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

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.


Basic Calculus   September 7, 2000
13 out of 15 found this review helpful

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.


A really different calculus textbook.   November 10, 1999
10 out of 12 found this review helpful

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.


Finally, a calculus text for the mature student!   November 25, 1999
Ken Christenson (Parry Sound, Ontario)
9 out of 11 found this review helpful

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.


A Wonderful Introduction to Calculus   February 18, 2002
George E. Hrabovsky (Madison, Wisconsin United States)
4 out of 5 found this review helpful

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.