## Chapter structure

- 1.1 Introduction: Guidobaldo, Galileo, and the Subalternate Sciences
- 1.2 A Quick Tour of the Subalternate Sciences
- 1.3 The Scope of Mechanics in the Sixteenth Century
- 1.4 The Mathematical and the Natural in Guidobaldo’s Paraphrasis
- 1.5 Demonstrating the Law of the Lever
- 1.6 Some Perspectives on the Problem of Mathematization

and the New Philosophy of Nature - Acknowledgements
- References
- Footnotes

## 1.1 Introduction: Guidobaldo, Galileo, and the Subalternate Sciences

How should we place Guidobaldo del Monte in the changing landscape of late sixteenth, early seventeenth-century knowledge?
At once a faithful Aristotelian and early patron of Galileo, Guidobaldo seems to defy some naïve conceptions about the nature of the so-called scientific revolution.^{1} As is well known, one of the ways in which the mathematician Guidobaldo can be considered to have been a faithful Aristotelian is exactly that he is almost completely silent on philosophical issues, thus respecting the disciplinary boundaries that had become deeply engrained in the fields of knowledge. But this leaves us with little to go on if we want to ascertain how he would have understood his own endeavors, and possibly what connected them to or separated them from those of his younger friend, Galileo.

In the present paper I will try to argue for what I take to be a crucial connection between Guidobaldo’s work on mechanics and Galileo’s aspirations in constructing a new, mathematical science of nature. In particular, I will analyze how Guidobaldo laid bare the conditions under which Archimedes’s*mathematical* science of mechanics could be established, and argue that this kind of focus on the conditions that allow for mathematization implicitly prepared the way for Galileo’s^{2}

Seeing the connection between Galileo and Guidobaldo in these terms is not unconnected with the claims that have been made for the importance of the category of the subalternate (or middle or mixed) sciences.^{3} Since the 1970s, a number of historians of science have pointed out that Galileo’s^{4} ^{5} (we will come back to the reason why in Section 1.2).

But if we grant this conclusion with respect to Galileo, what does this tell us about his connection to the enterprise of people like Guidobaldo? One of the major motivations for linking Galileo to the subalternate sciences has always been the idea that he stood in a well-established sixteenth-century tradition, and that his new sciences thus did not arise miraculously. This idea also lies behind a very thoughtful recent reconsideration of Galileo’s relation to the subalternate sciences by Biener (2004), *a philosophical tradition* and as *a practical one*. And whereas it might well be that the philosophical tradition could not have appealed at all to Galileo, the practical tradition nevertheless appears to have provided the model according to which Galileo did structure his own new (mathematical) sciences—as Biener’s subtle analysis of the first two days of the Discorsi shows. The present paper can be seen as a further elaboration of this perspective by investigating the approach of the mathematical practitioner with whom Galileo might most reasonably have been expected to share a tradition: Guidobaldo del Monte. This will also allow us to sharpen the distinction made by Biener by providing an understanding of how exactly the practical tradition provided some of the methodological insights that someone like Galileo was looking for, and which the philosophical tradition conspicuously missed.

Part of my conclusion will indeed be that Laird’s^{6} At first sight, this might be surprising since, as already mentioned, Guidobaldo is in many respects rightfully portrayed as a faithful Aristotelian. Nevertheless, the main reason why he did not engage directly with the philosophical discourse surrounding the category is that it would not have been helpful in answering the questions regarding mathematization which Guidobaldo *as a practitioner* of the mathematical sciences found pressing—rather that it would even have confused issues in an important way! I hope to show that the way in which he *did* answer these questions, moreover, *does* show us an important and very interesting link between Guidobaldo’s and Galileo’s work.

The structure of my paper is as follows. I will first give a quick overview of the concept of the subalternate sciences as it was being discussed by philosophers from Grosseteste*illuminating* way. In elaborating this point, I will then come back to the distinction between a practical and a philosophical tradition of subalternate sciences.

## 1.2 A Quick Tour of the Subalternate Sciences

The present section will be kept to an absolute minimum as good discussions already exist. I base my summary mainly on (Laird 1983)

Puzzled by some of Aristotle’s *Posterior Analytics* about some sciences “being under” other sciences, commentators on the Stagirite’s work elaborated and analyzed the category of the “subalternate” sciences. The context in which Aristotle had introduced the germs of this idea was in discussing how some sciences could use mathematical demonstrations to arrive at conclusions about physical things, apparently violating the essential Aristotelian requirement of homogeneity which states that all terms in valid scientific demonstrations must belong to the same genus. Nevertheless, sciences like astronomy or optics can use mathematical principles, he claimed, because they are related to mathematics as “one under the other.” He moreover added that the lower science (e.g., optics) knows the fact (*oti, quia*), while the reasoned fact (*dioti, propter quid*) belongs to the higher science (e.g., geometry). In *Physics* Aristotle also called astronomy, optics and harmonics the “more physical of the mathematical sciences” (to add to the confusion, in the oft-used translation of the Middle Ages by James of Venice,*Metaphysics*, finally, there is a passage where Aristotle makes the seemingly contrary claim that optics treats visual rays (i.e., physical lines), but only as mathematical lines.

It is clear that this is a rather scant basis on which to develop a full-fledged theory, moreover leaving ample room for disagreement, not least due to the apparent incoherence in Aristotle’s *what* the objects of the subalternate sciences are, *how* these objects are considered, and how this enables us to understand what is *quia*, and what is *propter quid* about the different demonstrations in which these objects figure. On the first question, there seems to have been general agreement that the subalternate sciences deal with some kind of composite subject which is basically a mathematical object to which an extra sensible condition is added (such as “visual” to line). The second and third questions received very different answers. It was both thought that the subalternate sciences basically consider these objects as somehow physical (e.g., by Grosseteste*resconsiderandi* (which according to Zabarella, for example, is the added *sensible* condition) and the *modus considerandi* (which again according to Zabarella is *mathematical*). The answer to this second question obviously influenced the widely differing interpretations of what exactly was demonstrated *quia*, and what was known *propter quid*.

I do not think it is necessary to enter here into any of the further details (although these are important in that they remind us that
there was not just one Aristotelian position on these issues). For now, I only want to draw attention to a fact that is already forcefully stressed by Laird^{7} When discussing, for example, the proof of the law of reflection in optics, the (empirical) principle stating that certain properties hold between the angles of incoming and reflected rays is simply assumed without further comment—the question is whether one can use further geometrical principles about triangles to analyze some of the consequences following from this principle.^{8}

## 1.3 The Scope of Mechanics in the Sixteenth Century

Again, I can be rather brief because the broad lines of the story are already rather well known. I will be mainly interested in stressing two considerations that are not always appreciated as clearly as they should be, but that are relevant for understanding the prospects of mechanics as a subalternate science. They should help us to better appreciate Guidobaldo’s complete silence on the issue of subalternation.

The title of the present section obviously refers to the influential paper by Laird *Mechanical Problems*. The mere existence of a treatise on mechanics thought to be written by Aristotle*topoi* that could be fruitfully exploited to strengthen this legitimacy. Laird summarizes these as follows:

first, it [mechanics] was a theoretical science rather than a manual art; second, it was mathematical, although its subject was natural; third, it concerned motions and effects outside of or even against nature; and fourth, it produced them for human ends. (Laird 1986, 45–46)

Let us try to focus a little more on the second and third aspects. The second aspect (a mathematical science of a natural subject) of course immediately brings the subalternate sciences to mind, especially as the Aristotelian preface also claims that “the how” of mechanical problems is known through mathematics, and “the about what” through physics. Thus, it is not surprising that most sixteenth-century commentators on the *Mechanical Problems* explicitly linked this to the philosophical discussions on the subalternate sciences referred to in the previous section. They also agreed that mechanics considered its subject matter through mathematical considerations. Tartaglia*scientia subalternata*; Maurolico*scientia media* between the mathematical and the natural (Tartaglia 1546, 82v; Baldi 1621, 4; Maurolico 1613, 7–8).^{9}

So far, this is the familiar story that clearly links
important practitioners, such as Tartaglia*praeter naturam*.^{10} As with the second aspect, this could also be
fruitfully linked to other places in the Aristotelian corpus. After all, the basic opposition between what is according to nature and what goes counter to it is one of the true cornerstones of Aristotle’s*Physics II* concerning the distinction between the natural and the artificial is obviously relevant to the case of mechanics and was accordingly often referred to. In the passages discussing this distinction, Aristotle famously claimed that art imitates nature, which was a statement that was frequently taken up by the commentators on the *Mechanical Problems*. The basic Aristotelian idea is that art does not simply overrule nature, but that it profits from the natural constitution of things to bring about useful effects that would not arise naturally. Moreover, it brings about these effects by imitating nature. As Aristotle claims: “if a house, e.g., had been a thing made
by nature, it would have been made in the same way as it is now by
art” (the underlying idea apparently being that art takes
all its clues from nature’s characteristic ways of organizing matter in
goal-oriented ways) (Aristotle 1930, 199). Guidobaldo
was especially explicit when he stated that the arts are able to bring about
effects that are *praeter naturam* exactly *because* they
imitate nature, but this idea seems to have been generally shared by most
commentators.^{11}

But this could be thought to leave us in a quandary. If it is true that art imitates nature, how could it be that mechanical demonstrations (supposedly explaining something about how art can achieve its goals) are wholly mathematical: nature certainly does not operate according to mathematical principles on an Aristotelian view. I do not want to overstate this point, or to make too much out of it, but it does seem that the second and third aspect identified by Laird*Paraphrasis* with an extended discussion of art’s imitation of nature, does not refer to the idea of subalternate sciences. I hope to make this claim more plausible in the next section, but before doing so, I would like to add a further consideration that is relevant to this issue.

The story about the legitimate scope of the mechanical sciences in the sixteenth century has often concentrated on the reception of the *Mechanical Problems*, of course coupled with the publication of the Archimedean treatises. Because of their highly abstract mathematical character, however, the latter do not seem to contain much material that is directly relevant to defining this scope. The goal of the next two sections will be to counter exactly this impression, but it is equally important to stress that writers on mechanics had more sources at their disposal than just these two in crafting an interesting image for their
science—sources that often offered a much more encompassing vision of the scope of mechanics. Vitrivius work was of course directly relevant, and by the second half of the century the eighth book of Pappus’s*Mathematical Collections* should also be added to the available classical background. The work was only published in 1588, but Guidobaldo knew its contents much earlier through his association with Commandino, who was responsible for the translation. It is thus not accidental that the
introduction to his *Mechanicorum liber*, and especially the dedicatory letter by Filippo Pigafetta to its Italian translation of 1581, contain much more substantial allusions to Pappus’s introduction of his eighth book than to the preface to the *Mechanical Problems* (Monte 1577; 1581).^{12} This is not only relevant because Pappus introduced the idea of systematizing a science explicitly devoted to the five simple machines, which could all be explained from a common principle (an idea for which he refers to Heron*Mechanicorum liber*), but also, and more importantly for our topic, because he explicitly stated that the theoretical part of mechanics makes use of “geometry, & arithmetic, & astronomy, & physics!”^{13}

Not only was there a potential tension between seeing art as imitating nature and considering mechanical demonstrations to be purely mathematical, there was also an alternative authoritative view that included physical considerations within the theoretical part of mechanics! We can thus have Baldi

## 1.4 The Mathematical and the Natural in Guidobaldo’s Paraphrasis

After having published his influential *Mechanicorum liber* in 1577, which as already mentioned saw its Italian translation in 1581, Guidobaldo in 1588 also published a detailed commentary on Archimedes*Equilibrium of plane figures* (Monte 1577; 1588). As Guidobaldo explains in his dedicatory letter, *Paraphrasis* is meant to answer criticisms that were made of his earlier
*Mechanicorum liber* by people who were maybe not so adept in
“the mechanical way of investigating the causes of
things” (Monte 1588, page 1 of the unnumbered dedicatory
letter). In this way, he immediately introduces one of the running themes of
his commentary—if not the most important message of the whole book—that
the mechanical science has a special way of demonstrating its propositions,
which must be grasped before one can truly understand any of its claims. In
*Mechanicorum liber* Guidobaldo had moreover *assumed* the
validity of the law of the lever, whereas this law actually contains the
true foundation of the science of mechanics (as he never tires of stressing in this work devoted to the demonstration of its validity).^{14} In *Paraphrasis*,
Guidobaldo accordingly shows that the validity of the law of the lever
is indeed grounded in a special method of demonstration—in the “argumentandi modus huius scientiae maximè proprius […]” (Monte 1588, 44)—implying that
whoever does not grasp this method of demonstration actually ignores the proper
foundations for the mechanical science.

The opening pages of the preface to *Paraphrasis*
are much more explicitly tied to the preface to the Aristotelian *Mechanical Problems* than was the preface to *Mechanicorum liber*. This means that Guidobaldo in turn touches on the admiration accorded to
mechanical effects, and especially the link of this admiration with their
*praeter naturam* character, and on the mechanical science having
both mathematical and physical characteristics. His discussion of the
*praeter naturam* effects is concise and very elegant. He begins by recalling
that Aristotle *Physics II* and the *Mechanical Problems*
had considered three ways in which art can operate: by imitating nature
(such as in sculpture), by finishing what nature could not achieve (such as
in medicine), and by operating *praeter naturam* (such as in
mechanics). But, he adds, on closer consideration it turns out that the
latter also happens through the imitation of nature, since “it is
through nature itself that nature is overcome.” This can be made clear as
follows, he explains. Suppose we have two bodies, A and B, of which A is
heavier than B. It would be in the nature of things that A would be able to
raise B but not the other way around. Consider however what happens if we
put them on some lever in such a way that their common center of gravity C
lies in-between B and the fulcrum D which by its nature can not move: the
center of gravity will by its own nature move down, which will have the
effect that (because of the presence of the fixed point D) A will be raised
and B will be lowered. So what is it that art brings about? Nothing other than
that it places things with respect to each other in such a way that
thereupon the desired effect follows by just letting nature follow its
course (Monte 1588, 2–3).

The basic explanatory scheme behind this discussion is of course the one expounded in great detail in Guidobaldo’s *Mechanicorum liber*, which explains the operation of all simple machines through the relative positions of a compound system’s center of gravity and a fulcrum.^{15}*overcoming* and *imitating* nature are inextricably interwoven. This becomes especially clear if we contrast it with the difficult-to-grasp Aristotelian identification of the *praeter naturam* aspect of mechanical phenomena with the part of circular motion that is supposedly *praeter naturam*.^{16}*praeter naturam* effect, since this was actually grounded in the *mathematical* nature of the circle!

In a passage immediately following the explanation of the *praeter naturam* nature of mechanical phenomena, Guidobaldo enters into the physical/mathematical issue, thus following the broad lines of the structure of the Aristotelian preface, but he does so in a rather unexpected way which sets the stage for the rest of his commentary. He reports that
Aristotle

In itself this is not a very enlightening statement, but we will see in the next section that it actually refers to a subtle understanding of the structure of Archimedes

It must be noted that Guidobaldo had also opened his *Mechanicorum liber* with a discussion of mechanics operating against nature, in which he claimed that mechanics comes from the union of geometry and physics. But at this point he did seem to understand this union more or less along the lines of the philosophical discussions on the subalternate
sciences, as is testified by the fact that he goes on to identify (implicitly) the physical part with the material substrate and the geometrical part with necessary demonstrations. He then further claimed that mechanics exerts control over physical things (presumably because it is
understood to apply necessary demonstrations to physical subject matter), by operating against the laws of nature. His short discussion is ended by claiming that mechanics “adversus naturam vel eiusdem emulata leges excercet” (Monte 1577, page 2 of unnumbered preface). This last characterization was translated by Stilman Drake ^{17}*Mechanicorum liber*, Guidobaldo had not yet really thought through how to understand this operation against, or possibly in imitation of, nature. This silence allowed him to introduce a clear echo of the characterization of subalternate
sciences in discussing the *praeter naturam* nature of mechanics. It is the conceptualization of mechanical phenomena as introduced in *Mechanicorum liber* itself, however, that provides the clue to understanding this operation in the passage from *Paraphrasis* discussed above. And at this point, “holding control over
physical things” no longer comes about by the simple application of geometrical arguments to physical subject matter, but by exploiting some of the physical properties of this subject matter in a cunning way—by also approaching “natural things through natural considerations, such as those relating to the nature of the center of gravity and motion up and down.”

Let us now return to Guidobaldo’s exposition of the Archimedean treatise. Before entering into the propositions and their proofs, Guidobaldo deems it necessary to explain two further things in his preface. First it needs to be understood what the proper definition of center of gravity is, and secondly it needs to be explained how Archimedes*of plane figures*.

The absence of a definition of center of gravity in Archimedes’s*Mechanicorum liber* by introducing the definition that Pappus

The centre of gravity of any body is a certain point within it, from which, if it is imagined to be suspended and carried, it remains stable and maintains the position which it had at the beginning, and is not set to rotating around that point. (Monte 1588, 9)^{18}

Following the example of Pappus^{19} *within*
a cosmological frame. As long as weight is something internal to bodies,
these kinds of problems are bound to crop up, and it is thus this point in
which we must think the natural propensity (which gives rise to a body’s
weight) to be concentrated—because it is actually this point which truly
wants to unite itself with the center of the world.

Guidobaldo next introduces a further consideration which is
relevant to the relation between mathematical and physical notions (Monte 1588, 11). He states that besides the
center of gravity, we can introduce three further centers in our
considerations: the center of a figure, the center of a magnitude, and the
center of the world. The distinction between the first two is not
immediately relevant to our purposes, for which it is only important to
notice that they are both mathematical notions. Guidobaldo then considers
different scenarios in which all four centers either coincide or differ in
different combinations. In the first scenario all centers coincide, which is the case if we consider the earth. This is of course a significant example. The fact that the earth is
supposed to have a mathematically determinate spherical form was one of the
staple examples in debating the nature of mathematical abstraction in the
Aristotelian tradition, and Guidobaldo introduces it by stating that it is
acknowledged by everybody. He further refers to Aristotle’s *De caelo*
*On Floating Bodies* for the fact that the center of
the earth’s circular form coincides with the center of the world; to which
he further adds that the stability of the earth in this position implies
that the earth’s center of gravity also coincides with these three centers.
The discussion of this kind of exemplary scenario brings out the essential
dual nature of a body’s center of gravity. It is a notion which can be
ascribed to every physical body having a natural tendency for motion, but
which at the same time is to be connected with some of the mathematical
accidents of this body, such as its geometrical form and position. It is
this double aspect that lies behind Guidobaldo’s earlier quoted assessment
that Archimedes’s considered mathematical things, such as distances and
proportions, through geometrical demonstrations; and that he considered
natural things through natural considerations, such as those relating to the
nature of the center of gravity and motion up and down. What is clarified
through this further discussion is that the notion of center of gravity
essentially binds together both kinds of considerations. It is connected
with physical properties, such as the equilibrium effects of weight, but at
the same time it is to be considered as a mathematical point which can thus
be introduced in geometrical demonstrations: just as we can abstract the
geometrical spherical form of the earth from its physical makeup, so we can
also abstract the geometrical point in which its physical center of gravity
is situated. In one of the crucial scholia in which Guidobaldo discusses the
Archimedean proof method he expands a bit further on this double nature (Monte 1588, 48–49). He stresses that insofar
as the notion is introduced in geometrical proofs, it is to be ascribed to
bodies considered as geometrical magnitudes, but that insofar as it is linked
directly with effects of equilibrium, it is to be ascribed to these same
bodies as heavy. But, he hastens to add, also when we consider bodies as
magnitudes, we have to understand that we are dealing with magnitudes
*to which heaviness is predicated*—as otherwise the notion of
center of gravity would lose all meaning.

In passages like these, Guidobaldo pays considerable attention
to the possibility of inscribing mechanics in an Aristotelian framework,
with its sophisticated understanding of abstraction as a mental operation,
and in doing so, comes close to certain positions defended in the debates on
the status of the subalternate sciences. Such an understanding of
mathematical abstraction was indeed one of the main factors that lay behind
the different views on the relation between subalternating and subalternated
sciences (Laird 1983).

It is thus to the extent that Guidobaldo wants to do justice to the Aristotelian theory of abstraction that his pronouncements fit very nicely with the philosophical debates on the status of subalternate sciences. But it must be stressed that nowhere does he connect this with the further issue of the status of the resulting demonstrations, nor does he claim these demonstrations to be essentially mathematical or physical—a point to be expanded upon in the next section. That is, the ontological status of mathematical entities is a possible point of concern in his mind, but the epistemological requirement of homogeneity (which, one will recall, was the main impetus behind the debate on the subalternate sciences) does not seem to attract his attention.

The fact that we must always consider geometrical magnitudes to
which the property of weight is predicated raises a further problem on which
Guidobaldo comments in his preface (Monte 1588, 15–18). Indeed, if this is the
case, it becomes hard to understand how Archimedes*plane figures* could be
well founded, since, as Guidobaldo puts it, such a predicate is completely
alien to the nature of plane figures. He tries to dismantle this objection
by admitting that insofar as they are plane figures, they indeed have no
weight at all, but by stressing that we can still “mentally
conceive” plane figures to be equilibrating and thus
showing the effects of gravity. He offers three reasons for this view.
Firstly, he explains that we can consider any plane figure to be the upper
surface of a solid body that is suspended in equilibrium, upon which we can
conclude that this plane figure is also in equilibrium. We can moreover
designate exactly one point in that surface as the center of gravity of that
plane figure, as there will only be one such point from which when suspended
the solid will remain at rest. There is thus nothing contradictory in also
ascribing a center of gravity to plane figures.^{20}
Secondly, he asks why it would be legitimate for a mathematician to consider
heavy bodies as if they had no weight (which is a point generally
acknowledged by Aristotelian philosophers), whereas it would not be
legitimate to consider things that have no weight as if they did have it.
And thirdly, he remarks that we can easily imagine that a greater plane
figure represents a greater weight than a smaller figure, and that we thus
can also imagine these plane figures to be in both equilibrium and disequilibrium.

Maybe the most important of Guidobaldo’s considerations,
however, is that the first eight propositions, which contain the true
foundation of the science of mechanics, need not be limited to plane figures (Monte 1588, 19). Guidobaldo thus stresses that
Archimedes

## 1.5 Demonstrating the Law of the Lever

It was already mentioned how Guidobaldo throughout his preface
and commentary stresses the fact that Archimedes*Mechanicorum liber*) the conditions of
equilibrium for any simple machine (and thus the ensuing multiplication of
force). Guidobaldo’s commentaries on the proof of this law accordingly focus
on the question of how it is possible to give such a *mathematical*
determination, and he painstakingly lays out what he sees as the crucial
factor in this respect: the peculiar nature of a body’s center of gravity as
defined by Pappus.*purely physical* nature, and as such leaves
its mathematical determination completely open. Giving such a precise
determination is thus exactly the task of the first eight propositions of
Archimedes’s treatise. Guidobaldo’s commentaries accordingly focus on
precisely this problem: how does this purely physical characterization allow
for a precise mathematical determination? In the preceding section we
already saw part of the answer: as a point that is situated in a body, it is
linked with some of the body’s physical properties (tendency toward motion
and equilibrium) but it can also be treated *as* a mathematical
point. The important question now becomes: how is it possible to determine
further mathematical properties of this point on the basis of nothing more
than its physical nature?

At first sight, this might be thought to resemble the crucial
question addressed by the philosophers discussing the status of the
subalternate sciences—how it is possible to apply mathematical
demonstrations to physical things characterized mathematically, so as to
demonstrate further mathematical properties of these things? Such a
similarity would only hold, however, if the determination of the further
mathematical properties (the law of the lever) were based only on the
center of gravity being a mathematical point (the only mathematical
characterization given at this point), which decidedly can do no justice to
Archimedes*sufficiently rich*
mathematical structure, which can then really ground a fruitful science. And
as will becomes clear, this involves both physical argumentation and
mathematical demonstration in a two-way interaction which is much too subtle
to be grasped by a crude opposition between considering the subject matter
*either* as physical or as mathematical.

A more interesting perspective on Guidobaldo’s exposition of
Archimedes

Archimedesmakes the action of two equal weights to be the same under all circumstances as that of the combined weights acting at the middle point of their line of junction. But, seeing that he both knows and assumes that distance from the fulcrum is determinative, this procedure is by the premises unpermissible, if the two weights are situated at unequal distances from the fulcrum. (Mach 1960, 20)

Moreover, Mach

It is exactly this insight that Guidobaldo had already
expounded at great length four hundred years earlier when he tried to
explain how the physical properties of a body’s center of gravity allow us
to give it a precise mathematical determination. Moreover, Guidobaldo
clearly recognized that the assumption singled out by Mach

if two equal magnitudes do not have the same center of gravity, then the magnitude that is composed of both magnitudes has its center of gravity in the middle of the line that connects the centers of gravity of the magnitudes. (Monte 1588, 42)

One of the comments Guidobaldo adduces to this proposition is that Archimedes*in
the middle* of the line connecting their centers of gravity. Of course, this
is still a rather meager result, and the crucial question raised by Mach

Guidobaldo notes that Archimedes*composed* from
the two equal magnitudes. Moreover, and this is absolutely crucial, he
stresses that since this is one magnitude, it also has one unique center of
gravity—and this must be completely independent of the form of the
composing magnitudes (Monte 1588, 43; Van Dyck 2006, 376–381)

That this really undercuts Mach’s^{21} The
form of this dependency in no way entered into the argument securing the
validity of the replacement, contrary to what is claimed in Mach’s analysis.
But the validity of the replacement is indeed enough to prove the law of the
lever, starting from the fifth proposition, which is basically an extension
of the situation described in the fourth proposition where we now consider
an arbitrary number of equal magnitudes placed at pair-wise equal distances.
The proof of the law of the lever comes down to showing that if the weights
of two magnitudes are inversely as the distances from which they are
suspended, then these weights can be distributed over different smaller
magnitudes along the line connecting the original magnitudes’ centers of
gravity in such a way that it follows directly from this fifth proposition
that the common center of gravity of all these smaller magnitudes taken
together—and *thus* also of the original magnitudes—coincides
with the point of suspension (i.e., we can transform the asymmetric case into
a symmetric one). This proof involves two essential ingredients: purely
geometrical facts about the relations holding between the distances and the
smaller magnitudes in which the original magnitudes are divided (facts which
show that it is possible to distribute the weights along the line in such a
way that the conditions of the fifth proposition will be satisfied); and the
assumption that there is a *mechanical equivalence* between the
original situation and the one in which the weights are divided and
distributed along the line according to the scheme first made clear in the
fourth proposition (but now extended to an arbitrary number of parts).

Guidobaldo’s care in laying out the conditions which underwrite
the validity of this proof is brought out nicely in some of the editorial
interpolations which he interjects in the Archimedean text of the proof
(interpolations which go to great lengths, but which, with Guidobaldo’s
characteristic scruples, are clearly marked by using a different typography).^{22} After having shown that the
weight of the magnitudes A and B can be distributed along the magnitudes
STVX and ZM in such a way that the latter’s respective centers of gravity will be
in E and D, he states the following (Guidobaldo’s interpolations are in italics):

Butmagnitudes STVX are equal to magnitude A, & ZM to B, thusmagnitude A isas it were [tanquam]placed at E, and B at D;

for certainly the magnitude A placed at E will behave the same way

as do the magnitudes STVX; and B will have the same behaviour at D as the magnitudes ZM. (Monte 1588, 63)

Guidobaldo thus tries to lay bare the “argumentandi
modos” (Monte 1588, 55) of Archimedes

It should be sufficiently clear by now why the category of the
subalternate sciences can offer no insight into the fundaments of Archimedes*simultaneously* physical and
mathematical. Both the mathematical properties of magnitudes and the
physical equivalence between different situations enter critically in the
proof of the law of the lever. Guidobaldo’s neglect of this category can
thus be further explained by the deep insight that he had into the nature of
the Archimedean proof procedure. It moreover shows that he indeed had very
good reasons to prefer the characterization of the nature of mechanical
demonstrations as given by Pappus.

## 1.6 Some Perspectives on the Problem of Mathematization

and the New
Philosophy of Nature

Let me now try to draw together the two main lines of argument
from the preceding sections, and then offer some reflections on the
significance of Guidobaldo’s analysis of the Archimedean proof procedure. To
start with, I argued that, at least in some possible interpretations, there
exists a potential tension between mechanics being a science of *praeter
naturam* effects and it being a science of essentially mathematical
demonstrations. We should now be able to see clearly how Guidobaldo’s
*Paraphrasis* offers an essentially different perspective: it is
exactly because in mechanics we exploit a *natural property*
(“art imitates nature”) that the
mathematization of its basic properties is possible (the essential role of
the natural properties of center of gravity in validating the law of the
lever). The concept of center of gravity thus plays a double role in
founding the science of mechanics: it provides the artificial effects with a
well-defined ontological place in the Aristotelian cosmos; and in doing so
it simultaneously allows the epistemological grounding of the law of the
lever.

The first role of the concept can be adduced as further proof
that Guidobaldo considered it important to inscribe the mathematical science
of mechanics within a broad Aristotelian framework, a concern which is also
further testified by his attention to the problems regarding the ontological
status of the objects of Archimedes

So, what could Galileo have learned from the work of people
like Guidobaldo? Not only that *it was possible* to give mathematical
explanations concerning physical phenomena, but also something about *what made this possible*.

Maybe the most interesting way to think of Guidobaldo’s
commentary is thus as an analysis of the conditions under which mathematical
principles can be considered to be true of physical things. The apparent
necessity of offering such an analysis immediately shows that a purely
empirical approach is not considered sufficient. This of course is perfectly
understandable, as the mathematical principles state very precise relations
which can only be approximated in reality. Moreover, this latter fact
implies that we will also be confronted with apparent counterexamples, which
implies that we must have further grounds to argue that the latter really
are indeed only apparent and that their divergence from the ideal case must
be ascribed to disturbances and the like. That this is also Guidobaldo’s
attitude is testified to by an interesting remark in a letter to Giacomo
Contarini^{23} This
betrays the role played by rational argumentation over and above direct
empirical information: we know that this aberrant situation (equilibrium for
unequal weights) must be due to impediments such as friction, because we
have the rational guarantee that the true cause of equilibrium is equality
in weight.

Although a purely empirical approach will thus not suffice to
lay the foundations of a mathematical science, it is still important to
stress that these foundations do crucially involve empirical input. In the
case of Archimedes*Mechanicorum liber* with a long discussion on the possibility and necessity
of indifferent equilibrium. This long passage by Guidobaldo has not always
fared well among historians of science and has often been badly
misinterpreted.^{24}*Mechanicorum liber*, claims to have been able to construct a balance
that exhibits indifferent equilibrium (Monte 1581, 28v).^{25} This proves empirically that bodies do indeed have a point situated
within them that shows the required property, contrary to Jordanus’s
misguided arguments.

The rigor that Guidobaldo searches is thus not absolute mathematical rigor, which would describe the empirical world in full and hideous detail, but the rigor of any well-founded applied mathematical science. And to ground such a science, one has to select—and possibly stabilize experimentally (e.g., by building a balance that shows indifferent equilibrium)—those properties of the empirical world that can be linked with fruitful mathematical demonstrations. This linkage then requires a second component next to this empirical underpinning. As again shown by Guidobaldo’s analysis of the Archimedean demonstrations, the empirical information must be processed in a specific type of conceptual argumentation before mathematical consequences can be drawn from it. We have indeed seen how Guidobaldo takes much care in explaining that the Archimedean proof rests on the device of replacing a body with another body having the same mechanical effect.

At one point, Guidobaldo uses a tantalizing choice of words to
express that this relation holds when he speaks about the fact that two
bodies, which are suspended at their common center of gravity, are
“aequipollent” (Monte 1588, 45). This is a term which in the
first place expresses the simple fact that the bodies have equal power, but
it was also a term with a well-engrained technical meaning within medieval
logic, where it expresses something like truth valued equivalence because of
the syntactic features of language.^{26} We should of course not make too
much out of Guidobaldo’s rather casual use of this term in just one place,
but even then it offers us a glimpse of things to come. The new mathematical
science of nature, which would be developed from Galileo onwards, can be seen
as an attempt to systematize these relations of causal equivalence by
introducing concepts to denote exactly these cases. These can be transformed
into each other without altering the effect, thus actually allowing the
construction of a logic that is supposed to do justice to the syntax of the
world. A prime example, directly linked to Guidobaldo’s use of
“aequipollence,” is of course the
introduction of the concept of “momento”
to express the equivalence holding in the case of bodies in equilibrium on a
balance.

It is precisely the assumed validity of this kind of substitution that also makes clear in what sense the science of mechanics implicitly offered a new philosophy of matter. Assuming bodies to be equivalent and thus substitutable for each other, on account of no more than
the fact that they have the same mechanical effect, cuts across most of the Aristotelian categories for judging the identity of objects. Most importantly, according to Archimedes

So finally, what does this tell us about sixteenth-century
mechanics as part of a practical tradition of “subalternate
sciences”? Let me first go back to Biener’s*Discorsi*
“does in fact fit the structure of arguments in the
subalternate sciences” (Biener 2004, 281). According to his convincing analysis,
the *First Day* establishes that all en-mattered bodies posses certain
properties that can be described mathematically, and the second day (which
thus constitutes the proper subalternate science) shows that these
mathematical properties entail additional mathematical properties (relevant
for establishing facts about the fracture of en-mattered bodies). This
analysis throws much light on the apparently confusing structure of the
arguments in the *First Day*; and Biener’s point that this mode of double
argumentation must be placed in some kind of practical tradition can indeed
be significantly strengthened by comparing it to the relation between
Guidobaldo’s *Paraphrasis* and his *Mechanicorum liber*: the
first establishes that all equilibrium situations can be described
mathematically, and the latter exploits this description to demonstrate
further mathematical facts which are relevant for establishing facts about
the multiplication of force in all simple machines (again implying that only
the *Mechanicorum liber* would constitute a subalternate science).
But, if this is the general outlook on the structure of applied mathematical
sciences that Galileo inherited from the practical tradition, my analysis
points to the fact that in this tradition itself there was already
considerable attention paid to the thorny question of how to establish a
sufficiently rich and fruitful minor premise (the subject of Guidobaldo’s
*Paraphrasis*, and later of Galileo’s *First Day*). It is here that the
true challenge arose for the establishment of mathematical sciences that
validly capture physical phenomena; and it is here that the philosophical
tradition remained silent, and moreover in this silence obscured the most
important insight that Guidobaldo already explicated: that this challenge
could only be met by clinching a “modo
argumentando” that was truly *sui generis*, being
simultaneously mathematical and physical.^{27}

## Acknowledgements

Part of the research on this paper was done while a visiting researcher at the Max Planck Institute for the History of Science in Berlin. I would like to thank Zvi Biener,

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## Footnotes

Bertoloni Meli (1992) gathers sufficient evidence to believe that Guidobaldo would have seen himself as a faithful Aristotelian. In what follows I will add some further evidence, when considering Guidobaldo’s views on mathematical abstraction as applied to mechanical concepts. In the conclusion, however, I will offer some reflections that could make us somewhat more careful in our use of such a denomination—as opposed to the question of how Guidobaldo would have seen himself—by pointing in which way he ignored important Aristotelian epistemological constraints, and in doing so helped prepare the way for a thoroughly anti-Aristotelian philosophy of nature.

I will not comment on the significant difference of opinion between Guidobaldo and Galileo on the conditions under which the phenomenon of *motion* could be legitimately mathematized. This difference has already been often commented upon, but in my opinion many of these comments suffer from an insufficient understanding of Guidobaldo’s quite sophisticated understanding of the problems involved. This paper can thus be seen as laying part of the groundwork that, by stressing the important convergence of thought on a number of issues, should make possible a much more nuanced understanding of what in the end separated Guidobaldo and Galileo.

In this paper, I will use “subalternate” consistently. Although usage among the Aristotelian commentators is not fixed, it is important to notice that “mixed” (or its cognates) only seems to have come into use in the seventeenth century (none of the sixteenth-century mathematical or philosophical authors that I am aware of use it), and as this might reflect some important changes in the understanding of the most fundamental characteristics of these sciences—as I will indicate in the concluding section of this paper—I believe it should be avoided when discussing the earlier incarnations of the notion.

See (Machamer 1978; Wallace 1984; Lennox 1986; Biener 2004).

See (Laird 1997).

In an earlier paper (Van Dyck 2006), I include Guidobaldo’s mechanics within the category of mixed sciences, without much ado. I do believe that most claims in that paper still stand, but that there are good reasons to be more careful with the use of the category.

One can see this, for example, in Grosseteste’s discussion of this proof, analyzed in (Laird 1983, 37).

Both the works of Baldi and Maurolico were written in the sixteenth century, but only published posthumously.

There were many ways in which this opposition was phrased, but the
expression *praeter naturam* seems to have been most used, and the one to
express the generally accepted line of thought on the issue most accurately. See (Festa and Roux 2001) for further references and for some of the issues surrounding this crucial aspect of the mechanical sciences. Popplow (1998, 154–168) also offers some further discussions on the theme.

For more detailed discussions, see (Monte 1588, 2; Piccolomini 1565, 7v; Maurolico 1613, 29; Monantheuil 1599, 8). The next section will analyze Guidobaldo’s particular interpretation.

Pigafetta wrote his translation in close association with Guidobaldo, so we can safely assume the almost literal extracts from Pappus to have been provided by Guidobaldo.

In the translation by Commandino: “rationalem quidem partem ex geometria, & arithemetica, & astronomia, & ex physicis rationibus constare” (Pappus 1660 [1588], 447).

In proposition V of *On the balance*, the
first book of *Mechanicorum liber*, Guidobaldo at first sight
does give a proof of the law of the lever, but he actually assumes its
validity also in the proof of that proposition.

For more details on this scheme, see (Van Dyck 2006).

Aristotle had claimed that all circular motion is composed of a natural and a praeternatural component. For a detailed analysis, see (Micheli 1995, 41–86).

In a famous definition, Cicero clearly links “aemulatio” with “imitation:” “imitatio virtutis aemulatio
dicitur” (Cicero 1971). A further interesting occurrence carrying a clear link to imitation is in Claudianus’s poem on Archimedes’s planetarium, which is cited by Henri Monantheuil in his commentary on the *Mechanical Problems* (Monantheuil 1599, 3–4 of unnumbered preface).

Translation from (Drake and Drabkin 1969, 259).

Translation from (Drake and Drabkin 1969, 259).

Luca Valerio would also pay much attention to this problem, more or less taking over Guidobaldo’s solution (Napolitani 1982). Baldi also mentions the problem, and again introduces the idea that plane figures are the surfaces of solids. This line of thought implies that the center of gravity of solid figures is prior to that of plane figures, which thus added extra importance to the enterprise of Maurolico, Commandino, Galileo and Valerio to study this topic on which no extant Archimedean writings exist.

That there must be some kind of dependency is implied by the second postulate, stating that if two equal weights are suspended at unequal distances, the one farther from the point of suspension will go down.

The Latin text of Archimedes used by Guidobaldo is taken from the 1544 Basel edition, but with some minor terminological changes (e.g., a consistent use of “aequeponderare,” whereas the Basel text also uses the expression “aequaliter ponderare”).

“La materia fa qualche resistenza […] la qual [materia] vuol la parte sua ancor lei, e quanto sono più grandi in materia tanto più resiste, sì come si prova tutto il giorno nelle libre che, per picole e giuste che le siano e che habbino pesi da tutte due le bande eguali e giusti, non di meno a un di loro se gli potrà metter sopra et aggiunger un peso di tanto poco momento, come un minimo pezzolino di carta che la bilancia starà senza andar giù da detta parte, né per questo la bilancia sarà falsa; 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; e con queste considerationi la troverà sempre che l’esperienza e la demonstrazione andaranno sempre insieme” (Gamba and Montebelli 1988, 76). The context of this remark is Guidobaldo’s claim that the rational principles which hold for a balance in rest can not simply be extended to the balance in motion. I will not discuss this issue further, since it demands, as already mentioned, a thorough treatment in its own right.

For more detailed discussions, see (Van Dyck 2006; Bertoloni Meli 2006).

This text is in the voice of Pigafetta, but Gianni Micheli (1995, 163–167) has published a letter of Guidobaldo to Pigafetta showing that it is actually due to the former.

To quote from a contemporary of Guidobaldo, who moreover is especially
well known because of his mathematical and mechanical work, Maurolico gives
the following definition: “Aequipollentia est inter
propositiones aequivalentia ut essent unum et idem significantes: ut ‘quoddam corpus non est animal’ et ‘non omne corpus est animal’
aequipollent” (§ 74 of *Dialectica Maurolyci*,
electronic edition on the webpage of the Maurolico project:
http://maurolico.free.fr/introen.htm).

One could plausibly argue that the semantic shift toward “mixed” as the name for the category of applied mathematics was at least helped by this insight.