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Based on the latest historical research, this is the first book to provide a course on the history of geometry in the 19th century. The book is a comprehensive resource with full background material and selections and translations from original sources. It provides both an overview of the growth of a major part of mathematics and offers practical guidance on the teaching of the history of mathematics.
- Sales Rank: #17257561 in Books
- Published on: 2011-04-09
- Original language: English
- Dimensions: 9.21" h x .84" w x 6.14" l,
- Binding: Paperback
- 412 pages
Review
From the reviews:
"Gray’s new book will become both a classic reference and a model on how to write a useful course text. With original source material woven in with historical context, this book is a fun read as it examines geometry historically as a connected sequence of diverse ideas … . it includes a useful index and extensive bibliography. If you enjoy mathematics, buy this and read it! Summing Up: Highly recommended. General readers; lower-division undergraduates through faculty." (J. Johnson, CHOICE, Vol. 44 (11), July, 2007)
"Gray (Open Univ., UK) does it again. Complementing his works on Jinos Bolyai or the world of different geometries, Gray's new book will become both a classic reference and a model on how to write a useful course text. With original source material in with historical context, this book is a fun read as it examines geometry historically as a connected sequence of diverse ideas: duality in projective geometry, the problem of parallels in non-Euclidean geometries, the nature of surfaces in differential geometry, and concern for geometry as a true "measure" of space (physical or philosophical). Documenting mathematicians' contributions within this historical sequence, the book becomes a fascinating Who's Who: Poncelet, Desargues, Pascal, Gergonne, Saccheri, Lambert, Legendre, Gauss, Bolyai, Lobachevski, M�bius, Pl�cker, Beltrami, Klein, Poincar�, Hilbert, and Einstein. Finally, the book's unusual twist is its inclusion of three chapters on writing and critical reading about the history of mathematics; these chapters alone make the book valuable to those helping students improve as writers of mathematics. An offshoot of a University of Warwick course, it includes a useful index and extensive bibliography. If you enjoy mathematics, but this and read it! Summing up: Highly recommended. General readers; lower-division undergraduates through faculty."�- J. Johnson, Western Washington University
"This is an outstanding text for the well-prepared student who has a basic knowledge of linear algebra, matrix theory, calculus of several variables, and geometry. … an important addition to the mathematical and historical literature. Albeit primarily a textbook for a one-semester upper level or graduate course in the history of mathematics, it will be appreciated and enjoyed by those interested in geometry, history, philosophy, and pedagogy. … The book contains an excellent bibliography. … it provides a useful reference for anyone interested in geometry." (James J. Tattersall, Mathematical Reviews, Issue 2008 b)
"The 19th century was certainly an exciting time in geometry. To chronicle all of that excitement in one place is a monumental task; to have done so with real clarity and attention to detail, as Jeremy Gray has done, is an impressive achievement. … Gray has succeeded on several levels: as a historical chronicler, as a mathematical scholar, and as an advisor to teachers. Worlds Out Of Nothing is a first-rate addition to the geometry enthusiast’s bookshelf." (Mark Bollman, MathDL, January, 2008)
From the Back Cover
Worlds Out of Nothing is the first book to provide a course on the history of geometry in the 19th century. Based on the latest historical research, the book is aimed primarily at undergraduate and graduate students in mathematics but will also appeal to the reader with a general interest in the history of mathematics. Emphasis is placed on understanding the historical significance of the new mathematics: Why was it done? How - if at all - was it appreciated? What new questions did it generate?
Topics covered in the first part of the book are projective geometry, especially the concept of duality, and non-Euclidean geometry. The book then moves on to the study of the singular points of algebraic curves (Pl�cker’s equations) and their role in resolving a paradox in the theory of duality; to Riemann’s work on differential geometry; and to Beltrami’s role in successfully establishing non-Euclidean geometry as a rigorous mathematical subject. The final part of the book considers how projective geometry, as exemplified by Klein’s Erlangen Program, rose to prominence, and looks at Poincar�’s ideas about non-Euclidean geometry and their physical and philosophical significance. It then concludes with discussions on geometry and formalism, examining the Italian contribution and Hilbert’s Foundations of Geometry; geometry and physics, with a look at some of Einstein’s ideas; and geometry and truth.
Three chapters are devoted to writing and assessing work in the history of mathematics, with examples of sample questions in the subject, advice on how to write essays, and comments on what instructors should be looking for.
The Springer Undergraduate Mathematics Series (SUMS) is designed for undergraduates in the mathematical sciences. From core foundational material to final year topics, SUMS books take a fresh and modern approach and are ideal for self-study or for a one- or two-semester course. Each book includes numerous examples, problems and fully-worked examples.
About the Author
Jeremy Gray is Professor of the History of Mathematics and Director of the Centre for the History of the Mathematical Sciences at the Open University in England, and is an Honorary Professor in the Mathematics Department at the University of Warwick. He is the author, co-author, or editor of 14 books on the history of mathematics in the 19th and 20th Centuries and is internationally recognised as an authority on the subject. His book, Ideas of Space, is a standard text on the history of geometry (see competitive literature).
Most helpful customer reviews
9 of 11 people found the following review helpful.
Interesting but somewhat irritating as well (3.5 stars)
By A. J. Sutter
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 _Trait�_"; 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 stuff "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., B�zout'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 M�bius's barycentric coordinates doesn't have any diagrams (Chap. 13), even though M�bius'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 M�bius'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, M�bius had a novel, entirely algebraic description of conic sections and their projective transformations." This sure makes it sound as if M�bius 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. What makes this kind of looseness especially damaging is that once a reader discovers one or two cases of it, it's hard not to be suspicious about a lot of the rest of what one has read. You might be more tolerant, e.g. if you're less interested in history of ideas, but this definitely detracted from my enjoyment of the book.
(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 G�del'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 Pl�cker, the the book is silent on the topic I'd hoped to learn about from it: Pl�cker 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.
8 of 10 people found the following review helpful.
interesting
By doodler
I'm actually reading this to gain insights to what those freaks were thinking when they invented projective geometry. All the projective geometry texts I have read are so dry and free of any motivational thoughts as to actually mystify the subject. Mathematicians are some of the worst writers on their own subject as history clearly shows. But this book really does help set the culture and motivations for this difficult and abstract field. There are typos in both text and diagrams. So you'll have to be wary as you read through. Luckly his other books can provide the correct formulas if you do a web search of the subject. For example "Plato's Ghost" has the correct text for the description of 4th Harmonic Point. And I do so hate typos in mathematics books.
10 of 14 people found the following review helpful.
Good history, especially on projective geometry
By Viktor Blasjo
Poncelet developed projective geometry in 1813. He saw that when dealing with geometric theorems that do not involve metric concepts we can allow ourselves greater freedom than Euclidean methods allow. The idea of projections and points at infinity enable us to give extremely swift proofs of previously cumbersome theorems. Consider for instance Pappus' theorem: we have a line ABC and a line A'B'C' and the theorem says that the three intersections AB'-A'B, BC'-B'C, AC'-A'C lie on a line. Projective geometry allows us to send the points AB'-A'B and BC'-B'C to infinity (i.e. make AB' parallel to A'B and BC' parallel to B'C) and then prove the theorem in this special case, which is a ridiculously easy application of similar triangles. Desargues' theorem can be proved in exactly the same way. Poncelet also studied dualisation with respect to a conic, which is a transformation that interchanges points and lines while preserving (in inverted form) all projective relations between them. Any theorem can be dualised to give a new theorem; for example, a simple derivative argument shows that the number of tangents to a curve of degree n through a given point is n(n-1), so by dualising Bezout's theorem we see that two curves have n(n-1)m(m-1) tangents in common.
"But it is one thing to realise that dualising a figure is a good way to obtain new theorems, which is what Poncelet did, and quite another thing to claim that points and lines are interchangeable concepts which must logically be treated on a par. This was the view that Gergonne put forward in 1825." (p. 55) This view would of course be clarified by the analytic approach of Plucker, but Plucker also solved a more important problem: we just saw that a curve has n(n-1) tangents; when we dualise these become n(n-1) points on the dual curve; but all those tangents met in a point, so their dual points all lie on a line, so the dual curve intersects a line n(n-1) times, so it has degree n(n-1). But, of course, dualising twice should get us back where we started. How can that be when dualising always increases the degree? This "paradox" was resolved by Plucker and his formulae.
"The argument presented here that Plucker's resolution of the duality paradox was decisive for the future of projective geometry seems to be new" (pp. vi-vii). The solution is that the tangent count n(n-1) fails when there are singularities involved. Consider the curve y^2=x(x+a)(x-b). It has two separate components: an infinite curve and a loop. Now, suppose we are standing so that we have a view through the opening between the two components; this field of view is bounded by two tangents to the curve. If we let b go to zero the two components come together to form a single curve that crosses itself at the origin. Our two tangents merge to a false tangent, which is counted twice by the n(n-1)-method when in fact it should not be counted at all. Now start over and consider the two tangents that bound our view of the loop. As a goes to zero the loop shrinks to a point and again the two tangents become one false tangent counted twice. These are examples of double points. If both a and b go to zero at once then all three tangents considered above come together to one false tangent counted three times. This is a cusp. Therefore, the number of tangents is not n(n-1) but rather n(n-1)-2d-3c. This almost makes the degree of the dual of the dual equal to the degree of the original curve, but not quite. A further correction must be made if the original curve has bitangents (same tangent for two points) which of course would become double points of the dual curve. But since we secretly know the answer to the "paradox" this in fact reveals a new theorem: the number of bitangents on a curve with no double points or cusps must be whatever number produces the correct correction term, which is n(n-2)(n^2-9)/2.
After the Plucker formulae we pick up a different thread. First we look at attempts to prove the parallel postulate, then the non-Euclidean geometry of Bolyai and Lobachevsky, and then the differential geometric view of non-Euclidean geometry of Riemann and Beltrami. This is all basically rehashed material from Gray's Ideas of Space book, although now, finally, with some reference to Stillwell's book. After this detour, projective geometry gets the last laugh, because Klein showed that projective geometry subsumes non-Euclidean geometry (since the metric of Beltrami's disc model may be defined in terms of cross-ratios), suggesting that the entire foundations of geometry should be attacked through projective geometry, which indeed turned out to be a very fruitful point of view, as we see (briefly) in the works of Hilbert and others.
In all, the individual chapters are very interesting, although at times the book as a whole feels somewhat sloppily composed. This feeling is reinforced by poor proof-reading. For example, the description of the construction of the fourth harmonic point (which comes with no picture) has "CR" where it should be "AR" (p. 27); the construction is then repeated on p. 33 ("Recall that ..." and then the whole thing again), this time with a different mistake ("PB" where it should be "BQ"); later on the same page we are asked to "repeat the construction but with the points A, B and C" which are the same points as before; clearly it should be C' instead of C.
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