(278 B.C.E. - 212 B.C.E.)
"The importance of the role played by Archimedes in the history of science can scarcely be exaggerated. He was emulated and admired in his own day and at successive periods in later times" (Clagett, 1).
"The importance of the role played by Archimedes in the history of science can scarcely be exaggerated. He was emulated and admired in his own day and at successive periods in later times" (Clagett, 1).
During the time period before Archimedes, Aristotle had already effectively drawn a line between philosophy and mathematics. After his date philosophy is carried on without mathematical inspiration. There is an outbreak, known as the Golden Age of Greek mathematics, that just happens to occur in Alexandria during the period 300 to 200 B.C.E.. This period lasted only a short time however because philosophic faith in mathematics gradually disappeared. Philosophers were more inclined to use their intellect to come up with explanations based on simply theoretical assumptions and by faith (Ginsburg, 57). Since philosophy had been around long before mathematics was thought up, it remained the more publicly acceptable way to answer questions. By not linking the two, "they missed a grand opportunity to blow open the secrets of the universe to science, and they bequeathed to posterity, a heavy obstacle to the progress of science as a whole" (Ginsburg, 58). Archimedes works at this time are therefore described as magical and mysterious, rather than explained in the words of a modern day scientist.
"Had he been born in another age... had he come into a different cultural inheritance and had the seeds of his own work fallen upon more favorable soil, he might also have ranked as the greatest of all time" (Ginsburg, 56).
In his time, Archimedes served as a mathematician, physicist, and inventor. Unlike other mathematicians of his day, Archimedes was able to achieve some fame during his lifetime, and many references are made to him in works of this time period. Although, it is important to point out that this acquisition of fame wasn't primarily owed to mathematical or enlightening discoveries, but rather his ability to develop advanced weapons of destruction that were used for warfare. The people of his day were more interested in practical purposes rather than their physical or mathematical implications. As told in a book concerning Greek history "Archimedes emerged as a figure larger than life in the popular imagination, legendary for the seeming miracles he performed through his mechanical inventions" (Brunschwig, 544). Aside from the creation of these weapons Archimedes achieved many advances in the various fields in which he was involved. One example of this is Archimedes' use of an exhaustion method, cutting up shapes into infinitely small pieces to discover their volumes. This method paved the way for what we now call integral calculus, which was later perfected by 16th and 17th century scientists such as Kepler and Newton (Grande, 240). Many of his exploits and achievements have been passed down for generations and oftentimes find themselves being retold in colorful and imaginative stories, some of which are not entirely truthful.
When studying ancient Greek mathematicians "the information available about the lives of those who were engaged in its pursuit is one of the weakest points" (Dijksterhuis, 9). The periods in which they lived and worked are, for the most part, only educated estimations. The story of Archimedes is believed to have began in 278 B.C.E. because of a statement made in a 12th century Byzantine work. It stated that he was 75 years old when he was killed in 212 B.C.E. during the Roman conquest of Syracuse (Dijksterhuis, 10). He grew up in Syracuse, Sicily and was presumably raised by a man named Phidias, an early astronomer. The information about Archimedes' father is also uncertain but is assumed due to Archimedes' own reference of him in his work The Sand-reckoner. From the ancient historians Polybius, Livy, and Plutarch we are told of some connection between Archimedes and the family of King Heiron II of Syracuse, including Gelon his son. Because of this connection Archimedes was called upon by King Heiron to develop weapons and fortification designs that would act as a defense for attacks on Syracuse. His military advancements proved to be very influential in the Second Punic War (215-212 B.C.E.) when Syracuse was attempting to fend off Marcellus, the Roman commander, and his army (Brunschwig, 544).
From the prefaces of his works we can also discover more about Archimedes' life and the people he interacted with. Archimedes is known to have spent some years of his life studying at Alexandria which was then the center of Greek science. Here it is believed that mathematics was placed in very high regard because it was where the famous geometer Euclid had taught. It is interpreted that he always maintained close relationship with a few scholars in Alexandria, especially Conon of Samos and Eratosthenes, to whom, he used to send mathematical discoveries before their publication (Dijksterhuis, 12). Many of Archimedes' works are dedicated to these colleagues. These colleagues were integral in keeping the mathematical and physical explanations of Aristotle alive. They were able to look beyond the narrow-mindedness of the past, just like they had done with Euclid's geometry, and realize important implications his work would have on the future.
To explain who Archimedes was as a person and how he felt about his achievements we can look to the historian Plutarch and his work Life of Marcellus. In this account he describes how Archimedes did not think highly of his technical activity as an occupation:
"he did not deign to leave behind him any written work on such subjects; he regarded as sordid and ignoble the construction of instruments, and in general every art directed to use and profit, and he only strove after those things which, in their beauty and excellence, remain beyond all contact with the common needs of life" (Dijksterhuis, 13).
Mathematics, particularly geometry, was where he was believed to have found his true passion and Plutarch explains he was so preoccupied with math that:
"he was continually bewitched by a Siren who always accompanied him, he forgot to nourish himself and omitted to care for his body; and when, as would often happen, he was urged by force to bathe and anoint himself, he would still be drawing geometrical figures in the ashes or with his finger would draw lines on his anointed body, being possessed by a great ecstasy and in truth a thrall to the Muses" (Dijksterhuis, 13).
This passion for mathematics
came at a time when philosophical explanations were more dominant.
Ginsburg, in his book The Adventure of Science believes that perhaps
"the first experimental scientist was so strange a sight in the ancient
world that people gazed at him in wonder but did not think of following
his example" (Ginsburg,
59). Advancements in science after Archimedes death took a while
to start up again. Some historians estimating that a gap of eighteen
hundred years separated the discovery of calculus and its more modern explanation
by Newton and his contemporaries (Ginsburg, 65).
The Surviving Works of Archimedes
Of the many works that came out of Greece at this time, 10 important treatises are attributed to Archimedes. These works are (Cooke, 115):
1. On the Equilibrium
of Planes (Part I)
2. Quadrature of the
Parabola
3. On the Equilibrium
of Planes (Part II)
4. On the Sphere and
the Cylinder (Parts I and II)
5. On Spirals
6. On Conoids and Spheroids
7. On Floating Bodies
8. Measurement of a
Circle
9. The Sand-reckoner
10. The Method
Much detail goes into the development of each of these books. My goal is just to clarify the main points of some of them.
The first work On the Equilibrium of Planes sets the basis for mechanics by using geometrical theory. He discovers fundamental theorems concerning the center of gravity of many plane figures, such as a triangle and a square (http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Archimedes.html).
The Quadrature of the Parabola says that if given 2 areas, some multiple of the first exceeds the second. This theory denies the presence of infinitesimal areas and is proved by both mechanical considerations of balance and the method of exhaustion, which now call calculus (Cooke, 118).
On the Sphere and the Cylinder
is known for stating "the surface of
any sphere is equal to four times the greatest circle in it" and Archimedes
shows that the volume of a sphere is 2:3 the volume of a circumscribed
cylinder (Cooke,
119). He also proves that ratio of the surface area of the sphere
to the surface area of a circumscribed cylinder, including each base, was
also 2:3. Because the volume of a cylinder was known, he was able
to discover the true volume of a sphere. Archimedes regarded this
proof to be such a remarkable achievement that, according to Plutarch:
"His discoveries were numerous and admirable; but he is said to have requested
his friends and relations that, when he was dead, they would place over
his tomb a cylinder containing a sphere, inscribing it with the ratio which
the containing solid bears to the contained" (http://www.mcs.drexel.edu/~crorres/Archimedes/contents.html).
In the treatise, Measurement of a Circle, Archimedes proposes "Every
circle exceeds three times its diameter by an amount less than one seventh
and more than 10 parts of 71 parts of the diameter" (Cooke,
117). A modern day representation of this statement is:
The greek symbol (pie) was not used by Archimedes in his proof but using modern day technology we have calculated this number to many significant figures. We now use this number in numerous geometrical calculations including; the Area of a Circle = 2*pie*radius squared and the Circumference of a Circle = 2*pie*radius.
In On Floating Bodies, Archimedes uses a hands on approach to develop what is probably his most famous theorem. Archimedes' principle, which it is called today, is the foundation for the study of hydrostatics in Physics. This principle states "that a body immersed in a fluid is buoyed up by a force equal to the weight of the displaced fluid" (http://www.encyclopedia.com/articles/00686.html) and "A solid heavier than a fluid will, if placed in it, descend to the bottom of the fluid, and the solid will, when weighed in the fluid, be lighter than its true weight by the weight of the fluid displaced" (Cooke, 146). This principle develops the concept of density, which is the weight per unit volume, and by comparing densities we can ultimately determine the buoyant force. Archimedes will use this principle practically in what is commonly known as the Gold Crown Problem (see Additional Accomplishments).
The Sand-reckoner uses a different approach from the aforementioned treatises. This treatise comes down as a letter to King Gelon, successor to his father King Heiron II. He proceeds to ask the question "Is the number of grains of sand required to fill up the Universe finite or infinite?" (Cooke, 123). To answer this question Archimedes proposes a number system that can symbolize very large numbers. This in itself is a great accomplishment, but he then proceeds to apply this number to the universe and make estimates. The actual implications of this theory started in motion two important mathematical questions (Cooke, 123):
In The Method Archimedes proved two geometric relationships.
First, the volume of a paraboloid of revolution is 1/2 the volume of the
circumscribing cylinder. Secondly, he explained the volume of a sphere
can be related to the volume of a cylinder and a cone, of which he proved
(Grande, 241).
Additional Accomplishments (or Legends) of Archimedes
In addition to his 10 treatises, Archimedes is also well known for certain instances of sheer brilliance. Whether these instances be profound statements, thought out proofs, or well constructed machines Archimedes' mind was always producing original ideas that were groundbreaking at the time. What makes these ideas so special was that as they began to surface many centuries later people started using them as a basis for the mathematics and physics we use today. They developed systems out of his proofs that we now teach in our schools to scholars who hope to uncover the same mysteries Archimedes did.
Archimedes and Simple Machines
The first accomplishment I would like to discuss is Archimedes use of simple machines to solve a problem presented to him by King Heiron II. On this occasion according to Plutarch, Heiron had ordered a ship of amazing weight, 4200 tons, to be constructed and launched towards Egypt as a gift to King Ptolemy. The main problem was that the ship was so large they could not launch it, even with mass amounts of human strength. Finally, Archimedes solved this problem by setting up simple machines, such as pulleys, and using them in unison to create enough force to move the ship. Adding to the amazement, he is said to have boasted (Dijksterhuis, 15):
Archimedes and the Golden Crown
Another great accomplishment that is also thought to have come about upon request of King Heiron is Archimedes' hydrostatic principle (see above). As we learn from an account retold by an architect Vitruvius in the preface to Book IX of De Architectura. It is believed that King Heiron had given an exact amount of gold to a craftsman who was to create an all gold crown equaling the exact weight given. First of all, I would like to correct the common misconception that what Vitruvius meant by gold crown in his book actually stands for a gold wreath, that was to be dedicated to the immortal gods (http://www.mcs.drexel.edu/~crorres/Archimedes/contents.html). After the wreath was made, it was believed that some of the gold going into the making of the wreath was replaced by a quantity of silver with the same weight. The craftsman would not confess to the implication, and the wreath could not be melted down because it was a gift for the gods so the problem was thrown into the hands of Archimedes.
As legend has it, Archimedes was bathing in a public bath when he made a shocking discovery. He suddenly realized that the deeper he went into the bath the amount of water that overflowed the tub was related to the amount of his body that was submerged. He is said to have been so excited that he immediately jumped out of the water, not even pausing to put his clothes on, and ran home naked through the streets of Syracuse yelling "Eureka! Eureka!" (which translates to I have found it!) (http://www.treasure-troves.com/bios/Archimedes.html).
"The
solution which occurred when he stepped into his bath and caused it to
overflow was to put a weight of gold equal to the crown, and known to be
pure, into a bowl which was filled with water to the brim. Then the gold
would be removed and the king's crown put in, in its place. An alloy of
lighter silver would increase the bulk of the crown and cause the bowl
to overflow" (http://www.mcs.drexel.edu/~crorres/Archimedes/contents.html).
Vitruvius' theory has had some criticisms. For example, when the bath was overflowing it does not explain how Archimedes discovered the upward thrust that acts on a body in water or the method of determining specific gravity, Vitruvius just said that knowledge was inferred (Dijksterhuis, 21).
Archimedes and Mechanical Inventions
Archimedes is believed to be the inventor of 3 mechanical devices, of which, were not used for war purposes.
Archimedes screw is one of those inventions. It is believed to have
been invented during one of his trips to Egypt, but may not have been invented
by him, although all historical accounts seem to say it was. This
device "consists of a cylinder inside of which is a continuous screw, extending
the length of the cylinder, forms a spiral chamber. By placing the
lower end in water and revolving the screw, the water is raised to the
top" (http://www.encyclopedia.com/articles/00686.html). This
tool is integral in not only the transportation of water for things like
irrigation and drainage, but also for the movement of grainy substances
such as sand and ash.
Archimedes has also been credited with the development of both the Planetarium, which replicates the motion of celestial bodies, and the Hydraulic Organ, where air was fed to pipes and then compressed to do work (Dijksterhuis, 23-26).
Archimedes and Wartime Developments
It is believed that Archimedes was asked to help with the defense of Syracuse against the Romans in the Second Punic War (215-212 B.C.E.) later in his life. These inventions consisted of both offensive and defensive ones.
Of the offensive ones, Archimedes is said to have set up huge ballistic machines, that could launch massive chunks of stone great distances. He used these machines and miniature ones like it particularly in defense of the Roman commander Marcellus and his fleet of ships. Because of his success, Archimedes was able to gain the respect of his enemy so much that it is believed according to Plutarch that Marcellus ordered his life to be spared once Syracuse was taken over.
Another great military accomplishment is known as the burning mirrors. The Greek writer Lucian quotes the story around 2nd century A.D.. It is believed that Aristotle had men line up mirrors to focus the sun's rays so intensely on the ships of the Romans that they caught fire (Dijksterhuis, 28). These mirrors, believed to be concave, explained how all rays coming in parallel to the surface will reflect to one single point known now as the focus of the mirror.
The Death of Archimedes
Three accounts of the death of Archimedes can be found in the works of Livy and Plutarch (Dijksterhuis, 30). All three have the death taking place in 212 B.C.E. during the siege of Syracuse in the Second Punic War.
Version 1 (found in both Livy's
and Plutarch's account) says Archimedes was so intensely working out a
problem by diagram that he failed to notice Syracuse had been taken over.
At this time a soldier came upon him and commanded him to follow to Marcellus.
He declined the request because he had not yet finished the problem, and
because of this the soldier became enraged and killed him. Although
in Livy's account the soldier did not know who he was and slayed him any
ways.
Version 2 (found only in Plutarch's account) says that the soldier already threatened him with death and Archimedes pleaded for his life till he could complete the proof. This guard also ends up killing him.
Version 3 (found only in Plutarch's account) explains that Archimedes was
carrying mathematical equipment back to Marcellus when he was killed because
the guards thought he may be carrying gold.
In Conclusion
Archimedes' legend is still alive to this day. His discoveries paved they way for many scientists to make their own important discoveries that, in the end, have changed our ways and views of life. It is unfortunate, as Ginsberg said, that Archimedes lived in a time where little was known about mathematical and experimental science. I believe his legend would indeed be greater if more people of his time would have been able to understand where his ideas were coming from. Even so, his ideas were rediscovered in the middle ages and, fortunately, they were built upon like many modern discoveries are today. We are indeed in debt to Archimedes for his perseverance and drive to understand phenomena at their mathematical and experimental levels.
Brunschwig, Jacques and Geoffrey E. R. Lloyd. Greek Thought. Cambridge: Harvard Press. 2000.
Clagett, Marshall. Archimedes in the Middle Ages Volume 1. Madison: University of Wisconsin Press. 1964.
Cooke, Roger. The History of Mathematics. New York. John Wiley and Sons, Inc. 1997. pp. 115
Dijksterhuis, E. J. Archimedes. Princeton: Princeton Univ. Press. 1987.
Ginsberg, Benjamin. The Adventure of Science. New York: Tudor Publishing Company. 1930.
Grande, John Del. "The Method of Archimedes," The Mathematics Teacher, Vol. 86, Num. 3 (Mar. 1993): 240-243.
The following websites were also used for references:
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Archimedes.html
http://www.mcs.drexel.edu/~crorres/Archimedes/contents.html
http://www.encyclopedia.com/articles/00686.html
http://www.treasure-troves.com/bios/Archimedes.html