1.1. Contents.

Preface
Introduction
Chapter I. Homogeneous Coordinates and the straight line.
Chapter II. One-one Algebraic Correspondence.
Chapter III. Cross-ratio and Harmonic Ranges.
Chapter IV. The Conic, treated parametrically.
Chapter V. Conic-Locus and Envelope.
Chapter VI. Special Forms of Equation.
Chapter VII. Correspondence on a Conic.
Chapter VIII. Quadrilateral, Quadrangle and related Results.
Chapter IX. Pencils of Conics.
Chapter X. Miscellaneous Properties.
Chapter XI. Relation to Euclidean Geometry.
Chapter XII. Applications to Euclidean Geometry.
General Examples
Answers to the Examples
Index

1.2. From the Publisher.

This is a book about powerful mathematical methods rather than a mere catalogue of the properties of conics. The treatment is elegant and refreshing as well as systematic, and throughout the book the author is concerned to lay the foundations of future work. Consequently (unlike too many textbooks) there is little for the student to unlearn when he goes on to more advanced courses. A valuable feature is the large number of carefully selected and graded problems for solution (about 400 in all).

1.3. From the Preface.

This book is an introduction to the methods of projective geometry, based on the use of homogeneous coordinates. It is intended for pupils in their last year at school and their first year at the university. It is written as a study of methods and not as a catalogue of theorems; and I hope that a student reading it will have nothing to unlearn as he proceeds to apply these methods to study the geometry of figures In three dimensions or in higher space.

The first three chapters introduce homogeneous coordinates, the equation of the straight line, duality, one-one algebraic correspondence and cross-ratio. The fourth chapter (which some teachers may prefer to leave until a later part of the course) deals with the conic, treated parametrically. In the fifth and sixth chapters, the standard properties of conics are obtained, care being taken to show the importance of an intelligent choice of the triangle of reference. The seventh chapter applies the theory of one-one correspondence to the study of conics, including Chasles's theorem. The eighth chapter gives an account of the quadrilateral and the quadrangle, and of pencils of conics through four pointe or touching four lines; in the ninth chapter more general pencils are considered. The methods given earlier in the book are applied in the tenth chapter to the study of various classical properties; these properties are not discussed in much detail, but it is hoped that the reader will be encouraged to read more advanced works.

In the first ten chapters the ideas of length and angle are not used at all. The eleventh chapter gives the rules for interpreting the projective results metrically, and in the twelfth chapter these rules are applied to a variety of problems. Here, again, there is no attempt to be exhaustive; the methods are the important things.

Though the book is mainly on the use of homogeneous coordinates, I have not hesitated to introduce the methods of Pure Geometry where they seemed most suited to my immediate purpose. The good geometer should move freely in both Pure and Analytical Geometry, and he will find here the use of both.

A word should be said about the examples. I have searched the Cambridge Examination Papers for many years back, and I have also used other sources when I could get a copy conveniently. I found the arrangement hard; the trouble with examples in geometry is that they can be tackled by many different methods, as any examiner knows. I have tried to place them where they arise most naturally, and I suggest that, as the reader progresses through the book, he should turn back to the examples in earlier chapters to see whether he can solve them more simply in the light of his increased experience. I have added some routine examples in the first few chapters till the reader gets the 'feel' of the subject, but later examples are almost entirely taken from papers; an exception had to be made in Chapter IV, where the topic seems to have eluded the examiner. There are also no examples to Chapter XI, as it is really an introduction to the following chapter.

1.4. Review by: Ernest Preston Lane.
Science, New Series 105 (2724) (1947), 296.

This is a textbook on analytic projective geometry in the plane. Plane analytic geometry as ordinarily taught in the freshman year in universities and colleges in the United States is a reasonable prerequisite for understanding it. Although the author insists that he is primarily interested in the methods of projective homogeneous coordinates rather than in the geometrical content of the volume, he nevertheless gives a very satisfactory account of the theory of configurations constructed of points and straight lines, and his discussion of conics might even be called elaborate. Numerous exercises, taken mostly from old examinations, are provided, appropriate reference to the examination which served as the source of each such problem being given.

Doubtless the author had reasons which seemed to him adequate for including no geometrical figures, but to the reviewer if would seem to be good pedagogical practice to illustrate such a treatment with ample drawings and diagrams.

This scholarly work at an elementary level should be supplemented by a similar book on the analytic projective geometry of ordinary space.

1.5. Review by: Alan Robson.
The Mathematical Gazette 31 (293) (1947), 58-59.

This is a book for pupils in their last year at school or their first year at the university. It is true to its title, being genuinely a book about methods and not merely about properties of conics, though these are sufficiently treated by way of illustrations of the methods. It is clearly written and ought to be useful to many students. Any difficulties that they may find will be due to lack of background rather than to intrinsic difficulties of the text. To say that "the reader should become familiar at the earliest possible moment with the idea of duality" is an understatement: he ought really to have been becoming familiar with it for the past year or two; because the acquisition of a properly constituted dual mind is not a process that can be rushed. For this and similar reasons, the reviewer feels that there are topics in this book which ought to be taken up at an earlier stage than the last year at school. On the other hand, if a student has been brought up with a hopelessly biased metrical outlook, the book will be an excellent corrective to be applied before the university stage.

There are chapters on homogeneous coordinates applied to the line, on one-one algebraic correspondence, and on cross-ratio and harmonic ranges. The conic is then treated parametrically and also from the locus and the envelope point of view, including the special equations referred to convenient triangles of reference. Then follow chapters on the quadrangle and quadrilateral, pencils of conics, and properties such as the harmonic locus and envelope, and out-polar and in-polar conics. The spirit of the book is well illustrated by the titles of the last two chapters which are "Relation to Euclidean Geometry" and "Applications to Euclidean Geometry"; and it is illustrated too by the fact that there are no diagrams. The author recognises, however, that the geometer should move freely in both pure ant analytical geometry, and he has not hesitated to use methods of pure geometry where they seemed most suitable to his purpose.
...

1.6. Review by: C N S.
Current Science 16 (2) (1947), 65.

The number of available books on the analytical geometry of conic sections is so large that one would naturally ask for himself whether it is worth while going through one more treatise on the subject. But the pre sent book is really a welcome addition. Whether one agrees or not with the arrangement and the method of treatment adopted by the author, the book is interesting reading. The treatment of the chapters "One-One Algebraic Correspondence" and "Cross-Ratio and Harmonic Ranges" is very satisfactory from the algebraic point of view, and one will feel sorry that the same exhaustive treatment is not kept up, and that important subjects such as the invariants of two conics are not touched. The author, it is true, plainly says that his book "is written as a study of methods and not as a catalogue of theorems" and hopes "that a student reading it will have nothing to un learn as he proceeds to apply these methods to study the geometry of figures in three dimensions or in higher space". But the treatment of the subject is so refreshing that we cannot help wishing for some more of these methods, and a more exhaustive treatment of many chapters would not have reduced the book to a catalogue of theorems. This is not a criticism against the book, but the reviewer feels that this is the best compliment that can be given for the book. In contrast with the above, the examples in the book are numerous and varied in character, and form a regular catalogue. Many of them are taken from various recent examinations, and hence provide welcome additions to the well-known stereo typed problems available in all books. The book can be strongly recommended to the student and to the teacher of the subject

1.7. Review of 1960 reprint by: Anon.
Current Science 31 (3) (1962), 125.

This book was first published in 1946 and has already gone through six reprints, which speaks for its popularity and demand amongst those for whom it has been intended. In this book the author has given an excellent introduction to the methods of projective geometry, based on the use of homogeneous coordinates. It is not merely a catalogue of theorems but an appreciation of the methods, and as such lays the foundations for extensions in the subject as for example, study of the geometry of figures in three dimensions or in higher spaces. A valuable feature of the book is the large number of carefully selected and graded problems for solution.

2.1. Review by: Alan Robson.
The Mathematical Gazette 34 (308) (1950), 144.

This book should be a help to teachers who wish to stimulate the study of pure geometry by able pupils at the period, just after the stage of the old-style school certificate, when there is a temptation to make the course too exclusively analytical.

It is arranged in three sections each dealing with about 16 configurations. There are about 260 examples along with the configurations and 120 other examples taken from examination papers.

The treatment is original and should, as the author hopes, rescue the theorems from the isolation in which they often seem to stand.

In the first section, which deals chiefly with properties of the triangle, the threads of more elementary work are gathered together and some interesting new matter is added. Section II leads the reader gently from metrical ideas to a quasi-projective geometry resting on certain metrical props. The topics considered include duality, the line at infinity, ranges, pencils, projection, section, Menelaus and Ceva, Desargues, Pappus and involution. These lead on in section III to properties of the circle, many of which apply without modification to conics, and the configurations include those of the Pascal and Brianchon theorems, pole and polar, coaxal circles, and figures obtained by inversion; but conics are regarded as just outside the scope of the book.

The author does not overlook the fact that there will usually be a concurrent course of analytical geometry, but believes that a short course of pure geometry has a value of its own in helping to trace a continuous argument from the simple beginnings into the richness afforded by projective geometry.

The reviewer finds the account of points at infinity quite unconvincing, and feels that this is because the subject (without the help of coordinates) is too difficult for a school class. While he welcomes the projective oases, he fears that the quasi-projective treatment may lead to misunderstandings that will be hard to remove at a later stage.

2.2. Review by: Howard F Fehr.
The Mathematics Teacher 43 (6) (1950), 301.

The pattern of modern Euclidean geometry for college students has been set by such texts as Casey's and Altshiller-Court's. It is refreshing to have a new text with a novel presentation of the subject, and with the inclusion of some projective geometry. The text covers all the usual content of college geometry, has a wealth of riders (original exercises) and a fair introduction to concepts of projective geometry. Each separate presentation begins with a configuration of which there are 47. Before the figuration is a development of important proper ties of the figure, without any formal enunciation of a theorem. There then follow a set of problems for solution, all relating to the given configuration. A student thus obtains a thorough, deep, and whole view of a large configuration rather than a catalogue of many varied relationships in many figures. The text promises to give a real insight into the nature of modern geometry as a closely integrated structure rather than a mere series of theorems.

3.1. From the Introduction.

The purpose of this book is at once modest and ambitious, namely, to provide a short introduction to algebraic geometry in space of three dimensions, to make clear its spirit, and to prepare the way for deeper study. I have in mind a reader who has just read my book on homogeneous coordinates in a plane (to which this stands as a second volume) and is in the early stages of his second-year work at the University. I have also in mind a class of reader who has read further in mathematics generally, but has found the existing detailed accounts of this work too full or too specialised for his own needs; I hope that such a reader will find here a temptation to get to grips with the subject.

In spite of the existence of a large number of textbooks on the geometry of space of three dimensions, I think it is true that there are few which deal with the subject in the essential spirit of projective geometry. The two accounts which seem to me to be of greatest importance for further study are, first, a well-established authority, the Principles of Geometry, Vol. III by Prof H F Baker, and, secondly, an important recent work, Projective and Analytical Geometry, by Dr J A Todd which for the first time (I think) establishes in readily available form the synthesis of projective geometry with modern matrix algebra. The aim of the present book will be fulfilled if it encourages the reader to turn to these two accounts and, perhaps, helps him a little along the way.

In place of the elementary examples which usually appear throughout each chapter in a book of this kind, I have included a fairly large number of Theorem-examples; these are almost entirely standard results, which should be known, and which follow directly from the preceding work. The conscientious solution of these examples is an essential part of the reading of this book, and I have added short hints which should remove any possible difficulty.

3.2. Review by: Alan Robson.
The Mathematical Gazette 36 (315) (1952), 62.

This is a sequel to the author's book on the corresponding plane geometry, which was reviewed in Gazette, XXXI, 58. It follows the same general plan. The point, line, plane, quadrics, generators, line-geometry and the twisted cubic are dealt with in six chapters. Chapter VII is concerned with Euclidean geometry, and Chapter VIII with the use of Matrices. There is no attempt to deal exhaustively with these subjects, and for that reason the book may prove particularly useful to young undergraduates who want a quick conspectus of solid geometry. The beginner may receive an unexpected jolt when he reads on p. 114 that he is expected to have had an introductory course of real cartesian solid geometry. Here, there is a real problem for the student or his teacher: to what extent can the old-fashioned metrical geometry be cut short or neglected? The author says enough to show that he does not want it to be done in old-fashioned detail; and the student will have already faced the same questions for plane geometry. The reviewer feels that the change from old point-of-view to new ought to be a gradual transition in sixth forms, and not a sudden one on arrival at the university. Probably it remains for teachers to work out exactly how the transition should be achieved.

3.3. Review by: R C Sanger.
Mathematics Magazine 25 (3) (1952), 166-167.

This text deals primarily with classical elementary analytic projective geometry of three dimensions, with special emphasis on the properties of quadric surfaces, the elements of line geometry, and the elementary properties of twisted cubic curves. No use is made of the metric notation, though an indication of its value is given in the final chapter.

The book is a sequel to the author's work on two-dimensional projective geometry, and references are made to this work. Some knowledge of this subject is necessary for an understanding of the text under consideration.

The work is tersely written, which would make it difficult reading for the average American student. There are no suggestive illustrations, few illustrative examples, and few simple exercises. However, many of the exercises deal with interesting properties of curves and surfaces, and some contain theory essential to the understanding of the subject.

It should be noted that this type of geometry is not at present popular in this country. However, this book should prove to be of value as supplementary reading for those interested in this subject.

3.4. Review by: J W A.
Science Progress (1933-) 40 (159) (1952), 530.

This book is a sequel to an earlier work, with a similar title and by the same author, on plane geometry. The chapter headings are: I, the point, the straight line and the plane; II, the quadric surface; III, the generators of a quadric surface; IV, line geometry; V, the twisted cubic; VI, systems of quadrics; VII applications to Euclidean Geometry; VIII, the use of matrices. The topics covered appear, more or less, in most honours courses on solid geometry.

The author declares that his purpose is to provide a short introduction to the subject, to make clear its spirit and to prepare the way for deeper study. If he had said that his object is to help students through examinations I should have agreed that it is achieved admirably. As it is the introduction is too short in some fundamental respects; and surely the spirit abroad in geometry today demands that first steps be carefully examined.

If the major part of this book is anything to go by, one must infer that Dr Maxwell is a lucid and successful teacher, whose wide experience of examinations is generously applied to the advantage of his readers. They must come to know what geometry is about and to perceive some of its form and grace. There can be no doubt that many students will value the book for the ease with which it can be read. But this ease is sometimes attained only by glossing over certain difficulties (of which the author is aware, as he indicates explicitly, for example, at the beginning of Chap V); the reader's intuition will usually help him through these difficulties (when he sees them), though it must suffer strain in the early part of Chap VII.

4.1. Review by: Charles E Springer.
The American Mathematical Monthly 67 (5) (1960), 489.

This is a sequel to the author's earlier work on the methods of plane projective geometry based on the use of general homogeneous coordinates (Cambridge University Press, 1946). A knowledge of the material in the earlier volume is assumed. The principal emphasis is on projective geometry of ordinary space. One chapter portrays applications to Euclidean geometry.

It is curious that the author replaces the usual statement concerning linear dependence of elements by the remark that the elements are in syzygy. It is regrettable that use is not made of a notation which can be extended easily to the case of n-dimensions. The eighth and last chapter introduces matrices, and contains a clever and novel treatment of line coordinates. The belated mention of matrices deprives the reader of an opportunity to gain practice in the use of this tool. Many standard exercises appear under the heading of Theorem- examples. More difficult exercises are found at the end of each chapter. Following a development of quadric surfaces, there is a treatment of line geometry including the linear and tetrahedral complexes. The twisted cubic and systems of quadrics are studied.

There is much of interest in this well-written account of a selection of topics from classical algebraic geometry. It should be a welcomed addition to the literature because of its uniqueness of approach and its value to one who aspires to read more advanced treatises on geometry.

4.2. Review by: Thomas L Wren.
Science Progress (1933-) 48 (190) (1960), 355.

This book provides a short and clear introduction to algebraic geometry in space of three dimensions, treated in the spirit of projective geometry. The first six chapters establish fundamental properties, in projective space, of quadric surfaces, twisted cubic curve, line geometry, and pencils and nets of quadrics. The next chapter deals with applications to metrical properties in Euclidean space. The last chapter gives a brief account of matrix algebra, showing how it may be used to obtain results already established in the preceding chapters. This book presents the subject in a manner which should encourage beginners to make further progress.

5.1. Review by: Francis R Brown.
The Mathematics Teacher 46 (5) (1953), 383.

American teachers will find this text by an English author interesting reading. The nature of the materials presented is indicated by the table of contents which lists the following twenty chapters: Coordinates, The Straight Line, Digression on Linear Equations, Elimination and Determinants, The Straight Line (continued), The Straight Line in "Perpendicular" Form, Introduction to Analytical Methods, Two Standard Curves, The Circle, The Ellipse, The Hyperbola, The Use of Calculus, Some Typical Curves, Envelopes, Focus and Directrix, Some Problems on Tangency for Conies, Geometrical Properties of the Parabola, Geometrical Properties of the Ellipse, Geometrical Properties of the Hyperbola, and Polar Coordinates. The author suggests various other sequences that may be followed.

The average class might find parts of the text a little difficult to read and to keep organised. In the first chapter, the author has introduced such topics as translation and rotation of axes, theorem of Menelaus, parameter equations (which are not defined until fifty pages later). The author considers determinants and the calculus as fundamental problems in equipment, but he does not insist on their use.

5.2. Review by: P M Hunt.
The Mathematical Gazette 37 (322) (1953), 300-301.

This book, at sixth form level, is intended for those who hope to become mathematical specialists.

The subject is developed from the beginning. Coordinates and straight lines are discussed in the first two chapters but in order to be able to deal more conveniently and compactly with certain aspects of the subject to follow later, chapter III is devoted to a discussion of the solution of linear equations and determinants. The remainder of the book is written, however, so that although determinants are actually used in the text they are not essential to the arguments involved.

Chapter IV has as its title "Introduction to Analytical Methods". The emphasis here is on the methods available and these are given with reference to a particular typical curve so that a thorough survey of the way in which the subject is to develop is obtained.

The portion of the book dealing with conics can be divided into two parts in the first of which the methods of analytical geometry are used. The curves, apart from the circle, are defined parametrically and appear in the following order; parabola, rectangular hyperbola, circle, ellipse, hyperbola, which is chosen since the circle although most familiar geometrically is less attractive analytically. In the second part, where the focus-directrix definitions are given, the methods of pure and analytic geometry are used and are closely interwoven, the emphasis being on the geometry of the figures. Between these two parts three typical curves possessing a point of inflexion, a cusp, and a double point are discussed parametrically and a chapter is devoted to envelopes in which the locus of the line $lx + my + n = 0$ where $l, m, n,$ are functions of $t$ is considered. The book concludes with a chapter on polar co-ordinates in which the polar equations of the conics are established.
...

6.1. Review of Vol 1 and Vol 2 by: N M H L.
Science Progress (1933-) 42 (168) (1954), 708-709.

These volumes are the first two of four which in the words of the author "present the subject in such a way that the more exact treatment, when it comes, can follow by natural development, without being forced to return to a fresh beginning which is often felt to be both unnecessary and even pointless." He states that, while he has tried to keep the standard of discussion at a level which the mathematical specialist will appreciate, the needs of the scientist and engineer have not been forgotten. The reviewer has a great deal of sympathy with this point of view but feels that the pace of the first volume is much too great for an introduction to the subject. By the time the student has reached the stage of the second volume, namely the Advanced and Scholarship levels of the General Certificate of Education, it may be presumed that he will be better able to appreciate the author's method of presentation, and it is felt that this volume is eminently successful.

6.2. Review of Vol 1 and Vol 2 by: Douglas Arthur Quadling.
The Mathematical Gazette 39 (327) (1955), 83-85.

This set of four volumes will, when completed, lead the student right through his study of calculus at school and well on into the university course. The first volume covers rather more of the subject than would normally be introduced in the first year in the sixth form. In the first three chapters the definitions, techniques and applications of differentiation are discussed in turn, and integration is developed along similar lines in the remaining three. This volume on its own would be very suitable for a sixth form pupil who wishes to study mathematics as a subsidiary interest; but it is clear that Dr Maxwell has primarily in mind the mathematical specialist, and he writes all the time with an eye to the future. The faster worker will find some strong meat in the optional sections at the ends of the chapters - the Cauchy mean value theorem, limits of "indeterminate forms ", reduction formulae and areas of surfaces of revolution (this last discussed, strangely, before length of arc, which is postponed until Volume II).

The second volume begins with a full account of the functions $\log x$ and $e^{x}$, based on the integral definition; this is followed by chapters on series, plane differential geometry, complex numbers (but not the complex variable), systematic integration and "integrals involving infinity". There is an appendix on partial differentiation, but this is only a brief sketch, presumably inserted in order to cover within this volume the syllabus of certain examinations. Functions of more than one variable are to be the principal subject of Volume III.

Dr Maxwell usually maintains a sensible standard of rigour.
...
These books can be wholeheartedly recommended for the mathematically abler pupil. They are written in the attractively informal style which we have come to associate with Dr Maxwell's school textbooks. The remaining volumes will be awaited with interest.

6.3. Review of Vol 1 and Vol 2 by: Myron F Rosskopf.
The Mathematics Teacher 48 (5) (1955), 353.

The two volumes present in a clear, concise fashion the fundamental processes of differential and integral calculus. A reader from the United States notices the sparing use of italics, the lack of heavy type and boxes to set off formulas, and the small number of exercises at the end of the sections.

The books are written for upper high school and beginning college students. The purpose is to present ideas with a rigour sufficient to the mathematical maturity of the students. Where it is impossible for the author to be mathematically rigorous at the level of his audience he does not hesitate to postulate results or to depend upon descriptive statements to make clear to a student a process. This reviewer was struck by the author's ability to make intuitively clear a knotty point in calculus and yet to write in such a way that no mathematician could legitimately criticise him.

There are no pictures in the two volumes. Where it is absolutely impossible in the opinion of the author to make a clear presentation otherwise, a line drawing is used. These drawings are of the classroom variety, the sort a good teacher would use in working with a group.

6.4. Review of Vol 3 by: N M H L.
Science Progress (1933-) 43 (170) (1955), 339.

This is the third of four volumes designed to bridge the gap between the Calculus as taught to beginners in schools and Analysis as taught in the Universities. It is chiefly concerned with the extension of the concepts of the first two volumes to functions of several variables. The author finds it necessary in places to relax the high standard of rigour which was set for the previous volumes but is always careful to point out where this relaxation occurs; in fact the student will have nothing to "unlearn" if and when he arrives at the stage of a specialist mathematical course. The book is intended also for students of science and engineering and these will, no doubt, find the degree of rigour adequate for their needs.

The volume deals with partial differentiation, maxima and minima of functions of several variables, Jacobians, multiple integrals and concludes with a chapter on the sketching of curves which, while excellent in itself, does not seem relevant to the rest of the book. The treatment is excellent throughout and many examples are worked in the text with very full and careful explanations. Each chapter concludes with a set of exercises; the student who masters these will undoubtedly have a thorough knowledge of the subject.

As with the previous volumes, the production of the book leaves nothing to be desired. It is a pity, however, that a work which sets such a high standard should have any misprints; they are the more misleading in that the reader expects to find complete accuracy. ... These are, however, minor blemishes which the reviewer feels bound to mention in order to show that he has read the book; they will, no doubt, be removed in future editions. The book will undoubtedly prove of great value to the type of student at which it is aimed and will give many useful ideas to the thoughtful teacher.

6.5. Review of Vol 3 by: Douglas Arthur Quadling.
The Mathematical Gazette 39 (329) (1955), 251.

The greater part of this volume is devoted to functions of more than one variable. It is unusual to find a textbook at this level which gives so much space to this subject, and the clear exposition to be found here is likely to prove popular with university students and scholarship candidates. The scope of the book is roughly that of a first year university course; there are chapters on maxima and minima (in which Lagrange's method of undetermined multipliers is clearly explained) and on Jacobians.

As in the preceding volumes, the standard of rigour is sensible, and the assumptions made are carefully stated. The formulae of partial differentiation are derived on the assumption of the continuity of the partial derivatives, but later on the author takes pains to show that the "negligible" terms in certain multiple summations do in fact tend to zero in the limit. Dr Maxwell never conceals the fact that there is a pill to be swallowed, but he provides a coating of sugar by introducing many topics with particular examples and by referring to geometrical illustrations where appropriate.

The rest of the book deals with the sketching of curves. This is an exciting subject, full of interest for the budding mathematician, and for this reason, if for no other, most teachers will wish to place it earlier in the course than this book suggests. There is, however, very little in the treatment given here which the pupil could not tackle at a much earlier stage. The subject is one which suffers from being on the fringe of several branches of mathematics. The approach of this book is purely analytical, and the teacher will probably wish to supplement it with reference to the methods of algebraic geometry.

The method described for finding successive approximations at the origin and at infinity is that of undetermined coefficients and exponents. This is quite general, though some pupils may find it rather difficult and tedious to apply in simple cases where other methods are available.
...
This volume can be recommended as heartily as its predecessors. It will be all the more welcome because its subject-matter receives scant attention from many existing textbooks.

6.6. Review of Vol 4 by: Pasquale Porcelli.
The American Mathematical Monthly 65 (7) (1958), 536-537.

This is the final volume in a series of classroom textbooks by the author covering the calculus and early stages of analysis. The volume under review is divided into three sections: 1) Ordinary differential equations, 2) Functions defined by infinite series and integrals, and 3) Laplace and related equations. The theme of the book is set forth in the first section and it is directed towards enabling the student to develop skill and confidence in solving differential equations. Among the assets of the book are 1) its spirit of liveliness and intellectual honesty, 2) selection of topics, 3) careful statements of the conditions under which the techniques used are correct, 4) numerous examples worked out in detail to illustrate when the techniques succeed or fail, and 5) an abundance of exercises for the student. His proof of theorems, when given, are well within the understanding of the calculus student. The emphasis on the first section is on the $n$th order differential equation with constant coefficients; in the second section, which is the largest, on a) handling of series including term-wise integration and differentiation, b) improper integrals and the differentiation and integration of functions defined by integrals, and c) an introduction to Fourier series; the third section contains an excellent introductory treatment of the Laplace, heat, and wave equations, Jacobians, and spherical harmonies. Special functions such as the Bessel, Beta, Gamma, and Legendre functions are discussed. The book is distinctive and well written.

6.7. Review of Vol 4 by: T M Flett.
The Mathematical Gazette 45 (351) (1961), 56-57.

This is the final volume of the author's series on the Calculus. The first two volumes cover the standard theory of differentiation and integration of functions of a single variable up to first year University level, and Volume III extends this theory to functions of several variables. This final volume is devoted to the study of differential equations and those parts of analysis concerned with functions defined by infinite series and integrals.

The book is divided into three sections, headed respectively Ordinary Differential Equations, The Definition of Functions by Infinite Series and Integrals, and Laplace's Equation and Related Equations.
...
The whole volume provides an excellent introduction to the problems which are encountered in the solution of differential equations. It does to a very considerable degree achieve the author's aim of "bridging the gap between the works used in schools and more advanced studies with their emphasis on rigour", and can be recommended as whole-heartedly as its predecessors.

6.8. Review of Vol 4 by: N M H L.
Science Progress (1933-) 46 (182) (1958), 348-349.

This is the last volume of a series of four. The first volume began with the simplest ideas of differentiation and the last ends with a chapter on spherical harmonics. The student is thus led from the beginning of the calculus to the end of a first year course at the University, or even beyond this. The aim throughout has been to make the exposition as rigorous as possible having regard to the type of student who is likely to make use of the book; the author does not hesitate to resort to "informal" methods on occasion, though he is always extremely careful to point out when this is done.

The volume is divided into three sections with the titles:- Ordinary Differential Equations, The Definition of Functions by Infinite Series and Integrals, Laplace's Equation and Related Equations. The first section is concerned chiefly with methods for solving linear differential equations with constant coefficients which are very clearly explained and illustrated by numerous examples. The second section begins with chapters on convergence and leads to the solution of differential equations by means of infinite series with a final chapter, which is very well written, on Fourier Series. The third section gives a sketch of some methods of solution of some of the partial differential equations of mathematical physics. As only fifty-four pages are devoted to this part of the work, including examples, the pace is necessarily fast and the reviewer feels that the students for whom the work is intended will find the matter somewhat indigestible.

7.1. Review by: Eric John Fyfe Primrose.
The Mathematical Gazette 43 (345) (1959), 211-212.

There are so many books on plane coordinate geometry that my first reaction was to doubt whether there could be any good reason for writing another. Having read the book I am converted. I cannot remember having read any other book in which the author takes so much trouble to anticipate the difficulties which a beginner may meet. To take one example, he explains that $xy^{2}$ means $x$ multiplied by the square of $y$, but $XY^{2}$ means the square of the distance between the points $X$ and $Y$. All through the book the author explains carefully what he is doing and (perhaps more important still) why he is doing it.

The early part of the book gives the impression that the author is hopping rapidly from one topic to another before finally settling down. However, this is deliberate: the intention is that the beginner should have an early introduction to some of the curves which he will study systematically later.

Most of the book is concerned with straight lines and conics (including the general equation of a conic). The last two chapters introduce some further curves. These are mostly curves which have simple geometrical definitions. These include familiar friends such as the Cissoid of Diocles and the Witch of Agnesi, but also some less familiar curves such as Watt's Lemniscate, which is defined by means of a simple linkage.

This is a book which can be strongly recommended to sixth-formers. It will be particularly valuable for those who have to do a good deal of the work by private study.

7.2. Review by: Earl LaFon.
The American Mathematical Monthly 67 (3) (1960), 308.

The author aims to present the minimum amount of coordinate geometry of three dimensions, which he considers necessary for the young mathematician, together with some vectors and tensors. Believing that coordinate methods are essential and that a background should be laid for the application and appreciation of vectors, he presents planes and straight lines first. This preparation permits a presentation of vectors with an algebraic basis. The chapter on tensors is largely confined to the very useful $\delta_{ij}$ and $\epsilon_{ijk}$. There follow chapters on spheres, central quadrics, paraboloids, and the general quadric. Such topics as Joachimstal's ratio equation, coaxial spheres, tangents, poles and polars, diameters, centres, and generators are treated. Two treatments of the general quadric are given, one ordinary and the other with the double suffix notation of tensors.

The work throughout is carefully organised. The level is above the elementary. The presentation is clear and rigorous. The exercise lists abound with excellent problems. The author accomplishes his stated aims.

8.1. From the Preface.

The aim of this book is to instruct through entertainment. The general theory is that a wrong idea may often be exposed more convincingly by following it to its absurd conclusion than by merely denouncing the error and starting again. Thus a number of by-ways appear which, it is hoped, may amuse the professional and help to tempt back to the subject those who thought they were losing interest.

The standard of knowledge expected is quite elementary; anyone who has studied a little deductive geometry, algebra, trigonometry and calculus for a few years should be able to follow most of the exposition with no trouble.

Several of the fallacies are well known, though I have usually included these only when I felt that there was something fresh to add. ... I have also tried to avoid a bright style; the reader should enjoy these things in his own way.

My original idea was to give references to the sources of the fallacies, but I felt, on reflection, that this was to give them more weight than they could carry. I should, however, thank the editor of the Mathematical Gazette for his ready permission to use many examples which first appeared there.

It was with pleasure that I received the approval of the Council of the Mathematical Association to arrange for the Association to receive one half of the royalties from the sale of this book. I welcome the opportunity to record my gratitude for much that I have learned and for many friendships that I have made through the Association.

I must express my thanks to members (past and present) of the staff of the Cambridge University Press, who combined their skill and care with an encouragement which, in technical jargon, became real and positive. I am also indebted for valuable advice from those who read the manuscript on behalf of the Press, and to my son, who helped me to keep the proofs from becoming unnecessarily fallacious.

8.2. Review by: Thomas Arthur Alan Broadbent.
The Mathematical Gazette 44 (349) (1960), 234.

We all know how to prove that any triangle is isosceles, and how to expose the flaw in the argument by drawing an accurate figure. But do we all know how to deal with the contumacious opponent who maintains that his diagram is just as likely to be right as ours? If we have not got Pasch's axiom at our finger-tips, we had better buy Dr Maxwell's book at once. He has not only made an interesting collection of fallacies, new and old, and exhibited the logical errors, he has in most cases traced these to their roots. The schoolboy at about A level, and his teacher, should find the book as useful as it is entertaining. The proper answer to the pupil who says "No doubt your method gives the right answer, but I still do not see what is wrong with my method" is a reference to basic principles and logical deductions, and it is this constant reference back to fundamentals which lifts Dr Maxwell's volume far above the usual level of the "Mathematics for fun" type of book.

From the text, we may deduce that the author is a humane and experienced examiner, for some of his items have surely been acquired in the course of patient and toilsome efforts to decipher and unravel complications in examination scripts. We might also deduce his mathematical nihilism, from the glee with which he proves that there are practically no numbers or lengths, that a circle has neither inside nor radius, that there are no variables, no quadrilaterals, almost no inverse points. The further deduction that his generosity is delicate and abundant may be made from one sentence in the preface: "It was with pleasure that I received the approval of the Council of the Mathematical Association to arrange for the Association to receive one half of the royalties from the sale of this book." Members of the Association now have a simple but effective way of showing their gratitude.

8.3. Review by: Philip Rabinowitz.
Science, New Series 130 (3388) (1959), 1570.

Almost everyone who has studied high-school mathematics has been confronted with proofs that 0 = 1 and that every triangle is isosceles. This book is concerned with these and many more fallacies, defined by the author as proofs which lead by guile and plausible reasoning to a wrong conclusion. Some of the fallacies are of a trivial nature; others lead to a deeper understanding of the mathematics involved. Examples of both kinds are given, but much more emphasis is placed on the nontrivial fallacies, most of which come from the domain of geometry.

Maxwell first gives a number of fallacious proofs from some discipline of mathematics such as geometry, algebra, differentiation, or integration, and asks the reader to discover the fallacious step in the argument. Then he provides a commentary on each fallacy; this may consist of a few words or a long discussion. Several of the discussions on the geometrical fallacies presuppose a sound knowledge of geometry, which an English college freshman may already have but which an American student acquires only if he takes college geometry.

The book ends with a series of howlers which are almost the opposite of fallacies; here we find solutions of problems by incorrect methods that lead to correct results. These howlers were taken from real life and provide a certain amount of amusement. However, much more enjoyment as well as enlightenment is provided by trying to detect the fallacies, or at least by reading the solutions given by the author of this lovely little work.

8.4. Review by: Editors.
Mathematical Reviews MR0099907 (20 #6343).

Many of the fallacies in this book will be interesting to American high school students; some require a knowledge of calculus. There are 11 chapters: Mistake, howler and fallacy; Four geometrical fallacies; Digression on elementary geometry; Analysis of the fallacy of the isosceles triangle; Analysis of other geometrical fallacies; Fallacies in algebra and trigonometry; In differentiation; In integration; Fallacies based on the circular points at infinity; Some 'limit' fallacies; Some miscellaneous howlers.

8.5. Review by: Bent Birkeland.
Nordisk Matematisk Tidskrift 7 (3) (1959), 125.

This is a collection of false mathematical reasoning and "evidence" for obviously wrong propositions. The errors are mostly old and well-known (division by zero, wrong sign for square roots, geometric evidence from wrongly drawn figures, and the use of divergent series and integrals), but they are neatly disguised, and give "evidence" that looks highly credible. The comments associated with each "fallacy" are very good. They clearly show where the error lies, and often contain some interesting additional remarks, which in a couple of cases are very profound and detailed. The book also contains some very funny examples of what the author calls "howlers", that is, obviously insane reasoning that gives the right result. The collection will certainly be difficult for high school students, but that does not prevent much of the material from being able to be used to advantage in teaching, both in high schools and higher secondary courses.

8.6. Review by: James R Newman.
Scientific American 202 (2) (1960), 178.

The author has gathered a number of relatively elementary fallacies in different branches of mathematics, demonstrations that lead by guile to wrong but plausible conclusions. In this category are proofs that every angle is a right angle, that every point inside a circle lies on its circumference, that every triangle is an isosceles triangle, that 0 is greater than zero, that all numbers are equal, that $\large\frac{1}{x}\normalsize$ is independent of $x$, that all lengths are equal, that plus one equals minus one, that $\pi$ equals zero, and so on. Anyone, says Maxwell, who has studied a little geometry, algebra and trigonometry for a few years, "should be able to follow most of the exposition with no trouble." But the principal merit of the book, namely a somewhat deeper analysis of the sources of certain fallacies, is accessible only to the more practiced and sophisticated mathematician.

8.7. Review by: T L Wren.
Science Progress (1933-) 48 (189) (1960), 115-116.

The author's aim in writing this book is "to instruct through entertainment. The general theory is that a wrong idea may often be exposed more convincingly by following it to its absurd conclusion than by merely denouncing the error and starting again."

He has dealt with a number of fallacies of various types. In each case an unbroken statement of the fallacious argument gives the reader an opportunity to detect the error unaided; this is followed later by an analysis sometimes of quite unexpected depth, tracing the error to its most elementary source.

This book will be instructive to beginners, as the standard of mathematical knowledge expected is quite elementary; and it contains much that will be interesting and entertaining even to experienced mathematicians.

8.8. Review by: Aubrey J Kempner.
The American Mathematical Monthly 67 (3) (1960), 309.

By definition, "the fallacy leads by guile to a wrong, but plausible conclusion; the howler leads innocently to a correct result." - This small book is devoted essentially to fallacies. At the end, a few howlers, apparently encountered in class, are included. ...
...
The fallacies are taken from the fields of algebra, trigonometry, analytic geometry and calculus. The theory of sets is on the whole disregarded, thus eliminating a fruitful source of good fallacies. The level of the fallacies, as well as the level of the explanations, varies within wide limits. The fallacy that every triangle is isosceles is taken up in three different forms. The analysis of one of these occupies nine pages and leads to Pasch's axiom, Ptolemy's theorem on four concyclic points, fourth order determinants and the equation in line coordinates of a conic through the vertices of the triangle of reference. ...
...
The circular points at infinity are introduced to explain the fallacies that the four points of intersection of two conics are collinear, and that concentric circles invert into concentric circles with respect to an arbitrary point. The brief explanation given requires a greater degree of knowledge of projective geometry in the complex plane than the preface of the book would seem to assume.

8.9. Review by: Charles W Trigg.
Mathematics Magazine 36 (2) (1963), 131-132.

The announced purpose of this little gem is to instruct through entertainment. The author classifies mathematical errors into MISTAKES, HOWLERS which lead innocently to correct results, and FALLACIES which lead by guile to wrong but plausible conclusions. The fallacies of the isosceles triangle, the right angle, the trapezium and the empty circle are analysed in depth. Then some unusual fallacies in algebra, geometry, differentiation, integration, circular points at infinity, and limits are discussed. The final chapter presents a collection of amazing howlers. Recommended to all who wish to test their logical processes and particularly to teachers who wish additional insight into the sources of student errors.

8.10. Review of 2006 paperback version by: Nick Lord.
The Mathematical Gazette 92 (524) (2008), 366.

All that many Gazette readers need to know is that this is an unchanged paperback re-issue of the charming classic which was originally published in 1959. As Douglas Quadling recalled in his affectionate and appreciative obituary in the March 1988 Gazette, Edwin Maxwell was a generous and devoted stalwart of the Mathematical Association which he served with distinction as Editor of the Gazette, Treasurer and President. Many of the items in this anthology of fallacies first appeared in the Gazette and approximately half of them are concerned with geometry. Some, such as those in Chapter IX involving the circular points at infinity, remind us of how much school mathematics syllabuses have changed since the 1950s; others, such as the multiple 'proofs' that all triangles are isosceles, have enduring appeal and nowadays are fun to explore with geometrical software packages. A real highlight is Maxwell's penetrating analysis in Chapter IV of one of the isosceles triangle 'proofs' which epitomises the truisms that not only are fallacies very rich starting-points for memorable lessons, but also that a full exposé can have unexpected depth. The rest of the fallacies come from algebra, trigonometry and calculus ...
...
Maxwell's original aim for this slim volume was that it would 'instruct through entertainment'. As we near the 50th anniversary of its first appearance, I am certain that this re-issue will attract and captivate a new generation of readers.

9.1. Review by: Eric John Fyfe Primrose.
The Mathematical Gazette 45 (351) (1961), 60.

Dr Maxwell, having already written textbooks on geometry and calculus, now turns to algebra, and the result is a book which should prove very useful in sixth forms of grammar schools. The subject matter corresponds roughly to the syllabus for advanced level examinations, though it starts a little below this and goes considerably beyond it in places. Everything is explained very clearly, and there are many examples, some worked by the author. It is difficult at this level to invent any new methods which are likely to be successful, but as far as I know Dr Maxwell's method for expressing a rational function in partial fractions, when the denominator contains repeated factors, is new and an improvement on existing methods. Several "warning examples" are given, to show the reader that what seems obvious may not be true. ...
...
The first volume should be a great success, and we shall await the second volume with interest.

9.2. Review by: Helen G Russell.
The American Mathematical Monthly 68 (2) (1961), 193-194.

This book deals with the usual topics of advanced algebra, - polynomials and related equations and inequalities, simultaneous equations, determinants, complex numbers, partial fractions, graphs of rational functions, permutations and combinations, the binomial theorem, summation of finite series, and infinite series. Partial fractions are studied in logical detail for both repeated linear and quadratic factors of the denominator. Methods for summing generalised arithmetic and harmonic series are provided. The binomial series lead to expansions of rational functions in power series, these, in turn, to exponential and logarithmic series. To discuss the latter series the author employs elementary calculus but of necessity states the basic assumptions about convergent series; the detailed discussion might better be postponed to a calculus text.

The book is carefully prepared and concisely written. The author emphasises the logical structure of algebra as much as he believes consistent with the mathematical maturity of the students for whom the text is intended. His illustrative material contains techniques for solving routine problems and develops methods which the intelligent student can adapt to the solution of numerous challenging examples. The reader must attend closely both to the theory and the illustrative examples if he is to solve the problems of the text. For such a reader the book contains much of immediate interest and provides valuable training for mathematical growth.

9.3. Review by: Edward H Whitmore.
The Mathematics Teacher 54 (2) (1961), 102.

According to Mr Maxwell this book "... ought to form a basis for most upper school requirements below full Scholarship level, and, in places, beyond." There is no question as to the placement of this book in our own system of education. It definitely contains "college algebra" material and beyond. ...
...
In general, this textbook is a sound and logical presentation of the area of elementary algebra. Though very traditional (the author says "fairly standard") it contains a wealth of in formation and many exercises. The cost of the book is an item that has merit in itself. Any student who completes this book will be prepared admirably for future mathematical work. No algebraic manipulation in the calculus will be beyond him and his work in matrix theory will have the necessary foundations.

On the negative side, the book is quite difficult and formidable. Without a large amount of help from the instructor, only the most able and mature student can handle the material. Although symbols are rather universal in nature, care must be taken with the exactly reversed placement, compared to use in the United States, of the decimal points and period for multiplication. Giving the book the true "foreign flavour" are those problems concerning the monetary system in England. Unless a person is quite familiar with this system the problems may have to be omitted. Some people claim that any coin problems should be omitted although probability naturally adapts well to "flipping pennies." On the other hand, whether an American penny or an English penny was flipped, the effect on the probability of heads is rather small.

10.1. From the Publisher.

Dr Maxwell has written this short book to introduce students (and teachers) to the ideas involved in abstract mathematics. In his preface he states, 'I have often felt that the present plunge into abstraction is too sudden and that there is a need for more elementary work to make the immersion less exhausting.' Dr Maxwell does this by taking various topics from elementary mathematics, and showing what happens when the rules are altered in quite simple ways. He first discusses digital arithmetic, then introduces the idea of a group and illustrates some properties of groups by examples from algebra and geometry.

10.2. From the Preface.

The subject matter of most of the topics developed in this book is believed to be essentially new, though, of course, the ideas have been both foreshadowed and overtaken by many other writers. The aim is to provide material, familiar in substance but unfamiliar in treatment, that may catch the interest of pupils (and, dare I say of teachers?) as they cross into the somewhat puzzling world of abstract mathematics. I have often felt that the present plunge into abstraction is too sudden and that there is a need for more elementary work to make the immersion less exhausting.

To carry the subject forward from this stage will be the work of others; the hope is that this book may help to ease the start.

10.3. Review by: Albert Geoffrey Howson.
The Mathematical Gazette 51 (376) (1967), 172-173.

A Gateway to Abstract Mathematics is a slim book intended to introduce students and teachers to the ideas involved in abstract mathematics. The greater part of the book is an introduction to groups - the approach being original and beautifully accomplished. The remainder is devoted to an entertaining, but bizarre, look at geometry. One's only criticism of this excellent book is that the reader is not informed that two-thirds of the way through he has left the mainstream of mathematics for a quiet tributary of the Cam!

11.1. Review by: Paul R Halmos.
Mathematical Reviews MR0177993 (31 #2251).

This is an elementary textbook. ... It begins with the concept of a binary operation and proceeds slowly to groups, Euclidean spaces, and vector spaces. Rings and fields are defined, and the divisibility properties of integers and of polynomials are discussed. About half the book is on matrices, their manipulation, and their relation to linear equations. There are many exercises. The language is sometimes vague and sometimes unorthodox. Vague: there is, for example, a paragraph on discrete and continuous sets that is unnecessary and confusing, and the treatment of polynomials is not completely satisfactory. ...

On the whole, however, the organisation and the exposition are good, and the unusually leisurely pace would probably make the book attractive to many beginners.

11.2. Review by: Albert Geoffrey Howson.
The Mathematical Gazette 51 (376) (1967), 172-173.

Work on algebraic structure is treated in the first section of this book. Here we meet sets, groups, vectors, rings and fields. The level of treatment can be judged from the fact that Lagrange's Theorem is the limit attained in group theory and that, although the word isomorphic is introduced, the author does not (at that stage) define isomorphisms (although the word occurs in a sub-section heading). The last three chapters of Section 1 contain interesting work on the theory of numbers: congruences, primes and Euclid's algorithm applied to positive integers and polynomial forms. The other two sections in the book are devoted to matrices (introduced with the aid of the simple transformations of rotation and reflexion (sic)) and to more advanced work. The matrix section contains a valuable (but, of necessity, rather difficult) introduction to vector spaces, linear dependence and rank. Quadratic forms, eigenvalues and eigenvectors are considered in the section on advanced work.

There is, of course, no need to comment on the way in which the work is presented. Every reader must have a favourite group of hotels or chain of stores on which he feels he can rely for good service and value for money, and I hope that Dr Maxwell does not think it too unflattering if I compare his books with such a group. It is possible that there is a better hotel or store in the town and there may be a better book on the subject, but one knows that one will not go far wrong in putting one's faith in a hard-earned reputation. Nevertheless, as Dr Maxwell says in his introduction, every reader familiar with the subject will regret some omission or other and will perhaps have alternative views on the presentation of material. Since so little work on "modern algebra" has been done in the class-room, what can and what cannot be done is largely a matter of opinion, but one feels that Dr Maxwell has been too cautious. Chapter 19 - "Some abstract conceptions" - gives the unfortunate appearance of being added to appease the enthusiasts. Is the fundamental idea of an equivalence relation so much more difficult to grasp than that of a field? That it should be relegated to the penultimate chapter, where it is grouped with "homomorphisms", "isomorphisms" and "isometries" (which is the odd man out?), seems to be a great pity. And why were isometries included here? Not, it seems, to provide further examples of groups or even of equivalence classes which were introduced only a few pages earlier. One feels that additional, and earlier, work on these topics would have proved more valuable and interesting than that on eigenvectors and eigenvalues which, divorced from its geometrical applications and separated from its differential equation analogue, seems somewhat arid.

However, any teacher who is considering adopting one of the "modern" A level syllabuses should not be dissuaded by these criticisms from buying this book. Also any undergraduate will derive enormous benefit from working through the many carefully selected examples.

12.1. Review by: John V Tyson.
The Mathematical Gazette 60 (413) (1976), 229-230.

I have been a fan of Dr Maxwell's books ever since I came across them - and him - in my undergraduate days. I was grateful then for their immensely clear exposition and logical structure; and it has always since been a real pleasure to study from them, or to recommend them to others.

On the cover of the latest production it is stated that Dr Maxwell claims to be "a slightly below average Scottish dancer and a slightly above average violinist". I would add that he has always been a very much above average textbook writer and that Geometry by transformations is well up to the high standards we have all come to expect.

The purpose of the book - one in the series of SMP Handbooks - is to put elementary transformation geometry on a more formal basis than is to be found in normal school texts. I would regard it to be most suitable as a book of background reading for teachers at middle and upper school level; and there is no doubt that it would also be useful to undergraduates. It is also an obvious candidate for school and college libraries.

There are three distinct sections. Part I, entitled Basic geometry, is largely a matter of fundamental definitions, with appeal to practical techniques such as paper folding. Sample property to be proved: "the diagonals of a rhombus are perpendicular" - which shows just how very basic is this first part of the book.

Part II is entitled Transformations and their algebra. We are taken through a thorough examination of the algebra associated with reflections, rotations, translations and glides, and then on to the relevant group theory. After the treatment of circles and enlargements there is an interesting chapter on 'spiral similarities' and 'stretch reflections'

Part III is concerned with the application of matrix methods to the geometry previously considered, taking things on to a look at eigenvectors and non-homogeneous similarities (in which the origin is not a self-corresponding point). The work is by no means all simple, but it is dealt with as lucidly as seems possible