*Summa Summarum* can best be categorized as either a monograph or a reference work. On page xi the author refers to it as a textbook, but no exercises are included, so it’s not really suitable for that purpose.

The book has the ambitious aim of helping the reader to “evaluate almost any finite algebraic sum.” Although such a goal cannot be met, the work goes far and includes both new results and general attacks that could be used on sums that aren’t covered. Infinite sums are not included and, as the qualifier “algebraic” indicates, one won’t find formulas for sums involving any of the transcendental functions. Also, hypergeometric forms are not used. The author provides a four chapter introduction and then a fifth chapter explaining his classification scheme, which must be understood by the reader in order to find the particular sum that he or she is seeking.

A comparison with other reference works in discrete mathematics that this reviewer is familiar with will help to give the reader a better idea of how *Summa Summarum* fits into the literature. *The Encyclopedia of Integer Sequences* [3] is very easy to use — one simply looks the sequence up (or, for the online version [1], enters enough terms of the sequence to distinguish it from other sequences that start the same way). If one is lucky, the sequence is present and the results may be immediately examined and followed up on in greater detail through the references. Nearly as easy to use is Rosen’s *Handbook of Discrete and Combinatorial Mathematics* [2]. It has a nice table of contents and index that can fairly quickly lead the user to what he or she is looking for.

*Summa Summarum* demands more from the user. In fact, it is likely to demand more than most undergraduates would have patience for. The classification scheme detailed in chapter five is very reasonable, but the non-specialist reader must invest some time in understanding it before the book can be used. Also, the sum will likely need to be manipulated to see how it fits into the classification scheme. There may also be some work for the reader after finding the sequence, if the solution takes the form of a technique that may be applied to find the sum.

Although it won’t serve as a textbook, *Summa Summarum* is far from a simple listing of formulas. A reader may study particular sections to gain a better understanding of, for example, difference equations (chapter four), but gentler, less terse introductions can be found. Larsen treats difference equations in a very theoretical manner. Examples are special cases rather than particular examples.

I’m very skeptical about the ability of someone with little previous exposure (i.e., undergraduates) to be able to learn much from this high level treatment of the material. The style is very terse. The notation could form a barrier for lower level readers. There are some slight inconsistencies, such as ( ) for combinations in the text, but { } for the same purpose in appendix B. The latter notation is sometimes used in other works to denote Stirling numbers.

The author was likely disappointed to find his first name is incorrectly given as “Morgens” on the title page and indicia. The index is not especially useful for a non-specialist — it is only two pages long and contains a high percentage of names. A student who, for example, wants to look up the formula for a finite geometric series won’t find “geometric series” indexed. To be fair, a student who cannot quickly derive this formula is far below the target audience, which is described on the back cover as “graduate and upper-level undergraduate students, researchers, and nonspecialists.”

A very nice feature is that, in contrast to many reference works, a very high percentage of results are proved. Those that aren’t often have an indication of how the proof goes. Many of the simpler proofs are simply listed as being “obvious.” For this reason, as well as the new results that are included, this work may be of great value to researchers and graduate student who find themselves working with finite algebraic sums.

**References**

1. The On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/~njas/sequences/

2. Rosen, Kenneth H., editor, *Handbook of Discrete and Combinatorial Mathematics* , CRC Press, Boca Raton, 2000.

3. Sloane, N. J. A., and Simon Plouffe, *The Encyclopedia of Integer Sequences* , Academic Press, San Diego, 1995. (This is way out of date compared to the website – only use it when the internet is down!)

Craig Bauer is an Associate Professor of mathematics at York College of Pennsylvania. He also serves as the editor-in-chief of Cryptologia, a quarterly journal devoted to all aspects of cryptology.