Geometry: Seeing, Doing, Understanding, 3rd Edition 3rd Edition
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Jacobs begins the text with a discussion of inductive reasoning and its limitations. He uses this discussion to stress the importance of deductive reasoning and proof before introducing Euclid's postulates. Jacobs covers lines and angles, congruence, inequalities, parallel lines, quadrilaterals, and transformations of the plane before a midterm review. Then he covers area, similarity, right triangle trigonometry, circles, concurrence theorems, regular polygons in relation to the circle, geometric solids, and non-Euclidean geometries before concluding with a final review. Each chapter contains a summary and a chapter review in addition to the problem sets at the end of each section.
Jacobs carefully develops the material, proving each result except in the more intuitive discussions in the chapters on transformations, solids, and non-Euclidean geometries. I found the chapter on concurrence theorems particularly fascinating.
The book contains a glossary, a list of formulas, and a list of postulates and theorems, making it useful as a reference. However, concepts discussed only in the problems are not included in these lists. There are answers to a few of the problems in the back of the text, but not enough of them to be useful if you want to check your answers systematically.
I recommend this book as a supplement rather than a text because it contains many intriguing problems but does not teach the reader how to write proofs. Those students who want to learn how to write proofs should consult the rigorous text Geometry by Edwin E. Moise and Floyd L. Downs, Jr.
I taught at Reed, Wellesley, and Bard Colleges and watched the reform mathematics program develop when I was associate director of the Mathematical Association of America, in Washington, DC. Geometry is my research area. I worked in publishing as an editor for more than 20 years. I have read every word of this book and worked all of the exercises because I was its freelance editor. I am a knowledgeable, interested party.
The third edition towers over the second edition, which is described by its most recent Amazon reviewer, Edward Lee, as "the best geometry text in existence, bar none" (January 25, 2003). Begin by noticing the use color throughout, then notice how color has been used to make key material in the text and diagrams stand out more clearly. Detailed comparisons will show you that every part of the book has been scrutinized and reworked, adding a host of new examples and exercises, fine-tuning the concepts and wording. Coordinates are used throughout, so that analytic methods are now another tool rather than the subject of a special chapter, late in the book.
Chapter 1, An Introduction to Geometry is completely new and shows the reader how geometry has been used from the dawn of history, in the East and the West, to design cities, measure the earth's circumference, design pyramids, and figure land taxes. This last brings us to the final lesson of this chapter, "We Can't Go on Like This." Here the student discovers that the Egyptian tax assessor's formula, though plausible, does not work. Something may look sensible and even be used, but we need to be careful and check things. Not everything that is plausible is true. And so we are off to Chapter 2 on deductive reasoning, and then on to all of geometry, including solid geometry (Chapter 15) and non-Euclidean geometry (Chapter 16) --- optional in most first courses.
Jacobs put all of his art into this revision. It is his best effort. Donald J. Albers begins his foreword "This is one of the great geometry books of all time. ... It is the finest example of instructional artistry I have ever encountered."
Geometry is a wild and beautiful subject. Think of it as a continent you might visit and explore. The lessons in this book are station stops on your tour. At each stop, Jacobs gives you a sense of what there is to see and explore. The exercise sequences are side trips for individuals or groups. It is these jaunts that give you a real feel for the place, they build the muscle you need for further exploration and show you small wonders or glimpses of distant peaks. Albers calls these exercises "the beating heart of the book."
Here is a side trip you can explore now: Take a lopsided quadrilateral and erect equilateral triangles on its sides so that their third vertices point alternatingly into and out of the quadrilateral. Connect these four new vertices in the order of the sides of the quadrilateral they are derived from. You will see that no matter what your original quadrilateral was, the new quadrilateral is of a very special sort. The exercise is straightforward, and the result is surprising. Some readers may want to understand the geometry that lies behind this observation. That goal is like the wish to scale a distant peak. Many may feel the call, but only some will set out and reach the summit. Geometric proofs, sometimes so mysterious, are our search for an answer to the question "Why?"
A Teacher's Guide with solutions to all the exercises, lesson plans, reduced size images of the transparency masters, and commentaries on the subject is available. There is also a separate Test Bank. The Transparency Masters, for teachers who use an overhead projector, are available on a CDROM.
In 10 years, I expect to see a crop of geometers who cut their teeth on this book. In the meantime, I expect to see many reviews from students and teachers on this site. Let this be the beginning.