Archimedes: What Did He Do Beside Cry Eureka? / Edition 1

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Many people have heard of two things about Archimedes: he was the greatest mathematician of antiquity and he ran from his bath crying, "Eureka, eureka!" few of us, layperson or mathematician are familiar with the accomplishments on which his reputation rests.

This book answers the questions by describing in detail his astonishing accomplishments: how he developed the theory of the lever and the center of gravity; how he used the center of gravity to study whether a floating object would tip over; how he summed a geometric series and the squares; and how he found the volume and surface area of a sphere. His ability to do so much with the few tools at his disposal is astonishing.

Archimedes was like a one-man Institute of Advanced Study making fundamental discoveries in the fields of geometry, mechanics, and hydrostatistics.

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Editorial Reviews

From The Critics
"Engaging account of some of Archimedes' discoveries, including centers of gravity, areas within spirals and parabolas, equilibria of floating bodies, surface area and volume of a sphere, approximations to [pi]. Includes exercises, much historical commentary. Excellent for calculus students and teachers."
American Mathematical Monthly
Australian Mathematics Teacher
"The author's writing style is elegant yet logical, and there are simple exercises to help guide the reader to get into the mathematics. Furthermore there are many diagrams to illustrate the text. In short, the book is eminently readable, not just for its own sake, but for the insights it provides into the amazing power of Archimedes' thinking. Highly recommended for all teachers of mathematics."
C. L. Kaller
"Stein gives an admirable presentation, in very attractive and readable form, of the accomplishments of Archimedes during the Golden Age of Greece.... This small book..should be found a useful addition to any senior high school, college and even university library."
Crux Mathematicurum
Charles Ashbacher
"The thought of a man running naked through the streets shouting with joy over a physical and mathematical discovery is one to warm the hearts of all those who value knowledge. When Archimedes experiences this flash of joy, little did he know that his actions would become the genesis of a legend that would last for thousands of years. However, he should be remembered for so much more than that and several of his significant mathematical contributions are explored in this book. It is really amazing to realize how close he was to inventing calculus 22 centuries ago, which was 18 before Newton and Leibniz.... Ten of [Archimedes'] most significant discoveries are presented and the solutions are those of Archimedes, although modern notation is used. While the proofs are generally easy to follow, one is often left in awe as to how he thought of how to approach some of these solutions. The explanations are succinct, yet thorough, which is the signature of a solid storyteller."
Charles Ashbacher Technologies
"Stein discusses Archimedes' significant discoveries in an informal manner accessible to anyone who has had high school algebra.... Great diagrams enrich the text and its mathematical arguments. Highly recommended to anyone interested in mathematics and its history, as it is eye-opening and a great read."
James M. Parks
"In this book Stein has achieved his stated purpose: "To make the discoveries of Archimedes easily accessible to a wide audience, from anyone with a background in high school algebra to a busy practicing mathematician."...The topics in this volume are treated carefully, clearly, and with many illustrations."
SB&F Magazine
Thomas Sonnabend
"This well-written book succeeds in presenting some of the most important discoveries of Archimedes in an accessible form. The opening chapter contains an interesting, well-researched discussion about Archimedes' life and the plausibility of various legends about him. The remainder of the book includes historical background and the intriguing story of an Archimedian work that was discovered in the twentieth century... The book will allow a wide audience to understand the nature of Archimedes' work. It is a useful resource for teachers who will need to explain more how Archimedes' work relates to their students' other knowledge about mathematics and physics."
The Mathematics Teacher
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Product Details

  • ISBN-13: 9780883857182
  • Publisher: Mathematical Association of America
  • Publication date: 10/1/1999
  • Series: Classroom Resource Materials Series
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 168
  • Product dimensions: 5.90 (w) x 8.80 (h) x 0.40 (d)

Read an Excerpt

The Life of Archimedes?
What we are told about Archimedes is a mix of a few hard facts and many legends. I will describe their sources so that you may decide for yourself how much truth lies in each one. Hard facts-the primary sources- are the axioms of history. Unfortunately, a scarcity of facts creates a vacuum that legends happily fill, and eventually fact and legend blue into each other. The legends resemble a computer virus that leaps from book to book, but are harder, even impossible to eradicate. Before we try to disentangle them in the case of Archimedes let's consider a simpler case, one closer to home, which goes back a mere two centuries rather than two millennia. This will put un in the proper mood to deal with the facts and fancies that surround Archimedes.
 Little is known of George Washington's boyhood, but that didn't stop Mason Weems, in his popular Life of George Washington, published in 1806, from introducing the now-famous incident of the hatchet and the cherry tree.

"George," said the father, "do you know who killed that beautiful little cherry-tree yonder in the garden?"
 "I can't tell a lie, Pa… I did cut it with my hatchet."
 "Run to my arms…," cried his father… "glad am I that you killed my tree…."

 As early as 1824 Weems was described as the "author of  A Washington's Life-not one word of which we believe." In 1887 a biographer of Washington could write, "The material [relating to his boyhood] is rather scanty. The story of the hatchet and the cherry sapling, whether true or not, is singularly characteristic…. Nobody would ever have thought of relating such a story in connection with the boyhood of Napoleon…."
 And we find, in a children's book published in 1954, "Stories about George Washington as a boy have been retold so often… that even though we're not sure they really did happen, they have become a part of the story of America. And they do tell us something of the kind of boy he was."
 Present-day historians must carefully back up their assertions by citing the evidence or the source. But, as P.G. Walsh, an authority on ancient history, points out in Livy, His Historical Aims and Methods, the historians in Archimedes' time did not all abide by such harsh standards: "the majority of history written in this period [400-200 B.C.] were not so laudable. [The] detailed reasons for such a retrogression… are undoubtedly connected with the growth of the schools of rhetoric…. [W]riters addicted to 'tragic' techniques sought to thrill their readers by evoking feelings of pity and fear…. Certain types of description are particularly amenable to this type of treatment, such as the fate of conquered cities, or the deaths of famous men."
 Clearly we yearn to discover the human side of our heroes, to find out what influences and circumstances shaped them into such unique figures. Perhaps subconsciously we are looking for the magic formula that will turn children into creative adults.
 We should be on guard about Archimedes, who invites far more invention than Washington does. There is no material at all about his boyhood and little about his adulthood. However, there is ample evidence that he lived in Syracuse (Syracusa, the city at the southeast corner of Sicily, then part of the Greek world) and was killed there by a Roman soldier in the year 212 B.C., at the end of a siege conducted by the Roman general Marcellus. Legends soon grew up about the manner of his death. Plutarch (about 46-120 A.D.) described it this way:
 "… He was alone, examining a diagram closely: and having fixed both his mind and his eyes on the object of his inquiry, he perceived neither the inroad of the Romans nor the taking of the city. Suddenly a soldier came up to him and bade him follow to Marcellus, but he would not go until he had finished the problem…. [t]he soldier became enraged and dispatched him. Others, however, say that the Roman came upon him with drawn sword intending to kill him at once, and that Archimedes…entreated him to wait…so that he might not leave the question…only partly investigated: but the soldier did not understand and slew him. There is a third story, that as he was carrying to Marcellus some of his mathematical instruments, such as sundials, spheres, and angles adjusted to the apparent size of the sun, some soldiers… under the impression that he carried treasure in the box, killed him."
 Neither Plutarch nor Livy (59 B.C.-17 A.D.), who also wrote about Archimedes, mentions that he told the soldiers, "Noli turbare circuos meos." "Don't disturb my circles.") The first time words were provided for Archimedes was about 30 A.D., when Valerius Maximus had him request, "Noli obsecro istum disturbare." ("Please don't disturb this.") Presumably Archimedes was referring to a diagram he had drawn on the ground or on a table. By the twelfth century, Archimedes is saying, with more feeling and in Greek, "Fellow, stand away from my diagram." I suspect that these variations are pure fiction. Who would have reported them? Would a soldier who had killed Archimedes, against orders from his commanding general, offer this incriminating evidence? Nor do Plutarch and Livy mention his drawing diagrams on the ground. This was a later "improvement."
 As for his date of birth we must depend on The Book of Histories, a twelfth-century work by the historian Tzetzes, who lived some fourteen centuries after Archimedes. Hardly his contemporary, Tzetzes wrote: "Archimedes, the wise, that famous maker of engines, was a Syracusan by race, and worked at geometry till old age, surviving five and seventy years." That statement, together with a bit of arithmetic, is the source for asserting that Archimedes was born in 287 B.C.
 The tale of Archimedes crying "Eureka, Eureka" goes back to the Roman architect, Vitruvius, who lived in the first century B.C. Archimedes had to find the volume of a sacred wreath (not, as is often said, a crown), allegedly made of pure gold, to determine whether the goldsmith had replaced some gold with silver.
 "While Archimedes was turning the problem over, he chanced to come to the place of bathing, and there, as he was sitting down in the tub, he noticed that the amount of water which flowed over the tub was equal to the amount by which his body was immersed. This showed him a means of solving the problems, and he did not delay, but in his joy leapt out of the tub and, rushing naked towards his home, he cried out with a loud voice that he had found what he sought. For as he ran he repeatedly shouted in Greek, 'Eureka, Eureka' (I have found, I have found)."
 I doubt that such an elementary observation would have impressed Archimedes enough to stage a celebration, especially when it is compared to his dazzling discoveries about the surface area and volume of a sphere, the center of gravity, and the stability of floating objects. In fact, Archimedes viewed his work on the sphere as his best, as Plutarch reported: "And though he had made many elegant discoveries, he is said to have besought his friends and kinsmen to place on his grave… a cylinder enclosing a sphere, with an inscription giving the proportion by which the volume of the cylinder exceeds that of the sphere."
 That must be a hard fact, for Cicero, about 75 B.C., saw the stone: "When I was questor I searched out his tomb, which was shut in on every side and covered with thorns and thickets. The Syracusans dis not know of it: they denied it existed at all. I knew some lines of verse which I had been told were inscribed on his gravestone, which asserted that there was a sphere and a cylinder on the top of his tomb.
 "Now while I was taking a thorough look at everything- there is a great crowd of tombs at Agrigentine Gate- I noticed a small column projecting a little way from the thickets, on which there was a representation of a sphere and cylinder, I immediately told the Syracusans-their leading men were with me- that I thought that was the very thing I was looking for. A number of men were sent in with sickles and cleared and opened the place.
 'When it had been made accessible, we went up to the base facing us. There could be seen the epitaph with about half missing where the ends of the lines were worn away. So that distinguished Greek city, once also a centre of learning, would have been unaware of the tomb of its cleverest citizen, if it had not learned of it from a man from Arpinum…
 "Who in the world is there, who has any dealings with… culture and learning, who would not rather be this mathematician that that tyrant [Dionysius, who ruled Syracuse for 38 years]? … [T]he mind of the one was nourished by weighing and exploring theories, along with the pleasures of using one's wits, which is the sweetest fod of souls, the other's in murder and unjust acts…"
 During his lifetime, Archimedes was famous for his military exploits, in particular for his contribution to the defense of Syracuse against the Romans. The earliest description of this side of Archimedes is in the Histories of Polybius (about 204-120 B.C.):
 "But in this the Romans did not take into account the abilities of Archimedes; nor calculate on the truth that, in certain circumstances, the genius of one man is more effective than any numbers whatever. However they now learnt it by experience… Archimedes had constructed catapults to suite every range: and as the ships sailing up were still at a considerable distance, he so wounded the enemy with stones and darts, from the …longer engines, as to harass and perplex them to the last degree. And when these began to carry over their heads, he used smaller engines graduated according to the range required. Finally, Marcellus was reduced in despair to bringing up his ships under cover of night. But when they had come too near to be hit by the catapults, they found that Archimedes had prepared another contrivance…"
 Polybius goes on to describe other devices Archimedes employed, including grappling hooks that lifted the Roman ships out of the water. With these devices Archimedes forced Marcellus to give up any hope of a successful frontal attack, and inaugurated the role of the warrior  scientist, who, especially in this century, has had a far greater impact on the nature of war then have the generals.
 The catapults, equipped with ropes and pulleys, exploited the principle of the lever, where distance is traded for force. Plutarch mentions Archimedes' appreciation of the lever: "Archimedes, a kinsman and friend of King Hiero, wrote to him that with a given force it was possible to move any given weight; and emboldened, as it is said, by the strength of the proof, he asserted that, if there were another world and he could go to it, he would move this one.
 By the fourth century A.D. this had become, in Pappus' version, the well-known epigram, "Give me a place to stand and I can move the Earth." As we will see in later chapters, Archimedes certainly appreciated the principle of the lever, applying it in his study of area, volume and, center of gravity.
 Another legend has it that he set the Roman ships on fire by using mirrors arranged in parabola to reflect the sunlight on a single burning point. However, there is no mention of this in the three earliest descriptions of the defense of Syracuse, those by Polybius, Livy, and Plutarc. Lucian, wiriting in the first century A.D., says only that Archimedes used "artifical means." The first mention of mirrors occurs in Galen, around the year 160 A.D., almost four centuries after the death of Archimedes.
 The more one thinks of trying to burn ships at a distance in the midst of a battle, the more implausible the project seems; The sun must be bright and at a convenient angle, the boat not bob on the waves, the people holding the mirrors able to focus on the exact same spot while dodging volleys of flying arrows. Moreover, had the mirrors done their work, they would have become a standard weapon; yet there is no sign that they were added to the armaments of the time.
 At the end of his study based on the ancient sources and the pertinent physics, D.L. Simms concludes, "The historical evidence for Archimedes' burning mirrors is feeble, contradictory in itself, and the principal and very late authorities are silent…. Modern experiments suggest a burning mirror is highly unlikely to produce ignition on a moving ship.
 Where does all this leave us? That Archimedes lived in Syracuse and applied mathematics to the so-called "real-world," and died in the assault on his city. Historians conjecture that he may have visited Alexandria, Egypt, which with its great library, was a major scientific center, and it is possible that he invented the device for raising water for irrigation, the "Archimedes screw." But these assertions are part of that ambiguous and hypothetical world of "perhaps," "maybe," and "it is possible that," where we are left not knowing what to think.
 For a more extensive discussion of the sources of information about the life of Archimedes I refer the reader to the opening pages of Dijksterhuis' book and Knorr's supplement at the end.
 When all is said and done, we do have many of Archimedes' writings, filtered though they may be through Arab, Latin, and English translations. These, after all, bear the most reliable witness of the man, and are the main reason we are interested in him. The best way to appreciate Archimedes is to follow his mathematical arguments, just as we best appreciate Giotto by viewing his frescoes and Mozart by listening to his music.

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Table of Contents


1. The Life of Archimedes?
2. The Law of the Lever
3. Center of Gravity
4. Big Literary Find in Constantinople
5. The Mechanical Method
6. Two Sums
7. The Parabola
8. Floating Bodies
9. The Sprial
10. The Sphere
11. Archimedes Traps [pi]

Appendix A Affine Mappings and the Parabola
Appendix B The Floating Paraboloid: Special Case
Appendix C Notation
Appendix D References

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