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Basic Calculus: From Archimedes to Newton to its Role in Science

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How can we say that introductory courses in calculus at universities are meaningful if the students are never involved in math as professionals? What motivation can we offer them for studying it? This textbook, by Alexander J. Hahn, provides an outstanding answer to the question backed up by the author's precious teaching experience at the University of Notre Dame.

After reading the text, the reader will start to see calculus as a gift by our ancestors that helps us to analyze practical daily problems: calculus as a culture to be passed on to the next generation. Firstly, as the author says, "this text could as well have the title The Story of Calculus." As we read it, we find ourselves reliving history with the great persons like Archimedes, Descartes, Leibniz and Newton. We feel the activity and wisdom of the characters close-up, and we even experience their joys and sorrows as if they were our own. In a way, this book is a historical novel. It shows what calculus looks like as a critical tool that has helped to clear up the mysteries of the universe. Secondly, "the purpose of this text is to demonstrate its broad and formidable informative power." As the author explains, calculus enables us to designing telescopes, to read nuclear clocks, to design suspension bridges, and to understand the interior ballistics of rifles, the rocket equation, gravity, and the expanding universe. Economic subjects, such as banking, CPI, market mechanisms, cost analysis are also covered with full explanations. Books with such range and depth are rare indeed. It is easy to understand why the author received an award for teaching excellence.

Of course, "the emphasis is always on the careful development of the mathematics and information that it provides", and most of the topics of first-year calculus courses (including differential equations) are covered (but partial differentials and double/triple integrals are not). The exercise section of each chapter contains advanced explanations of historical, scientific, and mathematical topics, and is organically integrated with the text. The total number of the problems in all the 15 chapters is close to 700. With its many figures and illustrations, as well as full derivations of the equations, this text is also suitable as a supplementary or a self-study manual.

I strongly recommend Basic Calculus to those who have doubts about "the usual math training" which sometimes makes us feel like machines (not humans), as a rare and engaging view of mathematics from a different angle. I have found the contents of Hahn's textbook ideal for my students in general physics and calculus courses at Hosei University, Tokyo, and I am now completing a translation of this book into Japanese in collaboration with my colleague Professor Ichimura.

If you are deeply curious about the amazing ability of mathematics to define, describe and predict the physical world and its behavior (incl. the solar system) you will be thrilled with this book. It concentrates on the essence of the matter, basic calculus, and includes real-world applications set within the context of some of history's most important scientific questions.

The author clearly demonstrates that he not only possesses a great curiosity, fluency, and appreciation for the subject but also thrives on imparting these things to others. He has provided a great deal of supplementary information on his web site including a detailed description of the contents, scope and focus of the book.

The Solution Manuals ARE available from the author simply by e-mailing him at: hahn.1@nd.edu

Most calculus books make some compromise in presenting the material. In the case of a thoroughly rigorous text, that compromise most often means sacrificing historical context, intuitive understanding, and real-world application (even though the book may be "exercise-rich" with contrived examples). Basic Calculus successfully navigates a difficult (and different) course, focusing on these commonly sacrificed areas and effectively presenting the pearls of calculus knowledge without delving too deeply into eye-glazing minutiae. In lucid and interesting style, it accomplishes exactly what it sets out to do - it imparts the essence of the matter, in context.

This splendid book aims to develop calculus from within its rich historical context and to demonstrate its power across a range of disciplines. The author succeeds admirably. Two hundred pages devoted to key ideas in the history of mathematics and science lead smoothly into calculus as we know it today. The remaining three hundred plus pages cover the usual topics, but with attention given to an extraordinary spread of interesting problems in science and business. The explanations of concepts and notation are as lucid as any I have encountered in a basic calculus book.

Because one of the distinguishing features of Basic Calculus from Archimedes to Newton to its Role in Science is its historical dimension, something should be said about the criticism of one reviewer that the book oversimplifies the history by using modern notation. Yes, Hahn does tidy things up. (Very nicely, I might add.) But what else can anyone really do? As Hahn notes, Leibniz's cryptic first work on calculus - Nova methodus pro maximis et minimis, itemque tangentibus...calculi genus - bewildered even his friends, the brothers Bernoulli. These famous mathematicians found Leibniz's article "an enigma rather than an explication." Hahn could try to unriddle the Nova methodus for us, explaining in detail all the fuzzy concepts and strange notation that baffled the Bernoullis. But that hardly seems the thing to do in a basic calculus book. Better to do just what Hahn does - seize on the essential ideas and use everything now at a mathematician's command to bring them into a clear light. Hahn has an excellent sense of just how far to go. The result is a truly extraordinary book that will amply reward readers looking for something special.

If you want a sound understanding and appreciation of basic calculus, have a look at Alexander Hahn's "Basic Calculus". An author presenting calculus could seek to impart (A) Proficiency in its techniques, (B) Justification of its methods, (C) Understanding of its principles or (D) Appreciation of its significance in the history of mathematics and science. Many pre-engineering calculus texts do a good job of drilling proficiency and building an appreciation of calculus by showing its utility in scientific applications. It is, however, easy to walk away from a pre-engineering calculus course without an intuitive grasp of the subject, or any sense of why calculus is important in the history of thought. There are also many mathematically oriented calculus texts that present the logical structure of calculus very well. These books stress precise concepts and proofs that were actually developed after calculus had revolutionized the physical sciences. Since the precise justification was developed after the fact, it is clear that one can understand the concepts without understanding the justification. Unfortunately many of formal calculus texts also show that it is possible to understand the justification (follow the proofs) without an understanding of the subject matter. History of mathematics books can convey an appreciation of why calculus was a great intellectual achievement, but they often tell the reader very little about the mathematics itself. There are many fewer books that attempt to convey an understanding of calculus and an appreciation of its significance in the history of mathematics and in science. Hahn's book addresses this neglected area.

Hahn's book has two parts, "From Archimedes to Newton" and "Calculus and the Sciences". The first part focuses on building an understanding of basic concepts and appreciation of calculus as a watershed in the history of mathematics. Hahn starts with Greek geometry and shows its application to astronomy. This serves as a review of some pre-calculus mathematics, as a historical preamble, and as an introduction to a set of problems that become a challenge, motivation, and later an important triumph of calculus. The first chapters paint a vivid "hands-on" picture of an intellectual world in which geometry was completely separate from the theory of numbers and algebra, and where astronomy and mechanics were thought to be literally as different as heaven and earth. These chapters are typical. They teach some mathematical ideas, show why they represent important advances inside mathematics, and illustrate major applications that make them important outside mathematics.

The narrative continues with advances of the late Hellenistic world: trigonometry, conic sections, and mechanics. Hahn then skips nearly all of the middle ages, which though beloved of historians, in the history of mathematics were truly dark ages. He takes up the story again with the early scientific revolution, Copernicus, Galileo, Kepler, Descartes. This discussion shows how the gap between geometry and algebra began to close with the introduction of analytic geometry, and introduces the work on the problems of tangents and quadrature that set the stage for calculus.

The story of the birth of calculus concludes with Leibniz, Newton, and Newton's masterwork, "The Principia", which closes the gap between our understanding of terrestrial and celestial motion. The problem with presenting ideas in the context of their history is that concepts are rarely clear at their inception. This can make the historical presentation murky and unnecessarily pedantic. Hahn avoids pedantry by developing what might be called historical reconstructions of important ideas. For example, Hahn reconstructs both Leibniz's and Newton's presentation of the Fundamental theorem of calculus. His narrative captures the distinctive differences in the two men's approach but the presentation does not do without the conceptual clarity of the notion of mathematical functions, in the interests of historical purity. There is also a good detailed account of Newton's derivation of Kepler's Laws. In this case Hahn's reconstruction goes as far as using Newton's actual diagrams and following closely the steps of Newton's derivation. Students of history of science will find this part of the book very enlightening, and may perhaps even forgive Hahn the practical expedient of using some modern notation.

The second part of the book covers the growth and maturation of calculus, and focuses on building an appreciation of the diverse applications of calculus in the sciences. Hahn starts by covering some of the ideas of nineteenth century calculus that made its concepts precise and its justification sound. However, he considers these ideas only in so far as they help to clarify concepts and aid understanding. You will, for example, find an exposition of why the idea of differentiability is a good analysis of the intuitive idea of the smoothness of a function, but you will not find a proof of L'Hospital's Rule. The remainder of the text discusses applications. Each chapter introduces some new mathematics and then exploits it along with the basic concepts of calculus to illuminate some application area. The applications are essentially the "interesting stuff" that is left out of most purely mathematical texts. The reader will be able to "see" the calculus in the ordinary furnishings of our technical society: Telescopes, Suspension Bridges, Nuclear Reactors, E. coli, Banking, Rockets and Guns.

Part Two builds on the foundation laid in Part One to not only give the reader understanding and appreciation of calculus, but does it in a way that is consistently interesting and enjoyable.

Calculus has traditionally been the window through which college students first glimpse the intimate relationship between mathematics and the physical sciences. Unfortunately, few calculus textbooks give more than an inkling of this relationship, perhaps because of their author's desire to present the mathematics in full detail and the lack of a common background of scientific knowledge among our students. Alexander Hahn's Basic Calculus is a delightful exception; here, calculus is developed as the thread uniting the solution of historically important scientific problems by the likes of Archimedes, Ptolemy, Leibniz, and Newton. The scientific applications are not only of historical interest; they include potassium-argon dating in geology, modeling the forces and the motion of a bullet in a gun barrel, examples of mathematics in microeconomics, and even a thoughtful discussion of cosmology. The scientific ideas are developed in far more detail than is customary in mathematics textbooks. Each is presented as a story---the problem is placed in a historical context, including vignettes of the people involved; then the student is led through the basic science and the associated mathematical analysis, using technically accurate data.

The exercises at the end of each chapter are unusual too. After a few routine skill-building problems, the remainder pose small scientific problems to extend the discussion in the text. Students who work through any one of the many scientific topics Hahn treats in this way, working the associated exercises, will feel a justified sense of accomplishment. And in the process they will gain an accurate sense of how calculus developed and how it continues to play a key role in science and engineering. This fall my college has been debating how to structure the science component of our general education requirements, and if our entering students had enough mathematical preparation to profit from a study of calculus, I could not think of a better way to introduce them to science and mathematics than through a course from Hahn's book. This book would certainly be my choice of text for an honors course. I would expect it to convert some students who had planned on another major field of study to change to science or mathematics, and those who did not change their plans would still have a lasting experience of the attraction of the scientific enterprise.

I've been wanting to learn calculus for several years now, but at 50 years old I need reasons to stuff even one new idea into an already crowded and reasonably well balanced head. I keep getting distracted looking for context: answers to the questions which always pop up in an older brain trying to make things FIT! And in the standard texts and even books for laymen I haven't been getting my questions satisfactorily answered. Until now.

Thank you Alexander J. Hahn for CONTEXT, CONTEXT, CONTEXT.

Basic Calculus: From Archimedes to Newton to Its Role in Science
is a beautifully done text. It is very clearly written and
logically organized, tracing the development of calculus with
many interesting examples from the physical world and man's
quest to understand the physical world. The text is concise
and so readily understood as to be elegant. Finally, all of the
solutions to the exercises are given at Professor Hahn's internet
site. Its address is www.nd.edu/~hahn/ One way to remember the
website is that the letters "nd" are for Notre Dame, where Dr. Hahn teaches.

What this book is not is a traditional calculus text. It covers a lot of traditional topics, but not in a familiar way. It is not terribly rigorous, nor does it need to be. It is designed to fill the first two semesters of calculus. There are a LOT of books that do this in the traditional way, that is they scare the life out of the student :-). It is my belief that this book will take a lot of the mystery out of calculus, since it develops the subject in the context of applications. I also think that most students will find the approach engaging. There are plenty of practice problems at the ends of the chapters, and some are quite challenging.

The focus of this book is not to present calculus as a theory, a thing which most students are simply not prepared for at this level. Rather it is to present calculus as the pragmatic development of methods to solve certain classes of problems. In this regard it does a fantastic job. Along the way the students's algebraic, geometric, and trigonometric skills are all tested and firmed up.

The notion of the limit, such a mystery to most freshmen (and, truth-be-told, to many upper-level undergrads) is given a strong intuitive thrust right from the beginning.

If you want more problems, get the Schaum's outline book and read them side-by-side.

I loved this book. I find Calculus extremely intimidating and I never have quite grasped the importance of this topic. This book seems to pull things together, presents material you would not find in a "regular" text.

It's a lovely book. What's especially nice about it is how it places calculus firmly in the world, first by embedding it in history and second by showing that its purpose is to solve interesting problems, physical and otherwise. That's different from, and much better than, ordinary calculus books, with their handed-down-from-Sinai exposition and phony "applications". A student who gets through the book will have a liberal education in mathematics, physics, and economics.