## Reviews of Dan Pedoe's books

Daniel Pedoe published a number of books. Perhaps his best known is the three volume treatise

*Methods of algebraic geometry*written with W V D Hodge. Extracts from reviews of this work are given on a separate page. Pedoe wrote a number of other books, several of which were republished with a different title from that of the first edition. This makes the list we give below rather more complicated than it might otherwise have been. We give short extracts from some reviews of these books below, trying to indicate when books are republished with a new title. The books are listed with further editions and reprints given separate numbers. However, we list the books in chronological order of first edition, listing further editions and reprints immediately below the first edition.**1. Circles (1957).**

**1.1. Review by: Nathan Altshiller Court.**

*Amer. Math. Monthly*

**66**(1) (1959), 72.

Pedoe is sure that the circle can be used to make friends for - and influence people in favour of - mathematics. He begins by treating his readers to a few choice bits (or bites?) of modern geometry of the triangle and the circle - and real titbits they are - served in a most attractive way. ... All is accomplished with elementary means in an elegant manner, in the space of a small and slender volume of a few dozen pages.

**1.2. Review by: E J F.**

*The Mathematical Gazette*

**42**(341) (1958), 237.

This interesting little book deals with various topics in mathematics in which circles play an important part. ... The book is intended primarily for university students. However, it would also serve the purpose of introducing some of the topics of higher mathematics to boys and girls before they go to a university, and for this reason it would be excellent for a school library.

**1.3. Review by: H S M Coxeter.**

*Mathematical Reviews*, MR0090058

**(19,761f)**.

The first chapter of this attractive little book contains three proofs of the existence of the nine-point circle and several applications of inversion, including a neat proof of Feuerbach's Theorem, Gergonne's construction for a circle touching three given circles, and a new treatment of the geometry of compasses. ... The third chapter deals with the Argand diagram, groups of Möbius transformations, and Poincaré's conformal model of the hyperbolic plane. In the final chapter, Steiner's proof of the isoperimetric property of the circle is amplified by a careful discussion of the precise meaning of perimeter and area.

**2. 'Circles' republished as: Circles: a Mathematical View (1995).**

**2.1. Review by: Ralph W Cain.**

*The Mathematics Teacher*

**89**(5) (1996), 434; 436.

This excellent little book is an invaluable addition to the library of any teacher of geometry at any level. Its formal presentations of theorems and problems are suit able form ore advanced treatments of the Euclidean geometry of the circle, whereas many of the topics covered and the form of their presentations can easily be adapted tom ore elementary treatments, down to the middle school level. Several classical theorems are presented in easily understandable forms, and such topics as the nine-point circle and the problem of Apollonius are included. Several topics are included, especially in later chapters, that are suitable only for advanced students or for those who wish to pursue them for special projects or enrichment purposes. ... The inclusion of unfamiliar, yet conceptually relatively simple, phenomena associated with circles furnishes examples that should be both interesting and challenging to students.

**3. The Gentle Art of Mathematics (1959).**

**3.1. Review by: Maths Reviews.**

*Mathematical Reviews*, MR0102468

**(21 #1261)**.

A short, interesting book for the layman. There are 9 chapters: mathematical games, chance and choice, where does it end (i.e., transfinite numbers), automatic thinking (logic, algebra of classes, etc.), two-way stretch, rules of play (elementary algebra, groups, etc.), an accountant's nightmare (infinite series), double talk (antinomies), what is mathematics.

**3.2. Review by: R L Goodstein.**

*The Mathematical Gazette*

**43**(344) (1959), 145.

This is a delightful book which makes a valuable contribution to the important task of giving the general reader a glimpse of the mathematicians' collection of jewels. Of special interest are two letters from Isaac Newton to Samuel Pepys discussing the relative chances of throwing a six with six dice or two sixes with twelve dice. ... The book may be warmly recommended to a wide circle of readers.

**3.3. Review by: Rothwell Stephens.**

*The Mathematics Teacher*

**52**(7) (1959), 575.

This thin book, in the traditional and enjoyment, continues a long succession of such books. The plan is largely one of collecting into a chapter well-known related puzzles which can be explained by developing a minimum of common theory. The reading is enlivened by the historical comments and the digressions. These range from Eliza Doolittle to two letters on probability from Isaac Newton to Samuel Pepys. ... Although the author states would like to know what mathematics is about, especially modern mathematics," the spirit of the book and the topics chosen are essentially nineteenth century, with only slight excursions into the twentieth century. The book should be useful in introducing a bright student to a variety of mathematical concepts, even though the theoretical development may be too elliptical.

**3.4. Review by: Philip Rabinowitz.**

*Science, New Series*

**131**(3396) (1960), 295-296.

This is a book for the intelligent layman who wants to know something about modern mathematics and is willing to work a little to attain this knowledge. It starts out entertainingly enough with a discussion of mathematical games. These are used to acquaint the reader with number systems other than the familiar decimal system ... Although there is some new material, presented rather pleasantly, one has the impression that he has seen most of this before in some other popular treatment of mathematics. In addition, there are other failings; these include a very detailed table of contents that promises more than it delivers and several errors, some rather serious.

**4. The Gentle Art of Mathematics (2nd edition) (1963).**

**4.1. Review by: Alonzo Church.**

*The Journal of Symbolic Logic*

**31**(4) (1966), 675.

A book of popular descriptive character about modern mathematics has two chapters in the field of this Journal: Automatic thinking, about algebra of classes; Double talk, about the Epimenides antinomy, Russell's antinomy, Russell's contributions to the foundations of mathematics, mathematical intuitionism.

**5. The Gentle Art of Mathematics (reprint) (1973).**

**5.1. Review by: James D Bristol.**

*The Mathematics Teacher*

**67**(4) (1974), 345.

The topics treated are mathematical games, chance and choice, infinity, sets and logic, topology, groups-rings-fields, series, and more logic. The pertinence of these topics probably depends upon whether or not the teacher has read the original some 15 years ago or other similar of every teacher's mental inventory of back ground information whether or not the material itself is explicitly taught. Here the various topics are treated deeply enough to be accurate and authentic, yet lightly enough for understanding and giving a taste of modern mathematics.

**6. An introduction to projective geometry (1963).**

**Review by: P Vincensini.**

*Mathematical Reviews*, MR0192375

**(33 #600)**.

In his preface the author states a truth (also general) that perhaps we tend to overlook, namely, that it would be very wrong to want to neglect the classical concepts in a modern exposition of projective geometry. We very much need these concepts, and they are the inspiration of much recent work, so we cannot ignore them. Presenting a rigorous modern introduction to projective geometry in the context of the fundamental results from the ordinary perspective, this is the goal that the author has offered. ... Although the book is mostly devoted to plane geometry, the author presents in a few pages in a special chapter (Chapter X), all the essentials of the axiomatic theory of projective spaces of n dimensions. A final appendix groups together all algebraic results used, and helps make the author's book a very useful working tool, an extremely enjoyable read, and easy to adapt to various levels of education.

**7. A Geometric Introduction to Linear Algebra (1963).**

**7.1. Review by: A Jaeger.**

*Mathematical Reviews*, MR0224624

**(37 #223)**.

This is a delightful text on the elements of linear algebra in which the coordinate geometry of two or three dimensions serves as motivation for the fundamental concepts. Among other basic topics, it includes the well-known methods for solving systems of linear equations as well as the manipulation and application of matrices and determinants, but excludes quadratic forms. The algebraic abstractions are gently introduced, the concept of a vector space and its basic properties are first treated in the middle of the book, and linear mappings and their connection with matrices do not appear until the last chapter. Since the author wants to emphasize the development of mathematical ideas, the style is sometimes deliberately informal. ... The reviewer shares the following opinion expressed by the author in the preface: "With the growth of interest in mathematics in high schools, there is every reason why this book should be studied in the higher grades, since the amount of mathematical preparation for its study is slight."

**7.2. Review by: Howard Levi.**

*Science, New Series*

**143**(3612) (1964), 1320.

This textbook for use in an undergraduate course in linear algebra is excellent with respect to the mathematics it sets forth and the manner in which the material is developed. That linear algebra is an essential subject in undergraduate mathematics education has become increasingly evident, and a great many textbooks for courses in this area have been published in the past few years. But few of these texts have all the virtues of Pedoe's book. ... The choice of topics, the clear exposition, the carefully designed and worked out examples, and the excellent exercises all contribute to the high quality of this fine book. The only reservation that I have is concerned with the place of such a course in the undergraduate curriculum. The author uses geometry as a familiar vantage point from which to survey and eventually study linear algebra. This is certainly the correct order of events from the point of view of history, but it is not necessarily the most effective order.

**7.3. Review by: Edwin Arthur Maxwell.**

*The Mathematical Gazette*

**48**(366) ( 1964), 456.

The cover tells us that Professor Pedoe "has taught mathematics at all levels in England, America, Africa and Asia" (the present work was born at Purdue University, Lafayette, Indiana). He has also acquired the happy knack of taking the common run of mathematical topics and making them his own by the freshness of his treatment. ... The book can be recommended without reservation for those who, having acquired a sound basis at school, look forward to seeing fresh views ahead as they approach the abstractions of modern mathematics. Both as textbook and as "interest-catcher" it should prove very useful.

**7.4. Review by: D C Murdoch.**

*Amer. Math. Monthly*

**73**(2) (1966), 216-217.

The title of this book should algebra and not a textbook on that subject. Moreover, the introduction is geometric in that it is firmly based on geometric concepts which are developed in the first four chapters ... In the reviewer's opinion the choice of subject matter makes good sense pedagogically and the book should be useful to at least two groups of students. For the mathematics major it provides a good background in the more concrete and computational aspects of matrix algebra as well as some insight into the generalizations of familiar geometric concepts that are possible in space of n dimensions. On completion of this book a student would be well equipped for a sophisticated course in linear algebra. On the other hand, for the increasing number of students whose major field is not mathematics, but who require some knowledge of matrix algebra and linear systems, an excellent terminal course could be based on this book, although some supplementation would be desirable.

**8. A Geometric Introduction to Linear Algebra (2nd edition) (1976).**

**8.1. Review by: H S M Coxeter.**

*American Scientist*

**65**(4) (1977), 505.

In this eminently readable textbook, each new concept is carefully motivated considered first, and thus the reader be comes prepared for more general situations. ... Since very little previous knowledge is required, the book can be recommended for high school students as well as college freshmen.

**9. A Course of Geometry for Colleges and Universities (1970).**

**9.1. Review by: H S M Coxeter.**

*Mathematical Reviews*, MR0267442

**(42 #2344)**.

This beautifully printed book, with large type and clear figures on large pages, fulfils its purpose admirably. It contains many instances of the author's individual view of geometry.

**9.2. Review by: A A Bruen.**

*Amer. Math. Monthly*

**79**(5) (1972), 532-533.

At first glance this handsome book seems like a most impressive and worthwhile piece of work. The impression is sustained through several subsequent glances. The author is well known as a fine expositor and, equally important, as a fine geometer. These qualities are reflected in this book which thus compares very favourably with most other geometry texts around. Another reason why this book is really good is quite simply the fact that, above all, it is interesting, and consequently very stimulating. I haven't yet used it in a course, but I eagerly look forward to doing so.

**10. 'A Course of Geometry for Colleges and Universities' republished as: Geometry: A Comprehensive Course (1988).**

**10.1. Review by: W J Satzer.**

*Mathematical Association of America*(2010).

http://www.maa.org/publications/maa-reviews/geometry-a-comprehensive-course

The subject is "elementary geometry", defined by the author to mean geometry up to, but not including the rigorous study of algebraic curves. It includes projective geometry in two and three dimensions as well as some geometry in

*n*dimensions. As the title indicates, it is a very comprehensive work, as complete a volume on the subject that I'm aware of. The emphasis here is on the use of algebraic methods to study geometry, and linear algebra in particular is used extensively. The author does use synthetic (or, as he says, Euclidean) geometric proofs on occasion, when he finds them more appealing. ... There is a great deal of material in this book, so much that an unguided student could easily get lost. The author has chosen comprehensiveness over selectivity, and the book - as a textbook - suffers for that. In addition, while the power of the algebraic approach to geometry is clear, it often seems inelegant. Much of the inherent beauty of the subject just doesn't come through.

**11. Geometry and the Liberal Arts (1976).**

**11.1. Review by: J F Rigby.**

*The Mathematical Gazette*

**62**(419) (1978), 61.

This book can be thoroughly recommended to all those interested in the visual and artistic, as well as the mathematical, aspects of geometry. There are chapters on the Roman architect Vitruvius, Dürer, Leonardo da Vinci, form in architecture, Euclid's 'Optics' and 'Elements', Cartesian and projective geometry, curves, and space of up to four dimensions, but these brief headings cannot give an adequate idea of the diversity of topics. ... This well written book is not a textbook, although there is plenty of mathematics in it. It provides admirable leisure reading, but have your ruler and compasses handy: there are practical exercises at the end of each chapter.

**12. 'Geometry and the Liberal Arts' republished as: Geometry and the Visual Arts (1983).**

**12.1. Review by: R Infante.**

*Mathematical Association of America*(2011).

http://www.maa.org/publications/maa-reviews/geometry-and-the-visual-arts

The book's original title was

*Geometry and the Liberal Arts*. Neither title is entirely appropriate. The first four chapters bear the titles "Vitruvius," "Albrecht Dürer," "Leonardo da Vinci," and "Form in Architecture." The bulk of the book, however, including these four chapters, is a presentation of a variety of geometrical facts with some proofs. An exception to this observation is the nice discussion of one-point perspective in the Dürer chapter. The central chapter is the sixth, "Euclid's Elements of Geometry." In a mere 32 pages, Pedoe manages to move from the five postulates through absolute geometry, the fifth postulate, and non-Euclidean geometry in a coherent manner. The unwritten subtext is the central position of Euclidean geometry in Western Civilization: think of Aquinas, Spinoza, and Newton and the explicit modeling of their work on a logico-deductive system. Until recently, we in the West withheld belief in philosophical pronouncements unsupported by an argument. A mere listing of maxims ala oriental mystics just wasn't our style. This mode of intellectual activity is a direct consequence of the impact of Euclid. ... "For whom is this book written?" In the preface, Pedoe makes a passing reference to the "general reader," but even thirty-five years ago the "general reader" would be thrown by the gratuitous mention of the polar equation of a cardioid ... The general reader Pedoe has in mind is someone with a mathematical background beyond high school, say an engineer or scientist or mathematician.

**13. (with Hidetosi Fukagawa) Japanese Temple Geometry Problems (1989).**

**13.1. Review by: L M Kelly.**

*Mathematical Reviews*, MR1044556

**(91g:01005)**.

This fascinating tract is the first publication in English dealing in detail with the intriguing collection of problems and/or theorems culled from the wooden tablets (plaques) displayed in the Shinto shrines and Buddhist temples throughout Japan during the so-called Edo period (1603-1867), when Japan was almost completely isolated from the western world. Many geometric theorems, associated in the west with such names as Casey, Poncelet, Malfatti, Descartes, and Soddy are found explicitly or implicitly among the hundreds of temple problems posed during this period. Approximately 900 tablets are extant, some with as many as 20 problems. The intricate and beautifully rendered configurations were seldom described nor were solutions offered. ... The importance, flavor and significance of this work defies an easy verbal description. It is, in essence, a pictorial essay requiring a persistent intense concentration on the simple and subtle beauty inherent in the plethora of diagrams. ... The publication provides further evidence, if any be needed, of the remarkable artistic flair and intense devotion to detail of the Japanese people. Problemists and competition committees can mine this lode for years.

**13.2. Review by: Dan Sokolowsky.**

*Amer. Math. Monthly*

**98**(4) (1991), 381-383.

The book is a joy to study, and difficult to put down. Certainly geometry-loving readers will find the problems hard to resist and should be warned not to approach this book without lots of paper at hand ... as well as time! I found myself marvelling at the sheer diversity of these problems, and the imagination and ingenuity of their creators. I recall becoming intrigued years ago by ,for example, the well-known Ancient Theorem of Pappus, yet here one finds page after page of such problems, the variations seem endless. From the pedagogical point of view, while not a textbook, the book provides very desirable collateral reading for an appropriate course, say in geometry or the history of mathematics, besides being a sourcebook for innumerable interesting problems. It is certainly a most valuable addition to our body of classical geometry and its history.