Well designed and executed, this historical account of mathematical ideas is a pleasure to read. Sondmeimer and Rogerson tell the story of the evolution, from ancient times to the nineteenth century, of two concepts: number and infinity, devoting roughly half of the book to each idea.
The text began as a series of lectures to beginning undergraduates, to whom mathematics is usually presented as a static given. The authors aim both to convey some sense of how mathematics is really done and to cast light on how the subject acquired its present structure and conventions. From the preface:
In fact many of the ideas about the nature of mathematics which are taken for granted today are of very recent origin, and it may reassure the student, as he grapples with concepts that he finds difficult and that are not at all ‘obvious’ to him, to know that these concepts were also not obvious to people in the past, and that it took much argument, confusion, and searching for the ‘right’ answer before the modern approach was created.
To display some of this argument, confusion, and searching, the authors have selected several themes within the concepts of number and infinity and discuss the motivations and personalities behind their historical development. This story of birth pangs frames the presentation of some revolutionary, profound, and enduring ideas.
To convey these ideas most effectively, the authors avoid an exposition dominated by either formal proof on the one hand or mere anecdote on the other. Instead, they rely on examples to craft their account, creating a comfortable narrative punctuated by equations and informal proofs. Readers interested in the development of mathematical concepts but daunted by mathematical formalism are invited to skip or take for granted the more technical passages. Indeed, the narration alone, lucid and direct, forms a stimulating chronicle, and such readers will still develop an intuition for the ideas.
The equations and proofs are not of the type requiring facility with the extreme abstraction of some modern mathematical constructs. On the contrary, most of the mathematical formalism is quite accessible to anyone willing to practice a little careful deduction. The effort is well spent, as working through the selected pioneering results of mathematicians past ennobles the reading experience and renders the text that much more engaging. (Examples are Archimedes’ quadrature of the parabola and Euler’s summing of the inverse squares to obtain π2 /6.) As an aid here, exactly the right amount of figures pepper the book, most of them uncluttered yet complete. Once readers finish this short Dover reprint, they will have acquired a sturdy understanding of numbers and infinity, discovered a new appreciation for mathematical ideas and methods, and begun to hunger for more. Few books deliver this much impact in a mere 159 pages.
The first chapter focuses on rational numbers (integers, then fractions) as they have been represented from prehistory to the digital computer age. In accord with their practice throughout the book, the authors are already offering the reader explicit invitations to pick up a pencil and try some mathematics. For example, the ancient Egyptians expressed all fractions as the sum of unit fractions. A demonstration by Fibonacci that this can always be done is presented almost in its entirety, the reader invited to supply the final touches.
The second chapter details some number theory. The proof by contradiction given by Euclid that there is an infinity of primes provides the authors an opportunity to comment on mathematical thinking. (The proof in brief: Assume p is the largest prime. Then 2(3)(5)(7)…(p) + 1 is a larger prime.) “This is an elegant proof, simple and tailored to the problem in hand. It does not attempt to do the ‘obvious’ thing, which would be to look for the next prime after p. That question presents a much harder problem because there is no simple law for the sequence of successive primes. Euclid ’s proof instead looks for some prime beyond p, and this is all that is needed for the purpose of the proof. Asking the right questions in the right way is often the secret of success in mathematics.” Similar observations are made throughout the book, to the great benefit of the student.
Chapters three to seven unfold to the reader a definition of number that becomes progressively broader (negative integers, fractions, real numbers, complex numbers) and more refined (rational/irrational, algebraic/transcendental). The very accessible chapter three should be required reading for prospective teachers. Subtraction and division prompt the introduction of negative integers and fractions, respectively. Then irrational numbers are presented in multiple ways, debuting via a few ancient Pythagorean geometrical arguments and also by an inspection of infinite decimals. This leads to the observation that “the definition of an irrational number necessarily involves in some way an infinite set of integers.” Now, aware that no discussion of number is complete without a discussion of infinity, the reader begins to anticipate chapters 8–11.
The book is repeatedly looking forward and backward. Several explicit cross-references act as guideposts. More fundamentally, the authors reiterate earlier observations in new contexts. As an example, the idea of showing existence that arose in the proof of the infinity of the primes is touched on again when illustrating one of Gauss’s proofs of the fundamental theorem of algebra in chapter 6. After a mention that the Greeks understood the necessity of demonstrating the existence of mathematical objects prior to trying to prove theorems about them, Gauss is attributed with recognizing that an existence proof for roots of the general n-th degree polynomial equation need not involve a formula for computing them. A differently flavored scenario concerning existence involves Liouville’s demonstration that transcendental numbers exist. He constructed a certain real number and showed that it is not algebraic; hence, it must be transcendental. The authors point out that the determination of whether a given number is transcendental is an altogether much more difficult problem, and they end the chapter by presenting simple examples of real numbers whose nature is of yet unknown.
Chapters 8–11 are devoted to the idea of infinity. The considerable contributions of the ancient Greeks receive excellent treatment in chapter 8, one of my favorites. Here are Zero’s paradoxes, Hippocrates’ squaring of the lune, the method of exhaustion, and some of Archimedes’ amazing accomplishments, including the quadrature of the parabola and the integral calculus in embryo. One cannot exit this chapter without an admiration of Archimedes’ genius. Though the story focuses primarily on geometry, the authors look back over their shoulders at discussions about the real numbers earlier in the book. After proving, via the method of exhaustion, that the areas of two circles are in the ratio of the squares on their radii, they reflect, “This is a logically rigorous proof (given by Euclid): all the mystery of the infinite process lies in the axiom of continuity!”
The theorems and proofs in this chapter are deep, yet accessible. I intend to have students in my geometry course for prospective teachers prove some of these results. For some proofs, I will provide hints. For others I will allow students to study the proofs, possibly in groups, and demonstrate them to the class. Most of the chapters are like this one in offering up such accessible and engaging historical milestones, satisfying problems with which students can interact at various levels.
Calculus students would derive much benefit from working through chapter 9, which illustrates some of the motivations and early results of the subject. Contributions of Cavalieri open the chapter. First is presented his view of plane areas being the sum of lines and solid volumes being the sum of planes. A trio of figures illustrate where this intuition leads: both to some correct conclusions and to some results that are clearly wrong. Cavalieri’s generalization of Archimedes’ quadrature of the parabola is given next, complete with a proof-without-words from the Arabs for the summation of the first n cubes, a formula usually assumed without inspection when first studying the definite integral. Fermat receives good coverage for his work on integration and differentiation. His approaches to the latter both motivate and supplement the perspective of the standard calculus course. The contributions of many other mathematicians are mentioned, the authors clearly intent to exhibit calculus not as the product solely of two men but as the discovery of many pioneers. Finally Newton and Leibniz are discussed along with their great accomplishments.
The evolution of the function concept is related in chapter 10. Fourier series were the crucible in which many ideas arose, in particular concerning limits of functions. A limit of curves can equal a line, and an infinite sum of continuous functions can have discontinuities, prompting the idea of uniform convergence. Refinements in the ideas of continuity and differentiability are next discussed, and the chapter ends with a look at fractals. This crowd-pleasing topic is followed by another in the final chapter: transfinite arithmetic. The usual ideas are here, including Cantor’s seminal developments. Mentioning contemporary resistance to Cantor’s ideas, the authors repeat once more an observation they have injected into the discussion various times throughout the book: that mathematics progresses by transcending the old way of looking at things and pursuing a fresh approach, one that will push back the horizon and reveal new realms.
This book provides a wealth of stimulation for its length. Advanced high school students, beginning undergraduates, and laypeople interested in mathematics should find it a real treat. More advanced undergraduates in technical disciplines will gain new perspectives and a greater coherence of the mathematics they have learned. All will want to delve deeper, for within this historical account of mathematical concepts dwells an unexpected vibrance, enchanting and galvanizing. Readers’ hunger for more is answered by a bibliography keyed to each chapter. A list of 34 possible essay topics suggests further paths of exploration for the student, while providing the instructor a ready list of subjects for writing assignments. Useful, informative, and engaging, this short history is a joy to read. I highly recommend it.
David A. Huckaby is an assistant professor of mathematics at Angelo State University.