Euclid alexandrian

 

EUCLID OF ALEXANDRIA – The Father of Geometry

Who is Euclid

The Greek mathematician Euclid lived and flourished in Alexandria in Egypt around 300 BCE, during the reign of Ptolemy I. Almost nothing is known of his life, and no likeness or first-hand description of his physical appearance has survived antiquity, and so depictions of him (with a long flowing beard and cloth cap) in works of art are necessarily the products of the artist’s imagination.

He probably studied for a time at Plato’s Academy in Athens but, by Euclid’s time, Alexandria, under the patronage of the Ptolemies and with its prestigious and comprehensive Library, had already become a worthy rival to the great Academy.

Euclid is often referred to as the “Father of Geometry”, and he wrote perhaps the most important and successful mathematical textbook of all time, the “Stoicheion” or “Elements”, which represents the culmination of the mathematical revolution which had taken place in Greece up to that time. He also wrote works on the division of geometrical figures into into parts in given ratios, on catoptrics (the mathematical theory of mirrors and reflection), and on spherical astronomy (the determination of the location of objects on the “celestial sphere”), as well as important texts on optics and music.

The “Elements” was a lucid and comprehensive compilation and explanation of all the known mathematics of his time, including the work of Pythagoras, Hippocrates, Theudius, Theaetetus and Eudoxus. In all, it contains 465 theorems and proofs, described in a clear, logical and elegant style, and using only a compass and a straight edge. Euclid reworked the mathematical concepts of his predecessors into a consistent whole, later to become known as Euclidean geometry, which is still as valid today as it was 2,300 years ago, even in higher mathematics dealing with higher dimensional spaces. It was only with the work of Bolyai, Lobachevski and Riemann in the first half of the 19th Century that any kind of non-Euclidean geometry was even considered.

The “Elements” remained the definitive textbook on geometry and mathematics for well over two millennia, surviving the eclipse in classical learning in Europe during the Dark Ages through Arabic translations. It set, for all time, the model for mathematical argument, following logical deductions from inital assumptions (which Euclid called “axioms” and “postulates”) in order to establish proven theorems.

Euclid’s five general axioms were:

1. Things which are equal to the same thing are equal to each other.
2. If equals are added to equals, the wholes (sums) are equal.
3. If equals are subtracted from equals, the remainders (differences) are equal.
4. Things that coincide with one another are equal to one another.
5. The whole is greater than the part.

His five geometrical postulates were:

1. It is possible to draw a straight line from any point to any point.
2. It is possible to extend a finite straight line continuously in a straight line (i.e. a line segment can be extended past either of its endpoints to form an arbitrarily large line segment).
3. It is possible to create a circle with any center and distance (radius).
4. All right angles are equal to one another (i.e. “half” of a straight angle).
5. If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.