This book gives a deep insight of the mathematics involved in the forecast of weather. Being addressed to the general public, with a very leisurely and friendly style, it starts by accounting the history and personalities involved in the developing of the scientific understanding of the weather processes over the Earth.

The author explains the equations that govern the weather, which include seven variables: velocity (of wind), time, pressure, moisture, and density of air. The mathematical bits are separated from the main text and are commented without much detail, not to deter the non-mathematical reader. All names of people involved and the contributions and lives of each of them appear along the way.

The equations are hard to analyze (which is the reason of the difficulty of forecasting), so until the appearance of efficient computers able to do these calculations, it was difficult to do reliable predictions. But the use of mathematics in weather dates back from the beginning of the twentieth century, with the pioneering work of Vilhelm Bjerknes, who analyzed the behaviour of the equations for the vorticity, which involve less number of variables. This together with the improvement of the recollection of empirical atmospheric data at locations, and the improvement of the transmission of data to a central point, allowed to depict the well-known meteorological charts that we are so familiarized to watch on the TV News. These have been used since the beginning of the XX century to predict qualitatively the weather for several days ahead. Nowadays, this is done by the use of computing power, by discretizing the set of differential equations involved and treating them numerically.

Large scale phenomena, like the effect of the rotation of the Earth in the circulation of great masses of air or the formation of cyclones, and more local phenomena, like the movement of clouds or the sea breeze, are treated in the book. The authors also explain the theory of chaos, which says that in non-linear problems, even small inaccuracies in initial data can lead to very large deviations in the evolution of the solution. This is inherent to the analysis of weather, making impossible to get accurate solutions for more than 10 days ahead even with the best of the actual supercomputers.

The second half of the XX century witnessed the raise of pure mathematical methods to analyze the equations of meteorology, together with the simultaneous appearance and use of computers. The study of hydrostatic and geostrophic phenomena gave a way to understand the qualitative behaviour of weather and make sense of the intractability of its non-linearity. On the other direction, there has been a feedback from the studies of meteorology to pure mathematical areas, like the Lorenz attractor appearing in Dynamical Systems.

This book is about the role of mathematics in explaining why it is possible to understand weather and climate, even in the presence of chaos. There are degrees of unpredictability, but there are also many stabilizing mechanisms and, most importantly, there is mathematics to quantify the rules. The authors have done a brilliant work to collect a huge amount of historical information, as well as mathematical information, but keeping always a level in the explanations that makes the text accessible to undergraduate students in the first years, and even to people not so familiar with mathematics. All in all, this is a very interesting and enjoyable reading.