Archimedes is commonly viewed as the greatest mathematician of antiquity. In his time he was well known, but more for his inventions of war than his mathematics, because war machines are much more exciting to the masses than mathematical theory. These machines gave him an enormous reputation, but to him they were merely amusements in geometry. He is said to have stated that he would spend no time wasted on writings describing things that were so unimportant. Presently Archimedes is known not only for his inventions, but is nearly worshipped for his studies on mathematics and science. In his lifetime he worked endlessly on the discovery of scientific ideas, and published them in treatises and other writings; from these writings came present day mathematical and scientific theorems. He planted and sewed many a mathematical seed that would merely be cultivated by mathematicians to follow him. His writings gave way to many theorems in mathematics and science. They set the stage for calculus, and opened our eyes in Physics, Geometry, and Astronomy. Though the details of Archimedes personal life are little known, and there are many questions to the validity of the ones that are, many of his works are still in existence and help us decipher the important components of his life. As for the holes, many have tried to fill them in, and from these men, I aim to gather the facts and relay the story of Archimedes.
Archimedes was born in Syracuse, Sicily when it was an independent city-state. Syracuse is located very near the mainland of Italy, and possesses a port that is easily accessible from the ocean. Its prime location made the city-state a target for those who sought to build empires during that time. Throughout most of Archimedes’ life Carthage and Rome played tug-of war to gain control of Syracuse during what were called the Punic Wars.1
According to Tzetzes, Archimedes died at the age of 75 during the capture of Syracuse by the Romans in 212 BCE, so it is agreed upon that he was born in 287 BCE. He was the son of Phidias, the astronomer, and as a young boy he also was interested in the heavens. It is conjectured that he studied at Alexandria for a period of time with successors of Euclid, and possibly Euclid himself. After leaving Alexandria, he is said to have kept in close correspondence with many of the mathematicians and friends he had there. He would send his results to them before he had them published. Among these friends were Conon of Samos, whom he held in high esteem, and Eratosthenes to whom it is said he sent the famous cattle-problem. This correspondence to Alexandria is why much of his material is still in existence2.
While in Egypt, Archimedes is credited with the invention of the Archimedes screw. This hydraulic screw was an invention of necessity, for the land along the Nile was fertile, but their current means of irrigating were tiresome. The Archimedes screw was a long cylindrical shaped device that would cycle water up from the river, by means of a screw shaped blade, to the land above for irrigation (see figure 1). This device was most useful because one person could operate it. It was also used to pump water from the hold of ships when they began to flood.3
Figure 1 After his studies in Alexandria he returned to Syracuse where he was great friends with and possibly related to King Hieron (also referred to as King Hiero II). Two of the most relayed stories about Archimedes stemmed from this relationship. The validity of these stories remains questionable, but it is necessary to relay the details so as to learn more about the personality of Archimedes, and the origin of some of his most important laws and theorems:
King Hieron, after commissioning a goldsmith to construct a crown from a lump of pure gold, heard rumors that made him doubt the honesty of this man. Hieron went to Archimedes with the problem, and asked him to find a way to test the legitimacy of the crown without sullying it. The problem perplexed Archimedes. As the story is told, one day while at the baths Archimedes lowered himself into the pool of water, and after noticing the displacement of the water, hit upon the solution. Without bothering to replace his clothing he ran naked through the streets screaming, “Eureka! Eureka!” (The Greek phrase for “I have found it”). Presumably after getting dressed, Archimedes took a lump of gold the same weight as the crown and measured the amount of water each displaced. The results were in, and a goldsmith was allegedly punished.
While the details of the story may have been subject to falsification, the fact is this discovery led way to his treatise On Floating Bodies in which he describes the fundamental principle of hydrostatics, and relays Archimedes Principle which gives the weight of a body immersed in liquid.4
Before Archimedes existed it was known that a given weight could be moved with a lever and a fulcrum, but this law was never officially applied in a mathematical model5. It is told that upon the discovery of this model, Archimedes uttered the phrase “Give me a place to stand and I will move the Earth”6. King Hieron, in disbelief, employed Archimedes to demonstrate this principle. As the story was told, there existed a ship made by King Hieron for the use of Ptolemy. This was a great vessel with three masts, loaded with freight. A multitude of Syracusans could not manage to budge the ship, so Archimedes devised a compound pulley system with which to move it. King Hieron looked on in amazement as Archimedes moved the ship along smoothly, without effort, and stated from that day on, Archimedes was to be believed in everything he might say.7
Though the actual quote by Archimedes “Give me a place to stand…” is represented in a multitude of ways in many different writings, the discovery of the Law of Levers at this point in time is universally relayed. “Magnitudes are in equilibrium at distances reciprocally proportional to their weights” this statement is given in Propositions 6 and 7 of Book I of On the Equilibrium of Planes. The theorem is also proved in this treatise. 8
Archimedes’ relationship King Hieron would prove to benefit not only the King, but the citizens of Syracuse as well. As was stated before Syracuse was sought by Rome at the time, and Roman armies, under the command of Marcus Claudius Marcellus were sent with the intent of its capture. Archimedes used his scientific knowledge to invent many devices, or engines of war as others like to call them, to stave off the Roman armies. Among these weapons of mass destruction were the catapult, the Archimedes Claw, and machines for discharging showers of missiles through holes in walls.
The Archimedes claw was a device mounted on the wall of the city, and it is said, would turn ships over completely, or it would rain showers of boulders upon their decks. Plutarch best describes these machines9:
... when Archimedes began to ply his engines, he at once shot against the land forces all sorts of missile weapons, and immense masses of stone that came down with incredible noise and violence; against which no man could stand; for they knocked down those upon whom they fell in heaps, breaking all their ranks and files. 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 (a dreadful thing to behold), 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.10
Figure 3 These weapons were said to have been so effective that it reached the point where, when a soldier would see a rope or a stick rise above the wall, he would scream, “there it is” and turn and run away.
As stated before Archimedes put no mind into the engines of war, and said that they were merely amusements in Geometry. His beliefs were that pure mathematics was the only worthy pursuit. In fact, his works in pure mathematics would prove to be exceedingly more appreciated today than any of his inventions.
Throughout his life Archimedes is said to have had an obsession with mathematics. He would often be spotted drawing diagrams anywhere he could; in the sand on the beach, in the ashes of the fireplace, and even in the ointment on his own skin. It is said that he would neglect his own hygiene and nutrition, as he was so deeply involved in his studies. Plutarch writes:
Oftimes 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 savours, with his fingers he drew lines upon his naked body, so far he was taken from himself, and brought into ecstasy or trance, with the delight he had in the study of geometry.11
This obsession of Archimedes’ brought upon amazing results. Among his treatises still existing today are: The Cattle-Problem, On The Equilibrium of Planes, Quadrature of the Parabola, On the Sphere and Cylinder, also two books, On Spirals, On Conoids and Spheroids, On floating bodies, in two books, Measurement of a Circle, The Sand-Reckoner, and Method.12 Most, if not all, of these treatises exist in the form of letters and most in correspondence to Alexandria. Let it be known that the names were not given by Archimedes; they were appropriately named, after his death, by their subject.
The Cattle-Problem was one of the famous correspondences to Alexandria from Archimedes. This problem, addressed to Eratosthenes of Cyrene, was posed in the form of an Epigram. In rather poetic prose he writes about the cattle of the sun, which exist in four different colors. He writes:
If thou art diligent and wise O Stranger, compute the number of cattle of the sun…
The question was to find eight unknowns; the number of female and male cattle of every color. He poses this problem with a long list of constraints, such as:
…the white bulls were equal to half of and a third of the black together with the whole of the yellow…
Today this problem could be solved simply on a computer using a matrix and a mathematical computing program, but then this problem was impossible. Needless to say it went unsolved for thousands of years until a computer found the solution in 1965. It was shown that the total number of “cattle of the sun” was an integer with 206,545 digits. It is questionable whether or not Archimedes himself knew the solution, probably not, but we will never know.13
On the Equilibrium of Planes, is a Book in two volumes. As discussed before it included the Law of Levers. Book I states seven different postulates and multiple propositions, in which he proposes certain mechanical properties of weights and planes and distances of weights on planes using techniques of geometry14. He goes on to prove such things stating what is now known as The Law of Levers: force x force applied = weight x weight applied15. He also finds the center of gravity of a parallelogram, a triangle, and a trapezium.16 Book II of On the Equilibrium of Planes is committed entirely to finding the center of gravity of a parabola.17 Quadrature of the Parabola: This work was sent to Dositheus in Alexandria and it is prefaced with an expression of sadness over the death of his great friend Conon of Samos. He discusses how he would have liked to send the proof to Conon due to his interest in such matters. This work discusses “squaring the segment bounded by a straight line and a section of right angled cone (parabola).”18 In it Archimedes proves a theorem to find the area of a segment of parabola cut off by any chord.19
On the Sphere and Cylinder is a treatise in two books. It also houses another of Archimedes’ most famous theorems. Book I proves many geometric properties such as the surface of a sphere is four times that of its greatest circle, how to find the area of any segment of sphere, the volume of a sphere is two-thirds that of the volume of a circumscribed cylinder, and the surface of a sphere is two-thirds that of a circumscribed cylinder including its bases. Book II describes how to cut any given sphere by a plane so the ratio of the volumes of the segments is prescribed20.
In On Spirals, Archimedes reflects many of the properties he has proven in previous writings, and goes on to define fundamental properties of spirals. He discussed area of spiral segments. He talked about tangents to the spiral. He also stated a property that connects the length of the radius vector with the angles that it revolves through. 21
On Conoids and Spheroids, includes a quantity of very impressive properties of surfaces of revolution pertaining particularly to volumes of their segments. He discussed what are now referred to as paraboloids, hyperboloids, and spheroids (or ellipsoids) made by rotating an ellipse about its major or minor axis.
On floating bodies was Archimedes’ famous work, which discusses elementary principles of hydrostatics. It includes Archimedes’ Principle, which states:
…that part [of the fluid] which is thrust the less s driven along by that which is thrust the more.
Of course this description is now known as displacement. It goes on to discuss principles of floating bodies (or, as it may be, sinking bodies) by discussing what is now referred to as density. Book II discusses such properties as the relationship between the immersed part of a floating body, and its weight.
Measurement of a Circle was also a major achievement. It states the relationship between the radius and circumference and its importance in finding the area of the triangle. He lists many other properties to be used in finding the area of a circle. In proposition three of this book he estimates the value of pi:
The ratio of the circumference of any circle to its diameter is less than 3 but greater than 3.
This is an amazingly accurate representation. In this text he also estimates the square root of three with greater than and less than values, and the square roots of several large numbers that are not complete squares are given. It is not clear how he arrived at these results, for he did not give a method. This text also contains Archimedes’ famous precursor to calculus, the Method of Exhaustion, in which you determine the area of a circle by inscribing it and circumscribing it with polygons bearing many sides. This bears close resemblance to a limit.
The Sand-Reckoner is addressed to King Gelon, King Hieron’s son. In this work he tries to geometrically prove that the amount of sand in the universe is not infinite. He gives the dimensions of the universe, and states as a description of the universe:
…the sphere whose centre is the centre of the earth and whose radius is equal to
the straight line between the centre of the sun and the centre of the earth.
He discusses the distance between heavenly bodies, and the perimeter and diameter of the earth. He also proposes a numerical system capable of expressing numbers up to 8x1063. 22
Finally we discuss Method, also known as the lost Archimedes’ Palimpsest. Method was discovered in 1906 by Johan Heiberg in Istanbul. It was a parchment copy of the lost Archimedes’ work, and had been scribed over with a Christian prayer book. The manuscript went lost again, but finally appears at Christie’s in Paris, where it is auctioned off for two million dollars, and then finally donated to the Walters Art Museum, in Baltimore, where it is being restored23.
In the work of Method Archimedes’ discusses his methods of finding the theorems of all the other writings. It is a miracle to have found this piece, for not many Greek mathematicians were willing to offer up their findings, let alone their methods. For this reason it is labeled, by some, as the most important work of Archimedes.24
It is not hard to see why Archimedes is held in such high regard for his mathematical ingenuity. His writings laid the groundwork for an amazing number of theorems and principles of geometry, physics, and calculus. It is a blessing that these works still exist today, and a miracle that Method has been found.
Archimedes died in 212 BCE, when the Romans invaded Syracuse. His death, like much of his life, is a mystery, but it is commonly told in three stories:
In the first version Archimedes’, as was common, was working out a geometrical problem by drawing diagrams in the sand. It is said he had no knowledge of the Roman invasion. A Roman soldier approached him and asked him to come to Marcellus (for it was said that Marcellus wanted the genius alive). Archimedes asked the soldier to wait until he was finished resolving his problem, and it is said that the soldier was so enraged that he killed him.
The second version says that Archimedes was drawing diagrams in the sand and a soldier approached him, threatening to kill. Archimedes is said to have exclaimed, “stand away, fellow, from my diagram,” and the soldier slew him.
The third version states that Archimedes was carrying mathematical instruments to Marcellus, and some soldiers spotted him and thought he had a load of gold, so they killed him.25
Whichever way it happened, his death was a crushing loss.
After Archimedes death it is said that he was forgotten. Perhaps it was the embarrassment of the Romans to have killed such a great man, or perhaps it was the fact that he was not able to amuse the people anymore by constructing engines of war. Not much Archimedes was discussed until 75 BC, when Cicero went to Sicily in search of information on this great man. He searched for his tomb and found it neglected and forgotten:
... and found it enclosed all around and covered with brambles and thickets; for I remembered certain doggerel lines inscribed, as I had heard, upon his tomb, which stated that a sphere along with a cylinder had been put on top of his grave. Accordingly, after taking a good look all around ..., I noticed a small column arising a little above the bushes, on which there was a figure of a sphere and a cylinder... . Slaves were sent in with sickles ... and when a passage to the place was opened we approached the pedestal in front of us; the epigram was traceable with about half of the lines legible, as the latter portion was worn away26
Though Archimedes’ works were studied at Alexandria, it wasn’t until the sixth century AD that Archimedes’ works became more widely known. His dedication to his work has made a profound impact on the world, and that impact is best summed up by Plutarch27.
It is not possible to find in all geometry more difficult and intricate questions, or more simple and lucid explanations. Some ascribe this to his natural genius; while others think that incredible effort and toil produced these, to all appearances, easy and unlaboured results. No amount of investigation of yours would succeed in attaining the proof, and yet, once seen, you immediately believe you would have discovered it; by so smooth and so rapid a path he leads you to the conclusion required.28
Archimedes, by E.J. Dijksterhuis, Ejnar Munksgaard, Copenhagen Ó1956
Journey Through Genius: The Great Theorems of Mathematics, by William Dunham, Penguin Books Ó1991
The Works of Archimedes, Edited in modern notation with introductory chapters by TL heath, New york, dover publications, inc Ó1897, supplement Ó1912
Infinte Secrets, “Archimedes”, PBS, Nova Website. Active URL: http://www.mcs.drexel.edu/~crorres/Archimedes/contents.html Katz
A History of Mathematics: An Introduction, “Archimedes” 2nd edition, by Victor J. Katz, p102-117, Addison, Wesley, Longman, Inc. Ó1998
Archimedes of Syracuse, MacTutor website, by J J O'Connor and E F Robertson, Active URL: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Archimedes.html
The Great Mathematicians, by Herbert Westren Ternbull, New York University Press, Ó1961