The Ancient Greeks contributed so much to rational thought and in so many diverse disciplines such as, natural science, engineering, philosophy and mathematics; tis enough to make your head swim. In many areas their contribution would not be equalled or excelled for nearly two thousand years. And indeed it was the rediscovery of ancient Greek treatises in the late middle ages which would act as an intellectual goad to stimulate the explosion of Western thought which would define the renaissance and scientific revolution of the 16th and 17th centuries.
What is even more surprising is that the Greeks made important advances in pure mathematics with a mathematical notion not particularly conducive to even the most basic arithmetical manipulation. Consider trying to multiply 36 by 42, quickly, using Roman numerals; the Greek system was similar to this. Therefore, even simple algebra (as we understand it) was unknown to the Greeks and in fact nearly their entire mathematics was based on geometry. All that they achieved was achieved with a straight edge, a compass and a lot of contemplation. And what they achieved was astonishing. From the ancient Greeks we obtain Pythagoras’ famous theorem concerning right angled triangles; 2D geometry of various many sided figures; conic solids; Pi; pesky irrational numbers and much more. We also obtain the concept of 'proofs' based on logical reasoning ultimately derived from self evident and impeachable axioms.
Pythagoras was probably one of the most intellectually gifted men who have ever lived. He flourished on earth about 532 BC (born ?570 BC) and he lived most of his later life in the Greek colony of Croton in southern Italy. It was here that he founded a school and attracted the greatest scholars of the day. I said a school, but perhaps I should have said an 'aesthetic community'. The Pythagoreans immersed themselves in pure mathematics to a fanatical degree. They believed all was 'number' and that mathematics had a beautiful unifying existence which underpinned and transcended everything (mathematics=God). This was not to last. Pythagoras for all his lust for rigorous/vigorous proofs was also a mystic. It is interesting to speculate why a man of such prodigious intellectual gifts should be drawn to the irrational and esoteric. Isaac Newton was also of this ilk. I could name others, but fundamentally, I'm an intellectually lazy man. His community famously eschewed beans and women. Nor were adherents allowed to pluck a garland, not allowed to sit on a quart measure and never look in a mirror beside a light. All very sound advice, I'm sure. I would never have been admitted to the hallowed halls of the Pythagoreans due to my fondness/weakness for women and beans, although on a good day I could probably give up the beans. Thus Pythagoras comes across as a mixture of Einstein and the Dali Llama on acid- go figure.
Any group espousing absolute and fanatical beliefs is heading for a fall. Sadly, or gladly, the world is not made that way. The Pythagorean love affair with simple, unsullied and sublime number came a cropper due to the discovery of 'irrational numbers'. Up to this time Pythagoras viewed numbers as perfect. The square root of 100 has a beautiful symmetry. On the basis of pure number theory, the author can indulge in rapture which is denied most mortal men. Sometimes mathematics is the only solace I can find for a turbulent mind; I'm starting to digress. But one dark day a Pythagorean student discovered the square root of 2, or at least the geometrical equivalent. For his diligence and contribution to pure thought he was drowned at sea. For his sake, I will expose the mathematical heresy here and try not to get wet: The square root of 2= 1.4142135623746....... It goes on forever with no elegant repeating sequence. As an aside, NASA has calculated the square root of 2 to over 10 million digits. You would think they would be better employed using their computing power sending ferrets to Mars or at least developing cold fusion/illusion. This simple mathematical truth dealt the death knell to the fundamentalist dogma of the Pythagoreans and therefore innocence was lost. Their philosophy would never be the same again. They emerged chastened and mayhap students developed a taste for the delights of beans and women. As the sect soon died out I suspect they indulged in the former only.
Archimedes (287 BC - 212 BC), a savant of Syracuse in Sicily, was another great intellectual of the ancient world. His contribution to knowledge was prodigious. He is mainly remembered for his 'Eureka moment' and a few of his engineering feats combating the fierce besieging Romans, to which he ultimately succumbed. However, in his day he also made substantial advances in mathematics, again using very simple geometric techniques. His most interesting contribution relates to the calculation of the area of a circle using many sided polygons. It is possible to calculate the area of a circle, to within very narrow defining limits, using inscribed and circumscribed polygons to the point of exhaustion. Fascinatingly, this methodology anticipated infinitesimal calculus which would be independently developed by Newton and Leibnitz in the 17th century. As many are aware, calculus 'is' the tool which formulates all fluid and dynamic functions of higher mathematics and therefore is essential for understanding our world in motion.
So there we have it: a brief foray into the exciting world of ancient Greek mathematics. They achieved much considering the limitations under which they laboured. I wonder how their mathematics would have evolved if they had had access to a flexible and powerful notation which, we today, take for granted. Greek genius halted when the Romans became Lords of the Mediterranean. The Romans had no interest in abstract mathematics and applied themselves wholeheartedly to war and politics, which for the Romans, was one of the same thing.