1.1. Review by: H E Vaughan.
The Journal of Symbolic Logic 5 (4) (1940), 155-157.

In the first six chapters the theory of partially ordered systems is developed along with its applications to algebra and geometry. Chapters VII, VIII, and IX are devoted to applications to function theory, logic, and probability, respectively. There is a list of unsolved problems, a short bibliography, and a subject as well as an author index.

1.2. Review by: Orrin Fink Jr.
Mathematical Reviews MR0001959 (1,325f).

This is the first comprehensive treatment of lattice theory and its applications. The chapter headings are: I. Partially Ordered Systems; II. Lattices; III. Modular Lattices; IV. Complemented Modular Lattices; V. Distributive Lattices; VI. Boolean Algebras; VII. Applications to Function Theory; VIII. Applications to Logic; and IX. Applications to Probability. Each successive chapter deals with a more special type of lattice, with examples and applications of each type. Lattice theory is thus made to serve as a unifying principle like group theory, emphasising similarities between different mathematical fields.

There is a foreword which reviews briefly the notions of metric space, $L$-space and topological space, and such concepts of universal algebra as homomorphism, congruence relations and direct product. In the first chapter are introduced the concepts of chain, duality, the maximal and minimal elements I and O, points or atoms, covering and Hasse diagrams. The dimension of a partially ordered system $P$ is defined to be the maximal length of a chain in $P$. The ascending and descending chain conditions are discussed in connection with well ordering. Sum, product and power of partially ordered systems are defined. In the second chapter the fundamental lattice operations of meet $x \bigcap y$ and join $x \bigcup y$ are defined in terms of greatest lower bound and least upper bound, and they are shown to be characterised by four algebraic laws. Examples of complete lattices are given. A theorem on representation by sets, and MacNeille's theorem on completion by cuts are proved.
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1.3. Review by: Adrian Albert.
Science 92 (2400) (1940), 606.

The past decade has seen a tremendous and continuing growth of interest in modern abstract algebra. Algebraic concepts and technique have been found to be frequently applicable to other branches of mathematics as well as other sciences, and their fundamental value has been emphasised thereby.

A major portion of algebraic study has been concerned with mathematical systems consisting of a set of elements which behave with respect to two operations very much as ordinary numbers do with respect to addition and multiplication. In studying subsystems of the same type as a given system one usually considers the meet and join of any two of them, and finds that these are again subsystems of the same abstract type. Thus one may again abstract and study a new type of mathematical system consisting of two operations and a set of elements whose behaviour with respect to the operations is now like those of subsystems with respect to meet and join. The application of algebraic research technique to this most recent phase of abstract algebra is a study which has strongly attracted many American algebraists and one of the most prolific workers in the field is Garrett Birkhoff, who has called the subject Lattice Theory.

Birkhoff's book is the first comprehensive treatment of the subject and its applications. It contains all the recent major developments in the subject in a unified form which will make the book an inspiring research reference for the relatively large number of research algebraists interested in the field. The exposition is clear and well written and should prove of great value in satisfying the demand of non-specialists in the subject -who have been anxious for several years to obtain a text by the use of which it may be possible to present the subject as a graduate course in modern mathematics.

1.4. Review by: L R Wilcox.
Bulletin of the American Mathematical Society 47 (3) (1941), 194-196.

This is the first book on the far-reaching subject of lattices. The author has succeeded in giving a comprehensive, yet not too terse, account of the theory of lattices and its relation to other branches of mathematics. The general plan is to devote six chapters to the abstract theory, and three to applications. Chapter I deals with partially ordered systems, Chapter II with lattices and their general properties; the important modular axiom is assumed in Chapter III, and the axiom of complementation is added in Chapter IV. Distributive lattices and Boolean algebras end the abstract theory in Chapters V and VI. Thus the author takes the reader through the most important general classes of lattices, by imposing successively restrictive conditions. The last three chapters apply lattice theory to function theory, logic and probability theory.
...
Perhaps the most striking features of the book are the homogeneity of presentation and the thoroughness with which so large a subject is covered in so small a space. At the end the author includes a list of seventeen unsolved problems, which should prove provocative to the reader. The reviewer was especially pleased to find the first lattice-theoretic treatment ever given of Moore's extensional attainability, a simple but extremely useful tool. The section on polarity in Chapter II is also worthy of notice because of its wide applicability.

2.1. From the Preface.

The most striking characteristic of modern algebra is the deduction of the theoretical properties of such formal systems as groups, rings, fields, and vector spaces. In writing the present text we have endeavoured to set forth this formal or "abstract" approach, but we have been guided by a much broader interpretation of the significance of modern algebra. Much of this significance, it seems to us, lies in the imaginative appeal of the subject. Accordingly, we have tried throughout to express the conceptual background of the various definitions used. We have done this by illustrating each new term by as many familiar examples as possible. This seems especially important in an elementary text because it serves to emphasise the fact that the abstract concepts all arise from the analysis of concrete situations. To develop the student's power to think for himself in terms of the new concepts, we have included a wide variety of exercises on each topic. Some of these exercises are computational, some explore further examples of the new concepts, and others give additional theoretical developments. Exercises of the latter type serve the important function of familiarising the student with the construction of a formal proof. The selection of exercises is sufficient to allow an instructor to adapt the text to students of quite varied degrees of maturity, of undergraduate or first year graduate level. Modern algebra also enables one to reinterpret the results of classical algebra, giving them far greater unity and generality. Therefore, instead of omitting these results, we have attempted to incorporate them systematically within the framework of the ideas of modern algebra. We have also tried not to lose sight of the fact that, for many students, the value of algebra lies in its applications to other fields: higher analysis, geometry, physics, and philosophy. This has influenced us in our emphasis on the real and complex fields, on groups of transformations as contrasted with abstract groups, on symmetric matrices and reduction to diagonal form, on the classification of quadratic forms under the orthogonal and Euclidean groups, and finally, in the inclusion of Boolean algebra, lattice theory, and transfinite numbers, all of which are important in mathematical logic and in the modern theory of real functions.

2.2. Review by: Nathan Jacobson.
Mathematical Reviews MR0005093 (3,99h).

This is a text on modern algebra that is particularly suited for a first year graduate course or for an advanced undergraduate course. A very striking feature of the book is its broad point of view. There are contacts with many branches of mathematics and so it can serve as an introduction to nearly the whole of modern mathematics. Thus there is a careful development of real numbers, such as Dedekind cuts, and such set-theoretic concepts as order, countability and cardinal number are discussed. Throughout the study of matrices and quadratic forms the geometric point of view is emphasised. There is also contact with the field of mathematical logic in the chapter on the algebra of classes and with the ideas of topology in the proof of the fundamental theorem of algebra.

The following is the table of contents by chapters: I. The integers. II. Rational numbers and fields. III. Real numbers. IV. Polynomials. V. Complex numbers. VI. Group theory. VII. Vectors and vector spaces. VIII. The algebra of matrices. IX. Linear groups. X. Rank and determinants. XI. Algebra of classes. XII. Transfinite arithmetic. XIII. Rings and ideals. XIV. Algebraic number fields. XV. Galois theory. As can be judged from this outline, concrete systems with which the student is familiar are studied first and then their properties are formulated as axioms leading thereby to the definition of important abstract algebraic systems. This has been done with great skill in the text and moreover the large number of excellent exercises should enable the student to keep a firm hold on the theory. There is a wide gap between the first chapter dealing with elementary properties of integers and the last chapter proving the insolvability of quintic equations by radicals. One is confident, however, that a student who has made the journey over the intermediate chapters will have achieved the maturity needed to follow the intricacies of the Galois theory.

The most serious criticism that the reviewer has to make is a rather minor one: The theory of determinants is developed only for matrices with entries in a field. Because of the applications we believe it would have been worthwhile to treat the general case in which the entries belong to any commutative ring. The book is very clearly written and seems to be free from error.

2.3. Review by: L W Griffiths.
National Mathematics Magazine 16 (5) (1942), 268-269.

This exposition of the elements of modem algebra has been planned with great skill, and the plan has been carried through very successfully. It is a unified and comprehensive introduction to modem algebra. The classical algebra is nicely embedded in this structure, as are also applications to other fields of thought.

This book is distinguished by its procedure from the concrete to the abstract. Familiar examples are carefully presented to illustrate each new term or idea which is introduced. Then the abstract definition appears simple, and the theoretical properties which are deduced from the definition exhibit the power of the concept.

This book is distinguished also by the great clarity with which all details have been presented. Statements of theorems and definitions are precisely phrased. Proofs are very carefully organised. Illustrations are penetratingly exhibited. Also, methods of procedure and general information are discussed as needed, when they are not easily and explicitly available elsewhere. The lists of graded. non-trivial exercises also serve to clarify the text.

The first five chapters give a postulational development of integers and integral domains. of rational numbers and fields. of real and complex numbers and rings. A wealth of material is included, ranging from divisibility of integers and de Moivre's theorem to the Peano postulates and Dedekind cuts.

The next five chapters, on abstract groups, linear spaces, matrices, linear groups. and determinants. constitute a central half of the book. A feature of the excellent chapter on abstract groups is the fact that only rarely is the discussion limited to finite groups. In the chapter on vector spaces, the emphasis on inner products is admirable. A noteworthy feature of the chapter on the algebra of matrices is the discussion of rectangular matrices. The long chapter on linear groups is especially commended. These chapters VI to X could be read independently of the preceding chapters.

A chapter on the algebra of classes, and a chapter on transfinite arithmetic, precede the final three chapters on rings and ideals, algebraic and transcendental extensions of a field and the Galois theory. The section on applications to algebraic geometry enhances the discussion of ideals. The section on adjunction of roots is outstanding in an excellent chapter on extension of fields.

2.4. Review by: Morgan Ward.
Science 95 (2467) (1942), 386-387.

The rejuvenation of algebra by the systematic use of the postulational method and the ideas and point of view of abstract group theory has been one of the crowning achievements of twentieth century mathematics. Although many of the basic results stem back to Kronecker, Dedekind and Steinitz, the present-day subject is largely the creation of the great woman mathematician, Emmy Noether. Modern Algebra, by one of her pupils, B L van der Waerden, will always remain the classical account of the subject as she conceived it.

Although two or three books on the new algebra have already appeared in English, the present volume appears to the reviewer to be the best all-round introduction to the subject, unique in its clarity, balance, generality and inclusiveness. The size and plan of the book preclude a comprehensive treatment of any one topic; in compensation, the authors are able to say something about nearly every important topic, and they usually succeed in saying the really important things. In addition the book is enlivened by striking applications of modern algebra to other branches of science and made eminently teachable by the inclusion of numerous excellent problems and exercises.

The power of the postulational method is emphasised from the onset by developing the properties of the integers, rationals, real and complex numbers along with the elements of ring theory and field theory from well-chosen postulates. There follow chapters on elementary group theory, vector spaces, linear groups, ideal theory, algebraic numbers, Galois theory and other topics. The geometrical treatment of matrices as linear operators over a vector space is a judicious innovation. The authors even find space for the fundamental ideas of lattice theory, a vigorously growing branch of algebra particularly cultivated by American mathematicians.

In conclusion, the book is emphatically recommended either as a text, an introduction to the literature or a bird's-eye view of one of the great branches of modern mathematics.

2.5. Review by: R M Thrall.
Bulletin of the American Mathematical Society 48 (12) (1942), 342-345.

In this book Professors Birkhoff and Mac Lane have made an important contribution to the pedagogy of algebra. Their emphasis is on the methods and spirit of modern algebra rather than on the subject matter for itself. The word "survey" in the title is quite accurate; for, although many topics are treated, none of them is really completely developed. The most important parts of each theory are included and that is all that can be asked of an introductory textbook.

Because of the authors' emphasis on "method" rather than "fact" the book will not be of much use as a reference work. But there is no dearth of good reference works in algebra, and in the reviewer's opinion the present textbook will prove more useful than another encyclopaedic treatise would have been.

A discussion of the topics included will help to indicate the authors' purposes.

The first three chapters of the book are ostensibly devoted to the development of number systems: starting with postulates for the integers, then defining rational numbers in terms of the integers, and next giving an outline of the Dedekind construction of real numbers. Actually much more is happening. Such fundamental concepts as congruence, residue class, isomorphism, and ordered and well ordered sets are introduced and applied in a natural manner to the theory. Also the generalisations from integers to integral domain, and from rational number to field are made at suitable stages of the development. This procedure of starting with properties of a familiar mathematical system and generalising to an abstract system is typical of the spirit of the book and is used in the development of almost every topic treated.
...
The authors' preface includes a summary of the purposes and contents of each chapter, and also indicates several levels of courses in which the book is designed to be used as a text. The most elementary of these is a one year undergraduate course based on only high school algebra. Provision is made for a semester course covering the tools used in physical applications. Finally, a semester course on abstract algebra can be carved out by selecting suitable chapters. The reviewer recommends the use of the book as a text in any of these ways.

3.1. From the Publisher.

Like its elder sister group theory, lattice theory is a fruitful source of abstract concepts, common to traditionally unrelated branches of mathematics. Both subjects are based on postulates of an extremely simple and general nature. Lattice theory has become a recognised branch of modern algebra, with a steady flow of contributions to it from numerous mathematicians. This revised edition takes these contributions into account.

3.2. Review by: Orrin Fink Jr.
Mathematical Reviews MR0029876 (10,673a).

This second edition of Lattice Theory is more than just a revision of the first edition of 1940. It has doubled in size. Eight of the sixteen chapters are new, and the other chapters have all been rewritten and expanded. This increase in size reflects the rapid growth of the subject during the intervening years. Titles of the new chapters are: Chains and chain conditions, Complete lattices, Applications to algebra, Semi-modular lattices, Applications to set theory, Lattice-ordered semigroups, Lattice-ordered groups, and Ergodic theory. A prominent new feature consists of hundreds of exercises distributed throughout the book, many of them containing new results both of the author and of others. Along with the exercises 111 unsolved problems are proposed, replacing the 17 problems of the original edition, 8 of which have since been solved.

There are two forewords, on algebra and topology. The first chapter, on partially ordered sets (or partly ordered sets, as they are called here) now treats the extended cardinal and ordinal arithmetic due to the author and studied also by M M Day. The second chapter on lattices includes results of the author on neutral elements and of Whitman on free lattices. Chapter III on chain conditions has a new treatment of Zorn's lemma, well-ordering, and the axiom of choice. The fourth chapter on complete lattices contains new material on polarities and Galois connections, as well as on representations and intrinsic topologies of lattices and fixed point theorems.

The sixth chapter on applications to algebra is new. It deals with quasi-groups and loops, new extensions of the Jordan-Hölder and Kurosh-Ore theorems, and the classification of groups by their subgroup lattices. A new formulation, due to the author, of the theorem of Ore is proved: in a modular lattice of finite length, the elements of any two representations of the unit element as a direct join of indecomposable elements are projective in pairs. Three theorems on algebras with commutative congruence relations, announced at the Princeton Bicentennial Conference in 1946, are published here for the first time. The author's theorem on subdirect unions of subdirectly irreducible algebras is also proved.
...
The bibliography has been enlarged, but still contains only the more important references. Hundreds of other references are given in footnotes. The subject index and author index have also been greatly expanded. The book as a whole is a useful, complete and up-to-date account of a new and active branch of abstract algebra.

3.3. Review by: Alonzo Church.
The Journal of Symbolic Logic 15 (1) (1950), 59-60.

Though not in the field of mathematical logic (the author does not undertake to use the logistic method, even in the chapter entitled Applications to logic and probability), this book has many points of contact, and as a standard reference work on lattice theory and Boolean algebra it will be often consulted by the logician. The first edition has been thoroughly revised, and nearly doubled in size. The added material is concerned generally with discoveries in lattice theory made during the last decade, including some previously unpublished results of the author, but in this review we confine attention to parts of the book which bear most directly on logic.
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In the section on the algebra of attributes the statement is made that 'or' and 'not' can be defined in terms of 'and' alone, and in an exercise the task is set to carry this out. Strictly taken, the task is demonstrably impossible. No doubt it is intended to allow besides 'and' also the use of the equality sign and of quantifiers; but it would have been desirable to give at least some indication of this important difference between the sense in which all the Boolean operations can be defined in terms of 'and' and that in which they can all be defined in terms of Sheffer's stroke.

In the reviewer's opinion, discussion of the logical antinomies and of the incompleteness theorem of Gödel has no place in a treatment otherwise devoted exclusively to such elementary branches of logic as propositional calculus and algebra of classes, and might better have been omitted. The brief account of the antinomies is very misleading as to the magnitude of this matter in its relation to the foundations of mathematics. There is also an error in making Burali-Forti's antinomy depend on the axiom of choice. On pages 194-195 the Gödel incompleteness theorem is misstated in a way that reduces it to a triviality. And the proof of the theorem is then erroneously said to be non-constructive, and to depend on admitting uncountable many propositions but only countably many proofs.

3.4. Review by: J L B Cooper.
The Mathematical Gazette 34 (309) (1950), 235-237.

A characteristic feature of modern mathematics is the tendency to study simple structures which occur, combined with structures of different sorts, in diverse branches of mathematics. The study of these simple structures, defined by suitable sets of axioms, tends to unify various mathematical theories. Best known of such structures are groups: abstract algebra and topology have developed many others. The book under review studies one of these abstract algebraic structures, lattices, and shows their power in many fields of thought.

Lattices are partially ordered sets such that every two elements have both a least upper bound and a greatest lower bound. They can also be described as algebraic systems with two operations - formation of greatest lower and least upper bound - obeying certain rules. In a particular case these rules go back to Boole, for his algebra of logic is the earliest study of a lattice, the lattice of subsets of a set, partially ordered by inclusion: here the two operations are set intersection and union. More general lattices were studied by Dedekind in his ideal theory: the integers ordered by divisibility are an example of his type. The general theory of lattices was developed mainly in the last thirty years, under various names - Verbände, structures, etc.

Garrett Birkhoff has been one of the main contributors to the subject. This is the second edition of his book, and it has been greatly expanded, and almost completely rewritten. It gives an account of the great development of the subject since 1940, much of it the work of the author and his school. In addition the presentation has been expanded and improved, so that the book is now much more self-contained and readable. A useful addition for the reader is the inclusion of a large number of interesting examples to be worked as exercises, in addition to much illustrative matter in the text. There are also given many unsolved problems. The book ends with a bibliography of the more important works on the subject and an index; copious references are also given in the text.

The reader of the book will require to know the elementary ideas of abstract algebra as well as whatever specialised knowledge is needed to understand the various applications. A foreword gives some of the less well-known algebraic concepts which are used, and another gives very briefly the fundamental ideas of topology in terms of the closure axioms.

The first chapter discusses partially ordered sets, giving, inter alia, an interesting account of a theory of cardinal and ordinal arithmetic developed by the author and his associates. Chapter II defines lattices and begins their algebraic theory. Chapter III deals with chains - totally ordered sets. It contains excellent accounts of the theory of well-ordered sets, of transfinite induction, of the classical arithmetic of totally ordered sets, and of the connections between the axiom of choice, the well-ordering theorem and Zorn's lemma; as far as I know this is the only place in the literature where this last subject is written out explicitly, though it is often taken for granted.

The following chapters are concerned with a series of types of lattices subjected to progressively more restrictive postulates: the modular law, the distributive law, the existence of complements, etc., culminating in the Boolean lattices. Each chapter gives an account of the logical connections between the postulates, and of the applications of the type of lattice dealt with. Most important of these are the applications to algebra, to which are given a separate chapter in addition to the sections in other chapters. This chapter develops a theory of the congruence relations on abstract algebras of a very general type, and discusses ideals by means of the congruences which define them. These congruences form a modular lattice, and the Jordan-Holder theorem is proved in a widely generalised form as a theorem on modular lattices. The theorem of Kurosh and Ore are next discussed. An important idea introduced by the author is that of a subdirect union of algebras: it is evidently a powerful tool, judging from its applications to the representation theory of distributive lattices and Boolean algebras in later chapters. Other sections deal with projective and affine geometries which, considered as they here are almost entirely in terms of the incidence axioms, are characterised by the modular lattices of their linear subspaces. Generalisations give infinite dimensional geometries, studied by the author and his school. There is also a brief account of the very interesting continuous geometries defined by von Neumann, for which the dimensions correspond to all numbers between 0 and 1.

The last six chapters deal with applications of the theory. Chapter XI deals with applications to set theory, giving an account of Stone's applications of Boolean algebra to topology, and of work of Wallman and Kaplansky which show that a compact topological space is characterised by the lattice of its closed sets, on the one hand, and by the lattice of its continuous functions on the other. An account of Carathéodory's measure theory on Boolean algebras follows. The next chapter deals with applications to logic and probability, rather discursively. There follow chapters on lattice ordered semigroups, which are important in ideal theory, lattice ordered groups, and vector lattices. The latter subject is one which is in rapid growth and important in analysis; it has been found fruitful to supplement the abstract concepts of linear space theory by taking into account the partial orderings which exist in all concrete spaces. A useful account of the algebra and topology of these spaces is given.

The final chapter deals with ergodic theory. This is a theory which slipped its moorings some eighteen years ago, and now keeps up only distant contact with its base in Statistical Mechanics. The chapter gives an account of the abstract theory, with indications of the connection with the older theory and with probability.

In the main, the style of the book is clear and brisk, but on occasion it tends to undue compression, particularly in the illustrative examples. Occasionally this involves actual omissions ...

The study of the simple abstract structures is certain to become an important part of mathematical education. It is illuminating to see which of the simple structures is involved in a given piece of reasoning about a composite structure. It is important, for example, to realise that the Cantor definition of the real numbers depends on the topological structure of the rationals, while the Dedekind definition depends on their order structure as an incomplete lattice; or to understand that the thermometric temperature depends only on the order structure of the real numbers, while the absolute temperature depends on the algebraic structure as well.

This book is the best introduction in our language to this way of thought. For this reason, and also for the width and wealth of mathematical culture it displays, the insights it gives into a variety of subjects, it will be of interest to all mathematicians. To students of abstract algebra it will be invaluable.

4.1. Review by: Anon.
The Military Engineer 47 (320) (1955), 498.

This book is devoted to the complicated logical relation between theory and experiment and to applications of symmetry concepts in fluid mechanics.

4.2. Review by: J Kravtchenko and Robert Gerber.
Mathematical Reviews MR0038180 (12,365m).

The modern science of fluids has become so vast that the title of this book does little to inform the reader of the matters covered in this volume. We are forced to analyse the content chapter by chapter.

In chapter I, the author briefly recalls the axioms which serve as the basis for the equations of hydrodynamics: he discusses the validity of these abstract premises by reviewing some of their paradoxical consequences, the study of which constitutes the best "introduction to the art of applying theories." These paradoxes are of several types. Some hold to the very nature of the basic axioms: the mathematician, for banal reasons of convenience, attributes to the bodies he studies properties of regularity that real bodies do not always possess. But it can happen, and it is here that the author's critical examination becomes truly suggestive thanks to the variety and, often, the originality of the facts invoked in support of his theses, that the singular consequences of calculations arise from the deep analytical properties of mathematical physics equations. We know, for example, that the valid formulas for the flows of viscous fluids are not reduced, when we make the coefficient of viscosity tend towards zero, to those which govern the movements of perfect fluids. On the other hand, the very structure of the hydrodynamic equations is such that the apparent symmetry of the causes is not necessarily found in the effects [cf. the well-known phenomenon of alternating Kármán vortices]. The author emphasises on this occasion the physical instability of certain symmetrical regimes theoretically acceptable for appropriate values of the Reynolds number. It is natural to reserve extensive developments for the so-called paradoxes of resistance. Here the author uses the notion which seems fruitful, of reversible theory; this process allows him to analyse, to a certain extent, the nature of the difficulties that the theory encounters; but there remain many enigmas which still await their solution. Note that the author discusses several paradoxes discovered recently [von Neumann, Kopal] and which, to our knowledge, have not yet been the subject of an overall didactic presentation.
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The brief analysis which precedes will have convinced the reader of the interest of Birkhoff's work. The subjects covered, both classic and new, are always considered from the perspective of modern mathematics. The very richness of the table of contents too often forces the author to limit himself to brief indications. Also, this small volume does not seem intended for a beginner. But it will contribute to making known some modern and capital theories of theoretical hydrodynamics; it will inspire researchers to fruitful reflections, sometimes objections, without ever leaving the reader indifferent.

4.3. Review by: T A A Broadbent.
The Mathematical Gazette 40 (334) (1956), 302.

Garrett Birkhoff's stimulating book appeared as recently as 1950. The author is concerned with the role of mathematics in bridging the gap between hydrodynamic theory and experiment, exposed by certain notorious paradoxes, and in establishing the foundations of dimensional analysis and of the theory of similitude and models. The volume is an admirable, even provocative, supplement to the classic treatises of hydrodynamic theory.

5.1. Review by: Irvin H Brune.
The Mathematics Teacher 47 (2) (1954), 128.

The present edition represents a refinement of an already highly useful text. The original comprehensive Survey has been reordered somewhat and augmented to the extent of approximately fifty pages. Only the last five chapters remain unchanged.

Instructors who have used the original edition with college classes appreciate its scope. Those desiring a text replete with possibilities for courses tailored to various kinds of students should welcome this new edition.

Teachers of mathematics in secondary schools may want this book in their personal libraries. Probably the best way to appreciate the vitality and growth of mathematics today is to study modern algebra. Nowhere can teachers better catch today's spirit of mathematics. Moreover, many of the examples in this text might help teachers to communicate this spirit to their students.

5.2. Review by: Editors.
Mathematical Reviews MR0054551 (14,939a).

Although some additions and rearrangements have been made for this edition, the content remains essentially the same as the 1941 first edition.

5.3. Review by: Kenneth O May.
Econometrica 22 (3) (1954), 391.

Occasionally a textbook makes possible a real leap forward in ways of learning. In 1941, when the first edition of this book appeared, the curriculum in algebra was the result of a hodge-podge accumulation. A Survey of Modern Algebra made it possible to teach an undergraduate course that reflected the richness, vigour, and unity of the subject as it is growing today. It provided a synthesis formerly obtainable only after much more advanced work. Moreover, it was written in a clear and enthusiastic style that conveyed to the reader an appreciation of the aesthetic character of the subject as well as its rigour and power. This reviewer can testify to its appeal to students. The revised edition differs only in minor rearrangements and additions.

For the social scientist whose mathematical studies have reached through the calculus, this book can confidently be urged as the thing to study next. While it can be used as a reference, it should rather be read through carefully over a period of years-one must think in terms of years if one wishes to absorb fully the material and to do the problems. It would be easy to point out those topics that happen to be most relevant to current preoccupations of economists (linear systems, matrices, forms, etc.), but it would be risky to predict that any particular items would be useless. Each is immediately useful in illuminating the other topics, if in no other way.

One of the best things about this book is the balanced approach to rigor and abstractness in relation to intuitive appreciation and concrete application. The authors are quick to indicate applications and careful to motivate and illustrate abstractions. Rigour is not allowed to become rigour mortis! This should be good medicine for those who have developed a scholastic preference ordering for research in which A is preferred to B if and only if A is more general, abstract, unmotivated, and incomprehensible than B! It is sometimes forgotten that science and mathematics have developed most rapidly when new theories have been developed boldly in close relation to empirical knowledge and practical application. There is no trace of mathematical snobbery in the writing of these authors.

This reviewer would like to see parts of Chapters XI and XII on set theory placed at the beginning of the book. Not only is this material easy, but it could be used for a categorical description of the integers, which are at present treated rather ambiguously as a known (but undefined) set and as axiomatically defined up to an isomorphism. There is also available a Brief Survey of Modern Algebra that differs by the omission of several chapters and sections and is not recommended to readers of Econometrica.

5.4. Review by: L M Graves.
The Scientific Monthly 78 (2) (1954), 118.

This well-known textbook has served, in the last twelve years, to introduce a great many students to the fundamental concepts of modern algebra in an extraordinarily effective way. It does this by discussing examples of mathematical systems or situations already partially familiar to the student, isolating important properties of these as postulates, and deducing some of the consequences of these postulates. These theorems are then applied to some familiar and to some less familiar examples, thus broadening the student's viewpoint without getting him lost in abstractions. The ratio of definitions to theorems and exercises is kept low. Interesting historical references appear in a number of places.

In the revised edition, chapters I to V which treat integers, integral domains, rational numbers, fields, polynomials, real numbers, and complex numbers, have been subjected to minor improvements in the way of additions, omissions, and rearrangements. A section on the Peano postulates for the positive integers has been added to chapter II. The chapter on polynomials has been placed before that on the real numbers, which seems a better arrangement for pedagogical purposes. In addition there have been included a section on the real roots of real polynomial equations and a paragraph on the trigonometric solution of the cubic equation. To chapter V has been added a brief section giving criteria for all roots of a quadratic or a cubic equation to have positive real part.

In chapter VI on group theory, Section 2, which is an introduction to transformation groups, has been considerably expanded, so as to clarify the basic ideas and notations for the student. The authors here succumb to the prevalent disease of using the preposition "onto" as an adjective.

Chapters VII to X, entitled "Vectors and Vector Spaces," "The Algebra of Matrices," "Linear Groups," and "Determinants and Canonical Forms," respectively, have been subjected to a thorough revision and rearrangement, and have been expanded by a total of 30 pages, so that now they afford a more adequate treatment of this aspect of algebra. Matrices and their row equivalence, linear forms and the notion of dual spaces, have been introduced into chapter VII. Chapter IX contains a more complete treatment of the full linear group and some of its subgroups, and of the invariants of linear, bilinear, and quadratic forms under some of these groups. It closes with a new section on projective geometry. The new chapter X is shorter than the old, because several of the topics have been incorporated into the earlier chapters. However, some new sections on invariant subspaces and canonical forms have been added.

The last five chapters, entitled "Algebra of Classes," "Transfinite Arithmetic," "Rings and Ideals," "Algebraic Number Fields," and "Galois Theory," are little changed from the first edition. They constitute good but brief introductions to their respective topics. The bibliography has been enlarged by the inclusion of quite a number of recent books. The index has also been enlarged, and a number of the lists of exercises have been revised.

The authors are to be congratulated on having improved an already excellent text.

6.1. From the Publisher.

Applied Mathematics and Mechanics, Volume 2: Jets, Wakes, and Cavities provides a systematic discussion of jets, wakes, and cavities. This book focuses on the general aspects of ideal fluid theory and examines the engineering applications of fluid dynamics. Organised into 15 chapters, this volume starts with an overview of the different types of jets and explores the atomisation of jets in carburetors in connection with gasoline engine design. This text then emphasises the formal treatment of special flows and examines the flows that are bounded by flat plates and free streamlines. Other chapters consider the flows that are bounded by the cavity behind a symmetric wedge. This book discusses as well the intuitive momentum and similarity considerations. The final chapter deals with several surprising physical complications. Mathematician, physicists, engineers, and readers interested in the fields of applied mathematics, experimental physics, hydraulics, and aeronautics will find this book extremely useful.

6.2. Review by: David Gilbarg.
Mathematical Review MR0088230 (19,486f).

This monograph is the first comprehensive review of a field that has been actively studied by hydrodynamicists and mathematicians for almost a century. It is an attempt to present systematically all sides of the subject and is sufficiently complete, especially with its exhaustive bibliography, to serve as a valuable reference work. However, it is apparent that the authors face a serious expository problem in the unevenness of the material - the level of the theory varies from the exact and mathematically mature (for cavitational flow) to the empirical (for wakes in real fluids) - and as a result they are led to remark, "the presentation retains much of the heterogeneous character of its original sources."

The volume seems to fall naturally into several parts which are distinguished from one another both in subject matter and in the type of theory involved. Chapters 2, 3, and 5 are concerned with exact solutions of plane free boundary problems obtained by the hodograph method. A more complete account is hard to imagine. An innovation here is the systematic study of classes of free surface flows according to their hodographs. Chapters 4, 6, and 7, which will be of most interest to mathematicians, present the very attractive qualitative theory of jet and cavity phenomena and several proofs of existence and uniqueness. The authors contribute, among other things, a new and relatively simple existence proof (based on Leray-Schauder) with a unified approach to several cavity models. The variational method is given in detail except for the essential proof of analyticity of the free streamline. A chapter on effective computation, based largely on the authors' own important work in this area, provides a useful collection of numerical results on plane free boundary problems. Chapters 8, 10, and 11 are devoted to the comparatively limited theoretical results on compressibility and gravity, and on axially symmetric and unsteady flows. The final chapters (12-15) discuss various developments - mostly empirical and semi-empirical - concerning the theory of wakes and related topics.

6.3. Review by: C V R.
Current Science 26 (9) (1957), 271-272.

By a jet is meant a stream of material which travels for many diameters in a nearly constant direction. Water issuing from a nozzle under pressure is the most familiar example, but we can also have gas jets in air and water jets in water, and they are also important. The spectacular recent development of jet-propelled aircraft is familiar to all. Very recently also, meteorologists have become aware of jet streams of air in the stratosphere travelling with a velocity of a few hundred miles per hour.

When an obstacle or barrier is held stationary in a moving stream, the flow usually separates from the obstacle along the so-called separating stream lines. The fluid between these stream lines constitutes the wake. Just behind the obstacle it is relatively peaceful. But far behind it, it consists of a train of eddies.

In the case of high-speed motion through a liquid, the wake becomes gaseous. Such a wake is called a cavity. For example, if a sphere is dropped into water at speeds of 25 ft per second or more, an air-filled cavity is formed. Again, if an obstacle is held in a stream of liquid moving at high speed, a vapour filled cavity is formed.

The phenomena presented by jets, wakes and cavities are of great interest from several different points of view and are of high practical importance. For example, wakes are important because they represent the main source of real fluid resistance or drag; no resistance would normally occur in a non-viscous fluid at subsonic velocities if it were not for flow separation and the attendant wake. The formation of vapour-filled cavity bubbles raises serious problems in marine propellers and in hydraulic turbines. Remarkably enough, the phenomenon had been anticipated in the eighteenth century by Euler.

The fascinating and complex field of fluid dynamics presented by jets, wakes and cavities have interested investigators who approached the subject from different points of view, viz. pure mathematicians, applied mathematicians, hydraulic engineers and finally physicists, and their investigations from these diverse points of view have been fruitful. As a remarkable instance, it may be mentioned that the study of the oeolian tones arising from the motion of air over thin rods gave the first indication that we had to do with aperiodic phenomenon which was later identified with the formation of parallel rows of regularly formed vortices behind the cylinder.

The book under review attempts to bring together into a coordinated whole the works of these various groups of specialists. It is devoted. to the quantitative scientific analysis of jets, wakes and cavities. Wherever possible, an attempt is made to predict their behaviour by solving an appropriate boundary value problem, using the partial differential equations of fluid motion. In the case of liquid-jets in air and of cavity flow, this programme can be carried through successfully, at least in simple cases, if the flow is rapid enough for gravity to be negligible and for viscosity effects to be confined to the boundary layer. In these cases, Euler's partial differential equations for non-viscous flow are approximately applicable. However, the cases of wakes, or gas jets in a gas of nearly equal density and of liquid-jets in a liquid, cannot be treated even approximately in this way, in spite of the enormous mathematical literature suggesting the contrary. Presumably, the flow is determined by the Navier-Stokes equations, but the complexity of the observed experimental phenomena makes the difficulties of rational prediction apparent.

Recognising the situation thus briefly stated, the authors treat first the application of Euler's differential equations to flows with free boundaries. The major portion of the book (Chapters 2 to 11) is devoted to these applications. In the next three chapters (12 to 14) the authors turn their attention to the cases of laminar viscous, periodic and turbulent jets and wakes. Here exact results are exceedingly rare and a quasi-empirical approach has therefore been adopted. Finally, in Chapter 15, they have summarised many important limitations on the deductions made in Chapters 2 to 11 due to neglect of surface tension, dissolved gas and other physical variables. This discussion is almost entirely empirical.

The reviewer feels that the idea of bringing the three subjects together in a single volume is an admirable one. The book draws on the resources of pure and applied mathematics, and of experimental physics, and it sheds light on numerous problems of hydraulics and aeronautics. Hence, as the authors remark in their preface, it would have the greatest interest for readers whose scientific curiosity spans all the fields just mentioned. However, others also would find it a useful and stimulating book to study in connection with their own problems. The book is illustrated. by numerous diagrams and excellent photographs. ... We heartily commend the book to all those interested in or working at these fascinating fields of fluid dynamics.

6.4. Review by: E J Watson.
Science Progress (1933-) 46 (181) (1958), 152.

The phenomena described by the title of this book form an important branch of fluid mechanics. Many of them have great importance in technical applications, and they have been studied extensively by mathematicians, physicists and engineers during the past century. The results of this work have been systematised by the authors of the present book, which is the second volume of a series of monographs on Applied Mathematics and Mechanics.

The first chapter presents a general introduction to the topics of the book, with several results obtained by elementary methods. The remainder falls into two groups. The former, containing ten chapters, deals with flows in which viscosity may be ignored, since its effects are confined to thin boundary layers on the fixed surfaces. A systematic account is given of steady two-dimensional potential flows with free streamlines, and it includes the mathematical problems of existence and uniqueness as well as many special flows of physical interest. This theory is extended to cover axisymmetric flow and the effects of compressibility and gravity. Methods for the computation of numerical results, and the authors' experience of these methods, are described, and the plates show the results graphically. Unsteady flows, such as those produced by the pulsations of a spherical cavity or by the entry of a body into water, are also treated.

The last four chapters are concerned mainly with problems in which viscous effects are important. The subjects of these chapters are, respectively, steady viscous flow, periodic wakes, turbulent flow, and miscellaneous experimental facts. The mathematical analysis is less elaborate than in the first part of the book, but each problem needs special treatment since there is little general theory. The ideas of boundary layer flow, and of similitude, are of importance here.

The book should be useful to many who are interested in fluid mechanics. The authors provide a great number of references in the footnotes and bibliography, and many useful figures and plates. Progress is still active in this field, and the final chapter shows that many phenomena still await detailed explanation.

6.5. Review by: L M Milne-Thomson.
Quarterly of Applied Mathematics 17 (2) (1959), 219-220.

This book is a noteworthy essay in bringing together the known theories and facts concerning the flows described in the title, all of which are characterised by the presence of free streamlines or streamlines of discontinuity. For this, if for no other reason, the authors are to be congratulated in filling a real gap in the existing literature of fluid motion.

To quote from the preface "our book draws on the resources of pure and applied mathematics and on experimental physics, and it sheds light on numerous problems of hydraulics and aeronautics. Therefore, it will perhaps have the greatest interest for readers whose scientific curiosity spans all the fields just mentioned. However, we hope that others will also find it a useful and stimulating reference in connection with many special questions."

Owing to the development of high-speed computing methods which enable many calculations to be performed which hitherto would have taken a prohibitive amount of time, approximations are now the order of the day. It may therefore be permissible to say that the present work falls, using the authors' delightful phrase, "into two roughly equal halves"; the first concerned with exact mathematical methods applied to the two-dimensional motion of an inviscid liquid, the second with the significant theory of axisymmetric jets and cavities which has developed in the last two decades, and the reinterpretation of the fundamental facts about vortex trails, and turbulent jets and wakes.

The exact two-dimensional theory of Chapters II-IX proceeds naturally by use of the complex variable and is well-classified to permit the application of uniform methods, but one would like to have seen at least one problem involving gravity flow pursued to a conclusion instead of stopping at the point where it becomes interesting. The most valuable contributions of these chapters are contained in Chapter IV where there are brought together accounts of recent work on comparison of flows originated by Lavrentieff and simplified in a series of papers by Gilbarg and Serrin with applications to uniqueness and minimum cavity drag, and in Chapter VII on existence and uniqueness.

The remaining chapters concern (X) axially symmetric flows, (XI) unsteady potential flows, (XII) steady viscous wakes and jets, (XIII) periodic wakes, (XIV) turbulent wakes and jets, (XV) miscellaneous experimental facts. These chapters treat generally increasingly complicated situations. At each stage of complication the authors try to give a picture of what is known, organised as rationally as possible. Here rigorous mathematical theory is not always forthcoming and hypothesis and empirical methods playa large part. Nevertheless, these chapters constitute an admirable and absorbing part of the book.

6.6. Review by: M J Lighthill.
Journal of Fluid Mechanics 3 (4) (1958), 437-440.

There can be little doubt that Helmholtz, with his paper "Uber discontinuerliche Flüssigkeitsbewegungen" (1868), laid one of the cornerstones of modern fluid mechanics. He himself, with the editors of The Philosophical Magazine, who published a translation of the paper in the same year, was confident of its importance-just as, thirty-six years later, Prandtl was confident in his paper on the boundary layer. Yet, Helmholtz seems not to have known which cornerstone he was laying, and probably he imagined that he was laying Prandtl's.

Certainly the discontinuous solutions of Laplace's equation constructed by Helmholtz, and by his successors such as Kirchhoff (who introduced the name 'free streamline') and Rayleigh, have found effective application almost exclusively to flows of liquid bordered by regions of gas or vapour - notably, liquid jets in a gaseous atmosphere, and liquid flows about obstacles with cavity formation-whereas the flows of a single homogeneous fluid, to which the authors originally intended the solutions to apply, have proved too complicated for detailed study by these means. Meanwhile, the theory and practice of constructing such discontinuous solutions have been developed to a high degree of virtuosity by an army of workers, and many interesting phenomena have been illuminated as a result. Nevertheless, too few of those writing on the subject have grasped its true relation to physical phenomena, and many have repeated almost word for word the errors of the founders of the theory, such as Helmholtz's ascription of flow separation from a solid surface to causes which really are those underlying cavitation. Especial harm has been done by the writers of hydrodynamical textbooks, who have actively propagated false views of the significance of free-streamline theory.

The vast structure built upon Helmholtz's original paper - comprising the mathematical theory of the existence and uniqueness of flows with free streamlines ; the techniques of obtaining analytical expressions for such flows in particular cases and then translating them into numerical results; the use of the solutions to throw light on jet collisions, on planing, on the effectiveness of shaped charges, on vapour-filled and air-filled cavities, on contraction coefficients and flows over weirs and waterfalls ; the experimental and theoretical study of the physical conditions in flows involving jets and cavities and the relation between these conditions and aspects of the mathematical solutions; the further study of possible relationships between separated flows of a homogeneous fluid and particular types of free-streamline solution - all this is a structure spanning the widely different disciplines of pure and applied mathematics, and involving substantial parts of physics and engineering. The need for 'a unified account of all this varied material, which would help those working on isolated parts of it to see the material as a whole, and reduce the danger of their misinterpreting its significance, has long existed, as has the difficulty of finding anyone competent to undertake such a synthesis.

Now Professors Garrett Birkhoff and E H Zarantonello, both distinguished workers in the field, have collaborated to produce just such a unified account, and it is a real pleasure to report that they have succeeded, achieving both the necessary breadth and the necessary depth. Professor Birkhoff is to be congratulated on having made the difficult transition from pure to applied mathematics, at least for the purpose of writing parts of this book; and those applied mathematicians who found their enjoyment of his Hydrodynamics spoilt by its oversimplified view of the relations between natural science and mathematics can be assured that this experience will not be repeated.

The authors are particularly strong adherents to the view that free streamlines are valuable only as representations of liquid-gas boundaries. With the principal object of emphasising this, they include three chapters on wakes and jets in the flow of homogeneous fluid (for example, air jets in air), chapters which summarise rather briefly the knowledge in these fields but at least make clear to any doubters that an approach to them by means of free-streamline solutions must be quite inadequate. With the exception of these chapters the book is a study of cavities and liquid jets, and of the theory of the mathematical determination of free-streamline solutions. The authors are careful not to give an oversimplified view of the cavitation phenomenon, and they devote their final chapter to its many complexities as well as to descriptions of the various instabilities which lead to the break-up of jets.
...
the appearance of this powerful critical survey of the present position in the whole subject may be regarded as bearing witness to the vitality of Helmholtz's original conception. Our thanks are due to the authors for making such a book as pleasant to read as it is comprehensive and instructive.

7.1. From the Publisher.

A complete revision of the first edition this book. The author has added a chapter on turbulence, and has expanded the work on paradoxes and modelling. W M Elsasser said of the first edition, "A book such as this, concentrating as it does on the boundaries of fundamental progress, should be indispensable to all those engaged in hydrodynamical research who are concerned with the type of generalisation that so often in the past has led to fundamental progress."

7.2. From the Preface.

The present book is largely devoted to two special aspects of fluid mechanics: the complicated logical relation between theory and experiment, and applications of symmetry concepts. The latter constitute "group theory" in the mathematical sense.

The relation between theory and experiment is introduced in Chapters I and II by numerous "paradoxes," in which plausible reasoning has led to incorrect results. In Chapter III this relation is studied more closely in the special case of flows with "free boundaries."

Chapter IV is devoted to analysis of "modelling" end its rational justification. Theory and practice are compared (or contrasted!), and the origins of modelling in symmetry concepts are described as well, thus providing a transition between the two main aspects of fluid mechanics studied in this book.

The rest of the book centres around applications of group-theoretic ideas. These are shown in Chapter V to motivate a large proportion of the known exact solutions to problems involving "compressible" and "viscous" flows. In Chapter VI they are shown to yield the classical theory of virtual mass, as a special case of the modern geometrical theory of "homogeneous spaces."

The overall organisation of material thus closely follows that of the first edition. It has, however, been very carefully revised in detail, and a number of interesting new developments of the past decade have been included.

The basic ideas were originally presented as Taft lectures at the University of Cincinnati in January 1947. These ideas matured and developed while the author was a John Simon Guggenheim Fellow in Europe in 1948. Similarly, the material in the first three chapters of the revised edition was presented as lectures at the Universidad Nacional de Mexico, and was carefully polished while the author was Visiting Professor there on sabbatical leave in 1958 under a Smith-Mundt grant. Finally, all the material was presented in a graduate course at Harvard University in 1959.

7.3. Review by: Keith Stewartson.
Mathematical Reviews MR0122193 (22 #12919).

This book is mainly concerned with two aspects of fluid mechanics: the inter-relation between theory and experiment, and the application of group theory to the subject. It is a revision in detail of the first edition, and includes much new material uncovered during the last decade. The chapter headings are as follows: (I) Paradoxes of non-viscous flow; (II) Paradoxes of viscous flow; (III) Jets, wakes and cavities; (IV) Modelling and dimensional analysis; (V) Groups and fluid mechanics; (VI) Added mass. ...
In the reviewer's opinion the whole of this book is of major importance to all writers on fluid mechanics who are not inclined to make sacred cows either of purely mathematical arguments, however elegant, or of fudged mathematics whose chief merit is apparent agreement with experiment.

Particularly important are the first three chapters in which the necessity of a close collaboration between theory and experiment is driven home by means of paradoxes and discussions of difficulties in the subject. Judging from many modern textbooks these are still not sufficiently well-known. To take one example, hardly any book mentions that there is a fundamental difficulty in compressible viscous flow; one does not know how to relate the thermodynamic pressure to the stress tensor in a general fluid.

In his final conclusions the author says that he has tried to throw two bridges across the widening gap between pure mathematics and physics. Whether he has been successful remains to be seen, of course, but there is every hope and expectation that he will be, and that many fluid dynamicists, finding this brilliant book stimulating, will adopt his creed as their own.

8.1. From the Preface.

The theory of differential equations is distinguished for the wealth of its ideas and methods. Although this richness makes the subject attractive as a field of research, the inevitably hasty presentation of its many methods in elementary courses leaves many students confused. One of the chief aims of the present text is to provide a smooth transition from memorised formulas to the critical understanding of basic theorems and their proofs.

We have tried to present a balanced account of the most important key ideas of the subject in their simplest context, often that of second-order equations. We have deliberately avoided the systematic elaboration of these key ideas, feeling that this is often best done by the students themselves. After they have grasped the underlying methods, they can often best develop mastery by generalising them (say, to higher-order equations or to systems) by their own
efforts.

Our exposition presupposes primarily the calculus and some experience with the formal manipulation of elementary differential equation". Beyond this requirement, only an acquaintance with vectors, matrices, and elementary complex functions is assumed throughout most of the book.

Chapters I through IV constitute a review of material to which, presumably, the student has already been exposed. ... This first part covers elementary methods of integration of first-order, second-order linear, and nth order, linear, constant-coefficient differential equations. ... It includes rigorous discussions of comparison theorems and the method of majorants. ...

Chapters V through VIII deal with systems of nonlinear differential equations. Chapter V includes theorems of existence, uniqueness, and continuity, both in the small and in the large, and introduces the perturbation equations. Chapter VI treats plane autonomous systems, including the classification of nondegenerate critical points, and introduces the important notion of stability and Liapunov's method, which is then applied to some of the simpler types of nonlinear oscillations. Chapters VII and VIII provide a brief survey of the theory of effective numerical integration.

Finally, Chapters IX through XI are devoted to the study of second-order linear differential equations. Chapter IX develops the theory of regular singular points, with applications to some important special functions. Chapter X is devoted to Sturm-Liouville theory and related asymptotic formulas, for both finite and infinite intervals. Chapter XI establishes the completeness of the eigenfunctions of regular Sturm-Liouville systems, without assuming the Lebesgue integral.

8.2. Review by: Nicholas D Kazarinoff.
Mathematical Reviews MR0138810 (25 #2253).

This is a well-written, well-balanced, teachable book which will bring about the modernisation of many an intermediate course. It is a suitable text for an honours first course. The only major part of the subject neglected is asymptotic solutions.

Unfortunately, parts of the book contain confusing slips.

8.3. Review by: J D Elder.
Pi Mu Epsilon Journal 3 (9) (1963), 481.

As stated in the preface, one of the chief objectives of this book is to bridge the gap between the usual material treated in a first course, and the study of advanced methods and techniques. The book amply meets this objective. The background expected of the reader is, in addition to the usual first course in differential equations, a thorough grasp of the major ideas and methods given in a sound course in advanced calculus, and some knowledge of vectors, matrices, and elementary complex variable theory.

The first four chapters review the methods usually covered in a first course, and also include careful discussions of many theoretical questions, and some new techniques. Chapters V through VIII deal with nonlinear systems, while Chapters IX through XI treat second order linear differential equations. An additional indication of the subject matter treated is given by the chapter headings of these latter chapters. V - Existence and Uniqueness Theorems. VI - Plane Autonomous Systems. VII - Approximate Solutions. VIII - Efficient Numerical Integration. IX - Regular Singular Points. X - Sturm-Liouville Systems. XI - Expansions in Eigenfunctions.

Important special functions are defined and studied by means of their defining differential equations and boundary conditions. The book is suitable for a year1s work; or parts of it, as suggested in the preface, can be used for a semester course.

The choice of subject matter is excellent and the exposition is clear. There is a thoroughly adequate set of problems. The authors are to be congratulated on having made a substantial educational contribution to the field involved.

8.4. Review by: Kern W Dickman.
Pi Mu Epsilon Journal 3 (9) (1963), 482.

The authors state that the book was not written to be used as a text book. No exercises are included. The book may be used, however, as a supplement for a course in multivariate analysis.

Perhaps the book will be most useful to research workers in the behavioural sciences at installations which have not yet acquired a library of behavioural science programs. These persons can merely copy the programs and thus obtain immediately a small basic library. The authors state that each program has been tested on an IBM 709 and proven to be correct.

8.5. Review by: C E Langenhop.
The American Mathematical Monthly 70 (7) (1963), 777-778.

As stated in their preface, one of the chief purposes of the authors in writing this text is to aid the student in making the transition from the elementary theory of differential equations to the study of advanced methods and techniques. The reader is assumed to have some experience with the formal manipulation of elementary differential equations and to have a knowledge of advanced calculus and elementary complex variable theory. Fortunately, the second printing corrects many of the slips and errors which tended to undercut the achievement of the authors' purpose in the first printing. The terse style of exposition combined with errors confound the reader in many places as if in an obstacle course through which he can barely move, and thus unusual fortitude on the part of the reader becomes an additional prerequisite.

Apart from such difficulties there is much to admire in this book. The scope of the book and the coverage of basic topics are excellent. Besides the standard existence and uniqueness theorems employing the Lipschitz condition, some of the less well-known ones are included. Power series solutions are given very thorough treatment and the method of majorants is used for a rigorous discussion of convergence. Sturm-Liouville systems are admirably treated with considerable emphasis on asymptotic formulas. There are two chapters in which numerical methods of integration and related topics are discussed in some detail and the last chapter provides a good introduction to the theory of eigenfunction expansions.

Throughout the book classical results and modern methods are well blended, as illustrated by the advantageous use of differential inequalities in various proofs. There is an ample supply of exercises, both the manipulative type and those requiring considerable mastery of the theory, but too few answers are provided. In view of the authors' stated purpose one must applaud the fact that many topics are not presented in complete generality but primarily in the context of second order equations or in the context of a system of two first order equations. Linear equations and related topics predominate, however, and the stability of critical points is about the only topic of nonlinear theory included. Compensating for some of the erroneous diagrams in this part is an interesting example showing that all solutions of an autonomous system can tend to a critical point as $t \rightarrow ∞$ and yet the critical point need not be asymptotically stable.

The kind of difficulty caused by the excess of errors in this text is well illustrated in the discussion of transfer functions. Inconsistencies begin to appear on page 87, but the real culprit seems to be in the definition of phase lag on page 85. The very concise exposition throughout this book makes such demands on the reader that he might almost as well begin with some of the "more advanced and systematic treatments" for which the text "can be used as an introduction." At any rate, one may expect that in such systematic treatments conciseness will be accompanied by a precision of statement and completeness of reasoning, which is too often lacking in this work. Not only are many of the guideposts barely adequate to point out the path of reasoning, but some could tend to encourage the less sophisticated reader to supply invalid arguments ...
...
One welcomes a book which includes so many diverse but basic topics, but one should be wary of the formidable challenge this text presents to the relative novice in the theory of differential equations. Unfortunately, the challenge is frequently attributable more to faulty composition than to difficulty of the material. This reviewer, for one, hopes that a careful revision of this book will not be too long in making its appearance.

8.6. Review by: J C Burkill.
The Mathematical Gazette 48 (364) (1964), 240-241.

Differential equations occupy a central position in mathematics, having a large and growing body of analytical theory and being the most-used tool of the applied mathematician. No one is better fitted to write a book about them than Garrett Birkhoff whose versatility has carried him from abstract algebra to hydrodynamics and who, besides being professor of pure and applied mathematics at Harvard, acts as consultant to some of the largest engineering corporations. Birkhoff and Rota have written an excellent book well adapted to the student who wants to go beyond formal manipulation to a more satisfying survey of the subject. They include existence theorems, solution in series, plane autonomous systems, approximate and numerical methods, Sturm-Liouville systems and expansions in eigenfunctions. They are able to cover so much ground in a lucid, unhurried way by cutting out unrewarding generality. If the essential features of a process are clear in a second-order equation, they omit at their discretion the explicit extension to the nth order equation.
...
There are several hundred well-chosen exercises with hints for some of the more difficult ones. The book is to be commended warmly to the third-year undergraduate and young graduate.

8.7. Review by: H A Antosiewicz.
SIAM Review 5 (2) (1963), 160-161.

This book is principally aimed at filling the gap between the numerous elementary texts on ordinary differential equations and the few rigorous up-to-date presentations of the more advanced parts of the subject. In this aim the authors have succeeded very well. The exposition, which begins with the usual elementary considerations of first order equations, is soon concerned with topics such as first order differential inequalities and the resulting comparison theorems that are not commonly found in books of this kind. Yet, throughout, only a fair knowledge of advanced calculus and of elementary complex function theory is presupposed, together with some facility for manipulating vectors and matrices. The style is easily readable and unhurried. A good number of examples are worked out as illustrations, and nearly all sections are provided with exercises of varying degrees of difficulty.

Chapters I through IV deal with first order equations, linear equations of second order with analytic coefficients and their solutions in the form of power series, and the general linear equation of nth order with constant coefficients. Included here are sections on transfer functions and the Nyquist diagram which do not usually find their way into mathematical texts. The standard existence, uniqueness and continuity theorems for nonlinear systems of first order are given in Chapter V. In Chapter VI autonomous equations are taken up, the phase portrait of two-dimensional systems is discussed in detail, and Lyapunov's method is sketched briefly for such systems. Chapters VII and VIII on approximate solutions and efficient numerical integration give the standard numerical quadrature formulas, finite difference methods and the Runge-Kutta method together with error bounds and a discussion of numerical stability. Chapters IX through XI are concerned with singular points of second order equations, the classical Sturm-Liouville theory and eigenfunction expansions.

The present book will no doubt contribute much to the improvement and modernisation of intermediate courses in ordinary differential equations.

9.1. Review by: Editors.
Mathematical Reviews MR0177992 (31 #2250).

This third edition of a standard text on modern algebra [first edition 1941; revised edition, 1953] is substantially the same as the revised edition. A section on bilinear forms and tensor products has been added to the chapter (7) on vector spaces, while Chapter 11, now entitled "Boolean algebras and lattices", contains a new introduction to Boolean algebras, as well as a section on the representation of such by sets. Beyond this, occasional sections have been revised and a few problems have been added to some of the exercises.

10.1. Review by: Editors.
Mathematical Reviews MR0180595 (31 #4829).

This is an abbreviated version [first edition, 1953] of A survey of modern algebra, first published in 1941. The main changes consist of the inclusion of a new chapter on Boolean algebras and lattices, and the exclusion of the last four chapters of the larger version. These were the chapters on transfinite arithmetic, rings and ideals, algebraic number fields, and Galois theory. Changes have also been made in details of the presentation.

10.2. Review by: Anon.
Quarterly of Applied Mathematics 24 (2) (1966), 176.

This is the second edition of an abbreviated version of the authors' well known Survey of Modern Algebra, first published twenty-five years ago. The new edition is revised and modernised in many details and, in particular, contains a new chapter on Boolean algebra and lattices. The work assumes only high school algebra on the part of the student and is admirably suited for a one-term course in modern algebra at the undergraduate level.

11.1. Review by: Charles W Curtis.
Mathematical Reviews MR0214415 (35 #5266).

The concrete examples which underlie undergraduate algebra courses - integers, real and complex numbers, polynomials, vectors, matrices, determinants - are neither modern nor abstract. A textbook in which these objects are presented becomes a book on "modern algebra" by the kind of scaffolding it erects to exhibit these objects. An earlier generation of mathematicians learned, for example in the authors' A survey of modern algebra that these objects are examples of groups, rings, fields, and vector spaces, and saw that the standard constructions in algebra often reduce to finding the right equivalence relation.

It is refreshing to see in this book the standard material of undergraduate algebra presented systematically from a new point of view. This time the axiomatic systems and equivalence relations are there, but are governed in turn by the language of categories, functors, universal objects, and dualities.

A brief survey of the contents is appropriate at this point. The chapter headings are the authors', the parenthetical sketch of contents the reviewer's: (I) Sets, functions, and universal elements; (II) The integers; (III) Groups; (IV) Rings (integral domains, polynomials, unique factorisation); (V) Special fields (ordered fields, the real and complex fields); (VI) Modules; (VII) Vector spaces; (VIII) Matrices; (IX) Determinants and tensor products; (X) Similar matrices and finite Abelian groups; (XI) Quadratic forms; (XII) Affine and projective spaces; (XIII) Structure of groups (Sylow's theorems; the Jordan-Hölder theorem); (XIV) Lattices; (XV) Categories and adjoint functors; (XVI) Multilinear algebra.

The book is clearly written, beautifully organised, and has an excellent and wide ranging supply of exercises, giving the student plenty of experience with the new ideas from category theory as well as traditional exercises on the concrete examples. The book contains ample material for a full year course on modern algebra at the undergraduate level. One subject sometimes included in undergraduate courses, and not in this book, is Galois theory.

The authors present a strong and enthusiastic case for the role of category theory in the organisation of algebra. The reviewer wishes to raise the pedagogical questions whether pure algebra is the proper domain of these ideas, and whether they should not be used, at the undergraduate level, to introduce portions of subjects such as algebraic topology, differential geometry, or even algebraic geometry, where the required blending of algebra, geometry, and analysis makes these powerful conceptual tools indispensable, whereas they are not indispensable for the topics in algebra taken up in the present book.

11.2. Review by: A C Mewborn.
The American Mathematical Monthly 74 (10) (1967), 1279.

It is natural to compare this book with the authors' A Survey of Modern Algebra since it is directed to the same audience. Most of the topics covered in the earlier book appear in some form in the present book; there is one notable exception: Galois Theory is omitted. The following chapters are primarily concerned with topics not included in Survey: VI Modules, XV Categories and Adjoint Functors, XVI Multilinear Algebra. Chapter I Sets, Functions, and Universal Elements has no real counterpart in the earlier book.

Despite the large intersection of topics covered in Survey and in Algebra, the new book is not just an updating of the old. The treatment of many topics and the general tone differ greatly from that of Survey. The notation and terminology are "categorical," as are many proofs. Universal properties and duality are introduced in the first chapter and play an essential role throughout most of the book. Many concepts are defined in terms of, or are related to, universal mapping properties; and these properties are used when the concepts are applied.

The linear algebra is distinctly module oriented, in marked contrast to Survey. For example, the rational and Jordan canonical forms of a matrix are deduced from the decomposition theorem for modules of finite type over a principal ideal domain.

The level of abstraction is much greater than that of Survey, and in the reviewer's opinion students will find it more difficult. But a course from this book should provide the talented and well-motivated student an excellent introduction to the study of algebra.

11.3. Review by: Dennis Sciama.
Scientific American 217 (3) (1967), 304-309.

Late in the 1930's this same pair of authors more or less formed the curriculum in algebra for American mathematics students with their work A Survey of Modem Algebra, whose third edition is now a couple of years old. In that work the idea of algebra was taken up on the basis of a general and axiomatic approach, and the student learned that discrete elements that could be combined according to several rules of combination formed the essence of this realm of mathematics. Groups, rings, fields and algebras be came common currency; vectors and matrices began to appear as mere portions of a broader topic. The style is quite abstract; one recalls the dismay with which a physics student would find the whole of the algebra of the Dirac matrices as a single exercise in one chapter of Mac Lane and Birkhoff. This book is a fresh start at the same task, aiming at advanced undergraduates, hoping to introduce them to the still more powerful generalisations of algebra in the 1960's. The old topics are there, taking their places in turn as special cases. Now the generality is contained in concepts as powerful as that of category-a category is a class of objects together with two particular functions mapping them in certain ways. There is a category of all sets, one of all groups, of all rings and so on. Another powerful concept is functor; such a construction builds sets from given sets and functions from given functions. It acts on all sets and all functions. The study of algebra can be seen once again as a multiple exemplification of a certain essence. Again this is likely to become a widely used text. There are problems every few pages. The algebras of the physicist are fading into special cases of the problems.

11.4. Review by: Wilhelm Magnus.
Mathematics of Computation 22 (103) (1968), 693-694.

In spite of the similarity of the titles and the coincidence of the names (although not the sequence of names) of the authors, this is not a new edition of the Survey of Modern Algebra (Macmillan Co., 1953) but a new book.

The motivation for it is summarised in the first paragraph of the Preface: "Recent years have seen striking developments in the conceptual organisation of mathematics. These developments use certain new concepts such as 'module,' 'category,' and 'morphism' which are algebraic in character and which indeed can be introduced naturally on the basis of elementary materials. The efficiency of these ideas suggests a fresh presentation of algebra."

As in the Survey of Modern Algebra, the concepts and basic facts of the theory of sets, integers, groups, rings, fields, matrices, and vector spaces are introduced and proved; in addition, modules, lattices, multilinear algebra and other topics have their own chapters and are treated either in greater detail or as new subjects. But most of these chapters are used now also for the purpose of introducing and illustrating the concepts of "category," "functor," and "universal element" which, in the penultimate chapter on Categories and adjoint functors become the main topic of the book. Functors on sets to sets are introduced on page 24. The definition of a universal element appears on page 26. Its description as "the most important concept in algebra" seems to refer to the present book rather than to algebra as a discipline (but this is indicated merely by the italicising of the word "algebra").

Concrete categories are introduced on page 64. However, most of the theorems and proofs in the book can be read without knowledge of the theory of categories.

The book is extremely well organised and very well written. Examples illustrating a new concept are given immediately after its definition. Chapters and even sections are preceded by brief, summarising statements. Theorems are followed frequently by elucidating comments. Proofs are chosen on the basis of transparency rather than brevity; for instance, the first Sylow theorem is proved in the traditional manner without using Wielandt's elegant combinatorial argument, and the Jordan-Hölder theorem is proved without using the powerful but difficult lemma of Zassenhaus.

The omission of Galois Theory (which had a brief but important chapter in the "Survey") is deplored by the authors and is easily explained by the fact that the present book has 598 pages versus 472 of the "Survey" which also had a smaller format. However, this very fact indicates a serious difficulty arising in the teaching of mathematics (or at least of algebra). Galois died 136 years ago. His theory of algebraic equations (or finite extensions of fields) is still a highly relevant and important part of algebra. But to get acquainted with it seems to require an increasing amount of studies. We know (Genesis 29, 30) that Jacob served for Rachel not only seven years but yet seven other years. Will it be the fate of our students to reach the goal of their studies only at the age granted to patriarchs?

12.1. From the Publisher.

Since its original publication in 1940, this book has been revised and modernised several times, most notably in 1948 (second edition) and in 1967 (third edition). The material is organised into four main parts: general notions and concepts of lattice theory (Chapters I-V), universal algebra (Chapters VI-VII), applications of lattice theory to various areas of mathematics (Chapters VIII-XII), and mathematical structures that can be developed using lattices (Chapters XIII-XVII). At the end of the book there is a list of 166 unsolved problems in lattice theory, many of which still remain open. It is excellent reading, and ... the best place to start when one wishes to explore some portion of lattice theory or to appreciate the general flavour of the field.

12.2. Review by: Peter A Fillmore.
Mathematical Reviews MR0227053 (37 #2638).

This expanded and thoroughly reorganised edition contains accounts of many important discoveries of the last twenty years, integrated with a streamlined and modernised treatment of the material of the previous editions. It amply attests to the growth and vitality of the subject during those years. With the size of the volume, the length of the list of unsolved problems continues to grow - 166 in this edition. Both suggest that lattice theory will continue to be the subject of increasing research activity.

In Chapters I-V the various types of lattices are introduced and studied. Chapter III deals with structure and representation theory, including new work of Grätzer and Schmidt, and Chapter IV with geometric lattices, including recent discoveries of Rota and others. Applications to algebra are considered in Chapters VI and VII; such topics as free algebras, the word problem, group theory, permutable congruence relations, and the structure lattice are treated.

In Chapters VIII-XII applications to set theory and analysis (including topology and measure theory) are discussed. Chapter VIII opens with a treatment of transfinite induction and related matters, and proceeds to a proof of the subdirect decomposition theorem. Algebraic lattices are introduced and studied (among them the subalgebra and structure lattices of an algebra). Up to this point the book is mainly concerned with lattices of finite length and their generalisation, the algebraic lattices. Chapters IX-XI deal with continuous lattices. In Chapter IX lattices of open (or closed) sets are considered, leading to a lattice-theoretic treatment of compactness and to Stone's representation theorem. Chapter X develops the theory of metric and topological lattices, including continuous geometries, and Chapter XI Borel and measure algebras, von Neumann lattices, and dimension theory.

The remainder of the book is devoted to partially ordered systems having an additional binary operation. The theory of lattice-ordered groups is systematically developed, as is that of vector lattices, both chapters containing much material new in this edition. There is a chapter on lattice-ordered monoids, with applications to the ideal theory of Noetherian rings and to relation algebras. An interesting chapter on positive linear operators includes various generalisations of Perron's theorem on positive matrices, as well as treatments of transition operators and ergodic theory. The final chapter introduces lattice-ordered rings, and contains a section on averaging operators.

12.3. Review by: Mary Katherine Bennett.
Bulletin of the American Mathematical Society 79 (1) (1973), 35-39.

Perhaps the best known book on lattices in general is Garrett Birkhoff's Lattice theory, first published in 1940, revised in 1948 and more recently in 1967. This is mainly a reference work - the field has grown too big to allow a complete and exhaustive treatment in one book. It is excellent reading, and the many references Birkhoff gives makes it the best place to start when one wishes to explore some portion of lattice theory or to appreciate the general flavour of the field.
...
It is natural that the largest single addition to the book is a chapter on universal algebra, where Birkhoff has made so many contributions. Aside from this, the various chapters have been reorganised, with developments made subsequent to the second edition included throughout the book - often in the form of exercises.

People working in lattice theory today tend to concentrate on special parts of the field - there are specialists in geometric lattices, orthmodular lattices, Noetherian lattices etc. It seems that this third edition is very useful as a reference book for lattice theorists who wish to know what sorts of things have been going on in branches of lattice theory other than their own.

Birkhoff's claim that the bibliography is incomplete is slightly misleading. While it is true that the list of papers he gives at the end of the book is short, there are many footnotes and references throughout the text. The reader who wishes to pursue some topic in more depth can easily get a starting point from these footnotes.

Birkhoff has organised his material into four main parts - general lattice-theoretic concepts, universal algebra, applications of lattice theory to various branches of mathematics, and mathematical structures which can be made into lattices.
...
The book seems too difficult for use as an introductory text; however it should be quite valuable to students working in the field. It does more than inform - it whets the appetite and arouses the reader's curiosity - what more can be said for a mathematics book.

13.1. Review by: Editors.
Mathematical Reviews MR0236441 (38 #4737).

This edition differs from the first [1962] primarily in that the fifth and sixth chapters have been interchanged and an appendix on linear systems (6 pages) has been added. The book has been completely reset.

14.1. Review by: E Klotz.
The American Mathematical Monthly 79 (5) (1972), 529-530.

Recent years have witnessed the development of a number of interesting applications of modern algebra (see, for example, Norman Levinson's article on coding theory). In fact it is quite timely to have texts appearing which are devoted to the applications of modern algebra and related subjects. (The book under discussion also contains bits of graph theory, combinatorics, and the like and might be more properly entitled "Discrete Mathematics;" at the very least the title ought to be "Applied Modern Algebra," since I don't think that "applied algebra" exists as a subject by itself, in any vintage.) The authors of the book are a distinguished mathematician and a well-known computer scientist; both are experienced authors. The subject is au courant. What could go wrong? Alas, quite a bit.

It is traditional to begin a modern algebra text with a chapter of sweepings from set theory, combinatorics, binary relations, and so on. This text has not one but two such chapters, apparently designed to simultaneously anesthetize the non-mathematician with excessive formalism, and to raise the dander of the mathematician with excessive informalism (see, for example, the definition and subsequent discussion of binary relations for both).

Throughout much of the book, the theoretical machinery is far heavier than required for the applications. The presentation is such that few rewards are reaped for the extra effort needed to assimilate the excess theory. There is a gratis chapter on lattices, which seems completely disconnected from the rest of the book, except for the proof of the representation theorem for finite Boolean algebras. The chapters on polynomial rings and finite fields are carried out in the full blaze of characteristic $p$, while really only applied to the world of characteristic two. All in all, the student is likely to come away with the impression that modern algebra is chiefly applied by introducing a great many irrelevant notions and proceeding with ponderous abstraction; and that its main purpose is to becloud simple situations with a heavy academic obscurantism.

The original printing was riddled with errors. Two subsequent printings have reduced the number to merely considerable. The exercises throughout the text seem to come in two flavours: trivial and impossible (the latter due either to errors or due to being placed before some of the relevant concepts have been introduced!). In a semester, a good class and I were able to weave through most of the first eight chapters and some of chapters 10-12. For reasons included above, Applied Modern Algebra was not favourably received by the students. It made me feel needed.

14.2. Review by: Werner C Rheinboldt.
Bulletin of the American Mathematical Society 78 (3) (1972), 383-385.

Over the past forty years or so, modern (abstract) algebra, as envisaged by van der Waerden in his classic book, has become a well-accepted, standard course topic in most college mathematics curricula. This corresponds, of course, to the increasing importance of algebraic thought in many branches of theoretical mathematics. The past two or three decades have now brought a surprising growth in the applications of abstract algebraic concepts and results in various outside areas. The best-known - but by no means only - examples of this are probably applications in electronic engineering and computer science, such as the uses of Boolean algebra in connection with switching networks, the development of algebraic coding theory, and, more recently, the algebraic study of finite state machines and of formal languages.

Very few of our college mathematics departments have taken much notice of these applications of abstract algebra, and special courses on the particular topics mentioned are now most often found in electrical engineering or computer science departments. However, the last years have seen a developing awareness of the need for the mathematics community to accept a responsibility for broader educational opportunities in a more encompassing "mathematical science" in which students may explore the areas of overlap between mathematics, its applications, and scientific computing.

The present book is an important contribution to this need for a broadening of mathematical education. Its aim is to present a sound introduction to basic ideas and techniques of modern algebra which have proved to be useful in certain applications while, at the same time, familiarising the reader with these applications themselves. It addresses itself first and foremost to mathematics students with interests in scientific computing although it could be very useful as well for students from, say, computer science or electrical engineering. At the same time, the work does not really constitute an applied mathematics text in the narrow sense since its emphasis is predominantly on theorem proving rather than on problem solving.

By necessity the authors had to concentrate on certain specific applications of algebra, and they chose for these some of the problem areas cited above related to data communication and the design of switching networks; in particular, a recurring problem considered throughout the text is that of optimal coding of binary information. Correspondingly, the principal algebraic structures discussed are Boolean algebras, monoids and groups, lattices, rings and finite fields. A better impression of the range of topics presented may be obtained from a brief summary of the content of the book.
...
In any book of this type, there is inevitably some separation between theoretically oriented sections and chapters and those devoted to applications. In general, the authors have clearly succeeded in minimising this separation, although a few instances appear to reflect some lack of correlation between theoretical results and their practical uses. In line with the overall aim of the book, the presentation in the theoretically oriented parts - in format and precision - is that of a modern mathematics text. At the same time, a clear effort was made in the applications oriented parts to avoid assumption of specialised prerequisite knowledge of electrical engineering or computer science. As a result, some of the applied parts seem to be comparatively more elementary and discursive than the theoretical ones.
...
All in all, this is a significant addition to the mathematics text market, which deserves widespread and very thoughtful attention and, hopefully, will stimulate in many institutions the introduction of courses following its ideas.

15.1. Review by: Olof Widlund.
Mathematics of Computation 26 (119) (1972), 802.

This book consists of the revised notes of a series of lectures given at an NSF sponsored Regional Conference in Applied Mathematics. It gives a concise, readable and up to date survey of most available methods for the numerical solution of elliptic equations. It contains many well-chosen references and should therefore also be quite useful as a guide to further studies.

There are nine lectures. The first describes typical elliptic problems, and, in the last, the author discusses some of his experiences with complicated practical problems. Lectures two and three are on classical analysis and finite difference methods, while the following two lectures survey the well-known successive overrelaxation, semi- iterative and alternating direction methods. The sixth lecture discusses the use of the classical integral equation approach, a topic often neglected in surveys of this kind. In addition, there are two sections on approximation theory and closely related variational methods.

A discussion of special methods for problems which can be solved by separation of variables such as Hockney's and Buneman's methods, of great importance in specialised applications, is missing. [cf. Hockney, Methods in Computational Physics 9 (1970)].

The finite element method which is now rapidly being developed (to perhaps the most important numerical method for elliptic problems) is discussed only briefly. (It should be noted, however, that the following regional NSF conference dealt exclusively with variational methods.

15.2. Review by: Jacob Burlak.
Mathematical Reviews MR0311125 (46 #10221).

This book is an outgrowth of a series of lectures given at the University of Missouri at Rolla. As the author remarks in his preface "These lecture notes are intended to survey concisely the current state of knowledge about solving elliptic boundary-value and eigenvalue problems with the help of a modern computer. To some extent, these notes also provide a case study in scientific computing, by which I mean the art of utilising physical intuition, mathematical theorems and algorithms, and modern computer technology to construct and explore realistic models of (perhaps elliptic) problems arising in the natural sciences and engineering."

Table of Contents: Acknowledgments; Preface; General References; Lecture 1, Typical elliptic problems; Lecture 2, Classical analysis; Lecture 3, Difference approximations; Lecture 4, Relaxation methods; Lecture 5, Semi-iterative methods; Lecture 6, Integral equation methods; Lecture 7, Approximation of smooth functions; Lecture 8, Variational methods; Lecture 9, Applications to boundary value problems.

The above may give some idea of the author's intention but it cannot convey the breadth of scope of this survey, the wealth of illustrative examples and illuminating discussion that the author brings to bear and the perspectives that he opens up, for example in Lecture 9 where he summarises his "impressions of the current 'state of the art' of solving elliptic problems numerically, as applied by engineers, physical chemists and other users." His remark on the last page, regarding problems that interested von Neumann in the very early days of computer science ("hardly a dent has been made [in the past 20-25 years] in any of the problems listed above"), is meant to be a challenge to the future and not a judgment of the past.

16.1. Review by: R Langlands.
American Scientist 63 (2) (1975), 239.

One can learn from this collection, which contains excerpts in English from papers that originally appeared in English, French, German, and Latin, much about the nineteenth-century development of familiar material - the foundations of real and complex analysis, Fourier series, ordinary differential equations, and potential theory - as well as topics not so widely studied but certainly no less important - abelian integrals, elliptic functions, the calculus of variations, and wave equations.

The articles dealing with the former are readily accessible, and the editor's comments aid in understanding their mathematical and historical back ground. Readers without the appropriate knowledge, however, will find some of the other articles difficult and even obscure, but the references in the foot notes should help. All in all, the editor has assembled a delightful introduction to the works of the nineteenth-century analysts; even those who prefer to examine the sources at more length or in the original will find this useful as a guide.

16.2. Review by: Frank Smithies.
The Mathematical Gazette 59 (409) (1975), 203-204.

This book belongs to a series entitled Source books in the history of the sciences in the General Editor's preface it is stated that "the point of this series is to make these texts easily accessible and to provide good translations of the ones that have not been translated at all, or only poorly".

The present volume is primarily devoted to texts illustrating the development of classical analysis during the nineteenth century. The topics covered include the foundations of real and complex analysis, infinite series and products, asymptotic expansions Fourier series and integrals, elliptic and Abelian integrals, elliptic and automorphic functions, ordinary and partial differential equations (including potential theory and wave propagation), the calculus of variations and integral equations. There are altogether ninety-one extracts from books and papers, their length varying from 1 page to 11 pages forty-six authors are represented. Apart from four early items (from Stirling, Laplace Legendre and Lagrange), the original publication dates lie in the century from 1811 to 1910. Cauchy appears most frequently, with fifteen extracts, and is followed by Abel and Riemann, who have six each.

Within the range of topics mentioned, the texts are well selected and the translations well executed. Each section is preceded by an editorial introduction, explaining the significance of the selected texts in the development of the subject, and giving other relevant references.

In reviewing such a selection, one inevitably notes certain omissions; for instance, there are no extracts illustrating the later development of the theory of the convergence of infinite series or the summation of divergent series. More serious, perhaps, is the fact that there is nothing on the late nineteenth century development of real analysis; one would like to have seen something from Dini on derivatives, from Hankel, Harnack and others on integration, and from Schwarz on the area of surfaces. Another unrepresented area is the gradual development of the notion of an analytically representable function, beginning with Weierstrass's polynomial approximation theorem for continuous functions and culminating in the work of René Baire.

One serious criticism has to be made: the bibliographical work can only be described as sloppy. The data given are often incomplete or confusing, and sometimes even positively misleading. For instance, it is not mentioned that Riemann's work on trigonometric series and integration, although dating from 1854, as stated, remained unpublished until 1867; the only reference given for the published paper is to the second edition (1892) of Riemann's collected works. No date is given for the first publication (actually 1812) of Gauss's work on the convergence of the hypergeometric series; the reference given is an undated one to the collected works (actually 1866). Weierstrass's work on power series is quoted as Werke, l (1841); the results do date from the 1840s, but they were first published in the Werke in 1895, although they had become widely known a good deal earlier through Weierstrass's lectures on analytic functions.

The book is not one for the professional historian of mathematics, who will always wish to get as close as possible to original sources. It should appeal to anyone who would like to see something of the background against which analysis has developed, and to learn something of the problems that have stimulated the growth of the subject; it could be used to correct the impression that is often given by university courses and textbooks that mathematics is a static and established collection of results. The editorial comments are interesting and often penetrating, but are not enough by themselves to give a properly balanced conspectus of the development of analysis during the period in question; for this purpose they would have to be supplemented by a more general historical account.

16.3. Review by: Elaine Koppelman.
Isis 66 (2) (1975), 284-285.

Source books are collections of excerpts from significant documents of the past, usually translated, if necessary, into English and presented in a historical context. Those which contain the writings of scientists can be helpful to the professional historian of science as a starting point; but their primary purpose is didactic. If the selections are well chosen and the translations accurate, they are useful not only in history of science courses but also as a tool to convey to working scientists a sense of the historical development of their subject. Therefore the Harvard series of Source Books in the History of the Sciences is a very welcome enterprise. The latest volume to appear is the Source Book in Classical Analysis.

The book covers analysis in the nineteenth century, though a few selections fall outside that period at either end. There are eighty-one articles by forty-seven mathematicians, divided by subject into thirteen chapters. The choices seem to be good, though there is somewhat of an overemphasis on Cauchy, who is represented by twelve selections while Riemann and Abel have six, Jacobi four, and Weierstrass three. Of those I found outstanding I will single out two: Riemann's introduction to the surfaces which bear his name is among the clearest I have seen, and Poincaré's on asymptotic series is a model of exposition from an author not always noted for his clarity.

The quality of the translations is generally high, though there is one problem I will mention later. The introductory sections are uneven, and though most of them are helpful, some are unorganised and fragmentary. There is, wisely, no attempt to present an overall history of the subject, but one aspect of such a history which does come out very clearly is the large part played in the development of classical analysis by applied problems. Furthermore, the way in which the problems changed throughout the period - from a search for answers to questions about nature to the study of mathematical objects for their own sake - is well illustrated, particularly in the selections on Fourier analysis and Dirichlet's principle.

I have, however, one serious reservation about the approach taken in the preparation of this volume. It concerns the modernisation of terminology and notation, a policy followed by Garrett Birkhoff, the editor. Certainly the problem of terminology can be a difficult one when it comes to translation, but I would like to state a case for sticking as closely to the original as possible. It seems to me that changing the notation and terminology falsifies history, by not showing when new concepts were introduced nor how fast they became accepted.

More important, the purpose of the changes is, I presume, to make the material easier to understand. But for the mathematician, to make it too easy loses some of the point, which should be the way great men had to struggle to formulate ideas that have now become commonplace. Also, ideally this book should be used in teaching, not only in the history of mathematics, but as an adjunct in general history of science courses, which often tend, wrongly, to omit mathematics. In this respect I found, to my surprise, that at times the original would have been easier for the nonmathematician than the modernised version. Mathematics has gained much in power and conciseness from its evolving language, but the effect has been to make it foreign to outsiders, a pattern that had not gone quite so far in the period covered in this book as it has now. For example, "lim sup" is short and sweet and conveys all it should to someone who has had a course in real analysis, but the original, "limit toward which the greatest values tend," can be comprehended by a wider class of readers. In all fairness I should say that the translator does often give both versions, at least the first time the concept is introduced. The changes made include such things as altering a writer's symbols, say an $i$ or $u$ to an $n$ or $V$, in order to conform with present standard usage. This may seem trivial, but one of the results one hopes to get from a source book is that it will inspire its readers to seek out the originals, read them in their entirety, and delve more deeply into the subject. Now, even though there is usually a "dictionary" given in a footnote, the task of reconciling the translations with the originals is often quite formidable and may well discourage a student. Furthermore the changes have not been made consistently, so that often in the middle of an argument the $n$ or $V$ is apt to disappear and be replaced by the original $i$ or $u$.

And that brings me to my final point. In these days of inflation one has become inured to a cost of $25 for a fairly thick volume, which this is. But I think one should still have the right to expect a certain minimal amount of care in the production of the book. This does not seem to be the case here. There are many editorial problems which detract from the readability of the work. The list of errata, assuredly incomplete, which I compiled has over 290 entries, most of the errors in mathematical formulas. In addition to those introduced by the changes in notation, there are misplaced and missing subscripts, superscripts, and integral signs, misprinted symbols, and so on. Even the references are at times incomplete or incorrect. "Poincare, Oeuvres" is not too helpful a citation, and one work by Weierstrass is not to be found in either of the two locations given for it (which happen to contain another work with the same title). I sincerely hope that the errors will be corrected in the next printing so that the book will be better able to achieve the editor's laudable aim of stimulating deeper and wider interest in the history of mathematics.

16.4. Review by: I Grattan-Guinness.
The British Journal for the History of Science 8 (2) (1975), 176-177.

This volume is the one of the latest in the Source Book series published by Harvard University Press. The book is divided into thirteen sections, which contain English translations or editions of excerpts from major works of nineteenth-century mathematical analysis preceded by short editorial introductions. These sections (in order) have the following titles: Foundations of real analysis; Foundations of complex analysis; Convergent expansions; Asymptotic expansions; Fourier series and integrals; Elliptic and Abelian integrals; Elliptic and automorphic functions; Ordinary differential equations (two sections); Partial differential equations; Calculus of variations; Wave equations and characteristics; and Integral equations.

For such a multiply connected bunch of topics there is no definitive order of presentation; but one may legitimately quarrel with the extreme disturbance of chronological order which the above sequence causes. For example, the discussion of foundations of real analysis includes passages from Cauchy (c. I820) and Riemann (I854; including, surprisingly, Riemann's history of trigonometric series, which is not definitive and in any case is not in itself nineteenth-century mathematics); but the intermediate work of Dirichlet (1829) receives emphasis only a hundred pages later, in the section on Fourier series. Again, Lipschitz's condition is cited from an I876 paper on ordinary differential equations, but its genesis in I864 as a means of extending Dirichlet's I829 conditions is not mentioned. Again, the section on partial differential equations occurs long after that on Fourier series, which might explain but not excuse the amazing claim (p. 3I2) that 'the theory of partial differential equations hardly existed before 1840' (whereas there were plenty of interesting things long before Fourier).

Let me turn to a few general points of scholarship. On checking the translations of some extracts at random I found no substantial errors and welcomed the regular quotation within square brackets of the original technical terms or important words. But by contrast, the footnotes are not so consistently handled. Occasionally square brackets are used to indicate interpolations, and sometimes the initials of original authors are indicated at the end of a footnote; but often there is no indication at all, and pages 5, 23, 39, 41, 98, 222, 236, 239, 315, 332, 338, and 381 contain footnotes whose authorship is not at all obvious. In addition, notations have often been changed. Sometimes the changes are minor (in which case the need is not obvious), but a few are radical modernisations, which make the text disturbingly ahistorical (especially pages 115-117). The editorial introductions are adequate within their very general purpose, but significant points are sometimes missed. For example, there is no discussion of Cauchy's definition of the differential on page 5, as opposed to the traditional conceptions of the idea. The references are usually sufficiently precise, though inevitably a few errors creep in. For example, Fourier's 1822 book on heat is cited on page 130 instead of Poisson's of 1835; the citation from Klein's Abhandlungen on page 236 (as should be printed on page xi, incidentally) is way off (read 'III, 630-710'); and so on. Finally, the index is slight for such a book.

On re-reading the above, my conscience is pricked by the tone of ingratitude for a book whose preparation obviously caused much effort for Birkhoff and his helpers. In essence, the criticisms exemplify a basic point; that classical analysis in the nineteenth century is too vast for précis into one volume. It is all too easy to pick almost any of the sections and point to texts of major importance which are not quoted or perhaps even mentioned. In the preface, Birkhoff announces that his aim is 'to give a panoramic view' of 'the magnificent development of so called "classical analysis" in the 19th century'. The danger of the panoramic view is that the impression gained is misleading and incomplete. On the other hand, the virtue of panoramas is that they show us countryside which previously was unknown, and for any such enlightenment we should be grateful.

17.1. Review by: Kenneth Bogart
The American Mathematical Monthly 92 (10) (1985), 743-745.

With a strict selection of topics (omitting entire chapters), Basic Algebra I seems ideally suited for a two semester or three quarter course which would cover all the standard topics (including Galois theory) in the groups-rings-fields-vector spaces tradition. However, separate study in linear algebra would be essential to fill out the traditional program.

18.1. Review by: Editors.
Mathematical Reviews MR0524398 (80d:00002).

The first edition, was published in 1967. The main changes in the second edition are indicated in the preface: "The treatment of many of the topics in the first edition has been simplified - and clarified - in this second edition. The material on universal constructions, formerly introduced at the end of the first chapter, has now been assembled in Chapter IV, at a point where there are at hand many more effective examples of these constructions. A great many points in the exposition have been clarified, for instance in a simpler construction of the integers, a more elementary description of polynomials, and a more direct treatment of dual spaces. The chapter on special fields now includes power series fields and a treatment of the $p$-adic numbers. There is a wholly new chapter on Galois theory; in exchange, the chapter on affine geometry has been dropped. New exercises have been added and some old slips have been excised."

18.2. Review by: Kenneth Bogart
The American Mathematical Monthly 92 (10) (1985), 743-745.

It is ironic that two of the most modem texts in abstract algebra, two of the best examples of the power of abstraction, have neither the word "modem" nor the word "abstract" in their titles. This modesty signals the beginning of a new style in abstract algebra texts. More than a decade ago, this reviewer lamented the difficulty of teaching to undergraduates from the first edition of Algebra, a book which was the first to depart significantly from the Van der Waerden tradition. The second edition maintains this departure but as a result of reorganisation is much more accessible to undergraduates.
...
Informally, a universal characterisation of an object describes a property of the object and then asserts that anything else with the property is related by a morphism to the original object. Typically, the property can be expressed in terms of morphisms as well. Thus, universality is an idea of category theory. (In much the same way that group theory abstracts the study of the automorphisms of a single mathematical object, category theory abstracts the study of homomorphisms among mathematical objects.) As we learn in Algebra, virtually all the familiar algebraic construction s- rings of quotients, homomorphic images, direct products, even the natural number 1 - are universal constructions. Chapter 4 of Algebra explains universality in terms of functors to the category of sets. When they are introduced in Chapter 4, categories and functors appear just as natural a part of algebra (and Algebra) as groups and homomorphisms.
...
Algebra follows one tradition from the Birkhoff and Mac Lane "Survey Series", namely linear algebra's role is more predominant than in other texts. Perhaps 40% of the book consists of topics we would normally associate with linear and multilinear algebra. This book is well suited to give all the advanced training in algebra an undergraduate needs. In fact, as one who teaches graduate courses in combinatorics using topics from widely separated branches of algebra, I would be delighted if my graduate students had mastered the concepts in Algebra

19.1. Review by: Kenneth Bogart
The American Mathematical Monthly 92 (10) (1985), 743-745.

In its two volumes, Basic Algebra comprises an entire education, advanced undergraduate and graduate, in algebra, an education deep enough to serve as a springboard to research in most traditional areas of algebra. It is unlikely that many graduate students, even those specialising in algebra, have learned such a breadth of ideas in such depth. To my knowledge, only Bourbaki has attempted as deep a survey of algebra as appears in Basic Algebra Volume 2. (Of course in their encyclopaedic way, the Bourbaki group covers even more.) ... Basic Algebra Volume 2 takes us on a tour of "core" algebra including commutative algebra, the structure theory of rings and algebras, representation theory for groups, and field theory in addition to its category theory, universal algebra and homological algebra. (Homological algebra, which arose from work on chains and cochains in algebraic topology, is based on the study of sequences of modules connected by homorphisms. Homological algebra has provided proofs of results of algebraic interest, some appearing in Basic Algebra Volume 2, as well as results of topological interest.

20.1. Review by: Robert S Anderssen.
Mathematical Reviews MR0769468 (86e:65001).

As the authors explain in the preface, their book is the successor to Birkhoff's book, The numerical solution of elliptic equations [1971]. The emphasis on problems that have important scientific and/or engineering applications remains. ...

Though the authors make no comment to this effect, an obvious goal is to make the technology for the accurate solution of linear two-dimensional boundary value problems at moderate cost accessible to as wide an audience of mathematically trained practitioners as possible. The book reads easily and lucidly to anyone with a basic training in the numerical analysis of partial differential equations. The book should therefore be seen as a reasonably well-rounded and up-to-date survey of the subject. Consequently, it is descriptive in nature with no undue emphasis on theory for theory's sake; it is largely self-contained, contains extensive cross-referencing to a comprehensive bibliography, and concludes with a discussion of a computer package for solving elliptic problems (ELLPACK).

Difference methods, efficient algorithms for some sparse linear algebraic equations, finite element methods and boundary integral methods are all examined in considerable detail. However, the book makes no claim to completeness. Eigenvalue problems are not treated except in a passing comment. Because of the obvious connection back to linear problems through linearisation and iteration, there is a short discussion of nonlinear problems. The book contains no discussion of adaptive grid refinement techniques and makes only passing comments about singular problems.

20.2. Review by: Eugene L Wachspress.
Mathematics of Computation 48 (178) (1987), 830-831.

As stated by the authors, "The aim of this monograph is twofold: first to describe a variety of powerful numerical techniques for computing approximate solutions of elliptic boundary value problems and eigenproblems on high speed computers, and second, to explain the reasons why these techniques are effective." An attempt is made "to provide a reasonably well-rounded and up-to-date survey of these methods." The authors succeed in striking a delicate balance between exposition of underlying theory and its application to numerical solution techniques, thereby providing an impressive addition to the literature, which could well become a classic reference on this subject. In a work of this breadth, the depth of treatment is limited. Nevertheless, the crucial theorems are given with reasonable outlines of basic arguments in their proofs and extensive reference to appropriate literature. Current research is centred on vector and parallel computation, a field which is still in its infancy. The practitioner would do well to use this monograph as a springboard from which to launch on new methods for the incredible variety of emerging parallel and vector architectures. A summary of the nine chapters follows.

Chapter 1. "Typical Elliptic Problems," in which a description of a variety of physical problems which lead to elliptic systems motivates this treatise.

Chapter 2. "Classical Analysis," wherein a concise overview of the most essential and commonly used classical results are given which serve as a guide to formulation and solution of discrete approximations.

Chapter 3. "Difference Approximations" is a thorough treatment of several differencing techniques with analysis of approximation error and interrelationship of associated properties with solution techniques.

Chapter 4. "Direct and Iterative Methods" and

Chapter 5. "Accelerating Convergence" contain in-depth reviews of many of the most prevalent numerical techniques for solving large elliptic systems. Relative advantages of direct and iterative methods as a function of type of problem are discussed along with methods which utilize a combination of both approaches.

Chapter 6. "Direct Variational Methods" and

Chapter 7. "Finite Element Approximations" deal with variational principles characterizing solution of boundary-value problems, application via patchwork finite element approximation, and error estimation with the aid of classical polynomial approximation theory.

Chapter 8. "Integral Equation Methods," in which there is a concise description of Green's functions, boundary element methods, conformal mapping, capacitance matrix methods and other techniques for solving elliptic equations. Methods discussed here are "quasi-analytic" in that numerical approximations are made in conjunction with extensive analytic reduction.

Chapter 9. "ELLPACK" describes some of the capabilities of the Purdue ELLPACK software package for solving elliptic problems.

21.1. From the Preface.

The present volume of reprints are what I consider to be my most interesting and influential papers on algebra and topology. To tie them together, and to place them in context, I have supplemented them by a series of brief essays sketching their historical background (as I see it). In addition to these I have listed some subsequent papers by others which have further developed some of my key ideas.

The papers on universal algebra. lattice theory, and general topology collected in the present volume concern ideas which have become familiar to all working mathematicians. It may be helpful to make them readily accessible in one volume. I have tried in the introduction to each part to state the most significant features of each paper reprinted there, and to indicate later developments.

The background that shaped and stimulated my early work on universal algebra, lattice theory, and topology may be of some interest. As a Harvard undergraduate in 1928-32, I was encouraged to do independent reading and to write an original thesis. My tutorial reading included de la Yallee-Poussin's beautiful Cours d'Analyse Infinitesimale, Hausdorff's Grundzüge der Mengenlehre, and Frechet's Espaces Abstraits. In addition, I discovered Carathéodory's 1912 paper "Uber das lineare Mass von Punktmengen" and Hausdorff's 1919 paper on "Dimension und Ausseres Mass," and derived much inspiration from them. A fragment of my thesis, analysing axiom systems for separable metrisable spaces, was later published. This background led to the work summarised in Part IV.

My undergraduate courses were intended for the career of a mathematical physicist, for which the reading just mentioned would today seem inappropriate. Functional analysis was at the time becoming an independent subject. Von Neumann. Stone, and Banach were writing their great classics, and the ergodic theorem had just been proved. Central to all these developments was the concept of a function space. First defined in Frechet's Thesis, this concept made point-set topology relevant to mathematical physics.

My courses at Harvard covered layerings of topics in analysis and mathematical physics. I was essentially a self-taught algebraist. My only training beyond analytic geometry and determinants was in a course on combinatorial topology given by Marston Morse. He explained the canonical form of a matrix of integers under row and column equivalence. I was fascinated by the definition of a group in Miller, Blichfeldt, and Dickson's Finite Groups, which I found in Harvard's departmental library. I then conceived the naive project of classifying finite groups.

Working alone in Munich the summer after graduation, I classified commutative groups, unaware that Kronecker had proved his fundamental theorem on Abelian groups 15 years earlier. There I had the good idea of calling on Carathéodory, whose paper I had perused; I mentioned to him my interest in group theory. He recommended that I read Speiser's Gruppentheorie and van der Waerden's Modern Algebra. I took his advice, and after a few months had the good fortune or being assigned Philip Hall as adviser at Cambridge University. It became apparent to me that I was more interested in algebra than in mathematical physics, as it was represented in Cambridge at the time by Dirac.

Always modest, Hall did not explain to me his own beautiful results on finite groups. Instead, at my request, he assigned me a problem in group theory whose solution he thought might be publishable. I solved it in a few days, and published the results in the paper "A Theorem on Transitive Groups". In the same year, I determined the characteristic subgroups of a finite Abelian group having found a mistake in an earlier paper by Miller on the subject. Although I continued to be fascinated by finite groups, 1 only published one other paper on group theory, co-authored with Philip Hall. We estimated the least common multiple of the orders of groups of automorphisms for all the groups with given order $g$.

Through the study of the structure of groups, and especially of six 1930-32 papers by Remak. I was led to formulate the concept of a lattice. My ideas about universal algebra arose from my interest in the structure of lattices. It became obvious to me that Emmy Noether's techniques for treating homomorphisms of rings and groups with operators were inadequate for the study of lattices, and would have to be replaced by equivalence relations having the substitution property, what are now called congruence relations. In turn my background in set theory and general topology proved valuable in developing lattice theory and universal algebra from that lime on.

My interest in groups soon led me to read about Lie groups. A decade later, my study of Lie groups bore fruit in my book Hydrodynamics: A Study of Logic, Fact, and Similitude, and in my solution of the Riemann-Helmholtz problem. Both of these problems fall outside the scope of this volume.

After the Second World War, I became involved in activities far removed from the work in algebra of 1932-40. A conclusion that emerges from my experience of that time is the value of serendipity. Every project of work that I undertook because of my intrinsic interest bore fruit, but often a very different fruit from that which I expected at the start of that project.

21.2. Review by: Manfred Stern.
Mathematical Reviews MR0895820 (88k:01062).

The present volume is valuable from several points of view. First of all, as Birkhoff states in his preface, it contains "what I consider to be my most interesting papers on algebra and topology. To tie them together, and to place them in context, I have supplemented them by a series of brief essays sketching their historical background (as I see it)."

These historical notes as well as comments by Birkhoff on his own scientific background are valuable sources for the history of the subject matter. The text is divided into 6 chapters in each of which significant contributions by Birkhoff are reprinted. The chapters are (in parentheses we write the number of papers reprinted): I. Lattices (12): II. Universal algebra (6); III. Topology (8); IV. Lie groups and Lie algebras (7); V. Lattice-ordered algebraic structures (4); VI. History of algebra (4).

The 41 papers reprinted here have all been reviewed elsewhere. In his comments, the author also relates his results to results obtained by others and to more recent further developments. After the preface there is a bibliography of Birkhoff's books and papers, 1933-1986, and a list of his Ph.D. students, 1936-1978.

The bibliography and the topics of the Ph.D. theses supervised show the remarkably broad outlook of Birkhoff, ranging from abstract algebra through hydrodynamics and potential theory to numerical mathematics and slowing down of neutrons, not to forget the history of mathematics.

Since the emphasis of this book is on algebra and topology, other topics treated by Birkhoff (as, for instance, hydrodynamics) fall outside the scope of this volume.

22.1. From the Preface.

This book aims to present modern algebra from first principles, so as to be accessible to undergraduates or graduates, and this by combining standard materials and the needed algebraic manipulations with the general concepts which clarify their meaning and importance.

The modern conceptual approach to algebra starts with the description of algebraic structures by means of axioms chosen to suit the examples at hand; as for instance with the axioms for groups, rings, fields, lattices, and vector spaces. This axiomatic approach, emphasised by David Hilbert and developed in Germany in the 1920's by Emmy Noether, Emil Artin, B L Van der Waerden, and others became available on the graduate level in the 1930's, and was then popularised on the undergraduate level in the 1940's and 1950's in part by our own A Survey of Modern Algebra. Since that time, algebra has expanded vigorously; this book is designed to present also the basic new concepts which have appeared.

In linear algebra, the notions of module and tensor product are emphasised, because of their use in topology, differential geometry, and elsewhere. Here the standard axioms used to describe a vector space with scalars from a field also apply when the scalars are elements of a ring; the axioms then define a module over that ring. By means of examples, we aim to make this notion easy to handle.

Homomorphisms are the means of comparing two algebraic structures of the same type. Such homomorphisms can be composed, and under this composition constitute the morphisms of a category, while many constructions on algebraic structures apply also to these morphisms, and so can be understood as functors ( = morphisms) on the category at issue.

The construction of a new algebraic object will often solve a specific problem in a universal way, in the sense that every other solution of the given problem is obtained from this one by a unique homomorphism. The basic idea of an adjoint functor arises from the analysis of such universals.

Our presentation starts with integers, groups, and rings. In Chapter I the integers are constructed from the natural numbers so as to provide the universal way to make subtraction possible, thus providing a fundamental explanation of this operation. Chapter II begins with the origin of groups in the study of symmetry. Given a group $G$ with a normal subgroup $N$, there is a universal construction of a homomorphism to another group $G/N$, so that the given $N$ is mapped to the identity element; this description of the quotient group is decisive, because it provides all the needed information about the behaviour of this $G/N$. Rings (Chapter III) involve both addition and multiplication; here the extension of a ring to a ring of polynomials in an indeterminate $x$ is described as the universal way of adding one new element (to wit, $x$) to the original ring. With these and other examples of universal constructions at hand, Chapter IV gives the general description of a universal element, plus the axiomatic description of categories (of morphisms) and of lattices (of subobjects).

The next three chapters treat linear algebra: modules, vector spaces, and matrices, with emphasis on the idea that matrices are tools used to represent and compute with linear transformations between vector spaces. The proof of the existence and properties of eigenvalues and eigenvectors for linear transformations in Chapter X is based on the special properties of the real and complex fields, as developed in Chapter VIII. There, other examples of fields (of power series and of $p$-adic numbers) are also given. The intermediate Chapter IX constructs the tensor product of two vector spaces conceptually in terms of universal bilinear functions, and includes related notions such as that of determinant and exact sequence- an idea important for topology.

Up to this point, the chapters fall in a natural sequence; thereafter, the chapters are largely independent of each other. Chapter XI uses the concept of module over a principle ideal domain to provide a unified treatment of both the rational canonical form for matrices under similarity and the structure theorem for finitely generated groups-a fine illustration of the use of conceptual unification. A second chapter on group theory develops notions such as that of a composition series, to be used in the subsequent chapter presenting Galois Theory. There are also chapters on Lattices, on Categories, and on Multilinear Algebra, including tensor and exterior algebra. These chapters were also present in our second edition, but we have also included the chapter on Affine and Projective Geometry, with an appropriate axiomatisation - a chapter from the first edition of our book, but omitted from the second. Moreover, a number of misprints from the second edition have now been corrected.

As in the prior editions, we extend our thanks to the many friends who have helped and commented on our work, together with our thanks to the Chelsea Publishing Company for making this new edition available.

22.2. Contents.

Front Cover
Preface to the Third Edition
From the Preface to the First Edition
From the Preface to the Second Edition
Contents
List of Symbols

CHAPTER I Sets, Functions, and Integers
1. Sets
2. Functions
3. Relations and Binary Operations
4. The Natural Numbers
5. Addition and Multiplication
6. Inequalities
7. The Integers
8. The Integers Modulo $n$
9. Equivalence Relations and Quotient Sets
10. Morphisms
11. Semigroups and Monoids

CHAPTER II Groups
1. Groups and Symmetry
2. Rules of Calculation
3. Cyclic Groups
4. Subgroups
5. Defining Relations
6. Symmetric and Alternating Groups
7. Transformation Groups
8. Cosets
9. Kernel and Image
10. Quotient Groups

CHAPTER III Rings
1. Axioms for Rings
2. Constructions for Rings
3. Quotient Rings
4. Integral Domains and Fields
5. The Field of Quotients
6. Polynomials
7. Polynomials as Functions
8. The Division Algorithm
9. Principal Ideal Domains
10. Unique Factorisation
11. Prime Fields
12. The Euclidean Algorithm
13. Commutative Quotient Rings

CHAPTER IV Universal Constructions
1. Examples of Universals
2. Functors
3. Universal Elements
4. Polynomials in Several Variables
5. Categories
6. Posets and Lattices
7. Contravariance and Duality
8. The Category of Sets
9. The Category of Finite Sets

CHAPTER V Modules
1. Sample Modules
2. Linear Transformations
3. Submodules
4. Quotient Modules
5. Free Modules
6. Biproducts
7. Dual Modules

CHAPTER VI Vector Spaces
1. Bases and Coordinates
2. Dimension
3. Constructions for Bases
4. Dually Paired Vector Spaces
5. Elementary Operations
6. Systems of Linear Equations

CHAPTER VII Matrices
1. Matrices and Free Modules
2. Matrices and Biproducts
3. The Matrix of a Map
4. The Matrix of a Composite
5. Ranks of Matrices
6. Invertible Matrices
7. Change of Bases
8. Eigenvectors and Eigenvalues

CHAPTER VIII Special Fields
1. Ordered Domains
2. The Ordered Field $\mathbb{Q}$
3. Polynomial Equations
4. Convergence in Ordered Fields
5. The Real Field $\mathbb{R}$
6. Polynomials over $\mathbb{R}$
7. The Complex Plane
8. The Quaternions
9. Extended Formal Power Series
10. Valuations and $p$-adic Numbers

CHAPTER IX Determinants and Tensor Products
1. Multilinear and Alternating Functions
2. Determinants of Matrices
3. Cofactors and Cramer's Rule
4. Determinants of Maps
5. The Characteristic Polynomial
6. The Minimal Polynomial
7. Universal Bilinear Functions
8. Tensor Products
9. Exact Sequences
10. Identities on Tensor Products
11. Change of Rings
12. Algebras

CHAPTER X Bilinear and Quadratic Forms
1. Bilinear Forms
2. Symmetric Matrices
3. Quadratic Forms
4. Real Quadratic Fonns
5. Inner Products
6. Orthonormal Bases
7. Orthogonal Matrices
8. The Principal Axis Theorem
9. Unitary Spaces
10. Normal Matrices

CHAPTER XI Similar Matrices and Finite Abelian Groups
1. Noetherian Modules
2. Cyclic Modules
3. Torsion Modules
4. The Rational Canonical Form for Matrices
5. Primary Modules
6. Free Modules
7. Equivalence of Matrices
8. The Calculation of Invariant Factors

CHAPTER XII Structure of Groups
1. Isomorphism Theorems
2. Group Extensions
3. Characteristic Subgroups
4. Conjugate Classes
5. The Sylow Theorems
6. Nilpotent Groups
7. Solvable Groups
8. The Jordan-Holder Theorem
9. Simplicity of $A_{n}$

CHAPTER XIII Galois Theory
1. Quadratic and Cubic Equations
2. Algebraic and Transcendental Elements
3. Degrees
4. Ruler and Compass
5. Splitting Fields
6. Galois Groups of Polynomials
7. Separable Polynomials
8. Finite Fields
9. Normal Extensions
10. The Fundamental Theorem
11. The Solution of Equations by Radicals