Mathematical Masterpieces: Further Chronicles by the Explorers (Undergraduate Texts in Mathematics / Readings in Mathematics)

Fully evolved teaching text, developed and class tested for over 15 years Uses original sources to teach the history of mathematics – an original yet fascinating approach Heavily illustrated with line drawings and half-tones, including many historical photographs Each chapter is self-contained and could be used independently Experience the discovery of mathematics by reading the original work of some of the greatest minds throughout history. Here are the stories of four mathematical adventures, including the Bernoulli numbers as the passage between discrete and continuous phenomena, the search for numerical solutions to equations throughout time, the discovery of curvature and geometric space, and the quest for patterns in prime numbers. Each story is told through the words of the pioneers of mathematical thought. Particular advantages of the historical approach include providing context to mathematical inquiry, perspective to proposed conceptual solutions, and a glimpse into the direction research has taken. The text is ideal for an undergraduate seminar, independent reading, or a capstone course, and offers a wealth of student exercises with a prerequisite of at most multivariable calculus. Content Level » Lower undergraduate Related subjects » Computational Science & Engineering - Geometry & Topology - History of Mathematical Sciences - Number Theory and Discrete Mathematics Cover......Page 1 Undergraduate Texts in Mathematics......Page 2 Garduate Texts in Mathematics & List of UTM Publications......Page 3 Mathematical Masterpieces Further Chronicles by the Explorers......Page 4 Library of Congress Control Number: 2006940178......Page 5 Preface......Page 6 Contents......Page 12 1.1 Introduction......Page 14 1.2 Archimedes Sums Squares to Find the Area Inside a Spiral......Page 31 1.3 Fermat and Pascal Use Figurate Numbers, Binomials, and the Arithmetical Triangle to Calculate Sums of Powers......Page 39 1.4 Jakob Bernoulli Finds a Pattern......Page 54 1.5 Euler’s Summation Formula and the Solution for Sums of Powers......Page 63 1.6 Euler Solves the Basel Problem......Page 83 2.1 Introduction......Page 96 2.2 Qin Solves a Fourth-Degree Equation by Completing Powers......Page 123 2.3 Newton’s Proportional Method......Page 138 2.4 Simpson’s Fluxional Method......Page 145 2.5 Smale Solves Simpson......Page 153 3.1 Introduction......Page 172 3.2 Huygens Discovers the Isochrone......Page 180 3.3 Newton Derives the Radius of Curvature......Page 194 3.4 Euler Studies the Curvature of Surfaces......Page 200 3.5 Gauss Defines an Independent Notion of Curvature......Page 209 3.6 Riemann Explores Higher-Dimensional Space......Page 227 4.1 Introduction......Page 242 4.2 Euler Discovers Patterns for Prime Divisors of Quadratic Forms......Page 264 4.3 Lagrange Develops a Theory of Quadratic Forms and Divisors......Page 274 4.4 Legendre Asserts the Quadratic Reciprocity Law......Page 292 4.5 Gauss Proves the “Fundamental Theorem”......Page 299 4.6 Eisenstein’s Geometric Proof......Page 305 4.7 Gauss Composes Quadratic Forms: The Class Group......Page 314 4.8 Appendix on Congruence Arithmetic......Page 319 References......Page 324 Credits......Page 336 Name Index......Page 338 Subject Index......Page 342 List of Pubications of Undergraduate Texts in Mathematics......Page 347 In introducing his essays on the study and understanding of nature and e- lution, biologist Stephen J. Gould writes: [W]e acquire a surprising source of rich and apparently limitless novelty from the primary documents of great thinkers throughout our history. But why should any nuggets, or even?akes, be left for int- lectual miners in such terrain? Hasn't the Origin of Species been read untold millions of times? Hasn't every paragraph been subjected to overt scholarly scrutiny and exegesis? Letmeshareasecretrootedingeneralhumanfoibles.... Veryfew people, including authors willing to commit to paper, ever really read primary sources—certainly not in necessary depth and completion, and often not at all.... I can attest that all major documents of science remain cho- full of distinctive and illuminating novelty, if only people will study them—in full and in the original editions. Why would anyone not yearn to read these works; not hunger for the opportunity? [99, p. 6f] It is in the spirit of Gould's insights on an approach to science based on p- mary texts that we o?er the present book of annotated mathematical sources, from which our undergraduate students have been learning for more than a decade. Although teaching and learning with primary historical sources require a commitment of study, the investment yields the rewards of a deeper understanding of the subject, an appreciation of its details, and a glimpse into the direction research has taken. Our students read sequences of primary sources.

this Book Traces The Historical Development Of Four Different Mathematical Concepts By Presenting Readers With The Original Sources. Although Primary Sources Can Be More Demanding, The Investment Yields The Rewards Of A Deeper Understanding Of The Subject, An Appreciation Of The Details, And A Glimpse Into The Direction Research Has Taken.

each Chapter Contains A Different Story, Each Anchored Around A Sequence Of Selected Primary Sources Showcasing A Masterpiece Of Mathematical Achievement. The Authors Begin By Studying The Interplay Between The Discrete And Continuous, With A Focus On Sums Of Powers. They Proceed To The Development Of Algorithms For Finding Numerical Solutions Of Equations As Developed By Newton, Simpson And Smale. Next They Explore Our Modern Understanding Of Curvature, With Its Roots In The Emerging Calculus Of The 17th Century, While The Final Chapter Ends With An Exploration Of The Elusive Properties Of Prime Numbers, And The Patterns Found Therein.

this Book Emerged From A Course Taught At New Mexico State University To Juniors And Seniors Majoring In Mathematics. The Intended Audience Is Juniors And Seniors Majoring In Mathematics, As Well As Anyone Pursuing Independent Study. The Authors Have Included Exercises, Numerous Historical Photographs, And An Annotated Bibliography.

Intended for juniors and seniors majoring in mathematics, as well as anyone pursuing independent study, this book traces the historical development of four different mathematical concepts by presenting readers with the original sources. Each chapter showcases a masterpiece of mathematical achievement, anchored to a sequence of selected primary sources. The authors examine the interplay between the discrete and continuous, with a focus on sums of powers. They then delineate the development of algorithms by Newton, Simpson and Smale. Next they explore our modern understanding of curvature, and finally they look at the properties of prime numbers. The book includes exercises, numerous photographs, and an annotated bibliography. Introduces the excitement of discovery, combined with an advanced level of mathematics. Explores great achievements in mathematics through original sources