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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?

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  • $\begingroup$ Have you seen Principia Mathematica ? $\endgroup$ – Rene Schipperus Jun 21 '14 at 19:53
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    $\begingroup$ See the first answer to this post for some modern references. $\endgroup$ – Mauro ALLEGRANZA Jun 21 '14 at 19:58
  • $\begingroup$ @ReneSchipperus Some parts of it, but I don't understand any of them. $\endgroup$ – Omicron Jun 21 '14 at 20:00
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    $\begingroup$ 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). $\endgroup$ – Mauro ALLEGRANZA Jun 21 '14 at 20:07
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    $\begingroup$ What is you motivation for wanting to read these works ? $\endgroup$ – Rene Schipperus Jun 21 '14 at 20:32
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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.

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