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^{1}^{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^{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mechanics because he held that Archimedean statics had superseded the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

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}*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

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^{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*Liber de
ponderibus* in 1533, and then by Tartaglia’s*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}*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*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}*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*Mathematical Collection* of Pappus^{9}*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*being* at that place, but only by its *departing from* that
place.^{10}*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

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
so.^{15} ^{16}

Fig. 2.1

In his treatises on the wedge and on the screw,
Guidobaldo cited Pappus’s^{17}^{18}

In the *Meditatiunculae*, Guidobaldo sketched his own version of
Pappus’s^{19}

Fig. 2.2

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*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.

Fig. 2.3

According to Guidobaldo, Archimedes*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}*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}*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,^{25}

Fig. 2.4

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*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^{26}

Fig. 2.5

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}

Fig. 2.6

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*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}*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^{30}

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## Footnotes

“Ac si aliquando, vel sine demonstrationibus geometricis, vel sine vero motu res mechanicae considerari possint” (Monte 1577, f. **1v; tr. Drake and Drabkin 1969, 245).

See (Duhem 1905-1906, I, 209–226; tr. Duhem 1991, 148–159). The phrases quoted are on p. 151 and 159.

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 the
sixteenth-century tradition of the *Mechanica*, see (Rose 1975; Rose and Drake 1971; De Gandt 1986; Laird 1986).

See (Nemorarius 1533 and 1565).

For Jordanus and the medieval science of weights, see (Moody and Clagett 1952; Tartaglia 1546; excerpts tr. Drake and Drabkin 1969, 104–143).

See (Maurolico 1613; Tucci 2004), on Maurolico’s mechanics, see (Laird in press).

On
Commandino’s edition and translation of Pappus’s *Mathematical
Collection*, and Guidobaldo’s role in its publication, see (Rose 1975, 209–213).

See (Monte 1577; tr. Monte 1581; tr. Drake and Drabkin 1969, 267–268).

“Eruditissimus tamen libellus de aequeponderantibus prae manibus hominum adhuc versatur, in quo tanquam in copiosissima poenu omnia fere mechanica dogmata reposita mihi videntur” (Monte 1577, f. **1r; tr. Drake and Drabkin 1969, 244).

“Tota mechanica facultas tanquam unico, praecipuoque fundamento innititur” (Monte 1588, 4); the Preface is edited and translated into German in (Frank 2007), the quotation is on p. 118; the translation quoted here is by Rose (1975, 234).

See (Monte 1577, ff. 30v–36r; tr. Monte 1581, ff. 29v–33v; tr. Drake and Drabkin 1969, 296, proofs omitted).

“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 [3], 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).

For Galileo’s statement of the principle, see (Galilei 2002, 45–47); tr. (Drabkin and Drake 1960, 147–149).

Guidobaldo, *Mechanicorum liber*, *De cuneo*, f.
115 r, tr. (Monte 1581, f. 110 r; tr. Drake and Drabkin 1969, 321); *De cochlea*, [Prop. 2], ff. 126 r–127 r; tr. (Monte 1581, ff. 121 r–122 r; tr. Drake and Drabkin 1969, 325–326).

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).

See (Monte 1577, f. 118 v–119 r; tr. Monte 1581, f. 114 r–v; tr. Drake and Drabkin 1969, 322–323); on the force of percussion, see (De Gandt 1987; Laird 1991; Roux 2010).

“Percioche essendo, che le possanze lo quali movono possano essere infinite, non sene puo assegnare ferma regola, come si farebbe della possanza, che sostiene, laquale è una sola e determinata.” See (Monte 1581, f. 127 v; tr. Drake and Drabkin 1969, 328); the insertion is mine.

See (Monte 1588, 4; Frank 2007, 118; see Rose 1975, 234).

Del Monte, *Meditatiunculae*, 30;
see (Tassora 2001, 266).

Del Monte,
*Meditatiunculae*, 31–32; see (Tassora 2001, 267–268).

Del Monte, *Meditatiunculae*, 55–56; see (Tassora 2001, 292–293); a similar proof is
given by Giuseppe Moletti in his unpublished *Dialogue on Mechanics* of
1576, for an edition and English translation of which see (Laird 2000, 97–99).

See (Catena 1556, 81–83; del Monte, *Meditatiunculae*, cit., 60–61; Tassora 2001, 298–300); that Guidobaldo heard Catena’s lectures,
see (Rose 1975, 222).

Del Monte,
*Meditatiunculae*, 59; see (Tassora 2001, 297–298).

“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).

On this point see (Van Dyck 2006, 398–399).

See,
for example, his notes on bodies falling through resisting media (Guidobaldo,
*Meditatiunculae*, 41–42) and on the path of
projectiles (op. cit., p. 236); see also (Tassora 2001, 90–93, 181–186, 281–283, 545–547).