1.1.Note.

Reprinted 1945, 1947, 1951, 1956, 1959, 1963, 1969, 1980.

1.2. Review by: Gaines B Lang.
National Mathematics Magazine 13 (4) (1939), 205-206.

The book is written primarily for university undergraduate students, and purposely excludes topics suitable for graduate work. No knowledge of mathematics beyond the elements of differential and integral calculus is presupposed, and Part I can be read without this.

In Part I the theory of convergence of series is developed to the extent usually found in the appropriate chapter of a typical algebra text, but with more detail.

Part II covers most of the remaining theory to be expected in an elementary text book, including absolute convergence, uniform convergence, multiplication of series, binomial, exponential, and logarithmic expansions, power series, and the integral tests. The order notation is introduced, and Tannery's Theorem is presented in a more general form than is usual. There are short chapters on double series and infinite products. Summation of divergent series is considered to the extent of defining and proving consistency for summation (C, 1). Part II closes with a brief chapter on Fourier Series. ...

Problems are numerous and have a considerable range of difficulty. Several features of the book are worthy of mention. A short- hand notation for such terms as "there exists", "such that", etc. is used consistently throughout the book. The word "divergent" is confined to series such that the sum of n terms becomes infinite as n increases without limit, and the term "non-convergent" is used to describe series which neither converge nor diverge. Certain of the hypotheses in some of the theorems are emphasised by bold type, while the form of other theorems are enlarged beyond their usual size by the inclusion of a statement that certain facts are not implied by the theorem. These last features seem unfortunate to the reviewer though perhaps they can be justified on pedagogical grounds. Altogether the book should be very useful.

1.3. Review by: J W A.
Science Progress (1933-) 33 (131) (1939), 574.

This is a delightful book. Teachers and students who have the chance to examine it will want to possess it.

The book is intended for the use of undergraduates. It is divided into two sections ; the first gives an introductory course which would be suitable for candidates for higher school certificates and scholarships ; the second part covers the usual course for honours students.

The development of the theory has been planned with a view to the student's probable state of knowledge about other branches of mathematics. For example, some proofs appeal to diagrams. It is right that the diagrams should be given for they put the gist of the matter clearly to the reader. But the method hides difficulties and the author sometimes gives alternative analytical proofs for the benefit of those who possess a sufficient knowledge of the calculus. Again, at an early stage the theory requires two properties of real numbers which seem reasonable enough yet which require careful proofs. The author has chosen to assume (explicitly) these properties and then to prove them in an appendix which gives an account of the theory of real numbers. This allows an uninterrupted treatment of the convergence theory. In general, the main principle has been to give a clear and elementary account of the fundamental ideas involved.
...
Mr Ferrar remarks that the book is based on lectures he has given at various times. This is very evident from the way in which the theory is presented and in which difficulties, logical and practical, are explained. The book is, in fact, not a cold recital of facts, one after another ; on the contrary, the teacher is there all the time ready for the queries which experience has shown must often occur in the student's mind.

1.4. Review by: Joseph L Walsh.
Science, New Series 89 (2299) (1939), 59-60.

As the title indicates, this book is a text on the theory of series. In the opinion of the reviewer it is clearly and carefully written, in a pleasant style, with good (albeit conventional) choice of material.

The subject-matter includes preliminary definitions of limit (especially well done), study of monotonic sequences, series of positive terms, comparison and ratio tests, alternating series, the Cauchy condition, uniform convergence, binomial and logarithmic expansions, integral tests, double series, Cesàro sums, Fourier series.

There are several unusual features of the book. ... There are a few matters in which the reviewer fails to see eye to eye with the author. ...

1.5. Review by: Lloyd Leroy Smail.
Amer. Math. Monthly 45 (8) (1938), 545-546.

In this book we have an excellent introduction to the theory of convergence of infinite processes. It is quite elementary and leisurely in treatment, and includes most of the fundamental topics for general use and is properly rigorous, but, as the author says, "excludes topics appropriate to post-graduate or to highly specialised courses of study." The chapters are short, with only a few fundamental related ideas in each chapter. One or more lists of exercises are included in each chapter; the early exercise lists are fairly easy and elementary, and the majority of the exercises a re "reasonably straightforward." Numerous details throughout the book show skilful ways of handling the tricks of technique. For most of the book, real numbers only are used; complex numbers are used occasionally, particularly in connection with power series.

The book is divided in to two parts: Part I, "A First Course in the Theory of Sequences and Series," containing seven chapters, and Part II, "The General Theory of Infinite Series," containing thirteen chapters. Part I opens with a chapter called "Preliminary Discussion," giving simple examples of convergent and divergent series. The next two chapters deal with infinite sequences: convergence, properties of convergent sequences, divergence, bounds of sequences, monotonic sequences. To avoid the necessity of a detailed discussion of the rigorous theory of irrational numbers at the beginning of the course, the treatment is based on two fundamental assumptions. These assumptions refer to the upper bound of a bounded set and to the representation of an irrational number as the limit of a sequence of rational numbers; they are proved in the Appendix. Also the Dedekind theory of real numbers is treated in an Appendix.

1.6. Review by: Glenn Thomas Trewartha.
Nature 142 (1938), 556.

In this book, the theory of convergence is developed on two fundamental assumptions. The first of these is concerned with upper bounds, namely, that a certain set of numbers has in it a least number; while the second refers to irrational number as the limit of a sequence of rational numbers, namely, that every irrational number is the limit of a monotonic increasing sequence of rational numbers. With the aid of these assumptions, the theory of convergence is developed without recourse to the properties of Dedekind cuts. The 'real' number appears only in the appendix, where the assumptions used in the body of the work are proved to be consequences of the definition of real number. In the appendix also, the first of the above-mentioned assumptions appears as a theorem, and a proof of the second is given. In fact, the appendix contains as much of the foundations of analysis as is necessary to justify the assumptions made in the initial chapters of the book. A short historical survey prefacing an examination of these 'foundations' shows why such a complex structure as the Dedekind cut is essential to the definition of number.

1.7. Review by: Thomas Arthur Alan Broadbent.
The Mathematical Gazette 22 (250) (1938), 314-315.

Mr Ferrar's book includes the convergence theory usually required for a university honours course in pure and applied mathematics. The first seven chapters deal with the elements of the theory of sequences and series; Chapters XIII-XX with the further theory of series, including uniform convergence, Tannery's theorem, double series, infinite products, and something about Cesaro summability and Fourier series. The theory is based on two fundamental assumptions, one concerning the existence of upper and lower bounds, the other asserting that every irrational number is the limit of a monotonic increasing sequence of rationals; these assumptions are discussed in more detail and justified in the Appendix.

2.1. Note.

Reprinted 1942, 1946, 1948, 1950, 1953.

2.2. Review by: Richard Hubert Bruck.
National Mathematics Magazine 17 (4) (1943), 188-189.

According to the preface, it was the author's intention to present in this book material "which might reasonably be reckoned as an essential part of an undergraduate's education" but to exclude "topics appropriate to post-graduate or to highly specialised courses of study." In the opinion of the reviewer the material has been happily chosen and charmingly presented.

Part I is devoted to the theory of determinants, and proceeds in a leisurely, careful manner (being aimed at approximately sophomore level) without sacrificing any of the standard theorems on determinants of order n. The presentation is not merely excellent, it exhibits a good deal of originality. Noteworthy, because they are unexpected, are brief sections on alternates, and the differentiation of determinants, and a short chapter on symmetric and skew-symmetric determinants, with mention of Pfaffians.

Part II deals with matrices. and includes, for example. linear equations and the characteristic equation. The rank of a matrix is defined in the classical manner, in terms of minors, but the approach to this concept through equivalence of matrices is also given in a final chapter.

In Part Ill, on linear and quadratic forms, invariants, and covariants, the tempo of the book is considerably accelerated. At two points in the chapter on orthogonal transformations the reviewer feels that the treatment is too brief. The material of this chapter-the expression of an orthogonal matrix in terms of a skew-symmetric matrix is well worth a more detailed exposition.

Elsewhere in Part III the presentation is uniformly good. Topics treated include Hermitian forms, positive definite real forms and simultaneous reduction of two real quadratic forms. The concluding chapter introduces the theory of invariants, using different processes but making no appeal to the symbolic notation or to representation theory.

Professor Ferrar's book should prove attractive to any student of higher algebra, but it ought to be a delight to the engineer or applied mathematician who happens to feel a need for matrices and determinants but does not care for the trappings of modern algebra. Little emphasis is placed on the concept of a field (in fact most of the proofs make little use of the nature of the underlying field) and the notation of a group is purposely omitted. References to more advanced texts are given at strategic points, and there are about two hundred exercises of various degrees of difficulty.

2.3. Review by: Robert F Rinehart.
Amer. Math. Monthly 49 (1) (1942), 51.

The purpose of this book is to provide material for an undergraduate or a first year graduate course in the elements of the theories of matrices and algebraic forms. The book is in three divisions: I Determinants, II Matrices, III Linear and Quadratic Forms. In a brief treatment, as this volume is, a careful selection of material is necessary. On the whole, the author's choice of subject matter is judicious, particularly in the chapter on invariants in Division III. The omission from Division II of the topics of similarity and matrices with polynomial elements may be questionable.

For the most part, the clarity of the presentation is commendable and the author in preparing the book has kept the student uppermost in his mind. How- ever, the statement of the separate corollaries to Theorem 7, p. 16, would be an insult to the intelligence of even a dull student. Following are listed several features which appealed to the reviewer: a novel definition of a determinant is used in Division I (which later affords a very neat proof of the theorem about the evenness or oddness of a permutation); an elegant proof of the theorem of Frobenius concerning the characteristic roots of a rational function of a matrix; a clear and elementary proof of the theorem of Darboux, that the principal minors of the matrix of a positive definite quadratic form are all positive; the well chosen problems (which do not include many arithmetical examples, however) particularly those in connection with form theory, invariant theory, and manipulation of determinants.

There are a number of features which most mathematicians will find objectionable. Those of a general occurrence are: sporadic use of dummy summation indices; lack of distinction between a square array and a determinant; use of X for matrix multiplication; use of ordinary as a synonym for non-singular; use of transformation of instead of transform of; failure to employ matrix theory to advantage in the development of form theory.

2.4. Review by: A Adrian Albert.
Science, New Series 94 (2450) (1941), 565.

This book was written to provide a text, principally for university undergraduates, on determinants, matrices and algebraic forms. The only prerequisite training required is that provided by the usual course in college algebra, and thus the author's Part I consists of a 59-page presentation of the classical theory of determinants. Part II, on the theory of matrices of complex numbers, presents in 52 pages the elementary matrix concepts, the notions of characteristic function and latent root, the definitions of elementary transformations over a number field F, and the theory of equivalence of rectangular matrices over F. The final part consists of 49 pages on real quadratic forms and 30 pages on invariants and covariants. The omission of the theory of similarity of square matrices is rather curious in a text presenting the theory of equivalence of pairs of real quadratic forms.

The author's sources include no modern treatment of his subject, and this probably accounts for his use of so much obsolete terminology. He states that his omission of any hint of abstract algebra is deliberate, but misses the point that even an elementary exposition of the theory of matrices with complex elements could profit by the adoption of the streamlining of the modern versions.

2.5. Review by: Joseph L Burchnall.
The Mathematical Gazette 25 (265) (1941), 184-185.

This book meets the need, acutely felt, for an algebra suitable to university students who, not desiring to be specialists, still require for their more serious pursuits some knowledge of the tools which algebra supplies and some acquaintance with its modern developments. Its perusal too will be of use to those who propose to undertake later that most abstract of studies, the modern higher algebra, for it will familiarise them with some of the ideas whence that study has sprung and enable them to interpret their abstractions in terms of the concrete. Though intended primarily for the undergraduate the book will be a suitable and useful addition to the school library.

In a work with a limited aim some selection of material was necessary: many excellent textbooks, ancient and modern, deal very adequately with the theory of equations and elimination and Mr Ferrar has ignored these topics. With perhaps less justification, though we appreciate the reasons he gives, he has omitted any mention of group theory and some might wish that he had sacrificed to this purpose the final chapter on invariants and covariants which is admittedly a sketch and stands apart from the remainder of the book. Those, however, who remember Elliott with affection and gratitude will rejoice in this evidence that in the university he served his studies are still pursued. Apart from this the book is divided into three portions of approximately equal length which deal respectively with determinants, matrices and linear and quadratic forms.

The definition of a determinant and the deduction of its characteristic properties abound in instances of the things which are easy to see being difficult to express. Mr Ferrar avoids some of these difficulties by defining a determinant as an expression having certain properties and proving the existence and uniqueness of the expression. The result is a curious and interesting blend of the permutation and inductive methods. We note with pleasure a special section on the interesting skew-symmetric determinants and other types receive treatment in the text or examples. The treatment of matrices is, within the limits set, full, careful and lucid: it terminates with the study of latent roots and the proof that a square matrix satisfies its canonical equation. Equivalent matrices and elementary transformations are defined but the study of canonical forms is beyond the scope of the book

3.1. Note.

Reprinted 1960, 1963.

3.2. Review by: Leon Mirsky.
The Mathematical Gazette 44 (348) (1960), 138.

The first edition of Dr Ferrar's book appeared in 1941 at a time when mathematical syllabuses in British universities were as yet almost innocent of any taint of algebra, and it may well have contributed to the great change of emphasis that has since taken place. The new edition of the work is to be warmly welcomed. The author has further enhanced its usefulness by adding a chapter on latent vectors.

Whereas in Dr Ferrar's traditional treatment matrices are regarded as rectangular arrays of numbers which obey certain algebraic rules, Professor Murdoch's approach is uncompromisingly modern. Starting from the notion of a vector space (which, for the sake of simplicity, is always taken as a subspace of the space of complex or real n-tuples), he develops the fundamental concepts of linear mappings of a vector space and of matrix representations of such mappings and then goes on to consider matrices and their simplest canonical forms. In particular, he bases the discussion of orthogonal and unitary matrices and of quadratic forms on a study of vector spaces for which an inner product has been defined. The book is designed as an introductory text for undergraduates and it succeeds admirably in its aim. It does not take the reader very far into linear algebra, but it provides him with all the necessary equipment for a more extensive journey. It could hardly be improved upon either in the choice of material or in the manner of exposition.

4.1. Note.

Reprinted 1948, 1952, 1956, 1959, 1965 (with additional examples.
Translated into German as Elemente der Algebra (R Oldenburg, Munchen, 1953).

4.2. Review by: Louise Weisner.
Amer. Math. Monthly 53 (5) (1946), 269.

The subject matter of this book corresponds roughly to that treated by numerous American textbooks bearing the title, College Algebra. However, the book differs from our college algebras in that it presents fewer topics but develops them at greater length.

Fully one-third of the book is devoted to the theory of polynomials and equations. There are two chapters on graphs, one on mathematical induction, and two chapters on infinite series with emphasis on the binomial series and the exponential and logarithmic functions. The concluding two chapters on linear equations and determinants do not go beyond determinants of order 4, but include such advanced topics as the rule for differentiating a determinant and the rule for multiplying two determinants.

It seems odd to an American that an algebra book which presupposes a knowledge of differentiation and integration of easy functions should omit De Moivre's theorem and allied topics. Again, the theory of permutations and combinations is treated briefly, and then primarily for the purpose of presenting the combinatorial proof of the binomial theorem; and there is no mention of probability. One also misses Descartes' rule and Horner's method, although Newton's method is included. These topics will presumably be treated in a subsequent volume promised by the author.

The book contains an abundance of illustrative examples and exercises involving polynomial identities, undetermined coefficients, relations among binomial coefficients, factorisation of determinants and many formal devices. An ambitious student will find these problems a valuable aid in cultivating algebraic techniques. Some of the exercises are labelled "hard" and others "harder." In the reviewer's opinion many of the exercises not so labelled would drive our average freshman to despair. But this remark is no reflection on the author, who has written an excellent book for a selected group of British students whose mathematical preparation and traditions are different from those of our freshmen.

5.1. Note.

Reprinted 1958, 1967.
Translated into German as Hohere Algebra (R Oldenburg, Munchen, 1954.

5.2. Review by: Louis Weisner.
Amer. Math. Monthly 56 (3) (1949), 194-195.

This book is a sequel to the author's Higher Algebra for Schools. Like its predecessor, it is addressed to a superior group of British students and is not adapted to the traditional algebra courses given in American colleges.

The book consists of ten chapters dealing with the following subjects: finite series, infinite series and approximations, complex numbers, difference equations and generating functions, theory of equations, partial fractions, inequalities, and continued fractions.

The topics are few in number but are treated in great detail, and some are developed at greater length than in comparable textbooks. Among these may be mentioned the exposition of linear functions of a complex variable, usually re- served for works on function-theory; the existence theorems for partial fractions; and the inequalities associated with the names of Maclaurin, Chebyshev, Weierstrass, Hölder, and Minkowski.

The book contains an abundance of well-chosen illustrative examples and classified problems, and some excellent advice on how to attack difficult problems. It is well adapted to individual study and is recommended to ambitious students desiring to cultivate algebraic techniques and to those preparing for competitive examinations in mathematics.

5.3. Review by: Alan Robson.
The Mathematical Gazette 32 (301) (1948), 269.

This is a sequel to the author's Higher Algebra for Schools and is intended for top mathematical forms in schools and first-year classes in universities. By a system of asterisks and other indications, the bookwork and examples are conveniently classified for ordinary sixth-form pupils, those approaching scholarship level, and first year undergraduates.
...
The general policy of the author is to explain methods, which he does very well, rather than to give formal proofs of general theorems. There are, however, a few of these proofs of special interest, e.g. a proof that a symmetric polynomial in the roots of an algebraic equation can be expressed in terms of the coefficients, using partitions; and the obtaining of an eliminant by a method which makes it clear that the condition is both necessary and sufficient.

The course provided by this book may be found not sufficiently ample for the best scholars. But it should be said that determinants and matrices have been deliberately omitted because the author feels that, after the stage of his more elementary book, these subjects should be studied in a separate text. The amount of ground to be covered at this stage depends also upon the views of the student and his teachers of the relative importance of algebra, analysis, geometry, and applied mathematics.

5.4. Review by: Ivan Niven.
Science, New Series 108 (2809) (1948), 487.

Although this book is a sequel to Higher algebra for schools, it is not written so that the earlier book is the only source of the needed preliminary knowledge. The material is planned for study in "top mathematical forms in schools and first year classes in universities." There are discussions of series, complex numbers, difference equations, theory of equations, partial fractions, inequalities, and continued fractions. There is no general discussion of convergence and divergence, the summing of finite and infinite series being stressed. A lucid introduction to complex numbers uses directed number, or vector, as the basic definition. Thus, De Moivre's theorem is readily obtained and is used in the next chapter to yield many trigonometric identities; bilinear transformations are another application of complex numbers. The relations between the coefficients of an equation and the symmetric functions of its roots, and the exact solutions of a cubic and a quartic, are the principal topics in the theory of equations. Several general inequalities are given, including Holder's and Minkowski's.
...
As a book in the celebrated English tradition of elementary mathematical texts, placing great emphasis upon, and replete with, manipulative problems, the volume is a success.

6.1. Review by: Cyrus C MacDuffee.
Amer. Math. Monthly 59 (6) (1952), 422-423.

This is a companion volume to the author's Algebra which was published in 1941. Many of the basic theorems concerning matrices are assumed with a footnote reference to the earlier volume. After an introductory chapter, equivalence over a field and equivalence of ª-matrices over a field are considered, but the field is by definition a subfield of the complex field. This third chapter contains the elementary divisor theory. The term c-equivalence is introduced for similarity, and in Chapter IV (entitled Collineation) the complex canonical form of a matrix under similarity transformations is obtained. From this the rational canonical form is obtained, valid only for a subfield of the complex field.

Chapter V is devoted to infinite series and functions of matrices, and regarding this material t he author claims in his preface that "only patches of it here and there appear to have been published before." It is not clear to the reviewer if Ferrar is familiar with Schwertfeger's monograph Les Fonctions de Matrices, Paris, 1938. This is certainly a connected treatise. At any rate Ferrar's Chapter V does not follow Schwertfeger very closely, and contains some interesting new material.

Chapter VI contains a treatment of the congruence of matrices, orthogonal and unitary matrices, quadratic and hermitian forms. The reviewer bridled a bit at the definition of a nilpotent matrix as one whose square is zero, but of course Wedderburn's definition is not patented. In Chapter VII some simple types of matrix equations are considered.

Chapter VIII, entitled Miscellaneous Notes, contains an exposition of the resolvent matrix, i.e., the theory of the Frobenius covariants: positive definite quadratic forms whose variables are subject to a linear condition; and the anti-commutative matrices of Eddington and Newman.

6.2. Review by: Sam Perlis.
Science, New Series 115 (2998) (1952), 659-660.

Interest in the theory of matrices has spread far and wide in the past few decades, but until recently very few texts were available in this field. Since the war many writers have been at work fulfilling this need. Finite Matrices is one of the latest in the post-war crop.

Various portions of the theory were included in an earlier work by Ferrar entitled Algebra. The author's aim in the present text is to complete the theory in such a way as "to make the argument simple and straightforward," so that it "can be read with reasonable ease." A fair measure of success in this purpose has been attained. On almost every page it is clear that the author was seeking out the difficulties in the subject and looking for lucid presentations. This makes it all the more surprising to find very often, in definitions, theorems, and proofs, that he lapses into a conversational style in which meanings are merely suggested rather than stated explicitly and accurately.

The definition of linear dependence (given by suggest ion after treating only the ease of three vectors) is wrong in that it requires a linear combination to vanish with at least two coefficients not zero. The greatest drawback of the book as a text is the broad knowledge of matrix theory it initially assumes. This includes the theory of rank, the Cayley-Hamilton theorem, and the reality of the roots of a Hermitian matrix, for all of which the reader is referred to the author's Algebra. It is clear that Finite Matrices can be used as a text only on condition that the students have first had a course in Ferrar's earlier text. After an introductory chapter summarising the assumed results, the book gets down to work with a chapter on equivalence. One preparatory chapter then leads to the major topic of collineation (similarity) in which the method employed for the most part is the use of suitable pairs of elementary transformations.

Later the author deals with the well-known theories of orthogonal and unitary similarity, but in the latter he studies only Hermitian matrices rather than normal matrices in general. Characteristic vectors are, of necessity, implicit in these discussions but are not treated explicitly. A long chapter is devoted to infinite series and functions of matrices. providing the only recent systematic Recount of this topic. The book closes with a brief treatment of matrix equations and a few miscellany.

6.3. Review by: Kurt A Hirsch.
The Mathematical Gazette 37 (321) (1953), 228.

The author continues his series of text-books on algebraic topics which now lead the reader from advanced school work to post-graduate level and almost to the threshold of original research. The volume under review contains selected material from the more advanced theory of matrices: equivalence of matrices in the complex field and of ª-matrices in the polynomial ring of one indeterminate, collineations including elementary divisors, congruence with orthogonal (unitary) equivalence and quadratic (Hermitian) forms, infinite series and functions of matrices, and matrix equations. Most of this is at present not usually included in an Honours course at our universities, although it is to be hoped that before long the standard theories of matrix equivalence, collineation and congruence, will be considered an indispensable part of the equipment of every Honours graduate.

The book makes easy reading as a result of the author's deliberate policy. His standpoint is "classical" rather than "modern". His treatment is direct and not axiomatic. His matrices have complex or real coefficients throughout. He does not strive for the greatest generality of his results nor for the shortest proofs. His guiding principle is to make things easy for the reader and to avoid subtle arguments which he believes are difficult to follow. To give an example: the chapter on collineations, the Jordan canonical form, and elementary divisors occupies nearly forty pages whereas van der Waerden covers about twice the material in half the space. Probably it must remain a matter of personal taste and of mathematical upbringing whether one prefers an appeal to a long series of (admittedly simple) calculations to concentrated mathematical reasoning.

The book is not quite free from minor blemishes. ...
...
The chapter on infinite series and functions of matrices is very interesting. It contains material which has never been in a text-book before, much of it the result of the author's own research

7.1. Note.

Reprinted 1962, 1967.

7.2. Review by: Reuben Louis Goodstein.
The Mathematical Gazette 40 (334) (1956), 307-309.

Ferrar's Differential Calculus is written more specifically for the University student. The second half of the book contains some excellent work on partial differentiation; this part of the book is sound and reliable and treats such questions as Euler's theorem on homogeneous functions, and maxima and minima in several variable with care and insight.
...
There are several errors, omissions and confusions, some perhaps of slight importance but some certainly serious. ...

8.1. Note.

Reprinted 1963.

8.1. Review by: Thomas M Flett.
The Mathematical Gazette 45 (354) (1961), 351-353.

This is a companion volume to the author's Differential Calculus (Oxford, 1956), and is aimed at first and second year undergraduates in both mathematics and science. It consists of three parts, on the indefinite integral, the definite integral, and double and curvilinear integrals.

The first part of the book, consisting of three chapters, contains a very thorough treatment of the problems of indefinite integration. After an introductory chapter, Chapter II gives the three standard devices normally used in indefinite integration, namely substitution, integration by parts, and reduction formulae, and adds a fourth which deserves to be better known, namely differentiation under the integral sign. Chapter III deals with systematic integration of rational functions and of functions which can be reduced to rational form by appropriate substitutions.

The second and third parts contain the more analytical material. The second part begins (Chapter IV) with an intuitive approach to the problem of definite integration, and then passes to the analytical definition of the Riemann integral by means of the upper and lower integrals of Darboux. The standard properties of the Riemann integral are then obtained, and (Chapter V) the methods of evaluating definite integrals are investigated. Chapter VI deals with the convergence of infinite and improper integrals, and Chapter VII contains an account of the problem of differentiation under the sign of integration in a definite integral. This part concludes (Chapter VIII) with a short introduction to the Riemann-Stieltjes integral.

Part III, consisting of six chapters, contains the standard material on double and curvilinear integrals-double integrals, reduction of double integrals to repeated integrals, the Gamma and Beta functions, change of variable in a double integral, surface integrals, curvilinear integrals, and the theorems of Green and Stokes.

The aim of the book appears to be to give the student a mastery of the computational techniques of integration, whilst at the same time stressing the conceptual aspects of the subject. The author carefully limits himself to a treatment which does not take him too far into analysis, but within these limits his approach is completely rigorous. In particular, the treatment of the material in Part III attains a much higher standard of rigour than is usual in books at this level. The style is leisurely and lucid, though it is possible that some readers may find the extreme thoroughness of the book rather intimidating.

It is easy to disagree with the limits which an author sets himself, but in several places in this book the reviewer felt that a more general treatment would have been no more difficult, and a good deal more natural, than the limited discussion given. This is particularly the case in the sections on lengths of curves, curvilinear integrals, and Green's theorem. The reviewer also found it somewhat disconcerting to find essentially different treatments given for the Riemann integral in one dimension, the Riemann-Stieltjes integral, and the Riemann double integral. It would seem to the reviewer much more satisfactory to show that all these integrals are particular instances of the same theory, than to display the various alternative treatments of the Riemann theory which are possible.

9.1. Review by: Edmund J Pinney.
Science, New Series 153 (3731) (1966), 52.

[This book] by W L Ferrar, is intended primarily as a textbook for colleges and colleges of advanced technology. The book is written to allow for several different programs of study at different levels, provided the material is appropriately selected.

Topics in trigonometry, analytic geometry, and calculus are covered. The writing style is terse, but the author's extensive teaching experience is evident. The abundantly supplied problems are of the "no-nonsense" type that develop ingenuity, not merely the ability to memorise definitions.

The approach is honestly and honourably old-style by American standards. The author's object is to teach material and technique. The emphasis is on vigour, not rigour. Delta-epsilonics are not mentioned. Many of the arguments would be regarded as strictly intuitive by mathematicians. Still, this approach has been historically more successful in training young students than the more rigorous but pedagogically poorer procedures in vogue in the United States since World War II.

The book, designed for a 1-year course, necessarily skips much calculus normal in a 2-year program. The selection of topics seems good for technical students whose mathematical training would end at this point, and could probably fit in well with a more advanced course for other students.

9.2. Review by: Edward Harrington Lockwood.
The Mathematical Gazette 51 (377) (1967), 251-252.

This is a companion volume to the author's Higher Algebra for Schools and the two works together provide most of what is wanted between "Additional Maths" and "A-level." There is not a great deal of overlap. The style of the present volume is similar to that of the Higher Algebra, but disparities of level are less evident.

As might be expected by those who know Dr Ferrar's other works, the bookwork is written with admirable care and clarity, and the author does not forget that he is writing for scientists, many of whom require first and fore- most a clear understanding of the basic minimum. The path of beginners might have been made smoother if results had been illustrated first by figures showing positive values, but the careful treatment of other cases is of course welcome. The suggestions for a first reading and a second reading are eminently practical and will be appreciated by teachers.

The examples are perhaps not as many or as varied as the teacher may wish. They are designed to illustrate the bookwork rather than to provide scope for the pupil's initiative; but it is easy to supplement them.

The first part of the book includes chapters on trigonometric and hyperbolic functions, complex numbers, inequalities, and the necessary minimum of analytical geometry. There is a chapter on vectors which includes scalar and vector products, differentiation and integration, with some applications to mechanics. The second part is devoted to calculus. While not suitable (or intended) for beginners, this provides a good summary and extension of the work, with more attention to the bookwork than is sometimes given. There are chapters on Maclaurin and Taylor Expansions (rather stiff reading), Differential Equations, and Partial Differentiation, as well as the usual work on differentiation, integration and applications.

The weaker A-level candidates will not find this an easy book but it is much to be recommended for those of university calibre.

10.1. Review by: Edwin Arthur Maxwell.
The Mathematical Gazette 52 (380) (1968), 202-203.

It is not easy to keep up with projects that are putting material into orbit at high frequency. Here, for example, are Book 3 of the " Scottish " project, Book 5 (Teachers' edition), and a set of 3-figure tables.

The range and sweep of this project are becoming well known, and anyone concerned to put his own teaching under review is warmly advised to consult these volumes, all of which are founded on a wealth of experience and experiment.

There is, in particular, a determined onslaught on the subject-matter of geometry, via translation and reflection in Book 3 and vectors in Book 5. (Some ambiguity is possible, though: to the question, "What is the position of Aberdeen in relation to Glasgow?" all unprejudiced Aberdonians would immediately reply, "Superior".)

The headings Arithmetic, Algebra, Geometry, Trigonometry of yesteryear remain, but the setting is changed beyond recognition. Topics like Relations and Mappings, Number Patterns and Sequences, Reasoning and Deduction, Logarithms and Calculating Machines, Estimation of Error, ... show how far the subject matter has progressed.

Printing and diagrams are excellent, and the books feel solid. The examples are based on matters of interest and (for example, Some Topics to Explore) designed to stretch the imagination. This is a series that can be recommended without reserve.

11.1. Review by: Edwin Arthur Maxwell.
The Mathematical Gazette 54 (389) (1970), 307.

Dr Ferrar's pen has for many years had a lively itch, to the great benefit of, surely, thousands of young mathematicians. The present volume inherits the accumulated expertise of its predecessors, both in subject matter and in presentation "in the form in which I would present (the topics) to a university or college scientific audience in its first or second year of specialist studies."

Section headings are: Differential equations; Matrices and determinants; Convergence, Fourier series, Orthogonal functions; and "Part IV," a miscellany of solid geometry, maxima and minima, and partial differential equations.

The book lives up to its title: there is a rich store of mathematics but little science. The author's intention is probably summed up in his own footnote to a discussion of Schrödinger's equation: "I quote, and with very little knowledge of the physical background of the problem; my readers will know, or will learn about, this aspect of the matter". And they can be assured that, when they do, the Foundations laid by Dr Ferrar will (if the metaphors are allowed together) smooth their path enormously.