1.1. From the Preface.

In this Monograph I have attempted an account of an aspect of dynamical meteorology which is now recognised as a study of major importance, not only because of its intrinsic interest and the fundamental part which it plays in the science of meteorology as a whole, but also because of its significance in economic, military and industrial spheres. The theory of atmospheric turbulence is more mathematically developed than those parts of dynamical meteorology which treat of large-scale motions of the atmosphere and I have considered the subject throughout as a branch of mathematical physics. This is a partial and perhaps a biased view of the subject, and it is with real regret that for reasons of space I have had to resist the temptation to include accounts of the special instruments which have been devised, with ingenuity and skill, for the highly accurate observations which the subject demands. I have also refrained from discussing turbulence in the upper air, but for quite another reason. This aspect of the subject is of the greatest importance both to meteorologists and to the designers of large aircraft, but the information available as yet is meagre and hardly in a state suitable for treatment in book form. The Monograph therefore deals exclusively with the properties of the atmosphere near the ground, undeniably the most important region for turbulence. It is obvious that a book of this size can make no claim to be exhaustive, and my aim throughout has been to indicate to the reader those parts of the subject which seem to me to be significant for future development rather than to provide a long list of researches.

1.2. Review by: P A S.
Science Progress (1933-) 38 (150) (1950), 360-361.

The word turbulence is commonly taken to connote an apparently irregular type of fluid flow so small in scale that its details are effectively beyond the wit of man to describe. Like Clerk Maxwell with molecules, the physicist regards the problem of turbulence as one to be solved by a statistical treatment, the physical consequences of a given field of turbulent flow, such as the transfer of heat, matter and momentum, being represented by expressions involving mean products of eddy velocities and the like. These expressions are then to be related to the properties of the mean motion and the conditions at the boundaries. There is, however, no necessity to consider turbulence as restricted to small-scale motions. It is more fruitful perhaps to regard it as a technique to be applied to any system possessed of a number of modes of motion. Then, in the study of any particular mode, the effects of smaller-scale modes are to be allowed for by certain of their statistical properties. The laws of turbulence will evidently depend on the physical nature of those smaller-scale flow patterns and can be expected in general to depend on scale. This approach would appear to be eminently suitable in meteorology since the atmosphere undoubtedly possesses many modes of motion of scale varying from the microscopic to the global.

Little, however, of this attitude is to be found in Prof Sutton's monograph, which is concerned almost exclusively with what may be called inner boundary- layer turbulence, and this he deals with in mathematical detail. The choice is understandable, for not only is he a leader in this particular branch but it has made more progress, albeit of a rather superficial kind, than the study of atmospheric turbulence generally.

Having introduced the reader to the salient features of turbulent flow on the laboratory scale and in the bottom 100 metres or so of the atmosphere, he proceeds to an examination of the mathematical treatments which have been adopted in the study of the latter region. Empirical $K$ (eddy viscosity) theory is quickly dismissed and we are then shown the products of Prandtl's mixing-length theory and of Sutton's formulation of the Taylor correlation-function, that function which expresses the rate at which an eddying mass of air loses its identity in motion through its environment. Formulae are derived for the turbulent transfer of heat, matter and momentum, and for the concentrations of diffusible entities from point, line and area sources which, by adjustment of constants, can be brought into good agreement with observation. The success is apparently considerable but it is attained without, at any stage, more than the slightest possible physical basis - much could be derived from simple dimensional analysis, a notable exception being Taylor's diffusion theorem. This is no criticism of Prof Sutton's matter; it represents the present state of the subject. But the reviewer feels that the severe limitations of the treatment should have been clearly exposed; they are not. The writer indicates that latterly a more fundamental approach has been developed but that the new methods, applied at present to motions in the laboratory, are not yet able to face the atmosphere with its complicating density gradients and multiplicity of motions.

Prof Sutton writes very clearly and acceptably, and the book is well printed and indexed. When a second edition is called for, may one hope for some attention to be given to the boundary-layer turbulence over the sea? The sea underlies four-fifths of the earth's atmosphere and it undoubtedly possesses qualities of specific interest.

1.3. Publisher's description of 2021 reprint.

Atmospheric Turbulence examines dynamic meteorology and the fundamental part it plays in the overall science of meteorology. The book examines the theory of atmospheric turbulence as a more mathematically developed area than largescale motions of the atmosphere and examines its significance in economic, military and industrial spheres. The book focuses on the effect and importance of atmospheric turbulence, not only to meteorologists, but the designers of large aircraft. The book addresses the effects of turbulence and the properties of the atmosphere that can be found closer to the ground. This book will be of interest to atmospheric physicists and meteorologists.

2.1. From the cover.

A simple and mainly non-technical account of aerodynamics, the science which lies at the root of the problem of mechanical flight and its development from the spears and arrows of the primitive hunter to the supersonic aircraft and giant rocket of today.

Oliver Graham Sutton is a native of Monmouthshire and, appropriately, of Welsh and English extraction. Educated at Pontywaun Grammar School, the University of Wales, Aberystwyth and Jesus College, Oxford, he has spent most of his life in Government scientific establishments. During the war he was in charge of research in chemical defence and later in tank armaments, and finally became Chief Superintendent of the Radar Research and development Establishment at Malvern. In 1947 he was appointed to the Chair of Mathematical Physics when the Military College of Science was reconstituted on university lines at Shrivenham. He was elected a Fellow of the Royal Society in 1949.

His wife was a fellow student at Aberystwyth and they have two sons. He is a Justice of the Peace for Berkshire and, apart from mathematics, his main interest is in book collecting.

2.2. From the Preface.

As its title indicates, this is a book about the theoretical aspects of flying. It is not concerned with the problem of aircraft design, which is the province of the engineer, and nowhere does it touch upon the actual technique of flying. In short, it is an attempt to explain to the layman something of a branch of applied mathematics, called aerodynamics, which lies at the root of all matters appertaining to mechanical flight. I have written the book because I believe there are many people who would like to know more of these matters. This means putting into plain English much which is usually expressed in the language and notation of mathematics, an interesting but never very easy task for the author who must, of necessity, be steeped in that notation himself. The difficulty here, I think, is precisely that which faces a judge when he has to sum up in an intricate case and explain the law to the jury . It is not that the legal argument is too involved for the jury to grasp but rather that law, like mathematics, takes the words of everyday speech and uses them for a special purpose and with limited and, frequently, archaic meaning.

Mathematical language has the same legacy from the past and at times seems to take a perverse delight in distorting the meanings of commonplace words - thus 'imaginary' numbers are as real as the figures on a cheque, a 'vortex' does not necessarily mean a whirlpool, and a 'shock wave' is not a wave in the familiar sense of the word, nor is there anything shocking about it. It seems to me that mathematics in general, and applied mathematics in particular, must be capable of being explained without recourse to its own peculiar symbols and language . This is not to say that the traditional notation can be dispensed with, for without it progress in mathematics would hardly be possible, but in any physical science equations and formulae must surely be less important than the ideas which they embody. From the reader of this book I ask for nothing more than an acquaintance, however, neglected, with the elements of algebraic notation, and then only when to express a simple mathematical formula in words would be pedantic and confusing.

2.3. Review by: Isidor Isaac Rabi.
Scientific American 184 (1) (1951), 58.

This little book will be a joy and delight to anyone desiring an easy physical introduction into the aerodynamic principles underlying flight, and can be read with great profit even by those who have already been introduced to the subject by heavier accounts. To many of the latter it will probably come as a great surprise that the subject can be dealt with in such a direct and simple way. The book should be required reading for every man in the Air Force. The first half of it is devoted to a clear discussion of the principles affecting lift and drag on airplane wings and propellers, and of the general questions of stability and control of aircraft in the usual region of subsonic flight. In the second half the reader is introduced to the special problems associated with flight at higher speeds, especially the transonic and supersonic, at which effects arising from the compressibility of the air become of dominating importance. There is also an elementary presentation of the physical principles underlying jet and rocket propulsion. and a sane and sensible account of the technical difficulties in the problem of building a rocket capable of leaving the earth.

3.1. From the Preface.

In this book I have attempted to meet the needs of meteorologists and of workers in other fields who require detailed information about physical processes in the regions of the atmosphere where life is most abundant. It is my hope that such an account will help to increase the number of micrometeorologists, and to this end the book has been planned, not as an encyclopaedic work of reference, but as an integrated course of reading, bringing together the main results of research ...

3.2. Review by: Howard Latimer Penman.
Nature 173 (4410) (1954), 841-842.

Dr Sutton has written the hook that only he could write; end there will be a warm welcome for this illuminating survey of territory that he himself has done so much to extend. A flippant reviewer will perhaps be forgiven for suggesting that the sub-title ought to be "Mathematical Processes in the Lower Atmosphere", for this is a mathematician's book with the scope and content of which there could be little quarrel were the design limited to producing a survey of postgraduate standard for mathematicians and physicists. The author, however, specifically states that he has attempted to meet the needs of workers in other fields who require detailed information about physical processes in the regions of the atmosphere where life is most abundant. To these other workers - biologists and the like who make up the great bulk of practising micro-meteorologists in the world - the book will be difficult where it is relevant, and disappointing where it is silent on what they regard as the important problems of micrometeorology: air movement. temperature and humidity gradients inside crops and forests; the physical environment of insects and animals, the dispersal of atmospheric plankton with non-negligible terminal velocities of fall, and wind erosion, to name only a few. At present, of course, the meteorologist can do no more than provide a qualitative discussion of the physics of some of these problems; but a lead from Dr Sutton would have been invaluable in stimulating thought and experiment.

The first of the eight chapters clears the way for later arguments in describing the static properties of the atmosphere. The next two deal with air moving in laminar flow and turbulent flow, the former primarily to set out and explain the basic aerodynamic equations and to introduce the concepts of shearing stress, viscosity and mean free path. Against this background there is an extended survey of mixing length theories in turbulent flow, first on the molecular analogy of Prandtl and then on the statistical basis initiated by Taylor. The analysis at this stage is mainly used to derive velocity profiles over smooth and rough surfaces, and the corresponding momentum transport constants. Here, as elsewhere, the mathematics is crisply set out and the argument uninterrupted: but at the expense of a discontinuity it would be helpful to have some discussion of the physical implications of these differing concepts of mixing length.

After considering the transport of heat in free convection, diffusive processes are studied in general terms that permit the analysis to be applied not only to heat transfer but also to imponderables in the atmosphere - water vapour, smoke and other pollution with negligible terminal velocities of fall.

The chapter on radiation is pure meteorology and includes an extended treatment of long-wave radiation leading to a discussion of the possibilities of predicting night minimum temperatures. As 'micro' meteorology this chapter might well have included some discussion of radiation exchanges between small bodies and their environments, a topic of wide importance in most branches of outdoor biophysics, particularly in the measurement of temperature. Almost every temperature measurement involves an uneasy compromise between adequate shielding from radiation and adequate exposure of the element to the environment. A section on the micro-meteorology of meteorological instruments would be a very welcome addition to a future edition of the book.

The remaining three chapters of the book take the mathematical analysis of the first part into the field to match formulae against facts. Temperature distribution and heat flow are considered after a preliminary survey of the heat balance at the earth/air boundary: from temperature, on to wind, with details of wind profiles over rough surfaces and the influence of atmospheric stability on them. Read in conjunction with his earlier chapter on turbulent flow, this chapter justifies Dr Sutton's statement that there has been much energy spent and ingenuity exercised in investigating problems of wind structure. Has it been overdone? The author's reply would doubtless be to ask the questioner to re-read his final chapter to find there how all that has gone before can be used in solving problems of dispersal of smoke and other forms of atmospheric pollution, and in providing complete, if complex, expressions for evaporation-rates from open water. These expressions would have much greater practical value if they were accompanied by short tables giving a range of values of the stability-dependent constants that appear in equations 8.55, 8.56 and elsewhere: not everyone who will want to use this chapter will possess tables of incomplete gamma functions or know where to find them.

3.3. Review by: Bernhard Haurwitz.
Science, New Series 117 (3050) (1953), 667-668.

Meteorology, the observational and theoretical study of our atmosphere, concerned itself at first merely with the large-scale aspects of weather and climate. In recent years increasing attention has been given to the special problems which arise in connection with the investigation of the atmospheric layers next to the ground. These layers are of particular importance both for meteorology in general and for practical reasons. It is these lowest layers which, by their roughness, provide the breaking action for atmospheric motion and which determine primarily the transfer of heat and water vapour from the solid ground and from the water surfaces to the atmosphere as a whole; hence their importance for meteorology in general. Furthermore, human activities take place almost exclusively in these lowest layers; hence their practical importance for such varied fields as agriculture and the investigation of atmospheric pollution. A peculiarity of this atmospheric boundary layer is the rapid change of the meteorological parameters, such as wind, temperature, and humidity over small distances, caused by changes in the properties of the underlying surface.

The term micrometeorology, as used by Professor Sutton, deals with the study of the physical phenomena taking place in these lowest atmospheric layers. A broader definition might also be taken to include such micrometeorological phenomena as the fine structure of upper atmospheric phenomena and the microphysics of clouds. But because of the large amount of information to be discussed the author wisely restricts himself to the more narrow field of the surface layers. Even here he does not touch at all on the importance of atmospheric effects on radio wave propagation, referring merely to existing accounts of this subject. Nevertheless, even specialists in this field will profit greatly by a study of Micrometeorology because the author presents an integrated picture of the present state of our knowledge of the distribution of meteorological parameters affecting electromagnetic wave propagation.

Micrometeorology is written so that it can be read by anyone who has acquired the "standard of an initial degree in mathematics and physics," and no initial knowledge of meteorology is assumed. Instead Sutton presents this, to the extent that it is required for the study of micrometeorology, in a concise and very readable fashion throughout his book. Accordingly the first chapter deals with "The Atmosphere at Rest." The next two chapters treat of "The Atmosphere in Motion" and discuss first laminar, then turbulent flow. Among the topics included in these chapters are Prandtl's boundary layer theory and the statistical and similarity theories of turbulence. Chapter IV takes up the discussion of heat transfer and diffusion; Chapter V surveys radiation and its micrometeorological significance. These first five chapters, slightly more than half of the book, thus lay the groundwork for the more detailed discussions in the latter part of the book - namely, the temperature field (Chap. VI), the wind structure (Chap. VII), and diffusion and evaporation (Chap. VIII).
...
The book is very well written and the presentation of the quantitative, mathematically formulated theories is clear and easy to follow. The author expresses the hope that the book will help to increase the number of micrometeorologists. Since the book sums up in a well-organised presentation our present knowledge of the subject it will not fail to do so. Meteorologists in general, and specialists in micrometeorology and in fields for which the physics of the lowest atmospheric layers is important, will be grateful to the author for providing them with an authoritative account of a very important branch of the science which is in rapid and vigorous development.

3.4. Review by: Anon.
The Military Engineer 45 (305) (1953), 248.

This is a study of the properties of the air layers near the ground. It provides instruction in the techniques evolved to solve problems arising in agricultural meteorology, hydrology, and air pollution.

3.5. Review by: E G R.
Science Progress (1933-) 41 (164) (1953), 702.

Meteorology as a branch of physics is a fairly new concept. Till recently its study was mainly in the hands of geographers and mathematicians, and the apparatus used, apart from that used in recording the "elements" at meteorological stations, was reminiscent of an earlier era in the physics laboratory. Now apparatus employing electronic devices and having much shorter time constants is commonly used in meteorological research, so that the modern physicist need fear no sensation of "going back" if he enters this fascinating field.

To attract such workers into meteorology is indeed one of the objects of Prof Sutton's interesting new book, which ably sets out the present position in the subject and exposes the main gaps in our knowledge of the behaviour of the atmosphere near the ground. As one who has contributed himself not a little to the theory of atmospheric motion, Prof Sutton does not neglect the theoretical side of his subject, which he presents in a straightforward and systematic way.

Theory and experiment in laminar and turbulent flow form the background of the book, to which are added later the problems of heat transfer and the diffusion of matter into the atmosphere. There is one chapter on thermal radiation, but one would have liked something on the propagation of other wave types - sound, light and radio-waves - which may help to elucidate the structure of the medium through which they pass. The book is well documented and is, in short, an excellent and invaluable exposé of the present state of meteorology and, incidentally, of the importance for our present-day existence of the problems with which the science deals.

3.6. Review by: Maurice Howard Halstead.
Bulletin of the American Meteorological Society 34 (5) (1953), 191.

When one considers that almost all atmospheric heat and moisture enters the air first through the earth atmosphere interface, it may seem strange that modern meteorology has concerned itself so little with the physical mechanism of heat and moisture transfer from the surface. While Geiger has covered this transfer process from the climatological point of view, and Lettau from the turbulence side, there has been no text which summarised, from the meteorological point of view, the results of the large number of scientific papers which deal with this subject.

Professor Sutton, professor of mathematical physics at the Military College of Science, England, has long been recognised as his country's leader in the field of atmospheric diffusion, and it is therefore fitting that he attempt to meet this need.

The book consists of chapters on still air, laminar flow, turbulent flow, heat transfer and diffusion, radiation, the vertical temperature field, wind structure in the vertical, and evaporation, in that order. Variations of meteorological parameters along the surface are not discussed, and the vertical distribution of atmospheric moisture receives little attention.

While the bulk of the material covered must of necessity be treated in mathematical fashion, Professor Sutton's technique of starting each chapter with a brief résumé of the work to follow is quite advantageous. This is especially true since the theoretical basis of the various chapters is seldom sufficient to derive the engineering procedures employed near the end.

It is because of the present lack of knowledge in the field and not a fault of the author, that Chapter 4 on "Heat Transfer and Problems of Diffusion" cannot be completely combined with Chapter 6, "The Temperature Field in the Lowest Layers of the Atmosphere," and again that Chapter 3, "The Atmosphere in Motion" should bear so little relation to Chapter 7, "Problems of Wind Structure near the Surface."

It should be pointed out that the field of micro meteorology is usually thought of as somewhat broader than this book would indicate, since there has been no attempt to deal with those variations in the horizontal which can be of such great importance. Here again, however, one must realise that it is still too early to expect that the basic theories which Professor Sutton presents could have been extended to yield quantitative results on horizontal variations.
...
The book as a whole fills a definite need ...

3.7. Review by: Mogens Koie.
Oikos 4 (2) (1952-1953), 201.

Although primarily the author addresses meteorologists and students of a post-graduate level in mathematics and physics attention should be paid to it by biologists concerned with microclimatology.

The book is divided into eight chapters, 1. The atmosphere at rest, 2. Laminar flow, 3. Turbulent flow, 4. Heat transfer and problems of diffusion, 5. Radiation, 6. The temperature field in the lowest layers of the atmosphere, 7. Problems of wind structure near the surface, 8. Diffusion and evaporation. - Each chapter opens with a general survey of the subject and thereafter develops a more detailed exposition in conjunction with a discussion of the problems in mathematical and physical terms. Special emphasis has been laid on the turbulence of the wind because, as claimed by the author, it is chiefly responsible for the spread of heat from the ground to the air, the exchange of carbon dioxide between plant and animal life, the scattering of pollen and the lighter seeds, and the cycle of water from the earth, seas, and rivers to the air and back again. As an example of the advanced stage already reached by micro- meteorological theories it can be mentioned that instead of measuring the evaporation with the so far unsatisfactory methods the author thinks it at least as accurate to calculate the evaporation from a formula based upon the known values of temperature, wind velocity, and humidity.

3.8. Review by: Walter T Wilson.
Eos, Transactions American Geophysical Union 35 (1) (1954), 174-175.

Sutton's Micrometeorology is an important and welcome contribution. It fills a definite gap in its field, and is written by an authority who has devoted much of his career to the subject. The book has eight pages of author and subject index. Each of the eight chapters is well documented mostly from British publications. Pertinent theories and methods of attacking the problem are presented in a progressive manner, both historically and logically. Each chapter is carefully organised and is introduced by a helpful discussion in simple language of what is to follow in more precise and mathematical terms. It would be helpful if each chapter also had a summary of conclusions. The few typographical errors noted by this reviewer are too obvious to be misleading. Some readers may experience frustration when they refer to the book for specific information on limited aspects of the subject and find that definitions of symbols and terms are scattered throughout the book and that much of the treatment relies heavily on what has preceded

4.1. From the Preface.

This book, written primarily for the layman, will prove, I hope, of interest also to students, especially those in the upper forms of schools or in their first years in the university. It is a view of the part played by mathematics in applied science, as seen by a mathematical physicist. The topics which are discussed in detail are all taken from subjects in which I have worked, professionally, from time to time. This explains what may appear otherwise to be a random selection of material from a very large field.

4.2. From the back cover of the 1984 Dover reprint.

According to Sir Graham Sutton, "The task of the applied mathematician is exactly that of using the tools provided by pure mathematics to clarify and extend the observations of the physicist." Phenomena must be measured and reduced to number in order to become part of the body of scientific knowledge. It is the purpose of this book to show that process in action. Unlike many texts in this area, this straightforward account is written for the layman, and is accessible to high school students and undergraduates - anyone with a grasp of rudimentary calculus. Moreover, its generalised view of the topic makes the book of special interest to young mathematicians, physicists and engineers. In illuminating the nature of applied mathematics and its influence on modern ideas concerning the physical nature of the universe, the author illustrates his points with examples from ballistics, automatic calculating machines, radio waves, atoms and electrons, theory of flight, statistics and meteorology. The book is divided into seven chapters: I. The Mathematician and his Task - II. Tools of the trade - III. Ballistics, or Newtonian dynamics in war - IV. An essay on waves - V. Mathematics of flight - VI. Statistics, or the weighing of evidence - VII. Mathematics and the weather. In the first two chapters, Sir Graham gives a lucid account of the role of the mathematician in applied science and the nature of his tools, covering such topics as theories of physics, mathematical techniques, complex numbers, new geometries and atomistic and field theories of physics. The remaining five chapters are devoted to specific applications in ballistics (gunnery as an exact science, calculation of trajectories, etc), waves (waves in the natural world, Fourier series, waves and particles, etc.) mathematics of flight (fundamentals of fluid motion theory, Joukowski's solution of the two dimension aerofoil problem, etc.), as well as applications in statistics and meteorology.

4.3. Review by: D R Davies.
Journal of the Royal Statistical Society. Series C (Applied Statistics) 3 (3) (1954), 207.

This book, written primarily for the lay person with some knowledge of the calculus, presents a very clear account of the methods and ideas of the applied mathematician. Beginning with a brief discussion of the development of the modern scientific approach in achieving an understanding of physical phenomena, Dr Sutton then turns to his main objective, which is to describe the particular task of the mathematician in gaining this understanding. In general terms this commences with the formulation of an ideal mathematical model based on the actual physical problem. This is followed by the solution of the mathematical problem involved with the aid of pure mathematics (guided frequently by physical insight, e.g. indicating suitable particular solutions of the basic differential equation), leading finally to a comparison of the results predicted by the model with those obtained by experiment. If the agreement is good and the concepts on which the mathematical model is based are physically reasonable, then an understanding of the physical phenomenon has been achieved, and the model may be used to make further predictions about the behaviour of the physical system concerned. Further experiments may show discrepancies in the theory, and this leads in turn to a refinement of the model by replacing the old concepts by newer and deeper ideas; in this way understanding grows.

Dr Sutton then illustrates this general theme by discussing several research topics in which he has himself actively participated. These include ballistics, numerical analysis, wave propagation in various physical contexts, the mathematics of flight, statistics, and lastly an admirable account of the problem of weather forecasting, a subject with which the author, who is now Director of the Meteorological Office, is of course intimately concerned. Naturally the discussions of these various branches of applied mathematics are not meant to be exhaustive, but enough is given to illustrate the main thesis of the book. A brief, but clear and interesting, description is included of the 'tools of the trade', viz. the calculus of differentiation and integration, differential equation theory, the method of treatment of the typical boundary value problem, etc.

Although intended primarily for the lay person, Dr Sutton's book, in giving considerable insight into the general methods and objectives of the applied mathematician, will also be particularly useful as general reading for students of mathematics and physics.

4.4. Review by: R Tiffen.
Science Progress (1933-) 43 (169) (1955), 123.

Having introduced the physical basis of applied mathematics and the pure mathematics required in subsequent developments, the author discusses briefly the subjects of ballistics, wave motion, subsonic and supersonic flow, statistics and atmospheric motion. Although the book is written primarily for the layman and the number of equations is kept to a minimum, the mathematics is treated with care. For example, the meaning of "differential" is explained more clearly than in many elementary textbooks on the Calculus. The author has not attempted an exhaustive account of applied mathematics but has confined his attention to those subjects to which he himself has made contributions. In this way the work has been given greater originality and interest. The text is enriched by details which bridge the gaps between idealised mathematical models and physical realities. The author leads the reader rapidly from elementary ideas to important mathematical theorems and thence to deeper physical concepts, as, for example, in the sequence: "wave motion - Fourier transforms - Heisenberg's uncertainty principle." This is done so skilfully that the non-mathematician is never embarrassed by the mathematics. A well-written section on statistics, a subject often neglected in texts on applied mathematics although of prime importance, is included. A few further mathematical details are given in the appendix, although the interested reader would be better advised to consult more detailed texts.

4.5. Review by: Charles George Paradine.
The Mathematical Gazette 39 (328) (1955), 159.

The theme of this book is the part played by mathematics in applied science. In a chapter on "the tools of the trade", Dr Sutton gives a brief introduction to complex numbers and calculus, including total and partial differential equations. He stops short of tensors and decides against the use of vector notation, as demanding experience in the reader for its comprehension. In fact the layman, for whom the book is primarily written, is likely to find the mathematical symbolism difficult, but he can appreciate the general picture of the solution of physical problems, often idealised or simplified, by means of mathematical equations.

The particular topics discussed at some length are ballistics, waves (including a descriptive account of wave mechanics), aerodynamics, statistics and meteorology, all of them fields in which the author has worked. An example is given in ballistics of the numerical calculation of the trajectory of a shell at intervals of one second from a differential equation which cannot be integrated. Modern computing machines will relieve us of the numerical drudgery, but there is much scope for mathematical knowledge and ability in the coding of problems. Moreover, the "mathematical technician" must understand and speak the language of the electronic engineer. It appears, however, that even electronic machines might fail to "keep up with the weather", had we the data and equations necessary for accurate forecasting.

A minor criticism is that the use of $dy$ for $\delta y$ on p. 45 is not consistent with the definition of the differential on p. 42. It should be explained, too, that in a test of significance using the binomial distribution, the relevant probability is the sum of the probabilities of the event tested and of all equally or less probable events. Even the layman might object, on mathematical grounds, to the statement that "seven heads and three tails ... is expected ... in at least three out of ten trials of ten tosses each."

Besides the layman, the author hopes to interest students in the upper forms of schools or in their first year at a university. From the point of view of the teacher of mathematics, the book may help to promote incentive, either by dispelling the idea, if it still persists, that specialisation in mathematics can lead to no other career than teaching, or by inspiring the type of pupil whose interest in mathematics is dormant until he realises the power and importance of the subject in practical applications. For this reason, as well as its general interest, the book is worth a place in the library of a grammar school or technical college.

4.6. Review by: Clifford Grobstein.
Scientific American 194 (1) (1956), 116.

Of the mathematician's endless fancies only a few are immediately applicable to what G H Hardy called this "stupidly constructed" universe in which we live. Yet somehow, by the formal juggling of symbols of our own creation according to rules we ourselves have invented, an enormous amount is learned about the physical world and even how to control it. In this unpretentious, attractively written book a leading British mathematical physicist explains how the abstract concepts of mathematics are turned into tools for extending the observations of the physicist and other experimental workers. He illustrates the process by examples from ballistics, meteorology, aerodynamics, electromagnetic theory, nuclear physics, statistics, astronomy. Sutton gives a rapid review of fundamentals-complex numbers, the calculus, differential equations - so that anyone who has had even a smattering of advanced mathematics can refresh himself sufficiently to follow other parts of the account. There are many simple, helpful diagrams. At an intermediate level this is one of the clearest and best surveys of its kind published in many years; indeed one must turn back half a century to Mellor's classic Higher Mathematics for Students of Physics and Chemistry to find a work of comparable lucidity.

5.1. From the Publisher.

This book provides the needs of two classes of readers: research workers who require a reference book of a theorem or a formulae to find out what conditions it holds and how to apply it; and undergraduates or technical students who want a summary of what is known in various branches of mathematics and physics. No proofs have been included and no attempt is made to define a number and the book does not include any topology.

5.2. Review by: R Tiffen.
Science Progress (1933-) 47 (187) (1959), 584.

The authors claim that this book is suitable for a wide class of readers, but, in fact, it is of little use to anyone. The standard of the material in the mathematics section varies from that of the middle-school, e.g. cyclic quadrilaterals, arithmetic means, etc., through elementary trigonometry of the sine and cosine formulae variety to that of the honours degree in mathematics, e.g. definitions of sets, cosets, homomorphisms, isomorphisms, etc. An inexperienced reader will be overwhelmed by the number and complexity of the topics which constitute mathematics. On the other hand, the material cannot honestly be described as suitable for the laboratory bench, for how often does an experimental worker feel the need for Ceva's and Menelaus' theorems, for plane projective geometry, or for reassurance that a bounded monotonic function is integrable in the Riemann sense, or that a bounded infinite set possesses at least one limit point? So many topics have been tackled that there is insufficient room for a reasonable account of any one.

A similar pattern is followed in the physics section. Indeed, this is little more than a collection of formulae. Only three pages are devoted to elasticity, one and a half to surface tension, two to wave motion and five to heat. The uselessness of much of the material may be illustrated by the statement of the second law of thermodynamics, viz. "A self-acting machine cannot transfer heat continuously from a colder to a hotter body, and produce no external effect." This assertion, originally due to Clausius, appears without previous preparation or subsequent explanation.

There is little to recommend in this volume. The authors would have been wiser to concentrate on fewer topics and treat them more thoroughly or, on the other hand, reduce the book to a set of formulae suitable for laboratory use.

5.3. Review by: Editors.
Mathematical Reviews MR0099282 (20 #5723).

The book is divided into two parts: I Pure mathematics; II Physics; with separate indexes. Part I has 22 sections, including arithmetic, statistics, groups, analytic and projective geometry, calculus, complex variables, and differential geometry with 7 tables; Part II has 15 sections, among which are mechanics, elasticity, viscosity, electricity and magnetism, heat, atomic physics. There are no proofs but, as the preface states, "the book is more than a mere collection of formulae, in that explanations are given as far as space permits". The authors have kept in mind both "research workers who require a reference book" and "undergraduates ... preparing for examinations. The account given stops short at a point a little beyond General Honours degree standard."

6.1. From the cover.

A simple account of the young and exciting science of meteorology, written with all the authority of the expert.

6.2. From the Preface.

This book is a series of essays on some aspects of weather which I believe are of interest to the general reader. It makes no claim to be a systematic account of meteorology and in a sense is a progress report on the science of the atmosphere. As such I hope it will help towards a better understanding, not only of weather, but also of the way in which the meteorologist approaches his problems.

6.3. Review by: L C W Bonacina.
The Geographical Journal 127 (1) (1961), 114-115.

In his preface Sir Graham Sutton, who is Director-General of the Meteorological Office, states that this book for the general reader is not designed as a systematic textbook of meteorology but rather as a progress report of the science of the atmosphere. With remarkable clarity he explains the exacting, complicated process, under time-limit, of preparing daily weather forecasts from synoptic charts based on surface and upper air observations, as well as certain new physical concepts which play a great part in the technique of modern forecasting. He foresees a future for mathematical methods of forecasting, which are still in the early stage, since electronic digital computers can solve the equations supplied to those fast enough to keep pace with the weather itself.

In a chapter on micro-meteorology, Sir Graham describes researches of his own into turbulent motion as affecting evaporation and smoke concentration; but, as he himself indicates, the problems of micro-climate are mostly concerned with the layers of air in contact with the soil where seedlings and insects are exposed to drastic vicissitudes of temperature and humidity at short range.

7.1. Review by: John W Miles.
Mathematics of Computation 15 (73) (1961), 92-93.

In the words of its author, "This book, written primarily for the layman, will prove ... of interest also to students, especially those in the upper forms of schools or in the first years in the university. It is a view of the part played by mathematics in applied science, as seen by a mathematical physicist."

Chapter 1, "The Mathematician and his Task," begins by discussing the meaning of theories in physics and the role of mathematics in the development of these theories. Chapter 2, "The Tools of the Trade," gives special attention to complex numbers and to the development of the calculus and the differential equations of mathematical physics. The approach is essentially that of the physicist ("an infinitesimal quantity [is] one which does not exceed the smallest change of which we can take cognisance in our calculations").

The remaining five chapters, entitled respectively, "Ballistics or Newtonian Dynamics in War," "An Essay on Waves," "The Mathematics of Flight," "Statistics or the Weighing of Evidence," and "Mathematics and the Weather," are essentially independent essays that not only provide illustrations of applied mathematics in action, but also serve to emphasise fundamentals in both classical and modern physics. An especially notable example is the approach to Heisenberg's uncertainty principle through the problem of the bandwidth required to resolve pulses of short duration. Perhaps the most stimulating chapter is the last, in which the author concludes that "Certain apparently sensible questions, such as the question of weather conditions ... several days ahead, are in principle unanswerable and the most we can hope to do is to determine the relative probabilities of different outcomes."

There may be some question as to whether the non-mathematical layman will be able to follow all of the development of the last five chapters, and the author is occasionally guilty of extravagance. (Few aerodynamicists, even in the United Kingdom, would be willing to admit that F W Lanchester's esoteric volumes Aerodynamics and Aerodonetics "played a part in aerodynamics not unlike that exercised by Newton's Principia in astronomy.") These things notwithstanding, the reviewer believes that the author has succeeded admirably in reaching the goal described in the opening quotation of this review. Indeed, he goes beyond this goal, and the book (especially the individual essays) is warmly recommended to practicing applied mathematicians, as well as to laymen and students.

7.2. Review by: R E Horton.
Mathematics Magazine 34 (1) (1960), 44-45.

This book is one of the Harper Torchbooks and brings to the reader a book of fundamental importance at a nominal cost. The cost factor is emphasised because I believe this excellent book is of great value in broadening the vision of our younger college students studying in the sciences and mathematics.

The book is an attempt to show how the basic disciplines of mathematics form the foundation for the theories of modern physical science. Going beyond this, the author shows how mathematics, both pure and applied, has influenced the discoveries of physicists and provided the scientists with models of the universe as well as the language in which their theories could be expressed. The book does this in language and a style that brings the most abstract of modern physical theories to the lay reader and college student at a level which is quite intelligible to him.

The first part of the book introduces the reader to the basic mathematical disciplines - arithmetic through calculus to probability and statistics. The author's clear explanations here could well be emulated by many of our textbook writers. Such subjects as non-Euclidean geometries, differential equations, the complex plane, and the Laplacian are covered in some 67 pages. What the reader gets from such an abbreviated treatment will depend upon his own mathematical background. Yet, the author does a remarkable job of presenting these ideas clearly and stripped of all unnecessary details.

The remainder of the book discusses the application to various fields of physical science of the mathematical disciplines which the early part of the book has developed. To cover all the major fields of physics would have required volumes, so some selection is necessary. The author chose those fields in which he was engaged in research. They form a varied, interesting, and quite representative set of scientific fields. They include: ballistics, wave mechanics, aerodynamics, and meteorology. These fields provide the opportunity to examine the classical Newtonian methods and contrast them with more recent mathematical methods. Of particular interest are his treatment of probability and statistics and the mathematics of weather forecasting.

I would recommend this book very highly for all secondary school teachers of science and mathematics. I have already suggested that my calculus students obtain it as a supplement to our text. And for those lay readers who are not afraid to face an occasional equation with some strange symbols, this book will provide the clearest explanation of mathematics in action that he is likely to find.

7.3. Review by: H Glenn Ayre.
The Mathematics Teacher 54 (2) (1961), 104-105.

This is one example where the title is consistent with the content of the work. The style is precise and accurate with an even flow that carries the reader along with a minimum of abstractions as the author reveals the role of mathematics in applied problems and especially the theories of the physical universe. James R Newman aptly states in the foreword, "It is a wonderfully exciting tour through the workshops of the mathematical physicist. ..."

The work belongs in the category of books popularising mathematics, since it deals with the triumph of mathematics in the physical sciences and man's search into the nature of the universe. The procedure for the mathematical solution of a problem in physics reveals the use of mathematical models and the scientific method of researchers.

There is first a brief discussion of the role of the mathematician in the study of the theories of the physical universe. This is followed by an equally brief statement of mathematics as a "tool of the trade." One unfamiliar with the calculus and some classical theory in physics might find it difficult to follow the discussion of topics such as the difference between the derivative and the differential, Laplacian equations, field theory in physics, mathematical waves, ballistics, Fourier series, fluid motion, and other classical topics. However, the author has done well in his brief, accurate, and concise exposition. It is abundantly clear that the body of knowledge about the physical universe is the result of a safari into applied mathematics.

The chapter on statistics is significant and timely in view of the increased demand for statistical analysis in so many areas of learning and the false conclusions presented by workers not familiar with the mathematics of statistics. The discussion on sampling should cause those to pause and think who treat all samples as if they fit the Gaussian curve and represent the parent population. This is particularly significant since statistics has become so important in the behavioural sciences.

This reviewer heartily recommends the work to all libraries, to sciences classes for careful supplementary reading, and to the layman with some intellectual curiosity in the area of science.

8.1. From the Preface.

This book is intended to give the reader a picture of the science of the atmosphere in its modern dress. It is written around the theme, familiar to meteorologists but possibly not to others, that climate and weather are manifestations of the activities of a complex of interlocking motion systems of varying size, driven by the energy of sunlight. The atmosphere presents to the meteorologist a challenge to produce an orderly account of this vast machine.
...
In this book an attempt is made to show meteorology as an analytic and deductive science, rather than a means of describing and cataloguing varieties of weather. The underlying theme is that in the atmosphere a hierarchy of motion systems ranging from the great semi-permanent rivers of air that flow around the globe to the tiny local eddies such as transfer water from the surface of the earth to the air and back again as rain and snow.

8.2. Review by: L C W Bonacina.
The Geographical Journal 128 (4) (1962), 550.

The scope and aim of this work by Sir Graham Sutton, Director-General of the Meteorological Office, is really implicit in its title. It is not a text-book of meteorology or climatology, and various topics are omitted which the reader, perhaps, might have expected to find. Rather is the book a statement of those advances in knowledge, from long-established physical laws to weather forecasting by mathematical methods and by electronic computers, which have already raised meteorology to the level of an exact science. The dynamics of hurricanes, tornadoes and thunderstorms are explained with masterly clarity, as also such modern concepts as upper thermal winds and thickness lines, divergence and convergence, vorticity and Rossby waves, which enter so largely into the technique of forecasting. This also applies to the treatment of the mathematical problems of micro-meteorology and turbulent motion to which the author himself has made such notable contributions, though in these sections the reader must expect to encounter difficult passages.

8.3. Review by: Sverre Petterssen.
Bulletin of the American Meteorological Society 44 (7) (1963), 475.

In reviewing this charmingly written book it may be well to remind the readers, particularly the non-meteorological ones, of the great complexity of the atmospheric processes. Essentially, the meteorologist is called upon to account for the behaviour of an open system of a thermally active substance which responds to energy radiation from the sun and to a number of physical and mechanical influences associated with the earth's surface. Typical phenomena exist in superimposition, with space-scales varying from turbulent eddies, through large migrating storms, to the global circulations, and with life-spans ranging from a few seconds to climatic periods. Yet, all these interwoven processes are important, for energy is shuttled back and forth between the various systems, and only by considering the whole spectrum of phenomena can we account for what is commonly called weather and climate.

Only a person with deep scientific insight and exceptional writing ability can do justice to the complexity of the individual processes as well as to the simplicity of the system as a whole. Sir Graham has succeeded in providing us with a book with these important attributes ; in comparison with this achievement it matters very little whether certain sections of the book could have had a somewhat different emphasis.

The strength of the book lies in the clarity and simplicity with which physical processes are explained. In areas where description rather than explanation is called for the book is, perhaps, less appealing; however, facts are rarely fascinating unless they are dressed in the garb of theory.

The book consists of a prologue and eight chapters, dealing with 1) climate and weather, 2) the atmosphere, 3) the physics of clouds and rain, 4) weather-producing systems, 5) hurricanes, tornadoes and thunderstorms, 6) the microscale of climate, 7) micrometeorology at work, and 8) forecasting: old and new. It also contains a glossary and an appendix with simple mathematical derivations.

The word challenge in the title is appropriate, for Sir Graham develops his arguments into intellectual challenges. These often reflect his personal views, and al though the reader may not agree, he will have enjoyed the encounter. Meteorologists will appreciate the discussion of turbulence and transfer processes, areas to which Sir Graham has contributed so much; they will also be impressed by the progress that has been made during the preceding two or three decades.

The reviewer, being particularly interested in meteorological predictions, finds it difficult to agree that all pre dictions of forthcoming weather are statements of the probability of an event. While he wishes that this were so, he doubts that many problems of meteorological predictions can fruitfully be formulated in probability terms.

The optimistic views on the future of meteorology, as a science as well as a service, are in good keeping with American thinking. The book will have a strong appeal to students in all branches of the physical sciences, and professional meteorologists will find it stimulating.

8.4. Review by: J L H S.
Geography 48 (1) (1963), 107-108.

The need has long been felt for a treatment of the basic concepts, methods and problems of meteorology and climatology at a level accessible to students with little background in mathematics or physics. This book goes a very long way towards meeting this need. Sir Graham Sutton throughout emphasises the fundamental concepts of atmospheric science and their application to the investigation of practical problems. Regional examples are introduced solely for purposes of illustration.

The first three chapters and the opening section of the fourth are concerned with the physical basis of the subject: the heat balance of the earth's surface and atmosphere, the laws of atmospheric motion, the general circulation, processes giving rise to condensation and precipitation, the identification of rotational and divergent components of motion. Consideration of the last leads to a discussion of the principal types of weather systems, a separate chapter being devoted to the study of the most violent and destructive phenomena: hurricanes, thunderstorms and tornadoes. The author next introduces the concept of turbulence and the study of the intricate motions observed in the air layers close to the ground. It is with this field that Sir Graham Sutton's name is particularly associated, and the progress made in the study of air pollution and evaporation, considered in the following chapter, owe much to theoretical advances in which he has taken a leading part. Forecasting forms the subject of the final section. The appeal, importance and limitations of short-range and extended-range forecasts are reviewed and an appraisal is given of the possibilities offered by the method of Numerical Forecasting, first conceived 40 years ago by L F Richardson, but made practical only by the development in recent years of electronic digital computers.

9.1. Review by: Sidney Barrington Gates.
The Aeronautical Journal 70 (666) (1966), 677.

Aerodynamics will never be without tears for those who lack a grounding in applied mathematics because many of its basic propositions are like the medium whose flow it describes: they will not stay still long enough to he grasped. Sir Graham Sutton's book is an interesting and largely successful attempt to mop up the tears (or perhaps more accurately, provide a prophylactic against their formation) by hanging the mathematics and physics of the subject very securely to a sketch of its historical development. He caters for the reader who will get comfort from being told that aerodynamics of a sort began with the cavemen's spear, that the bow and arrows of Agincourt could hardly be improved today, that men were firing supersonic missiles from guns long before Concorde came to trouble us, and that the nineteenth century mathematicians, uplifted in the hubris of their elegant classical solutions, were humbled by the guilty knowledge that the drag they calculated was always zero.

This earthy approach is a great help in maintaining the reader's momentum against the drag of the mathematical doses that are often quite sharply applied. Sir Graham has a gift for producing the homely, telling metaphor where it is most needed. To be told that laminar flow is an army on the march and turbulent flow is the lurching advance of a crowd under mob law may offend the purists but is just what a bemused reader wants. It enables Sir Graham to win his battle of the Reynolds number by a handsome margin and, as befits a meteorologist, to point out on the way that men are what they are only because the turbulence of the lower atmosphere is what it is. He is in rather more trouble in extracting lift from its mathematical seclusion, and only just manages to get away with the circulation theorem. One does not expect to meet Navier-Stokes in a book of this class, but Sir Graham evidently decided that a wary approach to these mountainous equations was the best way of introducing vorticity and viscosity en route to the essential features of Prandtl's boundary layer theory.

I would fault this gallant book only in this; that it tries too hard to say something about everything. The pace gets noticeably hotter, the writing rather more ragged, towards the end, just when extra coaxing is needed to stay the course. Supersonic theory is in some respects easier than the incompressible variety, but to reach it intellectually means taking a jump and then steadying oneself, rather like passing through a strong shock wave, After surviving this shocking experience to reach the Concorde, even the willing reader may burke at starting all over again with the elements of rocketry on the way to the satellites.

The photographs are excellent, and so are the thumbnail sketches of aerodynamic heroes. Sir Graham's favourites are Cayley and Lanchester, and to find Lanchester's rather sad story so well documented is a great pleasure. Lord Snow and others may be interested to learn that an aerodynamic celebrity of another sort - the Second Law of Thermodynamics - is not mentioned.