- Hardcover: 357 pages
- Publisher: Chelsea Publishing House; 1St Edition edition (January 1, 1952)
- Language: English
- ISBN-10: 0828400873
- ISBN-13: 978-0828400879
- Package Dimensions: 9.8 x 1.4 x 0.1 inches
- Shipping Weight: 1.7 pounds (View shipping rates and policies)
- Average Customer Review: 4.4 out of 5 stars See all reviews (13 customer reviews)
- Amazon Best Sellers Rank: #3,958,519 in Books (See Top 100 in Books)
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Geometry and the Imagination Hardcover – January 1, 1952
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In 1899, Hilbert proposed the Hilbert's axioms, substituting the traditional axioms of Euclid. They avoid weaknesses identified in those of Euclid, whose works at the time were still used textbook-fashion. Hilbert's approach signaled the shift to the modern axiomatic method. Axioms are not taken as self-evident truths. Geometry may treat things, about which we have powerful intuitions, but it is not necessary to assign any explicit meaning to the undefined concepts. The elements, such as point, line, plane, and others, could be substituted, as Hilbert is reported to have said, by tables, chairs, glasses of beer and other such objects. It is their defined relationships that are discussed. Hilbert first enumerates the undefined concepts: point, line, plane, lying on (a relation between points and lines, points and planes, and lines and planes), betweenness, congruence of pairs of points (line segments), and congruence of angles. The axioms unify both the plane geometry and solid geometry of Euclid in a single system. Hilbert put forth a most influential list of 23 unsolved problems at the International Congress of Mathematicians in Paris in 1900. This is generally reckoned the most successful and deeply considered compilation of open problems ever to be produced by an individual mathematician.
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Top Customer Reviews
There Hilbert says "...In this book, it is our purpose to give a presentation of geometry, as it stands today, in its visual, intuitive aspects. With the aid of visual imagination we can illuminate the manifold facts and problems of geometry, and beyond this, it is possible in many cases to depict the geometric outline ot the methods of investigation and proof, without necessarily entering into the details connected with the strict definitions of concepts and with the actual calculations."
A little further, he says "...This book was written to bring about a greater enjoyment of mathematics, by making it easier for the reader to penetrate to the essence of mathematics without having to weight himself down under a laborious course of studies."
As a reader of this book, I can say that the key words are "visual imagination" and "enjoyment of mathematics".
The purpose described by Hilbert is completely (and excellently) achieved.
The book is a masterpiece, written by one of the masters of Mathematics.
In an elegant and clear style, Hilbert explains the most beautiful geometrical concepts.
When reading it, you feel as if Hilbert was sit down besides you, just talking about geometry to you, (maybe with the aid of a sheet of paper and a pencil), and you can grasp the genius of the Göttingen Professor.
He does not use practically any formula or mathematical expression, however his prose is full of mathematical insights, geometrical facts, stimulating images and delicious "expository" proofs.
All the chapters of the book are structured in a similar way: Hilbert exposes at the beginning the most elementary concepts of the subject, with plenty of "visual imagination" and mathematical ideas. Then, step by step, he goes further and deeper, connecting these ideas and images and generalizing them.
The challenge for the reader is trying to follow Hilbert's thread of ideas until the end. It is not always easy but, after all, challenge is one of the ingredients of the "enjoyment of mathematics", isn't it?
However, one thing is sure, you will enjoy the path, and when you get lost, you can read again the last paragraph and try to retake the thread of the exposition.
I recommend this book very much. It is a joy to read, both for beginners and experienced mathematics.
On the other hand, how can one ignore the work of such a giant of mathematics who's willing to guide you into his imagination ?
It’s apparently a printed-on-demand copy, perhaps printed by Amazon directly, and the text is blurry throughout. The images are filled with gray splotches. I recently bought several Springer books which had the same problem, so it seems to be a trend across purchases of new math books from Amazon.
I decided to return the book and pick up a 50-year-old used copy instead and there is a night-and-day quality difference. The old book has crisp text and lovely illustrations. It was also $20 cheaper.
It is a shame that Amazon and/or the AMS is charging almost $50 for such a shoddy product, without anything on the product page to warn of the possibility of getting a print-on-demand book. Customers should watch out and be cautious about buying new hardcover technical books from Amazon.
However: The Preface states: "This book was written to bring about a greater enjoyment of mathematics, by making it easier for the reader to penetrate to the essence of mathematics without having to weight himself down under a laborious course of studies."
All I can say is that if you read this and find it "easy," then you have terrific mathematical talent! Yes, the drawings and the intuitive descriptions are helpful, but much of the book is so obscure that I have been told that one of the world's leading geometers is working on an annotated edition explaining what the authors were talking about. On topics which I had already studied elsewhere, I found the presentation illuminating.
I still recommend this book.
Anyone with a serious interest(occasional or consuming) in geometry should have this book handy.
Anyone who confuses math with the style of exposition known as "Definition, Theorem, Proof" should read any page of this book.
The only fault I see is that looking closely at the fonts their edges are not perfectly clear, and also the ink is not very dark, I think it's because of the mentioned reprint, the fonts look kind of old, but it's not a big issue to me in any case. The font size is perfect and can be read comfortably. The leaves, the text in general, the bookbinding, the cover, the quality of the pictures, are all good enough.
I attach some pictures to clarify it better.