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Lectures on Analytic and Projective Geometry

Dirk J. Struik
Dover Publications
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Bill Wood
, on

In his preface to Lectures on Analytic & Projective Geometry, Dirk Struik explains his motivation for writing the book: to highlight the important role geometry plays in mathematics curricula that were beginning to leave “old-fashioned” mathematics behind. That was in 1953. Nowadays, competition for curriculum space is even fiercer and it is not uncommon for mathematics majors to graduate with no formal exposure to classical geometry (this reviewer and geometer was one such mathematics major).

Indeed, I would recommend this book most strongly to scholars unexposed to topics like affine and projective geometry. It would be a lighter read for a mathematical veteran and the exposition comes paired with plenty of motivating historical context. Anyone planning to teach a course in this material should definitely give it a look even though it may not be appropriate for course adoption, given that this book is more briskly paced and challenging than many contemporary comparable texts on the subject.

The book has aged well. Struik’s course adopts the Kleinian transformational approach, first carefully developing the fundamental ingredients of projective geometry, including orientation, cross-ratio, and duality. Groups of geometric transformations are introduced with affine geometry, with Euclidean geometry emerging as a special case. Conics are introduced first as projective objects and then developed in the affine and Euclidean settings with the relationships among the three continually revisited. Hyperbolic and elliptic geometry are only briefly discussed, mostly in an historical context. The remainder of the book covers spatial geometry beginning with quadric surfaces. Most of the material is developed using algebraic methods.

Dirk Struik was an outstanding historian and mathematician and he comfortably wears both hats in crafting his narrative. The mathematics is thoughtfully presented with historical commentary that is relevant and enlightening. There are no examples but plenty of good exercises with solutions and hints to many in the appendix.

Prerequisites for the book include some linear algebra and sufficient mathematical maturity to accept the terse writing and lack of examples (which for this material in particular can be pedagogically valuable since it forces students to draw the algebraic and geometric connections themselves). Elementary topics in group theory and the complex plane are introduced in the text. The course was written for upper-level majors at MIT, which seems about right.

Lectures on Analytic & Projective Geometry is a valuable entry in the geometry literature written by a prolific and important mathematical expositor. Libraries should have it, practitioners and teachers of geometry should read it, and general mathematical audiences should consider it. This is also another example to making this reviewer grateful that Dover Publications is able to make books like this available inexpensively.

Bill Wood is an assistant professor of mathematics at the University of Northern Iowa.

1. Point Sets on a Line
2. Line Pencils
3. Line Coordinates. Homogeneous Coordinates
4. Transformations of the Plane
5. Projective Theory of Conics
6. Affine and Euclidean Theory of Conics
7. Projective Metric
8. Points, Lines, and Planes
9. Projective Theory of Quadrics
10. Affine and Euclidean Theory of Quadrics
11. Transformations of Space
Special Exercises
Collateral Reading
Answers to Exercises