Includes " Biographical history, as taught in our public schools, is still largely a history of boneheads: ridiculous kings and queens, paranoid political leaders, compulsive voyagers, ignorant generals — the flotsam and jetsam of historical currents. The men who radically altered history, the great scientists and mathematicians, are seldom mentioned, if at all." — Martin Gardner
[Why study mathematics? -- 12/31/06]

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Product Description

A classic book is back in print! It can be used as a supplement in a Calculus course, or a History of Mathematics course. The first half of Calculus Gems entitles, Brief Lives is a biographical history of mathematics from the earliest times to the late nineteenth century. The author shows that Science and mathematics in particular is something that people do, and not merely a mass of observed data and abstract theory. He demonstrates the profound connections that join mathematics to the history of philosophy and also to the broader intellectual and social history of Western civilization. The second half of the book contains nuggets that Simmons has collected from number theory, geometry, science, etc., which he has used in his mathematics classes. G.H. Hardy once said, A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. This part of the book contains a wide variety of these patterns, arranged in an order roughly corresponding to the order of the ideas in most calculus courses. Some of the sections even have a few problems. Professor Simmons tells us in the Preface of Calculus Gems: I hold the naïve but logically impeccable view that there are only two kinds of students in our colleges and universities, those who are attracted to mathematics; and those who are not yet attracted, but might be. My intended audience embraces both types. The overall aim of the book is to answer the question, What is mathematics for? and with its inevitable answer, To delight the mind and help us understand the world.

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Product Details

• Amazon Sales Rank: #259830 in Books
• Published on: 2007-01-12
• Released on: 2007-01-12
• Original language: English
• Number of items: 1
• Binding: Hardcover
• 376 pages

Review
This is a wonderful interdisciplinary source that makes connections among mathematics, humanities, social science, science, and philosophy. Appropriate for upper-division or graduate students or as as resource for secondary or or higher education mathematics or science teachers. --Jeanne Ramirez Mather, The Mathematics Teacher

About the Author
George Simmons has the usual academic degrees (CalTech, Chicago, Yale), and taught at several colleges and universities before joining the faculty of Colorado College in 1962, where he is (was) Professor of Mathematics. He is also the author of Introduction to Topology and Modern Analysis (McGraw-Hill, 1963), Differential Equations with Applications and Historical Notes (McGraw-Hill, 1972, 2nd edition 1991), Precalculus Mathematics Mathematics in a Nutshell (Janson Publications, 1981); Wipf & Stock, Eugene, Orego at present), Calculus with Analytic Geometry (McGraw-Hill, 1985, 2nd edition 1996), and with Steven Kranta) Differential Equations: Theory, Technique, Practice (McGraw-Hill 2006). When not working or talking or eatiing or drinking or cooking, Professor Simmons is likely to be traveling (Western and Southern Europe, Turkey, Israel, Egypt, Russia, China, Southeast Asia), playing pocket billiards (pool), or reading (literature, history, biography and autobiography, science, and enough thrillers to achieve enjoyment without guilt).

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Customer Reviews

A treasure for lovers of advanced math
This is a terrific book for anyone who is fascinated by the workings of great minds. In the first half of the book, Mr Simmons takes us through the lives of 33 of the most notable mathematicians on history, from pre-Archimedes, to the late 19th century. These are wonderful stories of great thinkers, and, to my relief, Mr. Simmons walks us through derivations of many famous formulas and discoveries. Do not fear that this is all calculus--much of the book is brilliant algebra, geometry, and number theory, and fully comprehendable by anyone with a good non-calculus high school education. But tasty calculus delights abound for those who are up to the challenge. The second half of the book, called Memorable Mathematics, are proofs and insights into some of the most wonderful discoveries of pre-twentieth century Mathematics. Topics include primes, irrational numbers, perfect numbers, proofs of infinite series involving e and pi, and a marvelous treatise on the cycloid and brachistochrone problems. Interspersed are interesting anecdotes about these great thinkers, including Newton, Euler, the Bernoulli Brothers, and Leibniz, just to name a few.

I loved this book, even though I am not a mathematician by profession. The best part about it is that not only are these famous formulas presented, but most are also proven, which goes along way in showing just how amazingly the brilliant minds of these historical geniuses work.

Yes, they are truly gems of exposition
Gems is the correct word to describe the tales in this book. These are some of the best stories of the people who made mathematics what it is today that you will ever find. The first stories are about the ancient Greeks and that amazing flowering of intellectual achievement that suddenly arose on the shores of the Aegean and eastern Mediterranean seas. We will probably never know what events fertilized this amazing garden, but suddenly the purely intellectual pursuits of geometry, number theory and logic became the pinnacle of civilization.
Unfortunately for us all, but an accurate reflection of historical reality, the first set of stories ends at 415 AD and the next does not begin until 1571 AD. However, the pent-up intellectual ferment led to many dramatic changes in a very short time. The germination of calculus could not occur until many philosophical viewpoints were overthrown. Geocentric views of the universe were completely incompatible with the ideas of Kepler and people had to once again believe that the pursuit of knowledge was a worthy task. It was also necessary for the opposition of the established churches to be reduced to a point where at least it was accepted for people to challenge doctrine. This process took over a century, and was not without many conflicts. Two of the greatest minds of the seventeenth century, Blaise Pascal and Isaac Newton, were emotionally unstable and it was manifested in some unusual religious writings. It is conceivable that a longer-lived and more focused Pascal would have invented calculus.
After the second start, the development of calculus then became an inexorable movement. Great intellects followed each other, each building a new section of the castle that is calculus. The author weaves the thread of how each required the achievements of those who preceded them. Personalities and their personal lives also form an integral part of the stories, which makes it much more lively to read. The people who created calculus were real people with sometimes unusual traits. What is striking is that while some were clearly known to be prodigies at an early age, others were quite ordinary in their youth. Newton's youth was quite undistinguished and Weierstrass did not blossom until his forties.
This is an ideal book for the study of the history of mathematics. Not only are the facts of development put forward in a sequential order, but you learn about the lives and personalities of the people who made it what it is today. They did not always succeed, were from widely different backgrounds and some of them led very unhappy lives. This should show us all that there is not one specific mathematical personality, but one mathematical discipline that can attract a wide variety of personalities.

Some good gems
The "memorable mathematics" part of this book treats many interesting things. One is "a simple approach to E=Mc^2". First we substitute the relativistic notion of mass m=m_0/sqrt(1-v^2/c^2) into F=ma=d/dt(mv) to get the relativistic F=ma, which is F=m_0a/(1-v^2/c^2)^(3/2). The work done by the force moving a particle from 0 to x is energy=integral from 0 to x of relativistic force=(change in mass)c^2. Another topic is rocket propulsion in outer space. Consider a rocket with no forces acting on it. Then mv is constant since d/dt(mv)=ma=F=0. The rocket moves forward by throwing out parts of its mass in the form of exhaust products with velocity -b relative to the ship. Since mv is constant we have mv at t=mv at t+dt, i.e. mv=(m+dm)(v+dv)+(-dm)(v-b), which reduces to dv=-b(dm/m) which we can integrate to get, e.g. the burnout velocity for given initial conditions b and fuel/m. But the best topics are two Euler classics. First the summation of the reciprocals of the squares. (sin x)/x has the roots pi, -pi, 2pi, -2pi, ..., which suggests that the "infinite polynomial" (sin x)/x=1-x^2/3!+x^4/5!-x^6/6!+... should factor as (1-x^2/pi^2)(1-x^2/4pi^2)(1-x^2/9pi^2)... Multiplying this out and equating coefficients of x^2 we get 1/pi^2+1/4pi^2+1/9pi^2+...=1/3!, so the sum of the reciprocals of the squares is pi^2/6. Also, as a bonus, if we put x=pi/2 in the infinite product for (sin x)/x we get Wallis's infinite product for pi. Euler's study of the reciprocals of the squares also led him to the zeta function zeta(s)=1+1/2^s+1/3^s+..., which he saw can also be written as a product: sum over all integers of 1/n^s = product over all primes of 1/(1-1/p^s), as we see by expanding each factor on the right hand side as a geometric series and multiplying out the product, which gives the reciprocal of each possible product of primes, to the power s, exactly once, i.e., by unique prime factorisation, the left hand side. This charming formula immediately pays off by yeilding a new proof of the old theorem that there are infinitely many primes: because zeta(1)=1+1/2+1/3+...=infinity we have also zeta(1) = product over all primes of 1/(1-1/p) = infinity, which is clearly possible only if there are infinitely many primes.