December 2002

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.

- []
- Newton, Isaac. ``De Motu Corporum in Gyrum''.
- []
- Westfall, Richard S.
*Never at Rest: A Biography of Isaac Newton*. Cambridge University Press, 1980. 402-437. - []
- Wilson, Curtis.
*From Kepler's Laws, So-called to Universal Gravitation: Empirical Factors*. 156-166. - []
- http://www.wikipedia.org/wiki/Isaac_Newton/Authoring_Principia
- []
- http://www.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion.

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On 18 Dec 2002, 17:39.