1.1. From the Preface.

An experience gained in preparing hundreds of students for the Inter. Arts and Sc. Examinations of the University of London, and correcting thousands of their papers on the Line and Circle, has given us an exceptional insight into the difficulties of beginners. Most of the existing textbooks assume that, before Coordinate Geometry is commenced, an accurate knowledge of Algebra and Trigonometry has been acquired; but, unfortunately, this is the case only with the minority of students. This book is written with the conviction that the subject is not too difficult for the ordinary undergraduate who does not aspire to mathematical honours, if the course is carefully graded, and divided into cosy stages.

We have tried to realise the position of the average learner, and have constantly borne in mind the needs of the private student: each time, after providing sufficient material for a new set of ideas, we have given illustrative examples, which, in addition to their ordinary use, may be taken as a reminder to review the whole of the ground covered since the last set. It is not sufficient to understand; it is necessary to fix the fresh facts in the mind, and to note their relation to previous ones as they occur.

The reader will, we are sure, appreciate the pains we have taken in beginning and ending on a page each important piece of bookwork, and the clearness gained by spacing out the lines of bookwork to be learnt for reproduction.

Different sizes of type have been used to indicate the relative importance of the matter, and paragraphs intended for those who wish to pursue the subject further have been marked by an asterisk.

Hints to the student are given in square brackets; these, of course, are not to be reproduced in answering questions.

Formulae, each of which must be firmly fixed in the memory before proceeding further, are displayed in bold type and reference numbers added: in the early part of the book the later have not been given each time the corresponding formula crops up; it is left to the student to discover their locality in so small an area.

We were fortunate enough, in sending out the advance copies of the part on the Point and Line, to gain the favour of many teachers in our principal schools, including the majority of the University Colleges, which have candidates for London Examinations; our thanks are due to several for corrections kindly sent. We cannot hope, in such a book, that there are no misprints, in spite of the care which several of our friends have taken, and we shall be obliged. to any of our readers who will let us know of any errors they may come across.

The beginner is urged to keep in mind that the subject is Geometry, and to represent his work graphically wherever possible. For this purpose paper ruler in small squares should be used, books of which can be obtained of most booksellers.

2.1. Preface to the First Edition.

For some time past it has been felt that a gap existed between the many excellent popular and non-mathematical works on Astronomy, and the standard treatises on the subject, which involve high mathematics. The present volume has been compiled with the view of filling this gap, and of providing a suitable text-book for such examinations as those for the B.A. and the B.Sc. degrees of the University of London.

It has not been assumed that the reader's knowledge of mathematics extends beyond the more rudimentary portions of Geometry, Algebra, and Trigonometry. A knowledge of elementary Dynamics will, however, be required in reading the last three chapters, but all dynamical investigations have been left till the end of the book, thus separating dynamical from descriptive Astronomy.

The principal properties of the Sphere required in Astronomy have been collected in the Introductory Chapter; and, as it is impossible to understand Kepler's Laws without a slight knowledge of the properties of the Ellipse, the more important of these have been collected in the Appendix for the benefit of students who have not read Conic Sections.

All the more important theorems have been carefully illustrated by worked-out numerical examples, with the view of showing how the various principles can he put to practical application. The authors are of opinion that a far sounder knowledge of Astronomy can be acquired with the help of such examples than by learning the mere bookwork alone.

2.2. Review by: Anon.
Nature 45 (1173) (1892), 579.

The task of writing a book on astronomy which shall enable a beginner to grasp all the fundamental principles and methods without entering into elaborate details of mathematics is by no means a light one. What the authors have done, and we may say very successfully too, has been to strike a mean between the numerous non-mathematical works and those which involve high mathematics, using just enough to enable the reader, if he wishes, to proceed to the more advanced treatises. To simplify matters further, all investigations which require a knowledge of the elements of dynamics have been collated together at the end under the title of dynamical astronomy, thus separating them from those of descriptive astronomy, which only needs elementary geometry, trigonometry, and algebra. Some of the chief properties of the ellipse which are of astronomical importance are contained in the appendix, while for the properties of the sphere an introductory chapter has been inserted.

This summary will give an idea of the range over which the student will have to extend his mathematical abilities, and after all it is by no means an extensive one, considering the ground which this work covers.

In the chapter on the Observatory a very good account is given of the transit circle and its accompanying errors; but of course, without spherical trigonometry, many points of great importance with regard to the reduction of observations have had to be omitted. The chapters on the earth, sun's apparent motion and time, all contain lucid and concise explanations, which are well illustrated by figures showing the great circles involved. Many interesting problems are worked out in the chapters on the moon and eclipses, while that on the planets contains a good account of the stationary points in their apparent motion.

"The Distances of the Sun and Stars" is the title of the chapter that concludes the non-dynamical section, and in it the interesting problems on finding the parallax of the sun are discussed, together with the various results that ensue from the aberration of light.

Coming now to the second part of the book, the rotation of the earth and the resulting consequences are first dealt with, in which the proofs of the former are clearly described; while many problems relating to pendulum oscillations, variation of gravity at different places, etc., are fully expounded. The following two and concluding chapters are devoted wholly to the laws of universal gravitation, and to the multiple applications to which I they are subjected. These chapters are perhaps the best in the volume, and contain, of course, some most important problems, such as the determination of the density of the earth, precession, tides, etc.

The examples and examination-papers, which are by no means few in number, will be found to be both original and well selected; and this is really important, for a sound knowledge of this subject can be obtained only by the continual practice of working them out.

In conclusion, we may state that altogether the work is one that is sure to find favour with students of astronomy, and will be found useful to those for whom it is especially intended. This is by no means the first volume that we have received which is published in this Tutorial Series, and the present work is a good example of the excellent text-books of which it is composed.

2.3. Review by: Anon.
The Journal of Education 40 (14) (988) (1894), 247.

Mathematical astronomy for a time seemed to be distanced in the race for position in the school curriculum ; but there is evidently a tendency at present to return it to its place. This new candidate for a place in the American secondary schools will be examined with much interest, and, if we mistake not, will find general use. It is comprehensive and scholarly, yet such is the arrangement of the matter that it will be very easy teaching. Under each chapter are a large number of examples for practice, miscellaneous questions for quickening thought, and a formal examination paper.

3.1. Preface to the Second Edition.

The first edition of Mathematical Astronomy having run out of print in less than eight months, we have hardly considered it advisable to make many radical changes in the present edition. We have, however, taken the opportunity of adding several notes at the end, besides answers to the examples, which latter will, we hope, prove of assistance, especially to private students; our readers will also notice that the book has been brought up to date by the inclusion of the most recent discoveries. At the same time we hope we have corrected all the misprints that are inseparable from a first edition. Our best thanks are due to many of our readers for their kind assistance in sending us corrections and suggestions.

4.1. Review by: H P C.
Nature 113 (2827) (1924), 7.

The teaching of astronomy is attended by many difficulties, and competes against other studies under disadvantages which cannot be denied. When facilities exist for practical work the weather may be trusted to play havoc with any prearranged time-table. When practical instruction is not attempted, astronomy when taken seriously demands so much knowledge of the elements of mathematics and the fundamental principles of physics that it may easily appear that the time would be more profitably spent in gaining familiarity with those primary sciences. Perhaps examinations and the requirements of candidates have a salutary effect in saving the subject from complete neglect. But its difficulties are not small, and even the student who has a fair mathematical equipment rarely finds it easy at first to acquire the jargon and the essential ideas of astronomy.

For the purpose of explaining the traditional terminology and fundamental notions of the science for the benefit of elementary students Barlow and Bryan's work is admirably adapted. It has been in existence for thirty years, and it is not surprising that it is still in demand. It is lucid, concise, and sound. It can scarcely be described as a stimulating book, but it fulfils a definite aim efficiently, and it is too well known to need description. This third edition has been revised by Dr Crommelin, and its general accuracy should be above suspicion.

In the circumstances it is strange to find the eye arrested by references to the Nautical Almanack. The publication in question is and always has been the Nautical Almanac. Curiously enough, if one consults the first issue in 1767, one finds on a left- and a right-hand page confronting one another the Act of Parliament ordering a Nautical Almanack to be prepared and the authority of the Commissioners (a galaxy of naval heroes!) to the printers to publish a Nautical Almanac. Whitaker's Almanack, of course, is a whole century younger.

5.1. Preface to the Fifth Edition.

For half a century "Barlow and Bryan" has remained one of the standard textbooks on elementary mathematical astronomy. During this time there have been great advances in physical astronomy, or astrophysics, as this branch of astronomy is commonly called. This subject is not treated in this book, which is concerned with the foundations of astronomy - the general mathematical and dynamical structure upon which everything else depends.

For the present edition the work has been completely revised and, though the original plan has been adhered to, there has been considerable rearrangement of the chapters. The conceptions of apparent and mean sidereal time. made necessary by the precision of modern time-keepers, have been introduced. The definition of the equation of time has been brought into accordance with the "Nautical Almanac" and common usage of today. The chapters on Refraction, Parallax and Aberration - the phenomena that affect the observed position of a celestial body - have been brought together. A chapter on Precession and Nutation, including the reduction from apparent to mean place of a star, has been introduced. A brief description of the bubble sextant, for observations in aircraft, is given and the section dealing with the position line method of determining the position of a ship or aircraft has been considerably expanded. An account is given of the arrangement of data in the "Air Almanac." The sections dealing with the obsolete method of finding longitude by the method of lunar distances and various other sections of little interest have been omitted.

It is hoped that in this new edition the work will be found of increased usefulness, both as a textbook and for reference purposes.

5.2. Review by: A Hunter.
The Mathematical Gazette 29 (283) (1945), 39.

In so far as the review of a book should indicate its scope, the reviewer's task with this volume is simple. "Barlow and Bryan" is in essentials the same book as it was on its first appearance over half a century ago: an excellent introduction to the general mathematical and dynamical structure of astronomy. The great advances which astronomy has seen since the publication of the first edition have not much affected that branch of the subject with which it deals, and the mathematical basis of physical astronomy is not yet touched upon in its pages. In this (the fifth) edition the Astronomer Royal has much improved the book by rearranging the order of the chapters somewhat, and by making condensations where the subject-matter has declined in importance. This has allowed a considerable expansion of the section on time - a distinction is now drawn between apparent and mean sidereal time-and the inclusion of a welcome chapter on the practical aspects of precession and nutation. The section on longitude determination, which had become distinctly out of date, is completely revised. Here the editor's close connection with the Nautical Almanac office will have proved invaluable in winnowing the grain from the chaff. Obsolete methods are dropped entirely, and the procedure described is brought into line with that actually used by the practical navigator. The special problems of celestial navigation from aircraft are recognised for the first time: the bubble sextant is described, and an account is given of the arrangement of data in the Air Almanac. The text is completely re-set in a more attractive fount, and the characteristic Egyptian section-headings of the Tutorial Press books have gone. In the few cases where the diagrams have been re-drawn, the improvement is so marked that one could wish the job completed. In view of the complete re-setting of the type, misprints are remarkably few, and none which the reviewer has noted will cause the reader any trouble. The Astronomer Royal's version of this standard textbook should be more useful than ever to the student who needs something less strenuous than the rigorous treatises on mathematical astronomy, to the teacher of mathematics as a fund of practical examples, and to both as a handy reference books.

6.1. Review by: Anon.
Nature 49 (1255) (1893), 52.

The examples which are here brought together are intended to serve as a graduated course on the right line and circle, forming thus a useful companion to the book on Co-ordinate Geometry already published by the same authors. The work line is specially designed for the private student, and this is why the problems have been dealt with in such detail, every step in their solution being clearly explained. The examination papers may fairly be taken as good test papers, for the questions seem to have been carefully selected, and the more important ones on book work are not lacking. For those teaching themselves this subject by working out the problems given, a good insight should be obtained, while the references to the author's work on co-ordinate geometry, above referred to, will be found very useful to those possessing that book.

7.1. From the Preface.

Of all the subjects usually included in an elementary mathematical course, Dynamics is undoubtedly the one over which the average beginner experiences by far the greatest difficulty. For this reason it has been the endeavour in the present book, in the first place, to give due prominence to the principles of the subject, and, in the second, to avoid the use of mathematical formulae, as far as possible, in the solution of problems and worked examples, and instead, to deduce the results by direct application of these principles themselves. Many of the methods have not, so far as we are aware, been given before, and we hope that in all such cases they will be found to simplify the subject matter proper by removing all mathematical difficulties which are not essential to it. Considerable attention is paid to "average velocity" and "relative velocity"; the first because it enables problems on uniformly accelerated motion to be solved without formulae; the second because it affords a tangible interpretation of component velocities, besides considerably simplifying certain problems relating to the approach of falling bodies, motion down incline planes, and the like. Due prominence is given to the metric system of units, and care is taken in all examples to point out what units are used at each stage of the process.

7.2. Review by: Anon.
The Journal of Education 41 (8) (1016) (1895), 131.

The University Correspondence College, represented in this country by W B Clive, 65 Fifth avenue, New York, began this school year with an enrolment of 740. This is one of the most helpful of the schemes for home culture of a high order. There are studentships and prizes amounting to $1,000. The staff of tutors comprise fifteen graduates who took first-class honours at the university examinations, and six honour men. The text-books, of which this on dynamics is one of the best, are all reliable, complete, and adapted to home study under correspondence direction.

7.3. Review by: J M C.
Amer. Math. Monthly 4 (4) (1897), 125.

We called attention in a previous number to the text-book on Hydro-Statics by the same authors. The treatise on Dynamics deserves the same commendation. Due prominence is given to the principles of the subject, and in the solution of problems results are deduced as far as possible from these principles themselves. Worked examples are freely inserted, and hints relating to special difficulties are given where needed. The examples are numerous and practical, the examination papers well selected, and the summary of results after each chapter of special value in reviews. The book may be open to criticism on some minor points, but there are few text-books on this subject which are so well suited to the needs of beginners.

8.1. From the Preface.

The ground covered by this book includes those portions of Statics which are generally read in an elementary course, and for which little or no knowledge of Trigonometry is required. It may, with advantage, be read after the same authors' Dynamics, as the latter book gradually leads up to the Parallelogram of Forces, which forms the basis of Statics. But this course is quite optional, for the present book assumes no previous knowledge of Dynamics, and the few dynamical facts and definitions required for the proof of the Parallelogram of Forces are briefly recapitulated in the first chapter.

Every endeavour has been made to remove, as far as possible, all difficulties usually experienced by beginners when reading Statics for the first time. For this reason, hints and explanations have been freely given wherever it seemed desirable and the important propositions have been profusely illustrated by worked-out examples. In several cases these examples precede instead of following the general investigations, and it is hoped that the reader will thus the better learn how to work out similar examples from first principles instead of merely obtaining their answers by quoting certain formulae from memory and substituting numerical values in them.

In most textbooks on Statics, the Mechanical Powers are collected together in a chapter near the end, and a separate chapter also is usually devoted to "Virtual" Work. After much careful deliberation we have decided to depart from this plan. The various machines are introduced as soon as the principles on which they depend have been explained, and it is hoped that the earlier chapters have thus been made more interesting. The Principle of Work is freely employed throughout the book as furnishing verifications or alternative proofs of results which have been established independently. Those who wish to omit all considerations relating to Work will have no difficulty doing so, as the sections in question have been kept distinct and can readily be picked out by their heading.

Among other special features of the book, we may, perhaps, call attention to the modification of Newton's proof of the Parallelogram of Forces, the simple expression for the resultant of two forces including any angle, the symmetric conditions of equilibrium of three parallel forces etc.

The general arrangement is the same as in the authors' Dynamics. Two sizes of type are used, the important bookwork being printed in the larger type while hints, explanations, examples, alternative proofs, and a few of the less important theorems are printed in smaller type Those articles which deal with the fundamental principles of the subject - such as the Parallelogram of Forces - have their numbers as well as their headings printed in dark type. These the student should be able to reproduce from memory.

9.1. Review by: Anon.
The Journal of Education 42 (23) (1056) (1895), 402.

The matter is carefully winnowed, the methods direct, the style clear, and the adaptation to students and teachers admirable.

10.1. From the Preface to the First Edition.

The ground covered by this book includes those portions of Hydrostatics and Pneumatics which are usually read by beginners and candidates for examinations of a standard such as that of the London Matriculation. In the illustrative and other examples, it has been our endeavour to deduce results from first principles and, as far as possible, discourage students from relying on memory for mathematical formulae. Where new departures have been thought desirable, they have generally been effected in such a way to allow teachers the opportunity of adhering to older methods of treatment if they so prefer. Thus according our arrangement, the student becomes familiar with specific gravity and the very important practical methods determining it, including the use of the Hydrostatic Balance before encountering the more theoretical considerations connected with pressure and its distinction from thrust. But any reader who prefers may pass on to Part II, after reading the first three or four chapters of part I to be read after Chapter XIII. Again, proofs involving the Principle Work have been introduced in several cases, but the possibility omitting them if desired has been pointed out.

We have given considerable attention to the illustrations, notably those of air and water pumps in which the up and down strokes are figured separately.

Where articles are marked with an asterisk they should certainly be omitted on first reading, and for many purposes they may be omitted altogether. Only a few articles are, however, "starred," as this matter depends so much on individual students that it is generally best left to the teacher.

We take the opportunity of pointing out, that where the distinction between pressure and thrust has not been consistently carried out in the examples, this has sometimes been purposely done in order that readers may become accustomed to such differences of nomenclature as commonly occur, even in different papers set for the same examination.

11.1. Review by: J M C.
Amer. Math. Monthly 4 (4) (1897), 126.

This little book contains the Sixth and Eleventh Books of Euclid, together with a summary of Book V., and many important additional propositions and applications relating to the Geometry of Similar Figures and the Plane. Euclid's order hats been closely followed, while the additional matter is mostly in the form of illustrative examples. The properties of centres of similitude and homologous points are collected in a supplement at the end of Book VI. In addition to the illustrative examples, numerous exercises for solution follow the propositions on which they depend. The feature of giving any alternative proofs enables the teacher to make his own choice of methods. It is a very satisfactory book in a useful series.

12.1. From the Preface to the Second Edition.

Although the main body of the original work has been untouched, we have taken the opportunity of a Second Edition to introduce several minor alterations and additions. Much of the chapter on Specific Gravities of Mixtures has been rewritten, and illustrative examples have been added, exemplifying the method of dealing with mixtures which contract. Further examples have also been worked out in the chapter on the Principle of Archimedes. Most of these new examples illustrate a plan of dealing with complicated problems which experience has proved to be very useful in the hands of the average student, namely, that of drawing up a table at the outset in which all the known data are entered in their proper place. In the practical working, instead of denoting the unknown quantities by letters, it will be found better to fill the blanks up in succession as these data are calculated, and a glance at the table at each stage of the process will suggest the next step to take. The use of the term "specific weight" to denote the weight of a unit volume of a substance has been discarded, as it was found to lead to confusion with specific gravity. We have to thank many readers for their kind suggestions and corrections.

13.1. Review by: F W H.
The Mathematical Gazette 1 (11) (1897), 120.

This book is intended to prepare students for the Elementary Examination of the Science and Art Department on the Theoretical Mechanics of Fluids, and on the whole it is well adapted for the purpose. The first nine chapters form a condensed and somewhat indigestible introduction to Dynamics, and constitute the most unsatisfactory portion of the book. Surely it is well for even the elementary student to be taught that the principles of Dynamics rest upon the laws of motion, and are not to be proved by experiment. The rest of the book, dealing with the Mechanics of Fluids, is good. The principles of the subject, and experiments for their verification, are well explained, and clear descriptions are given of the usual instruments and apparatus. The examples are numerous and well selected, and the ten examination papers on portions of the subject will serve to make the student pay some attention to the book-work.

14.1. Review by: G M M.
The Mathematical Gazette 1 (10) (1897), 95-96.

This book adds one more to the fairly long list of elementary works which are designed to help beginners in Dynamical Science. It falls into line with modern usage in basing the proof of the parallelogram of forces on Newton's kinetical definition of force; so that we may now congratulate ourselves on the disappearance of such "statical" methods as that of Duchayla. The present work is throughout extremely systematic, and it abounds in examples, examination papers, etc., all in their proper places as applications and illustrations of the text.

The principle of Virtual Work is introduced at an early stage and well explained; but in the simple case in which a particle is sustained by a horizontal force on a smooth inclined plane, while the value of this force in terms of the weight is easily deduced by giving (actually giving) the particle a displacement up the plane, we do not see why the authors take the trouble of specially saying that "it would be less easy to determine the Reaction by means of the Principle of Work." On the contrary, it is quite as easy to determine the Reaction as it is to determine the horizontal force by this principle; for we have merely to imagine the particle to receive a vertical displacement instead of one up the plane, and the Reaction is at once determined. The authors may, perhaps, justify themselves on the ground that they make a distinction between the "Principle of Work," which is introduced at p. 34, and the "Principles of Virtual Work and Virtual Velocities," which are not introduced until p. 216 is reached. The distinction, however, is not a valid one, because their "Principle of Work" is this: "If a particle, acted upon by any number of forces in equilibrium, is moved from one position to another, the algebraic sum of the works done by the several forces is zero" (The italics are ours.) A beginner may justly ask "How is the particle, which is in equilibrium, moved? It cannot be by the forces themselves: it must be, therefore, by some other agent, and hence the motion assumed can be only a virtual one." Moreover, if actual motion is contemplated, it must be one entirely unaccompanied by acceleration - which fact is certainly not properly explained in the above enunciation, and could not be understood by the student at such an early stage. It ought to be clearly understood that when no accelerations or gain of kinetic energy are involved, there is no distinction whatever between the "Principle of Work" and the "Principle of Virtual Work." The manner in which the authors introduce this latter is somewhat amusing. It is this:

"Virtual Work. - We often find it convenient for the sake of argument to suppose a particle displaced from one position A to another B, although the particle may have no tendency to move in this direction, or, perhaps, in any direction." Are there any other physical principles which exist "for the sake of argument?"

The authors have, we think, done well to treat of the equilibrium of forces acting on a particle before discussing the conditions of equilibrium of an extended rigid body; but we would point out that the crane is not properly a case of particle equilibrium, because the discussion assumes the forces at the extremities of the jib, to be directed along the jib, and this assumption ought to be justified by a consideration of the separate equilibrium of the jib itself.

The resultant of two parallel forces is deduced by a neat (and to us novel) method, viz., from the case of three forces acting at the middle points of the sides of any triangle, perpendicularly and proportional to those sides.

Moments are discussed and well illustrated in Chap. V. We have heard of the work of a couple; but a sudden and uncontrollable merriment was excited when we saw that Art. 73 was introduced to us in large type as the "WORK OF A MOMENT!"

We are glad to note that in the discussion of machines, which occupies a large part of the volume, the authors adopt the terms effort and resistance instead of the execrable "power" and "weight" of the old books, and alas, also of some of the new books and even modern examination papers!

One good feature of the book consists in the many numerical examples given - a kind of example hateful to the ordinary university mathematician. The figures, and especially those of machines, are throughout very good, and indicate much care and trouble on the part of the authors; but may we inquire the nature of the cargo conveyed by the cart on p. 163. Is it a load of glass wool? In conclusion, the work will be found extremely helpful to beginners on account of its clearness and thoroughness; and it will be no less useful to teachers on account of its abundance of excellent examples.

15.1. Review by: Anon.
The Journal of Education 47 (1) (1160) (1898), 11.

This volume is one of the University Tutorial Series. The first ten chapters deal mainly with the trigonometry of one angle. The next four chapters treat of the trigonometry of two or more angles; and the remainder of the book is devoted to logarithms and the trigonometry of triangles. The authors do not believe in deferring considerations of algebraic signs until a number of trigonometric identities for acute angles have been dealt with, as is the custom, hence the trigonometric functions are defined in this work at the earliest possible stage. Each preceding chapter contains just such introductory, matter as is necessary to pave the way for a general treatment of the subject which is to follow. The comparative importance of the subject-matter is indicated by the type used, and the fundamental propositions have the numbers of paragraphs, as well as their headings, in dark type. Special attention is given to graphic methods in science teaching. There can be no question as to the scholarly and accurate character of this work, and it is highly recommended.

15.2. Review by: F S M.
The Mathematical Gazette 1 (12) (1897), 143-144.

This Trigonometry is produced in the same style as other volumes in the Tutorial Series. The text is printed in many different degrees of size and thickness of type, with the object of indicating to the reader the precise degree of importance he should attach to each separate piece of bookwork. This results in the general appearance of the pages being far from attractive. We agree with the authors that the definitions of the trigonometric functions for all angles should be introduced early; but unfortunately their explanation of positive and negative is obscure and incorrect. The articles treating of the ratio of the circumference of a circle to the diameter, though leaving much to be desired, are a step in the right direction. We find, however, the old pretence of proving, what is in reality an axiom, that $\Large\frac{\sin\theta}{\theta}\normalsize = 1$ when $\theta$ is infinitely small. Some of the articles are written with exceptional clearness notably that on the ambiguous case in the solution of triangles.

16.1. Review by: Anon.
The Educational Times and Journal of the College of Preceptors 51 (1898), 505-506.

Mathematicians know Radhakrishnan's work, through our own pages if not otherwise. Messrs Briggs and Bryan have done well to adapt his Algebra, which is "the outcome of an intelligent digestion of the best English authorities, particularly of De Morgan, Clifford, and Chrystal," for English schools. This book may serve as a stimulating introduction to higher algebra; and, if we had any doubt as to this, we should be willing to defer to the judgement of the editors, who commend this volume for students at the intermediate stage.

17.1. From the Preface.

When I accepted an invitation to write the article for the Encyklopädie on the General Foundations of Thermodynamics, it was understood that the article should deal, as far as possible, exclusively with the laws of thermodynamics and consequences immediately deducible from them, and that all properties of particular substances and states which depended partially on experimental knowledge or other hypotheses should be left for another article. I had long felt the want of a book in which thermodynamics was treated by purely deductive methods, and it has been my object in the following pages to develop the subject still more on this line than was possible in an article professing to be to some extent an exposition of the history and actual state of knowledge of the subject.

It cannot be denied that the perfection which the study of ordinary dynamics has attained is largely due to the number of books that have been written on rational dynamics in which the consequences of the laws of motion have been studied from a purely deductive stand-point. This method in no way obviates the necessity of having books on experimental mechanics, but it has enabled people to discriminate clearly between results of experiment and the consequences of mathematical reasoning. It is maintained by many people (rightly or wrongly) that in studying any branch of mathematical physics, theoretical and experimental methods should be studied simultaneously. It is however very important that the two different modes of treatment should be kept carefully apart and if possible studied from different books, and this is particularly important in a subject like thermodynamics.

In most text books the treatment of the first and second laws is based more or less on the historic order, according to which a considerable knowledge of the phenomena depending on heat and temperature preceded the identification of these phenomena with energy-transformations. For a logical order of treatment it is better to regard the laws of thermodynamics as affording definitions of heat and temperature, just as Newton's laws afford definitions (so far as definitions are possible) of force and mass. But in any case there is great danger of assuming some property of temperature without realising that an assumption has been made, and of this danger we have an excellent illustration in the assumption commonly made, but rarely if ever explicitly stated, that the temperature of a body at any point is the same in all directions.

To lessen such risks and at the same time to carry the deductive method further back it appeared to me desirable to adopt the principles of conservation and degradation of energy as the fundamental laws of thermodynamics, and to deduce the ordinary forms of these laws from those principles. A paper was published by me on this subject in the Boltzmann Festschrift, and some criticisms on it sent to me by Mr Burbury have led to a more extended examination of the foundations upon which thermodynamics rests. Degradation of energy in some form or other is a necessary consequence of irreversibility of energy phenomena. We therefore go still further back and assume the principle of irreversibility as our starting point. When an irreversible transformation takes place the number of subsequent possible transformations is thereby from the very nature of the case reduced and we thus have a loss of availability in its most general sense. When we want to identify the more and less available forms of energy with those forms of energy which we see around us, an appeal to experience is necessary. It is in fact possible to conceive a universe in which irreversible phenomena tend in a different direction to what they do in our own. A mere reversal of the whole of the phenomena of our universe would give us one example, and if we want another we should only have to imagine ourselves of molecular dimensions when we should find that the whole progress of irreversible phenomena (whether regarded statistically or otherwise) would assume an entirely different aspect to that to which we are accustomed. The laws of thermodynamics are thus restricted to phenomena of a particular size in the scale of nature, and the lower limit of size is about the same as the limit involved in the applications of the infinitesimal calculus to the physical properties of material bodies e. g. in hydrodynamics, elasticity and so forth. The term "differential element" is introduced in the present book to represent the smallest element which can be regarded , for the purpose of these applications, as being formed of a continuous distribution of. matter, and the notion of temperature at a point is regarded as not more nor less justifiable than the corresponding conventions as to pressure and density at a point.

It is, however, in connection with entropy and with thermodynamic equilibria and stability, that the present method of treatment is found to be the most advantageous. A controversy on entropy between English mathematicians, physicists and electrical and other engineers took place in England in 1903 at the instigation of Mr Swinburne, an electrical engineer, who defined entropy by means of what he called "incurred waste". In the present book it is shown that if entropy be defined in terms of increase of unavailable energy this definition will apply not only in the case of entropy imparted to a system by heat conduction but also in the case of entropy produced by the irreversible changes within a system, of which a number of simple illustrations are given.

Moreover the available energy method possesses considerable advantages in the treatment of thermodynamical equilibria. If we assume that in a state of equilibrium the available energy of a system is a minimum it follows immediately that the conditions of equilibrium can be deduced from the equations of reversible thermodynamics and that it is only when the stability of the equilibrium is discussed that recourse must be had to the inequalities of irreversible thermodynamics.

18.1. From the Preface.

Up to the present time the problem of stability has received very inadequate attention in connection with aviation. From the point of view of the practical aviator, this is, perhaps, little to be wondered at. It would scarcely be possible for him to devote months of concentrated attention to long and laborious stability investigations when it is no exaggeration to say that very frequently his success or failure depends, above all things, on the names of the towns at which he starts and lands. If a prize is offered for flight from Folkestone to Flushing, it is useless for him to fly from Harwich to the Hook, even on much more stable machine than that used by the winner of the prize.

It hoped that the publication of this memoir will lead to aeroplane stability being made the subject of much more continuous study and investigation than has been possible in the past. A general abstract of the present investigation is contained in the introduction, which may be read with advantage before proceeding to details of a more mathematical character. The general conclusions show that there should be no difficulty in securing inherent stability, both longitudinal and lateral, in an aeroplane, by means of suitably placed auxiliary surfaces rigidly attached to the machine; but in order to achieve success the conditions of stability must be very carefully studied, and account must be taken of the effects of the inclination of the flight path to the horizon and other causes which may affect the result seriously.

There seems a general desire on the part of many writers to minimise the dangers of instability or defective stability, and to attribute accidents to other causes. But in reading the accounts of accidents, both fatal and otherwise, that appear every few days in the daily papers, it is difficult to avoid coming to the conclusion that much of this loss of life and damage could be avoided by a systematic study of stability and certain other problems regarding the motion of aeroplanes particularised in this book.

To the mathematician who is able to devote any time to original work, a wide region of unexplored ground is opened up, of which it has, in many cases, only been possible to exhibit glimpses in these pages. 'There is a freshness about this region which can scarcely be found to the same extent in searches for differential equations that have not been integrated, high primes, or further additions to the large existing collection of properties of triangles and circles. I do not think any mathematician who cares to take up the subject now will experience any difficulties comparable with those which have been encountered in the preparation of the present book.

There is abundant work, too, for the student who wishes to undertake "research" for educational purposes; and "aeroplane stability" seems to me a very useful alternative for some of the branches of applied mathematics now taught in our universities.

Quite recently, much has been written regarding so-called "automatic stability," depending on the use of gyrostats, pendulums, or other movable parts. Apart from the fact that movable parts are liable to get out of order, it must be remembered that they increase the number of degrees of freedom of the machine, thus further adding to the number of conditions which have to be satisfied for stability, a number quite large enough I anticipate that the successful aeroplane of the future will possess inherent, not "automatic" stability, movable parts being used only for purposes of steering.

The great difficulty in writing this book has been the newness of the subject, and the fact that the further one advances in it the greater appear the possibilities of still further exploration. The selection and arrangement of the text has thus been much more difficult than it would be in a text-book compiled from existing literature. I have tried, so far as possible, to avoid, introducing any investigation the results of which were certain to be only of academic interest; but this attempt has not made it much easier to draw the line. For example, the transformations of §30, which are, of course, particular cases of a more general transformation, were placed on the excluded list, but their application to the pendulum experiment suggested the desirability of retaining them.

18.2. Review by: W J Humphreys.
Science, New Series 35 (901) (1912), 543-544.

Any one seriously at work on the theory or the art of aviation would profit by reading what he can of this book, though precious few will have both the time and the ability wholly to master it and probably none, not even the author of it himself, fully to solve all the difficult problems it sets.

The first chapter gives, in 18 pages of concise non-mathematical language, a summary of existing knowledge on aeroplane stability, and incidentally forms a good mental appetiser for the very solid intellectual courses that follow. But the busy man and the man without mathematical training must take the statements of the next 8 chapters, or 146 pages, on faith. The essential equations in the greatest abundance are there and their meanings explained, but checking them all up would be no easy task.

The general conclusions all this mathematical work leads to are given in a short chapter of only 6 pages, and in another place 20 separate theses are proposed that should set many young men at work on problems that are both new and useful in applied mathematics.

But, the "practical" man will say, what's the use of all this theory and all these mathematical equations? The answer in this case as in all similar cases is: To tell the practical man what to practise, what experiments are needed and what are not, what general type of machine is likely to succeed and what is certain to fail; to save him from needless blundering and to assure him of results and how to obtain them that he never did and never would even dream of.

Professor Bryan's book is especially adapted to the needs of advanced students in physics, applied mathematics and certain branches of engineering, and richly deserves a place in both mathematical and physical libraries.

18.3. Review by: H Hilton.
The Mathematical Gazette 6 (99) (1912), 343-344.

The mathematical theory of aviation is rapidly being added to the lengthy list of subjects with which every mathematician should be acquainted. It is fortunate that the task of writing one of the first mathematical books on aviation has been undertaken by one who is not only a mathematician of first-rate eminence and a specialist in this particular line, but is also an experienced teacher. The reader will find the difficulties of the subject reduced to a minimum by the clearness of the exposition, and the care with which every assumption is emphasised, the notation explained, and the results summarised.

Prof Bryan is justified in his claim that he has opened out to the student a valuable field for research, and he gives further assistance by adding a brief account of some work done by other authors, and by suggesting problems which await solution; though these would probably prove more troublesome than those attacked in this monograph. His suggestion that the subject might be suitable for junior University students is not so convincing; it would probably be beyond the powers of the average undergraduate. I must make an exception in favour of the "graphic statics of longitudinal equilibrium," which is quite elementary and will afford interesting and useful illustrations for the lecturer on Statics.

A flying aeroplane may be considered as a rigid body in steady motion under the action of air-pressure, gravity, and propeller-thrust, having a plane of symmetry which is vertical. If it is slightly disturbed, it may oscillate about its position of steady flight, in which case its motion is said to be "inherently stable." The purpose of the book is to investigate the conditions for this stability. It is easy enough to write down equations of motion which are universally applicable; and we find that the oscillations in the plane of symmetry can be conveniently separated from other oscillations; in other words, we may discuss separately longitudinal and lateral (or symmetric and asymmetric) stability. Difficulties arise, however, when we attempt to manipulate the equations of motion so as to obtain results of any interest; partly owing to the cumbrousness of the algebra involved, and partly because the forces acting on the aeroplane are unknown.

Prof Bryan overcomes the difficulties by making certain assumptions; for instance, he supposes that the air-pressure on the planes is proportional to the square of the velocity, or he simplifies the algebra by taking the "angle of attack" as small. He is thus able to work out the conditions for longitudinal and lateral stability for all simple types of machine. The results can be extended to other types by making use of the "principle of independence of height" that longitudinal stability is only very slightly affected by raising or lowering the planes of a machine, and of the "principle of equivalence" that to every double lifting system of two narrow planes corresponds an equivalent single lifting system with the same conditions of equilibrium and longitudinal stability. For instance, the first principle shows that there is no practical difference between monoplane and biplane machines in respect of longitudinal stability.

Though the book will appeal mainly to the mathematician, the engineer will find the results suggestive; and he should have no difficulty in picking out what he wants without attempting to follow all the mathematical reasoning, especially as Prof Bryan assists him with an excellent summary. For instance, the information that a machine loses longitudinal stability when rising, or gains lateral stability by the addition of two raised vertical fins, should be of great value and interest to him. The practical man, however, will in all probability raise one serious criticism. In order to arrive at his results the author is obliged to make a series of assumptions - that the air-resistances on the planes are linear functions of the small changes in linear and angular velocities; that in steady motion they are proportional to the square of the velocity; that they are normal to the planes; that they are proportional to sin ± when the angle of attack ± is small; that the pressure on an element of a narrow plane is independent of the motion of neighbouring elements, etc. Methods of approximation are also at times employed to simplify the algebra. The accumulated effect of small inaccuracies in each assumption may be considerable; and the results obtained in this way must be verified by experiment before the assumptions can be considered justified. The author does not work out the numerical result obtained by applying his formulae to any actual aeroplane; and even if he had pronounced any given machine stable or unstable, it might be difficult to verify the accuracy of his prediction; for inherent instability of the machine may be counteracted in practice by skill on the part of the aviator.

Probably the verification or modification of the assumption will be (as the author suggests) the task of the experimental physicist. At present there is a fear lest the mathematician may be brought to a standstill by the lack of data on which to work. This does not in the least diminish the debt which scientists owe to Prof Bryan and his colleague, Mr Harper, for providing them with the outlines of a theory which will doubtless be the basis of much future work, whatever improvements in detail may eventually prove necessary.

19.1. Review by: R M Milne.
The Mathematical Gazette 7 (114) (1914), 433.

One cannot doubt from the success of this work that it serves admirably as a text-book for the London Matriculation Examination. At the same time it must be confessed that in some respects it is much behind the times. In their preface the authors state that one of their reasons for treating Statics without Trigonometry is because "the solution of most illustrative problems involving angles depends on the properties of two particular triangles." These two particular triangles are the ones with which we are familiar in the forms of the 45º and 60º set squares. Why illustrative problems should involve these angles we are not told. Nothing has tended more to make the subject of Mechanics unreal to beginners than the artificial importance given to the angles 30º, 45º and 60º.

19.2. Review by: Anon.
Nature 94 (2343) (1914), 88.

Advantage has been taken of the publication of a new edition of this well-known class book to add a collection of simple experiments to illustrate the fundamental principles of mechanics. This addition will certainly increase the usefulness of the book.

20.1. Review by: N M Gibbins.
The Mathematical Gazette 25 (264) (1941), 127-128.

"The Tutorial Algebra, which for many years has been regarded as a standard work and much appreciated by students and teachers alike, was originally written as a textbook suitable for the Intermediate and Degree examinations of the University of London and other examinations of similar standard. In view of changes which have taken place in recent years in these various examinations and in teaching methods, it was decided that the work should be subjected to a complete revision. It has in consequence been replanned, rewritten and extended in accordance with modern conceptions and teaching methods, and present-day examination requirements. Because of the large amount of new matter incorporated, the present edition is issued in two volumes. Volume I covers the algebra required by the syllabuses of the Intermediate examinations of the University of London, with the exception of Intermediate Economics. Taken in conjunction with Volume II, the syllabus of the latter examination, as well as that of Higher School Certificate examinations, are treated in full."

The foregoing extract from the Preface indicates clearly the aim of this volume, and it seems almost presumptuous to make comments on a work which is in such a full flood of success. But the present reviewer (to his shame be it said) has never seen the original book, and is therefore able to consider this edition entirely without prejudice.

The new work opens with a chapter on Theory of Indices in which reliance is placed on the Principle of the Permanence of Equivalent Forms (with the last four words in heavy type). It seems clear that, among other purposes, this book is written for those who may have to prepare for their examinations by private study alone; and so it is a great pity to be so heavy handed with them. ...

Next comes a chapter on Surds and Imaginary Quantities, and it is satisfactory to note the firm statement for this stage: " it is assumed that such imaginary numbers obey the same fundamental laws of algebra as do real numbers." A chapter of miscellaneous propositions, including the consideration of simple partial fractions and the remainder theorem, is sandwiched between chapters on ratio, proportion and variation, and chapters on equations in one, two and three variables.

We then come to the very important chapters on the idea of a limit and fundamental ideas on convergence and divergence. In these the limit ideas of zero and infinity are very carefully explained, and there are excellent diagrams to illustrate convergent, oscillating and less half-hearted divergent series. These two chapters amply support the claim in the Preface regarding modern conceptions and teaching methods.

The volume continues with chapters on arithmetic and geometric progressions, and other series; theory of quadratic equations and expressions, and of rational functions, with many excellent diagrams and hints in the worked-out examples. After chapters on logarithms, permutations and combinations there is one on mathematical induction with which very special pains have been taken; for as the author says it is "a branch of the subject which offers difficulty to many students." The last two chapters are on the Binomial Theorem for positive integral index, and on Interest and Annuities. In the former it is gratifying to note the introduction of the compact notation.

This is a long book, and, with the exception noted at the beginning of this review, it is very carefully and clearly written. There are 1177 examples to be worked with answers to them at the end of the book.

21.1. Review by: N M Gibbins.
The Mathematical Gazette 26 (272) (1942), 237-238.

It is necessary to recall the words of Dr Heywood, said at the Annual General Meeting of the Mathematical Association on 6th January 1925: "The teacher is given a number of students and is expected to prepare them for degrees, which means for examinations. If the students do not pass their examinations they will be dissatisfied, and the College authorities will be dissatisfied with the teacher. The teacher therefore, to save his own skin becomes a coach. He is none the less a coach because he and his college may pretend he is not. The textbooks used may or may not be written as cram books. But it is certain that if a book is not suitable for examination purposes it will not be used." It is proposed to examine this book in the light of these remarks, and we may at once gain support for Dr Heywood's contention by quoting a short extract from the Preface: "Volume II covers algebra necessary for the B.A. and B.Sc. General examinations of the University of London, and would serve as an introduction to the Honours examinations, since the ground covered extends beyond the scope of the General Degree. It is also hoped that this volume will be of material assistance to students who are reading for University Scholarship examinations as well as to those studying mathematics for the Natural Science and Mathematical Tripos examinations of the University of Cambridge." This is a big book, well printed, with full answers and a good index ... Strong exception must be taken to the following sentences, to be found at the top of p. 274: "As another example consider amp z. This is a multiple-valued function, but is made single-valued by considering the principal value of the angle. This is equivalent to restricting ourselves to the consideration of a single branch." The italics are as printed in the book, and it is not easy to disentangle the confusion of thought contained in these sentences. Suffice it to say that they ought never to have been written. ... In spite of the foregoing criticisms it is not to be concluded that a student, unassisted by oral instruction, will go far wrong in using this book. It is possible too that the author's somewhat pontifical style (as on p. 274) may give him a feeling of security which may tide him over his examination, and which he will not realise as false until afterwards - perhaps not until long, long afterwards. And so we come back to Dr Heywood again.

22.1. Review by: A Barton.
The Mathematical Gazette 39 (330) (1955), 336-337.

This well-established work, which was "fully revised, rewritten and extended" in the fifth edition (reviewed in The Mathematical Gazette 25 (1941), 127), has again been revised throughout with the addition of new material and examples, and two new chapters have been added at the end of the book in place of the chapter on interest and annuities. The first of these deals with Important Series including the binomial expansion, logarithmic and exponential series, and the expansions of sinh x, cosh x, sin x and cos x. The second, on Probability, deals with the addition and multiplication of probabilities, and with the binomial and Poisson distributions. Both are written with the clarity and attention detail which are a feature of the whole book.

Volume I claims to cover the algebra needed for the G.C.E. Advanced Level and the London Intermediate examinations, and the claim seems to be amply justified. It is a pity, however, that in a book which appears to be partly intended for the private student and includes a wealth of worked examples suited to that purpose, no indication is given of what parts should be taken at first and what at second reading.

The revision might perhaps have been carried further: Figs. 28 and 30 depict cubic curves which could be cut in five points by a straight line, and Chapter 1 still pins its faith to the mystical Principle of the Permanence of Equivalent Forms. But such blemishes are rare and there are other places (e.g. in the chapter on Induction) where one is tempted to put up a notice: Other writers please copy.

23.1. Review by: R L Goodstein.
The Mathematical Gazette 41 (338) (1957), 314-316.

This volume is intended for candidates for London B.A. and B.Sc. General Examination and University Entrance Scholarship candidates. The first third of the book deals with analysis, convergence of series, continuity and the elementary functions. This is followed by chapters on complex numbers, finite differences, summation of series, determinants, matrices, elimination and the theory of equations. The book contains an immense number of worked examples, and at the price is very good value indeed for the student in question. The chapter on matrices, written by Dr A Mary Tropper, is exceptionally good and provides a really first rate, easy to read introduction. The chapter on finite differences is also very good and interesting. Throughout the volume there is evidence of great care having been taken with definitions, but the quality of the book is uneven, and there is some want of coordination between the chapters ...