This is the first of two volumes designed to give an elementary exposition of certain basic ideas in numerical analysis in a manner suitable for applied scientists. The author has chosen to consider in some detail a limited selection of typical numerical methods with the object of illustrating the general underlying principles. Particularly pleasing is the emphasis laid throughout upon the estimation and practical control of errors.

The first chapter is concerned with the various types of error and the manner in which they occur in calculations, with particular regard to rounding-off, and includes a discussion of the possible ways in which the accumulation of rounding errors may be minimized when evaluating formulae on desk machines. This is followed by a general analysis of iterative procedures for solving algebraic and transcendental equations. The solution of systems of linear algebraic equations together with the evaluation of eigenvalues and eigenvectors occupy the second half of the book, and here the author has succeeded in covering the important topics remarkably well. One particularly interesting chapter is that concerned with elementary programming, in which a simple language, not based on any specific machine but typical of many autocodes currently in use, is developed and subsequently used to indicate how various computational procedures may be implemented on a digital computer. The text is supplemented by a valuable selection of exercises in each chapter.

As the author remarks, the book is both supplemented and
complemented by the already well known *Modern Computing Methods*
which is somewhat more of a working manual, so that in conjunction
these books will provide an excellent grounding for any scientist or
engineer engaged in non-trivial computations. The reviewer looks
forward with pleasant anticipation to volume 2, which will deal with
the numerical solution of ordinary and partial differential
equations.

J. OLIVER.

Some twenty-eight "proofs" of various "theorems" and half that number of howlers (false derivations of correct conclusions to given premises) from several branches of mathematics are collected together in this entertaining and instructive work. Dr. Maxwell's lucidly written account of the proofs and their errors is a delight to read, and the Syndics of the Cambridge University Press are to be warmly commended for issuing the work as a paperback (originally published in 1959).

D. J. DALEY.

The short guide to reading Russian which precedes the vocabulary contains a very concise summary of Russian grammar and word formation. This claims to be only "a key to the Russian 'code' rather than a Russian grammar," and the person who wishes to read Russian fluently will need to use a more comprehensive grammar, though this might be an adequate introduction. There is a strong emphasis on word formation, and the tables of prefixes, suffixes and final letters are useful for reference.

The vocabulary sets out to provide a complete coverage of terms used in pure mathematics and of most general terms which occur in mathematical works. It has about 11,000 entries with a liberal supply of common phrases and special usages. There are ample cross-references for irregular parts of verbs, and stress is indicated. This is an excellent dictionary for a person with some knowledge of Russian who wishes to read mathematics.

J. M. O. SPEAKMAN.

A conference entitled "New Directions in Mathematics" was held at Dartmouth, U.S.A., in 1961, and this l20-page book is an edited transcript of the proceedings. There are four sections: Secondary School, College, Applied and Pure Mathematics; each section consists of three or four prepared speeches, followed by spontaneous discussion.

As is common in such a situation, the spoken word, when written down, is not nearly so effective; what may have been serious discussion all too often comes through as an attempt to score a debating point. And was it really necessary to include everything - "Professor, do you have any further announcements? If not, this meeting is adjourned"?

Collectively, the many well-known mathematicians (Buck, Kac, Kaplansky are amongst them) produced several interesting prophesies and ideas. However, no attempt was made to integrate the suggestions, and at various points the speakers appeared to be at cross-purposes. Informative and relevant this book may well be, but it is hardly worth hard covers.

J. HAIGH.

This book is a translation of a very readable Russian book first published in 1959. In his foreword Novikov very aptly describes it as a textbook "to give a discussion of the fundamentals of mathematical logic in the most accessible form possible." It is on the whole very clear (despite a number of sentences which one hopes were more intelligible in the original Russian) and is always interesting. The implications of almost every theorem proved are discussed, and the reader sees why the theorem is necessary and what will follow from it.

There are six chapters and an Introduction. In the Introduction Novikov gives a general description of the problems and methods of mathemetical logic and the reasons for studying them. The first chapter presents the propositional calculus in a non-formal way; the second chapter develops it as a formal system and introduces the reader to metalogical reasoning. The following two chapters repeat this development for the predicate calculus. The fifth chapter on axiomatic arithmetic is basically an extension of the discussion of the predicate calculus to the system of arithmetic. In the final chapter Novikov proves the consistency of a system of restricted arithmetic.

This book is only an introduction to the field of mathematical logic, but as an introduction to the fundamentals of the subject, it is excellent.

P. B. GOLDSTEIN.

This book is a general introduction to the field of mathemetical logic. It is a concise survey of most of the principal branches of this field. The author assumes no prerequisites other than "a certain amount of experience in abstract mathematical thought."

Philosophical and general discussion are kept to an absolute minimum; the practical aspect of the enquiry dominates the book. Using this approach the author is able to cover a great deal of material, and he does this clearly and concisely. Mendelson begins with a truth-table discussion of the propositional calculus. This is followed by an axiomatic development. The second chapter presents quantification theory. The third chapter develops arithmetic as a formal system, introduces recursive functions and gives a proof of Gödel's incompleteness theorem. In the next chapter Mendelson presents the version of axiomatic set theory as developed by von Neumann, Robinson, Bernays and Gödel. The final chapter is devoted to the problem of effective computability. Mendelson discusses the approaches of Markov, Turing, and Herbrand and Gödel.

There are many exercises throughout the book, some of which are very interesting and point to applications and alternative developments of the subject being considered. These are often valuable in showing the reader new ways in which to view the material at hand.

For the reader with some familiarity with mathematical methods, and especially for one interested in the important results of the field, this book would be a very adequate introduction to mathematical logic.

P. B. GOLDSTEIN.

This book deserves to become a standard Part I text. It maintains the high tradition of clarity and rigour of Van Nostrand's University Series in Undergraduate Mathematics to which it is the latest addition, and gives a firm motivation for the study of abstract linear spaces.

The reader is aided by three welcome features: an index of symbols used; six appendices, placed at the end of the book to allow the text to be read smoothly, and very illuminating when consulted; and, most important, a summary after each chapter of the most fundamental ideas, definitions and results.

Chapters I and II, although meant as introductory, are quite essential, and must not be skipped; the main abstraction comes in Chapter III. The fourth chapter gives the theory of linear transformations and matrices, with simultaneous linear equations as motivation for both. Then come two independent accounts of the theory of determinants; one from the point of view of multilinear and alternating forms, and the other a more "traditional" treatment. The rest of the book is independent of the first treatment, and the final chapter is about (complex) inner product spaces; the text closes with a proof of the Spectral Theorem.

Both teacher and student will welcome the 200 or so exercises, and the diagrams at judicious points: a student who reads this book will have the firmest of foundations for later work.

J. HAIGH.

To write a book on Number Theory in less than 150 pages implies of necessity considerable selection of material, and might result in sketchiness of treatment. This, however, Dr. Hunter avoids, and gives a thorough basic introduction to the subject. He emphasises the algebraic basis for the subject, introducing the reader in the first chapter to the concepts of binary operation, equivalence class, group, ring, integral domain, and field, and draws his attention to their occurrence as we proceed with the theory of numbers; for instance, the existence of a primitive root (mod m) is related to the reduced set of residues forming a cyclic group.

In the first five chapters Dr. Hunter defines the natural numbers
by Peano's Axioms, and describes their extension to the ring of all
integers, and the field of rationals. Factorization and division
properties of integers are dealt with in detail, and the theory of
congruences leads on to a thorough discussion of primitive roots and
quadratic residues. In the final two chapters the reader is
introduced to the representation of integers by a binary quadratic
form (not necessarily positive definite), with that by the sum of two
squares as a special case, and to one or two Diophantine equations
such as *x*^{4} + *y*^{4} = *z*^{2}.

As is to be expected in a book of this size many interesting subjects receive no mention, but those that do are treated with thoroughness, and the examples at the end of each chapter are both interesting and instructive.

I. B. T.

The author of the first volume of Oliver and Boyd's new series of University Mathematical Monographs sets himself the task of providing "an introduction to the modern theory of probability and the mathematical ideas and techniques on which it is based." The first forty-four pages contain an account of those parts of the theory of sets, measure and integration which the author requires for his subsequent discussion of probability theory. The treatment of the latter includes the standard distributions used in statistics, a brief account of dependence and conditional probability, convergence of sequences of independent random variables, and finally a glimpse of the theory of stochastic processes. The results are given in theorem form throughout, and their proofs are made easier to follow by ample cross-referencing.

From the student's viewpoint it is a pity that there are no exercises set out as such, while a few comments on the "suggestions for further reading" would help the interested reader in continuing this studies. However, the book contains much that is normally given in an introductory course, and it is presented in a readable fashion.

Depending on one's point of view (statistician versus mathematician) one may or may not like the manner in which the author avoids the conventional definition of a random variable by defining the contexts in which the terms "random variable" and "probability" arise. From the first part of the book, as a result, essentially only the theory relating to Lebesgue-Stieltjes integration is used. Such an approach may be satisfactory within the confines of the book under review, but it hardly prepares the student to tackle more extensive treatises like Loève and Doob.

D. J. DALEY.

This work is an English translation, with certain modifications, of a compendium which, in its German version, Meyers Rechurduden, sold over over 200,000 copies in two years. It is aimed at "the Man in the Street, the harassed parent, and the technical or engineering student; or the scientist, engineer or accountant, for whom mathematics have not lost their fascination."

For these purposes the book is well suited: unpretentious definitions are supported by explanations, examples, properties and diagrams. In a work of this nature, whatever has to be omitted would cause some grumbles; this reviewer would have liked more cross-references - for instance, Pascal's and Brianchon's theorems are included, but not linked; and similarly with linear and affine transformations.

In addition to nearly 500 pages of alphabetically arranged "dictionary," we have 100 pages of tables, roots, powers, logarithms, trigonometric, hyperbolic and exponential functions; and a vast quantity of useful formulae, including tables of integrals, expansions of transcendental functions, and vector identities, among many others. Why not buy this book for the family at Christmas? They might let you borrow it sometime.

J. HAIGH.

A welcome re-issue, differing only slightly from the fourth edition, of a book which has established itself as a standard Part I text.

P. M. LEE.

This is a translation of the third (1961) edition of Khinchin's beautiful introduction to the theory of continued fractions. The main topics covered are the representation of numbers as continued fractions, convergents as best approximations, order of approximation, and the groundwork of the measure theory of continued fractions, including a treatment of various averaging operations connected with their elements. In a book of this length (95 pp.) which assumes very little mathematical knowledge on the part of the reader, only an outline of the subject can be given, but within the limitations he set himself, Khinchin achieved a clarity of exposition which it would be difficult to surpass.

P. M. LEE.

Reproduced from Eureka 27 pages 41-44.

HTML conversion Copyright © 2002-4 The Archimedeans.

Errata: "mathemetical" should read "mathematical" (twice), "this studies" should possibly read "his studies".

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