Letters
to
the
Editor
The Mathe~natwal Intelhgencer encourages comments about the material ~n thzs ~ssue Letters to the editor should be sent to the ed~tor-~n-chwf Chandler Dav~s
Interlaced?
To the caption labelhng a figure m "The Borromean Rings" (vol 20, no 1, see p 55) as "ten ~nterlaced patterns," let me demur (above emphasis mine) In your Figure 6, it is stud, Ten tnmties, interlaced, b e d - Yet actually not Only nine have a knot, And the tenth is unknotted instead v Charles Muses Mathemahcs & Morphology Research Centre Sardis BC V2R 1Y8 Canada
The ed~to~ ~ephes Is the tnv]al knot not a knot v The authors were hstmg the lot Of all ways three nngs lmk, They were quite nght, I think, To say one way to hnk is to not DNA Borromean R,ngs
LIGATE
+
PROMOTINGIN Z-DNACONDITIONS
The recent article "The Borromean Rings" by Peter Cromwell, Ehsabetta Beltraml, and Marta Rampichml (Math Intelhgence~ 20 (1998), no 1, 53-62) mentioned many occurrences of Borromean Rings and related mterlaced patterns m art, heraldry, mathemaUcs, physics, and theology We feel it would interest your readers to know that they can also be found m biochemistry We have recently constructed a Brunman hnk from three single-stranded o r c l e s of DNA [1] In order to do so, we have had to make use of two unusual motifs of DNA The first ;s branched DNA, winch is easy to ~/Z7 ~ prepare synthet]~ +"~ cally, although it f~-'~,~+r" c(log log d/log d) 3,
and
F * ( a ) = x d F ( x -1) A polynomial is called reciprocal if F * = F and nonreciprocal otherwise Obviously, being reciprocal implies a nice symmetry o n the coefficients of F, as in Lehmer's example (6) In his beautiful paper [25], Smyth p i o v e d T H E O R E M 3 I f F ( x ) ~ Z[x] F(0)F(1) r 0, then
~s nonrec~p)vcal
A torsion point of the circle is a root of unity, in other words, a complex n u m b e r r with ( g = 1 for some N E N An integral polynomial whose roots are all of this kind is called a cyclotomic polynomial The n a m e cyclotomlc m e a n s "circle dividing" literally and refers to the way the roots of unity divide up the unit circle m the complex plane
t b o o f o f Theorem 2 Roots of umty are all algebrmc mtegers of absolute ~alue 1 Thus, a cyclotoImc polynomial clearly has measure 0 Conversely, ff the right-hand side of (2) vanishes,
10
THE MATHEMATIC,~,LINTELLIGENCER
and (8)
The basic Idea b e h i n d the proof is extremely elegant, to illustrate it, I will prove Stewart's w e a k e r b o u n d
re(F) = 0 zf a n d only ~f all the roots o f F are toss,on p o i n t s o f the c~) cle
(7)
where c > 0 d e n o t e s an absolute c o n s t a n t (e g, c could be taken as 1/1200) In 1971, Smyth gave a positive a n s w e r to Lehmer's Problem in the "nonreciprocal" case To define this term, begin with the n a t u r a l involution z ~-> z - 1 o n the unit circle This Induces a n involution on one-variable polynomials F of degree d
m ( F ) >- )n(~"3 - x - 1) = 0 2811 T H E O R E M 2 Suppose F ( x ) E Z[~] ts p m m i h v e F(O) r 0 Then,
(6)
r e ( F ) >- 89
= 0 1115
(9)
P) oof o f (9) We use Parseval's Formula, whmh says that lff(z) is holomorphlc on the closed unit disc a n d has Taylor expansion f(z)=ao+alz+a2z
2+
,
(10)
then we have
fo
'lf(e~-~'~ ~- dO = ta012 +
laii
+ la212 +
(11)
To prove (9), a s s u m e F is irreducible We may assume F is tannic, otherwise r e ( F ) >- log 2, which is already greater than the right-hand side of (9) In a slm]lar vein, we may
a s s u m e F ( 0 ) = _+1 Write m = r e ( F ) , a n d M = e" Consider G = F(O)F/F* On the one hand, we see that G(z) has a p o w e r s e n e s e x p a n s i o n with integral coefficients
G(z) = l + aAz ~ +
,
laal->l
(12)
On the o t h e r hand, we m a y write G(z) in the form
F(z)
G(z) = F(o) F*(z---7 =
F(O) lit (z - ce,) 1-I, (1 - zT,)
As F can have no roots of a b s o l u t e value 1, we r e w n t e this last e x p r e s s i o n as
F(O) [ I
( . z--7-~-) [,~>i ( 1 - z c ~
-
I~, < t \ I - z %
Smyth's p r o o f uses J e n s e n ' s F o r m u l a as t h e b a s e step of an induction, with the induction step coming from a clever u s e o f the g e o m e t r y o f t h e e x p o n e n t p o l y t o p e The proofs o f Boyd and Lawton also go by induction, with the base step coming from a r e m a r k a b l e f o n m i l a I w a n t to r e c o r d this formula in the c a s e w h e r e n = 2, b e c a u s e it raises the possibility of higher-dimensional v e r s i o n s of Lehmer's Problem Suppose F ( z l , or2) E Z[otl, a2] Then, F(x, aa), for N ~ N, is a p o l y n o m i a l in one variable, and we could a s k what, if any, is the limiting b e h a w o u r of m (F(a, :~.v)) as N--+ ?c It d o e s not a p p e a r o b v m u s that there should be any Nonetheless,
g(z) h m m(F(x, ~ ) ) = m(F(xt, x2))
S u p p o s e f a n d g have been c h o s e n with f ( z ) = c + ctz + and g(z) = d + dtz + , notice that e = d = 1/M Then, c, = d~, ~ = 1, ,/, - 1, and ake + dA = ca W:thout loss of generality, w e a s s u m e [ca[-> idA] (If this is false, then the a r g u m e n t b e l o w w o r k s w:th g replacing f ) F r o m (12), w e have laAI >-- 1, so it follows that
c/2 0 denote a prime, with I Ip the usual p-achc valuatmn on Q Then, w e study a measure where log~Flp is rotegrated over a p-achc space Ttus b e c o m e s necessary m order to prove t h e o r e m s such as Theorem 6 In the dynanucal systems context also, ttus has been ~,ery frmtful In [15], Lmd and Ward gave a local-to-global a p p r o a c h to Yuz~lsk]'s formula The underlymg group, m t/us context, is a "solenoid," a kind of projective lmut of circles, wluch allows the basic mult]phcatmn m a p to be observed at all valuatmns sn~ neously We will n o w s e e h o w local d e f i n m o n s a n s e and p r o p o s e "correct" local defimtmns m the elliptic case In what follows, a s s u m e "undefined" t e r m s are o m i t t e d
T H E O R E M 7 Suppose c~~s any algebraic nmnbet
log+Jar = log max{l,
Io,I} = N--->o: hm 1 X~_llogiC" N r -
-
Then, ~1
_
W h e n f g r O, we say F is n o n r e o p r o c a l , all o t h e r c a s e s bemg called reciprocal To justify the name, think that the n a t u r a l revolution on the curve is P ~-> - P , which in t e r m s of the c o o r d i n a t e s for the Wemrstrass equatmn m e a n s (x, y) ~-) (x, - y ) If F*(P) d e n o t e s
P~ oof The f o r m u l a is pretty o b v m u s as long as we can clear
F*(P) = f ( x ) - yg(x),
I~ - ~ > CN-D,
14
THE MATHEMATICALIN:ELLIGENCER
up any p r o b l e m s which might arise from r o o t s of unity app r o x i m a t i n g a t o o well But we k n o w from dlophantlne app r o x i m a t i o n that
(28)
providing the left-hand side is nonzero, where C and D are positive constants winch d e p e n d on a only The big result beinnd tins kind of mequaltty is Baker's Theorem A small value on the left of (28) w o u l d indicate a good a p p r o x i m a tion to ~-/loga by a rational n u m b e r with d e n o m i n a t o r at m o s t N Baker's Theorem gwes a p r e c i s e estimate [the righth a n d side of (28)] of h o w well this can be achieved 9 Thus, the f o r m u l a is true b e c a u s e w e can approxamate the integral b y a sum over r o o t s of unity In general, we c o u l d t a k e this sum to define an integral, k n o w n as the S h m r e l m a n Integral Let the N th r o o t s of umty, t o g e t h e r with a, g e n e r a t e a field c o n t a m m g the p-adlc field Qp Then, the usual p-adm absolute value e x t e n d s to this field and w e can define
1
V l o g l ~ - o~1~
(29)
We can s h o w the analogue o f T h e o r e m 7, namely that the limit of the e x p r e s s i o n in (29) always exists and, m fact, equals log max{ 1, [alp ]
(30)
We can think of the closure of the group of all r o o t s of unity reside Q*p as a kind of p - a d m circle, written Tp We write
mTp(X -- a) = hm __1 ~ l o g ] ~ - alp If w e i n t e r p r e t the ordinary a b s o l u t e value as c o r r e s p o n d mg to p = ~, then this d e f i n m o n m a k e s sense and converges for every prime p = 2, 3, , ~ The equahty of the hmlt of (29) with (30) is w h a t I t h i n k of as the local version of J e n s e n ' s F o r m u l a (4)
LEMMA
LEMMA
re(F) = ~
mwp(F)
Proof sl~etch We m a y a s s u m e F is p n m m v e b e c a u s e the p r o d u c t f o r m u l a e n s u r e s that any nontn~ual g c d of the coefficients of F will be dwaded out Now, the total cont n b u t t o n from the non-arch~medean p r i m e s is zero 9 The first a d v a n t a g e of this a p p r o a c h is that w e do not n e e d to a s s u m e F m p n m l h v e The s e c o n d a d v a n t a g e is that it as obvious h o w to give a definition for p o l y n o m i a l s m several variables What is not so o b v m u s is the question of convergence m the several-variable c a s e In [7] a n d [20], the elliptic analogue of these alternative definitions is pursued In [6] and [8], w e defined local heights as Haar integrals rather t h a n S h m r e l m a n integrals F o r every prime p (including ~), w e c a n define a c o m p l e t e d curve Ep Thus, if p is mfimte, w e t a k e Ep to be the group of all c o m p l e x points ( i s o m o r p h i c to C/L for s o m e L) In every case, Ep is a c o m p a c t group, so the Haar m e a s u r e t.tEp exists, and w e can use th~s to define mEp
(x - a) = fEp log# -- alp d~Ep
~
mTp(X -- a~)
(33)
In the definition m (33), the variable x runs over the ~ coordinates of the p o i n t s on the c o m p l e t e d curve Ep It is a simple c o n s e q u e n c e [6] of the "parallelogram law" (detmled m [23] a n d [24]) that this mtegral r e p r e s e n t s the local c o n t n b u t m n to the global canonical height This fact w a s then e x p l o i t e d m [6] to r e w o r k the classmal theory of heights on elliptic curves m a w a y m o r e a n a l o g o u s to the t h e o r y of Mahler's m e a s u r e We can t h e n define
m$(F) = ~/. ~ m ( F ) = ~'
(32)
P
mEp(2 - o4)
p
(31)
p
Proof sketch The "product formula" for Q Is the simple
This agrees voth our earlier des only if all the P, have everywhere g o o d reduction, but the advantage to tins app r o a c h b e c o m e s clear when I write d o w n the final t h e o r e m
s t a t e m e n t that T H E O R E M 8 [6] Fo~ any F E Z[x], we have log[alp = 0, P
for any n o n z e r o ratmnal a Using the p r o d u c t f o r m u l a a n d the local v e r s m n of J e n s e n ' s Formula, we are interpreting the "log]a]" p a r t of the m e a s u r e [as m (2)] as the total cont n b u t l o n from the finite p r u n e s of Q 9 Conceptually, this a p p r o a c h to the m e a s u r e is very tmportant F o r example, m [15], Lind and Ward s h o w that e a c h of t h e s e local contributions can be i n t e r p r e t e d as a local e n t r o p y In a shghtly different direction, we could define the local m e a s u r e to be
mTp(F) = hm ~1 V~_, loglF(~Dlp N~m
(',=1
~n~(F)
=
o
~f and only *f all the P, are toss,on pmnts of E The p r o o f of T h e o r e m 8 is stmllar to its classical analogue In particular, m the g o o d reduction case, the "log[a]" part of the m e a s u r e ts r e a h s e d as the total contribution from the finite p r i m e s If there are "bad" p r i m e s p present, the local-to-global a p p r o a c h has the n e a t effect of replacing the p part of log]a i b y the c o r r e c t c o n t r i b u t i o n Conclusions Mahler's m e a s u r e :s ahve and well m several qmte d w e r s e c o n t e x t s The dfffenng points of vaew s e e m to generate a
VOLUME 20 NUMBER 3 1998
15
h e a l t h y friction If the general level of health is m e a s u r e d b y the quantity and quality o f unsolved p r o b l e m s , then it m a y help to list these 1 Lehmer's Problem 2 The elliptic analogue o f Lehmer, at least m t r a c t a b l e speclal cases 3 An explanation of Boyd's r e m a r k a b l e formulae It s e e m s that K-theory should p r o v i d e the c o n c e p t u a l f r a m e w o r k More generally, p e r h a p s values of the elhptlc Mahler m e a s u r e will a n s e as values o f L-functions o f higher-dimensional vanetms 4 It l o o k s almost certmn that the elhptm Mahler m e a s u r e s h o u l d arise as an e n t r o p y This would form a fascmatmg bridge b e t w e e n t w o large a r e a s of m t e r e s t Ward a n d I have begun to write a b o u t this [10] At the v e r y least, ttus would s h o w that the global canonical height o f an algebrmc p o m t on an elhpt~c curve arises as an e n t r o p y But of what, and w h a t d o e s tlus m e a n 9 5 There are m a n y o t h e r p r e t t y results a b o u t the classical Mahler m e a s u r e wtuch could be lifted to the elhpt~c settmg REFERENCES
1 D W Boyd, Kronecker s Theorem and Lehmer s problem for polynomials tn several variables, J Number Theory 13 (1981 ), 116-121 2 D W Boyd, Speculatrons concerning the range of Mahler's measure, Can Math Bull 24 (1981), 453-469 3 D Boyd, Mahler s measure and specral values of L-functqons to appear tn Jour of Expenmental Math (1998) 4 V Chothi, G Everest, and T Ward, S-integer dynamical systems Penod~c points, J Re/ne Angew Math 489 (1997), 99-132 5 E Dobrowolsk~, On a question of Lehmer and the number of ~rreducible factors of a polynomial Acta Anth 34 (1979), 391-401 6 G R Everest and Brid Ni Fhlathutn, The elhpt~c Mahler measure, Math Proc Camb Phil Soc 120 (1996), 13-25 7 G Everest and C P~nner Bounding the elhpt~cMahler measure, J London Math Soc to appear 1998 8 G R Everest, The elhpttc analogue of Jensen's Formula, J London Math Soc to appear 1999 9 G Everest and C Pinner, The elhpt~c analogue of Parseval's Formula (prepnnt) 10 G Everest and T Ward Speculations concerning a dynamical interpretatron of the elhpttc Mahler measure to appear in Jour of Expenmental Math 11 S L a n g , Introduction to Arakelov Theory, New York SpnngerVerlag (1988) 12 M Laurent, M~norat~on de la hauteur de Neron-Tate Semen Theone Nombres (1981-82) 137-152 13 W Lawton, A generalization of a theorem of Kronecker, J Sc~ Fac Ch/angmal Unlv 4 (1977), 15-23 14 D H Lehmer, Factonzatlon of certain cyclotomlc functions Ann Math 34 (1933), 461-479
16
THE MATHEMATICALINTELLIGENCER
15 D Dnd and T Ward, Automorph~sms of solenoids and p-adtc entropy, Ergodlc Theory & Dynamical Systems 8 (1988), 411-419 16 D Lind K Schm~dt, and T Ward, Mahler measure and entropy for commuting automorph~sms of compact groups Invent Math 101 (1990), 593-629 17 K Mahler, An apphcat~on of Jensen s formula to polynomials, Mathematlka 7 (1960), 98-100 18 K Mahler, On some ~nequalt~esfor polynomials ~nseveral vanables J London Math Soc 37 (1962), 341-344 19 D Masser, Counting points of small height on elhpt~ccurves Bull Soc Math France 117 (1989), 247-265 20 P Ph~hppon Surdeshauteursalternat~veslMath Ann 289(1991) 255-283 21 C Pinner, Bounding the elliptic Mahler measure Math Proc Camb Phil Soc to appear 1999 22 A J van der Poorten, Obituary of Kurt Mahler 1903-1988, J Austral Math Soc 51 (1991) 343-380 23 J H S~lverman, The Anthmetlc of Elliptic Curves, New York Spnnger-Verlag (1985) 24 J H Silverman, Advanced Topics in the Anthmet~c of Elliptic Curves, New York Spnnger-Verlag (1994) 25 C J Smyth, On the product of conjugates outside the unit c~rcle of an algebraic ~nteger,Bull London Math Soc 3 (1971), 169-175 26 C J Smyth, A Kronecker-type theorem for complex polynomials ~n several vanables, Can Math Bull 24 (1981) 447-452
PATRICIA AND JAMES ROTHMAN
estina ente ~
~
~
'd gwe anythzngfor a proof," Dr Faustzna Lente sazd to herself She had produced no ortgznal work sznce she returned to the Mathematws Department after her maternzty leave and felt her colleagues were murmuring that motherhood and mathematws dzd not mzx, matmces notwzthstandzng
A w o m a n with a kindly mr, w h o m F a u s t m a had never seen before, entered the room "Did I hear you think you'd give anything for a prooi v'' she asked "How did you knowg" exclmmed F a u s t m a "! k n o w m a n y things and m a n y esoteric proofs," stud the woman, "perhaps we can come to a n a r r a n g e m e n t If you give me what I want I'll help you with the proof of any m a t h e m a t i c a l proposition you care to n a m e " "That's very mterestlng," stud F a u s t m a doubtfully "Why should I believe youg" "All I w a n t is something you d o n ' t believe you have, so you've nothing to lose G1ve it a try If I can't help you, you can c o m m a n d me to do anything else you want Since number theory is your field there is n o r e a s o n we can't be perfectly amicable about this You c a n call me Luci Would you like m e to give you the proof F e r m a t had m m i n d ~ It's very short a n d considerably more elegant than Wiles's" "No", stud F a u s t m a emphatically, "I'd only be margmahsed for coming second I w a n t something really o n g m a l I bet you can't help me prove Goldbach's c o n j e c t u r e " "Please let me haze your definition of the proposmon," Lucl said in a b u s m e s s h k e tone, "Goldbach is not one of my clients " "There you are," said F a u s t l n a triumphantly, "I k n e w you c o u l d n ' t help Anyone who k n o w s anything about n u m b e r
theory k n o w s that Goldbach's conjecture is that every even n u m b e r can be expressed as the sum of at least one pmr of primes For example 18 can be expressed as the sum of the two primes 7 a n d 11 Fwe and 13 w o u l d do j u s t as well" F a u s t m a was surprised to see the very a s s u r e d Lucl wince at the m e n t i o n of 13 "I don't k n o w the n a m e s of your conjectures, b u t I can help you prove t h e m All I w a n t is something you don't beheve you h a v e - - y o u r soul But let's be clear, a conjecture ~s something that may or may not be true If it's true, then I'll help you prove it, but what do you w a n t me to do if it is not true '~" "Of course it's true," F a u s t m a stud ~mpatmntly "It's b e e n tested for every even n u m b e r up to 100 nulllon, a n d there are so m a n y different ways of sphttmg a larger n u m b e r into two that it's almost certmn that one of t h e m would consmt of two primes "In that case," asked Luci, ' why do you w a n t me to prove it )" "Only a n o n - m a t h e m a t i c i a n could ask such a question Even though the probablhty of there being a n u m b e r larger than 100 million which c a n n o t be p a r t m o n e d into two primes is neghglble, it isn't zero We c a n n o t be sure that there isn't such a n u m b e r until we prove It" "You can't be sure even then," objected Luci "There
9 1998 SPRINGER VERLAG NEW YORK VOLUME 20 NUMBER 3 1998
17
c o u l d be a mistake in y o u r p r o o f - - i t has h a p p e n e d before, y o u know." "You do n o t understand," said F a u s t i n a dismissively. "Let's not argue, said Luci. "It's j u s t that I do n o t like uncertainty in a contract. What do you w a n t m e to do if G o l d b a c h ' s conjecture isn't true?" "One of two things," said Faustina, "either help m e to p r o v e that there exists at least one even n u m b e r w h i c h cann o t b e e x p r e s s e d as the s u m of two p r i m e s o r give m e a c o u n t e r e x a m p l e a n d e n a b l e m e to d e m o n s t r a t e that it is n o t the sum of t w o primes, and of course I m u s t be able to a n n o u n c e the result." She blushed shyly at this somew h a t i m m o d e s t request. "Very well," said Luci, w h o by n o w was looking r a t h e r impressive, "just sign here." As s o o n as F a u s t i n a h a d signed, Luci said triumphantly, "Now that y o u r soul is mine, I'll tell you, y o u r G o l d b a c h ' s c o n j e c t u r e is n o t true." "How can you p o s s i b l y know?" q u e s t i o n e d Faustina. "Until a m o m e n t ago y o u h a d n ' t even h e a r d o f the p r o p o sition." "That's simple" said L u c i - - " G o d c o n c e i v e d t h e n u m b e r s a n d intended that all the even ones should be the s u m of at least one pair o f primes. H o w e v e r there w e r e so m a n y that She s t o p p e d paying attention, and I m a n a g e d to slip in an exception. Of c o u r s e it's very large." "How large?" a s k e d Faustina. "Well," said Luci, "it is larger than 12 (1212). But t h e g o o d n e w s is it's less than 13 (1313). I'm a bit s u p e r s t i t i o u s you see." "Can you help m e p r o v e the existence of that number?" a s k e d Faustina. "No," said Luci "that w o u l d take too long. The n u m b e r o f s t e p s in the p r o o f is g r e a t e r than the n u m b e r itself." "Ahha," c r o w e d Faustina, "if you cannot help m e with the proof, you m u s t do anything I w i s h - - r e l e a s e m e from t h e c o n t r a c t immediately!" "Not so fast! Let's l o o k at the c o n t r a c t first," said Luci. "Here it is. If the p r o p o s i t i o n is not true I either have to help you prove that it is untrue or I have to give you an even n u m b e r and e n a b l e y o u to d e m o n s t r a t e t h a t it is not t h e s u m of two primes, and that is w h a t I p r o p o s e to do." "You can't give m e s u c h a number," cried Faustina, "it's t o o huge, and w h a t ' s more, to enable m e to d e m o n s t r a t e t h a t it's not the s u m of t w o primes, you w o u l d have to give m e all the p r i m e s less t h a n the number." "I always keep m y promises," said Luci. "I can't give you the n u m b e r in your world but I can in mine; c o m e with me." F a u s t i n a felt a strange sensation as she m o v e d into other dim e n s i o n s that her colleagues had merely theorised. Her consciousness e x p a n d e d and with it h e r pain a n d misery, but she felt a glimmer of hope as she s a w what, in spite of its eleven-dimensional shape, a p p e a r e d to be a computer. "On that computer," said Luci, "you will find an even n u m b e r that c a n n o t b e e x p r e s s e d as the s u m of t w o p r i m e s a n d you will also find all the p r i m e s less t h a n it. All you have to do is s u b t r a c t e a c h p r i m e from the n u m b e r and s h o w that the r e m a i n d e r is n e v e r a n o t h e r prime. I a m s o r r y that it is not w h a t one w o u l d call an elegant proof, m o r e
18
THE MATHEMATICALfNTELLiGENCER
of a demonstration. I m u s t apologize t o o that the c o m p u t e r only h a s the s a m e clock s p e e d as one of your 386's, and the m o u s e sticks. But it's one hell o f a machine." "Hold on," said Faustina, "you p r o m i s e d I would be able to a n n o u n c e t h e result, h o w a m I going to be able to do that from here?" "I always k e e p my promises," Luci repeated. "When y o u have c o m p l e t e d the d e m o n s t r a t i o n y o u can go b a c k to y o u r sort of s p a c e a n d a n n o u n c e the result. Of course I didn't p r o m i s e that t h e r e would be a n y o n e left to h e a r you. After that I'll find s o m e t h i n g else for you to do." "What d o y o u m e a n 'after that'?" w a i l e d Faustina. "I'll never finish testing all these numbers." "Oh yes, you'll finish all right," g l o a t e d Luci Fer, "We,ve no end o f time." Patricia Rothman Department of Mathematics University College London London WC1E 6BT England e-mail:
[email protected] James Rothman E&MR 25, Norfolk Road London NW8 6HG England e-mail:
[email protected] E NEUENSCHWANDER
Documenting Riemann's Impact on tho Theory of Complex Functions
a
long w~th Euler, Gauss, and HzIbert, Bernhard R~emann ~s one of the most ~llustmous mathematicians of all t~me H~s prominence ~n the field of complex analys~s m a y be appreciated s~mply by noting that ~n the Mathemat~sches Worterbuch
by Naas-Schmld [1961, Vol 2,510-524], the entnes related to Rmmann's function-theoretical work (Rmmann mapping theorem, Rmmann differential equation, Rmmann surface, Rlemann-Roch theorem, Rmmann theta-functlon, Rmmann sphere, Rmmann zeta-function, etc ) take up almost as much space as those related to all the work of Euler or Gauss m total Rmmann's contnbutlon to complex analysis rests, on the one hand, on his publications, especially his inaugural dissertation on the foundations of complex analysis (1851) and his articles on the theory of hypergeometnc and Abehan functions (1856-57), but his several lecture courses in this field are also very important These were given in the years from 1855 to 1862, and regularly began with an introductory general part Rmmann then turned either to the theory of elliptic and Abehan functions or to hypergeometnc series and related transcendental functions Much of the more advanced parts of Rmmann's courses were published m his Collected Papers [Rmmann 1990, 599-692] and m a book by H Stahl [Rlemann 1899], but not his introductory lectures on general complex analysis Thus the latter gradually fell into obhwon, despite their lntnnslc interest, and despite their declsl've influence on later developments through the
closely related wntmgs of Dur~ge, Hankel, Koemgsberger, Neumann, Prym, Roch, and Thomae It therefore seemed appropnate to publish them m a cntmal edition [Neuenschwander 1996], m order to make them accessible to a broader circle of readers For this edition, I also prepared an extensive bibliography of the history of the m~pact and influence of Rmmann's function theory This newly assembled blbhography is mamly intended to close---at least m the field of complex analysis--the conslderable gap existing between the blbhograplues of Purkert and Neuenschwander, which were appended to the reprint of Bernhard R~emann's Gesammelte Mathemat~sche Werke [Rmmann 1990] The penod from 1892 to 1944, not systematically co~ered there, was scrutnuzed, usmg the indexes of names m B~blwtheca Mathematzca (1887-1914), and the Revue semestr~elle des Publwatwns mathdmat~ques (18931934/35), and exammmg the sections on "Gescluchte und Plulosoptue" and "Funktlonentheone" (or "Analysis") m the abstractmg journal Jahrbuch uber dze Fortschmtte der Mathe~nahk (1868/71-1942/44) Additionally, I consulted the holdings of older books in the libraries of the Institutes of Mathematics at the Umversmes of Gottmgen and Zurich Generally, I included only publlcanons which contmned explw~t reference to Rmmann's work (e g, quotations with
9 1998 SPRINGER VERLAG NEW YORK VOLUME 20 NUMBER 3 1998
19
e x a c t indication of location), but even this c r l t e n o n left m o r e than 1,000 out of t h o s e 8,000 titles provisionally sel e c t e d on the basis of the r e w e w s and the s y s t e m a t i c hb r a r y inspection, the o n g m a l s were then l o o k e d up, to m a k e sure that they m e t the c o n d m o n s for inclusion Naturally, the new blbhography Ls m no way comprehenswe The hterature w[uch refers to Rmmann's pioneenng w o r k is almost boundless Purkert, for example, examining about ten journals from the first 25 years after the death of Bernhard Riemann, already found more than 500 pubhcatmns referring to tus work According to database surveys, the continuation and extension of Purkert's research up to the present would have to take into account about 30,000 pubhcatmns, all of w[uch obwously cannot be exammed mchv~dually within a reasonable tune Nevertheless, I hope that the new blbhography will become a useful tool for further mvesUgat~ons In the following I will try to illustrate s o m e o f its possible applications, by s u r v e y m g the impact of R i e m a n n ' s function-theoretical w o r k m the four m o s t i m p o r t a n t E u r o p e a n countries Germany, France, Italy, a n d G r e a t B n t m n Special attention will be given to the d e v e l o p m e n t s in G r e a t Bntmn b e c a u s e t h e y have not yet b e e n a n a l y s e d in detail F o r m o r e specific mformaUon, the r e a d e r m a y turn to the bibliography itself
Germany: Early and Sustained Reception of Riemann's Methods F o r a prehmmary nnpresslon of the situation m Germany, let us first look at the references to Rmmann's w o r k m August Leopold Crelle's mfluentlal Journal f u r d~e re,he und angewandte Mathemat~k It ~s noteworthy that the first of these references goes b a c k to Helmholtz [Crelle 55 (1858), 25-55], who, as we know, later d i s c u s s e d Rlemann's h y p o t h e s e s c o n c e r m n g the f o u n d a t i o n s of g e o m e t r y Helmholtz w a s f o l l o w e d in chronological o r d e r by l_apsc[utz, Clebsch, Chnstoffel, Schwarz, Bnll, Fuchs, Gordan, Luroth, and Weber, all of whom, even t h o u g h they were n o t RmmaIm's i m m e d i a t e pupils, did a great deal to d~sseminate tus ideas, a s did [us own s t u d e n t s Roch, Thomae, and P r y m [Crelle 61 (1863)-70 (1869)] Clebsch and B r l l l - - h k e Klein a n d N o e t h e r - - p u b h s h e d their later w o r k p r i m a r i l y in the Mathemat~sche Annalen, w h i c h started to a p p e a r m 1869, thus, consulting Crelle's Journal alone p r o w d e s only mc o m p l e t e results for t h e m According to our bibliography, o t h e r i m p o r t a n t early p r o m o t e r s of Rmmann's i d e a s m the G e r m a n - s p e a k i n g w o r l d w e r e Cantor, Dedekmd, Du BolsReymond, Dur~ge, Hankel, Koemgsberger, Neumann, Schlafh, and, m [us later years, Schottky i
Italy: Enthusiastic Appreciation of Riemann's Work An ~mpressive pmture of the extent to w[uch Rmmann's w o r k was appreciated m Italy is gwen by a similar study of the references to [us writings m the Annal~ dz Matematwa pura ed apphcata, a journal which played an outstanding role m
the dmsemmation of Riemann's thoughts As early as 1859, Enrico Bettl, who was to b e c o m e Riemann's friend, translated [us & s s e r t a t m n [Annalz 2 (1859), 288-304, 337-356], to which he s o o n r e t u r n e d m an extensive article on the theory of elliptic functions [Annah 3 (1860), 65--159, 298-310, 4 (1861), 26-45, 57-70, 297-336] In the s a m e volumes, w e also find a r e p o r t by Betti on R i e m a n n ' s treatise concernmg the p r o p a g a t i o n of p l a n a r mr w a v e s [Annah 3 (1860), 232-241], and a r e p o r t by Angelo Genocc[u on Riemann's investigation of the n u m b e r of p r i m e s less than a given b o u n d [Annals 3 (1860), 52-59] In a later volume of the s a m e j o u r n a l [Annalz (2) 3 (1869-70), 309-326], we find a F r e n c h t r a n s l a t i o n of Rmmann's h y p o t h e s e s on the foundations o f g e o m e t r y by the F r e n c h m a t h e m a t i c i a n Jules Houel, w h o typically did not p u b h s h [us translatmn in a F r e n c h journal, b u t in t h e A n n a h We s h o u l d also m e n t i o n Eugemo Beltraml and Fehce Casorati, the latter havmg already p r e s e n t e d Rmmann's theories m 1868 m a b o o k [Casorati 1868] and m special lectures given m Milan [Armenante & Jung 1869, CasoraU & C r e m o n a 1869] In w e w of t[us excellent mtroduction and trml-blazmg, it is not surp n s m g that Rmmann's theories were widely known m Italy, and that they were quoted m m o r e than thu-ty articles m the Annalz alone up to 1890 2 As to Rmmann's own stays m Italy, and other followers of RIemann there, see the articles of Bottazznu, Dmudonn(!, Lena, Neuenschwander, Schermg, Tnconu, Volterra, and WeI1 cited m the bibliography
France: Hesitant Reception In France, the situalaon was ra(hcaUy different Skmmung through the pages of the Journal de Mathdmat~ques pures et apphqudes e&ted by Joseph Llouvllle, one finds almost no references to R]emann's papers before 1878, and m other French pubhcat~ons up to 1880 they also seem to be relaavely sparse F u r t h e r m o r e , a certmn cnt]cal r e s e r v e as to the usefulness of Rmmann's m e t h o d s quite often c o m e s through B n o t and Bouquet, for instance, write in the f o r e w o r d to the s e c o n d edition o f their Thdor~e des f o n c t w n s ell~pt~ques [Bnot & Bouquet 1875, I f ] In Cauchy's theory, the path of the ~mag~nary [complex] vartable ~s characterized by the movement of a point on a plane To represent thosefunctzons whzch assume several values f o r the same value of the vamable, Rzemann regarded the plane as formed of several sheets, supemmposed and welded together, ~n order to aUow the vartable to pass f r o m one sheet to another, wh~le t~avers~ng a connect~ng [branch] h n e The concept of a many-sheeted surface presents some d~ffienlt~es, despzte the beautiful results w h w h R~emann achieved by th~s method, ~t dzd not seem to us to offer any advantage f o r our own ob2ectwe Cauchy's zdea ~s very suitable f o r representing multivalued f u n c t w n s , ~t ~s suffw~ent to 2ozn to the value of the variable the corresponding value of the functzon, and
1For b~bl~ographrcal detatls on particular articles by these authors c~t~ng R~emann see [Neuenschwander 1996 131-232] 2The authors of other arttcles in the Annah containing references to R~emann are Ascoh Beltramt Casoratl Cesaro Chnstoffel Dtnl Ltpsch~tz Pascal Schlafh Schwarz Tonelh Volterra etc For details see [Neuenschwander 1996]
20
THE MATHEMATICALINTELLIGENCER
w h e n the v a ~ a b l e has described a closed c u r v e and the value o f the f u n c t w n has thereby changed, to ~ndzcate thzs change by an ~ndex Only Houel, m e n t m n e d above, tried quite early to p r o m o t e Rlemann's ~deas, and, tog e t h e r with Gaston Darboux, he several Umes d e p l o r e d their not being b e t t e r k n o w n m F r a n c e [Glspert 1985, 386-390 et p a s s i m ] A r e w e w o f Houel's p m n e e n n g b o o k m this field [Houel 1867-1874, parts 1 a n d 2 (1867/68)] concludes with the following, quite m s t r u c t w e p a s s a g e [Nouvelles A n n a l e s de M a t h e m a t~ques (2) 8 (1869), 136-143] M a y the ~eceptwn o f th~s w o r k encourage the autho~ to keep h~s prom~se to g~ve us, zn a thud part~ an exposztwn o f R z e m a n n ' s theories, w h i c h up to n o w have been almost u n k n o w n ~n out country, and w h w h out neighbours cult w a t e uuth so m u c h ardour and success ~
5~
dc./
4 Y3 7
g z/ / ,Z. /7 3 J/
1~ 3 .',/-,4-d 7
@
lO
/
In spite of Houel's efforts, the situation m ~ s ~ France seems to have changed fundamentally only after 1880, when a new generation of F r e n c h mathemahcmns t o o k over, the most / ~ / prominent among them were Henri Pomcare ~ (1854-1912), Paul Appell (1855-1930), Emile F,gure 1 D,scuss,on and representation of a branch point of an algebraic funct,on Plcard (1856-1941), and l~douard Goursat on a Rlemann surface, taken from Rmmann's lectures ,n the summer semester of (1858-1936) With these a u t h o r s the r e c e p t m n of the w o r k s of Rmmann's followers (mmnly 1861 The reproduced page comes from a set of lecture notes made by Eduard C l e b s c h a n d Fuchs) n o w b e g a n to lead to a Schultze w,th annotations by Hermann Amandus Schwarz, ,t ,s kept ,n the Schwarz Nachlass at the Archw der Ber|,n-Brandenburg,schen Akadem,e der W,ssenschaften m o r e t h o r o u g h r e a d m g o f Rmmann's own wntm g s - - a s numerous quotations m their pubhca- ,n Berl,n There ex, sts a s,m,lar representatmn m the (probably) or,g,nal notes by tmns prove As a mamfestat]on of the mcreas- Schultze wh,ch are now kept m the Rare Book and Manuscr, pt Library at Columb,a Un,verslty ,n New York Both manuscr, pts were d,scovered by the author, in 1979 mg appreclaOon of Rmmann's m e t h o d s among French mathematmmns at that tune, we may and 1986, respect,vely For further ,nformat, on and a transcr, pbon of the text, see take the fact that, m 1882, Georges Snnart wrote Neuenschwander 1988 and 1996, pp 75 f and 84, MS S and $s a dissertataon of 123 pages, n o w almost completely forgotten, m which he tried to acquaint F r e n c h mathM e m o i r s ~equ~res a knowledge o f R w m a n n ~ a n surfaces ematmmns w~th Rmmann's functmn-theoret~cal treatises In [and thus, f~naUy, o f R~emann's o w n wrtt~ngs], the u s e SLmart's mtroduetmn, we find the followmg statement, wtuch o f w h i c h has become standard at certain G e r m a n urnconfirms m y own assessment [Sunart 1882, 1] wets 1t~es We k n o w the m a g m f i c e n t results whzeh R z e m a n n a r r t v e d at ~n hzs two M e m o i r s concernzng the general theo~?] o f analytze f u n c t w n s a n d the theorq o f A b e h a n f u n c t w n s , but the m e t h o d s w h i c h he u s e d - - a n d explained p e r h a p s too suec~netly---a~e not well k n o w n ~n F~ance [ ] At the s a m e tzme, however, R ~ e m a n n ' s method c o n t i n u e s to have m a n y adherents ~n G e r m a n y , ~t serves as a bas~s f o r m a n y z m p o r t a n t p u b h c a t w n s by f a m o u s geometers, such as Koemgsberger, Carl N e w m a n n [s~c], Klezn, Dedekznd, Weber, P r y m , Fuehs, etc The reading o f these
Great Britain: A Forgotten Interest In c o n t r a s t to w h a t is k n o w n a b o u t the r e c e p h o n of Rmmann's t h e o r i e s m the three c o u n t r i e s a l r e a d y surveyed, their r e c e p t i o n m Great B n t m n is httle s t u d m d Let m e therefore p r o v i d e a m o r e detmled a c c o u n t In Great B n t m n Rmmann's t h e o r i e s s e e m to have become k n o w n a n d m o r e widely d i s s e m i n a t e d only after his death One of the first Bntmh m a t h e m a t i c i a n s who cited Rmmann frequently w a s Arthur Cayley, who, already m 1865/66, m e n t i o n e d the investigations o f Clebsch and
3Other scattered early ctta0ons of R~emann s wnttngs are to be found tn Bertrand Darboux Elhot Emmanuel Herm~te Jonqu~eres Jordan Mane and Tannery For further informa00n see [Neuenschwander 1996]
VOLUME20 NUMBER3 1998 21
R m m a n n on Abehan Integrals m his p a p e r s on the transf o r m a t i o n and higher smgular]tms of a plane curve Cayley also d i s c u s s e d Rmmann's w o r k in later y e a r s m great detml and with a p p r e c m t m n This can be seen from his note on Rmmann's p o s t h u m o u s l y p u b h s h e d early s t u d e n t p a p e r on generahzed integration a n d differentmtmn (1880), o r from his pres]dentml a d d r e s s to the Bntmh A s s o c l a t m n for the A d v a n c e m e n t of Science (1883) Other early c]tatmns of Rmmann's p a p e r s m Great Britain are by W T h o m s o n (1867 i f ) and J C Maxwell (1869 i f ) They o c c u r m conn e c t i o n with Helmholtz's p a p e r s on v o r t e x morton and w~th Lmtmg's studms on t o p o l o g y A few years later, W K Chfford, H J S Smith, and J W L Glmsher also thoroughly a n a l y s e d Rmmann's p a p e r s and, on several occasions, e m p h a s i z e d their great i m p o r t a n c e Clifford, for example, as early as 1873, translated Rlemann's f a m o u s p a p e r on the h y p o t h e s e s which lie at the b a s e of g e o m e t r y mto Enghsh for the j o u r n a l Nature, and m 1877 he p u b h s h e d his influential m e m o i r on the c a n o n m a l form a n d dmsectmn of a R m m a n n surface In N o v e m b e r 1876 Smith, m his presidential a d d r e s s On the Present State a n d Prospects o f s o m e B~anches o f Pu~e M a t h e m a t z e s to the L o n d o n Mathematical Socmty, r e p o r t e d m detml on R m m a n n ' s achievements, t a k m g mto a c c o u n t as well the n e w l y p u b h s h e d f r a g m e n t s m Rmmann's Collected Mathematical Works Smith's elaborate a d d r e s s s e e m s to m a r k an gutml h i g h p o m t in the a p p r e c m t m n of R m m a n n ' s w o r k s m Great B n t a m B e c a u s e Lt has not y e t b e e n studmd m this context, I will t r e a t It here m m o r e detml At the b e g m n m g of his a d d r e s s Smith outlined r e c e n t p r o g r e s s m n u m b e r t h e o r y a n d m p a r t m u l a r the mvest~gat m n s on the n u m b e r of p r i m e s less than a given b o u n d He w r o t e [Snuth 1876-77, 16-18] A s to out s o f the ser~es o f the p m m e n u m b e r s themselves, the advance s~nce the tzme o f E u l e r has been g~eat, z f w e th~nk o f the d~ffzculty o f the problem, but ve~:y s m a l l z f we compare w h a t has been done w z t h w h a t st~U r e m a i n s to do We m a y m e n t i o n , ~n the f i r s t place, the undemonst~ated, a n d ~ndeed congectu~al, theorems o f G a u s s and Legend~e as to the a s y m p t o t z c value o f the numbe~ o f p r i m e s ~nfer~or to a g~ven h m ~ t x [ ] The m e m o i r o f B e r n h a r d R~emann, "Ueber d~e A n z a h l de~ P m m z a h l e n u n t e r e~ner gegebenen Grosse," c o n t a i n s (so f a r as I a m a w a r e ) the only tnvestzgat~on o f the a s y m p -
totw f r e q u e n c y o f the p r i m e s whzch can be ~ega~ded as ~gorous [ ] No less ~mpo~tant than the ~nvestzgahon o f R~emann, but approaching the p~oblem o f the a s y m p totzc law o f the s e ~ e s o f p ~ m e s f l o r a a d~fferent s~de, ~s the celebrated memozr, "Sur les N o t a b l e s tbem~e~s," by M Tchgbychef [ ] The method o f M Tchdbychef, prof o u n d a n d ~n~m~table as st ~s, ~s ~n p o i n t o f f a c t o f a very e l e m e n t a r y character, and zn thzs respect c o n h a s t s strongly w~th that o f R~emann, whzch depends th~oughout on very abstruse theorems o f the Integral Calculus Simth then w e n t on to speak of some branches of analysis which a p p e a r e d to hnn to proimse much for the unmed]ate future Here he focussed on the advancement of the "Integral Calculus" He mentmned, among other works, Rmmann's m e m o i r on the hypergeometnc series and the unfimshed m e m o i r on hnear differential equatmus w~th algebrmc coefficients In domg so he stressed the "great beauty and ongmahty" of themann's reasonmg and the "fert~hty of the conceptlons of Cauchy and of Rmmann" In his closing remarks S~uth described the work of Enghsh mathemat]cmns m the field dunng the last ten years, nanung among others Glmsher, Cayley, and Chfford He was convmced that nothmg so hindered the progress of mathematmal scmnce m England as the want of advanced treatises on mathematmal subjects, and that there are at least three t~eat~ses w h i c h we g~eatly n e e d - - o n e on Defzn~te Integrals, one on the Theory o f F u n c t i o n s ~n the sense ~n w h i c h that p h r a s e zs understood by the school o f Cauchy and o f R~emann, and one (though he should be a bold m a n who would u n d e r t a k e the task) on the Hyperell~pt~c and Abel~an Integrals Smith called on his colleagues to close this gap, and to s o m e e x t e n t t h e y did 4 As early as a b o u t 1871 the B n t m h A s s o c m t m n for the A d v a n c e m e n t o f Scmnce set up a special Committee on Mathemat]cal Tables to which belonged, besides Smith, A Cayley, G G Stokes, W Thomson, a n d J W L Glmsher The p u r p o s e s for w h i c h the Committee w a s a p p o i n t e d were (1) to form as c o m p l e t e a catalogue as p o s s i b l e of exmtmg m a t h e m a t m a l tables, and (2) to r e p n n t o r calculate tables n e c e s s a r y for the p r o g r e s s of the m a t h e m a t i c a l s c m n c e s The C o m m i t t e e dec~ded to begm w~th the first task, and m 1873, p r e s e n t e d b y the care of Glmsher a huge catalogue of 175 p a g e s on m a t h e m a t i c a l t a b l e s w h i c h was p n n t e d in
4In a letter addressed to I Todhunter Smith g~ves a s~m~lar more dtrect and succinct estimate of the present state of Mathematics tn Great t3nta~n France and Germany It proves once agatn how highly he regarded the work of Rlemann and We~erstrass But I so heartily agree with much or rather with most of your book [Conflict of Studies 1873] that I should not have troubled you with this letter if it were not that I cannot wholly subscnbe to your estimate of the present state of Mathemattcs All that we have one may say comes to us from Cambndge for Dubhn has not of late qutte kept up the promtse she once gave Further I do not th~nk that we have anything to blush for ~n a companson wtth France but France ~s at the lowest ebb ~s conscious that she ~s so and ~s making great efforts to recover her lost place ~n Science Again ~n M~xed Mathematics I do not know whom we need fear Adams Stokes Maxwell Ta~t Thomson w~ll do to put against any hst even though ~t may conta~n Helmholtz and Claus~us But ~n Pure Mathematics I must say that I th~nk we are beaten out of s~ght by Germany and I have always felt that the Quarterly Journal ~s a m~serable spectacle as compared with Crelle [ s Journal] or even Clebsch and Neumann [Mathemat~sche Annalen] Cayley and Sylvester have had the hon s share of the modern Algebra (but even ~n Algebra the whole of the modern theory of equations substttut~ons etc is French and German) But what has England done rn Pure Geometry in the Theory of Numbers ~n the Integral Calculus ? What a tnfle the symbolic methods which have been developed ~n England are compared w~th such work as that of Rlemann and Weterstrass~ [H J S Smith Collected Mathematical Papers Introduction Vol 1 Ixxxv-lxxxw]
22
THE MATHEMATICALINTELLIGENCER
b e r 4, 1876 In these p a p e r s Glalsher t h a n k e d Smith for his help on the relevant hterature, z l ~ t ls a n d m a note referred e x p h c l t l y b a c k to Smith's a b o v e - m e n t i o n e d a d d r e s s to the London Mathematical Socmty m N o v e m b e r 1876 for specffic r e f e r e n c e s . s : . ~.=+1 , s~---! $=6 Further derails on the very intense later scientific mterchange b e t w e e n Snuth and ,-. 5 =~ , 2 = ~ :,-~-o Glmsher and their c o m m o n adm~ratlon for Rmmann can be gathered from Glmsher's P Introduction to tus e(fitlon of Snuth's Collected Mathematical Papens a n d from Smith's pap e r s t h e m s e l v e s 5 F r o m 1877 o n w a r d s J W L Glmsher's father J a m e s w a s engaged m the p r o j e c t of the c o n s t r u c t m n of factor tables of numbers from 3,000,000 up to 6,000,000, and he once again d o c u m e n t e d thereby the great superiority of Rmmann's formula for the numbers of primes as c o m p a r e d with those of Legendre and Tchebycheff J a m e s Glmsher's tables were cited m thetr ttu]l by various mathematmtans m Scandmav~an c o u n t r i e s (Opperm a n n 1882 f , G r a m 1884 ff, Lorenz 1891), w h e r e Rmmann's functmn-theoretmal w o r k F~gure 2 D~scussmn and representatmn of the branch points of the Rmmann surface h a d a l r e a d y e x c i t e d i n t e r e s t before, as is defmed by the equat,on s ~ - s + z = O from CasoratCs notes of conversatmns w,th s h o w n b y the p u b l i c a t i o n s of Bonsdorff, Gustav Roch m Dresden =n October 1864 The notes comprise 12 numbered pages At Mlttag-Leffler, and s o m e o t h e r s Moreover, m the end they gwe some mformatmn on h=s encounter w=th Roch (October 8-13) and 1884 W W J o h n s o n p u b h s h e d a detailed acCasoratCs w=shes to have a copy of R~emann's lectures The last page reproduced c o u n t of J a m e s Glalsher's f a c t o r tables and above from Casorat,'s notes of conversatmns w,th Roch contains one of the first the d l s t n b u t m n of p r i m e s m the American schematm representations of a R=emann surface in cross-section ("Sezlone normale j o u r n a l Annals of Mathemahcs, w h i c h mtroalle fronte") and v,ewed from above ("Veduta d= fronte") outs=de R,emann's own mand u c e d Rmmann's m v e s t i g a t m n s and the w o r k uscnpts Such representatmns later became very w,despread, as can be seen from of the British m a t h e m a t l c t a n s m the New s=m~lar drawings m Neumann 1865, Houel 1867-1874 Chfford 1877, Bobek 1884, World Amstem 1889, etc This way of representing a R~emann surface goes back m fact to Finally, attention s h o u l d be prod to R~emann h~mself, as can now be referred from page 52 from Schultze's lecture notes, A n d r e w Russell Forsyth, w h o w e n t to live m whmh =s pubhshed here for the first t,me (CasoratCs conversatmn notes were d,sthe t o w n of Cambridge a n d s t u d i e d nothing covered, studmd, and m part edited by the author during research m Pawa from Spring b u t m a t h e m a t m s m the s a m e year, 1876, that 1976 onwards ) For further mformatmn on CasoratCs notes and the Casorat= Nachlass, Smith a n d Glmsher s t a r t e d to d r a w a t t e n t m n see Neuenschwander 1978, especially pp 4, 19, 73 ff to Rlemann's w o r k In his o b i t u a r y notice, E T Whittaker d e s c r i b e d F o r s y t h ' s Theory of Functwns (1893) as h a w n g h a d a greater influence on British m a t h e m a t i c s t h a n any w o r k since the annual Report of the Br~ t~sh Assoc~a t~on In 1875 a conNewton's Pnnc~p,a According to W h i t t a k e r [1942, 218], t m u a t m n o f tius r e p o r t w a s p r e s e n t e d , c o m p i l e d b y Cayley p e r h a p s the m o s t original feature of th~s w o r k was the It included, u n d e r the heading "Art 1 DlvTsors a n d Prime melding of the t h r e e m e t h o d s a s s o c m t e d with the n a m e s Numbers," n e w additional r e f e r e n c e s to the table of the freof Cauchy, Rmmann, a n d Wemrstrass, w h i c h m the contiquency o f p r i m e s in Gauss's Collected Works and to the renental b o o k s w e r e r e g a r d e d as s e p a r a t e b r a n c h e s of mathlated a p p r o x m m t e formulae b y G a u s s and Legendre Both ematlcs F u r t h e r m o r e , one can r e a d m the Royal Socmty r e p o r t s c o n t a m e d no reference to Rmmann's f a m o u s p a p e r Bmgraphlcal M e m o i r of Henry F r e d e r i c k B a k e r by W V D on p r i m e n u m b e r s It ts only m e n t i o n e d m laterAssoc~atwn Hodge that F o r s y t h "rendered C a m b n d g e m a t h e m a t i c s Reports on Mathematical Tables (1877 ff ), and m a series great service by his efforts to b n n g a b o u t closer co-operaof p a p e r s on the e n u m e r a t i o n of the primes and on factor tion b e t w e e n mathemat~cians m this c o u n t r y a n d t h o s e on tables which Glmsher p r e s e n t e d to the Cambmdge the continent of Europe, and he m a d e it e a s m r for h~s Philosophical Society, the first one being r e a d on Decem5In addition to Smith s London address d~scussed above see also h~s paper On the ~ntegrat~on of d~scont~nuous functions [1875) w=th a deta=led analys~s of R~emann s memoir on the representation of a function by a tngonometnc senes h~s paper On some discontinuous senes considered by R~ernann (1881) and h~s Memoir on the
Theta and Omega Functsons which was wntten to accompany the Tables of the Theta Functions calculated by J W L Gla~sher
JOLUME 20 NUMBER3 1998 2 3
successors, mclu(hng Baker, to get the full benefit of the work of the great German masters of the late nmeteenth century" As can be seen from W[uttaker's obituary notice and Forsyth's own recollections of his undergraduate days [Forsyth 1935], there is a defimte link between Forsyth's later work m complex analysis and [us early studies in Cambridge, where he came mto repeated contact w~th Glmsher, Cayley, and the works of Smith, who taught at Oxford Cayley and Glmsher were also to be very instrumental in Forsyth's career in pure mathematics and helped [urn to overcome that "Cambridge atmosphere" m which "all were reared to graduation on apphed mathematics ,,6 Accorchng to Whittaker, it was Glmsher who suggested to Forsyth that he write his fLrst book, the Treatise on d~fferent~al e q u a t w n s (1885) His first two principal memoirs on theta functions (1881/1883) and on Abel's theorem and Abehan functions (1882/1884) were presented by Cayley, on the other hand, to the Royal Society In the first memoir Forsyth gives a list of 22 prmcipal papers in the field, mcluchng among others Rmmann's Theorte der Abelschen F u n c t w n e n as well as twelve other papers by the German mathematicians Jacobi, Rlchelot, Rosenhmn, Gopel, Wemrstrass, Koemgsberger, Kummer, Borchardt, and Weber The second memoir on Abel's theorem and Abehan functions contained a large section on Wemrstrass's approach, which clearly shows that around 1882 Forsyth was already fully aware of the achievements of the German mathematicians in this field Conclusion These bibhograplucal mvestlgaUons make it evident that Rmmann's ideas were more posltavely accepted m Great Britain, apparently even earher, than m France It seems they were among the more important stimuh wluch, through the efforts of Cayley, Clifford, Slmth, Glmsher, etc, later led to the flounshmg of Enghsh pure mathematacs and function theory under Hobson, Forsyth, Mathews, Baker, Barnes, Hardy, IAttlewood, T~tchmarsh, and many others In England and Italy Rmmann's theories entered open temtories, where they found mathematmal commumtms eager to catch up m pure mathematics In England this interest served to broaden to the "Cambridge atmosphere," which was nearly totally oriented towards natural philosophy and apphed mathematics, m Italy it fitted m w~th a desire to built up a modern mathematical education for the newly founded nation (cf Neuenschwander 1986) In France, on the other hand, Riemann's ideas first faced a cold receptaon from the well-estabhshed tradition of Cauchy and Briot & Bouquet
Allowing for a certmn politically motivated enthusiasm, Welerstrass was therefore probably right when, m a letter to Casorati dated 25 March 1867 (written m the early period before the publication of Rmmann's Collected Mathematical Works m 1876), he especially emphasized Italy's role m the dissemination of Rmmann's ideas In that letter (see [Neuenschwander 1978, 72]) 7 he writes The h a p p y r~se o f s c w n c e ~n y o u r f a t h e r l a n d wall nowhere else be f o l l o w e d wzth more vzvzd znterest than here zn North G e r m a n y , and you m a y be certain that the State o f Italy has nowhere else so m a n y s~ncere and d~s~nterested f r i e n d s Wzth pleca~ure w e are therefore ready to cont~nue the alliance between you and u s - - w h w h ~n the pol~twal f i e l d has had such good results---also ~n scwnce, so that also ~n that a~ea the barriers m a y m o r e and more be overcome, w~th w h w h u n h a p p y politics has f o r so long separated two generally congenial n a t w n s The pape~ whzch you sent to m e proves to m e once again that our s c i e n t i f i c endeavours are better understood and appreczated ~n Italy than ~n F~ance and England, partzcularly ~n the latter country, w h e r e an overwhelming f o r m a h s m threatens to stifle a n y feelzng f o r deeper ~nvest~gat~on How s~gn~ficant ~t ~s that o u r R w m a n n , whose loss we cannot s u f f i c i e n t l y deplore, ~s stud~ed and honoured outside Germany, only ~n Italy, ~n France he ~s certainly acknowledged externally but l~ttle understood, and ~n England [at least before 1867], he has r e m a i n e d almost u n k n o w n N o t e a d d e d t n p r o o f In a recent issue of this journal (vol 19, no 4, Fall 1997) there appeared an article by Jeremy Gray w[uch gives the reader a very welcome summary m Enghsh of Riemaim's introductory lectures on general complex analysis Gray's paper is largely based on my preprmt R ~ e m a n n s Vorlesungen z u r Funkt~onentheome Allgeme~ner Te~l, Darmstadt 1987 and on Roch 1863/65 (cf Neuenschwander 1996, p 15) but does not mention the substantially expanded version which appeared m Sprmg 1996 and which has been reviewed m MR 97d 01041 and Zbl 844 01020 The published version also contams, besides what was m the original prepnnt, the above-mentioned extensive bibliography of papers and treatises that were influenced by Rmmann's work, as well as a list of all known lecture notes of [us courses on complex analysis From this list [Neuenschwander 1996, 81 ff ], one can infer that Cod Ms Rmmann 37 comprises notes of three different courses (not just one, as suggested by Gray), and its first part is thereby quite easily datable to the Winter semester 1855/56 On page 111 of Cod Ms Rmmann 37 one
6Accordtng to Forsyth s own recollecttons as early as h~s student years he took an exceptional ~nterest tn pure mathematics and went for one term in hts third year to Cayley s lectures where at the beginning the very word plunged him ~nto complete bewilderment From other passages of Forsyth s recollections one may see that already at that t~me he made h~s first timid ventures outside the range of Cambridge textbooks and ploughed through among others a large part of Durege s Elhpt~sche Funct~onen Further details may be gathered from the following personal confession by Forsyth Something of d~fferenttal equations beyond mere examples the ele merits of Jacobean elhpttc functtons and the mathemattcs of Gauss s method of least squares I had learnt from Glatsher s lectures and by worktng at the matter of the one course by Cayley whtch had been attended I began to understand that pure mathemattcs was more than a collectton of random tools matnly fashioned for use ~n the Cambndge treatment of natural phtlosophy Otherwtse very nearly the whole of such knowledge of pure mathematics as ~s m~ne began to be acquired only after my Tnpos degree In that Cambndge atmosphere we all were reared to graduation on applied mathematics [Forsyth 1935 172] 7Nottce that h~s charactertzatton of the s~tuat~on ~n England ts strtktngly stm~lar to that of Smtth (see note 4) or Forsyth (see note 6)
24
THE MATHEMATICALINTELLIGENCER
Bottazzin~, U Rtemanns E~nflu6 auf E Bett~ und F Casoratt Archive
AUTHOI
for History of Exact Sciences 18 (1977/78), 27-37
Bottazztnl U II calcolo subhme stona dell'anahsl matematlca da Euler a We~erstrass Bonnghterl, Tonno 1981 Revised and enlarged English
ERWIN NEUENSCHWANDER Instrtut fur Mathemattk Abt Re~neMathemabk Un~vers~tatZunch-lrchel Wtnterthurerstrasss 190 8057 Zunch Switzerland Erw~n Neuenschwander ts Professor of the History of Mathematics at the University of Zurich He took his degrees under B L van der Waerden ~n Zurich He has published several papers on ancient, medieval, and n~neteenth-century mathematics, and especially on Rtemann, to whom h~s latest book ~s devoted Dunng the pest couple of years, he has also taken an ~nterest ~nthe h~story of the other exact so|ences and on the ~nteracttons between science, philosophy, and society The p~cture shows h~m before the tomb of Rtemann at Selesca (near Verban~a, Italy) on the occes~on of the Verban~a R~emann Memonal Conference ~n July 1991
reads "Fortsetzung ;m Sommersemester 1856 [Continuation m the Summer semester of 1856]," and on page 193 of the same manuscript "Theone der Functmnen complexer Groigen mlt besonderer Anwendung auf die Gauf~'sche Re;he F(a,/3, % x) und verwandte Transcendenten Wintersemester 1856/57" Furthermore, ;t should be noted that Rmmann's lectures on general complex analys;s developed gradually from 1855 to 1861 [Neuenschwander 1996, 11 f ], and that m 1861 he also treated We|erstrass's factonzatmn theorem for enUre functions w;th prescribed zeros, introducing to th;s end logarithms as convergence-producing terms [Neuenschwander 1996, 62-65], which ;s quite remarkable, although not mentioned by Gray 1997 Acknowledgments
I am indebted to Robert B Burckel (Manhattan, Kansas), Ivor Grattan-Gumness (Enfield), and Samuel J Patterson (Gottmgen) for various suggestmns concernmg both content and language The present art;cle was written under a grant from the Swiss National Foundation BIBLIOGRAPHY Armenante, A Jung, G [Relaztone sulle Leztoni complementali date nell Ist~tuto tecntco supenore a M~lano nello scorso anno (1868-69) dagl~ egreg~ Professort Br~osch~ Cremona e Casorat~] G/omale d~ Matemat/che 7 (1869), 224-234
version Springer, New York 1986 Bottazzin~ U Ennco Bett~ e la formaztone della seuola matemat~ca pisana In O Montaldo, L Grugnettl (eds) La stena delle matemat~che ~nItaha Att~ del Convegno, Caghar~, 29-30 settembre e 1 ottobre 1982 Unlverstta dl Cagharl, Cagharl 1983 229-276 Bottazztnt, U Rtemann tn Itaha In Conferenza Intemazlonale nel 125 ~ Ann~versano della morte d~ Georg Fnednch Bernhard R~emann 20 Lugho 1991 Verbanla Attl del convegno, 31-40 Bnot, C, Bouquet, J -C Theone des fonct~ons elhptlques Deux~eme edition Gauthter-Wllars Pans 1875 Brtot C Theone des fonct~ons abehennes Gauth~er-Vtllars Paris 1879 Casoratt F Teonca delle funz~onl d~ vanab~h complesse Fus~, Pavia, 1868 CasoratJ, F, Cremona L Intorno al numero det moduh delle equaz~on~o delle curve algebnche dL un dato genere Reale Ist~tuto Lombardo dl Sclenze e Lettere Rendlconb (2) 2 (1869), 620-625 = F Casoratl Opere vol 1,313-315 L Cremona, Opere matemat/che, vol 3, 128-132 Dteudonne J The beginnings of Italian algebraic geometry In Symposia mathemat~ca, vol 27, Academic Press, London/New York 1986, 245-263 Forsyth, A R Old Tnpos days at Cambridge The Mathemabcal Gazette 19 (1935), 162-179 GiIhsple, C C (ed) D~ctlonary of scientific biography, 16 vols Scnbner, New York 1970-1980 Gispert, H Sur la production mathemattque frangatse en 1870 (Etude du tome premier du Bullettn des Sciences mathemat~ques) Archives Intemabonales d'H~sto~re des Sciences 35 (1985), 380-399 G~spert, H La France mathemat~que La soctete mathemattque de France (1870-1914) Sutw de ctnq etudes par R Bkouohe, C Gtla~n C Houzel, J -P Kahane, M Zerner Cah~ersd'H~sto~reet de Ph~losoph~e des Sciences (2), No 34 Blanchard, Pans, 1991 Houel, J Theone elementalre des quant~tes complexes Gauth~erVdlars Pans, 1867-1874 Klein F R~ernann und seine Bedeutung fur d~e Entw~ckelung der mo dernen Mathemattk Jahresbencht der Deutschen Mathemat/kerVere~n~gung 4 (1894-95) [Berlin 1897], 71-87 = Gesammelte mathemat/sche Abhandlungen, Bd 3, 482-497 English translation Bulletin of the Amencan Mathematical Society 1 (1895), 165-180 Klein, F Vorlesungen uber die Entw~cklung der Mathemaflk ~m 19 Jahrhundert Tetle 1 und 2, Grundlehren der mathemat~schen Wtssenschaften, Bale 24 und 25 Spnnger Berltn, 1926-1927 Repnnt Berhn/Hetdelberg/New York 1979 Enghsh translation Brookhne, Mass 1979 Lona, G (ed) Lettere al Tardy d~ Genocchi Betti e Schlafh Atb della Reale Accademla de~ L~nce~ Sene qutnta Rendtcontt, Classe d~ sc~enze fis~che, matematiche e natural~ 24 1 (1915), 516-531 Lorta, G Stona delle matemabche 3 vol STEN Torino 1929-1933 Seconda ed~zione nveduta e agg~ornata Hoepl~, M~lano, 1950 Naas, J, Schm~d, H L (Hrsg) Mathemat/sches Worterbuch m~t Etnbeztehung der theorebschen Physlk, 2 Bde Akademie-Verlag/ Teubner, Berhn/Stuttgart, 1961 Neuenschwander, E Der Nachla8 ven Casoratt (1835-1890) ~n Pawa Archive for History of Exact Sciences 19 (1978), 1-89
VOLUME 20 NUMBER 3 1998 2 5
Neuenschwander E Uber dte Wechselwtrkungen zw~schen der franzosischen Schule, Rtemann und Wererstral3
E~ne Uberstcht m~t zwe~
Dedek~nd neu herausgegeben yon R Narastmhan Spnnger/Teubner, Berlin, etc/Leipzig, 1990
Quellenstudten Archive for History of Exact Sciences 24 (1981),
Schenng, E Zum Gedachtn~s an B R~emann In Gesammelte mathe-
221-255
mat/sche Werke von Ernst Schenng Bd 2 Berlin 1909 367-383
Enghsh version tn Bulletin of the Amencan Mathematical
Society, New Ser 5 (1981) 87-105 Neuenschwander, E Der Aufschwung der ~tahenlschen Mathematik zur
434-447 Partially reprinted ~n R~emann 1990, 827-844
Zest der pol~t~schen Etnigung Itallens und seine Ausw~rkungen auf
theone generale des fonct~ons et au pnnc/pe de D/nchlet Theses presen-
Deutschland In Symposia Mathemat~ca 27 (1986), 213-237
tees a la Faculte des Sciences de Pans Gauthler-V~llars, Paris 1882
Neuenschwander, E A bnef report on a number of recently d~scov-
Smith, H J S On the present state and prospects of some branches
Stmart G
Commenta/re sur deux Memo/res de R/emann relatlfs a la
ered sets of notes on R~emann's lectures and on the transmtsston of
of pure mathemabcs Proceedings of the London Mathematical Society
the R~emann Nachlass H~stona Mathemat~ca 15 (1988), 101-113
8 (1876-77), 6-29 = Collected Mathematical Papers, vol 2, 166-190
Reprinted ~n R~emann 1990, 855-867 Neuenschwander E Secondary hteratureon B R~emann In R~emann
Tncomi F G
1990, 896-910 Neuenschwander, E R/emanns E/nfuhrung ~n die Funkt/onentheone
Volterra, V
Bernhard Riemann e I'ltaha Un~vers/ta e Pol/tecn~co d~
Tonno, Rend~cont~del Sem~nano Matematlco 25 (1965-66), 59-72 Betty, BnoschJ, Casoratl, tro~s analystes ~tahens et trois
man~eres d'enwsager les quesbons d'analyse In Compte rendu du
Etne quellenknt/sche Edit/on se/ner Vorlesungen m/t e~ner B~bl~ograph~e
deuxleme Congres International des MathematJclens tenu a Pans du 6
zur
au 12 aoOt 1900 Gauth~er-Wllars, Pans, 1902, 43-57 = Opere matem-
W~rkungsgeschlchte
der
RJemannschen Funkt~onentheone
Abhandlungen der Akadem~e der W~ssenschaften
~n Gott~ngen,
Mathemabsch-Physikahsche Klasse, Dntte Folge Nr 44 Vandenhoeck
at~che, vol 3, 1-11 We~l A R~emann, Beth and the b~rth of topology Archive for H/story
& Ruprecht, Gotbngen 1996
of Exact Sciences 20 (1979), 91-96 21 (1979/80), 387
Purkert, W Arbe~ten b~s 1891 In Rtemann 1990, 869-895
WhFttaker, E T Andrew Russell Forsyth In Obituary Notices of Fellows
R~emann, B
Elhpt~sche Funct~onen
Vorlesungen von Bernhard
of the Royal Society 1942-1944, vol 4 no 11 (1942) 209-227
R~emann M~t Zusatzen herausgegeben von Hermann Stahl Teubner,
Wirtinger W
Leipzig 1899
und thre Bedeutung
Rtemann, B
Mathemat/ker-Kongresses /n Heidelberg vom 8 bls 13 August 1904,
Gesammelte mathemat/sche Werke w~ssenschaftl/cher
Nachlass und Nachtrage Nach der Ausgabe von H Weber und R
25
THE MATHEMATICALINTELLIGENCER
R~emanns Vorlesungen uber dte hypergeometnsche Re,he In Verhandlungen des dntten InternatJonalen
Teubner, Letpzlg, 1905 121-139 Repnnted in Riemann 1990 719-738
REINHARD BOLLING
An Unknown Photograph of Kovalevskaya
umng one of m y vzszts to the Mzttag-Leffler Institute (Djursholm, Sweden), I came across a photograph showzng the Russian m a t h e m a t w z a n Sofya Kovalevskaya (1850-1891) together wzth several other persons (Fzgure 1) Thzs m a y be the last photograph of Wezerstrass's f a m o u s pupzl to escape publwatzon With a little research, I was able to give a rather p r e c i s e dating and e x p l a n a t i o n of the c~rcumstances It Is easy to recogmze the standmg figure to the left o f Kovalevskaya Gosta Mlttag-Leffler (1846-1927), p r o f e s s o r since 1881 at the newly founded Umvemlty of Stockholm Left of hun we see lus young wife Slgne (1861-1921) Left of Slgne we see Anne Charlotte Leffier (1849-892), the sister of Mlttag-Leffier, a well-known author m her day, who was to write an unpresslve biography of Kovalevskaya m the short mterval b e t w e e n Kovalevskaya's death and her o w n But w h o is the m a n on the right side of the p h o t o g r a p h v The first clue is m the dating The m e m b e r s of the MtttagLeffler family are plmnly m m o u r n i n g Now Mlttag-Leffier's father J o h a n Olof Leffler (born 1813) dmd m July 1884, so we e x p e c t that the p~cture w a s t a k e n withm a few m o n t h s after that One finds on the p h o t o g r a p h that it was t a k e n at a studio m Sodertalje, a suburb o f S t o c k h o l m We ask, then, did K o v a l e v s k a y a and Mlttag-Leffler stay m Sodertalje d u n n g thin p e r i o d "~ Indeed they (rid, as w e fmd from her correspondence, and confirm from Anne Charlotte's biography "Mlttag-Leffier and a young German mathematician, whose acquaintance
Sonya had m a d e at Berlm durmg the summer, were with her at Soderta]je, and the young mathematician assisted her " ([3], p 215) The young German mathematician who came to Soderta]je was Carl Runge (1856-1927) Is he the man m our photograph v The only portrmt I k n o w of hm~ from ttus pemod ~s from the album p r e s e n t e d to Wemrstrass for the latter's 7Oth birthday m 1885 (Figure 2) [1], photograph 13 6 It s e e m s to m e that w e have found our many Kovalevskaya and Runge had m e t for the first tnne m Berlm m the s u m m e r of 1883 (not 1884, as Anne Charlotte writes) They w e r e very friendly in 1884, and Kovalevskaya arranged a visit to S w e d e n so that he could m e e t MlttagLeffler, a p p a r e n t l y 10 S e p t e m b e r to a b o u t 9 O c t o b e r [4], [5] Our p h o t o g r a p h m u s t have been t a k e n d u n n g this p e n o d The friendly relation b e t w e e n Kovalevskaya and Runge did not last very long By the time of the s e a r c h for a succ e s s o r to H Holmgren (1822-1885) at the umverslty and techmcal school m Stockholm, she told Mlttag-Leffler that she h o p e d Runge w o u l d not be c h o s e n b e c a u s e she had been on quite g o o d terms with htm but n o w h a d changed her oplmon [2] Although Mlttag-Leffier's favorite candidate for the position h a d always b e e n Runge, it w e n t to a S w e & s h a s t r o n o m e r and physicist, A Lmdstedt (1854-1939)
9 1998 SPRINGER VERLAG NEW YORK VOLUME 20 NUMBER 3 1998
27
Figure 1 (top). The S6dert~ilje Poctrait. (With kind permission of the Mittag-Leffler Institute.} Figure 2 (left). Carl Runge. (With kind permission of the Staatliche Museen zu Berlin, Kunstbibliothek.) REFERENCES [1] R. B611ing,A Photo Album for Weierstrass, Wiesbaden, Vieweg, 1994. [2] S.V. Kovalevskaya, 11 September 1885 letter to G. Mittag-Leffler, Mittag-Leffler Institute. See P.Ya. Kochina and E.P. Ozhigova, Correspondence between S.V. Kovalevskaya and G. Mittag-Leffler (Russian), Nauka, 1984, p. 124; and R. Cooke, The Mathematics of Sonya Kovalevskaya, New York, Springer, 1984, p. 34. [3] A.Ch. Leffler, Sonya Kovalevsky: Her Recollections of Childhood, with a biography by Anna Carlotta Leffler, Duchess of Cajanello, translated by Isabel F. Hapgood and A.M. Clive Bayley, New York, Century, 1895. [4] C. Runge, 3 September 1884 letter to S.V. Kovlevskaya, MittagLeffler Institute. [5] C. Runge, 15 October 1884 letter to G. Mittag-Leffler, Mittag-Leffler Institute.
Institut for Mathematik UniversitAt Potsdam Postfach 60 15 53 D-14415 Potsdam Germany
28
THE MATHEMATICALINTELLIGENCER
II,',Flil,t~,,vtlt.-rtll:lTllr
Probabilistic Proofs Th~s column ~s devoted to mathemahcs f o r f u n What better purpose ~s there f o r mathematws2 To appear here, a theorem orproblem or remark does not need to be profound (but ~t ~s allowed to be), zt may not be d~rected only at spec~ahsts, ~t must attract and fascinate We welcome, encourage, and frequently pubhsh c o n t n b u t w n s f r o m readers--ezther new notes, or rephes to past columns
Please send all subm)ss)ons to the Mathemattcal Entertatnments E@tor Alexander Shen, Instttute for Problems of Informat)on Transm~ss)on, Ermolovot 19, K-51 Moscow GSP-4, 101447 Russ)a e-rnatl shen@landau ac ru
Alexander
Shen,
Editor
n this zssue I p r e s e n t a collectmn of nice proofs that are based on some kind of a probabfllstlc argument, though the s t a t e m e n t doesn't m e n t m n any probabflmes First a simple geom e t n c example (1) It ~s k n o w n that ocean cove~s mo~e than one h a l f o f the Earth's surf a c e P~ove that the~e are two s y m m e t r i c p o i n t s covered by w a t e r Indeed, let X be a r a n d o m point Consider the events "X is covered by water" and " - X is covered by water" (Here - X denotes the point antzpodal to X) Both events have probability more than 1/2, so they c a n n o t be mutually exclusive Of course, the same (tnwal) argument can be explmned without any probabilities Let W C S 2 be the subset of the sphere covered by water, a n d let /.t(X) be the area of a region X C S 2 Then/x(W) + / x ( - W) > ~($2), so W A (-W) # 0 However, as we see m the following examples, probability theory may be more than a c o n v e m e n t language to express the proof (2) A sphere ~s colored ~n two colors 10% o f ~ts s u r f a c e ~s white, the remanning pa~t ~s black Prove that the)e ~s a cube ~nscr~bed ~n the sphere such that all ~ts 8 vertices are black Indeed, let us take a r a n d o m cube m s c n b e d in the sphere For each vertex the probability of the event "vertex is white" ~s 0 1 Therefore the event "there exists a wtute vertex" has probability at most 8 • 0 1 < 1, therefore the cube has 8 black vertices with a posmve probabflxty This a r g u m e n t a s s u m e s maphc~tly that there exxsts a r a n d o m variable (on some sample space) whose values are cubes wath n u m b e r e d v e m c e s and each vertex ~s uniformly dzstnbuted over the sphere The easmst way to construct such a varmble ~s to consider SO(3) with an m v a n a n t measure as a sample space It seems that here probability lan-
I
I
guage is more i m p o r t a n t if we dzd n o t have probabzhtms m mind, why should we consider an m v a n a n t measure o n SO(3) 9 Now let us switch from toy examples to more s e n o u s ones (3) In th~s e a a m p l e we w a n t to cons t r u c t a b~part~te g~aph w~th the foll o w i n g p~ operh es (a) both pa~ts L and R (called "left" and ")~ght") contain n vertices, (b) each ve~ tex on the left ~s connected to at most e~ght vertices on the r~ght, (c) f o r each set X C L that c o n t a i n s at least 0 5n vertices the set o f all neighbors o f all vertices ~n X conta~ns at least 0 7n vertices (These r e q m r e m e n t s are taken from the d e f i n m o n of "expander graphs", constants are c h o s e n to slmphfy calculaUons ) We want to prove that for each n there exists a graph that satisfies cond m o n s (a) - (c) For small n zt is easy to draw such a graph (e g , for n -< 8 we j u s t c o n n e c t all the verhces m L a n d m R), but it s e e m s that m the general case there is n o smaple constructxon wath an easy proof However, the following probabihstic a r g u m e n t proves that such graphs do exist For each left vertex x pink eight r a n d o m vertices on the n g h t (some of them may comclde) and call these vertices nezghbors of x All chomes are i n d e p e n d e n t We get a graph that satisfies (a) a n d Co), let us prove that it satisfies (c) wath positive probability F~x some X C L that has at least 0 5n vertzces and some Y C R that has less than 0 7n vertices What is the probability of the event "All neighbors of all elements of X belong to Y"9 For each fixed x E X the probablhty that all eight r a n d o m choices p r o d u c e an ele m e n t from Y, does n o t exceed (0 7) 8 For different elements of X choices are independent, so the resultmg probability is b o u n d e d by (0 7s) ~ = 0 74n
9 1998 SPRINGER VERLAG NEW YORK VOLUME 20 NUMBER 3 1998
29
T h e r e are f ewer than 2 '~ different possiblhtms for each of the sets X and Y, so the probabflgy of the e v e n t "there exist X and Y such that #X-->05n, #Y < 0 7n, and all n e ig h b o r s o f all vertices m X b e l o n g to Y" d o e s n o t e x c e e d 2 ~ X 2 n • 0 74n = 0 982n < 1 Thls event e m b o d m s the negation o f the requirem e n t (c), so we are d o n e All the e x a m p l e s a b o v e f o l l o w the s a m e s c h e m e We w a n t to p r o v e that an obJect with s o m e p r o p e r t y a exrots We c o n s i d e r a s t a t a b l e probability distribution and p r o v e that a rand o m obJect has p r o p e r t y ~ with n o n z e r o p r o b ab i l i t y Let us c o n s i d e r now two examples of a more general s c h e m e if the e x p e c t a t i o n o f a randora variable ~" is g r e a t e r t h a n s o m e n u m b e r ~, s o m e v a l u e s o f ( are g r e a t e r than (4) A p~ece o f pape~ has area 10 square centimeters Prove that ~t can be placed on the ~nteget g ~ d (the s~de o f whose square ~s 1 c m ) so that at least 10 g ~ d p o i n t s a~e covered Indeed, let us place a p i e c e o f p a p e r on the grid randomly The e x p e c t e d n u m b e r of grid points c o v e r e d by it is p r o p o m o n a l to its area ( b e c a u s e this e x p e c t a t i o n is an additive function) Moreover, for big p r a t e s the boundary effects are negligible, and the n u m b e r of c o v e r e d points is close to the area (relative error is small) So the coefficmnt is 1, and the e x p e c t e d n u m b e r of c o v e r e d points is equal to the area If the a r e a is 10, the e x p e c t e d n u m b e r is 10, so there must be at least o n e position w h e r e the n u m b e r o f c o v e r e d points is 10 or m o r e (5) A stone ~s convex, ~ts surface has a~ea S P~ove that the stone can be placed ~n the s u n h g h t ~n such a w a y that the shadow will have a~ea at least S/4 (We a s s u m e that h g h t ~s pe~pendwula~ to the plane where the s h a d o w ~s cast, ~f ~t ~s not, the shadow only becomes b~gge~ ) Let us c o m p u t e the e x p e c t e d area o f the s h a d o w Each p m c e o f the surface contributes to the s h a d o w exactly t w i c e (here co n v ex i t y is used), so the s h a d o w m half the s u m o f the s h a d o w s o f all pieces Taking into a c c o u n t that for e a c h piece all possible directions of light are eqmprobable, w e s e e that the e x p e c t e d area of the s h a d o w is pro-
30
THE MATHEMATICAL rNTELLIGENCER
p o m o n a l to the area of the st o n e surface To find the coefficmnt, take the s p h e r e as an ex am p l e it has a r e a 4~-r 2 and its s h a d o w has area wr 2, so the exp e c t e d s h a d o w area is S/4 (6) We fimsh our collection o f nice p i o b a b i h s t m proofs with a w e l l - k n o w n example, so race and u n e x p e c t e d that it c a n n o t be onutted It is the probabihstm p r o o f of the Welerstrass theor e m saying that a n y c o n t ~ n u o u s f u n c tton can be a p p r o x i m a t e d by a p o l y n o m i a l (AS far as I know, ttus p r o o f is due to S N B e r n s t e m ) Letf [0, 1] ~ ~ be a c o n t i n u o u s f u n c t m n Let p be a real n u m b e r m [0, 1] C o n s t r u c t a r a n d o m variable in t h e f o ll o w in g w ay Make n i n d e p e n d e n t trials, the probability of s u c c e s s in e a c h o f t h e m being p If the n u m b e r of s u c c e s s e s is /~, take f ( M n ) as the value o f the r a n d o m v a r i a b l e F o r e a c h p w e get a r a n d o m variable Its e x p e c t a t i o n is a function of p, let us call it f . ( p ) It is easy to see that for e a c h n the functlonf~t is a polynomml (What else can w e get ff the c o n s t r u c t m n uses only a finite n u m b e r of f-values~) On the o t h e r hand, fn is close to f, b e c a u s e for any p the ratm k/n m close to p with o v e r w h e l m m g probablhty (assuming n is big enough), so in most c a s e s the value o f f ( k / n ) is close to f ( p ) , since f ( p ) is uniformly continuous The formal argument r e q m r e s s o m e e s t m m t e s of probabilities ( C h e r n o f f b o u n d or whatever), but we omit the detmls
Colorings Revisited In the 1997, no 4 issue of The Intell~gencer I discussed a h o m o t o p m p r o o f of the following fact I f ~n a t ~ a n g u lat~on o f the sphere S 2 each vertea ~s ~nc~dent to an even nz~mbe~ o f edges, then the~e ~s a 3-coloring o f the vertices such that endpo~nts o f a n y edge have d~ffe~ent colo~s David Gale of Berkeley w n t e s in response As y o u m ay know, the c o n d m o n that e a c h v e r t e x lies on an e v e n number of edges is also a n e c e s s a r y and sufficient condition for the f aces to be 2-colorabte Using that fact I f o u n d the following homolog]cal, or rather cohomologmal, p r o o f of the t h e o r e m
Color the faces red and blue T h e n give the red triangles the clockwise, the blue ones the c o u n t e r - c l o c k w i s e orientation This gives a unique orientation to all ed g es o f the polyhedron, meaning w e can p u t arrows on the edges so that as o n e goes around a triangle, a r r o w s p o i n t in the same direction N o w use c o h o m o l o g y m o d 3 and define the 1-cocham w h i ch assigns 1 to each edge in the direction of its arr o w and - 1 m the o p p o s i t e direction By the p r o p e r t y a b o v e this is a 1-cocycle 0 t s c o b o u n d a r y on any t n a n g l e is 1 + 1 + 1 = 0), h e n c e b e c a u s e w e are on the s p h e r e it m u s t be the coboundary of a 0 - c o c h a m C, and thin C nmst assign different integers m o d 3 to adJacent v e r t i c e s Otherwise its c o b o u n d a r y w o u l d be zero on s o m e edge The p r o o f that the faces are 2-colorable is also h o m o l o g m a l Usmg m o d 2 homology, let c be the unit 1-chain that assigns 1 to all the edges By the evenness property, the boundary of this c h a m is zero, so it m a 1-cycle and h en ce b e c a u s e w e are on the sphere it must be the b o u n d a r y o f a 2-chmn This 2-chmn m u s t have distract values on adjacent faces, for if not, the boundary o p e r a t o r w o u l d assign 0 to their comm o n edge This finishes the p r o o f of the follov. mg t h e o r e m T h e o r e m 1 Let T be a triangulation o f the 2-sphere The f o l l o w i n g statements a~e equzvalent 1 Every vertex has even degree 2 The f a c e s can be 2-colored 3 The vertices can be 3-colored There is a s m u l a r t h e o r e m that also has a nice h o m o l o g l c a l p r o o f T h e o r e m 2 The vett~ces can be 4colored r the edges can be 3colored ( m e a n t n g all three colors appea~ a~ound every t~angle) The p r o o f uses cohomolog~j with coefficients in the Klein F o u r Group Ka (whereas the p r o o f o f T h e o r e m 1 used 7//2;7 and///37/) Assume that "~ertmes are 4-colored Identify the colors with the e l e m e n t s of K4 = {0, A, B, C} F o r any t na ngl e its c o l o n n g is e i t h e r of the form 0, A,
B or A, B, C, and the c o b o u n d a r y is either (A, A + B = C, B) or (A + B = C, B+C=A, C+A=B), so the edges are 3-colored with the n o n - z e r o elem e n t s of K4 Assume that edges are 3-colored Identify the colors with the non-zero elements of/{4 Then this is a 1-cocycle C1, since the s u m of the three nonzero e l e m e n t s of K4 ts zero Hence on the 2-sphere Ct is the c o b o u n d a r y of a 0-cochain Co, and this m u s t assign different colors to adJacent vertices a a n d b since C i ( a , b ) = Co(a) + Co(b) n m s t be non-zero
L e t t e r from Prof. Dr. Hanfr:ed L e n z
August 28, 1997 A p r o b l e m in mathematical entertmnment, w]thout any scientific value Fred two or more squares or higher powers of integers with the santo decimal digits in different order, such as 125 a n d 512, 256 and 625, 169, 196, and 961, 1024 and 2401, 1296, 2916, and 9216, 1728 and 2781 etc Such numbers can be constructed, e g, 10609, 16900, 19600, 90601, and 96100, ol else 30043=27,108,144,064 and 40033= 64,144,108,027 The following example shows a
method of c o n s t r u c t i o n Let a and b be two integers with one digit, and suppose a 2 and b 2 a r e e]thel both smaller than 10 o] both larger than 10 a n d ab Is smaller than 50 T h e n the two squares (100a + b) 2 = 10000a 2 + 200ab + b2 and (100b + a) 2 = 10000b2 + 200ab + a 2 ha~e the same dlg]ts It ]s easy to generalize this construction, see 30043 and 40033 above But I am more interested in r a n d o m examples such as the first example 125 and 512 Yours sincerely, H a n f n e d Lenz (Berlin and Mumch, Germany)
VOLUME 20 NUMBER 3 1998
31
WALTER GAUTSCHI
Ostrowski and the Ostrowski Prize EDITOR'S NOTE The Ostrowsk~ Pmze has been awa~ded five t~mes, at two-yea~ ,ntervals The f i r s t t~me st went to Louts de B~anges fo~ h~s proof of the B~eberbach con3ecture The second reczp~ent was Jean Bourga~n, who was ~ecogn~zed f o r h~s work on ergod,c theory The third award was g~ven jointly to M~klos Laczl~owch a~d Marzna Ratner fo~ their work on L,e groups The fourth award was made to Andrew W~les f o r hts proof of Fermat's Last Theorem The 1997 prize, fifth and most recent, was awarded lo~ntly to Yur~ V Neste~enko and G~lles Jean Geo~ges P~s,er for their work on algeb~aw number theory and operator theory, respectively Here ,s a brief ~ntroductwn to the Ib'tze and tts foundet, Professo~ Alexande~ Ost~owsk~ It ~s taken from the remarks made by Walte~ Gautsch~ at the ceremony of p~esent~ng the 1995 prize to Andrew W~les (which took place 4 June 1996 ,n Copenhagen) The p n z e was estabhshed by Professor and Mrs Ostrowsk] m the early 1980s with the stipulation that it should be a w a r d e d every two years after their deaths The p u r p o s e of the p n z e is " to p r o m o t e the mathematical sciences through penodlcally awarding an I n t e r n a t m n a l P n z e m order to recognize the best achievements made m the preceding five years in the areas of Pure Mathematics a n d the theoretical foundations of N u m e n c a l Mathemat-ms " It is c h a r a c t e n s t l c of Ostrowsl~'s w e w of mathematics as an international and universal science that he expressly stipulated that the award should be made "entirely without regard to poht~cs, race, religion, place of domicile, nationahty, or age of the a w a r d e e " The "Stfftungsrat" m a n a g e s the assets of the f o u n d a t m n a n d determines the p n z e s u m Selecting the p n z e winner(s) is a task given to a jury, whmh IS to be c o m p o s e d of one representative each of five scmntlfiC m s t l t u t m n s the mathematlcal institutes of the universities of Basel, Jerusalem, a n d Waterloo, and the scmnttfic academies of C o p e n h a g e n a n d Amsterdam One of these representatives acts, o n a rotational basis, as the president of the jury Let me n o w say a few w o r d s about the life a n d personality of the m a t h e m a t m l a n who created the p n z e Ostrowsl~ was b o r n on S e p t e m b e r 25, 1893 in Kmv Already
32
THE MATHEMATICAL INTELLIGENCER 9 1998 SPPINGER VERLAG NEW YORK
at the age of 18 he began to study privately with D m l t n i Aleksandrowd Grave, the founder of the Russian school of algebra, hLmself a former s t u d e n t of Ceby~ev m St Petersburg F r o m this association with Grave resulted O s t r o w s k f s first mathematical pubhcatlon, a long paper written in Russian on Galots fields He w e n t on to study m Marburg, b u t ended up a clvfl p n s o n e r w h e n World War I broke out Thanks to the intervention of Hensel, the r e s t n c t m n s o n IllS m o v e m e n t s were eased somewhat, and he was allowed to use the university library That was all he really n e e d e d D u n n g this p e n o d of isolation, Ostrowskl almost smgle-handedly developed his n o w famous theory of valuation on fields After the war was over and peace was restored b e t w e e n the Ukrmne a n d Germany, Ostrowskl m 1918 moved on to Gottingen, the world center of m a t h e m a t i c s at that t~me There, he s o o n stood out a m o n g the s t u d e n t s by his phen o m e n a l m e m o r y and his already vast a n d broadly based knowledge of the mathematical literature One s t u d e n t later recalled that the tedious task of hterature search, in Gottmgen, was extremely simple all one had to do was to ask the Russian s t u d e n t Alexander Ostrowskl and one got the a n s w e r - - i m m e d i a t e l ywHe could say something like oh yes, this you c a n find in a 1882 d~s~ertatlon of Mr so a n d
reer, e x c e p t for o c c a s i o n a l visits to the United States and Canada. It w a s h e r e in Basel w h e r e t h e bulk o f his mathematical w o r k unfolded. This is n o t t h e time and p l a c e to p r e s e n t an a c c o u n t of this work. Even if it were, it w o u l d be i m p o s s i b l e to give even a hint of the e n o r m o u s variety and d e p t h of his contributions. Suffice it to say that, beginning in the 1930s, and particularly after the 1950s, t h e r e was a n o t a b l e shift in his interests from P u r e Mathematics to m o r e Applied Mathematics, u n d o u b t e d l y reflecting the e m e r g e n c e of p o w e r f u l electronic c o m p u t e r s . Ostrowski r e m a i n e d m a t h e m a t i c a l l y active until his 80s, and at the age of 90 w a s still able to oversee t h e publication o f his Collected Works. These eventually a p p e a r e d in six volu m e s 1. Ostrowski in 1949 m a r r i e d Margret Sachs, a psychoanalyst from the s c h o o l of Carl Gustav Jung a n d at one time, as she o n c e told me, a s e c r e t a r y and c o n f i d a n t e of the Swiss p o e t and novelist Carl Spitteler. Her w a r m a n d charming p e r s o n a l i t y h e l p e d soften the severe lifestyle o f Ostrowski as a s c h o l a r a n d b r o u g h t in s o m e m e a s u r e o f joyfulness.
Professor Alexander Ostrowski
so in R o s t o c k - - a s o u r c e n o b o d y w o u l d have d r e a m e d of looking up. At one time, he even h a d to help out Hilbert during one of his lectures when he needed, as he put it, a beautiful t h e o r e m whose author he could not recall. It was Ostrowski w h o had to w h i s p e r to him: "But, Herr Geheimrat, it is one of your own theorems!" It is n o t surprising, then, t h a t Felix Klein t o o k on O s t r o w s k i as one of his a s s i s t a n t s and e n t r u s t e d him, tog e t h e r with Fricke, with editing the first volume o f his coll e c t e d works. In 1920, he g r a d u a t e d s u m m a cum laude with a thesis w r i t t e n u n d e r Hilbert a n d Landau. This, too, c a u s e d quite a stir, since it answered, in part, Hilbert's 18th p r o b lem. O s t r o w s k i s u c c e e d e d in proving that the Dirichlet zeta series d o e s n o t satisfy an algebraic differential equation. O s t r o w s k i ' s Habilitation t o o k p l a c e in Hamburg, with a w o r k also inspired b y Hilbert, dealing with m o d u l e s o v e r p o l y n o m i a l rings. In 1922, O s t r o w s k i returned to G6ttingen to t e a c h on r e c e n t d e v e l o p m e n t s in c o m p l e x function theory. This led to his w o r k on gap t h e o r e m s , o v e r c o n v e r g e n c e of p o w e r series, and b o u n d a r y b e h a v i o r of c o n f o r m a l maps. After a y e a r as a Rockefeller R e s e a r c h F e l l o w in Oxford, Cambridge, and Edinburgh, he r e c e i v e d - - a n d a c c e p t e d - a call to the University o f Basel in 1927. The local n e w s p a p e r could not help c o m m e n t i n g that 200 y e a r s earlier, the university lost Euler to St. P e t e r s b u r g by virtue o f the lottery t h e n in use to c h o o s e b e t w e e n c o m p e t i n g candid a t e s - - E u l e r lost! Now, however, the university hit the j a c k p o t by bringing b a c k O s t r o w s k i from Russia to Basel! O s t r o w s k i r e m a i n e d in Basel for his entire a c a d e m i c ca-
~Atexander Ostrowski, Collected Mathematical Papers, Vols. 1-6, BirkhAuser, Basel, 1983-1985.
VOLUME 20, NUMBER3, 1998
on the n e w e s t m a t h e m a t i c s and m a t h e m a t i c a l gossip The walls of the h b r a r y were filled with books, not all mathematical, but also a g o o d m a n y on s c m n c e fictmn and mystery stones, w h i c h p r o v i d e d his favored p a s t i m e reading Mrs Ostrowsk] p a s s e d a w a y m 1982, four years before Ostrowskl's d e a t h m 1986 They are b u r i e d m the lovely c e m e t e r y o f G e n n h n o , not far from the graze of Hermann Hesse, with w h o m t h e y were friends I r e m e m b e r O s t r o w s k l as a m a n totally c o m m i t t e d to his scmnce, e x t r e m e l y s t u b b o r n m [us p u r s m t o f problems, so much so that w h e n he was done with them, few ]f any quesn o n s r e m m n e d o p e n He could s i n c e r e l y marvel at the mgenmty m o t h e r p e o p l e ' s work, b u t at the s a m e time could e x p r e s s h]s d l s d a m at s h o d d m e s s and ineptitude You can well imagine, Ladms and Gentlemen, h o w dehghted he would have been, h a d he k n o w n a b o u t the b n l h a n t w o r k of tius y e a r ' s prize w i n n e r w
This w a s the p e r i o d w h e n I got to k n o w Ostrowskl, b o t h as one o f his students a n d as his assistant I well r e m e m b e r the m a n y p l e a s a n t evenings at their lovely old h o u s e m Basel, hstemng to their travel a d v e n t u r e s m the U m t e d States, a n d looking at the m a n y s h d e s they brought b a c k My b r o t h e r and I often e n t e r t m n e d w s m n g m a t h e m a t l c m n s b y playing four-hand p i a n o m u s i c at their h o m e After Ostrowskl's ret]rement m 1958, he and h]s wife t o o k up residence m Montagnola, w h e r e they e a r h e r h a d b m l t a beautiful v ] l l a - - A l m a r o s t (ALexander MARgret O S T r o w s k 0 , as they n a m e d l t - - o v e r l o o l a n g the Lake of Lugano They were always h a p p y to ~ecelve w s l t o r s at Almarost, and their gracious h o s p l t a h t y w a s l e g e n d a r y Mrs Ostrowskl, knowing the m c h n a t l o n s of m a t h e m a n clans, always led t h e m d o w n to O s t r o w s k f s h b r a r y m ord e r to leave them alone for awhile, so they could c a t c h up
34
THE MATHEMATICAL INTELLIGENCER
ii~,'~"'i;I,,][.~.]liIT'-:-II;i[,,'-Z-;-lil'e"J[,]lili,l'l,,l|;i[=.-]k-1
To Belong: The Role of Community in the Life and Work of J. J.
Sylvester Karen Hunger Parshall
This column is a foram for discussion of mathematical communities throughout the world, and through all time. Our definition of "mathematical community" is the broadest. We include
Marjorie Senechal,
Editor r
he atmosphere on 19 March 1897 was somber. England s last great Victorian mathematician, James Joseph Sylvester, had died of a stroke four days earlier in his rooms in Mayfair, and his friends had gathered to bury him. Percy Alexander MacMahon described the scene:
T
[T]hose who were present . . . at the simple, yet impressive ceremony at the Jewish cemetery at Dalston, must have realised that one of the giants of the Victorian era had been laid to rest. The Royal Society and the London Mathematical Society were represented at the funeral by Prof. Michael Foster, Sec. R.S., Major MacMahon, R.A., F.R.S., Prof. Forsyth, F.R.S., Prof. Elliott, F.R.S., Dr. Hobson, F.R.S., Prof. Greenhill, F.R.S., Mr. A. B. Kempe, F.R.S., and Mr. A. H. Love, F.R.S. There were also present Prof. Turner and the Sub-Warden of New College, Oxford [3, p. 494]. At least four of the communities to which Sylvester belonged saw him to rest: the Jewish community into which he had been born in 1814; the Royal
Society of which he had been a member since 1839; the London Mathematical Society which he had served as its second President from 1866 to 1868; and New College, Oxford where he had been named Savilian Professor of Geometry in 1883. One hundred years later, on 14 March, 1997, historians of mathematics under the rubric of the British Society for the History of Mathematics reunited groups to which Sylvester had belonged at the Jewish burial ground on Kingsbury Road in Islington, London, this time to commemorate and to celebrate his life and work. Representatives of each of the following communities testiffed to the place that Sylvester retains in their collective memories: University College, London where he was first a student (in 1828-1829) and then a colleague (1838-1841) of Augustus De Morgan; St. John's College, Cambridge where he formally studied for his degree but was debarred on religious grounds from taking it in 1837; the University of Virginia where he spent an unhappy four-and-a-half months on the faculty in 1841-1842; the Institute
"schools" of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.
Please send all submissions to the Mathematical Communities Editor, Marjorie Senechal, Department of Mathematics, Smith College, Northampton, MA 01063, USA; e-mail:
[email protected] Members of the Sylvester family (and Rabbi Tabick in robe at right of center) a t S y l v e s t e r ' s g r a v e , 14
March 1997.
9 1998 SPRINGER-VERLAGNEWYORK, VOLUME20, NUMBER3, 1998 3 5
o f Actuaries which he h e l p e d to found in the late 1840s and w h o s e wce-pres]dency he held, New College, Oxford, the Royal Society, and the London Mathematical Society, the Mathematical Associatmn, w]uch he served as President m the 1890s when it was still called the AssociaUon for the Improvement of Geometrical Teachmg, and two hnes of the Sylvester family A n u m b e r of c o m m u n i t i e s thus figu r e d p r o n u n e n t l y on t h e s e t w o occas i o n s s e p a r a t e d by one h u n d r e d years This suggests an e x a m i n a t i o n of Sylvester's published w o r k s a n d private c o r r e s p o n d e n c e to s e e lust what influence these c o m m u m t m s m a y have h a d on his d e v e l o p m e n t
Sylvester's early life w a s defined in large m e a s u r e by ]us J e w i s h heritage He received his earliest formal education at Neumegen's b o a r d i n g school for J e w i s h boys m Highgate a n d eventually p r o c e e d e d at the age o f fourteen to newly-founded, s e c u l a r London University (later University College London), the first Institution o f higher e d u c a t i o n in England to be explicitly m d e p e n d e n t of any organized religion There, a twenty-one-year-old Augustus De Morgan had r e c e n t l y b e e n n a m e d to the chmr of m a t h e m a t i c s , a n d the y o u n g Sylvester was a m e m b e r of the University's first entering class in 1828 [12] By virtue of his m a t h e m a t ] c a l talents, Syl~ester b y p a s s e d the j u m o r c l a s s e s and the lower-level s e m o r class a n d w e n t directly into the upper-level s e n i o r class, which c o n c e n t r a t e d on the differential and integral as well as a s p e c t s of the higher calculus In a c o m m i t t e e r e p o r t w r i t t e n on 18 N o v e m b e r 1837, De Morgan d e s c r i b e d Sylvester as "by far the first pupil" m that m o s t a d v a n c e d class [11, p 83] Unfortunately for b o t h s t u d e n t and t e a c h e r (De Morgan h a d p r e c i o u s few really talented s t u d e n t s in t h o s e early years), Sylvester w a s w i t h d r a w n from the Umverslty b y his farmly after only five m o n t h s on a c c o u n t o f his immaturity The official history of University College r e c o r d s that Sylvester "[took] a table-knife from the r e f e c t o r y with the intention of sticking it into a fellow s t u d e n t who had i n c u r r e d his displea-
36
THE MATHEMATICALINTELLIGENCER
sure" [1, p 180], the family m a y well it in his 1839 paper, "[t]he reflections have w i t h d r a w n the b o y to avoid an of- w]uch Sturm's r e m a r k a b l e t h e o r e m ficml e x p u l s m n had originally excited, were revived by In the fall of 1829, Sylvester m o v e d happening to be p r e s e n t [in 1839] at a on to a t t e n d the school a s s o c i a t e d with sitting of the F r e n c h Institute, where a the Royal Institution m Liverpool, letter was r e a d from the Minister o f where, agam, he failed to fit into the Public Instruction, requesting an opine d u c a t i o n a l c o m m u m t y during his t w o ion upon the e x p e & e n c y of forming tay e a r s of r e s i d e n c e "[R]endered extra- bles of elLmmatlon b e t w e e n two equaordinarily irritable by the c o n s t a n t ref- tions as high as the 5th or 6th degree erences, m a spirit of opposttmn, to his contmmng one repeatmg term [ l e , J e w i s h extraction" [4, p x]], he ulti- polynomial equations m 2 as high as the mately ran a w a y from the s c h o o l a n d 5th or 6th degree]" [20, p 44] Sylvester from Liverpool had j o u r n e y e d to Pans, the recogmzed The year 1831 finally found him center of m a t h e m a t i c a l research of the s o m e w h a t settled and apparently rea- day, to meet, hsten to, and talk v,l t h sonably h a p p y as a student at St John's mathematicians united w~thin the conCollege, C a m b n d g e Three bouts of ill- text of the Acade~,e des Scte~ces, and ness forced ]us prolonged and repeated this mteraction had stLrnulated ]us absence from his studms, but by 1837 he thoughts on Sturm's theorem Morehad taken the Mathematical T n p o s and over, b a c k at h o m e [us early a p p h e d recome m as Second Wrangler Had his searches had resulted m ]us election as nonconformity to the tenets of the Fellow of the Royal Socmty m 1839, and Church of England not prevented hun he personally dehvered ]us algebrmc from slgnmg the Church's reqmred ideas to the a s s e m b l e d savants of the Thu-ty-nme Amcles, Sylvester would, by British Association for the Advancev~tue of ]us ]ugh placement on the ment of Science at ~ts meeting m Tnpos, haze been a hkely candidate for Plymouth m 1841 These national coma college fellows]up As it was, however, mumtms served Sylvester not only as he could neither officially receive ]us de- avenues for c o m m u m c a t l o n but also as gree nor hold any sort of Cambridge po- velucles for the e s t a b h s h m e n t of pnors m o n Only nonsectarian Umverslty try and reputation [9] College offered employment possiblhAfter three p r o d u c t i v e years m ties for hun at the umverslty level, but London, Syl~ester resigned from UmDe Morgan contmued to hold its mathe- verslty College in 1841 to take a seemmatical chair When the professors]up of mgly m o r e congenial position, an acnatural philosophy did open up there m tual m a t h e m a t w s p r o f e s s o r s h i p at the 1837, Sylvester apphed, was appomted, Umverslty of Vlrgima It was a gamble and a s s u m e d the position the next year He would be t e a c h i n g mathematms, F r o m this London base, Sylvester but he w o u l d be far from England and p r o m p t l y set to w o r k estabhshlng hun- far from the s c h o l a r l y c o m m u n m e s self w~thin the b r o a d e r m a t h e m a t i c a l which had b e g u n to define him proand scmntlflC c o m m u n i t y D u n n g his fessionally The g a m b l e did not pay off first t w o y e a r s m the professorship, he Sylvester r e s i g n e d from the university p u b l i s h e d five p a p e r s (and two s h o r t in Charlottesville after four-and-a-half n o t e s ) in the Ph~losophwal Magazine, months, ]us c o l l e a g u e s had failed to four m m a t h e m a t i c s of a m o r e a p p h e d s u p p o r t him in his call for the expulnature c o n s o n a n t with his natural phi- sion of one of the s t u d e n t s in ]us class l o s o p h y post, and one on Sturm's the[8, pp 64-65, 2] o r e m for l o c a t m g the r o o t s of a polyNumerous failures to secure a teachnomial equation, f o r e s h a d o w i n g his mg position m the Umted States preclplifelong i n t e r e s t m algebraic t o p i c s If ]tated Sylvester's return to England and the f o r m e r w o r k u n d e r s c o r e s the in- ultimate a c c e p t a n c e of a j o b as actuary terplay b e t w e e n teachmg and r e s e a r c h and later also secretary at the Eqtuty and wtt]un the university community, the Law Life Assurance Socmty m London latter signals the u n p o r t a n c e of partic- The post was mathematical, tt provided ipating m the c o m m u n i t y defined by a comfortable mcome, but it left httle the scientific society As Sylvester p u t tune for original mathematical research
Kepler's Apostrophe J J Sylvester Yes v On t h e a n n a l s of my race, In c h a r a c t e r s o f flame, Which tune shall dun not n o r deface, I'll s t a m p m y d e a t h l e s s n a m e Minds d e s t i n e d to a g l o n o u s s h a p e Must first affhction feel, Wme o o z e s from the t r o d d e n grape, Iron's b h s t e r e d m t o steel, So gushes from affection b r u i s e d Ambition's p u r p l e tide, And s t e a d f a s t froth unlundly u s e d H a r d e n s to s t u b b o r n pride
Syl~ester confessed m a letter on 12 Apnl 1846 to his American friend, the physicist J o s e p h Henry "You ask me to give an account of myself and to state with what pursmts of science I a m occupied The question plants a dagger m m y conscience I have been too unhappy at one p e n o d - - t o o unsettled at ano t h e r - - t o o Intent upon securmg or makmg a posit]on at a t h i r d - - t o have given any regular attent]on to scmnt[fic purstats" [10, p 408] Indeed, he p u b h s h e d only three short notes m the five years from 1845 through 1849 Actuarial w o r k - - a n d from 1846 to 1850 studies at the Inner Temple preparatory for the Bar---occupmd him almost exclusively at this juncture and c~rcumscnbed a professlonal sphere that he w o r k e d to formalLze In the s u m m e r of 1848, Sylvester, together with fourteen other actuaries, crafted a plan for "a scientific and pract]cal Association amongst Actuaries, Secretaries and Managers" that became, on 8 July, the Institute of Actuaries Sylvester served as one of the society's first four Vice-Presidents and r e m a m e d qmte act]ve m the new group's affmrs through the end of the 1840s [15] F r o m his teenage y e a r s through 1850, Sylvester had doggedly sought but had h a d difficulty finding his niche He had w o r k e d to belong, first to the a c a d e m i c a n d scmntlfic c o m m u n i t i e s but then, w h e n w h a t he t e r m e d "a frmtless and h o p e l e s s struggle with an ad-
verce [slc] tide o f affmrs" [9, p 210] b l o c k e d t h o s e efforts, to a c o m m u m t y of actuaries he h e l p e d to organize The late 1840s thus found him actively moN ed with his fellow actuaries but Intellectually e s t r a n g e d from the mathematics he longed to develop He h a d been m nmthemat]cal isolation for almost a decade, he never tlmved m such a state By 1850, a n e w friendship had redirected hun t o w a r d mathematics and away from the actuarial sciences Arthur Cayley and Sylvester had come to comprise a mathematical commumty of t w o - - c o m p l e m e n t a r y yet surpnsmgly compat]ble personahtms devoted to the advancement of mathematical thought The two men met some tune prior to 1847, by that tune they had already begun the mathematical correspondence that would last until Cayley's death m 1895 Both mathematicians nnsplaced at the Inns of Court (Cayley was studying conveyancmg at Lmcoln's Inn), Cayley and Sylvester found m one another a mathematical soundmg board and fount of appreciation and encouragment F o r Sylvester, much m o r e than for Cayley, this relataonship provided a n e e d e d sUmulaUon and outlet for mathematical ideas The d y n a m i c s c o m e through clearly in Sylvester's early letters to his n e w friend "I think that I can put the substance of the latter p a r t of m y yesterday's c o m m u n i c a t i o n m a much c l e a r e r
p o i n t of ~new than I'd s u c c e e d e d m dorag," he wrote on 28 November, 1849 "Do you c o n s i d e r that t h e r e is a flagrant unposslblhty m the following result to which my investigations a p p e a r to conduct me 9" he a s k e d two days later And after another three weeks had passed, he declared hunself 'much obhged by your calling my attention to the defecUveness o f the Theorem" on 21 December, yet he confidently asserted that "it may however, I entertain httle apprehension, be verified ,,1 Cayley was a mathematical equal and partner who could cnt]cally evaluate new ideas and for whom results m e n t e d honmg and perfectmg His presence sufficed to create an enwronment of g~ve-and-take that nurtured and nourished Sylvester's mathematical p o w e r s The development of the theory of m v a n a n t s at the hands of Cayley and Sylvester m the early 1850s prowdes stunnmg evidence m support of this clmm [7] As these n e w m a t h e m a t i c a l ideas flooded the pages of BnUsh mathematreal journals, Sylvester's mathematical c o m m u m t y of two faurly qmckly exp a n d e d mto an international circle of mathematical c o r r e s p o n d e n t s that mc h i d e d the I n s h c l e r g y m a n a n d fellow l n v a n a n t theorist, George Salmon, F r e n c h geometers, Charles Hermlte a n d Michel Chasles, the Belgian stat]stician, Irenee-Jules Bienaym6, and the German mathemat]cian, Carl Borchardt, among others [6] Sylvester knew he was doing nnportant mathematical work, and it b e c a m e mcreasmgly unportant for him that that w o r k become k n o w n and recogmzed both at home and abroad In o r d e r to faclhtate this sort of commumcat~on, he began to b n n g ins work dtrectly before the b r o a d e r Bnt]sh scientific constituency as represented by the Royal Society and its Phffosophwal T~ansavtwns, the C o n t i n e n t a l - - a n d especially F r e n c h - mathematicians through pubhcat]on m the NouveUes Annales de Mathemat~ques, and the w~der mtemat]onal scientific audience through the dmsemman o n of r e p n n t s to, for example, the Pans Academ~e des Sciences [9] As he had m the late 1830s and early 1840s, Sylvester
1All three letters are Jn the James Joseph Sylvester Papers St John s College Cambndge Box 9 I thank the Master and Fellows of St John s for their permission to quote from these and the archwes quoted below It ~s always a pleasure to acknowledge their hospttal~ty and support
VOLUME20 NUMBER3 1998 37
once agam worked to establush his "membership" and r e p u t a t m n m the c o m m u n i t y of scmntusts Ttus time, he h a d a series of truly striking research a d v a n c e s to put forth in evidence for all to see He y e a r n e d for the opportum t y to devote himself m o r e exchis~vely to tim furtherance of mathematics, he w a s r e a d y to w i t h d r a w from the actuarial c o m m u m t y that h a d s u p p o r t e d hun for almost a d e c a d e After a failed attempt m 1854 to secure a post m higher educaUon, Sylvester succeeded the next year m obtaining the professorslup of mathematics at the Royal Military Academy m Woolwich As he put ~t m a letter written to Henry Brougham on 16 September 1855 m the euphoria of the m o m e n t m which he learned o f his appointment, "I hope henceforth (apart from all m e a n e r obj e c t s of ambltmn) to consecrate my life and p o w e r s to the study & humble lrmtatmn of the sprat of the great masters o f our science, to serve and ff possible advance which us m m y mind one of the noblest objects of a m b m o n which one can p i o p o s e to oneself "~ Sylvester clearly felt that the position at Woolwich w o u l d p r o v i d e him with m u c h desired time to d e v o t e to his research, the military p e r s o m m l in c h a r g e of the A c a d e m y had o t h e r ends in view, h o w e v e r Sylvester's first five y e a r s at Woolwich m a r k e d a s h a r p dechne in his r e s e a r c h p r o d u c t i v i t y and a dec~ded down-turn in his s p r a t s as he fought regularly with the a u t h o n t m s o v e r his teaching load and o'~erall time c o m m i t m e n t to the A c a d e m y , the next five w i t n e s s e d an upswing with hus (the first) p r o o f of N e w t o n ' s rule for d e t e n m n m g b o u n d s on the n u m b e r of p o s m v e and negative r o o t s of a polynomial equation, the f o n n a h z a t l o n of an Enghsh c o m m u n i t y of m a t h e m a t i cians m the form o f the London Mathematical Socmty ( f o u n d e d in 1865) [13] before which h e p r e s e n t e d this n e w result, m o r e fighting with the a u t h o n t m s , and his election as foreign c o r r e s p o n d e n t to the P a n s Academte des Sciences, the last five fotmd him groping for m a t h e m a t i c a l direction
p r i o r to hus p r e m a t u r e and f o r c e d i e t i r e m e n t from Woolwich at the age o f fifty-fi~e On 27 May, 1867, he a d m i t t e d d e j e c t e d l y to Cayley that I have done no mathematws--eve~ ~ntend~ng and e~e~ p u t t i n g it off Thus I have been too much ashamed to call upon or to wr~te to you but ha~,e been hoping all along to see m hear f~om you I f I thought tt would do a n y good I would ask you to p t a y f m m y rescue flora thts en.slawng tndolence and pa~alys~s of the wall f m such tt amounts to 3 The Woolwich years from 1855 to 1870 had f o u n d Sylvester m an a c a d e m i c setting, but one which p r o v i d e d him no r e s e a r c h respiration or e n c o u r a g e m e n t F o l l o w i n g s~x y e a r s unemplo:~ed m London he finally found h i m s e l f p a r t o f an a c a d e m m c o m m u m t y that n u r t u r e d him m a t h e m a t i c a l l y In 1876, he b e c a m e the first P r o f e s s o r o f M a t h e m a t i c s at The Johns H o p t a n s Umvers~ty m Baltimore, Maryland, a n d set a b o u t the p r o c e s s of a n i m a t i n g there w h a t w o u l d be America's first research-level training p r o g r a m m mathemaUcs [5, 8, pp 53-146] This n e x t p h a s e of Sylvester's car e e r found y e t anothe~ m a m f e s t a t m n of " c o m m u m t y " - - t h e m a t h e m a t i c a l res e a r c h s c h o o l - - p l a y i n g a crucial role m his hfe and w o r k F a c e d with research-level teaching for the first time, he p o u i e d h i m s e l f b a c k into the mvariant-theoretic ideas that h a d so capt ~ a t e d him in the emly 1850s and immediately made new breakthroughs of a c o m b m a t o n a l nature Hus s t u d e n t s s o o n found themselves s w e p t into their m e n t o r ' s n e w r e s e a r c h m~tmtlves through b o t h c l a s s r o o m lectures a n d what Sylvester called the "Mathematical S e m i n a r y " Ph:~sically a c o m m o n r o o m h n e d with shelves of m a t h e m a t ical b o o k s and journals, the s e m i n a r y also sern'ed as the venue for r e g u l a r p r e s e n t a t i o n s o f r e s e a r c h in p r o g r e s s Sylvester a n d his students s p o k e on their often i n c o m p l e t e l y f o r m u l a t e d ideas a n d s h a r e d a s p r a t of constrnc-
t i r e cnticusm m de~,elopmg and perfectmg n e w m a t h e m a t i c a l results a n d t h e o n e s As a p r i m e example of this sort of c o o p e i a t m n , Sylvester, togethez with hus s t u d e n t s F a b i a n Frankhn, Christine Ladd, Wllham Durfee, and others, d e ~ s e d a n d a r h c u l a t e d a socalled "constructive theory of p a r h tlons" t h r o u g h o u t the winter of 1883, they p u b h s h e d the frmts of their j o i n t labors m a m a j o r p a p e r m Sylvestes Amel~can Journal of Mathematics [17] The actrvltms within the Mathematical S e m i n a r y thus e x e m p h f y what can often b e the highly socml nature o f m a t h e m a t i c a l discovery Following seven-and-a-half amazingly p r o d u c t i v e y e a r s at Hopkins, Syl~ester left hus p o s t at the end o f De( e m b e r 1883 to t a k e the p r e s t l g m u s Savihan p r o f e s s o r s h i p o f g e o m e t r y m Oxford He h a d finally been a c c e p t e d by the O x b n d g e a c a d e m i c c o m m u n i t y that h a d b e e n c l o s e d to him professmnally before At Oxford, Sylvester tried to a n i m a t e y e t a n o t h e r r e s e a r c h school a r o u n d the n e w ideas on the theory o f dffferentml m v a r l a n t s - - o r r e c i p r o c a n t s - - t h a t he w o r k e d up for hus m a u g u l a l lecture m D e c e m b e r o f 1885 In that address, in fact, he s t a t e d his hope on thus s c o r e exphcltly When I lately had the pleasure of attendmq the new Slade P~ofebsor's ~naugu~al d~scou~se, I heard h~m promise to make hzs pupds parhc~patots ~n h~s work, by painting pwtures ~n the presence of h~s class I aspire to do mo~e than t h i s - - n o t o~dy to paint befme the members of m y class, but to ~nduce them to take the palette and brush and ~ontnbute w~th the~ own hands to the n,ork to be done upon the canvas Such u,as the plan l followed at the Johns Hopkins Untve~s~ty [18, p 297] hi a letter to Cayley dated 18 February, 1886, he seemed to gr~e an early mdlcatlon that this aspu'atlon was actually bemg reahzed, reporting that "I have a class o f 14 or 15 comprising several (5 or 6) of our college tutors to w h o m I lecture twice a week on Reclprocants ,,4 Among
2This letter is Sylvester J J 20240 in the Brougham Papers at Unwers~ty College London Archwes As always I thank University, College for permfss~on to quote from ~ts collechon 3St John s College Cambndge Sylvester Papers Box 10 4St John s College Cambndge Sylvester Papers Box 12
38
TNE MATHEMATICAL[NTELLIGENCER
the active participants were Edwin Bailey Elliott, Charles Leudesdorf, Leonard James Rogers, and James Hammond, and, just as with the published form of the work on partitions with his students at Hopkins, Sylvester (with Hammond's help) wrote up his "Lectures on the Theory of Reciprocants" crediting ideas and whole proofs to his various auditors [19]. Unfortunately, Sylvester's hope for a self-sustaining mathematical school at Oxford in the 1880s was premature. Not even his founding in 1888 of the Oxford Mathematical Society--in direct emulation of the Mathematical Seminary at Hopkins--succeeded in instilling in Oxford mathematicians a research-oriented mentality.
Marjorie Senechal cast a wide intellectual net when she articulated her conception of the InteUigencer's "Mathematical Communities" column. "Our definition of 'mathematical community' is the broadest," she explained. "We include 'schools' of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one" [14]. The life and work of James Joseph Sylvester suggest that, in each of these senses, "community" played a crucial role in the formation of a mathematical persona and the development of his mathematical ideas.
6. - - , James Joseph Sylvester. Life and Work in Letters, Oxford: Oxford University Press, to appear. 7. - - , Toward a History of NineteenthCentury Invariant Theory, in The History of Modem Mathematics, ed. David E. Rowe and John McCleary, 2 vols., Boston: Academic Press, 1989, 1 : 157-206. 8. Karen Hunger Parshall and David E. Rowe, The Emergence of the American Mathematical Research Community, 18761900: J. J. Sylvester, Felix Klein, and E. H. Moore, Providence: American Mathematical Society and London: London Mathematical Society, 1994. 9. Karen Hunger Parshall and Eugene Seneta, Building a Mathematical Reputation: The Case of J. J. Sylvester (1814-1897), The American Mathematical Monthly 104 (March 1997), 210-222. 10. Nathan Reingold and Mark Rothenberg, ed., The Papers of Joseph Henry, 7 vols., Washington, D. C.: Smithsonian Institution Press, 1972-1996, (see vol. 6). 11. Adrian C. Rice, Augustus De Morgan and the Development of University Mathematics in London in the Nineteenth Century, un-
12.
13.
14.
REFERENCES
1. H. Halle Bellot, University College London 1826-1926, London: University of London Press, Ltd., 1929. 2. Lewis S. Feuer, America's First Jewish Professor: James Joseph Sylvester at the University of Virginia, American Jewish Archives 36 (1984), 151-201. 3. Percy Alexander MacMahon, James Joseph Sylvester, Nature 55 (1897), 492-494. 4. - - , James Joseph Sylvester, Proceedings of the Royal Society of London 63 (1898), ix-xxv. 5. Karen Hunger Parshall, America's First School of Mathematical Research: James Joseph Sylvester at The Johns Hopkins University 1876-1883, Archive for History of Exact Sciences 38 (1988), 153-196.
15.
16.
17.
18.
Theory of the Singularities of Curves, Nature 33 (1886), 222-231, or Math. Papers JJS, 4:278-302. 19. m , Lectures on the Theory of Reciprocants, American Journal of Mathematics 8 (1886), 196-260; 9 (1887), 1-37, 113-
20.
161, 297-352; and 10 (1888), 1-16, or Math. Papers JJS, 4:303-513. m , On RationaI Derivation from Equations of Coexistence, That Is to Say, a New and Extended Theory of Elimination. Part I, Philosophical Magazine 15 (1839), 428-435, or Math. Papers JJS, 1:40-46.
published Ph.D. dissertation, Middlesex University, 1997. --, Mathematics in the Metropolis: A Survey of Victorian London, Historia Mathematica 23 (1996), 376-417. Adrian C. Rice, Robin J. Wilson, and J. Helen Gardner, From Student Club to National Society: The Founding of the London Mathematical Society in 1865, Historia Mathematica 22 (1995), 402-421. Marjorie Senechal, New Column on the Way, The Mathematical Intelligencer 18 (1996), 5. Hugh Stewart, Sylvester the Actuary, lecture given at New College, Oxford, 15 March, 1997. James Joseph Sylvester, The Collected Mathematical Papers of James Joseph Sylvester, ed. H. F. Baker, 4 vols., Cambridge: University Press, 1904-1912, reprint ed., New York: Chelsea Publishing Co., 1973 (abbreviated Math. Papers JJS). --, A Constructive Theory of Partitions, Arranged in Three Acts, an Interact, and an Exodion, American Journal of Mathematics 5 (1882), 251-330, or Math. Papers JJS, 4:1-81. --, Inaugural Lecture at Oxford 12 December, 1885, on the Method of Reciprocants as Containing an Exhaustive
VOLUME 20, NUMBER 3, 1998
39
The Liberal Arts Benno Artmann
n Plato's chalogue Theaetetus (145a), w r i t t e n a b o u t 370 BC, w e r e a d a b o u t the m a t h e m a t i c i a n T h e o d o r u s
I
Is Theodorus an eaToe~t zn geometry2 Of course he ~s, Socrates, ve~T] much so A n d also ~n astronomy and amthmet~c and m u s w and zn all the lzbe~al arts~ I am sure he ~s This m a y well be the first complete mentionmg of the Quadmwum Amthmet~c, Musw, Geometry, and Astronomy Presumably this canon of higher education goes b a c k to the Pythagoreans and sophists, who wrote m the years before Plato In Hellemstlc tames several other subjects were taught together wath the four above, but m the last century BC, wath the adchtzon o f Grammar, Rhetorzc, and Dzalectws, the seven hberal arts had been e s t a b h s h e d The subdlvaslon into the T m w u m (the latt e r three, m o r e elementary, t h a t ~s
"tnv3al," ones) a n d the Quadmvzum b e c a m e s t a n d a r d only m Carohngean times a r o u n d 800 AD (Man'ou Part II, Chap 8) Boetms ( a r o u n d 510 AD), "the last Roman," h a d a h e a d y classified the subjects o f the tn~Tum as the "tailing" and the ones o f the quadrlxamn as the "calculating" arts Music was considered as a p p h e d arithmetic ( h a r m o n i c s ) and a s t r o n o m y as applied g e o m e t r y The b o o k s by B o e t m s on ar~thraetlc and music (and t h e p s e u d o - B o e t i a n geometry) were very influential throughout m e d i e v a l trams D u n n g the R o m a n empire and on until a b o u t the y e a r 1000 the quadnvm m was c o n s i d e r e d less mlportant, if not ignored This changed w h e n the new restitutions o f higher e d u c a t i o n at the c a t h e d r a l s m~d m o n a s t e r i e s bec a m e e s t a b l i s h e d m the tenth c e n t u r y The decmwe p e r s o n m this p r o c e s s was G e r b e r t o f A u n l l a c ( - 9 4 0 - 1 0 0 3 ) He was f a m o u s for his k n o w l e d g e o f the anhque, Arabic, Byzantine, a n d Christian t r a d m o n s , b u t f o r e m o s t a s a
Does your hometown have any mathematwal toumst attrachons ~uch as statues, plaques, g~aves, the cafd where the famous conjecture was made, the desk where the famous zmtzals are scratched, b~rthplaces, houses, or memomals~ Have you encountered a mathemahcal szght on your travels~ If so, we mvzte qou to submit to thzs column a pwture, a descmptwn of its mathematwal s~gmficance, and e#her a map or d~rectzons so that others may follow zn your tracks
Please send all submtsstons to Mathematical Tour)st Edttor,
Dvrk Huylebrouck, Aartshertogstraat 42 8400 Oostende Belgium e-mall dtrk huylebrouck@pJngbe
From the town hall of Lemgo, Germany (Photos by B Artmann)
THE MATHEMATICALINTELLIGENCER9 1998SPRINGERVERLAGNEWYORK
Anthmebk und Geometne, a fifteenth-century broadside by Peter Wagner of Nurnberg Translation of Ar,thmetic I can count and calculate masterly and estimate With Boet,us I will begin wtth numbers Translation of Geometry I can build and well measure
The q u a d n w u m gradually lost its dominant positron m lugher e d u c a t m n when, starting from a b o u t 1100, a m o r e detailed k n o w l e d g e of the antique writrags b e c a m e wides p r e a d b e c a u s e of m o r e avatlable translations into Latin Starting from Caroh n g e a n tunes until well into the eight e e n t h century, the hberal arts w e r e a favorite subject of s c u l p t o r s and pmnters You will find s c u l p t u r e s all over w e s t e r n Europe, be it at the b a s e of the central pillar of the pulpit in the cathedral of Plsa, at the fountain in Perngm, o r m the e n t r a n c e
hall of the c a t h e d r a l of Frelburg, as well as at m a n y o t h e r c h u r c h e s and m m u s e u m s The p h o t o g r a p h s s h o w the s c u l p t u r e s of the p e r s o n i f i e d sciences a n t h m e t m , geometry, music, and ast r o n o m y from 1565 at the t o w n hall of Lemgo (some 25 km e a s t of Bmlefeld) m Germany
LITERATURE
Marrou, Henn Irenee A History of Education/n Antiquity New York Sheed and Ward 1956 (A very well wntten book w~th complete ~nformatlon about its subject ) L~ndgren, Uta Die Artes L~bera/es in AntJke und MJttelalter Munchen Instttut fur Gesch~chte der Naturw~ssenschaften (Reihe Algonsmus Heft 8) 1992 (Detatled ~nformation about the medieval t~mes whtch Marrou does not cover ) Pnestley W M
Mathematics and the L&eral
Arts Unpublished Manuscnpt
Techntsche Hochschule Darmstadt Fachbere~ch Mathemat~k SchloBgartenstr 7 D-64289 Darmstadt Germany
Therefore I will not forget Eucl,d
m a t h e m a t i c i a n He b e c a m e P o p e (and t o o k the n a m e Sylvester II) for the y e a r s 996-1003 (Is there any mathematician r e a d y to take over for the y e a r 2000 v) (See Lmdgren p 41-58 for detmls a b o u t the teachings of G e r b e r t )
Geometry, from the late twelfth-century Hortus Dehc~arum compiled by the Alsat,an abbess Herrad von Landsberg
VOLUME20 NUMBER3 1998 41
A Portrait of Brook Taylor in North Yorkshire Gunther Hammerhnl-
en you are touring m England, orth Yorkslure ts worth a ~lslt There are the austere beauty of the Yolksture Dales m the western part of the county, the North York Moors National Park m the east, and numerous places of lustorlc interest One of these places is Benmgbrough Hall, a m a n o r 8 miles northwest of York It is a rem a r k a b l e Georgzan hall, built in 1716, n o w a National Tz~st property Among other precious objects, zt c o n t a i n s a n exposition of o~ er 100 pictures o n loan from the National Portrait Gallery Of particular interest to m a t h e m a n c i a n s is pmture 81 The children of J o h n Taylor of Bffrons Park, Kent, by J o h n Closterman (1696) John Taylor was the father of Brook Taylor, his eldest son (1685-1731), whom we find seated at left m the picture together with his brothers a n d sisters B e m n g b r o u g h Hall is o p e n from 1 April to 31 October, Saturday to Wednesday D n ~ n g southwards about 100 miles from York on the A1 mto Lmcolnshzre, one passes Grantham, where y o u n g Isaac Newton attended the local gramm a r school One can stop at Woolsthorpe-by-Colstez~orth, 7 miles s o u t h of Grantham, to see the f a r m h o u s e
Woolsthorpe Manor, Newton's birthplace and fanuly home, including a pI obable d e s c e n d a n t of the famous apple tree in the orchard (The o n g m a l tree is behe~ed to have died in 1814 ) Math Inst der Un)vers)tat Therestenstr 39 D-80333 Munchen Germany
Figure 2 The apple tree at Woolsthorpe Manor
Figure 1 The Children of John Taylor of Bffrons Park, Kent, by John Closterman, 1696 Used by permission of the National Portrait Gallery
tThe editor rs sad to report that the author d)ed on 12 November 1997
42
THE MATHEMATICALJNTELLIGENCER9 1998SPRINGERVERLAGNEWYORK
Hilbert's 18th Problem and the GGtlingen Town Library Wtllt Mohnng
e w of the ~usltors to the Gottmgen town hbra~7 note the unusual tflmg, which can be seen above the mare entrance Even fewer are aware that these tries were designed by H e m n c h Heesch [1t and •ustrate the a n s w e r to a question raised b y Davad Hflbert the 18th p r o b l e m m [us fanlous lecture at the Intemahonal Congress m 1900 [2] With the design of t[us tale Heesch a d d e d a new pattern to the 17 tile patterns, wtuch are b a s e d on the wallpaper groups and wtuch go b a c k many centuries m Arab decoratzons [3,4]
F
.
/
X
/
Figure 1
Hribert asked, "is t h e r e m n-dlmens~onal e u c h d e a n s p a c e also only a firote n u m b e r of essentially different kinds of groups o f m o t t o n s wath a [compact] f u n d a m e n t a l region ~" [2] He refers to Fedorov, Schoenflms, a n d Bohn, who had given affirmative ans w e r s m two- a n d three-chmensmnal e u c h d e a n s p a c e A f u n d a m e n t a l region for a group is a d o m a m m s p a c e that c o v e r s the whole s p a c e wathout overlap ff all t r a n s f o r m a t i o n s o f the group a r e a p p h e d to ~t Obvmusly, these transformaUons g e n e r a t e a trimg of the
Z
Z
A t d m g w i t h t h e Heesch decagon Two independent translattons w i t h a related fun-
damental region are also shown
9 1998 SPRINGERVERLAGNEW'YORK VOLUME20 NUMBEP3 1998 4 3
-~--
d l m e n s m n s , I will consider only that case There Fedoro~ showed that every such group c o n t m n s two hn-
_2 _
i f
early i n d e p e n d e n t translations which generate a t r a n s l a t m n group A parallelogram s p a n n e d by these two t r a n s l a t i o n s is a poss]ble fundamental r e g m n for this group One obtains richer groups ff a fundamental regnon possesses a d d m o n a l symmetries One finds there are 5 groups without reflectmns a n d 12 with reflections, in total, the 17 wallpaper groups
[2,3,4,51
F,gure 2 The dodecagonal t,les above the Gott,ngen town I,brary
space fundamental r e g m n s can be used as tries With tlus m mmd, Hllbert asked, secondly, whether the converse ~s also true, ~ e , "Whether polyhedra also exist wluch do not appear as fundamental regions of groups of motmns, by m e a n s of wluch nevertheless by a statable juxtapos~t:on of congruent copies a complete filling up of all space is possible" He fmally asked a third quesUon, namely, what can be aclueved ff a complete trimg is not possible "How can one arrange most densely m space an mfimte n u m b e r of equal sohds of gwen form " The first q u e s t m n was a n s w e r e d affirmatively by Bmberbach m n dlmenstuns As Heesch's work refers to two
44"
THE MATHEMATICALINTELLIGENCER
The second question was answered m 1928 by Remhardt m three-dtmensmnal space He described a polyhedron such that the ma,n entrance to the whole space c a n be covered by c o n g r u e n t copras without o~ erlap a n d w h i c h is n o t a f u n d a m e n t a l reg m n of a group of m o t i o n s He also c o n j e c t u r e d that a p o l y g o n w~th thin p r o p e r t y does n o t exist in the p l a n e Such p o l y g o n s were t h e n f o u n d by Heesch m 1932 One such with 10 vertices is m Fig 1, which s h o w s h o w it c a n tile the plane Two i n d e p e n d e n t t r a n s l a t m n s are m a r k e d w h i c h transform the tflmg onto itself Two of the tiles form a f u n d a m e n t a l r e g m n of the t r a n s l a t i o n group The t r a n s f o r m a tion w i n c h t r a n s f o r m s o n e tile o n t o the o t h e r does n o t t r a n s f o r m the whole p a t t e r n onto itself, however, a n d there is n o other way of achlevmg thin S u b s e q u e n t l y m o r e s u c h tflmgs were f o u n d A c o n v e x p e n t a -
gon due to R B K e r s h n e r is s h o w n m
[4] When the G e r m a n ceramms company Vrileroy & B o c h learned of this discovery, it offered to produce a sample free of charge A statable tile shape with 12 corners was designed by Heesch The t]les were bmlt mto the newly erected extension of the Gottmgen Stadthaus, which n o w houses the Gottmgen town hbrary, where they can still be seen (Fig 2) Hribert's third question is still u n a n swered The tourist might be interested m a short biographical sketch (extracted from [1]) H Heesch was born m 1906 m Kml He studied the vmlm and mathematlcs m M u n c h e n and got hm doctor's degree m mathematics 1929 m Zurich In 1930 he went with H Weyl to Gottmgen He lost hm p o s m o n there m 1935 Durmg the next 20 years he had various, mostly bnef, positrons In 1961 he finally got a permanent p o s m o n m Hannover From 1947 on he worked on the four-color problem and actueved several very unportant results He realized as early as 1964 that computers might be needed, but he was not very successful m rmsmg funds, and could not get computer support such as went mto the proof m 1976 by Appel and Haken m Urbana. Heesch died m 1995 m Hannover Unfortunately he never attamed the acadenuc standmg one would have expected for such a gifted mathematician
REFERENCES
1 H G B~galke,Helnnch Heesch B~rkhauser Verlag 1988 2 F E Browder (editor), Mathematical Developments anslng from HiIbert Problems Proc Symp Pure Math XXVIII American Mathematical Society, 1976 3 I Stewart Concepts of Modern Mathematics, Penguin 1975 4 G E Martin Transformation Geometry, Spnnger-Verlag, 1982 5 V Dubrovsky, Ornamental groups, Quantum 2 (1991), 32-35
Max-Planck-lnstltut fur Stromungsforschung Bunsenstr 10 D-37073 Gottlngen Germany
In the Shadow
of Kepler and Galileo Gerard Buskes
Vzsltmg Prague, a city filled to the brim with history from m a n y ages, one e x p e c t s m o r e than a few m a t h e m a t i c a l remamders Antm~patmg relics of one of my heroes, Bolzano, I was v e r y h a p p y to see a beautiful lecture r o o m m the M a t h e m a t m s Departm e n t of Charles University dedicated to Bolzano But tlus n o t e from a tourist ~s n o t a b o u t h~m, F,gure 1 The Jew,sh cemetery ,n Prague n o r - - a t least n o t p r i m a r i l y - - a b o u t Tycho Brahe o r J o h a n n e s Kepler Indeed, Tycho Brahe ran his famous mark Only a few y e a r s later, movable a s t r o n o m m a l l a b o r a t o r y m Delmedlgo, still a student, h a d the opPrague b e t w e e n 1599 and 1601 His p o r t u m t y to s c a n the s k y through sepulchral m o n u m e n t c a n n o t be Galileo's n e w 1609 t e l e s c o p e m P a d u a m~ssed w h e n one ws~ts the Teynklrche Gahleo's first P a d u a lecture d a t e s from m the h e a r t of old Prague Tycho 1592 m whmh he w a r m l y p r a i s e d the Brahe's student, Kepler, m a d e tremenm s t r u m e n t a r l u m o f Tycho Brahe's laboratory This m a k e s Gans a n d dous a d v a n c e s m that same city for several y e a r s after Brahe's untunely Delmedlgo a s s o c m t e s o f a group of asdeath P o s t c a r d s with images of Kepler t r o n o m y rebels at the forefront of the or Tycho Brahe a r e c o m m o n tourist Copermcan Revolutmn fare m Prague What I r e p o r t h e r e is on The r e a c t m n of the Cathohc Church n a m e s of a different allure to this revolution is well k n o w n Lesser Not far from the magmficent k n o w n s e e m s to be the p o s i t i o n of the Teynklrche is the f a m o u s Jewish cemeJ e w s to the C o p e r m c a n w o r l d view In tery Among the graves are t h o s e of the m t e r e s t m g b o o k [1], N e h e r writes Dawd Gans (1541-1613) and J o s e p h of an instance w h e r e Tycho Brahe stud S o l o m o n Delmechgo (1591-1655) The to tus assmtant David Gans "Your Jewish m u s e u m at the c e m e t e r y hsts sages were w r o n g to s u b m i t to the nonthem as m a t h e m a t m m n s and RenmsJ e w i s h scholars They a s s e n t e d to a he, sance scholars W~th permtssmn of their for the truth lay w~th the Jewish s a g e s " Archives I have used some illustrations, David Gans strongly p r o p a g a t e d the along w~th some photographs that I took zdea of a long e a r h e r history of Jewish myself Who were these men whose astronomy, c o n n e c t i n g Pythagoras to names are not recorded m the EncycloCopernicus pedia Br~tannwa o r the Encyclopedia Interestingly, on G a n s ' s grave w e find the Star of Dawd, of S~2nt~fw Bwgraa first on the graves m phy "~ Their life s t o n e s the J e w i s h c e m e t e r y can be found m the It w a s h~s chome, not Encyclopedia Judawa j u s t for the n a m e One has to k e e p m David, b u t also as a m m d that m t h o s e s y m b o l o f the n e w asdays, a s t r o n o m y w a s t r o n o m y b a s e d on the s o m e t i m e s equated to a n c m n t G r e e k tradtm a t h e m a t i c s Kepler, taon o f geometry Gans for instance, was g~ven b e h e v e d zt was E u c h d the title of I m p e r i a l w h o h a d p r o v i d e d the Mathemat~cmn b y the l a d d e r to the heavens E m p e r o r Rudolf II (see [1]) That s y m b o l Dawd Gans, m Prague, b e c a m e an e m b l e m of assisted m the magnffthe J e w i s h p e o p l e as a ment laboratory that whole only much later, Tycho Brahe b r o u g h t m the n m e t e e n t h cenw~th hun from DenFigure 2 The grave of Delmed,go tury The httle bird-
V
9 1998 SPRINGER VER~J~,GNEWYORK VOLUME 20 NUMBER 3 1998 4 5
to c o m p o s e with a wealth of observational d a t a from Tycho B r a h e - - t h e socalled R u d o l p h m e Tables Notice that the Holy R o m a n E m p e r o r R u d o l f II died one y e a r before Gans d]d Rudolf, by s o m e d e s c n b e d as a n e o - p l a t o m s t mystic, had supported Tycho Brahe and Kepler, and h a d - - w i t h regard to Jews--b e e n t h e b e n e v o l e n t e x c e p t m n in a sequence o f rulers
F,gure 3 The grave of Gans
like figurme a b o v e the s t a r depicts a goose, as a r e m i n d e r that in G e r m a n gans m e a n s goose David Gans also was a cartographer and historian One of tus contributions s e e m s to have been a translation for Tycho Brahe of the so-called Alphonsme Tables from H e b r e w to G e r m a n (See h o w e v e r pages 178 a n d 179 m [1] ) T h e s e tables were m u c h o u t d a t e d a r o u n d 1600, which c o m p e l l e d Kepler
46
THE MATHEMATICALINTELLIGENCER
Delmedlgo, w h o spent most of hm hfe travelling a r o u n d Europe, Afrma, and Asta, had the e n c y c l o p e d i c knowledge of a R e n m s s a n c e s c h o l a r a n d a desire for o r g a m z a t l o n of science bey o n d geographic b o u n d a r i e s He pubhshed a b o u t the n e w vmws on the world of Galileo m A m s t e r d a m several years before G a h l e o ' s trial He acqmred s o m e f a m e as a physmlan He also u s e d s o m e o f the very first logarithmic t a b l e s for his calculations Their lives as d e s c r i b e d m [1] a n d the Encyclopedia Judawa, gwe Gans and Delmedlgo a role m the s h a d o w of the great r e v o l u t i o n a r i e s of their t i m e But while the graves of the f a m o u s have b e e n p h o t o g r a p h e d so often, w e m a y p a u s e to r e m e m b e r others REFERENCES
[1] Andre Neher Jewish Thought and the Scientific Revolution of the Sixteenth Century Oxford University Press 1986
Figure 4 Joseph Solomon Delmed,go
Department of Mathematics Unwers~ty of M~sslss~pp~ Unwerslty MS 38677 USA
to c o m p o s e with a wealth of observational d a t a from Tycho B r a h e - - t h e socalled R u d o l p h m e Tables Notice that the Holy R o m a n E m p e r o r R u d o l f II died one y e a r before Gans d]d Rudolf, by s o m e d e s c n b e d as a n e o - p l a t o m s t mystic, had supported Tycho Brahe and Kepler, and h a d - - w i t h regard to Jews--b e e n t h e b e n e v o l e n t e x c e p t m n in a sequence o f rulers
F,gure 3 The grave of Gans
like figurme a b o v e the s t a r depicts a goose, as a r e m i n d e r that in G e r m a n gans m e a n s goose David Gans also was a cartographer and historian One of tus contributions s e e m s to have been a translation for Tycho Brahe of the so-called Alphonsme Tables from H e b r e w to G e r m a n (See h o w e v e r pages 178 a n d 179 m [1] ) T h e s e tables were m u c h o u t d a t e d a r o u n d 1600, which c o m p e l l e d Kepler
46
THE MATHEMATICALINTELLIGENCER
Delmedlgo, w h o spent most of hm hfe travelling a r o u n d Europe, Afrma, and Asta, had the e n c y c l o p e d i c knowledge of a R e n m s s a n c e s c h o l a r a n d a desire for o r g a m z a t l o n of science bey o n d geographic b o u n d a r i e s He pubhshed a b o u t the n e w vmws on the world of Galileo m A m s t e r d a m several years before G a h l e o ' s trial He acqmred s o m e f a m e as a physmlan He also u s e d s o m e o f the very first logarithmic t a b l e s for his calculations Their lives as d e s c r i b e d m [1] a n d the Encyclopedia Judawa, gwe Gans and Delmedlgo a role m the s h a d o w of the great r e v o l u t i o n a r i e s of their t i m e But while the graves of the f a m o u s have b e e n p h o t o g r a p h e d so often, w e m a y p a u s e to r e m e m b e r others REFERENCES
[1] Andre Neher Jewish Thought and the Scientific Revolution of the Sixteenth Century Oxford University Press 1986
Figure 4 Joseph Solomon Delmed,go
Department of Mathematics Unwers~ty of M~sslss~pp~ Unwerslty MS 38677 USA
C. MUSES
De Morgan's Ramanujan: An Incident in Recovering OUF Endangered Cultural Memory of Mathematics
~
ur epoch, often called the "~nformatwn" age, could perhaps wzth even more 13ustwe be called the age of hbrary glut much chaff to go through for httle wheat And our threshzng processes, whereby pmmary (more often secondary or worse) sources are turned znto CD-ROMs or such, too often mzss vztal
c o n n e c t m n s - - w l t h the result that cultural memory is lost These paragraphs relate one stnkang example Maybe we will find a better way The prominent mathematlcmn Augustus De Morgan (t806-t871), still renowned today for his work m symbohc logm, played a lesser Hardy to his own Ramanujan called Ramchundra The story started an 1850 when De Morgan, hxmself born m India's Madras Presidency, received from h~s friend John Dnnkwater-Bethune (who dmd prematurely a year later) a work on m a ~ m a and minima by a 29-yearold self-taught mathematician, Ramchundra, then teaching science m Delh~ College Ramchundra's book had been pubhshed at Calcutta m 1850 at his own expense when h~s earnings were very meager Subsequently the author almost lost his hfe m the 1857 Indmn rebelhon agmnst then-xmperlahst England, but fortunately survived to see his book pubhshed w]th official approbatmn m London m 1859 F~gure 1 reproduces ~ts tlfie page The present writer was fortunate to be able to purchase the copy of this now rare volume that was presented to Lord Lyndhurst by the British Secretary of State for Indm De Morgan pubhshed the work w~th an mtroductaon contammg otherwase unobtainable details of Ramchundra's life
De Morgan recounts that when he presented Ramchundra's story there was u n a n i m o u s appreczatwn of Ramchundra's serwces to h~s country, and adm~sszon of the desirableness of encouragzng h~s efforts, wzth a request that I would p o i n t out how to bmng Ramchundra under the notwe of sczentzfic m e n ~n Europe In m y reply (March 18, 1858), I ~ecommended the c, rculatwn of the work ~n Europe De Morgan then well notes that m India the~ e stzll exzsts a body of hte~ ature and science w h w h m~ght well be the nucleus of a new czvd~zat~on, though every trace of Christian and Mohamedan c~whzatzon were blotted out of existence There exzsts ~n India, under c~rcumstances which p~ ove a very h~gh ant~quzty, a philosophical language [Sanskrtt] w h w h zs one of the wonders of the world, and w h w h ts a near collateral of the Greek, ~f not Its parent f o r m From those who wrote ~n th~s language we demve our s y s t e m of artthmet~c, and the algebra which ~s the most powerf u l ~ n s t ~ m e n t of modern analys~s In thzs language we f i n d a s y s t e m o f logw and of m e t a p h y s w s an astronomy worthy of comparison w~th that of Greece ~n
9 1998 SPRINGER VERLAG NEW YORK VOLUME 20 NUMBER 3 1998
47
TREATISE ON PROBLEMS Ot~
MAXIMA AND MINIMA, SOLVED BY ALGEBRA. BY RAMCHUNDRA, LATE TEACHER O~ SCIENCE~ DELHI COLLEGE
R E P R I N T E D B Y ORDEI~ O P T H E H O N O U R A B L E C O U R T OF D I R E C T O I ~ OE T H E EAST INDIA COMPA~IT F O R CIRCULATION Ilq E U R O P E & ~ D IN INDIA~ IN A C K N O W L E D G M E N T OF T H E M E R I T OF TIIE A E T H O R ~ A N D IN TESTIMONY OF T H E SENSE E N T E R T / ~ N E D OF TIIE I M P O R T A N C E OF I N D E P E N D E N T SPECULATION AS ILN INSTRUMENT
OP NATIONAL
PROGRESS IN INDIA
E t n ~ t~r ~t~mn~r~mc~ of
AUGUSTUS DE MORGAN, F R A , S of PRO~KflBOR
TRIlSrITT C O L L I O I .
FOPS
CAMIII~IDOI I
OP MATHIJlATICS I ~ LrNltlltlJ|TY COLLIOE. LONDON
LONDON WM H ALLEN & CO 7, LEADENHALL STREET 185q F,gure 1 Facs,m,le of the t,tle page of De Morgan's edit,on of Ramchundra's book
~ts best days, above comparison, P t o l e m y ' s Syntaxm be r e m o v e d
~f s o m e books o f
He quotes m support from S~r John Herschel The B r a h m a & d d h a n t a , the w o r k o f B~ahmagupta, an I n d m n a s t r o n o m e r at the beg~nn+ng o f the seventh century, contains a general method f o r the r e s o l u t w n o f zndeterm~nate p r o b l e m s o f the second degree, an ~nvest~gatwn w h w h actually baffled the s k d l o f e v e r y m o d e r n a n a l y s t tzU the tzme o f Lagrange's solution, not exceptzng the all-~nvent~ve E u l e r h ~ m s e l f
De Morgan contmues, R a m c h u n d r a ' s p r o b l e m - - a n d I th~nk +t ought to go by that n a m e , f o r I c a n n o t f i n d that zt w a s ever c u r r e n t as an exercise o f ~ngenuzty ~n E u r o p e - - ~ s to f i n d the value o f a vartable wh+ch wall m a k e an algebrawal f u n c t z o n a m a x i m u m or a m~n~mum, u n d e r the followzng c o n d z t w n s Not only +s the d~ffetentml calculus to be excluded, but even that g e r m o f ~t whzch, as
48
THE MATHEMATICAL INTELLIGENCER
g~ven by F e r m a t ~n h+s t r e a t m e n t o f th~s very p~oblem, m a d e s o m e th~nk that he w a s ent+tled to clazm the ~nventton The values o f ~ x and o f c~(x + h) are not to be compared, and no process ~s to be allowed w h w h + m m e d m t e l y po+nts out the r e l a t m n o f gox to the der~ved f u n c h o n c~'x A m a t h e m a t w ~ a n to w h o m I stated the cond+twned problem m a d e ~t, very naturally, hzs f i r s t remark, that he could not see h o w on earth I w a s to f i n d out w h e n ~t would be bzggest, ~f I would not let ~t g r o w The m a t h e m a t ~ c m n wall at last see that the q u e s t m n resolves ~tself ~nto the f o l l o w ~ n g - - R e q u z r e d a constant, ~, such that C x - r shall have a pa~r o f equal roots, w+thout a s s u m i n g the development o f r + h), o~ a n y o f +ts consequences R a m c h u n d r a , the author o f th~s wink, has transmztted to m e s o m e notes o f h~s o w n l~fe, f r o m w h w h I collect as f o l l o w s He w a s born ~n 1821, at Paneeput, about f i f t y mde~ f r o m Delh+ H~s f a t h e l , S o o n d u r Lall, w a s a H~ndoo Kaeth, and a n a t i v e o f Delhz, and w a s there e m p l o y e d unde~ the collector o f the revenue H e d~ed at Delh~ zn 1831-32, leaving a w t d o w Ovho s t d l surv+ves) a n d s z x son~s A f t e r s o m e educatzon ~n p m vate schools, R a m c h u n d r a entered the English G o v e r n m e n t school at Delht, to e v e r y pupal o f w h w h two rupees a m o n t h were gtven, a n d a scholarship o f f i v e rupees a m o n t h to all ~n the f i r s t and second classes I n thzs school he r e m a z n e d s ~ years It does not appea~ that a n y part+cula~ a t t e n t w n w a s pa~d to m a t h e m a t w s zn thzs school, but, shortly before leaving ~t, a taste f m that sczence developed ~tself zn R a m c h u n d r a , who s t u & e d at h o m e w z t h such books as he could p r o c u r e Afte~ l e a m n g school, he obtained emp l o y m e n t as a w r z t e r f o r two or three years In 1841, changes took place ~n the e d u c a t m n a l d e p a r t m e n t o f the Bengal preszdency, the school w a s f o r m e d +nto a college, a n d R a m c h u n d r a obtained, by compet~twn, a s e n w r scholarship, wzth t h i r t y rupees a m o n t h In 1844, he w a s a p p m n t e d teacher o f E u r o p e a n sczence ~n the Ortental d e p a r t m e n t o f the college, through the reed+urn o f the vernacular, w z t h f i f t y rupees a m o n t h add~tmnal
In Ramchundra's own words, as commumcated to De Morgan, tus potation had nnproved so much that In J a n u a r y , 1858, I w a s a p p m n t e d as nat+ve head m a s t e r zn the Thomason C+wl E n g , n e e n n g College at Roorkee, on 2 5 0 rupees a month, w h z c h s z t u a t w n I held f o r e~ght m o n t h s , and ,n the beg+nnzng o f the p r e s e n t month, September, 1858, I w a s a p p m n ted as head m a s ter o f the school (not a college) w h w h zs bmng orgarazed at Delh~
Ramchundra's preface, dated 16 February 1850 at Delhi, starts (emphasm ours) For the last fou~ or f i v e yea~s I w a s devoted to s o l m n g almost all p r o b l e m s o f m a x i m a a n d m ~ m m a by the pr~nc+ples o f Algebra, and not by those of the
Dffferentml Calculus
He shows great ingenuity throughout, and, without using the mf'mlteslmal calculus, successfully solves problems like finding the form of the rectangular paralleloplpedon of mmnnal surface, or the greatest parabolic segment in a given isosceles triangle, or the altitude of the greatest cylmder m s c n b a b l e m a given sphere (or cone), the greatest parallelopipedon mscnbable m a given elhpsold, and the list goes on in astonishing nclmess One of the most mterestmg of the problems he proposed for hLmself was confirming mathematically that the bees have learned to use a nummum of wax (and hence of labour) in constructing caps of the hexagonal-prismatic cells of their hives No mathematician before or since Ramchundra resolved this question without using some form of the mfimteslraal calculus The post-Ptolemmc
Alexandrian geometer Pappus (fl ca 285 B C E ), In the fifth book of his SynagSg~ (hterally "collection", the root of the later word "synagogue" as "assembly"), discusses the honey-cell problem, observing that the bees in theu natural wisdom realized that a hexagon not only tesselates the plane but will hold more honey for the same expenchtures of beeswax than the tnangular or square prisms Now follows Ramchundra's remarkable and purely algebraic resolution of this ancient question on pages 74-76 of De Morgan's edition, accompanied by Ramchundra's diagram (See box above ) Ramchundra then adds this interesting commentary
Th~s ~s the celebrated problem of the f o r m of the cells of bees Maraldz was the first who measured the an-
VOLUME 20 NUMBER 3 1998
49
gles of the faces of the terminating solzd angle, and he f o u n d them to be 109 ~ 28' and 70 ~ 32' ~espect~vely It occurred to Reaumu~ that thzs mtght be the f o r m
50
THE MATHEMATICAL INTELLIGENCER
which, f o r the same sohd content, g~ves the m~n~mum of surface, and he ~equested Kon~g to e~am~ne the question mathematwally That Geometer confirmed the
con3ectule, the ~esult of h~s calculatwns agreeing w~th Mamldl's measurements w i t h i n 2' Maclaunn and S Hu IlheL by d~ffe~ ent methods, vertfled the p~ ecedt ng ~esult, excephng that they showed that the d~fference of 2' was m w n g to an erto~ in the calculahons of Komg, and not to a mistake on the p a l t of the bees The reference IS to the first mathematical naturahst, Rene Antome Ferchault de Rdatmlur (1683-1757), who was elected to the F r e n c h Academy when only 24 years old He always tested for htmself generally received notions and thus demonstrated experimentally that the tensile strength of a rope is less than the sum of the strength of its separate strands He also s h o w e d that the old folk behef, n & c n l e d m his tnne, that crustacea could regenerate lost hmbs, was actually correct I-hs observations on bees occur m his nchly illustrated SLXvolume w o r k on msects [Mdmmres pore Se~,,r a l'Hlstooe des Insectes, Amsterdam, 1734-1742] m which, among many other things, he also p r o v e d - - a g a m e x p e r i m e n t a l l y - - t h a t corals are animals and not plants ha many ways he anticipated D'Arcy W Thompson's (hi G~owth and Form, wluch was also anticipated m even greater detail m the mag~stenal three-volume w o r k Destgn tn Nature b y the physlclan-naturahst J a m e s Bell Pettlgrew, pubhshed m London m 1908, and strangely uncre&ted by D'Arcy T h o m p s o n R a m c h u n d r a ' s solution o f the beehive question found the s t r u c t u r a l l y i m p o r t a n t angle 109 ~ 28' 16 39", not only the d~hedral o f the octagon b u t also t h e larger o f the t w o face angles of a rhombic d o d e c a h e d r o n , thus determining that the dihedral angle for that r e m a r k a b l e solid is 120 ~ The s a m e d i h e d r a l angle s e p a r a t e s any neighbourlng p m r of the 16 "surface"-cells of a 4-D octahedron, as well as any such p a r o f surface cells of the unique 24-celled regular 4-D h y p e r s o h d The t n h e d r a l cap o f a b e e h i v e ' s h o n e y cell t h u s t u r n s out to be the s h a p e of any three adjoining faces of a rhomblc d o d e c a h e d r o n That interesting form is also the 3-D "shadow" of a four-dimensional c u b e It is no w o n d e r that m ancient t i m e s b e e s were felt to be s a c r e d and also to slgmfy the p r i e s t e s s e s of the all-wise Great G o d d e s s m h e r various n a m e s of Artemis, Rhea, D e m e t e r ( = Thea Meter, Dea Mater, Dlwne Mother a n d M o t h e r of all dlvmttms) Again on page 184, he p o s e s the p r o b l e m of fmdmg x such that the p r o d u c t (rex + n)(ny + m ) is maximal, given am~b ny C He solves it without calculus m eight lmes On page 152 he tackles a p r o b l e m m o s t t e a c h e r s t o d a y w o u l d n o t be able to solve without calculus to m s c n b e the greatest p a r a l l e l o p t p e d within a given ellipsoid Agmn, vothout using calculus, R a m c h u n d r a s h o w s on p a g e s 60-61 of De Morgan's edition "that the altitude of the smallest cone that can be c i r c u m s c r i b e d a b o u t a given s p h e l e is equal to twice the d i a m e t e r of the s p h e r e " One o f the m o s t l n t n c a t e of his qnestions a p p e a r s on page 111 to p a g e 114, w h e r e he s e e k s (and finds) the angnlar "position of the p l a n e t Venus with r e s p e c t o f the Earth, w h e n h e r light is the g r e a t e s t " And one of the m o s t interesting a p p e a r s on pages 42 a n d 43 R a m c h u n d i a ' s s t a t e m e n t a n d solution, t o g e t h e r with his diagram (cor-
rectmg a misprint in the o n g m a l ) a p p e a r in the b o x on p 50 R a m c h u n d r a ends his b o o k on page 185 by saying he had m o r e to say, "but b e m g afraid of enlarging the w o r k too much, I c o n c l u d e t h e s e s h e e t s " We m a y well wish the a u t h o r h a d n o t so politely "feared" here, for he p l a c e d a j e w e l m our h a n d s Several years after finding Ramchundra, I c a m e across Ivan Nlven's well written-book M a x t m a and M~mma W~thout Calculus (Mathematical Association of Amencan, 1981) Though Niven addresses the s a m e ideas and even s o m e of the s a m e p r o b l e m s as R a m c h u n d r a a d d r e s s e d over a century earlmr, he has completely o v e r l o o k e d his Indmn predecessor, despite the lllustnous De Morgan's efforts to m a k e this r e m a r k a b l e Hmdu algebrmst k n o w n R a m c h u n d r a does not a p p e a r m Niven's text, mdex, o r blbhography It is clear that our twentieth-century "Information Highway" n e e d s a d d i t i o n s to its breadth, depth, and foundations to m a k e it m o r e than a cover-up for o u r eroding cultural m e m o r y In the m e a n t i m e j o u r n a l s hke Hlstorm Mathematwa and The Mathemahcal lntelhgencer come to o u r literally anti-lethal r e s c u e l~th# is the G r e e k r o o t for forgetting, which language's w i s d o m parallels with dying
=
VOLUME 20 NUMBER 3 1998
51
m,'~=-~ n ~ - ~ _,9 ,_[ . - -
Jeremy
Gray,
Hilbert on
Kinetic Theory and Radiation Theory (1912-1914) Leo Corry
Edrtor
I
Introduction and General Background In 1912 David H]lbert p u b l i s h e d his first article dealmg with physical issues, the foundations of the k m e t m t h e o r y o f g a s e s Over the c o m i n g y e a r s he w o u l d p u b h s h additional w o r k s on r a d m t i o n t h e o r y and on the g e n e r a l t h e o r y o f relativity Indeed, H d b e r t ' s interest m physics was n e i t h e r sporadzc n o r superficial, it was an o r g a m c c o m p o n e n t o f h~s overall scmntific worldv]ew 1 His interest in k m e t m theory a n d r a d l a t m n theory w a s only a small, often neglected, part o f a m o r e general attitude The p r e s e n t artmle is a brief a c c o u n t of this p o r t m n o f H d b e r t ' s scientific w o r k The s t r o n g c o n n e c t m n b e t w e e n Hflbert's p h y s i c s and mathematzcs is manifest, in part]cnlar, m his a x i o m a t i c a p p r o a c h Hflbert's axJomaUc c o n c e p tion a r o s e m c o n n e c t i o n with foundatmnal questions of projective geometry, b e g m m n g m 1894 2 But at the s a m e txme, he w a s curious a b o u t foundations o f mechanics, and k n e w the recent r e l e v a n t w o r k by H e m n c h Hertz in this d o m a m , this w o r k p r o v i d e d adr i m p e t u s to his p u r s u i t o f a systematic, a x m m a t t c analysis o f geo m e t r y F r o m the begmmng, Hllbert thought t h a t the m e t h o d should b e app h e d equally to physical t h e o r i e s The axlomatxc m e t h o d was n e v e r for Hllbert a starting p o i n t for res e a r c h Rather, it was a tool to e n h a n c e u n d e r s t a n d i n g of ex~stmg, e l a b o r a t e theories This c o n c e p t m n Is reflected in the followmg passage, t a k e n f r o m the lecture n o t e s of a course taught in 1905 The edtfice o f science ~s not razsed Izke a dwellzng, zn whzch t h e f o u n d a t z o n s
Column Editor's address Faculty of Mathemabcs, The Open Unrversrty Mrlton Keynes MK7 6AA, England
52
THE MATHEMATICALINTELLIGENCER9 1998 s
are f i r s t f i r m l y la~d and only then does one proceed to construct and to enlarge the ~ooms Science prefers to secu~ e as soon as possible comfortable spaces to w a n d e r around, and o n l y subsequently, w h e n s~gns appear here and there that the loose f o u n d a t w n s are not able to s u s t a i n the e x p a n s w n o f the rooms, does ~t seek to s u p p o r t and f o r t i f y t h e m Th~s ~s not a weakness, but r a t h e r the r~ght and healthy path o f d e v e l o p m e n t (Hllbert 1905, 102) 3 Hflbert was especially c o n c e r n e d about a sltuatmn he considered to be typxcal of the d e v e l o p m e n t of physical theories, m which n e w hypotheses are mtroduced to explain newly d~scovered phenomena, without properly checkmg whether an a d d e d hypothesis is consistent with the e m s t m g t h e o n e s The k m d of axmmatlc analysis he pursued app e a r e d to him as a p r o p e r tool to deal with ttus situation In lus well-known c o r r e s p o n d e n c e with Gottlob Frege, munechately following the pubhcaUon of lus Grundlagen der Geometme, Hflbert rinsed this p o m t very exphcltly A f t e r a concept has been f i x e d completely and unequivocally, ~t ts ~n m y w e w completely zlhc~t and ~llogwal to add an a x w m - - - a m~stake m a d e very f r e q u e n t l y , especially by p h y s w ~ s t s B y setting u p one n e w a x w m after another zn the course o f thezr ~nvest~gat w n s , w~thout c o n f r o n t i n g t h e m w~th the a s s u m p t w n s they m a d e earlier, and w z t h o u t s h o w z n g that they do not c o n t r a d w t a f a c t that foUows f r o m the a x w m s they set up earher, p h y s w ~ s t s often allow sheer n o n s e n s e to a p p e a r ~n their ~nvest~gatwns One o f the ma~n sources o f m~stakes and m~s-
1See Corry 1997 1998 2See Toepel11986 3Unless otherwuse stated translatuons from the German ongtnaJ are mtne Quotattons from Htlbert s lecture notes appear here by permtss~on of the hbrary of the Mathemahsches Insbtut Untversttat Gott~ngen I thank the hbraruan Mr Matheess for hts k~nd cooperatuon
VERLAGNEWYORK
u n d e r s t a n d i n g s ~n modern p h y s w a l ~nvest~gatwns is p~ectsely the procedure o f s e t t i n g up an aa~om, appeal~ng to ~ts truth, and ~nfe~9"~ng f l o r a th~s that ~t ~s compatible w~th the def i n e d concepts One o f the ma~n p u r poses o f m y Festschr~ft w a s to avoid th~s m~stake 4 By the turn of the century, the kinetic t h e o r y o f gases had a short, b u t a l r e a d y p a r t i c u l a r l y convoluted, history, that s e e m e d to furmsh an ideal e x a m p l e o f the situation d e s c r i b e d here by Hllbert In fact, from its inception, the t h e o r y gave u s e to h e a t e d c o n t r o v e r s i e s a r o u n d several issues, such as the so-called reversibility and r e c u r r e n c e p a r a d o x e s , the ergodlc hypothesis, and the atomistlc point o f vlew
5
In 1905 Htlbert taught a course m Gottmgen, on the axmmatm method and its apphcat:ons A considerable portion of the course was d e & c a t e d to the axiomatLzaUon of physics, and the notes of t[us course provide the earhest comprehensive account of Hflbert's picture of t[us subject 6 The kmetic theory appears m t[us course as a particular a p p h c a t m n of the calculus of probab~hUes, alongside the theory of compensations of errors (Ausglezchungs~eehnung), and msurance m a t h e m a t m s Hflbert a c c e p t e d w~thout reservations the controversial atommtm assumptions underlymg the classical a p p r o a c h to t[us theory as developed by Ludw:g Boltzmann He did stress, however, the problematic use of probabfllsUc arguments m physical theorles Even ff we know the exact position and velocities of the particles of a gas---I-hlbert e x p l a m e d - - l t ls :mpossible m practice to mtegrate all the differential equations d e s c n b m g the m o t i o n s of these p a r t i c l e s and their interactions We k n o w n o t h m g of the m o t i o n of m(hwdual particles, but rather c o n s i d e r only the average m a g n i t u d e s that constitute the s u b j e c t of the probabilistlc, kmetic t h e o r y o f gases In an oblique
reference to Boltzmann's replies to the objections r m s e d against his theory, Hllbert s t a t e d that the c o m b i n e d use of probabilities and infinitesimal calculus in this c o n t e x t was a very orlgmal contnbuUon o f m a t h e m a t i c s , which m a y lead to deep and interesting consequences, but w[uch at this stage had in no sense b e e n fully justified 7 Between 1898 and 1906, Hflbert lectured several times m Gottmgen on mechamcs, potential theory, and continuum m e c h a m c s Begmnmg m 1907, Hllbert's friend and colleague Hermann Minkowski p u b h s h e d a series of now famous w o r k s on the relatiwty pnnclple I-hlbert and Mmkowsk~ led two seminars on these issues m Gottmgen, m 1905 and m 1907, a n d it is evident that I-hlbert was closely mvol~ed in Mmknowski's current w o r k In fact, Mmkowski's w o r k is b e s t u n d e r s t o o d agmnst the b a c k g r o u n d of Hllbert's program for the ax:omatlzaUon of physics s Between 1903 and 1912, Hflbert's mathematical efforts concentrated on lmear integral equations At the same time, however, after Mmkowski's death, Hflbert returned to teach courses on physical issues He taught stat:sUcal mechamcs for the first Ume m the wmter semester of 1910-11 In December of 1911 he p r e s e n t e d to the Gottmgen Mathematical Society (GMG) an overvmw of [us recent mvestigations on the kmeUc theory of gases, w[uch were soon to be pubhshed 9 Begmmng in 1912, Hllbert p e r m a nently enrolled an assistant for physics, w h o was c o m m i s s i o n e d with the t a s k of keeping him a b r e a s t of current d e v e l o p m e n t s Paul P Ewald had recently finished his dissertation in Munich, and he w a s the first to hold this position Hllbert's involvement with physical i s s u e s b e c a m e increasingly b r o a d e r and deeper, and he devoted m u c h effort to rethmklng from a wider p e r s p e c t i v e the foundations of this discipline By 1910 Hilbert's app r o a c h had b e c o m e d o m i n a t e d by the
~uew that all p h y s i c a l p h e n o m e n a could be r e d u c e d to m e c h a n i c s This w e w was clearly manifest in the c o u r s e s he taught a n d in the w o r k s he p u b l i s h e d on kinetic t h e o r y a n d radiation t h e o r y In 1913, however, although his r e d u c t l o m s t i c inclinations did not change, he m o v e d from the m e c h a m s tic to the e l e c t r o m a g n e t i c p o m t of wew Electromagnetic reductlomsm d o m i n a t e d his a t t e m p t s to formulate a unified foundation for all of physics, b e g m n m g in 1915 Hilbert's Lectures on Kinetic Theory and Radiation Theory In the w m t e r of 1911-12 Hllbert taught a c o u r s e specifically d e v o t e d to the klneUc t h e o r y of gases for the first tnne In the introduction to the course, he d i s c u s s e d three p o s s i b l e w a y s of s t u d y m g different p h y s i c a l theories like hydrodynamics, electricity, etc F~rst, he m e n t i o n e d the "phenomenologmal perspective," often a p p l i e d to s t u d y the m e c h a n i c s of c o n t m u a Under flus perspective, the whole of physics is dlwded mto various chapters thermodynan~cs, electrodynanucs, optics, etc These can be a p p r o a c h e d usmg different assumptions, peculiar to each of them, and d e n v m g from these assumptions different mathematical consequences The m a m mathematical tool used m t[us a p p r o a c h is the theory o f partial differential equations A m u c h d e e p e r u n d e r s t a n d m g of the physical p h e n o m e n a involved In e a c h of these d o m a m s is r e a c h e d - Hilbert told [us s t u d e n t s - - w h e n the atommtm t h e o r y is i n v o k e d In t[us case, one a t t e m p t s to p u t f o r w a r d a s y s t e m of a m o m s which is valid for the w h o l e o f physics, and w h i c h can explain all physical p h e n o m e n a from a single, unified p o n t o f v m w The nmthe m a t i c a l m e t h o d s called for are obxlously quite different from those a d o p t e d in the p h e n o m e n o l o g i c a l pers p e c t l x e They can be subsumed, in general, u n d e r the t h e o r y of probabdl-
4Quoted ~n Gabriel et al (eds) 1980 40 5Two classical detailed accounts of the development of the ktnet~c theory of gases and the conceptual problems ~mplped by ~t (particularly dunng the late n~neteentr~ century) can be consulted Brush 1976 and Klein 1970 (esp 95-140) 6A detailed account of the contents of thqs course appears ~n Corry 1997 7HiIbert 1905 178-180 8See Corry 1997a ~See the announcement ~n the Jahresbencht der Deutschen Mathematlker Vere~n~gung (JDMV) Vol 21 p 58
VOLUME20 NUMBER3 1998 53
tins The most salmnt exanlples of this a p p r o a c h are found m the theory of gases and m radiahon t h e o r y Seen from t[us point of wew, Hllbert stated, the phenomenolog]cal perspective appears as a palhat]ve, a pnmlt]ve stage on the w a y to real knowledge, w[uch we must h o w e v e r pass through as s o o n as possible m order to gain entry mto the "real sanctuary of theoretical physics" (Hflbert 1911-12, 2) Unfortunately, he stud, mathematical analysis Is not yet so developed as to enable us to fulfill all the demands of t/us a p p r o a c h We must therefore do w~thout n g o r o u s logical deductions m t[us case, a n d temporarily be satisfied with rather vague mathematical formulas Still, Hilbert stud, It is amazing that usmg this method we nevertheless obtain ever n e w results that are m close a g r e e m e n t with expenence Yet a third a p p r o a c h , w[uch in Hilbert's vmw c o r r e s p o n d e d to the m a i n t a s k of physics, is the s t u d y of the m o l e c u l a r t h e o r y of m a t t e r itself The s t u d y of this t h e o r y s t a n d s above the k m e t i c theory m its d e g r e e of mathem a t i c a l sop[ustication a n d e x a c t i t u d e In the p r e s e n t course, Hilbert m t e n d e d to c o n c e n t r a t e on the kinetic theory, y e t he p r o m i s e d to c o n s i d e r the molec u l a r theory of m a t t e r m the following semester Hilbert's next course, d u n n g the s u m m e r s e m e s t e r of 1912, dealt with the theory of radiation Connectmg this topic with the p r o m m e i s s u e d at the beginning of the p r e c e d i n g semester, Hilbert d e c l a r e d that he n o w intended to a d d r e s s the "dommn o f phys]cs p r o p e r l y so-called," b a s e d on the a t o m i c theory Hilbert w a s clearly very m u c h i m p r e s s e d b y r e c e n t developm e n t s m quantum t h e o r y The sigmfic a n c e o f these d e v e l o p m e n t s was [ughhghted at the first Solvay Conference m O c t o b e r 191110, e c h o e s o f which had m o s t hkely r e a c h e d H]lbert "Never has t h e r e b e e n a m o r e p r o p i t i o u s and challenging time than now," he stud, "to und e r t a k e the study of the f o u n d a t i o n s of p h y s ] c s " What s e e m s to have lm-
~ Barkan 1993 11See Helllnger 1935 Toephtz 1922 ~2See Brush 1976 432-446 13See Born 1922 587-589
54
THE MATHEMATICALINTELLIGENCER
p r e s s e d Hllbert m o r e than anything else w e r e the deep m t e r c o n n e c t I o n s r e c e n t l y discovered m physics, "of w[uch formerly no one c o u l d have even dreamed, namely, that o p t i c s is nothing but a c h a p t e r of the t h e o r y of electnc]ty, that e l e c t r o d y n a n u c s and t h e r m o d y n a m ] c s a r e one a n d the same, that also energy p o s s e s s e s inert]al p r o p e r t m s , that physical m e t h o d s have b e e n i n t r o d u c e d mto c h e n u s t r y as well" (Hilbert 1912c, 2) And a b o v e all, the "atonuc theory," the " p n n c l p l e of dmcontmmty," as Hllbert said, "which t o d a y is n o t hypothesis anymore, but rather, h k e CopernICUS'S theory, a fact c o n f i r m e d by e x p e r i m e n t " Very m u c h like the umficat]on of a p p a r e n t l y distant m a t h e m a h c a l dommns, w i n c h p l a y e d a leadmg role t h r o u g h o u t Ills career, the umty of physical laws exe r t e d a strong attraction on Hllbert Hilbert's Publications on K i n e t i c T h e o r y I-hlbert's 1912 article on the kinetic theory o f gases a p p e a r e d as the last chapter o f [us treatise on the theory o f lmear integral equations (I-hlbert 1912), and it w a s also p n n t e d s e p a r a t e l y in the Mathemat~sche A n n a l e n (Hilbert 1912a) In developmg [us theory o f Integral equations, Hilbert was working on Ideas ongmall:~ Introduced by Henri Pomcare, Vlto Volterra, and Ivar F r e d h o l m I-hlbert treated the equations as 1kmlts of systems of an mfuute numb e r o f linear equations, using infLrute deternunants to solve them One k m d o f equations to w[uch Hflbert prod particular attention were those of the form
f ( s ) = ~p(s) +
s2
K(s,t)~(t)dt,
H e r e f ( s ) a n d K(s,t) are given a n d ~p(s) Is an u n k n o w n function When K(s,t), the "kernel," is a symmetric funct]on o f its arguments, Hilbert p r o v e d a s e r i e s of t h e o r e m s that greatly h e l p e d analyzing a n d solving the equation, meluding m a n y important t h e o r e m s o f e x i s t e n c e of solutions and convergence o f series 11
Hflbert's research into mtegral equations turned out to be strongly connected w~th a central Issue of the kmet]c theory the Maxwell-Boltzmann transport equation J a m e s Clerk Maxwell, who was the first to formulate t[us equation, had b e e n able to find only a partial solution of it, valid only for a very special case In 1872 Boltzmann reformulated Maxwell's equation m terms of a smgle mtegro-dffferent]al equation, m w[uch the u n k n o w n function represents the velocity distribution of the given gas The only exact solution Boltzmann was able to fmd was valid for the same part]cular case that Maxwell had treated m his own model 12 By 1912, s o m e progress had b e e n m a d e on the solution of the MaxwellBoltzmann equation The laws obtained from the partial k n o w l e d g e concernmg t h o s e solutions, describing the m a c r o s c o p ] c m o v e m e n t and t h e r m a l processes m gases, seemed to be quahtatively correct However, the mathematical m e t h o d s used m the denvat]ous s e e m e d ad hoc and unconvmcmg It was quite usual to d e p e n d on average magmtudes, and thus the calculated values of the coefficients of heat conduction and frmt]on a p p e a r e d unrehable A m o r e accurate estimation of these values remained a mare concern of the theory, and the techmques developed by I-hlbert offered the m e a n s to deal with it 13 Shortly after the p u b h c a t i o n o f has article on the kinetic theory, Hllbert orgamzed a s e m i n a r on t[us topic, together w]th hls f o r m e r student Erich Hecke The s e m i n a r was also a t t e n d e d by the Gottlngen d o c e n t s Max B o i l , Paul Hertz, T h e o d o r von K~irmfin, a n d E r w m Madelung The msues d i s c u s s e d m c l u d e d the following the ergodic hypothesm a n d its consequences, theorms of B r o w n i a n motion, the e l e c t r o n theory of m e t a l s m analogy to Hilbert's t h e o r y of gases, Hllbert's t h e o r y of gases, t e m p e r a t u r e spht by the walls, the t h e o r y of dilute gases using Hllbert's theory, the t h e o r y of chemical equlhbrlum, m c l u d m g a r e p o r t on the related w o r k of Sackur, dilute s o -
lutmns 14 The n a m e s of the y o u n g e r colleagues that p a m c i p a t e d m the seminar indicate that these d e e p p h y s i c a l Issues c o u l d not have b e e n d i s c u s s e d only superficially Especially m d m a tlve o f Hllbert's s u r p n s m g l y b r o a d s p e c t r m n o f interests is the r e f e r e n c e to the w o r k o f Otto S a c k u r S a c k u r w a s a physical chemast from Breslau, w h o s e w o r k dealt mainly with the laws o f c h e m i c a l e q m h b n u m an ideal gases and on Nernst's law of heat He also w r o t e a wadely used t e x t b o o k on therm o c h e m m t r y and t h e r m o d y n a m i c s ( S a c k u r 1912) His e x p e r i m e n t a l w o r k w a s of c o n s i d e r a b l e s~gmficance, a n d generally his w o r k was far from the lond of m a t h e m a t m a l physics whach Is usually a s s o c i a t e d w~th Hllbert a n d the G o t t m g e n s c h o o l ~5 Hllbert e w d e n t l y c o n s i d e r e d his mvestlgatlons to be m o r e than j u s t a maj o r c o n t r i b u t i o n to the d e v e l o p m e n t of the tonetlc t h e o r y as such As wath his m o r e p u r e l y m a t h e m a t i c a l works, Hllbert w a s always after the larger pacture, s e a r c h i n g for the underlying conn e c t l o n s a m o n g a p p a r e n t l y d~stant fields On m a n y o c c a s i o n s he s t r e s s e d the c o n n e c t i o n s of his w o r k on the lanetic t h e o r y with o t h e r physical domares, a n d in particular w~th r a d m t i o n theory, as in the following p a s s a g e
In m y treatise on the "Foundatwns of the k~netw theory of gases," I have showed, uszng the theory of h n e a r ~ntegral equatwns, that startzng alone f r o m the MaxwelI-Boltzmann fundamental f o r m u l a - - t h e so-called collzs~on formula--~t zs possible to construct the k~netw theory of gases systematwally Thzs constructwn ~s such that ~t only requires a consistent zmplementat~on of the methods of certain mathematical operations prescrtbed ~n advance, ~n order to obtain the proof of the second law of the~m o d y n a m w s , of Boltzmann's expression fo~ the entropy of a gas, of the equatwns of m o t w n that take ~nto ac-
count both ~nternal fr~ct~on and heat conductwn, and of the theory of d~ff u s w n of several gases L~kew~se, by further developzng the theory, we obtain the preczse condztwns unde~ whwh the law of equzpart~twn of energies ove~ the zntermolecula~ parameter ~s valid A new law ~s also obtained, concerning the m o t w n of compound molecules, accoTd~ng to which the continuity equatwn of hydrodynamws has a much more general meanzng than the usual one Meanwhde, there zs a second physwal domain whose p~tnc~ples have not yet been ~nvest~gated at all f r o m the mathen~atwal point of wew, and for the establzshment of whose foundahons--as I have ~ecently d~scove~ed--the same mathematwal tools prowded by ~ntegral equatwns a~e absolutely necessary I mean by th~s the elementary theory of ~ad~atzon, understanding by ~t the phenomenologwal aspect of the theory, whwh at the most ~mmed~ate level concerns the phenomena of em~sswn and abso~Tatwn, and on top of whwh stand K~ ~chhoffs laws concern~ ng the ~elatwns between emission and absorphon (Hllbert 1912b, 217-218) One m u s t always a p p r o a c h this kind of p o m p o u s d e c l a r a t i o n coming from Hilbert with a m o d i c u m of critical s p r a t But even if the self-evaluatmn of his w o r k s turns out to be e x a g g e r a t e d u n d e r c l o s e r scrntmy, one can be sure that Hflbert's i d e a s on the lonetm theory positively influenced slgmficant w o r k d e v e l o p e d b y several of has students F~rst w e r e two d o c t o r a l dissertataons written u n d e r his supervision on related assues, b y Hans Bolza and by Bernhard Baule 16 Second, o t h e r young G o t t m g e n scientists, like Max Born, T h e o d o r von K~rm~in, and Erich Hecke, who h a d a t t e n d e d Hllbert's seminar, p u b h s h e d an this field u n d e r its mfluence 17 But p e r h a p s of a m u c h greater ampact was the w o r k of the
Swedish physicist David Enskog, w h o a t t e n d e d Hflbert's l e c t u r e s of 191112 18 Building on ideas c o n t a i n e d m Hflbert's article, E n s k o g d e v e l o p e d w h a t has c o m e to constitute, together with the work of Sydney Chapman, the standard a p p r o a c h to the whole ~ssue of transport p h e n o m e n a m gases m Although a detailed analysis of Hflbert's mfluence on Enskog IS yet to be written, there can be httle doubt that it indeed goes back to the 1911-12 lectures Last is the possable influence of I-hlbert on the pubhcatlon of Paul and Tatyana Ehrenfest's famous artmle on the conceptual foundations of statIStical mechamcs (Ehrenfest 1959 [1912]) Paul Ehrenfest studmd m Gottmgen between 1901 and 1903, and returned there m 1906 for one year, befoae movmg with Ins wife Tatyana, w h o was also a Gottmgen-tramed mathematician, to St Petersburg The i d e a of w n t m g tlus article arose following a s e m m a r talk m Gottmgen, to wluch Paul Ehrenfest was mwted by Fehx Klem 2o The Ehrenfests' style of theory clarification, as manifest m tius artmle, IS s t n k m g l y remmmcent of Htlbert's lectures m many respects, and strongly suggests a du'ect influence Hilbert's P u b l i c a t i o n s on Radiation T h e o r y Hllbert's p u b h s h e d p a p e r s on radiation t h e o r y are mmnly c o n c e r n e d with axmmaUc denvaUon o f Klrchhoff's laws o f emission and a b s o r p t m n Gustav Klrchhoff had e s t a b l i s h e d the laws governing the energetac r e l a t m n s of rad m t m n m a state o f thermodynamac e q m h b r l u m A c c o r d m g to t h e s e laws, m the case of p u r e l y t h e r m a l r a d l a t m n the relat]on b e t w e e n the emission and a b s o r p t i o n c a p a c i t i e s of m a t t e r is a umversal function o f the t e m p e r a t u r e a n d the wavelength, m d e p e n d e n t of the n a t m e and the o t h e r characteristics of the b o d y m question In his w o r k on the theory of l a d m u o n , Planck substltuted for I~lchhoff's c o n c e p t s of enusslon and a b s o r p t i o n c a p a c i t y the
14References to this seminar appear ~n Lorey 1916 129 Lorey took this ~nformahon from the German students journal Semesterbenchte des Mathemat/schen ~,ete~ns ~SSee Sackur s obFtuary ~n Phystkaltsche ZeJtschnft Vol 16 1915 113-115 16The latter one was published as Baule 1914 17Cf for instance Bolza Born & van Karman 1913 Hecke 1918 Hecke 1922 ~SSee Mehra 1973 178 ~%ee Brush 1976 449-468 2~ Klein 1970 81-83
VOLUME20 NUMBER3 1998
coefficmnts of emission and absorption, 9 and a, respectively, defined for an e l e m e n t of volume P l a n c k s h o w e d that Karchhoff's law can be f o r m u l a t e d as follows the ratio q29 (q being the s p e e d o f light p r o p a g a t i o n in the body) is m d e p e n d e n t of the s u b s t a n c e of the b o d y involved, and is a u m v e r s a l function of the t e m p e r a t u r e a n d the freq u e n c y of radiation 21 In his first article on radiation theo r y (Hllbert 1912b), Hflbert a t t e m p t e d to p r o w d e the f o u n d a t i o n s o f this theory, while avoiding the kinds of slmp h f i c a t l o n s usually i n t r o d u c e d by physmmts (e g , that the b o d y is hom o g e n e o u s , simply limited, etc ) Hflbert a s s u m e d that the t h r e e param e t e r s 9, a, and q are given b y s o m e a r b i t r a r y functions o f their spatml location, and s h o w e d that the requirem e n t of energy e q u l h b n u m for each c o l o r leads to a separate, homogen e o u s integral equation of the s e c o n d t y p e for 9 w h o s e unique s o l u t m n ~s 9 = (a/q z) K (where K is a constant) Although Hflbert d e c l a r e d that his foundational study of r a d i a t i o n theory Was a x i o m a t i c , it w a s o n l y in an article p u b h s h e d the following y e a r (Hllbert 1913), and e s p e c i a l l y m his s e c o n d talk on the topic b e f o r e the Gottmgen Scientific Society (Hflbert 1913a), that he a r t i c u l a t e d the axaoms that lay at the basis of his t h e o r y and s t u d i e d their interrelations m o r e syst e m a t i c a l l y In the f o o t n o t e s a n d refere n c e s a p p e a n n g in his articles, Hflbert mentioned a considerable number o f w o r k s m the field b y Planck, E r n s t PringsheLm, W Behrens, Rudolf Ladenburg, Max Born, and S B o u g o s l a w s k l It w o u l d appear, however, that Halbert read s o m e of those works, if at all, only after a n u m b e r of obJections to his first article w e r e raised by Prmgsheun, wluch led to a s o m e w h a t heated debate between the two This d e b a t e is illustrative of the typical w a y in w h i c h a physicist could have r e a c t e d to Hfibert's a p p r o a c h to p h y s i c a l issues, and of how Hllbert's treatment, r a t h e r than presenting the s y s t e m a t i c
and finished structure c h a r a c t e r i s t i c o f the Grundlagen, was pmcewise, ad hoc, a n d s o m e t i m e s c o n f u s e d o r umllummatlng Pringshelm o b j e c t e d to the general a p p r o a c h a d o p t e d by Hllbert, a n d to m a n y o f the details of his a r g u m e n t s Pringshenn also s t r e s s e d the slgmficant differences b e t w e e n Hllbert's successive arhcles, m spite of the latter's m s l s t e n c e that there w e r e n o n e It is n o t e w o r t h y that also in his later w o r k in general relatiwty, Hilbert p u b h s h e d several v e r s i o n s and claimed that t h e y were essentially l d e n t m a l - - a claim that is not confLrmed b y a detailed exa m m a t l o n o f the various v e r s i o n s 22 At any rate, Pringshenn claimed that m foc u s m g on the inadequacies of all earher p r o o f s o f Kn.chhoffs t h e o r e m Hllbert w a s assuming, as g r o u n d s for his o w n proof, a fact that Kn.chhoff a n d all o t h e r physicists had c o n s i d e r e d to be in urgent n e e d of proof, namely, that the radiation at each wavelength separately is m e q u l h b n u m and no interchange o f energy takes place b e t w e e n different s p e c t r a l regions In fact, P n n g s h e t m claimed, a m a i n t a s k o f Kn.chhoffs w o r k was p r e c i s e l y to prove th~s assertion 23 Hllbert h a d to a d m i t the validity of these objections, and his successive articles w e r e att e m p t s to r e o r g a m z e his thoughts trying to t a k e c a r e of P r m g s h e l m ' s criticism Hllbert claimed t h r o u g h o u t the articles, however, that the main r e a s o n for applying the axiomatic m e t h o d to this p a r t i c u l a r physical t h e o r y w a s precisely the n e e d to introduce o r d e r into the e n t a n g l e m e n t of physical a s s u m p tions a n d m a t h e m a t i c a l d e r i v a t m n s that, in Hllbert's oplmon, affected it In o r d e r to prove the Lmposslbthty of d e n v m g the K~rchhoff-Planck equataons starting from the assumption of eqmh b n u m of total energy for all wavelengths, Hllbert had set the values of q and a equal to 1, mdependently of the value o f the wavelength A Prmgsheun considered tlus step madequate, because no actual body m nature has as its a b s o r p t i o n coefficient a = 1, and at
21For Planck s work see Kuhn 1978 22See Corry Renn and Stachel 1997 23pnngshelm published his objections in Pnngshe~m 1913 1913a 2'~See Born 1922 592-593
56
THE MATHEMATICALINTELLiGENCER
the same t~me no d~sperslon whatsoever (1 e , q = 1, the velocity of hght m vacuum) A s e c o n d obJection of Prmgsheun's was that Hllbert had not taken into a c c o u n t the effects of d~sperslon and reflection Hflbert's last article was written as an attempt to extend his p r o o f to t a k e these into consideration (Hflbert 1914) Htlbert also c l a i m e d to have provided a definitive p r o o f of the internal conststency o f his s y s t e m of a x i o m s and of Its c o n s t s t e n c y with the laws o f optics Halbert w a s gomg here m u c h farther than he h a d gone m the axiomatic analysis o f any other physical domain In the past, w h e n he d i s c u s s e d axmmat~c s y s t e m s for individual disclphnes, he n e v e r a c c o m p a m e d his discusslon with a d e t a i l e d analysis o f the kind he had p e r f o r m e d for geometry, though he often d e c l a r e d he had Tlus time he at least i n c l u d e d s o m e d e t a i l e d arguments c o n c e r n i n g the c o n s i s t e n c y of his system, although they are far from a c o m p l e t e p r o o f As m his earlier papers, H d b e r t ' s analysts of the logical m t e r r e l a t i o n s among the b a s i c c o n c e p t s and the p n n c l p l e s of the theory, and o f then" relations to o t h e r physical dommns, certainly p r o v i d e d a degree of clarity u n h k e that of his pred e c e s s o r s Still, tus d e c l a r a t i o n s a b o u t the strict logical c h a r a c t e r of his a x mmatlc a n a l y s i s - - a n d especmlly a b o u t its similarity with w h a t he had formerly done in g e o m e t r y - - o v e r s t a t e w h a t he actually did in the article Htlbert's articles on radiation theory a t t r a c t e d s c a n t attention from physmists Max Born attributed this neglect to n e w w o r k s a p p e a r m g s o o n after, dealing with d e e p e r p r o b l e m s of ra(fiatlon t h e o r y (especially the law of spectral energy dlstributton of the b l a c k body), w h i c h b e c a m e far m o r e tmportant than the issues dealt with in Hllbert's articles These n e w works, Born claimed, u n c o v e r e d m a n y interesting c o n n e c t i o n s with the foundattons of physics, w h i c h then led to a turning point in o u r understanding of radiation 24
Concluding Remarks F r o m the end of 1913 onwards, Hfibert's attention focused on the structure of matter, m o r e parttcularly, on the Lorentz-covanant, electromagnetac theory of m a t t e r developed by Gustav Mm This was to b e c o m e the basis for his w o r k on a unified foundation for physics, w[uch mcluded treatment of the field equations of grav~taUon m the general theory of relativity (Hflbert 1916, 1917) Hflbert's mterest m M~e's theory unphed a s~gnfficant change m [us basic physical conceptions, from a wholehearted s u p p o r t of an extreme mechanical reductlomstic approach throughout hls early career, to a similar support of an extreme electromagnetic reductionism Typically, Hflbert never mentioned tlns change, and of course he chd not explain what caused :t One m a y refer, with support froIn the [ustoncal evidence, that he increasingly reahzed the deep dlfficult~es revolved m the mathematical treatment of physical p h e n o m e n a b a s e d on the atommtm hypotheses Certainly, no one was better qualified than Hflbert to assess the degree of these dlfficullaes P n n g s h e l m ' s r e a c t i o n to Hilbert's articles on radiation is only one e x a m p l e o f the lack of e n t h u s i a s m aroused by Hflbert's many mcurslons into physics Emstem, for mstance, criticized Hfibert's approach to the general theory of relativity as bemg "childish m the sense of a c[uld that recogmzes no malice in the external world-25 Hermann Weyl constdered that Hflbert's w o r k m physics was of rather lmuted value, e s p e c i a l l y when c o m p a r e d to his w o r k in p u r e m a t h e m a t i c s Weyl thought that a valuable contribution to physics required a different kmd of skills than those m which Htlbert excelled 26 Max Born was one phys]cmt w h o exp r e s s e d a m o r e consistent e n t h u s i a s m for Hllbert's physics He s e e m s to have truly a p p r e c i a t e d the nature of Hllbert's p r o g r a m for axmmat~zmg physmal theo n e s and the potential contribution that p r o g r a m could m a k e Born exp l a m e d why, m [us opmlon, P n n g s h e u n had m i s u n d e r s t o o d Hflbert and w h y his r e p r o a c h e s were unjustified
Gustav M,e C o u r t e s y of Klaus M,e, K,el
The p h y s i c i s t sets out to explo~e h o w th,ngs are ,n nature, e x p e r i m e n t a n d theory are thus f o r h~m only a m e a n s to attain an a o n C o n s c w u s o f the ~nf i m t e c o m p l e x i t i e s o f the p h e n o m e n a w~th w h w h he ,s confronted ,n every e x p e r i m e n t , he ~es~sts the ,dea o f cons , d e m n g a theory as s o m e t h i n g def l m t ~ v e H e therefore abhors the wo~d " A x w m " , w h i c h ,n ~ts usual usage evokes the zdea o f d e f i n i t i v e h'ath The p h y s i c i s t ~s thus a c t i n g ,n accordance w~th h~s healthy ~nst~nct, that dogm a t , s m ~s the w o r s t e n e m y o f n a t u r a l science The m a t h e m a t w m n , on the contrary, has no bus~ness w~th f a c tual p h e n o m e n a , but rather w~th logwal ~ntertelat~ons In H~lbe~t's language the a x w m a t w t r e a t m e n t o f a d~sc~pl~ne ~mphes ~n no sense a def i m t ~ v e f o r m u l a t z o n o f specific axw i n s as eternal truths, but rather the follow~ng methodologwal d e m a n d s p e c i f y the a s s u m p t w n s at the beg, nno~g o f you~ dehbe~at~o~, stop fo~ a m o m e n t a n d ~nvest~gate whethe~ o7 not these a s s u m p t w n s a~e p a r t l y superfluous o~ conhad~ct each othe~ (Born 1922, 591)
o f axiomatic analysis he had done for geometry t h s derivations of the basic laws of the various chsclplmes from the a x m m s were rather sketchy, at best Many tunes he simply declared that such a derivation was possible Among [us p u b h s h e d works, [us last artmle on radmUon theory c o n t a m s - - p e r h a p s und e r the pressure of cntlc~sm--tus most detailed attempt to prove m d e p e n d e n c e and consistency of a s y s t e m o f a x m m s for a physical theory But w h a t is clear m every case is that Hllbert always consldered that an axmmatizatlon along the lmes he suggested was plausible and could eventually be fully p e r f o r m e d followmg the standards e s t a b h s h e d m the Grundlagen Whether o r not physicists should have l o o k e d m o r e closely at Hllbert's i d e a s than they actually did, and w h e t h e r or not Hllbert's p r o g r a m for the a x m m a t l z a t i o n o f p h y s i c s h a d any influence on s u b s e q u e n t d e v e l o p m e n t s m this discipline, it Is n e v e r t h e l e s s tmp o r t a n t to stress that a full picture of Hllbert's o w n c o n c e p t i o n of mathem a t i c s must include his v m w s on physIcal Issues and on the relations[up bet w e e n m a t h e m a t i c s a n d p h y s i c s More specifically, a p r o p e r u n d e r s t a n d i n g of Hllbert's c o n c e p t i o n o f the role of the axaoms in physical t h e o n e s - - a conc e p t i o n c o n d e n s e d m the a b o v e quoted p a s s a g e of B o r n - - h e l p s us u n d e r s t a n d his c o n c e p t i o n of the role of ayaoms m m a t h e m a t i c a l t h e o n e s as well The picture that a n s e s from such an understanding is o b v m u s l y very far a w a y from the w i d e s p r e a d image of Hilbert as the c h a m p m n of a f o r m a h s t l c conc e p t i o n of the n a t u r e o f m a t h e m a t i c s REFERENCES
Baule B
1914
Theoretlsche Behandlung der Er-
sche~nungen in verdunnten Gasen
Ann
Phys 44 145-176 Bolza, H, M Born and Th v Karman
1913
Molekularstromung und Tempera-
tursprung' Gott Nach (1913) 220-235 Born M
In fact Hflbert never performed for a physical theory exactly the same l~nd
1922
Hbert
und
d~e Physlk ' Die
Naturwlssenschaften 10 88-93 (Repr in
251n a letter of November 23 1916 Quoted rn Seellg 1954 200 26See Sigurdsson 1994 363
VOLUME20 NUMBER3 1998 57
verse Geometry and Physics, 18901930 Oxford Oxford University Press
t913a "Bemerkungen zur Begrundung der elementaren Strahlungstheone,' Gott
(forthcomtng) Corry L, J Renn and J Stachel 1997 Belated Decision ~n the H~lbertEinstein Pnority Dispute' Science 278, 1270-1273
Nach 1913, 409-416 Phys Z 14 592595 (GAVol 3, 231-237 ) 1914 Zur Begrundung der elementaren Strahlungstheone Dritte MJtte~lung " Gott Nach 1914 275-298, Phys Z 15, 878-
Ehrenfest, Paul and Ehrenfest Tatyana 1959 The Conceptual Foundations of the Stattst~cal Approach ~n Mechanlcs Ithaca, Cornell Unwers~ty Press (Enghsh translation by Mtchael J Moravcs~k of the German onglnal, Vol IV 2 II, #6 of the Encyclopadle der mathematlschen W/ssenschaften (1912) Leipzig, Teubner )
889 (GAVol 3, 231-257) 1916 "Die Grundlagen der Physlk (Erste M~tte~lung) ' Gott Nach 1916 395-407 1917 "D~e Grundlagen der Phys~k (Zwete Mrttellung)," Gett Nach 1917, 53-76 Klein, M 1970 Paul Ehrenfest Amsterdam NorthHolland Kormos Barkan, D 1993 The Witches' Sabbath The First International Solvay Congress in Physics,' Science in Context 6, 59-82 Kuhn, T S 1978 Black-Body Theory and the Quantum Dtscontlnulty, 1894-1912, New York Oxford Unwers~ty Press
Gabnel G, et al (eds) 1980 Gottlob Frege--Phdosoph~cal and Mathematical Correspondence, Chrcago The Unwerslty of Chicago Press (Abndged from the German edJtJon by Bnan McGuiness and translated by Hans Kaal) Hecke E 1918 "Uber orthogonal-~nvariante lntegralgleichungen, Math Ann 78, 398-404 1922 'Uber d~e Integralgleichung der klnet~schenGastheore "Math Z 12 274-286
M Born Ausgewahlte Abhandtungen, Gott~ngen, Vandenhoek & Ruprecht (1963), Vol 2 584-598) Brush S G 1976 The Kind of Mobon we Call Heat-A H~storyof the Ktnebc Theory of Gases in the t9th Century, Amsterdam-New YorkOxford, North-Holland Publishing House Corry L 1997 "David Htlbert and the Ax~omat~zat~on of Physics (1894-1905)' Arch H/st Ex Scl 51 83-198 1997a "Hermann M~nkowsk~ and the Postulate of Relattwty ' Arch H~st Ex So 51, 273-314 1998 'Hllbert and Physics (19001915), ~nJ Gray (ed), The Symbohc Un~-
58
THE MATHEMATICALINTELLIGENCER
Helhnger E 1935 "H~lbertsArbeten uber die Integralgle~chungssysteme und unendhche Gle~chungssysteme, in H~lbert GA Vol 3, 94-145 Htlbert, D GA Gesammelte Abhandlungen, 3 vols, Berltn Spnnger (1932-1935, 2d ed 1970) 1905 Logtsche Pnnc~plen des mathemabschen Denkens, Ms Vodesung SS 1905, annotated by E Helhnger Bibhothek des MathematJschen Seminars, Unwers~tat Gothngen 1911-2 K/net/sche Gastheone, WS 191112, annotated by E Hecke, Brbltothek des Mathemat~schenSeminars Unlvers~tat Gott~ngen 1912 Grundzuge elner allgeme/nen Theone der hnearen Integratgle~chungen Leipzig, Teubner 1912a "Begrundung der ktnetischen Gastheone,'Math Ann 72 562-577 1912b 'Begrundung der elementaren Strahlungstheorie," Gott Nach 1912, 773-789 Phys Z 13 1056-1064 (GA Vol 3, 217-230) 1912c Strahlungstheone Ms Vorlesung WS 1912, B~bl~othekdes Mathematischen Seminars Unwersttat Gott~ngen 1913 'Zusatz zur Begrundung der elementaren Strahlungstheone ' Jahrb DMV 22 16-20
Lorey, W 1916 Das Studlum der Mathemattk an den deutschen Un/versltaten sett Anfang des 19 Jahrhunderts Leipzig and Berhn Teubner Mehra, J 1973 The Physicist's Conception of Nature Boston, Re~del Prtngsheim E 1913 "Bemerkungen zu der Abhandlung des Herrn D H~lbert Bemerkungen zur Begrundung der elementaren Strahlungstheorle Phys Z 14 589-591 1913a 'Uber Herrn HIIberts axtomatlsche Darstellung der elementaren Strahlungstheone' Phys Z 14 847-850 Sackur, 9 1912 Lehrbuch der Thermochemte und Thermodynamlk Berlin, Spnnger Seehg, C 1954 Albert E~nste/n, Zunch Europa Verlag Sigurdsson S 1994 "Unification,Geometry and Ambivalence H#bect, Weyt and the Gottlngen Community, ' in K Gavroglu, et al (eds ), Trends /n the H~stonography of Science Dordreoht, Kluwer, 355-367 Toepell M M 1986 Uber die Entstehung yon David Hflberts "Grundlagen der Geometne," Gottrngen Vandenhoeck & Ruprecht Toephtz O 1922 Der Algebra~kerH~lbert D~eNaturw~ssenschaften 10 73-77
I I - - ~ a. . . .9 [ a ,,J--~ J e t
Wtmp,
Ed,tor
I
Emblems of Mind: The Inner Life of Music and Mathematics by Edwa~ d Rothste~n NEW YORK RANDOM HOUSE 1996 xx + 263 pp US $25 00 ISBN 0 81292 560 2
REVIEWED BY LEONARD GILLMAN
Feel hke wmtmg a review for The Mathematwal Intelhgencer~ You are welcome to submit an unsohc~ted review of a book of your chowe, or, ~f you would welcome being ass2gned a book to review, please wrote us, telhng us your expertise and your pred~lectwns
Column Edttor's address Department of Mathemat,cs, Drexel Untvers~ty, Phtladelphla, PA 19104 USA
his book is the h a r m o m c m e a n b e t w e e n fascinating a n d exasperating D u n n g the 1950s, I taught at Purdue Umvers~ty, wtuch had no music departm e n t (probably still the case) As a result, I was the musmal blgshot Tlus m e a n t among other thmgs that on several occasions I was mvlted to play at President Fred Hovde's preC o m m e n c e m e n t dmner for the Board of Trustees At one of them Mrs Hovde asked me, "How do you resolve your music and your mathematlcs'~" I looked her straight m the eye and rephed, "SomeUmes I respond to a pmce of music by thmkmg, 'That's as exqmslte as a beautiful t h e o r e m ' And s o m e t u n e s w h e n I contemplate a particularly arUsUc proof, I say to myself, 'That's as elegant as a Bach f u g u e ' " In a melhfluous voice and w~th a meltmg smile, she rephed, "You love your work, don't you'7"
T
Many people have pointed out that m a t h e m a t i c s and music are both abstract, a n d some will go so far as to c o n t e n d that mathematm~ans a n d theoretmal physicists tend to be more interested m music than are applied physlcmts and engineers But musicians as a group seem to show n o strong mterest In m a t h e m a t m s or theoretzcal p h y s i c s - - p r e s u m a b l y b e c a u s e although a person with no understanding of music can nevertheless enjoy a m u s m a l performance, it is unlikely that anyone can curl up with a m a t h e m a t m s book and enjoy it without u n d e r s t a n d i n g It
THE MATHEMATICAL INTELMGENCER 9 1998 SPRINGER VERLAG NEW YORK
Edward Rothstem has set out to zdentffy and explmn the s u m l a n t m s bet w e e n music a n d mathematics, a n d "to gzve some s e n s e of what zt is actually hke to be i m m e r s e d m both activities" He makes clear that he does n o t consider what he has done to be the last word, still, I find it hard to imagine very m a n y eager authors rushing m with a substitute or sequel At the tame the book was written, Rothstem was cluef music cntm for The New York Tzmes, he had stu&ed mathematacs, music, hterature, and philosophy at Yale, Brandeis, Columbia, and Chicago He is an engaging writer, though many mathematlczans will become unpatmnt with lus numerous forays mto the phflosoplucal, often bordering on the mystmal The subjects of the six chapters are
The need fo~ metaphor, The tuner hfe of mathematzcs, The ,nner hfe of musw, The pursuzt of beauty, The making of truth, The texture of thought, u n d e r the respectwe h e a d i n g s / b elude, Partita, Sonata, Theme and vamat~ons, Fugue, Chorale, wiuch are m e t a p h o n c a l allusmns ! might mentlon that a partita is a state of a halfdozen short pieces In classical d a n c e forms such as s a r a b a n d or gavotte, as put forth by Bach, who wrote seven such partitas for the piano, three for wohn, and one for flute And a fugue is a work for several "voices," which take turns stating the subject (theme) or one or more c o u n t e r s u b j e c t s The primary characteristic of a fugue is that it is contrapuntal, which m e a n s that the different voices state the various melodies s i m u l t a n e o u s l y rather than simply p r o w d e h a r m o m c accompanim e n t to a solo voice the structure is horizontal rather than vertical The actual form of a fugue is not rigid b u t tughly flexible, in keepmg w~th the derivation of the term from fuga, Latin for flight (cf "fugmve") Much of Rothstem's writing is eloquent, as w h e n he describes the process of a b s t r a c t m n m m a t h e m a t m s
and why ~t leads to progress, or w h e n ogy" IS a mistake for 'topography" or he reflects on what dehberatlons un"typography", they also know that alderlie aesthetic judgments, or m his gebra is algebra, a n d probably feel that s e n s m ~ e analyses of musmal works the only way there rmght be more than (Bach's prelude m C major from Book one is perhaps that in one you put all I of The Well-Tempered Clawed, Bach's the u n k n o w n s o n the left side and in fugue in D-sharp m i n o r from the same the other you put them on the n g h t book, Beethoven's "Appassmnata" Even such an apparently m n o c u o u s Sonata (first movement), Chopm's A term as ' the calculus," when thrown In m m o r Prelude, the Ma ~sedla ~se) or his without c o m m e n t , will puzzle the discourses on music in general Alas, reader who has heard only about ' calin the a b s e n c e of an a u d m tape, these culus "Musical terms left undefined inanalyses will m e a n httle to people who clude "measure, . . . . stave," "stretto," do not read music 'arpeggmted," a n d "fermata" The author's mare thesis is that M o ~ u t m e I f o u n d only a few mathstructures in m a t h e m a t m s a n d relaematical m l s p n n t s , none serious t m n s a n m n g them hay e their analogues enough to hold up the mathematical m music, but I would not say he m a k e s reader (or the u nmathemat~cal reader, a compelling case A v a n a t m n o n a who will be at sea in any case) theme is portrayed m terms of a map"Of S = {Pb , P~ }" should be "If ping of the theme into 'nmsmal space" S = {Pt, , P, }" (p 45 1 3b) (he loves mappings) But theme a n d "has a hmit of 0" should be "has a v a n a t l o n is a simple concept well unlimit of L" (p 59 1 8b) derstood by music-lovers, and g~ving ~t The c o n c l u s m n m the d e f i n m o n of a fancy n a m e does not lead to mc o n t m m t y reverses 9 and 8 (p 71) creased u n d e r s t a n d i n g or apprecmtion "2n parts" should be "2 ~ parts" (p of musmal structure or beauty, ~t thus 104) "S and t" should be "s and t" (twice) appears as mere v, o r d - d r o p p m g - - a s do the t e r m s ' musical space," "musical (p 130) surface," a n d "musical topology" I found one 3arnng solecism, unexSeveral of Rothstem's analogms pected from a cultured h u m a n i s t with music are strained, such as the ' [They] s e e m e d to lay beyond" instead of "[They] s e e m e d to h e beyond" (p n o t m n that the musical notes m our twelve-note scale form a set that has 50) the same propertms as the set of mtePhysicists Albert Einstein and gers m o d 12 (In the musical scale, Wolfgang Pauh should not be referred what are + and • to as m a t h e m a t i c i a n s (pp 13, 203) RothsteIn's mathematical discusHere n o w are some c o m m e n t s on selected chapters sions will be tough going for readers u n t r a i n e d m mathematics, as will the musmal d l s c u s s m n s for those unI. T h e N e e d for M e t a p h o r trained In music, moreover, w h e n he This chapter is an elegantly written mgets to Chapter III on music, he astroductmn to the book The author bes u m e s that the reader has assimilated gins by descnbmg Wflham Wo~ds~ orth's Chapter II o n n m t h e m a t m s He has a climb m the darkness to the top of a disconcerting predfiecnon for using motmtam and quotes sex eral lines from specmhzed terms without having de17~e Prelude, the poem It msptred, mfined them (although s o m e t i m e s he cludmg the line, "There I beheld the emdoes add a d e s c n p t m n retroactively) blem of a m i n d " He follows with an eloUndefined mathematical terms mclude quent passage I quote m full rather than "functmn," "space," "topology," and attempt to do justice to it in my o w n 'algebras" no one outside mathematwords ms can guess the technical m e a n i n g of "functmn", everyone outside matheThe j o u n ~ e y o f the poet, the labm~ous c h m b through d a K n e s s and sdence, matics k n o w s that a space is a gap, and will be mystified on e n c o u n t e n n g "theshould be f a m d m ~ to a n y o n e who has attempted to u n d e r s t a n d w h a t s e e m s o n e s of curves or spaces", n o n m a t h e matm]ans may well assume that 'topol- f o s t clouded tn m~st the dzscomfort
oJ the stall a o , the a w k w a r d pace on n a r r o w paths, the zsolated b t o o & n g s o f the c h m b Th~s book p r o m i s e s no less, but ~t hopes to p~omde sometho~g mo~e, s o m e hznt o f bmghten~ng by 3 o m ~ e y ' s end, s o m e w s w n o f the exp a n s e and v~stas that have opened to those who have m a d e such jomg~eys t h e o hfeworh, s o m e t n k h n g o f the p o w e r s and frowns that compose these e m b l e m s o f m~nd The~e are two p a t h s to be negotzated he~e--eaeh w t t h zts o w n t w i s t s a n d t~eacherous tmnzs, each wzth zts separate m a p s a n d ~esto~g places The p a t h s a~e those o f m u s i c and mathematzcs, and the claon that they a~e s~mda~, o7 at the ve~T least ~elated, has become a c o m m o n p l a c e - - a s has the claon fo~ the vast dlum~nat~on they offe~ to those who pledge themseh, es to the c h m b B u t ,t zs a commonplace shrouded o~ m y s t e r y Connections between the two have had a l m o s t no z m p m t a n c e f o r the develo p m e n t e~the~ o f m a t h or o f m u s w , m o s t l y a n y r e l a t m n s h z p s have been ~ e l e v a n t to t h e ~ p~actttmne~s a n d c~eatms, and m y s t w a l l y vague to everyone else They a~e s~mply acc o t e d w , t h o u t e x p l a n a t i o n m d~scusszon, w~thout even reahz~ng w h a t an u n h h e l y p a ~ o ~ g these two a t e Why should the~e be a n y h n k s at all ~ I have some c n t m i s m s I am glad to learn that the c o m p o s e r Jean-Phihppe Raineau felt ~t was only with the aid of mathematics that his ~deas became clear, but am left hanging v, hen I am not told details Inx o c a t m n s by contemporary c o m p o s e i s of set theory, Markov chains, or fractals do not necessaifly ~epresent a niatheniat~zatlon of nmslc, the c h a n c e s ai e that they just represent more word-dropping I doubt that n o n n i u s , c m n s ~fll u n d e r s t a n d what Is m e a n t by 'the strongest o~ ertones of the d o m m a n t rub against the tomc but also seek to meet m ' and I am not so sure a b o u t m u s l c m n s either II. T h e I n n e r Life of M a t h e m a t , c s In this introduction to nmtheinatlcs, the author p u i s u e s the laudable program of basing his d e v e l o p m e n t on our mtuiW, e concepts of space and s m o o t h n e s s This naturally requires the system of real numbers, which
VOLUME 20 NUMBER 3 1998
61
m e a n s coming to grips with the fundam e n t a l s of infinite s e q u e n c e s and irrational n u m b e r s Unfortunately, he is seriously confused a b o u t these concepts, as well as a b o u t Cantor's diagonal method A remark is in order a b o u t Euclid's proof by contradiction of the infinitude of primes The author m u s e s that one's first temptation is to keep "mal~ng" primes, lamentmg the fact that there is n o formula for them, from which he concludes that there is no way to make primes, let alone show that there are limitless n u m b e r s of them Remarkably, this program is exactly what Euclid c a m e s out (presumably without realizing It) For he finds (imphcltly) a prime greater t h a n the first N primes, thus mductl~ely producing a n increasing infinite sequence of primes Several of the topics seem inappropriate for the n o n m a t h e m a t i c a l reader The author goes big for the gee whiz stuff the equation e~n+ 1 = 0
that n o matter how far we proceed, we can ne~er express an irrational number e x a c t l y - - t h o u g h he n e v e r specifies what he m e a n s by exact e x p r e s s i o n Possibly he m e a n s expressible m a finite n u m b e r of arithmetic s y m b o l s - b u t t h e n "V~" qualifies, as perhaps does Llouvalle's t r a n s c e n d e n t a l number 10-" n= 1
In an a p p a r e n t about-face, he states that any irrational can be expressed by infinite rational sequences "reaching" a limit, the limit being "a simple m a t t e r of c a l c u l a t i o n " D e d e k m d cuts are m i s h a n d l e d royally First you consider any specified real n u m b e r (e g, 2) and look at the sets of all real n u m b e r s greater than it and all n u m b e r s smaller t h a n it (There is no m e n t i o n of the r a t i o n a l s ) "Dedekind decided to define that split as the n u m b e r itself" Cuts are thus tnvlallzed to be merely n e w n a m e s for n u m b e r s already known, a n d the revolutionary character of D e d e k m d ' s msight is entirely missed Cantor's diagonal m e t h o d is confused with the triwal fact that there is no smallest positive n u m b e r a n d Is confused with the fact that the real line is dense m itself In addition to e n t e r t m n m g these m i s c o n c e p t i o n s about the u n d e r p m nmgs of the real n u m b e r system, the author goes Into dramatic detml to d e m o n s t r a t e his significant lack of und e r s t a n d m g of just what Cantor did w h e n he showed that the set of reals c a n n o t be "listed"
(but the reader has little feeling for exp o n e n t s or complex n u m b e r s and n o n e at all for e), Cantor's diagonal method (whose logic often baffles math majors a n d whose c o n c l u s i o n confuses the author himself), Cantor's middle-third set (which will m e a n n o t h m g to a reader with no experience in manipulating p o i n t sets), and the existence of a contlnuous, nowhere dlfferentiable function (whose significance will be lost on any reader who has not already done considerable wrestling with derivatives) A gratuitous error due to carelessness ensues w h e n the author cites The very n o h o n o f o~derzng a n d lzstNewton's d e f m m o n of m s t a n t a n e o u s ~ng m e a n s znhe~ently that we can alvelocity as the quotient of two infim- w a y s f i n d a 'next one' a n d that we tesunals as being meaningless (0/0) but s o m e h o w ltnow h o w to p o c e e d f r o m ruins tiungs by writing "(0 = 0/0)", thus one to the othe~ It ~mphes a grounddecreemg that all i n s t a n t a n e o u s veloc- ~ng ~n s o m e p o i n t o f oregon a n d then ities are zero a rule o f s o m e A~nd that lets us m o v e After reading that "when there are f o r w a r d Th~s notion zs f u n d a m e n t a l c o n t i n u o u s mappings in both direc- to m a t h e m a t i c s and ~ts methods So to tions, the two spaces are called topo- p~ove, as Cantor dtd, that the~e can be logically equivalent," I have a n e w ap- no such h s h n g becomes qu~te p~opreciation of how difficult it is for f o u n d The ~esult o f h ~s p~ o o f opens the n o n m a t h e m a t l c l a n s to state mathe- door to an astonishing u n i v e r s e n~ matmal Ideas correctly w h w h the~e ~s no w a y to move ~n an The author can't get over the fact orderly f a s h w n f r o m one ob3ect to an-
62
THE MATHEMATICAL INTELLIGENCER
other Cantor d~amahcally changed the w a y we thznk about znfin~t~es A n d he d~d st by changing the w a y w e thznl~ about s o m e t h i n g as m u n d a n e as a h s t
While we do n o t expect Rothstem to recreate the theory of transfinite ordmals, we do expect him to u n d e r s t a n d that what Cantor did was to show that the set of real n u m b e r s c a n n o t be put into one-to-one correspondence with the set of n a t u r a l n u m b e r s
III. T h e I n n e r Life of M u s , c In this chapter the author is in his element He begins with a very mterestmg and thoughtful discussion of style, though I m u s t say that he leaves me w h e n he says that a musical style, like a mathematical one, may be a way of getting at the truth And he falls from grace with a thud when he suggests that a c r e s c e n d o is a climax, whereas every child struggling through music lessons k n o w s that crescendo m e a n s getting louder (typmally lead~ng to a chmax) I still d o n ' t know what a trope is despite his attempt to explmn it More important, Rothstem uses the two terms sonata (possibly familiar to the reader) a n d sonata f o r m (defined m the text) without ever tellmg us up front that they are entirely different things W n t m g a b o u t music runs the risk of descending into poetry and mystmmm, and Rothsteln de-escalates readily I have to w o n d e r how anyone u n t u t o r e d m music will extract a greater appreciation of it from reading Hegel's reference to its p o w e r "of penetrating with its m o t i o n s directly Into the mmost seat of all the motions of the soul," followed by the author's observation that "nmsic's great energies derive from the creation of continuity out of d l s c o n t i n u l t y - - a sort of inversion of the calculus, interested not m the mfmlteslmal a n d the i n s t a n t a n e o u s but m the ways they c o m b i n e into the gestural and fluid that resembles m some inchoate way o u r m n e r life " (It m e a n s little to me too, though I do k n o w that "resembles" should be " r e s e m b l e " - - a good example of a writer hoist by his own flowery prose ) The analysis of the opening of the
D-sharp m i n o r fugue from Book I of Bach's Well-Tempe)ed Clavte) is authoritative a n d excellent But I should like to put forth an alternative way of p h r a s i n g the theme A musical phrase is a musical thought corresponding to a literary clause or s e n t e n c e Thus it has no set length "Phrasing" a piece m e a n s deciding where phrases begin and end (and performing the piece accordingly) Phrasing a piece is therefore a m a t t e r of artistic interpretation (The word "phrase" also has a n o t h e r m e a n i n g in music, which need n o t concern us here ) My teacher at Jullhard was J a m e s F n s k m , a concert pianist and r e n o w n e d Bach scholar, the first pianist (I think) to perform both books of the Well-Tempered Clawer, comprising 48 Preludes and Fugues He was scholarly in everything he t u r n e d to, which is largely why he and I hit it off (When a n article in a Jullhard pubhcatlon referred to Bach as a Saxon on the apparently reasonable grounds that Elsenach, his birthplace, is in Saxony, Fnskan was the only p e r s o n to pomt out that it was not part of Saxony at the time Bach was b o r n ) F n s k l n ' s interpretive analyses were insightful and compelling The figure shows my copy of the fugue that Rothstem discusses (I used the B u s o m edition, in which both the prelude a n d fugue are p n n t e d in E-flat m m o r rather than (the equivalent) D-sharp m i n o r ) The arcs u n d e r the notes are B u s o n f s phrase marks, the same as Rothstem's The arcs above the notes show F n s k m ' s phrasing, Rothstem's comm e n t s a b o u t the spirit of the t h e m e still apply, though the role of the u p w a r d leap of the fourth has pretty m u c h disappeared Once one learns F n s k m ' s phrasing, it is hard to imagine interpretlng the passage any other way The c o m p o s e r Charles G o u n o d based a version of Ave M a i m on Bach's prelude m C major from Book I, the melody closely follov, mg the peak n o t e s that Rothstem highlights It is a bit surprising that Rothstem does not m e n t i o n this By the way, the score as p n n t e d in the book contains a n egregious m~sprmt that any c o m p e t e n t musician will hear at a glance namely, in m e a s u r e 29, each of the two low Cs in the bass should be G, a fifth higher, the
same as the bass note in the surl o u n d m g m e a s u r e s (The c o n d e n s e d version on page 129 has it right) The discussion of Beethoven's "Appasslonata" s o n a t a is enthralling and is the tugh p o i n t of the chapter
on which the c o n t i n u e d fraction is based [In the same way, every nonzero real n u m b e r z "contains" itself For, pick p and q such that z 2 = p z + q and q # 0, then
z=p+
q ] Z
IV: The Pursuit of Beauty The author has a love affair going with the Golden Rectangle and the Divine Proportion (or Golden Ratio), ~b = (1 + V5)/2, a n d hence with the number % 5 He is e n t r a n c e d by the fact that the decimal e x p a n s i o n of ~v'5 has all the seeming r a n d o m n e s s of that of w--[though almost any other irrational n u m b e r will do as well] (The v, ord ' seeming" is u n p o r t a n t the eye c a n n o t tell any difference, although Llouville's theorem a b o u t algebraic n u m b e r s suggests that there may be one ) Among the magical properties cited for r is that if ap r e p r e s e n t s the length of a line, then qb2 represents the area of a square [)] Pursuing this thread, Rothstem concludes that we can begin to see how the abstraction of n u m b e r s can be related to a more s e n s o r y relation, something available to the eye [has he never e n c o u n t e r e d analytic geometryg], and that the p o w e r of ap goes further than fractals in that it m a k e s a link b e t w e e n the act of addition and that of multiphcatlon The c o n t m u e d fraction expression for ap, wtuch consists entirely of l's, lea~ es him starry-eyed, as it shows that this n u m b e r "contains itself' The same conclusion was available earlier from the equation @
=
1+
1 -
(1)
-
Overwhelmed by the Divine Proportion, the author avers that the eye senses in it "a c o n t i n u o u s internal recurrence It also finds stability in its dimensions, a piquant restfulness, as the ratios can be Lmagmed reproducing themselves within it again and a g m n " [Can't we j u s t stare at a b l a n k wall and imagine anything we wash 9] The author repeats the c o m m o n misconceptions that the Golden Ratio inspired the design of the P a r t h e n o n and that the height of the h u m a n body is in this ratio to the height of the navel A scholarly analysis by George Markowsky in the College Mathematws J o u n m l [2] refutes these and other such claims, pointing out among other things that m e a s u r e m e n t s m the real world are necessarily imprecise, so that the numerologmt has a wade pallette of values to choose from Another myth he d e b u n k s is the centuries-old pron o u n c e m e n t that of all rectangles, the Golden Rectangle has the most pleasing shape Martm Gardner, in the chapter The G~eat P y r a m i d in his classic work [1], noting that a great deal had b e e n made of the relation of the number five to the Pyramid, reports that in a spur-of-the-moment investigation of the Washington M o n u m e n t based on the data in the World Almanac, he readily made out a strong case for the number five as found in the height of the m o n u m e n t in feet a n d inches, the dr-
Fuga VIII, ) a 8. Andante ~-~,,~ # f~'k- | : VII
e) """
pensieroso, -
non troppo ~"
accentato. -
g
p
. .-.2
ce
-'i j
/j"
-
'
i"
Openmg of the E-flat m,nor fugue from Book I o f the Well-Tempered Clavmr b y B a c h w , t h phrase markings of James Fr,skm (,ndmated by the upper arcs)
VOLUME 20 NUMBER3 1998
m e n m o n s of the base in feet, the weight of the capstone in pounds, the n u m b e r of letters in "Washington"(10) and "The Secretary of the Treasury" (25), and the date (in the Gregorian calendar) on which the Secretary armounced the decision to p n n t the Pyramid on onedollar bills
and, a little later, The central works of the Western repertory may be understood as tales which, m their highly metaphorical fashion, outhne our society's onglns, detail its passion, and trace the adventures of its heroes That is what makes the work of Wagner so powerflfl, and that of Beethoven so central to our understandmg of music
V. T h e Making of T r u t h This chapter starts with a n absorbing exposition of some lofty problems A book rexaew traditionally closes a b o u t mathematics, such as Wzgner's wath a brief assessment such as "This question of why it is so u n r e a s o n a b l y book belongs o n every scholar's bookeffective in the natural sciences Some shelf" or "This book wall add nothing of Rothstem's remarks are well-nigh to our k n o w l e d g e " Mine was pro,~aded epigrammatic, e g, "We nmy n o t know in the o p e n i n g sentence what 'pure thinking' is, but we do know it often comes in h a n d y " After quoting REFERENCES E i n s t e i n "Whoever u n d e r t a k e s to set 1 Gardner, Mart~n Fads & Fallacies in the himself up as a judge in the field of Name of Science Dover, 1956 Truth and Knowledge is shipwrecked 2 Markowsky, George M~sconcept~onsabout by the laughter of the gods," he rethe Golden Ratio College Math J 23 s p o n d s with, "Many of these problems (1992) 2-19 disappear, though, if we alter our concept]on of those gods " He justifiably 1606 The High Road nmrvels at and applauds the unshak- Aushn TX 7 8 7 4 6 - 2 2 3 6 able froth of Gahleo, Kepler, and USA N e w t o n m geometry a n d mathematical e-marl len@mathutexas edu fornmlas When he gets d o w n to rattygritty matters, though, he is hkely to strike out, as w h e n he asserts that the integer classes modulo 12 form a group u n d e r multiplication Turnmg to lofty considerations of by Lennart Be~gg~en, Jonathan music, he considers such matters as BomL,e~n, and Peter Borwe~n what the function is of "artistic form," NEW YORK SPRINGERVERLAG 199~ US $59 95 or m what way the construction of a ISBN 0 38794 924 0 composition affects us when we hsten He starts off wath fundamentals such as REVIEWED BY DAN SCHNABEL what a musical phrase is, which lead h u n to the "accented silence" that opens istories have traditionally b e e n the Beethoven Fifth Symphony Corretold by describing events, their spondmg roughly to Wlgner's question causes, effects, and interrelationships about mathematics Is his statement that Many authors have given their interthe "paradox at the heart of music is pretatzon of the history of mathematwhy it is possible to speak wath such ics m exactly this n m n n e r Almost all ease about music m spiritual or hterary mathematical e~ ents have the property or intellectual or technical t e r m s " that they can be said to have occurred When he gets to the societal nman- only w h e n they have b e e n c o m m u m lngs of music, the Ice thins cared The result Is a subject whose history is always already written, alThe music of the n i n e t e e n t h century though it mostly consists of a haphazwas the first music w n t t e n for and ard accretion of conjectures a n d a b o u t the emerging bourgeoisie, proofs In P, A Source Bool~, the ediwhich is why middle-class listeners tors ha',e deliberately resisted the today are still a b s o r b e d m its mes- t e m p t a t i o n to interpretation, choosing sage and m e a n i n g instead to gather o n g m a l and sec-
Pi: A SourceBook
H
THE MATHEMATICALrNTELLIGENCER
ondarv source m a t e n a l and let the documents speak for themseh,es Such an approach to history succeeds on the basis of the c o m p l e t e n e s s and coherence of the collection, and Berggren, Borwem, and B o r w e m ha~ e succeeded admzrably They have assembled a ~olume that fascinates a n d entertains It Is neither possible nor desmable to gather together all wrltmgs on pl The seventy pmces selected by the authors are sufficient to make this source book a hefty tome Yet nothing that is mchided seems superfluous The book has achieved a satisfymg sense of wholeness by havmg wrltmgs frequently interpreting, cross-referencing, and supportmg each other For example, not only do we read m Ranjan Roy's contribution, The D~scovery of the Ser~es
Formula fo~ 7r by Le~bn~z, G~egory and Ndal~antha, that the Scottish mathemat]clan James Gregory, wortung m Isolat]on, was kept a b r e a s t of the work of Isaac Newton by c o r r e s p o n d e n c e with John Collins, but we get a taste of that c o r r e s p o n d e n c e m the form of a reproduction of a letter from Gregory to Collins Gregory's letter, dated 1671, Is also interesting m that it Is m Enghsh and Latin, the latter reserved for the serious mathenmncal work Reproductions of excerpts from Vl~te's V a r w ~ m De
Rebus Mathemat~ezs Reponso~um Ltber VII of 1593, a n d Huygens's De C~culz Magn~tud~ne Inventa of 1724 further d e m o n s t r a t e the significance of Latin m the history of mathematics Berggren, Borweln, and Borwem are perhaps too true to the sprat of a source book w h e n they include these works without provadmg either translat]ons or references to translations The same is true of sxgmficant early proofs that pl is irrational and transcendental, which are included in their original F r e n c h a n d G e r m a n If your skalls m any of these languages are madequate, you wall miss some of the pleasure to be had from these works The formula for pl referred to m the title of Roy's article is ~1 1 --=1---+-----+ 4 3 5
1
7
This formula Is a direct consequence of the formula
X 3 alctan
3L - - - - - x - - - -
+
3L 5 --
3
--
5
The mfmlte series formula for arctan a, coupled with a formula which allows small values of x to be used, like Machin's formula
4
'
played a significant role m calculating extended decimal approxunatlons to pi Machm calculated pz to 100 d~g~ts m 1706 nsmg these formulae In 1949, the first computer calculation of pz produced over 2000 &g]ts using this method The details of this nulestone of computat]on are provided m a reprint of G W Re]tw~esner's An E N I A C Dete~m ~ n a t w n o f p t and e to 2000 Decimal Places In the mtroduct]on to Pz A Source Book, Berggren, Borwem, and Borwem hst Machm-hke arctangent foru~ulae as one of the three s~gmficant m e t h o d s used m the c o m p u t a t i o n of p~ References to this m e t h o d are e n c o u n t e r e d m various art]cles throughout their book R H Birch's arhcle A n A l g o m t h m For The Construction o f A~ctangent Relat i o n s h i p s d e m o n s t r a t e s a method that y~elds other Machm-hke formulae The article The B e s t (~) F o r m u l a fo~ Comp u t i n g 7r to a Thousand Decimal Places, by J P Ballantme, c o m m e n t s on a m e a s u r e of the suitability of ~anous a r c t a n g e n t formulae for extended decimal c o m p u t a t i o n s of pz This measure appears to have b e e n proposed by D H Lehmer m a 1938 article winch, given its szgmficance m the works of Birch a n d Ballantme, m strangely absent from P~ A Source Book This omission is m a d e more curious by ~ts u n i q u e n e s s Lehmer's m e a s u r e seems to be the only mathematical c o n c e p t or procedure m the entire book whose results are c~ted without details of the underlying m e t h o d Lehmer's article ~s not long and his method m not difficult He suggests that 1 logm~ is a statable m e a s u r e of the efficiency with which a~ a r c t a n ( ~ )
can be calculated, a n d he determines the values associated with several formulae D A Cox points out, m his amcle The A m t h m e t w - G e o m e t m c Mean o f Gauss, that "The mare difficulty m writing about the history of mathematics is that so much has to be left o u t " Lehmer's arUcle is only one among many works that were no doubt considered for inclusion m this book, but chd not make the final cut Fommately a comprehensive hibllography provides more than adequate compensatmn for the absences Offenng research hterature, historical studies, and whimsical d~versions, this chronolog]cally orgamzed book can be read and enjoyed from cover to cover Nevertheless, the editors identify a n u m b e r of uses for the book, and with each use they suggest a few relevant articles For example, Cox's article m r e c o m m e n d e d as an mtroduct]on to elhptlc integrals, a n d although it is slgmficantly longer t h a n any other article m the book, It fulfils that role excellently Elhptzc and m o d u l a r function methods constitute the third and most recent of the three szgmficant methods used in the c o m p u t a t m n of p] These n e w methods are the ones being used to calculate bllhons of decimal places of pl Articles co-authored by J o n a t h a n and Peter Borwem, s o m e t i m e s mcludmg the partm]patmn of additional coauthors, focus on these methods a n d their origins, especmlly m the works of R a m a n u j a n They indicate the leading edge of pl research and make ]t more accessible The first use of the symbol 7r for the ratm of a circle's circumference to its chameter is found m Wflham Jones's 1706 textbook A N e w Int~oductwn to the Mathematics, a select]on from which ]s reproduced in flus sourcebook The true value of Pt A Soutce Book is aptly summarized by Jones's description of his own work as intended
fo~ the use o f s o m e f m e n d s , u,ho had ne~the~ leisure, conven~ency, not, perhaps, patience, to search ~nto so m a n y d~ffe~ent authors, and tum~ over so m a n y t e d w u s volumes, as is unavoidably required to m a k e but a tolerable progress ~n the M a t h e m a t i c s
For their patience, a n d for this book that came of the]r patmnce, Berggren, Borwem, and Borwem will certainly make m a n y friends It should be m e n t i o n e d that the papers are photocopmd from originals (with varying degrees of quahty) with no editorial effort to correct occasional typos and logical lapses At least for the m o d e r n papers where this is feasible, such editorial m t e r v e n t m n w o u l d have b e e n helpful
REFERENCES
D H Lehmer "On Arctangent Relahons for ~-" Amencan Math Monthly 45 (1938) 657-664 6000 Yonge Street (#510) Toronto Ontario M2M 3Wl Canada e-mal schnabel@tnterlog com
An Introductionto Coding and InformationTheory by Steven R o m a n SERIES UNDERGRADUATE TEXTS IN MATHEMATICS NEW YORK SPRINGERVERLAG 1996 323 pp US $39 95 ISBN 0 38" 94704 3 REVIEWED BY S C
COUTINHO
any budding sclent]sts have their first ghmpse of what a hfe in scmnce is through reading a p o p u l a r book with biographical sketches of famous scmntmts In my t e e n s I used to keep one of these books by my bedside It had a n u m b e r of short blographms of famous c o n t e m p o r a r y scientists biologmts, physicists, astronomers, and even a m a t h e m a t i c i a n The m a t h e m a t i c i a n was Claude Shannon, the father of ~nfomnat~on theory Though I was already in lo~e ~ lth mathematics, I found the kind of mathemat]cs represented m that book ~ ery hard to fathom The book also mlphed that Shannon's theories v, ere so difficult that they could only be understood by very feb speclahsts, much as Emstem's were once stud to be That, of course, did not help much Luckily, here I am, 20 years later, revmwmg Steven Roman's lovely book It e x p l a m s - - t o everyone
M
VOLUME 20 NdMBER 3 1998
65
with a httle knowledge of p r o b a b l h t y - what Information theory is about, and it even includes theorems n a m e d after S h a n n o n himselP In fact, mformaUon theory accounts for less than h a g of the book, which really deals vflth codes Roughly speaking, w h e n we talk of a code we may be thinking of one of three different mathematical disciplines One of them IS ,nfo~vnatwn theory It deals with how one can code a message in the most efficmnt way before transmitting it thorough a channel, a telephone line, say This is also called data c o m p r e s s i o n Of course in real life no c h a n n e l is perfeet, they all suffer from " n o i s e " In other words, a z a n e t y of p r o b l e m s may o c c u r in the actual t r a n s m i s s i o n of the message, which m e a n s that it may be more or less garbled w h e n it arrives This leads one to coding theory, or how to encode a message so as to detect a n d correct errors that may have occurred during t r a n s m i s s i o n Finally there is cryptography, the science of coding a message so that it c a n n o t be u n d e r s t o o d in case it is intercepted by a n unauthorized p e r s o n The book u n d e r review is a beautifully written introduction to tnforma-
t~on theo~71 and coding theo~7], c~?Iptography is only m e n t i o n e d in passmg This is not surprising m o d e r n cryptography is founded u p o n n u m b e r theory, whilst the branches of pure mathematICS behind the material of this book are lmear algebra and probability What is a code, then ~ To keep things simple we will a s s u m e that we are dealing only with messages written in English that will be e n c o d e d using a hinary code The 26 letters of the alphab e t - - t o g e t h e r with the b l a n k s p a c e - form what is called the source alphabet We will encode each letter using a finite s t n n g of 0s and Is, in other words a b~na~y strong Thus a code C is a firote subset of the set of all binary s t n n g s The elements of C are called codewo~ds For exanlple, in ASCII, the letter A is encoded as 01000001, the letter B as 01000010, the letter C as 01000011, and so on Note that 8 binary digits are used in this example, this is called the length of the c o d e w o r d ff a code C is to be really useful, then we expect that, glzen a b i n a r y string,
66
THE MATHEMATICAL INTELLIGENCER
there will be at most one sequence of c o d e w o t d s that corresponds to that string Such a code is called umquely decipherable An even better kmd of code is the ~nstantaneous code For the latter codes, when a s e q u e n c e of codewords is transmitted, each codeword can be deciphered as s o o n as it is recelzed Of course an i n s t a n t a n e o u s code is always uniquely decipherable, but the c o n v e i s e does not hold Here Is a n example Consider the code C = {0, 10, 110, 1110, l l l l 0 , 11111} This is an i n s t a n t a n e o u s code the 0 tells you the codeword has ended, the n u m b e r of l s tells you which codeword it is However, the code D = {0, 01, 011, 0111, 01111, 11111} obtained by ieversmg the order of the d~glts in C, IS not i n s t a n t a n e o u s For example, w h e n you receive 0 you cannot tell w h e t h e r you have received the codeword 0 or only the first digit of one of the bigger codewords However this code is uniquely decipherable, once the whole message is receiz ed, you can read it backwards and get each codeword correctly w These matters are discussed in chapters 1, 2, and 3 Two hlghhghts are Huffman's method for p r o d u c i n g highly efficient i n s t a n t a n e o u s eneodlng s c h e m e s and S h a n n o n ' s Noiseless Codtng 17~eo~em of 1948 The latter gives an upper and a lower b o u n d on the m m m u r m average length of a codeword of a uniquely decipherable code in terms of a characteristic of the information source, its entropy The s e c o n d part of the b o o k (which a c c o u n t s for well over half its pages) deals with error detection and correctlon Whilst before we were a s s u m i n g the message am~ ed just as it had b e e n sent, we are n o w admitting that there may be noise in the channel One of the most famlhar errol-detectlng codes IS the ISBN code This is the 10-digit n u m b e r used to identify books, which in the case of the b o o k u n d e r rezaew is 0-387-94704-3 The first digit identifies the language (0 for English), the next three identify the publisher (387 for Spnnger-Verlag), the next 5 digits identify the book The finat digit is really r e d u n d a n t to the task
of identifying the book, i t is there to help detect errors that may have b e e n committed w h e n transcribing the ISBN n u m b e r of the b o o k It is easy to explain how this last digit is found Think of the 9 digits vt, , a9 that are sufficient to characterize the book as elements of 7/11 The (redundant) t e n t h digit a i0 is o b t a i n e d by solzlng the equation r i + 2a2 +
+ 9x9 + 10xi0 = 0
m Ei1 So this is like "casting out nines" to check that your s u m is right' The discussion of coding theory begins with the probability that an error may have b e e n committed when a message is t r a n s m i t t e d This leads to maxlmum-hkehhood decoding, nearestnelghbour decoding, m l m m u m distances, and sphere packings This fare is served in chapter 4, which ends with a brief discussion of S h a n n o n ' s theorem for noisy c h a n n e l s A binary code all of whose words haze length/~ c a n be thought of as a subset of 7/~ More generally, a code of length /~ is a s u b s e t of 7/p~, for some prime p If this s u b s e t Is also a linear subspace, then we haze a hnea~ code An example of such a code is the ISBN code described above Not surprisingly, when a code is linear the fact greatly simplifies the calculations required to check and correct possible errors These codes are introduced in chapter 5 Special examples of linear codes that have b e e n used in the real world can be found m chaptel 6 Besides the ISBN code, these Include the Golay codes used to transmit colour photographs of Jupiter and Saturn by the Voyager spacecraft, and the ReedMuller code used by Mariner 9 to transmit black a n d white photographs of Mars m 1972 Finally, chapter 7 is a very brief i n t r o d u c t i o n to cyclic codes This is a beautifully written book that makes a wonderful textbook for an elementary course on codes, aimed at students with a minimal background of linear algebra a n d probability The book is also strongly r e c o m m e n d e d to those who wish to learn the subject on their own Ezen though the book was written for undergraduate students of mathe-
mat~cs and c o m p u t e r science, the author found it necessary to mchide a brief m t r o d u c t m n to h n e a r algebra at the b e g m n m g of the chapter on h n e a r codes I have to admit that I would have done the same, but one c a n n o t help feehng sad when one reads m the m t r o d u c t m n to Weyl's book on quant u m m e c h a n m s that
~t ~s s o m e w h a t d~sDess~ng that the theory o f hnea~ algebra m u s t again and again be developed f l o r a the beg~nmng, fo~ the f u n d a m e n t a l concepts oJ th~s b~anch o f m a t h e m a t i c s crop up eve~ywhe~e ~n m a t h e m a t i c s and p h y s i c s , a n d a l~nowledge o f t h e m should be as w~dely d~ssem~nated as the e l e m e n t s o f d~fjerentml calculus That was m 1931 Some may well feel that we are losing on the calculus front, rather than w m n m g on the h n e a r algebra one I prefer to hope that, with more b o o k s hke th~s one being pubhshed, we are b o u n d to w m o n all fronts, even ff we have to wmt for another 70 years Departamento de Ctenc~ada ComputagAo Untvers,dade Federal do Rto de Jane,ro PO Box 68530 21945-970 R,o de Jane,ro, RJ Braz,I e-ma~l colher@,mpa br
The Art of Doing Science and Engineering; Learning to Learn by R z c h a t d W H a m m z n g NEWARK NJ GORDON AND BREACH 1997 x, + pp 364 US $27 95 (paDerback) ISBN 90 5699 5014 US $69 95 (hardbacK) /SBN 90 5699 5006 REVIEWED BY
ROGER PINKHAM
s one grows older, I thmk ~t comm o n to suffer a tmge of regret, regret that what one has worked so hard to learn or to understand c a n n o t be passed on, more or less effortlessly, to some deser~_ng young person or persons Perhaps this ~s why older members of the scmntffic c o m m u m t y often
A
do less research wath the years and tend to write about education, philosophy, or the broader issues m their field The book u n d e r revmw results from a lffetune of reflectmn on scmnce and the domg of scmnce The author has hved a n u m b e r of hves At Bell Telephone Laboratories m its heyday he was Mr Computmg consultant to engmeers, chenusts, physm~sts, and management He holds the key patents on errordetectmg, error-correctmg codes He wrote a ground-breakmg book on Numerical Methods which begins with a nmrvelous epigram, still too httle known among mathematm~ans "The purpose of computmg is insight, not numbers " I n that book, as m all else, he struggled to fend and exphcate general methods rather than detailed tricks He wrote a key book on the general design of filters, and lectured widely on the material Upon reurement from BTL he turned s e n o n s l y to teaching, leaving behmd numerical methods and active responsxbxhty for computmg In his 20 or so years at Naval Postgraduate School he has continued to turn out books and notes o n everything from probability to p r o g r a m m i n g style The c u r r e n t b o o k appears to be an attempt by the a u t h o r to dmcharge partlally his o b h g a t m n to the younger of the commumty, to tell it hke it is Here then are the reflectmns of s o m e o n e who has not b e e n a n idle bystander, but a doer, s o m e o n e who has questinned at each turn Now I have often made the o b s e r v a t m n that as a class, m a t h e m a t m m n s - - o f w h o m I am o n e - seem least hkely to carry mto their personal hves the tools of their trade logical consistency, unflagging curiosity, a tolerance for alternative vaews Hamming is not one such H~s o p m m n s will a n n o y some a n d anger a few He Is never dull, and he can back his views with observatmn, experience, and argument Whatever else, this book should make any m a t h e m a t i c i a n re-exanune his own behefs G~ven that the only numerical data that come through yore office door consist of computable numbers, and that computable numbers are countable, do the real numbers really make sense ~ Or m hght of the Axmm of Choice, how do you se-
lect an element from the set of noncomputable n u m b e r s ~ A refusal to take a n y t h m g on faith is a hallmark of H a m m m g ' s nature and success I r e m e m b e r attending a dinn e r party over 30 years ago given by the H a m m m g s (Wanda H a m m i n g is an excellent cook ) At the time I was cooking for five each day a n d almost exchis~vely Chmese I r e m a r k e d offhandedly that I p r o b a b l y used nearly a gallon of soy sauce every two months Qmck as a w m k H a m m i n g was mutt e n n g to himself, "Let's see, that's 4 quarts or 8 pmts h m m 256 tables p o o n s Do you use 4 tablespoons a day ~" I stud I thought I dxd He stud he guessed he'd beheve m y surmise of a gallon or so every two m o n t h s T He translated the p r o b l e m into a framework where one's m t m t m n was more likely to funcUon effectively Let me quote from the book, page 4
H e r e I m a k e a d~gress~on to dlusDate w h a t ~s often called 'back o f the envelope c a l c u l a t w n s ' I have f i e q u e n t l y observed great sc~enttsts a n d eng~needs do th~s m u c h m o r e often than 'the ~un o f the mall' people, hence zt r e q u o e s ~llustrat~on I wall take the above two s t a t e m e n t s , knowledge doubles every 17 years, and 90% o f the s c i e n t i s t s who eve7 h v e d are now ahve, and ask to w h a t e:~tent they a t e compatible Hamming then a s s u m e s a model of y ( t ) = exp(bt) and finds e x p ( - 1 7 b ) = 1/2, from the a s s u m e d d o u b h n g e~ery 17 years This determines b As to the 90~ statement, assuming a hfetlme of 55 years exclusl~e of childhood fo~ a scmntlst you find 1 - e x p ( - 5 5 b ) for the fractmn of all scmntmts past and p r e s e n t who hax e hved m the last 55 years Substituting for b in this last expressrun yields 0 894, which is certainly close to 90% They are mdeed consistent Hamming says, on page 271,
Ove~ the many yea~s, the~e ha~,e developed f i v e ma~n schools qt what M a t h e m a t w s ~s, a n d not one has p~oved to be sat~sfactot~t The oldest, and p~obably the one m o s t M a t h e m a t w m n s adhere to when
VOLUME 20 NUMBER 3 1998
67
they do not th~nl~ catefully about ~t, ~s the Platonic s c h o o l Plato ( 4 2 7 B C - 3 4 7 BC) cla,med the ~dea o f a chaz~ was mo~e ~eal than a n y pa~twu l a r chair Thus P l a t o n w Mathem a t ~ c m n s w,ll s a y they 'd~scove~ed' a result, not 'created' ~t The houble wzth P l a t o m s m ,s ~t f a z l s to be very behevable, and certainly cannot account fo~ how M a t h e m a h c s evolves, as d~st,nct f r o m e v p a n d , n g and elaborating, the b a s w ~deas and d e f i m t,ons o f Mathematics have g~adually changed ove~ the centuries, and th~s does not f i t well w~th the ,dea o f the , m m u t a b l e P l a t o m c ~deas I w a s a graduate student ~n M a t h e m a t t c s when th,s f a c t [Hdbert's ~nsert~on o f axzoms o f betweeness and ~ntersectzon ~nto E u c h d ' s postulates fo~ plane geometry] came to m y att e n h o n I read up on st a b,t, and then thought a great deal The~e a~e, I am told, some 4 6 7 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 m e one o f the reasons no theorem u, as false was that Hdbert 'knot,' the Euchdean theorems u, ete 'correct,' and he had p~clted h~s added postulates so th~s would be true But then I soon reahzed E u c h d had been ,n the same pos~hon, E u c h d k n e w the 'truth' o f the Pythagorean theo~em, and m a n y othe~ theorems, and had to f i n d a system o f postulates w h i c h would let h~m get the ~esults he k n e w ~n advance E u c h d dzd not lay down postulates and m a k e deduchons as ~t ,s commonly taught, he felt h,s w a y back f r o m 'known' results to the postulates he needed~
Richard Hamnung has served as a c o n s u l t a n t to the E l d e r s of the M o r m o n Church, s e r v e d on the Board of Directors of a large c o m p u t e r corporation, spent 30 y e a r s as a Member o f Technical Staff at Bell Telephone Laboratories, l e c t u r e d world-wide, received a n u m b e r of p r e s t i g i o u s m e d a l s a n d awards, and s p e n t 20 y e a r s at Naval P o s t g r a d u a t e School m the thick of educaUon His chatty, idiosyncratic, s o m e t ] m e s annoying, a l w a y s t h o u g h t provol~ng b o o k is one of a kind and a t e m b l y good r e a d
THE MATHEMATICAL tNTELLIGENCER
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 o b i t u a r y p a g e of the New York Times, Sunday, J a n 11, 1998, t h e r e a p p e a r e d an article u n d e r the headhne, 'Richard Hamming, 82 Dies, P i o n e e r in Digital T e c h n o l o g y " I quote from the article R~cha~d Wesley H a m m z n g , who d~scove~ed mathematzcal f o r m u l a s that allow computers to correct thez~ own errots, mako~g possible such ~nnovah o n s as modems, compact dzsks and satelhte c o m m u n t c a t w n s , d~ed on Wednesday at a hospital ,n Monterey, C a h f , where he heed He was 82 He d~ed o f a heart attach, h~s f a m ~ly sa~d
Feynman's Lost Lecture by David L Goodste~n and J u d i t h 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
Richard F,eynman was one of flus century s great physlctsts He shared the 1965 Nobel Prize for Physics with Juhan Schwmger a n d Shimchtro T o m o n a g a for the invention o f quant u m e l e c t r o d y n a m i c s Most p e o p l e with an interest in things scmnt]fic will recall that F e y n m a n served m 1986 on the p r e s i d e n t i a l c o m m m s m n investigating the Challenger s p a c e shuttle disa s t e r D u n n g a televised h e a n n g o f the commission, he d r a m a t i c a l l y d e m o n s t r a t e d that O-nng seal failure at low t e m p e r a t u r e s was a likely c a u s e o f the acodent In 1961 Feyrmlan agreed to t e a c h the two-year introductory physics course at the Cahforma Institute of Technology Ttus s e n e s of lectures was r e c o r d e d and transcribed, and the b l a c k b o a r d s p h o t o g r a p h e d F r o m this mformatmn, the mteruatmnally r e n o w n e d "Feynman
Lectures on Physics" were p r o d u c e d and p u b h s h e d In 1687 N e w t o n published his inverse-square l a w of grawty In the magnificent w o r k Phdosoph,ce N a t u ~ a h s P ~ n c , p ~ a M a t h e m a t w a , n o w commonly k n o w n as the Ib'~nc,pm The Pr~nc~pm is p r o b a b l y the greatest scientific w o r k e v e r published and has lnt n g u e d scmntmts and m a t h e m a t i c i a n s b e c a u s e o f the vast extent of the ground c o v e r e d a n d the beauty a n d difficulty of the p r o o f s zt c o n t a m s F e y n m a n ' s Lost Lecture is a reconstructaon of a lecture gqven by F e y n m a n which centered around attemptmg 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 w i n t e r quarter in 1964 as a guest lecture, not part of the ongmal lecture course It w a s ongmally r e c o r d e d on audio cassette, but the accompanymg photographs were nnslmd Thus, it had not been possible to reconstruct this lecture until m April 1992 Feynman's ongmal notes were dtscovered m the office of his colleague, Robert Lelghtman, followmg Leightman's death Once F e y n m a n ' s notes w e r e unearthed, D a w d Goodstem, a p h y s m s p r o f e s s o r at Caltech who w o r k e d with Feynman, w a s able to r e c o n s t r u c t b y sleuthhke d e d u c t i o n the lecture m its entirely It is n o t m a d e clear w h e t h e r it was ever a t t e m p t e d to locate n o t e s t a k e n b y a t t e n d e e s at the lecture for verification p u r p o s e s By w a y of an introduction to the subject, the b o o k p r o v i d e s b a c k g r o u n d m f o r m a t m n relating to the w o r k o f Tycho Brahe, Kepler, Newton, and others, t o g e t h e r with s o m e amusing anecdotal r e m ] m s c e n c e s of D G o o d s t e m ' s relatmnship with F e y n m a n Some photographs of F e y n m a n at the blackb o a r d are also r e p r o d u c e d The epilogue d i s c u s s e s b n e f l y the w o r k o f Maxwell and Rutherford, and describes how, after two h u n d r e d years, Einstein's t h e o n e s of relatlv~ty supers e d e d N e w t o n ' s t h e o r y of gravttatmn for s p e e d s a p p r o a c h i n g the s p e e d of light and for large c o n c e n t r a t i o n s of matter The r e c o n s t r u c t i o n is a bit l a b o r e d in places, p a r t i c u l a r l y m r e s p e c t o f
G t
F=gure 1 Construct,on Of Elhpse
Kepler's 2nd Law (equal areas swept out in equal time, which also implies c o n s e r v a t i o n of angular m o m e n t u m ) A more interesting part of the lecture is where F e y n m a n appeals to F e r m a t ' s Pnnclple, 1 e , light always takes the shortest path, in order to provide a s o m e w h a t novel proof of a property of an elhpse rather than adopting a purely geometrical approach, Figure 1 The p r o o f also c o n t a m s a very remarkable ~eloclty diagram, Figure 2, which was published previously by James Clerk Maxwell In his 1877 b o o k Matter and Motion Maxwell attributes the method to Sir William Hamilton, which goes to show how difficult It is to discover something completely o n g m a l F e y n m a n was apparently u n a w a r e of Maxwell's book, b e c a u s e he credits V Fano and L F a n o
with some of the Ideas in their discussions of the Rutherford Scattenng Law in the 1957 b o o k Basic Physws of Atoms and Molecules F e y n m a n shows rather cleverly that, as a result of Kepler's 2nd Law, orbit velocity diagrams subject to an mverse-square law of gravity must be circular The objective of the lecture was for F e y n m a n to prove to his students that elhpt~cal planetary orbits with the s u n at one focus are a direct c o n s e q u e n c e of Newton's reverse-square law However, close i n s p e c t i o n of the book re~eals that n e i t h e r F e y n m a n n o r the Goodstems have truly provided such a proof Nevertheless, the Goodstems present the F e y n m a n lecture as if It did actually c o n t m n a b n l h a n t proof, and this is a very real w e a k n e s s In the lecture given in c h a p t e r 4, F e y n m a n referred repeatedly to his "elementary demonstrations" and "demonstrations " F e y n m a n omits some crucial steps and r e f i n e m e n t s that would have to be Included for his d e m o n s t r a t i o n s to be acceptable as a proof Missing c o m p o n e n t s include 9 an explanation of the scalmg bet w e e n the hodograph velocity diagram and the orbit diagram, 9 a coherent a r g u m e n t why it is justified to use the p e r p e n d i c u l a r bisector of Op (diagram on page 162) to locate the corresponding point, P, on the orbit diagram, w h e n It Is not k n o w n a p r w m that the a n s w e r will turn out to be an ellipse, and, 9 an adequate e x p l a n a t i o n of how par-
\
IJ S
I
a) F,gure 2 a) Orb,t D,agram b) Velocity Dmgram
b)
abohc and hyperbolic orbits are identified, 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 F e y n m a n did d e m o n s t r a t e the existence of elhptlcal orbit solutions to the problem, what he did n o t demonstrate is the lmlqueness of these solutions Furthermore, he alludes to this situation on page 164 " is what I proved that the ellipse is a possible solution to the p r o b l e m " Unfortunately, F e y m n a n also made other statem e n t s apparently contradicting this view, so we will n e v e r really know how n g o r o u s he believed has lecture to be D u n n g his lecture, F e y n m a n confided to has s t u d e n t s that he had expermnced considerable difficulty with some of the comc-sectlon geometry F e y m n a n states " he [Newton] perpetually uses (for me) completely obscure properties of the conic sections," and " the r e m m n m g d e m o n s t r a t i o n is not one which c o m e s from Newton, because I found I c o u l d n ' t follow it myself very well, b e c a u s e it mvolves so m a n y properties of conic sections So I cooked up a n o t h e r o n e " As it happens, most the proofs in question were o n g m a l l y published in The Conws, Book III by Appolomus, circa 200 B C , a n d all were c o m m o n l y 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 s o m e o n e with such a ~ast knowledge of mathematics and physms as Feynman, it makes one w o n d e r how n m c h other useful knowledge has been dropped from the m o d e r n 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 Intell~gence~, ~ol 18, no 2), pro~ades a clear exposition from basic p n n c l p l e s 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
VOLUME20 NUMBER3 1998 69
the College Mathematws Jour~al 25, no 2 (March 1994), which pro~ades an excellent lntroductmn The Goodstems make a n assertion which is not umversally accepted by historians of scmnce " There ~s httle doubt that he [Newton] used these powerflfl tools [differential and integral calculus] 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 subsequently p u b h s h e d This was a result of his clash(es) with Robert Hooke
70
THE MATHEMATICALINTELLIGENCER
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 s a n s f y m g to see that a great physmmt like F e y n m a n was interested sufficiently m the h l s t o n c a l d e v e l o p m e n t of h~s sub3ect that he was p r e p a r e d to devote s l g m f m a n t n i n e to p r e s e n t i n g h~stoncal developments, s u c h as Newton's inversesquare law of gra~qty, to his s t u d e n t s I am c o m a n c e d that u n w e r s m e s will t u r n out b e t t e r educated s c m n n s t s m the future ff they encourage s t u d e n t s to a p p r e c m t e the p r o b l e m s that conf r o n t e d great scientists m the past, and to u n d e ] s t a n d how those s c m n n s t s solved them wath the tools available at the time It must be stud that ff the Goodstems had included an appendax providing a n over~aew of hodograph theory, the edu-
cauonal value of the book would have been greatly e n h a n c e d Nevertheless, this book has been produced to a high quahty and will be a'valuable addition to any library, and is recommended readmg for all students of Newton and Feymnan All the discussions should be readily understood by anyone famlhar with high school mathematms
Acknowledgments The re~aewer w o u l d hke to acknowledge useful and mformatlve discussions with Professor Robert Burckel (Kansas State) a n d Professor Robert Wemstock ( O b e r h n College) m connection with this rexaew Control Eng~neenngResearch Centre Oty University Northampton Square London EC1 0HB United Kingdom e-ma~l graham@sastco uk
~3~-,i;t,l.-t.],.,ta-.-
Robin
Isaac Newton
Please send all submtsstons to the Stamp Corner Editor Robin Wilson, Faculty of Mathemattcs and Computtng, The Open Unlverstty Milton Keynes, MK7 6AA England
72
Wilson,
Editor
I
n 1987 (Vol 9, no 4), the Stamp Corner c o m m e m o r a t e d the 300th a n m v e r s a r y of the p u b h c a t m n of Isaac Newton's Pr~nc~p~a Mathematwa w~th a two-page spread on Newton stamps rssued up to that tune We n o w update the story by presenting some Newton s ~ p s that have appeared m the past ten years, most of these commemorate the 3 5 ~ a m u v e r s ~ of his b~ m 1642---or 1~3, d e p e n ~ g on w h e n the country m quesUon adopted the Gregorian calendar Most notable was a set of five s ~ p s rssued by N o a h Kore~ of which t ~ e e are s h o w n here These stamps survey Newton's s o e n t ~ c work, from his contnbuUons to opUcs ~ s reflecting telescope), wa gra,ntaUon (not shown here), to the bmonual ~ e o r e m U ~ o r m n a t e l y , the b m o n u a l expansion gnven on the stamp rs the fimte versmn, where the exponent is a natural number, this had already been known for several h u n d r e d years before Newton HJS c o n t n b u t m n was the e x p a n s m n as an mfimte series
I
THE MATHEMATICAL INTELLIGENCER 9 1998 SPRINGER VERLAG NEW YORK
of (a + b)% where a may be non-mtegral and/or negative This he obtamed while still an undergraduate at Cambndge, although it dJd not appear m print until 1704, as an appendix to hrs
Opt~cks Newton's work m optics zs featured m the German stamp, incongruously with his second law of mohon, and on a 1994 Nmaraguan stamp as part o f a sen e s on astronomers I-hs c o n t n b u t m n s to the theory of gra~ataUon are commemorated on the Uruguayan stamp of 1996 and on a Bulgarian stamp featuring the reverse-square law
