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.