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Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century (Springer Undergraduate Mathematics Series)

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Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century (Springer Undergraduate Mathematics Series)

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    Available in PDF Format | Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century (Springer Undergraduate Mathematics Series).pdf | English
    Jeremy Gray(Author)
Based on the latest historical research, Worlds Out of Nothing is the first book to provide a course on the history of geometry in the 19th century. Topics include projective geometry, especially the concept of duality, non-Euclidean geometry, and more.

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Book details

  • PDF | 412 pages
  • Jeremy Gray(Author)
  • Springer; 2 edition (17 Dec. 2010)
  • English
  • 8
  • Science & Nature

Review Text

  • By A. J. Sutter on 14 November 2011

    As a "general reader," rather than a student using this book in a university course, I found the material highly interesting but the treatment often frustrating.On the plus side, I did come away from it with a better understanding of the substance and goals of projective geometry, of the intellectual context of non-Euclidean geometry, and of what might be called the struggle for co-existence (pace Darwin) of multiple geometries at the end of the 19th Century.Several aspects of the book were very frustrating, though:(1) This is a set of course materials, not a coherent monograph. The styles of the chapters change from biographical to mathematical and back again, and not always in chronological order. E.g., the sequence of Chaps. 2-4: "2. Poncelet (and Pole and Polar)" -> "3. Theorems in Projective Geometry" [sc., not by Poncelet] -> "4. Poncelet's _Traité_"; in a monograph the sequence of chapters 2 and 3 might have been reversed. The sequence of the last six chapters (26-31) also feels a bit jumbled, with the "Summary: Geometry to 1900" (i.e., to the end of the century that is the topic of the book) coming five chapters before the end. Three chapters on writing history, apparently directed to undergraduates, are dropped in from time to time. One of these is the final chapter, which describes an essay assignment, so that the book ends clumsily with a warning that you'll receive a failing grade if you plagiarize. The most charitable spin one can put onto it is that reading the book might be like reading a play: a lot depends on how it's used in action. E.g., maybe in the book we're reading material "in series" that in a class would be deployed "in parallel," to use an electrical analogy. Maybe -- but it doesn't make for a good read.(2) Maybe partly as a result of (1), some matters are referred to before they're defined. E.g., Bézout's theorem is relied on twice (@176, 186) before it's actually stated (@193). BTW it's also assumed you know what Menelaus's theorem is, which I imagine few people do who never took a university-level course in geometry. (It's a statement about the "cross ratio" of lengths of segments, AB·CD/AD·BC, of a line containing points A, B, C, D in that order.)(3) For a book on geometry, it could do with more illustrations -- and more accurate ones. The proof of the uniqueness of the fourth harmonic point, @38, mentions segments not shown in the accompanying figure. The discussion of Möbius's barycentric coordinates doesn't have any diagrams (Chap. 13), even though Möbius's original work used plenty of them; this trips up the author (JG) when he refers to a "horizontal plane" in a Cartesian coordinate system when it isn't clear whether that's, say the x-y plane, x-z, or what (@153). Most puzzling was that JG sometimes omits illustrations even when presenting an excerpt from an historical source that contained them, e.g. in his passages from Lobachevsky. (Both Möbius's and Lobachevsky's books are available online, though not necessarily in English.)(4) The professional standards of historians of mathematics seem to allow a lot of leeway for unannounced anachronisms: that is, the intrusion of modern concepts and notation into discussion of historical material without distinguishing the old from the new. As far as I could tell, most of these occur in the earlier, pre-Riemann portion of the book - but they made me so distrustful of what I'd been reading till then that I almost put the book back on the shelf, half-read.E.g., @155: "So a projective transformation emerges as a 3x3 matrix, with the proviso that it too is homogeneous (matrices A and kA have the same effect). As a result, Möbius had a novel, entirely algebraic description of conic sections and their projective transformations." This sure makes it sound as if Möbius used matrices, even though the complete passage stops short of explicitly stating that he did. But he didn't: I scrolled through all 300-plus pages of _Der baryzentrische Calcul_ and didn't find a single matrix. Which isn't surprising, since they were invented several decades later by Sylvester.E.g., in Chap. 3, where we are backtracking to cover some pre-Poncelet projective geometry in the lead-up to Poncelet's most famous proof, we're told that Menelaus's theorem means that the cross-ratio has a value of -1 (@37). Except that Menelaus didn't say any such thing. Firstly, he wrote in the Hellenistic era, before negative numbers were recognized; and secondly, I checked his proof (Lemma 3, book I), as translated into Latin by Edmond Halley in 1758. In fact, it's not clear when his theorem came to be generally known, since Halley was translating from Hebrew and Arabic (Menelaus's original Greek apparently being lost). Was it widely known at all before this Latin version? and when did the negative number version of it come to be known? Most importantly: *what was the version Poncelet knew?* -- the one published in Halley, or the one with negative numbers? If the latter, who came up with it?I don't have a problem in principle with an historian using modern notation or ideas to explain older math. Often the older symbols and narratives are very tedious to read, so some adaptation can be a relief. But even when such modernization is justified, I do expect a heads-up about which is which, especially when the historian is hanging a lot of weight on a specific nail. All it takes is a simple phrase like "To express so-and-so's argument in modern concepts, ... ," or something more graceful. I attribute these lapses to professional standards rather than singling out JG for blame, because he refers in several places to the book's referees -- so some of his peers must have thought this was OK. You might be more tolerant, but it drove me nuts.(5) Finally, while no book can cover anything, some of the book's omissions were surprising. Hilbert's axiomatization program is discussed at length, and a reader might finish the book thinking it was possible; not a syllable is uttered about Kurt Gödel's eventual proof to the contrary. Similarly, while there's some discussion of realism vs. formalism in mathematics and Hilbert's position on the issue, there isn't any mention of the related 20th Century debate between Hilbert and Brouwer. And ironically for me, while there's a lot of discussion of Julius Plücker, the the book is silent on the topic I'd hoped to learn about from it: Plücker coordinates (which are a stumbling block for the ignorant in an early chapter of Thomas Hawkins's history of Lie group theory). All in all, 3.5 stars, as a balance of the book's pluses and minuses.

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