1.1. From the Preface.

This treatise is intended to provide the English student with an intelligible outline of the Theory of Numbers which may serve as an introduction to the detailed study of the subject at first hand. No single work of reasonable size could possibly do justice to every part of the theory; and in the choice of material it is not easy to adopt any plan which is likely to approve itself to everyone. I have been guided principally by a desire to give a fairly complete account of the theories of congruences and of arithmetical forms, so far as they have been developed hitherto; to this I hope to be able to add a sketch of the different complex and ideal theories. Diophantine analysis proper, and questions of pure "tactic," have been omitted, except in so far as they have been subsidiary to the general scheme.

The range of this first volume is sufficiently indicated by the table of contents. It is hardly necessary to say that I have derived continual assistance from the works of Gauss and Dirichlet, and from H J S Smith's invaluable Report on the Theory of Numbers. I am also greatly indebted to Professor Dedekind for permission to make free use of his edition of Dirichlet's Vorlesungen über Zahlentheorie. So far as this present volume is concerned, the account of Dirichlet's researches has been taken primarily from his original memoirs; at the same time, I owe much to the study of the Vorlesungen, and I hope that I may eventually avail myself of Prof Dedekind's kindness more directly, by giving some account of his theory of ideals.

In the references at the ends of the chapters and elsewhere I have done my best to indicate fully the sources from which I have derived my information. No attempt has been made to give an exhaustive bibliography; even if I had been equal to the task of compiling it, the result would probably be merely embarrassing to the beginner, whose attention should be directed in the first place to the works of the great masters of the science.

Several friends, among whom I may mention Mr H F Baker, of St John's College, Cambridge, Mr R W Hogg, of Christ's Hospital, and Mr A G Greenhill, have kindly allowed me to send them proof-sheets; Mr J. Hammond has been good enough to revise my account of Professor Sylvester's researches on the distribution of primes; and I am indebted to my colleague, Mr A Gray, for advice and assistance in seeing the book through the press. To all of these my best thanks are due; and I may add that I shall be grateful for any criticisms or corrections that may be sent to me by any of my readers.

George Ballard Mathews.
University College of N Wales,
Bangor.

1.2. Review by: P A MacMahon.
Nature 47 (1893), 289-290.

The book under review is a great contrast in many ways to the "Théorie des Nombres" of M Edouard Lucas, the first volume of which has recently appeared under the aegis of Messrs Gauthier-Villars. The latter, reminding the reader much of the same author's "Récréations Mathématiques," exhales human interest from well-nigh every page. The former is on severe philosophical lines, and may be greeted as the first work of the kind in the English language. That this should be a fact is somewhat remarkable. When the late Prof. H. J. S. Smith died prematurely many years ago he left his fellow-countrymen a very valuable legacy. Fortunately he had been commissioned by the British Association to frame a report on the then present state of the Theory of Numbers, a subject with which he was pre-eminently familiar, and in which his own original researches had won for him a great and world-wide renown. The pages of the reports for the years 1864-66 inclusive yield as a consequence a delightful account of modern research in this recondite subject. It is, however, much more than a recital of victories achieved by many able men in many special fields. Prof Smith's fertile genius enabled him to marshal the leading facts of the theory, and to impress upon them his own personality in a manner that was scarcely within the reach of any other man. He contrived to impart a glamour to those abstract depths of the subject to which few mathematicians have sufficient faith and energy to penetrate. Since that day the scientific world has been yearly expecting his collected papers. There is no doubt that their appearance will greatly stimulate interest and research in Higher Arithmetic. The reports of the British Association are not sufficiently accessible. Doubtless the papers will soon emerge from the hands of those upon whom has devolved the responsibility of their production. In the meantime we welcome Part I. of the present work.

2.1. From the Preface.

This book has been written in view of the great and growing importance of the Bessel functions in almost every branch of mathematical physics; and its principal object is to supply in a convenient form so much of the theory of the functions as is necessary for their practical application, and to illustrate their use by a selection of physical problems, worked out in some detail.

Some readers may be inclined to think that the earlier chapters contain a needless amount of tedious analysis; but it must be remembered that the properties of the Bessel functions are not without interest of their own on purely mathematical grounds, and that they afford excellent illustrations of the more recent theory of differential equations, and of the theory of complex variable. And even from the purely physical point of view it is impossible to say that an analytical formula is useless for practical purposes; it may be so now, but experience has repeatedly shown that the most abstract analysis may unexpectedly prove to be of the highest importance in mathematical physics. As a matter of fact it will be found that little, if any, of the analytical theory included in the present work has failed to be of some use or other in the later chapters; and we are so far from thinking that anything superfluous has been inserted that we could almost wish that space would have allowed of a more extended treatment, especially in the chapters on the complex theory and on definite integrals.

With regard to that part of the book which deals with physical applications, our aim has been to avoid, on the one hand, waste of time and space in the discussion of trivialities, and, on the other, any pretension of writing an elaborate physical treatise. We have endeavoured to choose problems of real importance which naturally require the use of the Bessel functions, and to treat them in considerable detail, so as to bring out clearly the direct physical significance of the analysis employed. One result of this course has been that the chapter on diffraction is proportionately rather long; but we hope that this section may attract more general attention in this country to the valuable and interesting results contained in Lommel's memoirs, from which the substance of that chapter is mainly derived.

It is with much pleasure that we acknowledge the help and encouragement we have received while composing this treatise. We are indebted to Lord Kelvin and Professor J J Thomson for permission to make free use of their researches on fluid motion and electrical oscillations respectively; to Professor A Lodge for copies of the British Association tables from which our tables IV., V., VI., have been extracted; and to the Berlin Academy of Sciences and Dr Meissel for permission to reprint the tables of $J_{0}$ and $J_{1}$ which appeared in the Ahhandlungen for 1888. Dr Meissel has also very generously placed at our disposal the materials for Tables II. and III., the former in manuscript; and Professor J McMahon has very kindly communicated to us his formulae for the roots of $J_{n}(x) = 0$ and other transcendental equations. Our thanks are also especially due to Mr G A Gibson, M. A., for his care in reading the proof sheets. Finally we wish to acknowledge our sense of the accuracy with which the text has been set up in type by the workmen of the Cambridge University Press.

2.2. From the Introduction.

Bessel's functions, like so many others, first presented themselves in connexion with physical investigations; it may be well, therefore, before entering upon a discussion of their properties, to give a brief account of the three independent problems which led to their introduction into analysis.

The first of these is the problem of the small oscillations of a uniform heavy flexible chain, fixed at the upper end, and free at the lower, when it is slightly disturbed, in a vertical plane, from its position of stable equilibrium.
...
The oscillations of a uniform chain were considered by Daniel Bernoulli and Euler; the next appearance of a Bessel function is in Fourier's Théorie Analytique de la Chaleur in connexion with the motion of heat in a solid cylinder. It is supposed that a circular cylinder of infinite length is heated in such a way that the temperature at any point within it depends only upon the distance of that point from the axis of the cylinder. The cylinder is then placed in a medium which is kept at zero temperature; and it is required to find the distribution of temperature in the cylinder after a lapse of time t.
...
Bessel was originally led to the discovery of the functions which bear his name by the investigation of. problem connected with elliptic motion ...

2.3. Review by: Maxime Bôcher.
Bull. Amer. Math. Soc. 2 (8) (1896), 255-265.

The transcendental functions to which Bessel's name has been attached are not only of the highest importance in mathematical physics, second perhaps only to the trigonometric and exponential functions, they are also of great interest to the student of pure mathematics both from the formal side and from the point of view of the theory of functions. There has, however, up to this time been no connected treatment of these functions in the English language, with the exception of the utterly inadequate treatment contained in the last sixty-five pages of Todhunter's book, The Functions of Laplace, Lame and Bessel, published twenty years ago. The German monographs by C Neumann and Lommel make no attempt to cover more than small portions of the subject, and the same is true to an even greater extent of the sections devoted to Bessel's functions in Heine's Kugelfunctionen, Basset's Hydrodynamics, Rayleigh's Sound and elsewhere. Messrs. Gray and Mathews have therefore filled a real gap in mathematical literature.

The authors make it clear in their preface that their own interest is chiefly for the applications to mathematical physics. They have, however, none of the intolerance for and dislike of pure mathematics which is unfortunately too often manifested. ...
...
The book consists of fifteen chapters, followed by a few pages of miscellaneous matter and tables. Of the fifteen chapters, eight (87 pages) are devoted to the theory, while the remaining seven (135 pages) treat of the applications. This separation of theory and application is by no means complete, much analysis which might easily have been included in the earlier chapters being left until needed, while a few physical problems are introduced for the sake of illustration into the chapters on the theory of Bessel's functions. We think there can be no doubt that the authors have been wise in carrying this separation at least as far as they have done, as it conduces greatly to the clearness of the book, and makes it, what is very important, a convenient book of reference. The book might have been made even more useful in this respect than it is now by a somewhat extended collection of formulae, including in particular a table of definite integrals involving Bessel's functions, which could have been placed with the numerical tables at the end of the book.
...
While the treatment is too incomplete and even at times inaccurate to satisfy the pure mathematician, it is well adapted to the object the authors had in view of giving to the physicist a working knowledge of the more important properties of Bessel's functions. Even to the student of pure mathematics these chapters give, on account of the numerous sides of the subject on which they touch, an introduction to the theory of Bessel's functions which may fairly be considered satisfactory when supplemented by a certain amount of collateral reading. The presentation is clear and interesting and the references to the original memoirs useful.
...
Passing now to the chapters in which the applications to physical problems are considered, we are struck throughout by the very substantial knowledge of mathematical physics assumed on the part of the reader. This contrasts with the small amount of pure mathematics assumed in the earlier chapters. The simpler problems involving Bessel's functions, such for instance as are taken up in Byerly's text book on Fourier's Series, are usually omitted on the ground that the method of treatment is obvious. Here, as in the case of the earlier part of the book, the great diversity of subjects treated, borrowed from German as well as English writers, is an excellent feature.

3.1. Note.

The second edition was prepared by A Gray and T M MacRobert.

3.2. Review by: Anon.
The Mathematical Gazette 18 (231) (1934), 356-357.

A student who wishes to know how Bessel functions enter into the problems of applied mathematics and physics, and to appreciate how their properties are relevant to their use, cannot do better than consult this revised edition of a well-known treatise. Apart from changes in notation, the chapters which serve this purpose so admirably are almost untouched, and a short additional chapter of the same kind describes some results of Dougall's in the theory of elasticity.

The first part of the book, in which the purely mathematical theory of the functions is developed, has been recast so completely by Dr MacRobert that comparison with the earlier edition is almost impossible. The work is not ambitious in scope, and it has been done as well as limitations allowed. Sets of examples have been added.

The miscellaneous examples, and the appendix giving McMahon's formulae for zeros, remain. To another appendix Stokes' investigation of asymptotic formulae has been banished, to be replaced in the text by Gibson's. A few more tables are given, the British Association's undertaking, still unfulfilled after another ten years, being the reason why further tabulation is not attempted. The addition of a few items is hardly enough to bring the bibliography up to modern standards, from which qualitatively it falls short.

4.1. Note. The first edition of this book in 1880 was by Robert Forsyth Scott. It was revised by George Ballard Mathews and appears as a second edition with the both Scott and Mathews as authors.

4.2. Reviser's Preface.

The principal changes made in this edition are that some account has been given of infinite determinants, and of the elements of the theory of bilinear forms, together with the fundamental propositions about elementary divisors. I have intentionally refrained, as far as possible, from altering the character of the book, or increasing its size. The list of books and memoirs relating to determinants has been omitted, Dr Muir's bibliography being easily accessible; instead of this I have given a brief account of the earlier history of the subject. The new introductory chapter is intended for beginners, who are apt to feel discouraged if they first approach the theory in its most general form. For a similar reason the abbreviated notation employed in some chapters has not been used in those which are more elementary.

Besides original papers, I have consulted Pascal's excellent treatise in the Hoepli series, and Muth's Elemnentartheiler. The first volume of Kronecker's lectures on determinants appeared too late for me to consult it. I ought to say that for all the changes that have been made I am solely responsible, the revision having been left entirely in my hands by the author.

George Ballard Mathews.
May 1904.

4.3. Review by W H Metzler.
The Mathematical Gazette 3 (51) (1905), 182-183.

In this new edition an effort has been made to make the book more suitable for beginners. To this end an introductory chapter is given and difficult notations are omitted from the more elementary parts. Two new chapters are introduced, one on elementary divisors and the other on determinants of infinite order. The chapter on quadratic forms has been enlarged to include the elements of bilinear forms, and occasionally one finds a new theorem introduced into some of the other chapters. Theorems which might have found a place are Schwein's theorem, of which Laplace's expansion theorem is a particular case, the law of extensible minors, the law of complementaries, etc. By means of these many of the proofs might be shortened. Slight modifications occur here and there in the wording, due in part to the introduction of new matter and rearrangement of the old, and in part to clearer statements and simplified proofs. The book on the whole is comparatively little modified, but where changes have been made they are in general improvements. The theorem of article 17, chapter vi., attributed to Netto, was not new with him, and therefore should hardly bear his name. It is found in Reiss, which was published in 1867.

5.1. from the Preface.

This tract is intended to give an account of the theory of equations according to the ideas of Galois. The conspicuous merit of this method is that it analyses, so far as exact algebraical processes permit, the set of roots possessed by any given numerical equation. To appreciate it properly it is necessary to bear constantly in mind the difference between equalities in value and identities or equivalences in form; I hope that this has been made sufficiently clear in the text. The method of Abel has not been discussed, because it is neither so clear nor so precise as that of Galois, and the space thus gained has been filled up with examples and illustrations.

More than to any other treatise, I feel indebted to Professor H Weber's invaluable Algebra, where students who are interested in the arithmetical branch of the subject will find a discussion of various types of equations, which, for lack of space, I have been compelled to omit.

I am obliged to Mr Morris Owen, a student of the University College of North Wales, for helping me by verifying some long calculations which had to be made in connexion with Art. 52.

G. B. M.
BANGOR,
August, 1907.

6.1. From the Preface.

The two main objects of this work have been to develop the principles of projective geometry without making use of the theory of distance, and to give a satisfactory discussion of complex elements of space, up to three dimensions. No attempt has been made to start with the smallest sufficient number of indefinable terms and primitive propositions; there are ten primitive propositions, involving quite a large number of indefinables (besides point, plane, line, the indefinable entities ), and these propositions have been chosen because they seem to be obvious to the ordinary intuition, and fall into five correlative pairs. Thus the principle of duality appears at the outset, and is emphasised more and more as we proceed.

The proof of the fundamental theorem (after that of Lüroth and Zeuthen) involves the notion of a converging sequence of points, as well as that of order: so it is by no means the most rigorous that could be given, but I believe it is as near to a rigorous proof as an ordinary student is likely to appreciate. The beginner is recommended to assume the theorem of Art. 51, and treat the rest of Chap. VII as something to ruminate upon.

In dealing with complex elements I have constantly consulted von Staudt and Lüroth (Math. Ann. viii.); I hope it will be found that my treatment is in some respects more elementary than that of either of them. Of course, the facts are wholly due to von Staudt: it is only a question of how they should be presented.
It is very rash to claim any theorems in projective geometry as absolutely new; but I may venture to say that the theory of projective metrics in the last chapter but one is at any rate original, and seems to me to be one which can be logically defended on the base of a strictly limited set of projective axioms. It also affords a very interesting illustration of the principle of duality.

I am indebted to various friends for helpful criticisms and suggestions; to Dr F S Macaulay, who very carefully read the earlier part both in MS. and in print, and saved me from making some serious mistakes; to Prof F S Carey, who read some of the proofs, and made valuable suggestions about the scope of the book; to my colleague, Mr W E H Berwick, who has read nearly all the proofs and much of the MS. with great care, and has contributed a number of examples. With regard to the examples, those which Mr Berwick and I have not made up have been taken from various sources which seemed to me by this time common property, and I have seldom attempted to name the original authors. Messrs Veblen and Young and their publishers have very kindly allowed me to take examples from their Projective Geometry; these examples have been marked V. Y. In the same way, such references as G. 17 or B. 27 mean $17 of v. Staudt's Geometrieder Lage, or $27 of the same author's Beiträge respectively.

In preparing the diagrams for reproduction I have received most valuable help from my friend and former pupil, Mr W A Jones, B. A. (Wales), who ungrudgingly spent much time and trouble in constructing diagrams from my sketches and instructions. Every one who has tried knows that it is not an easy thing to draw such figures as these, in the space available, without either making them cramped or leaving them incomplete.

Finally, I should like to say that my first acquaintance with the true theory of projective geometry began when I attended Professor Henrici's lectures at University College in the session 1878-9. Whatever merits this book may possess are very greatly due to the interest in the subject which I then acquired; and I take this opportunity, as an old pupil, of expressing my gratitude to the teacher who first made me realise that mathematics is an inductive science, and not a set of rules and formulae.

G. B. M.

6.2. Review by: C.
Science Progress in the Twentieth Century (1906-1916) 9 (36) (1915), 696-697.

The publication of this book commemorates the centenary of the conception of the subject. For although Poncelet did not publish his Propriétés Projectives before 1822, he tells us that the ideas of the subject were conceived in his brain in a prison in Russia after the retreat from Moscow. In that most un- likely place he demonstrated the greatness of the human spirit by achieving an intellectual triumph more lasting in its results than the political success and failure of the millions who were engaged in that Russian campaign of 1813-4. In the hundred years which have succeeded, Projective Geometry has been elaborated by German, French, English, and Italian mathematicians, until it is now perhaps the most perfect monument in Pure Logic which has been raised by the genius of man.

It is one of the few omissions in Mr Mathews' book that he does not briefly unfold the story of the development of Projective Geometry, placing before us the stages by which the subject was first freed from the metrical conditions under which Poncelet conceived it, and then describing the parts which von Staudt, Laguerre, Cayley, and others played in bringing within its sphere the wider realms which it has conquered in the last sixty years.

In the English language we have in Projective Geometry a translation of one volume out of three of Reye's Geometrie der Lage, the first volume of a brilliant book by Messrs Veblen and Young, the second volume of which is awaited with eager expectancy, and two admirable tracts by Dr Whitehead upon the axioms. But this is the first book in the language which gives a serious account of the real scope of Projective Geometry.

The book is not one of those which follow the syllabus of examinations; examinations will, if English mathematicians wake up, follow it. But whether they follow it or not, no student will in future have to be satisfied with the jumble of disconnected theorems and haphazard results which have hitherto masqueraded as Projective Geometry in English text-books.

Mr Mathews has not started with the first verse of the first chapter of the first book of geometry as taught by the abstract logical school, and in this he is wise. He does not, however, shirk the genuine difficulties of the subject, as those who follow him will find when they arrive at Chap. VII. and tackle the Fundamental Theorem of Projectivity. In the elementary portions of the subject, which are covered by the first eighteen chapters, the author shows a wise discretion in avoiding the duplicity or quadriplicity of which the various theorems are susceptible. Students of the subject should, however, follow his precept and not his example, and state these theorems completely, as without much practice they will find it impossible to reach the true attitude of mind which, in geometry, is of more importance than knowledge of facts. Mr Mathews offers us an account, original often and brilliant always, of von Staude
s theory of complex elements; then he discusses the theory of casts and establishes homogeneous co-ordinates upon a projective basis. From this point the author gives a little too much play to his analysis. ...

In conclusion Mr Mathews is to be congratulated upon having written a book which cannot fail to have a wide and lasting influence upon the progress of geometrical knowledge in England. If the book receives the distinction of translation into foreign tongues, England will pay back a portion of the heavy debt which she owes to continental mathematicians who have done so much work in the field of Projective Geometry.

6.3. Review by: Anon.
The Mathematics Teacher 7 (4) (1915), 176.

In order to develop the principles of projective geometry without use of the theory of distance the author follows the lead of von Staudt, Reye, and other more recent authors, leaving all reference to measurement till the latter part of the book. Without attempting a rigorous development of the elementary principles he states ten theorems and quite a number of other principles which the reader is to accept as true. Upon these he bases the thirty-two chapters, each covering briefly some phase of the subject. The principle of duality, both in the plane and in space, is introduced very early and widely used throughout the work. About the middle of the book he introduces the study of complex elements by means of elliptic involutions. After this comes quite an extended chapter on the theory of casts. Metrical and quasi-metrical properties follow. After chapters on projectivities in space, quadric surfaces, null-systems, skew involutions, line geometry, etc., he concludes with a chapter on projective problems, an extended set of exercises, and an index.