Euclid: The Alexandrian School Of Mathematics

Euclid is a famous mathematician in history, nobody knows the exact date of his born and death, it has been said that he was born on 300 BC and died on 265 BC. Among all his contributions he became famous for Euclidean Geometry. Euclid was a math teacher and also he was the founder of the Alexandrian school of mathematics. “Euclid is known as a prominent mathematician of antiquity”. He is mostly known for his famous work ‘The Element’, those elements are divided into 13 books and on different field of mathematics. The first six books are written about Geometry, the next 7, 8, and 9 for Number Theory. While book ten is about irrational numbers and books 11, 1, and 13 are solid geometry. “Euclid proof that there is no greatest prime is an indirect

The famous theorem ‘Pythagoras’ was proved on his first book. On the other hand, he defined divisibility, prime numbers, GCD and also proved several properties of these notations. Euclid came up with the algorithm for finding gcd of two numbers using repeated division. Looking his work on “solid geometry he proved the existence of five regular polyhedral, tetrahedron, cube, octahedron and icosahedron”. Euclid is a famous Greek mathematician but he is “not a first rated but he was the first mathematician to realize that proof and reasoning were as important as the statement of the theorem”. The fifth postulate is a most debated topic in 19th century. The fifth postulate is known as parallel postulate, many geometers have tried to prove the fifth postulates from the other postulates, the process was to assume its falsehood, then derive a contradiction. “In 19th century Bolyai, Lobachevsky, and Gauss had found the way to deal with non-Euclidean geometry by means of analysis and accepted it as a valid kind of geometry”. Among all his work I want to discuss here about this four famous works. They are as

He began the number theory from book VII of elements, where he defined number as “a multitude composed of units”. Euclid has contributed many theorems and proof about the numbers. Theory of prime numbers has been studied and proved by different mathematicians over the years. Among them Euclid is one of them who prove the prime numbers, he showed that for “every finite set S of primes, the prime factors of the number 1+∏▒S are not in S”. Than S can always be extended to larger finite set of primes. Euclid mention in his thirteen book that there is no greatest prime number, the proof that lets to it is considered a glory of number theory. But Euclid himself was unable to prove it directly, instead he uses indirect method to proof that there is no greatest prime. When proving for infinitude of primes Euclid claims that “prime numbers are more than any assigned multitude of prime numbers”. Later the modern mathematician makes little different claim about the infinitude of primes, lets us look the Euclid proof for prime number, assuming prime numbers p=1,2,3,5,7,11,13,17,…N. Now considering the number 1+(1,2,3,5,…,N) i.e. P+1, here P+1 is not divisible by p. When we divide (p+1)/p we will get the remainder 1. Therefor we can conclude that P+1 is not divisible by any prime numbers. He makes some assumption and come up with two alternatives, Euclid figure it out that if N is prime, then there