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Sara Smollett Sara Smollett
December 2002

Influence of Newton's De Motu

In this paper I will discuss the effect Isaac Newton's November 1684 tract De Motu had on the evidential status of six (so-called) laws: Kepler's three laws and the laws of free fall, parabolic trajectory, and inertia. I will examine how Newton offered support for the status of these six laws. I will consider the impact of De Motu and, in particular, a handful of questions that it opened up for Newton and others to explore. I believe that Newton's work in De Motu had a profound influence on the studies of celestial and terrestrial mechanics. De Motu contributed both directly to the body of scientific theory and paved the way for further studies in motion. Like Copernicus' De Revolutionibus, De Motu marked the beginning of a revolution.

De Motu is a very short mathematical work (originally three definitions, four hypotheses, four theorems, and seven problems), upon which Newton embarked after a visit from Edmond Halley in August 1684. Halley, Christopher Wren, and Robert Hooke had all been entertaining the hypothesis of inverse square forces directed to a point in or near the sun. That is, the hypotheses that the magnitude of the centripetal force (a phrase coined by Newton) directed to the sun from any body is inversely proportional to the square of the distance from the sun. Halley posed the following question to Newton: what would the trajectory of the planets be under the influence of such an inverse square force? Newton, who had previously worked out the calculation, immediately responded that it would be an ellipse, and by November 1684 had reproduced his calculations of the specific ellipse in De Motu (Westfall, 403).

Although the work Halley received from Newton and subsequently registered with the Royal Society consisted of just a handful of geometric theorems and problems, it was immediately recognized by Halley and others who eagerly read it as revolutionary (Westfall, 420), for it advanced the inverse square problem that he and others had been working on. Halley saw in De Motu an answer to his trajectory question: the trajectory of a body under the influence of an inverse square force is a Keplerian ellipse.

De Motu is an extraordinary work which tied together not only Kepler's three laws of celestial mechanics, but also three laws of terrestrial mechanics: the laws of free fall, parabolic trajectory, and inertia. In addition to advancing the evidence for these laws, by treating these laws as a necessary result or epiphenomena of a physical cause, De Motu opened up a number of questions for further study. The importance of De Motu, then, lies not only in the answers it provided, but also in the effects it had on the studies of Halley, John Flamsteed, and others. I believe that in addition to influencing them, De Motu had a great influence on Newton himself. What might have started as a fairly abstract mathematical endeavor led Newton, step by step, from question to answer to question, to consider and develop a rich and powerful unified physical theory of orbital astronomy and local mechanics.

Kepler and Orbital Astronomy:

In the 1680's many astronomers were making use of Kepler's three laws, which were known to hold at least within observational accuracy. In De Motu Newton derives Kepler's three laws - that the orbits of the planets around the sun are ellipses with the sun at one focus; that a line joining a planet and the sun sweeps out equal areas in equal intervals of time (Kepler's area rule); and that the square of the sidereal period of a planet is directly proportional to the cube of the semimajor axis of the orbit (Kepler's 3/2 power rule) - from the inverse square principle.

In Theorem 1 of De Motu, Newton, assuming acceleration directed to a central point, showed that Kepler's area rule holds. In Problem 3 Newton showed that for a body revolving in an ellipse, there must be an inverse square centripetal force directed toward one focus of the ellipse. And he showed in Corollary 5 of Theorem 2 that the 3/2 power rule holds if and only if there is an inverse square centripetal force. Newton also provided a method for determining specific forces and orbits in the form of an initial value problem.

A significant contribution of De Motu was to offer support for Kepler's laws. According to Newton's theory, Kepler's laws are the observed consequences of an inverse square centripetal acceleration directed to a point which is the focus of an ellipse. The inverse square principle, if true, can be seen as evidence for Kepler's three laws. De Motu offers a physical explanation for elliptical orbits and Kepler's laws. The ellipse is not just a deviation from a circle requiring two forces, but is a specific ellipse as a direct consequence of one force: an inverse square centripetal force. As Kepler posed an answer as to how the planets move, Newton posed an answer as to why the planets move. Additionally, the coherence of the three laws with each other, as well as with the inverse square principle, can be taken as further evidence for them.

Perhaps De Motu's greatest contribution to Keplerian astronomy was that it tied together Kepler's three laws and, at the same time, tied them to the inverse square principle. This correlation offered both a physical explanation for Kepler's laws and the promise of improved calculations of planetary orbits. Newton's insight surpassed existing Keplerian theory by explaining celestial trajectories with the same force that can be used to explain terrestrial motion.

Having worked out the mathematics in De Motu, Newton was in a place to consider under what conditions Kepler's laws hold and whether they hold exactly, ideally, or only approximately. Newton supposed that the primary explanation for planetary orbit was a centripetal inverse square force pulling the planets toward the sun. If this is the only force, then the trajectory must be an exact ellipse and Kepler's other laws must also hold exactly. If there are other secondary forces, Kepler's laws are not exact, but they would still hold with a high degree of accuracy as idealizations which can be explained by a physical theory. Any observed deviations from a Keplerian ellipse, then, have meaning and can be used to identify secondary factors (perturbations of the orbits) and improve both the approximation and the physical theory.

While De Motu was a remarkable work which answered many questions, it also left a number of issues to be resolved. For instance, Newton's theory relied on an inverse square centripetal force, but it did not explain what causes such a force. Nor was there much evidence to accept the existence of inverse square centripetal forces in the universe. In the case of such a force directed to the Earth, there was evidence, but only the results of the moon test of the 1680's. Similarly, De Motu did not conclusively point to elliptical planetary orbits as a necessary conclusion of the inverse square principle. Questions also remained as to the location of the center of the universe, and in particular, if the inverse square principle could be used as part of a sufficient argument to distinguish between the Copernican and Tychonic system. I believe that someone reading Newton's De Motu at the time could very well have been led to consider and further investigate these ideas. These very questions were likely on Newton's mind as he wrote De Motu and soon thereafter, as well as on the minds of those who read De Motu.

There are two further questions that I think Newton began to seriously ponder while writing De Motu: ``What are the trajectories of the comets?'' and ``Are there multiple centripetal forces interfering with each other?'' Although these questions are not answered (nor in the case of the latter even addressed) in the first version of De Motu, I believe these questions were posited in part as a result of this work and, as such, can be indirectly attributed to De Motu.

The first of these questions is briefly explored in the Scholium to Problem 4 and shows that Newton is thinking about comets and the effects of centripetal force on them. The second of these is not discussed in De Motu but appears in a series of letters between Flamsteed and Newton in December 1684 and January 1685, just after Newton finished his draft but before Flamsteed had read it. Newton addresses Flamsteed with concerns about the interaction of forces on the motion of Jupiter, Saturn, and their satellites and the possibility of resultant perturbations in their orbits. The timing of this correspondence implies to me that Newton was thinking about the interaction of forces either as he was writing De Motu or immediately afterward as a result of the work he had done on De Motu.

De Motu was truly an important work. It supported Kepler's laws both by tying the three laws together and providing a simple physical explanation for Keplerian orbits. From the idea that deviations in the orbits could be used to improve the theory, it enabled astronomers and theorists to raise further questions based on observations. Further, Newton's attention to the inverse square principle led to the search for other consequences of inverse square centripetal forces and initiated the demand for a mechanical explanation for the existence of inverse square forces. I conclude that the importance of De Motu in the history of orbital astronomy lies in both the questions it answered and the questions it allowed to be asked by Newton and others.

Galileo and Local Motion:

Newton's contributions in De Motu extended beyond orbital astronomy; Problems 4-7 of De Motu describe the mechanics of local motion. In these problems, Newton considered two interrelated laws of Galileo: the law of free fall, which says that the rate of vertical fall is proportional to the square of time, and the law of parabolic trajectory, which describes the path of the motion of a projectile as a parabola. Newton did not merely reproduce the earlier work of Galileo. Unlike Galileo, Newton considered these two laws both in the absence of and in the presence of air resistance. Additionally, Newton's laws by relying not on uniform acceleration, but on the assumption that gravity is, as Newton says in the Scholium following Problem 5, ``one species of centripetal force''. Finally, although both Galileo and Newton arrived at projectile motion by combining uniformly accelerated motion (Problem 6) and vertical free fall, Galileo concluded that projectile motion is described by a parabolic arc, whereas Newton described the resultant trajectory as a very eccentric (nearly parabolic) ellipse.

Newton's De Motu can be seen as having significantly advanced the evidential status for the laws of free fall and parabolic trajectory (and to a lesser extent the law of inertia) both by tying all three together and by offering as a hypothesis a single cause from which all three laws would follow. As in celestial mechanics, the coherence of three laws with each other as well as their collective coherence with the inverse square principle (and Kepler's three laws) provided strong evidence for accepting these laws as nomological.

Newton's theory made further advances by reducing the level of calibration needed for calculating projectile motion. Whereas Galileo required different calibrations for variations in size, shape, and velocity of projectiles, Newton required only one calibration for resistance. Additionally, Newton advanced the concept of inertia (or rather of innate force). Following Descartes and Huygens, Newton stressed that it is not the continuation of motion, but changes in motion that require explanation in the form of an external cause. Finally, De Motu may be seen as marking the beginning of the inclusion of resistance and resisting forces in the science of dynamics.

Newton's work in De Motu, then, supports the conclusion that if the force we now refer to as terrestrial gravity is assumed to be an inverse square centripetal force, the laws of free fall and parabolic trajectory hold to a high degree of accuracy. However, Newton showed that they do not hold exactly even in the absence of air resistance; in particular, the motion of projectiles is not a parabola, but an ellipse approximating a parabola.

In addition to so establishing the status of these three laws, De Motu made advances by opening new directions and methods of inquiry in the studies of mechanics and kinematics. More work remained to be done on the laws of free fall, parabolic trajectory, and especially inertia. De Motu offered a conjectural theory, and it remained for Newton and others to amass evidence to support the theory (such as experiments determining the force of surface gravity). Newton's work also lent support to the idea that deviations from innate (inertial) and centripetal forces require explanations and that the study of such deviations can be used to devise and revise theories. Additionally, De Motu can be seen as a pioneering work in the science of resistance.

Conclusion:

Newton's De Motu was a revolutionary scientific work that made many advances in celestial and terrestrial mechanics. It tied together six laws - Kepler's laws of elliptical orbits, Galileo's laws of free fall and parabolic trajectory, and the law of inertia - and the previously separate studies of orbital astronomy and terrestrial mechanics with one single conjecture: the existence of inverse square centripetal forces directed toward the sun and planets with satellites. This conjecture offered a more reasonable alternative to vortex theory and provided an explanation for the laws of Kepler and Galileo, by enabling them to be viewed as consequences of physical forces. Additionally, De Motu provided support for all six laws under certain circumstances, while at the same time showing that they do not hold exactly.

Additional significance lies not in what De Motu accomplished, but in what it enabled. De Motu was primarily the result of a very powerful, but unsubstantiated, conjectural theory developed through elegant geometric proofs. It was not the result of experimentation nor a presentation of conclusive scientific evidence. The development of the hypothesis of inverse square centripetal force was only the first step in developing a new and richer mechanics. An important contribution of De Motu was in opening up new areas of study and providing the groundwork for making advances in these areas.

In conclusion, Newton's De Motu and the year 1684 mark a turning point in the history of science. It was a year when many old questions were answered and still many more new questions were posed. It was the beginning, not the peak, of a new scientific revolution.

## References

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Newton, Isaac. ``De Motu Corporum in Gyrum''.

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Westfall, Richard S. Never at Rest: A Biography of Isaac Newton. Cambridge University Press, 1980. 402-437.

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Wilson, Curtis. From Kepler's Laws, So-called to Universal Gravitation: Empirical Factors. 156-166.

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http://www.wikipedia.org/wiki/Isaac_Newton/Authoring_Principia

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http://www.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion.

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