Calculus Gems: Brief Lives and Memorable Mathematics

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.

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.

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.