PREFACE

“ . . . I consider that without understanding as much of the abstruser part of geometry, as Archimedes or Apollonius, one may understand enough to be assisted by it in the contemplation of nature; and that one needs not know the profoundest mysteries of it to be able to discern its usefulness. . . . I have often wished that I had employed about the speculative part of geometry, and the cultivation of the specious [symbolic] algebra I had been taught very young, a good part of that time and industry that I spent about surveying and fortification. . . .”

ROBERT BOYLE

I believe as firmly as I have in the past that a mathematics course addressed to liberal arts students must present the scientific and humanistic import of the subject. Whereas mathematics proper makes little appeal and seems even less pointed to most of these students, the subject becomes highly significant to them when it is presented in a cultural context. In fact, the branches of elementary mathematics were created primarily to serve extra-mathematical needs and interests. In the very act of meeting such needs each of these creations has proved to have inestimable importance for man’s understanding of the nature of his world and himself.

That so many professors have chosen to teach mathematics as an integral part of Western culture, as evidenced by their reception of my earlier book, Mathematics: A Cultural Approach, has been extremely gratifying. That book will continue to be available. In the present revision and abridgment, which has been designed to meet the needs of particular groups of students, the spirit of the original text has been preserved. The historical approach has been retained because it is intrinsically interesting, provides motivation for the introduction of various topics, and gives coherence to the body of material. Each topic or branch of mathematics dealt with is shown to be a response to human interests, and the cultural import of the technical development is presented. I adhered to the principle that the level of rigor should be suited to the mathematical age of the student rather than to the age of mathematics.

As in the earlier text, several of the topics are treated quite differently from what is now fashionable. These are the real number system, logic, and set theory. I tried to present these topics in a context and with a level of emphasis which I believe to be appropriate for an elementary course in mathematics. In this book, the axiomatic approach to the real numbers is formulated after the various types of numbers and their properties are derived from physical situations and uses. The treatment of logic is confined to the fundamentals of Aristotelian logic. And set theory serves as an illustration of a different kind of algebra.

The changes made in this revision are intended to suit special groups. Some students need more review and drill on elementary concepts and techniques than the earlier book provides. Others, chiefly those preparing for teaching on the elementary level, need to learn more about elementary mathematics than their high school courses covered. Teachers of twelfth-year high school courses and one-semester college courses often found the extensive amount of material in Mathematics: A Cultural Approach rather disconcerting because it offered so much more than could be covered.

To meet the needs of these groups I have made the following changes:

1.   Four of the chapters devoted entirely to cultural influences have been dropped. The size of the original book has thereby been reduced considerably.

2.   A few applications of mathematics to science have been omitted, primarily to reduce the size of the text.

3.   Some of the chapters on technical topics, Chapter 3 on logic and mathematics, Chapter 4 on number, Chapter 5 on elementary algebra, and Chapter 21 on arithmetics and their algebras have been expanded.

4.   Additional drill exercises have been added within a few chapters, and a set of review exercises providing practice in technique has been added to each of a number of chapters.

5.   Improvements in presentation have been made in a number of places.

With respect to use in courses, it is probably true of the present text, as it is of the earlier one, that it contains more material than can be covered in some courses. However, many of the chapters as well as sections in chapters are not essential to the logical continuity. These chapters and sections have been starred . Thus Chapter 10 on painting shows historically how mathematicians were led to projective geometry (Chapter 11), but from a logical standpoint, Chapter 10 is not needed in order to understand the succeeding chapter. Chapter 19 on musical sounds is an application of the material on the trigonometric functions in Chapter 18 but is not essential to the continuity. The two chapters on the calculus are not used in the succeeding chapters. Desirable as it may be to give students some idea of what the calculus is about, it may still be necessary in some classes to omit these chapters. The same can be said of the chapters on statistics (Chapter 22) and probability (Chapter 23).

As for sections within chapters, Chapter 6 on Euclidean geometry may well serve as an illustration. The mathematical material of this chapter is intended as a review of some basic ideas and theorems of Euclidean geometry and as an introduction to the conic sections. Some of the familiar applications are given in Section 6-3 (see the Table of Contents) and probably should be taken up. However the applications to light in Sections 6-4 and 6-6 and the discussion of cultural influences in Section 6-7 can be omitted.

Some of the material, whether or not included in the following recommendations for particular groups, can be left to student reading. In fact, the first two chapters were deliberately fashioned so that they could be read by students. The objective here, in addition to presenting intrinsically important ideas, was to induce students to read a mathematics book, to give them the confidence to do so, and to get them into the habit of doing so. It seems necessary to counter the students’ impression, resulting no doubt from elementary and high school instruction in mathematics, that whereas history texts are to be read, mathematics texts are essentially reference books for formulas and homework exercises.

For courses emphasizing the number concept and its extension to algebra, it is possible to take advantage of the logical independence of numerous chapters and use Chapters 3 through 5 on reasoning, arithmetic, and algebra and Chapter 21 on different algebras. To pursue the development of this theme into the area of functions one can include Chapters 13 and 15.

Courses emphasizing geometry can utilize Chapters 6, 7, 11, 12, and 20 on Euclidean geometry, trigonometry, projective geometry, coordinate geometry, and non-Euclidean geometry respectively. Some algebra, that reviewed in Chapter 5, is involved in Chapters 7 and 12. If knowledge of the material of Chapter 5 cannot be presupposed, this chapter must precede the treatment of geometry.

The essence of the two preceding suggestions may be diagrammed thus:

Of course, starred sections in these chapters are optional.

For a one-semester liberal arts course, the basic content can be as follows:

Chapter 2	on a historical orientation,
Chapter 3	on logic and mathematics,
Chapters 4 and 5	on the number system and elementary algebra,
Chapter 6	through Section 6–5, on Euclidean geometry,
Chapter 7	through Section 7–3, on trigonometry,
Chapter 12	on coordinate geometry,
Chapter 13	on functions and their uses,
Chapter 14	through Section 14–4, on parametric equations,
Chapter 15	through Section 15–10, on the further use of functions in science,
Chapter 20	on non-Euclidean geometry,
Chapter 21	on different algebras.

Any additional material would enrich the course but would not be needed for continuity.

Though the teacher’s problem of presenting material outside the domain of mathematics proper is far simpler with this text than with the earlier one, it may still be necessary to assure him that he need not hesitate to undertake this task. The feeling that one must be an authority in a subject to say anything about it is unfounded. We are all laymen outside the field of our own specialty, and we should not be ashamed to point this out to students. In contiguous areas we are merely giving indications of ideas that students may pursue further in other courses or in independent reading.

I hope that this text will serve the needs of the groups of students to which it is addressed and that, despite the somewhat greater emphasis on technical matters, it will convey the rich significance of mathematics.

I wish to thank my wife Helen for her critical scrutiny of the contents and her conscientious reading of the proofs. I wish to express, also, my thanks to members of the Addison-Wesley staff for very helpful suggestions and for their continuing support of a culturally oriented approach to mathematics.

New York, 1967

M.K.