Euclid’s Life and the Elements

Greek Mathematician Regarded Father of Geometry

Brief account of Euclid’s work in relation to geometry and the key postulates from his famous book the Elements.

Euclid’s greatest achievement was to create a coherent framework of basic theory and proofs by combining the geometrical theorems of his day. His best book, Elements, is a widely translated and studied mathematical book, one of the most influential of all time.

Euclid’s Life

Little is known about Euclid’s life. It is likely that he lived around 300BC in Alexandria, the Egyptian city founded by Alexander the Great on the shores of the Mediterranean. For this Euclid is sometimes referred to as “Euclid of Alexandria.”

Ptolemy Soter (c.367-283 BC), the first Greek ruler of Egypt, created Alexandria’s Museum and Library, which was claimed to be the most remarkable educational and intellectual institution in the ancient world. Euclid was probably a leading mathematics teacher there. Or, he may have been a student of Plato. There were few hints about his character, mainly from anecdotes, that he was a gentle and an encouraging teacher.

Euclid’s Geometry

Euclid’s great achievement was to consolidate the geometrical theorems during his day into a coherent framework of basic theory and proofs, which is the basis of all science to this day.

It should be noted that geometry — the mathematics of shape — was already well developed thousands of years earlier, considering the highly sophisticated pyramids of the ancient Egyptians. The word ‘geometry’ is coined from the Greek which means “earth measurement.”

In 1858, Scottish historian Alexander Rhind found a papyrus scroll written by an Egyptian scribe called Ahmes, around 1650BC. The Rhind papyrus and another papyrus now in Moscow showed that the ancient Egyptians knew a real deal about the geometry of triangles. What Euclid and the ancient Greeks did was to develop these practical techniques into purely theoretical system, something we might call today as “applied mathematics.”

The Elements: Postulates, Theorems and Proofs

Euclid and the Greeks gave mathematics extraordinary power by introducing the idea of proofs and the idea that rules could be logically worked out from certain postulates (Assumptions). For example, here’s something that is very much known today, that “A straight line is the shortest distance between two points.” Assumptions are then combined to make a basic idea for a rule called a theorem, which is then proved or disproved.

According to Farndon’s The Great Scientists, the center of Euclid’s the Elements are five key postulates or axioms. They are:

  1. Part of a line can be drawn between two given points.
  2. Such a part line can be extended indefinitely in either direction.
  3. A circle can be drawn with any radius with any given point at its center.
  4. All right angles are equal.
  5. If part of a line crosses two other lines so that the inner angles on the same side add up to less than two right angles, then the two lines it crosses must eventually meet.

Despite the limitations of Euclidean geometry in the 19th century, Euclid’s work of establishing these basic truths are still as powerful.

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