History
The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes.
Leucippus, Democritus, Antiphon, Eudoxus and Archimedes developed the method of exhaustion, which uses an infinite sequence of approximations to determine an area or a volume. Archimedes succeeded in summing what is now called a geometric series.
Newton dealt with series in his works on Analysis with infinite series (written in 1669, circulated in manuscript, published in 1711), Method of fluxions and infinite series (written in 1671, published in English translation in 1736, Latin original published much later) and Tractatus de Quadratura Curvarum (written in 1693, published in 1704 as an Appendix to his Optiks). In the latter work, Newton considers the binomial expansion of (x+o)n which he then linearizes by taking limits (letting o→0).
In the 18th century, mathematicians like Euler succeeded in summing some divergent series by stopping at the right moment; they did not much care whether a limit existed, as long as it could be calculated. At the end of the century, Lagrange in his Théorie des fonctions analytiques (1797) opined that the lack of rigour precluded further development in calculus. Gauss in his etude of hypergeometric series (1813) for the first time rigorously investigated under which conditions a series converged to a limit.
The modern definition of a limit (for any ε there exists an index N so that ...) was given by Bernhard Bolzano (Der binomische Lehrsatz, Prague 1816, little noticed at the time) and by Weierstrass in the 1870s.
Read more about this topic: Limit Of A Sequence
Famous quotes containing the word history:
“Books of natural history aim commonly to be hasty schedules, or inventories of God’s property, by some clerk. They do not in the least teach the divine view of nature, but the popular view, or rather the popular method of studying nature, and make haste to conduct the persevering pupil only into that dilemma where the professors always dwell.”
—Henry David Thoreau (1817–1862)
“The thing that struck me forcefully was the feeling of great age about the place. Standing on that old parade ground, which is now a cricket field, I could feel the dead generations crowding me. Here was the oldest settlement of freedmen in the Western world, no doubt. Men who had thrown off the bands of slavery by their own courage and ingenuity. The courage and daring of the Maroons strike like a purple beam across the history of Jamaica.”
—Zora Neale Hurston (1891–1960)
“A poet’s object is not to tell what actually happened but what could or would happen either probably or inevitably.... For this reason poetry is something more scientific and serious than history, because poetry tends to give general truths while history gives particular facts.”
—Aristotle (384–323 B.C.)