Read This!

The MAA Online book review column

Briefly Noted

October 2002

This sequence of booklets (which I'm referring to collectively as a book) reminds me of a problem that faces many parents. Suppose your 5 year old asks you, "Why is the sky blue?". We could either say, "because it is", or we could go into a lengthy discussion of the chemical composition of the sky, and perhaps throw some physics in for good measure. Of course, most people would do the former, simply because the child is (probably) not ready for the real reason. In the same way, if a college algebra student asks you why x + y = y + x, how would you respond?

My problem with this book is not with its mathematics, or with its text (which is written in a conversational style), but with the book's audience- Who exactly will be reading this book? It is stated in the preface that this book is intended for a student that has some algebra, but "does not really understand what is going on". So again I would ask the author exactly what level of understanding can we reasonably expect from a novice? Sometimes I think we, as mature mathematicians, forget that it took us years of training and a love of the abstract to even begin to understand the complexities of what we took for granted in our youth (i.e., arithmetic). I would allow students at the College Algebra level to be more or less procedural (although I may be in the minority with that opinion), and let their minds evolve a little before going into abstraction and subtlety that demands a level of maturity and experience that most algebra students do not have.

It is my opinion that this book might be fun for advanced undergraduates to read, but I would not recommend teaching a College Algebra course from this book. At that level, some things are better left unsaid. [Doug Hundley]

The preface of T. M. Mills' Problems in Probability explains the distinctive features of the book: long problems, verging on small projects, which would give students the flavor of research, and some major theorems, like the central limit theorem, presented as problems with hints. The first half of the book gives problems in chapters headed Sets, Measure and Probability, Elementary Probability, Discrete Random Variables, Continuous Random Variables, Limit Theorems and Random Walks. The solutions to the problems, given in the second half of the book, are not only very detailed and thorough, they are accompanied by comments which guide the reader towards generalizing what has been learnt from these problems. The comments also indicate how to follow up what has been learned by suitable reading from the bibliography. An excellent book, which should be in all libraries, and on the shelf of all instructors teaching courses in mathematical probability at the undergraduate or graduate level. [Ramachandran Bharath]

Isaac Todhunter's Plane Trigonometry for the Use of Colleges and Schools, With Numerous Examples was first published in London by Macmillan and Co. in 1874. The preface tells us that the book "contains all the propositions which are normally included in treatises on Plane Trigonometry, together with about a thousand examples for exercise. The design has been to render the subject easily intelligible..." One shudders to think what would happen if one attempted to use it today in either colleges or schools. In fact, I wonder if students ever found it "easily intelligible"!

Today, Todhunter's book is of interest mainly for historical reasons. It is interesting to see how much he includes (all the way to series expansions, complex exponentials, and methods for summing trigonometric series) and how densely packed with formulas his text is, making very few concessions to the student. Teachers looking for hard exercises are likely to find many of them in here.

This edition of Todhunter was published by Elibron Classics, a new company that is scanning old books (especially, it seems, from the library of the University of Moscow) and making them available in both electronic and paper editions. The books are print-on-demand editions and are photographically reproduced, so the quality of printing and binding is not always the best. Still, in many cases they provide the only easy way to own a copy of older books. [Fernando Q. Gouvêa]

Publication Data

Algebra from A to Z, Vols 1-5, by A.W. Goodman. World Scientific, 2001. Paperback, five volumes, 732 pages in all, $18.00 each. ISBN 981-02-4478-9.

Plane Trigonometry for the Use of Colleges and Schools, With Numerous Examples, by Isaac Todhunter. Elibron Classics, 2002. Softcover, 341pp., $15.95. ISBN 1402100337.
Elibron also makes this available in electronic (pdf) format for $9.45.

Problems in Probability, by T. M. Mills. World Scientific, 2001. Hardcover, 192pp. $28.00. ISBN 981-02-4598-X.

Doug Hundley (hundledr@whitman.edu) teaches at Whitman College in Walla Walla, Washington.

Fernando Q. Gouvêa (fqgouvea@colby.edu) is the editor of FOCUS and MAA Online.

Ramachandran Bharath (rbharath@colby.edu) is Visiting Professor of Mathematics at Colby College.

Back Issues:

• April 1999: Reprints of "classics", books on applied mathematics.
• May 1999: Books on math and physics, both popular and technical.
• July 1999: Various kinds of geometry, and classics in applied mathematics.
• August 1999: What's happening, Dirichlet, and Cryptonomicon.
• October 1999: Graph theory sources, John von Neumann, and Noncommutative Geometry.
• November 1999: Number theory in nine chapters, mathematics at work, and Jacques Hadamard.
• February 2000: Fallacies, Flaws, and Flimflam, measuring the universe, and primary sources at USMA.
• March 2000: Cardano and Manifold: Time.
• May 2000: Grassmann, Smarandache, and a really big prime.
• June 2000: A problem book, a history book, a collection of biographies, and a historical survey of reciprocity laws.
• August 2000: acrostics, field theory, categories, and partial differential equations.
• September 2000: excursions, history, and some high-level expository writing.
• November 2000: a festival of math videos and the Fourier-analytic proof of quadratic reciprocity.
• December 2000: Analysis problems, mathematical mysteries, and industrial mathematics.
• January 2001: Euler, Kolmogorov, and Honsberger.
• March 2001: history of mathematics, in three flavors: Harriot, prime numbers, and geometrical exactness.
• May 2001: three very different "applied mathematics" books.
• August 2001: introductions to algebraic number theory and to topology, and problem books.
• November 2001: Wilbur Knorr, John H. Conway, and vector analysis.
• February 2002: biological time, celestial mechanics, and things rarely talked about.
• March 2002: history of recent mathematics and differential geometry.
• June 2002: mathematics in other cultures, Archimedes, probabilistic thinking, and Dover's Phoenix.
• August 2002: Euclid's Elements, Hurewicz's Lectures, and Olympiads 2001.

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Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouv&ecric;a, Dept. of Math&CS, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.