they do not th~nl~ cateful ly about ~t,
~s the Platonic school Plato (427
B C - 3 4 7 BC) cla,med the ~dea o f a
chaz~ was mo~e ~eal than any pa~tw-
u lar chair Thus Pla tonw Mathe-
mat~cmns w,ll say they 'd~scove~ed' a
result, not 'created' ~t The houble
wzth P la tomsm ,s ~t faz ls to be very
behevable, and certainly cannot ac-
count fo~ how Mathemahcs evolves, as
d~st,nct f rom evpand ,ng and elabo-
rating, the basw ~deas and def im-
t ,ons of Mathematics have g~adually changed ove~ the centuries, and th~s
does not f i t well w~th the ,dea of the
,mmutable Pla tomc ~deas
I was a graduate student ~n
Mathemattcs when th,s f ac t [Hdbert's
~nsert~on of axzoms o f betweeness and ~ntersectzon ~nto Euchd 's postulates
fo~ plane geometry] came to m y at-
tenhon I read up on st a b,t, and then
thought a great deal The~e a~e, I am
told, some 467 theorems ,n Euchd,
but not one o f these theorems tu~v, ed
out to be false afte~ Halbert added h~s
postulates
It soon became emdent to me one of the reasons no theorem u, as false was
that Hdbert 'knot,' the Euchdean theo-
rems u, ete 'correct,' and he had p~clted
h~s added postulates so th~s would be
true But then I soon reahzed Euchd
had been ,n the same pos~hon, Euchd
knew the 'truth' o f the Pythagorean the-
o~em, and m a n y othe~ theorems, and
had to f i nd a system o f postulates which would let h~m get the ~esults he
knew ~n advance Euchd dzd not lay
down postulates and make deduchons
as ~t ,s commonly taught, he felt h,s
w a y back f rom 'known' results to the
postulates he needed~
Richard Hamnung has served as a consul tan t to the Elders of the Mormon Church, served on the Board
of Directors of a large compute r cor- porat ion, spent 30 years as a Member of Technical Staff at Bell Telephone Laboratories , lec tured world-wide, re-
ce ived a number of pres t ig ious medals and awards, and spen t 20 years at Naval Pos tgraduate School m the th ick of educaUon His chatty, idiosyncrat ic , somet ]mes annoying, a lways though t - provol~ng book is one of a kind and a t e m b l y good read
Department of Mathematical Sciences
Stevens Institute of Technology
Castle Potnt on Hudson
Hoboken NJ 07030
USA
e-mail rptnkham@stevens-tech edu
EDITOR'S NOTE On the obi tuary page of the New York Times, Sunday, Jan 11, 1998, there appeared an ar t ic le under
the headhne, 'Richard Hamming, 82 Dies, P ioneer in Digital Techno logy" I quote f rom the art icle
R~cha~d Wesley Hammzng, who d~s-
cove~ed mathematzcal f o r m u l a s that
allow computers to correct thez~ own
errots, mako~g possible such ~nnova-
hons as modems, compact dzsks and
satelhte commun tca twns , d~ed on
Wednesday at a hospital ,n Monterey,
C a h f , where he heed He was 82
He d~ed of a heart attach, h~s f a m -
~ly sa~d
Feynman's Lost Lecture by David L Goodste~n and
Judi th R Goodste~n
LONDON JONATHAN CAPE (1996)
ISBN 0 224 04394 3
REVIEWED BY GRAHAM W G R I F F I T H S
R i c h a r d F, eynman was one of flus century s great physlctsts He
shared the 1965 Nobel Prize for Physics with Juhan Schwmger and Shimchtro Tomonaga for the invention of quan- tum e lec t rodynamics Most peop le with an in teres t in things scmnt]fic will
recal l that Feynman served m 1986 on the pres ident ia l commmsmn investi- gating the Chal lenger space shut t le dis- as te r Dunng a televised heanng of the
commiss ion, he dramat ica l ly demon- s t ra ted that O-nng seal fai lure at low t empera tu re s was a l ikely cause of the
a c o d e n t In 1961 Feyrmlan agreed to teach the
two-year introductory physics course at the Cahforma Institute of Technology Ttus s enes of lectures was recorded
and transcribed, and the b lackboards photographed From this mformatmn, the mteruatmnal ly renowned "Feynman
Lectures on Physics" were p roduced and pubhshed
In 1687 Newton publ ished his in-
verse-square law of grawty In the mag- nificent work Phdosoph,ce Natu~ahs
P~nc,p~a Mathematwa, now com- monly known as the Ib'~nc,pm The
Pr~nc~pm is p robab ly the greates t sci- entific work ever publ ished and has ln- tngued scmntmts and mathemat ic ians
because of the vast extent of the ground covered and the beauty and dif- ficulty of the proofs zt contams
Feynman's Lost Lecture is a recon- structaon of a lecture gqven by Feynman which centered around at temptmg to
prove Newton's mverse-square law of gravity using only the mathematmal tools available to Newton Thts lecture
was gnven to freshmen at Caltech at the end of the win te r quar ter in 1964 as a
guest lecture, not part of the ongmal lec- ture course It was ongmally recorded on audio cassette, but the accompany- mg photographs were nnslmd Thus, it
had not been possible to reconstruct this lecture until m April 1992 Feyn-
man's ongmal notes were dtscovered m the office of his colleague, Rober t Lelghtman, fol lowmg Leightman's death Once Feynman ' s notes were un-
earthed, Dawd Goodstem, a physms professor at Cal tech who worked with Feynman, was able to recons t ruc t by
s leuthhke deduc t ion the lecture m its ent irely It is not made clear whe the r it was ever a t t empted to locate no tes
taken by a t t endees at the lecture for verif icat ion p u r p o s e s
By way of an in t roduct ion to the subject, the b o o k provides background mformatmn relat ing to the work of
Tycho Brahe, Kepler, Newton, and oth- ers, together wi th some amusing anec- dotal r em]mscences of D Goods tem ' s re la tmnship with Feynman Some pho-
tographs of Feynman at the black- board are also r ep roduced The epi- logue d iscusses bnef ly the work of Maxwell and Rutherford, and de-
scr ibes how, af ter two hundred years, Einstein 's t he one s of relatlv~ty super- seded Newton ' s theory of gravt ta tmn for speeds approach ing the speed of light and for large concent ra t ions of mat ter
The recons t ruc t ion is a bit l abored in places, par t icu lar ly m respec t of
THE MATHEMATICAL tNTELLIGENCER
G t
F=gure 1 Construct,on Of Elhpse
Kepler's 2nd Law (equal areas swept
out in equal time, which also implies
conservat ion of angular momen tum)
A more interesting part of the lecture
is where Feynman appeals to Fermat ' s
Pnnclple , 1 e , light always takes the
shortest path, in order to provide a
somewhat novel proof of a property of
an elhpse rather than adopting a purely
geometrical approach, Figure 1
The proof also contams a very re-
markable ~eloclty diagram, Figure 2,
which was published previously by
James Clerk Maxwell In his 1877 book
Matter and Motion Maxwell attrib-
utes the method to Sir William
Hamilton, which goes to show how dif-
ficult It is to discover something com-
pletely ongmal Feynman was appar-
ently unaware of Maxwell's book,
because he credits V Fano and L Fano
with some of the Ideas in their discus-
sions of the Rutherford Scat tenng Law
in the 1957 book Basic Physws of Atoms and Molecules Feynman shows
rather cleverly that, as a result of
Kepler's 2nd Law, orbit velocity dia-
grams subject to an mverse-square law
of gravity must be circular
The objective of the lecture was for
Feynman to prove to his s tudents that
elhpt~cal planetary orbits with the sun
at one focus are a direct consequence
of Newton's reverse-square law How-
ever, close inspect ion of the book re-
~eals that nei ther Feynman nor the
Goodstems have truly provided such a
proof Nevertheless, the Goodstems
present the Feynman lecture as if It did
actually contmn a bn lhan t proof, and
this is a very real weakness In the lec-
ture given in chapter 4, Feynman re-
ferred repeatedly to his "elementary
demonstrat ions" and "demonstra-
tions " Feynman omits some crucial
steps and ref inements that would have
to be Included for his demonstra t ions
to be acceptable as a proof Missing
components include
�9 an explanat ion of the scalmg be-
tween the hodograph velocity diagram
and the orbit diagram,
�9 a coherent a rgument why it is justi-
fied to use the perpendicular bisector
of Op (diagram on page 162) to locate
the corresponding point, P, on the or-
bit diagram, when It Is not known a
prwm that the answer will turn out to
be an ellipse, and, �9 an adequate explanat ion of how par-
\
I J
S I
a) F,gure 2 a) Orb,t D,agram b) Veloci ty Dmgram
b)
abohc and hyperbolic orbits are iden-
tified, using the hodograph method,
knowing only that the central force
obeys an reverse-square law, and that
equal areas are swept out m equal time
Whilst Feynman did demonstra te
the existence of elhptlcal orbit solu-
t ions to the problem, what he did not
demonstrate is the lmlqueness of these
solutions Furthermore, he alludes to
this si tuation on page 164 " is what
I proved that the ellipse is a possible
solution to the problem " Unfortu-
nately, Feymnan also made other state-
ments apparently contradicting this
view, so we will never really know how
ngorous he believed has lecture to be
Dunng his lecture, Feynman con-
fided to has s tudents that he had expe-
rmnced considerable difficulty with
some of the comc-sect lon geometry
Feymnan states " he [Newton] per-
petually uses (for me) completely ob-
scure properties of the conic sections,"
and " the remmnmg demonst ra t ion
is not one which comes from Newton,
because I found I couldn ' t follow it my-
self very well, because it mvolves so
many properties of conic sections So
I cooked up another o n e " As it hap-
pens, most the proofs in question were
ongmally published in The Conws, Book III by Appolomus, circa 200 B C,
and all were commonly included in
books on geometry until the early part
of this century, e g, An Elementa~?1 T~eatise On Conic Sectmns By 77~e Methods Of Co-ordinate Geometry by
C Smith, MacMillan, 1910 If the conic section properties were unfamiliar to
someone with such a ~ast knowledge
of mathematics and physms as
Feynman, it makes one wonder how
nmch other useful knowledge has been
dropped from the modern curncuhanl
in the name of progress
Those readers unfamiliar with the
f'mer points of Newton's derlxatlons
will find that S K Stem's article,
"Exactly How Did Newton Deal With
His Planets" (The Mathematwal In- tell~gence~, ~ol 18, no 2), pro~ades a
clear exposition from basic pnnclples
Slrmlarly, readers unfamthar x~lth the
use of velocity diagrams or hodographs
should refer to Andrew Lenard's paper,
"Kepler Orblts--Mo~e Geomeh wo," m
VOLUME 20 NUMBER 3 1998 69
the College Mathematws Jour~al 25,
no 2 (March 1994), which pro~ades an
excellent lntroductmn
The Goodstems make an assertion
which is not umversally accepted by
historians of scmnce " There ~s httle
doubt that he [Newton] used these pow-
erflfl tools [differential and integral cal-
culus] to make his great chscovenes"
This lmphes that Newton first worked
out his solutions usmg the Calculus, and
then recast them into a geometrical
form Whilst it is true, as R Westfall has
pointed out m his defmmve biography
of Newton, Neve~ at Rest, that Newton
confided to his frmnd Wflham Derhmn
that he deliberately made his Prmclpia
abstruse " to avoid bemg bmted by
httle Smatterers of Mathematxcks ,"
this apphed to the recasting of Book III
of the P~ ~ne~p~a from a prose style to
the mathematical format that he sub-
sequently pubhshed This was a result
of his clash(es) with Robert Hooke
D T Whites]de has made the point
forcibly that the mathematics used by
Newton to arrive at his d]scoverms is
the same mathematms he used m the
P~nc~pm It is extremely sansfymg to see that
a great physmmt like Feynman was in-
terested sufficiently m the h l s tonca l
deve lopment of h~s sub3ect that he
was prepared to devote s lgmfmant
n ine to present ing h~stoncal develop-
ments, such as Newton's inverse-
square law of gra~qty, to his s tuden ts
I am comanced that u n w e r s m e s will
turn out bet ter educated s cmnns t s m
the future ff they encourage s tuden ts
to apprecmte the problems that con-
f ronted great scientists m the past,
and to unde]s tand how those scmn-
ns t s solved them wath the tools avail-
able at the t ime
It must be stud that ff the Goodstems
had included an appendax providing an
over~aew of hodograph theory, the edu-
cauonal value of the book would have
been greatly enhanced Nevertheless,
this book has been produced to a high
quahty and will be a'valuable addition to
any library, and is recommended read-
mg for all s tudents of Newton and
Feymnan All the discussions should be
readily understood by anyone famlhar
with high school mathematms
Acknowledgments The re~aewer would hke to acknowl-
edge useful and mformatlve discus-
sions with Professor Robert Burckel
(Kansas State) and Professor Robert
Wemstock (Oberhn College) m con-
nect ion with this rexaew
Control Eng~neenng Research Centre
Oty University
Northampton Square
London EC1 0HB
United Kingdom
e-ma~l graham@sast co uk
70 THE MATHEMATICAL INTELLIGENCER