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Review from the Washington Post: 

One of the strangest things in a cosmos full of bewilderments is the power of mathematics to explain the way things are. The universe, after all, in its superlative indifference, is under no obligation to be comprehensible, much less explicable by calculation. Yet to a remarkable degree, the history of science is the chronicle of humanity's increasingly sophisticated ability to describe and then predict mathematically the nature of nature. 

Nowhere is this more apparent than in comprehending the shape of space. The pursuit originally called "geometry" when it merely concerned the Earth's surface is still at the brain-buckling frontier of research into four-dimensional Einsteinian "space-time" and even 10- (or more) dimensional superstring theory. As physicist Leonard Mlodinow puts it in this reader-friendly, high-spirited, splendidly lucid and often hilarious history, "Ever since the Greeks, mathematics has been at the heart of science, and geometry at the heart of mathematics." It took 2,600 years to arrive at our present condition. Mlodinow divides that progress into five "geometric revolutions" epitomized by Euclid, Descartes, Gauss (the guy who invented the "bell curve," thus becoming the patron saint of C students everywhere), Einstein and Witten. That last is Ed Witten of Princeton's Institute for Advanced Studies -- arguably the planet's paramount mathematical physicist and a maximum ayatollah of string theory. 

Even those readers who would still prefer a tax audit to solving the Pythagorean theorem will find that Mlodinow unfolds the story of geometry in a most inviting context. The mathematical essentials are all here, painstakingly explained in concrete physical and visual terms. But so is a good deal of relevant (as well as frequently irrelevant but fascinating) historical garnish. Thus we learn in detail how the iron postulates of Euclid's "Elements" applied a logical rigor to two-dimensional space, and how his notion of parallel lines eventually became untenable. But we also discover that property taxes "were perhaps the first imperative for the development of geometry," and that the word hypotenuse comes from the Greek for "stretched against" because surveying was conducted by teams of rope-stretchers who formed triangles with knotted cords. Pythagoras, it is revealed, was so weird that he bit a snake to death after it attacked him, and that a thief, after breaking into the mathematician's home, "saw such bizarre things that he fled empty-handed." Aristotle's influence on the idea of latitude, Newton's role in the invention of the sextant, Charlemagne's ego ("In one monastery alone, Charlemagne had 300 monks and 100 clerks praying continuously for him, in three shifts, around the clock. He died anyway.") and Descartes' opinion that his contemporary Galileo had a somewhat disorderly mind -- all are here. So are dozens of delightful stories such as the one about German geometrician Georg Riemann, who at the age of 19 was given an 859-page tome on number theory by his teacher. Riemann "returned it in six days with a comment along the lines of 'It was a good read.' " 

Many of these tales -- for example, Einstein's legendary underachievement in school -- are familiar. Others, such as the fact that Albert Michelson (of the eponymous Michelson-Morley experiment that killed the theory of "ether" in the universe) lived in madcap Virginia City, Nev., during the gold-rush mania, or the shocking anti-Semitism of some 20th-century Nobel laureates, may surprise even some science buffs. But the focus of the narrative always remains on the central issue: how science gradually moved from the conceptually magnificent certitude of Euclidean space to the non-Euclidean, "elliptical" geometry that, paradoxically, was so hard to understand and yet described empirical physical reality so surpassingly well. The basis of this evolution, as Mlodinow notes, is familiar to "any parent who has ever tried to wrap a round ball with flat gift wrap." That is, trying to map the real three-dimensional world by Euclidean methods is bound to fail. 

In Euclid's reckoning, the angles of a triangle always add to exactly 180 degrees. But picture a triangle on the Earth's surface with corners at the North Pole, Ecuador and the Congo. Its angles total about 270 degrees. Non-Euclidean concepts become particularly essential in depicting reality as uncovered by modern physics. That is the basic subject of the last two sections of "Euclid's Window." The general theory of relativity describes how mass bends space and time, and thus puts enormous demands on conventional views of geometry. Mlodinow handles this material with special care, including spectacularly cogent descriptions of the key concepts of relativity. 

String theory -- the idea that all elementary particles such as electrons or quarks are really just different vibrational modes of invisible strands about a trillionth of a trillionth of a trillionth of a meter long -- requires not only non-Euclidean geometry, but an additional six or seven dimensions as well. This is the only place where the general reader may find tough going. Not only is the subject complex and intrinsically forbidding, but the amount of space Mlodinow allots to explain it is simply too small. Nonetheless, it is more comprehensible than nine-tenths of what one reads in the popular press. As a result, by the end of this superbly imagined book, even the most math-averse will have experienced Mlodinow's -- and science's -- ultimate goal: "the joy of understanding."