Archimedes :: The Man Behind The Legend

Imagine being brought into the world at a time when Alexander the Great has just conquered Egypt. He dies and is replaced by King Ptolemy I who establishes the Library at Alexandria – the first intellectual Mecca to EVER be established in the history of mankind. This same King hires Euclid to compile all of the known mathematics in the world, which has recently been published. You are born to an astronomer anyway, and at the age of 4 you are sent from Syracuse to this great city of knowledge, Alexandria, to study for twelve years under many teachers, not the least of whom may have been Euclid himself! While you are there you get to see the Pyramids, Sphinx, the annual flooding of the Nile, the irrigation canals, and ALL of the splendor that was the height of, arguably, the greatest civilization to ever inhabit the Earth. He would have seen all of the mechanics of the wine and olive presses, the methods of leverage used by the Egyptians to construct such monumental pyramids, and learned all of the other royal subjects of philosophy, astronomy, and logic.

This was the childhood of The Greatest Mathematician of Antiquity. It is no wonder that Archimedes would advance mathematics so far that he would nearly invent Calculus within one generation of Euclid! In his lifetime he would:

• witness three full-scale wars involving the surrounding cities and provinces of the Mediterranean,
  
• be appointed as chief military advisor to King Hiero II,
  
• discover the laws of displacement,
  
• hack known mathematics to mimic calculus – thus greatly improving mankind’s approximation of pi after 4000 years,
  
• discover the exact area of a circle,
  
• the exact surface areas and volumes of the sphere and cylinder,
  
• discover mathematical representations for buoyancy and the properties of floating objects,
  
• create a mechanical way for moving water uphill,
  
• and devise countless war machines that would protect Syracuse for THREE years during the Roman siege in 214.
  

He would witness the death of his beloved King and was entrusted with the protection of all that was Syracuse. His legacy serves as a living testimony to the many joys, accomplishments, and rewards of a life spent dedicated to mathematics.

Archimedes lived to the ripe old age of 87, and was deeply respected by even his enemies, for they knew of his notorious abilities that absolutely annihilated enemy troops that would breech the peace at Syracuse. The ancient commentator, Plutarch, once wrote:

“In the meantime huge poles thrust out from the walls over the ships and sunk some by great weights which they let down from on high upon them; others they lifted up into the air by an iron hand or beak like a crane’s beak and, when they had drawn them up by the prow, and set them on end upon the poop, they plunged them to the bottom of the sea; or else the ships, drawn by engines within, and whirled about, were dashed against steep rocks that stood jutting out under the walls, with great destruction of the soldiers that were aboard them. A ship was frequently lifted up to a great height in the air … and was rolled to and fro, and kept swinging, until the mariners were all thrown out, when at length it was dashed against the rocks, or let fall.”

What was he like?

We don’t know a great deal about the type of person that he was in character. We can only infer the types of qualities that those of great intellectual strength tend to have and put it into historical perspective.

Syracuse was founded 8th century BC as an annex of Corinth at the epicenter of the Mediterranean. Because of its strategic position on the South-Eastern tip of the province of Sicily, it became a great city of trade and commerce, and a constant target for opposing forces. From the town’s inception until the rule of King Hiero II in 3rd century BC, constant turmoil swept the province as Rome constantly sought control of the strategic port. In 263 BC, Hiero II secured a peace treaty with Rome that lasted until his death in 215 BC. It was not until this time that the Great Mathematician of Antiquity came on the scene. Archimedes enjoyed peace and prosperity in his hometown for most of his life and this probably allowed him much of the time he needed to develop his discoveries.

Syracuse sits on the Southeast tip of Sicily

Sicily sits at the epicenter of the Mediterranean

Archimedes probably studied at the Library of Alexandria, Egypt where nearly all of the world’s written knowledge existed at that time. Egypt boasted the great Pyramids of Giza, the Sphinx, great architectural structures, irrigation canals, and a host of other technologically advanced mechanical applications from which Archimedes could observe and contemplate. Alexander the Great had recently (100 years since) conquered the Persians and mandated that a great learning center be constructed with the charge of amassing everything known to the Eastern and Western worlds.

We know that Archimedes was extremely dedicated to his craft and reaped great rewards because of it. Archimedes was, himself, very wealthy and likely wanted for nothing – for most certainly anyone with such critical responsibilities would have been very highly compensated. He probably woke up very early and went to bed later in the evening while taking periodic naps during the day as needed – just like many of our contemporary thinkers do.

It is said that, “Small minds talk about people, average minds talk about events, and great minds speak of ideas.” Because of this, Archimedes likely did not hold a great many true friends, but something more akin to a small band of intellectual thinkers with which he could leverage his own knowledge and bounce ideas off of (Napoleon Hill would call this a “mastermind group”). He probably shunned idle chatter, exhibited a great deal of sobriety, and spent a large portion of his time alone. To accomplish so much in one lifetime he must have been excellent in practicing time management, and likely was in great physical shape since it takes great strength and stamina to work upon engines and stay energized, let alone to live to 87.


What did he teach us?

The story of Archimedes is the testimony of a life spent dedicated to excellence, achievement, and innovation. Before his time there were no textbooks to be read on mechanics (though one existed for geometry), no internet how-to videos for fabricating a home-made potato bazooka, or electric power as we know it today. Archimedes lived by imagination – tying the abstract world of pressures, forces, and mathematics to the physical world of mechanics, pulleys, and levers. He brought us the great joy and deep satisfaction in understanding a quantified world. He brought enjoyment and application back to a study that, historically, has been dull, boring, and uneventful.

-Archimedes taught us that to truly accomplish anything we must immerse our minds in continuous thought. Plutarch wrote, “Often times Archimedes’ servants got him against his will to the baths, to wash and anoint him, and yet being there, he would ever be drawing out of the geometrical figures, even in the very embers of the chimney. and while they were anointing of him with oils and sweet savors, with his fingers he drew lines upon his naked body, so far was he taken from himself, and brought into ecstasy or trance, with the delight he had in the study of geometry.”

How many people ever think to apply something like this? An example might be if someone had the desire to learn Spanish. What is the best way to do this? Well, certainly the path most traveled to this end is going to college, spending $50,000 and four years of time, and then walking away with a degree, which may or may not mean you are completely fluent in a language. And anyway, while in college you are still speaking English – this is not immersion. Archimedes might not think too highly of this path for a number of reasons:

1.) In the first place, it takes a long time
2.) Secondly, that is a lot of money just to learn something
3.) Archimedes didn’t need the approval of others to value himself

Instead, he might have taken $5,000 down to Mexico, moved to a village where nobody spoke English, and lived life for six months; being sure to read their newspapers, listen to their music, and actively absorb the culture. He might have even found himself a Spanish Philly to date and learn from while he was there (of course, being entirely respectful). That is what it is like to be immersed in something.

-See with the mind … be creative to the point of invention. He often solved proofs through the “method of exhaustion”, sometimes referred the “indirect passage to the limit.” In this way, he would inexorably seek greater and greater accuracy in his work, being sure to think of every single detail. To him, being close enough wasn’t good enough. Before his time the closest estimation of pi was calculated by the Egyptians 4,000 years previously by estimating the area of a circle to be approximately equal to the area of a square whose side is eight ninths the length of the diameter. Archimedes needed greater accuracy so that he could be confident that all of the mathematics he spent so much time developing would be the absolute best it could possibly be. As a result, he didn’t just get it close – he nailed it.

By computing the area of a circle using this method
we get an estimation of pi that is a half of a percent off of its actual value.

The way that he did this was to cut the circle into an infinite number of slices and alternately align them tip to crust. The picture below only cuts the circle into eight pieces so it may be difficult to immediately discern the logic used to show this, but of course, as the circle is cut into more and more slices (to the point of infinity), the bumpiness of the line becomes perfectly straight. This concept became the birth of the infinitesimal, a numerical amount so incredibly small you literally cannot conceive of it’s value.

” … for certain things first became clear to me by a mechanical method,
although they had to be demonstrated by geometry afterwards
because their investigation by the said method did not furnish an actual demonstration.”
-Archimedes

**It is clear from this that, though he did not have the actual mathematics available to him, he trusted his gut that he was correct.**

- Archimedes accepted nothing less than perfection. It is said that if you are willing to accept anything less than the best, you often get it. We know that Archimedes used the method of exhaustion to check his work. In this way he would inscribe and circumscribe a polygon with n-sides about a circle and average the two together (below). By doing this, he certainly must have seen that, as he added additional side lengths to each polygon, each one began to look more and more like a circle. If he could add an infinite number of sides to the polygon, it would literally BE the circle. This realization that the number of sides could be, theoretically, infinite would create inconceivably small slices of the circle that could be reconstructed in such a way as to become the area of a rectangle. Archimedes pushed the boundaries of infinity to reach perfection.

This understanding of minuscule values is what led, ultimately, to the development of calculus. In fact, Archimedes probably would have invented Calculus himself if there had been ANY ground work for him to work from at all. On this topic however, there were no textbooks, no nomenclature (that is, existing symbolic representations he could use as shorthand), no class he could take at the local community college.

It is amazing to think of the fact that tens of thousands of students, every year, take and understand calculus before the age of 20 – a subject that Archimedes, himself, would never know, even at 87. It wouldn’t be for another 1800 years when Sir Isaac Newton and Wilhelm Leibniz would independently discover calculus (from the integral and derivative perspectives, respectively) that humanity would have mathematical methods to solve these equations. Nevertheless, he must have known he was incredibly close to developing the idea himself, and were his life extended an additional fifty years who knows what more would have been discovered.

– Archimedes must have taken personal responsibility for all things related to the protection of his city. At some point in every champion’s life there is a realization that success is up to the individual. Surely Archimedes realized that he, alone, had the mental capacity and ability to use known physics and mathematics to protect the city. Everything and everyone that he held so dear – his family, his friends, his home – counted on him to ensure their safety. After all, if he didn’t devise a new way to protect the harbor, if he couldn’t figure out how to leverage the forces that be to his advantage, if he didn’t develop and improve new and existing war machines, then the Roman Army, which beat at his doorstep, would sweep through and sack the city.

Who else could he or anyone else trust to oversee this task? Absolutely nobody, and that is why King Hiero II of Syracuse named Archimedes, a mathematician, as chief military advisor of the entire city! Is it really any wonder, then, how he could get so lost in his work that he forgot to even eat?


– Archimedes’ greatest lesson taught us to do more with less. Archimedes was clearly a holistic thinker. He made sure to see with his mind and not with his eyes – seeing machines before their manifestation, and he taught us to work smarter instead of harder.

In Stephen Hawking’s book, God Created The Integers he mentions how: