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Excerpts from the Stanford Encyclopedia of Philosophy
1. Overview: The Importance of the Work
Viewed retrospectively, no work was more seminal in the development of modern physics and astronomy than Newton's Principia. Its conclusion that the force retaining the planets in their orbits is one in kind with terrestrial gravity ended forever the view dating back at least to Aristotle that the celestial realm calls for one science and the sublunar realm, another. Just as the Preface to its first edition had proposed, the ultimate success of Newton's theory of gravity made the identification of the fundamental forces of nature and their characterization in laws the primary pursuit of physics. The success of the theory led as well to a new conception of exact science under which every systematic discrepancy between observation and theory, no matter how small, is taken as telling us something important about the world. And, once it became clear that the theory of gravity provided a far more effective means than observation for precisely characterizing complex orbital motions just as Newton had proposed in the Principia in the case of the orbit of the Moon physical theory gained primacy over observation for purposes of answering specific questions about the world.
The retrospective view of the Principia has been different in the aftermath of Einstein's special and general theories of relativity from what it was throughout the nineteenth century. Newtonian theory is now seen to hold only to high approximation in limited circumstances in much the way that Galileo's and Huygens's results for motion under uniform gravity came to be seen as holding only to high approximation in the aftermath of Newtonian inverse-square gravity. In the middle of the nineteenth century, however, when there was no reason to think that any confuting discrepancy between Newtonian theory and observation was ever going to emerge, the Principia was viewed as the exemplar of perfection in empirical science in much the way that Euclid's Elements had been viewed as the exemplar of perfection in mathematics at the beginning of the seventeenth century. Because of the extent to which Einsteinian theory was grounded historically on Newtonian science, the Principia has retained its unique seminal position in the history of physics in our post-Newtonian era. Perhaps more strikingly, because of the logical relationship between Newtonian and Einsteinian theory Einstein showed that Newtonian gravity holds as a limit-case of general relativity in just the way Newton showed (in Book 1, Section 10) that Galilean uniform gravity holds as a limit-case of inverse-square gravity even though the Principia can no longer be regarded as an exemplar of perfection, it is still widely regarded by physicists as an exemplar of empirical science at its best.
In spite of extravagant claims made about the Principia by some in the years after it first appeared he seems to have exhausted his Argument, and left little to be done by those that shall succeed him[ HYPERLINK "http://plato.stanford.edu/archives/win2008/entries/newton-principia/notes.html" \l "1" 1] the most positive view of it that anyone could have substantiated during the first half of the eighteenth century would have emphasized its promise more than its achievements. The theory of gravity had too many loose ends, the most glaring of which was a factor of 2 discrepancy in the mean motion of the lunar apogee, a discrepancy that undercut the claim that the Moon is held in orbit by an inverse-square force. No one knew these loose ends better than Newton himself, yet no one had a greater sense of the potential of the theory of gravity to resolve a whole host of questions in planetary astronomy which may well explain why he made these loose ends difficult to see except by the most technically skilled, careful readers. Between the late 1730s and the early 1750s the situation changed dramatically when several of the loose ends were tied up, in some cases yielding such extraordinary results as the first truly successful descriptive account of the motion of the Moon in the history of astronomy. During the second half of the eighteenth century the promise of the Principia was not only universally recognized by those active in empirical research, but a large fraction of this promise was realized. What we now call Newtonian mechanics emerged in this process, as did the gravity-based accounts of the often substantial divergences of the planets from Keplerian motion, the achievement of Newton's theory of gravity that ultimately ended all opposition to it.
During the eighteenth century the Principia was also seen as putting forward a world view directly in opposition to the broadly Cartesian world view that in many circles had taken over from the Scholastic world view during the second half of the seventeenth century. Newton clearly intended the work to be viewed in this way when in 1686 he changed its title to Philosophiae Naturalis Principia Mathematica, in allusion to Descartes's most prominent work at the time, Principia Philosophiae. (The title page of Newton's first edition underscored this allusion by placing the first and third words of the title in larger type.) The main difference in the world view in Newton's Principia was to rid the celestial spaces of vortices carrying the planets. Newtonians subsequently went beyond Newton in enhancing this world view in various ways, including forces everywhere expressly acting at a distance. The clockwork universe aspect of the Newtonian world view, for example, is not to be found in the Principia; it was added by Laplace late in the eighteenth century, after the success of the theory of gravity in accounting for complex deviations from Keplerian motion became fully evident.
In addition to viewing the theory of gravity as potentially transforming orbital astronomy, Newton saw the Principia as illustrating a new way of doing natural philosophy. One aspect of this new way, announced in the Preface to the first edition, was the focus on forces:
For the whole difficulty of philosophy seems to be to discover the forces of nature from the phenomena of motions and then to demonstrate the other phenomena from these forces. It is to these ends that the general propositions in books 1 and 2 are directed, while in book 3 our explication of the system of the world illustrates these propositions. For in book 3, by means of propositions demonstrated mathematically in books 1 and 2, we derive from celestial phenomena the gravitational forces by which bodies tend toward the sun and toward the individual planets. Then the motions of the planets, the comets, the moon, and the sea are deduced from these forces by propositions that are also mathematical. If only we could derive the other phenomena of nature from mechanical principles by the same kind of reasoning! For many things lead me to have a suspicion that all phenomena may depend on certain forces by which particles of bodies, by causes not yet known, either are impelled toward one another and cohere in regular figures, or are repelled from one another and recede. Since these forces are unknown, philosophers have hitherto made trial of nature in vain. But I hope that the principles set down here will shed some light on either this mode of philosophizing or some truer one. [P, 382][ HYPERLINK "http://plato.stanford.edu/archives/win2008/entries/newton-principia/notes.html" \l "2" 2]
A second aspect of the new method concerns the use of mathematical theory not to derive testable conclusions from hypotheses, as Galileo and Huygens had done, but to cover a full range of alternative theoretical possibilities, enabling the empirical world then to select among them. This new approach is spelled out most forcefully at the end of Book 1, Section 11:
I use the word attraction here in a general sense for any endeavor whatever of bodies to approach one another, whether that endeavor occurs as a result of the action of the bodies either drawn toward one another or acting on one another by means of spirits emitted or whether it arises from the action of ether or of air or of any medium whatsoever whether corporeal or incorporeal in any way impelling toward one another the bodies floating therein. I use the word impulse in the same general sense, considering in this treatise not the species of forces and their physical qualities but their quantities and mathematical proportions, as I have explained in the definitions. Mathematics requires an investigation of those quantities of forces and their proportions that follow from any conditions that may be supposed. Then, coming down to physics, these proportions must be compared with the phenomena, so that it may be found out which conditions of forces apply to each kind of attracting bodies. And then, finally, it will be possible to argue more securely concerning the physical species, physical causes, and physical proportions of these forces. [P, 588]
A third aspect of the new method, which proved most controversial at the time, was the willingness to hold questions about the mechanism through which forces effect their changes in motion in abeyance, even when the mathematical theory of the species and proportions of the forces seemed to leave no alternative but action at a distance. This aspect remained somewhat tacit in the first edition, but then, in response to criticisms it received, was made polemically explicit in the General Scholium added at the end of the second edition:
I have not as yet been able to deduce from phenomena the reason for these properties of gravity, and I do not feign hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy. In this experimental philosophy, propositions are deduced from the phenomena and are made general by induction. The impenetrability, mobility, and impetus of bodies and the laws of motion and law of gravity have been found by this method. And it is enough that gravity should really exist and should act according to the laws that we have set forth and should suffice for all the motions of the heavenly bodies and of our sea. [P, 943][ HYPERLINK "http://plato.stanford.edu/archives/win2008/entries/newton-principia/notes.html" \l "3" 3]
During most of the eighteenth century the primary challenge the Principia presented to philosophers revolved around what to make of a mathematical theory of forces in the absence of a mechanism, other than action at a distance, through which these forces work. By the last decades of the century, however, little room remained for questioning whether gravity does act according to the laws that Newton had set forth and does suffice for all the motions of the heavenly bodies and of our sea. No one could deny that a science had emerged that, at least in certain respects, so far exceeded anything that had ever gone before that it stood alone as the ultimate exemplar of science generally. The challenge to philosophers then became one of spelling out first the precise nature and limits of the knowledge attained in this science and then how, methodologically, this extraordinary advance had been achieved, with a view to enabling other areas of inquiry to follow suit.
2. The Historical Context of the Principia
The view is commonplace that what Newton did was to put forward his theory of gravity to explain Kepler's already established laws of orbital motion; and the universality of the law of gravity then ended up explaining the deviations from Keplerian motion by attributing them to gravitational interaction of the planets. This is wrong on several counts, the most immediate of which is that Kepler's laws were by no means established before the Principia. The rules for calculating orbital motion that Kepler put forward in the first two decades of the seventeenth century had indeed achieved a spectacular gain in accuracy over anything that had come before. Kepler's rules, however, did not yield comparable accuracy for the motion of the Moon, and even in the case of the planets the calculated locations were sometimes off by as much as a fourth of the width of the Moon. More importantly, by 1680 several other approaches to calculating the orbits had been put forward that achieved the same level of not quite adequate accuracy as Kepler's. In particular, Newton was familiar with seven different approaches to calculating planetary orbits, all at roughly the same accuracy. Only two of these, Kepler's and Jeremiah Horrocks's, used Kepler's area rule planets sweep out equal areas in equal times with respect to the Sun to locate planets along their trajectories. Ismal Boulliau and, following him, Thomas Streete (from whose Astronomia Carolina Newton first learned orbital astronomy) replaced the area rule with a geometric construction. Vincent Wing had adopted still another geometric construction in the late 1660s after having earlier used a point of equal angular motion oscillating about the empty focus of the ellipse; and Nicolaus Mercator in 1676 added still a further geometric construction.[ HYPERLINK "http://plato.stanford.edu/archives/win2008/entries/newton-principia/notes.html" \l "4" 4] Of these six alternative approaches, only Horrocks and, following him, Streete, took Kepler's 3/2 power rule the periods of the planets vary as the square root of the cube of their mean distances from the Sun seriously enough to use the periods rather than positional observations to determine their mean distances.[ HYPERLINK "http://plato.stanford.edu/archives/win2008/entries/newton-principia/notes.html" \l "5" 5]
All these approaches followed Kepler in using an ellipse to represent the trajectory. (The primary historical reason for this was Kepler's success in predicting the 1631 transit of Mercury across the Sun.) This, however, does not mean that the ellipse was established as anything more than a mathematically tractable close approximation to the true orbit. In fact, the planetary orbits known then are not all that elliptical. The minor axis of Mercury is only 2 percent shorter than the major axis, the minor axis of Mars, only 0.4 percent shorter, and in all other cases the difference between an ellipse and an eccentric circle was beyond detection. Newton had real grounds for claiming in a letter to Halley in June 1686 a right to the ellipse, remarking that Kepler knew the Orb to be not circular but oval, and guest it to be Elliptical [C, II, 436]. Entirely independently, the most judicious reader of the first edition of the Principia, Christiaan Huygens, wrote the following summary of the Principias achievement in his notebook upon reading the complimentary copy Newton had sent him:
The famous M. Newton has brushed aside all the difficulties together with the Cartesian vortices; he has shown that the planets are retained in their orbits by their gravitation toward the Sun. And that the excentrics necessarily become elliptical. [OH, XXI, 143]
So, all three of Kepler's rules that came to be called laws after the Principia were known to be nothing more than holding to high approximation when Newton started on the project in 1684. And the leading issue in orbital astronomy at the time was not why Kepler's rules hold, but rather which, if any, of the comparably accurate different approaches to calculating orbits was to be preferred.
The distinct possibility of the ellipse being only an approximation to the true trajectory explains the appropriateness of the question Hooke put to Newton in 1679 and Halley put to him again in 1684 what trajectory does a body describe when moving under an inverse-square force directed toward a central body? The inverse-square part of this question came from combining the mathematical theory of uniform circular motion, which Huygens had published in his Horologium Oscillatorium of 1673, with Kepler's 3/2 power rule: the force in a string retaining a body in a uniform circular orbit varies directly as the radius of the circle and inversely as the square of the period; but the squares of the periods of the planets vary as the cubes of their mean distances; and hence, at least to a first approximation, the forces retaining the planets in their orbits vary inversely with the square of the radii of their nearly circular orbits. But now allow the distance of the orbiting body from the center to vary rather than remaining constant, as in a circle. What trajectory would result if the force toward the center varies as the inverse-square of the distance from the center toward which the force is always directed? The answer in the nine page tract De Motu Corporum in Gyrum that Newton sent to Halley in November 1684 is, an ellipse, provided the velocity is not too high (and if it is, then instead a parabola or a hyperbola, depending on the velocity). The key step in developing this answer is a generalization of uniform circular motion to the case of motion under a centripetal force a term Newton coined from Huygens's centrifugal force, by which he meant the tension in the string keeping the body in a circle; and a key to this step was the discovery that a body moving under any form of centripetal force always sweeps out equal areas in equal times with respect to that center, so that the appropriate geometrical representation of time for generalizing uniform circular motion is area swept out rather than angle or arc length. The tract also confirms that Kepler's 3/2 power rule continues to hold for bodies orbiting in confocal ellipses governed by inverse-square centripetal forces.
These were remarkable steps forward at the time, but they and the questions behind them form only an initial part of the context in which Newton went on to write the Principia. Shortly after the De Motu tract went off to London, Newton revised the tract and added two further passages. The question precipitating this revision appears to have been about the effect the inverse-square centripetal forces directed toward Jupiter, as implied by its satellites, have on the Sun. Newton first added two principles that he first called hypotheses and then changed to laws:
Law 3: The relative motions of bodies enclosed in a given space are the same whether that space is at rest or moves perpetually and uniformly in a straight line without circular motion.
Law 4: The common center of gravity does not alter its state of motion or rest through the mutual actions of bodies. [U, 267]
The second of the two added passages concerns motion in resisting media; it provides a context in which to read Book 2 of the Principia.
The first added passage, which has become known as the Copernican scholium, we here quote in full because it, better than anything else, explains what led Newton into the further research that turned the nine-page tract into the five hundred page Principia. It occurs as a single long paragraph, but is here broken into three segments in order to facilitate commenting on it:
Moreover, the whole space of the planetary heavens is either at rest (as is commonly believed) or uniformly moved in a straight line, and similarly the common centre of gravity of the planets (by Law 4) is either at rest or is moved at the same time. In either case the motions of the planets among themselves (by Law 3) take place in the same manner and their common centre of gravity is at rest with respect to the whole space, and so it ought to be considered the immobile center of the whole planetary system. Thence indeed the Copernican system is proved a priori. For if a common centre of gravity is computed for any position of the planets, this either lies in the body of the Sun or will always be very near it.
By reason of this deviation of the Sun from the center of gravity the centripetal force does not always tend to that immobile center, and hence the planets neither move exactly in ellipses nor revolve twice in the same orbit. Each time a planet revolves it traces a fresh orbit, as happens also with the motion of the Moon, and each orbit depends upon the combined motions of all the planets, not to mention their actions upon each other. Unless I am much mistaken, it would exceed the force of human wit to consider so many causes of motion at the same time, and to define the motions by exact laws which would allow of an easy calculation.
Leaving aside these fine points, the simple orbit that is the mean between all vagaries will be the ellipse that I have discussed already. If any one shall attempt to determine this ellipse by trigonometrical computation from three observations (as is usual) he will be proceeding without due caution. For these observations will share in the very small irregular motions here neglected and so cause the ellipse to deviate somewhat from its actual magnitude and position (which ought to be the mean among all errors), and so there will be as many ellipses differing from each other as there are trios of observations employed. Very many observations must therefore be joined together and assigned to a single operation which mutually moderate each other and display the mean ellipse both as regards position and magnitude. [U, 280]
The first segment highlights a further component of the historical context in which the Principia was written and read. Galileo's discovery of the phases of Venus in 1613 had provided decisive evidence against the Ptolemaic system,[ HYPERLINK "http://plato.stanford.edu/archives/win2008/entries/newton-principia/notes.html" \l "6" 6] but it could not provide grounds favoring the Copernican over the Tychonic system. In the latter, Mercury, Venus, Mars, Jupiter and Saturn circumnavigate the Sun, and the Sun circumnavigates the Earth, with the consequence that these seven bodies are at all times in the same position in relation to one another as they are in the Copernican system. Whether any decisive empirical grounds could be found favoring the Copernican over the Tychonic system became one of the most celebrated issues of the seventeenth century. Kepler, Galileo, and Descartes all published major books in the first half of the century purporting to resolve this issue,[ HYPERLINK "http://plato.stanford.edu/archives/win2008/entries/newton-principia/notes.html" \l "7" 7] Kepler and Descartes basing their arguments on the physical mechanism each had proposed as governing the orbital motion. Nevertheless, the leading observational astronomer of the second half of the century, G. D. Cassini, was a Tychonist. In the first segment of the Copernican Scholium Newton identifies the center of gravity of the planetary system as the appropriate point to which all the motion should be referred the technical issue behind the issue over the two systems and then announces that the centripetal forces identified in the text of De Motu as governing the orbital motion open the way to establishing a slightly qualified form of the Copernican system.[ HYPERLINK "http://plato.stanford.edu/archives/win2008/entries/newton-principia/notes.html" \l "8" 8] Newton's discovery of this line of reasoning was surely a major factor urging him on to the Principia.
The second segment of the Copernican Scholium addresses an issue in orbital astronomy that forms a still further component in the historical context of the Principia. Separate from the question whether Kepler's or some other approach was to be preferred was the question whether the true motions are significantly more irregular and complicated than the calculated motions in any of these approaches. The complexity of the lunar orbit and the continuing failure to describe it within the accuracy Kepler had achieved for the planets was one consideration lying behind this question. Another came from Kepler's own finding, noted in the Preface to his Rudolphine Tables[ HYPERLINK "http://plato.stanford.edu/archives/win2008/entries/newton-principia/notes.html" \l "9" 9] and subsequently supported by others, that the true motions may involve further vagaries, as evidenced by apparent changes in the values of orbital elements over time. The most important consideration behind this question, however, came from Descartes' claim that, in keeping with the changing motions of his vortices over long periods of time, the orbits are not mathematically perfect and they are continuously changed by the passing of the ages [D, 3, 34]. In the second segment of the quoted Scholium, Newton concludes that, in contrast to the ellipse that answered the mathematical question put to him by Hooke and Halley, the true orbits are not ellipses, but are indeed indefinitely complex. This conclusion is nowhere so forcefully stated in the published Principia, but knowledgeable readers nonetheless saw the work as answering the question whether the true motions are mathematically perfect in the negative.
Finally, the second and third segments together not only point out that Keplerian motion is only an approximation to the true motions, but they call attention to the potential pitfalls in using the orbits published by Kepler and others as evidence for claims about the planetary system. For example, if the true motions are so complicated, then it is not surprising that all the different calculational approaches were achieving comparable accuracy, for all of them at best hold only approximately. Equally, the success in calculating the orbits could not serve as a basis to argue against Cartesian vortices, for the irregularities entailed by them could not simply be dismissed. The spectre raised was the very one Newton had objected to during the controversy over his earlier light and color papers: too many hypotheses could be made to fit the same data.[ HYPERLINK "http://plato.stanford.edu/archives/win2008/entries/newton-principia/notes.html" \l "10" 10] Worse, the multiplicity of tenable hypotheses was a spectre haunting mathematical astronomy as a discipline from the end of the sixteenth century forward.[ HYPERLINK "http://plato.stanford.edu/archives/win2008/entries/newton-principia/notes.html" \l "11" 11] So, the conclusion that calculated orbits can at most be mere approximations would have been seen as raising the possibility that truth and exactness were beyond the reach of mathematical astronomy. The main reason why the Principia includes so much beyond the De Motu tract is Newton's endeavor to reach conclusions that had claim to being exact and true in spite of the inordinate complexities of the actual motions.
The historical context in which Newton wrote the Principia involved a set of issues that readers of the first edition saw it as addressing: Was Kepler's approach to calculating the orbits, or some other, to be preferred? Was there some empirical basis for resolving the issue of the Copernican versus the Tychonic system? Were the true motions complicated and irregular versus the calculated motions? Can mathematical astronomy be an exact science? No reader of the Principia at the time had the benefit of seeing how Newton had these questions tied together in the Copernican Scholium because it did not appear in print until two hundred years later.[ HYPERLINK "http://plato.stanford.edu/archives/win2008/entries/newton-principia/notes.html" \l "12" 12] Nothing, however, brings out more clearly the extent to which the expanded scope of the Principia stemmed from Newton's preoccupation with the problem of reaching conclusions that had claim to being exact from evidence that, by his reckoning, held at best to high approximation. This is why the Copernican scholium provides the most illuminating context for reading the Principia. Equally, its being unknown for so long helps to explain why the Principia has generally been read so simplistically.
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