Archimedes

 


Joke de Backker
                           Katrien Mestdagh
                                                        Stéphanie Michiels

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"Give me a place to stand and I will move the earth
"

His Biography

Archimedes was and is the most famous ancient Greek mathematician and inventor. He was born between 290-280 before Christ in Syracuse, the principal city-state in Sicily. He probably spent some time in Egypt early in his career, but he resided for most of his life in Syracuse, where he was on intimate terms with its king, Hieron II. Archimedes published his works in the form of correspondence with the principal mathematicians of his time, including the Alexandrian scholars Conon of Samos and Eratosthenes of Cyrene. He played an important role in the defense of Syracuse against the siege laid by the Romans in 213 before Christ by constructing war machines so effective that they long delayed the capture of the city. But Syracuse was eventually captured by the Roman general Marcus Claudius Marcellus in the autumn of 212 or spring of 211 before Christ, and Archimedes was killed in the sack of the city. Far more details survive about his life than about any other scientist, but they are largely anecdotal, reflecting the impression that his mechanical genius made on the popular imagination.

Thus, he is credited with inventing the Archimedes Screw, a device for raising water, and he is supposed to have made 2 "spheres" that Marcellus took back to Rome-one a star globe and the other a device for mechanically representing the motions of the Sun, Moon, and planets. The story that he determined the proportion of gold and silver in a wreath made for Hieron by weighing it in water is probably true, but the version that has him leaping from the bath in which he supposedly got the idea and running naked through the streets shouting "heurêka!"is popular embellishment. Equally apocryphal are the stories that he used a huge array of mirrors to burn the Roman ships besieging Syracuse; that he said, "Give me a place to stand and I will move the earth"; and that a Roman soldier killed him because he refused to leave his mathematical diagrams- although all are popular reflections of his real interest in catoptrics, mechanics, and pure mathematics.

It will always be uncertain how far Archimedes prefered the theory to the practice. But there is no doubt about how his work causes a tension between theory and application, a tension that is still essential in the mathematics. Plutarchus writes : " And still Archimedes was a highstanded spirit, a profound soul and he possesed such a wealth of scientific knowledge that he , altough his inventions had given him a reputation of opperhuman, he rather didn’t want to be remembered of it". It will always be uncertain how far Archimedes himself prefered the theory to the practice. But there is no doubt about how his work causes a tension between theory and application, a tension that is still essential in the mathematics of 23 centuries later.

His Works

There are nine extant treaties by Archimedes in Greek. The principal results in in 2 books On the Sphere and Cylinder. Here in we read that the surface area of any sphere is 4 times the area of its greatest circle (S=4Pi r²) and that the volume of a sphere is 2/3  of the volume of the cylinder in which it is inscribed ( V=4/3 Pi r³). Archimedes was proud enough of the latter discovery to leave instructions for his tomb to be marked with a sphere inscribed in a cylinder. Cicero found the tomb , overgrown with vegetation, a century and a half after Archimedes’ death.

Measurement of the Circle is a short work in which Pi is shown to lie between the limits of 3+1/7 and 3+10/71. The approach to this problem devised by Archimedes, which consists of inscribing and circumscribing regular polygons with large numbers of sides, was the one followed by all those who subsequently dealt with the problem of determining Pi until the development of series expansions in the late 17th century. This work also contains accurate approximations to the square roots of 3 and several large numbers.

On Conoids and Spheroids deals with determining the volumes of the segments of solids formed by the revolution of a conic section about its axis. In modern terms, these are problems of integration. On Spirals develops many properties of tangents to the spiral of Archimedes.

On the Equilibrium of Planes is mainly concerned with establishing the centres of gravity of various rectilinear plane figures and segments of conics. The first book purports to establish the "law of the lever" (magnitudes balance at distances from the fulcrum in inverse ratio to their weights), and it is mainly on the basis of this treatise that Archimedes has been called the founder of theoretical mechanics.

Quadrature of the Parabola demonstrates, first by "mechanical" means and then by conventional geometrical methods, that the area of any segment of a parabola is 4/3 of the area of the triangle having the same base and height as that segment. This is a problem in integration.

The Sand-Reckoner is a small treatise that is a jeu d’esprit written for the layman that nevertheless contains some profoundly original maths. Its object is to remedy the inadequacies of the Greek numerical notation system by showing how to express a huge number. What Archimedes does is to create a place-value system of notation, with a base of 100.000.000. The work is also of interest because it gives the most detailed surviving description of the heliocentric system of Aristarchus of Samos and because it contains an account of an ingenious procedure that Archimedes used to determine the Sun’s apparent diameter by observation with an instrument.

Method Concerning Mechanical Theorems describes the process of discovery in maths. It is the sole surviving work from antiquity and one of the few from any period that deals with this topic. In it Archimedes recounts how he used a "mechanical" method to arrive at some of his key discoveries, including the area of a parabolic segment and the surface area and volume of a sphere. The technique consists of dividing each of two figures , one bounded by straight lines and the other by a curve , into an infinite but equal number of infinitesimally thin strips, then "weighing" each corresponding pair of these strips against each other on a notional balance, and summing them to find the ratio of the two whole figures. Archimedes emphasizes that, though useful as a heuristic method , this procedure doesn’t constitute a rigorous proof.

On Floating Bodies survives only partly in Greek, the rest in medieval Latin translation from the Greek. It’s the first known work in 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. In the first book various general principles are established, notably what has come to be known as Archimedes’ principle, which is : " A solid denser than a fluid will, when immersed in that fluid, be lighter by the weight of the fluid it displaces". The second book is a mathematical tour de force unmatched in antiquity and rarely equalled 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’ mathematical proofs and presentation exhibit great boldness and originality of thought on the one hand and extreme rigour on the other , meeting the highest standards of contemporary geometry. While Method shows that he arrived at the formulas for the surface area and volume of a sphere by "mechanical" reasoning involving infinitesimals, in his actual proofs of the results in Sphere and Cylinder he uses only the rigorous methods of passage to the limit that had been invented by Eudoxus of Cnidus in the 4th century before Christ. These methods, of which Archimedes was a master, are the standard procedure in all his works on higher geometry that deal with problems of integration. Their mathematical rigour stands in strong contrast to the "proofs" of the first practitioners of integral calculus in the 17th century, when infinitesimals were reintroduced into maths. Yet Archimedes’ results are no less impressive than theirs. The same freedom from conventional ways of thinking is apparent in the arithmetical field in Sand-Reckoner, which shows an understanding of the nature of the numerical system unparalleled before the modern era.

His influence

Given the magnitude and originality of Archimedes’ achievement , the influence of his mathematics in antiquity was rather small. Those of his results that could be simply expressed became mathematical commonplaces, and one of the bounds he established for Pi, 22/7, was adopted as the usual approximation to it in antiquity and the Middle Ages. But his mathematical work wasn’t continued or developed, as far as is known, in any important way in ancient times , despite his hope expressed in Method that its publication would enable others to make new discoveries. It wasn’t until some of his mathematical treatises were translated into Arabic in the late 8th or 9th century that attempts were made to extend his results , particularly in the determination of the volumes of solids of revolution. Several meritorious works by Arabic mathematicians of the early medieval period were inspired by their study of Archimedes. But the greatest effect of his work on that of later mathematicians came in the 16th and 17th centuries with the printing of texts derived from the Greek, and eventually of the Greek text itself, the editio princeps, in Basel in 1544. The Latin translation of many of Archimedes’ works by Federico Commandino in 1558 contributed greatly to the spread of knowledge of them, which was reflected in the work of the foremost mathematicians and physicists of the time, including Johannes Kepler and Galileo. David Rivault’s edition and Latin translation(1615) of the complete works, including the ancient commentaries, was enormously influential in the work of some of the best mathematicians of the 17th century, notably René Descartes and Pierre de Fermat. Without the background of the rediscovered ancient mathematicians, amongst whom Archimedes was paramount, the development of maths in Europe in the century between 1550 and 1650 is inconceivable. It is unfortunate that Method remained unknown to both Arabic and Renaissance mathematicians(it was only rediscovered in the late 19th century), for they might have fulfilled Archimedes’ hope that it would prove of use to his successors in discovering theorems.

The most important contribution of Archimedes to mathematics belongs to territory of what we call now : integralcalculus and the definition of the surface of plane figures and the contents of solids. He used the method of exhaustion which implies that one needs to have a good idea of the answer before the proof can be constructed.
For example in his Measurement of the Circle, he calculated values for Pi (between 3+1/7 and 3+10/71) approximating the circumference of the circle with the help of inscribed and circumscribed regular polygons.

The legend of his death goes like this :

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In 212 a Roman general Marcellus took possession of the harbour in Syracuse (Sicily).

So the king of the city asked Archimedes for help . Archimedes had just invented the lever and he combinated levers and pullies to make cranes to stop the enemy. He succeeded in shrinking back the enemy for about 3 years, but the 3th year, Marcellus attacted the town again and one of the soldiers of Marcellus killed Archimedes by sword.

At the moment of his death, he was drawing geometrical forms in the sand. Following Archimedes' wish a sphere inscribed in a cylinder, proudly showing his discovery that the volume of the sphere equals 2/3 of the volume of the cylinder, is marked on his tomb.

But what’s true of this legend ? There is a good reason to presume that designed mighty catapults could have dropped projectiles of 25 kg, 100 m far away. But recent investigations in the history of technology told us that it was impossible that he could have built cranes who could pull the ships of the enemy out of the water.

The problem of the horned beasts

In the 3th century before Chr. Appollonius van Perga couldn’t know what was waiting for him and many other generations of mathematicians, when he searched for the sollution of one of the problems of Archimedes about big numbers. Archimedes must have thought : " Let’s see who knows something about big numbers" and so he considered a mathematical problem about grazing horned beasts of which the solution contains such large numbers that the problem is solved only recently. Besides the problem isn’t solved by human beings, but by machines : the fastest computer in the world. Was Archimedes really the first who came up with the problem ? That’s still a question, but they are sure that he worked on it.

"Calculate, my friend, the herd of horned beasts of the sun. Apply your mind on that, so that you ‘ll get any wisdom.
Calculate the number of horned beasts who grazed ones on the planes of the Sicilian island Trinacriva and who were divided according to color into four groups. ..."

The problem goes on to establish 7 relationships between the bulls and the cows.The number of bulls in each herd was in the majority . In his mathematical essence the problem comes to this : to solve 7 equations with 8 unknowns. Whith only these 7 equations, the problem has an infinite number of solution, although the smallest solution is a total herd of 50 839 082 cows and bulls who could graze on Sicilies 2 573 225 ha.

But Archimedes didn’t leave the problems so easy, he made it much more difficult by charging it with 2 extra conditions :

1) the sum of the number of white bulls and the number of black bulls is a perfect square
2) the sum of the number of yellow bulls and the number of dappled bulls is a triangular number

When solved with the aid of modern computers, the solution has 206 505 digits !
No wonder that Archimedes suggested this problem was for those who occupy themselves with such things "out of love for maths".

The problem of the crown and the law of Archimedes

King Hieron had given a goldsmith a certain quantity of gold to make a crown of it. Having  received his crown, Hieron ordered Archimedes to see if the crown contained all the gold. If we believe the story of Vitruvius, a Romain architect of the 1° century B.C., Archimedes, as he was taking a bath, got the idea that the volume  of water, that was spoiled over the border was equal to the volume of the subject immersed under water. He realised that, whereas gold is a heavy metal, a total golden crown would have less volume than a mixture with a lighter metal. He filled a bath to the border with water, put the crown into it and collected the spoiled water. The crown(supposing being pure gold) deplaced more water, than his total weight implied, which proved the fact that the goldsmith had cheated king Hieron.

In his book « floating bodies », Archimedes stated his law saying that "a body, which is entirely or  partially immersed in a fluid, receives a lift in an upward direction from below with a force equal to the weight that the given body moves".

On this picture, the object moves 100 cm³ of water and as a result experiences an upwards force of
100 gf.

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Also the flight of an aerostat is essentially based on his law because a balloon immersed in the atmospheric fluid and filled with a gas lighter than the surrounding air will receive a lift upwards equal to the weight of an atmospheric quantity corresponding to its volume. If the lift is greater than the weight of the aerostat, it moves upwards inside the atmosphere, meaning it flies.

 

archbal.jpg (12216 bytes) Once the balloon has lifted off, it will rise to an altitude that corresponds to the new balance point it reaches in relation to the natural decrease of air density as the altitude increases.
Balloons with a variable volume are only partly filled with gass. When these balloons rise, the atmospheric pressure diminishes and the gas contained in the envelope expands, so the balloon inflates.
In balloons with a constant volume, the gas flows out freely through the envelope sleeve as the balloon rises in the atmosphere.

 

The screw of Archimedes

Archimedes invented the screw which can pump up water. It contains of a wide tube in which a screwplate is brought. By turning the screw, the water is pumped up. It’s used for irregation purposes.

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Spiral of Archimedes

It is the locus of a point P which starting from the origin O, moves uniformly along the length of a 'half line' OP, which itself
rotates uniformly about O. Setting OP = r and the angle with the initial position of the half line = q, then we get the polar equation r = aq since both r and q are proportional to the time of rotation. 

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Axioma of Archimedes

Also called axioma of Eudoxus :
" For every positive number a and every number b there exists a natural number n so that  n.a >b "

Archimedes' method to determine the centre of gravity of a lever

If we hang different weights on different places of a lever (A), where do we have to support the lever so that he is in balance? Archimedes solves this problem by thinking of these weights being divided in a big number of little(equal) weights. He puts the little weights near each other on the lever so that the centre of the gravity of the series stays at the same place as the centre of gravity of the original weights. (B)

The size of the little weights can always be chosen in a certain way so that the two series touch each other ( C). The middle of the total serie of weights is the center of gravity and is also the centre of support. (D). The distance between the centre of support and each of the original weights is inversely proportional to the size these weights.

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Sources

http://www.mcs.drexel.edu/~crorres/archimedes/contents.html
http://www.geocities.com/Athens/Acropolis/6681/archimed.htm

Encyclopedia Brittanica
Makers of Mathematics, Stuart Hollingdale, Penguin Books
Classic Math, History for the classroom, Art Johnson, Dale Seymour Publications
Natuurkunde, 1 Mechanica, Pergoot and Thys and Van Derstappen, Uitg. De Garve - Antwerpen

Ships of the Sky, Aerostats, fast as the wind, lighter than air, Marco Majrani, Edizioni dell'Ambrosino-Milano

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