# Great calculations: A surprising look behind 50 scientific inquiries

Many books have been written that are popularizing mathematics and science in general. The history is rich of anecdotes, oddities, and amazing facts so that one may easily fill up an entertaining book with these. The approach that Colin Pask has taken here is however original. Science started because people wanted to find answers to simple questions about the world they live in or about themselves. The progression of science depends on an interplay of theory and experiment, leading to models that are gradually modified and refined. Sometimes experiments detect anomalies that force the model to be changed, sometimes, the model predicts things that need to be verified by experiments. Whether it are experiments or theoretical predictions that are at the origin, there are always some calculations involved. So it makes sense to place these calculations at the center of major advances made in science. Selecting only 50 of them must be a big challenge, but this is what Pask has done, although he apologizes for the many others he has left out and that may have been closer to the heart of some readers.

When I say science, in this context, I mean the most formalized science. In this book, that is besides mathematics, mostly physics and astronomy and a bit of life-science. In this computerized age, it is difficult to imagine that these calculations were achieved with the tools available at that time. Think of the first estimation of pi by Archimedes. Taking into account that Fibonacci's *Liber Abbaci* popularized the Hindu-Arabic numerals only in 1202, then the richness of data in the *Almagest* of Ptolemy, or Kepler's astronomical computations are mindblowing achievements. And there are many obvious questions but with nontrivial answers that people have found answers for in the past. Why is the sky dark at night? How large and how old is the Earth? Why does the sun shine? Etc. And there are the more recent achievements in particle physics and cosmology. Just think of your own favorite major event in the history of science and you will notice that there is always some major calculation involved. Sometimes it is the number that is the answer, but sometimes it are not the numbers, but the pattern they represent that one is looking for.

Pask has organized his top 50 calculations in coherent chapters. Within the chapters, some historical chronology is maintained, as is loosely speaking also the case for the succession of the chapters. The first chapters are about mathematics (Pythagorean triples, logarithms, pi, prime numbers,...). The next chapters collect topics concerning our planet earth (its age, size, mass, tides,...), the solar system (heliocentric model, why the moon stays in orbit, detection of the planets, Halley's comet,...) and the universe (dark sky, its topology, the origin of chemical elements, dark matter, escape velocity,...). Then there is a chapter on life-science topics (blood circulation, population dynamics, annuity pricing, genetics, computer tomography, scaling for living species,...). The remaining chapters deal with physics. First there is optics (the speed of light, the colours of the rainbow, waves, electro-magnetism, photons, and relativity). Then we get the building blocks of our universe from atoms and Brownian motion to quantum and particle physics, fusion and fission. Dynamical systems is the subject of the penultimate chapter starting with Fourier analysis over Bessel functions to nonlinear dynamics and chaos.

From this skimmy survey it should be clear that the subjects are quite diverse and one can imagine that not all the mathematics and calculations are equally accessible to a broad audience that Pask is obviously addressing. So he has chosen to give for the simplest ones all the details, but not for others that are more involved. However, even if it becomes a bit technical, one van easily skip the details and get the general picture. And he always has some entertaining story to tell, and sometimes also he has to tell something `behind the scene'. It is interesting to read how scientists working on the same problem had different views. Pask gives many original citations and he gives detailed references in case one wants to know more about an event. Not so much about the mathematics or the technical content, but often advise is given like `an excellent introduction to the topic is...' `read all about the story behind this in...', 'my favorite book on this topic is...', 'the recent book ... uses pictures to illustrate...' etc. Even professional scientists will discover something they did not know. For example, G.H. Hardy, known for his praise of pure mathematics and thinking low of any application it may have, had chosen number theory as his main topic expecting it to be most detached from applications. However, we know that prime numbers are now essential for cryptography, but what I did not know was that his name is also attached to a well known Hardy-Weinberg law in genetics because of his 1908 paper in *Science*. Or this one: Heisenberg (deliberately or not) largely overestimated the amount of uranium needed to make the atomic bomb, which made Germany decide it was not feasible to proceed with the production of such a bomb. Or that John Adams had predicted the position of Neptune but how Leverrier and Galle got the glory. That the FPU project (named after Fermi, Pasta, and Ulam) on one of the first computers at Los Alamos to simulate the vibration of a string was programmed by Mary Tsingou, a major achievement in those days of emerging computers, but she never got proper recognition for it. Pask also apologizes that only three women are involved in his 50 calculations although during WW II just before computers were invented, the human computers were mostly female since they had the reputation of being more careful. Besides the many references and notes listed by chapter at the end, there are also many illustrations. Some are reproductions from the publications and others are produced by Annabelle Boag.

It is only in his last chapter that Pask explains his criteria for selecting his 50 favorite calculations, and then he continues by considering this as the shortlist for selecting his top 10. Depending on the criteria used and the personal preferences, this top list will differ for every reader. There will probably be overlap, but I am sure there will be many others that were not even mentioned in this book. So, since many relevant calculations are left untold, I am already looking forward to subsequent volumes with additional calculations. The questions asked (and answered) in the book, the level of exposition, the readability, the pleasant style, and the decorating stories make this book highly recommendable for anyone interested in mathematics and science and its history.

**Submitted by Adhemar Bultheel |

**20 / Jul / 2015