Geometry: A High School Course

by Serge Lang, Gene Murrow

For centuries, geometry has provided the core examples to introduce students to the world of higher mathematics, its ways of thinking, its beauties, its concepts of theorem and proof, and its standards of rigor. This role has been transferred to other areas of mathematics over the last few decades, leaving geometry a relatively secondary and often intellectually barren subject in the mathematical curriculum.

This text, inspired by the work and educational interests of a prominent research mathematician, and Gene Murrow's experience as a high school teacher, presents geometry in an exemplary and, to the student, accessibly and attractive form. The book emphasizes both the intellectually stimulating parts of geometry and routine arguments or computations in concrete or classical cases, as well as practical and physical applications. Neither is treated at the expense of the other. The book also teaches the student fundamental concepts and the difference between important results and minor technical routines. Altogether, the text presents a coherent high school curriculum for the geometry course. There are many examples and exercises.

Genres: Mathematics, Geometry

406 pages, Hardcover

First published February 19, 1987

About the author

Serge Lang was an influential mathematician in the field of number theory. Algebra is his most famous book.

Librarian Note: There is more than one author in the GoodReads database with this name. See this thread for more information.

Reviews

[Shahad Zawinski · Rating 4 out of 5]

I will surely accept that this may change you thinking as well

[Joseph Kulisics · Rating 4 out of 5]

I think that the book is a good treatment of the subject, but I'm guessing that the formalization could be handled with a little more care without sacrificing the important aspects of the presentation, the connection of geometry to analytic methods as explored in the dot product chapters and other sections of the book. For example, there seems to me to be some lack of rigor when comparing say, Hilbert's Foundations of Geometry, with the axiomatic basis in the book. (I could be missing something, but some facts, like the number of points in which a line can intersect a circle and the circumstances of each type of intersection, are either not a consequence of the book's axioms or require more sophisticated proofs than could be supplied in the book. In the first case, augmenting the axiomatic basis might have made sense and allayed any student concerns about missing axioms, and in the second case, at least acknowledging that the consequences, which were critically important to solving some problems because the problems basically required looking at a figure whose major features required justification, were provably true but required some skill.) In any case, I understand that there are similar problems in Euclid's original treatment of geometry. Maybe contemplating a model's shortcomings is of value, too. The peculiar value of the presentation of this book is not in its model perfection but in its introduction to rigorous proof and in the connections that the text makes between geometry and analytic topics. In making connections between the geometry and analytic methods and as a good introduction to proof, the text is a major success, and I can't recommend it enough.