1.1. From the Publisher of the 2012 reprint.

"Recommended with confidence" by The Times Literary Supplement, this lively survey starts with simple arithmetic and algebra and proceeds by gradual steps through graphs, logarithms, and trigonometry to calculus and the world of numbers. Generations of readers have found it the ideal introduction to mathematics, offering accessible explanations of how theory arises from real-life applications.

"The main object of this book is to dispel the fear of mathematics," declares author W W Sawyer, adding that "Many people regard mathematicians as a race apart, possessed of almost supernatural powers. While this is very flattering for successful mathematicians, it is very bad for those who, for one reason or another, are attempting to learn the subject." Now retired, Sawyer won international renown for his innovative teaching methods, which he used at colleges in England and Scotland as well as Africa, New Zealand, and North America. His insights into the pleasures and practicalities of mathematics will appeal to readers of all backgrounds.

1.2. From the Introduction.

The main object of this book is to dispel the fear of mathematics. Many people regard mathematicians as a race apart. possessed of almost supernatural powers. While this is very flattering for successful mathematicians, it is very bad for those who, for one reason or another, are attempting to learn the subject.

Very many students feel that they will never be able to understand mathematics, but that they may learn enough to fool examiners into thinking they do. They are like a messenger who has to repeat a sentence in a language of which he is ignorant full of anxiety to get the message delivered before memory fails, capable of making the most absurd mistakes in consequence. It is clear that such study is a waste of time. Mathematical thinking is a tool. There is no point in acquiring it unless you mean to use it. It would be far better to spend time in physical exercise, which would at least promote health of body.

Further, it is extremely bad for human beings to acquire the habit of cowardice in any field. The ideal of mental health is to be ready to face any problem which life may bring - not to rush hastily, with averted eyes, past places where difficulties are found. Why should such fear of mathematics be felt? Does it lie in the nature of the subject itself? Are great mathematicians essentially different from other people? Or does the fault lie mainly in the methods by which it is taught?

Quite certainly the cause does not lie in the nature of the subject itself. The most convincing proof of this is the fact that people in their everyday occupations - when they are making something do, as a matter of fact. reason along lines which are essentially the same as those used in mathematics: but they are unconscious of this fact. and would be appalled if anyone suggested that they should take a course in mathematics. Illustrations of this will be given later.

The fear of mathematics is a tradition handed down from days when the majority of teachers knew little about human nature, and nothing at all about the nature of mathematics itself. What they did teach was an imitation.

1.3. From the Text.

Nearly every subject has a shadow, or imitation. It would, I suppose, be quite possible to teach a deaf and dumb child to play the piano. When it played a wrong note, it would see the frown of its teacher, and try again. But it would obviously have no idea of what it was doing, or why anyone should devote hours to such an extraordinary exercise. It would have learnt an imitation of music, and it would fear the piano exactly as most students fear what is supposed to be mathematics.
...

Education consists in co-operating with what is already inside a child's mind. The best way to learn geometry is to follow the road which the human race originally followed: Do things, make things, notice things, arrange things, and only then reason about things.

1.4. Review by: Anon.
The Journal of Educational Sociology 20 (5) (1947), 320.

Mathematician's Delight is a mathematics teacher's attempt to give his subject interest and new life by relating it to the reader's everyday experiences, and to prove that it need not be the difficult subject students think it is. The reader is led carefully and accurately through simple discussions of the discovery and applications of geometry (practical and demonstrative), arithmetic, and algebra including graphs (thinking in pictures), trigonometry, and calculus.

The content is in storybook rather than textbook form, although there is sufficient illustrative and problem material to put across the author's ideas. It is an excellent book for the layman or for the teacher of mathematics who is looking for new inspiration and teaching material. I should, however, like to see it expanded into two or three volumes and published in a better edition. Its brevity is its chief weakness.

1.5. Review by: H P Evans.
The American Mathematical Monthly 54 (4) (1947), 240-242.

The book under review provides, in many respects, an excellent introduction to elementary mathematics. The title of the book is indicative of the author's ingenuity and originality in departing from traditional and time-worn patterns. In his own words, the book is intended to "try to show what mathematics is about, how mathematicians think, when mathematics can be of some use." Although intended chiefly for beginners, there is much in this book that will be of interest to the teacher as well.

In discussing the fear of mathematics which is held by so many laymen, the author contends that this is due not to the nature of the subject itself, but to the dull way in which it is so often taught. The thesis is developed that in many cases the student does not learn mathematics, but only an imitation of mathematics, which destroys his power to enjoy the real subject. ...

The book contains an interesting chapter on geometry in which the value of experimentation is emphasised and a number of rather ingenious geometrical experiments are proposed. A chapter, 'The Nature of Reasoning', leads gradually to mathematical reasoning and to the nature of mathematical abstraction. In making generalisations the author may occasionally be guilty of over-simplification. For example, the statement that "The more one studies the methods of the great, the more common-place do these methods appear," may require considerable qualification. An interesting discussion contrasting pure and applied mathematics is given and the point is made that all mathematics is tied, however remotely, to problems which arose originally in the study of the real world. This will seem rather academic to those mathematicians who believe that many of these ties are very remote indeed.
...

While in various places throughout the book the reader may not agree wholly with the author's contentions, nevertheless the reviewer regards this little book as a worthy addition to the growing list of survey type books, which deal in an elementary and intuitive way with the nature and the scope of mathematics while leaving its rigorous elaboration and methodology to the text-books and treatises.

1.6. Review by: Anon.
The Quarterly Review of Biology 22 (1) (1947), 103.

This is an attractive little book to place in the hands of students. The topics treated range from arithmetic and algebra to graphs, trigonometry, and calculus. The final chapter delivers a well placed fillip to that four hundred year old tool, the square root of minus one. There is a chapter on "ways of growing," but the biologist will regret the omission of statistics. The style of the book is planned for those without mathematical training. ...

1.7. Review by: H L Schmid.
zbMATH 0041.37408.

This little book is aimed at the layman and tries to convince him that mathematics is by no means a secret science. After some of the elementary arithmetic, the author also arrives at concepts such as the differential quotient, integral and infinite series; however, he always emphasises the sketchy character of the representation and emphatically points out the difficulties (such as questions of convergence) that arise in an exact treatment. In this way he avoids the impression of a "patent textbook". In the section on complex numbers, the number i is stripped of its mysterious character to the layman.

2.1. From the Introduction.

This is a book for boys and girls who like to invent and design things for themselves. Most boys and girls like being inventors, but often you want to invent and you cannot think of anything that needs inventing. Now, of course, if we just tell you something to make, and you copy our idea, you are not an inventor. So in writing this book, we have tried to leave the details to you.
...

Children often ask, 'Why do we have to learn arithmetic?' There are various answers - "You need it to go shopping', and so forth. The weakness of these that they appeal to the reason alone while the question is not really a request for information. It really means 'I find this dull', and the only effective answer is one directed to the feelings. The best answer of all is one not of words but of action - to let the child embark on some activity that is unquestionably exciting, and to let it discover at some stage that its progress is held up by lack of mathematical knowledge.
...

The more the children learn to organise their own lives, the more efficient the education will be. Things can go without the help of the teacher.

2.2. Review by: I R Vesselo.
The Mathematical Gazette 34 (310) (1950), 318.

The rise of the Secondary Modern School has brought with it the cry for mathematics which has meaning to the pupil, or is capable of immediate application to practical situations. The fashion of the moment is to select material which is likely to be of interest for its own sake and to extract some mathematics from it. This is the reverse of the treatment we have become accustomed to, in which the principle comes first, the application following, often almost as an afterthought; here the application is paramount.

As we have come to expect from Mr Sawyer, the choice of material is excellent. Aeroplanes, railways, bicycles, gadgets, gardens, sports and pastimes; if we have seen some of it before, yet there is enough that is new to be worth while. The boy or girl who is fortunate enough to obtain a copy of this book will spend pleasant hours reading and making, if not designing.

Those who have used this method in teaching will know its difficulties. The extraction ratio is low; what does come out is patchy. One would need a great pile of books of this type from which to develop a systematic course of elementary mathematics, even at this level. Nevertheless, as a stimulus to interest in mathematics, as an answer to the eternal why, or as a source of occasional material for the teacher, the book is invaluable.

3.1. From the Publisher of the 1979 reprint.

In this lively and stimulating account, noted mathematician and educator W W Sawyer (Professor Emeritus, University of Toronto) defines mathematics as "the classification and study of all possible patterns." It is a broad definition, but one that seems appropriate to the great scope and depth of the topic. Indeed, mathematics seems to have few boundaries, either in applications to practical matters or in its mind-stretching excursions into realms of pure abstraction.
Gearing his approach to the layman whose grasp of things mathematical may be a bit precarious, Professor Sawyer offers a lucid, accessible introduction to the mathematician's cast of mind. Five well-written preliminary chapters explore the beauty, power and mysticism of mathematics; the role of math as an adjunct in utilitarian matters; and the concepts of pattern, generalisation and unification as both tools and goals of mathematical thought.

After developing this conceptual groundwork, the author goes on to treat of more advanced topics: non-Euclidean geometry, matrices, projective geometry, determinants, transformations and group theory. The emphasis here is not on mathematics with great practical utility, but on those branches which are exciting in themselves - mathematics which offers the strange, the novel, the apparently impossible - for example, an arithmetic in which no number is larger than four.
Mathematicians will appreciate the author's grasp of a wide range of important mathematical topics, and his ability to illuminate the complex issues involved; laymen, especially those with a minimal math background, will appreciate the accessibility of much of the book, which affords not only a portrait of mathematics as a matchless tool for probing the nature of the universe, but a revealing glimpse of that mysterious entity called "the mathematical mind." Professor Sawyer has further enhanced this new Dover edition with updated material on group theory, appearing here in English for the first time.

3.2. From the Introduction.

This is a book about how to grow mathematicians. Probably you have no intention of trying to grow mathematicians. Even so, I hope you may find something of interest here. I myself have no intention of growing plants. I never do any gardening if I can possibly get out of it. But I like very much to look at gardens other people have grown. And I am still more interested if I can meet a man who will explain to me (what very few gardeners seem able to do) just how a plant grows; how, when it is a seed under the earth, it knows which way is up for its stem to grow, and which way is down for the roots; how a flower manages to face towards the light; what chemical elements the plant needs from the soil, and just how it manages to rearrange them into its own living tissue. The interest of these things is quite independent of whether one actually intends to go out and do some hoeing.

What I am trying to do here is to write not from the viewpoint of the practical grower, but for the man who wants to understand what growth is. I am not writing for the professional teacher of mathematics (though teachers may be able to make practical applications of the ideas given here) but for the person who is interested in getting inside the mind of a mathematician.

It is very difficult to communicate the things that are really worth communicating. Suppose, for instance, that you have spent some years in a certain place, and that these years are particularly significant for you. They may have been years of early childhood, or school days, or a period of adult life when new experiences, pleasant or unpleasant, made life unusually interesting. If you revisit this place, you see it in a special way. Your companions, seeing it for the first time, see the physical scene, a pleasant village, a drab town street, whatever it may be. They do not see the essential thing that makes you want to visit the place; to make them see it, you need to be something of a poet; you have to speak of things, but so as to convey what you feel about those things.

But such communication is not impossible. Generally speaking, we overestimate the differences between people. I am sure that if one could go and actually be somebody else for a day, the change would be much less than one anticipated. The feelings would be the same, but hitched on to different objects. Most human misunderstandings are due to the fact that people talk about objects, and forget the varying significance the same object can have for different people.

Generally speaking, teaching conveys thoughts about objects rather than living processes of thought. Suppose someone comes to me with some kind of puzzle; it may be a question in a child's arithmetic book, or a serious problem of scientific research. Perhaps I succeed in solving the puzzle. Then it is quite easy to explain the solution. Suppose I do so; I have shown the questioner how to deal with that particular problem. But if another problem, of a different kind. arises, I shall be consulted again. I have not made my pupil independent of me. What would be really satisfactory would be if I could convey, not simply the knowledge of how to solve a particular puzzle, but the living attitude of mind that would enable my pupil to attack puzzles successfully without help from anyone.

3.3. Review by: H S M Coxeter.
Mathematical Reviews MR0069135 (16,989j).

This sequel to "Mathematician's delight" was worked out while the author was head of the mathematics department of University College, Gold Coast, where, although "of course, there was no mathematical tradition in the country", he found the students "keen and of first-rate ability". Defining mathematics as "the classification and study of patterns," he makes a good attempt to show the general reader how vast the subject has become. He gives a glimpse of such varied topics as Hilbert's Finite-Basis Theorem, the hypergeometric function, quaternions and Cayley numbers, the Galois group of an equation, finite geometries and their application to statistics, Riemann's finite but unbounded space, and instances of "the help that geometry gives to algebra". Though avoiding technical details, he never sacrifices accuracy. His genial personality shows itself in many places. He follows the example of Sir Harold Jeffreys in beginning each chapter with a quotation, such as Cayley's advice in 1855 about the "theory of matrices which, it seems to me, ought to come before the theory of determinants;" or, setting the stage for projective geometry, "Infinity is where things happen that don't"; or, a propos of hyperbolic geometry represented by the interior of a sphere, "I could be bounded in a nutshell and count myself a king of infinite space".

3.4. Review by: Elizabeth Creak.
Mathematics Magazine 29 (3) (1956), 126.

In Mathematician's Delight one of the most popular Pelicans so far published, W W Sawyer described the traditional mathematics of the engineer and the scientist. In this new book the emphasis is not on those branches of mathematics which have great practical utility, but on those which are exciting in themselves: mathematics which is strange, novel, apparently impossible, for instance an arithmetic in which no number is bigger than four. These topics are preceded by an analysis of that enviable attribute "the mathematical mind." Professor Sawyer not only shows what mathematicians get out of mathematics, but also what they are trying to do, how mathematics has grown, and why there are new mathematical discoveries still to be made, His aim is to give an all-round picture of his subject, and he therefore begins by describing the relationship between pleasure-giving mathematics and that which is the servant of technical and social advance.

3.5. Review by: Viggo Brun.
Nordisk Matematisk Tidskrift 4 (3) (1956), 156-157.

To give an impression of what the author has wanted to present in this book, I will try to let him tell it himself with some characteristic quotes:

When we generalise a result, we make it more useful. It may strike you as strange that generalisation nearly always makes the result simpler too.

Very often, one of the greatest difficulties of learning is not a logical difficulty at all. One sees every step, and admits that the proof is logical, but one is left with an obstinate feeling of not really knowing what the new result is, what it is all about.

How are we to look for functions that will do what we want? In such cases, it is usually wise to take the simplest possible example and examine it carefully for hints of what happens in the more complicated cases.

The main difficulty in many modern developments of mathematics is not to learn new ideas but to forget old ones.

In a good proof, an illuminating proof, the result does not appear as a surprise in the last line; you can see it coming all the way.

I think these quotes are sufficient to show the tendency of the book. The author has faced the difficulty that he has wanted to make the book easy to read and elementary, at the same time as he has wanted to give certain advanced hints, especially those that lead to newer thoughts in mathematics. These hints can require quite a lot of knowledge, as when the author claims that over 95 percent of the functions currently studied by physical, technical and even mathematical students, can be covered by the symbol $F (a, b; c; x)$, the symbol of the hypergeometric series .... But mostly the book will also be read by people with little mathematical knowledge. As far as I can understand, very many could have benefited from reading the book.

4.1. From the Publisher of the 2018 reprint.

Brief, clear, and well written, this introduction to abstract algebra bridges the gap between the solid ground of traditional algebra and the abstract territory of modern algebra. The only prerequisite is high school-level algebra.

Author W W Sawyer begins with a very basic viewpoint of abstract algebra, using simple arithmetic and elementary algebra. He then proceeds to arithmetic and polynomials, slowly progressing to more complex matters: finite arithmetic, an analogy between integers and polynomials, an application of the analogy, extending fields, and linear dependence and vector spaces. Additional topics include algebraic calculations with vectors, vectors over a field, and fields regarded as vector spaces. The final chapter proves that angles cannot be trisected by Euclidean means, using a proof that shows how modern algebraic concepts can be used to solve an ancient problem. Exercises appear throughout the book, with complete solutions at the end.

4.2. From The Aim of the Book.

At the present time there is a widespread desire, particularly among high school teachers and engineers, to know more about "modern mathematics." Institutes are provided to meet this desire, and this book was originally written for, and used by, such an institute. The chapters of this book were handed out as mimeographed notes to the students. There were no "lectures"; I did not in the classroom try to expound the same material again. These chapters were the "lectures." In the classroom we simply argued about this material. Questions were asked, obscure points were clarified.

In planning such a course, a professor must make a choice. His aim may be to produce a perfect mathematical work of art, having every axiom stated, every conclusion drawn with flawless logic, the whole syllabus covered. This sounds excellent, but in practice the result is often that the class does not have the faintest idea of what is going on. Certain axioms are stated. How are these axioms chosen? Why do we consider these axioms rather than others? What is the subject about? What is its purpose? If these questions are left unanswered, students feel frustrated. Even though they follow every individual deduction, they cannot think effectively about the subject. The framework is lacking; students do not know where the subject fits in, and this has a paralysing effect on the mind.

On the other hand, the professor may choose familiar topics as a starting point. The students collect material, work problems, observe regularities. frame hypotheses, discover and prove theorems for themselves. The work may not proceed so quickly; all topics may not be covered; the final outline may be jagged. But the student knows what he is doing and where he is going; he is secure in his mastery of the subject, strengthened in confidence of himself. He has had the experience of discovering mathematics. He no longer thinks of mathematics as static dogma learned by rote. He sees mathematics as something growing and developing, mathematical concepts as something continually revised and enriched in the light of new knowledge. The course may have covered a very limited region, but it should leave the student ready to explore further on his own.

This second approach, proceeding from the familiar to the unfamiliar, is the method used in this book. Wherever possible, I have tried to show how modern higher algebra grows out of traditional elementary algebra. Even so, you may for a time experience some feeling of strangeness. This sense of strangeness will pass; there is nothing you can do about it; we all experience such feelings whenever we begin a new branch of mathematics. Nor is it surprising that such strangeness should be felt. The traditional high school syllabus - algebra, geometry, trigonometry - contains little or nothing discovered since the year 1650 A.D. Even if we bring in calculus and differential equations, the date 1750 A.D. covers most of that. Modern higher algebra was developed round about the years 1900 to 1930 A.D. Anyone who tries to learn modern algebra on the basis of traditional algebra faces some of the difficulties that Rip Van Winkle would have experienced, had his awakening been delayed until the twentieth century. Rip would only overcome that sense of strangeness by riding around in airplanes until he was quite blasé about the whole business.

4.3. Review by: Editors.
Mathematical Reviews MR0101185 (20 #7607).

This book was written for an institute for high school teachers. Fundamental facts regarding abstract fields and vector spaces are introduced by means of many familiar examples.

4.4. Review by: R L Goodstein.
The Mathematical Gazette 44 (348) (1960), 138-139.

The parts of modern algebra discussed in this book are the field axioms, field extension and polynomials over a field, and vector spaces; the results established are used to prove the impossibility of trisecting an angle by ruler and compass alone, by showing that any "constructible" number lies in a field of dimension $2^{n}$ over the rationals whereas the dimension the extension field of a root of the "trisection equation" $w^{3} - 3w - 1 = 0$ over the rationals is 3. The approach is concrete in the sense that it proceeds from the familiar to the new; everything that can be done by patient exposition to help the reader is done. Anyone familiar with school algebra who has tried - and failed - to read other books on modern algebra will almost certainly not fail with this one, if he conscientiously attempts the examples before looking up the solutions.

Only in discussing indeterminates does the author's standard of exposition fall below that of other recent accounts. Sawyer takes the indeterminate $x$ to be the polynomial (0, 1, 0, 0, ...), but it is surely clearer to take an indeterminate over a field $F$ to be an element of a superfield $G$ (containing $F$) which is transcendental over $F$ (i.e. does not satisfy a polynomial equation with coefficients in $F$). Moreover, we not need to introduce the concept of an indeterminate to justify the traditional proof of the remainder theorem; misled by the word division some (like the present reviewer) have mistakenly supposed that division in the sense of cancellation was involved, but in fact so-called long division is just repeated subtraction and is valid not only for polynomials but also for polynomial functions.

The book's value as an introduction would be greatly enhanced by the provision of a good index and bibliography.

4.5. Review by: Maurice L Hartung.
The Mathematics Teacher 53 (5) (1960), 390.

When you buy a book you do not, ordinarily, consider on which shelf in your bookcase you will later put it. You may have a little trouble in classifying W W Sawyer's new book A Concrete Approach to Abstract Algebra. It is about algebra - that much is clear. Although originally written for high school teachers studying mathematics at a National Science Foundation Institute, it is not a textbook in the usual style. On the contrary, Sawyer offers a rather informal discussion of the formal characteristics of modern algebra. At the same time, the book does contain "Questions" and "Exercises" for the student, with answers provided on the final pages, so that it can play the role of a textbook in suitable situations, such as institutes and some types of college courses which have similar aims. The book contains nothing explicit about methods of teaching, but a perceptive teacher can learn much about his art by studying not only the content itself, but also how the author has selected, organised, and presented it.

Sawyer wants, first of all, to communicate and develop the viewpoint of abstract algebra. Attention is focused upon the structure of various algebraic systems. The term "concrete" in the title has to be interpreted broadly. It is true that the author sometimes introduces structural ideas by describing some relatively simple calculating machines, electrical circuits, and other concrete objectives which constitute a realisation of the structure he has in mind. Arithmetic is abstract to an elementary school pupil, but becomes relatively concrete to the high school student who is studying a good algebra course. In similar fashion, high school algebra serves as a relatively concrete basis for the study of other systems in a program of more advanced study. It is these extended senses of the word "concrete" that apply to the approach of this book. Teachers and others who would like to take a tour of some of the high spots of modern algebra with Sawyer as an engaging guide should not be repelled by the possibly unfamiliar names of the places they will visit.
...

In the last fifty or sixty years a number of books with similar purposes have become avail able. Most of these have tended to be global in their coverage of mathematical content and points of view. In contrast, Sawyer has restricted himself to an exposition of algebraic ideas with the formal aspects dominant. With this limitation in scope, he has gone to extraordinary lengths to help the reader understand what the game is all about.

One of the most attractive features of this book is its low price. In these days of high prices for books of all kinds, and especially for technical books, the average reader should have no difficulty in getting his money's worth from this one, even if he doesn't get all the way through it. ...

If the book is to be used as a textbook for class instruction, let the teacher beware. He had better "know his stuff" rather thoroughly, for he may not be able to handle a class using this book in his customary way. When Sawyer wrote it, he put the material in the hands of his students, and "did not in the classroom try to expound the same material again. ... In the classroom we simply argued about this material. Questions were asked, obscure points were clarified." This method of teaching is dangerous for one who is not steeped in the subject matter. On the other hand, students in more conventional courses will surely find the book useful as an auxiliary textbook. It can substitute in part for the "bull sessions" of students through which much of their understanding of mathematics has customarily been acquired.

5.1. From the Introduction.

This book tries to show why nearly all mathematicians and scientists find their work beautiful. They see patterns in nature and in number. In this book, students will have the opportunity to discover some of these patterns for themselves.

No previous knowledge of science or mathematics is required, as this book can be read by anyone who knows the basic facts of arithmetic. A student can use it for individual study. It can, of course, be used for class study.
...

Most students will find that they can think their way successfully through this book. The work, however, is not spoon-feeding, and the student is given every opportunity to use his own judgement - to collect evidence, to make guesses, to observe, to invent.
...

[This book's] main purpose is to assemble material in which the student can see clearly what is happening and from which he can draw his own conclusions.

This is something which, as a rule, students enjoy doing.

6.1. From the Publisher.

In this book, the author tells what calculus is about in simple nontechnical language, understandable to any interested reader.

6.2. From the Introduction.

How should calculus be taught then? Should we bother the beginner with warnings that only become important in more advanced work? If we do so, then the beginner will be confused because he will not see any need for these warnings. If we do not, we shall be denounced by mathematicians for deceiving the young.

I believe the correct approach is to do one thing at a time. When you take a student into a quiet road to drive a car for the first time, he has plenty to do in learning which is the brake and which the accelerator, how to steer and how to park. You do not discuss with him how to deal with heavy traffic which is not there, nor what he would do if it were winter and the road were covered with ice. But you might very well warn him that such conditions exist, so that he does not overestimate what he knows.

In this book I begin with the simple ideas of calculus, with country driving. I do not look for awkward exceptions. In the main, I look at things as mathematicians did in the 17th century when calculus was being developed.

The theorem-proof-theorem-proof type of book does, in a certain limited sense, explain mathematics to the student. Theorem 1 is at least followed by a proof of Theorem 1, which may throw some light on why Theorem 1 is true. But very much is still left hidden. How did the writer decide that Theorem 1 should come first? How did he decide which theorems to include and which to omit? What is the book trying to do? What is the line of thought that lies behind it? How did all these theorems come to be discovered? What should the student do if he wishes to discover further theorems for himself? This last question is perhaps the most important of all. It is a very strange thing that many eminent mathematicians, who think the only thing really worth doing in life is to discover new theorems, often write books which give no hint at all of how a student should try to make his own discoveries.

There are at least four stages in mastering a mathematical result.

(1) You must see clearly and understand what the result states.

(2) You should collect evidence which shows that it is reasonable that this result should be so; you should feel that this result agrees with your evidence of mathematics.

(3) You should know what you can do with the result. It may have applications in science, or it may simply lead to other interesting theorems in pure mathematics. You ought to know what these are.

(4) You should know and understand the formal proof of the result.

I want to make it quite clear that this book does not attempt to provide formal proof of any result whatever. I have not attempted to deal with stage (4) at all. My concern is entirely with stages (1), (2), and (3). I want you to see that the ideas of calculus arise quite naturally, and indeed I want you to discover them for yourself. If we were in a room together, I would confine myself to asking you questions, and you would find that you arrived at calculus by clarifying ideas that you already have in a vague and shadowy form. Between the covers of a book I cannot follow that procedure. But I keep to it as nearly as I can. I am not trying to tell you any particular result. I am trying to call your attention to particular things that you can experiment with for yourself. The evidence that you collect will suggest certain conclusions to you. More than that, I do not claim. I do believe, however, that this experience will make it easier for you when you begin to learn calculus in real earnest. You will have some idea of the direction in which you are travelling.

6.3. Review by: W Eugene Ferguson.
Science New Series 134 (3489) (1961), 1514-1515.

What Is Calculus About? is sure to delight many students at the end of grade 9 and during the 10th grade. They should be able to do the first six chapters which end with simple maximum and minimum problems. Calculus is approached through the study of speed, velocity, and acceleration. Second-year algebra students have plenty of background for all of the material in this book.

6.4. Review by: J L Botsford.
The American Mathematical Monthly 69 (5) (1962), 444-446.

The writer answers the question of the title clearly and develops the ideas of the calculus using speed or velocity as an example of the derivative, and uses this analogy in developing the idea of the slope of the tangent. He does it well and with charming informality. His development of the "reverse problem" is, of course, skimpy but well done. The only criticism that might be directed at this book is that he moves too slowly, but the probabilities are that most high school students will welcome this feature of the book.

6.5. Review by: R L Goodstein.
The Mathematical Gazette 46 (56) (1962), 156-157.

Sawyer's volume covers familiar ground, derivatives of powers, curvature, and an introduction to integration, with a final chapter on curves without tangents.

6.6. Review by: Harry M Gehman.
Mathematics Magazine 35 (3) (1962), 182-183.

Sawyer's book is intended for the high school student unacquainted with the ideas of calculus. Many college students after having had a course in calculus would profit by reading this book and thereby re-examining calculus from an intuitive point of view. Sawyer bases the idea of a derivative upon the more familiar ideas of speed and of slope of a curve. Second derivatives are introduced thru acceleration and curvature, A simple maximum problem is solved. Finally the "reverse problem" of integration or anti-differentiation is briefly discussed and the calculation of areas and volumes is mentioned. In the final chapter, "Intuition and Logic" and in an appendix, "Guide to Further Study", the author emphasises that he has given the reader merely a brief intuitive survey of calculus. In order for him to master the subject and use its tools correctly, he will be required to do considerably more reading and studying.

6.7. Review by: Morris Kline.
Scientific American 206 (1) (1962), 157-162.

What Is Calculus About? is very well executed, considering that Sawyer apparently intends his book to be a first step. He wants to introduce young people to the calculus, and he presupposes that this introduction will be followed by further study of the subject. Hence he adopts a gradual, careful pace and succeeds in being intuitive and clear in what he has presented. In a short book one cannot get too far into the calculus; the layman who may be expecting a bird's-eye view with some attention to the range and power of the calculus and its place in mathematics and science may find the perspective a bit limited. Sawyer is capable of addressing sophisticated laymen, as readers of his books in the Penguin series well know, but the interests of such an audience seem to have been secondary in the present exposition.

6.8. Review by: Warren B White.
The Mathematics Teacher 55 (5) (1962), 408.

Any high-school mathematics teacher planning to include some informal calculus in his courses will find strong support for the project in this book. Professor Sawyer is not among those who believe that a student's first approach to calculus must be rigorous and in the modern tradition. He would have the student's training follow the historical development of the subject. This he considers in three stages: the happy-go-lucky stage, the epsilon-delta stage, and the stage of abstraction and generalisation.

The greater part of the book illustrates, with examples from physics and geometry, the first of these stages. Here the notion of speed is used as a unifying concept. Although the emphasis is on differentiation, two chapters introducing integration are included. A later chapter dis cusses some exceptional functions which suggests a need for a more rigorous treatment of the foundations of calculus. The development throughout is extremely patient and could be followed by any equally patient high-school student having a reasonable background in algebra and geometry.

The author works into the book at every opportunity items from his own philosophy of teaching. The following comment on discovery in mathematics is typical: "It is a very strange thing that many eminent mathematicians, who think the only thing really worth doing in life is to discover new theorems, often write books which give no hint at all of how a student should try to make his own discoveries."

Some questions and exercises are provided, and answers to them are supplied. An appendix discusses at some length a series of calculus texts to be studied sequentially by anyone who wants to acquire a more thorough knowledge of the subject and its evolution.

6.9. Review by Lee Lady.
University of Hawaii, 1996.
http://www.math.hawaii.edu/~lee/calculus/sawyer.html

This is a book about the ideas of calculus, not the techniques of calculus. You will not find the product rule, or quotient rule, or chain rule here. But you will find a rather detailed discussion of velocity, acceleration, and the slope (and direction of curvature) of graphs. The fundamental theorem of calculus is explained very clearly, but never named as such. This book does what, in my opinion, most current courses in calculus fail to do, viz. to show that calculus is something interesting and worthwhile.

7.1. From the Publisher of the 2012 reprint.

Here is a presentation of elementary mathematics that anyone can appreciate, especially those with imagination. As the title suggests, the author's technique relies on visual elements, and his approach employs the most graphic and least "forbidding" aspects of mathematics. Most people, he observes, possess a direct vision that permits them to "see" only the smaller numbers; with the larger numbers, however, vision fails and mental chaos ensues.

Sawyer addresses this difficulty, speaking both for those who like recreational mathematics and for those who teach, suggesting a variety of methods used by many effective teachers - techniques of visualising, dramatising, and analysing numbers that attract and retain the attention and understanding of students. His topics, ranging from basic multiplication and division to algebra, encompass word problems, graphs, negative numbers, fractions, and many other practical applications of elementary mathematics.

A valuable resource for parents and teachers, this book will captivate any reader seeking an improved understanding of mathematics.

7.2. From the Introduction.

This is a book about mathematics at a very low level. In recent years there seems to have been a sudden increase in the world's demand for mathematicians. A consequence of this has been that many people who do not feel themselves particularly qualified in mathematics find themselves called upon to teach it, either in an official or an informal capacity. A school's regular mathematics teacher leaves for a post in industry, and the French teacher has to take over his classes. A teacher of young children, who never liked arithmetic much and never did very well at it, finds that public attention is being focused on how she teaches it. Students at teacher training colleges have a similar experience. Parents find themselves unofficially involved in a similar dilemma. They want to help their children with mathematics but they fear that little good will come of the blind leading the blind.

7.3. Review by: R L Wilson.
Mathematics Magazine 39 (3) (1966), 185-186.

The author indicates in the preface that this book is designed to give background and assistance to the teacher of today who has had little mathematical training but is called upon to teach arithmetic and mathematics. As a result this volume, which is the first volume of a larger work, deals with very basic concepts in mathematics. It is refreshing, however, to find these concepts presented in a manner which should be clear and understandable even to those who may have had little training or bad training in mathematics. In many cases the introduction of concepts is by means of pictures. Such items as "casting out nines" are introduced at an early point, and, better yet, introduced with the logic presented in rather complete form. In fact, variations of the method of casting out nines are presented for other divisors.

Throughout the book there is the theme of looking for patterns in mathematics, be it in arithmetic, algebra, or geometry. Algebra and geometry are discussed together where this is appropriate. An attempt is made to present mathematics as a logical whole rather than in a fragmented form.

There are some who will quarrel with the book in view of the fact that terms such as "commutative" are not mentioned early in the text. The concepts are presented, however, and in a very clear manner. While the presentation is not one involving a large number of axioms and rigorous proofs, the flow of reason involved in establishing the proofs is apparent. The author obviously prefers to develop the logic without the framework of formal proof in his early presentations. It is the feeling of the reviewer that any one who worked through this book carefully would, even with a very poor background, be able to progress to more rigorous proofs without undue difficulty.

The format is good and there are numerous illustrations throughout the book. ... The person who has had some background in mathematics is likely to find new light shed on some items as he reads this book. On the other hand the person with little or no mathematical background should find this book within his comprehension.

7.4. Review by: G Matthews.
The Mathematical Gazette 49 (368) (1965), 212-213.

Anything Sawyer writes is well worth reading. His style is fresh and he always has a new mathematical slant to offer. This book is based on the premise that the usual O level algebra course has to be got through: simultaneous equations, factors, fractions, long division and multiplication. "There is a well-known type of problem, not of practical value in itself, but useful as an exercise in algebra. The problem might run: John is 10 years old and his father is 40. When will John's father be just twice as old as John? " It is not called into account whether this problem couldn't become a little less well-known. Could more algebra crop up naturally? For example, could

          $\large\frac{\{x+2\}}{\{x+3\}}\normalsize - \large\frac{x}{\{x+l\}}\normalsize$

wait till the sixth form, say in connection with chapter of this book is devoted to fractions. One main application of these is to the wills of eccentric Arabs, so it was not surprising to read "If I hear that someone has left me $\large\frac{2}{3}\normalsize$ of his estate ..."

But maybe the "puzzle" approach gives sufficient motivation." A man has two sons. The sons are twins and they are the same height. If we add the man's height ... ". This is copiously illustrated with the family in acrobatic postures. "Think of a number" is treated with pictures of bags and stones. $x$ as a number hidden behind a cloud is a nice touch. There are many sound teaching ideas, e.g. to be wary of saying "No, that's wrong", and the suggested sequence of making the children themselves suggest negative integers and the rules for multiplying them.

Four volumes are planned in this series and perhaps later ones will show how to be selective with the well-worn material and to introduce new topics judiciously. Meantime this first volume is full of good ideas for the struggling teacher clobbered with the traditional syllabus.

7.5. Review by: David M Clarkson.
The Arithmetic Teacher 12 (1) (1965), 68-69.

This book is the first of a projected series of four which aims, in the publisher's words, "to give readers the essentials of mathematical literacy - that is, to bring them to the stage where they can appreciate references to mathematics and . . . read technical and mathematical literature on their own." Volume 1 is more than an anthology of Professor Sawyer's recent articles and speeches; it is a valuable distillation of his experience in teaching mathematics to children. The book is primarily addressed to elementary teachers, but the topics discussed include the four basic operations on polynomials so that junior high and secondary teachers will find much of interest here also.

A distinctive feature of Sawyer's approach to elementary mathematics teaching is his consistent use of concrete models. Thus, he goes back to the Euclidean notion of number related to geometric shape, picturing numbers by triangular or rectangular arrays, as in his discussion of odd and even numbers in the first chapter. Sawyer's treatment of divisibility contains an original discussion of numeration systems and the properties of remainders. His chapter on the solution of simultaneous equations is unusually clear, and his introduction of algebraic variables by the model of stones in a bag must now be regarded as classic.

The development of the mathematical content of the book has been very carefully planned so that a moderately careful reader may acquire the techniques he needs to read through it. Very little previous mathematics background is assumed.

8.1. From the Introduction.

It is highly desirable that the opening pages of a book should give a potential reader some indication of the scope and purpose of the book, its level of difficulty and the knowledge that it presupposes.

First of all, it should he pointed out that, while this book follows Vision in Elementary Mathematics in time, it does not follow it in the development of the subject. Vision in Elementary Mathematics was aimed at the beginnings of education; it was intended to help the teacher or parent concerned with children between, say, five and thirteen years old; it did not assume any prior knowledge of mathematics apart from that minimum of arithmetic that most people have. This book does assume some background in mathematics. It supposes the reader to be fairly comfortable with the kind of topics covered in my earlier book Mathematician's Delight. This does Dot mean that every chapter requires an understanding of calculus - far from it. If Vision in Elementary Mathematics was meant to help the teacher of children five to thirteen years old, this book may be helpful to a teacher who is revising the syllabus for pupils between eleven and eighteen years old. The discussion must make suggestions for the Mathematics and Science Sixth, but it must also concern itself with the eleven-year-olds, and with classes that are Dot being taught by a mathematics specialist. Chapters One and Three, for example, develop an approach originally published in the Scientific American. If this approach is criticised, it will probably be on the grounds that it is too childish. Again, a very considerable part of Chapter Nine has been tried out in schools, and found to be intelligible and entertaining to pupils who knew just a little algebra and Pythagoras' Theorem. Wherever possible, an idea taken from modern mathematics has been explained in terms of quite elementary mathematics.

Should the whole book have been written within an elementary framework, with all references to calculus excluded? This was decided against for the following reason. I have seen many expositions of modern mathematics which were extremely mystifying. An idea was explained to the audience. The audience were not told where it came from, nor what could be done with it. They had to take it on trust that this was an important mathematical concept, though they could not for the life of them see why. Now mathematics is above all subjects that in which you do not take things on trust; you demand proof. A very poor way to start a campaign for mathematical reform is to brainwash teachers so that they are willing to abandon their critical thinking, and accept changes without knowing why. In no sense can it be said that you are teaching modern mathematics if you simply chip off a few ideas and words from recent mathematics and convey these in isolation, without showing their relationship to other parts of mathematics, the problems they enable you to solve, the reasons why mathematicians attach importance to them.

One would therefore wish to ten a connected story, to show the ideas that led a mathematician to some new concept and the further developments he expected this concept to produce. Now the mathematicians who made the decisive discoveries of the early twentieth century had all had a very thorough training in nineteenth-century mathematics. It was by this that their imaginations had been nourished. Their aims were to clear up those points of logic which the nineteenth century had left obscure. to solve those problems the nineteenth century had left unsolved, to provide neat answers to questions that had been answered clumsily, to penetrate deeply into matters that had been discussed superficially, to unify what had been left separate, to generalise what had been handled as something particular. A twentieth-century discovery would be recognised as significant because of the light it threw on a host of nineteenth-century problems. To present the mathematics of this century without any reference to the previous century is like presenting the third act of a play without any explanation of what is supposed to have happened in the first two acts.

8.2. Review by: Eric Barton.
The Mathematical Gazette 51 (378) (1967), 339.

This book is number 4, though only the second to appear, of the series Introducing Mathematics. The first, Vision in Elementary Mathematics, was the easiest of Professor Sawyer's books. The present work is of a more advanced character, comparable rather with A Concrete Approach to Abstract Algebra; perhaps easier, because less concerned with abstract situations, but the reader with no background of calculus and dynamics will have his difficulties. The author's aim is clearly stated in the Introduction: " ... to present some of those topics in recent mathematics that are likely to be of value to people who are not professional mathematicians. Most of the topics ... illustrate one of the recurring themes of modern mathematics - that algebra, geometry and calculus have a much wider scope than had formerly been imagined."

With the clarity and ingenuity that we have come to expect of him Dr Sawyer leads us through vector-spaces (with a cat-and-dog basis), matrices and their eigenvectors to a discussion of linearity and a glimpse of the ideas of Fréchet and Banach.

This book could be read with profit by mathematics students in colleges of education, by sixth-formers aiming to read mathematics at a university and by anyone who thinks he knows about modern mathematics because he has "done sets".

8.3. Review by: William Koenen.
The Mathematics Teacher 61 (6) (1968), 636-637.

In his Introduction the author says, "The object of this book has been to identify and to present some of those topics in recent mathematics that are likely to be of value to people who are not professional mathematicians." He also states his hope that by the end of the book his readers "will not merely have met some new ideas, but will have seen some of the uses to which these ideas can be put." Having thus stated his goals, the author reveals his commitment to a practical, rather than a purist, viewpoint. He offers a quotation from Dieudonné as an expression of the purist view point; he then contrasts this with his own by saying, "There do seem to be some strands in modern mathematics that are of practical as well as poetic interest."
...

Those who have heard Professor Sawyer address teachers' groups might possibly find in this book summaries and extensions of topics that he has presented.

Any teacher or student should find the book enjoyable. It contains many ideas for presentation of topics as well as an overall picture of the interrelations between topics. The author has achieved his goal of giving a connected account of mathematics. Even a reader who is familiar with the topics in the book might gain a new understanding of them because of the way that they have been presented.

9.1. From the Introduction.

Vision in Elementary Algebra, the first volume in the series Introducing Mathematics, tried to present the basic ideas of algebra in a form that most adults would find easy to understand and that children could be expected to learn at a fairly early age. That volume, which from now on we shall refer to briefly as Vision, gave an account of how algebra arises naturally from arithmetic and described how the fundamental routines of algebraic calculation were carried out. This naturally leads to the question, 'When you have mastered these fundamental processes, what can you do with them?' This question was very much in mind when the contents of the present volume were being decided.

Those who use mathematics (or who profitably could use it, if they understood it) fall into three groups. The first group is very high-powered; it consists of mathematicians themselves and of research workers in fields such as physics and the more abstruse parts of engineering, which are so permeated by mathematics as to be in effect branches of mathematics. It seems unlikely that any of this group (except perhaps at a very tender age) will be readers of a series entitled Introducing Mathematics; this volume accordingly ignores applications of mathematics which are only of interest to advanced workers. A second group consists of those who are doing original, and possibly advanced work, in some field such as biology, psychology, sociology, or linguistics, and are using a limited part of mathematics in their work. A third, very numerous and important group, consists of those who are not using mathematics at all to make calculations or to solve problems, but rather as a language in which to learn some subject: they find that books suddenly introduce a formula or equation in the course of an explanation. The reader does not have to calculate by means of the formula or solve the equation, but simply to understand what message these are intended to convey.

It seemed possible that this present volume might offer some help to members of the second and third groups which they could not obtain from a standard textbook.

The material in this book was collected in the following way. I browsed through a large number of books in a public library, dealing with subjects that used some, but not too much, mathematics. Various topics were thus extracted that would serve to illustrate actual uses of mathematics. These form the skeleton of this book. To them it has of course been necessary to add some purely mathematical chapters in order to make the applications intelligible.

The arrangement of the book is by mathematical themes. Some mathematical idea is taken and then illustrated, as far as possible, by its use in various practical investigations. A reader should not be unduly disappointed if he finds that some concept, which he needs for his particular interests, is not illustrated in terms of his favourite science. Mathematics owes its existence as a useful subject to the fact that it deals with various patterns that turn up in the utmost variety of situations. The Binomial Coefficients may appear in an Indian study of poetry in 200 B.C., a seventeenth century study of gambling, or in Mendel's theory of heredity. Somebody who understands them in one of these contexts understands them in all.

One guiding principle has been that, the easier material is, the earlier it should be discussed. Thus, as far as other considerations would permit, the less use a topic made of algebraic symbolism and manipulation, the earlier the chapter discussing it would be placed. In the same way, within the chapters I have tried to treat first the simplest matters, and keep anything involving some degree of complication for discussion at the end of the chapter. Thus a reader who finds some difficulty with a point raised at the conclusion of one chapter may well find that the beginning of the next chapter causes no trouble at all.

Science is continually developing and changing. If I report that a certain investigator produced a graph in support of some contention, my comments are concentrated on the meaning of that graph and the message it seems to bear. I have not tried to make sure that no other scientist has since produced more weighty evidence pointing in exactly the opposite direction; had I done so. this book would never have been finished.

9.2. Review by: Geoffrey Matthews.
The Mathematical Gazette 54 (390) (1970), 412.

We have already seen Vol. 1 in this series (Vision in Elementary Mathematics). It is a good puzzle now to guess the title of Vol. 2. The present volume is certainly at the heart of things, dealing with the appearance of the same mathematics in a variety of contexts. For example Chapter 6 is entitled "Resistances, Condensers, Springs and Baths"; the hallowed problem of taps and waste-pipes is traced back to the 12th century but shown to have the same mathematics underneath it as the problem of condensers in series.

There are plenty of good, "real" examples, the fruits of "browsing through a large number of books in a public library". Simple practical work (pulleys and linkages) is followed by excellent chapters on graphs including the use of semi-log and log-log paper. The method of discovery by induction (shades of Pólya) is applied to the binomial theorem and there is a nice chapter on Algebra and Statistics.

The book is addressed to two main groups of people - those doing work in other fields (such as biology or psychology) and those "who are not using mathematics at all to make calculations or to solve problems, but rather as a language in which to learn some subject". These are presumed to have had a rough passage at school, so there are some "remedial" chapters on equations, notation, fractions, coordinate geometry, etc. which will appeal less to those who are already converted. There is a very untypical lapse on p. 104 when $P(x)$ is divided by $x - a$ and to prove the Remainder Theorem it is suggested that "all that is now needed is to 'substitute $x = a$'."

Some parts of this book will be found more enjoyable than others by readers of the Gazette, but it would be impossible for Sawyer to be dull.

10.1. From the Publisher.

Professor Sawyer's book is based on a course given to the majority of engineering students in their first year at Toronto University. Its aim is to present the important ideas in linear algebra to students of average ability whose principal interests lie outside the field of mathematics; as such it will be of interest to students in other disciplines as well as engineering. The emphasis throughout is on imparting an understanding of the significance of the mathematical techniques and great care has therefore been taken to being out the underlying ideas embodied in the formal calculations. In those places where a rigorous treatment would be very long and wearisome, an explanation rather than a complete proof is provided, the reader being warned that in a more formal treatment such results would need to be proved. The book is full of physical analogies (many from fields outside the realm of engineering) and contains many worked and unworked examples, integrated with the text.

10.2. From the Introduction.

Is it justified to have a course in mathematics for engineers? Would it not be better to provide a course in electrical calculations for electrical engineers, chemical calculations for chemical engineers and so on through all the departments? The reason for teaching mathematics is that mathematics is concerned not with particular situations but with patterns that occur again and again. This is most obvious in elementary mathematics. In arithmetic, the number 40 may occur as $40, 40 horsepower, 40 tons, 40 feet, 40 atoms, 40 ohms and so on indefinitely. It would be most wasteful if we decided not to teach a child the general idea of 40, but left this idea to be explained in every activity involving counting or measurement.

Advanced mathematics cannot claim the universal relevance that arithmetic has. There are mathematical topics vital for aerodynamics that leave the production engineer cold. An engineer cannot simply decide to learn mathematics. He must judge wisely what mathematics will serve him best. His aim is not only to find mathematics that will help him frequently now, but also to guess what mathematics is most likely to help in industries and processes still to be invented.

Linear algebra qualifies on both counts; it is already used in most branches of engineering, and has every prospect of continuing to be.

Very many situations in engineering involve the mathematical concept of mapping. We shall restrict ourselves to the simplest class of these, namely linear mappings. We shall explain what these are and how to recognise an engineering situation in which they occur. And again in the interests of simplicity, we will not consider linear mappings in general, but restrict ourselves to linear mappings involving only a finite number of dimensions.

10.3. Review by: C C Goldsmith.
The Mathematical Gazette 58 (403) (1974), 68-69.

Professor Sawyer writes for college students in engineering "of average ability whose principal interests lie outside the field of mathematics", and assumes no previous knowledge of matrices, transformations or complex numbers. After introducing the ideas of mapping and linearity with a variety of well-chosen practical examples, he launches into the elementary 2-D work to be found in most recent British O level texts. This is well done, and the digressions on complex numbers and trigonometry are appropriate. There are rather few problems from engineering or any other application, but eigenvectors and eigenvalues are introduced through a simple example on normal modes of oscillation, and later an electrical network problem is discussed in an interesting way. The second half of the book is concerned with orthogonal transformations and quadratic forms, then lines, planes and linear equations, and finally the Simplex method for linear programming. General theorems are avoided, the examples are mostly in 3-D, and the explanations are full and clear.

This is a well produced book ideal for sixth formers or college students who have missed out on linear algebra in their earlier mathematics teaching. In school mathematics libraries it will help pupils to clarify their understanding of the fundamental ideas, and teachers who dip into it will find ways of introducing familiar topics in stimulating new ways.

10.4. Review by: Saul Birnbaum.
The Mathematics Teacher 66 (6) (1973), 543.

This book was written, not as a basis for a lecture course, but to be read by the student piece by piece as a basis for discussions in class and tutorials. The author feels that, with the invention of printing, the raison d'être for lecturing has ceased to exist. In his exposition he avoids the mortis of extreme rigour and the indigestion of the cookbook approach.

Instead, the student learns to picture or imagine the thing he is tackling, and then, because his imagination is working, he can reason about those things. Thus, in the very first pages, the author starts with the real world and shows how real situations lead to the basic concepts of linear algebra. The writing is simple and easy to understand. Try it - you'll like it.

11.1. From the Publisher of the 2010 reprint.

Functional analysis arose from traditional topics of calculus and integral and differential equations. This accessible text by an internationally renowned teacher and author starts with problems in numerical analysis and shows how they lead naturally to the concepts of functional analysis.

Suitable for advanced undergraduates and graduate students, this book provides coherent explanations for complex concepts. Topics include Banach and Hilbert spaces, contraction mappings and other criteria for convergence, differentiation and integration in Banach spaces, the Kantorovich test for convergence of an iteration, and Rall's ideas of polynomial and quadratic operators. Numerous examples appear throughout the text.

11.2. From the Introduction.

In the introduction to his book, Functional analysis and numerical analysis, L Collatz said that numerical analysis had been revolutionised by two things - the electronic computer and the use of functional analysis.

This statement is striking not only for its emphatic tone but also for the contras t of mathematical epochs it involves. Numerical analysis is an earthy subject concerned with questions involving numbers. Even if it uses very sophisticated modern equipment, essentially it is concerned with arithmetic, the oldest branch of mathematics, whose beginnings lie in prehistory. Functional analysis, on the other hand, while it has roots in nineteenth century mathematics, is a product of the present century.

In trying to cope with such a modern branch of mathematics, a student of numerical analysis encounters two difficulties. Any part of modern mathematics is the end-product of a long history. It has drawn on many branches of earlier mathematics, it has extracted various essences from them and has been reformulated again and again in increasingly general and abstract forms. Thus a student may not be able to see what it is all about, in much the same way that a caveman confronted with a vitamin pill would not easily recognise it as food.

The second difficulty is that functional analysis was not created with numerical applications in view. It arose from a great variety of sources - from the calculus of variations, from integral equations, from Fourier series, from mechanics, from the theory of real and complex variables, from number theory, and from other topics. It therefore has a great range of possible applications and a student cannot assume, because a theorem in functional analysis is generally regarded by mathematicians as of great importance, that it will necessarily help us in problems of numerical analysis .

The aim of this book is to make some contribution towards overcoming these two difficulties - by explaining the ideas of the subject and, as far as possible, by emphasising those ideas that have proved useful to numerical analysts. In the main the concepts of functional analysis will be introduced by discussing problems in numerical analysis from which these concepts could arise. References will be given so that readers can go back to original sources to clear up any obscurity or to judge for themselves the relevance of the topic to their own interests.

11.3. Review by: L Collatz.
Mathematical Reviews MR0500054 (58 #17764).

The author gives a clear and easily understandable, broadly written presentation of some general ideas of functional analysis with special emphasis on terms that are used in numerics. With the help of numerous examples from analysis and many instructive illustrations, he would like to build a bridge between abstract and concrete mathematics. Basic terms such as convergence, linear operators, norm, Fréchet's differentiation, Hilbert space, dual space, etc. are developed in detail and applied to contracting maps, Newton-Raphson method, Taylor and Fourier series and some other areas. The book concludes with reflections on compactness and vector products.

11.4. Review by: Johnston Anderson.
The Mathematical Gazette 63 (424) (1979), 135-137.

This is the fifth volume in the Oxford Applied Mathematics and Computing Science series. According to the author, "the aim of the book is to provide an intelligible introduction to functional analysis by giving samples of its applications to numerical analysis". Functional analysis is a very broad and diverse subject and while this book does, I think, succeed in its aim over such topics as metric and normed spaces, there are inevitably many areas of functional analysis which are ignored or barely mentioned, so that in some respects, therefore, the title is a little over-ambitious. A good deal of attention is naturally paid to questions of iteration and convergence with numerical examples well in evidence, but beside this the book seeks to enlighten the reader about the general ideas of functional (or linear) analysis.

These ideas are often introduced by considering classical results (e.g. the mean value theorem) and looking at generalisations, or by discussing situations in numerical analysis from which concepts can arise (for instance, defining more general and abstract notions of approximation and iteration). ...

The style of the book is crisp and the author shows he has lost none of his well known skill in putting across his ideas, frequently appealing to geometrical analogy and using any metaphor or imagery that he considers striking or useful in effect. The result is a lucidity that often illuminates very clearly what could otherwise be simply rather opaque symbolism. What is not so obvious, though, is the target readership. It is not, in my view, the kind of book that someone lacking a good grounding in classical analysis and a fairly mature mathematical outlook would find easy reading, despite the informality of the style. The book could be aimed at numerical analysts or students of numerical analysis who wish to know some- thing about functional analysis, or it could be aimed at people who do know some functional analysis and who might want to see some practical uses. To a person in the latter category, the book may well be not only interesting and informative but, as an informal exposition, extremely valuable. I can imagine many students with muddled ideas of analysis finding that this book puts these in perspective for the first time. However, while a student who knows some classical and numerical analysis might discover from reading this book what functional analysis is about (particularly in respect of numerical analysis), I don't think he will actually learn much functional analysis from it, apart from some of the vocabulary. In no sense is it a textbook of functional analysis and it does not claim to be; nor is it a textbook of numerical analysis, so there is a distinct danger that it will be ignored by both disciplines.

This would be a great pity, for the book does fulfil to a reasonable degree the author's aims. Probably it would best be read during and after introductory courses in functional and numerical analysis. Students should certainly be made aware of the book's existence, for it is they who are most likely to benefit from it, though lecturers too will glean pedagogic points of value. As a supplement to a lecture course, it could be invaluable, especially if the students were persuaded to follow up the references.