Euclid

Euclid was a greek mathematician,often referred to as the "Father of Geometry".
little is known about Euclid's actual life. He was living in Alexandria about 300 B.C.E. based on a passage in Proclus' Commentary on the First Book of Euclid's Elements. Indeed, much of what is known or conjectured is based on what Proclus says. After mentioning two students of Plato, Proclus writes

All those who have written histories bring to this point their account of the development of this science. Not long after these men came Euclid, who brought together the Elements, systematizing many of the theorems of Eudoxus, perfecting many of those of Theatetus, and putting in irrefutable demonstrable form propositions that had been rather loosely established by his predecessors. He lived in the time of Ptolemy the First, for Archimedes, who lived after the time of the first Ptolemy, mentions Euclid. It is also reported that Ptolemy once asked Euclid if there was not a shorter road to geometry that through the Elements, and Euclid replied that there was no royal road to geometry. He was therefore later than Plato's group but earlier than Eratosthenes and Archimedes, for these two men were contemporaries, as Eratosthenes somewhere says. Euclid belonged to the persuasion of Plato and was at home in this philosophy; and this is why he thought the goal of the Elements as a whole to be the construction of the so-called Platonic figures. (Proclus, ed. Friedlein, p. 68, tr. Morrow)

It is apparent that Proclus had no direct evidence for when Euclid lived, but managed to place him between Plato's students and Archimedes, putting him, very roughly, about 300 B.C.E. Proclus lived about 800 years later, in the fifth century C.E.
There are a few other historical comments about Euclid. The most important being Pappus' (fourth century C.E.) comment that Apollonius (third century B.C.E.) studied "with the students of Euclid at Alexandria."

thus we know almost nothing about Euclid's life. But we have more of his writings than any other ancient mathematician. Besides the Elements, there are the Data, On Divisions of Figures, the Phaenomena, and the Optics. All are included in the Euclidis opera omnia of Heiberg and Menge (see below) in Greek and translated into Latin. Other translations are listed below. Euclid also wrote other books which no longer exist but were mentioned by later writers. They include Surface Loci, Porisms, Conics, and the Pseudaria (that is, the Book of Fallacies).

Elements

Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later.

There is no mention of Euclid in the earliest remaining copies of the Elements, and most of the copies say they are "from the edition of Theon" or the "lectures of Theon", while the text considered to be primary, held by the Vatican, mentions no author. The only reference that historians rely on of Euclid having written the Elements was from Proclus, who briefly in his Commentary on the Elements ascribes Euclid as its author.

Although best-known for its geometric results, the Elements also includes number theory. It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers, Euclid's lemma on factorization (which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

The geometrical system described in the Elements was long known simply as geometry, and was considered to be the only geometry possible. Today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century.

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