Geometry: Seeing, Doing, Understanding

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.


Book and guide provide a thorough geometry course   February 25, 2005
Molly Crocker (Ferndale, WA United States)
51 out of 51 found this review helpful

There is some criticism that this textbook has lost the rigor of the 2nd edition. Having used that for 5 years and this for two years, I would argue that the current edition demonstrates far more care toward the reluctant student and causes the math nerds to stop and think about who is interested in knowing or needing geometry.

There are good reasons for every change. One is simply that there are only 180 days in the school year. I use this text with homeschoolers. We meet 72 days per year, and we do every chapter, every problem, plus a fair bit of other supplements.

I do not understand the criticism that the book is disorganized and chaotic. In the last 8 years I have taught math from more than 10 different texts, from pre-algebra to pre-calculus. Harold Jacobs sense of organization is a relief. I suspect that since the 2nd edition has been around since 1987 and has stood the test of time, that the criticism stems from the fact that even mathematicians dislike change. (What in the world is wrong with geometry students using a protractor?) I do not believe the book is a nod to political correctness defined by the NCTM. I think it rather corrects the course taken by other publishers in their interpretation of the NCTM standards.

Proofs from the 2nd edition are available online from the Freeman publisher website, so you can add that back in, as I do. The teacher guide that accompanies the text, written by Peter Renz (above reviewer), adds several more levels of richness and complexity. Use as much or as little as you want. You now have the flexibility to use this text with those enthralled by math as well as those resistant to math.

In my first review, since withdrawn, I was critical of the tests provided by the publisher as being for weenies. They are being replaced by a set that Harold Jacobs wrote himself. I have been given the opportunity to test drive some of them, and I am satisfied that this completes a first-year presentation of geometry.

Harold Jacobs is a master mathematician, and a master teacher. He clearly loves doing both. (Confessional:) I do not own a degree in mathematics, and I my greatest growth in math has been a result from working with texts by Harold Jacobs. This is not a text generated by a publisher to fill a hole in a lineup of texts. This is a successful presentation of a difficult subject.


Uncluttered, clear and concise   August 26, 2006
W. B. Moskalski (CT United States)
26 out of 26 found this review helpful

I am a retired chemist who teaches math to homeschooled students. Early in my career I was also a math teacher so I'm fortunate to be able to see how the approach to math has changed over the years. Jacob's book is clear and concise with an enormous number of problems after each lesson. I personally believe math is learned by doing problems. It is the constant application of concepts in solving problems that enables understanding to take place; the problems in Jacob's book reinforce the postulates and theorems by applying them in different situations. The problems are also practical, amusing and interesting which certainly helps to engage students.

The layout of the book is very consistent and well organized developing a pattern which makes it easy to thumb backwards to find previous information. Consistency is important in the learning process.

I think many textbooks today make the mistake of trying to sell the subject to the student with glitz and graphics. I believe this makes those books distracting and confusing. Jacob's approach is to state a few postulates or theorems clearly with a few examples. The problem sets demonstrate their use exhaustively.

I typically assign 150 problems to my students a week. This takes about 45-60 minutes a day. We typically go over the answers and discuss the next topic very briefly. Even if the number of assigned problems is cut in half, the arrangement of the problem sets enables competency to be attained. Of course in the world of today where homework is a thing of the past, many educators believe students need to be seduced with fluff but fluff doesn't drive concepts home. Practice makes perfect still works for me. It is sad to see how other countries have surpassed us by adhering to principals of hard work as hard work will almost always guarantee success.

My only complaint about the book is the slight inconsistency in stating definitions, postulates and theorems. When virtually every geometry text including Jacob's belabors conditional statements, I believe every theorem, etc. should be stated explicitly as a conditional statement. I restate them when they stray and have students consider the converse. I find students naturally restate the theorems more casually anyway.


Careful consideration needed   March 8, 2006
Andrew L. Brown (Jessup, MD USA)
20 out of 20 found this review helpful

I loved this book. Yes, it is true that the non-Euclidean geometries are somewhat excessive and most classes do not have time to study them, however, we must put it into perspective. I have just received a copy of Geometry from the Holt publishing company from my local high school and there is far more material in it than in Jacobs' book on Spherical and Hyperbolic geometries. This was college material for me.
I found Jacobs' book to be clear, concise, and more mature of a text than others I have had experience with. W.H. Freeman was very helpful in redirecting my questions to the proper authorities when asked where to find their teacher's guide, etc.
I have had an all-around positive experience with this book.


Geometry: Seeing, Doing, Understanding   October 10, 2003
Daniel S. Yates (Mechanicsville, VA United States)
25 out of 27 found this review helpful

"I have just spent a couple of hours browsing through Harold Jacob's GEOMETRY, and I think I've fallen in love again. This book is really lovely. What a treat. I didn't want to put it down. Harold has really done a masterful job of bringing in so much stuff -- culture, puzzles, challenges, current events, etc., and his use of cartoons and similar things that appeal to kids (like me!) is the best. This is a very rich and compelling guide through the central ideas of geometry. Harold has created a roadmap that will let learners experience the thrill and wonder and discovery of important geometrical truths. As a fellow author, I am inspired to work even harder to try to reach his standard of excellence. Congratulations to Harold and his editors on an OUTSTANDING contribution to mathematics learning." -