**1. Algebraic Projective Geometry (1952), by J G Semple and G T Kneebone.**

**1.1. Review by: William Franklin Atchison.**

*Amer. Math. Monthly*

**60**(5) (1953), 343-344.

The authors state that their "main purpose in this book is to construct and develop a systematic theory of projective geometry, and in order to make the system both rigorous and easily comprehensible we have chosen to build it on a purely algebraic foundation." The book is divided into two parts. Part I contains 39 pages and consists of two chapters of an historical and introductory character. Early in Chapter I the following statement is made: "The present chapter is devoted to a rather general consideration of the nature of mathematics and more specifically, of geometry, while Chapter II contains an outline of the intuitive treatment of projective geometry from which the axiomatic theory has gradually been disentangled by progressive abstraction." This gives a good picture of the contents of these two chapters. They are well written and make interesting reading. Chapter II however cannot be fully appreciated by the beginning student since it is just a summary. The formal development of abstract projective geometry starts with Part II, Chapter III. ... There is a wealth of excellent material in this text book.

**1.2. Review by: Alex Rosenberg.**

*The Mathematics Teacher* **46** (5) (1953), 382-383.

This book presents a very detailed exposition of the projective geometry of the line, plane and three space over the complex numbers. Algebraic methods are used throughout in preference to synthetic ones, although the underlying geo metrical ideas are brought out as much as possible. ... The usual topics: collineations and correlations, conies, quadrics and twisted cubics are treated in great detail, but there is also a brief treatment of such topics as quadratic transformations, invariants and cubic surfaces. The last chapter of the book contains a brief account of the geometry of four and five dimensional space. Many excellent exercises occur throughout the text.

**1.3. Review by: John Arthur Todd.**

*The Mathematical Gazette* **37** (320) (1953), 151-152.

It is not so long since it was usual, when lecturing on geometry to mathematical honours students, to apologise for the lack of an adequate text-book. The last few years have seen a great change: the difficulty today is rather one of selecting from the steadily growing number of books which have been written more or less expressly to fill an obvious gap in the literature. The present volume, none the less, will receive a warm welcome, since it provides an account of all the topics which should normally find place in an honours course, presented with a simplicity and clarity of treatment which should make it easy to use in a class of students of varied attainments. At the same time, it contains hints and suggestions which should attract the attention of future specialists. After two well-written informal chapters which set out to link the ideas underlying projective geometry with those of the elementary geometry with which the reader will already be familiar, the authors proceed to the formal development of their subject. In accordance with modern trends, the foundations are algebraic rather than synthetic in character; but, while algebraic methods are used freely, the authors always make it clear that they are discussing geometry, and the spirit of the book is geometrical throughout. After introducing the fundamental ideas of projectivity and cross-ratio the book deals with the usual properties of conics, quadrics, twisted cubics and line systems, with a good treatment of elementary correspondence theory. A notable feature is the attention given to collineations in the plane and space, though the general theory of invariant factors is not touched upon. Some emphasis is placed on affine and metrical specialisations of projective theorems, and the examples selected for this purpose are interesting and at times unusual. A very interesting final chapter deals briefly with the possibilities of geometry in higher space, which should serve to whet the appetite of the attentive reader.

**2. Algebraic Curves (1959), by J G Semple and G T Kneebone.**

**2.1. Review by: Eric John Fyfe Primrose.**

*The Mathematical Gazette*

**45**(352) (1961), 157-158.

Most branches of mathematics pass through two stages, the creative stage, when new results are obtained rapidly, and the critical stage, when the logical foundations are carefully examined. In algebraic geometry the great creative stage occurred during the nineteenth century and the beginning of the present century, and the critical stage began only quite recently, with the work of van der Waerden in the thirties. The present book gives a completely rigorous account of the theory of algebraic curves up to the Riemann-Roch Theorem. The authors assume that the reader has studied projective geometry (the notations of the authors' Algebraic Projective Geometry are used, so that this new book follows on naturally from the earlier one). They also assume that the reader either knows a good deal of modern algebra or is prepared to work quite hard to learn it; there is a very helpful appendix which gives all the algebraic results used in the book, with references to where proofs can be found. The treatment is algebraic throughout. ... I enjoyed reading the whole book, but particularly the last chapter ... The book will undoubtedly become the standard work on curves for post-graduate students of algebraic geometry.

**2.2. Review by: Margherita Piazzolla-Beloch.**

*Mathematical Reviews*, MR0124801 **(23 #A2111)**.

In the preface of this book the authors observe that the subject of algebraic geometry is one to which mathematicians have been powerfully drawn over a very long period of time. One of the reasons of this enduring influence of algebraic geometry is the fact that the subject makes a strong appeal to the imagination in that it not only illuminates properties of geometrical figures that we are all able to draw or visualize but also extends the range of geometrical thinking far beyond the bounds of intuition, and one can think about it in a much deeper sense than that of granting formal assent to its conclusions. Algebraic geometry has shown itself again and again to be immensely fertile, giving rise to profound and far-reaching investigations of the most general character, while at the same time suggesting innumerable particular problems and leading to a wealth of special results of great elegance and attractiveness, with a wide field of applications. The present volume on algebraic curves, according to the authors, is intended to be, in the spirit of the above-mentioned considerations, an introduction to the study of algebraic geometry. The book is designed specifically to provide the readers with a rigorous and reasonably self-contained account of the basic theory of algebraic curves, making full use throughout of the traditional language of geometry. In conformity with the general mathematical tendency of the present time, the formal theory is developed by purely algebraic means.

**3. Mathematical Logic and the Foundations of Mathematics (1963), by G T Kneebone.**

**3.1. Review by: Trevor Evans.**

*Amer. Math. Monthly*

**72**(2) (1965), 215.

The author has attempted to give a survey (at the junior-senior first-year graduate level) of the whole field of mathematical logic and foundations of mathematics up to the developments in the 1930s. If we put to one side for the moment the question of whether a survey at this level is a good thing, then the author has been moderately successful in what he set out to do. The book contains a large amount of information presented in a readable manner. ... The book is based on a course of lectures given ay the University of London for an audience containing not only graduate students beginning to specialise in mathematical logic but undergraduates majoring in fields other than mathematics. There are some difficulties in using this book as a textbook for a course. It seems to the reviewer that it is too technical for most non-mathematicians and it is questionable also whether a survey is the right way for a serious mathematics student to begin a study of mathematical logic.

**3.2. Review by: John C Shepherdson.**

*The British Journal for the Philosophy of Science* **15** (59) (1964), 268-270.

According to the jacket this book 'is designed as an introduction addressed primarily to two kinds of reader. The first includes honours undergraduates in mathematics, who need this topic as part of their degree course, high school mathematics teachers and those under training, and postgraduates specialising in some other branch of mathematics. All these require a book which covers the whole field, is informative, seriously written and substantial but not overloaded with technicalities. The second group consists of postgraduate students who want an introduction to mathematical logic or the philosophy of mathematics that will give them a correct orientation, a view of the entire subject and its literature, and quickly enable them to work from primary sources.' At present this audience is not very well catered for. Most existing books on mathematical logic (a notable exception being Beth's The Foundations of Mathematics) which attempt to be fairly comprehensive are either suitable only for the trained mathematician who can skim through pages of detailed calculation and extract the ideas behind them for himself or else represent a rather individual view of what is important rather than a balanced view of the whole field. Kneebone seems to have succeeded very well in producing a book for this audience. He has covered almost all branches of the subject and has been particularly careful to include historical notes and other background material which supply the reader with motivation. It would be difficult to produce a better book without restricting the audience ...

**3.3. Review by: J Tucker.**

*Mathematical Reviews*, MR0150021 **(27 #26)**.

The author has worked out a whole new philosophy of mathematics with unusual clarity and integrity. He conceives of mathematics as an evolving activity and not as the investigation of an absolute, objective, timeless realm. Mathematics, in his view, evolves in successive phases of dialectical and manipulative thought. Concepts evolve dialectically, and the dialectical phase of concept formation is followed by the manipulative phase. In dialectical thinking, the concepts involved are in process of formation or revision and so cannot be subsumed under a formal criterion of validity. Nor can they be handled combinatorially. In manipulative thinking the concepts are fixed and can be manipulated combinatiorially. The validity of these moves can be assessed in accordance with a formal criterion. The author applies this distinction to the natural sciences as part of his thesis that mathematics is the same sort of activity as the other sciences. For in the natural sciences there are, according to the author, two sorts of induction. Determinative induction is manipulative and is accordingly open to formalisation. Conceptual induction is dialectical and is not. Now since mathematics, like the other sciences, passes through dialectical and manipulative phases, no peculiar a priori certainty or absoluteness attaches to it. It is not to be distinguished from the other sciences by such frozen features. It evolves in the same way that they do. The security for which fundamentalists search is to be found, not in the quiescence of a completed system, but in continual dialectical revision. The author appeals throughout to Bourbaki's distinction between the foundational requirements of the working mathematician and those of the philosopher of mathematics. Mathematics is, in his view, autonomous and capable of providing its own standards of validity. The working mathematician is in the best position to know what these are. Nevertheless, what will do for the working mathematician will not necessarily do for the philosopher of mathematics. For this reason the problems of foundations lie wide open. For although axiomatic set theory, as in Bernays and Bourbaki, provides a mathematically satisfactory foundation, it fails, on account of its ineradicable relativity, to be philosophically satisfactory. Mathematics is highly charged with significance and one of the important questions of foundations is to show how this comes about; and since logic merely provides a criterion of validity, not of significance, it fails to explain this feature of mathematics. According to the author, a piece of mathematics has significance when and only when it can be 'realised' or applied. The strongest possible guarantee of significance is that of a finite realisation. Pure mathematics is concerned with structural patterns. The application or 'realisation' of a piece of pure mathematics amounts to the claim that the given structure is to be found in the given body of experience. From this he draws the odd conclusion that formal logic is to count as applied mathematics. It is applied Boolean algebra. Theoretical physics is entirely structural and has no objective reference other than to the numbers involved. The author has exceptional gifts as an expositor and gives a lucid account of mathematical logic and of the main movements in foundations.

**4. The Theory of Sets and Transfinite Numbers (1966), by B Rotman and G T Kneebone.**

**4.1. Review by: Reuben Louis Goodstein.**

*The Mathematical Gazette*

**51**(376) (1967), 179.

This is an attractively produced and well written introduction both to axiomatic set theory and to the theory of transfinite ordinal and cardinal numbers. About one half of the book is devoted to the arithmetic of ordinals and cardinals, and in this short space a number of important results are obtained. Following von-Neumann, ordinals are identified with well-ordered sets in which every element a is identical with the segment of the set preceding a, and the cardinal of a set is identified with the least ordinal of the same power as the set. The system of axiomatic set theory described is that of Zermelo-Fraenkel. The system is clearly explained and the axioms well motivated; the von-Neumann distinction between set and class is introduced and unobtrusively maintained in the later arithmetical developments. The book concludes with a discussion of several equivalents of the axiom of choice, Zorn's lemma, Hausdorff's principle and the Tukey-Teichmuller theorem on the existence of a maximal element in a set of finite character.

**4.2. Review by: Reuben Louis Goodstein.**

*American Scientist* **55** (2) (1967), 205A.

The senior of the two authors of this little volume, Dr Kneebone, is well known for his books on logic and geometry, and his collaboration with a specialist in ordinary theory has produced a well-written introduction to axiomatic set theory and transfinite numbers.

**4.3. Review by: Perry Smith.**

*The Journal of Symbolic Logic* **37** (3) (1972), 614.

This book is designed as a text for undergraduate and graduate students in mathematics. The first chapter covers the standard notions of the algebra of sets, and the second contains the simpler results about countable sets and sets of the power of the continuum, followed by Cantor's theorem |Al ≠ |P(A)l, the Schroder-Bernstein theorem, and some cardinal arithmetic. The third chapter is a semi-formal presentation of the Zermelo-Fraenkel axioms. These three chapters are a readable account of "what every mathematician should know" about set theory, and are written for freshmen and sophomores. Chapters four and five are a relatively advanced treatment of the theory of infinite ordinals and cardinals. The order of topics in chapter four is from the abstract to the concrete: well-ordered sets, the definition of the ordinals as the well-ordered sets such that each element equals the set of its predecessors, definition and proof by transfinite induction, finite and infinite sums and products of ordinal numbers, and only after all this the division of ordinals into successor and limit ordinals. Normal functions and ordinal exponentiation follow, and the discussion of the natural numbers as the finite ordinals closes the chapter. Chapter five discusses the second number class, normal functions on the set of countable ordinals, and the further development of cardinal arithmetic. Chapter six, on equivalents of the axiom of choice, is independent of chapters four and five except for the proof of Zorn's lemma, and could be included with the first three chapters in a less advanced course.

**4.4. Review by: Robert Roth Stoll.**

*American Scientist* **57** (3) (1969), 250A.

This book gives an erudite account of both intuitive set theory and those aspects of axiomatic set theory which should be understood by every serious student of mathematics. There are numerous exercises for the reader to test his understanding of the material; many are instructive or challenging. ... The reader who completes the book will be rewarded with an understanding of how the whole of pure mathematics may be regarded as a structure based on an adequate theory of sets.

**4.5. Review by: A H Kruse.**

*Mathematical Reviews*, MR0210599 **(35 #1485)**.

This book offers a presentation of set theory appropriate for undergraduate students and first year graduate students. ... Each chapter is composed of several sections, to each of which is appended a list of well-chosen and stimulating exercises, some with hints and some of which ask for proofs of well-known theorems (e.g., Ramsey's theorem and König's theorem). At the end of each chapter there is an informative list of notes consisting of historical, bibliographical, and technical comments. ... The book is in general well-conceived and well-written as a beginning textbook.