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Galilean Analogies: Imagination at the Bounds of SenseAuthor(s): Lorraine J. DastonSource: Isis, Vol. 75, No. 2 (Jun., 1984), pp. 302-310Published by: The University of Chicago Press on behalf of The History of Science SocietyStable URL: http://www.jstor.org/stable/231828Accessed: 16/09/2009 03:17

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LORRAINE J. DASTON LORRAINE J. DASTON

In the area of scientific discourse, however, Bacon ascribed to imagination a more positive and permanent place. The main interest of his treatment lies in his extension of standard rhetorical techniques from religion, politics, and law to the realm of science, a move wholly in keeping with his doctrine of the unity of knowledge. The analogies of scientific discourse, unlike those used in sci- entific discovery, were artificial similitudes concocted by an ingenious writer to illuminate a particular set of properties of an object or phenomenon-for ex- ample, the idea that the latent configuration of matter is in some respects like the texture of a piece of cloth. Less potent, but easier to control, their utility was ultimately more enduring.

But both discursive and inductive analogies, according to Bacon, relied on the power of imagination for their meaning and force. The aim of both was to move from object to image to concept, where the concept was interpreted as the highest and most accurate similitude of all. Thus Bacon's use of imagination illustrates his ties to the traditions of Renaissance writing on psychology while also pointing forward to the work of the young Descartes.

GALILEAN ANALOGIES: IMAGINATION AT THE BOUNDS OF SENSE

By Lorraine J. Daston

GALILEO'S WRITINGS ABOUND in analogies: the moon is compared to a bleached and burnished silver plate; the earth to a moving ship populated by butterflies, birds, and bowls of fish; iron rods and hemp ropes to "ropes" and "rods" of sand and water; the motion of the Jovian moons to the vibrations of a pendulum. Yet Galileo subscribed to a view of the imagination, derived largely from Aristotelian sources, that severely restricted his use of analogy. While Gal- ileo was a master of the expository analogy-decking out new scientific ideas in similitudes, examples, and diagrams in order to reach an audience beyond the university lecture hall-he employed explanatory analogies only rarely, and then with evident reluctance. A third type, mathematical analogies, superseded ex- planatory analogies in Galileo's works, linking mathematics and the physical world by translating "this grand book, the universe," into "the language of mathematics"' and also by enriching the language of mathematics with concepts and approaches imported from physics.

In this essay I will argue that Galileo's vision of a reformed natural philosophy and his distrust of the imagination conspired to all but exclude explanatory anal- ogies from his scientific writings, although he granted wide scope to analogies of both the expository and mathematical type. Taking Galileo's discussion of the continuum as my text, I will describe the special circumstances that brought all three types of analogy to bear on this subject, with special attention to Gal- ileo's use of physicalist analogies in mathematics as the obverse of his better known mathematical approach to physics.

1 Galileo, The Assayer (1623), in Discoveries and Opinions of Galileo, trans. Stillman Drake (Garden City, N.Y.: Doubleday, 1957), pp. 237-238.

In the area of scientific discourse, however, Bacon ascribed to imagination a more positive and permanent place. The main interest of his treatment lies in his extension of standard rhetorical techniques from religion, politics, and law to the realm of science, a move wholly in keeping with his doctrine of the unity of knowledge. The analogies of scientific discourse, unlike those used in sci- entific discovery, were artificial similitudes concocted by an ingenious writer to illuminate a particular set of properties of an object or phenomenon-for ex- ample, the idea that the latent configuration of matter is in some respects like the texture of a piece of cloth. Less potent, but easier to control, their utility was ultimately more enduring.

But both discursive and inductive analogies, according to Bacon, relied on the power of imagination for their meaning and force. The aim of both was to move from object to image to concept, where the concept was interpreted as the highest and most accurate similitude of all. Thus Bacon's use of imagination illustrates his ties to the traditions of Renaissance writing on psychology while also pointing forward to the work of the young Descartes.

GALILEAN ANALOGIES: IMAGINATION AT THE BOUNDS OF SENSE

By Lorraine J. Daston

GALILEO'S WRITINGS ABOUND in analogies: the moon is compared to a bleached and burnished silver plate; the earth to a moving ship populated by butterflies, birds, and bowls of fish; iron rods and hemp ropes to "ropes" and "rods" of sand and water; the motion of the Jovian moons to the vibrations of a pendulum. Yet Galileo subscribed to a view of the imagination, derived largely from Aristotelian sources, that severely restricted his use of analogy. While Gal- ileo was a master of the expository analogy-decking out new scientific ideas in similitudes, examples, and diagrams in order to reach an audience beyond the university lecture hall-he employed explanatory analogies only rarely, and then with evident reluctance. A third type, mathematical analogies, superseded ex- planatory analogies in Galileo's works, linking mathematics and the physical world by translating "this grand book, the universe," into "the language of mathematics"' and also by enriching the language of mathematics with concepts and approaches imported from physics.

In this essay I will argue that Galileo's vision of a reformed natural philosophy and his distrust of the imagination conspired to all but exclude explanatory anal- ogies from his scientific writings, although he granted wide scope to analogies of both the expository and mathematical type. Taking Galileo's discussion of the continuum as my text, I will describe the special circumstances that brought all three types of analogy to bear on this subject, with special attention to Gal- ileo's use of physicalist analogies in mathematics as the obverse of his better known mathematical approach to physics.

1 Galileo, The Assayer (1623), in Discoveries and Opinions of Galileo, trans. Stillman Drake (Garden City, N.Y.: Doubleday, 1957), pp. 237-238.

ISIS, 1984, 75 : 302-310 ISIS, 1984, 75 : 302-310

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GALILEAN ANALOGIES

TYPES OF ANALOGY

In the Third Day of the Discourses and Mathematical Demonstrations Con- cerning Two New Sciences (1638), Galileo's spokesman Salviati firmly steers the discussion of natural accelerated motion away from the "fantasies" of causal hypotheses, preferring "to investigate and demonstrate some attributes of a mo- tion so accelerated (whatever be the cause of its acceleration)."2 This passage epitomizes the Galilean program in natural philosophy (the best we are likely to have, since Galileo was not given to sustained methodological reflection): the mathematical redescription of natural phenomena. Three aspects of this program should be emphasized here because they greatly constrained Galileo's use of analogies: first, the program's primary aim was descriptive rather than explan- atory; second, the form of the description was mathematical, thus narrowing the domain of natural philosophy to regular phenomena;3 and, third, the definitions and demonstrated properties of these select phenomena remained in almost all cases at the level of observables. Like Descartes, Galileo was an opponent of Aristotelian natural philosophy, but he did not share Descartes's interest in ex- plaining macroscopic appearances by appealing to microscopic mechanisms. Whereas Descartes's critique of Aristotle centered on the unintelligibility of scholastic categories, Galileo's complaint was that mere plausibility was allowed to take the place of demonstrative certainty in scholastic arguments.4

Galileo's vision of natural philosophy as the mathematical redescription of phenomena, coupled with his suspicion of the imagination on traditional Aris- totelian grounds, narrowed the scope for the use of explanatory analogies in his work. Galileo accepted the Aristotelian view that the imagination was at best a combinatorial faculty, limited to permutations of elements drawn from sensation, and at worst a source of distortion and error in straying too far from sensation. In his writing he repeats the clich6 that centaurs, sirens, and other fantastic creatures are but "a composite of things and parts of things seen at different times," and he commonly lumps the imagination together with "hallucinations" and "fantasies," opposing them to the authentic testimony of the senses.5 Above all, Galileo warned that the faculty of imagination is an impoverished source of causal explanations in natural philosophy because nature invents far more causes for the same effect than the human imagination can fathom.6

Consequently, when Galileo hazards one of his rare hypotheses, such as that concerning the causes responsible for the appearance of comets, he employs explanatory analogy in the most tentative vein. First he describes the image of a candle flame reflected in a clean carafe as a dot of light, then as a dot with a

2 Galileo, Discourses and Mathematical Demonstrations Concerning Two New Sciences (1638), trans. Stillman Drake (Madison: Univ. Wisconsin Press, 1974), p. 159.

3 Galileo despaired, for example, of any true science of resistance, which, "by reason of its mul- tiple varieties," could not be "subjected to firm rules, understood, and made into science"; Dis- courses, p. 224. Apparently, not all natural phenomena were regular enough to be mathematized.

4 See, e.g., Galileo, Discourses, p. 15; Galileo, Dialogue Concerning the Twvo Chief World Sys- tems (1632), trans. Stillman Drake (2nd ed., Berkeley: Univ. California Press, 1967), pp. 122, 185. I do not mean to suggest that Galileo was unconcerned with intelligibility (see Assayer, p. 241) nor Descartes with certainty, but only to suggest that their emphases differed.

5 Galileo, Dialogue Concerning World Systems, p. 62; and (fantasies), e.g., Galileo, Dialogue on Motion (composed ca. 1590), in Mechanics in Sixteenth-Century Italy, trans. Stillman Drake and I. E. Drabkin (Madison: Univ. Wisconsin Press, 1969), p. 333.

6 Galileo, Assayer, pp. 256-258.

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luminous "tail" after the carafe has been streaked by an oily finger. But then he immediately, and characteristically, qualifies the analogy: "I do not mean to imply by this that there is in the sky a huge carafe, and someone oiling it with his finger, thus forming a comet; I merely offer this as an example of Nature's bounty and variety of methods for producing her effects. I could offer many, and doubtless there are still others that we cannot imagine."7 Because nature's ingenuity outstrips human imagination, explanatory analogies are at best singled out of a myriad of possibilities, many unknown to us, and therefore cannot as- pire to the certainty required of any natural philosophy worthy of the name. Galileo thus would have rejected the fourth rule of Descartes's method-to enu- merate all possibilities-as humanly impossible.

However, while explanatory analogies are few and far between in Galileo's scientific writings,8 casual analogies, illustrations drawn from experience, and diagrams that color and clarify the exposition appear frequently in Galileo's dia- logues. Almost all of these devices render the point under discussion more vi- sually vivid, for Galileo subscribed to the Renaissance commonplace that vision was "the sense eminent above all others in the proportion of the finite to the infinite . . . the illuminated to the obscure."9 Galileo's major scientific works were cast in the form of dialogues among characters drawn from the educated gentry as well as from the university, and written in the vernacular in order to reach a far broader audience than, for example, the technical Latin treatise from the pen of "the Academician" quoted by Salviati at length in the Discourses. The demands of the new genre and its mixed audience of lay and learned readers taxed Galileo with Bacon's injunction to set forth new scientific ideas through striking comparisons and images.

The perplexed interlocuter of a Galilean dialogue, be he the apt Sagredo or the obtuse Simplicio, is enlightened by a variety of analogical and graphic aids. Similes highlight salient points and help wage arguments: it is easier to count stars fixed on a solid sphere, just as it is easier to count tiles inlaid in a courtyard than the children running around it. More familiar examples of the principle in question illuminate the obscure: just as cut velvet appears darker than taffeta cut from the same silk, so a densely forested patch on the moon would look darker than surrounding areas. Such analogies are sometimes drawn from the operation of machines, as when Galileo uses a comparison to the workings of a clock to justify an astronomical assumption that it will take the same body longer to trace out a larger circle thn a smaller one.10 Experimental illustrations refocus the mind's eye on the relevant aspects of experience, as in his use of the inclined plane experiments to introduce the concept of inertia. Galileo sup- plies diagrams "so that we can more clearly comprehend" a geometric dem- onstration or picture a concrete object.11 Yet all these expository analogies, de-

7 Galileo, Assayer, p. 261. See also the very tentative tone in which Galileo compares sunspots to smoke or clouds, breaking off the analogy lest he "mix dubious things with those which are definite and certain"; Galileo, Letters on Sunspots (1613), in Discoveries and Opinions, p. 140.

8 Though not wholly absent: see William R. Shea's interesting discussion of the lever as a model for hydrostatic phenomena in the Discourse on Floating Bodies (1612), in Galileo's Intellectual Rev- olution (New York: Science History Publications, 1972), pp. 16-18.

9 Galileo, Assayer, p. 277. 10 Galileo, Dialogue, pp. 120, 99, 457. 11 See, e.g., Galileo's explanation with diagram of the strength of ropes in Discourses, p. 18.

Winifred Wisan argues that Galileo uses experience to render the mathematical principles of me- chanics, which require no further confirmation, "evident to the intellect"; "Galileo's Scientific

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signed to clarify and instruct, differ distinctly from explanatory analogies, which seek to probe the unobserved causes of observed effects through what Galileo deemed to be an unreliable exercise of the imagination.

Galileo's mathematical analogies constitute the third category alongside his rare explanatory analogies and his ubiquitous expository ones. They are of three kinds: those that translate a physical effect into the "characters" of "triangles, circles, and other geometric figures";12 those that regroup within a single cate- gory phenomena previously seen as disparate; and those that fortify and extend mathematical notions and techniques with physical parallels. The first type of mathematical analogy has been the subject of much comment since Koyre's studies, and I will not enlarge upon that literature here except to note that re- definition plays a critical role both in recasting physical effects in mathematical terms and in judging which physical effects will prove tractable to such treat- ment.13 The second type, the reclassifying analogy, need not be mathematical, but for Galileo it often is, because he uses it in tandem with the first type of mathematical analogy. Mathematical redescription provides the middle term in such alliances. For example, a silver rod used for gilding and a flatbottomed grain sack can both be abstracted into cylinders by the talented natural philos- opher, who "must deduct the material hindrances," and they then become two examples of the same demonstrated properties.14

The third type of mathematical analogy, which inverts the order of the first by interpreting mathematical entities in terms of physical ones, typically occurs when Galileo approaches the boundaries of contemporary mathematical knowl- edge (as in his analysis of instantaneous velocities) or what he believes to be the boundaries of human understanding itself. As we shall see, these boundary conditions coincide in Galileo's discussion of the continuum, obliging him to have reluctant recourse to the imagination. Despite the risks of a forced reliance upon the imagination, Galileo was compelled to take up the question of the con- tinuum, for it was at once key to his understanding of both accelerated motion and the internal cohesion of matter.15 Although Galileo, like most of his con- temporaries, showed little interest in pure mathematics for its own sake, he wielded it as a means toward developing a demonstratively certain mechanics. Thus his discussions of motion, resistance, and cohesion hinge on original math- ematical claims. The apparent invisibility of mathematics per se in Galileo's writings may stem not only from his predominantly physical interests but also from the fact that the mathematics is presented in physicalist dress.

As Katharine Park's essay demonstrates, Renaissance psychology of both the Aristotelian and Neoplatonist varieties would have supported Galileo's use of the imagination in this capacity. Mathematics and the imagination, which oc- cupied parallel positions between sensation and pure intellect, were thus natu- rally paired in both theories. Geometric figures, being extended and therefore

Method: A Re-examination," in New Perspectives on Galileo, eds. Robert E. Butts and Joseph Pitt (Dordrecht: Reidel, 1978), p. 43.

12 Galileo, Assayer, p. 238. 13 Galileo's comments on the importance of mathematical definition are scattered, but suggestive:

see, e.g., Assayer, p. 241; Discourses, p. 36. 14 Galileo, Dialogue, p. 207. 15 See Maurice Clavelin, "Le probleme du discontinu et les paradoxes de l'infini chez Galilee,"

Thales, 1959, 10:1-26, for a discussion of the influence of Galileo's view of acceleration as the infinite sum of instantaneous velocities on his treatment of the continuum.

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LORRAINE J. DASTON

divisible, properly belonged neither to the intellect, which trafficked only in pure and indivisible simples, nor to imperfect, mutable sensation. Rather, they were, according to Proclus's prologue to his commentary on Euclid, "projections" of the ideas of the intellect into sensory forms purified of imprecision and exhibiting "many likenesses of divine things and also many paradigms of physical rela- tions." That is, mathematics, through the agency of the imagination, could render sensible objects intelligible and pure ideas visible. Mathematical dem- onstrations reflect this double mission: "Proofs must vary with the problem han- dled and be differentiated according to the kinds of being concerned, since math- ematics is a texture of all these strands and adapts its discourse to the whole range of things."16 Mixed mathematics in the Galilean mode would have been wholly congenial to this account of the mathematical imagination.

Here perhaps a cautionary note is in order concerning the tendency to conflate the "mixed mathematics" of the seventeenth century with modern "applied mathematics." The latter presupposes a body of pure mathematics that is con- ceptually, if not historically, distinct from its applications. Indeed, it is precisely this independence that accounts for the broad range of such applications for a single mathematical technique. For certain classes of problems within the clas- sical canon of exact sciences-harmonics, optics, statics, astronomy-Galileo and his contemporaries gave precedence to arithmetic and geometry over such "subordinate sciences," as Aristotle called them.17 However, mixed mathe- matics also embraced areas into which mathematics had been only partially in- troduced and which were rapidly evolving, such as the study of local motion; here mathematical forms followed physical intuitions, as in Barrow's views on the generation of mathematical magnitudes and Roberval's and Torricelli's methods of finding tangents. Such physical notions as instantaneous velocities and interstitial vacua supplied not only the occasion but also the content for new mathematical techniques. This was particularly true of the embryonic stages in the development of such techniques, when they still bore the specific imprint of their immediate origins.18

THE PROBLEM OF THE CONTINUUM

Thus it is not surprising to find Galileo's discussion of the structure of continuum tightly interwoven with his treatment of the structure of matter in the First Day of the Discourses. The problems had been closely associated in Averroist dis- cussions of minima naturalia in the fourteenth century and in sixteenth-century discussions at the University of Padua.19 Galileo, however, went beyond the

16 Proclus, A Commentary on the First Book of Euclid's Elements, trans. Glenn R. Morrow (Princeton: Princeton Univ. Press, 1970), p. 29. Francis Barozzi (Barocius) translated Proclus into a Latin edition published at Padua in 1560.

17 Aristotle, Posterior Analytics, 1.7, 75b14- 17. 18 See Isaac Barrow, Geometrical Lectures (1670), trans. J. M. Child (Chicago/London: Open

Court, 1916), pp. 37-38, 40; and Evelyn Walker, A Study of the "Traite des indivisibles" of Gilles Persone de Roberval (New York: Teachers College, Columbia University, 1932), pp. 127-130, 137- 139. The incorporation of physical notions into mathematics was not uncontroversial at the time: Cardan for one objected to invoking physical postulates in what he viewed as the strictly mathe- matical problem of Aristotle's wheel; see I. E. Drabkin, "Aristotle's Wheel: Notes on the History of a Paradox," Osiris, 1950, 9:162-198.

19 See John Murdoch, "Superposition, Congruence, and Continuity in the Middle Ages," Melanges Alexandre Koyre (Paris: Hermann, 1964), Vol. I, pp. 416-441; A. Mark Smith, "Galileo's Theory of Indivisibles: Revolution or Compromise?" Journal of the History of Ideas, 1976, 37:571-588.

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medieval identification of physical minima and mathematical points into a theory of cohesion, rarefaction, and condensation, on the one hand, and a theory of a continuum neither actually nor potentially composed of indivisibles, on the other. Galileo's treatment of these problems is a particularly revealing example of his use of analogical reasoning in two ways: first, as a striking illustration of the interplay of mathematical physics and physical mathematics in both concept and method; second, as one of Galileo's rare explanatory analogies that sought a subsensory explanation for manifest effects.20 The continuum lies at the margin of Galilean science, delving into the forbidden domains of the subsensible and the infinite; and it is just this marginality that forces Galileo to enlist the imag- ination, however cautiously, in order to render the invisible and unintelligible visible and comprehensible.

Galileo's representative Salviati introduces a highly tentative solution to the problem of cohesion in guarded tones, echoing those in which Galileo couched his few other explanatory analogies: "I shall tell you what has sometimes passed through my mind [persato per l'imaginatione] on this; I do this not as the true solution, but rather as a kind of fantasy [fantasia] full of undigested things that I subject to your higher reflections."21 Salviati goes on to suggest that materials may be held together by innumerable tiny interstitial vacua-in fact, an infinite number of infinitesimal vacua-which act according to the principle that nature abhors a vacuum. Because of this very principle, the voids cannot be extended; but Galileo apparently believes that if they shrink to indivisibles each will be, so to speak, below nature's notice, even though their summed effect will be formidable, expressed as the resistance of solid bodies to dissolution.

By his own lights, Galileo is now doubly imperiled. He has descended to the subsensory level of microscopic causes to explain macroscopic effects, and he has, moreover, broached the "paradoxical" questions of infinites and indivisi- bles. As usual, Galileo takes a dim view of whatever aid the flawed human imag- ination might offer in such straits, for, as Salviati is made to remark apropos of such paradoxes: "These are among the marvels that surpass the bounds of our imagination, and that must warn us how gravely one errs in trying to reason about infinites by using the same attributes that we apply to finites; for the na- ture of these have no necessary relation between them."22 Yet Galileo perse- veres, despite his misgivings, and moreover draws heavily upon the resources of the imagination in order to make his points. As Descartes tartly observed to Mersenne concerning this passage: "He [Galileo] errs in all that he says about the infinite, for notwithstanding his admission that the finite human mind is not capable of understanding it, he goes right on to discuss it as if he did."23

In order to explain how finite extension could encompass an infinite number of such voids, Galileo turns to the venerable paradox of Aristotle's wheel, taken from the pseudo-Aristotelian De mechanica. Thus he conflates the physical and mathematical continua and surreptitiously adds the element of motion. Galileo's

20 This is one of only a handful of instances in which Galileo broached the question of atomism; see William R. Shea, "Galileo's Atomic Hypothesis," Ambix, 1970, 17:13-27.

21 Galileo, Discourses, p. 27; Opere di Galileo Galilei, ed. Antonio Favaro (Florence: G. Barbera, 1898), Vol. VIII, p. 66.

22 Galileo, Discourses, p. 46. 23 Marin Mersenne, Correspondance, ed. Cornelis de Waard (Paris: Editions du Centre National

de Recherche Scientifique, 1963), Vol. VIII, pp. 97-98; my translation. The letter is dated 11 Oct. 1638.

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LORRAINE J. DASTON

strategy is first to reduce the concentric, revolving circles, which he calls "poly- gons of infinitely many sides," to polygons of a finite number of extended sides, "the effect of which is intelligible and already understood" (see Fig. 1). He then uses this more intelligible case of the polygon to illustrate his view of the con- tinuum (both physical and mathematical) as a porous tissue of "full" and "empty" indivisibles.24

In essence, Galileo argues that the trajectory of the inscribed polygon, driven by the circumscribed one, contains gaps that increase in number and decrease in size as the number of sides multiplies. The hexagon HIKLMN traces five and a fraction gaps in its path as it is rolled around by the larger hexagon ABCDEF. Generalized to the case of the circle, which is pictured as a polygon in which each point on the circumference counts as a "side," the gaps become indivisibles and infinite in number, and the trajectories of the large and small circles become equal. These indivisible holes in the line correspond to the infinite and indivisible voids that riddle all matter. Thus Galileo attempts to simultaneously solve the mathematical paradox of Aristotle's wheel (how can the trajectories of the two

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circles be equal after one revolution of each when their circumferences are pat- ently unequal?) and the physical puzzles of rarefaction, condensation, and cohe- sion with a new model of the continuum that is neither unequivocally physical nor unequivocally mathematical. The new model accounts not only for how an ounce of gold might expand to fill the celestial spheres, or how the earth might contract to the radius of a nutshell, but also for the perplexing properties of lines and numbers (e.g., how integers and their squares may be put into one-to- one correspondence.)

This frontier case penetrates below the level of the senses, extends beyond the range of extant mathematics, and skirts the infinitely large and the infinitely small. Galileo must appeal constantly to the imagination to unfold his argument by forging analogies between the macroscopic and the microscopic, between the physical and the mathematical, and between the finite and the infinite, even over his own protests that such matters are "inherently incomprehensible." Subtle fire-particles go "snaking among the minimum particles of this or that metal" and dissociate them by filling the indivisible voids; the infinitude of unity dis- solves into "a single continuum, fluid perhaps, like water or mercury"; a line is

24 Galileo, Discourses, p. 33; see Drabkin, "Aristotle's Wheel," for a history and analysis of the paradox.

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"bent" into an infinite number of parts "at one fell swoop," thus sidestepping (or so Galileo hopes) the scholastic distinction between actual and potential in- finites.25 Galileo's new model of the continuum was motivated by physical prob- lems and imbued with physical notions: its dotted-line construction derives from a problem in mechanics; its porous fine structure expands and contracts like gunpowder or dew to solve problems of the correspondence of two unequal line segments, each composed of an infinite number of indivisibles; it dictates that the properties of unity should be those of a fluid. Conversely, this physicalized continuum served Galileo as a mathematical model for the physical effects of cohesion, condensation, and rarefaction.

ANALOGIES OF METHOD AND SUBSTANCE

Analogies of method also unite the mathematical and the physical in Galileo's writings, and the case of the continuum shows these connections to good ad- vantage as well. Galileo's conviction that mechanics should aspire to the de- monstrative certainty of mathematics through the demonstrative methods of mathematics is well known.26 The belief that mechanics and mathematics were both demonstrative sciences licensed Galileo to extend the parallel from methods of justification to those of discovery as well, but in this case from me- chanics to mathematics rather than the reverse. As Salviati tells Simplicio apropos of Aristotle's arguments for the incorruptibility of the heavens, in the demonstrative sciences a posteriori observations and experiments usually pre- cede a priori reasons because of the happy fact that true conclusions lead to true arguments, as well as the reverse. Aristotle and Pythagoras, physicist and mathematician, both profit from this symmetry of analytic and synthetic methods.27

Galileo supplemented this familiar complementarity of the methods of analysis and synthesis (in which analysis has been expanded to include observations as well as known true propositions as points of departure) with the distinctive method of approximative series.28 This method, which permitted him to extrap- olate from a carefully ordered sequence of physical effects to the unattainable ideal case, yielded some of Galileo's most ingenious physical arguments. It en- abled him, for example, to reason from inclined planes of ever steeper slopes to the perpendicular case of free fall, or from times of descent of unequal weights in media of decreasing density to the case of equal times in a vacuum. These series distill the essential from the merely accidental and provide "tan- gible evidence" that Galileo's mathematical idealizations are at least continuous with observed effects ranged in the proper order.

Galileo's reasoning on the nature of infinity runs along similar lines, despite his warnings, quoted earlier, of the hazards of assuming the continuity of finite and infinite cases, between which "no necessary relation" exists. In his writings Galileo seems to prefer paradox to discontinuity: when, for example, the ap- plication of the principle of continuity leads to an apparent equality of a point

25 Galileo, Discourses, pp. 38, 27, 47, 54. 26 For a comprehensive account, see Wisan, "Galileo's Scientific Method." 27 Galileo, Dialogue, p. 51. 28 See Neal Gilbert, Renaissance Concepts of Method (New York: Columbia Univ. Press, 1960),

Chs. 3, 7; Walter Ong, Ramus, Method, and the Decay of Dialogue (Cambridge, Mass.: Harvard Univ. Press, 1958), Ch. 11.

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and a line, he defends the procedure as "consistent," despite the "repugnance and contrariety of nature encountered by a bounded quantity in passing over to the infinite."29 Similarly, when Salviati observes that the number of squares and cubes proportionally decreases as we count higher-that is, that the series was not in some sense converging toward the intended goal-he reverses the direc- tion of his search for the infinite: "Hence it is manifest that to the extent that we go to the greater numbers, by that much and more do we depart from the infinite number. From this it follows that turning back (since our direction took us always farther from our desired goal), if any number may be called infinite, it is unity."30 From the observable to the ideal in mechanics, from the finite to the infinite in mathematics-in both spheres, continuity underwrote Galileo's generalizations from ordered specifics to limiting case, albeit with mixed results.

Analogies of substance and method thus link mathematics to physics, as well as the reverse, in Galileo's discussion of the continuum. Every successful analogy modifies both of its terms in connecting them: for example, when Gal- ileo likens a "rope" or "rod" of water or sand to iron rods and hemp ropes in his treatment of the strength of materials, we think differently of both water and ropes thereafter.31 In this essay I have emphasized the neglected side of the analogy that Galileo creates between mathematics and the physical world in order to redress the balance as well as to illustrate the full range of Galileo's analogical reasoning. As I mentioned earlier, I do not believe that Galileo's use of physicalist analogy was altogether exceptional among mathematicians of this period; Torricelli and Roberval, and to a lesser extent Cavalieri, seem to be particularly promising subjects of further study in this regard. The field for in- vestigation of analogies between the methods of mathematics and the physical sciences in the early seventeenth century is broader still. In addition to the ex- tension of mathematical analysis and synthesis to physical resolution and com- position, the kinship between indirect proof and crucial experiment might also be explored.32 Such physico-mathematical hybrids might be reasonably sought in attempts to extend mathematics to new kinds of physical problems.

For Galileo, the problem of cohesion provided just such an occasion. But, wedded as he was to a theory of the imagination and a vision of natural phi- losophy that restricted the legitimate uses of analogy to exposition, he could only regard such forays into the shadowy realm of infinites, indivisibles, and subsensibles with grave doubt. When Sagredo accepts Salviati's "fantasy" in the provisional spirit in which it was tendered, suspending judgment as to "whether in fact nature proceeds in any such way," he voices Galileo's own reservations concerning the trustworthiness of the imagination in natural philosophy. Thus Descartes unwittingly echoed Galileo's own worst fears when he wrote to Mer- senne that all of Galileo's speculations on the infinite were utterly false-that they were, in Descartes's words, nothing but "pure imagination."33

29 Galileo, Discourses, pp. 36-37, 46. 30 Ibid., p. 45. 31 Ibid., p. 23. 32 See A. I. Sabra, Theories of Light from Descartes to Newton (Cambridge: Cambridge Univ.

Press, 1981), p. 41. 33 Mersenne, Correspondance, Vol. VIII, p. 98. See Pierre Costabel, "La roue d'Aristote et les

critiques franqaises l'argument de Galilee," in Galilke: Aspects de sa vie et de son oeuvre (Paris: Presses Universitaires de France, 1968), pp. 277-288, for a complete account of French responses to Galileo's treatment of the continuum.

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