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From Divine order to Human approximation: Mathematics in Baroque Science


Ofer Gal

University of Sydney

1. Baroque Science?

Forceful and occasionally forced paradox; violent contrast; reliance on sensual detail, particularly color and touch, to indicate moral condition and religious theme; deliberate distortion of regular structures to produce the asymmetric effect of baroque art; and unity of thought more dependent on imagery than on logic. (White et al., Seventeenth-Century Verse and Prose, Vol. 1; 391)

These are the words used by the Macmillan’s anthology of seventeenth-century literature to describe the writing of Richard Crashaw (b. 1612 or 13, d. 1649), a poet of turbulent life and modest, though not completely negligible repute. They are interesting for two reasons. First, because Crashaw is the definitive ‘Baroque’ English poet. Literally so; here is the very definition of “baroque” in Hollander and Kermode’s The Literature of Renaissance England:

(1) originally (and still), an oddly shaped rather than a spherical pearl, and hence something twisted, contorted, involuted. (2) By a complicated analogy, a term designating stylistic periods in art, music and literature during the 16th and 17th centuries in Europe. The analogies among the arts [continue Hollander and Kermode] are frequently strained, and the stylistic periods by no means completely coincide. But the relation between the poetry of Richard Crashaw in English and Latin, and the sculpture and architecture of Gianlorenzo Bernini (1598-1680), is frequently taken to typify the spirit of the Baroque.” (Hollander and Kermode, The Literature of Renaissance England; 1049)

Secondly, the words used to characterize Crashaw’s style are interesting because they represent an exact mirror image of what we are accustomed to think of the great cultural achievement of Crashaw’s life time—the New Science: “Forceful and occasionally forced paradox” is a clear opposite of the strict logical coherence advocated by the ideologues of the new philosophy; “violent contrast” runs against the mathematical harmonies they champion; “sensual detail, particularly color and touch” are the secondary qualities they specifically taught should not be relied upon. “Deliberate distortion of regular structures to produce the asymmetric effect of baroque art” is diametrically opposed to the trust in mathematical generality and symmetry which we allegedly inherited from the New Science, “and unity of thought more dependent on imagery than on logic,” is contrary to the very essence of everything mathematical natural philosophy was striving to establish.

How are we to understand this discrepancy between the two primary cultural movements of the seventeenth century, the New Science and the Baroque, in as much as they are properly captured by the oppositions above? Historians of science, by and large, conveniently ignore it altogether. To the degree that they are motivated by the traditional, progressive conception of science developed in the analytic philosophy of last century, they are disposed to view seventeenth century natural philosophy as a harbinger of Newtonian Enlightenment (in which this very conception was originally bred). If they lean towards the historicist-skeptical attitude, they tend to describe it as the brainchild of Renaissance.

In either case, the question of science as a Baroque phenomenon, which should be suggested by simple chronology, seems to escape the attention even of historians of science engaged with those aspects of the 17th century culture most prone to the paradoxical, sensual and asymmetrical characterizing Crashaw’s poetry. Mario Biagioli, for example, in his iconoclastic studies of Galilean science as a product of court culture, does gesture rarely towards Baroque, but analyses the Florentine and Papal courts all the way to the 1630s in Renaissance terms. More important—what is in a name, after all—to the degree that he applies those courtly practices (which he takes as rather more Renaissance than Baroque) of “thought more dependent on imagery than on logic” directly to the study of Galileo’s science, it is strictly on those aspects which are in and of themselves ‘sensual’, namely: his astronomical observations. Biagioli pays much attention to legitimatory and epistemological debates about the status of mathematics, but Galileo’s mathematical and experimental practices, which set the most fundamental agenda for what we recognize as the new science, are barely touched by Biagioli’s analysis.

Biagioli, to be sure, is much more attentive to the cultural resources and constraints I am interested in than is common among historians of science. To the degree that the term ‘Baroque science’ does make a rare appearance, and not entirely as a chronological designator, (c.f. Eriksson and Høyrup), it is in reference to the obviously ‘sensuous’ realms of the new science—natural history, Wunderkammern and similar areas (c.f. Findlen and Rowland). It seems that only in these contexts can the discrepancy between the complimentary tone of ‘science’ and the slightly pejorative one of ‘Baroque’ be transcended. But the term itself seems problematic. Even Eileen Reeves, who studies Galileo’s artistic resources and whose main heroes, like Rubens and Velasquez, are clearly ‘Baroque painters’, refrains from using the term. Yet more interesting is the fact that even those historians of science who are of the critical approach and find the juxtaposition intriguing take it almost for granted that the new science, rather than being born by the Baroque, signals its demise. A particularly telling example in this regard is Paula Findlen, who boldly introduced themes from Bakhtin and Huizinga to the study of 17th century science. In her story, science provides the final blow to the “color and touch,” to the “deliberate distortion;” to the playful aspects of European culture.

One might think that the assumption that science was a way out of the Baroque reflects the instinctive arrogance of historians of science, but it is actually shared by those who approach the relation from the opposite direction. Baroque scholars like Maraval do discuss, if rarely at any length, the role of the new science in providing 17th century high culture with material and intellectual resources, but never consider science as born out of “Baroque’s stubborn ambiguity” (Guardiani in Scaglione et al.; 81) itself. They find in Mathematicized natural philosophy, rather, a cultural counter-trend or a harbinger of culture’s next turn.1

I think I have done more than enough to point at what I am seeking: a depiction of the new science—in its hard core, mechanical, mathematical incarnations—as a Baroque Phenomenon. And though there is, as I claimed, too little direct reference to this idea in the literature, there is also important scholarship that suggests that this may be an important and under-explored perspective on early modern science. For example: Margaret Jacob has analyzed 17th century science from the perspective of high culture, which wearily observes its appeal to naturalism and other destabilizing tendencies, and then transforms and assimilates a ‘rational’ and conservative version of it as a buttress against radicalism. Eileen Reeves, as I mentioned, explored the neo-Stoic currents common to early Baroque science and art, heralding change and mutability over eternity and substantiality. Peter Dear (Discipline and Experience) showed the intricate relations between single experiments and mathematical generalizations. Fridriech Steinle (in Weinert; 316-69) demonstrated the complexity and ambivalence of the notion of ‘law of nature’ in Newton in particular and seventeenth century science and philosophy in general. In the philosophical realm, Nancy Cartwright (How the Laws of Physics Lie) has been arguing for the superiority of local, causal laws over what she term ‘bridging laws’. The locus classicus of the approach I am trying to promote is of course Svetlana Alpers’ The Art of Describing, which analyzes representation as an assembly of concepts, sensibilities and practices common to Dutch Baroque paintings and to its contemporary science. I owe much to her work.

2. Kepler and Newton

A comprehensive characterization of ‘Baroque Culture’ is obviously beyond the scope of this paper. What I will attempt to do is clarify what it is that I mean by claiming that the New Science should be treated as part and parcel of the high culture of the 17th century. I would like to set my claim against the almost unchallenged conception of the New Science, namely, that its main achievement was the submission of all phenomena to a small set of exact mathematical laws, and that the pursuit of this achievement was motivated by a conviction that a simple, perfect and harmonious structure underlies all seemingly unruly phenomena, a structure decipherable by means of mathematics—the science of simple, perfect structures {ref.}. This common conception is wrong. It is a backward projection of enlightenment epistemology on the one hand, and an overextension of late Renaissance dreams and aspirations on the other. The great success of late 17th century mathematicized natural philosophy, I will argue, was predicated on treating mathematics not as an instrument for revealing the divine harmony of the universe, but as means to approximate the varieties and multitudes of nature into a human-scale, local order. This approach followed and entailed a tacit conviction that the universe is an imperfect machine, an erratic assemblage of isolated laws and constants. True, 17th century natural philosophers continued to herald the perfection, simplicity and beauty of creation, while increasingly treating nature as irreducibly complex. More than anything else, it is this very tension between public commitments to order and practical acknowledgement of discord that reflects the new science deep entrenchment in high Baroque culture.

Of these general claims, I will concentrate in the following in illustrating the change in the concept of mathematical order through the century. It can be nicely captured by comparing the fates of the following two diagrams:

70 year separate these diagrams, and they are different in audience and target. Kepler’s is public and in print—it opens the 1609 Astronomia Nova—Newton’s is private and hand-drawn—part of a 1679 letter to Robert Hooke. Kepler is aiming to convince the general astronomical public that the geostatic system, whether in its Ptolemaic or Tychonic version, is untenable. Newton is suggesting to his correspondent Hooke that his—Hooke’s—idea that planetary motions are a compound of inertial motion and solar attraction is fundamentally flawed. Kepler’s diagram is based on a careful calculation from the geostatic theory he thinks obsolete; Newton, on the other hand, feigns a quantitative theory he does not really have and fabricates a construction.

Yet the structure of the argument these diagrams embody is essentially the same. Both depict a hypothetical planetary orbit, suggested by the theory under consideration, and both expect their audience to immediately perceive the orbit as clearly and obviously absurd, and eschew the theory that produced it. And why is the orbit obviously absurd? Because it is chaotic. Because, and I quote Kepler, “These motions, continued farther, would become unintelligibly intricate, for the continuation is boundless, never returning to its previous path.” As Newton will put it later, Hooke’s idea means that “there are as many orbits to a planet as it has revolutions.” And for both writers the argument ends here—an “unintelligibly intricate” orbit is prima facie unacceptable.2

An absolute trust in simplicity and orderliness is entailed in these diagrams. This is the assumption, too clear and distinct to require explication, that turns both into arguments, and it is around this assumption that my argument revolves. For Kepler, I am going to claim, this is a genuine commitment: it is fundamental to the epistemology that guides him and is embedded in his work throughout his career. For Newton, on the other hand, it is a commonplace. He can use this type of ‘argument from order’ effectively, but by the time he sends his sketch to Hooke (1679-80), it represents no more than a rhetorical topos, which he easily forgoes once it is in conflict with his problem-solving strategies. For all practical intents and purposes, Newton’s universe is imperfect and far from simple. This is a metaphysical difference, but far from being free-wheeling ‘white mythology’, it has unmitigated effect on their mathematical work: for Kepler it is the perfection of mathematics which makes it the proper medium through which to express the beauty, majesty and absolute perfection of the Creator and His creation—the magnificent Harmony of the World. In diametric opposition, Newton’s work takes the turn that would lead him to the Principia once, following the correspondence of Hooke, he adopts the view expressed in the following Scholium:

The whole space of the planetary heavens either rests … or moves uniformly in a straight line, and hence the communal centre of gravity of the planets … either rests or moves along with it. In both cases … the relative motions of the planets are the same, and their common centre of gravity rests in relation to the whole of space, and so can certainly be taken for the still centre of the whole planetary system. Hence truly the Copernican system is proved a priori. For if the common centre of gravity is calculated for any position of the planets it either falls in the body of the Sun or will always be very close to it. By reason of this deviation of the Sun from the centre of gravity the centripetal force does not always tend to that immobile centre, and hence the planets neither move exactly in ellipse nor revolve twice in the same orbit. So that there are as many orbits to a planet as it has revolutions … and the orbit of any one planet depends on the combined motion of all the planets, not to mention the action of all these on each other. But to consider simultaneously all these causes of motion and to define these motions by exact laws allowing of convenient calculation exceeds … the force of the entire human intellect. Ignoring those minutiae, the simple orbit and the mean among all errors will be the ellipse ...3

Here is the “forceful and occasionally forced paradox,” the “violent contrast” of the Baroque. Newton is not promising to save the unruly appearances—he is proclaiming the disorderly behavior of the elements. It is metaphysical imperfection rather than epistemological incompetence he is lamenting, because it is not only that “calculation exceeds … the force of the entire human intellect,” but that “the planets neither move exactly in ellipse nor revolve twice in the same orbit. So that there are as many orbits to a planet as it has revolutions.” And this is not just a mysterious consequence of some esoteric theoretical move for Newton—it is a necessary and fundamental correlate of the metaphysics of force that he inherits from Hooke and deploys in the De Motu and the Principia. The disorder is necessitated by the underlying mechanisms: if attraction is the basic quality of matter, and if, therefore, all bodies attract each other, and if the parameters of this attraction are what accounts for the planetary motions, then the planetary orbits have to be complex. Or, to say it even more boldly, there are no real orbits at all; there is only a “mean among all errors.” There is no eternal, perfect infrastructure to the universe, to be captured mathematically, only “all these causes of motion,” which mathematics can help construct into “the simple orbit” by “ignoring minutiae.”

Newton’s “distortion of regular structures to produce the asymmetric effect of baroque” is not merely in the acknowledgement that “there are as many orbits to a planet as it has revolutions.” It also resides in the suppression of this metaphysical conviction. Albeit central to Newton’s De Motu Sphæricorum Corporum in Fluidis—one of the last drafts of the Principia—this Scholium has been carefully omitted from his opus magnum. The caution is completely understandable: it comprises more than a belated argument for Copernicanism, and much more than a technical consequence of the eccentricity of the sun. What Newton expresses here is a conversion of a religious magnitude, and indeed religious implications. Newton does not claim that we should strive at approximate and instrumental knowledge because of the inherent epistemic limitations of the human mind. Quite the opposite: the only hope for order, he declares, is in the structures imposed on the intrinsic metaphysical disorderliness of the world by “ignoring minutiae” and constructing “the mean among all errors.” And it “Baroque’s stubborn ambiguity” which resides in the carefully contained but ever present tension between Newton’s conviction, which, as I will demonstrate, is essential to his trailblazing work after the correspondence with Hooke, and celebratory proclamations such as “… Here ponder too the Laws which God / Framing the universe, set not aside / But made the fixed foundations of his work,” from Halley’s pen4, which he allowed to adorn his publication

There are good reasons to view the scholium as reflecting a personal conversion which Newton is going through. The concept of hands-on, personally involved God that Newton develops in the 1690s suggests that much. But my interests here are not in the biographical but rather in the general decline, through the 17th century, of the trust in mathematically-ordered nature and the institution, in its stead, of mathematics as a means for imposing a precarious human order on a fundamentally disordered nature. And I will sketch this by comparing the fate of the two diagrams I presented above.

3. Kepler and Perfection

For Kepler, I claimed above, the Astronomia Nova diagram represents a real commitment: it is embedded in a working metaphysics of universal harmony and an effective belief in the power of mathematics to reveal the divine infrastructure of the universe. In his Mysterium Cosmographicum, published 13 years earlier, Kepler provides his most explicit expression of both: the universe, he tells his readers there, is “complete, thoroughly ordered and most splendid” (23-4/95-97). It is simple, and its structure necessary. And Kepler’s mathematical inquiry is strictly structured by these assumptions. His question in the Mysterium is why are there exactly six planets, and his answer is that there are exactly five perfect solids. Thus, if the distances between the planets correspond to these solids, namely—if the proportions between their distances can be shown to correspond to the proportions between the solids (for there are no material solids in the heavens)—then the number of planets has been explained—the mathematical directly account for the physical.

But what kind of an explanation is this? Why should abstract mathematical proportions account for a material fact? Why should their aesthetic value be evidence for their truth? Guided by the metaphysics of order, Kepler suggests two complementary answers to this question. Either:

God, like one of our own architects, approached the task on constructing the universe with order and pattern, as if it were not art which imitated Nature, but God himself had looked to the mode of building of Man who was be


it is by some divine power, the understanding of the geometrical proportions governing their courses, that the stars are transported through the ethereal fields and air free of the restraints of the spheres

Either mathematics is God’s own blueprint for the universe, or the planets themselves are using the “geometrical proportions governing their courses” to navigate the empty vastness of the heavens. Kepler never gives up on the first possibility (that he would later use as a proof for creation and the existence of God). The assumption, that the rationality of the structure can only affect the material realm if that realm (or elements of it, like the celestial bodies) is also endowed with rationality is, however, an awkward one, and Kepler largely retreats from it in the Astronomia Nova. But, against common wisdom, Kepler never eschews the mathematical enthusiasm of the Mysterium for the physicalism of the Astronomia Nova. This is how he himself analyses his development in a note he adds to this paragraph in the second edition of the Mysterium, published unchanged (apart from the annotation) in 1621: “So indeed I supposed,” he says, concerning the rationality of the planets,

but later in my Commentaries on Mars [the Astronomia Nova] I showed that not even this understanding is needed in the mover. For although definite proportions have been prescribed for all the motions … by God the Creator, yet those proportions between the motions have been preserved … not by some understanding created jointly with the Mover, but by … the completely uniform perennial rotation of the sun [and] the weights and magnetic directing of the forces of the moving bodies themselves, which are immutable and perennial properties. (60/169)

Now note what it is that Kepler thinks he has defended in the Astronomia Nova. The notion of perfect proportions remains untouched; changed is only the mechanism by which they are followed. The mechanism itself is simple: the rotation of the sun and the magnetism of the planets are “completely uniform [and] perennial.” The mathematics—the analysis of the proportions between the solids and the consequents distances and periods—is left to safeguard the “complete, thoroughly ordered and most splendid universe.”

4. Newton and the Moving Apsides

It does not mean, of course, that Kepler is unaware of the difficulties in applying his grand mathematical scheme to the minute details of observation. Quite the contrary. The major part of the Mysterium is dedicated to this task; in particular, to finding a place for the eccentricities of the planetary orbits between the nesting polyhedra—this indeed was the pretext for his correspondence with Brahe, leading to their illustrious collaboration. The point, however, is exactly this: Kepler excuse the eccentricities by fitting them into the mathematical model constructed according to independent principles. For Kepler, the world has a universal, harmonic, perfect structure, which can be discovered by a-priori, mathematical considerations, and into which one then has to fit the embarrassing particularities of the empirical.

Compare all this, now, to Newton’s transformation of the argument against Hooke, which in its original form was so similar in structure to Kepler’s. Its final version is to be found in Props. 43-45 of the first book of the Principia:

If a body, under the action of a centripetal force that is inversely as the square of the height, revolves in an ellipse having a focus in the center of forces, and any other extraneous force is added to or taken away from this centripetal force, the motion of the apsides that will arise from that extraneous force can be found out … and conversely.

The suggestion that the consequent motion of the apsides invalidates Hooke’s proposal to “compound the celestiall motions of the planets,” then, has been turned on its head. Now, it is the ability to calculate this motion that validates for Newton the fantastic mathematical edifice he erected on the basis of Hooke’s proposal. There is nothing embarrassing anymore in the fact that the orbits themselves are revolving, anymore than there is in the “deviation of the Sun from the centre of gravity.” Both are contingent, particular facts about out world. For the post-Hooke Newton, the world is full with such contingent, particular constants for which we can construct more-or-less stable mathematical structures. Mathematics is a tool to manipulate and control the complexity of the universe, rather than a mirror of its underlying simplicity. The order and exactitude in the Newton’s physica coelestis is that introduced by his mathematics.

5. The ISL: from Geometrical Necessity to convinient approximation

Let me offer another example of this change, involving the same protagonists and a related issue—that of the law of decline with the square of distance, which Kepler formulated for light and Newton, famously, applied to universal gravity. This is a particularly interesting example not only because Newton’s indirect indebtedness to Kepler here allows some more continuity in my narrative, but because it touches exactly on the question of the place of mathematics in the order—or ordering—of the universe. Light, for Kepler, fulfills exactly this role: it is the conduit of God’s mathematical archetypes, as he explains in his Optical Part of Astronomy of 1604:5

the spherical is the archetype of light (and likewise of the world); the point of the center is in a way the origin of the spherical solid, the surface the image of the inmost point, and the road to discovering. The surface is understood as coming to be through an infinite outward movement of the point out of its own self, until it arrives at a certain equality of all outward movements. The point communicates itself into this extension, in such a way that the point and the surface, in a commuted proportion of density and extension, are equal. (Ad Vittelionem; 19)

Light is a substantiation of geometry. It is the embodiment of sphericity, from which follows the law governing its decline with distance:

just as [the ratio of] spherical surfaces, for which the source of light is the center, [is] from the wider to the narrower, so is the density or fortitude of the rays of light in the narrower [space], towards the more spacious spherical surfaces, that is, inversely. For … there is as much light in the narrower spherical surface, as in the wider, thus it is as much more compressed and dense here than there. (Ad Vittelionem, 10)

The assumption of the mathematical structure of nature, fortified by reflections about the essential geometrical nature of light, allows Kepler to infer the physical properties of light from purely geometrical considerations. Five years after completing the Optical Part of Astronomy he proceeds towards a full-blown physica cœlestis in the Astronomia Nova, and attempts to make use of his discoveries. The import of light, the courier of the sun’s powers to the planets, suggests it as a perfect analogy by which to conceptualize his virtus motrix—the solar force stirring the inertial planets. But instead of applying to the motive force the mathematical properties he so painfully derived for light, these very geometrical considerations finally convince Kepler that light can, at best, serve as analogy—it cannot be one an the same with the solar motive force. The velocity of the planets is inversely proportional to their distance from the sun, not to the square thereof, he reasons, not to mention that light is dispensed spherically, and the motive force, apparently—only in the plane of he ecliptic. The geometrical make-up of the two types of solar emanation—the mathematics embedded in their nature—is different, so the motive force cannot follow the inverse square law, hence cannot be light. The commitment to the geometrical infrastructure has a price as well as benefits.

Kepler’s torturous way of legitimizing his mathematical ‘physics of the heavens’ failed to impress even his popularizer and most ardent admirer, Ismaël Boulliau. Perhaps his Catholicism released Boulliau from Kepler’s protestant worries about God and mathematical perfections, or perhaps it was the benefit of another generation of mathematization of natural philosophy, but Boulliau simply could not see the point in Kepler’s vacillations. “On the rocks of these hesitations,” he exclaims in the Astronomia Philolaica of 1645, Kepler “crushes his very astronomy into shipwreck,”6 and suggests both geometrical and physical arguments to save the forsaken analogy between light and virtus motrix. Boulliau finds it almost hard to distinguish between them, using ‘species’—carefully assigned by Kepler to the solar moving force—to denote light, and assaults each and every one of Kepler’s diffident distinctions between the two.

But it was easy for Boulliau to patronize—he never took upon himself the daunting task of making mathematics explanatory, and his geometrical and physical speculations remained completely distinct from each other, even if adjacent. When, another twenty years later, Robert Hooke was attempting to follow Kepler’s footsteps, he found himself facing very similar difficulties and, like Kepler, was retrained from making full physical use of the inverse square law by the geometrical considerations from which this ratio was born. Not that Hooke had much patience for neo-Platonic worries. In his 1665 Micrographia he seamlessly imports the inverse square law from light to gravity in the following parenthesized remark:

[I say Cylinder, not a piece of a cone, because … that triplicate proportion of the shels of a Sphere, to their respective diameters, I suppose to be removed by the decrease of the power of Gravity]7

Hooke is concerned here with the Tychonic problem of the implications of atmospheric refraction on astronomical observations, and he conducts Torricelli-style experiments in order to calculate the size and density of the atmosphere. This off-hand argument allows him to approximate the height of the column of air above his mercury tubes: the decline of “the power of gravity”—necessarily by the square of the distance (which he doesn’t bother to explicate, but without which the argument does not work)—means that instead of truncated cone (in which the volume is proportional to the cube of height), he can calculate the column as a cylinder (namely—as if its volume is proportional to the height of the atmosphere).

This almost frivolous use of mathematical approximation is already quite removed from Kepler’s grave hesitations about the way the perfection of his geometry reflects the perfection of creation, the way perfect geometry is distributed into the imperfect physical realm (through light), and what all this allows by way of mathematical hypotheses. But Hooke’s application of the inverse square law actually has more in common with Kepler’s attitude than might be assumed. As I have shown in a different place8, the only justification he has for the move is exactly the geometrical analogy: like light, gravity, and with it the atmosphere, expands spherically. The image of spherical ‘explosion’ of agency or active principle from center towards periphery, which produced the inverse square law for Kepler, is exactly what is on Hooke’s mind when he inquires about the behavior of light in the atmosphere and how the atmosphere itself is constituted by gravity. Like Kepler, he treats the agency as operating on the enveloping “shells” and can thus easily apply the law for the decline of light to the decline of gravity. But, again like Kepler, these very considerations prevent him from making real physica cœlestis use of the inverse square law. In 1673 he promises a

System of the World ... answering in all things to the common Rules of Mechanics [which] depends on three Suppositions. First, That all Cœlestial Bodies Whatsoever, have an attraction or gravitating power towards their own Centers, whereby they attract not only their own parts ... but ... also ... all the other Cœlestial Bodies that are within the sphere of their activity; and consequently that not only the Sun and the Moon have an influence upon the body and motion of the Earth, and the Earth upon them, but that [all the planets], by their attractive powers, have a considerable influence upon its motion as in the same manner the corresponding attractive power of the Earth hath a considerable influence upon every one of their motions also. The Second Supposition is this, That all bodies whatsoever that are put into a direct and simple motion, will so continue to move forward in a streight line, till they are by some other effectual powers deflected and bent into a Motion, describing a Circle, Ellipsis, or some other more compound Curve Line. The third supposition is, That these attractive powers are so much the more powerful in operating, by how much the nearer the body wrought upon is to their own Centers9

Hooke’s mechanical “System of the World” was to be based on the suppositions of universal attraction, Cartesian inertia and a mathematical force law: “these attractive powers are so much the more powerful in operating, by how much the nearer the body wrought upon is to their own Centers.” All is ready to apply the inverse square law for the decline of “attractive powers” with distance, the law he so easily imported from light to gravitation eight years earlier in the Micrographia, but Hooke declines the opportunity: “what these several degrees [of decline]” he adds, “I have not yet experimentally verified.” The image of spherical ‘explosion’ that related gravity to light seems to him inapplicable to the notion “attractive power,” as it seemed to Kepler concerning his motive force, so, like Kepler, he refrains from applying the law for decline from solar illumination to solar attraction. In spite not being truly committed to Kepler’s notions of mathematical order, the similar geometrical reasoning which leads him to adopt the inverse square law prevents him from turning it into a flexible algebraic operator in the calculation of orbits.

And this is exactly what Newton does, starting from the same version of De Motu from which the ominous “Copernican Scholium” is taken, and he does it in a most Baroque way. Using a few fast-and-loose moves which Kepler would have hardly recognized as the “Mathematicals,” which “God the Creator had with him as archetypes from eternity” (Mysterium Cosmographicum, 1619 edition note to Ch. 11), Newton establishes a geometrical expression for the centripetal force holding a revolving body in a circular orbit:  AD2/R, where AD is an infinitesimal arc.10 He then adds five corollaries, all simple derivations from this expression. He assumes uniform motion, so AD is proportional to the body’s velocity. Thus, combining AD  V with  AD2/R, it follows that:

Corr. 1.  V2/R.

Since the velocity of rotation is inversely proportional to the period of revolution, i.e.,  1/T, this is equivalent to:

Corr. 2.  R/T2.

Combining these two proportions, Newton can construct a force law—a ratio between force and distance—for any given ratio between the radius of the orbit and the period of revolution, and he demonstrates this capacity by providing three different ones:

Corr. 3. if T2 R, then f is distance-independent,

Corr. 4. if T2 R2, then  1/R, and

Corr. 5. if T2 R3, then  1/R2.

“the case of the fifth corollary holds for the celestial bodies … astronomers are now agreed” he adds, almost as an afterthought.

Newton has no use for the geometrical imagery that provided Hooke and Kepler with justification for the mathematical-causal claims. But his seemingly non-committal presentation does not reflect any particular doubt on Newton’s part either in the mathematical expression or in what “astronomers are now agreed.” Nor is the curious run through the possible force laws produced by replacing the variables of his construct with imaginary data a draft trial to be perfected in the final opus with well-established real data. Quite the opposite: the five corollaries of the De Motu are expanded in Proposition 4 of the Principia to nine and the case of where “the periodic times are as the 3/2 powers of the radii” is just the sixth of them. Moreover, the language distinguishing this particular in the scholium to follow is hardly more excited than in De Motu:

The case of corol. 6 holds for the heavenly bodies (as our compatriots Wren, Hooke, and Halley have also found independently). Accordingly, I have decided that in what follows I shall deal more fully with questions relating to the centripetal forces that decrease as the squares of the distances from centers (Principia, 452)

Which force law to attend is, apparently, a matter of choice for Newton; he had “decided” on the Inverse Square Law. This idea becomes less surprising when one considers Newton’s previous proposition and its corollaries:

Proposition 3, Corollary 2: And if the areas are very nearly proportional to the times, the remaining force will tend towards body T very nearly.

Proposition 3, Corollary 3: And conversely, if the remaining force tends very nearly toward body T, the areas will be very nearly proportional to the times. (Principia, 448-9)

All that Newton requires, so these corollaries suggest, is that the mathematics and the observations fit quam proxime. But there is more than skepticism or careful induction here. The ‘very nearly’ does not mark for Newton the boundaries of inquiry. The precarious relation between the mathematics and the empirical findings that Newton establishes in De Motu represents a fundamental and carefully worked out principle and an efficient tool of physico-mathematical investigation: Newton works his way to the Inverse Square Law of gravity (ISL) by comparing what the data allows quam proxime.11

Newton’s approach to demonstrating the applicability of the hard-earned ISL to the force by which the sun shapes the planetary orbits is founded on the insight that it cannot be achieved by analyzing the empirical data and Kepler’s ‘laws’ directly: very different laws can produce heavenly motions which are “very nearly” identical. Using the same proto-infinitesimal techniques of the De Motu Newton will prove in Prop. 10 that for a body traveling in an elliptical orbit, “the law of the centripetal force tending towards the center of the ellipse” is as the (changing) distance of the body from the center of force (Principia, 459). And in the next proposition, no. 11, he will prove that if “the centripetal force [is] tending towards a focus of the ellipse,” it will be inversely as the square of the distance (Principia, 462-3). In other words, if the sun is in the center of the planets’ elliptical orbits, gravity increases with distance; if the sun is at the focus of these orbits, gravity declines as the square of this distance. For orbits which are deviate very little from the circular, as the orbit Kepler assigns to Mars, the sun is both “very nearly” at the center and very nearly at the focus; but obviously, gravity cannot be proportional to both the distance and its square.

Thus, the ellipse may provide a better geometrical depiction of natural motion—it is the orbit with the closest fit to the observations, Kepler demonstrated—but assuming it one cannot distinguish between the various possible laws of the force affecting this orbit. Instead, Newton remains with motion “in the circumference of a circle,” and develops a very complex theorem (proposition 7) which allows him “to find the law of centripetal force tending toward any given point” inside this circular orbit. Expanding it on the basis of the preceding propositions, George Smith transformed Newton’s geometrical proportion into modern algebraic notation in which force is inversely as:

Where S is the hypothetical position of the center of force (the sun in the solar system), P—the position of the moving body (the planet), a—the diameter of the orbit and —its eccentricity (the distance of center of force—the sun—from the geometrical center to the obit). As Smith acutely points out, “SP to the power of 2 is nowhere to be found in this expression” (40). The expression as a whole, however, converges towards SP2 the closer the eccentric circle can be seen as an approximation of an ellipse with the center of force at a focus—it provides that gravitation will be ‘very nearly’ proportional to 1/r2 if the planetary orbits are very nearly ellipses and the sun very nearly at their focus.

It is therefore completely clear that Newton’s “Quam proxime” does not express a failure to apply simple mathematics to complex nature. Rather, it is a particular constraint that Newton puts on both sides—nature and mathematics: the mathematical law is not exactly describe an idealization of the natural motion, but to approximate a trajectory that approximates a particular curve. For the ISL to be a demonstrated law of nature, it is not enough to deduce it from Kepler’s (idealized) first and third laws—the force law needs to converge towards ISL as the orbits converge to Kepler’s first law. The relation between the ISL and the ellipse fails this criterion. But there is no fact of the matter as to whether the orbit is an eccentric circle or ellipse. After all, “the planets neither move exactly in ellipse nor revolve twice in the same orbit,” so Newton is free to prove the ISL from the former, even though “the simple orbit and the mean among all errors will be the ellipse.” And indeed, the famous propositions concerning the motion of the planets at the beginning of “The System of the World” of Book 3 of the Principia are all based on the circular orbits theorems beginning Book 1.

6. Conclusion

The ISL has been given a brave hew meaning. From a feature of divine infrastructure of young Kepler—a member of the reified ‘mathematicals’—through a partially-flexible geometrical structure for Hooke, to a sophisticated means of approximation is Newton’s Principia, the import of the ISL captures what mathematical order has come to represents: a human solution to the challenges of a messy nature. Not a “deliberate distortion of regular structures,” but most definitely an acute insight that these structures are not furnished by nature, and had to “forced,” paradoxically, by human powers of approximation. Nature and Nature's laws lay hid in night; wrote Pope, God said, "Let Newton be!" and all was light. What Pope had in mind was Kepler’s Renaissance dream of divine order. Newton’s achievement was largely indebted to relinquishing this dream in the name of the enforced order of the Baroque.


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1 C.f. Maraval, e.g. 193 ff. Skine (e.g. 122 and 131) provides an interesting counter example, but his view on what is common to science and Baroque is “fundamental laws and transcendental order” (146)—contrary to what Guardiani (in Scaglione et al.; 81) calls “Baroque’s stubborn ambiguity,” and diametrically opposed to mine.

2 Ref to Ashgate

3 ref

4 Ref to Halley’s Ode to Newton

5 Ref to HOS

6 Ref to Bouliau

7 Ref to Micrographia.

8 Ref to Hunter and Cooper

9 ref

10 Ref. to book.

11 In the following I am much indebted to George Smith’s excellent analysis of these theorems in “From the phenomenon of the Ellipse to an Inverse-Square Force: Why Not?”

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