| 释义 |
infinitesimal, n. and a.|ɪnfɪnɪˈtɛsɪməl|
Also erron. -ess-.
[f. mod.L. infīnītesim-us, f. L. infīnīt-us (cf. cent-esimus hundredth, mill-esimus thousandth): cf. F. infinitésime (1752 in Dict. Trévoux), infinitésimal (1762 in Dict. Acad.).
The form. of the mod.L. word shows that it was orig. meant as an ordinal, viz. the ‘infiniteth’ in order, that which is at an infinite distance from the first; but the ordinals are also used to name fractions, e.g. hundredth (part), 1/100, thousandth (part) 1/1000; hence, infinitesima pars, infinitesimal part or infinitesimal, came to mean unity divided by infinity, ⎜ 1 / ∞ ⎟, and thus an infinitely small part or quantity. Although essentially an adj., our earliest example shows the word used absolutely as a n.]
A. n. (or absol.)
†1. As ordinal: The ‘infiniteth’ member of a series. Obs. rare.
1655H. More App. Antid. Ath. xiii. 391 But for us whose capacities are finite, if we would venture to name a first in infinite succession, we should call it πρῶτον ἀπειραστὸν, the first infinitessimal, and acknowledge our selves unable to go through, our understandings being finite.
2. a. (Chiefly Math.) As a fraction or fractional quantity. The inverse or reciprocal of an infinite quantity; † an infinitely small fraction or part of anything (obs.). Hence b. (Math.) An infinitely small quantity or amount, a quantity less than any assignable quantity.
Chiefly used of the infinitesimal differences or differentials treated of in the differential and integral calculus: see differential A. 3, B. 1 a. Mathematicians distinguish different orders of infinitesimals; thus, if we make x infinite, 1 / x is an infinitesimal of the first order, 1 / x2 ⎜being infinitely less than 1 / x ⎟ an infinitesimal of the second order, and so on.
[1704Hayes Fluxions 1 These infinitely little Parts of an infinitely little Part of a given Quantity are..called Infinitesimæ Infinitesimarum or Fluxions of Fluxions.] 1706Ditton Fluxions 20 Let m denote an infinite Quantity, d any finite one; then is d / m the Infinitesimal of d, according to Mr. Nieuwentyt. 1710Berkeley Princ. Hum. Knowl. §130 Some..not content with holding that finite lines may be divided into an infinite number of parts, do yet farther maintain that each of these infinitesimals is itself subdivisible into an infinity of other parts or infinitesimals of a second order, and so on ad infinitum. These, I say, assert there are infinitesimals of infinitesimals of infinitesimals, etc., without ever coming to an end. 1745Needham Microsc. Disc. Introd. 3 A little Ant-hill..would appear to its Inhabitants..an Infinitesimal of the terraqueous Globe.
b.1734Berkeley Analyst §6 An infinite succession of infinitesimals, each infinitely less than the foregoing. 1743Phil. Trans. XLII. 349 In the Method of Infinitesimals, the Element, by which any Quantity increases or decreases, is supposed to become infinitely small. 1816tr. Lacroix's Diff. & Int. Calculus 78 A very simple explanation of the various orders of infinitesimals admitted by Leibnitz. 1831Hind Princ. Differ. Calc. 116 The method of Infinitesimals adopted by Leibnitz as the foundation of his Differential Calculus. 1873B. Williamson Diff. Calculus (ed. 2) ii. §36.
3. In loose or hyperbolical use: An extremely small quantity or amount; something excessively minute or insignificant.
1840Hood Up Rhine 255 Hahnemann, having hit on the advantage of small doses, never rested till he had reduced them to infinitesimals. 1854Emerson Lett. & Soc. Aims, Resources Wks. (Bohn) III. 197 What power does Nature not owe to her duration of amassing infinitesimals into cosmical forces!
B. adj.
1. (Chiefly Math.)
a. Infinitely or indefinitely small; smaller than any assignable fraction or magnitude. (Correlative to infinite, and, with it, opposed to finite.)
b. transf. Relating to infinitesimal quantities; esp. in infinitesimal calculus, a name for the differential and integral calculuses considered as one (corresponding to the direct and inverse methods of fluxions).
1710Berkeley Princ. Hum. Knowl. §132 It will not be found..necessary to make use of or conceive infinitesimal parts of finite lines. 1770Horsley in Phil. Trans. LX. 435 note, The infinitesimal segments of that line. 1801W. Dickson (title) Reflections on the Infinitesimal Calculus. From the French of Carnot, with Notes. 1862H. Spencer First Princ. i. iii. §17 (1875) 57 It is quite possible to think of its motion as diminishing insensibly until it becomes infinitesimal. 1871Tyndall Fragm. Sc. (1879) I. ii. 58 The aqueous vapour it contains is of almost infinitesimal amount.
2. In loose or hyperbolical use (cf. A. 3): Too small to be measured or reckoned; extremely minute or insignificant.
1733Cheyne Eng. Malady iii. iv. (1734) 337, I was not reduc'd to such extreme Weakness, that infinitesimal Errors, could do great Hurt. 1748Hartley Observ. Man i. iii. 393 An Obstruction of the infinitesimal Vessels of the Nervous System. 1830Lyell Princ. Geol. I. 474 No river can push forward its delta without raising the level of the whole ocean, although in an infinitesimal degree. 1831Carlyle Sart. Res. ii. ix. (1858) 120 Were it but the pitifullest infinitesimal fraction of a Product, produce it in God's name! 1884Times (weekly ed.) 19 Sept. 6/4 Each infinitesimal right of grazing..had to be surveyed, examined into.
Hence infiniˈtesimalist, one who supports the method of infinitesimals (sense A. 2); infinitesiˈmality, an infinitesimally small matter; infiniˈtesimalness, infinitesimal smallness.
1863[see momentarian]. 1867Gd. Words 801/1 The infinitesimality (I am sorry to have to coin a word) of his influence. 1895Columbus (O.) Disp. 17 Oct. 11/3 It is well sometimes to let the mind dwell on such infinitesimalities. 1897N.Y. Voice 8 July 2/3 This infinitesimalness of the Church practise. 1937Mind XLVI. 227 Berkeley's penetrating criticism of the postulates of the fluxionists and infinitesimalists of his day. |