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Ve A Contextual History of Mathematics. ISBN Kdo byl kdo. Praha: Albatros, The MacTutor History of Mathematics archive.
Leden [cit. Cicero and Archimedes' Tomb. The Journal of Roman Studies. The world of mathematics. Svazek 1. Mineola: Courier Dover Publications, Phoenix Edition Series.
Kapitola Commentary on Archimedes, s. Praha: Panorama, Archimedes and the Roman imagination. Ann Arbor: University of Michigan Press, Kapitola Introducion The "Life of Archimedes", s.
The History of Cartography: Cartography in prehistoric, ancient, and medieval Europe and the Mediterranean.
Galileo considered it "probable that this method is the same that Archimedes followed, since, besides being very accurate, it is based on demonstrations found by Archimedes himself.
In a 12th-century text titled Mappae clavicula there are instructions on how to perform the weighings in the water in order to calculate the percentage of silver used, and thus solve the problem.
A large part of Archimedes' work in engineering arose from fulfilling the needs of his home city of Syracuse.
The Greek writer Athenaeus of Naucratis described how King Hiero II commissioned Archimedes to design a huge ship, the Syracusia , which could be used for luxury travel, carrying supplies, and as a naval warship.
The Syracusia is said to have been the largest ship built in classical antiquity. Since a ship of this size would leak a considerable amount of water through the hull, the Archimedes' screw was purportedly developed in order to 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 a low-lying body of water into irrigation canals.
The Archimedes' screw is still in use today for pumping liquids and granulated solids such as coal and grain. The Archimedes' screw described in Roman times by Vitruvius may have been an improvement on a screw pump that was used to irrigate the Hanging Gardens of Babylon.
The Claw of Archimedes is 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 arm from which a large metal grappling hook was suspended.
When the claw was dropped onto an attacking ship the arm would swing upwards, lifting the ship out of the water and possibly sinking it.
There have been modern experiments to test the feasibility of the claw, and in a television documentary entitled Superweapons of the Ancient World built a version of the claw and concluded that it was a workable device.
Archimedes may have used mirrors acting collectively as a parabolic reflector to burn ships attacking Syracuse. Centuries later, Anthemius of Tralles mentions burning-glasses as Archimedes' weapon.
In the modern era, similar devices have been constructed and may be referred to as a heliostat or solar furnace.
This purported weapon has been the subject of ongoing debate about its credibility since the Renaissance.
A test of the Archimedes heat ray was carried out in by the Greek scientist Ioannis Sakkas. The experiment took place at the Skaramagas naval base outside Athens.
On this occasion 70 mirrors were used, each with a copper coating and a size of around 5 by 3 feet 1. When the mirrors were focused accurately, the ship burst into flames within a few seconds.
The plywood ship had a coating of tar paint, which may have aided combustion. Flames broke out on a patch of the ship, but only after the sky had been cloudless and the ship had remained stationary for around ten minutes.
It was concluded that the device was a feasible weapon under these conditions. The MIT group repeated the experiment for the television show MythBusters , using a wooden fishing boat in San Francisco as the target.
Again some charring occurred, along with a small amount of flame. When MythBusters broadcast the result of the San Francisco experiment in January , the claim was placed in the category of "busted" i.
It was also pointed out that since Syracuse faces the sea towards the east, the Roman fleet would have had to attack during the morning for optimal gathering of light by the mirrors.
MythBusters also pointed out that conventional weaponry, such as flaming arrows or bolts from a catapult, would have been a far easier way of setting a ship on fire at short distances.
In December , MythBusters again looked at the heat ray story in a special edition entitled " President's Challenge ".
The show concluded that a more likely effect of the mirrors would have been blinding, dazzling , or distracting the crew of the ship.
While Archimedes did not invent the lever , he gave an explanation of the principle involved in his work On the Equilibrium of Planes.
Earlier descriptions of the lever are found in the Peripatetic school of the followers of Aristotle , and are sometimes attributed to Archytas.
The odometer was described as a cart with a gear mechanism that dropped a ball into a container after each mile traveled.
After the capture of Syracuse c. Cicero mentions similar mechanisms designed by Thales of Miletus and Eudoxus of Cnidus.
The dialogue says that Marcellus kept one of the devices as his only personal loot from Syracuse, and donated the other to the Temple of Virtue in Rome.
Marcellus' mechanism was demonstrated, according to Cicero, by Gaius Sulpicius Gallus to Lucius Furius 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 moved the globe, it happened that the Moon followed the Sun by as many turns on that bronze contrivance as in the sky itself, from which also in the sky the Sun's globe became to have that same eclipse, and the Moon came then to that position which was its shadow on the Earth, when the Sun was in line.
This is a description of a planetarium or orrery. Pappus of Alexandria stated that Archimedes had written a manuscript now lost on the construction of these mechanisms entitled On Sphere-Making.
Modern research in this area has been focused on the Antikythera mechanism , another device built c. While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics.
Plutarch wrote: "He placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life.
Archimedes was able to use infinitesimals in a way that is similar to modern integral calculus. Through proof by contradiction reductio ad absurdum , he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay.
In Measurement of a Circle , he did this by drawing a larger regular hexagon outside a circle then a smaller regular hexagon inside the circle, and progressively doubling the number of sides of each regular polygon , calculating the length of a side of each polygon at each step.
As the number of sides increases, it becomes a more accurate approximation of a circle. In On the Sphere and Cylinder , Archimedes postulates that any magnitude when added to itself enough times will exceed any given magnitude.
This is the Archimedean property of real numbers. The actual value is approximately 1. He introduced this result without offering any explanation of how he had obtained it.
This aspect of the work of Archimedes caused John Wallis to remark that he was: "as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results.
If the first term in this series is the area of the triangle, then the second is the sum of the areas of two triangles whose bases are the two smaller secant lines , and so on.
In The Sand Reckoner , Archimedes set out to calculate the number of grains of sand that the universe could contain.
In doing so, he challenged the notion that the number of grains of sand was too large to be counted. He wrote:. There are some, King Gelo Gelo II, son of Hiero II , who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited.
To solve the problem, Archimedes devised a system of counting based on the myriad. He proposed a number system using powers of a myriad of myriads million, i.
The works of Archimedes were written in Doric Greek , the dialect of ancient Syracuse. Pappus of Alexandria mentions On Sphere-Making and another work on polyhedra , while Theon of Alexandria quotes a remark about refraction from the now-lost Catoptrica.
The writings of Archimedes were first collected by the Byzantine Greek architect Isidore of Miletus c. There are two volumes to On the Equilibrium of Planes : the being is in fifteen propositions with seven postulates , while the second book is in ten propositions.
In this work Archimedes explains the Law of the Lever , stating, " Magnitudes are in equilibrium at distances reciprocally proportional to their weights.
Archimedes uses the principles derived to calculate the areas and centers of gravity of various geometric figures including triangles , parallelograms and parabolas.
This is a short work consisting of three propositions. It is written in the form of a correspondence with Dositheus of Pelusium, who was a student of Conon of Samos.
This work of 28 propositions is also addressed to Dositheus. The treatise defines what is now called the Archimedean spiral. It is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity.
This is an early example of a mechanical curve a curve traced by a moving point considered by a Greek mathematician. In this two-volume treatise addressed to Dositheus, Archimedes obtains the result of which he was most proud, namely the relationship between a sphere and a circumscribed cylinder of the same height and diameter.
The sphere has a volume two-thirds that of the circumscribed cylinder. Similarly, the sphere has an area two-thirds that of the cylinder including the bases.
A sculpted sphere and cylinder were placed on the tomb of Archimedes at his request. This is a work in 32 propositions addressed to Dositheus.
In this treatise Archimedes calculates the areas and volumes of sections of cones , spheres, and paraboloids. In the first part of this two-volume treatise, Archimedes spells out the law of equilibrium of fluids, and proves that water will adopt a spherical form around a center of gravity.
This may have been an attempt at explaining the theory of contemporary Greek astronomers such as Eratosthenes that the Earth is round.
The fluids described by Archimedes are not self-gravitating , since he assumes the existence of a point towards which all things fall in order to derive the spherical shape.
In the second part, 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, similar to the way that icebergs float.
Archimedes' principle of buoyancy is given in the work, stated as follows:. Any body wholly or partially immersed in a fluid experiences an upthrust equal to, but opposite in sense to, the weight of the fluid displaced.
Also known as Loculus of Archimedes or Archimedes' Box ,  this is a dissection puzzle similar to a Tangram , and the treatise describing it was found in more complete form in the Archimedes Palimpsest.
Archimedes calculates the areas of the 14 pieces which can be assembled to form a square. Research published by Dr.
Reviel Netz of Stanford University in argued that Archimedes was attempting to determine how many ways the pieces could be assembled into the shape of a square.
Netz calculates that the pieces can be made into a square 17, ways. This work was discovered by Gotthold Ephraim Lessing in a Greek manuscript consisting of a poem of 44 lines, in the Herzog August Library in Wolfenbüttel , Germany in It is addressed to Eratosthenes and the mathematicians in Alexandria.
Archimedes challenges them to count the numbers of cattle in the Herd of the Sun by solving a number of simultaneous Diophantine equations.
There is a more difficult version of the problem in which some of the answers are required to be square numbers. This version of the problem was first solved by A.
Amthor  in , and the answer is a very large number , approximately 7. In this treatise, also known as Psammites , Archimedes counts the number of grains of sand that will fit inside the universe.
This book mentions the heliocentric theory of the solar system proposed by Aristarchus of Samos , as well as contemporary ideas about the size of the Earth and the distance between various celestial bodies.
The introductory letter states that Archimedes' father was an astronomer named Phidias. The Sand Reckoner is the only surviving work in which Archimedes discusses his views on astronomy.
This treatise was thought lost until the discovery of the Archimedes Palimpsest in In this work Archimedes uses infinitesimals , and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume.
That is, again, a problem in integration. Its object is to remedy the inadequacies of the Greek numerical notation system by showing how to express a huge number—the number of grains of sand that it would take to fill the whole of the universe.
What Archimedes does, in effect, is to create a place-value system of notation, with a base of ,, That was apparently a completely original idea, since he had no knowledge of the contemporary Babylonian place-value system with base The work is also of interest because it gives the most detailed surviving description of the heliocentric system of Aristarchus of Samos c.
Method Concerning Mechanical Theorems describes a process of discovery in mathematics. It is the sole surviving work from antiquity, and one of the few from any period, that deals with this topic.
Archimedes emphasizes that, though useful as a heuristic method, this procedure does not constitute a rigorous proof.
On Floating Bodies in two books survives only partly in Greek, the rest in medieval Latin translation from the Greek.
It is the first known work on hydrostatics , of which Archimedes is recognized as the founder. Its purpose is to determine the positions that various solids will assume when floating in a fluid, according to their form and the variation in their specific gravities.
The second book is a mathematical tour de force unmatched in antiquity and rarely equaled since.
In it Archimedes determines the different positions of stability that a right paraboloid of revolution assumes when floating in a fluid of greater specific gravity , according to geometric and hydrostatic variations.
Archimedes is known, from references of later authors, to have written a number of other works that have not survived.
Those include a work on inscribing the regular heptagon in a circle; a collection of lemmas propositions assumed to be true that are used to prove a theorem and a book, On Touching Circles , both having to do with elementary plane geometry; and the Stomachion parts of which also survive in Greek , dealing with a square divided into 14 pieces for a game or puzzle.
These methods, of which Archimedes was a master, are the standard procedure in all his works on higher geometry that deal with proving results about areas and volumes.
The same freedom from conventional ways of thinking is apparent in the arithmetical field in Sand-Reckoner , which shows a deep understanding of the nature of the numerical system.
In antiquity Archimedes was also known as an outstanding astronomer: his observations of solstices were used by Hipparchus flourished c. Surprising though it is to find those metaphysical speculations in the work of a practicing astronomer, there is good reason to believe that their attribution to Archimedes is correct.
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