99i:01005 01A05

Laubenbacher, Reinhard(1-NMS); Pengelley, David(1-NMS)

Mathematical expeditions. (English. English summary)

Chronicles by the explorers. Undergraduate Texts in Mathematics. Readings
in Mathematics.

Springer-Verlag, New York, 1999. x+275 pp. $32.00. ISBN 0-387-98433-X

Born of a freshman honors course, this book reflects the belief of the
authors that students should

experience mathematics through the words of its creators. The five
independent chapters treat (1)

the parallel postulate and non-Euclidean geometry; (2) set theory and
infinity (Bolzano, Cantor,

Zermelo); (3) infinitesimal analysis (quadratures of Archimedes and
Cavalieri, calculus of Leibniz and

Cauchy, Robinson nonstandard analysis); (4) Fermat's last theorem (Euclid
to Euler, Germain and

Kummer); and (5) solvability by radicals (culminating in Galois). Following
an overview giving the

historical and mathematical setting, each chapter contains annotated
historical excerpts interspersed

with exercises to encourage further exploration. There is a large bibliography.
This book is suitable

for a general mathematical reader and would be an excellent text for
a history-based undergraduate

course, although some items, while not technical, require considerable
mathematical maturity.

Reviewed by E. J. Barbeau

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Teaching with Original Historical Sources in Mathematics