Adventures of a Mathematician - Softcover

Ulam, S. M. M.; Hirsch, Daniel; Mathews, William G.; Ulam, Françoise

Softcover

ISBN 10: 0520071549 ISBN 13: 9780520071544

Verlag: University of California Press, 1991

Inhaltsangabe

This autobiography of mathematician Stanislaw Ulam, one of the great scientific minds of the twentieth century, tells a story rich with amazingly prophetic speculations and peppered with lively anecdotes. As a member of the Los Alamos National Laboratory from 1944 on, Ulam helped to precipitate some of the most dramatic changes of the postwar world. He was among the first to use and advocate computers for scientific research, originated ideas for the nuclear propulsion of space vehicles, and made fundamental contributions to many of today's most challenging mathematical projects. With his wide-ranging interests, Ulam never emphasized the importance of his contributions to the research that resulted in the hydrogen bomb. Now Daniel Hirsch and William Mathews reveal the true story of Ulam's pivotal role in the making of the 'Super,' in their historical introduction to this behind-the-scenes look at the minds and ideas that ushered in the nuclear age. It includes an epilogue by Francoise Ulam and Jan Mycielski that sheds new light on Ulam's character and mathematical originality.

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Über die Autorin bzw. den Autor

S. M. Ulam (1909-1984) was born in Poland and was a key member of the now legendary Polish School of Mathematics. In the United States from 1935 on, he received many academic appointments and honors and authored many articles, essays, and mathematical books, including Analogies between Analogies (California, 1990). Daniel Hirsch is President of the Committee to Bridge the Gap, located in Los Angeles. William G. Mathews is Professor of Astrophysics at the University of California, Santa Cruz. Francoise Ulam is a resident of Santa Fe, New Mexico. Jan Mycielski is Professor of Mathematics at the University of Colorado.

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With his wide-ranging interests, Ulam never emphasized the importance of his contribution to the research that resulted in the hydrogen bomb. Now William Mathews and Daniel Hirsch reveal the true story of Ulam's pivotal role in the making of the 'Super, ' in their historical introduction to this behind-the-scenes look at the minds and ideas that ushered in the nuclear age.

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Adventures of a Mathematician

By S.M. Ulam

University of California Press

Copyright © 1991 S.M. Ulam
All right reserved.
ISBN: 0520071549
Chapter 1
Childhood
1909–1927

My father, Jozef Ulam, was a lawyer. He was born in Lwów, Poland, in 1877. At the time of his birth the city was the capital of the province of Galicia, part of the Austro-Hungarian Empire. When I was born in 1909 this was still true.

His father, my grandfather, was an architect and a building contractor. I understand that my great-grandfather had come to Lwów from Venice.

My mother, Anna Auerbach, was born in Stryj, a small town some sixty miles south of Lwów, near the Carpathian Mountains. Her father was an industrialist who dealt in steel and represented factories in Galicia and Hungary.

One of my earliest memories is of sitting on a window sill with my father and looking out at a street on which there was a great parade honoring the Crown Prince, who was visiting Lwów. I was not quite three years old.

I remember when my sister was born. I was told a little

girl had arrived, and I felt—it is hard to describe—somehow grown up. I was three.

When I was four, I remember jumping around on an oriental rug looking down at its intricate patterns. I remember my father's towering figure standing beside me, and I noticed that he smiled. I felt, "He smiles because he thinks I am childish, but I know these are curious patterns." I did not think in those very words, but I am pretty certain that it was not a thought that came to me later. I definitely felt, "I know something my father does not know. Perhaps I know better than my father."

I also have the memory of a trip to Venice with the family. We were on a vaporetto on a canal, and I had a balloon which fell overboard. As it bobbed along the side of the boat, my father tried to fish it out with the crooked end of his walking stick but failed. I was consoled by being allowed to select a souvenir model of a gondola made of Venetian beads and still remember the feeling of pride at being given such a task.

I remember the beginning of the first World War. As a boy, I was a Central Powers patriot when Austria, Germany, and Bulgaria—the "Central Powers"—were fighting against France, England, Russia, and Italy. Most of the Polish-speaking people were nationalistic and anti-Austrian, but nevertheless, at about the age of eight I wrote a little poem about the great victories of the Austrian and German armies.

Early in 1914, the Russian troops advanced into Galicia and occupied Lwów. My family left, taking refuge in Vienna. There I learned German, but my native language—the language we spoke at home—was Polish.

We lived in a hotel across from St. Stephen's cathedral. The strange thing is that even though I visited Vienna many times afterwards, I did not actually recognize this building again until one day in 1966 while I was walking through the streets with my wife. Perhaps because we were talking about my childhood I suddenly remembered it and pointed

it out to her. With this a number of other memories buried for over fifty years surfaced.

On the same visit, while walking through the Prater gardens, the sight of an outdoor café suddenly brought back the memory of how I had once choked in the wind with a sort of asthmatic reaction in front of that very café—a feeling that I was not to experience again until many years later in Madison, Wisconsin. Curiously the subsequent sensation did not make me recall the childhood episode. It is only when I was at that very spot many years later that this sensory memory returned as a result of the visual association.

I will not try to describe the mood of Vienna as seen through the eyes of a six-year-old. I wore a sort of military cap; when an officer saluted me on Kärntner Strasse (one of the main streets of Vienna) I remember vividly that I was absolutely delighted. But when somebody mentioned that the United States would have ten thousand airplanes (there was such a rumor) I began to have doubts about the victory of the Central Powers.

At about this time in Vienna I learned to read. Like so much of learning throughout my life, at first it was an unpleasant—a difficult, somewhat painful experience. After a while, everything fell into place and became easy. I remember walking the streets reading all the signs aloud with great pleasure, probably annoying my parents.

My father was an officer in the Austrian Army attached to military headquarters, and we traveled frequently. For a while we lived in Märisch Ostrau, and I went to school there for a time. In school we had to learn the multiplication tables, and I found learning arithmetic mildly painful. Once I was kept home with a cold just as we were at six times seven. I was sure that the rest of the class would be at twelve times fifteen by the time I went back. I think I went to ten times ten by myself. The rest of the time I had tutors, for we traveled so much it was not possible to attend school regularly.

I also remember how my father would sometime read to me from a children's edition of Cervantes' Don Quixote . Episodes that now seem only mildly funny to me, I considered hilarious. I thought the description of Don Quixote's fight with the windmills the funniest thing imaginable.

These are visual pictures, not nostalgic really but bearing a definite taste, and they leave a definite flavor of associations in the memory. They carry with them a consciousness of different intensities, different colors, different compositions, mixed with feelings which are not explicit—of well-being or of doubt. They certainly play simultaneously on many physically separate parts in the brain and produce a feeling perhaps akin to a melody. It is a reconstruction of how I felt. People often retain these random pictures, and the strange thing is that they persist throughout one's life.

Certain scenes are easier of access, but there are probably many other impressions which continue to exist: Experiments have re-created certain scenes from the past when areas of a patient's brain were touched with a needle during an operation. The scenes that can be summoned up from one's memory at will have a color or flavor which does not seem to change with time. Their re-creation by recollection does not seem to change them or refresh them. As far as I can tell when I try to observe in myself the chain of syllogisms initiated by these impressions, they are quite analogous now as to what they were when I was little. If I look now at an object, like a chair, or a tree, or a telegraph wire, it initiates a train of thought. And it seems to me that the succession of linked memories are quite the same as those I remember when I was five or six. When I look at a telegraph wire, I remember very well it gave me a sort of abstract or mathematical impulse. I wondered what else could do that. It was an attempt at generalization.

Perhaps the store of memory in the human brain is to a large extent already formed at a very early age, and external stimuli initiate a process of recording and classifying the im-

pressions along channels which exist in large numbers in very early childhood.

To learn how things are filed in the memory, it obviously helps to analyze one's thoughts. To understand how one understands a text, or a new method, or a mathematical proof, it is interesting to try to consciously perceive the temporal order and the inner logic. Professionals or even interested amateurs have not done enough in this area to judge by what I have read on the nature of memory. It seems to me that more could be done to elicit even in part the nature of associations, with computers providing the means for experimentation. Such a study would have to involve a gradation of notions, of symbols, of classes of symbols, of classes of classes, and so on, in the same way that the complexity of mathematical or physical structures is investigated.

There must be a trick to the train of thought, a recursive formula. A group of neurons starts working automatically, sometimes without external impulse. It is a kind of iterative process with a growing pattern. It wanders about in the brain, and the way it happens must depend on the memory of similar patterns.

Very little is known about this. Perhaps before a hundred years have passed this will all be part of a fascinating new science. It was not so long ago that scientists like John von Neumann began to examine analogies between the operation of the brain and that of the computer. Earlier, people had thought the heart was the seat of thought; then the role of the brain became more evident. Perhaps it actually depends on all the senses.

We are accustomed to think of thinking as a linear experience, as when we say "train" of thought. But subconscious thinking may be much more complicated. Just as one has simultaneous visual impressions on the retina, might there not be simultaneous, parallel, independently originated, abstract impressions in the brain itself? Something goes on in our heads in processes which are not simply strung out on

one line. In the future, there might be a theory of a memory search, not by one sensor going around, but perhaps more like several searchers looking for someone lost in a forest. It is a problem of pursuit and of search—one of the greatest areas of combinatorics.

What happens when one suddenly remembers a forgotten word or name? What does one do when one tries to remember it? Subconsciously something is turning. More than one route is followed: one tries by sound or letters, long words or short words. That must mean that the word is filed in multiple storage. If it were only in one place there would be no way to recover it. Time is a parameter, too, and although in the conscious there seems to be only one time, there may be many in the subconscious. Then there is the mechanism of synthesizer or summarizer. Could one introduce an automatic search system, an ingenious system which does not go through everything but scans the relevant elements?

But I have digressed enough in these observations on memory. Let me now return to this account of my life. I only wish that I could have some of Vladimir Nabokov's ability to evoke panoramas of memories from a few pictures of the past. Indeed one can say that an artist depicts the essential functions or properties of a whole set of impressions on the retina. It is these that the brain summarizes and stores in the memory, just as a caricaturist can convey the essentials of a face with just a few strokes. Mathematically speaking, these are the global characteristics of the function or the figure of a set of points. In this more prosaic account I will describe merely the more formal points.

In 1918 we returned to Lwów, which had become part of the newly formed Republic of Poland. In November of that year the, Ukrainians besieged the city, which was defended by a small number of Polish soldiers and armed civilians. Our house was in a relatively safe part of town, even though occasional artillery shells struck nearby. Because our house was safer, many of our relatives came to stay with us. There

must have been some thirty of them, half being children. There were not nearly enough beds, of course, and I remember people sleeping everywhere on rolled rugs on the floor. During the shelling we had to go to the basement. I still remember insisting on tying my shoes while my mother was pressing me to hurry downstairs. For the adults it must have been a strenuous time to say the least, but not for us. Strangely enough, my memories of these days are of the fun I had playing, hiding, learning card games with the children for the two weeks before the siege was lifted with the arrival of another Polish army from France. This broke the ring of besiegers. For children wartime memories are not always traumatic.

During the Polish-Russian war in 1920 the city was threatened again. Budenny's cavalry penetrated to within fifty miles, but Pilsudski's victory on the Warsaw front saved the southern front and the war ended.

At the age of ten in 1919 I passed the entrance examination to the gymnasium. This was a secondary school patterned after the German gymnasia and the French lycées. Instruction usually took eight years. I was an A student, except in penmanship and drawing, but did not study much.

One of the gaps in my education was in chemistry. We did not have much of it in school and fifty years later, now that I am interested in biology, this handicaps me in my studies of elementary biochemistry.

About this time I also discovered that I did not have quite normal binocular vision. It happened in the following way: the boys in the class had been lined up for an eye examination. Awaiting my turn to read the charts, I covered my eyes with my hand. I noticed with horror that I could only read the largest letters with my right eye. This made me afraid that I would be kept after school, so I memorized the letters. I think it was the first time in my life when I consciously cheated. When my turn came I "read" satisfactorily and was let off, but I knew my eyes were different, one was myopic. The other, normal, later became presbyotic. This

condition, rather rare but well known, is apparently hereditary. I still have never worn glasses, although I have to bend close to the printed text to read with my myopic eye. I am not normally aware which eye I use; once later in life a doctor in Madison told me that this condition is sometimes better than normal, for one or the other eye is resting while the other is in use. I wonder if my peculiar eyesight, in addition to affecting my reading habits, may also have affected my habits of thought.

When I try to remember how I started to develop my interest in science I have to go back to certain pictures in a popular book on astronomy I had. It was a textbook called Astronomy of Fixed Stars , by Martin Ernst, a professor of astronomy at the University of Lwów. In it was a reproduction of a portrait of Sir Isaac Newton. I was nine or ten at the time, and at that age a child does not react consciously to the beauty of a face. Yet I remember distinctly that I considered this portrait—especially the eyes—as something marvelous. A mixture of physical attraction and a feeling of the mysterious emanated from his face. Later I learned it was the Geoffrey Kneller portrait of Newton as a young man, with hair to his shoulders and an open shirt. Other illustrations I distinctly remember were of the rings of Saturn and of the belts of Jupiter. These gave me a certain feeling of wonder, the flavor of which is hard to describe since it is sometimes associated with nonvisual impressions such as the feeling one gets from an exquisite example of scientific reasoning. But it reappears, from time to time, even in older age, just as a familiar scent will reappear. Occasionally an odor will come back, bringing coincident memories of childhood or youth.

Reading descriptions of astronomical phenomena today brings back to me these visual memories, and they reappear with a nostalgic (not melancholy but rather pleasant) feeling, when new thoughts come about or a new desire for mental work suddenly emerges.

The high point of my interest in astronomy and an unforgettable emotional experience came when my uncle Szy-

mon Ulam gave me a little telescope. It was one of the copper- or bronze-tube variety and, I believe, a refractor with a two-inch objective.

To this day, whenever I see an instrument of this kind in antique shops, nostalgia overcomes me, and after all these decades my thoughts still turn to visions of the celestial wonders and new astronomical problems.

At that time, I was intrigued by things which were not well understood—for example, the question of the shortening of the period of Encke's comet. It was known that this comet irregularly and mysteriously shortens its three-year period of motion around the sun. Nineteenth-century astronomers made several attempts to account for this as being caused by friction or by the presence of some new invisible body in space. It excited me that nobody really knew the answer. I speculated whether the 1/r2 law of attraction of Newton was not quite exact. I tried to imagine how it could affect the period of the comet if the exponent was slightly different from 2, imagining what the result would be at various distances. It was an attempt to calculate, not by numbers and symbols, but by almost tactile feelings combined with reasoning, a very curious mental effort.

No star could be large enough for me. Betelgeuse and Antares were believed to be much larger than the sun (even though at the time no precise data were available) and their distances were given, as were parallaxes of many stars. I had memorized the names of constellations and the individual Arabic names of stars and their distances and luminosities. I also knew the double stars.

In addition to the exciting Ernst book another, entitled Planets and the Conditions of Life on Them , was strange. Soon I had some eight or ten astronomy books in my library, including the marvelous Newcomb-Engelmann Astronomie in German. The Bode-Titius formula or "law" of planetary distances also fascinated me, inspiring me to become an astronomer or physicist. This was about the time when, at the age of eleven or so, I inscribed my name in a notebook, "S.

Ulam, astronomer, physicist and mathematician." My love for astronomy has never ceased; I believe it is one of the avenues that brought me to mathematics.

From today's perspective Lwów may seem to have been a provincial city, but this is not so. Frequent lectures by scientists were held for the general public, in which such topics as new discoveries in astronomy, the new physics and the theory of relativity were covered. These appealed to lawyers, doctors, businessmen, and other laymen.

Other popular lecture topics were Freud and psychoanalysis. Relativity theory was, of course, much more difficult.

Around 1919–1920 so much was written in newspapers and magazines about the theory of relativity that I decided to find out what it was all about. I went to some of the popular talks on relativity. I did not really understand any of the details, but I had a good idea of the main thrust of the theory. Almost like learning a language in childhood, one develops the ability to speak it without knowing anything about grammar. Curiously enough, it is possible even in the exact sciences to have an idea of the gist of something without having a complete understanding of the basics. I understood the schema of special relativity and even some of its consequences without being able to verify the details mathematically. I believe that so-called understanding is not a yes-or-no proposition. But we don't yet have the technique of defining these levels or the depth of the knowledge of reasons.

This interest became known among friends of my father, who remarked that I "understood" the theory of relativity. My father would say, "The little boy seems to understand Einstein!" This gave me a reputation I felt I had to maintain, even though I knew that I did not genuinely understand any of the details. Nevertheless, this was the beginning of my reputation as a "bright child." This encouraged me to further study of popular science books—an experience

I am sure is common to many children who later grow up to be scientists.

How a child acquires the habits and interests which play such a decisive role in determining his future has not been sufficiently investigated. "Plagiarism"—the mysterious ability of a child to imitate or copy external impressions such as the mother's smile—is one possible explanation. Another is inborn curiosity: why does one seek new experiences instead of merely reacting to stimuli?

Inclinations may be part of the inherited system of connections in the brain, a genetic trait that may not even depend on the physical arrangement of neurons. Apparently headaches are related to the ease with which blood circulates in the brain, which depends on whether the blood vessels are wide or narrow. Perhaps it is the "plumbing" that is important, rather than the arrangement of the neurons normally associated with the seat of thinking.

Another determining factor may be initial accidents of success or failure in a new pursuit. I believe that the quality of memory develops similarly as a result of initial accidents, random external influences, or a lucky combination of the two.

Consider the talent for chess, for example. José Capablanca learned the game at the age of six by watching his father and uncle play. He developed the ability to play naturally, effortlessly, the way a child learns to speak as compared with the struggles adults have in learning new subjects. Other famous chess players also first became interested by watching their relatives play. When they tried, perhaps a chance initial success encouraged them to pursue. Nothing succeeds like success, it is well known, especially in early youth.

I learned chess from my father. He had a little paper-bound book on the subject and used to tell me about some of the famous games it described. The moves of the knight fascinated me, especially the way two enemy pieces can be

threatened simultaneously with one knight. Although it is a simple stratagem, I thought it was marvelous, and I have loved the game ever since.

Could the same process apply to the talent for mathematics? A child by chance has some satisfying experiences with numbers; then he experiments further and enlarges his memory by building up his store of experiences.

I had mathematical curiosity very early. My father had in his library a wonderful series of German paperback books—Reklam , they were called. One was Euler's Algebra . I looked at it when I was perhaps ten or eleven, and it gave me a mysterious feeling. The symbols looked like magic signs; I wondered whether one day I could understand them. This probably contributed to the development of my mathematical curiosity. I discovered by myself how to solve quadratic equations. I remember that I did this by an incredible concentration and almost painful and not-quite-conscious effort. What I did amounted to completing the square in my head without paper or pencil.

In high school, I was stimulated by the notion of the problem of the existence of odd perfect numbers. An integer is perfect if it is equal to the sum of all its divisors including one but not itself. For instance: 6 = 1 + 2 + 3 is perfect. So is 28 = 1 + 2 + 4 + 7 + 14. You may ask: does there exist a perfect number that is odd? The answer is unknown to this day.

In general, the mathematics classes did not satisfy me. They were dry, and I did not like to have to memorize certain formal procedures. I preferred reading on my own.

At about fifteen I came upon a treatise on the infinitesimal calculus in a book by Gerhardt Kowalevski. I did not have enough preparation in analytic geometry or even in trigonometry, but the idea of limits, the definitions of real numbers, the notion of derivatives and integration puzzled and excited me greatly. I decided to read a page or two a day and attempt to learn the necessary facts about trigonometry and analytic geometry from other books.

I found two other books in a secondhand bookstore.

These intrigued and fascinated me more than anything else for many years to come: Sierpinski's Theory of Sets and a monograph on number theory. At the age of seventeen I knew as much or more elementary number theory than I do now.

I also read a book by the mathematician Hugo Steinhaus entitled What Is and What Is Not Mathematics and in Polish translation Poincaré's wonderful La Science et l'Hypothèse, La Science et la Méthode, La Valeur de la Science , and his Dernières Pensées . Their literary quality, not to mention the science, was admirable. Poincaré molded portions of my scientific thinking. Reading one of' his books today demonstrates how many wonderful truths have remained, although everything in mathematics has changed almost beyond recognition and in physics perhaps even more so. I admired Steinhaus's book almost as much, for it gave many examples of actual mathematical problems.

The mathematics taught in school was limited to algebra, trigonometry, and the very beginning of analytic geometry. In the seventh and eighth classes, where the students were sixteen and seventeen, there was a course on elementary logic and a survey of history of philosophy. The teacher, Professor Zawirski, was a real scholar, a lecturer at the University and a very stimulating man. He gave us glimpses of recent developments in advanced modern logic. Having studied Sierpinski's books on the side, I was able to engage him in discussions of set theory during recess and in his office. I was working on some problems on transfinite numbers and on the problem of the continuum hypothesis.

I also engaged in wild mathematical discussions, formulating vast and new projects, new problems, theories and methods bordering on the fantastic, with a boy named Metzger, some three or four years my senior. He had been directed toward me by friends of' my father who knew that he too had a great interest in mathematics. Metzger was short, rotund, blondish, a typical liberated ghetto Jew. Later I saw a youthful portrait of Heine which reminded me of his

face. People of his type can still be found occasionally. They exhibit amateurism, even about the very foundations of arithmetic. We discussed "an iterative calculus" on the basis of practically no knowledge of the existing mathematical material. He was "crazy" and full of the urge to innovate which is so Jewish. Stefan Banach once pointed out that it is characteristic of certain Jews always to try to change the established scheme of things—Jesus, Marx, Freud, Cantor. On a very small scale Metzger showed this tendency. Had he had a better education he might have done good things. He obviously came from a very poor family and his Polish had a strong, guttural accent. After a few months he abruptly vanished from my ken. This is the first time I have thought about him in all these years. Perhaps he is alive. This memory of Metzger and our discussions brings back the very smell and color of the "abstractions" we exchanged.

Strangely enough, at this youthful and immature age I was also occasionally trying to analyze my own thinking processes. I tried to make myself more aware of them by periodically going back every few seconds to see what it was that molded the train of thought. Needless to say, I was fully aware of the fact that there is a danger in indulging too much and too frequently in such introspection.

So far, the image I had formed of astronomers and scientists, and of mathematicians in particular, came almost exclusively from my reading. I got my first "live" impressions when I went to a series of popular mathematics lectures in 1926. On successive days there were talks by Hugo Steinhaus, Stanislaw Ruziewicz, Stefan Banach, and perhaps others. My first surprise was to discover how young they were. Having heard and read of their achievements I really expected bearded old scholars. I listened avidly to their talks. Young as I was, my impression of Banach was that here was a homespun genius. This first impression—deepened, enriched, and transformed, of course—remained during my subsequent long acquaintance, collaboration, and friendship with him.

Then in 1927, Zawirski told me a congress of mathematicians was to take place in Lwów and foreign scholars had been invited. He added that a youthful and extremely brilliant mathematician named John von Neumann was to give a lecture. This was the first time I heard the name. Unfortunately, I could not attend these lectures for I was in the midst of my own matriculation examinations at the Gymnasium.

Still, my interests in science did not take all of my time. I avidly read Polish literature, as well as writers as diverse as Tolstoy, Jules Verne, Karl May, H. G. Wells, and Anatole France. As a boy I preferred biographies and adventure stories.

Besides these more cerebral activities, I engaged actively in sports. Beginning at about fourteen I played various positions in soccer with my classmates: goalie, right forward and others. I started playing tennis, too, and was active in track and field.

After school I played cards with my classmates. We played bridge and a simple variety of poker for small stakes. In poker the older boys won most of the time. One of the abilities that apparently does not decrease but rather improves with age is a primitive type of elementary shrewdness. I played chess also, two or three times a week. Although I don't think I ever had too much talent for the game, I certainly had a more than average feeling for positions, and I probably was one of the best players in my group. Like mathematics, chess is one of the things where constant practice, constant thinking, and imagining, and studying are necessary to achieve a mastery of the game.

In 1927 I passed my three day matriculation examinations and a period of indecision began. The choice of a future career was not easy. My father, who had wanted me to become a lawyer so I could take over his large practice, now recognized that my inclinations lay in other directions. Besides, there was no shortage of lawyers in Lwów. The thought of a university career was attractive, but professorial

positions were rare and hard to obtain, especially for people with Jewish backgrounds like myself. Consequently, I looked for a course of studies which would lead to something practical and at the same time would be connected with science. My parents urged me to become an engineer, and so I applied for admission at the Lwów Polytechnic Institute as a student of either mechanical or electrical engineering.

Continues...

Excerpted from Adventures of a Mathematicianby S.M. Ulam Copyright © 1991 by S.M. Ulam. Excerpted by permission.
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Details zur Ausgabe

• Verlag: University of California Press
• Erscheinungsdatum: 1991
• Sprache: Englisch
• ISBN 10: 0520071549
• ISBN 13: 9780520071544
• Einband: Tapa blanda
• Anzahl der Seiten: 388
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Nicht verfügbar