1.1. Review by: E H Neville.
The Mathematical Gazette 11 (161) (1922), 209-210.

Dr Weatherburn is known to readers of the Gazette as a firm believer in the use of vectors, and a text-book from him is very welcome. Since a knowledge of vectors may benefit many students who will never require the nabla or the dyadic, he has designed a volume complete in itself to cover the elementary parts of the subject, and here we have an admirable account of vector algebra and of differentiation and integration with respect to a single scalar variable, followed by a synopsis and preceded by a historical sketch that is well balanced and well written. ... let us conclude by saying emphatically that this is a good text-book, which should be useful a much wider range of students than the author seems to have had in mind.

We welcome this elementary treatise on Vector Analysis as a book that has been wanted in English for some time. The author confines himself entirely to the elementary parts of the subject, but treats them very fully. Many applications of vectors to geometry, kinematics, to dynamics, both of rigid bodies and of a particle, and to statics are given. To get the student of dynamics to think in terms of vectors is wholly to the good; for, apart from the greater elegance of vector methods, it gives him a much clearer realization of the physical realities of the problems than is attained by the ordinary analytical methods.

The author's object in this book is to present the simpler portions of vector analysis and to apply them to portions of mechanics. He adheres to the notation of Gibbs. He gets as far as differentials and integrals, but does not bring in the notions of curl, convergence, and other ideas that belong to the general study of fields. The definitions are geometric for the scalar and the vector products, the vectors being always thought of as lines, or geometric vectors. ... The geometry of curves in space is treated briefly, kinematics and dynamics of a particle, systems of particles, rigid kinematics, rigid dynamics, rigid statics. The book should serve as a simple introduction to these subjects treated by way of the vector methods, and for the purposes in view is admirably adapted to the student's needs.

2.1. Review by: Ludwik Silberstein.
The Mathematical Gazette 12 (174) (1925), 293.

This is a very useful and welcome supplement to the author's recent book on Elementary Vector Analysis. ... The presentation is very clear and pleasant, yet concise enough. The book may be warmly recommended to mathematicians and to physicists as well.

This book is a sequel to the author's Elementary Vector Analysis, which dealt with vector algebra and differentiation with respect to one scalar variable. Here we begin with partial differentiation and are introduced at once to the gradient of a scalar and the divergence and curl of a vector function. Then come the theorems connecting line, surface, and volume integrals, and, later, an introductory account of linear vector functions and dyadics. This general theory occupies four chapters; the rest of the book is concerned with the applications to potential theory, the conduction of heat, hydrodynamics, rigid dynamics, elasticity and electricity, not forgetting some account of the restricted principle of relativity. The treatment is admirably clear and interesting, and exhibits the advantages of the use of vector methods in mathematical physics, provided that they are kept in their proper place.

3.1. Review by: W V D Hodge.
The Mathematical Gazette 13 (190) (1927), 425-426.

One of the results of the recent work in the theory of relativity has been that a fresh impetus has been given to the study of differential geometry. Many of the problems that have arisen are problems in differential geometry, and in the solution of these there has grown up a new calculus, the calculus of tensors, which has had a remarkable success in this connection. This has resulted in the appearance of several new volumes on differential geometry, in which the methods of the standard works on the subject have been replaced by those which have proved so successful in satisfying modern requirements. In expressing an opinion on any of these volumes, we must consider primarily the success or failure of the use made of these new methods, for therein generally lies the reason for the existence of the book. ... Professor Weatherburn's volume is of a very elementary character. Great use is made of vector methods, and in this respect the author rather surprisingly claims some originality of treatment. These methods have however been superseded by the use of tensors, which have the advantage of generality, and are not open to the criticism that they do not lead easily to new results. This criticism of vector methods is one with which Professor Weatherburn disagrees, for he adds a special chapter in order to explain results which he has obtained by these means. ... As a textbook for elementary students, Professor Weatherburn's volume is marred by a certain looseness of expression.

Professor Weatherburn has shown himself to be an enthusiast for vectors, and so, as one would expect, his book makes great play with vector methods and is full of rather repulsive-looking dots and crosses and Clarendon type. It is much more geometrical than Mr Campbell's [A Course of Differential geometry (1926)], as is natural, and it is of course much more elementary, beginning at the beginning and keeping to three dimensions or less. The subject matter is, in fact, that usual in treatises on differential geometry, of which there is a fair choice. The book will certainly be of use to those who do not read German or Italian ; moreover, there are plenty of instructive examples. But the student must not neglect Darboux, as Prof Weatherburn has done.

"The objects of the present volume are to provide an introductory treatise on (metric) differential geometry, and to show how vector methods may be employed to advantage." The author shows excellent judgement in the way he goes about securing these two ends. He chooses his geometrical material well, and, in adopting Gibbs's notation for his vector treatment, he is undoubtedly in accord with the current preference. ... When a specialist in vector analysis turns his attention to geometry, it is too often the result that the geometry becomes merely a foil for the aggrandizement of the vector analysis. The present writer treats geometry more kindly. He keeps it clearly in mind as the first of the two objects he set out to achieve and does exceedingly well by it. The author's geometric insight is keen, clever, and instructive. But at times he is found offering, as rigorous proofs, intuitive geometric arguments which, though enlightening and to the point in their proper place, are lacking in substance. ... An able presentation of the elements of the subject by vector methods, a clearly written text with an abundance of good exercises, this book should prove a welcome addition to the literature in differential geometry.

4.1. Review by: Ernest P Lane.
Amer. Math. Monthly 38 (1) (1931), 36-38.

The second volume of this work on metric differential geometry continues the discussion of the subject along lines which are a natural extension of those followed in the first volume, which appeared in 1927. It seems appropriate, therefore, to make a few comments on certain characteristics which the two volumes have in common, before considering the second volume specifically. The first thing that strikes the reader on turning the pages of the two volumes is that consistent use of vector analysis is made throughout. The classical notation of Gibbs is employed, symbols for vectors being printed in Clarendon type. Certain economies are thus effected in the way of simplifying and condensing the presentation of the subject. Those who have been in the habit of lecturing to graduate students on the subject of metric differential geometry, using the conventional methods, might do well to consider the advisability of trying out a presentation by vector methods. To anyone who ventures on this undertaking these two volumes before us will be very useful; they should be in the hands of the students as well as on the lecturer's desk. Another commendable characteristic of the entire work is that the author has not allowed himself to forget that he is, before all, writing a treatise on geometry. The geometry is the thing that holds the centre of the stage, and the analytical machinery is relegated to a subordinate place in the background where it ought to be in a book on geometry. The author does not make the mistake of becoming so immersed in the intricacies of his machinery, or so occupied with juggling his tools, that he loses sight of his main undertaking. Moreover, it is worthy of remark in this connection that the author's geometric insight and intuition are as clear and penetrating as his geometric interest is dominant. The treatise lacks the profundity of such monumental works as those of Bianchi and Darboux. The more elementary parts of the subject are fairly adequately treated, but the more advanced portions are touched upon rather lightly. ... The book is well written. The author evidently understands the fundamental principles of good mathematical exposition.

The second volume of Professor Weatherburn's Differential Geometry of Three Dimensions should prove a very useful book to those familiar with the earlier volume. The branches of the subject discussed are not usually included in an honours course in English universities, but students who have studied the subject to degree standard will find this book an excellent introduction to further work. Much of the volume is devoted to subjects to which the author has himself contributed in the last few years, particularly in the theory of families of curves and surfaces, and of small deformations. Other topics are however included, with the result that the two volumes together give an account of most of the principal branches of classical Differential Geometry.

The first volume of this work (see this Bulletin, vol. 34 (1928), pp. 785-786) covered the fundamentals of metric differential geometry of curves and surfaces. The present volume, though containing certain classical material supplementing that of the first volume, is primarily devoted to a consequential exposition of the author's published contributions to the subject. The treatment in both volumes is in terms of vectors. But, whereas the first volume employs, except in the last chapter, on differential invariants, only the algebra of vectors, the second volume uses also, and to a great extent, the differential and integral calculus of vectors. Moreover, dyadics are introduced in the middle of the volume and are used to good effect throughout the later chapters. ... The tendency of the vector analysis to override the geometry is greater in this than in the previous volume. Only the author's strong interest in geometry and keen geometrical insight save the book from becoming too formal. As an extension of the first volume and an exposition of the author's widely published researches, it should make a strong appeal both to the student and to the specialist.

5.1. Review by: J A T.
The Mathematical Gazette 22 (251) (1938), 415.

This book is a short introduction to Riemannian Geometry. After an introductory chapter recalling certain results in algebra and analysis which are constantly used in the sequel, the author proceeds in three chapters to develop the elements of the tensor calculus, the operations of raising and lowering suffices by means of fundamental symmetric tensor of the second rank, and the theory of covariant differentiation. Applications follow to the theory of curves in a Riemannian space, to geodesics, to parallelism of vectors, to congruences of curves and to orthogonal ennuples. In the next chapter the curvature tensor is introduced and the rest of the book consists of applications to hypersurfaces and general subspaces of a Riemannian space; here the work is simplified by the use of a generalised form of covariant derivative which is most appropriate in this connection, when functions arise which involve the coordinates both in the subspace and in the enveloping manifold. The book ends with an interesting historical review of the subject and an extensive bibliography. The work is clearly and concisely written, and should prove a useful introduction to a very important branch of modern geometry.

In the author's words this is "a book which will bridge the gap between the differential geometry of euclidean space and the more advanced work on the differential geometry of generalized space." It is dedicated to Dean Eisenhart and Professor Veblen. Indeed it follows very closely the content, notation and arrangement of the former's Riemannian Geometry. But it is purposely more elementary, less rigorous and less complete. ... It will be a disappointment to some that the book was not constructed along more original and more stimulating lines although this is much to demand of an introductory text. Greater effort might have been made to lessen the emphasis on formalism which is so difficult to avoid in this field. Basic concepts could have been presented more carefully and given richer meaning. ... The book is neither for dilettantes nor advanced students of the subject. But to those who are seeking an introductory treatment of textbook character to serve perhaps as a sequel to the author's Differential Geometry it is to be recommended.

6.1. Review by: J Wolfowitz.
Mathematical reviews MR0019262 (8,392d).

This book is intended "to provide a mathematical text adapted to the needs of the student with an average mathematical equipment, including an ordinary knowledge of the integral calculus.'' It assembles the formal derivations of many results used in elementary statistics. The student will find it a convenient collection of them. ... This book is not in the modern manner. A test of significance is described as based on the occurrence of a very rare event. No mention is made of the idea of power, and hence of why certain very rare events are held to confirm the null hypothesis and others to contradict it.

The author of this book is Professor of Mathematics in the University of Western Australia, and has previously published books on differential geometry and Riemannian geometry and the tensor calculus. He has not; so far as is known, published any papers on the theory of statistics, and it is thus interesting to see the attitude of a mathematician, with some ability as a text-book writer, who has now turned to statistics as his theme. ... The book is based on a course of lectures given by the author, and first impressions are that the subject has been dealt with competently and well, so far as it is taken. ... A very good idea of what the subject is about generally, and of how it is dealt with by modern exponents, can be had from this book by anyone who has a moderate mathematical knowledge and who possibly finds Kendall's two volumes too formidable and time-consuming. It would be a good idea if mathematical teachers, who all too often know nothing about statistics, would read this text; they would find it written simply in their own language, and could not advance the same objections that they are inclined to raise with books written for biologists without mathematical proofs. The reason why this course is recommended is that the subject has yet to find an established place in the curriculum of many mathematical schools, and a wider knowledge of the field is likely to lead to the realization that more must be done to cater for the needs of students who are interested in the subject, either for its own sake or, more often, because interesting and lucrative avenues for employment are open to those who have been trained as statisticians. It should be emphasized, however, that the book is only an introduction; and if the teaching is entrusted, as it should be, to the specialist in statistics, such a man will be required to go beyond what is written here. Most of the content of the book should by now be standard knowledge.

Professor Weatherburn dedicates his book to Professor R A Fisher and to the memory of Professor Karl Pearson. The reader, it is believed, will agree that it is a worthy contribution in a distinguished tradition. The book is carefully written, and the result is a high order of clarity throughout. It is not a little astonishing to realize after reading the book how much both of theory and of example has been set comfortably in 271 pages. ... This is a book every statistician will want in his library.

The author makes little attempt at mathematically rigorous or precise definitions of his fundamental concepts. In fact, after several readings, the reviewer is not clear what definition of a frequency distribution would be consistent with the statements in the text ... The reviewer's general conclusion is that Weatherburn's book should be on the shelves available to students but that most teachers will probably feel that it is not completely satisfactory. This is perhaps a good thing; it will force the teacher to decide for himself what his students should be taught.

Professor Weatherburn has based his book on lectures given to students at the University of Western Australia. He presupposes some mathematical knowledge in his reader ... The need for a book of this kind has long been felt, and Professor Weatherburn is to be congratulated on filling it so admirably. The conventional subjects are dealt with, but special mention must be made of the author's treatment of small sampling distributions. He devotes a lengthy chapter to the properties of three distributions, and then shows that most of the sampling distributions in common use may be regarded as sums, products or quotients of these variates. The treatment is remarkably lucid and brings out well the connection between the different distributions, and the underlying unity of the subject.

This book has a good many points of similarity to the even more recent text by Hoel, reviewed above. It is written at about the same mathematical level and for the most part, as is natural, deals with the topics found in the other book. It is, however, even more concerned with the mathematics of statistics, so far as its mathematical level permits, and by not including a n account of some of the more recent developments dealt with in Hoel, space is found for a fuller treatment of some of its subjects. Greater mathematical clarity seems to be obtained at points, especially in the first half, but the discussion of the meaning of the results from the viewpoint of application to experimental work is even more brief. This is primarily a book for a comparatively mature student who already has a working knowledge of methods of statistical investigation, who wishes to gain a sounder understanding of their validity, and who has a good command of the calculus or a little more as his mathematical equipment. For students who are quite new to statistics, a good deal would need to be added by the instructor to keep a course based on this book from being quite formal.

An ideal text-book on statistics should effectively co-ordinate a lucid exposition of the basic logic and concepts of statistical theory, with a thorough (and, if possible, rigorous) elucidation of the auxiliary mathematics needed. Such ideal tests are, unfortunately, rare, and most of the standard books on statistics emphasize one of these aspects at the cost of the other. The author of the book under review has, however, explicitly restricted its scope to be "a mathematical text on the theory of statistics" only, and since well written books of this type also are comparatively scarce, it will be a welcome addition to the already existing texts. The book is based on a course of lectures on statistical mathematics delivered in the University of Western Australia. Within the self-imposed restrictions, the author-a reputed writer of mathematical texts has dealt with the subject admirably. Apart from the merits of the book as a text on Statistical Mathematics, it will also be a useful help-book for the students of the degree course in Statistics, when read in conjunction with the usual texts. ... The book presents a fair picture of the major domain of mathematical statistics, and is complete in itself. One misses, however, several more or less important topics, which, the reviewer feels, might find their in place even in an introductory text on statistical mathematics, as the book is designed to be. ... The book is written in a lucid and elegant style and does justice to its title. The reviewer has no doubt that it will be widely appreciated by the students of statistics.

Dr Weatherburn, Professor of Mathematics at Perth, Western Australia, will be known to many readers as the author of Differential Geometry of Three Dimensions. The present book is based on a course on "the mathematical foundations of the interpretation of statistical data", taking the form of some sixty lectures at the University of Western Australia to graduate and under- graduate students in agriculture, biology, economics, psychology, physics and chemistry. It is a sound and workmanlike volume. ... The book is dedicated to R A Fisher and to the Memory of Karl Pearson, and the former author especially is extensively quoted, both for ideas and for tables of functions, though there is, as suggested reading, a larger proportion than is common in Britain of text of U.S.A. treatises.

An outstanding feature of the present statistical time is the number of text-books which are being written, and each one from a slightly different point of view. It is this which makes statistical theory interesting to study, for there can be no rigid approach to a subject which is used and expounded by so many and diverse persons. Professor Weatherburn has taken a rather formal mathematical exposition of the subject, and mathematical students will find his book both interesting and profitable to read. Numerical examples are given for the reader to apply the appropriate mathematical technique. It is possible that these would have been of greater utility if they had contained the material in its and had not been streamlined so that the application of the technique is immediately obvious, but nevertheless many new examples are there. I am not sure whether this book will be entirely useful to students of other subjects than mathematics. While the mathematical analysis is undoubtedly clear it is possible that many will not be able to follow it in detail, and the conclusions of the analysis are not emphasized strongly. ... In spite of the criticisms which I make, however, I would recommend this book to students who have obtained some idea of the aims and objectives of statistical theory, and who are desirous of learning the development of the mathematical technique as well as its application. Professor Weatherburn's mathematical analysis makes pleasant reading and may well throw new light on old methods for those who have learnt the rudiments of the theory.

7.1. Review by: The editors.
Mathematical reviews MR0121888 (22 #12617).

This is a reprint of the 2nd edition (1949) which differed from the 1st edition [1946] only in the correction of a few errors and misprints, the addition of new references and of a section on the distribution of the range of a sample.

This is the paper-back edition of a book which first appeared in 1946, with a second edition in 1949, and which has been reprinted in 1947, 1952, 1957 and 1961. It is a useful exposition of such reading as might be necessary for the mathematics student beginning the study of statistics.