This impressive volume is styled after the rather successful *The Princeton Companion to Mathematics* (2008, edited by Timothy Gowers). The project started in 2009 under supervision of Nicholas Higham as the main editor. Assisted by five associate editors invitations for contributions were sent out in 2011 and 165 expert authors have accepted. So it has been a major project that eventually resulted in this amazing product.

The extra benefit of having all this material collected here in printed form, rather than scattered on the Web, is that it is more structured and self-contained. The level of specialization is smoothed out and all the authors have done a great effort in transmitting their expert knowledge to a broad audience. This is something you will not experience by just searching the Web. It does not claim to be exhaustive though. That would be impossible in just one volume, even if it has over 1000 page. Moreover, applied mathematics is not static, but still in full expansion. To keep this material in permanent evolution, it might not be a bad idea to have it also available in a hyperlinked form. Somewhat like the applied mathematics section in the Wikipedia portal on mathematics. In fact the guidelines with pros and cons of printed versus web material can be found in Part VIII of this very book where it deals with how to read, write, teach, and communicate about (applied) mathematics.

In principle the material should be accessible by junior university students with a good mathematical background. The subjects are however so diverse that some parts may be much more difficult than others. Although one should not read this cover to cover, the structure is initially a bit like a textbook where in a first part it is explained what should be understood by applied mathematics, what is dealt with, the language used, algorithms, solution techniques, and an extensive survey of the historical evolution of the subject.

The second part deals with many (applied) mathematical concepts (e.g. asymptotics, graph theory, Krylov subspaces, Markov chains, wave phenomena,... to name just a few) and part III has most of the famous equations, usually with some scientist's name attached to them (Black-Scholes, Navier-Stokes, Korteweg-de Vries, Riccati, Painlevé, Schrödinger, and many others). The items in these parts are treated briefly since many of these return at different instances in further contributions.

The items in parts IV, V, and VII are more substantial because they survey some topic. Part IV describes areas of applied mathematics (spectral theory, approximation theory, ordinary and partial differential equations, optimization, numerical linear algebra, soft matter, signal processing, algebraic geometry,...), and part V surveys modeling in several areas (sport, chemistry, biology, optics, turbulence,...). Part VII is again discussing application areas, but more specific ones, less broad than in part IV (aircraft noise, chip design, compressed sensing, text mining, voting systems,...). The intermediate part VI gives some shorter descriptions of example problems (foams, insect flight, web page ranking, random number generation, robotics,...).

The final part VIII was mentioned above and discusses dissemination, assimilation, and perception in (applied) mathematics and, perhaps equally important, how (applied) mathematics is conceived in the media.

What to think of this printed book? If you were convinced of the value of the *Companion to Mathematics* then you should definitely acquire this book as a most valuable addition. And I think that decent science libraries should have it on their shelves. I personally still love paper books, but I must admit that also electronic access to information has its advantages. Observing students or young researchers, I must conclude that they do not go very often to the library to look up something on paper anymore. So, having the e-version, would help them since they are certainly (part of) the target readership. Manipulating the printed version is heavy (physically). It is certainly a desktop book, that you would not conveniently take on your lap. Moreover, it is not like a dictionary where you look up something briefly, but you probably want to sit down and read through an entry completely.

What I do appreciate are the ample graphical illustrations and that references are restricted to the essentials for further reading and these are placed at the end of the entry, and not scattered through the text. In the text we do find pointers to other entries in the book, but also that is kept to a moderate level. It is obvious that some topics will show up at several places. Think of linear algebra, differential equations, or numerical techniques, that will repeatedly need discussion in different contexts. So for these topics you do not have a single structured entry. You will have to look it up in the elaborate index and compose your own survey by jumping back and forth.

As I said before, you do not read this book cover to cover, and yet many definitions and concepts are given in parts I and II that are used in the more advanced entries. For example if you read about quantum mechanics, you need to know about operators, statistics, relativity theory, Schrödinger equation, and many more. There is not always a cross reference, and if you are a beginning student, it might require many jumps to the index, diversions to other entries, swiping back and forth, that can make it difficult to keep track. That would not be different with an e-version, but it would reduce the unwieldy manipulation needed for flapping thick layers of pages.

If you are a more experienced scientist, you will have easy access to a comprehensive survey of some topic you might be less familiar with. As the editors state it themselves, they do not claim to be exhaustive, and it will not be difficult to find your favorite topic that is not (or not in sufficient detail) discussed here. For example support vector machines and neural nets are mentioned relatively briefly, machine learning is not explicitly discussed, and there is big activity in nano-science that one might consider not well represented, etc. But one could come up with many other topics as well. Some topics are deliberately left out. These are mentioned in the introduction: like wavelets, statistics, cryptography,... and all of their applications in different fields of science and engineering. For these the reader is referred to the Mathematics Companion (one more reason to have both volumes). Do not consider this a shortcoming. It is an unavoidable feature of such a project, since a limited time and space forces the editors to make a choice.

So, all in all, do not interpret my critique too hard. In fact, I quite like the concept and the book, which is the result of a lot of hard work by many. It must be possible though to work out a hyperlinked version with the same kind of ideas. It would be possible to make that more dynamic and keep it more easily up-to-date. That however, would require much higher investment of editors, authors and maintenance. For the moment, this is probably the best alternative available.