Newton’s “superb theorem” for the gravitational 1/r2 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 1/r2 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.

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.
Reference 1, pp.
1
3
.
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.
Reference 1, pp.
270
273
.
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.
Reference 1, pp.
272
and
273
.
14.
J. E.
Littlewood
,
A Mathematician’s Miscellany
(
Methuen
,
London
,
1953
), p.
97
.
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
);
L. D.
Landau
and
E. M.
Lifshitz
,
Mechanics
, 3rd ed. (
Elsevier
,
Amsterdam
,
2004
);
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.
Reference 1, pp.
269
270
.
20.
J. D.
Jackson
,
Classical Electrodynamics
, 3rd ed. (
Wiley
,
New York
,
1999
), Sec. I.2.
21.
Reference 3, Lemma 1, p.
433
, and Scholium on pp.
440
443
.
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.
H.
Wussing
,
6000 Jahre Mathematik
(
Springer
,
Berlin
,
2008
), p.
432
.
28.
Reference 20, Sec. 1.3.
29.
Reference 16, Sec. 4–5.
30.
Reference 3, Corollary 1 and Lemmas 2–4, pp.
433
435
.
31.
Reference 1, pp.
270
271
.
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.