Galileo Theory Indivisibles

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Galileo's Theory of Indivisibles: Revolution or Compromise? Author(s): A. Mark Smith Source: Journal of the History of Ideas, Vol. 37, No. 4 (Oct. - Dec., 1976), pp. 571-588Published by: University of Pennsylvania PressStable URL: http://www.jstor.org/stable/2709025Accessed: 06-05-2015 07:05 UTC

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GALILEO'S THEORY OF INDIVISIBLES: REVOLUTION OR COMPROMISE?

BY A. MARK SMITH

E. A. Burtt would have us believe that Galileo stripped matter of all but "being" and "being in motion," thus banishing man from a deter- minist Democritean world into a sensory limbo of secondary qualities.' Because Galileo's commitment to atomism led him to view Nature as a purely kinetic reality comprehensible in mathematical terms alone,2 he made of the world a "vast, self-contained, mathematical machine."3 So "at the price of a sort of denaturing of nature,"4 he supposedly de- stroyed the Aristotelian plenum, denying metaphysics its traditional wealth of qualities and relationships.

Alexander Koyre certainly followed Burtt's lead to some extent but shifted emphasis to Galileo's radical esprit de geometrie. In his view Galileo broke the stranglehold of Aristotelian metaphysics with his bold "substitution of the abstract space of Euclidean geometry for the concrete space of pre-Euclidean physics."5 And by ruthlessly geometrizing space (and time) he left himself no choice but to adopt Democritean atomism as the ontological basis for his kinetic world- view. The stage had thus been set by Galileo for the culminating Newtonian concept of an a priori space-time continuum to which matter was in a sense merely accidental.

Such an hypostasis of Euclidean space is the central issue in this metaphysical evaluation of Galileo's science. Implicit in Burtt's study, explicit in Koyre's, the notion of Pure Space as a real geometrical back- drop for the new kinetic analysis of Nature explains Galileo's total de- fection from the Aristotelian camp. But just how total was this defec- tion? While it is undeniable that he devoted a great portion of his life to undermining Aristotelian physics, Galileo seems never to have felt the same compulsion to uproot traditional metaphysics. Quite the contrary, he demurred when faced with the crucial and potentially wrenching

'Edwin A. Burtt, The Metaphysical Foundations of Modern Science (Garden City, N.Y., 1954). 2Ibid., 99. 3lbid., 104.

4G. Gusdorf, La Revolution Galileenne, Vol. I (Paris, 1969), 95-my translation. 5Koyre, Etudes Galileennes, Vol. I, "A l'aube de la science classique," Actualites

Scientifiques et Industrielles, no. 852 (Paris, 1939), 9-my translation. Cf. also A. Koyre, "The Significance of the Newtonian Synthesis," Newtonian Studies (Chicago, 1965), 6-7. It is interesting to note that, in dealing with Galileo's analysis of accelerated motion, Koyre states that "space is only a result, an accident, a symptom of an essentially temporal reality." Etudes Galileennes, Vol. II, in Actualites ... no. 853, p. 75-my translation.

571


572 A. MARK SMITH

problem of possible material discontinuity within the traditional world- plenum. And it was precisely his reliance on geometry in neutralizing the problem that permitted him to preserve, essentially intact, the Aris- totelian view of the continuum that he is supposed by Burtt and, more especially, Koyre to have supplanted.

To support this contention I shall offer a brief analysis not only of Galileo's theory of indivisibles in the "First Day" of the Discorsi6 but of its implications with regard to both Aristotelian metaphysics and the application of Euclidean geometry in interpreting the world. In so doing I hope to demonstrate three basic points:

1. that Galileo's theory of indivisibles did not take on the radical hue of Epicurean atomism;

2. that his much-touted devotion to geometrical analysis, far from leading him to reject the essential Aristotelian view of the world-plenum or continuum, was totally compatible with it;

3. and, consequently, that his metaphysics, though undoubtedly influenced by developing mathematical and scientific trends away from the Aristotelian tradition, offered no substantial break with that tradition.

Although Galileo's recorded speculations on atomism, spanning a period of some thirty years, appear to fall into three phases,7 we shall concern ourselves only with the final one developed in the "First Day" of the Discorsi. The choice is deliberate. Not only is this Galileo's last word; it represents his most extensive and metaphysical treatment of a topic in which "his interest ... was derivative"8 and consequently spurred only by specific problems within larger issues. Bearing this in mind, let us turn to the "First Day" and pick up the thread of argument with Sagredo's conclusion that a vacuum can indeed exist "at least for a very brief time."9

This caps the discussion provoked by Salviati's distinction between the resistance due to the "repugnance that nature has against allowing a void to exist" (adhesion) and that which "tenaciously connect[s] the particles" of any body (cohesion).10 In this instance adhesion is the natural resistance to the formation of a void that prevents polished plates when joined from separating easily under force. In opposing the

6Discorsi e dimostrazioni matematiche intorno a due nuove scienze in Le Opere di Galileo Galilei (Edizione Nazionale, Vol. VIII), ed. Antonio Favaro (Florence, 1898)-henceforth referred to as Ed. Naz. For my English citations I have relied almost exclusively on Two New Sciences, trans. Stillman Drake (Madison, 1974). In subsequent footnotes I shall first cite the Drake translation followed by the Ed. Naz. page reference in brackets.

7Cf. William R. Shea, "Galileo's Atomic Hypothesis," Ambix, 17 (1970), 13. 8Idem. 9Two New Sciences, 20 [Ed. Naz., 60]. '0Ibid., 19 [59]-my parentheses. I have inserted the terms "adhesion" and "cohe-

sion" to sort out Galileo's conceptions more easily.


GALILEO'S THEORY OF INDIVISIBLES 573

creation of such a vacuum between the surfaces, each plate, as a sort of occupying medium, clings to its mate to forestall what Nature so abominates. Yet when the force of separation is large enough, the sur- rounding air rushes almost immediately to fill any intervening space, breaking the resistance and allowing the plates to fall apart." How else can we explain this, asks Salviati, without admitting the actual existence, however short-lived and problematical, of a vacuum? But in accounting for adhesion in such a general way Galileo opens up the possibility of a broader application to cohesion. Why may we not reject Salviati's distinction and assume, with Sagredo, that the cause offered by the "void, which surely does exist, suffice[s] also for all resistances" to separation?12 In other words, why not explain both forms of resistance through the same causal mechanism?

In terms of the conditions already established for adhesion the proposition is absurd; the macroscopic force that holds smooth plates together is simply not powerful enough to account for the cohesion of homogeneous solids like copper. To demonstrate, Salviati cites the "adhesive" property of water,13 arguing that it is by this alone that a column of water will support itself to a height of eighteen cubits before "it breaks, just as if it were a rope."14 Taken generally, the adhesive force in a column of any material whatever is equivalent only to the weight of eighteen cubits of water in a column of equal diameter.15 Obviously a far stronger internal cement than this must be sought.

In fact, Sagredo has anticipated the direction of search in an apparent non sequitur that Salviati now puts to good use. If a general levy of farthings suffices where a fortune in gold does not, why not by analogy accept the notion of "tiny voids operating on the most minute particles

1Ibid., 19-20 [59]. This inrush takes time; hence, contrary to Aristotle's contention, not only can a vacuum exist (as a receptacle for incoming air), but motion through it cannot be instantaneous. Actually, this conclusion is not quite as unequivocal as it may at first appear, for Aristotle-via his notion of causality-is allowed the parting shot. "I cannot see," says Sagredo, "how the cause of adherence of the two slabs and their repugnance to being separated-effects that are actual-can be a void that does not exist [first], but which must follow" (21 [60]-author's brackets). Simplicio offers the solution that, "since void space is self-refusing, nature prohibits any action in consequence of which a void would follow, and such is the separation of the two surfaces" (Idem). In short, the vacuum exists as a potential, the actualization of which Nature is constantly on guard to thwart. Now, with this hedging compromise-apparently acceptable to all three disputants-the critical analysis is abruptly dropped. Henceforth "vacuum" is to be accorded a strictly essential (as opposed to existential) status only insofar as it denotes "resistance to separation"; but no matter-it is merely thefact, not the "how" of its existence that Galileo is so eager to establish. See also 71 [112].

'2Ibid., 21 [61]. '3lbid., 23-25 [62-64]. '4Ibid., 25 [64]. In short, the particles of water have no cohesive bonds whatever. '5Idem. Without the special notion of atmospheric pressure, Galileo is forced to view

this "adhesion" as a universal property-a force constantly added to that of "cohesion."


574 A. MARK SMITH

[of matter] so that the same coinage as that with which the parts are joined is used throughout"?16 Like countless farthings, these tiny interstitial vacua, so insignificant singly, might still add up to gargantuan sums. 17

So far we have been gently and cogently drawn by experience and reason through a train of apparently valid conclusions to precisely this understanding of cohesion. After all, adds Sagredo (who is rarely slow to seize the point): A number of ants might bring to land a ship loaded with grain, for our eyes daily show us that an ant can readily transport a grain, and it is clear that in the ship there are not infinitely many grains, but some limited number. We can take a number several times as great, and put that number of ants to work; and they will bring to land not only the grain, but the ship along with it. It is true that the number would have to be large, but in my opinion so is that of the voids that hold together the minimum particles of a metal.'8

But Salviati, at his Socratic best, turns reason to unreason; "if an infinitude [of voids] were required you would perhaps hold this to be impossible?"19 And with this query he sets us all to chasing the infinite and the infinitesimal through a tortuous course of not-quite logic.

Such an unexpected break in pursuit of two concepts that are "inherently incomprehensible to us"20 seems to signal a drastic switch in Galileo's approach. Yet there is method in his radical departure from the moderate path of experience, for, like the paradoxes he will attempt to unravel, Nature offers apparent dichotomies. On the one hand, matter acts and reacts in ways that strongly suggest structural discontinuity while, on the other, it presents a facade of seemingly perfect continuity. The problem is the age-old one of saving the phenomena without transgressing rational limits; to solve it Galileo will invoke the linear and "spatial" continuum of Euclidean geometry.

What must be done is to effect a smooth transition between the asymptotic approach of reason and its goal of infinity; somehow the actuality of composition, of an infinite aggregation of voids and indivisibles subsumed within a material continuum, must be rendered logical.21 So, "since paradoxes are at hand," as Salviati observes, "let us see how

"Ibid., 27 [66]. '7This is especially true as the effective surface area of solids vastly increases in pro-

portion to the volume and, consequently, the weight as those solids diminish in size. Ibid., 93-95 [133-34].

18Ibid., 27 [67]. '9Idem. 20Ibid., 38 [76]. 21"It will be precisely in attempting to solve the intrinsic contradiction of a world, in

which the ideal of resolving the continuum into elementary component parts is constantly undermined by the need to recognize the infinitude of those very parts, that Galileo will submit the coherence of his conception to the test of a semi-mathematical, semi-logical analysis." Maurice Clavelin, "Le probleme du continu et les paradoxes de l'infini chez Galilee," Thales, 10 (1959), 8-my translation.



it might be demonstrated that in a finite continuous extension it is not impossible for infinitely many voids to be found."22 The geometrical demonstration will center on the well-known phenomenon of Aristotle's wheel.

Let us begin with concentric polygons rolling and jouncing along their respective paths, the larger carrying the smaller. Regardless of the number of sides, the smaller polygon by jumping will always mark out a path approximately equal to that laid off by the larger. Take however many sides as you please, the situation remains invariant. So far the reasoning remains faultless. Now, let us add just one side more than "as many as you please"; let us make the conceptual shift to "polygons of infinitely many sides."23 Let us, in short, conceive of circles. By extension, the inner polygon/circle will still trace out a path "approximately equal" to that of the outer by an alternation of sides and spaces, for after all, "how . . . without skipping, can the smaller circle run through a line so much longer than its circumference?"24 And since these skips must always be commensurate to the sides, they must be infinite in number and so small as to be unquantifiable.

Thus, granted his assumption of a real transition from finite to infinite, Galileo has led us to concur that a determinate path, an Euclidean line ostensibly continuous, can yet consist of "infinitely many points, part of them filled points and part voids."25 Moreover, he has shown it to be equal and, to all appearances, perfectly similar to that fully continuous line traced by the larger circle. Briefly, he has demonstrated the existence of composition in a linear continuum, doing only the subtlest violence to reason in preparing the way for a final step into di- mensionality: What is thus said of simple lines is to be understood also of surfaces and of solid bodies, considering those as composed of infinitely many unquantifiable atoms.... In this way there would be no contradiction in expanding, for instance, a little globe of gold into a very great space without introducing quantifiable void spaces-provided, however, that gold is assumed to be composed of infinitely many indivisibles.26

In applying the paradox of Aristotle's wheel, in extending his conclusions to the material world, Galileo will account for a host of phenomena like expansion, contraction,27 gross impenetrability, subtle permeability, liquefaction and solidification-all within the context of a compositional continuum. However, for those of us like Simplicio to whom "this com- posing the line of points, the divisible of indivisibles, and quantified of

22Two New Sciences, 28 Ed. Naz. [68]. 23Ibid., 33 [71]. 24Ibid., 31 [70]. 25Ibid., 33 [71]. 26Ibid., 33-34 [72]. 27In "contraction" the circles switch roles, the smaller carrying the larger, while the

larger traces out a path "approximately equal" to the circumference of the smaller. Ibid., 55-57 [93-96].


576 A. MARK SMITH

unquantifiables" constitutes a "reef... hard to pass,"28 Galileo must first undertake a more profound and abstract theoretical analysis.

At Galileo's behest, Simplicio is laboring under the gross, but highly instructive, misapprehension that infinity is operationally extensive and that it represents a goal to be won either by division or addition. To the contrary, the infinity about which Galileo is concerned is never to be reached or built; it is fully contained within determinate bounds. For infinity is the measure of indivisibles; and, like points in lines, indivisibles cannot be juxtaposed to create what "cannot be constructed out of two or ten or a hundred or a thousand ... but requires an infinite number."29 Like the totality of points in a line or indivisibles in a body, infinity is "equally" and always contained without thereby rendering its container extensively infinite.

The obstacle to Simplicio's understanding lies in his inability to think in other than extensional terms; his is a frame of mind that will always prompt misapplications of finite criteria, such as comparatives, to the infinite. As Salviati warns: These are some of the difficulties that derive from reasoning about infinites with our finite understanding, giving to them those attributes that we give to finite and bounded things ... for I consider that the attributes of greater, lesser, and equal do not suit infinities, of which it cannot be said that one is greater, or less than, or equal to, another.30 To illustrate, he offers a proposition: .... If I say that all numbers, including squares and non-squares, are more [numerous] than the squares alone, I shall be saying a perfectly true proposition; is that not so?31

Without the slightest hesitation Simplicio agrees, leaving Salviati to continue: But if I were to ask how many roots there are, it could not be denied that those are as numerous as all the numbers, beause there is no number that is not the root of some square.32 The point is this: although operational analysis may reveal that in a passage to greater and greater whole numbers the proportion of perfect squares diminishes, logic demands that the totality of such squares be equal to the totality of whole numbers.33 Consequently, while progress through the whole-number series may seem to draw us ever closer to infinity, it actually carries us away by pushing the ultimate limit of the succession of perfect squares always further beyond reach. Ironically,

28Ibid., 34 [72-73]. 29Dialogues Concerning Two New Sciences, tr. Henry Crew and Alfonso de Salvio

(New York, 1914), 31 [Ed. Naz., 77]. 30Two New Sciences, 39-40 [Ed. Naz., 77-78]. 31Ibid., 40 [78]-translator's brackets. 32Idem. 33Ibid., 40-41 [78-79].



by imposing extensional criteria any process aiming at infinity will only guarantee its own deflection from the target. Just so, the search for a lower limit by division into ever-decreasing but finite parts will prove to be a futile chase after the indeterminate. The very act of approach vi- tiates its own purpose, rendering its object-infinity-a potential rather than an actual quantity, exactly as Aristotle would have it.34 In granting therefore "to the distinguished philosophers that the continuum contains as many quantified parts as they please," Salviati is merely allowing that "the quantified parts in the continuum ... [are] neither finite nor infinitely many, but so many as to correspond to every specified number."35 Thus an indefinite "intermediate term" (every assigned number), neither finite nor infinite, must serve to describe the Aris- totelian continuum of spatial and temporal process.

Having shown that addition and division bring us not to the infinite but rather to the indefinite, Galileo has cleared the path for one last paradox: that infinity, insofar as it is number, is a kind of unity. Unity-"a square, and a cube, a fourth power, and all the other powers"36 containing an infinite process without containing parts-is yet contained within all number. And by analogy infinity, insofar as it is a magnitude or quantity, cannot contain; it must be contained. For only in being contained or defined within a determinate whole can infinity make the "transition" from potency to act. Indeed, while "we understand well that there cannot be an infinite circle . . . that still less can a sphere be infinite; nor can any other solid or surface having a shape be infinite,"37 we must also understand that every circle, sphere, body, and surface contains infinity by virtue of its unity or completeness. And understanding this, we now know why Galileo so readily forsakes lofty speculation to reveal, by the simple manipulation of a line into a circle, "the method ... of distinguishing and resolving the whole infinitude at one fell swoop."38 At this point he has, for all intents and purposes, taken us me- thodically full circle back to the physical world; henceforth he will apply the theory that he has derived and vindicated with such art.39

34Ibid., 42-45 [80-82]. 35Ibid., 43 [81]. 36Ibid., 45 [83]. 37Ibid., 47 [85]. 38Ibid., 54 [93]. 39 It is clear that, taken at face value, the hypothesis of indivisibles has given Galileo

what he deems the best of two worlds. While, on the one hand, saving the continuum in order to avoid an embarrassing consequence of grape-shot atomism-undue pene- trability of bodies-his theory still offers the advantage of ultimate material discreteness, subtle permeability. Furthermore, it accounts for a wide variety of apparent quantitative and qualitative changes in Nature without recourse to a welter of explanatory devices tailored to specific problems. So, for example, melting of metals is accomplished by the "subtle fire-particles [which] ... by filling the minimum voids distributed between . . . minimum particles [of metal], free them from that force . . . forbidding their separation" (27 [66]). In this way solids become liquids, being perfectly "resolved into indivisibles, infinitely many," (p. 47 [p. 85]) without thereby truly losing quality or quantity. And, by the same token, liquids become solids, gaining consistency and increasing solidity with the interposition of cohesive vacua.


578 A. MARK SMITH

To preserve the heterogeneity of experience in a peculiarly homogeneous world Galileo has been, to some extent, "traveling along the road of those voids scattered around by a certain ancient philosopher"40 in an itinerary that has often been difficult to follow. And up to now our task has been to analyze this journey and its terminus, the theory of indivisibles, from an internal vantage, attempting to understand it in its own terms without meanwhile searching for ulterior significance. But now the time has come to pose the "why" where formerly we had asked the "how" of Galileo's theory, to explore its implications in terms of possible intellectual as well as historical precedents and motivations.

Historically atomism represented a major response to the Eleatic challenge of reconciling fixity and flux, Same and Other, in a world so often subject to lawless shift and variation. Conferring "being" on Void or "non-being" and filling it with indivisible particles, the first atomists attempted reconciliation by exploiting change, not by explaining it away. Thus a kinetic reality of myriad particulate contacts was created out of featureless space-"that within which"-and primordial, discontinuous substance-"that which"-to form a cosmos that accorded rather well with that of the Pythagoreans whose "number atomism had run itself into a dead end."4' In short, the atomists reduced the Many, if not to One, at least to three-matter, space, and motion-in a system that ought to have appealed to anyone willing to denude the world of real qualities to get at an underlying consistency.

But there were problems. Not only did the ontological status of the void present logical difficulties; the concept of extended material indivisibles was due to suffer, by association, the fate of the Pythagorean monads.42 For Greek philosophy still bore the scars of what Santillana calls "the crisis of the irrational."43 What once had been a bold foray into the metaphysics of numerical discreteness had turned by now into a discretionary retreat to geometry. And there was no place in the new geometry for spatial, material, or conceptual discontinuity.

Thus, within the new metaphysical pale of geometrical continuity the Parmenidean challenge was subjected to a fresh attack that midwifed two monumental solutions. Platonism, from its essentially static perch, opted to abandon the quest for a real, invariant material substratum in favor of a formal superstratum by virtue of which all worldly "images" might possess a modicum of participative being.44 Aristotelianism, far

40Ibid., 34 [72]. 41Giorgio de Santillana, The Origins ofScientific Thought (New York, 1961), 70. 42Although "Democritus seems to have discriminated clearly between physical and

mathematical atoms-as did his later follower, Epicurus," Aristotle apparently failed to note or credit the distinction and lumped him with the number atomists. See Carl B. Boyer, The History of the Calculus and its Conceptual Development (New York, 1949), 22. 43Santillana, The Origins ..., 69-73.

44"By all that is, this is a world of image and unreality, a world given not to reason but to sense. Hence, in view of the assumption that the world of sense experience is



less ethereal in its vantage, attempted to confront the problem by turn- ing directly to the mire of worldly flux and sounding for a bottom. Each "system" manifested a necessary response to the "crisis of the irrational," but the respective forms of that response were literally worlds apart. On the one hand, Platonism sought to make the irrational, the in- commensurable, reasonable but in so doing got bogged down in a geometrical conceptualism that, in many ways, matched the arithmo- logical mystique of the Pythagoreans. The world might well be reflective of geometry; yet, after all was said and done, it was only an imperfect reflection of that higher perfection. But the Aristotelian response-and this is what makes it so germane to our analysis-was to regard geometry as reflective of the world and its processes.45 Rather than permit geometry to divert him irremediably from sensible experience, Aristotle used it in a very fundamental way to interpret that experience. This point is so crucial that it warrants some discussion.

There is little question that Aristotle was well aware, at least in a general way, of contemporary mathematical trends46 without being seduced into speculative flights by their theoretical implications. His down-to-earth attitude forced him to regard geometry as a mere "para- digm of knowledge" to be used for ulterior analyses, not as an end in itself.47 However, in spite of his tendency to shun pure mathematics, he seems to have exerted some, if not a great deal of, influence on the procedural formation of Euclid's Elements.48 Hence, despite its ostensibly Platonic cast,49 the Elements bears witness to a demonstrative rigor, not to mention an axiomatic and definitional concern that springs directly from Aristotle's work with deductive logic. And, more important, it faithfully, albeit abstractly, reflects his view of the real world of extension and process.

Points, lines, and planes-from the Aristotelian point of view these

characterized by intrinsic instability and non-permanence, by imaging, Plato's approach to the principles which account for the world cannot be inductive. There is no potential intelligibility to be actualized from that which always becomes and never is, nothing within image-beings which the mind can draw out, no possible content in terms of which human reason can be able to organize them into the unity of science." Leonard J. Eslick, "The Material Substrate in Plato," The Concept of Matter in Greek and Medieval Philosophy, ed. Ernan J. McMullin (Notre Dame, 1963), 40.

45Carl B. Boyer, The History of The Calculus . . ., 38. Barring the assumption of perfect circular motion in the super-lunary region and perfect straight-line motion in the sub-lunar sphere, Aristotle, unlike Plato, tended to shy away from mystical geometrical or arithmetical apriorism.

46Carl B. Boyer, A History of Mathematics (New York, 1968), 108. 47G. E. R. Lloyd, Aristotle: The Growth and Structure of His Thought (Cambridge,

1968), 125. Mathematics was for Aristotle the analytic key to a clear conception of deductive reasoning.

48Euclid's Elements, trans. T. L. Heath (New York, 1956), "Introduction," I, 119-24 and 146-48. 49Ibid., 2.


580 A. MARK SMITH

are Euclidean abstractions of the reality manifested in this world of material relations and limits. "A line is a breadthless length,"50 but so is the actual boundary of any body. "A point is that which has no part,"'5 but so is any true intersection of lines. The world of Euclidean geometry can be regarded as a disembodied replica retaining bounds without the bounded. It is, in a way, our world with many of the kinks ironed out, possessing "perfections" like straightness, circularity, and equality that are rarely, if ever, reached in material reality. And, like this world, the Euclidean world is based on definition; it is not laboriously created by composition but is uncovered by the ever-widening light of definition and subsequent demonstration. Lines may be defined by moving points, planes by moving lines and solid figures by moving planes; but they can never be built or composed of one another. Euclidean Space? There can be no true Space in a geometrical system that unfolds itself so strictly by definition into a continuum of lines, planes, and figures. Wherever this "figurative" continuum does not reach is by definition (or, rather, the virtual lack of it) "nowhere."

Euclid's Elements represents the culmination of an inexorable geometrical shift in Greek mathematical thought occasioned by the full appreciation of incommensurability. Composition-be it in terms of points, lines, planes, or bodies-had been rendered logically insupport- able; so the world of absolute limit, of true infinity as an upper or lower boundary, had necessarily to be replaced by a strict definitional continuum.52 From now on there would be the finite and the indefinite, nothing more. And of course this exemplary caution in approaching the problem of extension, the hallmark of Euclidean geometry, carried over to physical thought, ultimately revealing itself in the common-sense metaphysics of Aristotle.

Instead of the conceptual Euclidean continuum, Aristotle fell back upon a continuum of prime matter which represents the unvarying, though purely potential, substratum over which the spirit of change is always moving.53 Nothing in itself, it is nonetheless fundamental to the qualitative and quantitative actualization of material reality. Given essential formative principles, transformed into "second substance" by primary individuating factors, it gives stability to reality while allowing for a world of differences.54 In short, it plays potential Same to the world's actual Other. Therefore the world of quality and quantity is ultimately to be understood by following its process of actualization through increasing formal individuation, because, like the figurative

50Ibid., Bk. I, Def. 1, 153. 5Ibid., Bk. I, Def. 2, 153. 52Boyer, The History of the Calculus ..., 23-27. 53Joseph Owens, "Matter and Predication in Aristotle," The Concept of Matter

... 92. 54E. J. Dijksterhuis, The Mechanization of the World Picture (Oxford, 1961), 19-20.



continuum of Euclidean geometry, the material continuum of Aris- totelian metaphysics is given reality only by actual definition. While Euclidean "space" is defined by the existence of conceptual form,55 Aristotelian "space" is defined by the existence of actualized form or in- dividuated substance. Thus wherever the actual does not reach is "nowhere."56

This physical conception of a quality-substrate continuum, which, to an almost exclusive extent, dominated the natural philosophy of the Middle Ages and the Renaissance, endured because it was so brutally sensible in spite of its shortcomings, because it meshed so well with the common-sense experience of material permanency and qualitative transience. But neither Aristotle nor his commentators after him could entirely avert the insidious entrance of composition into their material continuum. Vague Aristotelian references to "smallest parts" of or- ganic substances57 formed the exegetical basis for an almost immediate theoretical development of minima naturalia which, with Averroes, took on the character of doctrine. No longer would "natural minima mean ... a theoretical limit of divisibility"; they would henceforth be considered "something like physical realities" in all substances.58

Moreover, as the fourteenth century witnessed a reawakened interest-concomitant with the rise of critical logic-in the problem of the continuum, indivisibles, and infinites, atomist positions were variously proposed, defended, and assailed with typical scholastic

55Notice, then, that Euclidean "space" can be no-dimensional, one-dimensional, two- dimensional or three-dimensional, depending on whether it is being conceptually defined by a point, line, plane, or figure.

56This notion, reflective of the Aristotelian concept of "place," is what underlies his assertion that there can be no "outside" of the world sphere. De Caelo, I, 9, 279a, 12-13.

57Physics, I, 4, 187b, 14-23 and 28-36. "Since every body must diminish in size when something is taken from it, and flesh is quantitatively definite in respect both of great- ness and smallness, it is clear that from the minimum quantity of flesh no body can be separated out; for the flesh left would be less than the minimum of flesh." Physics, I, 4, 187b, 36-38 and 188a, 1-2, from The Works of Aristotle, Vol. II, trans. W. D. Ross (Ox- ford, 1930). "The natural minima, therefore, are purely potential parts. Nowhere do we find any indication that Aristotle conceives them as actualized by some process such as chemical reaction." A. G. Van Melsen, From Atomos to Atom, trans. H. J. Koren (Pittsburgh, 1952), 43.

58Idem. "It is worth noting that this Aristotelian-Averroist theory of minima naturalia differs fundamentally from ... Democritean-Epicurean atomistics. There are four essential differences: (1) the minima naturalia of different substances are qualita- tively different from one another ... (2) for every substance the minima have a charac- teristic size . . . (3) no supposition is made with regard to ... geometrical form ... (4) according to the Aristotelian conception, the different minima that have become juxtaposed act upon one another, as a result of which they undergo internal alterations and together produce the qualitas media . . ." E. J. Dijksterhuis, The Mechanization . . . 205.


582 A. MARK SMITH

gusto.59 Whether, in fact, the ensuing controversy which originated in England eventually found its way with any force or integrity beyond France into Italy is at present unclear.60 Yet it is certainly noteworthy that the Mertonians, especially Thomas Bradwardine, took an active and highly significant part in the debate.61

59Medieval atomists, while sometimes paying lip-service to Democritus, seem to have been unaware of the radical metaphysical aspects of the separation between Space and Matter in his theory. And the resulting scholastic positions (far from unequivocally atomist) as well as their refutations were often motivated by totally non-metaphysical considerations rooted in a wide variety of sources. Foremost, naturally, are the Aristotle commentaries; but Sentence commentaries figured in the debate as well. Furthermore, discussions of the general problem, though subsidiary, were often broached in the context of entirely different problems like, for instance, that of potencies-maximum quod sic and minimum quod non-which inevitably raised questions about actual bounds for discrete (thus medially divisible) sets or potentially (and thus infinitely continuously) merging, common termini ad quem. Yet perhaps the most significant trend in sources during this fourteenth century debate was the increasing appearance of treatises expressly devoted to the problem of the composition or non-composition of the continuum. See the following works by John E. Murdoch: 'Rationes mathematice:' un aspect du rapport des mathematiques et de la philosophie au moyen age (Paris, 1961); "Mathesis in Philosophiam Scholasticam Introducta: The Rise and Development of the Application of Mathematics in Fourteenth Century Philosophy and Theology," Actes du IVieme Congres International de Philosophie Medievale (Montreal, 1968), 215-54; "Superposition, Congruence and Continuity in the Middle Ages," L'Aventure de la science: Melanges Alexandre Koyre (Paris, 1964), 416-41; "Two Questions on the Continuum: Walter Chatton (?), O. F. M. and Adam Wodeham, O. F. M.," Franciscan Studies, 26 (1966), 212-88; also Curtis Wilson, William Heytesbury: Medieval Logic and the Rise of Mathematical Physics, Chap. 3 (Madison, 1960).

60Clearly the debate reached Paris not only with Nicolaus of Autrecourt, an atomist who had a singular capacity for arousing spirited opposition, but with others like Gre- gory of Rimini who, during his Paris sojourn, broached the subject in his Sentence commentary. The question, however, is not so much the existence of speculation on the continuum etc. but rather the status and pedigree of the problem as a truly live issue in late medieval Italy. Cf. Julius R. Weinberg, Nicolaus of Autrecourt. A Study in Fourteenth Century Thought (Princeton, 1948) and Murdoch, "Mathesis in Philosophiam ...."

61It is with the Mertonian mathematicians that a truly sophisticated application of geometry to the problem of continua and indivisibles-in terms of intensive-remissive motions and qualities as well as physical quantities-gained currency. And among these mathematicians it was Bradwardine who, according to Murdoch, offered the most devastating geometrical refutation of atomism in the Middle Ages. In fact, it seems that the anti-indivisiblists quickly gained the upper hand precisely because they were able to employ geometry so much more facilely and cogently than their atomist opponents. The potential significance of the Mertonian contribution, in regard to Galileo's work, can hardly be overstated, for the mathematical analysis of kinematics reached Padua, via Paris, in a number of direct as well as indirect ways. And it seems probable that Galileo was quite familiar with this general kinematic approach if not with specific first-hand sources. In fine, while it is not clear whether he actually read such works as Bradwardine's Tractatus de continuo, he nonetheless had access to a vital current of mathematical "physics" stemming from the Merton school. See John E. Murdoch, Geometry and the Continuum in the Fourteenth Century: A Philosophical Analysis of Thomas Bradwardine's "Tractatus de continuo," (Doctoral Dissertation: University of Wisconsin, 1957) and Marshall Clagett, The Science of Mechanics in the Middle Ages (Madison, 1959), 251-416.



Meanwhile, within the Italian university tradition, above all perhaps at Padua, the evolution of minima theories crested with commentators like Agostino Nifo in the sixteenth century.62 But Nifo was merely the harbinger of an interpretive trend that spread within Italy and well beyond into the rest of Europe to culminate in the chemical thought of the early seventeenth century.63 Clearly, within both the lifetime and the intellectual ambit of Galileo, the struggle for a chemical appreciation of Nature was provoking speculation about integral components and continuity. Whether it impinged effectively on him, through the school of Padua for instance, is an open question,64 but it is only plausible to assume that he was at least aware of it.

Obviously, in view of its many scholastic and extra-scholastic accre- tions and emendations, the Aristotelianism to which Galileo was heir could hardly be called pristine. Quite the contrary, it was a patchwork system of ad hoc sub-theories like that of the minima naturalia designed around a central core of Peripatetic physics and metaphysics. But still there was this central core, and it played a decisive role in determining the world-view of the time. To expect therefore that Galileo was not inevitably aware of that world-view would be to demand his total intellectual withdrawal from a milieu in which he was passionately involved.

For Galileo was not out to overthrow Aristotelian metaphysics but to tidy it up. Fully certain, like Aristotle, that the real world consisted in a material plenum-an essential substrate-continuum-he allowed that certitude to dictate in spite of almost insuperable difficulties. Knowing full well that atomism offered the simplest explanation for many hydro- static phenomena,65 he still clung obstinately to the notion of continuity in his theory of indivisibles because, like Aristotle, he viewed the world in a fundamentally geometrical way. And, like Aristotle, he recognized the need to face the problem of the irrational. Consequently, if he was to treat the world as truly geometrical, while at the same time meeting the Euclidean requisite of absolute continuity, he had to accept the basic

62Van Melsen, From A tomos to Atom, 60. 63Ibid., 65-73 and Dijksterhuis, The Mechanization ..., 277-84. Not only was the

school of Padua a supposed breeding-ground for such speculation, but on the periphery of this new speculative movement thinkers like Zabarella also considered the composition of matter according to indivisibles. Thus, within Italy (and in Germany as well) there was developing a new awareness of the problem of composition of the material continuum, partly as a result of a heightened interest in chemistry, that led invariably to compromise measures approaching atomism. However, we must interpret this trend with caution; as Van Melsen echoes Dijksterhuis' warning: "Although this obviously will put the minima in the foreground, nevertheless there is no question of an atomist conception, especially not in the sense of Democritus . ." From A tomos to A tom, 72.

64Neal W. Gilbert, "Galileo and the School of Padua," Journal of the History of Philosophy, 1 (1963), 223-31. Gilbert makes a telling case against J. H. Randall's un- compromising thesis concerning the intellectual contribution of the school of Padua to Galileo and, thus, modern science in the way of methodology.

65Shea, "Galileo's Atomic Hypothesis," 13-15.


584 A. MARK SMITH

Aristotelian position. However, unlike Aristotle, he could tap the mainstream of medieval physics which had, in its treatment of intension and remission, passed beyond the static geometrical formalism of Aris- totle's world-view to an inchoate notion of formalized process.66 The possibility of geometrizing change, of submitting temporal process to quantitative analysis, that was thus held out to Galileo may well have in- clined him toward a slightly different tack. The notion of growing temporal increments of speeds in accelerated motion may have led him to seek a valid compromise, demonstrable by means of Euclidean geometry, that would paradoxically permit of a substantial discontinuity.67 And, unlike Aristotle, he was able to avail himself of a current of mathematical thought that made this compromise ostensibly feasible.

It was a current started in the late Renaissance by a virtual deluge of Greek-more especially Archimedean-mathematical texts68 which, like the works of Aristotle some centuries before, created a stir of reverent commentary and emulation. But lacking the cultural heritage that drove the Greeks to a punctilious methodology in mathematics, Renaissance scholars often took the substance of Euclidean and Archimedean geometry without taking fully to heart the demonstrative rigor that went with it. Perhaps, as Boyer claims, it was a "pervasive Platonism [that] allowed to geometry what Aristotle's philosophy and Greek mathematical rigor had denied," but whatever the catalyst there was obviously a new and "free use of the concepts of infinity and the infinitesimal...."69 Circumventing the cumbersome but logically supe- rior "method of exhaustion,"70 many European mathematicians forsook nicety for convenience in postulating indivisibles as the foundation of geometrical demonstrations despite an inadequate logical basis.71 Thus

66It is worthwhile reiterating the fact that this sort of endeavor, especially at the hands of Galileo, was absolutely world-based; it did not smack of the theoretical in any Platonic sense whatever: "The student of today thinks of analytical geometry and the function concept [i.e., intension and remission of forms] as almost inseparable, but historically their origins were scarcely related, as a study of Galileo's work clearly shows. Descartes' geometry was essentially an outgrowth of pure mathematics, whereas the function concept was more directly related to problems in dynamics which first crystallized in medieval discussions ... concerning the application of geometry to the study of change. Galileo was involved only in the latter evolution." Carl B. Boyer, "Galileo's Place in the History of Mathematics," Galileo: Man of Science, ed. Ernan J. McMullin (New York, 1967), 234.

67Clavelin, "le probleme du continu .. .," 6. 68The Works of Archimedes, ed. Thomas Little Heath (New York, 1912), pp.

xxiii-xxx for a list and brief account of manuscripts and printed editions of Archimedes' works available in the Renaissance.

69Boyer, The History of the Calculus ..., 89. 70The "method of exhaustion" was a Greek limit-approach method for comparing

the areas of rectilinear and curvilinear figures. For details see The Works of Archi- medes, pp. cxlii-cliv and Boyer, The History of the Calculus .. ., 32-38.

7'Apparently even Archimedes was given to relaxing standards, to developing theorems through the use of actual infinitesimals in order to arrive more easily at the



well before Galileo's academic advent, the mathematical and philosophical precedent had been set for toying with actual infinites and infinitesimals.72 And, considering the appeal this precedent held for so many of his contemporaries, it is small wonder that Galileo should so readily have turned to it in dodging the many logical quandaries that confronted him.73

All in all, it seems that a confluence of powerful physical, metaphysical, and mathematical currents must have impelled Galileo toward his doctrine of indivisibles. At bottom it was an Euclidean-based world-view, founded on the notion of a substantial continuum, that so assured him of the inherent mathematical nature of the world that he regarded geometry not as a mere tool of application or as a perfect- ibility gauge of Platonic reality but as an absolute formal reality. And he may well have taken this essentially Aristotelian position, occupied so long ago against the mathematical and metaphysical onslaughts of Par- menides and Zeno, almost automatically.

conclusion. However, in his formal demonstrations of those conclusions he reverted to the rigorous "method of exhaustion" and a double reductio ad absurdum. See Boyer, A History of Mathematics, 140-54.

72A striking example of such a "precedent" very much in the Galilean style is Thomas Hariot. Sharing to a remarkable extent (although somewhat earlier) many of Galileo's bents-e.g., an interest in hydrostatics and free-fall-he also employed much the same mathematical reasoning as later (and presumably independently) did Galileo to arrive at a justification for atomism. However, while Hariot seems to have been more radically Democritean than Galileo, he was commensurately more discreet in airing his views. Hence, the chances of Galileo's having been thereby influenced seem accordingly slim. See Robert H. Kargon, Atomism in England from Hariot to Newton (Oxford, 1966), 23-28 for a fuller account of Hariot's views and his socio-scientific ambience.

73It should be noted that mathematically Galileo departed somewhat from the older conception of indivisibles as equidimensional with, though incomparably smaller than, the figures containing them by adumbrating the notion, later more fully developed by his disciple Cavalieri, that mathematical indivisibles are of one less dimension than their containers. However, it should also be noted that it was with Cavalieri, Torricelli, Fermat, Roberval (to mention only a few) that this new view took firm shape. Thus it was only as a catalyst for Cavalieri's later thought that Galileo played any real part in this trend which, evolving through the middle and late decades of the 17th century, eventually influenced both Leibniz and Newton. On the other hand, as a philosopher Galileo did accept equidimensionality for his physical indivisibles and, consequently, had to admit a paradoxical conceptual dichotomy between the physical and the purely mathematical continuum, an admission that reflects his basic ambivalence toward mathematics and physics. For a good thumbnail sketch of Cavalieri's role in the development of mathematical indivisiblist theory, see Ettore Carruccio's article, "Ca- valieri," Dictionary of Scientific Biography (New York, 1971), III, 149-53; for more detailed analyses of various aspects of Cavalieri's work and influence: Carl B. Boyer, "Cavalieri, limits and discarded infinitesimals," Scripta Mathematica, 8, (Dec. 1941), 79-91; A. Koyre, "Bonaventura Cavalieri et la geometrie des continus," Eventail de l'histoire vivante: Hommage a Lucien Febvre, I, 319-40; and G. Cellini, "Gli indivisibili nel pensiero matematico e filosofico di Bonaventura Cavalieri," Periodico di Ma- tematiche, 44, Ser. 4 (1966), 1 -21.


586 A. MARK SMITH

However, as we have seen, the Aristotelian and Euclidean traditions underwent gradual modifications over the almost two millennia that saw them into the Renaissance. In the first place, metaphysics had suf- fered logical dilution through scholastic half-measures, like the minima naturalia, that reflected an increased attraction for the notion of material composition. But brought to the verge of atomism, most Renaissance theorists ultimately balked, as did Galileo some years later, at the final Democritean step.74 In the second place, a spate of mathematical discoveries, textual as well as original, had aroused intense theoretical interest during the Renaissance. Eventually this interest stimulated an ostensibly Archimedean approach to many new problems while, unfortunately, provoking over-eager Latins to impose actual limits-indivisibles-where none by logic belonged. And finally, the fourteenth-century adumbration of the function concept had signalled a highly significant departure from the eminently static Greek approach to Nature-a departure that Galileo almost singlehandedly transformed into a scientific mainstream.

So Galileo had available to him a complex set of precedents from which to draw. With a full appreciation of Archimedean hydrostatics for example, he was early led to recognize the advantages of material discontinuity in explaining apparent qualities like density, fluidity, material resistance, etc. And he had a long-standing tradition of "minima" theories to offer him support. Yet at the same time he could only demur at any atomist solution which failed to preserve the actual geometrical continuity of the world, for there was absolutely no question of abandon- ing his deepest convictions for the sake of explanatory ease or elegance. Nonetheless he managed to find salvation in the very geometry he was so scrupulous to save. And having established a geometrical basis for indivisibles, having shunted operationally extensional infinity-the Aris- totelian nemesis-into the limbo of indeterminacy, he returned to the substantial world with a slightly altered view of continuity. His final task, then, was to prove the actual existence of vacua (no matter how fleeting or small) in order to introduce the grounds for material discontinuity. The result was a highly equivocal atomism that fully reflected Galileo's ambivalence. Rather than take the bold Democritean plunge into a space disrupted by matter, he clung to the notion of matter disrupted by space, for "his idea of the cosmos ... was too vivid for him to be satisfied by the notion of an infinite world in which from the whirl- ing motion of an infinite number of atoms worlds arise and perish again in endless succession.'75

74Even Francis Bacon seems to have flirted with atomist theory for several years (ca. 1603-1612) in the hope of finding metaphysical coherency; but, ultimately recoiling from the a priorist connotations of atomism, he returned to a more conservative view of the material continuum. Ibid., 43-47.

75Dijksterhuis, The Mechanization ..., 419.



Is Galileo's theory of indivisibles the manifestation of revolution or merely another in a long series of metaphysical half-measures taken within an inescapable Aristotelian tradition? This question of course brings the analytical focus back to the notion of the reification of space, back to the validity of the "metaphysical interpretation" of early modern science and Galileo's relation to it. At the very outset it is ob- vious that Galileo's atomic hypothesis offers little or no systematic foundation for a fully coherent kinetic interpretation of Nature.76 The basic spatial freedom requisite for such an approach is entirely lacking, and in its stead is a vestigial concern for preserving continuity and matter-medium intimacy, a concern that may well find its inspiration in hydrostatics. Behind it all lies the Aristotelian-Euclidean conception of "place," of the strict equivalence between "body" and "definitive being" in the world-plenum. Still bound to the Archimedean notion of displacement, Galileo was apparently unable to conceive logically the Epicurean view of totally free movement within an independently exist- ing void. At any rate, whatever his reasons he obviously took a middling course in the theory of indivisibles, a course that by itself hardly signalled a metaphysical revolution.

Furthermore, since space (except in the most restricted sense of "vacuum") was non-existent for Galileo, there is precious little reason to uphold the contention that he geometrized it in a revolutionary access of Platonism. For, unlike Plato, he regarded mathematics as reflective of the world, not the converse. Geometry was real and worthwhile only insofar as it served a formal function abstractable from, but not im- posed upon, physical reality. But, unlike Aristotle, Galileo had a knack for rendering the "book of nature" geometrically intelligible through a clear-sighted exclusion of irrelevant chapters: Just as the computer . .. must discount the boxes, bales, and other packings, so the mathematical scientist ... must deduct the material hindrances, and if he is able to do so, I assure you that things are in no less agreement than arithmetical computations. The errors, then, lie not in the abstractness or concreteness, not in geometry or physics, but in a calculator who does not know how to make a true accounting.77

761 say this despite the few passages from Galileo's writings that might be (and have been) adduced in support of the contention that Galileo was a dedicated "mechanical philosopher" who viewed Nature in terms of particles in various "qualitative" states of motion [e.g., "The Assayer," Discoveries and Opinions of Galileo, ed. Stillman Drake (Garden City, 1957), 274-76; and Dialogues Concerning the Two Chief World Systems, trans. Stillman Drake (Berkeley, 1967), 40.]. Actually these passages are not only highly conjectural, but in context they are absolutely unconnected with any systematic development of kinetic theory. Furthermore, it is clear that Galileo did not postulate the infinitesimal pockets of interstitial "space" to provide for atomic motions in wide variety. Therefore, to infer (as did Burtt) a full-fledged, coherent system of world mechanization from such meager evidence is totally unwarranted.

77Dialogues Concerning the Two Chief World Systems, 207-08.


588 A. MARK SMITH

There is no question of existential supremacy in a geometry that is so solidly rooted in, and dependent upon, physical reality as is Galileo's. Thus, to pretend to find Platonic overtones in such a functional abstrac- tion merely because it happens to be geometrical is somewhat forced; to pretend to find a metaphysical revolution there is totally implausible.

The long and short of it is that many historians have tended to misin- terpret Galileo's genius, to overrate his modernity in relation to his epoch.78 Because he did what he did so much better, so much more clearly and astutely than did his contemporaries, we have often been led to expect more of him than we can fairly demand. And consequently we have been brought to regard him as a prodigy comprehensible only in terms of upheaval or revolution. Because he undermined the Aris- totelian celestial/terrestrial dichotomy with such wit and vigor, because he made such a clear transition from traditional to "modern" dynamics and kinematics, we have even sought his intellectual motivations in an utter metaphysical overthrow of barren Aristotelian scholasticism. But in such a quest there is always the attendant danger of doing uninten- tional violence to a definitive context. It should always be remembered that Galileo, in practically every endeavor he undertook, attained a "not quite" point in approaching modern theory.

If, then, Koyre is correct in his conviction "that the rise and growth of experimental science is not the source, but, on the contrary, the result of [a] new metaphysical approach to nature that forms the content of the scientific revolution of the seventeenth century,"79 Galileo's role as a bellwether of the revolution is at best questionable. If, on the other hand, Galileo did indeed inaugurate a scientific revolution, the metaphysical genesis of that revolution is equally questionable.

University of Wisconsin. 78A prime example of this zeal for anachronism is the following: "The curvature of

inertial motion does not yet mean that Galileo enunciated the notion of curved, homogeneous space, [but] if we compare the Galilean idea to the current notion of the homogeneity of space, Galileo's idea will correspond to the notion of curved, homogeneous space." Boris G. Kuznetsov, "L'Idee d'homog6enite de l'espace dans le 'Dia- logo' de Galilee et son evolution posterieure," Actes du Symposium International des Sciences Physiques et Mathematiques dans la Premiere Moitie du XVIIieme Siecle (Pisa, 1958; Paris, 1960), 136-my translation.

79Koyre, "The Significance of the Newtonian Synthesis," 6.


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Issue Table of ContentsJournal of the History of Ideas, Vol. 37, No. 4, Oct. - Dec., 1976Volume Information [pp. 731 - 736]Front MatterGalileo's Theory of Indivisibles: Revolution or Compromise? [pp. 571 - 588]A Little Great Awakening: An Episode in the American Enlightenment [pp. 589 - 602]Irreversibility and Indeterminism: Fourier to Heisenberg [pp. 603 - 630]Malthus, Darwin, and the Concept of Struggle [pp. 631 - 650]Hans Kohn's Liberal Nationalism: The Historian as Prophet [pp. 651 - 672]NotesErasmus on the Antithesis of Body and Soul [pp. 673 - 688]Bacon's Empiricism, Boyle's Science, and the Jesuit Response in Italy [pp. 689 - 695]John Yates of Norfolk: The Radical Puritan Preacher as Ramist Philosopher [pp. 697 - 706]Locke's Analysis of Language and the Assent to Scripture [pp. 707 - 714]

Review-ArticleAdam Smith's Wealth of Nations [pp. 715 - 720]

Books Received [pp. 721 - 730]Back Matter [pp. 696 - 735]

Galileo Theory Indivisibles

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