Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Archimedes is generally considered to be the most celebrated mathematician and is regarded as one of the leading scientists in classical antiquity, as well as one of the greatest of all time.
Background
Archimedes was born around 287 B.C. in the seaport city of Syracuse, Sicily. Archimedes’ approximate birth date is conjectured on the basis of a remark by the Byzantine poet and historian of the twelfth century, John Tzetzes, who declared that Archimedes “worked at geometry until old age, surviving seventy-five years.” His father was the astronomer Phidias we know from Archimedes himself in his The Sandreckoner.
Education
Archimedes almost certainly visited Alexandria, where no doubt he studied with the successors of Euclid and played an important role in the further development of Euclidian mathematics.
Career
Archimedes was perhaps a kinsman of the ruler of Syracuse, King Hieron II (as Plutarch and Polybius suggest). At least he was on intimate terms with Hieron, to whose son Gelon he dedicated The Sandreckoner. At any rate Archimedes returned to Syracuse, composed most of his works there, and died there during its capture by the Romans in 212 B.C. There are picturesque accounts of Archimedes’ death by Livy, Plutarch, Valerius Maximus, and Tzetzes, which vary in detail but agree that he was killed by a Roman soldier. In most accounts he is pictured as being engaged in mathematics at the time of his death.
No surviving bust can be certainly identified as being of Archimedes, although a portrait on a Sicilian coin (whatever its date) is definitely his. A well-known mosaic showing Archimedes before a calculating board with a Roman soldier standing over him was once thought to be a genuine survival from Herculaneum but is now considered to be of Renaissance origin.
While Archimedes’ place in the history of science rests on a remarkable collection of mathematical works, his reputation in antiquity was also founded upon a series of mechanical contrivances which he is supposed to have invented and which the researches of A. G. Drachmann tend in part to confirm as Archimedean inventions. One of these is the water snail, a screwlike device to raise water for the purpose of irrigation, which, Diodorus Siculus tells us, Archimedes invented in Egypt.
We are further told by Atheneus that an endless screw invented by Archimedes was used to launch a ship. He is also credited with the invention of the compound pulley. Some such device is the object of the story told by Plutarch in his life of Marcellus. When asked by Hieron to show him how a great weight could be moved by a small force, Archimedes “fixed upon a three-masted merchantman of the royal fleet, which had been dragged ashore by the great labors of many men, and after putting on board many passengers and the customary freight, he seated himself at a distance from her, and without any great effort, but quietly setting in motion a system of compound pulleys, drew her towards him smoothly and evenly, as though she were gliding through the water.”
Much more generally credited is the assertion of Pappus that Archimedes wrote a book On Spheremaking, a work which presumably told how to construct a model planetarium representing the apparent motions of the sun, moon, and planets, and perhaps also a closed star globe representing the constellations. At least, we are told by Cicero that Marcellus took as booty from the sack of Syracuse both types of instruments constructed by Archimedes:
For Gallus told us that the other kind of celestial globe [that Marcellus brought back and placed in the Temple of Virtue], which was solid and contained no hollow space, was a very early invention, the first one of that kind having been constructed by Thales of Miletus, and later marked by Eudoxus of Cnidus ... with the constellations and stars which are fixed in the sky. ... But this newer kind of globe, he said, on which were delineated the motions of the sun and moon and of those five stars which are called the wanderers ... contained more than could be shown on a solid globe, and the invention of Archimedes deserved special admiration because he had thought out a way to represent accurately by a single device for turning the globe those various and divergent courses with their different rates of speed.
Finally, there arc references by Polybius, Livy, Plutarch, and others to fabulous ballistic instruments constructed by Archimedes to help repel Marcellus. One other defensive device often mentioned but of exceedingly doubtful existence was a burning mirror or combination of mirrors.
We have no way to know for sure of Archimedes’ attitude toward his inventions. One supposes that Plutarch’s famous eulogy of Archimedes’ disdain for the practical was an invention of Plutarch and simply reflected the awe in which Archimedes’ theoretical discoveries were held.
The mathematical works of Archimedes that have come down to us can be loosely classified in three groups (Arabic numbers have been added to indicate, where possible, their chronological order). The first group consists of those that have as their major objective the proof of theorems relative to the areas and volumes of figures bounded by curved lines and surfaces. In this group we can place On the Sphere and the Cylinder; On the Measurement of the Circle; On Conoids and Spheroids; On Spirals; and On the Quadrature of the Parabola.
The second group comprises works that lead to a geometrical analysis of statical and hydrostatical problems and the use of statics in geometry: On the Equilibrium of Planes, Book 1, Book II; On Floating Bodies; On the Method of Mechanical Theorems; and the aforementioned propositions from On the Quadrature of the Parabola. Miscellaneous mathematical works constitute the third group: The Sandreckoner; The Cattle-Problem, and the fragmentary Stomachion.
Archimedes also seems to have written a tract On Polyhedra, perhaps one On Blocks and Cylinders, certainly one on Archai or The Naming of Numbers (a work preliminary to The Sandreckoner), and a work on Optics or Catoptrics. Other works are attributed to Archimedes by Arabic authors, and, for the most part, are extant in Arabic manuscripts (the titles for which manuscripts are known are indicated by an asterisk; see Bibliography): The Lemmata, or Liber assumptorum (in its present form certainly not by Archimedes since his name is cited in the proofs), On Water Clocks, On Touching Circles, On Parallel Lines, On Triangles, On the Properties of the Right Triangle, On Data, and On the Division of the Circle into Seven Equal Parts.
In proving theorems relative to the area or volume of figures bounded by curved lines or surfaces, Archimedes employs the so-called Lemma of Archimedes or some similar lemma, together with a technique of proof that is generally called the “method of exhaustion,” and other special Greek devices such as neuseis, and principles taken over from statics. These various mathematical techniques are coupled with extensive knowledge of the mathematical works of his predecessors, including those of Eudoxus, Euclid, Aristeus, and others.
In the development of physical science, Archimedes is celebrated as the first to apply geometry successfully to statics and hydrostatics. In his On the Equilibrium of Planes, he proved the law of the lever in a purely geometrical manner. His weights had become geometrical magnitudes possessing weight and acting perpendicularly to the balance beam, itself conceived of as a weightless geometrical line. His crucial assumption was the special case of the equilibrium of the balance of equal arm length supporting equal weights. This postulate, although it may ultimately rest on experience, in the context of a mathematical proof appears to be a basic appeal to geometrical symmetry.
Views
In his On Floating Bodies, Archimedes emphasized once more largely on geometrical analysis, in Book I, a somewhat obscure concept of hydrostatic pressure is presented as his basic postulate: "Let it be granted that the fluid is of such a nature that of the parts of it which are at the same level and adjacent to one another that which is pressed the less is pushed away by that which is pressed the more, and that each of its parts is pressed by the fluid which is vertically above it, if the fluid is not shut up in anything and is not compressed by anything else."
As his propositions are analyzed, it is obvious that Archimedes essentially maintained an Aristotelian concept of weight directed downward toward the center of the earth conceived of as the center of the world. In fact, he goes further by imagining the earth removed and so fluids are presented as part of a fluid sphere all of whose parts weigh downward convergently toward the center of the sphere. The surface of the sphere is then imagined as being divided into an equal number of parts which are the bases of conical sectors having the center of the sphere as their vertex. Thus the water in each sector weighs downward toward the center. Then if a solid is added to a sector, increasing the pressure on it, the pressure is transmitted down through the center of the sphere and back upward on an adjacent sector and the fluid in that adjacent sector is forced upward to equalize the level of adjacent sectors. The influence on other than adjacent sectors is ignored. It is probable that Archimedes did not have the concept of hydrostatic paradox formulated by Stevin, which held that at any given point of the fluid the pressure is a constant magnitude that acts perpendicularly on any plane through that point. But, by his procedures, Archimedes was able to formulate propositions concerning the relative immersion in a fluid of solids less dense than, as dense as, and denser than the fluid in which they are placed.
Book II, which investigates the different positions in which a right segment of a paraboloid can float in a fluid, is a brilliant geometrical tour de force. In it, Archimedes returns to the basic assumption found in On the Equilibrium of Planes, On the Quadrature of the Parabola, and On the Method, namely, that weight verticals are to be conceived of as parallel rather than as convergent at the center of a fluid sphere.
Quotations:
“Give me a place to stand and I will move the earth.”
“Mathematics reveals its secrets only to those who approach it with pure love, for its own beauty.”
"Man has always learned from the past. After all, you can't learn history in reverse!'
"Those who claim to discover everything but produce no proofs of the same may be confuted as having actually pretended to discover the impossible."
"Many people believe that the grains of sand are infinite in multitude ... Others think that although their number is not without limit, no number can ever be named which will be greater than the number of grains of sand. But I shall try to prove to you that among the numbers which I have named there are those which exceed the number of grains in a heap of sand the size not only of the earth, but even of the universe."
"Equal weights at equal distances are in equilibrium and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance."
"I am persuaded that this method [for calculating the volume of a sphere] will be of no little service to mathematics. For I foresee that once it is understood and established, it will be used to discover other theorems which have not yet occurred to me, by other mathematicians, now living or yet unborn."
"Having been the discoverer of many splendid things, he is said to have asked his friends and relations that, after his death, they should place on his tomb a cylinder enclosing a sphere, writing on it the proportion of the containing solid to that which is contained."
Personality
Quotes from others about the person
"And yet Archimedes possessed such a lofty spirit, so profound a soul, and such a wealth of scientific theory, that although his inventions had won for him a name and fame for superhuman sagacity, he would not consent to leave behind him any treatise on this subject, but regarding the work of an engineer and every art that ministers to the needs of life as ignoble and vulgar, he devoted his earnest efforts only to those studies the subtlety and charm of which are not affected by the claims of necessity. These studies, he thought, are not to be compared with any others; in them, the subject matter vies with the demonstration, the former supplying grandeur and beauty, the latter precision and surpassing power. For it is not possible to find in geometry' more profound and difficult questions treated in simpler and purer terms. Some attribute this success to his natural endowments; others think it due to excessive labor that everything he did seemed to have been performed without labor and with ease. For no one could by his own efforts discover the proof, and yet as soon as he learns it from him, he thinks he might have discovered it himself, so smooth and rapid is the path by which he leads one to the desired conclusion." - Plutarch