Newton’s “superb theorem” for the gravitational force states that a spherically symmetric mass distribution attracts a body outside as if the entire mass were concentrated at the center. This theorem is crucial for Newton’s comparison of the Moon’s orbit with terrestrial gravity, which is evidence for the law. In this paper, we give an elementary geometric proof, which is simpler than Newton’s geometric proof and more elementary than proofs using calculus.
REFERENCES
1.
S.
Chandrasekhar
, Newton’s Principia for the Common Reader
(Clarendon
, Oxford
, 1995
), Secs. 1 and 15.2.
J. W. L.
Glaisher
, address on the occasion of the bicentenary of the publication of the Principia, quoted in Ref. 1, pp. 11
–12
.3.
I.
Newton
, The Principia: Mathematical Principles of Natural Philosophy. A New Translation
, by I. Bernard Cohen and Anne Whitman, Preceded by a Guide to Newton’s Principia
by I. Bernard
Cohen
(U. of California Press
, Berkeley
, 1999
), Book I, Sec. 12, p. 590
.4.
5.
D. T.
Whiteside
, “Before the Principia: The maturing of Newton’s thoughts on dynamical astronomy, 1664–1684
,” J. Hist. Astron.
1
, 5
–19
(1970
).6.
Reference 1, p.
12
, and Newton’s text after Proposition 8 of Book III.7.
W. W.
Rouse Ball
, An Essay on Newton’s Principia
(Macmillan
, London
, 1893
), pp. 156
–159
, and Ref. 1, p. 12
.8.
Reference 1, p.
12
.9.
Reference 1, p.
6
.10.
11.
R.
Weinstock
, “Newton’s Principia and the external gravitational field of a spherically symmetric mass distribution
,” Am. J. Phys.
52
(10
), 883
–890
(1984
).12.
J. T.
Cushing
, “Kepler’s laws and universal gravitation in Newton’s Principia
,” Am. J. Phys.
50
(7
), 617
–628
(1982
); see pp. 625–626, and footnote 62.13.
14.
15.
The Mathematical Papers of Isaac Newton, Vol. 6 (1684–1691)
, edited by D. T.
Whiteside
(Cambridge U. P.
, Cambridge
, 1974
), p. 183
.16.
R. P.
Feynman
, R. B.
Leighton
, and M.
Sands
, The Feynman Lectures on Physics
(Addison-Wesley
, Reading, MA
, 1963
), Vol. I
, Sec. 13-4.17.
J. B.
Marion
, Classical Dynamics of Particles and Systems
, 3rd ed. (Harcourt Brace Jovanovitch
, San Diego
, 1988
), pp. 161
–163
.18.
H.
Goldstein
, C.
Poole
, and J.
Safko
, Classical Mechanics
, 3rd ed. (Addison-Wesley
, San Francisco
, 2009
);T. W. B.
Kibble
and F. H.
Berkshire
, Classical Mechanics
, 5th ed. (Imperial College Press
, London
, 2005
);A. L.
Fetter
and J. D.
Walecka
, Theoretical Mechanics of Particles and Continua
, 2nd ed. (Dover
, New York
, 2003
);V. I.
Arnold
, Mathematical Methods of Classical Mechanics
, 2nd ed. (Springer
, New York
, 1989
).19.
20.
21.
22.
Reference 3, p.
442
.23.
Reference 3, p.
129
.24.
Reference 3, p.
441
.25.
V. J.
Katz
, A History of Mathematics
(Pearson
, Boston
, 2009
), Secs. 3.8, 4.2, and 4.3.26.
R.
Thiele
, “Antiquity
,” in A History of Analysis
, edited by H. N.
Jahnke
(American Mathematics Society
, Providence, RI
, 2003
), Secs. 1.3 and 1.4.27.
28.
Reference 20, Sec. 1.3.
29.
Reference 16, Sec. 4–5.
30.
31.
© 2011 American Association of Physics Teachers.
2011
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
