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## PowerPedia:Archimedes

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Archimedes (Greek: ?' ???????) (c. There was an error working with the wiki: Code[2] There was an error working with the wiki: Code[50], There was an error working with the wiki: Code[51], There was an error working with the wiki: Code[52], There was an error working with the wiki: Code[53], and There was an error working with the wiki: Code[54] born in the seaport colony of There was an error working with the wiki: Code[55], There was an error working with the wiki: Code[56], what is now current day There was an error working with the wiki: Code[57].

#### Inttroduction

Many consider him one of the greatest, if not the greatest, mathematicians in There was an error working with the wiki: Code[3]. There was an error working with the wiki: Code[58], himself frequently called the most influential mathematician of all time, modestly claimed that Archimedes was one of the three epoch-making mathematicians (the others being There was an error working with the wiki: Code[59] and There was an error working with the wiki: Code[60]). Apart from his fundamental theoretical contributions to math, Archimedes also shaped the fields of physics and practical engineering, and has been called "the greatest scientist ever".

He was a relative of the Hiero monarchy, which was the ruling family of Syracuse (Saracussia), a seaport kingdom. King There was an error working with the wiki: Code[61], who was rumored to be Archimedes' uncle, commissioned him to design and fabricate a new class of ships for his navy, which were crucial for the preservation of the ruling class in Syracuse. Hiero II had promised large caches of grain to the Romans in the north in return for peace. Faced with war when unable to present the promised amount, Hiero II commissioned Archimedes to develop a large luxury/supply/war barge in order to serve the changing requirements of his navy. It is rumored that the Archimedes Screw was actually an invention of happenstance, as he needed a tool to remove bilge water. The ship, coined Saracussia, after its nation, may be mythical. There is no record on foundry art, nor any other period pieces depicting its creation. It is solely substaintiated by a description from There was an error working with the wiki: Code[62], who said "it was the grandest equation ever to sail."

He is credited with many inventions and discoveries, some of which we still use today, like his There was an error working with the wiki: Code[63]. He was famous for his compound pulley, a system of pulleys used to lift heavy loads such as ships. He made several war machines for his patron and friend Hiero II. He did a lot of work in geometry, which included finding the surface areas and volumes of solids accurately. The work that has made Archimedes famous is his theory of floating bodies. He laid down the laws of flotation and developed the famous Archimedes principle.

#### Discoveries and inventions

Archimedes became a very popular figure as a result of his involvement in the defense of There was an error working with the wiki: Code[4] against the There was an error working with the wiki: Code[5] There was an error working with the wiki: Code[6] and There was an error working with the wiki: Code[7]!" ("I have found it!"). He has also been credited with the possible invention of the There was an error working with the wiki: Code[64] during the There was an error working with the wiki: Code[65]. One of his inventions used for military defense of Syracuse against the invading Romans was the There was an error working with the wiki: Code[66].

It is said that he prevented one Roman attack on Syracuse by using a large array of There was an error working with the wiki: Code[67]s (speculated to have been highly polished shields) to reflect sunlight onto the attacking ships causing them to catch fire. This popular legend, dubbed the "Archimedes death ray", has been tested many times since the Renaissance and often discredited as it seemed the ships would have had to have been virtually motionless and very close to shore for them to ignite, an unlikely scenario during a battle. A group at There was an error working with the wiki: Code[68] have performed their own tests and concluded that the mirror weapon was a possibility

although later tests of their system showed it to be ineffective in conditions that more closely matched the described siege <pesn type=. The television show There was an error working with the wiki: Code[8] with flaming bolts, led the team to believe that the heat ray was far too impractical to be used, and probably just a myth.

It can be argued that even if the reflections didn't induce fire, they still could have confused, and temporarily blinded the ship crews, making it hard for them to aim and steer. Making them hot and sweaty before primary battle may have also tired them faster. The effectiveness may have simply been exaggerated. Archimedes also has been credited with improving accuracy, range and power of the There was an error working with the wiki: Code[69].

Archimedes was killed by a Roman soldier during the sack of Syracuse during the There was an error working with the wiki: Code[9] that he was not to be harmed. The Greeks said that he was killed while drawing an equation in the sand engrossed in his diagram and impatient with being interrupted, he is said to have muttered his There was an error working with the wiki: Code[70] before being slain by an enraged Roman soldier: ?? ??? ???? ?????? ?????? ("Don't disturb my circles"). The phrase is often given in There was an error working with the wiki: Code[71] as "Noli turbare circulos meos" but there is no direct evidence that Archimedes ever uttered these words. This story was sometimes told to contrast the Greek high-mindedness with Roman ham-handedness however, it should be noted that Archimedes designed the siege engines that devastated a substantial Roman invasion force, so his death may have been out of retribution http://www.math.nyu.edu/~crorres/Archimedes/Death/Histories.html.

In creativity and insight, Archimedes exceeded any other European mathematician prior to the European There was an error working with the wiki: Code[10] proofs for his results. To what extent he actually had a correct version of integral calculus is debatable. He proved that the There was an error working with the wiki: Code[11] of the There was an error working with the wiki: Code[12] but he gave a procedure to approximate it to arbitrary accuracy and gave an approximation of it as between 3 + 10/71 (approximately 3.1408) and 3 + 1/7 (approximately 3.1429). He was the first There was an error working with the wiki: Code[13] mathematician to introduce There was an error working with the wiki: Code[14] with equal base and height. (See the illustration below. The "base" is any There was an error working with the wiki: Code[15] "the same base" means the same "horizontal" component of the length of the base "horizontal" means orthogonal to the axis. "Height" means the length of the segment parallel to the axis from the There was an error working with the wiki: Code[72] to the base. The vertex must be so placed that the two horizontal distances mentioned in the illustration are equal.)

In the process, he calculated the earliest known example of a There was an error working with the wiki: Code[73] summed to infinity with the There was an error working with the wiki: Code[74] 1/4:

: \sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3} \ .

If the first term in this series is the area of the triangle in the There was an error working with the wiki: Code[75] then the second is the sum of the areas of two triangles whose bases are the two smaller secant lines in the illustration, and so on. Archimedes also gave a quite different proof of nearly the same There was an error working with the wiki: Code[76] by a method using There was an error working with the wiki: Code[77].

He proved that the There was an error working with the wiki: Code[16], a result he was so proud of that he made it his There was an error working with the wiki: Code[78].

Archimedes is probably also the first There was an error working with the wiki: Code[17] and There was an error working with the wiki: Code[18]. He invented the field of There was an error working with the wiki: Code[19] of There was an error working with the wiki: Code[20]s. Using only There was an error working with the wiki: Code[21] There was an error working with the wiki: Code[79], he also gave the equilibrium positions of floating sections of paraboloids as a function of their height, a feat that would be taxing to a modern physicist using There was an error working with the wiki: Code[80].

Apart from general physics, he was also an There was an error working with the wiki: Code[22] brought two devices back to There was an error working with the wiki: Code[23] for constructing these devices. For some time this was assumed to be a legend of doubtful nature, but the discovery of the There was an error working with the wiki: Code[81] has changed the view of this issue, and it is indeed probable that Archimedes possessed and constructed such devices. There was an error working with the wiki: Code[82] of There was an error working with the wiki: Code[83] writes that Archimedes had written a practical book on the construction of such spheres entitled There was an error working with the wiki: Code[84].

Archimedes' works were not widely recognized, even in There was an error working with the wiki: Code[24]. He and his contemporaries probably constitute the peak of Greek There was an error working with the wiki: Code[25] There was an error working with the wiki: Code[26] was lost until around There was an error working with the wiki: Code[27] and There was an error working with the wiki: Code[28] centuries.

#### Writings by Archimedes

Several of archimededs works were briefly inspected in There was an error working with the wiki: Code[29] There was an error working with the wiki: Code[30] There was an error working with the wiki: Code[31] by There was an error working with the wiki: Code[85]. His works include:

"Equilibrium of Planes"

"Spiral Lines"

"The Measurement of the Circle"

"Sphere and Cylinder"

"On Floating Bodies" (only known copy in Greek)

"The Method of Mechanical Theorems" (only known copy)

"Stomachion" (only known copy)

##### On the Equilibrium of Planes

2 volumes

This scroll explains the law of the lever and uses it to calculate the areas and centers of gravity of various geometric figures.

##### On Spirals

In this scroll, Archimedes defines what is now called There was an error working with the wiki: Code[32]. This is the first mechanical curve (i.e., traced by a moving point) ever considered by a Greek mathematician.

##### On the Sphere and The Cylinder

In this scroll Archimedes obtains the result he was most proud of: the relation between the area of a sphere to that of a circumscribed straight cylinder is the same as that of the volume of the sphere to the volume of the cylinder (exactly 2/3).

##### On Conoids and Spheroids

In this scroll Archimedes calculates the areas and volumes of sections of cones, spheres, and paraboloids.

##### On Floating Bodies

2 volumes

In the first part of this scroll, Archimedes spells out the law of equilibrium of fluids, and proves that water will adopt a spherical form around a center of gravity. This was probably an attempt at explaining the observation made by Greek astronomers that the Earth is round. Note that his fluids are not self-gravitating: he assumes the existence of a point towards which all things fall and derives the spherical shape. One is led to wonder what he would have done had he struck upon the idea of universal Gravitation.

In the second part, a veritable tour-de-force, he calculates the equilibrium positions of sections of paraboloids. This was probably an idealization of the shapes of ships' hulls. Some of his sections float with the base under water and the summit above water, which is reminiscent of the way There was an error working with the wiki: Code[86]s float, although Archimedes probably was not thinking of this application.

##### The Quadrature of the Parabola

In this scroll, Archimedes calculates the area of a segment of a parabola (the figure delimited by a parabola and a secant line not necessarily perpendicular to the axis). The final answer is obtained by There was an error working with the wiki: Code[33] the area and summing the geometric series with ratio 1/4.

##### Stomachion

This is a Greek puzzle similar to a There was an error working with the wiki: Code[87]. In this scroll, Archimedes calculates the areas of the various pieces. This may be the first reference we have to this game. Recent discoveries indicate that Archimedes was attempting to determine how many ways the strips of paper could be assembled into the shape of a square. This is possibly the first use of There was an error working with the wiki: Code[88] to solve a problem.

##### Archimedes' Cattle Problem

Archimedes wrote a letter to the scholars in the Library of Alexandria, who apparently had downplayed the importance of Archimedes' works. In these letters, he dares them to count the numbers of cattle in the There was an error working with the wiki: Code[34] equations, some of them There was an error working with the wiki: Code[35] (in the more complicated version). This problem is one of the famous problems solved with the aid of a computer. The solution is a very large number, approximately There was an error working with the wiki: Code[1] (See the external links to the Cattle Problem.)

##### The Sand Reckoner

The Sand Reckoner (Greek: ???????? - psammites) is probably the most accessible work of Archimedes in some sense, it is the first research-expository paper. In this work, Archimedes sets himself to challenge the then commonly held belief that the number of grains of

sand is too large to count. In this scroll, Archimedes counts the number of grains of sand fitting inside the There was an error working with the wiki: Code[89]. This book mentions There was an error working with the wiki: Code[90]' theory of the There was an error working with the wiki: Code[91] (concluding that "this is impossible"), contemporary ideas about the size of the Earth and the distance between various celestial bodies. From the introductory letter we also learn that Archimedes' father was an astronomer.

Archimedes first has to invent a system of naming There was an error working with the wiki: Code[92] in order to give an upper bound, and he does this by starting with the largest number around at the time, a myriad myriad or one hundred million (a There was an error working with the wiki: Code[93] is 10,000). Archimedes' system goes up to

:10^{8 \times 10^{16}}

which is a myriad myriad to the myriad myriadth power, all taken to the myriad myriadth power. Another way of describing this number is a one followed by (There was an error working with the wiki: Code[36]) eighty quadrillion (8 1016) zeroes compared to this number the otherwise enormous There was an error working with the wiki: Code[94], or one followed by one hundred zeroes, seems paltry. This gives a good indication of the notational difficulties encountered by Archimedes, and one can propose that he stopped at this number because he did not devise any new There was an error working with the wiki: Code[95] (larger than 'myriad myriadth') to match his new There was an error working with the wiki: Code[96]. Archimedes also discovered and proved the law of exponents

: 10^a 10^b = 10^{a+b}

necessary to manipulate powers of 10. Archimedes then sets about estimating an upper bound for the number of grains of sand. Not wanting to be outdone, he counts not only the grains of sand on a beach, but on the entire earth, the earth filled with sand, and then in a universe filled with sand. He then estimates this for the largest model of the universe yet proposed, the There was an error working with the wiki: Code[97] of There was an error working with the wiki: Code[98] (in fact, this now lost work is known due to this reference). The reason for this is that a heliocentric model must be much larger if There was an error working with the wiki: Code[99] is not clearly measurable. Archimedes proceeds by giving upper bounds for the diameter of the earth, the distance from the earth to the sun, and the diameter of the universe. In order

to do this last step, he assumes that the ratio of the diameter of the universe to the diameter of the orbit of the Earth around the Sun, equals the ratio of Earth's solar-orbital diameter to the diameter of the Sun. This simply says that stellar parallax equals solar parallax, and one can interpret this as Archimedes' reason for using this assumption, which is not clearly explained in the text. The resulting estimate is that the radius of the universe is about one There was an error working with the wiki: Code[100], which is consistent with current estimates for the radius of the solar system. Archimedes' final estimate gives an upper bound of 10^{64} for the number of grains of sand in a filled universe.

Archimedes makes some interesting experiments and computations along the way. One experiment estimates the angular size of the sun, as seen from the earth. Archimedes' method is especially interesting as it may be the first known example of experimentation in There was an error working with the wiki: Code[101], the branch of There was an error working with the wiki: Code[102] dealing with the mechanics of human perception, and whose development is generally attributed to There was an error working with the wiki: Code[103] (this work of Archimedes is not well known in psychology). In particular, Archimedes takes into account the size and shape of the eye in his experiment measuring the angular diameter of the sun. Another interesting computation accounts for solar parallax, in particular, the differences in distance from the sun, whether taken from the center of the earth or from the surface of the earth at sunrise. Once again, this may be the first known computation dealing with solar parallax.

##### ''

In this work, which was unknown in the Middle Ages, but the importance of which was realised after its discovery, Archimedes pioneers the use of There was an error working with the wiki: Code[37] and other authors. Archimedes lived in the There was an error working with the wiki: Code[38], but the copy was made in the There was an error working with the wiki: Code[39] text. It was a book of nearly 90 pages before being made a palimpsest of 177 pages the older leaves folded so that each became two leaves of the liturgical book. The erasure was incomplete, and Archimedes' work is now readable using digital processing of There was an error working with the wiki: Code[104], There was an error working with the wiki: Code[105], and visible light. Archimedes probably considered these methods not mathematically precise, and he used these methods to find at least some of the areas or volumes he sought, and then used the more traditional There was an error working with the wiki: Code[106] to prove them.

##### What Archimedes did

Although the only mathematical tools at its author's disposal were what we might now consider secondary-school There was an error working with the wiki: Code[40]. Among those problems were that of calculating the There was an error working with the wiki: Code[41]'s Stereometria.

Some pages of the Method remained unused by the author of the Palimpsest and thus they are still&mdashprobably forever&mdashlost. Between them, an announced result concerned the volume of the intersection of two cylinders, a figure that Apostol and Mnatsakian have renamed n&nbsp=&nbsp4 Archimedean globe (and the half of it, n&nbsp=&nbsp4 Archimedean dome), whose volume relates to the n-polygonal pyramid. This is amusing because the collaboration on indivisibles between Galileo and Cavalieri&mdashranging between years 1626 to around 1635&mdashhas as a main argument the hull and pyramid of the n&nbsp=&nbsp? dome. So in some sense it is true that the Method is only a theorem behind the modern infinitesimal theory.

In Heiberg's time, much attention was paid to Archimedes' brilliant use of infinitesimals to solve problems about areas, volumes, and centers of gravity. Less attention was given to the Stomachion, a problem treated in the Palimpsest that appears to deal with a children's puzzle. Reviel Netz of There was an error working with the wiki: Code[107] has shown that Archimedes found that the number of ways to solve the puzzle is 17,152. This is perhaps the most sophisticated work in the field of There was an error working with the wiki: Code[108] in classical antiquity.

##### Use of infinitesimals

Archimedes was the first mathematician to make explicit use of There was an error working with the wiki: Code[42], and therefore said explicitly that his arguments fall short of being finished mathematical proofs. The proof of the first proposition in the palimpsest appears below.

##### The first proposition in the palimpsest

The curve in this figure is a There was an error working with the wiki: Code[109]. It the points A and B are on the curve, the line AC is parallel to the axis of the parabola. The line BC is There was an error working with the wiki: Code[110] to the parabola.

The first proposition states:

The area of the triangle ABC is exactly three times

the area bounded by the parabola and the secant line AB.

Proof: Let D be the midpoint of AC. The point D is the fulcrum of a lever, which is the line JB. The points J and B are equidistant from the fulcrum. As Archimedes had shown, the center of gravity of the interior of the triangle is at a point I on the "lever" so located that DI:DB = 1:3. Therefore, it suffices to show that if the whole weight of the interior of the triangle rests at I, and the whole weight of the section of the parabola at J, the lever is in equilibrium. If the whole weight of the triangle rests at I, it exerts the same torque on the lever as if the infinitely small weight of every cross-section EH parallel to the axis of the parabola rests at the point G where it intersects the lever. Therefore, it suffices to show that if the weight of that cross-section rests at G and the weight of the cross-section EF of the section of the parabola rests at J, then the lever is in equilibrium. In other words, it suffices to show that EF:GD = EH:JD. That is equivalent to EF:DG = EH:DB. And that is equivalent to EF:EH = AE:AB. But that is just the equation of the parabola. There was an error working with the wiki: Code[111].

##### Other propositions in the palimpsest

A series of other propositions of geometry are proved in the palimpsest by similar arguments. Some of them have the location of a center of gravity as the conclusion. One of those states that the center of gravity of the interior of a hemisphere is located 5/8 of the way from the pole to the center of the sphere.

##### The Palimpsest's modern career

From the There was an error working with the wiki: Code[43] versus There was an error working with the wiki: Code[44] grounds, and the palimpsest was bought for \$2 million by an anonymous information technology person.

The palimpsest is now at the There was an error working with the wiki: Code[45], where conservation continues (as it had suffered considerably from There was an error working with the wiki: Code[112]). A more accurate edition of the manuscript, including its drawn geometrical figures, is expected, possibly in 2007.

A team of imaging scientists from the There was an error working with the wiki: Code[113] and Johns Hopkins University has used computer processing of digital images from various spectral bands, including ultraviolet and visible light, to reveal the Archimedes text. Dr. Reviel Netz of [[Stanford University] has been trying to fill in gaps in Heiberg's account with these images.

Four pages that had been painted over with Byzantine-style religious images, which turned out to be 20th-century forgeries intended to increase the value of the prayer book, rendered the underlying text of Archimedes forever illegible, it appeared. Then, in May 2005, highly focused There was an error working with the wiki: Code[114] produced at the There was an error working with the wiki: Code[115] in Menlo Park, California were used to begin deciphering the parts of the 174-page text that have not yet been revealed. The production of x-ray There was an error working with the wiki: Code[116] was described by Keith Hodgson, director of SSRL. "There was an error working with the wiki: Code[117] is created when electrons traveling near the speed of light take a curved path around a storage ring&mdashemitting electromagnetic light in X-ray through infrared wavelengths. The resulting light beam has characteristics that make it ideal for revealing the intricate architecture and utility of many kinds of matter—in this case, the previously hidden work of one of the founding fathers of all science." http://news-service.stanford.edu/news/2005/may25/archimedes-052505.html.

#### Quotes

?'???. (Eureka!)

Translations: " I have found it!" or "I have got it!"

Said to be what he exclaimed as he ran from his bath (without clothes), realizing that by measuring the displacement of water an object produced, compared to its weight, he could measure its density (and thus determine the proportion of gold that was used in making a king's crown).

??? ??? ??? ??? ??? ???\omega ??? ??? (Dos moi pou sto kai kino taen gaen)

Doric Greek: ??? ??? ?' ?? ??? ??? ?? ??? ??????

Translations: "Give me the place to stand, and I shall move the earth." or "Give me a place to stand, and I shall move the world." or "Give me a fulcrum, and I shall move the world."

Said to be his assertion in demonstrating the principle of the lever.

"?? ??? ???? ?'????? ??'????!" (in Greek)

"Noli turbare circulos meos."

"There was an error working with the wiki: Code[118] circulos meos"

Translation: "Do not disturb my circles!"

Comment: Uttered to a Roman soldier who, despite being given orders not to, killed Archimedes at the conquest of Syracuse.

#### Related items

The There was an error working with the wiki: Code[46] on the There was an error working with the wiki: Code[119] which was named in his honor.

#### External articles and references

Sources

There was an error working with the wiki: Code[120], Archimedes, There was an error working with the wiki: Code[121], Princeton University Press, Princeton, ISBN 0-691-08421-1. Republished translation of the 1938 study of Archimedes and his works by an historian of science

Fred S. Kliner Christin J. Mamiya, "Gardener's Art Through the Ages" twelfth ed. Vol II 2005, Thompson Wadsworth- Los Angeles

Episode 55: Steam Cannon/Breakfast Cereal

Archimedes (287-212 B.C.), Greatest Scientist Ever" There was an error working with the wiki: Code[47] http://www.idsia.ch/~juergen/archimedes.html

There was an error working with the wiki: Code[48] and There was an error working with the wiki: Code[49] (There was an error working with the wiki: Code[122]) ISBN 0-387-98434-8 (Hardcover) ISBN 0-387-98433-X (Paperback) p. 95,

There was an error working with the wiki: Code[1], Wikipedia: The Free Encyclopedia. Wikimedia Foundation.

Original Greek text

The Sand Reckoner

The Sand Reckoner (annotated)

Archimedes, The Sand Reckoner, by Ilan Vardi, includes a literal English version of the original Greek text, as well as details of assertions made above

The Archimedes Palimpsest Project Web Page

The Archimedes Palimpsest web pages at the Walters Art Museum

The Nova Program outlined

The Nova Program teacher's version

May 2005 Stanford Report: Heather Rock Woods, May 19, 2005

The Greek Orthodox Patriarchate of Jerusalem v. Christies’s Inc., 1999 U.S. Dist. LEXIS 13257 (S.D. N.Y. 1999)

General

Archimedes' Book of Lemmas at There was an error working with the wiki: Code[123]

Archimedes and the Rhombicuboctahedron by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.

The Archimedes Palimpsest web pages at the There was an error working with the wiki: Code[124].

NOVA program on Archimedes Palimpsest

The Archimedes Palimpsest project at The Walters Art Museum in Baltimore, Maryland

Archimedes - The Golden Crown points out that in reality Archimedes may well have used a more subtle method than the one in the classic version of the story.

Archimedes' Quadrature Of The Parabola Translated by There was an error working with the wiki: Code[125].

Archimedes' On The Measurement Of The Circle Translated by There was an error working with the wiki: Code[126].

Archimedes' Cattle Problem

Archimedes' Cattle Problem

Angle Trisection by Archimedes of Syracuse (Java)

Archimedes'Triangle (Java)

An ancient extra-geometric proof

Archimedes' Squaring of Parabola (Java) at There was an error working with the wiki: Code[127]

Archimedes and his Burning Mirrors, Reality or Fantasy?

Biography of Archimedes

Squaring the circle History Topic at MacTutor