Archimedes

Unfortunately, Archimedes is one of those semi-mythical guys who wasn't written about heavily until hundreds of years after his death. What we do know about him is a combination of contemporary and later sources, all of which agree on at least one thing: he was crazy smart.

He approximated pi to an unprecedented degree, which by itself is one heck of an accomplishment. He built world-changing inventions, created new mathematical formulas which incorporated infinity and zero (two concepts which scared the daylights out of Aristotle), applied buoyancy, and created complicated pulley systems which allowed for the earliest cranes to be built. I have focused mostly on how his inventions applied to warfare and farming, but those were the two chief interests of the ancient Mediterranean, when you get right down to it.

Though he died an untimely death at the hands of a foolish Roman Legionnaire, his concepts and inventions continued to change history long after his death. That is, the inventions that they could duplicate continued to change the world. There were several items the plunderers found in his workshop which could not be re-created and were taken as trophies by Marcellus and his Captains. One of them was a sphere with gears and levers which could calculate the exact position of the known planets.

Pax vobiscum

Things That Archimedes Made

Following in the footsteps of other great eccentric Greek mathematicians, Archimedes seems to have gone about his life by trying to learn something new every day. To make an exhaustive list of everything he discovered, invented, or improved would take far more space than what should be contained in these virtual pages, so today we'll take a look at just a few of his mind-boggling creations.

The pulley itself is so ancient that its creator's name has been lost in the mist of time. Archimedes experimented heavily with pulleys and created the world's first block and tackle system (basically a mechanism in which two or more pulleys are utilized). He discovered that each pulley he added made the load seem lighter, since the force was being transferred equally among all the pulleys in the array. This allowed merchants to construct great cranes capable of lifting heavier cargo, which meant more money on arrival.

But his genius did not end at maritime physics. While trying to find an easier way to irrigate highland farms, he invented Archimedes' screw, a device capable of carrying water upward. Now, ground water could be raised to the surface and reused on the crops. It could even be carried up mountains, if the pipes and screws were long enough. This revolutionary invention was so useful that the Romans later built it into all their farmland and even used it to carry drinking water into high fortified camps. Here's a great image of how this works:

During the First Punic War, he invented a primitive odometer. It was an array of gears and levers which would drop a ball into a small box for each mile it traveled. To have done this during a time when people believed that disease was caused by the moon is not only surprising; it's mind-bogglingly astonishing. In addition to this, he created several astronomical implements, some of which were carried away as loot by the Roman commander who sacked Syracuse.

In between revolutionary and brilliant inventions, he did Calculus. But wait! Didn't Sir Isaac Newton invent Calculus? No more than Henry Ford invented the automobile. Newton greatly expanded the science, which laid dormant for more than a thousand years, but it was Archimedes himself who made the first foray into the measurement of the infinite. Unfortunately, there was no one around capable of understanding Archimedes' work, let alone carry it on. Though he served as a bright light for a time, the world went dark again at his passing, and the Greeks and Romans went back to their day-to-day business, being afraid of things they didn't understand. His work lives on, and he remains one of the most important mathematicians and inventors in all of history.

Culture Wednesday: More Than a Theorem

Anyone who's learned even the most rudimentary geometry has heard his name. Yes, in his own way, Pythagoras achieved the Greek ideal of fame that causes him to live forever, even if he is relegated to the pages of High School textbooks. But there is more to this mathematician than a simple ratio. Among other things, he believed himself to be the reincarnation of a Trojan hero, and refused to eat beans. His mathematekoi brotherhood was thought to be the most well-learned in the ancient world, but they did not hesitate to murder one man who would expose their most embarrassing secret: irrational numbers.

Though he seems harmless in his textbook depictions, Pythagoras was considered by many in the ancient world to be a dangerous cult leader, and a malcontent. He lived during that necessary twilight between oral tradition and written history, and thus his life and work is shrouded in an unfortunate cloud of mystery and myth. It is said that he traveled all throughout the world to gain knowledge of mathematics, science, medicine, astronomy, astrology, and mysticism from whomever would teach him. I think it likely that he traveled to Egypt, home of the famous ancient mathematician Thales, who accurately calculated the height of a pyramid by measuring its shadow. The Greek philosophy of math and science was that it was attainable, that they could actually observe and learn from what they saw to predict or manipulate future behavior. However, it was also steeped in Pagan mysticism, something which taints their learning somewhat and caused many, especially Pythagoras, to go off the rails.

Eccentric though he was, even by ancient weirdo standards, he built a school around this central tenet: All is number. He believed that numbers could be used not only to define all things, but could even give them greater meaning. His disciples took this motto to heart, and immediately began measuring angles and lengths to find the hidden constant ratios between them. In fact, most of the really boring parts of Geometry today (constructions, proofs) were what the Pythagoreans discovered when they were just playing around. They would challenge each other with number riddles like, “can you form a right triangle if given two points?”

One story claims that Pythagoras discovered the mathematical value of music. He was passing by a smithy one day, and the ringing of the anvils was sticking in his head. He noticed the relationship between their individual pitches, and examined three of them to discover that the middle one was one-third bigger than the smallest, and that the largest was one third bigger than the middle. Through experiments that the school performed on strings, bells, and other instruments, they created the octave as a means of dividing the musical notes, something we still do in Western music today.

As clever as these stories make him out to be, my belief is that Pythagoras was just the charismatic leader of some very bright young Greeks. I believe his school as a group made much of the discoveries that he is given credit for, just like professors will occasionally take credit for their students' findings today. In either case, he should at least be honored for cultivating an environment in which learning and discovery could take place.

However, this was long before the days of public education, and knowledge of every sort was a tightly guarded secret, particularly mathematics. As we will see in coming weeks, mathematics can kill people, and it often does so in great quantities with a minimal effort. For the Pythagoreans, the biggest secrets that they kept were the ones that they hated and couldn't explain.

Pythagoras and his followers were so convinced that everything could be expressed as a ratio of whole numbers that when they discovered ratios that didn't work out to nice, neat, clean whole numbers they were thrown into a crisis of faith. You see, this wasn't “just math” to them; this was their religion. Everything they held to be true, yes, the very order of the universe was at stake, and if it was just all random, irrational events, then knowledge itself was a fools game, a mirage in the desert of unpredictability.

Try this to see what I mean: draw yourself a square. Go ahead, draw one. Now, assign each side the number 1. Doesn't matter how big you've made it, just pretend you've created your own unit of measurement, name it after yourself if you like. Now that you have your square, draw a diagonal line from one corner to the other. Great. Now, using the Pythagorean Theorem, a^2 + b^2 = c^2, calculate the length of that diagonal. You should get 1^2 + 1^2 = c^2, and c^2 is equal to 2. So how long is that diagonal? The number √2, if you bother to plug it in, should make your calculator go crazy with an endless stream of numbers. However, it's a decimal answer, and therefore not a whole number. The Pythagoreans probably convinced themselves that there must be some sort of mistake in all of this, but their equations would continue to come out irrational, that is, being unable to express in a ratio of whole numbers. And since the square was one of their 'sacred shapes,' there's no doubt that they wanted to keep this a secret until they could find a way to make it work.

Well, enter Hippasus, a Pythagorean disciple who couldn't keep this cat inside its loosely constructed bag. He blew the whistle to the general public and alerted everyone in the Mediterranean that Pythagoras and his school were just a bunch of frauds. Little is truly known about this character as well, though some believe he created a rival school of mathematics. In either case, it is believed that he was murdered by Pythagorean zealots while trying to leave town. Yes, that's right, he was murdered for mathematics.

Pythagoras himself was killed during a political uprising in Croton in which he found himself on the wrong side. Supposedly, he ran from his assassins and was gaining ground until he came to a field of beans. He stopped and declared that he would not cross a field of beans, at which point I like to imagine his attackers stopping for a second to tilt their heads in unison before promptly dispatching this mathematical primadonna.

Pax vobiscum