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Understanding modern science through 10 headline equations
on 11 September 2011
This is an accessible science book, bringing together a collection of best known (and less well-known) equations and giving just a hint at the beauty of the mathematics that underpins science. Robert Crease opens with Pythagoras' Theorem, a^2 + b^2 = c^2, which most people with school-level mathematics will remember allows you to work out the third side of a right-angle triangle if you already know two sides. He shows some proofs (there are over 500 known proofs of the theorem), and leads on to a wider discussion, including its application to the 4-dimensional geometry of Einstein and spacetime.
Next, Newton's second law (F=ma), which relates applied force to acceleration, and is used as a practical tool by schoolchildren in Physics or Applied Maths. The book reveals that although Newton formulated it, it was not expressed as a formula until Euler came along some years later. The following chapter has the law of universal gravitation, which can be used to describe the motion of a planet around a star, or the gravitational effect of one planet upon another.
Chapter 4 shows the mathematical beauty of Euler's equation, which relates two natural constants, pi and via the so-called imaginary number i (the square root of -1). It goes on to discuss some of the important topics discussed in Euler's work, "Introductio".
Chapter 5 gives us the second law of thermodynamics, which in brief tells us that everything in the universe tends towards increasing entropy (i.e. becomes more disorganised). Chapter 6 brings us to Maxwell's equations. While other scientists (like Faraday) had found and documented interesting aspects of electricity and magnetism, it took Maxwell to express these mathematically. As Crease tells us, Richard Feynman (brilliant scientist and communicator) once said that Maxwell's equations were the most significant event of the 19th century, even including the American Civil War. This is a very interesting chapter, but if I have any criticism of the book, it would be that this chapter is not long enough. I wanted to know more about what the equations mean; what Heaviside's reformulation of the equations brings, and really how they are applied?
Chapter 7 reaches E=mc^2 with some inevitability, and Crease dubs this "the Celebrity equation", because of course nearly everyone knows this formula, even if they have no idea what it signifies. This leads us onto Chapter 8, which covers general relativity (i.e. how spacetime works). This section of the book covers the same material as the Brian Cox book Why Does E=mc2?, but I felt that Crease's version was more readable and less repetitive.
Chapter 9 calls on Schroedinger's equation, and by this point in the book, the equations have ceased to have the simple beauty of Euler, and now are using calculus. However, if you can stand to read on past the mathematics, we start to enter the strange world of quantum physics, where particles and waves can be the same thing, and the certainties of Newton's world are replaced by probabilities. Chapter 10 covers the Heisenberg Uncertainty principle, which means that you can know the position of a particle but only a probability of its momentum (or vice versa). The chapter discusses some of the events and politics among the top physicists of the day; this was an inflection point where the science changed fundamentally, and the personal and professional relationships of some scientists of the day went through incredible turmoil. Crease quotes from Michael Frayn's play Copenhagen [DVD]  [Region 1] [US Import] [NTSC], which covers one aspect of this, i.e. the relationship between Werner Heisenberg and Niels Bohr.
Stephen Hawking said that every equation you include in a book halves the potential audience, and ultimately included only one in his A Brief History Of Time: From Big Bang To Black Holes: From the Big Bang to Black Holes, namely E= mc^2. I hope that Hawking is wrong: not only for the sake of Crease's book, which is an excellent read, but because better familiarity with science would be a benefit to the whole of our society.