Archimedes
<~His Life and Times~>
His Life and Times

There are lots of unusual stories about how the world's great thinkers came up with their most important 
ideas. Isaac Newton allegedly discovered gravity while sitting under an apple tree, which dropped an 
apple nearby. Jean Jacques Rousseau's first theory came to him after he went into a semi-trance in which 
he could neither walk nor breathe. Galileo worked things out while dropping objects off the top of the 
leaning Tower of Pisa. And for Archimedes, the greatest scientist of ancient times, had one of his best 
ideas was inspired by a bathtub.
The question he was considering that day as he sat in a relaxing bath was the question of a crown. The 
King of Syracuse, where Archimedes lived, suspected that a goldsmith had substituted some of the gold in 
his crown with silver, and hoarded the remaining gold for himself. Knowing the intelligence of 
Archimedes, he brought the crown to him and asked him to try and determine whether it was pure gold, or 
whether the goldsmith had cheated him. He cautioned Archimedes to reach his conclusions without doing 
any damage to the crown itself. But short of melting the crown and comparing its volume with an equal 
weight of pure gold, Archimedes could think of no way to check the honesty of the crown-maker.
Distracted by his problem, Archimedes went to the public baths to relax and think it over. As he stepped 
into the full tub, some of the water overflowed. In an instant it came to him: the amount of water that 
overflowed must be equal to the volume of his body that he'd inserted into the water. So, if he dipped 
the crown into water he'd be able to tell the volume of it by the rise and fall of the water level. He 
could compare that with the volume of an equal weight of gold. If equal - the crown was pure gold. But 
if the crown contained silver, which was bulkier than gold, it would have a greater volume and the water 
level would be different.
Excited by this new idea - which is now called "the principle of buoyancy," Archimedes is said to have 
dashed out of the bathhouse and run completely naked through the city streets, yelling, "I've got it! 
I've got it!" - or, in his native tongue of Greek: "Eureka! Eureka!" And ever since that day two 
thousands years ago in Syracuse, the remark "Eureka! Eureka!" has been used over and over by people at 
the joyous moment of discovery or revelation. Fortunately for Archimedes, nudity wasn't nearly as 
shocking to the ancient Greeks as it is to us today. He made his way safely home where he could begin 
his experiments. Archimedes found that the crown displaced more water than pure gold did; therefore the 
crown contained silver and the king had been cheated. With that experiment, he attained lasting fame and 
also with that experiment, so it's said, the goldsmith lost his head. The theory on the weight of a body 
immersed in a liquid is still known today as "Archimedes' Principle."  Archimedes was probably only a 
teenager at the time of this discovery, even though artists portraying the event later liked to depict 
him as a wise old man.
This is only one of many colorful stories that have been told about Archimedes, most of which are 
believed to be true. Perhaps the most famous of all is his remark: "Give me a place to stand on and I 
can move the world." He was referring at the time to his discovery of the lever, but the king, who had 
heard this remark, challenged him to prove it. Archimedes, who had gained a good deal of fame in 
Syracuse for his outstanding feats, was glad to take up the challenge. He took the king to the beach 
where a group of men were straining to haul ashore a three-masted ship loaded with cargo. Between the 
prow of the ship and a nearby pier, Archimedes attached an arrangement of ropes and pulleys. He then 
pulled gently with his hand and the ship drifted smoothly across the sand toward him, "as if," wrote an 
ancient historian, "it were passing over the sea." The king is reported to have called out: "From this 
day forth Archimedes is to be believed in everything that he may say."
The king in both these stories was King Hiero II, who ruled Syracuse from 270 B.C. to 216 B.C., a span 
that covered most of Archimedes' adult life. The king and Archimedes were good friends - in fact many 
historians believe they were related, possibly even cousins. Hiero was dedicated to the progress and 
prosperity of his city and built many temples, theaters, and fortifications to protect it. During his 
entire 54 year reign, it's reported he ruled "without killing, exiling, or injuring a single citizen," 
an extraordinary achievement for the times but one that doesn't account for the fate of the wayward 
goldsmith. Hiero also gave full support and encouragement to his resident genius, Archimedes.

Archimedes was a wealthy aristocrat who never had to worry about a living and could devote his time to 
whatever interested him. His interests were science, mathematics, and invention. In an age of brilliant 
thinkers, he was one of the most brilliant of all - some believe his genius wasn't equaled again until 
two thousand years later when Sir Isaac Newton came along. The range of Archimedes' talents was awe-
inspiring - from the discovery of mathematical relations to the establishment of physical laws to the 
design of engineering and military equipment - and he seems to have had a flair for public relations 
that, although unmediated, was just as impressive.
Archimedes was born about 287 B.C. in Syracuse, a Grecian city/state and a very important metropolis. 
Its location on the eastern coast of Sicily put it close to both Rome and Carthage, two empires that 
were competing for power. In those days Syracuse was the largest Greek city in the world, with a 
population of over half a million.
Very little is known about Archimedes' life, except the dozens of legends that may or may not be 
trusted. Although he carefully recorded his mathematical work, he never wrote about himself. His father, 
Phidius, was an astronomer, and probably a good one, since Archimedes later used some of his 
calculations in his own work. He sent his son to Alexandria to study under a teacher who had been a 
pupil of the great Euclid - the ancient Greek who founded modern day geometry. Alexandria was then the 
scientific center of the world, and the adopted home of most of the great scientists of the third 
century B.C. They lived and worked together at the Museum, a sort of early university, and were paid 
salaries by the kings, or ptolemies. As far as we know, this was the first scientific organization 
founded and supported by a government. For some reason, Archimedes decided not to settle in Alexandria, 
perhaps because he had such a beneficial relationship with King Hiero back home, perhaps because he was 
independently wealthy and didn't need the support of the ptolemies. However, at least twice in his life 
he sailed back to Alexandria, staying for several months to confer with scientists there and to take 
advantage of its great library. He may also have journeyed to Spain, where it's said he invented a 
device for flinging burning pitch on enemy ships. But other than these occasional excursions, he seems 
to have spent his entire long life in Syracuse.
Another of his famous escapades in that city was his attempt to calculate the number of grains of sand 
it would take to fill the entire universe - of course, to do so, he had to guess at the size of the 
universe, but that was incidental. The main point he was trying to make is that nothing that existed was 
too large or too small to be measured - nothing material was infinite. To prove this he created a system 
for expressing large numbers that is almost exactly like our current one. He began his sand project on a 
small scale and meticulously worked his way up to a titanic scale. First he said, no more than 10,000 
grains of sand could fit in a poppy seed. No more than forty poppy seeds fill a thimble; no more than a 
certain number of thimbles would fill a Greek stadium. Step by step he went on, increasing the volume of 
sand.
It's said that Archimedes was so absent-minded when working on an idea that he never knew whether he'd 
bathed or eaten, a trait of many scientists and inventors. His friends would periodically drag him off 
to the baths where, after being cleansed and oiled, he'd proceed to trace diagrams on his oily body as 
he worked out a difficult math problem. Other times they would find him near the fireplaces used to heat 
water, drawing figures in the ashes with a stick. In spite of his absorption, Archimedes seemed to have 
had a humorous and playful side. He loved to make up games and puzzles with his mathematics, and he once 
inserted some false propositions into a paper he wrote just to play a joke on his friends. It served the 
additional purpose, so he said, of trapping plagiarists.
One of the problems he had to work out at some point was the request by King Hiero that he build a 
gigantic freighter. The king continuously called on Archimedes to solve one dilemma or another, and 
whether the scientist resented these interruptions in his studies or welcomed them as challenges, is 
unknown. Only Noah's rivaled the latest challenge, except that Archimedes didn't have to worry about 
animals. The exact dimensions of the ship he built have been lost but it was big enough to carry thirty-
five hundred tons of cargo and used the same amount of lumber and materials it would take to build sixty 
average ships. It was also one of the world's original luxury liners. It had beautifully tiled floors 
that told the entire story of Homer's Iliad. And there was more, lots more. In the words of an ancient 
historian:
"It had a gymnasium and promenades, garden beds of every sort luxuriant with plants of marvelous growth; 
bowers of white and purple grape vines; a shrine of Aphrodite; walls and ceilings of cypress wood, and 
doors of ivory and fragrant cedar; a library; ten stalls for horses; a water tank at the bow with a 
capacity of twenty thousand gallons; a fish tank; and the whole ship was adorned with appropriate 
paintings."

Archimedes probably had little to do with these decorations - his job was simply to design a vessel that 
could support it all. When he finished the hull, he then had to design a windlass - the first in history 
- to launch the hull into the water. A windlass is a device for hauling, which consists basically of a 
barrel, or drum on which the hoisting rope is wound, and turned by means of a crank.

After the hull was completed, Archimedes worked on the military features of the ship, designing a wall 
with battlements, eight turrets to hold warriors, and stone hurling machines for which he had to 
construct a special, elevated deck. Finally he added his famous Archimedes screw, which he'd invented 
years earlier, for pumping water out of the hold of the ship.

After all this work, King Hiero then discovered that no harbor on the Mediterranean was large enough to 
handle the "The Lady of Syracuse," as the ship had been christened. As a solution he loaded it with 
grains and salted fish and sent it off to the King of Egypt, whose country was in the middle of a 
drought. It took thousands of Egyptians to haul "The Lady of Alexandria," as it had been renamed, onto 
shore, and there it stayed. Egypt too had no idea what to do with the magnificent, but awkward ship.

<~Scientist and Warrior~>

Scientist and Warrior

In spite of possessing one of the most brilliant scientific minds in history, in his own time Archimedes 
achieved his greatest glory as a warrior. During his old age, Rome went to war against Carthage, a city 
in northern Africa that was ruled by the famous conqueror Hannibal. Hannibal invaded Italy, accompanied 
by his renowned herd of elephants, and Rome, which had a treaty with Syracuse, called on that city for 
help. When the ruler of Syracuse, now Hiero's grandson, decided to help Carthage instead, Rome sent a 
fleet to attack Syracuse and launched a war that lasted three years. It's been said that this war was 
really a battle between the entire fleet of the Roman general Marcellus and one man - Archimedes of 
Syracuse.

The Romans had the power to take the city quite easily, except for the ingenious devices Archimedes kept 
coming up with that not only drove the Romans back, but frightened them as well. One of these was a 
large concave lens, or group of lens, which Archimedes constructed. It could, so the story goes, focus 
the sun on the Roman ships and set them afire. The report of this mirror later caused a debate between 
two great thinkers of the 17th century - Galileo and René Descartes. When Galileo said he believed the 
story, he incurred the ridicule of Descartes, who said it was impossible to ignite a ship with a 
reflection. Neither man made any attempt to test the theory, but a century later a naturalist named 
Georges Buffon did and was successful. Then, in the 20th century another scientists tested it, using 
shields as the men of Syracuse may have had, and he too successfully ignited a boat.

Archimedes also built catapults that bombarded and sank the ships with missiles weighing a quarter of a 
ton. Still other catapults launched showers of rocks and arrows that had the Romans fleeing in panic. 
The Greek historian Plutarch wrote a vivid account of the battles:

"Marcellus proceeded to attack the city both by land and sea. Marcellus, with sixty galleys, each with 
five rows of oars, furnished with all sorts of arms and missiles, and a huge bridge of planks laid upon 
eight ships chained together, upon which was carried the engine to cast stones and darts, assaulted the 
walls. The Syracusians were struck dumb with fear, thinking that nothing would avail against such 
violence and power. But Archimedes began to work his engines and hurled against the land forces all 
sorts of missiles and masses of stone, which came down with incredible noise and speed; nothing at all 
could ward off their weight, but they knocked down in heaps those who stood in the way and threw the 
ranks in disorder. Furthermore, beams were suddenly thrown over the ships from the walls, and some of 
the ships were sent to the bottom by means of weights fixed to beams and plunging down from above. As 
for the engine which Marcellus was bringing up from the platform of ships...while it was still some 
distance away, as it was being carried to the wall, a stone some 1800 pounds in weight was discharged at 
it, and after this a second, and a third; some of these, falling upon it with a great crash and sending 
up a wave, crushed the base of the engine, shook the framework and dislodged it from the barrier, so 
that Marcellus in perplexity sailed away in his ships and passed the word to his land forces to retire."

According to stories written later, the Romans became so terrified that if they saw even a rope or a 
small beam hanging over the wall of the city, they would turn and run, fearing it was some new device 
Archimedes had designed to destroy them.

There are also stories about a crane Archimedes invented, with a grapple hook to lift the Roman ships 
out of the water and turn them upside down, or crash them against the cliffs, but most historians think 
this story was an exaggeration. There was a crane of some kind however, and Marcellus was reported to 
have complained that Archimedes was using his ships to ladle out the sea. Whatever he did, Archimedes 
managed to hold the Romans off until the year 212 B.C. when Syracuse was finally defeated, partly 
because of spies and traitors within the city, partly because the Romans attacked one night when the 
city was celebrating a festival and all its soldiers were drunk, and partly because the city, under 
siege for three years, was nearly out of food and supplies.

It's said that the Roman General Marcellus wept at the moment of victory because he knew what the fate 
of the beautiful city would be. It was a rule of war in those days that the victorious soldiers had the 
right to plunder, and it was a right with which even a General couldn't interfere.

In a sense Syracuse went down in glory - as the first city to apply scientific methods to the art of 
war. At the time, Archimedes was 75 years old. But after the Romans entered the city, he had but a few 
hours of life left.

The death of Archimedes is a famous story also - in fact it's two famous stories. The most commonly 
accepted version is that he was sitting in his yard, working out a mathematical problem in the sand, 
unaware that Syracuse had been defeated. If so, he was probably using the conventional method of 
spreading sand on a tray, making his calculations, and then wiping them away when he needed to start 
anew - an ancient form of the blackboard. As he was bent over his figures, distracted as usual, he was 
approached by a Roman soldier who ordered him to leave the beach. Archimedes reportedly gestured at the 
soldier and said, "Don't disturb my circles," or, depending on which account you read, "Stand back from 
my diagrams." The soldier, a man who had either little patience or much pride, took out his sword and 
killed Archimedes on the spot, probably not knowing whom he was. The Roman General, Marcellus, had 
earlier given orders that Archimedes was to be taken alive and treated with respect, so he was horrified 
when he heard what his soldier had done. He mourned for Archimedes, gave instructions for an honorable 
burial, and made sure his relatives were protected.

Another version of the death of Archimedes is that he was hurrying to Marcellus with boxes containing 
mathematical and astronomy equipment. Suspecting the boxes held valuables, the soldiers killed him just 
for the plunder. Whichever story is true, it seems the greatest scientist of the time was killed for the 
smallest reason.

A century after the death of Archimedes, the Roman statesman Cicero visited Syracuse and found the tomb 
of the renowned scientist, sadly neglected. Cicero reported that the tomb was easily found because of 
its decoration; Archimedes had asked his friends to inscribe it with what he considered the perfect 
geometric design: a cylinder holding a sphere within it, and written beside it the ratio 3/2. The ratio 
stands for both the area and volume of the two shapes and the discovery of it was one of Archimedes' 
proudest achievements. Despite energetic efforts, no trace of Archimedes' tomb has ever been found since 
Cicero's time.

The science of ancient Greece faded as the Roman Empire declined and interest in the works of Archimedes 
and others all but disappeared during the Dark Ages. From about the fifth to the 15th century, 
civilization became very interested in religion, and very disinterested in science. That interest wasn't 
revived until the Renaissance, when just about everything regarding ancient Greece was revived, 
including its art and philosophy.

<~His achievements~>
His Achievements

The inquiring and versatile mind of Archimedes ranged through both the theoretical and applied sciences 
- he was a mathematician, physicist, astronomer, inventor, military engineer, and nautical engineer.

In his experiments with the king's crown, Archimedes developed the principle of Specific Gravity. Before 
his time, scientists were puzzled as to how to describe the density of an object. Specific Gravity gave 
them the solution - it's simply the weight of a substance relative to water. For instance: Gold, being 
nineteen times heavier than water, has a specific gravity of nineteen. Silver has a Specific Gravity of 
eleven. His principles of buoyancy that came out of these experiments  - that a floating body loses in 
weight an amount equal to the weight of the water it displaces - is what's known as Archimedes' 
Principle.

Although the principle of the lever had been experimented with before, it was Archimedes who worked it 
out in full detail. He showed that a small weight at a distance from a fulcrum would balance a large 
weight near a fulcrum - and that the weights and distances were in inverse proportion. Thus he developed 
the notion of a center of gravity. His laws of the lever and balance were so accurate that no advances 
were made on them until the late 1500's. It was these discoveries that later helped inspire Galileo to 
make his own discoveries in the field.

Archimedes was also the first to use a COMPOUND pulley - he devised systems for using a number of 
pulleys to double the force of the pull over and over, an arrangement known today as a block and tackle. 
His experiments with levers and centers of gravity were the start of the science of mechanics - the 
study of forces and the effects they have.

His famous "Archimedes Screw" was a hollow cylinder with spirals like a screw, that, when rotated, could 
serve as a water pump. In this device, the screw-like worm meshes with the teeth of a gear, increasing 
the mechanical advantage. Archimedes developed his hydraulic screw while on a visit to Alexandria, 
probably at the request of Ptolemy II, who used it to control the devastating floods of the Nile River. 
Later it was used for such purposes as lifting water out of the holds of ships. And of course there was 
his windlass, which he used to haul the great ship he designed for Hiero.

As clever as these devices were, Archimedes himself always believed that his mechanical inventions were 
inferior to his theoretical work, so he published only his mathematical discoveries and not his designs. 
We know about them chiefly through the writings and stories of others of his time. The historian 
Plutarch wrote about Archimedes' indifference to his mechanical accomplishments, saying:

"He possessed so high a spirit, so profound a soul, and such treasures of scientific knowledge, that 
though these inventions had now obtained for him the renown of more than human sagacity, he yet would 
not design to leave behind him any writing on such subjects; but repudiating as sordid and ignoble, 
every sort of art that lends itself to mere use and profit, he placed his whole affection and ambition 
in those purer speculations where there can be no reference to the vulgar needs of life...studies in 
which the only doubt can be whether the beauty and grandeur of the subjects examined, or the precision 
of the methods, most deserve our attention."

The mathematics of Archimedes, which he carefully recorded in manuscripts throughout his life, was lost 
to the world for fifteen centuries. It wasn't until the Renaissance that other scientists eagerly tried 
to uncover his work and study it, and even then they found only ten documents. One of his works, 
entitled On the Method wasn't discovered until the year 1906, when a Danish scholar studying a church 
ritual written on a roll of parchment in the 1200's, noticed another layer of writing underneath. After 
treating the parchment with special chemicals, under light rays, scientists discovered a treatment by 
Archimedes that was hailed as "one of the most revealing documents in the history, not only of ancient 
science, but of science in general at all times." It contained his advanced mathematic systems. Another 
document was unearthed in the 1920's but since then there have been no more findings and the likelihood 
there will be, grows slimmer.

Even with the small amount of work uncovered, scientists and historians alike are astounded by what 
Archimedes discovered, with so few resources and information available. Rather like Mark Twain's 
Connecticut Yankee in King Arthur's Court, Archimedes operated in a time and place that was ages behind 
his own mind. It's not surprising that people of ancient Greece viewed him more as a wizard than a 
genius, and made few attempts, if any, to preserve his writings. The scroll discovered in 1906 had been 
written over by a monk who undoubtedly found the sophisticated mathematics on it nothing more than 
gobbledy gook.

In the time of Archimedes, mathematics consisted of a very elementary form of arithmetic and geometry. 
While trying to extend the scope of it, he was handicapped by the existing number system, which used the 
27 letters of the Greek alphabet for numerals. The Greeks named no number larger than our ten thousand, 
which they called a "myriad." Our million, for example, would have been a hundred myriads. To overcome 
this limitation, Archimedes invented a system for expressing large numbers. With that system he could 
express numbers as large as 1 followed by 80,000 million million zeros. Then, with a system not unlike 
modern calculus, he came up with a value for pi - the ratio of the length of the circumference of a 
circle, to its diameter -, which was the most accurate of any early mathematician. He said pi was 
between 3.1408 and 3.1429 - today we accept it as 3.1416.

Eventually Archimedes refined his methods so he could determine the area of any surface or the volume of 
any solid. An historian three centuries later said of his work: "It's not possible to find in all 
geometry more difficult and intricate questions, or more simple and lucid explanations." 

It was Sir Isaac Newton who later invented calculus itself, but many say Archimedes would have done it 
first, if only he had a decent system of mathematical symbols with which to work. Nevertheless, it was 
Archimedes who elevated mathematics from a mental exercise to a practical tool for solving problems. And 
use it he did.

Archimedes put his mathematical methods to work to solve many of the problems engineers still face today 
- such as the effects of gravity and weight distribution on buildings, dams, bridges and other 
structures. He also used it to try and determine the size of the universe; a figure needed to complete 
his famous mathematical exercise of how many grains of sand would fit in the universe. The ancient 
Greeks believed the universe did have a finite size - to them the sun and the stars rode on a giant 
sphere that enclosed that rest of the universe. So all Archimedes needed to do was figure out the 
distance to the sun, and to do that he needed to know the size of the sun. This led him on a series of 
mathematical calculations that proved inaccurate but were sound in method. He had no instruments and no 
previous research to call on and ended up deciding the sun was five times farther away than it actually 
is. He also concluded it was 30 times as large as the earth when in fact its 109 times larger. But the 
math he used to reach these figures was far more advanced than any used before him. His methods were 
recorded in his paper called Sand Reckoner, which historians believe was later studied by Copernicus.

Archimedes also pioneered the use of mathematics to plot the movement of planets and satellites in their 
journeys through space. He made a planetarium in which he represented the sun, the earth, the moon, and 
the five planets then known (Saturn, Jupiter, Mars, Venus, and Mercury) and arranged them so a person 
could turn a crank and set them in motion, all at differing speeds and directions.

One can't help wonder how mathematics and technology might have progressed, had the work of Archimedes 
not been buried in dust for fifteen hundred years. By the time it was discovered, many of the truths for 
which he so tirelessly labored, had been re-learned by others who followed him. Still he was an 
inspiration to many of the greats who came later, including Newton and Galileo, both of whom turned to 
him for guidance in mathematics and physics. Once Galileo, in answering an argument against his law for 
the path of projectiles, said in rebuttal: "The authority of Archimedes alone will satisfy everybody." 
The methods Archimedes discovered, at the very dawn of science, pointed the way to the advances of 
today, for they were modern in every basic principle. Archimedes did move the world after all, not with 
a pulley and a rope, but with ideas.