2 Guidobaldo del Monte and Renaissance Mechanics

To historians of mechanics, Guidobaldo del Monte presents something of a paradox. On the one hand, he attempted to found mechanics on the strictest principles of abstract, Archimedean statics. On the other, he insisted that mechanics was not a purely abstract, mathematical science, but rather was essentially concerned with actual machines. He vigorously criticized Tartaglia (among others) for vainly attempting to separate a mathematical from a physical mechanics, “as if mechanics could be considered apart from either geometrical demonstrations or actual motion.”¹ His practical interest in the workings of actual machines has been remarked on by a number of historians, including Alex Keller, Enrico Gamba, Gianni Micheli, and Mary Henninger-Voss; they and others, notably Paul Lawrence Rose, Stillman Drake, and Domenico Bertoloni Meli, have also called attention to Guidobaldo’s important role in the sixteenth-century Archimedean revival. These two features of Guidobaldo’s mechanics—the practical and the Archimedean—were perhaps his most significant contributions to the renaissance of mechanics in the sixteenth century.² But Guidobaldo has also come under considerable criticism from historians, both for his unreasonable demands for an excessive mathematical precision in mechanics, and for his failure to include principles of motion and dynamics. He is notorious, for example, for trying to take account of the convergence of the arms of the balance to the center of the earth, a convergence that is immeasurably small even in the largest balances. For this reason Pierre Duhem dismissed him as a narrow-minded geometer, whose “exaggerated regard for deductive rigour” (le souci exagéré de la rigueur déductive) and his “uncritical admiration of the Ancients” (l’admiration exclusive de anciens) blinded him to the promising results reached through more intuitive reasoning by Jordanus de Nemore, Girolamo Cardano, and Niccolò Tartaglia.³ More recently, Stillman Drake repeated Duhem’s criticisms, but specified that what Guidobaldo had missed in Jordanus and Tartaglia was the general principle that the products of force and virtual displacement are equal for systems in equilibrium. According to Drake, this was because Guidobaldo had insisted that a greater power was necessary to produce motion than equilibrium; and Guidobaldo had excluded all dynamical concepts such as work and virtual velocity from mechanics because he held that Archimedean statics had superseded the dynamical approach of the pseudo-Aristotelian Mechanical Problems (Drake and Drabkin 1969, 48). Paul Lawrence Rose went even further, to assert that for Guidobaldo, statics and dynamics were “two entirely separate sciences without common principles” (Rose 1975, 232; see also 233, 234–235, note 2). According to Rose, Guidobaldo

despaired [...] of there ever being a mathematical science of dynamics and himself erected unbridgeable barriers between dynamics and mathematical statics. (Rose 1975, 229)

But as Maarten Van Dyck has shown in a recent article, Guidobaldo was not so slavish a follower of the ancients, nor so blinded by a concern for mathematical rigour, as Duhem and others have thought. Guidobaldo did take account of the convergence of the ends of the balance, but only to refute the mechanical principles of Jordanus and Tartaglia, which was necessary to defend the coherence of what he saw as the sovereign principle of mechanics, the equilibrium of centers of gravity, understood within an earth-centered Aristotelian cosmos. Van Dyck has thus restored to Guidobaldo’s criticisms of Jordanus and Tartaglia their original purpose and intent (Van Dyck 2006). In a similar way, I should like to show how Guidobaldo’s so-called failure to include in his mechanics dynamical principles such as virtual velocities was the natural result of his adoption of the equilibrium of centers of gravity as its foundational principle. In other words, I should like to recover the original scope and intent of Guidobaldo’s mechanics from the expectations imposed upon it by historians looking back through later developments.

Given that Guidobaldo had adopted the equilibrium of centers of gravity as the sovereign principle in mechanics, how did he attempt to apply it to the actual motion of real machines? And how did his choice of this principle determine the nature and scope of the mechanics that followed from it? To answer these questions I shall first look at the application of the principle to the simple machines in Guidobaldo’s Mechanicorum liber (1577), his major mechanical work. His other published work on mechanics, the Paraphrase of Archimedes’s On Plane Equilibrium (1588), concerns the establishment and mathematical applications of this principle, and so has little to add concerning its mechanical applications, though its Preface contains some interesting comments on mechanics. But in addition to these two printed works, Guidobaldo also made a number of notes on mechanical matters that form part of his unpublished Meditatiunculae de rebus mathematicis (musings on mathematical topics), the manuscript of which Guglielmo Libri discovered in the Bibliothèque Nationale, in Paris, and from which he printed a few extracts in 1840.⁴ These notes on mechanics, at least some of which were written after the publication of the Mechanicorum liber, include both an attempt to recast the pseudo-Aristotelian Mechanica in an Archimedean mold, and Guidobaldo’s own treatment of the inclined plane. My argument will be that, because the principle of the equilibrium of the balance is Guidobaldo’s fundamental principle of mechanics, mechanical motions for him are fundamentally disequilibriums; this means that while equilibrium is a determinate state and thus subject to mathematical exactitude, disequilibrium produces motion, which is thus indeterminate and subject to unavoidable and unaccountable material disturbances. This explains, I think, both the source of his criticisms of Jordanus and Tartaglia, and his apparent neglect of motion and dynamics in his mechanics.

But before I turn to the Mechanicorum liber, I should like to sketch briefly the state of mechanics before Guidobaldo. The work that set the scope and program of mechanics and gave the first definitive content to the nascent science in the sixteenth century was the pseudo-Aristotelian Mechanica (or Quaestiones mechanicae). In its introduction, the Mechanica reduced mechanical marvels to the balance and ultimately to the marvelous properties of the circle. Analyzing the movement of the ends of the balance into a natural and a violent or preternatural component, it argued that a power is swifter and thus more effective the greater its natural component over its violent. For this reason a weight or a power is more effective the longer the arm of the balance, since the longer arm partakes more of the natural than the violent movement. This principle of circular movement was then applied to a number of questions, the first few concerning the balance and the lever, later ones taking up the wheel, the wedge, pliers, and the like, including a number of questions on topics such as the motion of heavy bodies, projectile motion, and whirlpools that have little or nothing to do with the principle of circular movement. The Mechanica was translated into Latin early in the sixteenth century and was the subject of several commentaries and paraphrases by mid-century, including an influential paraphrase and commentary by Alessandro Piccolomini. It was lectured on at the University of Padua by Pietro Catena in the 1560s, by Giuseppe Moletti in the 1580s, and by Galileo in the 1590s.⁵

At the same time as the pseudo-Aristotelian Mechanica was becoming more widely known, the medieval science of weights (scientia de ponderibus), represented especially by the works attributed to Jordanus de Nemore, was reintroduced in the sixteenth century, first by Peter Apian’s printing of the Liber de ponderibus in 1533, and then by Tartaglia’s printing of the magisterial De ratione ponderis in 1565.⁶ For Jordanus, the swiftness and thus the effectiveness of a weight depended on the directness or obliquity of its motion, where motion on the circumference of a larger circle is more direct than motion on a smaller. Significantly, Jordanus recognized that the speeds and the distances of the motions of weights were to be measured along their vertical descents, which led him to the correct solution of the inclined plane and historians to the conclusion that he was, in effect, appealing to the principle of virtual work. For Niccolò Tartaglia, the science of weights from Jordanus provided the principles of the mechanics found in the pseudo-Aristotelian Mechanica. Book 7 of Tartaglia’s Quesiti et inventioni diverse (Diverse Questions and Inventions, 1546) was thus devoted to a discussion of the Mechanica, while Book 8 established its principles using the science of weights.⁷ To the pseudo-Aristotelian Mechanica and the medieval science of weights, a third tradition in mechanics was added. By the mid-sixteenth century, the works of Archimedes were already being edited, translated, and assimilated into mathematics, notably by Francesco Maurolico in Messina and Federico Commandino in Urbino. Lacking Archimedes’s text, Maurolico (by his own account) reconstructed On the Equilibrium of Planes in his brilliant De momentis aequalibus (On Equal Moments); like Guidobaldo, as we shall see, Maurolico saw equilibrium as providing the foundation for mechanics, which he developed as a commentary on and an extension of the pseudo-Aristotelian Mechanica, although neither of his works on mechanics was to be published until the next century.⁸ Guidobaldo, on the other hand, knew the Equilibrium of Planes in the translation published by Federico Commandino in 1565. But the greatest difference between Maurolico and Guidobaldo was in the scope and content of mechanics: where Maurolico included in mechanics more or less everything in the pseudo-Aristotelian Mechanica, Guidobaldo restricted it to Heron’s five simple powers or machines, an account of which he had found in Commandino’s translation of the Mathematical Collection of Pappus of Alexandria.⁹ And from Pappus Guidobaldo also adopted Heron’s general challenge for mechanics: to move a given weight with a given power using a machine. Guidobaldo’s purpose in writing the Mechanicorum liber, then, was to demonstrate the principle of equilibrium of centers of gravity, exposing the errors of those like Tartaglia who relied on the science of weights, and then to apply this principle in turn to explain each of the five simple machines in order to answer Heron’s challenge.

The Mechanicorum liber thus has six parts or treatises, the first devoted to the demonstration of the principle of equilibrium of the balance, the subsequent five to the lever, pulley, wheel and axel, wedge, and screw. The first part has already been treated elsewhere in detail by Vico Montebelli (1988) and by Maarten Van Dyck (2006). To their accounts I should like to add only that the source and foundation of Guidobaldo’s criticism of Jordanus and Tartaglia seems to have been that, in attributing mechanical effects to the swifter or slower speeds of more or less direct or oblique motions, they had mistaken effects for causes. For according to Guidobaldo, they simply could not demonstrate that a weight on the end of beam is heavier when the beam is horizontal than at any other position, since its straighter or swifter movement at the horizontal position is merely a sign (i.e., a result) rather than a cause; nor do they prove that the weight is heavier by its being at that place, but only by its departing from that place.¹⁰ For the true foundation of mechanics is not direct or oblique motion, according to Guidobaldo, but Archimedes’s principle of the equilibrium of centers of gravity. In the Dedicatory Letter of the Mechanicorum liber, Guidobaldo stated that in Archimedes’s Equilibrium of Planes “all the theories of mechanics are gathered as in an abundant store.”¹¹ And in the Preface to his later Paraphase of On the Equilibrium of Planes, he wrote that “the whole of mechanics depends on this sole and foremost foundation,” that is, on the principle that in equilibrium the weights are inversely as the distances.¹²

That speed and motion are results, not the causes, of equilibrium and disequilibrium Guidobaldo states explicitly in the corollary to Proposition 6 of the first treatise, De libra (On the Balance). In Proposition 5 Guidobaldo had proved the central theorem of equilibrium, that weights are in equilibrium when their distances from the center are inversely as their weights. In Proposition 6 he then proves that equal weights weigh in proportion to their distances from the center. And from this follows the corollary that, since the farther a weight is from the center of the balance the heavier it will be, so its motion will be the swifter. Relegated to a corollary, speed and motion are thus the results, not the causes, of greater or lesser heaviness.¹³

Having established in these first propositions the principle of the equilibrium of the balance, Guidobaldo then applied it in turn to each of Heron’s five simple powers or machines—the lever, pulley, wheel and axel, wedge, and screw. In each case, he used the principle to find the power needed to sustain the load in equilibrium; he then assumed that actually to move the load would require a somewhat greater power. In the case of the lever, he stated this as follows:

For the space of the power has the same ratio to the space of the weight as that of the weight to the power which sustains the same weight. But the power that sustains is less than the power that moves; therefore the weight will have a lesser ratio to the power that moves it than to the power that sustains it. Therefore the ratio of the space of the power that moves to the space of the weight will be greater than that of the weight to the power.¹⁴

The conditions of equilibrium having been established, motion is produced only by the addition of some indefinite amount of power. Notice that Guidobaldo has no aversion to comparing the spaces moved by powers and weights, in exactly the way that Jordanus and Tartaglia did. But for Guidobaldo, these spaces and motions are the results of the disequilibrium caused by an indefinite increase of power; they are not themselves the causes.

In the treatises on the pulley, on the wheel and axel, and on the screw, Guidobaldo also introduced the time taken to move the weight and its speed, noting that the more easily a power can move a weight, the more slowly it does so.¹⁵ He thus fully understood the central principle of Galileo’s mechanics, but with this crucial difference: he saw it as an effect, rather than as a cause.¹⁶

In his treatises on the wedge and on the screw, Guidobaldo cited Pappus’s theorem on the inclined plane, since both the wedge and the screw can be reduced to inclined planes, and Pappus reduced the inclined plane to the lever and thence to the balance. But significantly, Guidobaldo did not present Pappus’s theorem, although Pigafetta added it in the commentary to his Italian translation; it is possible that Guidobaldo was not happy with Pappus’s proof, and that he omitted it from his Latin text for this reason.¹⁷ Pappus had assumed that a certain power C is needed to move a sphere of weight A on a horizontal plane, and then undertook to determine the power needed to sustain the same weight on an inclined plane (see Figure 2.1). This he did by placing one end of a horizontal balance at the center E of the sphere, the other on its circumference at G, and the fulcrum F directly above the point of contact L between the sphere and the plane. The weight of the sphere acts at its center E, so that the balance is in equilibrium and the sphere is at rest on the plane when this weight is counterbalanced by a second weight B applied at G such that the ratio of weights is the inverse ratio of their distances EF and FG from the fulcrum. Next, he assumed that the ratio of a power D, needed to move the second weight B on a horizontal plane, to the power C, needed to move the original weight A on a horizontal plane, will be equal to the ratio of the weights B to A. The power necessary to move the sphere up the plane, then, is the sum of these two powers C and D.¹⁸

In the Meditatiunculae, Guidobaldo sketched his own version of Pappus’s theorem on the inclined plane, although with several significant differences. Like Pappus, Guidobaldo reduced the inclined plane to the lever, though he extended his proof to include all three classes of levers (see Figures 2.2 a, b and c). Now, because he was interested only in finding the sustaining power, not the moving power, he does not assume, as did, that a certain power is needed to move the weight on a horizontal plane. Instead, he simply finds the ratio of the sustaining power to the weight of the body as Pappus did, by placing them on the unequal arms of the balance. And he completely omits Pappus’s crucial last step, of finding the power of moving the second weight on a horizontal plane and adding it to the power of moving the original weight. This means that the sustaining power actually has to be equal to or greater than the weight of the sphere itself for planes that place the fulcrum more than half way from the center of the sphere, and that on a vertical plane it becomes infinite. Pappus’s theorem at least gives an intuitive approximation of how much power is needed to move a body up an inclined plane, though it too implies that the power needed to move the body up a vertical plane is infinite; Guidobaldo’s theorem produces paradoxical results for all but the shallowest planes. His note at the end, that, if the fulcrum is located directly above or below the center of the sphere, the sustaining power should be equal to the weight of the sphere, suggests that he was aware of this consequence and wanted to correct it (see Figure 2.2d).¹⁹

Despite his general emphasis on equilibrium, motion and its effects do find their way into his mechanics, notably in his treatment of the wedge, but only as secondary causes. In the Mechanicorum liber, after attempting to reduce the wedge to a pair of levers and thus account for its effectiveness, Guidobaldo adds the power of the blow striking it as another explanation. The power of the blow, he explains, depends both on the weight of the hammer and the distance through which the hammer moves, which is greater the longer the handle. The longer the handle, then, effectively the heavier the hammer and so the stronger the impulse of the blow. So far these effects can be seen to arise from the properties of the lever and thus the balance. But then he adds that the effectiveness of the wedge also arises in part from the very strong force of percussion, citing Question 19 of the Mechanica, which in fact his explanation echoes. Here he has moved entirely away from equilibrium as the cause of a mechanical effect and invoked the unexplained power of percussion.²⁰

Each of the separate treatises on four of the five simple machines ends with Heron’s problem, that is, to find the conditions under which a given weight can be moved by a given power using each machine. In the case of the wedge, however, Heron’s program breaks down. With a wedge, according to Guidobaldo, any given power cannot move any given weight, since any given power cannot move any given weight by means of an inclined plane, though he does not explain exactly why. Further, since a wedge is in effect two opposing levers, as it splits the load the fulcrums of these levers themselves move and thus fail to maintain a constant ratio of load to power. In his general comment at the end of his translation of the Mechanicorum liber, Pigafetta explains that the wedge and the screw, unlike the other machines, are suitable only for moving weights, not for sustaining them; and,

Since the powers that move may be infinite [in number], one cannot give a firm rule for them as may be done for the power that sustains, which is unique and determined.²¹

In fact, this is true for all of the machines, for while the conditions for equilibrium in each case are determinate and subject to an exact mathematical rule, the conditions for motion are many and indeterminate and thus in principle are unknowable with any precision.

According to Guidobaldo, Archimedes had clarified the principles of mechanics by accepting the explanations in the pseudo-Aristotelian Mechanica for the power of the lever, but then went further to discover and to demonstrate the exact relation between weights and distances, which is the sole foundation of mechanics.²² In the Meditatiunculae, he in fact attempted to prove several of the questions from the Mechanica using Archimedean principles. The first two of these are headed Questiones Aristotelis de libra aliter demonstrate (Aristotle’s questions on the balance demonstrated in another way) and begin with a single supposition: centrum gravitatis deorsum tendere (the center of gravity tends downwards). In the two propositions that follow, Guidobaldo proves the stability of equilibrium of a balance supported from above, and the instability of one supported from below, by appealing to the position of the balance’s center of gravity; these proofs are in effect identical to those in the Mechanicorum liber. In the note that follows them, he criticises Alessandro Piccolomini’s Paraphrase in its Italian translation, and he refers to his own Mechanicorum liber of 1577, which shows that he was writing this after 1582, when the Italian Paraphrase was printed.²³ On the next two pages there follows a proposition effectively the same as Proposition 6 of the treatise on the balance of the Mechanicorum liber.²⁴ Some twenty pages later, Guidobaldo offered a fuller proof of Aristotle’s Question 1, why larger balances are more exact than smaller. This proof makes no appeal to centers of gravity, but relies entirely on considerations of motion. First he demonstrates as a lemma, citing the appropriate propositions from Euclid, that of two equal lines GB and HE dropped perpendicular from the diameter to the circumference, the line GB in the smaller circle has a smaller ratio to GA than the line HE in the larger to HD (see Figure 2.3). Then he applies this lemma to the balance by showing that when a longer and a shorter balance are deflected an equal vertical distance, in the motion of the longer arm there is a greater proportion of natural (vertical) motion to preternatural (radial) motion, so that the larger balance is easier to move and is moved more swiftly in the same time by the same power (see Figure 2.4).²⁵ What is significant about this proof is its appeal exclusively to motion and displacement rather than to centers of gravity.

Guidobaldo’s attempt to take into account the material resistance of real machines comes up in several notes inspired by questions in the Mechanica concerned with wheels. Question 11 of the Mechanica asked why weights are more easily moved on rollers than on wheels despite the fact that rollers are smaller in diameter than wheels; the answer there was because wheels are subject to friction at the axel. Pietro Catena, in his Universa loca of 1556—and presumably also in his now-lost lectures on the Mechanica, which Guidobaldo heard in Padua in 1564—had added to this “physical” explanation a “truly demonstrative” geometical proof supposedly showing that rollers, with their smaller diameter, make less contact with the ground than wheels do and so encounter less resistance to rolling. Guidobaldo came to the opposite conclusion: with the help of a geometrical lemma, he showed why it is in fact easier for a larger wheel to roll over an obstacle of the same size than for a smaller wheel (see Figure 2.5). Treating the obstacle effectively as an inclined plane, he reduced the problem to the lever, again invoking Pappus’s theorem on the inclined plane.²⁶

But in a variation of Question 9 of the Mechanica, Guidobaldo attempted to take account of the friction of the axel mentioned in Question 11. He asks why weights are in practice more easily moved with larger wheels, meaning in this case on a windlass. He imagines two equal weights suspended from A and C on the circumferences of two unequal wheels concentric around center G (see Figure 2.6). Since the weights at A and C are equal, the powers needed to sustain them in equilibrium at D and B will also be equal. But to move the weights, additional power must be added at D and B, since the axel resists motion because of contact and friction, which Guidobaldo represents as a load applied at E. Since the ratio of BG to FG is greater than the ratio of DG to FG, less power must be added at B to overcome this resistance than at D. Thus weights are moved more easily with larger wheels.²⁷

These fragments are apparently all that he wrote, or all that survive, in his attempt to reduce the pseudo-Aristotelian Mechanica to Archimedean principles, and they are, at best, a mixed success. But they show several important features of his approach to mechanics: they show his general determination to bring mechanical effects under Archimedean principles (though on occasion he resorted to motion and speed), and they show how he tried to take into account the material resistance of real machines. And the material resistance of real machines lies at the heart of Guidobaldo’s attempt to exclude motion from the causes and principles of mechanics. A letter he wrote to the mathematician Giacomo Contarini in 1580, the substance of which he repeated shortly afterwards in a letter to Filippo Pigafetta, the Italian translator of the Mechanicorum liber, offers a clue to this. Both Contarini and Pigafetta had raised doubts about theoretical results contained in the Mechanicorum liber, since they did not seem to conform to experience. In his reply to Contarini, Guidobaldo asserted that, if a balance in equilibrium fails to move when a slip of paper is added to one of its weights, it is not therefore inaccurate:

where one must consider that the resistance that the material makes is made when weights are to be moved and not when they are merely to be sustained, because then the machine neither moves nor turns.²⁸

Because resistance arises only when there is motion, according to Guidobaldo, a balance in equilibrium corresponds exactly to abstract mathematical theory; but to disturb that balance, to set it into motion, is to introduce all the irregularities and uncertainties of matter. And working machines are precisely such disturbed equilibria.²⁹ This view of motion as the result of disturbed equilibrium and as subject to unaccountable material hindrances seems to lie at the root of his rejection of the dynamical tradition of mechanics represented by Jordanus and Tartaglia. Since motion is the result of disequilibrium, it cannot be the cause of either equilibrium or disequilibrium. And once equilibrium is disturbed, the resulting motion is indeterminate because of the material hindrances it is subject to. However true their conclusions, then, the fundamental error of Jordanus and Tartaglia was to mistake effects for causes.

Guidobaldo’s main contribution to the renaissance of mechanics in the sixteenth century was to take the vague and wide-ranging scope of mechanics suggested by the pseudo-Aristotelian Mechanica and restrict it to Archimedean explanations of Heron’s five simple machines. In his attempt to found a demonstrative, mathematical science of mechanics, the sole principle he recognized was the principle of the equilibrium of centers of gravity as established by Archimedes. Only equilibrium is susceptible to exact mathematical treatment; motion and speed, since they are the results of disequilibrium and are subject to material hindrances, are in principle indeterminate and thus unknowable with any great precision. But they can be known to some extent, and mechanics is the science of knowing the actual motions and effects of real machines within these natural limits. This, I think, accounts for the apparently paradoxical nature of Guidobaldo’s mechanics, with its insistance on both extreme mathematical rigour and actual practical machines. As for impetus and percussion—themselves merely the results of motion—they seem to lie outside of Guidobaldo’s mechanics, and in this sense Rose’s conclusion about his unbridgeable barrier between statics and dynamics holds true. If there could be an exact science of motion that included such things—and Guidobaldo’s notes on projectile motion and falling bodies suggest that he did not entirely despair of one³⁰—it would be entirely separate from the science of real machines that he attempted to establish on Archimedean principles.

References

Footnotes