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Archimedes of Syracuse

Born: 287 BC in Syracuse, Sicily (now Italy)Died: 212 BC in Syracuse, Sicily (now Italy)

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Archimedes' father was Phidias, an astronomer. We know nothing else about Phidias otherthan this one fact and we only know this since Archimedes gives us this information in one ofhis works, The Sandreckoner. A friend of Archimedes called Heracleides wrote a biographyof him but sadly this work is lost. How our knowledge of Archimedes would be transformedif this lost work were ever found, or even extracts found in the writing of others.

Archimedes was a native of Syracuse, Sicily. It is reported by some authors that he visitedEgypt and there invented a device now known asArchimedes' screw. This is a pump, stillused in many parts of the world. It is highly likely that, when he was a young man,Archimedes studied with the successors ofEuclidin Alexandria. Certainly he was completelyfamiliar with the mathematics developed there, but what makes this conjecture much morecertain, he knew personally the mathematicians working there and he sent his results toAlexandria with personal messages. He regardedConon of Samos, one of the mathematiciansat Alexandria, both very highly for his abilities as a mathematician and he also regarded himas a close friend.

In the preface to On spirals Archimedes relates an amusing story regarding his friends inAlexandria. He tells us that he was in the habit of sending them statements of his latesttheorems, but without giving proofs. Apparently some of the mathematicians there hadclaimed the results as their own so Archimedes says that on the last occasion when he sent

them theorems he included two which were false [3]:-
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... 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.

Other than in the prefaces to his works, information about Archimedes comes to us from anumber of sources such as in stories fromPlutarch,Livy, and others. Plutarch tells us that

Archimedes was related to King Hieron II of Syracuse (see for example [3]):-

Archimedes ... in writing to King Hiero, whose friend and near relation he was....

Again evidence of at least his friendship with the family of King Hieron II comes from thefact that The Sandreckonerwas dedicated to Gelon, the son of King Hieron.

There are, in fact, quite a number of references to Archimedes in the writings of the time forhe had gained a reputation in his own time which few other mathematicians of this periodachieved. The reason for this was not a widespread interest in new mathematical ideas butrather that Archimedes had invented many machines which were used as engines of war.

These were particularly effective in the defence of Syracuse when it was attacked by theRomans under the command ofMarcellus.

Plutarch writes in his work on Marcellus, the Roman commander, about how Archimedes'engines of war were used against the Romans in the siege of 212 BC:-

... 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 themariners were all thrown out, when at length it was dashed against the rocks, or let fall.

Archimedes had been persuaded by his friend and relation King Hieron to build suchmachines:-

These machines [Archimedes] had designed and contrived, not as matters of any importance,but as mere amusements in geometry; in compliance with King Hiero's desire and request,

some little time before, that he should reduce to practice some part of his admirable

speculation in science, and by accommodating the theoretic truth to sensation and ordinary

use, bring it more within the appreciation of the people in general.

Perhaps it is sad that engines of war were appreciated by the people of this time in a way thattheoretical mathematics was not, but one would have to remark that the world is not a verydifferent place at the end of the second millenium AD. Other inventions of Archimedes suchas the compound pulley also brought him great fame among his contemporaries. Again we

quote Plutarch:-
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[Archimedes] had stated[in a letter to King Hieron] that given the force, any given weightmight be moved, and even boasted, we are told, relying on the strength of demonstration, that

if there were another earth, by going into it he could remove this. Hiero being struck with

amazement at this, and entreating him to make good this problem by actual experiment, and

show some great weight moved by a small engine, he fixed accordingly upon a ship of burden

out of the king's arsenal, which could not be drawn out of the dock without great labour andmany men; and, loading her with many passengers and a full freight, sitting himself the while

far off, with no great endeavour, but only holding the head of the pulley in his hand and

drawing the cords by degrees, he drew the ship in a straight line, as smoothly and evenly as if

she had been in the sea.

Yet Archimedes, although he achieved fame by his mechanical inventions, believed that puremathematics was the only worthy pursuit. Again Plutarch describes beautifully Archimedesattitude, yet we shall see later that Archimedes did in fact use some very practical methods todiscover results from pure geometry:-

Archimedes possessed so high a spirit, so profound a soul, and such treasures of scientificknowledge, that though these inventions had now obtained him the renown of more than

human sagacity, he yet would not deign to leave behind him any commentary or writing on

such subjects; but, repudiating as sordid and ignoble the whole trade of engineering, and

every sort of art that lends itself to mere use and profit, he placed his whole affection and

ambition in those purer speculations where there can be no reference to the vulgar needs of

life; studies, the superiority of which to all others is unquestioned, and in which the only

doubt can be whether the beauty and grandeur of the subjects examined, of the precision and

cogency of the methods and means of proof, most deserve our admiration.

His fascination with geometry is beautifully described by Plutarch:-

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 was he taken from himself, and

brought into ecstasy or trance, with the delight he had in the study of geometry.

The achievements of Archimedes are quite outstanding. He is considered by most historiansof mathematics as one of the greatest mathematicians of all time. He perfected a method ofintegration which allowed him to find areas, volumes and surface areas of many bodies.

Chaslessaid that Archimedes' work on integration (see [7]):-

... gave birth to the calculus of the infinite conceived and brought to perfection byKepler,

Cavalieri,Fermat,LeibnizandNewton.

Archimedes was able to apply themethod of exhaustion, which is the early form ofintegration, to obtain a whole range of important results and we mention some of these in thedescriptions of his works below. Archimedes also gave an accurate approximation to andshowed that he could approximate square roots accurately. He invented a system forexpressing large numbers. In mechanics Archimedes discovered fundamental theoremsconcerning the centre of gravity of plane figures and solids. His most famous theorem gives

the weight of a body immersed in a liquid, calledArchimedes' principle.
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The works of Archimedes which have survived are as follows. On plane equilibriums (twobooks),Quadratureof theparabola,On the sphere and cylinder(two books), On spirals, Onconoids and spheroids, On floating bodies (two books),Measurement of a circle, and TheSandreckoner. In the summer of 1906, J L Heiberg, professor of classical philology at theUniversity of Copenhagen, discovered a 10th century manuscript which included Archimedes'

workThe method. This provides a remarkable insight into how Archimedes discovered manyof his results and we will discuss this below once we have given further details of what is inthe surviving books.

The order in which Archimedes wrote his works is not known for certain. We have used thechronological order suggested byHeathin [7] in listing these works above, except for The

MethodwhichHeathhas placed immediately before On the sphere and cylinder. The paper[47] looks at arguments for a different chronological order of Archimedes' works.

The treatise On plane equilibriums sets out the fundamental principles of mechanics, usingthe methods of geometry. Archimedes discovered fundamental theorems concerning the

centre of gravity of plane figures and these are given in this work. In particular he finds, inbook 1, the centre of gravity of a parallelogram, a triangle, and a trapezium. Book two isdevoted entirely to finding the centre of gravity of a segment of a parabola. In the Quadratureof the parabola Archimedes finds the area of a segment of a parabola cut off by any chord.

In the first book ofOn the sphere and cylinderArchimedes shows that the surface of a sphereis four times that of agreat circle, he finds the area of any segment of a sphere, he shows thatthe volume of a sphere is two-thirds the volume of acircumscribedcylinder, and that thesurface of a sphere is two-thirds the surface of a circumscribed cylinder including its bases. Agood discussion of how Archimedes may have been led to some of these results usinginfinitesimalsis given in [14]. In the second book of this work Archimedes' most importantresult is to show how to cut a given sphere by a plane so that the ratio of the volumes of thetwo segments has a prescribed ratio.

In On spirals Archimedes defines a spiral, he gives fundamental properties connecting thelength of the radius vector with the angles through which it has revolved. He gives results ontangentsto the spiral as well as finding the area of portions of the spiral. In the work Onconoids and spheroids Archimedes examines paraboloids of revolution, hyperboloids ofrevolution, and spheroids obtained by rotating anellipseeither about its major axis or aboutits minor axis. The main purpose of the work is to investigate the volume of segments ofthese three-dimensional figures. Some claim there is a lack of rigour in certain of the results

of this work but the interesting discussion in [43] attributes this to a modern dayreconstruction.

On floating bodies is a work in which Archimedes lays down the basic principles ofhydrostatics. His most famous theorem which gives the weight of a body immersed in aliquid, calledArchimedes' principle, is contained in this work. He also studied the stability ofvarious floating bodies of different shapes and different specific gravities. InMeasurement ofthe CircleArchimedes shows that the exact value of lies between the values 310/71 and 3

1/7.This he obtained by circumscribing and inscribing a circle with regular polygons having 96sides.

The Sandreckoneris a remarkable work in which Archimedes proposes a number systemcapable of expressing numbers up to 8 1063 in modern notation. He argues in this work that
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this number is large enough to count the number of grains of sand which could be fitted intothe universe. There are also important historical remarks in this work, for Archimedes has togive the dimensions of the universe to be able to count the number of grains of sand which itcould contain. He states thatAristarchushas proposed a system with the sun at the centre andthe planets, including the Earth, revolving round it. In quoting results on the dimensions he

states results due toEudoxus, Phidias (his father), and toAristarchus. There are other sourceswhich mention Archimedes' work on distances to the heavenly bodies. For example in [59]Osborne reconstructs and discusses:-

...a theory of the distances of the heavenly bodies ascribed to Archimedes, but the corrupt

state of the numerals in the sole surviving manuscript[due to Hippolytus of Rome, about220AD] means that the material is difficult to handle.

In theMethod, Archimedes described the way in which he discovered many of hisgeometrical results (see [7]):-

... certain things first became clear to me by a mechanical method, although they had to beproved by geometry afterwards because their investigation by the said method did not furnish

an actual proof. But it is of course easier, when we have previously acquired, by the method,

some knowledge of the questions, to supply the proof than it is to find it without any previous

knowledge.

Perhaps the brilliance of Archimedes' geometrical results is best summed up by Plutarch, whowrites:-

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.

Heathadds his opinion of the quality of Archimedes' work[7]:-

The treatises are, without exception, monuments of mathematical exposition; the gradual

revelation of the plan of attack, the masterly ordering of the propositions, the stern

elimination of everything not immediately relevant to the purpose, the finish of the whole, are

so impressive in their perfection as to create a feeling akin to awe in the mind of the reader.

There are references to other works of Archimedes which are now lost.Pappusrefers to awork by Archimedes on semi-regular polyhedra, Archimedes himself refers to a work on thenumber system which he proposed in the Sandreckoner,Pappusmentions a treatise Onbalances and levers, and Theon mentions a treatise by Archimedes about mirrors. Evidencefor further lost works are discussed in [67] but the evidence is not totally convincing.

Archimedes was killed in 212 BC during the capture of Syracuse by the Romans in theSecond Punic War after all his efforts to keep the Romans at bay with his machines of warhad failed. Plutarch recounts three versions of the story of his killing which had come down

to him. The first version:-
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Archimedes ... was ..., as fate would have it, intent upon working out some problem by adiagram, and having fixed his mind alike and his eyes upon the subject of his speculation, he

never noticed the incursion of the Romans, nor that the city was taken. In this transport of

study and contemplation, a soldier, unexpectedly coming up to him, commanded him to

follow to Marcellus; which he declining to do before he had worked out his problem to a

demonstration, the soldier, enraged, drew his sword and ran him through.

The second version:-

... a Roman soldier, running upon him with a drawn sword, offered to kill him; and that

Archimedes, looking back, earnestly besought him to hold his hand a little while, that he

might not leave what he was then at work upon inconclusive and imperfect; but the soldier,

nothing moved by his entreaty, instantly killed him.

Finally, the third version that Plutarch had heard:-

... as Archimedes was carrying to Marcellus mathematical instruments, dials, spheres, andangles, by which the magnitude of the sun might be measured to the sight, some soldiers

seeing him, and thinking that he carried gold in a vessel, slew him.

Archimedes considered his most significant accomplishments were those concerning acylinder circumscribing a sphere, and he asked for a representation of this together with hisresult on the ratio of the two, to be inscribed on his tomb.Cicerowas in Sicily in 75 BC andhe writes how he searched for Archimedes tomb (see for example [1]):-

... 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

away.

It is perhaps surprising that the mathematical works of Archimedes were relatively littleknown immediately after his death. As Clagett writes in [1]:-

Unlike the Elements ofEuclid, the works of Archimedes were not widely known in antiquity.... It is true that ... individual works of Archimedes were obviously studied at Alexandria,

since Archimedes was often quoted by three eminent mathematicians of Alexandria:Heron,

PappusandTheon.

Only afterEutociusbrought out editions of some of Archimedes works, with commentaries,in the sixth century AD were the remarkable treatises to become more widely known. Finally,it is worth remarking that the test used today to determine how close to the original text thevarious versions of his treatises of Archimedes are, is to determine whether they haveretained Archimedes' Dorian dialect.
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7/22

Archimedes

From Wikipedia, the free encyclopedia

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For other uses, seeArchimedes (disambiguation).

Archimedes of Syracuse

(Greek:)

Archimedes ThoughtfulbyFetti(1620)

Born

c. 287 BC

Syracuse, Sicily

Magna Graecia

Diedc. 212 BC (aged around 75)

Syracuse

Residence Syracuse, Sicily

FieldsMathematics,Physics,Engineering,Astronomy,

Invention

Known for Archimedes' Principle,Archimedes' screw,Hydrostatics,
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Levers,Infinitesimals

Archimedes of Syracuse(Greek:;c. 287 BCc. 212 BC) was aGreekmathematician,physicist,engineer,inventor, andastronomer. Although few details of his lifeare known, he is regarded as one of the leadingscientistsinclassical antiquity. Among hisadvances inphysicsare the foundations ofhydrostatics,staticsand an explanation of theprinciple of thelever. He is credited with designing innovativemachines, including siegeengines and thescrew pumpthat bears his name. Modern experiments have tested claims thatArchimedes designed machines capable of lifting attacking ships out of the water and settingships on fire using an array of mirrors.[1]

Archimedes is generally considered to be the greatestmathematicianof antiquity and one ofthe greatest of all time.[2][3]He used themethod of exhaustionto calculate theareaunder thearc of aparabolawith thesummation of an infinite series, and gave a remarkably accurate

approximation ofpi.

[4]

He also defined thespiralbearing his name, formulae for thevolumesofsurfaces of revolutionand an ingenious system for expressing very large numbers.

Archimedes died during theSiege of Syracusewhen he was killed by aRomansoldierdespite orders that he should not be harmed.Cicerodescribes visiting the tomb ofArchimedes, which was surmounted by asphereinscribedwithin acylinder. Archimedes hadproven that the sphere has two thirds of the volume and surface area of the cylinder(including the bases of the latter), and regarded this as the greatest of his mathematicalachievements.

Unlike his inventions, the mathematical writings of Archimedes were little known in

antiquity. Mathematicians fromAlexandriaread and quoted him, but the first comprehensivecompilation was not made until c. 530 AD byIsidore of Miletus, while commentaries on theworks of Archimedes written byEutociusin the sixth century AD opened them to widerreadership for the first time. The relatively few copies of Archimedes' written work thatsurvived through theMiddle Ageswere an influential source of ideas for scientists during theRenaissance,[5]while the discovery in 1906 of previously unknown works by Archimedes intheArchimedes Palimpsesthas provided new insights into how he obtained mathematicalresults.[6]
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ki/Middle_Ageshttp://en.wikipedia.org/wiki/Middle_Ageshttp://en.wikipedia.org/wiki/Middle_Ageshttp://en.wikipedia.org/wiki/Renaissancehttp://en.wikipedia.org/wiki/Archimedes#cite_note-4http://en.wikipedia.org/wiki/Archimedes#cite_note-4http://en.wikipedia.org/wiki/Archimedes#cite_note-4http://en.wikipedia.org/wiki/Archimedes_Palimpsesthttp://en.wikipedia.org/wiki/Archimedes_Palimpsesthttp://en.wikipedia.org/wiki/Archimedes_Palimpsesthttp://en.wikipedia.org/wiki/Archimedes#cite_note-5http://en.wikipedia.org/wiki/Archimedes#cite_note-5http://en.wikipedia.org/wiki/Archimedes#cite_note-5http://en.wikipedia.org/wiki/Archimedes#cite_note-5http://en.wikipedia.org/wiki/Archimedes_Palimpsesthttp://en.wikipedia.org/wiki/Archimedes#cite_note-4http://en.wikipedia.org/wiki/Renaissancehttp://en.wikipedia.org/wiki/Middle_Ageshttp://en.wikipedia.org/wiki/Eutocius_of_Ascalonhttp://en.wikipedia.org/wiki/Isidore_of_Miletushttp://en.wikipedia.org/wiki/Alexandriahttp://en.wikipedia.org/wiki/Cylinder_(geometry)http://en.wikipedia.org/wiki/Inscribehttp://en.wikipedia.org/wiki/Spherehttp://en.wikipedia.org/wiki/Cicerohttp://en.wikipedia.org/wiki/Roman_Republichttp://en.wikipedia.org/wiki/Siege_of_Syracuse_(214%E2%80%93212_BC)http://en.wikipedia.org/wiki/Surface_of_revolutionhttp://en.wikipedia.org/wiki/Volumehttp://en.wikipedia.org/wiki/Archimedes_spiralhttp://en.wikipedia.org/wiki/Archimedes#cite_note-3http://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/Series_(mathematics)http://en.wikipedia.org/wiki/Parabolahttp://en.wikipedia.org/wiki/Areahttp://en.wikipedia.org/wiki/Method_of_exhaustionhttp://en.wikipedia.org/wiki/Archimedes#cite_note-1http://en.wikipedia.org/wiki/Archimedes#cite_note-1http://en.wikipedia.org/wiki/Mathematicianhttp://en.wikipedia.org/wiki/Archimedes#cite_note-death_ray-0http://en.wikipedia.org/wiki/Archimedes%27_screwhttp://en.wikipedia.org/wiki/Machinehttp://en.wikipedia.org/wiki/Leverhttp://en.wikipedia.org/wiki/Staticshttp://en.wikipedia.org/wiki/Fluid_staticshttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Classical_antiquityhttp://en.wikipedia.org/wiki/Scientisthttp://en.wikipedia.org/wiki/Astronomerhttp://en.wikipedia.org/wiki/Inventorhttp://en.wikipedia.org/wiki/Engineerhttp://en.wikipedia.org/wiki/Physicisthttp://en.wikipedia.org/wiki/Greek_mathematicshttp://en.wikipedia.org/wiki/Greek_mathematicshttp://en.wiktionary.org/wiki/%E1%BC%88%CF%81%CF%87%CE%B9%CE%BC%CE%AE%CE%B4%CE%B7%CF%82http://en.wikipedia.org/wiki/Ancient_Greekhttp://en.wikipedia.org/wiki/Archimedes%27_use_of_infinitesimalshttp://en.wikipedia.org/wiki/Lever


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Contents

1 Biography 2 Discoveries and inventions

o 2.1 The Golden Crowno 2.2 The Archimedes Screwo 2.3 The Claw of Archimedeso 2.4 The Archimedes Heat Rayo 2.5 Other discoveries and inventions

3 Mathematics 4 Writings

o 4.1 Surviving workso 4.2 Apocryphal works

5 Archimedes Palimpsest 6 Legacy 7 See also 8 Notes and references

o 8.1 Noteso 8.2 References

9 Further reading 10 The Works of Archimedes online 11 External links

Biography

This bronze statue of Archimedes is at theArchenhold ObservatoryinBerlin. It was sculpted by

Gerhard Thieme and unveiled in 1972.

Archimedes was born c. 287 BC in the seaport city ofSyracuse, Sicily, at that time a self-governingcolonyinMagna Graecia. The date of birth is based on a statement by theByzantine GreekhistorianJohn Tzetzesthat Archimedes lived for 75 years.[7]InThe Sand

Reckoner, Archimedes gives his father's name as Phidias, anastronomerabout whom nothingis known.Plutarchwrote in hisParallel Livesthat Archimedes was related to KingHiero II,the ruler of Syracuse.[8]A biography of Archimedes was written by his friend Heracleides butthis work has been lost, leaving the details of his life obscure.[9]It is unknown, for instance,

whether he ever married or had children. During his youth Archimedes may have studied inAlexandria,Egypt, whereConon of SamosandEratosthenes of Cyrenewere contemporaries.
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a.org/wiki/Eratostheneshttp://en.wikipedia.org/wiki/Conon_of_Samoshttp://en.wikipedia.org/wiki/Ancient_Egypthttp://en.wikipedia.org/wiki/Alexandriahttp://en.wikipedia.org/wiki/Archimedes#cite_note-mactutor-8http://en.wikipedia.org/wiki/Archimedes#cite_note-7http://en.wikipedia.org/wiki/Hiero_II_of_Syracusehttp://en.wikipedia.org/wiki/Parallel_Liveshttp://en.wikipedia.org/wiki/Plutarchhttp://en.wikipedia.org/wiki/Astronomerhttp://en.wikipedia.org/wiki/The_Sand_Reckonerhttp://en.wikipedia.org/wiki/The_Sand_Reckonerhttp://en.wikipedia.org/wiki/Archimedes#cite_note-6http://en.wikipedia.org/wiki/John_Tzetzeshttp://en.wikipedia.org/wiki/Byzantine_Greekshttp://en.wikipedia.org/wiki/Magna_Graeciahttp://en.wikipedia.org/wiki/Colonies_in_antiquityhttp://en.wikipedia.org/wiki/Syracuse,_Sicilyhttp://en.wikipedia.org/wiki/Berlinhttp://en.wikipedia.org/wiki/Archenhold_Observatoryhttp://en.wikipedia.org/wiki/File:Gerhard_Thieme_Archimedes.jpghttp://en.wikipedia.org/wiki/Archimedes#External_linkshttp://en.wikipedia.org/wiki/Archimedes#The_Works_of_Archimedes_onlinehttp://en.wikipedia.org/wiki/Archimedes#Further_readinghttp://en.wikipedia.org/wiki/Archimedes#Referenceshttp://en.wikipedia.org/wiki/Archimedes#Noteshttp://en.wikipedia.org/wiki/Archimedes#Notes_and_referenceshttp://en.wikipedia.org/wiki/Archimedes#See_alsohttp://en.wikipedia.org/wiki/Archimedes#Legacyhttp://en.wikipedia.org/wiki/Archimedes#Archimedes_Palimpsesthttp://en.wikipedia.org/wiki/Archimedes#Apocryphal_workshttp://en.wikipedia.org/wiki/Archimedes#Surviving_workshttp://en.wikipedia.org/wiki/Archimedes#Writingshttp://en.wikipedia.org/wiki/Archimedes#Mathematicshttp://en.wikipedia.org/wiki/Archimedes#Other_discoveries_and_inventionshttp://en.wikipedia.org/wiki/Archimedes#The_Archimedes_Heat_Rayhttp://en.wikipedia.org/wiki/Archimedes#The_Claw_of_Archimedeshttp://en.wikipedia.org/wiki/Archimedes#The_Archimedes_Screwhttp://en.wikipedia.org/wiki/Archimedes#The_Golden_Crownhttp://en.wikipedia.org/wiki/Archimedes#Discoveries_and_inventionshttp://en.wikipedia.org/wiki/Archimedes#Biography


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He referred to Conon of Samos as his friend, while two of his works (The Method ofMechanical Theoremsand theCattle Problem) have introductions addressed toEratosthenes.[a]

Archimedes died c. 212 BC during theSecond Punic War, when Roman forces under General

Marcus Claudius Marcelluscaptured the city of Syracuse after a two-year-longsiege.According to the popular account given byPlutarch, Archimedes was contemplating amathematical diagramwhen the city was captured. A Roman soldier commanded him tocome and meet General Marcellus but he declined, saying that he had to finish working onthe problem. The soldier was enraged by this, and killed Archimedes with his sword. Plutarchalso gives a lesser-known account of the death of Archimedes which suggests that he mayhave been killed while attempting to surrender to a Roman soldier. According to this story,Archimedes was carrying mathematical instruments, and was killed because the soldierthought that they were valuable items. General Marcellus was reportedly angered by thedeath of Archimedes, as he considered him a valuable scientific asset and had ordered that henot be harmed.[10]

A sphere has 2/3 the volume and surface area of its circumscribing cylinder. Asphereandcylinder

were placed on the tomb of Archimedes at his request.

The last words attributed to Archimedes are "Do not disturb my circles" (Greek:

), a reference to the circles in the mathematical drawing that he wassupposedly studying when disturbed by the Roman soldier. This quote is often given inLatinas "Noli turbare circulos meos," but there is no reliable evidence that Archimedes utteredthese words and they do not appear in the account given by Plutarch.[10]

The tomb of Archimedes carried a sculpture illustrating his favorite mathematical proof,consisting of asphereand acylinderof the same height and diameter. Archimedes hadproven that the volume and surface area of the sphere are two thirds that of the cylinderincluding its bases. In 75 BC, 137 years after his death, the Roman oratorCicerowas servingasquaestorinSicily. He had heard stories about the tomb of Archimedes, but none of thelocals was able to give him the location. Eventually he found the tomb near the Agrigentine

gate in Syracuse, in a neglected condition and overgrown with bushes. Cicero had the tombcleaned up, and was able to see the carving and read some of the verses that had been added
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as an inscription.[11]A tomb discovered in a hotel courtyard in Syracuse in the early 1960swas claimed to be that of Archimedes, but its location today is unknown.[12]

The standard versions of the life of Archimedes were written long after his death by thehistorians of Ancient Rome. The account of the siege of Syracuse given byPolybiusin his

Universal History was written around seventy years after Archimedes' death, and was usedsubsequently as a source by Plutarch andLivy. It sheds little light on Archimedes as a person,and focuses on the war machines that he is said to have built in order to defend the city.[13]

Discoveries and inventions

The Golden Crown

Archimedes may have used his principle of buoyancy to determine whether the golden crown was

lessdensethan solid gold.

The most widely knownanecdoteabout Archimedes tells of how he invented a method fordetermining the volume of an object with an irregular shape. According toVitruvius, avotivecrownfor a temple had been made for King Hiero II, who had supplied the pure goldto beused, and Archimedes was asked to determine whether somesilverhad been substituted bythe dishonest goldsmith.[14]Archimedes had to solve the problem without damaging the

crown, so he could not melt it down into a regularly shaped body in order to calculate itsdensity. While taking a bath, he noticed that the level of the water in the tub rose as he got in,and realized that this effect could be used to determine thevolumeof the crown. For practicalpurposes water is incompressible,[15]so the submerged crown would displace an amount ofwater equal to its own volume. By dividing the mass of the crown by the volume of waterdisplaced, the density of the crown could be obtained. This density would be lower than thatof gold if cheaper and less dense metals had been added. Archimedes then took to the streetsnaked, so excited by his discovery that he had forgotten to dress, crying "Eureka!" (Greek:"!," meaning "I have found it!"). The test was conducted successfully, proving that

silver had indeed been mixed in.[16]

The story of the golden crown does not appear in the known works of Archimedes.Moreover, the practicality of the method it describes has been called into question, due to the
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extreme accuracy with which one would have to measure the water displacement.[17]Archimedes may have instead sought a solution that applied the principle known inhydrostaticsasArchimedes' Principle, which he describes in his treatise On Floating Bodies.This principle states that a body immersed in a fluid experiences abuoyant forceequal to theweight of the fluid it displaces.[18]Using this principle, it would have been possible to

compare the density of the golden crown to that of solid gold by balancing the crown on ascale with a gold reference sample, then immersing the apparatus in water. If the crown wasless dense than gold, it would displace more water due to its larger volume, and thusexperience a greater buoyant force than the reference sample. This difference in buoyancywould cause the scale to tip accordingly.Galileoconsidered it "probable that this method isthe same that Archimedes followed, since, besides being very accurate, it is based ondemonstrations found by Archimedes himself."[19]

The Archimedes Screw

TheArchimedes screwcan raise water efficiently.

A large part of Archimedes' work in engineering arose from fulfilling the needs of his homecity of Syracuse. The Greek writerAthenaeus of Naucratisdescribed how King Hieron IIcommissioned Archimedes to design a huge ship, theSyracusia, which could be used forluxury travel, carrying supplies, and as a naval warship. The Syracusia is said to have beenthe largest ship built in classical antiquity.[20]According to Athenaeus, it was capable ofcarrying 600 people and included garden decorations, agymnasiumand a temple dedicated tothe goddessAphroditeamong its facilities. Since a ship of this size would leak a considerableamount of water through the hull, theArchimedes screwwas purportedly developed in orderto remove the bilge water. Archimedes' machine was a device with a revolving screw-shaped

blade inside a cylinder. It was turned by hand, and could also be used to transfer water from alow-lying body of water into irrigation canals. The Archimedes screw is still in use today forpumping liquids and granulated solids such as coal and grain. The Archimedes screwdescribed in Roman times byVitruviusmay have been an improvement on a screw pump thatwas used to irrigate theHanging Gardens of Babylon.[21][22][23]The world's first seagoingsteamshipwith ascrew propellerwas theSS Archimedes, which was launched in 1839 andnamed in honor of Archimedes and his work on the screw.[24]

The Claw of Archimedes

TheClaw of Archimedesis a weapon that he is said to have designed in order to defend the

city of Syracuse. Also known as "the ship shaker," the claw consisted of a crane-like armfrom which a large metal grappling hook was suspended. When the claw was dropped onto
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an attacking ship the arm would swing upwards, lifting the ship out of the water and possiblysinking it. There have been modern experiments to test the feasibility of the claw, and in 2005a television documentary entitled Superweapons of the Ancient Worldbuilt a version of theclaw and concluded that it was a workable device.[25][26]

The Archimedes Heat Ray

Archimedes may have used mirrors acting collectively as aparabolic reflectorto burn ships attacking

Syracuse.

The 2nd century AD authorLucianwrote that during theSiege of Syracuse(c. 214212 BC),Archimedes destroyed enemy ships with fire. Centuries later,Anthemius of Trallesmentionsburning-glassesas Archimedes' weapon.[27]The device, sometimes called the "Archimedesheat ray", was used to focus sunlight onto approaching ships, causing them to catch fire.

This purported weapon has been the subject of ongoing debate about its credibility since theRenaissance.Ren Descartesrejected it as false, while modern researchers have attempted torecreate the effect using only the means that would have been available to Archimedes.[28]Ithas been suggested that a large array of highly polishedbronzeorcoppershields acting asmirrors could have been employed to focus sunlight onto a ship. This would have used the

principle of theparabolic reflectorin a manner similar to asolar furnace.

A test of the Archimedes heat ray was carried out in 1973 by the Greek scientist IoannisSakkas. The experiment took place at theSkaramagasnaval base outsideAthens. On thisoccasion 70 mirrors were used, each with a copper coating and a size of around five by threefeet (1.5 by 1 m). The mirrors were pointed at a plywood mock-up of a Roman warship at adistance of around 160 feet (50 m). When the mirrors were focused accurately, the ship burstinto flames within a few seconds. The plywood ship had a coating oftarpaint, which mayhave aided combustion.[29]

In October 2005 a group of students from theMassachusetts Institute of Technologycarriedout an experiment with 127 one-foot (30 cm) square mirror tiles, focused on a mock-upwooden ship at a range of around 100 feet (30 m). Flames broke out on a patch of the ship,
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but only after the sky had been cloudless and the ship had remained stationary for around tenminutes. It was concluded that the device was a feasible weapon under these conditions. TheMIT group repeated the experiment for the television showMythBusters, using a woodenfishing boat inSan Franciscoas the target. Again some charring occurred, along with a smallamount of flame. In order to catch fire, wood needs to reach its autoignition temperature,

which is around 300 C (570 F).[30][31]

WhenMythBusters broadcast the result of the San Francisco experiment in January 2006, theclaim was placed in the category of "busted" (or failed) because of the length of time and theideal weather conditions required for combustion to occur. It was also pointed out that sinceSyracuse faces the sea towards the east, the Roman fleet would have had to attack during themorning for optimal gathering of light by the mirrors.MythBusters also pointed out thatconventional weaponry, such as flaming arrows or bolts from a catapult, would have been afar easier way of setting a ship on fire at short distances.[1]

In December 2010,MythBusters again looked at the heat ray story in a special edition

featuringBarack Obama, entitled President's Challenge. Several experiments were carriedout, including a large scale test with 500 schoolchildren aiming mirrors at a mock-up of aRoman sailing ship 400 feet (120 m) away. In all of the experiments, the sail failed to reachthe 210 C (410 F) required to catch fire, and the verdict was again "busted". The showconcluded that a more likely effect of the mirrors would have been blinding, dazzling, ordistracting the crew of the ship.[32]

Other discoveries and inventions

While Archimedes did not invent thelever, he gave an explanation of the principle involved

in his workOn the Equilibrium of Planes. Earlier descriptions of the lever are found in thePeripatetic schoolof the followers ofAristotle, and are sometimes attributed toArchytas.[33][34]According toPappus of Alexandria, Archimedes' work on levers caused himto remark: "Give me a place to stand on, and I will move the Earth." (Greek: )[35]Plutarch describes how Archimedes designedblock-and-tacklepulleysystems, allowing sailors to use the principle ofleverageto lift objects that would otherwisehave been too heavy to move.[36]Archimedes has also been credited with improving thepower and accuracy of thecatapult, and with inventing theodometerduring theFirst PunicWar. The odometer was described as a cart with a gear mechanism that dropped a ball into acontainer after each mile traveled.[37]

Cicero(10643 BC) mentions Archimedes briefly in hisdialogueDe re publica, whichportrays a fictional conversation taking place in 129 BC. After the capture of Syracuse c.212 BC, GeneralMarcus Claudius Marcellusis said to have taken back to Rome twomechanisms used as aids in astronomy, which showed the motion of the Sun, Moon and fiveplanets. Cicero mentions similar mechanisms designed byThales of MiletusandEudoxus ofCnidus. The dialogue says that Marcellus kept one of the devices as his only personal lootfrom Syracuse, and donated the other to the Temple of Virtue in Rome. Marcellus'mechanism was demonstrated, according to Cicero, byGaius Sulpicius GallustoLuciusFurius Philus, who described it thus:

Hanc sphaeram Gallus cum moveret, fiebat ut soli luna totidem conversionibus in aere illo quot

diebus in ipso caelo succederet, ex quo et in caelo sphaera solis fieret eadem illa defectio, et

incideret luna tum in eam metam quae esset umbra terrae, cum sol e regione. When Gallus
http://en.wikipedia.org/wiki/MythBustershttp://en.wikipedia.org/wiki/MythBustershttp://en.wikipedia.org/wiki/MythBustershttp://en.wikipedia.org/wiki/San_Franciscohttp://en.wikipedia.org/wiki/San_Franciscohttp://en.wikipedia.org/wiki/San_Franciscohttp://en.wikipedia.org/wiki/Autoignition_temperaturehttp://en.wikipedia.org/wiki/Autoignition_temperaturehttp://en.wikipedia.org/wiki/Autoignition_temperaturehttp://en.wikipedia.org/wiki/Archimedes#cite_note-29http://en.wikipedia.org/wiki/Archimedes#cite_note-29http://en.wikipedia.org/wiki/Archimedes#cite_note-29http://en.wikipedia.org/wiki/Archimedes#cite_note-death_ray-0http://en.wikipedia.org/wiki/Archimedes#cite_note-death_ray-0http://en.wikipedia.org/wiki/Archimedes#cite_note-death_ray-0http://en.wikipedia.org/wiki/Barack_Obamahttp://en.wikipedia.org/wiki/Barack_Obamahttp://en.wikipedia.org/wiki/Barack_Obamahttp://en.wikipedia.org/wiki/Archimedes#cite_note-death_ray2-31http://en.wikipedia.org/wiki/Archimedes#cite_note-death_ray2-31http://en.wikipedia.org/wiki/Archimedes#cite_note-death_ray2-31http://en.wikipedia.org/wiki/Leverhttp://en.wikipedia.org/wiki/Leverhttp://en.wikipedia.org/wiki/Leverhttp://en.wikipedia.org/wiki/Peripatetic_schoolhttp://en.wikipedia.org/wiki/Peripatetic_schoolhttp://en.wikipedia.org/wiki/Aristotlehttp://en.wikipedia.org/wiki/Aristotlehttp://en.wikipedia.org/wiki/Aristotlehttp://en.wikipedia.org/wiki/Archytashttp://en.wikipedia.org/wiki/Archimedes#cite_note-lever_rorres-32http://en.wikipedia.org/wiki/Archimedes#cite_note-lever_rorres-32http://en.wikipedia.org/wiki/Archimedes#ci

des of syracuse

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