Many common English words phrases have specialized meaning in mathematical writing that differ from their meaning in everyday usage. This specialized usage causes problems for students learning to write mathematics and presents even greater challenges to students who are not native English speakers. Usage in mathematical writing has been discussed by a number of authors including Krantz (1997), Higham (1998), and Wells (2003.)

Jerzy Trzeciak is the senior copy editor at the Institute of Mathematics of the Polish Academy of Sciences. In his book and a free companion web site , Trzeciak provides guidance on mathematical usage based on his experience as a copyeditor.

The first part of the book consists of a collection of examples of correct usage, organized by where they might appear within a mathematical paper. For example, there are sections with examples showing how to state theorems, how to introduce definitions and notation, and how to make references to the literature. Although these sections contain many examples, the author has provided very little in the way of explanation. A reader might well wonder why a the author selected a particular phrase.

In the second half of the book, Trzeciak discusses particular issues in English grammar, including the use of definite and indefinite articles, passive voice, word order, and how to avoid repetition. Again the sections consist primarily of examples, with very little explanation. A section on typical errors contains many examples of how not to do things.

The fundamental weakness of this book is the lack of any serious attempt to explain how or why particular words and phrases should be used. Instead, the reader is expected to generalize from the many examples presented by the author. Other books on this topic make a greater effort to explain mathematical usage to the reader. Most readers would be better off with one or more of these other books.

**References**

Krantz, Steven G. *A Primer of Mathematical Writing* , American Mathematical Society, 1997.

Nicholas J. Higham. *Handbook of Writing for the Mathematical Sciences* , SIAM, 1998.

Wells, Charles. *A Handbook of Mathematical Discourse* , Infinity Publishing, 2003.

Brian Borchers is a professor of Mathematics at the New Mexico Institute of Mining and Technology. His interests are in optimization and applications of optimization in parameter estimation and inverse problems.