Life and Works

Archimedes was almost a modern scientist, though he lived from 287 to 212 B.C. He was born and lived in Syracuse, but he may have traveled to Alexandria Egypt in order to study geometry and mathematics. Archimedes was friend to king Hieron of Syracuse and perhaps his family was somehow connected to the royal household. Whatever his connection to royalty, Archimedes enjoyed wide latitude and freedom to pursue his interests. There are more than a few stories about him, and as usual these tales are suspect, but they point to what his life may have been. King Hieron asked him to determine if a crown made by a possibly unscrupulous blacksmith was constructed from pure gold. The king suspected another metal had been mixed into the crown and consecrating a gold and silver crown as gold would be an insult to the gods. Thus, the issue was spiritual and politically important. Archimedes is famous for the method by which he determined the composition of the crown as well as his reaction when he discovered the solution. Ancient sources claim that he was in a bath and he noticed that when his feet entered the water, he displaced a certain volume of the liquid. Without concerning himself to don clothing he ran from the bath naked, saying “I have found it” or Eureka in ancient Greek. Archimedes is said to have been killed during the capture of Syracuse by the Roman general Marcellus, who wanted Archimedes to be taken unharmed, but perhaps because of the eccentric mathematician's dismissive disposition he died at the hands of a soldier. There are conflicting anecdotes about his death. One is that a soldier came in order to fetch him for Marcellus and when Archimedes saw a shadow on the proof he currently worked, he demanded the soldier get out of his way. Another story [i] tells of how a soldier came to Archimedes who was so absorbed in his work that he had not noticed Syracuse had fallen to the Romans. The soldier said to Archimedes that Marcellus awaits and Archimedes replied that he wanted to finish his proof first. In either case, the soldier grew angry and slayed the septuagenarian mathematician. It is likely that we will never know precisely how Archimedes died, but he almost certainly perished in the taking of Syracuse.

Several works of Archimedes survive extant, but a new codex has recently come to light and perhaps more of his work awaits rediscovery. On the Sphere and Cylinder is the work he was most proud to create, including a calculation for the surface area of a sphere and that the volume of a sphere is two-thirds that of the cylinder in which it is inscribed. [ii] Measurement of the Circle is a section of a longer work that has in it a calculation of π, still a mathematical constant. On Conoids and Spheroids determines the volumes of segments of solids in a conic section, which is a precursor to calculus. On Spirals deals with tangents to and areas related to the spiral of Archimedes.[iii] On the Equilibrium of Planes [iv] concerns centers of gravity of various figures; it establishes the “law of the lever.” Quadrature of the Parabola asserts that the area of any segment of a parabola is 4/3 of the area of a triangle having the same base and height as that segment, among other things. The Sand Reckoner attempts to express very large numbers in an effort to remedy Greek numerical notation. Method Concerning Mechanical Theorems describes mathematical discovery, demonstrating how Archimedes used a “mechanical” method in some of his primary discoveries. On Floating Bodies concerns hydrostatics; it determines positions solids will take while floating in fluid. Archimedes wrote other works that do not survive, but what scholars possess is formidable indeed. One, called the Stomachion, is a game consisting of a square cut into fourteen pieces.

First Principles

Archimedes was an eccentric, seemingly interested only in his calculations and his not insignificant insights into material reality, and so his main preoccupation was pure mathematics. One may be justified in calling him a theoretical mathematician, except that the specific modern difference between theoretical and applied mathematics did not exist during his lifetime, and Archimedes used theoretical mathematics to make discoveries about empirical aspects of material reality; he used material reality for insights into mathematics. He, like modern physicists, used mathematics and geometry to explain the “how” of material reality. Archimedes explains the geometric aspects of the effect of gravity upon a lever in Equilibrium of Planes, for example. Occasionally, he paused from his work in order to invent some practical device. These practical devices are the things for which he was popularly known in the ancient world: the water screw, possibly a grappling hook that was a siege-engine of war, a method for determining the density of an object placed in water. Archimedes, like Euclid, does not explicitly state his first principles. Numbers for Archimedes seem to have a spatial significance, though he was likely not a Pythagorean in the traditional sense. He applies geometry to relations between physical objects and motions and in this sense his work anticipates what is called classical mechanics in physics. Archimedes' lack of application of applied mathematics tells us that he was not necessarily seeking god or the origin of the universe in his work, but rather he seems to be observing and working out mathematical observations and relationships of physical happenstance. He may be thought of as having a first principle of mathematics as it is applied to motion, objects and physical relationships. He studied what was there and sought regular patterns that could be explained upon each manifestation through numbers. Seemingly unconcerned to articulate first principles, he believed in the ability of mathematics to explain how things operate and that is a fundamental aspect of modern science.

Causation and Cosmology

Archimedes may have believed in a heliocentric universe, but we may correctly suspect that he did not believe in the traditional pagan gods in the manner that his contemporaries did. His Sand Reckoner contains what may be a clue to Archimedes' s cosmology: a description of a heliocentric system and an attempt to determine the sun's diameter. Causation for Archimedes seems to lie at least partially in gravitation as well as in differences between shapes of objects interacting with one another in space, and not necessarily time. Thus, explanation of how objects operate in the universe is a fundamental aspect of what may be called his cosmology, which is the same as saying that he is a precursor to those early modern physicists who defined and labored over classical mechanics, classical mechanics being the concern over the physical laws that make up motions of objects that have been influenced by forces of the universe. The study of motion was not new, even to Archimedes, but he studied motion without the hindrance of seeking god or the prejudice of a preconceived notion of some kind of transcendence. He simply sought to explain how what is there operates physically. One cannot call his perspective objectivity, but one may want to say that he carried with him less preconceived notions than earlier thinkers. He is an applied mathematician and geometrician. Thus, the mechanical objects he discovered and his breakthroughs in geometrical relationships are manifestations of his sense of causation and how the universe around us is composed. Thus, he does not articulate a cosmology, and if he did he would not have adhered to a traditional one, thought it may have had the veneer of traditional belief.

Archimedes' water screw [v] is a simple mechanism still employed to drive water up from depths where perhaps it otherwise may be impossible to obtain. The device is rather simple in construction. It is a tube with blades lining the inside in the shape of a screw.