Customer Reviews

A Modern View of Geometry (Dover Books on Mathematics)

5 star:		(1)
4 star:		(1)
3 star:		(0)
2 star:		(0)
1 star:		(0)

 

 

This book provides a great intro to non-standard geometries by creating different axiomatic systems and finding models of them. It then constructs and analyzes operators (addition, multiplication, and the like) on the plane. It devotes a good section to discussing the Desargues and Pappus properties and their fundamentality in most geometries. A good treatise of projective geometry follows, and the book ends with a quick skim of metric geometries and non-euclidean geometries. This is not a good book if you are planning to study Hyperbolic, Spherical or Elliptical geometry, nor does it do a fair treatment of the effects of a metric on a geometry, but it does provide a short, comprehensive intro to axiomatic coordinate geometry.

Far away and long ago, I read this little big book. I borrowed it from a public library, with the hope to learn plane projective geometry. I'm still amazed I could understand almost everything there in one month. Years later I bought it. The first three chapters are addressed to anybody able to read, write and think, with little or null background in mathematics. The book starts with a glimpse on Euclid's fifth postulate and his history, Then it gives an overview of elementary logic and says what an axiomatic theory is. In chapter three, Hilbert axioms for the euclidean plane geometry are presented. Here, the less trained reader could consider a trip thru Euclid's Elements The Thirteen Books of Euclid's Elements before going further . If you have some mathematical education and keep faithful to Blumenthal, you face the affine plane and the problem of introducing cordinates (chapter four); to do so, Blumenthal follows Marshall Hall's concept of planar ternary ring (see the revered Hall's work The Theory of Groups (AMS Chelsea Publishing) to confirm). I was shocked when I knew finite geometries were possible. The book shows that the plane admits the Desargues configuration if and only if coordinates form a (maybe non commutative) field of "numbers", while the plane admits Pappus configuration if and only if the field is commutative (G. Hessenberg's theorem). Next, an axiomatic definition of projective plane is presented (curiously, we don't find here the name of Staudt, the author of the first axiomatic system for the projective geometry, published in his book "Der Geometrie der Lage", Nüremberg, 1847). More on coordinates and configurations on a projective plane is discussed in chapter six. At the end, there is a nice report on the elliptic plane, which is nothing but the real projective plane with a suitable metric (by the way, as a topological space, Möbius discovered that it is a non-orientable surface) and a mention of its differences with respect to the euclidean plane and with respect to the Poincaré half plane model of hyperbolic plane geometry. I regret that there is no bibliography, although many mathematitians are cited. What I appreciated most of this little book (in addition to its rich content) is the fact that it is both a model of mathematical reasoning and a very pleasant reading. Later, I had to study E. Artin's dry book Geometric Algebra (Wiley Classics Library), where a different approach is used to introduce coordinates and where the link between the projective geometry and the underlying linear algebra properties is quite well established, (with the so called "Fundamental Theorem of Projective Geometry", which essentially says that if coordinates are good enough, then evey transformation sending lines into lines arise from a linear map of a vector space underlying the projective space, a property which is not mentioned by Blumenthal). To be modern is nowadays much less important than it was in 1960. I don't know if the title of Blumenthal's work is still realistic. After Hilbert, between 1900 and 1970, many mathematicians felt forced to write its own books on the "foundations" of (projective) geometry. Among them, Coxeter's books are enjoyable, for example The Real Projective Plane. Modern may be not, but Elie Cartan's LECONS SUR LA GEOMETRIE PROJECTIVE COMPLEXE ("Leçons sur la Géométrie Projective complexe", Gauthier-Villars, Paris, 1950) deserves the highest attention.