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But Newton had a darker side. Despite the fame and recognition he had received, Newton refused to let anyone threaten to overshadow him or stand in his way of greater achievements.
Reverend John Flamsteed was the first Astronomer Royal - a position he held for 44 years serving under 6 kings. He spent his night in the observatory of Greenwich gazing through telescopes, cataloguing the stars. Newton wanted this information to figure out a better way to navigate to oceans, a major problem in his day. He was convinced Flamsteed was holding back the critical information he needed. For that, Newton used all the considerable power at his disposal to end the career of Flamseed. He almost suceeded. It was only because of the dedication of Flamsteed's widow that his 3-volumn Historia Coelestis Britannica was published.
Today, because of Flamsteed's work, we measure longitude from the place he accomlished his work - Greenwich.
The work of Stephen Gray is less known. A commoner trained as a dyer, he was a most unlikely member of the Royal Society.
Gray was a long time friend of Flamsteed. He carried on a regular coorespondence with the elder scientist, sharing with him his own celestial observations.
But it was Gray's pioneering work in using electricity for communications that earned him immortality. Work, that if not for Newton, may have been accomplished 20 years sooner.
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While many of the problems have been solved, it is a tribute to Hilbert that some are still unsolved and there appears to be no hope that they will be resolved soon. A few of the problems were solved relatively quickly, but most succumbed only after decades of intensive work. All of the problems that he put forward are explained in great detail, and if they were solved, the manner of solution demonstrated. Since these problems are hard, it is not possible to thoroughly describe them without resorting to some advanced mathematics. However, that is kept to a minimum, so it is possible for someone without detailed knowledge to understand most of the explanations.
The German universities were very powerful centers of mathematical progress during Hilbert's lifetime and the story about the interaction of the personalities and the split between pure and applied mathematics makes very interesting reading. Mathematics is in many ways just another human endeavor, subject to petty spats, nationalistic rivalries and personal biases. The saddest part of the book is the description of what happened to the once proud university system when the Nazi party rose to power. An incredible amount of talent was hounded away, which was fortunate for them as most of those who remained and had an incorrect heritage were killed. Hilbert was a firm believer in the value of applied mathematics, so he no doubt would have been frustrated over the split between the pure and applied camps that occurred after the end of the second world war. Given that he was so much of both, I wonder what tone his voice would have had.
Hilbert was an intellectual giant who is known most for his set of famous problems rather than his impressive work on resolving problems. While the emphasis is on the famous 23 problems, enough effort is expended on what else he did to make the book as much a biography of Hilbert as it is on the problems he posed. That alone would make it well worth reading.
Published in Journal of Recreational Mathematics, reprinted with permission.
It is easy to read and well explained, even if you don't grasp the full maths, still there is a story around every of the 23 problems that lets you understand the implication, and the full drama of its solution.
It is a nice biography of Hilbert 'the man', intertwined with the 23 problems, so it does not get boring like some biographies do with endless lists of calendar-facts.
There is even a full translation of the original speech he gave in Paris in 1900, which otherwise would be impossible to find.
The problems itself are well explained, as well in the timeframe of 1900, when first posed, as later in our time when maths was ready to solve them. The author did a good job also telling which of the problems really were important, really gave mahts further problems to think about, and which problems didnt give rise to new mathematical areas, and therefore became more or less curiosities after solution.
Reading this book gave me a feeling of how beautiful maths can be, how unexpectedly some problems can and cannot be solved, and evokes some of the drama of the worlds biggest minds at work.
If you are interested in maths and/oir in science and great minds: this is an excellent read!
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The book provided an excellent overview of this fabulous bird species, including their history, tips on raising, feeding, housing, breeding, caring for them etc. The book was very easy to read and full of very useful information.
The book is printed on quality paper, with a quality binding. The professionally taken photographs are outstanding. The printing technique of the book makes them look like actual glossy photographs - glued into the book! As a bibliophile with thousands of books, and an amateur photographer - I can honestly say I've never encountered such a well-printed book! The pictures alone are worth the price.
The book positively influenced my decision to add another member to my family - an African Grey Parrot.
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Basil Hallward is a merely average painter until he meets Dorian Gray and becomes his friend. But Dorian, who is blessed with an angelic beauty, inspires Hallward to create his ultimate masterpiece. Awed by the perfection of this rendering, he utters the wish to be able to retain the good looks of his youth while the picture were the one to deteriorate with age. But when Dorian discovers the painting cruelly altered and realizes that his wish has been fulfilled, he ponders changing his hedonistic approach.
Dorian Gray's sharp social criticism has provoked audible controversy and protest upon the book's 1890 publication, and only years later was it to rise to classical status. Written in the style of a Greek tragedy, it is popularly interpreted as an analogy to Wilde's own tragic life. Despite this, the book is laced with the right amounts of the author's perpetual jaunty wit.
Dorian Gray is beautiful and irresistible. He is a socialité with a high ego and superficial thinking. When his friend Basil Hallward paints his portrait, Gray expresses his wish that he could stay forever as young and charming as the portrait. The wish comes true.
Allured by his depraved friend Henry Wotton, perhaps the best character of the book, Gray jumps into a life of utter pervertion and sin. But, every time he sins, the portrait gets older, while Gray stays young and healthy. His life turns into a maelstrom of sex, lies, murder and crime. Some day he will want to cancel the deal and be normal again. But Fate has other plans.
Wilde, a man of the world who vaguely resembles Gray, wrote this masterpiece with a great but dark sense of humor, saying every thing he has to say. It is an ironic view of vanity, of superflous desires. Gray is a man destroyed by his very beauty, to whom an unknown magical power gave the chance to contemplate in his own portrait all the vices that his looks and the world put in his hands. Love becomes carnal lust; passion becomes crime. The characters and the scenes are perfect. Wilde's wit and sarcasm come in full splendor to tell us that the world is dangerous for the soul, when its rules are not followed. But, and it's a big but, it is not a moralizing story. Wilde was not the man to do that. It is a fierce and unrepressed exposition of all the ugly side of us humans, when unchecked by nature. To be rich, beautiful and eternally young is a sure way to hell. And the writing makes it a classical novel. Come go with Wotton and Wilde to the theater, and then to an orgy. You'll wish you age peacefully.
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It has lots of good color photos, although only one photo per bow.
The book covers reproductions of traditional bows from all over the world, beginning with prehistoric times. Naturally there are only a few samples from the different continents and ages, and most reproductions are made by people the author knows or have met in USA.
The book does not tell much about the people who used these bows, or how they were shot, rather why they designed and developed them as they did, and how they are made.
I definately recommend this book, especially for the great photos, but you will soon look for other books that go more in depth with a specific area or age. Still, this one is more of a survey, and a great one at that!
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Brannan, Esplen, and Gray's Geometry accomplish for math what those Scientific American articles did for physics: speaking at a level accessible to anyone with a good high school education, they bring the interested reader up to speed in affine, projective, hyperbolic, inversive, and spherical geometry. They provide the simple explanations, diagrams, and computational details you are assumed to know-but probably don't-when you take advanced courses in topology, differential geometry, algebraic geometry, Lie groups, and more. I wish I had had a book like this when I learned those subjects.
Individual chapters of about 50 pages focus on distinct geometries. Each one is written to be studied in the course of five evenings: a week or two of work apiece. Although they build sequentially, just about any of them can be read after mastering the basic ideas of projective geometry (chapter 3) and inversive geometry (chapter 5). This makes the latter part of the book relatively accessible even to the less-committed reader and an effective handbook for someone looking for just an overview and basic formulas.
The approach is surprisingly sophisticated. The authors do not shy away from introducing and using a little bit of group theory, even at the outset. (Scientific American, even in its heyday, never dared do that.) They present all geometries from a relatively modern point of view, as the study of the invariants of a transitive group of transformations on a set. Many explanations and proofs are based on exploiting properties of these transformations. This brings a welcome current of rigor and elegance to a somewhat static subject long relegated to out of date or sloppy authors (with the exception of a few standouts, such as Lang & Murrow's "Geometry").
One nice aspect is the authors' evident awareness of and appreciation for the history of mathematics. Marginal notes begin at Plato and wind up with Felix Klein's Erlangen program some 2300 years later. Although the text does not necessarily follow the historical development of geometry, its references to that development provide a nice context for the ideas. This is an approach that would improve the exposition of many math texts at all levels.
The authors are British and evidently write for students with slightly different backgrounds than American undergraduates. Obvious prerequisites are a mastery of algebra and a good high school course in Euclidean geometry. Synopses of the limited amounts of group theory and linear algebra needed appear in two brief appendices. However, readers had better be intuitively comfortable with matrix operations, including diagonalization and finding eigenspaces, because matrices and complex numbers are used throughout the book for performing computations and developing proofs. A knowledge of calculus is not needed. Indeed, calculus is not used in the first two-thirds of the book, appearing only briefly to derive a distance formula for hyperbolic geometry (a differential equation for the exponential map is derived and solved). During the last third of the book (the chapters on hyperbolic and spherical geometry), some basic familiarity with trigonometric functions and hyperbolic functions is assumed (cosh, sinh, tanh, and their inverses). Definitions of these functions are not routinely provided, but algebraic identities appear in marginal notes where they are needed.
Now for the quibbles. The book has lots of diagrams, but not enough of them. The problems are usually trivial, tending to ask for basic calculations to reinforce points in the text. The text itself does not go very deeply into any one geometry, being generally content with a few illustrative theorems. An opportunity exists here to create a set of gradually more challenging problems that would engage smarter or more sophisticated readers, as well as show the casual reader where the theories are headed.
This book is the work of three authors and it shows, to ill effect, in Chapter 6 ("non-euclidean geometry"). Until then, the text is remarkably clean and free of typographical and notational errors. This chapter contains some glaring errors. For example, a function s(z) is defined at the beginning of a proof on page 296, but the proof confusingly proceeds to refer to "s(0,c)", "s(a,b)", and so on.
The written-by-committee syndrome appears in subtler ways. There are few direct cross-references among the chapters on inversive, hyperbolic, and spherical geometry, despite the ample opportunities presented by the material. Techniques used in one chapter that would apply without change to similar situations in another are abandoned and replaced with entirely different techniques. Within the aberrant Chapter 6, some complex derivations could be replaced by much simpler proofs based on material earlier in the chapter.
The last chapter attempts to unify the preceding ones by exhibiting various geometries as sub-geometries of others. It would have been better to make the connections evident as the material was being developed. It is disappointing, too, that nothing in this book really hints at the truly interesting developments in geometry: differentiable manifolds, Lie groups, Cartan connections, complex variable theory, quaternion actions, and much more. Indeed, any possible hint seems willfully suppressed: the matrix groups in evidence, such as SL(2, R), SU(1,1, C), PSL(3, R), O(3), and so on, are always given unconventional names, for instance. Even where a connection is screaming out, it is not made: the function abstractly named "g" on pages 296-97 is the exponential map of differential geometry, for instance.
Despite these limitations, Brannan et al. is a good and enjoyable book for anyone from high school through first-year graduate level in mathematics.
The first chapter treats some basics about conics. The second chapter is on affine geometry. The third and fourth chapters are about projective geometry. In the fifth chapter you will be led through Inversive geometry which functions as a base for the sixth and seventh chapter. The sixth chapter has as itst title Non-Euclidean geometry, but it is in fact the Hyperbolic geometry of Boljay in a formulation of Henry Poincaré. The seventh chapter is about Spherical Geometry. In the eighth chapter all of these geometries are demonstrated to be special cases of the Kleinian vieuw of geometry: that is, every geometry can be seen as consisting of the invariants of a specific group of transformations of the 2 dimensional plane into itself. It is clearly demonstrated that this is less trivial than you would expect.
I learned two things from this book. The first is, that you can, in principle, prove every theorem of geometry by just using Euclidean geometry. But if you do this, the amount of work it takes can be very huge indeed. It is a far better strategy to try to determine what geometry is best suited for the problem at hand, and solve it within that geometry.
Since the book gives a very clear picture not only of the particular geometries, but also to how the geometries relate to each other, you have, as an extra bonus, insight in the level of abstraction and the scope of your theorem.
The second thing I learned is how you can use geometry to make concepts as simple as 'triangle' precise. What I mean is this: a right angle triangle is not the same as an equilateral triangle. But both are the same in the sense that they are both triangles. The question is this: how can two 'things' be the same and at the same time not 'the same'? The book gives an answer to this 'question about the meaning of abstractions'. It gives the following solution. Take a triangle, ANY triangle. Consider the group of all affine transformations A (which consists of an uncountably infinite set of transformations.) If you subject this one triangle Tr to every affine transformation in this group A, you will have created a set consisting of exactly ALL triangles. In other words, the abstract idea of 'triangle' consists of ONE triangle Tr together with the set of ALL affine transformations. You can denote this as the pair (Tr, A). In the same way you can express the abstract idea of ellipse by the pair (El, A), and the abstract idea of parabola by the pair (Par, A). And, by passing to the more abstract Projective geometry, you can express the abstract idea of 'conic' by giving just one quadratic curve, be it a parabola, ellipse or hyperbola, by the pair (Qu, P), whereby P is the group of all projective transformations.
The book presupposes some group theory and some knowledge of linear algebra. Furthermore you have to know a little calculus. I have very little knowledge of group theory, and I have just about enough knowledge and skill about linear algebra to know the difference between an orthogonal and unitary matrix, and to know what eigenvectors are. I have studied the first 5 chapters of CALCULUS from Tom M. Apostol, which does not go too deep into linear algebra. This proved to be enough.
I have only one point of critique. Virtually all problems in the book are of the 'plug in type', even those at the end of every chapter (from which, by the way, you cannot find the solutions at the end of the book, while the solutions of those in the text can be found in an appendix). If you have understood the text, you have no difficulties whatsoever to solve them. The problems are not challenging enough to give you a real skill in all of these geometries, although they do become more challenging in later chapters. They are only intended to help you to understand the basic principles of all of these geometries, no more, no less. So if you want to have a tool to help you in obtaining a greater skill in, say, the special theory of relativity by studying hyperbolic geometry, this is not a suitable book. That is why I have given it 4 stars, and not the full 5 stars.
I also have a piece of advise. Although the problems are, from a conceptual point of view, not challenging, a mistake is easily made. Therefore it is best to solve the problems by making use of a mathematical program like Maple or Mathematica. If you then have made a mistake, you can backtrack exactly where you have made it, and let the program take care of all of the tedious calculations. This has also stimulated to try to calculate some outcomes by following a different approach, and then to compare the results.
I have enjoyed studying this book immensely.