Ⅰ
The ancient Library of Alexandria was part of what was called the “Museum”. This was not, however, a museum in the contemporary meaning of this word, but rather a university, the first university in history. In those days if you were a mathematician and wanted your work to be read by others, you sent it to the Library of Alexandria. Meanwhile, many mathematicians includes Euclid, Archimedes, Eratosthenes, Heron were active or have their master piece in Library of Alexandria. It is them who build the foundation of contemporary mathematic.

ⅡFamous Mathematician
(a) Euclid
Euclid was the most distinguished mathematicians. When Ptolemy I, he was a teacher in school of Alexandria. Library of Alexandria includes many of the Euclid authentic originals, including 'The Elements'.
‘The Elements’ was finished around 300BCE in Alexandria. The historical significance of the epoch-making masterpiece is, in geometry, it sets an earliest model of the method using axiomatic deductive mathematical system. In ‘The Elements’, Euclid apply logical chain to expand whole geometry. At first, he made the appropriate selection of axioms and postulates, which are very difficult tasks, requiring extraordinary judgment and insight, and no one have done this before. Second, he carefully arranged these theorems, so that every theorem and the previous theorems are logically consistent. Finally, he made a supplement in sections where are lack of sufficient evidence and steps. Birth of ‘The Element’, marking geometry has become a theoretical system, a discipline with a relatively strict and scientific methods.
In aspect of Demonstration Methodology, Euclid proposed analysis, synthesis method and reductio ad absurdum. Analysis is assuming that we already got the result, and to analyze the conditions when the result has been got, achieving proven steps; synthesis method is proving facts from facts we have got before, and gradually deduce to prove; reductio ad absurdum is keeping the assumption of giving condition, and get a opposite conclusion, then starting from giving conclusion, deduce to fact contradictory with condition or facts that have been proved before, thus proving the original conclusion is correct.

Basically, all of the rules we use in Geometry today are based on the writings of Euclid, specifically 'The Elements'. Euclid's book the Elements also contains the beginnings of number theory.

(1) Example 1
Euclid’s proof of the fact that the set of prime numbers is infinite.
In the following, by “number” I mean a positive integer. We recall that prime numbers are those numbers which are only divisible by themselves and unity. In a sense they are the “building blocks” in the realm of numbers, because all other numbers are composite, being built by taking products of primes. Even the most primitive examination reveals that the primes thin out among numbers as we proceed to larger numbers. So the question arises: do they stop somewhere? That is, is there a last prime, all numbers after that being composite? Euclid was the first to ask this question and the first to answer it, and in a most perfect fashion. Observe that no computer could ever answer the question, because it is a question about infinity. Only the mind could. Here is Euclid’s proof. Suppose that on the contrary the set of prime numbers is finite, so we can count them in increasing order, omiting unity: p1, p2, · · ·, pn
Consider then the number: M = _ + 1 were _ is the product p1p2 · · · pn 1
This number being larger than the last prime, pn, must be composite. Then M must have a prime factor, say q. So q must be one of of the p1, p2, · · ·, pn But if q = pk for some k = 1, ..., n, then since q divides M as well the product _ it must divide their difference, which is unity. But this is absurd; for no number other than unity itself can divide unity, and we have omited unity. Consequently the contrary of our initial hypothesis must hold, that is, the set of prime