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The Great Equations: Breakthroughs in Science from Pythagoras to Heisenberg

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Chances were good that I would like this book, even as I read the title. Any book that looks to examine the history of math and science through some of the most important equations ever discovered automatically appeals to me. Still, I'm happy to report that the book pretty well lived up to its promise.

Dr. Crease does a number of things here I really like. Most importantly, he harps on one of the great, often ignored truths of discovery: they most often come about through the work of many minds, though we often attribute it to one. Many cultures discovered the "Pythagorean Theorem" independently. The form of "Maxwell's Equations" with which we're most familiar never appeared in Maxwell's work. The incredibly wide line-up that contributed to our understanding of entropy. In addition, his prose is very readable--an important consideration for someone like me, a teacher of math and physics, who is always looking for things that will help make things more understandable for my students. And, though his focus is more on history, sociology and philosophy than math and science, it works reasonably well when he keeps his prejudices at bay.

Of course, as clever as the premise is, it sets itself up for criticism. The choice of equations is personal. Some are missing and perhaps it's a stretch to include some. Though entropy and general relativity are both important and interesting, the equations are not really well known outside of the scientific world unlike the Pythagorean Theorem, the law of universal gravitation, or special relativity. These equations, great though they are, merely allow Crease to discuss topics in which he is interested. And where is the quadratic equation? I often joke with my students that, if they remember nothing else from high school math, they must remember the Pythagorean Theorem and the quadratic formula. I would have loved to have seen a discussion of this.

Still, these are quibbles. This book takes a fun premise and works through it well. Dr. Crease should be commended for his work and scholarship.

The author's writing style is quite lyrical for a technical book, his choice of words is sometimes beautiful, sometimes a bit long winded, however that's not too disturbing. I learned a few interesting things e.g. Maxwell's equations started with a mechanical model, unfortunately the author doesn't go very deeply into that and as a reader you are left behind wondering how exactly Maxwell went from that mechanical model to his equations which were written using quaternions. I also keep wondering which path Heaviside followed to reformulate these equations i.e. how did he make the transition from the quaternion-presentation to the vector differential representation, I'm also surprised the author didn't go deeper into the matter of the displacement current (e.g. google for Ivor Catt).

On p.150 the author becomes a bit unprecise, calling D and B currents and then presenting the equations without D

Chapter 5 fails to go into S = k log W and misses depth.

As the author knows of Richard Feynman's work, I thought Feynman's views on QED would have been discussed in the chapters on quantum theory, but unfortunately nothing about Feynman there.

I also missed clarifying illustrations now and then


a few minor disturbances:
On p.24 there is a picture which is printed unrecognizable
On p.138 In part II of the paper, _Faraday_ handled -->_Maxwell_ handled

As a whole this book is a nice read but it lacks depth

The Great Equations summarizes what the author feels are the ten greatest equations in the history of human intelligence, and who could argue with him? During the time reading it, I thought about Planck's resolution of the Blackbody Catastrophe, Dirac's QED eq, leading to anti-matter (actually Dirac was trying to unify Quantum Mechanics with Special Relativity, but 'stumbled' onto anti-matter),or Hubble's expanding universe (Einstein had postulated this much earlier, but recanted causing him to state that it was the greatest blunder of his life), but guess top 13 would be unlucky. Listed, not by import, but chronologically as follows:
1-Phythagoras's Theorem (~700 BC), which states that the sum of the squares of the lengths of the two sides whose vertex is a right angle, equals the square of the third side. This is not just some schoolchild postulate, but is the basis of ancient civilization. Structures and ideas were constructed on the basis of this deep insightful concept. But there is proof that in Asia, and South America, a similar theorem was proposed, evidenced by the level of their intellectual progress.

2-Newton's Second Law of Motion (1666), which states that any force produced by, or imposed upon a body, results from the product of its mass and its change in motion. This is the pillar upon which all explainations of motion and forces on any bodies, stand to this very day. Galileo was on the brink of this idea, but the concept of 'change in motion' was still too abstract then, plus he was having serious troubles with the Catholic Church. Sir Isaac was the first the understand this concept, and led him to discover differential calculus as an afterthought.

3-Newton's Universal Gravitation Law (1666). The main word here is Universal, implying that it is valid, not only on in merry ole England, or even the Earth, but anywhere in the heavenly skies. Remembering that we are not far removed from the medieval period, and to demonstrate this degree of dogmatic farsightness was nothing short of revolutionary.

4-Euler's Equation (1740's).(not Euler's eq for fluid dynamics, which is just a fluid version for #2) Here Euler combines the five most fundamental numbers in all of mathematics (base of the natural log, pi, square root of -1, one, and zero) into one simple arithmetic statement. I remember asking my math teacher if this was a coincidence, and he replied 'there are no coincidences in mathematics.', meaning that there is a deep interrelationship between the values of these numbers.

5-The Second Law of Thermodynamics (1840's - 1850's). Aka the Law of Entropy, it was the culmination of the work of many,and in the author's words, worthy of a 'Shakespearean drama'. This was the period of the Industrial Revolution, and the Steam engine was all the rage. It was left to the physicist to explain what is heat, and its effects on the environment. The plot and cast is as follows: Carnot examines heat engines; Joule and Kelvin quantify heat; Clasius introduces entropy; Boltzmann and Maxwell applies statistics to describe entropy, the first time probabalistic concepts entered science, and foretold much of what was to come. It also definitively defined the direction of time. Up until then, it was assumed that time traveled with the clock hand, sunrises or birthdays, but there was no proof.

6-Maxwell's Equations (1860's). From statistical thermodynamics, Maxwell moved on to electricity and magnetism. He was deeply influenced by Faraday's lines of force and proceeded to unite the seemingly unrelated laws of Gauss, Ampere and Faraday. Along the way, he proved that light was just another electromagnetic effect. He actually proposed 12, sometimes more equations, and it took the practical ingenuity of Heaviside in 1884 to give it the familiar, symmetric, iconic (and much simpler) form we know today.

7-E=mc^2 (1905). Who in this world has not seen this? It conjures up images of wild-eyed, eccentric, white-haired madness and the atomic mushroom. Einstein, during one of his famous thought experiments, imagined to be surfing on a light wave. What would happen to the laws of physics? He concludes that the speed of light is the same for all observers, and that it is time and the three physical dimensions (space-time) that are variant. This was earth shattering at that time, but he was not yet done.

8-General Theory of Relativity (1915), so called because #7 was only a 'special' case. This showed that "space-time tells matter how to move, and matter tells space-time how to curve"; that the universe is like a trampoline-it conforms to the objects that are on it. The extension of this theory includes time travel, black holes, multiple universes, and string theory.

9-Shrodinger's Equation (1926). In the early 1920's, the new quantum mechanics was the latest hot research topic. But who really understood it. There's an expression-'if you can't explain it, then you don't understand it.' Shrodinger was an 'old-time' physicist (in his late 30's), and was comfortable with the conservative approach. Everyone knows the wave equation, which he adapted to ever growing number of quantum observations. The scientific community applauded his effort, which enabled them to 'simply' solve the 'standard' equation which fit in nicely with experimental data.

10-Heisenberg's Uncertaincy Principle (1927). Although Heisenberg also shared the credit for #9, his approach was much too perplexing, utilizing obscure matrices, confounding many who tried to apply his method. But not to be forgotten, his uncertainty principle forever destroyed the notion of predictability of the universe. It was a century earlier, when Laplace declared, 'give me the mass and motions of every object in the universe, and I will predict the future.' We now know that there is a calculable probabilty, that you are not where you think you are right this second; that parallel universes can exist, that its possible for Scotty to 'beamed us up'.

The book appeared to be one I would like with some quotes from my personal favorite, Dr. Richard P. Feynman. Oh how I wished Cal Tech would have accepted female students in the 1960's. I recently taught high school physics and hoped for a book I could recommend to students. While I could not quibble with the choice of equations, I felt cheated of that awe I experienced in high school when my trig derivations fell into place, or when I took a course in Z-transforms and I could solve the problems with pencil and paper. Additionally, material was repeated, the plethora of endnotes were disappointing and rarely added to the text. I would have loved to have reviewed this book before the final edit because the idea was truly inspired.

I found the book a nice refresher for some of the most important discoveries in mathematics and physics. My main criticism is that several of the equations presented are not explained. The last equation that is actually explained is Euler's, and here the author does a good job. But the reader is left with hardly a clue of what Maxwell's, Schroedinger's, and Einstein's General Relativity equations actually mean, let alone what they were derived from. The author apparently deems these theories too complex for substantial explanation and so he leaves the reader with a historical account of their development. I expected somewhat more scientific content from the book.

This book is about the most fundamental equations mankind has developed in its perpetual search for the "truth". It shows how far we have come from the basic 1+1=2, via the Pythagorean theorem, Newton Laws, Euler' s "God' s Equation", the 2nd Law of Thermodynamics, through Maxwell' s Equations , Special & General Relativity, Schrodinger' s Equations to Heisenberg' s Uncertainty Principle. This took us form 500 BC to 1927. This book also shows that we still have a long way to go. Einstein, Schrodinger and Heisenberg all acknowledged that their theories still don't describe "reality" perfectly. Equations are not the "presence" they are only "re-presentations". Equations are the result of the human mind and conscience observing, processing, translating in symbols and establishing abstract relationships. This inward focused process still cannot fully grasp the truth; perhaps the ultimate truth is un-knowable. It is interesting to observe that the great "insight" leading to these great equations always occurred when the mind stepped "outside" of itself and was open to a complete different way of looking at reality. The concepts of the equations cannot been "seen" only "experienced"; whether it is gravity, the electromagnetic "field", "space-time" curvatures, quantum "probability waves".
Perhaps this eternal search was already revealed in Plato's dialogue "Meno" in which Socrates explains the Meno Paradox, which asks "how is it possible to learn anything at all? "; If you don't know what you are looking for, you won't be able to recognize it when you come across it , while if you do know, you don't bother looking for it. Socrates finds an answer by stating that Souls/Minds are immortal, and thus have seen everything under the sun and just forgotten it all and have to "relearn ". For this reason it makes sense to end this book with Werner Heisenberg and his Uncertainty Principle

My mathematics education was somewhat lacking. Growing up in Mississippi, I didn't have a teacher that really knew their stuff until college and I was always curious about what was over the next hill and yet I always seemed to be behind. I never was able to go as far in mathematics as I would have liked and have continued to study the field on my own.

One of the things that I was always curious about was where did those formulas come from? How were they derived?

Without going completely overboard, this book does a good job of answering those questions. It is well worth reading if only to put into perspective the the advance of mathematics and its place in history.

After reading the first fifty or so pages in this book, I admit that I was a bit disappointed: there was way too much philosophy than I was expecting and not enough science/mathematics. Fortunately, the proportions soon shifted more to my liking, turning this book into a very pleasant and informative read. At the start of each of the book's ten chapters, the author presents an equation (or a set of equations or an inequality) that has proved of tremendous importance in physics and/or mathematics through history. In each case, the author presents the cast of characters (or single individuals) that have been involved in the equation's development in light of the science and mathematics of the times. The chapters are separated by a few pages of "interlude" that delightfully expand on some of the material discussed in (usually) the preceding chapter. As can be expected from an author who is chairman of a university philosophy department, an odd bit of philosophical verbiage can be found here and there throughout the book. The writing style is remarkably clear in its explanations of the scientific principles. It is also relatively accessible, friendly, authoritative and, for science buffs like me, generally quite engaging. Although dedicated general readers may glean much through reading this book, it would likely be of most interest to science buffs and science historians.

I have enjoyed this book, which has proved to be an excellent gift item for those who entertain a bit of curiosity. The writing is vivid and accessible, as well as spot-on correct.

Great book about the history of science. If you are a guy that like to know about this you are really going to like it.