Customer Reviews

Excursions in Geometry

5 star:		(8)
4 star:		(1)
3 star:		(0)
2 star:		(0)
1 star:		(1)

 

 

I am very pleased to see there is a Dover edition of this excellent book, which might otherwise be out of print. I read this book when I was 14 years old. Most geometry books for people with very little prerequisite knowledge are boring. This one was fascinating to me when I read it, and still is now. The author's purpose is to show students with very little background how seductive the subject can be. He succeeds brilliantly.

The chapters on harmonic division and inversive geometry are a sort of preview of conformal mapping and (although Ogilvy doesn't say so, as far as I recall) of the geometry of the complex projective line. The chapter on the golden section is comprehensible to people who know no more math than what is known to almost everyone who can solve a quadratic equation. It shows clearly in only 13 pages how geometry, number theory, algebra, and analysis can be intimately connected with each other, along the way discussing pentagrams, spirals, knots, self-similarity, the five Platonic polyhedra, and the Fibonacci numbers (and quadratic equation, of course). The chapter on projective geometry is just as elementary even while it discusses topics that engage the attention of expert geometers (albeit at a more abstract level).

This is superb expository writing. Every 14-year-old who, as I did, has thoughts of becoming a mathematician, should read this book.

Is the previous reviewer right to say that "This book would only be reccomended to the top 2% of math students"? Perhaps. I would put it this way: No one who cannot understand, do, and enjoy mathematical reasoning will appreciate this book. So certainly this limits the market; as the previous reviewer said, it's not for the general public.

I am baffled by the previous reviewer's statement that this book tends to veer off course, or that the diagrams are not explained. Both statements are false and unjust.

In this slim little volume, Ogilvy sets the standard for the genre. His subject matter is gloriously organized and impeccably motivated; he proves every result thoroughly but without getting bogged down in the sort of tedious formalism that all too often cripples mathematics texts; and the results themselves are the very picture of geometric elegance. (In particular, the chapter on Soddy's Hexlet is a gem.)

Ogilvy leads his readers on an excursion through geometric inversion, projective geometry, and the conic sections. Some of the subjects (most notably, conics, and that unexpected and magical projective invariant, the cross-ratio) appear again and again throughout the various chapters, and even the seasoned mathematician is almost guaranteed to find a new presentation of a familiar topic. Hence, I presume, one reviewer's assertion that the author has a tendency to veer off topic, which would be true if geometry were a disjoint collection of unrelated ideas. Ogilvy shows definitively that it's not; he's not changing the subject when he brings up the cross-ratio in a chapter on inversion - rather, he's unifying two (or more!) ostensibly disconnected subjects.

This book is suitable for anyone with even the slightest interest in geometry. Everything is developed from scratch, and so the lapsed mathematics student shouldn't be intimidated. The bright high school student will be captivated by the elegance and accessibility of the results; and the graduate student or professor of mathematics will find this book to be a lesson in mathematics pedagogy - or just a perfect leisurely read.

The work describes many types of geometric challenges in
everyday life. For instance, the optimal angle theta is presented
in a movie theatre. The screen is depicted as the base and a
mid-point in the back of the theatre is the triangle peak.
Steiner's circles are shown so that equal circles can be
moved in an infinite combination of patterns. The work has
a variety of Euclidean topics to challenge the mathematically
inclined readers. This work is perfect for any class science
project. Some of the challenges presented could occupy a
graduate thesis.

This was an excellent little book which I thoroughly enjoyed and which taught me a lot of Geometry not commonly available in schools or Universities or on the WEB. For example the coverage of "Soddy's Hexlet" was one of the most concise and rewarding pieces of Mathematics I've ever read. It deserves its five stars.
The material was easy to follow and yet left plenty of gaps for the reader to fill in to consolidate their knowledge. Some traditional books on a subject like this simply assume too much knowledge and experience on the part of the reader and therefore quickly lose their appeal and remain largely unread outside of the Univerities. If more books like this were widely available I think they would spawn some very adept Geometers. Books like this hold one's interest, teaching the subject in a step by step unassuming fashion and indeed lead into more advanced work.

Congratulations to the author.

Dover is to be commended for reprinting both of Ogilvy's books
"Excursions in Number Theory" and "Excursions in Geometry". They
impressed me as a young high school student and I borrowed
them from the library many times. I hope they arouse interest
in mathematics for a new generation of young readers. Teachers
will find much material of interest for classes.

I also recommend another Dover reprint "Geometric Exercises
in Paper Folding" by T Sundara Row.

Also look at "Geometry Revisited" by H S M Coxeter and
S L Greitzer.

This is a beautiful book. It assumes essentially no knowledge of geometry, and is ideal for recreational mathematical reading. It's written--and illustrated with diagrams--so well that it can be read without clarifying things for yourself with a pencil and paper. But you'll want a pencil and paper just to play around with the ideas.

This is a pretty interesting collection of classical geometry. Much of it is well known, but some topics are shamefully neglected today, such as harmonic division and Apollonian circles, which is treated in the interesting chapter 2. Suppose there is a ship at A and a ship at B, and that the ship at A is k times faster. If the ships sail towards each other they will meet at C; if the B-ship tries to flee away from A, the A-ship will catch up with it at D. So we have AC/CB=AD/BD=k, and one says that the line segment AB has been "harmonically divided" by C and D. Now here is a theorem proved by Apollonius: the locus of points that the ships can reach at the same time, i.e. points P such that AP/BP=k, is a circle, the Apollonian circle. Proof: Consider such a point P. Bisect the angle APB, cutting AB at C. Extend AP and bisect the exterior angle, cutting the extended AB at D. One sees at once that CPD is a right angle. And by drawing the parallels to CP and PD through B and applying similar triangles twice we find that both AC/CB and AD/BD is equal to AP/BP=k, so C and D are the same C and D as above, and the points the ships can reach at the same time lie on the circle with diameter CD. A remarkable extension is the theorem that the diameter of a circle is divided harmonically by another circle if and only if the circles are orthogonal. So harmonic divisions of AB corresponding to different values of k come from a family of circles orthogonal to the circle with diameter AB, and for any one, cutting AB at C and D, say, we can draw the family of circles having their diameters harmonically divided by C and D. In this way we get a net of mutually orthogonal circles, illustrating a connection between the classical notion of harmonic division and the modern notion of harmonic function. There are also implications for naval pursuit. Suppose we are on a ship, hunting an enemy ship. Assuming that both we and the enemy are going in a straight line, the points that we could reach at the same time as the enemy lie on the Apollonian circle of points P such that AP/BP=ratio of velocities. If we are faster than the enemy, this circle will encircle him. Thus, if the enemy keeps a straight course (e.g., if we are a submarine), we should aim for the point where his course intersects the circle.

All the other reviews which rate this book highly are exactly correct. I am only going to add one thing: The discussion of the cross-ratio, in this book, is absolutely great.

I have read a number of books on projective geometry. In those books the cross-ratio is defined and its invariance is proved in a very dry manner, without any explanation of why it is interesting at all. Yes, it is the key invariant of projective geometry, but it seems to come from nowhere - it is just a definition out of the blue.

This book starts with a simple theorem on two chords of a circle, and then develops what it means to "divide a line harmonically", and then goes to the cross-ratio. This development, though simple, actually for the first time for me really motivated what the heck the cross-ratio was, and why anyone was initially interested in it. And then, chapter by chapter, the cross-ratio is used in interesting ways.

First, it is used in coaxial families of Apollonian circles, a simple example. Then, inversive geometry is developed and it - inversive geometry, and thus the cross-ratio - become the means to explain and/or solve really interesting and fun problems like Peaucellier's linkage, the Apollonius problem of finding a circle tangent to three other circles, and truely amazing things like Steiner chains and Soddy's hexlet.

Later in the book the discussion turns to projective geometry and there the cross-ratio reappears - and well it should as the cross-ratio is the key invariant in projective geometry. (I believe I already said that.) But the discussion here is so much better motivated than the discussion in the rigorous texts on projective geometry I've been studying.

At last I feel I understand the cross-ratio!

(By the way, if you like inversive geometry, look at Geometric Transformations IV: Circular Transformations (New Mathematical Library) - that's a really good text/problem book for it.)

I'm a physics PhD candidate and am in the final stage of finishing my thesis. But when I began to read this book, I can't let it go until I read through. In my junior school years, inversion and projective geometry were not taught although I did some problems. I was always eager to know more of these two subjects. This book fulfils my dream.
Also, I'll recommend Richard Courant's book, "What is mathematics?"

This guide is ambiguous and provides very little information about obscure topics such as inversive geometry. The text is difficult to comprehend not because of its vocabulary, but because it tend to veer off course. Although the subject matter is interesting, it is very difficult to expose when confronted to diagrams that are basically not explained. This could not be used as a classroom guide because it tends to confound even gifted math students. An interesting remark was made, which was, "I think he's going to start talking about his dog right now." This book would only be reccomended to the top 2% of math students, because it is not accesible to the general public. I should hope that someone who reads this book writes a revision of it.