Crossword clues for infinitesimal
Longman Dictionary of Contemporary English
The Collaborative International Dictionary
Infinitesimal \In`fin*i*tes"i*mal\, a. [Cf. F. infinit['e]simal, fr. infinit['e]sime infinitely small, fr. L. infinitus. See Infinite, a.] Infinitely or indefinitely small; less than any assignable quantity or value; very small.
Infinitesimal calculus, the different and the integral calculus, when developed according to the method used by Leibnitz, who regarded the increments given to variables as infinitesimal.
Infinitesimal \In`fin*i*tes"i*mal\, n. (Math.) An infinitely small quantity; that which is less than any assignable quantity.
Douglas Harper's Etymology Dictionary
1710 (1650s as a noun), "infinitely small," from Modern Latin infinitesimus, from Latin infinitus "infinite" (see infinite) + -esimus, as in centesimus "hundredth." Related: Infinitesimally.
Wiktionary
a. 1 incalculably, exceedingly, or immeasurably minute; vanishingly small. 2 (context mathematics English) Of or pertaining to values that approach zero as a limit. 3 (context informal English) Very small. n. (context mathematics English) A non-zero quantity whose magnitude is smaller than any positive number (by definition it is not a real number).
WordNet
adj. infinitely or immeasurably small; "two minute whiplike threads of protoplasm"; "reduced to a microscopic scale" [syn: minute, microscopic]
Wikipedia
In mathematics, infinitesimals are things so small that there is no way to measure them. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were quantitatively small. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the " infinite -th" item in a sequence. It was originally introduced around 1670 by either Nicolaus Mercator or Gottfried Wilhelm Leibniz. Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" means "extremely small". To give it a meaning, it usually must be compared to another infinitesimal object in the same context (as in a derivative). Infinitely many infinitesimals are summed to produce an integral.
Archimedes used what eventually came to be known as the method of indivisibles in his work The Method of Mechanical Theorems to find areas of regions and volumes of solids. In his formal published treatises, Archimedes solved the same problem using the method of exhaustion. The 15th century saw the work of Nicholas of Cusa, further developed in the 17th century by Johannes Kepler, in particular calculation of area of a circle by representing the latter as an infinite-sided polygon. Simon Stevin's work on decimal representation of all numbers in the 16th century prepared the ground for the real continuum. Bonaventura Cavalieri's method of indivisibles led to an extension of the results of the classical authors. The method of indivisibles related to geometrical figures as being composed of entities of codimension 1. John Wallis's infinitesimals differed from indivisibles in that he would decompose geometrical figures into infinitely thin building blocks of the same dimension as the figure, preparing the ground for general methods of the integral calculus. He exploited an infinitesimal denoted 1/∞ in area calculations.
The use of infinitesimals by Leibniz relied upon heuristic principles, such as the law of continuity: what succeeds for the finite numbers succeeds also for the infinite numbers and vice versa; and the transcendental law of homogeneity that specifies procedures for replacing expressions involving inassignable quantities, by expressions involving only assignable ones. The 18th century saw routine use of infinitesimals by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange. Augustin-Louis Cauchy exploited infinitesimals both in defining continuity in his Cours d'Analyse, and in defining an early form of a Dirac delta function. As Cantor and Dedekind were developing more abstract versions of Stevin's continuum, Paul du Bois-Reymond wrote a series of papers on infinitesimal-enriched continua based on growth rates of functions. Du Bois-Reymond's work inspired both Émile Borel and Thoralf Skolem. Borel explicitly linked du Bois-Reymond's work to Cauchy's work on rates of growth of infinitesimals. Skolem developed the first non-standard models of arithmetic in 1934. A mathematical implementation of both the law of continuity and infinitesimals was achieved by Abraham Robinson in 1961, who developed non-standard analysis based on earlier work by Edwin Hewitt in 1948 and Jerzy Łoś in 1955. The hyperreals implement an infinitesimal-enriched continuum and the transfer principle implements Leibniz's law of continuity. The standard part function implements Fermat's adequality.
Vladimir Arnold wrote in 1990:
Nowadays, when teaching analysis, it is not very popular to talk about infinitesimal quantities. Consequently present-day students are not fully in command of this language. Nevertheless, it is still necessary to have command of it.Usage examples of "infinitesimal".
Even admitting that experiments on animals have contributed to the relief of human suffering, such measure of relief is infinitesimal compared with the pain which has been inflicted to secure it.
There was an explosion and Sandy felt his flesh sizzling, his body slammed to the ground, and his soul shredded into infinitesimal pieces.
It takes a very moderate amount of erudition to unearth a charlatan like the supposed father of the infinitesimal dosing system.
There were infinitesimal differences about each of them, and there was an aura of easygoing good nature about Victoria that more than bordered on mischief, and yet one would have been hard-pressed to tell them apart if one had to.
Arriving at infinitesimals, mathematics, the most exact of sciences, abandons the process of analysis and enters on the new process of the integration of unknown, infinitely small, quantities.
But later discovery has shown that each of these several herbs contains lime, and earthy salts, in a subtle form of high natural sub-division: whilst, at the same time, the law of cure by medicinal similars has established the cognate fact that to those who inherit a strumous taint, infinitesimal doses of these earth salts are incontestably curative.
Because at last we had a nondestructive means of scanning the exact state of muons through infinitesimal passages of time, we were able to find some astonishing correlations between memory and the barely detectable muon states of slant and yaw.
There was a moment of strained silence while Tach stared inward at pictures he would rather forget, and David with elaborate concentration gave the pocket pegboard chess set an infinitesimal shift to the left.
The ground is composed of a black spongelike substance and flickers with a trillion infinitesimal glimmers.
Like a silence spreading over a room full of people, like an infinitesimal coolness on a sultry day, like a passing memory of some long-forgotten sound or scent, like all that is stillest and smallest and most hard to seize in nature, Oyarsa passed between his subjects and drew near and came to rest, not ten yards away from Ransom, in the centre of Meldilorn.
In this infinitesimal fraction of a second Byrne had the sense of being really on his quest, of having reached the turn of the path, almost within hail of Tom.
I use the word that Anaxagoras invented for the infinitesimal bits of matter that make up creation.
This theory, that apparitions come in an infinitesimal moment of sleep, while a man is conscious of his surroundings and believes himself to be awake was the current explanation of ghosts in the eighteenth century.
In light of the bloodiness of the crime, its hands-on physicality, and the number of victims and defendants, the discovery of a few mass-produced fibers from items available in Wal-Marts and other clothiers all over the country struck Lax as an infinitesimal amount of evidence, which was also highly circumstantial.
But assuming it to be a fact that diseases are cured by remedies capable of producing symptoms like their own, no manifest relation exists between this fact and the next assertion, namely, the power of the infinitesimal doses.