the mathematical intelligencer volume 22 issue 4

Letters to the Editor

The Mathematical Intelligencer

encourages comments about the

material in this issue. Letters

to the editor should be sent to the

editor-in-chief, Chandler Davis.

Unfair Dice

Dawson and Finbow have shown in [ 1] that it is impossible to load a cubic die so that it will stand on only one facet, and that though the regular octahedron, dodecahedron, and icosahedron are vulnerable to such loading, it is a practical impossibility. Whilst Dawson and Finbow's results do not have realworld utility, it is worth noting that the

real issue, in any game involving repeated throwing of dice of any description, is a small advantage that

remains unknown to opponents. In games where interest is placed on the

total score (as opposed to using the ciphers on the facets as mere labels),

there are such possibilities of accruing small advantages.

The traditional design for a cubic

die is that each of the pairs 1 and 6, 2

and 5, 3 and 4 goes on opposite facets. This allows two possible cases, mirror images; in manufactured dice one of them predominates. Imperfections in the material from which the die is manufactured might result in, say, greater density near the intersection of the facets with 4, 5, 6, thus skewing the expected distribution towards lower val

ues. In [2], with an emphasis on dice based on the five Platonic solids, a colleague and I sought an answer to the general question, "What distribution of the integers over the facets will minimise the effect of ... imperfections or of a deliberate bias?" We looked for simple criteria by which the set of integers { 1, 2, ... , n) may be distributed as uniformly as possible over the n faces for each of the Platonic solids and the semi-regular solids with 10 facets.

Given that a die that rolls one number too frequently would be easier to detect, we concentrated on more general and hence less detectable biases. Based on work by Singmaster [3] in an analysis for the design of dartboards, a simple interpretation of the minimi-

sation of the effect of irregularity in a die is to require larger numbers to lie adjacent to smaller ones, where adjacency means a common edge between facets. For dice this may be generalised as a requirement for the max

imisation ofS = L(ai - a1)2, where the sum is over all edges, and ai and a1 are the values on the facets sharing the edge.

With this criterion we noted that S is minimised for the standard cubic die, which thus is the one most sus

ceptibl� to potential distortion-at least by the criterion of total score over a number of throws. S is maximised when (6,5), ( 4,3), (1,2) are the opposite pairs. This design is thus proposed to replace the existing standard.

Several other criteria for the construction of dice have been envisaged. Rouse Ball [4] suggested numbering the faces of polyhedra in such a way that the faces around a vertex add to a constant sum. For regular polyhedra

this is achievable only for the octahe

dron, and three non-isomorphic cases may be identified. A generalisation of the idea is minimisation of the variance of the sums of facet values around each vertex. A further alternative is the minimisation of the variance of the sum of face values surrounding each facet. In each of these alternatives, the algebra is akin to that found in Singmaster's work on the design of dartboards, and it is necessary to introduce correlations amongst non-adjacent faces. The ensuing algebra appears intractable for more than six facets, although numerical approaches as in [2] could be employed.

If instead we minimise the variance of sums of opposite faces, the algebra

is simple; in contrast to the criterion in [2], this leads to favouring the standard die! Note that for the cube, this criterion agrees with the variance-ofsum-around-facet criterion just discussed.

© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 4, 2000 3

In [2] we also identified the maximising and minimising distributions

for the regular solids with n = 8, 12, 20 facets and also for the two semi-regu

lar solids with n = 10. Between the extremes lie other labellings whose vulnerability to loading is intermediate.

I am grateful to the various commentators on this note for their manyfaceted suggestions.

RI!F&RENCES

(1) Dawson R.J.M., and Finbow, W.A. "What

Shape is a Loaded Die?" The Mathematical

lntelligencer, 21, No.2 (1 999), 32-37.

(2] Blest, D.C. and Hallam, C.B. "The Design

of Dice", Bull. IMA, 32, Nos. 1/2 (1 996),

8-13.

[3) Singmaster, D., "Arranging a Dartboard."

Bull. IMA, 16, No. 4(1 980), 93-97.

[4) Rouse Ball, W.W. and Coxeter, H.S.M. ,

Mathematical Recreations and Essays,

12th edition. University of Toronto Press

(1974).

David C. Blest

School of Mathematics and Physics

University of Tasmania

Launceston, Tasmania

Australia 7250

e-mail: [email protected]

Parsing a Magic Square

Being a magic square enthusiast, I read with great delight "Alphabetic Magic Square in a Medieval Church" (lntelligencer, vol. 22 (2000), no. 1, 52-53), where A. Domenicano and I. Hargittai present and comment upon a stone inscription on a church near Capestrano, Italy. The stone is inscribed with the Latin text "ROTAS OPERA TENET AREPO SA TOR," arranged in the form of a 5 X 5 alphamagic square.

In their note, Domenicano and

Hargittai give the meaning and case of the words ROTAS, TENET, and SATOR, but

they are not sure about the grammatical case of OPERA because the word it

qualifies, AREPO, "is not Latin though it recalls the Latin word ARATRO = plough (ablative)". The authors then drop this aspect of their considerations on the square by commenting that "the meaning of the text remains obscure."

If one looks at the alphamagic square from the perspective of its author, however, it seems odd that he or she would

4 THE MATHEMATICAL INTELLIGENCER

introduce a strange word like AREPO, thereby ruining the intended cleverness of the whole exercise. So if AREPO is not one single Latin word, then it must be two or maybe even three Latin words, all grouped in a single line because the situation requires it. Anned with this Ansatz, let us look at the possibilities: "A REPO" and "ARE PO" are indeed divisions into two Latin words, but they do not fit the present context. This leaves the following division into three words: "A RE PO," each of which is Latin.

In fact this appears to be the solu

tion, for it gives the text a reasonable meaning. Rearranging the order of the words according to the rules of English, one gets "SATOR TENET ROTAS A RE PO OPERA." The word PO is an archaic form

of the adverb POTISSIMUM. With this interpretation, OPERA then is in the dative case, not the nominative or the ablative, as surmised by the authors. See Dictionnaire illustre Latin-Fra'n9ais by F. Gaffiot, Hachette, Paris, 1934. The

text means that the sower looks after the wheels because of their importance,

in particular for work. Finally, let me mention an astute ob

servation made by a physicist col

league who is an expert in optics, Dr. Jacques Gosselin. When one looks at the photograph of the stone (top, p.

53), one gets the impression that the letters are protruding. But the stone was set in the wall upside down, so to see the picture with the correct light

ing one should look at it with the page reversed. Now one sees at once that

the letters are indented, as was to be expected. This is a well-known illusion; I don't know whether to call it an optical or a neurological illusion.

Napoleon Gauthier

Department of Physics

The Royal Military College of Canada

Kingston, Ontario K7K 784

Canada


c.m mt.J,;

A Mathematician's View of Evolution Granville Sewell

The Opinion column offers

mathematicians the opportunity to

write about any issue of interest to

the international mathematical

community. Disagreement and

controversy are welcome. The views ..

and opinions expressed here, however,

are exclusively those of the author,

and neither the publisher nor the

editor-in-chief endorses or accepts

responsibility for them. An Opinion

should be submitted to the editor-in

chief, Chandler Davis.

In 1996, Lehigh University biochemist Michael Behe published a book enti

tled Darwin's Black Box [Free Press], whose central theme is that every living cell is loaded with features and biochemical processes which are "irreducibly complex"-that is, they require the existence of numerous complex components, each essential for function. Thus, these features and processes cannot be explained by gradual Darwinian improvements, because until all the components are in place, these assemblages are completely useless, and thus provide no selective advantage. Behe spends over 100 pages describing some of these irreducibly complex biochemical systems in detail, then summarizes the results of an exhaustive search of the biochemical literature for Darwinian explanations. He concludes that while biochemistry texts often pay lip-service to the idea that natural selection of random mutations can explain everything in the cell, such claims are pure "bluster," because "there is no publication in the scientific literature that describes how molecular evolution of any real, complex, biochemical system either did occur or even might have occurred."

When Dr. Behe was at the University of Texas El Paso in May of 1997 to give an invited talk, I told him that I thought he would fmd more support for his ideas in mathematics, physics, and computer science departments than in his own field. I know a good many mathematicians, physicists, and computer scientists who, like me, are appalled that Darwin's explanation for the development of life is so widely accepted in the life sciences. Few of them ever speak out or write on this issue, however-perhaps because they feel the question is simply out of their domain. However, I believe there are two central arguments against Darwinism, and both seem to be most readily appreciated by those in the more mathematical sciences.

I. The cornerstone of Darwinism is the idea that major (complex) improvements can be built up through many minor improvements; that the new organs and new systems of organs which gave rise to new orders, classes and phyla developed gradually, through many very minor improvements. We should first note that the fossil record does not support this idea, for example, Harvard paleontologist George Gaylord Simpson ["The History of Life," in Volume I of Evolution after Darwin, University of Chicago Prt;ss, 1960] writes:

It is a feature of the known fossil record that rrwst taxa appear abruptly. They are not, as a rule, led up to by a sequence of almost imperceptibly changing forerunners such as Darwin believed should be usual in evolution. . . . This phenomenon becomes. more universal and more intense as the hierarchy of categories is ascended. Gaps among known species are sporadic and often small. Gaps arrwng known orders, classes and phyla are systematic and almost always large. These peculiarities of the record pose one of the most important theoretical problems in the whole history of life: Is the sudden appearance of higher categories a phenomenon of evolution or of the record only, due to sampling bias and other inadequacies?

An April, 1982, Life Magazine article (excerpted from Francis Hitching's book, The Neck of the Giraffe: Where Darwin Went Wrong) contains the following report:

When you look for links between major groups of animals, they simply aren't there. . . . ''Instead of finding the gradual unfolding of life," writes David M. Raup, a curator of Chicago's Field Museum of Natural History, "what geologists of Darwin's time and geologists of the present day actually find is a highly uneven or jerky

© 2000 SPRINGER-VERLAG NEW YORK. VOLUME 22, NUMBER 4. 2000 5

record; that is, species appear in the fossil sequence very suddenly, show little or no change during their existence, then abruptly disappear." These are not negligible gaps. They are periods, in .aU the major evolutionary transitions, when immense physiological changes had to take place.

Even among biologists, the idea that new organs, and thus higher categories, could develop gradually through tiny improvements has often been challenged. How could the "survival of the fittest" guide the development of new organs through their · initial useless stages, during which they obviously present no selective advantage? (This is often referred to as the "problem of novelties.") Or guide the development of entire new systems, such as nervous, circulatory, digestive, respiratory and reproductive systems, which would require the simultaneous development of several new interdependent organs, none of which is useful, or provides any selective advantage, by itself? French biologist Jean Rostand, for example, wrote [A Biologist's View, Wm. Heinemann Ltd., 1956]:

It does not seem strictly impossible that mutations should have introduced into the animal kingdom the differences which exist between one species and the next . . . hence it is very tempting to lay also at their door the differences between classes, families and orders, and, in short, the whole of evolution. But it is obvious that such an extrapolation involves the gratuitous attribution to the mutations of the past of a magnitude and power of innovation much greater than is shown by those of today.

Behe's book is primarily a challenge to this cornerstone of Darwinism at the microscopic level. Although we may not be familiar with the complex biochemical systems discussed in this book, I believe mathematicians are well qualified to appreciate the general ideas involved. And although an analogy is only an analogy, perhaps the best way to understand Behe's argument is by comparing the development of the genetic code of life with the de-


velopment of a computer program. Suppose an engineer attempts to design a structural analysis computer program, writing it in a machine language that is totally unknown to him. He simply types out random characters at his keyboard, and periodically runs tests on the program to recognize and select out chance improvements when they occur. The improvements are permanently incorporated into the program while the other changes are discarded. If our engineer continues this process of random changes and testing for a long enough time, could he eventually develop a sophisticated structural analysis program? (Of course, when intelligent humans decide what constitutes an "improvement", this is really artificial selection, so the analogy is far too generous.)

If a billion engineers were to type at the rate of one random character per second, there is virtually no chance that any one of them would, given the 4.5 billion year age of the Earth to work on it, accidentally duplicate a given 20-character improvement. Thus our engineer cannot count on making any major improvements through chance alone. But could he not perhaps make progress through the accumulation of very small improvements? The Darwinist would presumably say yes, but to anyone who has had minimal programming experience this idea is equally implausible. Major improvements to a computer program often require the addition or modification of hundreds of interdependent lines, no one of which makes any sense, or results in any improvement, when added by itself. Even the smallest improvements usually require adding several new lines. It is conceivable that a programmer unable to look ahead more than 5 or 6 characters at a time might be able to make some very slight improvements to a computer program, but it is inconceivable that he could design anything sophisticated without the ability to plan far ahead and to guide his changes toward that plan.

If archeologists of some future society were to unearth the many versions of my PDE solver, PDE2D, which I have produced over the last 20 years, they would certainly note a steady in-

crease in complexity over time, and they would see many obvious similarities between each new version and the previous one. In the beginning it was only able to solve a single linear, steady-state, 2D equation in a polygonal region. Since then, PDE2D has de

veloped many new abilities: it now solves nonlinear problems, timedependent and eigenvalue problems, systems of simultaneous equations, and it now handles general curved 2D regions. Over the years, many new types of graphical output capabilities have evolved, and in 1991 it developed an interactive preprocessor, and more recently PDE2D has adapted to 3D and 1D problems. An archeologist attempting to explain the evolution of this computer program in terms of many tiny improvements might be puzzled to find that each of these major advances (new classes or phyla??) appeared suddenly in new versions; for example, the ability to solve 3D problems first appeared in version 4.0. Less major improvements (new families or orders??) appeared suddenly in new sub-versions; for example, the ability to solve 3D problems with periodic boundary conditions first appeared in version 5.6. In fact, the record of PDE2D's development would be similar to the fossil record, with large gaps where major new features appeared, and smaller gaps where minor ones appeared. That is because the multitude of intermediate programs between versions or subversions which the archeologist might expect to fmd never existed, because-for example-none of the changes I made for edition 4.0 made any sense, or provided PDE2D any advantage whatever in solving 3D problems (or anything else), until hundreds of lines had been added.

Whether at the microscopic or macroscopic level, major, complex, evolutionary advances, involving new features (as opposed to minor, quantitative changes such as an increase in the length of the giraffe's neck, or the darkening of the wings of a moth, which clearly could occur gradually),

also involve the addition of many interrelated and interdependent pieces. These complex advances, like those made to computer programs, are not

always "irreducibly complex" -sometimes there are useful intermediate stages. But just as major improvements to a computer program cannot be made 5 or 6 characters at a time, certainly no major evolutionary advance is reducible to a chain ortiny improvements, each small enough to be bridged by a single random mutation.

II. The other point is very simple, but also seems to be appreciated only by more mathematically-oriented people. It is that to attribute the development of life on Earth to natural selection is to assign to it-and to it alone, of all lmown natural "forces"-the ability to violate the second law of thermodynamics and to cause order to arise from disorder. It is often argued that since the Earth is not a closed system-it receives energy from the Sun, for example-the second law is not applicable in this case. It is true that order can increase locally, if the local increase is compensated by a decrease elsewhere, i.e., an open system can be taken to a less probable state by importing order from outside. For example, we could transport a truckload of encyclopedias and computers to the moon, thereby increasing the order on the moon, without violating the second law. But the second law of thermodynamics-at least the underlying principle behind this law-simply says that natural forces do not cause extremely improbable things to happen, and it is absurd

to argue that because the Earth receives energy from the Sun, this principle was not violated here when the original rearrangement of atoms into encyclopedias and computers occurred.

The biologist studies the details of natural history, and when he looks at the similarities between two species of butterflies, he is understandably reluctant to attribute the small differences to the supernatural. But the mathematician or physicist is likely to take the broader view. I imagine visiting the Earth when it was young and returning now to fmd highways with automobiles on them, airports with jet airplanes, and tall buildings full of complicated equipment, such as televisions, telephones, and computers. Then I imagine the construction of a gigantic computer model which starts with the initial conditions on Earth 4 billion years ago and tries to simulate the effects that the four lmown forces of physics (the gravitational, electromagnetic, and strong and weak nuclear forces) would have on every atom and every subatomic particle on our planet (perhaps using random number generators to model quantum uncertainties!). If we ran such a simulation out to the present day, would it predict that the basic forces of Nature would reorganize the basic particles of Nature into libraries full of encyclopedias, science texts and novels, nuclear power plants, aircraft

knows all about cancer. He's got it. Luckily, Adam has St. Jude Children's Research Hospital, where doctors and scientists are making progress on his disease. To learn how you can help, call:

1-800-877-5833.

AUTHOR

GRANVILLE SEWELL

Mathematics Department

University of Texas at E1 Paso

El Paso, TX 79968 USA


Granville Sewell completed his PhD at

Purdue Un iversity in 1972. He has

subsequently been employed by (in

chronological order) Universidad Sim6n

Bolivar (Caracas), Oak Ridge National

Laboratory, Purdue University, IMSL

(Houston), The University of Texas

Center for High Performance Com

puting (Austin), and the University of

Texas El Paso; he spent Fall 1 999 at

Universidad de Tucuman in Argentina

on a Fulbright grant. He has written

four books on numerical analysis.

carriers with supersonic jets parked on deck, and computers connected to laser printers, CRTs, and keyboards? If we graphically displayed the positions of the atoms at the end of the simulation, would we find that cars and trucks had formed, or that supercomputers had arisen? Certainly we would not, and I do not believe that adding sunlight to the model would help much. Clearly something extremely improbable has happened here on our planet, with the origin and development of life, and especially with the development of human consciousness and creativity.

Granville Sewell

Mathematics Department

University of Texas El Paso

El Paso, TX 79968

USA


VOLUME 22, NUMBER 4, 2000 7

MICHAEL EASTWOOD1 AND ROGER PENROSE

Drawing with Comp ex Nu m bers

• t is not commonly realized that the algebra of complex numbers can be used in an

~ elegant way to represent the images of ordinary 3-dimensional figures, orthograph

ically projected to the plane. We describe these ideas here, both using simple geome

try and setting them in a broader context.

Consider orthogonal projection in Euclidean n-space onto an m-dimensional subspace. We may as well choose coordinates so that this is the standard projection p : �n �

�m onto the first m variables. Fix a nondegenerate simplex I in �n. Two such simplices are said to be similar if one can be obtained from the other by a Euclidean motion together with an overall scaling. This article answers the following question. Given n + 1 points in �m, when can these points be obtained as the images under P of the vertices of a simplex similar to I?

When n = 3 and m = 2, then P is the standard orthographic projection (as often used in engineering drawing), and we are concerned with how to draw a given tetrahedron. We shall show, for example, that four points a, {3 , y, 8 in the plane are the orthographic projections of the vertices of a regular tetrahedron if and only if

(a + f3 + y + 8)2 = 4(a2 + {3 2 + y + SZ) (1)

where a, {3 , y, 8 are regarded as complex numbers! Similarly, suppose a cube is orthographically projected and normalised so that a particular vertex is mapped to the origin. If a, {3 , y are the images of the three neighbouring vertices, then

(2)

again as a complex equation. Conversely, if this equation is satisfied, then one can find a cube whose orthographic image is given in this way. Since parallel lines are seen as parallel in the drawing, equation (2) allows one to draw the general cube:

'Supported by the Australian Research Council.

8 THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK

In this example, a = 2 - 26i f3 = -23 + 2i 'Y = 14 + 7i

The result for a cube is known as Gauss's fundamental theorem of axonometry-see [3, p. 309] where it is stated without proof. In engineering drawing, one usually fixes three principal axes in Euclidean three-space, and then an orthographic projection onto a plane transverse to these axes is known as an axonometric projection (see, for example, [8, Chapter 17]). Gauss's theorem may be regarded as determining the degree of foreshortening along the principal axes for a general axonometric projection. The projection corresponding to taking a, {3 , y to be the three cube roots of unity is called isometric projection because the foreshortening is the same for the three principal axes.

In an axonometric drawing, it is conventional to take the image axes at mutually obtuse angles:

If la l =a, 1131 = b, I'Yi = c, then equation (2) is equivalent to

the sine rule for the triangle with sides a2, /32, y , namely

a2 b2 c2 sin 2A sin 2B sin 2C'

In this form, the fundamental theorem of axonometry is

due to Weisbach, and was published in Tiibingen in 1844 in the Polyte chnische Mitte ilunge n ofVolz and Karmasch.

Equivalent statements can be found in modem engineer

ing drawing texts (e.g., [7, p. 44]).

Equation (2) may be used to give a ruler-and-compass

construction of the general orthographic image of a cube.

If we suppose that the image of a vertex and two of its

neighbours are already specified, then (2) determines (up

to a two-fold ambiguity) the image of the third neighbour.

The construction is straightforward, except perhaps for the

construction of a complex square root, for which we ad

vocate the following as quite efficient:

First, C is constructed by marking the real axis at a distance

llzll from the origin. Then, a circle is constructed passing

through the three points C, 1, and z. Finally, the angle be

tween 1 and z is bisected and vZ appears where this bi

sector meets the circle.

In engineering drawing, it is more usual that the images

of the three principal axes are prescribed or chosen by the

designer and one needs to determine the relative degree

of foreshortening along these axes. There is a ruler-and

compass construction given by T. Schmid in 1922 (see, for

example, [8, § 17 . 17-17 .19]):

In this diagram, the three principal axes and a are given. By

drawing a perpendicular from a to one of the principal axes

and marking its intersection with the remaining principal

axis, we obtain P. The point Q is obtained by drawing a semi

circle as illustrated. The point R is on the resulting line and

equidistant with a from Q. Finally, f3 is obtained by drop

ping a perpendicular as shown. It is easy to see that this con

struction has the desired effect-in Euclidean three-space,

rotate the right-angled triangle with hypotenuse Pa about

this hypotenuse until the point Q lies directly above 0, in

which case R will lie directly above f3 and the third vertex

will lie somewhere over the line through 0 and Q. One may

verify the appropriate part of Weisbach's condition

a2 b2 sin 2A sin 2B (3)

by the following calculation. Without loss of generality we

may represent all these points by complex numbers nor

malised so that Q = 1. Then it is straightforward to check that

R = 1 + i - ia ,

and therefore that

P =·a (a +a)+ 2(1 - a - a)

- - ' a - a a (a + a) + 2(1 - a - a) . f3 = - �, 2 - a - -a

a2 + /32 = 4(a - 1)(a- 1)(a + a- 1)

. (a + a- 2)2

That a2 + f32 is real is equivalent to (3). To prove Gauss's theorem more directly, consider three

vectors in �3 as the columns of a 3 X 3 matrix. This ma

trix is orthogonal if and only if the three vectors are or

thonormal. It is equivalent to demand that the three rows

be orthonormal. However, any two orthonormal vectors in

�3 may be extended to an orthonormal basis. Thus, the

condition that three vectors

in �2 be the images under p : �3 � �2 of an orthonormal

basis of �3, is that

and (Yl Y2 Y3)

be orthonormal in �3. Dropping the requirement that the

common norm be 1, we obtain

x12 + xi + X32 = Y12 + Y22 + Y32 and

X1Y1 + X2Y2 + X3Y3 = 0.

Writing a = x1 + iy1, f3 = x2 + Y2, y = X3 + Y3, these two

equations are the real and imaginary parts of (2). To de

duce the case of a regular tetrahedron as described by

equation (1) from the case of a cube as described by equa

tion (2), it suffices to note that equation (1) is translation

invariant and that a regular tetrahedron may be inscribed

in a cube. Thus, we may take B = a + f3 + y and observe

that (1) and (2) are then equivalent.

It is easy to see that the possible images of a particular

tetrahedron 2: in �3 under an arbitrary Euclidean motion fol-


lowed by the projection P form a 5-dimensional space-the group of Euclidean motions is (klimensional, but translation orthogonal to the plane leaves the image unaltered. It therefore has codimension 3 in the 8-dimensional space of all tetrahe<h:al images (2 degrees of freedom for each vertex). Allowing similar tetrahedra rather than just congruent ones reduces the codimension to 2. Therefore, two real equations are to be expected. Always, these two real equations combine as a single complex equation such as (1) or (2). At first sight, this is perhaps smprising; and even more so when the same phenomenon occurs for P : !Rn � IR2 for arbitrary n.

For n = 3, there is a proof of Gauss's theorem which brings in complex numbers at the outset. Consider the space H of Hermitian 2 X 2 matrices with zero trace, i.e., matrices of the form

X = ( w.

u + iv ) u -w -w

for

We may identify H with !R3, and, in so doing, -det X becomes the square of the Euclidean length. The group G of invertible 2 X 2 complex matrices of the form

A= (� -:) acts linearly on H by X� AXN. Moreover,

det(AXAI) = (jaj 2 + j bj 2)2 det X ,

so G acts by similarities. It is easy to check that all similarities may be obtained in this way. (This trick is essentially as used in Hamilton's theory of quaternions and is well known to physicists. In modem parlance it is equivalent to the isomorphism of Lie groups Spin(3) = SU(2).) Therefore, an arbitrary orthographic image of a cube may be obtained by acting with A on the standard basis

and then picking out the top right-hand entries. We obtain

A(� �) N = (: A(�i

�) N = (: i(a 2 + b2)) ·c 2 b2) f3

* ��a + =

A(� �J N = (: 2:b)�2a b = y

and therefore a2 + {3 2 + y = 0, as required. Conversely, this is exactly the condition that a ,{3 , y may be written in this form. (Compare the half-angle formulae: if s2 + c2 = 1, then s = 2t/(1 + t2) and c = (1 - t2)/(1 + t2) for some t.) That Gauss [3, p. 309] makes the same observation concerning the form of a ,{3 , y suggests that perhaps he also had this reasoning in mind.

In general, the following terminology concerning the standard projection P : !Rn � !Rm is useful. We shall say that vh v2, . . . , Vn E !Rm are normalised e utactic if and only if there is an orthonormal basis u1 , u2, ... , Un of !Rn with v1 = Pu1 for


j = 1, 2, . . . , n. We shall say that vh v2, ... , Vn E !Rm are e utac tic if and only if J.LVb p,v2, . . . , J.LVn are normalised eutactic for some p, =F 0. The proof of Gauss's theorem using orthogonal matrices clearly extends to yield the following result

Theorem The points z1, z2, . . • , Zn E e = !R2 are e utac tic if a nd only if

Z 12 + Z 22 + · · · + Zn2 = 0

and not a U z1 are ze ro.

There is an alternative proof for n = 4 based on the isomorphism

Spin(4) = SU(2) X SU(2),

and, indeed, this is how we came across the theorem in the first place.

However, a more direct route to complex numbers, and one which applies in all dimensions, is based on the observation that Gr� (IR2), the Grassmannian of oriented two-planes in !Rn, is naturally a complex manifold. When n = 3, this Grassmannian is just the two-sphere and has a complex structure as the Riemann sphere. In general, consider the mapping

eiP'n-1 \ IRIP'n-1 � Gr�(!Rn)

induced by en 3 z � iz/\Z . In other words, a complex vector Z = X + iy E en is mapped to the two-dimensional oriented subspace of [Rn spanned by x andy, the real and imaginary parts of z. Let ( , ) denote the standard inner product on [Rn extended to en as a complex bilinear form. Then, (z, z) = 0 imposes two real equations

and (x, y) = 0

on the real and imaginary parts. In other words, x, y is proportional to an orthonormal basis for span{x, y). Hence, if z and w satisfy (z, z) = 0 = (w, w) and defme the same oriented two-plane, then w = >.z for some A E C\{0}. The nonsingular complex quadric

K = { [z] E e1P'n-1 s.t. (z, z) = 0)

avoids IRIP'n- 1 C e1P'n-b and we have shown that 11]Kis injective. It is clearly surjective. The isomorphism

7T : K.::::.,. Gr2 + (!Rn)

respects the natural action of SO (n) on K and Gr2 +c�Rn). The generalised Gauss theorem follows immediately, for, rather than asking about the image of a general orthonormal basis under the standard projection P : !Rn � !R2, we may, equivalently, ask about the image of the standard basis e 1, e 2, ... , en under a general orthogonal projection onto an oriented two-plane n c !Rn. Any such n is naturally complex, the action of i being given by rotation by 90° in the positive sense. If n is represented by [ZI, z2, ... , Zn] E K as above and we use x, y E TI to identify TI with e, then e1�z1and

Z12 + Z22 + · · · + Zn2 = (z, z) = 0,

as required. Conversely, a solution of this complex equation determines an appropriate plane n.

For the case of a general tetrahedron or simplex and for general m and n, it is more convenient to start with Hadwiger's theorem [4] or [2, page 251] as follows. The proof is obtained by extending our orthogonal matrix proof of Gauss's theorem.

Theorem (Hadwiger) Asse mble vb v2, . . • , Vn E [Rm as the columns of an m X n ma trix V. The se ve ctors a re norma lise d e utactic if and only if W1 = 1 (the m X m ide ntity ma trix ).

Proof If v1, v2, • . . , Vn are normalised eutactic, then assembling a corresponding orthonormal basis of !Rn as the columns of an n X n matrix, we have V = P U and fYU = 1 (the n X n identity matrix). Therefore, UU1 = 1 and

w = Puutpt = ppt = 1,

as required. Conversely, if W = 1, then the columns of V1 may be completed to an orthonormal basis of !Rn, i.e., V1 = U1P1 for UU1 = 1. Now, U1U = 1 and V = PU, as required. D

The case of a general simplex is obtained essentially by a change of basis as follows. Suppose a 1, a 2, . . . , an, an+l are the vertices of a non-degenerate simplex I in !Rn whose centre of mass is at the origin. In other words, the n x ( n + 1) matrix A has rank n and Ae = 0 where e is the column vector all of whose n + 1 entries are 1. Form the (n + 1) X (n + 1) symmetric matrix

Q = At(AAt)-2 A,

noting that rank A = n implies that the moment ma trix AA1 is invertible.

Theorem Given b1, b2, . . . , bm bn+ 1 E !Rm asse mble d as the columns of an m X (n + 1) ma trix B, the se ve ctors are the images under ort hogonal projection P : [Rn � [Rm of the vertices of a simplex congrue nt to I if and only if

(4)

Proof The vertices of a simplex congruent to I are the columns of a matrix UA + ae1 for some orthogonal matrix U and translation vector a E !Rn. Also, note that Qe = 0.

Thus, if B = P(UA + ael ), then

as required.

BQB1 = PUAQA1[Jlpt = PUAAt(AAtr2AAtUtpt = PUfYP1 = ppt = 1,

Conversely, Qe = 0 implies that (4) is translation invariant. So, without loss of generality, we may suppose that b1 + b2 + . . . + bn + bn+l = 0, that is to say,Be = 0. Writing out ( 4) in full gives

BAt(AAt)-l(BAt(MtrlY = 1

so, by Hadwiger's theorem, there is an orthogonal matrix Uso that

Thus,

and Be= 0.

Certainly, B = PUA is a solution of these equations; but it is the only solution, because A1(AA1)-1 A has rank nand e is not in the range of this linear transformation. D Corollary (case m = 2) Points zb z2, ... , Zn, Zn+l E C are the images unde r orthogonal proje ction of the vertice s of a simplex similar to I if and only if

z1Qz = 0

whe re z is the column vector with compone nts z1, z2, ... , Zn, Zn+l·

It is, of course, possible to compute Q explicitly for any given example. If the simplex I has some degree of symmetry, however, we can often circumvent such computation. Consider, for example, the case of a regula r simplex. From the corollary above, we know that the image of such a simplex in the plane is characterised by a complex homogeneous quadratic polynomial. The symmetries of the regular simplex ensure that this polynomial must be invariant under 9' n+ 1, the symmetric group on n + 1 letters. Hence, it must be expressible in terms of the elementary symmetric polynomials. Equivalently, it must be a linear combination of

(zl + Z2 + ... + Zn + Zn+l)2 and

Z12 + Z22 + "' + Zn2 + Zn+l2·

Up to scale, there is only one such combination that is translation-invariant, namely

(zl + Z2 + . . . + Zn + Zn+l)2-(n + 1)(zl2 + Z22 + . .. + Zn2 + Zn+l2). (5)

It follows that the vanishing of this polynomial is an equation that characterises the possible images of a regular simplex under orthogonal projection into the plane. The special case n = 2 characterises the equilateral triangles in the plane [1, Problem 15 on page 79].

Equation (2) characterising the orthographic images of a cube, may be deduced by similar symmetry considerations. If a particular vertex is mapped to the origin and its neighbours are mapped to a, {3, y, 'then, since each of these neighbouring vertices is on an equal footing, the polynomial in question must be a linear combination of (a + {3 + y)2 and a2 + {32 + y . To fmd out which linear combination, we need only consider a particular projection, for example:

In this example, (a + {3 + y)2 = 2i and a2 + {32 + y = 0. Up to scale, therefore, (2) is the correct equation.

The case of a regular dodecahedron is similar. Using the fact that a cube may be inscribed in such a dodecahedron [5], we may deduce a particular projection:


v'5- 1 v'5+ 1. 7=-----t 4 4

with ( a + f3 + y )2 = (7- 3v'5)! 2 and if+ {3 2 + y = (2 - v'5)!2. In this particular case,

(a + f3 + 'Y? + cV5- 1)(if + {3 2 + y ) = 0.

Therefore, this is the correct equation in the general case. It may be used as the basis of a ruler-and-compass construction of the general orthographic projection of a regular dodecahedron.

It is interesting to note that if aU the vertices of a Platonic solid are orthographically projected to z 1, z 2, ... , ZN E C, then necessarily

Cz 1 + z 2 + · · · + ZN )2 = N(z 12 + z 22 + · · · + z� ). (6)

Only for a tetrahedron, when (6) coincides with (1), is this condition also sufficient. To verify (6) for the other Platonic solids, first note that it is translation-invariant. Therefore, it suffices to impose z 1 + z 2 + · · · + ZN = 0 and show that z 12 + z 22 + ··· + z� = 0. The case of a cube now follows immediately, as its vertices may be grouped as two regular tetrahedra. The dodecahedral case may be dealt with by grouping its vertices into five regular tetrahedra. The regular octahedron is amenable to a similar trick, but not the icosahedron. Rather than resorting to direct computation, a uniform proof may be given as follows.

As before, assemble the vertices of the given solid I as the columns of a matrix A, now of size 3 X N, and consider the moment matrix M = AAt . Observe that

(I i 0) M G ) � z,' + z,' + · · · + zJ' .

The moment matrix is positive definite and symmetric. In other words, it defmes a metric on IR3, manifestly invariant under the symmetries of I. If I is regular-or, more generally, er\ioys the symmetries of a regular solid (e.g., a cuboctahedron or rhombicosidodecahedron)-then its symmetry group acts irreducibly on !R3. Thus, M must be proportional to the identity matrix and the result follows. For a general solid I, the two complex numbers

±V z 12 + z22 + ··· + z�

are the foci of the ellipse

(x y) R (=) = 1, where R is the inverse of the quadratic form obtained by restricting M to the plane of projection.

This reasoning also works in higher dimensions, where it shows (as cor\iectured to us by H.S.M. Coxeter) that the or-


thogonally projected images in the plane of the N vertices of any non-degenerate regular polytope, real or complex, will satisfy equation (6). This includes regular polygons in the plane, where the projection is vacuous. As already remarked, for polyhedra other than simplices, a quadratic equation such as (6) is no longer sufficient to characterise the orthogonal image up to scale. In general, there will also be some linear relations. For a non-degenerate N-tope in !Rn there will be N - n - 1 such relations. The simplest example is a square in !R 2, which is characterised by the complex equations (a + f3 + 'Y + 8)2 = 4( a2 + {3 2 + r + fJ2) and a+ 'Y = f3 + 8.

It is interesting to investigate further the relationship between a non-degenerate simplex I in !Rn and its quadratic form Q = At (AAI )-2 A. Recall that A is the n X (n + 1) matrix whose columns are the vertices of I. There are several other formulae for or characterisations of Q. LetS denote the ( n + 1) X ( n + 1) symmetric matrix

1- _1 ( � � �1.: ) · n+1 :

1 1

It is the matrix of orthogonal projection in !R n+ 1 in the direction of the vector e. We maintain that Q is characterised by the equations

QG =S and Qe = 0,

where G = AtA. Certainly, if these equations hold, then they are enough to determine Q, because the matrix G has rank n and e is not in its range. The second equation is evident, and the first equation with Q replaced by At (AAI )- 2 A and G by A tA reads

At (AAt )-1A = S.

To see that this holds it suffices to observe that it is clearly true after postmultiplication by At or e. We may equally well characterise Q by means of the equations

GQ =S and Qe = 0

These equations relate G and Q geometrically: both matrices annihilate e, whilst on the hyperplane orthogonal to e they are mutually inverse. This is to say that G and Q are generali sed inverses (6] of each other. Thus we write

Q = at = (AIA)t = AtAtt

where At is the generalised inverse of A. In this case, At = At (AAt )- 1. This also shows how to compute Q more directly in certain cases. The matrix G has direct geometric interpretation as the various inner products of the vectors a1, a2, . . . , an, an+l· In the case of a regular simplex, for example, we know that ll aill 2 is independent of i, that ll ai - aill 2 is independent of i * j, and that a1 + a2 + . . . + an + an+l = 0. We may deduce that, with a suitable overall scale, G = S. Since st = S, it follows that Q = S. This is a direct derivation of (5).

It is clear geometrically that G determines I up to congruency. Therefore, so does Q. In other words, the possi-

ble quadratic forms Q that can arise give a natural parametrisation of the non-degenerate simplices up to congruency. As to which Q do arise, certainly they eiijoy the following properties:

• Q is a real (n + 1) X (n + ll symmetric matrix. • Qe = 0, and only multiples of e are in the kernel of Q. • All other eigenvalues of Q are positive.

Conversely, these properties characterise the possible Q that can arise: given such a Q, we may take A1 to have as its columns a system of mutually orthogonal eigenvectors for the non-zero eigenvalues of Q, each being scaled to have length· equal to the square-root of the reciprocal of the corresponding eigenvalue. It follows easily that Q = At(AAI)-2 A.

It is also possible to repeat this analysis in pseudoEuclidean spaces. The only difference is that the condition that the non-zero eigenvalues of Q be positive is replaced by a condition on sign precisely reflecting the original signature of the inner product.

Finally we should mention some possible applications. There is much current interest in computer vision. In particular, there is the problem of recognising a wire-frame object from its orthographic image. The results we have described can be used as test on such an image, for example to see whether a given image could be that of a cube, or to keep track of a moving shape. It is clear that such

AUTHORS

MICHAEL EASTWOOD

Department of Mathematics

University of Adelaide

South Australia 5005 e-mail: [email protected]

Michael Eastwood, BA Oxford, PhD Princeton, has been at the

University of Adelaide since 1985. He is now a Senior Research

Fellow of the Australian Research Council. His mathematical in

terests are differential geometry, extending to several complex vari

ables, integral geometry, and twistor theory. His hobbies are rock

climbing, volleyball, and-when the aforementioned have not done

excessive damage to his fingernails-playing classical guitar.

2Computer Aided Drafting and Design.

tests could be implemented quite efficiently. Another possibility is in the manipulation of CADD2 data. Rather than storing an image as an array of vectors in IR3, it may sometimes be more efficient to store certain tetrahedra within such an image by means of the corresponding quadratic form. For orthographic imaging this may be preferable.

We would like to thank H. S.M. Coxeter for drawing our attention to Hadwiger's article, R. Michaels and J. Cofman for pointing out Gauss's and Weisbach's work, E.J. Pitman for many useful conversations, and the referee for suggesting several improvements.

REFERiiNCES

(1 ] S. Barnard and J .M. Child, Higher Algebra, MacMillan 1 936.

[2] H.S.M. Coxeter, Regular Polytopes, Methuen 1 948.

[3] C.F. Gauss, Werke, Zweiter Band, Koniglichen Gesellschaft der

Wissenschaften, Gottingen 1 876.

(4] H. Hadwiger, Ober ausgezeichnete Vectorsteme und regulare

Polytope, Comment. Math. Helv. 13 (1 940), 90--108.

(5] D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination,

Chelsea 1 952, 1983, 1 990. _

[6] R. Penrose, A generalised inverse for matrices, Proc. Camb. Phil.

Soc. 51 (1 955), 406-41 3.

(7] R.N. Roth and I.A. van Haeringen, The Australian Engineering Drawing

Handbook, Part One, The Institute of Engineers, Australia 1 988.

[8] R.P. Hoelscher and C.H. Springer, Engineering Drawing and

Geometry, Second Edition, Wiley 1961 .

ROGER PENROSE

Mathematical Institute

University of Oxford

Oxford OX1 3LB

England


Roger Penrose - since 1 994, Sir Roger Penrose-holds ap

pointments at the University of Oxford and at Gresham College

London, and a part-time appointment at the Pennsylvania State

University; earlier he was at Birl<beck College London. Among

his many honors are the 1988 Wolf Prize and the Royal Society Royal Medal. He is known especially for contributions to the study

of non-periodic tilings, and for his theory of twisters, which aims

to unite Einstein's general relativity with quantum mechanics.

Among his books are the best-selling The Emperor's NfJW Mind

(1989), and a recent novel (with Brian Aldiss), White Mars.


i,l,ijj:i§rr6hl¥11@i§#bii,'I,J§:id Alexander Shen , Editor I

This column is devoted to mathematics

for jun. JWtat better purpose is there

for mathematics? To appear here,

a theorem or problem or remark does

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.

Please send all submissions to the

Mathematical Entertainments Editor,

Alexander Shen, Institute for Problems of

Information Transmission, Ermolovoi 1 9,

K-51 Moscow GSP-4, 1 01 447 Russia;


Cl iques, the Cauchy Inequa l ity, and I nformation Theory

Here I present two rather unexpected solutions of a simple prob

lem:

Let G = (V,E) be an undirected graph having n edges; to prove that the number of 3-cliques in G does not exceed (v'2! 3)n312.

As usual, a 3-clique is a set of three vertices connected by three edges. It would be just as good to formulate the problem without any graph-theoretic terminology at all:

Assume given a collection of n segments in the plane; to prove that there are at most (v'2i3)n312 triangles whose sides belong to the collection.

This theorem is an easy consequence of the following inequality. Let A be a finite subset of the Cartesian product X X Y X Z. Defme the sets Axy, Axz, and Ayz as the projections of A onto X X Y, X X Z, and Y X Z, respectively. Then

(#A)2 ::::; #Axy · #Axz · #Ayz, (C)

where #8 stands for the cardinality of a fmite set 8.

How do we apply (C) to our prob-


lem? Denoting the set of vertices by V, the product V X V X V is the set of ordered triples of vertices. We are estimating m, the number of 3-cliques; each of these corresponds to 6 = 3! ordered triples. Let A, then, be the subset of V X V X V coming from 3-cliques. It has 6m elements, and we want to use (C) to estimate this. For this, we need to know the number of elements in the projections of A. Projection Axy contains only pairs (x,y) that are connected by an edge, and each edge gives two pairs, so the cardinality of the projection does not exceed 2n, where n is the number of edges. Therefore (C) says that

(6m)2 :5 (2n) · (2n) · (2n),

in agreement with the stated conclusion.

Now I present two proofs of inequality (C).

(First proof) I prefer to use the following geometric version of (C): if A is a (measurable) subset of IR3 having volume V, and 81, 82, 83 are the areas of its two-dimensional projections (onto the three coordinate planes), then

V2 ::::; 818�3. (G)

(If A is composed of unit cubes with integer vertices, this is exactly the statement (C), for the volume is the number of unit cubes and the area is the number of unit squares in the projection.)

To prove (G), first generalize it to say

(If I f(x,y)g(x, z )h(y, z ) dx dy d z )2 :5 :5 If j2(x,y) dx dy · If g2(x, z ) dx dz ·

I I h2(y, z ) dy dz (I)

for any non-negative!, g, and h. Iff, g, and h are equal to 1 inside the corresponding projections of A and to 0 outside, thenf(x,y)g(x, z )h(y, z ) = 1 for all (x,y,z ) E A (and maybe for some other points), so that (I) gives (G).

Now inequality (I) is a variation of the Cauchy inequality and may be reduced to it:

(JI I f(x,y)g(x,z )h(y,z ) dx dy dz )2 :::; :::; fi j2(x,y) dx dy · II (f g(x,z )h(y,z ) dz )2 dx dy :::; :::; II f2(x,y) dx dy · II (f g2(x,z )dz I h2(y,z )dz ) dx dy = = I I f2(x,y) dx dy · I I g2(x,z ) dx dz · I I h2(y,z ) dy dz .

(Second proof) This proof of (C) is completely different (and rather strange). It uses the notion of Shannon entropy of a random variable with finite range. If a random variable gtakes n values with probabilities p1, . . . , Pn, then the Shannon entropy of g is defined as

HW = -L Pi log2 Pi i

It does not exceed log2 n and is equal to log2 n when all values are equiprobable.

If g and 17 are both random variables with finite range, then so is the pair (g,17), and its Shannon entropy H((g,17)) is given by the general definition.

The conditional entropy H(g 1 17) of g when 17 is known can be defmed as

You can easily check that this agrees with the natural definition of conditional entropy of g given 17: namely, fix any value of 17 and compute H(g) using the conditional probabilities of the g values in place of the Pi; and then take the weighted average of the results, weighted by the probabilities of the various values of 11·

It is a standard fact that

H((g,17)) :::; HW + H(17).

Tne reader not acquainted with these matters will probably ef\ioy tackling this by straightforward analysis. So now we know that

(L)

for any g and 17· Now it is easy to prove that

2H((g,7],7)) :::; H((g,17)) + H((g,7)) + H((7],7)). (E)

Indeed, (E) can be rewritten as

H(7 I (g,7])) + H(7] I (g,7)) ::5 H((7],7))

where the right-hand side equals H( 7) + H( 17 I 7). It remains to note that H(7 1 . . . ) ::5 H(7), and H(17 I (g,7)) :::; H(17 I 7). (The first of these is fact (L), the second is the "conditionalized" version of it. The intuitive content of these is that any conditional entropy is smaller the more we know.)

Now to prove (C) using (E). Consider the random variable that is uniformly distributed in the set A c X X Y X Z. It can be considered as a triple of (dependent) variables (g, 17, 7), where g E X, 17 E Y, and � E Z. The entropy of the triple (g, 7],7) equals log2 #A. Using (E), we get that

2 log2 #A :::; H((g,7])) + H((g,7)) + H((7],7)).

The pair (g, 17) takes values inAxy, therefore its entropy does not exceed log2 #Axy· For the same reasons H((g,7)) ::5 log2 #Axz and H((7],7)) :5 log2 #Ayz· Therefore

2 log2 #A ::5 log2 #Axy + log2 #Axz + log2 #Ayz,

and we get (C) by exponentiation.

***

I have recei ved the foUowing letter, completi ng the pi cture sketched i n an earli er column.

Your column in The Mathemati cal InteUigencer for Spring, 2000, never mentions the name of the problem discussed. It is called the "majority problem" in the theoretical computer science literature, and the two most interesting papers on the subject (he says with a blush) are

L. Alonso, E. M. Reingold, and R. Schott, "Determining the Majority," Info. Proc. Let. 47 (1993), 253-255.

L. Alonso, E. M. Reingold, and R. Schott, "The AverageCase Complexity of Determining the Majority," SIAM J. Computi ng 26 (1997), 1-14.

Both papers deal with (exact, achievable) lower bounds.

Edward M. Reingold

Department of Computer Science

University of Illinois at Urbana-Champaign

Urbana, IL 61 801 -2987

USA


What are

your chances

of dying

on your

next flight,

being called

for jury duty,

or winning

the lottery?

In this collection of 21 puzzles, Paul Nahin applies the laws of probability as they apply to a fascinating a rray of situations.

Written in an informal way and containing a wealth of

interesting historical material, Duelling is ideal for those who are fascinated by mathematics and the role it plays in everyday life. 65 liM i/JunntionJ. 42 computer $imulillions. Cloth $24.95 ISBN 0·691 -00979-1 Due October

Princeton University Press (1 -243) 779777 U.K. • 777-4726 U.S. • WWW.PUP.PRINCETON.EDU

VOLUME 22. NUMBER 4. 2000 15

Bertrand 's Paradox Revis ited

Random Chords- Paradox Lost?

The classic form of Bertrand's paradox concerns "random chords" of a circle [B1907, §5]. Bertrand asks for the probability p that such a chord is longer than the side of the equilateral triangle inscribed in the circle. He then computes p in three different ways, obtaining three different values. His conclusion is that the question is mal posee. Indeed, it is not hard to believe that his computations implicitly interpret "random chord" in three different ways. The paradox, though disconcerting, appears to be easily resolved. Let us recall some details.

One can argue that p = 1/3, as follows. We may fix, or regard as known, one end P1 of the chord. By symmetry, or the "principle of indifference," this knowledge cannot affect the outcome. The angle t/J between the chord P1Pz and the tangent to the circle at P1 (see Figure 1) is uniformly distributed over [0,7T].

The chords that form sides of an inscribed equilateral triangle correspond to t/J = 7T/3 and t/J = 27T/3, so that favourable values of t/J lie in the middle third of the interval [0, 7T]. It follows that p = 1/3. A variant interpretation of "random chord" chooses the endpoints P1 and Pz of the chord independently and with uniform distribution over the circumference of the circle. The equivalence of these two models is clear upon noting that the angle subtended by the chord at the circle's centre is 2tfr (see Figure 1).

On the other hand, one can argue that p = 1/2, as fol-


JOHN HOLBROOK AND SUNG SOO KIM

lows. Again by symmetry, we may consider the direction of the chord fiXed. Let us represent a fiXed diameter of the circle by the interval [ -R,R], where R is the radius of the circle. A chord perpendicular to this diameter intersects it at some point x E [-R,R], and it is easy to see that sides of an inscribed equilateral triangle correspond to x = -R/2 or x = + R/2. The favourable longer chords are obtained for x E [ -R/2,R/2], so that p = 1/2. See Figure 2.

Bertrand also proposed a third model, implying that p = 114. We'll come to that in our discussion of PET-scans.

Cosmic Rays-Paradox Regained?

Suppose we present Bertrand's problem as a physical experiment, for which there is presumably just one correct interpretation. To be specific, imagine that cosmic rays produce tracks in some disc-shaped detector, perhaps a classic Wilson cloud chamber. If the chamber is a thin disc of radius R, it will detect, as chords of the corresponding circle, just those cosmic rays travelling in the plane of the disc. We seek the probability p that such a cosmic ray track is longer than v'3R (the length of a side of an inscribed equilateral triangle). It has been our observation that students presented with the problem in this form may respond with rather convincing arguments for both the p = 112 model and the p = 113 model. Now, however, only one (at most!) of these models can be correct.

On the one hand, those cosmic rays falling perpendicu-

Figure 1. Bertrand's first model, with 1/J uniformly distributed.

lady on a given diameter [ -R,R] of the chamber (which Figure 2 might also depict) will clearly be uniformly distributed over the interval. They make tracks of length greater than v3R exactly when they fall on the middle half of the interval; hence p = 112. On the other hand, each cosmic ray must enter the chamber at some point P1 (as in Figure 1), and it cannot matter which. Moreover, the isotropy of the cosmic radiation suggests that the angle of incidence should range uniformly over the interval [0, 1r] from one tangent direction to its opposite. But this angle i� essentially if; in Figure 1, and we may follow the corresponding argument to claim that p = 1/3.

Many puzzling probabilistic paradoxes have been invented (see Szekely's rich compendium [Sz1986], for example), and Bertrand's paradox has been examined from sev-

- R -L· ,_ :

X 0

Figure 2. Bertrand's second model, and a cosmic ray.

R.

.. · · ·.·. ·� e � f

.. ·

· .

. .. .

. . )

Figure 3. Cosmic rays enter a tangent window of width ,;.

eral points of view (see our final section for a brief account). Yet it appears that the more urgent form of the paradox that is posed by our physical model is not widely known. Of course, this form of the paradox can also be resolved.

One way to view its resolution is by refming our physical model. We may think of the incoming cosmic rays as entering the detection chamber through any one of a finite number of tiny windows, represented by the sides of a regular n-gon ( n large) approximating the circular boundary. It then becomes clear that the distribution of if; for rays entering a given window depends on the cross-section presented by the window to rays at angle if; (see Figure 3). Thus the distribution of if; E [0, 7T] has density proportional to sin if;. This leads to the calculation:

1 2 '

the same value obtained from the other argument (which the experienced reader will, perhaps, have picked as the "right" choice all along).

Other Rays, and PET -scans

Curiously, another well-known cloud chamber experiment does require the uniform distribution of if;. If a small radioactive sample is fixed to the wall of the chamber (at some point P1 as in Figure 4), then some of the particles emitted (a and {3 "rays") will form tracks with a uniform distribution of if; over [0,7T]. Thus, by the now-familiar argument, the proportion of tracks of length exceeding v3R will be 1/3.

In Figure 4 we have also suggested a model that has been studied abstractly (see [Szl986, §l . lld]) but could be viewed in terms of positron emission tomography. During


"1_

Figure 4. a-rays from radioactive sample at P1, and PET -scan y-rays.

a PET-scan, a positron-electron union may occur at any point M in the object under study. This results in the emission of a pair of y-ray photons, travelling in opposite directions away from M and defining a chord AB, as in Figure 4. If we assume that M is uniformly distributed with respect to area within the disc of radius 1, then the distance r from M to the centre 0 has density 2r. If we also assume that the angle 'P is uniformly distributed, we have yet another variant of our question: what, now, is the probability p that AB exceeds the length of a side of the inscribed equilateral triangle? This is just the probability that the midpoint of AB lies within distance 1/2 of the centre 0. It is an amusing exercise to verify that

P = JI sin-1(1/2r) 2r dr = .!. + v3 = 0.609, 0 7T/2 3 27T

where sin-1(1/2r) is interpreted as 1r/2 when r < 1/2. If, for some reason, the y-rays always trace out the short

est possible chord through M (i.e., if 'P is always 7T/2), we arrive back at the simple model proposed by Bertrand as his third interpretation of "random chord." In this case, each random choice of M determines the unique chord CD having M as midpoint, and the probability p is just the probability that M is within distance 1/2 from the centre, namely the relative area of the disc of radius 1/2. Thus Bertrand obtained his third answer: p = 1/4.

Crofton, Borel, Poincare, et al.

. . . geometric probabi li ti es have run into diffi culti es culmi nati ng in the paradoxes of Bertrand whi ch threatened the fledgli ng fi eld wi th bani shment from the home of Mathemati cs . . .

-Mark Kac, from his foreword to [81976]

While Kac may have been indulging in provocative overstatement, Bertrand's paradox is rather anxiously discussed


in many of the classic works of probability, and it is interesting to trace the various interpretations and attitudes found there. Crofton, in [C1885] (written even before the first edition of Bertrand's book), describes a model for random lines in the plane that corresponds to our cosmic ray analogy. Crofton's model is measure-theoretic but can be changed to a probability model by the device of regarding a random line as a set of parallel lines, making angle (} with some fixed direction and separated by 2R, where R is again the radius of our circle. If r is the distance from the centre of the circle to the closest line of the family, Crofton's model says that r and (} are independent and uniformly distributed on [O,R] and [0,27T], respectively, and it leads directly to the conclusion that p = 1/2 (see [C1885], [S1976]).

Edgeworth, on the other hand, writing in a later edition of the Encyclopaedi a Britanni ca [E 1911] than did Crofton, seemed willing to consider random lines as those defined by equations ax + by = 1, where the parameters a and b are independent and uniform over the real line. Moreover, von Mises [vM1964] felt that even in the famous Buffon needle problem the various possible models must be distinguished with care.

Borel discussed Bertrand's paradox in [Bo1909], but spent more energy on a related paradox from [B1907,§7] concerning the distribution of latitude for a point chosen at random on the surface of the earth. If we first use symmetry to claim that, as each longitude is equivalent to every other, we may restrict our attention to points on a single meridian, then we come to the wrong conclusion that latitude A will be uniformly distributed over [ -90,90]. Borel uses a device analogous to our infmitesimal windows (those that admit cosmic rays in an earlier section), namely the narrow sectors between nearby meridians. By this means, he explains that A will have a density proportional to cos A (see Figure 5). We note that an even more classical treatment of this problem may be based on Archimedes's theorem on the areas of spherical slices between parallel planes: these areas are proportional to the

e '\. I.A t\to '("

Figure 5. Meridians versus infinitesimal sectors.

AU T H O R S

JOHN HOLBROOK

Mathematics & Statistics Department

University of Guelph

Guelph, Ontario N1 G 2W1 Canada


John Holbrook studied mathematics at Queen's University in

Ontario and at Caltech. He has taught in the United States

and in Venezuela; for many years he has been at the University

of Guelph. He has worked mostly in operator theory and ma

trix analysis, but also in electron micrograph tomography, ran

dom balanced sampling, and computational experiment. Here

he is seen with a grandson, also John Holbrook.

thickness of the slices, i.e., the distance between the planes. This means that sin A is uniformly distributed, and that A has density cos A/2.

Poincare [P1912] resolves the Bertrand paradox more formally in terms of the transformation of coordinates. His view is that the model for random lines should be invariant under rotations and translations, so that if (r, O) are the polar coordinates of the point on the line that is closest to the origin, then r and 0 are independent and uniformly distributed (Crofton's model). Conclusion: p = 1/2, and if the variables 1/J (from Figure 1) and 0 are introduced with the Jacobian factor (sin 1/J) relating them to r and 0, we also get p = 112. This is tantamount to our "physical" argument, reflected in Figure 3.

Jaynes applied another invariance argument to Bertrand's problem (see [J1973], [Sz1986]). He argued that the size and location of the circle are not specified by Bertrand. The distribution of lines should be invariant under expansion/contraction, translation, and rotation, in order that the normalized distribution of chords should be identical for every circle. This leads to Crofton's model, so that according to Jaynes's invariance argument, Bertrand's problem is well-posed and p = 112.

Acknowledgements

(1) To the MPC2 students at the University of Guelph, for their spirited arguments about the physical implications of Bertrand's paradox.

SUNG SOO KIM

Hanyang University

Ansan. Kyunggi 425-791 Korea

[email protected]

Sung Soo Kim studied mathematics at Hanyang University,

and did his graduate work at Kaist (Korean Advanced Institute

of Science and Technology) under Kil Hyun Kwon. He has

been at Hanyang University since and is now an Associate

Professor. He and his wife Beanmoo have no children, but en

joy playing with twin nieces , Youkyung and Soobean.

(2) To John Milton, whose epic poems permit (perhaps) the pun: Paradox Lost/Regained. Mter putting this note together we came upon Ian Stewart's column in Scientific American (June 2000), where paradoxes are also lost and regained. One wonders when, in the 3.5-century interval since Milton's time, this little joke first occurred. We don't know, but a cursory search turned up the story "Paradox Lost" by SF writer Fredric Brown; it was first published in 1943.

REFERENCES

[81 907) J. Bertrand, Calcul des 'Probabilites, Chelsea (reprint of 1 907

edition)

[Bo1 909) E. Borel, Elements of the Theory of Probability (translation by

J. Freund), Prentice-Hall 1 965

[C1 885] M. W. Crofton, Probability, in Encyclopaedia Britannica, 9th

ed., v. 1 9, 768-788, 1 885

[E1 9 1 1 ] F. Y. Edgeworth, Probability, in Encyclopaedia Britannica, 1 1 th

ed., v. 22, 376-403, 1 91 1

[J1 973) E. T. Jaynes, The well-posed problem, Foundations of Physics

3, 477-493, 1 973

[vM1 964] R. von Mises, Mathematical Theory of Probability and

Statistics (H. Geiringer, ed.), Academic Press, 1 964

[P1 912) H. Poincare, Calcul des Probabilites, Gauthier-Villars, 1 91 2

[81 976) L. Santal6, Integral Geometry and Geometric Probability,

Addison-Wesley, 1 976

[Sz1 986) G. Szekely, Paradoxes in Probability Theory and Mathematical

Statistics, Reidel, 1 986

VOLUME 22. NUMBER 4, 2000 19

[email protected]§116'h¥1MQ.'I,'i,tl!:hh¥J Marjorie Senechal , Editor I

Impoverishment, Fem in ization, and Glass Ce i l ings: Women in Mathematics in Russ ia Karin Johnsgard

This column is a forum for discussion

of mathematical communities

throughout the world, and through all

time. Our definition of "mathematical

community" is the broadest. We include

"schools" of mathematics, circles of

correspondence, mathematical societies,

student organizations, and informal

communities of cardinality greater

than one. What we say about the

communities is just as unrestricted.

We welcome contributions from

mathematicians of all kinds and in

all places, and also from scientists,

historians, anthropologists, and others.

Please send all submissions to the

Mathematical Communities Editor,

Marjorie Senechal, Department

of Mathematics, Smith College,

Northampton, MA 01 063, USA;

e-mail: senechal@minkowski .smith.edu

From May 16th to 25th, 1998, a delegation of women in mathematics

visited St. Petersburg and Moscow, meeting with their female counterparts at secondary schools, universities, and research institutions of the Russian Academy of Science. The delegation consisted of eight Americans and one Norwegian-some professors, some high school teachers. (At the end of this article is information about the individual members of the delegation.) The purpose of the meetings was to exchange information about research and education mechanisms and issues in our countries, and specifically to discuss gender-related issues in education and professional advancement. 1

It should be understood that the participants in the meetings (on both sides) were mathematicians, not historians or political scientists. None of the delegates even spoke Russian (we relied on translators). This article, therefore, should be viewed in that light: One woman's attempt to collate a body of largely unverifiable and sometimes selfcontradictory data, impressions, and first-hand observations from an all-toobrief visit to a country from which more reliable data is not obtainable, from her own perspective as an intelligent but ignorant first-time visitor.

A Delegate's Perspective,

Prior to the Journey

Why should the mission focus on women mathematicians in particular? Why did we go to Russia and not some other nation? And why did anyone feel that the

best way to conduct the mission was personal contact, rather than just looking up information on the Internet or in a book?

The answers to these are interrelated. With regard to the choice of nation: I now know that many nations would be appropriate for an exchange on gender issues in mathematics. Russia, however, does have the distinction of not maintaining data on women in mathematics, making longdistance study impractical. Relations between Russia and the West were unusually good in 1998, making a journey to that nation appropriate and timely.

Readers uncertain as to why the delegates thought that there might be any gender-sensitive issues in mathematics should consider the following. That summer of 1998, the U.S. Census Bureau reported that the average earnings for an American woman were 7 4% of the average for men; this percentage was a new high for our nation. With regard to academia in particular, the AAUP Annual Report of 1997-98 reported disparities in academic salaries (appears at the bottom of the page).

In 1993, women made up 45% of the U.S. workforce, but only 16% of employed scientists and engineers. Among recent graduates in science and engineering, 35% of the men were actually employed in such occupations, while only 18% of the women were. There was also an earnings gap: among 1993 college graduates, women's median starting salary was 84.2% of men's over all subjects, and 84.7% in natural sciences and mathematics.

U.S. Academic Salaries: Women's salary as a percentage of men's [AAUP]

Public Institutions Private Institutions

Rank '82-'83 '87-'88 '92-'93 '97-'98 '82-'83 '87-'88 '92-'93 '97-'98

Prof. 90.5 89.4 89.3 87.7 84.9 85.2 85.5 87.2

Assoc. 94.2 93.1 93.4 92.9 92.1 91 .6 92.4 92.0

Asst. 93.6 91 . 1 92.7 93.9 91 .7 89.2 90.3 91 .9

(Notice that the gap has actually widened at public institutions, and remained essentially unchanged at private institutions.)

1The delegation and itinerary were organized by People-to-People International, a non-profit organization dedicated to increasing international understanding. See website http://www.ptpi.org for more information.


In 1996 at U.S. colleges and universities, women earned over half of all bachelor's and master's degrees and 40% of all doctorates. However, in science and engineering disciplines they received only 300!6 of bachelor's degrees and less than a quarter of advanced degrees2 [A, NCES].

% of U.S. degrees eamed by women In

'95-'96

Overall and in math-related fields [NCES]

All fields

Math

Phys. Sci.

Comp. Sci.

Eng.

BA/BS MAIMS Ph.D.

55.1 55.9 39.9

45.7 38.8 20.43

36.0 32.2 23.1

27.5 26.7 1 4.5

1 6. 1 1 7.2 1 2.5

The percentage of women receiving U.S. doctorates in mathematics has been increasing, especially among American recipients. (The figures shown include statistics degrees, among which 34.3% of the '97 -'98 recipients were female.)

Despite the fact that about one in four recent math Ph.D.'s for that last decade have been female, the proportion of women among the degree recipients who were hired by doctorategranting math departments in 1998 was ocly 18.5%. (This has been fairly constant over recent years.) Among the 48 departments ranked as excellent, the percentage of women among those hired was 16.2% (at private schools in this category it was 7.1%) [DMR]. Virtually all of the hires at the wellranked universities are temporary instructorships. At the time (1991) of the Jenny Harrison sex discrimination case against UC Berkeley, there were only four tenured women (to 303 tenured men) in the ten top-ranked math departments in the United States, and one untenured woman to 86 untenured men [S]. (I am unaware of any significant improvement since that time. )4

The standard justification for these inequalities, of course, is that "women are not as good at math," and the proof

% of U.S. math doctorates eamed by women, by year

Over all recipients and over U.S. recipients [NCES,DMR]

All

u.s.

'77-'78

1 4.9

1 4

'82-'83

1 6.4

20

of this assertion is invariably given as the Scholastic Aptitude Test, Math section. There has been a very persistent gap in the mean scores of the two genders, with girls' mean currently about 93% of the boys' (SAT]. The conclusion about their relative abilities is then clinched by examining the very highest scorers and noting they are disproportionately male. But in studies of total population, both the highest and the very lowest scorers are disproportionately male; the real difference is that the male scores have a larger standard deviation [S]. Almost none of the low tail of the curve is reflected by the SAT; in 1997, only 14% of exam-takers reported GPA's lower than a B [NECS]. Nevertheless, the 1997 U.S. Dept. of Education report [BS] uses precisely this same examination of exceptionally high scores to report, "Men score higher than women on the SAT mathematics . . . and . . . Advanced Placement (AP) examinations."

In fact, the SAT-M consistently reports a greater gender difference in math than other standardized exams. A remarkable cross-study analysis on this topic was carried out in the late 1980s [HFL]; the following is taken from the abstract of the results.

Reviewers lwve consistently concluded tlwt males perform better on mathematics tests tlwn females do. To make a refined assessment of the magnitude of gender differences in mathematics performance, we performed a metaanalysis of 100 studies . . . . Averaged over aU effect sizes and based on samples of the general population, d was -0. 05,5 indicating tlwt females outperformed males by only a negligible

'87-'88

16 . 1

21

'92-'93

23.8

28

'97-'98

24.4

28

amount. For computation, d was -0.14 (the negative value indicating superior performance by females). For understanding of mathematical concepts, d was -0.03; for complex problems solving, d was 0. 08. An examination of age trends indicated tlwt girls showed a slight superiority in computation in elementary school and middle school. There were no gender differences in problem solving in elementary or middle school; differences favoring men emerged in high school (d = 0.29) and in coUege [d = 0.32). Gender differences were smallest and actuaUyfavoredfemales in samples of the general population, flrew larger with increasingly selective samples, and were largest for highly selected samples and samples of highly precocious persons. The magnitude of the gender difference has declined over the years; for studies published in 1973 or earlier d was 0.31, whereas it was 0. 14 for studies published in 1974 or later. We conclude tlwt gender differences in mathematics performance are smaU.

The authors of the study found that if they included the SAT-M, d overall was 0.20, but if it was excluded, d was 0.15. For SAT-M itself that year, d = 0.40.6 "The �agnitude of the gender difference favoring males grew larger as the sample was more highly selected: d was 0.33 for moderately selected samples (such as college students), 0.54 for highly selected samples (such as students at highly selected colleges, or graduate students), and 0.41 for samples selected for exceptional mathematical precocity.'' The authors concluded that the SAT-M scores are an anomaly, even among moderately se-

2The percentages are significantly lower if the field of psychology is not included among these. 3247 women out of 1 209 total. The AMS annual suNey found 250 women out of 1 1 54 total, for a representation of 21 .7%.

4See the final section, "Closing thoughts, " for more on these issues. 5This d is the mean for males mi_nus the mean for females, divided by the mean of the two within-sexes standard deviations. 6Considered "moderate to large." I computed a figure of d = 0.32 for 1 999 SAT·M data [SAT], but lack the resources to do a comparable meta-analysis on other recent data.


lected samples. They obseiVed that more girls than boys take the SAT, and the girls are overall less advantaged in terms of family income and level of education, and less likely to attend private schools. 7 They also suggested that there may be problems with the test itself or its administration.

An international study of the same period [H] testing five subject areas of math in 20 countries found inconsistent results: girls out-performed boys in some areas in some countries but outcomes were reversed in other countries. Overall, differences between countries greatly exceeded gender differences, which were insignificant in three subjects and small in the remaining two.

Note that the atypical but widely publicized gender difference on the SAT-M and AP (both of which are selfselecting exams) occurs precisely where the cross-study says the male advantage should be greatest: at the extreme high end. Even the testing service that administrates the SAT and AP cautions that their test scores considered alone are an inadequate indicator of potential in undergraduate studies (far less a measure of future success in profession). In particular, the SATM underestimates freshman GPA for girls by 0. 10, and overpredicts for boys by 0.1 1 [SATR] . Girls continue to get better grades than boys do in college math courses such as calculus! In math courses, among students receiving the same course grade, girls had SAT-M scores 33 pts lower than the boys did. Standardized multiple choice math exams do not appear to reflect faithfully the qualities that instructors seek and evaluate in the classroom.

I am unaware of any studies linking professional mathematicians and scientists with exceptional SAT-M scores. (My own were high but not extraordinary, and my first GRE-M was quite low. Nevertheless, I won an award for original research in graduate school, and was both a Sloan Fellow and an

ally pursue training and careers in fields emphasizing these abilities. Males are still more likely to take math and physics courses in high school (although the gap has narrowed substantially), and more likely to take AP test in these areas [BS]. Female representation in math and science studies decreases astonishingly at every step from high school to doctorate. Many studies suggest women's flagging participating during this period reflects poor early training (often resulting from inappropriate academic advising), unequal classroom treatment, lessening levels of interest, or self-doubt, rather than innate lack of ability.

Percent of U.S. mathematically talented

students earning Ph.D.'s In math,

engineering, and physical sciences [A]

Math Eng. Phys.

Women 0.1 0.7 0.25

Men 0.5 3.4 3. 7

So in brief, in the U.S. women are scarce in math at all levels (but de-crease even further toward the higher levels), do not get hired at the best re-search universities, and earn less money than their male peers. Some have mentioned other problems-lack of professional opportunities, disrespect among one's colleagues, sexual harassment-but this quick sketch should suffice to give an impression of

why the delegates might be curious about conditions elsewhere. I, at least, did not know whether I should expect matters to be better in Russia, worse, or not significantly different; but I was eager to investigate.

A Brief Overview of Impressions

and Discoveries

The delegation's visit (May 16th to 25th) took place shortly before the July 1998 Russian economic disaster and ensuing political developments. It was a time in fact of unprecedented Russian prosperity. This circumstance should be kept in mind throughout.

My own subconscious expectations before the trip were wildly mixed. The most negative of these (derived from a high school visit to Hungary in coldwar conditions) were of peiVasive grayness of scenery and spirit: a general poverty of beauty or extravagance of any sort, rigorous public adherence to doctrine, and personal isolation and distrust. In opposition to these were mental images of modern western European cities, and almost a fear that I would fmd Russian cities virtually indistinguishable from these. Of course neither extreme proved accurate. On the whole, however, the Russian cities seemed more closely akin to Newark than to cold-war Budapest. The stores were full of goods of all kinds, even

NSF Postdoctoral Fellow.) The members of the delegation with the women of Moscow school #103: (1-r) Eleanor Jones,

Of far greater relevance than SAT-M Liv Berge, Olga Medovar [Moscow teacher], Mila Bolgak [translator], Sue Geller, Diana Vincent,

scores is the question of whether math- Audrey Leef, Karin Johnsgard, Elena Fyodorova [principal], Pamela Ferguson, Natalie

ematically gifted girls and women actu- Golubinsteva [assistant principal], Maureen Gavin, Lucy Dechene.

7These still hold; 7,529 more girls than boys took the SAT in 1 996 [SATR].


luxury items, and the streets were filled both with bustling business people in fashionable garb and with wandering panhandlers. The woefully archaic streets were jammed to bursting with cars, and the walls were �lastered with neon restaurant signs and billboard advertisements for cigarettes, liquor, and cosmetics. New banks, one woman told us, "spring up overnight like mushrooms." At the same time, actual production seems scant. The laws and taxes are so labyrinthine and severe that itis all but impossible to start or run a business legally. Farming as it is known in the West is purely impracticable; thus, it is easier to import even food than to produce it locally.

Everywhere we saw fabulously colorful old palaces and breath-taking churches, and the churches were really being used as places of worship, not as "cultural museums." (Religion, we were told, is "fashionable" currently.) A drive outside of St. Petersburg revealed both impoverished collective farms and the intimidating new brick dachas of "New Russians" (i.e., mobsters). People we spoke to were open about their problems, hopes, and fears; we encountered some indifference, a ltttle downright rudeness, a fair amount of perplexity over the purpose of our visit, and a great deal of curiosity about ourselves and willingness to cooperate with our questions.

In light of the then current Russian economic prosperity (however unfairly distributed and ephemeral), the delegation's basic fmding stands even more starkly illuminated. The overwhelming issue affecting our Russian counterparts is basic subsistence funding: 900,1, of Russian scientists and educators live in poverty, even though many hold second jobs. Library, computer, and technical funding at many institutions have shrunk to the point of being virtually non-existent, and are expected to decrease further. An inevitable corollary has been the Russian "brain leak": 80% of Russian scientists and teachers wish to go abroad, and 70,000 to 90,000 do so each year,8 with 52% of those departing being mathematicians or physicists. Those leaving

Some of the professors and graduate students of St. Petersburg State University. Photo by

Sue Geller.

the country (or abandoning science for more lucrative professions) have been disproportionately male, leading to "feminization" of the sciences and education: women accounted for 53% of Russian scientists and educators in 1992, up from 47% in 1970 and 42% in 1960 [AWSE]. (We will see that increasing female representation has not ensured equity, however.) These basic trends (except feminization) were discussed everywhere we visited.

With regard to gender issues: Many Russian women we spoke to denied that discrimination existed or was possible in their country, and saw no reason for the existence of any organization specifically for women in mathematics. These same women made statements (or heard without apparent reaction compatriots' statements) such as, "Well, girls aren't as good in math as boys are." Statistics provided by the [Russian] Association of Women in Science and Education (A WSE), employment information from the Central Economics and Mathematics Institute (CEMI), and a study of employees of the Academy of Science all reveal virtually impenetrable glass ceilings for women in their professional advancement. Thus there is a substantial discrepancy in earning power be-

tween men and women. In addition, because school choices with long-term effects are made for children while they are quite young, the system tends to perpetuate parental stereotypes about gender aptitudes.

Based on our classroom observations and our discussions of curriculum with Russian instructors, the quality of education appears to be extremely high for those who receive it, at least in the institutions visited by the delegation. The dedication and achievement of Russian educators in the face of dwindling resources is remarkable. This level of excellence will be unsustainable if Russia continues to lose itS best minds to other nations by withholding decent living wages; to deny its institutions of learning and research basic operating expenses; and to allow further erosion of a onceexcellent system of universal preschool training.

And the delegation's reactions to all of this? In my own case at least, my mood varied with the reactions we elicited at a given meeting. At our first meeting for example (at St. Petersburg University), I felt horror (over the appalling state of the facilities and lack of resources), shame (for my past disinterest and ignorance of these condi-

6This is as many as for the past decade in Latin America and Asia combined.


tions ), awkwardness (neither side seemed to be completely prepared for the meeting or to know how to conduct it), and above all, frustration: frustration over the sense that the two sides' purposes were at odds, that we were failing to communicate, that we were asking the wrong questions and that we were equally unable to elicit interest in our topics of greatest concern, that we were unwelcome, that our contacts were either lying about or were fundamentally unable to perceive what the deleg_ates saw as blatant institutionalized gender discrimination.

I was not the only delegate to find this first meeting difficult. In a discussion afterwards over lunch, the delegates dissected the event and our various dissatisfactions with it. I said plainly that I felt pessimistic about the trip in general, suspecting that "I had been a fool to think I would have anything to contribute." Another woman rebuked me, saying that it would be arrogant to suppose that we could "contribute" anything at all, and that we should be content simply to learn. We all were unhappy with the way our Russian contacts had seemed to view the purpose of the meeting simply as a forum for us to ask questions of them, while showing no interest in learning anything from us. We resolved that at the next meeting (to take place at a secondary school) we would volunteer information whether it was requested or not. We also resolved to word our own questions more carefully, giving illustrative examples instead of using terms subject to misinterpretation. As a probably consequence of this resolution, we did a great deal of the talking at our next meeting and elicited plenty of sympathy, but didn't learn much!

One recurring source of miscommunication was due to the different university degree structures within the U.S. and Russia (explained below). Both sides in the meetings were prone to assume that the degree systems were comparable. Thus, the delegates felt belittled when our doctorates were apparently being interpreted as mere Master's degrees, and were shocked at how few female Russian doctorates we met. On the reverse side, at least once a Russian was astonished that one of


my youthful appearance could possibly hold a doctorate.

The imperfect understanding of purposes in our professional meetings was a continual disappointment during our trip, as much as we cherished other experiences. I felt that only one of these exchanges represented a true meeting of minds and purposes: our meeting with the Association of Women in Science and Engineering (AWSE). This was the one time that both sides clearly shared basic working assumptions about being female and a mathematician. Finally, we had found contacts who understood our purposes and concerns; who had studied the issues themselves and were happy to share their data (unobtainable otherwise); who were eager to hear about our own experiences and ideas. Our relief and joy were profound. It was largely this meeting that makes me personally feel that our mission was not a failure in its most critical goal: fmding common ground for understanding.

One other meeting was notable. Immediately after our exchange with A WSE, the delegates (and one A WSE member) went on to the Central Economics and Mathematics Institute (CEMI). It was by this time late on a Friday afternoon; the women who had been assigned to meet with us there at first seemed bored, barely masking their impatience to leave for the weekend. For ourselves, we were starting to feel somewhat burnt out (it was our third meeting of the day, on the heels of an overnight train ride from St. Petersburg) and a bit let down over what appeared to be another indifferent reception right after our wonderful high-energy meeting with A WSE. For example, during the round of introductions one young woman's research was described to us as being so striking and original that there had been some talk of awarding her a doctorate. This paragon said nothing in response to this description, but just smiled dreadfully in a way that I could only interpret as cynical. Her silent reaction struck me; clearly, she felt our officially sanctioned meeting was meaningless, not even meriting the effort of honest participation. Our dis-

cussion continued in a stultified fashion until one of the delegates, Pam Ferguson, inquired about CEMI's system of employee ranks. She had hit upon the key point! The women seemed to open to the discussion one by one, in order to air their grievances on the glass ceiling conditions at CEMI. Ms. Vinokurova, who had accompanied us from the A WSE meeting, gave an impromptu but passionate address on obstacles to women's advancement in the mathematical sciences. We inquired whether any of the other women were affiliated with A WSE, and were simply stunned to discover that the rest had never heard of this organization-with a chapter located perhaps five minutes drive away! Even our translator seemed struck by the gulf between these backto-back meetings. The CEMI researchers no longer seemed bored; I would describe them as seeming somewhat shaken and beginning to question whether our visit to CEMI was so inappropriate and meaningless after all. They certainly did not seem sleepy and indifferent any more. Delegate Sue Geller described A WM (Association for Women in Mathematics) to them and explained that its origins were as humble and self-funded as those of A WSE, but that it now has government funding for its projects and carries real clout with other national mathematics organizations. We departed from this meeting feeling that at the very least, our visit had given them food for thought. It made a satisfying conclusion to our meetings!

In the following three sections I give more detail about conditions in Russia, separating these into economic effects on the scientific community, the Russian educational system, and gender differences. For more about my personal reactions, see the concluding section.

Poverty in the Russian Scientific

and Educational Community

Our delegation visited only St. Petersburg and Moscow. However, we were provided with some more general data by A WSE, recording the plummeting economic status of scientists and educators:

Salary of general science worker (as a % of the national average salary)

Year 1 940 1 960 1 980 1 990 1 991 1 993 1 994

Science salary avg. 1 42.3 1 37.3 1 06.3 77 65 66 76*

*(60 in Moscow)

Academy salary (as a % of the national average salary)

Year 1 970

Avg. salary (All univ. prof.) 1 68

Avg. salary (Full prof./Dept. Head) 41 0

In a survey of (mostly female) math

ematicians attending a conference, the

highest monthly salary reported was

2,500 rubles (about $440). For 27% of

respondents, monthly earnings were

less than 600 rubles (about $104), and

for 18% less than 300 rubles; some re

ported as little as 150 rubles. Our trans

lator in St. Petersburg told us that her

mother, a professor, earned a salary of

$30 a month.

A popular Russian witticism is that

since educators continue to teach no

matter how much their salaries are

slashed, the government is now trying

to devise a way to charge them for the

privilege of working. At CEMI, this is

very nearly true: a junior researcher

earns 340 rubles a month, but has to

pay 180 rubles each month for a "traf

fic card" in order to get to work

.. The most visible and distressing

signs of poverty witnessed by the dele

gation were at St. Petersburg State

University. The faculty of necessity has

been forced to use old textbooks, as

very few new ones can be obtained, and

even used ones cannot be supplied to every student. Library funding (particu

larly for journals) is all but non

existent; the faculty asked us to send

them even year-old copies of journals.

The library holdings are not computer

ized-students do not have access to stacks, but must search the card catalog

and make written requests for specific

volumes from the over-worked staff.

The library has no copiers. Computer

equipment at the university is outdated

and in short supply; there is no student

access to printers, and access is difficult

to obtain even for faculty. The tmiver

sity building interiors are chipped, bro

ken, and peeling, and only one in three

toilets actually functions. Enrollment is

1 979 1 989 1 994 1 996

1 50 1 08 55 45

290 240 95 89

falling (245 students at the main cam

pus, versus a previous enrollment of

400) and their best and brightest stu

dents go abroad afterwards. (It should

be emphasized that the programs are

academically rigorous-insofar as we

could determine by discussion of cur

riculum with the faculty-and the qual

ity of the education appears to be ex

cellent. Almost the first thing we were

told was that the school's computer

team had recently won second place in

an international competition in the

United States; considering students' lim

ited access to resources, this seems in

credible.)

Although the conditions appeared

worst at St. Petersburg University,

everyone encountered by the delega

tion spoke of funding problems, par

ticularly in terms of journals and equip

ment, and of an inability to provide

fellowships, summer programs, or

travel grants for young mathemati

cians. Travel outside the former Soviet

states, once prohibited for political

reasons, is now generally impossible

because of economic hardship. Dr.

Yuri Matiyasevich (of Hilbert's Tenth

Problem acclaim) asked us to inform

others of the existence of the Euler

Institute in St. Petersburg, an institu

tion that serves as a contact point for

Russian and foreign scientists. This

converted mansion has facilities for

small conferences (including beautiful

rooms and dining facilities, a library,

computers, and electronic equipment)

and small rooms for "tete-a-tete" pair

ings of Russian and foreign mathe

maticians doing research together.

This Institute, too, is underfunded, and

the conference facilities all too often

go unused. (For more information,

check http://www.pdmi.ras.ru/EIMI.)

Summary of the Russian

Educational System9

Expense

Although education is mandatory in

Russia, the agencies that formerly en

forced participation are no longer

funded, and many children are now

working or begging instead. Kinder

gartens were free under the Soviets,

but now free kindergartens are both

rare and of very dubious quality. Both

private and public primary and sec

ondary schools exist; lunches for pub

lic school students are funded by the

city government, but transportation is

not. Private schools may cost as much

as $600 a month (three times the aver

age monthly Russian salary). Never

theless cost does not guarantee quality

education, or even that a school is

legally entitled to present, graduates

with the official document of educa

tion. University tuition is free to stu

dents passing the admission exams,

but students must pay their own living

expenses.

Infancy through primary grades

Ideally, children aged one month to

three years attend nursery school, fol

lowed by kindergarten for ages three

to five (kindergarten classes are di

vided into three grades). Primary

schools (grades one through three)

have entrance exams: children should

be able to read, count to a hundred,

memorize and recite poetry, and

demonstrate set recognition (e.g.,

"Which one doesn't belong?"). In the

ory, children have the right to attend

their local school even if they fail the

exams, but in practice this is not the

case because students who perform

well are accepted in preference to

those who have not. Some of the pri

mary schools are "specialized." For ex

ample, a school may offer heavy con

centration in some subject such as

math or foreign languages; such

schools assign homework, perhaps an

hour's worth each day. Schools that

are not specialized are not appropriate

preparation for students who will

eventually enter institutions of higher

learning. (Thus, parents decide when

91nformation in this section was 'obtained primarily in briefings by the group's translators; we also obtained details directly from Russian educators in response to our questions.


their children are six or seven whether they will prepare for a college education, and even broadly which subjects will be potential majors.) Students are responsible for keeping the school and grounds clean. Classes include a "home craft" lesson, which consists of cooking and sewing for the girls, carpentry for the boys. At the end of each spring semester are exams. The scores are 1 to 5, with 5 being highest; a student who receives all 2's must repeat the grade, and a student who fails is expelled. (Such a student may of course 'seek enrollment elsewhere; see "Addressing specialized needs.") Students finishing primary school take graduation exams in math, Russian, literature (which is an oral exam), and physical education.

Secondary school

There is no "grade four" (by that name) due to an education reform. Students entering secondary school go directly into what is called "grade five" and continue to "grade eleven." Again, secondary schools may be ordinary or specialized. Homework is heavy and is referred to as "torture"; it consists (at a specialized school) of some three to four hours of work each evening, primarily math and literature. Children have seven to eight lessons daily, plus extracurricular subjects such as History of Art. Specialized schools may have arrangements with nearby research institutes to provide special advanced classes. There are graduation exams in the core subjects.

Higher education

Seventy percent of Russian children fmishing secondary school receive further education. This may consist of "college," which is really vocational school or educational training, or university study (which requires passing

entrance exams). University-level instruction is also available from research institutions such as CEMI.

The university degree structures in the U.S.10 and Russia are quite dissimilar, which led to confusion for the delegation on both sides. The Russian "candidate of science" is considered the terminal degree, although it falls between the American Master's and Doctoral degrees in terms of rigor. The Russian "doctorate" is actually an additional step that may be conferred on established professionals after years of work experience in their field, and is based on an overview of total accomplishments. This degree is necessary for those wishing to rise to certain high ranks in their professions, but is not a prerequisite for employment and therefore is often not sought. The very highest scientific honor is admittance to the Academy of Science; a member is distinguished by the title, "Academician."

Addressing specialized needs

There exist (often very expensive) private schools for children with learning or behavioral problems, or offering high security for wealthy parents who believe their children at risk for kidnapping. One of the secondary schools the delegation visited was one of four special government-subsidized boarding schools for the gifted (in this instance, for children gifted in the natural and mathematical sciences, literature, or history). A fairly recent innovation in Russian education is night classes: an adult who did not finish secondary school may earn graduation certification through evening instruction, and universities now offer night classes as well. (Written sources imply that technical night classes were common under Lenin and Stalin.) CEMI provides classroom instruction in English for foreign scientists.

Gender Disparities in Russian

Education and Promotion 1 1

Birth rate

One of our translators, who had been a teacher, said that the birth ratio of girls to boys in Russia is high and used to be higher-in her mother's day, the ratio was 3:2. A teacher we met later claimed that boys are disproportionately represented in her math class because more boys than girls were born that year. 12

Enrollment and distribution by

discipline

We were told of girls who had been denied entry to a primary school because the school already had a disproportionate number of female students. However, the secondary school for the gifted visited by the delegation has a 700;6 male enrollment. 13 Within a given classroom, representation by gender may be quite disproportionate. In one classroom observed by the delegation, six of 23 students present were girls, and in another (consisting of lOth grade biology students), eleven of the thirteen students present were female. 14 The director of the school for the gifted told us that the percentage of female students varies by major, being high in biology, about even in chemistry, and between 20 and 30 percent in math. According to "unofficial data," the percentage of girls leaving secondary school interested in mathematics and wishing to continue their studies is high, but over 700;6 of women in mathematical disciplines wind up in economics or other applied specializations; the speaker felt that these young women were "pushed" away from pure mathematics [V]. At the St. Petersburg State University campus for math, mechanics, astronomy, and computer science, we were told that "about a third" of the students in each discipline were

101n the U.S., there are some two-year degree programs suitable for low-end job training, usually offered by "community colleges" attracting only local students. A professional, however, needs a minimum of a four-year program or Bachelor's degree, and often requires a Master's degree (requiring typically two more years). The latter degree may involve require a thesis, but not usually original research. For professors and researchers, the Master's is optional, but they must earn a Doctorate. This "Ph.D." averages ten years (8 for math) beyond the Bachelor's of study and research work, typically seven of them at the university. This period culminates in a substantive written dissertation of original research, which must be deemed a significant contribution to the field by a committee of established experts. 1 1This section contains sometimes-contradictory reports received from different sources. 121n the United States, more male babies are conceived and bom. However, the incidence of miscarriage, infant mortality, and indeed death at all ages is higher for males. By age 1 7 , female representation in U.S. schools is typically 50.5%.

13Admission is by competitive examination; preparatory classes are available. 14Biology and medicine are primarily practiced by women in Russia, and doctors are not well paid.


female. At Moscow University, the mathematics students consisted of four women and 24 men; in economics, there were ten women and 20 men. We were also told that the proportion of women students there used to be higher.

Instructors

Over 809-6 of Russian schoolteachers are women, but in the secondary and or specialized schools women teachers fall to about 700A> [LF]. 15 (We were also told that at specialized schools, male instructors may be the majority.) At the Moscow secondary school visited by the delegation, 800A> of the instructors were female, and the principal and assistant principal were women. At the school for the gifted, the director was male, and he informed us that the school was very atypical because "at most schools, most of the math teachers are female graduates of teaching universities; here, teachers are actively involved in research." There are more men than women teaching at vocational schools, and this disparity is much greater at universities. At the St. Petersburg State University we were told that that about a fourth of the instructors in math-related disciplines were women, but only four of these faculty women possessed Russianlevel doctorates (no data was given for comparative purposes, making analysis difficult).

Completion of doctorate

The members of the delegation were shocked when we were told that the average time for a Russian candidate to earn a doctorate was ten to fifteen years for a man, seventeen to thirtyfive years for a woman [V] . Our first reaction was that here was blatant evidence of discrimination. (Because of our American model, we envisioned this period as time spent at the university actively seeking a terminal degree.) However, in light of the different meaning and significance -of the doctorate in Russia, it seems likely that

a great deal of the difference is attributable to women delaying seeking this additional degree, or that they are not as active in research, or both. This could be due to lower expectations, fewer opportunities, or less scope because of family responsibilities (see later section, "Awareness and attitudes"). In any case, however, this delay must profoundly affect a woman's lifetime earning capability.

Research and professional

advancement

Professional discrimination based on gender is illegal in Russia, and several of those interviewed felt that statement of this fact thoroughly exhausted the topic. However, according to a survey by A WSE, discrepancies in highest degree obtained and in job title account for an average salary for Russian women in mathematics that is just 700;-6 of the average for men. The A WSE president, Dr. Galina Yu. Riznichenko, also indicated that it is difficult for women to publish in respected journals and to present papers at prestigious conferences. Academician Olga Ladyzhenskaya of the Steklov Institute confirmed that there are very few Russian women in high positions in government and science, in particular in the Academy, and that the last honor appears to have been withheld inappropriately from deserving women mathematicians in the past. Dr. Ladyzhenskaya also told us that significant research results of women had been altered slightly and published by men as their own work.

Information on the composition of the Academy of Science was collected by Vitalina Koval in 1989 [LF]. Among Academy employees without an advanced educational degree, 41.7% were women; of those holding as highest degree candidate of the sciences (roughly equivalent to an American Ph.D.), 34.4% were women; of those holding Russian doctorates, 14.9% were women; of those holding professorships, 7. 7% were women. Of depart-

ment or laboratory heads, 1 1.6% were women, and Academic Boards of institutes had between 3% and 9% women members.

CEMI provided an illustrative example of promotion discrepancy. Employees are paid based on their rank. Even the lowest rank, "junior researcher," requires a high level of education. With sufficient achievement, one may after perhaps ten years of work achieve the rank of "researcher." Successfully defending a thesis (in math or economics) entitles one to the rank of "senior researcher." Defending a doctoral thesis confers the rank of "leader," roughly analogous to a full professor. The highest rank is "chief researcher." The "senior researcher" rank forms a glass ceiling beyond which no woman has passed. Of the 500 �rsonnel at CEMI, 70 hold doctorates, but none of these is female. (See also preceding , subsection, "Completion of doctorate.") Thus, not one of the CEMI administrators is female.

Awareness and attitudes

According to A WSE, there are no Russian government statistics on women in math and science. 16 The library of Moscow State University contains only three relevant articles, two of which are written in English. A WSE has consequently carried out its own survey; most of the Russian statistics in this paper were provided by A WSE.

On several occasions during the delegation's visit, a Russian woman would make ·some sweeping statement linking gender (or race) to aptitude, and none of the Russians hearing it seemed to fmd this at all unreasonable. "Political correctness" in this regard seems unknown in Russia. One secondary school teacher in Moscow claimed both that girls are not as good as boys in math, and that women make better teachers than men (even in math). Interestingly, one woman at CEMI claimed that women are better at pure than applied math, reversing

15For comparison, in 1 993 about 74% of U.S. teachers were women, in both public and private schools. At public schools, 47% of teachers held master degree's or higher; at private schools, 34% did. About 35% of principals were women [NCES]. In 1 995, about 40% of all U.S. college faculty (full- and part-time) were women [B]. 16The U.S. National Assessment of Education Programs dates from 1 967; its publications are free and easily accessible on the Internet. The American Association of University Professors has been collecting gender data since 1 975.


what is the apparently prevailing conception (I noticed that another woman there made a face of disagreement).

In St. Petersburg, both at the university and the secondary school, the women said that there is no discrimination at all, not even on the level of faculty discussions. Tl).ey did not understand the purpose of an organization like the A WSE nor see a need for it. The only particular obstacles the university faculty perceived as women in mathematics were that most men do not wish to marry mathematicians, and that those who do marry them expect their wives to do all of the child-rearing and cookingP However, it should be noted that the secondary school teachers allowed the (male) principal to lead the discussion and to do almost all of the talking, and that the university faculty did not contradict the (female) Registrar when she claimed that males are superior in math. (Both St. Petersburg institutions were also clearly catastrophically underfunded, and delegation members felt that the overwhelming need to focus on simple survival might contribute to the lack of interest in gender inequity.)

Women of the Steklov Mathematics Institute: (1-r) Natalia Kirpichnikova, Olga Ladyzhenskaya,

and a "candidate of sciences" (Ph.D. equivalent).

The Moscow secondary school was better funded and had female administration. The women interviewed (principal, assistant principal, and teacher) seemed calm and confident and showed no reluctance to speak up. Otherwise, they seemed similar to their St. Petersburg counterparts, showing no interest in gender issues.

The women of the Steklov and of CEMI (both of which are research institutions of the Academy of Science) showed some awareness of professional discrimination. However, the women at CEMI seemed as shocked as the delegates by the statistics cited by one of their own workers (Natalya Vinokurova, a member of A WSE), and had apparently been totally unaware of the existence of A WSE.

The membership of A WSE were the only Russian women encountered by the delegation to whom we did not

have to explain and justify the genderissue portion of our mission. A WSE's primary purposes are dissemination of information on financial support available for women in science and education, support for women's professional advancement by nominations to key administrative positions, promotion of scientific and educational events for women, and improved communication between women in science (both within Russia and internationally). There are two striking differences between the activities of the A WSE and those of the [American] Association for Women in Mathematics (AWM). First, the major activity of the A WSE is the organization of entire conferences intended primarily for female participants, and the publication of the attendant proceedings. This isapparently seen as necessary because of the rarity of women's papers being accepted by major conferences and journals in the former Soviet nations. (The A WM by contrast encourages its members to participate in existing conferences and journals. It hosts one-day workshops that are held in conjunction with the annual Combined Meetings, primarily as a travel-fund mechanism for young female mathematicians.) Second, A WSE

1 7 An AWSE survey showed 73% of the respondents were married, and 71 % had at least one child. In addition, 64% said that the woman did all of the housework, and 1 8% said that the housework was shared equally (no one claimed that the man did all the housework).


hopes to publish textbooks and research monographs that are not available in print because of the general poverty of the Russian scientific community. Since their own activity is virtually entirely provided by dues and by volunteer work (outside funding has been found only for their conferences), this project is still unrealized. It would

Elena Novikova and Yuri Matiyasevich, out

side the Euler International Mathe-matics

Institute. Photo by Sue Geller.

The Euler Institute is housed in a renovated mansion that was nationalized during the

Communist revolution.

be difficult to exaggerate the poverty of the A WSE; their first attempt to survey conference attendees was greatly hindered by the fact they could afford to make only fifty copies of the questions. (However, they received fiftyseven responses; some attendees copied the form by hand in order to answer the questions.)

The A WSE survey reports, "The overwhelming majority of the women admit that the officially stated equality of opportunities for men and women in science does not ·exist." Less than half believed there was discrimination in opportunity to defend the thesis. However, 54% felt that there was inequity in "opportunities for access to information," 7 4% felt that participation in major manage-

ment decisions was unequal, and 800;6 felt that chances for promotion were unequal. One of the questions posed was why women are a minority in leadership positions in Russian science and education. In their responses, 27% felt that women had less ambition, lower qualifications, and less "work capacity" than their male colleagues; 35% felt that home and family commitments inhibited women's professional advancements; and 38% blamed patriarchal traditions and the creation of negative images of female leadership in the mass media. However, virtually all the women (96%) said that they never consider leaving their jobs, citing love of their work and responsibility to future generations as their primary reasons [LF].

Closing Thoughts

Why the differences in belief?

One question that our trip left largely unanswered was why Russians, particularly female mathematicians, seemed so thoroughly convinced that women are inferior at mathematics. While the annual media fanfare for the SAT-M "gender gap" keeps that hoary old refrain alive in the American popular culture, any implication of male superiority is considered simply inadmissible in academia and in the technological business world. More than one male colleague in mathematics has assured me that he has observed that a significant majority of the best students he trains are female. American professors know their female students are as good as and often better than their mal� students; why isn't this obviomsto our Russian counterparts?

I am indebted to Marjorie Senechal for giving me some insight-into this puzzle. According to Dr. Senechal, who has had the opportunity to study honors programs for young mathematicians in Russia, almost the only method used in Russia for measuring mathematical excellence is highly competitive, very stressful exams. (Here again, we see an instance where a conclusion about innate gender ability in mathematics is based on a self-selecting sample at the extreme high end of ability.)

In America we do have some such exams, but for most students we look at grades, letters of reference, and (where possible) research to get a comprehensive picture of potential. Our experience in this' country is that young women seldom take highly competitive math exams if it is not mandatory. This may result from not being encouraged to do so by their teachers, discomfort because the existing profile of exam takers is overwhelmingly male, self-doubt of their abilities, or simple disinterest in such competitions. According to letters in the A WM Newsletter, young women who do participate in math competitions are sometimes belittled or harassed by male competitors.

What seems to me very peculiar is the assumption that future potential for research mathematics should necessarily be in any way correlated to performance on time-critical competitions! These


Delegate Audrey Leef enlists the assistance of translator Mila Bolgak to describe the

Association for Women in Mathematics (AWM) to Natalya Vinokurova and Zelikina Lyudmila

of CEMI.

competitions in no way simulate re

search conditions. If the Russian exams

are similar to American ones, exam-tak

ers have no access to references, but

many of the exam questions are never

theless based on advanced topics not

covered in a standard curriculum. Thus,

such exams sharply penalize any test

taker who has not had special opportu-

nities. The outstanding scores go to stu

dents who have received special train

ing and coaching sessions. Given the

Russian school systems' "specialized"

nature and the prevailing stereotypes,

probably few girls receive the appropri

ate mathematical training.

I may add that at least until recently,

the National Security Agency did much

Several of the members of the [Russian] Association of Women in Science and Education

(AWSE), waving good-bye to the delegation. The (foreground) woman in black is Galina

Reznichenko, the AWSE president whose term was just ending, and the woman on the far

right is Irina Gudovich, expected to be the next to hold that office.


of its recruiting based on the results of

such competitions. In 1994 I attended

a symposium on Women in Mathemat

ics hosted by the NSA, at which we

were asked: Why isn't the NSA at

tracting more women hires? We told

them: Stop putting so much credence

in those exams!

The news from MIT

In March 1999 (a year after our trip), the

Massachusetts Institute of Tech-nology

(MIT) stunned academia by publishing

a report admitting that there had been

gender discrimination towards the fac

ulty in their School of Science. The ef

fects, described as "marginalization" of

the women faculty, were observed by

each female faculty member increas

ingly as she became more senior. These

senior women described themselves as

being excluded from having any voice

in their departments and from positions

of any real power.

The MIT Dean of Science has been

acting on the recommendations of the

Committee, including increasing the

number of female faculty, and the re

port says that the results have already

been highly beneficial. (However, they

also point out that because of pipeline

considerations, even at the increased

rate of new hires it will be 40 years be

fore 400/0 of the faculty in the School of

Science are female.)

"Feminization" of higher education in

the United States?

One aspect of the delegation's discover

ies that I found difficult to understand

was the fact that women composed a

majority of Russia scientists and educa

tors, but were still not able to achieve

equitable treatment. In the U.S., the

common wisdom is that increasing rep

resentation will bring increased influ

ence and therefore, equity. But is this necessarily so? According to a recent

AAUP study [B], in 1995 women com

prised 40% of all U.S. faculty, up from

27% in 1975. However, salary gender disparities "not only remain substantial but

are greater in 1998 than in 1975 for half

of the categories, including 'all-institu

tion' average salaries for full, associate,

and assistant professors." Nor can these

disparities be attributed solely to dis

parity in average seniority within rank

As female participation in the profession increases, women remain more likely than men to obtain appointments in lower-paying types of institutions and disciplines. Indeed, even controUing for category of institution, gender disparities continue and in some cases increased, because women are more often found in tlwse specific institutions (and disciplines) that pay lower salaries . . . . The increasing entry of women into the profession has so jar exceeded the improvement in the positions that women attain that the proportion of aU female faculty wlw are tenured has actuaUy declined from 24 to 20 percent . . . .

The report observes that during the huge upswing in women in academia, male participation in the profession has been almost constant, and the (raw) number of men in tenure-track positions has actually dropped 28%. "Simply stated, fewer men are finding their professional futures in academe, whereas female participation continues to increase despite the declining terms and conditions of faculty employment. . . . universities can successfully offer women terms of agreement that would rtt>t be acceptable to similar numbers of similarly qualified men."

Personal responses

Regarding the effects of the journey on myself: I learned a great deal (basically, I spent an entire month after our return simply trying to record all that I had learned and seen and felt). I read news reports in a new way, and have followed unfolding events in Russia with entirely new interest. I sought data from other nations to put what I had learned in a larger perspective. I gave presentations to students at my own college, comparing the situations of women in mathematics in America and in Russia. And I wrote (and revised, and revised) this article, hoping to spread the effects of our journey, believing our experiences were important and should be shared with a wider audience.

The economic and political events of the past two years have led to substantially more acrimonious relations between Russia and ·the nations of NATO. It seems likely that conditions

for scientists and educators in Russia have worsened further, but a second delegation from the U.S. would be unlikely to fmd a particularly warm welcome at this time.

Composition of the Delegation

and Some of Its Contacts

The delegation

Leader: Dr. Pamela Ferguson, Past President of Grinnell College (lA). Dr. Ferguson had traveled in Russia before, and graciously consented to substitute for our originally intended leader, Dr. Alice Schafer, who was unable to attend.

Ms. Liv Berge, Upper Secondary School (Husnes, Norway). Ms. Berge, the author of several articles on gender and mathematics, was working on a project entitled "Mathematics, Gender, and Politics." She shared with us this data (from the Nordic Institute for Women's Studies and Gender Research): Women have 400Al representation in Norwegian government, but comprise only 7% of mathematics professors. (In Sweden, the percentage of female mathematicians is even lower.)

Dr. Lucy Dechene, Fitchburg State College (MA). Fitchburg State, a fouryear liberal arts institution, has exchange programs with graduate computer science programs in China and Russia and is developing one in India; there is also a special program for undergraduates in Bermuda. In addition to her teaching duties, Dr. Dechene supervises the mathematical skills center, as well as independent study projects and undergraduate research. She had been a past participant in Peopleto-People programs to other nations.

Ms. Maureen Gavin, Bodine High School for Int'l Affairs (PA). Bodine, a magnet school, was founded in cooperation with the World Affairs Council of Philadelphia to have as its primary focus global studies and geography. Ms. Gavin has traveled extensively (for example to Tibet) and had accompanied her students on a trip to Russia just a month before the delegation's journey.

Dr. Sue Geller, Texas A&M Univ. (TX). Dr. Geller, in addition to her teaching and research duties, directs the master of science program, mentors students and junior faculty, is involved in conflict resolution and mediation,

and runs tenure appeal hearings. She has also been active in the Association for Women in Mathematics (AWM); attendees at the Combined Meetings may be familiar with Dr. Geller from the "Micro-inequities" skits, illustrating small (and large) il\iustices against women in mathematics in our culture.

Dr. Karin Johnsgard, Richard Stockton College of New Jersey (NJ). I have been a registered Girl Scout for over 25 years, and can truly describe my interest in gender issues as life-long. In high school, I was a People-to-People student ambassador to several European nations. I was one of the women graduate students who benefited from the A WM's one-day workshops. On this delegation, I was the youngest and only untenured participant.

Dr. El�anor Jones, Norfolk State Univ.-(VA). Norfolk State 'has historically had predominantly AfricanAmerican enrollment. Dr. Jones's presence on the delegation helped keep us sensitive to the virtual absence of peo-

The author at Moscow school #103, in the

"sports museum" celebrating the Olympic

International Youth games hosted by the city

this year. (I am about to be handed an

Olympic torch.) Photo by Sue Geller.


ple of color we saw in Russia, and to the occasionally explicit racism. Her main focus on the mission was pedagogical.

Dr. Audrey Leef (emerita), Mont-clair State Univ. (NJ). Montclair is particularly noted for training secondary school teachers. Although retired, Dr. Leef still teaches as an acljunct and has supervised student teachers in their fieldwork Her world travels have included Antarctica!

Dr. Diana Vincent, Medical Univ. of South Carolina (SC). MUSC is a teaching hospital that trains health care professionals, conducts basic and clinical research, and provides patient care. Dr. Vincent described her work (in part) as a bridge between the mathematical and physical scientists and the medical staff.

Translators

Ms. Irina Alexandrova, St. Petersburg Center of International Programs. Ms. Mila Bolgak, Prospects Business Cooperation Center, Moscow.

Acknowledgments

The author is grateful to the members of the delegation and all who assisted our mission. In particular, I wish to thank Dr. Sue Geller for her encouragement and help with details. I also greatly

CARE plants the most wonderful seeds on earth. Seeds of self-sufficiency mar help

rarving people become healthy, productive people. And we

do it village by village by village. Please help us rurn cries for help

imo rhe laughter of hope.

appreciated my editor, Dr. Marjorie Senechal, for her guidance, insight, and patience. Dr. Mary Beth Ruskai sent me some very relevant material. I also thank my husband (Dr. Ami Silberman), both for accompanying me in Russia and for his feedback in editing this paper. Any errors herein are solely the author's.

REFERENCES

United States and multi-national

[A] Joe Alper, "Science education: The pipeline

is leaking all the way along, " Science, Vol.

260, 1 6 April 1 993, pp. 409-41 1 .

[AAUP] "Doing better: The annual report on the

economic status of the profession, 1 997-98,"

Academe: Bulletin of the American Associa

tion of University Professors, Vol. 84, No. 2 ,

March-April 1 998, pp. 1 3-1 06.

[BS] Yupin Bae and Thomas M. Smith, "The

Condition of Education, 1 997. No 1 1 : Women

in mathematics and science," U .S. Dept. of

Education, National Center for Education

Statistics, 1 997. (Available at website http://

nces.ed.gov/edstats)

[B] Ernst Benjamin, "Disparities in the salaries

and appointments of academic women and

men: An update of a 1 988 report of com

mittee W on the status of women in the aca

demic profession," AAUP, http://www.aaup.

org/Wrepup.html (1 999).

[DMR] Paul W. Davis, James W. Maxwell, and

Kindra M. Remick, " 1 998 Annual Survey of the

Mathematical Sciences (first report)," Notices

of the AMS, Vol. 46, No. 2, Feb. 1 999, pp.

224-235. (Available at website http://www.

ams.om/employmenVsurvey.html)

[H] G. Hanna, "Mathematics achievement of

girls and boys in grade eight: Results from

20 countries," Educ. Stud. Math. , Vol. 20,

1 989, pp. 225-232.

[HFL] Janet Shibley Hyde, Elizabeth Fennema,

and Susan J. Lamon, "Gender differences in

mathematics performance: A meta-analy

sis," Psychological Bulletin, Vol. 1 07, No. 2,

1 990, pp. 1 39-1 55.

[MIT] "A study on the status of women faculty in

science at MIT," The MIT Faculty Newsletter,

Vol. 1 1 , No. 4 (Special Edition), March 1 999.

(Available at the website http://web.mit.edu.

/fnl/women/women.html)

[NCES] "Digest of Education Statistics, 1 998

edition (NCES 1 99-032)," U.S. Dept. of

Education, National Center for Education

Statistics, 1 999. (Available at website http://

nces.ed.gov/edstats)

[R] Mary Beth Ruskai, "Guest comment: Are

A U T H OR

KARIN JOHNSGARD

NAMS DMsion

Richard Stockton Col lege

Pomona, NJ 08240-0195


Karin Johnsgard, in her teens, collab

orated with her ornithologist father Paul

Johnsgard in writing and illustrating a

book about dragons and unicorns. Her

publications since then have mostly

concerned knot groups, combinatorial

group theory, and geodesics in cell

complexes. She has been a Sloan

Doctoral Dissertation Fellow and an

NSF Postdoctoral Fellow. Photo by

Ralph Beam

there innate cognitive gender differences?

Some comments on the evidence in re

sponse to a letter from M. Levin," Am. J. Phys. , Vol. 59, No. 1 , Jan. 1 991 , pp. 1 1 -14 .

[S] Paul Selvin, "Does the Harrison case reveal

sexism in math?" Science, Vol. 252, 28 June

1 991 , pp. 1 781-83.

[SAT] "National report on college-bound seniors

1 999," College Entrance Examination Board.

(Available at website http://www.clep.com/saV

sbsenior/yr1 999/NA T /natsdm99.html)

[SATR] "Common sense about SAT score dif

ferences and test validity (RN-01 ), " Research

Notes, The College Board, June 1 997.

(Available at website http://www.college

board.org/research/html/m index.html)

Russia

[AWSE] Information copied from slides prepared

by the [Russian] Association of Women in

Science and Education; sources unspecified.

M Data as cited by Natalia A Vinokurova of

CEMI and AWSE; possible source an AWSE

survey she conducted with Nana Yanson. (A

preliminary report on this survey was con

tained in Lady Fortune.)

[LF] Lady Fortune, publication of AWSE.

OLEKSIY ANDRIYCHENKO AND MARC CHAMBERLAND

I terated Strings and Ce u lar Automata

• n 1996, Sir Bryan Thwaites [ 4] posed two open problems with prize money offered �for solutions. The first problem (with a £1000 reward) is the well-known 3x + 1 prob

lem which has received attention from many quarters. This easily-stated problem has

eluded mathematicians for about 50 years; for more information, see Lagarias [ 1]

and Wirsching [5] . Thwaites's other problem (with a £100 reward) has no clear origin. He states it as follows:

Take any set of N rational numbers. Form another set by taking the positive differences of successive members of the first set, the last such difference being formed from the last and first members of the original set. Iterate. Then in due course the set so formed will consist entirely of zeros if and only if N is a power of two.

Thwaites concludes his note by saying that "Although neither I, nor others who have been equally intrigued, have yet proved [the second problem], one's instinct is that here is a provable cof\iecture; and so the prize for the first successful proof, or disproof, is a mere hundred pounds." The present paper offers an elementary proof of this second problem. In the process, binomial coefficients and cellular automata are encountered.

We will let (a1, a2, : • • , an) represent a string of length n, where ai is rational for all i. Upon iteration, its succes-

sor will be Cla1 - a2l, la2 - asl, . . . , lan- 1 - ani, lan - ai l). A string containing only zeros will be called the zero-string, while a string containing only ones will be called the onestring. The way the problem was posed by Thwaites is somewhat imprecise: the one-string iterates to the zerostring regardless of the string's length. We restate the (proper) theorem to be proved formally:

Theorem 1.1 AU strings of length n will eventually iterate to the zero-string if and only if n = 2k for some k E Z+.

Half of the proof comes easily:

Theorem 1.2 If the string's length n is not a power of two, then there exist strings which will never iterate to the zero-string.

Proof: The problem considers 0-1 strings, strings whose terms take only the values 0 or 1. Since the set of 0-1 strings is forward-invariant under our iterative process, this will suffice.

First we prove the case when n is odd. Working backwards, note that the only predecessor of the zero-string is


the one-string. The only predecessor of the one-string has

terms which alternate between 0 and 1, which is impossi

ble since n is odd. Therefore the only 0-1 strings of odd

length iterating to the zero-string are the zero-string itself

and the one-string. This completes the proof when n is odd.

If n is an even number which is not a power of two, it

must have an odd prime factor, say p. Create a string of

length n by concatenating nip substrings of length p, each

of which is the string starting with a one then having all

zero terms. For example, if n = 12, take p = 3 and create

the string

1 0 0 1 0 0 1 0 0 1 0 0

The periodic nature of the iterative process implies that

each substring iterates as if it were the whole string:

1 0 0 � 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 � 1 0 1 1 0 1 1 0 1 1 0 1

Because each of the (odd-length) substrings will never it

erate to the zero string, neither will the whole string; which

completes the proof. D The previous proof needed only the set of 0-1 strings.

To prove the other half of Theorem 1.1, we argue that con

sidering only 0-1 strings is sufficient. First note that by scal

ing a string by a constant, the dynamics do not change, so

multiply each element in the string by the appropriate in

teger (the least common multiple of the denominators) to

yield an integer string. Also, one interaction on a string

yields a non-negative string, so we can assume from here

on that the string consists only of non-negative integers.

Next, we show that it is sufficient to consider strings

whose values are only 0 and possibly one other (positive)

value. To do this, we show that if the string contains at

least two distinct positive values, the maximum value (de

noted henceforth by m) will eventually decrease. If there

is no zero value, the maximum value will automatically de

crease after one iteration, so we may assume there is at

least one zero value. Consider any substring whose terms

are only zero or m (with at least one m), and assume this substring is maximal, so that it takes the form

where ak equals zero or m (with at least one m) for all k, and 0 < b, c < m. After one iteration, the substring has one

few term. Note that such substrings (with at least one m) cannot be created, so after a finite number of iterations,

these substrings all vanish. This process forces the maxi

mum of the whole string to decrease, leaving us ( dynami

cally) with two possibilities: either this descent continues

until all the terms are zero, or the string iterates until all

its terms are either 0 or possibly one positive value.

Dividing each term by this positive value (which leaves the

string dynamically unaltered) yields a string whose terms

are only 0 or 1. Figure 1 shows iterations of the string ( 1 1 0 0 1 1 0 0 ),

where the black dots represent 1 and the white dots rep

resent 0.


Figure 1 . Iterating the string ( 1 1 0 0 1 1 0 0 ).

At this point, it is worth pointing out that iterating a

0-1 string mirrors the dynamics used in generating the

Sierpinski Gasket with cellular automata. Consider the

"rules" in Figure 2. For each rule, the parity (black or

white) of the upper squares determines the parity of the

lower square. Starting with an infinite row with only one

black square, one generates the Sierpinski Gasket in a

stretched form. Figure 3 shows the first few rows. The

black cells in this figure correspond to the odd terms in

Pascal's triangle, where the top black cell corresponds to

the apex of the triangle. Details of the mathematics may

be found in Peitgen et al. [3] . The dynamics of our 0-1 strings are similar, with the important difference that the

string is periodic.

To finish the proof of Theorem 1.1, one is required to

show that 0-1 strings whose length is a power of 2 even

tually iterate to the zero string. The analysis is simplified

Figure 2. Cellular automata "Rules.�>

if we replace 0 (resp. 1) with 1 (resp. - 1), and instead of using the absolute value of the difference, simply consider the product. For example, before we had the successive terms (1 0) produce [1 - 0[ = 1, whereas now we have ( - 1 1) produce ( - 1)(1) = - 1. One may easily verify that the dynamics are equivalent; we are simply representing the group £:2 in a different way. The rest of the proof will work with this new system. We will let ai,j denote the value of the f11 element of the string after i iterations. For ease of notation, it will be understood that if kn <j ;::;; (k + 1)n for some k E z+, then ai,j = ai,j-kn·

Lemma 1.1 If a string has length n, then

a · . = a(�) a (f) . . . a(1) t,J O,j O,j+ 1 O,j+i

for 1 :5 i :5 n, 1 :5 j :5 n - i.

A U T H O R S

OLEKSIY ANDRIYCHENKO

Department of Mathematics and Computer Science

Grinnell College

Grinnell, lA 501 1 2-1 690

USA

Oleksiy Andriychenko is a current student at Grinnell College,

majoring in mathematics and economics. He graduated from

the Ukrainian National Mathematical-Physical Lyceum in Kiev in

1995. During high school and college years, he successfully

participated in a number of math competitions , including the

Ukrainian National Olympiads, the Putnam Competition, and the

Mathematical Contest in Modeling. Of al l the mathematical top

ics he has seen so far, he considers problem solving in num

ber theory as the most fascinating . His other interests include

chess, ping-pong , and consumer advertising.

The proof is by induction.

Lemma 1.2

ek; 1) is odd if 0 :5 j :5 2k - 1.

The proof is by expanding the binomial coefficient. We note that this lemma has a generalization (see, for example, [2]): for any prime p,

p does not divide ipk j- 1) if 0 :5 j :5 pk - 1.

These two lemmas lead to the last step in the proof of Theorem 1.1:

Theorem 1.3 If the string's length is n = 2k, then an,j =

1 for aUj.

MARC CHAMBERLAND

Department of Mathematics and Computer Science

Grinnell College

Grinnell, lA 501 1 2-1 690

USA e-mail: [email protected]

Marc Chamberland obtained his degrees from the University

of Waterloo and has been at Grinnell College since 1 997. His

research interests are principally in differential equations and

dlynamical systems, though a beautiful, tough problem Oike the

3x + 1 problem or the Jacobian Conjecture) can easily lead

him to other waters. Outside of mathematics, he spends time

with his wife and two young sons, fulfills his passion for mu

sic (voice, piano, and guitar) , and seeks quiet places for med

itation .


Figure 3. Generating the Sierpinskl Gasket.

Proof: The first step is to show

an-1,j = ao,1 ao,2 . . . ao,n

for j = 1, . . . , n. Using Lenunas 1 . 1 and 1 .2 successively, we have

a . = a(no1)a(nJ. 1) . . . a(�=O n-1,J O,j O,j+1 O,j+n-1

= ao,3tlo,j+1 · · · ao,j+n-1 = ao,1a0,2

.. . ao,n

If an-1,j = 1 for allj, we are finished. If all the terms are - 1, one more iteration forces an,j = 1 for all j. D

Epilogue

When we presented this work to Sir Bryan Thwaites, he informed us that the problem had been solved long since. However, he expressed admiration for our method, so even without the cash prize we felt he had given his blessing to our publishing it.

ACKNOWLEDGEMENT

The authors would like to thank Grinnell College for financially supporting O.A. to work with M.C. during the sununer of 1999.

REFERENCES

[1 ) J.C. Lagarias. The 3x + 1 Problem and its Generalizations.

American Mathematical Monthly 92:3-23, 1 985.

[2) I . Niven, H.S. Zuckerman and H.L. Montgomery. An Introduction to

the Theory of Numbers. Wiley, 1 991 .

[3) H.-0. Peitgen, H. Jurgens and D. Saupe. Chaos and Fractals.

Springer-Verlag, 1 992.

[4) B. Thwaites. Two Conjectures or how to win £1 1 00. Mathematical

Gazette 80:35-36, 1 996.

[5) G.J. Wirsching. The Dynamical System Generated by the 3n + 1

Function. Lecture Notes in Mathematics, 1 681 , Springer-Verlag,

1 998.

36 THE MATHEMATICAL INTELUGENCER

Ode to Andrew Wiles, KBE Tom M. Apostol

Fermat's famous scribble-as margin� noteLaunched thousands of efforts-too many to quote. Anyone armed with a few facts mathematical Can settle the problem when it's only quadratical. Pythagoras gets credit as first to produce The theorem on the square of the hypotenuse.

Euler's attempts to take care of the cubics Might have had more success if devoted to Rubik's. Sophie Germain then entered the race With a handful of primes that were in the first case.

Lame at mid-century proudly announced That the Fermat problem was finally trounced. But the very same year a letter from Kununer Revealed the attempt by Lame was a bununer.

Regular primes and Kununer's ideals Brought new momentum to fast-spinning wheels. Huge prizes were offered, and many shed tears When a thousand false proofs appeared in four years. Then high-speed computers tried more and more sam-

ples, But no one could find any counter examples.

In June '93 Andrew Wiles laid claim To a proof that would bring him fortune and fame. But, alas, it was flawed-he seemed to be stuckWhen new inspiration suddenly struck.

The flaw was removed with a change in approach, And now his new proof is beyond all reproach. The Queen of England has dubbed him a Knight For being the first to show Fermat was right.

1 -70 Caltech

Pasadena, CA 91 1 25

BY JOHN BRUNING, ANDY CANTRELL, ROBERT LONGHURST, DAN SCHWALBE, AND STAN WAGON

Rhapsody i n Wh ite : A Victory for Mathemat ics

• n 1999, the Breckenridge International Snow Sculpture Championships saw its first �mathematical surface: the Costa surface, whose production in snow was reported on

in [2]. That effort might have set the stage, for this year another minimal surface

took several awards at the same event.

Robert Longhurst has used the ideas of negative curvature in much of his sculpting work in wood, and his piece showing an Enneper surface (Figure 1) seemed ideal for realization in the hard snow that Breckenridge prepares. The team, again sponsored by Wolfram Research, Inc. (makers of Mathematica), consisted of Longhurst, a wood and stone sculptor from Chestertown, New York; Dan Schwalbe and Stan Wagon, who had learned the rudiments

of sculpting negative curvature from Helaman Ferguson at the 1999 event; Andy Cantrell, a sophomore at Macalester College; and John Bruning of the Tropel Corporation, the nonsculpting photographer for the team. It was through Bruning that the rest of the team was introduced to Longhurst's work

We sculpted an Enneper surface of degree 2 (see [1]), a minimal surface that was discovered by A Enneper in 1864.


Longhurst's wooden model of an Enneper surface, carved from bubinga wood.

Inspired by the swooping curves, we named it "Rhapsody in White." Unlike the Costa surface, Enneper's surface does not embed in 3-space (it has self-intersections) and is topologically dull (it is homeomorphic to the plane). But the process of truncating the infinite surface just before the self-intersections yields a surface that, at least from a sculptural perspective, has a very beautiful shape. One must drill and then expand several holes, which gives a topological flavor to the work. The openness of the result yields quite a pleasant view. And the negative curvature that occurs at each point gives the piece structural strength that allows it to be built out of snow.

The parametric representation [1 ) is simple:

.f(r, O) = (r cos (} - � r5 cos(58),

1 2 ) r sin (} + 5 r5 sin(58), 3 r3 cos(38)

Plottingjwith r varying from 0 to 1.4 and (} in [0, 2 7T] leads to Figure 1, though in that figure the x-axis is the vertical


axis and the viewpoint is on the positive z-axis. Figures 2 and 3 show two views of our sculpture.

There were 17 teams at the January, 2000, event, from England, The Netherlands, Germany, Switzerland, Finland, Russian, Mexico, Canada, and the U.S.; for images of almost all of the pieces see [3]. The audience's reaction told us that our work had succeeded and had the desired impact. The curves were swooping and graceful, the overhang exciting, and the surface smooth. But how would the art judges react to a purely mathematical shape? Would its entrancing form win them over, or would they find it unimaginative? When third place was announced to the Swiss team's punctured sphere, we became concerned, for we thought that the judges would not award two medals to geometric shapes. But then the silver medal was awarded to us. First place went to a soaring Russian structure illustrating human striving. The judging seemed fair; but the next night, we became convinced that our work had thoroughly won over all viewers, when we received both the People's Choice award (voting by the approxi-

/

Time to start: Plastic sheeting was used to mark out the initial pro

jection (photo by J. Bruning).

Rhapsody in White, side view, with crowd (photo by J. Bruning).

mately 10000 viewers who see the sculptures on the final

weekend) and the Artists' Choice award (voting by all the

sculptors).

Snow is a fantastic medium for sculpting mathematical

shapes. Readers interested in information on entering the

2001 event can contact Wagon for information. All that is

required is stamina, a good set of tools, an appealing de

sign, and an understanding of snow. Here are some com

ments from Wagon's acceptance speech.

"Julia Child has said," 'II faut mettre les mains dans Ia

pate': To be a baker, one must put one's hands in the dough.

Four members of our team are mathematicians and we

spend a lot of time looking at images on a computer screen.

But, both for us and for the viewers of our work, true un

derstanding can be obtained only by interacting with the

piece in a truly three-dimensional way. This is what snow

allows us to do. In a very short period of time and with a

minimum of tools we can sculpt a complicated shape and

so learn much more about it. It's a glorious opportunity and

tremendous fun."

REFERENCES

[1] A. Gray, Modern Differential Geometry of Curves and Surfaces with

Mathematica, 2nd ed., CRC Press, Boca Raton, Fla. , 1 998.

[2] C. and H. Ferguson, T. Nemeth, D. Schwalbe, and S. Wagon,

Invisible Handshake, The Mathematical lntelligencer 21 :4 (Fall 1 999),

30-35.

[3] D. Schwalbe's web page: www.math.macalester.edu/snow2000.


A U T HORS

L to R: Cantre l l, Longhurst, Schwalbe, Wagon, Brun ing

ANDY CANTRELL

Macalester College

St. Paul, MN 551 05

USA [email protected]

ROBERT LONGHURST

407 Potter Brook Road

Chestertown, NY 1 2871

USA

DAN SCHWALBE


Macalester College

Robert Longhurst got his degree in archi-

St. Paul, MN 551 05


Andy Cantrell hails from Fort Collins, Colora- lecture, Kent State University, 1 975; since

do. His interest in both mathematics and art

blossomed at Poudre High School and gets

plenty of scope as he continues study of

mathematics at Macalester College, and pur

sues ceramic art on the side.

1 976 he has had his own sculpture studio.

His distinctive and personal visual vocabu

lary has captivated a large audience, and his

pieces may be found in hundreds of private,

corporate, and museum collections interna

tionally.

Dan Schwalbe is co-author of the Maple

Flight Manual, and (with Stan Wagon) of

VisuaiDSolve. He oversees the computer

labs at Macalester College. His wife Kathy

just published her first book, Information

Technology Project Management. They have

three children, one of whom, the 1 3-year-old

son, came along on this snow-sculpting trip

to do some skiing with his father.

STAN WAGON


Macalester College

St. Paul, MN 55105


Stan Wagon makes a mission of disseminat

ing mathematical ideas to the public, and

hopes snow sculpture serves this. He is as

sistant editor of Mathematica in Education and

Research, for which he writes a regular col

umn. He got some notoriety recently by con

structing a square-wheeled bicycle that rolls

smoothly along a specially designed track. His

main non-mathematical interest is skiing; he

skied to near the top of Canada's highest

peak, Mt. Logan, in May 2000.


JOHN BRUNING

Trope! Corporation

60 O'Connor Road

Fairport, NY 1 4450 USA

[email protected]

John Bruning is President and CEO of Trope!

Corporation, a high-tech company that

makes specialized optics used in manufac

turing computer chips. He has degrees in

electrical engineering (BS Penn State, PhD

University of Illinois) and is a member of the

National Academy of Engineering. He ex

plores the common area of mathematics and

art, through Mathematica and woodworking.

M ath e m a t i c a l l y B e n t Colin Adams, Ed itor

The proof is in the pudding.

Opening a copy of The Mathematical

Intelligencer you may ask yourself

uneasily, "JfJI,at is this anyway--a

mathematical journal, or what?" Or you may ask, "JfJI,ere am /?" Or even

"JfJI,o am /?" This sense of disorienta

tion is at its most acute when you

open to Colin Adam's column.

Relax. Breathe regularly. It's

mathematical, it's a humor column,

and it may even be harmless.

Column editor's address: Colin Adams,

Department of Mathematics, Williams

College, Williamstown, MA '01 267 USA

e-mail: Colin [email protected]

The S.S. Riemann

The S.S. Riemarm embarked on its maiden voyage from the dock of

the Department of Mathematics, Yale University on April 2, 1999. Weighing in at 934 pages, and including a separate 50,000-line computer proof of the main lemma, she was the most massive theorem ever produced up to that time. There wasn't another theorem afloat on the mathematical ocean that compared.

She had a crew of over 31, including Captain Alphonse Huber, a full professor and Fields medalist, five other full professors, eleven associate professors, eight assistant professors, and six post-docs in steerage. Various grad students tagged along for the ride.

Yes, she was the crown jewel in the fleet of theorems that had come out of Yale. Designed to survive any catastrophe, she was built with expendable lemmas shielding her bow. There were back-up lemmas and back-up lemmas to those. The proof was constructed with a graph-like structure so that if an edge were to be destroyed, there would be another path to the same point in the proof. Mathematicians marveled at the intricacy of her design. The computer proof used interval arithmetic, making it as rigorous as if it had all been done by hand. They said she was unsinkable.

This first cruise was a shake-down run, to get the kinks out; just a quick trip to Berkeley and Stanford for a going over by the experts there, and then on to the University of Michigan for a week-long seminar. The subsequent voyage would be a straight shot to the Annals of Mathematics.

As she departed from the wharf in New Haven, the graduate students

cried out and waved exultantly, throwing streamers. Bands played exuberant marches. Administrators made promises that they knew they couldn't keep. It was a sight to behold.

Once in Berkeley, they put her through her paces. The crew cranked up the logical engines, and she forged ahead. Nothing could slow her down. She sliced through questions like a scull in the Charles River. The crew oiled a proposition here, tightened a corollary there, and she lived up to her reputation as the most po�erful theorem on the mathematical sea.

At Stanford, her reception was

grand. Wine and imported cheese on sesame crackers, and little spanikopita hors d'oeuvres. No expense was spared. The crew reveled in the attention. After a colloquium replete with standing ovation, they turned her and headed for Michigan.

Ann Arbor was cool at that time of year, but no one was overly concerned. After all, she was the queen of the ocean. They docked amid much fanfare. But the hubbub died · down quickly, and a week-long set oflectures in the analysis seminar began.

It started out fme. Huber remained at the helm at first. But soon he began to relax. She had proved herself in the Bay Area He could ease off and let other members of the crew pilot the craft. As the week wore on, the seminar shrank in size, and they were down to a handful of experts.

Late in the week, many of the crew had dozed off, and others had wandered out for coffee. It was a post-doc, Dimmick, who was on watch when he realized there was something off the port bow, something in a question asked by the diminutive Prof. Feisberg, an expert on holomorphic functions. At first, Dimmick wasn't sure that it would amount to anything, so he did not sound the alarm. But as the issue loomed larger in the darkening semi-

© 2000 SPRINGER· VERLAG NEW YORK, VOLUME 22, NUMBER 4, 2000 41

nar room, he realized how serious it was.

"Counterexample, counterexample, dead ahead," he screamed out. "Full reverse, fu1! reverse. All hands on deck" The professors leapt to their feet. Evecyone grabbed for chalk and erasers. But the theorem ground on toward the immovable object ahead. Nothing could stop their forward momentum in time. The counterexample loomed out of the darkness, tall, white, stark against the evening sky. Some of the graduate students remained oblivious to the impending disaster, as they played intramural soccer on an adjacent field.

Prof. Huber tried to convince the crew that it would be all right. "She can withstand it," he said. But crew members were leaping out the door of the seminar room at an alarming rate.

When the collision occurred, it seemed to happen in slow motion. There was a grinding crunch. Lemmas sloughed off the prow. Edges of the graph-like structure buckled under the impact. The hull seemed to crumple up like the ego of a jobless Ph.D.

Almost immediately, they realized she was going to go down.

Huber turned to the communications officer. "Reynolds," he said. "Use your cell phone to call the nearest functional analyst. I think it's Alder at Wisconsin. We are going to need help once we are in the water. There aren't enough hypotheses to go around."

As Reynolds dialed frantically, the Captain tried to quell the growing panic. "Evecyone, I ask you to remain calm. We have radioed for help."

But Reynolds turned to the Captain with tears running down his cheeks. "Captain, Captain!" he cried. "Alder is in Germany. The nearest functional analyst is in Utah, and there's no way she could get here in time. We're on our own."

There was pandemonium at the doorway to the seminar room, as the mob fought to get out in time.

"Please," shouted the captain, "let the grad students and post-docs have the hypotheses. Show some courage." But full professors were grabbing lemmas and claims as they pushed postdocs to the floor in their frantic haste to escape. Reynolds seized a corollacy but Huber stopped him. "Reynolds, that won't float."

MOVING? We need your new address so that you

do not miss any issues of

Dimmick turned to the captain. "Sir, we should get out before it's too late."

"I'm not getting out," replied the Captain gravely. "I'm going down with her."

"I'll go down with her, too, sir," said Dimmick, tcying not to look frightened.

"No," said Huber. "You have your whole career in front of you. Don't throw it away on this ship, as beautiful as she is. Abandon her. You will survive to crew another theorem."

Dimmick shook his head no, but Huber placed a firm hand on his shoulder. "That's an order," he said. Dimmick saluted one last time, and then scrambled out of the seminar room.

As the afternoon light dimmed, Huber and the crew members who hadn't managed to escape slowly disappeared beneath the waves, lost forever in the immeasurable ocean known as mathematics.

Some day a ship will leave port again, a ship with the name S.S. Riemann. And that ship will be truly indestructible. And mathematicians around the world will rejoice. But until then, remember to book your passage carefully, and bring along plenty of hypotheses.

THE MATHEMATICAL INTELLIGENCER.


Please send your old address (or label) and new address to:

Springer-Verlag New York, Inc.,

Journal Fulfillment Services

P.O. Box 2485, Secaucus, NJ 07096-2485 U.S.A.

Please give us six weeks notice.

DIDIER NORDON

Ethnog raph ic

"You spent more than 10 years in the field. In spite of that, scholars have given a very poor reception to your description of the Urematherpays.2 How do you account for the rejection you have suffered?"

People have their theories and they stick to them. If you bring back observations which don't fit their theories, they don't believe you. Anything to avoid having to reconsider their preconceptions.

"Nobody took the trouble to go and verify your observations?"

You have to remember that it's very difficult to get access to the Urematherpays. Very few have made it to their country; still fewer have returned.

"Are they such a cruel people?" Oh, no! Not specially. The problem is not that. It's just

that it takes such effort to get into their culture that nobody, or almost nobody, has enough energy left to get away again. That's why everyone is afraid of them. Acculturating to them leaves you inextricably involved with them.

"What is so compelling about Athermay culture? Their mores?"

No, the mores are pretty routine. Their system of chieftancy is a network of pyramids. The Urematherpays are divided into tribes. Each tribe has a chief, called an Andarinmay. Around each chief are sub-chiefs, called rofpays; then sub-sub-chiefs, and so on down to the lowest rank, the tudentsays. The andarinmay doesn't give orders

to a rofpay, nor does the rofpay give orders to a tudentsay. They work together with seemingly complete equality. Still, when you look closer you realize that everybody knows perfectly who is above, and the thought would never occur to any of them to challenge the hierarchy.

There is no special privilege associated with being an andarinmay. It is reward enough just to know that one is considered a chief by the others.

"And by the other tribes?" No. Every tribe is organized in this pyramidal structure,

but the andarinmay of one tribe is almost a nobody to another tribe. Every Urematherpay is free to choose a tribe. Once you have made your choice you have to stick to it. Relations between tribes �e intensely competitive.

"Are the tribes numerous?" Very. On the other hand, some have very few members. "What are their names?" The oldest ones are very ancient, and their names never

change: Eometryjay, Gebralay, Alysisnay . . . When a new tribe is formed, which happens often, the coining of its name goes by complicated, fluid rules, mostly by putting together names of existing tribes.

"Are there no exchanges between tribes?" As little as possible. One is proud to be an

Urematherpay, but one tries to avoid communicating with Urematherpays of other tribes. When circumstances, like proximity for example, result in too great a mutual com-

1 Reprinted by permission from Gazette des Mathematiciens 67 (1 996), 43--46. This article appeared also in Deux et deux font-its quatre?, a collection of the author's essays: Pour Ia Science, Belin,, 1 999.

2-franslator's Note: The author's tribal names are concocted using verlan, so I have used the closest English equivalent, Pig Latin. Thus where the author transforms mathematicien pur to matheux pur to purtheuma, I transform pure mathematician to pure mather to urematherpay.


prehension between tribes, they form a new tribe, which

hastens to break all bridges with the others.

"How do they go about it?"

The new tribe creates its own language which the other

tribes don't understand, and makes it as difficult as possi

ble to translate. For one thing, they manage not to talk

about the same objects. One tribe takes the gebra as its

totem animal, talks only about the gebra and makes it the

object of all its prayers, casting of auguries, etc. Another

tribe talks about the dime and the deg, and so on. 3 These

are mythical animals, of radically different natures. Those

who have seen the gebra have never seen the dime or the

deg, and vice versa. A statement about the gebra can not

be expre�sed in terms of the deg, for example; for these

two creatures have n9 simultaneous existence, evolve in

incompatible worlds with no communication between

them. Every attempt at translation is bound to fail. This

makes mutual incomprehension complete.

"Can it really be complete?"

People refuse to recognize the accomplishments of this

people. To be sure, all the tribes deal with the same mate

rial reality. If their different languages were designed to de

scribe that, then translation between them would be pos

sible. But it is not so. The Urematherpays consider the

material world a triviality not worth talking about. Each

tribe creates a purely imaginary world and pays no atten

tion to anything else. As these worlds have no common

point, there is no passing from one to another, and there

is no translating one Athermay language to another. That

is why I assert that the puwose of language among the

Urematherpays is to attain non-communication.

"That is the conclusion that has come under attack."

Naturally. It contradicts the accepted ideas about lan

guage. How can we imagine a language which is not in

tended for communication? All right, go and live among the

Urematherpays. I've done it for many long years, and I as

sure you that with them, the function of language is avoid

ing exchange. The mythical creatures of each tribe seal it

off perfectly from the others.

"But you claim more than that. You say that even within

each tribe mutual comprehension is poor."

Yes. That is a subtler question. My hypothesis is that the

breaking of communication between tribes leads to break

ing of communication within tribes. Not that this is neces

sarily sought by the Urematherpays. It can be an unin

tended side effect.

"Still, they do talk to each other within a tribe?"

Oh, sure, they talk. But they don't understand.

"Why not?"

Well, I told you that their language deals only with imag

inary things. Their statements don't purport to have any

agency: they are not followed by any action which would

give a check on whether such-and-such order had been fol-

-AU T HO R

DIDIER NORDON

47, rue du Sablonat

33800 Bordeaux

France

e-mail: [email protected]·bordeaux.fr

Didier Norden, bom 1 946, graduated at Paris Sud, Orsay. He

has been teaching mathematics at the University of Bordeaux

1 since 1 970. Still there remain people in Bordeaux who un

derstand nothing about mathematics. . . . A main interest of

his is the relationship between the world view of mathemati

cians and that of the surrounding society. Among his books

are Les mathematiques pures n 'existent pas! (new edition

Actes Sud, 1 993) and La droite amoureuse du cercle, a col

lection of fantasies, Editions Autrement, 1 997. He is a colum

nist for the monthly Pour Ia Science, the French version of

Scientific American.

lowed correctly. You're an Urematherpay, say, and you give

the order, "Horace! Blow up your method, and put the deg

on a subvariety"-a typical Athermay utterance. No word

in the sentence has a concrete referent. How could you

possibly verify whether Horace has understood your com

mand? You can't. Now it's a well-known psychological law

that if you have no way of confmning that your interlocu

tor understands your statement, you can be sure that in

fact he doesn't. Oh, perhaps he'll understand it once or

twice. Maybe by chance. But you are tending inexorably

toward incomprehension.

"And that is happening among the Uremathpays?"

Very likely. They permitted me to take part in their most

important ritual, the eminarsay. The eminarsay is a weekly

meeting of the whole tribe. Each tribe has its own. The

speaker, one of the members, gives an hour's incantation to

the gods of the tribe. All my observations indicate that the

communicants have little understanding of what the offi

ciant is saying. No response from them, little variation in

muscle tone or intellectual tone; visible langor; many doz

ing. Well, what do you suppose happens at the end of the

incantation? The listeners add little incantations of their

3J. Alexander & A. Hirschowitz, La methode d'Horace eclatee: application a ! ' interpolation en degre quatre, lnventiones Mathematicae 1 07 (1 992), 585-602: "Dans cette variants eclatee, on exploits une sous-variete de codimension quelconque: Ia dime est un enonce de rangement sur cette sous-variete, landis que Ia degue est un enonce de rangement sur Ia variete obtenue en eclatant cette sous-variete." J.-P. Serre, Gebres, L'Enseignement Mathematique 39 (1 993), 33-85: "Objet de ce texte, les enveloppes algebriques des groupes lineaires et leurs relations avec les differents types de gebres: algebres, cogebres et bigebres.'


own, in the form of questions which the officiant answers.

It shows they don't need to understand to talk to each other.

When they paid me the compliment of letting me give the

incantation, I spoke sentences without any meaning I could

see. The audience reaction at the end was the same as usual.

I even suspect that the aim-I_repeat, the aim-of verbal ex

change between Uremathpays is mutual incomprehension.

"For what purpose?"

At the foundation of the Athermay conception of the

world is the idea (maybe astonishing to you, but natural to

them) that an imaginary world is the richer for being diffi

cult to communicate. The ultimate richness, then, would

be found in incommunicability. But how subtle they are!

Contrary to what you might suppose, they detest circuitous

verbiage: that, they consider a gratuitous obstacle. Every

Urematherpay struggles to be as clear as possible. It's only

if in spite of these struggles he remains enigmatic to the

other Urematherpays that he knows he has attained the

summit of richness.

"And that is satisfying?"

Yes. Look at the matter from both sides. No doubt it is

unpleasant for an Urematherpay not to understand what is

said to him. But they're practical people: they realize that

this unpleasantness is a small price to pay for the gratifi

cation of being recognized as incomprehensible by others.

And permit me in conclusion to ask a question of you. Is

it really always all that satisfying when you do understand

what others are telling you?

EXPAN D YOUR MATHEMATICAL BOU N DARIES!

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Mathematics Without Borders A History of the International Mathematical Union

As told by Professor Olli Lehto, the history of the International Mathematical Union (IMU) is surprisingly compelling.

The twentieth century has been fraught with tremendous international conflict, but there has also been a great deal of

progress made tO\IIIard cooperation among nations. Ever since its inception in 1 897, the IMU has sponsored International

Corsresses throughout the 'NOI'ld, effectively bridging cultural and political gaps by uniting scientists through their genuine

love and appreciation for mathematics.

For anyone strictly interested in the mathematical and organizational details of the IMU Corsresses, this book will serve as an excellent resource. However, Mathematics Without Borders also takes time to focus on the individuals, many of

them leading mathematicians of the twentieth century, and their stories-told against the backdrop of world events.

Contents: Prologue to the History of the IMU • The Old IMU (1920-1932) • Mathematical Cooperation Without the IMU (1933-1939) • Foundation

of the New IMU (1945-1951 ) • The IMU Takes Shape (1952-1954) • Expansion of the IMU (1955-1958) • The IMU and International Corsresses

(1 958-1962) • Consolidation of the IMU (1 963-1970) • North-South and East-West Connections (1971-1978) • Politics Interferes with the IMU

(1 979-1986) • The IMU and Related Organizations • The IMU in a Changing World (1986-1990)

1 998/368 PP., 53 FIGS./HARDCOVER/$35.00/ISBN 0-387-98358-9

Order Today! Call: 1-800-SPRING£R • f'ax: (201)-348-4505 • VIsit: {http://www.sprlnger-ny.com}

f

'

';.0 :· Springer

VOLUME 22. NUMBER 4, 2000 45

The Fibonacci Ch imney Mats Gyllenberg

Karl Sigmund

Does yaur hometown have any

mathematical tourist attractions such

as statues, plaques, graves, the cote

where the famaus conjecture was made,

the desk where the famous initials

are scratched, birthplaces, houses, or

memorials? Have yau encauntered

a mathematical sight on yaur travels?

If so, we invite yau 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 yaur tracks.

Please send all submissions to

Mathematical Tourist Editor,

Dirk Huylebrouck, Aartshertogstraat 42,

8400 Oostende, Belgium


F ive years ago, residents and visi

tors in Turku were confronted with

a sequence of seven-foot-high digits

running along the smokestack of the

local power plant-an intellectual

challenge to all except, of course,

mathematicians. In due time, the

smokestack became the most salient

showpiece in the city fathers' attempt

to turn Turku-the oldest university

town in Finland and a major calling

port for midsummer cruises in the

Baltic-into a capital of Conceptual

Art. The artist responsible for the dis

play, Mario Merz (b. 1925) from Italy,

had been obsessed by the sequence for

almost 30 years. He has used it to dec

orate the Saint Louis chapel in Paris'

Salpetriere as well as a spire in Turin,

but nowhere to the same effect as in

Turku, where it dominates the water

front and the estuary of the Aura river.

It is entirely by accident that the se

quence reflects two of the major re

search fields of the University of

Turku, namely, number theory and

mathematical biology. As is well

known, Fibonacci introduced the se

quence at around AD 1200 to model the

growth of a rabbit population.


20014 University of Turku

Finland


University at Vienna

1 090 Vienna, Austria

The Turun Energia Power Plant in Central Turku. Photo courtesy of Turun Energia photogra

pher Seilo Ristimiiki.


Vi ln ius Between the Wars Stanisfaw Domoradzki and

Zofia Pawlikowska-Brozek

I n the beginning of the 20th century, Polish mathematics rapidly expanded,

in its two major mathematical centers, Lvov and Warsaw, and in Krakow, one of the oldest universities of Europe. Stanislaw Zaremba (1863-1942), Waclaw Sierpiriski (1882-1969), Hugo Steinhaus (1887 -1972), Stefan Mazurkiewicz (1888-1945), Stefan Banach (1892-1945) and Juliusz Schauder (1899-1943) are a few of the names linked to that epoch. A center also of some importance was Vilnius.

Vilnius is the name of the city in the Lithuanian language, and it is now the capital of independent Lithuania; between the wars it was in Poland and was usually designated by its Polish name, Wilno. It is a clean, beautiful, lively city, with spires of churches and newly renovated fa<;ades of historical buildings. In the present text, a picturesque route is described that a mathematical tourist may follow when

GERMANY

CZECHOSLOVAKIA

arriving by train in Vilnius, going to the University. Some details are given about well-known Vilnius mathematicians, reviving the glorious past of the city. Traces of these close colleagues of the so-called Polish school of mathematics are rare in Vilnius, because many fled the country, and only a few were buried in the city, as we shall discover.

From the Railway Station, we follow Gelezinkelio street (this is its Lithuanian name; Kolejowa is the Polish equivalent). Ausros Vartu street (or Ostrobramska) leads us to the Ostra Brama, a sanctuary in honor of St. Mary. Passing a Polish bookshop and St. Kazimierz's church, the visitor then reaches the Town Hall through a narrow street. Didiioji street (Wielka) leads to 11 group of buildings between Universiteto (Uniwersytecka) and Pilies (Zamkowa) streets.

There, St. John's church can be

LITHUANIA

USSR

Figure 1. A map of Poland in the years 1922-1939.


seen, attached to the University buildings surrounding the courtyards. They are named after the great poet Adam Mickiewicz and the rectors Piotr Skarga (1536-1612), the very first rector of the Vilnius Academy, and Marcin Poczobutt (1728-1810), who was in charge during reforms of the educational system. Nearby is the Papal Seminary, where refreshments are offered in a courtyard surrounded by 17th-century cloisters and arcades.

The University Library contains archives of Polish mathematicians who worked at the University in the period between the wa.rS, as well as old documents from the very beginning of the Vilnius Academy. They go back to 1570, when the Jesuits founded a college; later, King Stefan Batory of Poland promoted it to the rank of Academy. The Vilnius Academy became the cultural center radiating to the East, South, and North, as the Krakow Academy was spreading its influence from the West. The lecturers often came from abroad, but many Polish scientists also worked at the Vilnius Academy.

Towards the end of the 18th century, when the first Polish Ministry of Education was founded, the Academy flourished again. The lectures on higher mathematics were introduced and conducted by Franciszek Milikont Narwojsz (1742-1819), based on the works of Isaac Newton. The astronomical obseiVatory was equipped with modem instruments thanks to Rector Marcin Poczobutt, a well-known astronomer in Europe. Other obseiVations were made by Jan Sruadecki (1756-1830), author of Trygorwmetria kulista analitycznie wy/bZona (1817) (An Analytical Presentation of Spherical Trigonometry), which was translated into German. Originally from Krakow University, he had come to Vilnius to become professor of astronomy and rector, and he very actively contributed to the reforms in the Polish educational system. In 1803 the Academy changed into a University and flourished until the Russian occupation closed it in 1832.

After Poland regained independence in 1919, the University revived and be

isted until 1939, when World War II started. In that period, numerous math-ematics and natural sciences conferences took place in Poland, and in Vilnius in particular. The Congress of Polish Physicians and Natural Scien-tists, which included a mathematical section, was held in 1929. Two years later, the Second Congress of Polish Mathematicians was organized by the Polish Mathematical Society. The Society had been founded in 1919, and its first congress was held in 1927, in Lvov. All professors working in Vilnius at the time took part in the organization of the Congress: Wiktor Staniewicz (1866-1932), Stefan Kempisty (1892-1940), Juliusz Rudnicki (1881-1948),

came the Stefan Batory University. It ex- Figure 2. View of Vilnius streets.


Stanislaw Krystyn Zaremba (1903-1990), and Antoni Zygmund (1900-1992).

Zygmund had come from Warsaw University, where he got his doctoral degree in 1923. He was appointed assistant professor at the Philosophy Department of Warsaw University in 1926 and worked there until 1929, to become head of the Mathematics Department in Vilnius in 1930. Thanks to a Rockefeller Foundation fellowship, he had gone to Oxford and Cambridge, where he had met G. H. Hardy and J. Littlewood. There he also got in touch with R. E. A. C. Paley, an outstanding young English mathematician, with whom he would later publish several works. In Vilnius, Zygmund

Figure 3. Vilnius University today.

did not lose contact with his colleagues of the Warsaw school of mathematics. In 1938, his collaboration with Stanislaw Saks (1897-1942) led to their book Analytic Functions, which would be republished 14 years later in the Mathematical Monographs series.

Zygmund's arrival in Vilnius strengthened the mathematical circle. At that time he was already the author of over forty scientific works published abroad, as well as in the leading Polish mathematical journals, such as Fundamenta Mathematicae and Studia Mathematica. Thanks to friends such as J. Tamarkin, Zygmund would move to the USA in 1940, where his mathematical career flourished. After a short time at the University of Pennsylvania in Philadelphia, he would become a professor in the Mathematics Department of the University of Chicago, until his retirement in 1980. Zygmund was awarded honorary degrees in Washington, Toruri, Paris, and Uppsala, and was a member of the National Academy of Sciences of the USA, the Polish Academy of Sciences, the Royal Academy of Sciences of Madrid, and the Academy of Sciences of Palermo. In 1986, he received the National Medal of Science of United States, 6 years before his death.

One of Zygmund's students at the Stefan Batory University was J6zef Marcinkiewicz (1910-1940), who might have become a first-rank math-

ematician if destiny had been as favorable to him as it was to Zygmund. Already in 1933, he published two works: "On a theorem of trigonometric series" (Journal of the London Mathematical Society) and "On a class of functions and their Fourier series" (Reports of the Warsaw Science Society). At the age of 25, he took a

Figure 4. A drawing of Antoni Zygmund, by

L. Jesmanowicz.

doctor's degree at Vilnius University. He proved that it is possible to construct a continuous function whose Fourier series is uniformly convergent while the interpolation polynomials are almost everywhere divergent. J. Marcinkiewicz spent some time at the famous Lvov mathematical school and this led to several papers in the Lvov Studia Mathematica; and in 1937 he qualified as assistant professor of Stefan Banach. However, his scientific career was tragically cut short. At the beginning of the 1938-39 academic year, J. Marcinkiewicz left for London and Paris on a research grant from the National Culture Foundation. A� the end of August 1939 he interrupted the journey and came back to Poland because of the international political situation. ije took part in the defense of LvoV, and after the Soviet Army marched into the city, he was taken prisoner, to Starobielsk. Only 30 years old, he was murdered in 1940; his name would join the list of the Katyil victims.

World War II took a heavy toll of Polish mathematicians. Those who survived were scattered all over the world. J. Rudnicki became the head of the Mathematics Department at Lublm University, but when he met his colleagues at the University of Mikolaj Kopernik in Toruri, he moved there in 1946. Similarly, Leon Jesmanowicz (1914-1989), who worked for about two years at the University of Stefan Batory, came to the Kopemik University. He is probably best known for his caricatures (see figure 4). Miroslaw KrzyZariski (1907 -1965), who had taken a doctor's degree of mathematics at the University of Stefan Batory in 1934 and was a specialist on partial differential equations, became a professor at Krakow Technical University and the Jagiellonian University.

Let us come back to the route we were following in Vilnius. At the other side of Universiteto (or Universytecka in Polish) street, there is the presidential palace, called the Prezidentura (Palac Prezydencki). Then we continue our road crossing a flowery square, reaching the Arkikatedros aikste (Plac Katedralny), with the monumental St. Stanley's Cathedral. On the left-hand side we pass the Gedimino


Figure 5. The arms of Vilnius.

bokStas (G6ra Zamkowa) or Castle Hill to see the architectural contrast of the St. Ann's church and the Bernardine's church.

We cross the Vilnia (Wilejka) River,

which surrounds a part of Vilnius and joins the Neris (Willa), the main river on which the city has grown. A trolley bus will bring us to the Rossa cemetery, where some Vilnius professors of mathematics are buried. Just behind the gate, next to the fence, lies Wiktor Staniewicz (1866-1932), of the University of Stefan Batory, while a bit further is the grave of Tomasz Zycki, of the 18th-century Vilnius University. From there, one can conclude the journey by paying a visit to the Lituvos Centrinis Valstybes Archyvas, that is, the Lithuanian Central Record Office. There, documents and personal portfolios of the University of Stefan Batory professors are gathered.

BIBLIOGRAPHY

Bielir'\ski, J . , Uniwersytet Wileriski (The Vilnius

University) (1579-183 1) , Krakow, 1 899-

1 900.

Calderon, A.P., Stein, E., Antoni Zygmund

(1900-1992), Notices of the American

Mathematical Society, v.39, n.8, p.848-9,

October 1 992.

Fetterman, Ch. , Kahane, J.P. , Stein, E.M. , 0

dorobku naukowym Antoniego Zygmunda or

On Antoni Zygmund's scientific achieve

ments, bibliography and portrait, Wiado

moSc:i Matematyczne XIX.2, p.91 -1 26,

1 976.

Jesmanowicz, L. Caricatures of Polish mathe

maticians, 1 3th Congress of Polish Mathe

maticians, Toruli, 1 994. Zemajtis, Z. Fiziko-Mathematiczeskije nauki w

starom Wilniusskom Universitetie (1579-

1882), Utovskij Matematiceskij Sbornik 1 1 .2 ,

1 962.

Zygmund, A. J6zef Marcinkiewicz, Wiadomosci

Matematyczne VI, p.1 1 -41 , 1 96o-61 .

Zofia Pawlikowska-Broiek

University of Mining and Metallurgy

Department of Applied Mathematics

al. Mickiewicza 30

30-059 Krakow

Poland.

Stanistaw Domoradzki

Pedagogical University

Institute of Mathematics

ul. Rejtana 1 6a

35-31 0 Rzeszow

Poland.

Mathema t ica l O lymp iad Cha l l enges Titu Andreescu, American Mathematics Competitions, University of Nebra ka, Lincoln, NE

Razvan Gelca, University of Michigan, Ann Arbor. Ml

This is a comprehensive collection of problems written by two experienced and well-known

mathematics educators and coaches of the U.S. International Mathematical Olympiad Team.

H undreds of beautiful, challenging, and instructive problems from decades of national and

international competition are presented, encouraging readers to move away from routine exer

ci e and memorized algorithms toward creative solutions and non-standard problem-solving

techniques.

The work is divided into problem clu tered in self-contained sections with olution provid

ed eparately. Along with background material, each ection include representative examples,

beautiful diagrams, and li t of unconventional problems. Additionally, historical in ight and

a ide are pre ented to stimulate further inquiry. The empha i throughout is on stimulating

readers to find ingeniou and elegant olutions to problems with multiple approache .

Aimed at motivated high school and beginning college students and i nstructors, this work can

be used as a text for advanced problem-solving cour es, for self-study, or as a resource for teach

ers and students training for mathematical competitions and for teacher professional develop

ment, seminars, and workshops.

From the foreword by Mark Saul: "The book weave together Olympiad problems with a com

mon theme, so that insight become techniques, tricks become methods, and method build to

mastery . . . Much is demanded of the reader by way of effort and patience, but the investment is

greatly repaid."

2000 / 280 PP., 85 ILL S.

SOFTCOVER / ISB 0-8 1 76-41 55·6 / S29.95

HARDCOVER / ISB 0·8 1 76-4 1 90-4 / S59.95

To Order: Call: 1 -800-777-4643 • Fax: (20 1 ) 348-4505 • E-mail: [email protected] • Visit: www.birkhauser.com Textbook evaluation copies available upon request, call ext. 669. Price are valid in orth America only and subject to change without notice.

Birkhauser Boston - International Publisher for the Mathematical Sciences 7100 Promotion IY1097


JOHN A. EWELL

Cou nti ng Latt ice Po i nts on Spheres

roblems formulated in one branch of mathematics but solved with tools of another

branch have for a long time been a source of great satisfaction, especially for those

among us who would guard against the tendency toward overspecialization.

Noteworthy among such problems are those which lie on the boundary between

geometry and arithmetic (number theory). These can be traced back to Pythagoras, and perhaps beyond (e.g. , to the ancient Babylonians).

In this discussion, we let n run over N : = {0, 1, 2, . . . }, and for each such n construct in IR3 a sphere of radius Vn

centered about the origin (0, 0, 0), as in Fig. 1. We then set the problem of counting the lattice points (x, y, z) on the surfaces of these spheres. (A lattice point (x, y, z) is one all of whose coordinates are integers.) This problem is one of three-dimensional Euclidean geometry, where all of the numbers involved (including the radii Vn, n = 0, 1, 2, . . ·) are constructible. The tools for solving the problem come from number theory.

As there are two counting functions r3 and q0, defined on N and having values in N, I shall briefly describe these: for each n E N,

where 7L := {0, ::!:: 1 , ±2, . . . }; and q0(n) : = the number of partitions of n into distinct odd parts. By convention q0(0) : = 1. E.g. , among the five unrestricted partitions of 4, viz., 4, 3 + 1, 2 + 2, 2 + 1 + 1 and 1 + 1 + 1 + 1, there is exactly one into distinct odd parts, so that q0(4) = 1. The

reader should be easily convinced that the function q0(n), n E N, is generated by the infinite product expansion

00 00

fl (1 + _i!n-1) = L qo(n)xn, n�l n�o

which is valid for each complex number x such that P::l < 1.

For the sake of completeness put q0( n) : = 0 whenever n < O.

The Counting Algorithm

First of all, if on the one hand n E N and r3(n) > 0, then for each of the relevant points (x, y, z) E 7L3, n = x2 + y2 + z2 if and only if Vn = Y(x - 0)2 + (y - 0)2 + (z - 0)2. On the other hand, if n E N and r3(n) = 0, then there exists no point (x, y, z) E 7L3 such that n = x2 + y2 + z2, or equivalently, such that Vn = Y(x - 0)2 + (y - 0)2 + (z - 0)2. Hence, for each n E N, r3(n) is the count of all lattice points on the sphere of radius Vn centered about the origin (0, 0, 0). Next, I will state and discuss (but not prove) three arithmetical theorems.

Theorem 1 (Legendre) If S : = {n E N : n = 4k (8m + 7), for some k, m E N}, then for each n E S, r3(n) = 0, and for each n E N - S, r3(n) > 0.


Figure 1. Here presented are complete descriptions of the degenerate sphere of radius Yo and the unit sphere of radius v1, the six dots

representing all of the lattice points on the surface of the latter. To avoid clutter, descriptions of the spheres corresponding to the radii \12 and v3 are left incomplete. Of course, the reader must imagine the spheres having radii Vn, n E N - {0, 1 , 2, 3}.

Theorem 2 For each n E N,

I ( - 1)k(k+l)12qo(n - k(k + 1)/2) = kEN {C- 1r, if n = m(3m ± 1), Jor some m E N,

0, otherwise.

Theorem 3 lf !FD := N - {0}, then for each n E N,

r3(n) = qo(n) - I ( - 1)kC3k - 1Y2(6k - 1)qo(n - k(3k - 1)/2)

kElP'

+ I ( - 1)kC3k + 1)12(6k + 1)q0(n - k(3k + 1)/2) kElP'

= qo(n) + 5qo(n - 1) + llqo(n - 5) - 17 qo(n - 12) + · · ·

+ 7qo(n - 2) - 13qo(n - 7) - 19qo(n - 15) + · · ·

At the present time no simple proof of Legendre's Theorem 1 is known. E.g., see [3, p. 311]. Clearly, the theorem has a striking nonintuitive interpretation relative to our problem. For a proof of Theorem 2 see [1, pp. 1-2]; and for a proof of Theorem 3 see [2].


Our recursive two-step algorithm proceeds as follows: (i) Use the recursive determination of q0 in Theorem 2 to compile a table of values of q0, as in

n 0

1

2

3

4

5

6

7

1

0

8 2

9 2

1 0 2

1 1 2

1 2 3

TABLE 1 .

n qo(n) 1 3 3

1 4 3

1 5 4

1 6 5

1 7 5

1 8 5

1 9 6

20 7

21 8

22 8

23 9

24 1 1

25 1 2

(ii) In terms of these computed values of q0, utilize Theorem 3 to compile a table of values of r3, as in

TABLE 2. n r3(n)

0 1

6

2 1 2

3 8

4 6

5 24

6 24

7 0

8 1 2

9 30

1 0 24

1 1 24

1 2 8

n 1 3

1 4

1 5

1 6

1 7

1 8

1 9

20

21

22

23

24

25

r3(n) 24

48

0

6

48

36

24

24

48

24

0

24

30

A U T H O R

JOHN A. EWELL

Department of Mathematical Sciences

Northern Illinois University

DeKa/b, IL 601 1 5-2888

USA Each of these tables 1 and 2 can be indefinitely extended (in the stated order) with the aid of machine computation. For a fixed but arbitrary choice of n E N, the running time for each table is O(n312). Legendre's Theorem 1 here provides an excellent check on the accuracy of computation.

Concluding Remarks

The problem of lattice points on spheres is one of elementary geometry, easily visualized; the first few cases are easily computed. Intuition at that stage leads us to expect that the values r3(n) will rise steadily but irregularly with n. Legendre's Theorem steps in to show us the unexpected and striking exceptional values of n for which r3(n) = 0.

John Ewell earned his Ph.D. in 1 966 under the direction of

E.G. Straus. He has been on the professorial staff at Northern

Illinois University for many years, and though recently retired

he is still active there and still continues his interest in additive number theory and related fields. He is partial both to

opera and to baroque instrumental music.

ACKNOWLEDGMENTS

REFERENCES

1 . J.A. Ewell , "Recurrences for two restricted partition functions,"

Fibonacci Quarterly, 18 (1 980): 1 -2.

2. J. A. Ewell, "Recursive determination of the enumerator for sums of

three squares," Int. J. of Math. and Math. Sc. (to appear).

I \vould like to thank Eric Behr for producing the descriptive picture of Figure 1.

3. G. H . Hardy and E. M. Wright, An Introduction to the Theory of

Numbers, Fourth edition, Clarendon Press, Oxford, 1 960.

Clear, Simple, Stimulating Undergraduate Texts from the Gelfand School Outreach Program

Trigonometry 1. M. Gelfand, Rutgers University, New Brunswick, NJ & M. Saul, rhe BrofiXl!i/le School, Bronxville, NY

This new text in the collection of the Gelfand School Outreach Program is written in an engaging style, and approaches the material in a unique fashion that will moti· vate students and teachers alike. All basic topics are covered with an emphasis on beautiful illustrations and

'--------' examples that treat elemen· tary trigonometry as an outgrowth of geometry, but stimulate the reader to think of all of mathe· matics as a unified subject. The definitions of the trigonometric functions are geometrically motivated. Geometric relationships are rewritten in trigonometric form and extended. The text then makes a transition to a study of the algebraic and analytic properties of trigonometric functions, in a way that provides a solid foundation for more advanced mathematical discussions.

2000 I Approx. 280 pp., 185 illus. I Hardcover ISBN 0-8176-3914-4 I $ 1 9.95

Algebra I.M. Gelfand & A. Sben

"The idea behind teaching is to expect students to team why things are true, rather than have them memorize ways of solving a few problems . . . {This j same philosophy lies behind the current text . . . A serious yet lively look at algebra. "

-The American Mathematical Monthly

1993, 3rd prinling 2000 I 160 pp. I Softcover ISBN 0-81 76-3671-3 I S 1 9.95

The Method of Coordinates I.M. GeLrand, E.G. Glagoleva & A.A. Klrlllov

"High school students (or teachers) reading through these two books would learn an enor· mous amount of good mathematics. More importantly, they would also get a glimpse of how mathematics is done. "

-The Mathematical lnteUigencer

1 990, 3rd prinling 1 996 I 84 pp. I Softcover ISBN O-B 176-3533·5 I S 1 9.50

Functions and Graphs I .M. Gelfand, E.G. Glagoleva & E.E. Sbnol

"All through both volumes, one finds a careful description of the step-by-step thinking process that leads up to the correct definition of a concept or to an argument that dinches in the proof of a theorem We are . . . very fortunate that an account of this caliber has finally made it to printed pages. "

-The Mathematical lntelligencer

1 990, 51h prinling 1999 I 1 1 0 pp. / Softcover ISBH 0-81 76-3532-7 I $ 1 9.95

To Order: Call: 1-800-777-4643 • Fax: (201) 348-4505

E-mail: [email protected] Visit: www.birkhauser.com

Textbcd evaiU01ion cOjliel avoiloble upon request, call ext 669. Prim are 1111id in N011h Arneti<o only ond subject 10 dlonge wilhout noli<e. i Blrkhiiuser Boston

International Publisher for the Mathematical Sciences

SoW Promotioo IYI018

QfifW-J.t.i Jeremy G ray, Editor I

Kepler's Crit ique of Algebra Peter Pesic

Dedicated to Curtis Wilson

Column Editor's address:

Faculty of Mathematics, The Open University,

Milton Keynes, MK7 6AA, England

A lthough Johannes Kepler lived during the initial flowering of early

modem algebra, he did not make much use of it, preferring to work in the classic geometric manner. Kepler considered Euclidean geometry to be fundamental, and thought that algebra could not compete with geometry as a source of intelligible truths, although it was a valuable, if approximate, method. Kepler's conception of the world also rested on crucial musical assumptions. As with his work in physical astronomy, Kepler insisted on the importance of observation in judging musical questions. It is from such empirical observations of musical practice that he excludes intervals that have no geometrical expression, but only algebraic ones. In this way, his musical observations support his geometrical exposition and his rejection of algebra.

His critique of algebra was based on what he saw as its hidden reliance on infmite processes; his example of the heptagon shows that the solutions even of cubic equations rely on transcendental functions. In a striking anticipation of intuitionistic arguments, Kepler rejected infinite processes, and conceived of geometry as finite constructions. Though perhaps the earliest critic of symbolic algebra, Kepler already raised profound questions.

Kepler's "Geometrical Kabbala"

Kepler phrased his mathematical demonstrations in the classic geometric manner of Euclid. He did this for deep philosophical reasons. As he wrote: "Geometry is unique and eternal, a reflection from the mind of God. That mankind shares in it is because man is an image of God." [Cas 380] He was also fascinated by the new techniques of logarithms, which he independently rederived and extended in his Chilias logarithmorum (1624). He did this, however, in a Euclidean way. Although Kepler was aware of Franc;ois Viete, the "father of modem


algebra," he did not employ the new symbolic mathematics, and expressed critical reservations on the few occasions he mentioned it. [NA 256; KGW 9. 1 12-113, 1 7.258] While Viete was critical of the "contrivance" of the Copernican alternative Kepler so passionately espoused, Kepler expressed a certain attraction but even more skepticism towards the use of symbols as the central device of mathematics. [Swe] In a letter of May 12, 1608, written to his friend, the physician Joachim Tanckius in Leipzig, Kepler remarked that

I too play with symbols; I have planned a little work, Geometrical Kabbala, which is about the Ideas of natural things in geometry; but I play in such a way that I do not forget that I am playing. For nothing is proved by symbols, nothing hidden is discovered in natural philosophy through geometrical symbols; things already known are merely fitted [to them]; unless by sure reason it can be demonstrated that they are not merely symbolic but are descriptions of the ways in which the two things [i.e., the two terms of the analogy] are connected and of the causes of this connection. [Wal 55]

There may well be some irony here, for elsewhere Kepler remarked, "I hate all kabbalists." [Cas 292] Kepler noted dryly that "whoever wants to nourish his mind on the mystical philosophy . . . will not fmd in my book what he is looking for." [KGW 6.397; Fie2, Ros] The "Geometrical Kabbala" he mentions was not finished or has disappeared, but perhaps he is referring to some geometrical algebra of the sort that Descartes was later to introduce.

Kepler's reasoning is especially interesting: symbolic mathematics, while it may reveal hidden interconnections, cannot elucidate the inner nature of things in themselves. This goes far beyond his dislike of esoteric mystifica-

tions; it reflects his Platonic critique of mathematics. Plato had argued that mathematics restricts itself from a full inquiry into the merits of its own axioms, preferring instead to deduce further theorems and consequences of given presuppositions. Accordingly, Plato assigns the highest level of knowledge not to mathematics but to philosophy, which can inquire into the good of the axioms that mathematics takes as given. [Republic 509d-51le]. However, Kepler treats mathematics, particularly geometry, as a way of gaining access into nature's secrets, not merely deducing consequences of unexamined axioms. Here Kepler may follow other Platonic examples (notably the Timaeus ), which use geometric hypotheses as the staging ground for shaping a "likely story" to explain astronomical appearances.

Beyond these Platonic recollections, though, Kepler had another important point. In his view, mathematical science works by making connections between diverse phenomena, rather than by penetrating to what lies behind the phenomena His insight anticipates Newton's avoidance of "hypotheses" in favor of faithful mathematical prediction. Kepler had explicitly responded to Petrus Ramus's challenge to find an "astronomy without hypotheses"; in The Secret of the Universe (1596), Kepler claimed that he (and Copernicus) had answered this challenge, a claim he repeated in the New Astronomy (1609) and Rudolphine Tables (1627). [HW xi-xii] In contrast, elsewhere in his work, Kepler did try to form fundamental hypotheses about the cause of gravity, viewing it as closely analogous to a magnetic force. Indeed, in the New Astronomy (1609) such hypotheses were helpful to Kepler as he formulated his laws, if only as heuristic devices that allowed him a different, fruitful perspective on a familiar problem. [NA 271-280] Kepler drew much from the magnetic analogy, and yet the quote we have been considering shows that Kepler was able to take a more skeptical, detached view of his hypothesis, at least as a mathematical construction. This may also have been Kepler's way of indicating the primacy of physical considerations (the analogy with magnets) over mathematical representations (the way that analogy is deployed in his astronomical work).

Algebra and the Heptagon

There is only one extended passage using algebraic symbolism and methods in Kepler's works; it shows both Kepler's familiarity with the "cossa" (as he called algebraic technique) and his arguments against it. The passage occurs in Book I of The Harmony of the World (1619), in the context of his general discussion of the construction of regular figures. [HW 60-79] Kepler uses the case of the regular heptagon to examine the claims of algebra; in contrast to Cardano's approach to this problem, Kepler adhered to the ancient Greek notion of numbers as integers (counting numbers rather than "real" numbers, in the modern sense). [Kl, Fie3] Kepler's criterion is what he calls "knowability," correlated to the degree of irrationality of different regular polygons, following Euclid's Elements Book X. Kepler also set out the degrees of knowability, beginning with the first degree (a line equal to the diameter of a given circle,

or a surface equal to the square on that diameter). He begins with a theorem asserting that it is not possible to construct regular polygons with a prime number of sides greater than five; "it is on account of this result that the Heptagon and other figures of this kind were not employed by God in ordering the structure of the World, as He did employ the knowablefigures explained in our preceding sections." Though Kepler's result was later superseded by Gauss's proof that polygons of sides 22n + 1 can be constructed, his point about the heptagon remained. Here Kepler contradicts the claims of Cardano that the heptagon (having as many sides as there were known planets) has cosmological significance.

At this point, Kepler brought up the work of Jost Btirgi (1552-1632), a notable clock-builder and instrumentmaker, as well as a skilled mathematician, who used algebra to calculate the side of the heptagon. Kepler went through Btirgi's argument in detail, using Kepler's own simplified version of the Italian notation for the unknown (res or "cossa") and its powers: i, ij, iij, iiij, v, vj, . . . (which later writers would call x, x2, x3, x4, x5, x6, • • . ). Burgi had used a more awkward !Gillan notation; Kepler showed ingenuity in devising an apt symbolism reflecting his close study of what he calls Btirgi's "very ingenious and surprising achievements in this matter." [KGW 6.527-528] Kepler analyzed Btirgi's equation of the heptagon to show that it does not amount to a construction of the figure by comparison with the geometrical construction of the pentagon. He then took Burgi's result for the side of a pentagon (the root of the equation 5j - 5iij + 1v, which in modem form is 5 - 5x2 + x4 = 0, quadratic in x2), and noted that

Again, as for the heptagon, this does not tell us how to construct the continuous proportion for which this relationship will hold, nor does it express the lengths of the proportionals in terms of things already known, but it tells us, once the [system of continuous j proportion is set out, what relationship will follow.

Kepler may have been the first (1619) to advance such a critique of Viete's symbolic mathematics, the "analytical art," a critique often ascribed mainly to Thomas Hobbes (1656), but also closely related to Newton's reservations about algebra. [Pyc, Pes3] Yet Kepler's reliance on geometric construction still leaves open the possibility that algebraic solutions might "express the lengths of the proportionals in terms of things already known" in some other way. As J. V. Field points out, Cardano had already considered geometric and algebraic proofs to be interchangeable in 1570. [Fie3]

Kepler knew that algebraic equations can be solved approximately; he gave approximations for the regular nineand eleven-sided polygons and used approximation techniques in his astronomical calculations. He did not consider such approximations as knowledge, and (despite his keen awareness of their numerical utility) he denied them any fundamental status. Yet the equations that can be derived by Btirgi's methods for the side of the pentagon (x4 -


5x2 + 5 = 0) and octagon (x4 - 4x2 + 2 = 0) are soluble, for they are quadratic in x2, as are the equations for the square (x4 - 2x2 = 0) and triangle (x4 - 3x2 = 0). Furthermore, the equations for the hexagon (x3 - 3x + 2 = 0) and heptagon (x6 - 7x4 + 14x2 - 7 = 0) are cubic in x and x2, respectively. Kepler refers to Cardano, and it is likely that Kepler was aware of the solutions of Cardano, Tartaglia, and Ferrari for cubic and quartic equations. If so, he would have known that the heptagon equation should have a solution in closed form involving square and cube roots (the solution of the hexagon's cubic equation is simple (x = 1) because of its close relation to the triangle). Kepler could then have regarded this solution of the heptagon proplem as yielding "knowledge" even according to his own definition: "T() know in geometry is to measure in terms of some known measure. In this matter of inscribing figures in circles the known quantity is the diameter of the circle." [HW 18] After all, the solution x of the heptagon equation does measure its side in terms of the diameter of the circle, taken as equalling 2. Field notes that "it is rather difficult to convince oneself that he is not putting arbitrary limits to God's powers by restricting Him to using only a straight edge and compasses." [Fie1 122] Yet the fact that Kepler brings this algebraic treatment forward shows that he considered that it might offer a different kind of "knowability" besides that of geometry, for he notes that "in this art the sides of all kinds of Polygon seem to be determinable." [HW 66] In contrast, Galileo's complete silence on algebra shows that he did not know of it or disdained to compare it to synthetic geometry. [Boy1]

A closer examination of the solution of this equation shows, however, that Kepler realized an important limitation of algebra. Applying the Cardano-Tartaglia method to the heptagon equation yields

Cardano himself treated quantities like � as "false" or "fictitious" (ficta) or "sophistic negatives." [Car] However, Field notes that Cardano's treatment of the heptagon is flawed by incorrect and conflicting equations. [Fie3 232-235] Rafael Bombelli (1572) had been able to extract the cube roots of certain complex numbers that arise in special cases of the cubic equation, and, by so doing, to eliminate the imaginary parts. [Boy2, Wae] However, that cannot be done in general, or even in this specific case. [Tig] If Kepler reached the above solution for x, he may rightly have been perturbed by the �. In the general irreducible case, it turns out that an attempt to take this cube root by purely algebraic means will lead back to the very same cubic equation from which the sought-for cube root arose in the first place! Extracting the cube root of a complex quantity requires the use of trigonometry and the De Moivre identity, (cos e + i sin e)n = cos ne + i sin ne. Without the use of transcendental functions, the cubic equation remains generally insoluble, if one demands that the cube roots of complex quantities be expressible in


terms of their real aKd imaginary parts. After using the De Moivre identity, the solution turns out to be real and positive, x = 0.8678. . . . If "knowing the solution" means determining its value only through taking roots, without using trigonometric functions, the side of the heptagon is indeed not knowable.

Here Kepler also pointed towards a limitation of the algebraists' program. Viete proudly claimed that his analytical art solved "THE PROBLEM OF PROBLEMS: TO LEAVE NO PROBLEM UNSOLVED," including unknowns raised to arbitrarily high powers. [Pes2, Pes3] Kepler raised important objections well before Descartes's La Geometrie (1635), which seems to sidestep these difficult questions of whether indeed all equations are soluble, and whether transcendental curves can really be invoked in the same way as algebraic curves. Certainly the Cardano-Tartaglia solution should be a crucial instance of algebraic solvability, but Kepler indicated that, despite their claims, the algebraists have not even solved the cubic equation in general. This objection goes beyond the formal issue of whether algebra as an analytical art can treat an unknown as if it were already given. Kepler might have allowed this analytical method, but he would not allow algebra silently to invoke infinite processes in its solutions. He objects to the algebraists' claim to have achieved trisection on just these grounds: "as rough and unshaped matter is to something which has form and as an indeterminate and indefinite quantity is to a figure, so also is the analytic method to geometrical determination . . . , " while admitting that "this Analytic of Biirgi' s tells us something general, not only about these two unequal chords but also about many other chords of a circle, which is useful for expressing [their lengths] in numbers." [HW 83-84] He especially objected that

I am required to pass over by a single act or motion something which potentiaUy involves infinite division; so that by this passage something may be attained which is concealed in that potential infinity, without the light of perfect knowledge, which the problems the ancients dubbed Plane do have. This kind of postulate is used frequently by Franyois Viete, a Frenchman, and Dutch Geometers of our day, in solutions of their problems, which by their very nature are not soluble except in a way that goes against the rules of the art, such as numericaUy or by Geometrical motions whose changes need to be guided by some kind of infinity. [HW 87-88]

This makes explicit his objection to the hidden recourse to infinite processes implicit in modem algebra and its transcendental firnctions. "We correctly maintain that the side of the Heptagon is among Non-Entities and is not susceptible of knowledge. For a formal description of it is impossible; thus neither can it be known by the human mind, since the possibility of being constructed is prior to the possibility of being known: nor can it be known by the Omniscient Mind by a simple eternal act: because by its nature it is among unknowable things." In a marginal note, Kepler emphasized that "these formal ratios of geometri-

cal entities are nothing else but the Essence of God; because whatever in God is eternal, that thing is one inseparable divine essence." Kepler did not admit the possibility that God might grasp the heptagon through an infinite process, for he would not allow that divine geometry could diverge from the finite perfec_tion of a Euclidean proof. Though this precludes the powerful achievements of modem algebra, it is evidence of Kepler's conviction that in geometry the human mind merges with the divine, as both grasp knowledge in one "simple eternal act."

The Evidence of Music

Kepler found confirmation of these opinions in his musical experience. His musical judgments speak decisively against algebra, and they rely on contemporary sensibilities rather than the best known ancient authority. He explicitly considered himself as both restating ancient harmonic knowledge and also reforming and perfecting it. [W al] Although he had originally planned to publish a translation of Ptolemy's Harmonics, he decided that his new account superseded it. Specifically, Kepler followed the practice of his time in adopting just intonation, in which the major third is 4:5 and the minor third 5:6, rather than the far more complex ratios assigned these intervals by Boethius and Macrobius (64:81 and 2304:2144). These preeminent ancient authorities (the only sources known to medieval musical theorists) accordingly classified both these intervals as dissonances (along with the major and minor sixths). Kepler knew that these intervals were treated as consonances (albeit imperfect ones) by contemporary composers and by the theorist Gioseffo Zarlino (1558), and h� gave them precedence over Boethius.

Ptolemy had also defmed the thirds as 4:5 and 5:6, in contrast to Boethius; though Kepler in this matter agrees with Ptolemy, on other points he objects to the "poetic or rhetorical rather than philosophical or mathematical" character of Ptolemy's "symbolism." [HW 499-508] Here Kepler took a position in a lively contemporary debate, for Vincenzo Galilei (Galileo's father) had attempted to return to the ancient defmition of the major third (64:81) in order to recapture the lost powers of ancient music. Kepler knew and quotes Galilei, but clearly was guided by Zarlino and the empiricism of just intonation. At several points, Kepler emphasizes that the test of consonance is not reliance on ancient authority but rather on the judgment of the ear; he expressed this fundamental principle not only in The Harmony of the World but also as early as the The Secret of the Universe (1596):

Therefore even by reference to the sole evidence of my book The Secret of the Universe the hearing is sufficiently fortified against the distraction oj the sophists, and those who dare to disparage the trustworthiness of the ears on very minute divisions, and their very subtle discrimination of consonance-especially since the reader sees that I followed the evidence of my ears at a time when, in establishing the number of the divisions, I was still struggling over their causes, and did not do the same as

the ancients did. They advanced to a certain point by the judgment of their ears, but soon abandoning their leadership completed the rest of the journey by following erroneous Reason, so to speak dragging their ears astray by force, and ordering them outright to turn deaf [HW 164-165]

If Kepler were to acknowledge the algebraic solution of the heptagon as legitimate knowledge, he would have to admit into music highly dissonant intervals such as 3:7 derived from the heptagon, just as the major sixth (3:5) is derived from the pentagon. Indeed, Kepler had had to struggle with the case of the pentagon, whose side he considered knowable only in the eighth degree (x = Y5 - v'5! v'2), while the hexagon is knowable in the second degree (x = 1). [HW 53] Here Kepler parted company with Ptolemy, whom he often praised, for "Ptolemy still denies that the thirds and sixths, major and minor (which are covered by the proportions 4:5, 5:6, 3:5 and 5:8) are consonances, which all musicians of today who have good ears say they are. On the other hand he accepts the proportions 6:7, 7:8," which would emerge fronrthe heptagon, intervhls that are "utterly abhorrent to the ears of all men and the usages of singing, even though it may be possible for strings to be tuned in that way, seeing that as they are inanimate they do not interpose their own judgment but follow the hand of the foolish theorist without the least resistance." [HW 138]

Geometry as the general science of magnitude . is undoubtedly prior to harmony, both in Boethius's account and Kepler's. This does not mean, though, that harmony merely follows in geometry's traces. Harmony involves simple ratios, and so excludes the complex ratios and incommensurables that abound in geometry. In contrast to ancient numerological accounts, Kepler set for himself the problem of accounting for the harmonic ratios in terms of the properties of geometrical figures, not relying on vague assertions of the qualities of the smallest integers. [HW 137] Harmony is the goal toward which his geometrical exposition aspires. He was, after all, discoursing on the harmony of the world, notjustits geometry, and this harmonic goal deeply informs his proceedings. Here, as elsewhere in his works, Kepler approaches empirical data "theoryladen," as Curtis Wilson has put it. [Wil]

At the beginning of the Harmony of the World Kepler reminded us that "I am not a geometer working on philosophy, but a philosopher working on this part of geometry." [HW 14] As Walker emphasizes, Kepler "always gave absolute priority to empirical evidence; if the theoretical pattern, however beautiful, did not fit the facts, it was discarded." [Wall The sensual was crucial for Kepler, as in his description of musical cadence as a kind of orgasm. [HW 242]

Kepler was clearly expressing his own strong reactions to the polyphonic music of his time, which "with its thirds and sixths, excites and moves us deeply as does sexual intercourse because God has modelled both on the same geometric archetype," as Walker puts it. Field considers that


Kepler's musical preferences were "on the side of orthodoxy rather than standing up to be counted as a partisan of the avant garde," namely Claudio Monteverdi and the school that followed Vincenzo Galilei. [Fie3] Though that is in many ways true, by adhering to just intonation Kepler made a -pronounced departure from the best-known ancient sources. Kepler explicitly turned to "modem figured music" for the combined motions of the planets, even though at one point he compared the individual planets to the [monophonic] choral music of the ancients. [HW 430] No less than Vincenzo Galilei, Kepler was engaged in a dialogue between ancient and modem music. In the end, Kepler asserts that

it is no longer surprising that Man, aping his Creator, has at last found a method of singing in harmony which was unknown to the ancients, so that he might play, that is to say, the perpetuity of the whole of cosmic time in some brief fraction of an hour, by the artificial concert of several voices, and taste up to a point the satisfaction of God his Maker in His works by a most delightful sense of pleasure felt in this imitator of God, Music. [HW 447-448}

Kepler's decision to follow the evidence of the senses is important; it led him to search for an account of thirds and sixths that shows their intelligibility in terms of the proportions of the pentagon. It is consonant with his reliance on precise astronomical observations and physical data elsewhere in his work, even where those data challenged his assumptions. It is also consistent with his stress on the moral and political effects of music. Kepler referred to ancient discussions of the ethical implications of music, but relied on his own judgment. He refers to the political writings of Jean Bodin on this matter, but is careful to correct what he considers Bodin's misunderstandings of harmonic ratios. [HW 255-279] Beyond the technical definitions of the different kinds of proportion, Kepler's overriding concern was to show how the harmonic mean is far more appropriate than the arithmetic or geometric means in matters of justice, friendship, love, punishment, reparation, and the proper tempering of the state. These matters open a larger perspective on Kepler's whole project. Gerald Holton has emphasized that "so intense was Kepler's vision that the abstract and concrete merged." [Hol] In his most daring insights, Kepler joined astronomy with terrestrial physics, bridging the heavenly and the sensuous; likewise, he considered music the sexual congress of the soul with mathematical forms.

Given this underlying musical agenda, it is more understandable why Kepler was distrustful of algebra, for it conflicted with the sensual evidence that Kepler took as fundamental. If his task was to render the harmonic intervals intelligible, Kepler had to throw out algebra, which would have validated the dissonant heptagon. Considered this way, Kepler's treatment of algebra can be viewed as his way of keeping faith with the exact data of hearing, just as he kept faith with Tycho's precise observations in his plan-


etary theory. It is not coincidental that his rejection of algebra agrees with his geometrical predilections; such judgments are often overdetermined. As Kepler weighed the possibilities of algebraic knowledge, his musical judgments guided his mathematical choices. He must have been deeply moved that his intercourse with sensuous har

monies converged with the inward rapture of Euclidean geometry.

REFERENCES

[Boy1] Carl B. Boyer, "Galilee's place in the history of mathematics" in

Galileo Man of Science (Eman McMullin, ed.), New York: Basic Books

(1 967), 232-255.

[Boy2] Carl B. Boyer, A History of Mathematics, 2nd ed., New York:

John Wiley (1 991) , 287-289.

[Car] Girolamo Cardano, Ars Magna (T. Richard Witmer, tr.), New York:

Dover (1 993), 1 2-22, 21 9-221 .

[Cas] Max Caspar, Kepler, New York: Dover (1 993).

[Cox] H. S. M. Coxeter, "Kepler and Mathematics," Vistas in Astronomy

1 8 (1 975), 661 -670.

[Fie1 ] J. V. Field, Kepler's Geometrical Cosmology, Chicago: University

of Chicago Press (1 988).

[Fie2] Judith V. Field, "Kepler's rejection of numerology, " in [Vic

273-296].

[Fie3] J. V. Field, "The relation between geometry and algebra: Cardano

and Kepler on the regular heptagon" in Girolamo Cardano: Philosoph,

Naturforscher, Arzt (E. Kessler, ed.), Wiesbaden: Harrassowitz Verlag

(1 994), 21 9-242.

AU T H OR

PETER PESIC

St. John's College Santa Fe, NM 87501 -4599

USA e-mail: [email protected]

Peter Pesic received his doctorate in physics at Stanford. he

has been at St. John's College in Santa Fe since 1 980, where

he is Tutor and Musician-in-Residence. He writes on the his

tory and philosophy of science. His book Labyrinth: A Search

for the Hidden Meaning of Science has just been published

by MIT Press. As a concert pianist, he has performed cycles

of the complete sonatas of Beethoven and Schubert, and he

has underway a cycle of J.S. Bach's complete keyboard

works.

[Hoi] Gerald Holton, "Johannes Kepler's Universe" in his Thematic

Origins of Scientific Thought, 2nd ed. , Cambridge, MA: Harvard

University Press (1 988), 69-90.

[KGW] Johannes Keplers Gesammelte Werke (Max Caspar et. a/., eds.),

Munich: C. H. Beck (1 934-), giving volume number and page.

[HW] Johannes Kepler, The Harmony of the World (E. J. Aiton, A. M.

Duncan, and J. V. Field, trs.), Philadelphia: American Philosophical

Society (1 997).

[NA] Johannes Kepler, New Astronomy (William H. Donahue, tr.),

Cambridge: Cambridge University Press (1 992).

[KI] Jacob Klein, Greek Mathematical Thought and the Origin of Algebra

(Eva Brann, tr.), New York: Dover (1 992).

[Pes1 ] Peter Pesic, "Franr;;ois Viete, Father of Modern Cryptanalysis

Two New Manuscripts," Cryptologia 2 1 (1 ) (1 997), 1 -29.

[Pes2] Peter Pesic, "Secrets, Symbols, and Systems: Parallels between

Cryptanalysis and Algebra, 1 580-1 700," Isis 88(4) (1 997), 67 4-692.

[Pes3] Peter Pesic, Labyrinth: A Search for the Hidden Meaning of

Science, Cambridge, MA: MIT Press (2000) .

[Pyc] Helena M. Pycior, Symbols, Impossible Numbers, and Geometric

Entanglements, Cambridge: Cambridge University Press (1 997), 1 35-

1 48, 1 67-208.

[Rit] Frederic Ritter, "Franr;;ois Viete, inventeur de l'algebre moderne,

1 540-1 603," Revue Occidentale 1 0 (1 895), 234-274, 354-415.

[Ros] Edward Rosen, "Kepler's attitude towards astrology and mysti

cism" in [Vic 253-272].

[Swe] Noel M. Swerdlow, "The Planetary Theory of Franr;;ois Viete 1 .

The Fundamental Planetary Models," Journal for the History of

Astronomy 6 (1 975), 1 85-208.

[Tig] Jean-Pierre Tignol, Galois' theory of algebraic equations, New

York: Longman (1 988), 21-30.

[Vic] Occult and scientific mentalities in the Renaissance (Brian Vickers,

ed.), Cambridge University Press (1 984), 253-296.

[Wall D. P. Walker, "Kepler's Celestial Music," in his Studies in Musical

Science in the Late Renaissance, Leiden: E. J . Brill (1 978), 34-62.

[Wae] B. L. van der Waerden, A History of Algebra: From ai-Khwarizmi

to Emmy Noether, Berlin: Springer-Verlag (1 980), 60-62.

[Wil] Curtis Wilson, Astronomy from Kepler to Newton, London:

Variorum (1 989).

Revisit the Birth of Mathematics . . . EU C LI D


Goursat, Pringshe im, Walsh, and the Cauchy I ntegra l Theorem Jeremy Gray

The Cauchy Integral Theorem

The Cauchy Integral Theorem is the

gateway to complex function theory. It

is used to connect complex functions abstractly defmed with their power series expansions; it marks one of the ways in which complex function the

ory distinguishes itself from the theory

of functions of two real variables. In any modem presentation of the subject

it says:

Theorem: If a function f(z) is analytic in a domain D containing a region T, and C is a simple closed curve in D that is the boundary of T, then

J f(z)dz = 0. c

Here, as usual, the integral along a path C in the plane of complex numbers is

defmed to be

L f(z)dz = f f(z(t))z(t)dt, C a

where the path C is a piecewise continuously differentiable arc with equa

tion z = z(t), a :::;; t :::;; b. The Cauchy Integral Theorem has a

long history, and was given many proofs in the 19th century, starting,

happily, with those by Cauchy. There

are a number of obscurities attending its birth. It is not clear what sort of a path was contemplated-Cauchy him

self changed his mind. Today, it is

common to see the theorem stated for piecewise continuously differentiable

curves but accompanied by remarks

about its validity for any rectifiable curve homotopic to the given one. This

is correct, but the integral needs to be defmed in this more general setting. As for proofs, it is possible to argue by infmitesimal variation of the path

(Cauchy's method), to appeal to general theorems in the calculus of varia

tions, to invoke Green's Theorem (Cauchy again, and, perhaps independently, Riemann, whose method was followed by several German authors),

or to argue ad hoc. Throughout the 19th

century, all of these approaches were

tried, and all were found problematic.

In each case tacit or explicit use was made of the extra hypothesis that the

derivative f(z) is itself continuous. As every modem text reveals, an essential


simplification was found by Goursat,

who showed that it was unnecessary

to assume that the derivative is con

tinuous. As many a not-so-modem text

reveals, the actual nature of Goursat's achievement is more complicated.

Briot and Bouquet

We can start our story a mathematical

generation before Goursat, with the French mathematicians Charles Auguste

Briot and Jean Claude Bouquet. They

were the first to write a book on the theory of elliptic functions which attempted

to derive that theory from Cauchy's theory of complex functions. Their [1859]

was a success, and they brought out a greatly revised and enlarged second edi

tion in 1875. Only in this book did they

attempt to prove the Cauchy Integral

Theorem, and their argument was rather ingenious, if at times imprecise.

They considered a star-shaped region, R, of the plane, and supposed

without loss of generality that any

point z in the region can be joined to

the origin by a line segment lying en

tirely inside R. They then argued by

comparing the integral A: = fJCz)dz

with the integral A' :=J f(z)dz, where C'

the contour C' is obtained from the

contour C by scaling it by a factor a,

and also lies inside R. They let the con

tour C be defined by the function acf>( t) = z, 0 :::;; t :::;; l. As a increases from

a to a' = a + Lla, the contour moves from Ca defmed by acf>(t) = z to C� defmed by a'</>( t) = z', and the differ

ence in the corresponding integrals,

A' - A = L f(z)dz -J f(z)dz, vanishes c� c;,

with Lla. They deduced that the inte-

gral J f(z)dz is acontinuous function Ca

of a, which they denoted �(a). Then they considered the difference

quotient

A' - A

Lla

= I: ( af(z')

Ll� f(z)

+ f(z'))<t>'(t)dt.

They noted that f(z') - f(z)

f(z') - f(z) z ' - z • -A--. Now the function

z' - z ua

f(z) is holomorphic, so it has a derivative f' (z),which f(z') - f(z)

means that the quotient , tends to a limitas z' z - z

f(z') - f(z) tends to z. So = f'(z) + e,where etendsto zero

z' - z with .:la. Moreover,J(z') = f(z) + e', where e' tendsto zero

• A • f(z') - f(z) f( ') F'( ) , With u.a, so one can wnte a Aa + z = z + e ,

where F(z) = zj(z) is a holomorphic function and e" = e + e'. Consequently

A' - A z -- = J (F' (z) + e")f/>'(t)dt. .:la o

However, the integral ofF' along a closed crnve vanishes, so

A' - A z � = Jo e" f/>'(t)dt.

But as .:la tends to zero, so does the integral on the right hand side, and so the real function �:p( a) is differentiable with differential zero. Consequently the function cp( a) is constant, and its value is arbitrarily small when a is small, so this value must be zero. The Cauchy Integral Theorem is therefore proved.

In view of discussions later in this paper, the reader may er\ioy discovering where the above proof relies, tacitly, on the continuity of the derivative of the function!

Goursat's proof of 1 884 Goursat, quite correctly, saw his result as illuminating the general definition of a complex function, and he suggested that it just used the mere definition of the derivative, and the fact that the theorem is true for fc dz and fc zdz. He argued that a region A bounded by a simple or multiple contour C of finite length can be broken up into squares of side-length A by two families of parallel lines. Let Ci = abed be one of these squares, of area ai = A2, and zi a point inside it. If the square lies entirely inside C, then from the definition of the derivative it follows that

so

f(z) - f(zi) = f' ( ·) + . z, e,,

Z - Zi

J f(z)dz = [f(zi) - zif'(zi)l J dz ci ci + f' (zi) J zdz + J ei(z - zi)dz. ci ci

Of these integrals, the first two gn the right-hand side vanish trivially, and the remaining one satisfies

If the square contains part of the boundary, as in this fashion,

e d c

a

b c

then a similar argument shows that

it. f(z)dz [ = It ei(z - Zi)dz [ ' '

< 4eiaiv'2 + eiAv'2.arc(ae).

Adding these two kinds of contribution up, one fmds that

[ J f(z)dz [ < 'Y/ v'2 (4 A' + AS), ci where 'Y/ is the maximum of the ei, A ' is the area of the squares inside C, and S is the total length of C. But this quantity can be made arbitrarily small, and so the Cauchy Integral Theorem is proved.

At least, that is what Goursat claimed. But there is a gap, which the diligent reader may have spotted, and which Goursat went on to discuss in this fashion. In truth, he said, the proof supposes that the length A can always be taken small enough so that all the Bi can be made less than some arbitrarily small number given in advance. But this is true, because if the derivative is continuous in A and on C, then given any (J" > 0 there is a 8 > 0 such that lhl < 8 implies lf(z + h) - f(z) [ h - f'(z) < (J" for all z inside or on C. This

being the case, it is enough to take A < &v'2, and the theorem is proved "with complete rig our." Later readers were not to be so indulgent.

Pringsheim's first critique·

Goursat's proof, and many of its predecessors, were criticised by Alfred Pringsheim [1895a]. He objected to the assumption that Cauchy's Theorem was valid for the simple integrals I c dz and I c zdz, because this required a limiting argument no simpler than the general one he went on to present. The problem was that earlier writers, and Goursat in particular, had assumed that the differential quotient f(z + h) - f(z) :.._:_ _ ____::__::___:___ tended uniformly to the derivative!' (z) for h all z in the region T bounded by the path C. But this must be proved, and it turns out to be equivalent to the continuity of the derivative. Goursat's casual assumption at the end of his paper is equivalent to the uniform differentiability of the function!

Pringsheim's argument was very careful, entirely general, and backed up by a wide-ranging historical analysis (quoted partially above). He started by defining a path in-


tegral in a way that did not require the path to be differentiable, merely continuous. In particular, there was no restriction to rectifiable curves. Instead, he let P(g, YJ) be a single-valued function defmed on a curve C = { (g, YJ) : YJ = c/>(0) inside a region T. The curve has merely to be the image of a -real interval under a continuous function cf>. He then defmed the path integral as follows:

He then showed how to establish the Cauchy Integral Theorem for regions bounded by curves which are stepshaped (made up, piece-wise, of curves that are parallel to the coordinate axes). He then extended the proof to deal with continuous curves, C, that are the limits of stepshaped curves, using im analogue of the mean value theorem. He also considered when an integral of the form r(x,y)

), P(g,YJ)dg + Q(g,YJ)dYJ is independent of the path in a (xo,yo)

simply or multiply connected region, and found that the aP aQ

condition - = - was necessary. aYJ ag

In a second paper later the same year [ 1895b], he cleared up the relationship between the derivative and the differential quotient, by giving a simple proof that continuity of f'(z) implied that the differential quotient converged uniformly tof'(z). So, he said, Goursat's achievement was only a simpler proof of a major result, not a weakening of its conditions.

Three years later Pringsheim [1899a, b] returned to the question, and sought to investigate what happens when the function j(z) fails to be analytic at a number of points, indeed, even on a 2-dimensional set of points. He found that there were cases when the Cauchy Integral Theorem still held. He therefore concluded that it might be the case that the mere existence of the derivative, without any assumption of continuity, was enough to establish the Cauchy Integral Theorem. As long as this possibility was not excluded by counter-examples, the question, he said, must remain open.

The American debate

Pringsheim's papers provoked a flurry of comments in America. Bocher had already taken up the issue in his [1896]. Now Goursat, at Osgood's request, repeated his proof in the first issue of the Transactions of the American Mathematical Society. In fact, the proof was more sophisticated this time around. He now said that the closed contour C satisfies condition a with respect to a number e > 0, if there is a fixed z' inside or on C such that

if(z) -j(z') - (z - z')f'(z') j < lz - z' le

as z describes the contour C. He then established what came to be called Goursat's Lemma: given any e > 0, any region T bounded by a simple closed contour can be divided into portions satisfying condition a with respect to

1Jordan [1 893], §§ 1 93-6.


the number e. The proof was by contradiction. If the claim is false, successive subdivisions of T can be made yielding a sequence of subregions that never contain a region appropriately bounded. But any sequence so obtained converges to a limit point at which, however, the functionj(z) is differentiable. This implies a contradiction.

The proof of the Cauchy Integral Theorem followed immediately. The given region T is divided up into congruent squares so small that condition a applies to them for an arbitrary but fixed e > 0. Condition a allows the integral to

be estimated by estimating values of J dz and J zdz, which were bounded by the perimeter and area of the squares. But for them the theorem is trivially true, and so, allowing e to tend to zero, the full theorem is established.

In the same issue E.H. Moore gave his own proof. He defmed a continuous curve as the image in the complex plane of an interval under a continuous function, and J j(z)dz as the limit as 8k � 0 and n -"> oo of the sum c n I,J(?k)(zk+ l - zk), where Zn+ l = zo, ?k is any point on k=O the arc (zk, zk+I), and 8k = tk+l - tk. He observed that the path integral exists if the curve is rectifiable, quoting a theorem in Jordan's Cours d'analyse to that effect. 1 In a footnote he said he had done this because Pringsheim [1895]

had proved what were special cases of this result, and "seems to be unfortunately out of touch with the current notion of the general rectifiable curve" as treated by Scheeffer, Ascoli, and Study.

Moore now proved a theorem about any single-valued functionfwhich is continuous and has a single-valued derivative everywhere inside and on a region R bounded by a closed, continuous, rectifiable curve C, subject to the following conditions: 1) the curve C meets curves parallel to the x and y axes in only fmitely many points; and, to simplify the proof, 2) if a sequence of squares whose sides are parallel to the x and y axes converges to a point A on C, then the ratio of the total lengths of the arcs of C inside the squares to the perimeter of the squares is ultimately less than some constant Pc which may vary as ? traverses C (for the usual curves considered, P( = 1 for all points ?). Theorem (Moore [ 1900}) Under the above conditions, J f(z)dz = 0. c

The proof was the usual proof by contradiction, the various hypotheses being introduced to guarantee the existence of suitable estimates. The observation that for each z E R, f(z) = fW + (z - 0f'W + Ll(z), where ja(z) j < e jz - � wherever z is within a suitably small distance of ?, reduced the evaluation of the integral to estimating sums of inte-

grals of the form J Ll(z)dz around suitable contours. Con

ditions (1) and (2) control the lengths of the parts of the curve C to be considered and the behaviour of Pc· A compactness argument is at work here under the surface.

The Cauchy Integral Theorem followed immediately from Moore's theorem. As he observed, requiring the boundary curve to be rectifiable allowed him to avoid Goursat's Lemma.

Pringsheim's second critique

In May 1901 Pringsheim presented his reply to the Amer-ican Mathematical Society at its meeting in Ithaca; it was published in the second volume of the Transactions. He certainly did not agree that he was "out of touch." As a friend of the late Ludwig Scheeffer he could claim, he said, to be as well acquainted with the new ideas as anyone, and he referred any doubting reader to his recent articles in the Encyclopiidie der Mathematischen Wissenschaften (vol 2, p. 41).2 He now objected to Goursat's proof on the grounds that it was incautiously expressed: there was not only no need to use congruent squares, but if one were so restricted then only a restricted class of boundary curves could be admitted. It would be necessary to allow those that were only piecewise monotonic (so their coordinate functions have only finitely many extrema). Moore's condition (1) is insufficient, as the example of y = x2 sin(1/x) shows, to ensure that small enough squares meet the boundary curve C in at most two points. If that condition is not met, there may be curves that go back and forth through some of the squares. Pringsheim therefore proposed to subdivide only those squares for which Goursat's condition did not already hold, thus adapting the subdivision to the curve at hand, and to exclude curves for which there was no suitable assembly of squares. This gave him a proof of the Cauchy Integral Theorem for rectifiable curves based on his (new) proof of Goursat's Lemma

.. Pringsheim returned to the question in 1903, when he gave the proof his obituarist (Perron) was to regard as definitive3. He began by noting that Heffer4 had recently es-

tablished that the integral Jc P(x,y)dx + Q(x,y)dy vanishes

when taken along a closed curve, provided that P(x,y)dx + Q(x,y)dy is an exact differential and satisfies the condition aP aQ - = -. This result contains the Cauchy Integral Theorem ay ax as a special case. But Prinsgheim now wished to avoid his earlier use of step-shaped functions, and to give a proof immediately applicable to contours bounded by straight lines, such as triangles. To describe what he did, we need

to explain his notation. He wrote !1 for aj

and f2 for aj

, ax ay

and defmed

f(x,ylxo,Yo): = f(x,y) - f(xo,Yo) -fi(xo,Yo) · (x - xo) - f2(xo,Yo) · (y - Yo).

He said that a functionf(x,y) was (totally) differentiable at a point (xo,Yo) if and only if fr(xo,Yo) and f2(xo,Yo) have values there and Ve > 0, 3 8 > 0 such that lx - xol < 8 and

IY - Yol < 8 implies IJtx,ylxo,Yo) l < eCix - xol + IY - Yol). He observed that uniform differentiability was a stronger condition than this. He could now state and prove the following result.

Theorem (Pringsheim, [ 1903]) Let P( x,y) and Q( x,y) be differentiable in the interior and on the boundary of a triangle ..1, and suppose that P2(x,y) = Q1(x,y), then

{ (P(x,y)dx + Q(x,y)dy = 0.

Proof First, an observation. Letfbe a function differentiable at each point of a domain T. Consider a triangle Ll lying entirely inside T, and defme the integrals (taken in

the positive direction) J Jtx,y)dx and J Jtx,y)dy by re-a a

striction. Then:

L. f(x,y)dx = L. Jtx,y lxo,Yo)dx

+ (ftxo,Yo) - !I(xo,Yo) Xo - f2(xo,Yo) Yo) L. dx

+ fi(Xo,Yo) L. :glx 4- f2(xo,Yo) L. ydx

But since clearly J dx and J xdx both vanish, it follows that a a

f Jtx,y)dx = J ftx,ylxo,Yo)dx + f2(xo,Yo) J ydx. a a a

Similarly,

L. Jtx,y)dy = L. Jtx,ylxo,Yo)dy + fi(Xo,Yo) L. xdy .

Now, to prove the Theorem, subdivide Ll into four con

gruent similar triangles. Pick one for which the integral J ftx,y)dy is largest; if this is 111 then a

It (P(x,y)dx + Q(x,y)dy) l � ± IL. (P(x,y)dx + Q(x,y)dy) l .

Proceed successively in this manner. One fmds

IL. (P(x,y)dx + Q(x,y)dy) l ::::; 4n iJ.:l.n

(P(x,y)dx + Q(x,y)dy) l . The nth triangle Lln has perimeter sn and the perimeters

s halve at each stage, so if Ll has perimeter s then Sn = 2n . The triangles converge to a point (xo,Yo) inside or on Ll. Because P and Q are differentiable, for any e > 0, there is an n such that P(x,ylxo,Yo) and Q(x,y lxo,Yo) are each less

2Pringsheim [1 899]. This article, while acute in its criticisms and citing a wide range of recent literature, is about real analysis in general; p. 41 carries a reference to Scheeffer's work but is much more to do with the types of discontinuities a function can have. 3Perron [1 952]. 4Heffter [1 902].


than e(lx - xol + IY - Yol) for all (x,y) E an. The above observation applied to P and Q yields

J P(x,y)dx = J P(x,ylxo,Yo)dx + P2(xo,Yo) J ydx dn dn dn J Q(FC,y)dy = J Q(x,ylxo,Yo)dy + Qt(Xo,Yo) J xdy. dn dn dn

ButP2(x,y) = Q1(x,y), and J ydx + J xdy = J d(xy) = 0, so dn dn dn

l{n (P(x,y)dx + Q(x,y)dy) l < E J Clx - xol + IY - Yol)(dx + dy). dn

If now n is taken large enough so that fx - xol and IY - Yol < Y(x - xo)2 + (y - Yo)2 < 8;, then IJ (P(x,y)dx +

dn 2 Q(x,y)dy) l < e 82n f Cldxl + IdYl) < BSn • 2 Sn

= e · !____, dn 2 4n

By the inequality relating integrals around an and a, one deduces

{ (P(x,y)dx + Q(x,y)dy) < es2.

But since e can be arbitrarily small, the sought-for result follows.

The Cauchy Integral Theorem follows on letting P and Q be the real and imaginary parts of a complex function

f(z); the integrability condition is one of the CauchyRiemann equations. However, Pringsheim pointed out the above proof can easily be adapted directly to the complex case. By the inequality relating integrals around an and a, one deduces that IL f(z)dzl < 4n ILJCz)dzl . Define, as above,

fCzlzo) = f(z) - fCzo) - f'(zo)'(z - zo).

Then IJCzlzo) l < �z - zol for lz - zol < 8.

The above argument, combined with a direct proof that L dz = 0 = L zdz, now shows that IL f(z)dzl< Vn Vn Vn e L lz -zolldzl < �4n , so L f(z)dz < es2 for arbitrary e. In Vn V other words, J

vf(z)dz = 0.

End of Act 1 The route to Goursat's proof is surprisingly intricate, and closely related to what might be called the discovery of continuity: the realisation that once a curve is not smooth but merely continuous many expected properties may lapse, or at least be hard to establish. Familiar examples from the period include the Jordan Curve Theorem and Peano's space-filling curve. The acuity with which Pringsheim pounced on what seemed like a triviality to Goursat is a good example of what has to be done. In fact, the question of how to admit general, continuous bound-ary curves so that the integral J f makes sense is another story, barely begun in 1903. Y


Act 2, Pringsheim to Walsh

Problems with the Cauchy Integral Theorem flared up again after 1929, when Pringsheim returned to rebut a charge levelled at him by Mittag-Leffler. The point at issue was a published remark of Mittag-Leffler's (in MittagLeffler [ 1923]) that quoted Pringsheim out of context and seemingly in error. Pringsheim wrote to Mittag-Leffler, who agreed he had made a mistake and offered to correct his mistake at the first opportunity. That was in May 1925, but when Mittag-Leffler died in July 1927 restitution had not been made, so Pringsheim took up the issue himself.

Mittag-Leffler's mistake had been to confuse Pringsheim's remarks about the proof of the Cauchy Integral Theorem with the statement of the theorem itself. As Pringsheim saw it, the so-called Riemannian proof of the theorem, by a Green's Theorem argument, was due to Cauchy before Riemann, and Riemann should be credited with introducing the theorem itself into Germany. But Mittag-Leffler had gone on to remind readers that he and others had priority over Goursat. The first was the Swedish mathematician C.J. Malmsten (in Malmsten [ 1865], which I have not seen), then Mittag-Leffler himself (Mittag-Leffler [ 1873], [1875]), and independently Briot and Bouquet. These contributions seem to have been forgotten, and so he took the occasion of yet another proof appearing (this one by Borel) to remind readers of the earlier work.

Indeed, as a young man Gosta Mittag-Leffler had published a new proof of the Cauchy Integral Theorem in 1873. That article being in Swedish, he recapitulated the proof in German two years later, in the Gottinger Nachrichten. In 1895, Pringsheim had criticised it for tacitly assuming that

lf(z + h) - f(z) I the quantity h - f' (z) converged uniformly

to zero as h � 0. In the letter of 1925, Mittag-Leffler implied that the fault lay in the German translation, and referred Pringsheim to a new, more accurate French version (of which he enclosed a copy). Never one to be fobbed off, Pringsheim recruited a Swedish mathematician who spoke good German to make a new translation of the Swedish original. He found that the German edition amounted to the first half of the Swedish version, but where they overlapped they had only inessential differences. Both texts agreed in assuming

1) that the function f(x) was (in addition to being finite and continuous) such that it had a single-valued and finite derivative f'(x), and

2) in making no mention at all of the uniform convergence l f(x + h) - f(x) I of h - f'(x) .

However, in the new French version, and its German translation, matters were the other way round. Now the above assumption (1) was missing, but assumption (2) now appeared, in the form of an assumption that

lf(ptefhi) -fCr)(/hi) fCr)(/hi) - JCpefii) I (Pt - p)elhi - p(elhi - efii < e

held uniformly for IP - Pll < 8, j e - el l < 8, and for all z = peie in the annular domain Ro � p � R, 0 � 8 < 27T. This assumption crucially makes no reference to the existence and

equality of the two differential quotients, but only to the uniform vanishing of their difference (from which the Cauchy Integral Theorem can be derived).

Now, said Pringsheim, a quick look at the proof of 1873 shows that the assumptions about f' (x) are used only to establish the equality of this difference in the limit, which

means that it would have been enough to assume precisely

such a limiting equality. A more precise argument then

shows that it is sufficient to establish this result that the

limiting property holds uniformly.

What to make of this muddle? Pringsheim took the shrewd view that in 1873 the idea of uniform convergence

and the awareness of its indispensability was not yet in the shared lore of mathematicians. Even Weierstrass,

who had led the way in emphasising the importance of

the concept, had seen fit to explain the uniform convergence of a sequence of rational functions carefully in a

footnote to a paper of 1880, and in 1873 Mittag-Leffler

had yet to make his trip to Germany and hear Weierstrass

lecture for the first time. Thereafter he took the

W eierstrassian approach to analysis so firmly to heart

that he perhaps read into his earlier work arguments that were not in fact there. So Pringsheim was inclined to

credit Mittag-Leffler with being the first to have the idea

that the Cauchy Integral Theorem could be proved with

out assuming the function to be continuously differen

tiable, and for being the first to have some success in that

dl,rection. But priority could not be claimed for the proof

of 1923, for a rigorous proof of that kind had been given

by Lichtenstein in 1910. In that paper, Pringsheim explained, Lichtenstein had

shown how to push through a Green's Theorem approach to the Cauchy Integral Theorem, first with, and then-sur

prisingly-without, assumptions of uniformity. Pringsheim argued that Lichtenstein's proof fmally showed clearly

what lay behind Goursat's proof. Lichtenstein had consid

ered the (in Pringsheim's view inappropriately named)

Green's formula:

rr ( aQ -

aP) dxdy = i Pdx + Qdy, JJT ax ay aT

where P and Q are functions of real variables x and y, con

tinuous in the region T and on its boundary, and the partial derivatives are taken to be continuous and single-val

ued. The boundary aT of the region T is taken to be a

rectifiable Jordan curve, and the right-hand integral is

taken along it in the positive sense. He then defined

11xQ : = Q(x + 8,y) - Q(x,y) and 11yP : = P(x,y + 8) - P(x,y),

and observed that Green's formula was equivalent to the

claim that

LI lim � (11xQ - 11yP)dxdy = i Pdx + Qdy T 8->0 u aT

1 1 provided the two limits lim - ( 11xQ) and lim - ( 11xP) exist

8->0 8 8->0 8 and are continuous in T. This he showed by vindicating the exchange of the limits and integration, thus showing that Green's formula was equivalent to the claim that

lim � (I (11xQ - 11yP)dxdy = i Pdx + Qdy. 8->0 u )JT aT

Lichtenstein's crucial insight was that this argument could be reversed, and Green's formula deduced without requir-

1 1 ing that the two limits lim - (11xQ) and lim - (11xP) exist

8->0 8 8->0 8 and are continuous in T. Instead it was enough to show the weaker requirement that

was a continuous function of x and y in T. Lichtenstein

proved the theorem by reducing it to the special case where

the boundary of the region is a triangle.

Pringsheim noted that-the.

Cauchy Integral Theorem

now followed on setting x + iy = z and letting P and Q be

come complex functions: P(x,y) = f(z), Q(x,y) = if(z). Green's formula then says that

i f(z)dz = 0 if lim _!_(il1xf(z) - 11yf(z)) = 0. aT 8->0 8

This is the Cauchy Integral Theorem without any assump

tion about the differentiability of f(z), almost exactly as

Mittag-Leffler had proclaimed it.

Pringsheim's paper seems to have re-opened the question. In his paper [ 1932] Kamke astutely asked what it was

that the Cauchy Integral Theorem actually said. Which of the following was it?

1) If a function f(z) is regular in a simply-connected do

main bounded by a closed continuous, rectifiable curve

C, then J0

f(z)dz = 0;

2) If a function f(z) is regular in a domain bounded by a

closed, rectifiable Jordan curve C, and it is regular on

C, then Jc

f(z)dz = 0;

3) If a function f(z) is regular in a domain bounded by a

closed, rectifiable Jordan curve C, and it is continuous

inside and on C, then fc f(z)dz = 0.

He observed that proofs of the first version could be found in the books by Bieberbach [1930, p. 1 18] and Knopp [1930, p. 56], and of the second version also by Knopp [ 1930, p.

63]; he knew no proof of the third, although it was stated in that form in the books by Osgood [1928, p. 369] and Hurwitz-Courant [1929, p. 283]. However, Knopp's proof of

(2) seemed to need some more care. Knopp had reduced (2) to (1) by the Reine-Borel

Theorem, arguing that C and its interior can be covered by

finitely many circles inside each of which f(z) is regular, thus giving a larger region G containing C and for which


the first result was true. Accordingly version (2) followed. But Knopp felt this was a little glib. So he first showed that the function f extends to a function g which is regular on G. To do this he covered the boundary C by discs, took a finite subcover of the boundary, and then argued carefully that the analytic continuation of the individual function elements yielded a single-valued function. This still left version (3) without what Kamke presumably regarded as a satisfactory proof, although he did not specify what he found wrong with the published attempts.

His paper stimulated Del\ioy (see his [ 1933]) to prove (3) in the form: if a function f(z) is defined in a domain bounded by a closed, rectifiable Jordan curve C, has a finite derilmtive inside C, and is continuous inside and on C, then J f(z)dz = 0. To prove this result Del\ioy took an ar-c -bitrary plane set E, and considered the squares (closed, con-taining their boundaries, of side B) which have an interior or boundary point in common with E (which he called a polygonal approximation to E). He then showed that, if G is a simple, rectifiable Jordan curve of length L, any polygonal approximation to G having more than 8 sides had a perimeter less than 16L. (Del\ioy assumed that B is less than L.)

He argued that one could work round the boundary of the curve G and its polygonal approximation picking vertices N common to two squares and points M which are (among the) closest to N in the corresponding squares so that the points N and M occur in the same order.

N

It follows that the length of the polygonal curve defmed by the M's is at most L. Del\ioy then argued by contradiction that as a result, if N11 • • • , Ns are 8 consecutive vertices then at least 2 of the corresponding points M are at least B apart. Consequently there are at least two points M which are less than B apart. This in turn implies the claim about the sides of the polygonal approximation.

This done, Del\ioy took a region R bounded by a Jordan curve C of length L (in fact, Del\ioy considered fmitely many Jordan curves, but that makes for no extra difficulty). He let f(z) be a holomorphic function defined inside R which is continuous inside and on C. Then, if A is a point of R and B is sufficiently small, the boundary, u, of the domain formed by the squares in the polygonal approximation to C that also contain the point A is a polygonal approximation to the curve C having at least 10 sides and therefore a perimeter of length less than 16L. Now, Goursat's Theorem applies to the domain u. Consider the curve g defined by the line segment NM, the arc MM' of C, the segment M' N' (where N' is the next vertex of the polygonal approximation after N) and the side NN' of the polygonal approximation.


N

� M ....-----+-. A

Let w(g) be the oscillation ofjon g and w(B) the maximum value of w(g). Then

IJc f(z)dz l = IJc f(z)dz - L f(z)dz l = II{ f(z)dz l < w(B) · (I length(g)) < w(B) · (16 + 16v'2 + 1)L.

But as B tends to zero so does the largest dimension of each g and so w(B) tends to zero, and Goursat's Theorem is proved in the form stated.

What may be the last word on the matter was then given by the Harvard mathematician J.L. Walsh, in a one-page paper [1933]. His proof was, as he said, "much more immediate than that of Del\ioy, although not so elementary." Walsh began by observing that the Cauchy Integral Theorem (in the third of Kamke's forms) was true for a polynomial, because it was then possible to replace the contour C by a suitably chosen polygon. However, the given function can be represented in the closure of the interior of the contour as the limit of a uniformly convergent sequence of polynomials, because the function is analytic inside C and continuous inside and on C. This sequence can be integrated term by term, and so the result is established. The theorem can be extended to regions bounded by finitely many non-intersecting rectifiable Jordan curves by replacing the polynomial approximations with rational functions whose poles lie outside the regions considered.

More modem treatments of the Cauchy integral theorem naturally couch it in the language of homology theory (which derives more from Riemann than from Cauchy). Ahlfors's influential text (Ahlfors 1953, p. 1 18) states it in this form: If a function j(z) is analytic in a domain D and C is a cycle in D that is homologous to zero in D, then J0

f(z)dz = 0. A cycle is a formal sum of continuous arcs.

A closed cycle C is said to be homologous to zero if the winding number n(C,a) = 0 for all points a outside C. A key stage in the proof is showing that the integral of a locally exact differential is not altered if the given cycle is replaced by an approximation consisting of horizontal and vertical arcs. This modem formulation allows one to deal fmally with a vast panoply of curves that are all, somehow, equivalent to polygons. Jordan would have been pleased.

Acknowledgments

I am always grateful to Bob Burckel for his careful editing of this column, but in the case of this article I am particularly so. I am also grateful to Alan Beardon, who made many useful comments.

Continued on p. 77

OSMO PEKONEN

Gerbert of Au ri ac : Mathematician and Pope

0 ne thousand years ago-when the world was cringing before the imminent YJK

problem-an extraordinary man, Gerbert of Aurillac, was elected Pope. He is also

well known in the history of mathematics, as he is credited with introducing the

Arabic number system to Europe. He is the most significant mathematician who has ever occupied the Holy See.

Gerbert was born about 945 at or near Aurillac, in the mountainous region of Auvergne, in central France. Since neither his place of birth nor his parents were recorded, it seems likely that he was of peasant origin. He must have been a young man of unusual talent, because the Benedictines-the most successful headhunters of those days-recruited him at the age of 18 to the service of the Church. He received his first training at the monastery of Saint-Gerald at Aurillac. It was a part of the "archipelago of Benedictine monasteries" which dotted the map of medieval Europe. The Benedictine order was governed by the mighty abbots of Cluny, who themselves were subject only to the Pope.

Gerbert's freshman and sophomore education was the usual medieval routine: he learned his grammar, i.e., Latin, and rhetoric under the tutelage of Abbot Raymond de Lavaur, for whom he held a special affection for the rest of his life. On the other hand, the third topic of the trivium-dialectic, or logic-could only be touched upon. In 967, Count Borrell of Barcelona visited the monastery, and the Abbot asked the Count to take Gerbert back to

Catalunya with him so that the lad could study mathematics there. Gerbert's next school was to be the monastery of Santa Maria de Ripoll, which was famous for its library. Mathematics in those days meant the quadrivium-geometry, astronomy, arithmetic;, and music-which he studied under Bishop Atto of Vich.

Muslims then held most of Spain. Catalunya was a Christian frontier territory at the outskirts of the Muslim world, and there was considerable communication of ideas between the two civilizations. The largest Muslim city of Spain was Cordoba. With 250,000 inhabitants, it may have been the biggest city of the world at the tum of the millennium. It boasted, among other cultural attractions, a scientific library far better equipped than any of Christian Europe. The Muslims had fallen heir to both Greek and Persian science in their initial expansion, and had translated many classics of antiquity into Arabic. At the same time, Arabic traders and travelers were in contact with India and China, and had absorbed many of their advances.

Muslim astronomy was the most advanced in the world, and Muslim astronomers proficient in using the astrolabe had done much to map the skies. The whole world still uses the Arabic names of some major stars-Aldebaran, Altair,


Fomalhaut, etc.-and terms of astronomy, such as al

manac, azimuth, zenith. The Arabs were even further ad

vanced in arithmetic. They had adopted the concept of

zero, which had originally emerged in India, and used a po

sitional numeric system much like the modem system. The

cathedrai school of Vich was able to offer Gerbert some of

this knowledge, and he took advantage of the opportunity.

Popular literature about Gerbert is teeming with allu

sions to his "Arabic" or "Muslim" teachers. It makes a beau

tiful story for a future Pope to have been directly exposed

to Muslim scholarship, but yet, to the present author's

knowledge, there is no evidence for such a conclusion.

In 970, Count Borrell and the Bishop of Vich made a pil

grimage to Rome, and took young Gerbert with them. The

journey proved disastrous: the Bishop was assassinated in

Rome. Gerbert now found himself without an adviser. His

mathematical knowledge delighted Pope John XIII, who in

troduced him to the Holy Roman Emperor Otto I. The Pope

recommended Gerbert as a tutor for the Emperor's son,

the future Otto II, who was to marry a Greek princess. The

young monk attended the imperial wedding ceremony in

Rome in 972. The King of France was represented by

Archdeacon Gerann, a famous logic teacher from the

cathedral school of Reims. The two learned men were in

troduced to each other, and Gerbert got an invitation to

pursue his studies of logic at Reims. The Emperor allowed

him a leave of absence.

Gerbert soon made quite a name for himself in Reims.

He was invited by Archbishop Adalberon (who later or

dained him) to join the faculty. He reformed the teaching

of logic in Reims and introduced Boethius to the curricu

lum. An envious colleague from Magdeburg, Otric, de

nounced him to Emperor Otto II. In December 980 the

Emperor summoned both scholars to Ravenna and en

gaged them in a debate on the subject of classifying knowl

edge. In modem terms, the issue was whether physics is a

branch of mathematics or an independent subject. The ve

hement argument was terminated only when the Emperor

intervened. Otto was quite impressed by the intellectual

performance of his former teacher, and he bestowed upon

Gerbert the wealthy monastery of St. Columban of Bobbio

in Lombardy, Italy.

Bobbio was a major center of learning which possessed

one of the great libraries in Western Europe. It was close

to Genoa and had benefited from the trade and commerce

that were beginning to enrich all of northern Italy, but it

had fallen on hard times. Incompetent abbots had depleted

its treasury, local nobles had seized its lands, and its monks

had taken great liberties with their duties. Gerbert under

took to remedy these affairs, but he turned out to be inept

in administration and provoked outright mutiny among

monks, clerics, and nobles.

Otto II died in December 983, and Gerbert lost his pa

tron and protector. He had to flee from Bobbio and hasten

back to Reims. Despite his failure at Bobbio, his reputa

tion was so great that he could reclaim his position as the

master of the cathedral school of Reims and secretary to

the Archbishop. He became deeply involved in the power

politics of the times. As a loyal servant of the Ottonian dy

nasty, he defended the three-year-old Otto III against the

pretender duke Henry of Bavaria. In France, Gerbert

helped to raise Hugh Capet, the Count of Paris, to the

throne in 987, thereby replacing the old Carolingian line

with a new dynasty, to be called Capetian. These were non

trivial matters that consumed a fair share of his time and

drew him deep into the muddled waters of politics. Gerbert

found little time for teaching and research any more.

Having backed the right horses, though, he emerged as

Archbishop of Reims when the turmoil was over. He turned

out to be a singularly self-willed Archbishop who, centuries

later, was remembered as a forerunner of Gallicanism, i.e.,

self-assertion of the church of France.

Mter the death of Hugh Capet in 996, Gerbert clashed

with his successor, Robert II, whose marriage to a cousin

he judged illegal. A newly appointed bishop sided with the

King and refused to be consecrated by Gerbert. Pope

Gregory V summoned Gerbert to Rome, and stripped him

of his episcopal functions.

The unfortunate former logic teacher never returned to

Reims again, but approached the new German Emperor

Otto III, then 16. He seems to have offered Boethius's De arithmetica to the Emperor. The Emperor responded by

inviting him to teach the Franks mathematics, in order to

awaken in them the genius of the ancient Greeks. 1 Gerbert

wrote back, praising him for appreciating the universal im

portance of mathematics. 2 Gerbert's intelligence charmed the Emperor who en

gaged him into his court and chancellery in Aachen. He

started in 997 as Otto's combined advisor, teacher, scribe,

chaplain, and court musician. He impressed the court by

constructing a nocturlabium. The next year he was ele

vated Archbishop of Ravenna. When Pope Gregory V died

in 999, Otto decided to wrest control of the papacy, and

did so by appointing Gerbert pope. He was consecrated on

Easter day, April 9, 999. Gerbert was the first French Pope. He took the name

Sylvester II, Sylvester I having been the advisor of the

Roman Emperor Constantine. This reflected the newly

elected Pope's close cooperation with Otto's ideal of a re

newed Christian Roman Empire, Renovatio imperii Romanorum. There may have been some millennia! fever

about the sudden idea of re-establishing the greatness of

ancient Rome. At Pentecost 1000, Otto made a curious pil

grimage to the tomb of Charlemagne in Aachen: He had the

1Nous voulons que, sans faire violence a notre liberte, vous chassiez de nous Ia rudesse saxonne, mais surtout que vous reveliez Ia finesse hellenique qui est en nous. . . . Aussi nous vous prions de vouloir approcher de notre modeste foyer Ia !Iamme de votre intelligence et de cultiver en nous le vivace genie des Grecs, de nous enseigner le livre de l 'arithmetique, afin qu'instruits par ces enseignements, nous puissions comprendre quelque chose de Ia subtilite des Anciens. :Votre demande honnete et utile est digne de votre majeste. Si vous n'etiez pas si fermement convaincu que Ia science des nombres contient en elle ou produit les premices de toutes chases, vous ne montreriez tant d'ardeur a en prendre une connaissance entiere et parfaite.


tomb opened and divested the dead man of a golden cross, some gannents, and one tooth.

As a spiritual leader, Sylvester II was a morally vigorous one. He took energetic measures against the abuses in the life of the clergy represented by simony and concubinage, and was anxious that ouly men capable of spotless lives should receive the episcopal office. He turned out to be a shrewd diplomat, as well. His Ostpolitik was farreaching. He established the first independent archbishoprics of Poland and Hungary, and moreover granted the title of king to Stephen, ruler of Hungary, in the year 1000, and appointed him as Papal Vicar of his country. He also exchanged ambassadors with the newly converted Russia.

We may wonder whether Sylvester, as a mathematician, was particularly keen on exploiting the round figure of the year 1000 to embellish his diplomatic moves. Thanks to him, one thousand years later, the Hungarians now celebrate the millennium of their first Christian ruler, Saint Stephen, and his crown.

Many advances of science, like the construction of various astronomical instruments, were posthumously attributed to Gerbert. He was an avid collector of manuscripts, who left behind a substantial library and a legacy of learning. As for his own writings, the scholars are very much divided on which of the surviving texts are attributable to Gerbert himself. The genuinely Gerbertian mathematical corpus seems to be meager compared to his writings on other topics. For instance, a text on Roman land surveying, which is generally attributed to Gerbert, is rather uninteresting in its mathematical contents.

His writing on the abacus, Regulae de numerorum abaci r'ationibus, became a standard text, and included a presentation of Arabic numerals. Gerbert's abacus used the positional system up to 27 decimal places, which sounds amazing. One may wonder whether octillions were really needed in the administration of the Catholic church, or whether the Pope was merely showing off with his supercomputer.

And what about the end of the world? Despite a lot of later romantic history writing about the "great panic of the year 1000," there seems to have been hardly any panic at all at the tum of the first millennium, for the good reason that most of Europe's populace consisted of illiterate peasants who had no access to almanacs.

However, among the learned few, there may have actually been a mathematical Y1K problem in the air when the date suddenly shifted from the complicated DCCCCXCIX to the simple M. It would be amusing to conclude that a

Sipos Celentis Temenias Zen is Calctis I( \ I/ \ I \ I/ \ I/ \

(/J 9 s f.. Jo Figure 1 .

mathematician-pope solved the Y1K problem by introducing the zero. However, there exists no contemporary document where the date 1000 would appear written in Arabic numerals. The adoption of the zero in Europe was a much slower process. Nonetheless, it is appropriate to include a celebration of the millennium of the zero as a theme of the World Mathematical Year 2000.

Rarely has a mathematician shaped political history as much as Gerbert did. During his reign, the frontiers of the Catholic church were pushed to the Danube and to the Vistula, where they have stayed ever since. Otto III died on January 23, 1002, and Sylvester II on May 12, 1003. Their departure put an end to an early dream of unified Europewhose fulfillment we may be witnessing today.

According to an early biographer, Gerbert himself modestly summed up his career saying that he passed "from R to R to R" (meaning Reims, Ravenna, Rome). Just the kind of statement to be expected from a mathematician.

REFERENCES

N. Bubnov (ed.), Gerberti post� Silvestri II papae opera, mathematica

(972-1003), accedunt aliorum opera ad Gerberti libel/as aestiman

dos intelligendosque necessaria, Berlin, 1 899, repr. Hildesheim,

1 963.

Gerberto: scienza, storia e mito. Atti del "Gerberti Symposium", Bobbio

25-27 /uglio 1983, Bobbio, 1 985.

P. Riche, Gerbert d'Aurillac, le pape de /'an mil, Paris, 1 987.

P. Riche and J.-P. Callu (eds.), Gerbert d'Aurillac, Correspondance, 2

vols. , Paris, 1 993.

0. Guyotjeannin and E. Poulle (eds.), Autour de Gerbert d'Aurillac, le

pape de /'an mil, Ecole des Chartes, Paris, 1 996.

Pictures

Although Arabic numerals do not occur in any surviving manuscript directly attributable to Gerbert, they do appear in an 1 1th-century manuscript called "Geometry II" (Erlangen, Universitatsbibliothek, 379, fol. 35-v), whose unknown author, called Pseudo-Boethius, must have been much influenced by Gerbert. The figure below shows the Arabic numerals used there, with thej.r early names, whose etymology remains mysterious. Two of the names, "Arbas" (4) and "Temenias" (8), are identifiable as deformations of the respective Arabic names of numbers, and it might be the case of all of them. Our present word "zero" (as well as "cipher") is derived from the Arabic sifr, meaning void. The name "Sipos" (0) in the figure, however, might rather be related to the Greek word 1/Jfj</Jo� for bead. Indeed, in Pseudo-Boethius's

Quinas Arb as Ormis Andras I gin

1/ \ 1/ \ 1/ \ 1/ '\ 1/ \

q � � � 1 VOLUME 22, NUMBER 4, 2000 69

treatise the numerals are intended to be inscribed on the beads of an abacus. A bead carrying the numeral zero was not used in most of the early abaci; however, if this was the case in Gerbert's abacus, it would be an indication of a fullydeveloped positional number system.

The statue of Pope Sylvester IT at Aurillac, Gerbert's home town, was sculpted by Pierre-Jean David d'Angers and erected in 1851. Notice that the scientist-pope's right hand is not depicted in a customary blessing gesture; rather he seems to be lecturing. Indeed, the intention of the sculptor was to reconcile Religion and Enlightment. One of the reliefs of the pedestal perpetuates the ahistorical legend of Gerbert as the inventor of the mechanical clock. In reality, the first mechanical clocks were constructed only in the 13th century.

A 20th-century ll)._ural painting at the Benedictine monastery of Pannonhalma, Hungary, illustrates a scene where Pope Sylvester II hands over the Holy Crown to emissaries of the Hungarian King Stephen for his coronation at Christmas 1000. An ancient crown representing the Holy Crown (or according to the most fervent believers, the Holy Crown itself) is still venerated by Hungarians. It was recently transferred, within Budapest, from the Hungarian National Museum to the Houses of Parliament.

ACKNOWLEDGMENTS

Figure 1 is reproduced from the book: Olivier Guyotjeannin & Emmanuel Poulle (eds.), Autour de Gerbert d'Aurillac, le Pape de l'An Mil, Ecole des Chartes, Paris, 1996. Figure 2 is a postcard photograph by Yves Bos. Figure 3 is an original photograph taken by Osmo Pekonen.

Figure 2.


Figure 3.

A U T H O R

OSMO PEKONEN


University of Jyvaskyla 40351 Jyvaskyla

Finland e-mail: [email protected]

Osmo Pekonen was born in Mikkeli, Finland, in 1 960. He stud

ied mathematics at the universities of Jyvaskyla and Paris, and

wrote a PhD thesis at Jyvaskyla in 1 988. His main work has

been on Riemannian geometry, Teichmuller spaces, and string

theory. He is a member of the European Committee for the

World Mathematical Year 2000. Besides his mathematics,

Pekonen has pursued a career in poetry. His main achieve

ment is the first-ever verse translation of the Beowulf into

Finnish, written in collaboration with the Old English scholar

Clive Tolley.

I d§lj l§i.Jtj Jet W i m p , Editor I

Feel like writing a review for The

Mathematical lntelligencer? 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.

Column Editor's address: Department

of Mathematics, Drexel University,

Philadelphia, PA 1 91 04 USA.

Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture by David M. Bressoud

CAMBRIDGE: CAMBRIDGE UNIVERSITY PRESS, 1 999,

xv + 27 4 pp. HARDCOVER, US $7 4.95, ISBN 0 521 66170

6; SOFTCOVER, US $29.95 ISBN 0 521 66646 5.

REVIEWED BY MARTIN ERICKSON

An alternating sign matrix is a square array of Os, 1s and - 1s

such that the non-zero entries of each row and of each column alternate in sign, and each row sum and column sum is 1. These matrices generalize permutation matrices. Here are the seven alternating sign matrices of order three:

G 0 D G 0 D 1 0 0 1

G 1 D G 1 D O 1 D 0 0 - 1 0 0 1

G 0 D G 0 n 0 1 1 0

In the early 1980s, mathematicians William Mills, David Robbins, and Howard Rumsey wondered whether they could find a formula for An, the number of alternating sign matrices of order n.

Notice that the definition forces each of the "borders" (first and last columns and rows) of an alternating sign matrix to consist of a single 1 and all other entries 0. Letting An,k be the number of alternating sign matrices of order n in which the first row's 1 occurs in column k, we have An = An+l,l = An+l,n+l·

Based on knowledge of An,k for the first twenty values of n, the three researchers conjectured that

and therefore that

n-1 (3j + 1)! An = IJ ( ') 1 ·

i=O n + J .

(3j + 1)!

(n + J)!

The validity of the latter formula is known as the Alternating Sign Matrix Conjecture.

David M. Bressoud's book is the story of thiS conjecture, c�ting in its proof in 1995 by Doran Zeilberger. Along the way, the author searches out and explicmes the connections between alternating sign matrices and a host of other combinatorial topics, including generating functions, partitions, determinants, lattice paths, inversion numbers, plane partitions, symmetric functions, Schur functions, Young tableaux, hypergeometric series, and square ice (a model of H20 molecules frozen in a square lattice). The story also touches on the lives and contributions of many great mathematicians of the past, including Leibnitz, Euler, Lagrange, Gauss, Waring, Cauchy, Jacobi, Boole, Sylvester, and Ramanqjan, as well as contemporary researchers such as Ian Macdonald, Richard Stanley, Donald Knuth, George Andrews, John Stembridge, and Greg Kuperberg (who discovered a different proof of the conjecture in 1995). Even Lewis Carroll makes an appearance (via Dodgson's algorithm).

In the author's conception, the process by which the conjecture was investigated and eventually proved illustrmes a way of looking m mathematics that differs from the standard paradigms. It is not theorem-proof-corollary; it is not "scaling the peaks" (moving from accomplishment to greater accomplishment); and it is not random exploration. By analogy with archaeology, Bressoud suggests a new way of describing what mathematicians do:


"I would like to consider the doing of mathematics and the fmding of proofs as analogous to the work of the archaeologist. When Mills, Robbins, and Rumsey first discovered their conjecture, they were not dissimilar to the archaeologist who has just unearthed a strange and marvelous object of unknown provenance and purpose. What is it? What was it used for? Why is it here? What does it tell us about the people who once lived here? The real work of the archaeologist is to make connecti\ms: connections to other objects at other places_ at other times, connections to other facts that are known about this particular site. The goal of the archaeologist is to provide a context in which we can understand this object. As each object comes to be understood, it facilitates the interpretation of others, not just in this place, but also in other places and from other times. It provides a foundation upon which we construct our theories.

"This is the role of proof, to enrich the entire web of context that leads to understanding. The mathematician does not dig for lost artifacts of a vanished civilization but for the fundamental patterns that undergrid our universe, and like the archaeologist we usually find only small fragments. As archaeology attempts to reconstruct the society in which this object was used, so mathematics is the reconstruction of these patterns into terms that we can comprehend.:'

I believe that Bressoud successfully illustrates his thesis in this book The details are not always easy to follow, as there are many related conjectures and much mathematical machinery to keep track of. (Most of the conjectures have been proved, but the author refers to them as conjectures, rather than theorems, in order to preserve the historical point of view.) But the author does a good job of helping us keep the main points in mind, and his narration is quite clear.

This is enjoyable history of modem mathematics of the type found in the book From Error-Correcting Codes Through Sphere Packings to Simple Groups, by Thomas Thompson. As in that book, we are treated to quotes from the investigators, photographs,


lots of diagrams, and much background information. In fact, the main story of the Alternating Sign Matrix Conjecture is told in Chapters 1 and 6, while Chapters 2 through 5 and 7 provide related material.

The author also supplies Mathematica code so that the reader can obtain data and follow the discussion in an active way. This is an excellent idea, and the code is simple enough to copy and use in a few minutes. I enjoyed doing so immensely.

Proofs and Confirmations is a fascinating look at mathematics in the making. And it is a generous guide to additional information that is interesting in its own right.

Department of Mathematics and Computer

Science

Truman State University

Kirksville, MO 63501

USA


Physics from Fisher Information: A Unification by B. Roy Frieden

CAMBRIDGE: CAMBRIDGE UNIVERSITY PRESS (1 999),

318 pp.

US $74.95, ISBN: 0521631 67X

REVIEWED BY ROBERT GILMORE

I t is hard not to be seduced by a book whose very first equation is the

Cramer-Rao inequality. This is a profoundly beautiful and important result which may be written poetically as

!J.y!J.a 2= 1.

It is beautiful because it is simple. It is important because it is far-reaching: it manifests itself in many different ways in many branches of physics. It is known to sound engineers in the form of a time-frequency uncertainty relation

1 !J.w!J.t 2= -,

2

where !J.t is the time duration of a signal and !J.w = 27T!J.v is the precision to

which the angular frequency can be estimated.

It occurs in Quantum Mechanics as both the position (x)-momentum (p) uncertainty relation

and the time-energy uncertainty relation

1 tJ.EtJ.t 2= - n,

2

where h is Planck's constant and n = h/27T. It appears also in Statistical Mechanics in the form of uncertainty relations between extensive variables (U,V,N, ' ") (internal energy, volume, number of particles, · · ·) and their conjugate intensive variables (�, �' �' · · ·) (temperature, pressure, chemical potential, · . . ), one of which is

!J.U!J. ( �) 2= k,

where k is Boltzmann's constant. The Cramer-Rao inequality exhibits

the duality which exists between the two fields: Probability Theory and Statistics. Specifically, assume that P(yia) is a probability distribution function for the random variable Y. This probability distribution depends on the values of one or more parameters a. Then, given a, it is possible to estimate the various statistics of the random variable Y, such as its mean jj = (y) and standard deviation !J.y, where !J.y2 = ((y - y)2). Conversely, it is possible to estimate the value of the parameter a from the measurements

Yi· These estimates lead to a mean value a, and a standard deviation !!.a, defmed by !J.a2 = ((a - a)2). The product of these two variances, of the random variable and the parameter estimates, are bounded below by a nonnegative term which can often be normalized to + 1 by suitable change in the defmition of the random variable and/or the parameters. The term !J.y2 is called the Fisher information

!J.y2 = I(a) = J (� dP��a)yP(yia)dy.

Fisher information is the starting point of the journey on which the author of the present book embarks.

Frieden has certainly made a number

of useful observations:

1) Uncertainty relations play a fun

damental role in modern physics.

2) Fisher information, through the

Cramer-Rao inequalities, underlies all

uncertainty relations.

3) The dynamical laws of physics

can all be formulated as variational

principles. These involve integrals over

quadratic forms in the gradients of

suitable functions.

4) Fisher information has a similar

structural form.

On the basis of these observations,

Frieden attempts to create a vision of

physics as a stepchild of Information

Theory, specifically of Fisher informa

tion. His thesis is that " . . . all physical

law, from the Dirac equation to the

Maxwell-Boltzmann velocity disper

sion law, may be unified under the um

brella of classical measurement the

ory. In particular, the information

aspect of measurement theory-Fisher

Information-is the key to unifica

tion." Frieden's efforts have created a

considerable buzz in the world of sci

ence-a recent lead article in the jour

nal The New Scientist alluded very fa

vorably to Frieden's book. However,

speaking as a card-carrying physicist

who has derived the uncertainty rela

tions for Statistical Mechanics using

the Cramer-Rao inequality, I do not be

lieve that Frieden has succeeded in his

effort.

The variational formulation of the

laws of dynamics is a very powerful

tool in the physicists' bag of tricks. The

method itself dates back to the 17th

century, to Fermat's Principle of Least

Time, and to the 18th century, in the

form of Maupertuis's Principle of Least

Action. The original formulations had

a distinctly theological formulation.

Even today, the method smells faintly

sulfurous. Here I will give away one of

the tricks of our trade. We put exactly

"the right stuff" into a Lagrangian, that

is, whatever is necessary to recover the

desired dynamics once the button is

pushed and the machinery goes into

gear. The machinery involves a Taylor

expansion around the appropriate so

lution, one or more integrations by

parts, and then an argument about an

integrand necessarily being zero. The

final result is some equation of statics

or dynamics. What makes the varia

tional method so attractive is that

there are certain rules which severely

restrict the form of the Lagrangian

function. This drastically reduces the

guesswork required in deriving dy

namical laws to describe new physics.

Frieden encounters three problems

in trying to formulate physics from

Fisher information. I illustrate them

for a particularly simple case: the quan

tum-mechanical description of a parti

cle of mass m moving in one dimen

sion in a potential V(x). The variational

problem is

8 I L( �. :�)d.x =

8 I { 2� (:�Y + V(x)(�(x))2}d.x = 0.

For simplicity, I have taken the wave

function to be real. In the problem as

specified, there is always the trivial so

lution �(x) = 0. To eliminate the trivial

solution it is useful to impose the nor

malization constraint f(�(x))2d.x = 1.

This constraint is normally imposed

through the use of Lagrange multipli

ers, so that the constrained variational

problem becomes

8(f{ 2� (:�t + V(x)(�(x))2}d.x - AI(�(x))2d.x) = 0.

The Fisher Information is

I( aP)2 1 !(a) = aa P(yla) dy

At this point in the analysis we are

faced with three procedural problems:

1) The gradients are with respect

to the spatial coordinates and the pa

rameters of a probability distribution

function, respectively.

2) The physical Lagrangian has a po

tential term, V(x)(�(x))2, and a con

straint term, -A(�(x))2; the Fisher in

formation has neither.

3) The physical law involves a vari

ation, the Fisher information does

not.

Frieden attempts to solve the frrst

problem by assuming the probability

P(yla) is for the distribution of the spa

tial coordinate (x) (not the wave func

tion �(x)). Further, he assumes that

the probability distribution is transla

tion-invariant

P(y I a) space coordinate P( X I a) translation invariance P(x _ a).

He does this so that the derivative d/da which occurs in the Fisher information

can be replaced by the coordinate de

rivative -d/d.x. This brings the Fisher

information into a form more similar

to that of a physical Lagrangian.

Invariances in physics are closely

related to conservation laws. In the

present case, translation-invariance

suggests momentum conservation,

which requires V(x) = constant, or

dV/d.x = 0 (no forces). If V(x) is not

constant, is translation-invariance of

P(xia) reasonable to assume? I claim

not. Think of the harmonic oscillator,

with .V(x)· = tJc:r2. Is it likely that

P(x ia) has the same form when a - 0 as when lal is very large?. To put the

question in even more stark terms,

consider the particle in a box, so that

V(x) = 0, 0 < x < L, but V(x) = "oo",

x ::s; 0 and L ::s; x. Surely P(xla = 0) =/= P(xla = t L) =/= P(xla = L).

Frieden deals with the second and

third problems together. He introduces

a second type of information, a "bound

Fisher information" J. This functional

somehow describes the physical infor

mation which is intrinsic to the quantity

measured. Although he strenuously

tries to put great distance between J and constraints on physical systems

which are invariably treated with

Lagrange multipliers, there is in fact no

difference. He argues that the transfer

of information between I and J during

a measurement is the same: 81 = 8J.

More specifically, 8(/ - J) = 0. In summary, my very strong reser

vations about Frieden's technical pro

gram to reconstruct physics from

Fisher information are two in number:

the assumption of translation-invari

ance of the probability distribution

function P(xia) is incorrect, and the

constraint terms in the "bound infor

mation" J are put in in an ad hoc man

ner to guarantee that the appropriate

laws are recovered. A physicist would

admit this up front. It is difficult to dis

cern that the author of the present

work is guilty of this, but he is.


Frieden is forced to identify the parameters of the probability distribution

a with space-time coordinates (x,t) in order to introduce spatial (and tempo

ral) derivatives into the expression for

the Fisher information. As a result, he is forced to focus on uncertainties in position and time, rather than on the

amplitudes of the electric and magnetic fields, or the complex wave func

tions of Quantum Mechanics. As a con

sequence, his interpretations of physical phenomena differ in signifi

cant ways from the standard interpretations ol physicists.

This divergence. in viewpoints is well illustrated in Fig. 1.4 in this book

This shows a point source located in a screen at the left of the page, and a dif

fraction pattern produced by this point

source on a screen at the right side of

the page. Every physicist has done this experiment. To us, the physics lies in

the peaks and valleys (intensity max

ima and minima), and in particular, in

the ratios of heights of successive peaks, and the ratio of heights of adja

cent peaks and valleys. We know that

the pattern may be offset from its ide

ally predicted position because of slight displacements of the point

source or placement and/or disorien

tation of the intermediate lens. The off

set of the pattern is not important, this

is an "engineering" problem. For us, the physics lies in the intensity distribution. For Frieden, the physics lies in

the offset. There is more to physics than dy

namical laws of motion. Frieden finds that Newton's Second Law of Motion

is a consequence of variation of the

Lagrangian L(x,x) = �xL V(x) (provided F = -\lV), but where is Newton's First Law, whose purpose is to define the subset of reference frames which are inertial, in which the Second Law is true? Or Newton's Third Law, the conservation of momentum? Can the principles of unitarity, equivalence, covariance, or the conservation

of energy, momentum, angular momentum be consequences of informa

tion of any kind? There is in fact an important role

that information theory can play in the

formulation of physical theories. This

can be illustrated in terms of the elec-


tromagnetic field. The field can be formulated in two ways, called for simplicity the 19th-century formulation and the 20th-century formulation. ln the former the electric and magnetic

fields, E(x,t) and B(x,t), are intro

duced. Then a system of 4 equations is

introduced (by Maxwell) which these fields satisfy. This formulation has

been called "manifestly covariant." The

20th-century formulation regards the electromagnetic field as composed of

photons with two polarization (helicity) states. There is essentially a 1-1

correspondence between electromagnetic fields and superpositions of pho

ton states. The superpositions satisfy

no constraints. So: what role do Maxwell's equa

tions play? Maxwell's equations are ex

pressions of our ignorance. By introducing fields E(x,t) and B(x,t) we are

introducing mathematical functions some of which cannot represent real

physics. The function of Maxwell's equations is to eliminate all those

mathematical functions which describe nonphysical fields, and to allow only those functions which do describe

physically allowable fields. The simplest way to see this is as

follows. Resolve both the manifestly

covariant (19th-century) description and the quantum (photon) description in terms of their propagation direction

4-vectors (k,k4), where k-k - c2k� = 0 in free space. For each 4-vector (k, k4) there are 6 amplitudes Ei(k, k4), Bi(k,k4) in the manifestly covariant description and just 2, one for each

helicity, in the photon description.

Choose any particular 4-vector (k, k4) and compare the transformation properties under the Poincare group of the

6 amplitudes from the first description and the 2 amplitudes from the second. This comparison identifies the 4 linear

combinations of amplitudes from the first description which must vanish, and the 2 which describe the positive and negative photon helicities. Now

transform this identification to any other 4-vector (k', k4) by a Poincare transformation. VioUi-Maxwell's equa

tions result. The point is that every time we

write down an equation in physics we are expressing our ignorance. The only

purpose of an equation is to winnow

out the nonphysical from the physical. Wouldn't it be more elegant to build up

every allowable physical state from a small number of building blocks (e.g., photon states) which obey no con

straints, so that there is a 1-1 correspondence between linear superpositions and physically allowable states?

If there is a way that all/most/some

parts of physics could be formulated in

an information-theoretic way, it would be much more elegant to do it in this "building-up way" (Aujbauprinzip) than in the more classical "find-theequation-which-elirninates-the-nonphys

ical" approach (''tearing-down way").

It may be possible to formulate

some part of physics in an informationtheoretic setting. I do not believe the

formulation by Frieden is successful.

Department of Physics

Drexel University

Philadelphia, PA 1 91 04

USA


NURBS: From Projective Geometry to Practical Use by Gerald E. Farin

NATICK, MA; A K PETERS, 1 999, 267 PP

US $49.00, Hardcover, ISBN: 1 -56881 -084-9

REVIEWED BY LES PIEGL

Non-Uniform Rational B-splines, commonly referred to as NURBS,

have acquired a remarkable success in

only a decade and a half. It all started in the late 1940's with Schoenberg's in

vestigations into splines using truncated power functions. While research

on splines remained active throughout

the 50's and 60's, nothing really happened in the world of computational science of splines until the famous Cox-de Boor algorithm was published

in 1972 independently by Maurice Cox and Carl de Boor. The Cox-de Boor algorithm allowed fast and reliable eval

uation of B-splines without the truncated power functions and divided differences. Bill Gordon, then at

Syracuse, had Rich Riesenfeld look at

Bezier curves and surlaces to see how the new B-splines, defined by an easily computable recursive formula, can be used to defme curves and surfaces. Once Riesenfeld figured out the relationship between knots, nodes, and control points, he found a scheme that was far superior to anything used thus far. It also contained Bezier c1lTVes and surlaces as special cases. In 1973 Riesenfeld's thesis was publishedwhich marks the birth of B-spline curves and surlaces.

Though these entities were quite nice to represent free-form shapes, common curves such as the circle were not representable by integral Bsplines. In 1975 Versprille's thesis becarne available, investigating B-spline curves and surlaces in homogeneous space. Upon projection of the 4-D functions to 3-D, a rational form was obtained, which is what we call today a rational B-spline.

The work of Riesenfeld and Versprille became the basis of industrial research in the late 1970's. The CAD/CAM industry was looking for a mathematical form that was able to handle both freeform as well as specialized curves and surfaces. Companies such as Boeing aMI SDRC (Structural Dynamics Research Corporation) played a crucial role in pushing this technology forward. The frrst commercial product based entirely on rational B-splines, called GEOMOD, was released by SDRC in the early 1980's. The term NURBS was coined around that time (probably by Bob Blomgren, working for Boeing at the time). Today NURBS are the de facto standards for geometry representation and data exchange, and are used almost exclusively in the broad field of computer-aided design and manufacturing (CAD/CAM).

Farin's book on NURBS is a tremendous disappointment. In the 250 pages of text, he devotes exactly 14 pages to B-splines, not NURBS! He gives very brief (a page or two) discussions on such general topics as knot insertion, the de Boor algorithm, blossoms, and derivatives. There is nothing in these pages that the designer of a NURBS system can use. The chapter does not even teach the reader how to evaluate a NURBS curve or surlace.

The rest of the book is a collection of (again very short) chapters on projective geometry, conics, Bezier formulation, Pythagorean curves, rectangular patches, rational Bezier triangles, quadrics, and Gregory patches. For the mathematician who wants to learn the basis of NURBS for further research, this book is a dead end. For the serious implementer who needs algorithmic details, this book is a waste of time. Though it might provide entertaining reading for someone with a solid knowledge of high school algebra and may even create the impression that the reader has learned something, it gives nothing of substance to think about.

Department of Computer Science

University of South Florida

4202 Fowler Avenue

Tampa, FL 33620


A Panorama of Harmonic Analysis by Stephen Krantz

CARUS MATHEMATICAL MONOGRAPH NUMBER 27 WASHINGTON, D.C.: THE MATHEMATICAL ASSOCIATION

OF AMERICA, 1 999, 368 PP. US $39.95, ISBN: 088385031 1

REVIEWED BY MARSHALL ASH

I had some trouble with the topology section of the University of Chicago's

master of science exam in May 1963. But the analysis section went very well. No small part of the credit for the latter result was due to Antoni Zygrnund who had taught me real analysis I and II, and to Alberto Calderon who had taught me complex analysis I and II. In very short order, I abandoned my plan of specializing in point-set topology and picked Fourier series as the main topic for my next hurdle, the two-topic exam. Over the next year, Bill Connant, Larry Domoff, and I read Zygrnund's Trigonometric Series in preparation for the exam. After the exam, Professor Zygrnund accepted me as his student and my career as a harmonic analyst began.

Looking through Krantz's A Panorama of Harmonic Analysis feels like watching a horne movie production entitled "The Zygrnund School of Analysis: 1965-1999." Throughout most of this period, I was lucky enough to be near the University of Chicago, where the Monday 3:45 PM CalderonZygrnund seminar featured, among many other things, just about every development in harmonic analysis mentioned in Krantz's book. These were exciting times in harmonic analysis, and my connection with the University of Chicago's seminar placed me near the center of the action.

I have always been attracted by questions that have crisp, easily grasped statements. A good example of such was Lusin's conjecture that there could exist a real-valued square-integrable functitm defmed on the interval 1f =

[0,27T) whose Fourier series diverged at each point of a set of positive measure. Already in 1927 Kolmogorov had given an example where the function was integrable, but not square-integrable, and the Fourier series diverged at every point. Since giving an example seemed like it couldn't be very hard, I proposed to Zygrnund that I take the Lusin conjecture for my thesis problem. He immediately discouraged this idea, explaining that this problem might prove to be rather difficult. Zygrnund had realized that the square-integrable case was much deeper than the integrable case, even though his almost infallible intuition this time predicted the existence of an example. In the early fall of 1964 my fellow graduate student Lance Small came back from a summer visit to Berkeley carrying the news that Lennart Carleson had just proved that the Fourier series of a square-integrable function actually converges almost everywhere. He was closely questioned by Zygrnund and Calderon, who thought that he could not have gotten the story straight. But he had and Carleson had. Several years later when I spent two months working through Richard Hunt's careful exposition of his extension of Carleson's theorem, it became clear to me that although Zygrnund's guess about the outcome of the conjecture had been wrong, his assessment that an extremely high level


of mathematics would be required to decide the issue was quite accurate.

After finishing my degree in 1966, I was a Ritt instructor at Columbia. I was at a loss for how to begin my career as a research mathematician. My inertia was assisted by the cultural cornucopia that New York City provided, and also by the Columbia campus protest movement featuring Mark Rudd, SDS, and the occupation of the math building which contained my office. I wrote to Zygmund, who suggested that I get into partial differential equations. This certainly proved prophetic. The great bulk of harmonic analysiS- being done now seems to be in connection with partial differential equations. One of the ways to see this is to note that the preponderance of talks being given nowadays at the Calder6n-Zygmund seminar fits this profile. Nevertheless, most of my own interest never did move in that direction.

One thing I did to stay mathematically alive was to attend Stein's seminar at Princeton. One of the talks I heard there was by Stein's extremely young Ph.D. student Charles Fefferman. At the time, I did not have a sufficient overview of harmonic analysis to appreciate the depth and beauty of his mathematics, but fortunately I have heard him lecture many times since. It is also fortunate that his expository skills have improved from very good to extraordinary. For example, I consider it a high compliment when I say that Krantz's book does justice to the lectures I later heard in Chicago, wherein Fefferman explained the proof of his theorem that the characteristic function of the unit ball is not a multiplier on .LP(IR2) when p of. 2.

After three years at Columbia, I moved to DePaul and back to the Calder6n-Zygmund seminar. When I first arrived in Chicago, the harmonic analysts there were reading Igari's book on multiple Fourier series.[!] Grant Weiland and I immediately began doing research in this direction, and the main thrust of my mathematical career has been in this direction ever since. For this reason I have an especially strong interest in chapter 3 of A Parwrama of Harmonic Analysis, which is entitled Multiple Fourier Series.


Extending the work of Carleson, Richard Hunt proved that the Fourier series of an LP(T) function converges almost everywhere, provided that p > 1. What happens in dimension two? Krantz points out that if "converges" means that the partial sums include terms of the series with indices lying in the dilates of a fixed polygon, the analogue of Hunt's Theorem is true, whereas if "converges" means that the partial sums are taken to include the terms with indices lying in rectangles of variable eccentricity, then there is a counterexample, due to Charles Fefferman. (Larry Gluck and I later added a small "bell and whistle" to that example.) But the most important question of what happens when "converges" means that the partial sums include the terms with indices lying in the dilates of an origin-centered disk remains unsolved. Fefferman's Theorem that the unit ball is not a multiplier guarantees that it is not enough for p to be greater than 1, but gives no insight as to what happens when p = 2. This leaves open the question of whether the Fourier series of anL2(T2) function has circularly convergent partial sums almost everywhere. To my way of thinking, this question is the Mount Everest of multiple Fourier series.

An interesting question not dealt with in chapter 3 is the question of uniqueness. Is the trigonometric series with every coefficient equal to zero the only one that converges at every point to 0? I have spent much of my life working on this question and have been pleased to see an almost complete set of answers discovered. [AW] The only thing I want to say here is that uniqueness has been shown to hold in many cases, but here the situation is opposite to that for convergence of Fourier series mentioned above. We do know that uniqueness holds for circularly convergent double trigonometric series, but we don't know if it holds for square convergent double trigonometric series.

Speaking of chapter 3, one thing I would like to clarify is the definition of restricted rectangular convergence. I tried to explain this very subtle defmition in my 1971 paper with Weiland, and I will take another try at it here.

Fix a large number E >> 1. Let { amn lm� 1,2, . . . ;n� 1,2, . . . be a doubly indexed series of complex numbers and denote their rectangular partial sums by SMN = ��=1 �;i=1 amn· Then say that S = ��amn is E-restrictedly rectangularly convergent to the complex number s(E) if

lim SMN = s(E). M,N --7 oo

i, < ;, < E

Finally say that S is restrictedly rectangularly convergent if there is a single complex number s such that for every E, no matter how large, s(E) exists and is equal to s. An example may help to clarify this. For n = 2, 3, . . . , let an2,1 = n, an2,n = -n, and let amn =

0 otherwise. Notice that SMN of. 0 only if there is an n > N such that n2 ::::; M so that an2 1 is included in the partial sum, while

' an2,n is not. But then N2 < n2 ::::; M, so that MIN > N. Thus if any eccentricity E is given, as soon as N exceeds E, the condition MIN < E becomes incompatible with SMN of. 0. In other words, s(E) is 0 for every E, so that this series is restrictedly rectangularly convergent to 0. And this happens despite the fact that SN2,N-1 = N, so that limrnin{M,NJ _, oo SMN does not exist, which is to say that S is not unrestrictedly rectangularly convergent.

Krantz has made wonderful selection choices for all of his chapters. The chapter titles are: overview of measure theory and functional analysis, Fourier series basics, the Fourier transform, multiple Fourier series, spherical harmonics, fractional integrals singular integrals and Hardy spaces, modern theories of integral operators, wavelets, and a retrospective. In particular, I think that ending with a chapter on wavelets represents a correct analysis of which way a good part of the winds of harmonic analysis have been blowing for the past few years as well as a shrewd guess as to which way they will blow in the near future. A botanist recently asked me for some help in finding a good mathematical representation for ferns that she has been studying. Although my work is usually not very applied, I have looked into this a little bit and it seems likely that wavelets may prove to be the right tool.

A Panorama of Harmonic Analysis is Cams Mathematical Monograph number 27. The Publisher, the Mathematical Association of America, says that books in the series "are intended for the wide circle of thoughtful people familiar with basic graduate or advanced undergraduate mathematics . . . who wish to extend their knowledge without prolonged and critical study of the mathematical journals and treatises." Krantz has done an admirable job of carrying out the publisher's intentions. The right way to read this book is quickly, with-

out too much fussing over the details.

While other books, such as those by Zygmund[Z] and Stein and Weiss[SW], are probably better for a graduate student who will need to achieve techni

cal competence in the area, A Panorama of Harmonic Analysis provides an excellent way of obtaining a well-balanced overview of the entire subject.

Analysis and Nonlinear Differential Equations

in Honor of Victor L. Shapiro, Contemporary

Math. , 208(1 997), 35-71 .

[I] S. lgari, Lectures on Fourier Series in Several

Variables, University of Wisconsin, Madison,

1 968.

[SW] E. M. Stein and G. Weiss, Introduction to

Fourier Analysis on Euclidean Spaces,

Princeton Univ. Press, Princeton, 1 971 .

[Z] A. Zygmund, Trigonometric Series, 2nd rev.

ed., Cambridge Univ. Press, New York, 1 959.

REFERENCES

[AW] J. M. Ash and G. Wang, A survey of

uniqueness questions in multiple trigono

metric series, A Conference in Harmonic


DePaul University

Chicago, IL 6061 4

Continued from p. 66

BIBLIOGRAPHY

Bieberbach, L. (1 930) Lehrbuch der Funktionentheorie, Springer Verlag,

Berlin, (1 st ed. 1 921) .

Bacher, M. 1 896 Cauchy's Theorem on complex integration, Bulletin

of the American Mathematical Society (2) 2, 1 46-9.

Briot, C.A.A. and Bouquet, J.C. (1 859) Theorie des fonctions double

ment periodiques et, en particulier, des fonctions elliptiques, Paris.

Bl'iot, C.A.A. and Bouquet, J.C. (1 875) Theorie des fonctions elliptiques,

Paris.

Denjoy, A. (1 933), Sur les polygones d'approximation d'une courbe

rectifiable, Comptes Rendus Acad. Sci. Paris 1 95, 29-32.

Ahlfors, L.V. (1 953) Complex Analysis, McGraw-Hill, New York.

Goursat, E. (1 884) Demonstration du theoreme de Cauchy. Acta

Mathematica 4, 1 97-200.

Goursat, E. (1 900) Sur Ia definition generals des fonctions analytiques,

d 'apres Cauchy, Transactions of the American Mathematical Society,

1 , 1 4-1 6.

Heffter, L. (1 902) Reelle Curvenintegration, G6ttingen Nachrichten,

26-52.

Heffter, L. (1 930) Ober den Cauchyschen lntegralsatz, Mathematische

Zeitschrift 32, 476-480.

Hurwitz, A. and Courant, R. (1 922) Allgemeine Funktionentheorie und

elliptische Funktionen. Springer Verlag, Berlin.

Jordan, C. (1 893) Cours d'analyse, 3 vols, Gauthier-Villars, Paris.

Kamke, E. (1 932) Zu dem lntegralsatz von Cauchy, Mathematische

Zeitschrift 33, 539-543.

Knopp, K. (1 930) Funktionentheorie, Springer Verlag, Leipzig and Berlin.

Lichtenstein, L. (1 91 0) Ober einige lntegrabilitatsbedingungen zwei

gliedriger Differentialausdrucke mit einer Anwendung auf den

Cauchyschen lntegralsatz, Sitzungsberichte der Mathematischen

Gesellschaft, Berlin, 9.4, 84-1 00.

Malmsten, C.J. (1 865) Om definita integraler mellan imaginara granser,

Svenska Vetenskaps-Akademiens Handlingar, 6.3.

Mittag-Leffler, G. (1 922) Der Satz von Cauchy Ober das Integral einer

Funktion zwischen imaginaren Grenzen, Journal fur die reine und

angewandte Mathematik 152, 1 -5.

Mittag-Leffler, G. (1 873), Forsok tillett nytt bevis for en sats inom de definita

integralemas teori, Svenska Vetenskaps-Akademiens Handlingar.

Mittag-Leffler, G. (1 875) Beweis tor den Cauchy'schen Satz,

Nachrichten der Koniglichen Gesellschaft der Wissenschaften zu

G6ttingen, 65-73.

Moore, E. H. (1 900) A simple proof of the fundamental Cauchy-Goursat

Theorem, Transactions of the American Mathematical Society 1 ,

499-506.

Osgood, W.F. (1 928) Lehrbuch der Funktionentheorie, Teubner,

Leipzig, 5th ed (1 st ed. 1 907).

Perron, 0. 1 952 Alfred Pringsheim, Jahresbericht der Deutschen

Mathematiker Vereinigung 56, 1 -6.

Pringsheim, A. (1 895a) Ueber den Cauchy'schen lntegralsatz,

Sitzungsberichte der math-phys. Classe der K6nigliche Akademie der

Wissenschaften zu Munchen, . 25, 39-72.

Pringsheim, A. (1 895b) Zum Cauchy'schen lntegralsatz, as above,

295-304.

Pringsheim, A. (1 898) Zur Theorie der Doppel-lntegrale, Sitzungs

berichte . . . Munchen 28, 59-74.

Pringsheim, A. (1 899) Zur Theorie der Doppel-lntegrale, Green'schen und

Cauchy'schen lntegralsatzes, Sitzungsberichte . . . Munchen 29, 39-62.

Pringsheim, A. ( 1901) Ueber den Goursat'schen Beweis des

Cauchy'schen lntegralsatzes, Transactions of the American Mathe

matical Society 2, 41 3-421 .

Pringsheim, A. (1 903) Der Cauchy-Goursat'sche lntegralsatz und seine

Obertragung auf reelle Kurven-lntegrale, Sitzungsberichte .

Munchen 33, 673-682.

Pringsheim, A. (1 929) Kritische-historische Bemerkungen zur

Funktionentheorie, Sitzungsberichte Bayerische Akademie der

Wissenschaften zu Munchen, 281 -295.

Walsh, J.L. (1 933), The Cauchy-Goursat Theorem for rectifiable Jordan

curves, Proc. Nat. Acad. Sci, of USA 1 9, 54Q-541 .


k1f'I.I.M9.h.i§i Robin Wilson I

Ind ian Mathematics

A round 250 BC King Ashoka, ruler of most of India, became the first

Buddhist monarch. The event was celebrated by the construction of pillars carved with his edicts. These columns contain the earliest known appearance of what would eventually become our Hindu-Arabic numerals. Unlike the complicated Roman numerals, and the Greek decimal system in which different symbols were used for 1, 2, . . . , 9, 10, 20, . . . , 90, 100, 200, . . . , the Hindu number system uses the same ten digits throughout, but in a place-value system where the position of each digit indicates its value. This enables calculations to be carried out column by column.

Indian mathematics can be traced back to around 600 Be, and a number of Vedic manuscripts contain early work on arithmetic, permutations and combinations, the theory of numbers, and the extraction of square roots.

The two most outstanding mathe-

maticians of the first millennium AD were Aryabhata (b. 4 76) and Brahmagupta (b. 598). Aryabhata gave the first systematic treatment of Diophantine equations (algebraic equations where we seek solutions in integers), obtained the value 3.1416 for TT, and presented formulae for the sum of natural numbers and of their squares and cubes; the first Indian satellite was later named after him, and he is commemorated on an Indian stamp. Brahmagupta discussed the use of zero (another Indian invention) and negative numbers, and described a general method for solving quadratic equations. He also solved quadratic Diophantine equations such as 92J:2 + 1 = y2, obtaining the integer solution x = 120, y = 1 151.

In later years Indian mathematicians and astronomers became interested in practical astronomy, and built magnificent observatories such as the Jantar Mantar in Jaipur.

Vedic manuscript Indian Ashoka column Nepalese Ashoka column

Please send all submissions to

the Stamp Corner Editor,

Robin Wilson, Faculty of Mathematics,

The Open University, Milton Keynes,

MK7 6AA, England



Aryabhata satellite Jantar Mantar

the mathematical intelligencer volume 22 issue 4

Documents