How Many People Can the Earth Support? By Joel E. Cohen. Norton: 1995. Pp. 532. $30, £22.50. UK publication date, 27 March.

Book of numbers

In 1922 Barnes Wallis, who later invented the bouncing bomb immortalized in the movie The Dam Busters, fell in love for the first and last time, aged 35. The object of his affection, Molly Bloxam, was 17 and setting off to study science at University College London. Her father decreed that the two could correspond only if Barnes taught Molly mathematics in his letters. Mathematics with Love presents, for the first time, the result of this curious dictat: a series of witty, tender and totally accessible introductions to calculus, trigonometry and electrostatic induction that remarkably, wooed and won the girl. Deftly narrated by Barnes and Molly's daughter Mary, Mathematics with Love is an evocative tale of a twenties courtship, a surprising insight into the early life of a World War Two hero, and a great way to learn a little mathematics.

The history of mathematics: a reader, edited by John Fauvel and Jeremy Gray. Pp 628. £14·25 (paperback), £35·00 (hardback). 1987. ISBN 0-333-42790-4, 0-333-42791-4 (Macmillan Education and Open University)

The mathematics of the past: distinguishing its history from our heritage

Winner of the 2012 PROSE Award for Mathematics from The American Publishers Awards for Professional and Scholarly Excellence.

	"A great book, one that I will certainly add to my personal library."
	—Paul J. Nahin, Professor Emeritus of Electrical Engineering, University of New Hampshire

	Classic Problems of Probability presents a lively account of the most intriguing aspects of statistics. The book features a large collection of more than thirty classic probability problems which have been carefully selected for their interesting history, the way they have shaped the field, and their counterintuitive nature.

	From Cardano's 1564 Games of Chance to Jacob Bernoulli's 1713 Golden Theorem to Parrondo's 1996 Perplexing Paradox, the book clearly outlines the puzzles and problems of probability, interweaving the discussion with rich historical detail and the story of how the mathematicians involved arrived at their solutions. Each problem is given an in-depth treatment, including detailed and rigorous mathematical proofs as needed. Some of the fascinating topics discussed by the author include:

	
		Buffon's Needle problem and its ingenious treatment by Joseph Barbier, culminating into a discussion of invariance
	
		Various paradoxes raised by Joseph Bertrand
	
		Classic problems in decision theory, including Pascal's Wager, Kraitchik's Neckties, and Newcomb's problem
	
		The Bayesian paradigm and various philosophies of probability
	
		Coverage of both elementary and more complex problems, including the Chevalier de Mere problems, Fisher and the lady testing tea, the birthday problem and its various extensions, and the Borel-Kolmogorov paradox


	Classic Problems of Probability is an eye-opening, one-of-a-kind reference for researchers and professionals interested in the history of probability and the varied problem-solving strategies employed throughout the ages. The book also serves as an insightful supplement for courses on mathematical probability and introductory probability and statistics at the undergraduate level.

Classic Problems of Probability

We continue in this paper a comprehensive theory of antenna near fields inaugurated in Part I. The concept of near-field streamlines is introduced using the Weyl expansion in which the total field is decomposed into propagating and nonpropagating parts. This process involves a breaking of the rotational symmetry of the scalar Greens function that originally facilitated the derivation of the Weyl expansion. Such symmetry breaking is taken here to represent a key to understanding the structure of the near fields and how antennas work in general. A suitable mathematical machinery for dealing with the symmetry breaking procedure from the source point of view is developed in detail, and the final results are expressed in clear and compact form susceptible to direct interpretation. We then investigate the concept of energy in the near field where the localized energy (especially the radial localized energy) and the stored energy are singled out as the most important types of energy processes in the near-field zone of general antenna systems. A new devolvement is subsequently undertaken by generalizing the Weyl expansion in order to analyze the structure of the near field, but this time from the far-field point of view. A hybrid series combining the Weyl and Wilcox expansions is derived, after which only the radial streamline picture turns out to be compatible with the far-field description via Wilcox series. We end up with an explication of the general mechanism of far-field formation from the source point of view. It is found that the main formative processes in the antenna near-field zone are reducible to simple geometrical and filtering operations.

A Theory of Antenna Electromagnetic Near Field—Part II

Preface 1. Crisis in mathematics: Fourier's series 2. Infinite summations 3. Differentiability and continuity 4. The convergence of infinite series 5. Understanding infinite series 6. Return to Fourier series 7. Epilogue A. Explorations of the infinite B. Bibliography C. Hints to selected exercises.

A radical approach to real analysis, by David Bressoud. Pp328. $29. 1994. ISBN-0-88385-701-4 (Mathematical Association of America)

This is a discussion of number theory, covering factorization and primality testing. It is part of a series of undergraduate texts in mathematics.

Factorization and primality testing, by D. M. Bressoud. Pp 251. DM98. 1989. ISBN 3-540-97040-1 (Springer)

From the Preface: (...) The book is addressed to students on various levels, to mathematicians, scientists, engineers. It does not pretend to make the subject easy by glossing over difficulties, but rather tries to help the genuinely interested reader by throwing light on the interconnections and purposes of the whole. Instead of obstructing the access to the wealth of facts by lengthy discussions of a fundamental nature we have sometimes postponed such discussions to appendices in the various chapters. Numerous examples and problems are given at the end of various chapters. Some are challenging, some are even difficult; most of them supplement the material in the text. In an additional pamphlet more problems and exercises of a routine character will be collected, and moreover, answers or hints for the solutions will be given. This first volume of concerned primarily with functions of a single variable, whereas the second volume will discuss the more ramified theories of calculus (...).

Introduction to calculus and analysis, by Richard Courant and Fritz John. Second printing 1989. - Vol. 1. Pp 661. DM96. 1989. ISBN 3-540-97151-3 - Vol. 2. Pp 954. DM126. 1989. ISBN 3-540-97152-1 (Springer)

The Belgian Jesuit, Gregory of St Vincent (1584 - 1667), along with Cavaleri, Fermat and Descartes was amongst those who prepared the way for the calculus of Newton and Leibniz. He was the author of one long book Opus Geometricum (1647) in which he gave a more precise account of progressions and limits than had been previously available and which Leibniz particularly valued. The term ‘exhaustion’, for the limiting process of the Greeks, is due to Gregory of St Vincent. This book contains a section on the hyperbola in which Gregory studied the area between the curve and an asymptote and this section provided the basis for Anthony de Sarasa's claim in 1649 that the area here was like a logarithm.

Gregory of St Vincent and the rectangular hyperbola

References 1. D. F. Lawden, On the solution of linear difference equations, Math. Gaz. XXXVI (1952) pp. 193-196. 2. L. Moser, King paths on a chessboard, Math. Gaz. XXXIX (1955) p. 54. 3. R. Grassl, T. Mingus, Equivalence classes and a familiar combinatorial identity, Math. Gaz. 82 (1998) pp. 98-100. 4. S. Simons, A curious identity, Math. Gaz. 85 (2001) pp. 296-298. MICHEL BATAILLE 12 Rue Sainte-Catherine, 76000 Rouen, France e-mail: michelbataille @ wanadoo.fr

Bob Burn

Papers

A radical approach to real analysis, by David Bressoud. Pp328. $29. 1994. ISBN-0-88385-701-4 (Mathematical Association of America)

Factorization and primality testing, by D. M. Bressoud. Pp 251. DM98. 1989. ISBN 3-540-97040-1 (Springer)

Introduction to calculus and analysis, by Richard Courant and Fritz John. Second printing 1989. - Vol. 1. Pp 661. DM96. 1989. ISBN 3-540-97151-3 - Vol. 2. Pp 954. DM126. 1989. ISBN 3-540-97152-1 (Springer)

Gregory of St Vincent and the rectangular hyperbola

87.23 Triangles with a 60° angle and sides of integer length