# George Temple's books

George Temple wrote a number of books most of which received high praise from reviewers but some were rather heavily criticised. We give some extracts from Prefaces and reviews of his books below attempting to illustrate the aspects which were greatly liked but also showing areas where reviewers were critical.

Click on a link below to go to the information about that book.

An introduction to quantum theory (1931)

Rayleigh's principle and its applications to engineering. The theory and practice of the energy method for the approximate determination of critical loads and speeds (1933) with William G Bickley

The general principles of the quantum theory (1934)

The Classic and Romantic in Natural Philosophy: An Inaugural Lecture (1954)

An introduction to fluid dynamics (1958)

Turning points in physics (1958) with R J Blin-Stoyle, D ter Haar, K Mendelssohn, F Waismann, and D H Wilkinson

Cartesian tensors: An introduction (1960)

The structure of Lebesgue integration theory (1971)

100 years of mathematics. A personal viewpoint (1981)

1. An introduction to quantum theory (1931), by George Temple.
1.1. From the Preface.

Since the opening of the present century every department of fundamental physical inquiry has been invaded and reorganised by the principle of relativity and the quantum theory. It is characteristic of the critical temper of modern scientific thought that both of these principles relate to the nature of our knowledge of the physical world rather than to the nature of that world itself; and that both principles are more concerned with the correct formulation of physical problems than with their detailed solution.

Neither of these principles can be justly appreciated in the absence of historical perspective. The theory of relativity is only the latest phase of a critical principle which issued earlier in the method of dimensional analysis and in the applications of functional equations to the basic axioms of dynamics. The quantum theory has a much briefer history, for it is but a recent synthesis of two theories which previously divided between them the whole territory of physics, mechanics and wave theory. These monistic theories of the nature of matter and light have now been fused into a single dualistic theory; and it is precisely the success of the quantum theory in coordinating the corpuscular and undulatory properties of both light and matter that constitutes its claim on the attention of the physicist.

1.2. Review by: J Hargreaves.
The Mathematical Gazette 16 (220) (1932), 285-287.

The Introduction to the Quantum Theory, by G Temple, is a well-written book on the mathematical theory. The approach is from rather a novel, but interesting, standpoint. As an analogy to hydrodynamical ideas, an electric fluid is imagined, whose density is proportional to the square of the amplitude of the wave function, thus furnishing a mind-picture of the statistical distribution. The analogy proves very fruitful in developing the theory from Bernoulli's Equation, but it appears to be rather overstressed and its usefulness strained. There are good chapters on quantum algebra, matrix mechanics, relativistic wave mechanics, and numerous simple problems are worked out illustrating the general theory. The book is very clearly written and is a useful general survey of the subject without reference to any particular aspect of it. It is essentially mathematical in outlook.
2. Rayleigh's principle and its applications to engineering. The theory and practice of the energy method for the approximate determination of critical loads and speeds (1933), by George Temple and William G Bickley.
2.1. From the Preface.

In recent years engineers (and physicists) have begun to realise the utility of Rayleigh's Principle as an instrument for the rapid and direct calculation of the approximate values of critical loads and frequencies. It is well known that the approximate values determined in this way are necessarily in excess of the true value, and that, in all cases in which the true value is known, Rayleigh's approximation is well within the engineering degree of tolerance. To perfect this method of calculation all that is required is a means of calculating an approximation which is necessarily in defect of the true value, and which it as accurate as Rayleigh's upper estimate. In this book Rayleigh's energy method is developed in such a way as to furnish both upper and lower estimates of the true value required, so that it enables critical loads and frequencies to be determined with close and known degrees of approximation.

A purely mathematical account of this method, based on the theory of integral equations and Green's functions, was given a few years ago. Such topics as these: are not included in the course of study normally pursued by engineering students, and, for this reason, no knowledge of them is presupposed in the readers of this book. The mathematical apparatus is constructed piece by piece as required, in such a way as to exhibit its physical significance and to conform to Poinsot's counsel of perfection which has been taken as the motif of this work. The introductory chapter and the appendix to Chapters III and IV give a perspective view of the argument and may serve as an Ariadne's skein for these who might otherwise be lost in the mathematical labyrinth.

We wish to record our thanks to Professor Chapman who suggested the collaboration of the authors and made many helpful suggestions. And we owe a great debt of gratitude to Mr T W Dickson, M.C., B.A. of the City and Guilds (Engineering) College who read the manuscript and proofs with great care, verified all the calculations, and pointed out many obscurities and inaccuracies.

2.2. Review by: John Prescott.
The Mathematical Gazette 17 (226 (1933), 339-340.

In the book under review the authors, besides working out a number of problems in applied mathematics, have supplied the theory of Rayleigh's Principle. In fact, the two most important chapters in the book are devoted to the study of the functions arising in these vibration problems, and therein we are shown under what conditions Rayleigh's Principle will give good results. A further most important contribution to the subject is a method of successively approximating to the slowest frequency and of using the same set of calculations to give an approximation to the second frequency, the last being less accurate at each stage than the corresponding approximation to the slowest frequency. All this is done by a function here called an Induction Function, each problem having its own induction function. While this function is very useful in theory, nevertheless the authors very soon discard it in practice because, as it has always discontinuities, it leads to cumbersome integrals in the applications. These chapters on the rigour of the subject form a useful piece of work of which applied mathematicians themselves have felt the need. It is so easy, when we have found a simple trick that usually works and gives good results, to go on using it without thoroughly understanding it. It is generally only when it fails to work that we begin to investigate it properly - or turn for help to somebody who can do it for us.

Recognising the abstruseness of the mathematical arguments, the authors have done their best to illustrate the theory by simple applications, and have also given a resume of the mathematical arguments and conclusions at the end of Chapter 4 for the benefit of those who are not exactly mathematical experts.

If the book impresses on engineers and physicists the usefulness and the simplicity of this method of solving difficult problems of vibrations and of elastic stability by means of very simple mathematics, it will have been very well worth the labour of production.

2.3. Review by: Anon.
The Military Engineer 49 (330) (1957), 329.

A rapid method of determining the fundamental period of a vibrating system or the condition of stability of an elastic system within the accuracy demanded in engineering problems is explained.

2.4. Review by: A H T.
Mathematics of Computation 11 (60) (1957), 291-292.

The authors state: "The object of this book is to explain and justify a rapid method of determining the fundamental period of a vibrating system or the condition of stability of an elastic system, with the degree of accuracy usually demanded in engineering problems. ... The fundamental principle is due to the third Lord Rayleigh, and it applies not only to vibrating systems with a finite number of degrees of freedom, but also to continuous systems such as a stretched string or metal reed."

The authors only mention that the Rayleigh Principle is intimately related to the principle of least action and state that whenever the differential equations of a problem, dynamical or otherwise, are equivalent to a variational principle, the problem is always soluble by methods analogous to those discussed in this book. However, this intimate relationship is nowhere discussed nor is the relation between variational principle for the system under discussion and the Induction Function $G(x, s)$ (the Green's function of the equations describing the system mentioned).

An outstanding feature of this book is the use of this function $G(x, s)$ to generate sequences of functions $f_{n}(x)$ which approach the first (and higher) proper function of the equations describing the system and the use of these approximate proper functions for the determination of approximations to the proper values of the system. The major portion of the discussion is devoted to a method for determining the first proper function and proper value. However lower and upper bounds are given for the latter quantity and the lower bound is shown to depend on the ratio of the first two proper values. A method for estimating this ratio is also given.
...
The book is extremely well written and will be particularity useful to those who desire closed formulas for various approximations in engineering problems. Modern numerical analysts and other mathematicians will find this a thought-provoking book and very suggestive of methods for solving engineering problems. However it is an open question as to how suitable the methods discussed in this book are if numerical solutions are to be obtained by the use of automatic digital computers.
3. The general principles of the quantum theory (1934), by George Temple.
3.1. Review by: W W.
Science Progress (1933-) 31 (121) (1936), 161-162.

The author claims as his object to give an introductory account of the general principles which form the physical basis of the quantum theory.

The book undoubtedly provides a sound exposition of the mathematical apparatus of the quantum theory and we do not question the correctness of the equations and formulas. The actual physical principles, however, do not stand out as we should have wished.

Some of the statements might be criticised, although the author's meaning is perhaps correct, e.g., "macroscopic physics and microscopic physics differ widely in their formal objects." We venture to disagree with this statement. What is meant is that the mathematical expression of microphysical laws is very different from the older description of macrophysical laws but the older is really a limit approached by the newer. The author unfortunately conveys the impression that the operational calculus employed is something rather difficult - much more difficult than we feel it ought to be if it were expressed in the best way.

The book constitutes a sound and conscientious presentation but is rather better suited for those who have already some acquaintance with the quantum theory and its notation than for beginners. It is a good book but not a satisfactory introduction. One gratifying feature is the use of the English adjective "proper" instead of the German "eigen." This seems to indicate that the author knows more of the history of mathematical terminology than many writers, for the word "proper" in this sense was in use before the quantum theory was conceived.

We do not wish to be hypercritical; the book is undoubtedly to be recommended for all who wish to acquire a sound knowledge of the quantum theory.
4. The Classic and Romantic in Natural Philosophy: An Inaugural Lecture (1954), by George Temple.
4.1. Review by: L B.
Blackfriars 35 (411) (1954), 276-277.

Firmly setting aside the sublime and the prophetic styles of inaugural lecturing, Professor Temple chooses, so he tells us, the familiar. It was a wise choice, for he is master of this 'modest and friendly manner', a manner, surely, that is peculiarly Oxford's own. There are some newcomers not to be thought of as strangers; it is thus that Oxford will welcome her new Sedleian professor of natural philosophy.

The basis of his lecture is the fact of 'two great movements in natural philosophy - one leading from experiment to general principles and the other returning from general principles to experiment'. It is perhaps worth noting that there seem to be very few modes of thought in which a similar distinction is not to be found. In particular, students of Kant will recognise the contrast of analysis and synthesis. Professor Temple's application is to treatises of applied mathematics. Where the second movement predominates, and from a few principles are deduced in regular order a large number of particular disciplines, the language of the treatise may, he suggests, fitly be termed 'classical'. But where principles are being discovered from a tangle of new experimental data, where no ways are safe and intuition rules, the literary analogy must be with the romantic style.

With wit and learning Professor Temple proceeds to analyse, along these lines, the works of the masters in his subject, from Sir Isaac Newton to Sir Edmund Whittaker. And he concludes by putting in a plea for lecturing itself as a method of teaching able to provide something lacking in printed works, so long as it concentrates on the way of discovery, leaving precise and perfection should be reserved for the monograph: the successful lecture is almost inevitably a romantic adventure.' Wise words; which might well be pondered by lecturers even outside the school of applied mathematics.

4.2. Review by: William Keith Chambers Guthrie.
Philosophy 30 (114) (1955), 282-283.

Professor Temple apologises for venturing to appropriate some of the "noble concepts and terminology" of humane studies. I cannot tell how his scientific colleagues will view the procedure, but a student of the humanities can only be profoundly grateful when some of the fundamental characteristics of scientific research are revealed, and its appeal transmitted, in the terms of a familiar conceptual framework. Taking as the marks of a classic style unity, completeness and irreducibility, and of romance the adventurous seeking and accepting of whatever may occur (which entails perpetual incompleteness), he sees the one in the deductive, downward movement from general principles to particular experiment and the other in the upward, inductive path by which experiments lead to the discovery of first principles. (It is a refreshing thing to learn that "Saintsbury's characterisation of the romantic fits fluid dynamics like a glove.") The emphasis on one or the other varies from subject to subject and period to period, and also, of course, with the temperament of the individual researcher, as is made plain by the examples which follow. The contrast is delightful, for instance, between the English untidiness of Clerk Maxwell and the classic French precision of Poincaré, whose criticism of the Englishman Professor Temple quotes at length.

Finally the author suggests - and how one longs to see his advice more generally followed - that while the classic style may suit a monograph, the romantic is the only tolerable one for lectures. If the interest of the student is to be held, he must be shown not only the static perfection of the result, but the mental struggles, the false starts and wild surmises, that went to the making of it. "Classics may be models, but they are not guides."

The lecture is a pleasure to read, not least for the felicity of its quotations, and seldom can so much light have been shed in the space of twenty-two pages.
5. An introduction to fluid dynamics (1958), by George Temple.
5.1. From the Preface.

The object of this book is to provide an introduction to Fluid Dynamics, primarily for students reading for Honours in Mathematics and Theoretical Physics. There is an undoubted need for such a work, for although we have the comprehensive treatises of Lamb and of Milne-Thomson, there is no elementary account of the basic principles of modern Fluid Dynamics. The works of Basset and Ramsey still repay consultation, but the emphasis in Fluid Dynamics is no longer on analytical solutions of ingenious problems but on the development of the physical significance of the fundamental principles.

Our main purpose therefore is to introduce Fluid Dynamics as a branch of dynamics and to concentrate on the fundamental dynamical principles and their immediate applications to the types of fluid flow which are actually observed or produced, especially to the disturbance flow produced by the motion of a solid body through a fluid. Such an introduction presupposes some empirical knowledge of fluid flow as derived from observation of river flow, of ocean waves, of the flight of birds, of the man-produced flow over weirs, under sluices, and in wind tunnels. It is also incomplete without some knowledge of experimental technique, either acquired at first hand in the laboratory, or from the stimulating works of Prandtl.

5.2. Review by: Louis Melville Milne-Thomson.
Quarterly of Applied Mathematics 17 (3) (1959), 329.

The preface states that "the object of this book is to provide an introduction to fluid dynamics, primarily for students reading for honours in mathematics and theoretical physics." One cannot but approve the object and simultaneously acknowledge that the author has achieved it with brilliant success.

Keeping within the bounds of inviscid continuous fluid and with an eye firmly fixed on the physical problem, Temple has produced in 190 pages a fascinating account of hydrodynamics. Particular emphasis has been placed on the "fluid body" that is a portion of fluid which always consists of the same fluid particles. In this connection the argument of 1-3 concerning the application of the laws of motion appears to be unconvincing, for it starts from the tacit assumption that the internal forces form a self-equilibrating system.

The reader is led by easy stages through elementary notions to sources, doublets and vortices, to distributions of these singularities and the action on a body in a uniform stream. Here, in deriving the Kutta Joukowski theorem the author obtains a drag term due to the total source strength in Green's equivalent stratum. To the reviewer this term seems to be necessarily zero, for the normal velocity on the surface of a body at rest in a uniform stream vanishes and it is to this normal velocity that the stratum is due. The text then goes on to conformal mapping, free streamlines, design of wing profiles, axisymmetric flow, and finally slender body theory applied to solids of revolution and checked by exact solutions for the ovary ellipsoid.

The book can be heartily recommended.

5.3. Review by: Thomas MacFarland Cherry.
Mathematical Reviews MR0098526 (20 #4983).

The object of this work is to provide an introduction to fluid dynamics, with the emphasis on physical principles and such developments therefrom as are of practical significance in aerodynamics. The author has designed it for 'students reading for honours in mathematics and theoretical physics', and states that there is an undoubted need for such an introductory work; and the reviewer agrees with this opinion: the work will be valuable not merely for students but also for those who, like himself, were nurtured in the tradition of Lamb's classic treatise, but wish their teaching to be more modern and realistic.

The author considers only fluids that are incompressible and non-viscous; the emphasis is on two-dimensional steady irrotational flows, and of the specific flows that are discussed, the most complicated are the 'disturbance flow' produced by a circular cylinder in a uniform stream and flows produced from this by conformal mapping, and the discontinuous flow around a flat plate. A final chapter on slender-body theory is a welcome and novel feature. The reader is assumed to be familiar with the calculus of functions of several variables and with the elements of complex-variable theory and vector analysis. The modern point of view of the author is illustrated by his terminology: he speaks nearly always of the 'mass equation' rather than 'equation of continuity' and of 'potential flow' as often as 'irrotational flow', and the term 'Laplace's equation' is never used.

5.4. Review by: R Tiffen.
Science Progress (1933-) 48 (189) (1960), 120.

The aim of this book is to provide an introduction to modern fluid dynamics suitable for Honours Students of Mathematics or Theoretical Physics. Emphasis is placed on fundamental dynamical principles rather than on the solution of those problems which are of merely analytical interest. Such a book has been needed for some time and much satisfaction and pleasure will be felt that Professor Temple has undertaken to fill the gap. Although the volume is the best of its kind, it is open to criticism in one or two respects. Some readers would prefer the separation of purely analytical theorems from the discussion of physical concepts. Mathematical difficulties inherent in the theorems of Green and Stokes, the distinction between irrotational and rotational fields, etc. tend to magnify the difficulties of hydrodynamical theory. As a further improvement it could be suggested that a more wholehearted use of vector methods at the beginning would be preferable. Many of the equations are first introduced and manipulated in terms of rectangular cartesian coordinates. Although it would be simpler to formulate the theory in terms of vectors from the outset, the method adopted here enables the reader with no previous knowledge of vectors to go ahead.

An excellent feature of this text is the way in which assumptions such as continuity of the variables involved in the general theory are clearly stated. In addition, discussions of topics not generally found in an introductory course are included here. For example, there are accounts of flow with discontinuities over a surface, the design of wing profiles and flow past a slender body.

About a third of the book is devoted to two-dimensional motion. Here functions of a complex variable are freely used. The admirable balance between mathematical and physical ideas ensures that this volume will prove of equal interest to mathematicians and physicists.
6. Turning points in physics (1958), by R J Blin-Stoyle, D ter Haar, K Mendelssohn, G Temple, F Waismann, and D H Wilkinson, with an Introduction by A C Crombie.
6.1. Review by Jerome Raymond Ravetz.
The British Journal for the Philosophy of Science 11 (42) (1960), 167-168.

The essays making up this book are taken from the lectures given to Oxford undergraduates, by a group of five physicists and one philosopher. Dr Cromble's only explicit contribution is a brief introduction, but the essays speak for his skill in selecting and 'briefing' his speakers. The bulk of the book is devoted to explaining, to non-specialists, the conceptual and philosophical developments associated with the revolution in physics which took place in the first half of this century. ... Professor Temple's essay is the most stimulating and charming, probably because he concentrates on a conceptual, rather than a historical, analysis of relativity.

Scientific American 202 (5) (1960), 219-220.

This book presents a series of lectures, given at the University of Oxford in 1958, that was addressed to philosophers and scientists who were not physicists. The turning points described are the end of mechanistic philosophy and the rise of field physics, the quantum theory, the entry of probability into physics, relativity, the causality crisis, new concepts of elementary particles. The publishers say I that the book is for laymen as well as specialists, but most of the lectures call for both a background of knowledge and close attention. Conceding these requirements, the book is highly instructive as well as revealing. It explains a good deal of what has happened in and to physics since James Clerk Maxwell and Michael Faraday, and it also explains why the subject is so active, so fruitful, so magnificently successful and so nervously unsure of itself-not only moving from strength to strength, but lurching from crisis to crisis.
7. Cartesian tensors: An introduction (1960), by George Temple.
7.1. From the Preface.

The purpose of this book is to provide an introduction to the theory of Cartesian tensors for first-year students pursuing an Honours course in Mathematics or Physics.

Tensor analysis was first forced upon the attention of theoretical physicists by the publication of Sir Arthur Eddington's 'Report on the Relativity Theory of Gravitation'. In that report, however, tensor analysis was inevitably deployed on what was then the strange terrain of Riemannian geometry. In 1931 Sir Harold Jeffreys had the happy idea of displaying tensor analysis on the familiar stage of three-dimensional Euclidean space. In this setting tensor analysis is freed from all irrelevant complications and is manifested in all its simplicity and power.

The excuse for writing another book an Cartesian tensors is that in the last thirty years the subject has been developed in a number of different directions which are of interest and importance to theoretical physicists.

The original definition of a tensor as a set of variables which are transformed cogrediently with the coordinate system with which they are associated has been replaced by the simpler and deeper definition now codified in the work of Bourbaki. I have somewhat domesticated the native abstraction of this definition while preserving (I trust) its spirit and utility. Thus tensors are here defined as multilinear functions of direction, and this definition is found to simplify many theorems and to give a new unity to the subject.

The analysis of the structure of tensors (especially those of the second rank) in terms of spectral sets of projection operators is part of the very substance of quantum theory and therefore requires at least an elementary discussion.

The subject of isotropic tensors (whose components are the same in all orthogonal bases), always of fundamental importance in elasticity and hydrodynamics, has received new vigour from developments of the theory of group-representations and of abstract invariant theory.

The development of spinor analysis, both as an algebraic discipline and as an integral part of quantum theory, appears to demand an introductory account, even in a work restricted to three-dimensional space.

I have therefore attempted to provide some initiation into these topics without quitting the confines of Euclidean space. I have, however, resisted the temptation to trespass too deeply into the territory of the hydrodynamicist and elastician, the quantum theorist and the relativist. For physicists the theory of tensors in a spacetime manifold is so intimately associated with the special theory of relativity, and the theory of tensors in a Riemannian manifold with the general theory, that I have also held myself excused from these questions in this elementary introduction.

A number of examples have been devised to illustrate the general theory and to indicate certain extensions and applications.

A number of examples have been devised to illustrate the general theory and to indicate certain extensions and applications. For the pure mathematician this book can scarcely do more than encourage the study of linear algebras and of Riemannian geometry; for the applied mathematician and physicist it may foster an acquaintance with the theory and practice of that most useful language - Cartesian Tensors.

I am grateful to my colleague Professor C A Coulson who has kindly read this book in manuscript and has made many helpful suggestions.

7.2. Review by: R Tiffen.
Science Progress (1933-) 49 (195) (1961), 518-519.

This is a good introduction to the theory of Cartesian tensors for first-year students pursuing honours courses in mathematics or physics. Readers already familiar with the subject in its older form will find something new in the present approach, which is derived from the work of Bourbaki, but the simplification here is too great for the elegance of the latter's work to be really appreciated. Moreover, physicists and applied mathematicians who are already conversant with general tensor theory may be impatient with yet another discussion of Cartesian tensors.

The book contains a general account of vectors, bases, orthogonal transformations, tensor algebra and tensor calculus. Illustrations from physical theories such as those of elasticity addition there is a section on the theory of spinors, useful for a full understanding of tensor theory and essential for the quantum mechanist. The final section is concerned with orthogonal curvilinear coordinates, and many useful formulae are derived. However, the reviewer feels that, once the realm of Cartesian tensors has been left behind, the only satisfactory approach is that of general tensor theory, which would be unsuitable here. It would have been safer to omit this section altogether.

The real importance of the present work lies in its suitability for first-year students. Many courses for mathematicians and physicists contain no hint of this theory, which should rightly come near the beginning of the course. It is to be hoped that the value of early courses in the subject will be appreciated more widely, and proper use made of Professor Temple's labours.

7.3. Review by: J E Adkins.
Mathematical Reviews MR0116281 (22 #7076).

After a preliminary discussion of the elementary ideas on vectors, bases and orthogonal transformations, the author introduces a tensor as a multilinear function of direction and outlines the elementary operations of combination and differentiation, indicating some of the applications to classical and fluid dynamics and elasticity. The book also includes chapters on isotropic tensors, spinors, and orthogonal curvilinear coordinates.

In the section on isotropic tensors the author performs a service in drawing attention to Weyl's method for finding polynomial invariants of a system of vectors under the orthogonal group. The isotropic tensors of higher order are derived by the method used by G F Smith and R S Rivlin [Quart. Appl. Math. 15 (1957), 308-314] for the anisotropic tensors and an explicit reference to their work would perhaps have been useful.

7.4. Review by: A Pipkin.
Quarterly of Applied Mathematics 20 (2) (1962), 120.

This is an odd, irritating, and excellent book. The oddness is in insistence on modern points of view even when old-fashioned ones may be simpler. Irritation arises from the fact that the book does not appear to have been proof-read. Once through the distracting maze of misprints and errors, however, it becomes clear that there has been a gap in the literature, and that this book, in a second edition, may fill it.

The first four chapters present the basic theorems of tensor algebra and calculus, with scarcely a mention of transformation rules. Through a number of physical and geometrical examples, Professor Temple convinces the reader that tensors are multilinear functions of direction, and defines them as such. The transformation rules for components then appear as a simple and ignorable consequence of the definition. The stubborn and successful refusal to make any use of transformation rules accounts for a good deal of the interest of the book.

Isotropic tensors are derived by studying what in the older terminology would be called orthogonal invariants of vectors. In view of Professor Temple's definition of a tensor, this method of derivation is the natural one, and it is possibly for this reason that no reference is made to the work of Rivlin and Smith, who have used the method extensively. It is unfortunate that the full power of their method is not used, since the reader is left with the impression that derivation of isotropic tensors of any given rank is a separate and complicated job.

The structure of second-rank symmetric tensors is discussed in terms of the spectral theory of operators. The treatment of this subject is exceptionally clear and simple. Also in the line of subjects slanted toward quantum theory, there is a chapter on spinors which I am not qualified to comment on. The book closes with a chapter on orthogonal curvilinear coordinates. There are a few problems, always well-chosen, to prepare the student for material coming later in the book.

Some of the misprints are unnervingly consistent. For example, "Kronecker" is always spelled "Knonecker". The stress tensor is proven symmetric by a calculation which is uniformly wrong. The discussion of invariants of three vectors in three dimensions is misleading, although not literally false. The section on invariants of four vectors is so full of both misprints and real errors as to be useless. In the derivation of strain components in curvilinear coordinates, it is almost never clear, whether the summation convention is in force or not. It is a pity that such a basically good book should be so disfigured.

7.5. Review by: Ram Prakash Kanwal.
Technometrics 3 (4) (1961), 570.

This book furnishes an elegant introduction to cartesian tensors for students of applied mathematics and physics. All the topics discussed by Sir Harold Jeffery in his classical textbook on cartesian tensors are there, arranged in a very systematic manner. In addition, there are topics such as the structure of tensors and spinors. The tensors are defined, following Bourbaki, as multilinear functions of direction. The structure of tensors is explained in terms of eigenvectors. The theory of spinors is introduced by the way of isotropic tensors as well as by the way of Clifford algebra.

Every concept is illustrated with examples from elasticity, hydrodynamics, quantum theory and relativity. In addition to its simplicity and clarity, the book is self-contained. As such this book would be widely welcomed.

7.6. Review by: Thomas Arthur Alan Broadbent.
The Mathematical Gazette 45 (354) (Dec., 1961), 357-358.

Those to whom tensors meant alarming symbols in bizarre spaces were enlightened, 30 years ago, by Sir Harold Jeffreys' exposition of the simplicity and value of tensor analysis in Euclidean three-space. Professor Temple underlines and extends this lesson in his short monograph, which should be easily comprehended by any Honours student. Among the book's many merits, two in particular deserve notice. Temple follows Bourbaki in recognising a tensor as an invariant multilinear function of direction. This makes it easy to answer the novice's question "What is a tensor?" The more sophisticated student may understand that how a tensor behaves is more important than what a tensor is, but the definition of a tensor as a vague something which obeys a not very luminous law of transformation must have been a stumbling block to many a beginner. The concept of a multilinear function of direction is easy enough to grasp, particularly when the general notion is prefaced by concrete instances, and the transformation law becomes almost self-evident for tensors of the second rank and is easily generalised to tensors of higher rank. Secondly, there is a short and valuable chapter on spinors. These somewhat subtle entities can be approached either by considering the eigenvectors of a rotation, of which two are complex, or by way of the Pauli spin matrices, the unit rotors for the axes, which exemplify a Clifford algebra. Both methods are slightly devious, and the exposition here is terse, though clearer than that to be found in some other accounts. The reader might be well advised to brush up his knowledge of the Cayley-Klein rotation parameters before reading this chapter; he would then be more likely to appreciate the motivation.

The final chapter, on tensors in orthogonal curvilinear coordinates, is strictly outside the declared purpose of the book, but is a welcome compact discussion of component formulae in these systems.

Of several recent accounts of Cartesian tensors, this is quite the clearest and most elegant.

7.7. Review by: Francis D Murnaghan.
Mathematics of Computation 15 (75) (1961), 303-304.

This introduction to vector and tensor algebra is well planned and should prove of value to the better than average student. The amount of material covered in less than 100 pages is surprising; in addition to the usual topics, there are chapters on isotropic tensors, spinors, and orthogonal curvilinear tensors. The influence of Weyl is evident in the treatment of isotropic tensors and of Brauer and Weyl in the treatment of spinors. We heartily recommend the book, which is addressed to first year students "pursuing an Honours course in Mathematics or Physics" in England.

7.8. Review by: Charles Eugene Springer.
Amer. Math. Monthly 68 (8) (1961), 821.

In this small volume the author discusses briefly a number of topics (of particular interest and importance to theoretical physicists) which have expanded the field of cartesian tensors during the thirty years since the appearance in 1931 of the book by Jeffreys. Instead of defining a tensor as an invariant with a set of functions as components which transform cogrediently with the coordinate system with which they are associated, this author defines tensors in the Bourbaki fashion as multilinear functions of direction. Examples are chosen to illustrate the usefulness of the tensor notation in both the algebra and analysis of tensors. There appear concise treatments of the strain tensor, isotropic tensors, and spinors, all of which is accomplished within the confines of three-dimensional Euclidean space. A short discussion of tensors in orthogonal curvilinear coordinates is provided in the closing chapter.
8. The structure of Lebesgue integration theory (1971), by George Temple.
8.1. From the Preface.

The purpose of this work is to introduce the principles and techniques of the theory of integration in the general and simple form that we owe primarily to Lebesgue, de la Vallée-Poussin, and W H Young. It is addressed to those who are already familiar with the elementary calculus of differentiation and integration as applied to the standard functions of algebraic and trigonometric type. Some slight acquaintance with the topology of open and closed sets may also now be presumed in most first-year undergraduates, for whom the book is written, but it is not essential. I have endeavoured to provide an account of the essentials of the theory and practice of Lebesgue integration that are indispensable in analysis, in theoretical physics, and in the theory of probability in a form that can be readily assimilated by students reading for honours in mathematics, physics, or engineering. To realise this purpose is a serious and important pedagogical problem, for the theory of Lebesgue integration occupies a strange, ambivalent position in the minds of mathematicians confronted with the challenge of planning a syllabus for undergraduates. Then Lebesgue integration appears to be at once indispensable and unattainable, desirable and impracticable.

8.2. Review by: Thomas Muirhead Flett.
The Mathematical Gazette 56 (397) (1972), 264-265.

The object of this book is to provide an elementary introduction to the Lebesgue integral for undergraduates. The theory is developed via the notion of Lebesgue measure, and, except for one chapter, is set entirely in the real line $\mathbb{R}$. The whole discussion is carefully motivated, and the first three chapters explain the programme to be carried out in the rest of the book. The treatment is a good deal more concrete than that in many other recent books on the subject, and while this is particularly fitting in a first introduction, it has not been achieved without some cost in the length of the treatment. ...
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In his preface, Professor Temple states that the book has been written for first-year undergraduates. In contrast, I first learnt the ideas of Lebesgue integration as a first-year postgraduate student at Cambridge. Although a number of topics that were postgraduate subjects in my student days have now become commonplace to first-year undergraduates, this book does little to convince me that Lebesgue integration should be added to the list.

I must add that the book on Lebesgue integration which was recommended to me as a postgraduate student was de la Vallee Poussin's Integrales de Lebesgue, and it is this book which has most strongly influenced Professor Temple's treatment.

8.3. Review by: Editors.
Mathematical Reviews MR0435323 (55 #8283).

Table of contents: Chapter 1, Motivation (pp. 13-18); Chapter 2, The concept of an integral (pp. 19-31); Chapter 3, The techniques of Lebesgue theory (pp. 32-43); Chapter 4, Indicators (pp. 44-53); Chapter 5, Differentiation of monotone functions (pp. 54-70); Chapter 6, Geometric measure of out and inner sets (pp. 71-86); Chapter 7, Lebesgue measure (pp. 87-102); Chapter 8, The Lebesgue integral of bounded, measurable functions (pp. 103-125); Chapter 9, Lebesgue integral of summable functions (pp. 126-143); Chapter 10, Multiple integrals (pp. 144-162); Chapter 11, The Lebesgue-Stieltjes integral (pp. 163-173); Chapter 12, Epilogue (pp. 174-181); Index (pp. 182-184).
9. 100 years of mathematics. A personal viewpoint (1981), by George Temple.
9.1. Review by: F H C Oates.
The Mathematical Gazette 66 (436) (1982), 161-162.

On reading the title, my thoughts were as follows. "An historical account, presumably. Which century? If the twentieth, as seems likely, is it really possible, in the space of 300 pages, to do justice to a period during which 90% of all known mathematics was discovered? And who is qualified to undertake such a formidable task?"

To begin with the final question, the distinguished author is Sedleian Professor Emeritus of Natural Philosophy at Oxford. His book is indeed an historical survey of major developments in mathematics over the last hundred years. The best way to indicate the immense scope of the book is to quote the chapter headings, which are Real numbers, Infinitesimals, Cantor and transfinite numbers, Finite and infinite numbers, Vectors and tensors, Geometry and measurement, The algebraic origins of modern algebraic geometry, The primitive notions of topology, The concept of functionality, Derivatives and integrals, Distributions, Ordinary differential equations, Calculus of variations, Potential theory and Mathematical logic. However, even Professor Temple had to draw the line somewhere: the major omissions are abstract algebra (to my regret!), numerical analysis, probability and statistics.

It will already be obvious that this book is not, and is not intended to be, a discursive 'history' in the style of, say, Boyer's well-known book; indeed, the recentness of the material precludes such an approach. Professor Temple begins where most histories end. Within each chapter he reviews briefly the 'pioneer work' in his chosen subject and then describes in more detail the important results discovered between 1870 and 1970. To quote from the introduction, the book is "essentially an account of the discovery or invention of mathematical concepts". The material is presented "neither chronologically nor biographically, but philosophically". Thus there is considerable overlap between chapters- intentionally so, for the author's aim is to show that "whereas in 1870 mathematics appeared to be diversified into distinct 'branches' ..., at the present time the side-shoots of these branches are growing together and reuniting in a single trunk."

Potential buyers should be warned that this is a very demanding book; few concessions are made to the reader. ...
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This is an important work, quite unlike any other mathematical book that I have ever seen. Writing it has clearly been a labour of love for Professor Temple. The key word here is 'labour': the amount of research required must have been phenomenal. There are 700 references, most of them to research papers written this century. The author remarks that the days of the great universalists (such as Poincaré and Hilbert) seem to have gone; after reading this book I am not so sure! I find it quite incredible that one man should possess the breadth and depth of mathematical knowledge displayed here, and that he should produce a book such as this at the age of 80.

9.2. Review by: Robert B Burckel.
Mathematical Reviews MR0719374 (85j:01002).

Let us start with some paraphrases from the author's extensive introduction: This book has been written to appeal to those who desire a broad survey of the main currents of mathematical thought. It is primarily and essentially an account of the discovery or invention of mathematical concepts, and the historical material is therefore divided and classified, neither chronologically nor biographically, but philosophically. Thus, for example, limiting processes are studied from the convergent sequences of A Cauchy to the filters of H Cartan. The book is aimed at relative cognoscenti, not the mathematically uninitiated. An author can describe each change in mathematics as it was seen by the contemporary mathematicians or he can view these changes from the standpoint of the present day. The first alternative lends a certain vividness to the narrative but seems more appropriate to the biographies of mathematicians. The second alternative gives an interpretation of the past which should make the present picture more intelligible. The author has adopted the second alternative and has attempted to see the last hundred years of mathematics from the vantage-point of the 1970s. Among the topics not treated the author lists the theory of functions of a complex variable, modern abstract algebra, the later developments of algebraic geometry and topology, statistics, probability and numerical analysis. Save for these omissions, he feels that the corpus of material treated is "what every young mathematician should know" of the history of mathematics. The book is divided into three parts entitled Number, Space and Analysis. The theory of number is explored from the work of Meray, Weierstrass, Cantor and Dedekind to Peano, Frege and Zermelo, with an excursion into nonstandard analysis. Vectors and tensors are traced from Hamilton, Grassmann and Gibbs to Pauli, Dirac and the theory of spinors. The theory of "distance" in geometry is treated and the origins of modern algebraic geometry are examined. The primitive notions of topology are traced in the theories of convergence from Frechet to H Cartan and Weil, in the beginnings of functional analysis from Volterra to Banach, and in the fixed point theorems of Kronecker and Brouwer. The classical subject of derivatives and integrals is expounded, together with set theory, measure theory and the theory of distributions (or generalised functions). Analysis is represented in a long chapter on differential equations, in a chapter on the calculus of variations and in an extensive exposition of potential theory up to the modern abstract work of Deny and Brelot. A concluding chapter (which is a de facto Part 4) examines mathematical logic from Boole to Brouwer to Godel.

The author's knowledge is awesome, his prose lively and literate; the narrative flows remarkably smoothly considering the frequency with which gears must be shifted. A vast panorama of mathematics is portrayed and through the author's organisation and balanced infusion of mathematical detail some sense of the development of the ideas is conveyed.

Painting on such a large canvas must be difficult and one should not expect the depth and completeness which characterise some of the excellent recent works of more limited scope like those of J Lutzen [The prehistory of the theory of distributions, 1982] and J Dieudonne [History of functional analysis, 1981]. ...
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... we have a rich and exciting book which largely achieves its aims but is somewhat flawed in its execution.

9.3. Review by: Dale M Johnson.
The British Journal for the History of Science 16 (3) (1983), 293-294.

The author's aim in writing this book was to provide a broad historical survey of the mathematics of the last hundred years or so, roughly from 1870 to 1970. Much of contemporary mathematics developed during this wonderfully fruitful period. The book is intended as a guide for working mathematicians and students of mathematics to the topography of the immense world of mathematics of this time. The book is neither an encyclopaedia of all the mathematics nor an anecdotal history of the personalities who shaped the mathematics. Rather the author has attempted to give a fair summary of the mathematical problems, ideas, and general theories of this period. He openly says that he has taken the vantage point of the 1970s when writing about the mathematics of the last hundred years. Thus he has not tried to analyse in detail the changing pattern of development of the mathematical ideas.

Consequently, this book does not provide a real history of mathematical developments and growth. We do not get a picture of how mathematicians set about constructing and developing their ideas in an historical setting. What we do get are some good summaries of historically significant mathematical papers and works and ideas and general theories. The coverage of the book is quite vast, giving a good portrayal of the scope of modern mathematics.
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In general the book offers a panoramic view of recently developed mathematical theories. Many difficult topics are given good coverage. The material on fixed point theorems in topology and analysis, for example, is well motivated and presented. The reviewer does not regard this as an history of recent mathematics per se, because it does not really attempt to explore mathematical developments as they occurred. Nevertheless, mathematicians should find it well worth reading for its extensive treatment of the theories which help to make up the fabric of modern mathematics.

9.4. Review by: Jean Dieudonné.
Revue d'histoire des sciences 36 (3/4) (1983), 361-364.

The author of this volume takes as his subject the history of mathematics from 1870 to 1970. He rejects the point of view which concentrates interest on the life and works of some mathematicians (like the book by E T Bell, Men of mathematics) and intends to make a "history of mathematical concepts and ideas". He admits having had to eliminate many parts of mathematics, by choosing those which "have shown themselves to be fruitful and have become essential parts of contemporary mathematics", and by establishing for these theories "the family tree of discovery and research."

It remains to be seen how this program is accomplished. The author does not seem to have published any personal work outside of quantum mechanics and hydrodynamics; he has also written a number of popularising articles on distribution theory, which he seems to have been enthusiastic about. It is therefore not surprising to see that, of the 282 pages of text, more than half are devoted to Topology (almost exclusively General Topology) and to Analysis, the latter being represented by five chapters: theory of measurement, distributions, ordinary differential equations, calculus of variations, and potential theory. There is no mention of spectral theory, apart from half a page on the Fredholm equation; not a word on the calculus of probabilities nor on ergodic theory. Partial differential equations other than Laplace's equation are not mentioned, except for two brief allusions (without any explanation) to hyperbolic equations. Fourier's theory is also reduced to the congruent portion (there is no mention of Plancherel's theorem or of Poisson's formula), and there is no mention of locally compact commutative groups. In the chapter on distributions (his favourite theory), the author insists on the possible variants (of very limited interest for applications) of the definition of Sobolev-Schwartz, but completely neglects to show how the notion of distribution has become fundamental in today's Analysis, by the considerable progress it has made since 1950 in the theory of linear partial differential equations: although all prior to 1970, none of the work of Malgrange, Ehrenpreis, Hörmander, Calderón, Trèves, and Nirenberg is mentioned!
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In summary, we can say that this work is a medley of lecture notes chosen without much discernment, more or less well digested and collected in great disorder. It is difficult to understand why the publishers saw fit to publish such a poor and uninteresting book.

9.5. Review by: David E Rowe.

In the preface of his delightful Concise History of Mathematics (3rd ed., 1966), Dirk Struik wrote: "It is time that the history of mathematics from 1900 to 1950 be written, if only in the form of a 'concise history.'" Nearly twenty years have passed since Struik issued this plea, but the gap still remains to be filled. George Temple, formerly Sedleian Professor of Natural Philosophy at Oxford, has now written a book that purports to do so by surveying the main developments in mathematics from 1870 to 1970. In truth it presents a great deal more information about work that was done before 1870 than after 1950. One finds, for example, many more references to Cauchy, Riemann, and Weierstrass than to Weyl, Siegel (who is never even mentioned!), and Weil. Thus the hundred-year period 1850-1950 would be a more accurate description of its actual contents, with the emphasis falling on the first fifty years.

Temple has a decidedly British outlook on modern mathematics, as one might perhaps assume from the subtitle of his book - which is strong on analysis and mathematical physics, but weak in areas that are strongly rooted in modern algebra and topology. The work is arranged in three parts: (1) Number, (2) Space, and (3) Analysis. The first two give rather superficial summaries of the foundations of real analysis, geometry, topology, and vector algebra. The third is much more substantive and constitutes more than half of the book. Here one finds long sections devoted to ordinary differential equations, potential theory, the calculus of variations, and distribution theory, which give valuable synopses of many important publications in these fields. Other topics that also receive substantial attention include integration theory, set theory, mathematical logic, and nonstandard analysis.

Had the author confined himself to these fields, he might well have produced a substantial and coherent book. The material in the first and third parts dovetails fairly nicely and could easily have been expanded to a work of the present size. The chief problem with the present volume lies in the middle portion, which attempts to "throw in" vector and tensor analysis, foundations of geometry, differential geometry, topology, and algebraic geometry in fewer than 100 pages, making the whole thing read like a cut-and-paste collection of definitions and theorems. In accounting for the omission of such fundamental areas as complex variable theory, modern algebra, and (surprisingly enough, considering the importance of the British contribution) probability and statistics, Temple rightly points out that "the writer has to choose between intelligibility and comprehension, and it seemed better to treat some subjects in detail, rather than to attempt to include all subjects in one volume. ..." Unfortunately, he appears to have forgotten this maxim somewhere along the way, for his book is neither comprehensive nor always intelligible.

Obviously there is a very difficult selection problem that must be dealt with here, and an author should not be unduly castigated for sins of omission, immense as these may be. He should, however, be held somewhat accountable for what he chooses to include in such a survey, and in this particular case the selection criteria often seem to be out of step with the mainstream developments in mathematics. A large amount of space is devoted to philosophical or quasi-mathematical contributions made by figures like Alfred North Whitehead and Bertrand Russell. Thus the section on "Theories of Distance" contains lengthy accounts of a number of obscure ideas that never seem to have made a ripple in the great sea of modern mathematics. At the same time, the influence of the Bourbaki tradition is confined to a tiny paragraph on the penultimate page of the book. The "philosophical" remarks in the introduction and conclusion contain little that is substantive, and, in general, tend to blur the distinction between philosophising about mathematics (in the spirit of Whitehead and Russell) and actually doing mathematics.

Besides giving too much space to unimportant topics, the author overemphasises early developments that are familiar to nearly everyone with a nodding acquaintance with the history of mathematics. The eight pages devoted to integration theory prior to Lebesgue could easily have been reduced to two. Finally, from a technical standpoint, this volume leaves a great deal to be desired. There are faulty references, misspellings (H A Schwarz appears as Schwartz throughout), imprecise definitions (e.g., manifold on pp. 66-67), and a very inaccurate name index. Moreover, the historical literature (Kline, Hawkins, Grattan-Guinness et al.) is virtually ignored. From all this one can only conclude that, whatever merits it may have as a summary of developments in modern analysis, this was definitely not the book Dirk Struik had in mind.

Last Updated January 2021