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# A Brief Guide to the Great Equations (Brief Histories)

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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] [2002] [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.
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on 5 May 2017
A good primer and really makes the layperson think.

Get it as a present for your teenager.
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on 5 March 2010
As a teenager, you hear music for the first time. The excitement and revelation of new musical inventions is fantastic, a real `high' in life. Then over the years you get used to it all and although some of the music is still enjoyable, you can't `hear it for the first time' anymore. But, very occasionally, you hear something new that brings back some of that original excitement , some of that thrilling bafflement and intrigue.

This book has it in abundance. If you think everything in science is pretty much solved, understood and in fact fairly boring, I would recommend The Great Equations. By following, in some depth, the original journeys, the original struggles and blind alleys, Robert Crease captures the excitement and, in fact, the mystery of all these equations. This is not a straightforward analytical look at the known products, instead it is probably as close as you can get to following the paths of creation through the protagonists' eyes and thoughts. Philosophical issues are here, as they should be (scientists who try to dismiss these aspects are missing something) but the central stories are the personal stories.

At first sight The Great Equations looks like just another popular science book (and there is nothing wrong with that - there are a lot of good ones out there) but I think it is more. It goes deeper than the run of the mill popular science and is so much more rewarding for it. Having said that, it is very well written -I found I was carried along - and apart from chapter headings, equations are largely absent.

I learned a lot from this book, for every equation covered in fact. Some of the equations are quite familiar to me but it is like seeing them for the first time. The fact that they are not given laws of nature in the form written by God, but are contingent on how our minds perceive reality, is really brought home. Of course they have some deep connection with reality, but to me, the fact that there is still a mystery as to what that connection could be, restores the excitement.

Einstein's journey from the first inkling that mass should depend on the motion of the observer to the final famous form of his equation is well covered. It hadn't occurred to me before that the key bit of maths in the derivation of special relativity is the Pythagorean theorem! To mention another chapter where I was learning throughout - Heisenberg's uncertainty principle. I thought I knew the stories but I had little idea of the intense arguments over the approaches to quantum mechanics in Europe in the `20s. Again, this chapter was a bit of a revelation. I felt, after all these years, that I had a fresh insight into quantum mechanics after my struggles with it as an enthusiastic undergrad. It makes me want to have another go at some serious maths (another good book for rekindling excitement is The Art of the Infinite by R & E Kaplan.)

Finally, this book has the best system for looking up chapter notes I have seen. Highly recommended.

Oh, and Cameron Diaz has gone up no end in my estimation! (Robert Crease quotes something from The Biography of an Equation by David Bodanis - another cracking good read).
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on 28 July 2011
The author has selected 10 equations so perhaps the title should have been "Some of The Great Equations". However, as it is also called "A Brief Guide" then that's probably fair enough. There was no room for more than a passing reference to Gauss whom some would rate with Euler as the two greatest mathematicians of all time. I was expecting a detailed analysis of these great equations but if you are baffled by the form of Schrodinger's equation you will not find any enlightenment here. What this book does do is to provide the reader with something of the life and times of these outstanding scientists. Four of the chapters are devoted to Isaac Newton and Albert Einstein (two each) and I don't think anyone would complain about that. The author has culled biographies of each individual to provide a fascinating insight. Sadly some led tragic and short lives. Most appear to have had powerful personalities and did not welcome criticism of any kind. I was surprised to find the first chapter devoted to the so-called Pythagoras Theorem but was won over by the authors discussion. To find the theorem reappear in both chapters on Einstein made me look at this work with fresh eyes and greater respect. I have given this book a 5 Star rating even though it wasn't what I expected. The fact is that I thoroughly enjoyed the book and recommend it without reservation.
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on 17 December 2013
As other reviewers have noted, this book contains very little useful discussion to probe the meaning of the equations described. It is profusely wordy without making any useful contribution to the reader's understanding of the connection between the mathematics and the world(s) it describes.

Other reviewers have also noted the author's rather short-sighted critique of Maxwell's original form for the equations of electro-magnetism in favour of Heavyside's representation. This is an unfortunate and surprising feature of a book written by an historian of science who must surely be aware that all major physical theories are expressible in a variety of mathematical forms (Wave mechanics or Matrix mechanics, Newton's equations or the Maupertuis/Hamilton Principle of Least Action, QED Feynman diagrams or Schwinger/Tomonaga integrals, etc etc etc). A real beauty of Maxwell's electromagnetism is that the existence of the magneto- part follows directly and inevitably from the elctro- part *given* the principle of relativity. That is less than obvious from either Maxwell's or Heavyside's formulation of the equations, was perfectly understood by Einstein, but was made most clear by Herman Weyl who (in effect) re-wrote the equations in frame invariant Tensor form, thus proving them to obey the General Principle of Relativity. Space, Time, Matter (Dover Books on Physics)

Ian Stewart's book Seventeen Equations that Changed the World is far better in both quality and precison of writing and in modern relevance of the material chosen. Stewart also manages to find worthy candidates more recent than 1927 - which is where the present author runs out of ideas, with Schrodinger's equation and Heisenberg's 'Uncertainty principle'.
As all serious quantum physicists of the time, including Schrodinger himself, would have agreed, the 'Schrodinger Equation' was not consistent with the Special Theory of Relativity and, therefore, could not be regarded as the final answer. But within a couple of years Dirac had produced his equation for the relativistic electron field ... from which followed the revolutionary concept of anti-matter and most of modern Quantum Field Theory. To quote Werner Heisenberg " [Dirac's] discovery of anti-matter was perhaps the biggest jump of all the big jumps in physics of our [20th] century. It was a discovery of utmost importance because it changed our whole picture of matter". The lack of Dirac's equation in this role-call of 'Great Equations' is just the first of several that were missed by Robert Crease.
on 30 March 2011
The Great Equations: The hunt for cosmic beauty in numbers, by Robert C. Crease, W.W. Norton, New York, 2008; Constable & Robinson, London, 2009, 318 ff.

A philosophical journey through some classical equations
By Howard Jones

The author of this fascinating book is Professor and Chairman of the Department of Philosophy at Stony Brook University in New York, a man who has written many other books and articles. The book is about the journeys that we make in our increasing understanding of the world through the mathematical equations that represent some of the fundamental laws of physics . . . and therefore the laws of Nature. Crease makes the point that science is about the search for greater understanding of the way the universe works and that the equations are simply beautiful, but shorthand, expressions for some of the regularities we find.

The presentation begins, perhaps unsurprisingly, with the relation we know from high school as Pythagoras' Theorem, though how much Pythagoras himself had to do with it we don't know. There's an interesting philosophical aside in this first chapter about Meno's Paradox on the nature of learning, which Plato presented in one of his dialogues. This is intended as an allegory of the utility of mathematics. Each chapter is followed by an Interlude that gives additional commentary on the preceding chapter.

Chapter 2 takes us into physics with Newton's Second Law of Motion that defined the concepts of force, acceleration, and this strange quantity `mass' and gives us the relation between them. Newton was the first to introduce the ideas of `force' and `mass'. This treatment is followed by Newton's Law of Gravitation that illustrates the force that keeps the planets in orbit around the sun.

It may seem a strange thing to say about a book about equations but there is very little maths in this book at all, and certainly none that should deter readers. However, in Chapter 4 we do meet the first equation that we could describe as more advanced mathematics, and that is Euler's Equation. This concerns a relation between the base of natural logarithms (denoted by `e'), the unit of imaginary numbers (denoted by `i'), and the ratio of the perimeter to the diameter of a circle (denoted by pi), all symbolic notations that Euler created or standardized.

And so on in the remaining chapters, through the Second Law of Thermodynamics, Maxwell's Equations for Electromagnetism (these do look a bit scary, but the explanation in the chapter is easy to follow), Einstein's Mass-Energy and Relativity Equations, and Heisenberg and Schrödinger's basic equations of quantum physics (again, a bit challenging, but lucidly explained). Crease also brings their authors to life!

This book is accessible to anyone who has a curiosity about the fundamental relationships of physics and how these relations are expressed symbolically. It is 99% prose and 1% mathematics, so don't let a few symbols intimidate you! The description of each relation is both enlightening and intriguing. And if the background to each equation in each chapter is still not enough for you, there are 26 pages of Notes at the end of the book.

Dr Howard A. Jones is the author of The Thoughtful Guide to God (O Books, 2006), The Tao of Holism (O Books, 2008) and The World As Spirit (Fairhill Publishing, 2011)

Looking Glass Universe: The Emerging Science of Wholeness
The Arrow of Time: The Quest to Solve Science's Greatest Mysteries (Flamingo)
The Man Who Changed Everything: The Life of James Clerk Maxwell
Physics and Philosophy: The Revolution in Modern Science (Penguin Modern Classics)
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on 14 April 2012
This book deals with the interactions between scientists and the struggle not so much to come up with the new concept but to define it concisely. Many advances that are attributed to a particular person are due mainly to that person being able to concisely define (probably by an equation) what was the current thinking at the time. The book is pretty much a biography about the scientists not much about the equations.

I think Lyndal Hughes's review is pretty much spot on. It is certainly not what I was expecting either. It gives no explanations of the equations at all. Basically it just waffles. I got a feeling that the author may have felt it was too hard for a scientist to explain things in a layman's terms (i.e. you needed to be a physicist to understand) so he didn't bother. For this reason I give it one star not five because I wasn't after a soap story but an explanation of the equations and the maths and physics behind them.
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on 3 March 2012
I really enjoyed this book. I got it because I read the reviews and they looked positive. Its very well written and easy to follow. Excellent resourse for history of science. Ofcourse there is a lot missing but I guess when you want to squeeze 10 only its a difficult task.. may be 15 would have been better.. but the author had to draw a line somewhere on a good round number! I highly reccomend it to anybody who wants to know how we got here!
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on 25 October 2012
I bought this book because I had read THE SECOND CREATION - MAKERS OF THE REVOLUTION IN TWENTIETH CENTURY PHYSICS, a much -quoted book. Even that book in retrospect makes theorists out to be inventing theories at random, and does not explain that the plethora of them in the 1960s was due in part to the lack of precision experimental data that would have allowed one to distinguish between effects due to matter particles and those due to force -transmitters.

In my opinion too, as reviewer Colin said, much of this book is just waffle. In particular, criticising Maxwell for expressing his equations in terms of potentials shows an alarming lack of understanding of why they became the prototypical gauge theory.

The reader is better served by sticking to popularising books written by professional physicists.
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on 11 May 2013
The book has interesting parts, but.... it waffles a lot.

I did find myself struggling to keep interested in it, probably as I don't have any real interest in knowing about the "biography" of an equation. Certainly any information within can be summarised in a few sentences, rather than several pages.
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