History of mathematics IV

Greek mathematics refers to mathematics written in the Greek language between about 600 BC and AD 300.^[22] Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to North Africa, but were united by culture and language. Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics.

Thales of Miletus

Greek mathematics was more sophisticated than the mathematics that had been developed by earlier cultures. All surviving records of pre-Greek mathematics show the use of inductive reasoning, that is, repeated observations used to establish rules of thumb. Greek mathematicians, by contrast, used deductive reasoning. The Greeks used logic to derive conclusions from definitions and axioms.^[23]

Greek mathematics is thought to have begun with Thales (c. 624–c.546 BC) and Pythagoras (c. 582–c. 507 BC). Although the extent of the influence is disputed, they were probably inspired by the mathematics of Egypt, Mesopotamia and India. According to legend, Pythagoras traveled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests.

Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. Pythagoras is credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.^[24] In his commentary on Euclid, Proclus states that Pythagoras expressed the theorem that bears his name and constructed Pythagorean triples algebraically rather than geometrically. The Academy of Plato had the motto, "Let none unversed in geometry enter here".

The Pythagoreans proved the existence of irrational numbers. Eudoxus (408–c.355 BC) developed the method of exhaustion, a precursor of modern integration. Aristotle (384—c.322 BC) first wrote down the laws of logic. Euclid (c. 300 BC) is the earliest example of the format still used in mathematics today, definition, axiom, theorem, proof. He also studied conics. His book, Elements, was known to all educated people in the West until the middle of the 20th century.^[25] In addition to the familiar theorems of geometry, such as the Pythagorean theorem, Elements includes a proof that the square root of two is irrational and that there are infinitely many prime numbers. The Sieve of Eratosthenes (c. 230 BC) was used to discover prime numbers.

Archimedes (c.287–212 BC) of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave remarkably accurate approximations of Pi.^[26] He also studied the spiral bearing his name, formulas for the volumes of surfaces of revolution, and an ingenious system for expressing very large numbers.