Archimedes

Archimedes is universally acknowledged to be the greatest of ancient mathematicians. He studied at Euclid’s school (probably after Euclid’s death), but his work far surpassed the works of Euclid. His achievements are particularly impressive given the lack of good mathematical notation in his day. His proofs are noted not only for brilliance but for unequalled clarity, with a modern biographer (Heath) describing Archimedes’ treatises as “without exception monuments of mathematical exposition … so impressive in their perfection as to create a feeling akin to awe in the mind of the reader.” Archimedes made advances in number theory, algebra, and analysis, but is most renowned for his many theorems of plane and solid geometry. He was first to prove Heron’s formula for the area of a triangle. His excellent approximation to √3 indicates that he’d partially anticipated the method of continued fractions. He found a method to trisect an arbitrary angle (using a markable straightedge — the construction is impossible using strictly Platonic rules). Although it doesn’t survive in his writings, Pappus reports that he discovered theArchimedean solids. One of his most remarkable and famous geometric results was determining the area of a parabolic section, for which he offered two independent proofs, one using his Principle of the Lever, the other using a geometric series.

Archimedes anticipated integral calculus, most notably by determining the centers of mass of hemisphere and cylindrical wedge, and the volume of two cylinders’ intersection. Although Archimedes made little use of differential calculus, Chasles credits him (along with Kepler, Cavalieri, and Fermat) as one of the four who developed calculus before Newton and Leibniz. He was similar to Newton in that he used his (non-rigorous) calculus to discover results, but then devised rigorous geometric proofs for publication. His original achievements in physics include the principles of leverage, the first law of hydrostatics, and inventions like the compound pulley, the hydraulic screw, and war machines. His books include Floating Bodies, Spirals, The Sand Reckoner, Measurement of the Circle, and Sphere and Cylinder. He developed the Stomachion puzzle (and solved a difficult enumeration problem involving it). Archimedes proved that the volume of a sphere is two-thirds the volume of a circumscribing cylinder. He requested that a representation of such a sphere and cylinder be inscribed on his tomb.

Archimedes discovered formulae for the volume and surface area of a sphere, and may even have been first to notice and prove the simple relationship between a circle’s circumference and area. For these reasons, π is often called Archimedes’ constant. His approximation 223/71 < π < 22/7 was the best of his day, though Apollonius soon surpassed it. That Archimedes shared the attitude of later mathematicians like Hardy and Brouwer is suggested by Plutarch’s comment that Archimedes regarded applied mathematics “as ignoble and sordid … and did not deign to [write about his mechanical inventions; instead] he placed his whole ambition in those speculations the beauty and subtlety of which are untainted by any admixture of the common needs of life.”

In the 20th century, modern technology led to the discovery of new writings by Archimedes, hitherto hidden on a palimpsest, including a note that implies an understanding of the distinction between countable and uncountable infinities (a distinction which wasn’t resolved until Georg Cantor, who lived 2300 years after the time of Archimedes). Although Newton may have been the most important mathematician, and Gauss the greatest theorem prover, it is widely accepted that Archimedes was the greatest genius who ever lived. Yet, Hart omits him altogether from his list of Most Influential Persons: Archimedes was simply too far ahead of his time to have great historical significance.

Tags: Archimedes