Phillips G.M. Two Millennia of Mathematics: From Archimedes to Gauss

Springer, 2000. — xii, 223 p. — (CMS Books in Mathematics). — ISBN: 978-1-4612-7035-5, 978-1-4612-1180-8.This book is intended for those who love mathematics, including under graduate students of mathematics, more experienced students, and the vast number of amateurs, in the literal sense of those who do something for the love of it. I hope it will also be a useful source of material for those who teach mathematics. It is a collection of loosely connected topics in areas of mathematics that particularly interest me, ranging over the two millennia from the work of Archimedes, who died in the year 212 Be, to the Werke of Gauss, who was born in 1777, although there are some references outside this period. In view of its title, I must emphasize that this book is certainly not pretending to be a comprehensive history of the mathematics of this period, or even a complete account of the topics discussed. However, every chapter is written with the history of its topic in mind. It is fascinating, for example, to follow how both Napier and Briggs constructed their log arithms before many of the most relevant mathematical ideas had been discovered. Do I really mean "discovered"? There is an old question, "Is mathematics created or discovered?" Sometimes it seems a shame not to use the word "create" in praise of the first mathematician to write down some outstanding result. Yet the inner harmony that sings out from the best of mathematics seems to demand the word "discover."From Archimedes to Gauss
Archimedes and Pi
Variations on a Theme
Playing a Mean Game
Gauss and the AGMLogarithms
Exponential Functions
Logarithmic Functions
Napier and Briggs
2The Logarithm as an Area
Further Historical NotesInterpolation
The Interpolating Polynomial
Newton's Divided Differences
Finite Differences
Other Differences
Multivariate Interpolation
The Neville-Aitken Algorithm
Historical NotesContinued Fractions
The Euclidean Algorithm
Linear Recurrence Relations
Fibonacci Numbers
Continued Fractions
Historical NotesMore Number Theory
The Prime Numbers
Congruences
Quadratic Residues
Diophantine Equations
Algebraic Integers
The equation x3 + y3 = z3
Euler and Sums of CubesReferences
Index