Mathematics for the Million by Lancelot Hogben

First published: 1937, rev. ed., 1967; illustrated

Subjects: Education and science

Type of work: Science

Recommended Ages: 15-18

Form and Content

After an introduction indicating the general plan of the book and some specific questions to be discussed, Mathematics for the Million is organized as a historical narrative of the development of mathematics and its role as a tool for solving practical problems, particularly those dealing with measurement. Thus, a concept will be discussed as of the time that it was first studied, and only later does the reader see how it was developed by later thinkers, to the level at which it is understood today. In the course of the book, the reader is introduced to many of the main areas of mathematical study, including arithmetic, algebra, geometry, trigonometry, calculus, matrix algebra, and probability. The book is profusely illustrated, with drawings and diagrams of the problems. Each chapter concludes with a substantial series of test problems and a summary list of major concepts and rules to be memorized.

The narrative begins with a discussion of the first number systems and a look at the sort of questions that mathematicians have tried to solve. This information is followed by an analysis of Euclid’s geometry, which is seen not as a pure axiomatic system but as a way of applying observations about size and shape in a systematic fashion. Eschewing the rigors of formal proof, Lancelot Hogben demonstrates that Euclid’s major results do in fact match the experience of measurement in the real world,

The discussion of Euclid is followed by a treatment of the trigonometric functions, also seen as tools for measurement, leading to a discussion of the concept of pi (the ratio of the circumference of a circle to its diameter) and methods of approximating this number. Hogben then shows how the ancient Greeks used geometrical methods to solve arithmetical problems through the arrangement of numbers in triangular, square, and other regular patterns. This discussion introduces a more general treatment of the idea of number series, which will recur in the book.

Hogben demonstrates that a particularly important change in the Middle Ages was the concept of zero, which enabled the crude early number systems to be replaced by the positional notation used today. This system in turn enabled a general form for linear and quadratic equations. Hogben looks at the solutions of these problems and discusses methods for making sure that one has successfully translated the verbal problem to be solved into the language of equation. The exploration of quadratic equations leads to the concept of the square root of a negative number.

Mention of the great advances in sea travel after the Middle Ages serves as a springboard for a discussion of the use of mathematics in navigation, including a more thorough investigation of trigonometry and the geometry of the sphere. This examination is followed by an analysis of the conic sections—circle, ellipse, parabola, and hyperbola—emphasizing that they can be seen as tracing the movements of a point in accordance with the well-defined rules of quadratic equations. Consideration of the trigonometric and logarithmic functions leads to a treatment of them as the limits of infinite series, which in turn can be generalized and applied to the concept of complex numbers. Calculus is then presented as a way of measuring the rate of change of the sort of functions that have been studied.

Hogben goes on to cover the solution of series of linear equations by the use of determinants—square arrangements of the coefficients of these equations—and methods by which these determinants can be manipulated to make the work easier. As always, the emphasis is on specific practical problems. In the final chapter, the tools derived earlier in the book are applied to the study of probability, first in the sort of gambling problems that inspired the original development of the subject and then with indications of how these tools can be used by governments or insurers.

Critical Context

Mathematics for the Million and its companion volume, Science for the Citizen (1938), can be seen as the centerpiece of Lancelot Hogben’s career for a number of reasons: They are his best-known works, they were published at around the middle of his life, and they represent a synthesis of his interests. Hogben began as a zoologist, writing technical books on physiology and genetics, and wound up trying his hand at a number of areas—from language, in Essential World English (1963), to political reform, in Interglossa: A Draft of an Auxiliary for a Democratic World Order (1943). Perhaps his most common theme, however, and the one for which he is best known, is the idea of expressing technical subjects in ways that make them intelligible by, and relevant to the daily concerns of, as many people as possible. Mathematics for the Million was followed by more specialized mathematical and scientific books for adults, on probability, statistics, and document design, and by children’s books in the same areas, including The Wonderful World of Mathematics (1955) and Beginnings and Blunders: Or, Before Science Began (1970).

The response to Hogben has been predominantly favorable. Critics praised him for his thoroughness, his expository skills, and his ability to make technical material interesting and understandable, but some have made objections to the polemical nature of his writing. Hogben’s attempts to aim Mathematics for the Million at a general adult audience—rather than a specifically juvenile or young adult one—have limited the discussion of this text as a school book, but it is one whose ease of exposition makes it accessible to most adolescents.