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First of all, I want to master Geometry, I have knowledge on high school geometry and I was thinking of learning Euclidean Geometry. I bought a copy of Euclid's Elements, it is very interesting, however, it does have a fairly different method compared to the modern approach in teaching geometry. Can I ask if it is required in our modern mathematics to learn Euclid's Elements? Or is learning Euclid's elements just for intellectual exercise? Are there any modern textbook on Euclidean Geometry or plane geometry? I have no problem with the formal mathematical approach using Axioms and Postulates, I enjoy having a first exposure to them, actually.

In the future, I want to read Principia Mathematica by Isaac Newton, is it a must to learn Euclid's Elements to learn it? Or Descartes's Geometry is the basis of it? Or maybe there is a modern geometrical approach to explain it?

Omicron
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  • Have you seen Principia Mathematica ? – Rene Schipperus Jun 21 '14 at 19:53
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    See the first answer to this post for some modern references. – Mauro ALLEGRANZA Jun 21 '14 at 19:58
  • @ReneSchipperus Some parts of it, but I don't understand any of them. – Omicron Jun 21 '14 at 20:00
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    For Newton's Principia, a amstering of Euclidean geometry can be very useful. For a modern "companion", see Subrahmanyan Chandrasekhar, Newton's Principia for the Common Reader (1995). – Mauro ALLEGRANZA Jun 21 '14 at 20:07
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    What is you motivation for wanting to read these works ? – Rene Schipperus Jun 21 '14 at 20:32
  • Well, I read Newton and Einstein speak highly of this book. Then afterwards I bought it, (I do have the copy now). I will still read it but I just want to know if my hardwork would be beneficial to my future study in higher mathematics. – Omicron Jun 21 '14 at 20:39
  • IMVHO, Euclid is useful only for historians. If you want the modern equivalent, it would probably be the Tarski axioms for geometry. They are, however, extremely difficult to work with. I believe it took decades (?), for example, for his followers to prove the existence of the mid-point of a line segment. You might prefer something like differential geometry, topology or group theory. – Dan Christensen Jun 21 '14 at 20:41
  • Hmm, I must be very precise on my current position. I only know knowledge of high school geometry, most precisely, my knowledge of geometry is quite insufficient in all aspects. I certainly want to know higher mathematics. But first I must help myself get acquainted with geometry. Is there any other textbooks one can advice? – Omicron Jun 21 '14 at 20:50
  • If you look on the site michigan historical mathematics, there are some nice books, I like Askwith, A course in pure geometry. – Rene Schipperus Jun 21 '14 at 21:36
  • There is also Eves Survey of geometry and a book by Coxeter. If you are interested in foundations Hilbert's brilliant and readable Foundations of Geometry is a must. Newton is VERY difficult to read, its probably better to understand the mechanics from a modern book. Newtons geometrical approach isnt really that useful. – Rene Schipperus Jun 21 '14 at 21:41
  • Akswith's pure geometry will come in handy thanks for that. Hmmm, Foundations of Geometry, Do I need a knowledge other than the high school geometry I possess to understand Hilbert? By what means do you mean modern book on mechanics? You mean modern translation? http://www.ucpress.edu/book.php?isbn=9780520088177 like this? – Omicron Jun 21 '14 at 21:53
  • Some knowledge of projective geometry, which is in Askwith, and is helpful for Hilbert. I mean a modern book on analytical mechanics. I dont really understand the motive for reading Newton, its difficulty is out of proportion with the rewards, given that there are so many other books. Also the Lagrange and Hamiltonian formulations of mechanics are more important for modern physics. – Rene Schipperus Jun 21 '14 at 21:59
  • Well, please forgive my ignorance, I am only starting out as learning mathematics from bottom to up, and I must say I am only an inch higher from the bottom. Actually, I can admit that Euclid's proofs are quite insufficient in really demonstrating "how he gets to the conclusion" there is some leap of explanation happening in some of his theorems. Luckily, I am only at Book 1, so it is an advantage that I learned this quite early. – Omicron Jun 21 '14 at 22:41

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A more 'modern' way to study Euclidean geometry is to recast all theorems and prove them using methods of Linear Algebra, using coordinate space R^2 and R^3. I would be interested if there was an author who would accept this challenge. The nice thing about linear algebra is that you can verify results easily using a computer. There are advantages and drawbacks to using Linear Algebra. In linear algebra proofs tend to be more compact and involve more algebraic type manipulation. In synthetic geometry proofs involve the use of complicated diagrams and tend to be wordy. On the other hand, in synthetic geometry it is easier to draw such basic figures as a line segment, while in R^2 or R^3 we would have to use parametric equations. It is easier to 'discover' geometric relationships when you can draw lines and circles freely as we do in synthetic geometry. Still, proving these statements tends to be more compact using coordinate space or methods of Linear algebra. Personally i'm not a fan of reading synthetic geometry proofs, that is why i am interested in a different approach to Euclidan geometry. why not have the best of both worlds, the lean compactness of linear algebra proofs but discover them using the normal euclidean tools of points, lines, circles, etc.

dan richard
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