In the half-dozen or so years leading up to the publication of the Principia, Isaac Newton observed the comets of 1680 and 1682 and wrestled with the extent to which his law of gravitation could be applied. In time, he would see the connections between the four possible orbits of a satellite (circular, elliptical, parabolic, and hyperbolic) and the four curves produced by the careful carving of a cone. But if we look a little further into the conic sections, we find some interesting connections among the natural orbit of a satellite, ancient mathematics, and the roots of familiar words. Illuminating these connections for introductory physics students may help them to better understand the role of language and mathematics in the descriptions of science.

1.
Richard S. Westfall, The Life of Isaac Newton (Cambridge University Press, 1993), pp. 154–156.
2.
Ref.1, pp. 159–162.
3.
All quotes from the Principia are from I. Newton, The Principia: Mathematical Principles of Natural Philosophy, translated by I.B. Cohen (University of California Press, 1999).
4.
The orbital diagram shown is from Newton's A Treatise on the System of the World (1685, 1728). For an interesting discussion on what Newton may or may not have understood about this diagram, see
D.
Topper
and
D. E.
Vincent
, “
An analysis of Newton's projectile diagram
,”
Euro. J. Phys.
20
,
59
66
(
1999
), and
M.
Nauenberg
, “
Comment on ‘An analysis of Newton's projectile diagram,’
Euro. J. Phys.
21
,
L5
L6
(
2000
).
5.
See for example J.L. Coolidge, A History of the Conic Sections and Quadric Surfaces (Oxford Press, 1945).
6.
Ref. 1, p. 151. For an interesting problem illustrating the change from a circular orbit to an elliptical orbit due to a slight change in speed of a satellite orbiting Jupiter, see the 2001 AP Physics C exam free-response question Mechanics 2 at http://www.apcentral.collegeboard.com.
7.
A paperback copy of Apollonius's treatise On Conics can be purchased from Green Lion Press at http://www.greenlion.com.
8.
All Greek-English translations are taken from H.G. Liddell and R. Scott, A Greek-English Lexicon, 9th ed. (Clarendon Press, Oxford, 1996).
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