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THE problem of presenting the infinitesimal calculus to schoolboys and university students who seek to acquire a knowledge of it for applications in other fields is notoriously difficult. Methods based upon geometrical intuition lack rigour and to the thinking student produce an impression of mystery. Methods based on a rigorous theory of real numbers are suited only to the student who is to become a mathematical specialist. A Text-Book of Mathematical Analysis The Uniform Calculus and its Applications. By R. L. Goodstein. Pp. xii + 476. (Oxford : Clarendon Press ; London : Oxford University Press, 1948.) 30s. net.

THIS book gives a thorough introductory study of the properties of ordinary points in the differential geometry of curves and surfaces in 3-space. Chapter 1 gives an account of twisted curves, Chapter 2 describes the tensor calculus. In Chapter 3 this calculus is applied to investigate the intrinsic geometry of a surface, that is to say, those properties which depend on the first fundamental form. Chapter 4 develops those properties which derive also from the second fundamental form and therefore characterize the surface as viewed from the enveloping space. An Introduction to Differential Geometry with Use of the Tensor Calculus By Prof. Luther Pfahler Eisenhart. (Princeton Mathematical Series, 3.) Pp. x + 304. (Princeton, N. J.: Princeton University Press; London: Oxford University Press, 1940.) 21s. 6d. net.

THE memoir in which Bessel, the astronomer, examined in detail the functions which now bear his name was published in 1824, and was the outcome of his earlier researches concerning the expression of the radius vector in planetary motion. A Treatise on the Theory of Bessel Functions By Prof. G. N. Watson. Second edition. Pp. viii + 804. (Cambridge: At the University Press, 1944.) 60s. net.

THAT the foundations of mathematics are A important is a proposition which will find few opponents, for the science of mathematics is commonly regarded as man's securest intellectual possession. What constitutes these foundations is a subject on which agreement has not been reached. There are, however, three main directions into which the body of modern research has branched, namely, the logistic, the intuitionistic and the formalistic theories. Broadly speaking, the logistic theory regards mathematics as a branch of logic, the intuitionistic theory regards the theorems of mathematics as having actual significance, the formalistic theory regards mathe matics as devoid of meaning per se. The most powerful exponent of the formalistic attitude is Hilbert, and the present volume is the first part of a systematic exposition of mathematical foundations from the point of view of the school of thought which he has founded. This is probably the most important book on mathe-matical foundations which has appeared since Whitehead and Russell's “Principia Mathematical Grundlagen der Mathematik Band 1. Von D. Hilbert und P. Bernays. (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Beriick-sichtigung der Anwendungsgebiete, herausgegeben von R. Courant, Band 40.) Pp. xii + 471. (Berlin: Julius Springer, 1934.) 37.80 gold marks.

THESE two volumes are photo-lithoprints of the second edition of Vol. 1, issued in 1930, and Vol. 2, issued in 1937. The first volume contains seven chapters as follows: 1, the algebra of linear transformations and quadratic forms; 2, the development of an arbitrary function in series; 3, the theory of linear integral equations; 4, the basic principles of the calculus of variations; 5, the oscillation and eigenvalue problems of mathematical physics; 6, application of the calculus of variations to eigenvalue problems; 7, special functions defined by eigenvalue problems. Methoden der mathematischen Physik Von Prof. R. Courant Prof. D. Hilbert. (Published and distributed in the Public Interest with the consent of the Alien Property Custodian under License No. A. 82.) Band 1. Zweite verbesserte Auflage. Pp. xv + 469. 8 dollars. Band 2. Pp. xiv + 549. 8 dollars. (New York: Interscience Publishers, Inc.; London: H. K. Lewis and Co., Ltd., 1931–1937.) 2 vols., 14 dollars.

SINCE 1931, when the British Association started to publish mathematical tables in volume form, three sets of tables have appeared. The first comprised tables of general utility: trigonometric, hyperbolic, exponential, gamma and other functions. The second issue contained Emden functions and had therefore a less popular appeal, since the tables were designed to aid in certain astrophysical researches, and the cost of printing was undertaken by the International Astronomical Union. The present (third) volume, while not perhaps of such a specialised character, deals with a subject which from its nature must interest directly but a small body of scientific workers. British Association for the Advancement of Science. Mathematical Tables. Vol. 3: Minimum Decompositions into Fifth Powers. Prepared By Prof. L. E. Dickson. (Published under the supervision of the British Association Committee for the Calculation of Mathematical Tables.) Pp. vi + 368. (London: British Association, 1933.) 10s.

(1) THIS is a clearly printed table of logarithms and antilogarithms to five places with a four figure argument and mean differences. A seven place table of (l+r/100)n for n = 1 to 9 and r = 0.00 to 7.50 is appended. Femcifrede Logaritmer og Antilogaritmer (Five Figure Tables of Logarithms and Anti-Logarithms). By A. K. Erlang. Udgivet ved (edited by) R. E. H. Rasmussen. Pp. 48. (Kobenhavn: G. E. C. Gads Forlag, 1930.) n.p. (2) Addition-Subtraction Logarithms to Five Decimal Places. By L. M. Berkeley. Pp. xii + 134. (New York: White Book and Supply Co., 1930.) 3.25 dollars.

THE second section of the fourth volume of Borel's treatise on probability bears the title “Applications aux jeux de hasard”. The present work is due to the interest which this excited among bridge players and consists of a thorough investigation of the numerical probabilities which bridge presents. Such a study is of value not only to bridge players ; it contributes equally to the progress of the science of probability which already owes much to the study of games of pure chance, and of which many applications are related to the theory of games where chance and the psychology of the players both enter. Theorie mathematique du bridge a la portee de tous Par Prof. Emile Borel Andre Cheron. (Monographies des probabilites, Fascicule 5.) Pp. xx + 392. (Paris: Libr. Gauthier-Villars, 1940.) 175 francs.

Operational Calculus based on the Two-Sided Laplace Integral By Dr. Balth van der Pol and Dr. H. Bremmer. Second Edition. Pp. xiii + 415. (Cambridge: At the University Press, 1955.) 60s. net.

DR. GEORGE KAVAN was born at Prague in 1877. His parents were Francis Kavan, a musical conductor and a composer of merit, and his wife Miroslava, an ardent supporter of the civil rights of women. After studying mathematics, physics and astronomy, Kavan became in 1900 chief assistant at the Astronomical Institute of the Czech University at Prague, where he remained until his appointment in 1919 as director of the Observatory at Stara Dala. The Observatory being at that time not fully equipped with instruments, it was found possible to use part of the publication funds to cover the cost of printing the present handsome volume of tables. The calculation was started in 1917, and after the author's death in 1933 his wife, Madame Kavanova, undertook the reading and correction of the final proofs. Factor Tables giving the Complete Decomposition into Prime Factors of all Numbers up to 256,000 By Dr. George Kavan. Pp. xii + 514. (London: Macmillan and Co., Ltd., 1937.) 42s. net.

IN the preface to the first of the above books, the author mentions a question asked him by a young woman friend: “When you began this work you had no idea that anything so clever would come out of it, had you?” This quotation certainly sums up the impression conveyed after reading (1). Schrodinger, however, uses it to bring out the point that in a set of papers in which a single theme is developed, the results of the later papers were more or less unknown when the earlier ones were written. This must be borne in mind when judging these collections. (1) Collected Papers on Wave Mechanics. By Prof. E. Schrodinger. Translated from the second German edition. Pp. xiii + 146. (London and Glasgow: Blackie and Son, Ltd., 1928.) 25s. net. (2) Four Lectures on Wave Mechanics: delivered at the Royal Institution, London, on 5th, 7th, 12th, and 14th March 1928. By Prof. Dr. Erwin Schrodinger. Pp. viii + 53. (London and Glasgow: Blackie and Son, Ltd., 1928.) 5s. net. (3) Selected Papers on Wave Mechanics. By Louis de Broglie Dr. Leon Brillouin. Authorised translation by Winifred M. Deans. Pp. vi + 151. (London and Glasgow: Blackie and Son, Ltd., 1928.) 15s. net.

THE path of a material particle of mass m moving in a conservative field of force in which the potential energy V is a function of position (x, y, z) only is determined by Hamilton's Principle. If E is the constant total energy and A and B the points of departure and arrival, this may be expressed in the form

THE science of mathematics, the most fundamental of all intellectual activities, has been studied since the dawn of history, and almost certainly before that. The motives which impel this study are various, but it would scarcely be an exaggeration to say that everyone uses mathematics in one form or another. In the past, many of the advances of pure mathematics have been bound up with practical applications. For example, it is probably not often realised that the Gibbs phenomenon of Fourier series was brought into prominence by results obtained in using the harmonic analyser of Michelson and Stratton, and was at first attributed to an inaccuracy in the machine itself. Mathematics for Self-Study Series. By J. E. Thompson (1) Arithmetic for the Practical Man. Pp. xiii + 269. (2) Algebra for the Practical Man. Pp. xviii + 291. (3) Trigonometry for the Practical Man. Pp. x + 204. (4) The Calculus for the Practical Man. Pp. x + 323. (London: George Routledge and Sons, Ltd., 1931.) 7s. 6d. net each volume. (5) Mathematics: a Text-book for Technical Students. By Bevis Brunel Low. Pp. vii + 448. (London, New York and Toronto: Longmans, Green and Co., Ltd., 1931.) 12s. 6d. net.

THE book is based on a series of lectures on infinitesimal calculus given regularly for more than seventeen years in the University of Basle. This the first volume, contains a course intended for those whose main interest is in the applications of the calculus, and to this end it is freed as far as possible from what such students might reasonably regard as ‘unnecessary subtleties’. Also the proofs of certain theorems, when too lengthy, are deferred to the second volume. Nevertheless, the course is full, clear, and sound in its foundations. Vorlesungen uber Differential- und Integralrechnung Von Prof. A. Ostrowski. Band 1: Funktionen einer Variablen. (Lehrbucher und Monographien aus dem Gebiete der exakten Wissenschaften, 8.) Pp. xii + 373. (Basel: Verlag Birkhauser 1945.) 47.50 Swiss francs.

THE vector analysis of Gibbs and Heaviside and the more general tensor analysis of Ricci are now recognized as standard tools in mechanics, hydro-dynamics and electrodynamics. Their use not only materially simplifies and condenses the exposition, but also makes mathematical concepts more tangible and easy to grasp. Moreover, tensor analysis provides a simple automatic method for constructing invariants. Since a tensor equation has precisely the same form in all co-ordinate systems, the desirability of stating physical laws or geometrical properties in tensor form is manifest. The perfect adaptability of the tensor calculus to the theory of relativity was responsible for its original renown. It has since won a firm place in mathematical physics and engineering technology. Thus Sir Edmund Whittaker rates the tensor calculus as one of the three principal mathematical advances in the last quarter of the nineteenth century. Vector and Tensor Analysis By Prof. Louis Brand. Pp. xvi + 439. (New York: John Wiley and Sons, Inc.; London: Chapman and Hall, Ltd., 1947.) 33s. net.

THIS book is intended as an introduction to operational calculus based upon the transform of f(x) defined by that is to say, the Laplace transform multiplied by p. The function p(p) so defined is identical with what Heaviside called the operational form. The advantage of the present treatment over Heaviside's is that the theorems of operation can be placed upon an impeccable basis without the necessity for acts of faith. The contents are : 1, the Laplace transform ; 2, rules of the operational calculus ; 3, solution of ordinary linear differential equations with constant coefficients ; 4, solution of partial linear differential equations with constant coefficients ; 5, evaluation of integrals ; 6, derivation of Laplace transforms ; 7, Laplace transform for a finite interval ; impulses. Modern Operational Calculus With Applications in Technical Mathematics. By Dr. N. W. McLachlan. Pp. xiv+218. (London: Macmillan and Co., Ltd., 1948.) 21s. net.

THE preface states that this book was written at the suggestion of some students who were unable to find a book on the subject suitable for their requirements. The seven chapters deal with preliminary mathematics ; gravitational attraction and potential, simple applications ; attraction and potential internal points, spheres ; theorems of Laplace, Poisson and Gauss, general theory ; Green's theorem; harmonic functions ; attraction of ellipsoids. An Introduction to the Theory of Newtonian Attraction By A. S. Ramsey. Pp. ix + 184. (Cambridge: At the University Press, 1940.) 10s. 6d. net.

AWAY to solve linear differential equations by operational methods which avoid the introduction of arbitrary constants by taking direct account of the initial conditions was invented by Oliver Heaviside (1850-1925) for the purpose of discussing electromagnetic waves. In his hands the method led to correct results, but the mathematical foundation was so obscure as to cause it to be regarded with misgiving. Brornwich (1875-1930) was the first to explain Heaviside's method, and, by expressing the solution in the form of a contour integral, to justify its application to ordinary linear equations. Since then many systems, modifications and expositions of varying degrees of merit have appeared. Operational Methods in Applied Mathematics By Prof. H. S. Carslaw J. C. Jaeger. Pp. xvi + 264. (Oxford: Clarendon Press; London: Oxford University Press, 1941.) 17s. 6d. net.

BERNARD PARKER HAIGH, professor of applied mechanics in the Royal Naval College, Greenwich, who died on January 18, was born in 1884. Educated at Allan Glen's School, Glasgow, and afterwards at the University, he was apprenticed to Bruce Peeples, electrical engineers in Edinburgh, where he worked under J. L. la Cour, the inventor of the rotary converter which bears his name. This association took Haigh to Germany and to Sweden, where he obtained much valuable experience. From 1909 he was engaged for two years on the design of steering gears at the Edinburgh firm of Brown Bros., and then became assistant to Prof. Archibald Barr at the University of Glasgow. It was in 1913 that Haigh took the step which was to determine his career, when he became lecturer in applied mechanics at the Royal Naval College, where he succeeded in 1920 to the chair left vacant when Sir James Henderson retired in order to give his whole time to gunnery problems.