1.1 From the Preface.

The distinctive feature of this work is that the functions discussed are primarily not functions of a single variable direction but functions of several independent directions. Functions of a single direction emerge when the directions originally independent become related, and a large number of elementary theorems of differential geometry express in different terms a few properties of a few simple functions since one of the objects of the essay is to emphasise the coordinating power of the theory, the presence of many results with which every reader will be thoroughly familiar calls for no apology.

In the applications to the geometry of a single surface two functions thought to be new are described. The first, studied in Section 4, depends on two tangential directions, reduces to normal curvature when these directions coincide, and is called here bilinear curvature. I became acquainted with this function in 1911 and used it in lectures early in 1914. The second, the subject of Section 6, depends on three directions, and reduces to the cubic function associated with the name of Laguerre; the function is symmetrical, and because the equations of Codazzi can be read as asserting its symmetry I have called the general function the Codazzi function.

The theory of multilinear functions does not merely coordinate. It affords simple proofs of the relations between the cubic functions of Laguerre and Darboux and of formulae for the twist of a family of surfaces, and it leads naturally to expressions for the rates of change of the two principal curvatures of a variable member of a family of surfaces at the current point of an orthogonal trajectory of the family, expressions that are interesting because their existence was deduced by Forsyth in 1903 from an enumeration of invariants.

E H N
June, 1920.

NOTE For the sake of brevity, the space considered is real, but the restriction operates only to the same extent as in other branches of differential geometry. If it is removed, the intrinsic distinction between the positive square root and the negative square root of a given uniform function has to be replaced by a more artificial distinction based on a dissection that is to some extent arbitrary. And there is always a possibility that results need modification if isotropic lines or planes are involved as a rule, nul vectors are admissible as arguments but nul directions are not.

1.2. Review by: F P W.
Science Progress in the Twentieth Century (1919-1933) 17 (68) (1923), 657-658.

An important problem in elementary differential geometry is to associate the curvatures and torsions of curves on a surface with the form of the surface itself; it is a problem which may be investigated by means of "moving axes." This book extends and develops this method out of all recognition. The functions considered are primarily not functions of a single variable direction, but functions of several independent directions. By relating these originally independent directions we obtain functions of a single direction; and a large number of elementary theorems of differential geometry become co-ordinated and expressed by means of properties of a few simple functions. Of these we may mention the bilinear curvature, which depends on two tangential directions and reduces to normal curvature when these directions coincide; and the Codazzi function, depending on three directions and reducing to the cubic function of Laguerre when they coincide; it is so called because the equations of Codazzi can be interpreted as asserting that it is a symmetrical function. The theory is used to prove the relations between the cubic functions of Laguerre and Darboux, and to obtain formulae for the twist of a family of surfaces and for the rates of change of the two principal curvatures of a variable member of a family of surfaces along an orthogonal trajectory.

Prof Neville has a faculty for inventing new names and symbols as well as new methods; his book, in consequence, is terribly difficult to read. ...

2.1. From the Introduction.

To the general reader, the name of the fourth dimension brings reminiscences of Flatland and The Time Machine. On hearing that to the mathematician the extension from three dimensions to four or five is trivial, he thinks he is being told that a study of mathematics, if reasonably intense, creates physical faculties or powers of visualisation with which the uninitiated are not endowed. Learning that Minkowski and Einstein combine space and time into a single continuum, he tries to believe in the existence of a state of mind in which the sensations of space and time are confused, and naturally he fails.

The position of students of mathematical physics, and of all but a fortunate few of the students of pure mathematics, is little better. Accustomed to regard a Cartesian frame of axes as a scaffolding erected in the real space around them, they can attach no meaning to a fourth coordinate, but having used complex electromotive forces with success in the theory of alternating currents, and having treated a symbol of differentiation as a detachable algebraic variable even to the extent of resolving operators into partial fractions for the solution of differential equations, these students are prepared to give pragmatical sanction to the most fantastic language.

The pure mathematician makes no attempt to imagine a space of four dimensions; he lays no claim to visualising a world that is inconceivable to other men. Only he finds that certain notions in algebra are discussed most readily in terms adopted from geometry and given a meaning entirely algebraic, and since it is to the mathematician alone that algebraic problems are of concern in themselves, fear lest the man in the street should mistake the very subject of a mathematical conversation he might overhear has not prevented the mathematician from using the vocabulary he finds best suited to his own needs. Now it has happened that the talk of a few mathematicians has suddenly become of universal and absorbing interest, and a dictionary explaining the meanings they are in the habit of giving to some familiar words is required.

It is this dictionary that I have tried to write, and I have written it in the simplest terms I could find, in the hope that it will prove intelligible to anyone familiar with elementary trigonometry and with the solution of simultaneous linear equations in algebra; for this reason, I have not treated the point as indefinable, I have supposed the numbers used always to be real, and I have avoided Frege-Russell definitions. The reader's first feeling will be one of disillusion. Are Einstein and Eddington talking not about a new heaven and a new earth but about linear algebraic equations! To discuss the question is beyond the province of a lexicographer.

Perhaps even the mathematical student, if he can overcome a reasonable irritation at the restrictions, from his point of view arbitrary, to four dimensions and to real numbers, and at the absence of certain obvious forms of abbreviation, may derive some help from the pamphlet. The possibility of constructing an abstract 'space' is always assumed, but the details of the construction, even for two or three dimensions only, are either taken for granted or disguised as theorems on matrices or on linear equations. The idea of direction and the measurement of angles in a constructed plane demand careful consideration. The nature of pure rotation in four dimensions is by no means obvious; on the contrary, rotation is the most difficult of the elementary notions used in the theory of relativity, and with an account of rotation our formal work comes to a natural end.

2.2. Review by: Ebenezer Cunningham.
The Mathematical Gazette 11 (159) (1922), 127.

Professor Neville has fastened upon a fact which should have been obvious but has been seriously overlooked. While many have been talking at length upon Non-Euclidean geometry in four-dimensions, and particularly upon the very complex notions of the curvature of three-dimensional manifolds in four-dimensional space, or of four-dimensional manifolds in five-dimensional space, very few have had any preliminary preparation in the shape of a detailed consideration of the most elementary propositions in four-dimensional Euclidean geometry. The ordinary procedure in everyday geometry is to begin with the study of straight lines and planes, then to proceed to spheres, and 80 gradually to the differential geometry of surfaces in general. This may or may not turn out to be the proper order in which the systematic study of hyperspace should be undertaken; but at any rate it is worth trying. The first chapters of the text-book are here presented to us.

What are the meanings to be attached to the words "line," "plane," "space," when we come into the world of four-dimensions? This is the question which is answered in detail in the first half of this useful little book. It is shown how the geometry of four-dimensions is but algebra with the addition of some terminology adopted by analogy with three-dimensional geometry. Unfortunately, the dictionary which Prof Neville tells us he is writing, is faulty at one point. A line is 1-dimensional, a plane is 2-dimensional a space is 3-dimensional, but there is no distinctive name for the 4-dimensional domain within which these others may be described; and the adoption of the word "space" for a 3-dimensional variety satisfying one linear equation, brings us into confusion with the term "hyperplane," which has often been used in this sense.

These are small matters, however, and the student who will take the trouble to read these pages attentively will be well rewarded by finding his power of thinking in four-dimensions strengthened and clarified.

The later part of the book dwells particularly on the generalisation of the idea of "rotation." In ordinary space the displacement of a rigid body with one point fixed from anyone position to any other leaves a certain axis in the body fixed in position. Prof Neville proves in detail that in four-dimensions the displacement of any system of points, the intervals between each pair being unaltered, and one point being fixed, leaves all the points of a certain plane (2-dimensional manifold) undisturbed. He gives us, in fact, a rational and complete description of the idea which, noticed by Minkowski, was the germ-thought of all the analytic development of the theory of relativity. In 1908 Minkowski pointed out that the Lorentz-Einstein transformation was, formally, precisely a rotation in four-dimensions about a fixed plane. This led him to the unification of space and time into what is now called a space-time. It is greatly to Prof Neville's credit that he has perceived the need for an elementary exposition of the matter, and that he has had the courage and humility to set his hand to it. We are very grateful to him for doing so.

3.1. From the Preface.

The vitality of the mathematical form of speech is of a peculiar kind. Words grow, not by continuous and subtle variations in meanings already possessed, but by the acquisition of meanings entirely new; meanings that are outgrown are neither dead nor discarded, but survive unchanged to be used when they are appropriate, and it often happens that an assertion can be interpreted to give a number of different theorems that are all true.

The first half of the present work is an account of the principles underlying the use of Cartesian axes and vector frames in ordinary space. The second half describes ideal complex Euclidean space of three dimensions, that is, three-dimensional 'space' where 'coordinates' are complex numbers and 'parallel lines' do meet, and develops a system of definitions in consequence of which the geometry of this space has the same vocabulary as elementary geometry, and enunciations and proofs of propositions in elementary geometry remain as far as possible significant and valid.

The arrangement of the material has been dictated by convenience for the structure as a whole, without regard to the logical relation between the initial assumptions. For this reason I have not called the volume a treatise on the foundations or on the principles of analytical geometry. Either description would have suggested a discussion of axioms, and the field to which this work belongs is not part of the region that extends for English readers from Russell's earliest work to Baker's latest; as far as I call judge yet, this work has no ground in common with the Principles of Geometry, and my debt to Russell, great as it is, is for the methods of the Principles of Mathematics, not for the philosophy of the Foundations of Geometry. The title 'Principles of Analytical Geometry' would have been no less misleading if the word had been associated with the formal logic of Principia Mathematica, or with the wide survey of general methods to which Darboux gave that very name.

The discussion falls into five parts. There is a preparatory book, that deals first with such fundamental matters as the avoidance of ambiguity in the measurement of angles and the meaning of the sign attached to the volume of a tetrahedron, and afterwards with the simplest kind of projection.

The second book is an introduction to vector analysis, and was written only after many efforts to utilise one or other of the current textbooks. While ready to take for granted an acquaintance with the formal laws, I wished to protest both against the confusing notion that a vector is a piece of a line but is in some mysterious way indistinguishable from equal pieces of parallel lines, and against the assumption that lengths, curvatures, speeds, and the tensors of vectors generally, are intrinsically either signless or positive, a mischievous supposition which not only would encumber the geometry and mechanics of real space beyond endurance if it was not in practice ignored whenever it becomes inconvenient, but also is an insurmountable barrier to the extension of vector analysis from real to complex space. It seemed necessary to show that the formal development of vector analysis is not complicated by the view of the vector as duplex, and ultimately a compound of quotation and qualification gave place to a straightforward summary of the subject.

The beauty of the calculus of quaternions does not alter the fact that the geometer deals with the actual cosine of an angle and the actual square of a distance, not with the negatives of these numbers. I have therefore taken the line of Grassmann and Gibbs, and regarded as fundamental the negative of Hamilton's scalar product, and this I have ventured to call the projected product.

Some novelty will I think be found in the treatment of rotors. Much use has been made of the conception of the momental product of two rotors or of two sets of rotors, and the consideration of sets of rotors begin before couples have been mentioned, These are details of economy, not matters of principle, and everywhere I have refrained from lengthening the work by attack or defence.

The third book applies vector analysis to obtain formulae for use with Cartesian axes and with vector frames. The Cartesian frame is not assumed to be trirectangular, nor are the Hamiltonian unit vectors i, j, k mentioned. The claim that oblique frames are not more cumbersome than trirectangular in theoretical work is less extravagant than might be supposed. Vector frames are discussed partly because the discussion introduces in its simplest form a quantity of analysis that is fundamental in differential geometry, and partly because in complex space nul vectors are invaluable as vectors of reference but nul lines cannot serve as axes of a Cartesian frame. Problems that involve the locating of lines by means of frames of reference explain the range of the second book by illustrating the utility for analytical geometry of the idea of the vector product and of the elements of the theory of rotors.

3.2. Review by: Thomas Percy Nunn.
The Mathematical Gazette 12 (168) (1924), 27-30.

I could wish that the task of reviewing this important book had fallen into more competent hands; but I have had no difficulty in seeing that it is extraordinarily good, and feel that if my qualifications as a critic were higher its merits would appear to me still more conspicuous. It possesses, I think, all the qualities needed to give distinction to a work in this genre. An immense amount of laborious thinking must have gone to its making, yet it is never heavy or dull; it is based upon wide learning, but the learning is felt in the fibre, not exhibited for admiration; the diction is economical, lucid and precise; and the mathematical craftsmanship is of the highest order. Above all, the exactness in detail which is the special characteristic and in truth the special aim of the work expresses not the niceness of a pedant who worries too much de minimis, but the intellectual passion of a man who, like the ideal metaphysician, is obstinately determined to think clearly about everything he handles. Thus the work has a warmth and movement which must make its reading a delightful tonic to any one who has a sense for the aesthetic value as well as the ultimate practical importance of clear and precise notions about mathematical entities and their relations. It is not irrelevant to add that the text is well arranged and beautifully printed, and that the student's progress is facilitated by a most careful system of marginal numeration of the sections and smaller elements of the argument, and by full references from one part of the argument to another. The work, as a whole, falls into two main parts.
...
Book I deals with familiar and very elementary matters - directions and angles, measurement by steps, and parallel projection. But although the topics are simple, the precision and subtlety of the treatment prepare the reader's mind usefully for what is to be the prevalent temper of the work.
...
The second book, which contains the theory of vectors and rotors, is a most interesting and satisfying piece of work, and I think that Prof Neville must have enjoyed peculiar pleasure in writing it. He begins with a novelty at which I have already glanced. According to the Hamiltonian definition, a vector consists of a signless real number associated with a single direction, but mathematicians have often found it convenient to depart from the classical conception and to make use of the idea of a vector with a negative amount. In order to legitimise such practices Prof Neville defines a vector at the outset as consisting of two real numbers, one the opposite of the other, associated respectively with a direction and its reverse. This is the concept which he carries through his excellently lucid and illuminating discussion of the nature and fundamental properties of vectors, rotors and couples.
...
... it is a masterly treatise, full of instruction and inspiration for a serious student, abounding in technical excellencies and advances which an expert may study with interest and profit, and altogether a notable contribution to British mathematical scholarship.

3.3. Review by: Anon.
Nature 112 (1923), 582-583.

Were a Greek from the Academy of Plato to visit England, it would surely please him to find a title he could read without using a dictionary. Should he persist in acquainting himself with the first chapters, he would be delighted with the precision of language and thought and with the homeliness of the contents; indeed, it may be said that the number of readers of this beautifully executed work will be a fair measure of the Greek spirit among our geometers of the present day. To barbarians it will seem to cut right across the course of modern geometry with an independence which shows itself in nomenclature and notation, in absence of references, and most of all in the limitations which the author has placed upon himself in the selection of his material. This is partly accounted for by the fact that Prof Neville is avowedly a disciple of Mr. Russell, whose well-known aphorisms are scattered over the book, and it is scarcely to be expected that a subject written in the form which modern logic demands should develop itself along lines which appear fundamental in discovery.

4.1. From the Preface.

At one time the study of elliptic functions began with the inversion of Legendre's integral. Every young mathematician was familiar with sn $u$, cn $u$, and dn $u$, and algebraic identities between these functions figured in every examination. But a growing realisation that the inversion of a complex integral raises issues which are not all elementary brought about a change. Today, many a good teacher says nothing of the Jacobian functions until he can utilise theta functions, and many a good student learns nothing of them at all. ...

This book is an attempt to restore the Jacobian functions to the elementary curriculum by exhibiting them as functions constructed on a lattice. In the course of the general theory of doubly periodic functions, we find that the lowest order possible for such a function is the second, and that therefore the simplest functions have either one double pole or two simple poles in a primitive parallelogram. The investigation of the first possibility is the invariable method of introducing the Weierstrassian function ℘(z). It is seldom - the first edition of Modern Analysis was an honourable exception - that the investigation of the alternative is recognised as the natural sequel. This is our starting point. We associate with an arbitrary Weierstrassian function a symmetrical group of functions of the second kind, and this group becomes a Jacobian system by an appropriate specialisation of one of the parameters fundamental in the theory. So found, the Jacobian functions are known in advance to be doubly periodic, no parameters are restricted to be real, and simple functional proofs of addition theorems and of the transformations of Jacobi and Landen replace the algebraical proofs demanded by the inverted integral.

For a moment we are tempted to think that the problem of inverting an integral need not be faced. The classical functions have come easily into analysis, they display a multitude of fascinating properties, and their relations to their derivatives imply that they can be used for the evaluation of integrals of the forms with which they are traditionally associated. Let them be studied, and they will be available when wanted. But will they? ...
...
We dare not say that we understand the relation between the function and the integral unless we see how the double periodicity of the function is implicit in the integral form of the relationship, and in the discovery of double periodicity from this end the origin of the constants is not relevant. Since also definition by inversion of an integral is equivalent to definition by a simple form of differential equation and is not in itself a suspicious process, a mystery remains for the student unless we put a finger for him on the ultimate difficulty. In point of fact, the more precisely the problem of inversion is analysed, the narrower the crucial gap becomes and the less formidable the task of bridging it appears.

The design of this treatise will now be intelligible. There are three divisions of the subject, first the direct theory of functions with simple poles derived from a Weierstrassian function whose periods are arbitrary, then the theory of the inverted integral and the solution of the problem of inversion, and lastly the fertile theory of the classical system. To the writer the order of exposition is almost inevitable, but the reader impatient to make the acquaintance of Jacobi's functions can pass to Chapter X from Chapter IV or even from Chapter III, and he can return at any time to read Chapter VI, on the connexion between integration and periodicity, as an independent chapter and not necessarily as a stage in the inversion argument. ...

4.2. Review by: Philip Franklin
Science, New Series 101 (2624) (1945), 378.

In this book the author has reworked his lectures into a careful and logical presentation of the subject, elliptic functions regarded as a branch of the theory of functions of a complex variable. The reader is assumed to be familiar with the elements of the theory of functions, but not necessarily with doubly periodic functions, the general theory of which is developed from the Weierstrass point of view in a fifty-page introduction. With this as a basis, the author constructs functions on lattices with two simple poles in each cell, develops their general theory, and later specialises the discussion to Jacobi's functions sn u, cn u, dn u, and the nine related functions introduced by Glaisher, or rather the generalisations of these with complex parameters. This development avoids the artificiality, if also the brevity, of the treatments based on theta functions which here are introduced near the end of the book. On the other hand, most brief treatments from the lattice point of view which make any reference to elliptic integrals and the inversion problem for complex parameters are logically incomplete, and the overcoming of this difficulty is the main virtue of the treatise under review. The author has introduced several convenient bits of notation which make for an efficient wholesale derivation of formulas, and has usually indicated the alternative classical notation, and also pointed out that for deriving a particular result first principles are usually more convenient than the generalised notation.
...
While the treatment has made contact with applications, these are not discussed as fully as the basic theory. ... The author gives some sporadic references of a historical nature, but has not arranged these so as to he of much help to a reader who needs to be oriented in the literature of the subject. There is no index.

Thus this book is not suited to the reader who merely wishes to locate some particular result for a specific application. However, it will interest the pure mathematician as a systematic discussion from a unified point of view. ...

4.3. Review by: Miguel Antonio Basoco.
Mathematical Reviews MR0012658 (7,53g).

In a lecture in 1943, the author gave what amounts substantially to a preview of this volume. He remarked, "thirty-five years ago my contemporaries would have included in any such list [a list of mathematical topics which would have seemed as important to all future students as they did to the author's generation] the elements of the theory of the Jacobian elliptic functions; it never crossed our minds that the time would come when the ordinary mathematical undergraduate was to know less of these functions than we did.'' The author is of the opinion that the principal reason for this neglect of the subject can be attributed to "the unnatural way in which the theory is stifled by the multitude of special formulae to which its applications give rise." That this criticism is valid to a certain extent no one will deny. Yet it is doubtful that the lack of interest in the subject can be in a large measure ascribed to inadequate presentation. Mathematics, in common with other things of interest to the human race, is subject to fashions and styles; even the rigour in which it prides itself is relative and subject to change. A rapid examination of the mathematical abstracting and review journals indicates clearly that, over a period of fifty years, the theory of elliptic functions has become less and less fashionable. This is partly due to the fact that its theory was fairly thoroughly explored at an early stage and partly because other newer branches of analysis have forced it into the background, this in spite of its important contributions to geometry and to the theory of numbers. The volume under review is an attempt to restore the Jacobian functions to the university curriculum by introducing them in a manner which is appealing by reason of its directness, simplicity and novelty.

4.4. Review by: Wladimir Seidel.
Bull. Amer. Math. Soc. 52 (7) (1046), 604-607.

In the development of the theory of elliptic functions it is shown at an early stage that, as far as singularities are concerned, the simplest elliptic functions other than constants are those of order 2. This naturally leads to the classification of elliptic functions of order 2 into those with one double pole (of zero residue) and those with two simple poles in the parallelogram of periods. The Weierstrassian theory of doubly periodic functions, the theory which is still frequently included in a first course of complex variables, starts with a function ℘(z) of the first kind which has the double pole at the origin. By using Liouville's general theorems, elliptic functions with singularities, arbitrary within the permissible limits, are constructed.

The Jacobian theory starts out basically with functions of the second kind, and it is Professor Neville's merit to lay particular stress on this purely function-theoretic classification. ...
...
In the introduction the author states that his aim in writing the book was the restoration of the study of Jacobian functions to the elementary curriculum. The reviewer feels sceptical, for various reasons, of the possibility of such a reform in this country. Aside from this issue, however, the book is an excellent one and constitutes a valuable and important addition to the large and distinguished literature on the subject. The style of the author is vivid and picturesque, and the relatively few misprints cause no real difficulty in reading. The reviewer can heartily recommend it to any one with some background in complex variables who wishes to learn the Jacobian theory.

5. The Farey Series of Order 1025, Displaying Solutions of the Diophantine Equation $bx - ay = 1$ (1950), by Eric Harold Neville.

The series which occupies this volume was accepted for publication by the British Association Mathematical Tables Committee in 1947. In the following year the responsibility which the British Association has exercised since 1871 through that Committee was transferred to the Royal Society, which set up a Mathematical Tables Committee of its own to take over the work. The volume has therefore become the first volume of tables to be published for the Royal Society, but if the serial title is new, the succession is unbroken; the style is that of the British Association Tables, and the standard of production is maintained.

Farey series of low orders can be read in the columns of tables of decimal quotients or gear ratios, but a Farey series written in full for its own sake, or perhaps one should say for the sake of the uses to be made of it purely as an arrangement of fractions, is without precedent. If the spectacular applications of these series are in the deeper parts of the analytical theory of numbers, their elementary properties and their simplest uses can be  enjoyed by anyone who takes a delight in working with the common integers. The publication of a Farey series under the Cunningham bequest for the production of new tables in the theory of numbers is appropriate on these general grounds. And there is more specific ground. In 1927 Cunningham thought it worth while to include in a volume entitled Quadratic and Linear Tables, which is concerned principally with quadratic partitions and quadratic congruences, tables giving explicit solutions of the linear equation $bx - ay = 1$, and the equivalent compact tables giving solutions of the congruences $bx \equiv 1(\mod a)$ for prime values of $a$, in the one case with a limit of 100 on $a$ and $b$ in the other with a limit of 97 on $a$ and of $a - 1$ on $b$; there could be no stronger evidence that in applying part of the fund with which he entrusted the British Association to the publication of a table which incidentally raises the Diophantine limit to 1025, the Royal Society is perpetuating Cunningham's memory precisely as he would have wished.

5.2. From the Introduction.

To every mathematician of our time, Farey series recall the name of Srinivasa Ramanujan. For me they revive memories of the man himself, who became my friend in the golden twilight of our lost civility, the earlier half of the year 1914. Ramanujan's place in history was determined not when his first letter to Hardy showered fireworks in the Cambridge sky in 1913, for he resisted all immediate efforts to attract him to England, but when in Madras a year later he put into my hands the now famous Notebook and suggested that I should take it and examine it at my leisure. Only in a slight degree was the compliment personal to me; to Ramanujan then Hardy was a name on paper, and Walker and Littlehailes were parts of the governing machine, but I was a human being. Nevertheless, if Hardy could long remember with satisfaction that he could recognise at once what a treasure he had found, I too can be proud, for if I had failed to win the confidence of Ramanujan and his friends, Ramanujan would not have followed me to England. Nor could my failure have been remedied; before the autumn, that is, before any other visitor from Cambridge could have made contact with him, war was raging, and five years were to elapse before communication between India and England was again easy.

Ramanujan's first three months in England were spent in my house in Cambridge, and the friendship which developed ended only with his death.

5.3. Review by: Derrick Henry Lehmer.
Mathematical reviews MR0041934 (13,24e).

This volume is the first of a new series of British tables sponsored by the Royal Society. The main table gives the numerators and denominators of all reduced rational numbers

          $\Large\frac{P_n}{Q_n}$ with $0 < P_{n} < Q_{n} ≤ 1025$

arranged in increasing order. ... The introduction illustrates the application of the table to such problems as the rational approximation to irrationals and the solution of linear Diophantine equations.

5.4. Review by: Richard Rado.
The Mathematical Gazette 36 (315) (1952), 60-61.

In 1816 a geologist, J Farey, published, without any hint of a proof, a certain conjecture concerning rational numbers, not knowing that already in 1802 C Haros had actually published a proof of his conjecture. In all probability Farey was quite unable to prove his proposition, and yet his name was immortalized in the Farey Series, one of the life lines of the additive theory of numbers.

The Farey series of order $n$, denoted by $F_{n}$, is defined as the succession, in increasing order of magnitude, of all rational numbers. in the range $0 ≤ r ≤ 1$ whose denominators do not exceed $n$. Thus $F_{6}$ is the series

          $0, \large\frac{1}{6}\normalsize , \large\frac{1}{5}\normalsize , \large\frac{1}{4}\normalsize , \large\frac{1}{3}\normalsize , \large\frac{2}{5}\normalsize , \large\frac{1}{2}\normalsize , \large\frac{3}{5}\normalsize , \large\frac{2}{3}\normalsize , \large\frac{3}{4}\normalsize , \large\frac{4}{5}\normalsize , \large\frac{5}{6}\normalsize , 1$.

Farey's conjecture, or rather Haros' theorem, states that if $\Large\frac{a}{b}\normalsize, \Large\frac{a'}{b'}\normalsize, \Large\frac{a''}{b''}\normalsize$ are three successive members of $F_{n}$, in their reduced form, then

          $\Large\frac{a'}{b'}\normalsize = \Large\frac{a+a''}{b+b''}$.

The series $F_{n}$ possesses a certain importance for engineers as giving all gear ratios which are possible with two cog wheels of at most $n$ teeth each. Its interest for mathematicians derives from the use Hardy and Ramanujan made, in 1917, of $F_{n}$ in their famous method for dealing with problems in the additive theory of numbers. It must, however, be stated that in this connection not the arithmetical details of individual series $F_{n}$ are important, but only a certain asymptotic overall regularity of $F_{n}$ as $n$ becomes large.

In the volume under review Professor Neville has assembled the 319765 terms of $F_{1025}$. Although two members of the series $F_{n}$ which are the same number of terms away from the centre term $\large\frac{1}{2}\normalsize$ obviously add up to unity, the author has done well to print the whole series, thus reducing the labour of prospective users of the tables. By an ingenious arrangement of the entries this has only meant an increase of the length in the ratio 2 : 3 instead of the expected 1 : 2.

5.5. Review by: Paul Trevier Bateman.
Bull. Amer. Math. Soc. 57 (4) (1951), 325-326.

The Farey series of order $n$ is the table obtained if one arranges in order of magnitude the rational numbers having denominators not greater than $n$ and lying between 0 and 1 inclusive; each rational number is understood to be written in its "lowest terms," the numbers 0 and 1 being given in the forms $\large\frac{0}{1}\normalsize$ and $\large\frac{1}{1}\normalsize$ respectively. The table under review is by far the largest of its kind ever published in full and probably will remain so for some time to come, since the Farey series of order 1025 consists of 319,765 fractions.

Even though the fractions $\large\frac{a}{b}\normalsize$ and $\large\frac{b-a}{b}\normalsize$ are given together, the Farey series of order 1025 itself occupies 400 pages. Except for the last page, each page contains 400 pairs of fractions, arranged in lines of 20 each. Besides the main table there is an appendix containing the following smaller tables: (1) the Farey series of order 50 with decimal equivalents, (2) the Farey series of order 64, (3) the denominators of the Farey series of order 100. In addition there is a brief introduction in which the author discusses the background, construction, and use of the tables. ...

The principal interest of such a table, aside from its obvious usefulness in finding rational approximations to irrational numbers, lies in the fact that if $\large\frac{a}{b}\normalsize$ and $\large\frac{x}{y}\normalsize$ are consecutive fractions in the Farey series of order $n$, then $bx - ay = 1$. Thus the present table can be thought of as a table of solutions of the diophantine equation $bx - ay = 1$ for all pairs of integers $a, b$ such that $1 ≤ a ≤ b ≤ 1025$ and $(a, b) = 1$, where the entries are arranged in order of magnitude of the ratio $\large\frac{a}{b}\normalsize$. Of course it is possible to extend the usefulness of the table well beyond its apparent range.

Since the main table of the work under review does not give the decimal equivalents of the fractions listed, there is considerable difficulty in locating a given fraction in the series or in fixing a given irrational number (or rational number with denominator greater than 1025) between two fractions of the series. While it would clearly be out of the question to give the decimal equivalent of every fraction listed, it seems to the reviewer that it would have been quite feasible to give at the end of each line of the table the decimal equivalents of the last pair of fractions in the line. This would have enhanced the value of the table considerably, for it would have made the location problem relatively easy. As it is, the user of the table is expected to locate a given fraction in the series (or to determine in what interval a number not in the series would fall) by appealing to the uniform distribution of the Farey fractions. ... To be sure, the author gives a table of the locations of the fractions equivalent to $\large\frac{k}{1000}\normalsize (k = 0, 1, 2, ..., 500)$, but between two consecutive such fractions there are on the average about 320 other fractions.

This work is the first volume in a series of mathematical tables which is to be published by the Royal Society and which is intended as a continuation of the well known series of tables published by the British Association. The format and printing of this first volume are very satisfactory. Although this work will be a mere curiosity to most mathematicians and will certainly not have widespread use, number-theoretic experimenters will find it of considerable interest.

6.1. Review by: Editors.
Mathematical Reviews MR0041934 (13,24e).

Except for the addition of a chapter on "Integrals of the third kind," this is a photographic reprint of the first edition [1944] with corrections in the text.

6.2. Review by: Edward Thomas Copson.
The Mathematical Gazette 36 (315) (1952), 59-60.

The first edition of this important book was published in 1944, but, by a regrettable oversight, it was never reviewed in the Mathematical Gazette. In congratulating Professor Neville on the appearance of a second edition, we must also apologise for not doing honour to his work seven years ago. Forty years ago, everyone would have regarded the theory of the Jacobian elliptic functions as being part of the main body of mathematics which every student would always have to learn. Yet interest in the subject has lapsed; and it is Professor Neville's thesis that, not only are the reasons for this not hard to seek, but also that they are reasons which should not be allowed to persist.

"Briefly, the modern introduction of the classical functions is belated and unnatural, and in the complete absence of symmetry in the fundamental relations between the functions, every general principle is stifled by the multitude of special formulae to which its application gives rise." For example, a long list of the various forms of an addition theorem is to us "depressing beyond words. We may admire the principles by which addition theorems are constructed, but we would as soon learn by heart a table of sines as bother about the details of such a mass of formulae. Worse. A collection like this does not involve a perpetual return to first principles, but is compiled from one or two of its members by transformations and combinations. ... As a result, the most elementary algebra comes to masquerade as serious work on elliptic functions, and the subject falls into contempt." With this criticism, all who have ever tried to teach the subject will agree heartily. Professor Neville's aim is to try to persuade us "that there is a natural line of approach to the Jacobian functions, and that the functional treatment, far from merely making use of arguments already played out in their Weierstrassian applications, eliminates crude algebra from the theory."
...
Evidently in a short notice one cannot do more than say that the treatment is as thorough and elegant as one would expect from the author. In particular, we would draw attention to the careful discussion of the problem inverting an elliptic integral.

That a book published by the Clarendon Press is well-produced goes without saying. It is unfortunate that nowadays one has to be content with a reproduction by photo-lithography from corrected sheets of the first edition. This prevents the author from making a thorough revision; the only substantial change is an additional chapter on elliptic integrals of the third kind. Moreover, although the standard of photo-lithographic reproduction is far higher than it was before the war, it is still not entirely satisfactory. There is inevitably a coarsening of the type. There is also an unevenness from page to page: the lighter parts of the letters sometimes disappear - occasionally parts of workings fade out. ... All this makes the first edition far more pleasant to read. So, in congratulating Professor Neville on the success of his book, we also hope that the day when it will attain the honour of a real third edition is not far off.

7.1. From the Preface by Jeffrey Charles Percy Miller.

Professor E H Neville played an active part in the formulation and execution of the table-making activities of the British Association Mathematical Tables Committee. He was chairman during its more active years from 1931 to 1947, when the series of B.A. Mathematical Tables, Volumes I-IX and Part-volumes A and B - inspired largely by Dr L J Comrie, Secretary of the Committee from 1929 to 1937 - was produced. The part played by the Chairman was visible only in his lively and well-phrased prefaces to several of the volumes. It is thus very fitting that the Royal Society Mathematical Tables - successors to the B.A. Mathematical Tables should start with two volumes by Neville.

Volume I, The Farey Series of Order 1025 (1950), continues a series of tables connected with number theory, made possible by a bequest of Lt Col A J C Cunningham to the British Association in 1928. Volume 2, the present one, sets the style for another kind of table, one with values of high accuracy, for relatively few selected values of the arguments. These are designed primarily for  the use of table-makers, and for others who can often choose arguments so that interpolation is infrequent. It is pleasant to record an appreciation of the part Professor Neville has played in the discussions of both Committees on the many facets of the attempts to produce perfect tables, and to direct attention to the Introduction, where, in characteristic and stimulating style he discusses the production and use of this volume of Rectangular-Polar Conversion Tables.

7.2. Review by: Anon.
Biometrika 47 (1/2) (1960), 216.

Given $x = r \cos \theta$ and $y = r \sin \theta$, this table is concerned with the evaluation of $r$ and of $\theta$ (in degrees) and $\log r$ and $\theta$ (in radians) for different values of $x$ and $y$.

7.3. Review by: Alan Fletcher.
The Mathematical Gazette 41 (337) (1957), 234.

The construction of the tables (by many hands) is described at length. The table was planned in order to provide table-makers with accurate values at the lattice points mentioned, and the results may be regarded as definitive. Although interpolation was not primarily intended, and the compiler points out that rectangular-polar conversions can in many cases be effected directly with the aid of other standard tables, he proceeds to demonstrate that interpolation is, in fact, much easier than might have been expected.
...
The introduction is graced by the compiler's customary distinction of style, and the whole volume by the usual Cambridge excellence of production.

7.4. Review by: Keith Douglas Tocher.
Journal of the Royal Statistical Society. Series A (General) 119 (3) (1956), 346.

These are specialist tables prepared for the use of table computers.
...
The arithmetic technique used in developing the tables consisted in forming the Farey Series $F_{105}$. The angles corresponding to the tangents have first differences whose tangents are the reciprocals of integers (using Haros's result concerning Farey series). Using the addition theorem again, a set of tangents whose angles are the second differences can be formed and these are small enough to be evaluated by the Gregory series with very few terms. These angles were expressed in both radians and degrees and the resulting angles built up by repeated summation to form the two θ tables. By eliminating duplication of values in the two difference angle sets the amount of work was considerably reduced below that required to evaluate the 3,375 terms of $F_{105}$. The calculations were performed to 20 decimal places of which 15 are given in the table.

The introduction gives a thorough account of the methods used to ensure the complete accuracy of the table and this makes fascinating reading. Professor Neville and his co-workers have produced an outstanding example of the table-maker's art.

7.5. Review by: John Todd.
Mathematical Reviews MR0077245 (17,1011a).

There are various indices, a descriptive bibliography, and a detailed introduction, describing the construction and checking of the tables, and interpolation in them, illustrated by worked examples. ... Careful study of this volume, and in particular, the introduction, the format and printing is recommended to all table-makers: they will realise some of patient work of many collaborators which provided the solid basis for reputation of the group of table makers which was led by Neville for many years.

8.1. From the Editor's Preface.

It is strange that whereas every sixth-former, who has read a course in mathematics, will understand fully the periodicity of the trigonometric functions, very few of them know anything at all about doubly periodic functions. Again, most of them know that it is impossible to find the length of the arc of an ellipse in terms of algebraic, trigonometric or exponential functions, but few, even as undergraduates, have any acquaintance with elliptic functions, apart from the fact that these functions are so named because they are needed to evaluate the integral which results from the attempt to rectify the ellipse. This situation is regrettable, not only because the solution of many problems in applied mathematics demands the solution of simple elliptic integrals, but also because the theory of doubly periodic functions is by no means difficult and has a fascination all of its own.
...
The editor [W J Langford] of this volume was a pupil of the late E H Neville at the University of Reading in the early 1920s, and was fortunate, as a postgraduate, to be the first student to take the course of lectures which were later expanded into a book [Jacobian elliptic functions, 1944]. After the publication of Jacobian elliptic functions, it was represented to Neville that a more elementary treatment was desirable, particularly in view of the fact that he had developed an entirely new notation, one consequence of which is drastically to reduce the complications of the classical formulae. The primer was written soon after the major treatment was published, but no steps were taken to put it into print. After Neville's death, the editor was asked to deal with the papers that had been left, and the manuscript came to light. The student who reads this small book thoughtfully will be able to pass on to a study of Jacobian elliptic functions or to any of the classical treatises in the subject.
...
The student who reads this small book thoughtfully will be able to pass on to a study of Jacobian Elliptic Functions or to any of the classical treatises in the subject. It is the editor's hope that, in making the primer available to young mathematicians, he will have paid a tribute in gratitude to his friend and teacher, Eric Neville, and, at the same time, will have done a little to bring a fascinating branch of the theory of the complex plane back into popularity.

8.2. Review by: Ida Winifred Busbridge.
The Mathematical Gazette 56 (398) (1972), 355-356.

In 1944 Professor E H Neville published Jacobian Elliptic Functions (Oxford) and this was followed, a few years later, by a second edition (slightly revised). For anyone with an interest in elliptic functions, Neville's was an exciting book. By modifying the traditional notation, he made the subject into a coherent structure based on two sets of twelve allied functions, each set related to the other. Moreover his treatment of elliptic integrals was far superior to any other which I have seen. With the aid of translation rules from the old notation to the new, an undergraduate could make some use of Neville's book while following a traditional course, but the book as a whole was too advanced for undergraduates.

Elliptic Functions: A Primer was found among Neville's papers after his death. He wrote it in response to a demand for a more elementary treatment suitable for third-year undergraduates and first-year postgraduates, but he did not publish it. As a tribute to Professor Neville, the book has been prepared for publication by Mr W J Langford with considerable help from Dr M R Rees and Professor T A A Broadbent. It has been attractively produced, with clear printing and good diagrams, by the Pergamon Press.

It is a pity that the book was not published immediately after Neville had finished writing it. In the years which have elapsed, elliptic functions have gone out of fashion. They reappear as examples in books on meromorphic functions (i.e. functions whose only singularities in the finite complex plane are poles), but the subject is studied less and less for its own sake. Elliptic integrals arise in practical problems, but probably these are now computerised. In any case tables exist for the values of the most common integrals. Yet it is a pity to see such an elegant branch of complex analysis fading into oblivion and I hope that this book will do something to arouse renewed interest in the subject. Unlike Jacobian Elliptic Functions, about half the book is devoted to the general theory of elliptic functions and to the Weierstrassian ℘-function and related functions. The twelve "primitive functions" are introduced and studied in this part.
...
To sum up: This is a book which will teach a student a lot about elliptic functions. He could acquire his knowledge more rapidly elsewhere, but he would not end with the understanding of the whole subject which he will gain from this book.