Archimedes of Syracuse was an Ancient Greek mathematician, physicist, engineer, astronomer, and inventor from the city of Syracuse in Sicily. Although few details of his life are known, based on his surviving work, he is considered one of the leading scientists in classical antiquity, and one of the greatest mathematicians of all time. Archimedes anticipated modern calculus and analysis by applying the concept of the infinitesimals and the method of exhaustion to derive and rigorously prove many geometrical theorems, including the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral.
Biography
The details of Archimedes's life are obscure; a biography of Archimedes mentioned by Eutocius was allegedly written by his friend Heraclides Lembus, but this work has been lost, and modern scholarship is doubtful that it was written by Heraclides to begin with. Based on a statement by the Byzantine Greek scholar John Tzetzes that Archimedes lived for 75 years before his death in 212 BC, Archimedes is estimated to have been born c. 287 BC in the seaport city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. In the Sand-Reckoner, Archimedes gives his father's name as Phidias, an astronomer about whom nothing else is known; Plutarch wrote in his Parallel Lives that Archimedes was related to King Hiero II, the ruler of Syracuse, although Cicero and Silius Italicus suggest he was of humble origin.
It is also unknown whether he ever married or had children, or if he ever visited Alexandria, Egypt, during his youth; though his surviving written works, addressed to Dositheus of Pelusium, a student of the Alexandrian astronomer Conon of Samos, and to the head librarian Eratosthenes of Cyrene, suggested that he maintained collegial relations with scholars based there. In the preface to On Spirals addressed to Dositheus, Archimedes says that "many years have elapsed since Conon's death." Conon of Samos lived c. 280 – c. 220 BC, suggesting that Archimedes may have been an older man when writing some of his works.
Method of exhaustion
Archimedes developed arguments that compared curved figures with combinations of simpler shapes. His method of exhaustion used progressively more precise bounds rather than a modern algebraic formula. Earlier Greek mathematicians, including Eudoxus, had used related techniques to study areas and volumes. Archimedes extended this tradition in works on circles, parabolas, spheres, and other figures, combining rigorous demonstration with ingenious ways to discover a result before proving it.
In Measurement of a Circle, he compared a circle with regular polygons drawn inside and outside it. Repeatedly increasing the number of polygon sides narrowed the possible relationship between circumference and diameter. His calculation produced remarkably close upper and lower bounds for that ratio. The achievement illustrates the practical force of a geometric argument: an unknown curved length could be constrained by straight lengths that were easier to calculate.
Archimedes Palimpsest
In 1906, the Danish professor Johan Ludvig Heiberg visited Constantinople to examine a 174-page goatskin parchment of prayers, written in the 13th century, after reading a short transcription published seven years earlier (in 1899) by Papadopoulos-Kerameus. He confirmed that it was indeed a palimpsest, a document with text that had been written over an erased older work. Palimpsests were created by scraping the ink from existing works and reusing them, a common practice in the Middle Ages, as vellum was expensive. The older works in the palimpsest were identified by scholars as 10th-century copies of previously lost treatises by Archimedes. The palimpsest holds seven treatises, including the only surviving copy of On Floating Bodies in the original Greek.
It is the only known source of The Method of Mechanical Theorems, referred to by Suidas and thought to have been lost forever. Stomachion was also discovered in the palimpsest, with a more complete analysis of the puzzle than had been found in previous texts.