## 5.1 Introduction

In this essay I consider some perspectives from which Guidobaldo del Monte’s work on mechanics was viewed in the historical literature around his time and in subsequent centuries. At first sight it may seem peculiar to address the matter from this angle, since from the greatest part of the current historiography Guidobaldo’s chief contribution to mechanics—the 1577 *Mechanicorum liber*—

appears at best largely unrelated and at worst a hindrance to the tumultuous developments of this field, especially the science of motion right to the time of Isaac Newton*Dialogo* and especially the 1638 *Discorsi*. Several historians later in the century followed Duhem’s views.^{1}

However, there are many ways to look at Guidobaldo’s contributions to mechanics and historically there have been many ways to look at *Mechanicorum liber*. My contribution does not aim at an exhaustive survey but rather explores four moments of the fortune—or, one could say, at times misfortune—of Guidobaldo’s legacy in reverse chronological order. I start from a brief analysis of Pierre Duhem’s*Mécanique analitique* contain valuable analyses of the development of mechanics and of Guidobaldo’s contributions in particular. I continue my journey backwards in time moving to the seventeenth century, especially to the French mathematician Pierre Varignon

I consider different ways of practicing mechanics: one, with which perhaps we are more familiar, relies on principles—increasingly more abstract and general—from which the solution to different problems can be derived; the other way relies either on the established example of the lever, or on other examples in different fields, and seeks to employ them to solve more complex problems by showing that they can be reduced to simpler cases. Instances involve showing that the winch or the inclined plane can be reduced to a lever, or that the motion of projectiles can be reduced to a special case of falling bodies. In conclusion, I wish to argue that, despite significant shortcomings, from both perspectives Guidobaldo played a more significant role in the development of seventeenth-century mechanics and the science of motion than has been generally acknowledged.

## 5.2 Duhem and the Punctilious Scholar

The French historian and philosopher of science Pierre Duhem*Les origines de la statique* (Paris, 1905-1906), Duhem provided a comprehensive account of the discipline across the centuries. In his account Duhem made of Guidobaldo a mediocre pedant or, in his words, a “narrow-minded” and “punctilious” mind eager to quibble over matters of little or no significance while disregarding valuable insights provided by the intuition of his medieval predecessors. Overall, Duhem was eager to promote the Middle Ages over the Renaissance: Guidobaldo’s allegiance to the Greeks and dislike for medieval scholars such as Jordanus of Nemore*Les origines de la statique*, Duhem was quite interested in results, whereas Guidobaldo showed greater sensitivity to the rigor and coherence of proofs and methods, or the foundational aspects of mechanics: in the case of the problem of the equilibrium of weights on the inclined plane, for example, del Monte preferred the problematic solution by Pappus*result* by Jordanus, since his method has been considered problematic in that according to some he introduced in the proof the result he wished to demonstrate. Be that as it may, Jordanus did not rely on the lever in order to account for the inclined plane, as advocated by Pappus and del Monte, but rather sought an independent solution. I shall discuss del Monte’s solution below. Since I have dealt elsewhere with Duhem’s work and its pervasive influence, I can be rather brief here.^{2}

In *Mechanicorum liber* Guidobaldo discussed at great length the problem of the equilibrium of the balance in which the center of suspension and the center of gravity coincide. One may question what is the general significance of this problem, given that it does not seem to be of central importance to the history of mechanics. In fact, the issue is quite subtle because it does involve an important methodological point concerning the problem of rigor and of approximations in the transition from mathematics to *physica* or the study of nature: in the sixteenth century it was not immediately clear which factors had to be included and which ones could be neglected, what was a suitable and acceptable approximation, and which approximation introduced significant errors in the result. Thus I would argue that although the problem of the equilibrium of the balance in the panorama of studies of sixteenth-century mechanics cannot be seen as crucial in terms of results, it did have broader methodological implications.^{3}

At first del Monte’s lengthy discussion seems paradoxical: in the opening of his treatise, he had argued that the key notion to study the equilibrium of the balance is that of center of gravity, which does not change by rotation. Therefore, if we rotate a balance suspended by its center of gravity, the equilibrium conditions are not altered and the balance remains stable in any position in which it is left, or is in a position of indifferent equilibrium. Later, however, del Monte challenged the opinions of his predecessors Niccolò Tartaglia

Fig. 5.1: Del Monte and the convergence of the lines of gravity

We can gain a deeper understanding of del Monte’s strategy by considering that Tartaglia and some of his contemporaries and predecessors had attached a physical role to the so-called “angle”—the so-called angle of contact—between the circle and its tangent, a magnitude that we now consider strictly nil but that was not considered so at the time, although it was considered smaller than any given angle. Thus it appears that Guidobaldo was taking into account and comparing different magnitudes in his approximations, implying that it is not legitimate to ignore the tiny but finite angle of convergence of the weights of the balance toward the center of the earth while taking into account the angle of contact, which in any case is smaller. But in the end, Guidobaldo’s reasoning had a rhetorical stance, since he did not believe that that convergence played a role in the equilibrium of the balance anyway. In fact, Guidobaldo accepted the reasoning by Tartaglia^{4}

I wish to prevent a misunderstanding of my argument. Despite Duhem’s*Mechanica*, for example, John Wallis^{5} At the time of the French revolution, historian of mathematics Jean Etienne Montucla^{6}

## 5.3 Lagrange and the Principles of Mechanics

In *Mécanique analitique* Joseph-Louis Lagrange*Mechanicorum liber*, where the visual aspect was of central significance.^{7}

Lagrange^{8} We may wonder what role Lagrange attributed to Guidobaldo in his scheme. In the historical introduction to statics in the 1788 *editio princeps* of *Mécanique analitique*, Lagrange simply ignored Guidobaldo. In later editions, however, he inserted two references to *Mechanicorum liber*. In the first, he argued that Guidobaldo was unable to apply the principle of the equilibrium of the lever to the inclined plane and the machines that depend on it: indeed, as we know from Duhem’s*Mechanicorum liber*, one in which del Monte had followed the unsatisfactory approach of Pappus^{9}

Lagrange’s*Le mecaniche*, first published by Marin Mersenne*Mechanica* of 1670–1.^{10}

So far we have discussed Lagrange’s views on the principles of statics. With regard to the science of motion or, as Lagrange*la dynamique*, he attributed it entirely to the moderns, beginning with Galileo

## 5.4 Varignon, Descartes and the Rejection of Reduction

Let us move now to the remaining principle of statics according to Lagrange*Project d’une nouvelle mechanique*, in which he challenged del Monte’s *Mechanicorum liber*. There are striking differences between Lagrange and Varignon: Lagrange discussed Guidobaldo from a strictly historical standpoint and paid special attention to his formulation of a general principle, an aspect that played a secondary role in *Mechanicorum liber* but that was of great significance to mechanics at the time when Lagrange wrote. Varignon treated Guidobaldo as a major figure in the field of mechanics, a proponent of an approach still worth considering, and, unlike Lagrange, examined Guidobaldo not for his formulation of a general principle of mechanics but rather for his practice based on the primacy of the lever. Varignon’s publication raises several questions: In which sense was Varignon’s project new? Why challenge a work first published in 1577, one hundred and ten years earlier, by an author who had died in 1607, eighty years before? Was Varignon’s work related to broader concerns about the formulation and practice of mechanics at the time?

A reader of Varignon’s*Project *interested in new results, as opposed to methods, will be disappointed: Varignon’s treatise is strictly methodological and foundational. In the preface Varignon states that he came across a letter in Descartes’s*Mechanicorum liber*.^{11}*Explication des engines*, however, that Descartes addressed the question of the foundations of mechanics understood as the science of simple machines in a more direct fashion. The *Explication* was appended by Descartes to a letter dated 5 October 1637 and addressed to Constantijn Huygens*Le mecaniche*, first published by Marin Mersenne in 1634, and then in the 1656 edition of the *Discorsi*.^{12}

In *Project d’une nouvelle mechanique* Varignon*Mechanica, sive de motu tractatus geometrico* of 1670–1, Wallis had followed an approach derived from Galileo

Varignon*The equilibrium of planes* Archimedes*principle* of composition of motion to a study of simple machines (Figures 5.2 and 5.3).

As Descartes’s*Mechanicorum liber* was still a notable source from a methodological standpoint. Guidobaldo had identified and addressed the problem of the foundations of mechanics: in order to be a *science*, mechanics could not be a heterogeneous collection of problems and *ad hoc* solutions, but had to be structured as a coherent body of knowledge descending from sound and widely accepted principles. For del Monte this anchor of certainty was to be found not in a new abstract principle but in the classical tradition and Archimedes’s

There is also another aspect worth considering at this point. First Galileo,*Thinking with Objects*, the practice of relying on some objects or cases to explain other more complex ones was quite widespread in the entire domain of the science of motion. It was this practice that was becoming less popular at the time of Varignon

Fig. 5.2: Varignon’s principle

Fig. 5.3: Varignon’s principle

This departure from Guidobaldo’s way of proceeding is exemplified by Newton’s*Principia* *mathematica*—symbolically published in the very same year of Varignon’s

Fig. 5.4: Newton’s

## 5.5 Guidobaldo, Galileo and the Practice of Mechanics

Important as principles are, I believe that they should not be seen as the only way of practicing mechanics, especially at a time when mathematicians often reasoned by analogy—whether this was rigorous or not is not my primary concern here—and explicitly advocated a way of doing mechanics based on the conceptual and practical manipulation of objects. While del Monte did attempt to formulate a principle of mechanics anchored to the objects or devices under investigation, he also invoked and applied in practice a visual hands-on approach that eschewed abstract principles in favor of concrete techniques and methods of proceeding. It is to his own work that I now turn.

Guidobaldo was the heir of a tradition, going back to Pappus*Quaestiones mechanicae* then attributed to Aristotle and now considered to be the work of one of his early followers; that work, however, does not consider the simple machines and at times seems more concerned with the theme of wonder at the properties of the circle and balance than with rigorous proof. Moreover, *Quaestiones mechanicae* deal with a range of problems and not all of them can be reduced to the balance. It is in Book 8 of the *Collectiones mathematicae* by Pappus that one finds the most complete and influential treatment of mechanics based on the lever and it is no accident that Guidobaldo took Pappus as his master. The work had not been published yet but in all probability Guidobaldo had access to the translation by his teacher Federico Commandino that he was to see through the press in 1588, eleven years after *Mechanicorum liber* was published. First, one may ask, why start from the lever? The answer to this question was straightforward for both Pappus and Guidobaldo: the doctrine of the lever had been formulated and formalized by Archimedes*On the equilibrium of planes* and was therefore the bedrock of mechanics: it was impossible to go beyond “divine” Archimedes in terms of authority and certainty. But it seems fair to argue that besides relying on historical precedence and authority, del Monte and others saw in the lever the archetypal mechanical device conceptually as well.

Another question one may ask is in what way the lever was used. There is no better way than to look at a concrete example, such as the winch: in this instance, del Monte started from an engineering diagram of the actual device shown in perspective. Then he showed a geometric section of the same and through this geometric diagram he showed visually in a process that I have called “visual unmasking” that lurking inside the winch one could detect a lever: this is what Guidobaldo called “reduction” of the winch to the lever (Figure 5.5). Another celebrated and in this case problematic example is that of the inclined plane. Here too Guidobaldo sought to find a lever in disguise, but in this case his solution ended up being problematic at many levels.^{13}

Fig. 5.5: Guidobaldo on the winch

Fig. 5.6: Galileo

Guidobaldo talked explicitly of a “reduction” of simple machines to the lever and the cases we have just seen illustrate his important concept. Looking at *Mechanicorum liber*, one does not get the impression that Guidobaldo worked from abstract principles, like those at the center of Lagrange’s^{14}

Fig. 5.7: Galileo

Moving on to Galileo*cas* or *car *with bent arms, where the arm *as* is perpendicular to the inclined plane *hg*, and the arm *ar* to *tn*, rather than just uncovering or unmasking it; on the basis of simple geometry he could conclude that the weights of the bodies on the inclines are in equilibrium when they are inversely as the lengths of the inclines. As to the science of resistance of materials, Galileo’s method is strikingly similar to Guidobaldo’s: he sought to identify a lever lurking in the geometric diagram of a beam protruding from a wall (Figure 5.7), where B is the fulcrum and AB and BC are the arms.^{15}

Fig. 5.8: Galileo’s

Galileo’s*Discorsi* in 1638, aged seventy-four—seeking a so-called mechanical foundation for the science of motion. Then in 1638 he put forward a new science relying on a definition and an axiom or postulate that were unrelated to the lever. Other portions of the science of motion, however, relied on visual techniques similar to those used by Guidobaldo. In the case of projectiles, Galileo showed lurking inside (Figure 5.8) a parabolic trajectory *bifh* a falling body, identified by the odd-number rule *bogln*: thus he was able to show that the violent motion of a projectile and the natural motion of a falling body were just variants of each other, the difference being a horizontal projection.^{16}

Similar techniques can be found throughout the century, in different forms. From Robert Hooke

## 5.6 Concluding Comments

Guidobaldo was concerned with rigorous foundations of mechanics as opposed to a set of solutions to individual cases of this or that simple machine. In metaphorical terms, we can see Guidobaldo’s program like a glacier slowly moving forward, solidly secured to its Archimedean sources represented by the theory of the lever. Guidobaldo saw the virtue of keeping the glacier intact even at the cost of limiting the areas he could explore, thus excluding a mathematical science of motion, for example.

In seeking to extend Guidobaldo’s method to new areas while, at the same time, remaining committed to the primacy of the lever, Galileo

Thus, although Lagrange

In conclusion, Lagrange

## References

Benvenuto, E. (1991). *An Introduction to the History of Structural Mechanics*. New York: Springer.

Bertoloni Meli, D. (2006). *Thinking with Objects. The Transformation of Mechanics in the Seventeenth Century*. Baltimore: John Hopkins University Press.

- (2010a). Patterns of Transformation in 17th-Century Mechanics. *The Monist* 93: 578-595

- (2010b). The Axiomatic Tradition in 17th Century Mechanics. In: *Discourse on a New Method. Reinvigorating the Marriage of History and Philosophy of Science* Ed. by M. Dickson, M. Domski. Chicago: Open Court 23-41

Capecchi, D., A. Drago (2005). On Lagrange's History of Mechanics. *Meccanica* XL: 19-33

Damerow, P., J. Renn, J. R. (2001). Hunting the White Elephant: When and How Did Galileo Discover the Law of Fall?. In: *Galileo in Context* Ed. by J. Renn. 21-149

Descartes, R. (1897–1913). *Oeuvres*. Paris: Léopold Cerf.

Drake, S., I. E. Drabkin (1969). *Mechanics in Sixteenth-Century Italy*. Madison: University of Wisconsin Press.

Duhem, P. (1905-1906). *Les origines de la statique*. Paris: Hermann.

- (1991). *The Origins of Statics*. Dordrecht: Kluwer.

Galilei, G. (1890–1909). *Galileo Galilei, Opere*. Firenze: Barbèra.

- (1960). *On Motion and on Mechanics: Comprising De Motu (ca. 1590) and Le Meccaniche (ca. 1600)*. Madison: University of Wisconsin Press.

- (1974). *Two New Sciences, Including Centers of Gravity and Force of Percussion. Translated, with Introduction and Notes, by Stillman Drake*. Madison: University of Wisconsin Press.

- (2002). *Le mecaniche*. Firenze: Olschki.

Gamba, E., V. Montebelli (1988). *Le scienze a Urbino nel tardo Rinascimento*. Urbino: QuattroVenti.

Henninger-Voss, M. (2000). Working Machines and Noble Mechanics: Guidobaldo del Monte and the Translation of Knowledge. *Isis* 91: 233-259

Lagrange, J. L. (1788). *Méchanique analitique*. Paris: Desaint.

- (1811-1815). *Mécanique analytique*. Paris: Ve Courcier.

- (1867-1882). *Oeuvres*. Paris: Gauthier-Villars.

Laird, W. R. (1997). Galileo and the Mixed Sciences. In: *Method and Order in Renaissance Philosophy of Nature* D. A. Di Liscia, E. Kessler, C. Methuen Aldershot: Ashgate Publishing Limited 253-270

Monte, Guidobaldo del (1577). *Mechanicorum liber*. Pesaro: Hieronymum Concordiam.

- (1581). *Le mechaniche dell'illustriss. sig. Guido Ubaldo de' Marchesi del Monte: Tradotte in volgare dal sig. Filippo Pigafetta*. Venezia: Francesco di Franceschi Sanese.

Montucla, J. E. ([1799]-1802). *Histoire des mathématiques*. Paris: H. Agasse.

Palmieri, P. (2008). Breaking the Circle: The Emergence of Archimedean Mechanics in the Late Renaissance. *Archive for History of Exact Sciences* 62: 301-346

Rose, P. L. (1975). *The Italian Renaissance of Mathematics*. Genève: Droz.

Roux, S. (2004). Cartesian Mechanics. In: *The Reception of the Galilean Science of Motion in Seventeenth-Century Europe* Ed. by C. R. Palmerino, J. M. M. Hans Thijssen. Dordrecht: Kluwer 25-66

Van Dyck, M. (2006). Gravitating Towards Stability: Guidobaldo's Aristotelian-Archimedean Synthesis. *History of Science* 44: 373-407

Wallace, W. A. (1984). *Galileo and His Sources*. Princeton, N. J.: Princeton University Press.

Wallis, J. (1693-9). *Opera mathematica*. Oxford: Sheldonian Theater.

## Footnotes

For the pervasive nature of Duhem’s views see (Rose 1975, 233; Drake and Drabkin 1969, 46; Wallace 1984, 204–5, 241).

See (Bertoloni Meli 2006, 26–30) and P. Duhem, *The Origins of Statics*, published originally in 1905–6, transl. in (Duhem 1991, 151–152).

See (Bertoloni Meli 2006, 10–12, 32–35). On this topic see also (Gamba and Montebelli 1988, 213–250; Damerow et.al. 2001; Palmieri 2008, 302).

See (Bertoloni Meli 2006, 26–30). Van Dyck (2006) has independently reached similar conclusions. See also the contribution by Walter R. Laird in this volume. Paolo Palmieri has defended rather different views in (Palmieri 2008, 302).

J. Wallis, *Mechanica*, in (Wallis 1693-9, vol. 1, 619 and 630–2). See also (Drake and Drabkin 1969, p. 47; Gamba and Montebelli 1988, 241–242; Roux 2004, 36–52).

A critical analysis of Lagrange’s historical work written by non-historians is (Capecchi and Drago 2005).

See (Lagrange 1788, 8–11, at pp. 10–11). Lagrange provides a more sophisticated definition of this principle (Lagrange 1867-1882, vol. 11, 7, 19ff).

Lagrange, *Mécanique* (1867-1882, vol. 11, 18–19); Lagrange referred both to *Le mecaniche* and to the *Discorsi* in the expanded 1656 edition (Lagrange 1811-1815, vol. 1, 7, 20; Monte 1577, ff. 43r–v with quoted passage, and f. 104v; Monte 1581, ff. 39r–v, 97r).

See (Benvenuto 1991, vol. 1, ; Duhem 1991, 156–157; Galilei 1890–1909, vol. 2, 156–7, reprinted 1968; Galilei 1960, 148–149; Galilei 2002, 45–46).

“In trochlea autem ineptum mihi videtur vectem quaerere; quod si bene memini, Guidonis Ubaldi figmentum est” (Descartes 1897–1913, vol. 4, 696); see also (Duhem 1991, 421).

See (Lagrange 1788, 9). If this were so, it would be extremely interesting in view of the fact that Galileo’s principle was clearly inspired by Guidobaldo, as his *Le mecaniche* was inspired by *Mechanicorum liber*.

See (Monte 1577, f. 105v).