Moritz Pasch (1843-1930) is justly celebrated as a key figure in the history of axiomatic geometry. Less well known are his contributions to other areas of foundational research. This volume features English translations of 14 papers Pasch published in the decade 1917-1926. In them, Pasch argues that geometry and, more surprisingly, number theory are branches of empirical science; he provides axioms for the combinatorial reasoning essential to Hilbert's program of consistency proofs; he explores "implicit definition" (a generalization of definition by abstraction) and indicates how this technique yields an "empiricist" reconstruction of set theory; he argues that we cannot fully understand the logical structure of mathematics without clearly distinguishing between decidable and undecidable properties; he offers a rare glimpse into the mind of a master of axiomatics, surveying in detail the thought experiments he employed as he struggled to identify fundamental mathematical principles; and much more. This volume will: Give English speakers access to an important body of work from a turbulent and pivotal period in the history of mathematics, help us look beyond the familiar triad of formalism, intuitionism, and logicism, show how deeply we can see with the help of a guide determined to present fundamental mathematical ideas in ways that match our human capacities, will be of interest to graduate students and researchers in logic and the foundations of mathematics.

This collection of historical research studies covers the evolution of technology as knowledge, the emergence of an autonomous engineering science in the Industrial Age, the idea of scientific managment of production and operation systems, and the interaction between mathematical models and technological concepts.

The book is published with the support of the UNESCO Venice Office - Regional Office for Science & Technology in Europe as an activity of the Project: The evolution of events, concepts and models in engineering systems.

Galileo Galilei (1564-1642), his life and his work have been and continue to be the subject of an enormous number of scholarly works. One of the con- quences of this is the proliferation of identities bestowed on this gure of the Italian Renaissance: Galileo the great theoretician, Galileo the keen astronomer, Galileo the genius, Galileo the physicist, Galileo the mathematician, Galileo the solitary thinker, Galileo the founder of modern science, Galileo the heretic, Galileo the courtier, Galileo the early modern Archimedes, Galileo the Aristotelian, Galileo the founder of the Italian scienti c language, Galileo the cosmologist, Galileo the Platonist, Galileo the artist and Galileo the democratic scientist. These may be only a few of the identities that historians of science have associated with Galileo. And now: Galileo the engineer! That Galileo had so many faces, or even identities, seems hardly plausible. But by focusing on his activities as an engineer, historians are able to reassemble Galileo in a single persona, at least as far as his scienti c work is concerned. The impression that Galileo was an ingenious and isolated theoretician derives from his scienti c work being regarded outside the context in which it originated.

Philanthropies funded by the Rockefeller family have been prominent in the social history of the twentieth century for their involvement in medicine and applied science. This book provides the first detailed study of their relatively brief but nonetheless influential foray into the field of mathematics. The careers of a generation of pathbreakers in modern mathematics, such as S.Banach, B.L.van der Waerden and Andre Weil, were decisively affected by their becoming fellows of the Rockefeller-funded International Education Board in the 1920s. To help promote cooperation between physics and mathematics Rockefeller funds supported the erection of the new Mathematical Institute in Gottingen between 1926 and 1929, while the rise of probability and mathematical statistics owes much to the creation of the Institut Henri Poincare in Paris by American philanthropy at about the same time. This account draws upon the documented evaluation processes behind these personal and institutional involvements of philanthropies. It not only sheds light on important events in the history of mathematics and physics of the 20th century but also analyzes the comparative developments of mathematics in Europe and the United States. Several of the documents are given in their entirety as significant witnesses to the gradual shift of the centre of world mathematics to the USA. This shift was strengthened by the Nazi purge of German and European mathematics after 1933 to which the Rockefeller Foundation reacted with emergency programs that subsequently contributed to the American war effort. The general historical and political background of the events discussed in this book is the mixture of competition and cooperation between the various European countries and the USA after World War I, and the consequences of the Nazi dictatorship after 1933. Ideological positions of both the philanthropists and mathematicians mattered heavily in that process. Cultural bias in the selection of fellows and of disciplines supported, and the economic predominance of American philanthropy, led among other things to a restriction of the programs to Europe and America, to an uneven consideration of European candidates, and to preferences for Americans. Political self-isolation of the Soviet Union contributed to an increasing alienation of that important mathematical culture from Western mathematics. By focussing on a number of national cultures the investigation aims to represent a step toward a true inter-cultural comparison in mathematics."

At its meeting in April 1990 at the University of Cambridge, the Executive Committee of the International Mathematical Union (IMU) decided that the largely unorganized archives of the Union should be properly arranged and catalogued. Simultaneously, the Executive Committee expressed the wish that a history of the Union should be written [1). As Secretary of the Union, I had proposed that these issues be dis cussed at the Cambridge meeting, but without having had in mind any personal role in the practical execution of such projects. At that time, the papers of the IMU were stored in Zurich, at the Eidgenossische Technische Hochschule, and I saw no reason why they could not remain there. At about this time, Professor K. Chandrasekharan produced a handwritten article titled "The Prehistory of the International Mathematical Union" [2), and it seemed to me that this might serve as the beginning of a more compre hensive history. I had first thought that Tuulikki MakeUiinen, who during eight years as the Office Secretary ofthe IMU had become well acquainted with the Union, would do the arranging of the archives in Zurich. She had a preliminary look at the material there, but it soon became clear that the amount of work required to bring order to it was too great to be accomplished in a few short visits from Helsinki. The total volume of material was formidable.

Habent sua Jata colloquia. The present volume has its ongms in a spring 1984 international workshop held, under the auspices of the Israel Academy of Sciences and Humanities, by The Institute for the History and Philosophy of Science and Ideas of Tel-Aviv University in cooperation with The Van Leer Jerusalem Foundation. It contains twelve of the twenty papers presented at the workshop by the twenty-six participants. As Proceedings of conferences go, it is a good representative of the genre, sharing in the main characteristics of its ilk. It may even be one of the rare instances of a book of Proceed ings whose descriptive title applies equally well to the workshop's topic and to the interrelations between. the various papers it includes. Tension and Accommodation are the key words. Thus, while John Glucker's paper, 'Images of Plato in Late Antiqu ity, ' raises, by means of the Platonic example, the problem of interpreta tion of ancient texts, suggesting the assignment of proper weight to the creator of the tradition and not only to his many later interpreters in assessing the proper relationship between originator and commentators, Abraham Wasserstein's 'Hunches that did not come off: Some Prob lems in Greek Science' illustrates the long-lived Whiggish tradition in the history of science and mathematics. As those familiar with my work will undoubtedly note, Wasserstein's position is far removed from my stance on ancient Greek mathematics."

Although not so well known today, Book 4 of Pappus' Collection is one of the most important and influential mathematical texts from antiquity. The mathematical vignettes form a portrait of mathematics during the Hellenistic "Golden Age," illustrating central problems - for example, squaring the circle; doubling the cube; and trisecting an angle - varying solution strategies, and the different mathematical styles within ancient geometry.

This volume provides an English translation of Collection 4, in full, for the first time, including: a new edition of the Greek text, based on a fresh transcription from the main manuscript and offering an alternative to Hultsch's standard edition, notes to facilitate understanding of the steps in the mathematical argument, a commentary highlighting aspects of the work that have so far been neglected, and supporting the reconstruction of a coherent plan and vision within the work, bibliographical references for further study.

An innovative, dramatic graphic novel about the treacherous pursuit of the foundations of mathematics.

This exceptional graphic novel recounts the spiritual odyssey of philosopher Bertrand Russell. In his agonized search for absolute truth, Russell crosses paths with legendary thinkers like Gottlob Frege, David Hilbert, and Kurt Godel, and finds a passionate student in the great Ludwig Wittgenstein. But his most ambitious goal--to establish unshakable logical foundations of mathematics--continues to loom before him. Through love and hate, peace and war, Russell persists in the dogged mission that threatens to claim both his career and his personal happiness, finally driving him to the brink of insanity.

This story is at the same time a historical novel and an accessible explication of some of the biggest ideas of mathematics and modern philosophy. With rich characterizations and expressive, atmospheric artwork, the book spins the pursuit of these ideas into a highly satisfying tale. Probing and ingeniously layered, the book throws light on Russell's inner struggles while setting them in the context of the timeless questions he spent his life trying to answer. At its heart, "Logicomix "is a story about the conflict between an ideal rationality and the unchanging, flawed fabric of reality.Apostolos Doxiadis studied mathematics at Columbia University. His international bestseller "Uncle Petros and Goldbach's Conjecture" spearheaded the impressive entrance of mathematics into the world of storytelling. Apart from his work in fiction, Apostolos has also worked in film and theater and is an internationally recognized expert on the relationship of mathematics to narrative. Christos H. Papadimitriou is C . Lester Hogan professor of computer science at the University of California, Berkeley. He was won numerous international awards for his pioneering work in computational complexity and algorithmic game theory. Christos is the author of the novel "Turing: A Novel about Computation." Alecos Papadatos worked for over twenty years in film animation in France and Greece. In 1997, he became a cartoonist for the major Athens daily "To Vima." He lives in Athens with his wife, Annie Di Donna, and their two children. Annie Di Donna studied graphic arts and painting in France and has worked as animator on many productions, among them "Babar" and "Tintin." Since 1991, she has been running an animation studio with her husband, Alecos Papadatos. This innovative graphic novel is based on the early life of the brilliant philosopher Bertrand Russell. Russell and his impassioned pursuit of truth. Haunted by family secrets and unable to quell his youthful curiosity, Russell became obsessed with a Promethean goal: to establish the logical foundation of all mathematics. In his agonized search for absolute truth, Russell crosses paths with legendary thinkers like Gottlob Frege, David Hilbert, and Kurt Godel, and finds a passionate student in the great Ludwig Wittgenstein. But the object of his defining quest continues to loom before him. Through love and hate, peace and war, Russell persists in the dogged mission that threatens to claim both his career and his personal happiness, finally driving him to the brink of insanity. "Logicomix" is at the same time a historical novel and an accessible explication to some of the biggest ideas of mathematics and modern philosophy. With rich characterizations and expressive, atmospheric artwork, the book spins the pursuit of these ideas into a captivating tale. Probing and ingeniously layered, the book throws light on Russell's inner struggles while setting them in the context of the timeless questions he spent his life trying to answer. At its heart, "Logicomix" is a story about the conflict between an ideal rationality and the unchanging, flawed fabric of reality. "At the heart of Logicomix stands Sir Bertrand Russell, a man determined to find a way of arriving at absolutely right answers. It's a tale within a tale, as the two authors and two graphic artists ardently pursue their own search for truth and appear as characters in the book. As one of them assures us, this won't be 'your typical, usual comic book.' Their quest takes shape and revolves around a lecture given by Russell at an unnamed American university in 1939, a lecture that is really, as he himself tells us, the story of his life and of his pursuit of real logical truth. With Proustian ambition and exhilarating artwork, "Logicomix"'s search for truth encounters head-on the horrors of the Second World War and the agonizing question of whether war can ever be the right choice. Russell himself had to confront that question personally: he endured six months in jail for his pacifism. Russell was determined to find the perfect logical method for solving all problems and attempted to remold human nature in his experimental school at Beacon Hill. Despite repeated failures, Russell never stopped being 'a sad little boy desperately seeking ways out of the deadly vortex of uncertainty.' The book is a visual banquet chronicling Russell's lifelong pursuit of 'certainty in total rationality.' As Logic and Mathematics, the last bastions of certainty, fail him, and as Reason proves not absolute, Russell is forced to face the fact that there is no Royal Road to Truth. Authors Dosiadis and Papadimitriou perfectly echo Russell's passion, with a sincere, easily grasped text amplified with breathtaking visual richness, making this the most satisfying graphic novel of 2009, a titanic artistic achievement of more than 300 pages, all of it pure reading joy."--Nick DiMartino, "Shelf Awareness" "This is an extraordinary graphic novel, wildly ambitious in daring to put into words and drawings the life and thought of one of the great philosophers of the last century, Bertrand Russell. The book is a rare intellectual and artistic achievement, which will, I am sure, lead its readers to explore realms of knowledge they thought were forbidden to them."--Howard Zinn "This magnificent book is about ideas, passions, madness, and the fierce struggle between well-defined principle and the larger good. It follows the great mathematicians--Russell, Whitehead, Frege Cantor, Hilbert--as they agonized to make the foundations of mathematics exact, consistent, and complete. And we see the band of artists and researchers--and the all-seeking dog Manga--creating, and participating in, this glorious narrative."--Barry Mazur, Gerhard Gade University Professor at Harvard University, and author of "Imagining Numbers (Particularly the Square Root of Minus Fifteen)" "The lives of ideas (and those who think them) can be as dramatic and unpredicteable as any superhero fantasy. "Logicomix" is witty, engaging, stylish, visually stunning, and full of surprising sound effects, a masterpiece in a genre for which there is as yet no name."--Michael Harris, professor of mathematics at Universite Paris 7 and member of the Institut Universitaire de France

To mark the centenary of the 1910 to 1913 publication of the monumental Principia Mathematica by Alfred N. Whitehead and Bertrand Russell, this collection of fifteen new essays by distinguished scholars considers the influence and history of PM over the last hundred years.

This is a charming and insightful contribution to an understanding of the "Science Wars" between postmodernist humanism and science, driving toward a resolution of the mutual misunderstanding that has driven the controversy. It traces the root of postmodern theory to a debate on the foundations of mathematics, early in the 20th century then compares developments in mathematics to what took place in the arts and humanities, discussing issues as diverse as literary theory, arts, and artificial intelligence. This is a straight forward, easily understood presentation of what can be difficult theoretical concepts and demonstrates that a pattern of misreading mathematics can be seen on both the part of science and on the part of postmodern thinking. This is a humorous, playful yet deeply serious look at the intellectual foundations of mathematics for those in the humanities and is the perfect critical introduction to the bases of modernism and postmodernism for those in the sciences.

HIS STUDY OF GIROLAMO CARDANO is the T work of an amateur in the field of the history of science and the history of ideas. As a mathematical physi cist I lack the depth of training in philological-historical disciplines necessary to discuss the sources of Car dano's knowledge and trace the influences that shaped his views on science, medicine, and philosophy. What little recent literature on Cardano there is some times shows a lack of true understanding, or is primarily an appreciation of the mathematician. I relied, therefore, largely on his own writings, which are collected in the Opera Omnia. My excerpts and translations are taken di rectly from these works, and I hope that I have succeeded in capturing their essential meaning and spirit. MARKUS FIERZ Preface to the English Edition THANK the publisher Birkhauser Boston, Incorpo I rated, for the venture of this English edition of my essay on Cardano that originally appeared in the Poly Series published by Birkhauser Verlag, and for the care given to the translation. For this edition I have added two longer sections: one on Cardano's voyage to Scotland, and another on a rather interesting "mathematical theosophy" contained in his Liber de Proportionibus (Basel, 1570). Also included is a list of references quoted in the notes, and a record-as complete as possible-of the original editions of Cardano' s writings.

The ancient Greeks played a fundamental role in the history of mathematics and their ideas were reused and developed in subsequent periods all the way down to the scientific revolution and beyond. In this, the first complete history for a century. Reviel Netz offers a panoramic view of the rise and influence of Greek mathematics and its significance in world history. He explores the Near Eastern antecedents and the social and intellectual developments underlying the subject's beginnings in Greece in the fifth century BCE. He leads the reader through the proofs and arguments of key figures like Archytas, Euclid and Archimedes, and considers the totality of the Greek mathematical achievement which also includes, in addition to pure mathematics, such applied fields as optics, music, mechanics and, above all, astronomy. This is the story not only of a major historical development, but of some of the finest mathematics ever created.

A new translation makes this classic and important text more generally accessible. The text is placed in its contemporary context, but also related to the interests of practising mathematicians today. This book will be of interest to mathematical historians, researchers, and numerical analysts.

Als P. R. Fuss, ein Urenkel Leonhard Eulers und seinerzeit standi- ger Sekretar der Petersburger Akademie der Wissenschaften, das zweibandige Werk Mathematischer und physikalischer Briefwech- sel einiger bedeutender Geometer des 18. Jahrhunderts heraus- gab, reservierte er den erst en Band fiir den Briefwechsel seines 1 beriihmten UrgroBvaters mit dessen Freund Christian Goldbach. 1m zweiten Band nimmt die Korrespondenz Goldbachs mit zwei 2 Vert ret ern der Familie Bernoulli, mit Nikolaus II und Daniel , einen beachtlichen Platz - mehr als 300 Seiten - ein. AIle diese Gelehrten des 18. Jahrhunderts waren Mitglieder der Pe- tersburger Akademie der Wissenschaften. Euler und Daniel Ber- noulli genieBen Weltruhm, Nikolaus II Bernoulli starb ganz jung, ohne geniigend Zeit gehabt zu haben, sein offensichtlich vorhan- denes Talent zu entfalten. Goldbachs wissenschaftliche Verdienste sind weitaus weniger bekannt, obwohl jeder Mathematiker schon etwas von der Goldbachschen Vermutung in der Zahlentheorie gehort hat. Fuss auBerte sich tiber Goldbach folgendermaBen: "Sein Briefwechsel zeigt, daB es der graBen Breite seiner Kennt- nisse geschuldet ist, wenn er auf keinem Spezialgebiet beriihmt wurde. Bald sehen wir ihn mit Bayer knifRige Fragen der klassi- schen und orientalischen Philologie behandeln; bald laBt er sich auf endlose Streitereien liber Archaologie mit dem berlihmten Stosch ein; hier zieht ihn Biilfinger zu den damals in Mode kom- menden metaphysischen Spekulationen heran, die indessen zu rein gar nichts fiihrten; dort regen Euler und die Bernoulli ihn an, sich mit Mathematik zu beschaftigen und weihen ihn in die Geheimnisse der hoheren Analysis und der Zahlentheorie ein.

During Edmund Husserl s lifetime, modern logic and mathematics rapidly developed toward their current outlook and Husserl s writings can be fruitfully compared and contrasted with both 19th century figures (Boole, Schroder, Weierstrass) as well as the 20th century characters (Heyting, Zermelo, Godel). Besides the more historical studies, the internal ones on Husserl alone and the external ones attempting to clarify his role in the more general context of the developing mathematics and logic, Husserl s phenomenology offers also a systematically rich but little researched area of investigation. This volume aims to establish the starting point for the development, evaluation and appraisal of the phenomenology of mathematics. It gathers the contributions of the main scholars of this emerging field into one publication for the first time. Combining both historical and systematic studies from various angles, the volume charts answers to the question "What kind of philosophy of mathematics is phenomenology?""

Numerous scientists have taken part in the war effort during World War I, but few gave it the passionate energy of the prominent Italian mathematician Volterra. As a convinced supporter of the cause of Britain and France, he struggled vigorously to carry Italy into the war in May 1915 and then developed a frenetic activity to support the war effort, going himself to the front, even though he was 55. This activity found an adequate echo with his French colleagues Borel, Hadamard and Picard. The huge correspondence they exchanged during the war, gives an extraordinary view of these activities, and raises numerous fundamental questions about the role of a scientist, and particularly a mathematician during WW I. It also offers a vivid documentation about the intellectual life of the time; Volterra's and Borel's circles in particular were extremely wide and the range of their interests was not limited to their field of specialization. The book proposes the complete transcription of the aforementioned correspondence, annotated with numerous footnotes to give details on the contents. It also offers a general historical introduction to the context of the letters and several complements on themes related to the academic exchanges between France and Italy during the war.

The Twenty-First International Congress of Mathematicians (ICM) was held in Kyoto, Japan, from August 21 through 29, 1990, the first congress that has taken place in the Eastern hemisphere. On this occasion, Japanese historians of mathe- matics organized the History of Mathematics Symposium which was held at the Sanjo Conference Hall of the University of Tokyo on August 31 and September 1, as one of the related conferences of the Congress. The symposium was officially sponsored by the Executive Committee of the ICM 90, the History of Science So- ciety of Japan, and the International Commission on the History of Mathematics. The Executive Committee consisted of Murata Tamotsu (Chairperson, Momoyama Gakuin University), Sugiura Mitsuo (Vice-Chairperson, Tsuda College), Sasaki Chikara (Secretary, The University of Tokyo), Adachi Norio (Waseda University), Nagaoka Ryosuke (Tsuda College until 1990, now Daito Bunka University), and Hirano Yoichi (Treasurer, Tokai University). The symposium emphasized the following three fields of study: (1) mathe- matical traditions in the East, (2) the history of modern European mathematics, and (3) interaction between mathematical research and the history of mathemat- ics. These fields were chosen mainly because, first, the symposium was related to the ICM, the most important congress of working mathematicians, and, second, the Kyoto ICM was held in a non-Western country for the first time. The sym- posium consisted of the two Sessions: Session A for invited speakers and Session B for short communications.

Viewed as a flashpoint of the Scientific Revolution, early modern astronomy witnessed a virtual explosion of ideas about the nature and structure of the world. This study explores these theories in a variety of intellectual settings, challenging our view of modern science as a straightforward successor to Aristotelian natural philosophy. It shows how astronomers dealt with celestial novelties by deploying old ideas in new ways and identifying more subtle notions of cosmic rationality. Beginning with the celestial spheres of Peurbach and ending with the evolutionary implications of the new star Mira Ceti, it surveys a pivotal phase in our understanding of the universe as a place of constant change that confirmed deeper patterns of cosmic order and stability.

In the 18th century, purely scientific interests as well as the practical necessities of navigation motivated the development of new theories and techniques to accurately describe celestial and lunar motion. "Between Theory and Observations" presents a detailed and accurate account, not to be found elsewhere in the literature, of Tobias Mayer's important contributions to the study of lunar motion-including the creation of his famous set of lunar tables, which were the most accurate of their time.

Beginning in 1983, the Swedish Council for Planning and Coordination of Research has organized an annual workshop devoted to some aspect of the behavior and modeling of complex systems. These workshops have been held at the Abisko Research Station of the Swedish Academy of Sciences, a remote location far above the Arctic Circle in northern Sweden. During the period of the midnight sun, from May 4-8, 1987 this exotic venue served as the gathering place for a small group of scientists, scholars, and other connoisseurs of the unknown to ponder the problem of how to model "living systems," a term singling out those systems whose principal components are living agents. The 1987 Abisko Workshop focused primarily upon the general system-theoretic concepts of process, function, and form. In particular, a main theme of the Workshop was to examine how these concepts are actually realized in biological, economic, and linguistic situations. As the Workshop unfolded, it became increasingly evident that the central concern of the participants was directed to the matter of how those quintessential aspects of living systems-metabolism, self-repair, and replication-might be brought into contact with the long-established modeling paradigms employed in physics, chemistry, and engineering.

This publication was made possible through a bequest from my beloved late wife. United together in this present collection are those works by the author which have not previously appeared in book form. The following are excepted: Vorlesungen uber Differential und Integralrechnung (Lectures on Differential and Integral Calculus) Vols 1-3, Birkhauser Verlag, Basel (1965-1968); Aufgabensammlung zur Infinitesimalrechnung (Exercises in Infinitesimal Calculus) Vols I, 2a, 2b, and 3, Birkhauser Verlag, Basel (1967-1977); two issues from Memorial des Sciences on Conformal Mapping (written together with C. Gattegno), Gauthier-Villars, Paris (1949); Solution of Equations in Euclidean and Banach Spaces, Academic Press, New York (1973); and Stu- dien uber den Schottkyschen Satz (Studies on Schottky's Theorem), Wepf & Co., Basel (1931). Where corrections have had to be implemented in the text of certain papers, references to these are made at the conclusion of each paper. In the few instances where this system does not, for technical reasons, seem appropriate, an asterisk in the page margin indicates wherever a correction is necessary and this is then given at the end of the paper. (There is one exception: the correc- tions to the paper on page 561 are presented on page 722. The works are published in 6 volumes and are arranged under 16 topic headings. Within each heading, the papers are ordered chronologically according to the date of original publication.

To attempt to compile a relatively complete bibliography of the theory of functions of a real variable with the requisite bibliographical data, to enumer ate the names of the mathematicians who have studied this subject, exhibit their fundamental results, and also include the most essential biographical data about them, to conduct an inventory of the concepts and methods that have been and continue to be applied in the theory of functions of a real variable ... in short, to carry out anyone of these projects with appropriate completeness would require a separate book involving a corresponding amount of work. For that reason the word essays occurs in the title of the present work, allowing some freedom in the selection of material. In justification of this selection, it is reasonable to try to characterize to some degree the subject to whose history these essays are devoted. The truth of the matter is that this is a hopeless enterprise if one requires such a characterization to be exhaustively complete and concise. No living subject can be given a final definition without provoking some objections, usually serious ones. But if we make no such claims, a characterization is possible; and if the first essay of the present book appears unconvincing to anyone, the reason is the personal fault of the author, and not the objective necessity of the attempt."

Diese Arbeit enthiilt zwei grof3ere Fallstudien zur Beziehung zwischen theo- retischer Mathematik und Anwendungen im 19. Jahrhundert. Sie ist das Ergebnis eines mathematikhistorischen Forschungsprojekts am Mathemati- schen Fachbereich der Universitiit-Gesamthochschule Wuppertal und wurde dort als Habilitationsschrift vorgelegt. Ohne das wohlwollende Interesse von Herrn H. Scheid und den Kollegen der Abteilung fUr Didaktik der Mathema- tik ware das nicht moglich gewesen: Inhaltlich verdankt sie - direkt oder indirekt - vielen Beteiligten et- was. So wurde mein Interesse an den kristallographischen Symmetriekon- zepten, dem Thema der ersten Fallstudie, durch Anregungen und Hinweise von Herrn E. Brieskorn geweckt. Sowohl von seiner Seite als auch von Herrn J. J. Burckhardt stammen uberdies viele wert volle Hinweise zum Manuskript von Kapitel I. Herrn C. J. Scriba mochte ich fur seine die gesamte Arbeit betreffenden priizisen Anmerkungen danken und Herrn W. Borho ebenso fUr seine ubergreifenden Kommentare und Vorschlage. Beziiglich der in Kapitel II behandelten projektiven Methoden in der Baustatik des 19. Jahrhunderts gilt mein besonderer Dank den Herren K. -E. Kurrer und T. Hiinseroth fUr ihre zum Teil sehr detaillierten Anmerkungen aus dem Blickwinkel der Geschichte der Bauwissenschaften. Schliefilich geht mein Dank an alle nicht namentlich Erwiihnten, die in Gesprachen, technisch oder auch anderweitig zur Fertig- stellung dieser Arbeit beigetragen haben. Fur die vorliegende Publikation habe ich einen Anhang mit einer Skizze von in unserem Zusammenhang besonders wichtig erscheinenden Aspekten der Theorie der kristallographischen Raumgruppen hinzugefUgt. Ich hoffe, daB er zum Verstiindnis des mathematischen Hintergrunds der historischen Arbeiten des ersten Kapitels beitragt.