If you read *one* mathematics book in you whole life, this should be it. No, it will not help you do your taxes or win the lottery. It may, however, change the way you look at the world and give you a serious appreciation of what mathematics as a creative endeavor is about. "Lines and Curves" is an invitation to Euclidean geometry from a dynamic perspective. It will teach you how to think about points in motion rather than visualizing figures as static entities. The exposition is so clear and the examples so well chosen that the barest background will allow you to follow the entire exposition. If you half-remember the concepts of congruence and similarity of triangles you are well on your way to enjoying the intellectual ride of a lifetime (a very concise appendix summarizes the formal prerequisites). Hundred of exquisite exercises are a pleasure to try, varying in difficulty from easy to moderately difficult.
The style is engaging and entertaining. I invite anyone to read the Introduction (available free from Amazon) to get a taste of the material. To keep my comments concrete, consider Chapter 2, "The Alphabet": no fewer than six different interpretations of a straight line as a geometric locus are explained (and will consistently be used throughout the rest of the book). The same goes for the circle, for which at least four interpretations are given. Other conics (ellipses, parabolas, hyperbolas) are treated similarly in Chapter 6.
A further remarkable feature is the authors' willingness to employ analytic geometry at crucial places where resorting to purely synthetic methods would be cumbersome and not particularly illuminating. The best illustration of this is the "Theorem on the Squares of the Distances" in Chapter 2 (What is the locus of all the points in the plane whose weighted sum of squares of distances to given fixed points is equal to a constant?) Another instance is to be found already in section 0.2 of the Introduction (read it from the links above!) Exercise for the reader of this review: solve 0.2 using no analytic methods, but rather by modifying the argument of 0.1 and using the fact that the compression/dilation by a factor of b/a of a circle of radius a with respect to a diameter is an ellipse of semiaxis lengths a and b. The latter approach will seem natural enough to a reader who has absorbed the main lessons of "Lines and Curves".
I can only assume that readers of this little gem will want to go further. The book does not have a bibliography, but I can offer the following suggestions: "Geometric Transformations" (volumes I-III) by I.M. Yaglom, H.S.M. Coxeter's "Geometry Revisited", and the hard-to-get but delightful monograph "The Kinematic Method in Geometrical Problems" by Lyubich and Shor.
(Note: While my personal favorite is number theory, "Lines and Curves" still holds a special place in my heart fifteen years after reading Mir Publishers' Spanish translation. English readers should feel very fortunate indeed that this 2004 Birkhäuser translation is available.)
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