|© CAMBRIDGE UNIVERSITY PRESS 1999|
2. First Qualitative Physics: The Newton-Bentley Exchange
Isaac Newton needs no introduction.
Richard Bentley was one of England's leading theologians, with strong scientific interests and very worldly ambitions. Eventually he became Master of Trinity College, Cambridge, reigning for forty-two contentious years. Tyrannical and overbearing, Bentley tried to reform the College (as well as the University Press) and spent much of the College's income on new buildings, including a small observatory. To balance the College accounts he reduced its payments to less active Fellows, while increasing his own stipend. After ten years of this, some of the Fellows rebelled and appealed to the Bishop of Ely and Queen Anne, the ultimate College authorities, to eject Bentley from the mastership. Various ruses enabled Bentley to put off the trial for another four years. Finally the Bishop condemned Bentley in a public court. But before he could formally deprive Bentley of his mastership, the Bishop caught a chill and died. Queen Anne died the next day. Bentley now put his theological talents to work to convince his opponents that he had won "victory" by divine intervention. So he retained the mastership and raised his salary still higher. Some years later, another attempt to expel him by a fresh Bishop also failed, and he remained Master until dying in 1742 at the age of eighty. During the crucial period of these collegiate upheavals, about 1708-1713, Bentley had Newton's firm support; simultaneously he was seeing the second edition of Newton's Principia through the University Press.
As a young man Bentley was asked, possibly through Newton's maneuvering, to give the first Robert Boyle lectures. Although now mainly known for his result that the pressure ofa perfect gas is linearly proportional to its density at constant temperature, Boyle also left an endowment for lectures in defense of religion. Earlier, Bentley had studied much of the Principia, having obtained a list of preliminary readings in mathematics and astronomy from Newton. By late 1692, he had a few questions to ask Newton as he prepared the final manuscript of his eight Boyle Lectures.
These lectures were supposed to confute atheists and show that Natural Science still required a role for the Creator. In modern terms, the role that emerged was to provide initial conditions, boundary conditions, and divine interventions to account for those aspects of the Universe that could not be understood with the physics of Newton's day. Among other questions, Bentley asked why some matter in the Universe formed the luminous Sun and other matter became the dark planets, whether the motions of the planets could arise from natural causes, whether the lower densities of the outer planets were caused by their distance from the Sun, and what produced the inclination of the Earth's axis. The implication of these questions was that Newton's physics could offer no complete explanation, and so Newton, who was basically theistic, agreed that these phenomena were evidence for the deity.
In answering the first question, Newton (1692) made his famous prescient comments about the distribution of matter in the Universe:
As to your first query, it seems to me that if the matter of our sun and planets and all the matter in the universe were evenly scattered throughout all the heavens, and every particle had an innate gravity toward all the rest, and the whole space throughout which this matter was scattered was but finite, the matter on the outside of the space would, by its gravity, tend toward all the matter on the inside, and by consequence, fall down into the middle of the whole space and there compose one great spherical mass. But if the matter was evenly disposed throughout an infinite space, it could never convene into one mass; but some of it would convene into one mass and some into another, so as to make an infinite number of great masses, scattered at great distances from one to another throughout all that infinite space. And thus might the sun and fixed stars be formed, supposing the matter were of a lucid nature. But how the matter should divide itself into two sorts, and that part of it which is fit to compose a shining body should fall down into one mass and make a sun and the rest which is fit to compose an opaque body should coalesce, not into one great body, like the shining matter, but into many little ones; or if the sun at first were an opaque body like the planets, or the planets lucid bodies like the sun, how he alone would be changed into a shining body whilst all they continue opaque, or all they be changed into opaque ones whilst he remains unchanged, I do not think explicable by mere natural causes, but am forced to ascribe it to the counsel and contrivance of a voluntary Agent.
Bentley wrote back questioning whether matter in a uniform distribution would convene into clusters because by symmetry there would be no net force, either on the central particle in a finite system or on all particles in an infinite system. And so, five weeks after his first letter, Newton replied in January 1693:
The reason why matter evenly scattered through a finite space would convene in the midst you conceive the same with me, but that there should be a central particle so accurately placed in the middle as to be always equally attracted on all sides, and thereby continue without motion, seems to me a supposition as fully as hard as to make the sharpest needle stand upright on its point upon a looking glass. For if the very mathematical center of the central particle be not accurately in the very mathematical center of the attractive power of the whole mass, the particle will not be attracted equally on both sides. And much harder it is to suppose all the particles in an infinite space should be so accurately poised one among another as to stand still in a perfect equilibrium. For I reckon this as hard as to make, not one needle only, but an infinite number of them (so many as there are particles in an infinite space) stand accurately poised upon their points. Yet I grant it possible, at least by a divine power; and if they were once to be placed, I agree with you that they would continue in that posture without motion forever, unless put into new motion by the same power. When, therefore. I said that matter evenly spread through all space would convene by its gravity into one or more great masses, I understand it of matter not resting in an accurate poise.
... a mathematician will tell you that if a body stood in equilibrio between any two equal and contrary attracting infinite forces, and if to either of these forces you add any new finite attracting force, that new force, howsoever little, will destroy their equilibrium and put the body into the same motion into which it would put it were those two contrary equal forces but finite or even none at all; so that in this case the two equal infinities, by the addition of a finite to either of them, become unequal in our ways of reckoning; and after these ways we must reckon, if from the considerations of infinities we would always draw true conclusions.
Thus Newton recognized, qualitatively, the main ingredients of galaxy clustering (although he had the formation of the stars and solar system in mind): a grainy gravitational field produced by discrete objects, a multiplicity of cluster centers in an infinite (we would now say unbounded) universe, the impotence of the mean gravitational field in a symmetric distribution, and the finite force produced by perturbing an equilibrium state. Strangely, it would take nearly 300 years to quantify these insights and compare their results with the observed galaxy distribution. The difficulty was understanding nonlinear many-body gravitational clustering, whose delightful subtleties are all the more surprising for being based on a simple inverse square interaction.
Newton himself might have been able to begin the quantitative analysis. Indeed, perhaps he did. In an intriguing essay, Harrison (1986) suggests that Newton estimated the timescale (G )-1/2 ~ 108 years for gravitational instability, but it conflicted so strongly with the biblical timescale since the flood (~ 5,000 years), and with his own estimate in the Principia for the cooling of the Earth from a globe of red-hot iron (~ 50,000 years), that he thought it wiser not to publish. This could be behind his allusion at the end of the first letter to Bentley: "There is yet another argument for a Deity, which I take to be a very strong one; but till the principles on which it is grounded are better received, I think it more advisable to let it sleep." However, Westfall (1980) thinks this remark may refer to the development of history as foretold in the prophecies. We'll never know unless someone rediscovers old documents or finds a hidden message in Newton's writings.
Newton's insights had their precursors. We saw earlier that Lucretius recognized the essentially different nature of clustering in finite and infinite universes. Although most of the medieval church scholastics who succeeded him were mainly interested in reconciling the natural world with a literal interpretation of the Bible, some of the ancient Greek knowledge and independence of mind survived (Dreyer. 1905; Duhem. 1985) at a few scattered monasteries from Bremen in Germany to lona in the Inner Hebrides. Nicholas of Cusa (translation 1954) realized in the fifteenth century that if the universe did not have boundaries, it would have neither a fixed pole nor a center. Every point would be an equivalent center. This suggests Newton's view that many centers of clustering grow in an infinite universe. How much Newton knew of the earlier work is unclear, although in the Scholium of the Principia he used a straightforward example similar to one of Cusa's for discussing absolute motion. Thomas Diggs, one of England's leading mathematicians a century before Newton, had strongly promoted the idea of an infinite universe of stars surrounding a Copernican solar system, an idea also associated in a more mystical context with Giordano Bruno about the same time. But these were just pictures of the world until Newton sketched a dynamical explanation.
Fifty years after Newton, the idea of an infinite universe filled with structure whose primary cause was gravitation began to consolidate. Thomas Wright (1750) proposed that the distribution of light in the band of the Milky Way showed that the Sun lies in a great disk of stars at considerable distance from its center. The idea impressed Immanual Kant (1755) who noted: "It was reserved for an Englishman, Mr Wright of Durham, to make a happy step with a remark which does not seem to have been used by himself for any very important purpose, and the useful application of which he has not sufficiently observed. He regarded the Fixed Stars not as a mere swarm scattered without order and without design, but found a systematic contribution in the whole universe and a universal relation of these stars to the ground-plan of the regions of space which they occupy:" If there was one Milky Way, Kant reasoned, why could there not be many? He even identified them as the elliptical nebulae that the French philosopher Maupertuis thought were enormous rotating single stars. Then Kant went on to suggest that even the distribution of many Milky Ways may be structured as part of a grand hierarchy of systems:
The theory which we have expounded opens up to me a view into the infinite field of creation... . If the grandeur of a planetary world in which the Earth, as a grain of sand, is scarcely perceived, fills the understanding with wonder; with what astonishment are we transported when we behold the infinite multitude of worlds and systems which fill the extension of the Milky Way. But how is this astonishment increased, when we become aware of the fact that all these immense orders of star-worlds again form but one of a number whose termination we do not know. and which perhaps like the former, is a system inconceivably vast - and yet again but one member in a new combination of numbers! We see the first members of a progressive relationship of worlds and systems; and the first part of this infinite progression enables us already to reorganize what must be conjectured of the whole. There is no end but an abyss of a real immensity, in presence of which all capability of human conception sinks exhausted, although it is supported by the aid of the science of number.
This happy interplay among observations of the nebulae, Kant's essentially correct but unsupported interpretation of them as galaxies, and his speculative extrapolation to even larger scales of clustering was held together by the attraction of universal gravity. Kant also recognized that centrifugal forces were necessary for equilibrium: "The attraction which is the cause of the systematic constitution among the fixed stars of the Milky Way acts also at the distance even of those worlds, so that it would draw them out of their positions and bury the world in an inevitably impending chaos, unless the regularly distributed forces of rotation formed a counterpoise or equilibrium with attraction, and mutually produced in combination that connection which is the foundation of the systematic constitution."
At this stage Kant goes astray. Guided too strongly by analogies with the solar system, he suggests that all these Milky Ways and higher systems move around a common center despite his agreement with Cusa's view that an infinite universe has no center. To reconcile these views, he proposes a remarkably modem cosmogony:
Let us now proceed to trace out the construction of the Universal System of Nature from the mechanical laws of matter striving to form it. In the infinite space of the scattered elementary forms of matter there must have been some one place where this primitive material had been most densely accumulated so as through the process of formation that was going on predominantly there, to have procured for the whole Universe a mass which might serve as its fulcrum. It indeed holds true that in an infinite space no point can properly have the privilege to be called the center; but by means of a certain ratio, which is founded upon the essential degrees of the density of primitive matter, according to which at its creation it is accumulated more densely in a certain place and increases in its dispersion with the distance from it, such a point may have the privilege of being called the center; and it really becomes this through the formation of the central mass by the strongest attraction prevailing in it. To this point all the rest of the elementary matter engaged in particular formations is attracted; and thereby, so far as the evolution of nature may extend it makes in the infinite sphere of creation the whole universe into only one single system.
... the creation, or rather the development of nature, first begins at this center and, constantly advancing, it gradually becomes extended into all the remoter regions, in order to fill up infinite space in the progress of eternity with worlds and systems Every finite period, whose duration has a proportion to the greatness of the work to be accomplished, will always bring a finite sphere to its development from this center; while the remaining infinite part will still be in conflict with the confusion and chaos, and will be further from the state of completed formation the farther its distance is away from the sphere of the already developed part of nature.
If only Kant had taken the next bold step and realized that there could be many equivalent centers of developing structure, he would have been the very model of a modern astrophysicist. He could have reconciled the geometric property that all points in an infinite universe are equivalent to the center (i.e., none are geometrically distinguished) with the structural property that any point is the center of an equivalent structure. To do this, however, he would have had to take the short but subtle step to the idea of statistical homogeneity. This idea supposes that by sampling a large number of well-separated regions, that is, regions without common structure, one obtains average statistical properties (such as density, correlation functions, distribution functions) that are independent of the actual sample. Any one or several regions on any scale may have peculiar configurations of galaxies, but these configurations will have different positions, orientations, and forms from region to region and so give common averages over samples with many regions. In this sense, all points would be statistically equivalent with respect to large-scale structure. The initial conditions needed to produce Kant's form of large-scale structure could have been present at many places, remote from each other, instead of at just one as Kant thought. Had Kant known of Newton's unpublished first letter to Bentley and combined Newton's insight into multiple clustering with his own extrapolation of Wright's Milky Way structure, he just might have realized all the essential aspects of our modern understanding of galaxy clustering.
As it happened, Kant's views survived for over a century. The questions he addressed had excited discussion in fashionable French salons ever since Fontenelle's (1686) Conversation on the Plurality of Worlds first popularized the new astronomy. Now a much later but timely English edition (1769) reinforced this interest:
I confess it Madam; all this immense mass of matter, which composes the universe, is in perpetual motion, no part of it excepted; and since every part is moved, you may be sure that changes must happen sooner or later; but still in times proportional to the effect. The ancients were merry gentlemen to imagine that the celestial bodies were in their own nature unchangeable, because they observed no alteration in them; but they did not live long enough to confirm their opinion by their own experience; they were boys in comparison of us.
Perhaps Fontenelle was anticipating that he would himself live to be a hundred. Three decades later, Lambert (1800) thought he had identified Kant's center of motion of the Universe: It was the Orion nebula!
If ever astronomy needed further observations to get off the wrong track, this was the time. Newton and Kant had posed questions of the motion and distribution of matter in terms which, for the first time, could be interpreted quantitatively. There was no chance in the late eighteenth century of measuring motions at great distances, but Herschel was beginning to chart the nebulae and provide the next great clue.