«Archimedes of Syracuse (287 - 212 BCE), the most famous and probably the best mathematician of antiquity, made so many discoveries in mathematics and ...»
Archimedes of Syracuse1
Archimedes of Syracuse (287 - 212 BCE), the most famous and
probably the best mathematician of antiquity, made so many discoveries
in mathematics and physics that it is difficult to point to any of them
as his greatest.
He was born in Syracuse, the principal city-state of Sicily, the son
of the astronomer Phidias. He spent considerable time in Alexandria,
where he studied with Euclid’s successors. It is there he met Conon of
Samos (fl. 245 BCE) and Eratosthenes of Cyrene (c. 276 - 195 BCE), both leading mathematicians of their day. However, he resided most of his whole life in Syracuse, an intimate friend of the court of King Hieron II.
He was an accomplished engineer, indeed he is said to have dis- dained mechanical invention, who loved pure mathematics. With one exception, his only extant works are on pure mathematics. His methods of proof and discovery, though, were based substantially upon mechani- cal principles as revealed in his treatise Method Concerning Mechanical Theorems.
In fact, he seems to have disdained the source of his fame during his day, ingenious mechanical inventions, on which he left no written 1 °2000, c G. Donald Allen Archimedes 2 description. Said Plutarch, ”he possessed so high a spirit, so profound a soul, and such treasures of scientific knowledge that, thought these inventions had obtained for him the renown of more than human sagac- ity, he yet would not deign to leave behind him any written work on such subjects,....” Stories from Plutarch, Livy, and others describe machines invented by Archimedes for the defense of Syracuse. These include the cata- pult and the compound pulley. Also described is his instruments ap- plying “burning-mirrors.” His fascination with geometry is beautifully described by Plutarch.2 Often times 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 savors, with his figures 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.
During the siege of Syracuse in the Second Punic War, inventions by Archimedes such as a catapult equally serviceable at a variety of ranges, caused great fear to the Roman attackers. Another invention, the compound pulley, was so powerfully built as to lift Roman ships from the sea and drop them back into it. However, the story that he used an array of mirrors, burning-mirrors, to destroy Roman ships is probably apocryphal. So much fear did these machines instill in the Romans that general Marcus Claudius Marcellus, the Roman commander, gave up on frontal assault and placed his hopes in a long siege. When at last Syracuse did fall in about 212 BCE, Archimedes was killed during the capture of Syracuse by the Romans Plutarch recounts this story of his killing: As fate would have it, Archimedes was intent on working out some problem by a diagram, and having fixed both his mind and eyes upon the subject of his speculation, he did not notice the entry of the Romans nor that the city was taken. In this transport of study a soldier unexpectedly came up to him and commanded that he accompany him.
When he declined to do this before he had finished his problem, the enraged soldier drew his sword and ran him through. Marcellus was 2 Plutarch(c. 46 - 119 CE), was a Greek biographer and author whose works influenced the evolution
greatly saddened by this and arranged for Archimedes’ burial.
Archimedes’ Works 1 It was to Conon that Archimedes frequently communicated his results before they were published. There were no journals, as such, in that time. Major works were developed into books. There are nine extant books of Archimedes, that have come to us. Substantially in the form of advanced monographs, they are not works for students nor for the dilettante, as each requires serious study. Almost certainly, they were not as widely copied or studied as other works such as The Elements.
But how do we know about the works? Where did they come from?
For most of the second millennium, the earliest sources of Archimedes works date from the Latin translations of Greek works made by William of Moerbeke (1215 - 1286). He used two Greek manuscripts. Both have disappeared, the first before 1311 and the second disappears about the 16th century. No earlier versions were known until about 1899, when an Archimedes palimpsest was listed among hundreds of other volumes in a library in Istanbul. In 1906, the great Greek mathematical scholar was able to begin his examination of it.
A palimpsest is a document which has been copied over by another text. Two reasons are offered for doing this. First, parchment was expensive and reusing it was an economical measure. Second, at the time it was considered virtuous to copy over pagan texts. In the case at hand, the Archimedes palimpsest was covered over by a religious text.
Moreover, the original sheets were folded in half, the resulting book of 174 pages having a sown binding.
What Heiberg found were four books already known but which had been copied in the 10th century by a monk living in a Constantinople monastery. This version was independent of the two manuscripts used by William of Moerbeke. However, a new book was found. It was the Method concerning Mechanical Theorems. This book, though known to have been written, had not been found to that time. Its importance lies in that in this volume, Archimedes described his method of discovery of many of his other theorems.
The story of the Archimedes palimpsest over that past century is interesting with suggestions of theft and manuscript alteration. Having Archimedes 4 disappeared in 1922, it reappeared in 1998 as an auction item displayed by Christie’s in New York. It sold at auction for two million dollars in October of 1998 to an anonymous buyer. This buyer has agreed to make the manuscript available for scholarly research. For further details, the interested reader should consult http://www-history.mcs.st-and.ac.uk/history/HistTopics/Greek sources 1.html The works themselves are ² On Plane Equilibria, Volume I ² Quadrature of a Parabola ² On Plane Equilibria, Volume II ² On the Sphere and Cylinder, Volumes I and II ² On Spirals ² On Conoids and Spheroids ² The Sand-Reckoner ² On Floating Bodies, Volumes I and II ² On Measurement of the Circle ² Method Concerning Mechanical Theorems Another volume Stomachion, is known in fragments only. Yet another volume, a collection of Lemmas Liber Assumptorum comes down to us from the Arabic. In its present form, it could not been written by Archimedes as his name is referenced in it, though the results are likely due to Archimedes. Overall, we may say that he worked in the Geometry of Measurement in distinction to the Geometry of Form advanced by his younger colleague/competitor Apollonius (260 - 185 BCE). His methods anticipated the integral calculus 2,000 years before Newton and Leibniz. In the following subsections, we describe some of the results, recognizing the impossibility of rendering anything near an adequate description of the overwhelming depth and volume of his works.
Measurement of the Circle1.1
Among Archimedes’ most famous works is Measurement of the Circle, in which he determined the exact value of ¼ to be between the values 3 10 71 and 3 1. This result is still used today, and most certainly every reader of 7 these notes has used 22 = 3 1 to approximate ¼. He obtained this result 7 7 by circumscribing and inscribing a circle with regular polygons having up to 96 sides. However, the proof requires two fundamental relations about the perimeters and areas of these inscribed and circumscribed regular polygons.
The computation. With respect to a circle of radius r, let
Further, let b2 ; : : : ; bn denote the regular inscribed 6¢2 : : : 6¢2n polygons, similarly, B2 : : : Bn for the circumscribed polygons. The following formulae give the relations between the perimeters and areas of these 6 ¢ 2n polygons.
the familiar formula.
2 Vsphere = V circumscribed cylinder 3 This he considered his most significant accomplishments, requesting that a representation of a cylinder circumscribing a sphere be inscribed on his tomb. He established other fundamental results including Proposition 33. The surface of any sphere is equal to four times the greatest circle on it.
Similarly, but for cones, we have Proposition 34. Any sphere is four times the cone which has as its base equal to the greatest circle in the sphere and its height equal to the radius of the sphere.
From this of course follows Archimedes relation above. In Volume II, Archimedes proves a number of results such as Proposition 1. Given a cone or a cylinder, to find a sphere equal to the cone or to the cylinder.
Proposition 3. To cut a given sphere by a plane so that the surfaces of the segments may have to one another a given ratio.
Proposition 9. Of all segments of spheres which have equal surfaces the hemisphere is the greatest in volume.
On Conoids and Spheroids 1.3
Though easy to verify using calculus, this result requires a careful and lengthly proof using only the standard method of the day, i.e. double reductio ad absurdum.
In On Floating Bodies Archimedes literally invented the whole study of hydrostatics. In one particular result he was able to compute the maximum angle that a (paraboloid) ship could list before it capsized — and he did it without calculus! This result, a tour de force of computation, is not nearly as well known as the story which describes Archimedes crying “Eureka” after discovering whether a newly made crown was truly pure gold.
The case of the fraudulent gold crown. King Hieron II commissioned the manufacture of a gold crown. Suspecting the goldsmith may have substituted silver for gold, he asked Archimedes to determine its authenticity. He was not allowed to disturb the crown in any way. What follows is a quote from Vitruvius.3 The solution which occurred when he stepped into his bath and caused it to overflow was to put a weight of gold equal to the crown and know to be pure, into a bowl which was filled with water to the brim. Then the gold would be removed and the king’s crown put in, in its place. An alloy of lighter silver would increase the bulk of the crown and cause the bowl to overflow.
There are some technical exceptions to this method. A better solution applies Archimedes’ Law of Buoyancy and his Law of the Lever:
Suspend the wreath from one end of a scale and balance it with an equal mass of gold suspended from the other end. Immerse the balanced apparatus into a container of water. If the scale remains in balance then the wreath and the gold have the same volume, and so the wreath has the same density as pure gold. But if the scale tilts in the direction of the gold, then the wreath has a greater volume 3 Vitruvius’s comments can be found in his work De architectura (about 27 BCE) a comprehensive
than the gold. For more details, consult the Archimedes home page, http://www.mcs.drexel.edu./~crorres/Archimedes/index.html.
Sand-Reckoner 1.5 The Sand-Reckoner is a small treatise that is addressed to Gelon, son of Hieron. Written for the layman, it nevertheless contains some highly original mathematics. One object of the book was to repair the inadequacies of the Greek numerical notation system by showing how to express a huge number, in particular the number of grains of sand that it would take to fill the whole of the universe. Apparently independent of the Babylonian base 60 system, Archimedes devised a place-value system of notation, with a base of 100,000,000. He constructed numbers up to 8 £ 1017. The work also gives the most detailed surviving description of the heliocentric system of Aristarchus of Samos, the Ancient Copernican.
On the Equilibrium of Planes 1.6 In a treatise of two volumes Archimedes discovered fundamental theorems concerning the center of gravity of plane figures and solids. His most famous theorem gives the weight of a body immersed in a liquid, called Archimedes’ principal.
Quadrature of a Parabola 1.7
He also showed how to trisect angles using the spiral. Suppose the particular angle to be trisected is / AOB. Construct circles with center O that intersect the spiral. Construct the line segment OD and mark the point C. Trisect the segment CD and construct circles with center O with the respective radii at the trisection points. Since the spiral sweeps out the radius in exact proportion to the respective angle, the new circles will intersect the spiral at equal angles from the lines OA and OB. The angle between them will be the same, as well. Thus the angles / AOB is trisected.