menaechmus

Rene Descartes described analytic geometry in 1637, in an appendix to his Discourse of Method. The coordinate plane idea's origin is uncertain, and may date back to Menaechmus, a Greek mathematician from the 4th century B.C. Please see the related links for details.

Although Descartes is generally credited with inventing analytical geometry, several aspects had been used by the Greek mathematicians Menaechmus (4th century BC) and Apollonius of Perga (3rd century BC).

Much later (11th century), the Persian mathematician Omar Khayyam, used methods which modern mathematicians consider akin to analytical geometry.

In 1637 Rene Descartes published his work on the topic but it was incomplete and was in French. It took another decade or so before it was translated into Latin and some of the gaps filled that Descartes' ideas took off.

It is generally acknowledged that mathematics developed in the 7th and 6th centuries B.C., somewhat after the Greeks had developed a uniform alphabet. However, the mathematical knowledge existed before the Greeks. In fact, the Egyptians and Babylonians knew many empirical facts centuries before the rise of the Greek civilization. For example, they could solve quadratic equations, compute the areas of geometric figures such as, squares, rectangles, and triangles, and they possessed a good formula for the area of a circle, using the value of 3.16 for pi. They also knew how to compute volumes of cubes, cones, cylinders, and pyramids. The Greeks, who settled throughout the Mediterranean, must have played an important role in preserving and spreading the mathematical knowledge of the Egyptians and Babylonians. However, there were different formulas for the same areas or volumes. For example, the Babylonians had one formula for the volume of a pyramid with a square base, and the Egyptians had another. But, between two given formulas, it was clear that only one could be correct. So the necessity to find the right one, led to the development of mathematical proof and to the method of deductive reasoning. The person usually credited for the invention of rigorous mathematical proof was a merchant named Thales of Miletus (624 -548 B.C.), who is said to be the creator of Greek geometry. And it was this geometry (earth measure), as an abstract mathematical theory supported by rigorous deductive proofs, that was one of the turning points of scientific thinking. It led to the creation of the first mathematical model for physical phenomena. For example, one of the most beautiful geometric theories developed during antiquity was that of conic sections, whichs include the straight line, circle, ellipse, parabola, and hyperbola. Their discovery is attributed to Menaechmus, a member of the school of the great Greek philosopher, Plato.

Ancient Athens was a complex system, in many ways relatable, and in other ways remarkably alien.

We must remember the context of ancient Athens; they lived literally millenia before the concept of equal human rights; when everywhere else in the world was ruled by tyrants, religious godheads, and petty warlords. The idea that even a small group of individuals could decide policy was alien to any sort of thinking; the cultural capital of the world at that time was Egypt, which for nearly 2000 years had been ruled by a single ruler, whose word was absolute and his authority divine. Egyptian people would spend their entire lives saving up for a lavish funeral, and spend their free time memorizing religious verses to serve them in the afterlife. All over the world, rulers envied the fanatical devotion of the Egyptian people to their ruler and their religion, a tradition which was virtually unchanged for nearly 2,500 hundred years, and peoples from all over the world came to marvel at the vast cities of mausoleums which the Egyptians constructed in honor of their dead

However influenced by Egypt the Greeks were, their way of thinking was completely different. The Greeks were Hero-worshippers; they considered the ultimate glory to be great accomplishments on Earth, and not veneration of an unseen deity. A Greek was more likely to venerate Achilles or Herakles instead of Zeus or Poseidon. In fact, popularly known stories of such Gods were more of an afterthought in Greek religion than canon, because a Greek was wholly unconcerned with the afterlife, and instead focused on what he as a man could accomplish. For example, in nearly 70 years, the Greeks perfected human anatomy in sculpture making; a feat which the Egyptians failed at despite nearly 3000 years of making art.

In this way; Greeks are very similar to us. Religion was merely an aspect of their lives; they spent most of it tending to their business, engaging in spiritual or intellectual fulfillment, attending or playing sports, or debating politics.

And boy, did the Athenians loved to debate politics. It was considered proper for a man to debate politics at every opportunity. As Athens was a direct democracy, ever citizen voted on legislation, thus civic participation was encouraged among the people. Strangers making small talk on the street were as likely to comment about the hot button political issue as you would be making small talk with someone about sports or the weather. Athenians even came up with a word for people who had no interest in politics, or people who only voted for pet projects; "Idiot".

If it wasn't politics, it was sports. Athenians, like all Greeks, had a nearly religious devotion to athletics, and would devote a portion of their time each week to exercising and engaging in athletic competitions. Winners of one of the games, such as the Olympics, were the recipients of great honor, and Athenians revered those who excelled in athletics.

Of course, I refer to the men only. Women had no rights in Athens. A woman was the property of her father until she was married, where she would then become her husband's property. Attending a religious ceremony or a funeral were the only reasons a woman was allowed to leave her house. This is sharply contrasted with Spartan women, who were not only allowed to live independently of the men, but even own property, play sports, divorce their husband, and participate in civil matters.

Slavery did exist in Athens, but as Michael Foucault pointed out; people talk about things they are conflicted about. And the issue of slavery was an ever present issue in our records of Athenian thought. Remember that stoicism started in Athens, and Stoicism preaches eternal brotherhood with all men. But the conflict comes from reality; Greece was surrounded on all sides by barbarians; whose raids could penetrate deep into Greece. It was very common in Greek literature for slaves to be freed through acts of virtuous behavior, thus Greeks would have at least held on to some notion that slavery was a sort of "civilizing" process for the barbarian. However, most Greeks were ultra-nationalistic and highly xenophobic, and the controversy regarding slavery mostly stemmed from the idea that it takes jobs away from honest, hard working Greeks. Most of the farming done in the countryside around the city was actually done by Athenian citizens, and slaves were either servants, handmaidens, or prostitutes, or if they were particularly unlucky, they served in mines or rock quarries. The slave trade was profitable, and sometimes Prisoners of War could be turned into slaves. However, city-states often kept Prisoners of war as diplomatic bargaining chips, as fathers would go to understandably great lengths and would pay vast sums of money to see their sons safely home, and as city-states often had a small number of citizens who were eligible to be soldiers, it was in the states interest to have their soldiers returned to them, lest their military become paralyzed for an entire generation.

Education was the cornerstone of any Athenian Citizen's early life, and Athens was an epicenter of learning that continued for nearly a thousand years after the end of the classical era. Whether it was Plato's Academy or Aristotle's Lyceum, a young Athenian man could expect to learn three disciplines; philosophy, literature, and rhetoric. Philosophy was things regarding the world; religion, mathematics, science, astronomy/astrology, and engineering. Literature was things regarding the past, as the Greeks had no written records of the earlier, Mycenaean era, which had collapsed nearly 600 years before the start of the classical era, thus a legend and a historical fact were virtually indistinguishable (Aristotle even went so far as to say that poetry and literature was superior to history since they depict man as he should and could be, rather than as he is). Rhetoric is things that deal with people, talking to them, getting them to see things your way, obviously it was the craft of merchants and politicians. As I mentioned, Athens was a direct democracy, thus skilled orators held a considerable amount of political clout, making rhetoric a critical area of study.

I've already typed a lot, so I won't go on. A good introduction to ancient Greek way of life is wikipedia. Another good place to continue would be to read some Greek plays, such as The Pot of Gold or the Brothers Menaechmus. I know that suggestion may make you squirm, but one thing that you will learn is how similar their way of thinking is to ours when you realize how similar their sense of humor was to ours. Even Shakespearean comedies can be difficult to comprehend, but Ancient Greek comedies are slapstick, funny, timeless, and simple, thus easy to get a good laugh out of.