Archimedes: What Did He Do Beside Cry Eureka?

Archimedes is one of the greatest mathematicians to have lived. He worked in geometry, physics, and ballistics. His work has spanned the ages. What's more, he did his work without the tools we have today. With tools I don't mean calculators, or computers. I mean without the descriptive mathematical language of equations and the number systems of today. Archimedes held his ideas in total within his mind and was able to solve problems that would take calculus today. And he didn't have calculus either. This book describes many of the problems Archimedes solved and how he approached the problems. We find center of gravity and buoyancy, for example. Each chapter is a thorough discussion of the problems, Archimedes's solution, and, at times, the importance of the problem. The book is slim and handy to take with you for reading whenever you find a few minutes. In that sense, the chapters are short enough to read quickly (although you'll want to spend time going through the equations) so that you get a good view of the approach taken. Finally, the author takes you on a few tangents to explain how Archimedes viewed mathematics and here, too, you see the greatness of Archimedes. He was meticulous and precise in his work. He didn't publish his methods until he was certain of them and he corresponded with other mathematicians to work out problems and discuss various topics. What a good insight into the past and, for us, a good place to draw lessons for our own work.

The thought of a man running naked through the streets shouting with joy over a physical and mathematical discovery is one to warm the hearts of all who value knowledge. When Archimedes experienced this flash of joy, little did he know that his actions would become the genesis of a legend that would last for thousands of years. However, he should be remembered for so much more than that and several of his significant mathematical contributions are explored in this book. It is really amazing to realize how close he was to inventing calculus 22 centuries ago, which was 18 before Newton and Leibniz. With notation that was minimally expressive, he was able to solve problems using a technique that demonstrates at least a rudimentary understanding of the concept of a limit. While many different problems can be solved using calculus, it only takes one breakthrough solution to demonstrate how it can be applied to so many of the others. It can be plausibly argued that algebraic and decimal notations would have been the tools that would have allowed him to overcome those last barriers. One can only speculate on how that would have changed history. The book is not exhaustive and no attempt is made to make it that. Ten of his most significant discoveries are presented and the solutions are those of Archimedes, although modern notation is used. While the proofs are generally easy to follow, one is often left in awe as to how he thought of how to approach some of these solutions. The explanations are succinct, yet thorough, which is the signature of a solid storyteller. Given the answers to the question posed in the title of this book, one can pose another that logically follows. Was Archimedes the greatest mind of all time? If the legends are correct, then the answer is probably yes. However, even if the unconfirmed stories are false, the mathematical and mechanical discoveries should make him a legend for more than one short stint of becoming a 'natural man.' Published in Journal of Recreational Mathematics, reprinted with permission.

The author's aim is to make what he views "as Archimedes' most mathematically significant discoveries accessible to the busy people of the mathematical community." In this he succeeds admirably. The book is not only understandable by anyone who "recognizes the equation of a parabola," but is also very well written in a style that brings out the beauty of the mathematical ideas discussed, as well as the power of Archimesdes' creativity. As the author points out, the book treats most of Archimedes' mathematical discoveries. The presentation cleverly integrates Archimedes' own writing with the author's modern explanation of the ancient discoveries. Frequently, before a main idea is introduced, a quotation from Archimedes' own writing is presented in which the master reveals his thinking about what he had accomplished in that particular topic.In addition to providing the scientific community with a detailed account of Archimedes' main mathematical discoveries and an insight into the ancient master's thinking, this book, I believe, can be useful in the classroom in a variety of ways. The most obvious use, of course, would be in designating it as a textbook or a reference in courses on the history of calculus or, more generally, on the history of mathematics. But it would also make an excellent textbook for a course on axiomatic mathematics: the book starts with a few axioms from which Archimedes had developed the theory of center of gravity and used it throughout a good part of the material covered in the book, including the development of the volumes of a paraboloid and a sphere and the theory of floating bodies.In sum, this is an excellent book that should be within reach of any person interested in mathematics or science.