*Acta
Cryst.* (1995). A**51**,
590

**Pp. xi + 223.
New York and Berlin: Springer-Verlag, 1994
Price US $79.00.
ISBN 3-540-58115-4**

The second edition of
this book will certainly engage the lively
interest of crystallographers in various specialities - both those
who
know the first edition and those who are only now looking for a
mathematical guide to help them in their work.
To bring his book up to date, the author has added a new chapter
about the fast Fourier transform, so the present edition contains
the following chapters: 1. Matrices: definitions and fundamental
operations; 2. Symmetry of finite objects; 3. Symmetry of infinitely
repeated patterns; 4. Vectors; 5. Tensors; 6. Data fitting;
7. Estimation of uncertainty; 8. Significance and accuracy;
9. Constrained crystal structure refinement; 10.
The fast Fourier
transform. Moreover, improvements have been made to
the original chapters, and two new figures have
been added. As the first edition of the book has already been
reviewed in *Acta Crystallographica Section
A* [*Acta Cryst*.
(1984), A**40**, 86-87], I shall
concentrate mainly on the new Chapter 10.

Although the fast
Fourier transform (FFT) is an important
component of popular crystallographic program systems, probably
only a few of their users know what causes it to be fast. To
explain the mechanism of FFT calculations, the author starts
with the definition of the Fourier transform, one of the basic
concepts in crystallography, proceeds to the discrete Fourier
transform (DFT) in matrix notation, and so gets immediately at the
substance of the computational problem, avoiding the customary
considerations of conditions for existence and
other inessential (in this context) properties of the transform.
Next, the reader's
attention is directed to the somewhat complex transformations of
the DFT matrix. These are performed to express the DFT
matrix as a product of matrices that are both sparse (having many
zero elements) and have many elements equal to
either
±
1 or
±
*i*, which makes the
calculations easier and faster.
Progress in the subject is demonstrated by successive
solutions, such as the Good-Thomas, the Cooley-Tukey
and the Rader field algorithms.

Next, the discussion about economy of calculation in the Fourier transform is transferred to the native field of crystallography and to three-dimensional space. The author conveys to his readers, or reminds some of them of, the reduction in computation that results when one deals with real - in most cases - values of the density of scattering matter, and the presence of symmetry elements in the given crystal structure.

The proper selection of material, illustrated above, and the
didactic aspect of the book are worth emphasizing. The
author himself seems to be present during one's reading. He knows
how the reader is thinking - makes useful comments, places
numerical examples, takes care on the correct usage of certain
terms, gives some historical details and, sometimes, tells
relevant anecdotes.
Owing to its clear organization,
well aimed wording, pictorial comments and handy index, this book may
be
used for various purposes. The whole book can be highly
recommended for beginners in crystallography, as complementary
reading while they are going through the fundamentals. More
advanced
crystallographers may concentrate on selected problems that
interest them - to widen, brush up or systematize their
knowledge. For `outsiders', the book may serve as a dictionary of
basic mathematical and crystallographical terms. A short but
well chosen bibliography points the reader to the relevant literature.
The detailed Fortran programs, enclosed as
Appendix *G*, will certainly be
useful;
but it seems to me that the mathematical formulae alone, well
arranged within the main text, are very likely to inspire
crystallographers to write their own programs.
I found few errors or misprints.
The lack of the minus sign in one
exponent of the sequence of formulae on page 141 should be quickly
spotted,
but the equation accompanying Definition 15 on page 4 may cause a little
more trouble.

To sum up, the author's aim `to write . . . a *vade
mecum* for
active research workers' has been fully achieved. His approach
invites active crystallographers on a comfortable journey
through mathematical problems specially selected for them -
with a reasonably light luggage of necessary formulae and
definitions, and with a competent guide.

**Ewa
Galdecka**

*
Institute of Low
Temperature and Structure Research
Polish Academy of
Sciences
PO Box
937
50-950
Wroclaw 2
Poland
*

**Copyright © 1997 International Union of Crystallography**