Gregory Palermo, Suraj Uttamchandani, Michael Eisigner
INTD 288 - History of the Physical Sciences
Drs. McLean, Cope, and Towsley
April 17, 2012
In the late 16th and early 17th century, Johannes Kepler built on the observations of Tycho Brahe
to conjecture his three laws, the first of which states that planetary orbits are elliptical. At that time, the
conjecture was based only on observation and geometry. A proof was still lacking, possibly because there
did not yet exist a precise and mathematically rigorous definition of velocity and acceleration or a
quantitative explanation of their relation to these elliptical orbits. Newton developed calculus to find these
more precise definitions for velocity and acceleration. He was soon publicly challenged to prove some of
Kepler’s ideas about planetary motion. In response, he derived them from his Universal Law of
Gravitation and put them forward in his Principia. While he supposedly achieved this without calculus in
the published work, there exist derivations of Kepler’s First Law that are done using Newton’s Calculus
in its contemporary form. The discussion of elliptical orbits is a binary one; the problem can be discussed
with rigorous proof or astute observation, approached directly or inversely, and attacked with calculus or
with geometry. These types of dichotomies affected Kepler and Newton themselves, just as they affect
our own interpretation of these scientists’ work.
Kepler’s audience, for instance, held strongly to the notion that planets move in circular orbits.
Scientific reasoning suggested that certain constellations, such as Draco, Cepheus, and Cassiopeia
exhibited a kind of circular motion in observation from earth. A recurring theme was that “periodicity
and regularity,” present in much astronomical observation of the time, was explainable only by circular
motion. From a religious standpoint, heaven was home to divine beings, and no imperfect motion could
exist there. Philosophical thought supplemented this view: Aristotle, whose ideas were still influential at
Kepler’s time, asserted that circular motion was of the most perfect kind, since it never terminates
(Kozhamthadam 203). Additionally, many held on to the Euclidian idea that any “useful” figure could be
constructed with just a compass and a ruler – a property that ovals and ellipses do not have.
Kepler himself believed very strongly in these arguments for circularity. When attempting to
model the orbit of Mars, however, he found several discrepancies in his proposed values for Mars’
positions and the ones he observed from Earth. After attempting to account for these discrepancies by
considering observational error and contradictions between his collected data and the works of
Copernicus and Brahe, Kepler ruled out the possibility that the orbit of the planets could be circular. He
ultimately (and grudgingly) made the conjecture of elliptical orbits and set out to prove it, eventually
providing the framework for Newton to prove his Universal Law of Gravitation.
Newton, born thirteen years after Kepler’s death, quickly developed into an eager math student
who voraciously read all of the treatises he could find. Around 1665, he first began thinking about the
forces that kept celestial bodies in line. Newton tried to calculate the exact extent of these forces’ actions
on the moon, but, because of wrong assumptions, did not come to a correct answer. At this point, Newton
temporarily lost interest in the problem.
In a fateful conversation in August 1684, however, Edmund Halley inquired of Newton what path
a body would take under an inverse-square law of attraction. Newton responded by saying that he had
already proved the path to be elliptical, but had misplaced the proof. He promised to write another, and
within three months had written De Motu Corporum in Gyrum, which provided much of the core material
of the Principia. After submitting this treatise to the Royal Society of London, Newton began working in
earnest on his masterpiece; he originally intended the work to be a more detailed investigation of material
from his previous volume, but he soon began studying other problems that made it into the Principia,
such as comets and tides.
Instead of proving his theories with what we know today as calculus, however, Newton argues his
points using advanced geometry. While in his early years, Newton followed the contemporary trend of
using algebra in his research, he distanced himself from modernity in the 1670s, preferring instead to
work with the geometry of the classical mathematical giants. This explains Newton’s respect for
Huygens, who, though contemporary, was heavily involved in geometrical proofs and reasoning. The
reasons for this switch in preference boil down to Newton’s wish to further explore the geometry in his
mathematical techniques, as well as a proclaimed dislike for Descartes, who advocated the “New” school
of mathematics (Guicciardini 30). Additionally, Newton’s relative isolation at Cambridge from the
intellectual hub of the Royal Society added to his desire to use what he felt suited his purpose the best:
geometry. Without pressure from his contemporaries, Newton was free to pursue geometric arguments in
order to the Kepler problem.
One can solve the Kepler problem in either of two directions, referred to as the “direct problem”
and the “inverse problem.” In Newton’s time, solving the direct problem involved “determining the nature
of the gravitational force acting on the planets” (Brackenridge 15). Note what these labels connote: the
most “direct” path of inquiry was to follow in the footsteps of natural philosophers, trying to find the
cause of motion. Likewise, solving the (at the time less important) “inverse” problem was using the
known nature of a force to predict what the resulting motion would be.
Newton used Kepler’s preconceived laws and geometrical reasoning to find a mathematical
representation of the dependence of distance on his gravitational force; while he eventually proved the
relationship in both directions, Newton engaged first with the direct problem. Michael Raugh, a professor
of Mathematics at Harvey Mudd College who solves both problems using calculus in its current form,
offers a possible explanation for this approach. “By starting with an ellipse,” Raugh posits in his abstract,
“Newton avoided the more difficult...problem of starting with the inverse-square law and solving the
differential equations of motion to infer an ellipse.” In choosing to approach the direct problem first,
Newton also had the advantage of knowing, on a conceptual level, that the relationship he had in mind
held when he tackled the inverse problem.
When solving the problem in both directions, Newton focused on this mathematical relationship
and nothing more. As he explains in the Principia’s introduction, “the reader is not to imagine that by
those words I anywhere take upon me to define the kind, or the manner of any action, the causes or the
physical reason [of the forces]…” (Chandrasekhar 21-22). At Newton’s time, the burden of scientific
proof shifted from an emphasis on natural philosophy to one of mathematical modeling. In a conscious
departure from the preoccupations of Natural Philosophy, Newton chose to focus exclusively on what we
call Dynamics; rather than preoccupying himself, as Descartes and other natural philosophers did, with
the cause behind the force controlling planetary motion, or with “the Platonic challenge to seek celestial
circularity,” Newton was more concerned with describing in mathematical the nature of and cause of the
motion itself (Brackenridge 13).
“These mathematical derivations were without precedent in the history of science,” historian of
science Thomas Kuhn asserts in his book The Copernican Revolution. “From Newton’s inverse-square
law and the mathematical techniques that related it to motion,” he writes, “both the shape and the speed of
celestial and terrestrial trajectories could be computed for the first time with immense precision” (256).
Newton’s employment of rigorous mathematics throughout his proofs caused other scientists to
reconsider the relationship between mathematics and science.
How does this treatment of dynamics affect what physical scientists today set out to do? In the
words of science historian Niccolo Guicciardini, “How should natural philosophy be mathematized? Is it
legitimate to use in this context uninterpreted nature? Can we depart from the established Archimedean
or Galilean/Huygenian tradition of geometrizing nature” (11)? Consider, for instance, Dr. Pogozelski’s
course description for Physics I here at Geneseo. “‘Physics’,” he states, “is a description (not an
explanation) of the behavior of the physical world around us.” This epitomizes the influence of Newton’s
Dynamics in the physics discipline; we, too, are not as concerned with the cause of the relationship
between force and motion as with its description in mathematical language. And because of Newton’s
work in solving the direct problem, we can preoccupy ourselves with the inverse problem--finding the
path of motion associated with a given force--so much that the definition has switched: we now call it the
direct problem.
A similar reversal has occurred between the mathematical approaches of geometry and calculus. Which
feels more “physical” to us in modern science? “Today,” Guicciardini posits, “we take it for granted that
calculus is a more suitable tool than geometry, particularly in the case of applications to dynamics” (5).
We saw truth in Guicciardini’s words in our own research; we had trouble interpreting the geometrybased arguments of both Kepler and Newton because contemporary education does value calculus more
than geometry. These reversals are the binaries that emerge when discussing the elliptical orbits of
planets. Newton’s Principia--a proof so grounded in geometry--fostered an emphasis on calculus in
dynamics over three hundred years later.
Works Cited:
Brackenridge, J. Bruce. The Key to Newton's Dynamics: The Kepler Problem and the "Principia"
California: University of California, 1996.
Chandrasekhar, S. Newton's Principia for the Common Reader. Oxford: Clarendon, 1995.
Guicciardini, Niccolò. Reading the Principia: The Debate on Newton's Mathematical Methods
for Natural Philosophy from 1687 to 1736. Cambridge: Cambridge UP, 1999.
Katz, Victor J. A History of Mathematics: An Introduction. New York: HarperCollins.
Kozhamthadam, Job. The Discovery of Kepler's Laws: The Interaction of Science, Philosophy,
and Religion. Notre Dame, IN: University of Notre Dame, 1994.
Kuhn, Thomas S. The Copernican Revolution. New York: MJF, 1985.
Pogozelski, Edward. "Analytical Physics 1, Fall 2004." Dr. Edward Pogozelski. N.p., Aug. 2004.
Web. 1 Apr. 2012. <http://www.geneseo.edu/~pogo/Analyt1/p123home.htm>.