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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. Chicago: University of Chicago Press, 1.
He was also one of the first to apply mathematics to physical phenomena , founding hydrostatics and statics , including an explanation of the principle of the lever.
He is credited with designing innovative machines , such as his screw pump , compound pulleys , and defensive war machines to protect his native Syracuse from invasion.
Archimedes died during the Siege of Syracuse , where he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting the tomb of Archimedes, which was surmounted by a sphere and a cylinder , which Archimedes had requested be placed on his tomb to represent his mathematical discoveries.
Unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Mathematicians from Alexandria read and quoted him, but the first comprehensive compilation was not made until c.
The relatively few copies of Archimedes' written work that survived through the Middle Ages were an influential source of ideas for scientists during the Renaissance , while the discovery in of previously unknown works by Archimedes in the Archimedes Palimpsest has provided new insights into how he obtained mathematical results.
Archimedes was born c. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years.
During his youth, Archimedes may have studied in Alexandria , Egypt , where Conon of Samos and Eratosthenes of Cyrene were contemporaries. He referred to Conon of Samos as his friend, while two of his works The Method of Mechanical Theorems and the Cattle Problem have introductions addressed to Eratosthenes.
Archimedes died c. According to the popular account given by Plutarch , Archimedes was contemplating a mathematical diagram when the city was captured.
A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem.
The soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives a lesser-known account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier.
According to this story, Archimedes was carrying mathematical instruments, and was killed because the soldier thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable scientific asset and had ordered that he must not be harmed.
The last words attributed to Archimedes are "Do not disturb my circles", a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier.
This quote is often given in Latin as " Noli turbare circulos meos ," but there is no reliable evidence that Archimedes uttered these words and they do not appear in the account given by Plutarch.
Valerius Maximus , writing in Memorable Doings and Sayings in the 1st century AD, gives the phrase as " …sed protecto manibus puluere 'noli' inquit, 'obsecro, istum disturbare' " "…but protecting the dust with his hands, said 'I beg of you, do not disturb this ' ".
The tomb of Archimedes carried a sculpture illustrating his favorite mathematical proof, consisting of a sphere and a cylinder of the same height and diameter.
Archimedes had proven that the volume and surface area of the sphere are two thirds that of the cylinder including its bases.
He had heard stories about the tomb of Archimedes, but none of the locals were 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 tomb cleaned up, and was able to see the carving and read some of the verses that had been added as an inscription.
The standard versions of the life of Archimedes were written long after his death by the historians of Ancient Rome.
The account of the siege of Syracuse given by Polybius in his The Histories was written around seventy years after Archimedes' death, and was used subsequently as a source by Plutarch and Livy.
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.
The most widely known anecdote about Archimedes tells of how he invented a method for determining the volume of an object with an irregular shape.
According to Vitruvius , a votive crown for a temple had been made for King Hiero II of Syracuse , who had supplied the pure gold to be used, and Archimedes was asked to determine whether some silver had been substituted by the dishonest goldsmith.
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 the volume of the crown.
For practical purposes water is incompressible,  so the submerged crown would displace an amount of water equal to its own volume.
By dividing the mass of the crown by the volume of water displaced, the density of the crown could be obtained. This density would be lower than that of gold if cheaper and less dense metals had been added.
Archimedes then took to the streets naked, so excited by his discovery that he had forgotten to dress, crying " Eureka!
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 extreme accuracy with which one would have to measure the water displacement.
This principle states that a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces.
The difference in density between the two samples would cause the scale to tip accordingly. 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. In modern terms, those are problems of integration.
See calculus. On Spirals develops many properties of tangents to, and areas associated with, the spiral of Archimedes —i. It was one of only a few curves beyond the straight line and the conic sections known in antiquity.
On the Equilibrium of Planes or Centres of Gravity of Planes ; in two books is mainly concerned with establishing the centres of gravity of various rectilinear plane figures and segments of the parabola and the paraboloid.
Much of that book, however, is undoubtedly not authentic, consisting as it does of inept later additions or reworkings, and it seems likely that the basic principle of the law of the lever and—possibly—the concept of the centre of gravity were established on a mathematical basis by scholars earlier than Archimedes.
His contribution was rather to extend those concepts to conic sections. 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.Jetztspielen De four lines joining opposite vertices divide the cube into six congruent pyramids, each face of the cube being the base of a pyramid. Author s :. Archimedes concluded that the cone must have one-third the volume of the cylinder and, since the cone and sphere together equal the cylinder in volume, the sphere must Bayer Sv Wuppertal two-thirds the volume of the cylinder:. Genres 12 2 m. Leonhard was sent to school in Basel and during this time he lived with his grandmother on his mother's side.