1.1. Form the Introduction.

This book is a systematic treatise on the fundamental principles of the science of Mechanics: it also provides material which will enable students to become proficient in the application of these principles. It has been written in conformity with certain rules, the most distinctive of which is that concepts and principles are introduced one at a time, and that the introduction of any particular concept or principle is delayed until further progress without it would be inconvenient. In this way, the authors believe that a student will learn to appreciate the logical structure of the subject - an appreciation which is of prime importance, not only in the full understanding of the theory, but also in the solution of problems.

The second rule followed consists in proceeding from the concrete to the abstract, and from the more familiar to that which is less familiar. In pursuance of this rule, the gravitational measure of force is used before the absolute measure is introduced; and consequently the concept of mass is not discussed until a relatively late stage. This has the effect of making portions of the book conform to the practice in engineering instruction.

Statics and Dynamics are developed side by side with as much similarity as possible. The internal forces of a body in equilibrium are treated on the same lines as those adopted in the discussion of moving bodies. This is a prominent feature of the book, and is an extension of Newton's methods to Statics. It is, of course, the practice of books in which the principles of Statics are deduced from those of Dynamics, but in such books little attention is usually given to Statics; books in which Statics is dealt with as a separate section generally follow the pre-Newtonian tradition.

In the appeal to mechanical phenomena, familiar experience and a posteriori evidence are quoted more often that the results of specially designed laboratory experiments. This is in accord with the logical nature of the book, but the authors recognise the high value of work in a laboratory for a student of Mechanics.

1.2. Review by: Harry Bateman.
Amer. Math. Monthly 33 (4) (1926), 224.

This book by a professor emeritus in the University of Liverpool and a young professor who has made notable advances in the study of the tides is a good example of what can be accomplished when the experience of a great teacher is combined with the brilliance of an ardent investigator. The subject is well presented, one good feature being the frequent use of vectors. Many illustrations are given to assist the reader and numerous examples are worked out at the end of the book to help along any student who is reading the subject for the first time. Though the work is quite elementary, room is found for a useful chapter on frameworks and an elementary treatment of hydrostatics is included. We fully agree that instruction on these subjects should be given in a course on mechanics.

1.3. Review by: David E Smith.
The Mathematics Teacher 18 (7) (1925), 438-439.

This is the most recent volume in the Longmans' Modern Mathematical Series, a collection of books that has done much to set forth the modern purposes and methods in the teaching of pure and applied mathematics. It is written by Professor Emeritus Carey and Professor Proudman, both of the University of Liverpool. Professor Carey has long been known for his contributions to the study of the calculus and mechanics and for the high standard set by him when directing the work in these subjects in his university, and his collaborator has been prominent in the same field. The book represents, therefore, an author ship that assures a scholarly treatment of the subject.

The work is intended for use in the first year of the university, but it represents a standard that is hardly reached as early as this in our American colleges. It does not require a preliminary study of the calculus, but it presupposes what we do not generally have in this country, namely, a good course in mechanics in the secondary school.

While the chapter headings naturally represent the classical topics, the presentation of these topics represents a considerable departure from classical methods. The theory in general follows a statement of the problem to which it applies, proceeding from simple facts well known to the reader to the mathematical explanation of these facts. This is seen in the case of speed and its measurement, of velocity-acceleration, of gravity, and so on through the list. Much attention is also paid to simple laboratory practice, a feature which British teachers have been able to carry out with much more attention to mathematics than is usually the case with us.

In the matter of arrangement in presentation the authors have adopted a plan that is worthy of attention. In the chapters there is printed a general treatment of the topic; then the contents refers to "Worked Examples" and "Examples" in the second half of the book. In this way they combine a text of theory and a book of exercises. Whether this is as good as our custom of combining the examples with the text is probably answerable only by considering the habits of the users.

Perhaps the most striking feature of the treatment is to be found in the use of vectors and in the well-arranged set of exercises on the subject.

The English writers have for a long time excelled in their exercises in applied mathematics. This fact strikes an American reader as he looks over the work under review. There are more than a hundred pages devoted to applications. Since the type is small, they represent the equivalent of 150 to 200 pages of our ordinary textbooks. Such a supply, including a large amount of new material, will prove very helpful to American students.

The entire work is worthy of careful reading and it seems not improbable that it will find a place as a textbook in some of our colleges.

1.4. Review by: Charles Ronald MacInnes.
Bull. Amer. Math. Soc. 32 (2) (1926), 172.

The book starts with kinematics treated from a geometric point of view; a point of view which is carried so far that what they call speed-acceleration is defined as the slope of the time-speed graph, and velocity-acceleration as a velocity on the hodograph. Along this same line, in illustrating speed as the slope of a curve, the authors make the curious statement that "it is important that the student realise that in the concept of speed, we divide a distance by a time." This should give pause even to one who would accept the corresponding statement concerning the measure of a speed.

The early chapters cover the ideas of velocity, acceleration, projectiles, relative motion, and general kinematics. Then comes a chapter on vector addition and subtraction; multiplication is left to a later chapter, in fact to a chapter which comes after some of the ideas have been used in getting the moment of a force. Statics and dynamics of a particle and of a set of particles are developed side by side. Then comes the idea of moments, followed by centres of gravity and hydrostatics. The authors have quite justly left to this late date the difficult idea of mass. After momentum and impact are taken up, come work and energy. The book closes with a little celestial mechanics and a few interesting historical notes.

The book is not intended for use in a first course; on the other hand, no use is made of the calculus. The authors have gathered a large collection of problems, about a thousand, and many of them quite substantial enough to test the ingenuity of the best students. They are all gathered at the end, even the illustrative ones, arranged by chapters. Answers are furnished. Graphical methods are used a good deal, including Maxwell diagrams for the solution of truss problems.

2.1. From the Preface.

This book is a deductive treatise. Starting from the fundamental principles of dynamics and of thermodynamics, with the physical properties of seawater, and assuming also the conditions which arise outside the ocean, deductions are made relating to the movements of the waters of the oceans.

The main structure of the book is one of simple mathematical argument, in which the operations employed do not extend beyond elementary differentiation and integration. This remains true in the second half of the book, where partial differential equations are much used. But use is also made of alternative arguments involving only qualitative reasoning, and it is hoped that the book may be of service to readers with very little mathematical knowledge. Most of the mathematical parts of the book refer to conditions which have been ideally simplified.

In the early chapters, these simplifications consist in the absence of turbulence and often also in the absence of variations with time. But later, much attention is paid to type-problems; these are concerned with basins of a simple geometrical form so that the results can only be applied to the actual oceans and seas in a general way. In spite of these restrictions, the dynamical natures of all the chief kinds of ocean water-movements are elucidated and their characteristic relationships demonstrated. The theoretical relationships are compared with corresponding relationships based on observations made at sea, and in this connexion much literature on oceanographical research is quoted.

Each chapter has a section on the history of the development of the theory contained in the chapter, and in these sections an attempt has been made to quote the first occasions on which particular results were published. But the arguments of the book are usually very different from those of the original sources. Each chapter concludes with a list of references to the work quoted in the chapter, both on theory and on observations. The book is intended for the use of students and of those engaged in practical oceanography.

2.2. Review by: G E R D.
The Geographical Journal 120 (1) (1954), 105-106.

Oceanographers, like geographers, have passed beyond the comparative or descriptive stage to ask how?, when? and why? as well as where? and this extension of the study of the sea is difficult because the water movements, which have a bearing on all events in the sea, are governed by processes that are very complex as well as reasonable. Oceanographers have had plenty of warning: a well-known geographer told them eighty years ago that few subjects have excited more interest and attention than oceanic circulation, yet few are in a more imperfect and unsatisfactory condition because the question is one which properly belongs to the domain of physics and mechanics whereas few physicists of note have given the subject special attention. Little has been done since then to make the subject attractive to mathematicians and physicists: they look upon it as geographical exploration or a minor adjunct to marine biology and fishery research, in which the problems seem to remain the same for ever, and never to be satisfactorily cleared up. Their approaches to it are inevitably discouraged by the absence of basic data of the kind they can use, and the absence of an adequate guide to such theoretical work as exists among the mass of qualitative reasoning which characterises the literature.

Professor Proudman's book is the first comprehensive specialist approach to the subject in the English language, and it presents it in an orderly way which is likely to attract mathematicians and physicists. Most of today's physical oceanographers, recruited into the subject from chemistry, geology and geography, will not follow most of the arguments; and the fact that in such a systematic approach the emphasis must be more on type problems than actual conditions in the sea must cause some disappointment, but they have begun to find their problems so perplexing that they realise the need for a new approach and their welcome for the book will be sincere though subdued.

The book gives a concise account of what is known of the principles underlying the mixing of sea-water, generation of ocean currents, waves, water circulation, tides and seiches. There is an introduction at the beginning of each chapter sufficient to show its application to actual oceans and seas, and the theoretical relationships are compared with the results of observations wherever this is possible.

It is hardly a book for the general student, but little progress will be made in oceanography until more of the serious students of the subject have mastered its contents.

2.3. Review by: Henry Stommel.
Science, New Series 118 (3065) (1953), 365.

When one considers the extensive publicity given oceanographic expeditions by the press, the astonishing success of popular books concerning the marine sciences, the large sums expended in oceanographic research (about $5,000,000 a year), he will be surprised to realise that Professor Proudman has provided us with the first text on theoretical oceanography in the English language. It is now possible for any physical scientist to obtain an idea of the theoretical framework of oceanography in berms that he can appreciate. Until the publication of this work, a physicist seeking information about the ocean was without an adequate guide to such theoretical literature as does exist and might easily suffocate in the mass of qualitative reasoning that characterises so much of the professional literature.

The only prior English language treatise that does exist (The Oceans, Sverdrup et al., Prentice-Hall, 1942) is a compendium covering all branches of marine science (biology, chemistry, etc.) in which the physical portions are written in a form more suited for meteorologists and the professional oceanographer than for physical scientists.

The first three chapters of Dynamical Oceanography are an introduction to the hydrodynamical equations in a rotating reference frame. Chapter 4 is concerned with the standard gradient current approximation so much used for computing the field of motion from the observed density structure. Chapter 5 treats various examples of stationary accelerated current systems. Chapters 6 and 7 deal with the subject of ocean turbulence and mixing processes. A very original treatment of the role of friction in the dynamics of parallel cur rents is introduced in Chapter 8. The following chapter begins with an ingenious explanation of the asymmetry of the wind-driven surface circulation, discusses the classical wind-drift current theory of Ekman and the many subsequent studies inspired by Ekman's work. Chapter 10 is unique in the oceanographic literature: it grapples with the difficult thermodynamical circulations in the ocean.

In Chapters 11-14 Professor Proudman enters into the subject of tides, where he is the world's foremost authority. Chapter 15 is an exposition of internal tides and waves. The final chapter is a brief summary of the main results of classical surface wave theory. The exposition is clear and concise, and the book is so arranged that one may read any chapter at will without having to refer constantly to equations in previous chapters. Many readers will be glad to hear this. This book should prove particularly useful for teachers of oceanography because each subject is discussed in a series of examples of increasing complexity, and in most cases the proofs and demonstrations are entirely original and novel. There is a short history and bibliography at the end of each chapter. These references contain few works of recent date, but this was perhaps a necessary restriction in order to maintain a well-balanced presentation of the entire subject, many aspects of which are so difficult that little progress has been made f or many years. For example, a satisfactory model of the meridianal thermohaline circulation has never been investigated chiefly because of the essentially nonlinear nature of the transfer equations. Or, as another example, it has been impossible to integrate the linear tidal equations for the real oceans because the geometry of the ocean basins is so irregular. There is also a great deal to be said for giving references to original works, rather than to recent elaborations, no matter how old the original works happen to be.

The serious reader will find it desirable to supplement his study of Proudman's book with reading of a more descriptive nature. This reviewer recommends for this purpose the chapter on "water masses" in Sverdrup's The Oceans.

For years physical oceanography has been something of an ugly duckling among other prouder, established sciences. To many an academic scientist it seemed It species of geographical exploration, an expensive hobby for amateurs like the Prince of Monaco, or a minor adjunct to marine ecology. This unfortunate impression was largely due to the chaotic state of its theoretical framework. Professor Proudman's book presents this material in an orderly, understandable fashion and ought to do much in attracting the attention of capable mathematicians and physicists to the many perplexing theoretical problems of the ocean. Publication of this splendid volume makes one feel that oceanography has at last come of age.

2.4. Review by: David Vaux.
Science Progress (1933-) 42 (165) (1954), 129-130.

Prof Proudman's book is written from the mathematical standpoint, but its results and implications are in terms which should be understood by scientists engaged in marine, as well as in physical, oceanography. Before his appointment as Professor of Oceanography at Liverpool University twenty years ago, he had been professor in the department of Applied Mathematics for some fourteen years. His book, therefore, has a particularly sound hydrodynamical background. Results applicable to practical oceanography are deduced from the basic equations of dynamics. After each new concept has been introduced and the theory given, the author draws on his wide experience to quote data appropriate to his simplified type problems, and works through examples. Even if each step in the theory has not been fully understood, the resulting technique will be of use. Each chapter has a short and useful introduction to the type of problem about to be discussed, and a concluding paragraph on the history of the subject, with a well-selected list of references.

The first part of the book deals with the fundamental equations of motion, and leads to a study of currents. The following chapters on turbulence and mixing in sea water are highly important. Prof Proudman's approach is somewhat different from the published ideas of other investigators; it follows the theory of mixing by turbulent motion previously given in a paper to the Royal Society. Theory is again illustrated by examples. The estimation of mixing coefficients from TS diagrams as suggested by J P Jacobsen (1927) is treated at some length, and values for different sea areas are obtained. The effects of internal friction (viscosity) and wind conditions are next discussed, and these, of course, complicate the equations of motion already considered.

The last six chapters of the book deal with the theory of tides, waves, and sieches, both internal and surface. To the mathematically-minded reader, these chapters will be of special interest. Again, the equations are developed from those already used, but the non-mathematically-minded are not likely to appreciate the text throughout. There are, however, still more worked examples which can be followed fairly easily.

Name and subject indexes are included, and the production of the book is of the high standard one would expect. All who are branches of oceanography, and particularly in the physical, side, will be grateful to Prof Proudman for putting his long experience at the forefront of his subject.

The sudden death on 26 July, 1928, of Prof Frank Stanton Carey, who for thirty-seven years was professor of mathematics at Liverpool, first in University College, and then in the University, removed one who did much valuable pioneer work in the building up of a new university. Born in Somersetshire in 1860, F S Carey received his early education at Bristol Grammar School, and then proceeded to Trinity College, Cambridge. He was Third Wrangler in 1882, placed in Division I of Part II of the Mathematical Tripos in the same year, and elected to a Fellowship at Trinity in 1884.

In 1886, Carey was appointed to the chair of mathematics at Liverpool, which had been founded three years earlier, and already occupied by A R Forsyth and R A Herman. A born teacher, he was exceptionally able to impart knowledge to the dullest of his pupils, and, at the same time, to inspire the most brilliant of them. Both types of men continuously sought his advice long after they had left the University, and they were always amply rewarded. He himself never ceased to be an enthusiastic student of pure mathematics, always keeping a youthful outlook and fully appreciating the modern ideas in that subject, very different as they are from all that he was taught at Cambridge.

Some of Carey's qualities are indicated by his choice of colleagues and assistants. The list of mathematical appointments made at the University of Liverpool during his tenure of office is as follows: W H Young, R W H T Hudson, H Bateman, E Cunningham, J Mercer, H R Hasse, W J Harrison, P J Daniell, H W Turnbull, J Proudman, R O Street, A T Doodson, B M Wilson, E L Ince, and S F Grace.

Carey's original contributions to mathematics are not large; they consist of isolated papers on geometry and the theory of numbers. His textbooks are better known, and have been used by a large number of students; they are Solid geometry, Infinitesimal calculus, and The elements of mechanics (of which he was joint author). His latest publication (also a joint one) was Four-place tables with forced decimals. But of his writings perhaps that which shows him at his best is his chapter on mathematics in the volume on Modern France, published in 1922 by the Cambridge University Press. In this there occurs a sentence which reveals an admirable spirit for a university teacher: "Perhaps the new ways were invisible except to the eyes of youth". His culture was a wide one and he appears to have been able to enter intimately into the spirit of the scientific pioneers of the seventeenth and eighteenth centuries.

In the administration of his University, Carey took a prominent part, and on council, senate, and faculties he always judiciously upheld the claims of science and scholarship. He rendered vital help in the establishment of the Tidal Institute. The library, teachers' training college, finance committee, and athletic club all benefited by his active sympathy and sound judgment. His death will be deeply regretted by a wide circle of friends and former pupils, many of the latter being teachers and engineers.