# When Least Is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible

PAUL J. NAHIN
Pages: 400
https://www.jstor.org/stable/j.ctt7rh9x

1. Front Matter
(pp. i-viii)
(pp. ix-xii)
3. Preface to the Paperback Edition
(pp. xiii-xx)
4. Preface
(pp. xxi-xxviii)
5. 1. Minimums, Maximums, Derivatives, and Computers
(pp. 1-36)

This book has been written from the practical point of view of the engineer, and so you’ll see few rigorous proofs on any of the pages that follow. As important as such proofs are in modern mathematics, I make no claims for rigor in this book (plausibility and/or direct computation are the themes here), and if absolute rigor is what you are after, well, you have the wrong book. Sorry!

Why, you may ask, areengineersinterested in minimums? That question could be given a very long answer, but instead I’ll limit myself to just two illustrations (one serious and...

6. 2. The First Extremal Problems
(pp. 37-70)

Ancient mathematicians, the Greeks and the Egyptians of the several centuries before Christ, treated a number of questions of the type we are interested in. They included theisoperimetric problem(what closed curve of given length encloses the greatest area?), and such questions as how to determine the line of minimum length that joins a given point to a given curve. Apollonius of Perga (262–190 B.C.) gave many ingenious geometric constructions to the latter question in his workConics, but generally such problems are now handled easily with calculus. I’ll not discuss Apollonius’ solutions here, then, but if you...

7. 3. Medieval Maximization and Some Modern Twists
(pp. 71-98)

After the ancient isoperimetric problems discussed in the previous chapter, it seems that very little if anything new on minimization/maximization theory appeared in mathematics for a very long time. Indeed, not for anotherfifteen centuriesafter Christ! And then, in 1471, the German mathematician Johann Müller (1436–76), more commonly known today as Regiomontanus, posed a clever maximization problem totally unlike any that had come before. I’ll state it here in slightly more dramatic fashion than he did, but the basic problem itself is as Regiomontanus conceived it.

A somewhat confusing trait of some of the medieval mathematicians makes it...

8. 4. The Forgotten War of Descartes and Fermat
(pp. 99-139)

Modern students, when first introduced to the differential calculus, learn that it was the simultaneous and independent creation of the Englishman Isaac Newton (1642–1727) and the German Gottfried Leibniz (1646–1716). Perhaps they are told that Newton and Leibniz (and their respective followers) engaged in a lengthy and acrimonious debate over intellectual priority, and that Newton continued the battle even after Leibniz’s death, right up to the day of his own death. Almost certainly, however, they are not told anything about an equally nasty war of words between two French mathematicians a half-century earlier, on some of the same...

9. 5. Calculus Steps Forward, Center Stage
(pp. 140-199)

Starting with Fermat’s near miss of the derivative, and the later work by Newton and Leibniz, and others, in developing general differentiation formulas, the differential and integral calculus had, by 1700, becomethemathematics for solving many (but not all, as you’ll see when we get to later chapters) extrema problems. But noteverybodywas convinced that a quantum leap in mathematics had been achieved. As late as 1734, for example, the British philosopher George Berkeley (1685–1753) could rightfully pen an attack on the logical foundations of calculus, as he did inThe Analyst: or a discourse addressed to...

10. 6. Beyond Calculus
(pp. 200-278)

The story of Galileo Galilei (1564–1642), and of his research into the physics of free-fall by dropping various weights from the top of the Leaning Tower of Pisa, is too well known to be retold here. Whatever the truth of the details of that story, it is undeniable that the Italian astronomer was deeply interested in how things move under the influence of gravity. It was that interest that eventually led to what is generally thought to be the first solved problem in the calculus of variations, which was the next great step beyond the calculus of Newton and...

11. 7. The Modern Age Begins
(pp. 279-330)

With the development of the calculus of variations well under way as mathematics entered the nineteenth, attention was redirected to an old problem in geometry. The problem is deceptively simple, but it proved to be a signpost to the future for extrema studies: given a triangle, as shown in figure 7.1, where is the pointPinside that triangle that minimizes the sum of the distances fromPto the three vertices?Pis often calledSteiner’s point, after the nineteenth-century Swiss mathematician Jacob Steiner, whose geometric work on the isoperimetric problem was discussed in section 2.3. In fact, however,...

12. Appendix A. The AM-GM Inequality
(pp. 331-333)
13. Appendix B. The AM-QM Inequality, and Jensen’s Inequality
(pp. 334-341)
14. Appendix C. “The Sagacity of the Bees” (the preface to Book 5 of Pappus’ Mathematical Collection)
(pp. 342-344)
15. Appendix D. Every Convex Figure Has a Perimeter Bisector
(pp. 345-346)
16. Appendix E. The Gravitational Free-Fall Descent Time along a Circle
(pp. 347-351)
17. Appendix F. The Area Enclosed by a Closed Curve
(pp. 352-358)
18. Appendix G. Beltrami’s Identity
(pp. 359-360)
19. Appendix H. The Last Word on the Lost Fisherman Problem
(pp. 361-363)
20. Appendix I. Solution to the New Challenge Problem
(pp. 364-366)
21. Acknowledgments
(pp. 367-368)
22. Index
(pp. 369-372)