**Archimedes** was the greatest mathematician of his age. His contributions in geometry revolutionised the subject and his methods anticipated the integral calculus. He was a practical man who invented a wide variety of machines including pulleys and the Archimidean screw pumping device.

- We know nothing else about Phidias other than this one fact and we only know this since Archimedes gives us this information in one of his works, The Sandreckoner.
- A friend of Archimedes called Heracleides wrote a biography of him but sadly this work is lost.
- How our knowledge of Archimedes would be transformed if 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 visited Egypt and there invented a device now known as Archimedes' screw.
- It is highly likely that, when he was a young man, Archimedes studied with the successors of Euclid in Alexandria.
- In the preface to On spirals Archimedes relates an amusing story regarding his friends in Alexandria.
- Other than in the prefaces to his works, information about Archimedes comes to us from a number of sources such as in stories from Plutarch, Livy, and others.
- There are, in fact, quite a number of references to Archimedes in the writings of the time for he had gained a reputation in his own time which few other mathematicians of this period achieved.
- The reason for this was not a widespread interest in new mathematical ideas but rather that Archimedes had invented many machines which were used as engines of war.
- 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.
- Other inventions of Archimedes such as the compound pulley also brought him great fame among his contemporaries.
- Yet Archimedes, although he achieved fame by his mechanical inventions, believed that pure mathematics was the only worthy pursuit.
- The achievements of Archimedes are quite outstanding.
- Archimedes was able to apply the method of exhaustion, which is the early form of integration, to obtain a whole range of important results and we mention some of these in the descriptions of his works below.
- Archimedes also gave an accurate approximation to π and showed that he could approximate square roots accurately.
- In mechanics Archimedes discovered fundamental theorems concerning the centre of gravity of plane figures and solids.
- His most famous theorem gives the weight of a body immersed in a liquid, called Archimedes' principle.
- The works of Archimedes which have survived are as follows.
- In the summer of 1906, J L Heiberg, professor of classical philology at the University of Copenhagen, discovered a 10th century manuscript which included Archimedes' work The method.
- This provides a remarkable insight into how Archimedes discovered many of his results and we will discuss this below once we have given further details of what is in the surviving books.
- The order in which Archimedes wrote his works is not known for certain.
- Archimedes discovered fundamental theorems concerning the centre of gravity of plane figures and these are given in this work.
- In the Quadrature of the parabola Archimedes finds the area of a segment of a parabola cut off by any chord.
- In the first book of On the sphere and cylinder Archimedes shows that the surface of a sphere is four times that of a great circle, he finds the area of any segment of a sphere, he shows that the volume of a sphere is two-thirds the volume of a circumscribed cylinder, and that the surface of a sphere is two-thirds the surface of a circumscribed cylinder including its bases.
- In the second book of this work Archimedes' most important result is to show how to cut a given sphere by a plane so that the ratio of the volumes of the two segments has a prescribed ratio.
- In On spirals Archimedes defines a spiral, he gives fundamental properties connecting the length of the radius vector with the angles through which it has revolved.
- In the work On conoids and spheroids Archimedes examines paraboloids of revolution, hyperboloids of revolution, and spheroids obtained by rotating an ellipse either about its major axis or about its minor axis.
- On floating bodies is a work in which Archimedes lays down the basic principles of hydrostatics.
- His most famous theorem which gives the weight of a body immersed in a liquid, called Archimedes' principle, is contained in this work.
- There are also important historical remarks in this work, for Archimedes has to give the dimensions of the universe to be able to count the number of grains of sand which it could contain.
- There are other sources which mention Archimedes' work on distances to the heavenly bodies.
- There are references to other works of Archimedes which are now lost.
- Pappus refers to a work by Archimedes on semi-regular polyhedra, Archimedes himself refers to a work on the number system which he proposed in the Sandreckoner, Pappus mentions a treatise On balances and levers, and Theon mentions a treatise by Archimedes about mirrors.
- Archimedes was killed in 212 BC during the capture of Syracuse by the Romans in the Second Punic War after all his efforts to keep the Romans at bay with his machines of war had failed.
- 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.
- as Archimedes was carrying to Marcellus mathematical instruments, dials, spheres, and angles, 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 a cylinder circumscribing a sphere, and he asked for a representation of this together with his result on the ratio of the two, to be inscribed on his tomb.
- It is perhaps surprising that the mathematical works of Archimedes were relatively little known immediately after his death.
- individual works of Archimedes were obviously studied at Alexandria, since Archimedes was often quoted by three eminent mathematicians of Alexandria: Heron, Pappus and Theon.
- Only after Eutocius brought 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 the various versions of his treatises of Archimedes are, is to determine whether they have retained Archimedes' Dorian dialect.

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

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Algebra, Analysis, Ancient Arab, Ancient Greek, Ancient Indian, Astronomy, Geometry, Origin Italy, Number Theory, Physics, Puzzles And Problems, Special Numbers And Numerals, Topology

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**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive