial spheres around the centre of the universe; other circular motions, for Cardano, are either violent or mixed motions. In voluntary motions, the body as a whole remains in one place. Cardano considers such motions to be uniform and to be simpler than other motions. This discussion by Cardano is a probable source of Galileo’s reflection, added as a note to his De motu, concerning neutral motions. That reflection led ultimately to the inertial concept. But Cardano remained faithful throughout his works to impetus theory – possibly an example of the well-known importance of terminology in science, for the word ‘voluntary’ has animistic implications, whereas ‘neutral’ suggest indifference to motion.

“Cardano’s impetus discussions led him to some potentially fruitful reflections concerning weight and speed, and, though these turned out to be tautologous rather than physical, they may have been turned to good account by later writers who were able to discard the Euclidean theory of proportions and apply algebra to physical concepts

“Another valuable and probably original concept of Cardano’s is that of concealed motion in a resting weight. That concept he repeatedly applied, not only in a sense in which it adumbrates a sort of potential energy, but also in an action-reaction sense similar to that of Galileo’s De motu. Cardano was aware that a sphere on a horizontal plane could be moved by any force sufficient to divide the surrounding air. Like Nicholas of Cusa, he limited this argument to the sphere, whereas Galileo later extended it to horizontal motions of any body. Yet Cardano’s proposition is linked in another way to Galileo’s, for both men use the idea of constant distance of the moving body from the centre of the world …

“Cardano’s treatment of the inclined plane is curious. Accepting neither the correct theorem of the Jordanus tradition, nor the incorrect theorem of Pappus, he offers a proof that the effective weight of a body on any inclined plane is proportional to the ratio of the angle of the plane to a right angle. It is interesting that in the course of this proof he remarks that it is a matter of common knowledge that no (appreciable) force is required to move a body horizontally. Yet in none of these views is he truly consistent, for elsewhere (using the argument that seems to assume speed on an inclined plane is proportional to effective weight on that plane) he declares that the speed is not proportional to the angle, but increases more rapidly than the angle. And in several propositions he discusses the force needed to draw or push a body along the horizontal, relating this to the shape of the body and the position of the applied force. Still more curious is the proposition immediately preceding that of the inclined plane, a proposition that Duhem takes to refer to a screw-jack, though the diagram is hard to reconcile with that. But whatever its application, the proposition certainly relates the power used in raising a weight to the length of the path over which it is moved, a correct relation which is ignored in the inclined plane proposition that follows.

“Cardano offers many propositions concerning the speed of fall. I shall mention the last of these first, as it is the best known, most interesting, and most puzzling all. In it he seems to assert that all spheres of the same material falling from the same place through air will reach the horizontal plane at the same time. His argument is difficult to follow … but the gist seems to be this. If one sphere is triple the other, then their weights are as 27 to 1, the volumes of the cylinders of air beneath them are as 9 to 1, and the density of that air is as 3 to 1. The greater impetus of the larger sphere is thus able to drive away nine times as much air three times as dense as the smaller sphere needs to do, for which tasks the respective weights are exactly sufficient. The only difference in the time of their reaching the plane, he concludes, results from the difference of their diameters, and the same is said to hold true for fall through water.

“This curious argument may have been an isolated reflection of Cardano’s based on the fourth part of [Jordanus’] De ratione ponderis, or an attempt on his part to find an Aristotelian explanation in terms of density of medium for the observed fact of equal speed of fall …

“With regard to fall in different media, Cardano asserted that the weights of two bodies falling in the same time through the same interval will be as the squares of the rarities (that is, inversely of the squares of the densities) of the two bodies. As a scholium he adds that the argument does not apply to media as widely different as air and water, ‘for a ball of wood weighing 100 pounds no more descends in water than a wooden ball of one pound.’ Cardano attempted to determine the relative densities of air and water by the speeds of descent of the same body in both, obtaining an estimate of 50 to 1. It is noteworthy that Cardano did not use the Archimedean method of weighing alloys in air and water; for that he substituted the clumsier method of weighing water, metal, and container together in the required combinations.

“Cardano may have been the first writer on mechanics to attempt a discussion of impact. In some propositions on percussion, he multiplied weight by impetus, which he did not associate directly with velocity. Impetus, speed, force, and motion were left undefined by Cardano, and he often substituted one for another without apparent system. He gave several propositions on impacts between boats of different sizes and loads, and others regarding sails and wind directions, all suggested by [Aristotle’s] Questions of Mechanics” (Drake & Drabkin, pp. 27-31).

“The Ars Magna has always been highly praised as a milestone in the history of mathematics … Nowadays, the Ars Magna would be characterized as a text on algebraic equations. To Cardano’s contemporaries it was a breakthrough in the field of mathematics, exhibiting publicly for the first time the principles for solving both cubic and biquadratic equations, giving the roots by expressions formed by radicals [square and cube roots], in a manner similar to the method which had been known for equations of the second degree since the Greeks, or even the Babylonians. Cardano actually does not claim either of the two innovations entirely as his own; he rather considers the special third degree equation first solved by Scipione del Ferro, and the fourth degree equation solved by Lodovico Ferrari, Cardano’s secretary, as toe-holds which enable him to create his own general theory embracing all possible cases.

“These many cases, produced largely by the necessity for separating the arguments for positive and negative numbers and by the lack of an efficient algebraic notation, lead to elaborate lists of equation types. Cardano studies what he considered to be properties of general equations, for instance, relations between roots and coefficients, rules for the signs or the locations of the roots. He barely touches upon the numerical solution of equations, but here he brings little new. One notable aspect of Cardano’s discussion is the clear realization of the existence of imaginary or complex solutions. They appear as necessary consequences of the formulas, and he does not avoid them or brush them aside as unimportant, as often done by earlier writers. On the contrary, Cardano constructs examples for the express purpose of dealing with problems with imaginary roots (Chapter 37)” (Witmer, pp. vii-viii).

There is a deep, substantial difference between the quadratic and the cubic formulae: while the quadratic formula only involves imaginary numbers when all the solutions are imaginary too, it may happen that the cubic formula contains imaginary numbers, even when the three solutions are all real (and different). This means that a scholar of the time could stumble upon numerical cubic equations of which he already knew three (real) solutions, while Cardano’s formula for the roots contains square roots of negative numbers. This was later called the ‘casus irreducibilis’. Cardano’s De Regula Aliza is, at least in part, meant to try to overcome the problem entailed by it. He is not successful, however, which is only to be expected as it has been known since the work of Galois in the early nineteenth century that the appearance of imaginaries in the ‘casus irreducibilis’ is inevitable.

Adams C689; Wightman 135; Riccardi I 256 9;
Smith Rara 338f. Drake & Drabkin, Mechanics in Sixteenth-Century Italy, 1969. Witmer (tr. & ed.), Ars Magna or The Rules of Algebra, 1968.

Three parts in one vol., folio (299 x 191 mm), pp. [xvi], 271, [1, blank]; 163, [1, blank]; [viii], 111, [1], with many woodcut diagrams in text. Eighteenth-century vellum, spine gilt with two red lettering-pieces (a little soiled, worming to the front joint, small piece missing from the lower label).

Item #5904

Price: $15,000.00

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