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Aarish Shahab Archimedes’ life

Much of what we know about Archimedes is minuscule when compared to the amount we do not know. Living in the times of the Ancient Greeks, Archimedes was born in Syracuse, in 287 B.C. His father, Phidias was a man of great wealth during this era and self-proclaimed astronomer. He devoted his wealth and experience to ensure that his son received the most superior education of the time. Archimedes indeed grew up watching his father use calculations in order to explain the phenomena of the world, and used it as inspiration to become a and philosopher; however, much is unknown regarding the details of Phidias and the rest of Archimedes’ relatives. In fact, the concrete information we do know are only revealed in one of Archimedes’ works: The Sandreckoner. Archimedes’ friend, Her- acleides, once wrote a biography of him, however, it has truly been lost in history. Much of Archimedes’ inventions are told by the perspective of men like Plutarch and Livy, as they reveal the context in which he created each of these inventions, such as during times of warfare.

That being said, we glean clues of Archimedes’ life from the context of his environment. For instance, some proponents argue that Archimedes’ wealth was due to his relations with the King of Syracuse at the time, King Hiero II. This explains why Archimedes was granted the resources and luxury to explore mathematics to its very core, and reveal inventions that are still regarded as truly spectacular.

Syracuse, Sicily was a bustling city full of commerce, art, and science while Archimedes was growing up; however, its bustling economy was one of the primary reasons why other nations attacked it, wishing to claim its riches. King Hiero II spearheaded the revolt against the Mamertines, send- ing Archimedes away to , Egypt to further his studies. At the time, Alexandria had already earned a reputation for great learning and scholarship, a beautiful complement to Archimedes’ evident natural curios- ity and creative problem solving.

Archimedes was extremely keen to travel to Alexandria due to his fascination with one of the most renowned of the time, . This well-known scholar was credited for compiling a book that catalogued all existing Greek geometrical - a framework that forever changed the study of and influenced Archimedes’ later works. Archimedes thus happily traveled to Alexandria to study under the very teachers who tutored Euclid, hoping to leave an impact just as great, if not greater than Euclid himself.

In Alexandria, Archimedes corresponded with other scholars such as Conon of and of Cyrene. It is with these principal mathemati- cians of the time that he published many of his works.

Archimedes, equipped with newfound knowledge and intuition regarding how the world worked, travelled back to Syracuse, Sicily even though his hometown was being ravaged by warfare. It is there where he aided in the war effort, defending Syracuse against the Roman Siege of 213 BC, and invented ingenious solutions to the King’s various problems.

Indeed, Archimedes proved himself to be not only Sicily’s, but also human- ity’s greatest asset in the growth of mathematics.

2 Archimedes’ mathematical works

Archimedes’ brilliance cannot be contained to just his era, as his inventions are still being used thousands of years later. For instance, one of Archimedes’ most famous moments was where he coined the comical phrase ’Eureka!’

In this account, King Henry II had told his goldsmith that he required a new royal crown made of solid gold. The pure amount of gold in one setting tempted the goldsmith to cunningly steal some of it. He only used a small portion of the gold while making the crown and mixed it with enough silver in order to make it seem like the weight was the same as if it was all 100 percent gold. However, King Hiero II suspected that something was wrong with his crown and ordered Archimedes to intuitively discover a method to test his theory without damaging the crown in any way.

Archimedes remained stumped on this issue, until one day he decided to take a bubble bath. As he settled himself in the bath, he noticed that some of the water fell off the sides of the tub. Suddenly, he realized that he could measure the crown’s volume by measuring the amount of water that was displaced. With the volume in mind, all he had to do was find the weight of the crown and divide it by its volume in order to determine the crown’s unique density. If the density (a measure of the purity of the crown) was found to be different than pure gold, then the goldsmith would be held accountable. In this moment of celebrated discovery, Archimedes shouted ’Eureka!’ and journeyed across the streets of Syracuse to inform the king.

Today, this method is called ’Water Displacement’ and is a useful tool for mathematicians and chemists who need to measure using indirect methods.

For practical purposes, water is incompressible, so Archimedes’ assumption was that the crown submerged would displace the same amount of water as its volume. However, this story is not present in Archimedes’ works. In fact, the measurements regarding this Water Displacement method would have to be done very precisely in that even a small error would lead to an incorrect density and therefore a blame on an innocent goldsmith.

It is thus proposed that Archimedes may have used a different analysis to prove that the goldsmith was cunning: The Archimedes Principle. This theory involved the use of and is further explained in his work ’.’ This principle states that a body immersed in fluid

3 is subject to a buoyant force that is equal to the weight of the fluid that is displaced. This law could then be tested via the use of an apparatus that consisted of balancing the crown on a scale with the gold reference sample and placing this apparatus in water. If the crown was truly made of pure gold, then the scale would remain suspended equally, meaning that the wreath and gold would have the same volume and thus the same density. However, if the scale instead tips in favor of the side with gold, than the crown has a greater volume, a lower density, and is not pure.

This Law, though needing much more materials than regular Wa- ter Displacement, is indeed more accurate and therefore is more likely to have been used by Archimedes.

F is the buoyant force p is the density of displaced fluid

V is the volume of displaced fluid g is the acceleration due to gravity

g = 9.8 in meters per second squared

F = p × g

Note: The density and volume in this equation is referencing the displaced fluid and not the object that is submerged. For instance, if an object was submerged in water and displaces 1L of water, where 1 kg/l is the density, then the equation can be used to determine the buoyant force on the object since both the volume and density of the displaced fluid are known.

Plugging this in:

F = 1 × 1 × 9.8

Thus, if the weight of the object is more than 9.8N, then the object will

4 sink. If it is less than 9.8N, then the object will float. But if the weight is exactly 9.8N, then the object will neither sink nor float.

Another story regarding King Hiero II and Alexander detailed a problem with one of Hiero II’s ship designs. Although the ship design of the Greeks was superior, rainfall posed a clear and burdensome obstacle, as the rain- water would fill the hull of the ship. In order to emerge victorious in naval battle, Hiero called upon Archimedes to remedy the situation with another act of genius. Archimedes went on to create a machine consisting of a hollow tube with a that could easily be turned at one end. When it was set into the hull, the turning motion carried water up the tube out of the hull, and forever revolutionized the economy.

Today, this ingenious invention is called ’The Archimedes Screw’ and is a widely used irrigation technique in developing countries.

Another invention by Archimedes created in order to defend Syracuse was ’The Claw of Archimedes.’The weapon consisted of a grappling hook at the end of a long crane that would rip into nearby ships. The upward ’swing’ motion would lift the ship upwards and damage it, if not sink it. The realism of this device has been called into question, and only in 2005 was it declared that it truly functioned.

Another one of Archimedes’ widely debated inventions was his ’Archimedes Heat Ray.’ It is said that Archimedes used heat reflected from mirrors to burn ships attacking Syracuse. This device used the principle of a parabolic reflector, similar to the dynamics of a solar furnace. However, in 2005 stu- dents from MIT replicated this finding and determined that if certain con- ditions are met, the ship’s wood could indeed reach its melting point of 300 degrees Celsius, thus technically setting the ship on fire.

Archimedes other works were more varied. While he was theorizing about levers, he exclaimed ’give me a place to stand on, and I will move the .’He used combinations of levers, block-and-tackle pulleys, catapults, and odometers to aid Syracuse during the First Punic War. By all defini- tions, Archimedes was indeed a soldier of Syracuse and an essential asset to its military.

Archimedes works in Mathematics were more theoretical than practical, as opposed to his inventions. It is in Mathematics that Archimedes could ex-

5 plore his suspicions regarding how the world worked. One of his preferred problem solving techniques was proof by contradiction, bounding his argu- ments by limitations if they existed. ’The ’ used this principle, as he applied the theory to approximate the .

Further explained in his published work ’Measurement of a ,’ Archimedes graphically represented his findings by drawing a small hexagon within a cir- cle and a bigger hexagon outside of the circle. Each consecutive hexagon had double the sides of the previous one. As he measured the lengths of each of the polygons, he approximated Pi to be between 22/7 and 223/71.

His proof for this is as follows with respect to a circle of r: b1 represents an inscribed hexagon with perimeter p1 and a1

B1 represents an circumscribed hexagon with perimeter P1 and A1 bn represents the regular hexagon inscribed 6 · 2n polygons

Bn represents the number of circumscribed polygons.

P n + 1 = (2pn × P n) ÷ (pn + P n)

pn + 1 = ppn × (P n + 1)

√ (an + 1) = an × An

(An + 1) = (2 × (an + 1) × An) ÷ ((an + 1) + An) by using polygons up to 96 sides, he derived that the ratio of a circle’s to its diameter is between 22/7 and 223/71

In his ’,’ Archimedes attempted to revaluate and fix the problems with the established Greek number system; the numerical system at the time lacked the capacity to express large numbers, and there- fore did not allow mathematicians to contemplate ideas that exceeded the

6 Babylonian base 60 system. Archimedes thus devised a base 100,000,000 system (essentially a place-value system of notation), allowing him to finally use numbers up to such as:

8 × 1063 in modern notation.

It is in this book that Archimedes argues that this number is large enough to count the number of grains of sand that can fill the entire - a quantity that was thought to reach infinity. However, Archimedes cited the work of , saying that Aristarchus hypothesized that the earth revolves around the , while the sun and other stars remain stationary. Archimedes interpreted his findings to mean that the ratio of the earth’s size to the universe’s size is alike to the orbit of the earth compared to the of immovable stars in the universe. He came up with this number:

8 × 1063 by saying that it is ”one thousand myriads of eight numbers . . . and is therefore obvious that the number of grains of sand filling a sphere of fixed stars is less than one thousand myriad myriad of eighth numbers.” This the- ory was ahead of its time, as Archimedes was employing the understanding which modern astronomers use to conceptualize the current universe.

In his other book, ’On the Sphere and Cylinder,’ Archimedes proved that Pi multiplied by a circle’s radius squared was equivalent to its . In this work, he also postulated that ’any magnitude when added to itself enough times will exceed any given magnitude. Today, this is coined as the ’Archimedean Property of Real Numbers.’ In this volume Archimedes also discovered that the volume of the sphere is 2/3 the volume of the circum- scribed cylinder. In modern notation:

2 V = × V a 3

7 Where V is denoted as the volume of the sphere and Va is denoted as the volume of the circumscribed cylinder. From this he arrived at numerous propositions that we are essential to our understanding of practical geomet- ric shapes. For instance his Proposition 33 dictates that the surface of any sphere is equivalent to four times the greatest circle on it. To parallel this thought with , his Proposition 34 dictated that ’Any sphere is four times the which has its base equal to the greatest circle of the sphere and height equivalent to the sphere’s radius.’

Archimedes’ other mathematical achievements include closely approximat- ing the of 3, and doing so without revealing how he approx- imated the value. In his book ’Quadrature of the ,’ Archimedes proved than the area enclosed by a line segment intersecting a parabola is 4/3 the area of an inscribed triangle within that same space.

Although mentioned above, Archimedes’ other works include ’On the Equi- librium of Planes,’ ’On the , ’On ,’ ’On the Sphere and the Cylinder,’ ’,’ ’On Floating Bod- ies,’ ’The Quadrature of the Parabola,’ ’,’ ’The Sand Reckoner,’ and ’The Method of Mechanical Theorems.’

Collaboration with other scholars

As stated before, much of what we know of Archimedes is due to his in- teractions with the people in his travels. Often, he tells us about his man- nerisms with others in his published books. For instance, in ’,’ Archimedes would send his friends in Alexandria theories that were plagu- ing his mind while withholding the proofs. These theorems are still trying to be solved today by the world’s greatest mathematicians, as it is unclear whether Archimedes secretly proved them or not. His justification for doing this was ’so that those who claim to discover everything, but produce no proofs of the same, may be confuted as having pretended to discover the impossible.’

Archimedes often published his compositions in correspondence with two well known mathematicians: and Eratosthenes of Cyrene. Conon was an astronomer and a dear friend of Archimedes that he met around 245 BC. It is only with his assistance that Archimedes could dis- cover what is now known as ’The Spiral of Archimedes.’ Conon also worked

8 diligently on the subject of conic sections, which contributed to Archimedes’ theories regarding partial cones in the real world.

More heavily so than with Conon, Archimedes drew a much greater influence from Eratosthenes of Cyrene. Eratosthenes was also from Alexandria and is credited to truly lay the foundations of modern day geography. Archimedes drew a lot of his concepts from his work which is dated around 220 BC; these included Eratosthenes’ invention of the , his calculation of Earth’s circumference, and his measurement of the of elevation of the Sun at noon on the Summer among others. was amazed by Eratosthenes’ fervor and passion for mathematics that he dedicated his book ’The Method of Mechanical Theorems’ to Eratosthenes.

Historical events that marked Archimedes’ life.

When Archimedes was born in Syracuse, Sicily in 287 BC, the city was a self-governed colony in Magna Graecia. Since Archimedes’ conception, the area had constantly been pillaged by war. Much of the history of Syracuse can be gleaned from following the adventures of Hiero II.

Hiero was born in 306 BC and served as the co-commander of the military. He was a man of great stature and authority; his reputation preceded him among his fellow citizens. He used the power of his reputation and his family connections to launch a successful military coup in 275 BC. In order to consolidate his power over the region, he married the daughter of a powerful Syracusan and became one of the people rather than an unwanted outsider. People began to welcome him and peace ensued; however, the mercenaries whom he hired in the military coup acted out in these times of peace, causing Hiero to resort to ruthless measures in order to uphold the order. He led them into battle while withholding the reserves, silencing the mercenaries once and for all.

This was not the only time mercenaries troubled the land of Syracuse. In 265 BC, the Mamertines, a group of Italian pirates who had already taken the Sicilian city of Messana, attempted to also take over Syracuse. Fortunately, this was attempted when Hiero’s power was fully consolidated, so the victory was overwhelming. From then on, Hiero had sealed the loyalty of his citizens.

However, the victory was short lived, as it upset the balance of power among

9 The Greeks, Romans, and Carthaginians. It instigated the First Punic War in 264 BC, a time in which Syracuse was in great danger as all three powers wanted to control the majority of Sicily. Because the Mamertines allied with the Romans, Hiero had to side with Carthage. However, the might of Rome proved to be too fierce; as Carthage’s forces began to wane, Hiero quickly rethought his position and attempted to side with Rome instead. Thus, in 263 BC, Syracuse and Rome signed a treaty in which Syracuse would aid Rome with additional food and supplies in its fight against Carthage. It was agreed, thus, that as long as Rome and Carthage were in battle, the safety of Syracuse was guaranteed.

The Battle of Agrigentum marked the strength of the alliance between Rome and Syracuse. As the Romans were attempting to besiege Agrigentum, Carthage completely demolished the supplies coming from Syracuse to the Roman army; however, the Romans relied on brute force to overrun Agri- gentum, eventually destroying all of the city’s defenses. As the Carthage forces retreated, the Romans sacked the city and established their superior might. The remainder of the war followed a remarkable pattern in terms of Syracuse’s role; Rome spearheaded the war effort with Syracuse offering ad- ditional military and aid, eventually crushing Carthage and earning Prince Hiero a peaceful reign.

Although Syracuse emerged from the First Punic War relatively unscathed, the Second Punic War proved to be even more tumultuous. Instigated by Carthage’s King , this war was declared against Rome; however, this time, Syracuse’s newly appointed King Hieronymous revolted against Hiero II’s pro-Roman ideology and thus, provided military aid to Carthage rather than Rome, as he hoped to end Roman political control in the region. Hieronymous’ ambitions were mighty; however, his military was no match for Rome’s superior forces, eventually leading to the Siege of Syracuse from 214 BC onwards.

Syracuse, due to its superior location, was responsible for securing the sea routes for supply. The intense sea battle between Rome and Syracuse was credited to Archimedes’ ingenuity, as it was his inventions that subdued all Roman advances; however, Rome strategically went on to destroy all other battlefield, attempting to destroy Syracuse’s support. After the majority of the major and minor cities fell to the Romans, Hieronymous’ forces proved incapable of stopping the Rome’s attacks, even with the assistance of 20,000 Carthaginians. This fatal battle truly marked the downfall of Carthage, the

10 end of Syracuse, and the death of Archimedes.

The story of Archimedes’ death was deemed an unfortunate occurrence universally - even by the Romans. The inventions that Archimedes cre- ated reaped international terror, eventually forcing Roman General Marcus Claudius Marcellus to give up his frontal assault and instead pray for the success of a long siege. In 212 BC, the general did just that; during this time, Archimedes was reportedly intent in mind and body working on a diagram regarding , and thereby did not realize that Romans had invaded the city.

It is said that during this time, a soldier unexpectedly came upon Archimedes and ordered that he come with him to the Roman base. However, Archimedes refused to do so until he solved the puzzle he was working on. The soldier, clearly angered at his defiance, went against his orders to take Archimedes back alive, and used this opportunity to run his sword through Archimedes. Roman general Marcellus was thus angered and saddened by Archimedes’ death as he realized that Archimedes was not just an asset for Syracuse, but an asset for the entire world. He thus made sure to hold a special burial for Archimedes.

Significant historical events around the world during Archimedes’ life

In the 3rd Century, there was much going on around the world outside of the world of Archimedes and Syracuse. In India, one of the greatest emperors came to power: Ashoka the Great. His reign included everything from the Hindu Kush mountains to the Bengals in the East. This man ruled with an iron fist, ruthlessly owning a torture chamber built called ’Ashoka’s Hell.’ After he witnessed the mass casualties in the Kalinga War, Ashoka converted to Buddhism and preached the peaceful teachings of Gautama Buddha.

This century marked the end of the Warring States Period in China, as it marked a great political shift in Chinese History. Qin Shi Huang rose to power, becoming the first emperor of the Qin Dynasty. One of his major accomplishments was that he greatly expanded and consolidated the Chi- nese State; he added territories including the Yue lands of Hunan and the Ordos Loop. He worked with his ministers to execute both economic and political reforms in order to centralize his domain. In order to truly unify his kingdom, he employed means of censorship by banning books to ordinary

11 laymen and by executing dissidents. In an effort to ensure that his state was truly organized and safe, he had his citizens laboriously construct the Great Wall of China. It is said that the conditions for building it were so harsh that the wall includes bodies of workers who died from pure exhaustion.

Although the Second Punic War greatly affected Syracuse, the results of this war forever changed the dynamics of the world. The final battle culminated in the Battle of Zama; it is in this distinct battle that neither the Carthagini- ans nor the Romans had the upper hand. Indeed, The Carthaginians had a superior infantry while the Romans had superior military and battle tech- niques; however, King Hannibal’s lack of confidence is thought to be the root of the defeat. He expected his army to truly be crushed by Rome’s fight, and thereby initially refused to lead his troops onto the battlefield. This lack of morale ultimately gave the Romans the advantage to win, even when the Carthaginians used mighty elephants. By Hannibal’s urging, he surren- dered to the Romans to construct a peace treaty that established Rome’s worldwide dominance.

At the end of the war, Rome stripped Carthage of territories including the large land, Hispania. Carthage could no longer raise a military without the permission of Rome, and had its navy severely limited. Rome also forced Carthage to pay the equivalent of 660,000 pounds, further weakening Carthage’s economy. Carthage, evidently weakened and vulnerable to at- tack, was pillaged by the Numidians. It is this turn of events that instigated the Third Punic War half a century later.

Truly, it is evident that the world that Archimedes was born into was not the world that arose from his death.

Significant mathematical progress during the Archimedes’ lifetime

Other than Archimedes, the only major mathematician of the time was Eu- clid of Alexandria. He is regarded as the patriarch of Geometry, as his book, ’Elements,’ taught the world mathematics for the next seventeen centuries. This book clearly organized mathematical concepts by using a unique se- quence of proofs; it is this method of proving concepts that is still primarily used in modern mathematics. Although the book primarily conveyed the basics of , it also contained the foundation of number theory. It is in this field that he devised methods for factorization, basic

12 arithmetic, the greatest common divisor, and much more.

One of example of Euclid’s proofs includes his proof that there are infinitely number prime numbers; however, his proof is different form the ones mod- ern mathematicians use as his is using specific proofs as opposed to purely general ones. His proof is as follows:

Let A, B and C be prime numbers. Let there be a least number DE measured by A, B, and C and add the unit DF to DE. Then, the number EF has two possibilities: to be prime or not.

First, let EF be prime: then, the prime numbers A, B, and C, and EF have been found which are greater than A,B, and C.

Now, let EF not be prime: then, EF is measured by some prime number that is denoted as G (a prime number that is not measured by A, B, or C). Thus, A, B, C, and G all measure DE; but, G also measures EF. This means that G measures the remainder DF, and is therefore not the same as any of the numbers A, B, and C.

A, B, C, and G have been found that are greater than the assigned multitude of A, B and C. Therefore, prime numbers are more than any assigned group of numbers.

Euclid also came up with a method that is coined as ’Euclidean Division,’ - a computation process that allows one to rewrite a number as an integer combination of different numbers. His proof states that if ’a’ is an inte- ger number, and ’n’is a natural number, then there exists unique integer numbers ’u’ and ’r’ - where r lies between 0 and n, inclusive.

Thus, it can be said that:

a = nu + r

He went farther to state that if another integer number ’b’ is divided by the same number ’n’and has the same remainder ’r’ as the integer number ’a’ divided by ’n’, then:

13 ’a’ is congruent to ’b’ modulo ’n’.

Another work by Euclid includes his book ’On Divisions of Figures,’ which details that geometric figures, when taken apart, yield certain ratios. His book ’,’ involved the properties of mirrors. The bent nature of mirrors and its effect on light are thought to have influenced Archimedes to use mirrors to burn Roman ships during the Punic Wars. Further adding on to the idea of bent rays, Euclid cast most of his attention on the eye, theorizing that the spherical nature of the eye is caused by discrete rays that project from it. His exact rationale constituted that ’things seen under a greater angle appear greater, and those under a lesser angle less, while those under equal appear equal.’

Euclid’s theories have ultimately laid the foundation for centuries to come.

Connections between history and the development of mathematics

As seen throughout history, the huge advancements in mathematics were discovered in order to solve problems that were plaguing the real world. It is only due to Hiero’s need of a hands-off procedure to test the purity of his crown that Archimedes developed the Water Displacement Method. It is only because of Hiero’s need for a solution regarding water in his boat’s hull that Archimedes invented ’The Archimedes’ Screw’. It is only by Hiero’s persistence of needing a superior naval offensive that Archimedes developed ’The Archimedes Claw’ to sink incoming Roman ships. Indeed, history’s need for mathematics for survival and prosperity is credited for Archimedes’ brilliance.

Remarks

Though Archimedes is widely lauded in the modern era for his many contri- butions to mathematical thinking (both applied and pure), his work created very little impact during the ancient times. This could be explained my a number of causes, such as that the Romans had little interest in purely the- oretical mathematics and solely focused on the practical side. Yes, it can be said that the numerical approximation of Pi, or more thorough understand- ing regarding the geometry of shapes were used in all aspects of working life; however, his theories of hydrostatics and quadrature were never pur-

14 sued with the same fervor as he showed.

The true magnitude of his influence became prevalent in the 8th and 9th cen- turies with the Arab civilization, as some Arab contributions complimented where Archimedes had stopped working on his proofs; however, Archimedes received the greatest amount of recognition in the 16th and 17th centuries, where the texts printed by the Greeks passed all over the world. This led to the rise of other well known physicists and mathematicians: Kepler and Galileo. They used Archimedes’ thought processes regarding basic geomet- ric shapes and applied them to the field of astronomy. When Archimedes works were translated into , mathematicians such as Descartes and Fermat began to understand Archimedes’ various principles and use those to create their own theorems regarding the natural world.

The worldwide consensus is that if Archimedes’ contributions had never taken place, then the landscape of the greatest advances of the 16th century would have been incomprehensibly delayed. It’s even harder to imagine how 19th century mathematics would have changed, as the same results would have probably been made later; however, with a mechanical focus rather than a geometric one.

Truly, we are forever in Archimedes’ debt for forever changing the world in which we live in.

References

1. ”The Archimedes .” The History of Archimedes. N.p., n.d. Web. 05 Oct. 2015.

2. ”Archimedes of Syracuse.” Archimedes Biography. N.p., n.d. Web. 05 Oct. 2015.

3. ”Royal Family of Syracuse (Hiero II).” Royal Family of Syracuse (Hiero II). N.p., n.d. Web. 05 Oct. 2015.

4. ”What Did Archimedes Invent.” Historyrocket. N.p., n.d. Web. 05 Oct. 2015.

5. ”Archimedes — Greek Mathematician.” Encyclopedia Britannica Online.

15 Encyclopedia Britannica, n.d. Web. 05 Oct. 2015.

6. ”Archimedes: Early Years and Mathematics.” By Ron Kurtus. N.p., n.d. Web. 05 Oct. 2015.

7. http://www.math.tamu.edu/ dallen/masters/Greek/archimed.pdf

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