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Letters

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Editor

The Mathematical InteUigencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.

T h e B e n e f i t s of N o n - B l i n d n e s s

The U s e s of R e f e r e e s

The opinion piece by "Lemme B. Bourbaki" on blind refereeing (Winter 1999) imputes only evil to giving a referee the identity of the author. But a bias can be used in a positive way. A good illustration was given by Saunders Mac Lane at the time of the AMS experiment with blind refereeing: he pointed out that when he knows that the author is a beginner, he goes out of his way to be helpful. I was happy to read this, especially as I was already doing the same thing. For instance, in my referee's report, under comments for the author, I would tend to write: (1) I feel sure that in the main theorem, you can dispense with the hypothesis that A is normal, and I suggest you spend a month or two looking into this question. (2) The symbol t' that appears in Theorem 3 has a different meaning from the same symbol in Theorem 1. Suggestion: call the new one t*. About the same paper some smart-alec referees might write, "The results tend to be weak and the notation is sometimes inconsistent," or worse, "Weak results, inconsistent notation." In the other direction, if I recognize the author as an established expert in the field, or deduce the fact from the bibliography, then I expect his proofs to be correct, and do not take the time to check every detail. (This shortens the prepublication process, for which both author and editor are grateful-though I get my comeuppance when the editor loses little time sending me another manuscript to referee.)

In her/his opinion piece "On Blindness" published in The Mathematical InteUigencer, vol. 21, no. 1, Lemme B. Bourbaki argues that the referees should not know who the authors of a paper are before making a judgment about its publication. These arguments look irrefutable if we consider the referee as a judge who decides (or helps to decide) whether the paper should be made accessible to mankind or discarded.' However, nowadays e-mall, ftp, and www servers are (in most parts of the world) easier to access than printed journals, so in reality all papers are made accessible. Journals in coming years may have more the role of recommended reading lists. Maybe one should consider a different question: not whether to make the authors invisible, but whether to make the referees visible. If what is really happening is that somebody is recommending papers from the electronic archive that she/he finds interesting, then it seems natural that such a recommendation be signed.

Leonard Gillman Department of Mathematics University of Texas at Austin Austin, TX 78712 USA email: [email protected]

Alexander Shen Institute for Problems of Information Transmission Ermolovoi 19 K-51 Moscow GSP-4, 101447 Russia e-mail: [email protected]

L e m m e B. Bou~baki responds:

I wonder how Professor Gillman et al. recognize which manuscripts are by beginners? Just by unfamih'arity of the authors' names? It seems more likely that they try to recognize the author as a beginner from the content of his or her paper. Blind refereeing would not impair this ability. Perhaps he would appreciate it if referees received manuscripts which

O 1999 SPRrNGER-VERLAG NEW YORK, VOLUME 21, NUMBER 3, 1999

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REFERENCES [1] L~vy-Leblond, J.-M., 1997, Math. Intelligencer 19(4), 63. [2] Bracewell, R.N., 1986, The Fourier Transform and its Applications, 2nd ed. (New York: McGraw-Hill). [3] Glasser, L., 1987, J. Chem. Educ., 64, A228. L. Glasser Centre for Molecular Design Department of Chemistry University of the Witwatersrand Johannesburg South Africa e-mail: [email protected]

-0,5-

-

10

time

20

Figure 1. The Fourier kernel multiplied by an exponential decay, f(~

(with or without authors' names) indicated authors' level of seniority? But whatever the procedure, Professor Gillman's letter is not such as to make a beginner hope to get him as referee. By his own account, he doesn't check all the details of an experienced mathematician's proof, but reserves his mordant scrutiny for the beginner. Exactly. If he didn't understand from my original article why some of us favor blind refereeing, perhaps his own example will make it clearer. Dr Shen's interesting proposal seems independent of the position in my original Opinion piece. Fourier and Argand in 3D J.-M. L~vy-Leblond has recently [1] depicted Fourier series, sums, and integrals in two-dimensional "portraits" and "movies" as Argand diagrams in the complex plane. The "portrait" describes the series as a sequence of vectors at equal angles to one another, so that the Fourier sum is the vector connecting start to end of this chain; the "movie" is obtained by altering each of the equal angles between segments in concert with the time variable, so that the sum vector rotates correspondingly. These "portraits" and "movies" provide such an instructive geometrical view of the processes of Fourier analysis that L~vy-Leblond wonders "why they have not been put forward earlier." In point of fact, diagrams with much

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THE MATHEMATICALINTELLIGENCER

=

exp(-x),

the same motivation have been presented, sometimes in three dimensions [2] with the time axis (the argument) providing the third dimension. I have used such three-dimensional diagrams [3] in attempting to elucidate the relationship between the physical action of spectral analysis and the mathematics of Fourier transformation. This approach shows the Fourier kernel exp(-i2~rxs) as a vector of unit magnitude which, with the passage of time x, coils with a frequency s; plotting time normal to the Argand plane, this appears as a helix. Then if the functionf(x) to be transformed is real, multiplying the kernel by it generates a space curve which is altered in the course of rotating around the time axis (Fig. 1). With a little more effort, the 3-dimensional figures can be displayed stereoscopically (Fig. 2).

Still More Simple and Straightforward? In "The simple and straightforward construction of the regular 257-gon," vol. 21 (1999), no. 1, 31-37, Prof. C. Go~lieb states that "the construction of the regular 257-gon does not seem to have been performed very often" and goes on to cite Richelot's 1832 series of 4 papers dealing with that construction. A very simple construction, without any use of Galois theory, and in some points quite similar to the author's, was described by K. Haage, "Einfache Behandlung der 257-teilung des Kreises," Z. Math. Naturw. Untercicht 41 (1910), 448-458. The method Gottlieb describes for the construction of two numbers given their sum and their product is attributed to the famous 19th-century historian Carlyle by D. W. DeTemple, "Carlyle circles and the Lemoine simplicity of polygon constructions," Amer. Math. Monthly 98 (1991), 97-108, where a different construction of the regular 257-gon is outlined. Victor Pambuccian Department of Mathematics Arizona State University West P.O. Box 37100 Phoenix, AZ 85069-7100 USA e-mail: [email protected]

Figure 2. Stereo pair representation of a periodic signal, showing both the p l o t of the

kernel and the plot o f the kernel multiplied by the signal. From [3], by permission.

Christian Gottlieb replies: I did not mean to claim that nothing had happened since Richelot! I expect that still more references will be communicated to me. Avoiding Galois theory does make the proof more accessible. My inten-

tion w a s that m y p a p e r s h o u l d b e readable b y those unfamiliar with Galois theory. On the o t h e r hand, I b r o u g h t s o m e Galois t h e o r y in b e c a u s e it helps u n d e r s t a n d w h a t is going on. The c o n s t r u c t i o n of t w o n u m b e r s given their s u m and p r o d u c t w a s found first b y Descartes (as I m e n t i o n e d in passing, on p. 33). This w a s an import a n t a c h i e v e m e n t in D e s c a r t e s ' s time, t h o u g h by Carlyle's time it w a s a m e r e exercise. A C o r r e c t i o n T e r m for t h e Biblical ~ ? In the Fall issue of the Intelligencer, G e o r g e C. Bush tries to e x o n e r a t e the a n c i e n t H e b r e w s from having the very c r u d e estimate ~r - 3.

This r e m i n d s m e o f w h a t I l e a r n e d from m y late colleague, P r o f e s s o r Shlomo Breuer, concerning the circ u m f e r e n c e o f the t e m p l e ' s m o l t e n sea, as it is written in the s a m e Biblical passage (1 Kings 7:23) " . . . a n d a line o f thirty cubits did c o m p a s s i t . . . " It is the c u s t o m among the J e w s to use the 22 alphabet letters for numbers: aleph for 1, beth for 2, and so on for 3,4, . . . , 1 0 , 2 0 , . . . , 9 0 , 1 0 0 , . . . , till the last letter, tav, for 400. The w o r d for line, kav, which is 106 in this method, app e a r s in the traditional text with an ext r a letter. This s e e m s to be one of the few h u n d r e d spelling mistakes which were the result of copying the Bible through the ages until getting the canonical version (about the 6th century),

where the traditional reading (qere) differs from the written text (ktiv). However, it m a y be that this particular "spelling mistake" was done on purpose b y a Hebrew scholar who, like G.C. Bush, w a s troubled b y the mathematical mistake. The e x t r a letter has the numerical value 5. Thus, the "correction" is: "relJlace 106 b y 106 + 5." Applying this correction factor to 30/10, one gets 333/106 = 3 . 1 4 1 5 . . . , an excellent rational approximation in its time. Dan Amir School of Mathematical Sciences Tel Aviv University Tel Aviv 69978 Israel e-mail: [email protected]

VOLUME 21, NUMBER 3, 1999

5

EDWARD BERTRAM AND PETER HORAK

Somc Applications of Graph Theocy to Other Parts of Mathematics

any mathematicians are now generally aware of the significance of graph theory as it is applied to other areas of science and even to societal problems. These areas include organic chemistry, solid state physics and statistical mechanics, electrical engineering (communications networks and coding theory), computer science (algorithms and computation), optimization theory, and operations research. The wide scope of these and other applications has been well documented (e.g., [4, 11]). However, not everyone realizes that the powerful combinatorial methods found in graph theory have also been used to prove significant and well-known results in a variety of areas of pure mathematics. Perhaps the best known of these methods are related to a part of graph theory called matching theory. For example, results from this area can be used to prove Dilworth's chain decomposition theorem for fmite partially ordered sets. A well-known application of matching in group theory shows that there is a common set of left and right coset representatives of a subgroup in

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THE MATHEMATICALINTELLIGENCER9 1999SPRINGER-VERLAGNEWYORK

a finite group. Also, the existence of matchings in certain infinite bipartite graphs played an important role in Laczkovich's affirmative answer to Tarski's 1925 problem of whether a circle is piecewise congruent to a square. Other applications of graph theory to pure mathematics may be found scattered throughout the llteraturel Recently, a collection of examples [10] showing the application of a variety of combinatorial ideas to other areas has appeared. There, for example, matching theory is applied to give a very simple constructive proof of the existence of Haar measure on compact topological groups, but the other combinatorial applications do not focus on graph theory. The graph-theoretic applications presented here do not overlap with those in [10], and no attempt has been made at a survey. Rather, we present five examples, from

set theory, n u m b e r theory, algebra, a n d analysis, w h o s e s t a t e m e n t s are well k n o w n o r are easily u n d e r s t o o d b y m a t h e m a t i c i a n s w h o are n o t e x p e r t s in the area. Additional criteria for c h o o s i n g t h e s e five e x a m p l e s w e r e t h a t the s t a t e m e n t can b e f o r m u l a t e d using few definitions and that the p r o o f c a n b e e x p l a i n e d in a relatively s h o r t space, w i t h o u t t o o m u c h technical detail. The p r o o f s h o u l d exhibit the strength a n d elegance o f graph-theoretic m e t h o d s , although, in s o m e cases, one m u s t consult t h e lite r a t u r e in o r d e r to c o m p l e t e t h e proof.

Preliminaries F o r t h e c o n v e n i e n c e of o u r readers, w e recall the n e c e s s a r y definitions from graph theory. An (undirected) graph G = (V, E ) is a p a i r in w h i c h V is a set, the v e r t i c e s o f G, a n d E is a set o f 2-element subsets o f V, the edges of G. An edge e ~ E is d e n o t e d b y e = xy, x a n d y being the e n d v e r t i c e s of e. Here, e is i n c i d e n t with x (and e is incident with y). T h e degree of a v e r t e x v, deg(v), is the n u m b e r of e d g e s i n c i d e n t with v. In a d i r e c t e d graph, o r simply digraph, G = (V, E ) , the (directed) e d g e s a r e o r d e r e d p a i r s of v e r t i c e s o f V and a r e d e n o t e d b y e = (x, y). A t r a i l of length n in a g r a p h G (digraph G) is a s e q u e n c e o f v e r t i c e s Xo, x l , x2, 9 9 9 Xn (xi E V), such that for i = 0, 1, . . . , n - 1, x i x i + l is an e d g e o f G ((xi,xi+l) is an orie n t e d edge o f G). If x0 = x,~, t h e n the trail is said to b e closed. When all the vertices in the sequence are distinct, the trail is called a path. A c l o s e d trail, all of w h o s e vertices a r e distinct e x c e p t for Xo a n d Xn, is called a cycle. A g r a p h G is c o n n e c t e d if a n y t w o vertices of G are j o i n e d by" a p a t h in G. Otherwise, G is said to b e d i s c o n n e c t e d . T h e c o m p o n e n t s of G a r e t h e m a x i m a l c o n n e c t e d subg r a p h s o f G. A tree is a c o n n e c t e d graph w i t h o u t cycles. A g r a p h G = (V, E ) is said to b e b i p a r t i t e if V can b e partit i o n e d into t w o n o n e m p t y s u b s e t s A a n d B such t h a t e a c h edge o f G has one end v e r t e x in A and one end v e r t e x in B. Then, G is also d e n o t e d b y G = (A, B; E). If ( H , . ) is a group and S a s e t of g e n e r a t o r s o f H, n o t n e c e s s a r i l y minimal, the C a y l e y g r a p h G(H, S), o f ( H , . )

with r e s p e c t to S, h a s vertices x,y,... E H, and x y is an edge if a n d only if e i t h e r x = y . a o r y = x . a for s o m e a E S. If G is any g r a p h and e = x y an edge o f G, t h e n b y a cont r a c t i o n along e, w e m e a n t h e graph G ' which arises f r o m G b y identifying t h e vertices x and y ( s e e Fig. 1). We say that a graph G1 is c o n t r a c t i b l e onto a graph G2 if t h e r e is a s e q u e n c e of c o n t r a c t i o n s along e d g e s w h i c h t r a n s f o r m s G1 to G2. T h e a u t o m o r p h i s m g r o u p of a g r a p h G is t h e group o f all p e r m u t a t i o n s p o f the vertices o f G with t h e p r o p e r t y that p ( x ) p ( y ) is a n edge of G iff x y is a n edge o f G. A g r o u p H of p e r m u t a t i o n s acting o n a set V is called s e m i r e g u l a r ff for e a c h x ~ V, t h e stabilizer H x : = {h ~ H I xh = x} consists o f the identity only, w h e r e x h d e n o t e s the image o f x u n d e r h. If H is transitive a n d semiregular, t h e n it is regular.

Cantor-Schr6der-Bemstein Theorem Our first e x a m p l e is a graph-theoretical p r o o f o f the classical result of SchrOder a n d Bernstein. Actually, the theor e m w a s s t a t e d b y Cantor, w h o did n o t give a proof. The t h e o r e m w a s p r o v e d i n d e p e n d e n t l y b y S c h r ~ d e r [1896) a n d Bernstein (1905). The i d e a b e h i n d the p r o o f p r e s e n t e d h e r e can b e f o u n d in [8]. Theorem (Cantor-SchrUder-Bernstein):

L e t A a n d B be sets. I f there i s a n i n j e c t i v e m a p p i n g f: A --->B a n d a n i n j e c t i v e m a p p i n g g: B ~ A, t h e n there i s a b i j e c t i o n f r o m A onto B, t h a t is, A a n d B h a v e the s a m e c a r d i n a l i t Y .

Proof. Without loss of generality, we m a y a s s u m e that A a n d B are disjoint. Define a bipartite graph G = (A, B; E ) , w h e r e x y E E if and only if e i t h e r f ( x ) = y o r g ( y ) = x, x E A, y E B. By o u r hypothesis, 1 -< deg v -< 2 for each v e r t e x v of G. Therefore, each c o m p o n e n t o f G is either a one-way infinite p a t h (i.e., a path of the form x0, Xl, 9 9 9 Xn, 9 9 9 ), o r a twow a y infinite p a t h (of the f o r m . . . X - n , X-n+1, 9 9 9 x - l , Xo, xl, 9 9 9 Xn, 9 9 .), o r a cycle o f even length with m o r e than two vertices, o r a n edge. Note that a finite path of length -->2 cannot be a c o m p o n e n t of G. Hence, t h e r e is in each com-

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ponent a set of edges s u c h that each vertex in the component is incident with precisely one o f these edges. Hence, in each component, the s u b s e t of vertices f r o m A is o f the same cardinality as the s u b s e t of vertices from B. []

C o r o l l a r y . I f J is a subgroup o f a group H, then a n y G(H, S) is contractible onto G(J, T) f o r some set T o f generators o f J. P r o o f . H R , the regular r e p r e s e n t a t i o n of H, acts naturally

Fermat's (Little) T h e o r e m There are m a n y p r o o f s o f F e r m a t ' s Little Theorem, even s h o r t algebraic o r n u m b e r - t h e o r e t i c proofs 9 The first k n o w n p r o o f of the t h e o r e m was given by Euler, in his lett e r of 6 March 1742 to Goldbach. The i d e a o f t h e graphtheoretic one p r e s e n t e d b e l o w can be f o u n d in [5] where this method, t o g e t h e r w i t h s o m e n u m b e r - t h e o r e t i c results, w a s used to p r o v e E u l e r ' s generalization to n o n p r i m e modulus. T h e o r e m (FermaO: L e t p be a p r i m e such that a is not divisible by p. Then, a p a is divisible by p. -

-

Proof. Consider the g r a p h G = (V, E), w h e r e V is the set o f all sequences ( a l , a 2 , 9 9 9 , ap) of natural n u m b e r s bet w e e n 1 a n d a (inclusive), with a i r aj for s o m e i C j . Clearly, V h a s aP - a elements. F o r any u E V, u = (Ul, 9 9 9 Up-l, Up), let us s a y t h a t u v E E j u s t in c a s e v = (Up, Ul, 9 9 9 Up-i). Clearly, e a c h v e r t e x o f G is of d e g r e e 2, so each c o m p o n e n t o f G is a cycle, of length p. But then, the numb e r of c o m p o n e n t s m u s t b e (a p - a)/p, so pla p - a. []

Nielson-Schreier T h e o r e m Let H be a group and S b e a set o f g e n e r a t o r s o f H. Then, a p r o d u c t o f g e n e r a t o r s and their inverses w h i c h equals (the identity) 1 is c a l l e d a trivial relation a m o n g the gene r a t o r s in S if 1 can b e o b t a i n e d from that p r o d u c t by rep e a t e d l y replacing x x -1 o r x - i x b y 1, o t h e r w i s e such a p r o d u c t is called a nontrivial relation 9 A group H is f r e e if H has a set o f g e n e r a t o r s such t h a t all relations among the g e n e r a t o r s a r e trivial. In [1] Babai p r o v e d the N i e l s o n - S c h r e i e r T h e o r e m on s u b g r o u p s of free groups, as well as o t h e r results in diverse areas, from his "Contraction Leinma." The p a r t i c u l a r case of this l e m m a w h e n G is a tree, and its use in p r o v i n g the N i e l s o n - S c h r e i e r Theorem, w a s also o b s e r v e d b y Serre [12, Chap 9 1, Sec. 3]. The p r o o f o f the Contraction L e m m a b e l o w is s o m e w h a t technical, although it uses only t h e i d e a s from group t h e o r y a n d graph t h e o r y w e have a l r e a d y recalled, and is o m i t t e d here 9 C o n t r a c t i o n L e m m a . Let H be a s e m i r e g u l a r subgroup o f the a u t o m o r p h i s m group o f a connected g r a p h G. Then, G is contractible onto s o m e Cayley graph o f H. If H is a group a n d h E H, c o n s i d e r the p e r m u t a t i o n hR o f H o b t a i n e d b y multiplying all the e l e m e n t s o f H on the right b y h. The collection HR = {hR: h ~ H} is a regular group o f p e r m u t a t i o n s ( u n d e r c o m p o s i t i o n ) a n d is called t h e (right) regular p e r m u t a t i o n r e p r e s e n t a t i o n o f H. It is k n o w n [1] t h a t G is a Cayley graph o f t h e group H if and only if G is c o n n e c t e d and H R is a s u b g r o u p of the a u t o m o r p h i s m group o f G.

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as a r e g u l a r p e r m u t a t i o n group o n G(H, S), w h i c h is connected. Thus, the s u b g r o u p of H R c o r r e s p o n d i n g to the ele m e n t s o f J i s a semiregular s u b g r o u p of the a u t o m o r p h i s m group o f G(H, S). N o w apply t h e Contraction Lemma. [] T h e o r e m (Nielson-Schreier): A n y subgroup o f a f r e e group is free. Proof. We first s h o w that in a n y group H a n d for any set S of g e n e r a t o r s of H, the Cayley g r a p h G(H, S) contains a cycle of length > 2 if a n d only if t h e r e is a nontrivial relation among t h e generators in S. To s h o w this, s u p p o s e x0, Xl, 9 9 9 Xn = Xo is a cycle of G(H, S). Then, there are ai E S, 1-i-n, such that X i - l a ~ = x i , w h e r e e i E { 1 , -1}. Hence, X n = X n - l a n ~n ---- X n - 2 a nOn-1 _ 1 a ~ n = "'" _-- xoa e1l a e22 . . . an• n , i.e., the identity 1 = a~l a~2 ... agn. If this w e r e a trivial relation, t h e n there w o u l d exist an integer i, 1 -< i -< n, such that ai = a i + l and ei = - e / + t . However, this implies that xi-1 = xi+t, a contradiction. Similarly, if a~la~2 ... a~n = 1 is a nontrivial relation, then Xo, Xl, 9 9 9 Xn-1, Xn, w h e r e xi = x i - l a ~ , 1 <- i <- n, and Xo = Xn, is a c l o s e d trail in G(H, S), w h i c h m u s t contain a cycle. S u p p o s e n o w that H is a free group, S a minimal set o f g e n e r a t o r s o f H, a n d J a s u b g r o u p o f H. Since there is no nontrivial relation on the e l e m e n t s of S, G(H, S) d o e s n o t contain a cycle. Also, from t h e c o r o l l a r y above, G(H, S) is c o n t r a c t i b l e onto G(J, T) for s o m e set T o f g e n e r a t o r s o f J. B e c a u s e any c o n t r a c t i o n o f a cycle-free graph is again cycle-free, G(J, T) m u s t b e cycle-free, and, thus, t h e r e is no nontrivial relation on the e l e m e n t s of T. Hence, J m u s t be a free group, freely g e n e r a t e d b y T. [] In [7] the interested r e a d e r m a y further pursue the substantial use of elementary graph theory in giving simplified proofs o f important theorems in combinatorial group theory.

Existence of a N o n m e a s u r a b l e Set The following p r o o f of the e x i s t e n c e of a s u b s e t of the real n u m b e r s R which is n o n - m e a s u r a b l e in the Lebesgue s e n s e is due to T h o m a s [15]9 He w r o t e his p a p e r while an und e r g r a d u a t e student. We realize t h a t m a n y r e a d e r s m a y still p r e f e r Vitali's proof 9 However, it is quite u n e x p e c t e d that this t h e o r e m can be r e d u c e d to t h e t h e o r e m below, an easily p r o v e d result in m e a s u r e theory, b y using only d i s c r e t e mathematics 9 A simple, well-known result f r o m graph t h e o r y says t h a t a g r a p h (V, E ) is bipartite if a n d only if all its cycles are o f even length. F o r a proof, it suffices to p r o v e it for conn e c t e d g r a p h s only 9 Choose a n y x ~ V a n d defme V1 = {y E V: any p a t h connecting x a n d y is o f o d d length} and V2 = {y E V: a n y p a t h connecting x a n d y is of even length}. Since t h e r e are no o d d cycles in (V, E ) , any t w o p a t h s connecting x a n d y are of the s a m e parity, so V1 and V2 yield a p a r t i t i o n o f V. F r o m the definitions of Vt and V2, it fol-

lows t h a t no edge has b o t h v e r t i c e s in the s a m e Vi, s o the g r a p h is bipartite. The s e c o n d implication is obvious. C o n s i d e r n o w the graph T = (R, E), w h e r e x y E E if and only if Ix - y[ = 3 k, with k an integer. In o r d e r to s h o w that T i s bipartite, s u p p o s e that Xo, Xl, x2, 9 9 9 Xn-1, Xn = Xo is a c y c l e of T of length n. Then, b y definition of T, Xn = Xn-1 4- 3 kn = Xn-2 +- 3 kn-1 4- 3 ~ . . . . . X0 --+ 3 k~ --+ 3 k2 ---9"" -- 3 k~ and, thus,

In 1964, S h a r k o v s k y [13] gave a c o m p l e t e a n d amazing a n s w e r to this question w i t h the following T h e o r e m . Let f: R ---> R be a c o n t i n u o u s f u n c t i o n w i t h a k-periodic point. Then, f has a n m-periodic p o i n t i f k precedes m i n the f o l l o w i n g ordering (S) o f all the n a t u r a l numbers: 3, 5, 7, 9, ..., 2.3, 2.5, 2.7, 2.9, ..., 22. 3, 22. 5, 22. 7, 22. 9, ...

- 3 k~ +- 3 k2-+ 3 k ~ - "" -+ 3 k~-- 0, w h e r e [ki}~ is a set of integers. Multiplying b o t h sides b y 3N, w h e r e N is an integer s u c h that N + ki > 0, 1 - i < n, yields

2k. 3, 2k" 5, 2 k" 7, 2 k. 9, "", 2 k+l" 3, 2k+l" 5, 2 k+l" 7, 2 k+l" 9, ""

+-- 3kl+N +--3 k2+N -- "'" -- 3k~+N = 0,

This is b e s t possible, s i n c e w h e n e v e r k a n d m are natural n u m b e r s a n d m p r e c e d e s k, t h e r e exists a c o n t i n u o u s function fi R --) R with a k-periodic point, b u t no m-periodic point. The original p r o o f by S h a r k o v s k y is very complicated, and, later, s e v e r a l m a t h e m a t i c i a n s p r e s e n t e d m u c h s i m p l e r proofs. In s o m e o f them, g r a p h t h e o r y w a s used, with t h e m o s t i m p o r t a n t s t e p being m a d e by Straffin [14]. He def'med a digraph associated-with" a p e r i o d i c p o i n t ' o f a function a n d p r o v e d t h e crucial result. F o r this p u r p o s e , let x b e a k-periodic p o i n t o f a funct i o n f l Then, the distinct v a l u e s { x , f ( x ) , f 2 ( x ) , . . . , f k - l ( x ) } d e t e r m i n e k - 1 finite intervals 11, 12, 9 . . , Ik-1, l a b e l e d from left to right, after locating these n u m b e r s in their natural o r d e r on the x (and y ) axis (see, for example, Fig. 2). Define a digraph G = (V, E) b y V = { I 1 , . . . , Ik-1} with (Ii, Ij)~E E w h e n e v e r f(Ii) D_Ij. F o r e x a m p l e the digraph corr e s p o n d i n g to t h e 4-periodic p o i n t x = 0 o f f , s e e n in Fig. 2, is the graph given in Fig. 3. A c l o s e d trail in a digraph is said to b e nonrepetitive ff it d o e s n o t c o n s i s t entirely o f a cycle of s m a l l e r length t r a c e d several times. F o r example, the digraph in Fig. 3 h a s nonrepetitive trails of lengths 1 and 2 only. Now, w e a r e able to state Straffm's theorem, w h i c h t u r n s the p r o b l e m

w h i c h implies t h a t n is even, since otherwise the left side of t h e a b o v e equation is odd, a contradiction. Thus, T is bipartite. Hence, there are sets A a n d B with A n B = ;~, A U B = R, s u c h that e a c h edge of T is incident with one v e r t e x in A a n d the o t h e r v e r t e x in B. If b o t h A a n d B w e r e m e a surable, then at least one o f them, say A, w o u l d have p o s itive m e a s u r e . F u r t h e r m o r e , for each integer k, A + 3 k C_ B, w h i c h y i e l d s A n (.4 + 3 k) = O. Since 3k---~ 0 as k---) -0% this c o n t r a d i c t s the following theorem, w h i c h is a s t a n d a r d result in m e a s u r e theory. F o r the c o n v e n i e n c e o f the reader, w e include the p r o o f f r o m [15]. T h e o r e m . Let M be a set o f real n u m b e r s w i t h p o s i t i v e Lebesgue measure. Then, there exists a 8 > 0 such that f o r every x E R, IxI < & M U (M + x) r 0 . Proof. Find a c l o s e d set F a n d an open set G with F C_ M a n d F C G such that 3A(G) < 4A(F) ( w h e r e h is Lebesgue m e a s u r e ) . Since G is a c o u n t a b l e union o f disjoint o p e n intervals, t h e r e is one a m o n g them, say I, such that 3A(/) < 4A(F n I ) . Let 8 = 89 a n d s u p p o s e that Ixl < & Then, I U (x + I) is an interval o f length less than ~h(I) w h i c h contains b o t h F n I a n d x + ( F n I ) . The last two sets c a n n o t b e disjoint, since o t h e r w i s e

~h(I) = ~h(I) + ~A(I) < h((F n I ) u (x + (F n I))) -< ~(z u (x + I ) ) --- ~ ( 1 ) ,

. . . . . .

, 2 n, 2 n - l , "'', 22, 2, 1.

IGURE

f3(x) I

which is a contradiction. Hence, O r (F fq I ) f-) (x + (F fq I ) ) C_ M n (x + M), c o m p l e t i n g the proof.

[]

R e m a r k . It is well k n o w n t h a t a n o u m e a s u r a b l e set cannot b e c o n s t r u c t e d w i t h o u t using the a x i o m o f choice. Our graph T is not connected, and, in fact, e a c h c o m p o n e n t o f T h a s only a c o u n t a b l e n u m b e r o f vertices. Thus, to define A a n d B, w e n e e d to m a k e use o f this axiom.

Sharkovsky's Theorem Let f: R --> R b e a c o n t i n u o u s function. A p o i n t x E R is called a k-periodic p o i n t o f f i f f k ( x ) = X a n d f i ( x ) r x for i = 1, 2 . . . . , k - 1. Here, f n is t h e n t h iterate off, i . e . , f n =

fofn-1. fffhas a k-periodicpoint,is itnecessary thatfhave an m-periodic point for some m r k?

f (X)~

I, f2(x )'

II

x=O

ii

f2(x)

~2/

f (x) 13

f3(x)

VOLUME 21, NUMBER 3, 1999

9

IGURE!

FIGURE ;

11

13 of the existence of a periodic point into a problem about the corresponding digraph. Theorem, [14]. I f the digraph associated w i t h a k-periodic point of a f u n c t i o n f has a nonrepetitive dosed trail of length m, then f has an m-periodic point. Figure 4 shows the digraph associated with any 3-periodic point of a function. Clearly, this digraph contains a nonrepetitive closed trail of arbitrary length, showing that the existence of a 3-periodic point o f f implies t h a t f has periodic points of all orders. This special case, and other results on systems with 3-periodic points, were proved in 1975 by Li and Yorke [9], when Sharkovsky's theorem was still little noticed. The reader is referred to Straffin's one-page proof of his theorem above, which is modeled after Li and Yorke's. Straffm's proof makes essential use of two lemmas which are standard in analysis courses: L e m m a 1. Suppose I and J are dosed intervals, f continuous, and J C f ( I ) . Then there is a closed interval Q c I such that f(Q) = J. L e m m a 2. Suppose I is a closed interval, f continuous, and I C f(1). Then f has a f i x e d point in I. Using his theorem above, Straffm proved some parts of Sharkovsky's Theorem, and his approach subsequently allowed several authors to complete the proof (see [3, 6]). In the proof of Sharkovsky's Theorem presented in [2] graphs were used without applying Straffm's result. To give some of the flavor of the proofs in [3, 6] we sketch the proof of a partial result, showing that in the ordering S, all even integers lie after all the odd integers (see [6]). Theorem. I f a continuous function f: R---> R has a point of odd period 2n + 1 (n >- 1), then it has periodic points of all even periods. FIGURE 4

I1

10

THE MATHEMATICAL INTELLIGENCER

Proof (sketch). For n = 1, the proof was given above. Now, suppose n > 1 and assume by way of induction that the theorem is true w h e n e v e r f h a s a point of odd period 2m + 1, where 3 -< 2m + 1 < 2n + 1. Straffm proved generally that the digraph corresponding to a periodic point of period k contains a closed trail of length k in which some vertex is repeated exactly twice. In our case, k = 2n + 1, and this closed trail can, therefore, be decomposed into two closed nonrepetitive trails, one of which has odd length, say 2m + 1 < 2n + 1. If this closed trail is of length greater than one, the assertion follows by our induction assumption and the previous theorem. If not, then Straffm proved that our digraph must contain the directed subgraph given in Fig. 5. This subgraph has a cycle of length 2, and one of length 4. For any even number t > 4, we may begin a nonrepetitive closed trail of length t at the bottom right-hand vertex, traverse the 4-cycle once, and follow this by traversing the 2-cycle exactly (t - 4)/2 times. By the previous theorem, the existence of all even periods follows. [] REFERENCES

1. L. Babai, Some applications of graph contractions, J. Graph Theory 1 (1977), 125-130. 2. L. Block, T. Guckenheimer, M. Misiurewicz, and L.-S. Young, Periodic points and topological entropy of one dimensional maps, in Global Theory of Dynamical Systems, Lecture Notes in Mathematics vol. 819, Springer-Verlag, Berlin (1980), pp. 18-34. 3. U. Burkart, Interval mapping graphs and periodic points of continuous functions, J. Combin. Theory (B) 32 (1982), 57-68. 4. L. Caccetta and K. Vijayan, Applications of graph theory, in Fourteenth Australasian Conference on Combinatorial Mathematics and Computing (Dunedin, 1986); Ars. Combin. 23 (1987), 21-77. 5. K. Heinrich and P. Horn.k, Euler's theorem, Am. Math. Monthly 101 (1994), 260. 6. C.-W. Ho and C. Morris, A graph theoretical proof of Sharkovsky's theorem on the periodic points of continuous functions, Pacific J. Math. 96 (1981), 361-370. 7. W. Imrich, Subgroup theorems and graphs, in Combinatorial Mathematics V, Proceedings of the Fifth Australian Conference, Lecture Notes in Mathematics Vol. 622, Springer-Verlag, Berlin (1977), pp. 1-27. 8. D. K6nig, Theorie der endlichen und unendlichen Graphen, Akademische Verlagsgesellschaft, Leipzig (1936); reprinted by Chelsea, New York (1950).

9. T.-Y. Li and J. A. Yorke, Period three implies chaos, Am. Math. Monthly 82 (1975), 985-992. 10. L. L6vasz, L. Pyber, D. J. A. Welsh, and G. M. Ziegler, Combinatorics in pure mathematics, in Handbook of Combinatorics (R. L. Graham, M. Gr6tschel, and L. Lovasz, eds.), ElsevierScience B.V., Amsterdam, (1996). 11. F. S. Roberts, Graph Theory and Its Applications to the Problems of Society, CBMS-NSF Monograph 29, SIAM Publications, Philadelphia, 1978. 12. J.-P. Serre, Groupes Discretes, Extrait de I'Annuaire du College de France, Paris (1970). 13. A. N. Sharkovsky, Co-existence of the cycles of a continuous mapping of the line into itself, Ukr. Math. Zh. 16 (1964), 60-71 (in Russian). 14. P. D. Straffin, Periodic points of continuous functions, Math. Mag. 51 (1978), 99-105. 15. R. Thomas, A combinatorial construction of a non-measurable set, Am. Math. Monthly 92 (1985), 421-422.

IL'~:--~.~L.-~_'.:..--

Jeremy

Gray,

Sale of the Century?

Column Editor's address: Faculty of Mathematics, The Open University, Milton Keynes, MK7 6AA, England

12

Editor

I

n October 29, 1998 Christie's in New York auctioned the Archimedes Palimpsest for $2 million, which, with their commission, means that an as-yet-unknown buyer paid $2.2 million for it. What is this text--surely one of the most expensive mathematical manuscripts in existence---and what were the circumstances surrounding its sale? Archimedes (287?-212 BC) was one of the great Greek mathematicians; many would say the greatest. His high status in medieval times helped ensure the transmission of his works down the centuries, and today they fill a book of respectable size. He wrote on many things, but the volumes of pieces of the solids of revolution associated with conic sections was a major theme of his work, as was the study of their centres of gravity and of buoyancy. At the end of the nineteenth century the energetic and meticulous Danish scholar Johan Ludvig Heiberg embarked on a programme of producing the most accurate possible editions of surviving works of Greek mathematics. This involves tracking down all extant manuscripts and early printed books and sorting out the differences in the texts (and there are always some differences) by examining their known histories--not at all an easy task in the days before photocopiers, scanners, and the like. The aim is to determine which manuscripts were copied from others that have survived, and then what lost manuscripts they were ultimately copied from. The next job is to reconstruct the lost manuscripts, to the extent possible. In the case of Archimedes, however, the thread was rather tenuous (the textual basis for Heiberg's edition of Euclid's Elements is richer, for example). The existing manuscripts were all copies or translations of two manuscripts, and both were lost. The first of these was one of two sources used by William of Moerbeke in the early fourteenth century when

O

THE MATHEMATICAL INTELLIGENCER 9 1999 SPRINGER-VERLAG NEW YORK

he made his medieval Latin translation of a number of the Archimedean treatises. No trace of it is known after 1311. The second of Moerbeke's sources was copied several times during the Italian Renaissance, but it too disappeared, sometime in the 16th century; some of the copies survive. So the discovery of a new text, the Archimedes palimpsest, in 1899 caused real excitement when it was listed in a catalogue of the library of the Metochion of the Holy Sepulchre in Istanbul. Heiberg was able to examine the manuscript in 1906 and 1908, and he published the results of his study in the second edition of his critical text of Archimedes's works (in series of texts published by Teubner, 1910-15). Most excitingly, the new text is independent of the two lost manuscripts. The palimpsest not only gave alternative readings of four mathematical treatises, it included the original Greek text of On Floating Bodies, until then known only from the medieval Latin translation. Better yet, it contained the text of a treatise called the Method of Mechanical Theorems, in which Archimedes explained how he used mechanical means to discover the theorems for which he subsequently provided logical mathematical proofs. This provides exceptional insight into how Archimedes worked, and on the careful distinction he observed between discovery and subsequent proof in mathematics. It is unique among ancient scientific writings for its treatment of methodology. It also contains a fragment of the Stomachion, otherwise only known in Arabic, a treatise on transposing figures. The palimpsest itself carries the Archimedean text, as it was copied in Constantinople in the mid-10th century, on vellum leaves, originally 300 • 200 mm. These leaves had been washed clean in the twelfth century, folded in haft to make a smaller book, and covered with Greek religious texts; the lines of the second script run

p e r p e n d i c u l a r l y a c r o s s t h o s e of the first. Cicero's Republic survives in the s a m e w a y in the Vatican Library. The A r c h i m e d e s p a l i m p s e s t f o r m s a "book" 71/2 b y 6 inches, with 174 p a g e s of text, a n d to r e a d the A r c h i m e d e a n material one m u s t r o t a t e the pages, p~eer bet w e e n the lines, and grapple w i t h the fact t h a t the "Archimedean" o r d e r was s c r a m b l e d in t h e process. T h e r e the m a t t e r might h a v e rested, with a m o d e r n scholarly edition in place, and the p r e c i o u s m a n u s c r i p t h o u s e d in Istanbul. But the t e r r i b l e hist o r y o f t h e 20th century intervened. After the First World War, a brutal struggle e r u p t e d b e t w e e n G r e e c e and Turkey, innocent civilians w e r e mass a c r e d in great numbers, a n d a rich c o s m o p o l i t a n mix of c u l t u r e s a r o u n d the A e g e a n was, if the h i d e o u s m o d e m e u p h e m i s m m a y m a k e t h e s e events m o r e poignant, "ethnically cleansed." The survival o f the library of the M e t o c h i o n could not be assured, n o r i n d e e d could t h a t of the m o n k s . The catalogue of 1890 listed 890 Greek manuscripts, and in 1922 the Greek Orthodox Patriarch of Jerusalem sent a letter to the Exarchus of the Metochion, asking him to send them all secretly to the National Library of Greece for safekeeping. A total of 823 manuscripts w e r e sent, and they are n o w in the Library, the p r o p e r t y of the Patriarchate. The Archimedes palimpsest w a s not one of them. Officially it is one of the 67 that were lost, b u t the m y s t e r y s u r r o u n d i n g its j o u r n e y to New York is p a r t o f the drama. At all events, the m a n u s c r i p t w a s i n a c c e s s i b l e to s c h o l a r s from s o m e time in the 1920s until quite recently. R u m o u r s circulated to the effect t h a t it w a s being privately offered for sale, b u t t h e doubtful legal situation m a d e it i m p o s s i b l e for universities to p r o c e e d . Other r u m o u r s s u g g e s t e d t h a t t h e r e w e r e no legal o b s t a c l e s but t h a t v a r i o u s a t t e m p t s to b u y the manu s c r i p t c a m e to nothing. Meanwhile it s e e m e d m o r e d e e p l y lost t h a n it had b e e n for m a n y years. (In this connection it is interesting that four p a g e s of the m a n u s c r i p t n o w c a r r y icons. These w e r e n o t m e n t i o n e d b y Heiberg, and one o f t h e m is on t o p of a p a g e of the Method. One can only s p e c u l a t e on w h e t h e r these w e r e a d d e d recently,

Page from the Archimedes Palimpsest. 9 Christie's Images, Ltd. 1999.

p e r h a p s w i t h a view to increasing the value of t h e m a n u s c r i p t in a sale.) Christie's, w h o i n f o r m e d the G r e e k G o v e r n m e n t of the f o r t h c o m i n g sale on August 13, argued that the manuscript w a s b o u g h t in the 1920s b y an u n n a m e d F r e n c h m a n . The w a y in which he c a m e b y the m a n u s c r i p t is entirely unclear. The G r e e k Minister for Culture, Evangelos Venizelos, w a s quoted b y t h e Athens News A g e n c y on O c t o b e r 24 as saying that P a t r i a r c h Diodoros of Jerusalem had informed the Minister that t h e r e w a s no r e c o r d o f it ever having b e e n sold. If so, the possibilities w o u l d s e e m to be that the F r e n c h m a n , o r a n o t h e r collector, stole

it, o r t h a t it was illegally s o l d by one o f the monks, p e r h a p s to b u y his w a y out of the raging war. However, the m a t t e r is actually m o r e c o m p l i c a t e d than that. Christie's agreed t h a t p r o o f of theft w o u l d m a k e the sale i m p o s s i b l e a n d ff e s t a b l i s h e d they w o u l d w i t h d r a w it. But, t h e y said in a s t a t e m e n t on Octob e r 26, it was well k n o w n that manuscripts f r o m the M e t o c h i o n h a d left t h e library a p p a r e n t l y quite legally at various times. A p a g e o f the p a l i m p s e s t was acquired b y t h e G e r m a n c o l l e c t o r Tischendorf and on his death sold by his executors to Cambridge University Library in 1876. A r o u n d the turn of the century the Biblioth~que Nationale de

VOLUME 21, NUMBER 3, 1999

13

Page from On Floating Bodies, the Archimedes Palimpsest. 9 Christie's Images, Ltd. 1999.

France acquired another, and two more migrated to America in the 1920s and 1930s. At no time did the Jerusalem Patriarchate protest at their disappearance or allege theft, and it never claimed any of them back. Christie's concluded that they were convinced that "the consignors have proper title and every right to sell," adding, a little obscurely, that they hoped that "Greece, the fountain-head of democracy, Western art and science, will be able to purchase the codex." In fact, Greece itself has no title to the manuscript. At least in 1890 it be-

14

THE MATHEMATICALINTELLIGENCER

longed to the Greek Orthodox Patriarchate of Jerusalem. The Archimedean text was written in a monastery of that order in the tenth century--not, that is to say, byArchimedes. But it could only be the Greek government, and not the Patriarchate, that could afford to buy it if it came up for sale. Felix de Marez Oyens, the head of Christie's books and manuscripts department, held a news conference in Athens on October 26, and the Athens News (27 October 1998) reported him saying, "We hope it will end up in Greece or a great national or university institution else-

where where it can be properly studied and preserved. It would be a great pity for it to go back to private ownership." Sums of over $1 million were by now being talked about, and the Greek government began to look into its pockets. Meanwhile the legal matter went before the courts, and the Ministry of Culture appointed a lawyer, Stavros Dimas, to coordinate legal action. He had successfully opposed the sale in 1993 of 308 artefacts stolen from the Aidonia region near Corinth by antiquity smugglers in the 1970s. An out-ofcourt settlement with the Greek state in that case led to the Michael Ward gallery in New York donating the collection to the "Greek Heritage Protection Association" in Washington. The palimpsest was by now on display at Christie's in New York. It had gone on display on Monday 26 October, and interested parties, not all of them with the money to buy it, could take it out of its case and examine it. Because the binding had been broken open it was now possible to read Archimedean text that had been trapped in the spine when Heiberg saw it, although it seems very likely that his reconstructions of such passages will be found to be valid. There was not much literary licence in a Greek mathematical work. The diagrams are more interesting. Even in the catalogue two diagrams from On Floating Bodies can be seen that differ significantly from Moerbeke's, so they are presumably derived from a common predecessor. Some scholars have begun to wonder ff there are not two versions of On ~ o a t i n g Bodies. Heiberg's diagrams are not from the manuscripts. Late on Wednesday 28 October Judge Kimba Wood, the federal judge hearing the appeal, ruled against the Jerusalem Patriarchate, on the grounds that in French law someone who has had an object for 30 years acquires a permanent right of possession and is therefore entitled to sell it. Indeed, according to some accounts they may already have done so, for rumours circulate that the present owner, who wants to be anonymous, purchased the manuscript from the family of the collector not very long ago. The auction began the next day at

2 p.m. Bidding w e n t quickly from $400,000 to $1 million, with t h e c u r r e n t b i d as usual d i s p l a y e d (in lights) in dollars a n d several o t h e r currencies, including F r e n c h francs, Swiss francs, D e u t s c h m a r k s , pounds, a n d yen. Then it s e e m e d as if the smaller bidde~rs (pres u m e d to be A m e r i c a n universities) w e r e burned off, and the battle was bet w e e n a man at the front with a cordless p h o n e and s o m e o n e at the back. When the bidding reached $2 million, the man at the front asked the auctioneer to walt j u s t one minute while he consulted his client. The auctioneer agreed and the tussle for extra time was r e p e a t e d a few times until he closed the bidding, and the Archimedes palimpsest w e n t to the m a n at the back. This w a s Simon Finch, an u p m a r k e t L o n d o n b o o k d e a l e r . He w a s a s k e d the n a m e o f the n e w o w n e r a n d r e f u s e d to say, b u t he did s a y that the b u y e r und e r s t o o d h o w i m p o r t a n t it w a s for s c h o l a r s to r e a d and i n t e r p r e t the p a l i m p s e s t and that the d o c u m e n t w o u l d b e m a d e available to scholars. There, for the moment, the m a t t e r rests. How, and where, the n e w o w n e r

will m a k e the manuscript available for scholars to consult is n o t known. The palimpsest is our only source for the Method, a n d the only G r e e k source of On l~oating Bodies. Heiberg's scholarship does n o t leave much r o o m for n e w textual discoveries, b u t the whole nature of G r e e k mathematical diagrams is not well understood. It is currently being studied b y Reviel Netz, and the n e w diagrams in the palimpsest will call for a thorough examination. The text is also interesting to philologists: A r c h i m e d e s w r o t e in Doric, not Attic, Greek. Perh a p s m o s t important, it is easily o u r oldest a n d m o s t substantial c o n n e c tion to the writings of one of the greatest of all m a t h e m a t i c i a n s - - t r u l y a maj o r achievement of h u m a n history and one that belongs, somehow, to us all.

Postscript The Baltimore Sun, 11 March 1999, reported that Will Noel, curator of manuscripts and rare b o o k s at the Waiters Art Gallery, and Abigail B. Quandt, senior conservator of mannscripts and rare b o o k s at the Waiters, ~11 be taking care of the Palimpsest. It is on display there

from 20 June to 5 September 1999. Quandt will then conserve it before returning it to its presently still unidentified owner. She will spend a year and a half trying to arrest the deterioration o f the pages and stabilizing them, which involves taking the palimpsest apart, separating the leaves, and dealing with the fungus that affects many pages. Then she will tend the edges, which have b e e n damaged by fire and b e c o m e brittle. When the repairs are done, a decision will be taken about whether to rebind the book. Advanced methods of digital image enhancement will be used to produce the Archimedean text, which m a y well be presented as photographs.

Acknowledgments I w o u l d like to t h a n k David Fowler, Henry Mendell, and Reviel Netz for their helpful-conunents on an earlier draft, and for bringing m e up to date on the Archimedean sources, and Rosalind Mendell for providing an account of the sale. Of course all mistakes that remain are mine. I would also like to thank Christie's for supplying the illustrations of the Palimpsest.

OSMO PEKONEN

he

irst

Bourbak' ? Autograph he story of Andrd Weil's arrest in Finland in 1939 has been often told. The version he gave himself [2] has a certain acceptance, and is repeated f o r example in the obituary in The Times (12 August 1998). For m a n y years none of us, however curious, could do m u c h to fill in the details. No government easily releases records on suspected espionage, and Finland has been especially cautious. In 1991 it became possible---though still difficult--for me to see some of the dossier. The full story, a good deal more dramatic and even more ironic than Weil's own reconstruction, has been published [1] with an afterword by Andr~ Weft himself, confirming that my account is consistent with his recollection. Andr~ and Eveline Weft arrived in Finland 15 June 1939. They made long visits to Lars Ahlfors and Roll Nevanlinna, then a touristic jaunt to the North. Eveline Weil left for

16

THE MATHEMATICALINTELLIGENCER9 1999 SPRINGER-VERLAG NE'W YORK

France 20 October. Andr6 was arrested 30 November on suspicion of spying for the Soviet Union (which had opened hostilities that very day), and remained in detention until he was expelled 12 December. (The grounds for suspicion were incomprehensible references in his notes to the Poldevian Academy and such, and compromising correspondence such as the letter dated 31 August 1939 from P.S. Alexandrov ending, "I hope your illustrious colleague M. Bourbaki will continue sending me the proofs o f his magisterial work.") Pursuing the records of the Weils' visit, I consulted

SA:

Figure 1. The entry in Rolf Nevanlinna's guest book at Korkee. Reproduced by permission of Arne Nevanlinna.

Arne Nevanlinna, an architect, son of the mathematician Rolf Nevanlinna. He showed me the entry in the guest book for 21 July, which is reproduced here. May this be the earliest extant autograph of the great N. Bourbaki? Notes on documentation. Many records of the S6minaire Bourbaki were burned by Maoists who celebrated the centenary of the Paris Commune by occupying the Ecole Normale Sup6rieure 20-21 March 1971; possibly some early Bourbaki autographs would have been there. The Finnish dossier on Weil has now been opened and is at the National Archives. REFERENCES

1. O. Pekonen, "L'affaire Weil & Helsinki en 1939," Gazette des Math~maticiens 52 (April 1992), 13-20. 2. A. Weil, Souvenirs d'apprentissage, Birkh&user, 1991. Department of Mathematics University of Jyv&skyl& 40351 Jw&skylA Finland Figure 2. Andr6 and Eveline Weil in 1939. Photo courtesy of the National Archives of Finland.

VOLUME 21, NUMBER 3, 1999

17

J. W. NEUBERGER

C'ontinuous Newton's Method for Po ynomia s In the early 1980s, m a n y o f us w e r e s u r p r i s e d to hear of the fractal nature of d o m a i n s of a t t r a c t i o n for N e w t o n ' s m e t h o d a p p l i e d to polynomials. To recall this shock, t a k e p ( z ) = z3 - 1

zEC.

(1)

F o r 1, w, and oJ2, the t h r e e r o o t s o f p, c o l o r a p o i n t z0 E C r e d if {zn}~ = 0 c o n v e r g e s to 1, Zn+l = zn - p(zn)/p'(Zn),

n = 0, 1 , . . . .

(2)

Color z0 blue if {z,~}n=0 converges to ~o, a n d c o l o r it green if {Zn}n=O c o n v e r g e s to ~o2. Any remaining p o i n t gets colo r e d b l a c k [e.g., if p ' ( Z n ) = 0 for s o m e non-negative integer n]. Figures 1 a n d 2 were g e n e r a t e d b y the c o d e "croots.for" b y R o b e r t Renka. Figure 1 s h o w s (with the lightest shading s t a n d i n g for red, the d a r k e s t for green, and t h e i n t e r m e d i a t e shading for blue) d o m a i n s o f attraction for Eq. (2). Figure 2 indicates c o r r e s p o n d i n g d o m a i n s o f a t t r a c t i o n for the modified N e w t o n ' s m e t h o d (for ~ = 89 Zn+l=zn-~*p(Zn)/p'(zn),

n=0,1,....

(3)

As ~ is c h o s e n smaller, one gets c o r r e s p o n d i n g pictures with even s m a l l e r jewels. E n c o u r a g e d b y this, m a t h e m a t i cians naturally divided b y t~ and let ~ --> 0. This is the "continuous N e w t o n ' s method." In its b a s i c form, it consists of finding a function z on a s u b s e t o f R so that z'(t) = -p(z(t))/p'(z(t)),

t E D(z).

(4)

Then, t ---) ~ should give p ( z ( t ) ) --> O. F o r this note, I u s e an i m p r o v e d version o f c o n t i n u o u s N e w t o n ' s method: F o r z0 ~ C, a continuous function z from a / / o f R to C is s o u g h t so that

18

THE MATHEMATICAL INTELLIGENCER 9 1999 SPRINGER-VERLAG NEW YORK

p ( z ( O ) ) = Zo,

p(z)'(t) = -p(z(t)),

t E R.

(5)

The i m p r o v e m e n t is in t h e handling o f singularities. Solutions o f (5) m a y not b e s o l u t i o n s o f p ' ( z ) z ' = - p ( z ) , for t h e y m a y be such that p ' ( z ( t ) ) = 0 and z ' ( t ) d o e s n o t exist. The analysis o f (5) allows us to sail right t h r o u g h t h e s e singularities. N o w l o o k h o w m u c h b e t t e r c o n t i n u o u s N e w t o n (4) o r (5) d o e s w i t h the e x a m p l e p ( z ) = z 3 - 1. We w o u l d n ' t exp e c t c o n v e r g e n c e starting from the rays 0 = ~r/3, 0 = - ~r/3, 0 = 7r. However, let us start in M, the c o m p l e m e n t o f t h e union o f t h e s e three rays. F o r z0 in one of the t h r e e c o m p o n e n t s o f M and z satisfying (5), u = ~-mt--.~ z ( t ) exists a n d is t h e r o o t o f p in that c o m p o n e n t . These d o m a i n s o f a t t r a c t i o n are j u s t w h a t any right-thinking p e r s o n w o u l d (wrongly) s u s p e c t for Eq. (2). This n o t e gives a r a t h e r c o m p l e t e d e s c r i p t i o n of dom a i n s o f attraction for c o n t i n u o u s N e w t o n ' s m e t h o d for polynomials. Results are e x p r e s s e d in three theorems. We a r e seeking r o o t s o f p, a n o n c o n s t a n t c o m p l e x polynomial. The m e t h o d is to s e e k c o n t i n u o u s functions z f r o m R to C w h i c h solve the differential equation p(z)'(t) = -p(z(t)),

t E R.

(6)

Let Q b e t h e set of such functions. T h e o r e m 1. I f z ~ Q, t h e n u = lim z ( t ) e x i s t s a n d p ( u ) = O.

In short, continuous N e w t o n ' s m e t h o d d o e s find roots! The r a n g e of a m e m b e r of Q will b e called a trajectory.

f

Figure 1. Newton's method for roots of p:

p(z) = z 3 -

1, z E C .

T h e o r e m 2. E v e r y m e m b e r o f C is contained i n s o m e trajectory.

Figure 2. Damped Newton's method for p:

p(z) = z 3 -

1, z E C .

gleaned from these cited papers, in the presellt note I try to give a rather serf-contained treatment of the polynomial case,

A subset G of C is called an incoming trajectory if there are x E C, d ~ R, and z ~ Q so that p ( x ) r O,

p'(x) = O,

z(d) = x,

and

G = z ( ( - % d]).

Such an incoming trajectory is said to end at x. It connects to an outgoing trajectory starting at x, namely z([d, ~)). Denote by M the set of all members of C which belong to no incoming trajectory. T h e o r e m 3. E v e r y component ( m a x i m a l connected subset) o f M contains j u s t one root of p. I f z ~ Q and R ( z ) intersects S, a component o f M, then

u = t---> limoo z(t) is the root o f p w h i c h is i n S. The set of all continuous solutions z to Eq. (6) form a generalized flow in the sense of [1]. This concept helps in organizing situations like the present one, in which uniqueness under initial conditions doesn't hold. I am grateful to John Mayer and Hartje Kriete for leading me to a n u m b e r of recent references to Newton's methods. It is safe to say that any of the authors of [2, 3, 5, 6] could have written the present note had they decided to do so. All of these papers deal with discrete Newton's m e t h o d (3), and most c o n c e r n various aspects of continuous Newton's method as well. In [2], there is a treatment of h o w Julia sets connected with (3) converge to portions of the set M above. In [3, 5, 6] interesting connections between the two Newton's m e t h o d s are given, for polynomials as well as for analytic functions more general than polynomials. Even though some of the present results can be

R e m a r k . I have not treated the root structure of polynomial maps from C k to C k f o r K > 1. Continuous Newton's method might apply, but I expect both formulations and proofs would be different.

Next are some vector field pictures for five examples. These are done with Mathematic& The Mathematica commands for Fig. 3 are p[z_] := z 3 - 1, n[z_] := -p[z]/p'[z], n l [ x _ , y_] := Re[n[x + Iy]] n2[x_, y_,] := Im[n[x + Iy]] PlotVectorField [{nl[x, y], n2[x, y]}, {x, -1.2, 2}, {y, -1.5,1.5}1. Change the first line to get a vector field for another polynomial. Change limits on the last line to examine other regions in C. The reader is invited to sketch in the union of any incoming trajectories in each case. Perhaps the main points of interest in these pictures are the points of attraction, i.e., the roots, and the hyperbolic points. Hyperbolic points are precisely the ends of incoming trajectories. The example in Fig. 3 has three points of attraction and one hyperbolic point. Two of the roots are imaginary. The hyperbolic point is zero, the sole root of p'. It turns out that any incoming trajectory can be continued as a trajectory to converge to any of the three roots. This illustrates some typical behavior. The example in Fig. 4 has three roots (two complex) and two hyperbolic points. Note that either of the incoming trajectories converging to the rightmost root o f p ' can be con-

VOLUME 21, NUMBER 3, 1999

19

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w e can r e a s o n a s in T h e o r e m 1 to get a contradiction. Thus, limt___~b_ w ( t ) exists. Consequently, a = l i m t ~ b p ' ( w ( t ) ) also exists. But a = 0, o t h e r w i s e the solution w to (11) c o u l d be e x t e n d e d p a s t b, contradicting t h e maximality of (a, b). The s e c o n d conclusion to the l e m m a follows with a similar argument. []

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Next is the technique I p r o m i s e d for patching solutions t o g e t h e r (nonuniquely) at a singularity. L e m m a 3. S u p p o s e x E C, p ' ( x ) = 0 a n d p ( x ) r O. There i s ~ > 0 so that i f s E R a n d 0 < Isl < ~, then

~),z~ C.

is a r o o t of p. This says t h e t r a j e c t o r y m u s t shuttle infinitely often b e t w e e n distinct r o o t s of p. Let U b e a union of s e p a r a t e d o p e n disks e a c h containing e x a c t l y one root. Continuity of z obliges it to have infmitely m a n y v a l u e s in C\U. Then it m u s t have ~ l i m i t points there---a contradiction. [] Following a r e s o m e l e m m a s on w h i c h a r g u m e n t s for T h e o r e m s 2 a n d 3 depend. L e m m a 1. S u p p o s e x E C, q E R, and p "(x) r O. There is a m a x i m a l o p e n i n t e r v a l (a, b) c o n t a i n i n g q o n w h i c h there is a f u n c t i o n w so that w(q) = x

and

p ' ( w ( t ) ) r O, p(w)'(t) = -p(w(t)),

t E (a, b).

(10)

Moreover, s u c h a f u n c t i o n w i s u n i q u e o n this m a x i m a l interval.

P r o o f . Note that if (10) holds, then w is differentiable on (a, b) and, consequently, w(q) = x

and p ' ( w ( t ) ) r O, w'(t) = -p(w(t))/p'(w(t)),

(11)

The result follows from classical e x i s t e n c e and uniqueness t h e o r y for o r d i n a r y differential equations. [] L e m m a 2. S u p p o s e x E C, q E R, p ' ( x ) r 0 w i t h w a n d (a, b) as i n (10). I f b < ~, t h e n lim w ( t ) e x i s t s a n d lim p ' ( w ( t ) ) = 0. t---->bt-->bIf a ~ -% then

(12)

lim w ( t ) e x i s t s a n d lim p ' ( w ( t ) ) = 0. t-*a+ t-)a+

(13)

Proof. First, a s s u m e that b < ~. Then, {w(t): t E [q, b)} is b o u n d e d as n o t e d in the a r g u m e n t for T h e o r e m 1. S u p p o s e that

lim w ( t )

t-->b-

i f s > O, there i s f: [0, s] --->C so that f(O) = x a n d p ( f ) ' ( t ) = - p ( f ( t ) ) , t E [0, s],

(14)

and i f s < 0 a n d v = - s , t h e ~ is g: [0, v] --> C so that g(v) = x, p ( g ) ' ( t ) = - p ( g ( t ) ) , t ~ [0, v].

(15)

Proof. Select ro > 0 so that if y E C, 0 < lY - xl < to, t h e n p ( y ) r p ( x ) a n d p ' ( y ) r 0. F o r s n e a r 0, w e will s t u d y Ps, x as in (9). Denote b y x8 a r o o t of Ps,x c h o s e n to minimize IXs - x I. To see w h y

x = lim xs,

(16)

8---->0

factor Ps, x c o m p l e t e l y a n d n o t e that if (16) failed, it c o u l d not be that lim Ps,x(X) = 0;

8--->0

but that is true b e c a u s e Ps, x(X) = p ( x ) - e x p ( - s ) p ( x ) . F o r s r 0 s u c h that Ix8 - x I < ro, t a k e as, bs, and Ws so that ws(O) = xs,

t E (a, b).

t E (a, b);

p(Ws)'(t) = - P ( W s ( t ) ) ,

t E (as, bs),

with as < 0 < bs and (as, bs) m a x i m a l in the s e n s e of (10). Choose r > 0 s o that r < r0. A s s e r t i o n . There is ~ > 0 so that if 0 < s < & t h e n IWs(t) - x I < r

if as < t < O.

To p r o v e this assertion, a s s u m e it is n o t true. Denote b y {Sk}~=O a d e c r e a s i n g sequence of positive n u m b e r s converging to 0 s u c h that if k is a positive integer, there is t ~ (ask, 0) such that IWsk(t) - x I >- r.

F o r e a c h positive integer k, d e n o t e b y tk the largest numb e r t E (ask, 0) so that Iwsk(t) - x I = r,

a n d note t h a t if tk - t -< 0, t h e n Iwsk(t) - x I -< r a n d so p'(Wsk(t)) r 0,

p(Wsk(t)) r p ( x ) .

VOLUME21, NUMBER3, 1999 21

Then for all k, p(Wsk(t)) = e x p ( - t ) p ( x s k ) = e x p ( - ( t + Sk))P(X), ask --Sk, k = 1, 2, . . . . Hence, l i m ~ = (tk + Sk) = 0 since limk--,= Sk = 0. Thus, p(Wsk(tk)) = e x p ( - - ( t k + S~))p(x) --* p ( x )

as k --) ~.

However, IWs~(tk) -- XI = r, k = 1, 2 , . . . , so some subsequence of {Wsk(tk)}s = 1 converges to an e l e m e n t y E C so that [y - x l = r. But then, p ( y ) = p ( x ) , a contradiction. Thus, the assertion is true. Choose ~ so that the assertion holds. Observe that i f s E ( 0 , 6), then as = - s ; either of the assertions as < - s and as > - s leads to a contradiction. By Lemma 2, if 0 < s < ~, then y = limt-,a,+ Ws(t) exists, p ' ( y ) = 0 and lY - xl <- r since as = - s . This implies that y = x. For 0 < s < ~, define f on [0, s] by f(O) = x,

f ( t ) = Ws ( t - s ) ,

t E (0, s]

a n d observe that f satisfies (14). We proceed i n a n entirely similar way (possibly reducing our ~) to prove (15). [] The ideas in the p r o o f of Lemma 3 can be extended to show that if x ~ C, p ( x ) r 0, p ' ( x ) = 0, and x as a root o f p ' has multiplicity k, then there are at least k + i incoming trajectories ending at x a n d at least k + i outgoing trajectories starting at x. This follows starting with the fact that if s 0, x is a root of p, a n d x~ is a root of P,,x, then Po,x(X,) + Ps, x(X) = 0, where Ps,x and Po,x are defined using (9). P r o o f [Theorem 2]. Take any x E C. C a s e 1. If p ( x ) = 0, t h e n the function z so that z(t) = x, t E R, is in Q. C a s e 2. Suppose p ( x ) r 0 a n d p ' ( x ) r 0. Using Lemma 1, choose w satisfying (10) where (a, b) is m a x i m a l and q = 0, i.e., w(0) = x. If b < % t h e n by L e m m a 2, y = limt__)~ w ( t ) exists a n d p ( y ) r 0 and p ' ( y ) = 0. Using L e m m a 3, p i c k f satisfying (14) a n d e x t e n d w so that w ( b ) is the left limit of w at b a n d so that if t E (0, s), then w ( b + t) = f ( t ) . Alternately, using Lemmas 1, 2, a n d 3, we arrive at w defreed on (a, ~) so that p ( w ) ' ( t ) = - p ( w ( t ) ) , t E (a, ~). If a = - % we are fmished, ff a > - % repeat the extension process only going to the left. In any case the e n d result is a function z in Q. C a s e 3. Finally, s u p p o s e that p ( x ) r 0 and p ' ( x ) = O. P i c k f a n d g satisfying (14) and (15), respectively, and defree w o n I - v , s] so that w(0) = x, w ( t ) = g ( t + v) (t ~ [ - v , 0)), w ( t ) = f ( t ) (t ~ (0, s]). Note that p ( w ) ' ( t ) = - p ( w ( t ) ) , t ~ [ - v , s] (there is something to reflect u p o n c o n c e r n i n g the differentiability of

22

THE MATHEMATICALINTELLIGENCER

p ( w ) at 0). Extend w to the left a n d fight as n e e d e d using Lemmas 1, 2, and 3 to arrive at a n extension z ~ Q. [] P r o o f [Theorem 3]. By definitions, M contains every root ofp. Also, note that i f z E Q, S a c o m p o n e n t of M, and R ( z ) intersects S, then R(z) C S, so the root u o f p such that u = l i m t _ ~ z ( t ) must also be in S. It follows that every comp o n e n t S of M contains at least one root of p. Suppose a c o m p o n e n t S of M contains more t h a n one root of p, say ul, 9 9 9 Ub for s o m e integer b > 1. Partition S into S1 . . . . , Sb with S j = { x E S : z E Q, x E R ( z ) ,

u j = l~=z(t)},

j=

l, . . . , b,

and note that no two m e m b e r s o f $ 1 , 9 9 9 S b intersect. Since S is connected, there are integers m, n E { 1 , . . . b} s u c h that v = limk_~= Vk for some v E S m and vl, v2, 9 9 9 ~ Sn. Choose z, zl, z2, 9 9 9 E Q such that z(0) = v a n d Zk(O) = Vk, k = 1, 2 , . . . . Then, lira z ( t ) =Um,

b--~ oo

lim Zk(t) = Un,

t - - ) cr

k = 1, 2 . . . . .

There are tl, t2, . 9 9 E R such that lim Zk(tk) = Um and, consequently, ~

k--~

p(zk(tk)) = O.

However, for fixed positive integer k, lira zk(t) = Un,

t-..)~

~'m~=p ( z k ( t ) ) = O.

Because of the continuity of each of {Z k}k=l and the local c o m p a c t n e s s of C, we arrive at infmitely m a n y roots of p, a contradiction. Thus, S c o n t a i n s only one root of p. [] Using similar arguments b u t without using Lemma 3, one c a n prove the weaker result that through every x with p ( x ) #= 0 and p ' ( x ) r 0 goes a solution of z ' ( t ) -- p ( z ( t ) ) / p ' ( z ( t ) ) converging either to a root of p or to a root of p ' , and that for some x E C, the flint alternative holds. N o w Lemma 3 is the only place in the d e v e l o p m e n t which uses the F u n d a m e n t a l T h e o r e m of Algebra, so one can d e d u c e that result from the o n e j u s t stated.

Ulterior Motive The above development gives a w a y to tag a solution u to p ( u ) = 0 with a region in C, roughly its d o m a i n of attraction in the d e s c e n t process p(z)' = -p(z). This p r o b l e m is analogous to the following. Suppose each of H a n d K is a Hilbert space, F is a C (2) function from H to K, a n d

r

= I~(u)l~2,

u E H.

I have in mind, for example, cases in which H is a Sobolev space of functions on s o m e region in Euclidean space, K is a n L2 space on that region, a n d the p r o b l e m of finding u ~ H so that F(u) = 0

(17)

characterized by a constant struggle for a few crumbs of compactness. In contrast, in arguments for continuous Newton's m e t h o d for polynomials I could relax: there w a s plenty of compactness. I felt like a kid again. REFERENCES

t. J.M. Ball, Continuity properties and global attractors of generalized semifiows and the Navier-Stokes equations, Nonlinear Sci. 7 (1997), 475-502. 2. F. von Haeseler and H. Kriete, The relaxed Newton's method for rational functions. Random Computat. Dynam. 3 (1995), 71-92. 3. H. Jongen, P. Jonker, and F. Twilt, The continuous, desingularized Newton method for meromorphic functions, Acta AppL Math. 13 (1988), 81-121. 4. J.W. Neuberger, Sobolev Gradients and Differential Equations, Springer Lecture Notes in Mathematics Vol. 1670, Springer-Verlag, New York, 1997. 5. H. Peitgen, M. Prufer, and K. Schmitt, Global aspects of the continuous and discrete Newton method: A case study, Acta Appl. Math. 13 (1988), 123-202. 6. D. Saupe, Discrete versus continuous Newton's method: A case study, Acta Appl. Math. 13"(1988), 59-80.

represents a system of partial differential equations with some but perhaps not enough boundary conditions to imply existence of one and only one solution. It is this kind of root-fmding which has been a major focus of attention for me recently [4]. t. 4Y(u)h = (F'(u)h, F(u))K = (h, F'(u)*F(U))H,

u, h ~ H,

where F'(u)* E L(K, H ) is the Hilbert space adjoint of F ' ( u ) , u E H. This leads to a Sobolev gradient V~b for ~b satisfying the identity r

=(h, (Vr

U, h E H,

= F'(u)*F(u), u E H. by taking ( V r Seek u E H such that F ( u ) = 0 by means of continuous steepest descent, i.e., consider z: [0, oo) __) H so that z(O) = x E H,

z ' ( t ) = -(V~b)(z(t)),

t -- 0, (18)

in the hope that u = lim z(t) exists and F ( u ) = 0.

(19)

t---*~

In analogy with continuous Newton's method for polynomials, one says that x, y E H are equivalent relative to (18) provided that they lead, through (18) and (19), to the same element u E H. Granted that the limit exists for each x E H and z in (18), one has H p a r t i t i o n e d in such a way that each leaf in the partition contains precisely one solution. These leaves are analogous to the components of M in the first section above. Numerical, geometric, and algebraic studies of these leaves m a y provide an a p p r o a c h to the general boundary value problem for the system (17). There are some results in this direction in [4]. The study of partial differential equations, however, is

VOLUME 21, NUMBER 3, 1999

23

IIl.[.an~',eli,[~.,r-~l[.-~411-.],~-~--~l

The Hidden Pavements of Michelangelo's Laurentian Library .?_

Jay Kappraff

Does your hometown have any mathematical tourist attractions such

Dirk Huylebrouck,

Editor

since publishing m y b o o k

~nections: the Geometric Bridge between Art and Science [1], I have b e c o m e u s e d to playing t h e role o f a "mathematical tourist." I a m frequently c o n t a c t e d b y r e s e a r c h e r s keen on discussing s o m e d i s c o v e r y t h a t they have m a d e and seeking m y a d v i c e as to t h e m a t h e m a t i c a l a s p e c t o f their work. S o m e t i m e s the w o r k a p p e a r s to s h e d n e w light on ancient o r m o d e r n geometry; other times it s e e m s to lead nowhere. It is in this context that Ben Nicholson telephoned me three years ago. He had b e c o m e privy to a set of facsimiles of fifteen 8'6" • 8'6" pavement d e s i g n s - possibly created by Michelangelo-t h a t lay hidden beneath the floorboards of the Laurentian Library in Florence [2,3]. He was trying to decipher their geometries in order to enable him and an artist, Blake Summers, to reconstruct t h e m at full scale. This project definitely piqued m y interest. One thing led to another, and I soon found myseff a part of Nicholson's team of r e s e a r c h e r s devoted to the study of the p a v e m e n t s - -

I

Blake Summers; Rolf Bagemihl, a n archivist living in Florence; David Krell, a philosopher, ArieUe Salber, a graduate student at Yale specializing in Renaissance history; and Saori Hisano, a graduate student at I]linois Institute o f Technology, the college w h e r e Nicholson teaches. F r o m time to time w e have also s o u g h t the help of o t h e r students o f Nichoison's, Salvatore Camporeale, a F l o r e n t i n e theologian, a n d E r n e s t McClain, a musicologist. I s a w on m y first visit to the studio o f Blake S u m m e r s that he w a s recreating the p a v e m e n t designs with t h e aid of a giant aluminum b a r w h i c h s e r v e d as a compass. S u m m e r s a n d Nicholson w e r e using intuitive geometry very m u c h in the spirit of Boethius, w h o t r a n s l a t e d t h e Elements of Euclid into a language u n d e r s t a n d a b l e to t h e guilds of m a s o n s during Middle Ages. In the p r o c e s s o f analyzing the pavem e n t s w e feel t h a t we have d i s c o v e r e d a t a x o n o m y o f ancient g e o m e t r y t h a t c o m m i n g l e s all of the g e o m e t r i c syst e m s h a n d e d d o w n from antiquity into an i n t e g r a t e d whole. We have identi-

as statues, plaques, graves, the cqfd where the famous conjecture was made,

the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? I f so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.

Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42,

Figure 1. Entrance stairway at the Laurentian Library. (Figs. 1-7 are from Firenze Biblioteca

8400 Oostende, Belgium e-mail: [email protected]

Medicea Laurenziana, salone de Michelangelo. By permission of the Minister of Culture. Further reproductions are strictly forbidden.)

24

THE MATHEMATICALINTELLIGENCER9 1999 SPRINGER-VERLAGNEW YORK

fled six principai geometries upon which the pavements appear to be based: (1) the Vesica Pisces [1,2]; (2) the law of repetition of ratios popularized by the 20th-century designer Jay Hambridge under the name dynamic symmetry [4,5]; (3) the eight-pointed Bnmes star discovered by Tons Brunes, the late Danish engineer [6,7,8];'(4) a set of constructions based on ~ and referred to by Brunes as the sacred cut [2,5,9]; (5) the ad-quadratum squarewithin-a square; and (6) the golden mean [1,5]. Let me summarize what I have learned about this remarkable set of designs and briefly describe the structure of two of them. The Laurentian Library, which was designed by Michelangelo, is situated on the second floor of the San Lorenzo church complex in the heart of Florence. Work on the library was begun in 1523 by Pope Clement VII, alias Guilio Medici, the nephew of Lorenzo di Medici, as a monument to his uncle; it was opened to the public 48 years later by his distant cousin, Grand Duke Cosimo I. The Library was meant to be a home for the books from antiquity that survived to the Renaissance. The modest seeing of the Library leaves one utterly unprepared for what one encounters upon entering. First one is confronted with a massive staircase (Fig. 1) calculated to provoke a numerological trance: There are two steps to get into the building, then series of 3 steps, 7 steps, and 5 steps, with 9 steps to the left and right. After mounting the staircase one enters the Reading Room (Fig. 2). Here the seeming regularity and normalcy hides a frenzy of paradox and ambiguity. Just look at the walls. There is no predominant structure. The wall consists of seven planes, completely disorienting the viewer. In 1774 a portentous accident occurred in the Reading Room of the Laurentian Library. The shelf of desk 74, overladen with books, gave way and broke. In the course of its "repair, workmen found a red and white terracotta pavement which had lain hidden for nearly 200 years beneath the floorboards. The librarian had trapdoors, still operable today, built into the floor, so future generations could view these un-

usual pavements. Further details of the history and significance of the pavements can be found in Nicholson's CDROM, Thinking the Unthinkable House [101. Overall, the pavement consists of two side aisles and a figurative center aisle (Fig. 2). Desks situated on a raised wooden dais have been placed over the pavements. On the side of each desk are listed the books that were to be stored in it. Beneath the desks are a se-

ties of fifteen panels, of different designs, each about 8'6" x 8'6". The fifteen panels along one aisle mirror the ones on the other aisle, but differ in subtle ways. When juxtaposed, the 15 pairs of panels appear to tell a story about the essentials of geometry and number. In 1928 the pavements were photographed for the fLrSt time when the desks were removed temporarily whilst structural repairs were made to the subflooring (Fig. 3).

Figures 2 and 3. The Laurentian Library Reading Room--with and without desks.

VOLUME 21, NUMBER 3, 1999

25

Figure 4. a) The Index Panel 1; and b) the Cross Panel 15. (Figs. 4-7 are details of the pavements.)

The spatial conundrums, paradoxes, and errancies of the building fabric reappear in the geometry of the pavements. We think that the apparent raggedness of the panels can be explicated as accurate and premeditated interpretation of antique geometry in terms of the philosophical concerns of the 1500s. Michelangelo was working with themes well understood at the time, a "secret art of geometry" which could be read in the pavement by the knowledgeable, but which is much more inscrutable today. The books in the Library were organized with the sacred books to the East and the profane to the West, a throwback to the ancient "tree of knowledge". On the East side, tucked behind the projecting entrance door, is a single desk whose books include the Koran, Kaballah, Machiavelli, and books of magic. These subjects escape the tidy categories used to order the Laurentian collection, and they were placed out of sight. The sequence hits its stride with 13 desks containing the works of Italian, Latin, and Greek poets, and continues on through books devoted to the quadrivium (music, astronomy, geometry, and arithmetic), leading to 27 desks loaded with Latin and Greek books of theology, ending with books devoted to the Pentateuch. On the West side of the Reading Room,

~)6

THE MATHEMATICALINTELLIGENCER

the books make a counterpart to those to the East and follow the epistemological form devised by Aristotle. Across from the poets are the texts related to the trivium (granunar, rhetoric, and oratory), then on to logic, medicine, history, ethics, and metaphysics. Aristotle's books were not bound in a single compendium as we might fmd them in a bookstore today, but were found in six different locations in the library, according to the part of his epistemology that they addressed. To traverse the length of the Reading Room could be thought of as a journey through the hill extent of the world's wisdom and knowledge. Monsignior Comporeale feels that movement from Panel 1 (see Fig. 4a) near the door to Panel 15 (see Fig. 4b) on the other side of the Room may have represented the Christian's journey from baptism to enlightenment. Panel 1 consists of octagons and crosses symbolic of baptism, while Panel 15, adjacent to the Pentateuch, also contains crosses and 10 concentric sets of squares surrounding a central square. That the Hebrew Pentateuch is placed here may be deliberate metaphor for the 10-ness that pervades it: the 10 Commandments, 10 generations to Abraham, and 10 more to the Flood. Also Ernest McClain has found that much of the numerology of the Hebrew Bible can be re-

lated to an ancient musical scale based on the first 10 numbers [11]. Let's analyze two of the pavements. Panel 14 is referred to by Nicholson as the Timaens panel. It is composed of four diamonds set within circles that are cut with segments of circles, and the whole design is framed with a white border. In Fig. 5, this panel is shown juxtaposed with a reproduction by Fabbrini, from the circle of Michelangelo, of Michelangelo's system of proportions. Michelangelo felt that the system of proportions developed at the time by Dtirer was inadequate to describe the supple human body, and that his own system better allowed for flexible joints within the body. You will notice that Michelangelo provides a scale on the right subdivided first into 2 parts, then 4 and 8 parts, with each unit further subdivided into 3 parts for a total of 24 equal parts. This is reminiscent of the lambda figure 1

2 4 8

3 6

12

9 18

27

found in Plato's Timaeus and referred to there as the World Soul. This was one of the neo-Platonic ideas brought to the Renaissance by Ficino's academy. It forms the basis of the musical system studied by Pythagoras and writ-

Figure 5. The geometric construction of the Timaeus Panel 14.

t~n about by Nichomachus [12,13,14], and it was used by Alberti as the basis of his system of architectural proportions [5,9]. So we expect the number 27 in Michelangelo's system rather than 24. Sure enough, the head of the model projects higher, the foot projects beneath the floor plane, and an extra unit is intercalated at the hip. Now we have the 27 units of the World Soul.

Figure 6. The Medici Impressa.

The 27 units of Michelangelo's system of proportions can be found in Panel 14. The space from bottom to top

is subdivided into 27 equal parts. However, it is surely deliberate that the seemingly similar space from left to

Figure 7. The Cosimo Panel 2.

VOLUME21, NUMBER3, 1999

27

right has an extra, incommensurate interval. The four circles of the figure echo the coat of arms of Cosimo di Medici, which is also emblazoned on the pavements of the central aisle (Fig. 6). Each bf the 27 traits is exactly 3 soldis in width resulting in an 81-square grid for the entire design. The panel across from this on the other side of the library appears to be based on an 80square grid. Is it coincidental that the ratio 80:81, known in musical parlance as the syntonic comma, is exactly the ratio by v(hich the tones of the ancient scale, attributed to the followers of Pythagoras based o n t h e primes 2 and 3, differs from the Just scale, based on primes 2, 3, and 5? Such conundrums are found over and over in the structure of the pavements. Panel 2, the Medici panel (Fig. 7), is a rosette form typical of many such antique rosette forms that appeared at the time in Florence. The pavement is a rectangle of dimension 12 x 13. Nicholson feels that these numbers are significant as the number of months in the solar and lunar calendars. The small difference between a square and a rectangle is crucial to its construction. 1. In the first step in the construction, the rectangle is extended to a 13 x 13 square concentric with a 12 x 12 square, and the horizontal and vertical axes are placed in the squares. An equilateral triangle is drawn to a side of the 12 x 12 square. The distance from the center of the square to the vertex of the triangle is the radius of a standard circle of the construction called the pitch circle (Fig. 8a). Beginning where the pitch circle cuts the horizontal axis, six circles o f radius equal to the pitch circle are drawn (Fig. 8b). 2. Next six additional circles are drawn beginning where the pitch circle cuts the vertical axis. 3. Twelve additional circles are drawn by repeating steps 2 and 3 for the pair of perpendicular diagonals of the squares resulting in a 24-rosette pattern (Fig. 8c). 4. Twenty-four additional circles are drawn haft-way between the circles of the rosette. These will be widened into the white bands appearing in the

28

THE MATHEMATICAL INTELLIGENCER

panel. For that step, the small difference between the diagonal of the t2 x 13 rectangle and the 13 x 13 square is exploited. Circles with this small difference as radii are drawn where the 24 circles intersect the pitch circle to yield 48 n e w points on the pitch circle (Fig. 8d). 5. Forty-eight additional circles with radius equal to the pitch circle are n o w drawn. These demarcate the white bands of the Medici panel (Fig. 8e).

6. In the final step eight varieties of ellipses are created to fill the diamond shapes. Nicholson's and Summer's reconstruction of panel 2 is s h o w n in Fig. 9. What appeared as a kind of errancy in the deviation of the rectangle from a square exploded into the entire design. Furthermore, in the steps leading to its creation, a series of 3, 6, 12, 24, 48, and 96 circles are created. This is the series that led to the Titius-Bode

Figure 8. Construction of Panel 2. a) A triangle in a square establishes the pitch circle; b) a rosette of six circles; c) a rosette of 24 circles; d) the mismatch of the diagonals of the 12 x 12 square and the 12 x 13 rectangle generates 48 additional circles; e) ninety-six circles create a set of spiral bands in which eight classes of ellipses are placed.

Acknowledgments

As you c a n see, this p r o j e c t has materialized for m e into the ultimate o f m a t h e m a t i c a l tours. I wish to ack n o w l e d g e the fruitful c o l l a b o r a t i o n that I have u n d e r t a k e n with m y colleagues, Ben Nicholson a n d Saori Hisano, that has m a d e this w o r k a g r e a t pleasure. REFERENCES

Figure 9. Nicholson's reconstruction of the Cosimo Panel.

law t h a t p r e d i c t e d the p o s i t i o n s of the p l a n e t s up until Saturn. Again, is this c o i n c i d e n c e o r p r e s c i e n c e ? The pavem e n t mirroring Panel 2 is p l a c e d in an 11 • 12 square p e r h a p s symbolizing the 12 disciples and the 11 disciples o n c e J u d a s w a s excluded. The question begging to b e a s k e d is w h y 30 magnificent p a v e m e n t s w o u l d b e c o n s t r u c t e d and t h e n c o v e r e d up. P e r h a p s s o m e cryptic s y m b o l s w e r e c o n c e a l e d within the p a v e m e n t s in the m a n n e r of U m b e r t o E c o ' s Name of the Rose and then h i d d e n to p r e v e n t their revelation. Nicholson has a m o r e mund a n e hypothesis. When the library was conceived there were approximately 1000 b o o k s in existence. W h e n it w a s c o m p l e t e d , the Library c o n t a i n e d over 3000. The Library had to b e reconfigu r e d to a c c o m m o d a t e the n e w books. In the original plan for the Library, a triangular r o o m w a s s u p p o s e d to have b e e n c o n s t r u c t e d to h o u s e the r a r e s t o f t h e books. However, this r o o m w a s n e v e r built. Unfortunately we do not k n o w who created the pavements. The b o o k s listing the financial transactions in connection with them would m o s t likely have listed the designer, but they have been lost. However, surely Michelangelo w o u l d at least have had a major say concerning such a crucial c o m p o n e n t of his library.

1. Kappraff, J. Connections/ The Geometric Bridge between Art and Science. New York: McGraw-Hill Books. (1991). 2. Nicholson, B., Kappraff, J., and Hisano, S. "A Taxonomy of Ancient Geometry Based on the Hidden Pavements of Michelangelo's Laurentian Library." In Art and Science: The Proceedings of the Second Conference on Art and Science edited by J. Barrallo, San Sebastian, Spain: Univ. of the Basque Country Press (1998), and in Bridges: Mathematical Connections in Art, Music and Science: Conference Proceedings edited by R. Sarhangi, Arkansas City, KS: Gilliland Publ. (1998). 3. Nicholson, B., Kappraff, J., and Hisano, S. "The Hidden Pavements of the Laurentian Library." In Nexus I1: Architecture and Mathematics edited by K. Williams. Fuccechio, Italy: Edizioni dell'Erba (1998). 4. Hambridge, J. The Elements of Dynamic Symmetry. Originally published by Brentano's (1929) (rpt. By New York: Dover). 5. Edwards, E.B. Pattern and Design with

Dynamic Symmetry. Originally published by Dynamarhythmic Design (1932) (rpt. New York: Dover) 6. Brunes, T. The Secrets of Ancient Geometry and its Use. Copenhagen: Rhodos (1967). 7. Kappraff, J. "A Secret of Ancient Geometry." In Geometry at Work: A Collection of Papers in Applied Geometry edited by K. Gorini. Math. Assoc. of Amer. Notes (In press). 8. Kappraff, J. Mathematics Beyond Measure: A Guided Tour through Nature, Myth, and Number. New York: Plenum Press (In press). 9. Kappraff, J. "Musical Proportions at the Basis of Systems of Architectural Proportion both Ancient and Modern." In Nexus: Architecture and Mathematics edited by K. Williams. Fuccechio: Edizioni dell'Erba (1996). 10. Nicholson, B. "Architecture, Books + G-'~bmetry." In CD-Rom: 'Thinking the Unthinkable House. Renaissance Society at the University of Chicago (1997). 11. McClain, E. "The Star of David as Jewish Harmonical Metaphor." Intemational Journ. of Musicology. Vol. 6, pp. 24-49. (1997). 12. McClain, E. Private communication. 13. McClain, E. The Pythagorean Plato. York Beach, ME: Nicolas-Hays (1978). 14. McClain, E. "Temple Tuning Systems." InternationalJoum. of Musicology, 3 (1994). New Jersey Institute of Technology Newark, NJ 07102 USA

VOLUME 21, NUMBER 3, 1999

29

More or Less Mathematics Wanted in Engineering?

Two images, one of the medieval ceUarium at Fountains Abbey, England, and another of a tiled forecourt in Vernon, France, inspired two engineers to different conclusions. Thefirst author considers that the importance of abstract mathematics is somewhat overrated, and he even makes an appeal to the scientific community to support his view. The second author calls for another kind of intervention: he asks for some mathematical assistance by proposing an unsolved problem to readers. Perhaps engineers and mathematicians will have reactions, outraged or otherwise, to contribute to The Intelligencer's letters.--D.H.

A. Less Mathematics and More Numeracy Wanted in Engineering

Edward Reed To give academic respectability to a trivial piece of engineering research, it is standard practice to add some mathematics to make it appear more significant than it is. Mathematics is the silicon implants of academia. This is not to be confused with numeracy. Numeracy is highly relevant for engineers and unforttmately is often lacking (see Figure 1). Most readers will have looked at the drawing, read the caption and answered the question before even reading the first paragraph. I understand this is a question in Trivial Pursuit, although I cannot confirm it. The answer universally accepted is the Great Wall of China. The real question is, is this true? I ask students, colleagues, and friends this question. Some look it up in books, like Hutchinson's New Century Encyclopaedia (Helicon Publishing Ltd., 1995), or the Readers' Digest Book of Facts. They choose between the supporters of the c o m m o n positive answer, as proposed in the first reference, or the opponents, as in the latter. Few give a reasoned answer. What I am really after is some c o m m o n sense, ap-

30

THE MATHEMATICAL INTELLIGENCER 9 1999 SPRINGER-VERLAG NEW YORK

plied numeracy. We will return to this question later. At school I liked and was good at mathematics. Once the prettiest girl in the class allowed me to ldss her in exchange for letting her copy my maths homework. That is the only time in m y life, and I am a sexagenarian, that I've ever found a use for mathematics. More seriously, it has been my experience that very few engineers use mathematics in practice. They have of course to be numerate. The insistence on mathematics for all engineers has been a damaging deterrent to some highly capable, imaginative, and creative people. Engineering is after all about making things work. The empirical approach, trial and error, the ability to experiment quickly is of much more importance. Most problems cannot be solved by mathematics anyway, or they have to be reduced to too simple a model to be useful. I expressed this view in a rather light-hearted manner in an article published internally within my university. To my surprise, many colleagues said that what I had said needed saying again and again. The professor of mathematics got the vapours, refused to speak to me, and shortly afterwards took early retirement. I also likened mathematics in engineering to the wowing of ladies: as there is supposed to be a lot of it about and since m o s t men find little of it coming their w a y

Figure 1. What is the only man-made object that can be seen on Earth from the Moon?

Figure 2. The medieval cellarium at Fountains Abbey, England (photograph reproduced by kind permission of Yorkshire Post Newspapers).

t h e y believe s o m e b o d y else m u s t b e doing awfully well. After learning m y views on mathematics, m a n y o f m y critics tell m e that t h e y w o u l d n o t like to p a s s over a b r i d g e that I h a d designed. Neither w o u l d I: My a r e a is m e c h a t r o n i c s . However, bridge-building will serve as ~m example. No bridge w a s ever built b y mathematics. They are designed and built b y t e a m s of engineers requiring a v a s t range of skills; the m o r e grandiose the bridge, the bigger the range of

Figure 3. Map of France, iocating Vernon.

course numerate: they n e e d e d to calculate such m a t t e r s as t h e n u m b e r o f eggs required for the m o r t a r (see Figure 2). B a c k to the Great Wall problem. Here is one solution. Put y o u r thumb in front o f you and hold it up to the Moon. The Moon a p p e a r s as a small disc a b o u t one-sixth the size of your thumbnail. By estimation it can b e said that an o b j e c t as far a w a y and as big as the M o o n - which is about 3000 k m across---appears about 3 m m across to the eye. If we estimate that the wall is 5m across, we c a n calculate b y p r o p o r t i o n w h a t thickness it w o u l d a p p e a r to s o m e b o d y on the Moon. The a n s w e r is that it is much t o o small for even the most p o w erful o f telescopes, let alone the n a k e d eye. An often e x p r e s s e d view that the wall c a n be seen b e c a u s e it is over 2000 km long is patently daft. It is h o p e d that the wall e x a m p l e has d~monstrated what is meant b y numeracy.

skills. Bridges are built as something fit for the p u r p o s e at the m o s t economicai price, a n d they have b e e n since time immemorial. The medieval builder h a d only w o o d and stone, b u t he could construct s t o n e arches and k n e w that if a shape, k n o w n to us as a catenary, could b e d r a w n so as to go through every stone, t h e n his arch w o u l d s t a n d up. The great medieval bridges and cathedrals of E u r o p e were built without mathematicians, but the builders had to be imaginative, creative, sldlled, and of

Figure 4. The tiled forecourt in Vernon--is there a pattern?

VOLUME 21, NUMBER 3, 1999

31

As a teacher of engineering, I have problems with students and their numerical skills even if their mathematics is good. I recently did an overseas exchange with another teacher, which i n v o l v e d ' u s taking one another's classes. I found his students similarly poor at numerical work, so maybe this is a universal problem. Numeracy might be boring and mathematics exciting: I confess to having found this so at school. However it is numerical skills that are most lacking in engineering students. Perhaps mathematicians should address themselves to this problem. Leeds MetropolitanUniversity Leeds LS1 3HE United Kingdom

32

THE MATHEMATICAL INTELLIGENCER

B. More Mathematics Required to Explain an Ingenious Tiling Pattern in Vernon, France

R.J. Holroyd While on holiday in France I visited Vernon, a small town that stands on the Seine about midway between Paris and the sea. (See figure 3.) About a mile upstream is the small village of Giverny, with the house and gardens where Monet spent the last half of his life. While walking around the town, I came across a recently built arts centre (the Espace Culturel Philippe Auguste) with an intriguingly tiled forecourt. I was told that the workmen had a plan for laying the blocks, but no one could tell me how the plan was produced. The pattern, shown in figure 4, is made

up of five different blocks: 3 X 2, yellow; and 2 x 2 and 2 x 1 in both black and white, but their arrangement appears to be neither regular nor irregular. A mathematician suggested that the pattern might be an example of asymptotic periodicity and referred to me to some algebra textbooks, but so far as I could see these did not shed any light on the problem. Out of my depth about these mathematician's mathematics, apparently so different from the usual engineering computations, I would welcome any enlightenment about this fascinating pattern-or lack of pattern. 104 Arbury Road Cambridge CB4 2JF United Kingdom

AVRAHAM FEINTUCH AND ALEXANDER MARKUS

Tho Toeplitz-I lausdorff Theorem and Robust Stability Theory Introduction In his Hilbert Space Problem Book, Paul Halmos wrote [9],

a convex set? He then states that the result is false for the case n -> 3. Halmos reports these facts as follows [9, p. 110]:

p. 108]: In early studies of Hilbert space (by Hilbert, Hellinger, Toeplitz and others) the objects of chief interest were quadratic forms. Nowadays they play a secondary role. First comes an operator A on a Hilbert space Y~and then, apparently as an afterthought, comes the numerical-valued function f--* (Af, f ) on Y~. This is not to say that the quadratic point of view is dead; it still suggests questions that are interesting with answers that can be useful. As Halmos points out, the object of importance is the numerical range of an operator ("operator" throughout this article means a b o u n d e d linear operator on a c o m p l e x Hilbert space), the set

W(A) = {(Af, f ) : [[f[[ = 1} C C, where ( , ) denotes the inner p r o d u c t on Y~and II [Ithe n o r m induced by it. The fundamental property of this set, k n o w n as the Toeplitz-Hausdorff Theorem, is that it is convex. Toeplitz [13] first proved that the boundary of W(A) is a c o n v e x curve. Hausdorff [10], using a different approach, s h o w e d that, in fact, W(A) is a convex set. He also suggested looking at the theorem in the following way. Write A as A = B + i C , where B = (A+~*) and C = (A-A*) are 2 2i Hermitian operators. Then, the Hermitian forms (Bf, f ) and (Cf, f ) take on real values only~ and the point sets

W(A) C C

and

{((Bf, f ) , (Cf, f ) ) : Ilfll = 1} C R 2

are identical. This suggested to Hausdorff the following possible generalization: If A1..., An are Hermitian operatots, is the set {((Atf, f ) , (A2f, f ) , ..., (Anf, f)): IJfH = 1} C R n

(8)

It is a pity that it is so very false. It is false for n = 3 in dimension 2; counterexamples are easy to come by. The irony is that although false for dimension 2, it turns out to be true for all other dimensions, for n = 3 (for n > 3 Halmos's statement is precise). This was s h o w n by various authors from different points of view ([1], [3], [11]). In fact, the difference between the two-dimensional case and that of higher dimensions was already pointed out in [4]. Also, [5] proves the following result on "quasi-convexity" for the set (*): it contains, with any two points, an ellipsoid (perhaps degenerate) joining them. There are also variations of Hausdorff's suggested generalization which hold for all n (see the section Further Connections). Recently, connections 'have been made between these convexity results and the ideas arising in the theory of robust stability of systems. The purpose of this article is to highlight these relationships. In the next section, we present an elementary p r o o f of the generalized Toeplitz-Hausdorff T h e o r e m for three selfadjoint operators and a connterexample for the case n = 4, which is appropriate for any dimension. The main idea of the p r o o f given here is from [11] and is in the spirit of the original p r o o f of Hausdorff [10]. In order to shorten the proof, we assume a dimension of at least 4. The three-dimensional case needs additional efforts. In the following section, we give a brief discussion of robust stability of linear feedback systems and formulate a central problem. We go on to s h o w h o w the generalized Toeplitz-Hausdorff Theorem is used to solve this problem. The final section gives a brief survey of further generalizations and connections.

9 1999 SPRINGER-VERLAG NEW YORK, VOLUME 21, NUMBER 3, 1999

We complete this introduction with an exercise, which supplies the counterexample alluded to by Halmos. Let A]:[~

10],

A2=[~-~],

A3=[10

_~].

Show tha~ {((A1 f, f ) , (A2f, f ) , (A3f, f ) ) : Ilfl] = 1} is not convex (it is in fact the unit sphere in R3).

The Joint Numerical Range is a complex Hilbert space and ~ ( ~ ) denotes the bounded linear operators on Y~. The j o i n t n u m e r i c a l range W(A1, A2, A3) of three Hermitian operators A1, A2, A3 ~(Y~) is the set [((Alf, f ) , (A2f~f), (A3f, f)): ]]fil = 1}.

then h ._L A f and h ._L A * f imply (by the above) that h and f are path connected in Z(A). Also, (h, Ag - A ' g ) = 0 implies that I(Ag, h)l = [(Ah, g)[, so h and g are path connected in Z(A). Thus, so are f and g. Assume h ~_ Z(A) or (Ah, h) r 0. We show the existence of a path in Z(A) of the form s(t) = tf + (1 - t)g + u(t)h, where u(t): [0, 1]---) R satisfies u ( 0 ) = u ( l ) = 0. The assumptions on h gave I(Ag, h)l = I(Ah, g)l. Replacing h by ei*h, for s o m e #s E R, if necessary, allows us to assume that (Ag, h) + (Ah, g) = O. This, together with the assumption that h _L A f and h _L_A'f, gives that (As(t), s(t)) = t(1 - t){(Af, g) + (Ag, f ) } + u2(t)(Ah, h). Thus, (As(t), s(t)) = 0 if and only if

We assume that Y~ h a s dimension at least 3.

u2(t ) =

t(1 - t)[(Af, g) + (Ag, J)] (Ah, h)

T h e o r e m 1: W(A1, A2, As) is a convex set i n It 3. For Zl We m a k e the observation that if x, y E W(A1, A2, A3) and t E (0, 1), we can construct an aff'me transformation T on R 3 such that Tx = (0, O, 1 - t), Ty = (0, 0, - t ) , and T(tx + (1 - t)y) = (0, 0, 0). T m a p s the triple (A1, A2, As) to a new triple (B1, B2, B3) of Hermitian operators. Thus we can reduce the p r o o f of the t h e o r e m to showing that if p, /z > 0 with (0, 0, p), (0, O, - t z ) ~ W(A1, A2, A3), then (0, O, O) E W(A1, A2, A3). We give an alternate formulation. Write A E L(YEJ as A = A1 + iA2 and define Z(A) = { f E ~: Hf[I = 1, (Af, f ) = 0}. It is to be shown that if (A3f, f ) takes on positive and negative values on Z,(A), then it takes on the value zero as well. This will follow from showing Z(A) is connected or, equivalently, that Z(A) = {f r 0: (Af, f ) = 0} is. This was proved by Hausdorff [10] forA Hermitian and in [9] for any operator A with Y~ of dimension at least 3 (in case of dimension 2, A = [~

T h e o r e m 2. For A E D~(Y~) w i t h d i m e n s i o n ~. >--4, Z(A) is a connected set.

(Af~ and z ~ = (Ag.f) c h o o s e a real n u m b e r 0 such

that fl -- e i~ Zl + e -i~ z2 <- O. Replacing f by ei~ gives u(t) = ~x/-flt(1 - t). We sdll need to show that for 0 < t < 1, s(t) r 0. Suppose for 0 < to < 1, S(to) = O. Then, (S(to), Ag - A ' g ) = 0, and since (h, A g - A ' g ) = O, (g, Ag) = (g, A ' g ) = 0, we obtain to(f, Ag) - to(Af, g) = O. This contradicts I(Af, g)l r I(Ag, f ) l 9 [] E x a m p l e . Suppose dim Y~ = m (3 - m -< ~), and consider

~] provides a counter-

example). The p r o o f is significantly easier technically for dimension Y~ >- 4, so w e will m a k e this assumption.

=

0

1

B 1 =

1

0

1

0

B3 =

0-1

:J, :J,

B4 =

:]

'

~ =

0

o0 i1 0

0

9

0

Then, f o r f = (fl, . . . ,fro) E Y~, 3

(Bkf, f)2 = (If1[2 + If=l=)=, k=l

(S~,f,

f)

If~[ =.

-k=3

4

Proof. We show that any two vectors f, g E Z(A) are path connected in Z(A): there exists a continuous function s(t): [0, 1] ~ Z(A) with s(0) = g and s(1) = f . If f = (xg, this is obvious, so we can a s s u m e they are linearly independent. If f and g satisfy (Af, g) + (Ag, f ) = 0 (for example, g _LAf, g _L A~f), then for s(t) = t f + (1 - t)g, (As(t), s(t)) = t(1 - t){(Af, g) + (Ag, f ) } = O, so t h a t f and g are connected by a line in Z(A). We extend this argument to the condition I(Af, g)l = I(Ag, f)l. In this case, let Zl = (Af, g) and z2 = (Ag, f ) . Then, there exists a real n u m b e r ~ such that e i~ Zl + e -i~ z2 = O, or (A(ei~ f ) , g) + (Ag, eiv f ) = O. Thus, ei~ f and g are coimected in Z(A ) and so are f and g. Now suppose/(Af, g)l r I(Ag, f)[. Let h #= 0 be orthogonal to {Af, A ' f , Ag - A'g} (using dim Y~ -> 4). If h E Z(A),

THE MATHEMATICALINTELLIGENCER

Thus, for [if i[ = 1, ~. (Bkf~f) 2 > 0; SO 0 {~ W(B1, B2, B3, B4). k=l

On the other hand, f = (2 -v2, 2 - l e , 0 , . . . , 0) gives ((Bkf, f))~ = (1, 0, 0, 0), and f = (2 -1/2, - 2 -1/2, 0 , . . . , 0) gives ((Bkf, f))~ = ( - 1 , 0, 0, 0), so that W(B1, B2, B3, B4) is not convex. Robust Stability Theory A m a j o r p u r p o s e of control engineering is that of regulation: external disturbances act on a physical system, and one m u s t design m e c h a n i s m s that keep certain to-be-controlled variables within certain bounds. The temperature in houses is regulated by a t h e r m o s t a t so that the inside t e m p e r a t u r e remains within certain bounds. This thermostat illustrates a central concept of control, feedback. The value of one variable in the system is m e a s u r e d and is "fed back" in order to take appropriate action through a con-

trol variable at a n o t h e r p o i n t in the system. The t h e r m o s t a t s e n s e s the r o o m t e m p e r a t u r e , c o m p a r e s it with t h e des i r e d temperature, and f e e d s b a c k the result to the furnace, w h i c h then starts or shuts off, in o r d e r to raise o r l o w e r t h e t e m p e r a t u r e so that it will c o m p l y with w h a t is desired. We p r e s e n t a s t a n d a r d m a t h e m a t i c a l m o d e l for a linear f e e d b a c k system. The signals are m o d e l e d by v e c t o r s in a Hilbert s p a c e ~ and the s y s t e m s are linear o p e r a t o r s on Y~. Stable linear s y s t e m s c o r r e s p o n d (at this stage) to b o u n d e d linear o p e r a t o r s on Y~, and instability m e a n s that the relevant o p e r a t o r is n o t bounded. A f e e d b a c k s y s t e m is d e s c r i b e d b y the equations u

e

~

y

l y = Me, e = u + Ay.

Now, in o r d e r for the d e s c r i p t i o n to m a k e sense, the outp u t signal m u s t d e p e n d on the external input signal u, so it m u s t b e p o s s i b l e to solve t h e s e equations to o b t a i n y in t e r m s o f u. In o r d e r to do this, one m u s t b e able to invert I - AM, and if that is possible, w e obtain y = M ( I A M ) - l u . If M and A are s t a b l e and ( I - ~ ) - 1 is stable, w e s a y that the s y s t e m is internally stable. In the last t w o decades, c o n t r o l engineers have b e e n c o n c e r n e d with the fact t h a t modeling e r r o r s c a n highly d i s r u p t conclusions a b o u t stability of f e e d b a c k systems. T h e y have d e v e l o p e d a t h e o r y o f "robust stability": constructing f e e d b a c k s y s t e m s t h a t will r e m a i n stable in the p r e s e n c e of a p r e s c r i b e d class o f m o d e l i n g errors. One w a y of doing this is b y taking M to r e p r e s e n t a k n o w n stable s y s t e m and letting h r e p r e s e n t an uncertainty system which is k n o w n only to belong to a certain o p e r a t o r ball, s a y IIAII < 1. In this case, it is i m m e d i a t e that if NMII -< 1, then the given f e e d b a c k system is stable, for then IIAMII < 1 a n d ( I AM) -1 is given by the norm-convergent geometric series cc

~ . (AM) n. In fact, this c o n d i t i o n on M is also necessary. n=0

If IIMII > 1, t h e r e exists A with IIA]] < 1 such t h a t / - A M i s n o t invertible. This is s e e n as follows: let p = IIMII > 1 a n d T = (1/p2)M *. Then, IITII < 1 and I - T M = ( 1 / p 2 ) ( p 2 I M ' M ) . Because M * M is a non-negative o p e r a t o r a n d p2 -IIMH2 = HM*MII 9 o-(M*M) (the s p e c t r u m o f M ' M ) , I - T M is n o t invertible. However, w e naturally w a n t the t h e o r y to treat the situ a t i o n w h e n t h e r e are multiple inputs and outputs e a c h giving rise to its o w n uncertainties. Then M is an n • n opera t o r matrix, a n d the u n c e r t a i n t i e s m o d e l e d by h arise as an n X n diagonal o p e r a t o r matrix. Even for n -- 2, this s e t u p is challenging. In o r d e r to d i s c u s s this situation, we i n t r o d u c e s o m e term i n o l o g y and notation. The s t r u c t u r e d n o r m ( i n t r o d u c e d b y Doyle [6] and Safonov [12]) o f A 9 ~(Y~), relative to a given s u b a l g e b r a ~ C ~s with identity, is the n u m b e r

p~(A)

1

inf{llTIl: T 9 ~ , I - TA n o t invertible}"

This n u m b e r h a s p r o v e d t o b e a p o w e r f u l tool in the s t u d y o f r o b u s t stability. ~ gives a m e a s u r e of stability with res p e c t to the t y p e s of uncertainties d e s c r i b e d above. Using this term, w e c a n reformulate our p r e v i o u s d i s c u s s i o n a s the formula/z:~(vO (M) = NMII. It is easily s e e n that if ~ 1 C_ ~2, then t z ~ ( A ) - / ~ 2 ( A ) , and that for 50 = {M: A 9 C}, Ix~r(A) = p(A), the s p e c t r a l radius o f A. Thus, in general, p(A) <- Pat(A) <--[IAII. We will c o n s i d e r the c a s e where A is an n X n o p e r a t o r m a t r i x acting on the direct s u m Ye(n) o f n c o p i e s of ~ . The entries o f A a r e from ~(Y~). ~ n will d e n o t e the diagonal m a t r i c e s with entries from ~(Y~), a n d ~ n will d e n o t e t h e o p e n unit ball o f ~n. Also, w e will r e p l a c e ~ n by/-~n, emphasizing the r o w - c o l u m n size of A. Clearly, for all IIAII _ 1, T 9 ~ n , I - TA is invertible. S u p p o s e X is an invertible o p e r a t o r w h i c h c o m m u t e s with all m e m b e r s o f ~n (which w e indicate b y w r i t i n g X 9 ( ~ n ) ' ) , with IIXAX-111 -< 1. Since I - TA is invertible if a n d o n l y if X ( I - T A ) X - 1 = I TXAX -1 is, t h e n even if-~AII > 1, I - TA is irivertible for T 9 !~n. This implies that the sufficient condition IIAII _< 1 for r o b u s t stability is quite conservative, and this m o t i v a t e s the i n t r o d u c t i o n of a n o t h e r number:

f~(A) = ~{[IXAX-~II: X 9 (~n)', X invertible}. It is always t h e case that pro(A) - 4, there exists A s u c h

that ~ ( A ) < ~ ( A ) . The similarity in n u m b e r s b e t w e e n this fact a n d the generalized T o e p l i t z - H a u s d o r f f T h e o r e m is no c o i n c i d e n c e and is the s u b j e c t o f the n e x t section o f this article. In fact, w h e r e a s the c a s e n = 2 follows i m m e d i a t e l y f r o m the c a s e n = 3, it c a n also be o b t a i n e d directly from t h e classical T o e p l i t z - H a u s d o r f f Theorem. From Toeplitz-Hausdorff to /z = f~ Let A b A2, a n d A3 b e Hermitian o p e r a t o r s on Y~. The foll o ~ n g l e m m a is a simple c o n s e q u e n c e of the generalized T o e p l i t z - H a u s d o r f f Theorem. L e m m a [8]. I f there is no f s u c h that (Akf, f ) > Of o r k = 1, 2, 3, then there e x i s t n o n - n e g a t i v e n u m b e r s ~1, a2, a n d a3 s u c h that at + ol2 Jr ol3 > 0 a n d atA~ + c~2A2 + a3A 3 --< 0. Proof. Let S+ = {<xl, x2, x3> E R3: Xk > 0, k = 1, 2, 3}. By hypothesis, S+ A W(A1, A2, A3) -- ~ . Since W(A1, A2, A3) is convex, t h e r e exists a p l a n e alXl + a2x2 + aax3 = 0 in R 3 such t h a t W(AI, A2, A3) C {<Xl, x2, x3>: a l x t + a2x2 + c~3x3 --< 0}

and S + C [(Xl, x2, x3>: o/ix 1 -F o/2x 2 + o~3x3 ~ 0}.

VOLUME 21, NUMBER 3, 1999

The first of these conditions gives alA1 + a~A2 + a~A~ <0; the s e c o n d gives ak --> 0, k = 1, 2, 3. []

should be mentioned that in [7] computational algorithms for finding the distance from the origin to the convex set W(A1, A2, A3) are used to compute/~3(A).

T h e o r e m [8]. For Further Connections

I

-All A12 A131 A = A21 A22 A23/ LA31 A32 A33J

with Aij E ~(Y0,/~3(A) =/23(A). Proof. The diagonal algebra ~3 has a commutator (~ which is all scalar diagonal 3 x 3 matrices. For At, h2, h3 :# 0, define

.o11, 0 ,] i ,0 ]

AA =

h2 ]I

L0

00

A

h21

0

9

0

Since /~3(A) -
aj(f)

=

I1 , Ajkfkll 2 -IIf3ll

2, J

1, 2, 3,

=

f o r f = (fl, f2,f3), fi E Y~.

k=l

It is not hard to show that if I - AA is invertible for all A described above, then for no f i s ~ j ( f ) > 0 f o r j = 1, 2, 3. Then, applying the lemma, we obtain the existence of numbers al, O~2, Ot3 ~ 0 w i t h O~1 4- O~2 + OL3 > 0 such that o ~ t ~ l ( f ) + a 2 ~ 2 ( f ) + a 3 ~ 3 ( f ) ~ 0, or

il

9=

,il ll[J d.ill ozlt2

. 3

To simplify matters, we assume ai > 0, i = 1, 2, 3. (The cases where one or two of the ai are zero can be dealt with by a limiting process.) In this case, simply let gk = ak-1/2fk and rewrite the inequality as 1/2 OLk

73 J

j = l II

Taking hk = ak-1/2 gives the required result. R e m a r k s . In the pioneering w o r k of Doyle [6], another approach, which holds apparently only for dim Y~ < % is used. The validity of the equality pro(A) =/2n(A) is reduced to the problem of convexity of the numerical range of n 1 self-adjoint operators in a subspace ~0 whose dimension is the multiplicity of the maximal singular value of XoAXb-1 for which/2n(A) = JJXoAXolII..Thus when n = 3 the required result is derived from the classical Toeplitz-Hausdorff Theorem. When n = 4, one can construct an example where dim Y~0 = 2, and this is exactly where the joint numerical range is not convex for three operators. If the maximal singular value ofXoAX~ t has multiplicity 1 or -----3then /~4(A) =/24(A). Thus, the failure of the theorem for n = 4 is a restricted phenomenon, and more research is required to understand fully even the finite-dimensional case. It

T H E M A T H E M A T I C A L INTELLIGENCER

The importance of the joint numerical range for the study of/~ and/2 goes b e y o n d what we described in the previous section. Although the fact that/-r is continuous with respect to the n o r m topology on ~(Y0 is elementary, the continuity o f / 2 is nontrivial and follows from a generalized Toeplitz-Hausdorff Theorem [2]. Another direction is the study of causal systems. This means that we consider only operators which are lower triangular with respect to a given fLxed orthonormal basis {e~}. In this case,/z](A) = lim 11(I - Pk)AII, where Pk is the prok--~ao

jection onto {ei}k = t. It is interesting that in this case, the equality b e t w e e n / ~ a n d / ~ holds even for n >- 4. The reason for this is that although W(A1, A2 . . . . . An) is not, in general, convex, there is an appropriate "limit," the essential joint numerical range, which is convex. This is what is required for the appropriate analog of the theorem we presented (see [8]). We conclude by repeating the words of Halmos quoted earlier: "This is not to say that the quadratic point of view is dead; it still suggests questions that are interesting with answers that can be useful." REFERENCES 1. Y.-H. Au-Yeung and Y.-T. Poon, A remark on the convexity and positive definiteness concerning Hermitian matrices, Southeast Asian Bull. Math. 3 (1979), 85-92. 2. H. Bercovici, C. Foia~, and A. Tannenbaum, The structured singular value for linear input-output operators, SIAM J. Cont. 9 34 (1996), 1392-1404. 3. P. Binding, Hermitian forms and the fibration of spheres, Prec. AMS 94 (1985), 581-584. 4. Ch. Davis, The shell of a Hilbert-space operator. II, Acta Sci. Math. 31 (1970), 301-318. 5. Ch. Davis, The Toeplitz-Hausdorff theorem explained, Can. Math. Bull. 14 (1971), 245-246. 6. J. C. Doyle, Analysis of feedback systems with structured uncertainties, IEEE Proc. Pt. D, 129 (1982), 242-250. 7. M. Fan and A. Tits, m-Form numerical range and the computation of the structured singular value, IEEE Trans. Automatic Cont. 33 (1988), 284-289. 8. A. Feintuch and A. Markus, The structured norm of a Hilbert space operator with respect to a given algebra of operators, Oper. Theory: Adv. Appl. (in press). 9. P. R. Halmos, A Hilbert Space Problem Book, Van Nostrand, Princeton, NJ, 1967. 10. F. Hausdorff, Der Wertvorrat einer Bilinearform, Math. Z. 3 (1919), 314-316. 11. Yu. Lyubich and A. Markus, Connectivity of level sets of quadratic forms and Hausdorff-Toeplitz type theorems, Positivity 1 (1997), 239-254. 12. M. G. Safonov, Stability Robustness of Multivariable Feedback Systems, M.I.T. Press, Cambridge, MA, 1980. 13. 9 Toeplitz, Das algebraische Analogon zu einem Satze von Fejer, Math. Z 2 (1918), 187-197.

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VOLUME 21, NUMBER 3, 1999

37

ILAN VARDI

What is Ancient Mathematics? In m y opinion, it is not only the serious accomplishments of great and good m e n which are worthy o f being recorded, but also their amusements. XENOPHON,SYMPOSIUM The title of this paper is a result of comments on earlier drafts by mathematicians: "This is not mathematics, this is history!" and by historians of mathematics: "This is not history, this is mathematics!" After some reflection, I came to the conclusion that the historians were right and the mathematicians were w r o n g - - f o r example, I have found little difference between reading papers of Atle Selberg (1917-, Fields Medal 1950) and Archimedes (287-212 BC) (who both lived in Syracuse!). I believe that the mathematicians I spoke to were expressing a generally held belief that reading mathematical papers that are over a hundred years old is history of mathematics, not mathematics. Thus, the reconstruction of Heegner's solution to the class-number-one problem (1952) appeared in a mathematics journal [52], while a reconstruction of the missing portions of Archimedes's The Method (250 BC) appeared in a history journal [29]. To me, reading and proving results about a mathematical paper, whether it was written in 1950 or 250 BC, is always mathematics, though the latter case might be called "ancient mathematics." At least as to Greece, this is accepted by some eminent mathematicians [30, p. 21]: Oriental mathematics m a y be an interesting curiosity, but Greek mathematics is the real thing . . . The Greeks, as Littlewood said to me once, are not clever schoolboys or "scholarship candidates," but "Fellows of another college." So Greek mathematics is "permanent," more perhaps even than Greek literature. Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not.

38

THE MATHEMATICAL INTELLIGENCER 9 1999 SPRINGER-VERLAG NEW YORK

I am saying that ancient Greek mathematicians were in every essential way similar to modern mathematicians. In fact, some mathematicians might find more in common with Archimedes and Euclid than with many colleagues of their departments, and even reading the original Greek--a subject traditionally taught in High School [9]--seems easier than understanding, say, the proof that every semistable elliptic curve is modular [59]. Nineteenth-century mathematicians dedicated much of their research to elementary Euclidean geometry. It is possible that some mathematicians of that era felt that the influence of the past was too great, as Felix Klein wrote [38, Vol. 2, p. 189]: Although the Greeks worked fruitfully, not only i n geometry but also in the most varied fields of mathematics, nevertheless we today have gone beyond them everywhere and certainly also in geometry. For whatever reason, geometers recently tend to distance themselves from Euclidean geometry. For example, the book Unsolved Problems in Geometry [16], part of a series on "unsolved problems in intuitive mathematics," does not have a section devoted to classical Euclidean geometry, and with few exceptions, such as [1% articles on this subject are relegated to "lowbrow" publications. Yet earher in this century, Bieberbach, Hadamard, and Lebesgue all wrote books on elementary Euclidean geometry [13] [27] [44], and excellent books and articles on ancient mathematics are still being written [31] [55]. See [17] for further analysis of these issues. In this paper, I will give an example of ancient mathematics by using techniques that Archimedes developed in his paper The Method to derive results that he proved in his paper On Spirals. I will try to present these in a way that Archimedes might understand [57], in particular, the

diagrams are intended to conform to ancient Greek standards [46]. I will also indicate how ideas in these papers can lead to some surprising results (e.g., Exercise 4 below). The paper will include such exercises as may challenge the reader to understand concepts of Archimedes as he expressed them. (I have concentrated on the works of Archimedes because these are most similar to modern mathematical research papers, sharply focused on problems and their solution. By comparison, the works of Euclid read like a generic textbook; and so little is known about Euclid that it cannot be ruled out that he was actually a "consortium." Moreover, it seems likely that the works of Euclid are based on the efforts of earlier mathematicians [24] [39]. 1) The balance of the paper shows how a precise knowledge of ancient mathematics allows one to navigate in the sea of inaccuracies and misconceptions written about the history of mathematics. This also gives one perspective on cultural aspects of mathematics, as it forces one to understand ideas of first-rate mathematicians whose cultural background is very different from the present one. For example, it can help you read The New York Times [37]: "Alien intelligences m a y be so f a r advanced that their m a t h would simply be too hard for us to grasp," [Paul] Davies said. "The calculus would have baffled Pythagoras, but with suitable tuition he would have accepted it." Reading this paper should make it clear that Archimedes could have been Pythagoras's calculus tutor, thus refuting any implication that calculus was an unknown concept to ancient Greeks. It is my hope that I can convince mathematicians that there are many interesting and relevant ideas to be uncovered in ancient Greek mathematics, and that it might be worthwhile to take a first-hand look, being wary of popular accounts and secondary sources, this one included!

Extending Archimedes's Method In 1906 the Danish philologist J.L. Heiberg went to Constantinople to examine a manuscript containing mathematical writing which had been discovered seven years earlier in the monastery of the Holy Sepulchre at Jerusalem. What he found was a 10th-century palimpsest-a parchment containing works of Archimedes that, sometime between the 12th and 14th centuries, had been partially erased and overwritten by religious text. Heiberg managed to decipher the manuscript [33] and found that it included a text of The Method, a work of Archimedes previously thought lost. (The story of the transmission of Archimedean manuscripts given in [18] reads like a chapter from The Maltese Falcon. Late bulletin: Heiberg's

palimpsest was sold by Christie's for $2,000,000--see Jeremy Grey's article in this issue.) This discovery had a significant impact on the understanding of ancient Greek mathematics, for two reasons. The first is the aim of the paper, summarized by Archimedes 2 [54, Vol. 2, p. 221]: Moreover, seeing in you, as I say, a zealous student and a m a n of considerable eminence in philosophy, who gives due honour to mathematical inquiries when they arise, I have thought f i t to write out f o r you and explain in detail in the same book the peculiarity of a certain method, with which furnished you will be able to make a beginning in the investigation by mechanics of some of the problems in mathematics. I a m persuaded that this method is no less useful even f o r the proofs of the theorems themselves. For some things f i r s t became clear to me by mechanics, though they had later to be proved geometrically owing to the fact that investigation by this method does not amount to actual proof," but it is, of course, easier to provide the proof w h e n some knowledge of the things sought h ~ br acquired by ,this method rather than to seek it with no prior knowledge. This is a radical divergence from all other "extant Greek works, as T.L. Heath explains [6, Supplement, p. 6]: Nothing is more characteristic of the classical works of the great geometers of Greece, or more tantalising, than the absence of any indication of the steps by which they worked their ways to the discovery of their great theorems. As they have come down to us, tttese theorems are finished masterpieces which leave no traces of any rough-hewn stage, no hint of the method by which they were evolved. . . A partial exception is now furnished by The Method; for here we have a sort of lifting of the veil, a glimpse of the interior of Archimedes' workshop as it were. The other surprising aspect of The Method is the revelation that Archimedes worked with infinitesimals, for example, "The triangle FZA is composed of the straight lines drawn in FZA" [54, Vol. 2, p. 227], "The cylinder, the sphere and the cone being filled by circles thus taken." [8, vol. 3, p. 91], see [1] [42]. As every mathematician knows, infmitesimals were reinvented by mathematicians such as Cavalieri (1598-1647) and Leibniz (1646-1716), see [2] [21]. Archimedes used them to compute the area and volumes of various geometrical figures including what he considered his greatest achievement: 3 A n y cylinder having f o r its base the greatest of the circles in the sphere, and having its height equal to the diameter of the sphere, is one-and-a-half times the sphere,

1Hence I question the curriculum of St, John's College, which purports to educate its students by following an historical sequence of original sources, Its reading list also includes the ancient textbook [47]. 2Archimedes is addressing Eratosthenes of Cyrene (circa 284-194 B.C.), director of the library of Alexandria, famous for his accurate measurement of the circumference of the earth [14] and his sieve to compute prime numbers [47]. 3Archimedes requested that g diagram of a sphere inscribed in a cylinder along with their proportion be placed on his grave, which Cicero reported finding in 75 B.C. when he was treasurer of Sicily [54. Vol. 2, p. 33].

VOLUME21, NUMBER3, 1999 39

a result he subsequently proved, On the Sphere and Cylinder, I, Corollary to Proposition 34 [54, Vol. 2, p. 125]. Archimedes understood that his method does not produce valid proofs due to its use of infmitesimals, 4 though it is unclear if the same is true of his successors. In any case, it is easy "to make the arguments rigorous, given present knowledge. The basic ideas of The Method are still presented in contemporary calculus courses [35] [51, p. 709], and a physical model of Archimedes's argument has been built [25]. On the other hand, Archimedes's On Spirals is a masterpiece of rigorous mathematics. In this paper, Archimedes computes the area and tangent of a spiral, and, in doing sg, derives much of the Calculus I curriculum, including related rates, limits tangents, and the evaluation of Riemann sums. This is reflected by the fact that a number of contemporary Calculus texts outline the basic idea of Archimedes's computation of the area of a spiral [3, p. 3] [12, p. 75], though both these works avoid technical difficulties by substituting a parabola, but then incorrectly imply that Archimedes used such an approach for the parabola [3, p. 8] [12, p. 75]. The considerable length of the paper is a consequence of proving these results from basic principles. Unfortunately, it does not yet have a faithful English translation [56]; Heath's intent in [6] was to capture the modern flavor of Archimedes's works in order to make them more accessible. A generally faithful French translation, including the Greek text, is available [8]. The mechanical method does not seem to produce directly the area of a spiral, or even the area of a circle (also computed by Archimedes), so one might wonder how he fLrst derived them. W.R. Knorr [40] has suggested that the writings of Pappus of Alexandria (fourth century AD) indicate that Archimedes wrote an earlier version of On Spirals which used a different argument to compute the area of the spiral, but then rejected it as inelegant (this approach is developed in the solution to Exercise 1). The object of the next section is to show how a natural extension of the mechanical method easily produces these results.

Weighing a Spiral The Method relies on a mechanical analogy by using a balance to compare objects. This requires a few simple assumptions and facts about the properties of a lever, which are developed (sometimes implicitly) in Archimedes's On the Equilibrium of Planes I, [6] [18, Chapter IX]. These can be summarized by A s s u m p t i o n 1. Two objects will balance each other if the distances of their centers of gravity to the fulcrum are inversely proportional to their weights. The center of gravity of an object lies on an axis of symmetry.

IGURE c

When only the weight of an object is relevant to an argument, I will place it on a pan suspended from the balance. The object and any of its sections will then be assumed to have their centers of gravity at the point where the pan is suspended. I will also make extra assumptions not seen in Archimedes's works (however, see the solution to Exercise 4). A s s u m p t i o n 2. A plane figure is composed of circular arcs with common center, and each circular arc weighs the same as a line segment of equal length. E x e r c i s e 1. What happens if you instead decompose plane figures into radii with common center? I will first show how the mechanical method can be used to derive Archimedes's formula for the area of a circle given in Measurement of the Circle, Proposition 1 [54, Vol. 1, p. 317]. 5

Proposition 1. A n y circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the base is equal to the circumference. E x e r c i s e 2. Explain why Proposition 1 is equivalent to the familiar formula: Area of a circle = ~-R 2. Suspend two pans on opposite sides of a balance and at equal distances to the fulcrum. On one pan, place a circle with center at A and radius AB, on the other place a line segment CD of length AB. By Assumption 2, the circle is composed of circumferences with center A and radius AE for any E lying on AB. For each such circumference, place a line segment FG perpendicular to CD, of length the circumference through E, such that its endpoint F lies on CD and CF is equal to AE. By Assumption 2, the line segment FG is in equilibrium with the circumference through E. The resulting figure is a right triangle of height AB, base the circumference through B, and it balances a circle of radius AB, which is the statement of Proposition 1.

4In The Quadrature of the Parabola Archimedes gave what he considered to be a rigorous proof using the mechanical method of a result conjectured in a similar way in The Method, but using infinitesimals. 5A similar method was used by Rabbi Abraham bar Hiyya (1070-1136), see V.J. Katz, review of "Force and Geometry in Newton's Principia" by F. de Gandt, American Math. Monthly 105 (1998), 386-392 and F. Sanchez-Faba, Abraham Bar Hiyya and his "Libro de Geometria," (Spanish) Gac. Math., I 32 (1998), 101-115.

40

THE MATHEMATICALINTELLIGENCER

FIGURE

E x e r c i s e 3. Why is the resulting figure in this construction a triangle? E x e r c i s e 4. Generalize the following heuristic from The Method [6, Supplement]: " . . . judging from the fact that any circle is equal to a triangle with base equal to the circumference and height equal to the radius of the circle, I apprehended that, in like manner, any sphere is equal to a cone with base equal to the surface of the sphere and height equal to the radius."

IGURE z

I

A

B

M

Archimedes's definition of a spiral and its relevant comp,onents is given by [54, Vol. 2, p. 183]: 1. If a straight line drawn in a plane revolve uniformly any number of times about a fixed extremity until it return to its original position, and if, at the same time as the line revolves, a point move uniformly along the

:IGURE ..

straight line, beginning at the fixed extremity, the point will describe a spiral in the plane. 2. Let the extremity of the straight line which remains fixed while the straight line revolves be called the origin of the spiral. 3. Let the position of the line, from which the straight line began to revolve, be called the initial line of the revolution.

Proposition 2. The area inside a spiral anywhere within its first revolution is one third the sector of a circle with center at the origin of the spiral, radius equal to the distance of the point describing the spiral to the origin, and angle equal to the angle between the line and the initial line. (Archimedes gave areas for complete revolutions only, but his proof also applies to this case.) Consider a spiral with origin A, initial line AB, and C the position of the point describing the spiral. Consider also a balance arm DE of length twice AC and let the midpoint F of DE be the fulcrum. On this balance suspend a pan from D and place the spiral region in the pan. By Assumption 2, the spiral region is composed of arcs GH for each G lying on AC, where H is the intersection of the circle with center A and radius AG and the spiral. Extend AH to intersect the circle of center A and radius

VOLUME 21, NUMBER 3, 1999

41

A C a t / . Consider a line s e g m e n t J K of length equal to the arc CI a n d crossing D E at L s u c h that J K and D E are perpendicular, L is the m i d p o i n t o f JK, and F L is equal to AG. I claim that J K a n d t h e arc G H are in equilibrium. To s e e this note that, b y E x e r c i s e 3, the length o f a n arc is prop o r t i o n a l to its radius, so that arc G H : arc CI :: A G : AC, and the result follows from the a s s u m p t i o n t h a t the arc CI has its c e n t e r of gravity at D a n d from A s s u m p t i o n 1. N o w extend the arc CI to i n t e r s e c t AB at M; t h e n t h e arc CI is equal to the arc C I M m i n u s t h e arc IM, a n d b y the definition o f spiral, I M is p r o p o r t i o n a l to AH. Since the arc CIM remains c o n s t a n t in this argument, the s e c o n d p a r t o f Exercise 3 s h o w s t h a t the arc CIM m i n u s J K is proportional to FL, which:'lneans that the resulting figure is an isosceles triangle w h i c h b a l a n c e s the inside o f the spiral. The e x a c t s a m e a r g u m e n t s h o w s that the a r e a b e t w e e n the spiral and the initial line that lies within t h e s a m e sect o r b a l a n c e s the s a m e isosceles triangle, b u t r e v e r s e d so that its v e r t e x lies on t h e fulcrum. The crucial s t e p is to recall the following F a c t : The c e n t e r o f gravity of a triangle lies at the int e r s e c t i o n o f the medians, and the m e d i a n s o f a triangle intersect each o t h e r in a ratio of 2:1.

IGURI

t 42

THE MATHEMATICAL INTELLIGENCER

The first p a r t is s u g g e s t e d b y the observation that a median divides a triangle into t w o triangles o f equal weight, and its p r o o f is one of the m a i n results of On the E q u i l i b r i u m o f Planes L The s e c o n d p a r t is an easy e x e r cise [15, w and follows f r o m On the E q u i l i b r i u m o f Planes I, Proposition 15, generalized to trapezoids. This s h o w s t h a t reversing the first triangle p l a c e s t h e c e n t e r o f gravity twice as far f r o m the fulcrum, so the second triangle will b a l a n c e t w i c e the first. One c o n c l u d e s t h a t the inside of the spiral weight one half the outside o f t h e spiral a n d thus one third o f the s e c t o r of the circle, w h i c h is t h e s t a t e m e n t o f P r o p o s i t i o n 2. E x e r c i s e 5. Evaluate the a r e a of the spiral using t h e s a m e p r o c e d u r e as for the circle, i.e., b y only c o m p a r ing weights p l a c e d on pans. E x e r c i s e 6. Use the m e c h a n i c a l m e t h o d to c o m p u t e t h e c e n t e r o f gravity of a spiral region. A Modern Translation The b a s i c o b s e r v a t i o n is t h a t A s s u m p t i o n 2 e x t e n d s A r c h i m e d e s ' s m e t h o d to p o l a r c o o r d i n a t e s . C o n s i d e r a curve r = f(O) in p o l a r c o o r d i n a t e s , where, for simplicity, f ( r ) is an i n c r e a s i n g function, s o t h e r e is an i n v e r s e function 0 = g(r) (this n o t a t i o n is m o r e c o n v e n i e n t given t h e

IGURE ,

difficulties of E x e r c i s e 1). To c o m p u t e the area of a region lying inside the curve a n d having 0 ---- 0 -- O, one partit i o n s ~ into thin circular shells of width h > 0, as in Figure 6. Using the f o r m u l a 0r2/2 for the area o f a s e c t o r o f angle 0 a n d radius r, each shell h a s a r e a (O - O)rh + R(r,h), w h e r e the e r r o r R(r,h) is less than the area of the small shell e l e m e n t o f a r e a (r + h)h~q(r + h) - g(r)), s e e Figure 6, a n d this is less than Crh 2, for s o m e c o n s t a n t C, a s s u m ing t h a t g(r) is well behaved. It follows that, ignoring t e r m s of o r d e r h 2, the a r e a o f e a c h shell is (O - O)rh, w h i c h is the length of the bottom a r c o f the shell multiplied b y h. This s h o w s w h y the first p a r t o f A s s u m p t i o n 2 holds. All these shells have a r e a a linear function of h up to an e r r o r t e r m o f l o w e r order, a n d form a disjoint union of ~ , w h i c h s h o w s w h y the s e c o n d p a r t o f A s s u m p t i o n 2 holds. Letting h --> 0, it follows that the area o f ~ is

j

:

rR (O - O)rdr = Jo [g(R) - g(r)]rdr.

The s t a n d a r d derivation of this f o r m u l a uses the f o r m u l a rdrdO for the area e l e m e n t in polar c o o r d i n a t e s A r e a o f ~ = f f d x d y = f f rdrdO =

s;s; =0

rdOdr =

=g(r)

s;

[g(R) - g(r)lrdr.

A circle is simply g(r) = 0, w h i c h yields 21r f R r d r = 7rR29 A spiral, in p o l a r c o o r d i n a t e s , is given b y the equation r = aO, for s o m e constant a, which can b e written as O = kr, w h e r e k = 1/a. By the above, the area o f the spiral is (kR - k r ) r d r = k R

rdr - k kR 3 2

r2dr = k

R3 3

kR 3 1 OR 2 - -- 6 3 2 '

w h e r e the t e r m on the right is seen to be 1/3 the a r e a of the s e c t o r of the circle o f r a d i u s R and angle O, yielding Proposition 2. A n y p r o o f of this f o r m u l a is equivalent to evaluating s u c h integrals. A r c h i m e d e s e v a l u a t e d fR r2dr b y d e c o m p o s i n g it into Riemann s u m s a n d obtaining a c l o s e d form for the s u m 12 + ... + n 2. In P r o p o s i t i o n 2 this integral is c o m p u t e d b y realizing it as t h e m o m e n t of a triangle and evaluating this as its weight multiplied b y the d i s t a n c e o f its c e n t e r of gravity from t h e fulcrum.

The Way of Archimedes The Calculus Reform m o v e m e n t has e m p h a s i z e d experim e n t a t i o n over rigor in calculus e d u c a t i o n and h a s b e e n criticized as a result [53]9 To d e f e n d its p o s i t i o n t h a t physical p r o b l e m s should b e u s e d t o d i s c o v e r m a t h e m a t i c a l results, Harvard Calculus a p p e a l s to A r c h i m e d e s a n d The Method [35, p. vii]: T h e W a y o f A r c h i m e d e s : F o r m a l d e f i n i t i o n s a n d procedures evolve f r o m the i n v e s t i g a t i o n o f p r a c t i c a l problems.

This principle a c c u r a t e l y r e p r e s e n t s the w o r k s o f Archimedes, b u t a disparity arises in t h a t H a r v a r d Calculus p o s t p o n e s m a t h e m a t i c a l rigor indefinitely; A r c h i m e d e s ' s name should not b e a s s o c i a t e d with s u c h an endeavor. F o r example, the m e t h o d of e x h a u s t i o n u s e d by A r c h i m e d e s is essentially t h e E-ti a r g u m e n t a b a n d o n e d b y H a r v a r d Calculus, as B.L. van der W a e r d e n w r i t e s [58, p. 220]: 9 . . the e s t i m a t i o n s , w h i c h occur i n the s u m m i n g o f i n f i n i t e series a n d i n l i m i t operations, the 'epsilontics', as the calculation w i t h a n a r b i t r a r y s m a l l E i s s o m e t i m e s called, w e r e f o r A r c h i m e d e s an open book. I n this respect, h i s t h i n k i n g i s entirely modern. Moreover, A r c h i m e d e s h e l d in c o n t e m p t t h o s e w h o did not furnish p r o o f s o f their results. In t h e i n t r o d u c t i o n to O n Spirals, A r c h i m e d e s r e v e a l s that he intentionally ann o u n c e d false t h e o r e m s in o r d e r to e x p o s e s o m e of his cont e m p o r a r i e s [6]: 9 I w i s h n o w to p u t t h e m i n r e v i e w one by one, p a r ticularly as i t h a p p e n s t h a t - t h e r e are two a?nong t h e m w h i c h [are w r o n g and w h i c h m a y serve as a w a r n i n g to] those w h o c l a i m to discover e v e r y t h i n g but produce no proofs o f the s a m e m a y be confuted as hading actually pretended to discover the impossible. Harvard Calculus fails m i s e r a b l y w h e n m e a s u r e d against this Way of Archimedes. A p a r t from the p a s s a g e quoted above, the w o r d "theorem" a p p e a r s in [35] only in the n a m e " F u n d a m e n t a l T h e o r e m o f Calculus." C o m p a r e this with a s t a n d a r d calculus t e x t [22], which lists 130 t h e o r e m s in its index. Even m o r e revealing, the only instance o f the w o r d "proof" I l o c a t e d in [35] w a s in A r c h i m e d e s ' s i n t r o d u c t i o n to the m e t h o d q u o t e d above and u s e d in [35] to justify "The Way of Archimedes." In fact, this quote e m p h a s i z e s that d i s c o v e r y of t h e a n s w e r to a p r o b l e m leads to a t h e o r e m w h o s e p r o o f is facilitated b y k n o w l e d g e of the answer. My i n t e r p r e t a t i o n is n o t Calculus Reform b u t P r o b l e m - S o l v i n g : W h e n faced w i t h a p r o b l e m , use a n y m e t h o d t h a t allows y o u to c o n j e c t u r e the answer, t h e n find a r i g o r o u s proof. A r e c e n t d e v e l o p m e n t : The s e c o n d edition o f [35] h a s t a k e n a m o r e m o d e r a t e a p p r o a c h to Calculus Reform and n o w includes s o m e c o m p l e t e p r o o f s [35, 2nd Edition, p. 78] and the e-8 definition of a limit [35, 2nd Edition, p. 128]. However, this n e w edition no longer includes "The Way o f Archimedes."

Popular Misconceptions It m u s t be noted that the p e n u l t i m a t e r e m a r k o f the previous s e c t i o n p a r a p h r a s e s E.T. Bell [11, p. 31]: "In s h o r t he u s e d m e c h a n i c s to a d v a n c e his m a t h e m a t i c s . This is one o f his titles to a m o d e m mind: he used a n y t h i n g a n d everything that suggested i t s e l f as a w e a p o n to attack h i s problems." However, strong opinions s u c h a s t h o s e e x p r e s s e d

VOLUME 21, NUMBER 3, 1999

43

in [11] are fraught with danger, and it is instructive to include the continuation of this passage: To a modern all is f a i r in war, love, and mathematics; to m a n y of the ancients, mathematics was a stultified game to be played according to the p r i m rules imposed by the philosophically-minded Plato. According to Plato only a straightedge and a p a i r of compasses were to be permitted as the implements of construction in geometry. No wonder the classical geometers hammered their heads f o r centuries against 'the three problems of antiquity': to trisect an angle; to construct a cube having double the volume of a given cube; to construct a square equal to a circle.

This has~ since been discredited, see [24] [41] (better yet, look at original sources, e.g., as collected in [54, Vol. 1, Chapter 9]); and van der Waerden writes [58, p. 263], The idea, sometimes expressed, that the Greeks only permitted constructions by means of compasses and straight edge, is inadmissible 9 It is contradicted by the numerous constructions, which have been handed down, f o r the duplication o f the cube and the trisection of the angle9

In particular, Archimedes trisected the angle with rifler and compass in Proposition 8 of The Book of L e m m a s [6, p. 309], see [20] [31, Section 31]. The history of this misconception might prove an interesting subject for further study. Unfortunately, it is only one of a number of popular misconceptions about the limitations of Greek science [56]. For example, Isaac Asimov (1920-1992) has written [5], To the Greeks, experimentation seemed irrelevant. It interfered with and detracted f r o m the beauty of pure ded u c t i o n . . . To test a perfect theory with imperfect instruments did not impress the Greek philosophers as a valid w a y to gain k n o w l e d g e . . . The Greek rationalization for the "cult of uselessness" m a y similarly have been based on a feeling that to allow m u n d a n e knowledge (such as the distance f r o m Athens to Corinth) to intrude on abstract thought was to allow imperfection to enter the Eden of true philosophy. Whatever the rationalization, the Greek thinkers were severely limited by their attitude9 Greece was not barren of practical contributions to civilization, but even its great engineer, Archimedes of Syracuse, refused to write about his inventions and discoveries . . . to m a i n t a i n his amateur status, he broadcast only his achievements in pure mathematics.

This passage is contradicted by numerous examples of Greek scientific experiments, for example, Eratosthenes's measurement of the earth [4]9 Asimov may be excused for paraphrasing Plutarch's account of Archimedes in his Life of Marcellus, written circa 75 AD [49] [54, Vol. 2, p. 31]: Yet Archimedes possessed so lofty a spirit, so profound a soul, and such a wealth of scientific inquiry, that al-

44

THE MATHEMATICALINTELLIGENCER

though he had acquired through his inventions a n a m e and reputation f o r divine rather than h u m a n intelligence, he would not deign to leave behind a single writing on such subjects. Regarding the business of mechanics and every utilitarian art as ignoble or vulgar, he gave his zealous devotion only to those subjects whose elegance and subtlety are untrammeled by the necessities of life

Despite Plutarch's ancient credentials, he had no better insight into Archimedes's scientific contribution, which contradict his story. The reader is already aware that The Method shows that physical considerations played an important role in Greek mathematics. But Asimov and Plutarch are completely refuted by Archimedes in The Sand Reckoner [6] [ 18]: While examining this question I have, f o r m y part tried in the following manner, to show with the aid of instruments, the angle subtended by the sun, having its vertex at the eye. Clearly, the exact evaluation of this angle is not easy since neither vision, hands, nor the instruments required to measure this angle are reliable enough to measure it precisely. But this does not seem to me to be the place to discuss this question at length, especially because observations of this type have often been reported. For the purposes of m y proposition, it suffices to f i n d an angle that is not greater than the angle subtended at the sun with vertex at the eye and to then f i n d another angle which is not less than the angle subtended by the s u n with vertex at the eye. A long ruler having been placed on a vertical stand placed in the direction the rising sun is seen, a little cylinder was put vertically on the ruler immediately after sunrise. The sun, being at the horizon, can be looked at directly, and the ruler is oriented towards the sun and the eye placed at the end of the ruler. The cylinder being placed between the sun and the eye, occludes the sun. The cylinder is then moved further away f r o m the eye and as soon as a small piece of the s u n begins to show itself f r o m each side of the cylinder, it is fixed. I f the eye were really to see f r o m one point, tangents to the cylinder produced f r o m the end of the ruler where the eye was placed would make an angle less than the angle subtended by the sun with vertex at the eye. But since the eyes do not see f r o m a unique point, but f r o m a certain size, one takes a certain size, of round shape, not smaller than the eye and one places it at the extremity of the ruler where the eye was p l a c e d . . , the width of cylinders producing this effect is not smaller than the dimensions of the eye. 9 It is therefore clear that the angle subtended by the sun w i t h vertex at the eye is also smaller than the one hundred and sixty fourth part of a right angle, and greater than the two hundredth part of a right angle.

The correct value of the angular diameter of the sun is now known to average about 34' [26, p. 95], i9 the 159th part of a right angle. It is important to note that this shows not

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m

only that ancient Greeks frequently performed experiments, but that Archimedes dealt with experimental error and also compensated for the fact that the human eye is part of the observational instrument, thus anticipating scientists such as Hermann von Helmoltz (1821-1894) [34]. A translation and analysis of The Sand Reckoner is given in [56]. A n s w e r s to E x e r c i s e s E x e r c i s e 1. A naive approach leads to incorrect results, evidence of the dangers of using infinitesimals, and indicating why Archimedes did not consider his method to be rigorous. For example, taking the radii of a circle of radius R, with respect to the circumference, and reordering them to form a rectangle, yields area 2qrR2. For a general figure, it's not even clear how to pick the radii. To make sense of what is going on, one regards radii as limits of sectors, i.e., infinitesimal triangles. In the case of the circle, this means that the weight of a radius, with respect to the circumference, is equal to one half its length. This can be loosely interpreted as the argument Archimedes used to compute the area of the circle [1]. In the general case, the following is justified: A s s u m p t i o n 3. The weight of a radius is proportional to the square of its length. In modern notation, this is simply i f rdrdO = f :

=0

rf(~

rdrdO = l f : [f ( O)]2dO,

Jr=0

where the radii have been chosen with respect to the unit circle. Given Assumption 3, one can compute the area of the spiral by using Pappus's argument [48, Book 4, Proposition 21], see also [32, p. 377] [41, p. 162]. To compute the weight of a spiral region, take each radius of the spiral, starting from the fmal radius, and place a disk with diameter equal to this radius at height the current angle so the resulting figure is a cone. Similarly, for each radius of the sector place a disk with diameter equal to this radius at height the current angle, resulting in a cylinder with the same base and height as the cone. Since Euclid's Proposition 2 6f Book 12 proves that "circles are to one another as the squares on the diameter," Assumption 3 shows that the ratio of the weight of the spiral region to the weight of the sector is the same as the ratio of the volume of the cone to the volume of the cylinder. But Euclid's Proposition 10 of Book 12 proved that the volume of a cone is one third the cylinder with the same base

and height, so the spiral weighs one third of the sector, which is the statement of Proposition 2. (Note that equilateral triangles could have been used instead of circles resulting in a pyramid whose volume is easier to compute.) Knorr [40] comments that this appeal to three-dimensional figures might have been considered inelegant by Archimedes as it uses volumes to compute areas. On the other hand, reversing this argument and using the evaluation above shows that the volume of a cone can be computed by the mechanical method, a result which does not appear in The Method. E x e r c i s e 2. In modern notation, Archimedes'$formulation of Proposition 1 is Area of circle of radius R = f [ 2~'rdr, for the integral represents the area of a right triangle with base R and height 2~rR. E x e r c i s e 3. This is equivalent to the fact that the length of an arc of fixed angle is proportional to its radius. In particular, ~r exists, see [45] [56]. The proof is similar to [23, Book 12, Proposition 2] cited in Exercise 1, and is implicit in Archimedes's M e a s u r e m e n t o f the Circle. Similarly, the length of an arc of fixed radius is proportional to its angle. E x e r c i s e 4. By analogy with Assumption 2, consider a sphere as being composed of spherical shells centered at the center of the sphere, where each shell weighs the same as a circle of equal area. The justification follows exactly as in Proposition 2: Consider two pans suspended at equal distances from the fulcrum of a balance. On one pan, place a sphere of center A and radius AB and on the other a line CD of length equal to AB. For each E on AB there is a spherical shell passing through E, and consider a circle of area equal to this spherical shell with center at F lying on CD, where CF equals AE, and such that the circle is perpendicular to CD. The resulting figure is a cone with base the area of the sphere and height the radius of the sphere; since it balances the sphere, the claim is justified. The similarity of this argument to the one of Proposition 1 suggests that Archimedes may have been implicitly aware of the ideas of this paper. Moreover, the reader may verify that the heuristic of this exercise and its justification directly generalize to higher dimensions (a different generalization is given in [19]): P r o p o s i t i o n 3. The volume o f a n n - d i m e n s i o n a l ball is equal to the volume o f a cone whose base has n - 1-

VOLUME 21, NUMBER 3, 1999

REFERENCES

dimensional volume equal to the (n - 1)-dimensional volume of the boundary of the ball and height equal to the radius of the ball. E x e r c i s e 5. The procedure, when applied to the spiral, yields a section of a parabola. The general formula for such areas was computed by Archimedes in The Quadrature of the Parabola, and in this case it states that the resulting area is four-thirds the triangle with same base and height as the section of the parabola. Since the height and base are equal to the final radius and half the final radius, respectively, Proposition 2 follows. E x e r c i s e 6. Further extensions of Archimedes's method could be a subject for investigation. As Archimedes wrote in The Method [6, Supplement, p.13], I deem it necessary to expound the method partly because I have already spoken of it but equally because I a m persuaded that it will be of no little service to mathematics; f o r I apprehend that some, either of m y contemporaries or of m y successors, will, by m e a n s of the method when once established, be able to discover other theorems in addition, which have not yet occurred to me. Acknowledgment I would like to thank Main Herreman, Reviel Netz, and David Wilkins for helpful comments.

THE MATHEMATICAL INTELLIGENCER

[1] A. Aaboe and J. L. Berggren, Didactical and other remarks on some theorems of Archimedes and infinitesimals, Centaurus 38 (1996), 295-316. [2] K. Andersen, Cavalieri's method of indivisibles, Arch. Hist. Exact. Sci. 31 (1985), 291-367. [3] T. Apostol, Calculus, voL I, 2nd Edition, John Wiley & Sons, New York, 1967. [4] I. Asimov, How did we Find out that the Earth is Round?, Walker & Co., New York 1972. [5] I. Asimov, Asimov's new Guide to Science, Basic Books, New York 1984. [6] Archimedes, The Works of Archimedes, edited in modern notation with introductory chapters by T.L. Heath. With a supplement, The method of Archimedes, recently discovered by Heiberg, Dover, New York, 1953. Reprinted (translation only) in [36]. [7] Archimedes, Opera Omnia, IV vol., cum commentariis Eutocii, iterum edidit I.L Heiberg, corrigenda adiecit E.S. Stamatis, B.G. Teubner, Stuttgart, 1972. [8] Archim~de, Oeuvres, 4 vol., texte 6tabli et traduit par C. Mugler, Les Belles Lettres, Paris, 1970-72. [9] M. Balme and G. Lawall, Athenaze, An Introduction to Ancient Greek, 2 vols., Oxford University Press, New York, 1990. [10] A. Avron, On strict constructibility with a compass alone, J. Geometry 38 (1990), 12-15. [11] E.T. Bell, Men of Mathematics, Simon and Schuster, New York, 1937. [12] L. Bers, Calculus, vol. I, Holt, Rinehart and Winston, New York, 1969. [13] L. Bieberbach (1886-1982), Theorie der geometrischen Konstruktionen, LehrbQcher und Monographien aus dem Gebiete der exakten Wissenschaften, Mathematische Reihe, Band 13, Verlag BirkhAuser, Basel, 1952. [14] Cleomedes (circa 150 B.C.), De Motu Circulari Corporum Caelestium, 2 vols., H. Ziegler, editor, Teubner, Leipzig 1891. See [32, p. 106] [43] [54, Vol. 2, p. 267]. [15] H.S.M. Coxeter, Introduction to Geometry, John Wiley & Sons, New York, 1989. [16] H.T. Croft, K.J. Falconer, and R.K. Guy, Unsolved Problems in Geometry, Springer-Verlag, New York 1991. [17] P.J. Davis, The rise, fall, and possible transfiguration of triangle geometry: A mini history, Amer. Math. Monthly 102 (1995), 204-214. [18] E.J. Eijksterhuis, Archimedes, Princeton University Press, Princeton 1987. [19] M. Djori~ and L. Vanhecke, A Theorem of Archimedes about spheres and cylinders and two-point homogeneous spaces, Bull. Austral. Math. Soc. 43 (1991), 283-294. [20] U. Dudley, A budget of trisections, Springer-Verlag, New York, 1987. [21] C.H. Edwards, The Historical Development of the Calculus, Springer-Verlag, New York, 1979. [22] C.H. Edwards and D.E. Penney, Calculus and Analytic Geometry, second edition, Prentice Hall, Englewood Cliffs, NJ, 1988. [23] Euclid, The Thirteen Books of Euclid's Elements, translated with introduction and commentary by T.L. Heath, Dover 1956. [24] D.H. Fowler, The Mathematics of Plato's Academy: a New Reconstruction, Clarendon Press, Oxford 1990.

[25] S.H. Gould, The Method of Archimedes, Amer. Math. Monthly 62 (1955), 473-476. [26] R.M Green, Spherical Astronomy, Cambridge University Press, Cambridge 1985. [27] J. Hadamard (1865-1963), Legons de g~om6trie el#mentaire, 2 vols., A. Colin, Paris, 1937. [28] D. Hammett, The Maltese Falcon, Penguin, Middlesex, 1930. [29] E. Hayashi, A reconstruction of the proof of Proposition 11 in Archimedes method, Historia Scl. 3 (1994), 215-230. [30] G.H. Hardy (1877-1947), A Mathematician's Apology, Cambridge University Press, New York, 1985. [31] R. Hartshorne, A Companion to Euclid, a course of geometry based on Euclid's Elements and its modem descendents, AMS, Berkeley Center for Pure and Applied Mathematics, 1997. [32] T. Heath, A History of Greek Mathematics, vol. II, Dover, New York, 1981. [33] I.L. Heiberg, Eine neue Archimedes-Handschrifl, Hermes 42 (1907), p. 235. [34] H.L.F. von Helmholtz, Helmholtz treatise on physiological optics, translated from the 3d German ed., edited by J.P.C. Southall, Dover, New York, 1962. [35] D. Hughes-Hallett, A.M. Gleason, et al., Calculus, John Wiley & Sons, New York, 1994. Second edition, 1998. [36] Great Books of the Western World, voL 11, R.M. Hutchins, editor, Encyclopaedia Britannica, Inc., Chicago, 1952. [37] G. Johnson, The Big Question: Does the Universe Follow Mathematical Law? The New York Times, February 10, 1998. [38] F. Klein (1849-1925), Elementary Mathematics from an Advanced Viewpoint, 2 vols, Macmillan, New York, 1939. [39] W.R. Knorr, The evolution of the Euclidean elements, Synthese Historical Library 15, D. Reidei, Dordrecht-Boston, MA, 1975. [40] W.R. Knorr, Archimedes and the Spirals: The Heuristic Background, Historia Math. 5 (1978), 43-75. [41] W.R. Knorr, The Ancient Tradition of Geometric Problems, Birkh&user, Boston, 1986. [42] W.R. Knorr, The method of indivisibles in ancient geometry, in "Vita mathematica" (Toronto, ON, 1992; Quebec City, PQ, 1992), 67-86, MAA Notes 40, Math. Assoc. America, Washington, DC, 1996. [43] K. Lasky, The Librarian who Measured the Earth, Little, Brown & Co., Boston 1994. [44] H, Lebesgue, LeGons surles constructions g~ometriques, GauthierVillars, Pads, 1950. Reissued, Jacques Gabay, Paris, 1987. [45] E. Moise, Elementary Geometry from an Advanced Viewpoint, Addison-Wesley, Reading, MA, 1974. [46] R. Netz, personal communication. [47] Nicomachus of Gerasa (circa 100 A.D.), Introduction to Arithmetic, translated by M.L. D'Ooge, in [36]. See [54, Vol. 1, p. 101]. [48] Pappus, La Collection Math~matique, traduit avec une introduction et notes par P. Ver Ecke, Descl6e de Brouwer, Paris, 1933. [49] Plutarch, Lives, vol. 5, translated by B. Perrin, Loeb Classical Library 87, Harvard University Press, Cambridge, MA, 1917. Also, translated by John Dryden (1631-1700), http://www.oed.com/ plutarch.html.

[50] A. Seiberg, Collected Works, 2 vols, Springer Verlag, New York, 1989, 1991. [51] G.F. Simmons, Calculus with Analytic Geometry, McGraw Hill, New York, 1985. [52] H.M. Stark, On the "gap" in a theorem of Heegner, J. Number Theory 1 (1969), 16-27. [53] H. Swann, Commentary on rethinking rigor in calculus: The role of the mean value theorem, Amer. Math. Monthly 104 (1997), 241-245. [54] I. Thomas, Greek Mathematical Works, 2 vol., Loeb Classical Library 335, 362, Harvard University Press, Cambridge, MA, 1980. [55] G. Toussaint, A new look at Euclid's second proposition, Math. Intelligencer 15 (1993), 12-23. [56] I. Vardi, A classical reeducation, in preparation. [57] I. Vardi, APXIMHz~OY:SrlEPI TQ,N EMKD,N E~O&O2:, in preparation. [58] B.L. van der Waerden, Science Awakening I, Scholar's Bookshelf Press, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988. [59] A. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. 141 (1995), 443-551

VOLUME 21, NUMBER 3, 1999

47'

Ii'iFIli~li[~,,e-:ml[.-~-~m=rpli(~-'~-~.~..,=a.~ .

-

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This column is devoted to mathematics for fun. What better purpose is there for mathematics? To gppear here, a theorem or problem or remark does

-

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Alexander

Shen,

Editor

Unexpected Proofs

Now, a s s u m e w e have two b o x e s , one inside another. Then, the e-neighb o r h o o d o f t h e first b o x will be inside the e - n e i g h b o r h o o d of the second, so

n e o f the nice things a b o u t mathe m a t i c s is that s o m e t i m e s a question l o o k s very simple b u t t h e a n s w e r u s e s an u n e x p e c t e d and e l e g a n t argument. Let m e s h o w two e x a m p l e s .

This is true for any 6, even for a large one w h e n t h e e 2 t e r m is t h e m a i n t e r m (note that the e 3 t e r m s are the s a m e for b o t h n e i g h b o r h o o d s and cancel e a c h other). Therefore, 11 = w l + hi + d l d o e s n o t e x c e e d / 2 = w2 + h2 + d2. The s e c o n d p r o o f u s e s r a n d o m n e s s . Let X b e a c o n v e x set in R 3. C o n s i d e r a r a n d o m line m in R 3. The orthogonal p r o j e c t i o n o f X onto m is a segment. Let us d e n o t e by d(X) the e x p e c t e d length o f this segment. Let Xm b e a s e g m e n t of length m. Then, d(Xm) is p r o p o r t i o n a l to m, i.e., d(Xm) = e m for s o m e c. (In fact, c = 1/2, but the e x a c t value is n o t i m p o r t a n t

O

not need to be profound (but it is allowed to be); it may not be directed only at specialists; it must attract and fascinate. We welcome, encourage, and frequently publish contributions from readers---either new notes, or replies to past columns.

Boxes in a Train Rules o f the M o s c o w u n d e r g r o u n d say t h a t y o u are a l l o w e d to bring o n a rect a n g u l a r b o x o f size w • h • d only if w + h + d d o e s not e x c e e d 150 cm. Question: Is it p o s s i b l e to c h e a t by p a c k i n g one b o x into a n o t h e r ? The ans w e r is no: If a rectangular b o x w I X hi X dt c a n b e p l a c e d inside a n o t h e r one o f size w 2 x h 2 x d2, then Wl + hi + d l --< w2 + h2 + d2. We p r e s e n t t w o c o m p l e t e l y different p r o o f s of this fact. The first cons i d e r s t h e ~ n e i g h b o r h o o d of a b o x (inchiding the interior part). Its volume V(e) is defined for non-negative ~. It is e a s y to s e e that V(e) is a p o l y n o m i a l in

V(~) = V + Se + r

2 + (4/3)~-~.

Here, V is the volume of the box, S is the a r e a o f its surface, a n d 1 is t h e sum o f t h e d i m e n s i o n s (w + h + d). Indeed, the n e i g h b o r h o o d c o n s i s t s o f

Please send all submissions to the Mathematical Entertainments Editor, Alexander Shen, Institutefor Problems of Information Transmission, Ermolovoi 19, K-51 Moscow GSP-4, 101447 Russia; e-mail:[email protected]

48

I

9 t h e b o x itself (V) * six r e c t a n g u l a r b o x e s (of t h i c k n e s s e) coveting the faces and having total v o l u m e S e * t w e l v e p i e c e s n e a r the e d g e s that c a n b e c o m b i n e d into t h r e e cylind e r s o f radius e a n d lengths w, h, and d; t o t a l volume zre2(w + h + d) 9 eight p i e c e s n e a r the v e r t i c e s that f o r m a ball of radius e having total v o l u m e (4/3)~re3.

THE MATHEMATICALINTELLIGENCER9 1999 SPRINGER-VERLAGNEW YORK

Vl + SI~ § 7r/1~2 4- (4/3)zre3 -< V2 + S2e + zr/2*"2 + (4/3)~-e3.

nOW.) Now, let X b e a b o x o f size w x h x d. F o r e a c h line m, the p r o j e c t i o n o f X onto m h a s l e n g t h p w + Ph § Pal, w h e r e Pw, Ph, and Pd are p r o j e c t i o n s o f segm e n t s of length w, h, a n d d, the e d g e s o f the box. By averaging, w e get

d(X) = c(w + h + d). If a b o x 0(1) is p l a c e d inside a n o t h e r one (X2), t h e n the p r o j e c t i o n o f X1 o n t o a line m is i n c l u d e d in t h e p r o j e c t i o n o f X2 onto m, s o d(X1) ~ d(X2). Combining this o b s e r v a t i o n with the preceding one, w e see that w l + hi + dl -< w2 § h2 + d2. (End o f the s e c o n d proof.)

Square Split into Triangles It is e a s y to split a square into n equal triangles if n is even. However,

it is impossible to split a square into n triangles of equal area i f n is odd. However, t h e p r o o f o f this fact is n o t s t r a i g h t f o r w a r d and uses s o m e topoiogy and algebra.

We start with a special case where (a) the triangles form a triangulation and Co) all vertices have rational coordinates. (Later, we'll see h o w these assumptions can be removed.) For any rational number r, defme its 2-valuation 114]as follows: i f r = 2k(p/q), where p and q are odd, I1~]is 2 -k. By defmition, ]]0H= 0. In a sense, 1141measures "oddness" of ~. for example, 3/2 is "odder" than 1, and 2 is "odder" than 4. Now, divide all rational points (x, y) (both x and y are rational) into three classes. If both x and y (represented as irreducible fractions) have even numerators, the point (x, y) belongs to class A. If at least one of x and y has an odd numerator, compare the "oddness" o f x and y: when x is "more odd," we get a B point, otherwise a C point. Formally, A: Ilxl] < 1 and IlYll < 1 B: Ilxll > Ilyll and Ilxll-> 1

C: Ilxll---IlYll and IlYll-> 1 Let us return to our square 12 = [0, 1] • [0, 1] and its triangulation with rational vertices. L e m m a . There exists a triangle in the triangulation whose vertices are labeled w i t h all three labels A, B, and C. Proof. Our classification can be considered as a mapping a from the set of vertices into the set {A, B, C}. Imagine that A, B, and C are vertices of some triangle ABC. Then, a can be uniquely extended to a mapping of the whole square into the triangle ABC that is piecewise affme (affme on each triangle of the triangulation). N o w the statement of the lemma can be reformulated as follows: a covers the interior part of the triangle ABC. To prove this statement (a version of Sperner's lemma), let us consider the restriction of a to the boundary of the unit square. We k n o w its values on the square's vertices: (0, 0) has type A, while (1, 0) has type B, and both (1, 1) and (0, 1) have type C (see Fig. 1). Moreover, it is easy to see that any vertex on the lower side of the square has type A or B and any vertex on the left side has type A or'C, whereas all

(0, 1) C"

Bor

C

_ (1.1) 'C )

AorC

BorC

A (0,0) w

VA or B

B (1,0) v

Figure 1 vertices on the remaining two sides have type B or C. Therefore, the restriction a]0I 2 of a to the boundary of the s q u a r e / 2 maps it into the boundary of the triangle ABC and has degree 1. Therefore, alOI 2 is not homotopic to a constant mapping. On the other hand, if the image a ( I 2) were contained in the boundary of triangle ABC, a would provide a h o m o t o p y between c~]0I2 and a constant mapping. (End of the p r o o f of the lemma.) Now we k n o w that our triangulation contains a triangle whose vertices are labeled A, B, and C. Let their coordinates be (al, a2), (bl, b2), and (cl, c2), respectively. This triangle has area s=ldet

~:-al

al

b2-a2 C2 -- a2 '

and I]511> 1 (as we'll see). On the other hand, S = 1/n, because all n triangles of the triangulation have the same area. Therefore, n is even. It remains to prove that I~1 > 1. Recall two main properties of the 2valuation:

9 Ilabll

=

Ilall'llbll

9 Ila + bll--< max(llall, Ilbll); this inequality turns into equality if Nail r Ilbll Using these properties, it is easy to check that the point (b~, b~) = (bl - al, b 2 - a2) belongs to type B and the point (c~, c ~ ) = (C1 --al, C 2 - a2) belongs to type C (bl is "more odd" than al, so subtracting al we do not change "oddness" of bl, etc.) By definition of types B and C, we have

So the statement is proved for the case of triangulation with rational vertices. Let me say, briefly, what could be done for the general case. Assume that the triangles do n o t form a triangulation, e.g., vertex Q of one triangle lies on side P R of another one. (See Fig. 2.) What can we do? We can admit "degenerate" triangles like PQR, get a triangulation, apply our argument, and find a triangle that is ABClabeled. This triangle cannot be degenerate since for its area S, we have proved that I111 > 1. What should we do if the coordinates of the vertices are irrational? In this case, one can extend the 2-valuation to an extension of Q that contains all the coordinates, and use the same argument. (I omit the details.) I found.this problem (and its solution) iIi an article of B. Bel~ker and N. Netsvetaev; they attribute it to J o h n Thomas (A dissection problem, Math. Mag. 41 (1968), 187-190) for the case of rational coordinates and Paul Monsky (On dividing a square into triangles, Am. Math. Monthly 77 (1970), 161-164) for the general case. I.ette~

Concerning Poncelet's theorem and your article in the Intelligencer, I wonder if you know this. Poncelet's initial theorem concerned a pencil of circles, and he stated it like this: let I, II, and III be three circles in a pencil. Start from a point m in I, draw the tangent to II, get a second point n in I; from n draw a tangent to III, get another point p in I. Then, the line mp, when m runs through I, envelops a circle IV (from the initial pencil). All closure theorems follow from this one. Now, the p r o o f of the InteUigencer applies to this, one has only to remark that the lengths of tangents drawn from points of I to circles in the pencil are proportional with universal constants. Then, the associR

IIb;ll > IIb ll; IIb;ll >- 1; Ilchll-> IIc;ll; Ilchll -> 1. ' ' ' ' II and IIb'1c211-' Therefore, IIb,c=ll > flb2c, 1, so 112sll = IIb;c5 - b c;ll = IIb;c ll--- 1 and 1511 = 211211 > 1.

P

Figure 2

VOLUME21, NUMBER3, 1999 4 9

ated measures on the circle I, given by H, III, etc., are proportional. Otherwise stated: for such a measure the line joining two points differing by a translation of this measure always envelops some circle of the pencil. Elliptic functions are at the core of Poncelet's theorem; here the elliptic function is the new measure. Marcel Berger Institut des Hautes Etudes Scientifiques 91440 Bures-sur-Yvette France e-mail: ber'[email protected]

Preparing an article on the story of Poncelet's theorem, I read some of the original papers. My following notes sketch the historical background. 1. Poncelet's original theorem is not about a triangle inscribed in a circle and circumscribed around another circle, but about an n-gon inscribed in a conic section and circumscribed around another conic section. Moreover, the theorem in Poncelet's approach is a consequence of a more general theorem. This general theorem is about a pencil of conic sections. L e t C, cl, c2, 99 9 Cn-1 be the elements of this pencil. Poncelet states: there is a conic section Cn such that whenever points A1, A2, 9 9 A n

THE MATHEMATrCALINTELUGENCER

are on C, a n d line A1A2 touches Cl, l i n e A2A 3 touches c2, 99. , An-l, An touches Cn- 1, thenAnA1 will touch Cn. The first publication was in 1822. 2. Poncelet was an officer of Napoleon. He was imprisoned in Russia, in Saratov, for more than a year. At this time, without books and equipment, he created many notions of projective geometry: ideal and imaginary points, for example. One of his practical results: Circles are exactly the conic sections containing the points (1, i, 0) and (1, - i , 0). One can transform two conics into two circles simply by projecting two of their common points to (1, i, 0) and (1, - i , 0). 3. The proof you found in Prasolov and Tikhomirov's textbook goes back to Jacobi (CreUe J. M a t h . 3 (1828), 376). Here is the short history of his proof: In Bd. 2 of Crelle's J o u r n a l (1827), Steiner proposed the problem of finding the algebraic relation of the radii of circles Cl and c2, and the distance of their centers, if there is a 4-gon, 5-gon, 9 8-gon inscribed in cl and circumscribed around c2. (In fact, these problems had been partly solved previously by Fuss, the academic secretary of St. Petersburg. Euler solved the problem for n = 3, his student Fuss for n = 4, and Fuss was able to solve the prob-

lem for n = 5, 6, 7, 8 if the n-gon is symmetric about the center of the circles.) Steiner gave the appropriate equations without proof in Crelle's J o u r n a l of the same year. In this issue, Abel and Jacobi had many articles on elliptic integrals and on their inverse, the elliptic functions. In Bd. 3, Jacobi wrote three articles on this topic. Then, he wrote a fourth one: he proved Poncelet's theorem for two circles by integrals and he could even check the equations of Steiner (and Fuss). When the old Poncelet refers shortly to Jacobi's proof, he uses essentially your arguments. Jacobi's article is longer. As I can see, Jacobi tried to solve geometrically the problem proposed by Steiner. He could set up equations, and these equations reminded him of Legendre's addition formulas of elliptic integrals. If Jacobi could find the connection, he could set up an elliptic integral related to Poncelet's theorem. It was only after this that he perceived the geometric meaning of the integrand: the reciprocal of the length of the tangent. Andras Hrask6 Tornoc u. 17 1141 Budapest Hungary e-mail: [email protected]

RODRIGO DE CASTRO AND JERROLD W. GROSSMAN

rails to Paul Erdds ar-Rous

~

he notion of Erdds number has floated around the mathematical research community for more than thirty years, as a way to quantify the common knowledge that mathematical and scientific research has become a very collaborative process in the twentieth century, not an activity engaged in solely by isolated individuals. In this

p a p e r w e e x p l o r e s o m e (fairly short) c o l l a b o r a t i o n p a t h s t h a t one can follow from Paul E r d 6 s to r e s e a r c h e r s inside a n d outside of m a t h e m a t i c s .

An Outstanding Component of the Collaboration Graph The collaboration graph C h a s as vertices all r e s e a r c h e r s ( d e a d o r alive) from all a c a d e m i c disciplines, with an edge j o i n i n g vertices u and v if u a n d v have j o i n t l y p u b l i s h e d a r e s e a r c h p a p e r o r b o o k (with p o s s i b l y m o r e co-authors). As is t h e c a s e for any simple (undirected) graph, in C w e have a notion o f distance b e t w e e n t w o vertices u a n d v: d(u,v) is the n u m b e r of edges in the s h o r t e s t p a t h b e t w e e n u a n d v, if such a p a t h exists, ~ otherwise (it is u n d e r s t o o d t h a t d(u,u) = 0). In this p a p e r w e are c o n c e r n e d with the c o l l a b o r a t i o n s u b g r a p h c e n t e r e d at Paul E r d 6 s (1913-1996). F o r a res e a r c h e r v, the n u m b e r d ( P a u l Erd6s, v) is called the ErdSs n u m b e r of v. That is, Paul Erd6~ himself has E r d 6 s numb e r 0, a n d his co-authors have E r d 6 s n u m b e r 1. P e o p l e n o t having E r d 6 s nttmber 0 o r 1 w h o have p u b l i s h e d with s o m e one with E r d 6 s n u m b e r 1 have E r d 6 s n u m b e r 2, and so on. T h o s e w h o are not linked in this w a y to Paul E r d 6 s have E r d 6 s n u m b e r ~. The collection o f all individuals with a finite E r d 6 s n u m b e r constitutes the Erd6s component o f C.

The E r d 6 s c o m p o n e n t o f C is outstanding for its amazing size and for the m a n n e r in w h i c h it clusters a r o u n d Erd6s. A l m o s t 500 p e o p l e have E r d 6 s n u m b e r 1, and o v e r 5000 have E r d 6 s n u m b e r 2. In the h i s t o r y of scholarly publishing in m a t h e m a t i c s , n o one has e v e r m a t c h e d Paul Erd6s's number of collaborators or number of papers ( a b o u t 1500, a l m o s t 70% o f w h i c h w e r e j o i n t works). With his r e c e n t d e a t h the m a n w h o i n s p i r e d so m u c h mathematical thinking h a s - - t o use his t e r m i n o l o g y - - l e f t , but his legend lives on (see for e x a m p l e t w o r e c e n t b i o g r a p h i e s [20], [28]). A n d p a r t o f this legend, inside a n d o u t s i d e mathe m a t i c a l circles, is the notion o f E r d 6 s numbers. The first explicit m e n t i o n in the literature o f a p e r s o n ' s E r d 6 s n u m b e r a p p e a r s to b e [11], w h e r e the r e a d e r is ass u r e d that Paul E r d 6 s himself was, for a long time, u n a w a r e of this entertainment. But t h e first s y s t e m a t i c a t t e m p t to s t u d y the E r d 6 s c o m p o n e n t o f C w a s c a r r i e d o u t b y t h e s e c o n d a u t h o r in [16] and [18] and c o n t i n u e s on the E r d 6 s N u m b e r P r o j e c t World Wide Web site [13]. This Web site contains a list o f all p e o p l e with E r d 6 s n u m b e r I (currently 485) a n d their o t h e r co-authors with E r d 6 s n u m b e r 2 (currently 5337). The files (available also via a n o n y m o u s ftp, s e e [14]) are u p d a t e d annually. It h a s b e e n s u r m i s e d that m o s t s c i e n t i s t s m u s t have a finite E r d 6 s number, but the evidence offered in s u p p o r t

9 1999 SPRINGER-VERLAG NEW YORK, VOLUME 21, NUMBER 3, 1999

51

has not been really abundant. In [5] the first author contributed new information, and the present paper pursues the matter much further. By skimming through several bibliographic sources we have found that many important people in academic areas--other than mathematics proper-as diverse as physics, chemistry, crystallography, economics, fmance, biology, medicine, biophysics, genetics, meterology, astronomy, geology, aeronautical engineering, electrical engineering, computer science, linguistics, psychology, and philosophy do indeed have finite Erd6s numbers. We report on some of these intriguing connections here; others can be found in an expanded version of this paper, available on-line [13]. Of course, it cannot be immediately inferred that all people in the mentioned disciplines, "or related ones, must have fmite Erd6s numbers. But the names first :~esulting from this kind of browsing are among the most prominent and productive (including more than 60 Nobel Prize winners), and most have had many collaborators over the years. Thus one is led to believe that the majority of researchers in those fields, except for those working in total isolation, probably have finite Erd6s numbers. When referring to all academic or scientific fields, the last statement should be regarded as bold--though credib l e - g u e s s . If we restrict ourselves to authors publishing mathematical research, then the conjecture ( m o s t active mathematical researchers of (e) ~the twentieth century have a finite (and 1. rather small) Erd6s number seems so plausible that it has been accepted folklore. Looking at the list of those with Erd6s number -<2, one sees 5500 people belonging to numerous and varied areas of research in the mathematical sciences; therefore (e) shou/d be true. We intend to provide some more conclusive or "hard" evidence in support of (e). To do so we first select a rather high-class sample of the mathematical research community: the winners of the most prestigious awards, namely, the Fields Medal, the Nevanlinna Prize, the Wolf Prize in Mathematics, and the Steele Prize for Lifetime Achievement. By criss-crossing multiple bibliographic references, we have determined that all recipients of these prizes have indeed an Erd6s number -<5. Complete tables of upper bounds on these Erd6s numbers are presented below 1. The respective collaboration paths linking all awardees to Paul Erd6s are displayed in full detail for the interested reader in the Web site [13]. The individuals belonging to these exclusive lists are especially original, prolific, and influential; most of them have had many disciples, collaborators, and doctoral students. Their impact and influence is not limited to one institution or even to one country or particular epoch (Paul Erd6s himself was given the Wolf Prize in 1983-84). Furthermore, these distinctions are conferred

with no exclusion of research area (except the Nevanlinna Prize, which is in computer science). The fact that all these big names are in the Erd6s component of C is strong evidence for (e). Next, we go two steps further: we trace the subject matter of the papers by some researchers known to have a small Erd6s number, and we branch out into other academic disciplines. This will give us a more concrete idea of how far the Erd6s connection really extends within the mathematical sciences and beyond. Lastly, we pose some open questions. Obviously this work is incomplete (for instance, we have not traced any recent Nobel laureates in physics), and it should not be hard to establish further links with important mathematicians and scientists. For brevity, we say that a person is Erd6s-n if his or her Erd6s number is -< n. Thus Erd6s's co-authors are Erd6s1 and their co-authors who are not Erd6s-0 or Erd6s-1 are Erd6s-2. The list [14], containing all Erd6s-2 individuals and their respective Erd6s-1 co-authors, is referred to as the

Erdds-2 list. Interesting Connections Without intending to be 100% exhaustive, we have examined several bibliographic databases and historical accounts (e.g., [3], [4], [19], [22], [24], [25], [26], [31], [33], and Internet sites too numerous to list) and discovered that some very conspicuous thinkers and researchers from manifold academic branches are in the Erd6s component of C. The following examples evince the amazing diversity of the scientific collaboration network directly linked to the name of Paul Erdds, providing--in passing--an ample glimpse into the practice of academic collaboration, an aspect of scientific research that has become essential in the twentieth century and has not been systematically addressed in the literature. Due to space limitations we cannot list at the end of the present paper the 120+ bibliographic references corresponding to the cited collaborative works. We shall use double brackets [[ ]] for those references, which the reader can fmd in full detail on the Erd6s Number Project World Wide Web site [7]. 9

Albert Einstein has Erd6s number 2 due to the two joint papers with his Princeton assistant (in the years 1944-48) Ernst G. Straus, with whom Erd6s wrote 20 papers (the first in 1953) 2. Einstein wrote jointly with about 25 collaborators (see [27]), among them Nobel laureates in physics Wolfgang Pauli and Otto Stern. At age 20 Pauli had surprised the physics establishment with his 200-page encyclopedia article on the theory of relativity, a piece of which Einstein wrote a laudatory review. Not surprisingly, their joint paper [[55]] of 1943 (their only joint paper, written during Pauli's stay in Princeton) deals with technical aspects of the general theory of relativity. Pauli received the 1945 Nobel Prize

1Upper bounds for the Erd6s numbers for all Fields Medalists up to 1994 had already been presented by the authors in [5], [13], and [17}, but many bounds have been lowered for the present paper. 2A complete bibliography of Erdds's works through about 1996 has been prepared by the second author [15], with annual updates posted on the Erdds Number Project Web site [13].

52

THE MATHEMATICALINTELLIGENCER

for the so-called Pauli exclusion principle. With Stern, Einstein also wrote only one joint article [[57]], when they were both in Prague. Stern was awarded the 1943 Nobel Prize for his discovery of the magnetic moment of the proton. Einstein also published v~ith Russian Boris Podolsky and Austrian Paul Ehrenfest [[52]], one of his closest friends. The well-known Einstein-Podolsky-Rosen paradox, conceived as a thought experiment against the quantum-mechanical conception, originated in their 1935 joint paper [[56]]. Co-authors of Podolsky include at least two Nobel laureates: the great British theoretical physicist Paul Dirac [[48]] and American chemist Linus Pauling [[111]]; hence, they are both at most Erd6s-4. Pauling received the 1954 chemistry prize for his research on chemical bonding. As a result of his campaign for an international control of nuclear weapons, Pauling was awarded the 1962 Nobel Peace Prize. There are, moreover, two very curious non-technical joint publications by Einstein. The fLrst is a report about an international bureau of meteorology, written in 1927 with Marie Curie and Hendrik A. Lorentz and published in the journal Science [[39]]. The second is a booklet entitled Why War?, which he wrote in 1933 with Sigmund Freud [[53]] (see also [27]). It appeared in German, French, and English, and was published by the International Institute of Intellectual Cooperation of the League of Nations. 9 Hendrik A. Kramers, a Dutch physicist, was one of Pauli's collaborators [[88]] and also wrote with Danish Nobel laureate Niels Bohr [[22]], one of the pillars of twenti~eth-century scientific thought. Therefore, Bohr is at most Erd6s-5. In 1923 Bohr published jointly with Dirk Coster [[21]], another Dutch physicist, who in the same year coauthored a research paper with George C. De Hevesy3, a Hungarian chemist who went on to receive the Nobel Prize in chemistry in 1943 for his use of isotopes as tracers. A distinguished collaborator of Bohr was John A. Wheeler. In 1939 they wrote the seminal work The mechanism of nuclear fission [[23]], which made Wheeler the first American involved in the theoretical development of nuclear weapons; in that memoir uranium-235 was singled out for use in a possible atomic bomb. Another of Kramers's co-authors is Leonard S. Ornstein [[108][, in turn linked with fellow Dutchman Frits Zernike [[109]], winner of the 1953 Nobel Prize in physics (for his invention of the phase-contrast microscope). 9 J. Robert Oppenheimer is among Ehrenfest's co-authors [[50]], which collaboration makes him at most Erd6s-4. Oppenheimer is remembered as director of the Los Alamos laboratory dttring degelopment of the atomic bomb (1943-45) and as director of the Institute for Advanced Study at Princeton (1947-66). Robert Serber, Oppenheimer's former student and close collaborator [[106]], is linked to at least two Nobel laureates, American

nuclear physicists Ernest O. Lawrence and Edwin M. McMillan [[28]]. Serber, Lawrence, and McMillan were indispensable members of the Los Alamos scientific team. Lawrence was the winner of the 1939 Nobel Prize in physics for his invention of the cyclotron; chemical element 103, lawrencium, is named after him. McMillan shared the 1951 chemistry Nobel Prize for his discovery of element 93, neptunium, the first element heavier than uranium. The above links show that both Lawrence and McMillan have an Erd6s number of at most 6. Max Born, a Nobel laureate in physics (1954), is at most Erd6s-3 through his collaboration with Norbert Wiener, the creator of cybernetics, whose Erd6s number is 2. Their only joint paper [[27]] was written during Born's visit to MIT in 1925. Among Born's co-authors we fend fellow Germans Werner Heisenberg, Pascual Jordan [[24]]--the three are founders of modern quantum mec h a n i c s - a n d Max yon Laue [[26]] (the last collaboration might be considered a bit of a stretch, a jointly written technical obituary for Max Abraham). For his preeminent role in the foundation of quantum mechanics Heisenberg was the sol6"winner of the 1932 Nobel Prize in physics. Laue had been awarded the 1914 Nobel Prize for his research on the diffraction of X-rays'in crystals. Furthermore, Heisenberg published with the director of his doctoral dissertation, German Arnold Sommerfeld [[81]] (also Pauli's thesis advisor in Munich), who is remembered for his successful modifications of Bohr's atomic model. One of Sommerfeld's many co-authors is Peter J. Debye (also spelled Debije) [[45]], a Dutch scientist and Nobel laureate in chemistry (1936). Debye's Erd6s number is at least one lower than implied by this collaboration, however, since he wrote a joint paper with Pauling [[46]]. We should mention another very famous co-author of Born, Theodore von ~ a r m ~ [[25]], the Hungarian-born American research aeronautical engineer. John von Neumann and Erd6s never wrote jointly although they were both Hungarian by birth and just 10 years apart in age. Actually, von Neumann did not write with any of Erd6s's almost 500 co-authors; his Erd6s number stands at 3 through his varied collaborations with individuals in the Erd6s-2 list (e.g., Salomon Bochner, Paul Halmos, Herman H. Goldstine). In turn, von Neumann had very illustrious co-authors, notably David Hilbert, Oswald Veblen, Garrett Birkhoff, Pascual Jordan and Nobel laureate physicists Eugene Wigner and Subrahmanyan Chandrasekhar (see [30]). With Hilbert, yon Neumaim wrote about the mathematical foundations of quantum mechanics [[82]] shortly after Heisenberg had proposed his quantum scheme, known as matrix mechanics. Hilbert, yon Neumann, and Heisenberg were at G6ttingen at that time. Wigner was also a Hungarian and a friend of yon Neumann since childhood; most of their joint papers deal with quantum mechanics as well. Through Wigner we find a path to one of

3In that paper [[37]] they reported the discovery of a new chemical element, hafnium.

VOLUME21, NUMBER3, 1999

the biggest names in quantum physics, Austrian Erwin Schr6dinger, who shared the 1933 Nobel Prize with Dirac for their introduction of wave equations in quantum mechanics. This is the route: Wigner with R. F. O'Connell [[83]] with John Trevor Lewis [[65]] with James McConnell [[94]] with Schr(idinger [[100]]. Thus, SchrSdinger becomes at most Erd6s-8. Another of von Neumann's collaborators was the Austrian economist Oskar Morgenstern, with whom he wrote in 1944 the very influential work Theory of Games and Economic Behaviour [[126]]. This book stimulated a worldwide development of the mathematical aspects of game theory and its applications (see [29]). At least three Nobel Prizes in economics have been awarded to gametheorists; all of them are in the Erd6s component of C (as we shall demonstrate below), even though they are not directly linked with either von Neumann or Morgenstern. Morgenstern, in turn, wrote jointly with John G. Kemeny [[86]], the creator (along with Thomas E. Kurtz) in the mid-1960s of BASIC,a very popular general-purpose programming language. 9 George Uhlenbeck, the noted Dutch-American physicist, has Erd6s number 2. He is best known for having postulated, along with Samuel Goudsmit, the concept of electron spin, which led to major changes in atomic the-

THE MATHEMATICAL INTELLIGENCER

ory and quantum mechanics. Their famous joint paper [[125]] was published in 1925 when they were graduate students in physics at the University of Leiden in the Netherlands (both of them were pupils of Ehrenfest). Among Uhlenbeck's co-authors we encounter at least two Nobel physicists: American Willis E. Lamb (1955 prize) [[105]], whose experimental work spurred refinements in the quantum theories of electromagnetic phenomena, and Italian-born Enrico Fermi (1938 prize) [[61]], one of the chief architects of the nuclear age. Fermi had legions of co-authors and collaborators in Europe and the United States; one of them was his former student in Rome Emilio Segrr [[59]]. Segr~ and his colleague at the University of California, Berkeley, American Owen Chamberlain, discovered the antiproton in 1955 and for that feat were awarded the Nobel Prize in physics in 1959 (they also published jointly [[31]]). The above links show that the Erd6s numbers of Fermi, Segr~ and Chamberlain are at most 3, 4, and 5, respectively. (Actually, if we are willing to use technical reports in establishing collaboration links, then we can lower Fermi's Erd6s number to 2, for he published a Los Alamos technical report with Stanislaw Ulam.) Another co-author of Segr~ is the American nuclear chemist Glenn T. Seaborg [[120]], who received one half of the chemistry Nobel Prize in 1951 for

his research on transuranium elements (the other co-winner was McMlllan, whom we mentioned above). Edward Teller, the Hungarian-born American nuclear physicist who led the development of the world's first thermonuclear weapon, is another co-author of Fermi [[60]]. One of Teller's doctoral students in Chicago was the Chinese-born physicist Chen Ning Yang, who later became assistant to Fermi, publishing joint research work with him [[62]]. Yang and fellow Chinese Tsungdao Lee received the 1957 Nobel Prize in physics for their work [[91]], [[92]] in discovering violations of the principle of parity conservation, a major discovery in particle physics theory. The just cited collaborations make Yang at most Erd6s-4 and Lee at most Erd6s-5. * Freeman J. Dyson, the British-American physicist known by the general public for his writings on extraterrestrial civilizations and his advocacy of space exploration, is a conspicuous member of the Erd6s-2 list. He is linked, by way of Richard H. Dalitz [~40]], with the GermanAmerican physicist Hans A. Bethe [[42]], one of the main figures in twentieth-century atomic physics. Bethe was head of the Theoretical Physics Division of the Manhattan Project and was honored with the 1967 Nobel Prize for his explanation of the energy production in the Sun and other stars. The prominent Austrian astro-

physicist Edwin E. Salpeter is included in the large group of Bethe's co-authors and collaborators [[118]]. Bethe is one of the protagonists of a joint publication [[2]] which compels attention for the unique combination of names in its byline: Alpher, Bethe, Gamow. The third author is George Gamow, the Ukranian-born nuclear physicist and cosmologist who also made contributions to modern genetic theory; the first author is Ralph Alpher, one of his students. The paper itself (The origin of chemical elements) is actually very important; in it the authors advanced the idea that the chemical elements were synthesized by thermonuclear reactions which took place in a primeval explosion. It was Gamow who coined the expression "big bang." From Dalitz we find a path to another Nobel physicist, American Robert Hofstadter, a co-recipient of the 1961 prize for his investigations of protons and neutrons. This path makes him at most Erd6s-5: Dalitz with D. G. Ravenhall [[41]] with R. Hofstadter [[77]]. Sheldon Lee Glashow, an American theoretical physicist and Nobel laureate (1979), has Erd6s number 2 for his collaboration [[71]] with the Erd6s-1 combinatorialist Daniel Kleitman, his brother-in-law. Glashow shares with Einstein the distinction of being, up until now, the only Nobel-winning physicists with Erd6s number -< 2.

VOLUME 21, NUMBER 3, 1999

I i~"1nF'-'-'-'-'-'-'-Hll '-i j .~.~ mo[,:,t i[, ~.-E.].e :rL;[,~.-a,i, i, ,i .[:,]~'l.]fl.-r,] ,~[--i1~m~ l ~ ~;~,,,lt,t1|~~ I

Nobel Prize in physics

Year

Erd~s number

Nobel Prize in physics

Year

Erd6s number 4

Max von Laue

1914

4

Emilio Segr~

1959

Albert Einstein

1921

2

Owen Chamberlain

1959

5

Niels Bohro

1922

5

Robert Hofstadter

1961

5 4

Louis de Broglie

1929

5

Eugene Wigner

1963

Werner Heisenberg

1932

4

Richard P. Feynman

1965

4

Paul A. Dirac

1933

4

Julian S. Schwinger

1965

4 4

Erwin Schr6dinger

1933

8

Hans A. Bethe

1967

Enrico Fermi

1938

3

Luis W. Alvarez

1968

6

Ernest O. Lawrence

1939

6

Murray GelI-Mann

1969

3

Otto Stem

1943

3

John Bardeen

1972

5

Isidor I. Rabi

1944

4

Leon N. Cooper

1972

6

Wolfgang P~.uli

1945

3

John R. Schdeffer

1972

5

Frits Zemike

1953

6

Aage Bohr

1975

5

Max Born

1954

3

Ben Mottelson

1975

5

Willis E. Lamb

1955

3

Leo J. Rainwater

1975

7

John Bardeen

1956

5

Steven Weinberg

1979

4

Walter H. Brattain

1956

6

Sheldon Lee Glashow

1979

2

William B. Shockley

1956

6

Abdus Salam

1979

3

Chen Ning Yang

1957

4

S. Chandrasekhar

1983

4

Tsung-dao Lee

1957

5

Norman F. Ramsey

1989

3

Nobel Prize in economics

Year

number

Nobel Prize in chemistry

Year

number

Paul A. Samuelson

1970

6

Peter J. Debye

1936

5

Kenneth J. Arrow

1972

3

George De Hevesy

1943

7

Tjalling C. Koopmans

1975

4

Otto Diels

1950

7

Gerard Debreu

1983

3

Kurt Alder

1950

6

Franco Modigliani

1985

5

Edwin M. McMillan

1951

6

Robert M. Solow

1987

6

Glenn T. Seaborg

1951

5

Harry M. Markowitz

1990

2

Linus Pauling*

1954

4 4

Erd6s

Erdo"s

Merton H. Miller

1990

4

Walter Gilbert

1980

John C. Harsanyi

1994

8

Jerome Karle

1985

4

John F. Nash

1994

4

Herbert A. Hauptman

1985

3

Reinhard Selten

1994

7

Robert C. Merton

1997

7

physiology/medicine

Year

number

Francis H. C. Crick

1962

7

James D. Watson

1962

8

Nobel Prize in

Erd~s

A co-author of Glashow is another Nobel-winning physicist (1969): fellow American Murray Gell-Mann [[70]], who introduced the concept and the word quark for a basic subatomic particle. Gell-Mann also collaborated with the American physicist and Nobel Prize winner Richard Feynman [[63]]. Feynman participated in the Manhattan Project (he was only 25 when he was recruited) and wrote jointly with Bethe [[9]]. The two of them devised the formula for predicting the energy yield of a nuclear explosive. Feynman, who had been a pupil of Wheeler at Princeton and published with him [[127]], became a salient figure of postwar physics, receiving the Nobel

THEMATHEMATICALINTELLIGENCER

*Also received the 1962 Nobel Peace Prize

Prize in 1965 for his quantum electrodynamics theory. Sharing the prize with Feynman was Julian S. Schwinger, who independently formulated a theory of quantum electrodynamics, unaware that Feynman in the United States and Sin-Itiro Tomonaga in Japan were working on the same problem. The equivalent theories reconcile quantum mechanics with the special theory of relativity. We can link Schwinger to Erd6s via this path: Schwinger with Norman F. Ramsey [[112]] with W. H. Furry [[68]], the last named being Erd6s-2. Thus, Schwinger is at most Erd6s-4. Curiously enough, Feynman and Schwinger were born in the same year, 1918, in the same city, New York, received the Nobel Prize the same year for the

It

i ~ l I'] I ~ l [ / ~ l l I J t ~ , ['.1 I I i i l ] i h i 9 k l l , ] i l

: l ~ l [ , ] . ' ! I l I I i i l , I :~ t . 1 1 , 1 I I . ' ~ l ] i i [ : 4 1 , i k~l I h [* i i i k l ~

Main research field

~

~

B it l

Erd6s number

Walter AIvarez

Geology

7

Rudolf Carnap

Philosophy

4

Jule G. Charney

Meteorolog~

4

Noam Chomsky

Linguistics

4

Freeman J. Dyson

Quantum physics

2

George Gamow

Nuclear physics and cosmology

5

Stephen Hawking

Relativity and cosmology

7

Pascual Jordan

Quantum physics

4

Theodore von K ~ r m a n

Aeronautical engineering

4

John Maynard Smith

Biology

4

Qskar Morgenstern

Economics

4

J. Robert Oppenheimer

Nuclear physics

4

Roger Penrose

Relativity and cosmology

8

Jean Piaget

Psychology

3

Karl Popper

Philosophy

5

Edwin E. Salpeter

Astrophysics

5

Claude E. Shannon

Electrical engineering

3

Arnold Sommerfeld

Atomic physics

5

Edward Teller

Nuclear physics

4

George Uhlenbeck

Atomic physics

2

John A. Wheeler

Nuclear physics

5

same achievement, and--as far as we know--they also have the same Erd6s number, namely 4. The two co-authors of Schwinger in the above-cited l~aper [[112]], Americans Isidor I. Rabi and Ramsey, are themselves Nobel laureates. Rabi was given the 1944 Nobel Prize in physics for his 1937 invention of the magnetic resonance method. Ramsey received one half of the 1989 physics award for his development of a technique called the separated oscillatory fields method, which provides the basis for the cesium atomic clock. Hence, Ramsey is at most Erd6s-3 and Rabi at most Erdds-4. 9 David Pines, an American physicist who publishes on condensed matter theory and theoretical astrophysics, is a key figure in the collaboration graph. As a co-author of Gell-Mann [[58]], he is at most Erd6s-4. Pines has coauthored research papers with six Nobel Prize winners and is at distance 2 from eight more (see Figure 1). None of the scientists we mention in the present article (including Erd6s himself) clusters so closely around so many Nobel winners. Two of the co-authors of Pines are John Bardeen [[14]] and John Robert Schrieffer [[110]], who, along with fellow American Leon N. Cooper, received the 1972 Nobel Prize in physics for their joint theory [[12]], [[13]], known as BCS theory for their surname initials, which was the first successful microscopic theory of superconductivity. When he made his principal contribution to the BCS theory, Schrieffer was a 26-year-old graduate student at the University of Illinois, were Bardeen was a professor in the physics and electrical engineering departments. It should be recalled that Bardeen had been a co-win-

ner of another Nobel Prize in physics, that of 1956, which he shared with Walter H. Brattain and William B. Shockley for their research on semiconductors and their joint invention of the transistor. Therefore, both Brattain and Shockley become at most Erd6s-6 because of their joint papers with Bardeen [[11]], [[15]]. Two additional co-authors of Pines are Nobel winners Danish Aage Bohr (the son of Niels Bohr) and DanishAmerican Ben Mottelson [[19]]. They shared the 1975 physics award for work in the early 1950s in determining the asymmetrical shapes of certain atomic nuclei. Their experiments had been inspired by the theories of the American physicist Leo James Rainwater, who was also a co-recipient of the 1975 Nobel Prize. Rainwater's Erd6s number is fmite too, at most 7, via Tsung-dao Lee, who, as we saw, is at most Erdds-5: Lee with C. S. Wu [[93]] with Rainwater [[113]]. A co-author of Aage Bohr is Kurt Alder [[1]], a German chemist, former student and assistant of Otto Diels, along with whom he received the 1950 Nobel Prize in chemistry for their joint method of preparing cyclic organic compounds. Apart from father Niels and son Aage, there is another member of the Bohr family in the Erd6s component of C, namely, Niels's younger brother Harald, known especiaily for the theory of almost periodic functions. Harald's Erd6s number is 3 due to his joint paper [[20]] with Borge Jessen, whose Erd6s number is 2. (This shows that the brothers Niels and Harald are at distance _< 8 in the graph C, and the distance between the two Nobel laureates, Niels and his son Aage, is -< 10; most likely these bounds can be improved.) Abdus Salam and Steven Weinberg shared with the aforementioned Glashow the 1979 Nobel Prize in physics for their theoretical work linking the electromagnetic interaction and the so-called weak interaction. Salam became the first Pakistani to win a Nobel Prize (in any category);

I PA~-bEv.r~6s [

J S. GL&~'aOW*

W, SHOCKIJ~* I DAVIDPreSS }

* Nobel Prize in physics + Nobel Prize in chemis*.ry Figure 1. Clustarin 9 o f N o b e l l a u r e a t e s within d i s t a n c e 2 of David Pines.

VOLUME21, NUMBER3, 1999 57

he is at most Erd6s-3 due to his joint paper [[117]] with J. C. Ward who is Erd6s-2. Weinberg is at most Erd6s-4 for his many collaborations with Salam (e.g., [[73]]). Among Salam's many co-authors we are able to fmd another Nobel winner, American molecular biologist Walter Gilbert 4 [[116]], who shared (with Paul Berg and Frederick Sanger) the 1980 chemistry award for their chemical and biological analyses of DNAstructure. 9 Edward Witten, the outstanding American theoretical physicist and Fields Medalist in 1990, is at most Erd6s5, as will be shown later. Among his co-authors we fred the American physicist Luis W. Alvarez [[4]], who received the Nobel Prize in 1968 for his work on subatomic partic],es. Another co-author of Witten is Gary Horowitz [[30]], who in turn h a s collaborated with Stephen Hawking [[69]], the English theoretical physicist. Hawking has also published with fellow British mathematician and physicist Sir Roger Penrose [[80]]. 9 Jean-Pierre Vigier, a distinguished French physicist, can be linked to some important scientists; his Erd6s number is at most 4 as the following collaborations show: Vigier with Constantin Piron [[64]] with Stanley P. Gudder [[76]], Gudder being an Erd6s-2 researcher. Among Vigier's co-authors stands out fellow French physicist Prince Louis de Broglie [[29]], who in the mid1920s developed (in his doctoral dissertation) a revolutionary theory of electron waves, enthusiastically defended by Einstein. Experimental evidence of his theory came a few years afterwards, and de Broglie was awarded the Nobel Prize in 1929. Also a co-author of Vigier in the work we have just cited is the American physicist and philosopher David Bohm, the last doctoral student of Oppenheimer at Berkeley and the originator of the causal interpretation of quantum theory. Bohm is also acknowledged for his joint research with David Pines (see Figure 1); their joint papers [[18]], published under the title A collective description of electron interactions, underlie all current research in plasma-state physics. Sir Karl R. Popper, the eminent Austrian-born British philosopher, is another co-author of Vigier [[95]]. 9 Claude E. Shannon, an American electrical engineer, became famous for his elegant and general mathematical model of "communication," known today as information

theory. Shannon's Erd6s number is at most 3 because of his collaboration [[122]] with Elwyn R. Berlekamp, whose Erd6s number is 2. 9 Francis H. C. Crick, a British biophysicist, and James D. Watson, an American geneticist and biophysicist, determined the molecular structure of deoxyribonucleic acid (DNA)--as a double-helix polymer--for which accomplishment they were awarded the 1962 Nobel Prize for Physiology/Medicine.

4Gilbert's Ph.D. degree is actually in mathematics (from Cambridge University).

58

THE MATHEMATICALINTELLIGENCER

In 1957 Crick published a short paper [[38]] on information theory, thereby entering the Erd6s component of C. Indeed, Crick's Erd6s number is at most 7 due to the following chain of joint works: Crick with J. S. Griffith [[38]] with I. W. Roxburgh [[114]] with P. G. Saffman [[1151] with H. B. Keller [[36]] with K. O. Friedrichs [[66]]. The last named is in the Erd6s-2 list. Hence, James Watson would be Erd6s-8, and from this connection we can conclude that many other active researchers in genetics, biophysics, biochemistry, and related fields have finite Erd6s numbers as well. 9 Herbert A. Hauptman, a mathematician, and Jerome Karle, a chemist and crystallographer (both American), were awarded the Nobel Prize in chemistry in 1985 for their development of mathematical methods for deducing the molecular structure of biological molecules from the patterns formed when X-rays are diffracted by their crystals. Hauptman's Erd6s number is at most 3 for his joint publication [[75]] with Fred Gross, who appears in the Erd6s-2 list. This makes Karle an Erd6s-4 researcher by way of his numerous joint articles with Hauptman (e.g., [[79]]). 9 John Maynard Smith, a British biologist, initiated a whole new area of research by his unusual applications of game theory to animal behavior and evolution, with works like

The theory of games and the evolution of animal conflict [[98]] and Evolution and the Theory of Games [[97]]. It turns out that Maynard Smith has a small Erd6s number, at most 4, through Josef Hofbauer [[99]] and Hal L. Smith [[84]], who is Erd6s-2. 9 Harry M. Markowitz, the American finance expert, is in the Erd6s-2 list (because of his collaboration with Alan J. Hoffman [[85]]), and is the only Nobel economist with such a low Erd6s number. He shared the 1990 prize with Merton H. Miller and William F. Sharpe for their study of financial markets and investment decision-making. Miller is at most Erd6s-4, through Abraham Charnes [[32]] and Fred Glover [[33]], the latter being Erd6s-2. Miller is also linked [[102]], [[103]] with another Nobel laureate, the Italian economist Franco Modigliani, who received the 1985 award for his mathematical analysis of household savings and the dynamics of financial markets. 9 Herbert Scarf, an American economist with Erdds number 2, has published articles on economic analysis with many other renowned economists, such as Kenneth J. Arrow [[6]] and G6rard Debreu [[44]], both winners of the Nobel Prize in economics (1972 and 1983, respectively). A co-author of the latter is the Dutch economist Tjalling C. Koopmans [[43]], also a Nobel laureate (1975). Additionally, Scarf has published with Lloyd S. Shapley [[124]], one of the major contributors to the development of game theory and a co-author of the American mathematician John F. Nash [[104]], a co-recipient of the 1994 Nobel Prize in economics. Nash shared his prize with the

Hungarian-born economist John C. Harsanyi and the German mathematician Reinhard Selten, for their beneficial use of game theory in economics (more precisely, for "their pioneering analysis of equilibria in the theory of non-cooperative games"). As can be expected, Harsanyi and Selten also ha~e a finite Erd6s number. This is the path: Koopmans with Beckman [[87]] with Marschak [[16]] with Selten [[96]] with Harsanyi [[78]]. Therefore, Selten is at most Erd6s-7 and Harsanyi at most Erd6s-8. From Arrow we can fmd a path leading to two more Nobel laureates in economics, the American economists Paul A. Samuelson and Robert M. Solow. This is the path: Arrow with Edward W. Barankin [[5]] with Robert Dorfman [[10]] with Samuelson and Solow [[49]]. An additional and more recent Nobel connection should be cited: Through his collaboration with Samuelson [[101]], the American economist Robert C. Merton, one of the recipients of the 1997 Nobel Prize in economics, is at most Erdds-7. Merton expanded the work of Myron S. Scholes and Fisher Black, who had advanced in 1973 a pioneering formula for the valuation of stock options. Scholes shared the prize with Merton, not so Black, due to his untimely death in 19955. * Noam Chomsky, the American linguist and political activist, is one of the most influential figures of twentiethcentury linguistics. The following path shows that Chomsky is at most Erd6s-4: Chomsky with M. P. Schutzenberger [[35]] with S. Eilenberg [[51]], the latter being Erd6s-2. 9 Rudolf Carnap, the German-born philosopher and member of the Vienna Circle, is at most Erd6s-4, as shown by this path: Carnap with Yehoshua Bar-Hillel [[7]] with M. Perles [[8]], the latter being in the Erd6s-2 list. A student of Carnap in Prague was Willard V. Quine, an American logician and philosopher, known for undertaking a systematic constructivist analysis of philosophy. He is at most Erd6s-3 due to his collaboration with J. C. C. McKinsey [[89]], an Erd6s-2 individual and a renowned philosopher himself. ErdSs Numbers of the Fields Medalists The Fields Medal was established by John Charles Fields (1863-1932), a Canadian mathematician. It has always been granted to mathematicians not older than 40, although the age limit was neither demanded nor suggested by Fields himself (see [21]). A minimum of two and a maximum of four medals are awarded on the occasion of the quadrennial International Congress of Mathematicians. The first two medals were conferred at the Oslo (Norway) Congress in 1936 to Finnish mathematician Lars Ahlfors and New Yorker Jesse Douglas, but due to the Second World War no medals were awarded during the next 14 years. The academic distinction resumed in 1950; to date there have been 42 awardees from 14 different countries.

a b l e 3. U p p e r b o u n d s o n E r d S s n u m b e r s

Fields Medal

Year

of t h e F i e l d s M e d a l i s t s

Country of origin

Erd6s number

Lars Ahlfors

1936

Finland

5

Jesse Douglas

1936

USA

4

Laurent Schwartz

1950

France

5

Atle Selberg

1950

Norway

2

Kunihiko Kodaira

1954

Japan

2

Jean-Pierre Serre

1954

France

3

Klaus Roth

1958

Germany

2

Rene Thom

1958

France

4

Lars Hormander

1962

Sweden

3

John Milnor

1962

USA

3

Michaei Atiyah

1966

Great Britain

4

Paul Cohen

1966

USA

5

Alexander Grothendieck

1966

Germany

5

Stephen Smale

1966

USA

5

Alan Baker

1970

Great Britain

2

Heisuke Hironaka

1970

Japan

4

Serge Novikov

1970

Russia

3

John G. Thompson

1970

USA

3

Enrico Bombieri

197,~

Italy

2

David Mumford

1974

Great Britain

2

Pierre Deligne

1978

Belgium

3

Charles Fefferman

1978

USA

2

Gregori Margulis

1978

Russia

5

Daniel Quillen

1978

USA

3

Alain Connes

1982

France

5

William Thurston

1982

USA

4

Shing-Tung Yau

1982

China

2

Simon Donaldson

1986

Great Britain

5

Gerd Faltings

1986

Germany

4

Michael Freedman

1986

USA

4

Vladimir Drinfeld

1990

Russia

5

Vaughan Jones

1990

New Zealand

Shigemufi Mori

1990

Japan

4 '3

Edward Witten

1990

USA

3

Pierre-Louis Lions

1994

France

4

Jean Christophe Yoccoz

1994

France

5

Jean Bourgain

1994

Belgium

2

Efim Zelmanov

1994

Russia

4

Richard Borcherds

1998

S. Africa/Great Britain

2

William T. Gowers

1998

Great Britain

4

Maxim L. Kontsevich

1998

Russia

4

Curtis McMullen

1998

USA

3

Table 3 shows that although Erdds never wrote jointly with any of the 42 Medalists (a fact perhaps worthy of further contemplation), 10 of them have Erd6s number 2 and for none is the number greater than 5. The collaboration paths from which these numbers have been obtained are presented in the Web site [6]. It is possible that some paths can be lowered still more, but with these data the average Erd6s number of the Fields Medalists is 3.52.

5For the mathematics behind the 1997 Nobel Prize in economics the reader is referred to [8] and [9],

VOLUME21, NUMBER3, 1999 59

T a b l e 5, U p p e r bounds on Erdi~s n u m b e r s of t h e w i n n e r s of t h e Wolf Prize in M a t h e m a t i c s .

Nevanlinna Prize

Year

Country of origin

Erd~s number

Wolf Prize in Mathematics

Year

Country of origin

Erd~s number

Robert Tarjan

1982

USA

2

Leslie Valiant

1986

Hungary/Great Britain

3

Izrail M. Gelfand

1978

Russia

4

Carl L. Siegel

1978

Germany

3

Alexander Bazborov

1990

Russia

2

Avi Wigderson

1994

Israel

2

Jean Leray

1979

France

3

2

Andre Weil (SP)

1979

France

4

Henri Cartan

1980

France

3

Andrei N. Kolmogorov

1980

Russia

5

Peter Shor

1998

USA

Erd(~s Numbers of the Steele, Nevanlinna, and Wolf Prize Winners The Fields Medal carries the prestige of a Nobel Prize, but there are many other important international awards for mathematicians. Perhaps the three most renowned, which are acquiring more and more prominence over the years, are the Roll Nevanlinna Prize, the Wolf Prize in Mathematics, and the Leroy P. Steele Prizes. These prizes were established within a span of 12 years, beginning in 1970. 9 Since 1982 the Rolf Nevanlinna P r i z e has been presented, along with the Fields Medal, at the International Congress of Mathematicians every four years [21]. The funds for the award are granted by the University of Helsinki. This distinction is given only to young mathematicians who deal with the mathematical aspects of information science, and only one prize is bestowed per congress. 9 The Wolf P r i z e is awarded by the Wolf Foundation, based in Israel [32]. Each year (since 1978) it gives prizes of $100,000 for outstanding achievements in agriculture,

Lars Ahlfors (FM)

1981

Finland

5

Oscar Zariski (SP)

1981

Poland

3

Hassler Whitney (SP)

1982

USA

2

Mark G. Krein

1982

Ukraine

4 2

Shiing Shen Chern (SP)

1983-84

China

Paul ErdSs

1983-84

Hungary

0

Kunihiko Kodaira (FM)

1984-85

Japan

2

Hans Lewy

1984-85

Germany

3

Samuel Eilenberg (SP)

1986

Poland

2

Atle Selberg (FM)

1986

Norway

2

Kiyoshi Ito

1987

Japan

3

Peter D. Lax (SP)

1987

Hungary/USA

3

Friedrich E, Hirzebruch

1988

Germany

3

Lars H6rmander (FM)

1988

Sweden

3

Alberto Calderon

1989

Argentina

3

John Milnor (FM)

1989

USA

3

Ennio De Giorgi

1990

Italy

3

Ilya Piatetski-Shapiro

1990

Russia

5

Lennart A. Cafieson

1992

Sweden

4

John G. Thompson (FM)

1992

USA

3

Mikhael Gromov

1993

Russia

3

Jacques Tits Jurgen K Moser

1993

Belgium

4

1994-95

Germany

3

Robert Langlands

1995-96

Canada

2

Andrew Wiles

1995-96

Great Britain

3

Joseph B. Keller

1997

USA

3

Yakov G. Sinai

1997

Russia

4

(FM): FieldsMedalist (SP): Steele Prize

Paul ErdSs and Alfred R~nyi, who collaborated on 32 papers, in Aarhus in 1957.

THEMATHEMATICALINTELLIGENCER

chemistry, medicine, and the arts, as well as mathematics and physics. The Wolf Prize in Mathematics was conferred (in 1984) on Paul Erd6s himself, and in addition to his contributions to many fields, the citation extols him "for personally stimulating mathematicians the world over." * The L e r o y P. Steele P r i z e s are awarded by the American Mathematical Society. From 1970 to 1976 one or more prizes were awarded each year for outstanding published mathematical research; in 1977 the Council of the AMS modified the terms under which the prizes are awarded (see [1]). Since then, up to three prizes have been awarded each year in the following categories: (1) Lifetime Achievement (for the cumulative influence of the total mathematical work of the recipient), (2) Mathematical Exposition (for a book or substantial survey or expository-research paper), and (3) Seminal Contribution to Research (for a paper, whether recent

Steele Prize (Lifetime Achievement)

Year

Country of origin

Erd~s number

Salomon Bochner

1979

Poland

Antoni Zygmund

1979

Poland

2

Andre Weil

1980

France

4

Gerhard P. Hochschild

1980

Germany

4

Oscar Zariski

1981

Poland

3 4

2

Fritz John

1982

Germany

Shiing Shen Chem

1983

China

2

Joseph L. Doob

1984

USA

2

Hassler Whitney

1985

USA

2

Saunders Mac Lane

1986

USA

3

Samuel Eilenberg

1987

Poland

2

Deane Montgomery

1988

USA

3

Irving Kaplansky

1989

Canada

1

Raoul Bott

1990

Hungary

3

Armand Borel

1991

Switzerland

4

Peter D. Lax

1992

Hungary/USA

3

Eugene B. Dynkin

1993

Russia

3

Louis Nirenberg

1994

Canada

3

John T. ]-ate

1995

USA

3

Gore Shimura

1996

Japan

2

Ralph S. Phillips

1997

USA

2

Nathan Jacobson

1998

LISA

3

or not, that has proved to be of fundamental or lasting importance in its field. We have compiled tables of Erd6s numbers for all recipients of the Nevanlinna Prize (Table 4), the Wolf Prize in Mathematics (Table 5), and the Steele Prize for IMetime

Achievement (Table 6), and have found that all these numbers are -< 5. Again, one may wonder why Kaplansky is the only recipient of any of these prizes who collaborated with Patti Erd6s. (As before, the collaboration paths from which these numbers have been obtained are presented in the Web site [6].) 6 How Far Does t h e Erdb's Connection Extend? In this section we first look at the various branches of mathematics to see how they are represented in the Erd6s component of C. Next, we consider many other academic disciplines in an effort to determine the scope of the Erd6s connection beyond mathematics. Both Mathematical Reviews (MR) [25] and Zentralblatt fi~r Mathematik (Zbl) [33] assign a number to each published work representing its primary subject area. For example, combinatorics is 05 and number theory is 11 (to mention the areas in which about 80% of Paul Erd6s's works appear). A total of 6 broad categories are currently in use [2]. It turns out--not surprisingly--that all 61 subject classifications are repl:esented in the ErdSs component of C. In fact, we can say much more: Erd6s himself published in at least 27 of these categories, his co-authors published in at least 32 more, and there are people with Erd6s number 2 who have published in the remaining two (K-theory and geophysics). We showed above that some outstanding scientists from myriad fields have fmite Erd6s numbers. We now extend the reach even further. Sophisticated mathematical models and tools have become standard in many fields outside the natural sciences, computer science, and engineering. It is not hard to f'md researchers with fairly small Erd6s numbers publishing in social sciences. For example, Scott A. Boorman, who has Erd6s number at most 7, has papers

Five Nobelists with collaborative links to Erd6s. (L-R) Edwin McMillan, Emilio Segr~, Isidor Rabi, Hans Bethe, and Luis Alvarez. (AlP Emilio Segr~) Visual Archives, Segr~ Collection.)

eNote added in proof: The obvious mutability of this article's results, and the consequent usefulness of the regularly updated Web sites, have been illustrated most happily by the recent awarding of the 1999 Wolf Prize to L&szl6 LovAsz, with Erdds number 1 (the same Lov~sz who is a Correspondent of The Mathematical Intelligencet).

VOLUME21, NUMBER3, 1999 61

in both the Journal of Mathematical Psychology and the Journal of Mathematical Sociology. Certainly hundreds, if not thousands, of statisticians have small Erd6s numbers, and they often become co-authors on papers growing out of their consulting work. As another example, Peter C. Fishburn, whose Erd6s number is 1 and who works in a variety of mathematical disciplines, has published in Management Science and Theory and Decision. Frank Harary, a co-author of Paul Erd6s who himseff has over 270 co-authors, reports 6 that he has published with anthropologists, architects, biologists, chemists, economists, engineers, geographers, journalists (including the grand-nephew of the writer James Joyce), philosophers, physicians, physicists (including George Uhlenbeck), political scientists, psychologists, scientific writers (including Martin Gardner), and sociologists, among others. Clearly much work remains to be done in exploring collaborations in other disciplines.

ponent of C it is necessary to have many co-authors. But one of the conclusions we draw from the compilation of data for this article is that what really matters is not how many people you publish with but whom you publish with. A more dramatic example than any presented thus far is the great Austrian logician Kurt G6del. In regard to the number of joint papers, G6del is at the other end of the spectrum from Erd6s: He wrote only one (see [10]), and that is a one-page note (in German) with Karl Menger and Abraham Wald [[72]] concerning Menger's approach to differential and projective geometry. It turns out that Wald's Erd6s number is 2. Hence, despite his paucity of joint papers, G6del still makes his way into the Erd6s component of C with a rather small Erd6s number. We close with some open questions which, even in the era of supercomputing and worldwide information networks, are extremely difficult to answer. The first two were already put forward in [18], but no hint to their possible solution has yet surfaced.

Final Remarks and Open Questions

9 In the collaboration graph C, what is the second largest component (measured by the number of its vertices)? If we restrict ourselves to looking only at mathematicians,

It could be thought a p r i o r i that in order for a mathematician to make his or her entrance into the Erdds com6pnvate communication.

62

THE MATHEMATICALINTELLIGENCER

t h e n the s e c o n d largest c o m p o n e n t is p r o b a b l y n o t v e r y large, but it is conceivable that there are large c o m p o n e n t s in o t h e r disciplines. 9 W h a t are the r a d i u s and t h e d i a m e t e r o f the E r d 6 s comp o n e n t of C (in graph-theoretical t e r m s ) ? Again, the question w o u l d be interesting b o t h as applied to all res e a r c h e r s a n d w h e n r e s t r i c t e d to mathematicians. 9 The Nobel-Erdds n u m b e r is, at a given moment, the numb e r o f Nobel Prize l a u r e a t e s having a finite E r d 6 s number. This n u m b e r changes as n e w prizes are a w a r d e d a n d m o r e p e o p l e e n t e r into the E r d 6 s c o m p o n e n t o f C. We have e s t a b l i s h e d that the Nobel-Erdds n u m b e r is ~> 63 b u t its e x a c t value (as of, say, the e n d of 1998) is unknown. Surely o u r b o u n d is n o t nearly the b e s t possible. 9 The Erd6s span m e a s u r e s h o w far b a c k in time the conn e c t i o n with Paul E r d 6 s extends. More precisely, w e can define the E r d 6 s s p a n as t h e s m a l l e s t n u m b e r representing the y e a r of birth o f a p e r s o n with a finite E r d 6 s number. All w e can s a y for n o w is that this n u m b e r is no g r e a t e r than 1849, w h i c h is the y e a r of birth of Georg F r o b e n i u s (1849-1917), t h e G e r m a n algebraist w h o m a d e m a j o r contributions to group theory. He d e v e l o p e d the t h e o r y of finite groups of linear substitutions m o s t l y in c o l l a b o r a t i o n with Issai S c h u r (1875-1941) [[67]]. It t u r n s o u t t h a t Schur's E r d 6 s n u m b e r is 2 b e c a u s e o f his 1925 j o i n t p a p e r [[119]] with G a b o r SzegS, a c o - a u t h o r o f Erd6s. We do n o t k n o w w h e t h e r the E r d 6 s s p a n can be t r a c e d furt h e r b a c k into the early 1800s. What w e can b e s u r e o f is that t h e E r d 6 s c o n n e c t i o n will e x t e n d forever into t h e future. REFERENCES

[1] American Mathematical Society, The Leroy P. Steele Prizes, Internet page: http://www.ams.org/secretary/prizes.html#steele. [2] American Mathematical Society, 1991 Mathematics Subject Classification, Internet page: http://www.ams.org/msc/. [3] Nicolas Bourbaki, Elements d'histoire des mathematiques, Hermann Editeurs, 1969. [4] Britannica CD-97, Encyclopaedia Britannica, Inc., 1997. [5] Rodrigo De Castro, Sobre el ndmero de Erdds, Lect. Mat. 17 (1996), 163-179. [6] Rodrigo De Castro & Jerrold W. Grossman, Collaboration paths for this paper, Internet page: http://www.oakland.edu/-grossman/collabpaths.html. [7] Rodrigo De Castro & Jerrold W. Grossman, Primary references for this paper, Internet page: http://www.oakland.edu/-grossman/erdosrefs.html. [8] Keith Devlin, A Nobel formula, Internet page: http://www.maa.org/ devlin/devlin 11 97.html. [9] Guillermo Ferreyra, The Mathematics Behind the 1997 Nobel Prize

in Economics, Internet page: http:l/www.ams.orglnew-in-math/

black-scholes-ito.html. [10] Kurt G6del, Collected Works, edited by S. Feferman, Oxford University Press, 1986. [11] Casper Goffman, And what is your Erdds number?, Amer. Math. Monthly 76 (1969), 791. [12] Ronald L. Graham & Jaroslav Ne~et~il, editors, The Mathematics of Paul Erdds, vols. I-II, Algorithms and Combinatorics 13-14, Springer-Vedag, 1997. [13] Jerrold W. Grossman, The ErdOs Number Project World Wide Web Site, http://www.oakland.edu/-grossman/erdoshp.html. [14] Jerrold W. Grossman, List of people with Erdds Number at most 2, available in [13] and via anonymous ftp to vela.aes.oakland.edu in directory pub/math/erdos, Oakland University, Rcohester, MI, 1998 (updated annually). [15] Jerrold W. Grossman, preparer, List of publications of Paul ErdSs, in [12], pp. 477-573. [16] Jerrold W. Grossman, Paul Erdds: the master of collaboration, in [12], pp. 467-475. [17] Jerrold W. Grossman, Review of [5], Mathematical Reviews, 98h:01041. [18] Jerrold W. Grossman & Pat~ck D. F. Ion, On a portion of the weftknown collaboration graph, Proc. 26th Southeastern Inter. Conf. on Combinatorics, Graph Theory and Computing (Bdca Raton, FL, 1995), Congr. Numer. 108 (1995), 129-131. [19] Steve J. Heims, John von Neumann and Norbert Wiener, MIT Press, 1980. [20] Paul Hoffman, The Man Who Loved Only Numbers, Hyperion, 1998. [21] International Mathematical Union, Fields Medals and Rolf Nevanlinna Prize, Internet page: http://elib.zib.de/IMU/medals. [22] Jahrbuch (~berdie Fortschritte der Mathematik, 1868-1942, Berlin. [23] John H. Kagel & Alvin E. Roth, editors, The Handbook of Experimental Economics, Princeton University Press, 1995. [24] MacTutor History of Mathematics, Internet page: http://wwwgroups.dcs.st-and.ac.uW-history/. [25] Mathematical Reviews, American Mathematical Society, 1940-. [26] Gert H. Muller, Wolfgang Lenski, et al., editors, s of Mathematical Logic, vols. I-Vl, Springer-Verlag, 1987. [27] Abraham Pais, 'Subtle is the L o r d . . . ': The Science and Life of Albert Einstein, Oxford University Press, 1982. [28] Bruce Schechter, My Brain Is Open: The Mathematical Journeys of Paul Erdds, Simon & Schuster, 1998. [29] L. C. Thomas, Games, Theory and Applications, Ellis Horwood Ltd., 1984. [30] John von Neumann, Collected Works, Pergamon Press, 19611963. [31] Edmund Whittaker, A History of the Theories of Aether and Electricity, Dover Publications Inc., 1989. [32] The Wolf Foundation, The Wolf Foundation, Internet page: http://www.aquanet.co.il/wotf. [33] Zentralblatt for Mathematik und Ihre Grenzgebiete, SpringerVerlag, 1931-.

VOLUME 21, NUMBER 3, 1999

BY SHALOSH B. EKHAD, XIV*

Plane Geometry: An Elementary School Textbook (ca, 2050 AD)

#INTRODUCTION: #Dear Children, #Do you know that until fifty years ago most of m a t h e m a t i c s was #done by h u m a n s ? Even more strangely, they u s e d h u m a n language #to state and p r o v e m a t h e m a t i c a l theorems. Even w h e n they s t a r t e d #to use c o m p u t e r s to p r o v e theorems, they always t r a n s l a t e d the #proof into the imprecise h u m a n language, because, ironically, c o m p u t e r #proofs were c o n s i d e r e d of q u e s t i o n a b l e rigor! ## ##Only thirty y e a r s ago, w h e n m o r e and more m a t h e m a t i c s was g e t t i n g #done by computer, p e o p l e r e a l i z e d how silly it is to go b a c k - a n d - f o r t h #from the p r e c i s e p r o g r a m m i n g - l a n g u a g e to the i m p r e c i s e humanese. #At the h i s t o r i c a l ICM 2022, the IMS (International M a t h Standards) #were introduced, and Maple was chosen the o f f i c i a l language for m a t h e m a t i c a l #communication. T h e y also r e a l i z e d that once a t h e o r e m is s t a t e d precisely, #in Maple, the proof p r o c e s s can be started right away, by r u n n i n g the # p r o g r a m - s t a t e m e n t of the theorem. ## #All the theorems that were k n o w n to our grandparents, and most of what they #called conjectures, can now be p r o v e d in a few n a n o - s e c o n d s on any PC. #As you p r o b a b l y know, c o m p u t e r s have since d i s c o v e r e d m u c h d e e p e r theorems #for w h i c h we o n l y have s e m i - r i g o r o u s proofs, b e c a u s e a c o m p l e t e proof w o u l d #take too long. ## #All the theorems in this t e x t b o o k were a u t o m a t i c a l l y d i s c o v e r e d (and #of course proved) by computer. The d i s c o v e r y p r o g r a m s t a r t e d w i t h 3 #generic p o i n t s in the plane, and i t e r a t i v e l y c o n s t r u c t e d new points, lines, #and circles u s i n g a few primitives. W h e n e v e r a n e w point (or line, or circle, #or whatever) c o i n c i d e d w i t h an old one, a ~theorem" was born. T h e n a search *Downloaded

64

from

the

Future

by

Doron

THE MATHEMATICALINTELLIGENCER9 lg99 SPRINGER-VERLAG NEW YORK

Zeilberger

([email protected])

#in Eric's M a t h T r e a s u r e Trove, v e r s i o n 1998, was performed, to see w h i c h of #the theorems that w e r e d i s c o v e r e d b y the D i s c o v e r y P r o g r a m w e r e a n t i c i p a t e d #by humans; the p r o g r a m then a u t o m a t i c a l l y a t t a c h e d the human names #to the theorems. Not surprisingly, all the theorems that t u r n e d out to #be a n t i c i p a t e d by humans, and that are p r e s e n t e d in this v e r y e l e m e n t a r y #textbook, were of v e r y low c o m p l e x i t y and depth. # ##HOW #

TO USE

THIS

TEXTBOOK:

#You don't have to read it cover-to-cover. Pick any t h e o r e m #in Part I, and then look up, in Part II, the d e f i n i t i o n s u s e d in the # s t a t e m e n t - p r o g r a m of that theorem. #These definitions, in turn, m a y involve other definitions, etc. #Don't worry, none of the d e f i n i t i o n s is circular. ## ##Example: Look up N a p o l e o n ' s Theorem. It involves two definitions: # I t I s E q u i and CET. L o o k up ItIsEqui. It involves DeSq. DeSq is #primitive. N o w look up CET. It uses Circumcenter. C i r c u m c e n t e r involves #the p r i m i t i v e d e f i n i t i o n s Ce and Center. Hence to u n d e r s t a n d the s t a t e m e n t #of N a p o l e o n ' s t h e o r e m you only n e e d to look up the definitions: Ce, Center, #CET, Circumcenter, D e S q and ItIsEqui, and get a c o m p l e t e l y s e l f - c o n t a i n e d # s t a t e m e n t of the s o - c a l l e d N a p o l e o n Theorem. Then to prove A t , get into ' #Maple by typing: ~maple" (without the quotes); once inside Maple, #type: ~read text;" (without the quotes), and then ~ N a p o l e o n ( ) ; < C R > " # ( w i t h o u t the quotes). You should i m m e d i a t e l y get the r e s p o n s e of the #computer: true. ## ## ##Note: A point is r e p r e s e n t e d as a list # # r e p r e s e n t e d as a*x+b*y+c. ##WARNING: x and y are global variables! ###

of

length

2.

Lines

are

# # # N o t e From the D o w n l o a d e r (DZ): ##IMPORTANT: THIS T E X T B O O K IS ALSO A V A I L A B L E O N - L I N E AS A M A P L E P A C K A G E # # C A L L E D "RENE", F R O M h t t p : / / w w w . m a t h . t e m p l e . e d u / - z e i l b e r g / # # ( c l i c k on "Maple p a c k a g e s and programs", then click on "RENE"). ##This t e x t b o o k was tested for M a p l e V, ver. 5 and p r e v i o u s versions. ##Unfortunately, e v e r y year or so, M a p l e comes out w i t h a new v e r s i o n # # ( b i g g e r but not always better, and sometimes buggier) ##that is not fully c o m p a t i b l e w i t h code w r i t t e n for p r e v i o u s versions. ##The p a c k a g e RENE will be c o n s t a n t l y u p d a t e d to c o n f o r m to future ##versions, but let's hope that M a p l e will start to b e c o m e u p w a r d - c o m p a t i b l e . ##RENE will also be c o n t i n u o u s l y e n l a r g e d to include n e w proofs, in p a r t i c u l a r ##of M o n t h l y p r o b l e m s and IMO p r o b l e m s (it a l r e a d y has a few now). ##########################Part AreaFormula:=proc()

local

I:

THEOREMS############

A,B,C:

ItIsZero(DeSq(A,B)*DeSq(C,Ft(C,Le(A,B)))/4-AREA(A,B,C)^2):end: B r i a n c h o n : = p r o c ( ) local Li,t,i,c,d,P: for i from 0 to 5 do L i [ i ] : = T a n g e n t T o E l l i p s e ( c , d , t [ i ] ) : od: for i from 0 to 5 do P [ i ] : = P t ( L i [ i ] , L i [ i + l m o d 6]): od: Concurrent(Le(P[0],P[3]),Le(P[l],P[4]),Le(P[2],P[5])):end: B u t t e r f l y : = p r o c ( ) local P , t , i , R , L i , M , X , Y : f o r i from 1 to 4 do P[i]:=ParamCircle([0,0],R,t[i]) od:M:=Pt(Le(P[l],P[3]),Le(P[2],P[4])): Li:=PerpPQ([0,0];M):X::Pt(Le(P[l],P[4]),Li):Y::Pt(Le(P[2],P[3]),Li): ItIsZero(DeSq(M,X)-DeSq(M,Y)):end:

VOLUME 21, NUMBER 3, 1999

65

CentroidExists:=proc() local A,B,C: Concurrent(Le(MidPt(A,B),C),Le(MidPt(A,C),B)

Le(MidPt(B,C),A)):end:

Ceva:=proc() local A,B,C,O,D,E,F: A:=[0,0]: B:=[I,0]: D:=Pt(Le(B,C),Le(A,O)):E:=Pt(Le(A,C),Le(B,O) :F:=Pt(Le(A,B),Le(C,O)):ItIsZero( DeSq(B,D)*DeSq(C,E)*DeSq(A,F)-DeSq(D,C)*DeSq(E,A)*DeSq(F,B)): end: Desargues:=proc() local A,B,i,m,t,s: for i from 1 to 3 do A[i]::ParamLine(m[i],0,t[i]): B [ i ] : : P a r a m L i n e ( m [ i ] , 0 , s [ i ] ) : o d : Colinear(Pt(Le(A[I],A[2]),Le(B[I],B[2])),Pt(Le(A[I],A[3]),Le(B[I],B[3])), Pt(Le(A[2],A[3]),Le(B[2],B[3]))):end: EulerLineExists:=proc()

local

A,B,C:

Colinear(Orthocenter(A,B,C),Circumcenter(A,B,C),Centroid(A,B,C)):end: EulerTetrahedronVolumeFormula:=proc() local PI,P2,P3,P4,P,Q,R,A,B,C,Vol, p31,p32,p41,p42,p43: with(linalg):Pl:=[0,0,0]: P 2 : = [ l , 0 , 0 ] : P 3 : = [ p 3 1 , p 3 2 , 0 ] : P4:=[p41,p42,p43]:P:=DeSqG(PI,P4,3):Q:=DeSqG(P2,P4,3):R:=DeSqG(P3,P4,3): A:=DeSqG(P2,P3,3):B:=DeSqG(P3,PI,3):C:=DeSqG(PI,P2,3): VoI:=AREA([p31,p32], [i,0], [ 0 , 0 ] ) * p 4 3 / 3 : e v a l b ( n o r m a l ( d e t ( a r r a y ( [ [ 0 , P , Q , R , l ] ,

[P,O,C,B,I], [Q,C,O,A,I], [R,B,A,O,I], [l,l,l,l,O]]))/Vol^2/288)=l):end: EulerTriangleFormula:=proc() local T,m,n,A,B,C,d,R,r,O, II: T:=Te(m,n):A:=T[l]:B:=T[2]:C:=T[3]:O:=Incenter(m,n):

Ii:=Circumcenter(A,B,C):r:=Inradius(m,n):R:=Circumradius(A,B,C): d:=sqrt(DeSq(O, Ii)): Feuerbach:=proc()

ItIsZero((d^2-R^2)^2-4*r^2*R^2):end:

local m , n : T o u c h C e ( N i n e P o i n t C i r c l e ( T e ( m , n ) )

Incircle(m,n)):end:

FoxTalbot:=proc() local q,L,i,a,b,c,j,M: L[l]:=x: for i from 2 to 5 do L[i] :: a[i]*x+b[i]*y+c[i]: od: for i from 1 to 5 do q : = Q u a d ( s e q ( L [ j ] , j = l . . i - 1 ) , s e q ( L [ j ] , j : i + l . . 5 ) ) :M[i] ::Le(MidPt(q[l] ,q[3]) , MidPt(q[2],q[4])) :od: Concurrent(seq(M[i] ,i:l..5)) :end: Herron:=proc() local q , a , b , c , s , A , B , C , b l , a 2 , b 2 , A r e a : A : = [ 0 , 0 ] : B : = [ 0 , b l ] : C:=[a2,b2]:a:=sqrt(DeSq(B,C)):b:=sqrt(DeSq(A,C)):c:=sqrt(DeSq(A,B)): s:=(a+b+c)/2: I t I s Z e r o ( A R E A ( A , B , C ) ^ 2 - s * ( s - a ) * ( s - b ) * ( s - c ) ) : e n d : IncenterExists:=proc() local m,n,A,B,C,T:T:=Te(m,n) :A:=T[I] :B:=T[2] : C:=T[3] : C o n c u r r e n t ( y - m * x , y + n * x - n , y - C [ 2 ] - ( x - C [ l ] ) * T S ( m , i / n ) ) :end: Johnson:=proc() local C,t,i,R,P: for i from 0 to 2 do C[i]:=ParamCircle( [0,0],R,t[i]): od: for i from 0 to 2 do P [ i ] : : M i r R e f O f 0 ( C [ i ] , C [ i + l mod 3]) od: I t I s Z e r o ( R a d i u s ( C e ( P [ 0 ] , P [ l ] , P [ 2 ] ) ) ^ 2 - R ^ 2 ) : end: Lehmus:=proc() local N , M , m , n : N : = P t ( y - T S ( m , m ) * x , y + n * ( x - 1 ) ) : M:=Pt(y-m*x,y+TS(n,n)*(x-1)) :factor(DeSq([l,0],N)-DeSq([0,0],M)) :end: Menelaus:=proc() local A,B,C,X,Y,Z,L,m,b: L:=y-m*x-b: X:=Pt(Le(B,C) ,L) : Y:=Pt(Le(A,C) ,L) :Z:=Pt(Le(A,B),L) :ItIsZero( D e S q ( B , X ) * D e S q ( C , Y ) * D e S q ( A , Z ) - D e S q ( C , X ) * D e S q ( A , Y ) * D e S q ( B , Z ) ) : end: Morley:=proc() local m,n,A,B,C,D,E,F:A:=[0,0] :B:=[I,0] : C: =Pt (y-TS (m,m,m) *x,y+TS (n, n,n) * (x-l)) :D: =Pt (y-m*x,y+n*x-n) : E::Pt(y-TS(m,m)*x,y-C[2]-(x-C[l])*TS(m,m,-n,sqrt(3))) : F: =Pt (y+TS (n, n) * (x-l) ,y-C [2 ] + (x-C [i] ) *TS (n, n, -m, sqrt (3)) ) :ItIsEqui (D, E, F) :end: Napoleon:=proc()

local

A,B,C:

ItIsEqui(CET(A,B) ,CET(B,C),CET(C,A)) : end:

N i n e P o i n t C i r c l e E x i s t s : : p r o c ( ) local A , B , C , O , D , E , F , G , H , I , K , L , M : D:=Ft(A, L e ( B , C ) ) : E : = F t ( B , L e ( A , C ) ) : F : = F t ( C , L e ( A , B ) ) : G:=MidPt(A,B):H:=MidPt(A,C):I:=MidPt(B,C):O:=Orthocenter(A,B,C):

THE MATHEMATICALiNTELLIGENCER

K:=MidPt(O,A):L:=MidPt(O,B):M:=MidPt(O,C):Concyclic(D,E,F,G,H,I,K,L,M):end: OrthocenterExists:=proc()

local A,B,C:

Concurrent(Altitude(A,Le(B,C)),Altitude(B,Le(A,C)),Altitude(C,Le(A,B))):end: Pappus:=proc() local t,s,m,b,ml,bl,i,P,Q: for i from 1 to 3 do-P[i]:=ParamLine(m,b,t[i]): Q[i]:=ParamLine(ml,bl,s[i]): Colinear(Pt(Le(P[l],Q[2]),Le(P[2],Q[l])),Pt(Le(P[l],Q[3]),Le(P[3],Q[l])), Pt(Le(P[2],Q[3]),Le(P[3],Q[2]))):end:

od:

Pascal:=proc() local c,d,s,t,i,P,Q: for i from 1 to 3 do P[i]:=ParEllipse( c,d,t[i]) :Q[i] :=ParEllipse(c,d,s[i]) :od:Colinear(Pt(Le(P[l],Q[2]) ,Le(P[2],Q[I])), Pt(Le(P[I],Q[3]),Le(P[3],Q[I])),Pt(Le(P[2],Q[3]),Le(P[3],Q[2]))) :end: Poncelet:=proc() local t0,tl,t2,t3,A,B,C,t,R,Q,QI,Q2,P,c,LI,L2,L3,eq: print('Works, on Maple V, 0-5, except ver.3"): Ll:=TangentToEllipse([0,0], [l,l],tl):L2::TangentToEllipse([0,0], [l,l],t2): L3:=TangentToEllipse([0,0], [l,l],t3):Ll:=subs(tl=l,Ll):A:=Pt(Li,L2):

B:=Pt(Li,L3):C:=Pt(L2,L3):R:=Circumradius(A,B,C):c:=Circumcenter(A,B,C): R:=simplify(R,symbolic): P:=ParamCircle(c,R, t0):Q:=ParamCircle(c,R,t): eq:=solve(TouchCeLel(x^2+y^2-l,Le(P,Q)),t):Ql:=subs(t=eq[l],Q): Q2:=subs(t=eq[2],Q):TouchCeLe(x^2+y^2-l,Le(Qi,Q2)):end: Ptolemy:=proc() local P,i,t,R: for i from 1 to 4 do P[i]:=ParamCircle([0,0],R,t[i]): od: Sqabc(DeSq(P[l],P[2])*DeSq(P[3],P[4]), DeSq(P[2],P[3])*DeSq(P[4],P[l]),DeSq(P[l],P[3])*DeSq(P[2],P[4])):end: Simson:=proc() local t,i,P,R:for i from 1 to 4 do P[i]::ParamCircle([0,0],R,t[i]): od: Colinear(Ft(P[4],Le(P[l],P[2])), Ft(P[4],Le(P[2],P[3])) ,Ft(P[4] ,Le(P[3],P[l]))) :end: Soddy:=proc()

local q,el,e2,e3,e4,c,d,e,TC,R,r,s,t,p:with(grobner):

R%=l:TC:=TcCesOut:c:=[r+R,O]:q:=gbasis({TC(c,r,d,s),TC(c,r,e,t),TC(d,s,e,t), TC([0,0],R,c,r),TC([0,0],R,d,s),TC([0,0],R,e,t)}, [d[l],e[l],d[2],e[2],r,s,t], tdeg):e4:=i/R:el:=I/r:e2:=i/s:e3:=i/t: p:=-2*(el^2+e2^2+e3^2+e4^2)+(el+e2+e3+e4)^2:p:=numer(nor~al(p)): ItIsZero(normalf(p,q, [d[l],e[l],d[2],e[2],r,s,t],tdeg)):end: ########################PART

II:DEFINITIONS##########################

#Def (The perpendicular to line Lel that passes through point Ptl) Altitude:=proc(Ptl,Lel): expand(coeff(expand(Lel),x,l)*(y-Ptl[~])coeff(expand(Lel),y,l)*(x-Ptl[l])): end: #Def (Area of triangle ABC) AREA::proc(A,B,C) :normal (expand((B[I]*C[2]-B[2]*C[I]-A[I]*C[2]+A[2]*C[I] -B[I]*A[2]+B[2]*A[I])/2)) :end: #Def (The Circumcircle of the input points) Ce:=proc() local eq,a,b,c,i,q:eq:=x^2+y^2+a*x+b*y+c:q:=solve({seq(subs( {x:args[i] [l],y=args[i] [2]},eq),i:l..nargs)} ,{a,b,c}):expand(subs(q, eq)):end: #Def

(The center of a circle Circ)

Center:=proc(Circ):[-coeff(expand(Circ),x,l)/2,-coeff(expand(Circ) #Def

y,l)/2]:end:

(The intersection of the three medians)

Centroid:=proc(A,B,C):Concurrency(Le(MidPt(A,B),C),Le(MidPt(A,C),B Le(MidPt(B,C),A)):end: #Def B)

(Circumcenter of the equilateral

triangle

two of whose vertices

are A and

VOLUME21, NUMBER3, 1999 6 7

CET:=proc(A,B) :Circumcenter(A, B, [(B[I]+A[I])/2-(A[2]-B[2])*3^(I/2)/2, B[2]/2+(A[I]-B[I])*3^(I/2)/2+A[2]/2]):end: #Def (The circumcenter of the triangle ABC) Circumcenter:=proc(A,B,C):Center(Ce(A,B,C)):end: #Def "(Circumradius of the triangle ABC) Circumradius:=proc(A,B,C):Radius(Ce(A,B,C)):end: #Def (Are the input points all on the same line?) Colinear:=proc() local i: if nargs<2 then ERROR('Need at least two Pts'): fi: for i from 3 to nargs if AREA(args[l],args[2],args[i])<>0 then RETURN(false):fi:od:true:end: #Def (The common point of the input lines) Concurrency:=proc() local q: q:=solve({args),{x,y}):[subs(q,x),subs(q,y)]: #Def (Are t h e input lines concurrent?) Concurrent:=proc(): not evalb(solve({args},{x,y}):NULL): #Def

end:

(Are the

input points all on the same circle?) local i,Cl:Cl:=Ce(args[l],args[2],args[3]): for i from 4 to nargs do if Cl
Concyclic:=proc()

true:

end:

#Def (Square of distance of point P to line L) DePtLeSq:=proc(P,L):DeSq(Pt(Altitude(P,L),L),P):end: #Def (Square of the distance of points A and B) DeSq:=proc(A,B):(A[I]-B[I])^2+(A[2]-B[2])^2: end: #Def (Square of the distance of points A and B, in dim-dimensional DeSqG:=proc(A,B,dim) local i :sum((A[i]-B[i])^2,i:l..dim): end: #Def

(The line

through

Circumcenter,

Orthocenter,

space)

and Centroid

EulerLine:=proc(A,B,C):Le(Orthocenter(A,B,C),Circumcenter(A,B,C)

:end:

#Def (Projection of point P~tl on line Lel) Ft:=proc(Ptl,Lel) :Pt(Altitude(Ptl,Lel),Lel):end: #Def (The incenter of the triangle with vertices A(0,0), B(I,0 , and with #slopes of AC and BC equal TS(m,m) and TS(n,n), resp.) Incenter:=proc(m,n) local C: C::Te(m,n) [3]: if m=n then [i/2,m/2] :else Concurrency(y-m*x,y+n*x-n,y-C[2]-expand((x-C[l])*TS(m,I/n))) fi:end: #Def (The eq. of the incircle of the standard triangle) Incircle:=proc(m,n) local C,R:R:=Inradius(m,n):C:=Incenter(m,n): expand((x-C[l])^2+(y-C[2])^2-R^2):end: #Def (The inradius of the standard triangle) Inradius:=proc(m,n) local A,B,C,T,O: T:=Te(m,n):A:=T[l]:B:=T[2]:C:=T[3]:O:=Incenter(m,n):sqrt(normal( {DePtLeSq(O,Le(A,B)),DePtLeSq(O,Le(A,C)),DePtLeSq(O,Le(B,C))}) [l]):end: #Def (Is the triangle ABC equilateral?) ItIsEqui:=proc(A,B,C):evalb(normal ({DeSq(A,B)-DeSq(A,C),DeSq(B,C)-DeSq(C,A)}):{0)):end: #Def (Is it zero?) ItIsZero:=proc(a):evalb(normal(a)=0):end: #Def (The eq. of the line joining Le:=proc(A,B) AREA(A,B, [x,y]) :end:

68

THE MATHEMATICALINTELLIGENCER

A and B)

do

end:

#Def (The midpoint between A and B) MidPt:=proc(A,B):[(A[l]+B[l])/2,(A[2]+B[2])/2]:end: #Def (Mirror reflection of the origin w.r.t, the line AB) MirRefOf0:=proc(A,B) local q: q:=Ft([0 0],Le(A,B)): [2*q[l],2*q[2]]:end: #Def (Mirror reflection of the origin w.r.t, the line Le) MirRefPtLe:=proc(P,l) local q: q:=Ft([0,0],subs({x=x+P[l],y=y+P[2]},l) : [2*q[l]+P[l],2*q[2]+P[2]]:end: #Def

(Euler's nine-point

circle for triangle ABC) Ce(MidPt(A,B),MidPt(A,C),MidPt(B,C)):end:

NinePointCircle:=proc(A,B,C): #Def

(The intersection

of the three perpendicular projections)

Orthocenter:=proc(A,B,C):Concurrency(Altitude(A,

Le(B,C)),Altitude(B,Le(A,C)),

Altitude(C,Le(A,B))):end: #Def

(Generic point

on a parametric

circle center

[c[i],c[2]]

and radius R)

ParamCircle:=proc(c,R,t):[c[l]+R*(t+i/t)/2,c[2]+R*(t-i/t)/2/I]:end: #Def

(Generic point

on a parametric

ellipse,

center

[c[i],c[2]])

ParEllipse:=proc(c,d,t):[c[l]+d[l]*(t+i/t)/2,c[2]+d[2]*(t-i/t)/2/I]:end: #Def (Generic point on a parametric line) ParamLine:=proc(m,b,t):[t,m*t+b]:end: #Def (Line through midpoint of PQ perpendicular PerpMid:=proc(P,Q):PerpPQ(P,MidPt(P,Q)):end:

to PQ)

#Def (Line through Q perpendicular to PQ) PerpPQ:=proc(P,Q):expand((y-Q[2])*(P[2]-Q[2])+(x-Q[l])*(P[l]-Q[l])):end: #Def (The point of intersection of lines Lel and Le2) P~:=proc(Lel,Le2) local q:q:=solve( {numer(normal(Lel)),numer(normal(Le2))},{x,y}): [normal(simplify(subs(q,x))),normal(simplify(subs(q,y)))]:end: #Def (Quadrilateral through four lines LI,L2,L3,L4) Quad:=proc(LI,L2,L3,L4):Pt(LI,L2),Pt(L2,L3),Pt(L3,L4),Pt(L4,LI):end: #Def

(The radius

of a circle Circ)

Radius:=proc(Circ) local q: q:=Center(Circ):sqrt(normal(subs({x=q[l],y=q[2]},-Circ))):end: #Def

(The slope of the

line joining points A and B)

Slope:=proc(A,B):normal((B[2]-A[2])/(B[l]-A[l])):end: #Def

(Is it true that

sqrt(a)+sqrt(b)=sqrt(c)

?)

Sqabc:=proc(a,b,c):ItIsZero((c-a-b)^2-4*a*b):end: #Def (Given a circle Cel and a point Ptl on it, find the eq. of the tangent) Tangent:=proc(Cel,Ptl) local A, x0,y0: A:=coeff(Cel,x,2):x0:=Ptl[l]: y0:=Ptl[2]: numer(normal((y-y0)*(2*A*y0+coeff(Cel,y,l))+(x-x0)*(2*A*x0+coeff(Cel,x,l)))): end: #Def (Tangent to the parametric ellipse at the parametric point) TangentToEllipse:=proc(c,d,t) local P: P:=ParEllipse(c,d,t): diff(P[1],t)*(y-P[2])-(x-P[l])*diff(P[2],t):end: #Def (Do the two circles with given centers and radii touch?) TcCesOut:= proc(CI,Ri,C2,R2):expand((Rl+R2)^2-(Cl[!]-C2[l])^2-(Cl[2]-C2[2])^2):end:

VOLUME 21, NUMBER 3, 1999 6 9

#Def (Standard triangle whose vertices are A(0,0) and B(I,0) #CAB and CAB given) Te:=proc(m,n):[0,0], [l,0],Pt(y-TS(m,m)*x,y+TS(n,n)*(x-l)):end:

and with angles

#Def (Given two circles C1 and C2, do they touch?) TouchCe:=proc(Ci,C2) local gu: gu:=expand(subs(y=solve(Cl-C2,y),Cl)): ItIsZero(4*coeff(gu,x,2)*coeff(gu,x,0)-coeff(gu,x,l)^2): end: #Def (The expression whose vanishing guarantees that the symbolic #circles touch) TouchCel:=proc(Cl,C2) local gu: gu:=expand(subs(y=solve(Cl-C2,y),Cl)): numer(normal(4*coeff(gu,x,2)*coeff(gu,x,0)-coeff(gu,x,l)^2)): end: #Def

iGiven a circle Cl and a line LI, do they touch?) local gu: gu:=expand(subs(y=solve(Ll,y),Cl)): ItIsZero(4*coeff(gu,x,2)*coeff(gu,x,0)-coeff(gu,x,l)^2): end:

TouchCeLe:=proc(Cl,Ll)

#Def (The expression whose vanishing guarantees that the symbolic #circle Cl and line L1 touch) TouchCeLel:=proc(Cl,Ll) local gu: gu:=expand(subs(y=solve(Ll,y),Cl)): numer(normal(4*coeff(gu,x,2)*coeff(gu,x,0)-coeff(gu,x,l)^2)): end: #Def (Tangent of a sum of given angles) TS:=proc(li) local i,t:if nargs=l then RETURN(args[I]) else t:=TS(seq(args[i],i=2..nargs)): RETURN((args[l]+t)/(l-t*args[l])):fi:end:

7~

THE MATHEMATICALINTELUGENCER

II:,:a,A[:a,,,J-'~ J e t W i m p ,

Editor

I

Feynman's Lost Lecture; The Motion of Planets Around the Sun by David L. Goodstein and Judith R. Goodstein NEW YORK: W. W, NORTON & COMPANY, INC., 1996. 224 pp. US $35.00, ISBN 0-39303-918-8 REVIEWED B Y ROBERT WEINSTOCK

Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.

E d i t o r ' s Note: This department received two reviews of the book based on Richard Feynman's lecture about the inverse-square law. Although they draw similar conclusions about the book, they are complementary and it was always intended to publish both. The fn'st, by Graham W. Griffiths, appeared in Vol. 20, No. 3, pages 68-70. The second, giving more of the mathematical argument, appears below. We regret that its publication was delayed until now. In the meantime, the book attracted some press attention (see Los Angeles Times for 28 July 1998). The heirs of Richard Feynman sued the authors, Caltech, and the publisher, maintaining that the physicist had never ceded rights to profits from his lectures to Caltech, that they the heirs held these rights, and that monetary loss to them exceeded $500,000. n March 1964, the eminent physicist Richard Feynman presented a lecture on the inverse-square orbit problem to an undergraduate class at the California Institute of Technology. More specifically, the lecture presented a technique devised by Professor Feynman for the purpose of proving, without use of calculus, that each planet in orbit about the Sun under the sole influence of an inverse-square force toward that body must trace an ellipse having one focus at the Sun--in brief, a technique for proving Kepler's first law as a consequence of inverse-square gravitation.

I

Column Editor's address: Department of Mathematics, Drexel University, Philadelphia, PA 19104 USA.

There are at least a dozen proofs that inverse-square central attraction implies conic-section orbit, of which Kepler's first law is a special case. 1The remarkable distinguishing feature of what Feynman offered to his class is its total avoidance of the use of calculus; his method is purely geometric and algebraic. Unfortunately, it did not prove what he claimed it to prove. The English-reading world now has access to the Feynman lecture in a book by physicist David L. Goodstein, friend and colleague of the late Professor Feynman, and his historian-ofscience;, archivist wife, Judith R. Goodstein. In it, the Goodsteins present what is essentially Feymm~'s March 1964 lecture, but with many more reader-friendly explanatory details, including several more diagrams than Feynman evidently used. They also include a word-for-word transcription of Feynman's lecture as the book's fmal chapter; they recommend reading it while listening to Feynman's recorded delivery of it from the compact disc that comes in the package in which their book is sold. It is the burden of this review to show that the method devised by Professor Feynman and presented by the Goodsteins actually falls to provide a proof of Kepler's fLrst law. In what follows, the symbol "i~/~ 2means "Feynman, as conveyed by Goodstein and/or Goodstein." ( a ) After considerable rather ingenious preparatory effort--including the unmentioned use (twice) of a particular critical assumption specified below--~/~3 2 achieves a simple descriptive characterization of each planetary orbit based on the circle whose center is C and in which O is a fixed interior point, as seen in Fig. 1: Let P (not shown) be any point (except C) of a planet's orbit that lies on the ray from C that intersects the circumference at p; then, the tangent to the orbit at P is perpendicular to the segment connecting p to fixed O. This must be so what-

9 1999SPRINGER-VERLAGNEWYORK,VOLUME21, NUMBER3, 1999

71

Figure 1

ever the angle 0--i.~, for all orientations of radius Cp relative to the faxed reference line OCj. Does the tangent orientation at each point thus expressed enable one to describe a geometrical construction of the planet orbit? (fl) Yes, declares ~ / ~ 2; one does it as follows: "To get the orbit, simply construct [for a given 0] the perpendicular bisector of the line from [O] to p [which] we k n o w . . , is parallel to [the tangent] at point P on the orbit. At some point, the perpendicular bisector crosses the line connecting p to the center, C." (See Fig. 2.) This crossing is then taken to represent the orbit point P; and the perpendicular bisector of Op is tagged, in consequence of the result reported in (a) above, as the tangent at P to the curve representing the orbit. Now, continues ~/~2, ,,As the point p moves around the circle, the intersection [P] of pC and the perpendicular bisector [of Op] moves around in a curve of its own"--thereby, ~/u32 claims, "creating the orbit." This

J

Figure 2

72

THE MATHEMATICALINTELLIGENCER

is followed by the assertion that "the size of the orbit will be arbitrary, but all the directions, and therefore the shape of the orbit, will be correct." ~ / ~ 2 is speaking, one must realize, of a single orbit. (T) With the curve traced out by P as described in (/3) identified as a properly shaped representation of an orbit, ~ / ~ 2 reminds the reader that the same construction procedure that yields it is proved earlier in the book to yield an ellipse whose foci are O and C, and whose tangent at P is the perpendicular bisector of Op. Indeed, ~/u5 2 proceeds to repeat the proof that the construction performed in (/3) does yield the ellipse endowed with the specified set of tangents: the set of points the sum of whose distances from O and C has the fLxed value Cp, the radius of the given circle. "The proof is now complete," ~ / ~ 2 declares, "... we have accomplished what we have set out to show. Newton's laws, together with an R - 2 force of gravity toward the sun, result in elliptical orbits for the planets." ~2, at least, regard the achievement as an "heroic feat." What ~ / ~ 2 has accomplished is truly amazing,.., but . . . . As alluded to in (a), Feynman's proffered proof as expounded by the Goodsteins gives no mention of a particular significant assumption: namely, that the radius vector from the sun to the planet sweeps out a complete revolution. This silent assumption eliminates parabolic and hyperbolic orbits, which are also, in addition to ellipses, physically allowed. It depends on the algebraic sign of the planet's total energy, about which nothing can be stated within the geometric framework erected by ~;/~ 2. Yet, this is a relatively minor fault in Feynman's argument; let us grant him and the Goodsteins the unexpressed assumption that leads to the descriptive characterization of each orbit presented above in (a). Worse is to come. Let us go on to scrutinize the procedure described in (/3) as the means for "creating the orbit." The procedure does visually trace a set of points, a simple closed curve, of which each point P is the intersection of a segment Cp with the perpendicular bisector of the corresponding segment Op. A n d - -

as twice proved in the book by 0;/~ 2 the specified perpendicular bisector is tangent to the traced curve at the intersection P, and the curve is an ellipse having foci at O and C. 2 From this, ~ / ~ 2 concludes that the curve generated by the procedure described in (/3) is the elliptical orbit of the planet under the sole influence of an inversesquare attraction toward the Sun. This conceals the tacit assumption that the condition necessary for a curve to be an orbit--namely, that the tangent to it at a point lying on Cp in Fig. 1 be perpendicular to Op for each value of 0--is also sufficient for it to be an orbit. However, there exist infinitely many other curves that satisfy the necessary (and presumed sufficient) condition for their serving as orbits. It can be shown, in fact, that through each point (exclu-

Surely you're kidding, Mr. Feynman. sive of C) of the ray drawn from C through j there passes a unique orbit having its tangent at the point perpendicular to the ray. Only one of these possible orbits--the one having as its point on the ray the midpoint of Oj--is acknowledged by the ~/~32 procedure, which misses all the rest. That all of these happen to be ellipses possessing C as one focus and the unique eccentricity (OC/Cj) has no bearing on the efficacy of the present criticism. For not only does ~/u32 miss all but one of the possible orbits; the likelihood of the Feynman method's being adapted so as to achieve an appropriate identification of any one of those missed is close to zero. It can, therefore, be stated as a fact that ~/(g 2 does not, in Feynman's Lost Lecture, present a proof that closed orbits under an inverse-square attraction are ellipses. Postscripta

1. A statement by Professor Feynman in his actual lecture might be conceived as a cryptic acknowledgment of the substance of the foregoing assertion. "Therefore," he says, after observing that the procedure described in

(fl) is that which yields an ellipse, each t a n g e n t to w h i c h is the p e r p e n d i c u l a r b i s e c t o r of the a p p r o p r i a t e Op, "the solution of the p r o b l e m is an ellipse---or the o t h e r w a y around, really, is w h a t I proved: that the ellipse is a p o s s i b l e solution to the problem 9 And it is this solution." But then he continues, w i t h o u t an intervening word, to a s s e r t this non sequitur: "So the orbits are e l l i p s e s . " - with no further explanation. 2. In their Epilogue, the G o o d s t e i n s write, "Richard F e y n m a n c o n j u r e d up his o w n brilliant p r o o f 9 b u t he was n o t t h e first to think of it. The s a m e p r o o f 9 a p p e a r s in a little b o o k called Matter and Motion, w r i t t e n b y J a m e s Clerk Maxwell and first published in 187793 Maxwell attributes the m e t h o d of p r o o f to Sir William Hamilton." F e y n m a n ' s purported proof, however, does not a p p e a r in Matter and Motion9 It is the converse--elliptical orbit with force directed t o w a r d one focus implies inverse-square force m a g n i t u d e - - t h a t Maxwell p r o v e s using the ellipse properties e x p o u n d e d b y ~ / ~ 2 . Furthermore, Maxwell d o e s not a t t r i b u t e the m e t h o d of his p r o o f to Hamilton. He m e n t i o n s m e r e l y that it was Hamilton 4 w h o i n t r o d u c e d the h o d o g r a p h - - " t h e p a t h t r a c e d out b y the e x t r e m i t y of a v e c t o r which continually represents, in d i r e c t i o n and magnitude, t h e velocity o f a moving b o d y " - - a s a m e t h o d of investigating the m o t i o n o f a body. The G o o d s t e i n s also attribute the hodog r a p h i n t r o d u c t i o n to Hamilton. (The circle a p p e a r i n g in all the figures of this r e v i e w is arrived at by ~ / ~ 2 as the h o d o g r a p h - - i . e . , the trace in its velocity p l a n e - - o f the planet.) NOTES AND REFERENCES

1. The earliest solution of the inverse-square orbit problem was evidently constructed by John Keill and published in the 1708 volume of the Philosophical Transactions. The latest published solution, so far as I am aware, can be found in Robert Weinstock, "Inverse-square orbits: Three little-known solutions and a novel integration techSique,"

Am. J. Phys. 60 (1992) 615-619. 2. If, instead of erecting the perpendicular to Op at its midpoint in Fig. 1 (as does ~/q32), we constructed the perpendicular to Op at, say, 0.75 of its length from O, and considered the point P' at which this perpendicu-

lar intersects Cp, then, as 0 increases from 0 to 2~-, the intersection P' would still trace a closed smooth curve. However, you will find the tangent to this curve at each P' is not (except for 0 = 0 and 0 = ~r) perpendicular to the segment Op. Only the choice of the halfway point P works. 3. J. Clerk Maxwell, Matter and Motion, D. Van Nostrand, New York (1878), Chap. VIII; reprinted from Van Nostrand's Magazine. 4. T.L. Hankins, Sir William Rowan Hamilton, Johns Hopkins University Press, Baltimore (1980), pp. 326-333. Department of Physics Oberlin College Oberlin, OH 44074 USA e-mail: [email protected]

Challenges by Serge Lang NEW YORK: SPRINGER, 1998, ix + 816 pp, US $34.95, ISBN 0-387-94861-9

REVIEWED BY STEVEN G. KRANTZ

id y o u k n o w that the s u b j e c t o f m a t h e m a t i c s has its o w n Don Quixote? But o u r Don d o e s n o t w a s t e his time tilting at windmills. He is m o r e likely to d y n a m i t e y o u r sluice o r to lace y o u r mill with plastique. Our Don Quixote h a s no Sancho Panza; he is a genuine single c o m b a t warrior. And he d o e s not m e r e l y engage one goliath at a time; he t a k e s on entire p h a l a n x e s of foes 9 He is Serge Lang, and if he is p r e p a r i n g a file on y o u then you h a d b e t t e r l o o k out. Serge Lang is a t o p - n o t c h mathematician, a m e m b e r o f the National A c a d e m y o f Sciences, and a P r o f e s s o r at Yale. He is a prolific w r i t e r of b o o k s a n d papers. (At last count, t h e r e w e r e 34 m a t h b o o k s and three b o o k s on politics; m a n y o f the b o o k s have b e e n through multiple editions. He has even written a calculus book! And he has written m o r e than 85 papers 9 But he is also a political warrior. By his o w n telling:

D

I don't like the nonsense that passes f o r rational discourse so often in our society. I a m very much bothered by

the inaccuracies, ambiguities, code words, slogans, catch phrases, public relation devices, sweeping generalizations, and stereotypes, which are used (consciously or otherwise) to influence people. I a m bothered by the w a y people fudge issues, or are unable to clarify them, sometimes because they are inhibited by "collegiality" and other f o r m s of intimidation (sometimes subtle, sometimes not). I a m bothered by the misinformation which is created and disseminated uncritically through the educational system and through the media, and by the obstructions which prevent correct information f r o m being disseminated. These obstructions come about i n m a n y ways--personal, institutional,, through self-imposed inhibitians, through extern'al inhibitions, through incompetence, and sometimes through outright dishone s t y - t h e list is a long one. I do not claim that the creation of false information and its dissemination is due to "evil" intent or to a conspiracy. I am not concerned with such motivation 9 In some cases, I simply note that some people give no evidence that they can tell the difference between a fact, a perception of a fact, an opinion, and what is none of the above9 In a n o t h e r p a s s a g e Lang asks,

9 . . how f a r do we submit to the higher-ups or how f a r do we press our in~)estigation of the higher-ups to make them accountable, possibly in the face of evasions, stonewalling and intimidation? Serge goes on, a n d it is wonderful. Surely e a c h and e v e r y one o f us recognizes here a c h a i r m a n o r a dean o r a m a y o r o r an a c q u a i n t a n c e o r an adversary. Serge h a s got the h u m a n condition nailed d e a d to rights. I a d m i r e him. I a d m i r e his courage, I admire his unflagging honesty, I a d m i r e his herculean energy a n d dedication, and I can s a y with s o m e c e r t a i n t y that he is usually on the side o f the angels. The b o o k u n d e r review is a comp e n d i u m of b a t t l e s that Serge has

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fought. Some he has won, and some he has lost, and for some the jury is still out. Most of us are not afraid to engage in gentle combat with colleagues in our own department; some of us have the chutzpah to fight with other mathematicians on the national scene. But Serge Lang's sword knows no bounds. He fights with members of college administrations, with high-ranking members of the medical establishment, with distinguished social scientists, with Nobel Laureates in the life sciences, with esteemed nabobs of AIDS researclt, and with the entire National Academy of Sciences. I like a good fight myself; but my physicians will only allow me one per decade. Serge Lang usually has three or four pitched battles going at any one time. Given anyone who doesn't fit the usual mold, who is a muckraker, who jousts with pomp and pretension, there will always be those who dismiss him with facile criticism: "what does he know about medicine?" "It is well established that HIV causes AIDS." "If he spent more time on his mathematics then he might be better at it." More fools they. This is who Serge Lang is. He has the energy of three hardy souls, and he applies those energies to three separate careers. Criticize Lang ff you will, but he does his homework. He usually quotes only primary sources, and he sticks to the facts. Many an adv e r s a r y - f r o m Robert Gallo to Samuel Huntington--has squirmed under Lang's intense and unyielding scrutiny. Serge Lang holds us to a higher standard of honesty than the comfortable one to which most of us have become accustomed. It can be quite disconcerting to spend time with Serge Lang. But he usually knows what he is talking about. For those unfamiliar with the three decades of political feuding that Serge Lang has conducted, it should be said that Lang's lingua f r a n c a is the 'Tile." He prefers that his interlocutors provide their statements in writing, over their signatures. This insistence is partly for the purpose of verifiability, and partly because Lang wants to put their words into one of his files. A La~g file is frequently 1" or more thick. It contains voluminous correspondence,

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documentation, vituperation, reports, newspaper clippings, and anything else that might bear on the issue at hand. To read a Lang file straight through is at once a vicarious experience (as though the reader himself has conducted the battle) and a long, tough haul. But the reader needs only to read it; Lang has had to live through it. To give a more complete picture of what Lang is about, I will now provide a sketch of some of Lang's principal fights. I give the first two in some detaft, in part because they are special to Lang. I shall treat the others only briefly.

The Huntington File In Lang's words, "The present book is an offshoot of the Huntington file." Samuel P. Huntington is a very distinguished political scientist and a professor at Harvard University. Huntington's books are frequently used in political science classes, even at Lang's institution Yale. Huntington is a past president of the American Political Science Association and chair of the Government Department at Harvard. Lang asserts that many of Huntington's studies amount to "political opinions passed off as science." He demonstrates that Huntington shows "an appalling lack of historical knowledge, which amounts to gross professional incompetence..." He goes on to cite "defective scholarship." when Huntington was nominated to the National Academy of Sciences in 1986 (and again in 1987), Lang conducted a successful campaign to oppose Huntington's induction into that august organization. You can imagine that Lang has raised some hackles in political science circles, what is the basis for Lang's opposition to and criticism of Samuel P. Huntington? A partial answer lies in more than 200 pages of the book under review, and that material is but a subset of Lang's full file on Huntington. Here I can only hint at the issue. Huntington's famous book Political Order i n Changing Societies (Yale

University Press, 1968) studies "systematic frustration" and "political instability". In the book, Huntington an-

alyzes an underdeveloped country by using the following three equations

(1)

(2)

(3)

Social mobilization Economic development

Social = frustration

Social frustration Mobility oppor~nities

Political participation

Political participation = Political institutionalization

Political instability

As a result of his analysis Huntington concludes that, in the early 1960s, South Africa was a "satisfied society." To amplify this conclusion, Huntington notes elsewhere that there were "no major riots, strikes, or disturbances [in South Africa at that time]." It is not difficult to imagine how a critical thinker could have a field day with this material. Let A B C D E F G

= Social m o b i l i z a t i o n = E c o n o m i c development = Social f r u s t r a t i o n = M o b i l i t y opportunities = Political p a r t i c i p a t i o n = Political i n s t i t u t i o n a l i z a t i o n = Political i n s t a b i l i t y

Then elementary manipulation of the given equations shows that A =B .D .F.G.

What does this mean? What are the units used for each of these quantities? How does one establish the three equations (1), (2), (3)? By his own telling, Lang created a collage of more than 50 pages of New York T i m e s articles which report on riots, strikes, police firings on crowds, and other disturbances in South Africa in the 1950s and on up to 1960. Among the most disturbing of these is the "Sharpeville massacre" in which 50 innocent civilians were killed when police fired on a crowd. Lang expresses particular concern that Huntington has served as an advisor to the United States government. One wonders whether a person who conducts these sorts of analyses should have that kind of responsibility.

The Ladd-Lipset File The study "A Survey of the American Professoriate," conducted in 1977, became one of Serge Lang's biggest projects. In fact the book The F/le, also published by Springer-Verlag, docktments Lang's campaign against the Ladd-Lipset survey. The survey, conducted by Everett C. Ladd of the University of Connecticut and Seymour Martin Lipset of Stanford University, professed that

The primary, reason for this faculty survey is to collect information useful to the formation of sound educational policy. I f intelligent responses are to be forthcoming, they m u s t be based on an adequate reading of faculty p r e f e r e n c e s . . . Lang's analysis of the survey is predicated on an interesting point, important in its own right: The way that a question is formulated will strongly influence the way that it is answered. I will now present a cluster of questions from the Ladd-Lipset survey. The respondent was asked to choose from among three answers.

Question 3: The statements below " relate to teaching and student performance. Does each correctly reflect your personal judgment? (1) Definitely yes. (2) Only partly. (3) Defmitely no. Question ( 3 a ) The students with whom I have close contact are seriously underprepared in basic skills--such as those required for written and oral communication. Question ( 3 b ) "Grade inflation" is a serious academic standards problem at my institution. Question ( 3 c ) American higher education should expand the core curriculum, to increase the number of basic courses required of all undergraduates. Lang analyzes the questions, 1 in part, as follows:

Lang Remarks on Question (3a): "In 1977, for example, there were about 900 students taking the fLrst three terms of calculus at Yale. Of these, about 150 freshmen had trouble with ninth-grade algebra. On the other hand, about 180 freshmen were qualified to take third-term (or second-year) calculus, and they made up about twothirds of the class . . . . "I have taught for 27 years. During this period there has been a substantial improvement in the mathematical skills of a large group of students. Another large group still performs disastrously in, say, ninth-grade algebra. But 27 years ago the performance of a comparable group would have been even worse. The emergence of a large number of well-prepared advanced students was the result of reforms dating back to the late 1950s . . . . While it would be useful to ask college professors their views on current high school training, and to gather concrete suggestions on how to improve it, this cannot be achieved by Question 3a, which lacks the necessary precision and is intellectually at the level of a TV panel. It can only mislead people, or be improperly interpreted, thus preventing the 'formation of sound education policy.'"

Lang Remarks on Question (3b): "The question does not deserve serious consideration because an answer can be interpreted in several ways. Suppose that a person chooses answer (3), "Definitely no." The respondent could mean that grade inflation is not a problem per se; that it is a problem per se, but not at the respondent's institution; that it is a problem at the institution, but not a serious problem; that it is a serious problem but not at the respondent's institution, etc."

Lang Remarks on Question (3c): "The question poses a generalization so sweeping that it does not make sense. Universities differ; their functions differ; there are plausible reasons for having many choices among them. Who

would want to answer a question like this one in an absolute way? What do the questioners mean by 'American higher education'? The university of Michigan? Chicago? Ohio State? Kent State? Berkeley? U. C. Riverside? I know little about the basic courses required of undergraduates at institutions other than my own . . . . How can I, lacking that knowledge, answer whether these should be increased or not? "Notwithstanding the vagueness of such questions, statistical conclusions based on them quickly find their way into the press, as in the 24 October

Newsweek:..." Lang goes on to say, "The Chronicle of Higher Education, in its 23 January issue, reported some of the recent protes~ against the survey( but took account of the detailed analysis I have made only in this comment: 'The specific criticisms of the faculty survey,' Mr. Ladd says, 'were the kind of criticisms that could be lodged against any kind of survey research.' "I do not wish to condemn all questionnaires. But if other questionnaires are subject to the same criticisms, that does not make this one any better." On the one hand, the Ladd-Lipset case does not beat one over the head. On the other hand, their study was intended to have a major impact on university life. It was heavily funded, and it has been widely quoted. The file is bound to be of interest.

The Baltimore File Lang includes in this chapter the text of an article that he published in Ethics and Behavior. He n o t e s - - a n d this is one of Lang's recurring themes---the difficulty that he had publishing the piece, and the number of times it failed publication elsewhere. Most of us are familiar--by way of the extensive press coverage given to this case---with the scandal surrounding the paper published in Cell by Thereza Imanishi-Kari, David Baltimore, and David Weaver in 1986. Post-doc

1I'm not crazy about any of these questions; I would particularly object to the use of the m-dash in the first question, to the phrase "academic standards problem" in the second, and to the use of the comma in the third.

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Margot O'Toole studied the paper and determined that the experimental data for that paper had been presented in a misleading fashion. Genuine scientific fraud was subsequently uncovered. The case g~mered particular attention because of (i) the effects that becoming a whistle-blower had on O'Toole's life and career, and (ii) the fact that Baltimore is a Nobel Laureate. Lang's analysis of the case is perhaps best summarized by Lang's own prdcis of his article: Part I gives mostly a historical background of the ?- early phases of t h e Baltimore case. Part II presents a discussion of certain scientific responsibilities based on that background, specifically: the responsibility of answering questions about one's work, and the responsibility as to whether to submit to authority. Part III summarizes the two NIH investigations. Part IV deals with the responsibilities of a Congressional Committee vis-h-vis science. Part V goes into an open-ended discussion of many issues of responsibility facing scientists, vis-hvis themselves and vis-h-vis society at large, including Congress. The list is long, and readers can look at the section and paragraph headings to get an idea of their content. Part VI deals with the factor of personal credibility and the shift at the scientific grass roots. The conclusion is an appeal to the scientific community to re-assert the traditional standards of science. Lang's outrage is certainly understandable and, to my mind, it is justified. The Gallo File This case centers around the discovery of the HIV virus in 1983/849 There ensued a controversy between Robert Gallo, head of a major laboratory at the National Institutes of Health (NIH), and scientists of the Institut Pasteur in Paris. Among the issues were (i) Who discovered this virus, (ii) How the virus was developed in a form usable for a blood test. The French subse-

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quently initiated as many as five sepa- 9 To Fund or Not to Fund, That is the rate legal proceedings in the matter. Question I find this controversy to be particu- 9 Journalistic Suppression and Manilaxly interesting. Of course it is of parapulation mount importance to find a cure for the 9 Richard Horton in the N e w York dreadful scourge of AIDS, and to see Review that it is efficiently disseminated to those who need it. But it is also clear Lang's introductory remarks to this that whoever is dubbed the "discoverer chapter are particularly telling: of the AIDS virus" will go down in his- 9 H a v i n g n a m e d this v i r u s " H I ~ ' - tory, and whoever discovers a cure will H u m a n I m m u n o d e f i c i e n c y V i r u s - become fabulously wealthy (and go contributes to m a k i n g people accept down in history as well)9 All these issues that "HIV is the cause o f A I D S . " are at play in the Gallo controversy. One However, to an e x t e n t that underquestion is whether Gallo discovered m i n e s classical s t a n d a r d s o f science, the AIDS virus; another is whether he s o m e p u r p o r t e d s c i e n t i f i c results conborrowed certain samples of virus c e r n i n g "HIV" a n d "AIDS" have been strains from the French Institute and handled by p r e s s releases, by d i s i n then represented them as his own; and f o r m a t i o n , by low-quality studies, a third is whether his subsequent appliand by s o m e s u p p r e s s i o n o f i n f o r m a cation for a patent was legitimate. tion, m a n i p u l a t i n g the m e d i a a n d In the end, and after a great deal of people at large9 I a m . . . concerned testimony and investigation, the inves9 . . w i t h s c i e n t i f i c standards, espetigation of Gallo was abandoned. But ciaUy the a b i l i t y to tell the difference emotions still run high over this case, between a fact, an opinion, a hypothand Lang tells a fascinating story. esis, a n d a hole i n the ground. A s w e shall see shortly, there does not even exist a single proper definition of The HIV and AIDS File The issue of who discovered HIV and "AIDS" on w h i c h discourse can reliThus a m o r a s s AIDS, and who deserves the patents, is ably be based . . . . one circle of questions. Another is about H I V a n d A I D S has been created. whether HIV actually causes AIDS at I f i n d i t d i f f i c u l t to w r i t e s y s t e m a t i all. This is the subject matter of cally about this m o r a s s w i t h o u t beChapter 6 of Lang's book. It is also one c o m i n g p a r t o f the morass. of his current passions. Although Lang may be in a small mi- The Shafarevich File nority in thinking that the causal con- This is the last topical chapter of the nections between HIV and AIDS have book. By Lang's own telling: not been scientifically established, he is Igor S h a f a r e v i c h is one o f the m a certainly not alone in this belief. Some j o r m a t h e m a t i c i a n s o f the century, as prominent laboratory scientists, with m u c h by h i s great p a p e r s as by h i s strong credentials in medical research, i n f l u e n c e v i a h i s s t u d e n t s a n d the have gone public with a similar conschool o f m a t h e m a t i c s he created tention, and have suffered accords i n c e the 1950s. H e w a s a d i s s i d e n t ingly--loss of grants, missed promoi n the S o v i e t Union, a n d spoke out tions, papers rejected, and so forth. throughout h i s life at t i m e s w h e n i t Lang's Chapter 6 tells this history. w a s dangerous to do so. H e received Since the story (and the contention) the a d m i r a t i o n a n d p u b l i c s u p p o r t o f will come as a great surprise--and the i n t e r n a t i o n a l m a t h e m a t i c a l comperhaps a shock--to most readers, it m u n i t y i n the 1960s a n d 1970s. makes particularly fascinating reading. A f t e r the collapse o f the Soviet Again, Lang's Table of Contents for Union, he adopted an increasingly naChapter 6 give an idea of his concerns: tionalist viewpoint, and supported cer9 HIV and AIDS: Questions of Scien- t a i n m o v e m e n t s and parties i n R u s s i a tific and Journalistic Responsibility i n a w a y w h i c h caused consternation 9 Updates, the Mess in S c i e n c e and the a m o n g those who had admired h i m Gutknecht-Shalala Exchange and supported h i m i n the p a s t . . . .

I reproduce three pieces I have written in connection with his case: m y original letter to h i m objecting to his book Russophobia,- a letter to the Council of the National Academy of Sciences objecting to the w a y this institution got involved in the SImfarevich case; and a general analysis of the meaning of membership in the ~Academy, comparing all the cases which have been discussed in this book . . . .

The Shafarevich case really tears at the human condition: Shafarevich is plainly a great scientist and teacher. Yet the anti-Semitic views contained in his book Russophobia are repugnant. Most of us are at a loss as to how to deal with such a situation. Lang lays the case out in a compelling manner, and offers some well-researched views. Conclusions Let me stress that the six "Files" that I have described here represent but a small subset of the totality of Lang's battles over the past 30 years. These are the ones that Lang chose to treat in his book, and they certainly number among the ones that have held widest interest. In his fmal seventh chapter, Laag endeavors to draw together the various cases in the book, and to enunciate some common themes. As I have listened to Serge over the past three decades, I have frequently been tempted to think that one should choose one's fights more carefully. Huntington's ideas are too easy to attack. Ladd and Lipset are not sufficiently interesting to attack. Baltimore and Gallo and HIV/AIDS are probably too important for a single individual to attempt to treat. Shafarevich is another ivory tower type, and if certain intellectuals want to play games crossing swords with him, then let them spend their time that way. I have better things to do. I could go fttrther, were I so inclined, and observe that Lang often prevails because others run out of steam long before he does. That doesn't make Lang right; it just makes him more obstinate. Sometimes, after you've locked horns with an aggressive opponent enough times, you just say "the hell with it." This seems to happen with Lang's opponents fairly frequently. But it should

be understood that, in many cases, their saying "the hell with it" amounts to stonewalling. Lang's facts, and his cases, tend to stick. They are airtight, and do not spring leaks. When people are caught red-handed, then what can they do? However, it has been cathartic for me to read this book. I now see Lang as a great teacher. Of course none of us is under any obligation to agree with Serge, nor even to listen to him. But he has done a lot of legwork for us, and he has assembled a great deal of important information. He has put himself at risk in order to do battle with some mighty cultural icons, and he has shared with us the fruits of his labors. Serge's battles, and his subsequent files, are a fascinating record of the way that institutions protect their own, of the way that highly placed people hide behind their rank, and of how information can be suppressed. Lang says repeatedly in his book that we should not take his words at face value. He gives copious additional references, and he is as open as anyone could want him to be. The pages of Challenges are laced equally with the text of attacks on Lang and texts containing praise of Lang. In no instance does Lang say at the end of a chapter, "See, I won! I was right all along." He is far too honest for that. He lets his readers draw their own conclusions. Serge Lang's goal is to educate the rest of us, and in that pursuit he has done a splendid job. Department of Mathematics Washington University St. Louis, MO 63130-4899 USA e-mail: [email protected]

The Cambridge Quintet: A Work of Scientific Speculation by John L. Casti READING, MASSACHUSETTS: ADDISON-WESLEY (HELIXBOOKS), 1998, xxiii + 181 pp. US $23.00, ISBN 0-201-32828-3 REVIEWED BY DON FALLIS A N D K A Y MATHIESEN

ohn Casti's "The Cambridge Quintet" is a work of speculative fiction somewhat in the style of the old Steve Allen show, "Meeting of Minds." In Casti's book, five famous mathematicians, scientists, and philosophers (J. B. S. Haldane, Erwin SchrSdinger, C.P. Snow, Alan Turing, and Ludwig Wittgenstein) come together to discuss whether it is theoretically possible for machines to think. As a work of fiction, the book is fun to read. Particularly enjoyable are Casti's descriptions of the personality and appearance of the various discussants. Turing is shy and stuttering. SchrSdinger is handsome and vain. The setting--a dinner in C.P. Snow's rooms at Cambridge with each chapter corresponding to a course of the meal--is attractive. The book is not as successful, however, a~ a n intellectual work. Ultimately, it becomes clear that the dialogue is to be a battle between a staunch defender of artificial intelligence (AI), Turing, and an opponent of AI, Wittgenstein, with the other characters playing a supporting role---asking questions of clarification, bringing up objections, etc. (See [6] for an actual exchange between Turing and Wittgenstein--although on a different issue.) Thus, Casti is trying to carry out t w o tasks: (1) to imagine what these five people would say to each other if they met and (2) to present the major arguments for and against the proposition that machines might think. These two tasks are not always compatible, and the attempt to combine them leads to two problems. First, it is all too clear which side Casti is on. It is difficult not to favor the consistent, timid, reasonable Turing over the inconsistent, bombastic, rabid Wittgenstein. Turing is inquisitive and seems to learn from what the others say, while Wittgenstein does not learn from the dialogue and simply shifts ground when any one of his arguments is questioned. This ultimately leads to the second problem: in an attempt to present all of the main arguments against AI, Casti has Wittgenstein present some arguments developed since his death (most notably, John Searle's

J

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"Chinese Room" argument). As we will argue below, this leads to a serious incoherency. Casti is up-front about not giving a perfectly faithful presentation of the discussants' views. He tells us in the introduction that "it should not come as a surprise to the reader that the tictional account of the hypothetical-but possible---gathering presented here will on occasion see the participants making statements that in ways depart from what we might imagine they would have said on the basis of their published works" (p. xii). Admittedly, this maY-be unavoidable if the book is not to be a series of quotes and paraphrases. However, that can not excuse Casti for providing a deeply contradictory and inaccurate presentation of Wittgenstein's views (and of the anti-AI position). It does not help to give someone extra arguments for his position ff those arguments are inconsistent with the basic assumptions of other arguments that he makes. Thus, Casti actually weakens--rather than strengthens--Wittgenstein's position by putting new arguments against AI in his mouth. At one point early on in the dialogue, Haldane thinks of Wittgenstein, "the man is a bundle of contradictions . . ." (p. 20). He may have been, but Casti's presentation makes Wittgenstein's position a bundle of contradictions. Consider Searle's Chinese Room argmnent against AI. (Casti calls it the "Hiero-glyphic room," but otherwise it is identical to Searle's argument.) Searle's Chinese Room argument is supposed to be a counter-example to the "Turing Test." In [4], Turing claimed that, ff a machine could convincingly imitate a human being, then we would have good reason to say that it thinks. Thus, we can test whether a computer thinks by having the computer play what he calls the "Imitation Game" (what is now called the "Turing Test"): A questioner sits in one room and in a separate room there is a computer. Messages are sent back and forth between the two rooms, and the questioner can ask the entity in the other room any questions she likes. If the questioner becomes convinced that the entity in the other room is a human

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THE MATHEMATICALINTELLIGENCER

being, then, according to T m ~ g , the computer thinks. In [2], Searle proposed a modified version of the Turing Test where the goal is for the questioner to become convinced that the entity in the other room understands Chinese. In Searle's version, a Chinesespeaking questioner sits in one room and in a separate room is a person who does not understand Chinese. However, this person has a book which contains a set of rules for putting together strings of Chinese symbols ("answers') in response to other strings of Chinese symbols ("questions"). Using this book, the person in the room convinces the questioner that she understands Chinese. Searle claims that, even if the person in the Chinese Room can convince the questioner that she understands Chinese, this does not mean that she really understands Chinese. According to Searle, she does not understand Chinese, because she does not have a particular kind of mental content, viz., semantic content. The person can follow the rules in the book, but she does not know what the strings of Chinese symbols that she produces mean. Similarly, the computer can follow its program, but it does not know what the strings of symbols that it produces mean. Thus, according to Searle, the computer doesn't think. As Casti has Wittgenstein conclude, "no amount of syntactic shuffling of symbols can ever give rise to semantics. And since all that Turing and his machine are capable of doing is syntactic symbol manipulation, there can be no thinking by such a machine" (p. 81). (Note that the Chinese Room argument is not supposed to show that a computer could not pass the Turing Test, but just that if it did, it would not be thinking.) What would the real Wittgenstein think of Searle's Chinese Room argument? First, Wittgenstein might not grant that there could be such a room. The existence of the Chinese Room (or the existence of a computer that could pass the Turing Test) would imply that there is a finite set of rules that when followed mechanically will allow one to speak a natural language. But in [5], Wittgenstein argued that the proper use of language cannot be captured by a set

of rules. Second, even if Wittgenstein accepted the possibility of such a room, he would not deny that the person in the room understands Chinese for the reason that Searle gives. Wittgeustein rejected the view of meaning espoused by Searle. As Casti notes earlier in the book, Wittgenstein held that "the meaning of a statement is the sum total of all the ways the statement can be used" (p. 17). In other words, meanings aren't "mental contents." And, if meaning is use, then, as long as the person in the room uses statements of Chinese correctly, she "means what she says" and this is all there is to "meaning" something. (Interestingly, Searle himself does not think that he and Wittgenstein are on the same side of this issue. In [3], Searle groups Wittgenstein with those who, like Turing, reduce thought to intelligent behavior. Searle criticizes Wittgenstein's claim that "an inner process stands in need of outward criteria" ([5], p. 153) on much the same grounds that he criticizes the Turing Test.) Thus, in attempting to provide the central arguments in the debate on AI, Casti has given Wittgeinstein arguments that directly contradict, not only what the real Wittgenstein would have said, but the other things that Casti has Wittgenstein say in the dialogue. This makes Wittgenstein's view, and the anti-AI position that he defends, incoherent. This is unfortunate because it is certainly possible to make compelling arguments against AI from a purely Wittgensteinian perspective. In fact, this is precisely what H. M. Collins does in [1]. Wittgensteinian arguments against AI could take at least two forms. First, a Wittgensteinian might argue that it is not possible for a computer to pass the Turing Test. Wittgenstein held that language is a social phenomenon. As we noted above, Wittgenstein argued that the proper use of language cannot be captured by a set of rules. Instead, the meaning of a word is determined by how those who share a way of life use it. Since the computer does not share our way of life, it would not be able to speak our language. (In fairness to Casti, he does bring up the Wittgensteinian point about ways of life. However, by failing

to make the crucial distinction between Wittgenstein's view of meaning and Searle's, he seriously damages the plansibility of the argument.) Second, a Wittgensteinian might find the whole debate over whether a computer can "think" misguided. Since meaning is use, if we want to know whether it would be correct for us to say that a computer thinks, we should carefully consider how we use the word "think." And, if we end up saying that a computer can think (or that it cannot), we will not have captured some deep metaphysical fact about the nature of thought or consciousness; we will simply have said something about the way that we use words. Now these arguments are compelling only to those sharing certain Wittgeusteinian views---hence probably not to the vast majority of the opponents of M. Thus, it is not clear that

it was really a good idea for Casti to make Wittgenstein the spokesperson for the anti-AI position in the first place. We have focussed here on Casti's presentation of Wittgeustein. However, this example is indicative of a problem with the book as a whole. The theories and arguments of the discussants are not presented with a high degree of clarity and consistency. (To some extent, this is a result of the dialogue form itseff which makes a sustained treatment of a theory rather difficult.) As a result, someone who is already familiar with these arguments is unlikely to gain any new insights here; and, for someone who is not already familiar with these arguments, this is probably not a good book to start with. REFERENCES

[1] H. M. Collins, Artificial Experts (Cambridge, Massachusetts: MIT Press, 1990).

[2] John Searle, Minds, Brains and Science (Cambridge, Massachusetts: Harvard University Press, 1984). [3] John Seade, The Rediscovery of the Mind (Cambridge, Massachusetts: MIT Press, 1992). [4] Alan Turing, "Computing Machinery and Intelligence," in The Mind's I, edited by Douglas Hofstader and Daniel Dennett (New York: Basic Books, 1981). [5] Ludwig Wittgenstein, Philosophical Investigations, translated by G. E. M. Anscombe (New York: Macmillan Publishing, 1958). [6] Ludwig Wittgenstein, Wittgenstein's Lectures on the Foundations of Mathematics, edited by Cora Diamond (Chicago: University of Chicago Press, 1976). School of Information Resources and Department of Philosophy University of Arizona Tucson,.AZ 85719 USA

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I[--1,P:-lnl;,|q,]dt[:-]dl Robin

Wilson,

Editor

I

Roman Numerals

Switzerland (1984)

Hessel Pot

Portugal (1959)

Germany (1990)

Korea (1964) Czechoslovakia (1968)

Netherlands (1954)

Please send all submissionsto the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics, The Open University,Milton Keynes, MK7 6AA, England

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oman numerals have appeared on many stamps; a delightful example is the Czechoslovak stamp (1968), illustrating an allegorical figure. When the Roman symbols I (one), V (five), X (ten), a (fifty), C (hundred), LD (five hundred), and M (thousand) are used to represent a number, it is common practice to avoid rows of four identical symbols; this is normally done by replacing IIII by IV and VIIII by IX; XXXX by XL and LXXXX by XC; CCCC by CD and DCCCC by CM. But this convention is not always followed. Exceptions may be found on stamps of the Netherlands (1954) [M DCCCC LIV], Portugal (1959) [M DCCCC LIX], Korea (1964) [XXXX V], Switzerland (1984) [IIII], and Germany (1990) [XXXX I]. Even more exceptional are the forms MDCC IC and MCM IL, to be found on two Italian stamps (1949) commemorating Alessandro Volta's invention of the Voltaic Pile. These are shorter than MDCC XCIX and MCM XLIX, but lose the separation of the thousands, hundreds, tens and units. In 1999 the question of 'proper'

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THE MATHEMATICAL INTELLIGENCER9 1999 SPRINGER-VERLAG NEWYORK

Italy (1949)

Italy (1949)

Roman notation is of particular interest. Should we write M DCCCC LXXXX VIIII, or the conventional M CM XC IX, or the convenient M IM? Tournosveld 67, 3443 ER Woerden, Netherlands