Euclid

It is little known about the life of the "father of geometry". According to Proclus, Euclid lived during the reign of Ptolemy in Alexandria, Egypt. When Ptolemy asked if there was a shorter path to study the geometry than the Elements, Euclid replied "there is no royal road to geometry". Proclus also mentioned that Euclid is the founder of the first school of mathematics in Alexandria, the city known for the greatest library of the antiquity.

He wrote a book about conics, more specifically, the intersections of conics with parallel planes. The book, now lost, was used by Apollonius for the work "On Conics". Euclid also published the earliest surviving Western study on visible light, entitled Optica, which provided the background for the later study of astronomy.

The thirteen volume Elements written on rolls of parchment, was considered the preeminent treatise on mathematics for over 2,000 years. Elements is the most studied textbook of all time and the best selling book in history next to the Bible. The originals were also lost but survived through later editions and almost disappeared in the Dark Ages. Euclid didn't claim the proofs of the theorems are his own; in fact some of results belong to earlier scholars such as Hippocrates of Chios .

The principal merit of Euclid is that he organized the geometry in a coherent systematic way. He introduced twenty-three definitions, five geometric postulates and five additional postulates called "common notions". From this basis 465 theorems were deducted. He defined rigorously terms such as point, line, straight line, circle, right angle, surface and plane.

The "common notions" are axioms of logic and not specific to geometry:

1. Two things which are both equal to a third thing are also equal to each other.

2. If equals are added to equals, the wholes are equal.

3. If equals are subtracted from equals, the remainders are equal.

4. Things which coincide with one another are equal to one another.

5. The whole is greater than the part.

The five geometric postulates are:

1. A straight line may be drawn from any point to any other point.

2. A finite straight line may be produced to any length in a straight line.

3. A circle may be described with any center at any center from the center.

4. All right angles are equal

5. If a straight line meets two other lines, so as to make the two interior angles on one side of it together less than two right angles, the other straight lines will meet if produced on that side on which the angles are less than two right angles.

The fifth postulate, called the "parallel postulate", is not as obvious as the others. Later mathematicians unsuccessfully tried to derive it from the other four postulates; they only could replace it with other equivalent forms. The most common is this axiom: Given a line and a point not on the line, there is exactly one line in the same plane that passes through the point and is parallel to the given line. The geometries that do not satisfy the Euclid's postulates are called non-Euclidean.

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