Why is Euclid and Euclidean geometry still studied to this day? Why do you think this book has been so important (and incredibly popular) over centuries?
Euclid and Euclidean geometry is still studied today because he wrote "The Elements". "The Elements" provides the basics of geometry. Within the 13 books, the books 1-6 are about plane geometry, books 7-9 deals with number theory and books 11-13 deal with third dimensional geometry. "The Elements" also begin with definitions and five postulates that set the ground rules for the following books. This book has been really important over centuries because it not only provides the basics of geometry, but the "standard of [proof] rigour was to become a goal for the inventors of the calculus centuries later."
Is there beauty in the Euclidean postulates, common notions and principles for proofs? How can we define beauty if these are considered beautiful?
I am not sure if there is beauty in the Euclidean postulates, common notions and principles for proofs, but I can say that it is extremely useful for setting the ground rules for writing geometric proofs. For example, it specifies that you can draw a line between 2 points and assumes the existence of points, lines and circles. The second definition of beautiful in the dictionary is that something is "of a very high standard". In the supplementary article, it states that "Euclid's Elements is remarkable for the clarity with which the theorems are stated and proved. The standard of rigour was to become a goal for the inventors of the calculus centuries later." So, his proofs were of a high standard that became a role model for later mathematicians, which is beautiful.
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