Meet the Math Greeks

So far this week, we've met the three biggest names in ancient Greek philosophy--Socrates, Plato, and Aristotle. Now let's pluck the strings and pull the levers of three great Greek mathematicians: Pythagoras, Euclid, and Archimedes.

If, hopping out of your time machine, you told any of these three that he's remembered as a remarkable mathematician, he'd actually be surprised--and not just at the sight of your hip time-traveling togs. In ancient Greece, math wasn't a specialty. It was part of a philosophical education, valued as a means of discerning the nature of reality.

Pythagoras and the Triangle of Truth

Pythagoras used math mainly to make a point about religion. Not much is known about his life, except that he was born around 580 BC on the island of Samos. After early philosophical study, sources say he traveled in Egypt and Babylonia, learning what he could of their science and practical mathematics. Later, he founded a religious school in the city of Croton in southern Italy. There he began preaching that numbers are the building blocks of existence.

To prove his point, Pythagoras and his followers examined numerical phenomena closely. Along the way, they discovered that the sum of the angles of a triangle add up to 180. They also discovered that, in a right triangle, the square of the hypotenuse (the long side) is equal to the sum of the squares of the other two sides--the famed Pythagorean theorem. Pythagoras's favorite discovery, though, might have been the numerical basis of musical harmony.

If you take two harp strings of equal thickness, a ratio in length of 2:1 will produce the musical interval of an octave (if the strings are held under equal tension). A ratio in length of 3:2 will produce the musical interval of a fifth. Harmonious arrangements were critical to Pythagoras's thought. He even hypothesized that the different planets emit harmonious notes (the "harmony of the spheres"). Political harmony, however, eluded Pythagoras. The Pythagoreans got involved in Croton's internal politics, and in 496 a rival party chased them out. Pythagoras died soon after in a neighboring city.

Euclid and the Timeless Textbook

As little as we know about Pythagoras, we know even less about Euclid. We know that he lived around 300 BC, probably studied in Athens at the Academy founded by Plato, and worked in the Library of Alexandria. Otherwise, all we know is that he wrote one truly excellent textbook--so good that he's now called "the father of geometry."

Euclid's textbook, the Elements, was in many ways unoriginal. He organized and systematized the work of previous mathematicians, filling in holes where he found them, and arranging geometrical concepts and proofs from simple to complex. He did his job so well, though, that afterwards no one bothered making copies of those earlier geometrical works, which are now completely lost.

Euclid's Elements was so influential that we now use the phrase "Euclidean geometry" to refer to both two-dimensional (or "plane") geometry and three-dimensional (or "solid") geometry. (Don't even ask about the non-Euclidean sort.)

Archimedes and the World-Lifting Lever

Pythagoras and Euclid were great, but Archimedes was even greater. The son of an astronomer, Archimedes was born around 290 BC in the Sicilian city of Syracuse. He became famous in his day for inventing marvelous machines. For Archimedes, though, such machines were useful mainly as a means to discover mathematical theorems.

Archimedes worked in practically every area of mathematics. His study of how submerging an object affects its weight is the foundation of modern hydrostatics (though the related story about him rushing down the street stark naked, shouting "Eureka," is likely embellished). Archimedes' work on determining the area of shapes bounded by a curve anticipated modern calculus by 2,000 years. He also did ground-breaking work on pulleys and levers, allegedly pronouncing, "Give me a place to stand and I will move the world!" He's even credited with inventing the "Archimedes screw," a device for raising water still used today.

Unfortunately, much of this came to nothing. Archimedes died in 212 or 211, when the Romans sacked Syracuse (though his war machines made things difficult for Rome's legions). Later generations mostly forgot he had ever lived. The West rediscovered his genius centuries later, via Arabic translations of his work. The rediscovery came too late to advance the cause of science much, but still in time for us to celebrate the achievements of a remarkable mathematician.