involutionn.

/ɪnvəˈl(j)uːʃən/

Etymology: < Latin involūtiōn-em, noun of action from involvĕre to involve v.: compare French involution (13–14th cent. in Hatzfeld & Darmesteter).

a. The action of involving or fact of being involved; implicit comprehension or inclusion; implication; also, quasi-concrete, that which is involved.

ΘΚΠ

the world > relative properties > wholeness > mutual relation of parts to whole > incorporation or inclusion > [noun] > implicit inclusion

involution1611

involvement1706

introvolution1829

implication1873

the world > action or operation > doing > activity or occupation > [noun] > fact or action of being concerned with

meddlingc1390

implicationc1430

mellingc1440

intermeddling1531

participation1582

mashing1607

trucka1625

concern1643

involvedness1654

interest1660

involvement1706

business1759

immixture1859

involution1886

a piece (also share) of the action1938

1611 R. Cotgrave Dict. French & Eng. Tongues Involution, an inuolution, enwrapping, infoulding.

1642 Bp. J. Taylor Of Sacred Order Episcopacy (1647) 136 Often..a Bishop nay an Apostle is called a Presbyter..by reason of the involution or comprehension of Presbyter within Episcopus.

1790 E. Gibbon Misc. Wks. (1814) III. 489 According to the philosophers, who can discern an endless involution of germs or organized bodies, the future animal exists in the female parent.

1798 S. T. Coleridge Satyrane's Lett. ii. 223 Aristotle has..required of the poet an involution of the universal in the individual.

1886 W. Stubbs 17 Lect. Study Hist. i. 17 From his own involution in the matter of which he is to judge.

1892 N. Smyth Christian Ethics ii. iii. 420 The instinct to discover the deeper moral involutions of current political questions is a power of great ethical value.

b. concrete. Something that involves or enwraps; an envelope, covering, etc.

ΘΚΠ

the world > space > relative position > condition of being external > covering > wrapping > [noun] > enfolding or enveloping > that which or one who

enfolder1545

inwrapper1553

involucre1578

involument1578

burse1601

involvement1632

investment1646

involution1646

mantling1652

involucruma1676

tunicle1678

enveloping1693

envelope1715

enveloper1883

1646 Sir T. Browne Pseudodoxia Epidemica v. xxi. 269 The involution or membranous covering..called the silly how, that sometimes is found about the heads of children upon their birth.

2. An involved or entangled condition; entanglement, complication; intricacy of construction or style (as in a literary work or the arrangement of words in a sentence); also concrete, something complicated; an intricate movement, a tangle, etc.

ΘΚΠ

the world > relative properties > order > disorder > confusion or disorder > entanglement or entangled state > complication or complexity > [noun]

intricatenessa1586

perplexednessa1586

involution1611

mixture1614

complicateness1656

puzzledness1662

complexedness1690

complexation1707

complexness1727

complexity1790

complication1793

complicacy18..

complicatedness1818

complicity1847

involvedness1867

multiplexity1938

society > leisure > the arts > literature > style of language or writing > obscurity > [noun] > complexity

involution1820

tortuosity1837

trickiness1885

1611 R. Cotgrave Dict. French & Eng. Tongues Anfractueux, full of turnings, compasses, involutions.

1647 T. May Hist. Parl. i. i. 73 All their acts and actions are so full of mixtures, involutions, and complications.

1751 S. Johnson Rambler No. 168. ⁋7 Mackbeth proceeds to wish..that he may, in the involutions of infernal darkness, escape the eye of providence.

a1763 W. Shenstone Oeconomy iii, in Wks. Verse & Prose (1764) I. 303 Such the clue Of Cretan Ariadne ne'er explain'd! Hooks! angles! crooks! and involutions wild!

1820 W. Hazlitt Lect. Dramatic Lit. 156 The style of the first act has..more involution, than the general style of Fletcher.

1837 H. Hallam Introd. Lit. Europe I. viii. 591 He introduced..a sort of involution into his style, which gives an air of dignity and remoteness from common life.

1858 G. MacDonald Phantastes xiv The whole place..swam with the involutions of an intricate dance.

3. Anatomy. A rolling, curling, or turning inwards; concrete a part of a structure formed by this action.

ΘΚΠ

the world > life > biology > biological processes > movement > [noun] > retraction or involution

retraction?a1425

involution1851

the world > life > biology > physical aspects or shapes > physical arrangement or condition > [noun] > turning inwards or inside out > introverted part

involution1851

introvert1883

1851 W. B. Carpenter Man. Physiol. (ed. 2) 494 A cavity..which is subsequently rendered more complex by the prolongation and involution of its walls in various parts.

1870 G. Rolleston Forms Animal Life Introd. 36 The peripheral apparatus retains its typical character as an involution of the integument in the olfactory..organs.

1873 St. G. Mivart Lessons Elem. Anat. ix. 392 The ear like the eye is formed by an involution of the skin.

1880 St. G. Mivart Cat viii. 230 Glands..are..complex involutions of an epithelial surface.

4. Physiology. ‘The retrograde change which occurs in the body in old age, or in some organ when its permanent or temporary purpose has been fulfilled’ ( New Sydenham Soc. Lexicon). Also attributive.

ΘΚΠ

the world > health and disease > ill health > a disease > disorders associated with age > [noun] > of old age

superannuation1655

caducity1769

climacteric disease1813

involution1860

1860 T. H. Tanner On Signs & Dis. Pregnancy ii. 93 The whole process of degeneration and reconstruction is spoken of as the involution of the uterus.

1878 A. Gamgee tr. L. Hermann Elements Human Physiol. (ed. 2) 530 The close of the period of fecundity and the arrest of menstruation are associated with certain bodily changes, especially of the generative apparatus, which are comprehended in the term ‘involution’.

1887 New Sydenham Soc. Lexicon Senile Involution, the shrinking of the whole body which accompanies old age.

1887 New Sydenham Soc. Lexicon Involution cysts, the cysts found in the shrivelled mammary glands of old women, being dilated acini or ducts of the gland filled with a thick fluid.

1898 G. E. Herman Dis. Women ix. 87 During the last few days of pregnancy, and the first few days of involution, giant cells with many nuclei are to be seen.

5. Biology. A retrograde process of development; the opposite of evolution; degeneration. Chiefly in combinations, as involution-form.

ΘΚΠ

the world > life > biology > biological processes > evolution > [noun] > degeneration

devolution1882

involution1896

1896 T. C. Allbutt et al. Syst. Med. I. 761 Involution forms [of bacilli] being pretty constantly developed.

1897 T. C. Allbutt et al. Syst. Med. II. 90 Evidence that the clubs are involution-forms.

6. Mathematics.

a. Arithmetic and Algebra. The multiplication of a quantity into itself any number of times, so as to raise it to any assigned power. Hence, in extended sense, the raising of a quantity to any power, positive, negative, fractional, or imaginary.

ΘΚΠ

the world > relative properties > number > arithmetic or algebraic operations > [noun] > multiplication > into or by itself

quadratec1450

squaring1579

affection?1683

figurationa1690

involution1706

exponentiation1903

1706 W. Jones Synopsis Palmariorum Matheseos 51 By the Involution of the Binomial Root.

1806 C. Hutton Course Math. (ed. 5) I. 197 Involution is the raising of powers from any proposed root; or the method of finding the square, cube, biquadrate, &c., of any given quantity.

b. Geometry. A system of pairs of points on a right line, so situated that the product of the distances of the two points of each pair from a certain fixed point on the line (the centre of involution) is equal to a constant quantity. Hence in various extended uses (see quot. 1847).

ΘΚΠ

the world > relative properties > number > geometry > point > [noun] > sets or groups of points

umbilic point1586

involution1847

triad1850

range1859

point group1887

tetrad1889

tristigm1889

neighbourhood1891

trinode1891

trigraphy1895

Cantor set1902

web1909

limit cycle1918

Leech lattice1968

1837 Chasles Aperçu Hist. 77 Desargues appelait la rélation qui constitue son beau théorème involution de six points.]

1847 Cayley in Cambr. & Dublin Math. Jrnl. II. 52 When three conics have the same points of intersection, any transversal intersects the system in six points, which are said to be in involution. It appears natural to apply the term to the conics themselves; and then it is easy to generalize the notion of involution so as to apply it to functions of any number of variables.

1879 G. Salmon Conic Sections (ed. 6) xvi. 311 The reciprocal of six points in involution is a pencil in involution.

1885 C. Leudesdorf tr. L. Cremona Elements Projective Geom. 101 In an involution the elements are conjugate to one another in pairs.

c. A function or transformation that is equal to its inverse.

ΘΚΠ

the world > relative properties > number > arithmetic or algebraic operations > transformation > [noun] > correspondence > inverting

alternate proportion1570

inversion1645

transposition1664

reversion1698

involution1916

1916 E. Kasner in Amer. Jrnl. Math. 38 177 It is easy to determine all regular transformations of period 2. In the direct type Z = f(z) the functional equation is f(f(z)) ≡ z, that is, f² = 1; in the reverse type Z = f(z₀) the functional equation is f(f₀(z)) ≡ z, that is, ff₀ = 1, where f₀ denotes the series whose coefficients are the conjugates of the coefficients of series f. We shall call a transformation of the former type (excluding the identical transformation) a conformal involution, and one of the latter type a conformal symmetry.

1969 F. M. Hall Introd. Abstr. Algebra II. ii. 31 If θ is a 1–1 correspondence between elements of A and itself such that θ= θ^—1, then θ is said to be an involution.

This entry has not yet been fully updated (first published 1900; most recently modified version published online March 2019).

单词	involution
释义	involutionn. /ɪnvəˈl(j)uːʃən/ Etymology: < Latin involūtiōn-em, noun of action from involvĕre to involve v.: compare French involution (13–14th cent. in Hatzfeld & Darmesteter). 1. a. The action of involving or fact of being involved; implicit comprehension or inclusion; implication; also, quasi-concrete, that which is involved. ΘΚΠ the world > relative properties > wholeness > mutual relation of parts to whole > incorporation or inclusion > [noun] > implicit inclusion involution1611 involvement1706 introvolution1829 implication1873 the world > action or operation > doing > activity or occupation > [noun] > fact or action of being concerned with meddlingc1390 implicationc1430 mellingc1440 intermeddling1531 participation1582 mashing1607 trucka1625 concern1643 involvedness1654 interest1660 involvement1706 business1759 immixture1859 involution1886 a piece (also share) of the action1938 1611 R. Cotgrave Dict. French & Eng. Tongues Involution, an inuolution, enwrapping, infoulding. 1642 Bp. J. Taylor Of Sacred Order Episcopacy (1647) 136 Often..a Bishop nay an Apostle is called a Presbyter..by reason of the involution or comprehension of Presbyter within Episcopus. 1790 E. Gibbon Misc. Wks. (1814) III. 489 According to the philosophers, who can discern an endless involution of germs or organized bodies, the future animal exists in the female parent. 1798 S. T. Coleridge Satyrane's Lett. ii. 223 Aristotle has..required of the poet an involution of the universal in the individual. 1886 W. Stubbs 17 Lect. Study Hist. i. 17 From his own involution in the matter of which he is to judge. 1892 N. Smyth Christian Ethics ii. iii. 420 The instinct to discover the deeper moral involutions of current political questions is a power of great ethical value. b. concrete. Something that involves or enwraps; an envelope, covering, etc. ΘΚΠ the world > space > relative position > condition of being external > covering > wrapping > [noun] > enfolding or enveloping > that which or one who enfolder1545 inwrapper1553 involucre1578 involument1578 burse1601 involvement1632 investment1646 involution1646 mantling1652 involucruma1676 tunicle1678 enveloping1693 envelope1715 enveloper1883 1646 Sir T. Browne Pseudodoxia Epidemica v. xxi. 269 The involution or membranous covering..called the silly how, that sometimes is found about the heads of children upon their birth. 2. An involved or entangled condition; entanglement, complication; intricacy of construction or style (as in a literary work or the arrangement of words in a sentence); also concrete, something complicated; an intricate movement, a tangle, etc. ΘΚΠ the world > relative properties > order > disorder > confusion or disorder > entanglement or entangled state > complication or complexity > [noun] intricatenessa1586 perplexednessa1586 involution1611 mixture1614 complicateness1656 puzzledness1662 complexedness1690 complexation1707 complexness1727 complexity1790 complication1793 complicacy18.. complicatedness1818 complicity1847 involvedness1867 multiplexity1938 society > leisure > the arts > literature > style of language or writing > obscurity > [noun] > complexity involution1820 tortuosity1837 trickiness1885 1611 R. Cotgrave Dict. French & Eng. Tongues Anfractueux, full of turnings, compasses, involutions. 1647 T. May Hist. Parl. i. i. 73 All their acts and actions are so full of mixtures, involutions, and complications. 1751 S. Johnson Rambler No. 168. ⁋7 Mackbeth proceeds to wish..that he may, in the involutions of infernal darkness, escape the eye of providence. a1763 W. Shenstone Oeconomy iii, in Wks. Verse & Prose (1764) I. 303 Such the clue Of Cretan Ariadne ne'er explain'd! Hooks! angles! crooks! and involutions wild! 1820 W. Hazlitt Lect. Dramatic Lit. 156 The style of the first act has..more involution, than the general style of Fletcher. 1837 H. Hallam Introd. Lit. Europe I. viii. 591 He introduced..a sort of involution into his style, which gives an air of dignity and remoteness from common life. 1858 G. MacDonald Phantastes xiv The whole place..swam with the involutions of an intricate dance. 3. Anatomy. A rolling, curling, or turning inwards; concrete a part of a structure formed by this action. ΘΚΠ the world > life > biology > biological processes > movement > [noun] > retraction or involution retraction?a1425 involution1851 the world > life > biology > physical aspects or shapes > physical arrangement or condition > [noun] > turning inwards or inside out > introverted part involution1851 introvert1883 1851 W. B. Carpenter Man. Physiol. (ed. 2) 494 A cavity..which is subsequently rendered more complex by the prolongation and involution of its walls in various parts. 1870 G. Rolleston Forms Animal Life Introd. 36 The peripheral apparatus retains its typical character as an involution of the integument in the olfactory..organs. 1873 St. G. Mivart Lessons Elem. Anat. ix. 392 The ear like the eye is formed by an involution of the skin. 1880 St. G. Mivart Cat viii. 230 Glands..are..complex involutions of an epithelial surface. 4. Physiology. ‘The retrograde change which occurs in the body in old age, or in some organ when its permanent or temporary purpose has been fulfilled’ ( New Sydenham Soc. Lexicon). Also attributive. ΘΚΠ the world > health and disease > ill health > a disease > disorders associated with age > [noun] > of old age superannuation1655 caducity1769 climacteric disease1813 involution1860 1860 T. H. Tanner On Signs & Dis. Pregnancy ii. 93 The whole process of degeneration and reconstruction is spoken of as the involution of the uterus. 1878 A. Gamgee tr. L. Hermann Elements Human Physiol. (ed. 2) 530 The close of the period of fecundity and the arrest of menstruation are associated with certain bodily changes, especially of the generative apparatus, which are comprehended in the term ‘involution’. 1887 New Sydenham Soc. Lexicon Senile Involution, the shrinking of the whole body which accompanies old age. 1887 New Sydenham Soc. Lexicon Involution cysts, the cysts found in the shrivelled mammary glands of old women, being dilated acini or ducts of the gland filled with a thick fluid. 1898 G. E. Herman Dis. Women ix. 87 During the last few days of pregnancy, and the first few days of involution, giant cells with many nuclei are to be seen. 5. Biology. A retrograde process of development; the opposite of evolution; degeneration. Chiefly in combinations, as involution-form. ΘΚΠ the world > life > biology > biological processes > evolution > [noun] > degeneration devolution1882 involution1896 1896 T. C. Allbutt et al. Syst. Med. I. 761 Involution forms [of bacilli] being pretty constantly developed. 1897 T. C. Allbutt et al. Syst. Med. II. 90 Evidence that the clubs are involution-forms. 6. Mathematics. a. Arithmetic and Algebra. The multiplication of a quantity into itself any number of times, so as to raise it to any assigned power. Hence, in extended sense, the raising of a quantity to any power, positive, negative, fractional, or imaginary. ΘΚΠ the world > relative properties > number > arithmetic or algebraic operations > [noun] > multiplication > into or by itself quadratec1450 squaring1579 affection?1683 figurationa1690 involution1706 exponentiation1903 1706 W. Jones Synopsis Palmariorum Matheseos 51 By the Involution of the Binomial Root. 1806 C. Hutton Course Math. (ed. 5) I. 197 Involution is the raising of powers from any proposed root; or the method of finding the square, cube, biquadrate, &c., of any given quantity. b. Geometry. A system of pairs of points on a right line, so situated that the product of the distances of the two points of each pair from a certain fixed point on the line (the centre of involution) is equal to a constant quantity. Hence in various extended uses (see quot. 1847). ΘΚΠ the world > relative properties > number > geometry > point > [noun] > sets or groups of points umbilic point1586 involution1847 triad1850 range1859 point group1887 tetrad1889 tristigm1889 neighbourhood1891 trinode1891 trigraphy1895 Cantor set1902 web1909 limit cycle1918 Leech lattice1968 1837 Chasles Aperçu Hist. 77 Desargues appelait la rélation qui constitue son beau théorème involution de six points.] 1847 Cayley in Cambr. & Dublin Math. Jrnl. II. 52 When three conics have the same points of intersection, any transversal intersects the system in six points, which are said to be in involution. It appears natural to apply the term to the conics themselves; and then it is easy to generalize the notion of involution so as to apply it to functions of any number of variables. 1879 G. Salmon Conic Sections (ed. 6) xvi. 311 The reciprocal of six points in involution is a pencil in involution. 1885 C. Leudesdorf tr. L. Cremona Elements Projective Geom. 101 In an involution the elements are conjugate to one another in pairs. c. A function or transformation that is equal to its inverse. ΘΚΠ the world > relative properties > number > arithmetic or algebraic operations > transformation > [noun] > correspondence > inverting alternate proportion1570 inversion1645 transposition1664 reversion1698 involution1916 1916 E. Kasner in Amer. Jrnl. Math. 38 177 It is easy to determine all regular transformations of period 2. In the direct type Z = f(z) the functional equation is f(f(z)) ≡ z, that is, f² = 1; in the reverse type Z = f(z₀) the functional equation is f(f₀(z)) ≡ z, that is, ff₀ = 1, where f₀ denotes the series whose coefficients are the conjugates of the coefficients of series f. We shall call a transformation of the former type (excluding the identical transformation) a conformal involution, and one of the latter type a conformal symmetry. 1969 F. M. Hall Introd. Abstr. Algebra II. ii. 31 If θ is a 1–1 correspondence between elements of A and itself such that θ= θ^—1, then θ is said to be an involution. This entry has not yet been fully updated (first published 1900; most recently modified version published online March 2019). < n.1611
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