| 释义 |
integral, a. and n.|ˈɪntɪgrəl|
[ad. late L. integrāl-is, f. integer, integr-: see integer and -al1. Cf. F. intégral (Oresme, 14th c.); It. integrale ‘entire, consisting of entirenesse’ (Florio).
‘Integralis pars’, and ‘partium integralium, quæ si conveniant, totum exstat’, occur in a 6th c. Comment. on Cicero de Invent. Rhet., in Suringar Hist. Crit. Scholiast. Latin. (1834) pp. 248, 222.]
A. adj.
1. Of or pertaining to a whole. Said of a part or parts: Belonging to or making up an integral whole; constituent, component; spec. necessary to the completeness or integrity of the whole; forming an intrinsic portion or element, as distinguished from an adjunct or appendage. (Cf. integrant.) (Formerly distinguished from essential: see quots. 1697, 1727.)
1551T. Wilson Logike 39 b, The integral partes, which make perfect the whole, and cause the bignesse thereof. a1639W. Whately Prototypes ii. xxvi. (1640) 43 The parts integrall, viz. as the severall members of the matter, head, heart, &c. in man. 1651N. Bacon Disc. Govt. Eng. ii. xv. (1739) 79 In a mixt Commonwealth they [kings] are integral Members. 1697Locke 2nd Vind. Reas. Chr. 247 (Seager) Integral parts..are contradistinguished to essential; and signify such parts, as the thing can be without, but without them will not be so complete and entire as with them. 1727–41Chambers Cycl., Integral, or Integrant, is applied by the schoolmen, to those parts which are necessary to the integrity of a whole... In which sense they stand contradistinguished from essential parts... The arms, legs, etc. are integral parts; body and soul essential parts of a man. 1786Burke W. Hastings Wks. 1842 II. 220 Forming no legal or integral part of the government. 1862Goulburn Pers. Relig. iii. x. (1873) 238 Recreation must form an integral part of human life. 1867Freeman Norm. Conq. I. ii. 69 A dependency of the British Crown..not an integral part of the United Kingdom. 1923Glazebrook Dict. Appl. Physics V. 165/1 This cylinder has an open-ended steel barrel with integral fins. 1958Chambers's Techn. Dict. 987/1 Integral stiffeners, the stiffening ridges left when an aircraft skin panel is machined from a solid billet. 1968Gloss. Formwork Terms (B.S.I.) 16 Integral facing, a special facing concrete or mortar cast simultaneously with the backing concrete so as to be monolithic with it. 1972[see integrally adv. b].
2. Made up of component parts which together constitute a unity; in Logic, said of a whole consisting of or divisible into parts external to each other, and therefore actually (not merely mentally) separable. Now rare or Obs. exc. in technical use.
1588Fraunce Lawiers Log. i. vi. 33 The whole Integrall cannot bee affirmed of any one of his parts, for a part is not the whole. 1628T. Spencer Logick 203 An Integrall whole is not in each part, neither according to their whole essence, nor vertue, and therefore it is no wayes predicated of the singular parts. 1649Jer. Taylor Gt. Exemp. ii. vi. §19 In every Christian there are three parts concerning this integral Constitution, body and soul, and Spirit. 1725Watts Logic i. vi. §10 As an integral Whole is distinguish'd into its several Parts by Division, so the Word Distribution is most properly used when we distinguish an universal Whole into its several Kinds of Species. Ibid., Logicians have sometimes given a mark or sign to distinguish when it is an integral whole, that is, divided into its parts and members, or when it is a genus, an universal whole, that is, distributed into its species and individuals. 1836–7Sir W. Hamilton Metaph. xxxvii. (1859) II. 340 The Integral or, as it ought to be called Integrate whole (totum integratum), is composed of integrant parts (partes integrantes) which are either homogeneous, or heterogeneous. 1864Bowen Logic iv. 67 note, The Essential or Physical whole is that which consists of Matter and Form, or substance and accident, as its essential parts. The characteristic of this whole is that, as its parts do not exist out of each other, they cannot be separated except in Thought..The Mathematical or Integral whole, on the other hand, has parts which are external to each other, so that they can be divided asunder. 1945H. D. Smyth Gen. Acct. Devel. Atomic Energy Mil. Purposes xii. 132 Two ‘integral experiments’ (experiments on assembled or integrated systems comprising fissionable material, reflector, and perhaps moderator also) may be described. 1953C. Wallace Photographer's Pocket-Bk. 112 In modern colour materials the colours are achieved by building up on a suitable base..an ‘integral tri-pack’ of three separate emulsions.
3. a. Having no part or element separated, taken away, or lacking; unbroken, whole, entire, complete. Now somewhat rare. [= mod.F. intégral.]
1611Florio, Integrale, whole or integrall. 1626Bacon Sylva §344 All Locall Motion keepeth Bodies Integrall, and their Parts together. 1651Biggs New Disp. ⁋238 Thorow the integral porous pelt. 1659D. Pell Impr. Sea 484 Their hearts are not integral, and entire in prayer. 1794Mathias Purs. Lit. (1798) 157 Excerpta of Writers whose integral works are lost for ever. 1862Lytton Str. Story II. 15 Who could expect that every link in a madman's tale would be found integral and perfect?
b. Of things immaterial.
1651Jer. Taylor Serm. for Year i. iv. 51 Repent with an integral, a holy and excellent repentance. 1656Earl of Monmouth Advt. fr. Parnass 281 They are thought by them to merit their Princes integral love. 1847R. W. Hamilton Sabbath v. (1848) 181 It is felt that, if we would retain Christianity, we must hold fast the full, the integral, sabbath.
†c. Gram. Applied by Wilkins to a word or part of speech denoting a complete notion; see B. 3. Obs.
1668Wilkins Real Char. 305 They supply the room either, 1. Of some Integral word, as Pronouns, or 2. Of some Sentence or complex part of it, as Interjections.
4. Math.
a. That is, or is denoted by, an integer, or involves only integers; consisting of a whole number or undivided quantity; not fractional, or not involving a fraction.
1658Phillips s.v., In Arithmetick integral numbers are opposed to fraction[s]. 1674S. Jeake Arith. (1696) 15 To express the true content of any Number Integral. 1812J. Smyth Pract. of Customs (1821) 286 The fractional part of a foot..is to be given up in favour of the importer, and the duties to be charged only upon the integral feet. 1816tr. Lacroix's Diff. & Int. Calculus 185 Q being a rational and integral function of x. 1875Todhunter Algebra (ed. 7) xxxvi. §516 When n has any value positive or negative, integral or fractional.
b. Relating to or involving integrals (see B. 4); obtained by, belonging to, or proceeding by integration.
integral calculus: the calculus of integrals (see B. 4); that branch of the infinitesimal calculus which deals with the finding and properties of integrals of functions (in this restricted sense, the inverse of the differential calculus, and corresponding to the ‘inverse method of fluxions’ in the Newtonian calculus), also used to include the solution of differential equations, and parts of the theory of functions and other branches of the higher mathematics. integral sign = sign of integration: see B. 4 a, and integration 2.
1727–41Chambers Cycl. s.v. Calculus, The integral Calculus..is the inverse of the differential one. Ibid., Suppose ∫ the sign of the sum, or integral quantity. 1802Woodhouse in Phil. Trans. XCII. 95 Expressions deduced from the true integral equations. 1875C. P. Buckingham Diff. & Int. Calc. (1880) §157 The..problem of the integral calculus is to pass from a given differential of a function to the function itself. 1881Maxwell Electr. & Magn. I. 21 In the expression under the integral sign only the finite values..are to be considered. 1887R. A. Roberts Int. Calc. 1 The principal object of the Integral Calculus is to find the value of a function of a single variable when its differential coefficient is given.
c. Applied to the entire or total amount of a continuous quantity (e.g. curvature) taken between definite limits, and thus expressible by a definite integral (see B. 4 a).
1879Thomson & Tait Nat. Phil. I. i. §10 The integral curvature, or whole change of direction of an arc of a plane curve, is the angle through which the tangent has turned as we pass from one extremity to the other.
d. integral domain: see domain n. 4 d.
B. n.
1. Something entire or undivided; a whole, either as wanting no part, or as made up of parts: see A. 2, 3. Obs. exc. as transf. from 4 = total sum.
1620T. Granger Div. Logike 177 A tree, a body, an house..are totall Integrals, whose integritie, or wholenesse..is made of their parts. 1657Tomlinson Renou's Disp. 9 In the third genus are contained all Animals whether Integrals or In-parts. 1784J. Barry in Lect. Paint. iv. (Bohn 1848) 152 Any other conjunction of parts forming an integral or whole. 1834Landor Exam. Shaks. Wks. 1846 II. 299/2 No more..than breaking an eggshell is breaking an egg, the shell being a part, and the egg being an integral. 1881Nature No. 625. 582 What is seen in a sun-spot is the integral, as it were, of all that is taking place..in many thousand miles of solar atmosphere.
†2. An integral part or element; a constituent, component: see A. 1. Obs.
1658–9Burton's Diary (1828) III. 557 We must, therefore, be very circumspect in the materials of the other House. Let us, therefore, look to the integrals in this building. a1677Hale Prim. Orig. Man. i. i. 21 Anatomy can give us the Position..of all the several Integrals of the Body of Man or Beast. Ibid. iv. viii. 372 They all make up a most magnificent and stately Temple, and every Integral thereof full of wonder. 1680Baxter Answ. Stillingfl. 82 Doth not every good Law and Rule distinguish between Essentials, Integrals, and Accidents, and make more Accidents than are Integrals, and Integrals, than are Essentials? 1685― Paraphr. N.T., 1 Cor. xii. 14 So wise, as besides the Essentials of Christianity, to know all the Integrals.
†3. Gram. Applied by Wilkins to those words or parts of speech which of themselves express a distinct notion, as distinct from those which express relations between notions. Obs.
1668Wilkins Real Char. iii. i. §2 By Integrals or Principal words, I mean such as signifie some entire thing or notion. 1688R. Holme Armoury iii. 251/2. 1845 Stoddart Gram. in Encycl. Metrop. I. 124/1 Wilkins includes under the term integral both the noun and the verb.
4. Math.
a. (of a function): That quantity of which the given function is the differential or differential coefficient (corresponding to the fluent of a given fluxion in Newton's method); so called because it may be regarded as the whole sum of a series of consecutive values assumed by an infinitesimal function (differential) of the variable while the latter changes continuously from any one value to any other. When such limits of variation are fixed or determinate, it is called a definite integral: see quot. 1877. An integral is denoted by the sign ∫ (originally a long s, for L. summa sum); in a definite integral the inferior and superior limits are indicated at the bottom and top of the sign, thus ∫ba . (Formerly sometimes applied to the quantity from which a given ‘finite difference’ or ‘increment’ is derived, as in quot. 1763; cf. quot. 1831 s.v. integrate v. 3.)
b. (of a differential equation, or a system of such equations): An equation or system of equations from which the given equation or system can be derived by differentiation. (In relation to a system of equations, any quantity which that system makes constant is sometimes called its integral.)
1727–41Chambers Cycl. s.v. Calculus, ∫ydx will denote the sum, or integral of the differential ydx. 1763Emerson Increments p. vii, Some Increments have no integrals, but what infinite series afford. 1802Woodhouse in Phil. Trans. XCII. 90 The integral or fluent of Px· is that function from which Px· is derived. 1877B. Williamson Int. Calc. (ed. 2) vi. §91 The expression ∫ X / x0 ϕ(x)dx is called the definite integral of ϕ(x)dx between the limits x0 and X, and represents the limit of the sum of the infinitely small elements ϕ(x)dx, taken between the proposed limits... In contradistinction, the name indefinite integrals is often applied to integrals..in which the form of the function is merely taken into account, without regard to any assigned limits. 1881Maxwell Electr. & Magn. I. 27 The double integrals destroy each other. |