Since ancient times humans have been fascinated with the sun, the moon, the other planets of our solar system and all the the stars out there. Meticulous observations were collected in almanacs and the celestial movements were listed. The *Almagest* is an astronomical and mathematical treatise written by the Greek Ptolemy in the 2nd century. It was originally called *Mathematical Syntaxis*, and later *He Megale Syntaxis*. These were lost, but fortunately, via an Arab translation the *al-majisti*, it came back to us as the *Almagest*. It illustrates that mathematics and astronomy were very close in those days, and they were going hand in hand in the days of Kepler and Galileo, and now that we can `see' so much further, mathematics is still a core instrument for astrophysicists and cosmologists. Mathematicians were forced to design models that explain the data, and the other way around, experimentalists are forced by predictions made by mathematical models to look for objects or phenomena that had not been observed before. There is nothing new under the sun and there is nothing new beyond it.

Ian Stewart who has written so many books on popular mathematics, engages here in guiding us through the playground of the extraterrestrial universe and points us to where and how mathematics were instrumental in obtaining insights in the laws of physics that govern the whole cosmological pinball machine. The subtitle of this book *How mathematics unveils the universe* is of course correct, but if you would be curious of the mathematical formulas and details, you would be disappointed. There are practically no formulas! There are diagrams and illustrations though. The mathematics are present on nearly every page but they are not explicit.

In a more or less chronological order, Stewart takes the reader through the development of several topics that are relevant in this context. Take for example gravity. That is directly connected with the switch from a geocentric to a heliocentric system in which gravitation is modelled by mathematical formulas such as the laws by Newton and Kepler. Later, to resolve some deficiencies that showed up in extreme circumstances, Einstein's relativistic approach brought the necessary corrections. In different geometries, a shortest path is no more a straight line, and bodies move differently. Stewart explains next how our solar system came about and he explores the different theories that may explain how we got our moon. Assumptions are made, checked with ever increasingly accurate simulations, which are verified against observations. What was originally considered a planetary system that ran like a clockwork turns out to be anything but consisting of fixed regular movements. As soon as three or more bodies are involved, chaotic behavior is the rule as Poincaré has shown. The geometric progression of distances of satellites from the core body predicted the existence of missing planets and of satellites around these planets. So one started looking for them and they were eventually found indeed. That's how Uranus and Neptune were detected. Pluto is a small trans-Neptunion object that only recently was degraded to the status of a dwarf-planet together with Ceres and Eris. Yet smaller objects are found in the astroide belt. For the origin of the rings of Saturn we only have several speculations available. The chaotic looking trajectory of the Rosetta space probe, launched in 2004, illustrates the slingshot manoeuvre, a very indirect trajectory to meet the comet 67P at a speed of 55,000 kph. It is obtained by using the gravity field of Mars and the Earth several times.

Outside our solar system there is our galaxy consisting of a dazzling number of stars, and this is just one galaxy among so many others. Stars are mainly observable by a spectral analysis of the light they emit. Astrophysicists investigate the properties of all the different stars and how they are organized in galaxies. There are different forms of galaxies and different theories exist about how their shape evolves or just how they came about. Not all of them have the familiar spiral form. Again mathematical simulations are used to reject or back up some of the theories. A realistic discussion is given about how many habitable exoplanets one can reasonably expect, and how, if ever, one could possibly try to reach them. Science-fiction has many solutions like wormholes, traveling faster than light, etc. This may assume the existence of parallel universes and how they relate to black holes. Once more, with current technology. these can only be discussed in a mathematical context. The origin and the evolution of the universe as the Big Bang and inflation theory, is generally accepted, but it is not the only theoretical explanation. These require to assume the existence of dark matter and dark energy, whatever these may be. The possible futures of the universe is in the hands of topology and the geometry that can come out of the equations. The same holds for the different multiverse theories and the basic question about why we are so special. Do all these parameters have to be indeed exactly as they are, within very tight limits, to make life possible? Nothing is absolutely certain. Maverick scientists propose alternative theories outside the mainstream. For them it is difficult to get funding and hence these theories are perhaps too easily left behind. This is a viewpoint put forward by Penrose in his book Fashion, Faith, and Fantasy in the New Physics of the Universe (reading it requires however much more mathematics than this book).

The nice thing is that Stewart not only gives the evolution of different historical trials, mistakes, additions, and generalizations, into what is now mainstream belief, he also discusses alternatives that are also possible, with a slight bias towards what he believes himself. One thing is for sure, after this guided tour on a dazzling large space-time scale, it becomes clear that we learned very much in the last hundred years, but all models designed so far needed sooner or later corrections, modifications, and generalizations so that many theories are still very shaky and unstable. There are still so many questions unanswered forcing us to be very humble and admit that we only understand just a tiny bit. But Stewart is a marvelous guide on this voyage to places where many have been before, showing us the myth and fallacies of the past and what is realistic enough to be generally accepted today. The text is remarkably up-to-date with the latest results up to 2015 and the first months of 2016. Even data from June 2016 are given 'at the moment of writing' while the book is in the bookshop only two months later. It illustrates how hot the topic is.

The level is just right for a general public: just enough scientific details but not too technical that it would prevent smooth reading. No mathematics, but many pointers to it, and there are just enough funny phrases. Some examples of Stewart's sense of humour. Poincaré had to prove that he was wrong in his first opinion and Stewart remarks: *unlike politicians, mathematicians do that kind of thing*; after describing some cosmic event of galaxies floating on the surface of bubbles in a cosmic foam he has the laconic remark: *don't try this at home*; when explaining superposition in quantum theory and referring to the faith of Schödinger's cat: *this is not more mysterious than not knowing what's in your Christmas present from Auntie Vera until you unwrap it. Yes she always sends either socks or a scarf, but that doesn't mean her present is a superposition of the two*; and when discussing the precise values of some cosmic constants: *they could have been 2.7742, or 842,006,444.998, or 42 for different physics*. The latter is a reference to the Hitchhiker's Guide to the Galaxy, where 42 is the answer to the ultimate question of life, the universe and everything.

I already mentioned there are many illustrations and a number of color plates. There are of course appropriate references to the literature for the interested reader. I also appreciated the few pages listing briefly the most common jargon that has been used and the carefully compiled index to find your way in this enormous pile of information. A marvelous breathtaking read this is. It brings you in one boost from the historical preliminary models to all the current up-to-date models and the beliefs and conjectures about our solar system and the universe.