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