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
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
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
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
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
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
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
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
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
In his treatises on the wedge and on the screw,
Guidobaldo cited Pappus’s
In the Meditatiunculae, Guidobaldo sketched his own version of
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
Each of the separate treatises on four of the five
simple machines ends with Heron’s
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
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
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
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
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
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Paris, Bibliothèque Nationale, fonds lat. ms 10246. See (Drake and Drabkin 1969, 48). The Meditatiunculae has been edited by Roberta Tassora (2001), a partial copy of which Pier Daniele Napolitani, who directed the thesis, kindly made available to me after this paper was written; the passages quoted from the Meditatiunculae are in my own transcriptions. The mechanical pages of the Meditatiunculae are discussed by Tassora (2001, 75–100).
On Commandino’s edition and translation of Pappus’s Mathematical Collection, and Guidobaldo’s role in its publication, see (Rose 1975, 209–213).
“Percioche lo spatio della possanza allo spatio del peso ha la medesima proportione, che il peso alla possanza, che sostiene il detto peso. Ma la possanza, che sostiene è minore della possanza che move, però haurà proportione minore alla possanza che lo move, che alla possanza, che lo sostiene. Lo spatio dunque della possanza che move allo spatio del peso haurà proportione maggiore, che il peso all’istessa possanza.” See (Monte 1577, f. 43r–v; tr. Monte 1581, f. 39v; tr. Drake and Drabkin 1969, 300).
See (Monte 1577); De trochlea, Prop. 28, Cor. 2, f. 107 v; tr. (Monte 1581, 101 v; tr. Drake and Drabkin 1969, 317); De axe in peritrochio, Prop. I, Corollary , f. 110r; tr. (Monte 1581, f. 106 r; tr. Drake and Drabkin 1969, 319); De cochlea, Prop. 2, Corollary, f. 128 r; tr. (Monte 1581, f. 125 r; tr. Drake and Drabkin 1969, 326).
See (Monte 1581, ff. 121 r–v; tr. in part in Drake and Drabkin 1969, 325–326); as Bertoloni Meli pointed out (Bertoloni Meli 1992, 25, note 42), Drake’s translation mistakenly prints H instead of G throughout; Pappus’s full proof can be found in translation in (Cohen and Drabkin 1958, 194–196); a discussion of Guidobaldo’s use of Pappus’s proof can be found in (Bertoloni Meli 2006, 35–37).
The proof is as follows: “Ducatur GH horizonti equidistans, cui ad rectos angulos ducatur CH, DK, sitque in hoc circulo constituenda potentia spheram sustinens in G. Sphera vero secetur per centrum et per C, plano horizonti erecto, quod quidem in sphera circulum efficiat maximum ABC. Sphera enim ABD habeat centrum D, que subiectum planum EF horizonti inclinatum in C contingat. Potentia invenire que datam spheram subjectum planum horizonti inclinatum tangentem in dato puncto sustineat. Oportet vero potentiam ita in sphera constituere ut circulus maximus per potentiam, et tactum transiens sit horizonti erectus. Intelligatur itaque GH vectis, cuius fulcimentum est in H, cum planum EF spheram tangat in C. Pondus vero in K esset appensum. Cum enim D sit centrum gravitatis sphere, erit perinde, ac si in K esset appensum ex dictis in tractatu de vecte nostrorum mechanicorum. Quam vero proportionem habet GH ad HK, ita fiat gravitas sphere ad potentiam in G. Potentia igitur in G cognita erit. Ac in prima quidem figura erit primus modus vecte, in secunda secundus, in tertia tertius. Notandum tamen quod si potentia esset in G, ita ut ducta horizonti perpendicularis per centrum sphere D transiret, ut DG tunc potentia totam sustineret spheram ac propterea ipsi equalis existeret. Veluti in puncto quoque B ob eandem causam” (Guidobaldo del Monte, Meditatiunculae de rebus mathematicis, Paris, BNF, fonds lat. ms 10246, 64; see Tassora 2001, 302–303).
“dove è da considerare che la resistanza che fa la materia lo fa quando si hanno da mover i pesi e non quando se hanno da sostenere solamente, perché all’hora l’instrumento non si move né gira.” Guidobaldo del Monte to Giacomo Contarini, Pesaro, 9 October and 18 December 1580, ed. Antonio Favaro (1899-1900); quoted in part in (Gamba and Montebelli 1988, 75–76); for the letter to Pigafetta, see (Keller 1976, 28).