Mechanicorum liber.

Pesaro: Hieronymus Concordia, 1577.

First edition of the author’s first work, generally regarded as the most important treatise on mechanics since Archimedes and a critical influence on Monte’s friend and disciple Galileo."From the time of its publication in 1577, [it was] the most authoritative treatise on statics to emerge since antiquity, and it remained pre-eminent until the appearance of Galileo’s Two New Sciences in 1638. It marks the high point of the Archimedean revival of the Renaissance. Not only did Guidobaldo establish statics on the rigorous mathematical procedures of Archimedes, but he also introduced an historiography of mechanics which was designed to legitimise the Archimedean revival" (Rose, The Italian Renaissance of Mathematics, p. 222). Stillman Drake, in his extensive discussion of this work (Mechanics in sixteenth century Italy, pp. 44-52), singles out for particular notice the section dealing with pulleys, which analyses them in terms of the lever, and which was taken over in its entirety by Galileo. Drake adds that “his Mechanics was the first truly systematic attempt at a rigorous treatment of the field, and it paved the way for Galileo’s synthesis of all the traditions that influenced mechanics in the sixteenth century.” “Guidobaldo’s book on mechanics pioneered the attempts to give a systematic account of mechanical knowledge following the model of Euclid and Archimedes. In order to achieve this goal he used the classification of simple machines ascribed to Heron and transmitted by Pappus. As no such systematic treatment of mechanics from antiquity was extant, Guidobaldo’s book may be considered to represent the autonomous continuation of the Greek tradition. It integrates Archimedean techniques with notions such as the concept of center of gravity – the Aristotelian framework in which weight is always to be referred to the center of the earth, the reduction according to Heron and Pappus of complex to simple machines, as well as the reduction of some machines to balance and lever as in the Aristotelian Problemata mechanica” (Renn & Damerow, pp. 10-11). Monte’s teacher Federico Commandino was the prime mover in the restoration of the works of Archimedes in the Renaissance, and Monte's allegiance to Archimedes is immediately clear from the title page: the globe and lever device is an illustration of the saying ascribed to Archimedes, ‘Toleret quis si consisteret’ – give me a place to stand and I will move the earth. But the text which influenced Monte more than any other was Pappus' Collectiones Mathematicae, which was translated by Commandino and eventually seen through the press by Monte in 1588. “The crucial passage in Pappus’ Collectiones can be found at the end of Chapter 8, where he claimed that, following Philo of Byzantium and Hero of Alexandria, he was seeking to systematize mechanics by identifying in the lever a common denominator for all simple machines … Since Archimedes had established the axiomatic doctrine of the lever, when a machine was shown to be a lever in disguise, its operations were fully understood and the problem of foundations was solved … Despite some shortcomings in dal Monte’s analysis, the program he so thoroughly pursued in a geometric and visual fashion left a deep mark on Galileo” (Meli, p. 24).

Provenance: Signature of Agostino Gabriel on the front paste-down, another signature crossed out on title page.

Guidobaldo’s purpose in writing the Mechanicorum liber 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. 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 axle, wedge, and screw.

“The Mechanicorum liber opens with Pappus’s definition of centre of gravity, accompanied by the corresponding definition due to Commandino, followed by a few obvious axioms about weight as a magnitude, and three suppositions, which read as follows:

  1. Every body has but a single centre of gravity.
  2. The centre of gravity of any body is always in the same place with respect to that body.
  3. A heavy body descends according to its centre of gravity.

“The first section of the treatise concerns the stability of the balance. The first propositions concern the stability of an equal arm balance with equal weights as it is sustained respectively above, under, and at its centre of gravity. All proofs combine a straightforward application of the Archimedean determination of the centre of gravity with the supposition that a body descends according to its centre of gravity (and the implicit acknowledgement that the fulcrum is a fixed point which must remain stationary). If the balance is sustained from above and removed from the horizontal position, the centre of gravity will be raised, and hence if the balance is released the centre will be able to descend until the balance is again in horizontal position. The two other cases can be treated in a completely similar way (the centre of gravity will be respectively lowered – and will be able to keep on descending – and remain stationary). Hence, we have respectively stable, unstable, and indifferent equilibrium. Immediately after the proof of indifferent equilibrium, Guidobaldo enters into a sustained polemical discussion of Jordanus and other writers who want to base mechanics on the notion of positional gravity” (Van Dyck, pp. 384-5).

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.

“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’ …

“In the treatises on the pulley, on the wheel and axle, 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. 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.

“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 … it is possible that Guidobaldo was not happy with Pappus’s proof, and that he omitted it from his Latin text for this reason …

“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 … Here he has moved entirely away from equilibrium as the cause of a mechanical effect and invoked the unexplained power of percussion.

“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.’

“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 …

“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 axle. 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’ geometrical 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. Treating the obstacle effectively as an inclined plane, he reduced the problem to the lever, again invoking Pappus’s theorem on the inclined plane …

“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.’

“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. 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 insistence 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 one—it would be entirely separate from the science of real machines that he attempted to establish on Archimedean principles” (Laird, pp. 39-49).

Monte (1545-1607) studied mathematics at Padua and later at Urbino became the friend and pupil of Federico Commandino, whose translation of Pappus he edited and published. Galileo wrote to him, and thus began an exchange of correspondence on scientific matters. Monte secured for Galileo an appointment to the chair of mathematics at Pisa and later at Padua. “Guidobaldo was Galileo’s patron and friend for twenty years and possibly the greatest single influence on the Mechanics of Galileo. In addition to giving Galileo advice on statics, Guidobaldo discussed projectile motion with him, and both scientists reportedly conducted experiments together on the trajectories of cannonballs” (DSB).

Adams U-7; Riccardi II.178 (“Raro”); DSB IX.487-89. Drake, Galileo at Work (“the most important book on mechanics published in the sixteenth century”, p. 13). Laird, ‘Guidobaldo del Monte and Renaissance mechanics,’ pp. 35-50 in: Guidobaldo del Monte (1545-1607) (Becchi, Meli & Gamba, eds.), 2013. Meli, Thinking With ObjectsThe Transformation of Mechanics in the Seventeenth Century, 2006. Renn & Damerow, Guidobaldo del Monte’s Mechanicorum liber, 2010. Van Dyck, ‘Gravitating towards stability: Guidobaldo’s Aristotelian-Archimedean synthesis,’ History of Science 44 (2006), pp. 373-407.

Folio (313 x 218 mm), ff. [viii] (the last blank), 131, [1, colophon], with woodcut title vignette of globe and lever device and numerous woodcut diagrams and illustrations in text (light spotting, mostly toward the end of the book). Contemporary vellum. Light damp stain to upper margin. A fine and completely restored copy.

Item #5398

Price: $28,500.00