CALCULUS Understanding Its Concepts and Methods

Joseph L. F. Bertrand (1822--1900) -- Historical sketch

The late Professor Paul Erdös, about whom the book The
Man Who Loved Only Numbers by Paul Hoffman
was written, penned the following couplet:

Chebyshev said it, and I say it again,

There
is always a prime between
and
**.**

Erdös had just devised a new proof of Bertrand's Postulate, which is described in the second line of his couplet. Bertrand's Postulate was conjectured by Joseph Bertrand in 1845 and was first proved in 1850 by the Russian mathematician Pafnuty Chebyshev (1821--1894). It says, for example, that there is a prime between your age and twice your age. Try it. ( Problem 1)

Bertrand was a formidable mathematician. Although he didn't prove his conjecture, it was a very good one. Many people have tried to improve on it with little success. Nothing so simple as, say, that there is always a prime between and . Bertrand had good reason to believe his conjecture: he had verified it for all numbers less than three million!

Joseph Bertrand was born in 1822 in Paris. His father, a science writer, died when Joseph was but nine, and he was reared by a sister of his father and her husband, the mathematician Jean-Marie Duhamel. Young Joseph was a prodigy. At age 11, he unofficially attended lectures at the illustrious École Polytechnique, obtaining a degree when he was 16, and he received his doctorate in mathematics at age 17 with a thesis on thermodynamics. He could then attend classes officially! He spent the rest of his life in various academic posts, including 50 years at the École Polytechnique, except when he was in the National Guard during the French Revolution.

Bertrand is famous for his two-volume work, Treatise on Differential and Integral Calculus. Work on a third volume was lost in a wartime fire that burned down his house. He wrote other books and papers on topics such as thermodynamics, probability theory, and mechanics. He was reportedly a beloved teacher.

Bertrand was a member of the Paris Academy of Sciences, serving as one of its officers from 1874 to the time of his death in 1900 in Paris.

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Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.