variation

variation
  (vɛərɪˈeɪʃən)
  Forms: 5 varyacyoune, -cio(u)n, 6 -cyon; 5 variacioun, 5–6 -cion, 6 -cyon, -tioun, 6– variation.
  [a. OF. variation, -acion (F. variation, = Sp. variacion, Pg. varia{cced}ão, It. variazione), a. L. variātiōn-, variātio, n. of action f. variāre to vary.]
  I. † 1. Difference, divergence, or discrepancy between two or more things or persons. Obs.

c 1386 Chaucer Knt.'s T. 1730 In al the world..So even withoute variacioun Ther nere suche companyes tweye. 1426 Lydg. De Guil. Pilgr. 20066, I sey also..That ther be..Many constellaciouns And many varyaciouns. 1460 J. Capgrave Chron. (Rolls) 48 Here is for to noten that their is grete variacion amongst auctoures, both of ȝeres and of Kyngis names. 1480 Caxton Myrr. ii. i. 65 This present fygure is..demonstraunce certayne and trewe, without ony variacion ne doubtaunce. 1553 Eden Treat. New Ind. (Arb.) 42 A clyme is a porcion of the worlde betwene South and North, wherein is variacion in length of the daye, the space of halfe an houre. 1628 T. Spencer Logick 68 Health..dissenteth from a man that is sicke, by reason of that distance, or variation, which ariseth from sicknes. a 1637 B. Jonson Discoveries Wks. 1640 II. 106 There is a great variation betweene him, that is rais'd to the soveraignity by the favour of his Peeres, and him that comes to it by the suffrage of the people.

  † 2. Discord, variance, dissension; an instance of this. Obs. rare.

c 1485 Digby Myst. (1882) iii. 923 Be-twyx yow and me be never varyacyounes. 1523 Ld. Berners Froiss. I. cccxlvi. 548 Thus the Christen realmes were in variacyon, and the churches in great dyfference, bycause of the popes.

  † 3. a. Uncertainty, doubt. Obs.^—1

1471 Caxton Recuyell (Sommer) 28 In this sorow and in this payne and varyacion..Vesca, Abell and the damoysel were a longe tyme.

  † b. Inconstancy; variableness.

1509 Hawes Past. Pleas xix. (Percy Soc.) 88 My heart shall be without variacion Wyth you present, in perfite sykernes. c 1530 Crt. of Love 1340, I..depely swere as mine power to bene Faithful deuoide of variacion.

  II. 4. a. The fact of varying in condition, character, degree, or other quality; the fact of undergoing modification or alteration, especially within certain limits.

1502 Ord. Crysten Men (W. de W. 1506) v. vi, In shynynge varyacyon of dyuers coloures. 1513 Bradshaw St. Werburge i. 1340 This present lyfe..How dredefull it is, full of varyacyon. 1555 Eden Decades (Arb.) 45 Paralleles, are lines whereby the sonne passynge causeth variation of tyme. 1579 Fenton Guicciard. i. (1599) 15 Let vs looke somewhat into the variation of times and things of the world. 1637 Nabbes Microcosmus 11, Two kisses more will cloy me; nought can relish But variation. 1674 Boyle Excell. Theol. ii. v. 214 According to the varying gravity of the atmosphere; which variation has..a very considerable influence on the weather-glass. 1750 tr. Leonardus' Mirr. Stones 53 As is held by many learned men who have written of the variation of the air. 1785 G. A. Bellamy Apology (ed. 3) I. 67 Lest you accuse me of a want of variation in the conclusion of my letters, I shall end this in the good old⁓fashion way. 1822 M. A. Kelty Osmond I. 36 In this variation of feeling the morning..wore away. 1845 G. E. Day tr. Simon's Anim. Chem. I. 246 From these data, it appears, that..the variation is the most striking with regard to the fibrin and globulin. 1885 Watson & Burbury Math. Th. Electr. & Magn. I. 96 In order to effect this object the charge upon the conductor must be capable of variation.

  b. The action of making some change or alteration.

a 1704 T. Brown Satire Antients Wks. 1730 I. 14 They used in other words the same variation of the letter u into i, as maxumus, maximus. 1711 in Nairne Peerage Evidence (1874) 133 The said parties having in order therto agreed..in the terms of the two former contracts..without change or variation. 1885 Law Rep. 29 Chanc. Div. 542 The powers reserved to Wilson Lomer..to control the variation of investments. 1913 Act 3 Geo. V, c. 3 §1 Where a resolution is passed..providing for the variation of any existing tax.

  5. a. variation of the compass, († lodestone,) or needle, the deviation or divergence of the magnetic needle from the true north and south line; the amount or angular measure of this; = declination 8 b.

1556 Burrough in Hakluyt (1886) III. 126, I went on shoare and obserued the variation of the Compasse, which was three degrees. 1571 Digges Pantom. i. xxix. I ij b, Drawing a right line making an angle..equall to the variation of the compasse in your region. 1613 Purchas Pilgrimage (1614) 49 Cabot first found out the variation of the Compasse. 1679 Moxon Math. Dict. 160 Variation of the Needle, the Turning or Deviation of the Needle in the Mariners Compass [etc.]. 1687 A. Lovell tr. Thevenot's Trav. ii. 156 The variation of the Loadstone. 1774 M. Mackenzie Maritime Surv. 62 How to find the Sun's Azimuth, and from thence to find the Variation of the Needle. 1834 M. Somerville Connex. Phys. Sci. (1840) xxix. 338 The variation of the compass. 1851 Greenwell Coal-trade Terms, Northumb. & Durh. 16 The diurnal variation of the needle being far from inconsiderable.

  b. ellipt. in the same sense.

1594 Davis Seaman's Secrets (1601) 17 If your Compasse be good and without variation. 1597 W. Barlowe Navigator's Supply A 2, By the Variation is vnderstood the difference in the Horizon betweene the true and the magneticall Meridian. 1627 Capt. Smith Seaman's Gram. ii. 12 There is also..a Compasse for the variation. 1669 Sturmy Mariner's Mag. iv. i. 138 The Points of the Needle..are subject to be drawn aside by the Guns.., or any Iron neer it, and liable to Variation, and doth not shew the true North. 1703 W. Dampier Voy. III. i. 100, I found that the Variation did not always increase or decrease in proportion to the Degrees of Longitude East or West. 1769 Falconer Dict. Marine (1780) s.v., The highest variation..appears to be 17°1/4 W. and the least 16°½ W. 1846 A. Young Naut. Dict. 81 The variation is in practice ascertained by comparing the sun's true and magnetic amplitude or azimuths. 1878 [see declination 8].

  c. variation of the variation (see last quot.).

1706 Phillips (ed. Kersey), Variation of the Variation, is so call'd, because the Variation of the Needle is not always the same in the same Place. 1839 Noad Electricity 201 The variation of the variation, that is, the fact that the variation was not a constant quantity, but varied in different latitudes, was first noticed by the discoverer of America. 1867 Smyth Sailor's Word-Bk. 710 Variation of the Variation, is the change in the declination of the needle observed at different times in the same place.

  6. The fact, on the part of the mercury, of standing higher or lower in the tube of a barometer or thermometer; the extent or range of this.

1719 Quincy Phys. Dict. (1722) 11 The greatest Variation of the Height of the Mercury being 3 Inches. 1748 Anson's Voy. ii. v. 183 The variation of the thermometer at Petersburgh is at least five times greater..than..at St. Catherine's. 1815 J. Smith Panorama Sci. & Art II. 28 Had the tube been straight, Q would have been the limit of the scale of variation. 1858 Lardner Hand-bk. Nat. Phil. 168 A rise or fall of the mercury in the tube, within the usual limits of barometric variation.

  7. Astr. a. The libration of the moon; = libration 2.

1704 J. Harris Lex. Techn. I, Variation is, according to Tycho, the third Inequality in the Motion of the Moon. 1728 Pemberton Newton's Philos. 199 This inequality of the moon's motion about the earth is called by astronomers its variation. 1812 Woodhouse Astron. (1823) I. ii. 682 The Variation is occasioned by the other resolved part, that which acts in the direction of the tangent to the Moon's orbit. 1879 Newcomb & Holden Astron. 163 The disturbing action of the sun [upon the moon] produces a great number of the other inequalities, of which the largest are the evection and the variation.

  b. (See quot.)

1867 Smyth Sailor's Word-Bk. 43 Annual Variation, the change produced in the right ascension or declination of a star by the precession of the equinoxes and proper motion of the star taken together.

  8. Math. † a. = permutation 3 b. Obs.

1710 J. Harris Lex. Techn. II, Variation, or Permutation of Quantities, is the changing any number of given Quantities, with respect to their Places. 1728 Chambers Cycl. s.v. Combination, Suppose the Quantities 3, and the Exponent of Variation 3; the Number of Changes is found 27 = 33.

  b. Change in a function or functions of an equation due to an indefinitely small increase or decrease in the value of the constants.

1743 W. Emerson Fluxions 3 The Velocity, Variation, or Quickness of Increase (or Decrease) of any Fluxion is called the second Fluxion. 1810 R. Woodhouse Treat. Isoperimetrical Probl. ii. 23 If problems involving merely one property, the maximum, require the variation of two, and those involving two properties, the variation of three elements [etc.]. 1843 Penny Cycl. XXVI. 136/2 Variation. Under this head comes the explanation of a part of the language of proportion which is much used... We refer to such phrases as the following:—A varies as B.—A varies inversely as B. 1847 Cambr. & Dublin Math. Jrnl. I. 264 We have, in this case, by Lagrange's theory of the variation of the arbitrary constants, the formulæ da/dt = [etc.]. 1885 Watson & Burbury Math. Th. Electr. & Magn. I. 6 Then..u will, on arriving again at O, have assumed by continuous variation the value u0 + H. 1918 H. C. Plummer Introd. Treat. Dynamical Astron. xii. 134 This is the foundation of Lagrange's method of the variation of arbitrary constants. 1966 H. Pollard Math. Introd. Celestial Mech. iv. 91 We shall begin with the undisturbed system x 1 = - x2, x. 2 = k2x1, and apply the method of variation of parameters.

  c. variation of curvature: (see quot. 1842).

a 1727 Newton Meth. Fluxions & Inf. Ser. (1736) 76 The Inequability or Variation of Curvature is required at any Point of a Curve. 1842 Francis Dict. Arts, Variation of curvature, the change made on a curve, so as to occasion it to be flatter or sharper in each succeeding part.

  d. Algebra. The following of a + sign after a - sign, or vice versa, in a row of signs.

1891 in Cent. Dict.

  e. The difference between the values of a function at either end of a subinterval; the sum of such differences for all the non-overlapping subintervals into which a given interval is divided; the upper bound (if any) of this sum when all possible modes of subdividing the interval are considered.

1905 J. Pierpont Lect. Theory Functions Real Variables I. xii. 349 An important class of limited integrable functions is formed by functions with limited variation. 1911 Q. Jrnl. Pure & Appl. Math. XLII. 57 Although in forming the positive (negative) variation over (a, b) we considered all possible sets of non-overlapping intervals, we may without loss of generality confine ourselves to sets consisting of a finite number of intervals only. 1946 H. & B. S. Jeffreys Methods Math. Physics i. 23 The total variation is of interest since it is related to the condition for existence of a Stieltjes integral..and to the determination of the total length of a curve. 1971 E. R. Phillips Introd. Anal. & Integration Theory ix. 251 Let us assume that the set of nonnegative variations..is bounded from above.

  9. Mus. (See quot.)

1730 Treat. Harmony 34 There is another sort of Division called Variation, which may also be upon a Division.

  10. Biol. Deviation or divergence in the structure, character, or function of an organism from those typical of or usual in the species or group.

1859 Darwin Orig. Spec. i. 11 There are many laws regulating variation, some few of which can be dimly seen. 1867–8 Lyell Princ. Geol. iii. xliii. (ed. 10) II. 488 If some modification of an organ, or instinct, be produced by what is called ‘Spontaneous Variation’. 1871 Tyndall Fragm. Sci. (1879) II. ix. 176 No naturalist could tell how far this variation could be carried. 1882 Vines tr. Sachs' Bot. 925 The characters of many of these varieties are perfectly hereditary, and all the organs show the greatest degree of variation.

  III. 11. a. An instance of varying or changing; an alteration or change in something, esp. within certain limits.
  Sometimes in specific senses: cf. 5–10 above.

1611 Cotgr., Muance, change, alteration; and particularly, a variation, or change of notes in singing. 1659 Pearson Creed (1839) 525 The natural course of variations in the creature. 1665 Phil. Trans. I. 31 A Baroscope, or an instrument to show all the Minute Variations in the Pressure of the Air. 1719 De Foe Crusoe ii. (Globe) 486 Variations of the Compass. 1758 Johnson Idler No. 11 ¶11 The most variable of all variations: the changes of the weather. 1786 A. M. Bennett Juvenile Indiscr. IV. 231 They..contrived to fill a long summer's day, or winter's evening, by an agreeable variation of female amusements. 1832 H. Martineau Weal & Woe vii. 94 Seasons are sometimes stormy and our commerce liable to variations. 1844 Proc. Philol. Soc. I. 196 We may therefore be disposed to consider all marked variations of dialect as evidences of difference of date. 1874 tr. Lommel's Light 181 The variations of light and shade are alone visible.

  b. A difference due to the introduction or intrusion of some change or alteration.

1699 Bentley Phal. 36 We have the firmer ground to go upon for this little Variation. 1727 T. Innes Anc. Inhab. Scot. (1879) 87 Variations which the negligence as well as the ignorance of transcribers is ordinarily the cause of. 1861 Paley æschylus (ed. 2) Supplices 842 note, The other MSS. present only slight variations. 1869 H. F. Tozer Highl. Turkey II. 272 The..stories have evidently come from the same original, but present curious variations in the form under which the youth is born. 1878 Huxley Physiogr. 192 In different specimens..the lava exhibits great variations.

  c. Biol. A slight departure or divergence from a type. (Cf. 10.)

1835 Lyell Princ. Geol. iii. ii. (ed. 4) II. 428 The phenomenon, that some individuals are made to deviate widely from the ordinary type... How far..may such variations extend in the course of indefinite periods of time? 1859 Darwin Orig. Spec. Introd. 4 We shall see how great is the power of man in accumulating by his Selection successive slight variations. 1871 R. H. Hutton Ess. I. 65 An accidental variation only means a variation of which you cannot determine the direction. 1882 Vines tr. Sachs' Bot. 777 Changes in these hereditary peculiarities, or variations, are never brought about by direct external influences.

  d. A different form or species; a variety, variant.

1863 Huxley Knowl. Org. Nat. 99 If, by crossing a variation with the original stock, you multiply that variation, and then take care to keep that variation distinct from the original stock, and make them breed together. 1868 Boy's Own Bk. 593 The Matadore Game..is a variation of All Fives. 1878 Browning Poets Croisic 5 Try a variation of the game!

  12. A deviation or departure from something.

1647 Clarendon Hist. Reb. i. §178 Besides that any Variation from it..would make the Uniformity the less. a 1662 Heylin Laud i. 223 It was best to take the English Liturgie, without any variation from it. 1782 J. Brown Nat. & Rev. Relig. iii. ii. 246 There often befalls it a deforming variation from the original happy constitution. 1818 Cruise Digest (ed. 2) I. 208 He did not think fit to make any variation from what was then determined.

  13. Math. a. (Cf. 8 a.)

1728 Chambers Cycl. s.v. Combination, Suppose two Quantities, a and b; their Variations will be 2; consequently, as each of those may be combined, even with it self, to these there must be added two Variations.

  b. The amount by which some quantity changes in value, or the addition made to the quantity; esp. the change in a function when there is a small change in the variables or constituent functions of the function; calculus of variations, a form of calculus applicable to expressions or functions in which the law relating the quantities is liable to variation.
  In the calculus of variations the function concerned is usu. an integral, and the aim is to find what relation between the variables in the integrand makes the integral a maximum or a minimum.

1810 Woodhouse (title), A Treatise on Isoperimetrical Problems, and the Calculus of Variations. Ibid. iii. 45 The general form of these equations is, P.bg - Q.ci + R.dδ, dδ being a variation of the ordinate similar to the variations bg and ci. 1814 J. Toplis tr. Laplace's Treat. Analytical Mech. ii. 46 Of all the curves along which a moving body, subjected to the forces P, Q, and R, can pass from one given point to another given point, it will describe that in which the variation of the integral ∫vds is nothing, and in which, consequently this integral is a minimum. 1834 [see least a. 1 e]. 1845 Encycl. Metrop. II. 209 To obtain the variation of a function of y we must write y + δy for y, and having expanded the new function according to the powers of δy, subtract from it the original function, and the first term of the difference will be the variation required. 1855 Brewster Newton I. xiii. 349 The calculus of variations discovered by Lagrange in 1760, was the greatest step in the improvement of the infinitesimal calculus which was made in the last century. 1861 Todhunter (title), A History of the Progress of the Calculus of Variations during the Nineteenth Century. 1934 W. V. Houston Princ. Math. Physics v. 56 The variation of the integral is defined as its value when α has the infinitesimal value δα, minus its value when α is zero. 1972 M. Kline Math. Thought xxiv. 578 By equating the variation of the integral to zero and by using a crude limiting process to transform the resulting difference equation, he obtained the differential equation which must be satisfied by the minimizing arc.

  14. a. Mus. A modification with regard to the tune, time, and harmony of a theme, by which on repetition it appears in a new but still recognizable form; esp. in pl., embellishments in an air for giving variety on repetition after playing it in its simple form.

1801 Busby Dict. Mus. (1811), Variations, or Var, the name given to certain ornamented repetitions, in which, while the original notes, harmony, and modulation, are..so far preserved as to sustain the parent subject, the passages are branched out in flourishes. 1820 Scott Let. in Lockhart (1837) IV. xi. 371 She ran a set of variations on ‘Kenmure's on and awa'’, which I told her were enough to raise a whole country-side. 1873 H. C. Banister Music 216 In some Sonatas, etc., one of the Movements is a Theme with Variations.

  b. elegant variation: in writing, the stylistic fault of studiedly avoiding repetition by using different words for the same thing. Also transf. and in ironic use.

1906 H. W. & F. G. Fowler King's English iii. 178 The locking of arms is..only an elegant variation for clinging. 1926 H. W. Fowler Mod. Eng. Usage 131/1 It is the second-rate writers..that are chiefly open to the allurements of elegant variation. 1947 Partridge Usage & Abusage 16/1 Mr Herd then picks on the device known as ‘elegant variation’. ‘If,’ he says, ‘the mayor has been mentioned, he makes further appearance as ‘the civic chief’, ‘the leader of our official life’..and so on.’ 1981 Guardian Weekly 5 July 21 Most of the costumes are..elegant variations on dancers' practice dress. 1982 Washington Post 7 May d2/6 {ddd} rarely occurs less than twice per page, and often as many as five, with a ‘f---’ or two stuck in for elegant variation.

  c. Ballet. A solo dance.

1912 Dancing Times Aug. 420/2 [Grahn's] career was interrupted by an accident while rehearsing a variation which she was to perform at [a] benefit. 1948 Ballet Ann. II. 49 She attacks the formidable difficulties of the variation and adagio with an ease and confidence. 1980 ‘M. Fonteyn’ Magic of Dance 65 He makes the preparation for his ‘variation’, or solo, with utmost care and accuracy.

  15. attrib. in sense 5 b, as variation-chart, variation compass, variation instrument; also variation method Physics, a method for finding an approximate solution to Schrödinger's equation by varying the trial solutions to find which gives the lowest value for the energy and is therefore closest to the true solution; variation order, an order authorizing a change in an original order or contract (see quots.); variation principle Physics, the principle (employed in the variation method) that the energy corresponding to an arbitrary wave function cannot be less than the actual lowest energy of the system under consideration.

1727 Bailey (vol. II), Variation Chart, a Chart design'd by Dr. Halley. 1748 Anson's Voy. Introd., A new variation-chart lately published. 1867 Smyth Sailor's Word-Bk. 710 The admiralty variation chart has been brought to great perfection.

1669 Sturmy Mariner's Mag. ii. vi. 67 The Use of the Quadrant and Variation-Compass. 1769 Phil. Trans. LIX. 483 The variation compass..was..a very good one.

1837 Lloyd in Rep. Brit. Assoc. VI. App. 21 The variation instrument will be placed in the magnetic meridian, with respect to the theodolite.

1935 Pauling & Wilson Introd. Quantum Mech. vii. 182 The variation method is..very frequently used to obtain approximate wave functions as well as approximate energy values. 1960 [see quantum-chemical adj. s.v. quantum 7 a]. 1974 P. W. Atkins Quanta 96/2 With these approximations in hand the variation method is applied to determine the best linear combination of atomic π-orbitals to describe the structure of the molecule.

1940 Chambers's Techn. Dict. 886 Variation order,..a document giving authority for some alteration in work being done under contract. 1975 E. B. Ceadel in Barr & Line Essays on Information & Libraries iv. 58 Where a change is made in the contract arrangements, it is listed by the architect as a variation order, which is subsequently costed by the quantity surveyors. 1977 Daily Tel. 19 Nov. 3/2 An order was prepared..requiring him to leave by Nov. 17... The Home Secretary..had considered whether, instead of issuing a ‘variation order’, he should proceed under another provision of the Immigration Act 1971 with a view to deportation.

1923 Variation principle [see quantum condition s.v. quantum 7 a]. 1975 H. F. Hameka Quantum Theory Chem. Bond i. 28 We shall discuss the two methods of approximation that are most widely used in quantum chemistry, namely the variation principle and perturbation theory.

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