## Reviews of Hans Wussing's books

**Die Genesis des abstrakten Gruppenbegriffes (1969), by Hans Wussing.****1.1. Review by: Thomas Hawkins.**

*Isis***61**(3) (1970), 418-419.Although the concept of an abstract group was introduced by Arthur Cayley in 1854, it was not until the last two decades of the nineteenth century that a number of mathematicians began to develop group theory from the abstract viewpoint - the viewpoint that has become the hallmark of mathematics in the twentieth century. During, roughly, the period 1850-1880 a fuller appreciation of the significance of groups had arisen: the group-theoretic foundation of Galois' work on the theory of equations came to be more fully understood, and group-theoretic ideas were found to be relevant in many other areas of mathematics. Hans Wussing's interesting and well-documented book is largely devoted to tracing the growth of this fuller appreciation and the manner in which it led to the adoption of the abstract viewpoint in group theory. ... the book represents an important and much-needed contribution to the history of mathematics since 1800. It is highly recommended to anyone interested in the historical background of modern mathematics.

**1.2. Review by: Bernhard Neumann.**This is not, of course, a history of the development of group theory, though much of this development, approximately to the end of the 19th century, can be found in the book. The author has set out to trace the process of abstraction that led finally to the axiomatic formulation of the abstract notion of group. His main thesis, ably defended and well documented, is that the roots of the abstract notion of group do not lie, as frequently assumed, only in the theory of algebraic equations, but that they are also to be found in the geometry and the theory of numbers of the end of the 18th and the first half of the 19th centuries.

**The genesis of the abstract group concept. A contribution to the history of the origin of abstract group theory (1984), by Hans Wussing.****2.1. Review by: Jeremy J Gray.**

*British Journal for the History of Science***18**(3) (1985), 360.This is an English translation of

*Die Genesis des abstrakten Gruppenbegriffes*(Berlin 1969). In the intervening 15 years the book has become one of the most widely read works on the history of mathematics, and its appearance in English is most welcome. It should certainly be read by all mathematicians interested in the history of their subject, as well as by the historians of science curious about the rise of abstract mathematics. Wussing traces the origins of group theory to three subject-areas: number theory, geometry, and the theory of equations. His discussions here are extremely thorough (the book has a bibliography of 747 items) and generally persuasive. He pursues his subject through a phase of permutation-theoretic ideas (when the theory of equations was central) to its emergence as an abstract theory (under the impetus of geometrical ideas).**2.2. Review by: Karen Hunger Parshall.**

*Amer. Math. Monthly***93**(10) (1986), 823-826.In his book, 'The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory', originally appearing in German in 1969 and now appearing for the first time in English translation, Hans Wussing traced the evolution of the group concept from its disparate sources in the eighteenth and early nineteenth centuries through its full-blown emergence at the beginning of the twentieth century. ... The group concept grew not from one but from three distinct yet intertwined roots: the traditionally recognized theory of algebraic equations, number theory, and geometry. Further- more, each of these roots shared equally in feeding the growing idea. As Wussing explains in his introduction: "The existence of two additional roots of abstract group theory has been obscured mainly by the fact that the group-theoretic modes of thought in number theory and geometry remained implicit until the end of the middle third of the nineteenth century; they made no use of the term 'group' and, in the beginning, had virtually no link to the contemporary development of the theory of permutation groups". Wussing's self-proclaimed task, then, was to locate ... . those paths of development of implicit group theory that have made a causal contribution to the rise of explicit group theory". This may sound like so much double-talk, but it really is not. Wussing realized that tracing the development of the group concept involves more than searching the literature for occurrences of the word "group". Patterns of thought and combinations of properties cropped up over and over again long before they became universally distinguished by a special nomenclature. In his study Wussing pursued these much more elusive phenomena and provides convincing accounts of their roles in the development of the group concept.

**Nicolaus Copernicus (1973), by Hans Wussing.****3.1. Review by: Mathematical Reviews.**One of the most useful features of this book is the author's (handsomely illustrated) description of the social, political, economic and scientific context in which the life and work of Copernicus must be set. Another is his description of the influence of Copernicus and of the work that has been done over the centuries in elucidating and assessing that influence. Thus, he has given a truly historical account, without, however, losing sight of the mathematical and astronomical details.

**Isaac Newton (1977), by Hans Wussing.****4.1. Review by: J S Joel.**This brief biography of Newton concentrates mostly on his works, which are viewed synchronously (but not synchronically), as a product of their times. The purely biographical material is only 40 pages, and the description of his life and works is put into a general diachronic perspective. There are numerous quotations from works by and about Newton, to flesh out aspects of his personality that are somewhat hidden in the purely scientific works.

**Vorlesungen zur Geschichte der Mathematik (1979), by Hans Wussing (with the collaboration of S Brentjes, H-J Ilgauds, K-H Schlote, P Schreiber, R Siegmund-Schultze, R Tobies and J Wilke).****5.1. Review by: Dirk J Struik.**This book arose from lectures given since 1960 at the University of Leipzig, first to student-teachers of mathematics and physics, and after 1978 to all students of mathematics. The presentation and selection of material is in accordance with directives issued by the government authorities of the German Democratic Republic on the study of the history of science in the training of mathematicians. It is a book written by a man who has tested the exposition of his rich knowledge in years of lecturing and discussions with colleagues and students. The result is one of our best classroom textbooks on the history of mathematics, as well as a book for general orientation.

**Geschichte der Naturwissenschaften. (1983), Hans Wussing (Editor).****6.1. Review by: L Pearce Williams.**

*Isis***75**(2) (1984), 384-385.The subject of this large book is itself very large. The volume is nothing less than a Weltgeschichte of science. Moreover, the editor has defined science to include applied science and technology. Thus, the contributions range in space from Pre-Columbian America to China and in time from primitive man to the mid-twentieth century. Physics, chemistry, biology, medicine, geology, mathematics, and technology are treated within the context of a somewhat mushy Marxist view. The results are necessarily superficial, and it is difficult to know who, outside of German- speaking countries, will wish to own this work. For Germans, it does provide a rather nice coffee-table piece that offers a rapid course in the history of science on a fairly low level. The virtues of this volume are simple. It does give a rapid and generally accurate survey of the history of science. ... What is wrong with the book [is] obvious. There is no room for analysis of either concepts or discoveries.

**Adam Ries (1989), by Hans Wussing.****7.1. Review by: Dirk J Struik.**The author gives us, in this small and very readable book, a fair account of the development of practical arithmetic and of algebra up to the times of Ries, together with a detailed report on the type of problems Ries tackled, and this in Ries's original language (with modern commentary) with illustrations from the texts.

**Wissenschaftsgeschichte en miniature: Neun Kapitel aus der Entwicklung der Mathematik und der Naturwissenschaften (1989), by Hans Wussing and Horst Remane.****8.1. Review by: Gregg De Young.**

*Isis***82**(2) (1991), 345.This book attempts to explain the historical and scientific significance of some personages, scientific and technological devices, and scientific or technological symbols appearing on postage stamps of the world. Through these philatelic images, the authors hope to stimulate interest in the historical development of science and to illustrate that development. ... In short, this book, although a fascinating evening's browsing, will be of use primarily to philatelists wishing to learn more about the personages and symbols found on postage stamps.

**Vom Zdhlstein zum Computer: Mathematik in der Geschichte. Volume 1: Uberblick und Biographien (1997), by Hans Wussing et al.****9.1. From the Preface.**Why study the history of mathematics when mathematics is such a dreaded subject for many students and most people only shudder when recalling classes and tests in mathematics? The mathematical historian Wussing gives a detailed answer to this question in the first chapter of this introduction: The occupation with the history of mathematics is an intellectual adventure, in which one can experience the suspense of how mathematics has evolved, how much hardship and error it took people from the first beginnings in the dim and distant past to erect - over the millennia - the magnificent mental structure whose contents and methods have become the foundation and indispensable apparatus for the development of all technology, the sciences, medicine, business and industry, and which, in the form of the computer, have practically embraced every aspect of modern life.

**9.2. Review by: Catherine Goldstein.**

*Isis***89**(4) (1998), 704-705.This textbook is the first volume of a projected series intended as a basic introduction to the history of mathematics and designed for independent study, in particular for students and high school teachers. ... The authors' goal is to present mathematics as a human adventure, tied to many aspects of cultural and social life. ... it is extremely rare to find a textbook that sheds light on the work of the historian, behind the narrative presentation of the mathematics. This emphasis, which is everywhere visible here, seems very useful, and I would recommend that it be adopted in other texts to help foster a more complete understanding of the history of science. ... the lively, clear, and simple style nicely conveys its main message: that mathematics is a human pursuit whose aims and motivations can be understood by everyone.

JOC/EFR October 2013

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