Gravitating towards Stability: Guidobaldo's Aristotelian-Archimedean Synthesis

Duhem P. , Les origines de la statique, i (Paris, 1905), 226.

Dugas R. , Histoire de la mécanique (Paris, 1950), 99; Costabel P. , Centre de gravité et équivalence dynamique (Paris, 1954), 10.

Rose P. R. , The Italian Renaissance of mathematics: Studies on humanists and mathematicians from Petrarch to Galileo (Geneva, 1975), chap. 10; Drake S. , “Introduction” in Mechanics in sixteenth-century Italy: Selections form Tartaglia, Benedetti, Guido Ubaldo, & Galileo, transl. and annotated by Drake S. Drabkin I. E. (Madison, 1969), 3–60.

Rose , op. cit. (ref. 3), 233; Drake , op. cit. (ref. 3), 48.

Drake , op. cit. (ref. 3), 46.

Gamba E. Montebelli V. , Le scienze a Urbino nel tardo Rinascimento (Urbino, 1988); Biagioli M. , “The social status of Italian mathematicians”, History of science, xxvii (1989), 41–95; Meli D. Bertoloni , “Guidobaldo dal Monte and the Archimedean revival”, Nuncius, vii (1992), 3–34; Micheli G. , “Guidobaldo del Monte e la meccanica”, Appendice II in Le origini del concetto di macchina (Florence, 1995); Henninger-Voss M. , “Working machines and noble mechanics: Guidobaldo del Monte and the translation of knowledge”, Isis, xci (2000), 233–59.

Drake , op. cit. (ref. 3), 13.

Biagioli , op. cit. (ref. 6), 57.

Gamba Montebelli , op. cit. (ref. 6).

Meli Bertoloni , op. cit. (ref. 6).

Micheli , op. cit. (ref. 6).

Henninger-Voss , op. cit. (ref. 6).

A useful summary of the landscape of sixteenth-century positions on this issue is provided in Laird W. R. , “The scope of Renaissance mechanics”, Osiris, n.s., ii (1986), 43–69.

Gamba Montebelli , op. cit. (ref. 6), part II, provides an exception but as the concept of centre of gravity is not further analysed there, the author misses an essential part of the fine-structure of Guidobaldo's conceptualization of mechanical phenomena.

Cf. especially Clagett M. , The science of mechanics in the Middle Ages (Madison, 1959), 3–23.

For some further historiographical reflections on the gradual process through which the distinction between statics and dynamics took its present-day shape, see Gabbey A. , “Between ars and philosophia naturalis: Reflections on the historiography of early modern mechanics”, in Field J. V. James F. A. J. L. (eds), Renaissance and revolution: Humanists, scholars, craftsmen and natural philosophers in early modern Europe (Cambridge, 1993), 133–45.

This is especially true with regard to Duhem's and Costabel's treatments of the status of the centre of gravity ( Duhem , Les origins de la statique, ii (Paris, 1906), chaps, xv, xvi; Costabel , op. cit. (ref. 2)). Their criticisms clearly pinpoint in what sense Guidobaldo's understanding of this notion must differ essentially from a modern understanding. However, this need not be taken as a sign of Guidobaldo's incoherence (as they frequently suggest). It can also be taken as a warning notice that if we want to understand the coherence of his science on his own terms, we certainly will have to make sense of these differences.

Koertge N. , “Galileo and the problem of accidents”, Journal of the history of ideas, xxxviii (1977), 389–408, p. 393.

Wallace W. A. , Galileo and his sources: The heritage of the Collegio Romano in Galileo's science (Princeton, 1984), 241.

del Monte Guidobaldo , Mechanicorum liber (Pesaro, 1577); Le mechaniche, transl. by Pigafetta F. (Venice, 1581); In duos Archimedis Aequeponderatium libros paraphrasis (Pesaro, 1588). English translation of the first two books, when available, will be given from Mechanics in sixteenth-century Italy (ref. 3).

I am not aware of any other detailed study of Guidobaldo's paraphrasis. The only partial exception is Micheli , Le origini… (ref. 6), which has many references to Guidobaldo's understanding of specific aspects of Archimedes's treatise dispersed throughout the book.

For a sample of the modern literature on Archimedes, see Dijksterhuis E. J. , Archimedes (Princeton, 1987); Drachmann A. G. , “Fragments from Archimedes in Heron's Mechanics”, Centaurus, viii (1963), 91–146; Knorr W. R. , Ancient sources of the medieval tradition of mechanics (Florence, 1982).

“Cùm itaquè supponat, nos exquisitam habere notitiam centri gravitatis.” In duos… (ref. 20), 8.

Pappi Alexandrini Collectionis quae supersunt, ed. and transl. by Hultsch F. (Berlin, 1878).

“Haec igitur doctrinae centrobaricae summa esse videtur, cuius elementa ediscas, si Archimedis de aequilibriis libros et Heronis mechanica adieris….” Collectionis (ref. 24), 1035.

Mechanics (ref. 3), 244.

Collectionis (ref. 24), 1069; cf. the works of Dijksterhuis, Drachmann, and Knorr cited in ref. 22.

In duos… (ref. 20), 8–9; translation taken from Mechanics (ref. 3), 259.

This is explained in great detail in Archimedes's Method, to which Guidobaldo obviously had no access.

“etenim in his semper loquitur vel de gravibus simpliciter, veluti in primis tribus theorematibus; vel de magnitudinibus, ut in reliquis quinque quod quidem nomen tam planis, quàm solidis quibuscunque est comune, ut etiam ij, qui parùm in Mathematicis versati sunt, satis norunt.” In duos… (ref. 20), 20.

In duos… (ref. 20), 19–21.

In duos… (ref. 20), 14–16.

Mach E. , The science of mechanics: A critical and historical account of its development, transl. by McCormack T. J. (LaSalle, 1974), 20.

“At verò quoniam demonstrationes ibi allatae indigent, quae Archimedes in sequenti sexta propositione demonstravit, idcirco demonstrationes illae huic loco non sunt oportunae.” In duos… (ref. 20), 59. (Guidobaldo is referring to demonstrations of some propositions in his Liber mechanicorum.).

In duos… (ref. 20), 55–58.

“Quocumque enim modo eadem gravia sese habent, eodem semper modo in eius gravitatis centra gravitant.” In duos… (ref. 20), 56.

In duos… (ref. 20), 9–11; 43–44.

Note that this involves a subtle shift of reasoning on Guidobaldo's part. To make this point, he turns to Commandino's definition of centre of gravity, which Guidobaldo always presents as completely equivalent to Pappus's definition (he calls it a “descriptionem” of the notion, rather than a definition, presumably implying that Commandino gives a further explanation of how we should understand the actual definition, which is due to Pappus). Commandino's definition, however, nowhere mentions suspension, but only states that the parts of the body on all sides of its centre of gravity will have equal moment (“Centrum gravitatis uniuscuisque solidae figurae est punctum illud intra positum, circa quod undique partes aequalium momentorum consistent”, In duos… (ref. 20), 9). Pappus's definition, with its emphasis on suspension, is rather ill-suited to establish this cosmological connection, since it seems improper to think of the role of the centre of the universe as a point of suspension.

“Quare dum asseritur, grave quodcumque naturali propensione sedem in mundi centra appetere, nil aliud significantur, quàm quòd eiusmodi grave proprium centrum gravitatis cum centro universi coaptere expetit, ut optimè quiescere valeat…. Ex iis omnibus, quae hactenus de centro gravitatis dicta sunt, perspicuum est, unumquodque grave in eius centro gravitates propriè gravitare…. Praeterea quando aliquod pondus ab aliqua potentia in centro gravitatis sustinetur; tunc pondus statim manet, totaquè ipsius ponderis gravitas sensu percipitur.” In duos… (ref. 20), 10.

“Quoniam scilicet recta linea AB eas [magnitudines AB] coniungit; ideo Archimedes considerat unam tantùm esse magnitudinem…. Neque magis una est magnitudo quadrilaterum, pentagonum, cubus, & huiusmodi aliae, quam sit magnitudo, quae componitur ex magnitudinibus AB unà cum linea AB. quòd si est una tantùm magnitudo, ergo unum habet centrum gravitatis.” In duos… (ref. 20), 43.

Rose P. L. Drake S. , “The Pseudo-Aristotelian questions of mechanics in Renaissance culture”, Studies in the Renaissance, xviii (1971), 65–104.

Micheli G. , Le origini… (ref. 6), especially chap. 3, is a recent and erudite study aimed at a more precise understanding of the Mechanical problems, which moreover pays much attention to Renaissance commentaries on the work.

Aristotle , Minor works, transl. by Hett W. S. (Cambridge, MA, 1963), 333. I will use this twentieth-century translation, without paying attention to the sixteenth-century translations and paraphrases, as the differences are irrelevant to my purposes. See Micheli , Le origini… (ref. 6), chap. 3, for discussions of some of these differences.

Aristotle , Minor works (ref. 43), 337.

I think it is clear from pseudo-Aristotle's own explanation that this force is not to be identified in general with the action of a weight, but with a tangentially applied force generating the motion of the radius and hence the ‘nature’ of the circle. All this is part of a general investigation of the properties of a circle, not of the behaviour of weights. Only at the end of his explanation, when actually answering the first problem, does pseudo-Aristotle identify the equal force on both a large and a small radius with the weight in a balance. It is of course a conspicuous aspect of a balance that its arms are placed horizontally, and that the action of the weight is thus indeed working tangentially. The general properties of a circle can thus be invoked to explain the behaviour of a balance near equilibrium (and this is all the author is interested in at this point).

Aristotle , Minor works (ref. 43), 341–3.

Aristotle , Minor works (ref. 43), 353.

Aristotle , Minor works (ref. 43), 353.

“Punctum autem illud, quod Archimedes accipit, unde sumuntur distantiae, ex quibus gravia suspenduntur, … Aristoteles centrum appellat.” In duos… (ref. 20), 24.

Yet it must be noted that it is not by accident that Guidobaldo most probably found the inspiration for his explanation in the Mechanical problems, where the parallelogram rule for the composition of motion is expounded and moreover lies at the centre of the explanatory structure. An important difference remains that Guidobaldo would have to consider the tangential force/motion as the resultant of the perpendicular free force (i.e. the weight) and the constraining force, which is normal to the circumference, whereas the Greek author considers the circumference itself as the result of the composition of motions which result from a force directed towards the centre and a tangentially applied force.

One reason why one may suppose that Guidobaldo never consciously analysed the details of such decomposition is that it would almost directly have led him to the correct solution of the inclined plane problem. The main reason why he did not take this route, and instead adopted Pappus's treatment of the inclined plane, is probably that he conceived an inclined plane as a wedge upon which a body is forced to move. As Guidobaldo himself did not include Pappus's proof in his own treatise (it was only added in Pigafetta's translation), as his references to Pappus's treatment are rather sloppy (the balance involved in Pappus's proof has e.g. its fulcrum in the point of contact between the body and the inclined plane, whereas the lever to which Guidobaldo wants to assimilate the wedge has its fulcrum in the tip of the wedge), and as he only uses the qualitative result that more force is needed as the plane is more oblique (in conformity with his belief that no exact proportions could be given for problems involving motion — See the last section of this paper), I think we can safely assume that he did not pay close attention to the conceptualization of the inclined plane problem. The references to Pappus rather seem to be added to justify his inclusion of the wedge and the screw in his mechanical treatise. Accordingly, I will not further treat the inclined plane in the present paper. Revealing questions can be posed about Guidobaldo's decision to refer to it in his treatise, but these fall outside the limited perspective I have adopted here.

“Supponit autem Archimedes hoc postulatum respiciens fortasse ad ea, quae Aristoteles in principio quaestionum mechanicarum ostendit, ubi colligit Aristoteles idem pondus celeriùs ferri, quò magis à centra distat….” In duos… (ref. 20), 26.

Drake , op. cit. (ref. 3), 15.

“demonstratum est enim in Archimedis libra <for> sive de stateris et in Philonis Heronisque mechanicis, a maioribus circulis superari minores circulos, si circa idem centrum conversio eorum fiat.” Collectionis (ref. 24), 1069.

Mechanicorum liber (ref. 20), lv; Mechanics (ref. 3), 259.

I will not give references to the places where the relevant passages in Tartaglia Cardano Jordanus can be found, since these are already noted in the translation of Guidobaldo's treatise in Mechanics (ref. 3).

A third argument described by Guidobaldo does not truly involve the notion of positional gravity.

The cogency of this critique was denied by Duhem, who stresses that according to Jordanus the positional gravity has to be calculated for an arc smaller than any assigned value ( Duhem , op. cit. (ref. 1), 215).

Mechanicorum liber (ref. 20), 26v; Mechanics (ref. 3), 290.

Mechanicorum liber (ref. 20), 26v; Mechanics (ref. 3), 290.

Mechanicorum liber (ref. 20), 20r; Mechanics (ref. 3), 282.

It is true that the present explanation introduces some problems of its own; it is especially hard to understand what happens with the bodies' tendencies to descend at the point when their centre of gravity coincides with the centre of the world. This situation reappears in Fermat's discussion of the geostatic question, and shows its problematic character in that context.

See the transcription of a letter of Guidobaldo to Pigafetta in an appendix to Micheli, op. cit. (ref. 6).

Le mechaniche (ref. 20), 28r; Mechanics (ref. 3), 294.

Mechanicorum liber (ref. 20), 5v; Mechanics (ref. 3), 262.

Cf. Duhem , op. cit. (ref. 17); Costabel , op. cit. (ref. 2); Roux S. , “Cartesian mechanics”, in Palmerino C. R. Thijssen J. M. M. H. (eds), The reception of the Galilean science of motion in seventeenth-century Europe (Dordrecht, 2004), 25–66.

This is especially so if we take into account that he had earlier criticized Tartaglia et al. because their arguments concerning the differences in positional gravity would imply a change in centre of gravity with the inclination of a balance.

Le mechaniche (ref. 20), 28r; Mechanics (ref. 3), 295.

I claimed in the introduction (cf. ref. 17 and the accompanying text) that some of the conclusions of Duhem and Costabel could be used as a kind of hermeneutic benchmarks, because they allow us to pinpoint in what respects Guidobaldo's conceptualization of mechanics is essentially different from a modern one. Let me quickly summarize these conclusions, and leave it to the reader to compare them with the foregoing discussions. Both Duhem and Costabel make a lot out of the presumed fact that Guidobaldo's conception of centre of gravity had to be incoherent because it involved both the definition due to Pappus, and the one due to Albert of Saxony. The first presumably involves parallel lines of descent (because, as we saw, this is a precondition for indifferent stability), whereas the second essentially involves the centre of the universe (it is broadly speaking the idea that in any body there is one point which strives to unite itself with the centre of the universe), and hence brings with it convergence of lines of descent. On this ground, they criticize Guidobaldo on two scores: That he does not realize this incoherence, and that he cannot possibly overcome it. According to them, this incoherence could only be overcome by leaving behind the overtly physical connotations of both definitions, and by introducing a purely geometrical definition. Such a definition would allow the centre of gravity (which would become an ill-suited name for the concept) to play its truly fruitful role: To be a centre of dynamical equivalence; i.e. one can derive from this geometrical definition that it is the point where one can conceive all the mass of a system of bodies to be concentrated and the geometrical resultant of all the forces on these bodies to be applied. If we take these forces to be forces of weight, and if these are considered to be parallel, then it follows that we can always replace the system of bodies by its centre of gravity. That we have indifferent equilibrium if we hold a body in its centre of gravity is merely a physical consequence of this fact, but it is no part of the defining characteristics of the concept.

“Riduco le cinque machine alia leva, è vero, ma non però riduco la bilancia alia leva, essendo che esse siano una med.ma cosa e fra loro non vi è altra differenza, se non che con la bilancia si considerano li pesi, e con la leva si considerano la forza e il peso insieme….” Quoted in Micheli, op. cit. (ref. 6), 161.

Drake's translation in Mechanics (ref. 3) skips almost all the proofs of the propositions concerning the lever, hence actually hiding the transformations that govern Guidobaldo's understanding of the lever.

Already in Proposition 5 on the balance Guidobaldo states that suspended weights (“pondera”) have gravity (“gravitate”) in proportion to the distance from the fulcrum. The closeness to our notion of static moment is explicit in Commandino's version of the definition of centre of gravity. It is important, however, to keep in mind that static moment depends not only on the length of the lever arm, but also on the direction of the applied force.

Mechanicorum liber (ref. 20), 42r; again a passage not included in Drake's translation.

Guidobaldo explicitly notices that in his pulley systems “the power will always move the weight as with a lever parallel to the horizon”. Mechanicorum liber (ref. 20), 77r; Mechanics (ref. 3), 311.

Duhem , op. cit. (ref. 1), 219–23.

Mechanicorum liber (ref. 20), 108r; Mechanics (ref. 3), 318.

Compare especially with the discussions at Mechanicorum liber (ref. 20), 29v–30r; Mechanics (ref. 3), 293.

“tunc eademmet potentia, vel in F, vel in T constituta idem pondus k sustinere poterit; cùm semper in cuiuscunque: Extremitate scytalae ponatur, ab eodem centro C aequidistans fuerit, ac secundum eandem circumferentiam ab eodem centro aequaliter semper distantem perpensionem habeal.” Mechanicorum liber (ref. 20), 109r.

At least, I have not been able to locate other places in Guidobaldo's writings where he would directly apply the insight that it is only the perpendicular component that must be taken account. Proposition 5 of the section on the lever in the Mechanicorum liber is certainly not a case, as is claimed by Montebelli (Gamba and Montebelli, op. cit. (ref. 6), 239–40). One only has to notice that Guidobaldo nowhere considers the projection of the arm on which the power is applied to see the inappropriateness of the figure that is provided by Montebelli (his Figure 14). Guidobaldo in this proposition is not discussing the need to project the lines of force on a perpendicular arm, but the place where we should consider the force of the weight to be applied to the lever arm (which need not result in a perpendicular projection).

Henninger-Voss , op. cit. (ref. 6), 255.

I borrow the apt expression “bringing into operative act” from Henninger-Voss , op. cit. (ref. 6), 247, which, notwithstanding the confusion just pointed out in the text, is undoubtedly the best analysis of the hybrid nature of this double exploitation.

Cf. especially In duos… (ref. 20), 48, where Guidobaldo stresses the fact that centre of gravity is a mathematical notion, defined for mathematical objects, which allows its introduction in the Archimedean proofs of Propositions 6 and 7.

Duhem , op. cit. (ref. 1), 209–26; Drake , op. cit. (ref. 3), 44–48; Rose , op. cit. (ref. 3), 233.

It is noteworthy that in Guidobaldo's own preface to the Mechanicorum liber, which stresses both the utility and the nobility of mechanics, he has only a scornful remark for Jordanus's “disastrous errors”; whereas Pigafetta's preface, which is almost exclusively devoted to the utility of mechanics, has a much more friendly reference to Jordanus, “who wrote of the science of mechanics” and “began to resuscitate it somewhat”. (Mechanicorum liber (ref. 20), unnumbered preface; Le mechaniche (ref. 18), unnumbered preface; Mechanics (ref. 3), 246, 252. For an analysis of the differences between the Latin work and its vernacular translation, see Henninger-Voss, op. cit. (ref. 6).) Guidobaldo's gibe occurs in the context of his stressing that he has tried to build up his work “from it foundation to its very top” — The most important problem with Jordanus is clearly not that he had made some easily correctable errors, or that he had introduced different concepts, but that he threatened these essential foundations.

Cf. e.g. the letter to Contarini cited in Gamba Montebelli , op. cit. (ref. 6), 86.

The structure of the mixed, middle, subordinate, or subalternate sciences has received a considerable amount of attention in the literature. Cf. e.g. McKirahan R. D. Jr , “Aristotle's subordinate sciences”, The British journal for the history of science, xi (1978), 197–220; Wallace , op. cit. (ref. 19), chap. 3; Lennox J. G. , “Aristotle, Galileo, and ‘mixed sciences’”, in Wallace W. A. (ed.), Reinterpreting Galileo (Washington, DC, 1986), 29–51; Laird W. R. , “Robert Grosseteste on the subalternate sciences”, Traditio, xxxiii (1987), 147–69; Dear P. , Discipline & experience: The mathematical way in the Scientific Revolution (Chicago and London, 1995), chap. 2.

That any body has a centre of gravity; that it descends according to its centre of gravity; etc.

Tartaglia , Quesiti et inventioni diverse (Venice, 1546), 76–78; transl. from Mechanics in sixteenth-century Italy (ref. 3), 106–7.

Mechanicorum liber (ref. 18), unnumbered preface; Mechanics in sixteenth-century Italy (ref. 3), 245.

Henninger-Voss M. , “How the ‘new science’ of cannons shook up the Aristotelian cosmos”, Journal of the history of ideas, lxiii (2002), 371–97, p. 382.

Le mechaniche (ref. 20), 28r; Mechanics (ref. 3), 295.

That this claim is not due to the fact that he “refused to countenance the use of insensibilia in mechanics, because they were not susceptible of precise mathematical definition” (as is claimed by Rose, op. cit. (ref. 3), 233) is proven by his discussion of the argument concerning smallest angles.

“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 provo tutto il giorno nelle libre che, per picole e guiste 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.” (Quoted in Gamba and Montebelli, op. cit. (ref. 6), 76.).

This does leave open the question of the grounds on which Galileo nevertheless chose to take the steps that Guidobaldo consciously refused to take.

The formulation is a little too concise: It is not necessary that the centre of gravity coincides with the fulcrum; it is enough that it lies on a straight line connecting the fulcrum with the centre of the world — This is of course exactly the difference between on the one hand indifferent and on the other hand stable and unstable equilibrium.

The proof of the first proposition in the Mechanicorum liber, which is skipped in Drake's translation, provides a nice illustration of this mode of argumentation.

Mechanicorum liber (ref. 20), 24v; Mechanics (ref. 3), 287.

Biagioli , op. cit. (ref. 6), 65.

I already quoted Drake's judgement that this was “a curious theory of the history of mechanics”. Knorr , op. cit. (ref. 22), provides convincing arguments for the exciting thesis that this might actually be the best history of mechanics available. He shows how the medieval so-called dynamical treatments of the balance in all probability derive directly from a lost work of Archimedes, pre-dating the Equilibrium of planes and the introduction of the concept of centre of gravity, and he adds the suggestion that Archimedes's interest in this kind of problems might have been triggered by the pseudo-Aristotelian treatment (ibid., 100–2). It hence appears that what most historians of science have construed as two entirely different traditions actually have a common root in closely related efforts that took place in one and the same context.