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For one thing, there are a lot of funny stories about him. Several portray him as the original math geek. For example, the historian Plutarch tells us that Archimedes could become so engrossed in geometry that it “made him forget his food and neglect his person.” (That certainly rings true. For many of us mathematicians, meals and personal hygiene aren’t top priorities.) Plutarch goes on to say that when Archimedes was lost in his mathematics, he would have to be “carried by absolute violence to bathe.” It’s interesting that he was such a reluctant bather, given that a bath is the setting for the one story about him that everybody knows. According to the Roman architect Vitruvius, Archimedes became so excited by a sudden insight he had in the bath that he leaped out of the tub and ran down the street naked shouting, “Eureka!” (“I have found it!”)
Other stories cast him as a military magician, a warrior-scientist / one-man death squad. According to these legends, when his home city of Syracuse was under siege by the Romans in 212 BCE, Archimedes — by then an old man, around seventy — helped defend the city by using his knowledge of pulleys and levers to make fantastical weapons, “war engines” such as grappling hooks and giant cranes that could lift the Roman ships out of the sea and shake the sailors from them like sand being shaken out of a shoe. As Plutarch described the terrifying scene, “A ship was frequently lifted up to a great height in the air (a dreadful thing to behold), and was rolled to and fro, and kept swinging, until the mariners were all thrown out, when at length it was dashed against the rocks, or let fall.”
In a more serious vein, all students of science and engineering remember Archimedes for his principle of buoyancy (a body immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced) and his law of the lever (heavy objects placed on opposite sides of a lever will balance if and only if their weights are in inverse proportion to their distances from the fulcrum). Both of these ideas have countless practical applications. Archimedes’s principle of buoyancy explains why some objects float and others do not. It also underlies all of naval architecture, the theory of ship stability, and the design of oil-drilling platforms at sea. And you rely on his law of the lever, even if unknowingly, every time you use a nail clipper or a crowbar.
Archimedes might have been a formidable maker of war machines, and he undoubtedly was a brilliant scientist and engineer, but what really puts him in the pantheon is what he did for mathematics. He paved the way for integral calculus. Its deepest ideas are plainly visible in his work, but then they aren’t seen again for almost two millennia. To say he was ahead of his time would be putting it mildly. Has anyone ever been more ahead of his time?
Two strategies appear again and again in his work. The first was his ardent use of the Infinity Principle. To probe the mysteries of circles, spheres, and other curved shapes, he always approximated them with rectilinear shapes made of lots of straight, flat pieces, faceted like jewels. By imagining more and more pieces and making them smaller and smaller, he pushed his approximations ever closer to the truth, approaching exactitude in the limit of infinitely many pieces. This strategy demanded that he be a wizard with sums and puzzles, since he ended up having to add many numbers or pieces back together to arrive at his conclusions.
His other distinguishing stratagem was blending mathematics with physics, the ideal with the real. Specifically, he mingled geometry, the study of shapes, with mechanics, the study of motion and force. Sometimes he used geometry to illuminate mechanics; sometimes the flow went in the other direction, with mechanical arguments providing insight into pure form. It was by using both strategies with consummate skill that Archimedes was able to penetrate so deeply into the mystery of curves.
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Mathematicians don’t come up with the proofs first. First comes intuition. Rigor comes later. This essential role of intuition and imagination is often left out of high-school geometry courses, but it is essential to all creative mathematics.
Archimedes concludes with the hope that “there will be some among the present as well as future generations who by means of the method here explained will be enabled to find other theorems which have not yet fallen to our share.” That almost brings a tear to my eye. This unsurpassed genius, feeling the finiteness of his life against the infinitude of mathematics, recognizes that there is so much left to be done, that there are “other theorems which have not yet fallen to our share.” We all feel that, all of us mathematicians. Our subject is endless. It humbled even Archimedes himself.
